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

## TL;DR

This paper derives formulas to determine the exponents of the longitudinal structure function at small $x$, revealing their independence from $Q^2$ at NNLO and predicting non-linear effects like shadowing at LHeC.

## Contribution

It introduces new formulas for extracting structure function exponents from Regge-like behavior and demonstrates their $Q^2$ independence at NNLO, with implications for non-linear QCD effects.

## Key findings

- Exponents are independent of $Q^2$ at NNLO.
- Reduced cross section exponents vary at different $x$ values.
- Good agreement with HERA data at small $x$.

## Abstract

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 $\frac{F_{2}}{\sigma}$ 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 $x$.