Abstract
In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators D2k Fn and D2k Rn in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension α′. Similar to one-loop ambitwistor computations, intermediate steps feature non-standard linearized Feynman propagators which eventually recombine to conventional quadratic propagators. With linearized propagators the loop integrand of the matrix elements obey one-loop versions of the monodromy and KLT relations. We express a variety of four- and five-point examples in terms of quadratic propagators and formulate a criterion on the underlying genus-one correlation functions that should make this recombination possible at all orders in α′. The ultraviolet divergences of the one-loop matrix elements are crosschecked against the non-separating degeneration of genus-one integrals in string amplitudes. Conversely, our results can be used as a constructive method to determine degenerations of elliptic multiple zeta values and modular graph forms at arbitrary weight.
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Z. Bern, J. J. M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Z. Bern, J. J. M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [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].
E. D’Hoker and D. H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
E. D’Hoker and D. H. Phong, Conformal Scalar Fields and Chiral Splitting on SuperRiemann Surfaces, Commun. Math. Phys. 125 (1989) 469 [INSPIRE].
J. J. M. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 Super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].
T. Adamo and E. Casali, Scattering equations, supergravity integrands, and pure spinors, JHEP 05 (2015) 120 [arXiv:1502.06826] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].
C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, P. H. Damgaard and B. Feng, Integration Rules for Loop Scattering Equations, JHEP 11 (2015) 080 [arXiv:1508.03627] [INSPIRE].
S. He and E. Y. Yuan, One-loop Scattering Equations and Amplitudes from Forward Limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [INSPIRE].
C. Cardona and H. Gomez, Elliptic scattering equations, JHEP 06 (2016) 094 [arXiv:1605.01446] [INSPIRE].
C. Cardona and H. Gomez, CHY-Graphs on a Torus, JHEP 10 (2016) 116 [arXiv:1607.01871] [INSPIRE].
S. He and O. Schlotterer, New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].
S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, Nucl. Phys. B 930 (2018) 328 [arXiv:1706.00640] [INSPIRE].
Y. Geyer and R. Monteiro, Gluons and gravitons at one loop from ambitwistor strings, JHEP 03 (2018) 068 [arXiv:1711.09923] [INSPIRE].
A. Edison, S. He, O. Schlotterer and F. Teng, One-loop Correlators and BCJ Numerators from Forward Limits, JHEP 09 (2020) 079 [arXiv:2005.03639] [INSPIRE].
A. A. Tseytlin, Vector Field Effective Action in the Open Superstring Theory, Nucl. Phys. B 276 (1986) 391 [Erratum ibid. 291 (1987) 876] [INSPIRE].
D. J. Gross and E. Witten, Superstring Modifications of Einstein’s Equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].
Y. Kitazawa, Effective Lagrangian for Open Superstring From Five Point Function, Nucl. Phys. B 289 (1987) 599 [INSPIRE].
A. A. Tseytlin, On nonAbelian generalization of Born-Infeld action in string theory, Nucl. Phys. B 501 (1997) 41 [hep-th/9701125] [INSPIRE].
P. Koerber and A. Sevrin, The NonAbelian D-brane effective action through order αJ4 , JHEP 10 (2002) 046 [hep-th/0208044] [INSPIRE].
L. A. Barreiro and R. Medina, 5-field terms in the open superstring effective action, JHEP 03 (2005) 055 [hep-th/0503182] [INSPIRE].
D. Oprisa and S. Stieberger, Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums, hep-th/0509042 [INSPIRE].
S. Stieberger, Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory, Phys. Rev. Lett. 106 (2011) 111601 [arXiv:0910.0180] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
L. A. Barreiro and R. Medina, Revisiting the S-matrix approach to the open superstring low energy effective lagrangian, JHEP 10 (2012) 108 [arXiv:1208.6066] [INSPIRE].
M. R. Garousi, Effective action of type-II superstring theories at order α′3: NS-NS couplings, JHEP 02 (2021) 157 [arXiv:2011.02753] [INSPIRE].
M. R. Garousi, On NS-NS couplings at order α′3, Nucl. Phys. B 971 (2021) 115510 [arXiv:2012.15091] [INSPIRE].
M. B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].
M. Cederwall, B. E. W. Nilsson and D. Tsimpis, D = 10 superYang-Mills at O(α′2), JHEP 07 (2001) 042 [hep-th/0104236] [INSPIRE].
