Extraction of gluon distributions from structure functions at small x in holographic QCD

We investigate the nucleon and pion gluon distribution functions in the framework of holographic QCD, focusing on the small Bjorken x region. Based on an approximate relation, the gluon distributions are extracted from structure functions of the unpolarized deep inelastic scattering which can be calculated with a holographic QCD model, assuming the Pomeron exchange. All the adjustable parameters of the model are determined with the HERA data of the proton structure functions. We explicitly show that the extracted proton gluon distribution is consistent with results of the recent global QCD analysis. The structure functions of the pion can be computed without any additional parameter, which enables us to predict its gluon distribution also. We find that the resulting pion gluon density is smaller than the proton's, and agrees with the recent global QCD analysis result within the uncertainties.


INTRODUCTION
Understanding the nucleon structure is one of the most important research topics in fundamental science, and tremendous efforts have been done to deepen our knowledge over several decades. The quark-gluon structure of the nucleon is usually encoded into the parton distribution functions (PDFs) which are expressed with two kinematic variables, the Bjorken scaling variable x and the probe scale Q 2 . The PDFs are required as inputs to make theoretical predictions or perform numerical simulations for high energy scattering phenomena. The study of PDFs in the wide x and Q 2 regime has provided us with great opportunities to improve our understandings about the various aspects of QCD.
To determine the PDFs, the global QCD analysis, utilizing high energy scattering data, is the most straightforward and reasonable way. In particular, high statistics and high precision date of the deep inelastic scattering (DIS) have played a crucial role in such analysis. Via the unpolarized lepton-nucleon DIS, the two independent structure functions, F 2 (x, Q 2 ) and F L (x, Q 2 ), can be measured, and from these one can extract the information for PDFs, utilizing the formulae derived based on the perturbative technique of QCD. Taking into account the recent high energy scattering data, including the DIS at HERA and the relevant events at LHC, various collaborations have performed the global QCD analysis to determine the proton PDFs [1][2][3][4][5].
Those efforts have gradually improved our knowledge about PDFs. However, pinning down the gluon distribution, which is dominant in the small x region, is still extremely difficult due to the nonperturbative nature of QCD. To extract the gluon distribution, the precise measurement of the longitudinal structure function F L is es-sential. In leading twist and next to leading order (NLO) QCD, F L can be expressed by the sum of two terms, which include F 2 and the gluon distribution function, respectively, and the contribution from the latter term is dominant at small x [6]. Since both F L and the gluon distribution function are highly nonperturbative physical quantities, in principle they are not calculable by the direct use of QCD. Furthermore, although there are available F L data, those have large errors. These facts cause the huge uncertainties which can be seen in the preceding studies based on the global QCD analysis.
In our model setup, the Pomeron exchange is described by the BPST kernel, and the wave functions of the U(1) vector field, which were derived in Ref. [22], is applied arXiv:1910.10008v1 [hep-ph] 22 Oct 2019 to describe the virtual photon in the five-dimensional AdS space. For the Pomeron-nucleon coupling, we adopt the nucleon gravitational form factor which can be obtained from the bottom-up AdS/QCD model [40,41]. There are four adjustable parameters in the model, which are to be determined with the HERA data for both the F 2 and F L structure functions, focusing on the highly nonperturbative kinematic regime, where x ≤ 10 −2 and Q 2 ≤ 10 GeV 2 . We will explicitly show that the considered data are well reproduced within the model. Then, utilizing the approximate relation proposed by the authors of Ref. [42], we extract the gluon distribution from the resulting structure functions. This extraction can be done without any additional parameter. It will be presented that our prediction is consistent with results of the recent global QCD analysis.
Furthermore, we also investigate the gluon distribution of the pion in the framework. Once all the model parameters are determined by the fit with the proton DIS data, the pion structure functions can be calculated without any additional parameter in the chiral limit. Hence, the pion gluon distribution can also be predicted, and we find that the resulting gluon density is smaller than the proton's. Recently, the first Monte Carlo global analysis of the pion PDFs was performed [43]. It will be shown that our prediction agrees with their result within the uncertainties.

