M. Tokarev * & I. Zborovsky **

ISMD'07, August 4-9, 2007, Berkeley, USA

1

High-pT Spectra from RHIC & QCD test of z-Scaling

*Joint Institute for Nuclear Research, Dubna, Russia

**Nuclear Physics Institute, Řež near Prague,Czech Republic

M. Tokarev* & I. Zborovsky**


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Motivation & goals z-Scaling (ideas, definitions, properties,…) RHIC high-pT data & z presentation QCD test of z-scaling Conclusions

Contents


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Motivations & Goals

Development of a universal phenomenological description

of high-pT particle production in inclusive reactions to search for:

- new physics phenomena in elementary processes (quark compositeness, fractal space-time, extra dimensions, ...) - signatures of exotic state of nuclear matter (phase transitions, quark-gluon plasma, …) - complementary restrictions for theory (nonperturbative QCD effects, Standard Model, ...).

Analysis of new pp experimental data obtained at RHIC to verify z-scaling observed at U70, ISR, SppS, and Tevatron

in high-pT particle production and predictions for LHC.

–


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Principles & Symmetries

Relativity (special, general, scale,…) Gauge invariance (U(1), SU(2), SU(3),…) Self-similarity (hydro & aerodynamics, point explosions, critical phenomena,...) Fractality (scale dependence,…) Locality (constituent level of interactions,…) …….

Guiding principles to discover new laws in Nature


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Locality in inclusive reactions

Locality of hadron interactions: at sufficiently high energies hadrons and nuclei interact via interactions of their constituents (partons, quarks and gluons,...). Gross features of an inclusive particle distribution can be described in terms of the kinematic characteristics of the corresponding constituent subprocesses (V.S. Stavinsky 1979).


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Self-similarity principle

Self-similarity of hadron interactions reflects a property that hadron constituents, their interactions, and formation of the produced particles are similar.

The self-similarity is connected with dropping of certain dimensional quantities out of the description of physical phenomena.

Multiple interaction of the constituents is an ensemble of mutually similar individual sub-processes. These properties are common to various interactions of hadrons and nuclei at high energies.


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Constituent subprocess (x1M1) + (x2M2 ) m1/y1 + (x1M1+x2M2+m2 /y2)

(x1P1+x2P2 –p/y1)2 = (x1M1+x2M2+m2/y2)2

is subject to the kinematic condition:

Hadron/nucleus collisions at a constituent level

M.T. & I.ZborovskyPart.Nucl.Lett.312(2006)

PRD75,094008(2007)

inclusive particle

colliding object

colliding object

recoil particle


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Scaling variable z

1

0ch

1/2

m)| /d(dN

sz c

is transverse kinetic energy of the constituent subprocess consumed on production of m1 & m2

Ω-1 is minimal resolution at which the subprocess can be singled out of the inclusive reaction dNch /dη|0 is multiplicity density of charged particles at η = 0 c is a parameter interpreted as “heat capacity” of the created medium m is arbitrary normalization (we fix it at the value of nucleon mass)

1/2s

M.T. & I.ZborovskyPhys.At.Nucl.70,1294(2007)Phys.Rev.D75,094008(2007)


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Ω & momentum fractions x1, x2, y1, y2

Principle of minimal resolution: The momentum fractions x1, x2

and y1, y2 are determined in a way to minimize the resolution Ω-1 of the fractal measure z with respect to all constituent sub-processes taking into account momentum conservation:

2222211

212211 )/ymMxM(x )p/yPxP(x

0|y / 0|x / 0|x /

)1()y(1)x(1)x-(1

)y,x,(xyy2

)y,x,(xyy2

)y,x,(xyy1

212

21

1

22111

22111

22111

21

y

Kinematic condition:


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Transverse kinetic energy consumed on production of m1 & m2

1/2s

energy consumed for the inclusive particle m1

energy consumed for the recoil particle m2

22211 )PP(s

)P(P

mM,

)P(P

p)(P ,,

,,

12

21221

12

1221

22211 )PP(s

21212

212

2121 ,/

,,, )( 2,1

1,2021

121,2 -1

-1)(

222111/2

2122111/2

11/2 m)MM(sym)MM(sys

2,12,12,1 x

22,111,21,2 // yy

210

2200 // yy

)(5.0

,)(

5.0

21

21

021

22

0 PPm

PPm )]1)(1/[()( 21021

2

Decomposition:

1

22,12,1 ,

2

1,

UU


11

3

31

inel dp

dEJ

) (dN/d

s(z)

Scaling function z

1P p

2P X

33 /dpEd

s1/2 is the collision energy dN/d is the pseudorapidity multiplicity density

inel is the inelastic cross section

is the inclusive cross section

J is the corresponding Jacobian

The variable z and the function Ψ(z) are expressed via momenta and masses of the colliding and produced particles, multiplicity

density, and inclusive cross section.

