RS model with a small curvature and inclined air showers
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Transcript of RS model with a small curvature and inclined air showers
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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sum of gravi-reggeons
+ …+
jj
ii
Gravitational amplitudeGravitational amplitude
- SM fields (q, g, l, ......)i,ji,j
1
(for all SM fields)~
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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
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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
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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
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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
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(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)
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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) )
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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
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Back Up Slides
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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
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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
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Change of variables:rx z e x
2 2 ( | |)| 2r yds e dz dz dy