make contact with this formalism, we recapitulate the main results and definitions of [@Pas:1999fc; @Pas:2000vn] in this section. The total amplitude of [$0\nu\beta\beta$ ]{}is most conveniently divided into two parts: Long-range and short-range contributions, see Fig. \[Fig:LongShort\].

Long-range contributions
------------------------

(400,110) (10,10)(40,10) (40,10)(70,10) (40,10)(40,30)[2]{}[5]{} (40,30)(70,30) (40,30)(40,50) (40,50)(70,50) (40,50)(40,70)[2]{}[5]{} (10,70)(40,70) (40,70)(70,70) (5,10)\[r\][$d$]{} (5,70)\[r\][$d$]{} (75,70)\[l\][$\overline{u}$]{} (75,10)\[l\][$\overline{u}$]{} (75,50)\[l\][$e^-$]{} (75,30)\[l\][$e^-$]{} (35,40)\[r\][$\nu$]{} (35,60)\[r\][$W^-$]{} (35,20)\[r\][$W^-$]{} (40,5)\[t\][(a)]{} (110,10)(140,10) (140,10)(170,10) (140,10)(140,30)[2]{}[5]{} (140,30)(170,30) (140,30)(140,50) (110,60)(140,60) (140,60)(170,70) (140,60)(170,50) (140,60)[10]{}[0.5]{} (105,10)\[r\][$d$]{} (105,60)\[r\][$d$]{} (175,70)\[l\][$\overline{u}$]{} (175,10)\[l\][$\overline{u}$]{} (175,50)\[l\][$e^-$]{} (175,30)\[l\][$e^-$]{} (135,40)\[r\][$\nu$]{} (135,20)\[r\][$W^-$]{} (140,5)\[t\][(b)]{} (210,20)(240,20) (240,20)(270,30) (240,20)(270,10) (240,30)(240,50) (210,60)(240,60) (240,60)(270,70) (240,60)(270,50) (240,60)[10]{}[0.5]{} (240,20)[10]{}[0.5]{} (205,20)\[r\][$d$]{} (205,60)\[r\][$d$]{} (275,70)\[l\][$\overline{u}$]{} (275,10)\[l\][$\overline{u}$]{} (275,50)\[l\][$e^-$]{} (275,30)\[l\][$e^-$]{} (235,40)\[r\][$\nu$]{} (240,5)\[t\][(c)]{} (310,30)(340,30) (310,50)(340,50) (350,45)(380,70) (350,45)(380,45) (350,35)(380,35) (350,35)(380,10) (345,40)[15]{}[0.5]{} (305,30)\[r\][$d$]{} (305,50)\[r\][$d$]{} (385,70)\[l\][$\overline{u}$]{} (385,10)\[l\][$\overline{u}$]{} (385,50)\[l\][$e^-$]{} (385,30)\[l\][$e^-$]{} (345,5)\[t\][(d)]{}

Consider first the long-range part. Here, we can sub-divide the amplitudes into parts (a)-(c) as shown in the figure. In case (a), a massive Majorana neutrino is exchanged between two SM charged current vertices, while cases (b) and (c) contain one and two (unspecified) non-standard interactions respectively, indicated by the black blobs.

At low energy, we can write the relevant part of the effective Lagrangian with the leptonic ($j$) and hadronic ($J$) charged currents as $$\begin{aligned}
\nonumber
\mathcal{L}^{\text{4-Fermi}}
&=& \mathcal{L^{\rm SM}} + \mathcal{L^{\rm LNV}} \\  
           & =& \frac{G_F}{\sqrt{2}}
           \left[
           j^{\mu}_{V-A} J_{V-A,\mu}
           