Features of warped geometry in presence of
Gauss-Bonnet coupling 1 2 3
Sayantan Choudhury
Physics and Applied Mathematics Unit
INDIAN STATISTICAL INSTITUTE, Kolkata,India
SUSY 2013
ICTP,Trieste,Italy
Date : 29/08/2013
1 Sayantan
Soumitra SenGupta & Soumya Sadhukhan,
Choudhury,
arXiv:1308.1477[hep-ph]
2 Sayantan Choudhury, Soumitra SenGupta, arXiv:1306.0492[hep-th]
3 Sayantan
Choudhury,
Soumitra SenGupta,
arXiv:1301.0918[hep-th],
JHEP 02 (2013) 136
Outline of My Talk
⋆Highlights....
⋆Einstein Gauss-Bonnet warped geometry model
in presence of string loop corrections and dilaton
couplings ....
⋆Warp factor and brane tension .....
⋆Analysis of bulk Kaluza-Klien (KK) spectrum.....
⋆Bulk Graviton & brane SM field interaction in
presence of Gauss-Bonnet coupling.....
⋆Collider constraints on the Gauss-Bonnet
coupling from Higgs phenomenology....
⋆Bottom lines....
1
HIGHLIGHTS
◮Determining the modified warp factor, the brane tensions
and addressing the gauge hierarchy issue.
2
HIGHLIGHTS
◮Determining the modified warp factor, the brane tensions
and addressing the gauge hierarchy issue.
◮Study of different bulk fields and the profile of the wave
functions to examine their overlap on the visible brane as well
as various KK mode masses for these bulk fields.
2- A
HIGHLIGHTS
◮Determining the modified warp factor, the brane tensions
and addressing the gauge hierarchy issue.
◮Study of different bulk fields and the profile of the wave
functions to examine their overlap on the visible brane as well
as various KK mode masses for these bulk fields.
◮Examining the interaction with the brane fields to evaluate
their possible signatures.
2- B
HIGHLIGHTS
◮Determining the modified warp factor, the brane tensions
and addressing the gauge hierarchy issue.
◮Study of different bulk fields and the profile of the wave
functions to examine their overlap on the visible brane as well
as various KK mode masses for these bulk fields.
◮Examining the interaction with the brane fields to evaluate
their possible signatures.
◮Comparing our results with that obtained through the usual
RS analysis.
2- C
HIGHLIGHTS
◮Determining the modified warp factor, the brane tensions
and addressing the gauge hierarchy issue.
◮Study of different bulk fields and the profile of the wave
functions to examine their overlap on the visible brane as well
as various KK mode masses for these bulk fields.
◮Examining the interaction with the brane fields to evaluate
their possible signatures.
◮Comparing our results with that obtained through the usual
RS analysis.
◮Stringent collider constraints on GB coupling from different
phenomenological probes obtained from 125 GeV Higgs.
2- D
THE BACKGROUND WARPED GEOMETRY MODEL
◮ 5D MODEL ACTION
3
THE BACKGROUND WARPED GEOMETRY MODEL
◮ 5D MODEL ACTION
S(5) = SEH + SGB + Sloop + SBulk + SBrane
where
3
M(5)
5 p
d
x −g(5) R(5)
2
h
i
α(5) M(5) R 5 p
ABCD(5) (5)
AB(5) (5)
2
SGB =
d x −g(5) R
RABCD − 4R
RAB + R(5)
2
h
α(5) A1 M(5) R 5 p
ABCD(5) (5)
θ1 φ
R
RABCD
−g
e
d
x
Sloop = −
(5)
