To compare the HSS unintegrated gluon distribution to data, we need to determine first PDFs, which will yield the total number of partons through
$$\begin{aligned} \left\langle n\left( \ln \frac{1}{x}, Q \right) \right\rangle = xg(x, Q) + x\Sigma (x, Q), \end{aligned}$$
(2)
where \(g(x, \mu _F)\) (\(\Sigma (x, Q)\)) denotes the gluon (seaquark) distribution function at the factorization scale \(\mu _F\). To this end we use the Catani-Hautmann procedure [26] for the determination of high energy resummed PDFs. At leading order, the prescription is straightforward for the gluon distribution function, which is obtained as
$$\begin{aligned} xg(x, \mu _F)&= \int _0^{\mu _F^2} d {\varvec{k}}^2 {\mathcal {F}}(x, {\varvec{k}}^2), \end{aligned}$$
(3)
where \(\mu _F\) denotes the factorization scale which we identify for the current study with the photon virtuality Q, and \({\mathcal {F}}(x, {\varvec{k}}^2)\) the unintegrated gluon distribution, subject to BFKL evolution. To obtain the seaquark distribution, we require a transverse momentum dependent splitting function [26],
$$\begin{aligned} \tilde{P}_{qg}\left( z,\frac{{\varvec{k}}^2}{{\varvec{\Delta }}^2} \right)&= \frac{\alpha _s 2n_f}{2 \pi } T_F \frac{{\varvec{\Delta }}^2}{[{\varvec{\Delta }}^2 + z(1-z) {\varvec{k}}^2]^2}\nonumber \\&\quad \times \left[ z^2 + (1-z)^2 + 4 z^2 (1-z)^2 \frac{{\varvec{k}}^2}{{\varvec{\Delta }}^2} \right] , \end{aligned}$$
(4)
where \({\varvec{k}}\) denotes the gluon momentum and \({\varvec{\Delta }}= {\varvec{q}}-z{\varvec{k}}\) with \({\varvec{q}}\) the t-channel quark transverse momentum; \(T_F=1/2\). Note that this splitting function reduces in the collinear limit \({\varvec{k}}\rightarrow 0\) to the conventional leading order DGLAP splitting function \( {P}_{qg}(z) = \frac{\alpha _s 2 n_f}{2 \pi } T_F \left[ z^2 + (1-z)^2 \right] \). The integrated seaquark distribution is then obtained as [26]
$$\begin{aligned} x\Sigma (x, Q)&\! = \!\!\int _0^\infty \! \frac{d {\varvec{\Delta }}^2}{{\varvec{\Delta }}^2} \int _0^\infty \!\! d{\varvec{k}}^2 \int _0^1 dz\Theta \left( Q^2 - \frac{{\varvec{\Delta }}^2}{1-z} \!-\! z {\varvec{k}}^2 \right) \nonumber \\&\quad \times \tilde{P}_{qg}\left( z,\frac{{\varvec{k}}^2}{{\varvec{\Delta }}^2} \right) \mathcal {F}(x, {\varvec{k}}^2). \end{aligned}$$
(5)
Note that in [27, 28] a corresponding off-shell gluon-to-gluon splitting function has been determined. Within the current setup, this would allow in principle for the determination of the gluon distribution at next-to-leading order. The use of this splitting function for the determination of the gluon distribution function at NLO has however not been worked out completely so far. Moreover, the HSS fit is based on a leading order virtual photon impact factors, which suggests the use of the leading order prescription Eq. (3) also for this study. The HSS unintegrated gluon density reads [29]
$$\begin{aligned} \mathcal {F}\left( x, {\varvec{k}}^2, Q\right)&= \frac{1}{{\varvec{k}}^2}\int \limits _{\frac{1}{2}-i\infty }^{\frac{1}{2} + i \infty } \frac{d \gamma }{2 \pi i} \; \; \hat{g}\left( x, \frac{Q^2}{Q_0^2}, \gamma \right) \, \, \left( \frac{{\varvec{k}}^2}{Q_0^2} \right) ^\gamma , \end{aligned}$$
(6)
where \(\hat{g}\) is an operator in \(\gamma \) space,
$$\begin{aligned} \hat{g}\left( x, \frac{Q^2}{Q_0^2} \gamma \right)&= \frac{\mathcal {C}\cdot \Gamma (\delta - \gamma )}{\pi \Gamma (\delta )} \; \cdot \; \left( \frac{1}{x}\right) ^{\chi \left( \gamma , Q, Q \right) } \, \nonumber \\&\quad \cdot \Bigg \{1 + \frac{\bar{\alpha }_s^2\beta _0 \chi _0 \left( \gamma \right) }{8 N_c} \log {\left( \frac{1}{x}\right) }\nonumber \\&\quad \times \Bigg [- \psi \left( \delta -\gamma \right) + \log \frac{{Q}^2}{Q_0^2} - \partial _\gamma \Bigg ]\Bigg \}\;, \end{aligned}$$
