Kaon generalized parton distributions and light-front wave functions in the Nambu–Jona-Lasinio model

Kaon generalized parton distributions (GPDs) and the leading Fock state light-front wave functions are investigated in the framework of Nambu–Jona-Lasinio model with proper time regularization. In addition, we compared the form factors, parton distribution functions, and generalized form factors obtained from them, respectively. The first Mellin moments of GPDs result in the form factors of local currents. The second Mellin moments of vector GPDs are related to gravitational form factors, the quark mass distribution θ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _2$$\end{document} and the quark pressure distribution θ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _1$$\end{document}. When taking a Fourier transform of GPDs in impact parameter space, we can get the mean-squared impact parameter for the quarks of the kaon: ⟨b⊥2⟩Ku=0.149\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle {\varvec{b}}_{\bot }^2\rangle _K^u=0.149$$\end{document} fm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}, ⟨b⊥2⟩Ks=0.088\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle {\varvec{b}}_{\bot }^2\rangle _K^s=0.088$$\end{document} fm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}. This means that the kaon s quark is nearer to the center of transverse momentum than the u quark. We also give the light-cone energy radius for the quarks of the kaon from the mass distribution θ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _2$$\end{document}: rE,LCu,K=0.187\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{E,LC}^{u,K}=0.187 $$\end{document} fm, rE,LCs,K=0.167\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r_{E,LC}^{s,K}=0.167$$\end{document} fm, and the light-cone charge radius from quark form factors of the kaon: rc,LCu,K=0.390\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{c,LC}^{u,K}=0.390 $$\end{document} fm, rc,LCs,K=0.296\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{c,LC}^{s,K}=0.296 $$\end{document} fm, which means that the s quark has a smaller extent than the u quark. The light-front transverse-spin distributions ρu1b⊥,s⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _u^1\left( {\varvec{b}}_{\bot },{\varvec{s}}_{\perp }\right) $$\end{document} and ρu2b⊥,s⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _u^2\left( {\varvec{b}}_{\bot },{\varvec{s}}_{\perp }\right) $$\end{document} show distortions, the average shift are ⟨b⊥y⟩1u=0.116\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle b_{\bot }^y\rangle _1^u=0.116$$\end{document} fm and ⟨b⊥y⟩2u=0.083\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle b_{\bot }^y\rangle _2^u=0.083$$\end{document} fm. On the kinematic domain associated with the valence-quark dominance, the unpolarized Wigner distribution from light-front wave functions is sharply peaked. It extends as the transverse position variable increases in magnitude and has a domain of negative support. Through the comparison of distributions from the two methods, we find that they give the same multi-dimensional mapping of the kaon in the Nambu–Jona-Lasinio model.


Introduction
Generalized parton distributions (GPDs) [1][2][3][4][5][6][7][8][9] were proposed in the 1990s to describe the multi-dimensional pictures of hadrons. GPDs include a mass of inaccessible information about the partonic structure of hadrons; therefore, GPDs have been under thorough theoretical and experimental studies ever since. Experimentally, GPDs offer a solid formal basis to link information from multifarious inclusive, semiinclusive and exclusive processes in an efficient, explicit way. Hard exclusive processes, such as deeply virtual Compton scattering (DVCS) [2,[10][11][12] and deeply virtual meson production (DVMP) [13,14], in the asymptotic regime, can be expressed in terms of GPDs. The publications of new experimental data [15,16] have increased the interest in the study of GPDs; the upgraded Jefferson Laboratory facility 12 can offer very precise experimental data in the near future, this will challenge the present understanding of GPDs and furthermore the present models of GPDs. Theoretically, the proper description of GPDs plays an important role in obtaining insight into the internal structure of hadrons. Comparing with the available experimental data, using the phenomenological models, the framework of GPDs are well built now, providing a three-dimensional picture of the hadrons [4,6]. GPDs provide more than the ordinary parton distribution functions (PDFs) [17][18][19][20][21][22][23][24][25][26] and form factors (FFs) [27][28][29][30][31][32] in understanding the internal structure of hadrons. Besides the longitudinal-momentum fraction x and the momentum transfer q 2 , GPDs also depend on the skewness parameter ξ . GPDs contain information of both the spatial and the momentum quark distributions in a hadron. GPDs can also provide information that can define the internal structure and dynamics of the hadron: the mechanical properties of the angular momentum [2,33], pressure, and shear forces [34,35]. Therefore, GPDs can be treated as a useful tool to get the transverse spatial distribution of partons.
LFWFs and GPDs are hard to calculate from QCD directly, the phenomenological parametrizations are successful, but still need to be improved. Thus, several modeling frameworks have been used in the last few decades, each one having its advantages. For instance, we have the Nambu-Jona-Lasinio (NJL) model [63][64][65][66][67][68], the constituent quark model [69][70][71], the light-front holographic model [72][73][74], the light-front quark-diquark model [75][76][77], the AdS/QCD model [78][79][80], the chiral quark model [81,82] and the Dyson-Schwinger equations (DSEs) [7,8,83]. The NJL model has an effective Lagrangian of relativistic fermions interacting through local fermion-fermion couplings, and especially, it preserves one of the most important fundamental symmetries of QCD, namely, chiral symmetry. NJL results are to be identified as qualitatively sound initial conditions for use in exploring the domain of physical possibilities in areas where more realistic frameworks are not yet able to provide insights or predictions, so that is widely used in many fields [63,[84][85][86][87]. In this paper, we will use the NJL model to derive LFWFs and GPDs. This paper is constructed as follows: In Sect. 2, we give a brief introduction of the NJL model, then the definition and calculation of the kaon GPDs, in addition, the fundamental properties of kaon GPDs will be examined in this section. In Sect. 3, kaon LFWFs in the NJL model will be discussed, the relationship of LFWFs with the PDFs, FFs, PDAs and GPDs will be given. Finally, we give a brief summary and outlook in Sect. 4.

