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Constraining self-interacting ultrahigh-energy muon neutrinos by cosmic microwave background spectral distortion

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Pith's one-line read The paper derives upper bounds on muon-neutrino self-interactions from CMB spectral distortions, reaching about 3.1e-5 with PIXIE-projected sensitivities and arguing future measurements would improve on FIRAS by one to three orders of…

desk verdict A clean but fragile application of known machinery: the PIXIE bounds are interesting, but they hinge on an unjustified maximal neutrino abundance and a dimensional typo in the central cross-section. read the letter →

arxiv 2506.24110 v2 pith:62HCHKA4 submitted 2025-06-30 astro-ph.CO hep-ph

Pravin Kumar Natwariya , Shibsankar Si , Alekha C. Nayak , Tripurari Srivastava This is my paper
classification astro-ph.COhep-ph
keywords neutrinoself-interactionsmuonneutrinosCMBspectraldistortionmu-typey-typePIXIEsuperheavydarkmattercosmicbackground
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 sets out to show that a standard cosmological observable, the shape of the cosmic microwave background spectrum, can be repurposed as a sensitive probe of muon-neutrino self-interactions. Its scenario is that superheavy dark matter decays into ultrahigh-energy muon neutrinos, and those neutrinos scatter radiatively off the cosmic neutrino background through a light scalar mediator, producing photon pairs that heat the early plasma. The heating creates two well-known kinds of CMB spectral distortion, a $\mu$-type distortion from energy injected at redshifts $5\times10^4 \lesssim z \lesssim 2\times10^6$ and a $y$-type distortion from later injection, and the size of either distortion is controlled by the self-coupling constant $g_{\nu_\mu}$. Comparing the predicted distortions with COBE/FIRAS upper limits and with the projected PIXIE sensitivities, the paper derives upper bounds on $g_{\nu_\mu}$ as a function of mediator mass, the strongest being about $3.1\times10^{-5}$ for 1 PeV neutrinos with PIXIE's $\mu$-distortion sensitivity. If the bounds are right, future CMB missions would not just measure the early thermal history; they would constrain neutrino physics beyond the Standard Model in parameter-space regions that BBN, kaon-decay, and IceCube probes currently leave open.

What carries the argument

The load-bearing object is the energy-injection rate $$\frac{dE}{dVdt}=f_\chi\Omega_\chi\rho_c n_{\nu_\mu,0}\,\$\sigma$(2m_{\nu_\mu}E_\nu)(1+z)^6,$$ which combines the assumed density of ultrahigh-energy muon neutrinos from dark-matter decay with the one-loop radiative-scattering cross-section $\sigma$ of a UHE neutrino off a nonrelativistic cosmic-background neutrino, mediated by a scalar $\phi$ and producing a photon pair through a muon loop. The cross-section depends on the scalar Passarino-Veltman integral $C_0(s,m_\mu)$, a standard one-loop function, and the rate is inserted into the standard evolution equations for the chemical-potential distortion $\mu$ and the Compton $y$ parameter. Because the rate scales as $(g_\mu g_{\nu_\mu})^2$, the observed or projected distortion limits become statements about the fourth power of the couplings. The resonance structure of the cross-section gives the bounds their characteristic shape: flat below $m_\phi\ll\sqrt{s}$, a kink at $m_\phi\approx\sqrt{s}$ where the mediator's width dominates, and a relaxation proportional to $m_\phi$ at higher masses.

What would settle it

A concrete check: normalize the present-day UHE neutrino density to the diffuse flux actually observed by IceCube and KM3NeT, and include a finite dark-matter lifetime so that only the fraction of parents decayed by redshift $z$ contributes, then recompute the $\mu$- and $y$-distortion integrals. If the resulting distortions fall below the FIRAS and PIXIE limits across the parameter space, the claimed bounds disappear; if they survive, the bounds are robust to the normalization worry.

