Abstract
We study finite N aspects of the O(m) × O(N − m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown [1, 2] to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d = 4 − ϵ dimensions. The large rank limit of the biconical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N . In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, Thermal Order in Conformal Theories, Phys. Rev. D 102 (2020) 065014 [arXiv:2005.03676] [INSPIRE].
N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, Symmetry Breaking at All Temperatures, Phys. Rev. Lett. 125 (2020) 131603 [INSPIRE].
G. Parisi, On self-consistency conditions in conformal covariant field theory, Lett. Nuovo Cim. 4 (1972) 777 [INSPIRE].
A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [Sov. Phys. JETP 39 (1974) 9] [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].
S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics : New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., 1–26 June 2015, pp. 1–74 [arXiv:1602.07982] [INSPIRE].
G. Mack, Conformal Field Theory in D > 2 dimensions, representations and harmonic analysis, arXiv:1902.03812 [INSPIRE].
A. N. Vasiliev, Y. M. Pismak and Y. R. Khonkonen, Simple Method of Calculating the Critical Indices in the 1/N Expansion, Theor. Math. Phys. 46 (1981) 104 [Teor. Mat. Fiz. 46 (1981) 157] [INSPIRE].
A. N. Vasiliev, Y. M. Pismak and Y. R. Khonkonen, 1/n Expansion: Calculation of the Exponents η and ν in the Order 1/n2 for Arbitrary Number of Dimensions, Theor. Math. Phys. 47 (1981) 465 [Teor. Mat. Fiz. 47 (1981) 291] [INSPIRE].
S. Chaudhuri, C. Choi and E. Rabinovici, Thermal order in large N conformal gauge theories, arXiv:2011.13981 [INSPIRE].
W. A. Bardeen, M. Moshe and M. Bander, Spontaneous Breaking of Scale Invariance and the Ultraviolet Fixed Point in O(N)-Symmetric \( \left({\phi}_3^6\right) \) Theory, Phys. Rev. Lett. 52 (1984) 1188 [INSPIRE].
E. Rabinovici, B. Saering and W. A. Bardeen, Critical Surfaces and Flat Directions in a Finite Theory, Phys. Rev. D 36 (1987) 562 [INSPIRE].
R. D. Pisarski, Fixed point structure of ϕ6 in three-dimensions at large N, Phys. Rev. Lett. 48 (1982) 574 [INSPIRE].
M. Goykhman, V. Rosenhaus and M. Smolkin, The background field method and critical vector models, JHEP 02 (2021) 074 [arXiv:2009.13137] [INSPIRE].
L. Fei, S. Giombi and I. R. Klebanov, Critical O(N ) models in 6 − ϵ dimensions, Phys. Rev. D 90 (2014) 025018 [arXiv:1404.1094] [INSPIRE].
S. Giombi, R. Huang, I. R. Klebanov, S. S. Pufu and G. Tarnopolsky, The O(N) Model in 4 < d < 6: Instantons and complex CFTs, Phys. Rev. D 101 (2020) 045013 [arXiv:1910.02462] [INSPIRE].
N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158 (1967) 383 [INSPIRE].
S. R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [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. Elitzur, A. Giveon, M. Porrati and E. Rabinovici, Multitrace deformations of vector and adjoint theories and their holographic duals, JHEP 02 (2006) 006 [hep-th/0511061] [INSPIRE].
S. Elitzur, A. Giveon, M. Porrati and E. Rabinovici, Multitrace deformations of vector and adjoint theories and their holographic duals, Nucl. Phys. B Proc. Suppl. 171 (2007) 231 [INSPIRE].
S. Rychkov and A. Stergiou, General Properties of Multiscalar RG Flows in d = 4 − ε, SciPost Phys. 6 (2019) 008 [arXiv:1810.10541] [INSPIRE].
H. Osborn and A. Stergiou, Heavy handed quest for fixed points in multiple coupling scalar theories in the ϵ expansion, JHEP 04 (2021) 128 [arXiv:2010.15915] [INSPIRE].
M. Hogervorst and C. Toldo, Bounds on multiscalar CFTs in the ϵ expansion, JHEP 04 (2021) 068 [arXiv:2010.16222] [INSPIRE].
A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rept. 368 (2002) 549 [cond-mat/0012164] [INSPIRE].
A. Aharony, Old and New Results on Multicritical Points, J. Stat. Phys. 110 (2003) 659 [cond-mat/0201576].
M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: A Review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].
A. N. Vasiliev and M. Y. Nalimov, Analog of Dimensional Regularization for Calculation of the Renormalization Group Functions in the 1/n Expansion for Arbitrary Dimension of Space, Theor. Math. Phys. 55 (1983) 423 [Teor. Mat. Fiz. 55 (1983) 163] [INSPIRE].
S. E. Derkachov and A. N. Manashov, The Simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion, Nucl. Phys. B 522 (1998) 301 [hep-th/9710015] [INSPIRE].
J. A. Gracey, Large Nf quantum field theory, Int. J. Mod. Phys. A 33 (2019) 1830032 [arXiv:1812.05368] [INSPIRE].
M. Goykhman and M. Smolkin, Vector model in various dimensions, Phys. Rev. D 102 (2020) 025003 [arXiv:1911.08298] [INSPIRE].
S.-k. Ma, Scaling Variables and Dimensions, Phys. Rev. A 10 (1974) 1818 [INSPIRE].
K. Lang and W. Rühl, Critical nonlinear O(N) sigma models at 2 < d < 4: The Degeneracy of quasiprimary fields and it resolution, Z. Phys. C 61 (1994) 495 [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
S. Caron-Huot, Y. Gobeil and Z. Zahraee, The leading trajectory in the 2 + 1D Ising CFT, arXiv:2007.11647 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2011.06003
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chai, N., Rabinovici, E., Sinha, R. et al. The bi-conical vector model at 1/N. J. High Energ. Phys. 2021, 192 (2021). https://doi.org/10.1007/JHEP05(2021)192
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)192