1221

zz

J


12

3

31

inel dp

dEJ

) (dN/d

s(z)

0

1(z)dz

Normalization equation

1P p

2P X

The scaling function z is probability density to produce the inclusive particle

with the corresponding fractal measure z.

N pdyddp

dE inel

23

3


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Fractality of hadron matter

Fractality is a specific feature connected with sub-structure of the interacting objects (hadrons and nuclei). Fractal compositeness includes sub-structure of hadron constituents over a wide scale range.

Fractality of soft processes concerning the multiparticle production was investigated by many authors (A.Bialas, R.Peshchanski, I.Dremin, E.DeWolf,…).

Fractality in hard processes regards fractal structure of the colliding objects and fractal character of particle formation. This aspect was specifically built into the definition of the scaling variable z.

The variable z is a fractal measure which can be attributed to any inclusive reaction.

1 resolution if ) z(


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Properties of z-presentation in pp

Energy independence of Ψ(z) (s1/2 > 20 GeV)

Angular independence of Ψ(z) (θcms>3-50,..)

Power law, Ψ(z) ~ z-β (z >4)

Multiplicity independence of Ψ(z) (dNch/dη=1.5-26.) Flavor independence of Ψ(z) (π,K,…)

M.T., I.ZborovskyPhys.At.Nucl. 70,1294(2007)Phys.Rev. D75,094008(2007)

These properties reflect self-similarity, locality, and fractality of the hadron interaction at constituent level.

It concerns the structure of the colliding objects, interactions of their constituents, and fragmentation process.


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Spectra of charged hadrons in pp

Energy independence of Ψ(z) Power behavior of Ψ(z) for z>4 RHIC data are compatible with data from FNAL, ISR

FNAL, ISR & RHIC

STAR

STARJ.Adams et al.,

PRL91,172302(2003)


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Spectra of π mesons in pp

Energy independence of Ψ(z) Power behavior of Ψ(z) for z > 4 RHIC data are compatible with data from FNAL, ISR

FNAL, ISR & RHIC

STAR

STARJ.Adams et al.,

PL B637,161 (2005)


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Spectra of K mesons in pp


FNAL, ISR & RHIC

STAR

R.Witt & STAR J.Phys.G31,S863, (2005)

STARB.I.Abelev et l.,

PRC75064901(2007)


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Spectra of antiprotons in pp


FNAL, ISR & RHIC

STAR

STARJ.Adams et al.,

PL B616,8 (2005)


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θ0 Spectra of π mesons in pp

Angular independence of Ψ(z) strong sensitivity to m2 & ε: m1=m2=mπ

Power behavior of Ψ(z) for z > 4

ISR

BS B.Alper et al.,

Nucl.Phys.B100,237(1975)


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θ0 Spectra of K mesons in pp

Angular independence of Ψ(z) strong sensitivity to m2 & ε: m1=m2=mK

Power behavior of Ψ(z) for z > 4 RHIC data are compatible with data from ISR

ISR & RHIC

STAR

BS B.Alper et al.,

Nucl.Phys.B100,237(1975)CHLM

M.G.Albrow et al., Nucl.Phys.B56,

333(1973)


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Multiplicity dependence of pp spectra Why is it interesting ?

Measured multiplicity density dNch/d in pp & pp is much more larger than dNch/d/(0.5Np) in central AA collisions at AGS, SppS, and RHIC

¯

¯

Multiplicity density is a characteristic of medium (<pT>, εBj) Regulator of modification of particle spectrum (high pT ) Search for sensitive indicators of phase transition


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Experimentally measurable quantities: σ, s1/2, N, dN/dη, …

Model dependent quantities: T, p, V, c, μ, …

The quantities c and dNch/dη|0 have physical meaning of “heat capacity” and “temperature” of the produced medium. Entropy S of the system depends on the resolution Ω-1. Maximal entropy S minimal resolution Ω-1.

z-Scaling & Entropy S

])y(1)y(1)x(1)x(1ln[)d/dN(lnc 2121

22

110ch

S

Wsz /~ 2/1

WS lnEntropy

VRTS lnlncV

W is proportional to all parton and hadron configurations of he colliding system which can contribute to production of the inclusive particle with mass m1 and momentum p1


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KS0 Spectra vs. Multiplicity

Multiplicity independence of Ψ(z) Power behavior of Ψ(z) for z > 4 RHIC (STAR) data confirm Tevatron data (E735)

STAR & RHIC

3.05.025.0

c

STAR nucl-ex/0403020B.I.Abelev et al.,

PRC75064901(2007)


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Λ Spectra vs. Multiplicity

Multiplicity independence of Ψ(z) sensitivity to “heat capacity” c Power behavior of Ψ(z) for z > 4 RHIC data allow to fix the value of c

STAR & RHIC

STAR nucl-ex/0403020B.I.Abelev et al.,

PRC75064901(2007)


