Computing integrals for divergent loop amplitudes?

Question

I am trying to compute the cross-section for the diagram below with a divergent triangle loop: $\qquad\qquad\qquad\qquad\qquad$

where $X^0$ and $X^-$ are some fermions with zero and negative charge respectively. I am interested in low energy limits, so you can consider W-propagator as $\frac{i\eta_{\mu\nu}}{M_w^2}$.

When computing the amplitude, ignoring the external wave functions, you end of with an integration of the form: $$ \int \frac {k_\mu \gamma^\mu +m_-} {k^2 -m_-^2 +i\epsilon} \frac {d^4 k} {(2\pi)^4} $$ where $m_-$ is mass of $X^-$.

Any ideas how to solve this integral in terms of kinematic parameters, masses etc?

Answer 1

Agree with marnot that you can't get rid of W propagators in the loop.

That being said, the integral per se $$ \int \frac {k_\mu \gamma^\mu +m_-} {k^2 -m_-^2 +i\epsilon} \frac {d^4 k} {(2\pi)^4} $$ is a typical single fermion loop (e.g. a vacuum bubble diagram), where the first term related to $$ k_\mu \gamma^\mu $$ drops out since it's odd in $k_\mu$.

The second term related to $$ m_- $$ is quadratic divergent, which amounts to zero however in dimensional regularization (one of the peculiarities of DR).