Gluon Momentum Research Articles

Impressions of Parton Distribution Functions

Abstract Parton distribution functions (DFs) have long been recognised as key measures of hadron structure. Today, theoretical prediction of such DFs is becoming possible using diverse methods for continuum and lattice analyses of strong interaction (QCD) matrix elements. Recent developments include a demonstration that continuum and lattice analyses yield mutually consistent results for all pion DFs, with behaviour on the far valence domain of light-front momentum fraction that matches QCD expectations. Theory is also delivering an understanding of the distributions of proton mass and spin amongst its constituents, which varies, of course, with the resolving scale of the measuring probe. Aspects of the pion DF and proton spin developments are sketched herein, along with some novel perspectives on gluon and quark orbital angular momentum.

Gluon moment and parton distribution function of the pion from Nf=2+1+1 lattice QCD

We present the first calculation of the pion gluon moment from lattice QCD in the continuum-physical limit. The calculation is done using clover fermions for the valence action with three pion masses, 220, 310 and 690 MeV, and three lattice spacings, 0.09, 0.12, and 0.15 fm, using ensembles generated by MILC Collaboration with 2+1+1 flavors of highly improved staggered quarks (HISQ). On the lattice, we nonperturbatively renormalize the gluon operator in RI/MOM scheme using the cluster-decomposition error reduction (CDER) technique to enhance the signal-to-noise ratio of the renormalization constant. We extrapolate the pion gluon moment to the continuum-physical limit and obtain ⟨x⟩g=0.394(58)stat+NPR(39)mixing in the MS¯ scheme at 2 GeV, with first error being the statistical error and uncertainties in nonperturbative renormalization, and the second being a systematic uncertainty estimating the effect of ignoring quark mixing. Our pion gluon momentum fraction has a central value lower than two recent single-ensemble lattice-QCD results near physical pion mass but is consistent with the recent global fits by JAM and xFitter and with most QCD-model estimates. Published by the American Physical Society 2024

Yang-Mills as a Liouville theory

We propose a description of the gluon scattering amplitudes as the inverse Mellin transforms of the conformal correlators of light operators in two-dimensional Liouville theory tensored with WZW-like chiral currents on the celestial sphere. The dimensions of operators are Mellin dual to gluon light cone energies while their positions are determined by the gluon momentum directions. Tree-level approximation in Yang-Mills theory corresponds to the semiclassical limit of Liouville theory. By comparing subleading corrections, we find b2=(8π2)−1β0g2(M), where b is the Liouville coupling constant, g(M) is the Yang Mills coupling at the renormalization scale M and β0 is the one-loop coefficient of the Yang-Mills beta function.

Spin-1 quarkonia in a rotating frame and their spin contents

We propose a new way of studying the spin content of a hadron by looking at its response in a rotating frame. By collecting all responses of quarks and gluons in a rotating frame, we describe the spin-rotation coupling of spin-1 quarkonia and thereby reveal their spin contents in a relativistic formalism. We demonstrate that both the perturbative and non-perturbative contributions in the operator product expansion follow a universal formula that identifies the spin-rotation coupling with unit strength. This allows us to recognize the total spin-1 of the vector and axial vector quarkonia in terms of the total angular momentum of quarks and gluons. Specifically, we find the spin contents of J/ψ, χc1, ϒ(1S), and χb1 are slightly different from the naive quark model picture. For example, the J/ψ is traditionally considered as an S-wave particle, but we find quarks do not carry all of the total spin.

Parton decomposition of nucleon spin and momentum: Gluons from dressed quarks

The lowest two Mellin moments of hadronic Generalized Parton Distributions are explored within a model that allows investigation of the inter-related quark and gluon contributions. For light quarks their dynamical connection is strong due to quark dressing. Our principal focus is the angular momentum J of the nucleon. This work employs and extends dynamical insights obtained from our recent model results for quark and gluon momentum fractions 〈x〉q/g in both pion and nucleon. The employed model is based on the Rainbow-Ladder truncation of the Dyson-Schwinger equations of QCD. The special case of a 1-loop treatment of a single hadronic quark is used to motivate several insights and obtain initial estimates such as the Wilson line correction to the established Landau gauge model (−7% for both Jq and 〈x〉q), and the “binding gluon” contribution to 〈x〉g (≤10%). We obtain the proton J within 1% after inclusion of the pion cloud mechanism to produce the sea. The gluon second Mellin moments of the proton reflect similar dynamics; (〈x〉g, Jg) are (26%, 24%) at model scale, and (40%, 38%) at 2 GeV. We also find that Jtot is shared almost equally between the total orbital and total intrinsic spin contributions.

Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. This extends previous works [J. A. Gracey et al., Phys. Rev. D 100, 034023 (2019); N. Barrios et al., Phys. Rev. D 102, 114016 (2020)] that considered similarly the two-loop ghost and gluon two-point functions as well as the two-loop ghost-antighost-gluon vertex (with vanishing gluon momentum). With the parameters of the model being adjusted by fitting the two-point functions to available lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case as already seen in previous studies. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the Faddeev-Popov framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [G. T. R. Catumba et al., EPJ Web Conf. 258, 02008 (2022)]. Finally, our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [F. Pinto-G\'omez et al., arXiv:2208.01020].

Unintegrated gluon distributions from the color dipole cross section in the BGK saturation model

We evaluate the unintegrated gluon distribution of the proton starting from a parametrization of the color dipole cross section including Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution and saturation effects. To this end, we perform the Fourier-Bessel transform of σ(x,r)/α(r). At large transverse momentum of gluons we match the so-obtained distribution to the logarithmic derivative of the collinear gluon distribution. We check our approach by calculating the proton structure function FL(x,Q2) finding good agreement with HERA data.

Physical characteristics of glasma from the earliest stage of relativistic heavy ion collisions

We present analytic results that describe the gluon field, or glasma, at very early times after a collision of relativistic heavy ions at proper time $\tau=0$. We use a Colour Glass Condensate approach, and perform an expansion in $\tau$. The full details of our method are described in our previous paper [1]. In this paper we present an analysis of various physical quantities that can be obtained from the energy-momentum tensor. We show that the expansion to order $\tau^6$ can be trusted to about $\tau=0.05$ fm. For times small enough that the expansion converges, the transverse and longitudinal pressures move towards their equilibrium values of one third of the energy density. The Fourier coefficients of the azimuthal flow are larger than expected, which contradicts the usual assumption that anisotropy is mostly generated during the hydrodynamic evolution of the plasma. We find a significant correlation between the elliptic flow coefficient and the eccentricity, which indicates that the spatial asymmetry introduced by the initial geometry is effectively transmitted to the azimuthal distribution of the gluon momentum field, even at very early times. This result is interesting because correlations of this kind are characteristic of the onset of hydrodynamic behaviour. We show that the angular momentum of the glasma is orders of magnitude smaller than the angular momentum of the initial system of ions colliding with non-zero impact parameter. This indicates that most of the angular momentum carried by the valence quarks is not transmitted to the glasma, and contradicts the picture of a rapidly rotating initial glasma state that has been proposed by several authors, but agrees with the current lack of experimental evidence for a significant polarization effect of the hyperons and vector mesons produced in heavy ion collisions at the highest accessible energies.

Proton generalized parton distributions from lattice QCD

Momentum and spatial distributions of quarks and gluons inside hadrons are typically encoded in the so-called generalized parton distributions (GPDs). GPDs are multi-dimensional quantities that are very challenging to extract, both experimentally and within lattice QCD. We present the first lattice results on the x-dependence of isovector unpolarized, helicity and transversity GPDs of the proton, obtained from lattice QCD using an ensemble of Nf = 2 + 1 + 1 maximally twisted mass fermions, with pion mass Mπ = 260 MeV and lattice spacing a ' 0.093 fm. Our calculations use the quasi-distribution formalism and the final distributions are presented in the MS scheme at a renormalization scale of 2 GeV.

Energy dependence of the proton geometry in exclusive vector meson production

The gluon radius of the proton is expected to increase at small gluon momentum fractions $x$, an effect which has hitherto not been considered in the dipole model framework. We investigate the energy dependence of exclusive $J/\ensuremath{\psi}$, $\ensuremath{\phi}$, and $\ensuremath{\rho}$ production by introducing three models for $x$ dependence of the gluon thickness function. We allow the transverse width of the proton to increase as $x$ decreases, using novel parametrizations in the spherical proton and the hotspot model. We compare these with a model where the number of hotspots increases as $x$ decreases and confront the models with HERA data. The models exhibit clear differences in the slope of the $t$-spectra and in the cross section ratios between coherent and incoherent events. Comparisons to $t$-slopes and ${W}_{\ensuremath{\gamma}p}$ measurements show a preference for models where the proton's size increases as $x$ decreases.

