Kaon GTMDs in the Dyson–Schwinger equations using contact interaction

Abstract An array of the kaon twist-two, three, four generalised transverse momentum dependent parton distributions (GTMDs) has been investigated within the framework of covariant and confining Dyson–Schwinger equations using contact interaction. GTMDs are of great significance as they encompass information regarding both the generalised parton distributions (GPDs) and the transverse momentum dependent parton distributions (TMDs), thus being considered as the parent distribution. From GTMDs, we can derive the twist-two, three, four GPDs and TMDs. GPDs are obtained through the integration of $$\varvec{k}_{\perp }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msub> </mml:math> from GTMDs, with a focus on the twist-two GPDs. The first Mellin moments of GPDs yield the form factors of local currents. The second Mellin moments of vector GPDs are related to gravitational form factors. The Wigner distribution can be obtained from a Fourier transform in the transverse space of the GTMDs at skewedness parameter $$\xi =0.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The Wigner distributions of an unpolarized, longitudinally polarized, and transversely polarized quark inside the kaon have been calculated. Through the three twist-two Wigner distributions, we study the dynamical spin effects of the quarks inside kaon to reveal its multidimensional structure. The spin-orbit correlations between a hadron and a quark can be explained based on the phase-space average of Wigner distributions. We investigate the correlation between the longitudinal spin and orbital angular momentum of valence quarks within the pion and kaon. Our findings reveal that $$C_z^{u,K}=-0.336,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>z</mml:mi> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>0.336</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$C_z^{s,K}=0.242,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>z</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>0.242</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$C_z^{u,\pi }=-0.374.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>z</mml:mi> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>0.374</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The parton distribution function in impact parameter space can be derived from the Wigner distribution. The study focuses on the light-front transverse-spin distributions $$\rho _u^1\left( \varvec{b}_{\bot },\varvec{s}_{\perp }\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>⊥</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> and $$\rho _u^2\left( \varvec{b}_{\bot },\varvec{s}_{\perp }\right) ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>⊥</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msub> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which exhibit distortions, and we calculate their average shift. This comprehensive analysis will enhance our understanding of the parton distribution picture of kaons. While there have been a few theoretical studies investigating the GTMDs in experiments, no experimental data is currently available. The results of our model calculation offer qualitative insights into these distributions.