The orbital angular momentum of quarks and gluons contributes significantly to the proton spin budget and attracted a lot of attention in the recent years, both theoretically and experimentally. We summarize the various definitions of parton orbital angular momentum together with their relations with parton distributions functions. In particular, we highlight current theoretical puzzles and give some prospects.
We present a recent calculation of the Wigner distributions of quarks and gluons in a perturbative model. We also present the results for the orbital angular momentum and the spin-orbit correlations.
We suggest a method of constructing gauge invariant quark and gluon distributions that describe an abstract QCD observable and apply this method to analyze angular momentum of a hadron. In addition to the known quark and gluon polarized structure functions, we obtain gauge invariant distributions for quark and gluon orbital angular momenta, and consider some basic properties of these distributions and their moments.
The assumption of angular ordering in gluon radiation is essential to obtain quantitative results concerning gluonic behaviors. In order to prove the validity of this assumption, we have applied our momentum space flux-tube formalism to check out the angular dependences of gluon radiation. We have calculated the probability amplitudes to get new gluon, and have found that the new gluon is generally expected to have the maximum amplitude when it is produced between the momentum directions of the last two partons.
The fundamental program in high energy spin physics focuses on the spin structure of the nucleon. The gluon and orbital angular momentum components of the nucleon spin are virtually unknown. The J_z=1/2 sum rule involves the integrated parton densities and can be used to extract information on the orbital angular momentum and its evolution. To avoid any bias on a model of Delta G, we assume that the gluon asymmetry, A=Delta G/G can be used to extract Delta G over a reasonable kinematic region. Combining the results for Delta G with the evolution equations, we can determine a theoretical expres…
A method to measure the suppression of soft-gluon radiation by a heavy quark ("dead cone") is discussed. We analyse this QCD phenomenon in the framework of the HERA experiment using Monte Carlo simulations.
The orbital angular momentum is one of the least understood of the spin characteristics of a proton. There are no direct ways to model the orbital angular momentum. However, the Jz=1/2 sum rule includes an angular momentum component and can provide indirect access to its properties. One of the other unknowns in the sum rule is the gluon polarization. We can define the gluon spin asymmetry in a proton as the ratio of the polarized to unpolarized gluon distributions. This can be written as a sum of a scale-invariant piece and a small scale-dependent term. The x-dependence of the asymmetry can be…
We present a general, model independent argument demonstrating that gluons produced in high energy hadronic collision are necessarily correlated in rapidity and also in the emission angle. The strength of the correlation depends on the process and on the structure/model of the colliding particles. In particular we argue that it is strongly affected (and underestimated) by factorized approximations frequently used to quantify the effect.
A new debate has recently arisen on the subject of orbital angular momentum in QCD, in particular on its observability and on its partonic interpretation. Orbital momentum can be defined in QCD using two different decomposition schemes that yield a kinetic and a canonical definition, respectively. We argue that kinetic orbital angular momentum is intrinsically associated with twist three generalized parton distributions, and it is therefore more readily observable, while, due to parity constraints, canonical angular momentum, if defined as suggested in the literature in terms of generalized tr…
We derive all order soft gluon resummation in dijet azimuthal angular correlation in hadronic collisions at the next-to-leading logarithmic level. The relevant coefficients for the Sudakov resummation factor, the soft and hard factors, are calculated. The theory predictions agree well with the experimental data from D0 Collaboration at the Tevatron. This provides a benchmark calculation for the transverse momentum dependent QCD resummation for jet productions in hadron collisions and can be readily applied at the CERN LHC.
