Kaon quark distribution functions in the chiral constituent quark model

We investigate the valence $u$ and $\overline{s}$ quark distribution functions of the $K^{+}$ meson, $v^{K(u)}(x,Q^2)$ and $v^{K(\overline{s})}(x,Q^2)$, in the framework of the chiral constituent quark model. We judiciously choose the bare distributions at the initial scale to generate the dressed distributions at the higher scale, considering the meson cloud effects and the QCD evolution, which agree with the phenomenologically satisfactory valence quark distribution of the pion and the experimental data of the ratio $v^{K(u)}(x,Q^2)/v^{\pi (u)}(x,Q^2)$. We show how the meson cloud effects affect the bare distribution functions in detail. We find that a smaller $SU(3)$ flavor symmetry breaking effect is observed, compared with results of the preceding studies based on other approaches.

Figure 1

The three diagrams characterizing the dressing corrections to a constituent quark. The wavy, thick, and dashed lines denote the virtual photons, the constituent quarks, and the Goldstone bosons, respectively.

Figure 2

The cross-hatched area represents the 68% C.L. uncertainty of our dressed valence $u$ quark distribution function of the kaon at $Q^2=27{\mathrm{GeV}}^2$. The circles are the empirical values of $x{v}^{K(u)}(x,Q^2)$ from the combination of the experimental data of the ratio $v^{K(u)}/v^{\pi (u)}$ [30] and the valence quark distribution function of the pion taken from Ref. [29]. The squares represent the results obtained from the combination of the ratio and the GRS parametrization [12]. The error bars for those empirical values are also provided here.

Figure 3

The bare valence quark distribution functions of the pion and kaon at $Q^2=Q_0^2$. The solid lines depict the pion’s distribution, and the dashed and dotted lines denote the $u$ and $\overline{s}$ valence quark distributions of the kaon, respectively. The left panel is for $v^{\pi ,K}$ and the right one is for $x{v}^{\pi ,K}$.

Figure 4

The dressed valence quark distributions as functions of Bjorken $x$. Left: The solid line shows the result taken from Ref. [29]. The dashed line represents the pion’s distribution with some improved treatments (see the text for details). The dotted and dash-dotted lines denote the results for the $u$ and $\overline{s}$ quarks of the kaon, respectively. These four results are obtained at $Q^2=Q_0^2$. Right: The solid, dashed, and dotted lines depict the results for the pion’s $u$ quark, the kaon’s $u$ quark, and the kaon’s $\overline{s}$ quark distributions at $Q^2=27{\mathrm{GeV}}^2$, respectively.

Figure 5

The dressed valence $u$ quark distributions of the pion and kaon as functions of Bjorken $x$. The cross-hatched and shadowed bands show the 68% C.L. $v_{\text{dressed}}^{K(u)}(x)$ uncertainties at $Q^2=Q_0^2$ and $27{\mathrm{GeV}}^2$, respectively. The dashed and dotted curves denote the distributions for the pion at $Q^2=Q_0^2$ and $27{\mathrm{GeV}}^2$, respectively.

Figure 6

The dressed valence quark distributions of the kaon as functions of Bjorken $x$. The cross-hatched and shadowed bands show the 68% C.L. $v_{\text{dressed}}^{K(\overline{s})}(x)$ uncertainties at $Q^2=Q_0^2$ and $27{\mathrm{GeV}}^2$, respectively. The dashed and dotted curves denote the $u$ quark distributions at $Q^2=Q_0^2$ and $27{\mathrm{GeV}}^2$, respectively.

Figure 7

The contributions of each term in Eq. (14) to $x{v}_{\text{dressed}}^{K(u)}$ (left panel) and those from the terms in Eq. (19) to $x{v}_{\text{dressed}}^{K(\overline{s})}$ (right panel) at $Q^2=Q_0^2$ as functions of Bjorken $x$.

Figure 8

The ratios of the valence quark distribution functions, $R^{K(u)}(x,Q^2)$ (left panel) and $R^{K(\overline{s})}(x,Q^2)$ (right panel) defined in Eqs. (23) and (24), as functions of Bjorken $x$ for various $Q^2$. The solid, dashed, dotted, dash-dotted, and dash-double-dotted curves depict the $Q^2=Q_0^2$, 1, 4, 16, $27{\mathrm{GeV}}^2$ cases, respectively.