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Log expansions from combinatorial Dyson-Schwinger equations. (English) Zbl 1446.81032
Summary: We give a precise connection between combinatorial Dyson-Schwinger equations and log expansions for Green’s functions in quantum field theory. The latter are triangular power series in the coupling constant \(\alpha\) and a logarithmic energy scale \(L\) – a reordering of terms as \(G(\alpha ,L) = 1 \pm \sum_{j \ge 0} \alpha^j H_j(\alpha L)\) is the corresponding log expansion. In a first part of this paper, we derive the leading log order \(H_0\) and the next-to\(^{(j)}\)-leading log orders \(H_j\) from the Callan-Symanzik equation. In particular, \(H_j\) only depends on the \((j+1)\)-loop \(\beta \)-function and anomalous dimensions. In two specific examples, our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature: for the photon propagator Green’s function in quantum electrodynamics and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e., fixed-point relations for the Green’s functions. Combining the arguments, our work provides a derivation of the log expansions for Green’s functions from the corresponding Dyson-Schwinger equations.
MSC:
81T18 Feynman diagrams
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
35J08 Green’s functions for elliptic equations
81V80 Quantum optics
81V10 Electromagnetic interaction; quantum electrodynamics
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[1] Kißler, H., On the gauge dependence of quantum electrodynamics, PoS (2018)
[2] Brown, F.; Kreimer, D., Angles, scales and parametric renormalization, Lett. Math. Phys., 103, 933-1007 (2013) · Zbl 1273.81164
[3] Kreimer, D., Panzer, E.: Renormalization and Mellin transforms. In: Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions. Proceedings, LHCPhenoNet School: Linz, Austria, 9-13 July 2012, pp. 195-223 (2013). 10.1007/978-3-7091-1616-6_8. arXiv:1207.6321 · Zbl 1308.81137
[4] Peskin, ME; Schroeder, DV, An Introduction to Quantum Field Theory (1995), Reading: Addison-Wesley, Reading
[5] Ward, JC, An identity in quantum electrodynamics, Phys. Rev., 78, 182 (1950) · Zbl 0041.33012
[6] Green, HS, A pre-renormalized quantum electrodynamics, Proc. Phys. Soc. A, 66, 873-880 (1953) · Zbl 0053.17104
[7] Takahashi, Y., On the generalized ward identity, Nuovo. Cim., 6, 371 (1957) · Zbl 0078.20202
[8] Krüger, O.; Kreimer, D., Filtrations in Dyson-Schwinger equations: next-\(to^j\)-leading log expansions systematically, Ann. Phys., 360, 293-340 (2015) · Zbl 1360.81244
[9] Delage, L.: Leading log expansion of combinatorial Dyson-Schwinger equations. arXiv:1602.08705
[10] Courtiel, J., Yeats, K.: Next-\(to^k\) leading log expansions by chord diagrams. arXiv:1906.05139 · Zbl 1447.81169
[11] Kreimer, D.; Yeats, K., An Etude in non-linear Dyson-Schwinger equations, Nucl. Phys. Proc. Suppl., 160, 116-121 (2006)
[12] Kreimer, D.; Yeats, K., Recursion and growth estimates in renormalizable quantum field theory, Commun. Math. Phys., 279, 401-427 (2008) · Zbl 1156.81033
[13] Kreimer, D., On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 2, 303-334 (1998) · Zbl 1041.81087
[14] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 210, 249-273 (2000) · Zbl 1032.81026
[15] Kreimer, D., Anatomy of a gauge theory, Ann. Phys., 321, 2757-2781 (2006) · Zbl 1107.81038
[16] van Suijlekom, WD, The structure of renormalization Hopf algebras for gauge theories. I: Representing Feynman graphs on BV-algebras, Commun. Math. Phys., 290, 291-319 (2009) · Zbl 1207.81083
[17] Kreimer, D.; Sars, M.; van Suijlekom, WD, Quantization of gauge fields, graph polynomials and graph homology, Ann. Phys., 336, 180-222 (2013) · Zbl 1327.81282
[18] Foissy, L.: Mulitgraded Dyson-Schwinger systems. arXiv:1511.06859 · Zbl 1443.81055
[19] Chetyrkin, KG; Kuhn, JH; Steinhauser, M., RunDec: a mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun., 133, 43-65 (2000) · Zbl 0970.81087
[20] Kißler, H.: Systems of linear Dyson-Schwinger equations · Zbl 1423.81126
[21] Ruijl, B., Ueda, T., Vermaseren, J.: FORM version 4.2. arXiv:1707.06453
[22] van Suijlekom, W., The Hopf algebra of Feynman graphs in QED, Lett. Math. Phys., 77, 265-281 (2006) · Zbl 1160.81432
[23] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The beta function, diffeomorphisms and the renormalization group, Commun. Math. Phys., 216, 215-241 (2001) · Zbl 1042.81059
[24] Yeats, K.A.: Growth estimates for Dyson-Schwinger equations. PhD thesis, Boston University, Massachusetts, (2008). arXiv:0810.2249
[25] van Suijlekom, W.D.: Renormalization of gauge fields using Hopf algebras. In: Quantum Field Theory: Competitive Models. Proceedings, 3rd Workshop on Recent developements in quantum field theory, Leipzig, Germany, July 20-22, 2007, pp. 137-154 (2008). 10.1007/978-3-7643-8736-5_8. arXiv:0801.3170 · Zbl 1159.81392
[26] Borinsky, M., Feynman graph generation and calculations in the Hopf algebra of Feynman graphs, Comput. Phys. Commun., 185, 3317-3330 (2014) · Zbl 1360.81012
[27] Kißler, H., Hopf-algebraic Renormalization of QED in the linear covariant Gauge, Ann. Phys., 372, 159-174 (2016) · Zbl 1380.81230
[28] Bergbauer, C.; Kreimer, D., Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lect. Math. Theor. Phys., 10, 133-164 (2006) · Zbl 1141.81024
[29] Foissy, L., Faà di bruno subalgebras of the hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math., 218, 136-162 (2008) · Zbl 1158.16020
[30] Prinz, D.: Gauge symmetries and renormalization. arXiv:2001.00104 · Zbl 1413.81035
[31] Kißler, H.: Computational and diagrammatic techniques for perturbative quantum electrodynamics. PhD thesis, Humboldt Universität zu Berlin, Berlin (2017)
[32] Kreimer, D.; van Suijlekom, WD, Recursive relations in the core Hopf algebra, Nucl. Phys. B, 820, 682-693 (2009) · Zbl 1196.81199
[33] Prinz, D.: Algebraic structures in the coupling of gravity to Gauge theories. arXiv:1812.09919 · Zbl 1413.81035
[34] Rotheray, L.: Hopf subalgebras from green’s functions. Master’s thesis, Humboldt-Universität zu Berlin (2014)
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