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Register a new userLeptonic Sum Rules from Flavour Models with Modular Symmetries
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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.
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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.
We use QCD Laplace sum-rules to predict masses of open-flavour heavy-light hybrids where one of the hybrids constituent quarks is a charm or bottom and the other is an up, down, or strange. We compute leading-order, diagonal correlation functions of several hybrid interpolating currents, taking into account QCD condensates up to dimension-six, and extract hybrid mass predictions for all $J^Pin{0^{pm},,1^{pm}}$, as well as explore possible mixing effects with conventional quark-antiquark mesons. Within theoretical uncertainties, our results are consistent with a degeneracy between the heavy-nonstrange and heavy-strange hybrids in all $J^P$ channels. We find a similar mass hierarchy of $1^+$, $1^{-}$, and $0^+$ states (a $1^{+}$ state lighter than essentially degenerate $1^{-}$ and $0^{+}$ states) in both the charm and bottom sectors, and discuss an interpretation for the $0^-$ states. If conventional meson mixing is present the effect is an increase in the hybrid mass prediction, and we estimate an upper bound on this effect.
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.
Neutrino mass sum rules have recently gained again more attention as a powerful tool to discriminate and test various flavour models in the near future. A related question which was not yet discussed fully satisfactorily was the origin of these sum rules and if they are related to any residual or accidental symmetry. We will address this open issue here systematically and find previous statements confirmed. Namely, that the sum rules are not related to any enhanced symmetry of the Lagrangian after family symmetry breaking but that they are simply the result of a reduction of free parameters due to skillful model building.
We investigate a model QCD sum rule for the pion wave function $varphi_{pi}(x)$ based on the non-diagonal correlator whose perturbative spectral density vanishes and $Phi(x,M^2)$, the theoretical side of the sum rule, consists of condensate contributions only. We study the dependence of $Phi(x,M^2)$ on the Borel parameter $M^2$ and observe that $Phi(x,M^2)$ has a humpy form, with the humps becoming more and more pronounced when $M^2$ increases. We demonstrate that this phenomenon reflects just the oscillatory nature of the higher states wave functions, while the lowest state wave function $varphi_{pi}(x)$ extracted from our QCD sum rule analysis,has no humps, is rather narrow and its shape is close to the asymptotic form $varphi_{pi}^{as}(x) = 6x(1-x)$.
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