Non-Extensive Aspects of Gluon Distribution and the Implications for QCD Phenomenology

Abstract

This study presents new insights into gluon transverse momentum distributions through non-extensive statistical mechanics, addressing their implications for QCD phenomenology. The saturation physics and scaling laws present in high-energy collision data are investigated as a consequence of gluon distribution modification in a high-density regime. This analysis explores how these modifications influence observables across different collision systems, such as proton–proton, proton–nucleus, and relativistic heavy-ion collisions. Both particle high- and low-transverse-momentum regions are successfully described in hadron production.

1. Introduction

At high energies, the proton can be viewed as a system with exceptionally high density of gluons, making the gluon distribution a fundamental aspect in the description of both ’hard’ and ’soft’ processes in the collisions at Large Hadron Collider (LHC) [1]. Although the traditional collinear factorization framework, based on the perturbative resummation of large logarithms of the momentum-transfer squared scale $Q^2$, provides suitable description of processes characterized by a large hard scale, this approach has limitations in several contexts. Initially, it was expected that the hard-scale dynamics of Quantum Chromodynamics (QCD) at high energies would simplify, allowing collinear factorization to generate highly accurate predictions. However, novel effects arising from cold and hot nuclear matter in TeV-scale collisions have challenged this assumption.

High-multiplicity (much higer than the event-average multiplicity) events in proton–proton (pp) collisions exhibit characteristics that are complicated to reconcile with collinear factorization, particularly at a moderate or small transverse momentum, $p_T$ below 2 GeV, of particles [2,3]. Similar challenges occur in hadron production in proton–nucleus (pA) collisions, where cold nuclear matter effects become significant and there is a need for the additional parameterization of nuclear parton distribution functions (PDFs) to fit experimental data. The most complex scenario arises in heavy-ion collisions AA where collective effects and quark–gluon plasma (QGP) thermalization are considered to lead to notable deviations from the collinear model. Furthermore, small-x (fraction of incident particle transverse momentum much below 1) processes, such as diffractive production, require alternative factorization mechanisms, even when a semi-hard momentum scale is involved. add the review/book ref. to justify, the readers to address.

Given the above mentioned limitations, a more appropriate description of the proton is expected to come from $k_T$-factorization [4,5,6,7,8] where the parton distribution depends on the transverse momentum $k_T$ of proton constituents $\varphi (x,k_T,Q^2)$. Various models have been proposed for these distributions based on different physical assumptions [6,9,10,11,12,13,14,15].

The present study investigates thermal aspects of gluon distributions, especially at high energies, and their implications for calculating observables in different systems, including electron–proton (ep) electron–nucleus (eA), pp, and AA collisions. In previous study studies [16], it was demonstrated that the transverse-momentum-dependent gluon distribution, parametrized as a power law [10], can be interpreted using non-extensive statistical mechanics, which was used to extract information about high-multiplicity processes at the LHC. It was shown that both the parameter interpreted as the temperature and the power-law index q, the latter associated with Tsallis entropy, can be parametrized in terms of partonic variables x and $Q^2$, resulting in cross-sections that exhibit scaling behavior. These scaling laws can then be employed to predict gluon distributions and other observables.

The importance of such investigations lies in the feature that parton distributions are essential for determining any observable in hadronic processes. Precise knowledge of these quantities reduces uncertainties in the particle physics Standard Model predictions and allows for a deeper understanding of QCD dynamics in different regimes. Moreover, controlling the nuclear modifications of parton distributions is crucial for heavy-ion physics and the study of QGP properties.

Experimental data for different observables, in a wide range of partonic variables ($x,k_T$), support the scaling hypothesis, where data from different collider energies $\sqrt{s}$ and hard scales $Q^2$ collapse into a single scaling line related to the variable $\tau =Q^2/Q_s^2\left(x\right)$ [10,17,18,19,20,21]. The onset of the gluon saturation phenomenon is driven by a dynamically generated transverse momentum scale, the saturation scale $Q_s$. The growth of cross-sections as a function of the interaction area and multiplicity can also follow this scaling by varying the scale $Q_s\left(x\right)$ with multiplicity [16,22,23]. Furthermore, hadron production in the $p_T$-spectra is characterized by a power law, partially attributed to the behavior of parton distributions.

