Chiral dynamics and partonic structure at large transverse distances

TL;DR

This study uses generalized parton distributions in the impact-parameter representation to isolate the universal, large-distance component of the nucleon’s parton densities driven by chiral dynamics. By modeling soft-pion exchange with finite-size form factors, the authors quantify how much of the sea—particularly the isovector antiquark asymmetry—originates from $b \gtrsim 0.5$ fm and $x \lesssim M_\pi/M_N$, finding about one-third of $\bar d - \bar u$ at $x \sim 0.1$ comes from this region. The analysis also shows the strange sea is predominantly short-distance, the singlet quark size can exceed the gluon size at small $x$, and the large-$N_c$ limit is respected through $N$ and $\Delta$ degeneracy. The work provides a framework for clean chiral observables in the nucleon and outlines experimental avenues (e.g., DVCS, exclusive $J/\psi$ production, and EIC studies) to test these predictions, while acknowledging limitations from diffusion and small-$x$ corrections that require future refinement.

Abstract

We study large-distance contributions to the nucleon's parton densities in the transverse coordinate (impact parameter) representation based on generalized parton distributions (GPDs). Chiral dynamics generates a distinct component of the partonic structure, located at momentum fractions x ~< M_pi/M_N and transverse distances b ~ 1/M_pi. We calculate this component using phenomenological pion exchange with a physical lower limit in b (the transverse "core" radius estimated from the nucleon's axial form factor, R_core = 0.55 fm) and demonstrate its universal character. This formulation preserves the basic picture of the "pion cloud" model of the nucleon's sea quark distributions, while restricting its application to the region actually governed by chiral dynamics. It is found that (a) the large-distance component accounts for only ~1/3 of the measured antiquark flavor asymmetry dbar - ubar at x ~ 0.1; (b) the strange sea quarks, s and sbar, are significantly more localized than the light antiquark sea; (c) the nucleon's singlet quark size for x < 0.1 is larger than its gluonic size, <b^2>_{q + qbar} > <b^2>_g, as suggested by the t-slopes of deeply-virtual Compton scattering and exclusive J/psi production measured at HERA and FNAL. We show that our approach reproduces the general N_c-scaling of parton densities in QCD, thanks to the degeneracy of N and Delta intermediate states in the large-N_c limit. We also comment on the role of pionic configurations at large longitudinal distances and the limits of their applicability at small x.

Figures (11)

• Figure 1: Parametric region where the pion distribution in the nucleon is governed by chiral dynamics. The variables are the pion longitudinal momentum fraction, $y$, and transverse position, $b$.
• Figure 2: The pion GPD in the nucleon. (a) Transition matrix element of the density of pions with longitudinal momentum fraction $y \sim M_\pi / M_N$ and transverse momentum transfer $|\bm{\Delta}_\perp | \sim M_\pi$, Eq. (\ref{['H_pi_number']}). (b) Invariants used in modeling finite--size effects with form factors. $t_{1, 2}$ are the pion virtualities in the invariant formulation, Eq. (\ref{['t_12']}); $s_{1, 2}$ the invariant masses of the $\pi B$ systems in the time--ordered formulation, Eq. (\ref{['s_12']}).
• Figure 3: The transverse spatial distribution of pions in the nucleon, $f_{\pi N} (y, b)$, as a function of $b$, for values $y = 0.07$ and 0.3. Shown is the radial distribution $2\pi b \, f_{\pi N} (y, b)$, whose integral over $b$ (area under the curve) gives the pion momentum distribution. Solid lines: Pion cloud model with virtuality cutoff (exponential form factor, $\Lambda_{\pi N} = 1.0 \, \textrm{GeV}$) Koepf:1995yh. Dashed line: Distribution for pointlike particles, regulated by subtraction at $\bm{\Delta}_\perp^2 = 0$; the integral over $b$ does not exist in this case. The estimated "core" radius, Eq. (\ref{['b_core']}), is marked by an arrow.
• Figure 4: The median pion virtuality in the unregularized integral, Eqs. (\ref{['I_8_10']})---(\ref{['phi_10']}), as a function of $b$, for $y = 0.07$ (solid line) and $0.3$ (dotted line). It is defined as the value of the virtuality cutoff, $\Lambda^2_{\text{virt}}$, for which $f_{\pi N} (y, b)$ reaches half of its value for $\Lambda^2_{\text{virt}} \rightarrow \infty$, corresponding to the unregularized integral.
• Figure 5: Effective momentum distribution of pions in $\pi N$ configurations with impact parameters $b > b_0$, Eq. (\ref{['fy_bint']}), in the pion cloud model. Solid lines: Distributions obtained with a virtuality cutoff, Eq. (\ref{['ff_virt']}) (exponential form factor, $\Lambda_{\text{virt}} = 1.0 \, \textrm{GeV}$), for $b_0 = 0$ (full integral), $b_0 = 0.55 \, \text{fm}$ and $b_0 = 1.1 \, \text{fm}$. Dashed lines: Same for distributions obtained with an invariant mass cutoff, Eq. (\ref{['ff_virt']}) (exponential form factor, $\Lambda_{\text{virt}} = 1.66 \, \textrm{GeV}$). The value of $\Lambda_{\text{virt}}$ was chosen such that it produces the same total number of pions ($y$--integral) for the full distribution as the given virtuality cutoff. The value $y = M_\pi / M_N$ is indicated by an arrow.