# Pauli-Villars (PV) regularisation breaks supersymmetry. How to see that?

Does the PV regulator breaks SUSY?

Take for instance the 1-loop (top/stop loops) correction to the Higgs squared-mass parameter in the MSSM, and you'll get something like,

$$\delta m^2_{h_u} = - 3Y_u^2/(4 \pi^2) m_{\tilde{t}}^2 ln (\frac{\Lambda_{PV}^2}{m_{\tilde{t}}^2})$$

Where, $\Lambda_{PV}$ is the PV regulator/cutoff, and $m_{\tilde{t}}$ is the stop-quark mass.

In my mind, as the calculation is performed before ElectroWeak Symmetry Breaking (EWSB) (i.e. no mass for the top), but at the same time it's considering softly broken susy (i.e. there is mass for the stop-quark), therefore we don't get perfect cancelation. But I heard someone saying that there's no perfect cancelation because PV regulator breaks SUSY!

I don't see where the PV breaking SUSY argument fits. Can anyone enlighten me?

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The question is unclear. If your question is about the PV regulator, why did you bring up that reference? It has nothing to do with PV, and it is not even mentioned once in the paper. – Zohar Ko Nov 26 '11 at 0:37
Because I took the equation from there. Umm, Ok let's say: Does the Pauli-Villars regularisation break supersymmetry? And how to see that? – stupidity Nov 26 '11 at 0:46
I still don't see why you copied a random equation from a random paper and proclaimed that their cutoff is a PV cutoff. So I still dont understand the question. It would be very helpful if you rephrase it. Also note that there is no problem to write a ghost kientic term in SUSY, so the bottom component of the chiral multiplet is a ghost while the fermion is fine. So the right question is not whether SUSY allows ghosts (it does) but whether this is enough to cancel the divergences. – Zohar Ko Nov 26 '11 at 0:53
I did similar calculation, as an exercise. I used PV and I got a similar result. But then, someone was saying something about PV regulator breaking susy and I was lost. :p Also, I didn't understand your answer. – stupidity Nov 26 '11 at 0:55
Perhaps the statement you are trying to reproduce has to do with dimensional regularization instead? Just a shot in the dark... – user566 Nov 26 '11 at 5:35
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## 1 Answer

I think that I am confused by the question and even more confused by the last remark. When you use PV regulator, you necessarily encounter for the ghosts. When you add to a propagator of a physical field another part, which looks like a propagator with a minus sign and a mass $\Lambda_{UV}$, you pretend that there is a heavy ghost "particle" in the theory (the wrong sign of the kinetic term in the Lagrangian translates into the sign flip in front of the propagator). Then if you ask the question whether this ghost can be supersymmetrized, the answer is "yes". So in this sense PV regulator does not violate SUSY. (As it was previously mentioned, the question whether it is enough to cancel all the divergences remains though).

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Welcome Andrey, according to the picture, you have grown a lot of hair since the last time I saw you. – Zohar Ko Nov 26 '11 at 13:17
Ok, thanks for your answer. The reason for the confusion is that I'm confused myself. I was following the PV scheme without knowing the physics behind it :/ But I think I'm getting a feeling of what the PV regulator does. Thanks again to you and to Zohar Ko whom I annoyed :p – stupidity Nov 26 '11 at 18:19
I was not annoyed at any point, just tried to make the question clearer for other commentators who might wanna take a shot at it. – Zohar Ko Nov 26 '11 at 18:46
Thanks! So can I ask then, is the PV regulator enough to cancel divergences? If not, then why? – stupidity Nov 28 '11 at 0:50

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It depends on what are you doing. If you regularize a Yukawa coupling, it is enough to introduce one PV regulator for you scalar to cancel the divergence. But there are more subtle examples, e.g. you need several sets of PV fermions to cancel consistently the divergences in QED photon self energy (the PV method becomes cumbersome and therefore Peskin switches to dim.reg.; you can see how it is done with PV regulators in Bjorken and Drell). I am not aware about any explicit example of a renormalizable theory, where the PV regularization fails, regardless how you introduce the regulators. – Andrey Katz Nov 28 '11 at 3:35
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