Tribimaximal Mixing From Small Groups

Bottom line: the enumeration is real work and worth having; the comparative claim in the abstract is softer than the abstract makes it sound. The two are separable, and the paper would be better if it separated them too. Note first that the classifier says the authors state no limitations. That's wrong — §4.6 owns the 0/1 vev choice and §5.3 owns the non-exhaustiveness. Give them credit for both; the problem is what they do with the numbers anyway. What's new and good: 76 groups, 439,820 Lagrangians, exact TBM identified exactly, and the A4×C3 case reproduces Altarelli–Feruglio as a sanity check. The van den Broek–Cornwell CGC algorithm is implemented and cross-checked against the literature, including catching that GAP's A5 representation matrices aren't unitary. Appendix C's subgroup-breaking algorithm is a reusable piece of machinery. This is careful computational bookkeeping, and the tables in Appendix A will get used. Where it's soft: the T7/T13 ranking. The numerator is 'models for which at least one vev gives TBM' — a maximum over 8 to 512 draws, depending on whether the flavons land in singlets or triplets. The denominator is 'inequivalent Lagrangians,' a bookkeeping granularity that varies by two orders of magnitude across groups (288 for T13, 22,932 for A4×C3). Exact TBM is measure-zero in continuous vev space; on a 0/1 lattice you hit it when a grid point happens to land on the TBM locus. So 59% versus 18.5% is not apples-to-apples, and 'on equal footing' is doing more work than the data supports. The authors half-see this — they remark that the TBM rate doesn't scale with model count — but they don't run the sensitivity check that would settle it. The qualitative claim, that TBM is achievable in many groups including T7 and T13, holds up fine. The θ13 = 0 'prediction' is the mode of their own histogram; fine as a statement about the ensemble, thin as a prediction. One more: for a paper whose entire result is a scan, the code isn't released. That's the first thing I'd ask for. Who should read it: anyone building discrete flavor models, and anyone who wants the group tables. It deserves referees — the enumeration is sound and the fixable problems are fixable. Ask for a vev-grid sensitivity study on the TBM fractions and for the scan code, and let the rest stand.

The claim at [abstract:s6] is comparative: T13 (59% TBM) and T7 are 'as suited' as A4×C3 (18.5% TBM). That comparison is a ratio whose numerator is 'models for which at least one vev configuration gives TBM' and whose denominator is 'inequivalent Lagrangians with non-singular mass matrices'. Both parts are grid- and counting-dependent in ways that differ across groups, so the ranking is not a property of the groups. Three concrete mechanisms: (i) Exact TBM is a measure-zero condition in continuous vev space (it requires θ13 = 0 exactly plus the two TBM angles). On the 0/1 grid of §4.6 it is hit only when a grid point happens to land on the TBM locus, so the numerator measures how the group's Clebsch–Gordan coefficients happen to map the 0/1 lattice onto that locus, not how 'natural' TBM is for the group. (ii) The number of vevs sampled per model varies from 2^3 to 2^9 (§4.6), and the numerator is a maximum over those draws. A group whose flavon assignments are preferentially triplets gets up to 512 draws per Lagrangian; one whose flavons are mostly singlets gets 8. The hit probability therefore rises mechanically with the number of draws, confounding the TBM fraction with the irrep spectrum rather than with any TBM-relevant structure. (iii) The denominator counts 'inequivalent Lagrangians' under an equivalence defined by Lagrangian equality after contraction (§5), a bookkeeping choice that differs in granularity across groups (288 models for T13 vs 22,932 for A4×C3). The authors acknowledge the 0/1 grid (§4.6) and non-exhaustiveness (§5.3), and even note that 'the chances of finding TBM does not scale with the total number of models', but they do not address the max-over-draws bias or the measure-zero character of exact TBM, which is what makes the T13 > A4 ordering fragile. The qualitative conclusion — that TBM is achievable in many groups, including T7 and T13 — is robust; the quantitative ranking that licenses 'on equal footing' is not.

