Problem with Dimensional Regularization

[baranas]

Good day to everyone. I am trying to apply dimensional regularization to divergent integral
\int\frac{d^{4}l}{\left(2\pi\right)^{4}}\frac{4\, l_{\mu}l_{\nu}}{\left[l^{2}-\triangle+i\epsilon\right]^{3}}.
I am very new to these thing. The first question is how should i apply Wicks rotation to the term l_{\mu}l_{\nu}As i understand it should be done before going to d dimensions. I need to avoid substitution l_{\mu}l_{\nu}\to\frac{1}{4}g_{\mu \nu}l^2
Would it work to rewrite
l_{\mu}l_{\nu}=\frac{g_{\mu \nu}}{g_{\mu \nu}g^{\mu \nu}}g^{\mu \nu}l_{\mu}l_{\nu}
Which gives in d dimensions
\frac{g_{\mu\nu}}{d}l^2
I would appreciate any help.

 

[fzero]

The way to usually determine these integrals is to start with the result for

\int \frac{d^N l}{(l^2 + a^2)^A} = \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{\pi^{N/2}}{(a^2)^{A-N/2}}

shift l=l'+p, so that

\int \frac{d^N l'}{((l')^2 + 2 p\cdot l' + a^2+p^2)^A} = \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{\pi^{N/2}}{(a^2+p^2)^{A-N/2}}.

Now we can generate factors of l'_\mu in the numerator by differentiating with respect to p, setting p=0 at the end as needed.