Quarkonium Spectroscopy and Perturbative QCD: A New Perspective

TL;DR

Brambilla, Sumino, and Vairo test the applicability of perturbative QCD to heavy quarkonia by reformulating quarkonium energies in terms of short-distance $\overline{MS}$ masses via an $\varepsilon$-expansion that cancels renormalons. Using $\alpha_s^{(5)}(M_Z)=0.1181 \pm 0.0020$ and fixing $m_b^{\overline{MS}}(m_b^{\overline{MS}})$ from the Υ(1S) mass, they compute the bottomonium spectrum and compare to data, finding good agreement for states where $\alpha_s(\mu)$ remains below unity and providing upper bounds on non-perturbative effects of order $\Lambda_{\rm QCD}^3$-driven renormalons. The analysis yields a qualitative picture in which level spacings remain nearly constant with increasing principal quantum number due to state-dependent self-energy corrections, with the dominant physics governed by the region $1/a_X \lesssim q \lesssim \overline{m}$ in the heavy-quark self-energies. The work supports a perturbative description of several bottomonium levels (up to $n\approx 3$) and suggests that non-perturbative effects are encoding in non-local condensates that reduce to local forms for ground states, while charmonium predictions remain more limited.

Abstract

We study the energy spectrum of bottomonium in perturbative QCD, taking alpha_s(Mz)=0.1181 +/- 0.0020 as input and fixing m_b^{MSbar}(m_b^{MSbar}) on the Upsilon(1S) mass. Contrary to wide beliefs, perturbative QCD reproduces reasonably well the gross structure of the spectrum as long as the coupling constant remains smaller than one. We perform a detailed analysis and discuss the size of non-perturbative effects. A new qualitative picture on the structure of the bottomonium spectrum is provided. The lowest-lying (c,cbar) and (b,cbar) states are also examined.

Figures (2)

• Figure 1: The bottomonium level structure as given by a pure Coulombic potential, by experiments and by the present analysis: the solid and dashed lines represent the $S$-states and $P$-states, respectively. The input parameter of the perturbative QCD calculation is $\alpha_{\rm s}^{(5)}(M_Z)=0.1181$ (see Tab. \ref{['table:spectra']}). We show only those levels that we can compute reliably. The Coulomb levels are calculated with $m_{\rm pole}=5.105$ GeV and $\alpha_{\rm s}=0.5752$ such that they reproduce the $1S$ and $2S$ levels.
• Figure 2: The support functions $f_X(q)$ vs. $q$ for $X = 1S$, $2S$ and $3S$ (solid lines). $f_X(q)$ is calculated using $m_{\rm pole}=5$ GeV and a different $\alpha_{\rm s}(\mu_X)$, taken from Tab. \ref{['table:spectra']}, for each $X$. Vertical lines represent the corresponding scales $\mu_X$ taken from the same table. Also $\alpha_{\rm s}^{(4)}(q)$ is shown by a dashed line.