Basis light front quantization for the charged light mesons with color singlet Nambu–Jona-Lasinio interactions

We apply the basis light front quantization (BLFQ) approach to describe the valence structures of the charged light meson ground states. Specifically, the light front wave functions of ${\pi }^{\pm }$, ${\rho }^{\pm }$, $K^{\pm }$, and $K^{*\pm }$ are obtained as the eigenvectors of the light front effective Hamiltonians with confinement potentials supplemented by the color singlet Nambu–Jona-Lasinio (NJL) interactions. We adjust our model such that the spectrum of these ground states and the charge radii of the pseudoscalar states agree with experimental results. We present the elastic form factors and parton distribution amplitudes (PDAs) as illustrations of the internal structures of the pseudoscalar pions and kaons in terms of valence quarks.

Figure 1

The elastic form factor for the ${\pi }^{+}$. The red solid line is the result from the BLFQ-NJL model. The blue dashed line is the dipole form with a charge radius of $0.672\mathrm{fm}$. The orange stars, yellow plus signs, purple crosses, green boxes, and light blue diamonds with error bars correspond to experimental measurements from Refs. [61, 62, 63, 64, 65] respectively. The elastic form factor of the ${\pi }^{+}$ is plotted on the left panel. On the right panel, the same form factor is multiplied by $Q^2$.

Figure 2

The elastic form factor for the $K^{+}$. The red solid line is the result from the BLFQ-NJL model. The blue dashed line is the dipole form with a charge radius of $0.560\mathrm{fm}$. The orange plus signs and yellow boxes with error bars correspond to experimental measurements from Refs. [66, 67] respectively. The elastic form factor of the $K^{+}$ is plotted on the left panel. On the right panel, the same form factor is multiplied by $Q^2$.

Figure 3

Parton distribution amplitude for the ${\pi }^{+}$ at the scale $\mu =935.9\mathrm{MeV}$. On the left panel, the blue solid line is the fit to the BLFQ-NJL model using Eq. (45) with $a=b=0.62$. The red dot-dashed line corresponds to the same functional form with the $a$ and $b$ parameters extrapolated to $L_{\max }\to +\infty$. The orange dashed line represents the $6x(1-x)$ from perturbative QCD. On the right panel, PDAs for the ${\pi }^{+}$ with $L_{\max }=8,16,32$ are plotted in dashed line, together with the fit to the $L_{\max }=32$ case.

Figure 4

Parton distribution amplitude for the $K^{+}$ at the scale $\mu =1138\mathrm{MeV}$. On the left panel, the blue solid line is the fit to the BLFQ-NJL model using Eq. (45) with $a=0.65$ and $b=0.70$. The red dot-dashed line corresponds to the same functional form with the $a$ and $b$ parameters extrapolated to $L_{\max }\to +\infty$. The red dashed line represents the $6x(1-x)$ from perturbative QCD in the $\mathrm{SU}\left(3\right)$ flavor symmetric limit. On the right panel, PDAs for the $K^{+}$ with $L_{\max }=8,16,32$ are plotted, together with the fit to the $L_{\max }=32$ case.

Figure 5

Extrapolations of the dependence of fitting parameters on the longitudinal basis cutoff using quadratic functions of $L_{\max }^{-1}$. On the left panel, the red stars are the parameters $a=b$ from fitting the BLFQ-NJL PDA for the ${\pi }^{+}$ using the functional form given by Eq. (45). The blue solid line is the fitted quadratic function of $L_{\max }^{-1}$. The right panel shows the extrapolation of fitting parameters for the $K^{+}$ PDA obtained using the BLFQ-NJL model. The orange stars are the parameters $a$ from fitting BLFQ-NJL using the functional form given by Eq. (45). The blue solid line is the fitted quadratic function of $L_{\max }^{-1}$ for the parameter $a$. The purple stars are the parameters $b$ from fitting the BLFQ-NJL PDA using the functional form given by Eq. (45). The red dashed line is the fitted quadratic function of $L_{\max }^{-1}$ for the parameter $b$.

Figure 6

Left panel: parton distribution amplitude for the ${\rho }^{+}$ at the scale $\mu =935.9\mathrm{MeV}$. Right panel: parton distribution amplitude for the $K^{*+}$ at the scale $\mu =1138\mathrm{MeV}$. Legends are identical on both panels and therefore are only shown on the right. The green dashed line corresponds to $6x(1-x)$ from perturbative QCD in the $\mathrm{SU}\left(3\right)$ flavor symmetric limit. The blue, red, and orange solid lines are PDAs obtained at $L_{\max }=8,16,32$ respectively. The purple dot-dashed line corresponds to the functional form in Eq. (45) with parameters $p=(\beta +1)/2=4.91,q=(\alpha +1)/2=4.91$ for the ${\rho }^{+}$, and $p=(\beta +1)/2=3.54,q=(\alpha +1)/2=4.90$ for the $K^{*+}$. These $\alpha$ and $\beta$ are calculated by Eq. (14) with parameters in Tables 1 and 3 to indicate the result with only the leading basis state contributing to the PDAs.