An analysis of the gluon distribution with next-to-leading order splitting function in small-$x$

An approximated solution for gluon distribution from DGLAP evolution equations with NLO splitting function in the small-$x$ limit is presented. We first obtain the simplified forms of LO and NLO splitting functions in the small-$x$ limit. With these approximated splitting functions, we obtain the analytical gluon distribution by using the Mellin transform. The free parameters in the boundary conditions are obtained by fitting the CJ15 gluon distribution data. We find that the asymptotic behavior of the gluon distribution is consistent with the CJ15 data; however, the NLO results with the consideration of ``ladder" structure of gluon emission are slightly better than those from LO. These results indicate that the corrections from NLO has a significant influence for the behavior of the gluon distribution in small-$x$ region. In addition, we investigate the DGLAP evolution of the proton structure function by using the analytical solution of the gluon distribution. The differential structure function shows that our results have a similar tendency with CJ15 at small-$x$. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.