Investigation of the difference in the angular distributions of $Z\to \ell^{+}\ell^{-}$ events produced in quark-antiquark, quark-gluon and gluon-gluon collisions
University of Rochester（罗切斯特大学）；Seoul National University（首尔大学）
We study the leading twist gluon generalized parton distributions (GPDs) and the gluon angular momentum inside the proton within a light-cone spectator model. Using the light-cone wave functions derived from the model, we provide the expressions of these distributions at the particular kinematical point $ξ=0$ in the overlap representation. The numerical results of the $H^g$, $E^g$, $\tilde{H}^g$, $H_T^g$ and $E_T^g$ as functions of $x$ at different $Δ_T$ are presented. Particularly, $H^g$, $\tilde{H}^g$ at non-zero $Δ_T$ are different from their forward counterparts, the unpolarized distributi…
Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, are exhibit a significant deviation from axial symmetry when the target and/or quark is transversely polarized. From this deformation, we present an intuitive derivation of the Ji relation. In a scalar diquark model and in QED, we compare the Ji and Jaffe-Manohar decompositions of the nucleon spin. Using the MIT bag model, we estimate spectator effects through the presence of the gluon vector potential in the definitions of the quark orbital angular momentum.
The evolution of orbital angular momentum distributions within the radiative parton model is studied. We use different scenarios for the helicity weighted parton distributions and consider a broad range of input distributions for orbital angular momentum. In all cases we are lead to the conclusion that the absolute value of the average angular momentum per parton peaks at relatively large $x\approx 0.1$ for perturbatively accessible scales. Furthermore, in all scenarios considered here the average orbital angular momentum per parton is several times larger for gluons than for quarks which favo…
We have calculated the Orbital Angular Momentum of quarks and gluons in the nucleon. The calculations are carried out in the next to leading order utilizing the so-called valon model. It is found that the average quark orbital angular momentum is positive, but small, and the average gluon orbital angular momentum is negative and large. We also report on some regularities about the total angular momentum of the quarks and the gluon, as well as on the orbital angular momentum of the separate partons. We have also provided partonic angular momentum, $L^{q,g}$ as a function of $Q^2$.
The induced momentum spectrum of soft gluons radiated from a high energy quark produced in and propagating through a QCD medium is reexamined in the BDMPS formalism. A mistake in our published work (Physical Review C60 (1999) 064902) is corrected. The correct dependence of the fractional induced loss $R(θ_{\rm cone})$ as a universal function of the variable $θ^2_{\rm cone} L^3 \hat q$ where $L$ is the size of the medium and $\hat q$ the transport coefficient is presented. We add the proof that the radiated gluon momentum spectrum derived in our formalism is equivalent with the one derived in t…
The Sivers mechanism for the single-spin asymmetry in unpolarized lepton scattering from a transversely polarized nucleon is driven by the orbital angular momentum carried by its quark and gluon constituents, combined with QCD final-state interactions. Both quark and gluon mechanisms can generate such a single-spin asymmetry, though only the quark mechanism can explain the small single-spin asymmetry measured by the COMPASS collaboration on the deuteron, suggesting the gluon mechanism is small relative to the quark mechanism. We detail empirical studies through which the gluon and quark orbita…
In the absence of a concrete discovery of new physics at the LHC, global analyses of the standard model effective field theory (SMEFT) are important to find and describe the impact of new physics beyond the energy reach of the LHC. Among the SMEFT operators that can be constrained via various measurements, the dimension six triple gluon operator involves neither the Higgs boson nor the top quark, yet its variation can have measurable effects on top and Higgs production. Without independent constraints on its impact, the sensitivity of measurements in the top and Higgs sectors to new physics is…
We propose a novel observable for the experimental detection of the gluon orbital angular momentum (OAM) that constitutes the proton spin sum rule. We consider longitudinal double spin asymmetry in exclusive dijet production in electron-proton scattering and demonstrate that the $\cos ϕ$ azimuthal angle correlation between the scattered electron and proton is a sensitive probe of the gluon OAM at small-$x$ and its interplay with the gluon helicity. We also present a numerical estimate of the cross section for the kinematics of the Electron-Ion Collider.
We compare the orbital angular momentum of the 'quark' in the scalar diquark model as well as that of the electron in QED (to order α) obtained from the Jaffe-Manohar de- composition to that obtained from the Ji relation. We estimate the importance of the vector potential in the definition of orbital angular momentum.
