Proton Structure Functions from Holographic Einstein-Dilaton Models

Deep inelastic scattering (DIS) is a powerful tool to understand nucleon structure. In the late 1960s, experiments at the Stanford Linear Accelerator (SLAC) Breidenbach et al. (1969); Bloom et al. (1969) (see also Whitlow et al. (1992) for a more recent account) used electron-proton ($ep$) scattering to obtain the first estimates for the structure functions of nucleons. Quantum chromodynamics (QCD) then emerged Gross and Wilczek (1973); Politzer (1973) and was consolidated as the fundamental theory of strong interactions between quarks and gluons inside protons and neutrons. A QCD-based formalism known as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations Dokshitzer (1977); Gribov and Lipatov (1972); Altarelli and Parisi (1977) was then developed to describe how parton distribution functions¹¹1Parton distribution functions describe the longitudinal momentum distribution of quarks and gluons (collectively called partons) inside nucleons, and are closely related to structure functions. However, in this work, we will only consider structure functions, not PDFs. (PDFs) evolve with momentum transfer $Q$ and large Bjorken $x$. This allowed QCD to be used to analyze experimental data (see, e.g. Gross et al. (1977); Jaroszewicz (1982); Manohar (1992); Gelis et al. (2010); Accardi et al. (2016)). Later, experiments at HERA Habib (2010) enabled unpolarized $ep$ scattering to be analyzed at wider kinematic ranges. Future facilities, such as the Electron Ion Collider Accardi et al. (2016), are expected to greatly improve our understanding of nucleon structure in the next decade.

Because of confinement, perturbative calculations become unreliable at low transverse-momentum scales, where QCD becomes strongly coupled. This requires alternative approaches to describe QCD phenomena in that particular regime. Of particular relevance to this work are holographic models constructed within the framework of the gauge/gravity duality Maldacena (1998); Gubser et al. (1998); Witten (1998); Aharony et al. (2000); Gubser et al. (2002), which have been extensively used to obtain insight into general phenomena displayed by strongly coupled gauge theories. In that context, following the pioneer work of Polchinski and Strassler Polchinski and Strassler (2003), different approaches to DIS in holography have been proposed using both top-down and bottom-up models (see, for example, Borsa et al. (2023); Kovensky et al. (2018a); Hatta et al. (2008); Ballon Bayona et al. (2008a, b); Cornalba and Costa (2008); Pire et al. (2008); Albacete et al. (2008); Ballon Bayona et al. (2008c); Gao and Xiao (2009); Taliotis (2009); Yoshida (2010); Hatta et al. (2009); Avsar et al. (2009); Cornalba et al. (2010a); Bayona et al. (2010); Cornalba et al. (2010b); Brower et al. (2010); Gao and Mou (2010); Ballon Bayona et al. (2010); Braga and Vega (2012); Koile et al. (2014, 2015a); Gao and Mou (2014); Folco Capossoli and Boschi-Filho (2015); Folco Capossoli et al. (2020a); Koile et al. (2015b); Jorrin et al. (2016a, b); Kovensky et al. (2018b); Amorim et al. (2018); Watanabe et al. (2020); Jorrin et al. (2020); Amorim and Costa (2021); Mamo and Zahed (2021); Tahery et al. (2023); Jorrin and Schvellinger (2022); Bigazzi and Castellani (2024); Mayrhofer (2024)). In particular, such models are devised to reproduce confinement, among other properties of strong interactions.

A particular set of holographic bottom-up models, originally investigated in Gursoy and Kiritsis (2008); Gursoy et al. (2008a), involve the metric and a dynamical dilaton scalar field in 5-dimensional asymptotically anti-de Sitter (AdS) spacetime. In this dynamical framework, a bulk action describing the general interactions between the five-dimensional metric $g_{mn}$, dual to the energy-momentum tensor of the boundary gauge theory, and a scalar field $\phi$, responsible for breaking conformal invariance and dual to the glueball operator $\textit{Tr}\,F^{2}$, is used to determine the coupled set of equations of motion describing the bulk geometry and the scalar, which may then be solved to determine the properties of the dual gauge theory. This is different than other AdS/QCD approaches, such as Erlich et al. (2005); Karch et al. (2006), where the metric is given and it is not necessarily a solution of equations of motion stemming from a well-defined action. The unknown in this bottom-up model is the dilaton potential $V(\phi)$, which is chosen to reproduce the properties of the problem at hand, such as linear confinement and glueball spectra, see Gursoy and Kiritsis (2008); Gursoy et al. (2008a) (see also Refs. Ballon-Bayona et al. (2018, 2023); Ballon-Bayona and Junior (2024) for related work on hadronic spectroscopy with or without chiral symmetry breaking at zero temperature). Once the potential is chosen, no other parameters are needed, and the corresponding results obtained from this approach can be seen as predictions of the model. Einstein-Dilaton models have also been very useful to investigate thermodynamic and transport properties of QCD at nonzero temperature Gursoy et al. (2009, 2008b); Gubser and Nellore (2008); Gubser et al. (2008a, b); Noronha (2010); Gursoy et al. (2011); Finazzo and Noronha (2014a, b) and baryon density²²2Nonzero baryon density requires including in the holographic model a dynamical gauge field in the bulk. DeWolfe et al. (2011a, b); Rougemont et al. (2016, 2015); Critelli et al. (2017); Ballon-Bayona et al. (2020); Grefa et al. (2021, 2022); Hippert et al. (2024), for a comprehensive review, see Rougemont et al. (2024). Finally, we remark that a generalization of the Einstein-Dilaton model to describe strong-coupled gauge theories in the Veneziano limit (where both the number of colors, $N$, and the number of flavors, $N_{f}$, are taken to infinity, though $N_{f}/N$ is finite), known as V-QCD, were developed in Jarvinen and Kiritsis (2012) (see Deng and Hou (2025) for an application in the context of nuclear structure).

