Advances in High Energy Physics

Volume 2019, Article ID 3036904, 68 pages

https://doi.org/10.1155/2019/3036904

## A Guide to Light-Cone PDFs from Lattice QCD: An Overview of Approaches, Techniques, and Results

^{1}Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland^{2}Department of Physics, Temple University, Philadelphia, PA 19122 - 1801, USA

Correspondence should be addressed to Martha Constantinou; ude.elpmet@cahtram

Received 17 November 2018; Accepted 15 January 2019; Published 2 June 2019

Guest Editor: Alexei Prokudin

Copyright © 2019 Krzysztof Cichy and Martha Constantinou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Within the theory of Quantum Chromodynamics (QCD), the rich structure of hadrons can be quantitatively characterized, among others, using a basis of universal nonperturbative functions: parton distribution functions (PDFs), generalized parton distributions (GPDs), transverse momentum dependent parton distributions (TMDs), and distribution amplitudes (DAs). For more than half a century, there has been a joint experimental and theoretical effort to obtain these partonic functions. However, the complexity of the strong interactions has placed severe limitations, and first-principle information on these distributions was extracted mostly from their moments computed in Lattice QCD. Recently, breakthrough ideas changed the landscape and several approaches were proposed to access the distributions themselves on the lattice. In this paper, we review in considerable detail approaches directly related to partonic distributions. We highlight a recent idea proposed by X. Ji on extracting quasidistributions that spawned renewed interest in the whole field and sparked the largest amount of numerical studies within Lattice QCD. We discuss theoretical and practical developments, including challenges that had to be overcome, with some yet to be handled. We also review numerical results, including a discussion based on evolving understanding of the underlying concepts and the theoretical and practical progress. Particular attention is given to important aspects that validated the quasidistribution approach, such as renormalization, matching to light-cone distributions, and lattice techniques. In addition to a thorough discussion of quasidistributions, we consider other approaches: hadronic tensor, auxiliary quark methods, pseudodistributions, OPE without OPE, and good lattice cross-sections. In the last part of the paper, we provide a summary and prospects of the field, with emphasis on the necessary conditions to obtain results with controlled uncertainties.

#### 1. Introduction

Among the frontiers of nuclear and particle physics is the investigation of the structure of hadrons, the architecture elements of the visible matter. Hadrons consist of quarks and gluons (together called partons), which are governed by one of the four fundamental forces of nature, the strong force. The latter is described by the theory of Quantum Chromodynamics (QCD). Understanding QCD can have great impact on many aspects of science, from the subnuclear interactions to astrophysics, and, thus, a quantitative description is imperative. However, this is a very challenging task, as QCD is a highly nonlinear theory. This led to the development of phenomenological tools such as models, which have provided important input on the hadron structure. However, studies from first principles are desirable. An ideal* ab initio* formulation is Lattice QCD, a space-time discretization of the theory that allows the study of the properties of fundamental particles numerically, starting from the original QCD Lagrangian.

Despite the extensive experimental program that was developed and evolved since the first exploration of the structure of the proton [1, 2], a deep understanding of the hadrons’ internal dynamics is yet to be achieved. Hadrons have immensely rich composition due to the complexity of the strong interactions that, for example, forces the partons to exist only inside the hadrons (color confinement), making the extraction of information from experiments very difficult.

Understanding internal properties of the hadrons requires the development of a set of appropriate quantities that can be accessed both experimentally and theoretically. The QCD factorization provides such formalism and can relate measurements from different processes to parton distributions. These are nonperturbative quantities describing the parton dynamics within a hadron and have the advantage of being universal, that is, do not depend on the process used for their extraction. The comprehensive study of parton distributions can provide a wealth of information on the hadrons, in terms of variables defined in the longitudinal direction (with respect to the hadron momentum) in momentum space, and two transverse directions. The latter can be defined either in position or in momentum space. These variables are as follows: the longitudinal momentum fraction carried by the parton, the longitudinal momentum fraction obtained via the longitudinal momentum transferred to the hadron, and the momentum transverse to the hadron direction of movement. Parton distributions can be classified into three categories based on their dependence on , , and the momentum transferred to the hadron, , as described below.

*Parton distribution functions* (PDFs) are one-dimensional objects and represent the number density of partons with longitudinal momentum fraction while the hadron is moving with a large momentum.

*Generalized parton distributions* (GPDs) [3–7] depend on the longitudinal momentum fractions and and, in addition, on the momentum transferred to the parent hadron, . They provide a partial description of the three-dimensional structure.

*Transverse momentum dependent parton distribution functions* (TMDs) [8–12] describe the parton distribution in terms of the longitudinal momentum fraction and the transverse momentum . They complement the three-dimensional picture of a hadron from GPDs.

