generalized solution for metric in randall-sundrum scenario alexander kisselev ihep, nrc...

Generalized solution for metric in Randall-Sundrum scenario

Alexander KisselevIHEP, NRC “Kurchatov Institute”,

Protvino, Russia

The XIII-th International School-Conference“The Actual Problems of Microworld Physics”

Gomel, Belarus, July 29, 2015

2XIII-th School-Conference, Gomel, Belarus,

July 29, 2015

Plan of the talkPlan of the talk

Randall-Sundrum (RS) scenario with two branes

Original RS solution for the warp factor of the metric

Generalization of the RS solution Role of the constant term. Different

physical schemes Conclusions


July 29, 2015

Space-time with one extra spatial compact dimension


July 29, 2015

Randall-Sundrum scenario Randall-Sundrum scenario

22 dydxdxgds

Periodicity: (x, y ± 2πrc) = (x, y) Z2-symmetry: (x, y) = (x, –y)

Background metric (y is an extra coordinate)

orbifold S1/Z2 0 ≤ y ≤ πrc

Two fixed points: y=0 and y= πrc

two 3-dimensional branes


July 29, 2015

0y cy r

Planck brane TeV brane Gravity

SM

Planck brane

AdS5 space-time with two branes


July 29, 2015

21 SSSS g

-2 (5)35

4 RMGdyxdSg

)2(1)2(1)2(14

)2(1 LgxdS

Five-dimensional action

Induced brane metrics

),(),0,( )2()1(cryxGgyxGg

]

[

2)2()2(

1)1()1(

)(

)(4

1

2

1||

35

cNM

NMMNMNMN

rygg

yggGGM

RGRG

Einstein-Hilbert’s equations


July 29, 2015

2)(22 dydxdxeds y

Five-dimensional background metric tensor

10

0gG MN )](2exp[ yg

E-H’s equations for the warp function σ(y):

)]()([12

1)(''

24)('

2135

35

2

yryM

y

My

c


July 29, 2015

(Randall & Sundrum, 1999)

3521

235 24)()(,24 RSRSRS MM

The RS solution:

- does not explicitly reproduce the jump on brane y=πrc

- is not symmetric with respect to the branes, although both branes (at points y=0 and y=πrc) must be treated on an equal footing

- does not include a constant term

||)(RS yy

)()('RS yy

Randall-Sundrum solution

Randall-Sundrum solution

)(2)('' RS yy


July 29, 2015

Solution of 2-nd E-H’s equation for

constant 4

)( 2121 cc ryyryyy

221 where

constant)( yy

For the open interval the solution is trivial

Let us define: 2,1352,1

235 ,24 MM 12

221

0 < y < πrc

0 ≤ y ≤ πrc

Generalization of RS solution

Generalization of RS solution

(to guarantee σ(y)=κy + C for 0 < y < πrc)

10

two possibilities:

- brane tensions Λ1,2 have the same sign

021 this case cannot be realized

- brane tensions Λ1,2 have opposite signs

021

121 As a result: 2)()(4

1cryy

Symmetry with respect to the branes

The periodicity condition means:

)()()( ccc ryryry

XIII-th School-Conference, Gomel, Belarus, July 29, 2015

11

Expression for the warp function (A.K., 2014-15)

Cr

ryyy cc

2) (

2)(

3521

235 12,24 MM

with the fine tuning

)()(2

)(' cryyy

is anti-symmetric in y:

0 ≤ C ≤ κπrc

Derivative of σ(y) at |y| < πrc

)(')(' yy


factor of 2 different than that of RS


July 29, 2015

σ(y)+ C

y2πrc-πrc 0 πrc

κπrc

yCy )(

)0()]()([)('' cc ryryyy

cc rryCy )()

)(

0( cry

yCy

orbifolding

periodicity


July 29, 2015

σ(y)+ C

y2πrc-πrc 0 πrc

κπrc

00 )( Cyy

Crryy cc )(

Two equivalent RS-like solutions


July 29, 2015

00 )( Cyy

Cr

ryyyyy cc

2) (

2)()(

2

1)( 0

Crryy cc )(

Solution symmetric with respect to the branes:

If starting from the point y=0

If starting from the point y=πrc

cccc rryyryr yy )2(Since

Neither of two branes/fixed points is preferable provided orbifold and periodicity conditions are taken into account

solution is consistent with orbifold symmetry


July 29, 2015

Randall-Sundrum solution: ||)(RS yy

Does it mean that σ(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), …


July 29, 2015

)(2

|)||(|2

2|)||(|

2)(

000

000

yrr

yyr

ryyrCyr

cc

c

ccc

cr 2π

Examples: definition of σ(y) outside this interval

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


July 29, 2015

Our solution for σ(y):

