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Register a new userMultiple modular symmetries as the origin of flavour
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We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries $S_4^A$, $S_4^B$ and $S_4^C$, associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bi-triplet scalars to the diagonal $S_4$ subgroup. The low energy effective theory involves the three independent moduli fields $tau_A$, $tau_B$ and $tau_C$, which preserve the residual modular subgroups $Z_3^A$, $Z_2^B$ and $Z_3^C$, in their respective sectors, leading to trimaximal TM$_1$ lepton mixing, consistent with current data, without flavons.
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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.
Sum rules in the lepton sector provide an extremely valuable tool to classify flavour models in terms of relations between neutrino masses and mixing parameters testable in a plethora of experiments. In this manuscript we identify new leptonic sum rules arising in models with modular symmetries with residual symmetries. These models simultaneously present neutrino mass sum rules, involving masses and Majorana phases, and mixing sum rules, connecting the mixing angles and the Dirac CP-violating phase. The simultaneous appearance of both types of sum rules leads to some non-trivial interplay, for instance, the allowed absolute neutrino mass scale exhibits a dependence on the Dirac CP-violating phase. We derive analytical expressions for these novel sum rules and present their allowed parameter ranges as well as their predictions at upcoming neutrino experiments.
We analyse how $U(3)^5$ and $U(2)^5$ flavour symmetries act on the Standard Model Effective Field Theory, providing an organising principle to classify the large number of dimension-six operators involving fermion fields. A detailed counting of such operators, at different order in the breaking terms of both these symmetries, is presented. A brief discussion about possible deviations from these two reference cases, and a simple example of the usefulness of this classification scheme for high-$p_T$ analyses at the LHC, are also presented.
I propose the use of CP-odd invariants, which are independent of basis and valid for any choice of CP transformation, as a powerful approach to study CP in the presence of flavour symmetries. As examples of the approach I focus on Lagrangians invariant under $Delta(27)$. I comment on the consequences of adding a specific CP symmetry to a Lagrangian and distinguish cases where several $Delta(27)$ singlets are present depending on how they couple to the triplets. One of the examples included is a very simple toy model with explicit CP violation with calculable phases, which is referred to as explicit geometrical CP violation by comparison with previously known cases of (spontaneous) geometrical CP violation.
Modular flavor symmetries provide us with a new, promising approach to the flavor problem. However, in their original formulation the kinetic terms of the standard model fields do not have a preferred form, thus introducing additional parameters, which limit the predictive power of this scheme. In this work, we introduce the scheme of quasi-eclectic flavor symmetries as a simple fix. These symmetries are the direct product of a modular and a traditional flavor symmetry, which are spontaneously broken to a diagonal modular flavor subgroup. This allows us to construct a version of Feruglios model with the Kaehler terms under control. At the same time, the starting point is reminiscent of what one obtains from explicit string models.
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