Non standard neutrino interactions: current status and future prospects

Any extension of the SM local gauge symmetry ${SU}{(2)}_{L}\otimes U{(1)}_{Y}$, in general, introduces new gauge bosons that modify the vector and axial coupling. Typical examples are E₆ string inspired models, that at low energies introduce the extra groups $U{(1)}_{\chi }\otimes U{(1)}_{\psi }$, leading to an additional Z' neutral gauge bosons. A similar situation happens with the left-right symmetric model ${SU}{(2)}_{L}\otimes U{(1)}_{Y}\otimes {SU}{(2)}_{R}\otimes U(1{)}_{Y}^{\prime }$ where one extra neutral and one extra charged gauge bosons, Z' and W', appear.

Within the NSI formalism, there is an easy direct relation between the phenomenological parameters and the parameters coming from these models. We can consider, for instance, the neutrino dispersion off nuclei, described in the SM Lagrangian

$$\begin{eqnarray}&&{{\mathcal{L}}}_{\nu N}^{{NC}}=-\displaystyle \frac{{G}_{F}}{\sqrt{2}}\displaystyle \sum _{q=u,d}\left[{\bar{\nu }}_{e}{\gamma }^{\mu }(1-{\gamma }^{5}){\nu }_{e}\right]\left\{{f}^{{qL}}\left[\bar{q}{\gamma }_{\mu }(1-{\gamma }^{5})q\right]+{f}^{{qR}}\left[\bar{q}{\gamma }_{\mu }(1+{\gamma }^{5})q\right]\right\},\end{eqnarray} \tag{ 5 }$$

where ${f}^{{qL},R}$ are the SM coupling constants defined elsewhere [8]. For the case of E₆ models, where two additional neutral vector bosons arise, there will be an additional contribution to these couplings constants given by [49]

$$\begin{eqnarray}{\varepsilon }^{{uL}} & = & -4\displaystyle \frac{{M}_{Z}^{2}}{{M}_{{Z}^{\prime }}^{2}}{\mathrm{sin}}^{2}{\theta }_{W}{\rho }_{\nu N}^{{NC}}\left(\displaystyle \frac{\mathrm{cos}\beta }{\sqrt{24}}-\displaystyle \frac{\mathrm{sin}\beta }{3}\sqrt{\displaystyle \frac{5}{8}}\right)\left(\displaystyle \frac{3\mathrm{cos}\beta }{2\sqrt{24}}+\displaystyle \frac{\mathrm{sin}\beta }{6}\sqrt{\displaystyle \frac{5}{8}}\right)\\ {\varepsilon }^{{dR}} & = & -8\displaystyle \frac{{M}_{Z}^{2}}{{M}_{{Z}^{\prime }}^{2}}{\mathrm{sin}}^{2}{\theta }_{W}{\rho }_{\nu N}^{{NC}}{\left(\displaystyle \frac{3\mathrm{cos}\beta }{2\sqrt{24}}+\displaystyle \frac{\mathrm{sin}\beta }{6}\sqrt{\displaystyle \frac{5}{8}}\right)}^{2},\\ {\varepsilon }^{{dL}} & = & {\varepsilon }^{{uL}}=-{\varepsilon }^{{uR}},\end{eqnarray} \tag{ 6 }$$

where M_Z is the mass of the SM neutral gauge boson; ${M}_{{Z}^{\prime }}$ accounts for the mass of an additional, heavier, new gauge boson; ${\theta }_{W}$ is the weak mixing angle; ${\rho }_{\nu N}^{{NC}}$ is the parameter which accounts for the radiative corrections; and the angle β describes the mixing between the two extra gauge bosons that arise from the $U{(1)}_{\chi }$ and $U{(1)}_{\psi }$ symmetries. These models have yet another extra gauge boson that is considered to be heavier than the Z' and decoupled from the above Lagrangian.

An extension of the SM fermion content can give rise to a rich phenomenology, especially when it contains extra neutral leptons, usually assumed to be heavy. Within this framework, it is possible to work in the standard ${SU}(2)\otimes U(1)$ gauge symmetry and consider the additional mixing of isodoublet and isosinglet neutral leptons [88]. As a result, the $V-A$ couplings will deviate from the SM prediction. For example, for the case of the CC, the ordinary light isodoublet neutrinos will mix with the extra heavy isosinglets; this mixing will be described by a matrix

$$\begin{eqnarray}&&K=({K}_{L},{K}_{H}),\end{eqnarray} \tag{ 7 }$$

where K_L and K_H describe, respectively, the mixing of ordinary isodoublet light neutrinos and extra heavy isosinglets. Notice that the signals of new physics coming from this matrix might appear in charged currents through the deviations from the standard interactions (that is, in the detection) as well as in oscillation experiments, since the mixing matrix K_L is no longer unitary. Moreover, regarding the NC interactions, they will be described by the interaction

$$\begin{eqnarray}&&{\mathcal{L}}=\displaystyle \frac{{{ig}}^{\prime }}{2\mathrm{sin}{\theta }_{W}}{Z}_{\mu }\bar{{\nu }_{L}}{\gamma }_{\mu }{K}^{\dagger }K{\nu }_{L},\end{eqnarray} \tag{ 8 }$$

where the matrix ${K}^{\dagger }K$ can be considered as a natural source for NSI in the neutral sector [88].

