Abstract
Modular graph functions (MGFs) are SL(2, ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2, ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.B. Green and P. Vanhove, The Low-energy expansion of the one loop type-II superstring amplitude, Phys. Rev. D 61 (2000) 104011 [hep-th/9910056] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type-II superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion, JHEP 08 (2015) 041 [arXiv:1502.06698] [INSPIRE].
F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Num. Theor. Phys. 10 (2016) 703 [arXiv:1512.05689] [INSPIRE].
E. D’Hoker, M.B. Green, O. Gürdogan and P. Vanhove, Modular Graph Functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].
E. D’Hoker and M.B. Green, Identities between Modular Graph Forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].
E. D’Hoker and J. Kaidi, Hierarchy of Modular Graph Identities, JHEP 11 (2016) 051 [arXiv:1608.04393] [INSPIRE].
J.E. Gerken and J. Kaidi, Holomorphic subgraph reduction of higher-point modular graph forms, JHEP 01 (2019) 131 [arXiv:1809.05122] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE].
J.E. Gerken, Basis Decompositions and a Mathematica Package for Modular Graph Forms, J. Phys. A 54 (2021) 195401 [arXiv:2007.05476] [INSPIRE].
J.E. Gerken, Modular Graph Forms and Scattering Amplitudes in String Theory, Ph.D. Thesis, Humboldt-Universität zu Berlin (2020) [DOI] [arXiv:2011.08647] [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus, J. Number Theor. 196 (2019) 381 [arXiv:1509.00363] [INSPIRE].
A. Basu, Poisson equation for the Mercedes diagram in string theory at genus one, Class. Quant. Grav. 33 (2016) 055005 [arXiv:1511.07455] [INSPIRE].
A. Basu, Poisson equation for the three loop ladder diagram in string theory at genus one, Int. J. Mod. Phys. A 31 (2016) 1650169 [arXiv:1606.02203] [INSPIRE].
A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].
A. Basu, Eigenvalue equation for the modular graph Ca,b,c,d, JHEP 07 (2019) 126 [arXiv:1906.02674] [INSPIRE].
A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].
F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
F. Zerbini, Modular and Holomorphic Graph Functions from Superstring Amplitudes, in KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, pp. 459–484 (2019) [DOI] [arXiv:1807.04506] [INSPIRE].
F. Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes, Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2017) [arXiv:1804.07989] [INSPIRE].
J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, JHEP 01 (2019) 155 [arXiv:1803.00527] [INSPIRE].
J. Broedel and O. Schlotterer, One-Loop String Scattering Amplitudes as Iterated Eisenstein Integrals, in KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, pp. 133–159 (2019) [DOI] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, JHEP 07 (2020) 190 [arXiv:2004.05156] [INSPIRE].
E. D’Hoker and W. Duke, Fourier series of modular graph functions, J. Number Theor. 192 (2018) 1 [arXiv:1708.07998] [INSPIRE].
O. Ahlén and A. Kleinschmidt, D6R4 curvature corrections, modular graph functions and Poincaré series, JHEP 05 (2018) 194 [arXiv:1803.10250] [INSPIRE].
E. D’Hoker and J. Kaidi, Modular graph functions and odd cuspidal functions. Fourier and Poincaré series, JHEP 04 (2019) 136 [arXiv:1902.04180] [INSPIRE].
D. Dorigoni and A. Kleinschmidt, Modular graph functions and asymptotic expansions of Poincaré series, Commun. Num. Theor. Phys. 13 (2019) 569 [arXiv:1903.09250] [INSPIRE].
A. Basu, Zero mode of the Fourier series of some modular graphs from Poincaré series, Phys. Lett. B 809 (2020) 135715 [arXiv:2005.07793] [INSPIRE].
E. D’Hoker, Integral of two-loop modular graph functions, JHEP 06 (2019) 092 [arXiv:1905.06217] [INSPIRE].
R. Rankin, Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions, Math. Proc. Cambridge Phil. Soc. 35 (1939) 351.
A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940) 47.
D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. 1 A Math. 28 (1982) 415.
