Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited

Abstract

Parallel to the construction of gauge invariant spin and orbital angular momentum for QED in paper (I) of this series [1], 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.PACS numbers: 11.15.-q, 13.88.+e, 14.20.Dh,14.70.DjAs a composite particle, the nucleon naturally gets its spin from the spin and orbital motion of its constituents: quarks and gluons. From a theoretical point of view, the first task in studying the nucleon spin structure is to find out the appropriate operators for the spin and orbital angular momentum of the quark and gluon fields. Given these operators, one can then study their matrix elements in a polarized nucleon state, and investigate how these matrix elements can be related to experimental measurements. Pitifully and surprisingly, after 20 years of extensive discussions of the nucleon spin structure [2,3,4,5], this first task was never done, and even largely eluded the attention of the community.At first thought, it seems an elementary exercise to derive the quark and gluon angular momentum operators. From the QCD Lagrangianone can promptly follow Nöther's theorem to write down the conserved QCD angular momentum:

Glue Spin and Helicity in the Proton from Lattice QCD

“…The gluon spin operator proposed in Ref. [15,16] with the non-abelian transverse condition belongs to this universality class and has been proven to be equivalent to the GIE of E × A in the Coulomb gauge ∂ · A = 0 [17,18],…”

“…The gluon spin operator proposed in Ref. [15,16] with the non-abelian transverse condition belongs to this universality class and has been proven to be equivalent to the GIE of E × A in the Coulomb gauge ∂ · A = 0 [17,18],where the factor 2 is from the normalization of the SU (3) The major task of this work is calculating the matrix element of S g in the proton, which will be indicated as S G , in the rest and moving frames. The results are then renormalized at 1-loop order in lattice perturbation theory and matched to the MS scheme at µ 2 =10 GeV 2 , to investigate their frame dependence and address the matching to the helicity.…”

“…[3]. The final result is S G (∞, µ 2 = 10 GeV 2 ) = 0.251(47)(16) with two errors from the statistical and systematic uncertainties.Summary and outlook: In this work, we calculated the glue spin in the proton for the first time based on E × A in the Coulomb gauge [15,16], with various quark masses, lattice spacings, volumes, and proton momenta. The results show mild dependencies on these quantities.…”

Orbital Angular Momentum: How to Define it and How to Measure it

“…There are two kinds of variants: (1) there is a difference between canonical and kinetic angular momentum and (2) there is a difference, depending on the form of dynamics being used i.e. between instant form and light-front a e-mail: e.leader@imperial.ac.uk dynamics The difference between canonical and kinetic has nothing to do with Field Theory.…”

“…The same is true for gluons, yet many experimental groups believe they are measuring the gluon spin! In 2008 Chen, Lu, Sun, Wang and Goldman [1] claimed, effectively, that all the QED textbooks were wrong, and that it was possible to split the photon angular momentum, in a gauge-invariant way, into a spin part and an orbital part. This unleashed a deluge of papers, most of them very technical, and causing confusion by effectively implying that there are an infinite number of ways of defining the AM.…”