Abstract
We consider the \( \mathbb{C}{\mathbb{P}}^{\left({N}_f-1\right)} \) Non-Linear-Sigma-Model in the dimension 4 < d < 6. The critical behaviour of this model in the large Nf limit is reviewed. We propose a Higher Derivative Gauge (HDG) theory as an ultraviolet completion of the \( \mathbb{C}{\mathbb{P}}^{\left({N}_f-1\right)} \) NLSM. Tuning mass operators to zero, the HDG in the IR limit reaches to the critical \( \mathbb{C}{\mathbb{P}}^{\left({N}_f-1\right)} \). With partial tunings the HDG reaches either to the critical U(Nf)-Yukawa model or to the critical pure scalar QED (no Yukawa interactions). We renormalize the HDG in its critical dimension d = 6. We study the fixed points of the HDG in d = 6−2ϵ and we calculate the scaling dimensions of various observables finding a full agreement with the order O(1/Nf) predictions of the corresponding critical models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
R.L. Stratonovich, On a Method of Calculating Quantum Distribution Functions, Dokl. Akad. Nauk S.S.S.R.115 (1957) 1097.
J. Hubbard, Calculation of partition functions, Phys. Rev. Lett.3 (1959) 77 [INSPIRE].
C. Domb and M. Green eds., Phase Transitions and Critical Phenomena, Volume 6, Academic Press (1977).
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 [INSPIRE].
A.N. Vasiliev, Y.M. Pismak and Y.R. Khonkonen, 1/N Expansion: Calculation of the Exponents η and Nu in the Order 1/N2for Arbitrary Number of Dimensions, Theor. Math. Phys.47 (1981) 465 [INSPIRE].
A.N. Vasiliev, Y.M. Pismak and Y.R. Khonkonen, 1/N expansion: calculation of the exponent eta in the order 1/N3By the conformal bootstrap method, Theor. Math. Phys.50 (1982) 127 [INSPIRE].
G. Parisi, The Theory of Nonrenormalizable Interactions. 1. The Large N Expansion, Nucl. Phys.B 100 (1975) 368 [INSPIRE].
R. Percacci and G.P. Vacca, Are there scaling solutions in the O(N)-models for large N in d > 4?, Phys. Rev.D 90 (2014) 107702 [arXiv:1405.6622] [INSPIRE].
P. Mati, Vanishing β-function curves from the functional renormalization group, Phys. Rev.D 91 (2015) 125038 [arXiv:1501.00211] [INSPIRE].
P. Mati, Critical scaling in the large-N O(N ) model in higher dimensions and its possible connection to quantum gravity, Phys. Rev.D 94 (2016) 065025 [arXiv:1601.00450] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Three loop analysis of the critical O(N) models in 6 − 𝜖 dimensions, Phys. Rev.D 91 (2015) 045011 [arXiv:1411.1099] [INSPIRE].
J.A. Gracey, Four loop renormalization of 𝜙3theory in six dimensions, Phys. Rev.D 92 (2015) 025012 [arXiv:1506.03357] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QEDd, F -Theorem and the E Expansion, J. Phys.A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
J.A. Gracey, Six dimensional QCD at two loops, Phys. Rev.D 93 (2016) 025025 [arXiv:1512.04443] [INSPIRE].
I.F. Herbut and L. Janssen, Critical O (2) and O(3) 𝜙4theories near six dimensions, Phys. Rev.D 93 (2016) 085005 [arXiv:1510.05691] [INSPIRE].
A. Eichhorn, L. Janssen and M.M. Scherer, Critical O(N) models above four dimensions: Small-N solutions and stability, Phys. Rev.D 93 (2016) 125021 [arXiv:1604.03561] [INSPIRE].
C. Brust and K. Hinterbichler, Free □kscalar conformal field theory, JHEP02 (2017) 066 [arXiv:1607.07439] [INSPIRE].
D. Roscher and I.F. Herbut, Critical O(2) field theory near six dimensions beyond one loop, Phys. Rev.D 97 (2018) 116019 [arXiv:1805.01480] [INSPIRE].
J.A. Gracey, I.F. Herbut and D. Roscher, Tensor O(N) model near six dimensions: fixed points and conformal windows from four loops, Phys. Rev.D 98 (2018) 096014 [arXiv:1810.05721] [INSPIRE].
