The exponent of the longitudinal structure function $F_{L}$ at low $x$

We present a set of formula to extract exponents of the longitudinal structure function and reduced cross section from the Regge-like behavior at small $x$. The exponents are found to be independent of $Q^{2}$ at NNLO analysis. As a result, we show that the reduced cross section exponents do not have the same behavior at some values of $x$. This difference predicts the non-linear effects and some evidence for shadowing and antishadowing at LHeC. Also the ratio $\frac{F_{2}}σ$ is calculated and compared with the corresponding HERA data. Our calculations show a very good agreement with the DIS experimental data throughout the small values of $x$.

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