2
SEH =
R
− 4R
AB(5)
(5)
RAB
+
2
R(5)
h
i
p
f ield
θ2 φ
SBulk = d x −g(5) LBulk − 2Λ(5) e
i
R 5 P2 q (i) h f ield
θ2 φ
SBrane = d x i=1 −g(5) L(i) − T(i) e
δ(y − y(i) )
R
5
3- A
i
THE BACKGROUND WARPED GEOMETRY MODEL
◮ 5D MODEL ACTION
S(5) = SEH + SGB + Sloop + SBulk + SBrane
where
3
M(5)
5 p
d
x −g(5) R(5)
2
h
i
α(5) M(5) R 5 p
ABCD(5) (5)
AB(5) (5)
2
SGB =
d x −g(5) R
RABCD − 4R
RAB + R(5)
2
h
α(5) A1 M(5) R 5 p
ABCD(5) (5)
θ1 φ
R
RABCD
−g
e
d
x
Sloop = −
(5)
2
SEH =
R
− 4R
AB(5)
(5)
RAB
+
2
R(5)
h
i
p
f ield
θ2 φ
SBulk = d x −g(5) LBulk − 2Λ(5) e
i
R 5 P2 q (i) h f ield
θ2 φ
SBrane = d x i=1 −g(5) L(i) − T(i) e
δ(y − y(i) )
R
5
◮ BACKGROUND METRIC
ds2(5) = gAB dxA dxB = e−2A(y) ηαβ dxα dxβ + rc2 dy 2
◮S 1 /Z2 orbifold points are yi = [0, π] and PBC in −π ≤ y ≤ π.
3- B
i
THE BACKGROUND WARPED GEOMETRY MODEL
◮From
a theoretical standpoint, like RS the
proposed warped geometry model has its
underlying motivation in the backdrop of
string theory where the throat geometry
(Klevanov- Strassler) solution exhibits
warping character.
4
THE BACKGROUND WARPED GEOMETRY MODEL
◮From
a theoretical standpoint, like RS the
proposed warped geometry model has its
underlying motivation in the backdrop of
string theory where the throat geometry
(Klevanov- Strassler) solution exhibits
warping character.
◮The higher curvature perturbative
corrections to usual RS model originate
naturally in string theory where power
expansion in terms of inverse string
tension yields the higher order corrections
to pure Einstein’s gravity.
4- A
THE BACKGROUND WARPED GEOMETRY MODEL
5
WARP FACTOR AND BRANE TENSION
◮ EINSTEIN GAUSS BONNET EQN
p
α
(5)
(5)
−g(5) GAB + M(5)
1 − A1 eθ1 φ HAB =
2
(5)
q
P
p
θ
φ
(5)
(i) (i) α β
e 2
Λ(5) −g(5) gAB + 2i=1 T(i) −g(5) gαβ δA
δB δ(y − y(i) )
−M
3
(5)
◮ GRAVIDILATON EQN
θ2
2
M(5)
P2
q
(i)
−g(5) eθ2 φ δ(y − y(i) ) =
i=1 T(i)
i
h
n
p
2
AB(5) (5)
ABCD(5) (5)
RAB + R(5)
RABCD − 4R
−g(5) α(5) A1 θ1 R
Λ(5)
✷
φ
+ 2 M 2 θ2 eθ2 φ + M(5)
(5)
(5)
6
WARP FACTOR AND BRANE TENSION
◮ EINSTEIN GAUSS BONNET EQN
p
α
(5)
(5)
−g(5) GAB + M(5)
1 − A1 eθ1 φ HAB =
2
(5)
q
P
p
θ
φ
(5)
(i) (i) α β
e 2
Λ(5) −g(5) gAB + 2i=1 T(i) −g(5) gαβ δA
δB δ(y − y(i) )
−M
3
(5)
◮ GRAVIDILATON EQN
θ2
2
M(5)
P2
q
(i)
−g(5) eθ2 φ δ(y − y(i) ) =
i=1 T(i)
i
h
n
p
2
AB(5) (5)
ABCD(5) (5)
RAB + R(5)
RABCD − 4R
−g(5) α(5) A1 θ1 R
Λ(5)
✷
φ
+ 2 M 2 θ2 eθ2 φ + M(5)
(5)
where
H
(5)
(5)
(5)
(5)
− 1g
= R
G
R(5)
AB
AB
2 AB
(5)
(5) C(5)
(5)
CDE(5)
(5)
(5)
+ 2R(5) R
R
RCD(5) − 4R
− 4R
R
= 2R
AB
AC B
ACBD
ACDE B
AB
(5)
(5)
(5)
− 1g
+ R2
− 4RAB(5) R
RABCD(5) R
AB
ABCD
2 AB
(5)
6- A
WARP FACTOR AND BRANE TENSION
◮ DILATON FIELD
φ(y) =
◮ WARP FACTOR
wheres
k± =
P2
p=1
|y|
5
2
θp
+
1
θp
A(y) := A± (y) = k± rc |y|
2
3M(5)
16α(5) (1−A1 eθ1 φ )
1±
r
1+
4α(5) (1−A1
e θ1 φ
5
9M(5)
)Λ(5)
e θ2 φ
7
WARP FACTOR AND BRANE TENSION
◮ DILATON FIELD
φ(y) =
◮ WARP FACTOR
wheres
k± =
P2
p=1
|y|
5
2
θp
+
1
θp
A(y) := A± (y) = k± rc |y|
2
3M(5)
16α(5) (1−A1 eθ1 φ )
1±
r
1+
4α(5) (1−A1
e θ1 φ
5
9M(5)
⋆RANDALL-SUNDRUM (RS) LIMIT:
s


