\section{Introduction}
The Standard Model Higgs boson -- or something like it -- has been
found \cite{:2012gk,:2012gu} and the race is now on to determine its
detailed properties.
Of particular interest is the $\tau \tau$ decay
channel, where, despite sensitivity to a signal comparable to that predicted in
the Standard Model with $m_h \simeq 125$ GeV,
only weak evidence somewhat below the Standard Model expectation has been seen~\cite{Einsweiler:HCP, Paus:HCP}.

This may be evidence for physics beyond the Standard Model or it may be
a statistical fluctuation.
We should dearly like to get to the bottom of the mystery, but progress is slowed by a number of
complications in the $h \rightarrow
\tau \tau$ search channel.

One complication is
the problem of triggering and identifying $\tau$ leptons, which decay
in a variety of ways, some of which resemble
common-or-garden QCD jets. A second is that, since a $\tau$ decay always
involves one or more invisible neutrinos,
it is only possible to reconstruct the resonance in an approximate way.
A third is the problem of
distinguishing $h \rightarrow
\tau \tau$ signal events from the dominant backgrounds, namely
$Z \rightarrow
\tau \tau$ decays (which, moreover, lie nearby in $\tau \tau$ invariant mass)
and production of QCD jets, with or without a $W$ boson, in which
other leptons or jets are mistakenly identified as $\tau$ leptons.

The third issue, of distinguishing the signal from backgrounds, is
compounded by the second issue, that we are {\em a priori} unable to
reconstruct the  invariant mass of the $\tau \tau$ pair.
As a result, and as is readily apparent from the data shown in \cite{CMS11}, the signal and backgrounds lie close to each other in distributions of the observables that are currently available to us.
Not only does this
increase the integrated luminosity required to be sure that an
apparent signal is not a mere background
fluctuation, but it also increases our vulnerability to systematic
errors.

To be explicit, imagine a hypothetical limit in which we have
a signal (the Higgs) that consists of a narrow peak in
invariant mass, a background (the $Z$) that is also a narrow
peak, but centred elsewhere, and additional backgrounds (like $W$ plus
jets and QCD) that are approximately flat in invariant mass. The signals and backgrounds are otherwise roughly
indistinguishable in their dynamics.
Now, if the observables that we have available are uncorrelated with the invariant
mass, then we are essentially reduced to counting events in order to
try to discover the signal and our ability to do so is greatly limited
by the total statistics available. We are, moreover, completely at the
mercy of systematic uncertainties in the overall background
normalization, which we have no way to measure in data.
Even if we were able to make a discovery in this way, we could at most
make one measurement of the signal properties (its overall size) and
here too we would be exposed to the systematic uncertainty in the
background normalization.

Conversely, if we find a way to reconstruct, more or less, the invariant
mass, the first benefit is that we are no longer limited by the
overall statistics, but rather by the number of signal and background
events in a region of invariant mass of our choosing (near 125 GeV
being the obvious choice for $h \rightarrow \tau \tau$).
Moreover, we
now have a clean separation between the signal and background and
indeed between the different backgrounds themselves.
This opens up the possibility of using the extra information to constrain
the uncertainties on the background yields and shapes via data-driven techniques.
Indeed, a simple sideband analysis would
suffice, in which the $Z$ background is measured in a `control' region near 90
GeV and the other backgrounds are measured in a control region away
from the peaks
near 90 GeV and 125 GeV. Finally, independent measurements of the signal mass and
cross-section times branching ratio become possible.

Needless to say, the real situation is rather more complicated for $h\rightarrow \tau \tau$ at the LHC, with the current performance falling somewhere between the two extremes of perfect and imperfect mass resolution. Nevertheless, the basic principle remains the same:
the more we are
able to separate the signal and the different background components
from each other, the less we shall find ourselves at the mercy of
statistical and systematic uncertainties.

So, how could we reconstruct something like the $\tau \tau$ invariant
mass? Several approximate methods or observables have previously been suggested in the literature
(see, for example, \cite{Ellis:1987xu,CMS2,Elagin:2010aw,Gripaios:2011jm,Barr:2011he,Barr:2011si}).
Some of these suffer from being rather poorly correlated with the invariant mass (some provide, for example, only an upper or lower bound on it), while others suffer from the fact that they turn out to be ill-defined for a significant fraction of events, with a consequent loss of statistics. As examples, the collinear approximation used in \cite{Barr:2011he} fails for one in three events, whereas the observable used in \cite{Ellis:1987xu, Barr:2011si} does not exist for a similar fraction of events.

Here we wish to propose yet another method, which differs
significantly in that  we focus on the subset of events in which a $\tau$ lepton undergoes a 3-prong
decay. This implies an immediate disadvantage in the form
of a reduced number of signal events overall for a given integrated luminosity: a
$\tau$ lepton has a branching ratio of 15 \% for a
3-prong decay (of which 9.3 \% are to $\pi^-\pi^+\pi^-\nu_\tau$
and 4.6 \% are to $\pi^-\pi^+\pi^- \pi^0 \nu_\tau$) \cite{Amsler:2008zzb}, meaning that only 28 \% of di-$\tau$ events feature at least one
3-prong decay.
However, the hope is that this disadvantage is more than compensated by the advantages.

These advantages all stem from the fact that the presence of a 3-prong
decay allows us to
reconstruct the $\tau \tau$ invariant mass, if the invariant mass of
the neutrino or neutrinos from the other $\tau$ decay is known.
Thus, for hadronic decays of the other $\tau$, we can fully reconstruct events (up to a
discrete ambiguity and in the absence
of detector mismeasurements, both of which we shall deal with below);
for leptonic decays of the other $\tau$, we are able to partially
reconstruct events. As a result we hope to benefit from a reduced
exposure to statistical and systematic uncertainties as argued above.

The extra kinematic information needed to reconstruct comes from the
location of the secondary (3-prong $\tau$-decay) vertex: if one can
measure with reasonable accuracy the impact parameter of each of the
three charged tracks, defined as the shortest distance between the
track and the primary vertex, then the intersection of these impact parameters
gives the location of the secondary vertex.\footnote{The fact that
  there are three intersections means that we have an indication of
  the quality of the vertex reconstruction in an individual event. This
  information could, in principle, be fed into the likelihood function
that we shall use to account for detector response, but we do not do
so here.}

Now, the location of the secondary vertex tells us the direction of
the $\tau$ momentum; the mass-shell constraint for the
$\tau$ then allows to reconstruct the magnitude of the $\tau$
momentum, up to a possible two-fold ambiguity. Given the measured
missing transverse momentum, we are then able to reconstruct the
momentum of the other $\tau$ (up to a further possible two-fold ambiguity), provided we know the invariant mass of
the neutrino(s) produced in the other $\tau$ decay.

The reconstruction process just described can only be expected to work
if things are well measured. For example, if they are not, we may end
up with no real solutions to the kinematic constraints.
We account for this by
defining an {\em ad hoc} likelihood function in which we convolute
the observed quantities with a function parameterizing the detector
response. The maximum of this likelihood function is an event
observable (albeit one with an obscure definition) and it is this
observable that we propose to use for signal discrimination.
The fact that we invoke a likelihood function also allows to deal with
the unknown invariant mass of the two neutrinos produced in a leptonic
$\tau$ decay: we marginalize
with respect to the unknown invariant mass,
including the matrix element for the $\tau$ decay.

Yet another advantage of focussing on 3-prong decays is that the fake backgrounds (coming from, e.g. W+ jets and QCD) will be reduced, as jets and other leptons are presumably less likely to fake a 3-prong decay (with a reconstructed secondary vertex) than they are to fake a generic hadronic tau or leptonic tau decay.\footnote{
Unfortunately, it is not possible for us to reliably
estimate the size of this effect, since neither the full details of the
experimental $\tau$ reconstruction algorithm nor the resulting
efficiencies or fake rates for 3-prong decays are public.}

The outline of the paper is as follows. In the next Section, we describe the algebraic details of the
reconstruction procedure. In Section \ref{sec:results}, we present the
results of our numerical simulations and in Section
\ref{sec:conclusions}, we draw our conclusions.
\section{The method \label{sec:method}}
As described in the introduction, events are reconstructed using the decay
vertex information.  If one of the $\tau$ leptons decays to a 3-prong
hadronic system and a $\tau$ neutrino, then the displacement $\mathbf{r}$ from the primary interaction point to the $\tau$ decay vertex
should be measurable with useful precision.\footnote{The ATLAS
CSC~\cite{Aad:2009wy} estimates a resolution of $0.6$ and $0.01$ mm in the
directions parallel and perpendicular to the displacement,
respectively.}

Let us begin by considering the limit of perfect detector resolution. Denoting the energy, momentum and mass of the 3-prong decay
products by $E_j$, $\mathbf{p}_j$ and $m_j$, respectively, and the angle
between $\mathbf{r}$ and $\mathbf{p}_j$ by $\theta$,  the momentum of that
$\tau$ lepton can be reconstructed, with a twofold ambiguity,
as $\mathbf{p}_\tau = p_\tau\mathbf{r}/|\mathbf{r}|$, where (neglecting the neutrino mass)
\begin{equation} \label{eq:reconptau}
p_\tau=\frac{(m_\tau^2+m_j^2)p_j\cos\theta\pm E_j
\sqrt{(m_\tau^2-m_j^2)^2-4m_\tau^2p_j^2\sin^2\theta}}
{2(m_j^2+p_j^2\sin^2\theta)}\,.
\end{equation}

The other $\tau$ lepton may decay either hadronically or leptonically,
into a visible system $j'$ (a hadronic jet or a charged lepton) and an
invisible system $i'$ (a $\tau$ neutrino or a pair of neutrinos).  The
transverse momentum of the invisible system is found from the missing
transverse momentum, $\slashed{\vp}_T$, and the reconstructed momentum
of the first $\tau$ via
\begin{equation}
\mathbf{p}_{Ti'} = \slashed{\vp}_T+\mathbf{p}_{Tj}-\mathbf{p}_{T\tau}\,.
\label{eq:pmiss}
\end{equation}
Given the invariant mass of the invisible system, $m_{i'}=m_\nu=0$ for
a hadronic and $m_{i'}=m_{\nu\nu}\geq 0$ for a leptonic decay, one can then solve for
the invisible longitudinal momentum, again with a twofold ambiguity:
\begin{equation} \label{eq:reconpl}
p_{L {i'}} = \frac 1{\mu_{j'}}\left(\alpha\,p_{Lj'}\pm E_{j'}
\sqrt{\alpha^2-\mu_{i'} \mu_{j'}}\right)\,,
\end{equation}
where
\begin{equation}
\mu_{i'} = p_{T {i'}}^2+m_{i'}^2\,,\;\;
\mu_{j'} = p_{Tj'}^2+m_{j'}^2\,,\;\;
\alpha=\frac 12 (m_\tau^2-m_{j'}^2-m_{i'}^2)  +  \mathbf{p}_{T {i'}}\cdot\mathbf{p}_{Tj'}\,.
\end{equation}
The momentum of the second $\tau$ can now be reconstructed as
$\mathbf{p}_{\tau'} = \mathbf{p}_{i'}+\mathbf{p}_{j'}$, and hence the invariant mass of the
$\tau\tau$ system, $m_{\tau \tau}$ is determined, up to a fourfold ambiguity.

