Hopf algebras and Dyson–Schwinger equations

Stefan Weinzierl

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PDF(367 KB)
Front. Phys. ›› 2016, Vol. 11 ›› Issue (3) : 111206. DOI: 10.1007/s11467-016-0562-9
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Hopf algebras and Dyson–Schwinger equations

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Abstract

In this paper I discuss Hopf algebras and Dyson–Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson–Schwinger equations.

Keywords

Feynman integrals / Hopf algebras / Dyson–Schwinger equations

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Stefan Weinzierl. Hopf algebras and Dyson–Schwinger equations. Front. Phys., 2016, 11(3): 111206 https://doi.org/10.1007/s11467-016-0562-9

References

[1]
H. Hopf, Über Die Topologie der Gruppen-Mannigfaltigkeiten und Ihre Verallgemeinerungen, Ann. Math. 42(1), 22 (1941)
CrossRef ADS Google scholar
[2]
S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111(4), 613 (1987)
CrossRef ADS Google scholar
[3]
D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2, 303 (1998)
CrossRef ADS Google scholar
[4]
A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199(1), 203 (1998)
CrossRef ADS Google scholar
[5]
S. Weinzierl, Hopf algebra structures in particle physics, Eur. Phys. J. C 33(S1), s871 (2004)
CrossRef ADS Google scholar
[6]
M. Sweedler, Hopf Algebras, New York: Benjamin, 1969
[7]
C. Kassel, Quantum Groups, New York: Springer, 1995
CrossRef ADS Google scholar
[8]
S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. J. Mod. Phys. A 05(01), 1 (1990)
CrossRef ADS Google scholar
[9]
D. Manchon, Hopf algebras, from basics to applications to renormalization, arXiv: math/0408405, 2004
[10]
A. Frabetti, Renormalization Hopf algebras and combinatorial groups, in Geometric and Topological Methods for Quantum Field Theory Proceedings of the 2007 Villa de Leyva Summer School, pp. 159–219, Cambridge University Press, 2010, arXiv: 0805.4385
[11]
R. Ehrenborg, On posets and Hopf algebras, Adv. Math. 119(1), 1 (1996)
CrossRef ADS Google scholar
[12]
P. Schupp, Quantum groups, noncommutative differential geometry and applications, Ph.D. thesis, UC, Berkeley (1993)
[13]
J. Ecalle, ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan, Journal de Théorie des Nombres de Bordeaux 15(2), 411 (2003)
CrossRef ADS Google scholar
[14]
C. Reutenauer, Free Lie Algebras, Oxford: Clarendon Press, 1993
[15]
S. Weinzierl, Feynman graphs, in: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, pp. 381–406, Vienna: Springer, 2013, arXiv: 1301.6918
[16]
C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A 25(13), 2585 (2010)
CrossRef ADS Google scholar
[17]
W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Commun. Math. Phys. 15(3), 208 (1969)
CrossRef ADS Google scholar
[18]
K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras in renormalization of perturbative quantum field theory, Fields Inst. Commun. 50, 47 (2007)
[19]
T. Krajewski and R. Wulkenhaar, On Kreimer’s Hopf algebra structure of Feynman graphs, Eur. Phys. J. C 7(4), 697 (1999)
CrossRef ADS Google scholar
[20]
D. Kreimer, On overlapping divergences, Commun. Math. Phys. 204(3), 669 (1999)
CrossRef ADS Google scholar
[21]
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem (1): The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210(1), 249 (2000)
CrossRef ADS Google scholar
[22]
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem (2): The beta function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216(1), 215 (2001)
CrossRef ADS Google scholar
[23]
W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Commun. Math. Phys. 276(3), 773 (2007)
CrossRef ADS Google scholar
[24]
K. Ebrahimi-Fard and F. Patras, Exponential Renormalization, Ann. Henri Poincare 11(5), 943 (2010)
CrossRef ADS Google scholar
[25]
K. Ebrahimi-Fard and F. Patras, Exponential renormalization (II): Bogoliubov’s R-operation and momentum subtraction schemes, J. Math. Phys. 53(8), 083505 (2012)
CrossRef ADS Google scholar
[26]
A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5(4), 497 (1998)
CrossRef ADS Google scholar
[27]
A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, arXiv: math.AG/0103059, 2001
[28]
J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353, 907 (2001)
CrossRef ADS Google scholar
[29]
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167(3), 177 (2005)
CrossRef ADS Google scholar
[30]
