generalized solution for rs metric and lhc phenomenology alexander kisselev ihep, nrc “kurchatov...
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Generalized solution for RS metric and LHC phenomenology
Alexander KisselevIHEP, NRC “Kurchatov Institute”,
Protvino, Russia
The Third Annual Conference on Large Hadron Collider Physics (LHCP2015)
St. Petersburg, Russia, September 2, 2015
Plan of the talkPlan of the talk
Randall-Sundrum scenario with two branes. Original RS
solution Generalization of RS solution Solution with explicit orbifold
symmetries Role of constant term. Different
physical schemes Dilepton production at LHC in RS-like scenario Conclusions
2LHCP2015, St. Petersburg, Russia,
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Randall-Sundrum scenario Randall-Sundrum scenario
Periodicity: (x, y ± 2πrc) = (x, y) Z2-symmetry: (x, y) = (x, –y)
Background metric (y is extra coordinate)
orbifold S1/Z2 0 ≤ y ≤ πrc
Two fixed points: y=0 and y= πrc
two (1+3)-dimensional branes
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2)(22 dydxdxeds y
warp factor
21 SSSS g
-2 (5)35
4 RMGdyxdSg
)2(1)2(1)2(14
)2(1 LgxdS
5-dimensional action
Einstein-Hilbert’s equations:
4LHCP2015, St. Petersburg, Russia,
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)]()([12
1)(''
24)('
2135
35
2
yryM
y
My
c
(brane terms)
(gravity term)
(Randall & Sundrum, 1999)
3521
235 24)()(,24 RSRSRS MM
The RS solution:
- does not explicitly reproduce the jump of derivative on TeV brane (at y=πrc)
- is not symmetric with respect to both branes (located at y=0 and y=πrc)
- does not include a constant term
||)(RS yy
)()('RS yy
Original Randall-Sundrum solution
Original Randall-Sundrum solution
)(2)('' RS yy
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σ(y)+ C
y2πrc-πrc 0 πrc
κπrc
yy )(0
cc rryy )(
Two equivalent solutions related to different branes
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Generalization of RS solution
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Cr
ryyCyyy cc
2) (
2)()(
2
1)( 0
3521
235 12,24 MM
with fine tuning
)()(2
)(' cryyy
(0 ≤ C ≤ κπrc)
1-st derivative of σ(y)
Generalized solution
)()()('' cryyy 2-nd derivative of σ(y)
factor of 2 different than that of RS
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11,)(Arccos 0 zz
cryx /
Cr
xxr
y cc 2
cosArccoscosArccos2
)(
Explicit account of periodicity and Z2-
symmetrySolution for the warp function in variable
Arccos(z) is principal value of inverse cosine
Arccos(cos x) = { x - 2nπ, 2nπ ≤ x ≤ (2n+1)π -x + 2(n+1)π, (2n+1)π ≤ x ≤ 2(n+1)π
In particular, σ(y) = κy for 0 ≤ y ≤ πrc
(see, for instance, Gradshteyn &Ryzhik)
(A.K., 2015)
(x)sin sign)(' y
(y ≠ πnrc , n = 0, ±1, ±2, …)
)(')(' yy
nc
nxnxr
y 22)(''
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1-st derivative of σ(y):
2-nd derivative of σ(y):
)()2( yry c
)()( yy
(periodicity)
(Z2 symmetry)
Orbifold symmetries:
Hierarchy relation )2exp(352
Pl C
MM
Interaction Lagrangian(massive gravitons only)
)()(
1)(
1
)( xTxhxLn
n
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Different values of C result inquite diverse physical models
Masses of KK gravitons mn and coupling Λπ depend on C via M5 and κ
2/3
5
Pl
1)2exp(
Mrκπ
Mxm
cnn
Masses of KK gravitons (xn are zeros of J1(x))
I. C = 0 cc rr )(,0)0(
)exp( cnn rxm
352
Pl
MM that requires
Pl5 ~~ MM
Masses of KK resonances
RS1 model
Graviton spectrum - heavy resonances,with the lightest one above 1 TeV
(Randall & Sundrum, 1999)
Different physical scenarios
Different physical scenarios
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II. C = κπrc 0)(,)0( rrc
nn xm
)2exp(352
Pl crM
M
Masses of KK resonances
RSSC model: scenario with small curvature of 5-dimensional space-time
For small κ, graviton spectrum is similar to that of the ADD model
nn xm
5M MeV 100 TeV,1for5.9 5 Mrc
(Giudice, 2005Petrov & A.K., 2005)
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III. C = κπrc /2 2/)()0( cc rr
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)2/exp( cnn rxm Masses of gravitons
Let
)(7.3 MeVnn xm
GeV 10 GeV,102 495 M
Almost continuous spectrum of KK gravitons
)2sinh(2 3
52Pl cr
MM
(A.K., 2015)
“symmetric” scheme
Virtual Gravitons at the LHCVirtual Gravitons at the LHC
pp-collisions at LHC mediated by KK graviton exchange in s-channel
Xjetsllpp ),( 2
,, )( ffhggqq n
Processes:
Sub-processes:
h(n)
Σ n≥1
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KK gravitons
SAM
fαβ
μναβinμν TPTAwhere
Tensor part of graviton propagator
Energy-momentum tensors
122
11)(S
n nnn mimss
Matrix element of sub-process
and
1.0,/ 23 nn mGraviton widths
(process independent)
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Dilepton production at LHC in scenario with small curvature
cutcut dp
)(dp,
dp
)(dp
SMgrav
pBpS
dN
dN
Number of events with pt > ptcut
Interference (SM-gravity) contribution is negligible
