Abstract
The low energy limit of the closed-string 3-loop amplitude (including its overall coefficient) is computed for four external massless states using the pure spinor formalism and the result is compared with a prediction of Green and Vanhove based on SL(2, \( \mathbb{Z} \)) duality. Agreement is found provided the three-loop amplitude prescription includes a symmetry factor 1/3. We argue for its inclusion in order to compensate a \( {{\mathbb{Z}}_3} \) symmetry present in genus-three Riemann surfaces.
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References
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089 [hep-th/0509120] [INSPIRE].
E. D’Hoker and D. 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, 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].
N. Berkovits and N. Nekrasov, Multiloop superstring amplitudes from non-minimal pure spinor formalism, JHEP 12 (2006) 029 [hep-th/0609012] [INSPIRE].
N. Berkovits, New higher-derivative R 4 theorems, Phys. Rev. Lett. 98 (2007) 211601 [hep-th/0609006] [INSPIRE].
E. Witten, More on superstring perturbation theory, arXiv:1304.2832 [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].
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].
M.B. Green and P. Vanhove, Duality and higher derivative terms in M -theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].
E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
E. D’Hoker and D. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
W. Siegel, Classical superstring mechanics, Nucl. Phys. B 263 (1986) 93 [INSPIRE].
I. Oda and M. Tonin, Y-formalism and b ghost in the non-minimal pure spinor formalism of superstrings, Nucl. Phys. B 779 (2007) 63 [arXiv:0704.1219] [INSPIRE].
R. Lipinski Jusinskas, Nilpotency of the b ghost in the non-minimal pure spinor formalism, JHEP 05 (2013) 048 [arXiv:1303.3966] [INSPIRE].
E. Witten, Twistor-like transform in ten-dimensions, Nucl. Phys. B 266 (1986) 245 [INSPIRE].
E. D’Hoker, M. Gutperle and D. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [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].
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].
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library Edition Published, Wiley-Interscience, Hoboken U.S.A. (1994).
E. D’Hoker and D. Phong, Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B 269 (1986) 205 [INSPIRE].
N. Sakai and Y. Tanii, One loop amplitudes and effective action in superstring theories, Nucl. Phys. B 287 (1987) 457 [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
C.L. Siegel, Symplectic geometry, Amer. J. Math. 65 (1943) 1.
N. Berkovits, Explaining pure spinor superspace, hep-th/0612021 [INSPIRE].
C.R. Mafra, PSS: a FORM program to evaluate pure spinor superspace expressions, arXiv:1007.4999 [INSPIRE].
J. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
H. Gomez and C.R. Mafra, work in progress.
C.R. Mafra and O. Schlotterer, The structure of n-point one-loop open superstring amplitudes, arXiv:1203.6215 [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].
E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, arXiv:1308.4597 [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, arXiv:1205.1516 [INSPIRE].
M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].
M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [INSPIRE].
M.B. Green, H.-H. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [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].
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].
J. Hoogeveen and K. Skenderis, Decoupling of unphysical states in the minimal pure spinor formalism I, JHEP 01 (2010) 041 [arXiv:0906.3368] [INSPIRE].
C. Stahn, private communication (2008).
N. Berkovits and N. Nekrasov, The character of pure spinors, Lett. Math. Phys. 74 (2005) 75 [hep-th/0503075] [INSPIRE].
Y. Aisaka, E.A. Arroyo, N. Berkovits and N. Nekrasov, Pure spinor partition function and the massive superstring spectrum, JHEP 08 (2008) 050 [arXiv:0806.0584] [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. Stahn, Fermionic superstring loop amplitudes in the pure spinor formalism, JHEP 05 (2007) 034 [arXiv:0704.0015] [INSPIRE].
M.B. Green, C.R. Mafra and O. Schlotterer, Multiparticle one-loop amplitudes and S-duality in closed superstring theory, arXiv:1307.3534 [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].
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ArXiv ePrint: 1308.6567
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Gomez, H., Mafra, C.R. The closed-string 3-loop amplitude and S-duality. J. High Energ. Phys. 2013, 217 (2013). https://doi.org/10.1007/JHEP10(2013)217
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DOI: https://doi.org/10.1007/JHEP10(2013)217