Non-Standard Neutrino Interactions — Testing the PMNS Framework

The PMNS mixing matrix is a powerful and well-tested description of neutrino oscillation. Its four physical parameters — three mixing angles and a CP-violating phase — account for the flavor transformations observed in solar, atmospheric, reactor, and accelerator neutrino experiments. Yet the PMNS framework rests on an assumption that is rarely stated explicitly: that the only mechanism by which neutrinos interact with matter during propagation is the Standard Model weak force.

This assumption is testable, and testing it is one of the active frontiers of oscillation physics. The framework for doing so goes by the name non-standard interactions, abbreviated NSI. The idea is both simple and consequential: if neutrinos couple to matter fermions through new physics not present in the Standard Model, those additional couplings would modify the effective Hamiltonian governing propagation — potentially shifting measured mixing parameters, introducing fake CP-violation signals, or generating new matter-induced flavor transitions with no Standard Model origin.

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The Standard Matter Hamiltonian

To understand NSI, it is useful to first recall the standard picture. A neutrino beam propagating through ordinary matter — rock, water, or the Earth’s mantle — encounters a medium of electrons, protons, and neutrons. The neutrino interacts with these particles via the weak force. Elastic forward scattering contributes a coherent phase to the neutrino wavefunction that modifies the oscillation Hamiltonian.

For a basis of three neutrino flavor states $({\nu }_e,{\nu }_{\mu },{\nu }_{\tau })$, the Hamiltonian in matter is:

$H=\frac{1}{2E}U\left(\begin{array}{ccc}0& 0& 0\\ 0& \Delta m_{21}^2& 0\\ 0& 0& \Delta m_{31}^2\end{array}\right)U^{†}+\left(\begin{array}{ccc}{V}_{CC}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

where $U$ is the PMNS matrix and the matter potential $V_{CC}=\sqrt{2}{G}_F{n}_e$ arises from coherent ${\nu }_e$-electron scattering via W exchange. Only ${\nu }_e$ receive this contribution at tree level; ${\nu }_{\mu }$ and ${\nu }_{\tau }$ have no Standard Model forward-scattering amplitude on electrons at leading order.

This is the MSW effect, and it is solidly established by solar and atmospheric neutrino measurements. The question NSI physicists ask is: what if there are additional terms?

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Parameterizing Non-Standard Interactions

Non-standard interactions are parameterized as effective four-fermion operators, added to the Standard Model Lagrangian:

${\mathcal{L}}_{NSI}=-\frac{G_F}{\sqrt{2}}{\sum }_{\alpha ,\beta ,f,P}{\epsilon }_{\alpha \beta }^{f,P}\left({\bar{\nu }}_{\alpha }{\gamma }^{\mu }L{\nu }_{\beta }\right)\left(\bar{f}{\gamma }_{\mu }Pf\right)$

Here $\alpha ,\beta \in \{e,\mu ,\tau \}$ label neutrino flavors, $f\in \{e,u,d\}$ labels the matter fermion (electron, up quark, down quark), and $P\in \{L,R\}$ labels the chirality of the matter current. The dimensionless parameters ${\epsilon }_{\alpha \beta }^{f,P}$ measure the strength of each new coupling relative to the Fermi constant $G_F$.

The Standard Model itself contributes ${\epsilon }_{ee}^{e,L}=1+{\epsilon }_{ee,\text{NSI}}^{e,L}$ and small contributions from neutral-current Z exchange, but the new-physics content is entirely in the ${\epsilon }_{\alpha \beta }$ deviations from zero.

For neutral matter with equal numbers of protons and neutrons (a reasonable approximation for the Earth), the effective matter Hamiltonian with NSI becomes:

$H_{\text{mat}}=\sqrt{2}{G}_F{n}_e\left(\begin{array}{ccc}1+{\epsilon }_{ee}& {\epsilon }_{e\mu }& {\epsilon }_{e\tau }\\ {\epsilon }_{e\mu }^{\ast }& {\epsilon }_{\mu \mu }& {\epsilon }_{\mu \tau }\\ {\epsilon }_{e\tau }^{\ast }& {\epsilon }_{\mu \tau }^{\ast }& {\epsilon }_{\tau \tau }\end{array}\right)$

where the ${\epsilon }_{\alpha \beta }$ are combinations of the fundamental couplings weighted by the local fermion densities. The diagonal terms shift individual flavor potentials. The off-diagonal terms — absent in the Standard Model — introduce direct flavor mixing driven by the medium itself, independent of vacuum mixing angles or mass differences.

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Why NSI Are Expected in Many Beyond-SM Theories

Non-standard interactions are not merely a phenomenologist’s bookkeeping device. They appear generically in many well-motivated extensions of the Standard Model.

Seesaw models with light mediators. The most economical explanation for neutrino mass involves extending the SM with heavy right-handed neutrinos. If, however, the mediating field has a mass at or below the TeV scale, integrating it out leaves effective neutrino-matter operators that contribute to the NSI $\epsilon$ parameters. Models of this type arise naturally in left-right symmetric theories, B-L extensions, and gauged lepton-number models.

