Refractive Neutrino Mass: Coherent Matter Effects

• Refractive neutrino mass is a medium-dependent effective mass term arising from coherent forward scattering in matter, relic neutrino backgrounds, or dense neutrino gases.
• It modifies standard oscillation parameters by introducing phase shifts and effective mass-squared differences, as seen in supernova neutrinos and solar dark-matter halos.
• Dark-sector interactions demonstrate that refractive effects can mimic vacuum mass contributions, offering testable predictions in neutrino oscillation experiments.

Refractive neutrino mass denotes a medium-dependent effective mass, or mass-squared contribution, induced by coherent forward scattering in a background rather than by a vacuum mass term alone. In the most conservative usage, it is a modification of the neutrino dispersion relation or propagation Hamiltonian in matter, relic neutrino backgrounds, or dense neutrino media; in more radical constructions, neutrinos are taken to be massless in vacuum and all oscillation phenomena are attributed to refractive effects generated by dark matter or by a refractive quantum vacuum (Kisslinger, 2012, Diaz et al., 2015, Sen et al., 2023, Oksanen et al., 2024).

1. Formal definition and mathematical structure

A common propagation-level definition starts from an in-medium dispersion relation

$E_i(p)=\sqrt{p^2+m_i^2}+V_i,$

or, equivalently, from a Hamiltonian in which coherent forward scattering contributes an additive potential. In the cosmic neutrino background (CNB) treatment, the fundamental kinetic mass parameters $m_i$ are unchanged, while the energy of each mass eigenstate is shifted by a diagonal potential $V_{ii}$; the associated oscillation phase can then be written with an effective refractive mass-squared difference

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$

which modifies propagation but not the bare mass parameters (Diaz et al., 2015).

A second, stronger identification is used for neutrinos at rest in dense matter. In the neutrinosphere calculation, the effective masses are obtained by taking $p=0$, so that

$m_{i,\mathrm{eff}}\equiv E_i(p=0),$

and the in-medium rest energies are directly interpreted as effective masses in matter (Kisslinger, 2012). This is the sense in which a matter potential can be read as a genuine “refractive mass” rather than merely a phase shift.

Dark-sector realizations introduce a particularly compact notation: $\tilde m^2 \equiv 2EV.$ When $V\propto 1/E$, $\tilde m^2$ is effectively constant and is operationally indistinguishable from an ordinary vacuum mass-squared in oscillation experiments (Sen et al., 2023). This usage is central to ultralight-dark-matter models, where refractive masses can reproduce the observed oscillation pattern even if neutrinos are fundamentally massless in vacuum (Sen et al., 2024).

These definitions are mathematically related but conceptually distinct. Some formulations treat refractive mass as a convenient encoding of medium-induced phase evolution; others treat it as an in-medium rest-energy eigenvalue; still others elevate it to the sole origin of neutrino oscillations.

2. Ordinary matter, relic backgrounds, and dense neutrino media

In ordinary matter the standard Wolfenstein potential is

$V=\sqrt{2}\,G_F N_e.$

Its most extreme realization in the supplied literature is the neutrinosphere of a core-collapse supernova. There the nucleon density is approximately nuclear matter density, $m_i$0, implying an electron density of order $m_i$1, and a matter potential $m_i$2, about $m_i$3 times larger than the terrestrial value $m_i$4. Using Freund’s cubic eigenvalue formalism, the dimensionless eigenvalues change from

$m_i$5

for $m_i$6 to

$m_i$7

for a neutrinosphere matter parameter $m_i$8. In that normalization, the splitting between the first two eigenstates becomes about $m_i$9 times larger than in vacuum, while the third eigenvalue is only weakly affected (Kisslinger, 2012).

