Analytical evolution of nucleon structure functions with power corrections at twist-4 and predictions for ultrahigh energy neutrino-nucleon cross section

In this paper we present an analytic result for the evolution in $Q^2$ of the structure functions for the neutrino-nucleon interaction, valid at twist-2 in the region of small values of the Bjorken $x$ variable and for soft nonperturbative input. In the special case of flat initial conditions, we include in the calculation also the contribution of the twist-4 gluon recombination corrections, whose effect in the evolution is explicitly determined. Finally, we estimate the resulting charged-current neutrino-nucleon total cross section and discuss its behavior at ultrahigh energies.

Figure 1

Comparison at $Q^2=50{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ between $F_2^{\nu N}(x,Q^2)$ as obtained from the ZEUS PDFs [38] and from our theoretical calculation (the vertical bars represent the uncertainties coming from the error in the fitted parameters). The values of the parameters of Eqs. (63) entering our results were obtained by a fit performed in the region $2.5{\mathrm{G}\mathrm{e}\mathrm{V}}^2\le Q^2\le 20{\mathrm{G}\mathrm{e}\mathrm{V}}^2$, ${10}^{-4}\le x\le 5\times {10}^{-3}$.

Figure 2

Slope $d{F}_2^{\nu N}(x,Q^2)/d\ln Q^2$ at $Q^2=10{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ according to our results. The uncertainties on the data come from the errors in the values of the fitted parameters of Eq. (63).

Figure 3

Cross section ${\overline{\sigma }}^{\nu N}$ as a function of $s$ according to our results (filled diamonds). Data do not include the uncertainties coming from the errors on the values of the fitted parameters of Eq. (63) (see, instead, Fig. 4). For comparison, data from Ref. 18 (open circles) and from Ref. 19 (open triangles) are also shown. The isolated point at $\sqrt{s}=306.42\mathrm{G}\mathrm{e}\mathrm{V}$ (filled square) represents the HERA measurement.

Figure 4

Cross section ${\overline{\sigma }}^{\nu N}$ as a function of $s$ according to our results (filled diamonds, same as Fig. 3, but with the inclusion of error bars) and without the contribution from the recombination (open squares).