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REVIEW 3 major objections 4 minor 60 references

A new four-loop QCD calculation gives inclusive semihadronic decay widths for heavy sterile neutrinos, including the previously missing tau-lepton channel, and yields constraints on neutrino mixing from tau decays.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 19:28 UTC pith:7I2FIVGI

load-bearing objection The genuinely new and useful piece is the m_tau≠0 phase-space integrals at O(alpha_s^4); the tau-lifetime constraints are the soft part, mainly because of an unquantified ~40% charm-mass error at m_N=3 GeV and a 1-sigma statistical presentation. the 3 major comments →

arxiv 2512.00476 v2 pith:7I2FIVGI submitted 2025-11-29 hep-ph

QCD corrections to charged-current decays with Heavy Sterile Neutrinos in initial or final state and their impact on τ decays

Tim Kretz , Ulrich Nierste This is my paper
classification hep-ph
keywords heavy sterile neutrinosinclusive semihadronic decaysQCD correctionstau decaysmixing angle constraintsphase-space integralsquark-hadron dualityperturbative QCD
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes that the inclusive semi-hadronic decay width of a heavy sterile neutrino N into a charged lepton and hadrons can be computed to fourth order in the strong coupling, with closed analytic expressions that cover the previously missing tau-lepton channel. The same calculation, crossed to tau decays, provides reliable perturbative predictions for tau→N+hadrons when the sterile neutrino mass is below about 600 MeV. Using these widths, the authors extract bounds on the N–nu_tau mixing angle from the tau lifetime and from tau branching fractions, and they give spectra that could reveal a sterile neutrino in tau decays.

Core claim

The central claim is that Eq. (3.16), built from the O(alpha_s^4) W-boson correlator and the new phase-space integrals I_0, I_1, I_2 plus the series representation for higher I_k, correctly gives the inclusive semi-hadronic charged-current decay width for N→ℓ+hadrons, including the nonzero lepton mass case ℓ=τ. The identical crossing to τ→N+hadrons is perturbatively reliable for m_N ≲ 600 MeV. From the tau lifetime, the paper derives |sinθ| ≤ 0.2 at m_N = 600 MeV (with weaker bounds for lighter N), and combining τ→πν and τ→Kν data yields |sinθ| = (9.1^{+3.7}_{-7.8})×10^{-2} for m_N > m_τ. It also reports that the measured Γ(τ→ℓ+nothing) sits about one sigma above the SM prediction, yielding

What carries the argument

The load-bearing machinery is the vector–axial-vector correlator of two charged quark currents, known to O(alpha_s^4) (five loops), combined with new phase-space integrals I_k(x_ℓ^2,(1-x_ℓ)^2) that appear when one of the final-state leptons is massive. For k≤2 the integrals are analytic in dilogarithms and trilogarithms; for k≥3 the paper gives a convergent series representation. A contour-integral step (quark-hadron duality) replaces the integral over the resonance region by an integral far from resonances, so the inclusive width is obtained even where the differential spectrum is not perturbative.

Load-bearing premise

The calculation assumes quark-hadron duality and the chiral limit: it neglects all quark masses, including charm, whose effects are estimated at 40% for m_N around 3 GeV, the very edge of the claimed reliability window.

