Gluon matter distribution in the proton and pion from extended holographic light-front QCD

The holographic light-front QCD framework provides a unified nonperturbative description of the hadron mass spectrum, form factors and quark distributions. In this article we extend holographic QCD in order to describe the gluonic distribution in both the proton and pion from the coupling of the metric fluctuations induced by the spin-two Pomeron with the energy momentum tensor in anti--de Sitter space, together with constraints imposed by the Veneziano model{\color{blue},} without additional free parameters. The gluonic and quark distributions are shown to have significantly different effective QCD scales.


I. INTRODUCTION
The gluonic composition of hadrons plays a key role in understanding the confining phase of quantum chromodynamics (QCD), which is still an unresolved issue in modern particle physics. A key nonperturbative feature of color-confining hadron dynamics is the intrinsic gluon distribution which exists in hadrons over a time scale independent of the resolution of the external probe. The coupling of the rank-two energy-momentum tensor (EMT), the tensor which couples to gravity [1][2][3], provides fundamental constraints on the quark and gluon generalized parton distribution functions (GPDs) of a hadron [4][5][6]. Gravitational form factors (GFFs), the hadronic matrix elements of the EMT, describe the coupling of a hadron to the graviton and thus provide information on the dynamics of quarks and gluons within hadrons due to the internal shear forces and pressure distributions of the quarks and gluons [7][8][9]. In this article, we present an extended holographic light-front (LF) QCD framework for studying the gluon GFFs and provide predictions for the intrinsic gluon distributions of hadrons without introducing additional parameters.
In addition to the role of gluons as fundamental constituents and as the glue binding quarks into hadrons, the knowledge of gluon distributions within hadrons is also essential for the understanding of Higgs boson production [10] and particle production cross sections at smallmomentum fraction x. The near threshold production of heavy vector quarkonium [11], such as the J/ψ and Υ, is dominated by multiple gluon interaction identified with Pomeron exchange [12]: It is expected to shed light on the QCD trace anomaly at the origin of the proton mass [13]. On the other hand, the intrinsic gluon parton distribution function (PDF) of the pion, the lightest QCD bound state, is of particular theoretical interest for understanding nonperturbative aspects of QCD, such as the connection of the spontaneous breaking of chiral symmetry and confinement [14]. Because of the special role of the pion in QCD, there have been sustained efforts and proposals to explore its gluon distribution [15], along with that of the nucleon, one of the main goals of the upcoming Electron Ion Collider [16,17].
Holographic light-front QCD (HLFQCD), a nonperturbative framework based on the gauge/gravity correspondence [18] and its light-front holographic mapping [19][20][21], has the remarkable feature that it reproduces, within its expected precision, the hadronic spectra with the minimal number of parameters: the confining scale λ and effective quark masses. The effective confining interaction for mesons and baryons is determined by an underlying superconformal algebraic structure [22][23][24], which leads to unexpected connections across the full hadron spectrum [25]. It reproduces the structure of hadronic spectra as predicted by dual models, most prominently the Veneziano model [26] with its typical features; linear Regge trajectories with a universal slope and the existence of "daughter trajectories".
The form factors (FFs) obtained within the HLFQCD framework can be expressed by Euler-Beta functions [27,28], a feature also predicted by the generalized Veneziano model [29,30], which includes the electromagnetic (EM) current and the FFs; F EM (t) ∼ B (γ, 1 − α ρ (t)), where α ρ (t) is the trajectory of the ρ-vector-meson, coupling to the quark current in the hadron. We emphasize that, in HLFQCD, the parameter γ, related to the falloff of the EM FF at large momentum transfer t, is not arbitrary, but fixed by the twist of the Fock state, γ = τ −1, consis-tent with the exclusive counting rules [31,32]. The twist τ is the number of constituents N , τ = N (τ = N + L for LF orbital angular momentum L).
The form of the quark distributions in the hadrons is heavily constrained by the analytic properties of the FFs. Furthermore, very natural assumptions, such as the incorporation of the inclusive-exclusive relation at large longitudinal momentum fraction x [33], allowed us to predict, in a very satisfactory way, the quark distributions in mesons and nucleons [27,34], as well as the strange-antistrange and the charm-anticharm asymmetries [35,36] in the nucleon. Notably, the preliminary NNPDF4.1 global analysis [37] indicates a valencelike intrinsic charm contribution |uudcc in the nucleon, consistent with the intrinsic charm-anticharm asymmetry computed in [36]. In addition to higher twist, the sea quark distributions have distinctive Regge trajectories from the EM current couplings to the sea: α φ (t) and α J/Ψ (t), respectively for the strange and charm sea in the proton [35,36]. Likewise, the |uudg and |udg Fock states should provide the leading contributions to the intrinsic gluon distributions of, respectively, the proton and pion coupled to a spin-two current with its characteristic Reggeized t-channel particle exchange.
