# Studies on gluon evolution and geometrical scaling in kinematic constrained unitarized BFKL equation: application to high-precision HERA DIS data

## Abstract

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution \(f(x,k_T^2)\) towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions (\(F_2\) and \(F_L\)) as well as double differential reduced cross section \((\sigma _r)\). The validity of our theory in the low \(Q^2\) transition region is established by studying virtual photon–proton cross section in light of HERA data.

## 1 Introduction

In perturbative QCD the two well-known parton evolution equations DGLAP [1, 2, 3] and BFKL [4, 5] serve as the basic tools for prediction of parton distribution functions (PDFs) on their respective kinematic domains. The DGLAP approach is valid for the large interaction scale \(Q^2\) since it sums higher order \(\alpha _s\) contributions enhanced by the leading logarithmic powers of \(\ln Q^2\). On the other hand, BFKL evolution deals with the small-*x* and semi-hard \(Q^2\) kinematic region by summing the leading logarithmic contributions of \((\alpha _s \ln 1/x)^n\). However, both the linear evolutions turn out to be inadequate to comply with the unitary bound or the conventional Froissart bound [6]. To restore the unitarity, a higher order pQCD correction is required to shadow the rapid parton growth which eventually leads to the saturation phenomena at very small-*x*. In this respect, several nonlinear equations have been proposed in recent years to entertain this shadowing correction in DGLAP and BFKL equation.

In pQCD interactions, the origin of the shadowing correction is primarily considered as the gluon recombination (\(g+g\rightarrow g\)) which is simply the inverse process of gluon splitting (\(g\rightarrow g+g\)). The modification of gluon recombination to the original DGLAP equation was first performed in GLR-MQ equation [7, 8, 9, 10] proposed by Gribov–Levin–Ryskin and Mueller–Qui. It is considered that the interferant cut diagrams of the recombination amplitude are the origin for the negative shadowing effects in the gluon recombination processes [7, 11]. The GLR-MQ equation was originally derived using AGK-cutting rules [12] to compute the contributions from these interference processes. However, GLR-MQ does not possess any antishadowing correction term since in this equation momentum conservation is violated in the gluon recombination processes. The shadowing and antishadowing effects are expected to happen in a complementary fashion which in fact ensure the momentum conservation during the process of gluon recombination [13]. In consequence of gluon recombination, a part of gluon momentum lost due to shadowing is believed to be compensated in terms of new gluons with comparatively larger *x*. This causes an enhanced density of larger momentum gluons responsible for the antishadowing effect. In this respect, to take care of momentum conservation and antishadowing effects in gluon evolution, a modified DGLAP equation was proposed [11] to replace the GLR-MQ equation. In the derivation of the modified DGLAP equation, the author has used TOPT-cutting rules based on time ordered perturbation theory (TOPT) to calculate the contribution of the gluon recombination, instead of using AGK-cutting rules unlike in the derivation of GLR-MQ equation. This equation naturally comes with two separate nonlinear terms, quadratic in gluon distribution function which are identified as the positive antishadowing and negative shadowing components in the gluon evolution. This suggests that negative shadowing effects are suppressed by antishadowing effects in the gluon evolution process.

However, above mentioned modified DGLAP approach cannot predict the evolution of unintegrated gluon distribution because DGLAP is based on collinear factorization scheme while the later deals with \(k_T\)-factorization. In this respect, a new unitarized BFKL equation was proposed by Ruan et al. [14] incorporating both shadowing and antishadowing correction. This modified BFKL (MD-BFKL) equation allows one to study antishadowing effects in unintegrated gluon distribution platform.

However, there are several other BFKL based nonlinear evolution equations that consider corrections from gluon fusion. One of the most widely studied models is Balitsky–Kovchegov (BK) equation [15, 16] which is originally derived in terms of scattering amplitude. It can be explained by the dipole model in which the nonlinear term comes from the dipole splitting while the screening effect origins from the double scattering of the dipole on the target [17]. The solution of the BK equation suggests an intense shadowing leading to the so-called saturation of the scattering amplitude showing a complete flat spectrum. However, like GLR-MQ equation the BK equation is also irrelevant to the gluon antishadowing. On the other hand, besides inclusion of antishadowing correction term, the nonlinear terms in MD-BFKL equation hold a simple form which is quadratic in unintegrated gluon distribution as well as running coupling constant with some constant coefficients. These unique features of MD-BFKL equation motivate our current studies on small-*x* physics.

*x*and \(k_T^2\) evolution followed by three-dimensional realization of gluon evolution in \(x\text {-}k_T^2\) phase space (Sect. 2.2). We have also extracted collinear gluon distribution \(xg(x,Q^2)\) from unintegrated gluon distribution \(f(x,k_T^2)\) and compared our prediction with that of global parameterization groups NNPDF 3.1sx and CT 14. In Sect. 3 we suggest a differential geometric approach towards finding the equation of the critical boundary between high and low gluon density regime. In this section, a phenomenological insight of the geometrical scaling is discussed as well.

