Analytic Form of the QCD Instanton Determinant for Small Quark Mass

Abstract

We use a novel method to calculate analytically the QCD instanton prefactor due to a quark field carrying a small mass parameter $m$. In the SU(2) instanton background of size $ρ$, the spinor effective action $Γ^F$ (in the minimal subtraction scheme), which gives rise to the prefactor $\exp (-Γ^F)$, is shown to have the small-$m ρ$ behavior Γ^F = -\ln (m /μ) - \ln (μρ)/3 -2 α(1/2) -(m ρ)^2 \{\ln (m ρ/2) +γ+1/2\} -2 (m ρ)^4 \{-\ln ^2(m ρ)/4+\ln (m ρ) (1/2-γ+\ln 2)/2+C \} +O((m ρ)^6), where $γ=0.577216...$, $α(1/2)=0.145873...$, and our numerically evaluated value for the constant $C$ is $C=-0.382727...$. A good agreement between this form and the numerically exact calculation is found if $(m ρ) \lesssim 0.8$.