[a]C. T. H. Davies

# Precision bottomonium properties and b quark mass from lattice QCD

, D. Hatton  , J. Koponen  , G. P. Lepage  and A. T. Lytle  for the HPQCD
collaboration

###### Abstract

As tests of QCD in the bottomonium system, we give the most accurate results
to date for the ground-state hyperfine splitting and the $\Upsilon$ leptonic
width from full lattice QCD. These quantities are both accurately known from
experiment, so can provide a good test of $b$ physics, but previous lattice
results have been rather imprecise. We also test the impact on these
quantities of the $b$ quark’s electric charge. Our results are:
$M_{\Upsilon}-M_{\eta_{b}}=57.5(2.3)(1.0)\,\mathrm{MeV}$ (where the second
uncertainty comes from neglect of quark-line disconnected correlation
functions) and decay constants, $f_{\eta_{b}}=724(12)$ MeV and
$f_{\Upsilon}=677.2(9.7)$ MeV, giving $\Gamma(\Upsilon\rightarrow
e^{+}e^{-})=1.292(37)(3)\,\mathrm{keV}$. We also give a new determination of
the ratio of the masses for $b$ and $c$ quarks that is completely
nonperturbative in lattice QCD and includes the calculation of QED effects for
the first time. This gives a result for the $b$ quark mass of
$\overline{m}_{b}(\overline{m}_{b},n_{f}=5)=4.202(21)$ GeV.

##  1 Introduction

Lattice QCD calculations of weak decay matrix elements for hadrons containing
$b$ quarks are critical to the flavour physics programme. It is therefore
important to have stringent tests of lattice QCD results for $b$ physics in
other settings to make sure systematic errors are under control. Here we
provide such tests in the bottomonium system with the ground-state hyperfine
splitting and the $\Upsilon$ leptonic width, both accurately known from
experiment. We give the most accurate lattice QCD results to date for these
quantities and also test the impact on them of the $b$ quark’s electric charge
[1]. Accurate masses for heavy quarks are important for high-precision
searches for new physics in Higgs decay [2]. We give here a new determination
of the ratio of the masses for $b$ and $c$ quarks that is completely
nonperturbative in lattice QCD and includes the calculation of QED effects for
the first time [3]. This allows an accurate determination of the $b$ quark
mass using our earlier accurate results for $m_{c}$ in QCD+QED [4].

##  2 Lattice QCD calculation

We use the Highly Improved Staggered Quark (HISQ) action [5] on ensembles of
gluon field configurations that include 2+1+1 flavours of HISQ quarks in the
sea, generated by the MILC collaboration [6]. The ensembles have a range of
lattice spacing values from 0.09 fm down to 0.03 fm and with $u/d$ sea masses
varying from 1/5 that of the $s$ quark ($M_{\pi}\approx$ 300 MeV) down to
their physical value. On each ensemble we calculate pseudoscalar and vector
2-point correlators for HISQ valence quarks with masses $m_{h}$ from that of
the $c$ quark up to $am_{h}=$ 0.8 or 0.9. On the finest lattices we can reach
the $b$ quark mass with $am_{h}=0.65$. We calculate correlators including
quenched QED on a subset of ensembles. Note that we do not include quark-line
disconnected correlators. More details are given in [1].

We determine the masses and amplitudes of the ground state mesons (denoted
$\eta_{h}$ and $\phi_{h}$) on each ensemble using standard correlator fits.
The hyperfine splitting is the difference of masses, $M_{\phi}-M_{\eta}$. The
vector and pseudoscalar decay constants are determined from the amplitudes in
the standard way; both are normalised accurately using lattice Ward identities
[7]. We then perform a model-independent fit to these results as a function of
lattice spacing and $\phi_{h}$ mass, using cubic splines. This allows us to
determine results in the continuum limit with physical sea quark masses for
the case where the heavy quark is the $b$ quark, i.e. where the $\phi_{h}$ has
the experimental mass of the $\Upsilon$. We use this criterion for tuning the
$b$ quark mass both in pure QCD and in QCD+QED.

