Spectral Function of the Photon Propagator—Mass Spectrum and Timelike Form Factors of Particles

Abstract

We consider contributions to the spectral function of the photon propagator from the states which contain a particle (charge $e$ and mass ${m}_{J}$) of arbitrary spin $J$, and its antiparticle. We express the contributions in terms of timelike form factors ${\mathcal{F}}_{J}(a)$ (where $a=\ensuremath{-}{P}^{2}g0$, and $P$ is the momentum of the photon) with the normalization ${\mathcal{F}}_{J}(0)=(2J+1){e}^{2}$. The unitarity limit of the spectral function can be transformed into the asymptotically bounded condition ${\mathcal{F}}_{J}(a)\ensuremath{\lesssim}O(a)$. The experimental information about the anomalous magnetic moment of the muon gives a restriction on the sum of all the contributions. Using the restriction, we examine various mass spectra of charged particles and obtain simple results. For example: If there is an infinitely rising mass spectrum ${m}_{J}=f(J)$, then the asymptotic form of the mass formula must be bounded by condition ${m}_{J}gO(J)$ (case I or II) or ${m}_{J}gO({J}^{\frac{1}{(2l+1)}})$ (case III), where $l$ is a parameter in the form factors, assumed to be ${\mathcal{F}}_{J}(a)=(2J+1){e}^{2}(\frac{a}{4{{m}_{J}}^{2}\ensuremath{-}1}){|1\ensuremath{-}\frac{a}{{\ensuremath{\mu}}^{2}}|}^{\ensuremath{-}2l}$ for $J=0,1,2,\ensuremath{\cdots}$, and ${\mathcal{F}}_{J}(a)=(2J+1){e}^{2}{|1\ensuremath{-}\frac{a}{{\ensuremath{\mu}}^{2}}|}^{\ensuremath{-}2l}$ for $J=\frac{1}{2},\frac{3}{2},\ensuremath{\cdots}$. For the purpose of experimental observations of the timelike form factors and the spectral function of the photon, the colliding-beam experiments ${e}^{+}+{e}^{\ensuremath{-}} (or {\ensuremath{\mu}}^{+}+{\ensuremath{\mu}}^{\ensuremath{-}})\ensuremath{\rightarrow}\ensuremath{\lambda}+\overline{\ensuremath{\lambda}}$ (where $\ensuremath{\lambda}$ is a particle of arbitrary spin) are discussed in some detail.