Leptoquarks. Theories with leptoquarks — particles that couple both to quarks and leptons — generate NSI between neutrinos and quarks at tree level. Many leptoquark models proposed to explain B-physics anomalies also predict NSI in the few-percent range, potentially observable in near-future experiments.

Extra dimensions. In theories with extra spatial dimensions, the higher-dimensional gauge bosons that propagate in the bulk generate effective four-fermion operators when the extra dimensions are integrated out, producing NSI at the level of $(m_W/M_{\text{KK}}{)}^2$ where $M_{\text{KK}}$ is the Kaluza-Klein mass.

Dark sector mediators. Light $Z^{\prime }$ bosons coupled to a combination of baryon number and lepton number ($B-L$) can mediate new neutrino-matter interactions, producing NSI whose strength depends on the $Z^{\prime }$ mass and coupling. Direct detection experiments and precision electroweak tests constrain such models independently, allowing the NSI constraints to be interpreted in model-specific terms.

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The Parameter Degeneracy Problem

The chief experimental challenge with NSI is not detection sensitivity — it is degeneracy. Certain regions of NSI parameter space produce oscillation signals that are nearly identical to standard oscillation with different PMNS parameters. The most studied example is the LMA-Dark degeneracy in the solar sector.

In the standard analysis, the large mixing angle (LMA) solution — with ${\theta }_{12}\approx 33°$ and $\Delta m_{21}^2\approx 7.5\times 10^{-5}$ eV² — explains all solar neutrino data. But there exists a second solution, LMA-Dark, with ${\theta }_{12}>45°$ (which would be in the second octant, normally excluded for solar neutrinos by matter effects) combined with a large diagonal NSI parameter ${\epsilon }_{ee}^d-{\epsilon }_{\mu \mu }^d\approx -0.36$. This combination flips the sign of the effective matter potential and, with sufficiently large NSI, reproduces the same day-night asymmetry and energy-dependent survival probability as the standard LMA solution.

The LMA-Dark degeneracy cannot be broken by any single solar neutrino measurement. It requires comparing solar neutrino data with reactor antineutrino data (which propagate through vacuum, with no matter effects) or with CEνNS measurements that constrain the quark NSI parameters directly.

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Experimental Constraints

CEνNS at COHERENT

The COHERENT experiment at the Oak Ridge Spallation Neutron Source provided the first measurement of coherent elastic neutrino-nucleus scattering (CEνNS) in 2017, followed by improved results using a CsI target and a liquid argon target. CEνNS is mediated predominantly by the Z boson in the Standard Model and is therefore directly sensitive to neutral-current NSI parameters ${\epsilon }_{\alpha \beta }^f$.

The COHERENT constraints are currently among the most stringent model-independent bounds on quark NSI. The CsI and LAr datasets together constrain:

$|{\epsilon }_{ee}^{u,d}|\lesssim 0.2,|{\epsilon }_{e\mu }^{u,d}|,|{\epsilon }_{e\tau }^{u,d}|\lesssim 0.1$

at 90% confidence level, after marginalizing over Standard Model nuclear physics uncertainties. Reactor-based CEνNS experiments — CONNIE, CONUS, NUCLEUS, RED100 — provide complementary data at lower neutrino energies and different target nuclei, where the nuclear form factor suppression is less severe.

Solar Neutrino Experiments

Solar neutrino experiments are sensitive to matter NSI through the propagation potential inside the Sun and, for ${\nu }_e$ specifically, through charged-current production NSI at the solar core. Borexino’s precision measurement of the ${}^7$Be, ${}^8$B, and CNO neutrino fluxes at different energies and SNO’s three-channel measurement provide the principal solar-sector NSI constraints.

The combination of Borexino, SNO, and KamLAND data breaks the LMA-Dark degeneracy at approximately $3\sigma$ confidence — sufficient to disfavor but not definitively exclude the alternative solution. JUNO’s projected sensitivity to the spectral shape of reactor antineutrinos will tighten this further.

Long-Baseline Accelerator Experiments

T2K and NOvA probe matter NSI through the 295 km and 810 km baselines, respectively. Neutrinos traveling through rock experience the Earth’s mantle potential, and off-diagonal NSI parameters ${\epsilon }_{e\mu }$, ${\epsilon }_{e\tau }$, ${\epsilon }_{\mu \tau }$ introduce flavor transitions that can mimic or obscure the CP-violation signature from ${\delta }_{CP}$.

Current T2K and NOvA constraints on matter NSI are weaker than the solar and CEνNS bounds for most parameters — the baselines and energies are not optimized for NSI sensitivity. DUNE, with its 1,300 km baseline, broad-band beam, and near detector complex (PRISM technique allowing $L/E$ scans), is designed with NSI sensitivity as an explicit secondary goal. DUNE’s projected sensitivity should improve bounds on $|{\epsilon }_{e\mu }|$ and $|{\epsilon }_{e\tau }|$ by roughly an order of magnitude beyond current limits.

