ニュートリノのCPの破れからフレーバーの対称性を探る
Morimitsu Tanimoto
Niigata University

November 11, 2022
岡山理科大学1

Outline of my talk1 Overview of Flavors
2 Modular symmetry of flavors
2.1 Modular group
2.2 Finite modular group
2.3 Residual symmetry at fixed points
3 CP violation of leptons
4 Summary2

1Overviews of Flavors
Generation 世代
Family家族Flavor⾹り

3

標準模型でのフレーバーの破れ

クォークのCKM混合

質量固有状態で書き直す• フレーバー固有状態と質量固有状態のずれ ⇒ フレーバーを変化させる• 1つの複素位相 ⇒ CP対称性を破れを引き起こす

標準モデルではニュートリノの質量はゼロ、フレーバー混合はない！！
レプトン

カミオカンデクォーク

スーパーカミオカンデ

カムランド

K2K T2Kクォーク

CKM行列Super-Kamによるニュートリノ質量の発⾒PMNS行列

W+0.15

ニュートリノ振動実験

e-νe, νμ, ντUei

Tribi-maximal 混合

Harrison, Perkins, Scott (2002)

Tri-bimaximal Mixing (TBM) は簡単な質量行列から導かれる

A4 symmetric7

Integer (inter-family related) matrix elementssuggest Non-Abelian Discrete Flavor Symmetry.

E. Ma, G. Rajasekaran 2001

フレーバーを対称性で理解する試み
DynamicsがわからないときにはSymmetryが有効なアプローチ（例：クォークモデル）
どのような対称性？

S3

連続群

U(1), U(2), SU(3)

離散群

A4

フレーバーの対称性は破れる ⾃発的破れ!!

Subgroups of SU(3)by S.F King

9C.Luhn, JHEP 1103, 108(2011) arXiv:1101.2417
A.Merle, R. Zwicky, JHEP, 1202 128(2012) 1110.4891

Another origin of finite discrete groups
Modular Group
It is well known thatthe superstring theory on certain compactifications
lead to non-Abelian finite groups.

Indeed, torus compactification leads toModular symmetery,which derives S3, A4, S4, A5 …… .
10

2 Modular Symmetry of Flavors
2.1 Modular group
Superstring theory 10D
Our universe is4D

Torus compactification

11The extra 6D
should be compactified.

by Feruglio

Two-dimensional torus T2 is obtained asα2α1(x,y)～(x,y)+n1α1+n2α2T2 = ℝ2 / ΛΛ is two-dimensional lattice,which is spanned by two lattice vectors

α1=2πRand α2=2πRττ =α2 /α1 is a modulus parameter (complex).
The same lattice is spanned by other bases under the transformation12

ad-bc=1
a,b,c,d are integer SL(2,Z)

ad-bc=1
a,b,c,d are integerτ =α2 /α113ɤModular transformation

The modular transformation is generated by S and T .

ɤduality

Dicrete shift symmetry

==α2

S
α1

α’1α’2τ =α2/α114T
α2α’2α1Duality
Dicrete shift symmetry±1 is identified

generate infinite discrete group

Modular group15

2.2Finite modular group

Modular group has subgroups
Imposecongruence condition

called principal congruence subgroups (normal subgroup)ΓN ≡ Γ / Γ(N) quotient group finite group of level N

ΓN
Γ216S3

Γ3A4

Γ4

isomorphic

S4

Γ5A5

We can consider effective theories with ΓN symmetry.
ΓN

modular form
In cases of ΓN ( N=2,3,4,5 ) (S3, A4, S4, A5)
explicit forms of f(τ) have been obtained.

ɤ= S, T

Automorphy factor

Modular transformation

modular forms of weight krepresentation matrix

K is modular weight

Chiral superfields

17

Modular forms are explicitly given if weight k is fixed.
On the other hand, chiral superfields are not modular forms andwe have no restriction on the possible value of weight kI, a priori.

Consider
Automorphy factor
vanishes if k = kI + kJ

Leff is modular invariant if sum of weights satisfy ∑kI=k .
Simplest Modular invariant kinetic terms of matters

18

ΓN
Taking T3=1 (N=3), we have

Γ3A4 group.

