Fermion masses, critical behavior and universality

Abstract

We look for signals of critical behavior in the Yukawa sector. By reviewing a set of models for the fermion masses, we select those where a symmetry-breaking order parameter sits at a transition point between a disordered phase and an ordered one. Many models based on ordinary flavor symmetries are formulated in terms of small corrections to a symmetric limit, which can hardly be interpreted unambiguously as a sign of near-criticality. Different is the case of nonlinearly realized flavor symmetries when the system is always in the broken phase. By inspecting a large number of modular and CP invariant models of lepton masses, we find that most of them cluster around the fixed point τ = i, where the system enjoys enhanced symmetry. Since a priori all values of the modulus τ are equally acceptable to describe the fermion spectrum, we regard this preference as a hint of near-criticality. We analyze in detail these models in the vicinity of all fixed points, showing that only one possibility provides a good description of neutrino masses and mixing angles. Near the fixed points the models exhibit a universal behavior. Mass ratios and mixing angles scale with appropriate powers of the order parameter, independently of the details of the theory, a feature reminiscent of systems belonging to the same universality class in second-order phase transitions. The observations of this work are inspired by the role near-criticality might play in solving the naturalness problem and are motivated by the fascinating possibility that most of the free parameters of the Standard Model could find a common explanation.

CP symmetry and symplectic modular invariance

“…The pattern in eq. ( 93) coincides with the one occurring in single-modulus SL(2, )invariant theories close to the fixed point τ 0 = ω [44]. We refer the reader to the literature for more details 9 .…”

“…SL(2, )-invariant theories close to the fixed point τ 0 = i [43,44]. We briefly recapitulate the results here and we refer the reader to the literature for more details 8 .…”

“…Such a preference is even more pronounced for the subset of CP-invariant models, where the violation of CP is spontaneous and entirely controlled by the modulus τ . Indeed, if electroweak lepton doublets are assigned to irreducible triplets of the finite modular group, a typical choice allowing to minimize the number of free parameters, the behavior of modular invariant models of lepton masses near the fixed points τ 0 = i and τ 0 = −1/2 + i √ 3/2 is universal [44]. All physical quantities scale with the distance |τ − τ 0 | in a way that is largely independent of the level n, the weight k I , the specific representations ρ I and even the form of kinetic terms.…”

“…The widest such a region has complex dimension two and provides a nontrivial extension of the framework studied in ref. [43,44].…”

“…In particular, the predictions for the lepton masses and lepton mixing matrix as well as mixing parameters have been presented in Appendices D.1.1 and D.1.2 and Section 6 of ref [44]…”