---
title: 'Neutral-current NSIs: Beyond Standard Oscillations'
url: https://www.emergentmind.com/topics/neutral-current-non-standard-interactions-nsis
type: topic
---

# Neutral-current NSIs: Beyond Standard Oscillations

Searching arXiv for recent and foundational papers on neutral-current NSIs to support the encyclopedia entry.
Neutral-current non-standard interactions (NSIs) are effective beyond-Standard-Model neutrino interactions that modify neutral-current amplitudes involving neutrinos and matter fermions. In the conventional oscillation framework, they are encoded as four-fermion operators of the form \(\mathcal{L}_{\rm NSI} = -2 \sqrt{2} G_F \varepsilon_{\alpha\beta}^{fP} \left( \overline{\nu_\alpha} \gamma^\mu P_L \nu_\beta \right) \left( \overline{f} \gamma_\mu P_C f \right)\), with \(f=e,u,d\), and they affect neutrino propagation through coherent forward scattering in matter [1209.2710]. In this standard sense, only the vector combination is relevant for matter effects in unpolarized media, and the effective matter Hamiltonian takes the form \(H_{\rm mat}=\sqrt{2} G_F N_e \left( \begin{matrix} 1+\epsilon_{ee} & \epsilon_{e\mu} & \epsilon_{e\tau} \cr \epsilon_{e\mu}^* & \epsilon_{\mu\mu} & \epsilon_{\mu\tau}\cr \epsilon_{e\tau}^*& \epsilon_{\mu\tau}^*&\epsilon_{\tau\tau}\end{matrix}\right)\) [1602.07099]. The same quark-level couplings also appear in neutral-current scattering, including coherent elastic neutrino–nucleus scattering (CE\(\nu\)NS), deep-inelastic scattering, and collider missing-energy signatures, so neutral-current NSIs sit at the interface of oscillation phenomenology, neutrino scattering, and new-physics model building [1907.00991].

## 1. Effective operator framework and matter-potential formulation

The conventional neutral-current NSI operator is written as
\[
\mathcal{L}_{\mathrm{NC-NSI}} = -2\sqrt{2}G_F \varepsilon_{\alpha\beta}^{fC} \bigl(\overline{\nu_\alpha}\gamma^\mu P_L \nu_\beta\bigr) \bigl(\overline{f}\gamma_\mu P_C f\bigr),
\]
with \(\alpha,\beta=e,\mu,\tau\), \(f=e,u,d\), and chirality \(C=L,R\) [2008.04161]. Hermiticity requires \(\varepsilon_{\beta\alpha}^{fC}=(\varepsilon_{\alpha\beta}^{fC})^*\) [1607.01745]. For oscillations in matter, the relevant effective quantities are matter-composition-weighted combinations,
\[
\varepsilon_{\alpha\beta} = \sum_{f,P}\varepsilon_{\alpha\beta}^{fP}\,\frac{n_f}{n_e},
\]
or equivalently \(\varepsilon_{\alpha\beta}\simeq \varepsilon_{\alpha\beta}^{e} +3\,\varepsilon_{\alpha\beta}^{u} +3\,\varepsilon_{\alpha\beta}^{d}\) for approximately neutral, isoscalar Earth matter [0907.0097]. In the same approximation, \(N_n\simeq N_p=N_e\), \(N_u\simeq N_d\simeq 3N_e\), and \(Y_e\simeq 0.5\) [2008.04161].

The propagation Hamiltonian is the sum of the vacuum term and the matter term,
\[
H \;=\; U \begin{bmatrix} 0 & 0 & 0 \\ 0 & k_{21}  & 0 \\ 0 & 0 & k_{31} \end{bmatrix} U^\dagger + V_{\mathrm{CC}} \begin{bmatrix} 1 + \varepsilon_{ee}  & \varepsilon_{e\mu}      & \varepsilon_{e\tau}   \\ \varepsilon_{e\mu}^*  & \varepsilon_{\mu\mu}    & \varepsilon_{\mu\tau} \\ \varepsilon_{e\tau}^* & \varepsilon_{\mu\tau}^* & \varepsilon_{\tau\tau} \end{bmatrix},
\]
with \(V_{\mathrm{CC}}=\sqrt{2}G_F N_e\) [2008.04161]. Only differences of diagonal NSI parameters are physical in oscillations, because adding a term proportional to the identity does not affect flavor transitions; accordingly, analyses commonly set \(\epsilon_{\mu\mu}=0\) and interpret \(\epsilon_{ee}\) and \(\epsilon_{\tau\tau}\) relative to it [1602.07099]. Off-diagonal entries such as \(\varepsilon_{e\mu}\), \(\varepsilon_{e\tau}\), and \(\varepsilon_{\mu\tau}\) are generally complex and introduce new phases beyond the standard Dirac phase [1209.2710].

