Generalized solution for RS metric and LHC phenomenology Alexander Kisselev IHEP, NRC “Kurchatov Institute”, Protvino, Russia The Third Annual Conference

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,

September 2, 2015

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


September 2, 2015

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:


September 2, 2015

)]()([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


September 2, 2015

σ(y)+ C

y2πrc-πrc 0 πrc

κπrc

yy )(0

cc rryy )(

Two equivalent solutions related to different branes


September 2, 2015

Generalization of RS solution


September 2, 2015

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


September 2, 2015

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)(''


September 2, 2015

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


September 2, 2015

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


September 2, 2015

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)


September 2, 2015

III. C = κπrc /2 2/)()0( cc rr


September 2, 2015

)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


September 2, 2015

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)


September 2, 2015

1616LHCP2015, St. Petersburg, Russia, September 2, 2015

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


September 2, 2015

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)


September 2, 2015

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


September 2, 2015

Conclusions


September 2, 2015

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)


September 2, 2015

Thank you for your attention

Back-up slides


September 2, 2015


September 2, 2015

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


September 2, 2015

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)


September 2, 2015


September 2, 2015

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


September 2, 2015

Graviton contributions in RSSC to the process

pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 14 TeV


September 2, 2015


September 2, 201530

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┴)


September 2, 2015

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


September 2, 2015

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), …


September 2, 2015

)(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


September 2, 2015

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


September 2, 2015


September 2, 2015

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


September 2, 201537

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Generalized solution for RS metric and LHC phenomenology Alexander Kisselev IHEP, NRC “Kurchatov Institute”, Protvino, Russia The Third Annual Conference