P. Koerber and A. Sevrin, The NonAbelian Born-Infeld action through order alpha-prime 3, JHEP 10 (2001) 003 [hep-th/0108169] [INSPIRE].
A. Collinucci, M. De Roo and M. G. C. Eenink, Supersymmetric Yang-Mills theory at order α′3, JHEP 06 (2002) 024 [hep-th/0205150] [INSPIRE].
N. Berkovits and V. Pershin, Supersymmetric Born-Infeld from the pure spinor formalism of the open superstring, JHEP 01 (2003) 023 [hep-th/0205154] [INSPIRE].
J. M. Drummond, P. J. Heslop, P. S. Howe and S. F. Kerstan, Integral invariants in N = 4 SYM and the effective action for coincident D-branes, JHEP 08 (2003) 016 [hep-th/0305202] [INSPIRE].
G. Policastro and D. Tsimpis, R4, purified, Class. Quant. Grav. 23 (2006) 4753 [hep-th/0603165] [INSPIRE].
P. S. Howe, U. Lindström and L. Wulff, D=10 supersymmetric Yang-Mills theory at α′4, JHEP 07 (2010) 028 [arXiv:1004.3466] [INSPIRE].
M. Cederwall and A. Karlsson, Pure spinor superfields and Born-Infeld theory, JHEP 11 (2011) 134 [arXiv:1109.0809] [INSPIRE].
Y. Wang and X. Yin, Constraining Higher Derivative Supergravity with Scattering Amplitudes, Phys. Rev. D 92 (2015) 041701 [arXiv:1502.03810] [INSPIRE].
J. Broedel and L. J. Dixon, Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators, JHEP 10 (2012) 091 [arXiv:1208.0876] [INSPIRE].
Y.-t. Huang, O. Schlotterer and C. Wen, Universality in string interactions, JHEP 09 (2016) 155 [arXiv:1602.01674] [INSPIRE].
T. Azevedo, M. Chiodaroli, H. Johansson and O. Schlotterer, Heterotic and bosonic string amplitudes via field theory, JHEP 10 (2018) 012 [arXiv:1803.05452] [INSPIRE].
C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, S. Caron-Huot, P. H. Damgaard and B. Feng, New Representations of the Perturbative S-matrix, Phys. Rev. Lett. 116 (2016) 061601 [arXiv:1509.02169] [INSPIRE].
R. P. Feynman, Quantum theory of gravitation, Acta Phys. Polon. 24 (1963) 697 [INSPIRE].
S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, From loops to trees by-passing Feynman’s theorem, JHEP 09 (2008) 065 [arXiv:0804.3170] [INSPIRE].
H. Gomez and E. Y. Yuan, N-point tree-level scattering amplitude in the new Berkovits‘ string, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].
N. Kalyanapuram, Ambitwistor integrands from tensionless chiral superstring integrands, JHEP 10 (2021) 171 [arXiv:2103.07943] [INSPIRE].
N. Kalyanapuram, On Chiral Splitting and the Ambitwistor String, Phys. Rev. D 104 (2021) 086027 [arXiv:2103.08584] [INSPIRE].
Y.-t. Huang, W. Siegel and E. Y. Yuan, Factorization of Chiral String Amplitudes, JHEP 09 (2016) 101 [arXiv:1603.02588] [INSPIRE].
M. M. Leite and W. Siegel, Chiral Closed strings: Four massless states scattering amplitude, JHEP 01 (2017) 057 [arXiv:1610.02052] [INSPIRE].
Y. Li and W. Siegel, Chiral Superstring and CHY Amplitude, arXiv:1702.07332 [INSPIRE].
E. Casali and P. Tourkine, On the null origin of the ambitwistor string, JHEP 11 (2016) 036 [arXiv:1606.05636] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [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].
R. R. Metsaev and A. A. Tseytlin, On loop corrections to string theory effective actions, Nucl. Phys. B 298 (1988) 109 [INSPIRE].
M. B. Green, J. G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].
B. Pioline, String theory integrands and supergravity divergences, JHEP 02 (2019) 148 [arXiv:1810.11343] [INSPIRE].
E. D’Hoker and M. B. Green, Exploring transcendentality in superstring amplitudes, JHEP 07 (2019) 149 [arXiv:1906.01652] [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, 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].
N. Berkovits and M. Lize, Field theory actions for ambitwistor string and superstring, JHEP 09 (2018) 097 [arXiv:1807.07661] [INSPIRE].
