Neutrino Mass via the Zee Mechanism
in the 5D Split Fermion Model
Phys. Rev. D 83, 025017 (2011)
CS2011, Apr. 4th, 2011
We-Fu Chang,Yi-Ting Chen, and Siao-Cing Liou
Department of Physics, National Tsing Hua University, HsinChu 300, Taiwan
2011年4月4日星期一
Outline
Introduction
Goal
Zee Model (FCNC, Parameters)
Extra dimensions & split fermions
Zee Model + 5-dim Split Fermion Model
—Reduce Free Parameters
—FCNC Suppressed
Model Setup
Neutrinos get Majorana masses from Zee model in 5D
Numerical Result
Phenomenology
Conclusions
2011年4月4日星期一
Introduction
Goal
Neutrino oscillation
neutrino masses
Zee model: a 2-higgs doublets model
A. Zee. Physics Letters B, 93(4):389 – 393, 1980.
FCNC problem
Extra-dimensions
mass hierarchy: wavefunction overlapping
FCNC problem: BC for higgs in extra Dim
2011年4月4日星期一
Introduction
Zee model A. Zee. Physics Letters B, 93(4):389 – 393, 1980.
neutrino masses through radiative correction.
Give the neutrino Majorana mass by introducing a new higgs
singlet h+ and a new higgs doublet Φ2 in SM.
QEDcharge
new singlet
�h
�
φ+
1
higgs doublet Φ1 =
0
φ
1 �
� +
φ2
higgs doublet Φ =
2
0
φ
� 2 �
νL
lepton doublet-L
ΨL =
−
lL
−
l
charge lepton-R
R
2011年4月4日星期一
Q
� +1 �
+1
� 0 �
+1
� 0 �
0
−1
-1
SU(2)L charge
hyper charge
I3
� 01 �
�
�
2
− 12
1
2
− 12
1
2
−1
2
0
�
�
Y
2
+1
1
2
1
2
− 12
-1
Introduction
Zee model
Neutrino mass matrix (Majorana)
9+9+3
too many parameters(21 complex Yukawa couplings...)
FCNC problems(Two higgs doublets)
A. Zee. Physics Letters B, 93(4):389 – 393, 1980.
2011年4月4日星期一
Introduction

The 5th extra dimensions are compactified in a very small
regions, like circles with radii R, so the Lagrangian and fields are
modified.
0
Lagrangian in (4+1) �dimensions:
thus

Extra dimensions
S=
L4D =
4
�
d x dy L(4+1)D
dy L(4+1)D
πR(−πR)
Kaluza-Klein(KK) decomposition :
the extra dimensional part of a field can be expanded in a
complete set fn (y)
(5)
Φ(x , y) =
µ
�
φ (x )fn (y)
n
µ
n
We can choose
� a orthonormal basis:
∗
fn (y)fm (y)dy
2011年4月4日星期一
= δnm
Introduction
Extra dimensions
The 5th extra dimensions are compactified in a very small
regions, like circles with radii R, so the Lagrangian and fields are
modified.

Kaluza-Klein(KK) Particles:
example:
4D
L
=
�
1
1
1 2 2
µ
y
dy ∂µ Φ∂ Φ + ∂y Φ∂ Φ − m Φ
2
2
2
Φ(x , y) =
µ
�
n
4D
L
2011年4月4日星期一
=
�1
n
n
φ (x )fn (y) ∼ cos( y)
R
n
µ
1 n 2 n 2 1 2 n 2
∂µ φ ∂ φ − ( ) (φ ) − m (φ )
2
2 R
2
n µ n
kk mass
Introduction
Extra dimensions
The 5th extra dimensions are compactified in a very small
regions, like circles with radii R, so the Lagrangian and fields are
modified.
Kaluza-Klein(KK) Particles:
L
=
n
1 n 2 n 2 1 2 n 2
∂µ φ ∂ φ − ( ) (φ ) − m (φ )
2
2 R
2
n µ n
n
4
3
2
1
0
2011年4月4日星期一
2
mn
...
4D
...
ex
�1
...