 k− → kRS = − Λ(5)
24M3(5)
(α(5) , A1 , θ1 , θ2 ) → 0 ⇒


 k →∞
+
)Λ(5)
e θ2 φ
with Λ(5) < 0
7- A
12
12
10
10
8
8
kHin MPL L
kHin MPL L
WARP FACTOR AND BRANE TENSION
6
4
k-
6
4
k+
2
k+
k-
2
kRS
0
0.0
kRS
0.2
0.4
0.6
0.8
0
1.0
0.0
0.2
Α5 H5 D Gauss - Bonnet CouplingL
0.4
0.6
0.8
1.0
Α5 H5 D Gauss - Bonnet CouplingL
(a) Λ(5) > 0 and A1 > 0
(b) Λ(5) > 0 and A1 < 0
14
12
12
10
kHin MPL L
kHin MPL L
10
8
6
k+
6
k+
4
k-
k-
4
2
0
-1.0
8
kRS
2
kRS
-0.8
-0.6
-0.4
-0.2
0.0
Α5 H5 D Gauss - Bonnet CouplingL
(c) Λ(5) < 0 and A1 > 0
0.2
0
-0.3
-0.2
-0.1
0.0
0.1
0.2
Α5 H5 D Gauss - Bonnet CouplingL
(d) Λ(5) < 0 and A1 < 0
8
WARP FACTOR AND BRANE TENSION
40
40
POINT OF NO WARPING IN THE BULK
POINT OF NO WARPING IN THE BULK
Α5 = 0.0046
Α5 = 0.0051
30
30
20
Α5 = 0.001
Α5 = 0.03
Α5 = 0.03
A+ @yD
A+ @yD
Α5 = 0.0009
Α5 = 0.3
10
20
Α5 = 0.3
10
0
-3
-2
-1
0
y
1
2
(e) Λ(5) > 0 and A1 > 0
3
0
-3
-2
-1
0
y
1
2
3
(f) Λ(5) > 0 and A1 < 0
9
WARP FACTOR AND BRANE TENSION
40
40
POINT OF NO WARPING IN THE BULK
POINT OF NO WARPING IN THE BULK
Α5 = 0.00064
Α5 = 0.0031
30
30
Α5 = 0.0007
A+ @yD
A+ @yD
Α5 = 0.0034
20
Α5 = 0.03
Α5 = 0.003
20
Α5 = 0.01
Α5 = 0.1
10
10
0
-3
-2
-1
0
y
1
2
(g) Λ(5) < 0 and A1 > 0
3
0
-3
-2
-1
0
y
1
2
3
(h) Λ(5) < 0 and A1 < 0
10
WARP FACTOR AND BRANE TENSION
40
40
POINT OF NO WARPING IN THE BULK
POINT OF NO WARPING IN THE BULK
Α5 = -0.00032
Α5 = -0.0016
30
30
Α5 = -0.00034
A- @yD
A- @yD
Α5 = -0.0017
20
20
Α5 = -0.005
Α5 = -0.04
10
10
Α5 = -0.05
Α5 = -0.1
0
-3
-2
-1
0
y
1
2
(i) Λ(5) > 0 and A1 > 0
3
0
-3
-2
-1
0
y
1
2
3
(j) Λ(5) > 0 and A1 < 0
11
WARP FACTOR AND BRANE TENSION
40
40
POINT OF NO WARPING IN THE BULK
Α5 = -0.00045
POINT OF NO WARPING IN THE BULK
Α5 = -0.0022
30
Α5 = -0.0005
30
Α5 = -0.005
A- @yD
A- @yD
Α5 = -0.0026
Α5 = -0.05
20
Α5 = -0.05
20
Α5 = -0.5
10
10
0
-3
-2
-1
0
y
1
2
(k) Λ(5) < 0 and A1 > 0
3
0
-3
-2
-1
0
y
1
2
3
(l) Λ(5) < 0 and A1 < 0
12
WARP FACTOR AND BRANE TENSION
◮
BRANE TENSION
±
∓
Tvis
=
−T
hid = 3 −θ2 φ
± 6k± M(5)
e
4−
4α(5) (1−A1 eθ1 φ ) 2 2
k± rc
2
3M(5)
13
WARP FACTOR AND BRANE TENSION
◮
BRANE TENSION
±
∓
Tvis
=
−T
hid = 3 −θ2 φ
± 6k± M(5)
e
4−
4α(5) (1−A1 eθ1 φ ) 2 2
k± rc
2
3M(5)
⋆RANDALL-SUNDRUM
(RS) LIMIT:-