Generalized parton distributions
Different from the diagonal parton distributions, GPDs are determined by matrix elements of quark and gluon lightcone operators for distinct initial and final states of hadron. GPDs include richer information than the ordinary parton distributions, thus they cannot be interpreted as probabilities directly. We will express GPDs in a symmetric manner as in Refs. [4,6], with a skewing parameter ξ . In this section, we will calculate kaon vector and tensor GPDs in the NJL model using proper time regularization (PTR).

Nambu-Jona-Lasinio model
The SU(3) flavor NJL Lagrangian in theqq interaction channel we take in the form [88] where the quark field has the flavor components ψ T = (u, d, s), λ a , a = 0, . . . , 8 are the eight Gell-Mann matrices in flavor space where λ 0 = √ 2/3 .m = diag (m u , m d , m s ) is the current quark mass matrix. G π is an effective coupling strength of the scalar (qλ a q) and pseudoscalar (qγ 5 λ a q) interaction channel.
The dressed-quark propagator in the NJL model is obtained by solving the gap equation in Fig. 1, where The interaction kernel of the gap equation in Fig. 1 is local so we get a constant dressed-quark mass M q , which satisfies where the trace is over Dirac indices. From the above equation we can see that in contrast with the SU(2) flavor case, flavor mixing is absent in the SU(3) flavor case [85,89]. Moreover, dynamical chiral symmetry breaking can take place only when the coupling constant G π > G critical , which gives a nontrivial solution M q > 0. In the NJL model, the description of mesons asqq bound states is obtained through the Bethe-Salpeter equation (BSE). The solution of the BSE in each meson channel is given by a two-body t-matrix that depends on the nature of the interaction channel, where the reduced t-matrices for the kaon meson reads where the bubble diagram P P (q 2 ) is defined as where the traces are over Dirac and isospin indices, for the kaon i, j= 4, 5, 6, 7. The mass of the kaon is given by the pole in the reduced t-matrix, Expanding the full t-matrix about the pole gives the homogeneous Bethe-Salpeter vertex for the kaon, the normalization factor is given by This residue can be explained as the square of effective meson-quark-quark coupling constant. Homogeneous Bethe-Salpeter vertex functions are an essential ingredient in, for example, triangle diagrams that determine the meson form factors. The NJL model is a non-renormalizable model, so a regularization scheme is needed. We will use the PTR scheme [90][91][92], where X represents a product of propagators that have been connected using Feynman parametrization. UV is the ultraviolet cutoff. The NJL model does not contain confinement, the infrared cutoff is used to mimic it, therefore it should be of the order QCD and we choose IR = 0.240 GeV. For the light-quark dressed masses we choose M u = M d = 0.4 GeV; the ultraviolet cutoff UV and the coupling G π are constrained by the empirical values of pion decay constant and pion mass. The kaon is regarded as a relativistic bound state of a dressed quark and a dressed antiquark whose properties are determined by solving theqq Bethe-Salpeter equation in the pseudoscalar channel. In Table 1, we give the parameters used in this paper. In the following, we will use the C functions and formulas as explained in the appendix. In the NJL model, kaon GPDs are given by Fig. 2, where p is the initial and p is the final kaon momentum, the kinematics and related quantities of this process are defined as where ξ is the so called skewness parameter, n is the lightcone four-vector defined as n = (1, 0, 0, −1), in the lightcone coordinate for any four-vector, v + in the light-cone coordinate can be defined as The vector and tensor quark GPDs of the kaon are defined as where x is the longitudinal-momentum fraction.
is the no spin flip or vector GPD, E q (x, ξ, t) is the spin flip or tensor GPD.
In Fig. 2, we give the Feynman diagrams of the kaon GPDs, where the operators in the diagram have the form where the first is for the vector GPD and the second for the tensor GPD. In the NJL model, the u quark vector and tensor GPDs of K + meson are defined as where Here we use the notations we can get the following reduced formulas: after some calculation we arrive at and where θ is the step function, and θ function is 1 in the corresponding region, otherwise it is zero. These results are in the region ξ > 0, under ξ → −ξ : θξ 1 ↔ θ ξ 1 ; and θξ ξ /ξ , θ αξ /ξ are invariant. The diagrams of vector and tensor GPDs are plotted in Figs. 3, 4, 5 and 6. From the diagrams we can see that H u (x, ξ, 0) = 0 when x < −ξ , the vector GPD is continuous but not differentiable at x = ±ξ except ξ = 0, the vector GPD is not continuous at ξ = 0, the tensor GPD is similar. The properties of GPDs, for example, in the forward limit it should reduce to PDF, we will check these properties separately in this section.