We investigate the impact of soft gluon resummation on the azimuthal angle correlation between the total and relative momenta of two energetic final state particles (jets). We show that the initial and final state radiations induce sizable $\cos(ϕ)$ and $\cos (2ϕ)$ asymmetries in single jet and dijet events, respectively. We numerically evaluate the magnitude of these asymmetries for a number of processes in collider experiments, including diffractive dijet and dilepton production in ultraperipheral $pA$ and $AA$ collisions, inclusive and diffractive dijet production at the EIC and inclusive d…
Using the MIT bag model, we study the contribution from the gluon vector potential due to the spectators to the orbital angular momentum of a quark in the bag model. For $α_s = {\cal O}(1)$, this spectator contribution to the quark orbital angular momentum in the gauge-covariant Ji decomposition is of the same order as the non gauge-covariant quark orbital angular momentum and its magnitude is larger for $d$ than for $u$ quarks and negative for both.
Quark spin and orbital angular momentum in the nucleon are calculated in symmetry-breaking chiral quark model. The results are compared with data and other models.
The induced momentum spectrum of soft gluons radiated from a high energy quark propagating through a QCD medium is derived in the BDMPS formalism. A calorimetric measurement for the medium dependent energy lost by a jet with opening angle $θ_{\rm cone}$ is proposed.The fraction of this energy loss with respect to the integrated one appears to be the relevant observable.It exhibits a universal behaviour in terms of the variable $θ^2_{\rm cone} L^3 \hat q$ where $L$ is the size of the medium and $\hat q$ the transport coefficient. Phenomenological implications for the differences between cold an…
We discuss the orbital angular momentum of partons inside a longitudinally polarized proton in the recently proposed framework of spin decomposition. The quark orbital angular momentum defined by Ji can be decomposed into the `canonical' and the `potential' angular momentum parts, both of which are represented as the matrix element of a manifestly gauge invariant operator.
We derive an inequality for the distribution of quarks with non-zero orbital angular momentum, and thus demonstrate, in a model-independent way, that a non-vanishing anomalous magnetic moment requires both a non-zero size of the target as well as the presence of wave function components with quark orbital angular momentum L_z>0.
Definitions of orbital angular momentum based on Wigner distributions are used as a framework to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar.We find that the difference between these two definitions can be interpreted as the change in the quark orbital angular momentum as it leaves the target in a DIS experiment. The mechanism responsible for that change is similar to the mechanism that causes transverse single-spin asymmetries in semi-inclusive deep-inelastic scattering.
We compute the gluon distribution in deep inelastic scattering at small x by solving numerically the angular ordering evolution equation. The leading order contribution, obtained by neglecting angular ordering, satisfies the BFKL equation. Our aim is the analysis of the subleading corrections. Although not complete - the exact next-to-leading contribution is not yet available - these corrections are important since they come from the physical property of coherence of QCD radiation. In particular we discuss the subleading correction to the BFKL characteristic function and the gluon distribution…
We demonstrate that there exist two kinds of gauge-invariant decompositions of the nucleon spin, which are physically inequivalent. The quark and gluon orbital angular momenta (OAMs) appearing in one decomposition are basically the canonical OAMs, while the quark and gluon OAMs appearing in another decomposition are called the dynamical OAMs. It is shown that the dynamical OAMs of quarks and gluons in the nucleon can be related to definite high-energy deep-inelastic-scattering observables. On the other hand, we conjecture that the canonical OAMs of quarks and gluon in the nucleon are model-dep…
We developed a new and unified scheme for describing both quark spin and orbital angular momenta in symmetry-breaking chiral quark model. The loss of quark spin in the chiral splitting processes is compensated by the gain of the orbital angular momentum carried by quarks and antiquarks. The sum of both spin and orbital angular momenta carried by quarks and antiquarks is 1/2. The analytic and numerical results for the spin and orbital angular momenta carried by quarks and antiquarks in the nucleon are given. Extension to other octet and decuplet baryons is also presented. Possible modification …
