Abstract
We propose a next-to-leading Lüscher-like formula for the finite-size corrections of the excited states energies in integrable theories. We conjecture the expressions of the corrections for both the energy and the particles’ rapidities by interpreting the excited states as momenta-dependent defects. We check the resulting formulas in some simple relativistic model and conjecture those for the AdS 5 /CFT 4 case.
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References
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].
R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling: the giant magnon, Phys. Rev. D 76 (2007) 126008 [arXiv:0708.2208] [INSPIRE].
N. Gromov, S. Schäfer-Nameki and P. Vieira, Quantum wrapped giant magnon, Phys. Rev. D 78 (2008) 026006 [arXiv:0801.3671] [INSPIRE].
N. Gromov, S. Schäfer-Nameki and P. Vieira, Efficient precision quantization in AdS/CFT, JHEP 12 (2008) 013 [arXiv:0807.4752] [INSPIRE].
Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].
Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [INSPIRE].
M.P. Heller, R.A. Janik and T. Lukowski, A New derivation of Lüscher F-term and fluctuations around the giant magnon, JHEP 06 (2008) 036 [arXiv:0801.4463] [INSPIRE].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].
G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].
D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N =4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [INSPIRE].
A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5 /CFT 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].
A. Cavaglia, D. Fioravanti, M. Mattelliano and R. Tateo, On the AdS 5 /CF T 4 TBA and its analytic properties, arXiv:1103.0499 [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, JHEP 07 (2012) 023 [arXiv:1110.0562] [INSPIRE].
A. Hegedus, Discrete Hirota dynamics for AdS/CFT, Nucl. Phys. B 825 (2010) 341 [arXiv:0906.2546] [INSPIRE].
N. Gromov, V. Kazakov and Z. Tsuboi, PSU(2, 2|4) character of quasiclassical AdS/CFT, JHEP 07 (2010) 097 [arXiv:1002.3981] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and Z. Tsuboi, Wronskian solution for AdS/CFT Y-system, JHEP 01 (2011) 155 [arXiv:1010.2720] [INSPIRE].
R. Suzuki, Hybrid NLIE for the mirror AdS 5 × S 5, J. Phys. A 44 (2011) 235401 [arXiv:1101.5165] [INSPIRE].
J. Balog and A. Hegedus, AdS 5 × S 5 mirror TBA equations from Y-system and discontinuity relations, JHEP 08 (2011) 095 [arXiv:1104.4054] [INSPIRE].
J. Balog and A. Hegedus, Quasi-local formulation of the mirror TBA, JHEP 05 (2012) 039 [arXiv:1106.2100] [INSPIRE].
J. Balog and A. Hegedus, Hybrid-NLIE for the AdS/CFT spectral problem, JHEP 08 (2012) 022 [arXiv:1202.3244] [INSPIRE].
R. Suzuki, Contour deformation trick in hybrid NLIE, JHEP 07 (2012) 152 [arXiv:1202.3947] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for AdS 5 /CF T 4, arXiv:1305.1939 [INSPIRE].
N. Gromov, Quantum algebraic curve for N = 4 SYM, talk given at Gauge/gravity duality 2013, http://ggd2013.mpp.mpg.de/, Max Planck Institute for Physics, Munich Germany (2013).
D. Volin, Quantum spectral curve for AdS 5 /CF T 4 spectral problem, talk given at IGST 2013, http://testweb.science.uu.nl/IGST13/files/program.html, University of Utrecht, Utrecht The Netherlands (2013).
S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar N = 4 SYM, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].
S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett. 109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].
M. Lüscher, On a relation between finite size effects and elastic scattering processes, lecture given at Cargese Summer Inst., Cargese France September 1-15 1983 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
T.R. Klassen and E. Melzer, On the relation between scattering amplitudes and finite size mass corrections in QFT, Nucl. Phys. B 362 (1991) 329 [INSPIRE].
Z. Bajnok and R.A. Janik, Six and seven loop Konishi from Lüscher corrections, JHEP 11 (2012) 002 [arXiv:1209.0791] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
Z. Bajnok and A. George, From defects to boundaries, Int. J. Mod. Phys. A 21 (2006) 1063 [hep-th/0404199] [INSPIRE].
Z. Bajnok and Z. Simon, Solving topological defects via fusion, Nucl. Phys. B 802 (2008) 307 [arXiv:0712.4292] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].
C. Destri and H. de Vega, Light cone lattice approach to fermionic theories in 2D: the massive Thirring model, Nucl. Phys. B 290 (1987) 363 [INSPIRE].
C. Destri and H. de Vega, Light cone lattices and the exact solution of chiral fermion and σ models, J. Phys. A 22 (1989) 1329 [INSPIRE].
D. Fioravanti, A. Mariottini, E. Quattrini and F. Ravanini, Excited state Destri-De Vega equation for sine-Gordon and restricted sine-Gordon models, Phys. Lett. B 390 (1997) 243 [hep-th/9608091] [INSPIRE].
G. Arutyunov and S. Frolov, The S-matrix of string bound states, Nucl. Phys. B 804 (2008) 90 [arXiv:0803.4323] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].
G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].
R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
R. Hernandez and E. Lopez, Quantum corrections to the string Bethe ansatz, JHEP 07 (2006) 004 [hep-th/0603204] [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].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS 5 × S 5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].
G. Arutyunov, M. de Leeuw and A. Torrielli, The bound state S-matrix for AdS 5 × S 5 superstring, Nucl. Phys. B 819 (2009) 319 [arXiv:0902.0183] [INSPIRE].
N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [INSPIRE].
M.C. Abbott, I. Aniceto and D. Bombardelli, Real and virtual bound states in Lüscher corrections for CP 3 magnons, J. Phys. A 45 (2012) 335401 [arXiv:1111.2839] [INSPIRE].
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Bombardelli, D. A next-to-leading Lüscher formula. J. High Energ. Phys. 2014, 37 (2014). https://doi.org/10.1007/JHEP01(2014)037
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DOI: https://doi.org/10.1007/JHEP01(2014)037