Generalized solution for metric in
Randall-Sundrum scenario
Alexander Kisselev
IHEP, NRC “Kurchatov Institute”,
Protvino, Russia
The XIII-th International School-Conference
“The Actual Problems of Microworld Physics”
Gomel, Belarus, July 29, 2015
Plan 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
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Space-time with one extra
spatial compact dimension
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Randall-Sundrum scenario
Background metric (y is an extra coordinate)
ds 2  g  dx  dx  dy 2
Periodicity: (x, y ± 2πrc) = (x, y)
Z2-symmetry: (x, y) = (x, –y)
orbifold S1/Z2
0 ≤ y ≤ πrc
Two fixed points: y=0 and y= πrc
two 3-dimensional branes
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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AdS5 space-time with two branes
Gravity
Planck brane
TeV brane
SM
y0
y   rc
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Five-dimensional action S  S g  S1  S 2

 
S g  d 4 x dy G 2M 53 R (5) - 



S1(2)  d 4 x g1(2) L1(2)  1(2)

Induced brane metrics
( 2)
g 
 G ( x, y   rc )
(1)
g 
 G ( x, y  0),
Einstein-Hilbert’s equations


1
1

| G | RMN  GMN R   


3
2


4M 5
[
(1)  
G GMN   g (1) g 
 M  N  ( y )1
( 2)  
 g ( 2) g 
 M  N  ( y   rc ) 2
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
]
6
Five-dimensional background metric tensor
G
MN
 g 
 
 0
0 
 1
g   exp[2 ( y)] 
ds 2  e 2 ( y )  dx  dx  dy 2
E-H’s equations for the warp function σ(y):
 '2 ( y )  
 ' ' ( y) 

24 M 53
1
12 M 53
[1 ( y )   2 ( rc  y )]
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Randall-Sundrum solution
(Randall & Sundrum, 1999)
 RS ( y)   | y |
RS  24M53 2 , (1)RS  (2 )RS  24M53
 'RS ( y)    ( y)
 ' 'RS ( y)  2  ( y)
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
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Generalization of RS solution
For the open interval 0 < y < πrc the solution is trivial
 ( y)   y  constant
Let us define:
  24 M 53 2 , 1,2  12M 53 1,2
Solution of 2-nd E-H’s equation for 0 ≤ y ≤ πrc
 ( y) 


1  2  y  y   rc   1  2  y  y   rc  constant
4
where 11  2  22 (to guarantee σ(y)=κy + C for 0 < y < πrc)
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Symmetry with respect to the branes
two possibilities:
- brane tensions Λ1,2 have the same sign
1  2  0
this case cannot be realized
- brane tensions Λ1,2 have opposite signs
1  2  0
As a result:
1  2  1

1
4
 ( y)   ( y   rc )2
The periodicity condition means:
 ( y   rc )   ( y   rc )   ( y   rc )
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Expression for the warp function (A.K., 2014-15)
 ( y) 

2
( y  y   rc ) 
 rc
2
 C 0 ≤ C ≤ κπrc
with the fine tuning
  24M53 2 , 1  2  12M53
Derivative of σ(y) at |y| < πrc
 ' ( y) 

2
factor of 2 different
than that of RS
 ( y)   ( y   rc )
is anti-symmetric in y:  ' ( y)   ' ( y)
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 ( y)  C   y
(0  y   rc )
σ(y)+ C
 ( y )  C   y   rc   rc
κπrc
y
-πrc
0
πrc
2πrc
 ( y)  C   y
 ' ' ( y)   [ ( y)   ( y   rc )] (0  y   rc )
XIII-th School-Conference, Gomel, Belarus,
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Two equivalent RS-like solutions
σ(y)+ C
  ( y )   y   rc   rc  C
κπrc
y
-πrc
0
πrc
2πrc
 0 ( y )   y  C0
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Neither of two branes/fixed points is preferable provided
orbifold and periodicity conditions are taken into account
If starting from the point y=0
 0 ( y )   y  C0
If starting from the point y=πrc
  ( y )   y   rc   rc  C
Solution symmetric with respect to the branes:
 ( y) 
1
 0 ( y)    ( y)   ( y  y   rc )   rc  C
2
2
2
Since  y   rc  y   rc  ( y  y  2 rc )  y   rc
solution is consistent with orbifold symmetry
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Randall-Sundrum solution:  RS ( y)   |y |
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), …
XIII-th School-Conference, Gomel, Belarus,
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Examples: definition of σ(y) outside this interval
Let y = πrc + y0 , where 0 < y0 < πrc
(that is, πrc < y < 2πrc)
 (rc  y0 )  C 


2

2
(| rc  y0 |  | y0 |) 
(| rc  y0  2π r c |  | y0 |) 
rc
2
rc
2
  (rc  y0 )
σ(y)+ C
κπrc
κ(πrc - y0)
πrc+ y0
0
πrc
2πrc
y
XIII-th School-Conference, Gomel, Belarus,
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Our solution for σ(y):
- is symmetric with respect to the branes
y  y   rc ,   
- makes the jumps on both branes
 ' ' ( y)  [ ( y)   ( y   rc )]
- is consistent with the orbifold symmetry
yy
- depends on the constant C
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Hierarchy relation
(depends on C)
Masses of KK gravitons
(xn are zeros of J1(x))
Interaction Lagrangian
on the TeV brane
(massive gravitons only)
2
M Pl

mn  x n

exp( 2C )
  



exp( 2π κ rc )  1  M 5 
M Pl
1
L( x)  

