$\nu\bar\nu$ production, annihilation, and scattering at MeV temperatures and NLO accuracy

G. Jackson; M. Laine

REVIEW 3 major objections 5 minor 15 references

$\nu\bar\nu$ production, annihilation, and scattering at MeV temperatures and NLO accuracy

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Neutrino production, annihilation, and scattering rates in a hot QED plasma can be computed at NLO in QED, shifting N_eff only at the fourth decimal.

desk verdict Strong double-differential NLO rate computation with a transparent but real caveat on the N_eff estimate. read the letter →

arxiv 2412.03958 v2 pith:RDDQ2D6D submitted 2024-12-05 hep-ph astro-ph.CO

G. Jackson , M. Laine This is my paper · ORCID

classification hep-phastro-ph.CO PACS 12.20.-m95.30.Cq98.80.-k

keywords neutrinodecouplingeffectivenumberofspeciesQEDplasmanext-to-leadingorderdouble-differentialratesthermalfieldtheoryenergytransferBoltzmannequations

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

This paper shows that the double-differential rates for neutrino–antineutrino pair production, annihilation, and scattering in a QED plasma at MeV temperatures can be defined in full thermal field theory and evaluated at next-to-leading order in QED. These rates are the momentum-resolved building blocks behind the collision terms that determine how neutrinos decouple in the early universe, and hence they feed the Standard Model prediction for the effective neutrino number $N_{\rm eff}$. In the massless-electron limit the NLO corrections are about one percent, shifting $N_{\rm eff}$ from $3.04859$ at leading order to $3.04867$ at NLO. The electron mass, treated at leading order, is a larger effect: it lowers $N_{\rm eff}$ to $3.04510$. The paper also tabulates all double-differential rates with a fast interpolation routine so they can be used in non-averaged kinetic equations.

What carries the argument

The object that carries the argument is the function $F(k;p_0,p)$ in eqs. (3.25), (3.26) and (3.29). It packages the dynamical information—transverse and longitudinal photon spectral functions $\rho_T,\rho_L$ and their NLO corrections, vertex corrections $\chi_{T,L}$, and HTL-resummed photon propagators $R^*_{T,L}$—together with flavour and coupling factors, and it determines all three rates because the neutrino momentum $k$ enters only as a kinematic weight. The paper factorizes $F$ into coefficients $A_i,B_i,C_i,D_i$ that are precomputed on a wedge-shaped grid in the $(p_+,p_-)$ plane and linearly interpolated, making the otherwise costly two-loop spectral integrals available inside a kinetic-equation solver. The Wightman-correlator definition enforces detailed balance between production and annihilation, and reduces to the known NLO neutrino interaction rate in the literature when integrated with the appropriate measure.

What would settle it

A direct solution of the full quantum kinetic equation for neutrinos at $T_\gamma\sim1\text{--}3\,{\rm MeV}$, using the same NLO spectral data but without the factorized Pauli-blocking approximation of eqs. (4.2)–(4.4), would settle the central claim: if the resulting $N_{\rm eff}$ differs from $3.04867$ already in the third decimal, the factorization assumption fails at NLO.

Watch

Extended reading notes

Core claim

The central claim is that the integrand of a previously computed NLO neutrino interaction rate is itself a physical quantity: the double-differential production rate $\Psi(\mathbf{k}_\nu,\mathbf{q}_{\bar\nu})$, the annihilation rate $\tilde{\Psi}$, and the scattering rate $\Theta(\mathbf{q}_\nu\to\mathbf{k}_\nu)$ can be defined directly from Wightman correlators of the weak current and computed to $O(\alpha_{\rm em})$ for $T_\gamma \ge m_e$. Weighted integrals of these rates reproduce the energy transfer rates used in averaged kinetic equations, and in the massless limit the NLO corrections to those transfer rates are of order $\sim 1\%$. Inserting the NLO rates into the averaged decoupling equations gives $N_{\rm eff}=3.04867$, against $3.04859$ at full LO and $3.04858$ in the improved Maxwell–Boltzmann approximation; the finite electron mass at LO gives $3.04510$, a third-decimal effect. The full momentum-dependent rates are tabulated so that future non-averaged kinetic equations can carry the same NLO accuracy.

