Bottomonium spectrum at order $v^6$ from domain-wall lattice QCD: Precise results for hyperfine splittings

The bottomonium spectrum is computed in dynamical $2+1$ flavor lattice QCD, using nonrelativistic QCD for the $b$ quarks. The main calculations in this work are based on gauge field ensembles generated by the RBC and UKQCD Collaborations with the Iwasaki action for the gluons and a domain-wall action for the sea quarks. Lattice spacing values of approximately 0.08 fm and 0.11 fm are used, and simultaneous chiral extrapolations to the physical pion mass are performed. As a test for gluon-discretization errors, the calculations are repeated on two ensembles generated by the MILC Collaboration with the Lüscher-Weisz gauge action. Gluon-discretization errors are also studied in a lattice potential model using perturbation theory for four different gauge actions. The nonperturbative lattice QCD results for the radial and orbital bottomonium energy splittings obtained from the RBC/UKQCD ensembles are found to be in excellent agreement with experiment. To get accurate results for spin splittings, the spin-dependent order-$v^6$ terms are included in the nonrelativistic QCD action, and suitable ratios are calculated such that most of the unknown radiative corrections cancel. The cancellation of radiative corrections is verified explicitly by repeating the calculations with different values of the couplings in the nonrelativistic QCD action. Using the lattice ratios of the $S$-wave hyperfine and the $1P$ tensor splitting, and the experimental result for the $1P$ tensor splitting, the $1S$ hyperfine splitting is found to be $60.3\pm {5.5}_{\mathrm{stat}}\pm {5.0}_{\mathrm{syst}}\pm {2.1}_{\exp }\mathrm{MeV}$, and the $2S$ hyperfine splitting is predicted to be $23.5\pm {4.1}_{\mathrm{stat}}\pm {2.1}_{\mathrm{syst}}\pm {0.8}_{\exp }\mathrm{MeV}$.

Figure 1

Square of the speed of light, calculated with the $v^4$ action (points from the two different lattice spacings are offset horizontally for legibility). The points at ${\mathit{n}}^2=12$ correspond to a meson momentum of about 1.6 GeV. Note that the very small deviation of $c^2$ from 1 at $a\approx 0.11\mathrm{fm}$, seen for ${\mathit{n}}^2=2$ and ${\mathit{n}}^2=3$, disappears at $a\approx 0.08\mathrm{fm}$.

Figure 2

Chiral extrapolation of radial and orbital energy splittings (calculated with the $v^4$ action). Extrapolated points are offset horizontally for legibility.

Figure 3

Chirally extrapolated results for the radial and orbital energy splittings, calculated with the $v^4$ action. The ${\rm Y}(1S)$ and ${\rm Y}(2S)$ masses are used as input in the lattice calculation. The errors on the lattice results shown here are statistical/fitting/scale setting only.

Figure 4

Chiral extrapolation of spin-dependent splittings from the $L=24$ and $L=32$ ensembles, part I. Extrapolated points are offset horizontally for legibility.

Figure 5

Chiral extrapolation of spin-dependent splittings from the $L=24$ and $L=32$ ensembles, part II. Extrapolated points are offset horizontally for legibility.

Figure 6

Chirally extrapolated results for the spin-dependent energy splittings, from the $L=24$ and $L=32$ ensembles. Left panel: $1P$ spin splittings (energies relative to $\overline{1{}^{3}P}$). Right panel: $S$-wave hyperfine splittings [energies relative to the ${\rm Y}(1S)$ and ${\rm Y}(2S)$ states, respectively]. The errors shown are statistical/fitting/scale setting only; see Sec. 3c1 for a discussion of systematic errors and Table  for the final results that include estimates of the systematic errors.

Figure 7

Analysis of autocorrelations in the two-point functions of the ${\rm Y}(1S)$, ${\rm Y}(2S)$, and ${\rm Y}(3S)$ interpolating fields. The data are from the $L=16$ ensemble with $a{m}_l=0.01$, the $L=24$ ensemble with $a{m}_l=0.005$, and the $L=32$ ensemble with $a{m}_l=0.004$.

Figure 8

Kinetic mass of the ${\eta }_b(1S)$ meson plotted as a function of the bare heavy-quark mass (both in lattice units). The lines and error bands are from correlated fits using the functional form $a{M}_{\mathrm{kin}}=A·a{m}_b+B$. Left panel: $L=24$, $a{m}_l=0.005$. Right panel: $L=32$, $a{m}_l=0.004$.

Figure 9

Heavy-quark mass dependence of the spin splittings on the $L=24$, $a{m}_l=0.005$ and $L=32$, $a{m}_l=0.004$ ensembles. Results are shown for both the $v^4$ and the $v^6$ actions. The dashed lines and gray error bands are correlated fits using the function $A/(a{m}_b)$. The data for the $1S$ hyperfine splittings are incompatible with this form, and additional fits using the function $A/(a{m}_b)+B$ are shown, which describe the data very well.

Figure 10

Shift in the lattice potential model $1S$ and $1P$ energies as a function of the lattice spacing, for different levels of Symanzik improvement in the Laplace operator.

Figure 11

Lattice potential model wave functions $\Psi (r_1,r_2,r_3)$ at $r_1=0$ for the $1S$ and $1P$ states. Data are shown for the tree-level Lüscher-Weisz gluon action ($c_1=-1/12$) and the DBW2 gluon action ($c_1=-1.40686$), at two different lattice spacings, using the $\mathcal{O}(a^4)$-improved Laplacian in all cases.

Figure 12

Shift in the lattice potential model energy levels and splittings as a function of the lattice spacing. All data shown in this figure were generated with the $\mathcal{O}(a^4)$-improved Laplacian, so that the shifts are dominated by gluon-discretization errors.

Figure 13

Shift in the lattice potential model energy splittings, rescaled using the $2S-1S$ splitting, as a function of the lattice spacing. All data shown in this figure were generated with the $\mathcal{O}(a^4)$-improved Laplacian, so that the shifts are dominated by gluon-discretization errors.

Figure 14

Heavy-quark mass dependence of the $1S$ hyperfine splittings on the MILC ensembles (see Fig. 9 for the data from the RBC/UKQCD ensembles).