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I'm not familiar with supergravity so here's my question: I've heard in talks that if one finds divergence for five-loop 4-graviton scattering amplitudes in five dimensions this translates to a divergence in $D=4$ at seven loops.

How do I see this? What's the argument behind this?

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This looks very much like the "experimental" bound that you can find in the paper by Zvi Bern (arXiv:1210.7709) eq. (1.1). It tells you that the theory is expected to be finite if $$D < 4 +\frac{6}{L}$$ for D-dimensions and L-number of loops. These come from looking at possible counter terms allowed by different symmetries and explicit calculations of different amplitudes. Introduction of the above paper should give you quite a good idea.

This bound apparently works for both N=8 Supergravity and N=4 Super-Yang-Mills. You have not specified the number of SUSY generators in your supergravity, so I cannot be more specify.

It is worth noting the relation between the amplitudes, that you need to calculate to find if they diverge, between Yang-Mills theory and SUGRA. It is called KLT relation. It is used as a nice calculation tool in this business

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Hi, thanks for your answer. I'm thinking about N=0 sugra. Does this particular sugra somehow imply this connection between divergence at 5 and 4 dimensions at 5, 7 loops, respectively? –  A friendly helper Jan 23 '13 at 17:18
    
Sorry, i meant N=8 Sugra...typo... –  A friendly helper Jan 23 '13 at 19:59
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N=8 Sugra is 4 dimensional theory. If you look at possible counter terms that may arise like R^2, R^3 i.e. different contractions of Riemann tensor. Some of them will be not allowed by different symmetries e.g. susy. The statement would be that on dimensional grounds counter terms at 7 loops in D=4 N=8 SUGRA, 5 loops in Maximally Supersymmetric SUGRA in 5 dimensions, will be allowed by the same symmetries. Therefore, if you find that one of them in absent, then there must be some new/unknown symmetry that kills it, which should also kill the other. –  Jakub Jan 24 '13 at 10:26
    
Thanks a lot for your illuminating answer :) –  user20200 Jan 24 '13 at 17:03
    
Hi friendly helper, you can upvote (and accept) Jakub s answer if you like it. This friendly comment in the form of an answer could earn you unneeded downvotes from people who dont see that you just do not have enough rep yet to comment ... :-/ –  Dilaton Jan 24 '13 at 17:31
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