As is clear from the above classification, PDFs, GPDs, and TMDs provide complementary information on parton distributions, and all of them are necessary to map out the three-dimensional structure of hadrons in spatial and momentum coordinates. Experimentally, these are accessed from different processes, with PDFs being measured in inclusive or semi-inclusive processes such as deep-inelastic scattering (DIS) and semi-inclusive DIS (SIDIS); see e.g., [13], for a review of DIS. GPDs are accessed in exclusive scattering processes such as Deeply Virtual Compton Scattering (DVCS) [14], and TMDs in hard processes in SIDIS [10, 11]. Most of the knowledge on the hadron structure is obtained from DIS and SIDIS data on PDFs, while the GPDs and TMDs are less known. More recently, data emerge from DVCS and Deeply Virtual Meson Production (DVMP) [15]. This includes measurements from HERMES, COMPASS, RHIC, Belle and Babar, E906/SeaQuest, and the 12 GeV upgrade at JLab. A future Electron-Ion Collider (EIC), that was strongly endorsed by the National Academy of Science, Engineering and Medicine [16], will be able to provide accurate data related to parton distributions and will advance dramatically our understanding on the hadron tomography. Together with the experimental efforts, theoretical advances are imperative in order to obtain a complete picture of hadrons. First, to interpret experimental data, global QCD analyses [17–26] are necessary that utilize the QCD factorization formalism and combine experimental data and theoretical calculations in perturbative QCD. Note that these are beyond the scope of this review and we refer the interested Reader to the above references and a recent community white paper [27]. Second, theoretical studies are needed to complement the experimental program and, in certain cases, provide valuable input. This is achieved using models of QCD and more importantly calculations from first principles. Model calculations have evolved and constitute an important aspect of our understanding of parton structure. An example of such a model is the diquark spectator model [28] that has been used for studies of parton distributions (for more details, see Section 4). The main focus of the models discussed in Section 4 is the one-dimensional hadron structure (-dependence of PDFs), but more recently the interest has been extended to the development of techniques that are also applicable to GPDs and TMDs (some aspects are discussed in this review). Let us note that there have been studies related to TMDs from the lattice, and there is intense interest towards that direction (see, e.g., [29–31], and references therein).

Despite the tremendous progress in both the global analyses and the models of QCD, parton distributions are not fully known, due to several limitations: global analysis techniques are not uniquely defined [22]; certain kinematic regions are difficult to access, for instance, the very small -region [32–34]; and models cannot capture the full QCD dynamics. Hence, an* ab initio* calculation within Lattice QCD is crucial, and synergy with global fits and model calculations can lead to progress in the extraction of distribution functions.

Lattice QCD provides an ideal formulation to study hadron structure and originates from the full QCD Lagrangian by defining the continuous equations on a discrete Euclidean four-dimensional lattice. This leads to equations with billions of degrees of freedom, and numerical simulations on supercomputers are carried out to obtain physical results. A nonperturbative tool, such as Lattice QCD, is particularly valuable at the hadronic energy scales, where perturbative methods are less reliable, or even fail altogether. Promising calculations from Lattice QCD have been reported for many years with the calculations of the low-lying hadron spectrum being such an example. More recently, Lattice QCD has provided pioneering results related to hadron structure, addressing, for instance, open questions, such as the spin decomposition [35] and the glue spin [36] of the proton. Another example of the advances of numerical simulations within Lattice QCD is the calculation of certain hadronic contributions to the muon , for example, the connected and leading disconnected hadronic light-by-light contributions (see recent reviews of [37, 38]). Direct calculations of distribution functions on a Euclidean lattice have not been feasible due to the time dependence of these quantities. A way around this limitation is the calculation on the lattice of moments of distribution functions (historically for PDFs and GPDs) and the physical PDFs can, in principle, be obtained from operator product expansion (OPE). Realistically, only the lowest moments of PDFs and GPDs can be computed (see, e.g., [39–44]) due to large gauge noise in high moments, and also unavoidable power-divergent mixing with lower-dimensional operators. Combination of the two prevents a reliable and accurate calculation of moments beyond the second or third, and the reconstruction of the PDFs becomes unrealistic.

