Partial $A_4$ flavor symmetry of the leptonic 3HDM

When the Higgs doublets in the 3HDM transform as a flavor triplet of the $A_4$ group, the lepton mass matrices accommodate the experimental neutrino mixing angles at arbitrary precision while maintaining the correct mass ordering of the charged and neutral leptons, the latter being Dirac neutrinos in the normal spectrum. Under $A_4$ symmetry, also agreement of the lepton masses with experimental data is obtained for Higgs vacua differing from those for fitting $U_{\text{PMNS}}$. For groups of order equal or less than 600 no contractions different from the one found for $A_4$ yield better agreement with experimental data, and the solution structure presented is unique within the set of groups studied.

💡 Research Summary

This paper, titled “Partial A4 flavor symmetry of the leptonic 3HDM,” presents a comprehensive investigation into the application of discrete flavor symmetries to explain lepton masses and neutrino mixing within a Three-Higgs-Doublet Model (3HDM). The core premise is extending the Standard Model with two additional Higgs doublets and three right-handed neutrinos (assuming Dirac neutrinos), and then imposing a finite flavor symmetry group G under which not only the left-handed lepton doublets and right-handed singlets but also the three Higgs doublets transform as triplet representations.

The authors’ methodology is rooted in group representation theory. For a candidate group G with 3D representations, the condition for the Yukawa Lagrangian to be invariant under G translates into a system of linear equations. The dimensionality of the solution space for the Yukawa coupling matrices, denoted n_l for charged leptons and n_ν for neutrinos, is determined by the number of times the trivial singlet (1) appears in the tensor product decomposition of the relevant representations. The researchers systematically scanned all groups of order |G| ≤ 600 (over 3200 groups) to find invariant pairs of mass matrices {M_l, M_ν} and test their viability against experimental data.

The key finding is that only solutions where both n_l and n_ν equal 2 can potentially accommodate the observed lepton mass hierarchies and neutrino mixing angles simultaneously. All 729 such inequivalent solutions originate from the group A4 (and isomorphic higher-order groups), generating mass matrices M_l and M_ν with an identical cyclic structure. These matrices are monomial, with their non-zero entries being linear combinations of the Higgs vacuum expectation values (VEVs) v_i and two independent Yukawa couplings for each sector (λ_l1, λ_l2 and λ_ν1, λ_ν2).

The analysis demonstrates that this A4-driven structure can successfully reproduce the correct ratios of charged lepton masses (m_μ/m_e, m_τ/m_μ) for a specific curve in the parameter space of |v2/v1| and |v3/v1|. For normally ordered Dirac neutrinos, it also yields neutrino mass squared differences (Δm^2_21/Δm^2_32) in close agreement with experimental values, and predicts absolute neutrino masses consistent with current bounds. Furthermore, wide regions of parameter space exist that produce neutrino mixing angles (sin^2 θ_12, sin^2 θ_23, sin^2 θ_13) within 3σ of their experimental measurements.

However, the paper clearly identifies limitations, justifying the “Partial” in its title. The parameter regions that yield correct lepton mass ratios do not overlap with those producing correct mixing angles, meaning no single parameter set can fully explain both simultaneously. Additionally, the model predicts a CP-violating phase δ_CP in the range of approximately 1.5 to 1.9 radians, which is in tension with the current experimental best-fit value near -1.9 radians. The study concludes that among all finite groups of order 600 or less, the solution structure provided by A4 symmetry in the 3HDM is unique and offers the maximal possible agreement with data, capturing major features of the lepton sector but falling short of a complete fit. This work establishes a promising benchmark and suggests directions for future exploration, such as considering reducible representations or incorporating Majorana mass terms.