In Ref. [10], we proposed a model that parameterizes the gluon distributions, incorporating scaling and a power-law parameter characterizing the high-momentum spectrum. This parameterization has been used to describe diffractive processes [13,24,25,26,27,28], $p_T$-spectra in pp [10] and heavy-ion [29,30] collisions. In Ref. [16], a connection with non-extensive statistical mechanics was established, showing that experimental data on multiplicity variation can be interpreted by analyzing partonic entropy and its dependence on the transverse overlap area of protons.

Although the thermal properties of transverse momentum hadronic spectra have been explored for long time in electron–positron (e⁺e⁻), pp, pA, AA collisions [31,32,33,34,35,36,37,38,39,40,41,42], our approach investigates the thermal-like appearance of the spectra as a consequence of transverse momentum distribution (TMD) and $k_T$-factorization, which exhibit universal behavior that can be used to generate predictions of different observables and improve our understanding of soft aspects of QCD.

One of the key objectives of using the statistical mechanics description is that the maximum entropy method can be used to obtain some information about the hadron structure. For instance, in Refs. [43,44], the pion valence quark content at a low resolution $Q^2\sim 0.1$ GeV² is determined using the maximum entropy plus Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) [45,46,47] evolution with nonlinear correction [48,49]. Moreover, entanglement entropy has been used as a tool to analyze QCD at soft scales, as has color confinement, which can be used to understand hadron production in deep inelastic scattering and small-x parton cascades [50,51,52,53].

Although new phenomenological results are presented in this paper, in addition we also briefly review the topic and suggest feasible future research directions. Section 2 discusses how traditional PDFs described by DGLAP evolution at large $Q^2$ can be parametrized in a power-law form, offering insights into the non-perturbative aspects of these distributions. In Section 3, we calculate the hadronic spectrum for the production of pions and kaons as a function of the multiplicity class, demonstrating that both can be placed on a scaling line. In Section 4, we present a possible extension to nuclear cases, considering Glauber’s multiple independent interaction model, and provide results for the nuclear modification factor in eA collisions in the small-x region. Finally, Section 5 discusses heavy-ion collisions showing that an additional modification is necessary to describe the LHC data.

2. Gluon Distribution and Power-like Parametrization

The thermal interpretation of parton distributions offers a unified framework for describing the infrared regime of QCD. Traditionally, one is compelled to separate the non-perturbative part of the PDF from that evolved by perturbative dynamics. Therefore, different frameworks are used to study these regimes. However, one can make use of the thermal picture to extend the unintegrated gluon distribution to the soft part of the partonic spectra. The accuracy of this extension can only be measured by the posterior ability to make sense of the data when using it. Thus, for now, here the extension is presented as a possibility.

Collinear PDFs [54,55,56,57,58] are usually obtained as a result of the global analysis of high $Q^2$ data where distributions evolve from the initial scale $Q_0^2$ to high $Q^2$ via DGLAP evolution equations. The gluon-integrated PDF, $xG(x,Q^2)$, can be related to the TMD or unintegrated gluon distribution (UGD) in the small-x limit as

$${\displaystyle \frac{\partial }{\partial k_T^2}}xG(x,k_T)=\varphi (x,k_T).$$

(1)

The relationship (1) can be used to obtain an unintegrated counterpart of the distribution in the small-x limit. TMDs are more sensitive to partonic dynamics than their integrated counterparts. The UGDs obtained from collinear PDF have power-like tails that result from gluon splitting and have an energy-dependent behavior. Let us define the rapidity of the gluon as $Y=log(1/x)$; then, at some rapidity, $Y_0$, the gluon distribution is expected follows a behavior given by its leading-order (LO), ${\varphi }_{\mathrm{LO}}(x_0,k_T)\propto 1/k_T^2$. However, the distinct gluon splitting process in different approaches like DGLAP [45,46,47], Balitskii–Fadin–Kuraev–Lipatov (BFKL) [59,60], and Balitsky–Kovchegov (BK) [61,62] ones changes this power-like behavior in the form $\varphi (x,k_T)\propto 1/k_T^{2+2·\delta n}$, with $\delta n$ representing a power-like index depending on the scaling variable.

