Quantum Field Theory – External Momenta in Renormalizing Pseudoscalar Yukawa Theory

fermionsfeynman-diagramslagrangian-formalismquantum-field-theoryrenormalization

This is a follow-up question to my earlier post here:

Now suppose we have the pseudoscalar Yukawa Lagrangian:
$$
L = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2+\bar\psi(i\not\partial-m)\psi-g\gamma^5\phi\bar\psi\psi.
$$
We can find its superficial degree of divergence as $D= 4-\frac{3}{2}N_f-N_s$. From this manual (p.80), we can find all divergent amplitudes as follows:
enter image description here
We do have other divergent graphs with odd scalar external lines. However, the author ignored them, and claimed they are potentially divergent diagrams that actually vanish. I wonder is there a straightforward way to see they vanish?

And as a consequence, does that imply we will need to add $\phi^4$ term in the Lagrangian and its counterterm $-i\delta_4$ to make the theory normalizable, but don't need to add $\phi^3$ term and its counterterm $-i\delta_3$ to the entire Lagrangian? Does this have anything to do with the fact that this Lagrangian is invariant under the parity transformation?

Best Answer

  1. $\gamma_5$ likes to be sandwiched between $\bar{\psi}$ and $\psi$. So, the interaction term should read $-g \phi \bar{\psi} \gamma_5 \psi$.

  2. $\bar{\psi} \gamma_5 \psi$ is a pseudoscalar, consequently also $\phi$ has to be a pseudoscalar.

  3. As a consequence, an interaction term like $\phi^3$ is forbidden by parity invariance.

  4. In order to formulate a consistent renormalizable theory (in 4 space-time dimensions) ALL possible terms up to (operator) dimension 4 invariant under space time symmetries and possibly also other symmetries have to be included. As the interaction term $\phi^4$ is even under parity, it must be included in the Lagrangian you are starting with.

Related Question