Review Article Some Aspects of Anisotropic Quark-Gluon ...parton moves very fast. Later, Chakraborty et al. [ ]also found the oscillatory behavior of the induced charge wake in the

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 371908 20 pageshttpdxdoiorg1011552013371908

Review ArticleSome Aspects of Anisotropic Quark-Gluon Plasma

Mahatsab Mandal and Pradip Roy

Saha Institute of Nuclear Physics 1AF Bidhannagar Kolkata 700064 India

Correspondence should be addressed to Mahatsab Mandal mahatsabmandalsahaacin

Received 5 April 2013 Revised 5 July 2013 Accepted 19 July 2013

Academic Editor Edward Sarkisyan-Grinbaum

Copyright copy 2013 M Mandal and P Roy This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We review the various aspects of anisotropic quark-gluon plasma (AQGP) that have recently been discussed by a number of authorsIn particular we focus on the electromagnetic probes of AQGP inter quark potential quarkonium states in AQGP and the nuclearmodifications factor of various bottomonium states using this potential In this context we will also discuss the radiative energyloss of partons and nuclear modification factor of light hadrons in the context of AQGP The features of the wake potential andcharge density due to the passage of jet in AQGP will also be demonstrated

1 Introduction

Ever since the possibility of creating the quark-gluon plasma(QGP) in relativistic heavy ion collision was envisagednumerous indirect signals were proposed to probe the prop-erties of such an exotic state of matter For example electro-magnetic probes (photon and dilepton) [1] 119869120595 suppression[2] jet quenching vis-a-vis energy loss [3ndash10] and manymore In spite of these many properties of the QGP arepoorly understood The most pertinent question is whetherthe matter produced in relativistic heavy ion collisions is inthermal equilibrium or not Studies on elliptical flow (up toabout 119901

119879sim15GeV) using ideal hydrodynamics indicate that

thematter produced in such collisions becomes isotropicwith120591iso sim 06 fmc [11ndash13] On the other hand using second-order transport coefficients with conformal field theory it hasbeen found that the isotropizationthermalization time hassizable uncertainties [14] leading to uncertainties in the initialconditions such as the initial temperature

In the absence of a theoretical proof favoring the rapidthermalization and the uncertainties in the hydrodynamicalfits of experimental data it is very hard to assume hydro-dynamical behavior of the system from the very beginningThe rapid expansion of the matter along the beam directioncauses faster cooling in the longitudinal direction than inthe transverse direction [18] As a result the system becomesanisotropic with ⟨119901

119871

2⟩ ≪ ⟨119901

119879

2⟩ in the local rest frame At

some later time when the effect of parton interaction rate

overcomes the plasma expansion rate the system returnsto the isotropic state again and remains isotropic for therest of the period If the QGP just after formation becomesanisotropic soft unstable modes are generated characterizedby the exponential growth of the transverse chromomagneticchromoelectric fields at short timesThus it is very importantto study the collective modes in an AQGP and use theseresults to calculate relevant observables The instability thusdeveloped is analogous to QEDWeibel instability The mostimportant collective modes are those which correspond totransverse chromomagnetic field fluctuations and these havebeen studied in great detail in [19ndash26] This is known aschromo-Weibel instability and it differs from its QED ana-logue because of nonlinear gauge self-interactions Because ofthis fact anisotropy driven plasma instabilities in QCD mayslow down the process of isotropization whereas in QED itcan speed up the process [27]

To characterize the presence of initial state of momentumspace anisotropy it has been suggested to look for someobservables which are sensitive to the early time after the col-lision The effects of preequilibrium momentum anisotropyon various observables have been studied quite extensivelyover the past few years The collective oscillations in anAQGP have been studied in [28 29] Heavy quark energyloss and momentum broadening in anisotropic QGP havebeen investigated in [30 31] However the radiative energyloss of partons in AQGP has recently been calculated in [32]Another aspect of jet propagation in hot and dense medium

2 Advances in High Energy Physics

is the wake that it creates along its path First Ruppert andMuller [33] have investigated that when a jet propagatesthrough the medium a wake of current and charge densityis induced which can be studied within the framework oflinear-response theory The result shows the wake in boththe induced charge and current density due to the screeningeffect of the moving parton In the quantum liquid scenariothe wake exhibits an oscillatory behavior when the chargeparton moves very fast Later Chakraborty et al [34] alsofound the oscillatory behavior of the induced charge wakein the backward direction at a large parton speed using HTLperturbation theory In a collisional quark-gluon plasma it isobserved that the wake properties change significantly com-pared to the collisionless case [35] Recently Jiang and Li [3637] have investigated the color response wake in the viscousQGP with the HTL resummation technique It is shown thatthe increase of the shear viscosity enhances the oscillation ofthe induced charge density as well as the wake potential Theeffect of momentum space anisotropy on the wake potentialand charge density has recently been considered in [38]

Effects of anisotropy on photon and dilepton yields havebeen investigated rigorously in [16 39ndash43] Recently theauthors in [44] calculated the nuclear modification factor forlight hadrons assuming an anisotropic QGP and showed howthe isotropization time can be extracted by comparing withthe experimental data

The organization of the review is as follows In Section 2wewill briefly discuss variousmodels of space-time evolutionin AQGP along with the electromagnetic probes which canbe used to extract the isotropization time of the plasmaSection 3 will be devoted to discuss the works on heavyquark potential and related phenomena (such as gluon 119869120595

dissociation cross-section in an AQGP) that have beeninvestigated so far We will also discuss the radiative energyloss of partons in AQGP and nuclear modification factor oflight hadrons due to the energy loss of the jet in Section 4In Section 5 the effect of momentum anisotropy on wakepotential and charge density due to the passage of a jet willbe presented Finally we summarize in Section 6

2 Electromagnetic Probes

Photons and dileptons have long been considered to be thegood probes to characterize the initial stages of heavy ioncollisions as these interact ldquoweaklyrdquo with the constituents ofthe medium and can come out without much distortion intheir energy and momentum Thus they carry the informa-tion about the space-time point where they are producedSince anisotropy is an early stage phenomena photons anddileptons are the efficient probes to characterize this stageThe yield of dileptons (henceforth called medium dilep-tonsphotons) in an AQGP has been calculated using aphenomenological model of space-time evolution in (1 + 1)dimension [42 43] This model (henceforth referred to asmodel I) introduces two parameters 119901hard and 120585 which arefunctions of time The former is called the hard momentumscale and is related to the average momentum of the particlesin themedium In isotropic case this can be identifiedwith thetemperature of the system The latter is called the anisotropy

parameter and minus1 lt 120585 lt infin Furthermore the modelinterpolates between early-time 1 + 1 free streaming behavior(120591 ≪ 120591iso) and late-time ideal hydrodynamical behavior (120591 ≫

120591iso) We first discuss various space-time evolution modelsof AQGP In model I the time dependence of 120585 is given by[42 43]

120585 (120591 120575) = (120591

120591119894

)

120575

minus 1 (1)

where the exponent 120575 = 2(23) corresponds to free streaming(collisionally broadened) preequilibrium state momentumspace anisotropy and 120575 = 0 corresponds to thermalization 120591

119894

is the formation time of the QGP For smooth transition fromfree streaming to hydrodynamical behavior a transitionwidth120574minus1 is introduced The time dependences of various relevant

parameters are obtained in terms of a smeared step function[42 43] as follows

120582 (120591) =1

2(tanh[

120574 (120591 minus 120591iso)

120591119894

] + 1) (2)

For 120591 ≪ 120591iso (≫ 120591iso) we have 120582 = 0(1) which corre-sponds to free streaming (hydrodynamics) Thus the timedependences of 120585 and 119901hard are as follows [42 43]

120585 (120591 120575) = (120591

120591119894

)

120575(1minus120582(120591))

minus 1

119901hard (120591) = 119879119894U13

(120591)

(3)

where

U (120591) equiv [R((120591iso120591

)

120575

minus 1)]

3120582(120591)4

(120591iso120591

)

1minus120575(1minus120582(120591))2

U equivU (120591)

U (120591119894)

R (119909) =1

2[

1

(119909 + 1)+tanminus1radic119909

radic119909]

(4)

and 119879119894is the initial temperature of the plasma In our calcu-

lation we assume a fast-order phase transition beginning atthe time 120591

119891and ending at 120591

119867= 119903

119889120591119891 where 119903

119889= 119892

119876119892

119867is the

ratio of the degrees of freedom in the two (QGP phase andhadronic phase) phases and 120591

119891is obtained by the condition

119901hard(120591119891) = 119879119888 which we take as 192MeVWe also include the

contribution from the mixed phaseThe other model (referred to as model II hereafter) of

space-time evolution of highly AQGP is the boost invariantdissipative dynamics in (0 + 1) dimension [45] This modelcan reproduce both the hydrodynamics and the free stream-ing limits The time evolution of the phase space distribution119891(119905 119911 p) can be described by Boltzmann equation Thus asa starting point we write the Boltzmann equation in (0 + 1)

dimension in the lab frame as follows

119901119905120597119905119891 (119905 119911 p) + 119901

119911120597119911119891 (119905 119911 p) = minusC [119891 (119905 119911 p)] (5)

Advances in High Energy Physics 3

where homogeneity in the transverse direction is assumedand C[119891(119905 119911 p)] is the collision kernel We assume that thephase-space distribution for the anisotropic plasma is givenby the following ansatz [28 29]

119891 (p 120585 (120591) 119901hard (120591)) = 119891iso ([p2 + 120585 (120591) (p sdot n)]

2

1199012hard (120591)) (6)

where n is the direction of anisotropy Note that in sub-sequent sections this distribution function will be used tocalculate various observables Now it is convenient to write(5) in the comoving frame Introducing space-time rapidity(Θ) particle rapidity (119910) and proper time (120591) one can write(5) in terms of the comoving coordinates as [45]

(119901119879cosh (119910 minus Θ)

120597

120597120591+

119901119879sinh (119910 minus Θ)

120591

120597

120597120591)

times 119891 (p 120585 119901hard)

= minusΓ119901119879cosh (119910 minus Θ) [119891 (p 120585 119901hard) minus 119891eq (p 119879 (120591))]

(7)

where Γ = 2119879(120591)(5120578) and 120578 = 120578119904 120578 is the shear viscositycoefficient

The zeroth-order and first-order moments of the Boltz-mann equation give the time dependence for 120585 and 119901hardas described in [45] Without going into further details wesimply quote the coupled differential equations that have tobe solved to get the time dependence of 120585 and 119901hard [45] asfollows

1

1 + 120585120597120591120585 =

2

120591minus 4ΓR (120585)

R34radic1 + 120585 minus 1

2R (120585) + 3 (1 + 120585)R1015840 (120585)

1

1 + 120585

1

119901hard120597120591119901hard

=2

120591minus 4ΓR

1015840(120585)


2R (120585) + 3 (1 + 120585)R1015840 (120585)

(8)

The previous two coupled differential equations have to besolved numerically The results are shown in Figure 1 Itis seen that the anisotropy parameter falls much rapidlycompared to the case whenmodel II is usedThere is a narrowwindow in 120591 where 120585 dominates in case of model I Thecooling is slower in case of model II as can be seen from theright panel of Figure 1 These observations have importantconsequence on various observables

The assumption of boost invariant in the longitudinaldirection can be relaxed and such a space-time model (theso called AHYDRO) has been proposed in [46] As beforethe time evolutions of various quantities can be obtained bytaking moments of the Boltzmann equation However in thiscase instead of two one obtains three coupled differentialequations The third variable is the longitudinal flow velocity(see [46] for details) The observations of this work are asfollows It removes the problem of negative longitudinal pres-sure sometimes obtained in 2nd-order viscous hydrodynam-ics and this model leads to much slower relaxation towardsisotropy In this review for the sake of simplicity the observ-ables of AQGP will be calculated using space-time model

I The same can also be calculated using other space-timemodels of AQGP and the results may differ from case to case

21 Photons We first consider the medium photon produc-tion from AQGP The detail derivation of the differentialrate is standard and can be found in [16 39ndash41] Here wewill quote only the final formula for total photon yield afterconvoluting with the space-time evolutionThe total mediumphoton yield arising from the pureQGPphase and themixedphase is given by

119889119873120574

1198891199101198892119901119879

= 1205871198772

perp[int

120591119891

120591i

120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

+int

120591119867

120591119891

119891QGP (120591) 120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

]

(9)

where 119891QGP(120591) = (119903119889minus 1)

minus1(119903119889120591119891120591minus1

minus 1) is the fraction ofthe QGP phase in the mixed phase [47] and 119877

perp= 12119860

13 fmis the radius of the colliding nucleus in the transverse planeThe energy of the photon in the fluid rest frame is given by119864120574= 119901

119879cosh(119910 minus Θ) where Θ and 119910 are the space-time and

photon rapidities respectivelyThe anisotropy parameter andthe hard momentum scale enter through the differential ratevia 119889119873

120574119889

4119909119889119910119889

2119901119879(see [16 39ndash41] for details)

We plot the total photon yield coming from thermalQGP thermal hadrons and the initial hard contribution inFigure 2 and compare it with the RHIC data for variousvalues of 120591iso In the hadronic sector we include photons frombaryon-meson (BM) and meson-meson (MM) reactionsTwo scenarios have been considered (i) pure hydrodynamicsfrom the beginning and (ii) inclusion of momentum stateanisotropy We observe that (i) photons from BM reactionsare important (ii) pure hydro is unable to reproduce the datathat is some amount ofmomentumanisotropy is needed and(iii) exclusion of BM reactions underpredict the dataWenotethat the value of 120591iso needed to describe the data also lies inthe range 15 fmc ge 120591iso ge 05 fmc for both values of thetransition temperatures

22 Dileptons The dilepton production from AQGP hasbeen estimated in [17 42 43] using the same space-timemodel It is argued in [17] that the transverse momentumdistribution of lepton pair in AQGP could provide a goodinsight about the estimation of 120591iso We will briefly discussthe highmass dilepton yield along with the 119901