The enriched structure that arise from these models can be parameterized in different forms that must take into account all the new mixing angles and phases. One of the most studied schemes in this context is the seesaw model [88–92], which gives a natural explanation for the smallness of the neutrino mass. In these models, however, the sizeable signals at low energies are expected to be negligible and, therefore, the effective NSI should be negligible. There are, however, other models where the low energy effects, though small, may be sizeable in the near future, such as the so-called inverse seesaw model [93, 94].

Although it is not common in the literature to consider these analyses in term of the NSI formalism, it is possible to include them in the formalism. For example, for the simple case of only one extra neutral heavy lepton, the effects on a neutrino electron scattering off electrons will be a global factor in the Lagrangian, due to the non-unitarity of the mixing matrix. In this case the corresponding NSI parameters for an electron neutrino experiment will be given as

$$\begin{eqnarray}&&{\varepsilon }_{{ee}}^{{eL}}=-{g}_{L}{\mathrm{sin}}^{2}{\theta }_{14},\ {\varepsilon }_{{ee}}^{{eR}}=-{g}_{R}{\mathrm{sin}}^{2}{\theta }_{14},\end{eqnarray} \tag{ 9 }$$

where ${\theta }_{14}$ is the mixing angle between the light neutrino and the extra heavy fermion and ${g}_{L,R}$ are the SM coupling constants for the neutrino electron scattering process.

Despite the NSI formalism preserves the $V-A$ structure of the theory, it is also possible to consider the impact of scalar couplings. For example, if we consider the case of low energy supersymmetry with broken R-parity [95–97] where one has trilinear L violating couplings of the form

$$\begin{eqnarray}&&{\lambda }_{{ijk}}{L}_{i}{L}_{j}{E}_{k}^{c},\ \ {\lambda }_{{ijk}}^{\prime }{L}_{i}{Q}_{j}{D}_{k}^{c}\end{eqnarray} \tag{ 10 }$$

with L and Q super-fields that contain the usual lepton and quark SU(2) doublets, E^c, and D^c super-fields that contain the singlets, and $i,j,k$ the generation indices. These couplings give rise, for example, to the following four-fermion effective Lagrangian for neutrino interactions with d-quark

$$\begin{eqnarray}&&{{\mathcal{L}}}_{\mathrm{eff}}=-2\sqrt{2}{G}_{F}\displaystyle \sum _{\alpha ,\beta }\;{\varepsilon }_{\alpha \beta }^{{dR}}{\bar{\nu }}_{L\alpha }{\gamma }^{\mu }{\nu }_{L\beta }{\bar{d}}_{R}{\gamma }^{\mu }{d}_{R},\end{eqnarray} \tag{ 11 }$$

where we have, among others, flavor-conserving and changing NSI, given, respectively, by

$$\begin{eqnarray}&&{\varepsilon }_{\mu \mu }^{{dR}}=\displaystyle \sum _{j}\;\displaystyle \frac{{| {\lambda }_{2j1}^{\prime }| }^{2}}{4\sqrt{2}{G}_{F}{m}_{{\tilde{q}}_{{jL}}}^{2}}\;\ \mathrm{and}\;\;{\varepsilon }_{\mu \tau }^{{dR}}=\displaystyle \sum _{j}\;\displaystyle \frac{{\lambda }_{3j1}^{\prime }{\lambda }_{2j1}^{\prime }}{4\sqrt{2}{G}_{F}{m}_{{\tilde{q}}_{{jL}}}^{2}}.\end{eqnarray} \tag{ 12 }$$

Here, ${m}_{{\tilde{q}}_{{jL}}}$ denotes the masses of the squarks, while $j=1,2,3$ stands for ${\tilde{d}}_{L},{\tilde{s}}_{L},{\tilde{b}}_{L}$, respectively. This is just one example, but other NSI couplings can be studied in this context [98]; moreover, constraints on generic scalar NSI can also be studied by using Fierz transformations [99]

Unless otherwise stated, the standard three-flavor picture of neutrinos is assumed, ${\nu }_{e}$, ${\nu }_{\mu }$ and ${\nu }_{\tau }$ and corresponding anti-particles. In vacuum, the mixing of neutrinos is supposed to be described by the usual flavor mixing without NSI,

$$\begin{eqnarray}&&| {\nu }_{\alpha }\rangle =\displaystyle \sum _{i=1}^{3}\;{U}_{\alpha i}^{*}\ | {\nu }_{i}\rangle \ \ (\alpha =e,\mu ,\tau ),\end{eqnarray} \tag{ 13 }$$

where U is the 3 × 3 matrix which describes the flavor mixing [100] of neutrinos. In this review, we use the standard parameterization found, e.g, in [8],

