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REVIEW 2 major objections 5 minor 44 references

Lorentz Breaking and SU(2)_L x U(1)_Y Gauge Invariance for Neutrino Decays

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A fully gauge-invariant electroweak model still lets superluminal neutrinos decay, so the tight astrophysical bounds on neutrino Lorentz violation cannot be dismissed as gauge-breaking artifacts.

desk verdict The central claim holds up: a fully gauge-invariant model can still produce neutrino NPCR/LPCR decays, and the paper's soft spots are presentational, not load-bearing. read the letter →

arxiv 1908.01389 v2 pith:RL3B3V3H submitted 2019-08-04 hep-ph hep-th

U. D. Jentschura , I. Nandori , G. Somogyi This is my paper · ORCID
classification hep-phhep-th PACS 11.30.Cp12.15.-y
keywords Lorentzviolationneutrinodecayelectroweakgaugeinvarianceneutrino-pairCerenkovradiationlepton-pairsuperluminalneutrinosStandard-ModelExtensionflavor-dependent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses a standing objection to the tightest astrophysical bounds on Lorentz violation in the neutrino sector: those bounds come from decay processes—neutrino-pair Cerenkov radiation (NPCR, a neutrino splitting into three neutrinos) and lepton-pair Cerenkov radiation (LPCR, a neutrino emitting an electron-positron pair)—that are usually modelled with kinematically superluminal (faster-than-light) neutrinos but ordinary charged leptons, a combination that seems to break the electroweak gauge symmetry $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ under which neutrinos and left-handed charged leptons form doublets. If the decay models are gauge-breaking, one could try to argue the bounds away. The paper shows they cannot be argued away: it exhibits a model with fully preserved $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge invariance, flavor-dependent Lorentz violation, and nonvanishing NPCR and LPCR decay rates. The key construction is to give the neutrino and its charged partner within each generation the same Lorentz-violating velocity, while letting different generations move at different velocities; in the high-energy limit, where flavor and mass eigenstates coincide, a faster generation decays into a slower one through $Z^0$ exchange. The conclusion is that electroweak gauge invariance does not protect superluminal neutrinos from decay and energy loss, and the astrophysical bounds stand.

What carries the argument

The machinery is a modified Dirac algebra built on a constant pseudo-metric $\tilde g^{\mu\nu}(v) = \mathrm{diag}(1,-v,-v,-v)$: modified gamma matrices $\tilde\gamma^\mu = e^\mu_A \gamma^A$ obey $\{\tilde\gamma^\mu,\tilde\gamma^\nu\} = 2\tilde g^{\mu\nu}(v^2)$ and yield the superluminal dispersion relation $E^2 = \vec p^{\,2} v^2 + m^2$. The load-bearing move for gauge invariance is the choice of generation-dependent but doublet-uniform modified gamma matrices, so that the same velocity $v$ applies to the neutrino and its charged partner inside each $\mathrm{SU}(2)_L$ doublet; this makes the modified vertex matrix proportional to the unit matrix and leaves the gauge group intact. In the high-energy limit the PMNS matrix unitarizes ($U_{k\ell} \to \delta_{k\ell}$), so flavor and mass eigenstates coincide, and the $Z^0$-mediated four-fermion vertices use the effective velocity $v_{\mathrm{int}} = v_i v_f = v^{(f)}_i v^{(f)}_f$. The decay-rate formulas—for instance $\Gamma_{\nu_i \to \nu_i e^- e^+} = a_{\mathrm{GI}}\, G_F^2 k_1^5 / (192\pi^3)$ with $a_{\mathrm{GI}} \propto (\delta_i - \delta_f)[(\delta_i - \delta_f)^2 + (7/17)(\delta_i + \delta_f)^2]$—carry the argument to its physical conclusion: the rates are nonvanishing whenever the initial generation is faster than the final one, and the earlier bounds are reproduced.

