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 $\rho$, the spinor effective action ${\Gamma }^F$ (in the minimal subtraction scheme), which gives rise to the prefactor $\exp (-{\Gamma }^F)$, is shown to have the small-$m\rho$ 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 $\gamma =\mathrm{0.577216...}$, $\alpha (1/2)=\mathrm{0.145873...}$, and our numerically evaluated value for the constant $C$ is $C=-\mathrm{0.382727...}$. A good agreement between this form and the numerically exact calculation is found if $(m\rho )\lesssim 0.8$.

25 pages, 2 figures

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Cited by in corpus (4)

• Small Mass Expansion of Functional Determinants on the Generalized Cone
• Nonperturbative results for the mass dependence of the QED fermion determinant
• New asymptotic techniques for the partial wave cut-off method for calculating the QED one loop effective action
• Calculation of renormalized fermion effective actions in radially symmetric non-Abelian backgrounds