Non-Standard Neutrino Oscillations

• Non-standard neutrino oscillations are deviations from the standard three-flavour paradigm caused by new interactions, sterile neutrinos, or Lorentz-violating effects.
• They modify the effective Hamiltonian through non-standard interactions, altering neutrino production, propagation, and detection with energy-dependent CP-violating signatures.
• Current experiments like T2K, NOνA, and DUNE use advanced statistical methods and varied baselines to constrain these effects and probe for new physics beyond the Standard Model.

Non-standard neutrino oscillations refer to deviations from the standard three-flavour oscillation paradigm induced by new interactions or effects beyond those predicted by the Standard Model. These phenomena can arise from new effective interactions with matter—termed non-standard interactions (NSI)—entanglement effects during neutrino production or detection, new sources of CP violation, Lorentz violation, additional (sterile) neutrino states, or other extensions. Non-standard oscillations can profoundly alter the extraction of fundamental oscillation parameters, introduce degeneracies, and provide unique signatures or tensions among experimental results across short and long baselines.

1. Theoretical Formulation of Non-Standard Oscillations

A general approach to non-standard neutrino oscillations is to extend the propagation and interaction Hamiltonian to include new effective operators. The typical effective NSI Lagrangian takes the form: $$\mathcal{L}_{\rm NSI} = -2\sqrt{2} G_F \sum_{f,f',\alpha,\beta} \varepsilon_{\alpha\beta}^{ff'X} (\bar{f} P_{L,R} \gamma^\mu f') (\bar{\ell}_\alpha P_L \gamma_\mu \nu_\beta) + \text{h.c.}$$ where $\varepsilon_{\alpha\beta}^{ff'X}$ parameterize the strength of the NSI relative to the standard electroweak interaction, $f,\,f'$ represent the external fermions (e.g., $e,u,d$), and $X=S,D$ denotes whether the interaction occurs at the source or detector (Akhmedov et al., 2010).

For propagation in matter, NSI modify the effective Hamiltonian: $$H_\text{eff} = \frac{1}{2E} U \,\mathrm{diag}(0, \Delta m_{21}^2, \Delta m_{31}^2)\,U^\dagger + A[I+\varepsilon^m]$$ where the Hermitian matrix $\varepsilon^m$ contains propagation NSI, and the matter potential is $A = 2\sqrt{2}G_F N_e E$ (Ohlsson, 2012).

Production and detection NSI are encoded by replacing flavor pure states with admixtures: $$| \nu_\alpha^X \rangle = C_\alpha^X [ |\nu_\alpha\rangle + \sum_\beta \varepsilon^{X}_{\alpha\beta} |\nu_\beta\rangle ]$$ for $X = S,D$, with $C_\alpha^X$ normalization factors (Akhmedov et al., 2010), leading to non-orthogonal flavor states and “zero-distance” effects.

Beyond the effective operator approach, non-standard oscillations can arise from other mechanisms:

• Lorentz violation (LV): Here, the oscillation Hamiltonian is augmented with terms violating Lorentz invariance (e.g., from the SME), often parameterized with a LV Hamiltonian $\delta H_{LV}$ producing energy-dependent mixing angles and phases (Ma, 2011).
• Sterile neutrinos: Additional mass eigenstates (e.g., in (3+1) or (3+2) schemes) can induce new mixing and mass splittings, as well as NSI-like interference terms in appearance and disappearance probabilities (Akhmedov et al., 2010).

2. Experimental and Phenomenological Implications

Non-standard oscillations have been invoked to address anomalies and tensions in several experimental contexts:

• MiniBooNE and LSND: The (3+1)+NSI framework introduces CP-violating terms via NSI at production/detection that can reconcile the positive signal in antineutrino mode with the absence of signal in neutrino mode (Akhmedov et al., 2010). In this scenario, the complex phase in the interference term

$$P_{\alpha\beta}(L) = 2[|\alpha_{\alpha\beta}|^2 - \text{Re}(\beta^*_{\alpha\beta}\alpha_{\alpha\beta})](1-\cos\Delta) + |\beta_{\alpha\beta}|^2 + 2\text{Im}(\beta^*_{\alpha\beta}\alpha_{\alpha\beta})\sin\Delta$$

generates CP violation even at short baselines that would be forbidden in a standard (3+1) oscillation (Akhmedov et al., 2010).

• MINOS anomaly: Attempted explanations using NSI-generated CP violation to address the apparent difference in neutrino and antineutrino disappearance probabilities require large NSI couplings ($\mathcal{O}(0.1)$) incompatible with bounds from other processes (Akhmedov et al., 2010).
• Long-baseline constraints: Experiments such as T2K, NO\nuA, and DUNE probe NSI effects both through their impact on oscillation probabilities and via the degeneracies they introduce with standard parameters. For example, model-independent bounds on $\varepsilon^m_{e\tau}$ and $\varepsilon^m_{\tau\tau}$ have been obtained by comparing oscillation probabilities with measured appearance and disappearance rates (Adhikari et al., 2012).

