Wick rotation

  • Thread starter lornstone
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I am trying to calculate the first order photon self-energy.

At a point, I must calculte the following integral :
[tex] \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} [/tex]

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

Can anybody help me?

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

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

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


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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.

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