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

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, K.Takagi (Hiroshima U.) 1�

1 Introduction We have a big question since the discovery of Muon “Who orderd that ?” 1937 Isidor Issac Rabi What is the principle to control flavors ? Now, neutrino experiments provide important clues for Beyond SM of flavors. 2�

Symmetry Approach for Flavor A 4 group Non-Abelian Discrete Symmetry Irreducible representations: 1, 1’, 1”, 3 The minimum group containing triplet It could be adjusted to Family Symmetry. 3 : ( le, lµ, l τ ) , 1: e R , 1”: µ R 1’: τ R Symmetry of tetrahedron Flavor symmetry should be broken ! We should know how to break the flavor symmetry. Possible Origin : Superstring theory 10 dim. ⇒ 4 dim. Phenomenology : neutrinos L arge mixing angles, 45° 3�

2 Modular Group It is well known that the superstring theory on certain compactifications lead to non-Abelian finite groups. R.Toorop, F.Feruglio, C.Hagedorn, Nucl.Phys. B858 (2012) 437, arXiv:1112.1340; F.Feruglio, arXiv:1706.08749; A 4 T.Kobayashi, K.Tanaka, T.H.Tatsuishi, Phys.Rev.D98(2018)016004, arXiv:1803.10391; S 3 J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292, arXiv:1806.11040; S 4 P.P.Novichkov, J.T.Penedo, S.T.Petcov, A.V.Titov, JHEP 1904(2019) 174, arXiv:1812.02158; A 5 G.J.Ding, S.F.King, X.G.Liu, arXiv:1903.12588; A 5 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 4�

The extra 6D Superstring theory 10D should be compactified. Our universe is 4D Torus compactification 5�

Two-dimensional torus T 2 is obtained as α 2 T 2 = ℝ 2 / Λ α 1 Λ is two-dimensional lattice, which is spanned by two lattice vectors α 1 =2 π R and α 2 =2 π R τ (x,y) ～ (x,y)+n 1 α 1 +n 2 α 2 τ = α 2 / α 1 is a modulus parameter (complex). The same lattice is spanned by other bases under the transformation ad-bc=1 a,b,c,d are integer SL(2,Z) 6�

ad-bc=1 a,b,c,d are integer τ = α 2 / α 1 Modular transformation 7�

The modular transformation is generated by S and T . Dicrete shift symmetry duality = = α 2 T S α ’ 2 α ’ 2 α 1 α ’ 1 α 2 α 1 τ = α 2 / α 1 8�

generate infinite discrete group Modular group has finite subgroups Impose congruence condition Γ N Γ 2 S 3 Γ 3 A 4 Γ 4 S 4 Γ 5 A 5 9�

3 Modular A 4 invariance as flavor symmetry Γ N Imposing T 3 =1, we have A 4 modular group ( Γ 3 ). Modular transformation is the transformation of modulus τ Yukawa couplings depend on mosulus τ , and they construct A 4 triplet ! T transformation S transformation weight 2 for A 4 triplet 10�

F. Feruglio, arXiv:1706.08749 A 4 triplet of modular forms with weight 2 Flavor symmetry acts non-linealy (Modular forms). Dedekind eta-function |q| ≪ 1 11�

How to predict mass matrices in A 4 modular symmetry Left-handed 3 of A 4 : ( le, lµ, l τ ) , Right-handed 1: e R , 1”: µ R , 1’: τ R Weinberg Operator (‘)(‘’) × 3 L × 3 Y 1 1 R A 4 3 L × 3 L × 3 Y 1 α , β , γ are fixed by the charged lepton masses M E = M ν = If τ (complex) is fixed, modular symmetry is broken, and flavor structure of mass matrices is determined including CP violation ! Modulus τ controls flavor mixing and CP phase ??? In this case, the prediction is too large θ 13 unfortunately ! 12�

Seesaw model Introduce right-handed neutrinos: A 4 Triplet M E ← A 4 1 1 3 3 1” 1 3 3 1’ 1 3 3 M N ← M D ← A 4 3 3 3 A 4 3 1 3 3 Seesaw 13�

ν L ν R symmetric x 3 Y anti-symmetric x 3 Y 14�

Consider the case of Normal neutrino mass hierarchy m 1 < m 2 < m 3 A 4 triplet 3 ( Le, Lµ, L τ ) 3 ( ν eR , ν µR , ν τ R ) A 4 singlets e R 1 ; µ R 1” ; τ R 1’ Parameters: α , β , γ , g 2 /g 1 =g , τ m e , m µ , m τ fix α , β , γ . Δ m 2 sol / Δ m 2 atm and θ 23 , θ 12 , θ 13 fix g 2 /g 1 =g and τ . 15�

best-fit CP violation best-fit Predicted <m ee > ∑ m i ~ 140 meV Planck 2018 results < 0.12 eV@ Λ CDM model 16�

Is lepton mass matriices are the uniqe one in A 4 modular symmetry ? Freedom of weights ! Modular transformation modular forms of weight k ρ ( ɤ ) = S, T 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 k I , a priori. L 1 is modular invariant if weights satisfy ∑ k I =k . Sum of weights should vanish. 17�

Let us consider Modular forms with higher weights k=4, 6 … # of modular forms is k+1 Weight 2 3 Modular forms Modular forms with higher weights are constructed by the tensor product of modular forms of weight 2 J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292 Weight 4 5 Modular forms Weight 6 7 Modular forms 18�

Model with weight 4 modular forms Sum of weights should vanish in Lagrangian Weinberg operator by using weight 4 modular forms After removing parameters α e , β μ , γ τ by inputting charged lepton masses, we have 2 complex parameters in addition to τ . 19�

4 Modular S 3 , S 4 and A 5 symmetries Γ 2 S 3 group Irreducible representations: 1, 1’, 2 T.Kobayashi, K.Tanaka, T.H.Tatsuishi, wight k=2 Phys.Rev.D98(2018)016004, arXiv:1803.10391 k/2+1 modular forms Γ 4 S 4 group Irreducible representations: 1, 1’, 2, 3, 3’ J.T.Penedo, S.T.Petcov, Nucl.Phys.B939(2019)292, arXiv:1806.11040 wight k=2 2k+1 modular forms Γ 5 A 5 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 wight k=2 20�

Okada-Tanimoto result of Γ 3 =A 4 F.Feruglio 21�

5 Summary and Prospect ● A 4 , A 5 , S 3 , S 4 … are subgroups of the modular group, which comes from superstring theory on certain compactifications. ● Mass matrices of A 4 model are determined by fixing modular parameter τ (Symmetry Breaking). Predictive power !! Minimal A 4 model predicts 22�

● 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 Z 2 : τ = i Z 3 : 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 ● Fundamental domain of τ A lot of physics on SL(2,Z) in the modular invariant approach !! ● Is Modulus τ common in both quarks and leptons ? Leptons τ Quarks τ� 23�

Back up Slides 24�

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duality Dicrete shift symmetry generate infinite discrete group Modular group has interesting subgroups Impose congruence condition called principal congruence subgroups Γ N ≡ Γ / Γ (N) quotient group finite group Γ N Γ 2 S 3 Γ 3 A 4 Γ 4 S 4 Γ 5 A 5 26�

Normal hierarchy of neutrino masses Input of 4 observed values: θ 12 , θ 23 , θ 13 , Δ m 2 sol / Δ m 2 atm Output : δ CP , <m ee >, ∑ m i Common τ for quarks and leptons ・ H.Okada, M.T, arXiv:1905.13421 27