I'm trying to understand how infinite mass corrections are cancelled for a particle that is massless at tree level. In short the problem is that we have infinite diagrams, but we don't have a counterterm for them since they don't exist at tree level. As a simple example consider a theory with three charged Weyl fermions, $ \chi _{ ++ }, \psi _- , \psi _+ $, as well a complex scalar, $ \phi _- $ (The $\psi_+ $ isn't really important, but just there to give a large Dirac mass for $\psi_-$)

Furthermore, assume that charge is approximately broken in another sector such that one of the fermions, $ \psi _- $, gets a small Majorana mass (this is very similar to the situation I'm actually interested in, a R symmetric SUSY model with R breaking through anomaly mediated SUSY breaking, so its not as far fetched as it may sound). The Lagrangian takes the form, \begin{equation} {\cal L} = {\cal L} _{ kin} - M ( \psi _- \psi _+ + h.c. ) - g ( \phi _- \chi _{ + + } \psi _- + h.c. ) - V ( \phi ) - \underbrace{ m ( \psi _- \psi _- + h.c. ) }_{\mbox{sym. breaking}}\end{equation}

where $ V ( \phi ) $ is the scalar potential.

Due to the symmetry breaking from $ \psi _- $ we can get a symmetry breaking Majorana mass for the $ \chi _{++} $ under loop corrections. However, we don't have a counterterm for it! For example to first order we have (we use chirality Feynman arrow notation),

$\hspace{2cm}$

Usually the $ 1 / \epsilon $ is harmless as we hide it in the counterterm. But in this case since we don't have a tree-level contribution to the Majorana mass for $ \chi _{ + + } $, we also don't have a counterterm for it. How is this issue resolved?

Edit:

1. I've found a related topic in the context of the weak interaction discussed in the appendix of arXiv:1106.3587. Here, if I understand correctly, they use the $Z$ boson to cancel the infinity. However, I don't understand how that would work here since this is not even a gauge theory.
2. Weinbeg also discusses a similar topic in the context of the weak interaction in "Perturbative Calculations of Symmetry Breaking", Phys Rev D Vol 7 Num 10.