The kaon off-shell generalized parton distributions and transverse momentum dependent parton distributions

TL;DR

This work extends the three-dimensional picture of hadron structure to kaons by computing off-shell generalized parton distributions (GPDs) and transverse momentum dependent distributions (TMDs) within the Nambu–Jona-Lasinio framework using proper time regularization. It derives vector and tensor off-shell kaon GPDs, analyzes forward limits, polynomiality, and impact-parameter dependent PDFs, and extracts off-shell electromagnetic and gravitational form factors. A key finding is that off-shell effects modify kaon GPDs by about $10$–$25\%$ and induce xi-odd terms in Mellin moments, yielding new off-shell form factors; the off-shell kaon TMD shows a stronger $x$-dependence with its peak at $x<0.5$, unlike the symmetric pion case. By comparing to on-shell results and the pion, the study clarifies how off-shellness influences the 3D partonic structure and provides a baseline for future investigations of vector mesons and more realistic interactions.

Abstract

We investigate the off-shell generalized parton distributions (GPDs) {and transverse momentum dependent parton distributions (TMDs)} of kaons within the framework of the Nambu--Jona-Lasinio model, employing proper time regularization. {Compared to the pion case, the off-shell effects in kaons are of similar magnitude, modifying the GPDs by about $10\%$--$25\%$, which is notable. The absence of crossing symmetry leads to odd powers in the $x$-moments of the off-shell GPDs, giving rise to new off-shell form factors.} We analyze the relations among these kaon off-shell form factors by analogy with electromagnetic form factors. Our results extend the off-shell GPDs from pions to kaons and simultaneously address the associated off-shell form factors. We also compare the off-shell and on-shell gravitational form factors of the kaon. In addition, the off-shell kaon TMD shows a stronger dependence on the momentum fraction $x$ than its on-shell counterpart.

Figures (9)

• Figure 1: The NJL gap equation in the Hartree--Fock approximation is illustrated in the figure, where a thin line denotes the elementary quark propagator and a shaded circle represents the $\bar{q}q$ interaction kernel. This kernel does not incorporate higher-order contributions, such as those from meson loops.
• Figure 2: Diagrams of off-shell GPDs for kaons, where $p^2 \neq p^{\prime 2} \neq m_K^2$.
• Figure 3: Kaon off-shell vector GPD: $H^u(x,\xi,t,p^2,m_K^2)$ in Equation (\ref{['agpdf1']}) with $\xi=0.5$ for $t = 0$ (upper panel) and $t = -1$ (lower panel). Case $p^2 = 0$ is represented by the red solid line, $p^2 = m_K^2$ black dotted line, $p^2 = 0.4$ GeV$^2$ blue dashed line, and $p^2 = 0.6$ GeV$^2$ purple dot-dashed line.
• Figure 4: Kaon off-shell tensor GPD: $E^u(x,\xi,t,p^2,m_K^2)$ in Equation (\ref{['agtpdf']}) with $\xi=0.5$ for $t = 0$ (upper panel) and $t = -1$ (lower panel). Case $p^2 = 0$ is represented by the red solid line, $p^2 = m_K^2$ black dotted line, $p^2 = 0.4$ GeV$^2$ blue dashed line, and $p^2 = 0.6$ GeV$^2$ purple dot-dashed line.
• Figure 5: Off-shell $u$ quark PDFs in the kaon: $u_K\left(x,p^2\right)$, for different virtualities: $p^2=0$ GeV$^{2}$---orange dotted curve, $p^2=m_K^2=0.495^2$ GeV$^{2}$---red solid curve, $p^2=0.4$ GeV$^{2}$---blue dashed curve, $p^2=0.6$ GeV$^{2}$---purple dot-dashed curve, and $p^2=0.8$ GeV$^{2}$---black solid thick curve.