Form factors and generalized parton distributions of heavy quarkonia in basis light front quantization

We calculate the electromagnetic (charge, magnetic, and quadrupole) form factors and the associated static moments of heavy quarkonia (charmonia and bottomonia) using the basis light front quantization (BLFQ) approach. For this work, we adopt light front wave functions (LFWFs) generated by a holographic QCD confining potential and a one-gluon exchange interaction with fixed coupling. We compare our BLFQ results with the limiting case of a single BLFQ basis state description of heavy quarkonia and with other available results. These comparisons provide insights into relativistic effects. Using the same LFWFs generated in the BLFQ approach, we also present the generalized parton distributions (GPDs) for selected mesons including those for radially excited mesons such as ${\psi }^{\prime }$ and ${\mathrm{{\rm Y}}}^{\prime }$. Our GPD results establish the foundation within BLFQ for further investigating hadronic structure such as probing the spin structure of spin-one hadrons in the off-forward limit.

Figure 1

$-t{G}_C\left(t\right)$ vs $-t$ for (pseudo)scalar mesons (a) and (axial) vector mesons (b) in the BLFQ approach.

Figure 2

$-t{G}_C\left(t\right)$ vs $-t$ for ${\eta }_c^{\prime }$ and ${\eta }_b^{\prime }$ (a) and $J/\psi$ and $\mathrm{{\rm Y}}$ (b) with different $N_{\text{max}}=L_{\text{max}}$ in the BLFQ approach.

Figure 3

The magnetic FFs $G_M\left(t\right)$ [Eq. (5)] for selected vector charmonia and bottomonia states in the BLFQ approach.

Figure 4

The quadrupole FFs $G_Q\left(t\right)$ [Eq. (6)] for selected vector mesons in the BLFQ approach.

Figure 5

The comparison of magnetic FFs $G_M\left(t\right)$ (a) and the quadrupole FFs $G_Q\left(t\right)$ (b) for $1^3{S}_1$ ($J/\psi$ and $\mathrm{{\rm Y}}$) with different $N_{\text{max}}=L_{\text{max}}$ in the BLFQ approach.

Figure 6

The comparisons of the EM FFs for $1^3{S}_1$ $[J/\psi$ (a) and $\mathrm{{\rm Y}}$ (b)] in the BLFQ and SBL approaches.

Figure 7

Helicity nonflip (pseudo)scalar GPDs $H(x,\xi =0,t=-{\mathrm{\Delta }}_{\perp }^2)$ [Eq. (18)] for charmonia (a) and bottomonia (b) at $\left|t\right|=0.765{\text{GeV}}^2$ in the BLFQ approach. Note $t\equiv {\mathrm{\Delta }}^2=-{\mathrm{\Delta }}_{\perp }^2$, where ${\mathrm{\Delta }}_{\perp }$ is the transverse momentum transfer between the initial and final states of the meson and $x$ is the average momentum fraction carried by the quark in the longitudinal direction.

Figure 8

3D plot of helicity nonflip vector meson GPDs, $H_i(x,\xi =0,t=-{\mathrm{\Delta }}_{\perp }^2),i=1,2,3,5$ [Eqs. (11), (12), (13), and (15)] for $J/\psi \left(1^3{S}_1\right)$ in the BLFQ approach.

Figure 9

3D plot of helicity nonflip vector meson GPDs, $H_i(x,\xi =0,t=-{\mathrm{\Delta }}_{\perp }^2),i=1,2,3,5$ [Eqs. (11), (12), (13), and (15)] for $\mathrm{{\rm Y}}\left(1^3{S}_1\right)$ in the BLFQ approach.

Figure 10

3D plot of helicity nonflip vector GPDs, $H_i(x,\xi =0,t=-{\mathrm{\Delta }}_{\perp }^2),i=1,2,3,5$ [Eqs. (11), (12), (13), and (15)] for ${\psi }^{\prime }\left(2^3{S}_1\right)$ in the BLFQ approach.

Figure 11

3D plot of helicity non-flip vector GPDs, $H_i(x,\xi =0,t=-{\mathrm{\Delta }}_{\perp }^2),i=1,2,3,5$ [Eqs. (11), (12), (13), (15)] for ${\mathrm{{\rm Y}}}^{\prime }\left(2^3{S}_1\right)$ in the BLFQ approach.