 T− → TRS = 24M3 kRS
(5)
vis
vis
(α(5) , A1 , θ1 , θ2 ) → 0 ⇒
 T+ → ∞
vis
with Λ(5) < 0
13- A
WARP FACTOR AND BRANE TENSION
0
0
-100
Tvis +
-300
Tvis Iin MPL5 M
Tvis Iin MPL5 M
-50
Tvis -
-200
Tvis RS
Tvis +
Tvis -100
Tvis RS
-150
-400
-500
-0.005
0.000
0.005
-200
-0.002
0.010
Α5 H5 D Gauss - Bonnet CouplingL
0.004
0.006
0.008
0.010
(n) Λ(5) > 0 and A1 < 0
0
0
-100
-100
-200
-200
-300
Tvis
Tvis
Tvis Iin MPL5 M
Tvis Iin MPL5 M
0.002
Α5 H5 D Gauss - Bonnet CouplingL
(m) Λ(5) > 0 and A1 > 0
-400
0.000
+
-
Tvis RS
-500
-300
Tvis +
Tvis -
-400
Tvis RS
-500
-600
-600
-700
-0.002
0.000
0.002
0.004
0.006
Α5 H5 D Gauss - Bonnet CouplingL
(o) Λ(5) < 0 and A1 > 0
0.008
-700
-0.0020 -0.0015 -0.0010 -0.0005
0.0000
0.0005
0.0010
0.0015
Α5 H5 D Gauss - Bonnet CouplingL
(p) Λ(5) < 0 and A1 < 0
14
ANALYSIS OF BULK KK SPECTRUM
◮GRAVITON MASS SPECTRUM:
15
ANALYSIS OF BULK KK SPECTRUM
◮GRAVITON MASS SPECTRUM:
•Spin-2 transeverse and traceless graviton d.o.f. are
generated via tensor perturbation in the warped metric.
(n)
P∞
χ±;G (y)
(n)
•KK reduction ansatz: hαβ (x, y) = n=0 hαβ (x) √rc
where

(mG
p
n )± k r |y|

± c
2k
r
|y|

±
c
e
)
J
(
2

k
r
e
k
±
c
±

(n)
(mG
n )± k± rc π
ek± rc π
χ±;G (y) =
)
J
(
2

k± e


 p
k± rc
•4D effective action:SG ⊃
R
4
d x
P∞
n=0
h
αβ (n)
for n > 0
for n = 0.
(n)
(x)hαβ (x)
G 2
mn ±
15- A
ANALYSIS OF BULK KK SPECTRUM
◮GRAVITON MASS SPECTRUM:
•Spin-2 transeverse and traceless graviton d.o.f. are
generated via tensor perturbation in the warped metric.
(n)
P∞
χ±;G (y)
(n)
•KK reduction ansatz: hαβ (x, y) = n=0 hαβ (x) √rc
where

(mG
p
n )± k r |y|

± c
2k
r
|y|

±
c
e
)
J
(
2

k
r
e
k
±
c
±

(n)
(mG
n )± k± rc π
ek± rc π
χ±;G (y) =
)
J
(
2

k± e


 p
k± rc
•4D effective action:SG ⊃
R
4
d x
•KK graviton mass spectra:
P∞
n=0
mG
n ±
h
αβ (n)
≈ n+
for n > 0
for n = 0.
(n)
(x)hαβ (x)
1
2
∓
1
4
G 2
mn ±
−k± rc π
πk± e
15- B
ANALYSIS OF BULK KK SPECTRUM
◮GRAVITON MASS SPECTRUM:
0.35
0.6
0.30
0.20
mG
1 @in MPL D
0.25
mG
1 @in MPL D
0.5
mG1 RS
mG1 -
0.15
0.10
0.4
mG1 -
0.3
0.2
mG1 +
mG1 +
0.1
0.05
0.00
mG1 RS
0.0
0.2
0.4
0.6
0.8
Α5 H5 D Gauss-Bonnet CouplingL
(q) Λ(5) > 0 and A1 > 0
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Α5 H5 D Gauss-Bonnet CouplingL
(r) Λ(5) > 0 and A1 < 0
16
ANALYSIS OF BULK KK SPECTRUM
◮GRAVITON MASS SPECTRUM:
0.30
mG1 RS
0.25
0.25
0.20
0.20
mG
1 @in MPL D
mG
1 @in MPL D
0.30
mG1 +
0.15
mG1 -
mG1 +
0.15
0.10
0.10
0.05
0.05
0.00
-0.6
-0.4
-0.2
0.0
Α5 H5 D Gauss-Bonnet CouplingL
(s) Λ(5) < 0 and A1 > 0
0.2
mG1 RS
mG1 -
0.00
-0.20
-0.15
-0.10
-0.05
0.00
0.05
Α5 H5 D Gauss-Bonnet CouplingL
(t) Λ(5) < 0 and A1 < 0
17
ANALYSIS OF BULK KK SPECTRUM
◮SCALAR MASS SPECTRUM:
18
ANALYSIS OF BULK KK SPECTRUM
◮SCALAR MASS hSPECTRUM:
i
R
p
→
−
→
−
•SΦ = 12 d5 x −g(5) g AB ∂ A Φ(x, y) ∂ B Φ(x, y) − m2Φ Φ2 (x, y)
(n)
P∞
χ±;Φ (y)
(n)
•KK reduction ansatz:Φ(x, y) = n=0 Φ (x) √rc
where