$$\begin{eqnarray}U & = & \left[\begin{array}{ccc}1 & 0 & 0\\ 0 & {c}_{23} & {s}_{23}\\ 0 & -{s}_{23} & {c}_{23}\end{array}\right]\left[\begin{array}{ccc}{c}_{13} & 0 & {s}_{13}{e}^{-i{\delta }_{{CP}}}\\ 0 & 1 & 0\\ -{s}_{13}{e}^{i{\delta }_{{CP}}} & 0 & {c}_{13}\end{array}\right]\left[\begin{array}{ccc}{c}_{12} & {s}_{12} & 0\\ -{s}_{12} & {c}_{12} & 0\\ 0 & 0 & 1\end{array}\right],\\ & = & \left[\begin{array}{ccc}{c}_{12}{c}_{13} & {s}_{12}{c}_{13} & {s}_{13}{e}^{-i{\delta }_{{CP}}}\\ -{s}_{12}{c}_{23}-{c}_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{{CP}}} & {c}_{12}{c}_{23}-{s}_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{{CP}}} & {s}_{23}{c}_{13}\\ {s}_{12}{s}_{23}-{c}_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{{CP}}} & -{c}_{12}{s}_{23}-{s}_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{{CP}}} & {c}_{23}{c}_{13}\end{array}\right],\end{eqnarray} \tag{ 14 }$$

where ${s}_{{ij}}\equiv \mathrm{sin}{\theta }_{{ij}}$, ${c}_{{ij}}\equiv \mathrm{cos}{\theta }_{{ij}}$, and ${\delta }_{{CP}}$ is the Kobayashi-Maskawa [101] type CP phase for neutrinos.

In addition to the mixing angles and CP phase, the mass squared differences of neutrinos, ${\rm{\Delta }}{m}_{{ij}}^{2}\equiv {m}_{i}^{2}-{m}_{j}^{2}$ with m_i (i = 1-3) being the neutrino masses, are the relevant parameters to describe neutrino oscillation. So far, all of these parameters have been measured with reasonably good accuracies except for the value of the CP phase and sign of ${\rm{\Delta }}{m}_{31}^{2}$ (${\rm{\Delta }}{m}_{32}^{2}$). The positive (negative) sign of ${\rm{\Delta }}{m}_{31}^{2}$ corresponds to the normal (inverted) mass ordering, often referred to as normal (inverted) mass hierarchy.

Different groups have carefully studied the neutrino data and obtained accurate values for most of the three-flavor neutrino oscillation parameters [103–105]. Their most important results are summarized in table 1, where a reasonable agreement can be seen.

Table 1.  Summary of the standard three-flavor picture parameters, as reported from three different groups, denoted as C (Capozzi el al [103], second and third column, F (Forero et al [104], fourth and fifth column), and G (Gonzalez-Garcia et al [105], the last two columns). The parameter ${\rm{\Delta }}{m}_{3l}^{2}$ has a slightly different definition in each case, being ${\rm{\Delta }}{m}_{3l}^{2}\equiv {m}_{3}^{2}-({m}_{1}^{2}+{m}_{2}^{2})/2$ for [103], ${\rm{\Delta }}{m}_{3l}^{2}\equiv {m}_{3}^{2}-{m}_{1}^{2}$ for [104] and ${\rm{\Delta }}{m}_{3l}^{2}\equiv {m}_{3}^{2}-{m}_{1}^{2}$ for normal hierarchy (NH) and ${\rm{\Delta }}{m}_{3l}^{2}\equiv {m}_{3}^{2}-{m}_{2}^{2}$ for inverted hierarchy (IH) for [105]

	C [103]		F [104]		G [105]
Parameter	Best fit	$3\sigma$ range	Best fit	$3\sigma$ range	Best fit	$3\sigma$ range
${\rm{\Delta }}{m}_{21}^{2}/{10}^{-5}$ eV2	7.54	6.99-8.18	7.60	7.11-8.18	7.50	7.02-8.09
${\rm{\Delta }}{m}_{3l}^{2}/{10}^{-3}$ eV2 (NH)	2.43	2.23-2.61	2.48	2.30-2.65	2.457	2.317-2.607
$-{\rm{\Delta }}{m}_{3l}^{2}/{10}^{-3}$ eV2 (IH)	2.38	2.19-2.56	2.38	2.20-2.54	2.449	2.307-2.590
${\mathrm{sin}}^{2}{\theta }_{12}/{10}^{-1}$	3.08	2.59-3.59	3.23	2.78-3.75	3.04	2.70-3.44
${\mathrm{sin}}^{2}{\theta }_{23}/{10}^{-1}$ (NH)	4.37	3.74-6.26	5.67	3.93-6.43	4.52	3.82-6.43
${\mathrm{sin}}^{2}{\theta }_{23}/{10}^{-1}$ (IH)	4.55	3.80-6.41	5.73	4.03-6.40	5.79	3.89-6.44
${\mathrm{sin}}^{2}{\theta }_{13}/{10}^{-2}$ (NH)	2.34	1.76-2.95	2.26	1.90-2.62	2.18	1.86-2.50
${\mathrm{sin}}^{2}{\theta }_{13}/{10}^{-2}$ (IH)	2.40	1.78-2.98	2.29	1.93-2.65	2.19	1.88-2.51
$\delta {/}^{o}$ (NH)	250	0-360	254	0-360	306	0-360
$\delta {/}^{o}$ (IH)	236	0-360	266	0-360	254	0-360