E. D’Hoker and M.B. Green, Exploring transcendentality in superstring amplitudes, JHEP 07 (2019) 149 [arXiv:1906.01652] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys. B 661 (2003) 19 [Erratum ibid. 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
M. Beccaria and V. Forini, Four loop reciprocity of twist two operators in N = 4 SYM, JHEP 03 (2009) 111 [arXiv:0901.1256] [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. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995) 24 [hep-th/9410152] [INSPIRE].
D. Dorigoni, A. Kleinschmidt and O. Schlotterer, Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems, arXiv:2109.05017 [INSPIRE].
D. Dorigoni, A. Kleinschmidt and O. Schlotterer, Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms, arXiv:2109.05018 [INSPIRE].
E. D’Hoker and M.B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, Commun. Num. Theor. Phys. 14 (2020) 315 [arXiv:1904.06603] [INSPIRE].
M.B. Green, C.R. Mafra and O. Schlotterer, Multiparticle one-loop amplitudes and S-duality in closed superstring theory, JHEP 10 (2013) 188 [arXiv:1307.3534] [INSPIRE].
E. D’Hoker, C.R. Mafra, B. Pioline and O. Schlotterer, Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors, JHEP 08 (2020) 135 [arXiv:2006.05270] [INSPIRE].
E. D’Hoker, C.R. Mafra, B. Pioline and O. Schlotterer, Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality, JHEP 02 (2021) 139 [arXiv:2008.08687] [INSPIRE].
E. D’Hoker and O. Schlotterer, Two-loop superstring five-point amplitudes. Part III. Construction via the RNS formulation: even spin structures, JHEP 12 (2021) 063 [arXiv:2108.01104] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].
S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].
O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: A genus-zero derivation, J. Phys. A 52 (2019) 045401 [arXiv:1808.00713] [INSPIRE].
P. Vanhove and F. Zerbini, Single-valued hyperlogarithms, correlation functions and closed string amplitudes, arXiv:1812.03018 [INSPIRE].
F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, Commun. Math. Phys. 382 (2021) 815 [arXiv:1910.01107] [INSPIRE].
D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, Commun. Num. Theor. Phys. 14 (2020) 413 [arXiv:1906.12339] [INSPIRE].
J.E. Gerken, A. Kleinschmidt, C.R. Mafra, O. Schlotterer and B. Verbeek, Towards closed strings as single-valued open strings at genus one, J. Phys. A 55 (2022) 025401 [arXiv:2010.10558] [INSPIRE].
P. Vanhove and F. Zerbini, Building blocks of closed and open string amplitudes, in MathemAmplitudes 2019: Intersection Theory and Feynman Integrals, (2020) [arXiv:2007.08981] [INSPIRE].
S. Abel and K.R. Dienes, Calculating the Higgs mass in string theory, Phys. Rev. D 104 (2021) 126032 [arXiv:2106.04622] [INSPIRE].
A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
N. Benjamin, S. Collier, A.L. Fitzpatrick, A. Maloney and E. Perlmutter, Harmonic analysis of 2d CFT partition functions, JHEP 09 (2021) 174 [arXiv:2107.10744] [INSPIRE].
D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2), A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler eds., Basel, pp. 497–512, Birkhäuser Basel (1994) [DOI].
J. Zhao, Analytic continuation of multiple zeta functions, Proc. Am. Math. Soc. 128 (2000) 1275.
S. Akiyama and H. Ishikawa, On analytic continuation of multiple L-functions and related zeta-functions, in Analytic Number Theory, C. Jia and K. Matsumoto eds., Boston, MA, pp. 1–16, Springer US (2002) [DOI].
D. Borwein, J.M. Borwein and D.M. Bradley, Parametric Euler sum identities, J. Math. Anal. Appl. 316 (2006) 328 [Math0505058].
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: 2110.06237
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
D’Hoker, E., Geiser, N. Integrating three-loop modular graph functions and transcendentality of string amplitudes. J. High Energ. Phys. 2022, 19 (2022). https://doi.org/10.1007/JHEP02(2022)019
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2022)019