Pauli-villars regularization in simple phi^4 case

[copernicus1]

I'm trying to work through the one-loop, one-vertex diagram in $\phi^4$ 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 $q$, so I end up with two large parameters, $M$ and whatever cutoff $\Lambda$ 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 $M$? Can anyone explain what I'm missing?

 

[Einj]

I'm not really familiar with the Pauli-Villars regularation but I think that the idea is that you can always tune M such that the divergent part is exactly canceled. In other words, at the end of all your calculation you will end up with just one large parameter, $\Lambda$, which will enter the definition of your mass M.

 

[Chopin]

I've never done PV regularization on phi-4 theory before, so I'm not exactly sure what the math is going to look like, but you're right that there should be no momentum dependence in the seagull diagram. Looking at your integral, though, my naive thought is that you've got a minus sign in your second term, so shouldn't that cause the q logarithms to cancel? That would feel in keeping with how PV usually manifests in the examples I've done before.

 

[Avodyne]

PV is not sufficient to regulate this divergence. I'm not completely sure, but I think this single closed loop of one propagator is the only failure. This loop then has to be assigned some constant value every time it appears. This value will ultimately be absorbed into the renormalization of the mass.

 

[vanhees71]

The tadpole self-energy diagram in $\phi^4$ theory is quaddratically divergent. Thus you need to subtract twice. If $\Pi(M^2)$ is the unrenormalized tadpole (it's independent of the external four-momentum since it's effectively a one-point function), then the minimal subraction to make it finite is
$$\Pi_{\text{ren}}(M^2)=\Pi(M^2)-\Pi(\mu^2)-(M^2-\mu^2) \Pi'(\mu^2)+C_1 +C_2 M^2,$$
where the constants $C_1$ and $C_2$ are determined by the appropriate renormalization conditions. $\mu^2$ is the mass-subtraction scale. Indeed, using your schematic expression leads to
$$\Pi(M^2)-\Pi(\mu^2)-(M^2-\mu^2) \Pi'(\mu^2)=\int_0^{\infty} \mathrm{d} q \frac{q^3(M^2-\mu^2)^2}{(q^2+\mu^2)^2(q^2+M^2)},$$
which is convergent.

 

[copernicus1]

Thanks all for your input. vanhees71, thanks a lot, this is making more sense now. I've seen PV regularization done with multiple regulator fields but I'm not quite used to it yet. It seems you are doing something slightly different though, with a derivative $\Pi'(\mu^2)$. Is there a source you recommend for doing it this way?

 

[vanhees71]

This is a slightly modified BPHZ renormalization technique, using a mass-independ renormalization scheme. There's a bit about renormalization theory in my QFT manuscript:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Pauli-Villars is one of many regularization methods, which is usually quite tedious. A much more convenient regularization is dimensional regularization.