Higher-order multipole amplitudes in charmonium radiative transitions

TL;DR

This work analyzes higher-order multipole contributions in charmonium radiative transitions using a large CLEO-c ψ' dataset, extracting M2 and possible E3 admixtures via unbinned maximum-likelihood fits to five-angle angular distributions. The measured normalized M2 amplitudes for ψ'→γχ_{c1,2} and χ_{c1,2}→γJ/ψ agree with theoretical expectations to first order in Eγ/m_c, yielding precise ratios that are largely independent of m_c and κ_c. The results confirm nonzero M2 components in χ_{c1} and χ_{c2} transitions, provide a determination of κ_c through a2^{J=1}, and offer meaningful comparisons with lattice QCD, while resolving several discrepancies from earlier experiments and setting stringent constraints on ψ'→γχ_{c2}. Overall, the findings support the traditional E1+M2 picture in these systems and enhance our understanding of charm-quark magnetic properties in radiative charmonium decays.

Abstract

Using 24 million $ψ' \equiv ψ(2S)$ decays in CLEO-c, we have searched for higher multipole admixtures in electric-dipole-dominated radiative transitions in charmonia. We find good agreement between our data and theoretical predictions for magnetic quadrupole (M2) amplitudes in the transitions $ψ' \to γχ_{c1,2}$ and $χ_{c1,2} \to γJ/ψ$, in striking contrast to some previous measurements. Let $b_2^J$ and $a_2^J$ denote the normalized M2 amplitudes in the respective aforementioned decays, where the superscript $J$ refers to the angular momentum of the $χ_{cJ}$. By performing unbinned maximum likelihood fits to full five-parameter angular distributions, we determine the ratios $a_2^{J=1}/a_2^{J=2} = 0.67^{+0.19}_{-0.13}$ and $a_2^{J=1}/b_2^{J=1} = -2.27^{+0.57}_{-0.99}$, where the theoretical predictions are independent of the charmed quark magnetic moment and are $a_2^{J=1}/a_2^{J=2} = 0.676 \pm 0.071$ and $a_2^{J=1}/b_2^{J=1} = -2.27 \pm 0.16$.

Figures (7)

• Figure 1: Charmonium energy levels. Only the transitions studied in this article are shown.
• Figure 2: Reference frames defining the angles used in this analysis. In the $\psi'$ frame, the angles $\theta'$, $\phi'$ are the polar and azimuthal angles of the beam pipe (specifically, the positron's direction) relative to $\gamma'$ defining the $z'$-axis, and $\gamma$ lying in the $x'$-$z'$ plane (with a positive$x'$-component). In the $\chi_c$ frame, the angle $\theta_{\gamma \gamma'}$ is the angle between the two photons. In the $J / \psi$ frame, the angles $\theta$, $\phi$ are the polar and azimuthal angles of the two leptons (specifically, the positive lepton's direction) relative to $\gamma$ defining the $z$-axis, and $\gamma'$ lying in the $x$-$z$ plane (with a negative$x$-component).
• Figure 3: Maximum reduced $\chi^2$ in all kinematic fits (including vertex fits) in generic Monte Carlo and data. Events with a maximum reduced $\chi^2$ below 16 (the dashed vertical line) are kept. Cumulative totals for the number of signal and impurity background events are also plotted for each potential value of a maximum reduced $\chi^2$. (a) $J_\chi = 1$ and (b) $J_\chi = 2$.
• Figure 4: Plot of the $\chi_{cJ}$ mass as calculated from subtracting the four-vector of the $\gamma'$ from the $\psi'$ four-vector. This variable is not used as a selection criterion because the 1C and 4C kinematic fits ensure that this criterion is redundant with the $\chi_{c1}$ mass selection criterion generated by adding the ${J/\psi}$ and $\gamma$ four-vectors. (a) $J_\chi = 1$ and (b) $J_\chi = 2$.
• Figure 5: (a) $J_\chi = 1$ and (b) $J_\chi = 2$ log likelihood contours as functions of $(a_2,b_2)$ for two-parameter fits. The fitted values (the solid squares) are $(a_2, b_2) = (-0.0611, 0.0281)$ for $J_\chi = 1$ and $(a_2, b_2, a_3, b_3) = (-0.093, 0.010, 0,0)$ for $J_\chi = 2$. These are, respectively, $11.1 \sigma$ and $6.2 \sigma$ from pure $E1$ (the solid circles). The theoretical values to first order in $E_\gamma/m_c$ with $\kappa_c = 0$ are given by the dashed lines.