Energies and stability of four-particle mesonic molecules

The energies and the stability of the four-particle Coulomb mesonic molecules (quadrions) are investigated in their dependence on the particles masses. It is found from the dependence of the energies of two kinds of symmetric quadrions X+X+Y-Y- and X+Y+X-Y- on the particles masses that the high-symmetric positronium molecule e+e+e-e- plas the role of a standard in the theory of four-particle Coulomb systems. Its energy determines the energies of all quadrions with an error of the fourth order only in the spread of the values of the particles masses. The analytical formulas are constructed for the energies of the symmetric mesonic molecules X+X+Y-Y- and X+Y+X-Y- with arbitrary particles masses based on their calculated energy values and on the rule determining the behaviour of the energy under inversion of the particles masses ratio. Using these formulas the following energy values E (in Hartree's atomic units) for six symmetric mesonic molecules are calculated: K+K+e-e-, E = -1.15985; π+π+e-e-, E = -1.14572; μ+μ+e-e-, E = -1.14098; K+K+μ-μ-, E = -179.76; K+K+π-π-. E = -223.22; d+d+p-p-, E = -1269.3. It is shown that all symmetric quadrions X+X+Y-Y- are stable with respect to dissociation. The upper bounds of the energies of asymmetric quadrions are constructed based on the energy values of the symmetric ones. It is shown that out of all 406 quadrions consisting of electrons, muons, pions, kaons, protons, deuterons, tritons and their antiparticles, 92 asymmetric mesonic molecules are definitely stable with respect to dissociation, in addition to the 28 stable symmetric quadrions X+X+Y-Y-