Watch

Extended reading notes

Core claim

The paper's central claim is that radiative scattering of ultrahigh-energy muon neutrinos on the cosmic neutrino background, mediated by a sub-GeV scalar that couples to both muons and muon neutrinos, injects enough energy into the early intergalactic medium to produce observable $\mu$- and $y$-type CMB spectral distortions, and that the measured or projected limits on those distortions translate into the strongest cosmology-based upper bounds on the flavor-specific self-coupling $g_{\nu_\mu}$ in the sub-GeV mediator-mass range. Concretely, at $E_\nu=1$ PeV with the projected PIXIE $\mu$-distortion sensitivity of $5\times10^{-8}$, the bound is $g_{\nu_\mu}\lesssim3.1\times10^{-5}$ in the scenario $g_\mu=5\times10^{-4}$ that addresses the muon $g-2$ anomaly; in the equal-coupling case $g_{\nu_\mu}=g_\mu$, the PIXIE $\mu$- and $y$-distortion bounds are $1.3\times10^{-4}$ and $1.5\times10^{-4}$. The bounds stay flat for mediator masses below the center-of-mass energy $\sqrt{s}\approx\sqrt{2m_{\nu_\mu}E_\nu}$, show a resonance kink near $m_\phi\approx\sqrt{s}$, and then relax in proportion to $m_\phi$ at higher masses. The paper concludes that CMB spectral distortion is a strong and complementary probe, with PIXIE improving on FIRAS by one to three orders of magnitude.

Load-bearing premise

The results assume that the entire dark-matter density is converted, with unit efficiency and no delay, into ultrahigh-energy muon neutrinos whose density scales as $(1+z)^3$ at every redshift, and that the one-loop cross-section formula is used at face value; if the parent lifetime is finite, the conversion fraction is lower, or the cross-section normalization is corrected, all quoted bounds shift.

Editorial extensions

If this is right

  • At $E_\nu=1$ PeV and with $g_\mu=5\times10^{-4}$, the PIXIE $\mu$-distortion sensitivity of $5\times10^{-8}$ gives $g_{\nu_\mu}\lesssim3.1\times10^{-5}$, roughly two orders of magnitude tighter than the FIRAS-based limit.
  • The limits are flat for $m_\phi\ll\sqrt{s}$, show a resonance kink at $m_\phi\approx\sqrt{s}$, and relax in proportion to $m_\phi$ above resonance, so the probe is strongest for sub-GeV mediators.
  • Raising the UHE neutrino energy from 1 PeV to 100 PeV shifts the resonance to heavier mediators and relaxes the coupling bounds by roughly an order of magnitude, e.g. the PIXIE $\mu$-distortion bound becomes $3.6\times10^{-4}$ at 100 PeV in the $g_\mu=5\times10^{-4}$ scenario.
  • Because the energy injection scales as $(g_\mu g_{\nu_\mu})^2$, a null PIXIE measurement constrains the product of the muon and neutrino couplings, not just the neutrino coupling alone.
  • The resulting constraints occupy parameter space complementary to BBN, rare kaon decay, and the Hubble-tension and muon $g-2$ explanations, so PIXIE could discriminate among those scenarios.

Reading between the lines

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

  • Read properly, the bounds constrain $f_\chi(g_\mu g_{\nu_\mu})^2$ rather than $g_{\nu_\mu}$ alone: if the UHE neutrino density is normalized to the observed IceCube or KM3NeT diffuse flux instead of $f_\chi=1$, the early-universe injection rate drops and all quoted bounds weaken by a factor $(f_\chi)^{1/4}$.
  • The same machinery should apply to electron- or tau-flavored self-interactions; replacing the muon mass and couplings in the loop would give a parallel set of PIXIE-era bounds, though the charged-lepton mass changes the loop function and the resonance position.
  • A dedicated computation of the distortion in the intermediate epoch $z\sim10^5$, where neither the pure $\mu$ nor the pure $y$ approximation holds, could add independent leverage and resolve how the two bound regimes connect.
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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 / 6 minor

Summary. The paper computes the CMB spectral distortions (μ- and y-type) generated by radiative scattering of ultra-high-energy muon neutrinos with the cosmic neutrino background, mediated by a light scalar boson. The UHE neutrinos are assumed to originate from super-heavy dark matter decay. Using COBE/FIRAS limits and PIXIE projected sensitivities, the authors derive upper bounds on the scalar-neutrino coupling g_νμ as a function of mediator mass for the cases g_νμ=g_μ and g_νμ≠g_μ. The headline results are bounds of order 10^-4 for 1 PeV neutrinos with PIXIE, improving on FIRAS by one to two orders of magnitude, with the strongest quoted bound g_νμ ~ 3×10^-5 for the g_μ=5×10^-4 benchmark.