Phenomenologically, the evolution equation of neutrinos in the flavor basis in the presence of propagation NSI in unpolarized matter can be generically written as,

$$\begin{eqnarray}&&i\displaystyle \frac{d}{{dr}}\left[\begin{array}{c}{\nu }_{e}\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right]=\left\{U\left[\begin{array}{ccc}0 & 0 & 0\\ 0 & {{\rm{\Delta }}}_{21} & 0\\ 0 & 0 & {{\rm{\Delta }}}_{31}\end{array}\right]{U}^{\dagger }+\displaystyle \sum _{f}{V}_{f}\left[\begin{array}{ccc}{\delta }_{{ef}}+{\varepsilon }_{{ee}}^{f} & {\varepsilon }_{e\mu }^{f} & {\varepsilon }_{e\tau }^{f}\\ {{\varepsilon }^{f}}_{e\mu }^{*} & {\varepsilon }_{\mu \mu }^{f} & {\varepsilon }_{\mu \tau }^{f}\\ {{\varepsilon }^{f}}_{e\tau }^{*} & {{\varepsilon }^{f}}_{\mu \tau }^{*} & {\varepsilon }_{\tau \tau }^{f}\end{array}\right]\right\}\left[\begin{array}{c}{\nu }_{e}\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right],\end{eqnarray} \tag{ 15 }$$

where ${\nu }_{\alpha }\equiv \langle {\nu }_{\alpha }| \nu (r)\rangle \ (\alpha =e,\mu ,\tau )$ denotes the probability amplitude to find neutrino as ${\nu }_{\alpha }$ at the position r, ${{\rm{\Delta }}}_{{ij}}\equiv {\rm{\Delta }}{m}_{{ij}}^{2}/(2E)$, E being the neutrino energy. ${V}_{f}={V}_{f}(r)\equiv \sqrt{2}{G}_{F}{N}_{f}(r)$ where N_f ($f=e,u$ or d) denotes the fermion number density along the neutrino trajectory in matter. Note that ${V}_{e}(r)$ is the standard matter potential [16] which induces the usual MSW effect [16, 102].

Since NSI effects in propagation enter only through the vector couplings, ${\varepsilon }_{\alpha \beta }^{f}$ must be interpreted as ${\varepsilon }_{\alpha \beta }^{f}={\varepsilon }_{\alpha \beta }^{{fL}}+{\varepsilon }_{\alpha \beta }^{{fR}}$. For simplicity, throughout this review, we consider the case where only d-quark has the propagation NSI with neutrinos, and write NSI parameters simply as ${\varepsilon }_{\alpha \beta }$ by omitting the fermion superscript. Note that the case of u-quark NSI is very similar in most cases to be discussed in this section because in the usual matter, ${N}_{u}\sim {N}_{d}\sim 3{N}_{e}$.

In this review, unless otherwise stated, for certainty, we use the following values of the mixing parameters as our reference values; ${\mathrm{sin}}^{2}{\theta }_{12}=0.31$, ${\mathrm{sin}}^{2}{\theta }_{13}=0.023$, ${\mathrm{sin}}^{2}{\theta }_{23}=0.5$, ${\rm{\Delta }}{m}_{21}^{2}=7.5\times {10}^{-5}$ eV² and $| {\rm{\Delta }}{m}_{31}^{2}| =2.4\times {10}^{-3}$ eV², and ${\delta }_{{CP}}=0$, which are consistent at 2σ with the results obtained by the recent global analysis [103–105].

Equation (15) defines the framework for neutrino propagation in matter with NSI. The parameters ${\varepsilon }_{\alpha \beta }$ ($\alpha ,\beta =e,\mu ,\tau$) describe the magnitude of NSI. The diagonal NSI parameters, ${\varepsilon }_{\alpha \alpha }(\alpha =e,\mu ,\tau )$, could play a role similar to the terms of the standard MSW matter potential, or could be interpreted as the NSI induced mass squared difference, mimicking the ones that contain ${\rm{\Delta }}{m}^{2}$, which could induce new resonance even if neutrinos were massless [18, 20]. On the other hand, off-diagonal NSI parameters, ${\varepsilon }_{\alpha \beta }(\alpha \ne \beta )$ could play a role similar to the mixing angle. Even if there is no mixing in vacuum, the flavor transitions ${\nu }_{\alpha }\to {\nu }_{\beta }$ can occur in matter due to the presence of the off-diagonal NSI [18–20]. The complex phases of the off-diagonal elements ${\varepsilon }_{\alpha \beta }$ could be a new source of CP violation, see e.g., [31, 64].

