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Modular invariance approach to masses and mixing of neutrino flavors - - PowerPoint PPT Presentation

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Modular invariance approach to masses and mixing of neutrino flavors Morimitsu Tanimoto Niigata University September 10, 2019 TAUP 2019, TOYAMA Collaborated with H. Okada (APCTP) T. Kobayashi, N. Omoto, T. Tatsuishi (Hokkaido U.) Y. Shimizu,

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Morimitsu Tanimoto

Niigata University

September 10, 2019

TAUP 2019, TOYAMA

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Modular invariance approach to masses and mixing of neutrino flavors

Collaborated with H. Okada (APCTP)

  • T. Kobayashi, N. Omoto, T. Tatsuishi (Hokkaido U.)
  • Y. Shimizu, K.Takagi (Hiroshima U.)
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1 Introduction

Now, neutrino experiments provide important clues for Beyond SM of flavors. We have a big question since the discovery of Muon “Who orderd that ?” 1937 Isidor Issac Rabi What is the principle to control flavors ?

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Symmetry Approach for Flavor

Non-Abelian Discrete Symmetry Possible Origin : Superstring theory 10 dim.⇒ 4 dim. Phenomenology : neutrinos Large mixing angles, 45° Irreducible representations: 1, 1’, 1”, 3 The minimum group containing triplet

A4 group

It could be adjusted to Family Symmetry.

Symmetry of tetrahedron

3 : ( le, lµ, lτ) , 1: eR , 1”: µR 1’: τR

Flavor symmetry should be broken ! We should know how to break the flavor symmetry.

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2 Modular Group

It is well known that the superstring theory

  • n certain compactifications lead to

non-Abelian finite groups.

T.Kobayashi, K.Tanaka, T.H.Tatsuishi, Phys.Rev.D98(2018)016004, arXiv:1803.10391; S3 J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292, arXiv:1806.11040; S4 P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov, JHEP 1904(2019) 174, arXiv:1812.02158; A5 G.J.Ding, S.F.King, X.G.Liu, arXiv:1903.12588; A5 X.G.Liu and G.J.Ding, arXiv:1907.01488 [hep-ph]; T’ P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov,JHEP1907(2019)165, arXiv:1905.11970 CP

R.Toorop, F.Feruglio, C.Hagedorn, Nucl.Phys. B858 (2012) 437, arXiv:1112.1340; F.Feruglio, arXiv:1706.08749; A4

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Superstring theory 10D Our universe is 4D

The extra 6D

should be compactified. Torus compactification

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α2 α1

(x,y)～(x,y)+n1α1+n2α2 Two-dimensional torus T2 is obtained as

T2 = ℝ2 / Λ

Λ is two-dimensional lattice, which is spanned by two lattice vectors α1=2πR and α2=2πRτ

The same lattice is spanned by other bases under the transformation ad-bc=1 a,b,c,d are integer SL(2,Z)

τ =α2 /α1 is a modulus parameter (complex).

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Modular transformation

τ =α2 /α1

ad-bc=1 a,b,c,d are integer

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The modular transformation is generated by S and T .

α1 α2 α’2

T S

α’2 α2 α1 α’1 = =

τ =α2/α1

Dicrete shift symmetry duality

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generate infinite discrete group Modular group has finite subgroups Impose congruence condition ΓN Γ2 S3 Γ3 A4 Γ4 S4 Γ5 A5

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3 Modular A4 invariance as flavor symmetry

A4 modular group ( Γ3 ). Imposing T3=1, we have Yukawa couplings depend on mosulus τ, and they construct A4 triplet !

ΓN

S transformation T transformation

for A4 triplet Modular transformation is the transformation of modulus τ

weight 2

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A4 triplet of modular forms with weight 2

|q|≪1

  • F. Feruglio, arXiv:1706.08749

Flavor symmetry acts non-linealy (Modular forms).

