Abstract
In this article we investigate one-loop amplitudes in maximally supersymmetric superstring theory. The non-anomalous part of the worldsheet integrand is presented for any number of massless open-string states. The polarization dependence is organized into the same BRST-invariant kinematic combinations which also govern the leading string correction to tree-level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling number \( {S}_3^{n-1} \) of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang-Mills theory and on the other hand the color tensors at order α′2 of the color-dressed tree amplitude.
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References
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [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].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N -point superstring disk amplitude II. Amplitude and hypergeometric function structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM n-point tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [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].
E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
M.B. Green and J.H. Schwarz, Covariant description of superstrings, Phys. Lett. B 136 (1984) 367 [INSPIRE].
M.B. Green and J.H. Schwarz, Properties of the covariant formulation of superstring theories, Nucl. Phys. B 243 (1984) 285 [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Vol. 2: loop amplitudes, anomalies and phenomenology, Cambridge Monographs On Mathematical Physics, Cambdrige Univ. Pr., Cambridge U.K. (1987).
C.R. Mafra, PSS: a FORM program to evaluate pure spinor superspace expressions, arXiv:1007.4999 [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
M. Tentyukov and J.A.M. Vermaseren, The multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [INSPIRE].
C.R. Mafra, Towards field theory amplitudes from the cohomology of pure spinor superspace, JHEP 11 (2010) 096 [arXiv:1007.3639] [INSPIRE].
C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, Six open string disk amplitude in pure spinor superspace, Nucl. Phys. B 846 (2011) 359 [arXiv:1011.0994] [INSPIRE].
E. Witten, Twistor-like transform in ten-dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].
E. D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [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].
S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [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].
L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [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].
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].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
I. Oda and M. Tonin, Y-formalism in pure spinor quantization of superstrings, Nucl. Phys. B 727 (2005) 176 [hep-th/0505277] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, H. Johansson and T. Sondergaard, Monodromy-like relations for finite loop amplitudes, JHEP 05 (2011) 039 [arXiv:1103.6190] [INSPIRE].
T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
A. Bilal, Higher derivative corrections to the non-Abelian Born-Infeld action, Nucl. Phys. B 618 (2001) 21 [hep-th/0106062] [INSPIRE].
M.A.A. van Leeuwen, A.M. Cohen and B. Lisser, LiE, a package for Lie group computations, Computer Algebra Nederland, Amsterdam The Netherlands (1992) [ISBN:90-74116-02-7].
D. Lüst, O. Schlotterer, S. Stieberger and T.R. Taylor, The LHC string hunter’s companion (II): five-particle amplitudes and universal properties, Nucl. Phys. B 828 (2010) 139 [arXiv:0908.0409] [INSPIRE].
P.S. Howe, Pure spinors lines in superspace and ten-dimensional supersymmetric theories, Phys. Lett. B 258 (1991) 141 [Addendum ibid. B 259 (1991) 511] [INSPIRE].
N. Berkovits, Explaining pure spinor superspace, hep-th/0612021 [INSPIRE].
N. Berkovits, Cohomology in the pure spinor formalism for the superstring, JHEP 09 (2000) 046 [hep-th/0006003] [INSPIRE].
N. Berkovits and B.C. Vallilo, Consistency of super-Poincaré covariant superstring tree amplitudes, JHEP 07 (2000) 015 [hep-th/0004171] [INSPIRE].
N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP 01 (2006) 005 [hep-th/0503197] [INSPIRE].
N. Berkovits and C.R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006) 011602 [hep-th/0509234] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
F.A. Berends and W.T. Giele, Recursive calculations for processes with n-gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
G. Policastro and D. Tsimpis, R 4 , purified, Class. Quant. Grav. 23 (2006) 4753 [hep-th/0603165] [INSPIRE].
J.P. Harnad and S. Shnider, Constraints and field equations for ten-dimensional super Yang-Mills theory, Commun. Math. Phys. 106 (1986) 183 [INSPIRE].
P.A. Grassi and L. Tamassia, Vertex operators for closed superstrings, JHEP 07 (2004) 071 [hep-th/0405072] [INSPIRE].
N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059 [INSPIRE].