Load-bearing premise

The load-bearing premise is that NLO rates computed from equilibrium Wightman correlators can be inserted into Boltzmann equations as simple gain and loss terms multiplied by Pauli blocking factors, even though the paper states that Boltzmann equations themselves cannot be directly extended to NLO.

Editorial extensions

If this is right

A full NLO computation of $N_{\rm eff}$ from momentum-dependent kinetic equations is now in principle possible, since the rates that parametrize those equations have been evaluated at NLO without kinematic approximations.
In the massless-electron limit, NLO QED corrections change $N_{\rm eff}$ only in the fourth decimal, from $3.04859$ to $3.04867$, placing the third decimal beyond the reach of this particular correction.
A finite electron mass moves the third decimal at leading order, $N_{\rm eff}=3.04510$, so the mass dependence is a larger systematic than the NLO QED correction.
The released tabulation reproduces the full leading-order energy transfer rates to better than $0.05\%$ and can also reconstruct the previously computed NLO neutrino interaction rate, giving independent cross-checks.
NLO results agree with one earlier partial estimate and disagree with another on the size of the shift, with the complete set of diagrams keeping the NLO effect small.

Reading between the lines

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

Because the paper itself notes that Boltzmann equations cannot be directly extended to NLO, the largest remaining uncertainty may be the factorization of NLO rates into gain and loss terms with simple Pauli blocking; the tabulated rates are exactly what would be needed to test this in a full quantum kinetic equation.
The same Wightman-correlator construction should apply to other light weakly-interacting species coupled to a thermal electromagnetic current, provided the relevant current correlators and spectral decompositions are known.
The electron-mass effect at LO exceeds the NLO QED shift, so a finite-mass NLO computation is the natural next step; until it is done, the third decimal of the Standard Model $N_{\rm eff}$ prediction rests on a leading-order mass treatment.
The single-temperature average used for the paper's $N_{\rm eff}$ numbers may wash out momentum-dependent NLO effects; feeding the tabulated rates into a momentum-resolved solver is a direct way to check whether the fourth-decimal conclusion survives.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the double-differential rates are the real result; the N_eff shift is a conditional estimate, and the paper mostly says so itself. The central QFT derivation is in good shape. Jackson and Laine show that the integrand of the previously computed neutrino interaction rate is a physical, gauge-invariant double-differential rate, and they compute production, annihilation, and scattering rates at NLO in QED for massless electrons. They verify the construction by recovering the known interaction rate of ref. [6], and they ship a tabulation and interpolation routine. That is a solid, reproducible piece of work, and it clears up the earlier confusion between the partial NLO estimates in refs. [5] and [7].

The soft spots are real but mostly located where the authors point. The NLO calculation is done for me=0; the LO electron-mass effect on N_eff is about ten times larger than the NLO shift, so the quoted 3.04867 is not a complete SM prediction. More importantly, the N_eff numbers come from inserting the quantum-mechanical rates into the averaged Boltzmann equations with the usual (1-f)(1-f) and f f factors, while the paper itself notes in Sec. 2 that Boltzmann equations cannot be directly extended to NLO. That step is an assumption, not a derivation from the full quantum kinetic equation. The N_eff shift should therefore be read as an estimate within the averaged single-temperature scheme, not as a controlled NLO prediction. The paper is transparent about this, so I would not call it a fatal flaw, but it could be stated more prominently in the abstract or conclusions.

The tabulation is a strong asset: the authors make the full momentum-dependent rates available, which is exactly what full kinetic-equation solvers need. The reproduction of the ref. [6] interaction rate and the agreement with ref. [7] on the sign and rough size of the NLO correction are good sanity checks.

Who is this for: anyone working on precision N_eff, neutrino decoupling, or thermal field theory rate computations. It deserves a serious referee. The main referee asks should be whether the Wightman-function definition of the rates is fully rigorous with respect to the Boltzmann-factorization step, and whether the published version makes the approximate nature of the N_eff estimate unmistakable. I would engage with it.