Significance. The mechanism is plausible and the spectral-distortion framework is standard; if the normalization were robust, the derived constraints would make CMB spectral distortion a competitive probe of muon-philic neutrino self-interactions. The paper is transparent about its assumptions, explicitly setting f_χ=1, and the constraints are genuine inequality inversions of the forward calculation rather than fits, so there is no circularity. However, the central numbers depend on an unphysical normalization of the UHE neutrino abundance in a decaying-DM model, and the quoted bounds are likely to shift when the decay lifetime is included. The qualitative conclusion that spectral distortions can probe these couplings is defensible, but the specific headline values need revision.

major comments (2)
  1. [Sec. III, Eq. (8)] The energy injection rate is computed with f_χ=1 and a UHE neutrino density that scales as (1+z)^3 at all redshifts. This is inconsistent with the stated origin of the neutrinos as decay products of super-heavy dark matter. For a two-body decay with lifetime τ, the daughter-neutrino density at redshift z is proportional to [1−exp(−t(z)/τ)], not to the assumed (1+z)^3 factor. At the μ-distortion epoch z∼10^5, t(z)∼10^2 yr, so for τ∼10^10 yr (typical in decaying-DM interpretations of IceCube) the density is suppressed by ∼10^-8 relative to Eq. (8). Because σ∝g_νμ^4, the μ-type bounds weaken by roughly two orders of magnitude and the y-type bounds by one order. The headline bound in the abstract (3.1×10^-5) is therefore not a robust prediction of the decaying-DM scenario. The authors should either include the decay factor in Eq. (8) or explicitly qualify the results as bounds on a hypothetical, maximally abundant UHE neutrino population that is not produced by the decaying DM described in Sec. II.
  2. [Sec. III, Eq. (8)] The assumed present-day UHE neutrino density with f_χ=1 and m_χ≳PeV corresponds to an isotropic local flux many orders of magnitude above the IceCube-observed diffuse flux, and the relativistic decay products cannot act as cold dark matter. The manuscript does not check consistency with IceCube or with structure-formation requirements. Without this check, the bounds in Figs. 4–6 are best interpreted as constraints on the product f_χ σ rather than on g_νμ alone. Please add a discussion of how the assumed abundance relates to current neutrino-flux measurements, or fold the allowed f_χ into the reported limits.
minor comments (6)
  1. [Sec. II, Eq. (2)] The decay width definition Γ_ϕ=m_ϕ g_μ g_νμ/(4π) is unusual; for a scalar with Yukawa couplings to muons and neutrinos, the total width should be the sum of quadratic partial widths for each open channel. Please clarify whether this is an effective width or a typo, as it affects the on-resonance cross-section.
  2. [Sec. II, Eq. (2)] The cross-section formula in Eq. (2) has a factor s in the numerator, giving the correct mass dimension GeV^-2 as used in Fig. 2; if the typesetting in the published version masks this, please make the division explicit.
  3. [Abstract] The abstract quotes g_νμ ∼ 3.1×10^-5 without specifying the benchmark g_μ = 5×10^-4; since the bounds vary by orders of magnitude with g_μ (e.g., g_νμ ≲ 1.7×10^-7 for g_μ=0.1), the abstract should state the assumed value of g_μ.
  4. [Sec. IV, Eq. (11)] The sentence 'The evolution of µ with redshift z can be written as [75]' appears twice in the vicinity of Eqs. (11)-(12); the duplicate should be removed.
  5. [Sec. IV, Eq. (14)] The y-type distortion is plotted only down to z=10^3 in Fig. 2b, while the integral in Eq. (14) formally extends to low redshift; please specify the lower integration limit used in the numerical calculation.
  6. [Sec. II] The choice m_νμ ≈ 0.1 eV for the cosmic neutrino background mass should be justified with an explicit reference, since current mass limits allow a lighter heaviest mass eigenstate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bounds are forward-model inversions against external FIRAS/PIXIE limits; the sole self-citation supplies standard mu/y kernels also attributed to external references.

