Angular Distributions and a Selection Rule in Charge-Pole Reactions

In the scattering of electrically and magnetically charged particles, it is found that, besides the orbital and spin angular momentum of each particle, there is a residual angular momentum in the electromagnetic field of the in or out scattering states given by ${M}^{\ensuremath{\mu}\ensuremath{\nu}}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\Sigma}{igj}^{}{\ensuremath{\mu}}_{\mathrm{ij}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\epsilon}_{}^{\ensuremath{\mu}\ensuremath{\nu}}{}_{\ensuremath{\kappa}\ensuremath{\lambda}}{}^{}{p}_{i}^{\ensuremath{\kappa}}{p}_{j}^{\ensuremath{\lambda}}{[{({p}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{p}_{j})}^{2}\ensuremath{-}p_{i}^{}{}_{}{}^{2}p_{j}^{}{}_{}{}^{2}]}^{\ensuremath{-}\frac{1}{2}}$, where ${\ensuremath{\mu}}_{\mathrm{ij}}={(4\ensuremath{\pi})}^{\ensuremath{-}1}({e}_{i}{g}_{j}\ensuremath{-}{g}_{i}{e}_{j})$ and ${p}_{i}$, ${e}_{i}$, ${g}_{i}$ are the 4-momentum and electric and magnetic charges of the ith particle. Because of the addition of this ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$ to the generator of Lorentz transformations, the scattering states do not transform like free-particle states, but the modification has a simple group-theoretical description. For each pair of particles $i$ and $j$, ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$ generates a one-dimensional representation of the little group of the pair of 4-vectors ${p}_{i}$, ${p}_{j}$. This is the subgroup of the Lorentz group which leaves both 4-vectors invariant and is isomorphic to the one-parameter group of rotations about the $z$ axis. The problem of constructing scattering amplitudes satisfying the new kinematics is solved. For two-body decay processes 1\ensuremath{\rightarrow}2+3, there results the selection rule ${s}_{1}+{s}_{2}+{s}_{3}\ensuremath{\ge}|{\ensuremath{\mu}}_{23}|={(4\ensuremath{\pi})}^{\ensuremath{-}1}|{e}_{2}{g}_{3}\ensuremath{-}{g}_{2}{e}_{3}|$ relating the spins ${s}_{i}$ of the particles to their electric and magnetic charges. Parity- and time-reversal-violating angular distributions are found. For example, in the decay 1\ensuremath{\rightarrow}2+3, if particle 1 has spin one and polarization vector $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}$ and particles 2 and 3 are spinless with ${\ensuremath{\mu}}_{23}=1$, the center-of-mass angular distribution is $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}}^{*}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{q} {\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{q}+i \stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}\ifmmode\times\else\texttimes\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{q}$, where $\stackrel{^}{q}$ is the direction of particle 2. It is found that a consistent Lorentz transformation law requires ${\ensuremath{\mu}}_{\mathrm{ij}}$ to take on integral or half-integral values, but the usual connection between spin and statistics further limits ${\ensuremath{\mu}}_{\mathrm{ij}}$ to integral values only.