Study of the conformal hyperscaling relation through the Schwinger-Dyson equation

TL;DR

This paper investigates how mass and finite-volume effects influence the conformal hyperscaling relation in large Nf QCD using the ladder Schwinger-Dyson equation, revealing significant corrections and proposing modifications for lattice data analysis.

Contribution

It provides an analytical study of mass and finite-volume corrections to the hyperscaling relation using the ladder SD equation, offering practical modifications for lattice analyses.

Findings

Mass corrections lower the anomalous dimension estimate compared to the fixed point value.

Finite-volume effects are negligible compared to mass corrections in the hyperscaling relation.

Near the critical point, finite-size hyperscaling remains valid with small dynamical mass.

Abstract

We study corrections to the conformal hyperscaling relation in the conformal window of the large Nf QCD by using the ladder Schwinger-Dyson (SD) equation as a concrete dynamical model. From the analytical expression of the solution of the ladder SD equation, we identify the form of the leading mass correction to the hyperscaling relation. We find that the anomalous dimension, when identified through the hyperscaling relation neglecting these corrections, yields a value substantially lower than the one at the fixed point \gamma_m^* for large mass region. We further study finite-volume effects on the hyperscaling relation, based on the ladder SD equation in a finite space-time with the periodic boundary condition. We find that the finite-volume corrections on the hyperscaling relation are negligible compared with the mass correction. The anomalous dimension, when identified through the finite-size hyperscaling relation neglecting the mass corrections as is often done in the lattice analyses, yields almost the same value as that in the case of the infinite space-time neglecting the mass correction, i.e., a substantially lower value than \gamma_m^* for large mass. We also apply the finite-volume SD equation to the chiral-symmetry-breaking phase and find that when the theory is close to the critical point such that the dynamically generated mass is much smaller than the explicit breaking mass, the finite-size hyperscaling relation is still operative. We also suggest a concrete form of the modification of the finite-size hyperscaling relation by including the mass correction, which may be useful to analyze the lattice data.