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I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential contains $\mu^2\phi^2$.

Bardeen argues that

  1. Setting $\mu^2=0$ would increase the symmetry of the Standard Model by a conformal symmetry.

    But the things we can measure - which are basically the Green's functions - are no more symmetric with $\mu^2=0$ than with $\mu^2\neq0$! Because the symmetry is anomalously broken.

  2. The conformal symmetry is broken anomalously by non-zero beta-functions.

  3. Quadratic divergences do not contribute to the beta-functions.

    Whilst its true that they don't contribute to the running of the renormalised parameters, they contribute to the running of the bare mass.

  4. Quadratic divergences must therefore be a separate (explicit) source of symmetry breaking.

    Not true if you consider that quadratic divergences contribute to the running of the bare mass.

  5. Quadratic divergences must therefore be an non-physical artefact of our re-normalisation procedure, and we must remove them with, for example, counter-terms, and this is not a fine-tuning.

I have added my comments. My main question is,

If conformal symmetry is broken anomalously, why should our Lagrangian respect conformal symmetry? The Green's function are not more symmetric with $\mu^2=0$. I am interested in this for any symmetry, but especially conformal symmetry.

Also, I don't understand 3. and 4. Quadratic divergences would contribute to the running of the bare mass. Wouldn't that break conformal symmetry anomalously? or is it only the renormalised parameters that must have vanishing beta-functions? The distinction seems artificial. I can't understand why quadratic divergences are an explicit source of symmetry breaking, whereas logarithms etc are anomalous. This is key to solving the naturalness problem, and I can't follow the argument.

My feeling is that these arguments are faulty (which makes me think I must be making mistakes because Bardeen is a real expert who has surely thought a lot about it!). I'm certainly unconvinced. Have they been confirmed/refuted at length in the literature?

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    $\begingroup$ Conformal symmetry is not a symmetry at all in a theory with a conformal anomaly. I think I pretty much agree with all of your comments. $\endgroup$ – Matt Reece Mar 27 '14 at 19:43
  • $\begingroup$ @matt thanks that gives me confidence :-) if you have any other corrections/insights/thoughts on this it would be great to hear them. $\endgroup$ – innisfree Mar 27 '14 at 20:48
  • $\begingroup$ the paper is only a write up of a talk, not in a journal or even on the arxiv. I can't find a follow up paper. did Bardeen also suspect the argument was faulty and abandon it? $\endgroup$ – innisfree Mar 27 '14 at 20:52
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Unfortunately, Bardeen seems to misunderstand the naturalness problem that has nothing to do with quadratic divergences per se. In the strict SM, there is no naturalness problem because the running Higgs mass squared is proportional to itself. But this is not the setup that people care about when talking about the actual naturalness problem that emerges instead as soon as one extends the SM to include some new scale $\Lambda$ (where new particles start propagating). Any such a deformation of the UV of the SM introduce the naturalness problem, and you can not screen the Higgs mass from large corrections $\sim\Lambda^2$ by invoking conformal symmetry simply because it is not a good symmetry at $\Lambda$ whose very existence represents in fact a large explicit symmetry breaking. It is this UV sensitivity of the SM, through the Higgs potential, that is the actual naturalness problem.

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