What would settle it

Recompute the NPCR and LPCR rates in the fully gauge-invariant model while keeping the full, non-diagonal PMNS matrix at high energies instead of setting $U_{k\ell} \to \delta_{k\ell}$: if the rates vanish or change sign once flavor mixing is retained at the highest neutrino energies, the decay picture behind the astrophysical bounds would not hold in the regime where the bounds are actually set. A separate check is to bound the charged-lepton Lorentz-violating parameters, since each charged lepton must share its generation's velocity; a bound on muon or tau lepton Lorentz violation tighter than the neutrino bounds would squeeze the required hierarchy $v_f < v_i$ and could eliminate the decays.

Watch

Extended reading notes

Core claim

The paper establishes that $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge invariance does not preclude the neutrino decay and energy-loss processes (NPCR and LPCR) that underlie the tight astrophysical bounds on neutrino Lorentz violation. Concretely, it constructs a three-generation model in which the modified Dirac matrices—and hence the maximum attainable velocities—are uniform inside each electroweak doublet but differ between generations ($\tilde\gamma^\rho_{\nu_e} = \tilde\gamma^\rho_e \neq \tilde\gamma^\rho_{\nu_\mu} = \tilde\gamma^\rho_\mu \neq \tilde\gamma^\rho_{\nu_\tau} = \tilde\gamma^\rho_\tau$). Because the modified gamma matrix is proportional to the unit matrix within each doublet, full $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ gauge invariance is preserved. In the high-energy limit (Eq. (45)), flavor eigenstates approximate mass eigenstates and the PMNS matrix (the neutrino flavor-mixing matrix) effectively becomes the identity, so a faster generation decays into a slower one, with the effective interaction parameter $v_{\mathrm{int}} = v_i v_f$. The paper computes the decay and energy-loss rates (Eqs. (61)–(66)) in this gauge-invariant setting and finds the same parametric dependence—positive powers of $(\delta_i - \delta_f)$—as in the kinematics-only approaches, so the quantitative conclusions of the earlier bounds are unchanged.

Load-bearing premise

The analysis assumes that at the energies of the astrophysical neutrinos, flavor and mass eigenstates coincide so the PMNS matrix effectively becomes the identity ($U_{k\ell} \to \delta_{k\ell}$), and that within each generation the neutrino and its charged partner share the same Lorentz-violating velocity; if either condition fails, the decay rates and the bounds built on them would need revision.

