Abstract
We calculate the two-loop correction to the dispersion relation for worldsheet modes of the BMN string in AdS n × S n × T 10−2n for n = 2, 3, 5. For the massive modes the result agrees with the exact dispersion relation derived from symmetry considerations with no correction to the interpolating function h. For the massless modes in AdS3 × S 3 × T 4 however our result does not match what one expects from the corresponding symmetry based analysis. We also derive the S-matrix for massless modes up to the one-loop order. The scattering phase is given by the massless limit of the Hernández-López phase. In addition we compute a certain massless S-matrix element at two loops and show that it vanishes suggesting that the two-loop phase in the massless sector is zero.
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References
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS 3 /CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].
D. Sorokin, A. Tseytlin, L. Wulff and K. Zarembo, Superstrings in AdS 2 xS(2)xT(6), J. Phys. A 44 (2011) 275401 [arXiv:1104.1793] [INSPIRE].
A. Cagnazzo, D. Sorokin and L. Wulff, More on integrable structures of superstrings in AdS 4 × CP 3 and AdS 2 × S 2 × T 6 superbackgrounds, JHEP 01 (2012) 004 [arXiv:1111.4197] [INSPIRE].
L. Wulff, On integrability of strings on symmetric spaces, JHEP 09 (2015) 115 [arXiv:1505.03525] [INSPIRE].
K. Zarembo, Worldsheet spectrum in AdS 4 /CFT 3 correspondence, JHEP 04 (2009) 135 [arXiv:0903.1747] [INSPIRE].
N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].
J.M. Maldacena and I. Swanson, Connecting giant magnons to the pp-wave: An interpolating limit of AdS 5 × S 5, Phys. Rev. D 76 (2007) 026002 [hep-th/0612079] [INSPIRE].
T. Klose and K. Zarembo, Reduced σ-model on AdS 5 × S 5 : One-loop scattering amplitudes, JHEP 02 (2007) 071 [hep-th/0701240] [INSPIRE].
T. Klose, T. McLoughlin, J.A. Minahan and K. Zarembo, World-sheet scattering in AdS 5 × S 5 at two loops, JHEP 08 (2007) 051 [arXiv:0704.3891] [INSPIRE].
V. Giangreco Marotta Puletti, T. Klose and O. Ohlsson Sax, Factorized world-sheet scattering in near-flat AdS 5 × S 5, Nucl. Phys. B 792 (2008) 228 [arXiv:0707.2082] [INSPIRE].
J. Murugan, P. Sundin and L. Wulff, Classical and quantum integrability in AdS 2 /CFT 1, JHEP 01 (2013) 047 [arXiv:1209.6062] [INSPIRE].
P. Sundin and L. Wulff, Worldsheet scattering in AdS 3 /CFT 2, JHEP 07 (2013) 007 [arXiv:1302.5349] [INSPIRE].
L. Bianchi, V. Forini and B. Hoare, Two-dimensional S-matrices from unitarity cuts, JHEP 07 (2013) 088 [arXiv:1304.1798] [INSPIRE].
O.T. Engelund, R.W. McKeown and R. Roiban, Generalized unitarity and the worldsheet S matrix in AdS n × S n × M 10−2n, JHEP 08 (2013) 023 [arXiv:1304.4281] [INSPIRE].
L. Bianchi and B. Hoare, AdS 3 × S 3 × M 4 string S-matrices from unitarity cuts, JHEP 08 (2014) 097 [arXiv:1405.7947] [INSPIRE].
M.C. Abbott, J. Murugan, P. Sundin and L. Wulff, Scattering in AdS 2 /CFT 1 and the BES Phase, JHEP 10 (2013) 066 [arXiv:1308.1370] [INSPIRE].
P. Sundin, Worldsheet two- and four-point functions at one loop in AdS 3 /CFT 2, Phys. Lett. B 733 (2014) 134 [arXiv:1403.1449] [INSPIRE].
R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The one-loop worldsheet S-matrix for the AdS n × S n × T 10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].
P. Sundin and L. Wulff, One- and two-loop checks for the AdS 3 × S 3 × T 4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS 5 × S 5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE].
R. Roiban and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly from quantum superstring, JHEP 11 (2007) 016 [arXiv:0709.0681] [INSPIRE].
R. Roiban and A.A. Tseytlin, Spinning superstrings at two loops: Strong-coupling corrections to dimensions of large-twist SYM operators, Phys. Rev. D 77 (2008) 066006 [arXiv:0712.2479] [INSPIRE].
S. Giombi, R. Ricci, R. Roiban, A.A. Tseytlin and C. Vergu, Quantum AdS 5 × S 5 superstring in the AdS light-cone gauge, JHEP 03 (2010) 003 [arXiv:0912.5105] [INSPIRE].
S. Giombi, R. Ricci, R. Roiban, A.A. Tseytlin and C. Vergu, Generalized scaling function from light-cone gauge AdS 5 × S 5 superstring, JHEP 06 (2010) 060 [arXiv:1002.0018] [INSPIRE].
S. Giombi, R. Ricci, R. Roiban and A.A. Tseytlin, Two-loop AdS 5 × S 5 superstring: testing asymptotic Bethe ansatz and finite size corrections, J. Phys. A 44 (2011) 045402 [arXiv:1010.4594] [INSPIRE].
Y. Iwashita, R. Roiban and A.A. Tseytlin, Two-loop corrections to partition function of Pohlmeyer-reduced theory for AdS 5 × S 5 superstring, Phys. Rev. D 84 (2011) 126017 [arXiv:1109.5361] [INSPIRE].
B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS 3 × S 3 × T 4 with mixed flux, Nucl. Phys. B 879 (2014) 318 [arXiv:1311.1794] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefański Jr., The complete worldsheet S matrix of superstrings on AdS 3 × S 3 × T 4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
B. Hoare, A. Pittelli and A. Torrielli, Integrable S-matrices, massive and massless modes and the AdS 2 × S 2 superstring, JHEP 11 (2014) 051 [arXiv:1407.0303] [INSPIRE].
N. Beisert, R. Hernandez and E. Lopez, A Crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, The complete AdS 3 × S 3 × T 4 worldsheet S matrix, JHEP 10 (2014) 66 [arXiv:1406.0453] [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS 5 × S 5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].
L. Wulff, The type-II superstring to order θ 4, JHEP 07 (2013) 123 [arXiv:1304.6422] [INSPIRE].
L. Wulff, Superisometries and integrability of superstrings, JHEP 05 (2014) 115 [arXiv:1402.3122] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, Dressing phases of AdS3/CFT2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
M. Beccaria, F. Levkovich-Maslyuk, G. Macorini and A.A. Tseytlin, Quantum corrections to spinning superstrings in AdS 3 × S 3 × M 4 : determining the dressing phase, JHEP 04 (2013) 006 [arXiv:1211.6090] [INSPIRE].
M.C. Abbott, The AdS 3 × S 3 × S 3 × S 1 Hernández-López phases: a semiclassical derivation, J. Phys. A 46 (2013) 445401 [arXiv:1306.5106] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
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Sundin, P., Wulff, L. The AdS n × S n × T 10−2n BMN string at two loops. J. High Energ. Phys. 2015, 154 (2015). https://doi.org/10.1007/JHEP11(2015)154
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DOI: https://doi.org/10.1007/JHEP11(2015)154