Alexandre Kisselev IHEP (Protvino, Russia) Scattering in the RS model Scattering in the RS model with the small curvature with the small curvature IXth INTERNATIONAL SCHOOL-SEMINAR “THE ACTUAL PROBLEMS OF MICROWORLD PHYSICS ” Belarus, Gomel, July 23 - August 3, 2007
Scattering in the RS model with the small curvature. Alexandre Kisselev IHEP ( Protvino , Russia). IXth INTERNATIONAL SCHOOL-SEMINAR “THE ACTUAL PROBLEMS OF MICROWORLD PHYSICS ” Belarus, Gomel, July 2 3 - August 3 , 200 7. SUMMARY. Warped extra dimension (ED) - PowerPoint PPT Presentation
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Alexandre Kisselev
IHEP (Protvino, Russia)
Alexandre Kisselev
IHEP (Protvino, Russia)
Scattering in the RS model Scattering in the RS model with the small curvaturewith the small curvature
Scattering in the RS model Scattering in the RS model with the small curvaturewith the small curvature
IXth INTERNATIONAL SCHOOL-SEMINAR
“THE ACTUAL PROBLEMS OF MICROWORLD PHYSICS ”Belarus, Gomel, July 23 - August 3, 2007
SUMMARY SUMMARY SUMMARY SUMMARY Warped extra dimension (ED)
with the small curvature AdS5 metric vs. flat metric with one compact ED Virtual graviton effects in the Bhabha scattering Trans-Planckian scattering on the brane Neutrino-nucleon interactions at ultra-high energies Conclusions
WarpedWarped Extra Dimension Extra Dimension Scenario with the Small Scenario with the Small
CurvatureCurvature
WarpedWarped Extra Dimension Extra Dimension Scenario with the Small Scenario with the Small
is the radius of EDis the radius of EDr , 0,1,...4M N
PlM - - Planck massPlanck mass
ryr
10r
2|)||(22 )( dydxdxedzdzzds yrNMMN
SM fields are confined to the TeV braneSM fields are confined to the TeV braneSM fields are confined to the TeV braneSM fields are confined to the TeV brane
SM 0y cy r
Planck brane
Planck brane
TeV brane
TeV brane
Gravity
Gravity lives in the bulkGravity lives in the bulkGravity lives in the bulkGravity lives in the bulk
Gravitational 5-dimensional fieldGravitational 5-dimensional fieldGravitational 5-dimensional fieldGravitational 5-dimensional field
Radion (scalar) fieldRadion (scalar) fieldRadion (scalar) fieldRadion (scalar) field
44( , ) ( )h x y x
graviton massesgraviton massesgraviton massesgraviton masses n nm x
Interaction Lagrangian on the TeV braneInteraction Lagrangian on the TeV braneInteraction Lagrangian on the TeV braneInteraction Lagrangian on the TeV brane
(0) ( )1 1 13
1Pl
nM
n
h h T T
L =-
physical scalephysical scale3
2 5M
Large curvature optionLarge curvature optionLarge curvature optionLarge curvature option
5 1 TeVM
series of massive resonancesseries of massive resonances
1( , 1 TeV)m
Small curvature optionSmall curvature optionSmall curvature optionSmall curvature option
5 1 TeVM
narrow low-mass resonancesnarrow low-mass resonanceswith the small mass splittingwith the small mass splitting ( )m
Formal relation to gravity in flat Formal relation to gravity in flat space-time with one compact ED space-time with one compact ED Formal relation to gravity in flat Formal relation to gravity in flat space-time with one compact ED space-time with one compact ED
1 ,c PlR M cR is the radius of EDis the radius of ED
5
5
10 0.1M
(Giudice et al., 2004,(Giudice et al., 2004,Kisselev & Petrov, 2005/6)Kisselev & Petrov, 2005/6)
RS model with the small curvature is not RS model with the small curvature is not equivalent to a model with equivalent to a model with oneone large ED large EDof the size of the size
