A Field-Theory Formulation of the Parton Model

A formulation of field theory given by Van Hove in 1955 is shown to be useful for putting "parton-model-like ideas" on firm theoretical grounds. This is done by discussing electron-proton deep-inelastic scattering, electron-positron annihilation, and the proton elastic form factor. A basic requirement that must be imposed on any field theory in order to discuss the concept of hadronic constituents is given. To put the formulation on firm ground (with respect to renormalizability), we assume that the wave-function renormalization constants of the theory are finite. This assumption satisfies the above-mentioned basic requirements, though it probably is not necessary. Within this framework we prove that $\nu W_2(q^2, \nu )$ is the same as that derived in the parton model. In this formalism, electron-positron annihilation is quite different in character from deep-inelastic electron-proton scattering. The constancy of the ratio $\frac{\sigma (e^{+}{e}^{-}\to \mathrm{hadrons})}{\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})}$ can be derived only if much stronger assumptions than the one mentioned above are adopted. The price paid for simplicity in the renormalization procedure is two physically undesirable results: (a) The proton elastic form factor is probably finite at large momentum transfer, and (b) $\nu W_2$ is finite and nonzero at $x=1\mathrm{}$. It is conjectured that these problems can be solved without changing other results by allowing wave-function renormalization constants to become infinite.