H. Flores and M. Lize, On the Spectrum and Spacetime Supersymmetry of Heterotic Ambitwistor String, JHEP 08 (2019) 094 [arXiv:1903.05792] [INSPIRE].
H. Gomez, S. Mizera and G. Zhang, CHY Loop Integrands from Holomorphic Forms, JHEP 03 (2017) 092 [arXiv:1612.06854] [INSPIRE].
H. Gomez, Quadratic Feynman Loop Integrands From Massless Scattering Equations, Phys. Rev. D 95 (2017) 106006 [arXiv:1703.04714] [INSPIRE].
H. Gomez, C. Lopez-Arcos and P. Talavera, One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations, JHEP 10 (2017) 175 [arXiv:1707.08584] [INSPIRE].
N. Ahmadiniaz, H. Gomez and C. Lopez-Arcos, Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators, JHEP 05 (2018) 055 [arXiv:1802.00015] [INSPIRE].
J. Agerskov, N. E. J. Bjerrum-Bohr, H. Gomez and C. Lopez-Arcos, One-Loop Yang-Mills Integrands from Scattering Equations, Phys. Rev. D 102 (2020) 045023 [arXiv:1910.03602] [INSPIRE].
J. A. Farrow, Y. Geyer, A. E. Lipstein, R. Monteiro and R. Stark-Muchão, Propagators, BCFW recursion and new scattering equations at one loop, JHEP 10 (2020) 074 [arXiv:2007.00623] [INSPIRE].
E. Plahte, Symmetry properties of dual tree-graph n-point amplitudes, Nuovo Cim. A 66 (1970) 713 [INSPIRE].
N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
P. Tourkine and P. Vanhove, Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett. 117 (2016) 211601 [arXiv:1608.01665] [INSPIRE].
S. Hohenegger and S. Stieberger, Monodromy Relations in Higher-Loop String Amplitudes, Nucl. Phys. B 925 (2017) 63 [arXiv:1702.04963] [INSPIRE].
A. Ochirov, P. Tourkine and P. Vanhove, One-loop monodromy relations on single cuts, JHEP 10 (2017) 105 [arXiv:1707.05775] [INSPIRE].
P. Tourkine, Integrands and loop momentum in string and field theory, Phys. Rev. D 102 (2020) 026006 [arXiv:1901.02432] [INSPIRE].
E. Casali, S. Mizera and P. Tourkine, Monodromy relations from twisted homology, JHEP 12 (2019) 087 [arXiv:1910.08514] [INSPIRE].
E. Casali, S. Mizera and P. Tourkine, Loop amplitudes monodromy relations and color-kinematics duality, JHEP 03 (2021) 048 [arXiv:2005.05329] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String One-Loop Amplitudes, arXiv:2105.06888 [INSPIRE].
C. R. Mafra and O. Schlotterer, Double-Copy Structure of One-Loop Open-String Amplitudes, Phys. Rev. Lett. 121 (2018) 011601 [arXiv:1711.09104] [INSPIRE].
C. R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics, JHEP 08 (2019) 091 [arXiv:1812.10970] [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].
M. Bianchi and A. V. Santini, String predictions for near future colliders from one-loop scattering amplitudes around D-brane worlds, JHEP 12 (2006) 010 [hep-th/0607224] [INSPIRE].
M. Bianchi and D. Consoli, Simplifying one-loop amplitudes in superstring theory, JHEP 01 (2016) 043 [arXiv:1508.00421] [INSPIRE].
M. Berg, I. Buchberger and O. Schlotterer, From maximal to minimal supersymmetry in string loop amplitudes, JHEP 04 (2017) 163 [arXiv:1603.05262] [INSPIRE].
M. Berg, I. Buchberger and O. Schlotterer, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459] [INSPIRE].
J. J. M. Carrasco, R. Kallosh, R. Roiban and A. A. Tseytlin, On the U(1) duality anomaly and the S-matrix of N = 4 supergravity, JHEP 07 (2013) 029 [arXiv:1303.6219] [INSPIRE].
Z. Bern, C. Cheung, H.-H. Chi, S. Davies, L. Dixon and J. Nohle, Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences, Phys. Rev. Lett. 115 (2015) 211301 [arXiv:1507.06118] [INSPIRE].
Z. Bern, H.-H. Chi, L. Dixon and A. Edison, Two-Loop Renormalization of Quantum Gravity Simplified, Phys. Rev. D 95 (2017) 046013 [arXiv:1701.02422] [INSPIRE].
