Abstract
In this paper we consider ϕ 4 theory in 4 − ϵ dimensions at the Wilson-Fisher fixed point where the theory becomes conformal. We extend the method in [1] for calculating the leading order term in the anomalous dimensions of some operators with spin. This method involves mostly symmetry arguments and reduces the process for calculating anomalous dimensions to some Wick contractions in the corresponding free theory. We apply this method in the case of operators with two and three fields whose twist is equal to the number of fields they contain, and we rederive known results for their anomalous dimensions. We also calculate the leading term in the anomalous dimensions of operators with spin two and three. In addition, we find expressions for the primary operators of the free theory, for arbitrary spin and number of fields, whose twist remains equal to the number of fields.
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References
S. Rychkov and Z.M. Tan, The \( \epsilon \) -expansion from conformal field theory, J. Phys. A 48 (2015) 29FT01 [arXiv:1505.00963] [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal Bootstrap in Mellin Space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory, arXiv:1611.01500 [INSPIRE].
L.F. Alday, Solving CFTs with Weakly Broken Higher Spin Symmetry, arXiv:1612.00696 [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].
P. Basu and C. Krishnan, \( \epsilon \) -expansions near three dimensions from conformal field theory, JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE].
K. Nii, Classical equation of motion and Anomalous dimensions at leading order, JHEP 07 (2016) 107 [arXiv:1605.08868] [INSPIRE].
C. Hasegawa and Yu. Nakayama, \( \epsilon \) -Expansion in Critical \( \phi \) 3 -Theory on Real Projective Space from Conformal Field Theory, Mod. Phys. Lett. A 32 (2017) 1750045 [arXiv:1611.06373] [INSPIRE].
V. Bashmakov, M. Bertolini, L. Di Pietro and H. Raj, Scalar Multiplet Recombination at Large-N and Holography, JHEP 05 (2016) 183 [arXiv:1603.00387] [INSPIRE].
S. Ghosh, R.K. Gupta, K. Jaswin and A.A. Nizami, \( \epsilon \) -Expansion in the Gross-Neveu model from conformal field theory, JHEP 03 (2016) 174 [arXiv:1510.04887] [INSPIRE].
A. Raju, \( \epsilon \) -Expansion in the Gross-Neveu CFT, JHEP 10 (2016) 097 [arXiv:1510.05287] [INSPIRE].
D. Anselmi, The N = 4 quantum conformal algebra, Nucl. Phys. B 541 (1999) 369 [hep-th/9809192] [INSPIRE].
A.V. Belitsky, J. Henn, C. Jarczak, D. Mueller and E. Sokatchev, Anomalous dimensions of leading twist conformal operators, Phys. Rev. D 77 (2008) 045029 [arXiv:0707.2936] [INSPIRE].
E.D. Skvortsov, On (Un)Broken Higher-Spin Symmetry in Vector Models, arXiv:1512.05994 [INSPIRE].
A.N. Manashov and M. Strohmaier, Conformal constraints for anomalous dimensions of leading twist operators, Eur. Phys. J. C 75 (2015) 363 [arXiv:1503.04670] [INSPIRE].
S. Giombi and V. Kirilin, Anomalous dimensions in CFT with weakly broken higher spin symmetry, JHEP 11 (2016) 068 [arXiv:1601.01310] [INSPIRE].
K. Sen and A. Sinha, On critical exponents without Feynman diagrams, J. Phys. A 49 (2016) 445401 [arXiv:1510.07770] [INSPIRE].
F. Gliozzi, A. Guerrieri, A.C. Petkou and C. Wen, Generalized Wilson-Fisher Critical Points from the Conformal Operator Product Expansion, Phys. Rev. Lett. 118 (2017) 061601 [arXiv:1611.10344] [INSPIRE].
R. Guida and J. Zinn-Justin, Critical exponents of the N vector model, J. Phys. A 31 (1998) 8103 [cond-mat/9803240] [INSPIRE].
J. Zinn-Justin, Precise determination of critical exponents and equation of state by field theory methods, Phys. Rept. 344 (2001) 159 [hep-th/0002136] [INSPIRE].
E. Brézin, J.C. Le Guillou and J. Zinn-Justin, Perturbation Theory at Large Order. 1. The \( \phi \) 2N Interaction, Phys. Rev. D 15 (1977) 1544 [INSPIRE].
S. Kehrein, F. Wegner and Y. Pismak, Conformal symmetry and the spectrum of anomalous dimensions in the N vector model in four epsilon dimensions, Nucl. Phys. B 402 (1993) 669 [INSPIRE].
V.M. Braun, G.P. Korchemsky and D. Mueller, The Uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys. 51 (2003) 311 [hep-ph/0306057] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Unitarity violation at the Wilson-Fisher fixed point in 4- \( \epsilon \) dimensions, Phys. Rev. D 93 (2016) 125025 [arXiv:1512.00013] [INSPIRE].
Yu. M. Makeenko, Conformal operators in quantum chromodynamics, Sov. J. Nucl. Phys. 33 (1981) 440 [INSPIRE].
V.M. Braun, S.E. Derkachov and A.N. Manashov, Integrability of three particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998) 2020 [hep-ph/9805225] [INSPIRE].
V.M. Braun, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baryon distribution amplitudes in QCD, Nucl. Phys. B 553 (1999) 355 [hep-ph/9902375] [INSPIRE].
V.M. Braun, G.P. Korchemsky and A.N. Manashov, Evolution equation for the structure function g 2(x, Q 2), Nucl. Phys. B 603 (2001) 69 [hep-ph/0102313] [INSPIRE].
S.K. Kehrein and F. Wegner, The Structure of the spectrum of anomalous dimensions in the N vector model in (4- \( \epsilon \) )-dimensions, Nucl. Phys. B 424 (1994) 521 [hep-th/9405123] [INSPIRE].
A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].
G. Mack, All Unitary Ray Representations of the Conformal Group SU(2,2) with Positive Energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
S.E. Derkachov and A.N. Manashov, The Spectrum of the anomalous dimensions of the composite operators in \( \epsilon \) -expansion in the scalar \( \phi \) 4 field theory, Nucl. Phys. B 455 (1995) 685 [hep-th/9505110] [INSPIRE].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
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Roumpedakis, K. Leading order anomalous dimensions at the Wilson-Fisher fixed point from CFT. J. High Energ. Phys. 2017, 109 (2017). https://doi.org/10.1007/JHEP07(2017)109
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DOI: https://doi.org/10.1007/JHEP07(2017)109