\mathcal{L}^{\text{eff}}
=\frac{G_F^2}{2}m_{P}^{-1} 
 \left[    \epsilon_1 JJj+\epsilon_2 J^{\mu\nu}J_{\mu\nu}j
           +\epsilon_3 J^{\mu}J_{\mu}j+\epsilon_4 J^{\mu}J_{\mu\nu}j^{\nu}
           +\epsilon_5J^{\mu}Jj_{\mu}
 \right]\, ,
\label{eps_short}\end{aligned}$$ where $m_{P}$ is the mass of proton. Here we omitted the indices for clarity. However, if chirality changes play a role in the value of the [$0\nu\beta\beta$ ]{}decay rate, one needs to maintain the chirality indices and define $\epsilon_{i}=\epsilon_{i}^{xyz}$, with $x,y,z \in \{L,R\}$; cf., App. \[app:decay\] for details. Since $(j_{R})^{\mu}=-(j_{L})^{\mu}$, we define $(j)^{\mu}=(j_{R})^{\mu}=-(j_{L})^{\mu}$.

Current limits on short-range type $\epsilon_i$ are summarised in Tab. \[Tab:LimitsShort\]. Note that, different from the $\epsilon_{\alpha}^{\beta}$ of the long-range part, here $\epsilon_i$ scale as $\epsilon_i \propto (\lambda^{\rm eff}_{\rm LNV})^4/\Lambda^5$. For $\lambda^{\rm eff}_{\rm LNV} \simeq g_L$ the limits given in Tab. \[Tab:LimitsShort\] then correspond to $\Lambda \gsim (1-3)$ TeV.

    Isotope       $|\epsilon_1|$        $|\epsilon_2|$      $|\epsilon_3^{LLz (RRz)} |$   $|\epsilon_3^{LRz(RLz)}|$     $|\epsilon_4|$        $|\epsilon_5|$
  ------------ --------------------- --------------------- ----------------------------- --------------------------- --------------------- ---------------------
   $^{136}$Xe   $2.6 \cdot 10^{-7}$   $1.4 \cdot 10^{-9}$       $1.1 \cdot 10^{-8}$          $1.7 \cdot 10^{-8}$      $1.2 \cdot 10^{-8}$   $1.2 \cdot 10^{-7}$

  : *Limits on effective short-range interactions. These limits are taken from [@Deppisch:2012nb] and are derived assuming only one $\epsilon$ is different from zero at a time.*[]{data-label="Tab:LimitsShort"}

General decomposition of the $\boldsymbol{d=9}$ $\boldsymbol{0\nu\beta\beta}$ decay operator {#Sec:decom}
============================================================================================

(420,110) (10,10)(50,50) (50,50)(10,90) (50,50)(90,50)[3]{}[4]{} (90,50)(90,90) (90,50)(130,50) (130,50)(130,90) (130,50)(170,50)[3]{}[4]{} (170,50)(210,10) (170,50)(210,90) (50,50)[3]{} (90,50)[3]{} (130,50)[3]{} (170,50)[3]{} 