Z. Bern, A. Edison, D. Kosower and J. Parra-Martinez, Curvature-squared multiplets, evanescent effects, and the U(1) anomaly in N = 4 supergravity, Phys. Rev. D 96 (2017) 066004 [arXiv:1706.01486] [INSPIRE].
Z. Bern, J. J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].
Z. Bern, J. J. M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].
Z. Bern et al., Ultraviolet Properties of N = 8 Supergravity at Five Loops, Phys. Rev. D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].
G. Mogull and D. O’Connell, Overcoming Obstacles to Colour-Kinematics Duality at Two Loops, JHEP 12 (2015) 135 [arXiv:1511.06652] [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. 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. 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, A. Kleinschmidt and O. Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, JHEP 07 (2020) 190 [arXiv:2004.05156] [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 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].
E. Panzer, Modular graph functions as iterated Eisenstein integrals, talk given at the workshop Elliptic Integrals in Mathematics and Physics Ascona, Switzerland (2018) [https://indico.cern.ch/event/700233/contributions/3112451/attachments/1712442/2761239/elliptic.pdf].
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].
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].
F. Cachazo, S. He and E. Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
A. Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories, Phys. Rev. D 39 (1989) 1626 [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].
C. R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure, JHEP 08 (2019) 092 [arXiv:1812.10971] [INSPIRE].
J. A. Minahan, One Loop Amplitudes on Orbifolds and the Renormalization of Coupling Constants, Nucl. Phys. B 298 (1988) 36 [INSPIRE].
H. Johansson and A. Ochirov, Pure Gravities via Color-Kinematics Duality for Fundamental Matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, Explaining Pure Spinor Superspace, hep-th/0612021 [INSPIRE].
C. R. Mafra and O. Schlotterer, Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortsch. Phys. 63 (2015) 105 [arXiv:1410.0668] [INSPIRE].
E. Bridges and C. R. Mafra, Local BCJ numerators for ten-dimensional SYM at one loop, JHEP 07 (2021) 031 [arXiv:2102.12943] [INSPIRE].
C. R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
T. Terasoma, Selberg integrals and multiple zeta values, Compos. Math. 133 (2002) 1.
F. C. S. Brown, Multiple zeta values and periods of moduli spaces \( {\overline{\mathfrak{M}}}_{0,n} \) (ℝ), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].
F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [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].
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].
J. M. Drummond and É. Ragoucy, Superstring amplitudes and the associator, JHEP 08 (2013) 135 [arXiv:1301.0794] [INSPIRE].
C. R. Mafra and O. Schlotterer, Non-abelian Z -theory: Berends-Giele recursion for the α′-expansion of disk integrals, JHEP 01 (2017) 031 [arXiv:1609.07078] [INSPIRE].
C. Mafra and O. Schlotterer, https://repo.or.cz/BGap.git.
J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α′-expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE].
A. Kaderli, A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory, J. Phys. A 53 (2020) 415401 [arXiv:1912.09406] [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, O. Schlotterer and S. Stieberger, http://mzv.mpp.mpg.de.
S. Stieberger and T. R. Taylor, Multi-Gluon Scattering in Open Superstring Theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].
R. H. Boels, On the field theory expansion of superstring five point amplitudes, Nucl. Phys. B 876 (2013) 215 [arXiv:1304.7918] [INSPIRE].
G. Puhlfürst and S. Stieberger, Differential Equations, Associators, and Recurrences for Amplitudes, Nucl. Phys. B 902 (2016) 186 [arXiv:1507.01582] [INSPIRE].
C. R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
R. R. Metsaev, M. Rakhmanov and A. A. Tseytlin, The Born-Infeld Action as the Effective Action in the Open Superstring Theory, Phys. Lett. B 193 (1987) 207 [INSPIRE].
M. Shmakova, One loop corrections to the D3-brane action, Phys. Rev. D 62 (2000) 104009 [hep-th/9906239] [INSPIRE].
H. Elvang, M. Hadjiantonis, C. R. T. Jones and S. Paranjape, All-Multiplicity One-Loop Amplitudes in Born-Infeld Electrodynamics from Generalized Unitarity, JHEP 03 (2020) 009 [arXiv:1906.05321] [INSPIRE].
H. Elvang, M. Hadjiantonis, C. R. T. Jones and S. Paranjape, Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order, JHEP 04 (2021) 173 [arXiv:2006.08928] [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].