What would settle it

A measurement of the inclusive τ→N+hadrons rate or of the hadronic mass spectrum in tau decay with an N of mass near 600 MeV, precise enough to test the O(α_s^4) prediction, would settle the calculation; alternatively, a lattice-QCD computation of the hadronic vacuum polarization at these low q^2 could verify the duality assumption.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The inclusive N→ℓ+hadrons width enables precise predictions of heavy sterile neutrino lifetimes and branching fractions in the mass range from about 0.6 GeV to 3 GeV, a window not covered by current collider searches.
  • For tau decays, the calculation constrains N–ν_τ mixing: |sinθ| ≤ 0.2 for m_N = 600 MeV, and |sinθ| ≈ 0.09 from τ→πν/τ→Kν data when m_N > m_τ.
  • The predicted lepton-energy and hadronic-mass spectra in tau decays with a light sterile neutrino give a concrete search strategy: for m_N = 300 MeV, the spectra are modified at the percent level for sinθ ~ 0.3.
  • The small observed excess in Γ(τ→ℓ+nothing) over the SM translates into an upper bound on decays τ→ℓ X_dark or τ→ℓ X_dark X_dark.
  • The analytic expressions for the integrated spectrum (Appendix D) allow experimentalists to cut at a chosen hadronic invariant mass and still trust the inclusive prediction via duality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same phase-space integral technique could be extended to include charm and bottom quark mass effects as an expansion in m_c^2/q^2, which would remove the 40% uncertainty the authors admit at m_N = 3 GeV and widen the reliable mass window.
  • The constraint on Γ(τ→ℓ+nothing) suggests that a dedicated search for τ→ℓ+missing mass with a lepton-energy spectrum could discriminate between a sterile neutrino and a single dark boson (majoron) by the shape of the missing-mass distribution.
  • If the extracted mixing angle from τ→πν and τ→Kν holds, it predicts an N–ν_τ coupling that should also appear in neutral-current processes, offering a cross-check in future high-intensity tau facilities.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper computes inclusive charged-current semi-hadronic decay widths for heavy sterile neutrinos up to O(alpha_s^4), using the five-loop W-boson correlator of Baikov, Chetyrkin and Kuhn. The new technical element is the treatment of a massive charged lepton in the final state: the phase-space integrals I_0, I_1, I_2 are given in closed form and I_k is represented by a Taylor-type series. The resulting formula is applied to N -> tau + hadrons (claimed reliable for m_N >= 3 GeV) and, by crossing, to tau -> N + hadrons (claimed reliable for m_N <= 600 MeV). These widths are then used to constrain the N-nu_tau mixing angle from the tau lifetime and from tau -> pi nu / tau -> K nu branching ratios, and to make predictions for HSN branching ratios and spectra. The massless limit of the width reproduces the known tau hadronic series of Ref. [10], which is a nontrivial consistency check of the integrals.

Significance. If the claimed reliability windows are correct, the paper supplies genuinely useful ingredients for HSN searches: analytic O(alpha_s^4) widths for N -> tau + hadrons and tau -> N + hadrons, plus quantitative constraints on V_Ntau in the mass range not covered by LHC searches. The massless-limit check in Eq. (4.2) is a real strength and gives confidence that the core phase-space calculation is sound. The paper is also honest in stating several limitations, in particular the omission of quark-mass/charm corrections. However, those limitations affect the central claim that the tau-tagged channel is reliably predicted for m_N just above 3 GeV, so the significance of the N -> tau X application is currently qualified by an unquantified systematic. The tau -> N X and tau-lifetime applications are less affected but share the same low-invariant-mass issue in part. Overall the paper is a solid technical contribution that needs revision before the phenomenological claims are made quantitative.

major comments (3)
  1. [Sec. 4.1, Figs. 5, 8, 9; Sec. 4.2] The claimed reliability window for the tau-tagged channel begins at m_N = 3 GeV, where the hadronic invariant mass is limited to q^2 <= (m_N - m_tau)^2 = 1.5 GeV^2. The authors state in Sec. 4.1 that charm-mass corrections are omitted and estimate an O(m_D^2/q^2) = O(40%) error for m_N = 3 GeV, 'which is not shown' in the plotted bands. Even if this estimate refers primarily to the muon curve beyond charm threshold, for the tau curve the same low-q^2 region makes the OPE and quark-hadron duality assumptions unsettled: the contour radius in the sense of Appendix B is only 1.5 GeV^2, not m_N^2. The scale-stability plots do not probe this error. The statement that the series is 'stable' for m_N >= 3 GeV is therefore insufficient to establish the quoted numerical reliability of Gamma(N -> tau X) at the lower edge of the window, and the branching-ratio predictions in Sec. 4.2 for tau-tagged H
  2. [Eq. (3.13) and Eq. (3.9)] The displayed definition of A_n(x_l^2) is inconsistent with the derivative formula in Eq. (3.9) for n = 1. From Eq. (3.9), d sqrt(lambda)/dx at x=0 equals lambda'(0)/(2 sqrt(lambda(0))) = -1, since lambda'(0) = -2(1-x_l^2) and lambda(0) = (1-x_l^2)^2. The displayed sum in Eq. (3.13) instead gives (+1) for n=1: the sign factor should be (-1)^j, not (-1)^(n-j), after the factor (-2)^(n-2j) is combined with 2^j. Because I_k is computed through the A_n series and I_3 enters the O(alpha_s^4) term in Eq. (3.16), the printed formula is load-bearing for reproducibility. The successful massless check in Eq. (4.2) suggests the authors' actual computation used the correct expression, but the published formula must be corrected.
  3. [Sec. 4.3, Eq. (4.8)] The tau-lifetime constraint for m_N <= 600 MeV uses Gamma_N = Gamma(tau -> N X) computed in the chiral limit. For m_N = 600 MeV the hadronic invariant mass is also bounded by q^2 <= (m_tau - m_N)^2 = 1.5 GeV^2, which is not far above the scales where quark-mass and nonperturbative corrections are potentially sizable. The paper does not provide an error estimate for this quantity, despite the fact that the derived bound on sin(theta) comes from a small difference between the measured and SM tau lifetimes (about 0.6%). A 20-30% shift in Gamma_N changes the extracted mixing angle by a comparable relative amount. The authors should quantify this systematic for the tau -> N X window, or at minimum state clearly that the lifetime bound ignores it.
minor comments (4)
  1. [Abstract vs. Sec. 4.4] The abstract gives |sin theta| = (9.1^{+3.7}_{-7.8}) * 10^{-2}, while Eq. (4.21) and the text quote (9.09 +/- 3.56) * 10^{-2}. Please reconcile these numbers and define how the asymmetric errors in the abstract were derived.
  2. [Sec. 2.1] The name 'Bodarenko' in the text should read 'Bondarenko' (Ref. [7]).
  3. [Eq. (3.9)] The exponent of lambda in Eq. (3.9) appears to be missing a minus sign; for n=1 the formula as printed gives lambda^{+1/2}, whereas the derivative of sqrt(lambda) requires lambda^{-1/2}. Please check the typeset formula.
  4. [Fig. 12] The caption says the 1-sigma and 3-sigma regions are shown and that the theoretical uncertainty is not tripled. Please specify exactly how the theory uncertainty (tau_SM = 288.59 +/- 2.31 fs) is propagated into the allowed band, since this affects the interpretation of the |sin theta| <= 0.2 bound.