Referee Report

3 major / 5 minor

Summary. This paper defines double-differential production, annihilation, and scattering rates for neutrinos and antineutrinos in a hot QED plasma, deriving them from thermal Wightman functions rather than from a perturbative expansion of Boltzmann equations. The central results are the NLO expressions for the double-differential rates in Eqs. (3.25), (3.26), and (3.29), the energy transfer rates built from them in Eqs. (4.2)-(4.4), and a tabulation/interpolation package for the momentum-dependent rates. The NLO spectral functions are taken from previous two-loop computations, and the integrated neutrino interaction rate is shown to reproduce the earlier result of Ref. [6]. In the massless-electron limit the NLO corrections to energy transfer rates are about one percent, and when inserted into the averaged kinetic scheme of Refs. [8,9] the quoted N_eff changes from 3.04859 (LO) to 3.04867 (NLO). A LO treatment of finite electron mass, in Appendix B, changes N_eff to 3.04510.

Significance. If the double-differential NLO rates are correct, the paper provides a valuable and more complete object than the single integrated interaction rate of Ref. [6]: it supplies the momentum-resolved integrands needed for non-averaged kinetic equations, and it establishes a direct connection between the earlier rate computation and the energy transfer rates used in averaged codes. The cross-check against Ref. [6], the use of previously published NLO spectral functions from Refs. [12,13], and the public tabulation and interpolation routine are concrete strengths. The quoted N_eff shift, however, should be read as an estimate within a specific averaged kinetic scheme, not as a complete NLO Standard Model prediction; the paper itself says in Sec. 2 that Boltzmann equations cannot be directly extended to NLO, while Eqs. (4.2)-(4.4) adopt exactly the LO collision-term factorization.

major comments (3)

[Sec. 2 and Eqs. (4.2)-(4.4)] The paper states in Sec. 2 that "Boltzmann equations themselves cannot be directly extended to the NLO level", yet Eqs. (4.2)-(4.4) insert the NLO Wightman-based rates into the leading-order gain-loss collision terms with simple Pauli blocking factors. The step from the quantum-mechanical rate, computed from an initial state with no neutrino population, to a Boltzmann collision term with factorized (1-f)(1-f) or f f dependence is an additional assumption at NLO, not a derived consequence of the QKE. This makes the N_eff numbers in Sec. 6 model-dependent estimates within the averaged kinetic scheme rather than a complete NLO prediction. The central rate computation is not undermined, but the N_eff claim needs to be explicitly labeled as such, or a derivation/justification of the factorization at NLO must be provided.
[Sec. 4.4 and Sec. 6] The NLO rates are computed only for massless electrons, while the paper itself shows that the LO electron-mass correction changes N_eff in the third decimal (3.04859 -> 3.04510, Sec. 6). The quoted NLO-to-LO shift (3.04859 -> 3.04867) is therefore a massless-electron result, and the conclusion that "NLO effects influence the fourth decimal" is not a complete Standard Model statement once O(me/T) effects are admitted to be larger at LO. This should be stated more prominently in the abstract and conclusions, since otherwise readers may take the N_eff value as the full NLO SM result.
[Eq. (4.27) and Sec. 5] The finite-temperature domain is restricted to T_gamma >= m_e, and the NLO expressions rely on the factorization of the hadronic uncertainty through the coefficient C_a. The uncertainty estimate in Eq. (4.28) is given as fixed percentages, but the overall reliability of the NLO interpolation is tested only against the integrated Q_LO and against the neutrino interaction rate of Ref. [6]. It would be useful to state explicitly whether the <0.05% accuracy quoted in Sec. 5 applies to all integrated quantities over the full T_nu/T_gamma range used in the N_eff estimate, or only to the specific test cases shown.

minor comments (5)

