[Paper Review] Radiative transitions of charmonium states in a constituent quark model

Weijun Deng, Li-Ye Xiao+2 more|arXiv (Cornell University)|Oct 28, 2015

Quantum Chromodynamics and Particle Interactions54 references3 citations

TL;DR

This paper investigates electromagnetic radiative transitions of charmonium states using a constituent quark model, incorporating both electric (E1) and magnetic (M2) multipole transitions. It finds significant M2 interference effects in E1-dominated processes and provides precise predictions for higher charmonium states, with a predicted ratio of Γ[X(3872)→ψ(2S)γ]/Γ[X(3872)→J/ψγ] ≈ 4.0, consistent with BaBar data, supporting X(3872) as χc1(2P).

ABSTRACT

We study the electromagnetic (EM) transitions of the $nS$, $nP$ ($n\leq 3$), and $nD$ ($n\leq 2$) charmonium states with a constituent quark model. We obtain a reasonable description of the EM transitions of the well-established charmonium states $J/ψ$, $ψ(2S)$, $χ_{cJ}(1P)$, $h_c(1P)$ and $ψ(3770)$. We find that the M2 transitions give notable corrections to some E1 dominant processes by interfering with the E1 transitions. Our predictions of EM decay properties for the higher charmonium states are also presented and compared with other model predictions. In particular, we discuss the EM decay properties of some $"XYZ"$ states, such as $X(3823)$, $X(3872)$, $X(3915)$, $X(3940)$ and $X(4350)$ as conventional charmonium states. Assuming $X(3872)$ as the $χ_{c1}(2P)$ state, our predicted ratio $Γ[X(3872) o ψ(2S)γ]/Γ[X(3872) o J/ψγ]\simeq 4.0$ is consistent with BaBar's measurement.

Motivation & Objective

• To provide a reliable description of electromagnetic transitions in charmonium states using a constituent quark model.
• To test the validity of conventional charmonium assignments for newly observed XYZ states, particularly X(3872).
• To investigate the role of M2 transitions in correcting E1-dominated radiative decays.
• To predict decay properties of higher radial and orbital excitations (nS, nP, nD) beyond established states.
• To compare model predictions with experimental data and other theoretical approaches, especially for X(3872), X(3823), X(3915), X(3940), and X(4350).

Proposed method

• Adopts a constituent quark model with a non-relativistic potential to describe the internal structure of charmonium states.
• Uses a specialized electromagnetic transition operator that includes the effects of the binding potential and allows for higher multipole contributions (e.g., M2).
• Calculates partial widths for radiative transitions (E1, M2) using matrix elements derived from the quark wave functions and photon coupling.
• Takes experimental masses for well-established states (e.g., J/ψ, ψ(2S)) and model-predicted masses for unestablished states from NR, GI, and SNR potential models.
• Compares predictions across three quark model variants (QM1, QM2, QM3) corresponding to different potential models.
• Performs interference calculations between E1 and M2 amplitudes to assess their combined impact on transition rates.

Experimental results

Research questions

• RQ1 Can the constituent quark model accurately describe the electromagnetic decays of established charmonium states like J/ψ, ψ(2S), and χcJ(1P)?
• RQ2 To what extent do M2 transitions contribute to E1-dominated radiative decays, and how do they affect the predicted branching ratios?
• RQ3 Is the assignment of X(3872) as χc1(2P) consistent with electromagnetic decay data, particularly the ratio of its decays to ψ(2S)γ and J/ψγ?
• RQ4 What are the predicted partial widths for radiative transitions of higher charmonium states (e.g., 2D, 3P, 3S) not yet fully established?
• RQ5 How do the model predictions for XYZ states like X(3823), X(3915), X(3940), and X(4350) compare with other theoretical models and experimental observations?

Key findings

• The model provides a reasonable description of electromagnetic transitions for well-established charmonium states, including J/ψ, ψ(2S), χcJ(1P), hc(1P), and ψ(3770).
• M2 transitions provide notable corrections to E1-dominated processes through destructive and constructive interference, significantly altering predicted branching ratios.
• The predicted ratio Γ[X(3872)→ψ(2S)γ]/Γ[X(3872)→J/ψγ] ≈ 4.0 is in good agreement with the BaBar measurement, supporting the assignment of X(3872) as χc1(2P).
• For unestablished D-wave states like ψ3(1D), ψ2(1D), and ψ1(1D), the model predicts partial widths ranging from 0.01 keV to several keV, with significant variation across different potential models.
• Predictions for radiative decays of higher states such as χc2(3P), χc1(3P), and ηc2(2D) show strong dependence on the choice of potential model (NR, GI, SNR), with differences in partial widths up to a factor of 2.
• The model predicts non-zero M2 contributions for transitions like ψ2(2D)→χc1(1P) and ηc2(2D)→hc(1P), with M2 widths reaching up to 10 keV, indicating non-negligible higher-multipole effects.