Abstract
We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. A. Lappo-Danilevsky, Théorie algorithmique des corps de Riemann, Rec. Math. Moscou 34 (1927) 113.
A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
J. Ablinger, J. Blumlein and C. Schneider, Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].
C. Duhr, Mathematical aspects of scattering amplitudes, in Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014), Boulder, Colorado, June 2–27, 2014, pp. 419–476 (2015) [DOI] [arXiv:1411.7538] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
S. Buehler and C. Duhr, CHAPLIN — Complex Harmonic Polylogarithms in Fortran, Comput. Phys. Commun. 185 (2014) 2703 [arXiv:1106.5739] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Lin and Li2,2 and on the evaluation thereof, JHEP 03 (2016) 189 [arXiv:1601.02649] [INSPIRE].
J. Ablinger, J. Blümlein, M. Round and C. Schneider, Numerical Implementation of Harmonic Polylogarithms to Weight w = 8, Comput. Phys. Commun. 240 (2019) 189 [arXiv:1809.07084] [INSPIRE].
L. Naterop, A. Signer and Y. Ulrich, handyG —Rapid numerical evaluation of generalised polylogarithms in Fortran, Comput. Phys. Commun. 253 (2020) 107165 [arXiv:1909.01656] [INSPIRE].
A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
A. V. Kotikov, Differential equations method: The Calculation of vertex type Feynman diagrams, Phys. Lett. B 259 (1991) 314 [INSPIRE].
A. V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. 295 (1992) 409] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
A. Sabry, Fourth order spectral functions for the electron propagator, Nucl. Phys. 33 (1962) 401.
D. J. Broadhurst, The Master Two Loop Diagram With Masses, Z. Phys. C 47 (1990) 115 [INSPIRE].
S. Bauberger, F. A. Berends, M. Böhm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams, Nucl. Phys. B 434 (1995) 383 [hep-ph/9409388] [INSPIRE].
S. Bauberger and M. Böhm, Simple one-dimensional integral representations for two loop selfenergies: The Master diagram, Nucl. Phys. B 445 (1995) 25 [hep-ph/9501201] [INSPIRE].
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
B. A. Kniehl, A. V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines, Nucl. Phys. B 738 (2006) 306 [hep-ph/0510235] [INSPIRE].
U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys. B 789 (2008) 45 [arXiv:0705.2616] [INSPIRE].
M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result, Nucl. Phys. B 824 (2010) 111 [arXiv:0811.4119] [INSPIRE].
F. Brown and O. Schnetz, A K3 in ϕ4, Duke Math. J. 161 (2012) 1817 [arXiv:1006.4064] [INSPIRE].
S. Müller-Stach, S. Weinzierl and R. Zayadeh, A Second-Order Differential Equation for the Two-Loop Sunrise Graph with Arbitrary Masses, Commun. Num. Theor. Phys. 6 (2012) 203 [arXiv:1112.4360] [INSPIRE].
S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
R. Huang and Y. Zhang, On Genera of Curves from High-loop Generalized Unitarity Cuts, JHEP 04 (2013) 080 [arXiv:1302.1023] [INSPIRE].
F. Brown and O. Schnetz, Modular forms in Quantum Field Theory, Commun. Num. Theor Phys. 07 (2013) 293 [arXiv:1304.5342] [INSPIRE].
D. Nandan, M. F. Paulos, M. Spradlin and A. Volovich, Star Integrals, Convolutions and Simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].
J. Ablinger et al., Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys. 59 (2018) 062305 [arXiv:1706.01299] [INSPIRE].
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys. 57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].
L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys. 57 (2016) 122302 [arXiv:1607.01571] [INSPIRE].
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 (2014) 102301 [arXiv:1405.5640] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].
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 (2015) 072303 [arXiv:1504.03255] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].
A. Beilinson and A. Levin, The Elliptic Polylogarithm, in Proc. of Symp. in Pure Math. 55, Part II, J.-P. S. U. Jannsen and S. L. Kleiman ed.s, pp. 123–190, AMS (1994).
A. Levin and G. Racinet, Towards multiple elliptic polylogarithms, math/0703237.
F. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].
J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].
J. Broedel, N. Matthes, G. Richter and O. Schlotterer, Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes, J. Phys. A 51 (2018) 285401 [arXiv:1704.03449] [INSPIRE].
L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys. 12 (2018) 193 [arXiv:1704.08895] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].
Y. I. Manin, Iterated integrals of modular forms and noncommutative modular symbols, in Algebraic geometry and number theory, vol. 253 of Progr. Math., Boston, pp. 565–597, Birkhäuser Boston (2006) [math/0502576].
F. Brown, Multiple modular values and the relative completion of the fundamental group of ℳ1, 1, arXiv:1407.5167.
N. Matthes, On the algebraic structure of iterated integrals of quasimodular forms, Alg. Numb. Theor. 11-9 (2017) 2113 [arXiv:1708.04561].
F. Brown, From the Deligne-Ihara conjecture to Multiple Modular Values, arXiv:1904.00178.
N. Matthes, Iterated primitives of meromorphic quasimodular forms for SL2(ℤ), Trans. Am. Math. Soc. 375 (2022) 1443 [arXiv:2101.11491].
C. Duhr and L. Tancredi, Algorithms and tools for iterated Eisenstein integrals, JHEP 02 (2020) 105 [arXiv:1912.00077] [INSPIRE].
M. Walden and S. Weinzierl, Numerical evaluation of iterated integrals related to elliptic Feynman integrals, Comput. Phys. Commun. 265 (2021) 108020 [arXiv:2010.05271] [INSPIRE].