The reader correctly identifies the 0/1 vev grid (§4.6) as the weakest assumption and correctly notes that Appendix C itself says mixing angles depend on vev alignment and magnitude, not only on the residual symmetry. That is the right neighborhood. My refinement is that the sharper problem is not merely 'coarse grid' but that the TBM fraction is a maximum over a variable number of draws (2^3 to 2^9 vevs per model) on a lattice that intersects a measure-zero TBM locus only by coincidence. That makes the cross-group ordering — the specific claim that T13 and T7 outrank or match A4 — the least secure part of the argument, while the existence claims (TBM is achievable in 38 groups; T7 is the smallest such group; a θ13 ≈ 5° model exists) are supported by explicit constructions and are not at risk. The reader's other points are fair but secondary: the θ13 = 0 'prediction' is indeed a histogram mode of the ensemble rather than a first-principles prediction, and the abstract does state the T7/T13 conclusion more crisply than §6, which disclaims a probabilistic reading. Neither of those is load-bearing in the way the ranking is. I do not think the paper should be rejected: the enumeration is careful, the A4×C3 case reproduces Altarelli–Feruglio, the CGCs are cross-checked against the literature and against the transformation properties of C_{ijk} p_i q_j for all 90 groups, and the limitations are stated in the text. The appropriate verdict remains CONDITIONAL, with the sensitivity study above as the specific thing that would settle whether the comparative claim lands. I would not ask the authors to re-prove any cited result; the concern is entirely internal to their own counting procedure.

The enumeration itself is a genuine and useful contribution: a large, reproducible, GAP/Python-driven survey of lepton-sector Lagrangians across essentially all small groups with a triplet representation, with the Clebsch–Gordan machinery implemented generally rather than case by case (Appendix D) and cross-checked against the literature, including an honest note that the GAP representation matrices for A5 are not unitary. The broadening of the set of viable family symmetries beyond A4 is of real value to model builders, and the θ13 statement is in principle falsifiable. The paper also ships its methodology in enough detail (GAP scripts, running times, equivalence classes) that the computation could be reproduced. These strengths are real and should be weighed alongside the concerns below.

The paper asks whether the tetrahedral group $A_4$ really deserves its reputation as the natural home of tribimaximal neutrino mixing, or whether that reputation is an artifact of where model builders happened to look. The authors answer by brute force: for each of 76 finite groups of order at most 100 that possess a three-dimensional irreducible representation, they enumerate every way of assigning the Standard Model leptons, two Higgs doublets and up to three flavon fields to representations, keep all operators of mass dimension at most six, and compute the resulting lepton mass matrices and mixing angles. Out of 439,820 distinct Lagrangians, 44 groups admit at least one vacuum whose angles fall inside the measured $3$-$3$sigma ranges, and 38 admit exact tribimaximal mixing. The smallest such group is $T_7$, of order 21, and the group with the highest fraction of tribimaximal models is $T_{13} = C_{13} rtimes C_3$, where the tribimaximal set and the experimentally viable set nearly coincide. If the counting means what the authors take it to mean, $A_4$ is one option among many rather than the privileged one.

What carries the argument. The load-bearing device is a fully automated enumeration pipeline rather than a single identity. For each group, GAP supplies the character table and explicit representation matrices; the van den Broek-Cornwell algorithm converts these into Clebsch-Gordan coefficients for arbitrary tensor products, which lets the code contract family indices without group-specific handiwork. The code then builds every invariant Lagrangian with exactly two leptons, at most two Higgses and at most one flavon, discards duplicate Lagrangians by lexicographic comparison, substitutes a grid of vacuum expectation values, and extracts mixing angles from $U_{text{PMNS}} = D_L U_L^dagger$ using standard parametrization formulae. Because $e$, $mu$ and $tau$ are treated as interchangeable, all row permutations of the mixing matrix are examined before angles are recorded. The counting of inequivalent charge assignments uses a multiset coefficient $C_r(N,k) = binom{N+k-1}{k}$, which is what keeps the $A_4 rtimes C_3$ case at 14,594,580 assignments rather than a much larger number.

