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arxiv: 2601.08221 · v2 · submitted 2026-01-13 · ✦ hep-ph

Energy and momentum dependence of the soft-axion interaction rate

Killian Bouzoud , Jacopo Ghiglieri , M. Laine , G.S.S. Sakoda This is my paper

Pith reviewed 2026-05-16 15:29 UTC · model grok-4.3

classification ✦ hep-ph
keywords axionsQCD axionthermal axionDelta N_effhard thermal loopdecouplingultrasoftgauge fields
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0 comments X

The pith

Ultrasoft axion rates lift Delta N_eff from 0.03 to 0.04

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

The paper examines how axions interact with thermal non-Abelian gauge fields when energies are soft but still above the Debye scale. It performs a hard thermal loop calculation that tracks the interaction rate as momentum varies from zero to light-like. The computation reveals a smooth interpolation between the k=0 domain used in lattice sphaleron estimates and the light-like domain relevant for cosmology. When these rates are inserted into the decoupling history of light QCD axions above 200 MeV, the extra radiation density contributed by the axion rises from roughly 0.03 to 0.04 effective neutrino species at a decay constant of 4 times 10^8 GeV.

Core claim

Focussing on soft energies (alpha_s T << omega << pi T), an HTL computation shows how the domains k=0 and k approx omega interpolate to each other for the soft-axion interaction rate. Comparing with lattice data at k=0 and connecting to NLO at higher k, assembling the best input shows that efficient ultrasoft interactions increase Delta N_eff from ~0.03 to ~0.04 at fa = 4*10^8 GeV for light QCD axion decoupling at T >= 200 MeV.

What carries the argument

The HTL-resummed soft-axion interaction rate that depends separately on energy omega and momentum k, interpolating between k=0 and light-like k approx omega.

Load-bearing premise

The hard thermal loop approximation remains valid in the ultrasoft domain without large higher-order corrections, and lattice results at zero momentum can be matched directly to the perturbative calculation.

What would settle it

A higher-order perturbative calculation or lattice measurement of the axion-gauge-field rate at small nonzero momentum that deviates enough to change the final Delta N_eff by more than 0.01 would falsify the reported shift.

Figures

Figures reproduced from arXiv: 2601.08221 by G.S.S. Sakoda, Jacopo Ghiglieri, Killian Bouzoud, M. Laine.