16
2
+
m
R2
9
2
+
m
R2
4
2
+
m
R2
1
2
+
m
R2
m2
Introduction
Split Fermions
Nima Arkani-Hamed* and Martin Schmaltz,
PHYSICAL REVIEW D, VOLUME 61,
033005 (2000)
•Background field:
•KK zero mode, localized in 5th D
fermions, massless! in 4D
•zero mode chiral
(separate L & R)
Φ(y) = ±2µ2 y
background field
(5)
L5D = Ψ
2011年4月4日星期一
(xµ , y)(iΓµ ∂µ + iΓ5 ∂y − Φ(y))Ψ(5) (xµ , y)
Introduction
Split Fermions
Nima Arkani-Hamed* and Martin Schmaltz,
•Background field: Φ(y) = ±2µ y
•KK zero mode, localized in 5th D
fermions, massless! in 4D
•zero mode chiral
(separate L & R)
PHYSICAL REVIEW D, VOLUME 61, 033005 (2000)
2
identify as SM fermions in 4D
background field
(5)
L5D = Ψ
2011年4月4日星期一
(xµ , y)(iΓµ ∂µ + iΓ5 ∂y − Φ(y))Ψ(5) (xµ , y)
Introduction
Split Fermions
Nima Arkani-Hamed* and Martin Schmaltz,
•Background field: Φ(y) = ±2µ y
•KK zero mode, localized in 5th D
•zero mode chiral fermions, massless! in 4D
PHYSICAL REVIEW D, VOLUME 61, 033005 (2000)
2
identify as SM fermions in 4D
background field
(5)
L5D = Ψ
(xµ , y)(iΓµ ∂µ + iΓ5 ∂y − Φ(y))Ψ(5) (xµ , y)
Γµ =
5D Dirac equation
KK expansion
2011年4月4日星期一
�
0
σ̄ µ
µ
σ
0
�
,
Γ5 = i
�
1 0
0 −1
�
(iΓµ ∂µ + iΓ5 ∂y − Φ(y))Ψ(5) (xµ , y) = 0
� �∞
�
µ
χn (x )gn (y)
(5) µ
n=0
�
Ψ (x , y) =
∞
µ
η
(x
)fn (y)
n
n=0
Gamma matrices in 5D
Introduction
Solutions
Split Fermions
back ground field Φ(y) = ±2µ2 y
Ψ0 (x ) =
µ
�
χ0 (x )
η0 (xµ )
µ
�
µ1/2
−µ2 y 2
g0 (y) = f0 (y) =
e
(π/2)1/4
Gaussian packet in 5th D
Φ(y) = 2µ2 y
L4D =
√
mn = 2µ n
�
Φ(y) = −2µ2 y
L4D =
dyL5D ⊃ iψ0L γ µ ∂µ ψ0L
Left zero mode, massless in 4D!
Identify as SM Left-handed leptons
�
dyL5D ⊃ iψ0R γ µ ∂µ ψ0R
Right zero mode, massless in 4D!
Identify as SM Right-handed leptons
2011年4月4日星期一
Introduction
Solutions
Split Fermions
back ground field Φ(y) = ±2µ2 y
Ψ0 (x ) =
µ
�
χ0 (x )
η0 (xµ )
µ
�
µ1/2
−µ2 y 2
g0 (y) = f0 (y) =
e
(π/2)1/4
√
mn = 2µ n
Gaussian packet in 5th D
1.0
0.8
�0.10
�0.05
different positions in y
1.0
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.05
0.10
y → (y − aL )
�0.10
�0.05
0.05
0.10
y → (y − bR )
1.0
0.8
0.6
Yukawa: mass hierarchy in 4D
higgs zero mode
0.4
0.2
�0.10
2011年4月4日星期一
�0.05
aL bR
0.05
0.10
Introduction
Zee model + Split Fermion in 5D
0
our model
πR(−πR)
mass hierarchy: wavefunction overlapping
FCNC problem: assign BC for Higgs in extra Dim.
2011年4月4日星期一
S 1 /Z2
Introduction
Zee model + Split Fermion in 5D
•Zee Model + Compactified-5D Split-fermion:
11 parameters and order 1 Yukawa couplings in 5D