Circularity Check

0 steps flagged

No circularity: Eq. (3.16) is a new phase-space convolution of external O(alpha_s^4) correlator coefficients; the tau constraints use external data, and the acknowledged charm-mass caveat is an accuracy limitation, not an input-output equivalence.

full rationale

The derivation chain is self-contained rather than circular. Eq. (3.16) is obtained by inserting the external O(alpha_s^4) charged-current correlator coefficients (Eqs. (2.12)-(2.21), credited to Baikov-Chetyrkin-Kuhn and others) into the phase-space integral Eq. (3.2) and evaluating the new integrals I0-I2 (Eqs. (3.6)-(3.8)) plus the series representation (Eq. (3.13)). No parameter is fitted to the observables later constrained; the m_l=0 limit in Eq. (4.2) is an explicit consistency check against Ref. [10], not a separate prediction. The tau-lifetime and branching-ratio constraints (Secs. 4.3-4.4) use external PDG/HFLAV/FLAG data and do not feed back into Eq. (3.16). The self-citations [9] and [59] are contextual and not load-bearing. The genuine caveats are limitations, not circularity: Sec. 2.2 states "From now on we adopt the chiral limit with zero quark masses" and Sec. 4.1 admits "we did not include charm mass corrections we expect an additional error of O(m_D^2/q^2)=O(40%) for m_N=3 GeV, which is not shown." These affect the precision claimed for the N->tau X window and should be weighed as correctness risk; likewise Appendix B's quark-hadron duality/contour integration is a standard external assumption. I therefore find no input-output equivalence and no fitted-input-called-prediction step.

Axiom & Free-Parameter Ledger

1 free parameters · 7 axioms · 0 invented entities

The central calculation is parameter-free in the strongest sense: given the α_s input from PDG averages and the external five-loop correlator, the new integrals and widths are fully determined analytic results with no fitted constants. The only hand-chosen freedom is the renormalization scale μ, whose variation defines the error bands. All seven axioms are either standard math or honestly stated physics assumptions; the chiral-limit axiom carries the largest quantitative risk. The paper invents no new entities: the HSN is a pre-existing hypothetical (νMSM literature), and the dark-sector final state X_dark is borrowed from the majoron/BSM literature (Refs. [57–59], one self-citation) and used only to reinterpret a bound.