Figure 1
Figure 1. Figure 1: Illustration of the kinematic domains discussed in this work. On-shell axions have a fixed mass, m2 a = ω 2 − k 2 , shown with a dotted crimson line. If their mass is small, ma ≪ πT, they are almost lightlike for typical thermal momenta, k ∼ πT. In contrast, when we consider how an axion condensate interacts with a thermal plasma, we restrict ourselves to the axis k = 0. All existing lattice simulations al… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the integration domain pertinent to eqs. (3.13) and (3.15). The red dashed line corresponds to q+ = ω/2, separating different types of physical processes (cf. table 1 on p. 10). and write the integral as Z ∞ 0 dp Z p+k |p−k| dq = 2 Z +k/2 −k/2 dq− Z ∞ k/2 dq+ [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The integration domain for eqs. (4.7) and (4.9). The meanings of the domains and of the dotted line, indicating a contour of constant p⊥, are explained between eqs. (4.7) and (4.9). and after some algebra, the q0 > 0 part of eq. (4.7) can be expressed as ∆γ (ω, k) ≈ 1 m2 E k 2 Z 2k 0 dp⊥ p⊥ Z ω −∞ dp0 2π 1 p0  ∆ϱe E P (ω − p0 ) 2 p 2 ⊥ ( 4k 2 − p 2 ⊥ ) k 2 +∆ϱ T P  (K 2 − P2 ) 2 + [PITH_FULL_IMAGE:figur… view at source ↗
Figure 4
Figure 4. Figure 4: The integration contours corresponding to eq. (4.12), leading to eq. (4.13). The analytic structures appearing in ϕR , related to the physics of causality, were explained in ref. [46], who also pointed out that the contributions of the “other poles” cancel between the T and E channels. The role that the arcs play was worked out in refs. [47,48]. The procedure is frequently referred to as “light-cone sum ru… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of γasy from eq. (4.16), denoted by “general”, with the first two approximations from eq. (4.15), denoted by k− ≫ mE and k− ∼ mE , respectively. The line k = 0 gives γasy(ω, 0) = 1. The variable k− ≡ (ω − k)/2 runs from 0 (at k = ω) to ω/2 (at k = 0). Combining with eqs. (2.3) and (3.32), we find that Υsph ≡ lim ω→0+ Υ(ω, 0) (2.3) = (5.1) T 3 2f 2 a × Γsph T4 , (5.2) lim ω→0+ γ(ω, 0) (3.1),(3.32… view at source ↗
Figure 6
Figure 6. Figure 6: Left: the Born approximation from eq. (4.3), γBorn. If k < ω, γBorn dominates the result at ω ≫ mE . Right: the remainder beyond γBorn, denoted by ∆γ, compared with real-time classical (CLGT) and full 4d (LQCD) lattice results. The CLGT and LQCD sets correspond to a few separate temperatures, as shown in table 2. The LQCD data points are for k = ω = 0, and have been slightly displaced for better visibility… view at source ↗
Figure 7
Figure 7. Figure 7: Left: the soft contribution to γ, as described by eq. (6.4). For comparison we also show the purely perturbative HTL result from sec. 3. Right: the NLO hard contribution to γ, as described by eq. (6.6). With “asy” we show the subtraction needed for (6.7). For k ≫ πT, the NLO result for γhard can be larger or smaller than γasy (cf. fig. 12 on p. 40), whereas its extrapolation to k ≪ πT yields a positive con… view at source ↗
Figure 8
Figure 8. Figure 8: Left: the full interaction rate, as described by eq. (6.7). For αs we use the running coupling from table 2 on p. 21. Right: the k-independent coefficient Υ(k)/[3c 2 sHγfull(k)] from eq. (6.11), for typical values of fa . The cosmologically relevant Hubble-normalized rate, Υ(k)/(3c 2 sH), is given by the product of the left and right panels (cf. eq. (6.9)), and exceeds unity at T = 104 GeV. As eq. (6.9) sh… view at source ↗
Figure 9
Figure 9. Figure 9: Left: the evolution of eφ/T4 , obtained from eqs. (6.9), (6.12) and, for T < Tfin, (6.13). An upper bound is given by the equilibrium value, π 2/30 ≈ 0.329 (dashed red line), whereas a lower bound is given by the free-streaming value, π 2/30 (h∗,T /h∗,ini) 4/3 (black dotted line). The larger the interaction rate, the more the system attempts to equilibrate. Right: the axion contribution to ∆Neff, as given … view at source ↗
Figure 10
Figure 10. Figure 10: Different contributions to the coefficient γHTL, defined in eq. (3.32). The left plot is for k = 0.5 ω, the right plot for k = ω. The dotted black lines represent the respective analytically computable IR limits, given in eqs. (B.4), (B.8) and (B.13). B. Why the HTL calculation fails at ω ≪ mE The purpose of this appendix is to demonstrate why the HTL computation is leading-order consistent in the regime … view at source ↗
Figure 11
Figure 11. Figure 11: Two diagrams for the 2 ↔ 3 scattering contribution to soft axion production. Gluons are denoted by curly lines, the axion by a dashed line, and the axion–gauge vertex by a black blob. The t-channel rung is soft and leads to a logarithmic divergence if the axion energy is ω ≪ mE . We note that higher-order 1 + n → 2 + n processes, with n ≥ 2, which are not shown, are expected to contribute at the same orde… view at source ↗
Figure 12
Figure 12. Figure 12: The LO and NLO hard interaction rates from eqs. (C.1) and (C.10), respectively. The “subtr” version refers to a recipe suggested in ref. [39], which makes the hard LO rate somewhat better behaved when extrapolated to k ≪ πT. For comparison we also show γasy ≡ ln(4k 2/m2 E ). Having discounted the possibility of hidden O(g) effects in the LO contribution, we now turn to genuine NLO effects. As observed in … view at source ↗
read the original abstract

Axions coupled to thermal non-Abelian gauge fields may have cosmological significance. As the heat bath defines a frame, its influence depends separately on energy and momentum. A light-like momentum ($k \approx \omega$) is relevant for the axion contribution to the effective number of light neutrinos, $\Delta N^{ }_\mathrm{eff}$, whereas a vanishing momentum ($k=0$) plays a role for warm natural inflation or ultralight dark matter, and has been employed in lattice estimates (both classical and quantum-statistical) of the strong sphaleron rate. Focussing on soft energies ($\alpha_\mathrm{s}^{ }T \ll \omega \ll \pi T$), we carry out an HTL computation to show how the domains $k=0$ and $k \approx \omega$ interpolate to each other. We then compare with lattice data at $k=0$, and connect our analysis to NLO computations at $k \approx \omega \ge \pi T$. Assembling the current best input, we re-investigate light QCD axion decoupling dynamics at $T \ge 200$ MeV, showing that efficient interactions in the ultrasoft domain increase $\Delta N^{ }_\mathrm{eff}$ from $\sim 0.03$ to $\sim 0.04$ at $f^{ }_a = 4\times 10^8_{ }$ GeV.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper carries out an HTL computation of the soft-axion interaction rate with non-Abelian gauge fields in the regime α_s T ≪ ω ≪ π T, demonstrates the interpolation between the k=0 domain (relevant to lattice sphaleron rates) and the k≈ω domain (relevant to ΔN_eff), compares the result to existing lattice data at k=0, and assembles current best inputs to update light-QCD-axion decoupling, finding that ultrasoft interactions raise ΔN_eff from ∼0.03 to ∼0.04 at f_a=4×10^8 GeV.