119879distribution in

AQGP Here we consider only the QGP phase as in the highmass region the yield from the hadronic reactions and decayshould be suppressed The dilepton production from quark-antiquark annihilation can be calculated from kinetic theoryand is given by

119864119889119877

1198893119875

= int1198893p

1

(2120587)3

1198893p

2

(2120587)3119891119902(p

1) 119891

119902(p

2) V

119902119902120590119897+119897minus

119902119902120575(4)

(119875 minus 1199011minus 119901

2)

(10)


0

2

4

6

8

10

0 2 4 6 8 10120591 (fmc)

120585

Ti = 044GeV 120591i = 015 fmc

(a)

01

02

03

04

05

0 2 4 6 8 10120591 (fmc)

pha

rd(G

eV)

Model IModel II

Ti = 044GeV 120591i = 015 fmc

(b)

Figure 1 (Color online) Time evolutions of (a) the anisotropy parameter 120585 and (b) the hard momentum scale 119901hard in the two space-timemodels described in the text The graphs are taken from [15]

(a)

0 1 2 3 4 5 6 7pT (GeVc)

10minus6

10minus4

10minus2

100

dNd

2pTdy

(GeV

minus2)

MM (pure hydro Tc = 192MeV)MM + BM (pure hydro Tc = 192MeV)

MM (120591iso = 1 fmc Tc = 192MeV)

Ti = 440MeV 120591i = 01 fmc

MM + BM (120591iso = 1 fmc Tc = 192MeV)120591iso = 05 fmc120591iso = 15 fmc

Figure 2 (Color online) Photon transverse momentum distribu-tions at RHIC energies The initial conditions are taken as 119879

119894=

440MeV 120591119894= 01 fmc and 119879

119888= 192MeV [16]

where 119891119902(119902)

is the phase space distribution function of themedium quarks (anti-quarks) V

119902119902is the relative velocity

between quark and anti-quark and 120590119897+119897minus

119902119902is the total cross-sec-

tion Consider

120590119897+119897minus

119902119902=

4120587

3

1205722

1198722(1 +

21198982

119897

1198722)(1 minus

41198982

119897

1198722)

12

(11)

Using the anisotropic distribution functions for the quark(antiquark) defined earlier the differential dilepton produc-tion rate can be written as [43]

119889119877

1198894119875

=5120572

2

181205875times int

1

minus1

119889 (cos 1205791199011)

times int

119886minus

119886+

1198891199011

radic1205941199011119891119902(radicp2

1(1 + 120585cos2120579p1) 119901hard)

times 119891119902(radic(E minus p

1)2+ 120585(p

1cos 120579p1 minus P cos 120579P)

2

119901hard)

(12)

The invariantmass and119901119879distributions of lepton pair can

be obtained after space-time integration using the evolutionmodel described earlier The final rates are as follows [17]

119889119873

1198891198722119889119910= 120587119877

2

perpint119889

2119875119879int

120591119891

120591119894

int

infin

minusinfin

119889119877

1198894119875120591119889120591 119889120578

119889119873

1198892119875119879119889119910

= 1205871198772

perpint119889119872

2int

120591119891

120591119894

int

infin

minusinfin

119889119877ann1198894119875

120591119889120591 119889120578

(13)

The numerical results are shown in Figure 3 for theinitial conditions 120591

119894= 088 fmc and 119879

119894= 845MeV cor-

responding to the LHC energies For 120591iso sim 2 fmc it isobserved that the dilepton yield from AQGP is comparableto Drell-Yan processThe 119901

119879distribution shows (Figure 3(b))

that the medium contribution dominates over all the othercontributions upto 119901

119879sim 9GeV The extraction of the

isotropization time can only be determined if these resultsare confronted with the data after the contributions from


2 4 6 8 10M (GeV)

minus9

minus8

minus7

minus6

minus5

Heavy quarksDrell YanJet conversion

Medium (120591iso = 0088 fmc)Medium (120591iso = 2 fmc)

log10

(dN

e+eminus

dydM

2(G

eVminus2))

(a)

2 4 6 8 10PT (GeV)

minus9

minus8

minus7

minus6

minus5

minus4

minus3

Drell YanJet conversion


log10

(dN

e+eminus

dydp2 T

(GeV

minus2))

(b)

Figure 3 (Color online) Invariant mass (a) and momentum (b) distribution of midrapidity dileptons in central Pb + Pb collisions at LHCThe figures are taken from [17]

the semileptonic decay from heavy quarks and Drell-Yanprocesses are subtracted from the total yield

3 Heavy Quark Potential andQuarkonium States in AQGP

In this section we will discuss the heavy quark potential inAQGP that has been calculated in [48] It is to be noted thatthis formalism will enable us to calculate the radiative energyloss of both heavy and light quarks and this will be discussedin Section 4 To calculate the interquark potential one startswith the retarded gluon self-energy expressed as [49]

Π120583]

(119870) = 1198922int

1198893119901

(2120587)3119875120583 120597119891 (p)

120597119875120573

(119892120573]

minus119875]119870120573

119870 sdot 119875 + 119894120576) (14)

This tensor is symmetric Π120583](119870) = Π

]120583(119870) and transverse

119870120583Π120583](119870) = 0 The spatial components of the self-energy

tensor can be written as

Π119894119895(119870) = minus119892

2int

1198893119901

(2120587)3V119894120597119897119891 (p) (120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598) (15)

where 119891(p) is the arbitrary distribution function To includethe local anisotropy in the plasma one has to calculate thegluon polarization tensor incorporating anisotropic distribu-tion function of the constituents of the medium We assumethat the phase-space distribution for the anisotropic plasma isgiven by (6) Using the ansatz for the phase space distributiongiven in (6) one can simplify (15) to

Π119894119895(119870) = 119898

2

119863int

119889Ω

(4120587)V119894

V119897 + 120585 (v sdot n) 119899119897

(1 + 120585(v sdot n)2)2(120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598)

(16)

where119898119863is the Debye mass for isotropic medium represent-

ed by

1198982

119863= minus

1198922

21205872int

infin

0

1198891199011199012119889119891iso (119901

2)

119889119901 (17)

Due to the anisotropy direction the self-energy apartfrom momentum k also depends on the anisotropy vectorn with 119899

2= 1 Using the proper tensor basis [28] one can

decompose the self-energy into four structure functions as

Π119894119895(119896) = 120572119860

119894119895+ 120573119861

119894119895+ 120574119862

119894119895+ 120575119863

119894119895 (18)

where

119860119894119895= 120575

119894119895minus

119896119894119896119895

1198962 119861

119894119895= 119896

119894119896119895

119862119894119895=

119899119894119899119895

1198992 119863

119894119895= 119896

119894119899119895+ 119899

119894119896119895

(19)

with 119899119894= 119860

119894119895119899119895 which obeys 119899 sdot 119896 = 0 120572 120573 120574 and 120575 are

determined by the following contractions

119896119894Π119894119895119896119895= k2120573 119899

119894Π119894119895119896119895= 119899

2k2120575

119899119894Π119894119895119899119895= 119899

2(120572 + 120574) TrΠ119894119895

= 2120572 + 120573 + 120574

(20)

Before going to the calculation of the quark-quark potentiallet us study the collective modes in AQGP which have beenthoroughly investigated in [28 29] andwe briefly discuss thishere The dispersion law for the collective modes of aniso-tropic plasma in temporal axial gauge can be determined byfinding the poles of propagator Δ119894119895 as follows

Δ119894119895(119870) =

1

[(k2 minus 1205962) 120575119894119895 minus 119896119894119896119895 + Π119894119895 (119896)] (21)


Substituting (19) in the previous equation and performing theinverse formula [28] one findsΔ (119870) = Δ

119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]

(22)The dispersion relation for the gluonic modes in anisotropicplasma is given by the zeros of

Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)

Let us first consider the stable modes for real 120596 gt 119896 inwhich case there are atmost two stablemodes stemming fromΔminus1

119866= 0 The other stable mode comes from zero of Δ

minus1

119860

Thus for finite 120585 there are three stablemodes Note that thesemodes depend on the angle of propagation with respect tothe anisotropy axis The dispersion relation for the unstablemodes can be obtained by letting 120596 rarr 119894Γ in Δ

minus1

119866= 0 and Δ

minus1

119860

leading to two unstablemodes and thesemodes again dependthe direction of propagation with respect to the anisotropyaxis

The collective modes in a collisional AQGP have beeninvestigated in [50] using Bhatnagar-Gross-Krook collisionalkernel It has been observed that inclusion of the collisionsslows down the growth rate of unstable modes and the insta-bilities disappear at certain critical values of the collisionfrequency

In order to calculate the quark-quark potential we resortto the covariant gauge Using the previous expression forgluon self-energy in anisotropic medium the propagator incovariant gauge can be calculated after some cumbersomealgebra [48] as follows

Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)

The structure functions (120572 120573 120574 and 120575) depend on 120596k 120585 and on the angle (120579

119899) between the anisotropy vector

and the momentum k In the limit 120585 rarr 0 the structurefunctions 120574 and 120575 are identically zero and 120572 and 120573 are directlyrelated to the isotropic transverse and longitudinal self-energies respectively [28] In anisotropic plasma the two-body interaction as expected becomes direction dependentNow the momentum space potential can be obtained fromthe static gluon propagator in the following way [32 48]

119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962

119911tanminus1 (radic120585) minus

119896119911k2

radick2 + 1205851198962perp

times tanminus1 (radic120585119896

119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2

119863((radic120585 + (1 + 120585) tanminus1 (radic120585) (k2 + 120585119896

2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)

times tanminus1 (radic120585119896119911radick2 + 1205851198962

perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp

radic120585tanminus1 (radic120585)

+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892

24120587 is the strong coupling constant and we

assume constant couplingThe coordinate space potential canbe obtained by taking Fourier transform of (26)

119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)

which under small 120585 limit reduces to [48]119881 (r 120585) asymp 119881iso (119903)

minus 1198922119862119865120585119898

2

119863int

1198893119896

(2120587)3119890minus119894ksdotr 23 minus (k sdot n)2k2

(k2 + 1198982

119863)2

(29)

where 119881iso(119903) = minus1198922119862119865119890minus119898119863119903(4120587119903) As indicated earlier the

potential depends on the angle between r and n When r nthe potential (119881

) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863


For arbitrary 120585 (28) has to be evaluated numerically Ithas been observed that because of the lower density of theplasma particles in AQGP the potential is deeper and closerto the vacuum potential than for 120585 = 0 [48] This means thatthe general screening is reduced in an AQGP

Next we consider quarkonium states in an AQGP wherethe potential to linear order in 120585 is given by (29) which canalso be written as [51]

119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)

where cos 120579 = 119903 sdot 119899 and the functions are given by [51]

1198910 (119903) =

6 (1 minus 119890119903) + 119903 [6 minus 119903 (119903 minus 3)]

121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)

With the previous expressions the real part of the heavyquark potential in AQGP after finite quark mass correctionbecomes [51]

119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]

minus 120590119903 exp (minus120583119903) minus08

1198982

119876119903

(34)

where 120583119898119863

= 1 + 120585(3 + cos 2120579)16 and the previousexpression is the minimal extension of the KMS potential[52] in AQGP The ground states and the excited states ofthe quarkonium states in AQGP have been found by solvingthe three-dimensional Schrodinger equation using the finitedifference time domain method [53] Without going intofurther details we will quote the main findings of the work of[51]The binding energies of charmonium and bottomoniumstates are obtained as a function of the hardmomentum scaleFor a fixed hard momentum scale it is seen that the bindingenergy increases with anisotropy Note that the potential (34)is obtained by replacing119898

119863by 120583 in KMS equation [52] For a

given hard momentum scale 120583 lt 119898119863 the quarkonium states

are more strongly bound than the isotropic caseThis impliesthat the dissociation temperature for a particular quarkoniumstate is more in AQGP It is found that the dissociationtemperature for 119869120595 in AQGP is 14119879

119888 whereas for isotropic

case it is 12119879119888[51]

Quarkonium binding energies have also been calculatedusing a realistic potential including the complex part in [54]by solving the 3D Schrodinger equation where the potentialhas the form

119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)

where the real part is given by (34) and the imaginary partis given in [55] The main results of this calculations are asfollows For 119869120595 the dissociation temperature obtained inthis case is 23119879

119888in isotropic case It has been found that

with anisotropy the dissociation temperature increases as thebinding of quarkonium states becomes stronger in AQGP

Thermal bottomonium suppression (119877119860119860

) at RHIC andLHC energies has been calculated in [56 57] in an AQGPTwo types of potentials have been considered there comingfrom the free energy (case A) and the internal energy (caseB) respectively By solving the 3D Schrodinger equationwith these potentials it has been found that the dissociationtemperature forΥ(1119904) becomes 373MeV and 735MeV for thecases A and B respectively whereas in case of 120585 = 0 thesebecomes 298MeV and 593MeV Thus the dissociation tem-perature increases in case of AQGP irrespective of the choiceof the potential In case of other bottomonium states thedissociation temperatures increase compared to the isotropiccase The nuclear modification factor has been calculated bycoupling AHYDRO [46] with the solutions of Schrodingerequation Introduction of AHYDRO into the picture makes119901hard and 120585 functions of proper time (120591) transverse coordinate(xperp) and the space-time rapidity (Θ) As a consequence both

the real and imaginary part of the binding energies becomefunctions of 120591 x

perp and Θ Now the nuclear modification

factor (119877119860119860

) is related to the decay rate (Γ) of the state inquestion where Γ = minus2I[119864] Thus 119877

119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)

where 120577(119901119879 xperp Θ) is given by

120577 (119901119879 xperp Θ) = 120579 (120591

119891minus 120591form)int

120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)

(37)Here 120591form is the formation time of the particular state inthe laboratory frame and 120591