A recurring point in the literature is that the phenomenologically relevant NC NSI coefficients are environment-dependent effective combinations, not fundamental couplings by themselves. The same microscopic theory therefore maps differently onto Earth-like and solar-like matter [1209.2710]. The review literature also stresses that the operators written in this form are low-energy effective interactions; in the heavy-mediator limit they are understood as arising after integrating out new degrees of freedom, whereas light-mediator realizations require separate treatment [1907.00991].

## 2. Flavor structure, diagonal ambiguities, and standard bounds

The flavor structure of neutral-current NSIs divides naturally into diagonal non-universal terms and off-diagonal flavor-changing terms. Diagonal entries \(\varepsilon_{ee}\), \(\varepsilon_{\mu\mu}\), and \(\varepsilon_{\tau\tau}\) modify flavor-dependent forward scattering, while off-diagonal entries \(\varepsilon_{e\mu}\), \(\varepsilon_{e\tau}\), and \(\varepsilon_{\mu\tau}\) induce flavor-changing matter effects [0907.0097]. Because only diagonal differences are observable, bounds on diagonal matter NSIs are intrinsically less direct than bounds on off-diagonal ones [1209.2710].

A useful compiled benchmark for effective matter NSIs in Earth-like matter is
\[
|\varepsilon^{\oplus}|< \begin{pmatrix} 4.2 & 0.33 & 3.0\\ 0.33 & 0.068 & 0.33\\ 3.0 & 0.33 & 21 \end{pmatrix},
\]
and for solar-like matter
\[
|\varepsilon^{\odot}|< \begin{pmatrix} 2.5 & 0.21 & 1.7\\ 0.21 & 0.046 & 0.21\\ 1.7 & 0.21 & 9.0 \end{pmatrix},
\]
with the explicit caveat that these are approximate effective combinations built from bounds obtained assuming one nonzero NSI coupling at a time [0907.0097]. Interpreted entry by entry, the strongest bounds apply to \(\mu\)-sector parameters such as \(\varepsilon_{\mu\mu}\), while \(\varepsilon_{ee}\) and especially \(\varepsilon_{\tau\tau}\) can remain comparatively large [0907.0097].

The same hierarchy is echoed in the broader review literature. Robustly constrained matter NSIs are mainly those involving \(\mu\)-flavor, especially \(\varepsilon_{\mu\mu}\), and to a lesser extent \(\varepsilon_{e\mu}\) and \(\varepsilon_{\mu\tau}\), whereas weakly constrained matter NSIs remain in the \(e\)- and especially \(\tau\)-diagonal sectors [1209.2710]. A further atmospheric relation,
\[
\varepsilon_{\tau\tau}^{f} \sim \frac{|\varepsilon_{e\tau}^{f}|^2}{1+\varepsilon_{ee}^{f} \pm \mathcal{O}(0.1)},
\]
illustrates that matter-NSI bounds are often shaped by correlations rather than by independent one-parameter exclusions [0907.0097].

A common misconception is to treat “NSI” as automatically meaning the usual vector matter operator. One study explicitly broadens the term to the full set of dimension-6 neutrino–quark four-fermion operators built from scalar, pseudo-scalar, vector, axial-vector, and tensor bilinears, but also notes that only the vector-current interaction corresponds most directly to the usual neutral-current NSI of oscillation and CE\(\nu\)NS phenomenology [1904.11214]. This distinction matters because the scalar, axial-vector, pseudo-scalar, and tensor interactions belong to a broader class of exotic neutrino interactions even when they are discussed under the NSI label [1904.11214].

