zbMATH — the first resource for mathematics

Iterative structure of finite loop integrals. (English) Zbl 1333.81217
Summary: In this paper we develop further and refine the method of differential equations for computing Feynman integrals. In particular, we show that an additional iterative structure emerges for finite loop integrals. As a concrete non-trivial example we study planar master integrals of light-by-light scattering to three loops, and derive analytic results for all values of the Mandelstam variables \(s\) and \(t\) and the mass \(m\). We start with a recent proposal for defining a basis of loop integrals having uniform transcendental weight properties and use this approach to compute all planar two-loop master integrals in dimensional regularization. We then show how this approach can be further simplified when computing finite loop integrals. This allows us to discuss precisely the subset of integrals that are relevant to the problem. We find that this leads to a block triangular structure of the differential equations, where the blocks correspond to integrals of different weight. We explain how this block triangular form is found in an algorithmic way. Another advantage of working in four dimensions is that integrals of different loop orders are interconnected and can be seamlessly discussed within the same formalism. We use this method to compute all finite master integrals needed up to three loops. Finally, we remark that all integrals have simple Mandelstam representations.

81S40 Path integrals in quantum mechanics
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Goncharov, AB, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497, (1998) · Zbl 0961.11040
[2] Brown, FCS, Multiple zeta values and periods of moduli spaces \( {\mathfrak{M}}_0 \), _{n}, Annales Sci. Ecole Norm. Sup., 42, 371, (2009) · Zbl 1216.11079
[3] Kotikov, AV, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett., B 254, 158, (1991)
[4] Remiddi, E., Differential equations for Feynman graph amplitudes, Nuovo Cim., A 110, 1435, (1997)
[5] Gehrmann, T.; Remiddi, E., Differential equations for two loop four point functions, Nucl. Phys., B 580, 485, (2000) · Zbl 1071.81089
[6] Argeri, M.; Mastrolia, P., Feynman diagrams and differential equations, Int. J. Mod. Phys., A 22, 4375, (2007) · Zbl 1141.81325
[7] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[8] Chen, K-T, Iterated path integrals, Bull. Amer. Math. Soc., 83, 831, (1997) · Zbl 0389.58001
[9] Henn, JM; Smirnov, AV; Smirnov, VA, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP, 07, 128, (2013) · Zbl 1342.81352
[10] Henn, JM; Smirnov, VA, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP, 11, 041, (2013)
[11] Henn, JM; Smirnov, AV; Smirnov, VA, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP, 03, 088, (2014)
[12] J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, arXiv:1402.7078 [INSPIRE].
[13] Alday, LF; Henn, JM; Plefka, J.; Schuster, T., Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP, 01, 077, (2010) · Zbl 1269.81079
[14] Drummond, JM; Henn, J.; Smirnov, VA; Sokatchev, E., Magic identities for conformal four-point integrals, JHEP, 01, 064, (2007)
[15] Drummond, JM; Henn, JM; Trnka, J., New differential equations for on-shell loop integrals, JHEP, 04, 083, (2011) · Zbl 1250.81064
[16] Dixon, LJ; Drummond, JM; Henn, JM, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N = 4 SYM, JHEP, 06, 100, (2011) · Zbl 1298.81168
[17] S. Caron-Huot and J.M. Henn, A toy model for light-by-light scattering, to appear.
[18] Anastasiou, C.; Beerli, S.; Bucherer, S.; Daleo, A.; Kunszt, Z., Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP, 01, 082, (2007)
[19] Henn, JM, Dual conformal symmetry at loop level: massive regularization, J. Phys., A 44, 454011, (2011) · Zbl 1270.81137
[20] A.I. Davydychev, Standard and hypergeometric representations for loop diagrams and the photon-photon scattering, hep-ph/9307323 [INSPIRE].
