The Nuclear Level Shift of Proton-Antiproton Atom

Abstract

In connection with the future experiments at LEAR on high-precision pp-atoms spectroscopy, we want to discuss some physical consequences for the pp̄-system, if the result of positive and rather large shift ΔE_1S will be confirmed. There is evidence for such a level shift of the 1S-state of that kind¹. It is worth mentioning that up to now this result has not been confirmed by other experiments and that is why one has to consider the result as a preliminary one. In Ref. 1 two possible solutions were given for the shift of the 1s-level: 1) ΔE_1S = (3.04 ± 0.06) keV and 2) ΔE_1S = 2.10 keV, the width Γ_1S ≲ 300 keV in both cases. For these rather large values of the shift one has to calculate the atomic spectrum using not perturbation theory, but the exact equation, which has the form:

$$ \lambda + 2[\psi (1 - {\lambda ^{ - 1}}) + en\;\lambda ] = \frac{1} {{{a_{cS}}}} + \frac{1} {2}{\tau _{cS}}{\lambda ^2} $$

This equation connects the atomic spectrum with a scattering length a_cS and an effective radius τ_cS. Here, λ = ik is the imaginary momentum, the energy E = — λ²/2, a_cS is the nuclear-Coulomb scattering length, ψ (z) = Γ (z)/Γ(z) is a logarithmic derivative of the gamma-function. We use atomic units n̄ = m = e = 1, mis the reduced mass. Using the experimental value of the shift and the width of the ground state, one can use Eq. (1) to calculate the energies of the other