119891is the time when the hard

momentumscale reaches119879119888 To study the nuclear suppression

of a particular state one needs to take into account the decayof the excited states to this particular state (so-called feeddown) For RHIC energies using the value of 119889119873ch119889119910 = 620

and various values of 120578119878 (note that 120578119878 is related to theanisotropy parameter 120585) the corresponding initial temper-atures are estimated with 120591

119894= 03 fmc 119901

119879integrated 119877

119860119860

values of individual bottomonium states (direct production)have been calculated using both the potentials A and B [5657] as functions of number of participants and rapidity It isseen that for both potentials more suppression is observedin case of anisotropic medium than in the isotropic caseHowever the suppression is more in case of potential modelA There is also indication of sequential suppression [56 57]Similar exercise has also been done in case of LHC energies(radic119904 = 276TeV) using a constant value of 120578119878 It is againobserved that the suppression in case of potential A is moreThese findings can be used to constrain 120578119878 by comparingwith the RHIC and LHC data

Next we discuss the effect of the initial state momentumanisotropy on the survival probability of 119869120595 due to gluondissociationThis is important as we have to take into accountall possibilities of quarkonium getting destroyed in the QGPto estimate the survival probability and hence 119877

119860119860 In con-

trast to Debye screening this is another possible mechanismof 119869120595 suppression in QGP In QGP the gluons have muchharder momentum sufficient to dissociate the charmonium


Such a study was performed in an isotropic plasma [58] Inthis context we will study the thermally weighted gluon dis-sociation cross of 119869120595 in an anisotropic media

Bhanot and Peskin first calculated the quarkonium-hadron interaction cross-section using operator productexpansion [59]The perturbative prediction for the gluon 119869120595

dissociation cross-section is given by [60]

120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)

where 1199020 is the energy of the gluon in the stationary 119869120595

frame 1205980is the binding energy of the 119869120595 where 119902

0gt 120598

0

and 119898119876is charm quark mass It is to be noted that we have

used the constant binding energy of the 119869120595 in AQGP atfinite temperature We assume that the 119869120595 moves with fourmomentum 119875 given by

119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752

119879is the 119869120595 transverse mass and 119910

is the rapidity of the 119869120595 A gluon with a four momentum119870 = (119896

0 k) in the rest frame of the parton gas has energy

1199020= 119870sdot119906 in the rest frame of the 119869120595 The thermal gluon 119869120595

dissociation cross-section in anisotropic media is defined as[15 58]

⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)

where Vrel = 1 minus (k sdot P)(1198960119872119879cosh119910) is the relative velocity

between 119869120595 and the gluon Now changing the variable (119870 harr

119876) one can obtain by using Lorentz transformation [15] thefollowing relations

1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579

119902) In the rest frame of

119869120595 numerator of (40) can be written as

int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)

while the denominator of (40) can be written as [30]

int1198893119896119891 (119896

0 120585 119901hard) = int119889

3119896119891iso (radick2 + 120585(k sdot n)2 119901hard)

=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)

where 120577(3) is the Riemann zeta functionThemaximumvalueof the gluon 119869120595 dissociation cross-section [60] is about 3mbin the range 07 le 119902

0le 17GeV Therefore high-momen-

tum gluons do not see the large object and simply passesthrough it and the low-momentum gluons cannot resolvethe compact object and cannot raise the constituents to thecontinuum

To calculate the survival probability of 119869120595 in an aniso-tropic plasma we consider only the longitudinal expansion ofthe matter The survival probability of the 119869120595 in the decon-fined quark-gluon plasma is

119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591

119894are the QGP formation time

119899119892(120591) = 16120577(3)119901

3

hard(120591)[1205872radic1 + 120585(120591)] is the gluon density

at a given time 120591 Now the 119869120595 will travel a distance in thetransverse direction with velocity v

119869120595given by

119889 = minus119903 cos120601 + radic1198772119860minus 1199032 (1 minus cos2120601) (45)

where cos120601 = V119869120595

sdot 119903 The time interval 120591120595= 119872

119879119889119875

119879is the

time before 119869120595 escapes from a gluon gas of transverse exten-sion 119877

perp The time evaluation of 120585 and 119901hard is determined by

(3) and (4)We now first discuss the numerical result of the thermal

averaged gluon dissociation cross-section in the anisotropicsystem The results are displayed in Figure 4 for 119875

119879= 0 and

119875119879= 8GeV for a set of values of the anisotropy parameter It

is seen that the velocity averaged cross-section decreases with120585 for 119901hard up to sim500MeV and then increases as comparedto the isotropic case (120585 = 0) (see Figure 4(a)) For higher119875119879 a similar feature has been observed in Figure 4(b) where

the cross-section starts to increase beyond 119901hard sim 200MeVFor fixed 119901hard the dissociation cross-section as a functionof 119875

119879shows that the cross-section first decreases and then

marginally increases [15]Equation (44) has been used to calculate the survival

probability Figure 5 describes the survival probability of 119869120595for various values of the isotropization time 120591iso at RHICenergy Left (Right) panel corresponds to 120579

119901= 1205872 (120579

119901=

1205873) It is observed that the survival probability remainsthe same as in the isotropic case upto 119875

119879= 4GeV in the

central region Beyond that marginal increase is observedwith the increase of 120591iso In the forward rapidity the results arealmost the same as the isotropic case throughout thewhole119875

119879

region For this set of initial conditions the argument of theexponential in (44) becomes similar to that of the isotropiccase Also the dissociation cross-section first decreases with119875119879and then increases Because of these reasons we observe

minor change in the survival probability [15] Thereforethe survival probability is more or less independent of thedirection of propagation of the 119869120595 with respect to theanisotropy axis whereas in the case of radiative energy loss of


0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)

Figure 4 (Color online) The thermal-averaged gluon 119869120595 dissociation cross-section as function of the hard momentum scale at centralrapidity (120579

119901= 1205872) for 120585 = 0 1 3 5 (a) corresponds to 119875

119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873

iso120591iso = 1 fmc

120591iso = 2 fmc120591iso = 3 fmc

(b)

Figure 5 (Color online)The survival probability of 119869120595 in an anisotropic plasma at central and forward rapidity regionThe initial conditionsare taken as 119879

119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc

fast partons in anisotropic media the result has been shownto depend strongly on the direction of propagation [32] Theresults for the LHC energies for the two sets of initial condi-tions at two values of the transition temperatures are shownin Figure 6 For set I we observe marginal increase as in thecase for set I at RHIC energies However for set II substantialmodification is observed for the reason stated earlier

4 Radiative Energy Loss and 119877119860119860

ofLight Hadrons

In this section we calculate the radiative energy loss in aninfinitely extended anisotropic plasma We assume that an

on-shell quark produced in the remote past is propagatingthrough an infinite QCD medium that consists of randomlydistributed static scattering centers which provide a color-screened Yukawa potential originally developed for theisotropic QCD medium given by [61]

119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909

119899is the location of the

119899th scattering center 119879 denotes the color matrices of theparton and the scattering center It is to be noted that the


0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)

Figure 6 (Color online) The survival probability of 119869120595 in an anisotropic plasma for the two sets of initial conditions at LHC energies Left(right) panel corresponds to 119879

119888= 200(170)MeV

potential has been derived by using hard thermal loop (HTL)propagator in QGP medium In anisotropic plasma the two-body interaction as expected becomes direction dependentThe two-body potential required to calculate the radiativeenergy loss has been calculated in the previous section

Now the parton scatters with one of the color center withthe momentum 119876 = (0 119902

perp 119902119911) and subsequently radiates a

gluon with momentum 119870 = (120596 119896perp 119896

119911) The quark energy

loss is calculated by folding the rate of gluon radiation Γ(119864)

with the gluon energy by assuming 120596 + 1199020asymp 120596 In the soft-

scattering approximation one can find the quark radiativeenergy loss per unit length as [62]

119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)

Here 119863119877is defined as [119905

119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862

119877= 43 119909 is the longitudinal

momentum fraction of the quark carried away by the emittedgluon

Now in anisotropic medium we have [32]

119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904

times int1198892119896perp

120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)

In the present scenario we assume that the parton ispropagating along the 119911 direction and makes an angle 120579

119899

with the anisotropy direction In such case we replace 119902perp

and 119902119911in (26) by 119902

perprarr radic1199022

perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr

|q| sin 120579119899cos120601 For arbitrary 120585 the radiative energy loss can

be written as [32]Δ119864

119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp

times radic1199024perp+ 21199022

perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)


In the previous expression 120582 denotes the average mean-freepath of the quark scattering and is given by

1

120582=

1

120582119892

+1

120582119902

(50)

which depends on the strength of the anisotropy In the lastexpression120582

119902and120582

119892correspond to the contributions coming

from 119902-119902 and 119902-119892 scattering respectively Consider

1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)

where 1198622(119894) is the Cashimir for the 119889

119894-dimensional represen-

tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888

for gluon scatterers 120588119894is the density of the scatterers Using

120588119894= 120588

iso119894

radic1 + 120585 we obtain

1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)

where119873119865is the numbers of flavors

The fractional energy loss in anisotropy medium for thelight quark is shown in Figure 7 We consider a plasma ata temperature 119879 = 250MeV with the effective number ofdegrees of freedom 119873

119865= 25 120572

119904= 03 and the length of

the medium is 119871 = 5 fm The energy loss in the anisotropicmedia depends on the angle of propagation of the fast partonwith respect to the anisotropy axis (n) We see that thefractional energy loss increases in the direction parallel to theanisotropy axis With the increase of anisotropy parameter120585 the fractional energy loss subsequently increases for 120579

119899=

1205876 However away from the anisotropy axis (120579119899

= 1205872)the fractional energy loss decreases because the quark-quarkpotential is stronger in the anisotropy direction [32] In theperpendicular direction to the anisotropy axis the fractionalenergy loss is quite small The fractional energy loss forthe heavy quarks (ie for charm and bottom) is shown inFigure 8 The fractional energy loss is enhanced in thedirection parallel to anisotropy axis as well as for 120579

119899= 1205876

However for 120579119899= 1205872 the fractional energy loss decreases

for both heavy and light quarksNext we consider the nuclear modification factor of light

hadrons incorporating the light quark energy loss in AQGPdiscussed in the previous paragraphs When a parton ispropagating in the direction of anisotropy it is found that thefractional energy loss increases In this section we will applythis formalism to calculate the nuclear modification factorof the light hadrons Starting with two-body scattering atthe parton level the differential cross-section for the hadronproduction is [63]

119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876

Figure 7 Color online fractional energy loss for the light quark for120585 = (05 1)

The argument of the 120575 function can be expressed in terms of119909119886and 119909

119887 and doing the 119909

119887integration we arrive at the final

expression as follows

119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910

) and the factor 119870 is introduced to takeinto account the higher-order effects It should be noted thatto obtain single-particle inclusive invariant cross-sectionthe fragmentation function 119863

ℎ119888(119911 119876

2) must be included

To obtain the hadronic 119901119879spectra in A-A collisions we

multiply the result by the nuclear overlap function for agiven centralityThe inclusion of jet quenching as a final stateeffect in nucleus-nucleus collisions can be implemented intwo ways (i) modifying the partonic 119901

119879spectra [64] and

(ii) modifying the fragmentation function [65] but keepingthe partonic 119901

119879spectra unchanged In this calculation we

intend to modify the fragmentation function The effectivefragmentation function can be written as

119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)

where 119911⋆ = 119911(1minusΔ119864119864) is themodifiedmomentum fractionNow we take into account the jet production geometry Weassume that all the jets are not produced at the same pointtherefore the path length transversed by the partons before


0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

Isotropic120579n = 0

120579n = 1205876120579n = 1205872

(b)

Figure 8 (Color online) Same as 7 for charm quark (a) and bottom quark (b) with 120585 = 1 and 119879 = 500MeV

the fragmentation is not the same We consider a jet initiallyproduced at (119903 120601) and leaves the plasma after a proper time(119905119871) or equivalently after traversing a distance 119871 (for light

quarks 119905119871= 119871) where

119871 (119903 120601) = radic1198772perpminus 1199032 sin1206012 minus 119877

perpcos120601 (56)

where119877perpis the transverse dimension of the system Since the

number of jets produced at r is proportional to the numberof binary collisions the probability is proportional to theproduct of the thickness functions as follows

P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)

In case of hard sphereP(119903) is given by [66]

P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)

where int1198892119903P(119903) = 1 To obtain the hadron 119901

119879spectra we

have to convolute the resulting expression over all transversepositions and the expression is

1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)

The quantity 119864(1198891198731198893119901119891) is the initial momentum distribu-

tion of jets and can be computed using LO-pQCD Here weuse average distance traversed by the partons ⟨119871⟩ is given by

⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)

where ⟨119871⟩ sim 58(62) fm for RHIC (LHC) Finally the nuclearmodification factor (119877

119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]

0corresponds to the hadron 119901

119879

distribution without the energy lossFor an expanding plasma the anisotropy parameter 119901hard

and 120585 are time dependent The time evaluation is againgiven by (3) and (4) In the present work it is assumedthat an isotropic QGP is formed at an initial time 120591

119894and

initial temperature 119879119894 Rapid longitudinal expansion of the

plasma leads to an anisotropic QGP which lasts till 120591iso For0ndash10 centrality (relevant for our case) we obtain 119879

119894=

440(350)MeV for 120591119894= 0147(024) fmc at RHIC energy [44]

Figure 9 describes the nuclearmodification factor for twodifferent initial conditions with various values of isotropiza-tion time 120591iso along with the PHENIX data [67] It is quiteclear from Figure 9(a) that the value of 119877

119860119860for anisotropic

medium is lower than that for the isotropic media as theenergy loss in the anisotropy medium is higher [32] It isalso observed that as 120591iso increases the value of 119877119860119860 decreasescompared to its isotropic value [44] This is because the hardscale decreases slowly as compared to the isotropic case thatis the cooling is slow For reasonable choices of 120591iso theexperimental data is well described It is seen that increasingthe value of 120591iso beyond 15 fmc grossly underpredict the dataWe find that the extracted value of isotropization time lies inthe range 05 le 120591iso le 15 fmc [44] This is in agreementwith the earlier finding of 120591iso using PHENIX photon data[16] In order to see the sensitivity on the initial conditions wenow consider another set of initial conditions 119879