Now consider a real detector and let
$\mathbf{q} =(\mathbf{r},E_j,\mathbf{p}_j,\mathbf{p}_j',\slashed{\vp}_T)$ correspond to the measured
quantities. These do not coincide with
their true values in an event, which we now denote by $\tilde\mathbf{q}$, but rather are shifted by amounts depending on the
detector resolution, which
we describe by a response function,
$f(\mathbf{q},\tilde\mathbf{q})$.  Then the
likelihood, as a function of the true invariant mass
$\tilde{m}_{\tau \tau}$, for an event with measured
quantities $\mathbf{q}$, may be written as
\begin{equation}\label{eq:like}
{\cal L}(\tilde{m}_{\tau \tau}|\mathbf{q}) = \int
d\tilde\mathbf{q}\,f(\mathbf{q},\tilde\mathbf{q})\,{\cal
  M}(\tilde\mathbf{q})\,\delta[\tilde{m}_{\tau \tau} - m_{\tau \tau} (\tilde\mathbf{q})]
\end{equation}
where ${\cal M}(\tilde\mathbf{q})$ is the matrix-element squared for the decay and
$m_{\tau \tau} (\tilde\mathbf{q})$ is the invariant $\tau \tau$ mass
reconstructed from the true quantities $\tilde\mathbf{q}$ according to the
recipe described above.
Here ${\cal M}(\tilde\mathbf{q})$ should also include the jacobian factor relating the
final-state phase space to the quantities $\tilde\mathbf{q}$. We find, in
most cases, that including these effects gives, at best, a marginal
improvement in the mass resolution. Indeed, some effects
(such as
the exponential distribution of the $\tau$-decay lifetimes),
lead to large fluctuations in the likelihood integrand and hence to
large errors in the numerical integration, worsening the mass
resolution.
Thus we do not include these effects, in general.

There is, however, one such effect which we do include. In the case of
leptonic decay of the second $\tau$,  the matrix elements also depend
on the momenta of the two invisible neutrinos, and the right-hand side
of eq.~(\ref{eq:like}) should include an integration over their phase
space, weighted by the expected distribution of the  $\nu\nu$
invariant mass. This is conveniently expressed as
 $P(m_{\nu\nu}^2)\,d\Phi_{\nu\nu}$, with
\begin{equation}
d\Phi_{\nu\nu} = \frac{d^3\mathbf{p}_{\nu\nu}}{2(2\pi)^3E_{\nu\nu}}
\frac{d\Omega^*}{(4\pi)^3}dm_{\nu\nu}^2\,,
\end{equation}
where $d\Omega^*$ is the element of solid angle in the $\nu\nu$
centre-of-mass frame and
\begin{equation}
P(m_{\nu\nu}^2) = \frac{2}{m_\tau^2}
\left(1-\frac{m_{\nu\nu}^2}{m_\tau^2}\right)^2
\left(1+2\frac{m_{\nu\nu}^2}{m_\tau^2}\right)\,.
\end{equation}
At each phase-space point,
the value of $m_{\nu\nu}$ is then used, with this weight, for the reconstruction of the decay.

In the integral over the delta function in eq.~(\ref{eq:like}),
we include all real
solutions to the equation $\tilde{m}_{\tau \tau} = m_{\tau \tau} (\tilde\mathbf{q})$. In contrast to
\cite{Gripaios:2011jm}, complex solutions should be discarded here,
since they cannot correspond to true values for genuine $\tau \tau$
resonance events.
Note, however, that measured values $\mathbf{q}$ that would correspond to
complex values of $m_{\tau \tau}$ lying close to the real axis if
reconstructed directly, which correspond to real solutions shifted
slightly by detector resolution, will be included in the integration
at neighbouring values of $\tilde\mathbf{q}$. For fake backgrounds, we often
find that no nearby values of $\tilde\mathbf{q}$ lead to real solutions,
allowing the event to be rejected.

We perform the integrations in eq.~(\ref{eq:like}) by a Monte Carlo
method similar to that adopted in \cite{Kawagoe:2004rz}, generating
a large number of points $\tilde\mathbf{q}$ distributed around each measured
point $\mathbf{q}$ according to a smearing function $f(\mathbf{q},\tilde\mathbf{q})$ deduced
from detector simulations.
The jet masses are generated according to certain probabilities that
we describe in the Appendix.
Each real solution $\tilde{m}_{\tau \tau} = m_{\tau \tau} (\tilde\mathbf{q})$ is
entered into a histogram with the corresponding weight.
Because the Monte Carlo method generates
only a finite number of points, all histogram bins are given a small
positive offset, to avoid multiplications by zero.

Since our likelihood function does not encode the matrix element in
its entirety, we cannot expect to be able to make statistical
inferences directly from it in the usual way. Doing so might lead, for
example, to us wrongly rejecting the Standard Model Higgs boson
hypothesis, or obtaining a biased measurement of its mass.
Instead, we use our {\em ad hoc} likelihood function to define an
event observable in the following way: for each event,
we extract the smallest value of $\tilde{m}_{\tau \tau}$ that gives a
local maximum of the event likelihood and define this to be the
event value of the observable $m_{\mathrm{SV}}$.\footnote{Since we
  have to solve a quartic equation, and since
  real roots thereof come in pairs, we invariably find multiple local maxima
  in the event likelihood.} Our simulations suggest that this
observable gives distributions for Higgs and $Z$ boson event samples
 whose peak locations provide

 a good
 determination of the
 corresponding boson mass, with small tails. In any case,
 the presence of such effects can be mitigated by comparing
 experimental distributions of
 observables to template Monte-Carlo samples, as we do in simulations
 of pseudo-experiments below.

\section{Simulations and results \label{sec:results}}
Our study is based on a sample from the
{\tt Herwig++} event generator~\cite{Bahr:2008pv},
version 2.52~\cite{Gieseke:2011na} and corresponds to an integrated
luminosity of 20 fb$^{-1}$,
which is very similar to what has been achieved at the LHC
by the end of 2012.

As regards the signal, recent Standard Model predictions for Higgs production at the LHC may
be found in Ref.~\cite{Dittmaier:2011ti}.   For a Higgs mass of 125
GeV, at a collision energy of 8 TeV, the expected total
cross section is 19.52 pb in the gluon fusion channel and 1.58 pb in
the vector boson fusion channel, with a probable uncertainty of around 10\%.
The predicted SM branching ratio for $\tau\tau$ decay, also given in
Ref.~\cite{Dittmaier:2011ti}, is 6.4\%.
Higgs production followed by $\tau\tau$
decay thus
corresponds to a cross section of 1.34 pb.

For the $Z$ background, CMS \cite{CMS10} reports a flavour-averaged
prediction of $\sigma (pp \rightarrow Z \rightarrow ll) = 1.13$ nb and
a measurement of 1.12 nb for the 8 TeV LHC. We take $\sigma
(pp \rightarrow Z \rightarrow \tau \tau ) = 1.13$ nb. For the $W+j$
background, we use  $\sigma (pp \rightarrow W j) = 2.15$ nb, taken from
{\tt Herwig++}.

All of our simulations are carried out at the parton level, without
showering or hadronization effects,
apart from hadronic tau decays which we simulate using
{\tt Herwig++}~\cite{Grellscheid:2007tt}.
The detector response was modelled
as follows.

Firstly we assume identification efficiencies of 0.4 and 0.3 for the 1- and 3-prong hadronic taus, respectively \cite{Aad:2011kt}.
The lepton identification efficiency is assumed to be 0.9 in our analysis.
To estimate the number of $W+j$ events in which the jet mimics a 1- or 3-prong hadronic tau, we use the fake rates of 0.01 and 0.002 for 1- and 3-prong taus, respectively.
These numbers are obtained from the simulation of $W + j$ events \cite{Chatrchyan:2012zz}.

Secondly, we parameterize detector mismeasurements by smearing the energy component of jets and leptons
with $\sigma(E)/E_j = 0.5\,{\rm GeV}^{\frac{1}{2}} / \sqrt{E_j}$ and  $\sigma(E)/E_\ell = 0.05\,{\rm GeV}^{\frac{1}{2}} / \sqrt{E_\ell}$,
respectively.
For the missing transverse momentum, we smear each component with $\sigma_x = \sigma_y = 5$\,GeV.

For the $\tau$ decay vertex, the Monte-Carlo truth position is smeared
by Gaussian distributions of widths $0.613 \pm 0.008 $ mm and
$10.5 \pm 0.2 \,\mu$m, in the directions parallel and perpendicular to the
3-prong tau-jet, respectively. For jets that fake taus, we take
the truth vertex position to be zero and then smear as above.

We then apply event selection cuts to purify the signal, for which we impose
\begin{equation}
p_{Tj} > 20\,{\rm GeV},~~~~
p_{Tj'} > 20\,{\rm GeV},~~~~
\slashed{p}_T > 20\,{\rm GeV}.
\end{equation}
For the hadron-lepton mode, we further impose
\begin{equation}
m_T \equiv \sqrt{ 2 |{\bf p}^{\ell}_T| |{\slashed{\vp}_T}| (1 - \cos\Delta \phi)  } < 40\,{\rm GeV}
\end{equation}
to reduce the background involving $W$s, where $\Delta \phi$ is the azimuthal difference between the lepton and the missing transverse momentum.
The cross section for each process/channel after taking account of the efficiencies of (mis)identification and the selection cuts
is listed in Table \ref{tab:xsec}.
We use the 3 prong-hadron\,(-lepton) channel for $m_{\mathrm SV}$ and
the hadron-hadron\,(-lepton) channel for $m_{\mathrm vis}$ and $m_{\mathrm eff}$.

\begin{table}[t!]
\begin{center}
\begin{tabular}{|c||c|c||c|c|}
\hline
& 3pr-had & 3pr-lep & had-had & had-lep \\  \hline
$H$ & 6.3 & 5.7 & 17.2 & 28.5 \\ \hline
$Z$ & 528.8 & 416.5 & 1451.4 & 2036.0 \\ \hline
$Wj$ & 50.5 & 32.4 & 135.3 & 161.9 \\ \hline
\end{tabular}
\caption{ The cross section times efficiency of each process/channel in fb.}
\label{tab:xsec}
\end{center}
\end{table}

In Figure \ref{fig:distros} we compare the signal and background
distributions of our variable $m_{\mathrm{SV}}$ with the existing
variables $m_{\mathrm{eff}}$ and
$m_{\mathrm{vis}}$. $m_{\mathrm{vis}}$ is simply the invariant mass of
the visible products of both $\tau$ decays, while $m_{\mathrm{eff}}$
includes the missing transverse momentum (an explicit definition may
be found in \cite{Barr:2011he}).
We show results for the lepton-hadron modes and hadron-hadron modes
separately.
\begin{figure}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/msv_comp_lep.eps}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/msv_comp_had.eps}
\end{minipage}

\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/mvis_comp_lep.eps}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/mvis_comp_had.eps}
\end{minipage}

\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/meff_comp_lep.eps}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/meff_comp_had.eps}
\end{minipage}
\caption{Signal and background distributions of $m_{\mathrm{SV}}$
  (top), $m_{\mathrm{vis}}$ (centre) and  $m_{\mathrm{eff}}$ (bottom), for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC, for $m_h =125$
  GeV. Left: lepton-hadron modes; right: hadron-hadron modes. \label{fig:distros} }
\end{figure}
The better separation between signal and
backgrounds that we expected to obtain using $m_{\mathrm{SV}}$ is
clear to see in the Figure. Indeed, $m_{\mathrm{vis}}$ has
distributions for the Higgs signal and $Z$ background which are
strongly peaked, but the two peaks sit on top of each
other. $m_{\mathrm{eff}}$
incorporates extra information in the form of the missing transverse
momentum, and slightly increases the separation between the maxima of
the Higgs and $Z$ boson peaks, but at the cost of introducing large
tails (from the smearing of the missing transverse momentum
measurement). As a result, the Higgs signal is easily hidden in the
large tail of the $Z$ background. Moreover, the shapes of all
components become similar, making discrimination difficult when the
overall normalizations are uncertain.
In contrast $m_\mathrm{SV}$ provides
a good separation between the narrow Higgs and $Z$ boson peaks, which appear at the true mass values.
These
peaks are, furthermore, very different in shape from the continuum $W+$
jet background.