In Sect. 4 we explore the phenomenological implication of our solution for unintegrated gluon distribution towards prediction of DIS structure function \(F_2\) and longitudinal structure function \(F_L\) at HERA. We show the comparisons of reduced cross sections (Sect. 4.1) as well as virtual photon–proton cross sections in the transition region (Sect. 4.2) with HERA H1 and ZEUS data. Finally, in Sect. 5 we summarize and outline our conclusion as well as possible future prospects.

## 2 Nonlinear effects in kinematic constraint improved MD-BFKL equation

### 2.1 Construction of kinematic constraint improved MD-BFKL

*x*(LLx) accuracy. A diagrammatic representation for probing gluon content of the proton at high photon virtuality \(Q^2\) is sketched in Fig. 1a. The BFKL kernel corresponds to the sum of gluon ladder diagram Fig. 1b generated by squaring the amplitude of Fig. 1a. The 1st term in the BFKL kernel, involving \(f(x,k_T^{'2})\) corresponds to diagrams with real gluon emission while the second term takes care of diagrams with virtual corrections. Note that the apparent singularity that is observed at \(k_T^{'2} =k_T^2\) cancels between real and virtual contributions.

*R*arises in the QCD cut diagram coupling 4 gluons to 2 gluons (see Fig. 2), the value of which depends on how exactly the gluon ladders couple to hadron. If the gluon ladder couples to two different constituents of hadron,

*R*is characterized by hadronic radius i.e. \(R\approx R_H=5 \text { GeV}^{-1}\) whereas if we consider the possibility of ladder coupling to the same parton then the appropriate choice of

*R*is the radius of valence quark i.e. \(R=2\text { GeV}^{-1}\). In the latter case \((R=2\text { GeV}^{-1})\) we have “hot spots” [8, 23].

*x*regime where the BFKL is valid, the gluon virtuality along the chain must be dominated by the transverse components of the gluon momentum,

*x*(LLx) accuracy. However, higher order corrections to the BFKL equation are already been evaluated up to NLLx accuracy, which turns out to be quite large [26]. Implementation of the constraint (8) on the evolution of BFKL based equations makes the theory more realistic in the sense that this ensures the participation of only the LLx part of higher order correction in the evolution. This, in fact, portrays the importance NLLx correction in BFKL kernel.

*x*. Moreover if we consider \(f^{(0)}\) is particularly contributed by the diagrams shown in Fig. 1c, d then \(f^{(0)}\) would be independent of

*x*(or, \(\frac{\partial f^{(0)}}{\partial x}=0\) ) at small-

*x*limit [19, 27].

*x*region is considered to have higher possibility towards exploring Regge limit of pQCD. Note that the BFKL dynamics itself is based on the concept of pomeranchuk theorem or pomeron: the Regge-pole carrying the quantum-numbers of the vacuum. However, the BFKL pomeron (also called hard pomeron) should be contrasted with non perturbative description of pomeron (soft pomeron) in the sense that they pose relatively different magnitude of intercept. For BFKL-pomeron the intercept is

*x*behavior of the parton distribution is supposed to be controlled by intercept of appropriate Regge trajectory. Regge model provides parametrizations of DIS distribution functions, \(f_i(x,Q^2)=A_i(Q^2)x^{-\lambda _i}\) [i = \(\sum \) (singlet structure function) and g (gluon distribution)], where\(\lambda _i\) is the pomeron intercept minus one \((\alpha _{P}(0)-1)\). At small-

*x*, the leading order calculations in \(\text {ln}(1/x)\) with fixed \(\alpha _s\) predicts a steep power law behavior of \(\textit{f} (x,k^2 )\sim x^{-\lambda _\text {BFKL}}\) [22, 29]. This motivates us to consider a simple form of Regge factorization as follows,

*x*and \(k_T^2\) [34]. This type of Regge behavior is considered to be valid only in the vicinity of the saturation scale, \(Q_s^2\) where scattering amplitude depends only on a single dimensionless variable, \(Q^2/Q_s^2\). The Regge theory is applicable if the quantity invariant mass,

*W*\((=\sqrt{Q^2(1-x)/x})\) is much greater than all other variables. Therefore, we expect it to be valid if

*x*is enough small, for any value of \(Q^2\).

Before going into in depth phenomenological analysis and consistency of the equation towards experimental result, we want to highlight two important features of the evolution Eq. (15). As we follow, in the region far below saturation limit, for the canonical choice of \(\lambda \sim 0.5\), the quadratic term in (15) tends to vanish since \((1-2^{2\lambda -1}) \rightarrow 0\). This indicates that below saturation region both the contribution from shadowing correction \(\frac{\partial f_{\text {shad}}}{\partial \ln {\frac{1}{x}}}\) and antishadowing correction \(\frac{\partial f_{\text {antishad}}}{\partial \ln {\frac{1}{x}}}\) coexists but seem to balance each other for which nonlinear effect becomes negligible. Therefore, in the below saturation regime, we can revert (15) back to the original BFKL equation choosing \(\lambda = 0.5\).