## 3. Propagation effects in oscillation experiments

Long-baseline, atmospheric, and solar oscillation experiments probe neutral-current NSIs through propagation in matter. The qualitative reason is that NSIs modify the flavor structure of the matter potential and therefore alter the interference pattern between vacuum oscillations and matter effects [1907.00991]. In long-baseline accelerator appearance channels, the key parameters are usually the \(e\)-flavor off-diagonal entries \(\varepsilon_{e\mu}\) and \(\varepsilon_{e\tau}\), because the \(\nu_\mu\to\nu_e\) channel is directly sensitive to them [2008.04161].

For \(\nu_\mu\to\nu_e\), the appearance probability can be written approximately as
\[
P_{\mu e} \simeq  P_{0}+ P_{1}+   P_{2},
\]
with a standard atmospheric term \(P_0\), a standard interference term \(P_1\), and an NSI interference term
\[
P_{2} \simeq  8 s_{13} s_{23} v |\varepsilon| \left[ a f^2 \cos(\delta_{\mathrm{CP}} + \phi) + b f g\cos(\Delta + \delta_{\mathrm{CP}} + \phi) \right],
\]
where \(v = V_{\mathrm{CC}}/k_{31}\) [2008.04161]. Because \(P_2\propto v\), matter-sensitive experiments such as NO\(\nu\)A and DUNE are substantially more affected than shorter-baseline experiments such as T2K [2008.04161]. This is the basis of the NO\(\nu\)A–T2K NSI explanation: with \(|\varepsilon_{e\mu}|\sim 0.2\) or \(|\varepsilon_{e\tau}|\sim 0.2\) and phases near \(3\pi/2\), the discrepancy in the inferred \(\delta_{\mathrm{CP}}\) between the two experiments can be alleviated in normal ordering [2008.04161].

The same mechanism produces degeneracies in other parameters. At DUNE, the appearance probability acquires an additional NSI-induced interference term carrying the phase \(\phi_{e\mu/e\tau}\), and for \(|\varepsilon_{e\mu}|=0.05\) or \(|\varepsilon_{e\tau}|=0.05\), the discovery potential of the octant of \(\theta_{23}\) can be completely lost for unfavorable combinations of \(\delta\) and \(\phi\) [1607.01745]. At the DUNE first oscillation maximum, the standard octant-sensitive splitting \(\Delta P_0\), the standard interference amplitude \(|A|\), and the NSI interference amplitude \(|B|+|C|\) are numerically comparable when \(|\varepsilon|\sim 0.05\), which is the quantitative reason the octant sensitivity can collapse [1607.01745].

Shorter-baseline, lower-energy experiments can instead act as degeneracy breakers. For MOMENT, the inequalities
\[
\sqrt{2} G_F N_e L \ll 1, \qquad \Delta m_{31}^2/E_\nu \gg \sqrt{2}G_F N_e
\]
make both standard and non-standard matter effects small, so the determination of \(\delta\) is comparatively robust against propagation NSIs [1602.07099]. Combining MOMENT with T2K and NO\(\nu\)A removes wrong-\(\delta\) solutions and improves some NSI bounds, including \(|\epsilon_{e\mu}|<0.15\) at \(3\sigma\) in the combined analysis [1602.07099].

The general review literature consistently emphasizes that NC NSIs can mimic or obscure CP violation, weaken mass-ordering and octant sensitivity, and generate generalized degeneracies in long-baseline fits [1907.00991]. A plausible implication is that robust extraction of standard oscillation parameters requires external information from scattering, atmospheric, and matter-insensitive measurements.

## 4. Neutral-current scattering, CE\(\nu\)NS, and direct-detection implications

Neutral-current NSIs also modify scattering cross sections. In CE\(\nu\)NS, the differential cross section is
\[
\frac{d\sigma_\beta}{dT}(E_\nu,T) \simeq \frac{G_F^2 M}{\pi}\, Q_{W\beta}^2 \left(1-\frac{MT}{2E_\nu^2}\right) F^2(q^2),
\]
with the weak charge
\[
Q_{W\beta }^{2} = \left[ Z(g_{p}^{V}+2\varepsilon _{\beta \beta}^{uV}+\varepsilon _{\beta \beta }^{dV})+ N(g_{n}^{V}+2\varepsilon _{\beta\beta}^{dV}+\varepsilon _{\beta \beta }^{uV}) \right]^{2} + \sum_{\alpha \neq \beta} \left| Z(2\varepsilon _{\alpha \beta}^{uV}+\varepsilon _{\alpha \beta }^{dV})+ N(2\varepsilon _{\alpha \beta}^{dV}+\varepsilon _{\alpha \beta }^{uV}) \right|^{2}
\]
for flavor \(\beta\) [2104.00425]. This expression makes clear that flavor-diagonal amplitudes interfere with the Standard Model, whereas flavor-changing detection amplitudes contribute incoherently in the summed final state [2104.00425].

