text). The mean mass spreads, i.e. the average heights of the boxes, are 17 MeV and 8 MeV for charmonium and bottomonium, respectively. The r.m.s. deviations of the masses from the PDG values are 31 MeV and 38 MeV for charmonium and bottomonium, respectively. See text for details. []{data-label="fig:spectroscopy"}](Figures/bottom_spectrum_a0p60_kap1p39_mb4p90_NL32_style9.pdf "fig:"){width="46.00000%"}

The reconstructed spectra at $N_{\max}=L_{\max}=32$ are presented in Fig. \[fig:spectroscopy\]. In these figures, we use boxes to indicate the spreads of the mass eigenvalues from different $m_j$. The mean values, marked by dashed bars, are defined as: $$\overline M \equiv \sqrt{\frac{M^2_{-j} + M^2_{1-j} + \cdots + M^2_j}{2j+1}},$$ where $M_{m_j}$ is the mass eigenvalue associated with the magnetic projection $m_j$. This definition is motivated by the covariant light-front analysis of relativistic bound states in Refs. [@Carbonell:1998rj; @ManginBrinet:2003nm]. On the other hand, the mass spreads $\delta_j M \equiv \max M_{m_j} - \min M_{m_j}$ measure the violation of the rotational symmetry. We also introduce the mean spread: $$\overline{\delta_jM} \equiv \sqrt{ \frac{1}{N_h} \sum_{h}^{j\ne0} (\delta_jM_h)^2 }.\qquad \Big(N_h\equiv\sum_h^{j\ne0} 1\Big)$$ For charmonium (bottomonium) states evaluated by PDG below the threshold, the mean mass spread is 17 MeV (8 MeV), improving our previous results [@Li:2015zda] by a factor of $\sim$3 ($\sim$2). More comparison between the results of this 