119894= 350MeV

and 120591119894= 024 fmc The result is shown in Figure 9(b) It is

observed that to reproduce the data a larger value of 120591iso is


0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13

1205870 PHENIX data1205780 PHENIX dataIsotropic

120591iso = 05 fmc120591iso = 15 fmc120591iso = 1 fmc

(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA

1205870 PHENIX data1205780 PHENIX data Isotropic

120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)

Figure 9 (Color Online) Nuclear modification factor at RHIC energies The initial conditions are taken as (a) 119879119894= 440MeV and 120591

119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc

needed as compared to the case of higher initial temperatureWe extract an upper limit of 120591iso = 2 fmc [44] in this case

5 Plasma Wakes

It is mentioned earlier that when a jet propagates through hotanddensemedium it loses energymainly by the radiative pro-cess As mentioned earlier it also creates wake in the chargedensity as well as in the potential Now we calculate the wakein charge density and the wake potential due to the passageof a fast parton in a small 120585 limit Dielectric function containsessentially all the information of the chromoelectromagneticproperties of the plasma The dielectric function 120598(k 120596) canbe calculated from the dielectric tensor using the followingrelation

120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)

where the dielectric tensor 120598119894119895 can be written in terms of thegluon polarization tensor as follows (given by (18))

120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)

Therefore the dielectric function is directly related to thestructure functions mentioned in Section 3 through (18)(63) and (62) To get the analytic expressions for the structurefunctions one must resort to small 120585 limit To linear order in120585 we have [28]120572 = Π

119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911

2minus 1) sin2120579

119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)

where 119911 = 120596119896In the presence of the test charge particle the induced

charge density and the wake potential depend on the velocityof the external charged parton and also on the distribution


of the background particle [68] When a static test charge isintroduced in a plasma it acquires a shielding cloud As aresult the induced charge distribution is spherically symmet-ricWhen a charge particle is inmotion relative to the plasmathe induced charge distribution no longer remains symmet-ric As a result spherical symmetry of the screening cloudreduces to ellipsoidal shape

The passage of external test charge through the plasmaalso disturbs the plasma and creates induced color chargedensity [69]Therefore the total color charge density is givenas

120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)

where 119886 represents the color index However the total colorcharge density is linearly related to 120588

119886

ext through the dielectricresponse function (120588119886tot(k 120596) = 120588

119886

ext(k 120596)120598(k 120596)) Thereforethe induced color charge density is explicitly written as

120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)

Nowwe consider a charge particle119876119886movingwith a constantvelocity v and interacting with the anisotropic plasma Theexternal charge density associated with the test charge parti-cle can be written as [33 34 38]

120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)

The delta function indicates that the value of 120596 is real andthe velocity of the charge particle is restricted between 0 lt

V lt 1 which is known as the Cerenkov condition forthe moving parton in the medium Therefore the collectivemodes are determined in the space-like region of the 120596-119896plane [34 38] According toCerenkov condition therewill betwo important scenarios which occur due to the interactionof the particle and the plasmon wave first the modes whichare moving with a speed less than the average speed ofthe plasmon modes can be excited but the particle movingslightly slower than the wave will be accelerated While thecharge particle moving faster than the wave will decreaseits average velocity [69] The slowly moving particle absorbsenergy from the wave the faster moving particle transfersits extra energy to the wave The absorption and emission ofenergy result in a wake in the induced charge density as wellas in the potential Second when the charge particle movingwith a speed greater than the average phase velocity V

119901 the

modes are excited and they may not be damped Such excitedmodes can generateCerenkov-like radiation and aMach stem[70] which leads to oscillation both in the induced chargedensity and in the wake potential

Substituting (68) into (67) and transforming into r-119905space the induced charge density becomes

120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)

times 120575 (120596 minus k sdot v)

(69)

First we consider the case where the fast parton is movingin the beam direction that is v n In spherical coordinatesystem k = (119896 sin 120579

119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579

119899) and the cyl-

indrical coordinate for r = (120588 0 119911) therefore the inducedcharge density can be written as

120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)

where 120594 is represented as cos 120579119899 1198690is the zeroth-order Bessel

function Γ = 119896120594(119911 minus V119905)119898119863 and Δ = (Re 120598(k 120596))

2+

(Im 120598(k 120596))2 To get the previous equation we use the simple

transformation 120596 rarr 120596119898119863and 119896 rarr 119896119898

119863 It is seen that the

charge density 120588119886

ind is proportional to1198983

119863

Numerical evaluation of the previous equation leads tothe contour plots of the induced charge density shown inFigure 10 with two different speeds of the fast parton Thecontour plot of the equicharge lines shows a sign flip along thedirection of the moving parton in Figure 10 The left (right)panel shows the contour plot of the induced color chargedensity in both isotropic and anisotropic plasma with partonvelocity V = 055(099) It is clearly seen that because ofanisotropy the positive charge lines appear alternately in thebackward space which indicates a small oscillatory behaviorof the color charge wake (see Figure 10(c)) When the chargeparticle moves faster than the average plasmon speed theinduced charge density forms a cone-like structure whichis significantly different from when the parton velocity isV = 055 It is also seen that the induced charge densityis oscillatory in nature The supersonic nature of the partonleads to the formation of Mach cone and the plasmon modescould emit a Cerenkov-like radiation which spatially limitsthe disturbances in the induced charge density [71] In thebackward space ((119911 minus V119905) lt 0) induced color charge densityis very much sensitive to the anisotropic plasma than that inthe forward space ((119911 minus V119905) gt 0) Due to the effect of theanisotropy the color charge wake is modified significantlyand the oscillatory behavior is more pronounced than theisotropic case It is also seen that the oscillatory natureincreases with the increase of the anisotropic parameter 120585

Next we consider the case when the parton moves per-pendicular to the anisotropy direction in which case theinduced charge density can be written as

120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD

minus10 minus5 0 5 10

minus000025minus00006minus00012minus0002

(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5

minus00015minus00008minus00005

minus00015minus00008

minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD

minus10 minus5 0 5 10(z

minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)

Figure 10 (Color online) Left Panel the plot shows equicharge line with parton velocity V = 055 for different 120585(0 08) Right Panel same asleft panel with parton velocity V = 099

with Ω = 119896(119911120594 + (120588 minus V119905)radic1 minus 1205942 cos120601)119898119863 Numerical

results of the equicharge lines are shown in Figure 11 Theleft (right) panel shows the contour plots of the inducedcharge density for the parton velocity V = 055(099)When V = 099 the number of induced charge lines thatappear alternately in the backward space is reduced for theanisotropic plasma in comparison to the isotropic plasmaTherefore the anisotropy reduces the oscillatory behavior ofthe induced color charge density when the parton movesperpendicular to the anisotropy direction

According to the Poisson equation the wake potentialinduced by the fast parton reads as [33]

Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)

Substituting (68) into (72) and transforming to the configu-ration space the wake potential is given by [38]

Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)

Using similar coordinate system as before the screeningpotential turns into

Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)

We solve the wake potential for the two special cases (i) alongthe parallel direction of the fast parton that is r v and also120588 = 0 and (ii) perpendicular to direction of the parton thatis r perp v The potential for the parallel case is obtained as

Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)

whereas that for the perpendicular case we have

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD

minus10 minus5 0 5 10minus10

minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)

Figure 11 (Color online) The left (right) panel shows the equicharge lines for V = 055(099) In this case the parton moves perpendicular tothe direction of anisotropy [38]

times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863

Figure 12 describes the scaled wake potential in twospecified directions In these figures the scaled parameterΦ119886

0is given by (2120587

2119898

119863)Φ

119886 The left panel shows the wakepotential along the direction of the moving color chargeIn the backward direction the wake potential for isotropicplasma decreases with the increase of 119911 minus V119905 and exhibitsa negative minimum when V = 055 With the increaseof the anisotropic parameter 120585 the depth of the negativeminimum decreases for V = 055 At large 120585(08) there isno negative minimum and the wake potential behaves likea modified Coulomb potential For V = 099 the wakepotential is Lennard-Jones potential type which has a shortrange repulsive part as well as a long range attractive part[34 38] in both isotropic and anisotropic plasma It is alsoseen that the wake potential is oscillatory in nature in thebackward direction It is clearly visible that the depth of thenegative minimum is increased compared to the case whenV = 055 Because of the anisotropy effect the oscillation ofthe wake potential is more pronounced and it extends to alarge distance [38] In the forward direction the screening

potential is a modified Coulomb potential in both types ofplasma Figure 12(b) describes the wake potential along theperpendicular direction of the moving parton It can be seenthat thewake potential is symmetric in backward and forwarddirection no matter what the speed is [38] In presence ofthe moving charge particle the wake potential is Lennard-Jones type When V = 055 the value of negative minimumis increased with increase of 120585 but in case of V = 099 itdecreases with 120585 However with the increase of 120585 the depth ofnegative minimum is moving away from the origin for boththe jet velocities considered here

Next we consider the case when the parton movesperpendicular to the beam direction The wake potential in(74) is also evaluated for the two spacial cases (i) along thedirection of the moving parton that is r v and (ii) perpen-dicular direction of the parton that is r perp v The wakepotential for the parallel case can be written as [38]

Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)


minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10(z minus t)mD

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)

Figure 12 (Color online) (a) scaled wake potential along the motion of the fast parton that is 119911-axis for different 120585with two different partonvelocity V = 055 and V = 099 (b) same as (a) but perpendicular to direction of motion of the parton

whereΩ1015840= 119896(120588minusV119905)radic1 minus 1205942 cos120601119898

119863 For the perpendicular

case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840

= 119896(119911120594 minus V119905radic1 minus 1205942 cos120601)119898119863 The left panel

in Figure 13 shows screening potential along the paralleldirection of the moving color charge The behavior of thewake potential is more like a modified Coulomb (Lennard-Jones) potential at parton velocity V = 055(099) For V =

099 the wake potential shows an oscillatory behavior inan isotropic plasma [34] but in anisotropic case oscillatorystructure of the wake potential is smeared out for 120585 = 05 andv perp n [38] But the depth of the negative minimum increasesin the case of anisotropic plasma for both the parton velocitiesconsidered here The behavior of the wake potential in theperpendicular direction of themoving parton is shown in theright panel of Figure 13 At V = 055 the anisotropy modifiesthe structure of the wake potential significantly that is itbecomes modified Coulomb potential instead of Lennard-Jones potential For V = 099 the wake potential is a Lennard-Jones potential type but the depth of the minimum decreasesin anisotropic plasma [38]

6 Summary and Discussions

Wehave reviewed the effect of initial statemomentum anisot-ropy that can arise in an AQGP on various observables Itis shown that electromagnetic probes could be a good signalthat can be used to characterize this anisotropic state as thiscan only be realized in the early stages of heavy ion collisionsIt has been demonstrated that the isotropization time of theQGP can be extracted by comparing the photon yieldwith theexperimental data We further estimate the radiative energyloss of a fast moving parton (both heavy and light flavours)in an AQGP and show that the it is substantially differentfrom that in the isotropic QGP Moreover it depends on thedirection of propagation of the parton with the anisotropicaxis Related to this is the nuclear modification factor of lighthadrons that is produced due to the fragmentation of lightpartons which lose energy in the mediumThuswe have alsodiscussed the nuclear modification factor in the context ofAQGP and compared it with the RHIC data to extract theisotropization time The extracted value is compatible withthat obtained from photon data

It might be mentioned here that the presence of unstablemodes in AQGP may affect radiative energy loss Howeverin [72 73] the authors have shown that the polarization loss(collisional loss) remains unaffected by the unstable modesbut in a recent paper [74] it is shown that the polarisation lossindeed has strong time and directional dependence and alsothe nature of the loss is oscillatory Such effect may be presentin radiative energy loss In this review we did not consider it

The heavy quark potential and the quarkonium states inAQGP have also been reviewed with both real and complexvalued potential In all these calculations it has been foundthat the dissociation temperature of various quarkoniumstates increases in comparison with the isotropic case Wehave also focused on the nuclear modification factors of var-ious bottomonium states which have been calculated by


minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10(120588 minus t)mD

minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10zmD

minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)

Figure 13 (Color online) The left (right) panel shows scaled wake potential for 120585 = 0 05 with parton velocity V = 055(099) In this casethe parton moves perpendicular to the direction of anisotropy [38]

combining hydrodynamics and solutions of 3D Schrodingerequation using two types of complex valued potentials

Apart from the energy loss of a jet in a medium the jetalso creates wake in the plasma We have demonstrated thatdue to the jet propagation in anAQGP thewake potential andthe charge density are significantly modified in comparisonwith the isotropic case

We end by mentioning that the ADSCFT calculationof the electromagnetic correlator has been performed instrongly coupled N = 4 super Yang-Mills theory usinganisotropicmomentumdistribution [75] Photon productionrate is then estimated and it is concluded that in the weakcoupling limit the rate is consistent with that in [41] with anoblate phase space distribution in momentum space Thereare other models that deal with the gravity dual theory foranisotropic plasma with additional bulk fields [76 77] Thusa comparative study of various observables in gravity dualtheory should be done in the future

References

[1] J-E Alam S Sarkar P Roy T Hatsuda and B Sinha ldquoThermalphotons and lepton pairs from quark gluon plasma and hothadronic matterrdquo Annals of Physics vol 286 no 2 pp 159ndash2482000

[2] H Satz and T Matsui ldquoJ120595 suppression by quark-gluon plasmaformationrdquo Physics Letters B vol 178 no 4 pp 416ndash422 1986

[3] J D Bjorken ldquoPartons lose energy due to interactions with themediumrdquo FERMILAB-Pub-8259-THY Erratum 1982

