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Bootstrap bound for conformal multi-flavor QCD on lattice

Yu Nakayama

TL;DR

The paper uses the conformal bootstrap to bound the mass anomalous dimension $γ_m$ of conformal multi-flavor QCD at a fixed point under lattice regularizations. By formulating four-point functions of bifundamental operators in $SU(N_f)_L \times SU(N_f)_R$-symmetric CFTs and numerically solving crossing constraints, it obtains $γ_m < 1.31$ for $N_f=8$ (from the symmetric-traceless channel) and $γ_m < 1.29$ for $N_f=16$, with a complementary bound $γ_m < 1.79$ from singlet channels. The results are closely aligned with, but slightly stronger than, previous staggered-fermion bounds, and imply that realizing larger $γ_m$ fixed points on Wilson/domain-wall lattices would require careful tuning of effective four-Fermi interactions. The findings guide lattice QCD approaches by clarifying symmetry-based limitations on conformal fixed points and signaling where more refined measurements or observables could further sharpen the bounds.

Abstract

The recent work by Iha et al shows an upper bound on mass anomalous dimension $γ_m$ of multi-flavor massless QCD at the renormalization group fixed point from the conformal bootstrap in $SU(N_F)_V$ symmetric conformal field theories under the assumption that the fixed point is realizable with the lattice regularization based on staggered fermions. We show that the almost identical but slightly stronger bound applies to the regularization based on Wilson fermions (or domain wall fermions) by studying the conformal bootstrap in $SU(N_f)_L \times SU(N_f)_R$ symmetric conformal field theories. For $N_f=8$, our bound implies $γ_m < 1.31$ to avoid dangerously irrelevant operators that are not compatible with the lattice symmetry.

Bootstrap bound for conformal multi-flavor QCD on lattice

TL;DR

The paper uses the conformal bootstrap to bound the mass anomalous dimension of conformal multi-flavor QCD at a fixed point under lattice regularizations. By formulating four-point functions of bifundamental operators in -symmetric CFTs and numerically solving crossing constraints, it obtains for (from the symmetric-traceless channel) and for , with a complementary bound from singlet channels. The results are closely aligned with, but slightly stronger than, previous staggered-fermion bounds, and imply that realizing larger fixed points on Wilson/domain-wall lattices would require careful tuning of effective four-Fermi interactions. The findings guide lattice QCD approaches by clarifying symmetry-based limitations on conformal fixed points and signaling where more refined measurements or observables could further sharpen the bounds.

Abstract

The recent work by Iha et al shows an upper bound on mass anomalous dimension of multi-flavor massless QCD at the renormalization group fixed point from the conformal bootstrap in symmetric conformal field theories under the assumption that the fixed point is realizable with the lattice regularization based on staggered fermions. We show that the almost identical but slightly stronger bound applies to the regularization based on Wilson fermions (or domain wall fermions) by studying the conformal bootstrap in symmetric conformal field theories. For , our bound implies to avoid dangerously irrelevant operators that are not compatible with the lattice symmetry.

Paper Structure

This paper contains 3 sections, 3 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bootstrap program and results
  3. Discussions

Figures (5)

  • Figure 1: Bounds on the scaling dimension of operators in the singlet representation.
  • Figure 2: Bounds on the scaling dimension of operators in the symmetric traceless $\times$ symmetric traceless representation.
  • Figure 3: Bounds on the scaling dimension of operators in the anti-symmetric $\times$ anti-symmetric representation.
  • Figure 4: Change of the bounds as we increase the search space dimension $\Lambda$
  • Figure 5: The asymptotic behavior of the bound on the scaling dimensions of the scalar operators in the symmetric traceless $\times$ symmetric traceless representation in $SU(10000)_L \times SU(10000)_R$ symmetric conformal field theories as a function of $\Lambda^{-1}$.