VEV
�
kk contributions
πR
dy
−πR
effective 4-D
2011年4月4日星期一
�
n
0
πR(−πR)
0
Introduction
Zee model + Split Fermion in 5D
•Zee Model + Compactified-5D Split-fermion:
πR(−πR)
11 parameters and order 1 Yukawa couplings in 5D
3+3 Leptons in the 5th-dim
near the origin
2+1 Higgs in the 5th-dim
1.0
1.0
0.8
0.5
0.6
0.4
�3
0.2
�0.10
�0.05
Gaussian
2011年4月4日星期一
�2
�1
1
�0.5
0.05
0.10
�1.0
Sin or Cos
2
3
Introduction
Zee model + Split Fermion in 5D
Charge lepton masses
Even parity in 5th -> SM Higgs
5th-dim
0
Effective 4D
�
0.6
0.2
�0.10
�0.05
πR
−πR
0.4
0.05
0.10
Split fermion & SM Higgs :
Gaussian distribution, constant & overlapping
2011年4月4日星期一
πR(−πR)
1.0
0.8
πR(−πR)
0
dy
charge lepton
mass hierarchy
Introduction
Zee model + Split Fermion in 5D
Neutrino masses
No VEV
5th-dim
Effective 4D
1.0
�
0.8
0.6
0.4
�0.10
�0.05
0.05
0.10
�0.2
�0.4
Split fermion & 2 Extra higgs :
Gaussian distribution, Sin(y) & overlapping
2011年4月4日星期一
πR
−πR
0.2
πR(−πR)
πR(−πR)
Odd parity in 5th
0
0
dy
small Yukawa coupling
Small neutrino
masses
Introduction
Zee model + Split Fermion in 5D
Neutrino masses
No VEV
5th-dim
Effective 4D
1.0
�
0.8
0.6
0.4
�0.10
�0.05
0.05
0.10
�0.2
�0.4
Split fermion & 2 Extra higgs :
Gaussian distribution, Sin(y) & overlapping
2011年4月4日星期一
πR
−πR
0.2
πR(−πR)
πR(−πR)
Odd parity in 5th
0
0
dy
small Yukawa coupling
Small neutrino
masses
Introduction
Zee model + Split Fermion in 5D
Neutrino masses
No VEV
5th-dim
Effective 4D
1.0
�
0.8
0.6
0.4
�0.10
�0.05
0.05
0.10
�0.2
�0.4
Split fermion & 2 Extra higgs :
Gaussian distribution, Sin(y) & overlapping
2011年4月4日星期一
πR
−πR
0.2
πR(−πR)
πR(−πR)
Odd parity in 5th
0
0
dy
small Yukawa coupling
Small neutrino
masses
Introduction
Zee model + Split Fermion in 5D
Neutrino masses
No VEV
5th-dim
Effective 4D
1.0
�
0.8
0.6
0.4
�0.10
�0.05
0.05
0.10
�0.2
�0.4
Split fermion & 2 Extra higgs :
Gaussian distribution, Sin(y) & overlapping
2011年4月4日星期一
πR
−πR
0.2
πR(−πR)
πR(−πR)
Odd parity in 5th
0
0
dy
small Yukawa coupling
Small neutrino
masses
Introduction
Zee model + Split Fermion in 5D
Neutrino masses
No VEV
5th-dim
Effective 4D
1.0
�
0.8
0.6
0.4
�0.10
�0.05
0.05
0.10
�0.2
�0.4
Split fermion & 2 Extra higgs :
Gaussian distribution, Sin(y) & overlapping
2011年4月4日星期一
πR
−πR
0.2
πR(−πR)
πR(−πR)
Odd parity in 5th
0
0
dy
small Yukawa coupling
Small neutrino
masses
Model Setup
0
Zee model in 5 dimensions
πR(−πR)
2011年4月4日星期一
Model Setup
0
Zee model in 5 dimensions
πR(−πR)