(7)
with \(\bar{\alpha }_s = \alpha _s N_c/\pi \), \(N_c\) the number of colors and \(\chi (\gamma , Q, Q)\) the next-to-leading logarithmic (NLL) BFKL kernel which includes a resummation of both collinear enhanced terms as well as a resummation of large terms proportional to the first coefficient of the QCD beta function, see Appendix A for details. Equations (3) and (5) is now used to calculate through Eq. (2) the partonic entropy Eq. (1); the result is then compared to H1 data [19]. To calculate entropy for the H1 \(Q^2\) bins, we employ the following averaging procedure,
$$\begin{aligned}&\bar{S}(x)_{Q_2^2,Q_1^2} \nonumber \\&=\ln \frac{1}{Q_2^2-Q_1^2} \int _{Q_1^2}^{Q_2^2}dQ^2\left[ xg(x,Q^2)+x\Sigma (x,Q^2)\right] . \end{aligned}$$
(8)
The results of our study are shown in Fig. 1, where we evaluate all expressions for \(n_f=4\) flavors.
We find that the partonic entropy obtained from the total number of partons gives a very good description of H1 data [19] in case of the HSS fit. As anticipated in [1], the purely gluonic contribution is clearly dominant and amounts to approximately 80% of the total contribution; nevertheless the seaquark contribution is needed for an accurate description of H1 data. Given the approximations taken in the derivation of Eq. (2) as well as the possibility that sub-leading corrections are relevant for the determination of hadronic entropy form the multiplicity distribution, we believe that the above result provides an impressive confirmation of Eq. (2) and the results of [1] in general.
In [19] the data shown in Fig. 1 have been compared to Eqs. (1) and (2). Based on the original proposal of [1], only the gluon PDF has been used, for which the LO gluon distribution of the HERAPDF 2.0 set [30] has been chosen. While the use of a LO gluon PDF is somehow natural, since Eq. (1) does at the moment clearly not address questions related to collinear factorization at NLO and beyond, it is well known that the convergence of the gluon distribution is rather poor in the low x region; differences between the LO and NLO gluon amount up to 100% in the low x region, see e.g. Fig. 26 of [30]. While there are still noticeable differences between NLO and NNLO gluon distribution (of the order of 30% at \(x=10^{-4}\)), one can nevertheless argue that the gluon distribution starts to converge beyond leading order and the values provide by the NLO gluon might be taken as a more realistic reflection of the true gluon distribution. To substantiate this point, we show in Fig. 1 also results based on an evaluation of Eqs. (1) and (2) with NNPDF collinear PDFs at NNLO [31]. We further show results obtained using NNLO NNPDF with next-to-leading logarithmic (NLL) low x resummation [32]. In both cases we assume \(\mu _F=Q\). While both PDF sets allow for an approximate description of data and may therefore serve as an additional confirmation of the correctness of Eq. (2), a satisfactory description of the x-dependence is only possible using the low x resummed NNPDF PDF set, which provides a very good description of the shape, with a slight off-set in normalization.
A different description of these data has been provided in [33] which uses the sea quark distribution only. The authors use however for their LO BFKL description the quark-to-gluon splitting function instead of the required gluon-to-quark splitting. The former is enhanced in the low x limit and yields an incorrect sea quark distribution which is presumably of the order of the gluon distribution. We also could not reproduce the description which is based on the collinear NNLO sea quark distribution.