Gluon-Gluon Interactions in the Bag Model
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Carlson, C.↗
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Two-gluon correlations in heavy–light ion collisions: Energy and geometry dependence, IR divergences, and k T -factorization
Here, we study the properties of the cross section for two-gluon production in heavy–light ion collisions derived in our previous paper in the saturation/Color Glass Condensate framework. Concentrating on the energy and geometry dependence of the corresponding correlation functions we find that the two-gluon correlator is a much slower function of the center-of-mass energy than the one- and two-gluon production cross sections. The geometry dependence of the correlation function leads to stronger azimuthal near- and away-side correlations in the tip-on-tip U + U collisions than in the side-on-side U + U collisions, an exactly opposite behavior from the correlations generated by the elliptic flow of the quark–gluon plasma: a study of azimuthal correlations in the U + U collisions may thus help to disentangle the two sources of correlations. We demonstrate that the cross section for two-gluon production in heavy–light ion collisions contains a power-law infrared (IR) divergence even for fixed produced gluon momenta: while saturation effects in the target regulate some of the power-law IR-divergent terms in the lowest-order expression for the two-gluon correlator, other power-law IR-divergent terms remain, possibly due to absence of saturation effects in the dilute projectile. Finally we rewrite our result for the two-gluon production cross-section in a k T -factorized form, obtaining a new factorized expression involving a convolution of one- and two-gluon Wigner distributions over both the transverse momenta and impact parameters. Lastly, we show that the two-gluon production cross-section depends on two different types of unintegrated two-gluon Wigner distribution functions.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Toward the first gluon parton distribution from the LaMET
Abstract We present progress towards the first unpolarized gluon quasi-parton distribution function (PDF) from lattice quantum chromodynamics using high-statistics measurements for hadrons at two valence pion masses M π ≈ 310 and 690 MeV computed on an a ≈ 0.12 fm ensemble with 2 + 1 + 1-flavors of highly improved staggered quark generated by the MILC collaboration. In this study, we consider two gluon operators for which the hybrid-ratio renormalization matching kernels have been recently derived and a third operator that has been used in prior pseudo-PDF studies of the gluon PDFs. We compare the matrix elements for each operator for both the nucleon and pion, at both pion masses, and using two gauge-smearing techniques. Focusing on the more phenomenologically studied nucleon gluon PDF, we compare the ratio and hybrid-ratio renormalized matrix elements at both pion masses and both smearings to those reconstructed from the nucleon gluon PDF from the CT18 global analysis. We identify the best choice of operator to study the gluon PDF and present the first gluon quasi-PDF under some caveats. Additionally, we explore the recent idea of Coulomb gauge fixing to improve signal at large Wilson-line displacement and find it could be a major help in improving the signal in the gluon matrix elements. This work helps identify the best operator for studying the gluon quasi-PDF, shows higher hadron boost momentum is needed to implement hybrid-ratio renormalization reliably, and suggests the need to study more diverse set of operators with their corresponding perturbative calculations for hybrid-ratio renormalization to further gluon quasi-PDF study.
Good, William (ORCID:0000000184281003)↗
One-loop central-emission vertex for two gluons in $\mathcal{N}$ = 4 super Yang-Mills theory
A necessary ingredient for extending the BFKL equation to next-to-next-to-leading logarithmic (NNLL) accuracy is the one-loop central emission vertex (CEV) for two gluons which are not strongly ordered in rapidity. Here we consider the one-loop six-gluon amplitude in $\mathcal{N}$ = 4 super Yang-Mills (SYM) theory in a central next-to-multi-Regge kinematic (NMRK) limit, we show that its dispersive part factorises in terms of the two-gluon CEV, and we use it to extract the one-loop two-gluon CEV for any helicity configuration within this theory. This is a component of the two-gluon CEV in QCD. Although computed in the NMRK limit, both the colour structure and the kinematic dependence of the two-gluon CEV capture much of the complexity of the six-gluon amplitudes in general kinematics. In fact, the transcendental functions of the latter can be conveniently written in terms of impact factors, trajectories, single-emission CEVs and a remainder, which is a function of the conformally invariant cross ratios which characterise the six-gluon amplitudes in planar $\mathcal{N}$ = 4 SYM. Finally, as expected, in the MRK limit the two-gluon CEV neatly factorises in terms of two single-emission CEVs.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
The gluon structure of hadrons and nuclei from lattice QCD
I discuss recent lattice QCD studies of the gluon structure of hadrons and light nuclei. After very briefly highlighting new determinations of the gluon contributions to the nucleon's momentum and spin, presented by several collaborations over the last year, I describe first calculations of gluon generalised form factors. The generalised transversity gluon distributions are of particular interest since they are purely gluonic; they do not mix with quark distributions at leading twist. In light nuclei they moreover provide a clean signature of non-nucleonic gluon degrees of freedom, and I present the first evidence for such effects, based on lattice QCD calculations. The planned Electron-Ion Collider, designed to access gluon structure quantities, will have the capability to test this prediction, and measure a range of gluon observables including generalised gluon distributions and transverse momentum dependent gluon distributions, within the next decade.
