\tau} | < 0.5.
\end{equation}

Finally, the tau neutrino case have less stringent constraints. For the interaction with electrons they are given by~\cite{Barranco:2007ej,Bolanos:2008km}
\begin{equation}
	-0.12 <  \epsilon^{e,L}_{\tau\tau} < 0.06, \quad
	-0.25 < \epsilon^{e,R}_{\tau\tau} < 0.43,
\end{equation}
and~\cite{Davidson:2003ha}
\begin{equation}
	| \epsilon^{e,L}_{\mu\tau} | < 0.1, \quad
	| \epsilon^{e,R}_{\mu\tau} | < 0.1.
\end{equation}
For the interaction with quarks, the flavor-diagonal  constraint is~\cite{GonzalezGarcia:2011my}
\begin{equation}
	|\epsilon^{q,V}_{\tau\tau}| < 0.037, 
\end{equation}
while for the flavor-changing case~\cite{Escrihuela:2011cf}
\begin{equation}
	| \epsilon^{q,L}_{\mu\tau} | < 23\times10^{-3}, \quad
	| \epsilon^{q,R}_{\mu\tau} | < 36\times10^{-3}.
\end{equation}

Once we have overviewed the current status for the NSI parameters, we
can turn our attention to the scalar, pseudoscalar, and tensor GNI
parameters. We start by defining the relevant cross-sections for our
study and their modification when a GNI parameter is present. 

\subsection{Cross-section for $e^{+}e^{-}\rightarrow\nu\bar{\nu}\gamma$}
The neutrino-antineutrino pair produced together with a single photon
is a clean signal measured with precision at LEP~\cite{Barate:1997ue,
	Barate:1998ci, Heister:2002ut,Abreu:2000vk,Acciarri:1997dq,
	Acciarri:1998hb, Acciarri:1999kp,Ackerstaff:1997ze, Abbiendi:1998yu,
	Abbiendi:2000hh}. This experimental result has been previously
studied to put constraints on NSI~\cite{Berezhiani:2001rs,Barranco:2007ej,Forero:2011zz} and non-unitarity~\cite{Escrihuela:2019mot}, for instance. 

To compute the GNI contributions for this observable, we must consider
that its cross-section is written as the product~\cite{Nicrosini:1988hw, Barranco:2007ej,Berezhiani:2001rs}
\begin{equation}\label{eq:xsec_nunug}
	\frac{d^2\sigma}{dx\,dy}= H(x,y;s)\, \sigma_0(s(1 -x)),
\end{equation}
where $H(x,y;s)$ is the radiator function 
\begin{equation}

	H(x,y;s) = \frac{2\alpha \left[(1-\frac{1}{2}x)^2 + \frac{1}{4}x^2y^2\right]}{\pi x(1-y^2)},
\end{equation}
with
\begin{equation}
	x = 2 E_\gamma/\sqrt{s}, \quad \quad y = \cos\theta_{\gamma}, 
\end{equation}
where $\sqrt{s}$ is the center-of-mass energy and $E_\gamma (\theta_\gamma)$ is the energy (angle) of the produced photon. The term $\sigma_0$ is the ``reduced'' cross-section for  $e^+e^-
\to \nu\bar{\nu}$,
\begin{align}\label{eq:reducedXS}
	\sigma_0(s) &= \sigma_{W}(s) + \sigma_{Z}(s) + \sigma_{W-Z}(s), \nonumber \\
	\sigma_0(s) &= \frac{{G_F}^2 s}{12 \pi} \left[2 + \frac{N_{\nu} (g_V^2 + g_A^2)}{(1 - s / {M_Z}^2)^2 + \Gamma_Z^2 / M_Z^2} + \frac{2 (g_V + g_A) (1 - s / {M_Z}^2)}{(1 - s / M_Z^2)^2 + \Gamma_Z^2 / M_Z^2}\right],
\end{align}
where $\Gamma_Z$ and $M_Z$ are the $Z$ boson decay width and mass, respectively, and $N_\nu$ is the number of neutrino species. 
The first term in the previous equation comes from the $W$
contribution to the cross-section while the second one from the $Z$
contribution, as is illustrated in the Feynman diagrams in Fig.~(\ref{fig:feynDiag_nng}). The third contribution appears from the interference between both
gauge bosons.

\begin{figure}[h]
	\begin{minipage}{0.05\textwidth}
		(a)
	\end{minipage}
	\begin{minipage}{0.3\textwidth}
		\includegraphics[scale=0.6]{CC_ee_to_nunuGamma_cropped.pdf}
	\end{minipage}
	\begin{minipage}{0.3\textwidth}\vspace{1.3cm}
		\includegraphics[scale=0.6]{CC_ee_to_nunuGamma_3_cropped.pdf}
	\end{minipage}
	\begin{minipage}{0.3\textwidth}
		\includegraphics[scale=0.6]{CC_ee_to_nunuGamma_2_cropped.pdf}
	\end{minipage}
	\begin{minipage}{0.05\textwidth}
		(b)
	\end{minipage}
	\begin{minipage}{0.4\textwidth}
		\includegraphics[scale=0.6]{NC_ee_to_nunuGamma_cropped.pdf}
	\end{minipage}
	\begin{minipage}{0.4\textwidth} \vspace{0.6cm}
		\includegraphics[scale=0.6]{NC_ee_to_nunuGamma_2_cropped.pdf}
	\end{minipage}
	\caption{Contributions to the $e^-e^+ \to \nu\bar{\nu}\gamma$ process at tree level, from the $W$ (a) and $Z$ (b) bosons.}
	\label{fig:feynDiag_nng}
\end{figure}

Considering only neutral current GNI, we notice that the corrections
to this interaction will come from diagrams that in the SM are
related to the $Z$ bosons (Fig.~\ref{fig:feynDiag_nng}). From
Eq.~\eqref{equation_1} we can obtain the GNI cross-section
contribution for the process 
\begin{equation}\label{equation_2}
	\sigma_{0}=\frac{G^{2}_{F}s}{48\pi}\sum_{\alpha,\beta}\left(8|\epsilon^{e,L}_{\alpha\beta}|^{2}+8|\epsilon^{e,R}_{\alpha\beta}|^{2}+3|\epsilon^{e,S}_{\alpha\beta}|^{2}+3|\epsilon^{e,P}_{\alpha\beta}|^{2}+32|\epsilon^{e,T}_{\alpha\beta}|^{2}\right)\,.
\end{equation}
In Eq.~\eqref{equation_2}, we can see that tensor interactions contribute one order of magnitude more to the cross-section than the equally-contributing scalar and pseudoscalar interactions.

With this information we can proceed to compute the total cross-section 
\begin{equation}
	\sigma(s)=\int_{x_\mathrm{min}}^1 dx \int_{-\cos\theta_{\mathrm{min}}}^{\cos\theta_{\mathrm{min}}} dy H(x,y;s)\, \sigma_0(s(1 -x)).
	\label{eq:nng-totalXS}
\end{equation}

In the case of energies above the $Z$ resonance, it is important to
consider finite distance effects on the $W$ propagator. They require
the substitution~\cite{Hirsch:2002uv,
	Barranco:2007ej,Berezhiani:2001rs}:
\begin{align}\label{eq:reducedCrossSec2}
	\sigma_W(s) \; & \to \; \; \sigma_{W}(s) F