A particularly specific result concerns CP-violating phases in CE\(\nu\)NS. Observable phase dependence requires simultaneously nonzero \(\varepsilon_{\alpha\beta}^{uV}\) and \(\varepsilon_{\alpha\beta}^{dV}\) for the same \((\alpha,\beta)\), because the rate depends only on the relative phase
\[
\Delta\phi_{\alpha\beta}=\phi_{\alpha\beta}^{uV}-\phi_{\alpha\beta}^{dV}
\]
through an interference term proportional to \(\cos(\Delta\phi_{\alpha\beta})\) [2104.00425]. A single isolated off-diagonal NSI parameter has no observable phase dependence in CE\(\nu\)NS [2104.00425]. This sharply contrasts with oscillation propagation, where a single complex off-diagonal matter NSI can already affect the inferred CP structure.

Neutral-current scattering also extends beyond CE\(\nu\)NS. A DUNE-like experiment using NC deep-inelastic scattering can probe axial NC NSI of neutrinos with \(u\), \(d\), and \(s\) quarks, which can hide from oscillations and CE\(\nu\)NS because those standard probes are dominantly sensitive to vector couplings [2312.12420]. Using both near and far detectors, such a DUNE-like setup can significantly improve present bounds on all the flavor elements studied there, and for the first time the couplings to strange quarks become accessible in a direct way [2312.12420].

NSIs also propagate into dark-matter direct-detection phenomenology through the neutrino background. In the effective-operator treatment of neutrino–quark interactions, vector-current and scalar-current NSIs can significantly change the neutrino floor, pseudo-scalar NSIs can raise it by about \({\cal O}(20\%)\), while axial-vector and tensor NSIs alter it only at the \({\cal O}(1\%)\) level [1904.11214]. The reason is structural: vector NSIs use the coherent \(W_M\) response and interfere with the Standard Model, scalar NSIs are coherently enhanced but non-interfering, and axial/tensor effects depend on spin-sensitive responses that are small or vanish for some targets [1904.11214]. This has led to the explicit suggestion that dark-matter direct-detection experiments should be combined with CE\(\nu\)NS measurements to test whether a recoil excess is due to exotic neutral-current neutrino interactions rather than WIMP scattering [1904.11214].

## 5. Collider probes and long-baseline neutral-current event samples

Collider missing-transverse-momentum searches provide a complementary high-energy probe of neutrino–quark NC NSIs. In a standard NC NSI parameterization,
\[
\mathcal{L}_{\mathrm{NSI}}=-2 \sqrt{2} G_{F} \epsilon_{\alpha \beta}^{f, Y} \left(\bar{\nu}^{\alpha} \gamma_{\mu} P_{L} \nu^{\beta}\right) \left(\bar{f} \gamma^{\mu} P_{Y} f\right),
\]
the LHC is effectively flavor blind because final-state neutrino flavor is not identified [2009.06668]. Recasting 13 TeV ATLAS and CMS mono-object data with NLO QCD matched to parton showering yields a final EFT-style collider bound \(\epsilon \lesssim 0.010\) at \(95\%\) CL for \(\epsilon_u=\epsilon_d=\epsilon\), while in a simplified \(Z'\) model with \(\Gamma_{Z'}/M_{Z'}=0.1\) the combined bounds are \(\epsilon \lesssim 0.042\) for \(M_{Z'}=0.2\) TeV and \(\epsilon \lesssim 0.0028\) for \(M_{Z'}=2\) TeV [2009.06668]. The paper emphasizes that the collider limits are especially useful for heavy mediators or contact-like interactions and that the EFT approximation requires the new-physics scale to exceed the typical LHC partonic energy [2009.06668].

