One-loop corrections to vacuum polarization with a specific Lagrangian

Question

I'm having some difficulties regarding this problem in QFT I'm doing to prepare for an exam. For the following problem I consider the theory described by the Lagrangian: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}(\partial_\mu A^\mu)^2+(\partial_\mu-ieA_\mu)\phi^-(\partial^\mu+ieA^\mu)\phi^++\frac{1}{2}\partial_\mu \chi\partial^\mu\chi-m^2_\phi\phi^+\phi^--\frac{1}{2}m^2_\chi\chi^2-\mu\phi^+\phi^-\chi$$

where $\chi$ is a neutral spin $0$ field, $\phi^\pm$ is a charged spin $0$ field and $A^\mu$ is the photon. The constant $e$ is the proton charge and $\mu$ has the dimensions of mass in natural units. This theory has the following propagators and vertices:

I'm considering the one-loop corrections in this model and only 1PI diagrams. I want to draw, for example, the diagrams that contribute to vacuum polarization of the photon and discuss its superficial degree of divergence. Here's where I want to check if I'm correct: for QED, we considered that electron-positron pairs are created as a "virtual" dipole that polarizes vacuum. In this case, since there are no vertices for charged fermions, but only for charged spin 0 bosons, then the diagram contributing for vacuum polarization in this case is only:

Am I correct? My second question is regarding the contribute to the one-loop correction to the vertex $\chi AA$, but I have no ideia how one corrects a vertex in this case: do we need to connect the photons, but with what?

Answer 1

First off, you have missed a of the necessary vertex in this theory. For scalar QED, you also have a $(\phi^{+}\phi^{-})^{2}$ vertex, which is needed for renormalizability. However, this will not directly affect the one-loop photon self-energy, the main topic of your question.

As to the photon self-energy, there are actually two diagrams. (A Google led me to this image showing both diagrams, which came from another question on this stack: Why does normal ordering violate the Ward identity?.)

If you try to just to just do the calculation with the first diagram, you will find that you cannot satisfy the Ward identity. The second diagram is a tadpole, which provides a momentum-independent divergence which is exactly what you need to cancel off the Ward identity violation from the first term.

The first diagram has the same kind of straightforward interpretation, as a virtual particle-antiparticle polarization of the vacuum, as in spinor QED. The interpretation of the second diagram is not as clearcut, but the diagram needs to be included nonetheless.

For the one-loop radiative generation of $\chi AA$ effective interaction vertex, you just attach a $\chi$ line to one of the internal charged scalar lines in those same two diagrams. While there is no tree-level way to attach a $\chi$ to a photon line, the three-scalar vertex allows you to attach a $\chi$ to any $\phi$ line.