INTRODUCTION

Determination of parton (quark and gluon) distribution functions (PDFs) in a proton and nuclei is a rather important task for modern high energy physics. In particular, detailed knowledge on the gluon densities is necessary for experiments planned at the Large Hadron Collider (LHC) and future colliders, such as Electron-Ion Collider (EIC), Future Circular hadron-electron Collider (FCC-he), Electron-Ion Collider in China (EicC) and Nuclotron-based Ion Collider fAcility (NICA) [1–6]. For unpolarized cases, there are a distribution of unpolarized gluons, denoted as \({{f}_{g}}(x,{{Q}^{2}})\), and a distribution of linearly polarized gluons \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\), which corresponds to interference between ±1 gluon helicity states.Footnote 1 Compared to \({{f}_{g}}(x,{{Q}^{2}})\), function \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\) is currently poorly knownFootnote 2 and depends on the gluon transverse momentum kt (so called Transverse Momentum Dependent, or TMD gluon density). A theoretical upper bound for \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\) was obtained [7, 8].

Previously, we have derived an analytical expressions for linearly polarized gluon density in a proton and investigated its behavior at low x [9]. Our analysis was based on the small-x asymptotics for sea quark and gluon densities calculated in the generalized double asymptotic scale (DAS) approach [10–14] (see also [15]) and was done with leading order (LO) accuracy. In the present note we extend the consideration [9] for nuclei.

As it is known, the study of deep inelastic scattering (DIS) of leptons on nuclei reveals the appearance of a significant nuclear effect, which excludes the naive idea of the nucleus as a system of quasi-free nucleons (see, for example, [16–20] for review). This effect was first discovered by the European Muon Collaboration in the domain of valence quark dominance, so it was called the EMC effect [21]. Currently, there are two main approaches to study the nuclear PDFs (nPDFs). In the first, which is currently more common, nPDFs are extracted from global fits (see the recent review [22] and references therein) to nuclear data using empirical parametrization of their normalizations and numerical solution of Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations [23–27] to describe the corresponding QCD evolution (dependence on scale Q2). The second strategy is based on some nPDF models (see [28, 29] for more information). Here we follow the rescaling model [30–32] based on the assumption [33, 34] that the effective size of gluon and quark confinement in the nucleus is greater than in the free nucleon. Within the perturbative QCD, it was pointed out [30–34] that this confinement rescaling predicts that nPDFs and PDFs can be connected by simply scaling the argument Q2 (see also a review [35]). Thus, one can say that the rescaling model demonstrates the features inherent in both approaches: there are certain relationships between conventional and nuclear PDFs that arise as a result of shifting the values of the kinematic variable Q2 and, at the same time, both densities obey DGLAP equations.

Initially, the rescaling model was proposed for the domain of valence quarks dominance, \(0.2 \leqslant x \leqslant 0.8\), where x is the Bjorken variable. Recently it was extended to a small x [36–38], where certain shadowing and antishadowing effectsFootnote 3 were found for the sea quark and gluon densities. Our main goal is to apply the rescaling model to linearly polarized gluon density \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\) and show its nuclear modification for small x values.

APPROACH

At LO of perturbation theory the linearly polarized gluon density in a proton \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\) at low \({{k}_{t}}\) values has the form [48] (all PDFs are multiplied by x)

$${{h}_{g}}(x,{{Q}^{2}}) \equiv \frac{{xk_{t}^{2}}}{{2{{M}^{2}}}}{\kern 1pt} h_{1}^{{ \bot g}}(x,k_{t}^{2},{{Q}^{2}})$$
$$ = \frac{{2{{a}_{s}}({{Q}^{2}})}}{{\pi {{M}^{2}}}}\int\limits_x^1 {\frac{{d{{x}_{1}}}}{{{{x}_{1}}}}\left( {1 - \frac{x}{{{{x}_{1}}}}} \right)} $$
(1)
$$ \times \;[{{C}_{A}}{{f}_{g}}({{x}_{1}},{{Q}^{2}}) + {{C}_{F}}{{f}_{q}}({{x}_{1}},{{Q}^{2}})] + ...,$$

where \({{C}_{A}} = {{N}_{c}}\), \({{C}_{F}} = (N_{c}^{2} - 1){\text{/}}(2{{N}_{c}})\) for the color \(SU({{N}_{c}})\) group. Here, \(M = 0.938\) GeV is proton mass,

$${{a}_{s}}({{Q}^{2}}) = \frac{{{{\alpha }_{s}}({{Q}^{2}})}}{{4\pi }} = \frac{1}{{{{\beta }_{0}}\ln ({{Q}^{{\text{2}}}}{\text{/}}\Lambda _{{{\text{LO}}}}^{2})}},$$
(2)

is related with the conventional strong coupling constant \({{\alpha }_{s}}({{Q}^{2}})\) and \({{\beta }_{0}} = 11 - 2f{\text{/}}3\) is the first coefficient of QCD \(\beta \)-function in the \(\overline {{\text{MS}}} \)-scheme and f is the number of active quarks.

