Abstract
We use renormalization group methods to study composite operators existing at a boundary of an interacting conformal field theory. In particular we relate the data on boundary operators to short-distance (near-boundary) divergences of bulk two-point functions. We further argue that in the presence of running couplings at the boundary the anomalous dimensions of certain composite operators can be computed from the relevant beta functions and remark on the implications for the boundary (pseudo) stress-energy tensor. We apply the formalism to a scalar field theory in d = 3−𝜖 dimensions with a quartic coupling at the boundary whose beta function we determine to the first non-trivial order. We study the operators in this theory and compute their conformal data using 𝜖 −expansion at the Wilson-Fisher fixed point of the boundary renormalization group flow. We find that the model possesses a non-zero boundary stress-energy tensor and displacement operator both with vanishing anomalous dimensions. The boundary stress tensor decouples at the fixed point in accordance with Cardy’s condition for conformal invariance. We end the main part of the paper by discussing the possible physical significance of this fixed point for various values of 𝜖.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.R. Coleman, D.J. Gross and R. Jackiw, Fermion avatars of the sugawara model, Phys. Rev. 180 (1969) 1359 [INSPIRE].
K.G. Wilson, On products of quantum field operators at short distances, LNS-64-15 (1965).
K.G. Wilson and W. Zimmermann, Operator product expansions and composite field operators in the general framework of quantum field theory, Commun. Math. Phys. 24 (1972) 87 [INSPIRE].
K.G. Wilson, Operator product expansions and anomalous dimensions in the Thirring model, Phys. Rev. D 2 (1970) 1473 [INSPIRE].
G. Martinelli and C.T. Sachrajda, A lattice calculation of the pion’s form-factor and structure function, Nucl. Phys. B 306 (1988) 865 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and resonance physics. Theoretical foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
V. Prochazka and R. Zwicky, Finiteness of two- and three-point functions and the renormalization group, Phys. Rev. D 95 (2017) 065027 [arXiv:1611.01367] [INSPIRE].
H.W. Diehl, The theory of boundary critical phenomena, Int. J. Mod. Phys. B 11 (1997) 3503 [cond-mat/9610143] [INSPIRE].
H.W. Diehl and S. Dietrich, Field-theoretical approach to static critical phenomena in semi-infinite systems, Z. Phys. B 42 (1981) 65.
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274.
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
K. Symanzik, Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1 [INSPIRE].
D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFTd , JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and interface CFTs from the conformal bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
A. Bissi, T. Hansen and A. Söderberg, Analytic bootstrap for boundary CFT, JHEP 01 (2019) 010 [arXiv:1808.08155] [INSPIRE].
A. Kaviraj and M.F. Paulos, The functional bootstrap for boundary CFT, arXiv:1812.04034 [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, An analytic approach to BCFTd , JHEP 12 (2019) 004 [arXiv:1812.09314] [INSPIRE].
C.P. Herzog, K.-W. Huang, I. Shamir and J. Virrueta, Superconformal models for graphene and boundary central charges, JHEP 09 (2018) 161 [arXiv:1807.01700] [INSPIRE].
L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu, 3d abelian gauge theories at the boundary, JHEP 05 (2019) 091 [arXiv:1902.09567] [INSPIRE].
G. Grignani and G.W. Semenoff, Defect QED: dielectric without a dielectric, monopole without a monopole, JHEP 11 (2019) 114 [arXiv:1909.03279] [INSPIRE].
A. Cappelli, G. D’Appollonio and M. Zabzine, Landau-Ginzburg description of boundary critical phenomena in two-dimensions, JHEP 04 (2004) 010 [hep-th/0312296] [INSPIRE].
D. Giuliano and P. Sodano, Effective boundary field theory for a Josephson junction chain with a weak link, Nucl. Phys. B 711 (2005) 480 [cond-mat/0501378] [INSPIRE].
D. Kutasov, M. Mariño and G.W. Moore, Some exact results on tachyon condensation in string field theory, JHEP 10 (2000) 045 [hep-th/0009148] [INSPIRE].
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514.
V. Prochazka, The conformal anomaly in bCFT from momentum space perspective, JHEP 10 (2018) 170 [arXiv:1804.01974] [INSPIRE].
H.W. Diehl and S. Dietrich, Field-theoretical approach to static critical phenomena in semi-infinite systems, Z. Phys. B 42 (1981) 65 [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, JHEP 08 (2019) 066 [arXiv:1807.02522] [INSPIRE].
L.S. Brown, Dimensional regularization of composite operators in scalar field theory, Annals Phys. 126 (1980) 135 [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
L.S. Brown and J.C. Collins, Dimensional renormalization of scalar field theory in curved space-time, Annals Phys. 130 (1980) 215 [INSPIRE].
J.C. Collins, A. Duncan and S.D. Joglekar, Trace and dilatation anomalies in gauge theories, Phys. Rev. D 16 (1977) 438 [INSPIRE].
K.A. Meissner and H. Nicolai, Effective action, conformal anomaly and the issue of quadratic divergences, Phys. Lett. B 660 (2008) 260 [arXiv:0710.2840] [INSPIRE].
H.W. Diehl and A. Ciach, Surface critical behavior in the presence of linear or cubic weak surface fields, Phys. Rev. B 44 (1991) 6642.
I. Brunner, J. Schulz and A. Tabler, Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models, JHEP 06 (2019) 046 [arXiv:1904.07258] [INSPIRE].
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].
C. Herzog, K.-W. Huang and K. Jensen, Displacement operators and constraints on boundary central charges, Phys. Rev. Lett. 120 (2018) 021601 [arXiv:1709.07431] [INSPIRE].
M.R. Gaberdiel and A. Recknagel, Conformal boundary states for free bosons and fermions, JHEP 11 (2001) 016 [hep-th/0108238] [INSPIRE].
C.G. Callan, I.R. Klebanov, A.W.W. Ludwig and J.M. Maldacena, Exact solution of a boundary conformal field theory, Nucl. Phys. B 422 (1994) 417 [hep-th/9402113] [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].
A. Karch and D. Tong, Particle-vortex duality from 3d bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
N. Seiberg, T. Senthil, C. Wang and E. Witten, A duality web in 2 + 1 dimensions and condensed matter physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].
S. Giombi and H. Khanchandani, O(N ) models with boundary interactions and their long range generalizations, arXiv:1912.08169 [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary conformal field theory and a boundary central charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
M. Billó et al., Line defects in the 3d Ising model, JHEP 07 (2013) 055 [arXiv:1304.4110] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
P. Liendo, Y. Linke and V. Schomerus, A Lorentzian inversion formula for defect CFT, arXiv:1903.05222 [INSPIRE].
A. Söderberg, Anomalous dimensions in the WF O(N ) model with a monodromy line defect, JHEP 03 (2018) 058 [arXiv:1706.02414] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
M. Billò, V. Gon¸calves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
T. Huber and D. Maître, HypExp: a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [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: 1912.07505
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
Procházka, V., Söderberg, A. Composite operators near the boundary. J. High Energ. Phys. 2020, 114 (2020). https://doi.org/10.1007/JHEP03(2020)114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2020)114