[4] M Gyulassy P Levai and I Vitev ldquoJet quenching in thin quark-gluon plasmas I formalismrdquoNuclear Physics B vol 571 pp 197ndash233 2000

[5] B G Zakharov ldquoOn the energy loss of high energy quarks in afinite-size quark-gluon plasmardquo JETP Letters vol 73 no 2 pp49ndash52 2001

[6] M Djordjevic and U Heinz ldquoRadiative energy loss in a finitedynamical QCD mediumrdquo Physical Review Letters vol 101 no2 Article ID 022302 4 pages 2008

[7] G-Y Qin J Ruppert C Gale S Jeon G D Moore and M GMustafa ldquoRadiative and collisional jet energy loss in the quark-gluon plasma at the BNL relativistic heavy ion colliderrdquo PhysicalReview Letters vol 100 no 7 Article ID 072301 2008

[8] R Baier Y LDokshitzer AHMueller andD Schiff ldquoQuench-ing of hadron spectra in mediardquo Journal of High Energy Physicsvol 2001 article 033 2001

[9] A K Dutt-Mazumder J Alam P Roy and B Sinha ldquoStoppingpower of hot QCD plasmardquo Physical Review D vol 71 no 9Article ID 094016 9 pages 2005

[10] P Roy J-E Alam and A K Dutt-Mazumder ldquoQuenching oflight hadrons at RHIC in a collisional energy loss scenariordquoJournal of Physics G vol 35 no 10 Article ID 104047 2008

[11] PHuovinen P KolbUHeinz and PV Ruuskanen ldquoRadial andelliptic flow at RHIC further predictionsrdquo Physics Letters B vol503 no 1-2 pp 58ndash64 2001

[12] T Hirano and K Tsuda ldquoCollective flow and two-pion correla-tions from a relativistic hydrodynamic model with early chem-ical freeze-outrdquo Physical Review C vol 66 no 5 Article ID054905 14 pages 2002

[13] M J Tannenbaum ldquoRecent results in relativistic heavy ion colli-sions from lsquoa new state of matterrsquo to lsquothe perfect fluidrsquordquo Reportson Progress in Physics vol 69 no 7 2005

[14] M Luzum and P Romatschke ldquoConformal relativistic viscoushydrodynamics applications to RHIC results at radic119904

119873119873=

200GeVrdquo Physical Review C vol 78 no 3 Article ID 03491517 pages 2008

[15] MMandal and P Roy ldquoGluon dissociation of J120595 in anisotropicquark-gluon plasmardquo Physical Review C vol 86 no 2 ArticleID 024915 7 pages 2012

[16] L Bhattacharya and P Roy ldquoMeasuring the isotropization timeof quark-gluon plasma fromdirect photons at energies availableat the BNL Relativistic Heavy Ion Collider (RHIC)rdquo PhysicalReview C vol 79 no 5 Article ID 054910 7 pages 2009


[17] M Martinez and M Strickland ldquoDilepton production as ameasure of QGP thermalization timerdquo Journal of Physics G vol35 no 10 Article ID 104162 2008

[18] R Baier A H Muller D Schiff and D T Son ldquolsquoBottom-uprsquothermalization in heavy ion collisionsrdquo Physics Letters B vol502 no 1ndash4 pp 51ndash58 2001

[19] EWebel ldquoSpontaneously growing transverse waves in a plasmadue to an anisotropic velocity distributionrdquo Physical ReviewLetters vol 2 no 3 p 83 1959

[20] S Mrowczynski ldquoPlasma instability at the initial stage ofultrarelativistic heavy-ion collisionsrdquo Physics Letters B vol 314no 1 pp 118ndash121 1993

[21] S Mrowczynski ldquoColor collective effects at the early stage ofultrarelativistic heavy-ion collisionsrdquo Physical Review C vol 49no 4 pp 2191ndash2197 1994

[22] S Mrowczynski ldquoColor filamentation in ultrarelativistic heavy-ion collisionsrdquoPhysics Letters B vol 393 no 1-2 pp 26ndash30 1997

[23] S Mrowczynski ldquoInstabilities driven equilibration of thequarkmdashgluon plasmardquoActa Physica Polonica B vol 37 pp 427ndash454 2006

[24] P Arnold J Lenghan G D Moore and L G Yaffe ldquoapparentthermalization due to plasma instabilities in the quark-gluonplasmardquo Physical Review Letters vol 94 no 7 Article ID072302 4 pages 2005

[25] A Rebhan P Romatschke and M Strickland ldquoHard-loopdynamics of non-abelian plasma instabilitiesrdquo Physical ReviewLetters vol 94 no 10 Article ID 102303 4 pages 2005

[26] P Romatschke and R venugopalan ldquoCollective non-abelianinstabilities in amelting color glass condensaterdquoPhysical ReviewLetters vol 96 no 6 Article ID 062302 4 pages 2006

[27] M Strickland ldquoThe chromo-weibel instabilityrdquo Brazilian Jour-nal of Physics vol 37 no 2 p 702 2007

[28] P Romatschke and M Strickland ldquoCollective modes of ananisotropic quark-gluon plasmardquo Physical ReviewD vol 68 no3 Article ID 036004 8 pages 2003

[29] P Romatschke and M Strickland ldquoCollective modes of ananisotropic quark-gluon plasma IIrdquo Physical Review D vol 70no 11 Article ID 116006 9 pages 2004

[30] P Romatschke and M Stricland ldquoCollisional energy loss of aheavy quark in an anisotropic quark-gluon plasmardquo PhysicalReview D vol 71 no 12 Article ID 125008 2005

[31] P Romatschke ldquoMomentum broadening in an anisotropic plas-mardquo Physical Review C vol 75 no 1 Article ID 014901 8 pages2007

[32] P Roy and A K Dutt-Mazumder ldquoRadiative energy loss in ananisotropic quark-gluon plasmardquo Physical Review C vol 83 no4 Article ID 044904 6 pages 2011

[33] J Ruppert and B Muller ldquoWaking the colored plasmardquo PhysicsLetters B vol 618 no 1ndash4 pp 123ndash130 2005

[34] P ChakrabortyMGMustafa andMHThoma ldquoWakes in thequark-gluon plasmardquo Physical Review D vol 74 no 9 ArticleID 094002 16 pages 2006

[35] P Chakraborty M G Mustafa R Ray and M H ThomaldquoWakes in a collisional quark-gluon plasmardquo Journal of PhysicsG vol 34 no 10 p 2141 2007

[36] B-F Jiang and J-R Li ldquoThewake potential in the viscous quark-gluon plasmardquo Nuclear Physics A vol 856 no 1 pp 121ndash1332011

[37] B-f Jiang and J-r Li ldquoThe polarized charge distributioninduced by a fast parton in the viscous quark-gluon plasmardquoJournal of Physics G vol 39 no 2 Article ID 025007 2012

[38] M Mandal and P Roy ldquoWake in anisotropic quark-gluonplasmardquo Physical Review D vol 86 no 11 Article ID 1140029 pages 2012

[39] L Bhattacharya and P Roy ldquoPhotons from anisotropic quark-gluon plasmardquo Physical Review C vol 78 no 6 Article ID064904 2008

[40] L Bhattacharya and P Roy ldquoRapidity distribution of photonsfrom an anisotropic quark-gluon plasmardquo Physical Review Cvol 81 no 5 Article ID 054904 2010

[41] B Schenke and M Stricland ldquoPhoton production from ananisotropic quark-gluon plasmardquo Physical ReviewD vol 76 no2 Article ID 025023 5 pages 2007

[42] M Martinez and M Strickland ldquoMeasuring quark-gluon-plas-ma thermalization timewith dileptonsrdquo Physical Review Lettersvol 100 no 10 Article ID 102301 4 pages 2008

[43] M Martinez and M Strickland ldquoPre-equilibrium dileptonproduction from an anisotropic quark-gluon plasmardquo PhysicalReview C vol 78 no 3 Article ID 034917 19 pages 2008

[44] M Mandal L Bhattacharya and P Roy ldquoNuclear modificationfactor in an anisotropic quark-gluon plasmardquo Physical ReviewC vol 84 no 4 Article ID 044910 10 pages 2011

[45] MMartinez andM Strickland ldquoDissipative dynamics of highlyanisotropic systemsrdquo Nuclear Physics A vol 848 no 1-2 pp183ndash197 2010

[46] M Martinez and M Strickland ldquoNon-boost-invariant aniso-tropic dynamicsrdquo Nuclear Physics A vol 856 no 1 pp 68ndash872011

[47] K Kajantie J Kapusta L McLerran and A Mekjian ldquoDileptonemission and the QCD phase transition in ultrarelativisticnuclear collisionsrdquo Physical Review D vol 34 no 9 pp 2746ndash2754 1986

[48] A Dumitru Y Guo and M Strickland ldquoThe heavy-quark po-tential in an anisotropic plasmardquo Physics Letters B vol 662 no1 pp 37ndash42 2008

[49] J P Blaizot and E Iancu ldquoThe quark-gluon plasma collectivedynamics and hard thermal loopsrdquo Physics Reports vol 359 no5-6 pp 155ndash528 2002

[50] B Schenke M Strickland C Greiner and M Thoma ldquoModelof the effect of collisions on QCD plasma instabilitiesrdquo PhysicalReview D vol 73 no 12 Article ID 125004 12 pages 2006

[51] ADumitru Y Gao AMocsy andM Strickland ldquoQuarkoniumstates in an anisotropicQCDplasmardquoPhysical ReviewD vol 79no 5 Article ID 054019 10 pages 2009

[52] F Karsch M T Mehr and H Satz ldquoColor screening anddeconfinement for bound states of heavy quarksrdquo Zeitschrift furPhysik C vol 37 no 4 pp 617ndash622 1988

[53] I D Sudiarta and D J W Geldart ldquoSolving the Schrodingerequation using the finite difference time domain methodrdquoJournal of Physics A vol 40 no 8 pp 1885ndash1896 2007

[54] M Margotta K McCarty C McGahan M Strickland andD Yager-Elorriaga ldquoQuarkonium states in a complex-valuedpotentialrdquo Physical Review D vol 83 Article ID 105019 2011

[55] A Dumitru Y Gao and M Strickland ldquoImaginary part ofthe static gluon propagator in an anisotropic (viscous) QCDplasmardquo Physical Review D vol 79 no 11 Article ID 114003 7pages 2009

[56] M Strickland ldquoThermalΥ(1119904) and 120594

1198871Suppression atradic(119904

119873119873) =

276 TeV Pb-Pb Collisions at the LHCrdquo Physical Review Lettersvol 107 no 13 Article ID 132301 4 pages 2011

[57] M Strickland and D Bazow ldquoThermal bottomonium suppres-sion at RHIC and LHCrdquo Nuclear Physics A vol 879 pp 25ndash582012


[58] X-M Xu D Kharzeev H Satz and X-N Wang ldquoJ120595 suppres-sion in an equilibrating parton plasmardquo Physical Review C vol53 no 6 pp 3051ndash3056 1996

[59] G Bhanot andM E Peskin ldquoShort-distance analysis for heavy-quark systems (II) ApplicationsrdquoNuclear Physics B vol 156 no3 pp 391ndash416 1979

[60] D Kharzeev and H Satz ldquoColour deconfinement and quarko-nium dissociationrdquo in Quark-Gluon Plasma II R C Hwa EdWorld Scientific Singapore 1995

[61] M Gyulassy and X N Wang ldquoMultiple collisions and inducedgluon bremsstrahlung in QCDrdquo Nuclear Physics B vol 420 no3 pp 583ndash614 1994

[62] M Djordjevic ldquoTheoretical formalism of radiative jet energyloss in a finite size dynamical QCD mediumrdquo Physical ReviewC vol 80 no 6 Article ID 064909 27 pages 2009

[63] J F Owens ldquoLarge-momentum-transfer production of directphotons jets and particlesrdquo Reviews of Modern Physics vol 59no 2 pp 465ndash503 1987

[64] P Roy A K Dutt-Mazumder and J-E Alam ldquoEnergy loss anddynamical evolution of quark p

119879spectrardquo Physical Review C

vol 73 no 4 Article ID 044911 4 pages 2006[65] X NWang ldquoEffects of jet quenching on high p

119879hadron spectra

in high-energy nuclear collisionsrdquoPhysical ReviewC vol 58 no4 pp 2321ndash2330 1998

[66] S Turbide C Gale S Jeon and G D Moore ldquoEnergy lossof leading hadrons and direct photon production in evolvingquark-gluon plasmardquo Physical Review C vol 72 no 1 ArticleID 014906 15 pages 2005

[67] S S Adler S Afanasiev C Aidala et al ldquoCommon suppressionpattern of 120578 and 120587

0 mesons at high transverse momentumin Au+Au collisions at radic(119904

119873119873) = 200GeVrdquo Physical Review

Letters vol 96 no 20 Article ID 202301 6 pages 2006[68] N A Krall and A W Trivilpiece Principles of Plasma Physics

McGraw-Hill New York NY USA 1973[69] S Ichimaru Basic Principles of Plasma Physics W A Benjamin

Menlo Park Calif USA 1973[70] L D Landau and E M Lifshitz Fluid Mechanics Butterworth-

Heinemann Woburn Mass USA 2nd edition 1987[71] W J Miloch ldquoWake effects and Mach cones behind objectsrdquo

Plasma Physics and Controlled Fusion vol 52 no 12 Article ID124004 2010

[72] P Romatschke and M strickland ldquoEnergy loss of a heavyfermion in an anisotropic QED plasmardquo Physical Review D vol69 no 6 Article ID 065005 13 pages 2005

[73] P Romatschke and M strickland ldquoCollisional energy loss ofa heavy quark in an anisotropic quark-gluon plasmardquo PhysicalReview D vol 71 no 12 Article ID 125008 12 pages 2005

[74] M E Carrington K Deja and S Mrowczynski ldquoEnergy loss inunstable quark-gluon plasma with extremely prolate momen-tum distributionrdquo Acta Physica Polonica B vol 6 supplementp 545 2013