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π,K,Λ,.. Spectra vs. Flavor

FNAL, ISR & RHIC

PHENIX

Particle ratio is flat vs. pT

Flavor independence of Ψ(z) Power behavior of Ψ(z) for z > 4 More convincing confirmation is needed

ω/π0 = 0.81± 0.02±0.07 η/π0 = 0.48± 0.02±0.02 KS

0 /π0 = 0.45±0.01±0.05 pT >2-3 GeV/c

nucl-ex/0702046

pp


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QCD test of z-scaling

QCD is basic theory for calculations of hadron interactions in terms of quarks and gluons. Perturbative expansion is under control (LO, NLO, ...). Non-perturbative effects – PDFs, FFs, μR, μF, μH, are partially under control. Correct extrapolation in low and high (x,pT) range is restricted by available data (e+e–, DIS,…). Additional constraints on PDFs and FFs are needed to confirm their universality (gluons, flavor, …). Soft regime (multiple interactions, … ).

A lot of data are analyzed in framework of z–presentation. New confirmations from RHIC and Tevatron are obtained. Can NLO QCD describe z-scaling in soft and hard regime ? …..

),,(

),,,,(),,(

11

2121

11

Hhq

Rqq

Fq

h

pzD

kkxxkxF

Hadron interaction at a constituent level


27

NLO QCD ingredients NLO QCD hadron production code (h±,π,K,…) F.Aversa, P.Chiappetta, M.Greco, J.Ph.Guillet Parton Distribution Functions CTEQ5m – H.L.Lai et al., Pumplin et al., MRST99 – A.D.Martin, R.G.Roberts, W.J.Stirling, R.S.Thorne Fragmentation Functions KKP – B.A.Kniehl, G.Kramer, B.Potter BKK – J.Binnewies, B.A.Kniehl, G.Kramer Scales μ = c · pT, c = 0.5, 1., 2. – Renormalization μR

– Factorization μF

– Hadronization μH

),(

),,,,(),(

1

2121

1

Hhq

Rqq

Fq

h

zD

kkxxxF


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h ± NLO QCD spectra in z-presentation

Strong dependence of spectra on energy s1/2 at high pT

Sensitivity to PDFs & FFs Sensitivity to μR, μF, μH scales NLO QCD results are in agreement with exp. data Different extrapolation of spectra predicted by NLO QCD and z-scaling


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π± NLO QCD spectra in z-presentation

Features of π and h± spectra are similar Available data are in agreement with NLO QCD z-presentation of NLO QCD calculated results indicates deviation from asymptotic behavior of Ψ(z) predicted by z-scaling


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K± NLO QCD spectra in z-presentation

Features of K and h±, π spectra are similar Available data are in agreement with NLO QCD Asymptotic behavior of the scaling function Ψ(z) is not reproduced by NLO QCD evolution of spectra


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Conclusions (I) New analysis of FNAL, ISR, and RHIC data on high-pT hadron

spectra in the framework of z-scaling is performed. Properties of z-presentation are confirmed. STAR data on multiplicity dependence of KS

0 & Λ spectra in pp

collisions give new insight on “heat capacity” c and fractal dimension ε.

z-Scaling is tested by NLO QCD:

- Self-similar features of particle production dictated

by z-scaling give restriction on the asymptotic behavior

of inclusive spectra in high-pT region.

- They are not reproduced by NLO QCD evolution of spectra with available PDFs and FFs in TeV energy range.


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Conclusions (II)

z-scaling in pp collisions is a regularity which reflects self-similarity, locality, and fractality of the hadron interactions at a constituent level. It concerns the structure of colliding objects, interactions of their constituents, and fragmentation process.

New experimental data on particle spectra over a wide range of collision energy, transverse momenta, production angle, and multiplicity density in pp collisions allow us to search for new phenomena in extreme conditions at RHIC.


33

Thank You for Your Attention


34

Spectra ratio vs. pT & multiplicity

The ratio of multiplicity binned pT spectra to multiplicity- integrated spectra scaled by mean multiplicity for each bin

for KS0 and Λ is sensitive to dNch/dη for high pT (Rpp > 10)

),(/

),(/

pmbiasdpdN

pmultdpdNFR scalepp

)(

)(

)(

)(

multN

mbiasN

multN

mbiasNF

ch

ch

evnt

evntscale

STAR B.I.Abelev et al.,

PRC75064901(2007)

KS0


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Scaling analysis in high energy interactions

z-Scaling: it provides universal description of inclusive particle cross sections over a wide kinematical region

(central+fragmentation region, pT > 0.5 GeV/c, s1/2 > 10 GeV )

Scaling variables

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2TT mpm

*max

*R /EEx

*max||

*||F /ppx

/pk light-cone variable

radial scaling variable

Feynman variable

transverse mass

/2(Pq)qx 2Bj Bjorken variable

These scaling regularities have restricted range of validity Violation of the scaling laws can be indication of new physics

nn / KNO variable

M. Tokarev * & I. Zborovsky **

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