The ratio of quark masses is scheme and scale independent in pure QCD, but not
in QCD+QED if the quarks have different electric charges. We calculate the
ratio of $b$ and $c$ masses at 3 GeV in the $\overline{\text{MS}}$ scheme
using a 3 step procedure (for full details see [3]). The first step is to fit
results for $m_{h}/m_{c}$ as a function of the $\phi_{h}$ mass and lattice
spacing in pure QCD. $m_{c}$ here is the tuned $c$ quark mass, obtained on
each ensemble using the experimental $J/\psi$ meson mass [4]. Evaluating the
fit function in the physical-continuum limit at the $\Upsilon$ mass then gives
the pure QCD $m_{b}/m_{c}$ value. The second step is to calculate how much
this ratio changes if BOTH the $b$ and $c$ quarks have charge $Q=e/3$ so that
the ratio is still scale-invariant. The third step corrects the final ratio in
the $\overline{\text{MS}}$ scheme at 3 GeV to take the $c$ quark charge to
2e/3. This is the largest QED effect and is calculated from the results of
[4].

##  3 Results - hyperfine splitting

Figure 1: The left-hand plot shows the heavyonium hyperfine splitting as a
function of the vector heavyonium mass from $n_{f}=2+1+1$ lattice QCD [1]. The
range covered is from the $c$ quark mass to the $b$ quark mass. Points show
lattice QCD results at a range of lattice spacings from 0.09 fm (denoted f),
through 0.06 fm (denoted sf) and 0.045 fm (denoted uf) to 0.03 fm (denoted
ef). Sea $u/d$ quark masses are either $1/5$ of the $s$ quark mass (for points
denoted $-5$) or physical. The purple band shows a model-independent fit to
these results using cubic splines. The black cross marks the experimental
average value [8] for the ground-state bottomonium hyperfine splitting. The
right-hand plot compares our result (denoted ‘This work’) to recent
experimental values. The pink band shows the experimental average [8] and the
blue band carries our result down the plot. Note the particularly good
agreement between our value and the Belle result [9], although we also agree
well with the experimental average that includes in addition results from CLEO
[10] and BaBar [11, 12].

Figure 1 shows our results [1] for the hyperfine splitting as a function of
$\phi_{h}$ mass along with our fit curve. Our result at the $b$, i.e. the mass
difference between $\Upsilon$ and $\eta_{b}$, is 57.5(2.3)(1.0) MeV, where the
second uncertainty comes from the neglect of quark-line disconnected diagrams.
Our result is compared to experimental results in the right-hand plot in the
Figure. We see good agreement, particularly with the most recent result from
Belle. QED effects here (and in the decay constant below) are tiny.

##  4 Results - decay constants

Figure 2: The left-hand plot shows the decay constant of the vector heavyonium
meson as a function of its mass from $n_{f}=2=1+1$ lattice QCD [1]. The points
are the lattice results (with symbols as in Figure 1) and the purple band
shows a model-independent fit using cubic splines. The right-hand plot shows
the ratio of vector to pseudoscalar decay constants as a function of the
vector heavyonium mass. Some systematic errors, such as discretisation
effects, largely cancel in this ratio. This makes it very clear that the ratio
changes from being greater than 1 to being less than 1 as the heavy quark mass
is increased from $c$ to $b$.

Figure 2 shows the vector meson decay constant and the ratio of vector to
pseudoscalar decay constants as a function of the $\phi_{h}$ mass [1]. Note
that, interestingly, the ratio is larger than 1 at $c$ and falls to less than
1 at $b$. This is unambiguous for our HISQ results because we are able to
normalise the currents accurately and fully nonperturbatively. At the $b$
quark mass our results are: $f_{\eta_{b}}=724(12)$ MeV and
$f_{\Upsilon}=677.2(9.7)$ MeV.

The result for $f_{\Upsilon}$ can be used to derive a leptonic width for the
$\Upsilon$ of $\Gamma(\Upsilon\rightarrow
e^{+}e^{-})=1.292(37)(3)\,\mathrm{keV}$ which can be compared to experiment.
Equivalently, Figure 4 shows the good agreement between our result for the
$\Upsilon$ decay constant and that inferred from the experimental $\Upsilon$
leptonic width.