[75] A Rebhan and D Steineder ldquoElectromagnetic signatures of astrongly coupled anisotropic plasmardquo Journal of High EnergyPhysics vol 2011 article 153 2011

[76] D Mateos and D Trancanelli ldquoAnisotropic119873 = 4 Super-Yang-Mills Plasma and Its Instabilitiesrdquo Physical Review Letters vol107 no 10 4 pages 2011

[77] D Mateos and D Trancanelli ldquoThermodynamics and instabil-ities of a strongly coupled anisotropic plasmardquo Journal of HighEnergy Physics vol 2011 article 54 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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Computational Methods in Physics

Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


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Biophysics


ThermodynamicsJournal of


is the wake that it creates along its path First Ruppert andMuller [33] have investigated that when a jet propagatesthrough the medium a wake of current and charge densityis induced which can be studied within the framework oflinear-response theory The result shows the wake in boththe induced charge and current density due to the screeningeffect of the moving parton In the quantum liquid scenariothe wake exhibits an oscillatory behavior when the chargeparton moves very fast Later Chakraborty et al [34] alsofound the oscillatory behavior of the induced charge wakein the backward direction at a large parton speed using HTLperturbation theory In a collisional quark-gluon plasma it isobserved that the wake properties change significantly com-pared to the collisionless case [35] Recently Jiang and Li [3637] have investigated the color response wake in the viscousQGP with the HTL resummation technique It is shown thatthe increase of the shear viscosity enhances the oscillation ofthe induced charge density as well as the wake potential Theeffect of momentum space anisotropy on the wake potentialand charge density has recently been considered in [38]

Effects of anisotropy on photon and dilepton yields havebeen investigated rigorously in [16 39ndash43] Recently theauthors in [44] calculated the nuclear modification factor forlight hadrons assuming an anisotropic QGP and showed howthe isotropization time can be extracted by comparing withthe experimental data

The organization of the review is as follows In Section 2wewill briefly discuss variousmodels of space-time evolutionin AQGP along with the electromagnetic probes which canbe used to extract the isotropization time of the plasmaSection 3 will be devoted to discuss the works on heavyquark potential and related phenomena (such as gluon 119869120595

dissociation cross-section in an AQGP) that have beeninvestigated so far We will also discuss the radiative energyloss of partons in AQGP and nuclear modification factor oflight hadrons due to the energy loss of the jet in Section 4In Section 5 the effect of momentum anisotropy on wakepotential and charge density due to the passage of a jet willbe presented Finally we summarize in Section 6

2 Electromagnetic Probes

Photons and dileptons have long been considered to be thegood probes to characterize the initial stages of heavy ioncollisions as these interact ldquoweaklyrdquo with the constituents ofthe medium and can come out without much distortion intheir energy and momentum Thus they carry the informa-tion about the space-time point where they are producedSince anisotropy is an early stage phenomena photons anddileptons are the efficient probes to characterize this stageThe yield of dileptons (henceforth called medium dilep-tonsphotons) in an AQGP has been calculated using aphenomenological model of space-time evolution in (1 + 1)dimension [42 43] This model (henceforth referred to asmodel I) introduces two parameters 119901hard and 120585 which arefunctions of time The former is called the hard momentumscale and is related to the average momentum of the particlesin themedium In isotropic case this can be identifiedwith thetemperature of the system The latter is called the anisotropy

parameter and minus1 lt 120585 lt infin Furthermore the modelinterpolates between early-time 1 + 1 free streaming behavior(120591 ≪ 120591iso) and late-time ideal hydrodynamical behavior (120591 ≫

120591iso) We first discuss various space-time evolution modelsof AQGP In model I the time dependence of 120585 is given by[42 43]

120585 (120591 120575) = (120591

120591119894

)

120575

minus 1 (1)

where the exponent 120575 = 2(23) corresponds to free streaming(collisionally broadened) preequilibrium state momentumspace anisotropy and 120575 = 0 corresponds to thermalization 120591

119894

is the formation time of the QGP For smooth transition fromfree streaming to hydrodynamical behavior a transitionwidth120574minus1 is introduced The time dependences of various relevant

parameters are obtained in terms of a smeared step function[42 43] as follows

120582 (120591) =1

2(tanh[

120574 (120591 minus 120591iso)

120591119894

] + 1) (2)

For 120591 ≪ 120591iso (≫ 120591iso) we have 120582 = 0(1) which corre-sponds to free streaming (hydrodynamics) Thus the timedependences of 120585 and 119901hard are as follows [42 43]

120585 (120591 120575) = (120591

120591119894

)

120575(1minus120582(120591))

minus 1

119901hard (120591) = 119879119894U13

(120591)

(3)

where

U (120591) equiv [R((120591iso120591

)

120575

minus 1)]

3120582(120591)4

(120591iso120591

)

1minus120575(1minus120582(120591))2

U equivU (120591)

U (120591119894)

R (119909) =1

2[

1

(119909 + 1)+tanminus1radic119909

radic119909]

(4)

and 119879119894is the initial temperature of the plasma In our calcu-

lation we assume a fast-order phase transition beginning atthe time 120591

119891and ending at 120591

119867= 119903

119889120591119891 where 119903

119889= 119892

119876119892

119867is the

ratio of the degrees of freedom in the two (QGP phase andhadronic phase) phases and 120591

119891is obtained by the condition

119901hard(120591119891) = 119879119888 which we take as 192MeVWe also include the

contribution from the mixed phaseThe other model (referred to as model II hereafter) of

space-time evolution of highly AQGP is the boost invariantdissipative dynamics in (0 + 1) dimension [45] This modelcan reproduce both the hydrodynamics and the free stream-ing limits The time evolution of the phase space distribution119891(119905 119911 p) can be described by Boltzmann equation Thus asa starting point we write the Boltzmann equation in (0 + 1)

dimension in the lab frame as follows

119901119905120597119905119891 (119905 119911 p) + 119901

119911120597119911119891 (119905 119911 p) = minusC [119891 (119905 119911 p)] (5)




2

1199012hard (120591)) (6)


(119901119879cosh (119910 minus Θ)

120597

120597120591+

119901119879sinh (119910 minus Θ)

120591

120597

120597120591)

times 119891 (p 120585 119901hard)


(7)



1

1 + 120585120597120591120585 =

2

120591minus 4ΓR (120585)


2R (120585) + 3 (1 + 120585)R1015840 (120585)

1

1 + 120585

1

119901hard120597120591119901hard

=2

120591minus 4ΓR

1015840(120585)


2R (120585) + 3 (1 + 120585)R1015840 (120585)

(8)





119889119873120574

1198891199101198892119901119879

= 1205871198772

perp[int

120591119891

120591i

120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

+int

120591119867

120591119891

119891QGP (120591) 120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

]

(9)

where 119891QGP(120591) = (119903119889minus 1)

minus1(119903119889120591119891120591minus1


perp= 12119860




120574119889

4119909119889119910119889






119864119889119877

1198893119875

= int1198893p

1

(2120587)3

1198893p

2

(2120587)3119891119902(p

1) 119891

119902(p

2) V

119902119902120590119897+119897minus

119902119902120575(4)

(119875 minus 1199011minus 119901

2)

(10)


0

2

4

6

8

10

0 2 4 6 8 10120591 (fmc)

120585

Ti = 044GeV 120591i = 015 fmc

(a)

01

02

03

04

05

0 2 4 6 8 10120591 (fmc)

pha

rd(G

eV)

Model IModel II

Ti = 044GeV 120591i = 015 fmc

(b)


(a)

0 1 2 3 4 5 6 7pT (GeVc)

10minus6

10minus4

10minus2

100

dNd

2pTdy

(GeV

minus2)


MM (120591iso = 1 fmc Tc = 192MeV)

Ti = 440MeV 120591i = 01 fmc



119894=

440MeV 120591119894= 01 fmc and 119879

119888= 192MeV [16]

where 119891119902(119902)





tion Consider

120590119897+119897minus

119902119902=

4120587

3

1205722

1198722(1 +

21198982

119897

1198722)(1 minus

41198982

119897

1198722)

12

(11)


119889119877

1198894119875

=5120572

2

181205875times int

1

minus1

119889 (cos 1205791199011)

times int

119886minus

119886+

1198891199011

radic1205941199011119891119902(radicp2

1(1 + 120585cos2120579p1) 119901hard)


1)2+ 120585(p


2

119901hard)

(12)



119889119873

1198891198722119889119910= 120587119877

2

perpint119889

2119875119879int

120591119891

120591119894

int

infin

minusinfin

119889119877

1198894119875120591119889120591 119889120578

119889119873

1198892119875119879119889119910

= 1205871198772

perpint119889119872

2int

120591119891

120591119894

int

infin

minusinfin

119889119877ann1198894119875

120591119889120591 119889120578

(13)


119894= 088 fmc and 119879

119894= 845MeV cor-







2 4 6 8 10M (GeV)

minus9

minus8

minus7

minus6

minus5



log10

(dN

e+eminus

dydM

2(G

eVminus2))

(a)

2 4 6 8 10PT (GeV)

minus9

minus8

minus7

minus6

minus5

minus4

minus3



log10

(dN

e+eminus

dydp2 T

(GeV

minus2))

(b)





Π120583]

(119870) = 1198922int

1198893119901

(2120587)3119875120583 120597119891 (p)

120597119875120573

(119892120573]

minus119875]119870120573

119870 sdot 119875 + 119894120576) (14)





Π119894119895(119870) = minus119892

2int

1198893119901

(2120587)3V119894120597119897119891 (p) (120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598) (15)


Π119894119895(119870) = 119898

2

119863int

119889Ω

(4120587)V119894

V119897 + 120585 (v sdot n) 119899119897

(1 + 120585(v sdot n)2)2(120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598)

(16)


ed by

1198982

119863= minus

1198922

21205872int

infin

0

1198891199011199012119889119891iso (119901

2)

119889119901 (17)




Π119894119895(119896) = 120572119860

119894119895+ 120573119861

119894119895+ 120574119862

119894119895+ 120575119863

119894119895 (18)

where

119860119894119895= 120575

119894119895minus

119896119894119896119895

1198962 119861

119894119895= 119896

119894119896119895

119862119894119895=

119899119894119899119895

1198992 119863

119894119895= 119896

119894119899119895+ 119899

119894119896119895

(19)

with 119899119894= 119860



119896119894Π119894119895119896119895= k2120573 119899

119894Π119894119895119896119895= 119899

2k2120575

119899119894Π119894119895119899119895= 119899

2(120572 + 120574) TrΠ119894119895

= 2120572 + 120573 + 120574

(20)


Δ119894119895(119870) =

1

[(k2 minus 1205962) 120575119894119895 minus 119896119894119896119895 + Π119894119895 (119896)] (21)



119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]


Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)



minus1

119860


minus1

119866= 0 and Δ

minus1

119860




Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)




119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962


119896119911k2

radick2 + 1205851198962perp


119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2


2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)


perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp


+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892



119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)


minus 1198922119862119865120585119898

2

119863int

1198893119896


(k2 + 1198982

119863)2

(29)



) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863




119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






2

1199012hard (120591)) (6)


(119901119879cosh (119910 minus Θ)

120597

120597120591+

119901119879sinh (119910 minus Θ)

120591

120597

120597120591)

times 119891 (p 120585 119901hard)


(7)



1

1 + 120585120597120591120585 =

2

120591minus 4ΓR (120585)


2R (120585) + 3 (1 + 120585)R1015840 (120585)

1

1 + 120585

1

119901hard120597120591119901hard

=2

120591minus 4ΓR

1015840(120585)


2R (120585) + 3 (1 + 120585)R1015840 (120585)

(8)





119889119873120574

1198891199101198892119901119879

= 1205871198772

perp[int

120591119891

120591i

120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

+int

120591119867

120591119891

119891QGP (120591) 120591 119889120591int119889120578119889119873

120574

11988941199091198891199101198892119901119879

]

(9)

where 119891QGP(120591) = (119903119889minus 1)

minus1(119903119889120591119891120591minus1


perp= 12119860




120574119889

4119909119889119910119889






119864119889119877

1198893119875

= int1198893p

1

(2120587)3

1198893p

2

(2120587)3119891119902(p

1) 119891

119902(p

2) V

119902119902120590119897+119897minus

119902119902120575(4)

(119875 minus 1199011minus 119901

2)

(10)


0

2

4

6

8

10

0 2 4 6 8 10120591 (fmc)

120585

Ti = 044GeV 120591i = 015 fmc

(a)

01

02

03

04

05

0 2 4 6 8 10120591 (fmc)

pha

rd(G

eV)

Model IModel II

Ti = 044GeV 120591i = 015 fmc

(b)


(a)

0 1 2 3 4 5 6 7pT (GeVc)

10minus6

10minus4

10minus2

100

dNd

2pTdy

(GeV

minus2)


MM (120591iso = 1 fmc Tc = 192MeV)

Ti = 440MeV 120591i = 01 fmc



119894=

440MeV 120591119894= 01 fmc and 119879

119888= 192MeV [16]

where 119891119902(119902)





tion Consider

120590119897+119897minus

119902119902=

4120587

3

1205722

1198722(1 +

21198982

119897

1198722)(1 minus

41198982

119897

1198722)

12

(11)


119889119877

1198894119875

=5120572

2

181205875times int

1

minus1

119889 (cos 1205791199011)

times int

119886minus

119886+

1198891199011

radic1205941199011119891119902(radicp2

1(1 + 120585cos2120579p1) 119901hard)


1)2+ 120585(p


2

119901hard)

(12)



119889119873

1198891198722119889119910= 120587119877

2

perpint119889

2119875119879int

120591119891

120591119894

int

infin

minusinfin

119889119877

1198894119875120591119889120591 119889120578

119889119873

1198892119875119879119889119910

= 1205871198772

perpint119889119872

2int

120591119891

120591119894

int

infin

minusinfin

119889119877ann1198894119875

120591119889120591 119889120578

(13)


