Abstract
In the non-supersymmetric γi-deformed \( \mathcal{N} \) = 4 SYM theory, the scaling dimensions of the operators tr[Z L] composed of L scalar fields Z receive finite-size wrapping and prewrapping corrections in the ’t Hooft limit. In this paper, we calculate these scaling dimensions to leading wrapping order directly from Feynman diagrams. For L ≥ 3, the result is proportional to the maximally transcendental ‘cake’ integral. It matches with an earlier result obtained from the integrability-based Lüscher corrections, TBA and Y-system equations. At L = 2, where the integrability-based equations yield infinity, we find a finite rational result. This result is renormalization-scheme dependent due to the non-vanishing β-function of an induced quartic scalar double-trace coupling, on which we have reported earlier. This explicitly shows that conformal invariance is broken — even in the ’t Hooft limit.
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References
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional \( \mathcal{N} \) =1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
C. Sieg and A. Torrielli, Wrapping interactions and the genus expansion of the 2-point function of composite operators, Nucl. Phys. B 723 (2005) 3 [hep-th/0505071] [INSPIRE].
T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B 376 (1996) 53 [INSPIRE].
N. Beisert and R. Roiban, Beauty and the twist: the Bethe ansatz for twisted \( \mathcal{N} \) = 4 SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].
N. Beisert, The complete one loop dilatation operator of \( \mathcal{N} \) = 4 super Yang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].
R. Roiban, On spin chains and field theories, JHEP 09 (2004) 023 [hep-th/0312218] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
K. Zoubos, Review of AdS/CFT integrability. Chapter IV.2: deformations, orbifolds and open boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed \( \mathcal{N} \) = 4 super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Twisted Bethe equations from a twisted S-matrix, JHEP 02 (2011) 027 [arXiv:1010.3229] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Finite-size effects in the superconformal β-deformed \( \mathcal{N} \) = 4 SYM, JHEP 08 (2008) 057 [arXiv:0806.2103] [INSPIRE].
C. Sieg, Superspace computation of the three-loop dilatation operator of \( \mathcal{N} \) = 4 SYM theory, Phys. Rev. D 84 (2011) 045014 [arXiv:1008.3351] [INSPIRE].
D.J. Gross, A. Mikhailov and R. Roiban, Operators with large R charge in \( \mathcal{N} \) = 4 Yang-Mills theory, Annals Phys. 301 (2002) 31 [hep-th/0205066] [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].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Single impurity operators at critical wrapping order in the β-deformed \( \mathcal{N} \) = 4 SYM, JHEP 08 (2009) 034 [arXiv:0811.4594] [INSPIRE].
J. Gunnesson, Wrapping in maximally supersymmetric and marginally deformed \( \mathcal{N} \) = 4 Yang-Mills, JHEP 04 (2009) 130 [arXiv:0902.1427] [INSPIRE].
D.Z. Freedman and U. Gürsoy, Comments on the β-deformed \( \mathcal{N} \) = 4 SYM theory, JHEP 11 (2005) 042 [hep-th/0506128] [INSPIRE].
T.J. Hollowood and S.P. Kumar, An \( \mathcal{N} \) = 1 duality cascade from a deformation of \( \mathcal{N} \) = 4 SUSY Yang-Mills theory, JHEP 12 (2004) 034 [hep-th/0407029] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i -deformed \( \mathcal{N} \) = 4 SYM theory, arXiv:1308.4420 [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, The complete one-loop dilatation operator of planar real β-deformed \( \mathcal{N} \) = 4 SYM theory, JHEP 07 (2014) 150 [arXiv:1312.2959] [INSPIRE].
S. Penati, A. Santambrogio and D. Zanon, Two-point correlators in the β-deformed \( \mathcal{N} \) = 4 SYM at the next-to-leading order, JHEP 10 (2005) 023 [hep-th/0506150] [INSPIRE].
S. Frolov and R. Suzuki, Temperature quantization from the TBA equations, Phys. Lett. B 679 (2009) 60 [arXiv:0906.0499] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, The spectral problem for strings on twisted AdS 5 × S 5, Nucl. Phys. B 860 (2012) 339 [arXiv:1201.1451] [INSPIRE].
S. Frolov, private communication.
Q. Jin, The emergence of supersymmetry in γ i -deformed \( \mathcal{N} \) = 4 super-Yang-Mills theory, arXiv:1311.7391 [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].
M. de Leeuw and S.J. van Tongeren, Orbifolded Konishi from the mirror TBA, J. Phys. A 44 (2011) 325404 [arXiv:1103.5853] [INSPIRE].
D.J. Broadhurst, Evaluation of a class of Feynman diagrams for all numbers of loops and dimensions, Phys. Lett. B 164 (1985) 356 [INSPIRE].
G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].
V.V. Khoze, Amplitudes in the β-deformed conformal Yang-Mills, JHEP 02 (2006) 040 [hep-th/0512194] [INSPIRE].
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New approach to evaluation of multiloop Feynman integrals: the Gegenbauer polynomial x space technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
A.A. Vladimirov, Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys. 43 (1980) 417 [INSPIRE].
S.J. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace or one thousand and one lessons in supersymmetry, hep-th/0108200 [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Clarendon Press, Oxford U.K. (1996).
J.C. Collins, Renormalization. An introduction to renormalization, the renormalization group, and the operator product expansion, Cambridge University Press, Cambridge U.K. (1984).
W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
R. Mertig and W.L. van Neerven, The calculation of the two loop spin splitting functions P (1) ij (x), Z. Phys. C 70 (1996) 637 [hep-ph/9506451] [INSPIRE].
W. Vogelsang, A rederivation of the spin dependent next-to-leading order splitting functions, Phys. Rev. D 54 (1996) 2023 [hep-ph/9512218] [INSPIRE].
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Fokken, J., Sieg, C. & Wilhelm, M. A piece of cake: the ground-state energies in γ i -deformed \( \mathcal{N} \) = 4 SYM theory at leading wrapping order. J. High Energ. Phys. 2014, 78 (2014). https://doi.org/10.1007/JHEP09(2014)078
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DOI: https://doi.org/10.1007/JHEP09(2014)078