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

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.