The orbital angular momentum operator expansion turns to be a powerful tool to construct the fully covariant partial wave amplitudes of hadron decay reactions and hadron photo- and electroproduction processes. In this paper we consider a useful development of the orbital angular momentum operator expansion method. We present the differential technique allowing the direct calculation of convolutions of two orbital angular momentum operators with an arbitrary number of open Lorentz indices. This differential technique greatly simplifies calculations when the reaction subject to the partial wave …
A Glasma flux-tube model has been proposed to explain strong elongation on pseudorapidity $η$ of the same-side 2D peak in minimum-bias angular correlations from $\sqrt{s_{NN}} = 200$ GeV \auau collisions. The same-side peak or "soft ridge" is said to arise from coupling of flux tubes to radial flow. Gluons radiated transversely from flux tubes are boosted by radial flow to form a narrow structure or ridge on azimuth. In this study we test the conjecture by comparing predictions for particle production, spectra and correlations from the Glasma model and conventional fragmentation processes with…
We introduce gauge-invariant quark and gluon angular momentum distributions after making a generalization of the angular momentum density operators. From the quark angular momentum distribution, we define the gauge-invariant and leading-twist quark {\it orbital} angular momentum distribution $L_q(x)$. The latter can be extracted from data on the polarized and unpolarized quark distributions and the off-forward distribution $E(x)$ in the forward limit. We comment upon the evolution equations obeyed by this as well as other orbital distributions considered in the literature.
Order-alpha_s QCD corrections in the soft-gluon approximation to angular distributions of decay charged leptons in the process e+e- -> t tbar, followed by semileptonic decay of t or tbar, are obtained in the e+e- centre-of-mass frame. As compared to distributions in the top rest frame, these have the advantage that they would allow direct comparison with experiment without the need to reconstruct the top rest frame or a spin quantization axis. Analytic expressions for the distribution in the charged-lepton polar angle and triple distribution in the polar angle of t and polar and azimuthal angl…
We reply to the Comment of X. Ji [arXiv:0810.4913] on our paper [PRL 100:232002 (2008)], concerning angular momentum algebra, locality, Lorentz covariance, and measurability of our gauge-invariant description of the spin and orbital angular momentum of quarks and gluons.
We calculate photon production from a quark-gluon plasma which is anisotropic in momentum space including the Compton scattering and quark/anti-quark annihilation processes. We show that for a quark-gluon plasma which has an oblate momentum-space anisotropy the photon production rate has an angular dependence which is peaked transverse to the beam line. We propose to use the angular dependence of high-energy medium photon production to experimentally determine the degree of momentum-space isotropy of a quark-gluon plasma produced in relativistic heavy-ion collisions.
We present a calculation of Wigner distributions for gluons in light-front dressed quark model. We calculate the kinetic and canonical gluon orbital angular momentum and spin-orbit correlation of the gluons in this model.
The difference between the quark orbital angular momentum (OAM) defined in light-cone gauge (Jaffe-Manohar) compared to defined using a local manifestly gauge invariant operator (Ji) is interpreted in terms of the change in quark OAM as the quark leaves the target in a DIS experiment.
We propose a new method of generating gamma rays with orbital angular momentum (OAM). Accelerated partially-stripped ions are used as an energy up-converter. Irradiating an optical laser beam with OAM on ultrarelativistic ions, they are excited to a state of large angular momentum. Gamma rays with OAM are emitted in their deexcitation process. We examine the excitation cross section and deexcitation rate.
We derive the consequences of the Myhrer-Thomas explanation of the proton spin problem for the distribution of orbital angular momentum on the valence and sea quarks. After QCD evolution these results are found to be in very good agreement with both recent lattice QCD calculations and the experimental constraints from Hermes and JLab.