To see how well each variable can reconstruct the resonance, we list
the peak location of the each variable's distribution and the input mass of the resonance in
Tables~\ref{tab:bias_had} and \ref{tab:bias_lep} in the hadron-hadron and hadron-lepton modes, respectively.
The pure $Z \to \tau \tau$ and $h \to \tau \tau$ samples with several Higgs masses are used.
The peak location is calculated as the weighed average of the three highest bins.
The error is estimated by using 10 independent samples.
The tables show clearly that $m_\mathrm{SV}$ reconstructs the masses of resonances very well compared to the other variables.  Although this correlation is not the basis of our method  for mass determination, it helps to separate the signal from the background.

Another remark is that the $S/B$ for the $W+$ jet background is better for $m_\mathrm{SV}$,
as can be seen from Fig.~\ref{fig:distros}.
This is because a fraction of the $W+$ jet background events do not produce real solutions.
The fact that the tau mass is much smaller than the typical momentum scale of the reconstructed objects
(jets and leptons) implies that the neutrino momenta are inferred to be very close
to those objects (see e.g. Eq.~(\ref{eq:reconptau})).
However the direction of this inferred momentum tends to conflict with the direction of the observed missing transverse momentum, since the neutrino from the $W$ decay is generally not collimated with respect to those objects, leading to no real solution.

Nevertheless, it is also apparent that the statistics available using
$m_\mathrm{SV}$ are lower than for the other variables, even in the region of maximum signal.
We need, therefore, to make a quantitative comparison of the three variables. To
do so,
we generate distributions of them (for $m_h = 125$ GeV signal and backgrounds) in
ten pseudo-experiments, each corresponding to an integrated
luminosity of 20 fb$^{-1}$ of 8 TeV LHC
data. Each pseudo-experiment is then compared to template model distributions
with different values of $m_h$ and with different normalization
factors $f_h$, $f_W$, and $f_Z$ for the Higgs signal and $W+j$ and $Z$
backgrounds, respectively.
(The values  $f_{h,W,Z} =1$ correspond to the leading order Monte-Carlo prediction.)
Allowing the model distribution normalizations to float in this way allows us not only to take into
account some of the most important systematic effects\footnote{
We assume uniform probability distribution functions for $f_{h,W,Z}$ in the likelihood calculation.
In the actual experimental situation, these probabilities are not uniform but localised around $f_{h,W,Z} = 1$
to avoid too large/small values.
In this sense, our treatment of the systematic uncertainty is conservative.
}
(arising from the uncertainties in the luminosity, the Monte-Carlo predictions, and data-driven extrapolations), but also provides a means to measure the
cross-section times branching ratio for Higgs production followed by
decay to $\tau \tau$.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $Z$(91) & $h$(119) & $h$(125) & $h$(131) \\ \hline
$m_{\rm SV}$ & $92.10 \pm 0.01$ & $117.81 \pm 0.73$ & $124.75 \pm 0.75$ & $130.84 \pm 1.00$  \\ \hline
$m_{\rm eff}$  & $89.97 \pm 0.02$ & $110.55 \pm 1.09$ & $115.12 \pm 1.19$ & $115.12 \pm 1.19$ \\ \hline
$m_{\rm vis}$  & $60.27 \pm 0.01$ & $73.87 \pm 0.84$ & $77.82 \pm 1.05$ & $78.42 +- 0.98$  \\ \hline
\end{tabular}
\caption{ Peak position vs.~input mass in the hadron-hadron mode.
The numbers in the parentheses are the input mass of the bosons.
Pure background and signal samples are used.
\label{tab:bias_had} }
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $Z$(91) & $h$(119) & $h$(125) & $h$(131) \\ \hline
$m_{\rm SV}$ & $91.54 \pm 0.40$ & $116.12 \pm 0.82$ & $122.15 \pm 0.84$ & $129.33 \pm 0.68$  \\ \hline
$m_{\rm eff}$  & $89.90 \pm 0.02$ & $108.64 \pm 1.12$ & $113.27 \pm 0.96$ & $116.38 \pm 1.20$ \\ \hline
$m_{\rm vis}$  & $59.87 \pm 0.01$ & $70.82 \pm 0.83$ & $73.59 \pm 0.77$ & $75.52 \pm 1.28$  \\ \hline
\end{tabular}
\caption{
Peak position vs.~input mass in the hadron-lepton mode.
The numbers in the parentheses are the input mass of the bosons.
Pure background and signal samples are used.
\label{tab:bias_lep} }
\end{center}
\label{default}
\end{table}%

We make a cut on the observable of interest itself, so as
to maximize its discovery potential. Roughly
speaking, this cut selects the region that contains the bulk of
the signal events for that observable. Thus we choose
$m_\mathrm{SV} \in  [110, 150]$ GeV,
$m_\mathrm{vis} \in  [75, 110]$ GeV, and
$m_\mathrm{eff} \in  [100, 150]$ GeV.

We then compute, for
each pseudo-experiment, a binned Poisson log-likelihood, $\log \mathcal{L}
= \Sigma_i \log
\mathcal{L}^P\left( n^\mathrm{trial}_i; x_i \right)$,
where $n^\mathrm{trial}_i$ is the number of events observed in
histogram bin $i$ in a given pseudo-experiment and $x_i = x_i (m_h,f_h, f_Z, f_W)$ is the number of events expected in a
given model, parameterized by $m_h,f_h, f_Z, f_W$.

To assess the discovery potential, we then compute the difference in
log-likelihood between models with and without a Higgs signal. In the
model without a signal, we maximize the log-likelihood with respect to
$f_Z,$ and $f_W$, whereas in the model with a signal, we additionally
maximize with respect to $m_h$ and $f_h$. In Fig.~\ref{fig:discovery},
we show $-2$ times the difference in log-likelihood for the three
variables. The centre of each bar shows the mean value over trials,
while the width of each bar gives the root-mean-square deviation over
trials.
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_lep.eps}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_had.eps}
\end{minipage}
\caption{Discovery potential using $m_{\mathrm{SV}}$, compared to
  $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$, for lepton-hadron (left)
  and hadron-hadron (right) modes, for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC. The centre of each bar shows the mean value over trials,
while the width of each bar gives the root-mean-square deviation. \label{fig:discovery} }
\end{figure}
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_lep_fix.eps}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_had_fix.eps}
\end{minipage}
\caption{
  The same as Fig.~2, however $f$s are fixed at 1 in the likelihood calculation.
  }
  \label{fig:discovery_fix}
\end{figure}
\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_lep_w2.eps}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/logL_had_w2.eps}
\end{minipage}
\caption{
	The same as Fig.~2, however
  the analysis uses a $W+j$ background sample which is twice as large as the expected one.
  }
  \label{fig:discovery_w2}
\end{figure}

We observe a significant improvement using $m_\mathrm{SV}$ in the
hadron-hadron channel, along with a more modest improvement (compared
to $m_\mathrm{vis}$) in the leptonic channel.
The different performances of $m_\mathrm{vis}$ in hadron-lepton and hadron-hadron modes can be understood from the distributions in Fig.~\ref{fig:distros}.
Unlike the hadron-lepton mode, the $m_\mathrm{vis}$ distribution in the hadron-hadron mode shows that both $Z$ and $W+j$ backgrounds as well as the signal have falling shapes in the signal region $m_\mathrm{vis} \in  [75, 110]$\,GeV.
This suggests that in the hadron-hadron mode, fitting the signal + background distribution with only the $Z$ and $W+j$ backgrounds by floating $f_Z$ and $f_W$ can be more possible compared to the hadron-lepton mode.
This feature can be seen explicitly in Fig.~\ref{fig:discovery_fix}, which show the same likelihoods as in Fig.~\ref{fig:discovery}
but with the $f$s fixed at 1.
Floating the $f$s brings significant degradation for $m_\mathrm{vis}$ in the hadron-hadron mode.

The absolute values of the discovery significance are
exaggerated, since we have neglected sub-dominant backgrounds and many
uncertainties, but the relative performance of the different variables
should be meaningful.

We have estimated the size of $W + j$ background using the reported tau fake rates in $W + j$ events.
The tau fake rate is generally dependent on the tau identification algorithm and the jet $p_T$.
To check the robustness of our result, in Fig.~\ref{fig:discovery_w2}
we show the same discovery potential plots as Fig.~\ref{fig:discovery}
but containing a $W + j$ background twice as large as the one in Fig.~\ref{fig:discovery}.
The discovery potentials are degraded slightly but the qualitative features are unchanged.

\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/mass_fit_lep.eps}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/mass_fit_had.eps}
\end{minipage}
\caption{The log-likelihood function near $m_h = 125$ GeV obtained
  using the $m_{\mathrm{SV}}$,
  $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$ distributions, for lepton-hadron (left)
  and hadron-hadron (right) modes, for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC.\label{fig:mass1} }
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c}
& $m_{\rm SV}$ & $m_{\rm vis}$ & $m_{\rm eff}$ \\ \hline
had-lep & 4.7 & 11.0 & 23.3 \\ \hline
had-had & 4.7 & 11.8 & 23.1  \\
\end{tabular}
\caption{Uncertainty in the Higgs boson mass (in GeV), as measured
  $m_{\rm SV}$, $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$ distributions, for lepton-hadron
  and hadron-hadron modes, for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC.\label{tab:mass2} }
\end{center}
\label{default}
\end{table}%
To assess the expected resolution in the Higgs mass measurement, we first show, in Fig.~\ref{fig:mass1}, the variation in
$-2\log\mathcal{L}$, averaged over pseudo-experiments, for values of the model Higgs mass, $m_h$,
in the neighbourhood of 125 GeV, after maximizing $\mathcal{L}$ with respect to
$f_h, f_Z$, and $f_W$. In the figure, we have placed the minimum of
$-2\log\mathcal{L}$
at height zero.
Because the pseudo-experimental data and templates are prepared in the same way, we cannot estimate any biases that might occur when each variable is used to determine the mass from real data. However, for each variable, we can estimate the precision of the mass measurement from a quadratic fit to the log-likelihood.
The resulting fractional uncertainties are shown in Table.~\ref{tab:mass2} and are seen to be much smaller for $m_{\mathrm{SV}}$ than for the other two observables, in both
hadron-hadron and lepton-hadron modes.
This is easily
explained by the fact that the Higgs signal peak is sharpest for $m_{\mathrm{SV}}$.
For the measurement of the product of the production cross section and
the branching ratio for the decay, the procedure is exactly analogous,
except that now we vary $f_h$, after maximizing with respect to
$m_h$. The resulting fit and fractional uncertainties are shown in
Fig.~\ref{fig:sigmabr1} and Table~\ref{tab:sigmabr}, respectively.
Again, there is always an improvement when using $m_{\mathrm{SV}}$.
For $m_{\mathrm{vis}}$, the difference in resolution between the hadron-lepton and hadron-hadron modes
can be blamed on different shapes in the $W+j$ background in these modes,
as we have discussed earlier in the connection with the different discovery potentials for $m_{\mathrm{vis}}$
between these modes.