Building upon the foundational framework of unintegrated gluon distributions, let us now examine the thermal interpretation of the distribution $\varphi (x,k_T$) emphasizing the power-like behavior and connection to non-extensive statistical mechanics. The UGD parametrizations are better understood in the color dipole picture where the scattering probability, $P(x,k_T)$, of the dipole–proton interaction in the momentum space is related to UGD as follows:

$$\varphi (x,k_T)={\displaystyle \frac{3{\sigma }_0}{4{\pi }^2{\alpha }_s}}{k}_T^{2}P(x,k_T),$$

(2)

where ${\alpha }_s$ is the strong couping constant.

The constant factor ${\sigma }_0$ is related to the transverse area of the proton, and the relationship (2) establishes that the fundamental element in the description of proton structure is the scattering probability. As we have proposed in Ref. [10], the amplitude $P(x,k_T)$ can be written in the power-like form

$$P^{\mathrm{MPM}}(\delta n,x,k_T)={\displaystyle \frac{1+\delta n}{\pi Q_s^2\left(x\right)}}{\displaystyle \frac{1}{{\left(1+k_T^2/Q_s^2\left(x\right)\right)}^{2+\delta n}}}.$$

(3)

In Ref. [16], we argue that Equation (3) can be obtained by maximizing the Tsallis entropy [63,64],

$$S_q=\int d^2{k}_T{\displaystyle \frac{1-{\left[P(x,k_T)\right]}^q}{q-1}},$$

(4)

and interpret the Lagrange parameter as the inverse of the temperature: ${\beta }^{-1}=T$, while using the scaling hypothesis:

$${\langle k_T^2\left(x\right)\rangle }_q\sim {\beta }^{-1}{(x_s/x)}^{\lambda },$$

(5)

where one has a generalization of the Einstein relation for anomalous diffusion [65,66]. Here $x_s$ represents the value of x where the saturation scale is equal to unit and the averaging ${\langle \dots \rangle }_q$ is taken over different multiplicity configurations. Thus, one can relate the quantities $\delta n$ and $Q_s\left(x\right)$ to non-extensive quantities q and T:

$$q={\displaystyle \frac{3+\delta n}{2+\delta n}}\mathrm{and}T=Q_s^2(q-1).$$

(6)

Based on the data from deep inelastic scattering (DIS) and inclusive pion production in pp collisions, we determine the following scaling-form parametrization for $\delta n$ and $Q_s$: (see [10] for details):

$$\delta n(Q^2/Q_s{\left(x\right)}^2)=0.075{\left({\displaystyle \frac{Q^2}{Q_s^2\left(x\right)}}\right)}^{0.188}\mathrm{and}{Q}_s^2\left(x\right)={\left({\displaystyle \frac{x_s}{x}}\right)}^{1/3}.$$

(7)

Now, let us focus on the discussion of collinear PDFs and their predicted behavior with the quantities (7). In order to investigate the non-extensive aspects of those PDFs, we use MMHT2014 gluon PDF in its leading-order version (MMHT2014–LO) [54]. Then, we obtain the unintegrated version by its derivative (1) for $k_T>Q_0=2$ GeV at different values of Y, as shown in Figure 1 by open squares. At large Y, the PDF becomes less steep and is close to its leading-order value.

We then apply Equation (3) to calculate the UGD from Equation (2) to find the parameters $Q_s$ and $\delta n$ by fitting the distribution at each selected value of Y. This way, we can extract the thermal parameters T and q from the MMHT2014 distribution considering the identification using Equation (6). The fit results are presented in Figure 1 in the solid lines from high $k_T$ up to $k_T=0$, where the infrared limit is naturally regulated by the saturation of the distribution. Let us notice that different models for the partonic distribution to be characterized by their own parameters T and q, based on the theoretical assumptions that underlie each model. The resulting parameters at each value of Y are presented in

Table 1. Parameters (temperature T and index $\delta n$) associated with the power-like form applied to the collinear MMHT2014-LO gluon PDF given by Equations (2) and (3) for different values of gluon rapidity Y.

Y	T	$\mathit{\delta }\mathit{n}$
3.0	0.368	0.968
4.0	0.516	0.466
5.0	0.761	0.282
6.0	1.083	0.157
7.0	1.557	0.061
8.0	2.523	0.059
9.0	3.904	0.024
10.0	6.021	−0.006
11.0	9.307	−0.033
12.0	14.39	−0.0579
13.0	22.41	−0.0806

for x in the range $2.3\times {10}^{-6}$ to $5\times {10}^{-2}$ and $k_T<100$ GeV. Within the range $x>{10}^{-4}$, where experimental data are available, there is no substantial distinction between the extrapolation and the true values.

