Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in ϕ3 theory up to six loops and 561 of 607 Feynman periods at seven loops analytically. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that ϕ3 and ϕ4 theory share the same number content.
Similar content being viewed by others
G. 't Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
S. G. Gorishnii, S. A. Larin and F. V. Tkachov, The algorithm for OPE coefficient functions in the MS scheme, Phys. Lett. B 124 (1983) 217 [INSPIRE].
A. V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
M. Chan, S. Galatius and S. Payne, Tropical curves, graph complexes, and top weight cohomology of ℳg, J. Amer. Math. Soc. 34 (2021) 565 [arXiv:1805.10186].
O. Schnetz, Seven loops ϕ4, in preparation (2022).
A. B. Zamolodchikov, ‘Fishnet’ diagrams as a completely integrable system, Phys. Lett. B 97 (1980) 63 [INSPIRE].
B. Basso, L. J. Dixon, D. A. Kosower, A. Krajenbrink and D.-L. Zhong, Fishnet four-point integrals: integrable representations and thermodynamic limits, JHEP 07 (2021) 168 [arXiv:2105.10514] [INSPIRE].
N. I. Usyukina and A. I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [INSPIRE].
F. Brown and O. Schnetz, Single-valued multiple polylogarithms and a proof of the zig-zag conjecture, J. Number Theor. 148 (2015) 478 [INSPIRE].
M. Kontsevich and D. Zagier, Periods, in Mathematics unlimited — 2001 and beyond, B. Engquist and W. Schmid eds., Springer (2001), p. 771.
F. Brown, Mixed Tate motives over Z, Ann. Math. 175 (2012) 949 [arXiv:1102.1312].
F. Brown, Motivic periods and the projective line minus three points, arXiv:1407.5165.
K. G. Chetyrkin and F. V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
D. I. Kazakov, Calculation of Feynman integrals by the method of ‘uniqueness’, Theor. Math. Phys. 58 (1984) 223 [Teor. Mat. Fiz. 58 (1984) 343] [INSPIRE].
K. G. Chetyrkin, A. L. Kataev and F. V. Tkachov, New approach to evaluation of multiloop Feynman integrals: the Gegenbauer polynomial x space technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
M. Borinsky, J. A. Gracey, M. V. Kompaniets and O. Schnetz, Five-loop renormalization of ϕ3 theory with applications to the Lee-Yang edge singularity and percolation theory, Phys. Rev. D 103 (2021) 116024 [arXiv:2103.16224] [INSPIRE].
O. Schnetz, HyperlogProcedures, Maple package, https://www.math.fau.de/person/oliver-schnetz/ (2022).
E. Speer, Ultraviolet and infrared singularity structure of generic Feynman amplitudes, Ann. Inst. H. Poincaré Phys. Theor. 23 (1975) 1.
J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 457.
D. I. Kazakov, The method of uniqueness, a new powerful technique for multiloop calculations, Phys. Lett. B 133 (1983) 406 [INSPIRE].
A. E. Kennelly, The equivalence of triangles and three-pointed stars in conducting networks, Elec. World Eng. 34 (1899) 413.
B. D. McKay and A. Piperno, Practical graph isomorphism, II, J. Symb. Comput. 60 (2014) 94.
M. Borinsky, Tropical Feynman quadrature, https://github.com/michibo/tropical-feynman-quadrature.
Google, Abseil C++ library, https://abseil.io/.
T. Granlund et al., GNU multiple precision arithmetic library, https://gmplib.org/.
J. M. Boyer and W. J. Myrvold, On the cutting edge: simplified O(n) planarity by edge addition, J. Graph Algorithms Appl. 8 (2004) 241.
J. M. Boyer et al., The edge addition planarity suite, https://github.com/graph-algorithms/edge-addition-planarity-suite.
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 (2000) 249 [hep-th/9912092] [INSPIRE].
M. Borinsky, G. V. Dunne and M. Meynig, Semiclassical trans-series from the perturbative Hopf-algebraic Dyson-Schwinger equations: ϕ3 QFT in 6 dimensions, SIGMA 17 (2021) 087 [arXiv:2104.00593] [INSPIRE].
D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Annalen 286 (1990) 613.
J. L. Bourjaily, Y.-H. He, A. J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau manifolds: scattering amplitudes beyond elliptic polylogarithms, Phys. Rev. Lett. 121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
E. R. Gansner and S. C. North, An open graph visualization system and its applications to software engineering, Softw. Pract. Exp. 30 (2000) 1203.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2206.10460
About this article
Cite this article
Borinsky, M., Schnetz, O. Recursive computation of Feynman periods. J. High Energ. Phys. 2022, 291 (2022). https://doi.org/10.1007/JHEP08(2022)291