In this article we extend our previous framework by incorporating gluonic matter with significantly different quarkonic and gluonic scales. Several models for nonperturbative QCD, see e.g. [38], predict that the static potential between quarks, Vq q , or between gluons, V gg , are substantially different, and scale with N C like the quadratic Casimir operator between the fundamental and adjoint representations, C −1 , a scaling confirmed by lattice calculation [39]. It is in qualitative accordance with the smaller slope of the Pomeron as compared to the Reggeons, α P α ρ , therefore with different scales λ q and λ g for quarkonic and gluonic matter with λ g λ q .
The rest of this paper is organized as follows. In Sec. II we study the perturbation of the anti-de Sitter (AdS) metric by an external spin-two current which couples to the EMT in the bulk. It allows us to identify the tensor Pomeron in the asymptotic physical boundary space with the induced metric fluctuations in AdS, and thus to determine the scale of the gluonic matter from the infrared deformation of the AdS metrics in terms of the Pomeron slope. By further imposing the structure of the generalized Veneziano amplitudes for a spin-two current, we are able to incorporate the Pomeron intercept in the analytic expression for the gravitational form factor A g (Q 2 ). Our results for the GFF for the proton and the pion are compared with recent lattice QCD computations. In Sec. III we extend the relation between FFs and GPDs [4][5][6] found in [27] to compute the gluon distribution functions. Our DGLAP evolved predictions for the proton and pion are then compared with global fits. Some final comments are included in Sec. IV.

II. POMERON EXCHANGE AND GRAVITATIONAL FORM FACTORS
As pointed out in [40], soft interactions play an important role in high-energy collisions. Especially, the trajectory for diffractive processes, the Pomeron, has a dominant role at small-angle high-energy scattering, which is beyond the applicability of perturbative QCD. Since the early days of QCD, Pomeron exchange was associated with two (or more) gluons [41][42][43]. The Pomeron couples as a rank-two tensor to hadrons [44][45][46][47][48] and couples strongly to gluons. It remains unclear whether there exists a relation between the soft [49] and hard [50][51][52] Pomerons and their crossover regime. It may be sufficient to consider only the soft Pomeron if one looks into the intrinsic gluon component of the nucleon structure functions, except, perhaps, for extremely small x [50][51][52][53][54]. Therefore we use the soft Pomeron of Donnachie and Landshoff [49] with the effective Regge trajectory, which is interpreted in QCD as a J P C = 2 ++ bound state of two gluons with intercept α P (0) 1.08 and slope α P 0.25 GeV −2 [40]. Pomeron exchange is identified as the graviton of the dual AdS theory [55][56][57][58][59][60][61] and the first hadronic state on the Pomeron trajectory is identified with the 2 ++ glueball [62]. We start with the AdS gravity action with five- modified by an exponential dilaton term e ϕg(z) in the string frame. The metric determinant is g, R is the scalar curvature and Λ = −6/R 2 , with R the AdS radius. We consider the perturbation of the action (2) by an arbitrary external source at the AdS asymptotic boundary which propagates inside AdS space and couples to the EMT [63,64]. By performing a small deformation of the AdS metric about its AdS background, where Φ represents the matter field content. In deriving (3), we use the harmonic gauge The interaction term (4) describes the coupling of the EMT with the graviton probe in AdS-which we identify here with the Pomeron.