RS model with the small curvature is not RS model with the small curvature is not equivalent to a model with equivalent to a model with oneone large ED large EDof the size of the size 1
cR
For instance,For instance,For instance,For instance,
can be realized only forcan be realized only forcan be realized only forcan be realized only for 1 50 MeV 1 GeVcR
7 10d
AdSAdS5 5 Metric vs. Flat Metric Metric vs. Flat Metric
with One Compact ED with One Compact ED
AdSAdS5 5 Metric vs. Flat Metric Metric vs. Flat Metric
with One Compact ED with One Compact ED
1d cR solar distancesolar distance
2 3d - strongly limited by - strongly limited by astrophysical bounds astrophysical bounds
- strongly limited by - strongly limited by astrophysical bounds astrophysical bounds
is the number of ED’sis the number of ED’sd
3
2 12 2 3551 2rr
Pl
MM e r M
2 1r 2
5
5
PlM M
M
Hierarchy relation in d flat ED’s (D=4+d)Hierarchy relation in d flat ED’s (D=4+d)Hierarchy relation in d flat ED’s (D=4+d)Hierarchy relation in d flat ED’s (D=4+d)
Limiting case of a similar relation Limiting case of a similar relation for the warped metricfor the warped metricLimiting case of a similar relation Limiting case of a similar relation for the warped metricfor the warped metric
2 22d d
Pl c DM R M
eV10 22
Virtual s-channel KK Virtual s-channel KK GravitonsGravitonsVirtual s-channel KK Virtual s-channel KK GravitonsGravitons
Scattering of SM fields mediated Scattering of SM fields mediated by graviton exchangeby graviton exchangeScattering of SM fields mediated Scattering of SM fields mediated by graviton exchangeby graviton exchange
For instanceFor instanceFor instanceFor instance
)(
)2(
ffee
Xjetsorllpp
bbGaa n )(
Energy region:Energy region:Energy region:Energy region:
5~ Ms
Matrix element of the process:Matrix element of the process:Matrix element of the process:Matrix element of the process:
SAM
aa TPT
Awherewhere
Tensor part of thegraviton propagator
Energy-momentum tensor
122
11)(
n nnn mimssS
Zero width continuous massZero width continuous massdistribution approximationdistribution approximation
Zero width continuous massZero width continuous massdistribution approximationdistribution approximation
sM
isS
352
)(
(Giudice et al., 2005)(Giudice et al., 2005)
Widths of the KK gravitons:Widths of the KK gravitons:
1.0,2
n
n
n m
m
S(s) can be calculated analyticallyS(s) can be calculated analyticallyby usingby usingS(s) can be calculated analyticallyS(s) can be calculated analyticallyby usingby using
)(
)(
2
11 1
122
, zJ
zJ
zzzn n
2235 sinhcos
2sinh2sin
4
1)(
A
iA
sMsS
223
5 sinhcos
2sinh2sin
4
1)(
A
iA
sMsS
wherewhere3
52,
M
ssA
(Kisselev, 2006)(Kisselev, 2006)
Some Some consequences:consequences:Some Some consequences:consequences:
2sinh
1
)(~
Im
)(~
Re
sinh21
1)(
~Re
tanh)(~
Imcoth
2
sS
sS
sS
sS
withwith )(2)(~ 3
5 sSsMsS
In trans-Planckian energy region In trans-Planckian energy region zero width result is reproducedzero width result is reproducedIn trans-Planckian energy region In trans-Planckian energy region zero width result is reproducedzero width result is reproduced
53Ms namely, atnamely, at
Is the mass splitting small enough for Is the mass splitting small enough for mass distribution to be continuous?mass distribution to be continuous?Is the mass splitting small enough for Is the mass splitting small enough for mass distribution to be continuous?mass distribution to be continuous?
