RS model with a small curvature RS model with a small curvature and inclined air showersand inclined air showers

RS model with a small curvature RS model with a small curvature and inclined air showersand inclined air showers

Alexandre Kisselev

(IHEP, Russia)

Alexandre Kisselev

(IHEP, Russia)

XXth Rencontres de Blois “Challenges in Particle Astrophysics”

Chateau de Blois, France, 18th-23rd May 2008

SUMMARY SUMMARY SUMMARY SUMMARY

Randall-Sundrum model with the small curvature

Trans-Planckian scattering of SM fields on the brane Cosmic neutrino-nucleon

interactions at ultra-high energies

Conclusions

WarpedWarped Extra Extra Dimension with the Dimension with the

Small CurvatureSmall Curvature

WarpedWarped Extra Extra Dimension with the Dimension with the

Small CurvatureSmall CurvatureBackground (AdSBackground (AdS5 5 ) metric ) metric

Background (AdSBackground (AdS5 5 ) metric ) metric

- Minkowski tensor

352 2( 1)M r

PlM e

- gravity scale in 5 dimensions5M

Hierarchy relationHierarchy relationHierarchy relationHierarchy relation

- radius of EDr

(Randall & Sundrum, 1999)

PlM - Planck mass

r y r 2 2 | | 2y rds e dx dx dy

10r

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 y 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

n nm x - graviton masses

(M, N = 0, 1…4)

1( ( ) 0)nJ 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 scale3

2 5M

Large curvature optionLarge curvature optionLarge curvature optionLarge curvature option

5 1 TeVM

series of massive resonances

1( , 1 TeV)m

Small curvature optionSmall curvature optionSmall curvature optionSmall curvature option

5 1 TeVM

narrow low-mass resonanceswith 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 ED

5

5

10 0.1M

(Giudice et al., 2004,Kisselev & Petrov, 2005)

Lower bounds on gravity scale MLower bounds on gravity scale M55

e e E

5 0.92TeVM (DELPHI Coll., 2006)

p p X (Tevatron)

5 0.81TeVM

LHC search limit:

(Kisselev, 2008)

(LEP)

p p X

5 5.1TeVM (for L = 30 fb-1)

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

t - 4-momentum transfer

2( ) 2n g g nt t m

(Kisselev & Petrov, 2005)

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

- string tension2g sM

sum of gravi-reggeons

+ …+

jj

ii

Gravitational amplitudeGravitational amplitude

- SM fields (q, g, l, ......)i,ji,j

1

(for all SM fields)~

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

B ( , )pA s t

gravA gravA

21,

16 p

dA s t

dy s

0

0

, 4 1 exp ,pA s t is dbb J bq i s b

B0

0

1, ,

4 ps b dq q J qb A s ts

/y t s

q t

- inelasticity

Eikonal

Differential cross section

Amplitude

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)

if x - standard distributions

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

B ( , ) ( , ) ( , )ip grav i

i

A s t A sx t F x t

eff

1, sh

dN ddE dy E E A E E

dt E dy

eff , sh sh p shA E E att E P E A E

shE y E

Effective aperture for neutrinos

- shower energy

detector efficiency flux attenuation

Number of neutrino induced air showersNumber of neutrino induced air showersNumber of neutrino induced air showersNumber of neutrino induced air showers

E - shower energy

Neutrino bound (Pierre Auger Coll., 2007)

2 7 -2 -1 -1/ 1.3 10 GeV cm s srE dN dE

(detector efficiency for Auger)

910( ) 0.654 log /1GeV 10 10.9P E E

0.151had 2( ) 1.475 /1eV kmpA E E

0.208em 6 2( ) 7.037 10 /1eV kmpA E E

(Anchordoqui et al., JCAP, 2005)

(for hadronic showers)

(for electromagnetic showers)

75 90 , 7th 5 10 GeV,E

Flavor ratio on Earth: 1:1:1

Inclined neutrino induced air showersInclined neutrino induced air showers

SM: 0.08 events per year (W-B)

12max 10 GeVE

(Berezinsky, Zatsepin & Smirnov, 1969/1975)

Expected rate at the Auger Observatory (in yrExpected rate at the Auger Observatory (in yr-1-1) )

CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS In the RS model with the small

curvature KK graviton contributions could be comparable or even larger than SM contributions

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

Gravity effects from ED are large enough to be measured in ultra-high-energy neutrino-nucleon inclined events at the Pierre Auger Observatory

Back Up Slides

RS model with the small curvature is not RS model with the small curvature is not similar to a model with one large EDsimilar to a model with one large EDof the sizeof the size

RS model with the small curvature is not RS model with the small curvature is not similar to a model with one large EDsimilar to a model with one large EDof the sizeof 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

1d cR solar distance

2 3d

d - - is the number of ED’s

- strongly limited by astrophysical bounds

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

(Randall & Sundrum, 1999)

2 2 | | 2yds e dx dx dy

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:

( ) ( ) 2 ( ) ( )1eff 4

0

n n n nn

n

S dx h h m h h

(Boos et al., 2002)

Expression is not covariant: indices are raised Expression is not covariant: indices are raised with the Minkowski tensor while the metric iswith the Minkowski tensor while the metric is

2( , ) rx r e

Change of variables:rx z e x

2 2 ( | |)| 2r yds e dz dz dy