We consider the propagation of h M N with components along Minkowski coordinates and impose the additional where h µν couples to the transverse and traceless part of the EMT in (4). The boundary limit of the graviton probe is a plane wave solution with polarization indices along the physical coordinates h ν As an example, we represent the quark matter content of a pion by a scalar field Φ with action modified by the dilaton term e ϕq(z) . It represents a pion with physical mass P µ P µ = M 2 π and AdS mass (µR) 2 = −4 [21]. The EMT follows from (6), with It leads from the interaction term (4) to the transition amplitude The action S q [Φ] (6) is modified by the dilaton term e ϕq(z) = e λqz 2 with λ q = 1/4α ρ (0.5 GeV) 2 , specific to the light front mapping to physical (3 + 1)-dimensional space in HLFQCD [21] and the constraints imposed by the superconformal algebraic structure [25]. On the other hand, the spin-two action S g [h] (3) describing the propagation of a gravitational fluctuation is only constrained by gauge invariance, therefore it is modified by a negative soft-wall dilaton profile [65] e ϕg(z) = e −λgz 2 with λ g = 1/4α P 1 GeV 2 from the Pomeron slope. The interaction term S i [h, Φ] (4) has no deformation term and does not introduce an additional scale. For a soft-wall profile ϕ g (z) = −λ g z 2 the solution to (5) is given by = a (2 + a) where a = Q 2 /4λ g , ξ = λ g z 2 , and U (a, b, z) is the Tricomi confluent hypergeometric function. The usual expression of the GFF from the hadronic matrix elements of the EMT follows from extracting the delta function from momentum conservation in (8). We obtain for A(Q 2 ) [63, 64] with Φ τ (z), the twist-τ hadron bound-state solution [21] and normalization A τ (0) = 1.
To compute the gluon GFF, A g τ (Q 2 ), the Pomeron is assumed to couple mainly to the constituent gluon [44][45][46][47][48]. The lowest twist is the τ = 4 Fock state |uudg in the proton and the twist τ = 3 state |udg in the pion, both containing an intrinsic gluon. The effective physical scale for this process is the scale of the Pomeron which couples to the constituent gluon over a distance ∼ 1/ α P , therefore described by the wave function Φ g τ (z) ∼ z τ e −λgz 2 /2 [21]. Upon substituting in (11) the x-integral representation of the bulk to boundary propagator (9) we find with N τ = B (τ − 1, 2 − α P (0)), the result of the generalized Veneziano model [26,29,30] for a spin-two current.
In writing (12) we have assumed that only the dilaton profile describing Pomeron exchange sets the scale when computing the gluon GFFs and GPDs, while only the dilaton corresponding to Reggeon exchange needs to be considered when finding the EM FFs and quark GPDs [27]. In (12) we have also shifted the Pomeron intercept to its physical value α P (0) ≈ 1, since the expression which follows from the holographic result (11) leads to a zero intercept. This procedure, which has been successfully applied to our treatment of the EM FF [27], maintains the analytic properties of the GFF: It amounts to a displacement of the timelike poles of the exchanged particles in the t-channel [27,66]. In fact, for integer twist the GFF (12) is expressed as a product of τ − 1 timelike poles located at the radial excitation spectrum of the spin-two Pomeron. The lowest state in this trajectory, the 2 ++ , has the mass M 1.92 GeV, compared with the lattice results M (2.15 − 2.4) GeV [40,[67][68][69].
The predictions for the GFF, A g (Q 2 ), for the nucleon and pion are presented in Fig. 1. We find for the gluon mass squared radius r 2 g p = 2.93/λ g = (0.34 fm) 2 and r 2 g π = 2.41/λ g = (0.31 fm) 2 for the proton and pion, indicating a gluonmass distribution concentrated in a rather small region compared with the spread of the charge [40], and also smaller than the proton mass radius found in [63,[72][73][74]. The normalization used in Fig. 1 is discussed below.

III. GLUON DISTRIBUTION FUNCTIONS
Recent calculations of the gluon distribution functions in the hadrons have been performed using holographic approaches, such as in [75][76][77][78]. Following the unified approach advanced in [27,34] we determine in the present work the unpolarized gluon distributions in the nucleon and pion without introducing any new free parameters. In the nonperturbative domain, low virtuality gluons interact strongly with each other to generate the color confinement potential and one cannot distinguish individual gluon quanta. At higher virtualities constituent gluons appear as new degrees of freedom. The lowest gluonic Fock state of the proton is |uudg and, for simplicity, we consider this Fock state to be the dominant contribution to the intrinsic gluon distribution.
Using (12) and the integral representation of the Beta function, the gluon GFF A τ (t) can be written in the reparametrization invariant form provided that w(x) satisfies the constraints w(0) = 0, w(1) = 1 and w (x) ≥ 0 [27,34]. The GFF can also be expressed as the first moment of the gluon GPD at zero skewness, H g τ (x, t) ≡ H g τ (x, ξ = 0, t), where f (x) is the profile function and g τ (x) is the collinear gluon PDF of twist-τ . Comparing (16) with the holographic expression (15) we find that both functions, f (x) and g τ (x), are determined in terms of the reparametrization function, w(x), by where g τ (x) is normalized by 1 0 dx xg τ (x) = 1 . If we identify x with the gluon LF momentum fraction, physical constraints on w(x) are imposed at small and large-x [79]. At x → 0, w(x) ∼ x from Regge theory [80], and at x → 1 from the inclusive-exclusive counting rule [33], g τ (x) ∼ (1 − x) 2τ −3 , which fixes w (1) = 0. The leading (1−x)-exponent determined in [81] by fitting the NNPDF gluon distribution [82] is consistent with the large-x counting rule.
