The unintegrated gluon distribution from the GBW and BGK models

Abstract

The gluon distribution is obtained from the Golec-Biernat–W$\ddot{\textrm{u}}$sthoff (GBW) and Bartels, Golec-Biernat, and Kowalski (BGK) models at low x. We derive analytical results for the unintegrated color dipole gluon distribution function at small transverse momentum, which provides useful information to constrain the $k_{t}$-shape of the unintegrated gluon distribution in comparison with the unintegrated gluon distribution (UGD) models. The longitudinal proton structure function $F_{L}(x,Q^2)$ from the $k_{t}$ factorization scheme, using the unintegrated gluon density, is computed. We compare the predictions for the on-shell and twist-2 corrections with the HERA data and the CJ15 parameterization method for $F_{L}$. We show that this method describes very well the experimental data within the on-shell and twist-2 framework. Effects of parameters on $F_{L}$, where charm contribution is taken into account, are investigated. These results are in good agreement with the data at fixed W.

Appendix

•  $\mathbf {\textrm{ABIPSW}~ \textrm{model}:}$

  This is a x-independent model of the UGD which has been proposed by the authors in Ref. [36] as

  $$\begin{aligned} f(x,k_{t}^{2})=\frac{A}{4\pi ^{2}M^{2}}\Big {[}\frac{k_{t}^{2}}{M^{2}+k_{t}^{2}}\Big {]}, \end{aligned}$$

  (28)

  which merely coincides with the proton impact factor. Here, M is a characteristic soft scale, and A is the normalization factor.

•  $\mathbf {\textrm{IN}~ \textrm{model}:}$

  In the large- and small-$k_{t}$ regions, a UGD soft–hard model (where the soft and the hard components are defined in [35]) has been proposed by the authors in Ref. [35] as

  $$\begin{aligned} f(x,k_{t}^{2})= & {} f_{\textrm{soft}}^{(B)}(x,k_{t}^{2})\frac{k_{s}^{2}}{k_{s}^{2}+k_{t}^{2}}\nonumber \\{} & {} +f_{\mathrm {\textrm{hard}}}(x,k_{t}^{2})\frac{k_{t}^{2}}{k_{h}^{2}+k_{t}^{2}}. \end{aligned}$$

  (29)

•  $\mathbf {\textrm{HSS}~ \textrm{model}:}$

  This model [37] is used in the study of DIS structure functions and takes the form of a convolution between the BFKL gluon Green$^{,}$s function and a leading-order (LO) proton impact factor, which has been employed in the description of single-bottom quark production at LHC and to investigate the photoproduction of $J/\Psi$ and $\Upsilon$, by the following form:

  $$\begin{aligned}{} & {} f(x,k_{t}^{2},M_{{h}})\nonumber \\ {}{} & {} \quad =\int \limits _{-\infty }^{+\infty }\frac{\text {d}\nu }{2\pi ^{2}}\mathcal {C}\frac{\Gamma (\delta -i\nu -\frac{1}{2})}{\Gamma (\delta )}\left( \frac{1}{x}\right) ^{\chi (\frac{1}{2}+i\nu )} \left( \frac{k_{t}^{2}}{Q_{0}^{2}}\right) ^{\frac{1}{2}+i\nu }\nonumber \\{} & {} \qquad \left\{ 1+\frac{\overline{\alpha }_{s}^{2}\beta _{0}\chi _{0}\left( \frac{1}{2}+i\nu \right) }{8N_{c}}\log \left( \frac{1}{x}\right) \right. \nonumber \\{} & {} \qquad \left. \times \left[ -\psi (\frac{1}{2}+i\nu )-\log \frac{k_{t}^{2}}{M^{2}_{{h}}}\right\} \right] . \end{aligned}$$

  (30)

  In the above equation (i.e., Eq. 30), $\chi _{0}(\frac{1}{2}+i\nu )$ and $\chi (\gamma )$ are respectively the LO and the next-to-leading order (NLO) eigenvalues of the BFKL kernel, and $\beta _{0}=11-\frac{2}{3}n_{f}$ with $n_{f}$ the number of active quarks. Here, $\overline{\alpha }_{s}=\frac{3}{\pi }\alpha _{s}(\mu ^{2})$ with $\mu ^{2}=Q_{0}M_{h}$, where $M_{h}$ plays the role of the hard scale which can be identified with the photon virtuality, $\sqrt{Q^{2}}$.

•  $\mathbf {\textrm{WMR}~ \textrm{model}:}$

  The WMR model [38] depends on an extra-scale $\mu$, fixed at Q, by the following form:

  $$\begin{aligned} f(x,k_{t}^{2},\mu ^{2})= & {} T_{g}(k_{t}^{2},\mu ^{2})\frac{\alpha _{s}(k_{t}^{2})}{2\pi } \int \limits _{x}^{1}\text {d}z\nonumber \\{} & {} \times \left[ \sum _{q}P_{gq}(z)\frac{x}{z}q\left( \frac{x}{z},k_{t}^{2}\right) + P_{gg}(z)\right. \nonumber \\{} & {} \left. \times \frac{x}{z}g \left( \frac{x}{z},k_{t}^{2}\right) \Theta \left( \frac{\mu }{\mu +k_{t}}-z\right) \right] ,\nonumber \\ \end{aligned}$$

  (31)

  where $T_{g}(k_{t}^{2},\mu ^{2})$ gives the probability of evolving from the scale $k_{t}$ to the scale $\mu$ without parton emission, and $P_{ij}^{,}$s are the splitting functions.

•  $\mathbf {\textrm{GBW}~ \textrm{model}:}$

  This model [1,2,3,4] derives from the effective dipole cross section $\sigma (x,\textbf{r})$ for the scattering of a $q\overline{q}$ pair of a nucleon by the following form:

  $$\begin{aligned} f(x,k_{t}^{2})= & {} k_{t}^{4}\sigma _{0}\frac{R_{0}^{2}(x)}{2\pi }e^{-k_{t}^{2}R_{0}^{2}(x)}, \end{aligned}$$

  (32)

  with $R_{0}^{2}(x)=\frac{1}{\textrm{GeV}^{2}}\left( \frac{x}{x_{0}}\right) ^{\lambda }$.

A novel UGD parameterization derived on the basis of a spectator model input, which is widely employed in the context of exploratory studies on gluon TMDs, is summarized in Refs. [45] and [66]. The model is based on the assumption that a nucleon can emit a gluon, and what remains after the emission is treated as a single spectator particle.