Abstract
The nonlinear corrections to the longitudinal structure function is derived at low values of the Bjorken variable x by using the Laplace transforms technique. The nonlinear behavior of the longitudinal structure function is determined with respect to the Gribov–Levin–Ryskin–Mueller–Qiu and Altarelli–Martinelli equations. These results show that the nonlinear longitudinal structure function can be determined directly in terms of the parametrization of \(F_{2}(x,Q^{2})\) and the derivative of the proton structure function with respect to \(\ln {Q^{2}}\). These corrections improve the behavior of the longitudinal structure function at low values of \(Q^{2}\) in comparison with other parametrization methods.
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Notes
The standard parametrization of the gluon distribution function at low x introduced by
$$\begin{aligned} G(x,Q^{2})=f(Q^{2})x^{-\delta } \end{aligned}$$where the low x behavior could well be more singular. By considering the variable change \(\nu {\equiv }\ln (1/x)\), one can rewrite the gluon distribution in s-space as
$$\begin{aligned} {\mathcal {L}}[\widehat{G}^{2}(\nu ,Q^{2});s]{=}\frac{f(Q^{2})^{2}}{(s-2\delta )},\\ {\mathcal {L}}[\widehat{G}(\nu ,Q^{2});s]^{2}{=}\frac{f(Q^{2})^{2}}{(s-\delta )^{2}}. \end{aligned}$$We observe that the function \({\mathcal {L}}[\widehat{G}^{2}(\nu ,Q^{2});s]\) is always lower than \({\mathcal {L}}[\widehat{G}(\nu ,Q^{2});s]^{2}\) for low s values in a wide range of \(Q^{2}\) values. According to this result, we use from this limited approach for solving the quadratic equation in s-space.
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The author is thankful to the Razi University for financial support of this project
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Boroun, G.R. Nonlinear corrections to the longitudinal structure function \(F_{L}\) from the parametrization of \(F_{2}\): Laplace transform approach. Eur. Phys. J. Plus 137, 371 (2022). https://doi.org/10.1140/epjp/s13360-022-02558-1
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DOI: https://doi.org/10.1140/epjp/s13360-022-02558-1