Modular A 5 symmetry for flavour model building
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Springer
Received: December 26, Revised: March 3, Accepted: April 26, Published: April 30,
2018 2019 2019 2019
P.P. Novichkov,a J.T. Penedo,b S.T. Petcova,c,1 and A.V. Titovd a
SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italy b CFTP, Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal c Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa, Japan d Institute for Particle Physics Phenomenology, Department of Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: In the framework of the modular symmetry approach to lepton flavour, we consider a class of theories where matter superfields transform in representations of the finite modular group Γ5 ' A5 . We explicitly construct a basis for the 11 modular forms of weight 2 and level 5. We show how these forms arrange themselves into two triplets and a quintet of A5 . We also present multiplets of modular forms of higher weight. Finally, we provide an example of application of our results, constructing two models of neutrino masses and mixing based on the supersymmetric Weinberg operator. Keywords: Beyond Standard Model, Discrete Symmetries, Neutrino Physics, CP violation ArXiv ePrint: 1812.02158
1
Also at Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)174
JHEP04(2019)174
Modular A5 symmetry for flavour model building
Contents 1 Introduction
1
2 The framework 2.1 Modular symmetry and modular-invariant theories 2.2 Generators of modular forms of level N = 5
3 3 5 7 8 9 10
4 Summary and conclusions
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A q-expansions A.1 Miller-like basis for the space of lowest weight forms A.2 Expansions for the lowest weight A5 multiplets
14 14 15
B A5 group theory B.1 Basis B.2 Clebsch-Gordan coefficients
15 15 16
C Higher weight forms and constraints
21
D Correspondence with the Dedekind eta function
29
1
Introduction
Understanding the origins of flavour remains one of the major problems in particle physics. The power of symmetries in governing laws of particle interactions does not need to be advocated. In this regard, it is rather natural to expect that symmetry(ies) also hold the key to the solution of the flavour problem. The fact that two out of three neutrino mixing angles are large [1–3] suggests the presence of a new flavour symmetry (at least in the lepton sector) described by a non-Abelian discrete (finite) group (see, e.g., [4–7]). While unifying the three known flavours at high energies, this symmetry may be broken at lower energies to residual symmetries of the charged lepton and neutrino mass terms, which correspond to Abelian subgroups of the original flavour symmetry group. In the bottom-up approach, starting from residual symmetries, one can successfully explain
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