### 2.2 Analytical solution of KC improved MD-BFKL

In this section, we present an analytical solution of our KC improved MD-BFKL equation, particularly in the saturation region where shadowing effect is the dominant one. Since our calculations are limited to fixed strong coupling \(\alpha _s\), so do fix \(\lambda \), therefore, the solution for both (15) and (16) will exhibit the same form only differ by the coefficients of their respective quadratic terms.

Now we are set to solve our original Eq. (18) which is indeed a 1st order semilinear (nonlinear) PDE. Our analytical approach of solving the same involves two steps: first we will express the nonlinear PDE in terms of a linear PDE then we will solve the linear PDE via. change of coordinates.

*x*-\(k_T^2\) plane. This will allow us to solve (28) for

*x*and \(k_T^2\) as functions of \(\sigma \) and \(\eta \). The only requirement is that we should ensure the Jacobian of the transformation does not vanish i.e. \(\text {J}=\left| \begin{array}{cc} \sigma _x &{} \eta _x \\ \sigma _{k_T^2} &{} \eta _{k_T^2} \end{array} \right| \ne 0\) in D.

#### 2.2.1 *x* and \(k_T^2\) evolution

In this section, we study the small-*x* dependence of gluon distribution by picking an appropriate input distribution at some high *x*, as well as the \(k_T^2\) dependence of gluon distribution setting input distribution at some low \(k_T^2\).

*x*is fixed throughout the evolution. At \((x,k_0^2)\) let the argument of G be denoted by \(t=\frac{e^{-\frac{k_0^{-2\lambda }}{l \lambda }}}{x}\).

*t*with any other argument will give us the value of G at that particular argument. We put \(t=\frac{e^{-\frac{k_T^{-2\lambda }}{l \lambda }}}{x}\) in (36) which gives us the l.h.s. of (35) and then we solve (35) for \(f(x,k_T^2)\). The solution for \(k_T^2\) evolution turns out to be

*x*evolution (38) and \(k_T^2\) evolution (37) in Figs. 4 and 6. Our prediction of unintegrated gluon distribution \(f(x,k_T^2)\) is contrasted with that of modified BK equation [25],

Our prediction of both unintegrated and collinear gluon distribution is obtained for two different form of shadowing: conventional \(R= 5 \text { GeV}^{-1}\) (order of proton radius) where gluons are spread throughout the nucleus and extreme \(R= 2 \text { GeV}^{-1}\) where gluons are expected to concentrated in hotspots within the proton disk, recalling that \(\pi R^2\) is the transverse area within which gluons are concentrated inside proton. From Figs. 4, 5, 6 and 7 it is clear that shadowing correction are more prominent when gluons are concentrated in hotspots within proton.

The \(k_T^2\) (and \(Q^2\)) evolution in Figs. 4 and 5 is studied for the kinematic range \(1 \text { GeV}^2\le k_T^2(\text {or }Q^2)\le 10^3 \text { GeV}^2\) corresponding to four different values of *x* as indicated in the figure. Our evolution for both \(f(x,k_T^2)\) and \(xg(x,Q^2)\) shows a similar growth as modified BK as well as datasets respectively for all *x*. It is also observed that the growth of \(f(x,k_T^2)\) (or \(xg(x,Q^2)\)) is almost linear for the entire kinematic range of \(k_T^2\) (or \(Q^2\)). This is expected since the net shadowing term in (16) has \(1/k_T^2\) dependence, which suppresses the contribution from the shadowing term at large \(k_T^2\).

The *x* evolution of \(f(x,k_T^2)\) and \(xg(x,Q^2)\) is shown in Figs. 6 and 7 for two \(k_T^2\) values viz. \(5 \text { GeV}^2\), \(50 \text { GeV}^2\) and two \(Q^2\) values viz. \(35 \text { GeV}^2\), \(100 \text { GeV}^2\). The input is taken at higher *x* value \(x=10^{-2}\) and then evolved down to smaller *x* value upto \(x=10^{-6}\) thereby setting the kinematic range of evolution \(10^{-6}\le x\le 10^{-2}\). We observed that the singular \(x^{-\lambda }\) growth of the gluon is eventually suppressed by the net shadowing effect. KC improved MD-BFKL seem to poses a more intense shadowing then modified BK. However, it is hard to establish the existence of shadowing for \(x\ge 10^{-3}\). The obvious distinction between the two form of shadowing \(R= \text {5 GeV}^{-1}\) (conventional) and \(R= 2\text { GeV}^{-1}\) (“hotspot”) is also observed towards small-*x* \((x\le 10^{-3})\). Interestingly towards very small-*x* \((x\le 10^{-5})\), at small \(k_T^2\) (or \(Q^2\)) values (viz. \(k_T^2=5\text { GeV}^2\), \(Q^2=35 \text { GeV}^2\)) the gluon distribution becomes almost irrelevant of change in *x*. This could be a strong hint for the possible saturation phenomena in the small-*x* high density regime.