Currently, almost all the neutrino data are consistent with the standard three flavor scheme of massive and mixed neutrinos. Therefore, NSI, if they exist, is expected to manifest only as a subdominant effect. NSI in propagation has been constrained mainly by the oscillation data of solar and atmospheric neutrinos as well as neutrinos produced by accelerators. Reactor neutrinos do not constrain the propagation NSI because the matter effect is expected to be very small though they can constrain the detection NSI.

Roughly speaking, for a given neutrino energy, and the matter density, ρ, the impact of propagation NSI in neutrino oscillation (modification of the standard oscillation due to NSI) is essentially determined by the magnitude of the following dimensionless quantity,

$$\begin{eqnarray}&&{\eta }_{\alpha \beta }\equiv \displaystyle \frac{{\varepsilon }_{\alpha \beta }{V}_{f}}{{{\rm{\Delta }}}_{{ij}}}\approx 0.1\times {\varepsilon }_{\alpha \beta }\left[\displaystyle \frac{E}{\mathrm{GeV}}\right]\ \left[\displaystyle \frac{2.4\times {10}^{-3}\ {\mathrm{eV}}^{2}}{{\rm{\Delta }}{m}_{{ij}}^{2}}\right]\ \left[\displaystyle \frac{\rho }{{\rm{g}}\;{\mathrm{cm}}^{-3}}\right],\end{eqnarray} \tag{ 16 }$$

where the NSI with d or u quarks are assumed; ${\rm{\Delta }}{m}_{{ij}}^{2}$ is the relevant mass squared difference in the corresponding oscillation channel, and the baseline L is assumed to be large enough, or ${{\rm{\Delta }}}_{{ij}}L\gtrsim O(1)$. Larger the value of ${\eta }_{\alpha \beta }$, larger the impact of NSI in propagation.

In this subsection we discuss the NSI effect for atmospheric neutrinos. The impact of NSI on atmospheric neutrinos have been considered by many authors, see e.g., [26, 29, 35, 41, 42, 44, 70].

Let us first consider the impact of NSI on the ${\nu }_{\mu }-{\nu }_{\tau }$ sector and assume that all the NSI parameters coupling to electron flavor neutrino, namely, ${\varepsilon }_{e\beta }$ are zero. In this scenario, ${\varepsilon }_{\mu \tau }$ and ${\varepsilon }_{\tau \tau }-{\varepsilon }_{\mu \mu }$ can be constrained mainly by the higher energy samples of the atmospheric neutrino data as will be seen below.

In the limit of ${\rm{\Delta }}{m}_{21}^{2}L/E\to 0$, with the constant matter density approximation, and ignoring ${\theta }_{13}$, the ${\nu }_{\mu }\to {\nu }_{\mu }$ survival probability is expressed as [32, 42]

$$\begin{eqnarray}&&P({\nu }_{\mu }\to {\nu }_{\mu })=1-P({\nu }_{\mu }\to {\nu }_{\tau })\simeq 1-{\mathrm{sin}}^{2}2{\theta }_{\mathrm{eff}}{\mathrm{sin}}^{2}\left[\xi \ \displaystyle \frac{{{\rm{\Delta }}}_{31}L}{2}\right],\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&{\mathrm{sin}}^{2}2{\theta }_{\mathrm{eff}}\equiv \displaystyle \frac{{| \mathrm{sin}2{\theta }_{23}\pm 2{\eta }_{\mu \tau }| }^{2}}{{\xi }^{2}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\xi \equiv \sqrt{{| \mathrm{sin}2{\theta }_{23}\pm 2{\eta }_{\mu \tau }| }^{2}+{\{\mathrm{cos}2{\theta }_{23}+({\eta }_{\mu \mu }-{\eta }_{\tau \tau })\}}^{2}},\end{eqnarray} \tag{ 19 }$$

and the $+(-)$ sign in front of ${\eta }_{\mu \tau }$ corresponds to the normal (inverted) mass ordering. For anti-neutrino channel, the sign of ${\eta }_{\alpha \beta }$ must be changed.

For the atmospheric neutrinos, the sensitivity to the NSI parameters can be estimated by studying the muon neutrino and anti-neutrino survival probabilities for different zenith angles, $\mathrm{cos}{\theta }_{z}$. Figure 1 shows the muon neutrino survival probabilities (for the normal mass ordering) for $\mathrm{cos}{\theta }_{z}=-0.3$ (left panels), $-0.6$ (middle panels) and $-1$ (right panels) for the cases without NSI (by solid lines) and with NSI (by non-solid lines). For this calculation, the neutrino evolution equation (15) was solved numerically (without ignoring neither ${\rm{\Delta }}{m}_{21}^{2}$ nor ${\theta }_{13}$) using the Earth matter density profile predicted in the Preliminary Reference Earth Model (PREM) model [106].

