Abstract
We investigate massless n-point one-loop amplitudes of the open RNS superstring with two external fermions and determine their worldsheet integrands. The contributing correlation functions involving spin-1/2 and spin-3/2 operators from the fermion vertices are evaluated to any multiplicity. Moreover, we introduce techniques to sum these correlators over the spin structures of the worldsheet fermions such as to manifest all cancellations due to spacetime supersymmetry. These spin sums require generalizations of the Riemann identities among Jacobi theta functions, and the results can be expressed in terms of doubly-periodic functions known from the mathematics literature on elliptic multiple zeta values. On the boundary of moduli space, our spin-summed correlators specialize to compact representations of fermionic one-loop integrands for ambitwistor strings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, Origin of the Pure Spinor and Green-Schwarz Formalisms, JHEP 07 (2015) 091 [arXiv:1503.03080] [INSPIRE].
N. Berkovits, Untwisting the pure spinor formalism to the RNS and twistor string in a flat and AdS 5× S 5 background, JHEP 06 (2016) 127 [arXiv:1604.04617] [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].
P. Ramond, Dual Theory for Free Fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].
A. Neveu and J.H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971) 86 [INSPIRE].
A. Neveu and J.H. Schwarz, Quark Model of Dual Pions, Phys. Rev. D 4 (1971) 1109 [INSPIRE].
J. Cohn, D. Friedan, Z.-a. Qiu and S.H. Shenker, Covariant Quantization of Supersymmetric String Theories: The Spinor Field of the Ramond-Neveu-Schwarz Model, Nucl. Phys. B 278 (1986) 577 [INSPIRE].
A. Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories, Phys. Rev. D 39 (1989) 1626 [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].
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].
F. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
B. Enriquez, Analogues elliptiques des nombres multizétas, arXiv:1301.3042.
N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089 [hep-th/0509120] [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].
N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [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].
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].
H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].
H. Gomez, C.R. Mafra and O. Schlotterer, Two-loop superstring five-point amplitude and S-duality, Phys. Rev. D 93 (2016) 045030 [arXiv:1504.02759] [INSPIRE].
C.R. Mafra and O. Schlotterer, One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, JHEP 04 (2016) 148 [arXiv:1603.04790] [INSPIRE].
C.R. Mafra and O. Schlotterer, The double-copy structure of one-loop open-string amplitudes, arXiv:1711.09104 [INSPIRE].
C.R. Mafra and O. Schlotterer, to appear.
C.R. Mafra and O. Schlotterer, The Structure of n-Point One-Loop Open Superstring Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [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].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [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].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [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].
S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity, arXiv:1706.00640 [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].
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].
D. Friedan, S.H. Shenker and E.J. Martinec, Covariant Quantization of Superstrings, Phys. Lett. B 160 (1985) 55 [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B 271 (1986) 93 [INSPIRE].
R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Springer (2013).
V.A. Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, Conformal Techniques, Bosonization and Tree Level String Amplitudes, Nucl. Phys. B 288 (1987) 173 [INSPIRE].
I.G. Koh, W. Troost and A. Van Proeyen, Covariant Higher Spin Vertex Operators in the Ramond Sector, Nucl. Phys. B 292 (1987) 201 [INSPIRE].
W.-Z. Feng, D. Lüst and O. Schlotterer, Massive Supermultiplets in Four-Dimensional Superstring Theory, Nucl. Phys. B 861 (2012) 175 [arXiv:1202.4466] [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology, Cambridge Monographs On Mathematical Physics, Cambridge University Press, Cambridge, U.K. (1987).
J.J. Atick and A. Sen, Covariant One Loop Fermion Emission Amplitudes in Closed String Theories, Nucl. Phys. B 293 (1987) 317 [INSPIRE].
Z.H. Lin, L. Clavelli and S.T. Jones, Five Point Function In The Covariant Formulation Of The Type I Superstring Theory, Nucl. Phys. B 294 (1987) 83 [INSPIRE].
Z.-h. Lin, One Loop Closed String Five Particle Fermion Amplitudes In The Covariant Formulation, Int. J. Mod. Phys. A 5 (1990) 299 [INSPIRE].
E. Witten, Superstring Perturbation Theory Revisited, arXiv:1209.5461 [INSPIRE].
A. Sen, Off-shell Amplitudes in Superstring Theory, Fortsch. Phys. 63 (2015) 149 [arXiv:1408.0571] [INSPIRE].
E. D’Hoker and D.H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
J.J. Atick and A. Sen, Correlation Functions of Spin Operators on a Torus, Nucl. Phys. B 286 (1987) 189 [INSPIRE].
