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arXiv:2001.07015 · Physical review. D/Physical review. D.

Operator product expansion coefficients in the exact renormalization group formalism

Carlo Pagani, Hidenori Sonoda

Published May 11, 2020• 17 citations
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Abstract

We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $\ensuremath{\epsilon}$ expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the Lee-Yang model in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.

DOI
10.1103/physrevd.101.105007

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We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansions, for the Wilson-Fisher fixed points of the real scalar theory in d=4\ensuremath−\ensuremathϵd=4\ensuremath{-}\ensuremath{\epsilon}d=4\ensuremath−\ensuremathϵ dimensions and the Lee-Yang model in d=6\ensuremath−\ensuremathϵd=6\ensuremath{-}\ensuremath{\epsilon}d=6\ensuremath−\ensuremathϵ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.
We consider the critical relaxation of the Ising model, the so-called model A, and study its operator product expansion. Within perturbation theory, we focus on the operator product expansions of the two-point function and the response function. At the fixed point, we normalize the coefficients and the scaling variables so that the result displays universality. The role of the fluctuation-dissipation theorem is also discussed, and it is shown that it provides non-perturbative relations among the operator product expansion coefficients. Finally, the large N limit is considered.
We consider the critical relaxation of the Ising model, the so-called model A, and study its operator product expansion. Within perturbation theory, we focus on the operator product expansions of the two-point function and the response function. At the fixed point, we normalize the coefficients and the scaling variables so that the result displays universality. The role of the fluctuation-dissipation theorem is also discussed, and it is shown that it provides nonperturbative relations among the operator product expansion coefficients. Finally, the large N limit is considered.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.
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.