2
(mΦ
(mΦ
n )± k r c π √

n )± k rc |y|
2k± rc |y|
±

k
r
e
e
e ±
)
(
J
c
±

k±
νΦ
k2

±
±
 v
for n > 0
u
2
mΦ
(n)
(
)
u (mΦ
n
k r π
n )± 2k rc π
t
χ±;Φ (y) =
Φ 2
J Φ( k ± e ± c )
e ±
+4−(ν±
)
ν
2
±

±
k

±
q


k ± rc

for n = 0.
−2k± rc π
1−e
•4D effective action:
h
SΦ =
R
4
d x
P∞
n=0
→ (n) −
→ (n)
µν −
η ∂ µ Φ (x) ∂ ν Φ (x)
−
2
(n)
(mΦ
(x))2
n )± (Φ
i
18- A
ANALYSIS OF BULK KK SPECTRUM
◮SCALAR MASS hSPECTRUM:
i
R
p
→
−
→
−
•SΦ = 12 d5 x −g(5) g AB ∂ A Φ(x, y) ∂ B Φ(x, y) − m2Φ Φ2 (x, y)
(n)
P∞
χ±;Φ (y)
(n)
•KK reduction ansatz:Φ(x, y) = n=0 Φ (x) √rc
where

2
(mΦ
(mΦ
n )± k r c π √

n )± k rc |y|
2k± rc |y|
±

k
r
e
e
e ±
)
(
J
c
±

k±
νΦ
k2

±
±
 v
for n > 0
u
2
mΦ
(n)
(
)
u (mΦ
n
k r π
n )± 2k rc π
t
χ±;Φ (y) =
Φ 2
J Φ( k ± e ± c )
e ±
+4−(ν±
)
ν
2
±

±
k

±
q


k ± rc

for n = 0.
−2k± rc π
1−e
•4D effective action:
h
SΦ =
R
4
d x
P∞
n=0
→ (n) −
→ (n)
µν −
η ∂ µ Φ (x) ∂ ν Φ (x)
•KK scalar mass spectra:
mΦ
n ±
−
≈ n+
2
(n)
(mΦ
(x))2
n )± (Φ
1 Φ
2 ν±
−
3
4
i
πk± e−k± rc π
18- B
ANALYSIS OF BULK KK SPECTRUM
◮U (1) GAUGE FIELD MASS SPECTRUM:
19
ANALYSIS OF BULK KK SPECTRUM
◮U (1) GAUGE FIELD MASS SPECTRUM:
•SA =
− 14
R
−
→
p
MN
d x −g(5) FM N (x, y)F
(x, y), FM N := ∂ [M AN ] (x, y)
5
(n)
χ±;A (y)
(n)
√
n=0 Aµ (x)
rc
P∞
•KK reduction ansatz:Aµ (x, y) =
where

(mA
n )± k rc |y|

√
±

k
r
|y|
)
J
(
c
±

e
k± rc 1 k ± e


k ± rc π
mA
(n)
(
e
n)
χ±;A (y) =
J 1 ( k ± ± ek ± rc π )



1

 √
2π
•4D effective action: h
SA = −
R
4
d x
P∞
n=0
η
νλ
(n)
1 µκ (n)
(x)
η
F
(x)F
µν
κλ
4
+
for n > 0
for n = 0.
(n)
(n)
A 2
1
(m
(x)A
)
A
ν
n ±
λ (x)
2
i
19- A
ANALYSIS OF BULK KK SPECTRUM
◮U (1) GAUGE FIELD MASS SPECTRUM:
•SA =
− 14
R
−
→
p
MN
d x −g(5) FM N (x, y)F
(x, y), FM N := ∂ [M AN ] (x, y)
5
(n)
χ±;A (y)
(n)
√
n=0 Aµ (x)
rc
P∞
•KK reduction ansatz:Aµ (x, y) =
where

