Heavy mesons with dynamical gluon on the light front

Abstract

We investigate the structure of charmonium, bottomonium, and $\rm B_c$ meson systems within the Basis Light-Front Quantization (BLFQ) approach, including both the quark-antiquark ($|q\bar{q}\rangle$) and quark-antiquark-gluon ($|q\bar{q}g\rangle$) Fock sectors. Our input light-front Hamiltonian incorporates a confining potential inspired by light-front holography, as well as the quark-gluon interaction from Quantum Chromodynamics. By adjusting model parameters to reproduce the mass spectra for low-lying states, we obtain the light-front wave functions for the heavy meson states. Based on these wave functions, we calculate electromagnetic form factors, decay constants, parton distribution amplitudes (PDAs), and parton distribution functions (PDFs) of the quarks and gluons in the heavy mesons. Our results for the charge radii and decay constants reasonably agree with experimental data and other theoretical approaches. The PDAs are consistent with the predictions from the earlier BLFQ calculations with an effective one-gluon exchange interaction. Furthermore, we present the first predictions within the BLFQ framework for the gluon PDFs in heavy mesons based on the light-front wave function in the $|q\bar{q}g\rangle$ sector.

Figures (11)

• Figure 1: The reconstructed mass spectrum of charmonium, bottomonium and $\rm B_c$ meson states from the $M_J=0$ sector. The horizontal axis labels the quantum numbers $J^{PC}$, and the vertical axis is invariant mass in GeV. We denote the BLFQ calculation including the $|q\bar{q}\rangle+|q\bar{q}g\rangle$ sectors as Dynamical Gluon (DG), to distinguish it from the BLFQ results based on the One-Gluon Exchange (OGE) effective interactions. The black lines are the results from BLFQ DG, and we compare with data from particle data group (PDG) pdg (dotted red line), BLFQ OGE with truncation parameters $N_{\max} = L_{\max} = 32$y.li1s.tang (dashed blue line) and the results from Lattice QCD HadronSpectrum:2012gicAarts:2014cdaPhysRevLett.94.172001PhysRevLett.104.022001DAVIES1996131 (orange band). The PDG values serve as the inputs to fix the model parameters. For charmonium and bottomonium systems, the results are marked by boxes representing the spreads of the mass eigenvalues in different $M_J$ sectors. For $\rm B_c$ meson, only the results in $M_J =0$ sector are shown due to the difficulty in identifying the multiplets with the same $J$, see text for details.
• Figure 2: Normalized momentum-space LFWFs in the $|q\bar{q}\rangle$ Fock sector for heavy meson states with $M_J=0$. Panels (a)-(h): the charmonium states; panels (i)-(p): the bottomonium states; and panels (q)-(x): the $\rm B_c$ meson states. For $\rm B_c$ meson states, $x_b$ is the longitudinal momentum fraction of the $b$ quark. See text for details on normalization.
• Figure 3: The electromagnetic FFs and $\rm Q^2$FFs of $J=0$ charmonium, bottomonium and $\rm B_c$ meson states with combined contributions from both $|q\bar{q}\rangle$ and $|q\bar{q}g\rangle$ Fock sectors. The solid, dashed, and dotted lines represent different states. For charmonium and bottomonium states, $F(Q^2)= F^{c(b)}(Q^2)$, see text for details. For $\rm B_c$ meson states, the electromagnetic FFs are weighted by the charge of $b$ and $\bar{c}$ quarks: $F(Q^2)=\frac{1}{3}F^b(Q^2)+\frac{2}{3}F^{\bar{c}}(Q^2)$.
• Figure 4: The charge radii of (pseudo)scalar mesons (solid circles), compared with results from BLFQ OGE (squares) y.li1s.tang, lattice QCD (rhombuses) PhysRevD.73.074507 and DSE (triangles) Maris:2006ea. The BLFQ DG results are obtained from the derivative of the charge form factors (shown in Fig. \ref{['ff']}) at zero momentum transfer, see Eq. \ref{['rc']}. The BLFQ OGE results are obtained with $N_{\max}=L_{\max}=32$ for charmonium and bottomonium states. For charmonium and bottomonium states, $\langle r^2_c \rangle$ = $\langle (r^{c(b)}_c)^2 \rangle$. For $\rm B_c$ meson states, the charge radii are weighted by the charge of $b$ and $\bar{c}$ quarks, $\langle r^2_c \rangle$= $\frac{1}{3} \langle (r^b_c)^2 \rangle + \frac{2}{3} \langle (r^{\bar{c}}_c)^2 \rangle$.
• Figure 5: The decay constants for pseudoscalar and vector states of charmonium and bottomonium, and vector $\rm B_c$ meson states (circles). $M_J=0$ states are used for vector states, see text for details. Results are compared with the BLFQ OGE (regular triangles) y.li1s.tang, lattice QCD (rhombuses) PhysRevD.86.074503PhysRevD.82.114504PhysRevD.86.094501PhysRevD.91.074514, Dyson-Schwinger Equation (squares) PhysRevD.84.096014, and the experimental data from PDG (inverted triangles) pdg.