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DGLAP Evolutions and cross-sections of Neutrino-Nucleon interaction at Ultra High energy

We determine the UHE neutrino-nucleon cross-sections analytically both for charged and neutral current in the leading order by using the solutions of non-singlet DGLAP equation as well as singlet DGLAP equation respectively. Next we determine the cross-sections for both charged and neutral current at one loop level by carring out numerical integration of the double differential cross-section. For analytical solution, we take the standard MRST 2004 f4 LO parton distributions at $Q_0^2=1$ \,GeV$^2$ as input and for numerical solution (exact), we consider the GRV 94 parton distributions. Our analytical as well as numerically determined results are compared with the NLO results of various authors. We find our numerically determined result in sufficient agreement with the results obtained by those authors. On the other hand, our analytical result matches better with other results at lower end of the energy spectrum of the UHE neutrino rather than the higher end.