Convergence of the Adiabatic Nuclear Potential. II

Abstract

A conjecture made in a previous paper concerning the non-convergence of the series of adiabatic nuclear potentials for meson pair theory obtained by means of perturbation methods is shown to be incorrect. The correct series is derived and summed and is in agreement with a result given previously by Wentzel. The same methods suffice for the derivation and summation of two additional series of potentials of the pseudoscalar theory with pseudoscalar coupling. One of these has as its leading term the one-pair potential of fourth order, and the other begins with the leading term of sixth order. Each series has the same radius of convergence which is determined by the condition $x{e}^x>2\mathrm{}\alpha$, where $x$ is the separation of the nucleons in units of the meson Compton wavelength and $\alpha =\left(\frac{g^2}{4\pi }\right)\left(\frac{\mu }{2M}\right)$. With $\left(\frac{g^2}{4\pi }\right)=15\mathrm{}$, perturbation theory converges for $x>\mathrm{}0.85\mathrm{}$; with $\left(\frac{g^2}{4\pi }\right)=10\mathrm{}$, for $x>\mathrm{}0.57\mathrm{}$. The convergence for $x\lesssim 1$ is in any case very slow for these values of the coupling constant. The possibility remains that for substantially smaller values of the coupling constant, as are suggested by the inclusion of radiative corrections, perturbation calculations of adiabatic potentials may yield a meaningful first approximation when used in conjunction with a suitable cut-off.