RS model as a Framework for Flavor Physics
John N Ng
Triumf
NTHU Taiwan , 2008
JNN Nov 08 – p.1/34
Outline
Brief introduction of Extra Dimensional (XD) Models and the RS1 model
(A) XD with flat internal space
(a) Scalar fields in higher dimenionaltheories
(b) Kaluza-Klein (KK) decomposition
(c) Boundary conditions
(B) Warped internal space
(a) Scalar fields
(b) Fermions in RS location ,location, location
(c) Gauge Bosons
. Bulk symmetry is
(d) The need for custodial
(e) Quark Mass Matrices : symmetrical or asymmetrical
(f) KK Fermions mixing
Some Phenomenology
Conclusions
JNN Nov 08 – p.2/34
Intro to XD
Compactification of internal dimensions always involve a scale R or ’volume’,V
Two equivalent descriptions
(A) At distances large compare to R
KK modes
4D langage is more more appropriate
(B) At distances smaller than R
Takes into account of all KK modes
Higher dim language is better
Use the KK langauage since it is more relevant to phenomenology.
JNN Nov 08 – p.3/34
The KK decomposition
"
!
via
KK mode
We allows us to think of
$)
$ 0
/
.
. #
%
$ #
-
to be orthonormal (basis)functions
,
Choose
$ #
)$
*
+
$
('
)$
$ #
&
We can always expand any function in any complete set of functions
$ #
%
parametrize the compact space
dimenions
Quantum fields in
as independent d.o.f
JNN Nov 08 – p.4/34
KK decomposition II
How to choose the basis is model dependent.
In general assume perturbation philosophy
Understand the ’free’ part of the XD Lagrangian
Add interactions later
JNN Nov 08 – p.5/34
Scalar Fields
4
6
5
7
1
2
1$
6
6
5
3
3
1
2
1$
4
The action of a free scalar field is
6
3
5
3
3
3
5
7
6
and
$ #
$ 8 6
$ #
6
5
3
3
Use the eigenfunctions of the XD part
This is linear partial differentail eqn we can solve
Choose appropriate boundary conditions which is part of the definition of the
theory
JNN Nov 08 – p.6/34
:9. 0
.
;
.=#
;
. #
@
1$
;
. 8 6
;
).
3
3
).
1
2
.=#
;
.%#
1$
;
?
>
.
:9.
; ).
6
.=#
;
5
3
3
).
).
.=#
;
3
3
.%#
1
2
1$
;
;
<
Use the KK expansion of
(
.
9.
( ( Scalar fields II
insert into S
We get
Implies
JNN Nov 08 – p.7/34
Physics of KK decomposition
$ 8 6
)$
)$
3
3
we have a massless 4D field
For 1 XD the eigenfunctions
$ #
A8
If
A free XD scalar is equivalent to an infinite tower of 4D scalars with masses
$8
$
1
2
The theory can be written as
are circular functions
Need to impose boundary conditions to specify these functions.
JNN Nov 08 – p.8/34
Boundary Conditions
D
C
the KK masses are
$
. They are called fixed
$ #
Compactify on an interval i.e extends from
to
points where branes are situated. At the fixed points
(A) Dirichlet b.c
D
$8
(B) If
5
6
Compactification on a circle
(A) Impose periodic b.c
B
Illustrate by considering 5D or 1XD examples.
#
3
(B) Neumann b.c.
(C) or a mixture
JNN Nov 08 – p.9/34
Bulk Gauge Fields
Use 5D QED compactify on a circle as an example to bring out the physics
H N
4
3
4
H
N
3
4
H G
J
!#
!
JM
JKL
4
4
IHG
$ #
term
$
O
N
N$
KK expand the gauge field as in the scalar case. Look at the
N
A
1
2
GH
FE6
The action is
The zero mode has a constant profile
A #
P 3
Q
Becuase of bulk gauge invariance the zero mode is by
4D gauge invaraince.
Other KK modes similar to scalars
JNN Nov 08 – p.10/34
Bulk Fermions
R
. The action is
The bulk field
R
X
R
8 U
R
R
H
Notice a bulk mass term is included and sign of
8
H
RW
3
H
R
RW
3
U
H
VUT
1
S
!
Again consider a minimal 5D Lagrangian for a fermion
is not determine.
is a 4-component Dirac spinor.
Z ^ [ZY
8 \U
8
Y
`\U
Y
P 3
c
\bU
3
Y
`_U !
_ !
3
Pa
gives
3
Bulk equation of motion from
0
]\U
R
Decompose under 4D Lorents subgroup into a pair of Weyl spinors
JNN Nov 08 – p.11/34
Bulk Fermions II
[dY
$J
$
b\U
$
$ #
b\U
$
Y
KK expand the 5D wavefunctions
$
$Y
$8
$8
$
b\U
_ !
3
$Y
_ U!
3
`\U
These fermions obey the 4D Dirac equations
$ #
$J
$ #
$J
8
8
$8
$8
$J
$ #
P 3
c
P 3
c
Substituting back into 5D EOM we get a set of coupled eigenvalue equations
JNN Nov 08 – p.12/34
A -
and the field takes the b.c. Dirichlet
Y
Consider the fix point
Bulk Fermions Boundary Conitions
Y
8
Pa3 The other component must satify
i.e. the Neumann condition
A #
8
AJ
A #
8
AJ
A #
then
D
e
C M
8
e
8
f
JNN Nov 08 – p.13/34
AJ
.P
M
.6
\
If we choose
A -
P 3
a
P 3
Q
For the zero mode the eigen equations decouple:
Introduction to the Randall-Sundrum Model
There are more than 4 dim. Indeed RS assumes
conformal metric. AdS
dim with a warp or
6
t
1
1
u
vw6
1
)
qp
rsp
M
1
1
)
(UV) and
C
Two branes are located at
)
t
hg1
Iji
6
6
nomlk
j
5D interval is given by
(IR).
v )-
z
_
u
y
Cx
t
r
ij
€
|
nl ‚ B
u
v~6
€
|
Iji
t
r
M
M
F}6

