# Problem with Dimensional Regularization

1. Jan 18, 2012

### 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.

2. Jan 18, 2012

### 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.