R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L. J. Dixon, M. Perelstein and J. S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
J. E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [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, arXiv:2010.10558 [INSPIRE].
M. B. Green, J. H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [INSPIRE].
S. Stieberger and T. R. Taylor, NonAbelian Born-Infeld action and type 1. — heterotic duality 2: Nonrenormalization theorems, Nucl. Phys. B 648 (2003) 3 [hep-th/0209064] [INSPIRE].
C. R. Mafra and C. Stahn, The One-loop Open Superstring Massless Five-point Amplitude with the Non-Minimal Pure Spinor Formalism, JHEP 03 (2009) 126 [arXiv:0902.1539] [INSPIRE].
C. R. Mafra and O. Schlotterer, The Structure of n-Point One-Loop Open Superstring Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].
C. R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153 [arXiv:1404.4986] [INSPIRE].
C. R. Mafra and O. Schlotterer, Berends-Giele recursions and the BCJ duality in superspace and components, JHEP 03 (2016) 097 [arXiv:1510.08846] [INSPIRE].
S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].
Y. Geyer, R. Monteiro and R. Stark-Muchão, Superstring Loop Amplitudes from the Field Theory Limit, Phys. Rev. Lett. 127 (2021) 211603 [arXiv:2106.03968] [INSPIRE].
L. Dolan and P. Goddard, Current Algebra on the Torus, Commun. Math. Phys. 285 (2009) 219 [arXiv:0710.3743] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1998) [DOI].
Z. Bern and D. A. Kosower, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett. 66 (1991) 1669 [INSPIRE].
Z. Bern and D. A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].
Z. Bern and D. A. Kosower, The Computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].
Z. Bern and D. C. Dunbar, A Mapping between Feynman and string motivated one loop rules in gauge theories, Nucl. Phys. B 379 (1992) 562 [INSPIRE].
Z. Bern, D. C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett. B 312 (1993) 277 [hep-th/9307001] [INSPIRE].
D. C. Dunbar and P. S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
P. Tourkine, Tropical Amplitudes, Annales Henri Poincaré 18 (2017) 2199 [arXiv:1309.3551] [INSPIRE].
M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge University Press, Cambridge (1988).
R. H. Boels and R. S. Isermann, New relations for scattering amplitudes in Yang-Mills theory at loop level, Phys. Rev. D 85 (2012) 021701 [arXiv:1109.5888] [INSPIRE].
R. H. Boels and R. S. Isermann, Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts, JHEP 03 (2012) 051 [arXiv:1110.4462] [INSPIRE].
Y.-J. Du and H. Lüo, On General BCJ Relation at One-loop Level in Yang-Mills Theory, JHEP 01 (2013) 129 [arXiv:1207.4549] [INSPIRE].
A. Primo and W. J. Torres Bobadilla, BCJ Identities and d-Dimensional Generalized Unitarity, JHEP 04 (2016) 125 [arXiv:1602.03161] [INSPIRE].
F. Brown, A class of non-holomorphic modular forms I, Res. Math. Sci. 5 (2018) 7 [arXiv:1707.01230] [INSPIRE].
F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, Forum Math. Sigma 8 (2020) 1 [arXiv:1708.03354].
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].
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].
B. Enriquez, Analogues elliptiques des nombres multizétas, Bull. Soc. Math. Fr. 144 (2016) 395 [arXiv:1301.3042].
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].
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].
V. A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1 [INSPIRE].
C. R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].
C. Vafa and E. Witten, A One loop test of string duality, Nucl. Phys. B 447 (1995) 261 [hep-th/9505053] [INSPIRE].
M. B. Green and C. Wen, Modular Forms and SL(2, ℤ)-covariance of type IIB superstring theory, JHEP 06 (2019) 087 [arXiv:1904.13394] [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].
P. H. Frampton and T. W. Kephart, Explicit Evaluation of Anomalies in Higher Dimensions, Phys. Rev. Lett. 50 (1983) 1343 [Erratum ibid. 51 (1983) 232] [INSPIRE].
P. H. Frampton and T. W. Kephart, The Analysis of Anomalies in Higher Space-time Dimensions, Phys. Rev. D 28 (1983) 1010 [INSPIRE].
M. B. Green and J. H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
M. B. Green and J. H. Schwarz, The Hexagon Gauge Anomaly in Type I Superstring Theory, Nucl. Phys. B 255 (1985) 93 [INSPIRE].
