Matter Neutrino Resonance (MNR)

• Matter Neutrino Resonance (MNR) is a phenomenon where the matter potential nearly cancels with neutrino self-interaction potential, leading to significant flavor transformation.
• Multi-angle effects and spontaneous symmetry breaking in neutrino emissions result in non-uniform, often partial, flavor conversion across different angular bins.
• Hydrodynamical simulations reveal that MNR influences r-process nucleosynthesis in merger remnants by altering neutrino fluxes that determine the neutron-to-proton ratio.

Matter Neutrino Resonance (MNR) refers to a class of collective neutrino flavor transformation phenomena that occur in dense astrophysical environments when the potential due to coherent forward scattering of neutrinos on electrons (the so-called matter potential) is nearly canceled by the potential arising from neutrino–neutrino self-interactions. Unlike the Mikheyev–Smirnov–Wolfenstein (MSW) resonance, where the matter potential alone determines the resonance condition, MNR requires both matter and neutrino self-interaction potentials to be significant and oppositely signed—a situation realized in environments such as neutron star merger remnants and compact merger accretion disks. This resonance can profoundly alter neutrino flavor distributions, potentially affecting nucleosynthesis and the interpretation of neutrino signals in extreme astrophysical events.

1. Theoretical Formulation and Resonance Condition

The flavor evolution of neutrinos in dense environments is governed by a Hamiltonian comprising three terms: the vacuum term (encoding mass differences and mixing angles), the matter term (from neutrino–electron forward scattering), and the neutrino–neutrino self-interaction term (from coherent neutrino–neutrino scattering). In the two-flavor approximation, the total Hamiltonian for a neutrino of momentum $\vec{p}$ is expressed as: $$H(\vec{p}, t) = H_{\rm vac} + H_{\rm mat} + H_{\nu\nu}(\vec{p}, t)$$ where

$$H_{\rm vac} = \frac{\omega}{2} \begin{pmatrix} -\cos2\theta_V & \sin2\theta_V \ \sin2\theta_V & \cos2\theta_V \end{pmatrix}\,,\quad \omega = \frac{\Delta m^2}{2E}$$

$$H_{\rm mat} = \lambda\,{\rm diag}(1, 0)\,,\quad \lambda = \frac{\sqrt{2}\,\rho_B}{m_N}Y_e$$

$$H_{\nu\nu}(\vec{p}, t) = \mu \int d\vec{v}' [\rho(\vec{v}', t) - \bar{\rho}(\vec{v}', t)] (1 - \vec{v}\cdot\vec{v}')$$

with $Y_e$ the electron fraction, $\rho_B$ the baryon density, $m_N$ the nucleon mass, and the density matrices $\rho, \bar{\rho}$ for neutrinos and antineutrinos, respectively.

The MNR occurs when the combined diagonal (ee) contributions from the matter and neutrino self-interaction potentials nearly cancel: $$H^{\nu\nu}_{ee}(\vec{x}, \vec{p}) - H^{\nu\nu}_{xx}(\vec{x}, \vec{p}) + \lambda(\vec{x}) \approx 0$$ Under this resonance condition, the flavor mixing becomes maximal, even if the vacuum mixing angle is small.

2. Multi-Angle Effects and Symmetry Breaking

In realistic settings, neutrinos are emitted over a range of directions ("multi-angle" distributions). The self-interaction potential depends on the angle via $(1 - \vec{v}\cdot\vec{v}')$, so the resonance is realized locally in angle space. The analysis distinguishes between isotropic and anisotropic initial angular distributions:

• Isotropic distributions: The ensemble-averaged potential can satisfy the MNR condition globally. However, even infinitesimal initial anisotropies (such as a perturbation of order $10^{-22}$ in the angular distribution) are amplified, leading to spontaneous breaking of isotropy. Higher multipole moments (dipole, quadrupole, etc.) in the angular distributions grow, and the flavor evolution exhibits rich multipolar structure, deviating markedly from the single-angle (SA) prediction. Complete flavor conversion is generally obstructed by this instability.
• Anisotropic (forward-peaked) distributions: Forward-peaked angular profiles, motivated by the flux factor distributions extracted from hydrodynamic merger simulations, result in only a subset of angular bins (those with directions close to the local disk normal) encountering resonance simultaneously. In these bins, flavor conversion may occur nearly synchronously, while other bins undergo little or no transformation. The net effect is a non-uniform conversion pattern, strongly controlled by the population and shape of the angular modes.

3. Consequences of Neutrino Angular Distributions

The critical dependence of the flavor conversion outcome on the detailed angular structure of the neutrino field emerges as a central theme. Key consequences as established by simulation:

• For fixed physical conditions (matter density, electron fraction, neutrino densities), the degree and spatial extent of flavor conversion are dictated by the initial angular profiles.
• Even with the matter and neutrino potentials cancelling in the mean, if the angular distribution skews toward forward emission, only those modes participate in resonance-driven transformation.
• The development of multipole structure through spontaneous symmetry breaking is a generic feature in any system where even tiny seed anisotropy is present.

