One-loop diagrams for the Yukawa pseudo-scalar interaction

Question

I have a Lagrangian describing a pseudo-scalar Yukawa interaction. This Lagrangian has a dimension $d=4-2\eta$. Here it is:

$$\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2}M^2 \phi^2 + \bar{\psi}(i \not{\partial} - m)\psi - ig\mu^\eta \bar{\psi}\gamma^5\phi\psi.\tag{1}$$

I have to determine the Feynman rules of this theory, then I give all potentially divergent one-particle irreducible (1PI) diagrams at one loop then I have to give the leading behavior of each of the diagrams in $\Lambda$ for $\Lambda \rightarrow \infty$.

For Feynman rules it is quite easy, we have a massive Dirac field $\psi$ interacting with a massive pseudo scalar field $\phi$ via a chiral interaction term $-ig\mu^{\eta}\gamma^5$. Here are the propagators of the theory:

for $\psi$: $$iS_F(k) = \frac{i(\not{k}+m)}{k^2-m^2+i\epsilon} \sim \frac{1}{k}.\tag{2}$$

for $\phi$: $$i\Delta_F(k)=\frac{i}{k^2-m^2+i\epsilon} \sim \frac{1}{k^2}.\tag{3}$$

Now to draw the one-loop irreducible diagrams I have a doubt. For me there are only the 3 diagrams attached. I know that there can be several variants for each of these diagrams (rotational or axial symmetries). Normally I should also put the 'triangle' diagrams or the other odd fermionic loop diagrams but according to Furry's theorem these diagrams do never contribute. Then there are the even fermionic loop diagrams like the box, these diagrams contribute? Do they diverge? Are there other one-loop irreductible diagrams? I would also know how does the counterterms serve in the lagrangian to eliminates the divergences. You have below what I found for the 3 diagrams that I have:

$\Pi(p_3) =-\int^\Lambda \frac{d^4 k}{(2\pi)^4} (-ig\mu^{\eta}\gamma^5)iS_F(p_3+k)(-ig\mu^{\eta}\gamma^5)iS_F(k) \sim \int^{\Lambda} \frac{d^4k}{k^2} \sim \Lambda^{2} $ (diverge)

$\Sigma(p_1) = -\int^\Lambda \frac{d^4 k}{(2\pi)^4} (-ig\mu^{\eta}\gamma^5)iS_F(k-p_1)(-ig\mu^{\eta}\gamma^5)i\Delta_F(k) \sim \int^{\Lambda} \frac{d^4k}{k^3} \sim \Lambda$ (diverge)

$\Lambda(p_1,p_2) = -\int^\Lambda \frac{d^4 k}{(2\pi)^4} (-ig\mu^{\eta}\gamma^5)iS_F(p_2-k)(-ig\mu^{\eta}\gamma^5)iS_F(k-p_1)(-ig\mu^{\eta}\gamma^5)i\Delta_F(k) \sim \int^{\Lambda} \frac{d^4k}{k^4} \sim \ln(\Lambda)$ (diverge)

Answer 1

1. OP's Lagrangian (1) should include a $\phi^4$ interaction term to be generic.

   Then there are 6 counterterms (2 wave function renormaliztions + 2 mass terms + 2 coupling constants).

   All other relevant & marginal terms not already in OP's Lagrangian (1) are excluded, since they break symmetry.

2. The 1PI $\phi^n$ vertex functions for $n$ odd vanish since $\phi$ is a pseudo-scalar, cf. Furry's theorem.

3. The 1PI $\phi^4$ vertex function $V$ contains a UV-divergent fermion 1-loop box diagram.

4. In total, there are 4 UV-divergent 1PI $n$-point functions (2 self-energies and 2 vertex functions).

   This can be seen by calculating the superficial degree of (UV) divergence, cf e.g. this Phys.SE post.

5. For details, see e.g. Ref. 1.

References:

1. M. Srednicki, QFT, 2007; Chapter 51. A prepublication draft PDF file is available here.