Within oscillation experiments themselves, neutral-current event samples have emerged as a direct NSI probe. One recent line of work shows that NC event rates at long-baseline experiments become oscillation sensitive once NSIs make NC cross sections flavor dependent [2412.08712]. Existing NOvA NC data then yield one-parameter-at-a-time 90% C.L. bounds such as
\[
\varepsilon_{\mu\mu}^A \in [-0.25,\ 0.25], \qquad
\varepsilon_{\tau\tau}^A \in [-0.41,\ 0.41], \qquad
\varepsilon_{e\mu}^A \in [-0.26,\ 0.26],
\]
for axial-vector NSIs under the isospin-conserving assumption, together with competitive vector constraints and a disfavoring of large diagonal vector NSI associated with the LMA-Dark region [2412.08712]. The paper emphasizes that these NC long-baseline data are highly complementary to SNO because SNO is not sensitive to the isospin-conserving axial direction \(\varepsilon_u^A=\varepsilon_d^A\) in the same way [2412.08712].

An even more specific complementarity has been developed in terms of isoscalar and isovector quark combinations. Charged-current oscillation analyses constrain NSIs through matter effects in propagation and probe almost exclusively the isoscalar combination of up- and down-quark couplings, while the orthogonal isovector combination is suppressed by a factor of \(\sim 100\) in Earth matter [2604.16176]. NC event samples, by contrast, depend on modified NC cross sections and the flavor composition of the beam at near and far detectors, and therefore probe both isoscalar and isovector couplings with comparable weight [2604.16176]. Using NOvA data and DUNE projections, this framework yields the first bounded long-baseline constraints on isovector NSI and resolves the individual quark couplings only when CC and NC information are combined [2604.16176].

## 6. UV completions, loop generation, and theoretical caveats

Large neutral-current NSIs are nontrivial to realize in realistic models. In the low-energy EFT, one expects \(\varepsilon \propto m_W^2/m_X^2\), so a mediator scale \(m_X\sim 1\,{\rm TeV}\) suggests \(\varepsilon\sim 10^{-2}\), whereas \(m_X\sim 10\,{\rm TeV}\) suggests \(\varepsilon\sim 10^{-4}\) [1209.2710]. This scaling already indicates that order-one matter NSIs are difficult in heavy-mediator theories unless special structures are present.

The most developed successful constructions therefore exploit light mediators. One class of viable models introduces a \(U(1)'\) gauge boson \(Z'\) with mass \(O(10~{\rm MeV})\) and tiny couplings \(g\sim 5\times 10^{-5} - 10^{-4}\), so that forward scattering in matter can be large while finite-momentum-transfer scattering bounds are suppressed [1612.04971]. In one model aimed at the LMA-Dark pattern, the generated quark NSIs satisfy
\[
\epsilon_{\mu\mu}^q=\epsilon_{\tau\tau}^q = \frac{b\,g'^2}{3\sqrt{2}\,G_F\,m_{Z'}^2},
\]
with \(\epsilon_{ee}^q=0\) and negligible off-diagonal entries [1612.04971]. A second model generates diagonal and off-diagonal NSI through mixing of active neutrinos with a new Dirac fermion \(\Psi\), leading to
\[
\epsilon_{\alpha\beta}^{u} = \epsilon_{\alpha\beta}^{d} \simeq \frac{g_B g_\Psi\,\kappa_\alpha^*\kappa_\beta} {6\sqrt{2}\,G_F\,M_{Z'}^2},
\]
and allowing complex off-diagonal matter NSIs relevant for CP-violating propagation effects [1612.04971].

A related light-\(Z'\) framework focused on lepton-flavor-violating matter NSIs shows that sizable propagation NSIs are realistically possible only in the \(\mu\)-\(\tau\) sector, because \(\mu e\) and \(e\tau\) cases are excluded by charged-lepton flavor-violating limits such as \(\mu\to e Z'\) and \(\tau\to e Z'\) [1512.09147]. In the viable \(\mu\)-\(\tau\) case, the model predicts correlated propagation parameters
\[
\tilde{\epsilon}_{\mu \mu} = 2A \sin (2\theta_{L}), \qquad \epsilon_{\mu \tau} = A \cos(2 \theta_{L}),
\]
with
\[
A = \sqrt{\epsilon_{\mu\tau}^2+\frac{\tilde\epsilon_{\mu\mu}^2}{4}},
\]
so that long-baseline data can in principle determine both the overall strength and the flavor angle [1512.09147].