Final-state gluon emission in deep-inelastic scattering at next-to-leading twist

A detailed reanalysis of the single gluon emission rate at next-to-leading twist is carried out. As was the case in prior efforts, the problem is cast in the framework of deep-inelastic scattering (DIS) of an electron off a large nucleus. The quark produced in the interaction propagates through the remaining nucleus and engenders scattering and gluon radiation, which is calculated in the limit of one re-scattering. This medium induced single gluon emission rate forms the basis of several energy loss calculations in both DIS and heavy-ion collisions. Unlike prior efforts, a complete transverse momentum gradient expansion of the hadronic tensor, including $N_c$ suppressed terms, phase terms and finite gluon momentum fraction terms, ignored previously, is carried out. These terms turn out to be surprisingly large. In contrast to prior efforts, the full next-to-leading twist gluon emission kernel is found to be positive definite and slowly increasing with the exchanged transverse momentum. Phenomenological consequences of these new contributions are discussed.

Many-Body Correlation Effects on the <i>x</i><sub>Bjorken</sub>-Dependence of Cross Section Ratios off Nuclei for 1 < <i>x</i><sub>Bjorken</sub> < 2

Many-body correlations in nuclei determine the behavior of Deep-Inelastic-Scattering (DIS) and Quasi-Elastic Scattering (QES) cross section ratios off heavy over light nuclei especially for xBjorken > 1, obtained at Jefferson Lab. They can be described in terms of quark-cluster formation in nuclei due to wave-function overlapping, manifesting itself when the momentum transfer is high so that the partonic degrees of freedom are resolved. In clusters (correlated nucleons) the quark and gluon momentum distributions are softer than in single nucleons and extend to xBjorken > 1. The cluster formation probabilities are computed using a network-defining algorithm in which the initial nucleon density is either standard Woods-Saxon or is input from lower energy data while the critical radius for nucleon merging is an adjustable parameter. The exact choice of critical radius depends on the specific nucleus and it is anti-correlated to the rescaling of the xBjorken needed for bound nucleons. The calculations show that there is a strong dependence of the cross section ratios on the xBjorken in agreement with the data and that four-body correlations are needed to explain the experimental results even in the range 1 xBjorken < 2. The dependence on the specific exponents of parton distributions in high-order clusters is weak.

Intrinsic glue and Wilson lines within dressed quarks

We construct a quark target model (QTM) to incorporate intrinsic glue into effective low-energy models of QCD, which often contain only quark degrees of freedom. This method guarantees the gauge invariance of observables order by order in the strong coupling. The quark and gluon PDFs for the dressed quarks are obtained in the QTM at leading order. We demonstrate gauge invariance of the results by comparing both covariant and light cone gauges, with the former including an explicit Wilson line contribution. A key finding is that in covariant gauges the Wilson line can carry a significant amount of the light cone momentum. With coupling strength ${\ensuremath{\alpha}}_{s}=0.5$ and dressed quark mass ${M}_{q}=0.4\phantom{\rule{4pt}{0ex}}\mathrm{GeV}$, we find quark and gluon momentum fractions of ${\ensuremath{\langle}x\ensuremath{\rangle}}_{q}=0.81$ and ${\ensuremath{\langle}x\ensuremath{\rangle}}_{g}=0.19$, where the Wilson line contribution to the quark momentum fraction is $\ensuremath{-}0.18$. We use the on-shell renormalization scheme and find that at one loop this Wilson line contribution does not depend on the covariant gauge but does vanish in light cone gauge as expected. This result demonstrates that it is crucial to account for Wilson line contributions when calculating quantum correlation functions in covariant gauges. We also consider the impact of a gluon mass using the gauge invariant formalism proposed by Cornwall, and combine these QTM results with two quark-level models to obtain quark and gluon PDFs for the pion.

Treating divergence in quark matter by using energy projectors

We calculate the gluon self-energy using quark energy projectors in a general quark-gluon plasma. By separating the quark field into positive- and a negative-energy modes, the quark loop constructed with the same mode is always convergent, and the divergence appears only in the mixed loop with different modes and is medium independent. After removing the divergence in vacuum, we obtain the one-loop gluon self-energy at finite temperature, chemical potential, and quark mass without approximation. With the method of quark-loop resummation, we calculate nonperturbatively the gluon Debye mass and thermodynamic potential. In the limit of small gluon momentum in comparison with temperature, chemical potential, and quark mass, our calculation comes back to the known HTL/HDL results in literature.

Quark-gluon vertex from Nf=2 lattice QCD

We study the quark-gluon vertex in the limit of vanishing gluon momentum using lattice QCD with two flavors of $\mathcal{O}(a)$ improved Wilson fermions, for several lattice spacings and quark masses. We find that all three form factors in this kinematics have a significant infrared strength and that both the leading form factor ${\ensuremath{\lambda}}_{1}$, multiplying the tree-level vertex structure, and the scalar, chiral symmetry breaking form factor ${\ensuremath{\lambda}}_{3}$ are significantly enhanced in the infrared compared to the quenched (${N}_{f}=0$) case. These enhancements are orders of magnitude larger than predicted by one-loop perturbation theory. We find only a weak dependence on the lattice spacing and quark mass.

