We compute the slope and curvature, at vanishing four-momentum transfer squared, of the disconnected contribution of the leading order hadron vacuum polarization function, using lattice QCD. Calculations are performed with $2+1+1$ flavors of staggered fermions directly at the physical values of the quark masses and in volumes of linear extent larger than $6$~fm. The continuum limit is carried out using five different lattice spacings.