In Fig. 8a we have shown a comparison between the linear BFKL equation, MD-BFKL without kinematic constraint and MD-BFKL with kinematic constraint for *x* evolution. The three solutions are compared for two different values of \(k_T^2\) i.e. \( 5 \text { GeV}^2\) and \(50 \text { GeV}^2\). Similarly in Fig. 8b we have shown similar comparison but for \(k_T^2\) evolution for two different values of *x* i.e. \(x=10^{-4}\) and \(x=10^{-6}\). In Fig. 8 the singular \(x^{-\lambda }\) behavior of \(f(x,k_T^2)\) is distinct for unshadowed linear BFKL equation. On the other hand, the deviation of the two different forms of the MD-BFKL equation from linear BFKL equation reflects the underlying shadowing correction. It is also observed that shadowing effect is more intense in MD-BFKL with KC than MD-BFKL without KC.

#### 2.2.2 Complete solution of KC improved MD-BFKL

In this section we implant a functional form of the input distribution (more likely a dynamic one) on the general solution of our KC improved MD-BFKL equation (34) and try to obtain a parametric form of the solution. The underlying motivation towards doing so is that this allows us to evolve our solution for both *x* and \(k_T^2\) simultaneously in \(x\text {-}k_T^2\) phase space which help us to portray a three-dimensional realization of the gluon evolution.

Equation (50) serves as the parametric form of the solution for KC improved MD-BFKL equation. The input distribution is inclusive in the solution (50) itself, therefore, we do not have to depend on the experimental data fits for input

distribution unlike we did in the previous section of *x* and \(k_T^2\) evolution. This parametric form of the solution actually helps us to explore the three-dimensional insight of the gluon evolution in *x*-\(k_T^2\) phase space. However, using (50) one may also study *x* and \(k_T^2\) evolution separately by setting *x* fixed for \(k_T^2\) evolution or \(k_T^2\) fixed for *x* evolution.

In Fig. 9 we have shown our solution of KC improved MD-BFKL equation (50) in three dimension. The kinematic region for our study is set to be \(10^{-6}\le x \le 10^{-2}\) and \(1 \text { GeV}^2\le k_T^2 \le 10^3\text { GeV}^2\). In the 3D surface, the suppression in the rise of the gluon distribution towards small *x* due to shadowing correction is visible. However, a linear rise of the surface in the \(k_T^2\) direction can be seen, attributed to the \(1/k_T^2\) factor in the nonlinear term, which offset the effect of shadowing at large *x*.

In Fig. 10 we have shown density plots of \(f(x,k_T^2)\) in \(x\text {-}k_T^2\) domain to examine the sensitivity of \(f(x,k_T^2)\) towards the parameter \(\lambda \). Density plot allows us to visualize and distinguish the kinematic regions with high/low \(f(x,k_T^2)\) in \(x\text {-}k_T^2\) plane which is more informative then any ordinary 3D plot. In Fig. 10 the plots are sketched for two canonical choices of \(\lambda \) viz. \(\lambda =0.4\) and 0.6 corresponding to two \(\alpha _s\) values 0.15 and 0.23. Our solution seems to be very sensitive towards a small change in \(\lambda \). An apparent 50% change in \(\lambda \) (0.4–0.6) suggests approximately around one order of magnitude rise in gluon distribution \(f(x,k_T^2)\) for an approximate limit of *x* and \(k_T^2\): \(10^{-6}\le x\le 10^{-5}\) and \(50 \text { GeV}^2 \le k_T^2\le 100 \text { GeV}^2\). It is also observed that the range of high \(k_T^2\) and very small-*x* is the high gluon distribution \(f(x,k_T^2)\) region where gluons are mostly populated. In Fig. 11 we have shown *R* sensitivity of our solution for two choices of the shadowing parameter *R* viz. \(R= 5 \text { GeV}^{-1}\) and \(2 \text { GeV}^{-1}\). A satisfactory shadowing effect is observed from the comparison of the two plots. The extreme form of shadowing \((R=2 \text { GeV}^{-1})\) is found to suppress atleast 10–20% magnitude of gluon density than the conventional shadowing \((R=5 \text { GeV}^{-1})\) in the high \(k_T^2\) and small-*x* region.

## 3 Equation of critical line and prediction of saturation scale: a differential geometric approach

The so-called saturation physics allows one to study the high parton density region in the small coupling regime. The transition from the linear region to saturation region is characterized by the saturation scale. The saturation momentum scale \(Q_s\) is the threshold transverse momentum for which non-linearity becomes visible. The boundary in \((x, Q^2)\) or \((x,k_T^2)\) plane along which saturation sets in is characterized by the critical line. An important feature of our analytical solution to the KC improved MD-BFKL equation is the finding of the equation of the critical line which is supposed to mark the boundary between dilute and dense partonic system in \(x\text {-}k_T^2\) phase space.

*x*separately, the gluon distribution depends on a single dimensionless variable \(\frac{k_T^2}{Q_s^2}\) i.e.