3. Non-Standard Oscillation Formalism and CP Violation

Non-standard oscillations can give rise to novel, energy-dependent CP-violating effects:

• In the (3+1)+NSI models, the interference between the “oscillatory” and constant “NSI-induced” amplitudes generates a phase $\delta = \arg(\alpha_{\mu e}\beta_{\mu e}^*)$ responsible for the CP-violating difference between neutrino and antineutrino appearance at short baselines (Akhmedov et al., 2010). The amplitude has the structure:

$$\mathcal{A}_{\mu e}(L) = \alpha_{\mu e}(e^{-i\Delta}-1) + \beta_{\mu e}$$

• Generalized CP violation also arises in frameworks with complex NSI parameters in the matter potential, e.g., through off-diagonal $\varepsilon_{e\mu}$ or $\varepsilon_{e\tau}$ terms (Adhikari et al., 2012, Denton et al., 2020).
• In LV-induced oscillations, the mixing angles and oscillation phases become energy-dependent, and the amplitude may vanish (or “freeze”) at high energy, in sharp contrast to the standard mass-induced oscillation regime (Ma, 2011).

4. Interplay with Experimental Constraints and Global Fits

Experimental analyses must account for the degeneracies and freedom introduced by NSI and non-standard oscillations:

• Disappearance and appearance channels are affected differently, and the (3+1)+NSI scenario can partially decouple appearance signals (e.g., at LSND) from disappearance constraints, especially if NSI parameters are allowed to differ between purely leptonic and semi-leptonic channels (“general” NSI model) (Akhmedov et al., 2010).
• The NSI-induced enhancement or suppression of oscillation probabilities can mimic standard parameter values. For instance, even small NSI values ($\varepsilon\sim 0.03$) can affect the effective $\theta_{13}$ extracted from reactor experiments (Ohlsson, 2012).
• In a bimagic baseline configuration, special values of energy and path length can make $P(\nu_e\rightarrow\nu_\mu)$ nearly independent of the CP phase and $\theta_{13}$ for one mass hierarchy, remaining sensitive for the other, and this property holds even in the presence of NSI (Adhikari et al., 2012). At higher energies, NSI effects become further enhanced compared to standard vacuum terms.
• Advanced statistical methods (e.g., binned log-likelihood, χ² minimization) have been used to project future sensitivities and evaluate the improvement in NSI parameter constraints, with large-scale experiments such as KM3NeT-ORCA and PINGU promising up to an order of magnitude stronger bounds (Chowdhury, 2020, Feng et al., 2019).

5. Quantum Field Theory, Entanglement, and Density Matrix Approaches

A comprehensive QFT treatment of non-standard oscillations reveals that NSI generically entangle neutrino mass, spin, and flavor with properties of the external particles in production and detection. In this framework:

• The neutrino state is described by a density matrix,

$$\rho_{\lambda,i;\eta,k}(E,\theta) = \mathcal{N} \int \text{dLips}\, M^P_{i,\lambda} M^{P*}_{k,\eta}$$

where $M^P_{i,\lambda}$ are production amplitudes and the trace normalization embodies the mixture induced by NSI (Ochman et al., 2010).

• The resulting transition amplitude no longer factorizes into distinct production, propagation, and detection contributions. The inclusion of NSI, especially those with non-trivial Lorentz structure (e.g., tensor or scalar), can break the equivalence between the full QFT and the “effective quantum mechanical” NSI formalism unless certain process-dependent matching conditions are satisfied (Falkowski et al., 2019).

6. Future Prospects and Theoretical Developments

Prospects for discovery or strong exclusion of non-standard oscillation effects hinge on tailored experimental designs and multifaceted theory approaches:

• Next-generation experiments with long baselines (DUNE, Hyper-K, PINGU) and new types of sources (muon accelerators for multi-TeV neutrino beams) leverage increased matter effects and high event rates to probe NSI couplings and sterile neutrino mixing to much greater precision (Feng et al., 2019, Kamp et al., 12 Aug 2025).
• The ability to test new sources of CP violation, distinguish NSI-induced effects from standard parameter shifts, and resolve parameter degeneracies grows with coverage across energies, baselines, and complementary detection channels.
• Theoretical developments include fully geometric treatments (leptonic unitarity triangles), density matrix evolution including entanglement, and extended parametrizations for Lorentz- and CPT-violating effects (Masud et al., 2021, Ma, 2011).

A global, process-dependent, and model-flexible approach is essential for both robust interpretation of experimental data and the identification of any non-standard features in neutrino oscillation phenomenology. Non-standard neutrino oscillations remain a focal point for efforts to probe new physics in the lepton sector and to extend the reach of precision tests beyond the Standard Model (Ohlsson, 2012, Miranda et al., 2015, Farzan et al., 2017).