Forward limit
In the case that the initial and final kaon have the same momentum p = p , which means ξ = 0, t = 0, the vector GPD reduces to the kaon PDF, which satisfies In Fig. 3, when ξ = 0, H u (x, 0, 0) is 0 when x ∈ [−1, 0], in the region x ∈ [0, 1], this u quark PDF of the kaon is different from the pion PDF in the NJL model [9].

Symmetry properties
In general, GPDs are neither even nor odd of x, the combinations, are useful, the first equation corresponds to charge conjugation C = +1 in the T-channel. The second equation corresponds to C = −1 in the T-channel, the tensor GPD Our results preserve the time reversal invariance property of GPDs, we can establish this straightforwardly using the results following Eqs. (23) and (A2).

Polynomiality condition
When integrating x n moments of GPDs, one can get the polynomiality condition which is the remarkable polynomiality property of GPDs. When n = 0, we get the FFs where F u K (Q 2 ) and F u T (Q 2 ) are the u quark vector FF and tensor FF. F u K (Q 2 ) is the electromagnetic form factor of the kaon, F u T (0) is the tensor anomalous magnetic moment for n = 0 [93]. We have When n = 1, the GPDs should follow the sum rule: where θ u 2 relates to the u quark mass distribution within the kaon and θ u 1 is associated with the u quark pressure distribution. The polynomial only has the ξ 0 and ξ 2 term, this satisfies Eq. (28). The n = 2 case has also been checked, it also satisfies the polynomiality condition. The generalized FFs for n = 1 are for the u quark tensor GPD E u (x, ξ, t) of the kaon in the NJL model, B u 2,2 (Q 2 ) = 0. We have plotted these FFs in Fig. 7; from the diagram we can see that θ u 1 is harder than θ u 2 and both of them are harder than the vector FF, the tensor anomalous magnetic moment B u 1,0 is softer than B u 2,0 . A light-cone energy radius can be defined in relation to the generalized FF A 2,0 (Q 2 ) defined as [94] the second equation is analogous to the light-cone charge radius, which can be compared with the light-cone energy radius. We get r u,K E,LC = 0.187 fm, r s,K E,LC = 0.167 fm, r u,K c,LC = 0.390 fm and r s,K c,LC = 0.296 fm. In these two cases, the s quark has a smaller extent than the u quark. Comparing with the results of pion r u,π E,LC = 0.188 fm, r u,π c,LC = 0.374 fm, then r u,π E,LC /r u,π c,LC = 0.502. In Ref. [95], their ratio is 0.56 from the leading Fock state LFWF of pion using DSEs, and our former results demonstrate that the pion A 2,0 from the leading Fock state LFWF is the same as our direct calculation from the vector GPD [9]. Therefore, the values are the same, our ratio is smaller than their results. The kaon A 2,0 is similar in the NJL model, r u,K E,LC /r u,K c,LC = 0.479, which is smaller than the pion case, while r s,K E,LC /r s,K c,LC = 0.564 is bigger. These radii will be impacted by higher Fock states.

