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\(\hbox {Next-to}{}^k\) Leading Log Expansions by Chord Diagrams

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Abstract

Green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter. The leading logs are given by the main diagonal of this expansion, i.e. the subseries where the coupling and the scale parameter appear to the same power; then the next-to leading logs are listed by the next diagonal of the expansion, where the power of the coupling is incremented by one, and so on. We give a general method for deriving explicit formulas and asymptotic estimates for any \(\hbox {next-to}{}^k\) leading-log expansion for a large class of single scale Green functions. These Green functions are solutions to Dyson–Schwinger equations that are known by previous work to be expressible in terms of chord diagrams. We look in detail at the Green function for the fermion propagator in massless Yukawa theory as one example, and the Green function of the photon propagator in quantum electrodynamics as a second example, as well as giving general theorems. Our methods are combinatorial, but the consequences are physical, giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories.

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Notes

  1. Here primitive means primitive in the renormalization Hopf algebra, or equivalently having no proper subdivergences.

  2. Note that because of an unfortunate sign convention originating in [18] and carried throughout related work, the sign of L is opposite to what would be in most of the physics literature, and so it is perhaps best to view L as \(-\log (q^2/\mu ^2)\). In particular this leads to a sign difference in the comparison with the work of Krüger and Kreimer in Sect. 6.1.

  3. An analysis not working in a quotient algebra should also be possible but would involve machinery similar to that which will be needed for extending the chord diagram expansions to systems of Dyson–Schwinger equations. In both cases this remains work for the future.

References

  1. Brown, F.: On the periods of some Feynman integrals. arXiv:0910.0114

  2. Brown, F., Kreimer, D.: Angles, scales and parametric renormalization. Lett. Math. Phys. 103, 933–1007 (2013). arXiv:1112.1180

    Article  ADS  MathSciNet  Google Scholar 

  3. Courtiel, J., Yeats, K.: Terminal chords in connected chord diagrams. arXiv:1603.08596

  4. Courtiel, J., Yeats, K., Zeilberger, N.: Connected chord diagrams and bridgeless maps (2017). arxiv:1611.04611

  5. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  6. Le Guillou, J.-C., Zinn-Justin, J. (eds.): Large-Order Behaviour of Perturbation Theory, Volume 7 of Current Physics Sources and Comments. North-Holland, Amsterdam (1990)

    Google Scholar 

  7. Hihn, M., Yeats, K.: Generalized chord diagram expansions of Dyson–Schwinger equations. arXiv:1602.02550

  8. Kontsevich, M., Zagier, D.: Periods. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited–2001 and Beyond, pp. 771–808. Springer, Berlin (2001)

    Chapter  Google Scholar 

  9. Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757–2781 (2006). arXiv:hep-th/0509135v3

    Article  ADS  MathSciNet  Google Scholar 

  10. Krüger, O., Kreimer, D.: Filtrations in Dyson–Schwinger equations: \(\text{ next-to }{}^{j}\) -leading log expansions systematically. Ann. Phys. 360, 293–340 (2015)

    Article  Google Scholar 

  11. Krüger, O.: Log-expansions from combinatorial Dyson–Schwinger equations. In preparation (2019)

  12. Marie, N., Yeats, K.: A chord diagram expansion coming from some Dyson–Schwinger equations. Commun. Numb. Theory Phys. 7(2), 251–291 (2013). arXiv:1210.5457

    Article  MathSciNet  Google Scholar 

  13. McKane, A.J.: Perturbation expansions at large order: results for scalar field theories revisited. J. Phys. A 52(5), 055401 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  14. Schnetz, O.: Quantum periods: a census of \(\phi ^4\)-transcendentals. Commun. Numb. Theory Phys. 4(1), 1–48 (2010). arXiv:0801.2856

    Article  MathSciNet  Google Scholar 

  15. van Baalen, G., Kreimer, D., Uminsky, D., Yeats, K.: The QED beta-function from global solutions to Dyson–Schwinger equations. Ann. Phys. 234(1), 205–219 (2008). arXiv:0805.0826

    MATH  Google Scholar 

  16. van Baalen, G., Kreimer, D., Uminsky, D., Yeats, K.: The QCD beta-function from global solutions to Dyson–Schwinger equations. Ann. Phys. 325(2), 300–324 (2010). arXiv:0805.0826

    Article  ADS  Google Scholar 

  17. van Suijlekom, W.D.: Renormalization of gauge fields: a Hopf algebra approach. Commun. Math. Phys. 276, 773–798 (2007). arXiv:hep-th/0610137

    Article  ADS  MathSciNet  Google Scholar 

  18. Yeats, K.: Growth estimates for Dyson–Schwinger equations. PhD thesis, Boston University (2008)

  19. Yeats, K.: Rearranging Dyson–Schwinger equations. Mem. Am. Math. Soc. 211, 1–82 (2011)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Karen Yeats.

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Communicated by H. Duminil-Copin

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J.C. is supported by the French INS2I JCJC Grant ASTEC. K.Y. is supported by an NSERC Discovery grant and the Canada Research Chair program. Thanks to the referee for their insightful comments.

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Courtiel, J., Yeats, K. \(\hbox {Next-to}{}^k\) Leading Log Expansions by Chord Diagrams. Commun. Math. Phys. 377, 469–501 (2020). https://doi.org/10.1007/s00220-020-03691-7

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  • DOI: https://doi.org/10.1007/s00220-020-03691-7

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