Our approach towards studying geometrical scaling and critical line is primarily based on the basic understanding from differential geometry, in particular gradient of a function which is considered as the direction of steepest ascent of that function. Each component of the gradient gives us the rate of change of the function with respect to some standard basis i.e. it gives us an idea about how fast our function grows or decays or saturates with respect to the change of the variables. One important advantage of choosing gradient is that it is a two dimensional object since it does not possess any component along \(f(x,k_T^2)\) axis in \(\mathbb {R}^3\). This ensures that it does not have any direct dependence on the magnitude of gluon density \(f(x,k_T^2)\), rather it depends on the rate at which \(f(x,k_T^2)\) changes with respect to *x* and \(k_T^2\) change. This property of gradient actually helps us in distinguishing out the saturation region and linear region although the distribution function \(f(x,k_T^2)\) has large variation in order of magnitude for different regions in \(x\text {-}k_T^2\) phase space.

*x*, gluon evolution is suppressed due to shadowing effect as sketch in Fig. 9 which motivates us to evaluate the gradient of \(f(x,k_T^2)\) particularly along

*x*basis. For simplicity we consider an unit vector \(\mathbf {\nu }\) along \(\tilde{\eta } \text { }( = 1/x)\) basis, we can project the gradient \(\nabla f(\tilde{\eta },k_T^2)\) along \(\mathbf {\nu }\) via dot product \(\nabla f(\tilde{\eta },k_T^2)\).\(\mathbf {\nu }\). This scalar quantity can also be interpreted as the directional derivative along the direction \(\mathbf {\nu }\),

We obtained a family of contours (or level curves) \(\tilde{\eta }(k_T^2)\) in \(\tilde{\eta }-k_T^2\) plane solving (55) as shown in Fig. 12a. Each contour depicts a constant gradient *g* along the curves. The set of contours can also be identified as some set of possible saturation scales. As sketch in Fig. 12a the \(\tilde{\eta }-k_T^2\) plane is divided into two regions: low gluon density region where the spacing between two consecutive contours is very small and high gluon density region where the spacing becomes very large compared to previous. This distinction in contour spacing in the two regions comes from the fact that the gradient changes very fast until the saturation is reached and then after reaching the saturation boundary gradient changes very slowly or almost freezes for further increase in \(\tilde{\eta }(k_T^2)\) as can be seen in Fig. 12a. In high gluon density region, the contour curves tend to \(\tilde{\eta }(k_T^2)\rightarrow \infty \) towards high \(k_T^2\). For low gluon density region, shadowing effects are negligible and the contours become almost parallel straight lines.

*k*is a constant \((k\in \mathbb {R})\). Therefore, the equations of the level curves are given by

*k*by plugging the point \((Q_{s0}(\tilde{\eta }_0),\tilde{\eta }_0)\) into (56)

*x*dependence than the one from GBW model \(Q_s^{'2}\). The saturation scale has direct dependence on partons per unit transverse area. Smaller

*x*suggests larger parton density giving rise to a larger saturation momentum scale, \(Q_s^2\). In other words the saturation scale \(Q_s\) depends on

*x*in such a way that with decreasing

*x*one has to probe smaller distances or higher \(Q^2\) in order to resolve the dense partonic structure of the proton which is clear from our analysis.

## 4 KC improved MD-BFKL prediction of HERA DIS data

### 4.1 DIS structure functions and reduced cross section

*x*, the sea quark distribution is driven by gluons via. \(g\rightarrow q\bar{q}\) process. This component from sea quark distribution can be calculated in perturbative QCD. The relevant diagram for this QCD process is shown in Fig. 13 and the contribution to the transverse and longitudinal components of the structure functions can be written using the \(k_T\)-factorization theorem [47, 48]

*p*and \(q^{'}\equiv q +xp\), where \(x=\frac{Q^2}{2p.q}\) and \(Q^2=-q^2\) like as usual (see Fig. 13). Now we can decompose quark and gluon momentum in terms of sudakov variables

*x*provided that the gluon distribution \(f(x,k_T^2)\) is known. An analytical approach towards the calculation of \(F_2\) and \(F_L\) can be found in literature [55, 56] for fixed coupling case using linear BFKL equation. In the literature [28], structure functions are calculated taking gluon distribution \(f(x,k_T^2)\) from the numerical solution of unitarized BFKL equation for running coupling consideration. It has been seen that the analytical approach [56] considerably overestimates the actual (numerical) solution [28]. This is because of the fact that analytical approach neglects terms down only by powers of \(\ln (1/z)^{-1}\) as well as it does not accommodate running of strong coupling. However, our approach is a semi-analytical one, in the sense that we take gluon distribution from our analytical solution while further integrations are performed numerically. This indeed allows us to check the feasibility of our analytical solution in describing DIS data.

*x*, towards higher

*x*their effect becomes weak and we cannot neglect other non-BFKL contribution. For instance, we assume that \(F_\text {i}^{\text {BG}}\) evolves like \(x^{-0.08}\) motivated from soft pomeron intercept \(\alpha _{P}(0)=1.08\) [58]. To be precise we use

*x*and then take \(F_\text {i}^{\text {BG}}(x_0,Q^2)\) from data [52] which is also listed in the figure caption. The recent HERA DIS data taken for comparison with our result can be found in [50, 57] from H1 collaboration and in [51] from ZEUS collaboration. In the text [50] from H1 collaboration inclusive neutral current \(e^{\pm }p\) scattering cross section data collected during the years (2003-2007) is presented. The beam energies \(E_p\) of corresponding H1 experiment run are 920, 575 and \(460 \text { GeV}^2\). Corresponding \(F_L\) data of H1 is taken from [57] where measurement are performed at c.m.s. energies \(\sqrt{s}=\) 225 and 252 \(\text {GeV}\). On the other hand, in [51] from ZEUS collaboration reduced cross sections for

*ep*scattering for different c.m.s. energies viz. 318, 251 and \(225\text { GeV}\) is presented. The fixed target data from NMC [52] and BCDMS [53] collaboration which exist for \(x>10^{-2}\) is also shown in the Fig. 14. Finally, we have included data from global parameterization groups viz. NNPDF3.1sx [37] and PDF4LHC15 [54] for comparison.