nKKm nKKm
relevanrelevantt
)/(~ sn
3
2
)( s
0r, equivalently,0r, equivalently, 53Ms
ADD model:ADD model:ADD model:ADD model:
3/1/214
)/22( dd
dPl MMs
Search for one ED (DELPHI)Search for one ED (DELPHI)(Klein, ICHEP 2006)(Klein, ICHEP 2006)
Photon energy spectrum in single Photon energy spectrum in single photon events:photon events:Photon energy spectrum in single Photon energy spectrum in single photon events:photon events:
Main background:Main background:
Gee
ee
CL % 95at TeV92.05 M
((reducedreduced gravity scale) gravity scale)
MeVTeVMGeVs 100,1,200 5
(gravity+ SM )/SM ratio in two schemes (gravity+ SM )/SM ratio in two schemes (A.K. vs. zero width approximation)(A.K. vs. zero width approximation)
(gravity+ SM )/SM ratio in two schemes (gravity+ SM )/SM ratio in two schemes (A.K. vs. zero width approximation)(A.K. vs. zero width approximation)
(Kisselev, 2007)(Kisselev, 2007)
MeVTeVMTeVs 100,4.1,1 5
The same as on previous slide but The same as on previous slide but collision energy and gravity scale collision energy and gravity scale
are largerare larger
The same as on previous slide but The same as on previous slide but collision energy and gravity scale collision energy and gravity scale
are largerare larger
GeVTeVMTeVs 1,4.1,1 5
The same as on previous The same as on previous slideslide
except for the curvatureexcept for the curvature
The same as on previous The same as on previous slideslide
except for the curvatureexcept for the curvature
MeVTeVMTeVs 900,1,1 5
The same energy as on The same energy as on previous slide but the gravity previous slide but the gravity
scale and curvature are scale and curvature are differentdifferent
The same energy as on The same energy as on previous slide but the gravity previous slide but the gravity
scale and curvature are scale and curvature are differentdifferent
GeVTeVMTeVs 994,1,1 5
The same as on previous slide but The same as on previous slide but the curvature is the curvature is slightly slightly differentdifferentThe same as on previous slide but The same as on previous slide but the curvature is the curvature is slightly slightly differentdifferent
eikonal approximation
Trans-Planckian Scattering on the BraneTrans-Planckian Scattering on the BraneTrans-Planckian Scattering on the BraneTrans-Planckian Scattering on the Brane
5,s M s t
Born amplitude is the sum of theBorn amplitude is the sum of thereggeized gravitons in t-channelreggeized gravitons in t-channelBorn amplitude is the sum of theBorn amplitude is the sum of thereggeized gravitons in t-channelreggeized gravitons in t-channel
Kinematical regionKinematical regionKinematical regionKinematical region
- SM fields (q, g, l, - SM fields (q, g, l, n…n…))i,ji,j
1
(for all SM fields)(for all SM fields)~
2 225
Im ( , ) exp / 4 ( )g
ss b b R s
M
( ) lng gR s s
Imaginary part of the eikonalImaginary part of the eikonalImaginary part of the eikonalImaginary part of the eikonal
One has to calculate the sumOne has to calculate the sumOne has to calculate the sumOne has to calculate the sum
- gravitational radius- gravitational radius
2
0
exp lng nn
m s
1
4nm n withwithwithwith
At At << M << M5 ,5 , we obtain we obtainAt At << M << M5 ,5 , we obtain we obtain
eikonal with no explicit dependence on the brane scale and curvature
3/ 2
5100 /1 TeVM TeV
for smallfor small 100 MeV
Fields are weakly coupled to gravity:Fields are weakly coupled to gravity:Fields are weakly coupled to gravity:Fields are weakly coupled to gravity:
The summation of t-channel The summation of t-channel reggeized gravitonsreggeized gravitonsThe summation of t-channel The summation of t-channel reggeized gravitonsreggeized gravitons
Radion production is strongly suppressedRadion production is strongly suppressedRadion production is strongly suppressedRadion production is strongly suppressed
3
Neutrino-nucleon Neutrino-nucleon Interactions Interactions at Ultra-high at Ultra-high EnergiesEnergies
Neutrino-nucleon Neutrino-nucleon Interactions Interactions at Ultra-high at Ultra-high EnergiesEnergies
( , )iF x t ( , )iF x t
pppp pppp
iiii iiii iiii iiii
( , )pA s t
gravA gravA
Skewed (t-dependent) parton distributionSkewed (t-dependent) parton distributionSkewed (t-dependent) parton distributionSkewed (t-dependent) parton distribution
- standard distributionstandard distribution of parton of parton ii
parameters of the hard Pomeron is usedparameters of the hard Pomeron is usedparameters of the hard Pomeron is usedparameters of the hard Pomeron is used
Four-dimensional gravitational action of Four-dimensional gravitational action of gravitational field in the RS model:gravitational field in the RS model:Four-dimensional gravitational action of Four-dimensional gravitational action of gravitational field in the RS model:gravitational field in the RS model:
dxhhmhhSn
nnn
nneff
0
)()(2)()(41
(Boos et al., 2002)(Boos et al., 2002)
Expression is not covariant: the indices are raised Expression is not covariant: the indices are raised with the Minkowski tensor wheares the metric iswith the Minkowski tensor wheares the metric is
rerx 2),(
Change of variables:Change of variables:
xezx r
2|)||(22 dydzdzeds yr
Hierarchy relation:Hierarchy relation:
rPl eM
M
2
352 1
PlMM 5
Hierarchy problem remains unsolved ! Hierarchy problem remains unsolved !