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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 $Gamma_7$, which is isomorphic to $PSL(2,Z_{7})$, the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as $PSL_2(7)$ or $Sigma(168)$. At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of $Gamma_7$. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on $Gamma_7$ 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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We study the spontaneous $CP$ violation through the stabilization of the modulus $tau$ in modular invariant flavor models. The $CP$-invaraiant potentential has the minimum only at ${rm Re}[tau] = 0$ or 1/2. From this prediction, we study $CP$ violation in modular invariant flavor models. The physical $CP$ phase is vanishing. The important point for the $CP$ conservation is the $T$ transformation in the modular symmetry. One needs the violation of $T$ symmetry to realize the spontaneous $CP$ violation.
The idea of modular invariance provides a novel explanation of flavour mixing. Within the context of finite modular symmetries $Gamma_N$ and for a given element $gamma in Gamma_N$, we present an algorithm for finding stabilisers (specific values for moduli fields $tau_gamma$ which remain unchanged under the action associated to $gamma$). We then employ this algorithm to find all stabilisers for each element of finite modular groups for $N=2$ to $5$, namely, $Gamma_2simeq S_3$, $Gamma_3simeq A_4$, $Gamma_4simeq S_4$ and $Gamma_5simeq A_5$. These stabilisers then leave preserved a specific cyclic subgroup of $Gamma_N$. This is of interest to build models of fermionic mixing where each fermionic sector preserves a separate residual symmetry.
We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $Gamma(N)/Gamma(N)$, and the modular group $SL(2,mathbb{Z})$ is extended to a principal congruence subgroup $Gamma(N)$. The original modular invariant theory is reproduced when $N=1$. We perform a comprehensive study of $Gamma_6$ modular symmetry corresponding to $N=1$ and $N=6$, five types of models for lepton masses and mixing with $Gamma_6$ modular symmetry are discussed and some example models are studied numerically. The case of $N=2$ and $N=6$ is considered, the finite modular group is $Gamma(2)/Gamma(6)cong T$, and a benchmark model is constructed.
The electric dipole moment (EDM) of electron is studied in the supersymmetric $rm A_4$ modular invariant theory of flavors with CP invariance. The CP symmetry of the lepton sector is broken by fixing the modulus $tau$. Lepton mass matrices are completely consistent with observed lepton masses and mixing angles in our model. In this framework, a fixed $tau$ also causes the CP violation in the soft SUSY breaking terms. The elecrton EDM arises from the CP non-conserved soft SUSY breaking terms. The experimental upper bound of the electron EDM excludes the SUSY mass scale below $2-6$ TeV for five cases of the lepton mass matrices. In order to see the effect of CP phase of the modulus $tau$, we examine the correlation between the electron EDM and the decay rate of the $mu rightarrow e gamma$ decay, which is also predicted by the soft SUSY breaking terms. The correlations are clearly predicted in contrast to models of the conventional flavor symmetry. The SUSY mass scale will be constrained by the future sensitivity of the electron EDM, $|d_e/e| simeq 10^{-30}$. Indeed, it could probe the SUSY mass range of $10-20$ TeV in our model. Thus, the electron EDM provides a severe test of the CP violation via the modulus $tau$ in the supersymmetric modular invariant theory of flavors.
Doubly charged excited leptons give rise to interesting signatures for physics beyond the standard model at the present Large Hadron Collider. These exotic states are introduced in extended isospin multiplets which couple to the ordinary leptons and quarks either with gauge or contact effective interactions or a combination of both. In this paper we study the production and the corresponding signatures of doubly charged leptons at the forthcoming linear colliders and we focus on the electron-electron beam setting. In the framework of gauge interactions, the interference between the $t$ and $u$ channel is evaluated that has been neglected so far. A pure leptonic final state is considered ($e^{-} , e^{-} rightarrow e^{-} , e^{-} , u_{e} , bar{ u}_{e}$) that experimentally translates into a like-sign dilepton and missing transverse energy signature. We focus on the standard model irreducible background and we study the invariant like-sign dilepton mass distribution for both the signal and background processes. Finally, we provide the 3 and 5-sigma statistical significance exclusion curves in the model parameter space. We find that for a doubly charged lepton mass $m^*approx 2 $ TeV the expected lower bound on the compositeness scale at CLIC, $Lambda > 25$ TeV, is much stronger than the current lower bound from LHC ($Lambda > 5$ TeV) and remains highly competitive with the bounds expected from the run II of the LHC.
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