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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),

$\hspace{2cm}$enter image description here

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.
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You simply have to include the Majorana mass term (into the most general Lagrangian compatible with the given symmetries) at a tree level because the masslessness of $\chi_{++}$ isn't guaranteed by any symmetry that is valid at the quantum level. In fact, as I will argue in a minute, the would-be symmetry is violated even at the classical level.

Then the counterterm for this Majorana mass term will swallow the divergent contribution, too. Note that in many other cases, the Weyl fermions have the chiral symmetry

$$ \chi \to \chi \times \exp(i\phi\gamma_5) $$

where $\gamma_5$ is omitted (set to one) in the two-component formalism, a symmetry that protects the masslessness even at the quantum level (the reason why it's natural for the higgsinos to assume that they stay light, for example, so why SUSY makes the light Higgs - the bosonic partner of the higgsinos - natural).

But the cubic term involving one factor of the $\chi_{++}$ field explicitly violates this chiral symmetry in your example. Note that the violation of the chiral symmetry may already be seen classically. There are other cases in which the violation of the symmetry would only occur at the quantum level – i.e. through an anomaly.

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