(mA
n )± k rc |y|

√
±

k
r
|y|
)
J
(
c
±

e
k± rc 1 k ± e


k ± rc π
mA
(n)
(
e
n)
χ±;A (y) =
J 1 ( k ± ± ek ± rc π )



1

 √
2π
•4D effective action: h
SA = −
R
4
d x
P∞
n=0
η
νλ
•KK mass spectra:
(n)
1 µκ (n)
(x)
η
F
(x)F
µν
κλ
4
mA
n ±
≈ n∓
1
4
+
for n > 0
for n = 0.
(n)
(n)
A 2
1
(m
(x)A
)
A
ν
n ±
λ (x)
2
πk± e−k± rc π
i
19- B
ANALYSIS OF BULK KK SPECTRUM
◮MASSIVE FERMION MASS SPECTRUM:
20
ANALYSIS OF BULK KK SPECTRUM
◮MASSIVE FERMION
MASS SPECTRUM:
n
R 5
←
→
µ
•Sf = d x [Det(V)] iΨ̄L,R (x, y)γ α VαM Dµ ΨL,R (x, y)δM
where ←
→
←
→
•Dµ := ∂µ +
Â
µ
− sgn(y)mf Ψ̄L,R (x, y)ΨR,L (x, y) + h.c.
gN P
16
+
[Â
B̂]
VN ∂[µ VP ]
Ĉ D̂
1 T S [Â B̂]
∂
V
V
g
V
[S
P ] Vµ ηĈ D̂
N T
2
, γ5 := 4!i ǫµναβ γ µ γ ν γ α γ β
ÂB̂
ÂB̂
[ΓÂ , ΓB̂ ] + igf Aµ
= iγ4
•Γ = γ
•{ΓÂ , ΓB̂ } = 2η , η = diag (−1, +1, +1, +1, +1)
•V44 = 1, VµÂ = eA±(y) δµÂ , Det(V) = e−4A± (y)
Â
B̂
•gM N := VM
⊗ VN
ηÂB̂
•ΨL,R (x, y) ≡ PL,R Ψ(x, y), PL,R = 21 (1 ∓ γ5 )
•PR + PL = 1, PR PL = PL PR = 0
20- A
ANALYSIS OF BULK KK SPECTRUM
◮MASSIVE FERMION MASS SPECTRUM:
2A
P∞
(n)
e ± (y) ˆ(n)
•KK reduction ansatz:ΨL,R (x, y) = n=0 ΨL,R (x) √rc fL,R (y)
where



L,R
mn
(
)

(mL,R
)± k± rc |y| J 1 mf  k± ± ek± rc |y| 

n

+
∓
e

k±
2
k±


q


for n > 0
L,R

k± r c π
L,R
m

(
)
n
e
(m
)
r
c
±
n

± k± r c π 

J e
mf
k±
(n)
1
+
∓
2
k±
f̂L,R (y) =
v
"
#

u (1±2 mf )k± rc π


k±
u2 e

−1
m
u

± k f k± rc |y|

t 
e ±
for n = 0.

