# zbMATH — the first resource for mathematics

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
##### Software:
FORM ; feyncop; RunDec; feyngen
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.