D.I. Kazakov, Ultraviolet fixed points in gauge and SUSY field theories in extra dimensions, JHEP03 (2003) 020 [hep-th/0209100] [INSPIRE].
E.A. Ivanov, A.V. Smilga and B.M. Zupnik, Renormalizable supersymmetric gauge theory in six dimensions, Nucl. Phys.B 726 (2005) 131 [hep-th/0505082] [INSPIRE].
G. Bossard, E. Ivanov and A. Smilga, Ultraviolet behavior of 6D supersymmetric Yang-Mills theories and harmonic superspace, JHEP12 (2015) 085 [arXiv:1509.08027] [INSPIRE].
L. Casarin and A.A. Tseytlin, One-loop β-functions in 4-derivative gauge theory in 6 dimensions, JHEP08 (2019) 159 [arXiv:1907.02501] [INSPIRE].
A.N. Vasiliev and M.Y. Nalimov, The C PN−1model: calculation of anomalous dimensions and the mixing matrices in the order 1/N , Theor. Math. Phys.56 (1983) 643 [INSPIRE].
A.N. Vasiliev, M.Y. Nalimov and Y.R. Khonkonen, 1/N expansion: calculation of anomalous dimensions and mixing matrices in the order 1/N for N × P matrix gauge invariant σ-model, Theor. Math. Phys.58 (1984) 111 [INSPIRE].
S. Hikami, Renormalization Group Functions of C PN−1Nonlinear σ-model and N Component Scalar QED Model, Prog. Theor. Phys.62 (1979) 226 [INSPIRE].
S. Benvenuti and H. Khachatryan, Easy-plane QED3’s in the large Nflimit, JHEP05 (2019) 214 [arXiv:1902.05767] [INSPIRE].
A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti and Y. Shen, The Equivalence of the top quark condensate and the elementary Higgs field, Nucl. Phys.B 365 (1991) 79 [INSPIRE].
J. Zinn-Justin, Four fermion interaction near four-dimensions, Nucl. Phys.B 367 (1991) 105 [INSPIRE].
A.N. Vasiliev, S.E. Derkachov, N.A. Kivel and A.S. Stepanenko, The 1/n expansion in the Gross-Neveu model: Conformal bootstrap calculation of the index eta in order 1/N3, Theor. Math. Phys.94 (1993) 127 [INSPIRE].
J.A. Gracey, Electron mass anomalous dimension at \( O\left(1\right./\left({N}_f^2\right) \)in quantum electrodynamics, Phys. Lett.B 317 (1993) 415 [hep-th/9309092] [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 [INSPIRE].
A.A. Vladimirov, Method for Computing Renormalization Group Functions in Dimensional Renormalization Scheme, Theor. Math. Phys.43 (1980) 417 [INSPIRE].
A.N. Vasil’ev, The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Boca Raton, U.S.A., Chapman and Hall/CRC (2004) [INSPIRE].
S. Benvenuti and H. Khachatryan, QED’s in 2+1 dimensions: complex fixed points and dualities, arXiv:1812.01544 [INSPIRE].
S. Gukov, RG Flows and Bifurcations, Nucl. Phys.B 919 (2017) 583 [arXiv:1608.06638] [INSPIRE].
Y. Nakayama and T. Ohtsuki, Five dimensional O(N )-symmetric CFTs from conformal bootstrap, Phys. Lett.B 734 (2014) 193 [arXiv:1404.5201] [INSPIRE].
S.M. Chester, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models in 4 < d < 6, Phys. Rev.D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE].
Z. Li and N. Su, Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models, JHEP04 (2017) 098 [arXiv:1607.07077] [INSPIRE].
H. Osborn and A. Stergiou, CTfor non-unitary CFTs in higher dimensions, JHEP06 (2016) 079 [arXiv:1603.07307] [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1907.11448
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Khachatryan, H. Higher Derivative Gauge theory in d = 6 and the \( \mathbb{C}{\mathbb{P}}^{\left({N}_f-1\right)} \) NLSM. J. High Energ. Phys. 2019, 144 (2019). https://doi.org/10.1007/JHEP12(2019)144
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)144