[Footnote 2, Sec. 3.2] The footnote says the electron mass could have been kept non-zero, but the paper then proceeds with massless electrons for all NLO results; a brief statement near Eq. (3.13) clarifying that Appendix B only covers the LO mass dependence would help avoid confusion.
[Fig. 2] The loss-term curves labelled "MB-LO" are multiplied by 10^{-1} as stated in the caption, but the corresponding legend entry does not show this rescaling; please make the rescaling explicit in the legend or in the axis annotation.
[Tables 1 and 2] The mX/pX notation is explained, but the tables would be easier to read if the power-of-ten exponent were separated from the mantissa, or if the entries were written in scientific notation consistently.
[Sec. 5] The interpolation grid is described for the first quadrant, and the text says the shaded region is obtained by reflecting p0 -> -p0; it would be useful to state explicitly that the tabulated data therefore covers all domains of Fig. 3 used in the integrations.
[Sec. 6] The sentence "we find N_eff = 3.04858 with the improved Maxwell-Boltzmann approximation" uses a different number of digits than the later values; please harmonize the significant digits in the quoted N_eff values.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the NLO rates are derived from thermal-field-theory correlators with independent spectral-function inputs, and the N_eff application is an acknowledged model-dependent use of a Boltzmann ansatz, not a fitted prediction.

full rationale

The central derivation is self-contained against first-principles inputs. Equations (3.12), (3.18), and (3.25)-(3.29) define the production, annihilation, and scattering rates directly from transition matrix elements and thermal Wightman correlators; no target rate or value of N_eff is used as an input. The NLO spectral functions δρ^{NLO} are taken from refs. [12,13], which are a published calculation and an accompanying numerical code, and the matching coefficient C_a in eq. (4.21) is taken from ref. [6] with an explicit hadronic uncertainty. These are stated inputs, not quantities fitted to the paper's own output. The cross-check in sec. 3.5, where integrating Ψ and Θ reproduces the neutrino interaction rate of ref. [6], is a validation of the new double-differential rates rather than a load-bearing derivation step: the numerical value of that interaction rate is never used to define Ψ or Θ. Similarly, the energy-transfer rates in sec. 4 are literal moments of the derived double-differential rates, and the N_eff numbers in sec. 6 are obtained by inserting those coefficients into the externally developed code of refs. [8,9]. The only calibrated quantities, the factors f^{FD}_a and f^{FD}_s in eq. (4.17), are diagnostics for an improved Maxwell-Boltzmann approximation benchmarked against full leading order; they are not used to produce the full NLO rates or the quoted NLO N_eff shift. The paper explicitly acknowledges that Boltzmann equations themselves cannot be directly extended to the NLO level, and that the rates are used within averaged kinetic equations; this is a physical approximation and a limitation, but not a circular reduction. The self-citations present, notably refs. [12,13,15] sharing an author, are backed by code-reproduced and published results, so they do not form a circularity chain. No step has been found in which a prediction reduces by construction to its own input, a fitted parameter is renamed as a prediction, or a load-bearing premise depends solely on an unverified self-citation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central NLO computation uses standard thermal field theory and prior two-loop spectral functions as external input. The only hand-fitted numbers are the Maxwell-Boltzmann correction factors, which do not affect the full LO or NLO results. No new particles, forces, or entities are introduced.

free parameters (2)

f_FD^a = 0.884
Fitted in eq. (4.17) so that the Maxwell-Boltzmann annihilation rate matches the full LO slope as T_nu approaches T_gamma. Used only in the MB approximation, not in the full LO or NLO central results.
f_FD^s = 0.829
Fitted in eq. (4.17) so that the Maxwell-Boltzmann scattering rate matches the full LO slope as T_nu approaches T_gamma. Used only in the MB approximation, not in the full LO or NLO central results.

assumptions (5)

domain assumption Fermi effective theory with the Fierz-transformed operator of eq. (3.1), including the matching coefficient C_a, describes neutrino-QED plasma interactions at NLO.
The derivation begins from this operator and relies on the NLO matching coefficient from refs. [6,14]. The hadronic uncertainty in C_a enters the final rates.
standard math The Wightman correlator is related to the spectral function by the standard thermal relation Pi^< = 2 f_B(p0) rho(p0).
Invoked in eqs. (3.12) and (3.15). This is a textbook Kubo-Martin-Schwinger relation, not derived in the paper.
domain assumption Only soft t-channel photons need HTL resummation; electrons are hard with momenta ~pi T and use tree-level propagators.
Stated in Sec. 3.1. This is the standard hard-thermal-loop power counting and is needed for the NLO calculation to be IR finite.
domain assumption The NLO spectral functions delta_rho_NLO^T,L from refs. [12,13] are correct and applicable.
The paper does not rederive the two-loop thermal spectral functions; it imports them via eq. (4.20). These are the dynamical input for the NLO energy transfer rates.
domain assumption The Boltzmann collision term factorizes into a rate multiplied by Pauli blocking factors at NLO.
The paper motivates this from LO Boltzmann equations in Sec. 2, but explicitly notes Boltzmann equations cannot be directly extended to NLO. The use of eqs. (4.2) to (4.4) relies on this factorization.

