Spin densities in the transverse plane and generalized transversity distributions

TL;DR

This work develops a density interpretation of generalized parton distributions in the nucleon's transverse plane at zero skewness, revealing how polarization maps to impact-parameter densities and exposing a rich structure of chiral-even and chiral-odd contributions. It derives positivity bounds that tighten known constraints, connects GPD-based densities with transverse-momentum dependent distributions, and establishes a framework of relations between twist-two and twist-three operators via QCD equations of motion. The results underscore the practical potential of lattice QCD to access generalized transversity moments and twist-three form factors, and they illuminate how spin-dependent spatial densities encode fundamental aspects of nucleon structure. The work thus links transverse spin phenomenology, lattice calculations, and the dialed interplay between twists in a coherent, gauge-invariant formalism.

Abstract

We show how generalized quark distributions in the nucleon describe the density of polarized quarks in the impact parameter plane, both for longitudinal and transverse polarization of the quark and the nucleon. This density representation entails positivity bounds including chiral-odd distributions, which tighten the known bounds in the chiral-even sector. Using the quark equations of motion, we derive relations between the moments of chiral-odd generalized parton distributions of twist two and twist three. We exhibit the analogy between polarized quark distributions in impact parameter space and transverse momentum dependent distribution functions.

Figures (6)

• Figure 1: Density plots in the impact parameter plane for the functions $b \exp[-b^2 /b_0^2] \sin\phi$ (left) and $b^2 \exp[-b^2 /b_0^2] \cos(2\phi)$ (right), with $b_0 = 0.5$ fm. These functions illustrate the terms in the quark density (\ref{['trans-distr']}) which break rotational symmetry, as explained after (\ref{['density-terms']}). Dark areas represent high densities.
• Figure 2: The vector fields $\epsilon^{ij} b^j \exp[-b^2 /b_0^2]$ (left) and $(2 b^i b^j - b^2 \delta^{ij}) S^j \exp[-b^2 /b_0^2]$ (right) with $S^i$ along the $x$-axis and $b_0 = 0.5$ fm. They illustrate the form of two terms in the decomposition of $F_T^i(x,\hbox{\boldmath{$b$}})$, which describes the transverse polarization of quarks in the impact parameter plane. The third term in the decomposition is a field parallel to $S^i$.
• Figure 3: Left: Illustration of the first moment $\int_{-1}^1 dx\, F(x,\hbox{\boldmath{$b$}})$ of the impact parameter density for unpolarized $u$-quarks in a proton with transverse spin vector $\hbox{\boldmath{$S$}}=(1,0)$. Right: The same for the first moment ${\frac{1}{2}} \int_{-1}^1 dx\, [F(x,\hbox{\boldmath{$b$}})+s^i F_T^i(x,\hbox{\boldmath{$b$}})]$ of the distribution of $u$-quarks with transverse spin vector $\hbox{\boldmath{$s$}}=(1,0)$ in an unpolarized proton. Dark areas represent the highest and light areas the lowest values of the density. Further explanation is given in the text.
• Figure 4: Illustration of the lowest moment ${\frac{1}{2}} \int_{-1}^1 dx\, [F(x,\hbox{\boldmath{$b$}}) + s^i F^i_T(x,\hbox{\boldmath{$b$}})]$ for $u$-quarks in a proton with transverse spin vector $\hbox{\boldmath{$S$}}=(1,0)$. The transverse quark spin vector is $\hbox{\boldmath{$s$}}=(1,0)$ in the left plot and $\hbox{\boldmath{$s$}}=(0,1)$ in the right plot.
• Figure 5: Illustration of the lowest moment $\int_{-1}^1 dx\, F^i_T(x,\hbox{\boldmath{$b$}})$ of the vector field describing the transverse polarization of $u$-quarks in an unpolarized proton (left) and in a proton with transverse spin in the $x$-direction (right).