From noninteracting to interacting picture of quark–gluon plasma in the presence of a magnetic field and its fluid property

We have attempted to build a parametric-based simplified and analytical model to map the interaction of quarks and gluons in the presence of magnetic field, which has been constrained by quark condensate and thermodynamical quantities like pressure, energy density, etc., obtained from the calculation of lattice quantum chromodynamics (QCDs). To fulfill that mapping, we have assumed a parametric temperature and magnetic field-dependent degeneracy factor, average energy, momentum and velocity of quarks and gluons. Implementing this QCD interaction in calculation of transport coefficient at finite magnetic field, we have noticed that magnetic field and interaction both are two dominating sources, for which the values of transport coefficients can be reduced. Though the methodology is not so robust, but with the help of its simple parametric expressions, one can get a quick rough estimation of any phenomenological quantity, influenced by temperature and magnetic field-dependent QCD interaction.

Longitudinal double-spin asymmetry for inclusive jet and dijet production in polarized proton collisions at s=200 GeV

We report high-precision measurements of the longitudinal double-spin asymmetry, ALL, for midrapidity inclusive jet and dijet production in polarized pp collisions at a center-of-mass energy of s=200 GeV. The new inclusive jet data are sensitive to the gluon helicity distribution, Δg(x,Q2), for gluon momentum fractions in the range from x≃0.05 to x≃0.5, while the new dijet data provide further constraints on the x dependence of Δg(x,Q2). The results are in good agreement with previous measurements at s=200 GeV and with recent theoretical evaluations of prior world data. Our new results have better precision and thus strengthen the evidence that Δg(x,Q2) is positive for x>0.05.Received 9 March 2021Accepted 28 April 2021DOI:https://doi.org/10.1103/PhysRevD.103.L091103Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasHadron-hadron interactionsPerturbative QCDQuark & gluon jetsPhysical SystemsGluonsProtonsPropertiesPolarizationSpinTechniquesElectromagnetic calorimetersHadron collidersTime-projection chambersNuclear PhysicsParticles & Fields

Parton Energy Loss in the Generalized High-Twist Approach

We calculate the radiative parton energy loss in the deeply inelastic scattering (DIS) off a large nucleus within a generalized high-twist approach. The final gluon radiation spectrum is a convolution of the hard partonic part and the transverse momentum dependent (TMD) quark-gluon correlation function, without the twist expansion in the transverse momentum of the initial gluons used in original high twist approach. The TMD quark-gluon correlation function can be factorized approximately as the product of initial quark distribution and TMD gluon distribution which can be used to define the generalized or TMD jet transport coefficient. The radiation spectrum will recover the Gylassy-Levai-Vitev (GLV) result in the first order of the opacity expansion, under the static scattering center and soft gluon radiation approximation. We also investigate numerically the difference as a result of the soft gluon radiation approximation, under the static scattering center approximation.

Electromagnetic radiation from the pre-hydrodynamics stage of the quark-gluon plasma

Photons produced in the pre-hydrodynamics stage of the quark-gluon plasma created in relativistic heavy-ion collisions were computed using momentum distribution functions of partons obtained from solutions of the Boltzmann equation. The effect of the initial gluon momentum anisotropy ξ and the dependence on the saturation momentum Qs was investigated; we report on results obtained by varying the latter in this work. We see that small Qs results in a photon yield enhancement, whereas a larger Qs results in a pre-equilibrium photon suppression, owing to the strict constraint of matching to experimental energy density used in this work.

The fully-differential gluon beam function at NNLO

The fully-differential beam function (dBF) is a universal ingredient in resummed predictions of hadron collider observables that probe the full kinematics of the incoming parton from each colliding proton — the virtuality and transverse momentum as well as the light-cone momentum fraction x. In this paper we compute the matching coefficients between the unpolarized gluon dBF and the usual parton distribution functions (PDFs) at the two-loop order. For observables probing both the virtuality and transverse momentum of incoming gluons, our results provide the part of the NNLO singular cross section related to collinear initial-state radiation, and are required for the resummation of large logarithms through N3LL. Further to this, the dBF is closely linked to the beam function appearing in a generalized version of threshold factorization, via a simple integration. By performing this integration for the two-loop gluon matching coefficients, we also obtain the corresponding quantities for the generalized threshold beam function.