\begin{figure}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[ width=0.9\textwidth]{eps_files/fs_fit_lep.eps}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=0.9\textwidth]{eps_files/fs_fit_had.eps}
\end{minipage}
\caption{The log-likelihood as a function of the signal strength $f_h$, obtained
  using the $m_{\mathrm{SV}}$,
  $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$ distributions, for lepton-hadron (left)
  and hadron-hadron (right) modes, for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC. $f_h=1$ corresponds to
  the truth value. \label{fig:sigmabr1} }
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c}
& $m_{\rm SV}$ & $m_{\rm vis}$ & $m_{\rm eff}$ \\ \hline
had-lep & 0.33 & 0.44 & 0.65 \\ \hline
had-had & 0.32 & 0.93 & 0.73  \\
\end{tabular}
\caption{Resolution for measurement of the production cross section times
 branching ratio
 for $pp \rightarrow h \rightarrow \tau \tau$ normalized by the leading order prediction
 using $m_{\mathrm{SV}}$, compared to
  $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$, for lepton-hadron (left)
  and hadron-hadron (right) modes, for an integrated
  luminosity of 20 fb$^{-1}$ at the 8 TeV LHC.\label{tab:sigmabr}}
\end{center}
\label{default}
\end{table}%

\section{Conclusions  \label{sec:conclusions}}
As shown in Figs.~\ref{fig:discovery}-\ref{fig:sigmabr1} and Tables~\ref{tab:mass2}-\ref{tab:sigmabr}, our
simulations suggest that a significant improvement in discovery
potential, Higgs boson mass resolution, and measurement of production
cross section times branching ratio can be obtained by focussing on
3-prong $\tau$ decays. The performance of $m_{\mathrm{SV}}$  is
roughly comparable irrespective of whether the other $\tau$ meson decays leptonically or
hadronically, leading to greater gains in the hadron-hadron
channel, where the other observables perform more poorly.
One possible reason for this is that
the detector resolution is invariably poorer in this channel, in which final-state leptons are replaced by jets.
Thus the performance of $m_{\mathrm{vis}}$ and $m_{\mathrm{eff}}$ is degraded.
However, these events are also
fully reconstructible using the vertex information, in the absence of smearing. As a result,
the likelihood that
defines the observable
$m_{\mathrm{SV}}$ is able to correct for the extra smearing to a certain extent, by insisting that the
unsmeared quantites consistently reconstruct the event.

The gains are greatest for the mass measurement, which is perhaps not
surprising since our method provides a means to reconstruct the mass
whilst partially correcting for the uncertainties that are introduced by
the detector resolution.

The results of our simulations are encouraging, but they should be
taken with a pinch of salt. The simulations themselves are rudimentary, and we have only performed a comparison
with the basic variables $m_{\mathrm{eff}}$ and
$m_{\mathrm{vis}}$. Both collaborations now employ more sophisticated
likelihood-based analyses. Unfortunately the full details of these
have not been made public, so it is difficult for us to make a fair
comparison.
CMS do say that their likelihood method gives a Higgs mass resolution of around 21$\%$ compared to 24$\%$ using mvis \cite{Chatrchyan:2011nx}.

We have also not made a full study of the
backgrounds, of which many are relevant for this search. However, the two backgrounds we did consider are very different in their
nature (one being a genuine, resonant background and the other being a
fake, continuum background). We hope therefore, that the other
backgrounds will be similar to one or other of these in their
behaviour. The recent CMS results suggest, moreover, that our two
backgrounds are the dominant ones in the
signal region in most of the $\tau \tau$ sub-channels (along with pure
jets, which we expect to be similar to $W+$jets).

Finally, we have only considered the most obvious systematic effect,
namely the uncertainty associated with the normalization of the signal
and backgrounds. Nevertheless, our qualtitative argument that a better
mass reconstruction gives a better separation between the signal and the
different backgrounds, means that many systematic uncertainties are expected
to be reduced.

We hope, at least, that our qualitative arguments and quantitative
simulations are enough to convince the collaborations to explore the
suitability of this method.
Even if it then turns out that a significant improvement is not obtained using our method
alone, we remark that an overall improvement can still be expected if it is combined
with existing approaches. Our method is complementary and, as is clear
from Fig~\ref{fig:correlation}, the observable we extract is not
strongly correlated with the existing variables  $m_{\mathrm{eff}}$
and  $m_{\mathrm{vis}}$.
It thus provides independent information and may be used to increase
the significance of searches in the $\tau \tau$ channel.
\begin{figure}
\centering
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=0.95\textwidth]{eps_files/msv_vs_mvis_lep.eps}
\includegraphics[width=0.95\textwidth]{eps_files/msv_vs_meff_lep.eps}
\includegraphics[width=0.95\textwidth]{eps_files/mvis_vs_meff_lep.eps}
\end{minipage}
\hspace{3mm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=0.95\textwidth]{eps_files/msv_vs_mvis_had.eps}
\includegraphics[width=0.95\textwidth]{eps_files/msv_vs_meff_had.eps}
\includegraphics[width=0.95\textwidth]{eps_files/mvis_vs_meff_had.eps}
\end{minipage}
\caption{Correlation between $m_{\mathrm{SV}}$,
  $m_{\mathrm{eff}}$ and  $m_{\mathrm{vis}}$, for the Higgs signal, for lepton-hadron (left)
  and hadron-hadron (right) modes.\label{fig:correlation}}
\end{figure}
Our simulations apply to the current 8 TeV run and our hope is that
application of our method to data being produced now will allow us to
clear up the mystery of the observed deficit of $h \rightarrow \tau
\tau$ decays. Nevertheless, we expect our
method to become even more relevant in the subsequent stages of the
LHC programme. For one thing, the method is limited by statistics,
but gains in reducing systematic uncertainties. It is the
latter that will
limit our ultimate ability to make precision measurements of the Higgs
sector at the LHC. What is more, both
ATLAS and CMS are planning upgrades or replacements of their vertex detectors, with an
improvement of a factor of a few expected in the vertex
resolution. The associated improvement in the $\tau \tau$ mass
reconstruction using our method should reduce the uncertainties even further.

\section*{Acknowledgments}
BMG thanks A.~Barr, M.~Klute,  M.~Mulders, A.~de Roeck and the
Cambridge SUSY working group for discussions. He also acknowledges the
support of King's College, Cambridge and thanks Perimeter Institute
and the CERN Theory Group for hospitality.
This work is in part supported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan (Nos. 22540300, 23104005 for MN) and World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
KS thanks K. Rolbiecki for helpful discussions.
BW acknowledges the support of a Leverhulme Trust Emeritus Fellowship,
and thanks IPMU, the CERN Theory Group, the CCPP at New York
University, the Galileo Galilei Institute and the Pauli Institute at ETH/University of Zurich
for hospitality.  He also thanks the INFN and the Pauli Institute for support during
parts of this work.


\section{Introduction}

Observations of stars in the solar neighbourhood show an increase of the
velocity dispersion (the random motion) proportional
to $t^{0.3\dots 0.6}$, where $t$ is the stellar age (Wielen 1977; Holmberg, Nordstr\"om \& Andersen 2007).
This effect can be interpreted as a
heating of the Galactic disc with time.
However, it is unclear what is the cause of this heating. Several mechanisms have been proposed:
Massive black holes in the galactic halo as a source of heating were
investigated by \cite{lo85}, \cite{wf90}, \cite{rl93} and \cite{hf02},
but seem to be
excluded by observations. \cite{atb03} considered
massive clumps of dark matter as a possible heating agent. Giant molecular
clouds were found to cause mostly vertical heating by \cite{l84} and later
\cite{hf02}. \cite{cs85} and \cite{c87} show that transient spiral waves
heat efficiently in the Galactic plane. Lastly, \cite{qhf93} and \cite{vw99} simulate
merging with small satellites and find both radial and vertical heating. This
last mechanism is discussed in this paper. It should be mentioned, however, that
radial mixing may influence the age-velocity dispersion relation significantly
(Sch\"onrich \& Binney 2009; Sellwood \& Binney 2001; Ro\v{s}kar et al. 2008).

The merging of galaxies can be divided into three categories according to their mass ratios.
Major mergers with mass ratios 1:1 to 1:3 of the total galaxy masses usually destroy discs
and result in an early type remnant \citep[][and references therein]{njb06}. Minor mergers
with mass ratios 1:3 to 1:10 of the total galaxy masses destroy a thin disc \citep{pbk10}.
The minor merger events are discussed in the context of the formation of thick discs
\citep{vh08,vh09} and substructure in the halo \citep{n02,pbk10}. The third category is the merging
of satellite galaxies (or dark matter clumps) with
smaller mass ratios. These merger or interaction events can be characterized as a
perturbation of the primary galaxy and it is more convenient to quantify the mass ratio in units
of the disc mass of the primary. In this context the survival of a thin disc over 10 Gyr in a
$\Lambda\rmn{CDM}$ universe lead to serious constraints on the clumpiness of the DM halo \citep{atb03}.
\cite{h08} compared the efficiency of the disc by
satellite galaxy mergers of different authors and argued that it scales with the square of the
satellite mass. This is consistent with the interpretation heating by dynamical friction during
each disc crossing event. As a consequence, the upper mass end of the perturbers are most
efficient and determine the survival of a thin disc. Kazantzidis et al. (2008, 2009) performed a
comprehensive study of the impact of DM subhaloes on a Milky Way like disc galaxy in a
realistic cosmological context. They investigated the resulting substructures and kinematic
features in disc and halo.

In numerical simulations the long-term (or secular) dynamical evolution of a thin galactic disc
is very sensitive to the spatial resolution, the level of numerical noise, and the dynamical
feedback with the DM halo. A live dark matter halo is important for two reasons. \cite{vw99}
have shown that the reaction of the disc on the perturbation of a satellite
galaxy is significantly higher with a live halo compared to a rigid dark matter potential.
Secondly, dynamical friction in the dark matter is important for the orbital evolution and thus
energy loss of satellites with masses exceeding $\sim 10^8~\msun$. This is exactly the mass range
dominating the heating rate of the disc by satellite galaxies. Additionally, the mass of the dark
matter particle must not be too large in order to avoid numerical heating of the disc. In
most simulations the disc is represented by less than 1 million particles leading to a significant
thickening of the disc by numerical two-body relaxation, which is much stronger in direct $N$-body
codes and Tree codes compared to particle mesh-codes. In \cite{vw99} and in \cite{hc06}
the numerical heating of the unperturbed disc was
shown explicitly. \cite{vw99} used a differential method to quantify the
dynamical heating of infalling satellites. This may underestimate the heating rate, since a
more realistic thinner, dynamically cooler, disc is more sensitive to perturbations. On the other
hand, the feedback on the satellite galaxy may be larger by the stronger disc shocking event.
Particle-mesh codes are predestined for the simulation of collisionless systems such as
galaxies, because two-body relaxation is strongly suppressed by the partial decoupling of the
point-like structure in orbital motion of the particles and the gravitational potential based on
the grid. \cite{k07} have shown that the growth of global mode perturbations
are easily damped out by noise due to an insufficient particle number. If the excitation of spiral
structure or bar-like perturbations are an important mechanism for the energy transfer from
the satellite to the disc, then a very high resolution well above 1 million particles is needed to
reproduce a realistic heating rate.