Number of modular forms depend on weight kk+1 for A4(2k+1 for S4)

There are 3 linealy independent
modular forms for k=2 (weight 2)
A4 Triplet を構成する

19

Modular transformation is the transformation of modulus τweight 2; k=23 modular forms==S transformation

T transformationcτ+d = -τ20

Flavor symmetry acts non-linealy (Modular forms).

cτ+d = 1

F. Feruglio, arXiv:1706.08749

A4 triplet of modular forms with weight 2

Dedekind eta-function

|q|≪1
21

2.3 Residual symmetry at fixed points● When, τ is fixed, the modular symmetry is broken.

What is a principle of fixing modulus τ ?

☆ Modulus stabilization (non-perturbative effect, model dependent)
Some models indicate the potential minimum
at or near the boundary of fundamental domain.

☆ Fixed point of τ

Residual symmetryZN↑● Z2 : τ = i

S symmetry ST symmeteryT : τ → τ+1

S : τ → -1/τ●●●Fundamental domain ofτ
on SL(2,Z)
22● Z3: τ = -1/2+√3/2i

ST : τ → -1/(τ+1)

P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov,
JHEP04(2019)005,arXiv:1811.04933
P.P.Novichkov, S.T.Petcov and M.T,
PLB 793 (2019) 247, arXiv:1812.11289
P.P.Novichkov, J.T.Penedo, S.T.Petcov,JHEP03(2022)145,arXiv:2201.02020

3 fixed points in A4
N=3. Z3: {1,T,T2}

T (τ → τ+1) preserved : <τ>= ∞ i (q=0) (Y1 , Y2 , Y3)=(1, 0, 0 )● S (τ → -1/τ) preserved : <τ>=i (q=e-2π )● ST preserved : <τ>= ω(Y1 , Y2 , Y3)=Y1(i) (1, 1-√3, -2+√3 ) Z2: {1,S}Z3: {1,ST,(ST)2}(Y1 , Y2 , Y3)=Y1(i) ( 1, ω, -1/2ω2 )ω3=1Consider S and T transformations ofmass matrix (M )RL , where L is 3.

●●Z3 {1,T,T2}:●

Z2 {1,S}:●Z3 {1,ST,(ST)2}：23

Mixing matrices which diagonalise (M†M) also diagonalize
T, S and ST, respectively !

Construct Modular invariant mass matrix of quark/lepton
Example of A4 symmetry of quarks
Non-Abelian Discrete Symmetry

A4 symmetry

Irreducible representations: 1, 1’, 1”, 3The minimum group containing triplet
It could be adjusted to Family Symmetry.

3: ( uL, cL, tL) , 1: uR , 1”: cR , 1’: tR

Symmetry of tetrahedron

3: ( dL, sL, bL) , 1: dc , 1”: sc , 1’: bc
Flavor symmetry should be broken !
We should know how to break the flavor symmetry.

24

Key : Modulus τ and Modular forms

Quarks

Mass matrix in SM :Let us introduce Modulus (in the case of A4).
The modulus τ plays the role of a spurion
Up-type quark sector

1 x1 x 3x 3 ⇒1A4 representation

VEV of H and τ give quark mass terms

Then, up-type quark masses are given as25

We can construct a simple mass matrix by using weight 2 modular forms
A4 assignments: left-handed doublet 3, right-handed singlets 1, 1”, 1’f

Typical mass matrix of fermions by using weight 2 modular forms
for both up- and down-quarks and charged leptons
6 real prarameters αq , βq , γq and τ are responsible for quark mass matrices.
However,the simplest matrices for up and down quarks is inconsistent with Vub .
26

One can construct Modular forms with higher weights k=4, 6 …
# of modular forms is k+1
Weight 2
3 Modular forms

Modular forms with higher weights areconstructed by the tensor product ofmodular forms of weight 2J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292

Weight 4
5 Modular forms

Weight 6
7 Modular forms
27Modulus τ as the source of CPV ?
At present,observed CP violation is in CKM matrix.
In the near future, CP violation will be observedby the neutrino oscillation experiments. Hyper-K
In addition,
T violation may be observed in EDM of quarks/leptons.

High energy theory is often CP invariantWhat is the source of CP (T) violation ?
VEV of modulus τ is possibly the source of CP violation !
28

3 CP violation of leptons
CP transformation in modular invariant theory

We can construct CP invariant mass matrices
in modular invariant flavor theory.
CP violation realized by fixing τ, which is consistent withthe observed lepton mixing angles and neutrino masses.
29

P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov,
JHEP 07(2019)165 [arXiv:1905.11970 [hep-ph]].

