Vanishing of Renormalized Charges in Field Theories with Point Interaction

Abstract

A short survey of the results obtained by applying the theory of Landau, Abrikosov, and Khalatnikov to pseudoscalar meson theory is presented. An independent deduction of the explicit expressions for the Green's functions and vertex part is obtained on the basis of simple renormalizability considerations. The relation thus obtained between ${{g}_{0}}^{2}$, ${{g}_{c}}^{2}$ and the momentum cutoff $\ensuremath{\Lambda}$ is such that the theory inevitably leads to the result that for point interaction (i.e., in the limit $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$) the renormalized charge ${g}_{c}$ must equal zero.It is shown that if two cutoffs ${\ensuremath{\Lambda}}_{p}$ and ${\ensuremath{\Lambda}}_{k}$ (corresponding to the nuclear and meson momenta) are introduced, this result can rigorously be proved for any value of ${{g}_{0}}^{2}$, provided that the limits are moved apart sufficiently rapidly when ${\ensuremath{\Lambda}}_{k}\ensuremath{\rightarrow}\ensuremath{\infty}$. In the course of the proof an estimation is made of the terms neglected in the zero approximation in the vertex part equation, these terms corresponding to diagrams with intersecting meson lines and nucleon loops.It is shown that for two different ways of carrying out the limiting process, namely, (a) for ${\ensuremath{\Lambda}}_{k}\ensuremath{\rightarrow}\ensuremath{\infty}$, and $[\mathrm{ln}(\frac{{{\ensuremath{\Lambda}}_{k}}^{2}}{{m}^{2}})]{[\mathrm{ln}(\frac{{{\ensuremath{\Lambda}}_{p}}^{2}}{{{\ensuremath{\Lambda}}_{k}}^{2}})]}^{\ensuremath{-}1}\ensuremath{\ll}1$, (b) for ${\ensuremath{\Lambda}}_{k}\ensuremath{\rightarrow}\ensuremath{\infty}$, and only ${[\mathrm{ln}(\frac{{{\ensuremath{\Lambda}}_{p}}^{2}}{{{\ensuremath{\Lambda}}_{k}}^{2}})]}^{\ensuremath{-}1}\ensuremath{\ll}1$, the contribution of these diagrams is vanishingly small for any ${{g}_{0}}^{2}$.In the second case, the contributions from an infinite set of meson-meson scattering diagrams are summed and it is found that the total contribution is of the same order as that from the simplest diagrams of this process.The theory with pseudovector coupling, which is not renormalizable when expanded into the usual perturbation theory series, is also considered. It is shown that renormalization can be carried out without expanding into a series; the renormalized charge in this case also vanishes. This result has been rigorously obtained only for a special type of limiting process ${\ensuremath{\Lambda}}_{k}\ensuremath{\rightarrow}\ensuremath{\infty}$, namely if the inequality (c)$\frac{{{\ensuremath{\Lambda}}_{k}}^{2}}{{m}^{2}}{[\mathrm{ln}(\frac{{{\ensuremath{\Lambda}}_{p}}^{2}}{{{\ensuremath{\Lambda}}_{k}}^{2}})]}^{\ensuremath{-}1}\ensuremath{\ll}1$ is obeyed.In conclusion, a short discussion on the possibility of an experimental proof of the inconsistency of field theory is presented.