The gluon distribution of the proton can be expressed as the sum of contributions from all Fock states, g(x) = τ c τ g τ (x), where the coefficients, c τ , represent the normalization of each Fock component. In practice, one has to apply a truncation up to some value of τ . In this study we only keep the leading term, τ = 4, and determine the coefficient c τ =4 using the momentum sum rule where q runs over all quark flavors. It also corresponds to the sum rule of the helicity-conserving GFF A(t), which is a measure of the momentum fraction carried by each constituent. Similarly the helicity-flip GFF B(t) provides a measure of the orbital angular momentum carried by each constituent of a hadron at t = 0. The constraint B g (0)+ q B q (0) = 0 was originally derived from the equivalence principle [83] and can be formally derived Fock state by Fock state in LF quantization [84]. Taking the quark distributions and the reparametrization function w(x) from Ref. [34], are compared in Fig. 2 with global analyses of gluon PDFs [82,85,86]. Contributions from higher Fock states are expected to be suppressed at large x and may affect the overall normalization through the momentum sum rule, thus suppressing the gluon distribution at large x while enhancing the distribution at small x. Our nonperturbative results falloff as (1 − x) 5 at large x from the leading twist-four term in (19). In contrast, the perturbative QCD results incorporate a (1 − x) 4 falloff from hard gluon transfer to the spectator quarks [87].
Incorporating the universality of our approach, we now compute the gluon distribution in the pion. Similar to the case of the proton, we only consider the lowest τ = 3 Fock state |udg with one constituent gluon. The coupling of the Pomeron to the hadrons depends on the vertex, but the trajectory α P (t) (1) is the same and unique to the Pomeron. Considering the valence quark distributions determined in Ref. [27], we express the total quark distribution as Using the momentum sum rule we obtain the gluon distribution of the pion as g π (x) = (0.429 ± 0.007) g τ =3 (x). We show in Fig. 2 the results evolved to µ 2 = 10 GeV 2 and a comparison with global analyses [88,89]. We note that the overall normalization of the gluon distribution from our calculation seems overestimated in comparison with some recent global analyses, which may arise from neglecting higher Fock states for the gluon GFF.

IV. CONCLUSION AND OUTLOOK
The light-front holographic extension presented in this article allows us to give a simultaneous description of the intrinsic gluon distributions, GPD and PDF, and the gluon GFF A g (t), within a unified nonperturbative framework, for both the nucleon and pion. The actual computations are based on the holographic coupling of the spin-two soft Pomeron to the hadron energymomentum tensor, constrained by the Veneziano structure. The comparison of our theoretical predictions, after DGLAP evolution, with global analyses and recent lattice computations clearly demonstrates the predictive power of this new framework.
In this article we have only used the parameters of the soft Pomeron. For hard processes the contribution of a hard Pomeron with a much larger intercept and smaller slope also seems to be necessary [43]. However, after evolution, our results describe the full gluon distribution in accordance with phenomenological determinations, see Fig. 2. This supports attempts to use the gauge/gravity duality [18,55,90] to describe hard and soft diffractive processes within a unified framework. The properties of the Pomeron depend then on the kinematic regime of the scattering process [55,90,91].
The gluonic and quark distributions of hadrons are shown to have significantly different effective QCD scales and sizes compared to their electromagnetic distributions. The twist-three Fock state |qqg in the pion (|qqqg for the nucleon) is responsible for the intrinsic gluon distribution at the initial evolution scale where the Pomeron probe couples strongly with the constituent gluon. However, this Fock component does not contribute significantly to the quark GPD since the EM probe does not resolve the deeply bound constituent gluon [92], and thus it is effectively included in the twist-two qq state (qqq for the nucleon). Our results for A(t) can be extended to the other two invariant GFFs B(t) and C(t) [3]. Of particular relevance is the coupling of the scalar Pomeron trajectory-with the similar slope, but different intercept, to determine the form factor C(t) [74,75]. It can allow us to gain further insights into the distribution of internal shear forces and pressure inside the proton and therefore of its dynamical stability.