Recent pioneering work of X. Ji [45] has changed the landscape of lattice calculations with a proposal to compute equal-time correlators of momentum boosted hadrons, the so-called quasidistributions. For large enough momenta, these can be related to the physical (light-cone) distributions via a matching procedure using Large Momentum Effective Theory (LaMET) (see Sections 3.1 and 8). This possibility has opened new avenues for direct calculation of distribution functions from Lattice QCD and first investigations have revealed promising results [46, 47] (see Section 3.2). Despite the encouraging calculations, many theoretical and technical challenges needed to be clarified. One concern was whether the Euclidean quasi-PDFs and Minkowski light-cone PDFs have the same collinear divergence, which underlies the matching programme. In addition, quasi-PDFs are computed from matrix elements of nonlocal operators that include a Wilson line. This results in a novel type of power divergences and the question whether these operators are multiplicatively renormalizable remained unanswered for some time. While the theoretical community was addressing such issues, the lattice groups had to overcome technical difficulties related to the calculation of matrix elements of nonlocal operators, including how to obtain reliable results for a fast moving nucleon, and how to develop a nonperturbative renormalization prescription (see Section 7). For theoretical and technical challenges, see Sections 5-6. Our current understanding on various aspects of quasi-PDFs has improved significantly, and lattice calculations of quasi-PDFs have extended to quantities that are not easily or reliably measured in experiments (see Sections 9-10), such as the transversity PDF [48, 49]. This new era of LQCD can provide high-precision input to experiments and test phenomenological models.

The first studies on Ji’s proposal have appeared for the quark quasi-PDFs of the proton (see Sections 3.2 and 9). Recently, the methodology has been extended to other hadrons, in particular mesonic PDFs and distribution amplitudes (DAs). Progress towards this direction is presented in Section 10. Other recent reviews on the -dependence of PDFs from Lattice QCD calculations can be found in [27, 50, 51]. The quasi-PDFs approach is certainly promising and can be generalized to study gluon quasi-PDFs, quasi-GPDs, and quasi-TMDs. In such investigations, technical difficulties of different nature arise and must be explored. First studies are presented here. Apart from the quasidistribution approach, we also review other approaches for obtaining the -dependence of partonic functions, both the theoretical ideas underlying them (see Section 2) and their numerical explorations (Section 11).

The central focus of the review is the studies of the -dependence of PDFs. We present work that appears in the literature until November 10, 2018 (published, or on the arXiv). The discussion is extended to conference proceedings for recent work that has not been published elsewhere. The presentation is based on chronological order, unless there is a need to include follow-up work by the same group on the topic under discussion. Our main priority is to report on the progress of the field, but also to comment on important aspects of the described material based on theoretical developments that appeared in later publications, or follow-up work. To keep this review at a reasonable length, we present selected aspects of each publication discussed in the main text and we encourage the interested Reader to consult the referred work. Permission for reproduction of the figures has been granted by the Authors and the scientific journals (in case of published work).

The rest of the paper is organized as follows. In Section 2, we introduce methods that have been proposed to access the -dependence of PDFs from the lattice, which include a method based on the hadronic tensor, auxiliary quark field approaches, quasi- and pseudodistributions, a method based on OPE, and the good lattice cross-sections approach. A major part of this review is dedicated to quasi-PDFs, which are presented in more detail in Section 3, together with preliminary studies within Lattice QCD. The numerical calculations of the early studies have motivated an intense theoretical activity to develop models of quasidistributions, which are presented in Section 4. In Section 5, we focus on theoretical aspects of the approach of quasi-PDFs, that is, whether a Euclidean definition can reproduce the light-cone PDFs, as well as the renormalizability of operators entering the calculations of quark and gluon quasi-PDFs. The lattice techniques for quasi-PDFs and difficulties that one must overcome are summarized in Section 6. Recent developments on the extraction of renormalization functions related to logarithmic and/or power divergences are explained in Section 7, while Section 8 is dedicated to the matching procedure within LaMET. Lattice results on the quark quasi-PDFs for the nucleon are presented in Section 9. The quasi-PDFs approach has been extended to gluon distributions, as well as studies of mesons, as demonstrated in Section 10. In Section 11, we briefly describe results from the alternative approaches presented in Section 2. We close the review with Section 12 that gives a summary and future prospects. We discuss the -dependence of PDFs and DAs, as well as possibilities to study other quantities, such as GPDs and TMDs. A glossary of abbreviations is given in the Appendix.

#### 2. -Dependence of PDFs

In this section, we briefly outline different approaches for obtaining the -dependence of partonic distribution functions, in particular collinear PDFs. We first recall the problem by directly employing the definitions of such functions on the lattice, using the example of unpolarized PDFs. The unpolarized PDF, denoted here by , is defined on the light cone:where is the hadron state with momentum , in the standard relativistic normalization^{1}, the light-cone vectors are taken as , and is the Wilson line connecting the light-cone points and , while the factorization scale is kept implicit. Such light-cone correlations are not accessible on a Euclidean spacetime, because the light-cone directions shrink to one point at the origin. As discussed in the Introduction, this fact prevented lattice extraction of PDFs for many years, apart from their low moments, reachable via local matrix elements and the operator product expansion (OPE). However, since the number of moments that can be reliably calculated is strongly limited, alternative approaches were sought for to yield the full Bjorken- dependence.

