Non-Standard Neutrino Interactions (NSI)

• NSI are hypothetical four-fermion interactions of neutrinos with matter that extend beyond Standard Model weak processes.
• NSI modify neutrino oscillation dynamics by altering production, detection, and propagation, thereby influencing experimental observations.
• NSI studies provide insights into UV completions through heavy mediator or scalar exchanges, linking collider, oscillation, and cosmological probes.

Non-Standard Interactions (NSI) refer to hypothetical four-fermion interactions of neutrinos with matter fields beyond the Standard Model (SM) weak interactions. NSI modify coherent forward scattering, neutrino propagation in matter, neutrino production and detection, and can arise in a variety of new physics scenarios, typically as higher-dimensional operators in effective field theory (EFT). The NSI phenomenology extends across terrestrial oscillation experiments, astrophysical and cosmological environments, and high-energy collider searches, providing a unique window into sub-weak new interactions and the ultraviolet (UV) completions that generate them.

1. Operator Formalism and Theoretical Structure

NSI are formulated as dimension-six or dimension-eight four-fermion operators of the general form: $$\mathcal{L}_{\rm NSI} = -2 \sqrt{2} G_F \sum_{f=e,u,d} \sum_{P=L,R} \varepsilon^{fP}_{\alpha\beta} \; (\overline{\nu}_\alpha \gamma^{\mu} P_L \nu_\beta) \;\; (\overline{f} \gamma_\mu P f),$$ where $\alpha,\beta = e,\mu,\tau$ denote neutrino flavors, $f$ is a first-generation SM fermion, and $P$ is the left/right-handed chiral projector. The dimensionless coefficients $\varepsilon^{fP}_{\alpha\beta}$ encode the strength and flavor structure of the new interactions relative to the Fermi constant $G_F$ (Ohlsson, 2012, Dev et al., 2019).

Gauge invariance at the electroweak scale generally mandates that any neutral-current NSI operator $$(\overline{\nu}_\alpha \gamma^\mu P_L \nu_\beta)(\overline{f} \gamma_\mu P f)$$ arises from $SU(2)_L \times U(1)_Y$-invariant combinations, leading to the unavoidable presence of charged-lepton partners at dimension-6 (Davidson et al., 2011, Davidson et al., 2019). In some UV completions, these dangerous dimension-6 terms can be suppressed by cancellations, leaving only higher-dimensional (dimension-8) momentum-dependent NSI operators of the schematic form: $$O_8 = \frac{1}{\Lambda_8^4} (\overline{q} \gamma^\mu P_{L} q)(H \overline{\ell} \gamma_\mu H \ell) \rightarrow \frac{v^2}{\Lambda_8^4} (\overline{q} \gamma^\mu P_{L} q)(\overline{\nu} \gamma_\mu \nu),$$ after Higgs symmetry breaking (Davidson et al., 2011, Davidson et al., 2011). For TeV-scale mediators ($\Lambda_8 \lesssim 2$ TeV), this yields $\varepsilon \sim v^2/\Lambda_8^4 \gtrsim 10^{-4}$.

2. NSI in Neutrino Propagation and Oscillation

In a medium, the flavor evolution Hamiltonian in the presence of NSI is (Coloma, 2015, Gouvêa et al., 2015, Ohlsson, 2012): $$H = H_{\rm vac} + H_{\rm matter}^{\rm SM} + H_{\rm NSI}$$

$$= \frac{1}{2E}\, U\,\text{diag}(0, \Delta m_{21}^2, \Delta m_{31}^2)\,U^\dagger + \sqrt{2} G_F N_e\,\text{diag}(1,0,0) + \sqrt{2} G_F N_e\,\varepsilon,$$

where $\varepsilon$ is the matrix of effective NSI parameters in matter,

$$\varepsilon_{\alpha\beta} = \sum_{f,P} \frac{N_f}{N_e} \varepsilon^{fP}_{\alpha\beta}$$

with $N_f$ the density of fermion $f$. Diagonal elements, $\varepsilon_{\alpha\alpha}$, alter the potential experienced by each flavor; off-diagonal terms, $\varepsilon_{\alpha\beta}$ ($\alpha \neq \beta$), induce new flavor transitions.

