Consider the t-channel diagram of phi-4 one loop diagrams. Evaluated it is, with loop momenta p,
$frac{lambda^2}{2}displaystyleintfrac{d^4p}{(2pi)^4}frac{1}{(p+q)^2-m^2}frac{1}{p^2-m^2}$
If I want to regularize this using Pauli-Villars regularization, which is the correct method? The procedure is to make the replacement $frac{1}{p^2-m^2}rightarrow frac{1}{p^2-m^2}-frac{1}{p^2-Lambda^2}$.
My question is do I apply the reguarization to one term in the integral or both terms?
I’ve seen variations where the propagators become
$frac{1}{p^2-m^2}frac{1}{(p+q)^2-m^2}rightarrow frac{1}{p^2-m^2}frac{1}{(p+q)^2-m^2}-frac{1}{p^2-Lambda^2}frac{1}{(p+q)^2-Lambda^2}$
and also where we have
$frac{1}{p^2-m^2}frac{1}{(p+q)^2-m^2}rightarrow (frac{1}{p^2-m^2}-frac{1}{p^2-Lambda^2})(frac{1}{(p+q)^2-m^2}-frac{1}{(p+q)^2-Lambda^2})$
In the latter case one ends up with four terms and each term is then evaluated using a Feynman parameter and integrating over wick rotated momenta, obtaining a logarithmic expression.
I’m pretty sure I’ve also seen where it was only applied to one of the terms.
Which is correct? (or are they equivalent?)