Editorial extensions

If this is right

  • The astrophysical bounds on neutrino Lorentz violation survive the gauge-invariance objection: the decay and energy-loss rates behind those bounds exist in a fully gauge-invariant model, so the limits cannot be argued away on the grounds that the decay models break electroweak symmetry.
  • In the gauge-invariant model each charged lepton carries the same Lorentz-violating velocity as its same-generation neutrino, so measurements that bound charged-lepton Lorentz violation directly constrain the corresponding neutrino parameters.
  • The gauge parameter $\xi$ of the $R_\xi$ gauge (a standard covariant gauge for the massive $Z^0$ propagator) drops out of the physical decay rate, making the computed NPCR and LPCR rates unambiguous observables rather than gauge artifacts.
  • The decay picture requires a velocity hierarchy across generations ($v_f < v_i$), predicting that high-energy neutrinos of faster generations convert into fermion pairs of slower generations, a pattern that could be probed through the flavor and energy dependence of the highest-energy neutrino events.
  • The transition from the low-energy (mass-basis) to the high-energy (flavor-equals-mass) regime occurs at the momentum scale $\sqrt{\delta m^2/\delta f}$, estimated near $10^8$–$10^9$ eV for $\delta f \sim 10^{-20}$ and $\delta m^2 \sim 10^{-3}$ eV$^2$, which defines where the simple decay rates apply.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because independent measurements already force electrons very close to luminal propagation, the generation-locked construction effectively pins electron neutrinos to the same near-luminal velocity; the decays that set the neutrino bounds would then need to be driven by muon or tau neutrinos being faster, a flavor pattern that could be tested in the flavor composition of ultra-high-energy neutrino
  • The same doublet-uniform, generation-differential construction applies to the quark sector, giving a template for gauge-invariant Lorentz violation in quarks in which each quark's Lorentz-violating parameter is tied to its same-generation partner.
  • The high-energy limit $U_{k\ell} \to \delta_{k\ell}$ is an approximation whose validity is set by the transition scale $\sqrt{\delta m^2/\delta f}$; whether the astrophysical bounds are fully robust depends on that scale lying below the energies of the observed neutrino events, which a calculation retaining full flavor mixing could confirm or overturn.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper examines whether the tight astrophysical bounds on neutrino Lorentz violation, derived from neutrino-pair Cerenkov radiation (NPCR) and lepton-pair Cerenkov radiation (LPCR), necessarily rely on models that break electroweak gauge invariance. The authors analyze three constructions: (i) the Cohen-Glashow / Bezrukov-Lee model I, which is shown to break SU(2)_L x U(1)_Y; (ii) a one-generation model in which the neutrino and charged lepton inside the same SU(2)_L doublet carry different modified Dirac matrices, preserving only a restricted U(1)_L x U(1)_Y subgroup and leading to the effective velocity replacement v_int = v_i v_f; and (iii) a three-generation model in which each generation has a common Lorentz-violating velocity for the neutrino and charged lepton, so that the modified gamma matrices are proportional to the identity in doublet space and full SU(2)_L x U(1)_Y gauge invariance is preserved. In the third model, a faster generation can decay into a slower one through NPCR and LPCR. The paper computes the decay and energy-loss rates in the R_xi gauge, asserts that the gauge parameter cancels, and concludes that electroweak gauge invariance does not protect superluminal neutrinos from these decays, so the astrophysical bounds cannot be dismissed on gauge-invariance grounds.

Significance. If the central claim holds, the paper closes a potential loophole: the tight bounds on neutrino Lorentz violation from NPCR and LPCR would not be invalidated merely because earlier effective models broke electroweak gauge invariance. The explicit construction of a fully SU(2)_L x U(1)_Y invariant model with flavor-dependent Lorentz-violating parameters is a nontrivial and useful result, and the identification of the effective velocity parameter v_int = v_i v_f clarifies the relation between earlier phenomenological models and gauge-invariant formulations. The paper is theoretical: no data are fitted, and the decay rates depend on the assumed Lorentz-violating parameters as inputs. The algebraic structure of the modified Dirac matrices and the gauge-invariant decay formulas are internally consistent, but two points need attention: the incorrect wording about the PMNS matrix in the high-energy limit, and the lack of a detailed demonstration of the R_xi gauge-parameter cancellation. These are fixable and do not, at this stage, invalidate the main conclusion.