mf

k ± rc
1±2
k±
•Sf =
R
4
d x
(n)
Ψ̄
L,R (x)
p=0
P∞ P∞
n=0
+
igf
√
rc
h ←
→
i δ np ∂ α
(nmp) (m)
I
Aα (x)
L,R
m=0
P∞
•KK mass spectra: mL,R
≈ n+
n
±
1
2
h
mf
k±
α
γ −
±
1
2
i
L,R np
mn
δ
−
1
4
i
(p)
ΨL,R (x)
πk± e−k± rc π
21
BULK GRAVITON & BRANE SM FIELD INTERACTION
•
K(5) R 5 p
SSM−G = − 2
d x −g(5) Tαβ
SM (x)hαβ (x, y)δ(y − π)
√
h
P∞
k± rc K(5) R 4
(0)
αβ
k± rc π
=−
(x)
+
e
d
x
T
h
(x)
SM
αβ
n=1
2
(n)
hαβ (x)
i
22
BULK GRAVITON & BRANE SM FIELD INTERACTION
•
K(5) R 5 p
SSM−G = − 2
d x −g(5) Tαβ
SM (x)hαβ (x, y)δ(y − π)
√
h
P∞
k± rc K(5) R 4
(0)
αβ
k± rc π
=−
(x)
+
e
d
x
T
h
(x)
SM
αβ
n=1
2
(n)
hαβ (x)
i
•The graviton KK mode couplings decrease due to GB
interaction leading to the decrease in their detection signature
in collider experiments unless one modifies the value of rc to
resolve the gauge hierarchy problem.
•The graviton KK mode masses decrease from their counter
part in RS model and the GB coupling make the detectability
of the signature of KK mode graviton through H → τ τ̄ more
pronounced.
•The detectability of graviton KK mode can also be achieved
by studying gravidilatonic interaction via the model
parameters A1 , θ1 and θ2 obtained from string loop correction.
22- A
COLLIDER CONSTRAINTS ON THE GB COUPLING
•We consider (A1 , θ1 , θ2 ) = 0 in the proposed model.
•Modified Warp factor:
s
1
3M52
4α5 Λ5 2
A(y) = kα rc |y| = 16α5 1 − 1 + 9M 5
rc |y|.
5
•SSB is occuring in the bulk and after KK reduction the zeroth
mode scalar is compared with SM Higgs in the brane.
Z
p
1 AB
5
g ∂A H(x, y)∂B H(x, y)
SSSB ⊃ d x −g(5)
2
2 2
+
−
+mh H (x, y) + mf ΨΨ̄ + yη HΨΨ̄ + hw HW W
•Zeroth mode scalar mass:
ms ≈ MH ≈
q
1
2
4+
m2h
2
kα
−
3
4
πkα e−kα rc π
23
COLLIDER CONSTRAINTS ON THE GB COUPLING
◮HIGGS MASS CONSTRAINT:
+0.5
CM S
AT LAS
MH
= (125.7 ± 0.3)+0.3
GeV,
M
=
(125.5
±
0.2)
−0.3
−0.6 GeV
H
130
LHC ATLAS ALLOWED REGION
LHC CMS ALLOWED REGION
128
MH
126
124
122
120
2. ´ 10-7
4. ´ 10-7
6. ´ 10-7
8. ´ 10-7
1. ´ 10-6
Α5 HGauss-Bonnet CouplingL
24
COLLIDER CONSTRAINTS ON THE GB COUPLING
◮µ PARAMETER CONSTRAINT FROM H → (γγ, τ τ̄ ) CHANNELS:
with
Σ(pp → H0 )
•µ =
Σ(pp → HSM )

1.6 ± 0.3




 0.77 ± 0.27
=

0.8 ± 0.7




 1.10 ± 0.4
BR(H0 → l1 l2 )
BR(HSM → l1 l2 )
for ATLAS l1 = l2 = γ
×
for CMS l1 = l2 = γ
for ATLAS l1 = τ, l2 = τ̄
for CMS l1 = τ, l2 = τ̄
Γ(X → l1 l2 )
•BR(X → l1 l2 ) =
Γ(X → all decay channels)
25
COLLIDER CONSTRAINTS ON THE GB COUPLING
•Total decay width:
Γtotal ≈ Γ(H0 → W + W − ) + Γfer
where
•Γ(H0 → W + W − ) =
with T
m2W , m2s
=
m4s
48m2W
2
FW
FG2 T
2
R
s
P4
¯
i=1 Γ(H0 → fi fi ) =
(k2 −m2W )2 +Γ2W m2W
2
4m2W k2
k2
− m4 .
m2s
s
Nc ms
8π
•Diphoton decay width:
Γ(H0 → γγ) =
m3
s
5
3456π m4
W
1
3
[λ 2 (m2W ,k2 ,m2s )+λ 2 (m2W ,k2 ,m2s )
d(k )
m2W
2
2
2
and λ(mW , k , ms ) = 1 − m2 −
•Γfer ≈
m2W , m2s
2 2
FW
FA


2 +
P4
i=1
12m2
W
2
ms
2
FQ
1−
i
+
12m2
W
2
ms
12k2 m2
W
m4
s
4m2f
i
2
ms
]
23



2
4mW
sin−1 s 1 2
2 − m2
s
4m

W

2
ms
26
2  2






COLLIDER CONSTRAINTS ON THE GB COUPLING
•Dilepton decay width:
Γ(H0 → gg) =
m2
f Ng
4
256π 5 m
4
2
F
F
G
Q
g
4
s