[21] Goncharov, AB; Spradlin, M.; Vergu, C.; Volovich, A., Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett., 105, 151605, (2010)
[22] Chetyrkin, KG; Tkachov, FV, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys., B 192, 159, (1981)
[23] Smirnov, AV, Algorithm FIRE — Feynman integral reduction, JHEP, 10, 107, (2008) · Zbl 1245.81033
[24] Smirnov, AV; Smirnov, VA, FIRE4, litered and accompanying tools to solve integration by parts relations, Comput. Phys. Commun., 184, 2820, (2013) · Zbl 1344.81031
[25] Anastasiou, C.; Lazopoulos, A., Automatic integral reduction for higher order perturbative calculations, JHEP, 07, 046, (2004)
[26] A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE]. · Zbl 1219.81133
[27] R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
[28] Lee, RN; Pomeransky, AA, Critical points and number of master integrals, JHEP, 11, 165, (2013) · Zbl 1342.81139
[29] F. Brown, Iterated integrals in quantum field theory, http://www.math.jussieu.fr/∼brown. · Zbl 1295.81072
[30] J. Zhao, Multiple polylogarithms, http://www.maths.dur.ac.uk. · Zbl 1164.33004
[31] Smirnov, VA, Four-dimensional integration by parts with differential renormalization as a method of evaluation of Feynman diagrams, Theor. Math. Phys., 108, 953, (1997) · Zbl 0962.81526
[32] Kravtsova, GA; Smirnov, VA, Evaluation of three-loop Feynman diagrams by four-dimensional integration by parts and differential renormalization, Theor. Math. Phys., 112, 885, (1997) · Zbl 0978.81514
[33] Argeri, M.; Vita, S.; Mastrolia, P.; Mirabella, E.; Schlenk, J.; etal., Magnus and Dyson series for master integrals, JHEP, 03, 082, (2014) · Zbl 1333.81379
[34] C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, cs/0004015. · Zbl 1017.68163
[35] Vollinga, J.; Weinzierl, S., Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun., 167, 177, (2005) · Zbl 1196.65045
[36] Remiddi, E.; Vermaseren, JAM, Harmonic polylogarithms, Int. J. Mod. Phys, A 15, 725, (2000) · Zbl 0951.33003
[37] Mandelstam, S., Analytic properties of transition amplitudes in perturbation theory, Phys. Rev., 115, 1741, (1959)
[38] Mandelstam, S., Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. general theory, Phys. Rev., 112, 1344, (1958)
[39] Kniehl, BA, Dispersion relations in loop calculations, Acta Phys. Polon., B 27, 3631, (1996)
[40] Czakon, M.; Mitov, A., Inclusive heavy flavor hadroproduction in NLO QCD: the exact analytic result, Nucl. Phys., B 824, 111, (2010) · Zbl 1196.81231
[41] Smirnov, AV; Tentyukov, MN, Feynman integral evaluation by a sector decomposition approach (FIESTA), Comput. Phys. Commun., 180, 735, (2009) · Zbl 1198.81044
[42] A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, arXiv:1312.3186 [INSPIRE]. · Zbl 1351.81078
[43] Eden, B.; Heslop, P.; Korchemsky, GP; Sokatchev, E., Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys., B 862, 450, (2012) · Zbl 1246.81363
[44] Drummond, J.; Duhr, C.; Eden, B.; Heslop, P.; Pennington, J.; etal., Leading singularities and off-shell conformal integrals, JHEP, 08, 133, (2013) · Zbl 1342.81574
[45] Drummond, JM; Henn, J.; Korchemsky, GP; Sokatchev, E., Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys., B 828, 317, (2010) · Zbl 1203.81112
[46] Caron-Huot, S., Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP, 12, 066, (2011) · Zbl 1306.81082
[47] M. Bullimore and D. Skinner, Descent Equations for Superamplitudes, arXiv:1112.1056 [INSPIRE].
[48] Freedman, DZ; Johnson, K.; Latorre, JI, Differential regularization and renormalization: A new method of calculation in quantum field theory, Nucl. Phys., B 371, 353, (1992)
[49] J.M. Gracia-Bondía, H. Gutiérrez-Garro and J.C. Várilly, Improved Epstein-Glaser renormalization in x-space. III. Versus differential renormalization, arXiv:1403.1785 [INSPIRE].
[50] S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, arXiv:1309.5865 [INSPIRE]. · Zbl 1319.81044
[51] Caron-Huot, S.; He, S., Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory, JHEP, 08, 101, (2013) · Zbl 1342.81560
[52] Bern, Z.; Freitas, A.; Dixon, LJ; Ghinculov, A.; Wong, HL, QCD and QED corrections to light by light scattering, JHEP, 11, 031, (2001)
[53] Binoth, T.; Glover, EWN; Marquard, P.; Bij, JJ, Two loop corrections to light by light scattering in supersymmetric QED, JHEP, 05, 060, (2002)
[54] G. Puhlfürst, The evaluation of master integrals via differential equations, M.Sc. Thesis (2012), http://qft.physik.hu-berlin.de/theses/master-theses.
[55] Simmons-Duffin, D., Projectors, shadows and conformal blocks, JHEP, 04, 146, (2014) · Zbl 1333.83125
[56] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Trnka, J., Local integrals for planar scattering amplitudes, JHEP, 06, 125, (2012)
[57] Weinberg, S., Six-dimensional methods for four-dimensional conformal field theories, Phys. Rev., D 82, 045031, (2010)
[58] Costa, MS; Penedones, J.; Poland, D.; Rychkov, S., Spinning conformal correlators, JHEP, 11, 071, (2011) · Zbl 1306.81207
[59] S. Weinzierl, Tutorial on loop integrals which need regularisation but yield finite results, arXiv:1402.4407 [INSPIRE].
[60] Laporta, S., High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys., A 15, 5087, (2000) · Zbl 0973.81082
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.