In this work, we present for the first time a framework to study DIS of (unpolarized) targets in Einstein-Dilaton models. The background fields are obtained by solving the equations of motion following from the Einstein-Dilaton model. We follow Polchinski and Strassler Polchinski and Strassler (2003) and introduce probe fermionic and gauge fields in this dynamically determined bulk. The probe fermionic fields in this model obey a Schrödinger-like equation with a confining potential, as demonstrated in this work. We then show how to use the solution of the equations of motion of the probe fields to calculate the $F_{1}$ and $F_{2}$ structure functions of the proton. As an application of this approach, we use the particular realization of the Einstein-Dilaton model worked out in Ref. Finazzo and Noronha (2014b), which provided a good description of the tensor glueball spectrum Lucini and Teper (2001); Lucini et al. (2010) and the thermodynamic properties Panero (2009) of large $N$ pure Yang-Mills theory. This way, the background is fixed and no new parameters are introduced in that regard. We numerically solve the equations of motion of the background, and use this background to numerically solve the equations of motion of the probe fields needed for the holographic DIS calculation. This particular realization of the model can successfully describe the proton mass and provide results for $F_{2}$ at large $x$ that are in qualitative agreement with experimental data, though quantitative agreement is not found. The model also reproduces some other general features common to other holographic models previously used to study DIS.

This paper is organized as follows. In Section II, we review how the structure functions of an unpolarized fixed target show up from the DIS amplitude through the definition of the hadronic tensor. In Section III, we briefly describe the holographic approach of Polchinski and Strassler (2003) to compute the hadronic tensor through a probe interaction action in the bulk spacetime. Next, in Sect. IV, we present the details of the Einstein-Dilaton model we use, and how we solve its equations of motion. Section V is dedicated to studying the fermionic and gauge probe fields in this geometry, in particular their equations of motion and their solutions. We also present the potentials and wave functions of the fermionic fields involved. Section VI is devoted to the evaluation of the interaction action between those fields and to the computation of the hadronic tensor and the further extraction of the desired structure function. In Section VII we present our numerical results for the proton structure function $F_{2}$, and the comparison to experimental data, for the specific choice of the dilaton potential used in this work. Finally, in VIII we discuss these results and draw conclusions regarding the limitations of Einstein-Dilaton models used to model DIS problems at large $x$.

Our notation is the following: $g_{mn}$ is the five-dimensional metric of the asymptotic AdS spacetime with determinant $g=\det(g_{mn})$ and, $\eta_{\mu\nu}$ is the Minkowski metric for the flat four-dimensional spacetime, with $\mu,\nu=0,1,2,3$. We use natural units $\hbar=c=k_{B}=1$.

Deep inelastic scattering plays an essential role in the description of nucleon structure. In particular, to study the internal structure of a proton, one can make use of lepton-proton inelastic scattering in which an incoming lepton hits the proton and interacts with it through the exchange of a virtual photon of momentum $q$. With energies high enough above $1$ GeV, the virtual photon can probe the internal substructure of a proton. This process is depicted in Fig. 1. The state $X$ represents the final state, which is no longer the proton.

The internal structure of a proton can be described by the hadronic tensor in four dimensions, defined as

where $J^{\mu}$ represents the electromagnetic hadronic current and $s$ the spin of the proton. In this work we are going to deal with an unpolarized target, for which the most general tensor decomposition of $W^{\mu\nu}$ can be written as Manohar (1992)

where $x=-{q^{2}}/{(2P\cdot q)}\,$ is the momentum fraction carried by the parton (i.e., the Bjorken $x$ parameter), and $F_{1,2}(x,q)$ are the unpolarized DIS structure functions, with $0\leq x\leq 1$ and $q^{2}>0$. The lower the value of $x$, the more inelastic the process is. The limit $x\to 1$ corresponds to the elastic regime.