119894= 088 fmc and 119879

119894= 845MeV cor-







2 4 6 8 10M (GeV)

minus9

minus8

minus7

minus6

minus5



log10

(dN

e+eminus

dydM

2(G

eVminus2))

(a)

2 4 6 8 10PT (GeV)

minus9

minus8

minus7

minus6

minus5

minus4

minus3



log10

(dN

e+eminus

dydp2 T

(GeV

minus2))

(b)





Π120583]

(119870) = 1198922int

1198893119901

(2120587)3119875120583 120597119891 (p)

120597119875120573

(119892120573]

minus119875]119870120573

119870 sdot 119875 + 119894120576) (14)





Π119894119895(119870) = minus119892

2int

1198893119901

(2120587)3V119894120597119897119891 (p) (120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598) (15)


Π119894119895(119870) = 119898

2

119863int

119889Ω

(4120587)V119894

V119897 + 120585 (v sdot n) 119899119897

(1 + 120585(v sdot n)2)2(120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598)

(16)


ed by

1198982

119863= minus

1198922

21205872int

infin

0

1198891199011199012119889119891iso (119901

2)

119889119901 (17)




Π119894119895(119896) = 120572119860

119894119895+ 120573119861

119894119895+ 120574119862

119894119895+ 120575119863

119894119895 (18)

where

119860119894119895= 120575

119894119895minus

119896119894119896119895

1198962 119861

119894119895= 119896

119894119896119895

119862119894119895=

119899119894119899119895

1198992 119863

119894119895= 119896

119894119899119895+ 119899

119894119896119895

(19)

with 119899119894= 119860



119896119894Π119894119895119896119895= k2120573 119899

119894Π119894119895119896119895= 119899

2k2120575

119899119894Π119894119895119899119895= 119899

2(120572 + 120574) TrΠ119894119895

= 2120572 + 120573 + 120574

(20)


Δ119894119895(119870) =

1

[(k2 minus 1205962) 120575119894119895 minus 119896119894119896119895 + Π119894119895 (119896)] (21)



119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]


Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)



minus1

119860


minus1

119866= 0 and Δ

minus1

119860




Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)




119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962


119896119911k2

radick2 + 1205851198962perp


119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2


2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)


perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp


+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892



119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)


minus 1198922119862119865120585119898

2

119863int

1198893119896


(k2 + 1198982

119863)2

(29)



) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863




119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

2

4

6

8

10

0 2 4 6 8 10120591 (fmc)

120585

Ti = 044GeV 120591i = 015 fmc

(a)

01

02

03

04

05

0 2 4 6 8 10120591 (fmc)

pha

rd(G

eV)

Model IModel II

Ti = 044GeV 120591i = 015 fmc

(b)


(a)

0 1 2 3 4 5 6 7pT (GeVc)

10minus6

10minus4

10minus2

100

dNd

2pTdy

(GeV

minus2)


MM (120591iso = 1 fmc Tc = 192MeV)

Ti = 440MeV 120591i = 01 fmc



119894=

440MeV 120591119894= 01 fmc and 119879

119888= 192MeV [16]

where 119891119902(119902)





tion Consider

120590119897+119897minus

119902119902=

4120587

3

1205722

1198722(1 +

21198982

119897

1198722)(1 minus

41198982

119897

1198722)

12

(11)


119889119877

1198894119875

=5120572

2

181205875times int

1

minus1

119889 (cos 1205791199011)

times int

119886minus

119886+

1198891199011

radic1205941199011119891119902(radicp2

1(1 + 120585cos2120579p1) 119901hard)


1)2+ 120585(p


2

119901hard)

(12)



119889119873

1198891198722119889119910= 120587119877

2

perpint119889

2119875119879int

120591119891

120591119894

int

infin

minusinfin

119889119877

1198894119875120591119889120591 119889120578

119889119873

1198892119875119879119889119910

= 1205871198772

perpint119889119872

2int

120591119891

120591119894

int

infin

minusinfin

119889119877ann1198894119875

120591119889120591 119889120578

(13)


119894= 088 fmc and 119879

119894= 845MeV cor-







2 4 6 8 10M (GeV)

minus9

minus8

minus7

minus6

minus5



log10

(dN

e+eminus

dydM

2(G

eVminus2))

(a)

2 4 6 8 10PT (GeV)

minus9

minus8

minus7

minus6

minus5

minus4

minus3



log10

(dN

e+eminus

dydp2 T

(GeV

minus2))

(b)





Π120583]

(119870) = 1198922int

1198893119901

(2120587)3119875120583 120597119891 (p)

120597119875120573

(119892120573]

minus119875]119870120573

119870 sdot 119875 + 119894120576) (14)





Π119894119895(119870) = minus119892

2int

1198893119901

(2120587)3V119894120597119897119891 (p) (120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598) (15)


Π119894119895(119870) = 119898

2

119863int

119889Ω

(4120587)V119894

V119897 + 120585 (v sdot n) 119899119897

(1 + 120585(v sdot n)2)2(120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598)

(16)


ed by

1198982

119863= minus

1198922

21205872int

infin

0

1198891199011199012119889119891iso (119901

2)

119889119901 (17)




Π119894119895(119896) = 120572119860

119894119895+ 120573119861

119894119895+ 120574119862

119894119895+ 120575119863

119894119895 (18)

where

119860119894119895= 120575

119894119895minus

119896119894119896119895

1198962 119861

119894119895= 119896

119894119896119895

119862119894119895=

119899119894119899119895

1198992 119863

119894119895= 119896

119894119899119895+ 119899

119894119896119895

(19)

with 119899119894= 119860



119896119894Π119894119895119896119895= k2120573 119899

119894Π119894119895119896119895= 119899

2k2120575

119899119894Π119894119895119899119895= 119899

2(120572 + 120574) TrΠ119894119895

= 2120572 + 120573 + 120574

(20)


Δ119894119895(119870) =

1

[(k2 minus 1205962) 120575119894119895 minus 119896119894119896119895 + Π119894119895 (119896)] (21)



119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]


Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)



minus1

119860


minus1

119866= 0 and Δ

minus1

119860




Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)




119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962


119896119911k2

radick2 + 1205851198962perp


119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2


2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)


perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp


+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892



119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)


minus 1198922119862119865120585119898

2

119863int

1198893119896


(k2 + 1198982

119863)2

(29)



) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863




119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




2 4 6 8 10M (GeV)

minus9

minus8

minus7

minus6

minus5



log10

(dN

e+eminus

dydM

2(G

eVminus2))

(a)

2 4 6 8 10PT (GeV)

minus9

minus8

minus7

minus6

minus5

minus4

minus3



log10

(dN

e+eminus

dydp2 T

(GeV

minus2))

(b)





Π120583]

(119870) = 1198922int

1198893119901

(2120587)3119875120583 120597119891 (p)

120597119875120573

(119892120573]

minus119875]119870120573

119870 sdot 119875 + 119894120576) (14)





Π119894119895(119870) = minus119892

2int

1198893119901

(2120587)3V119894120597119897119891 (p) (120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598) (15)


Π119894119895(119870) = 119898

2

119863int

119889Ω

(4120587)V119894

V119897 + 120585 (v sdot n) 119899119897

(1 + 120585(v sdot n)2)2(120575

119895119897+

V119895119896119897

119870 sdot 119881 + 119894120598)

(16)


ed by

1198982

119863= minus

1198922

21205872int

infin

0

1198891199011199012119889119891iso (119901

2)

119889119901 (17)




Π119894119895(119896) = 120572119860

119894119895+ 120573119861

119894119895+ 120574119862

119894119895+ 120575119863

119894119895 (18)

where

119860119894119895= 120575

119894119895minus

119896119894119896119895

1198962 119861

119894119895= 119896

119894119896119895

119862119894119895=

119899119894119899119895

1198992 119863

119894119895= 119896

119894119899119895+ 119899

119894119896119895

(19)

with 119899119894= 119860



119896119894Π119894119895119896119895= k2120573 119899

119894Π119894119895119896119895= 119899

2k2120575

119899119894Π119894119895119899119895= 119899

2(120572 + 120574) TrΠ119894119895

= 2120572 + 120573 + 120574

(20)


Δ119894119895(119870) =

1

[(k2 minus 1205962) 120575119894119895 minus 119896119894119896119895 + Π119894119895 (119896)] (21)



119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]


Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)



minus1

119860


minus1

119866= 0 and Δ

minus1

119860




Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)




119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962


119896119911k2

radick2 + 1205851198962perp


119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2


2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)


perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp


+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892



119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)


minus 1198922119862119865120585119898

2

119863int

1198893119896


(k2 + 1198982

119863)2

(29)



) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863




119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119860 [A minus C]

+ Δ119866[(k2 minus 120596

2+ 120572 + 120574)B + (120573 minus 120596

2)C minus 120575D]


Δminus1

119860(119896) = 119896

2minus 120596

2+ 120572 = 0

Δminus1

119866(119896) = (119896

2minus 120596

2+ 120572 + 120574) (120573 minus 120596

2) minus 119896

211989921205752= 0

(23)



minus1

119860


minus1

119866= 0 and Δ

minus1

119860




Δ120583]

=1

(1198702 minus 120572)[119860

120583]minus 119862

120583]]

+ Δ119866[(119870

2minus 120572 minus 120574)

1205964

1198704119861120583]

+ (1205962minus 120573)119862

120583]+ 120575

1205962

1198702119863120583]] minus

120582

1198704119870120583119870

]

(24)

whereΔminus1

119866= (119870

2minus 120572 minus 120574) (120596

2minus 120573) minus 120575

2[119870

2minus (119899 sdot 119870)

2] (25)




119881 (119896perp 119896

119911 120585) = 119892

2Δ00

(120596 = 0 119896perp 119896

119911 120585)

= 1198922

k2 + 1198982

120572+ 119898

2

120574

(k2 + 1198982

120572+ 1198982

120574) (k2 + 1198982

120573) minus 1198982

120575

(26)

where

1198982

120572= minus

1198982

119863

21198962perpradic120585

times[[

[

1198962


119896119911k2

radick2 + 1205851198962perp


119911

radick2 + 1205851198962perp

)]]

]

1198982

120573= 119898

2


2

perp)

+ 120585119896119911(120585119896

119911+ (k2 (1 + 120585) radick2 + 1205851198962

perp)


perp)))

times (2radic120585 (1 + 120585) (k2 + 1205851198962

perp))

minus1

)

1198982

120574= minus

1198982

119863

2(

k2

k2 + 1205851198962perp

minus1 + 2119896

2

119911119896

2

perp


+119896119911k2 (2k2 + 3120585119896

2

perp)

radic120585(k2 + 1205851198962perp)32

1198962perp


119911

radick2 + 1205851198962perp

))

1198982

120575= minus

1205871198982

119863120585119896

119911119896perp |k|

4(k2 + 1205851198962perp)32

(27)

where 120572119904= 119892



119881 (r 120585) = minus1198922119862119865int

1198893119896

(2120587)3119890minus119894ksdotr

119881 (119896perp 119896

119911 120585) (28)


minus 1198922119862119865120585119898

2

119863int

1198893119896


(k2 + 1198982

119863)2

(29)



) is given by [48]

119881 (r 120585) = 119881iso (119903) [1 + 120585(2

119890119903minus 1

1199032minus

2

119903minus 1 minus

119903

6)] (30)

whereas [48]

119881perp (r 120585) = 119881iso (119903) [1 + 120585(

1 minus 119890119903

1199032+

1

119903+

1

2+

119903

3)] (31)

where 119903 = 119903119898119863




119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119881 (r 120585) = 119881iso (119903) [1 minus 120585 (1198910 (119903) + 119891

1 (119903) cos 2120579)] (32)


1198910 (119903) =


121199032= minus

119903

6minus

1199032

48+ sdot sdot sdot

1198911 (119903) =

6 (1 minus 119890119903) + 119903 [6 + 119903 (119903 + 3)]

121199032= minus

1199032

16+ sdot sdot sdot

(33)


119881 (r) = minus120572

119903(1 + 120583119903) exp (minus120583119903) +

2120590

120583[1 minus exp (minus120583119903)]


1198982

119876119903

(34)

where 120583119898119863






case it is 12119879119888[51]


119881 (r 120585) = 119881119877 (r 120585) + 119894119881

119868 (r 120585) (35)








factor (119877119860119860


119860119860is given by [56 57]

119877119860119860

(119901119879 xperp Θ) = 119890

minus120577(119901119879xperpΘ) (36)


120577 (119901119879 xperp Θ) = 120579 (120591


120591119891

max(120591form 120591119894)119889120591 Γ (119901

119879 xperp Θ)






119894= 03 fmc 119901


119860119860



119860119860 In con-






120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







120590 (1199020) =

2120587

3(32

3)

2

(16120587

31198922

)1

1198982

119876

(11990201205980minus 1)

32

(11990201205980)5

(38)



0gt 120598

0



119875 = (119872119879cosh119910 0 119875

119879119872

119879sinh119910) (39)

where 119872119879

= radic1198722

119869120595+ 1198752






⟨120590 (119870 sdot 119906) Vrel⟩119896 =int119889

3119896120590 (119870 sdot 119906) Vrel119891 (119896

0 120585 119901hard)

int 1198893119896119891 (1198960 120585 119901hard) (40)




1198960=

(1199020119864 + 119902 (sin 120579

119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902))

119872119869120595

k = q +119902119864

|P|119872119869120595

times [(119902119872119879cosh119910 minus 119872

119869120595)

times (sin 120579119901sin 120579

119902sin120601

119902+ cos 120579

119901cos 120579

119902) + |P|] vJ120595

(41)where vJ120595 = P119864 119875 = (119864 0 |P| sin 120579

119901 |P| cos 120579

119901) and q =

(119902 sin 120579119902cos120601

119902 119902 sin 120579

119902sin120601

119902 119902 cos 120579



int1198893119902119872

119869120595

119864120590 (119902

0) 119891 (119896

0 120585 119901hard) (42)


int1198893119896119891 (119896

0 120585 119901hard) = int119889


=1

radic1 + 1205858120587120577 (3) 119901

3

hard

(43)