N. E. J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand Oxidation and One-Loop Colour-Dual Numerators in N = 4 Gauge Theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
A. Brandhuber, B. Spence, G. Travaglini and G. Yang, Form Factors in N = 4 Super Yang-Mills and Periodic Wilson Loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE].
A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. Yang, Harmony of Super Form Factors, JHEP 10 (2011) 046 [arXiv:1107.5067] [INSPIRE].
L. V. Bork, D. I. Kazakov and G. S. Vartanov, On MHV Form Factors in Superspace for N = 4 SYM Theory, JHEP 10 (2011) 133 [arXiv:1107.5551] [INSPIRE].
L. V. Bork, On NMHV form factors in N = 4 SYM theory from generalized unitarity, JHEP 01 (2013) 049 [arXiv:1203.2596] [INSPIRE].
O. T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].
D. Nandan, C. Sieg, M. Wilhelm and G. Yang, Cutting through form factors and cross sections of non-protected operators in N = 4 SYM, JHEP 06 (2015) 156 [arXiv:1410.8485] [INSPIRE].
L. Bianchi, A. Brandhuber, R. Panerai and G. Travaglini, Form factor recursion relations at loop level, JHEP 02 (2019) 182 [arXiv:1812.09001] [INSPIRE].
R. H. Boels, B. A. Kniehl, O. V. Tarasov and G. Yang, Color-kinematic Duality for Form Factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
G. Yang, Color-kinematics duality and Sudakov form factor at five loops for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 117 (2016) 271602 [arXiv:1610.02394] [INSPIRE].
G. Lin, G. Yang and S. Zhang, Three-Loop Color-Kinematics Duality: A 24-Dimensional Solution Space Induced by New Generalized Gauge Transformations, Phys. Rev. Lett. 127 (2021) 171602 [arXiv:2106.05280] [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, JHEP 02 (2021) 069 [arXiv:1912.08707] [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary Geometries, Generalized Particles and Strings, and Cluster Algebras, arXiv:1912.11764 [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Cluster Configuration Spaces of Finite Type, SIGMA 17 (2021) 092 [arXiv:2005.11419] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
J. J. M. Carrasco, L. Rodina, Z. Yin and S. Zekioglu, Simple encoding of higher derivative gauge and gravity counterterms, Phys. Rev. Lett. 125 (2020) 251602 [arXiv:1910.12850] [INSPIRE].
J. J. M. Carrasco, L. Rodina and S. Zekioglu, Composing effective prediction at five points, JHEP 06 (2021) 169 [arXiv:2104.08370] [INSPIRE].
H.-H. Chi, H. Elvang, A. Herderschee, C. R. T. Jones and S. Paranjape, Generalizations of the Double-Copy: the KLT Bootstrap, arXiv:2106.12600 [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-Loop Scattering Amplitudes from the Riemann Sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].
Y. Geyer and R. Monteiro, Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere, JHEP 11 (2018) 008 [arXiv:1805.05344] [INSPIRE].
Y. Geyer, R. Monteiro and R. Stark-Muchão, Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality, JHEP 12 (2019) 049 [arXiv:1908.05221] [INSPIRE].
B. Pioline, A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, J. Number Theor. 163 (2016) 520 [arXiv:1504.04182] [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].
E. D’Hoker, M. B. Green and B. Pioline, Asymptotics of the D8 R4 genus-two string invariant, Commun. Num. Theor. Phys. 13 (2019) 351 [arXiv:1806.02691] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, The Ultraviolet Structure of Half-Maximal Supergravity with Matter Multiplets at Two and Three Loops, Phys. Rev. D 88 (2013) 065007 [arXiv:1305.4876] [INSPIRE].
H. Johansson, G. Kälin and G. Mogull, Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP 09 (2017) 019 [arXiv:1706.09381] [INSPIRE].
Z. Bern, L. J. Dixon and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
M. G. Kozlov and R. N. Lee, One-loop pentagon integral in d dimensions from differential equations in ϵ-form, JHEP 02 (2016) 021 [arXiv:1512.01165] [INSPIRE].
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Edison, A., Guillen, M., Johansson, H. et al. One-loop matrix elements of effective superstring interactions: α′-expanding loop integrands. J. High Energ. Phys. 2021, 7 (2021). https://doi.org/10.1007/JHEP12(2021)007
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DOI: https://doi.org/10.1007/JHEP12(2021)007