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Orbital angular momentum in the pion and kaon: rest-frame and light-front

Y. -Y. Xiao, Z. -N. Xu, Z. -Q. Yao, C. D. Roberts, J. Rodríguez-Quintero

TL;DR

This work investigates how orbital angular momentum (OAM) in pseudoscalar mesons, notably the pion and kaon, is framed within a Poincaré-covariant QCD bound-state approach. Using continuum Schwinger function methods, the authors compute Bethe-Salpeter wave functions with RL and EHM-improved bRL kernels to reveal substantial intrinsic OAM that is frame-dependent, and they map rest-frame OAM onto light-front OAM through projections to yield LFWFs with significant L=0 and L=1 components. They demonstrate that the pion is roughly a 50/50 mix of light-front OAM zero and one, while the kaon is about 60/40, and that the LFWF OAM structure is essential for charge normalization and leptonic decay constants. These results underscore that NG bosons are complex bound states whose OAM content must be incorporated when computing observables, with broader implications for all hadrons and for the interpretation of OAM in QCD.

Abstract

Orbital angular momentum (OAM) is not a Poincaré invariant quantity; so, its value is observer dependent. Notwithstanding that, in quantum chromodynamics, a Poincaré-invariant theory, OAM is part of every hadron wave function. Using continuum Schwinger function methods, we elucidate both the subjective character of in-hadron OAM and expose some of its impacts on pion and kaon structure and observables. For instance, working with light-front projections of their Bethe-Salpeter wave functions, it is found that the pion is a roughly 50/50 mix of light-front OAM zero and one components and the kaon is a 60/40 system. The overall picture is that (near) Nambu-Goldstone modes are complex bound states, each with significant intrinsic OAM, independent of the observer's reference frame. This feature must be accounted for in the calculation of observables. Inductively, the same is true for all hadrons.

Orbital angular momentum in the pion and kaon: rest-frame and light-front

TL;DR

This work investigates how orbital angular momentum (OAM) in pseudoscalar mesons, notably the pion and kaon, is framed within a Poincaré-covariant QCD bound-state approach. Using continuum Schwinger function methods, the authors compute Bethe-Salpeter wave functions with RL and EHM-improved bRL kernels to reveal substantial intrinsic OAM that is frame-dependent, and they map rest-frame OAM onto light-front OAM through projections to yield LFWFs with significant L=0 and L=1 components. They demonstrate that the pion is roughly a 50/50 mix of light-front OAM zero and one, while the kaon is about 60/40, and that the LFWF OAM structure is essential for charge normalization and leptonic decay constants. These results underscore that NG bosons are complex bound states whose OAM content must be incorporated when computing observables, with broader implications for all hadrons and for the interpretation of OAM in QCD.

Abstract

Orbital angular momentum (OAM) is not a Poincaré invariant quantity; so, its value is observer dependent. Notwithstanding that, in quantum chromodynamics, a Poincaré-invariant theory, OAM is part of every hadron wave function. Using continuum Schwinger function methods, we elucidate both the subjective character of in-hadron OAM and expose some of its impacts on pion and kaon structure and observables. For instance, working with light-front projections of their Bethe-Salpeter wave functions, it is found that the pion is a roughly 50/50 mix of light-front OAM zero and one components and the kaon is a 60/40 system. The overall picture is that (near) Nambu-Goldstone modes are complex bound states, each with significant intrinsic OAM, independent of the observer's reference frame. This feature must be accounted for in the calculation of observables. Inductively, the same is true for all hadrons.
Paper Structure (5 sections, 17 equations, 4 figures, 4 tables)

This paper contains 5 sections, 17 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. NG bosons via the Bethe-Salpeter equation
  3. Rest-frame OAM
  4. Light-front OAM
  5. Perspective

Figures (4)

  • Figure 1: RL truncation valence dof OAM pairing contributions to the normalisation of a charged pseudoscalar meson, defined via Eq. \ref{['LIJRL']}. There are both positive (above plane) and negative (below plane) contributions, the total sum of which is unity.
  • Figure 2: RL truncation valence dof OAM pairing contributions to the normalisation of a charged pseudoscalar meson, defined via Eq. \ref{['DefLIJ']}. As in Fig. \ref{['figomega6']}, there are both positive (above plane) and negative (below plane) contributions -- see Table \ref{['C1OAM']}, the total sum of which is unity, notwithstanding the marked differences between related images.
  • Figure 3: bRL truncation valence dof OAM pairing contributions to the normalisation of a charged pseudoscalar meson, defined via Eq. \ref{['DefLIJ']}. As in Fig. \ref{['figomega6']}, there are both positive (above plane) and negative (below plane) contributions -- see Table \ref{['C1OAM']}, the total sum of which is unity, notwithstanding the marked differences between related images.
  • Figure 4: Zeroth Chebyshev moment of the ${\mathpzc E}_{\mathsf 5}(k;Q)$ term in Eq. \ref{['XOAM2']} as calculated with RL and bRL kernels. Using the RL kernel, $M(0)=0.4\,$GeV, and with the bRL interaction, $M(0)=0.45\,$GeV.