F}6
{

{

C
x
)
Metric is
JNN Nov 08 – p.14/34
RS model as 5D field theory
6
8
3
6
34
iH
)
1
i
1
2
(1)
‡
6
M 6
}2
8
†
3
M
}2
3
3
r
M
Integrate over
)
E
u
v„6
t
v
q
q
u
)
1
3
t
F}6
1
2
E
ƒ
E
H
S '
E
The action is generalized to 5D e.g. the bulk scalar field we have
to give a 4D effective theory
)
‰
and the KK eigen-mode
. 0
$
)
.
)
E
$
)
1
E
$
$
‰
ˆ
One recovers the canonical 4D scalar field
.
is normalized by
$ ˆ
$
v
u
'
$
$ )
M
}
Do KK decomposition.
JNN Nov 08 – p.15/34
satisfies the eigenvalue eqn
$ 8 6
$ 6
‹
$ 8 6
$
$
M 6
}6
8
†
$
q
M
}
M
}2
3
v 6 3
q
u
M
}
$
The
)
RS:Scalars
$8
$ 3
t
$
$ r Š
t
3
1
2
The 4D effective action becomes
are the zero modes. Identify them as SM fields.
Ž
6
6

q
onmlk
AŒ
8
y
z
t
A 
M
A
M
M <
t
onmlk
q
q
onmlk
The solutions are exponentials
JNN Nov 08 – p.16/34
More RS
$ 8 6
$ 6
‹
$ k
‚

the solutions are given by Bessel functions of order
‰
$
t ”
“
$
t
C
$ 
$8
and
are determined by boundary conditions at
are continuous.
)
$ Œ
$ 
“
’ˆ
)
$
M
}
‘
For
$8
.
'
‚
$
t
$ 3
r Š
t
3
1
2
After integrating out the extra the dimension the 4D effective action is
. The derivatives
JNN Nov 08 – p.17/34
Gauge Fields in RS
4
GH
4
•HG
i
)
1
and KK decompose the gauge field
)
$Y
$
v
u
'
N$
)
N
2
Choose the unitarity gauge
N
E
S 1
2
'
E
Take QED as the toy model. The action is
$
.Y
)
E
$Y
1
. 0
E
with a normalization
?
3
$Y
q
q
u
v„6
$
9.
E
.Y
)
1
3
t N.
N$
2
r
—
–
t
E
M>
F}6
The 4D Lagrangian is
JNN Nov 08 – p.18/34
RS Gage Fields II
C'
AY
The zero mode has a flat profile
This preserves charge universality
‰
$
$
C
are determined by b.c.c at the fixed points
)
and
$8
$
K
$ Œ
z
$
B”
“
K
“
B
$Y
’ˆ
M
$8
}
The solutions for KK excitaions are Bessel functions of order unity
JNN Nov 08 – p.19/34
Fermions in 5D Bulk

€
˜
S
!
™
™
The Dirac matrices in 5D are
™ H
\ |
R
\ {
5D fermions are 4-component spinors i.e. vector-like fermions