Zoom In Zoom Out Reset image size

Note that, by comparing the upper and lower panels, the dependence on the sign of NSI parameters for neutrinos and anti-neutrinos are opposite. Note also that, with a good approximation, the survival probabilities are invariant under the simultaneous transformation ${{\rm{\Delta }}}_{31}\to -{{\rm{\Delta }}}_{31}$ and ${\varepsilon }_{\mu \tau }\to -{\varepsilon }_{\mu \tau }$. In the limit of the 2 flavor approximation, what is relevant is only the relative sign of these quantities. As can be seen from figure 1, the impact of NSI is small for lower energies, for $E\lesssim 5$ GeV, even for the case where neutrino pass through the center of the Earth. On the other hand, the impact of NSI for energy $\gtrsim 10$ GeV, could be quite large for the NSI parameters considered in figure 1 and it is expected that these values could be disfavored or excluded.

Here we quote the bounds on these NSI parameters obtained by the Super-Kamiokande collaboration [107], $| {\epsilon }_{\mu \tau }| \lt 1.1\times {10}^{-2}$ and $-4.9\times {10}^{-2}\lt {\epsilon }_{\tau \tau }-{\epsilon }_{\mu \mu }\lt 4.9\times {10}^{-2}$ at 90% CL. More recently, by using the IceCube-79 and DeepCore data, authors of [108] obtained somewhat better bounds, $| {\epsilon }_{\mu \tau }| \lesssim 6\times {10}^{-3}$ and $| {\epsilon }_{\tau \tau }-{\epsilon }_{\mu \mu }| \lesssim 3\times {10}^{-2}$ at 90% CL.

For the ${\nu }_{e}-{\nu }_{\tau }$ sector, besides the probability for ${\nu }_{\mu }\to {\nu }_{\mu }$, it is also useful to consider the ${\nu }_{\mu }\to {\nu }_{e}$ case. The computation of these probabilities, shown in figure 2, was done in an analogous way to the case shown in figure 1. For this computation different NSI parameters have been considered: ${\varepsilon }_{{ee}}$, ${\varepsilon }_{\tau \tau }$ and ${\varepsilon }_{e\tau }$. It is important to notice that, in this case, the ${\varepsilon }_{e\tau }$ parameters, could play a role similar to ${\theta }_{13}$. Therefore, there is some impact on the ${\nu }_{\mu }\to {\nu }_{e}$ channel as it is possible to see in the lower panels of figure 2.

Zoom In Zoom Out Reset image size

When ${\varepsilon }_{{ee}}$, ${\varepsilon }_{\tau \tau }$ and ${\varepsilon }_{e\tau }$ are assumed to be simultaneously nonzero, it is known [44] that the allowed combinations of the NSI parameters are approximately given by the parabolic relation,

$$\begin{eqnarray}&&{\varepsilon }_{\tau \tau }\sim \displaystyle \frac{3{| {\varepsilon }_{e\tau }| }^{2}}{1+3{\varepsilon }_{{ee}}}.\end{eqnarray} \tag{ 20 }$$

This feature can be also confirmed in figures 8 and 9 in [107] which show the constraints on these NSI parameters obtained by the Super-Kamiokande atmospheric neutrino data. It is also pointed out in [44] that atmospheric neutrino data alone can not essentially constrain ${\varepsilon }_{{ee}}$ parameter. Therefore, in general one can obtain the allowed regions of ${\varepsilon }_{e\tau }$ and ${\varepsilon }_{\tau \tau }$ for given values of ${\epsilon }_{{ee}}$ as done in [44, 107]. For example, from figure 9 of [107], we see that for ${\mathrm{sin}}^{2}{\theta }_{23}=0.5$, for ${\varepsilon }_{{ee}}=-0.5$, 0, and 0.5, ${\varepsilon }_{e\tau }\lesssim$, 0.08, 0.11 and 0.18, respectively, at 90% CL.

So far the bounds on propagation NSI from accelerator neutrinos mainly come from the ${\nu }_{\mu }\to {\nu }_{\mu }$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_{\mu }$ channels. For these channels, at first approximation, the relevant NSI parameters are ${\varepsilon }_{\mu \tau }$, ${\varepsilon }_{\mu \mu }$ and ${\varepsilon }_{\tau \tau }$. The ${\nu }_{\mu }\to {\nu }_{\mu }$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_{\mu }$ survival probabilities as a function of neutrino energy for the MINOS baseline, L = 730 km, are shown in figure 3. The computations were done without the presence of NSI (solid lines) and with ${\varepsilon }_{\mu \tau }=\pm 0.1$ (dotted and dashed lines). As it is possible to see from figure 3 the impact of NSI for ν and $\bar{\nu }$ channels are opposite.

Zoom In Zoom Out Reset image size

The bounds obtained by the MINOS collaboration [109], translated to the notation used in this review, can be stated as $-0.067\lt {\varepsilon }_{\mu \tau }\lt 0.023$ at 90% CL.

