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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:

enter image description here

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:

enter image description here

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?

RicardoP
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1 Answers1

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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?.)

Photon self-energy

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.

Buzz
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