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I have seen the statement that the Standard Model has a Landau pole, or at least it its believed that it does at $\sim 10^{34}$ GeV. Has this actually been proven (at least in perturbation theory, as in QED) or what kind of evidence is there to support this?

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I'm not sure if it's the case, and it's possible that the answer is sensitive to measured constants (the Higgs mass, the top mass, alpha-strong). A well-known recent paper that runs relevant SM couplings up to the Planck scale is inspirehep.net/record/1116539 - it hints that the Higgs coupling doesn't blow up but actually almost vanishes at the Planck scale. –  Vibert Feb 19 '13 at 15:29
@Vibert: The $\lambda$ result in that paper is interesting, but the $U(1)$ coupling shows no signs of slowing down. I think right now I'd bet that the Standard Model has at least one Landau pole. –  user1504 Feb 20 '13 at 13:16

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