Generalised CP symmetry in modular-invariant models of flavour

We analyze CP symmetry in symplectic modular-invariant supersymmetric theories. We show that for genus g\ge 3g≥3 the definition of CP is unique, while two independent possibilities are allowed when g\le 2g≤2. We discuss the transformation properties of moduli, matter multiplets and modular forms in the Siegel upper half plane, as well as in invariant subspaces. We identify CP-conserving surfaces in the fundamental domain of moduli space. We make use of all these elements to build a CP and symplectic invariant model of lepton masses and mixing angles, where known data are well reproduced and observable phases are predicted in terms of a minimum number of parameters.

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Modular invariant A4 models for quarks and leptons with generalized CP symmetry

Journal of High Energy Physics ◽

10.1007/jhep05(2021)102 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Chang-Yuan Yao ◽

Jun-Nan Lu ◽

Gui-Jun Ding

Keyword(s):

Vacuum Expectation ◽

Vacuum Expectation Value ◽

Modular Invariant ◽

Quark Masses ◽

Mass Matrices ◽

Flavor Mixing ◽

Mixing Matrix ◽

Free Parameters ◽

The Masses ◽

Cp Symmetry

Abstract We perform a systematical analysis of the A4 modular models with generalized CP for the masses and flavor mixing of quarks and leptons, and the most general form of the quark and lepton mass matrices is given. The CP invariance requires all couplings real in the chosen basis and thus the vacuum expectation value of the modulus τ uniquely breaks both the modular symmetry and CP symmetry. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find 20 models with 7 real free parameters that can accommodate the experimental data of lepton sector. We then apply A4 modular symmetry to the quark sector to explain quark masses and CKM mixing matrix, the minimal viable quark model is found to contain 10 free real parameters. Finally, we give two predictive quark-lepton unification models which use only 16 real free parameters to explain the flavor patterns of both quarks and leptons.

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CP symmetry and its violation

AccessScience ◽

10.1036/1097-8542.166177 ◽

2020 ◽

Keyword(s):

Cp Symmetry

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CP symmetry

Physics Subject Headings (PhySH) ◽

10.29172/ddef0ddc-6f3d-4d34-93c8-5c52964a1704 ◽

2018 ◽

Keyword(s):

Cp Symmetry

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Modular invariant of rank 1 Drinfeld modules and class field generation

Journal of Number Theory ◽

10.1016/j.jnt.2020.11.005 ◽

2020 ◽

Author(s):

L. Demangos ◽

T.M. Gendron

Keyword(s):

Drinfeld Modules ◽

Modular Invariant ◽

Class Field ◽

Field Generation

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New modular invariants in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory

Journal of High Energy Physics ◽

10.1007/jhep04(2021)212 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Shai M. Chester ◽

Michael B. Green ◽

Silviu S. Pufu ◽

Yifan Wang ◽

Congkao Wen

Keyword(s):

Eisenstein Series ◽

Mass Parameter ◽

Flat Space ◽

Superstring Theory ◽

Modular Invariant ◽

Yang Mills ◽

Laplace Eigenvalue ◽

Modular Invariants ◽

Tensor Multiplet ◽

Super Yang Mills Theory

Abstract We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the $$ \mathcal{N} $$ N = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the $$ \mathcal{N} $$ N = 2∗ theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the $$ \mathcal{N} $$ N = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ,$$ \overline{\tau} $$ τ ¯ ). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5.

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Poincaré series, 3d gravity and averages of rational CFT

Journal of High Energy Physics ◽

10.1007/jhep04(2021)267 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Viraj Meruliya ◽

Sunil Mukhi ◽

Palash Singh

Keyword(s):

Poincaré Series ◽

Simons Theory ◽

Partition Functions ◽

Moody Algebra ◽

Modular Invariant ◽

Poincare Series ◽

Single Genus ◽

3D Gravity ◽

Chern Simons ◽

Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

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Modular invariant models of leptons at level 7

Journal of High Energy Physics ◽

10.1007/jhep08(2020)164 ◽

2020 ◽

Vol 2020 (8) ◽

Cited By ~ 4

Author(s):

Gui-Jun Ding ◽

Stephen F. King ◽

Cai-Chang Li ◽

Ye-Ling Zhou

Keyword(s):

Modular Forms ◽

Galois Field ◽

Special Linear Group ◽

Type I ◽

Time Level ◽

Projective Special Linear Group ◽

Modular Invariant ◽

Seesaw Mechanism ◽

Linearly Independent ◽

First Time

Abstract We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

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