$B_c$ and $\Upsilon$ Spectra from Lattice NRQCD - Results at $\beta$ = 5.7

TL;DR

This paper presents updated lattice NRQCD calculations of heavy-heavy meson spectra at different lattice spacings, including new predictions for the $B_c$ meson, and compares results between two beta values.

Contribution

It provides the first lattice predictions for the $B_c$ spectrum and compares heavy-heavy spectra at different lattice spacings using NRQCD.

Findings

Updated heavy-heavy spectrum results at $eta$=5.7

First lattice predictions for $B_c$ spectrum

Scaling comparison with $eta$=6.0 results

Abstract

We update our results for the heavy-heavy spectrum at $\beta$ = 5.7, using NRQCD. This includes a scaling comparison with the $\Upsilon$ spectrum at $\beta$ = 6.0 and first lattice predictions for the $B_c$ spectrum.

Tables

Table 1. Table 1: Results for static and kinetic masses and the difference between them for heavy-heavy mesons, using NRQCD at β 𝛽 \beta =5.7. Values are in lattice units.

	$E_0$	$M_{kin}$	shift
${\rm Y}$	0.5030	7.04	6.54
${\eta }_c$	0.618	2.430	1.812
$B_c$	0.6052	4.79	4.18

Equations

$$E(p)=E_0+p^2/2M_{kin}.$$

Full text

1 Introduction

We have recently given accurate results for bottomonium and charmonium spectroscopy on the lattice [1, 2], exploiting the non-relativistic nature of these systems by using a non-relativistic effective theory, NRQCD[3]. This can be systematically matched to full QCD, order by order in $\alpha_{s}$ and $v^{2}/c^{2}$, where $v$ is the velocity of the heavy quark in the bound state. A limited number of terms in the NRQCD action suffice provided that $v^{2}$ is small and the lattice heavy quark mass, $Ma$ is not too small. The latter condition prevents $a$ being taken to zero in this effective theory, but it should still be true that physical results scale in a limited region of $a$. This is all that is necessary to obtain continuum values. We test the scaling of our $\Upsilon$ spectrum results by comparing results at $\beta$ =5.7 with those at $\beta$ = 6.0, an updated picture of which is presented here in the talks by Paul McCallum and Junko Shigemitsu.

There is much current interest in the spectrum of mixed bound states of bottom and charm quarks with the hope that they will soon be observed experimentally. Here we present results for $B_{c}$ spectroscopy from NRQCD and use our $\Upsilon$ and $\Psi$ results to give good indications of the size and direction of systematic errors. We compare our results to recent calculations in potential models [4].

2 Lattice NRQCD

The NRQCD action, details of how heavy quark propagators are evaluated, the smearing functions and the fitting procedures used are described in detail in several publications [5, 1, 2] and are reviewed in this Proceedings in the talks by Paul McCallum and Junko Shigemitsu.

The starting point of NRQCD is to expand the original QCD Lagrangian in powers of $v^{2}$, the typical quark velocity in a bound state. For the $J/\Psi$ system $v^{2}\sim 0.3$ and for $\Upsilon$ $v^{2}\sim 0.1$. For $B_{c}$ $v^{2}\sim 0.15$ for the single particle with mass equal to the reduced mass of the $b$, $c$ system, but the kinetic energy will be shared unequally between the $b$ and $c$. $v^{2}\sim 0.5$ for the $c$ quark so relativistic corrections will be more important than for $c\overline{c}$ systems. NRQCD enables these corrections to be added systematically.

The action used here is the same one used in refs. [1] and [2]. Relativistic corrections ${\cal O}(M_{Q}v^{4})$ have been included for both $b$ and $c$ quarks. Other sources of systematic error are discretisation errors and errors from the absence of virtual quark loops because we use quenched configurations generated with the standard plaquette action. Finite volume errors should be negligible because of the relatively small size of heavy-heavy systems.

To calculate masses for $b\overline{c}$ bound states we define $b$ and $c$ quark Green functions on the lattice, in the standard way. We used 200 quenched gluon field configurations at $\beta=6/g^{2}$ = 5.7 generously supplied by the UKQCD collaboration [6], and fixed to Coulomb gauge, using a Fourier accelerated steepest descents algorithm [7]. The bare masses of the $c$ and $b$ quarks were chosen from fits to the kinetic mass of the $\eta_{c}$[2] and the $\Upsilon$, giving $am_{c}$ = 0.8 (with stability parameter, $n$ = 4) and $am_{b}$ = 3.15 ($n$ = 2).

We combine a $b$ propagator with a $\overline{c}$ propagator to make $B_{c}$ states. From the evolution equations, it is more efficient to have numerous $b$ quark propagators with small values of the stability parameter $n$ and only one $c$ quark propagator. We therefore fixed the $c$ quark with a $\delta$ function source and used various smearing functions as sources for the $b$ quark. This enabled us to also map out the $\Upsilon$ spectrum at the same time.

