$φ^3$ scalar field one-loop diagram calculation via dimensional regularization

Question

I have been trying to understand how to compute the following loop diagram in $frac{lambda}{3!}phi^3$ theory, with $mneq0$ and I am not getting anywhere. From what I understood I am supposed to calculate it using a process called dimensional regularization, which I have understood in theory but cannot apply it in the current situation. The integral I get from the diagram is $$int frac{1}{k^{2}+m^{2}} frac{1}{left((p+k)^{2}+m^{2}right)} d^{D} k.$$

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Michele Grosso · Accepted Answer

In order to use the dimensional regularization scheme, you have to work out the original integrand via the Feynman parameter trick, that is
$frac{1}{A B} = int_0^1 dx frac{1}{[A + (B - A) x]^2}$
where:
$A = (p + k)^2 + m^2$
$B = k^2 + m^2$
Then you complete the square as $[(k + cdot cdot cdot)^2 - Delta]^2$, next you shift $k^mu to k^mu + cdot cdot cdot$, the measure $d^dk$ being unchanged, and you get an expression to which you can apply the dimensional regularization. In this case
$int frac{d^dk}{(2 pi)^d} frac{1}{(k^2 - Delta)^2} = frac{i}{(4 pi)^{d/2}} frac{1}{Delta^{2 - d/2}} Gamma (2 - d/2)$
Then you expand as $d = 4 - epsilon$, with $epsilon to 0$.

$φ^3$ scalar field one-loop diagram calculation via dimensional regularization

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