full rationale

The derivation chain is self-contained in the relevant sense: no parameter is fitted to the distortion data. The cross-section (Eq. 2) is imported from external references [29,48,64,65]; the interaction rate (Eq. 5) follows from n_nu_mu sigma; the energy injection (Eq. 8) multiplies by the assumed UHE neutrino density n_nu,0 = f_chi Omega_chi rho_c / m_chi, with f_chi = 1 stated explicitly as an assumption; and the mu- and y-type distortions are computed with the standard kernels (Eqs. 9-14), attributed to external Refs. [51,52] as well as to the authors' own Ref. [70]. The comparison against COBE/FIRAS and PIXIE is a genuine inequality inversion, not a definitional identity. Ref. [70] is by three of the present authors, but it is cited alongside external Refs. [51,52] for conventional mu/y evolution equations; it supplies no uniqueness theorem, does not forbid alternatives, and its content is not the target neutrino-coupling bound, so it is not load-bearing. The f_chi = 1 normalization and the mass dimension of Eq. (2) are physical assumptions or possible correctness issues, not circular reductions; correcting them would shift the quoted bounds, but no output is equivalent to an input by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on two imported normalizations: the loop cross-section of Eq. (2) and the f_chi = 1 UHE-neutrino density of Eqs. (7) and (8), together with standard cosmological parameters and the standard mu-type and y-type distortion kernels. Neither imported normalization is re-derived, and the cross-section as printed is dimensionally inconsistent with its usage, so the absolute value of every quoted bound inherits that uncertainty. The free parameters are scenario choices (E_nu, g_mu, m_nu_mu) plus the extremal f_chi, rather than fitted quantities.

free parameters (4)
  • f_chi (fraction of dark matter converted into UHE neutrinos) = 1 (set to its maximum, not inferred)
    Eqs. (7) and (8) in Sec. III: the full dark matter density becomes the UHE neutrino population scaling as (1+z)^3. This maximal choice maximizes the energy injection and therefore strengthens the g_nu_mu bounds, and no dependence on f_chi < 1 is discussed.
  • m_nu_mu (mass of the background muon neutrino) = 0.1 eV
    Sets s = 2 m_nu_mu E_nu. Chosen at the high end of the allowed range and attributed to Ref [66], which does not support it. A smaller mass would shift the kink position in m_phi and weaken the bounds.
  • E_nu (UHE neutrino energy) = 1 PeV and 100 PeV
    Corresponds to dark matter mass m_chi = 2 E_nu. The center-of-mass energy sqrt(s) sets where the bounds stop being constant and become proportional to the mediator mass.
  • g_mu (muon-scalar coupling) = g_nu_mu (equal case), 0.1, or 5x10^-4
    Three benchmark scenarios: equal couplings, a large muon coupling, and the value 5x10^-4 taken from the (g-2)_mu explanation of Ref [28]. The quoted bounds scale roughly as g_nu_mu^4 through the cross-section, so this choice strongly affects the headline numbers.
assumptions (5)
  • domain assumption The radiative neutrino-neutrino to gamma-gamma cross-section of Eq. (2), imported from Refs [29,48,64,65], is correct as printed with dimensionless couplings g_nu_mu and g_mu.
    This formula sets the normalization of every bound. As printed it has mass dimension GeV^-4 while it is used as GeV^-2, and the scaling relations stated in Sec. V do not reproduce it, so the paper rests on an unverified imported normalization.
  • ad hoc to paper The UHE neutrino population tracks (1+z)^3 at all epochs with f_chi = 1 and no parent decay-width factor.
    Sec. III, Eqs. (6) to (8): yields the (1+z)^6 scaling of the energy injection rate. It is inconsistent with a dark matter lifetime long enough to evade the IceCube-measured diffuse neutrino flux, since the implied local neutrino density is orders of magnitude too large unless the decay is slow.
  • domain assumption The cosmic neutrino background is non-relativistic with s = 2 m_nu_mu E_nu and Moller velocity v_M = 1.
    Sec. III: valid for m_nu_mu = 0.1 eV against T_nu of about 1.65x10^-4 eV, but the mono-energetic treatment averages over the thermal spread of the background neutrinos.
  • domain assumption All energy injected as photons distorts the CMB spectrum with unit efficiency, and the standard mu-type and y-type Green's functions of Sec. IV apply with no intermediate-redshift interpolation.
    Sec. IV, Eqs. (9) to (14): standard in the literature (Refs [52,70]), but no photon cascade or deposition efficiency is computed for PeV-scale photons, and the transition epoch between the mu and y windows is not modeled.
  • standard math Standard cosmological parameters (Omega_b h^2 = 0.0224, z_dc = 1.11x10^7, the critical density, and the Hubble rate) enter the distortion integrals.
    Used in Eqs. (8), (10) and (12). Values are standard and are not varied, so no uncertainty is propagated into the quoted bounds.
invented entities (1)
  • Light scalar mediator phi with flavor-specific couplings to muon neutrinos (g_nu_mu) and to muons (g_mu) independent evidence
    purpose: Mediates UHE muon neutrino scattering on the cosmic neutrino background and the one-loop production of photon pairs that heat the plasma and distort the CMB.
    The scalar is not introduced by this paper; it is the muonphilic scalar hypothesis studied in Refs [28,57,84]. Independent falsifiable handles exist outside the paper, including rare kaon decays (K to mu nu phi), BBN constraints, supernova cooling, and the muon (g-2) anomaly. These are plotted as comparison regions in Figs. 4 to 6. Only the neutrino-side coupling g_nu_mu is weakly constrained elsewhere, which is the gap this paper targets.

