Particle Size and Contraction at High Velocity

Abstract

The question of size and contraction of size at high velocity is considered in the context of particle physics. Size is defined through a simultaneous interaction with an external potential. To second order in the external potential, one is led to consider matrix elements of the form $〈p\left|j_0\right(\mathrm{x}, x_3, 0\left)j_0\mathrm{}\right(0\left)\right|p〉$. For large $p$ such matrix elements are found to approach $p\delta \left(x_3\right)F\left(\mathrm{x}\right)$ if there are no Regge singularities at $J=1\mathrm{}$ when $t=0\mathrm{}$. If there are such singularities at $J=1\mathrm{}$ when $t=0\mathrm{}$ and if they recede below $J=1\mathrm{}$ for negative $t$, then matrix elements analogous to the one above, but for $t<\mathrm{}0\mathrm{}$, approach $\delta \left(x_3\right)$ at high velocity. $F\left(\mathrm{x}\right)$ is related to the residue of a wrong-signature fixed pole at $J=1\mathrm{}$ in a virtual Compton amplitude. $F\left(\mathrm{x}\right)$ is also shown to be equal to the second-order impact factor in the operator droplet model. These results are then generalized to an arbitrary number of interactions with the external potential. More singular interactions, where the above analysis breaks down, are considered. It is found that for a certain strength of singularity on the light cone, the particle size may shrink to zero at high velocities. In a large class of models which give the scaling law for deeply inelastic electroproduction and have a constant asymptotic total cross section for electroproduction, it is found that the particle size does not shrink. A converse statement is also found, in that a simple argument shows that if electromagnetic particle size at $t=0\mathrm{}$ shrinks, then the total asymptotic electroproduction cross section vanishes at high energy.