Dedekind eta-function

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How to predict mass matrices in A4 modular symmetry

A4 3L×3L×3Y 1 Ifτ(complex) is fixed, modular symmetry is broken, and flavor structure of mass matrices is determined including CP violation ! 1R

(‘)(‘’)×3L× 3Y 1

α, β, γ are fixed by the charged lepton masses

ME= Mν=

Weinberg Operator Left-handed 3 of A4: ( le, lµ, lτ) , Right-handed 1: eR , 1”: µR , 1’: τR

Modulus τ controls flavor mixing and CP phase ??? In this case, the prediction is too large θ13 unfortunately !

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MD ← MN ←

A4 3 1 3 3 A4 3 3 3 Seesaw

ME ←

A4 1 1 3 3 1” 1 3 3 1’ 1 3 3

Seesaw model

Introduce right-handed neutrinos: A4 Triplet

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νL νR

symmetric x 3Y anti-symmetric x 3Y

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Consider the case of Normal neutrino mass hierarchy

A4 triplet 3 ( Le, Lµ, Lτ ) 3 ( νeR, νµR, ντR ) A4 singlets eR 1 ; µR 1” ; τR 1’

me, mµ, mτ fix α, β, γ . Δm2

sol /Δm2 atm and θ23, θ12, θ13 fix g2/g1=g and τ .

m1 < m2 < m3 Parameters: α, β, γ, g2/g1=g , τ

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best-fit CP violation best-fit

∑ mi ~ 140 meV

Predicted <mee>

Planck 2018 results < 0.12 eV@ΛCDM model

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Freedom of weights !

ρ(ɤ) = S, T

Modular transformation modular forms of weight k Chiral superfields Modular forms are explicitly given if weight k is fixed (k=2,4,6,,,) . On the other hand, chiral superfields are not modular forms and we have no restriction on the possible value of weight kI, a priori.

L1 is modular invariant if weights satisfy ∑kI=k .

Sum of weights should vanish.

Is lepton mass matriices are the uniqe one in A4 modular symmetry ?

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Let us consider Modular forms with higher weights k=4, 6 …

Modular forms with higher weights are constructed by the tensor product of modular forms of weight 2

# of modular forms is k+1

Weight 2 3 Modular forms Weight 4 5 Modular forms Weight 6 7 Modular forms J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292

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Model with weight 4 modular forms

Weinberg operator by using weight 4 modular forms

After removing parameters αe, βμ, γτ by inputting charged lepton masses, we have 2 complex parameters in addition to τ. Sum of weights should vanish in Lagrangian

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Γ2 S3 group

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

4 Modular S3, S4 and A5 symmetries

Irreducible representations: 1, 1’, 2

Γ4 S4 group Irreducible representations: 1, 1’, 2, 3, 3’

2k+1 modular forms

wight k=2 J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292, arXiv:1806.11040 wight k=2

Γ5 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 k/2+1 modular forms

wight k=2

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F.Feruglio

Okada-Tanimoto result of Γ3=A4

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  • A4, A5, S3, S4 … are subgroups of the modular group,

which comes from superstring theory on certain compactifications.

  • Mass matrices of A4 model are determined by fixing

modular parameter τ (Symmetry Breaking). Predictive power !! Minimal A4 model predicts

5 Summary and Prospect

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A lot of physics in the modular invariant approach !!

  • What is a Principle of fixing modulus τ ?

☆ Modular stabilization (model dependent) need modulus dependent superpotential induced by non-perturbative effect ☆ Fixed point of τ Residual symmetry Z2: τ = i Z3: τ = -1/2+√3/2i

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

  • Is Modulus τ common in both quarks and leptons ?

Fundamental domain of τ

  • n SL(2,Z)
  • Leptons τ

Quarks τ

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Back up Slides

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Modular group has interesting subgroups

Impose congruence condition

ΓN ≡ Γ / Γ(N) quotient group finite group

Γ2 S3 Γ3 A4 Γ4 S4 Γ5 A5

called principal congruence subgroups

ΓN

generate infinite discrete group

duality Dicrete shift symmetry

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Normal hierarchy of neutrino masses H.Okada, M.T, arXiv:1905.13421

Common τ for quarks and leptons

・

Input of 4 observed values: θ12 , θ23 , θ13 , Δm2

sol /

Δm2

atm

Output : δCP , <mee>, ∑mi