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The Pauli Villars regularization involves replacing every propagator in a divergent diagram by a "subtracted propagator", where we subtract a fictitious, heavy particle propagator from the original propagator.

$\begin{equation} \frac{1}{p^2 - m^2 + i\epsilon} \rightarrow \frac{1}{p^2 - m^2 + i\epsilon} - \frac{1}{p^2 - \Lambda^2 + i\epsilon} = \frac{m^2 - \Lambda^2}{(p^2 - m^2 + i\epsilon)(p^2 - \Lambda^2 + i\epsilon)} \end{equation}$

I want to evaluate the order $\lambda^2$ contribution to the $\phi\phi\rightarrow\phi\phi$ scattering in $\phi^4$ theory using this method. For the loop when I use the P-V regulator I get cross terms, which are not nice, but can be evaluated.

However, in Peskin and Schroeder (pg. 218 and 194), when they use this method, they just replace the whole loop by a "ghost" loop, instead of replacing every propagator. They don't get cross terms, and there the calculation is almost trivial. Yet when they define this regulator, they talk about replacing each propagator individually.

Even in this question: Pauli Villars Regularization, the author of the answer uses this, without explanation how it is equivalent to replacing each propagator.

So what am I missing? Are these two equivalent? (I don't see at all why they should be) If not, why P&S disregards the cross terms? Or is it when they say replace each propagator by the subtracted propagator, they actually mean entire loops?

By the way, in my way of calculating, I got $\log(\Lambda)$ divergence, which is correct, but also a correction of $\mathcal{O(1/\Lambda^2)}$, which should not be correct, because it would mean P-V is better than other regularization methods, in that it suppresses the small scale physics to a higher order. So, either I made a mistake or P-V really is more efficient (I don't think so), or I misunderstood the whole thing and the "correct" way to do it would be to replace loops instead of individual propagators?

I hope I made my inner confusion manifest :D Anyway, a detailed answer would be very helpful. (Like explaining why what P&S does is justified in spite of the definition they suggest, and/or comparing both ways of calculating and showing both are same)

Thank you.

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