Scalar-Mediated Neutrino Interaction

• Scalar-mediated NSI are new interactions introduced by scalar or pseudoscalar fields that perturb the neutrino mass matrix via Yukawa-type couplings.
• They distinctly alter neutrino flavor evolution in dense astrophysical environments, modifying collective oscillation patterns in supernovae.
• These interactions provide a unique probe into neutrino nature, potentially distinguishing Majorana from Dirac neutrinos through observable flavor transitions.

Scalar-mediated non-standard neutrino interaction (sNSI) refers to scenarios in which new scalar or pseudoscalar degrees of freedom introduce effective four-fermion operators coupling neutrinos to other neutrinos or to matter via Yukawa-type interactions. Unlike the conventional non-standard interactions generated by the exchange of vector mediators—which affect the neutrino matter potential—scalar mediation typically acts as a perturbation to the neutrino mass matrix itself. This difference results in distinctive phenomenological consequences for neutrino propagation, especially in dense astrophysical environments such as core-collapse supernovae.

1. Theoretical Structure of Scalar-Mediated NSI

Scalar-mediated NSI in the neutrino sector are described by an effective Lagrangian of the form

$$-\mathcal{L}_\text{int} = \frac{1}{2} g_{\alpha\beta} (\bar{\nu}_\alpha \nu_\beta) \phi + \frac{i}{2} h_{\alpha\beta} (\bar{\nu}_\alpha \gamma^5 \nu_\beta) \chi$$

where $\phi$ ($\chi$) is a scalar (pseudoscalar) mediator and $g_{\alpha\beta}$, $h_{\alpha\beta}$ are real, symmetric coupling matrices in flavor space (Yang et al., 2018). In the limit where the mediator mass $m_\phi$ (or $m_\chi$) is much larger than the neutrino energy, this gives rise—upon integrating out the heavy field—to an effective four-fermion contact interaction,

$$\mathcal{H}_\text{int} \simeq \frac{1}{8m_{\phi}^2} g_{\alpha\beta}g_{\xi\eta} (\bar{\nu}_\alpha \nu_\beta)(\bar{\nu}_\xi \nu_\eta) - \frac{1}{8m_{\chi}^2} h_{\alpha\beta}h_{\xi\eta} (\bar{\nu}_\alpha \gamma^5 \nu_\beta)(\bar{\nu}_\xi \gamma^5 \nu_\eta)$$

The mean-field reduction of these four-neutrino operators, accounting for anti-symmetrization and spinor structure, is essential for connecting the effective Hamiltonian to concrete physical processes such as flavor transformation above the neutrinosphere.

2. Dirac–Majorana Distinction and Ultrarrelativistic Limit

A crucial result is that in the ultrarelativistic limit, scalar/pseudoscalar mediated NSSI vanish for Dirac neutrinos due to the absence of right-handed components in the forward scattering matrix elements. For Dirac neutrinos, right-handed weak currents are suppressed and expectation values involving right-handed fields—necessary for generating a nonzero effective Hamiltonian—vanish. Conversely, for Majorana neutrinos, the field contraction structure (including charge-conjugated left-handed states) survives, so the scalar-mediated effective interaction is nonzero in forward scattering even at high energies (Yang et al., 2018). The mean-field exchange terms relevant for Majorana neutrinos thus yield an effective contribution: $$(\bar{\nu}_\alpha \nu_\beta)(\bar{\nu}_\xi \nu_\eta) \rightarrow -\frac{1}{2}\langle \bar{\nu}_{\alpha L} \gamma^\mu \nu_{\eta L}\rangle \left( \bar{\nu}_{\xi L}^C \gamma_\mu \nu_{\beta L}^C \right) + (\alpha\eta \leftrightarrow \xi\beta)$$ This is key for experimental tests: any observable effect of scalar NSSI in the collective oscillation signal of supernova neutrinos would be an indirect signature that the neutrino is a Majorana fermion.

3. Formulation of the Effective Single-Particle Hamiltonian

The mean-field evaluation leads to a single-particle Hamiltonian for scalar-mediated NSSI,

$$H_S({\bf r},{\bf \hat{q}}) = 4 \int (1 - \mathbf{\hat{p}} \cdot \mathbf{\hat{q}})\left\{ \tilde{\mathfrak{g}} [\rho^*({\bf r},p)dn_\nu({\bf r},p) - \bar{\rho}({\bf r},p)dn_{\bar{\nu}}({\bf r},p)] \tilde{\mathfrak{g}} \right\} dE_p$$

where $\rho$ and $\bar{\rho}$ are density matrices for neutrinos and antineutrinos, $dn_{\nu}$ is the momentum-space volume element, and the coupling matrix $\tilde{\mathfrak{g}}$ is constructed in flavor space as: $$\tilde{\mathfrak{g}} = \left( \frac{\sqrt{2}}{4} G_F \right)^{1/2} \begin{pmatrix} \alpha_1 & \alpha_2 & \alpha_2 \ \alpha_2 & \alpha_1 & \alpha_2 \ \alpha_2 & \alpha_2 & \alpha_1 \end{pmatrix}$$ with $\alpha_1$ controlling flavor-preserving (FP) NSSI and $\alpha_2$ flavor-violating (FV) NSSI. The total Hamiltonian for flavor evolution combines standard vacuum, matter, and (V–A)-type self-interaction terms with this additional scalar-mediated NSSI contribution: $$i \frac{d}{dr} S(r) = [H_V + H_M + H_{SI}] S(r), \quad H_{SI} = H_{(V-A)} + H_S$$