Impact parameter dependent PDFs
In order to localize the hadron in the transverse direction, we introduce where | p + , p ⊥ are the light-cone helicity eigenstates. N is a normalization factor, R ⊥ is the center of transverse momentum, which is defined as where b ⊥ is the distance from the center of transverse momentum, and we sum over all partons in the hadron. The impact parameter dependent PDFs are given by This means that the impact parameter dependent PDFs defined above are the Fourier transform of H u x, 0, −q 2 ⊥ , therefore, when we get H u x, 0, −q 2 ⊥ , parton distribution as a function of the distance b ⊥ and the light-cone momentum fraction x can be determined.
When ξ → 0 and t = 0, the GPDs become where x is defined in the region x ∈ [0, 1], then we can get when integrating b ⊥ we can get the u quark PDF in Eq. (24). We hope H u (x, 0, −q 2 ⊥ ) becomes independent of −q 2 ⊥ as x → 1. It should be highlighted that this behavior is in accordance with "Feynman picture" for hadron form factors approximating x = 1. In this picture, when one parton carries x ≈ 1 and all others are "tiny" so that they can be easily scattered elastically, because the wee partons do not have a direction and therefore can be easily "sheer off". Thus, in the Feynman picture, the active quark takes a larger fraction of the hardron's overall momentum contributing to the FF and we have the tendency to a weaker t dependence [96]. The width distribution of partons in the kaon for a given momentum fraction x is when x → 1, this impact parameter should be zero. This is because the struck quark becomes nearer to the center of momentum with the increasing of its weight. For our results, is the u quark PDF in Eq. (24), when x → 1, Fig. 8 The width of the distribution of partons in the kaon for a given momentum fraction x defined in Eq. (44) it is a definite value. For , for the first term of Eq. (42), which means this term is zero, it is x independent. The integration of the second term in Eq. (42) is restricted to the region α ∈ [0, 1 − x]; when x → 1, the integration will vanish, so vanishes when x → 1. Our results satisfy r 2 ⊥ x→1 = 0. This also can be seen from Fig.  8, the width distribution of partons in the kaon r 2 ⊥ x become zero when x → 1. From the diagram we can also see that in the small momentum fraction region, the u quark is wider than the s quark, with the increasing of x, the two lines almost coincide, which means the two quarks have almost the same width.
From Eq. (40) we can see that the first term is −q 2 ⊥ independent, for the second term, the range of the integration is α ∈ [0, 1 − x], when x → 1, this term will vanish, so H u (x, 0, −q 2 ⊥ ) becomes −q 2 ⊥ independent, as expected. The mean-squared b ⊥ is defined as then we obtain b 2 ⊥ u K = 0.149 fm 2 , b 2 ⊥ s K = 0.088 fm 2 , for pion b 2 ⊥ u π = 0.139 fm 2 . This means that in the kaon the s quark is closer to the center of transverse momentum than the u quark.

Light-front transverse-spin distributions
The light-front transverse-spin distributions are defined as [97] where m H is the hadron mass, B n, ) and B n,0 (b 2 ⊥ ) are the two-dimensional Fourier transform of A n,0 (q 2 ⊥ ) and B n,0 (q 2 ⊥ ). In Fig. 9, we plot the light-front transverse-spin distribu- , when the quark is polarized in the light-front-transverse +x direction, the transversespin density is not symmetric around in the y direction for a transverse quark spin s ⊥ = (1, 0) in the x direction. Our results give b y ⊥ u 1 = 0.116 fm and b y ⊥ u 2 = 0.083 fm, which are larger than the pion results, but smaller compared with the lattice results in Ref. [97]. Their results for the pion are larger than our kaon results. That is because the NJL model is a contact interaction, FFs A u n,0 (q 2 ⊥ ) and B u n,0 (q 2 ⊥ ) are harder than the experimental results.

Light-front wave functions
As mentioned before, LFWFs offer the underlying connection between QCD and its fundamental degrees of freedom: quarks, gluons, and their asymptotic hadronic states. In phenomenology, LFWFs are particularly important because all hadronic properties can be deduced from them in principle.
In this section, we will derive the lowest-order qq valence LFWF for the kaon. Using the properties of LFWF, we generate kaon PDF, PDA, decay constant, TMD, and GPD, and compare these distributions with those obtained from the vector GPD, respectively.

