1/N_c corrections to the magnetic susceptibility of the QCD vacuum

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: $\chi< i\psi^\dagger \psi>_0 \simeq 35-40 \mathrm{MeV}$ with the quark condensate $<i\psi^\dagger \psi>_0$. 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.