This procedure can give us a different interpretation of the collinear-like PDF in the infrared regime. First, from Figure 1, one observes that the peak (maximum) moves towards high $k_T$ at small-x, where a large number of soft gluons dominate the proton wavefunction, indicating a larger temperature regime. At the same time, $\delta n$ approaches the zero limit, which indicates an entropic index of $q=3/2$. For large $k_T$ and $x<{10}^{-5}$, where PDFs do not have data to be constrained, $\delta n$ becomes negative. This can be understood due to an extrapolation of the error predicted by a collinear model. Figure 1 calrifies on the role played by the saturation physics effects: it prevents the infrared growth of the collinear parton distribution at high energies. In summary, the trends observed in gluon distribution are the same as those observed in multiplicity-dependent $p_T$-spectra for hard–soft scales [67], which indicates that the modification of the gluon function is the main source of the multiplicity-dependent effects in charged-particle spectra.

Figure 2 compares the collinear PDF extrapolated model (open squares) with the MPM scaling curve (solid lines) for the temperature parameters T and the indices q and $\delta n$ as a function of $Y=ln(1/x)$. These quantities are defined in Equations (6) and (7). First, the temperature growth is approximated by a power law $T\sim x^{-\lambda }$ as a function of rapidity. However, the predicted behavior from the collinear PDF toward large rapidity is much faster compared to that of the MPM scaling, where $\lambda =0.33$. The non-extensive index q obtained from the relations (6) shows much faster growth towards its leading-order value $3/2$.

In Refs. [68,69], a thermodynamically consistent Tsallis distribution is proposed. It is argued that, in order to reproduce thermodynamic relations, a modification to the method of identifying the non-extensive parameters, as defined in Equation (6), is necessary. In this approach, the multiplicity is given by the phase-space integration that includes an extra factor of q in the power index:

$$N=g{\sigma }_0\int {\displaystyle \frac{d^2{k}_T}{{\left(2\pi \right)}^2}}{\left[1+(q-1){\displaystyle \frac{k_T^2}{T}}\right]}^{{\displaystyle \frac{-q}{q-1}}},$$

(8)

where g is a degeneracy factor and ${\sigma }_0$ is the proton area. In this context, the identification of the non-extensive parameter q is modified:

$$q={\displaystyle \frac{2+\delta n}{1+\delta n}},$$

(9)

which results in $q\approx 2$ and a higher effective temperature T. This possibility is presented in Figure 2 and denoted with ’(B)’ (dashed lines and solid squares). At this point, some comments to be made. The expressions in Equation (6) are derived considering the q-exponential Tsallis distribution, and the results presented in Figure 2 (upper) show the potential impact of choosing alternative descriptions based on non-extensive approaches.

In Figure 2 (lower), the relation between the temperature and power index $\delta n$ is presented for collinear model MMHT2014–LO and MPM parametrization. For $x<{10}^{-2}$ ($Y>4$), the parameters are close in the two models and the power $\delta n$ decreases slowly, starting from $\delta n\approx$ 0.3. However, both transverse momentum spectra shown become harder as the temperature increases. This phenomenon is consistent with that observed in the LHC data for the production of hadrons at high energies: the hadronic spectra become harder as the temperature of the final hadron system increases. The relation between the non-extensive parameter q (or $\delta n$) and temperature distinguishes the models from each other, and, actually, principle, throughout the paper. Please consider. A.: OK, Agree! this analysis can be made for a large class of gluon distributions available in the literature. Therefore, their parameters can be studied in this non-extensive statistical perspective. The relationship between the observable temperature of the final hadron spectrum and the temperature defined here for the gluon system should also be determined.

The $p_T$-spectra of different identified produced hadrons in high-energy collisions is especially sensitive to the q-parameter, and the large enough amount of data available on this observable indicates that this is the best system with which to investigate its behavior, as described in Section 3 just below.