major comments (2)
  1. [Sec. 3.3, Eq. (45)] The statement that 'the PMNS matrix approaches the unit matrix, U_kℓ → δ_kℓ (high-energy limit)' is incorrect as written: the PMNS matrix is a fixed, energy-independent unitary matrix and cannot approach the identity. What appears to be intended is that, in the high-energy limit, the mass term is negligible compared with the Lorentz-violating term, so that the effective Hamiltonian in the flavor basis is approximately diagonal and the flavor eigenstates approximate the propagation (mass) eigenstates. Because Eq. (45) is used to justify the subsequent high-energy formulation in Eqs. (49) and (51), this statement should be rephrased carefully and the distinction between the fixed PMNS matrix and the effective propagation basis should be clarified.
  2. [Sec. 4, around Eq. (59)] The cancellation of the R_xi gauge parameter is asserted ('an explicit calculation shows', 'we have explicitly checked') but the calculation is not shown. Since the gauge-parameter independence of the squared matrix element is a central ingredient in the claim that the fully gauge-invariant model yields consistent decay rates, the authors should present the explicit cancellation, or at least provide a complete and accessible derivation (or a detailed reference) showing that the (xi-1) part of the Z-boson propagator in Eq. (59) vanishes when contracted with the modified currents of on-shell massless spinors satisfying the generalized Dirac equation. Without this, the gauge-invariance claim is not fully verifiable from the manuscript.
minor comments (5)
  1. [Abstract] There is a missing space in 'gauge invarianceThe first of these' in the abstract; this should be corrected.
  2. [Eq. (56)] The subscript notation in this effective Lagrangian is inconsistent: the first current uses γµ_j for the neutrino, while the second current uses γν_j for the Ψ field, which obscures which modified gamma matrix acts on which fermion. The intended structure is likely γµ_i for the initial neutrino and γν_f for the final-state fermions. In addition, the equality between the contraction with g_µν and that with ~g_µν(v_j v_k) is not immediate and should be either derived explicitly or stated as a shorthand with a reference to the derivation.
  3. [Eq. (61)] The derivation of the decay-rate formula is compressed; a few intermediate steps showing how the phase-space integration and the spin sums of Eq. (60) lead to the polynomial in δ_i and δ_f would improve readability and verifiability.
  4. [Sec. 3.3, after Eq. (51)] There is a typo: 'it fully preserves preserves gauge invariance' should read 'it fully preserves gauge invariance'.
  5. [Eq. (52)] It would be helpful to state explicitly the sign convention for the δ parameters so that the transition momentum |p| = √(δm²/δf1f2) is manifestly real, and to clarify that δf1f2 denotes the difference of the delta parameters of two flavors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: decay rates are conditional formulas in assumed LV parameters, and the gauge-invariant construction is self-contained.

full rationale

No circular step is present. The paper never fits a parameter and then re-presents that fit as a prediction: the Lorentz-violating velocities δ_i and δ_f are inputs throughout, and the decay rates of Sec. 4 are explicitly functions of these inputs (e.g., Eqs. 61 and 64), not extractions of them. The central gauge-invariance construction in Sec. 3.3 is self-contained: Eqs. (35)-(37) define generation-dependent modified gamma matrices that are proportional to the unit matrix within each SU(2)_L doublet, so full gauge invariance follows directly from commutativity with the gauge generators rather than from an imported uniqueness theorem. The replacement v_int = v_i v_f (Eqs. 34 and 51) is a product of modified vertex factors, and the ξ-independence of the squared matrix element is attributed to the on-shell spin-sum identity stated explicitly in Eq. (60), which is parameter-free and checkable from the modified Dirac algebra; citing the authors' earlier Ref. [3] for this identity is technical support, not a load-bearing circular citation. The paper does contain presentational weaknesses that should not be mistaken for circularity: Eq. (45) states that the PMNS matrix 'approaches the unit matrix' in the high-energy limit, which is imprecise because the PMNS matrix is constant and what becomes negligible is mass-induced mixing; Eq. (56) has a subscript typo (the second ~γ should carry the final-state index k for the equality to ~g(v_i v_k) to hold); and the decay-rate formulas of Eqs. (61)-(66) are summarized without the full intermediate algebra. These items affect clarity and rigor, but none of them makes a claimed result equivalent to its own input by construction. The final conclusion is best read as conditional: if flavor-dependent Lorentz violation with v_f < v_i is assumed, then SU(2)_L × U(1)_Y gauge invariance alone does not forbid NPCR and LPCR, and the paper exhibits an explicit gauge-invariant model in which those decays proceed.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central model rests on the standard electroweak gauge structure plus modeling assumptions specific to Lorentz violation: a constant isotropic pseudo-metric, equal LV velocity for neutrino and charged lepton within a generation, and the high-energy flavor-equals-mass approximation. The LV velocities are free inputs, not derived. No new particles or entities are introduced.

