Scaling of the Upsilon Spectrum in Lattice NRQCD

TL;DR

The paper investigates discretisation errors in the lattice NRQCD calculation of the bottomonium spectrum in the quenched approximation, using three lattice spacings to test scaling. It shows that spin-independent radial and orbital splittings are largely lattice-spacing independent, while spin-dependent splittings exhibit significant a-dependent effects due to missing higher-order corrections; the authors quantify scale setting, quark-mass tuning via Υ kinetic mass, and the impact of relativistic and radiative corrections. The results highlight substantial systematic uncertainties in spin splittings under quenched NRQCD and indicate that higher-order spin terms and unquenched simulations are necessary to approach experimental bottomonium data. The work emphasizes the need to match scale-setting and dynamical flavor content when extrapolating to real QCD with 2+1 flavors and provides a framework for incorporating future corrections and unquenching efforts.

Abstract

We present results for the spectrum of b-bbar bound states in the quenched approximation for three different values of the lattice spacing, in the range 0.05fm to 0.15fm. We find our results for spin-independent splittings in physical units to be independent of the lattice spacing, indicating the absence of systematic errors from discretisation effects. Spin-dependent splittings are more sensitive to the lattice spacing and higher order corrections to the action; we discuss the size of these effects and what can be done to arrive at a physical result.

Figures (8)

• Figure 1: Effective mass plots for the $(^1S_0)_{1l}$ correlator at all three values of $\beta$, in order with $\beta$ = 5.7 at the top. The time axis has been converted to physical units of ${\rm GeV}^{-1}$ using the $\overline{\chi_b} - \Upsilon$ splitting to set the scale (Table \ref{['scales']}).
• Figure 2: Effective mass plots for the $(^1P_1)_{1l}$ correlator at all three values of $\beta$, in order with $\beta$ = 5.7 at the top. The time axis has been converted to physical units of ${\rm GeV}^{-1}$ using the $\overline{\chi_b} - \Upsilon$ splitting to set the scale (Table \ref{['scales']}).
• Figure 3: Dimensionless ratios of the $\overline{\chi_b} - \Upsilon$ splitting to the parameter $\Lambda_V$ (diamond) and to the UKQCD $\rho$ mass ukqcd (circle) against the lattice spacing in fm (set by the $\overline{\chi_b} - \Upsilon$ splitting). Results using the GF11 $\rho$ mass gf11 are given by squares. The burst represents the experimental value for $\Delta (\overline{\chi_b} - \Upsilon)/m_{\rho}$.
• Figure 4: Dimensionless ratios of various splittings to the $\overline{\chi_b} - \Upsilon$ splitting against the lattice spacing in fm (set by the $\overline{\chi_b} - \Upsilon$ splitting). Circles represent the ratio for the $\Upsilon^{"} - \Upsilon$ splitting (experiment short dashes) and crosses for the $h_b^{'} - \Upsilon$ (experiment using $\overline{\chi_b^{'}}$ for $h_b^{'}$ dot-dash). The diamonds show the $\Upsilon^{'} - \Upsilon$ ratio with $a^2$ gluonic corrections (as described in the text) and the squares uncorrected results (experiment dashed line). The squares and crosses have been offset slightly in the horizontal direction for clarity.
• Figure 5: The hyperfine splitting in MeV using the $\Upsilon^{'} - \Upsilon$ splitting to set the scale, vs $a^2$ in ${\rm fm}^2$. Plain squares indicate our results from Table \ref{['table_energies']}. The diamonds indicate the results from ref. manke using a higher order action. They are at matching values of $\beta$ but offset slightly for clarity. Fancy squares indicate our results, rescaled by the square of $c_4$ calculated to $\cal{O}$$(\alpha)$ in trottier2. The fancy diamonds indicate the result from ref. manke shifted by the same amount as our results to account for radiative corrections to the $\sigma\cdot B$ term. The error bars shown are statistical only and include some of the error from the uncertainty in the scale (see text). The $x$ axis errors from uncertainty in the scale are shown only for the squares for clarity.