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Cite this review

Pith. "Pith review of $\nu\bar\nu$ production, annihilation, and scattering at MeV temperatures and NLO accuracy." pith.science (2026). https://pith.science/paper/RDDQ2D6D

@misc{pith2026241203958,
  author       = {Pith},
  title        = {Pith review of: $\nu\bar\nu$ production, annihilation, and scattering at MeV temperatures and NLO accuracy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDDQ2D6D}},
  note         = {Machine review of arXiv:2412.03958}
}

abstract

Interaction rates of neutrinos and antineutrinos within a QED plasma determine the dynamics of their decoupling in the early universe. We show how to define the relevant double-differential production, annihilation, and scattering rates at NLO. Integrating over these rates with specific weights, other quantities from the literature can be obtained, such as energy transfer rates, or a neutrino interaction rate. In the limit of massless electrons, we show that NLO corrections to the energy transfer rates are as small as those that enter the previously determined neutrino interaction rate, and only have a small influence on the neutrino decoupling parameter, $N_{\rm eff}\,$. For comparison, the influence of a finite electron mass is quantified at LO. Finally we provide a tabulation and fast interpolation routine for all double-differential rates, in order to allow for their use in non-approximate kinetic equations, which may further reduce the systematic uncertainties of the Standard Model prediction for $N_{\rm eff}\,$.

Figures

Figures reproduced from arXiv: 2412.03958 by the authors.

**Figure 1.** Percentual accuracy of the improved Maxwell-Boltzmann approximation from eq. (4.17) (MB), compared with the full LO evaluation based on eqs. (4.5)–(4.9) (LO), on the energy transfer rates from eqs. (4.2)–(4.4). The results are similar to those found in ref. [11]. where the numerically determined factors f FD a = 0.884 (“annihilations”) and f FD s = 0.829 (“scatterings”) guarantee that QMB gain − QMB loss and QMB sca… view at source ↗

**Figure 2.** Percentual influence of various effects on the energy transfer rates from eqs. (4.2)–(4.4). “MB” refers to the improved Maxwell-Boltzmann approximation from eq. (4.17), “LO” to the full LO evaluation based on eqs. (4.5)–(4.9), and “NLO” to the full NLO result originating from eqs. (3.25) and (3.29). The arrows show the uncertainly from low-energy hadronic input, as specified in eq. (4.28). For illustration we have s… view at source ↗

**Figure 3.** The domains of four-momentum transfer that are relevant for a given neutrino spatial momentum k, as found in eqs. (3.31) and (3.32). The spectral functions are antisymmetric in p0 → −p0 , so the full information is already contained in the first quadrant p0 > 0. The idea is then to interpolate, from the tabulated values, the p0 and p dependence of the integrand for the quantity of interest (for example, the energy d… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Grid for tabulation (circles and squares) and interpolation schemes. At the blue circles, we record the coefficients in eq. (4.26). The red squares “pad” the boundary, and are useful because the factor in eq. (5.1) results in the integrand being zero on these points (e…

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Jackson and M

G. Jackson and M. Laine, QED corrections to the thermal neutrino interaction rate, JHEP 05 (2024) 089 [2312.07015]

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Drewes, Y

M. Drewes, Y. Georis, M. Klasen, L.P. Wiggering and Y.Y.Y. Wong, Towards a precision cal- culation of Neﬀ in the Standard Model. Part III. Improved estimate of NLO cor rections to the collision integral, JCAP 06 (2024) 032 [2402.18481]

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K. Akita and M. Yamaguchi, A precision calculation of relic neutrino decoupling, JCAP 08 (2020) 012 [2005.07047]

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