Atmospheric Neutrinos

IceCube-DeepCore and Super-Kamiokande atmospheric analyses have sensitivity to NSI across a wide range of $L/E$, from sub-GeV events with baselines of tens of kilometres to multi-TeV events that traverse the Earth’s core. The matter density along each trajectory can be computed from seismic Earth models (PREM), allowing NSI to be tested against a precisely known matter potential profile.

IceCube’s analysis of three years of DeepCore data constrains $|{\epsilon }_{\mu \tau }|<0.02$ at 90% CL — among the tightest bounds on any off-diagonal NSI parameter. This result is interesting because ${\epsilon }_{\mu \tau }$ appears in the $({\nu }_{\mu },{\nu }_{\tau })$ subspace of the matter Hamiltonian and directly modifies atmospheric oscillation without any Standard Model analogue.

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NSI and the Mass Ordering

One of the most important practical consequences of NSI for near-future experiments concerns the mass ordering determination. The standard strategy — used by JUNO via reactor antineutrino spectroscopy, and by DUNE and Hyper-Kamiokande via matter effects in long-baseline beams — assumes the Standard Model matter potential.

If NSI are present at the level of $|\epsilon |\sim 0.05$, they can in principle flip the apparent mass ordering: a scenario with true inverted ordering plus appropriately chosen NSI can mimic the oscillation pattern expected for normal ordering in the Standard Model. This degeneracy is known as the sign-$\Delta m^2$–$\epsilon$ degeneracy.

Breaking it requires independent constraints on the NSI parameters — specifically from CEνNS, solar neutrinos, or a short-baseline measurement that is insensitive to matter effects. The combination of JUNO (vacuum-dominated spectral measurement) with DUNE (matter-effect-dominated appearance and disappearance) provides complementary handles that largely resolve the degeneracy. This is one of the principal motivations for running multiple experiment types simultaneously rather than relying on a single flagship measurement.

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NSI at Production and Detection

The discussion so far has focused on propagation NSI — modifications to the matter Hamiltonian. But the NSI operator also affects how neutrinos are produced in weak decays (source NSI) and how they are detected (detector NSI).

Source NSI arise when the four-fermion operator mediating pion or muon decay differs from the Standard Model prediction. Detector NSI modify the cross-sections for charged-current and neutral-current detection processes. In practice, source and detector NSI often enter oscillation analyses in combination with propagation NSI, and certain combinations are unobservable because they amount to a redefinition of the flavor basis.

A systematic treatment distinguishes physical NSI combinations — those that cannot be removed by unitary redefinition — from unphysical ones. The number of genuinely constrained combinations is smaller than the full set of ${\epsilon }_{\alpha \beta }$ parameters, which is why careful statistical analyses must account for the parameter space structure.

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Current Status and Outlook

A global fit to all available neutrino data in the presence of NSI — performed by several groups including NuFIT and the Valencia group — yields the following broad conclusions as of 2025:

Most diagonal NSI parameters (${\epsilon }_{ee}$, ${\epsilon }_{\mu \mu }$, ${\epsilon }_{\tau \tau }$) are constrained to below 20–50% of the weak coupling at 90% CL. Off-diagonal parameters are generally better constrained: $|{\epsilon }_{e\mu }|\lesssim 0.10$, $|{\epsilon }_{e\tau }|\lesssim 0.15$, $|{\epsilon }_{\mu \tau }|\lesssim 0.015$. The LMA-Dark solution remains disfavored but not excluded in combined analyses.

The next decisive steps will come from DUNE’s near detector (which can measure neutrino cross-sections with unprecedented precision, separating source from propagation NSI), from improved CEνNS measurements at spallation sources and reactors, and from JUNO’s vacuum-dominated spectral measurement. Together, these should push NSI sensitivity into the percent-level regime across most of the parameter space.

If NSI are discovered — if some ${\epsilon }_{\alpha \beta }$ is found to be non-zero at high significance — it would be the first direct evidence for physics beyond the Standard Model in the neutrino sector beyond the mass-and-mixing structure itself. The mediating particle’s identity, mass, and coupling would then become the subject of dedicated searches at colliders and in precision experiments.

If nothing is found, the upper bounds constrain the allowed parameter space of beyond-SM models, closing off whole classes of theories that would otherwise be difficult to test at collider energies.

Either outcome is informative. The PMNS framework has been tested to perhaps the five-percent level in most parameters. The next decade will test it at the one-percent level — and at that precision, the universe may not cooperate with the assumption of simplicity.

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Related reading: the MSW matter effect article covers the Standard Model matter potential in detail. The COHERENT result at Oak Ridge discusses the CEνNS measurement that currently provides the strongest model-independent NSI bounds. DUNE’s physics programme outlines the long-baseline sensitivity expected from the experiment.