free parameters (1)
  • renormalization scale μ = varied over 0.8–3.5 GeV
    The unphysical scale μ is hand-chosen (μ = m_N for the central results); the quoted width bands σ_Γ = (max Γ − min Γ)/2 over μ ∈ [0.8, 3.5] GeV are the paper's own measure of truncation uncertainty, and the reliability windows (m_N ≥ 1.5/3 GeV, m_N ≤ 600 MeV) are defined by the flatness of this dependence.
axioms (7)
  • domain assumption HSN interacts with SM particles only through N–ν_ℓ mixing, parametrized by a single angle θ with V_Nℓ = sinθ
    Sec. 1 and Sec. 4.3; mixing with ν_e/ν_μ is excluded and deferred to a full electroweak fit (Ref. [40]). Without this, the interpretation of every derived constraint changes.
  • domain assumption Chiral limit for the correlator: all quark masses set to zero; Π^(0) (∝ m_q²) neglected
    Sec. 2.2 ('From now on we adopt the chiral limit with zero quark masses'). The authors state the consequence: O(m_D²/q²) = O(40%) error at m_N = 3 GeV (Sec. 4.1), exactly where the ℓ=τ window begins.
  • domain assumption Quark–hadron duality via Poggio–Quinn–Weinberg smearing / contour integration turns the inclusive quark-level calculation into the hadronic width across resonance regions
    Appendix B; standard τ-physics methodology, but an assumption that fluctuates at the few-percent level for the lightest hadronic invariant masses probed (τ→NX with m_N up to 600 MeV).
  • standard math The O(α_s^4) coefficients c_{n,k} of the W correlator from Refs. [10, 11, 19] are correct
    External literature input transcribed in Eqs. (2.12)–(2.21); not re-derived here. If these were wrong, the central calculation inherits the error.
  • standard math The optical theorem relates the inclusive width to the imaginary part of the two-point function; longitudinal part neglected
    Eqs. (3.1)–(3.2), standard QFT.
  • domain assumption Experimental inputs (τ lifetime, branching fractions, CKM elements, decay constants, α(m_τ)) are correct and their quoted uncertainties are Gaussian and uncorrelated in the combinations used
    Sec. 4.4: the authors 'roughly estimated the errors by assuming they are all uncorrelated and distributed Gaussian'; applied to Eqs. (4.15)–(4.21).
  • domain assumption Flat μ-dependence of the O(α_s^4) series is a valid criterion for perturbative reliability
    Sec. 4.1 uses the μ-behavior to declare the trusted mass windows (m_N ≥ 1.5 GeV for e/μ; ≥ 3 GeV for τ; ≤ 600 MeV for τ→NX). A heuristic; tells nothing about non-perturbative corrections that are μ-independent.

pith-pipeline@v1.3.0-alltime-deepseek · 25565 in / 37962 out tokens · 361912 ms · 2026-08-03T19:28:43.016096+00:00 · methodology

0 comments
read the original abstract

Searches for a Heavy Sterile Neutrino $N$ profit from precise predictions of inclusive decay rates, entering predictions for branching fractions and lifetime. Once decay channels into semi-hadronic final states are open, a reliable calculation of inclusive decay rates is only possible if $N$ is heavy enough to permit a perturbative calculation. We adopt the scenario in which $N$ only interacts with SM particles through $N$-$\nu_\ell$ mixing, where $\ell=e,\mu,\tau$. Using literature results for $W$ boson correlators calculated to $\mathcal{O}(\alpha_s^4)$, we study the quality of the perturbation series for $N\to \ell +\mbox{hadrons}$ to determine mass ranges for which inclusive decay widths can be predicted robustly. We present novel analytic results for the decay rate $N\to \tau +\mbox{hadrons}$ in terms of $m_\tau/m_N$. Our expressions equally apply to $\tau \to N +\mbox{hadrons}$, perturbatively calculable for $m_N\lesssim 600\,$MeV. Applying our result to the $\tau$ lifetime, we determine the allowed parameter space for the $N$-$\nu_\tau$ mixing angle $\theta$ and $m_N$. We find $|\sin\theta| \leq 0.2 $ for $m_N=600\,$MeV and weaker bounds for a lighter $N$. For $m_N\geq m_{\tau}$ we find constraints from the dependence of $\tau$ decay rates on $\cos\theta$. Combining $\tau \to \pi^- \nu_\tau$ and $\tau \to K^- \nu_\tau$ data gives $|\sin\theta| = (9.1^{+3.7}_{-7.8}) \cdot 10^{-2}$ while $N$-$\nu_\tau$ mixing does not improve the agreement between theory and data for $\tau \to \ell \bar\nu_\ell \nu_\tau $. We find current data for the decay rate $\Gamma(\tau \to \ell+\mbox{nothing})$ about 1$\sigma$ above the SM prediction for $\Gamma(\tau \to \ell \bar\nu_\ell \nu_\tau)$, which leads to useful constraints on $\Gamma(\tau \to \ell X_{\mathrm{dark}})$ with dark-sector particles $X_{\mathrm{dark}}$ and might stimulate additional experimental effort on $\tau \to \ell+\mbox{nothing}$.

discussion (0)

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