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 $M^{\mu \nu }=\pm \Sigma {i>j}^{}{\mu }_{\mathrm{ij}}\times \epsilon _{}^{\mu \nu }{}_{\kappa \lambda }{}^{}{p}_i^{\kappa }{p}_j^{\lambda }{[{\left(p_i·p_j\right)}^2-p_i^{}{}_{}{}^{2}p_j^{}{}_{}{}^2]}^{-\frac{1}{2}}$, where ${\mu }_{\mathrm{ij}}={\left(4\mathrm{}\pi \right)}^{-1\mathrm{}}(e_i{g}_j-g_i{e}_j)$ 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 $M^{\mu \nu }$ 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 $j$, $M^{\mu \nu }$ 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→2+3, there results the selection rule $s_1+s_2+s_3\ge \left|{\mu }_{23}\right|={\left(4\mathrm{}\pi \right)}^{-1\mathrm{}}|e_2{g}_3-g_2{e}_3|$ 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→2+3, if particle 1 has spin one and polarization vector $\overrightarrow{\epsilon }$ and particles 2 and 3 are spinless with ${\mu }_{23}=1\mathrm{}$, the center-of-mass angular distribution is $\overrightarrow{\epsilon }·{\overrightarrow{\epsilon }}^{*}-\overrightarrow{\epsilon }·\hat{q} {\overrightarrow{\epsilon }}^{*}·\hat{q}+i \overrightarrow{\epsilon }\times {\overrightarrow{\epsilon }}^{*}·\hat{q}$, where $\hat{q}$ is the direction of particle 2. It is found that a consistent Lorentz transformation law requires ${\mu }_{\mathrm{ij}}$ to take on integral or half-integral values, but the usual connection between spin and statistics further limits ${\mu }_{\mathrm{ij}}$ to integral values only.