\
{

€
\
|
€
\ |
œ
\ {
6š
›
Project out the L,R chiral states by boundary conditions or orbifold parities ,i.e.
how the field transforms under
JNN Nov 08 – p.20/34
Fermions in Warp Space
R
j i
žŸj
U
R
™ž
X
B
}
M
}
M
}
M
M¡T
}
J
Fnl
ž

where
K !1
j
1
2
1
)
'
5D action for fermions is
)
¤
9
)
$ £
(2)
. ) £
)
. 0
$
)
¦
The profile of the wavefunction is controlled by
Bessel fn.
z
E
)
. Enters into the order of
1
8
$ ) ¥ £
E
. ) £
)
)
$ ) `¥ £
)
E
1
E
$
v
u
'
)
9 R
$ 9R M
‚} ¢
Do the usual KK decomposition:
JNN Nov 08 – p.21/34
Bulk Fermions II
$ ) £
}
$8
$8
M
(3)
$ ) £
}
M
.
v
u
q
8
3
$ ) £
z
>
?
v
u
q
8
3
$ ) £
z
?
>
The equations are
A
Œ
A
Œ
6
t
B

6
, only one of the two is allowed by the
)
even at


ª

f


LH zero mode resides close to UV (IR) brane for
ª


f
The RH chiral zero mode lives near the UV (IR) brane if

6š
Since both solutions are
.
6š
M
t
6
t
6
nFmlk
E
©
©
¨
B
q
§
z
u
v §

B
¨
M
(4)
©
¨
A
MŒ
onmlk
A
MŒ
onmlk
§
q
A ) £
A ) £
6
onlmk
z
E vu
t §  
B
¨

6©
k

¦
y
The zero modes which we identify as SM fermions
.
JNN Nov 08 – p.22/34
Profiles of bulk fermions
4
φ
2.9 3.0
3
π
3.1
φ̂L1
4
φ̂R
1
3
2
2
1
1
0
φ̂R,L
n
−1
φ̂L1
φ̂R
1
φ̂R
3
φ̂L3
−1
−2
−2
ν = (1/2 − 0.05)
−3
−4
π
3.1
0
φ̂R
3
φ̂L3
φ
2.9 3.0
0
0.2
0.4
R
φ̂L2 φ̂2
t
0.6
0.8
ν = (1/2 + 0.05)
−3
1
−4
0
0.2
0.4
R
φ̂L2 φ̂2
t
0.6
0.8
1
©
q
E§
My
«
nomlk
The thick red lines are the zero mode wave function of RH chiral fermion.
JNN Nov 08 – p.23/34
Fermion Masses in RS
9
¦¬
The coefficients
control the zero modes i.e peaks at UV or IR
Localize the Higgs at the IR brane
Have the zero modes i.e. the SM chiral fermions localize near UV brane
The overlap after SSB will be very samll
No need to fine tune Yukawa’s.
Quark masses are naturally small.

t-quark or
«
If all the fermions both LH doublet and RH singlets are localized near UV then
t-quark comes out too light
must not be too far from IR brane
JNN Nov 08 – p.24/34
Quark Masses in RS
1
L
#
µh ® ¦ h´® G
G
:¶±
·
v
GeV.
» ºv
M
½©
½©
n`mlk
q
¨
6¹
B
§
9 ¦
¸
6
B¹
¼»º v
M
nomlk
E
§
)
9 ¦
¤
v
Cu
z
denotes up-type or down-type quark species.
9 #A
¦ °
³
S
z 9
Cu
hµ® y
±²
´h® #A
C
¦ ¦ #A
C
°
where the label
#
v
Cu
±²
z 9
®¯­
° 5
S
³
®
®
The quark masses are given by
¾
!
are not necessarily symmetric in
shows that the masses are control by values of
9 ¦
9 #
¤
The Yukawa couplings
° ³
where the upper (lower) sign applies to the LH (RH) zero mode
The task is to configurations that fits the CKM matrix.
Added bonus : both LH and RH quark configurations are given for each solution
Ã
(Á Â
H
À¿
(
In the SM only LH rotations are detectable.
(
Both LH and RH rotations are given for each solution
.
JNN Nov 08 – p.25/34
Æ
where we have used
È
Mz
Fnmlk
E
Æ Ï
˜
Ë
Æ Æ
·
ƘÇ
€ÎÐ È
Ï
Æ Æ ˜
Æ Ï
Æ ˜
·
à 5 - ƘÏ
·
/
Ç
·
{