Local meson operators are tabulated in [1]. Using the notation ${}^{2S+1}L_{J}$, we have looked at meson propagators for the following states: ${}^{1}S_{0}$, ${}^{3}S_{1}$, ${}^{1}P_{1}$, ${}^{3}P_{0}$, ${}^{3}P_{1}$ and ${}^{3}P_{2}$ for both the E and T representation. Since $C$ is not a good quantum number for the $B_{c}$ system the ${}^{1}P_{1}$ and ${}^{3}P_{1}$ mesons will mix and so in addition we calculated the cross-correlation between these two. For the $s$ states, smearing functions both for the ground and first radially excited state were used as well as a local $\delta$ function. From this all possible combinations of smearing at the source with a local sink were formed. For the $p$ states only the ground state smearing function was used at the source. We calculated the dispersion relation for the $\Upsilon$ and for the ($B_{c}$) by looking at the meson propagator for small non-zero momenta. Because of the relatively small size of these systems it is possible to use more than one starting site for the mesons. We used 8 different spatial origins and 2 different starting times to increase our statistics, but we bin correlation functions over each configuration.

3 Simulation results

The most accurate quantities we calculate are spin-independent splittings such as the spin-averaged 1P-1S splitting. A comparison between this splitting at $\beta$ = 5.7 and $\beta$=6.0 shows good scaling behaviour once \cal{O}$$(a^{2}) errors in the gluon field configurations are removed by estimating their effect on 1S levels either perturbatively or non-perturbatively [8] (a 10% effect for the $\Upsilon$ at $\beta$ =5.7). Figure 1 shows a scaling plot for the ratio of the 1P-1S splitting to $m_{\rho}$. To see scaling we must use results for $m_{\rho}$ which have \cal{O}$$(a) errors removed. The results in Figure 1 come from UKQCD [9] using a tadpole-improved clover action for light propagators. The $\Upsilon$ splitting at $\beta$ = 6.2. is from ref. [10] using a lowest order NRQCD action only and therefore has a potential 10% systematic uncertainty.

The difference with experiment for this ratio is similar to that we find when comparing splittings for $b\overline{b}$ and $c\overline{c}$ at $\beta$=5.7 [2] and interpret it as a result of incorrect running of the coupling constant between scales in the quenched approximation. The light hadron spectrum may still have large systematic errors arising from other sources such as finite volume so it is too early to put this result down entirely to quenching.

We currently determine spin splittings only to leading order with no discretisation corrections. This has a large effect at $\beta$ = 5.7 as can be seen from Figure 2. The discretisation corrections tend to reduce the overall size of splittings and also tend to equalise the ${}^{3}P_{2}-^{3}P_{1}$ and ${}^{3}P_{1}-^{3}P_{0}$ splittings. This is already evident at $\beta$ = 6.0 and is worse at $\beta$ = 5.7. This can be improved by systematically adding discretisation corrections to the $E$ and $B$ fields.

Figure 3 shows results for the $B_{c}$ spectrum. We are able to predict the overall mass scale as well as the splittings since the $c$ and $b$ masses were fixed from the $\Upsilon$ and $\eta_{c}$ masses. The fact that $a^{-1}$ is different for $c\overline{c}$ and $b\overline{b}$ in the quenched approximation means that we must fix $a^{-1}$ for $b\overline{c}$ separately. We do this by setting the 1P-1S splitting to be the same as that for $b\overline{b}$ and $c\overline{c}$ (it is $M_{Q}$ independent). This gives an $a^{-1}$ between the two previous values and means that the quark masses in physical units are not the same in the three systems. This is an unavoidable artefact of quenching and adds to the systematic error.

The $B_{c}$ mass is obtained by calculating correlation functions at non-zero momenta to derive the dispersion relation. The mass is the kinetic mass in the equation:

[TABLE]

$M_{kin}$ will not agree with the static mass, $E_{0}$, because the quark mass has been removed from the Hamiltonian. There is then a shift between $E_{0}$ and $M_{kin}$ that must be applied per quark in a meson. We can check non-perturbatively that the shift for the $B_{c}$ is the average of that for $c\overline{c}$ and $b\overline{b}$. This is shown in Table 1 and works very well. For the clover action at large mass it does not work, and instead the shift is dependent, incorrectly, on the meson itself [11].

The comparison of our results with a potential model shows fairly good agreement (Figure 3) and indicates that relativistic effects may fortuitously not be very large. We have included the first relativistic corrections but potential model calculations include relativistic effects only through the phenomenological potential. Better estimates of these corrections need to be made.

Our experience with heavy-heavy spectra in the quenched approximation indicates that our calculation is likely to overestimate the 2S-1S splitting and to underestimate the hyperfine and $p$ states splittings. Discretisation errors also reduce our spin splittings, as above. Unquenching and discretisation corrections would then bring us into better agreement with the potential model results except for the $p$ fine structure where we already have larger splittings. We believe as a result that the potential model calculation is underestimating these splittings.

Our preliminary value for the $B_{c}$ decay constant is 460 MeV. This calculation includes only the leading order term in the heavy-light current. There is a $1/M_{c}$ correction for $B_{c}$ whereas in $b\overline{b}$ and $c\overline{c}$ the first correction appears at $1/M_{Q}^{2}$.

After this talk was given we learned that the ALEPH collaboration has candidate $B_{c}$ events [12].

4 Summary

We present heavyonium spectrum results at $\beta$ = 5.7, including first lattice predictions for the $B_{c}$ spectrum. Future calculations will include further relativistic and discretisation corrections as well as a comparison to unquenched results at similar lattice spacing.

This work is supported in part by the U.S. DOE, the NSF and by the UK PPARC. The numerical computations were carried out at NERSC and the Atlas Centre. We thank UKQCD for providing the quenched configurations