E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15(05), 725 (2000)
CrossRef ADS Google scholar
[31]
T. Gehrmann and E. Remiddi, Two-loop master integrals for jets: The planar topologies, Nucl. Phys. B 601(1-2), 248 (2001)
CrossRef ADS Google scholar
[32]
N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen, Nova Acta Leopoldina (Halle) 90, 123 (1909)
[33]
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theory 148, 328 (2015)
CrossRef ADS Google scholar
[34]
L. Adams, C. Bogner, and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55(10), 102301 (2014)
CrossRef ADS Google scholar
[35]
L. Adams, C. Bogner, and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56(7), 072303 (2015)
CrossRef ADS Google scholar
[36]
K. T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83(5), 831 (1977)
CrossRef ADS Google scholar
[37]
F. Brown, Iterated integrals in quantum field theory, in Geometric and Topological Methods for Quantum Field Theory Proceedings of the 2009 Villa de Leyva Summer School, pp. 188–240, Cambridge University Press, 2013
[38]
J. Ablinger, J. Blümlein, and C. Schneider, Harmonic sums and polylogarithms generated by cyclotomic polynomials,J. Math. Phys. 52(10), 102301 (2011)
CrossRef ADS Google scholar
[39]
M. E. Hoffman, Quasi-shuffle products, J. Algebr. Comb. 11(1), 49 (2000)
CrossRef ADS Google scholar
[40]
L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150(1), 117 (2000)
CrossRef ADS Google scholar
[41]
S. Moch, P. Uwer, and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multi-loop integrals, J. Math. Phys. 43(6), 3363 (2002)
CrossRef ADS Google scholar
[42]
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128(2), 209 (2005)
CrossRef ADS Google scholar
[43]
C. Duhr, Mathematical aspects of scattering amplitudes, arXiv: 1411.7538, 2014
[44]
I. Bierenbaum, D. Kreimer, and S. Weinzierl, The next-to-ladder approximation for linear Dyson-Schwinger equations, Phys. Lett. B 646(2-3), 129 (2007)
CrossRef ADS Google scholar
[45]
C. Bergbauer and D. Kreimer, Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lect. Math. Theor. Phys. 10, 133 (2006)
[46]
D. Kreimer and K. Yeats, Recursion and growth estimates in renormalizable quantum field theory, Commun. Math. Phys. 279(2), 401 (2008)
CrossRef ADS Google scholar
[47]
L. Foissy, General Dyson-Schwinger equations and systems, Commun. Math. Phys. 327(1), 151 (2014)
CrossRef ADS Google scholar
[48]
O. Krüger and D. Kreimer, Filtrations in Dyson-Schwinger equations: Next-to j-leading log expansions systematically, Ann. Phys. 360, 293 (2015)
CrossRef ADS Google scholar
[49]
I. Bierenbaum and S. Weinzierl, The massless two-loop two-point function, Eur. Phys. J. C 32(1), 67 (2003)
CrossRef ADS Google scholar
[50]
D. Kreimer and E. Panzer, Renormalization and Mellin transforms, in: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, pp. 195–223, Vienna: Springer, 2013, arXiv: 1207.6321
[51]
E. Panzer, Renormalization, Hopf algebras and Mellin transforms, arXiv: 1407.4943, 2014
[52]
F. Brown, The massless higher-loop two-point function, Commun. Math. Phys. 287(3), 925 (2009)
CrossRef ADS Google scholar
[53]
D. J. Broadhurst and D. Kreimer, Exact solutions of Dyson-Schwinger equations for iterated one loop integrals and propagator coupling duality, Nucl. Phys. B 600(2), 403 (2001)
CrossRef ADS Google scholar
[54]
D. Kreimer and K. Yeats, An etude in non-linear Dyson-Schwinger equations, Nucl. Phys. B Proc. Suppl. 160, 116 (2006)
CrossRef ADS Google scholar
[55]
G. van Baalen, D. Kreimer, D. Uminsky, and K. Yeats, The QED beta-function from global solutions to Dyson-Schwinger equations, Ann. Phys. 324, 205 (2009)
CrossRef ADS Google scholar
[56]
G. van Baalen, D. Kreimer, D. Uminsky, and K. Yeats, The QCD beta-function from global solutions to Dyson-Schwinger equations, Ann. Phys. 325(2), 300 (2010)
CrossRef ADS Google scholar
[57]
M. P. Bellon and P. J. Clavier, Higher order corrections to the asymptotic perturbative solution of a Schwinger-Dyson equation, Lett. Math. Phys. 104(6), 749 (2014)
CrossRef ADS Google scholar
[58]
M. P. Bellon and P. J. Clavier, A Schwinger-Dyson equation in the Borel plane: Singularities of the solution, Lett. Math. Phys. 105(6), 795 (2015)
CrossRef ADS Google scholar
[59]
P. J. Clavier, Analytic results for Schwinger-Dyson equations with a mass term, Lett. Math. Phys. 105(6), 779 (2015)
CrossRef ADS Google scholar

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