The most updated analysis of solar neutrinos in the context of the propagation NSI comes from [74]. For solar neutrinos, under the so-called one mass scale dominance approximation, $| {\rm{\Delta }}{m}_{31}^{2}| \to \infty$, the neutrino evolution can be effectively reduced to that of the 2 flavor system [110], as follows,

$$\begin{eqnarray}&&i\displaystyle \frac{d}{{dr}}\left[\begin{array}{c}{\nu }_{e}^{\prime }\\ {\nu }_{\mu }^{\prime }\end{array}\right]=\left\{{U}_{{\theta }_{12}}\left[\begin{array}{cc}0 & 0\\ 0 & {{\rm{\Delta }}}_{21}\end{array}\right]{U}_{{\theta }_{12}}^{\dagger }+\displaystyle \sum _{f}{V}_{f}\left[\begin{array}{cc}{c}_{13}^{2}{\delta }_{{ef}}-{\varepsilon }_{D}^{f} & {\varepsilon }_{N}^{f}\\ {{\varepsilon }^{f}}_{N}^{*} & {\varepsilon }_{D}^{f}\end{array}\right]\right\}\left[\begin{array}{c}{\nu }_{e}^{\prime }\\ {\nu }_{\mu }^{\prime }\end{array}\right],\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&\left[\begin{array}{c}{\nu }_{e}^{\prime }\\ {\nu }_{\mu }^{\prime }\\ {\nu }_{\tau }^{\prime }\end{array}\right]\equiv \left[\begin{array}{ccc}{c}_{12} & {s}_{12} & 0\\ -{s}_{12} & {c}_{12} & 0\\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{c}{\nu }_{1}\\ {\nu }_{2}\\ {\nu }_{3}\end{array}\right]=\left[\begin{array}{ccc}{c}_{12} & {s}_{12} & 0\\ -{s}_{12} & {c}_{12} & 0\\ 0 & 0 & 1\end{array}\right]{U}^{\dagger }\left[\begin{array}{c}{\nu }_{e}\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right],\end{eqnarray} \tag{ 22 }$$

with ${\nu }_{\tau }^{\prime }$ being decoupled from the system [110], and diagonal ${\varepsilon }_{D}$ and off-diagonal ${\varepsilon }_{N}$ NSI parameters are related to ${\varepsilon }_{\alpha \beta }$ as [74],

$$\begin{eqnarray}{\varepsilon }_{D} & = & {c}_{13}{s}_{13}\mathrm{Re}\left[{{\rm{e}}}^{i{\delta }_{{CP}}}({{\rm{s}}}_{23}\;{\varepsilon }_{e\mu }+{{\rm{c}}}_{23}\;{\varepsilon }_{e\tau })\right]-(1+{{\rm{s}}}_{13}^{2}){{\rm{c}}}_{23}{{\rm{s}}}_{23}\mathrm{Re}({\varepsilon }_{\mu \tau })\\ & & -\displaystyle \frac{{c}_{13}^{2}}{2}\left({\varepsilon }_{{ee}}-{\varepsilon }_{\mu \mu }\right)+\displaystyle \frac{{s}_{23}^{2}-{s}_{13}^{2}{c}_{23}^{2}}{2}\left({\varepsilon }_{\tau \tau }-{\varepsilon }_{\mu \mu }\right),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\varepsilon }_{N}={c}_{13}\left({c}_{23}\;{\varepsilon }_{e\mu }-{s}_{23}\;{\varepsilon }_{e\tau }\right)+{s}_{13}{e}^{-i{\delta }_{{CP}}}\left[{s}_{23}^{2}\;{\varepsilon }_{\mu \tau }-{c}_{23}^{2}\;{\varepsilon }_{\mu \tau }^{*}+{c}_{23}{s}_{23}\left({\varepsilon }_{\tau \tau }-{\varepsilon }_{\mu \mu }\right)\right].\end{eqnarray} \tag{ 24 }$$

In this approximation, the survival probability is given as

$$\begin{eqnarray}&&P({\nu }_{e}\to {\nu }_{e})={s}_{13}^{4}+{c}_{13}^{4}{P}_{2\nu }({\nu }_{e}^{\prime }\to {\nu }_{e}^{\prime }),\end{eqnarray} \tag{ 25 }$$

where ${P}_{2\nu }({\nu }_{e}^{\prime }\to {\nu }_{e}^{\prime })$ is calculated for the effective 2 flavor system described by (21). According to [74], the bounds on these parameters are $-0.25\lt {\varepsilon }_{D}\lt -0.02$ and $-0.14\lt {\varepsilon }_{N}\lt 0.12$ at 90% CL assuming NSI with d-quark.

In table 2 we show the summary of the bounds on the propagation NSI.

Table 2.  Constraints on the matter (propagation) NSI parameters at 90% C L. for the interaction of neutrinos with d type quark. The other NSI parameters are set to zero.