The definition and calculation of the kaon LFWF
In the NJL model, the Bethe-Salpeter vertex for the kaon has the form The kaon LFWF is defined in Refs. [98,99], Eq. (22) of Ref. [98]: wherex = 1 − x, the definition of the basis spinors u and v are defined in Ref. [99] u u xq + , k, λ = 1 where where β = γ 0 , α i = γ 0 γ i , i = 1, 2. Using the relationships in Ref. [99] The lowest-order u quark LFWF of the kaon can be given by in the NJL model, α 1 = √ Z K , we do not consider the mixing of the pseudoscalar and the pseudovector interaction terms in the Bethe-Salpeter equation for the kaon, so α 2 = 0; then we arrive at where λ is the quark spin, k λ ≡ k 1 + iλk 2 , ψ 2↑↑ and ψ 2↓↓ is the spin parallel to S z = ±1, ψ 2↑↓ and ψ 2↓↑ is the spin anti-parallel S z = 0. For the ψ 2↑↑ term λ = 1, for the ψ 2↓↓ term λ = −1. Both spin alignments should contribute to the kaon wave function, which was pointed out by Leutwyler [100,101]. The LFWF can be reduced to In the following calculation, we will use the PTR scheme.

Normalization
The normalization condition is this means the probability of finding kaon in its valence state is 1. We obtain where Z K = 20.47, the same as Z K obtained from Eq. (8).

Kaon decay constant
Similar to the pion, the constraint offered by the kaon decay constant f K on the wave function is for our result

Constraint from K 0 → 2γ
This constraint assumes the very simple form and we obtain f K = 0.094 GeV.

Kaon form factor
The kaon electromagnetic form factor is given by the overlap of two wave functions, where k ⊥ = k ⊥ + (1 − x)q ⊥ is the transverse momentum of the struck quark. We have comparing with the result in Eq. (31), it is hard to see their relation only through the equations; instead, we have checked that they are exactly the same numerically. The kaon charge radius r K uses the Drell-Yan formula, which is given by the gradient of form factor at zero momentum transfer, the charge radius r K is defined as and we obtain r u K = 0.477 fm, r s K = 0.363 fm.

Transverse size
The r.m.s. transverse momentum is given by which gives a transverse size scale R ⊥ : which is slightly smaller than the charge radius r u K . The dimensionless quantity C = f K R ⊥ [102] connects the "core radius" R ⊥ with the decay constant f K . Our result is C = 0.189, this is coincident with the result C 0.2 in Ref. [103].

Kaon structure function
The valence structure function of the kaon is given by the squares of the LFWF: when integrating x we get the normalization condition, which means that the probability to find a valence quark in the kaon is unity, as it should. For the mean value of the momentum fraction x carried by the u quark thus we have x s = 0.569. This is different from the pion value 1/2 [9], which means that the quarks do not share an equal amount of longitudinal momentum because of the difference of light quark and strange quark. In the NJL model, in the chiral limit m π = 0, the pion PDF q(x) = 1 [5,9,104]. When m π = 0, our results show the pion q(x) ≈ 1, it fluctuates around 1. Reference [5] plotted the pion PDF with different values of m π . For comparison, we plot the diagrams of the u quark PDFs of pion and kaon with different values of m π and m K in Fig. 10. From the diagrams we can see that different from the pion PDFs, the kaon PDFs do not show symmetry any more, the peaks shift to x < 0.5.

Parton distribution amplitude
The PDA was introduced to describe the hard exclusive processes. The PDA from LFWF is defined as and satisfies the relationship The pion PDA, in the chiral limit, similar to the pion PDF, is φ u π (x) = 1, when m π = 0, it fluctuates around 1, but the kaon is different. We do not plot the diagrams of the comparison pion and kaon PDAs because they are similar to the pion and kaon PDFs diagrams in Fig. 10, the peaks of the kaon PDAs shift to x < 0.5.

Transverse momentum dependent parton distribution
The unpolarized u quark TMDs of the kaon is given by when integrating k ⊥ , we get the u quark PDF in Eq. (24). The s quark TMD in kaon can be simply obtained from the u quark distribution by momentum conservation, the quark TMDs of the kaon are plotted in Fig. 11.

Generalized parton distribution
The relationship between LFWF and GPD in the region ξ ≤ x ≤ 1 is given by Ref. [105]: where one can obtain In the forward limit ξ = 0 and t = 0 we get the quark PDF in Eq. (24), when integrating over x we get the FF in Eq. (68).

