Ground state pseudoscalar mesons on the light front: From the light to heavy sector

We extract the leading Fock state light-front wave functions (LF-LFWFs) of both the light and heavy pseudoscalar mesons, e.g., the pion (at masses of 130, 310, and 690 MeV), ${\eta }_c$ and ${\eta }_b$, from their covariant Bethe-Salpeter wave functions within the rainbow-ladder truncation. It is shown that the LF-LFWFs get narrower in $x$ (the longitudinal momentum fraction of the meson carried by the quark) with the increasing current quark mass, and the leading twist parton distribution amplitudes inherit this feature. In the case of the pion, the LF-LFWFs only contribute about 30% of the total Fock state normalization, indicating the presence of significant higher Fock states within. In contrast, in the cases of the ${\eta }_c$ and ${\eta }_b$, the LF-LFWFs contribute more than 90%, suggesting the $Q\overline{Q}$ valence Fock state truncation as a good approximation for heavy mesons. We thus study the three-dimensional parton distributions of the ${\eta }_c$ and ${\eta }_b$ with the unpolarized generalized parton distribution function (GPD) and the transverse momentum-dependent parton distribution function. Through the gravitational form factors in connection with the GPD, the mass radii of the ${\eta }_c$ and ${\eta }_b$ in the light-cone frame are determined to be $r_{E,\mathrm{LC}}^{{\eta }_c}=0.150$ and $r_{E,\mathrm{LC}}^{{\eta }_b}=0.089\mathrm{fm}$, respectively.

Figure 1

The ${\psi }_0(x,{\mathit{k}}_T^2)$ and ${\psi }_1(x,{\mathit{k}}_T^2)$ of pion at $m_{\pi }=130$ (top row), 310 (middle row), and 690 MeV (bottom row).

Figure 2

The ${\psi }_0(x,{\mathit{k}}_T^2)$ and ${\psi }_1(x,{\mathit{k}}_T^2)$ of ${\eta }_c$ (top row) and ${\eta }_b$ (bottom row).

Figure 3

The PDA of the pion at masses of $m_{\pi }=130$ (solid), 310 (dashed), and 690 MeV (dotted). The colored bands are results from lattice QCD [22] at the same masses of $m_{\pi }=130$ (red), 310 (green), and 690 MeV (blue).

Figure 4

A comparison of our PDA of ${\eta }_c$ (green solid) and ${\eta }_b$ (red dashed) and sum rule results (blue dotted and purple dot-dashed, respectively) using background field theory [7].

Figure 5

The unpolarized GPD $H_M^q(x,\xi =0,t)$ of ${\eta }_c$ (yellow surface lower at $x=0.5$) and ${\eta }_b$ (purple surface upper at $x=0.5$) at scales around their masses.

Figure 6

The impact parameter-dependent GPD ${\rho }_M^q(x,{\mathit{b}}_T^2)$ of ${\eta }_c$ and ${\eta }_b$. The ${\rho }_{{\eta }_c}^c(x,{\mathit{b}}_T^2)$ is on the left column and ${\rho }_{{\eta }_b}^b(x,{\mathit{b}}_T^2)$ on the right.

Figure 7

Upper: the transverse spatial distribution of the charm quark within ${\eta }_c$ (blue solid) and the bottom quark within ${\eta }_b$ (purple dashed) on the light front. Lower: the density plots of the spatial distribution, with ${\eta }_c$ on the left and ${\eta }_b$ on the right.

Figure 8

The gravitational form factor $A(t)$ [defined in Eq. (23)] of ${\eta }_c$ (blue solid) and ${\eta }_b$ (purple dotted) in the leading Fock state truncation.

Figure 9

Density plots of the unpolarized TMD PDF $f_1^q(x,{\mathit{k}}_T^2)$ of ${\eta }_c$ (left) and ${\eta }_b$ (right).

Figure 10

The $|{\mathit{k}}_T|$ dependence of ${\eta }_c$’s and ${\eta }_b$’s unpolarized TMD at different $x$ values. The line styles are indicated in the plot. The accompanying gray dotted curves are fitting curves in Gaussian form.