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 ${\mathcal{F}}_J\mathrm{}\left(a\right)\mathrm{}$ (where $a=-P^2>0\mathrm{}$, and $P$ is the momentum of the photon) with the normalization ${\mathcal{F}}_J\mathrm{}\left(0\right)=(2J+1)\mathrm{}{e}^2$. The unitarity limit of the spectral function can be transformed into the asymptotically bounded condition ${\mathcal{F}}_J\mathrm{}\left(a\right)\mathrm{}\lesssim O\left(a\right)\mathrm{}$. 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 $m_J=f\left(J\right)\mathrm{}$, then the asymptotic form of the mass formula must be bounded by condition $m_J>O\left(J\right)\mathrm{}$ (case I or II) or $m_J>O\left(J^{\frac{1}{\mathrm{}(2l+1)\mathrm{}}}\right)$ (case III), where $l$ is a parameter in the form factors, assumed to be ${\mathcal{F}}_J\mathrm{}\left(a\right)=(2J+1)\mathrm{}{e}^2\left(\frac{a}{4{m_J}^2-1\mathrm{}}\right){|1-\frac{a}{{\mu }^2}|}^{-2l}$ for $J=0,1,2,\cdots$, and ${\mathcal{F}}_J\mathrm{}\left(a\right)=(2J+1)\mathrm{}{e}^2{|1-\frac{a}{{\mu }^2}|}^{-2l}$ for $J=\frac{1}{2},\frac{3}{2},\cdots$. For the purpose of experimental observations of the timelike form factors and the spectral function of the photon, the colliding-beam experiments $e^{+}+e^{-} (or {\mu }^{+}+{\mu }^{-})\to \lambda +\overline{\lambda }$ (where $\lambda$ is a particle of arbitrary spin) are discussed in some detail.

• Received 13 September 1968

DOI:https://doi.org/10.1103/PhysRev.177.2159