Neutrino Polarizability Operator

• The neutrino polarizability operator is a dimension-7 interaction that couples neutrinos to two photons via light scalar or pseudoscalar mediators.
• It induces distinct scattering processes, such as monophoton signatures and two-shower events, observable in both terrestrial and collider experiments.
• Experimental constraints from MiniBooNE, XENONnT, and projected DUNE sensitivities provide practical limits on the operator's coupling strength and mediator mass.

Neutrino polarizability refers to the effective electromagnetic coupling of neutrinos to two photons, encoded in a dimension-7 operator that generically arises when neutrinos interact with new, light states—typically (pseudo)scalar mediators. This operator is severely suppressed in the Standard Model due to the small neutrino masses, but can be significantly enhanced in the presence of such mediators, leading to potentially observable signatures in terrestrial and astrophysical settings. The structure, model realization, collider phenomenology, and experimental status of the neutrino polarizability operator are summarized below.

1. Operator Structure and Field-Theoretic Definitions

At energies below the electroweak scale, the leading gauge-invariant interactions coupling two neutrinos to two photons are given by the so-called Rayleigh operators. For Majorana neutrinos, these take the form

$$\mathcal{L}_{\text{pol}} = \tfrac12 \,\alpha_{\nu,ij}\,(\bar\nu^c_i P_L \nu_j) F_{\mu\nu} F^{\mu\nu} + \tfrac12\,\tilde\alpha_{\nu,ij}\,(\bar\nu^c_i P_L \nu_j) F_{\mu\nu} \tilde{F}^{\mu\nu} + \text{h.c.}$$

where $\nu_i$ are neutrino mass eigenstates, $F_{\mu\nu}$ is the electromagnetic field strength, and $$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$ is its dual. The Wilson coefficients $\alpha_{\nu,ij}$ (CP-even) and $\tilde\alpha_{\nu,ij}$ (CP-odd) are dimensionful ($[\text{mass}]^{-3}$) and encode new physics effects.

In the presence of sterile neutrinos, a generalized form couples active and sterile states: $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$ where $C_{ij}, C'_{ij}$ are dimensionless Wilson coefficients and $\Lambda$ is the new physics scale (Gehrlein et al., 8 Dec 2025, Gehrlein et al., 17 Jun 2025).

The corresponding nonrelativistic polarizabilities are $\nu_i$0, where $\nu_i$1 (Carey et al., 22 Aug 2025).

2. Ultraviolet Completion via Light Scalar or Pseudoscalar Mediators

Tree-level enhancement of the polarizability operator is realized by introducing a light pseudoscalar (or scalar) field, $\nu_i$2, with Lagrangian terms

$\nu_i$3

where $\nu_i$4 is the $\nu_i$5-photon coupling (GeV$\nu_i$6) and $\nu_i$7 is the neutrino–$\nu_i$8 Yukawa (Carey et al., 22 Aug 2025, Bansal et al., 2022, Gehrlein et al., 17 Jun 2025). After integrating out $\nu_i$9 for $F_{\mu\nu}$0

$F_{\mu\nu}$1

The matching to the effective operator is

$F_{\mu\nu}$2

where $F_{\mu\nu}$3 and $F_{\mu\nu}$4 denote the neutrino and photon couplings, respectively (Gehrlein et al., 17 Jun 2025).

In many Majoron or ALP-motivated models, $F_{\mu\nu}$5, allowing the enhancement from a light $F_{\mu\nu}$6 to compensate the suppression from $F_{\mu\nu}$7 (Bansal et al., 2022). In UV completions such as the inverse-seesaw Majoron model, with $F_{\mu\nu}$8 TeV, and suitably chosen triplet fermions, effective scales $F_{\mu\nu}$9 MeV–GeV are accessible.

3. Phenomenology: Scattering Processes and Signal Topologies

The neutrino polarizability operator induces neutrino scattering processes with an additional hard photon in the final state. The relevant signatures are:

a. Neutrino–Electron Scattering

$$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$0

The cross section is

$$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$1

with enhancement for hard photons ($$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$2 large) and small mediator virtuality ($$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$3) (Carey et al., 22 Aug 2025).

b. Coherent Neutrino–Nucleus Scattering

$$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$4

The squared amplitude yields

$$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$5

where $$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$6 is the Helm or another empirical nuclear form factor (Gehrlein et al., 17 Jun 2025, Carey et al., 22 Aug 2025).

These processes give rise to two primary experimental topologies in liquid argon detectors (Carey et al., 22 Aug 2025, Gehrlein et al., 17 Jun 2025):

• 1EM: A single monophoton-like electromagnetic (EM) shower from coherent $$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$7–Ar scattering, no hadrons.
• 2EM: Two well-separated EM showers (photon + $$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$8), typical of $$\tilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$9–$\alpha_{\nu,ij}$0 scattering, no hadrons.

Backgrounds include SM $\alpha_{\nu,ij}$1 CC events, NC $\alpha_{\nu,ij}$2, and elastic $\alpha_{\nu,ij}$3–$\alpha_{\nu,ij}$4 with bremsstrahlung. Monte Carlo implementations use MadGraph5 for signals and NuWro for neutrino–argon backgrounds.

