Single-quark electromagnetic form factors of charmonium up to $J=2$

TL;DR

The paper presents a systematic calculation of single-quark electromagnetic form factors for a broad set of charmonium states (up to J=2) within a relativized quark model using relativized mock meson states. It defines the RMMS framework, analyzes frame dependence as a computational error, and compares results against LQCD, DSE, and BLFQ where available, finding general agreement with a few notable discrepancies. The study provides detailed multipole form factors (G_C, G_M, G_Q, G_M3, G_E4) and derived moments (μ and Q) for η_c, J/ψ, χ_{cJ}, and h_c states, including predictions for χ_{c0}(2P), χ_{c1}(2P), h_c(1P/2P), and χ_{c2}(1P/2P) where data are lacking. A key outcome is that most RMMS results align with other theoretical approaches, while a puzzling difference in G_M for J/ψ(1S) indicates a target for further theoretical refinement. The framework establishes a coherent, predictive scheme for charmonium EM structure and provides concrete benchmarks for future lattice or continuum QCD studies.

Abstract

We calculate the single-quark electromagnetic form factors of a broad subset of charmonium, including $η_c(1S)$, $η_c(2S)$, $χ_{c0}(1P)$, $χ_{c0}(2P)$, $J/ψ(1S)$, $J/ψ(2S)$, $χ_{c1}(1P)$, $χ_{c1}(2P)$, $h_c(1P)$, $h_c(2P)$, $χ_{c2}(1P)$ and $χ_{c2}(2P)$, via a relativized quark model. The reference frame dependence of the results is estimated as the computational error. We compare our results with those of the lattice quantum chromodynamics (LQCD), the Dyson-Schwinger equation (DSE) and the basis light front quantization (BLFQ) approaches where available and we find that most of our results agree with the other results. We also predict the single-quark electromagnetic form factors of $χ_{c0}(2P)$, $χ_{c1}(2P)$, $h_c(1P)$, $h_c(2P)$, $χ_{c2}(1P)$ and $χ_{c2}(2P)$, where no direct comparisons are available.

Figures (9)

• Figure 1: Upper panel: Comparison of $F_{\eta_c(1S)}$ computed in the static frame using: (top) MMS with the temporal component current $j^0$, (middle) RMMS, (bottom) MMS with the spacial component current $j^i$ (i=1,2,3). Lower panel shows the results of $F_{\eta_c(1S)}$ in different reference frame using RMMS. From top to bottom, the Breit frame, the static frame and a frame where $\bm{P}_1 = 0.1*\bm{P}_2$ holds.
• Figure 2: Single-quark electromagnetic form factor of $\eta_c(1S)$ (upper) and $\eta_c(2S)$ (lower). The solid black line (RMMS) is our result via the relativized mock meson state, and the shadow is the computational error. The dotted blue line (DSE2007) is the Dyson-Schwinger equation result Maris2007. The dashed orange line (BLFQ2019) is the basis light front quantization result Adhikari2019. The dots with error bars are the lattice QCD results: Dudek2006 Dudek2006, Chen2011 Chen2011 and Delaney2024 Delaney2024.
• Figure 3: Single-quark electromagnetic form factor of $\chi_{c0}(1P)$ (upper) and $\chi_{c0}(2P)$ (lower). The caption is identical to that of FIG. \ref{['fig:effetac1S2S']}.
• Figure 4: Single-quark electric charge (upper), magnetic dipole (middle) and electric quadrupole (lower) form factor of $J/\psi(1S)$. The caption is identical to that of FIG. \ref{['fig:effetac1S2S']}.
• Figure 5: Single-quark electric charge (upper), magnetic dipole (middle) and electric quadrupole (lower) form factor of $J/\psi(2S)$. The caption is identical to that of FIG. \ref{['fig:effetac1S2S']}.