H. Gomez, One-loop superstring amplitude from integrals on pure spinors space, JHEP 12 (2009) 034 [arXiv:0910.3405] [INSPIRE].
H. Gomez and C.R. Mafra, The overall coefficient of the two-loop superstring amplitude using pure spinors, JHEP 05 (2010) 017 [arXiv:1003.0678] [INSPIRE].
Y. Aisaka and N. Berkovits, Pure spinor vertex operators in Siegel gauge and loop amplitude regularization, JHEP 07 (2009) 062 [arXiv:0903.3443] [INSPIRE].
R.M. Schabinger, One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops, J. Phys. A 44 (2011) 454007 [arXiv:1104.3873] [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].
S. Stieberger and T.R. Taylor, Amplitude for N -gluon superstring scattering, Phys. Rev. Lett. 97 (2006) 211601 [hep-th/0607184] [INSPIRE].
S. Stieberger and T.R. Taylor, Multi-gluon scattering in open superstring theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].
Z. Bern, G. Chalmers, L.J. Dixon and D.A. Kosower, One loop N -gluon amplitudes with maximal helicity violation via collinear limits, Phys. Rev. Lett. 72 (1994) 2134 [hep-ph/9312333] [INSPIRE].
S.J. Parke and T.R. Taylor, An amplitude for N -gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
C.R. Mafra, Pure spinor superspace identities for massless four-point kinematic factors, JHEP 04 (2008) 093 [arXiv:0801.0580] [INSPIRE].
J. Bjornsson and M.B. Green, 5 loops in 24/5 dimensions, JHEP 08 (2010) 132 [arXiv:1004.2692] [INSPIRE].
J. Bjornsson, Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, JHEP 01 (2011) 002 [arXiv:1009.5906] [INSPIRE].
N. Berkovits, M.B. Green, J.G. Russo and P. Vanhove, Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory, JHEP 11 (2009) 063 [arXiv:0908.1923] [INSPIRE].
N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089 [hep-th/0509120] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in twistor space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].
C.R. Mafra, Four-point one-loop amplitude computation in the pure spinor formalism, JHEP 01 (2006) 075 [hep-th/0512052] [INSPIRE].
L.A. Barreiro and R. Medina, 5-field terms in the open superstring effective action, JHEP 03 (2005) 055 [hep-th/0503182] [INSPIRE].
A. Tsuchiya, More on one loop massless amplitudes of superstring theories, Phys. Rev. D 39 (1989) 1626 [INSPIRE].
J.J. Atick and A. Sen, Covariant one loop fermion emission amplitudes in closed string theories, Nucl. Phys. B 293 (1987) 317 [INSPIRE].
S. Stieberger and T.R. Taylor, Non-Abelian Born-Infeld action and type I heterotic duality 2: nonrenormalization theorems, Nucl. Phys. B 648 (2003) 3 [hep-th/0209064] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [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.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].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N =4 super Yang-Mills theory, JHEP 05 (2009) 046[arXiv:0902.2987] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [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].
N. Berkovits and C.R. Mafra, Some superstring amplitude computations with the non-minimal pure spinor formalism, JHEP 11 (2006) 079 [hep-th/0607187] [INSPIRE].
L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].
M.B. Green and J.H. Schwarz, Supersymmetrical dual string theory. 3. Loops and renormalization, Nucl. Phys. B 198 (1982) 441 [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 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].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop selfdual and N = 4 super Yang-Mills, Phys. Lett. B 394 (1997) 105 [hep-th/9611127] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The complete four-loop four-point amplitude in N = 4 super-Yang-Mills theory, Phys. Rev. D 82 (2010) 125040 [arXiv:1008.3327] [INSPIRE].
J.J. 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].
B.E.W. Nilsson, Pure spinors as auxiliary fields in the ten-dimensional supersymmetric Yang-Mills theory, Class. Quant. Grav. 3 (1986) L41 [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].
M.B. Green, private communication.
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Mafra, C.R., Schlotterer, O. The structure of n-point one-loop open superstring amplitudes. J. High Energ. Phys. 2014, 99 (2014). https://doi.org/10.1007/JHEP08(2014)099
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DOI: https://doi.org/10.1007/JHEP08(2014)099