119878 (119875119879)

=

int 1198892119903 (119877

2

119860minus 119903

2) exp [minus int

120591max

120591119894119889120591 119899

119892 (120591) ⟨120590 (119870 sdot 119906) Vrel⟩119896]

int 1198892119903 (1198772119860minus 1199032)

(44)

where 120591max = min(120591120595 120591119891) and 120591


119899119892(120591) = 16120577(3)119901

3



119869120595given by


where cos120601 = V119869120595


119879119889119875

119879is the





119879= 0 and







119901= 1205872 (120579

119901=


119879= 4GeV in the


119879




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 02 04 06 08 10

02

04

06

08

1

12

14

phard (GeV)

120579p = 1205872 PT = 0GeV

⟨120590 r

el⟩

(mb)

(a)

0 02 04 06 08 1phard (GeV)

120585 = 0120585 = 1120585 = 3

0

02

04

06

08

1

12

14

120579p = 1205872 PT = 8GeV

⟨120590 r

el⟩

(mb)

(b)



119879= 0 and (b) is for 119875

119879= 8GeV

0

005

01

015

02

0 2 4 6 8 10PT (GeV)

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205872

(a)

0 2 4 6 8 10PT (GeV)

0

005

01

015

02

S(PT)

Ti = 550MeV 120591i = 07 fmc

Tc = 200MeV 120579p = 1205873


120591iso = 2 fmc120591iso = 3 fmc

(b)


119894= 550MeV 119879

119888= 200MeV and 120591

119894= 07 fmc



ofLight Hadrons



119881119899= 119881 (119902

119899) 119890

119894q119899 sdotx119899

= 2120587120575 (1199020) V (119902

119899) 119890

minus119894q119899 sdotx119899119879119886119899

(119877) otimes 119879119886119899

(119899)

(46)

with V(q119899) = 4120587120572

119904(119902

2

119899+ 119898

2

119863) 119909




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

002

004

006

008

0 2 4 6 8 10PT (GeV)

Ti = 820MeV 120591i = 05 fmc

Tc = 200MeV 120579p = 1205872

S(PT)

(a)

0 2 4 6 8 10PT (GeV)


120591iso = 2 fmc120591iso = 3 fmc

0

01

02

03

04

S(PT)

Ti = 830MeV 120591i = 008 fmc

Tc = 170MeV 120579p = 1205872

(b)


119888= 200(170)MeV









119889119864

119889119871=

119864

119863119877

int119909119889119909119889Γ

119889119909 (47)


119886 119905119888][119905

119888 119905119886] = 119862

2(119866)119862

119877119863119877 where

1198622(119866) = 3 119863

119877= 3 and 119862




119909119889Γ

119889119909=

119862119877120572119904

120587

119871

120582

1198982

119863

161205872120572119904


120587

1198892119902perp

120587

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

times [119896perp+ 119902

perp

(119896perp+ 119902

perp)2+ 120594

minus119896perp

1198962perp+ 120594

]

2

(48)

where 120594 = 1198982

1199021199092+119898

2

119892with119898

2

119892= 119898

2

1198632 and119898

2

119902= 119898

2

1198636 and

119881(119902perp 119902119911 120585) is given by (26)


119899


and 119902119911in (26) by 119902


perpminus 1199022

perpsin2120579

119899cos2120601 and 119902

119911rarr



119864

=119862119877120572119904

1205872

1198711198982

119863

120582

times int1198891199091198892119902perp

1003816100381610038161003816119881 (119902perp 119902119911 120585)

10038161003816100381610038162

1612058721205722119904

times [minus1

2minus

1198962

119898

1198962119898+ 120594

+1199022

perpminus 119896

2

119898+ 120594

2radic1199024perp+ 21199022

perp(120594 minus 1198962

119898) + (1198962

119898+ 120594)

2

+1199022

perp+ 2120594

1199022perpradic1 + 41205941199022

perp

times ln(1198962

119898+ 120594

120594

times (((1199022

perp+ 3120594) + radic1 + 41205941199022

perp(119902

2

perp+ 120594))

times ((1199022

perpminus 119896

2

119898+ 3120594) + radic1 + 41205941199022

perp


perp(120594 minus 1198962

119898)

+ (1198962

119898+ 120594)

2

)

minus1

))]

(49)



1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1

120582=

1

120582119892

+1

120582119902

(50)


119902and120582



1

120582119894

=1198621198771198622 (119894) 120588 (119894)

1198732

119888minus 1

int1198892119902perp

(2120587)2

1003816100381610038161003816119881 (119902perp 0 120585)

10038161003816100381610038162 (51)



tation and 1198622(119894) = (119873

2

119888minus 1)(2119873

119888) for quark and 119862

2(119894) = 119873

119888


120588119894= 120588

iso119894


1

120582=

18120572119904119901hard120577 (3)

1205872radic1 + 120585

1

119877 (120585)

1 + 1198731198656

1 + 1198731198654

(52)



119865= 25 120572



119899=



119899= 1205876




119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int119889119909119886119889119909

119887119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)119904

120587

119889120590

119889(119886119887 997888rarr 119888119889) 120575 (119904 + + )

(53)

0

01

02

03

04

05

06

ΔE

E

0 10 20 30 40 50p (GeVc)

T = 250MeV 120585 = 05 120572s = 03

Isotropic120585 = 05 120579n = 0

120585 = 05 120579n = 1205876

120585 = 05 120579n = 1205872

120585 = 1 120579n = 1205876






119864119889120590

1198893119901(119860119861 997888rarr jet + 119883)

= 119870sum

119886119887119888119889

int

1

119909min

119889119909119886119866119886ℎ119860

(119909119886 119876

2)119866

119887ℎ119861

times (119909119887 119876

2)2

120587

119909119886119909119887

2119909119886minus 119909

119879119890119910

119889120590

119889(119886119887 997888rarr 119888119889)

(54)

where119909119887= (119909

119886119909119879119890minus119910

)(2119909119886minus119909

119879119890119910)119909

119879= 2119901

119879radic119904 and119909min =

119909119879119890119910(2 minus 119909

119879119890minus119910


ℎ119888(119911 119876

2) must be included







119863ℎ119888

(119911 1198762) =

119911⋆

119911119863ℎ119888

(119911⋆ 119876

2) (55)



0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

02

04

06

08

1

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03

(a)

0

01

02

03

04

05

0 10 20 30 40 50p (GeVc)

ΔE

E

T = 500MeV 120585 = 1 120572s = 03


120579n = 1205876120579n = 1205872

(b)



quarks 119905119871= 119871) where


perpcos120601 (56)



P ( 119903) prop 119879119860 ( 119903) 119879119861 ( 119903) (57)


P (119903) =2

1205871198772perp

(1 minus1199032

1198772perp

)120579 (119877perpminus 119903) (58)


119879spectra we


1198891198731205870(120578)

1198892119901119879119889119910

= sum

119891

int1198892119903P (119903) int

119905119871

119905119894

119889119905

119905119871minus 119905

119894

times int119889119911

11991121198631205870(120578)119891

(119911 1198762)10038161003816100381610038161003816119911=119901119879119901

119891

119879

119864119889119873

1198893119901119891

(59)



⟨119871⟩ =int119877119879

0119903119889119903 int

2120587

0119871 (120601 119903) 119879

119860119860 (119903 119887 = 0) 119889120601

int119877119879

0119903119889119903 int

2120587

0119879119860119860 (119903 119887 = 0) 119889120601

(60)


119860119860) becomes [44]

119877119860119860

(119901119879) =

1198891198731205870(120578)

119860119860119889

2119901119879119889119910

[1198891198731205870(120578)

1198601198601198892119901

119879119889119910]

0

(61)

where [1198891198731205870(120578)

119860119860119889

2119901119879119889119910]


119879



119894and



119894=





119894= 350MeV




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 5 10 15pT (GeV)

0

02

04

06

08RAA

120572s = 035 C2s = 13



(a)

0 5 10 15pT (GeV)

0

02

04

06

08

RAA


120572s = 035 C2s = 13

120591iso = 15 fmc

120591iso = 2 fmc

(b)


119894=

0147 fmc and (b) 119879119894= 350MeV and 120591

119894= 024 fmc


5 Plasma Wakes


120598 (k 120596) =119896119894120598119894119895(k 120596) 119896119895

1198962 (62)


120598119894119895= 120575

119894119895minus

Π119894119895

1205962 (63)


119879 (119911)

+ 120585 [1199112

12(3 + 5 cos 2120579

119899)119898

2

119863minus

1

6(1 + cos 2120579

119899)119898

2

119863

+1

4Π119879 (119911) ((1 + 3 cos 2120579

119899) minus 119911

2(3 + 5 cos 2120579

119899)) ]

120573 = 1199112[Π

119871 (119911)

+ 120585 [1

6(1 + 3 cos 2120579

119899)119898

2

119863

+ Π119871 (119911) (cos 2120579119899 minus

1199112

2(1 + 3 cos 2120579

119899))]]

120574 =120585

3(3Π

119879 (119911) minus 1198982

119863) (119911


119899

120575 =120585

3119896[4119911

21198982

119863+ 3Π

119879 (119911) (1 minus 41199112)] cos 120579

119899

(64)

with

Π119879 (119870) =

1198982

119863

21199112

times [1 minus1

2(119911 minus

1

119911) (ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2))]

Π119871 (119870) = 119898

2

119863[119911

2(ln

1003816100381610038161003816100381610038161003816

119911 + 1

119911 minus 1

1003816100381610038161003816100381610038161003816minus 119894120587Θ (1 minus 119911

2)) minus 1]

(65)






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120588119886

tot (k 120596) = 120588119886

ext (k 120596) + 120588119886

ind (k 120596) (66)


119886


119886


120588119886

ind (k 120596) = (1

120598 (k 120596)minus 1) 120588

119886

ext (k 120596) (67)


120588119886

ext = 2120587119876119886120575 (120596 minus k sdot v) (68)



119901 the



120588119886

ind (r 119905) = 2120587119876119886int

1198893119896

(2120587)3

times int119889120596

2120587exp119894(ksdotrminus120596119905) ( 1

120598 (k 120596)minus 1)


(69)


119899cos120601 119896 sin 120579

119899sin120601 119896 cos 120579



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos Γ(Re 120598 (k 120596)

Δminus 1)

+ sin ΓIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(70)



2+






119863



120588119886

ind (r 119905) =1198761198861198983

119863

21205872int

infin

0

119889119896 1198962

times int

1

0

119889120594int

2120587

0

119889120601

2120587[cosΩ(

Re 120598 (k 120596)

Δminus 1)

+ sinΩIm 120598 (k 120596)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(71)


000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




000035 00007 0001

120588mD



(zminust)m

D

minus10

minus5

0

5

10 = 055 120585 = 0

(a)

0001

000100025

0010005

120588mD


(zminust)m

D

minus10

minus5

0

5



minus00005

minus0005

= 099 120585 = 0

(b)

00007000035

00003500007

0001

0001

120588mD


(zminust)m

D

minus10

minus5

0

5

10



minus0002

minus000025

= 055 120585 = 08

(c)

0001 000500025

0010005

minus0005

000250001

000100025

120588mD


minust)m

Dminus10

minus5

0

5


minus00015

minus0005


minus00005

= 099 120585 = 08

(d)





Φ119886(k 120596) =

120588119886

ext (k 120596)

1198962120598 (k 120596) (72)


Φ119886(r 119905)

= 2120587119876119886int

1198893119896

(2120587)3int

119889120596

2120587exp119894(ksdotrminus120596119905) 1

1198962120598 (120596 k)120575 (120596 minus k sdot v)

(73)


Φ119886(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)

times [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(74)


Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 [cos ΓRe 120598 (120596 k)Δ

+ sin ΓIm 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(75)


Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896

times int

1

0

119889120594 1198690(119896120588radic1 minus 1205942119898

119863)


0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0000800009

0001

(120588 minus t)mD


minus5

0

5

10zm

D

minus00025

minus0005

minus00075

= 055 120585 = 0

(a)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D


minus0003

minus00015

= 099 120585 = 0

(b)

0000900008

0000800009

00008

(120588 minus t)mD


minus5

0

5

10

zm

D

minus00025

minus0005

minus00075

= 055 120585 = 04

(c)

000500075

001

(120588 minus t)mD


minus5

0

5

10

zm

D

minus0004minus0003

minus00015

minus00015

= 099 120585 = 04

(d)


times [cos Γ1015840Re 120598 (120596 k)Δ

minus sin Γ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(76)

with Γ1015840= 119896120594V119905119898

119863



2119898

119863)Φ




Φ119886

(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ1015840Re 120598 (120596 k)Δ

+ sinΩ1015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(77)



minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

(a)


minus02

minus015

minus01

minus005

0

005

01

Φa 0(120588

=0

zminust)

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 08

= 099 120585 = 08

(b)




case it is given by

Φ119886

perp(r 119905) = 119876

119886119898119863

21205872int

infin

0

119889119896int

1

0

119889120594

times int

2120587

0

119889120601

2120587

times [cosΩ10158401015840Re 120598 (120596 k)Δ

+ sinΩ10158401015840 Im 120598 (120596 k)

Δ]

10038161003816100381610038161003816100381610038161003816120596=ksdotv

(78)

with Ω10158401015840










minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





minus01

minus005

0

005

01

Φa 0(z

=0

120588)

(Qam

D)

(a)


minus01

minus005

0

005

01

= 055 120585 = 0

= 099 120585 = 0

= 055 120585 = 05

= 099 120585 = 05

Φa 0(z

120588=0

)(Q

am

D)

(b)





References















119873119873=














































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













































119873119873) =



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Review Article Some Aspects of Anisotropic Quark-Gluon ...parton moves very fast. Later, Chakraborty et al. [ ]also found the oscillatory behavior of the induced charge wake in the

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