In three-flavor propagation, these modifications lead to changes in the location and nature of the Mikheyev–Smirnov–Wolfenstein (MSW) resonances, effective mixing angles, and can introduce new degeneracies in parameter extraction—particularly between the mass hierarchy, the octant of $\theta_{23}$, and the CP phase $\delta$, and with the appearance of the so-called LMA-dark solution (Coloma, 2015, Farzan et al., 2017). Equation (3.9)-(3.12) in (Ohlsson, 2012) gives the matter Hamiltonian including NSI: $$H_{\rm mat} = \sqrt{2} G_F N_e \begin{pmatrix} 1+\varepsilon_{ee} & \varepsilon_{e\mu} & \varepsilon_{e\tau} \ \varepsilon_{e\mu}^* & \varepsilon_{\mu\mu} & \varepsilon_{\mu\tau} \ \varepsilon_{e\tau}^* & \varepsilon_{\mu\tau}^* & \varepsilon_{\tau\tau} \end{pmatrix}.$$

3. Experimental Probes and Phenomenology

Long-Baseline Oscillation Experiments: The Deep Underground Neutrino Experiment (DUNE) will probe NSI parameters substantially below current bounds. DUNE is projected to reach $|\varepsilon_{e\mu}| < 0.073$, $|\varepsilon_{e\tau}| < 0.25$, $|\varepsilon_{\mu\tau}| < 0.035$ at 90% CL, and will definitively exclude the LMA-dark solution (characterized by $$\varepsilon_{ee} - \varepsilon_{\tau\tau} \approx -3$$), breaking degeneracies with mass ordering and CP phase (Coloma, 2015, Gouvêa et al., 2015).

Near Detector Measurements: Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) at DUNE ND is expected to probe vector NSI parameters $|\varepsilon_{\mu\mu}^u|, |\varepsilon_{\mu\mu}^d|$ down to $\sim 10^{-4}$ (statistical limit), though systematics at the per-mille level are necessary to reach this (Freitas et al., 2 May 2025).

Short-Baseline and Solar Experiments: NSI can shift survival probabilities, induce or mask spectral upturns in the solar $P_{ee}$, and particular NSI parameters can lead to the LMA-dark region solution (Coloma, 2015, Dutta et al., 2017). Solar and reactor experiments provide orthogonal constraints on the combinations $\varepsilon_{ee}-\varepsilon_{\mu\mu}$ and $\varepsilon_{e\tau}$.

High-Energy Colliders: Collider searches access NSI at high momentum transfers. LEP2 places strong bounds on NSI involving electrons, with $\varepsilon \lesssim 10^{-2} - 10^{-3}$ (Davidson et al., 2011, Davidson et al., 2011). At the LHC, $pp \to W^+W^- \ell\ell$ processes probe quark NSI coefficients, with expected sensitivity down to $|\varepsilon| \gsim 3 \times 10^{-3}$ for the contact-interaction limit at 14 TeV, $100\,\rm{fb}^{-1}$ (Davidson et al., 2011, Freitas et al., 2 May 2025).

Cosmology: NSI involving electrons can modify neutrino decoupling, leaving imprints as shifts in the effective number of relativistic degrees of freedom, $N_{\rm eff}$, in the early Universe. The maximal NSI-induced shift from allowed $\varepsilon$ values is $\Delta N_{\rm eff}\lesssim 0.02$ for $|\varepsilon|\sim0.3$, mostly via changes to scattering/annihilation rates; current and future CMB+LSS measurements can probe certain flavor and Lorentz structures, especially involving tau, that are challenging for lab experiments (Salas et al., 2021, Martínez-Miravé, 2021).

4. Ultraviolet Completion and Theoretical Constraints

NSI arise in EFT as remnants of heavy mediator exchange or loop processes. Simple UV completions include:

• Heavy $Z'$ bosons: $U(1)'$ gauge extensions, with anomaly-free charge assignments (e.g., $B-L$, $L_{\mu}-L_{\tau}$), generate NSI with characteristic flavor structures. The effective NSI parameters are $$\varepsilon_{\alpha\beta}^f= g'^2 Q^f Q^{\nu_\alpha}/(2\sqrt{2} G_F M_{Z'}^2)$$. Collider and CE$\nu$NS bounds typically dominate (Heeck et al., 2018, Freitas et al., 2 May 2025).
• Leptoquarks and Heavy Neutral Leptons: Scalar or vector leptoquarks, and singlet $N$ states, can generate flavor-specific NSI (e.g., large muon-philic interactions or via sterile mixing), sometimes escaping direct charged-lepton bounds if the mediators are heavy and flavor-aligned (Freitas et al., 2 May 2025).
• Loop-induced Mechanisms: NSI can be generated radiatively by hidden-sector scalars, allowing sizable $\varepsilon$ in flavor off-diagonal channels while evading charged-lepton flavor-violation (CLFV) constraints, provided appropriately light and secluded mediators (Bischer et al., 2018, Bellazzini et al., 2010).

However, generic gauge-invariant NSI models at dimension-6 induce charged-lepton contact interactions, tightly constrained by CLFV searches. The flavor-changing (e.g., $\varepsilon_{\mu e}$) NSI are limited to $|\varepsilon|\lesssim 10^{-5}$–$10^{-4}$ by $\mu\to e$ conversion for heavy mediators; $\tau\to\ell$ channels are limited at $|\varepsilon|\lesssim 10^{-2}$ unless tuned cancellations or light mediator scenarios are invoked (Davidson et al., 2019).