D. Haertl, O. Schlotterer and S. Stieberger, Higher Point Spin Field Correlators in D = 4 Superstring Theory, Nucl. Phys. B 834 (2010) 163 [arXiv:0911.5168] [INSPIRE].
O. Schlotterer, Higher Loop Spin Field Correlators in D = 4 Superstring Theory, JHEP 09 (2010) 050 [arXiv:1001.3158] [INSPIRE].
D. Haertl and O. Schlotterer, Higher Loop Spin Field Correlators in Various Dimensions, Nucl. Phys. B 849 (2011) 364 [arXiv:1011.1249] [INSPIRE].
L. Kronecker, Zur Theorie der elliptischen Funktionen, Math. Werke IV (1881) 313.
J. Broedel, N. Matthes, G. Richter and O. Schlotterer, Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes, arXiv:1704.03449 [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. 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].
J.J. Atick and A. Sen, Spin Field Correlators on an Arbitrary Genus Riemann Surface and Nonrenormalization Theorems in String Theories, Phys. Lett. B 186 (1987) 339 [INSPIRE].
P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Phys. 47 (1928) 631.
I.B. Frenkel and V.G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23.
L. Álvarez-Gaumé, J.B. Bost, G.W. Moore, P.C. Nelson and C. Vafa, Bosonization on Higher Genus Riemann Surfaces, Commun. Math. Phys. 112 (1987) 503 [INSPIRE].
L. Álvarez-Gaumé, G.W. Moore, P.C. Nelson, C. Vafa and J.b. Bost, Bosonization in Arbitrary Genus, Phys. Lett. B 178 (1986) 41 [INSPIRE].
L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta Functions, Modular Invariance and Strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].
L. Clavelli, P.H. Cox, B. Harms and H. Konno, Bosonization of odd spin structure amplitudes, Phys. Rev. D 43 (1991) 3998 [INSPIRE].
J.D. Fay, “Theta Functions on Riemann Surfaces, Lect. Notes Math. 352, Springer (1973).
D. Mumford, M. Nori and P. Norman, Tata Lectures on Theta I, Progress in Mathematics, Birkhäuser Boston (1983).
D. Mumford, M. Nori and P. Norman, Tata Lectures on ThetaII, Progress in Mathematics, Birkhäuser Boston (1984).
K.A. Roehrig and D. Skinner, A Gluing Operator for the Ambitwistor String, JHEP 01 (2018) 069 [arXiv:1709.03262] [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].
A.G. Tsuchiya, On new theta identities of fermion correlation functions on genus g Riemann surfaces, arXiv:1710.00206 [INSPIRE].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
O. Schlotterer, Higher Spin Scattering in Superstring Theory, Nucl. Phys. B 849 (2011) 433 [arXiv:1011.1235] [INSPIRE].
T. Banks, L.J. Dixon, D. Friedan and E.J. Martinec, Phenomenology and Conformal Field Theory Or Can String Theory Predict the Weak Mixing Angle?, Nucl. Phys. B 299 (1988) 613 [INSPIRE].
T. Banks and L.J. Dixon, Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B 307 (1988) 93 [INSPIRE].
S. Ferrara, D. Lüst and S. Theisen, World Sheet Versus Spectrum Symmetries in Heterotic and Type II Superstrings, Nucl. Phys. B 325 (1989) 501 [INSPIRE].
N. Berkovits, Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background, Nucl. Phys. B 431 (1994) 258 [hep-th/9404162] [INSPIRE].
N. Berkovits, A new description of the superstring, in Proceedings, 8th J.A. Swieca Summer School on Particles and Fields: Rio de Janeiro, Brazil, February 5-18, 1995, pp. 390–418, hep-th/9604123 [INSPIRE].
N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].
N. Berkovits, Quantization of the superstring with manifest U(5) superPoincaré invariance, Phys. Lett. B 457 (1999) 94 [hep-th/9902099] [INSPIRE].
N. Berkovits and B.C. Vallilo, One loop N point superstring amplitudes with manifest d = 4 supersymmetry, Nucl. Phys. B 624 (2002) 45 [hep-th/0110168] [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: 1710.07353
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
Lee, S., Schlotterer, O. Fermionic one-loop amplitudes of the RNS superstring. J. High Energ. Phys. 2018, 190 (2018). https://doi.org/10.1007/JHEP03(2018)190
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2018)190