A critique of the angular momentum sum rules and a new angular momentum sum rule

Abstract

We present a study of the tensorial structure of the hadronic matrix elements of the angular momentum operators $\mathbf{J}$. Well known results in the literature are shown to be incorrect, and we have taken pains to derive the correct expressions in three different ways, two involving explicit physical wave packets and the third, totally independent, based upon the rotational properties of the state vectors. Surprisingly it turns out that the results are very sensitive to the type of relativistic spin state used to describe the motion of the particle i.e. whether a canonical (i.e. boost) state or a helicity state is utilized. We present results for the matrix elements of the angular momentum operators, valid in an arbitrary Lorentz frame, both for helicity states and canonical states. These results are relevant for the construction of angular momentum sum rules, relating the angular momentum of a nucleon to the spin and orbital angular momentum of its constituents. It turns out that it is necessary to distinguish carefully whether the motion of the partons is characterized via canonical or helicity spin states. Fortunately, for the simple parton model interpretation, when the proton moves along $OZ$, our results for the sum rule based upon the matrix elements of $J_z$ agree with the often used sum rule found in the literature. But for the components $J_x,J_y$ the results are different and lead to a new and very intuitive sum rule for transverse polarization.

The results of our previous version are not changed here, but the presentation has been reorganized to make the discussion easier to follow

References in corpus (8)

• Gauge-Invariant Decomposition of Nucleon Spin and Its Spin-Off
• Time-reversal odd distribution functions in leptoproduction
• The complete tree-level result up to order $1/Q$ for polarized deep-inelastic leptoproduction
• Transverse Polarisation of Quarks in Hadrons
• Light-Cone Representation of the Spin and Orbital Angular Momentum of Relativistic Composite Systems
• The Spin Structure of the Nucleon in the Asymptotic Limit
• Probing transverse quark polarization via azimuthal asymmetries in leptoproduction
• The Gauge-Invariant Angular Momentum Sum-Rule for the Proton

Cited by in corpus (60)

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• Phenomenological assessment of proton mechanical properties from deeply virtual Compton scattering
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• The relativistic center of mass in field theory with spin
• Comparative analysis of the transversities and the longitudinally polarized distribution functions of the nucleon
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• Angular momentum and generalized parton distributions for the proton with basis light-front quantization
• The role of the chiral anomaly in polarized deeply inelastic scattering II: Topological screening and transitions from emergent axion-like dynamics
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• An introductory lecture on Generalised Parton Distributions
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• Transverse spin sum rule of the proton
• Partonic angular momentum in the nucleon's chiral periphery
• Boundary terms in quantum field theory and the spin structure of QCD
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• Exploring the polarization of gluons in the nucleon
• Proton spin after 30 years: what we know and what we don't?
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• Mapping the transverse spin sum rule in position space
• Analytic constraints on the energy-momentum tensor in conformal field theories
• Towards an understanding of nucleon spin structure: from hard to soft scales
• Measurement of the Transverse Single-Spin Asymmetry for Mid-rapidity Production of Neutral Pions in Polarized p+p Collisions at 200 GeV Center-of-Mass Energy
• Energy, angular momentum and pressure force distributions inside nucleons
• The gauge-invariant canonical energy-momentum tensor
• Nucleon spin decomposition at twist three
• Transverse energy-momentum tensor distributions in polarized nucleons
• Parton promenade into the nucleon