3. Multiplicity-Dependent ${\mathit{p}}_{\mathit{T}}$-Spectra of Hadrons

Higher-multiplicity events at the LHC are often interpreted as evidence of collective behavior. In such events, characterized by a large hadron multiplicity $N_i$, where i denotes the multiplicity class, several observables are measured such as the enhancement of strangeness [70,71,72,73], the hadron average transverse momentum, $\langle p_{Th}\rangle$ [67,74], correlations [75] and the production of quarkonium [76]. The multiplicity dependence of some observables challenges current models based on QCD. While collectivity and hydrodynamic models are expected to well describe central heavy-ion collisions, the appearance of this multiplicity dependence in the collisions of small systems is questionable to be explained.

Traditional models of perturbative QCD (pQCD) cannot reveal the bulk properties of the low-$p_T$-spectra, and understanding of the soft–hard interface is fundamental. Some models include a multiparton interaction [77], where partons can have more than one collision in order to increase multiplicity. Another possibility is to consider gluon saturation dynamics, where strong gluon fields lead to nonlinear effects [78]. In order to understand these new trends presented in the multiplicity dependence data, one employs caling analysis [10,16,20,21,22,23]. In this Section, we show how the concepts considered in Section 2 to be used phenomenologically to understand the multiplicity-dependent $p_T$-spectra.

The $p_T$-spectra of produced hadrons are characterized by a power-law behavior at high transverse momentum ($p_T$ > 1 GeV), which can be traced back to the gluon TMD index n or to the non-extensive parameter. In the $k_T$-factorization for pp collisions, one can express the differential cross-section as the convolution of the UGDs from target projectile protons [4]:

$$\begin{array}{ccc}\hfill E{{\displaystyle \frac{d^3\sigma }{d{\overrightarrow{p}}^3}}}^{\mathrm{ab}\to g+\mathrm{X}}& =& {\displaystyle \frac{\mathcal{A}}{p_T^2}}\int d^2{k}_T\varphi (x_a,k_T)\varphi (x_b,q_T)\hfill \\ & \equiv & f\left(\tau \right),\hfill \end{array}$$

(10)

for a and b objects collision producing gluons (g) and identified particles (X), where $\tau =p_T^2/Q_s^2$ is a scaling variable. For the low-$p_T$ behavior, we closely follow Ref. [79], where the transverse momentum of the gluon, $k_T$, is replaced by $k_T^2\to k_T^2+m_{\mathrm{jet}}^2$, with $m_{\mathrm{jet}}\simeq 0.5$ GeV being an effective minijet mass. This procedure naturally regulates the denominator in Equation (10) due to the presence of a non-zero jet mass (see Ref. [10] for details).

There is a dependence of both the energy and the multiplicity on $\tau$, which can be parametrized as a variation in the saturation momentum at each multiplicity class and the spectra can be calculated from equation (10) with the rescaling of the saturation momentum $f\left(\tau \right)\to f\left({\tau }_i\right)$, where we define

$${\tau }_i={\displaystyle \frac{Q^2}{{\left[X_i{Q}_s\left(x\right)\right]}^2}}.$$

(11)

In the expression (11), we consider the variation in the saturation scale in each multiplicity class (i) in relation to its minimum bias value, $X_i=Q_{si}\left(x\right)/Q_s\left(x\right)$. The energy dependence is, as usual [80], determined by the x-dependent scale. However, there can also be a dependence on the geometry of the overlap area and fluctuations, as investigated in Ref. [16]. The quantity $X_i$ is obtained from high-multiplicity data at the LHC.

The differential cross-section for hadron production can be obtained by considering the (average) overlap area $\langle A_T\rangle$, which is fitted at each multiplicity class:

$$E{{\displaystyle \frac{d^3{\sigma }_i}{d{\overrightarrow{p}}^3}}}^{\mathrm{ab}\to g+\mathrm{X}}\sim {\displaystyle \frac{\langle A_T\rangle }{\langle A_{T_{\max }}\rangle }}f\left({\tau }_i\right).$$

(12)

If one assumes that the entropy is extensive with respect to the area of interaction, then the following relation for the Tsallis entropy can be obtained at relatively low multiplicities:

$$S_{3/2}\left(X_i\right)\sim \langle A_T\rangle \sim {(d{N}_i/dy)}^{1/3},$$

(13)

which is in reasonable agreement with experimental data from the ALICE experiment [67,81].

