Boson-boson mixtures and triplet correlations

The hypernetted-chain formalism for boson-boson mixtures described by an extended Jastrow correlated wave function is derived, taking into account elementary diagrams and triplet correlations. The energy of an ideal boson ${}_{}{}^3{}_{}{}^{}\mathrm{H}\mathrm{e}-_{}^4{}_{}{}^{}$He mixture is computed for low values of the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration. The zero-${}_{}{}^3{}_{}{}^{}\mathrm{He}$-concentration limit provides a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ chemical potential in good agreement with the experimental value, when a McMillan two-body correlation factor and the Lennard-Jones potential are adopted. If the Euler equations for the two-body correlation factors are solved in presence of triplet correlations, the agreement is again improved. At the experimental ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ equilibrium density, the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ chemical potential turns out to be -2.58 K, to be compared with the experimental value, -2.79 K.