Finite temperature $0^{-+}$ glueball spectrum from non-susy D3 brane of Type IIB string theory

Here, we calculate the pseudo-scalar glueball mass at finite temperature from the holographic QCD in $3+1$ dimensions. The decoupled geometry of the non-supersymmetric (non-susy) D$3$ brane at the finite temperature which is the solution of the type-II supergravity is considered as the dual theory of QCD. We calculate the mass spectrum from the axion fluctuation in this gravity background using WKB approximations. Approximating the WKB equation for various orders of mass, we derive the analytical expressions of the mass spectrum of $0^{-+}$ at the finite temperature. Finally we evaluate the masses numerically as $m_{-+}=2.5$GeV and $m_{-+}^*=3.8$GeV from the complete WKB equation. The mass of a given state is found to decrease with increasing temperature and becomes zero at confinement -deconfinement transition temperature which is consistent with the idea of deconfinement and also matches with some recent lattice results. From this temperature variation, the QCD transition point is found to be about $186$MeV.