4. Experimental Sensitivities and Limits

Existing Bounds

Current constraints derive from both terrestrial and astrophysical sources:

• MiniBooNE (monophoton): for $\alpha_{\nu,ij}$5: $\alpha_{\nu,ij}$6–$\alpha_{\nu,ij}$7, $\alpha_{\nu,ij}$8 (light-mediator limit).
• NOMAD: $\alpha_{\nu,ij}$9 at $\tilde\alpha_{\nu,ij}$0 GeV (Gehrlein et al., 17 Jun 2025).
• XENONnT (solar $\tilde\alpha_{\nu,ij}$1): $\tilde\alpha_{\nu,ij}$2 independent of $\tilde\alpha_{\nu,ij}$3 (Carey et al., 22 Aug 2025).
• BaBar: constrains via $\tilde\alpha_{\nu,ij}$4, $\tilde\alpha_{\nu,ij}$5; limits on $\tilde\alpha_{\nu,ij}$6, and thus on $\tilde\alpha_{\nu,ij}$7 (Bansal et al., 2022, Gehrlein et al., 17 Jun 2025).
• Astrophysical/Cosmological: Supernova SN1987A cooling, BBN, and CMB set stringent limits for $\tilde\alpha_{\nu,ij}$8–100 MeV and constrain $\tilde\alpha_{\nu,ij}$9, $[\text{mass}]^{-3}$0 over broad parameter space (Bansal et al., 2022).

Projected Sensitivities

DUNE Near Detector (ND) will provide leading sensitivity:

• For $[\text{mass}]^{-3}$1 MeV, the 90% CL upper limits are (Carey et al., 22 Aug 2025):
  • 1EM channel
  • 1 yr, 10% syst: $[\text{mass}]^{-3}$2
  • 10 yr, 3% syst: $[\text{mass}]^{-3}$3
  • 10 yr, statistics-only: $[\text{mass}]^{-3}$4
  • 2EM channel is weaker by $[\text{mass}]^{-3}$5–30×.

These correspond to sensitivities on $[\text{mass}]^{-3}$6 ($[\text{mass}]^{-3}$7 MeV, 1 yr). Varying $[\text{mass}]^{-3}$8 from 10 MeV to 1 GeV, DUNE-ND probes untested parameter space above $[\text{mass}]^{-3}$9 MeV, substantially beyond existing terrestrial or supernova bounds (Gehrlein et al., 17 Jun 2025).

Future SBN (SBND, ICARUS) will approach $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$0 ($$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$1 GeV) (Gehrlein et al., 17 Jun 2025).

5. Extensions, Sterile Neutrinos, and Anomalies

Active–sterile neutrino polarizability generalizes the operator to involve one active and one sterile neutrino. The relevant effective Lagrangian is (Gehrlein et al., 8 Dec 2025): $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$2 Realizations via a light mediator can explain the MiniBooNE low-energy excess through softened monophoton kinematics, with best-fit model points at $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$3 MeV, $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$4 MeV, $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$5 GeV$$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$6 (Gehrlein et al., 8 Dec 2025).

Alternative UV completions include loop-induced operators via SM charged leptons, singly-charged scalars (Zee-type), and dark-pion or ALP variants mixing with $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$7. However, one-loop models are highly suppressed and not phenomenologically relevant at current accelerator sensitivities (Bansal et al., 2022, Gehrlein et al., 8 Dec 2025).

6. Implications for Axion-Like and Majoron Physics

The polarizability operator is sensitive to ALP and Majoron model parameters. For ALPs, $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$8, $$\mathcal{L}_{\text{pol}} = \frac{C_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{C'_{ij}}{\Lambda^3}\,\overline{N}_j \nu_i F_{\mu\nu} F^{\mu\nu} + \text{h.c.}$$9 (modulo loop factors), probing $C_{ij}, C'_{ij}$0–$C_{ij}, C'_{ij}$1 GeV for MeV–GeV mediator masses (Gehrlein et al., 17 Jun 2025). In Majoron models, the neutrino coupling arises from lepton-number breaking, with possible $C_{ij}, C'_{ij}$2 up to $C_{ij}, C'_{ij}$3, making photonic searches complementary to traditional Majoron searches (Bansal et al., 2022).

7. Summary Table: Representative Sensitivity and Constraints

Experiment	Observable	Limit on $C_{ij}, C'_{ij}$4 ($C_{ij}, C'_{ij}$5)	Limit on $C_{ij}, C'_{ij}$6 ($C_{ij}, C'_{ij}$7)
MiniBooNE	Monophoton	$C_{ij}, C'_{ij}$8	$C_{ij}, C'_{ij}$9–$\Lambda$0
XENONnT	DM/nuclear recoil	$\Lambda$1	$\Lambda$2
DUNE-ND (1 yr)	Monophoton/1EM	$\Lambda$3	$\Lambda$4
DUNE-ND (10 yr, stat-only)	Monophoton/1EM	$\Lambda$5	$\Lambda$6

The above summarizes only selected channels and $\Lambda$7 MeV—for full coverage, see (Carey et al., 22 Aug 2025, Gehrlein et al., 17 Jun 2025, Bansal et al., 2022).

Neutrino polarizability thus provides a rare, gauge-invariant handle on new physics coupling neutrinos to photons, with significant implications for both terrestrial intensity-frontier experiments and fundamental axion/Majoron-extended neutrino sectors. The most sensitive future probes are monophoton-like channels with suppressed hadronic activity, achievable at DUNE-ND and related detectors (Carey et al., 22 Aug 2025, Gehrlein et al., 17 Jun 2025, Bansal et al., 2022, Gehrlein et al., 8 Dec 2025).