A much weaker but conceptually clean case is neutrino refraction by the CNB. There the relic background has present temperature

$V_{ii}$0

and number density

$V_{ii}$1

per helicity state. The induced potential is diagonal in mass space, so the vacuum mixing angles are unchanged; only the eigenvalues are shifted. The potential decomposes into a flavor-blind term $V_{ii}$2, which affects kinematics such as beta decay, and a species-dependent term $V_{ii}$3, which modifies oscillation phases (Diaz et al., 2015). Quantitatively, however, the CNB effect is negligible for current phenomenology: a typical scale is

$V_{ii}$4

with associated oscillation length

$V_{ii}$5

far beyond terrestrial or astrophysical baselines (Diaz et al., 2015).

Dense neutrino gases generalize the refractive concept further, since the background is itself made of neutrinos. In that case the self-interaction term scales schematically as

$V_{ii}$6

and enters the flavor Hamiltonian alongside the vacuum mass term and ordinary matter potential. With nonstandard neutrino-neutrino couplings, flavor-changing refractive terms can trigger collective transformations even when the vacuum mixing angle vanishes exactly; in the inverted hierarchy, very small nonstandard couplings are sufficient to trigger the usual collective flavor evolution, while larger couplings can drive flavor equilibration when the ordinary matter potential is subdominant (0810.2297).

3. Dark-matter-induced refractive masses

A distinct class of models attributes neutrino oscillations to coherent forward scattering on ultralight scalar dark matter. The representative interaction is

$V_{ii}$7

with $V_{ii}$8 an ultralight dark-matter boson and $V_{ii}$9 a light fermionic mediator. In this setup neutrinos are taken to be fundamentally massless in vacuum, and the refractive potential is

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$0

where

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$1

and $$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$2 encodes a dark-matter charge asymmetry (Sen et al., 2024).

Above resonance, $$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$3, the potential reduces to

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$4

so that

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$5

an energy-independent quantity with the same oscillation-level role as an ordinary vacuum mass-squared (Sen et al., 2024). This regime is the reason such models can fit oscillation data. Below resonance, however, the behavior changes qualitatively. For nonzero asymmetry,

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$6

whereas for symmetric dark matter,

$$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$7

The effective mass-squared then decreases with energy and cannot be treated as a standard constant mass parameter; the group velocity and cosmological impact must be computed directly from the modified dispersion relation rather than by inserting $$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$8 into the usual massive-particle formulas (Sen et al., 2023, Sen et al., 2024).

The cold-gas and coherent-field realizations are not identical at low energy. In the cold-gas picture, $$\Delta \tilde m_{ij}^2=\Delta m_{ij}^2+2E\,\Delta V_{ij},$$9 has a resonance structure and a strong low-energy suppression. In the coherent classical-field picture, the refractive mass does not depend explicitly on energy but can depend on time, and it coincides with the cold-gas refractive mass in the high-energy limit (Sen et al., 2023). This time dependence motivates direct searches for modulation of oscillation parameters.

The cosmological consequence emphasized in the DESI-focused analysis is that relic neutrinos can remain ultrarelativistic during structure formation even when oscillation experiments see nonzero refractive mass-squared splittings. For

$p=0$0

and

$p=0$1

the model yields $p=0$2 and $p=0$3, so neutrinos behave as effectively massless during the epoch relevant for large-scale structure. This allows oscillation-scale masses to coexist with cosmological inferences such as $p=0$4 at $p=0$5 C.L. and even $p=0$6 at $p=0$7, while oscillation data imply $p=0$8 for normal ordering and $p=0$9 for inverted ordering (Sen et al., 2024).

4. Solar dark-matter halos and the dark-LMA question

A more elaborate refractive framework adds two sterile neutrinos and studies propagation in a non-uniform solar dark-matter halo. In that model the effective $m_{i,\mathrm{eff}}\equiv E_i(p=0),$0 Hamiltonian in the basis $m_{i,\mathrm{eff}}\equiv E_i(p=0),$1 can be diagonalized by a unitary matrix $m_{i,\mathrm{eff}}\equiv E_i(p=0),$2 parametrized by six mixing angles and one complex phase (Chattopadhyay et al., 24 Nov 2025). When the active-sterile mixing angles are small, propagation in a uniform DM background reduces to an effective two-flavor problem, with the usual solar angle $m_{i,\mathrm{eff}}\equiv E_i(p=0),$3 carrying the dominant radial evolution.