5. Supernova and Astrophysical Implications

In the supernova environment, NSI can substantially alter the pattern of flavor conversion, resonance structure, and even the observable neutronization-burst signal. The three-flavor Hamiltonian becomes

$$H(r) = H_{\rm vac} + V_e(r) \left[ \operatorname{diag}(1,0,0) + \varepsilon \right]$$

with matter profiles $V_e(r)$ varying from nuclear to low density (Jana et al., 2024, Stapleford et al., 2016). Key phenomena include:

• Level order inversion: Sufficiently large diagonal NSI ($\varepsilon_{\tau\tau}\gtrsim 0.33$) can invert the high-density energy eigenstate ordering, swapping the normal and inverted ordering predictions for the time-binned $\nu_e$ signal at DUNE (Jana et al., 2024).
• Inner "C" resonance: Off-diagonal NSI can trigger new level crossings deep in the core, controlled by the effective $\varepsilon_{e\tau}(r)$ and density profile (Jana et al., 2024).
• Neutronization-burst signatures: NSI can cause the re-appearance or disappearance of the neutronization burst peak, potentially mimicking an inverted ordering in the presence of large $\varepsilon_{\tau\tau}$ even for a normal hierarchy (Jana et al., 2024, Das et al., 2011).
• Explosive dynamics and nucleosynthesis: NSI-induced flavor swaps can impact the $\nu_e$ and $\bar{\nu}_e$ spectra at small radii, affecting shock reheating and the electron fraction $Y_e$, with implications for supernova explosion mechanics and heavy-element yields (Stapleford et al., 2016).

Table: Representative NSI Sensitivities and Constraints

Experiment	Channel / Observable	Typical $\varepsilon$ Reach
DUNE (LBL osc.)	Oscillation ($\nu$-Earth)	$|\varepsilon_{e\mu}|<0.073$, $|\varepsilon_{e\tau}|<0.25$ (Coloma, 2015)
LEP2	$e^+e^-\to \ell^+\ell^-$	$\varepsilon \lesssim 10^{-2}-10^{-3}$ (Davidson et al., 2011)
LHC (14 TeV, 100 fb$^{-1}$)	$pp\to W^+W^-\ell\ell$	$\varepsilon \gtrsim 3 \times 10^{-3}$ (Davidson et al., 2011)
CE$\nu$NS (COHERENT)	$\nu$-nucleus	$$|\varepsilon_{ee}|,|\varepsilon_{\mu\mu}|\lesssim 0.1$$ (Dev et al., 2019)
Cosmology (CMB-S4)	$N_{\rm eff}$	$|\varepsilon| \gtrsim 0.3$ (Salas et al., 2021, Martínez-Miravé, 2021)

6. Scalar NSI and Alternative Structures

Beyond the canonical vector-type NSI, scalar NSI have received recent attention for their qualitatively different phenomenology. Scalar NSI (arising from exchange of a light scalar) generate matter-dependent corrections to the neutrino mass matrix: $$\delta M_{\alpha\beta} = \sum_f \frac{n_f y_f Y_{\alpha\beta}}{m_\phi^2},$$ which shift both the effective mass splittings and mixing angles in matter by an energy-independent amount (Ge et al., 2018). Unlike vector NSI, scalar NSI affect low-energy experiments (e.g., reactor and solar) as strongly as high-energy accelerator experiments, and can potentially "fake" Dirac CP-violating effects. The Borexino phase-II data favor a best-fit scalar NSI parameter $\eta_{ee} \simeq -0.16$ at solar densities (Ge et al., 2018). Disentangling scalar from vector NSI requires dedicated cross-experiment and density-variation strategies.

7. Outlook and Future Directions

NSI remain among the most robust and testable manifestations of physics beyond the SM in the neutrino sector. Upcoming oscillation experiments (DUNE, Hyper-K), high-statistics CE$\nu$NS data, and direct searches at colliders will converge on the sub-percent level in relevant NSI parameters for many channels. The interplay between terrestrial, astrophysical, and cosmological probes enables stringent cross-checks and exclusion of degeneracies such as the LMA-dark branch. Model-building efforts increasingly focus on avoiding charged-lepton flavor violation and other indirect constraints, requiring either tuned UV structures or new light mediators. The supernova environment and neutronization-burst observation at DUNE are uniquely sensitive to flavor-diagonal NSI—even below current laboratory reach—potentially allowing the discovery of NSI down to the $\mathcal{O}(0.1)$ level and highlighting the strong complementarity between experimental frontiers (Jana et al., 2024, Stapleford et al., 2016, Freitas et al., 2 May 2025).