Effect of a Transverse Momentum Distribution in the Parton Model

The parton model for the inelastic lepton-nucleon scattering is generalized to include a realistic momentum distribution of the partons. In this formalism each parton is given a component of momentum which is orthogonal in a four-vector sense to the nucleon momentum. An approximation scheme is developed to take into account the effect of this orthogonal (transverse) momentum distribution of the partons. It is found to generate an additional scale-invariant contribution to the structure function $\nu W_2(\nu , Q^2)$ as well as a non-scale-invariant contribution, as is expected. The scale-invariance-breaking term is found to be a power series in $\frac{Q^2}{{\nu }^2}$ and vanishes as $\nu$ or $Q^2$ goes to infinity for a fixed $\omega =\frac{2M\nu }{Q^2}$. The effect of the transverse momentum distribution is explicitly displayed and discussed for a few momentum distribution functions. The data on inelastic $e-p$ scattering are then analyzed for any significant deviations from scale invariance, and it is concluded that there is evidence for such deviations. On the assumption that the systematic errors in the data are small, we make some fits which display the dependence on $Q^2$. Finally, an attempt is made to fit the data with the formalism developed in the initial part of the paper. For this purpose use is made of a model for the partons, proposed previously by the authors to explain the data in the approximation that scale invariance is satisfied. Some comments are made on the properties of partons which are necessary to fit the data and on the present status of scale-invariance breaking.