Loop generation provides an additional route. A one-loop extension of the NSI formalism shows that effective matter NSIs arise from loop-corrected \(Z\nu\nu\) couplings plus dimension-six four-fermion operators,
\[
\varepsilon^{m,f}_{\mu\tau}= \left( I^{f}_{3L} - 2 Q_f s^2_W \right) \left( \eta^{Z}_{L} - \eta^{\nu}_{L} \right)_{\mu\tau} + (\varepsilon^{m,f}_{\mu\tau})^{\rm dim-6},
\]
with gauge invariance enforcing finiteness through cancellations among wave-function and vertex corrections [1012.3759]. In the MSSM example developed there, the largest loop-induced effects found numerically are mostly source/detector rather than matter NSIs, but the formalism makes clear how genuine neutral-current matter NSIs emerge at one loop [1012.3759].

Several caveats recur across the literature. The heavy-mediator EFT description can fail for light mediators or large momentum transfer [2009.06668]. Oscillation matter effects constrain vector combinations, not axial ones [2312.12420]. Many quoted bounds assume one nonzero NSI parameter at a time, and correlated multiparameter fits can be substantially weaker [0907.0097]. A plausible implication is that any single quoted interval must be interpreted together with its flavor, mediator, and operator assumptions.

## 7. Open issues, complementarity, and current status

The current status of neutral-current NSIs is defined less by a single bound than by a network of complementary probes. Oscillation experiments constrain effective matter combinations and are especially sensitive to propagation phases and degeneracies [1907.00991]. CE\(\nu\)NS constrains the same quark-level vector couplings through scattering and is directly relevant to oscillation-motivated scenarios such as the NO\(\nu\)A–T2K tension and the LMA-Dark region [2104.00425]. Collider missing-energy searches constrain heavy-mediator neutrino–quark NSIs in a way that is flavor blind but sensitive to absolute coupling magnitudes [2009.06668]. Long-baseline NC event samples now probe both vector and axial structures, including directions that are nearly invisible to CC-only oscillation fits [2412.08712].

The literature repeatedly emphasizes complementarity rather than replacement. One status report frames this broadly in terms of oscillation data, scattering data, colliders, and model-building constraints all being necessary to map the parameter space [1907.00991]. More recent long-baseline work sharpens that point by showing that CC analyses probe almost only the isoscalar quark direction through matter effects, whereas NC event analyses access both isoscalar and isovector directions through cross sections [2604.16176]. This suggests that “neutral-current NSI” is best understood not as a single observable effect, but as a family of operator deformations whose phenomenology depends strongly on whether one studies propagation, scattering, or both.

Several controversies or common misunderstandings are now clear. Neutral-current NSI in the standard oscillation sense means the vector neutrino–matter interaction entering the propagation Hamiltonian, not the entire space of scalar, pseudo-scalar, axial, or tensor neutrino operators [1904.11214]. Large diagonal effective NSIs are not automatically excluded in a model-independent sense, although they are difficult to embed in gauge-invariant UV completions [1209.2710]. Apparent NSI-like effects in oscillation experiments can also arise from source and detector corrections or non-unitarity, so identifying a propagation NSI requires correlating oscillation and scattering information [1012.3759].

The most conservative synthesis is that neutral-current NSIs remain one of the standard EFT descriptions of sub-leading new physics in neutrino flavor transitions, but their interpretation is highly assumption dependent. Vector matter NSIs are tightly constrained in many directions yet still phenomenologically relevant because of degeneracies in long-baseline fits and because diagonal sectors remain comparatively weak [1907.00991]. Axial neutral-current NSIs can evade the standard oscillation and CE\(\nu\)NS probes and therefore motivate dedicated NC scattering analyses at DUNE-like facilities [2312.12420]. Light-mediator constructions show that large observable matter NSIs are not automatically ruled out, but they require specific flavor structures and are tied to broader laboratory, astrophysical, and cosmological signatures [1612.04971].

Source: https://www.emergentmind.com/topics/neutral-current-non-standard-interactions-nsis