S. Weinzierl, Modular transformations of elliptic Feynman integrals, Nucl. Phys. B 964 (2021) 115309 [arXiv:2011.07311] [INSPIRE].
H. Frellesvig, C. Vergu, M. Volk and M. von Hippel, Cuts and Isogenies, JHEP 05 (2021) 064 [arXiv:2102.02769] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
L. Adams and S. Weinzierl, The ε-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett. B 781 (2018) 270 [arXiv:1802.05020] [INSPIRE].
C. Bogner and F. Brown, Feynman integrals and iterated integrals on moduli spaces of curves of genus zero, Commun. Num. Theor. Phys. 09 (2015) 189 [arXiv:1408.1862] [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys. 21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].
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].
J. L. Bourjaily, A. J. McLeod, M. von Hippel and M. Wilhelm, Bounded Collection of Feynman Integral Calabi-Yau Geometries, Phys. Rev. Lett. 122 (2019) 031601 [arXiv:1810.07689] [INSPIRE].
A. Klemm, C. Nega and R. Safari, The l-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods, JHEP 04 (2020) 088 [arXiv:1912.06201] [INSPIRE].
K. Bönisch, F. Fischbach, A. Klemm, C. Nega and R. Safari, Analytic structure of all loop banana integrals, JHEP 05 (2021) 066 [arXiv:2008.10574] [INSPIRE].
K. Bönisch, C. Duhr, F. Fischbach, A. Klemm and C. Nega, Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives, arXiv:2108.05310 [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante and L. Tancredi, An analytic solution for the equal-mass banana graph, JHEP 09 (2019) 112 [arXiv:1907.03787] [INSPIRE].
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].
F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
K. G. Chetyrkin, M. Faisst, C. Sturm and M. Tentyukov, epsilon-finite basis of master integrals for the integration-by-parts method, Nucl. Phys. B 742 (2006) 208 [hep-ph/0601165] [INSPIRE].
R. N. Lee and A. I. Onishchenko, E-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process e+ e− → \( 2\left(Q\overline{Q}\right) \), JHEP 12 (2019) 084 [arXiv:1909.07710] [INSPIRE].
A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].
H. Frellesvig and C. G. Papadopoulos, Cuts of Feynman Integrals in Baikov representation, JHEP 04 (2017) 083 [arXiv:1701.07356] [INSPIRE].
M. Harley, F. Moriello and R. M. Schabinger, Baikov-Lee Representations Of Cut Feynman Integrals, JHEP 06 (2017) 049 [arXiv:1705.03478] [INSPIRE].
J. Bosma, M. Sogaard and Y. Zhang, Maximal Cuts in Arbitrary Dimension, JHEP 08 (2017) 051 [arXiv:1704.04255] [INSPIRE].
C. F. Doran, Picard-Fuchs uniformization: Modularity of the mirror map and mirror moonshine, in Proceedings of NATO-ASI and CRM Summer School on the Arithmetic and Geometry of Algebraic Cycles, (1998) [math/9812162] [INSPIRE].
F. Diamond, B. Sturmfels, J. Shurman and S. S. Media, A First Course in Modular Forms, Graduate Texts in Mathematics, Springer (2005) [DOI].
Y. Yifan, Transformation formulas for generalized dedekind eta functions, Bull. London Math. Soc. 36 (2004) 671.
K. S. Chua, M. L. Lang and Y. Yang, On Rademacher’s conjecture: congruence subgroups of genus zero of the modular group, J. Algebra 277 (2003) 408.
D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms, Springer (2008) [DOI].
E. Royer, Quasimodular forms: an introduction, Ann. Math. Blaise Pascal 19 (2012) 297.
P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences, Proc. Am. Math. Soc. 136 (2008) 3051.
M. Deneufchâtel, G. H. E. Duchamp, V. H. N. Minh and A. I. Solomon, Independence of hyperlogarithms over function fields via algebraic combinatorics, in Algebraic informatics, vol. 6742 of Lecture Notes in Comput. Sci., pp. 127–139, Springer, Heidelberg (2011) [DOI].
K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].
G. Bol, Invarianten linearer Differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949) 1.
E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].
W. Stein and D. Joyner, SAGE: System for algebra and geometry experimentation, ACM SIGSAM Bull. 39 (2005) 61 [http://www.sagemath.org/files/sage_stein2005.pdf].
R. S. Maier, On Rationally Parametrized Modular Equations, math/0611041.
G. Joyce, On the simple cubic lattice Green function, Phil. Trans. Roy. Soc. Lond. A 273 (1973) 583.
H. R. P. Ferguson and D. H. Bailey, A polynomial time, numerically stable integer relation algorithm, RNR Technical Report RNR-91-032 (1992).
H. A. Verrill, Root lattices and pencils of varieties, J. Math. Kyoto Univ. 36 (1996) 423.
T. J. Fonseca and N. Matthes, Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension, RIMS Kokyuroku 2160 (2020) 114 [arXiv:2009.10433].
S. Abreu, M. Becchetti, C. Duhr and R. Marzucca, Three-loop contributions to the ρ parameter and iterated integrals of modular forms, JHEP 02 (2020) 050 [arXiv:1912.02747] [INSPIRE].
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor — planar case, Nucl. Phys. B 698 (2004) 277 [hep-ph/0401193] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2109.15251
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Broedel, J., Duhr, C. & Matthes, N. Meromorphic modular forms and the three-loop equal-mass banana integral. J. High Energ. Phys. 2022, 184 (2022). https://doi.org/10.1007/JHEP02(2022)184
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2022)184