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Pith. "Pith review of Constraining self-interacting ultrahigh-energy muon neutrinos by cosmic microwave background spectral distortion." pith.science (2026). https://pith.science/paper/62HCHKA4

@misc{pith2026250624110,
  author       = {Pith},
  title        = {Pith review of: Constraining self-interacting ultrahigh-energy muon neutrinos by cosmic microwave background spectral distortion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62HCHKA4}},
  note         = {Machine review of arXiv:2506.24110}
}
abstract

The neutrino telescopes have firmly established the existence of ultrahigh-energy neutrinos. Observations of these neutrinos offer a unique probe of neutrino self-interactions. This work investigates how the self-interacting neutrinos, mediated by scalar bosons, inject energy into the medium through radiative scattering with the cosmic neutrino background, leaving an imprint on the cosmic microwave background (CMB) spectrum. The energy injection into plasma in redshift ranges, $5\times10^4\lesssim z\lesssim2\times10^6$ and $ z\lesssim5\times10^4$, leads to $\mu$-type and $y$-type CMB spectral distortions, respectively. Using observational constraints from Cosmic Background Explorer/Far Infrared Absolute Spectrophotometer (COBE/FIRAS) and projected sensitivities from Primordial Inflation Explorer (PIXIE) experiments for $\mu$-type and $y$-type CMB distortions, we derive the stringent upper bounds on the self-interaction coupling strength as a function of mediator mass for neutrino interactions. We focus on flavor-specific self-interaction related to muon neutrinos and sub-GeV mass mediators ($m_{\phi}$). We find the upper bound on the self-interaction coupling strength to be $\sim 2.8\times 10^{-4}$ for the muon neutrino, considering ultrahigh-energy muon neutrino energy to be 1 PeV and PIXIE projected upper bounds on $y$-type CMB spectral distortion. The bound remains constant till the mediator mass reaches the center-of-mass energy, and after that, it gets relaxed and becomes proportional to the mediator mass. We have also compared our results with existing bounds in the literature. Our findings indicate that CMB spectral distortion could play a decisive role in exploring neutrino physics beyond the standard model of particle physics, and future missions like PIXIE can provide valuable insights.

Figures

Figures reproduced from arXiv: 2506.24110 by the authors.

Figure 1
Figure 1. FIG. 1: Feynman diagrams (a) for scattering of ultra-high energy neutrinos ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Evolution of CMB spectral distortions, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Evolution of CMB spectral distortions as a function of the mass of mediator ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Upper on the self-interacting neutrino coupling ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Upper bound on the self-interacting neutrino coupling ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Upper bound on the self-interacting neutrino coupling [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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