In other question, Ron Maimon says that he thinks string theory is the physical regulator. I did not know that string theory regularize divergencies.

So, Q1: How does string theory regularize the ultra-violet divergencies of the "low-energy" (Standard Model) fields? And Q2: Why it doesn't regularize its own divergencies.

By regularization I mean that the theory is ultra-violet finite before removing the regulator.

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    $\begingroup$ Supersymmetric string theories don't have a perturbative regulator, they have no ultraviolet divergences. There are infrared divergences, but these are understood from soft modes in the theory. $\endgroup$ – Ron Maimon Sep 14 '12 at 6:06

The answer to both questions is that string theory is completely free of any ultraviolet divergences. It follows that its effective low-energy descriptions such as the Standard Model automatically come with a regulator.

An important "technicality" to notice is that the formulae for amplitudes in string theory are not given by the same integrals over loop momenta as in quantum field theory. Instead, the Feynman diagrams in string theory are Riemann surfaces, world sheets, and one integrates over their possible conformal shapes (moduli).

Nevertheless, if one rewrites these integrals in a way that is convenient to extract the low-energy limit of string theory, one may see that the stringy diagrams boil down to the quantum field theory diagrams at low energies and the formulae are the same except for modifications that become large, $O(1)$, at energies of order $m_{\rm string}\sim \sqrt{T}$. The string scale is where perturbative string theory's corrections to quantum field theory become substantial and that's where the typical power-law increasing divergences in QFT are replaced by the exponentially decreasing, ultra-soft stringy behavior.

The reason/proof why/that string theory has no UV divergences has been known for decades. UV divergences would arise from extreme corners of the moduli space of Riemann surfaces in which the "length of various tubes" inside the degenerating Riemann surface would go to zero. But all such extreme diagrams are equivalent to diagrams with "extremely thin tubes" and may therefore be reinterpreted as IR divergences: it's the only right interpretation of these divergences and no "extra UV divergences" exist because it would be double-counting.

Bosonic string theory has infrared divergences due to the tachyon and dilaton and their long-range effects. However, in 10-dimensional superstring theory, one may prove that all the IR divergences – and there are just several a priori possible candidates that could be nonzero to start with – cancel, essentially due to supersymmetry. It follows that superstring theory is free of all divergences.

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  • $\begingroup$ Hi Lubos, it's a nice answer +1, but one has to mention that certain infrared divergences remain in diagrams when you have massless modes being produced in time-dependent processes, which are analogous in every way to QED infrared divergences, and no more worrisome. $\endgroup$ – Ron Maimon Sep 14 '12 at 7:21
  • $\begingroup$ Thanks, +1. So string theory is a regulator of the SM divergencies in the sense that it is an ultraviolet completion of the SM that is free of ultraviolet divergencies, right? It is not that it regulates the loop integrals of QFT providing a sort of cut-off (?) $\endgroup$ – Diego Mazón Sep 14 '12 at 7:31
  • $\begingroup$ Nice explanation of cute cool things :-)! $\endgroup$ – Dilaton Sep 14 '12 at 8:53
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    $\begingroup$ Thanks, Ron, very true. Thanks, Dilaton! ;-) Drake: string theory does regulate the loop integrals, it's just doing so in a way that would be far from obvious in any field theory approach. But this new way of regulation is somewhat similar to brutal cutoffs with $\Lambda=m_{\rm string}$, at least when it comes to various estimates. $\endgroup$ – Luboš Motl Sep 14 '12 at 13:26
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    $\begingroup$ @drake: The best reference is to Regge theory--- this is sketchily done in Green-Schwarz-Witten ch-1 and completely in Gribov's book "Theory of Complex Angular Momentum". String exchange can be thought of as an exchange of families of particles in ever higher angular momentum and mass, Regge trajectories, which together sum to an analytic amplitude which is soft in the large angle high-energy limit where field theory is hard. This is the 1960s motivation for strings, and it is the reason for no divergences. The modern interpretation is infrared/ultraviolet duality, which is some holography. $\endgroup$ – Ron Maimon Sep 15 '12 at 4:34

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