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arXiv:1710.05601 · Nuclear Physics B

Action principle for OPE

Stefan Hollands

Published Nov 24, 2017• 26 citations
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Abstract

We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.

DOI
10.1016/j.nuclphysb.2017.11.013

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We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.
Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NR-CFT, in particular the mapping between operators and states in a non-relativistic “radial quantization” Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2, d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.
Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {Δi,Cijk,bjO^}\lbrace Δ_i,C_{ijk},b^{\hat{\mathcal{O}}}_j\rbrace{Δi​,Cijk​,bjO^​}. These are the scaling dimensions of boundary operators ΔiΔ_iΔi​, the boundary Operator Product Expansion (OPE) coefficients CijkC_{ijk}Cijk​ and the Boundary Operator Expansion (BOE) coefficients bjO^b^{\hat{\mathcal{O}}}_jbjO^​ that characterize how each bulk operator O^\hat{\mathcal{O}}O^ can be expanded in terms of boundary operators Oj\mathcal{O}_jOj​.For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.
Using the nonperturbative functional renormalization group (FRG) within the Blaizot-M\'endez-Galain-Wschebor approximation, we compute the operator product expansion (OPE) coefficient c112c_{112}c112​ associated with the operators O1∼φ\mathcal{O}_1\sim\varphiO1​∼φ and O2∼φ2\mathcal{O}_2\sim\varphi^2O2​∼φ2 in the three-dimensional O(N)\mathrm{O}(N)O(N) universality class and in the Ising universality class (N=1N=1N=1) in dimensions 2≤d≤42 \leq d \leq 42≤d≤4. When available, exact results and estimates from the conformal bootstrap and Monte-Carlo simulations compare extremely well to our results, while FRG is able to provide values across the whole range of ddd and NNN considered.
A bstract Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {∆ i , C ijk , bjO^ {b}_j^{\hat{\mathcal{O}}} bjO^​ b j O ̂ }. These are the scaling dimensions of boundary operators ∆ i , the boundary Operator Product Expansion (OPE) coefficients C ijk and the Boundary Operator Expansion (BOE) coefficients bjO^ {b}_j^{\hat{\mathcal{O}}} bjO^​ b j O ̂ that characterize how each bulk operator O^ \hat{\mathcal{O}} O^ O ̂ can be expanded in terms of boundary operators Oj {\mathcal{O}}_j Oj​ O j . For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.
We explore the consequences of conformal symmetry for the operator product expansions in nonrelativistic field theories. Similar to the relativistic case, the OPE coefficients of descendants are related to that of the primary. However, unlike relativistic CFTs the 3-point function of primaries is not completely specified by conformal symmetry. Here, we show that the 3-point function between operators with nonzero particle number, where (at least) one operator has the lowest dimension allowed by unitarity, is determined up to a numerical coefficient. We also look at the structure of the family tree of primaries with zero particle number and discuss the presence of conservation laws in this sector.