##  5 Results - $m_{b}$

Figure 3: The left-hand plot shows lattice QCD (without QED) results for
$m_{h}/m_{c}$ plotted agains the heavyonium meson mass. The two lines
correspond to using $\eta_{h}$ and $\phi_{h}$ and show best fit values for the
dependence obtained, using a cubic spline fit [3]. The line thickness shows
the $1\sigma$ uncertainty in the fit. The lattice results are shown by squares
for the pseudoscalar case (top line) and by circles for the vector case
(bottom line). Different colours denote different gluon configurations sets
covering the same range as in Figure 1. The right-hand plot shows the ratio
$R$ of $\overline{m}_{h}/\overline{m}_{c}$ computed with QED charge $e/3$ to
that without QED. The quark masses are tuned to give the same result for
heavyonium meson masses, with the red curve using the $\eta_{h}$ and the blue
curve using the $\phi_{h}$. The blue and red shaded areas show the $\pm
1\sigma$ error bands on the fits to the data [3].

Figure 4: The left-hand plot compares our result (filled blue hexagon) for the
$\Upsilon$ decay constant to an earlier HPQCD result using NRQCD $b$ quarks
[13] (green open square). Our result, carried down as the blue band, is also
compared to the experimental average result [8], inferred from the $\Upsilon$
leptonic width, given by the pink open circle. We see good agreement with
experiment at the level of our 1.4% uncertainty. The right-hand plot compares
our new value for $m_{b}$ in QCD+QED to earlier results that used lattice QCD
calculations, estimating QED effects. These earlier results are from [14, 15,
16, 13, 17].

Figure 3 (left-hand plot) shows the ratio of HISQ masses $m_{h}/m_{c}$ as a
function of heavyonium meson mass (either $\phi_{h}$ or $\eta_{h}$) in pure
QCD [3]. Our fit curves (using cubic splines) are also shown. The right-hand
plot shows the relative change in this curve if both quarks have electric
charge $e/3$. This is a tiny (< 0.04%) effect. A larger, 0.13%, QED effect
comes from changing the $c$ quark electric charge to $2e/3$. We do this at
scale 3 GeV using [4]. We obtain, for the ratio of $\overline{\text{MS}}$
masses at 3 GeV

$\frac{\overline{m}_{b}(3\,\mathrm{GeV})}{\overline{m}_{c}(3\,\mathrm{GeV})}\Bigg{|}_{\begin{subarray}{c}\mathrm{QCD}\\\
\mathrm{QED}\end{subarray}}\\!=\,4.586(12).$ (1)

We can convert this into a value for $m_{b}$, using our $m_{c}$ result in
QCD+QED of 0.9841(51) GeV [4], giving

$\overline{m}_{b}(\overline{m}_{b})\big{|}_{\begin{subarray}{c}\mathrm{QCD}\\\
\mathrm{QED}\end{subarray}}=\begin{cases}4.209(21)\,\mathrm{GeV}&n_{f}=4\\\
4.202(21)\,\mathrm{GeV}&n_{f}=5,\\\ \end{cases}$ (2)

from calculations that include QED for the first time [3]. Figure 4 compares
this to earlier pure QCD results (that estimated QED effects).

##  6 Conclusions

Following our earlier high precision QCD+QED charmonium calculations using
HISQ [4], we give here accurate results for $\Upsilon$ and $\eta_{b}$
properties. Good agreement with experiment is seen, with uncertainties now a
few % [1]. A new, 0.6%-accurate, QCD+QED determination of $m_{b}$ is also
given [3].

Acknowledgements We are grateful to the MILC collaboration for the use of
their gluon field configurations and for the use of MILC’s QCD code. Computing
was done on the Darwin supercomputer at the University of Cambridge High
Performance Computing Service as part of the DiRAC facility, jointly funded by
the Science and Technology Facilities Council, the Large Facilities Capital
Fund of BIS and the Universities of Cambridge and Glasgow. We are grateful to
the Darwin support staff for assistance. Funding for this work came from the
Science and Technology Facilities Council and the National Science Foundation.

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