The relation (13) between the overlap area and the multiplicity can be used to calculate the multiplicity-dependent $p_T$-spectra of differently produced final-state hadrons. Figure 3 shows the scaling function ${\tau }_{i}f\left({\tau }_i\right)$ for pions and kaons at $\sqrt{s}=7$ TeV and the comparison with the ALICE experiment data [81] for several multiplicity classes. In order to translate $p_T$-spectra from gluon production to the identified hadrons, one needs to model the soft hadronization process. In the case considered in this study, a simplified local-hadron–parton duality (LHPD) [82] is used, as done in Refs. [10,16]. We introduced an effective jet mass $m_{\mathrm{jet}}$ and a momentum fraction from hadron to parton $\langle z\rangle$ which can depend on the hadronic species. However, these two quantities are not multiplicity-dependent, and all modifications to the spectra are attributed to the initial-state gluon distribution. At ${\tau }_i\sim {10}^4$, one sees a deviation from the data compatible with the expected scaling violations at high $p_T$.

In this Section, we showed how the rescaling procedure can be used to reveal geometric quantities and the multiplicity dependence of the spectrum. In Section 4, we show an alternative way of introducing modifications to the multiplicity distribution given by the Glauber multiple-scattering formalism in order to include modifications due to the nuclear medium.

4. Nuclear Effects

The phenomenology of nuclear collisions reveals a range of anomalous effects that have been observed across various experiments over recent decades. Initially, such effects were not anticipated by the parton model of QCD, and their origins continue to be the focus of intense research. It is well understood that the nuclear medium modifies the PDFs of bound nucleons, and that the dynamics of strong interactions can be probed in a distinct regime, where cross-sections are amplified due to multiple interactions with the nuclear target.

The precise control of initial cold nuclear matter effects is of fundamental importance to untangle initial/final-state nuclear effects due to collectivity in QGP phase of heavy-ion collisions. It has been shown that part of the nuclear shadowing phenomena taken into account in the gluon TMD can change final state distribution parameters in the calculation of azimuthal anisotropy $v_2$ [30] or nuclear modification factors of high-$p_T$ produced pions in pA [83] and AA collisions [29]. Nuclear modification to TMDs is considered be of a key importance in future electron–ion colliders (EICs) [84].

At a small-x, gluon PDFs are shadowed, i.e., the distributions are reduced compared to the free per-nucleon PDF. Different models have been used to describe this behavior [29,85,86,87,88,89,90,91]. The most straightforward way to incorporate nuclear modification is to consider the behavior of independent interactions between the projectile nucleon and the target with A nucleons. In this model, the probability of finding k nucleons is given by a binomial-like distribution and the probability of at least one interaction of a nucleon N with the target A is given by the Poisson limit:

$$\begin{array}{cc}\hfill P_{\mathrm{NA}}\left(b\right)=1-{\left(1-{\sigma }_{\mathrm{NN}}{\displaystyle \frac{T_{\mathrm{A}}\left(b\right)}{A}}\right)}^A& \approx 1-e^{-{\sigma }_{\mathrm{NN}}{T}_{\mathrm{A}}\left(b\right)},\hfill \end{array}$$

(14)

where the approximation ${\sigma }_{\mathrm{NN}}{T}_{\mathrm{A}}\left(b\right)/A\ll 1$ implies, with $T_{\mathrm{A}}\left(b\right)$ the the nuclear thickness function (number of nucleons) at impact parameteter b and ${\sigma }_{\mathrm{NN}}$ the nucleon-nucleon cross-section. Even for relatively small mass nuclei, at $b=0$, one can obtain $T_A\left(b\right)<1{\mathrm{fm}}^{-1}$. At high energies, one should obtain ${\sigma }_{NN}\sim {\mathrm{fm}}^2$, that is, the ratio ${\sigma }_{\mathrm{NN}}{T}_{\mathrm{A}}\left(b\right)/A$ is often smaller than unity, which justifies the approximation.

In the color dipole framework, Alfred Mueller [92] implemented Glauber’s multiple-scattering formalism [93] to model the dipole’s interaction with a nuclear target. In the target rest frame, the incoming photon dissociates into a quark–antiquark pair. When the probability of interacting with a single nucleon becomes significant, the probability of subsequent scatterings increases, and the dipole then interacts with approximately $A^{1/3}$ nucleons, exchanging two gluons with each.