Improving the resolution of discs in vertical direction has always been
a challenge.
There is, indeed, a further problem which is inherent in any numerical
technique that is used for the simulation of disc galaxies. Stellar discs
have relaxation times larger than a Hubble time, i.\,e. they are collisionless
systems. This implies that in simulations the thickness of an unperturbed
disc indicates how well the code would model a collisionless gravitating
system.
The term ``unperturbed'' denotes the absence of perturbations such as
stellar bars, spiral arms, or molecular clouds.
One often notes, in simulations, that discs become thicker with time.
This is caused
by the graininess of the particle distribution and the limited spatial
resolution. We refer to this effect as ``numerical heating''.

For two decades {\sc Superbox} is used as a successful tool to simulate
the dynamics of isolated and interacting galaxies.
A first description was given by Bien, Fuchs \& Wielen (1991),
together with a consideration of the direct $N$-body technique and a
tree-code. For a more detailed discussion the reader is referred to \cite{f00}.
We also mention Fellhauer's lecture notes \citep{fln08}.
The code was evolving from the conventional particle-mesh technique
\citep[see, e.\,g.,][]{he88}
by applying grids and sub-grids in regions of high particle
density. The advantage of {\sc Superbox} is that the code can run on any
workstation or PC, giving reliable results. The code thus proves to be a
serious alternative to direct $N$-body schemes and tree-codes.

Initially, \cite{mb93} applied {\sc Superbox} to compare
numerical experiments to observations of the high-velocity encounter of
NGC 4782/4783. They found a convincing description of the morphological and
kinematical structure, and estimated the time elapsed since the closest
approach of the two interacting galaxies. This was the beginning
of a series of research projects using {\sc Superbox} as a tool. We note
as examples the dynamical evolution of a satellite ga\-la\-xy
\citep{kk98}, the decay of dwarf ga\-la\-xies in dark
matter haloes \citep{p02}, and dynamical friction, compare
\cite{p04} and \cite{jp05}.  It should be mentioned that
\cite{s03} studied the inspiral of a black hole to the Galactic
centre, and \cite{k07} considered unstable modes in a
disc. \cite{f08} published on the dynamics of the
Bootes dwarf galaxy. \cite{p08} investigated the tidal
evolution of dwarfs of the Local Group. Most recently, a study was
made on the dynamical friction of massive objects in galactic centres by \cite{j11}.
Our list is by far not complete.

So far, {\sc Superbox} was not able to adapt the sub-grids to discs.
A higher resolution in vertical direction can be
achieved by introducing flattened grids, see \cite{b08}. This
new feature and the application to dynamical heating of galactic
discs are the main topics of the present paper. The improved code
is called {\sc Superbox-10}.
As {\sc Superbox}, {\sc Superbox-10} is rather free of numerical heating.

\begin{figure}
\vspace{0.0cm}
~
\hspace{-0.80cm}
\includegraphics[width=80.00mm,height=73.06mm]{Fig1d.ps}
~
\vspace{0.25cm}
\caption{Projection of a simplified galaxy model in a 3D box. Shown are the basic parameters $R_{\rmn{core}}$,
$R_{\rmn{out}}$ and $R_{\rmn{sys}}$.}
\label{label1}
\end{figure}

Below we first describe {\sc Superbox} in detail and then extend to {\sc Superbox-10}.
It follows a discussion of the parallelised version which results in a significant speed-up.
Then our test of improved vertical resolution is described. As an important
application we consider disc heating by satellite galaxies. In particular, we address
the ratio of radial and vertical velocity dispersion and the transfer of energy and
angular momentum. Special attention is given to the interaction of the infalling
satellite with the dark matter halo.

  In a conference proceedings publication \citep{b08} we presented preliminary results
  on an improved version of {\sc Superbox} with increased vertical resolution of the disc:
Negligible disc thickening of an isolated disc after 2.5 Gyr. The present paper should be seen
as a pilot study for high resolution simulations of disc heating by satellite galaxy mergers.
Two new aspects are presented: (1) No correction for numerical heating is needed and we
can measure the heating rate directly. Dynamical friction in the dark matter halo is included
automatically. (2) We analyse for the first time additionally the transfer of angular momentum.
This is important for understanding the physical processes responsible for the disc heating.
For this pilot study we select a satellite with a relatively high mass on eccentric orbits, which is
destroyed after a few disc crossing events.

\section[]{A review of {\sc Superbox}}

\subsection{Basic concepts}

The conventional particle-mesh technique considers a set of massive particles,
often called ``superstars'', in a three-dimensional Cartesian grid (``box'') consisting
of $N \times N \times N$ cells which represents a relevant part of the universe.
In this section, the length of each cell is the unity.
We suppose that $\rho_{a,b,c}$ is the mean density in the cell with indices
$a$, $b$, $c$ which are integers running from 0 to $N-1$. Then the potential
can be obtained by solving Poisson's equation
\begin{equation}\label{eq:poisson}
  \Phi_{a,b,c} = G \sum_{\xi = 0}^{N-1} \sum_{\eta = 0}^{N-1} \sum_{\zeta = 0}^{N-1}
\rho_{\xi,\eta,\zeta} H_{\xi-a,\eta-b,\zeta-c}
\end{equation}
Here, $H_{\xi-a,\eta-b,\zeta-c}$ (i.\,e. Green's function) is the inverse distance
between the points having indices $a$, $b$, $c$ and $\xi$, $\eta$, $\zeta$, namely
\begin{equation}
\label{eqn:greens}
  H_{\xi-a,\eta-b,\zeta-c} = [(\xi-a)^2 + (\eta-b)^2 + (\zeta-c)^2]^{-1/2}
\end{equation}
We note that the term $G \rho_{\xi,\eta,\zeta} H_{\xi-a,\eta-b,\zeta-c}$
accounts for the contribution of the cell $\xi$, $\eta$, $\zeta$ to the
cell $a$, $b$, $c$. Thus $\Phi_{a,b,c}$ gives the correct potential
assigned to cell $a$, $b$, $c$. The evaluation of all $N \times N \times N$ values
$\Phi_{a,b,c}$ takes a time proportional to the number of cells squared,
$(N \times N \times N)^2$.

\begin{figure}
\vspace{0.0cm}
~
\hspace{-1.0cm}
\includegraphics[width=90.00mm,height=82.19mm]{Fig2f.ps}
~
\vspace{0.25cm}
\caption{The five grids of {\sc Superbox}. Grid 1 has the highest resolution
and resolves the core. Grid 1, grid 2 and grid 3 move through the local universe,
defined by grid 4 and grid 5.}
\label{label2}
\end{figure}

The direct convolution can be replaced by discrete Fourier transforms where $N$
is supposed to be a power of 2. First $\rho_{a,b,c}$ and $H_{a,b,c}$ are
transformed, resulting in
\begin{equation}
  \hat{\rho}_{k,l,m} = \sum_{a = 0}^{N-1} \sum_{b = 0}^{N-1} \sum_{c = 0}^{N-1}
  \rho_{a,b,c} \exp[-i{2\pi \over N}(ka+lb+mc)]
\end{equation}
and $\hat{H}_{k,l,m}$, analogously. The potential is then obtained by the
inverse transformation of the product $\hat{\rho}_{k,l,m} \hat{H}_{k,l,m}$,
\begin{eqnarray}
\Phi_{a,b,c} = \nonumber\\
{G \over N^3} \sum_{k = 0}^{N-1} \sum_{l = 0}^{N-1} \sum_{m = 0}^{N-1}
\hat{\rho}_{k,l,m} \hat{H}_{k,l,m} \exp[+i{2\pi \over N}(ka+lb+mc)]
\end{eqnarray}
The procedure can dramatically be accelerated when a Fast Fourier Transform
(FFT) is applied. The computing time is then roughly proportional to
$N^3\log N$.

The potential, as derived so far by Fourier transforms, is correct for
periodic systems only. The exact potential of isolated systems can be obtained
by doubling the number of cells in each coordinate direction, i.\,e. a
grid is considered which contains $2N \times 2N \times 2N$ cells.
A rigorous proof can be found in the paper by \cite{eb79}.
As before,
the $N \times N \times N$ sub-grid contains the particles (``active region'').
The remaining space is left empty. Likewise, $H$ is extended over the
$2N \times 2N \times 2N$ cells such that it satisfies
\begin{eqnarray}\label{eq:symmetry}
H_{2N-a,b,c} = H_{2N-a,2N-b,c} \nonumber\\
= H_{2N-a,b,2N-c} = H_{2N-a,2N-b,2N-c} = H_{a,2N-b,c} \nonumber\\
= H_{a,2N-b,2N-c} = H_{a,b,2N-c} = H_{a,b,c}
\end{eqnarray}
for $0 \leq a,b,c \leq N$. Having transformed the extended functions
$\rho$ and $H$, the correct potential $\Phi$ can be found in the active
grid. Outside, $\Phi$ is unphysical.

Numerical differentiation of the potential with respect to the coordinates
leads for each particle $j$ to the acceleration $(a_{x})_j$, $(a_{y})_j$,
 $(a_{z})_j$. The leap-frog scheme is applied to integrate the
equations of motions. In $x$-direction the algorithm reads
\begin{eqnarray}
\dot{x}_j(t+ \Delta t/2) &=& \dot{x}_j(t- \Delta t/2) + \Delta t (a_{x})_j \nonumber \\
x_j(t+ \Delta t) &=& x_j(t) + \Delta t \dot{x}_j(t+ \Delta t/2)
\end{eqnarray}
where $t$ is the time and $\Delta t$ denotes the constant integration step.

The particle-mesh technique holds some subtle twists.
For instance for $H_{a,b,c}$ it suffices a $(N+1) \times (N+1) \times (N+1)$
grid. From the symmetry conditions (\ref{eq:symmetry}) follows that the function is then
known on the whole $2N \times 2N \times 2N$ grid. The sub-grid is overwritten
by $\hat{H}$ which shows similar symmetries. $\hat{H}$ is calculated only once,
since it is fixed over the integration period. For each integration step the grid
containing the density $\rho$ is overwritten by the Fourier transform $(\hat{\rho}$
which, in turn, is overwritten by $\hat{\rho}\hat{H}$). This product is finally
overwritten by $\Phi$. The advantages of the method are obvious.
The conventional particle-mesh technique was
orders of magnitude faster than direct $N$-body codes at the time. This allows the
integration of a large number of particles. Thus, statistical noise is extremely
low. Particles leaving the outermost grid are lost. The user should thus carefully consider the number of
particles involved in the computation.

The greatest disadvantage, however, is the low spatial resolution in regions
of higher particle densities, e.\,g. the cores of spherical galaxies. An
improvement is {\sc Superbox} which treats the particles self-consistently
in a nested system of grids. Here, the term ``self-consistent'' means that
all particles have masses which define the total potential. For simplicity,
we consider a spherical galaxy of radius $R_{\rmn{out}}$. Fig. 1 shows how the
particles are subdivided into a central region (``core'') where the distance
$r$ from the centre is $< R_{\rmn{core}}$ and a region with
$R_{\rmn{core}} < r < R_{\rmn{out}}$. The local universe.is
given by $R_{\rmn{sys}}$.