4. Phenomenological Impact on Supernova Collective Oscillations

Numerical simulations within the neutrino bulb model, incorporating the full multi-angle flavor evolution, demonstrate that inclusion of scalar-mediated NSSI alters the flavor transformation patterns above the neutrinosphere in several key ways (Yang et al., 2018):

• Flavor-Preserving (FP) NSSI ($\alpha_1$): These terms act to suppress or delay the onset of collective oscillations, in some cases resulting in a near-complete suppression of spectral swaps or flavor transitions, especially in the high-energy part of the spectrum.
• Flavor-Violating (FV) NSSI ($\alpha_2$): FV terms can counteract the suppression from FP interactions; sufficiently large $\alpha_2$ can actually promote flavor transformation even in scenarios where standard V–A interactions alone give little or no collective motion.
• Dirac–Majorana Effect: As noted, only Majorana neutrinos manifest these scalar NSSI modifications. Detection of prominent suppression or enhancement in collective oscillations, as compared to Standard Model expectations, would point directly to the Majorana nature of the neutrino.

The practical consequence is that the collective evolution of flavor states in a supernova can be “shut down” or “re-awakened” by tuning the relative strengths of $\alpha_1$ and $\alpha_2$.

5. Constraints and Discrimination of Neutrino Properties

Supernova neutrino detection offers a unique probe of scalar-mediated NSI:

• If observable flavor transformation patterns are found to deviate—exhibiting either unexpected suppression or enhancement consistent with scalar NSSI—this would indicate that the underlying neutrino–scalar coupling is at least of order the weak scale.
• Given the strict Dirac–Majorana dichotomy in the ultrarelativistic limit, supernova signals have the potential to distinguish between these fundamental neutrino natures.
• Current laboratory—terrestrial—constraints on scalar couplings (for $m_\phi$ above a few hundred MeV) are weak, so bounds from collective oscillation features in supernova bursts can cover unexplored parameter space.

This suggests that a future high-statistics Galactic supernova neutrino burst, if accompanied by precise modeling of the expected flavor evolution, could both tighten bounds on or even discover new scalar neutrino interactions inaccessible to accelerator or scattering experiments.

6. Summary of Core Equations

Quantity	Formula	Context
Interaction	$$-\mathcal{L}_\text{int} = \frac{1}{2} g_{\alpha\beta} (\bar{\nu}_\alpha \nu_\beta) \phi + \cdots$$	Basic scalar–neutrino Yukawa coupling
Four-fermion	$$\mathcal{H}_\text{int} \simeq \frac{1}{8m_\phi^2} g_{\alpha\beta}g_{\xi\eta} (\bar{\nu}_\alpha \nu_\beta)(\bar{\nu}_\xi \nu_\eta)$$	Heavy mediator, contact approximation
Effective $H_S$	$$H_S(r, \hat{q}) = 4 \int (1-\hat{p}\cdot\hat{q}) \{\tilde{g}[\rho^*-\bar{\rho}] \tilde{g} \} dE_p$$	Additional term in evolution Hamiltonian
Coupling matrix	$\tilde{g} = (\sqrt{2}/4 G_F)^{1/2} \times\left[\begin{smaLLMatrix}\alpha_1&\alpha_2&\alpha_2\\alpha_2&\alpha_1&\alpha_2\\alpha_2&\alpha_2&\alpha_1\end{smaLLMatrix}\right]$	Flavor-preserving ($\alpha_1$), flavor-violating ($\alpha_2$) interaction

7. Concluding Remarks

Scalar-mediated NSSI in core-collapse supernovae—if of strength comparable to Standard Model V–A neutrino self-interactions—introduce qualitatively new effects in collective flavor transformation, with suppression or promotion of oscillations controlled by the flavor symmetry of the coupling matrix. Definitive observation of such effects in a high-precision Galactic supernova neutrino burst could provide the first indirect but robust evidence that neutrinos are Majorana particles, simultaneously constraining models of neutrino mass and new scalar or pseudoscalar interactions at a level not accessible with current terrestrial experiments. This establishes supernova neutrino astronomy as a premier avenue for probing scalar NSSI and the underlying nature of the neutrino (Yang et al., 2018).