The common feature of all the approaches is that they rely to some extent on the factorization framework. For a lattice observable that is to be used to extract PDFs, one can generically write the following:^{2}where is a perturbatively computable function and is the desired PDF. In the above expression we distinguish between the factorization scale, , and the renormalization scale, . These scales are usually taken to be the same and, hence, from now on we will adopt this choice and take . Lattice approaches use different observables that fall into two classes: (1)Observables that are generalizations of light-cone functions such that they can be accessed on the lattice; such generalized functions have direct -dependence, but does not have the same partonic interpretation as the Bjorken-.(2)Observables in terms of which hadronic tensor can be written; the hadronic tensor is then decomposed into structure functions like and , which are factorizable into PDFs.

Below, we provide the general idea for several proposals that were introduced in recent years.

##### 2.1. Hadronic Tensor

All the information about a DIS cross-section is contained in the hadronic tensor [52–56], defined bywhere is the hadron state labeled by its momentum and polarization , is virtual photon momentum, and is electromagnetic current at point . The hadronic tensor can be related to DIS structure functions and hence, in principle, PDFs can be extracted from it. is the imaginary part of the forward Compton amplitude and can be written as the current-current correlation functionwhere the subscript denotes averaging over polarizations. The approach has been introduced as a possible way of investigating hadronic structure on the lattice by K.-F. Liu and S.-J. Dong already in 1993. They also proposed a decomposition of the contributions to the hadronic tensor according to different topologies of the quark paths, into valence and connected or disconnected sea ones. In this way, they addressed the origin of Gottfried sum rule violation.

A crucial aspect for the implementation in Lattice QCD is the fact that the hadronic tensor , defined in Minkowski spacetime, can be obtained from the Euclidean path-integral formalism [54–58], by considering ratios of suitable four-point and two-point functions. In the limit of the points being sufficiently away from both the source and the sink, where the hadron is created or annihilated, the matrix element receives contributions from only the ground state of the hadron. Reconstructing the Minkowski tensor from its Euclidean counterpart is formally defined by an inverse Laplace transform of the latter and can, in practice, be carried out using, e.g., the maximum entropy method or the Backus-Gilbert method. Nevertheless, this aspect is highly nontrivial and improvements thereof are looked for. As pointed out in [59], a significant role may be played by power-law finite volume effects related to the matrix elements being defined in Euclidean spacetime. A similar phenomenon was recently observed also in the context of mixing [60]. Another difficulty of the hadronic tensor approach on the lattice is the necessity to calculate four-point correlation functions, which is computationally more intensive than for three-point functions, the standard tools of hadron structure investigations on the lattice. However, the theoretical appeal of the hadronic tensor approach recently sparked renewed interest in it [61–63]. We describe some exploratory results in Section 11.1.

##### 2.2. Auxiliary Scalar Quark

In 1998, a new method was proposed to calculate light-cone wave functions (LCWFs) on the lattice [64]. This finds its motivation from the fact that LCWFs enter the description of many processes, such as electroweak decays and meson production. LCWF is the leading term of the full hadronic wave function in the expansion, where is the hadron momentum. For concreteness, we write the defining expression for the most studied LCWF, the one of the pion, , where is the momentum fraction:with being boosted pion state, being vacuum state, and being pion decay constant. is the Wilson line that ensures gauge invariance of the matrix element.

The essence of the idea is to “observe” and study on the lattice the partonic constituents of hadrons instead of the hadrons themselves [65, 66]. As shown in [64], the pion LCWF can be extracted by considering a vacuum-to-pion expectation value of the axial vector current with quark fields separated in spacetime. Gauge invariance is ensured by a scalar quark propagator with color quantum numbers of a quark, and at a large momentum transfer. The relation between the Fourier transform of this matrix element, computed on the lattice, and the pion LCWF, , is given by the following formula:where is momentum transfer, is scalar colored propagator, and is discrete set of partonic momentum fractions (allowed by the discretized momenta in a finite volume). The spacetime points are explained in Figure 1, which shows the three-point function that needs to be computed. The interval needs to be large to have an on-shell pion. To extract the LCWF, several conditions need to be satisfied: injected pion momentum needs to be large (to have a “frozen” pion and see its partonic constituents), the scalar quark needs to carry large energy, the time (time of momentum transfer and “transformation” of a quark to a scalar quark) has to be small (to prevent quantum decoherence and hadronization), and the lattice volume has to be large enough (to minimize effects of discretizing parton momenta). We refer to the original papers for an extensive discussion of these conditions. An exploratory study of the approach was presented in [65, 66] and later in [67], both in the quenched approximation. Naturally, the conditions outlined above are very difficult to satisfy simultaneously on the lattice, due to restrictions from the finite lattice spacing and the finite volume. However, the knowledge of the full hadronic wave function from first principles would be very much desired and further exploration of this approach may be interesting. In particular, integrals of hadronic wave functions over transverse momenta yield distribution amplitudes and PDFs.