The halo introduces a spatially varying overdensity

$m_{i,\mathrm{eff}}\equiv E_i(p=0),$4

which rescales the refractive masses in the deep-halo region. In that limit the eigenvalues behave as

$m_{i,\mathrm{eff}}\equiv E_i(p=0),$5

while the active-sterile angles become

$m_{i,\mathrm{eff}}\equiv E_i(p=0),$6

This produces a region of halo dominance (RHD), and within it an RHD core where $m_{i,\mathrm{eff}}\equiv E_i(p=0),$7 and $m_{i,\mathrm{eff}}\equiv E_i(p=0),$8 can approach $m_{i,\mathrm{eff}}\equiv E_i(p=0),$9, and an RHD periphery where they remain small (Chattopadhyay et al., 24 Nov 2025).

The electron-neutrino survival probability in the adiabatic limit is then

$\tilde m^2 \equiv 2EV.$0

which reduces to the standard MSW form only when the active-sterile angles are negligible (Chattopadhyay et al., 24 Nov 2025). In the RHD core,

$\tilde m^2 \equiv 2EV.$1

whereas outside the RHD one recovers the standard solar expression

$\tilde m^2 \equiv 2EV.$2

A central result is that the survival probability depends strongly on the neutrino production radius even at fixed energy, so flux-averaged solar predictions must be weighted over the radial production profiles of pp, $\tilde m^2 \equiv 2EV.$3Be, $\tilde m^2 \equiv 2EV.$4B, CNO, pep, and hep neutrinos (Chattopadhyay et al., 24 Nov 2025).

This structure is used to revisit the dark-LMA solution, with $\tilde m^2 \equiv 2EV.$5. The analysis finds that a solar halo can make dark-LMA phenomenologically viable if the RHD boundary satisfies

$\tilde m^2 \equiv 2EV.$6

while the RHD core remains relatively compact,

$\tilde m^2 \equiv 2EV.$7

A benchmark dark-LMA realization uses

$\tilde m^2 \equiv 2EV.$8

and yields a high-energy $\tilde m^2 \equiv 2EV.$9B survival spectrum that is approximately flat, with features aligned with the Super-Kamiokande measurements (Chattopadhyay et al., 24 Nov 2025).

5. Observational probes and experimental sensitivities

The most explicit absolute-mass probe in the supplied material is time-of-flight from a Galactic core-collapse supernova at DUNE. In the ultralight-dark-matter scenario, the effective refractive mass scale is

$V\propto 1/E$0

and the supernova delay is

$V\propto 1/E$1

so the observable depends on the line-of-sight averaged dark-matter density. A density spike near the Galactic Center can enhance the delay by up to an order of magnitude relative to the smooth-halo case (Pompa et al., 14 Aug 2025).

Scenario	IO bound on $V\propto 1/E$2	NO bound on $V\propto 1/E$3
Vacuum mass comparison	$V\propto 1/E$4	$V\propto 1/E$5
Refractive mass, no DM spike	$V\propto 1/E$6	$V\propto 1/E$7
Refractive mass, spike with fixed $V\propto 1/E$8	$V\propto 1/E$9	$\tilde m^2$0
Refractive mass, spike with $\tilde m^2$1 marginalized	$\tilde m^2$2	$\tilde m^2$3

These are the $\tilde m^2$4 C.L. sensitivities quoted for a supernova at $\tilde m^2$5 kpc (Pompa et al., 14 Aug 2025). The same study also finds $\tilde m^2$6 C.L. bounds on the spike slope parameter,

$\tilde m^2$7

when neutrinos are assumed massless in reality and the fit marginalizes over $\tilde m^2$8 and the burst time offset (Pompa et al., 14 Aug 2025).

