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I'm trying to work through the one-loop, one-vertex diagram in [itex]\phi^4[/itex] theory using Pauli-Villars regularization, and I'm having trouble. Specifically, I can't get the momentum dependence to fall out after integrating, which I think it should. In computing the "seagull" diagram (two external legs, and one loop that begins and ends on the same vertex), assuming my steps up to this point are correct, I end up, after Wick rotation to 4D spherical coordinates, with an integral (with some factors out front) $$\int_0^\infty\left(\frac{q^3}{q^2+m^2}-\frac{q^3}{q^2+M^2}\right)dq.$$ But this doesn't converge. It is logarithmically dependent on the momentum [itex]q[/itex], so I end up with two large parameters, [itex]M[/itex] and whatever cutoff [itex]\Lambda[/itex] I impose on the integral.
I thought the whole point was that with Pauli-Villars regularization you can perform the integral over the momentum and be left with a single parameter [itex]M[/itex]? Can anyone explain what I'm missing?
I thought the whole point was that with Pauli-Villars regularization you can perform the integral over the momentum and be left with a single parameter [itex]M[/itex]? Can anyone explain what I'm missing?