Multipole decomposition of the nucleon transverse phase space

TL;DR

This work delivers the first complete leading-twist study of quark Wigner distributions in the nucleon by carrying out a multipole decomposition in the transverse phase space. It introduces a general, symmetry-consistent parametrization in terms of basic multipoles $B^{(m_k,m_b)}_{X}$ and coefficient functions $C^{(m_k,m_b)}_{X}$, clarifying how spin-spin and spin-orbit correlations populate the phase space and how certain multipoles map to GPDs or TMDs while others expose new information from naive $\mathsf{T}$-odd components linked to initial-/final-state interactions. A novel representation of the transverse phase space is proposed to visualize simultaneous structures in $\boldsymbol{k}_T$ and $\boldsymbol{b}_T$, and results (within a light-front quark model) illustrate the multipole content for both naive $\mathsf{T}$-even and $\mathsf{T}$-odd sectors. The work establishes how 5D (or 6D when including $b_L$) phase-space distributions reduce to traditional GPDs and TMDs upon integration and highlights the rich angular correlations that encode nucleon spin structure with potential implications for phenomenology and lattice studies.

Abstract

We present a complete study of the leading-twist quark Wigner distributions in the nucleon, discussing both the $\mathsf T$-even and $\mathsf T$-odd sector, along with all the possible configurations of the quark and nucleon polarizations. We identify the basic multipole structures associated with each distribution in the transverse phase space, providing a transparent interpretation of the spin-spin and spin-orbit correlations of quarks and nucleon encoded in these functions. Projecting the multipole parametrization of the Wigner functions onto the transverse-position and the transverse-momentum spaces, we find a natural link with the corresponding multipole parametrizations for the generalized parton distributions and transverse-momentum dependent parton distributions, respectively. Finally, we show results for all the distributions in the transverse phase space, introducing a representation that allows one to visualize simultaneously the multipole structures in both the transverse-position and transverse-momentum spaces.

Figures (12)

• Figure 1: Simple illustration of the decomposition \ref{['Decomposition']} at fixed $x$ and $\boldsymbol{b}_T$. The phase-space distribution $\rho$ can be written as a product of a basic multipole $B$ (here a dipole in $\boldsymbol{k}_T$-space) with an oval-shaped coefficient function $C$.
• Figure 2: Representation of the transverse phase space. The circle represents the points in impact-parameter space at a fixed distance $|\boldsymbol{b}_T|$ from the center of the target. To each point on this circle is associated a distribution in transverse-momentum space. See text for more details.
• Figure 3: Naive $\mathsf T$-even (left) and $\mathsf T$-odd (right) contributions to the transverse phase-space distribution $\rho_{UU}$. See text for more details.
• Figure 4: Naive $\mathsf T$-even (left) and $\mathsf T$-odd (right) contributions to the transverse phase-space distribution $\rho_{UL}$. See text for more details.
• Figure 5: Naive $\mathsf T$-even (left) and $\mathsf T$-odd (right) contributions to the transverse phase-space distribution $\rho_{UT}$ for the quark polarization $\vec{S}^q_T=\vec{e}_x$ (red arrow). See text for more details.