Fermion mixings vs d=6 proton decay

The useful thing here is narrow and real: in any GUT with symmetric Yukawa couplings, the d=6 gauge contributions to nucleon decay into antineutrinos collapse to two constants, k1 and k2. Measure p → K+ ν̄ and p → π+ ν̄, and every other antineutrino channel is a fixed ratio (eqs. 45–48). That is a genuine extension of De Rujula–Georgi–Glashow [26], which did two generations and dropped the mixings from neutrino diagonalization. Pavel does three generations with all CP phases, and the algebra is clean: unitarity of V_EN kills the lepton-flavor sum in going from eq. 35 to eq. 37, and the cross terms drop out by the delta structure. I checked eqs. 46–48 against the rate formulas and they are consistent. The SU(5) section is honest in a way I appreciate — it works through the operators and then says plainly that you cannot test SU(5) this way, which is a negative result stated as such rather than buried. The abstract's own caveats (s2, s3) are accurate, not decorative. Citations look normal; nothing self-serving, nothing cited that isn't there. The soft spot is the scope of the 'clear test' (s4). What is actually derived is a test of the d=6 gauge sector, and only where that sector dominates or can be isolated. Section I calls the Higgs d=6 operators 'less important' without an estimate, and Section V then says the analysis is valid in supersymmetric and non-supersymmetric scenarios — that does not sit well with d=5 dominating in SUSY, where the ratios would be distorted. The operator analysis is right; the claim is just wider than the derivation. Minor point: eq. 45 reads like a prediction but is really a consistency condition on Q2 > 1. This is for people doing GUT proton decay phenomenology and SO(10) model building, not a broad audience. It deserves a referee, mainly to check the index contractions in eqs. 33–37 line by line and to push the author to state the dominance assumption explicitly. I would cite it if I were working on d=6 channels.

The central structural result is that in symmetric-Yukawa GUTs the antineutrino channels depend only on k1 and k2, giving the parameter-free ratios of eqs. 45-48. The derivation (eqs. 35-37) is clean for the d=6 gauge operators: unitarity of V_EN collapses the sum over neutrino flavors, leaving only known CKM elements, and the cross terms vanish by the delta-function structure. I checked eqs. 46-48 against eqs. 13-21 and they are consistent. The load-bearing concern is not the algebra but the scope of the 'clear test' claim (abstract:s4). The test requires that the d=6 gauge operators be the dominant, or cleanly isolable, contribution to the measured p to K+ antineutrino and p to pi+ antineutrino channels. Section I asserts that Higgs-sector d=6 operators are 'less important' without quantifying their size, and it acknowledges that d=5 operators are important in SUSY, yet Section V concludes 'Our analysis is valid in supersymmetric and non-supersymmetric GUT scenarios.' In SUSY GUTs d=5 operators typically dominate the total decay rate, so the d=6 gauge contribution would be subdominant and the ratios of eqs. 46-48 would be distorted by the d=5 and Higgs d=6 contributions. The paper does not estimate either correction, so the 'clear test' is only clean in a regime (non-SUSY, or Higgs d=6 suppressed) that the paper does not establish. This is the reader's concern, and it is legitimate.

The reader's weakest_assumption identifies the right load-bearing concern: the test assumes gauge d=6 dominance, and the paper does not quantify the competing Higgs d=6 and d=5 contributions. I verified the key algebraic steps independently. The unitarity collapse in eq. 37 is sound: summing over the three light neutrino flavors gives a delta function that removes V_EN and V_DE, leaving only CKM elements. The cross terms between different i vanish because the delta functions force incompatible index values, so no CP phases survive. I checked eqs. 46-48 against eqs. 13-21 and they are consistent, including the A1/A3 placement in eq. 46. The paper is also honest about its scope in Section I ('In general we study in detail the gauge d=6 contributions') and about what it cannot do (abstract:s2, abstract:s3). The remaining tension is the abstract's 'clear test' framing (s4) and the Section V claim of SUSY validity, which are not supported without quantifying the d=5 and Higgs d=6 contributions. This warrants CONDITIONAL rather than ACCEPT, matching the reader. The contribution is a genuine but modest extension of De Rujula, Georgi and Glashow [26] to three generations with all CP phases. No circularity: the ratios are fixed by the operator structure, not by fitted constants. I did not find a sharper internal inconsistency that would change the verdict; the possible missing k2^4|V_12|^2 and k2^4|V_13|^2 terms in eq. 42 would affect only the extraction of k2, not the parameter-free ratios of eqs. 46-48, so they are not load-bearing for the central claim.

The structural result is genuinely useful: it isolates a class of GUTs in which the d=6 gauge contributions to the antineutrino channels are parameter-free, and it provides falsifiable ratios that could in principle distinguish symmetric-Yukawa SO(10)-type models from other GUTs. The effective-operator treatment is clean, and the derivation of Eq. (37) from Eq. (35) via the unitarity of V_DE V_EN is correct. The paper is a useful contribution to the proton-decay literature, and its negative result for generic SU(5) is a helpful caution.