The *x* dependence of the structure functions \(F_L\) and \(F_2\) is shown in Figs. 14 and 15 for the four \(Q^2\) values \(5 \text { GeV}^2\), \(15 \text { GeV}^2\), \(35 \text { GeV}^2\) and \(45 \text { GeV}^2\). The nonperturbative contribution, \(F_{2/L}^{\text {npert.}}\) (or \(F_{2/L}^{\text {BG}}\)) given by (68) contrasted with total (nonpert. +pert.) contribution, \(F_{2/L}^{\text {npert.+pert.}}\). QCD prediction shows a satisfactory agreement with DIS data for both structure functions \(F_2\) and \(F_L\). It is clear that perturbative contribution is insignificant towards large *x* (\(\ge 10^{-2}\)), while for small *x* around \(x\le 10^{-3}\) this contribution becomes visibly important. Interestingly both data and theory for \(F_2\) structure function seem to preserve the \(x^{-\lambda }\) singular behavior rather than showing taming due to net shadowing effect in the asymptotic limit \(x\rightarrow 0\). It is difficult to observe the existence of any sizable shadowing effect for \(x>10^{-3}\). However, for \(x<10^{-3}\), recalling that the shadowing term is proportional to \(1/R^2\) it is seen that for \(R=2\text { GeV}^{-1}\) (gluons in hotspots) the rise of structure function becomes slower than that of \(R=5\text { GeV}^{-1}\) (\(R=R_H\), hadron radius) as expected. But even the extreme form of the gluon shadowing (\(R=2\text { GeV}^{-1}\)) suppresses \(F_2\) only by about \(10\%\) or less at \(10^{-3}\). It just depicts that shadowing has the negligible impact on structure functions even towards smaller *x*. This is because of the fact that in this low *x* regime gluons are expected to drive the sea quark distribution via \(g\rightarrow q\bar{q}\). Therefore, a similar sea quark contribution to the structure function \(F_2\) in addition to the gluon contribution can be expected in low *x* regime. On the other hand from Fig. 15 it is clear that for \(x>10^{-3}\), the size of the longitudinal structure function is negligibly small while for very small-*x* regime (\(x<10^{-3}\)) , \(F_L\) grows eventually. This is in accord with our expectation since measurement of \(F_L\) directly probes the gluonic content of the proton which is dominant in small-*x* regime.

In Fig. 16 a phenomenological comparison between data and theory is shown to illustrate \(Q^2\) dependence of the ratio \(R(x,Q^2)\). The HERA data taken for comparison can be found in [57] where *ep* cross section data measured for two center of mass energies \(\sqrt{s}=225\) and \(252\text { GeV}\) is enlisted. From Fig. 16 it is clear that the HERA data is in general agreement with the QCD expectation. Available H1 data for very low \(Q^2\) \((Q^2\le 5\text { GeV})\) is also sketched in Fig. 16. We have excluded the very low \(Q^2\) region in our calculation of *R* since we are bounded to stick in the perturbative region. However, a more realistic study in this transition region is performed in Sect. 4.2.

*ep*scattering process from the knowledge of structure functions that we have discussed above. The inclusive deep-inelastic differential cross section for

*ep*scattering can be represented in terms of three structure functions \(F_2\), \(F_L\) and \(xF_3\). The structure functions have direct dependence with DIS kinematic variables,

*x*and \(Q^2\) only, while the cross section has additional dependence with inelasticity \(y=Q^2/s x\). The inclusive cross section for neutral current

*ep*scattering is given by

*x*dependence of \(e^{\pm }p\) reduced cross section \(\sigma _r\) calculated for center of mass energy \(\sqrt{s}=318\text { GeV}\) is shown in Fig. 17. Our theoretical expectation is compared with HERA H1 measurement [59, 60] and ZEUS (\(\sqrt{s}=318\text { GeV}\)) [51]. The available H1 data for low \(Q^2\) (\(\le 12\text { GeV}^2\)) measured at \(\sqrt{s}= 318\) \(\text {GeV}\) (SVX) and 300 \(\text {GeV}\) (NVX-BST) is taken from [59], while the same for high \(Q^2\) (\(>12\text { GeV}^2\)) is taken from [60]. The cross section measurement of SVX is found to be slightly higher than that of NVX-BST as expected because of the increase in center of mass energy. Both theory and data agree well for our phenomenology range \(Q^2\le 100\text { GeV}^2\). The distinction in \(\sigma _r\) due to the two forms of shadowing \(R=5\text { GeV}^{-1}\) and \(R=2\text { GeV}^{-1}\) is more prominent for smaller \(Q^2\) values. For each \(Q^2\), starting at some high