Mass distribution from LFWF GPD
When ξ = 0, the GPD becomes in the region x ∈ [0, 1], we examined that it is the same as Eq. (40), so we will get the same impact parameter dependent PDF u K (x, b 2 ⊥ ) in Eq. (42). The u quark mass distribution is in Fig. 12, we compare this with A u 2,0 in Eq. (33). From the diagram, we can see that it is exactly the same, so we get the same light-cone energy radius and light-cone charge radius from the two methods.

Wigner distribution
The definition of the unpolarized Wigner distribution [106][107][108][109][110][111][112][113][114][115][116][117], according to Refs. [74,117], is where b ⊥ is the transverse momentum transfer, the Wigner correlator is given by where and one obtains Using our LFWF, we get the following expression of the u quark unpolarized Wigner distribution of the kaon: for ρ UU (x, k ⊥ , b ⊥ ) when integrating k ⊥ we obtain Eq. (42), when integrating b ⊥ we can get TMD in Eq. (77). We plot the unpolarized Wigner distribution in Fig. 13. The diagrams show that, firstly, we have a sharp peak at (x = 1, k ⊥ = 0, b ⊥ = 0); secondly, the Wigner distribution shows a powerlaw suppression as k 2 ⊥ and/or b 2 ⊥ increasing; thirdly, it is negative on a neighborhood (x 0, k 2 ⊥ 0). The calculation of the unpolarized Wigner function with a realistic interaction is expected to display a similar behavior.

Summary and outlook
In this paper, we calculate kaon generalized parton distributions (GPDs) and leading Fock state light-front wave functions (LFWFs) in the Nambu-Jona-Lasinio (NJL) model using proper time regularization. We obtained form factors (FFs), generalized form factors, parton distribution functions (PDFs), impact parameter dependent PDFs, parton distribution amplitudes (PDAs) and transverse momentum dependent parton distributions (TMDs) from GPDs and LFWFs, then comparing these distributions from these two different methods, and checking their properties. For the vector GPD, we find that the u quark mass distribution θ u 2 and pressure distribution θ u 1 form factors are harder than its electromagnetic form factor F u K . We also study the mean-squared impact parameter for the quarks in the kaon: b 2 ⊥ u K = 0.149 fm 2 , b 2 ⊥ s K = 0.088 fm 2 , which means in the kaon the s quark is closer to the center of transverse momentum than the u quark. For the light-cone energy radius of quarks in kaon we have r u,K E,LC = 0.187 fm, r s,K E,LC = 0.167 fm. For the light-cone charge radius of the quarks in the kaon r u,K c,LC = 0.390 fm, r s,K c,LC = 0.296 fm, from the results we see that in each case the s quark has a smaller extent than the u quark.
For the tensor GPD, we plot the diagrams of the tensor anomalous magnetic moment B 1,0 and the tensor generalized form factor B 2,0 , we find that B 2,0 is harder than B 1,0 .
We also study the light-front transverse-spin distributions ρ 1 u (b ⊥ , s ⊥ ) and ρ 2 u (b ⊥ , s ⊥ ), the diagrams show distortions for transversely polarized quarks. For a quark polarized in the light-front-transverse +x direction, the transverse-spin density is no longer symmetric around b ⊥ = (b x = 0, b y = 0); the peaks shift to the (b x = 0, b y > 0) region. The average shift in the NJL model is b y ⊥ u 1 = 0.116 fm and b y ⊥ u 2 = 0.083 fm. From the kaon LFWF, we obtain the kaon decay constant, FF, TMD, PDF, PDA, impact parameter dependent PDF, and GPD in the region x ∈ [ξ, 1]. We obtain the same u quark FF, PDF, impact parameter dependent PDF u K (x, b 2 ⊥ ) and mass distribution θ u 2 from LFWF as from vector GPD. Thus we get the conclusion that the two methods in the NJL model give the same picture of the kaon structure. In addition, we give the unpolarized Wigner distribution of the u quark in the kaon. At the hadronic scale, the unpolarized Wigner function of the u quark in kaon is sharply peaked in the vicinity of (x = 1, k ⊥ = 0, b ⊥ = 0), it becomes widened as the transverse position variable b ⊥ , which conjugates to the probing momentum transfer, increases in magnitude. A similar behavior should be expected of unpolarized Wigner distributions calculated with a realistic interaction.
In summary, the NJL model is proved to be quite good in describing the pion and kaon structure. In the future, on the one hand, we can generalize these calculations to other hadrons which have a more complex structure, such as the ρ meson and K * meson, furthermore, protons and neutrons. On the other hand, we can repeat the results calculated herein using models with more realistic interactions; this may yield additional insights.