As shown in Fig. 2 {\sc Superbox} utilises five grids. They all have the same number
of cells, $N^3$.
\begin{enumerate}
  \item Grid 1 has the highest resolution and resolves the core which contains
        all particles with $< R_{\rmn{core}}$.
  \item Grid 2 contains the core, too, but the resolution is intermediate.
  \item Grid 3 is equal to grid 2, but contains only particles with
        $R_{\rmn{core}} < r < R_{\rmn{out}}$.
  \item Grid 4 is the fixed global grid. It contains only particles inside
        $R_{\rmn{out}}$.
  \item Grid 5 is equal to grid 4 with all particles outside $R_{\rmn{out}}$.
\end{enumerate}
As the galaxy moves, grid 1, grid 2 and grid 3 move through the local universe
(i.\,e. grid 4 and grid 5, respectively). This is done by centreing the inner and intermediate
grids on the density maximum of the galaxy in question. Alternatively, the centre of mass
can be considered. Particles outside $R_{\rmn{sys}}$
are lost, compare the discussion above. The code is not restricted to one galaxy. Since the potentials, and
thus the accelerations, are additive, all galaxies are treated sequentially
in the same five grids. The corresponding five total potentials are $\Phi_1$,
$\Phi_2$, $\Phi_3$, $\Phi_4$, and $\Phi_5$
\begin{enumerate}
  \item For a particle with $r < R_{\rmn{core}}$, the correct potential is
        $\Phi_1 + \Phi_3 + \Phi_5$.
  \item For a particle with, $R_{\rmn{core}} < r < R_{\rmn{out}}$ the sum
        $\Phi_2 + \Phi_3 + \Phi_5$ is taken.
  \item When the particle is outside $R_{\rmn{out}}$ then $\Phi_4$
        and $\Phi_5$ are used.
\end{enumerate}
In principle, number and type of the galaxies is arbitrary.

Two features should be emphasized. A particle at the point $(x,y,z)$
is assigned to the cell with indices
\begin{equation}
\label{eqn:cell}
  a=\left[\gamma ~x + {N \over 2}\right],
    ~ b=\left[\gamma ~y + {N \over 2}\right],
    ~ c=\left[\gamma ~z + {N \over 2}\right]
\end{equation}
where $[~]$ denotes the nearest integer function. The factor $\gamma$ enhances (or shrinks)
an area. If $\gamma = 10$ is assigned to grid 1, the resolution increases by a factor of 10.
Inside the cell with indices $i, ~ j, ~ k$, let the particle's coordinates be $\Delta x$,
$\Delta y$, and $\Delta z$. Then its acceleration, e.\,g. in $x$-direction is found by the expression
\begin{eqnarray}
\label{eqn:diff}
a_x = \nonumber\\
{\Phi_{i+1,j,k} - \Phi_{i-1,j,k} \over 2 l_x} \nonumber\\
+{\Phi_{i+1,j,k} + \Phi_{i-1,j,k}  - 2 \Phi_{i,j,k}  \over l_x^2} \Delta x \nonumber\\
+{\Phi_{i+1,j+1,k} - \Phi_{i-1,j+1,k}  +  \Phi_{i-1,j-1,k} - \Phi_{i+1,j-1,k} \over 4 l_x l_y} \Delta y \nonumber\\
+{\Phi_{i+1,j,k+1} - \Phi_{i-1,j,k+1}  +  \Phi_{i-1,j,k-1} - \Phi_{i+1,j,k-1} \over 4 l_x l_z} \Delta z
\end{eqnarray}
where $l_x = l_y = l_z$ is the length of the cell.
This scheme requires two empty cells at each boundary, i. e.,
only $N-4$ cells per dimension are considered.
We note that in {\sc Superbox}, the
three lengths are equal. When all velocities are updated by applying these accelerations, then new positions
of the particles are calculated. After that, a new integration cycle starts.

\subsection[]{Extensions to {\sc Superbox-10}}

For version 10, {\sc Superbox}'s original {\sc Fortran 77} code has been ported to
{\sc Fortran 95} and arranged into several modules, to make future extension easier.

\subsubsection{Individual masses}

The conventional particle-mesh technique considers only a single mass for all particles.
In {\sc Superbox-10}, an individual mass can be assigned to each particle. In practice,
however, the particles of each subgroup (i.\,e. halo, disc, etc.) carry identical mass.

\subsubsection{Improved vertical resolution}

{\sc Superbox} poorly resolves stellar discs in vertical direction. {\sc Superbox-10}
overcomes this shortcoming. We explain the basic idea by taking the example of a
disc-bulge-halo galaxy. First,
the potential of the disc-bulge-halo galaxy is calculated. The intermediate grids
grid 2 and grid 3 are flattened along the corresponding $z§$-axis. When, for instance,
the flattening is $q = 1/4$ the resolution is improved by a factor of 4.
As a consequence, equations \ref{eqn:greens}, \ref{eqn:cell} and \ref{eqn:diff}
need to be changed.
Yet, there is
a restriction to $q$. In order to cover grid 1 and at least two cells of grid 4 (and 5,
respectively), $q$ should not be smaller than
\begin{equation}
  q_{\rmn{crit}} = \max\left({R_\rmn{core} \over R_\rmn{out}},{4 \over
{N-4}}{R_\rmn{sys} \over R_\rmn{out}} \right)
\end{equation}
Otherwise, spurious results are obtained. If more than one galaxy is considered, $q < 1$
applies only to the first galaxy.

\cite{b08} simulated a disc-bulge halo galaxy using {\sc Superbox-10}.
The authors made a comparison with a code based on the TREE-GRAPE scheme \citep{fu05}.
For 2{,}577{,}235 particles the CPU time turned out to be about three days on a single
GRAPE6a board. The CPU time of {\sc Superbox-10} (which is still called {\sc Superbox}
in that paper) on a customary PC is of the same order. This result shows impressively
the efficiency of the code. \cite{b08} came to the conclusion that the extended code
\begin{enumerate}
  \item is very fast,
  \item does not produce noticeable numerical disc heating, and
  \item allows an improved vertical resolution.
\end{enumerate}

\subsubsection{Parallelisation}

The computationally most intensive part of {\sc Superbox-10} is the
calculation of the potential using the Fast Fourier Transform.
The non-parallelised version of {\sc Superbox-10} applies the 1D-FFT routine \mbox{\sc realft},
taken from {\it Numerical Recipes} \citep{p92}. The 3D-FFT is calculated
by doing $N\times N$ 1D transforms in each direction.
In the parallelised version, this scheme has been replaced by a 3D-FFT
from the {\sc fftw} library (version 2.1.5), see \cite{fj05}.
This routine divides the 3D array into slices
along one dimension and distributes these slices among multiple processors.
The processors then jointly calculate the 3D-FFT, using the Message Passing
Interface to communicate.
The modular design of {\sc Superbox-10} allows us to replace
the FFT routine without changing much of the rest of the code. The non-parallelised
FFT is still available.

In the following we introduce the fraction $f = t_{\rmn{FFT}}/t_{\rmn{tot}}$ in order
to analyse the non-parallelised version in more detail.
Here $t_{\rmn{FFT}}$ is the time spend in the FFT routine per integration step
and $t_{\rmn{tot}}$ is the total time used for the integration step.
According to
Amdahl's law \citep{a67}, the maximum achievable speed-up by improving the FFT is then given by
\begin{equation}
  S_{\rmn{max}} = \frac{1}{1 - f}
\end{equation}
Table~\ref{tbl:timediv_old} lists $f$, $S_{\rmn{max}}$ and the
speed-up $S$ achieved in benchmarks.

The values depend on the grid size $N$ as well as on the particle
number $n$. For $n \ga N^3$, the integration of the
particles' orbits starts to dominate over the potential calculation.
Thus, the higher $N$, the greater the expected speed-up will be.

\begin{table}
\centering
\caption{Maximum and achieved speed-up, $S_{\rmn{max}}$ and $S$, as function of grid
size $N$ and particle number $n$. $f =t_{\rmn{FFT}}/t_{\rmn{tot}}$, see text.}
\label{tbl:timediv_old}
\begin{tabular}{@{}cccccc}
\hline
$N$ & $n$ & $f$ & $S_{\rmn{max}}$ & $S$ & $S/S_{\rmn{max}}$ \\\hline
64 & 500k & 0.62 & 2.61 & 2.48 & 0.95\\\hline
128 & 500k & 0.92 & 12.58 & 8.96 & 0.71\\
& 2M & 0.74 & 3.80 & 3.26 & 0.86\\\hline
256 & 500k & 0.99 & 71.56 & 23.49 & 0.33\\
& 2M & 0.95 & 20.20 & 11.90 & 0.59\\
& 10M & 0.78 & 4.48 & 4.05 & 0.90\\\hline
\end{tabular}
\end{table}

Benchmarks were run on the Titan cluster at the Astronomisches Rechen-Institut,
which has 32 nodes, each containing a Xeon 3.2 Ghz dual core CPU and a nVidia GeForce 9800 GTX
Graphics Processing Unit (GPU).
However, we used only one core per CPU in order to
have more memory available, allowing for higher grid sizes. The GPUs
are currently not used by {\sc Superbox-10}, but a version suitable
for GPUs is in preparation.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{speed-up.eps}
\caption{Speed-up $S$ compared to non-parallelised {\sc Superbox} as function
of the number of processors for varying $N^3$, the number of grid cells, and particle number $n$.}
\label{fig:speed-up}
\end{figure}

Figure~\ref{fig:speed-up} shows the measured speed-up $S$ as function of the
number of processes in comparison to the
non-parallelised version of {\sc Superbox} with the old FFT when
varying both the grid cell number $N$ and the particle number $n$.
The application of {\sc FFTW} alone already results in
a speed-up between 1.8 and 2.4. In cases where the curves reach their
maximum ($N = 256$ for $n = 10\times10^6$ and $N = 128$ for all $n$)
the speed-up lies between $0.7$ and $0.9~S_{\rmn{max}}$.
In the remaining two cases ($N = 256$ for $n = 0.5\times10^6$ and $n = 2\times10^6$)
it lies between $0.3$ and $0.6~S_{\max}$. When more processors are applied
one can expect a further increase.

The fact that the speed-up almost reaches $S_{\rmn{max}}$ shows that
there is only a small overhead due to communication between the
nodes. Thus, from the values of $S_{\rmn{max}}$ in Table~\ref{tbl:timediv_old} we can formulate the following
rule of thumb: When using $p$ processors and
a grid of size $N^3$, the number of particles should not be greater
than $4 N^3 / p$. Conversely, when simulating $n$ particles with a grid
of size $N^3$, no more than $4 N^3/n$ processors should be used.

\section{Models}

In this section we will describe the models used in our simulations of isolated
disc-bulge-halo galaxies (section \ref{sec:isosim}) and merging of galaxies with satellites
(section \ref{sec:mergesim}).

\subsection{Galaxy model}

Our galaxy model consists of a disc, a bulge and a dark matter halo component.
The density profile of the disc is exponential in
radial and isothermal in vertical direction for $R<R_{\max}$ and $|z|<z_{\max}$:
\begin{equation}
  \varrho(R,z) = \varrho_0~e^{-R/h}~\rmn{sech}^2(z/z_0)
\end{equation}

The disc has a scale length $h = 2.5~\rmn{kpc}$ and a thickness $z_0 = 0.6~\rmn{kpc}$.
The profile is cut off at $R_{\rmn{max}} = 10~h$ and $z_{\rmn{max}} = 10~z_0$.
This excludes $0.05$ per cent of the mass an infinite profile would have.
At $R = 8~\rmn{kpc}$ the Toomre parameter has a value of $Q = 2$.

Both bulge and halo have a cropped Hernquist profile \citep{h90}
\begin{equation}
  \varrho(r) = \varrho_0~\frac{a^4}{r(r+a)^3}
\end{equation}
with scale radius $a$ and cutoff radius $r_c$.

The scale radius of the bulge is $a = 0.5~\rmn{kpc}$ and is cut off at $r_c = 14~a$.
The halo's scale radius is $a = 16.8~\rmn{kpc}$ and it is cut off at $r_c = 5~a$.