*x*the reduced cross section first increases as \(x\rightarrow 0\) and then an abrupt fall in cross section can be observed in both theory and data at very small-

*x*region (\(x<10^{-4}\)) . For all \(Q^2\), this region of

*x*corresponds to the highest inelasticity \(y\approx 0.65\) (\(y=Q^2/sx\)) and thus characteristic turn over of cross section at \(y\approx 0.65\) can be attributed to the influence of \(F_L\). In simple words, towards very small-

*x*(or high

*y*) the monotonic rise of \(F_2\) is suppressed by the contribution from longitudinal structure function \(F_L\) thereby causing an overall fall in the cross section. For low inelasticity \(y<0.65\), the contribution from the longitudinal structure function is small on the other hand, structure function \(F_2\) exhibits a steady increase as \(x\rightarrow 0\). Therefore, in the region where

*x*is not so small, the growth of the cross section is found to be power like as expected.

### 4.2 Virtual photon–proton cross section prediction in transition region

Traditionally the photon–proton interaction is classified into two separate processes depending upon the photon virtuality \(Q^2\) viz. photoproduction (at low \(Q^2\)) and deep inelastic scattering (at high \(Q^2\)). DIS is considered as a basic tool for exploring pQCD where the point like virtual photon directly probes the partonic contents of the proton. On the other hand, photoproduction is completely nonperturbative phenomena defined in the limit of vanishing \(Q^2\) where real (or quasi-real) photons interact with the proton more likely a hadron-hadron collision. The experiment at HERA collider provides a unique opportunity to study both the processes photoproduction and DIS on their respective kinematic domains. The \(Q^2\) dependence of the proton structure functions are well described by pQCD over a wide range of *x* and \(Q^2\) [61, 62] in accordance with HERA data. However, for \(Q^2\lesssim 2\text { GeV}^2\) (photo production region) the pQCD breakdowns since the higher order contributions to the perturbative expansion becomes very large. In this region, data can be only described by non-perturbative phenomenological models [63]. Our present study is especially focused on the transition region \((2\text { GeV}^2\le Q^2\le 10\text { GeV}^2)\) from photoproduction to deep inelastic scattering. To measure the photon–proton cross section in the transition region two dedicated runs were performed in the years 1999 (Nominal vertex “NVX’99”) and 2000 (Shifted Vertex “SVX’00”) by H1 experiment at HERA. The published data can be found in [59].

*ep*double differential cross section formula (71) defined in the region \(Q^2\ll M_Z^2\). In this region massive boson \((M_Z)\) exchange is neglected and only one photon exchange is considered, thereby the role of incoming electron reduces to be a source of virtual photon interacting proton. Thus, we can recast the formula (71) in terms of photon–proton reaction. In fact the structure function \(F_2\) and \(F_L\) in (71) related to the longitudinally and transversely polarized photon–proton scattering cross sections \(\sigma _L\) and \(\sigma _T\) by the relations

*x*. Considering (73) and (74) the reduce cross section in (71) can be written as

*y*i.e. \(\sigma _{\gamma ^*p}^{\text {eff}} \xrightarrow {y\rightarrow 0}\sigma _{\gamma ^*p}^{\text {tot}}\) which differ only in the region of high y.

Figure 18 shows the measurement of the virtual photon–proton cross section \(\sigma _{\gamma ^{*}p}^\text {eff}\) as a function of \(Q^2\) corresponding to different values of *W*. The total cross section \(\sigma _{\gamma ^*p}^{\text {tot}}\) is often expressed as a function of \(Q^2\) and invariant mass *W*. The standard relation between *W*, *x* and \(Q^2\) is \(W=\sqrt{Q^2(1-x)/x}\). Since \(Q^2\approx syx\), for small-*x* we can have an approximate relationship between *W* and *y* i.e. \(W^2\simeq sy\) which we have used in our calculations. HERA measurement for \(\sigma ^{\text {eff}}_{\gamma ^{*}p}\) from H1 [59] (\(\sqrt{s}= 318\text { GeV}\)) and ZEUS [64] (\(\sqrt{s}= 300\text { GeV}\)) are included for comparison with our theoretical prediction. We have chosen \(\sqrt{s}= 318\text { GeV}\) for our calculation analogous to H1 measurement. The precision of the data is such that their errors are hardly visible. Both the HERA data and theoretically measured cross sections for different values of *W* are multiplied by the factors multiple of 2 as indicated in the figure. For \(Q^2\gtrsim 3 \text { GeV}^2\), an excellent agreement between the theory and HERA data can be observed for the wide range of *W*, while for \(Q^2<3 \text { GeV}^2\), the slop of the QCD prediction seems unsatisfactory. This indicates the inadequacy of perturbative QCD at very low \(Q^2\) (\(<3 \text { GeV}^2\)) and roughly provides a lower bound of \(Q^2\) to our theory.