Shanahan, Phiala A.↗
Gluon distributions and their applications to Ioffe-time distributions
We investigate unpolarized and polarized gluon distributions and their applications to the Ioffe-time distributions, which are related to lattice QCD calculations of parton distribution functions. Guided by the counting rules based on the perturbative QCD at large momentum fraction x and the color coherence of gluon couplings at small x, we parametrize gluon distributions in the helicity basis. By fitting the unpolarized gluon distribution, the inferred polarized gluon distribution from our parametrization agrees with the one from global analysis. A simultaneous fit to both unpolarized and polarized gluon distributions is also performed to explore the model uncertainty. The agreement with the global analysis supports the (1 - x) power suppression of the helicity-antialigned distribution relative to the helicity-aligned distribution. The corresponding Ioffe-time distributions and their asymptotic expansions are calculated from the gluon distributions. Our results of the Ioffe-time distributions can provide guidance to the extrapolation of lattice QCD data to the region lacking precise gluonic matrix elements. Therefore, they can help regulate the ill-posed inverse problem associated with extracting the gluon distributions from discrete data from first-principle calculations, which are available in a limited range of the nucleon momentum and the spatial separation between the gluonic currents. Given various limitations in obtaining lattice QCD data at large Ioffe time, phenomenological approaches can provide important complementary information to extract the gluon distributions in the entire momentum fraction region, especially at small x. The possibility of investigating higher-twist effects and other systematic uncertainties in the contemporary first-principle calculations of parton distributions from phenomenologically well-determined Ioffe-time distributions in the large Ioffe-time region is also discussed.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Gluon splitting at small x: a unified derivation for the JIMWLK, DGLAP and CSS equations
We revisit the calculation of the next-to-leading order (NLO) corrections to dijet production in electron-ion collisions at small x. We focus on the back-to-back configuration where the relative transverse momentum P ⊥ of the measured jets is much larger than both their momentum imbalance K ⊥ and the target saturation momentum Q s (x, A). In this regime, we present for the first time a complete calculation of the real NLO corrections, associated with gluon emissions outside the jet cones, at leading power in 1/P ⊥ . Our result exhibits TMD factorisation, with the same hard factor as at tree-level and a NLO correction to the Weiszäcker-Williams (WW) gluon transverse momentum dependent (TMD) distribution which involves four Wilson-line operators. By studying different kinematical regimes for K ⊥ and for the radiated gluon, we recover all the quantum evolutions that were previously identified for this process at NLO: the B-JIMWLK high-energy evolution and the CSS evolution of the gluon WW TMD, and the DGLAP evolution of the gluon PDF. When both K ⊥ and the transverse momentum transferred by the target are large compared to Q s , all the Wilson-line operators boil down to the unintegrated gluon distribution and our NLO result for the gluon TMD can be used to isolate the transverse-momentum dependent gluon splitting function.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Mining for Gluon Saturation at Colliders
Quantum chromodynamics (QCD) is the theory of strong interactions of quarks and gluons collectively called partons, the basic constituents of all nuclear matter. Its non-abelian character manifests in nature in the form of two remarkable properties: color confinement and asymptotic freedom. At high energies, perturbation theory can result in the growth and dominance of very gluon densities at small-x. If left uncontrolled, this growth can result in gluons eternally growing violating a number of mathematical bounds. The resolution to this problem lies by balancing gluon emissions by recombinating gluons at high energies: phenomena of gluon saturation. High energy nuclear and particle physics experiments have spent the past decades quantifying the structure of protons and nuclei in terms of their fundamental constituents confirming predicted extraordinary behavior of matter at extreme density and pressure conditions. In the process they have also measured seemingly unexpected phenomena. We will give a state of the art review of the underlying theoretical and experimental tools and measurements pertinent to gluon saturation physics. We will argue for the need of high energy electron-proton/ion colliders such as the proposed EIC (USA) and LHeC (Europe) to consolidate our knowledge of QCD knowledge in the small x kinematic domains.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Scalar-scaffolded gluons and the combinatorial origins of Yang-Mills theory
