Flavor dependence of the pion and kaon form factors and parton distribution functions

The separate quark flavor contributions to the pion and kaon valence quark distribution functions are studied, along with the corresponding electromagnetic form factors in the space-like region. The calculations are made using the solution of the Bethe-Salpeter equation for the model of Nambu and Jona-Lasinio with proper-time regularization. Both the pion and kaon form factors and the valence quark distribution functions reproduce many features of the available empirical data. The larger mass of the strange quark naturally explains the empirical fact that the ratio $u_{K^{+}}\left(x\right)/u_{{\pi }^{+}}\left(x\right)$ drops below unity at large $x$, with a value of approximately $M_u^2/M_s^2$ as $x\to 1$. With regard to the elastic form factors we report a large flavor dependence, with the $u$-quark contribution to the kaon form factor being an order of magnitude smaller than that of the $s$-quark at large $Q^2$, which may be a sensitive measure of confinement effects in QCD. Surprisingly though, the total $K^{+}$ and ${\pi }^{+}$ form factors differ by only 10%. In general we find that flavor breaking effects are typically around 20%.

Figure 1

The NJL gap equation in the Hartree-Fock approximation. The thin line is the bare quark propagator, $S_0^{-1}\left(k\right)=k/-m+i\varepsilon$, whereas the thick line is the dressed quark propagator $S\left(k\right)$. The $\overline{q}q$ interaction kernel is given by Eq. (2).

Figure 2

The BSE in the NJL model, illustrated here for quark and antiquark scattering.

Figure 3

Diagrammatic representation of the electromagnetic current of the pion or kaon.

Figure 4

Illustration of the inhomogeneous BSE which gives the dressed quark-photon vertex. The large shaded oval represents the solution of the inhomogeneous BSE, the small dot is the inhomogeneous driving term ($\widehat{Q}{\gamma }^{\mu }$), and the double-dots represent the $q\overline{q}$ interaction kernel given in Eq. (2).

Figure 5

The dressed quark form factors obtained as solutions to the inhomogeneous BSE.

Figure 6

Feynman diagrams for the valence quark distributions in the pion or kaon. The red cross is the operator insertion ${\gamma }^{+}\delta \left(k^{+}-x{p}^{+}\right){\widehat{P}}_q$, where ${\widehat{P}}_q$ is the projection operator for quarks of flavor $q$.

Figure 7

Our results for the pion form factor are given as the solid line and a comparison is made to the DSE results of Ref. [31]. The empirical result (dotted line) has the form $F_{\pi }\left(Q^2\right)={[1+Q^2/{\mathrm{\Lambda }}_{\pi }^2]}^{-1}$, where the mass parameter is chosen to reproduce empirical radius found in Ref. [67], giving ${\mathrm{\Lambda }}_{\pi }^2=0.54{\mathrm{GeV}}^2$. The experimental data is from Refs. [67, 68, 69, 70, 71, 72].

Figure 8

Results for $Q^2{F}_{\pi }\left(Q^2\right)$. See caption to Fig. 7 for the nomenclature.

Figure 9

The $K^{+}$ form factor (solid line) together with the up (dashed-dotted line) and strange (dashed line) quark sector contributions. The dotted-line is the fit to data using the form $F_K\left(Q^2\right)={[1+Q^2/{\mathrm{\Lambda }}_K^2]}^{-1}$, giving ${\mathrm{\Lambda }}_K^2=0.687{\mathrm{GeV}}^2$, and the insert compares our results with existing data taken from Ref. [74].

Figure 10

Results for $Q^2{F}_{K^{+}}\left(Q^2\right)$ together with the charge-weighted quark-sector contributions and the empirical result obtained from Ref. [74]. This result clearly illustrates that the $s$ quark dominates the form factor at large $Q^2$.

Figure 11

We illustrate various pion and kaon form factor ratios, including for the quark sector form factors, to ascertain a measure of flavor breaking and environment sensitivity effects as a function of $Q^2$. Note, all ratios would be unity for all $Q^2$ in the $SU\left(3\right)$ flavor limit.

Figure 12

Results for the valence (minus-type) quark distributions of the ${\pi }^{+}$ and $K^{+}$, evolved from the model scale using the NLO DGLAP equations [78, 79, 80, 81]. The solid line represents the valence $u$ or $\overline{d}$ PDF in the ${\pi }^{+}$, the dot-dashed line is the valence $\overline{s}$ quark, and the dashed line the valence $u$ quark in the $K^{+}$. The experimental data is for the ${\pi }^{-}$ and are from Ref. [34]. The dotted line is a reanalysis of this data from Ref. [82].

Figure 13

The solid line gives the ratio of the $u$ quark distribution in the $K^{+}$ to the $u$ quark distribution in the ${\pi }^{+}$, after NLO evolution to $Q^2=16 {\mathrm{GeV}}^2$. The dashed line gives the ratio of the $u$ quark to $\overline{s}$ quark distributions in the $K^{+}$ at $Q^2=16 {\mathrm{GeV}}^2$. The data are from Ref. [33] and correspond to the ratio ${\overline{u}}_{K^{-}}/{\overline{u}}_{{\pi }^{-}}$, which is related to our ${\pi }^{+}$ and $K^{+}$ results by charge symmetry.