Model of CP violation in Lepton Sector
H.Okada, M.Tanimoto, JHEP 03(2021),010 [arXiv:2012.01688 [hep-ph]]Input of 3 masses
Weinberg operator by using weight 4 modular forms

5 parameters

3x3x3L30

LYrealImpose CP invariance
6 parameters and τ
input data are 831

Input of 8 observed values: me , mμ , mτ , θ12 , θ23 , θ13 , Δm2sol , Δm2atm
output : δCP , <mee>, ∑miAt fixed point of τ=i with weight 2 (k=2),
the charged lepton mass matrix is:

There remains a residual symmetry {I, S},
which is a subgroup of A4 .

32

Modular forms are given in the base ofMove to the diagonal base of generator SThen, the mass matrix is transformed as:33

is not diagonal because eigenvalues of S is (1, -1, -1).
A massless charged lepton appears !
2-3 flavor mixing angle is realized !

A large mixing angle is reproduced.
34

Neutrino mass terms
Weight (L3)= -2

The simplest case is k=4

At fixed point τ=i,35

Tribi

Diagonal base of S

2-3 mixing angle
If Re[g2]=2Re[g1], we obtain θ23ν＝４５°.
36Modular forms at nearby fixed point τ = i37
X=↑•

38

Fundamental domain ofτ
on SL(2,Z)

NuFIT5.139

We can construct Seesaw Model of Neutrinos
H.Okada, Y.Shimizu, M.Tanimoto,T.YoshidaJHEP 07(2021) 184 [arXiv:2105.14292 [hep-ph]].

k=1+ke,μ,τ

ge and gD are real.

40

●prediction by Weinberg operator

41

4 Summary● CP violation is realized by VEV of modulus τ,

which causes the breking of modular symmetry.
Predictions are clearly tested in the future.

● Quark/lepton mass hierarchy is realized at nearby

fixed point of τ.

● Is Modulus τ common in both quarks and leptons ?
One modulus or multi-modulei (Simplectic modular invariance) ?
How is GUT approach of modular symmetry ?
Modular symmetry of flavors could be applied to other flavor
phenomena, SMEFT, EDM …..
42Flavor theory of modular symmetry is developing !

Multiplication rule of A4 group
Irreducible representations: 1, 1’, 1”, 3for triplet

A4 invariant Majorana neutrino mass term(LL)1 = L1L1 + L2L3 + L3L2
3x3

43

A4 invariant

Generalized CP Symmetry

Neufeld, Grimus, Ecker,1987

CP symmetry is non-trivial when flavor symmetry is set in the Yukawa sector.
Transformations of Irreducible representation in flavor group G
under a Flavor Symmetry

If, we have canonical CP transformation.
However, under the CP invariance of Lagrangian,
Xr is not necessary element of flavor group G
Xr should satisfy a condition

自己同型 automorphism of Gf
for S3, S4, A4, A5
44

Mu-Chun Chen, Fallbacher, Mahanthappa, Ratz, Trautner, Nucl.Phys. B883 (2014) 267-305

In CP conserved modular invariant theory

The four-dimensional CP can be embedded to higher dimensional symmetry.
A six-dimensional proper Lorentz symmetry is the combination of
the four-dimensional CP and a transformation with negative determinant in theextra two dimensions.
Such extra dimensional transformations correspond to z →z* and z →-z*z = x + τy, (x and y are real coordinates)
The transformation z →-z* maps the upper half plane Im[τ ] > 0
to the same half plane.
Hence, we consider the transformation z →-z* as the CP symmetry.
That means

See also
P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov,
JHEP 07(2019)165 [arXiv:1905.11970 [hep-ph]].
45

Fundamental domain of τ on SL(2,Z)

CP transformation of modular multiplets

By using consistency condition
we obtain
So we haveSince
Matrix, we getis symmetric.
satisfies consistency conditionin the symmetric base46

T2Modular S3, S4 and A5 symmetries
Γ2S3 group Irreducible representations: 1, 1’, 2

T.Kobayashi, K.Tanaka, T.H.Tatsuishi,
Phys.Rev.D98(2018)016004, arXiv:1803.10391

wight k=2

k/2+1 modular forms

Γ4S4 group Irreducible representations: 1, 1’, 2, 3, 3’

J.T.Penedo, S.T.Petcov,Nucl.Phys.B939(2019)292, arXiv:1806.11040

2k+1 modular forms

Γ5wight k=2

A5 group

Irreducible representations: 1, 3, 3’, 4, 5
P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov,
JHEP 1904(2019) 174, arXiv:1812.02158

5k+1 modular forms
47

wight k=2