- makes the jumps on both branes

- is symmetric with respect to the branes

- is consistent with the orbifold symmetry

- depends on the constant C

,cryy

yy

)]()([)('' cryyy


July 29, 2015

Hierarchy relation(depends on C)

)2exp(352

Pl CM

M

2/3

5

Pl

1)2exp(

Mrκπ

Mxm

cnn

Masses of KK gravitons (xn are zeros of J1(x))

1)2exp(Pl

crκπ

M

Interaction Lagrangian on the TeV brane

(massive gravitons only)

Coupling constant(on TeV brane)

)()(

1)(

1

)( xTxhxLn

n

Different values of this constant result in quite diverse physical models

(A.K., 2014)

Parameters M5, κ, Λπ depend on constant C


July 29, 2015

The very expression for mn is independent of C,but values of mn depend on C via M5 and κ


July 29, 2015

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 schemes

Different physical schemes


July 29, 2015

II. C = κπrc 0)(,)0( rrc

nn xm

)2exp(352

Pl crM

M

Masses of KK resonances

RSSC model: RS-like scenario with the 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, Petrov & A.K., 2005)


July 29, 2015

III. C = κπrc /2 2/)()0( cc rr

In variable z = y - πκrc/2: Czr

zr

z cc

222)(

σ(z)- C

z

0

κπrc/2

-κπrc/2

πrc/2

-πrc/2

0

“Symmetric scheme” (A.K., 2014)


July 29, 2015

)2sinh(2 3

52Pl cr

MM

)2/exp( cnn rxm Masses of gravitons

Let 5M

MeV)(7.3 nn xm

GeV 10 GeV,102 495 M

GeV103 5

Almost continuous spectrum of the gravitons


July 29, 2015

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


July 29, 2015

nccnn xrrxm )exp()]exp([

cr Example: C = πκrc (RS1 → RSSC)

The end is near!


July 29, 2015

Virtual s-channel GravitonsVirtual s-channel Gravitons

Scattering of SM fields mediated by graviton exchange in s-channel

Scattering of SM fields mediated by graviton exchange in s-channel

qlfffee

Xjetsllpp

,),(

),( 2

,, )( ffhggqq n

Processes:

Sub-processes:

h(n)

Σ n≥1


July 29, 2015

SAM

fin TPT

Awhere

Tensor part of graviton propagator

Energy-momentum tensors

122

11)(

n nnn mimssS

Matrix element of sub-process mediated by s-channel KK gravitons

and

1.0,/ 23 nn mGraviton widths:

(process independent)


July 29, 2015

)(

)(

2

1)(

1

22/3

53 zJ

zJM

ssS

2/35

Mszwith

For chosen values of parameters:

ssS

3)TeV1(

)1(O|)(|

TeV physics at LHC energies (in spite of large Λπ)

In symmetric scheme (A.K., 2014)

(relation of mn with xn was used)


July 29, 2015

ConclusionsConclusions

Generalized solution for the warp factor σ(y) in RS-like scenario is derived

New expression: - is symmetric with respect to the branes - has the jumps on both branes - is consistent with the orbifold symmetry σ(y) depends on constant C (0 ≤ C ≤

κπrc) Different values of C result in quite

diverse physical schemes Particular scenarios are: RS1 model, RSSC model, “symmetric” model

Thank you for your attention


July 29, 2015


Back-up slides

31


July 29, 2015

RSSC model is not equivalent to the ADD model with one ED of the 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

Dilepton production at LHC

Pseudorapidity cut: |η| ≤ 2.4Efficiency: 85 %K-factor: 1.5 for SM background 1.0 for signal


July 29, 2015


July 29, 2015

Graviton contributions in RSSC to the process pp → μ+μ- + X (solid lines)

vs. SM contribution (dashed line) for 7 TeV


July 29, 2015

Graviton contributions in RSSC to the process

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


July 29, 2015

cutcut dp

)(dp,

dp

)(dp

SMgrav

pBpS

dN

dN

Statistical significance

Number of events with pt > ptcut

BS

S

NN

NS

Interference SM-gravity contribution is negligible

Lower bounds on M5 at 95% level S=5


July 29, 2015

Statistical significance in RSSC for pp → e+e- + X as a function of 5-dimensional reduced Planck scale

and cut on electron transverse momentumfor 7 TeV (L=5 fb-1) and 8 TeV (L=20 fb-1)


July 29, 2015

No deviations from the CM were seen at the LHC

M5 > 6.35 TeV (for 7 + 8 TeV, L = 5 fb-1 + 20 fb-1 )

LHC search limits on M5:

8.95 TeV (for 13 TeV, L = 30 fb-

1 )

(A.K., 2013)

generalized solution for metric in randall-sundrum scenario alexander kisselev ihep, nrc...

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