We present a new formulation for Yang-Mills scattering amplitudes in any number of dimensions and at any loop order, based on the same combinatorial and binary-geometric ideas in kinematic space recently used to give an all-order description of Tr Φ 3 theory. We propose that in a precise sense the amplitudes for a suitably “stringy” form of these two theories are identical, up to a simple shift of kinematic variables. This connection is made possible by describing the amplitudes for n gluons via a “scalar scaffolding”, arising from the scattering of 2n colored scalars coming in n distinct pairs of flavors fusing to produce the gluons. Fundamental properties of the “u-variables”, describing the “binary geometry” for surfaces appearing in the topological expansion, magically guarantee that the kinematically shifted Tr Φ 3 amplitudes satisfy the physical properties needed to be interpreted as scaffolded gluons. These include multilinearity, gauge invariance, and factorization on tree- and loop-level gluon cuts. Our “stringy” scaffolded gluon amplitudes coincide with amplitudes in the bosonic string for extra-dimensional gluon polarizations at tree-level, but differ (and are simpler) at loop-level. We provide many checks on our proposal, including matching non-trivial leading singularities through two loops. The simple counting problem underlying the u variables autonomously “knows” about everything needed to convert colored scalar to gluon amplitudes, exposing a striking “discovery” of Yang-Mills amplitudes from elementary combinatorial ideas in kinematic space.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Gluon dipole factorisation for diffractive dijets
Within the colour dipole picture for deep inelastic scattering at small Bjorken x , we study the production of a pair of relatively hard jets via coherent diffraction. By “relatively hard” we mean that the transverse momenta of the two jets — the quark ($q$) and the antiquark ($\overline{q}$) generated by the decay of the virtual photon — are much larger than the target saturation momentum Q s (Y P ) evaluated at the rapidity gap Y P . We argue that the typical final-state configurations are such that the hard $q$$\overline{q}$ dijets are accompanied by a semi-hard gluon jet, with a transverse momentum of the order of Q s (Y P ). The presence of this third jet ensures that the scattering is strong and thus avoids the strong suppression of exclusive (hard) dijet production due to colour transparency. For such “2+1” jet configurations, we demonstrate that both the emission of the semi-hard gluon and its scattering with the hadronic target can be factorised in terms of an effective gluon-gluon dipole. This effective description, originally proposed in [1–7], builds a bridge between the colour dipole picture and the transverse-momentum dependent (TMD) version of the collinear factorisation: the cross-section for diffractive 2+1 jets can be written as the product between a hard factor describing the $q$$\overline{q}$ dijets and a semi-hard factor expressing the unintegrated gluon distribution of the Pomeron. The latter is controlled by gluon dipole scattering in the black disk limit and hence is strongly sensitive to gluon saturation. By integrating out the kinematics of the 3 jets, we obtain the $q$$\overline{q}$$g$ contribution to the diffractive structure function in collinearly-factorised form.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
The analytic structure of non-global logarithms: Convergence of the dressed gluon expansion
Non-global logarithms (NGLs) are the leading manifestation of correlations between distinct phase space regions in QCD and gauge theories and have proven a challenge to understand using traditional resummation techniques. Recently, the dressed gluon ex-pansion was introduced that enables an expansion of the NGL series in terms of a “dressed gluon” building block, defined by an all-orders factorization theorem. Here, we clarify the nature of the dressed gluon expansion, and prove that it has an infinite radius of convergence as a solution to the leading logarithmic and large-N c master equation for NGLs, the Banfi-Marchesini-Smye (BMS) equation. The dressed gluon expansion therefore provides an expansion of the NGL series that can be truncated at any order, with reliable uncertainty estimates. In contrast, manifest in the results of the fixed-order expansion of the BMS equation up to 12-loops is a breakdown of convergence at a finite value of α s log. We explain this finite radius of convergence using the dressed gluon expansion, showing how the dynamics of the buffer region, a region of phase space near the boundary of the jet that was identified in early studies of NGLs, leads to large contributions to the fixed order expansion. We also use the dressed gluon expansion to discuss the convergence of the next-to-leading NGL series, and the role of collinear logarithms that appear at this order. Finally, we show how an understanding of the analytic behavior obtained from the dressed gluon expansion allows us to improve the fixed order NGL series using conformal transformations to extend the domain of analyticity. Furthermore, this allows us to calculate the NGL distribution for all values of α s log from the coefficients of the fixed order expansion.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Quarkonium semiclassical transport in quark-gluon plasma: factorization and quantum correction
