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Hopf algebra of Feynman graphs and critical exponents

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There is a nice reformulation of the renormalization procedure in QFT due to Connes and Kreimer, which makes use of the Hopf algebra of Feynman graphs. I wonder if this technique was shown to be helpful for the real computations in QFT like higher loops, say $4-\epsilon$-expansion for $\phi^4$-theory?

asked Mar 20, 2015 in Theoretical Physics by Meroman [ revision history ]
edited Mar 23, 2015 by Dilaton

1 Answer

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The paper 

D.J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation, Physics Letters B 475.1 (2000): 63-70. (http://arxiv.org/pdf/hep-th/9912093.pdf

presents QFT computations to 30 loops for Yukawa theory in 4D, which is probably impossible without the compact Hopf representation. Followup work by the same authors derives Schwinger-Dyson equations http://arxiv.org/pdf/hep-th/0012146.pdf to 500 loops for the same QFT.

Applications to resummed QED are given in http://arxiv.org/pdf/0805.0826.pdf (based on http://arxiv.org/pdf/hep-th/0605096) . Here most interesting is the fact that some of the resulting nonperturbative approximations do not have a Landau pole. The existence of the Landau pole in low order perturbation theory is frequently used as the argument to claim that QED probably cannot exist as a fundamental theory but only as an effective field theory. Thus the fact that (some of) the nonperturbative results here don't give a Landau pole shows that the problem of the existence of QED is in fact wide open.

http://arxiv.org/pdf/0906.1754 contains applications to QCD. 

Note that everywhere in QFT one must make approximations, and the nature of approximations depends on the method used. In the epsilon-expansion one only considers a fixed number of loops and neglects everything else. The Hopf algebra approach accommodates many loop orders but neglects complicated diagrams. Which neglect is worse is not easily answered. To estimate the effect of neglecting something one must make two calculations, one neglecting more, one less, and the difference in the results gives an indication of the accuracy. 

In some cases it is known that only fairly simple diagrams are relevant. For example, ladder diagrams are the only one's relevant when a nonrelativistic approximation is adequate, and renormalization amounts to resumming all repetitive diagrams into one sum. The resumming achievable through Hopf techniques amounts to something similar, hence it is quite likely an improvement compared to RG improved perturbation theory. But I don't know whether systematic studies of the errors have been made. 

answered Mar 20, 2015 by Arnold Neumaier (12,355 points) [ revision history ]
edited Mar 21, 2015 by Arnold Neumaier

A quick look at the first paper reveals that the authors used some truncation of the full set of diagrams nicely implemented via the Hopf algebra. Therefore, it is a counterexample in the sense that no honest higher loops computation was done, only those diagrams were taken into account whose nested combinatorics is simple. Please correct me if I am wrong.

@Meroman: Yes, you are right; they don't make a complete higher loop computation. But they take into account many more diagrams than one can do with other techniques. The same is the case in the second paper. It is not really clear to me which approximations they make in the QED paper. But as I understood your question it was primarily about real applications, not necessarily about higher loops. 

Indeed, I am interested in applications to computing some observable numbers. Therefore,

it seems to be difficult to say anything when only a subset of diagrams is taken into account - such truncation 'defines' some other theory with different indices

@Meroman: The traditional expansion to k loops also takes into account only a subset of diagrams, namely only those with at most k loops - hence defines a different theory. 

I added a paragraph at the end of my answer. 

Many thanks for the comments! I agree that a systematic study of contributions of terms being neglected is needed. When we expand in loops there is at least a parameter to expand and then one can estimate the error and it looks like people have been quite successful in turning various expansions into convergent ones after Borel resummation. On contrary when only the simplicity of combinatorics of some subset of diagrams is taken into account there is no well-defined expansion parameter, but you are right that this still can make a good approximation. It looks like the question is still open.

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