NSI parameters	Bounds	Reference
${\varepsilon }_{{ee}}^{m}-{\varepsilon }_{\mu \mu }^{m}$	(0.02, 0.51)	[74]
${\varepsilon }_{\tau \tau }^{m}-{\varepsilon }_{\mu \mu }^{m}$	($-0.01,0.03$)	[74]
${\varepsilon }_{\tau \tau }^{m}-{\varepsilon }_{\mu \mu }^{m}$	($-0.049,0.049$)	[107]
${\varepsilon }_{\tau \tau }^{m}-{\varepsilon }_{\mu \mu }^{m}$	($-0.036,0.031$)	[108]
${\varepsilon }_{e\mu }^{m}$	($-0.09,0.04$)	[74]
${\varepsilon }_{\mu \tau }^{m}$	($-0.01$, 0.01)	[74]
${\varepsilon }_{\mu \tau }^{m}$	($-0.011,0.011$)	[107]
${\varepsilon }_{\mu \tau }^{m}$	($-6.1\times {10}^{-3}$, $5.6\times {10}^{-3}$)	[108]
${\varepsilon }_{e\tau }^{m}$	($-0.13,0.14$)	[74]
${\varepsilon }_{e\tau }^{m}$ (for ${\varepsilon }_{{ee}}^{m}=-0.50$)	($-0.05,0.05$)	[107]
${\varepsilon }_{e\tau }^{m}$ (for ${\varepsilon }_{{ee}}^{m}$ = 0.50)	($-0.19,0.13$)	[107]

Different experimental set-ups (proposed to increase the precision for the determination of the neutrino oscillations parameters) have been considered in recent years. The need for a better knowledge of the Kobayashi-Maskawa type CP phase in the lepton sector as well as the mass ordering is certainly a major motivation for these proposals. Currently ongoing experiments such as T2K and NOvA may improve somewhat the current bounds on some NSI parameters but probably not so much, especially for that coming from the ${\nu }_{e}({\bar{\nu }}_{e})$ appearance mode due to relatively small statistics.

Hyper-Kamiokande [138–140] is an interesting proposal that expects to improve the sensitivity to NSI, by using the atmospheric neutrino data [141–143]; their expectations for the normal hierarchy case are particularly appealing. the LBNE [144] and LBNO [145] proposals, the expected sensitivity to NSI is also encouraging, especially for the flavor changing case of ${\epsilon }_{\mu \tau }$ and ${\epsilon }_{e\mu }$ [146].

Finally, important constraints are expected from the IceCube Deep Core and PINGU experiments. These are extensions of the IceCube experiment focused on a lower energy range. In this case, the expectations to constrain the flavor changing NSI parameter ${\epsilon }_{\mu \tau }$ could reach the one percent level. For the flavor diagonal case, it might be possible to obtain information about the elusive parameter ${\epsilon }_{\tau \tau }$ [108, 147, 148].

The future neutrino oscillation experiments will also be sensitive to the NSI parameters through a detection effect. For instance, for the case of the proposed JUNO [134] and RENO-50 [135] an improvement to the constraints on ${\epsilon }_{e\mu }$ and ${\epsilon }_{e\tau }$ is expected [77].

For the case of the interaction of electron neutrinos with electrons, both ISODAR [149] and LENA [150] proposals could give complementary information if both proposals are done in the future. In the case of ISODAR, it is proposed to use an intense anti-neutrino ⁸Li source, with an anti-neutrino energy ranging up to 14 MeV, in combination with the KamLAND liquid scintillator [149]. The LENA proposal plans to use a neutrino Chromium source, providing a monochromatic neutrino flux of energy, ${E}_{\nu }=0.747$ MeV, located at the top of a 100 kTon liquid scintillator cylindrical detector [150]. These are not the only proposals for neutrino electron scattering, but they illustrate the future potential of this experiments.

The use of either a neutrino or anti-neutrino source leads to a better determination of the left or right-handed couplings, respectively. Therefore, if both neutrino and anti-neutrino experiments are done in the future, there could be a good room for the improvement of the NSI parameters.

We illustrate this by showing, in figure 6, the expected sensitivity for the case of a neutrino artificial source in combination with the proposed LENA detector as has already been calculated in [151], where an expected total number of $1.9\times {10}^{5}$ neutrino events and a 5 % systematic error was considered. The case of an anti-neutrino source is also shown in figure 6. In this last case the analysis developed in [149] has been closely followed. The expected result from the combined analysis of both future experiments is shown by the region filled by the magenta color. We show in the same figure one of the current constraints on NSI, coming from the solar neutrino analysis [60]. It is possible to see that there is room for improving these constraints by almost one order of magnitude, especially if both experiments are realized.

Zoom In Zoom Out Reset image size

Regarding the NSI of neutrinos with quarks, there are severals proposals that could improve current bounds. An interesting proposal is that of neutrino coherent scattering off nuclei [152]. After seminal works on the construction of these type of detectors [153, 154], there was a renewed interest in the previous decade [155, 156]. At the same time, the sensitivity to NSI parameters and new physics searches was also noted [49, 157]. Different experimental set-ups have already been considered, either using a reactor anti-neutrino flux [156], and spallation source [158–160, 166], beta beams [161, 162], or pion decay [163–165]. These experiments could have an excellent sensitivity to the NSI. In particular, the TEXONO proposal has been studied in the past, using either a ⁷⁶Ge or ²⁸Si as a detector; an improvement of even one order of magnitude could be achieved in this case [157]. Other possible nuclei have also been studied [130] such as ⁴⁸Ti and ²⁷Al.