We discuss in detail the relativistic spatial distribution of transverse angular momentum, including both orbital and intrinsic spin contributions. Using the quantum phase-space formalism, we begin with the definition of the three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin in a generic Lorentz frame. By integrating these three-dimensional spatial distributions over the longitudinal axis, we derive for the first time the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum for spin-0 and spin-1/2 targets in the transverse plane. We verify the transverse spin sum rule about the relativistic center of spin for spin-0 and spin-1/2 systems, and find that the transverse total angular momentum distribution is non-trivial, even for spin-0 targets. We also show how the distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum change with the target momentum.

France

Cédric Lorcé

India

Asmita Mukherjee

Ravi Singh

Ho-Yeon Won

Mapping the transverse spin sum rule in position space

<p>We study in detail the relativistic distributions of energy, longitudinal momentum, longitudinal energy flux, and axial momentum flux inside nucleons based on the quantum phase-space formalism. Similar to recent studies on the electromagnetic current, we include the effects of the nucleon polarization and show that the latter are essential for understanding how the Breit frame distributions transform under a longitudinal Lorentz boost. We also explicitly demonstrate that, in the infinite-momentum frame, these distributions allow one to recover not only the “good” but also the “bad” components of the light-front energy-momentum tensor distributions.</p>

Relativistic energy-momentum tensor distributions in a polarized nucleon

The spin-orbit correlation in spin-0 hadrons can be investigated through the kinetic energy-momentum tensor form factor <math altimg='si1.svg'><msup><mrow><mover accent='true'><mrow><mi>F</mi></mrow><mrow><mo stretchy='false'>&#732;</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></math>. We observe that the latter is also related to a torque about the radial direction, which we interpret as a chiral stress. If we neglect the quark mass contribution, then <math altimg='si1.svg'><msup><mrow><mover accent='true'><mrow><mi>F</mi></mrow><mrow><mo stretchy='false'>&#732;</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo></math> is simply proportional to the electromagnetic form factor for spin-0 hadrons, and the spin-orbit correlation is equal to minus half of the valence quark number. Given the extensive studies on the electromagnetic form factor for spin-0 hadrons such as pions, kaons, and the <italic>&#945;</italic> particle, we present the spatial distributions of chiral stress and kinetic spin-orbit correlation based on current parametrizations of the pion electromagnetic form factor.

China

Qin-Tao Song

Spin-orbit correlation and spatial distributions for spin-0 hadrons

We present the first systematic study of the relativistic axial-vector four-current distributions inside a nucleon. We show in particular that the slope of the axial form factor <math><msub><mi>G</mi><mi>A</mi></msub><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> in the forward limit—conventionally denoted as <math><msubsup><mi>R</mi><mi>A</mi><mn>2</mn></msubsup></math> in the literature—does not represent the three-dimensional mean-square axial radius in the Breit frame, but corresponds instead to a contribution to the mean-square spin radius. We derive explicit expressions for the latter in different frames and find in general additional contributions that depend on both the nucleon mass and the forward values of the axial-vector form factors <math><msub><mi>G</mi><mi>A</mi></msub><mo>(</mo><mn>0</mn><mo>)</mo></math> and <math><msub><mi>G</mi><mi>P</mi></msub><mo>(</mo><mn>0</mn><mo>)</mo></math>. This provides an additional key motivation for ongoing lattice quantum chromodynamics (QCD) calculations and future experimental measurements of the induced pseudoscalar form factor <math><msub><mi>G</mi><mi>P</mi></msub><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>.

Yi Chen

Yang Li

Qun Wang

Nucleon axial radius

In the Standard Model of particle physics, the axial current is not conserved, due both to fermion masses and to the axial anomaly. Using perturbative quantum chromodynamics, we calculate matrix elements of the local and non-local axial current for a gluon target, clarifying their connection with the axial anomaly. In so doing, we also reconsider classic results obtained in the context of the nucleon spin sum rule as well as recent results for off-forward kinematics. An important role is played by the infrared regulator, for which we put a special emphasis on the nonzero quark mass. We highlight cancellations that take place between contributions from the axial anomaly and the quark mass, and we elaborate on the relation of those cancellations with the conservation of angular momentum.

United States

Ignacio Castelli

Adam Freese

Andreas Metz

Italy

Barbara Pasquini

Germany

Simone Rodini

Perturbative results of matrix elements of the axial current and their relation with the axial anomaly

We study the energy-momentum tensor of spin-0 and spin-<math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> hadrons in momentum space. We parametrize this object in terms of so-called gravitational transverse-momentum distributions, and we identify in the quark sector the relations between the latter and the usual transverse-momentum distributions. Focusing on particular components of the energy-momentum tensor, we study momentum densities, flux of inertia and stress distribution in momentum space, revealing part of the wealth of physical information that can be gained from higher-twist transverse-momentum distributions.

