A proposal to avoid the ambiguity in the identification of the scale energy parameter $\tilde{Q}^{2}$ of the PDFs in the $Z$-production in $eP$-dis
We discuss $Z$ -production in the deep inelastic process $e + P \to e + Z + X$ in the context of the Standard Model using the Parton Model. In contrast to the deep inelastic $eP$-scattering ($e + P \to e + X$), where the choice of $\tilde{Q}^{2} $ is unambiguous, in this case is not clear since the momentum transfer square, at the quark level, depends on the reaction mechanism. Our aim in this work is to show that the total cross section depends strongly on the prescription used for the scale parameter $\tilde{Q}^{2}$ and on a different option that we have taken for making the convolution of the parton distribution functions and the amplitude of the quark processes. We present results for the total cross section as a function of the total energy $\sqrt{s}$ of the system $eP$, in the range $300 \leq \sqrt{s} \leq 1300$ GeV. We find differences of up to 25\% in the rates of the total cross section for the different prescriptions that we have taken, for $\sqrt{s} \approx 1300$ GeV, which is the expected maximal total energy of the system $eP$ to be reached at the LHeC. Taking an integrated luminosity of $ab^{-1}$, such a difference (25\%) corresponds to $\approx 10^4$ produced $Z$ bosons. Finally, we propose to modify slightly the Parton Model in order to do a simple and practical prescription for the calculation without ambiguities of the $Z$-production in the deep inelastic $eP$ scattering.