We emphasize that both bulge and halo are represented by live particles.
The importance of treating the dark matter halo of galaxies as a live component, i.e.,
dynamically evolving and in mutual interaction with the baryonic component, has been
stressed in self-consistent $N$-body simulations of isolated disc galaxies
(e.g., Athanassoula 2002; Dubinski, Berentzen \& Shlosman 2009).
The angular momentum exchange between
the disc and dark matter halo in such cases is mediated by dynamical resonances.
These processes cannot be resolved when using a static, e.g., analytic
prescription for the halo potential.

The model has been implemented with the programme
{\sc MaGaLie} \citep{b01}. We extended {\sc MaGaLie} to allow for up to
10 million particles.

\begin{table}
\centering
\caption{The three components of our galaxy model and the satellite.
Shown are the number of particles $n$, the
total mass $M$ and the mass per particle $m$.}
\label{tbl:gal_param}
\begin{tabular}{@{}c|ccc}
\hline
& $n~[10^6]$ & $M~[10^{10}~\msun]$ & $m~[10^3~\msun]$ \\
\hline
disc & 5.19 & 4.82 & 9.30 \\
bulge & 2.30 & 2.14 & 9.30 \\
halo & 2.15 & 20.0 & 93.0 \\
\hline
satellite & 0.50 & 0.54 & 10.08 \\
\hline
\end{tabular}
\end{table}

The particle numbers and masses of the three components are listed in Table~\ref{tbl:gal_param}.
In order to reduce the total number of particles,
the particles of the halo are chosen to be ten times as massive as those of disc and bulge
This does not pose a problem though, because they are still light enough not to
cause numerical heating.

Initially, the model is not completely in equilibrium. Therefore,
it is evolved in isolation with high resolution ($N = 256$, $q = 0.25$)
until it reaches equilibrium. This resolution is high enough not to introduce
any numerical heating.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{rot-curve.eps}
\caption{The disc's mean rotational velocity $\langle v_{\Phi}\rangle$,
radial velocity dispersion $\sigma_R$, and vertical velocity dispersion $\sigma_z$.
Also shown is the circular velocity $v_c$. Note that the galaxy has already
reached equilibrium.}
\label{fig:rot-curve}
\end{figure}

Figure~\ref{fig:rot-curve} shows the disc's mean rotational velocity
$\langle v_{\Phi}\rangle$ and its velocity dispersions $\sigma_R$ and $\sigma_z$ in
radial and vertical direction after this initial relaxation. Also shown is
the circular velocity $v_c$. The stellar disc has a flat
rotation curve and an exponentially decreasing velocity dispersion.

\subsection{Satellite model}

The satellite for the merger simulations has a Plummer
profile
\begin{equation}
  \rho(r) = \frac{3M}{4\pi a^3}\left(1 + \frac{r^2}{a^2}\right)^{-\frac{5}{2}}
\end{equation}
with mass $M = 5.4\times10^9~\msun = 0.11\times M_{\rmn{disc}}$ and scale radius $a = 1.5~\rmn{kpc}$.
Its profile is cut off at $r_c = 10~a$. It consists of $5\times10^5$ particles (see
Table~\ref{tbl:gal_param}).
Like the galaxy, it is first evolved in isolation.

\section{Isolated galaxy models}
\label{sec:isosim}

\cite{b08} showed that {\sc Superbox} intrinsically has a very low level of
numerical heating. We discuss now in detail the dependence of the numerical
heating on the vertical flattening factor $q$. To that end we simulate the
previously discussed disc-bulge-halo galaxy model in isolation with various
resolutions and measure the change in vertical thickness of the disc component.

\subsection{Simulation parameters}

\begin{table}
\centering
\caption{Vertical resolution in parsec for various combinations of $q$ and $N$.
Note that $R_{\rmn{out}} = 28~\rmn{kpc}$.
The radial resolution is independent
of $q$ and has the same value as the vertical resolution for $q=1$.}
\label{tbl:vert_res}
\begin{tabular}{@{}c|cccc}
\hline
$q\backslash N$ & 64 & 128 & 256 & 512 \\
\hline
1 & 933 & 452 & 222 & 110\\
1/2 & 467 & 226 & 111 & 55\\
1/4 & 233 & 113 & 56 & 28\\
1/8 & - & 56 & 27 & 14\\
\hline
\end{tabular}
\end{table}

In all simulations, the grids have
$R_{\rmn{core}} = 3.5~\rmn{kpc}$, $R_{\rmn{out}} = 28~\rmn{kpc}$
and $R_{\rmn{sys}} = 105~\rmn{kpc}$.
Depending on the number of cells $N^3$ and flattening parameter $q$, the
simulations have vertical resolutions listed in Table~\ref{tbl:vert_res}. The radial
resolution for a certain $N=N_0$ has the same value as the vertical resolution
for $N=N_0$ and $q=1$.

We run low-resolution simulations with $N=64$ and $q=1,~0.5,~0.25$. The radial
resolution is 933 pc and the vertical resolution amounts to 933, 467 and 233 pc, respectively.
These simulations are compared to a medium-resolution simulation ($N=128$, $q=0.25$)
where the radial resolution is 452 pc and the vertical resolution is 113 pc.
Additionally, a comparison to the initial, relaxed, system is made.
A time step of $0.4~\rmn{Myr}$ is used and the length of the integration corresponds to
1 Gyr.

\subsection{Results}

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{z.stddev.eps}
\caption{Standard deviation of the $z$-coordinate of disc particles as function of $R$
for varying $N$ and $q$ at $t = 1~\rmn{Gyr}$, together with the initial curve. The values are
calculated in radial bins of equal particle numbers.}
\label{fig:z.stddev}
\end{figure}

Figure~\ref{fig:z.stddev} shows the root mean square (RMS) of the $z$-coordinate
of all disc particles, $\sqrt{\langle z^2\rangle}$, as a function of radial distance $R$.
We used radial bins of equal particle numbers. The RMS can be
taken as a measure of the thickness of the disc.
The low-resolution simulation ($N=64$, $q=1$) over-estimates the
thickness significantly. The maximum deviation from
the initial thickness is about 16 per cent.
Flattening the intermediate grid by a factor of 4, brings the
thickness down to within about 4 per cent of the initial values, without increasing the
computation time. The medium-resolution simulation ($N=128$, $q=0.25$) deviates at
most $\approx 1.6$ per cent from the initial values, but is closer than $1$ per cent for most
of the radial range.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{z.stddev.comp.eps}
\caption{Standard deviation of the $z$-coordinate of disc particles as function of $R$
for varying $N$ and $q$ at $t = 1~\rmn{Gyr}$, together with the initial curve. The values are
calculated in radial bins of equal particle numbers.}
\label{fig:z.stddev.comp}
\end{figure}

 To demonstrate the effect of flattening, Figure~\ref{fig:z.stddev.comp} directly
  compares two simulations with the same vertical resolution -- $N=128$ with $q = 0.25$
  and $N=256$ with $q=0.25$ -- corresponding to a vertical cell length of about 112 pc.
  As a reference the case of $N=128$ without flattening is
also shown. As can be seen, introducing flattening in the case of medium resolution
diminishes thickness, down to about the same value as in the case of high resolution.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{zprofa.eps}
\includegraphics[width=0.47\textwidth]{zprofb.eps}
\caption{Volume density $\rho$ as function of $z$ (averaged over the radial range
$7.5~\rmn{kpc}\le R\le 8.5~\rmn{kpc}$) for varying $N$ and $q$ at $t = 1~\rmn{Gyr}$.
The solid line shows the initial values. {\bf (a)} Whole range. {\bf (b)} Right tail.}
\label{fig:zprof}
\end{figure}

Figure~\ref{fig:zprof} displays the vertical density profile averaged
over the radial interval $7.5~\rmn{kpc} \le R \le 8.5~\rmn{kpc}$.
The initial profile (solid line) and the profile of the medium-resolution simulation
(dash-dotted line) are almost identical. In the case of the
low-resolution simulations (dashed line), the profile is
significantly widened for $|z|>1~\rmn{kpc}$ (see Figure~\ref{fig:zprof}~(b)). This is a
sign of numerical heating. Reducing $q$ brings the profile closer to the initial
one. For $q=0.25$, deviations are only visible beyond 2 kpc.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{z.stddev.evo.eps}
\caption{Standard deviation of the $z$-coordinate of disc particles as function of
time for varying grid size $N$ and flattening $q$.
The standard deviation is calculated in a region around $R = 8~\rmn{kpc}$. The solid line
shows the initial value.}
\label{fig:z.stddev.evo}
\end{figure}

Figure~\ref{fig:z.stddev.evo} shows the time evolution of the RMS
for varying grid size $N$ and flattening factor $q$ in the region around $R = 8~\rmn{kpc}$.
In the first 200 Myr, the model adapts to the new grid structure, which causes a
change in the RMS. After that, the RMS increases slightly with time in the case of the
low-resolution simulations. It reaches values that
are $8.7$ per cent ($q=1$), $3.6$ per cent ($q=0.5$) and $2.4$ per cent ($q=0.25$) greater
than the initial
value. In the medium-resolution simulation however, it remains stable at a deviation of
about $0.9$ per cent.

\section{Merger simulations}
\label{sec:mergesim}

One application that benefits from the increased z-resolution
is the study of the dynamical heating of galactic discs
caused by the merging with satellite galaxies. We simulate the merging
of a small satellite with a disc-bulge-halo galaxy for varying initial
positions and velocities of the satellite.

\subsection{Simulation parameters}

All satellites are initially at a distance of $R_{\rmn{A}}=25~\rmn{kpc}$ from the
centre of the galaxy and have a velocity of either $115~\kms$ or $81.3~\kms$.
If all the mass of the galaxy inside $R_{\rmn{A}}$ were concentrated in a
point mass, these velocities would result in elliptical orbits with apocentre distance
$R_{\rmn{A}}$ and eccentricities
$\epsilon = 0.56$ and $\epsilon = 0.78$, respectively. Here, the eccentricity is defined
as
\[ \epsilon = {\sqrt{a^2-b^2} \over a}\]
with semi-major axis $a$ and semi-minor axis $b$. In the two-body problem,
the pericentre distance of these orbits would be $R_{\rmn{P}} \approx 7~\rmn{kpc}$ and
$R_{\rmn{P}} \approx 3~\rmn{kpc}$, respectively.

\begin{table}
\centering
\caption{Satellite parameters $\epsilon$ (eccentricity), $v$ (velocity), $R_{\rmn{A}}$ (apocentre),
$R_{\rmn{P}}$ (pericentre) and $i$ (orbital inclination).}
\label{tab:satparam}
\begin{tabular}{@{}c|cccc}
\hline
$\epsilon$ & $v~[\rmn{km}/\rmn{s}]$ & $R_{\rmn{A}}~[\rmn{kpc}]$ & $R_{\rmn{P}}~[\rmn{kpc}]$ & $i~[\rmn{\degr}]$ \\\hline
$0.56$ & $115$ & 25 & 7.05 & 0, 10, $\dots$, 180\\
$0.78$ & $81.3$ & 25 & 3.08 & 0, 10, $\dots$, 120\\\hline
\end{tabular}
\end{table}

In addition to the eccentricity, we vary the orbital inclination $i$ relative to the plane
of the disc between $0\degr$ and
$180\degr$ for $\epsilon = 0.56$ in steps of $10\degr$, and likewise between $0\degr$ and $120\degr$ for $\epsilon=0.78$.
For $i < 90\degr$ the satellite is on a prograde orbit as compared to the galactic rotation,
while for $i > 90\degr$ the orbit is retrograde. For $i=90\degr$ the satellite's initial velocity
is perpendicular to the bulk motion of the disc.
Table~\ref{tab:satparam} summarises these parameters.