QCD corrections to order alpha_s in the soft-gluon approximation to angular distributions of decay charged leptons in the process e+e- -> t tbar followed by semileptonic decay of t or tbar, are obtained in the e+e- centre-of-mass frame. As compared to distributions in the top rest frame, these have the advantage that they would allow direct comparison with experiment without the need to reconstruct the top rest frame. The results also do not depend on the choice of a spin quantization axis for t or tbar. Analytic expression for the triple distribution in the polar angle of t and polar and azim…
The nucleon spin problem raises experimental and theoretical questions regarding the contribution of the orbital angular momentum of the quarks to the total spin of the nucleon. In this article we examine the commutation relationships of various operators that contribute to the total angular momentum of the nucleon. We find that the sum of the orbital plus gluon field angular momentum should satisfy the angular momentum commutators, at least up to the one-loop level. This requirement on the sum of these operators imposes a nontrivial restriction on the form of the color electric and magnetic f…
Wigner distributions contain combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We report on a recent model calculation of the Wigner distributions for the quark and their relation to the orbital angular momentum.
Parallel to the construction of gauge invariant spin and orbital angular momentum for QED in paper (I) of this series, we present here an analogous but non-trivial solution for QCD. Explicitly gauge invariant spin and orbital angular momentum operators of quarks and gluons are obtained. This was previously thought to be an impossible task, and opens a more promising avenue towards the understanding of the nucleon spin structure.
We present the numerical solution of the leading order QCD evolution equation for the orbital angular momentum distributions of quarks and gluons and discuss its implications for the nucleon spin sum rule. We observe that at small-$x$, the gluon helicity and orbital angular momentum distributions are roughly of the same magnitude but with opposite signs, indicating a significant cancellation between them. A similar cancellation occurs also in the quark sector. We explain analytically the reason for this cancellation.
The spin and orbital angular momentum carried by different quark flavors in the nucleon are calculated in the SU(3) chiral quark model with symmetry-breaking. The model is extended to all octet and decuplet baryons. In this model, the reduction of the quark spin, due to the spin dilution in the chiral splitting processes, is transferred into the orbital motion of quarks and antiquarks. The orbital angular momentum for each quark flavor in the proton as function of the partition factor $κ$ and the chiral splitting probability $a$ is shown. Although the total amount of the quark spin reduction i…
The leading order evolution of parton orbital angular momenta is treated in the probabilistic manner. As a result, the splitting probability matrix, while coinciding with the recent explicit calculations, is expressed in the terms of the standard spin-dependent and spin-averaged kernels. The implied relations between spin and orbital angular momenta evolution kernels may be interpreted as resulting from the ambiguity in the separation of the angular momenta to the spin and orbital contributions and are likely to be valid for higher orders and twists. Consequently, the orbital and total angular…
The modification and amplification of the gluon angular distribution produced along with hard jets in nuclear collisions is computed. We consider the limit of a thin quark-gluon plasma, where the number of rescatterings of the jet and gluons is small. The focus is on jet quenching associated with the formation of highly off-shell partons in hard scattering events involving nuclei. The interference between the initial hard radiation amplitude, the multiple induced Gunion-Bertsch radiation amplitudes, and gluon rescattering amplitudes leads to an angular distribution that differs considerably fr…
The Regge behaviour of the solutions of a Dirac hamiltonian describing a heavy quark-light quark system in high orbital angular momentum states is analyzed. It is found that the solutions of a scalar confining potential are physically admissible while those of a vector confining potential are not. It is argued that with a Dirac hamiltonian a scalar confining potential is preferred over a vector confining potential for any value of the orbital angular momentum.