We study quarkonium transport in the quark-gluon plasma by using the potential nonrelativistic QCD (pNRQCD) effective field theory and the framework of open quantum systems. We argue that the coupling between quarkonium and the thermal bath is weak using separation of scales, so the initial density matrix of the total system factorizes and the time evolution of the subsystem is Markovian. We derive the semiclassical Boltzmann equation for quarkonium by applying a Wigner transform to the Lindblad equation and carrying out a semiclassical expansion. We resum relevant interactions to all orders in the coupling constant at leading power of the nonrelativistic and multipole expansions. The derivation is valid for both weakly coupled and strongly coupled quark-gluon plasmas. We find reaction rates in the transport equation factorize into a quarkonium dipole transition function and a chromoelectric gluon distribution function. For the differential reaction rate, the definition of the momentum dependent chromoelectric gluon distribution function involves staple-shaped Wilson lines. For the inclusive reaction rate, the Wilson lines collapse into a straight line along the real time axis and the distribution becomes momentum independent. The relation between the two Wilson lines is analogous to the relation between the Wilson lines appearing in the gluon parton distribution function (PDF) and the gluon transverse momentum dependent parton distribution function (TMDPDF). The centrality dependence of the quarkonium nuclear modification factor measured by experiments probes the momentum independent distribution while the transverse momentum dependence and measurements of the azimuthal angular anisotropy may be able to probe the momentum dependent one. We discuss one way to indirectly constrain the quarkonium in-medium real potential by using the factorization formula and lattice calculations. The leading quantum correction to the semiclassical transport equation of quarkonium is also worked out. The study can be easily generalized to quarkonium transport in cold nuclear matter, which is relevant for quarkonium production in eA collisions in the future Electron-Ion Collider.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Quasiclassical Gluon Fields and Low’s Soft Theorem at Small Momentum-Fraction x
In the high-energy limit, soft gluons can be approximately described by quasiclassical gluon fields. It is well known that the gluon field is a pure gauge field on the transverse plane at eikonal order. We derived the complete next-to-eikonal order solutions of the classical Yang-Mills equations for soft gluons in the dense nuclear regime. Utilizing these solutions, it is shown that Low’s soft theorem at small x can be obtained by considering off-diagonal matrix elements of quasiclassical chromoelectric field between single-gluon states in the dilute regime. We further propose on extending Low’s soft theorem at small x to incorporate the effects of gluon saturation in the dense regime.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Numerical analysis of the unintegrated double gluon distribution
We present detailed numerical analysis of the unintegrated double gluon distribution which includes the dependence on the transverse momenta of partons. The unintegrated double gluon distribution was obtained following the Kimber-Martin-Ryskin method as a convolution of the perturbative gluon splitting function with the collinear integrated double gluon distribution and the Sudakov form factors. We analyze the dependence on the transverse momenta, longitudinal momentum fractions and hard scales. We find that the unintegrated gluon distribution factorizes into a product of two single unintegrated gluon distributions in the region of small values of x, provided the splitting contribution is included and the momentum sum rule is satisfied.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
The quantum evolutions of the diffractive transverse-momentum dependent gluon distribution
Using the Colour Glass Condensate description of electron-nucleus collisions at high energy, we study the diffractive production of a pair of jets with transverse momenta much larger than the nuclear saturation momentum Q s . At leading order in the QCD coupling, the di-jet cross-section exhibits transverse-momentum dependent (TMD) factorisation, with a gluon diffractive TMD distribution (DTMD) which is controlled by gluon saturation and describes the transverse-momentum imbalance between the produced jets. The next-to-leading corrections generate the various quantum evolutions of the diffractive gluon distribution. We focus on the Collins-Soper-Sterman (CSS) evolution which describes the change in the gluon DTMD when increasing the “hard scale” (the typical transverse momentum of the di-jets). We consider two different representations for this equation, one in transverse-momentum space, the other one in transverse-coordinate space. They are not fully equivalent with each other (despite being related by a Fourier transform) because of the respective boundary conditions. These conditions encode the essential physics of gluon saturation together with the effects of two other types of quantum evolution: the BK/JIMWLK evolution over the rapidity gap (“inside the Pomeron”) and the DGLAP evolution outside the rapidity gap (“within the diffractive system”). We demonstrate that, due to gluon saturation, one can compute both the boundary conditions and the CSS solutions mostly from first principles, without the need for a non-perturbative Sudakov. We numerically find a good agreement between the CSS solutions in the two aforementioned representations.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Linear Boltzmann transport for jet propagation in the quark-gluon plasma: Inelastic processes and jet modification