The paper asks a blunt question: can proton decay actually test a grand unified theory, given that the decay amplitude depends on how quarks and leptons mix, including CP phases nobody knows independently? Working through the effective dimension-six operators that mediate proton decay, the author writes the coefficients for every two-body channel and counts them: only seven independent measurable combinations exist, so a theory is testable only if it carries fewer than seven unknown physical parameters. For generic SU(5) and SO(10) models that count fails, and no clean prediction follows, which is the paper's negative result. The exception is any theory whose Yukawa couplings are symmetric, the realistic SO(10) models built on $10_H$ and $126_H$: there the decays into a meson plus an antineutrino lose every trace of mixing matrices and CP phases and depend only on two overall coefficients, $k_1$ and $k_2$. Measure two such rates, and the remaining antineutrino channels become fixed numerical ratios, which is a test rather than a fit.

What carries the argument. The load-bearing device is the cancellation in the summed antineutrino coefficient (eq. 37). Symmetric Yukawa couplings force the diagonalization matrices into the special forms $U^C = U K_u$, $D^C = D K_d$, $E^C = E K_e$ with the $K$'s diagonal phase matrices, which makes $V_1 = K_u^*$ and $V_4 = K_d^*$ pure phases. Substituting these into the general coefficients of eqs. 9 to 12 leaves the antineutrino amplitudes proportional to CKM elements and phases only, and those combine into magnitudes that cancel in the rate. A second device is a counting argument: eqs. 13 to 21 give only seven independent relations among all gauge dimension-six coefficients, so a model with seven or more unknown physical parameters cannot be pinned down by nucleon decay data at all.

Core claim. For theories with symmetric Yukawa couplings the mixing matrices cancel out of the summed antineutrino coefficients (eq. 37), so the rates for $p$ and $n$ decaying into a meson plus an antineutrino depend only on $k_1 = g_{GUT}/M(X,Y)$ and $k_2 = g_{GUT}/M(X',Y')$. The decay $p$ to $K^+$ plus antineutrino then determines $k_1$ by itself (eq. 38), and $p$ to $pi^+$ plus antineutrino determines $k_2$ (eq. 43). Everything else in that family is then fixed: $tau(n$ to $pi^0$ antineutrino$)/tau(p$ to $pi^+$ antineutrino$) = 2m_p/m_n$, explicit ratios for $n$ to $K^0$ antineutrino and $n$ to $eta$ antineutrino (eqs. 46 and 48), and a lower bound on $tau(p$ to $K^+$ antineutrino$)/tau(p$ to $pi^+$ antineutrino$) (eq. 45). The author stresses that the charged-antilepton channels do not share this property: they retain dependence on $K_2$ and $V_{DE}$ (eq. 51), so they separate competing models of fermion masses rather than test unification itself. The paper also notes that an earlier claim of model independence for $p$ to $l^+ $K^{0}$$ does not survive once all three generations and the extra CP phases are included.

The premise it rests on. The test assumes that the gauge dimension-six operators are the dominant source of nucleon decay, while the paper sets aside Higgs-sector dimension-six operators as less important and does not estimate their size, nor the size of the dimension-five operators that usually dominate in supersymmetric unified models.

In any unified theory with symmetric Yukawa couplings, a measurement of both $p$ to $K^+$ antineutrino and $p$ to $pi^+$ antineutrino determines $k_1$ and $k_2$ and thereby predicts the neutron antineutrino channels with no free parameters.

The ratio $tau(n$ to $pi^0$ antineutrino$)/tau(p$ to $pi^+$ antineutrino$) must equal $2m_p/m_n$ in these theories, an integer-free number fixed by masses alone.

The inequality of eq. 45 gives a lower bound on $tau(p$ to $K^+$ antineutrino$)/tau(p$ to $pi^+$ antineutrino$); a measured ratio below it would contradict the symmetric-Yukawa scenario.

Charged-antilepton channels such as $p$ to $e^+ K^0$ and $p$ to $mu^+ K^0$ remain mixing-dependent, so they can be used to discriminate among fermion-mass models rather than to test unification.

In SU(5), decays into a charged antineutrino are absent even for a Dirac neutrino (eq. 28), which offers a channel-level way to separate SU(5) from SO(10).

The same cancellation logic should apply to any unified model whose fermion mass matrices are symmetric in flavour space, not only to $10_H$/$126_H$ SO(10); the author's criterion is symmetry of the Yukawa matrices, and the SO(10) case is the worked example rather than the boundary of the argument.

Because the antineutrino ratios are mass-fixed, a future measurement of, say, $n$ to $pi^0$ antineutrino together with $p$ to $pi^+$ antineutrino would be a nearly background-free consistency check: the two-body kinematics differ, but the predicted ratio has no theoretical uncertainty from mixing at all.

The seven-relation counting bound suggests a practical filter for model builders: any proposed GUT whose dimension-six predictions introduce seven or more undetermined parameters is, by this argument, untestable through nucleon decay regardless of how elegant its structure is.

If Higgs-sector dimension-six operators or supersymmetric dimension-five operators contribute at a comparable size, the clean ratios would be shifted by an amount the paper does not estimate, which makes an independent estimate of those contributions the natural next calculation.

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