+
            \hspace{-0.3cm}
           \sum_{\begin{minipage}{1.5cm}
                    {\tiny 
                      $\alpha,\beta \neq V-A$ 
                    }
                  \end{minipage}} 
            \hspace{-0.3cm}
            \epsilon_{\alpha}^{\beta} 
            \hspace{0.1cm}
            j_{\beta}J_{\alpha}\right]\,.\label{eq:defLR}\end{aligned}$$ Here, we follow the notations of $j$ and $J$ adopted in [@Pas:1999fc], which are[^10] $$\begin{gathered}
\label{eq:CurrLR}
J^{\mu}_{V\pm A}
=
(J_{R/L})^{\mu}
\equiv
\overline{u}\gamma^{\mu}(1\pm\gamma_5)d\,,
\qquad
j_{V\pm A}^{\mu}
\equiv
\overline{e}\gamma^{\mu}(1\pm\gamma_5)\nu\,,
\\ \nonumber
J_{S\pm P}
=
J_{R/L}
\equiv
\overline{u}(1\pm\gamma_5)d\,,
\qquad
j_{S\pm P}\equiv\overline{e}(1\pm\gamma_5)\nu\,,
\\ \nonumber
J^{\mu\nu}_{T_{R/L}}
=
(J_{R/L})^{\mu \nu}
\equiv
\overline{u}\gamma^{\mu\nu}(1\pm\gamma_5)d\,,
\qquad
j_{T_{R/L}}^{\mu\nu} \equiv\overline{e}\gamma^{\mu\nu}(1\pm\gamma_5)\nu\ ,\end{gathered}$$ where the Lorenz tensor matrix $\gamma^{\mu \nu}$ is defined as $\gamma^{\mu \nu} = \frac{\rm i}{2} [\gamma^{\mu}, \gamma^{\nu}]$. Recall that $P_{L/R} =\frac{1}{2}(1 \mp \gamma_5)$ and we will use the short-hand notation $L$ and $R$ for left-handed and right-handed fermions, respectively. The first term of Eq.  is the SM charged current interaction, and the second term contains the new physics contributions, which do not take the Lorenz structure of the standard four-Fermi interaction $(V-A)(V-A)$. The coefficients $\epsilon_{\alpha}^{\beta}$ for the exotic four-Fermi interactions are normalised to the SM charged current strength $G_F/\sqrt{2}$. If these dimensionless coefficients take numbers smaller than one, diagram (c) in Fig. \[Fig:LongShort\] is of order $\epsilon^2$ and becomes immediately sub-dominant in the [$0\nu\beta\beta$ ]{}amplitudes.

The neutrino propagator in diagrams (a)-(c) contains two terms: $m_{\nu} + {q\!\!\!/}$, the mass and the momentum terms. Since the charged current 