We study the quark angular momentum distribution in the nucleon within a light-front covariant quark model. Special emphasis is put into the orbital angular momentum: a quantity which is very sensitive to the relativistic treatment of the spin in a light-front dynamical approach. Discrepancies with the predictions of the low-energy traditional quark models where relativistic spin effects are neglected, are visible also after perturbative evolution to higher momentum scales. Orbital angular momentum distributions and their contribution to the spin sum rule are calculated for different phenomeno…
A theoretical prediction is given for the spin and orbital angular momentum distribution functions of the nucleon within the framework of an effective quark model of QCD, i.e. the chiral quark soliton model. An outstanding feature of the model is that it predicts fairly small quark spin fraction of the nucleon $ΔΣ\simeq 0.35$, which in turn dictates that the remaining 65% of the nucleon spin is carried by the orbital angular momentum of quarks and antiquarks at the model energy scale of $Q^2 \simeq 0.3 {GeV}^2$. This large orbital angular momentum necessarily affects the scenario of scale depe…
We study effects of imposing various forms of the kinematical constraints on the full form of the CCFM equation and its non-linear extension. We find that imposing the constraint in its complete form modifies significantly way the shape of gluon density as compared to forms of the constraint used in the numerical calculations and phenomenological applications. In particular the resulting gluon density is suppressed for large values of hard scale related parameter and kT of gluon. This result might be important in description studies of jet correlations at Large Hadron Collider within CCFM appr…
In the last decade, it has been realized that the orbital angular momentum of partons inside the nucleon plays a major role. It contributes significantly to nucleon properties and is at the origin of many asymmetries observed in spin physics. It is therefore of paramount importance to determine this quantity if we want to understand the nucleon internal structure and experimental observables. This triggered numerous discussions and controversies about the proper definition of orbital angular momentum and its extraction from experimental data. We summarize the present situation and discuss rece…
It is well known that in gauge theories, the spin (and orbital angular momentum) of gauge particles is not gauge invariant, although the helicity is; neither are the canonical momentum and canonical angular momentum of charged particles. However, the simple appeal of these concepts has motivated repeated attempts to resurrect them as physical descriptions of gauge systems. In particular, measurability of the gluon-spin-contribution to the proton helicity in polarized proton scattering has generated many theoretical efforts in generalizing it and others as gauge-invariant quantities. In this wo…
Gluon PDFs, GPDs and TMDs play a significant role in an array of scattering processes, including SIDIS, DVCS, exclusive meson electroproduction and p - p scattering. Spin dependent gluon distributions can lead to distinctive features in the angular dependences and asymmetries of the scattering processes. Of particular interest are heavy quark production processes, wherein spin observables of the heavy quarks adumbrate the underlying gluon spin dependences. Top pair production at LHC is a prime example that proceeds primarily via gluon fusion. Decays of polarized top pairs through various chann…
A controversy is raging concerning the most physically meaningful definition of quark and gluon angular momentum, a question of importance in understanding the internal structure of the nucleon. There are, in my, opinion, two principal candidates: the Belinfante version, $\bm{J}_{bel}$, which is gauge invariant, and the Canonical version, $ \bm{J}_{can}$, which is non-gauge invariant, but which has a physically intuitive structure. I demonstrate the wholly unexpected result that for quarks in a spin 1/2 nucleon the expectation value of $\bm{J}_{bel}(\textrm{quark})$ is equal to the expectation…
In this work we obtain all the allowed values of the angular momentum and parity for two massless non-Abelian fields. In particular, we show that, unlike the two photons system, two gluons can be in a state with $J^P=1^\pm$, $(2k)^-$, and $(2k+1)^-$, $k\geq1$. This is a general result since it was obtained using only the laws of addition of the angular momentum.
We present the numerical solution of the one-loop QCD evolution equation for the genuine twist-three part of the orbital angular momentum (OAM) distributions of quarks and gluons inside a longitudinally polarized nucleon. This is based on the observation that the evolution is identical to that of the Efremov-Teryaev-Qiu-Sterman function for transverse single spin asymmetry. Together with the known evolution of the Wandzura-Wilczek part, the one-loop evolution of OAM distributions is now practically under control. We also study, for the first time, the scale dependence of the potential angular …