Gravitational transverse-momentum distributions

We study in detail the energy–momentum tensor of the scalar diquark model at the one-loop level using two different regularization methods. We extract the perturbative expressions for the gravitational form factors and we check explicitly that all the Poincaré sum rules are satisfied. We confirm in particular that, like in quantum electrodynamics, the symmetric energy–momentum tensor is finite. Finally, we discuss the results for the trace of the energy–momentum tensor and its relation to the mass of the system.

Arturo Amor-Quiroz

William Focillon

Energy–momentum tensor in the scalar diquark model

As a follow up of our work on the electromagnetic four-current, we study for the first time the relativistic polarization and magnetization spatial distributions inside a spin-<math><mfrac><mn>1</mn><mn>2</mn></mfrac></math> target within the quantum phase-space approach. While the polarization-magnetization tensor is usually defined in terms of the Gordon decomposition of the electromagnetic four-current, a Breit frame analysis reveals that a physically simpler and more natural picture of the system arises when the polarization-magnetization tensor is instead defined in terms of a Sachs decomposition. Relativistic polarization and magnetization distributions for a moving target are compared with their light-front counterparts. In particular, we show that the genuine light-front magnetization distributions are defined in terms of Fourier transforms of the Sachs magnetic form factor, rather than in terms of the Pauli form factor as suggested earlier in the literature. We finally illustrate our results in the case of a nucleon using the electromagnetic form factors extracted from experimental data.

Nucleon relativistic polarization and magnetization distributions

The quantum phase-space approach allows one to define relativistic spatial distributions inside a target with arbitrary spin and arbitrary average momentum. We apply this quasiprobabilistic formalism to the whole electromagnetic four-current operator in the case of spin-0 and spin-<math><mfrac><mn>1</mn><mn>2</mn></mfrac></math> targets, study in detail the frame dependence of the corresponding spatial distributions, and compare our results with those from the light-front formalism. While former works focused on the charge distributions, we extend here the investigations to the current distributions. We clarify the role played by the Wigner rotation and argue that electromagnetic properties are most naturally understood in terms of Sachs form factors, contrary to what the light-front formalism previously suggested. Finally, we illustrate our results using the pion and nucleon electromagnetic form factors extracted from experimental data.

Pion and nucleon relativistic electromagnetic four-current distributions

We estimate kinematical higher-twist (up to twist 4) corrections to the <math><msup><mi>γ</mi><mo>*</mo></msup><mo>(</mo><msub><mi>q</mi><mn>1</mn></msub><mo>)</mo><mi>γ</mi><mo>(</mo><msub><mi>q</mi><mn>2</mn></msub><mo>)</mo><mo>→</mo><mi>M</mi><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo><mover><mi>M</mi><mo>¯</mo></mover><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></math> amplitudes at large <math><msup><mi>Q</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><msubsup><mi>q</mi><mn>1</mn><mn>2</mn></msubsup></math> and small <math><mi>s</mi><mo>=</mo><mo>(</mo><msub><mi>q</mi><mn>1</mn></msub><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><msup><mo>)</mo><mn>2</mn></msup></math>, where <math><mi>M</mi></math> is a scalar or pseudoscalar meson. This process is known to factorize at leading twist into a perturbatively calculable coefficient function and generalized distribution amplitudes (GDAs). The kinematical higher-twist contributions of order <math><mi>s</mi><mo>/</mo><msup><mi>Q</mi><mn>2</mn></msup></math> and <math><msup><mi>m</mi><mn>2</mn></msup><mo>/</mo><msup><mi>Q</mi><mn>2</mn></msup></math> turn out to be important in the cross section, considering the kinematics accessible at Belle and Belle II. We present numerical estimates for the cross section for <math><msup><mi>γ</mi><mo>*</mo></msup><mi>γ</mi><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup></math> with the <math><mi>π</mi><mi>π</mi></math> GDA extracted from Belle measurements and with the asymptotic <math><mi>π</mi><mi>π</mi></math> GDA as inputs to study the magnitude of the kinematical corrections. To see how the target mass corrections of order <math><msup><mi>m</mi><mn>2</mn></msup><mo>/</mo><msup><mi>Q</mi><mn>2</mn></msup></math> affect the cross section, we also perform the calculation for <math><msup><mi>γ</mi><mo>*</mo></msup><mi>γ</mi><mo>→</mo><mi>η</mi><mi>η</mi></math> by using a model <math><mi>η</mi><mi>η</mi></math> GDA. In the range <math><mi>s</mi><mo>&gt;</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></math>, the kinematical higher-twist corrections account for <math><mo>∼</mo><mn>15</mn><mo>%</mo></math> of the total cross section, an effect which is not negligible. Since <math><mi>π</mi><mi>π</mi></math> GDAs are the best way to access the pion energy-momentum tensor (EMT), our study demonstrates that an accurate evaluation of EMT form factors requires the inclusion of kinematical higher-twist contributions.

Bernard Pire

Kinematical higher-twist corrections in <math><msup><mi>γ</mi><mo>*</mo></msup><mi>γ</mi><mo>→</mo><mi>M</mi><mover><mi>M</mi><mo>¯</mo></mover></math>