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Register a new userSpin and orbital angular momentum in gauge theories (II): QCD and nucleon spin structure
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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.
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This two-paper series addresses and fixes the long-standing gauge invariance problem of angular momentum in gauge theories. This QED part reveals: 1) The spin and orbital angular momenta of electrons and photons can all be consistently defined gauge invariantly. 2) These gauge-invariant quantities can be conveniently computed via the canonical, gauge-dependent operators (e.g, $psi ^dagger vec x timesfrac 1i vec abla psi$) in the Coulomb gauge, which is in fact what people (unconsciously) do in atomic physics. 3) The renowned formula $vec xtimes(vec Etimesvec B)$ is a wrong density for the electromagnetic angular momentum. The angular distribution of angular-momentum flow in polarized atomic radiation is properly described not by this formula, but by the gauge invariant quantities defined here. The QCD paper [arXiv:0907.1284] will give a non-trivial generalization to non-Abelian gauge theories, and discuss the connection to nucleon spin structure.
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 $Delta Sigma simeq 0.35$, 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 dependence of the nucleon spin contents in a drastic way.
We develop a general framework to analyze the two important and much discussed questions concerning (a) `orbital and `spin angular momentum carried by light and (b) the paraxial approximation of the free Maxwell system both in the classical as well as quantum domains. After formulating the classical free Maxwell system in the transverse gauge in terms of complex analytical signals we derive expressions for the constants of motion associated with its Poincar{e} symmetry. In particular, we show that the constant of motion corresponding to the total angular momentum ${bf J}$ naturally splits into an `orbital part ${bf L}$ and a `spin part ${bf S}$ each of which is a constant of motion in its own right. We then proceed to discuss quantization of the free Maxwell system and construct the operators generating the Poincar{e} group in the quantum context and analyze their algebraic properties and find that while the quantum counterparts $hat{{bf L}}$ and $hat{{bf S}}$ of ${bf L}$ and ${bf S}$ go over into bona fide observables, they fail to satisfy the angular momentum algebra precluding the possibility of their interpretation as `orbital and `spin operators at the classical level. On the other hand $hat{{bf J}}=hat{{bf L}}+ hat{{bf S}}$ does satisfy the angular momentum algebra and together with $hat{{bf S}}$ generates the group $E(3)$. We then present an analysis of single photon states, paraxial quantization both in the scalar as well as vector cases, single photon states in the paraxial regime. All along a close connection is maintained with the Hilbert space $mathcal{M}$ that arises in the classical context thereby providing a bridge between classical and quantum descriptions of radiation fields.
We determine within lattice QCD, the nucleon spin carried by valence and sea quarks, and gluons. The calculation is performed using an ensemble of gauge configurations with two degenerate light quarks with mass fixed to approximately reproduce the physical pion mass. We find that the total angular momentum carried by the quarks in the nucleon is $J_{u+d+s}{=}0.408(61)_{rm stat.}(48)_{rm syst.}$ and the gluon contribution is $J_g {=}0.133(11)_{rm stat.}(14)_{rm syst.}$ giving a total of $J_N{=}0.54(6)_{rm stat.}(5)_{rm syst.}$ consistent with the spin sum. For the quark intrinsic spin contribution we obtain $frac{1}{2}Delta Sigma_{u+d+s}{=}0.201(17)_{rm stat.}(5)_{rm syst.}$. All quantities are given in the $overline{textrm{MS}}$ scheme at 2~GeV. The quark and gluon momentum fractions are also computed and add up to $langle xrangle_{u+d+s}+langle xrangle_g{=}0.804(121)_{rm stat.}(95)_{rm syst.}+0.267(12)_{rm stat.}(10)_{rm syst.}{=}1.07(12)_{rm stat.}(10)_{rm syst.}$ satisfying the momentum sum.
We calculate the leading-twist helicity-dependent generalized parton distributions (GPDs) of the proton at finite skewness in the Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics (QCD). From these (and previously calculated helicity-independent GPDs) we obtain the spin decomposition of the proton, including predictions for quark intrinsic spin and orbital angular momentum. The inclusion of multiple species of diquarks is found to have a significant effect on the flavor decomposition, and resolving the internal structure of these dynamical diquark correlations proves essential for the mechanical stability of the proton. At a scale of $Q^2=4,$GeV$^2$ we find that the up and down quarks carry an intrinsic spin and orbital angular momentum of $S_u=0.534$, $S_d=-0.214$, $L_u=-0.189$, and $L_d=0.210$, whereas the gluons have a total angular momentum of $J_g=0.151$. The down quark is therefore found to carry almost no total angular momentum due to cancellations between spin and orbital contributions. Comparisons are made between these spin decomposition results and lattice QCD calculations.
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