## 5 Conclusion

In conclusion, we have presented a phenomenological study on the behavior of unintegrated gluon distribution at small-*x* and moderate \(k_T^2\) region particularly \(10^{-6}\le x\le 10^{-2}\) and \(2 \text { GeV}^2\le k_T^2\le 1000 \text { GeV}^2 \) which is also the accessible kinematic range to experiments performed at HERA *ep* collider. In the beginning, we have improved the MD-BFKL equation supplementing so-called kinematic constraint on it. Then we have solved this unitarized BFKL equation analytically in order to study *x* and \(k_T^2\) dependence of unintegrated gluon distribution function. Our prediction of gluon distribution is contrasted with that of modified BK equation as well as global datasets NNPDF 3.1sx and CT 14. The *x* evolution of gluon distribution shows the singular \(x^{-\lambda }\) type behavior of gluon evolution tamed by shadowing correction. We found that for intense shadowing \((R= 2 \text { GeV}^{-1})\), towards smaller x, certainly from \(x=10^{-4}\), the gluon distribution emerges from rapid BFKL growth which indicates the dominance of gluon shadowing. While in case of conventional shadowing \((R= 5 \text { GeV}^{-1}) \) an appreciable modification of BFKL behavior can only be seen from \(x= 10^{-5}\). The \(k_T^2\) dependence of unintegrated gluon distribution is also studied. Although obvious suppression due to net shadowing correction is seen in our studies, however no significant saturation phenomenon is observed. The reason is the nonlinear contribution term is suppressed by the factor \(1/k_T^2\) at large values of \(k_T^2\).

We have obtained a more general solution of KC improved MD-BFKL equation implementing a pre-defined input distribution on it. This has allowed us to visualize the gluon evolution in three dimensions \(\mathbb {R}^3\) into \(x\text {-}k_T^2\) phase space. We have also shown the sensitiveness of unintegrated gluon distribution towards *R* and \(\lambda \) using density plots in \(x\text {-}k_T^2\) plane.

An important achievement of obtaining an analytical solution in this work is its implication on qualitative studies on geometrical scaling which is presented in Sect. 3. Starting from a basic concept of differential geometry and knowledge of level curves we have managed to obtain an equation of the critical boundary which is supposed to separate low and high gluon density regions.

In Sect. 4 we have studied the small-*x* dependence of the structure function \(F_2\) and \(F_L\) obtained via \(k_T\)-factorization formula \(F_i=f\otimes F_i^{(0)}\). Here the unintegrated gluon distribution \(f(x,k_T^2)\) is taken from our analytical solution of KC improved MD-BFKL equation setting boundary condition at \(x_0=0.01\). Surprisingly the quantitative size of the shadowing correction to \(F_2\) and \(F_L\) is found to be very small and the structure functions seem to hold the singular \(x^{-\lambda }\) behavior. Even at intense shadowing condition \((R= 2 \text { GeV}^{-1})\) the \(F_2\) structure function is found to be suppressed only by \(10\%\). In addition we have measured \(e^{\pm }p\) reduced cross section as well as equivalent \(\gamma ^*p\) longitudinal to transverse cross section ratio \(R(x,Q^2)\) from the knowledge of \(F_2\) and \(F_L\). Our results are compared with recent high precision HERA measurements. The comparison reveals a good agreement between our theory and DIS data.

In Sect. 4.2 we have examined the virtual photon–proton effective cross section, particularly in the transition region from photoproduction to deep inelastic scattering. The quantity \(\sigma _{\gamma ^{*}p}^\text {eff}\) serves the role of the total cross section if small inelasticity *y* is concerned. We have compared our predicted \(\sigma _{\gamma ^{*}p}^\text {eff}\) with the HERA data of two dedicated runs “NVX’99” and “SVX’00” by H1 as well as ZEUS for the transition region. Our theoretical prediction shows well-consistency with HERA data particularly upto \(Q^2\sim 3 \text { GeV}^2\) in the transition region.

In summary, there are several attractive features of our present study. First, we have able to predict a wide range of physical quantities, starting right from our solution for unintegrated gluon density. Secondly, all analysis is performed in terms of relatively small numbers of parameters. Moreover, the idea developed in this work for studying geometrical scaling can be implemented in any other framework. Finally, a very significant feature of this analysis is that we have considered two extreme possibilities of shadowing which can be distinguished by experimental data. In the end, we conclude that, as per feasibility towards HERA DIS data is concerned, the KC improved MD-BFKL equation could be a reliable framework for exploring high energy physics over a wide range of *x* and \(k_T^2\) which is also relevant for LHC probe and future collider phenomenology.

## Notes

### Acknowledgements

We thank Prof. Johannes Bluemlein, DESY (Hamburg, Germany) and Prof. Dieter Schildknecht, Bielefeld university (Germany) for their valuable comments. Two of us (P. P. and M. L.) acknowledge Department of Science and Technology (DST), India (Grant DST/INSPIRE Fellowship/2017/IF160770) and Council of Scientific and Industrial Research (CSIR), New Delhi (Grant 09/796(0064)2016-EMR-I) respectively for the financial assistantship.

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