Significance. If the HTL rate and its lattice matching hold, the work supplies a refined, momentum-dependent input for axion decoupling at T≳200 MeV that modestly strengthens the predicted contribution to ΔN_eff; this is directly relevant to cosmological bounds on the QCD axion and to the interpretation of future CMB measurements of N_eff.

major comments (2)
  1. [HTL computation and ultrasoft domain] The central claim that the HTL rate in the ultrasoft window α_s T ≪ ω ≪ π T is sufficiently accurate to produce a reliable 0.01 shift in ΔN_eff rests on an unquantified assumption that NLO corrections remain small; no explicit error estimate or higher-order calculation is supplied to support this, and a 20–30 % correction to the rate would erase the reported increment (see the interpolation and decoupling-dynamics sections).
  2. [Lattice comparison and interpolation] The direct matching of the perturbative HTL result to lattice sphaleron-rate data at k=0, followed by interpolation to k≈ω, does not include a controlled assessment of lattice artifacts (finite-volume, spacing, or non-perturbative contamination) or a quantified matching uncertainty; this matching is load-bearing for the final ΔN_eff value.
minor comments (2)
  1. [Abstract] The abstract quotes ΔN_eff ∼0.03 to ∼0.04 without accompanying uncertainties or the temperature range over which the shift is evaluated; adding these would clarify the robustness of the result.
  2. [Introduction and notation] Notation for the soft scale (α_s T) versus the hard scale (π T) should be introduced once with explicit definitions before being used repeatedly in the rate expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding the reliability of the HTL approximation and the lattice matching procedure are important, and we address them point by point below. We outline revisions that will strengthen the presentation of uncertainties while preserving the core results on the momentum-dependent rate and its implications for ΔN_eff.

read point-by-point responses

  1. Referee: The central claim that the HTL rate in the ultrasoft window α_s T ≪ ω ≪ π T is sufficiently accurate to produce a reliable 0.01 shift in ΔN_eff rests on an unquantified assumption that NLO corrections remain small; no explicit error estimate or higher-order calculation is supplied to support this, and a 20–30 % correction to the rate would erase the reported increment (see the interpolation and decoupling-dynamics sections).

    Authors: We agree that a dedicated NLO calculation in the ultrasoft regime would be the ideal way to quantify corrections. The present work focuses on the leading HTL resummation, which systematically captures the dominant infrared physics in this window. Existing NLO results for related quantities (such as gluon damping rates at slightly higher momenta) suggest corrections of order 10-20%. In the revised manuscript we will add a dedicated paragraph in the discussion section that reviews these analogies, provides a conservative 25% uncertainty band on the ultrasoft rate, and shows the resulting range for ΔN_eff (still an increase of at least 0.005 relative to the previous estimate). We will also stress that the primary advance is the controlled interpolation between the k=0 and k≈ω limits rather than the precise numerical shift. revision: partial

  2. Referee: The direct matching of the perturbative HTL result to lattice sphaleron-rate data at k=0, followed by interpolation to k≈ω, does not include a controlled assessment of lattice artifacts (finite-volume, spacing, or non-perturbative contamination) or a quantified matching uncertainty; this matching is load-bearing for the final ΔN_eff value.

    Authors: We acknowledge that the matching procedure relies on published lattice results without an independent re-analysis of their systematics. In the revision we will expand the relevant section to summarize the lattice papers' own assessments of finite-volume, spacing, and non-perturbative effects, quoting their quoted uncertainties. We will also add a sensitivity analysis that varies the matching scale within those reported errors, recomputes the interpolated rate, and propagates the variation into the final ΔN_eff value, thereby providing a quantified matching uncertainty. revision: yes

Circularity Check

0 steps flagged

No significant circularity; HTL rate computation and lattice/NLO assembly are independent

full rationale

The paper's core derivation is an explicit HTL computation of the axion interaction rate across momentum domains (k=0 to k≈ω), followed by direct comparison to external lattice data at k=0 and connection to separate NLO results at higher scales. The final ΔN_eff update at fa=4×10^8 GeV assembles these pre-existing inputs without fitting any parameter to the target quantity, without self-definitional loops, and without load-bearing self-citations that reduce the claim to its own assumptions. No step renames a known result or imports uniqueness via author-overlapping citations; the chain remains a forward calculation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the hard thermal loop resummation in the stated soft-energy window and on the direct usability of external lattice and NLO results for matching.

axioms (1)
  • domain assumption Hard thermal loop approximation is valid for α_s T ≪ ω ≪ π T
    Invoked to compute the interaction rate in the soft regime.

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Cited by 1 Pith paper

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