Abstract
It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of uniform transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region \( \mathrm{\mathcal{M}} \)R surrounding the cusp and its complement \( \mathrm{\mathcal{M}} \)L, and an exact expression is obtained for the contribution to the amplitude from \( \mathrm{\mathcal{M}} \)R. The low-energy expansion of the \( \mathrm{\mathcal{M}} \)R contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from \( \mathrm{\mathcal{M}} \)L is computed in terms of modular graph functions up to order D12\( \mathrm{\mathcal{R}} \)4 in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the \( \mathrm{\mathcal{M}} \)L contributions. Uniform transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sum functions, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. D’Hoker, M.B. Green and P. Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion, JHEP08 (2015) 041 [arXiv:1502.06698] [INSPIRE].
E. D’Hoker, M.B. Green, Ö. Gürdogan and P. Vanhove, Modular Graph Functions, Commun. Num. Theor. Phys.11 (2017) 165 [arXiv:1512.06779] [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants and their exact asymptotics, Commun. Math. Phys.366 (2019) 927 [arXiv:1712.06135] [INSPIRE].
N. Kawazumi, Johnson’s homomorphisms and the Arakelov-Green function, arXiv:0801.4218.
S.-W. Zhang, Gross-Schoen cycles and dualising sheaves., Invent. Math.179 (2010) 1.
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, JHEP11 (2016) 051 [arXiv:1608.04393] [INSPIRE].
E. D’Hoker and D.H. Phong, Lectures on two loop superstrings, Conf. Proc.C 0208124 (2002) 85 [hep-th/0211111] [INSPIRE].
E. D’Hoker and D.H. Phong, Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys.B 715 (2005) 3 [hep-th/0501197] [INSPIRE].
N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP01 (2006) 005 [hep-th/0503197] [INSPIRE].
E. D’Hoker, M. Gutperle and D.H. Phong, Two-loop superstrings and S-duality, Nucl. Phys.B 722 (2005) 81 [hep-th/0503180] [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, arXiv:1308.4597 [INSPIRE].
E. D’Hoker, M.B. Green, B. Pioline and R. Russo, Matching the D 6R 4interaction at two-loops, JHEP01 (2015) 031 [arXiv:1405.6226] [INSPIRE].
E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys.B 440 (1995) 24 [hep-th/9410152] [INSPIRE].
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, JHEP02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
E. D’Hoker and M.B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, arXiv:1904.06603 [INSPIRE].
D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, arXiv:1906.12339 [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.B 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
M. Beccaria and V. Forini, Four loop reciprocity of twist two operators in N = 4 SYM, JHEP03 (2009) 111 [arXiv:0901.1256] [INSPIRE].
E. D’Hoker, Integral of two-loop modular graph functions, JHEP06 (2019) 092 [arXiv:1905.06217] [INSPIRE].
M.A. Virasoro, Alternative constructions of crossing-symmetric amplitudes with Regge behavior, Phys. Rev.177 (1969) 2309 [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys.A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
D. Zagier, Values of zeta functions and their application, in First European Congress of Mathematics (Paris, 1992), Vol. II, Birkhäuser, Progr. Math.120 (1994) 497.
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. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
F. Brown, Single-valued multiple polylogarithms in one variable, C.R. Acad. Sci. Paris Ser. I338 (2004) 527.
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].
M.B. Green and J.H. Schwarz, Supersymmetrical String Theories, Phys. Lett.109B (1982) 444 [INSPIRE].
N. Sakai and Y. Tanii, One Loop Amplitudes and Effective Action in Superstring Theories, Nucl. Phys.B 287 (1987) 457 [INSPIRE].
J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, JHEP01 (2019) 155 [arXiv:1803.00527] [INSPIRE].
J.A. Shapiro, Electrostatic analog for the virasoro model, Phys. Lett.33B (1970) 361 [INSPIRE].
L.F. Alday, A. Bissi and E. Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP06 (2019) 010 [arXiv:1809.10670] [INSPIRE].
E. D’Hoker and J. Kaidi, Modular graph functions and odd cuspidal functions. Fourier and Poincaré series, JHEP04 (2019) 136 [arXiv:1902.04180] [INSPIRE].
F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Num. Theor. Phys.10 (2016) 703 [arXiv:1512.05689] [INSPIRE].
O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills, JHEP11 (2016) 074 [arXiv:1608.00130] [INSPIRE].
A. Basu, Low momentum expansion of one loop amplitudes in heterotic string theory, JHEP11 (2017) 139 [arXiv:1708.08409] [INSPIRE].
A. Basu, A simplifying feature of the heterotic one loop four graviton amplitude, Phys. Lett.B 776 (2018) 182 [arXiv:1710.01993] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP01 (2019) 052 [arXiv:1811.02548] [INSPIRE].
K.-T. Chen, Iterated path integrals, Bull. Am. Math Soc.83 (1977) 831 [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited — 2001 and Beyond, B. Enquist and W. Schmid eds., Springer, Berlin-Heidelberg-New York (2001), pp. 771–808 [https://doi.org/10.1007/978-3-642-56478-9 39].
C. Duhr, Mathematical aspects of scattering amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders (TASI 2014), Boulder, Colorado, 2–27 June 2014, pp. 419–476 (2015) [https://doi.org/10.1142/9789814678766_0010] [arXiv:1411.7538] [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].
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, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys.61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP07 (2015) 112 [arXiv:1412.5535] [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].
F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, arXiv:1810.07682 [INSPIRE].
P. Vanhove and F. Zerbini, Closed string amplitudes from single-valued correlation functions, arXiv:1812.03018 [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, arXiv:1509.00363 [INSPIRE].
A. Basu, Proving relations between modular graph functions, Class. Quant. Grav.33 (2016) 235011 [arXiv:1606.07084] [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. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP09 (2017) 155 [arXiv:1706.01889] [INSPIRE].
F. Brown, A class of non-holomorphic modular forms I, 2017, arXiv:1707.01230 [INSPIRE].
F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, arXiv:1708.03354.
D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Tokyo28 (1982) 415.
A. Erdelyi, Higher transcendental Functions, Bateman manuscript project, Vol. 2, page 143, R.E. Krieger Publishing (1981).
Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys.B 530 (1998) 401 [hep-th/9802162] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1906.01652
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
D’Hoker, E., Green, M.B. Exploring transcendentality in superstring amplitudes. J. High Energ. Phys. 2019, 149 (2019). https://doi.org/10.1007/JHEP07(2019)149
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)149