By contrast, CNB refraction is far below detectability in both oscillations and beta decay. The flavor-blind CNB contribution can formally shift the beta-decay endpoint, but the characteristic scale $\tilde m^2$9 is negligible compared with present experimental resolution, so it does not obscure direct neutrino-mass measurements (Diaz et al., 2015). In the dark-matter refractive scenario, beta-decay kinematics can also be modified, with an effective endpoint parameter

$V=\sqrt{2}\,G_F N_e.$0

which can even be negative; for $V=\sqrt{2}\,G_F N_e.$1, $V=\sqrt{2}\,G_F N_e.$2, and $V=\sqrt{2}\,G_F N_e.$3, the inferred scale is $V=\sqrt{2}\,G_F N_e.$4, still below current KATRIN sensitivity (Sen et al., 2024).

The cleanest generic discriminator of a refractive origin is nontrivial dependence on energy or time. The ultralight-dark-matter analysis emphasizes that the refractive nature of neutrino mass can be tested by searches for its dependence on energy and time, since below resonance $V=\sqrt{2}\,G_F N_e.$5 departs from a constant and in coherent-field realizations the effective parameters can oscillate in time (Sen et al., 2023).

6. Conceptual status, alternative formulations, and open issues

The literature represented here uses “refractive neutrino mass” in at least three non-identical senses. First, it can denote a propagation-induced correction to the eigenvalues of the Hamiltonian while leaving intrinsic masses untouched, as in CNB refraction (Diaz et al., 2015). Second, it can denote an in-medium rest-energy eigenvalue, obtained by evaluating the dispersion relation at $V=\sqrt{2}\,G_F N_e.$6, as in the neutrinosphere calculation (Kisslinger, 2012). Third, it can denote the entire effective mass-squared structure measured by oscillation experiments in models where neutrinos are massless in vacuum and all nonzero $V=\sqrt{2}\,G_F N_e.$7 arise from dark-sector refraction (Sen et al., 2023, Sen et al., 2024).

A more radical alternative treats vacuum itself as refractive. In the “refractive quantum vacuum” proposal, flavor neutrinos are massless particles propagating through the Brout–Englert–Higgs vacuum, and the wave equation contains a forward-scattering term proportional to $V=\sqrt{2}\,G_F N_e.$8. In the high-energy limit the averaged flavor wave has a universal effective refractive mass

$V=\sqrt{2}\,G_F N_e.$9

and group velocity

$m_i$00

The standard oscillation probability in vacuum is then recovered from differences of refractive indices rather than from superposition of free massive particles with different group velocities (Oksanen et al., 2024). This is a conceptually distinct interpretation rather than a reformulation accepted by consensus.

Several limitations recur across the field. CNB refraction is too small to observe with current or foreseeable experiments (Diaz et al., 2015). In cold-gas ultralight-dark-matter models, the perturbative derivation of the refractive potential can break down at low energies, and the relevant perturbativity threshold can lie above resonance for parameters chosen to reproduce the observed oscillation scales (Sen et al., 2023). Supernova time-of-flight studies further note that recent analyses using time-modulation constraints disfavor refractive mass as the dominant source of neutrino mass if the ultralight field preserves coherence and exhibits $m_i$01 fluctuations, although the viability of that conclusion depends on assumptions about halo coherence and substructure (Pompa et al., 14 Aug 2025).

Taken together, these results establish refractive neutrino mass as a technically precise but theory-dependent notion. In standard environments it is an effective propagation parameter; in dark-sector models it can be promoted to the apparent origin of oscillation masses; and in all cases its defining feature is that neutrino mass-like behavior is generated by coherent interaction with a background rather than read directly from a vacuum mass term.