Nonvariational calculation of the hyperfine splitting and other properties of the ground state of the muonic helium atom

Abstract

The properties of the ground state of the muonic Helium atom $e\ensuremath{\mu}{ }^{4}{\mathrm{He}}^{2+}$ have been calculated nonvariationally. The correlation function hyperspherical harmonic method utilizing a nonlinear parametrization of the correlation function has been used. Up to $N=561$ coupled second-order differential equations were taken into account. Although all parametrizations of the correlation function accelerate the convergence with respect to linear parametrizations by several orders of magnitude, an especially fast converging parametrization was found. All parametrizations make the observables converge to the same values in the limit of large $N$. The lowest-order hyperfine splitting obtained, 4454.206(3) MHz, has error margins smaller than the differences in the literature. One variational value is 0.023 MHz lower and another 0.013 MHz higher, after the adjustment for the different masses used. The expectation value of the distance between the electron and the muon also differs slightly from that in the literature, while the energy obtained was below the variational values.