Inserting a complete set of final states $\ket{X}$ in (1), one can write $W^{\mu\nu}$ as

The task is then to compute the matrix element in Eq. (3). In the following section, we describe the holographic prescription put forward by Polchinski and Strassler (2003) to perform this calculation. Later on, in Sec. VI, we proceed with the full calculation of this quantity in our Einstein-Dilaton model.

As discussed in the previous section, the hadronic tensor is given in terms of the matrix element

To calculate (4) using holographic tools, we follow steps similar to the approach used by Polchinski and Strassler in Polchinski and Strassler (2003). The idea is to consider the matrix element above given by an interaction action defined as

where $z$ is the holographic coordinate, and $g_{V}$ is the coupling vertex of the photon field ${\cal A}^{m}$ with the initial and final fermionic states $\Psi_{i}$ and $\Psi_{X}$, involved the DIS process, respectively. All these fields live in the bulk of the five-dimensional asymptotic AdS space, and $\Gamma_{m}$ are the four-component Dirac matrices in those dimensions (see Section V.2 for details).

Polschinski and Strassler Polchinski and Strassler (2003), and other related works (see, e.g. Borsa et al. (2023); Kovensky et al. (2018a); Hatta et al. (2008); Ballon Bayona et al. (2008a, b); Cornalba and Costa (2008); Pire et al. (2008); Albacete et al. (2008); Ballon Bayona et al. (2008c); Gao and Xiao (2009); Taliotis (2009); Yoshida (2010); Hatta et al. (2009); Avsar et al. (2009); Cornalba et al. (2010a); Bayona et al. (2010); Cornalba et al. (2010b); Brower et al. (2010); Gao and Mou (2010); Ballon Bayona et al. (2010); Braga and Vega (2012); Koile et al. (2014, 2015a); Gao and Mou (2014); Folco Capossoli and Boschi-Filho (2015); Folco Capossoli et al. (2020a); Koile et al. (2015b); Jorrin et al. (2016a, b); Kovensky et al. (2018b); Amorim et al. (2018); Watanabe et al. (2020); Jorrin et al. (2020); Amorim and Costa (2021); Mamo and Zahed (2021); Tahery et al. (2023); Jorrin and Schvellinger (2022); Bigazzi and Castellani (2024); Mayrhofer (2024)) compute (5) by first solving the equations of motion for the fields, derived from actions of the form

where $\mathcal{L}$ includes the probe fermionic and gauge fields, and then insert the solutions back into the matrix element in order to read off the structure functions. In previous works, the form of the asymptotic $AdS_{5}$ space metric was given, with properties such that some relevant QCD phenomenology could be reproduced, e.g., confinement. In this work, however, we go further and consider a new approach in this context. Instead of choosing an ad hoc particular geometry to reproduce some QCD property, we start from an Einstein-Dilaton general action involving the metric and the scalar field and solve the equations of motion for these fields to determine the bulk geometry. Once the on-shell solution for the background is known, we solve the equations for the probe fields on this background to extract the structure functions. This way, one has greater systematic control about the properties of the bulk geometry, and its regime of validity (for example, concerning its behavior near singularities Gubser (2000)). In the following section, we derive the equations of motion of the background Einstein-Dilaton model, and explain how we solve them numerically.

In this section we give the details about the Einstein-Dilaton model used in this work. We follow Ref. Finazzo and Noronha (2014b), which investigated how the choice of the dilaton potential affects the Debye mass computed from Einstein-Dilaton models. The particular choice of potential used in this work corresponds to model B1 (see Sect. IV.3) in Finazzo and Noronha (2014b), which led to a good description of the tensor glueball spectrum and the thermodynamic properties of large $N$ pure Yang-Mills theory.

We start with a general Einstein-Dilaton action (with at most two derivatives) in the Einstein frame to describe the bulk

where $\kappa_{5}^{2}=8\pi G_{5}$, with $G_{5}$ being Newton’s constant in five dimensions. We consider here the case at zero temperature, and follow the Gubser-Nellore Ansatz Gubser and Nellore (2008) for the metric given by

Above, the holographic radial coordinate is given by the scalar field $\phi$ itself. The functions $A$ and $B$ depend on $\phi$. The potential $V(\phi)$ will be specified in Sect. IV.3. In the following, we will derive the equations of motion and solve them in order to determine the functions $A$ and $B$ to fully specify the background spacetime (8).

In the DIS process, the particles involved are probe fields in the bulk geometry we have obtained in the last section. We now proceed to obtain their equations of motion. This is done considering the actions for each field, separately.

In this section, we will explicitly compute the DIS interaction action in Eq. (5), repeated here for convenience

Using the definitions of the vielbein given by Eqs. (52) and (54), associated with the geometry of the metric (8), one can rewrite the interaction action $S_{\rm int}$, disregarding the component ${\cal A}^{\phi}$ as discussed in Sect. V.1, as follows