Energy levels of mesonic helium in quantum electrodynamics

On the basis of the variational method we study the energy levels of pionic helium $\left(\pi \text{−}e\text{−}\text{He}\right)$ and kaonic helium $\left(K\text{−}e\text{−}\text{He}\right)$ with an electron in the ground state and a meson in the excited state with principal and orbital quantum numbers $n\sim l+1\sim 20$. Variational wave functions are taken in the Gaussian form. Matrix elements of the basic Hamiltonian and corrections to vacuum polarization and relativism are calculated analytically in a closed form. We calculate some bound-state energies and transition frequencies which can be studied in the experiment.