Constituent transverse-momentum fluctuations and the hard-scattering expansion

We show that the cross section for large-transverse-momentum reactions $A+B\to C+X$ can be expanded in terms of a sum of incoherent hard-scattering reactions where groups of interacting constituents have small transverse momenta relative to $A$, $B$, or $C$. The effects of large-transverse momentum of the constituents cannot be represented in terms of simple convolution integrals, but are correctly incorporated in terms of a sum of subprocesses which, in physical processes, usually correspond to nonleading terms. This hard-scattering expansion yields a series in inverse powers of $\overrightarrow{P}_T^{}{}_{}{}^2$ in the case of ${\phi }^3$ field theory or the constituent-interchange model, and a series in inverse powers of $log\left(logp_T^{}{}_{}{}^2\right)$ in the case of asymptotically free field theories.