A linear Boltzmann transport (LBT) Monte Carlo model has been developed to describe jet propagation and interaction with the quark-gluon plasma (QGP) in relativistic heavy-ion collisions. A complete set of elastic-scattering processes and medium-induced gluon emissions based on the higher-twist formalism are incorporated for both jet shower and medium recoil partons. It has been employed to describe experimental data on large transverse momentum hadron and jet spectra, correlation and jet substructures in high-energy heavy-ion collisions. Here, we document in detail the structure of the model and validation of the Monte Carlo implementations of the physics processes in LBT, in particular, the inelastic process of medium-induced gluon radiation. We carry out a comprehensive examination of the jet-medium interaction as implemented in LBT through energy loss and momentum broadening of a single hard parton, the energy and transverse momentum transfer from leading partons to medium-induced gluons and jet-induced medium excitation, and medium modification of reconstructed jets in a static and uniform medium. With realistic and event-by-event hydrodynamic medium in heavy-ion collisions, we compute and compare with experimental data on the jet cone-size dependence of the single inclusive jet suppression at both the BNL Relativistic Heavy-Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC), the dijet asymmetry at the LHC and γ-jet correlation at RHIC. Effects of medium-induced gluon emissions and jet-induced medium excitation on jet observables are systematically examined. Rescatterings of the radiated gluons and recoil partons with the QGP are found essential to account for the enhancement of soft particle yield toward the edge of the jet cone.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Quasiclassical evaluation of gluon saturation induced helicity effects
At sub-eikonal order in the high-energy limit, helicity dependences are generally contained in the transverse components of the gluon field. In the gluon saturation regime, novel helicity effects arise from the nonlinear interaction between the eikonal-order longitudinal gluon field and the sub-eikonal-order transverse gluon field. We derive this saturation induced helicity-dependent field both by solving the classical Yang-Mills equations and through direct diagrammatic calculations. Our analysis shows that the saturation induced helicity effect is intrinsically a two-particle (or multiparticle) correlation effect, rather than a single-particle distribution effect. Furthermore, we evaluate this effect in the context of double-spin asymmetry for incoherent diffractive dijet production in longitudinally polarized electron-nucleus collisions, using a simplified helicity-extended McLerran-Venugopalan model. We find that the saturation induced helicity effect further suppresses the back-to-back peak in the dijet azimuthal angle correlation, with a magnitude comparable to that of the direct helicity effect. This gluon saturation induced helicity effect may offer a novel avenue to probe gluon saturation in polarized collisions. Published by the American Physical Society 2025
Li, Ming (ORCID:0000000195509691)↗
Fluctuations of the gluon distribution from the small- x effective action
The computation of observables in high-energy QCD involves an average over stochastic semiclassical small-x gluon fields. The weight of various configurations is determined by the effective action. We introduce a method to study fluctuations of observables, functionals of the small-x fields, which does not explicitly involve dipoles. We integrate out those fluctuations of the semiclassical gluon field under which a given observable is invariant. Thereby we obtain the effective potential for that observable describing its fluctuations about the average. Here, we determine explicitly the effective potential for the covariant gauge gluon distribution both for the McLerran-Venugopalan (MV) model and for a (nonlocal) Gaussian approximation for the small-x effective action. This provides insight into the correlation of fluctuations of the number of hard gluons versus their typical transverse momentum. We find that the spectral shape of the fluctuations of the gluon distribution is fundamentally different in the MV model, where there is a pileup of gluons near the saturation scale, versus the solution of the small-x JIMWLK renormalization group, which generates essentially scale-invariant fluctuations above the absorptive boundary set by the saturation scale.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