Considering low x asymptotics for gluon density \({{f}_{g}}(x,{{Q}^{2}})\), we have [12]:

$${{f}_{g}}(x,{{Q}^{2}}) = f_{g}^{ + }(x,{{Q}^{2}}) + f_{g}^{ - }(x,{{Q}^{2}}),$$
$$f_{g}^{ + }(x,{{Q}^{2}}) = A_{g}^{ + }{{I}_{0}}(\sigma ){{e}^{{ - {{{\overline d }}_{ + }}s}}} + O(\rho ),\quad A_{g}^{ + } = {{A}_{g}} + \frac{4}{9}{\kern 1pt} {{A}_{q}},$$
$$f_{g}^{ - }(x,{{Q}^{2}}) = A_{g}^{ - }{{e}^{{ - {{d}_{ - }}s}}} + O(x),\quad A_{g}^{ - } = - \frac{4}{9}{\kern 1pt} {{A}_{q}},$$
(3)

where Ag and Aq are magnitudes of gluon and (sea) quark densities at some initial scale \(Q_{0}^{2}\), I0 is modified Bessel function. Here also

$$\begin{gathered} s = \ln \left( {\frac{{{{a}_{s}}(Q_{0}^{2})}}{{{{a}_{s}}({{Q}^{2}})}}} \right), \\ \sigma = 2\sqrt {{\text{|}}{{{\hat {d}}}_{ + }}{\text{|}}s{\text{ln}}\left( {\frac{1}{x}} \right)} ,\quad \rho = \frac{\sigma }{{2\ln (1{\text{/}}x)}}, \\ \end{gathered} $$
(4)

and

$$\begin{gathered} {{{\hat {d}}}_{ + }} = - \frac{{12}}{{{{\beta }_{0}}}}, \\ {{\overline d }_{ + }} = 1 + \frac{{20f}}{{27{{\beta }_{0}}}}, \\ {{d}_{ - }} = \frac{{4Cf}}{{3{{\beta }_{0}}}} = \frac{{16f}}{{27{{\beta }_{0}}}}, \\ \end{gathered} $$
(5)

are the singular and regular parts of the anomalous dimensions.

The linearly polarized gluon density in a proton \({{h}_{g}}(x,{{Q}^{2}})\), which is important mostly at low x region, has the form [9]

$$\begin{gathered} {{h}_{g}}(x,{{Q}^{2}}) = h_{g}^{ + }(x,{{Q}^{2}}) + h_{g}^{ - }(x,{{Q}^{2}}),\quad h_{g}^{ - }(x,{{Q}^{2}}) = 0, \\ h_{g}^{ + }(x,{{Q}^{2}}) = \frac{{2{{a}_{s}}({{Q}^{2}})}}{{\pi {{M}^{2}}}}\left\{ {{{C}_{A}}A_{g}^{ + }\left( {\frac{1}{\rho }{{I}_{1}}(\sigma ) - {{I}_{0}}(\sigma )} \right)} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}} + \;{{C}_{F}}A_{g}^{ + }\left( {{{I}_{0}}(\sigma ) - \rho {{I}_{1}}(\sigma )} \right)} \right\}{{e}^{{ - {{{\overline d }}_{ + }}s}}}, \\ \end{gathered} $$
(6)

where Ii is modified Bessel function.

RESCALING MODEL

In the rescaling model [30–32], the valence part of quark densities is changed in the case of a A nucleus at intermediate and large values of variable x \((0.2 \leqslant x \leqslant 0.8)\) as follows

$$f_{V}^{A}(x,{{Q}^{2}}) = {{f}_{V}}(x,Q_{{A,V}}^{2}),$$
(7)

where the new scale \(Q_{{A,V}}^{2}\) is related to Q2 by [36]

$$s_{V}^{A} \equiv \ln \left( {\frac{{\ln (Q_{{A,V}}^{2}{\text{/}}{{\Lambda }^{2}})}}{{\ln (Q_{0}^{2}{\text{/}}{{\Lambda }^{2}})}}} \right) = s + \ln (1 + \delta _{V}^{A}) \approx s + \delta _{V}^{A},$$
(8)

i.e., the kernel modification of the main variable s depends on the Q2-independent parameter \(\delta _{V}^{A}\) having small values (see Tables 2 and 3 in [36]).