free parameters (3)
  • v_e (first-generation LV velocity, delta_e = v_e^2 - 1)
    Free input describing the common maximal velocity of the electron and electron neutrino in the gauge-invariant model; constrained externally, not fitted here.
  • v_mu (second-generation LV velocity, delta_mu = v_mu^2 - 1)
    Free input describing the common maximal velocity of the muon and muon neutrino; enters the cross-generational decay rates.
  • v_tau (third-generation LV velocity, delta_tau = v_tau^2 - 1)
    Free input describing the common maximal velocity of the tau and tau neutrino; enters the cross-generational decay rates.
assumptions (5)
  • domain assumption Standard Model electroweak gauge group SU(2)_L x U(1)_Y with left-handed lepton doublets and the usual covariant derivative.
    Invoked throughout Sec 3 and Sec 4 as the framework for gauge invariance; the paper modifies the Dirac algebra but not the gauge structure.
  • domain assumption Lorentz violation is described by a constant, spatially isotropic pseudo-metric tilde_g^{mu nu}(v) = diag(1,-v,-v,-v) in the fermion kinetic term, with modified Dirac matrices satisfying an anticommutator relation.
    Eqs (3)-(6) and (53); this is the effective-field-theory ansatz adopted from the SME literature rather than derived in the paper.
  • ad hoc to paper In the fully gauge-invariant three-generation model, the neutrino and charged lepton within a generation share the same Lorentz-violating velocity, e.g., tilde_gamma^mu_nu_e = tilde_gamma^mu_e.
    Eqs (35)-(36) and (46)-(48); this equality is required for the modified gamma matrix to be proportional to the identity inside the SU(2)_L doublet, preserving full gauge invariance. It is a modeling assumption that forces charged leptons to share the LV parameters.
  • domain assumption In the high-energy limit, flavor eigenstates approximately coincide with mass eigenstates and the PMNS matrix effectively becomes the identity.
    Eq (45) in Sec 3.3; the cross-generational decay picture and the replacement v_int = v_i v_f rely on this approximation. It is stated as a limit, not derived.
  • domain assumption Lepton masses and Yukawa couplings are negligible in the decay-rate calculation, so the neutral currents are conserved and the R_xi gauge-parameter terms cancel.
    Sec 4, paragraph after Eq (59); the massless limit is used to set c_V and c_A and to justify the single-Z-diagram gauge invariance.

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Pith. "Pith review of Lorentz Breaking and SU(2)_L x U(1)_Y Gauge Invariance for Neutrino Decays." pith.science (2026). https://pith.science/paper/RL3B3V3H

@misc{pith2026190801389,
  author       = {Pith},
  title        = {Pith review of: Lorentz Breaking and SU(2)_L x U(1)_Y Gauge Invariance for Neutrino Decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RL3B3V3H}},
  note         = {Machine review of arXiv:1908.01389}
}
read the original abstract

Conceivable Lorentz-violating effects in the neutrino sector remain a research area of great general interest, as they touch upon the very foundations on which the Standard Model and our general understanding of fundamental interactions is laid. Here, we investigate the relation of Lorentz violation in the neutrino sector in light of the fact that neutrinos and corresponding left-handed charged leptons form SU(2)_L doublets under the electroweak gauge group. Lorentz-violating effects thus cannot be fully separated from questions related to gauge invariance. The model dependence of the effective interaction Lagrangians used in various recent investigations is investigated, with a special emphasis on neutrino splitting, otherwise known as neutrino-pair Cerenkov radiation, NPCR, and vacuum pair emission (electron-positron-pair Cerenkov radiation, LPCR). We investigate two scenarios in which Lorentz violating effects do not necessarily also break electroweak gauge invarianceThe first of these involves a restricted set of gauge transformation, a subgroup of SU(2)_L x U(1)_Y, while in the second, differential Lorentz violation is exclusively introduced by the mixing of the neutrino flavor and mass eigenstates. Our study culminates in a model which fully preserves SU(2)_L x U(1)_Y gauge invariance, involves flavor-dependent Lorentz-breaking parameters, and still allows for NPCR and LPCR decays to proceed.

Figures

Figures reproduced from arXiv: 1908.01389 by the authors.

Figure 1
Figure 1. Feynman diagrams for the LPCR [Fig. (a)] and NPCR [Fig. (b) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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