cutcut dp
)(dp,
dp
)(dp
SMgrav
pBpS
dN
dN
Statistical significance BS
S
NN
NS
Scenario II (C = κπrc)
No deviations from the CM were seen at the LHC
M5 > 6.4 TeV for 7 + 8 TeV, L = 5 fb-1 + 20 fb-1
M5 > 9.0 TeV
for 13 TeV, L = 30 fb-1
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Week dependence of limits on parameter κ
(ptcut
= 200 GeV)LHC search limits on M5
)(
)(
2
1)(S
1
22/3
53 zJ
zJM
ss
2/35
Mszwith
For chosen values of parameters
ssS
3TeV)1(
O(1)|)(|
TeV physics at LHC energies
Scenario III (C = κπrc /2)
(relation of mn with xn was used)
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Generalized solution for metric in RS-like scenario is derived
This solution σ(y): - is symmetric with respect to the branes - has the jumps of its derivative on both branes - is consistent with the orbifold symmetries - depends on constant C (0 ≤ C ≤ κπrc) Different values of C result in quite
diverse physical scenarios: RS1, RSSC, “symmetric” models
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Conclusions
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All these schemes lead to TeV physics at LHC, but with different experimental signatures
LHC limits on gravity scale M5 are obtained in RS-like scheme with small curvature
Conclusions (continued)
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Thank you for your attention
Back-up slides
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ADD model :
MD > 4 TeV (for nED = 4) real graviton production
MS > 7 TeV (for nED = 4) (HLZ convention)
virtual graviton exchange(CMS, PLB 746 (2015) 79)
(CMS, EPJC 75 (2015) 235)
RS model :
mG* > 2.7 TeV (for κ/MPl = 0.1 )(ATLAS , PRD 90 (2014) 052005)
Extra dimensions: LHC limits
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Pseudorapidity cuts: |η| ≤ 2.4 (muons)|η| ≤ 1.44, 1.57 ≤ |η| ≤ 2.4 (electrons)
Efficiency: 85 %
K-factor: 1.5 for SM background 1.0 for signal
Dilepton production
Uncertainties in search limit (L = 30 fb-1)
Uncertainties in search limit (L = 30 fb-1)
5| | 17 GeVM
PDF set (2.7%)
ΔL (3%)
5| | 23 GeVM
5| | 26 GeVM
μ/2 →μ→2μ
Statistical significance in RSSC for pp → e+e- + X as a function of 5-dimensional reduced Planck scale
M5 and cut on electron transverse momentum ptcut
for 7 TeV (L=5 fb-1) and 8 TeV (L=20 fb-1)
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Statistical significance for the process pp → e+e- + X
as a function of 5-dimensional reduced Planck scale
M5 and cut on electron transverse momentum pt
cut
for 13 TeV (L=30 fb-1)
Graviton contributions in RSSC to the process pp → μ+μ- + X (solid lines)
vs. SM contribution (dashed line) for 7 TeV
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Graviton contributions in RSSC to the process
pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 14 TeV
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3/2
( ) 1( , ln )
t
SMdf x s
dp s
35
( ) 1( , ln )
t
d gravg x s
dp M
3
5
( ) /
( ) /t
tSM
d grav dp s
d dp M
Weak logarithmic dependence on energy comes from PDFs
dσ(grav): week dependence on curvature κ
(for fixed x┴)
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dσ(pp→h(n)→γγ) = 2 ∙ dσ(pp→h(n)→l+l-)
universal for all subprocesses
4)(
42)(
cos1:
cos4cos31:
llGgg
llGqqn
n
2cos1while in SM:
0
)()(2)()(4eff )]()()()([
4
1
n
nnn
nn xhxhmxhxhxdS
xxx C e'
nC
nnnCnn mmmhhh e',e' )()()(
Effective gravity action
C Shift is equivalent to the change
Invariance of the action rescaling of fields and masses:
Massive theory is not scale-invariant
From the point of view of 4- dimensional observer these two theories are not physically equivalent
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Randall-Sundrum solution ||)(RS yy
Does it mean that σRS(y) is a linear function for all y?
The answer is negative: one must use the periodicity condition first, and only then estimate absolute value |y|
In other words, extra coordinate y must be reduced to the interval -πrc ≤ y ≤ πrc (by using periodicity condition) before evaluating functions |x|, ε(x), …
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)(2
|)||(|2
2|)||(|
2)(
000
000
yrr
yyr
ryyrCyr
cc
c
ccc
cr 2π
Examples: definition of σ(y) outside interval |y| ≤ πrc
Let y = πrc + y0 , where 0 < y0 < πrc (that is, πrc < y < 2πrc)
σ(y)+ C
2πrc0 πrc
κπrc
y
κ(πrc - y0)
πrc+ y0
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RSSC model is not equivalent to the ADD model with one ED of size R=(πκ)-1 up to κ ≈10-20 eV
)2(1)2exp( 35
12352
Pl cr
c rMrM
M c
Hierarchy relation for small κ
But the inequality 12 cr means that
eVTeV1
1017.03
5182Pl
35
M
M
M
RSSC model vs.ADD model
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0)(,)(
)(
2
11,
1
122
,
n
n n
zJzJ
zJ
zzz
where3
52,
M
ssA
(A.K, 2006)
36
2235 sinhcos
2sinh2sin
4
1)(
A
iA
sMsS
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