{

2
Ï
Ë
Æ Ï ,- ˜
Ç
€ÎÐ Ë
Æ
È
È
Æ ˜
È |ÎÍ
Æ ˜
Æ ˜
Ï
Æ Ç Î
|˜
Æ Í
/
Æ ÇÏ
˜
Ç
5
Á -
Æ ˜
,-
Ç
Ç
Ç
Ç
Æ É
Æ ˜
Å
È
É
Ç
Æ Ë
˜
È
·
È
Æ Æ Æ˜
Ç
Å
¦Ê
Ƙ
¦Ì
Ç
Å
Ç
É
È
Ƙ
È
È
Ç
Æ Æ ˜
Ħ
General Configurations
In general quark mass matrices are not symmetrical in RS. Several configurations
found. One example:
(5)
The u and d quark mass matrices (at TeV scale)
TeV.
JNN Nov 08 – p.26/34
RS Quark Masses contd
vÒ Æ ˜
·
Ï
·
Ï (-
vÒ Æ Ë
˜
Á
Ò Æ˜
Á
ÁÑ
Ƙ
-
(-
(-
((-
Ò Æ˜
Ï Ç ÁÑ
Ƙ
-
(-
The CKM matrix elements for the above
(6)
Note the RH rotations are larger than the LH ones.
Appears to be true from the numerical searches we found
How to test it?
JNN Nov 08 – p.27/34
Symmetrical Mass Matrices in RS
Most of the ’constructions’ start from conjecture assuming that they are
symmetrical
Put zeros ( 1 to 3) in appropriate places to fit CKM and the observed mass
heirarchies.
Can RS accomodate these without fine tuning the Yukawa couplings
By construction
Only ONE texture zero structures are allowed.
JNN Nov 08 – p.28/34
All is not well
is protected by a custodial
It is known that
Ô
parameters will receive tree level corrections
The
Ó
The main problem is that the new KK modes will modify EWPT
symmetry
Promote that to a bulk gauge symmetry.
Õ
e
×
Take
Ö
The gauge symmetry is now
Tree level KK gauge effects are suppressed
JNN Nov 08 – p.29/34
D
6
×
and
šß
M !#
ghÝ1
JM
JKL
u
by vev on UV brane. We have a
SJ ß
×Ù
S J ß6
ÚÙ
š
ß
SJ
S J 6
à
Þ
Õ Û«
MÜ
ÚÙ
9
B
Ø
by orbifold b.c.
(
Break
Custodial RS model
×Ù
SJ
×
is the SM hypercharge gauge boson and broken with
Higgs.
S J ß6
S J 6
×
SJ ß
ÚÙ
à
and
on IR brane by
JNN Nov 08 – p.30/34
Zero modes have parity
Quark Representations
is a zero mode and
is broken on UV
Ó
«

×
å
«

â
ãä
must have their own (- +) partners
and
«
1
«
because


«
«
á
Usual assignment
They don’t affect the quark mass matrix.
JNN Nov 08 – p.31/34
ç
À šß
š
and
À
À š
FCNC
š
f¯
È
Æ
Besides the direct production of the KK Z
æ
FCNC in the Minimal Constrained RS Model
TeV is tree level FCNC
mixing
<H>
<H >
XKK
Z
f
KK-fermion mixings
<H >
<H >
Z
f¯
Going to the mass basis the unitarity is broken
fKK f¯KK
f
FCNC
JNN Nov 08 – p.32/34
Compare the decays in
O
c.f SM
vs single
L¦
«
f
D
×
channels.
by
¦
L
«
õ
Á
à Á
Ò
Äø
É
Å
®
ø f
ø LH and RH decays are different becuase
O
¢
¢
Á
ž Â
à
Äø
ž øh£
Ò Á
ÿ ‚
.
6ô
ö õ
ù
¶
®
ó
õ
#
õ
Ó
#
à
.
‚
.
ý ‚
ÿ ‚ þ
.
and
à
B
S
š
.
Ú
«
L¦
š
and
«
«U
The BR is
ù
where
O
D
«
×
«
ûöõ -
6
ù øh£ v§
Á©
6
Á©
«
ø£
ùü
v§
·
·
X
ûöõ - T
Ç
Æ Ë
ù
²
ñòó 6ô
š
«
÷öõ -
6
ùø£
v§
Á©
6
Á©
«
hø£ ùú
v§
ù
X
÷öõ - T
L¦
×
u
«
>
?
>
6
The BR is
in the config we found
.
JNN Nov 08 – p.33/34
]ëê
ðïè
îíì
é
è
Conclusions
We have found that the RS model can have good quark mass matrices without fine
tuning Yukawas
BR is
O
M¾
g«
S
«
Tree level FCNC best probe in
š
f
For asymmetrical conf
It can accomodate symmtricall mass matrices if there is only one texture zero and
not more
makes it very exciting at the LHC
Predicts RH decays are dominant.
JNN Nov 08 – p.34/34