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

CurvatureCurvatureBackground (AdSBackground (AdS5 5 ) metric ) metric

Background (AdSBackground (AdS5 5 ) metric ) metric

- - Minkowski tensorMinkowski tensor

352 2( 1)M r

PlM e

- - gravity scale in 5 dimensionsgravity scale in 5 dimensions5M

Hierarchy relationHierarchy relationHierarchy relationHierarchy relation

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

3/ 25

2( ) ( ) ( )MN MN MNMg z z h z

Kaluza-Klein (KK ) gravitons Kaluza-Klein (KK ) gravitons Kaluza-Klein (KK ) gravitons Kaluza-Klein (KK ) gravitons

( )

0

( , ) ( ) ( )nn

n

h x y h x y

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

is 4-momentum transferis 4-momentum transfer

2( ) 2n g g nt t m

(Kisselev & Petrov, 2005/6)(Kisselev & Petrov, 2005/6)

Gravi-Reggeons:Gravi-Reggeons:Gravi-Reggeons:Gravi-Reggeons:

- string tension- string tension2g sM

t

Gravi-Reggeon

++ ……++ ++

jj

ii

Gravitational amplitudeGravitational amplitude

- 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

20( , ) exp lni iF x t f x t b x P

(Petrov & Prokudin, 2002)(Petrov & Prokudin, 2002)

if x

2

35

ˆˆ ˆ, exp

ˆg

grav gg

sA s t t R s

M R s

Gravitational amplitude Gravitational amplitude Gravitational amplitude Gravitational amplitude

- 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

s sx

Differential cross section Differential cross section Differential cross section Differential cross section

- neutrino energy fraction - neutrino energy fraction deposited to the protondeposited to the proton

y

Inelastic neutrino-proton Inelastic neutrino-proton cross sectionscross sections

Inelastic neutrino-proton Inelastic neutrino-proton cross sectionscross sections

dotted curve:dotted curve: SM (cc) cross SM (cc) cross sectionsection

Detection of quasi-horizontal showers by the Auger Observatory

Detection of quasi-horizontal showers by the Auger Observatory

(Berezinsky, Zatsepin, Smirnov, 1969/75)(Berezinsky, Zatsepin, Smirnov, 1969/75)

5 1 TeVM

5 2 TeVM 4.9 per year4.9 per year4.9 per year4.9 per year1.6 per year1.6 per year

Waxman-Bahcall neutrino fluxWaxman-Bahcall neutrino flux

SM:SM: 0.22 per year 0.22 per year

Expected rate for quasi-horizontal airExpected rate for quasi-horizontal air showers at the Auger Observatoryshowers at the Auger Observatory

(Anchordoqui et al., 2005)(Anchordoqui et al., 2005)

( Zenith angle > 70º )

11282 106/ srscmGeVdEdNE

CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS In the RS model with the small

curvature virtual graviton contributions to the Bhabha scattering at the LC could be significantly larger than SM ones

Trans-Planckian gravity induced scattering of the brane fields is given by an infinite sum of t-channel reggeized gravitons (gravi-Reggeons)

Gravity effects from extra dimension are large enough to be measured in ultra-high-energy neutrino-nucleon events by the Auger detectors

р. Сож

Back Up Slides

Background RS metric Background RS metric Background RS metric Background RS metric

(Randall & Sundrum, 1999)(Randall & Sundrum, 1999)

2||22 dydxdxeds y

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 !

Neutrino-proton interaction Neutrino-proton interaction Neutrino-proton interaction Neutrino-proton interaction

solid curves:solid curves:dashed curves:dashed curves:

5 0.25, 0.5, 1 TeVM min 0.5,1, 2 TeVBHM

vs. black hole productionvs. black hole productionvs. black hole productionvs. black hole production