Let's consider the theory given by the following lagrangian
$$
\mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu \phi – \frac{1}{2} M^2 \phi^2 + \bar\psi (i\gamma^\mu\partial_\mu – m)\psi + \frac{1}{4!} \lambda \phi^4 + g \phi \bar{\psi}\psi.
$$
Superficial degree of divergence $D$ of 1PI diagram depends as follows on the number of external fermionic $E_f$ and bosonic $E_b$ lines:
$$
D = 4 – E_b – 3/2E_f.
$$
Thus, there are 7 superficially 1PI diagrams:
- vacuum diagram,
- diagrams with one, two, three or four external bosonic line,
- diagram with two fermionic lines,
- diagram with two fermionic lines and one bosonic.
If I treat the fields and couplings in the original lagrangian as bare, then some counterterms appear. There is, however, no counterterm of the form $\phi^3$ and $\phi$.
Are the 1PI diagrams with one or three external bosinic lines really divergent? If yes, should I add by hand additional counterterms which cancel these divergencies?
In comparison in the pseudoscalar Yukawa theory
$$
\mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu \phi – \frac{1}{2} M^2 \phi^2 + \bar\psi (i\gamma^\mu\partial_\mu – m)\psi + \frac{1}{4!} \lambda \phi^4 + i g \phi \bar{\psi}\gamma^5\psi.
$$
there parity symmetry makes the problematic superficially divergent diagrams with only one or three bosonic lines (for which there are no counterterms in the lagrangian) vanish identically.
Best Answer
In 1988 I published a paper (Relativistic Quantum Field Theory of the $\sigma$ + $\omega$ Model, Proceedings of the Internatiional Workshop on Relativistic Nuclear Many-Body Physics, 6-9 June 1988, World Scientific Press) that utilized the scalar Yukawa theory (ie your Lagrangian) for a calculation of infinite nuclear matter. I had begun this calculation in 1975 but did not complete it until 1988. While not quite as general as your question, my work did address the renormalization issues of this Lagrangian and my findings shed some light on the answer to your question. Briefly, there was no need for counter terms for the $\phi$ and $\phi^3$ fluctuations in the specific problem that I addressed. For the $\phi$ term, the fluctuation vanished by definition (since I was expanding about the finite expectation value $<\phi>$).
The $\phi^3$ fluctuation term also vanished, but the explanation was somewhat more involved. I treated the $\phi^4$ fluctuation via a harmonic oscillator motivated approximation ($<\phi^4>=3<\phi^2>^2$) and in that approximation the $\phi^3$ fluctuation was also easily shown to vanish. I used this approach to perform a nonperturbative calculation that relied upon a variational technique to sum some of the boson loop terms to infinite order. The approach that I employed was called quantum hadrodynamics (QHD) but it had already fallen out of favor by the time I completed the work due to the more fundamental QCD approach.