THEORETICAL FRAMEWORK
In the quark-parton model, the longitudinal structure function F L (x, Q 2 ) vanishes, which means that the origin of the experimentally observed finite F L is higher order effects of QCD. In leading twist and NLO QCD, F L is expressed in terms of F 2 and the gluon distribution function g(x, Q 2 ) by [6] F L x, Q 2 = α s Q 2 4π where G(y, Q 2 ) = yg(y, Q 2 ) and α s and e i are the QCD coupling and the quark charge, respectively. Substituting y = x/(1−z), Eqs. (2) and (3) are rewritten as respectively. When x is small enough, F 2 and G in the above equations can be expanded. Hence, following the procedures presented in Ref. [42], I F and I G are expressed as I F ≈ F 2 (2x)/2 and I G ≈ G (x/0.4) /5.9, respectively. Therefore, one obtains the approximate relation between the gluon distribution function and the structure functions: Due to the relatively large magnitude of the coefficient multiplying F L , it is understood that the gluon distribution is mainly dependent on F L . However, since the contribution from F 2 term is not negligibly small, we consider the both structure functions to numerically evaluate the gluon distribution in this study. Those DIS structure functions at small x can be calculated in the framework of holographic QCD, assuming the Pomeron exchange in the five-dimensional AdS space. Employing the BPST kernel denoted by χ [24], the scattering amplitude of a two-body process, 1 + 2 → 3 + 4, is expressed in the eikonal representation as where s and t are the Mandelstam variables, b is the twodimensional impact parameter, z(z ) are the fifth coordinates for the incident(target) particles, and P 13 (z) and P 24 (z ) represent the density distributions of the involved two particles in the AdS space.
To obtain the structure functions, we consider the total cross section of the forward scattering by applying the optical theorem. Keeping only the leading contribution from the kernel, the Pomeron exchange contribution is expressed by the imaginary part of the kernel. In the conformal limit, the analytical form of Imχ can be obtained, and the impact parameter integration in Eq. (7) can also be performed analytically [24,28]. Hence, the structure functions are written as where i = 2, L and τ = log(ρzz s/2). g 2 0 and ρ are adjustable parameters which control the magnitude and the energy dependence of the structure functions, respectively.
In the preceding studies [32,33,35], it was shown that the inclusion of the confinement effect in QCD is necessary to reproduce the proton structure function data with the BPST kernel, unless we consider the high Q 2 region. Therefore, instead of the conformal kernel, to numerically evaluate the structure functions we employ the modified kernel: where z 0 (z 0 ) are the cutoffs of the fifth coordinates. Note that z 0 is one of the adjustable parameters of the model, but z 0 is uniquely fixed with hadron masses. The first term in the right-hand side of Eq. (10) is exactly the same as the conformal kernel, and the second term mimics the confinement effect with the same functional form.
To numerically evaluate the structure functions, one needs to specify the density distributions, P (i) 13 (z, Q 2 ) and P 24 (z ) in Eq. (8). For the probe photon, we apply the wave functions of the five-dimensional U(1) vector field with a weight w on its longitudinal component: where K 0(1) are the modified Bessel functions of the second kind. When w = 1, the results presented in Ref. [22], which satisfy the Maxwell equation in the bulk AdS space, are recovered. It has been known that their results are useful to reproduce the experimental data of proton F 2 , however, the resulting F L is somewhat larger than the data. This can be seen in the observations that the experimentally favored value of the longitudinalto-transverse ratio, R = F L /F T = 0.26 [44], is obviously smaller compared to the theoretical predictions, 0.3 R 0.5 [31,32,35,37]. Hence, as a simple ansatz, we introduce the weight, which is to be determined with the data, in this study.
