Calculating First Order Photon Self-Energy Integral

[lornstone]

Hi,

I am trying to calculate the first order photon self-energy.

At a point, I must calculte the following integral :
$$\int d^4k \frac{(k+q)^\mu k^\nu+(k+q)^\nu k^\mu - g^{\mu \nu}(k \cdot(k+q) - m^2}{k^2 + 2x(q\cdot k) + xq^2 -m^2}$$

I know that I must wick rotate and that $$k^2$$ will become $$-k_E^2$$.
But I don't know what terms like $$(k+q)^\mu k^\nu$$ will become.

Can anybody help me?

Thank you

 

[TriTertButoxy]

Those terms will just stay the same. All you have to do is keep in mind that in the $k^\mu k^\nu$ tensor, $(\mu,\nu)=(0,0)$ component will acquire a minus sign and the $(0,i)$ and $(i,0)$ components for $i=\{1,2,3\}$ will have a factor of $i$.

But you do not need to worry about these changes. Just proceed with your calculation.

 

[lornstone]

Thank you!

But now I wonder if I can also wick rotate q so that after the change of variable $$k' = k+ qx$$ I will get no linear term in k in the denominator.

 

[vanhees71]

Of course, you Wick rotate all four-vectors, i.e., both the integration variables (loop momenta) and the external momenta that are not integrated out. After that, of course, you can do any manipulations like substitutions etc.