The probability in momentum space is now dependent of the impact parameter and is given by the Fourier transform [94]:

$$P_{\mathrm{A}}(x,k_T,b)={\displaystyle \frac{1}{2\pi }}{\nabla }_k^2\mathcal{F}\left\{{\displaystyle \frac{1-{\tilde{P}}_{\mathrm{A}}(x,r,b)}{r^2}}\right\},$$

(15)

where r denotes the color dipole transverse size, ${\tilde{P}}_{\mathrm{A}}(x,r,b)$ is color dipole amplitude in coordinate space, and ${\nabla }_k^2$ is the Laplacian operator with respect to gluon transverse momentum $k_T$.

One of the advantages of expression (15) is that in the dilute regime $k_T\gg Q_s\left(x\right)$, it trivially approximates the free nucleon distribution and the nuclear modification factor tends toward unity. This limit is essential for describing the nuclear modification factors in pA and AA collisions in the region $p_T>Q_s$.

Another advantage of this formalism is that it is independent of the extra parameterization of the nuclear distribution, i.e., given the UGD of the proton, one can obtain the nuclear distribution without the need to include any extra dependence on the saturation scale. It should be noted that the distribution applicable to protons also holds for neutrons. For these reasons, we believe that, for the purposes of this study, this is the most appropriate way to obtain the nuclear distribution to be used in the following analyses.

The Glauber–Mueller model was calculated for the Li, Be, C, Al, Ca, Fe, Sn, Au, and Pb nuclei, which is used in the analysis of nuclear DIS in Section 5 below. In this case, no analytical form exists for the A-dependence of the distribution, with each nuclear effect calculated from the nuclear density parametrized from elastic eA collision data [95,96]. For details of the Glauber model and the nuclear overlap function calculations, see, e.g., Refs. [97,98].

The nuclear ratio between two different nuclei in DIS off nuclei is defined as follows:

$$R\left(\mathrm{A}/\mathrm{B}\right)={\displaystyle \frac{{\sigma }^{{\gamma }^{*}\mathrm{A}}(x,Q^2)}{{\sigma }^{{\gamma }^{*}\mathrm{B}}(x,Q^2)}}{\displaystyle \frac{B}{A}},$$

(16)

where ${\sigma }^{{\gamma }^{*}A}(x,Q^2)$ represents an interaction of a virtual photon with nucleus.

Figure 4 shows the ratio (16) for the nuclei Li, C, Ca and Pb to the nucleus D compared to the data in Refs. [99,100]. There is a reasonable agreement between the calculations and the data for all nuclei up to $x<0.05$ where the influence of effects related to anti-shadowing becomes significant. Furthermore, one can see that the discrepancy in this limit for light nuclei is considerably smaller than that for lead. One explanation for this is that the suppression of nuclear UGD increases with the atomic number and is therefore larger for lead. In other words, the predicted shadowing for the lead nucleus is larger than expected in the region of higher x.

For higher values of x, there is a slightly larger suppression than that one observes in the data, since the tin nucleus, like lead, has a large number of nucleons, $A=118$. This effect is better present in Figure 5, where the results of our calculations are compared with the data from Ref. [101] as a function of A. For $x=0.035$ and $A>100$, the calculations slightly underestimates the data.

In summary, the Glauber–Mueller framework effectively captures key nuclear effects such as shadowing and multiple scattering. These findings are essential for exploring how nuclear effects influence hadron production in heavy-ion collisions, as discussed in the next section.

5. Nuclear Modification Effects in Heavy-Ion Collisions

The nuclear modification factor $R_{\mathrm{AA}}$ is a key observable in high-energy heavy-ion collisions, used to quantify deviations from the expected results in independent nucleon–nucleon collisions to modifications on a per-nucleon basis. $R_{\mathrm{AA}}$ is defined as follows:

$$R_{\mathrm{AA}}\left(b\right)={\displaystyle \frac{\frac{d{N}_{\mathrm{AA}}\left(b\right)}{d^2{p}_{Th}dy}}{\int d^{2}s{T}_{\mathrm{A}}\left(s\right)T_{\mathrm{B}}(b-s)\frac{d{\sigma }_{pp}}{d^2{p}_{Th}dy}}},$$

(17)

where $d{N}_{\mathrm{AA}}/d^2{p}_{T}dy$ is the double-differential multiplicity distribution in AA collisions.

