Abstract
We study operators with large charge j in the d-dimensional O(N) model with long range interactions that decrease with the distance as 1/rd+s, where s is a continuous parameter. We consider the double scaling limit of large N, large j with \( j/N=\hat{j} \) fixed, and identify the semiclassical saddle point that captures the two-point function of the large charge operators in this limit. The solution is given in terms of certain ladder conformal integrals that have recently appeared in the literature on fishnet models. We find that the scaling dimensions for general s interpolate between \( {\Delta }_j\sim \frac{\left(d-s\right)}{2}j \) at small \( \hat{j} \) and \( {\Delta }_j\sim \frac{\left(d+s\right)}{2}j \) at large \( \hat{j} \), which is a qualitatively different behavior from the one found in the short range version of the O(N) model. We also derive results for the structure constants and 4-point functions with two large charge and one or two finite charge operators. Using a description of the long range models as defects in a higher dimensional local free field theory, we also obtain the scaling dimensions in a complementary way, by mapping the problem to a cylinder in the presence of a chemical potential for the conserved charge.
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References
M.E. Fisher, S.-k. Ma and B.G. Nickel, Critical Exponents for Long-Range Interactions, Phys. Rev. Lett. 29 (1972) 917 [INSPIRE].
J. Sak, Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions, Phys. Rev. B 8 (1973) 281.
J. Sak, Low-temperature renormalization group for ferromagnets with long-range interactions, Phys. Rev. B 15 (1977) 4344.
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
S.S. Gubser, C. Jepsen, S. Parikh and B. Trundy, O(N) and O(N) and O(N), JHEP 11 (2017) 107 [arXiv:1703.04202] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, Long-range critical exponents near the short-range crossover, Phys. Rev. Lett. 118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys. A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
C. Behan, Bootstrapping the long-range Ising model in three dimensions, J. Phys. A 52 (2019) 075401 [arXiv:1810.07199] [INSPIRE].
S.S. Gubser, C.B. Jepsen, Z. Ji, B. Trundy and A. Yarom, Non-local non-linear sigma models, JHEP 09 (2019) 005 [arXiv:1906.10281] [INSPIRE].
S. Giombi and H. Khanchandani, O(N ) models with boundary interactions and their long range generalizations, JHEP 08 (2020) 010 [arXiv:1912.08169] [INSPIRE].
N. Chai, M. Goykhman and R. Sinha, Long-range vector models at large N, JHEP 09 (2021) 194 [arXiv:2107.08052] [INSPIRE].
S. Chakraborty and M. Goykhman, Critical long-range vector model in the UV, JHEP 10 (2021) 151 [arXiv:2108.10084] [INSPIRE].
N. Chai, A. Dymarsky, M. Goykhman, R. Sinha and M. Smolkin, A model of persistent breaking of continuous symmetry, SciPost Phys. 12 (2022) 181 [arXiv:2111.02474] [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
L. Alvarez-Gaume, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
G. Badel, G. Cuomo, A. Monin and R. Rattazzi, The Epsilon Expansion Meets Semiclassics, JHEP 11 (2019) 110 [arXiv:1909.01269] [INSPIRE].
L. Alvarez-Gaume, D. Orlando and S. Reffert, Large charge at large N, JHEP 12 (2019) 142 [arXiv:1909.02571] [INSPIRE].
G. Badel, G. Cuomo, A. Monin and R. Rattazzi, Feynman diagrams and the large charge expansion in 3 ε dimensions, Phys. Lett. B 802 (2020) 135202 [arXiv:1911.08505] [INSPIRE].
G. Cuomo, OPE meets semiclassics, Phys. Rev. D 103 (2021) 085005 [arXiv:2103.01331] [INSPIRE].
G. Cuomo, A note on the large charge expansion in 4d CFT, Phys. Lett. B 812 (2021) 136014 [arXiv:2010.00407] [INSPIRE].
O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang and C. Zhang, Charging the O(N) model, Phys. Rev. D 102 (2020) 045011 [arXiv:2003.13121] [INSPIRE].
G. Cuomo, M. Mezei and A. Raviv-Moshe, Boundary conformal field theory at large charge, JHEP 10 (2021) 143 [arXiv:2108.06579] [INSPIRE].
R. Moser, D. Orlando and S. Reffert, Convexity, large charge and the large-N phase diagram of the φ4 theory, JHEP 02 (2022) 152 [arXiv:2110.07617] [INSPIRE].
D. Orlando, S. Reffert and T. Schmidt, Following the flow for large N and large charge, Phys. Lett. B 825 (2022) 136881 [arXiv:2110.07616] [INSPIRE].
N. Dondi, I. Kalogerakis, R. Moser, D. Orlando and S. Reffert, Spinning correlators in large-charge CFTs, Nucl. Phys. B 983 (2022) 115928 [arXiv:2203.12624] [INSPIRE].
L.A. Gaumé, D. Orlando and S. Reffert, Selected topics in the large quantum number expansion, Phys. Rept. 933 (2021) 1 [arXiv:2008.03308] [INSPIRE].
S. Giombi and J. Hyman, On the large charge sector in the critical O(N) model at large N, JHEP 09 (2021) 184 [arXiv:2011.11622] [INSPIRE].
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, arXiv Mathematics e-prints (2006) math/0608640 [math/0608640].
A.P. Isaev, Multiloop Feynman integrals and conformal quantum mechanics, Nucl. Phys. B 662 (2003) 461 [hep-th/0303056] [INSPIRE].
S. Derkachov, G. Ferrando and E. Olivucci, Mirror channel eigenvectors of the d-dimensional fishnets, JHEP 12 (2021) 174 [arXiv:2108.12620] [INSPIRE].
A.V. Kotikov and S. Teber, Multi-loop techniques for massless Feynman diagram calculations, Phys. Part. Nucl. 50 (2019) 1 [arXiv:1805.05109] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, DAMTP-11-64 (2011) [arXiv:1108.6194] [INSPIRE].
F.J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12 (1969) 91 [INSPIRE].
J.M. Kosterlitz, Phase Transitions in Long-Range Ferromagnetic Chains, Phys. Rev. Lett. 37 (1976) 1577 [INSPIRE].
M. Aizenman, J.T. Chayes, L. Chayes and C.M. Newman, Discontinuity of the magnetization in one-dimensional 1/|x − y|2 Ising and Potts models, J. Statist. Phys. 50 (1988) 1 [INSPIRE].
M. Aizenman and R. Fernández, Critical exponents for long-range interactions, Lett. Math. Phys. 16 (1988) 39.
M. Aizenman, H. Duminil-Copin and V. Sidoravicius, Random Currents and Continuity of Ising Model’s Spontaneous Magnetization, Commun. Math. Phys. 334 (2015) 719 [arXiv:1311.1937].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
F. Bastianelli, O. Corradini and P.A.G. Pisani, Worldline approach to quantum field theories on flat manifolds with boundaries, JHEP 02 (2007) 059 [hep-th/0612236] [INSPIRE].
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Giombi, S., Helfenberger, E. & Khanchandani, H. Long range, large charge, large N. J. High Energ. Phys. 2023, 166 (2023). https://doi.org/10.1007/JHEP01(2023)166
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DOI: https://doi.org/10.1007/JHEP01(2023)166