i=0,\ldots,5$.
\begin{table}[h]
  \centering
  \begin{ruledtabular}
  \begin{tabular}{c||ccccccc}
    Set & $S_0$ & $S_1$ & $S_2$ & $S_3$ & $S_4$ & $S_5$\\
    \hline
    $N_{\rm HYP}$ & 10 & 5 & 1 & 0 & 3 & 3
    \\
    $\alpha_1$ & 0.75 & 0.75 & 0.75 & 0.75 & 0.75 & 0.6
    \\
    $\alpha_2$ & 0.75 & 0.75 & 0.75 & 0.75 & 0.4 & 0.6
    \\
    $\alpha_3$ & 0.75 & 0.75 & 0.75 & 0.75 & 0.65 & 0.6
  \end{tabular}
  \end{ruledtabular}
  \caption{Parameters used in HYP-smearing the heavy quark action.}
  \label{tab:smear}
\end{table}

To extract the lattice energies, $E_h$, of the various hadrons, $h$,
we compute the meson and baryon two-point correlation functions
\begin{eqnarray}
  \label{eq:5}
  C_f(t,t_0) &=& \sum_{{\bf x}} {\rm tr}\left[S_Q({\bf x},t;t_0)
   S_f^\dagger({\bf x},t;{\bf x},t_0) \right]\,,
\\
\label{eq:6}
  C^\Gamma_{f,g}(t,t_0) &=& \sum_{{\bf x}} 
    S^{k^\prime k}_{Q;\sigma\rho}({\bf x},t;t_0)\epsilon^{ijk}
    \epsilon^{i^\prime j^\prime k^\prime}
    \left(S_f^{i i^\prime}({\bf x},t;{\bf x},t_0) \Gamma\right)_{\rho\alpha} 
    \left(\Gamma S_g^{j j^\prime}({\bf x},t;{\bf x},t_0)\right)_{\sigma\alpha}  
\,,
\end{eqnarray}
for the various light flavour combinations, $f$($g$), and then combine
them appropriately.  The trace in the meson correlator is over color
and spinor indices, and where explicit, the upper(additional lower)
indices on the propagators correspond to colour(spin). For the baryons
with the spin of the light degrees of freedom, $s_\ell$, being zero we
choose $\Gamma=C\, \gamma_5$ where $C$ is the charge conjugation
matrix, while for the baryons with $s_\ell=1$, we measure the three
polarisations, corresponding to $\Gamma=C\,\gamma_{1,2,3}$, and
average them in extracting the energies as they are degenerate in the
infinite statistics limit. We combine the various correlators in
Eq.~(\ref{eq:6}) as appropriate for the specific
SU(3) representation. For example, the $\Xi_b^{0}$ belongs to the flavour
anti-triplet and is a $s_\ell=0$ state, consequently
\begin{equation}
  \label{eq:1}
  C_{\Xi_b^{ 0}}= C^{\gamma_5}_{u,s} -C^{\gamma_5}_{s,u}
\end{equation}
and similarly,
\begin{eqnarray}
  \label{eq:2}
    C_{\Lambda_b^{ 0}}= C^{\gamma_5}_{u,d} -C^{\gamma_5}_{d,u}, \quad \quad
    C_{\Sigma_{b,i}^{0}}= C^{\gamma_i}_{u,d} +C^{\gamma_i}_{d,u}, \quad\quad
    C_{\Xi_{b,i}^{\prime,0}}= C^{\gamma_i}_{u,s} +C^{\gamma_i}_{s,u}, \quad\quad
    C_{\Omega_{b,i}^{-}}= C^{\gamma_i}_{s,s} \,.
\end{eqnarray}
As we work in the isospin limit, the other singly-heavy hadrons are
degenerate with those above. In the static limit of the heavy quark,
the $J^P=\frac{1}{2}^+$ states ($\Sigma_b,\,\Xi_b^\prime,\, \Omega_b$)
and $\frac{3}{2}^+$ states ($\Sigma_b^\ast,\,\Xi_b^{\prime\ast},\,
\Omega_b^\ast$) are degenerate and our results correspond to
this. Future calculations for $m_b\ne\infty$ will enable the spin
splittings to be investigated, see Ref.~\cite{Lewis:2008fu} for recent
work.

The transfer matrix formalism immediately shows that in the limit of
large temporal separation, these correlators are dominated by
exponentially decaying signal of the ground-state hadron energy. That
is,
\begin{equation}
  \label{eq:3}
  C_h(t,t_0)\stackrel{t\gg t_0}{\longrightarrow} A_h \exp{[-E_h(t-t_0)]}\,,
\end{equation}
where $E_h$ is the lattice energy. In the case of the $s_\ell=1$
baryons of strangeness $<2$, this is complicated by the fact that, at
the physical quark masses and in infinite volume, they decay strongly
to lighter $s_\ell=0,1$ baryons and a pion. For example,
$\Sigma_b^+\to\Lambda_b\, \pi^+$ is the dominant decay mode
\cite{Amsler:2008zz}. However, at the unphysical masses and finite
volumes used (the pion couples derivatively, requiring non-zero
lattice momentum) here, these decays are forbidden. Additionally, the
finite temporal extent (non-zero temperature) of the lattice geometry
used in our calculations results in the appearance of exponentially suppressed states
that decay with ``lighter masses'' that may pollute the signals of the
zero temperature ground-state \cite{Detmold:2008aaa} (this is
asymmetric analog of the constant term expected in a two meson
correlation function from one meson propagating forward in time and
one propagating backward in time
\cite{Detmold:2008yn,Prelovsek:2008rf}).  We will return to this issue
below.

\section{Bottom Hadron Mass Splittings}
\label{sec:su3-mass-splittings}

In the static limit, the masses of bottom hadrons are infinite,
however, the residual mass, $\overline{\Lambda}_h$, of a heavy hadron
$h$ is well defined. In the lattice regularisation used here, the
lattice energies, $E_h$ diverge as the continuum limit is taken and a
perturbative subtraction is required to convert them to the
corresponding residual mass, for example, in the modified minimal
subtraction ($\overline{\rm MS}$) scheme. At one loop, this can be
performed straightforwardly for typical lattice actions, and with more
effort for the HYP-smeared gauge links used herein
\cite{Loktik:2006kz,Christ:2007cn}. However, there are significant
issues with higher order corrections and renormalons in the
subtraction procedure \cite{Martinelli:1998vt, Martinelli:1995vj}. 
In contrast, the differences
between the residual masses of two hadrons containing the same heavy
quark (that is, the same lattice discretisation of a static quark)
have well-defined continuum limits in which they are observable (the
perturbative corrections and renormalisation scale dependence cancel
in the difference). In our work, we concentrate on these differences
and thereby avoid the issues mentioned above.

\subsection{Heavy quark action}
\label{sec:heavy-quark-action}

We first study the effects of the choice of the heavy quark action via
the application of the various HYP-smearings shown in
Table~\ref{tab:smear}. For each energy or energy difference, the
results for all the choices of smearing are found to be consistent within
uncertainties. Figure~\ref{fig:s2n} shows the noise-to-signal ratios
for the correlators corresponding to the $\Sigma_b^+$---$B_u$,
$\Omega_b^-$---$B_s$ and $\Xi_b^0$---$\Lambda_b$ mass splittings for
the various smearing parameter sets as a function of Euclidean time
using the QCD propagators of ensemble B in Table
\ref{tab:ensemble}. As can be seen, the parameter set $S_0$ clearly
has the strongest signals. The other sets of propagators display
similar behaviour and we focus on set $S_0$ for the remainder of our
study.
\begin{figure}
  \cent