Coupling constant
(on TeV brane)
M 53


3/ 2
(n)
h
( x)T ( x)  
n 1
 
M Pl
exp( 2π κ rc )  1
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Parameters M5, κ, Λπ depend on constant C
The very expression for mn is independent of C,
but values of mn depend on C via M5 and κ
Different values of this constant result
in quite diverse physical models
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July 29, 2015
(A.K., 2014)
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Different physical schemes
I. C = 0
 (0)  0,  ( rc )   rc
M Pl2 
M 53

that requires M 5 ~  ~ M Pl
Masses of KK resonances
mn  xn exp(  r c )
RS1 model (Randall & Sundrum, 1999)
Graviton spectrum - heavy resonances,
with the lightest one above 1 TeV
XIII-th School-Conference, Gomel, Belarus,
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 (0)    rc ,  ( r)  0
II. C = κπrc
M Pl2 
  M 5
M 53

exp( 2  rc )
 rc  9.5 for M 5  1TeV,   100 MeV
Masses of KK resonances mn  xn
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
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
(Giudice,
Petrov & A.K.,
2005)
21
III. C = κπrc /2
 (0)    ( rc )   rc / 2
“Symmetric scheme” (A.K., 2014)
 ( z) 
In variable z = y - πκrc/2:

   rc
r

 z  c  z   C
2 2
2

σ(z)- C
κπrc/2
-πrc/2
0
0
πrc/2
z
-κπrc/2
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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2
M Pl

2M 53

Masses of gravitons
Let   M 5
sinh(2  rc )
mn  xn exp(  r c / 2)
M 5  2 109 GeV,   104 GeV
  3 105 GeV
mn  3.7 xn (MeV)
Almost continuous spectrum
of the gravitons
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Effective gravity action
S eff 
Shift
1

4
(n)
(n)

2 (n)
(n)
 
d
x
[

h
(
x
)

h
(
x
)


m
h
(
x
)
h
(
x
)]



 
 
n 

4 n 0
   C
is equivalent to the change
Invariance of the action
x   x'  eC x 
rescaling of fields and masses:
( n)
n)
( n)
h
 h'(
 eC h
, mn  m'n  eC mn
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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Example: C = πκrc (RS1 → RSSC)
      rc
mn  [ xn exp( r c)]  exp( r c)  xn
The end is near!
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July 29, 2015
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Virtual s-channel Gravitons
Scattering of SM fields mediated by
graviton exchange in s-channel
Processes:
pp  l l  ( , 2 jets)  X
e e  ff ( ), f  l , q
 
Sub-processes:
qq , gg  h
( n)
 ff , 
Σ
n≥1
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h(n)
26
Matrix element of sub-process
mediated by s-channel KK gravitons
M  AS
where
Tensor part of
graviton propagator
in 
A  T
P
Tf
Energy-momentum tensors
and
1
S ( s)  2


1

2
s

m
n 1
n  i mn n
(process
independent)
Graviton widths: n   mn3 / 2 ,   0.1
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July 29, 2015
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In symmetric scheme (A.K., 2014)
S ( s) 
1
2 
3
 M5 


s  
3/ 2
J 2 ( z)
J1 ( z )
with
s  M5 
z


   
3/ 2
(relation of mn with xn was used)
For chosen values of parameters:
O(1)
| S ( s ) |
(1TeV ) 3 s
TeV physics at LHC energies
(in spite of large Λπ)
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Conclusions





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
XIII-th School-Conference, Gomel, Belarus,
July 29, 2015
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Thank you for your attention
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Back-up slides
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RSSC model vs.ADD model
RSSC model is not equivalent to the ADD model
with one ED of the size R=(πκ)-1 up to κ ≈10-20 eV
Hierarchy relation for small κ
2
M Pl

M 53

  r  1
exp( 2  rc )  1 2
 M 53 (2 rc )
c
But the inequality 2  rc  1 means that
 
M 53
2
M Pl
3
18  M 5 
 eV
 0.17 10 
 1TeV 
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Dilepton production
at LHC
Pseudorapidity cut: |η| ≤ 2.4
Efficiency: 85 %
K-factor:
1.5 for SM background
1.0 for signal
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July 29, 2015
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33
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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July 29, 2015
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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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July 29, 2015
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Number of events with pt > ptcut
d (grav)
d (SM )
N S   cut dp 
, N B   cut dp 
p
p
dp 
dp 
Interference SM-gravity
contribution is negligible
Statistical significance
S 
S=5
NS
NS  NB
Lower bounds on M5 at 95% level
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July 29, 2015
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Statistical significance in RSSC for pp → e+e- + X
as a function of 5-dimensional reduced Planck scale
and cut on electron transverse momentum
for 7 TeV (L=5 fb-1) and 8 TeV (L=20 fb-1)
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No deviations from the CM
were seen at the LHC
M5 > 6.35 TeV (A.K., 2013)
(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 )
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