Leptons in 5D:
ψ̂a (x , y) ⊃ ψa0L (x ) ×
µ
µ
SM L-lepton doublets in 4D
ˆla (x , y) ⊃ la0R (x ) ×
µ
µ
SM R-charged-lepton singlets in 4D

kk Higgs:
√
µ
π 14
(2)
√
µ
( π2 )
1
4
2
−µ2 (y−cL
a)
e
2
−µ2 (y−cR
)
a
e
SM higgs in 4D
(even in y)
(odd in y)
(odd in y)
2011年4月4日星期一
Model Setup
4D Effective
Effective 4D zee model with kk particles:
�
πR
dy
−πR



higgs cubic term
charge lepton mass matrix
2011年4月4日星期一
Model Setup
Neutrino mass
kk contributions
Neutrino mass matrix
2011年4月4日星期一
Numerical Result
2011年4月4日星期一
Numerical Result
parameters:
µ, R ,
 Choose
,
the parameters:
µ = 2 × 103 TeV, R−1 = 1TeV


(5D Yukawa couplings: no hierarchy)
ρab
Let their components to have relative random coefficients in [0.5 ∼ 1.5]
and random phases in [0 ∼ 2π] θab
Search
2011年4月4日星期一
, range: [ 0, 20] (1/μ), and
Numerical Result
3-higgs coupling
4 sets are all
Inverted hierarchy.
experiment:
2011年4月4日星期一
0.51
105
1.776
0.87
> 0.92 < 0.19
Particle Data Group
Numerical Result
neutrinoless double beta decay
4 sets are all
Inverted hierarchy.
|mνee | ∼ 0.01eV
experiment:
< 10−12
< 3.6 × 10−8
< 3.7 × 10−8
< 2.7 × 10−8
< 3.2 × 10−8
Particle Data Group
We-Fu Chang and John N. Ng,, Phys. Rev. D 71, 053003 (2005)
2011年4月4日星期一
Phenomenology
FCNC suppressed
2011年4月4日星期一
Phenomenology
FCNC suppressed
ex: µ → 3e
Original Zee Model
Zee Model in 5D with kk modes
1
σ=
µ
width of the split fermion
2011年4月4日星期一
Conclusions
•Instead of 21 complex parameters in Zee model, our model has 11
plus order 1 Yukawa couplings with random phases.
•In this model, the idea of split fermion transform the mass hierarchy
for leptons into the position-differences in the 5th dimension.
•Smallness of neutrino mass & FCNC suppressed: the wavefunction
overlap & BC for Higgs in Extra-Dim.
•Through roughly estimating, the ranges of the parameters cover the
experimental data of charge lepton masses, the three mixing angles in
PMNS matrix, and the differences of neutrino mass squares.
•we found a inverted mass hierarchy solution for neutrinos.
•Predictions of nonzero
2011年4月4日星期一
θ13 , |mνee | ∼ 0.01eV
End
Thank you for listening!
2011年4月4日星期一
2011年4月4日星期一
2011年4月4日星期一
Appendex
Solutions
Split Fermions
back ground field Φ(y) = ±2µ2 y
Ψ0 (x ) =
µ
�
χ0 (x )
η0 (xµ )
µ
�
µ1/2
−µ2 y 2
g0 (y) = f0 (y) =
e
(π/2)1/4
Gaussian packet in 5th D
Φ(y) = 2µ2 y
L4D =
Φ(y) = −2µ2 y
L4D =
√
mn = 2µ n
�
dyL5D = iψ0L γ µ ∂µ ψ0L +
∞
�
n=1
iψ˜n γ µ ∂µ ψ˜n − mn ψ˜n ψ˜n
Left zero mode, massless in 4D!
Identify as SM Left-handed leptons
�
dyL5D = iψ0R γ µ ∂µ ψ0R +
∞
�
iψ˜n� γ µ ∂µ ψ˜n� + mn ψ˜n� ψ˜n�
n=1
Right zero mode, massless in 4D!
Identify as SM Right-handed leptons
2011年4月4日星期一