m2

f
1 + 1 − 4 m2s4





sin−1
s1
2
•Differential production cross section of the scalar:
m2
f4
m2
s
2  2






π 2 Γ[H0 → gg]
dσ
2
2
(pp̄ → H0 + X) =
g
(x
,
m
)g
(x
,
m
p
p
p̄
p̄
s
s)
dy
8m3s
we define gluon momentum fraction as:
ms e−y
ms ey
xp = √ , xp̄ = √
s
s
•gp (xp , Ms2 ) is the gluon distribution function in proton at the
gluon momentum fraction xp .
√
• s ⇒ BEAM ENERGY
27
COLLIDER CONSTRAINTS ON THE GB COUPLING
•VERTEX FACTORS IN 4D EFFECTIVE THEORY:
FQ
=
FG
=
FW
=
yη Q000
2Nf2 Ns
=
sinh (mf rc π),
mf rc
gf Nf2 sinh (kα − 2mf )πrc
gf G 000
= √
,
√
rc
(k
−
2m
)r
2πrc
α
f c
ge
ge 000
000
= p
hw W
= hw Ns , FA = 3 A
2πrc3
rc2
with the following overlapping integrals:
R +π
4A(y) (0)
ˆ(0)⋆ (y)fˆ(0) (y),
dy
e
χ
(y)
f
H
L
L
−π
R
+π
(0)⋆
(0)
(0)
G 000 := −π dy eA(y) fˆL (y)χA/A (y)fˆL (y),
a
R
+π
(0)
(0)
(0)
W 000 := −π dy χH (y)χAa (y)χAa (y),
R +π
(0)
(0)
(0)
000
A
:= −π dy χA (y)χAa (y)χAa (y).
Q000 :=
28
COLLIDER CONSTRAINTS ON THE GB COUPLING
2.0
LHC ATLAS ALLOWED REGION
LHC CMS ALLOWED REGION
LHC ATLAS ALLOWED REGION
1.5
LHC CMS ALLOWED REGION
ΜΤΤ
ΜΓΓ
1.5
1.0
0.5
0.5
0.0
2. ´10-7
1.0
-7
4. ´10
6. ´10
-7
8. ´10
Α5 HGauss-Bonnet CouplingL
-7
-6
1. ´10
0.0
2.´10-7
4.´10-7
6.´10-7
8.´10-7
1.´10-6
Α5 HGauss-Bonnet CouplingL
•Combined constraint from (MH + µγγ + µτ τ̄ )⇒
4.8 × 10−7 < α(5) < 5.1 × 10−7
29
BOTTOM LINES
•Need to address all phenomenological aspects which can
explore various hidden features of string phenomenology in
presence of GB coupling.
•Detection of graviton KK mode in future collider experiments
are more pronounced in presence of GB coupling via stringy
gravidilatonic interaction.
30
BOTTOM LINES
•Need to address all phenomenological aspects which can
explore various hidden features of string phenomenology in
presence of GB coupling.
•Detection of graviton KK mode in future collider experiments
are more pronounced in presence of GB coupling via stringy
gravidilatonic interaction.
•Stringent bound on the GB coupling obtained from the
collider constraints in pure EHGB warped geometry model are
in good agreement with solar system constraint.
•Need to address various cosmological aspects of inflation
and other alternative proposals, CMB physics including
different types and features of primordial non-Gaussianity.
30- A
BOTTOM LINES
•For all the cases the estimated GB coupling is below the
upper bound of GB coupling (α5 < 1/4) as obtained from the
viscosity entropy ratio in presence of GB coupling.
•By studying the phenomenological/cosmological features in
presence of GB coupling various features of string theory can
be explored from its low energy effective counterpart.
31
BOTTOM LINES
•For all the cases the estimated GB coupling is below the
upper bound of GB coupling (α5 < 1/4) as obtained from the
viscosity entropy ratio in presence of GB coupling.
•By studying the phenomenological/cosmological features in
presence of GB coupling various features of string theory can
be explored from its low energy effective counterpart.
•Supergravity, as the low energy limit of heterotic string
theory also yields the GB term as the leading order correction
and therefore became an active area of interest as a modified
theory of gravity from which we can also study the various
hidden aspects of beyond SM physics.
•By studying the phenomenological consequences of radion
with vev in the range 1-1.5 TeV we can also explain the
consistency with the first graviton excitation mass > 3 TeV.
31- A
REFERENCES
•L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370
[hep-ph/9905221].
•L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690
[hep-th/9906064].
•P. Dey, B. Mukhopadhyaya and S. SenGupta, Phys. Rev. D 81
(2010) 036011 [arXiv:0911.3761].
•B. Mukhopadhyaya, S. Sen, S. Sen and S. SenGupta, Phys.
Rev. D 70 (2004) 066009 [hep-th/0403098].
•B. Mukhopadhyaya, S. Sen and S. SenGupta, Phys. Rev. Lett.
89 (2002) 121101 [hep-th/0204242].
•U. Maitra, B. Mukhopadhyaya and S. SenGupta,
arXiv:1307.3018 [hep-ph].
32
THANKS FOR YOUR TIME.........
33