The Upsilon spectrum and m_b from full lattice QCD

Abstract

We show results for the Upsilon spectrum calculated in lattice QCD including for the first time vacuum polarization effects for light u and d quarks as well as s quarks. We use gluon field configurations generated by the MILC collaboration. The calculations compare the results for a variety of u and d quark masses, as well as making a comparison to quenched results (in which quark vacuum polarisation is ignored) and results with only u and d quarks. The b quarks in the Upsilon are treated in lattice Nonrelativistic QCD through NLO in an expansion in the velocity of the b quark. We concentrate on accurate results for orbital and radial splittings where we see clear agreement with experiment once u, d and s quark vacuum polarisation effects are included. This now allows a consistent determination of the parameters of QCD. We demonstrate this consistency through the agreement of the Upsilon and B spectrum using the same lattice bare b quark mass. A one-loop matching to continuum QCD gives a value for the b quark mass in full lattice QCD for the first time. We obtain m_b^{\bar{MS}}(m_b^{\bar{MS}}) = 4.4(3) GeV. We are able to give physical results for the heavy quark potential parameters, r_0 = 0.469(7) fm and r_1 = 0.321(5) fm. Results for the fine structure in the spectrum and the Upsilon leptonic width are also presented. We predict the Upsilon - eta_b splitting to be 61(14) MeV, the Upsilon^{\prime} - eta_b^{\prime} splitting as 30(19) MeV and the splitting between the h_b and the spin-average of the chi_b states to be less than 6 MeV. Improvements to these calculations that will be made in the near future are discussed.

Figures (19)

• Figure 1: Dependence of $\Upsilon$ energy splittings on change in NRQCD stability parameter $n$ and choice of $u_0$: (a) $n=2$, $u_{0L}$ (b) $n=4$, $u_{0L}$ (c) $n=2$, $u_{0P}$. The results are from the 0.01/0.05 2+1 flavor coarse ensemble. $2S-1S$ etc indicates the splitting between radial excitations of the $\Upsilon$ and $1P-1S$ the splitting between the appropriate $^1P_1$ state and the $\Upsilon$.
• Figure 2: $\Upsilon$ energies in lattice units as a function of number of exponentials used in a $3\times 3$ matrix fit for results on the 0.01/0.05 coarse ensemble. $\chi^2/\hbox{dof}$ values indicate goodness of fit.
• Figure 3: The ratio of lattice spacing values obtained from the $2S-1S$ and $1P-1S$ splittings in the $\Upsilon$ system as a function of bare light quark mass in the lattice QCD calculation. Notice the very magnified $y$-axis scale. The crossed square is from the super-coarse ensemble, the filled squares from the coarse 2+1 flavor ensembles and the open squares from the fine 2+1 flavor ensembles. The filled diamond is from the coarse $n_f=2$ ensemble. The open and filled triangles on the right of the plot are results from the quenched fine and coarse ensembles respectively. Errors are statistical only.
• Figure 4: The value of $r_0$ in fm obtained on different super-coarse, coarse and fine ensembles, using the $2S-1S$ splitting in the $\Upsilon$ system to fix the lattice spacing. The crossed square is from the super-coarse ensemble, the filled squares from the coarse 2+1 flavor ensembles and the open squares from the fine 2+1 flavor ensembles. The open and filled triangles on the right of the plot are results from the quenched fine and coarse ensembles respectively. The $y$-axis scale is expanded to make error bars visible. $r_0$ values in lattice units are taken from milc1milc2milcprivate. Errors include the statistical errors from determining the lattice spacing and from determining $r_0$.
• Figure 5: The value of $r_1$ in fm obtained on different super-coarse, coarse and fine ensembles, using the $2S-1S$ splitting in the $\Upsilon$ system to fix the lattice spacing. The crossed square is from the super-coarse ensemble, the filled squares from the coarse 2+1 flavor ensembles and the open squares from the fine 2+1 flavor ensembles. The open and filled triangles on the right of the plot are results from the quenched fine and coarse ensembles respectively. The $y$-axis scale is expanded to make error bars visible. $r_1$ values in lattice units are taken from milc1milc2milcprivate. Errors include the statistical errors from determining the lattice spacing and from determining $r_1$.