Thursday, May 11, 2006

Lance Dixon: transcendentality expansions

These days, Lance Dixon - a co-discoverer of orbifolds in string theory, among other things - is an expert in multi-loop calculations in gauge theory. In his seminar, he has shown many technical points about the similarities of Feynman diagrams for scattering amplitudes in "N=4" supersymmetric gauge theory and in the pure QCD. The "N=4" theory is conformal and the infrared divergences make the interpretation of the loop amplitudes problematic, but the formulae for these amplitudes still reveal interesting mathematical patterns.

The most attractive method underlying Dixon's talk - for most of the audience - were his transcendentality expansions. Write down all L-loop Feynman diagrams and evaluate the relevant integrals. The coefficients of different terms will be rational multiples of polynomials in "pi" and "zeta(3)" and similar values of the zeta function. Trust me that zeta(3), zeta(5), zeta(7) are transcendental numbers. Even if you don't believe me, you may abuse the language and define the degree of transcendentality "DT" of a number so that
  • DT(p/q) = 0 ... for rational p/q
  • DT(pi^k) = DT(zeta(k)) = k ... for positive "k"
  • DT(uv) = DT(u)+DT(v)
For example, "zeta(3)^2" has "DT=6". Zeta of an even positive number "k" is a rational multiple of "pi^k" which makes the definitions above self-consistent. The polynomials in "zeta(k)" with rational coefficients are the typical prefactors that appear in various Feynman diagrams. Dixon argued that there are nice regularities and a correspondence between QCD and "N=4" supersymmetric Yang-Mills theory that one recovers up to the "leading transcendentality" order.

Some coefficients don't have the form of polynomials in "zeta(k)". For example, "ln(2)" can occur as a prefactor, too. I've convinced Dixon to consider "ln(2)" to have "DT=1" until proved otherwise because
  • ln(2) = 1 - 1/2 + 1/3 - 1/4 + ...
looks, up to the alternating signs, as a formula for the (divergent) "zeta(1)". Note that for higher positive integers "k", the zeta functions with the constant and/or with the alternating sign are rational multiples of each other.

The homework exercise for the readers of the Reference Frame is to find a physical interpretation of the large transcendentality expansion - for example using the language of the AdS bulk description. In what modified theory or for what kind of processes do the leading transcendentality diagrams dominate? A mathematician has found correlations between the appearance of "zeta(3)" and "knots" in various Feynman diagrams - but is there a truly physical understanding of the diagrams? The solution to this homework could be analogous to, but more subtle than, 't Hooft's topological expansion of the Feynman diagrams in the large N limit.

Relevant references for this work of Dixon are mentioned by informed physicists in the fast comments.

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