1/Nccorrections to the magnetic susceptibility of the QCD vacuum

Abstract

We investigate the magnetic susceptibility of the QCD vacuum with the $1/{N}_{c}$ corrections taken into account, based on the instanton vacuum. Starting from the instanton liquid model, we derive the gauged light-quark partition function in the presence of the current quark mass as well as of external Abelian vector and tensor fields. We consider the $1/{N}_{c}$ meson-loop corrections which are shown to contribute to the magnetic susceptibility by around 15% for the up (and down) quarks. We also take into account the tensor terms of the quark-quark interaction from the instanton vacuum as well as the finite-width effects, both of which are of order $\mathcal{O}(1/{N}_{c})$. The effects of the tensor terms and finite width turn out to be negligibly small. The final results for the up-quarks are given as: $\ensuremath{\chi}⟨i{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\psi}{⟩}_{0}\ensuremath{\simeq}35--40\text{ }\text{ }\mathrm{MeV}$ with the quark condensate $⟨i{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\psi}{⟩}_{0}$. We also discuss the pion mass dependence of the magnetic susceptibility in order to give a qualitative guideline for the chiral extrapolation of lattice data.