Several cold nuclear matter effects may contribute to both enhancement and suppression in $R_{\mathrm{AA}}$. (i) Modification of PDFs. Changes in nuclear PDFs are significant in shaping the final hadron spectrum. For instance, the suppression observed in the nuclear cross-section at a low $p_T$ can often be explained by nuclear shadowing. In some papers [102,103], it is suggested that most nuclear modifications can be effectively described through adjustments in the collinear PDFs. (ii) Multiple scattering effects. Since its discovery, the Cronin effect has been understood as an increase in transverse momentum due to multiple scatterings of partons in the proton with the nuclear target. This approach introduces an intrinsic transverse momentum that varies with the centrality of the collision. (iii) Energy loss in the medium. Another mechanism contributing to nuclear modification is parton energy loss. Models like those proposed in Refs. [104,105,106,107] incorporate this effect by assuming a reduction in parton energy through gluon bremsstrahlung induced by interactions with the medium.

Although nuclear modifications adequately describe the nuclear structure function observed in DIS as commented to Equation (14), this approach alone is insufficient to capture entirely the complexity of heavy-ion collisions. To estimate the pion spectrum, we employ the same parametrization as that in pp collisions from Equation (10) with nuclear modification given by Equation (14). The nuclear modification is given as in the Glauber model in Equation (17).

Figure 6 illustrates the nuclear modification factor for two distict centralities (most central and peripheral collision samples) as a function of the transverse momentum $p_{Th}$ of the charged pions produced at $\sqrt{s}=2.76$ TeV [108]. One can observe a suppression at low $p_{Th}$ followed by an increase near $Q_s\left(x\right)$ resembling the behavior of UGD. Multiple interactions produces a Cronin-like peak as a result of increasing parton transverse momentum $k_T$. In central collisions, where the hot medium modifications are expected to be of higher importance, the suppression is larger than predicted by UGD modification (dotted–dashed lines). On the other hand, in peripheral collisions, the difference becomes smaller. To overcome this limitation, see Refs. [29,30]. Let us use a modified UGD calculation as the initial particle distribution function in the Boltzmann transport equation (BTE) in the relaxation time approximation (RTA):

$$\begin{array}{c}\hfill f_{\mathrm{fin}}=f_{\mathrm{eq}}+\left(f_{\mathrm{in}}-f_{\mathrm{eq}}\right)e^{-{\displaystyle \frac{t_f}{t_r}}},\end{array}$$

(18)

A Boltzmann–Gibbs blast wave (BGBW) function is taken as an equilibrium function $f_{\mathrm{eq}}$ to obtain the final distribution $f_{\mathrm{fin}}$ to describe the particle production in AA collisions. This procedure, at the same time, correctly describes the $p_T$-spectra, the nuclear modification factors, and the elliptic flow measurements at the LHC for different centralities. In Figure 6, the result obtained using such a procedure is represented by the solid lines.

In light of non-extensive-like parametrization, one can conclude that while temperature changes indicate modifications in the soft component of the spectrum, the high-$p_T$ region retains the same power-law behavior. The nuclear modification factor $R_{\mathrm{AA}}$ remains constant and is consistent with the experimental data. This characteristic of power-law parametrization contrasts that in exponential-like UGDs, where $R_{\mathrm{AA}}$ exhibits exponential growth or suppression.

The process by which the equilibrium distribution of the full spectrum develops from its initial configuration remains challenging to the full understanding. We believe that investigations like the one presented here can help bridge the gap between the initial partonic configuration and the final hadronic spectrum emerging from QGP-like evolution.

6. Summary and Conclusions

In summary, the present study, explored the non-extensive aspects of gluon distributions and their significance in QCD phenomenology. By employing $k_T$-factorization and incorporating non-extensive statistical mechanics, we provided novel insights into the thermal behavior of gluons and the impact of these distributions in high-energy collisions. The analysis presented here highlighted how modifications to gluon distributions influence observables across different collision systems, such as pp, pA, and AA collisions, and emphasized the relevance of scaling behavior in predicting multiplicity-dependent spectra. Future research in this area can further refine the understanding of partonic entropy, improve models of nuclear interactions, and support the development of more precise predictions for hadron collider experiments. The findings contribute to the broader effort to bridge theoretical models with experimental observations, particularly in the context of QGP formation and nuclear modifications in the small-x regime.