Core claim. The central claim is that tribimaximal mixing is not a special property of $A_4$. The authors scan 76 finite groups systematically, constructing every charge assignment consistent with the stated particle content, contracting family indices with Clebsch-Gordan coefficients, and diagonalizing the charged-lepton and neutrino mass matrices by singular value decomposition. They report that 44 of the 76 groups (58%) can accommodate mixing angles inside the $3$-$3$$\sigma$ experimental ranges and 38 (50%) can produce exact tribimaximal mixing for at least one vacuum configuration. The smallest group doing so is $T_7 = G(21,1)$, which after $A_4$ is the smallest group with a three-dimensional irreducible representation; the group with the largest tribimaximal fraction is $T_{13} = G(39,1)$, where 171 of 288 inequivalent models are tribimaximal and every model with a $3$-$3$$\sigma$ vacuum also has a tribimaximal vacuum. The authors conclude that $T_7$ and $T_{13}$ should be considered on equal footing with $A_4$, if not more promising. A secondary claim is that the recent tentative hints of a non-zero $theta_{13}$ are easy to accommodate: an explicit $A_4 rtimes C_3$ charge assignment with vevs $(1)$, $(1,0,1)$ and $(1,1,1)$ gives $theta_{12} = 33.9^circ$, $theta_{23} = 40.9^circ$ and $theta_{13} = 5.1^circ$, all inside the $1$-$1$$\sigma$ intervals.

The premise it rests on. Every flavon vacuum expectation value is fixed to entries of $0$ or $1$, so the counts of tribimaximal and $3$-$3$sigma models, and therefore the ranking of $T_7$ and $T_{13}$ against $A_4$, describe a coarse discrete grid rather than the groups themselves.

Model builders receive a ranked list of 38 candidate family groups that can produce tribimaximal mixing, most of which had not previously been used for lepton model building, with $T_7$ as the most economical entry by group order.

$T_{13}$ becomes a strong alternative to $A_4$ because in that group the set of models fitting the measured angles at $3$-$3$sigma and the set of tribimaximal models nearly coincide, so the two criteria need not be tuned against each other.

The $A_4 rtimes C_3$ scan yields a testable statement: with $theta_{12}$ and $theta_{23}$ restricted to their measured ranges, the most likely $theta_{13}$ is $0^circ$ and the window $0^circ < theta_{13} lesssim 12^circ$ is sparsely populated.

The explicit charge assignment $(L, e, mu, tau, h_u, h_d, phi_T, phi_S, xi) sim (3, 1, 1^{(4)}, 1''', 1', 1^{(7)}, 1^{(5)}, 3'', 3'')$ demonstrates that a $theta_{13}$ near the current best-fit value follows from the same framework rather than requiring a special construction.

The appendices supply reusable tools: an implementation of an algorithm for Clebsch-Gordan coefficients of any finite group, and a procedure for finding every vacuum expectation value that breaks a group to each of its subgroups, including the case of three flavons acting jointly.

The ranking of groups is a statement about a discrete grid of vacuum expectation values with entries $0$ or $1$, not about the groups themselves; the paper's own Appendix C notes that mixing angles depend on the alignment and magnitude of the vevs, not only on the residual subgroup, so a scan over continuous complex vevs could reshuffle the ordering.

If $theta_{13}$ is measured well inside $5^circ$-$12^circ$, the $A_4 rtimes C_3$ prediction that this window is almost depopulated becomes a discriminating test between this class of models and constructions that naturally place $theta_{13}$ there.

The same enumeration could be repeated with only two flavon fields, or with the flavon sector charged under a continuous abelian factor, to check whether $T_7$ remains the smallest viable group; if it does, $T_7$ is the cheapest experimental target among the 38.

Counting models is not counting likelihoods: a weighting over couplings and vacuum directions would be needed before 'as suited as $A_4$' becomes a quantitative statement rather than a statement about relative frequencies in a catalogue.

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