For the density distributions of the target hadrons, P 24 (z ) in Eq. (8), we apply the gravitational form factors which can be obtained with the bottom-up AdS/QCD models [40,41]. As discussed in Refs. [45][46][47][48], in the model the nucleon is described as a solution to the fivedimensional Dirac equation. The density distribution of the proton is given in terms of the Bessel functions by [33] where ψ L(R) are the left-handed and right-handed components of the Dirac field, respectively, and the cutoff parameter z p 0 is fixed with the proton mass m p by the condition, J 1 (m p z p 0 ) = 0. Utilizing the bottom-up AdS/QCD model of mesons [11], the analytical form of the pion wave function can be obtained in the chiral limit. The density distribution of the pion is expressed with Bessel functions as [33] where f π is the pion decay constant, α = 2πσ/3, and σ = (332 MeV) 3 is used in this study. z π 0 is the cutoff parameter which is uniquely fixed with the ρ meson mass m ρ by the condition, J 0 (m ρ z π 0 ) = 0.

NUMERICAL RESULTS AND DISCUSSION
In the model setup, there are four adjustable parameters in total, which are determined by a numerical fit, considering the HERA data for both the proton F 2 [49] and F L [44] structure functions simultaneously in the kinematic regime, where x ≤ 10 −2 and Q 2 ≤ 10 GeV 2 . For this procedure, the MINUIT package [50] is used. The resulting best fit values of the parameters are found to be: g 2 0 = 162.15 ± 6.71, ρ = 0.7798 ± 0.0021, z 0 = 3.059 ± 0.079, w = 0.6198 ± 0.0923, and the chi-square value per degree of freedom is χ 2 d.o.f. = 1.295 (240.8/186). We show in Fig. 1 and Fig. 2 the resulting F 2 and F L structure functions of the proton, respectively, compared to the HERA data. It is seen that the both F 2 and F L data in the whole considered kinematic regime are well described by the model.
Once all the four model parameters are determined with the data, we can extract the proton gluon distribution via Eq. (6) from the obtained structure functions. To do this, we consider the four flavor case, and utilize the approximate NLO solution to the renormalization group equation to calculate α s (Q 2 ) (see Ref. [51] for instance) in this study. We display in Fig. 3 the x dependence of the resulting proton gluon distribution at Q 2 = 10 GeV 2 , compared to the recent global QCD analysis results. One can see from the figure that our calculation is consistent with those results in the whole considered region within the uncertainties.
Using Eq. (17), instead of Eq. (15), as the target hadron density distribution, we can obtain the pion structure functions, and then extract its gluon distribution xg π (x, Q 2 ) without any additional parameter. We present our prediction at Q 2 = 10 GeV 2 in Fig. 4. It is seen from the comparison that our prediction agrees with the JAM result in the whole considered x region within  [49] are depicted by circles with error bars. For a display purpose, the absolute magnitudes are scaled by a factor 3 i . the uncertainties, although the GRS result is substantially larger than the others. Also, it should be noted that in our results the mean value of xg π is obviously smaller than that of xg p in the considered x range.
Finally, we make remarks for further improvements. Although this is the first study of the gluon distributions in the framework of holographic QCD, we obtained reasonable results which are consistent with the recent global QCD analysis. However, improving the model itself may be possible. In this work, we only focused on the single-Pomeron exchange, and adopted the hard-wall AdS/QCD models to obtain the target hadron density distributions. The multi-Pomeron exchange, the softwall versions, and the physical pion case shall be considered. Also, selecting the considered Q 2 range may affect the results. In fact, if we cut the low Q 2 ( 1 GeV 2 ) data and consider the slightly larger Q 2 ( 20 GeV 2 ) region, the resulting χ 2 d.o.f. is significantly improved. Further studies would also help to better understand the applicable limit of the model. Moreover, the importance of the further measurements of the longitudinal structure function should be emphasized. Since the errors of F L data considered in this study are large, the lepton-nucleon DIS experiment is still necessary to deepen our understand- ings about the nucleon structure in the small Bjorken x region. The pion gluon distribution xg π (x, Q 2 ) as a function of the Bjorken x at Q 2 = 10 GeV 2 . The yellow and blue bands represent our calculation and the global QCD analysis result of the JAM collaboration [43], respectively, with the 68% C.L. uncertainties. The red curve depicts the result of GRS [52].