This critical sensitivity means that predictions for nucleosynthetic yields and (anti)neutrino capture rates above merger remnants demand a self-consistent treatment of neutrino transport and radiation field angular structure.

4. Hydrodynamical Simulation Inputs and Realistic MNR Scenarios

The paper employs data from a 2D hydrodynamical simulation of a black hole–torus remnant reflecting typical post-merger conditions (BH mass $3\,M_\odot$, torus mass $0.3\,M_\odot$, snapshot at 15 ms). Extracted local baryon densities, electron fractions, and neutrino number fluxes are used to:

• Map the regions in the $(x, z)$-plane above the accretion torus where the MNR condition is satisfied (i.e., where the total diagonal potential for certain modes crosses zero).
• Construct angular distributions informed by local flux factors, ranging from isotropic to forward-peaked sigmoid parametrizations.
• Evolve flavor quantum kinetic equations along specific trajectories passing through identified resonance layers.

Key findings from this procedure include:

• In physically motivated forward-peaked scenarios, only the most populated (forward) angular bins undergo substantial MNR-facilitated conversion.
• In regions above the disk where the matter and neutrino potentials cancel, flavor conversion can begin, but in many trajectories full conversion is truncated by the rapid growth of multipole structure or insufficient angular overlap with the resonance.
• The hydrodynamical context, reflecting the merger geometry and emission characteristics, is thus essential to any robust prediction of flavor transformation due to MNR.

5. Comparison to Single-Angle Models and Limiting Cases

In prior (single-angle) treatments, the MNR condition typically triggered global and often complete transformation of electron neutrinos, with antineutrino states largely reverting to their progenitor flavor after passage through resonance. The multi-angle simulation, even for initially isotropic distributions, produces flavor evolution that matches the SA solution only if perfect isotropy is maintained and no spontaneous symmetry breaking occurs. However, under any real-world perturbation or in anisotropic conditions, the evolution can deviate dramatically from the SA paradigm.

A summary comparison:

Initial Distribution	Resonance Realization	Outcome
Single-angle (SA)	Global (all modes simultaneously)	Complete $\nu_e\to\nu_x$; $\bar{\nu}_e$ unaffected
Isotropic (multi-angle, perfect)	Locally, with possible symmetry breaking	Onset of multipole growth; incomplete or non-uniform conversion
Anisotropic/forward-peaked	Resonance for specific angles only	Partial conversion in most populated modes; depends on emission profile

6. Astrophysical Implications: Nucleosynthesis and Heavy Elements

The rate of (anti)neutrino capture on nucleons in the outflows above neutron star merger remnants directly impacts the neutron-to-proton ratio ($Y_e$)—a key parameter for r-process (rapid neutron-capture) nucleosynthesis responsible for producing elements heavier than iron. MNR-facilitated flavor transformation can:

• Increase or decrease the $\nu_e$ flux, altering the $n/p$ ratio available for r-process.
• In scenarios with strong forward-peaking, drive partial conversion in precisely those trajectories that most efficiently impact the outflow composition.
• Inhibit or facilitate r-process depending on the interplay between flavor evolution and wind properties.

The outcome is therefore acutely dependent on accurate modeling of the neutrino field's angular distribution, which is set by the microphysics of neutrino emission, transport, and matter geometry in the merger environment. Prediction of nucleosynthetic yields thus requires multi-dimensional, multi-angle, time-dependent neutrino transport coupled with flavor evolution.

7. Summary and Outlook

Matter Neutrino Resonance in post-merger environments is a sensitive function of the cancellation between the matter and neutrino self-interaction potentials, modulated by the angular structure of the neutrino emission. Multi-angle and hydrodynamics-informed studies demonstrate that:

• Complete global flavor conversion by MNR is unlikely except under highly idealized or perfectly isotropic circumstances.
• Spontaneous breaking of isotropy and the restriction of resonance to select angular bins are generic features, fundamentally limiting the scope of MNR-driven transformation.
• Astrophysical predictions for nucleosynthesis and transient observables from mergers must incorporate detailed angular and temporal evolution of the neutrino field to assess the impact of MNR.
• Accurate treatment of the multi-angle flavor evolution, with consistent inputs from hydrodynamic simulations, is mandatory for quantitative modeling and interpretation of heavy-element synthesis in neutron star merger ejecta (Padilla-Gay et al., 22 Mar 2024).

The refined understanding of MNR underscores the importance of multidimensional, angle-resolved simulations for reliably capturing collective neutrino flavor conversion phenomena in merger astrophysics and their far-reaching consequences for cosmic nucleosynthesis.