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Proposed method for laser spectroscopy of pionic helium atoms to determine the charged-pion mass

Metastable pionic helium ($π{\rm He}^+$) is a three-body atom composed of a helium nucleus, an electron occupying the $1s$ ground state, and a negatively charged pion $π^-$ in a Rydberg state with principal- and orbital angular momentum quantum numbers of $n\sim \ell+1\sim 16$. We calculate the spin-independent energies of the $π{\rm ^3He}^+$ and $π{\rm ^4He}^+$ isotopes in the region $n=15$--19. These include relativistic and quantum electrodynamics corrections of orders $R_{\infty}α^2$ and $R_{\infty}α^3$ in atomic units, where $R_{\infty}$ and $α$ denote the Rydberg and fine structure constants. The fine-structure splitting due to the coupling between the electron spin and the orbital angular momentum of the $π^-$, and the radiative and Auger decay rates of the states are also calculated. Some states $(n,\ell)=(16,15)$ and $(17,16)$ retain nanosecond-scale lifetimes against $π^-$ absorption into the helium nucleus. We propose to use laser pulses to induce $π^-$ transitions from these metastable states, to states with large ($\sim 10^{11}$ s$^{-1}$) Auger rates. The $π{\rm He}^{2+}$ ion that remains after Auger emission of the $1s$ electron undergoes Stark mixing with the $s$, $p$, and $d$ states during collisions with the helium atoms in the experimental target. This leads to immediate nuclear absorption of the $π^-$. The resonance condition between the laser beam and the atom is thus revealed as a sharp spike in the rates of neutrons, protons, deuterons, and tritons that emerge....(continued)