In [36], the PDF asymptotics shown in (3) were applied to the small x region of the EMC effect, using the simple fact that the sea quark and gluons densities increase with increasing Q2. Thus, in the case of nuclei, the PDF evolution scale is less than Q2, and this can directly reproduce the shadowing effect observed with global fits. Since there are two components for each gluon density, see (3) and (6), we have two free parameters \(Q_{{A, \pm }}^{2}\), which can be determined from the analysis of experimental data for the EMC effect at low x values.

Note that it is usually convenient to study the ratio

$$R_{t}^{{AD}}(x,{{Q}^{2}}) = \frac{{t_{g}^{A}(x,{{Q}^{2}})}}{{t_{g}^{D}(x,{{Q}^{2}})}},$$
(9)

for both conventional \({{f}_{g}}(x,{{Q}^{2}})\) and linearly polarized \({{h}_{g}}(x,{{Q}^{2}})\) gluon densities (here \(t = f,h\)). Taking advantage of the fact that the nuclear effect in the deuteron is very smallFootnote 4: \(t_{g}^{D}(x,{{Q}^{2}}) \approx {{t}_{g}}(x,{{Q}^{2}})\), we can assume that

$$\begin{gathered} t_{g}^{A}(x,{{Q}^{2}}) = t_{g}^{{A, + }}(x,{{Q}^{2}}) + t_{g}^{{A, - }}(x,{{Q}^{2}}), \\ t_{g}^{{A, \pm }}(x,{{Q}^{2}}) = t_{g}^{ \pm }(x,Q_{{AD, \pm }}^{2}). \\ \end{gathered} $$
(10)

The expressions for \(f_{g}^{ \pm }(x,{{Q}^{2}})\) and \(h_{g}^{ \pm }(x,{{Q}^{2}})\) are given in (3) and (6), respectively, and the corresponding values of \(s_{ \pm }^{{AD}}\) turned out to be

$$s_{ \pm }^{{AD}} \equiv \ln \left( {\frac{{\ln (Q_{{AD, \pm }}^{2}{\text{/}}{{\Lambda }^{2}})}}{{\ln (Q_{0}^{2}{\text{/}}{{\Lambda }^{2}})}}} \right) = s + \ln (1 + \delta _{ \pm }^{{AD}}),$$
(11)

where the results for \(\delta _{ \pm }^{{AD}}\) can be found [36].

RESULTS

Our numerical results obtained for \(R_{g}^{{AD}}(x,{{Q}^{2}})\) and \(R_{h}^{{AD}}(x,{{Q}^{2}})\) are shown in Fig. 1. As it was discussed [50], our predictions for \(R_{g}^{{AD}}(x,{{Q}^{2}})\) are very close to the ones [36] obtained at \(x \leqslant {{10}^{{ - 2}}}\), since the parameters \(\delta _{ \pm }^{{AD}}\) are taken from this article. Moreover, in the low \(x\) our calculations are also close to the recent results [22] obtained by fitting experimental data. We see that predicted \(R_{h}^{{AD}}(x,{{Q}^{2}})\) are quite similar to the \(R_{g}^{{AD}}(x,{{Q}^{2}})\), but they are less affected by nuclear effects. Indeed, in both cases: in the shadowing area (\(x\, \leqslant \,0.05\)) and in the antishadowing area (\(0.05\, \leqslant \,x\, \leqslant \,0.1\)), the values of \(R_{h}^{{AD}}(x,{{Q}^{2}})\) less differ from 1 than the corresponding values of \(R_{g}^{{AD}}(x,{{Q}^{2}})\). So that, the derived expressions could be useful for subsequent phenomenological applications.

Fig. 1.
figure 1

(Color online) x dependence of \(R_{t}^{{AD}}(x,{{Q}^{2}})\) at \({{Q}^{2}} = 10{\kern 1pt} \)GeV2. The blue and yellow solid lines are for \(R_{h}^{{AD}}(x,{{Q}^{2}})\) obtained in the present paper for 7Li and 16O, respectively. The purple dotted and red dashed lines are for \(R_{g}^{{AD}}(x,{{Q}^{2}})\) obtained in [50]. The blue dash-dotted and green lines and bands are borrowed from [22].

Full size image

In our forthcoming studies, we plan to extend our analysis for \(x \geqslant 0.1\) using recently obtained PDF set [53]. It is a combination of exact analytical solutions of DGLAP equations for small and large values of x and, therefore, could be applicable over the entire range of x. We also plan to apply other modification models for nPDFs [16–20, 22, 28, 29, 54, 55]. Studies in a wide x range could be supplemented with IMParton framework [56]. In addition, we will study nuclear modifications of transverse momentum dependent PDFs (or TMD gluon and quark densities) [53, 57, 58], which are close relatives of \({{h}_{g}}(x,k_{t}^{2},{{Q}^{2}})\). Now these quantities become very popular (see [59] and references and discussions therein).