A circular wire loop of radius $R$, is placed in an $X-Y$ plane, Centre at the origin $O$. A square loop of side $a \left(a<<R\right)$, having two turns, is placed with its center at $z=\sqrt{3}R$, along the axis of circular wire loop, as shown in the figure. The plane of the square loop makes an angle of ${45}^{\circ }$ with respect to the $z$-axis. If the mutual inductance between the loops is given by $\frac{{\mu }_0{a}^2}{2^{p/2}R}$, then the value of $p$ is