As shown in the previous section, a grid cell number of $N=128$ with flattening
factor $q=0.25$ causes no significant numerical heating. We adopt these
values for our merger simulations.
All simulations run for 1 Gyr. After that, the satellites are completely dissolved.
As a control, the galaxy model is also evolved in isolation.

At the end of the simulation, the plane of the disc is tilted by a few degrees.
In the following, the data are evaluated in a coordinate system where $x$
and $y$ define the galactic plane and $z$ is perpendicular to it.

\subsection{Vertical profile}

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{merge-zprof.eps}
\includegraphics[width=0.47\textwidth]{merge-zprof-inner.eps}
\caption{Volume density of the disc in the region $7.5~\rmn{kpc}\le R\le 8.5~\rmn{kpc}$
after the merging ($t=1~\rmn{Gyr}$) as function of $z$ for different satellite orbital inclinations $i$.
All satellites show have initial orbital eccentricity $\epsilon = 0.56$. The solid line corresponds
to the isolated simulation, i.e., without satellite.
{\bf (a)} Whole range. {\bf (b)} Inner part.}
\label{fig:merge-zprof}
\end{figure}

Figure~\ref{fig:merge-zprof} shows the vertical density profile of the disc
after merging. Only satellites with significant effect on the disc
are shown ($e = 0.56$, $i = 0\degr\dots 30\degr$).
Compared to the isolated simulation (i.e. without satellite), the central density is decreased
by a factor of 2 at the most ($i = 20\degr$).
In the inner part ($|z|<1~\rmn{kpc})$, the original $\rmn{sech}^2(z/z_0)$ profile
remains a good fit, albeit with a greater thickness $z_0$. The isolated galaxy has
$z_0 = 0.6~\rmn{kpc}$, while for $i = 20\degr$ the thickness is increased to
$0.8~\rmn{kpc}$.
In the outer part, however, the density is increased as compared to a $\rmn{sech}^2(z/z_0)$ law.
While the thickness of $0.8~\rmn{kpc}$ corresponds to a scale height of $0.4~\rmn{kpc}$,
the outer part is better fitted by an exponential function with scale height $\approx 0.65~\rmn{kpc}$.

  \cite{hc06} provide a formula for the increase of the thickness of the disc caused by cold
  dark matter subhaloes, which should also be applicable to satellite mergers.
  In our notation their formula is given by
  \[ \Delta z_0 \approx 8 h \left(\frac{M_{\rmn{sat}}}{M_{\rmn{disc}}}\right)^2 \approx 0.25~\rmn{kpc}~\rmn{.}\]
  This is compatible with an increase of $\Delta z_0\ga 0.2~\rmn{kpc}$ in the case of $i = 20\degr$
  as described above.

\subsection{Heat increase}

We define the total heat in the disc as the kinetic energy of random motion
\begin{equation}
  E_{\rmn{heat}} = \sum_i {1 \over 2} m_i \left|\mathbf{v}_i-\overline{\mathbf{v}}_c\left(R_i\right)\right|^2
\end{equation}
where the sum goes over all particles in the disc, $m_i$ is the mass of the $i$th particle,
$\mathbf{v}_i$ is the velocity of the $i$th particle,
and $\overline{\mathbf{v}}_c\left(R_i\right)$ is the mean circular velocity at the radial
distance $R_i$ of the $i$th particle. $\bmath{v}_c$ is first calculated in radial
bins of equal particle numbers and then interpolated to the individual distances $R_i$.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{heat.eps}
\includegraphics[width=0.47\textwidth]{sigma.eps}
\caption{{\bf (a)} Total heat of the disc after merging ($t=1~\rmn{Gyr}$) as a
function of satellite's orbital inclination $i$ for varying eccentricity $\epsilon$.
$E_{\rmn{heat}}$ is measured in units of the heat of the isolated galaxy,
$E_{\rmn{heat, iso}}$, at $t=1~\rmn{Gyr}$.
{\bf (b)} Squared total velocity dispersion at $t = 1~\rmn{Gyr}$ in the region around 8 kpc as a
function of orbital inclination $i$ and for varying eccentricity $\epsilon$.
$\sigma_{\rmn{tot}}^2$ is measured in units of the squared total velocity dispersion of the
isolated galaxy.}
\label{fig:heat}
\end{figure}

Figure~\ref{fig:heat} (a) shows $E_{\rmn{heat}}$
in units of the heat in the isolated disc $E_{\rmn{heat,iso}}$ at the end of the simulation
(i.e. at $t = 1~\rmn{Gyr}$).
$E_{\rmn{heat,iso}}$ increases only by $0.18$ per cent as compared to the initial value,
demonstrating
that our simulations are free of numerical heating. Figure~\ref{fig:heat} (b) displays
the same for the square of the total velocity dispersion in the solar neighbourhood.

The heating efficiency of
the satellite depends on its orbital inclination as well as on its orbital eccentricity.
For low inclinations, satellites on less elliptical orbits heat more effectively than
those on more elliptical ones. For $i > 40\degr$, however, satellites on orbits
with $\epsilon = 0.78$ are more effective.
Prograde satellites heat the most with an maximum increase of 22 per cent, while
retrograde ones only have a small effect of about 2 per cent. These results
are consistent with those obtained by \cite{vw99}.

Observations show that the velocity dispersion in the solar neighbourhood
increases from about 30 $\kms$ to about 60 $\kms$ in 8 Gyr
approximately proportional to $\sqrt{t}$ \citep[see, e.g.,][]{hf02}.
This means, that the specific heat
increases linearly by about $1350~\rmn{km}^2~\rmn{s}^{-2}$. If we assume that
the heat increase is completely due to satellite mergers, then the required
merger rate $n$ is
\begin{equation}
  \nu = \frac{1}{\Delta h}\times \frac{1350}{8}\frac{\rmn{km}^2}{\rmn{s}^2~\rmn{Gyr}}
\end{equation}
where $\Delta h$ is the specific heat increase imparted by a single merger.

In our simulations,
we found that for low inclinations the heat increase is approximately uniform
over the whole disc (except for the bulge-dominated inner part) and lies
between 100 and 1000 $\rmn{km}^2~\rmn{s}^{-2}$ per merger. For high inclinations, the
heat increase is mainly due to flaring of the outer parts of the disc ($R > 15~\rmn{kpc}$),
while the solar neighbourhood is basically unaffected. Only considering low
inclinations, the required merger rate then lies between $\nu=0.17$ and $\nu=1.69$ mergers per Gyr,
depending on inclination and eccentricity of the orbit.

\subsection{Velocity dispersion ratio}

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{sigrat.eps}
\caption{Velocity dispersion ratio $\sigma_z/\sigma_R$ at $t = 1~\rmn{Gyr}$ in the
region around 8 kpc as function of orbital inclination $i$ and for varying eccentricity $\epsilon$.}
\label{fig:sigrat}
\end{figure}

Observations show that the current ratio of the radial and vertical velocity dispersions
in the solar neighbourhood, $\sigma_z/\sigma_R$, is approximately $0.5\pm 0.1$ and
increases slightly with stellar age, $\propto t^{0.16}$ \citep{h07}.
Figure~\ref{fig:sigrat} shows this ratio after
merging as a function of orbital inclination $i$.
The satellites with low prograde orbits decrease the ratio to at most about 0.4, i.e.,
they heat more efficiently in radial than in vertical direction. In one case ($i=20\degr$,
$\epsilon=0.56$) there is an increase to approximately 0.6.
For inclinations above $30\degr$ the ratio remains mostly unchanged in the solar neighbourhood.

\subsection{Energy and angular momentum transfer}

An interesting question is where the initial energy and angular momentum of the
satellite end up in the final system.

\begin{figure}
\centering
\includegraphics[width=0.46\textwidth]{energy-transf.eps}
\includegraphics[width=0.46\textwidth]{ekin-transf.eps}
\includegraphics[width=0.46\textwidth]{epot-transf.eps}
\caption{Change in total (a), kinetic (b) and potential (c) energy of the four
components with respect to the initial energies as a function of orbital inclination $i$.
The energy change is measured in units of the total energy of the isolated galaxy.
Only the data for $\epsilon = 0.56$ is shown.}
\label{fig:energy-transf}
\end{figure}

Figure~\ref{fig:energy-transf} shows the change in energy of the various components.
The energy is
measured in units where the total energy of the isolated galaxy is $-1$. ``Satellite'' designates
those particles that once made up the satellite, but now
are distributed over the whole system.

The initial galaxy-satellite system is not in equilibrium.
As a consequence of virialisation, both the satellite and the halo gain
kinetic and lose potential energy. This results in a deeper potential well
in the centre, which reduces the potential energy of bulge and disc.
During its in-spiral, the satellite transfers part of its kinetic energy
to the halo particles. This is due to dynamical friction.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{angmom-transf.eps}
\caption{Change in total angular momentum of the four components with respect to the initial
angular momenta as a function of orbital inclination $i$. The change is measured in units
of the total angular momentum of the isolated galaxy. Only the data for $\epsilon = 0.56$ is shown.}
\label{fig:angmom-transf}
\end{figure}

This effect can also be seen in Figure~\ref{fig:angmom-transf},
which shows the change in angular momentum as a function of inclination, measured
in units of the total angular momentum of the isolated galaxy. The angular
momentum of the bulge barely changes. This is not surprising since the satellites are already
mostly destroyed before reaching the inner part of the galaxy.
Our data reveal that the radial distribution of angular momentum in the disc changes:
the disc slows down and expands. Its {\em total}
angular momentum, however, remains approximately nearly constant. It only increases slightly for
satellites on prograde orbits and decreases slightly for satellites on retrograde orbits.
The satellite's initial orbital angular momentum is $0.31$ in
these units. Accordingly, it loses between 15 per cent and 20 per cent of its angular momentum.
Of that, about 80 per cent goes into the halo, while the rest is imparted onto the disc.

\subsection{Comparison with higher resolution}

To ensure that the choice of resolution does not influence the results we
simulate a case with $\epsilon = 0.56$ and $i = 0\degr$, and higher resolution.
We choose $N=256$ and $q=0.125$, as this is the highest possible resolution
one can achieve on our present hardware.

We calculate the Fourier transform of the density distribution of the disc
at the end of the simulation in order
to find possible perturbations caused by low resolution. The $m$th complex
Fourier coefficient is given by

\[
  \tilde{A}_m(R_i) = \frac{1}{S_i} \sum_j M_j e^{-im\varphi_j}
\]

where $R_i$ and $S_i$ are the central radius and area of the $i$th bin, the index
$j$ runs over all particles in bin $i$, and $M_j$ and $\varphi_j$ are the mass and
polar angle of the $j$th particle. The amplitude of $\tilde{A}_m$ is denoted by
$A_m$ and the phase by $\theta_m$.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{modes.eps}
\caption{Amplitudes of the first and second spiral mode as a function of radius
  at the end of the simulation.}
\label{fig:modes}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{phases.eps}
\caption{Phase of the first mode as a function of radius
  at the end of the simulation.}
\label{fig:phases}
\end{figure}

Figure~\ref{fig:modes}
shows the amplitudes of the first and second mode normalised by the mean density,
while Figure~\ref{fig:phases} displays the phase of the first mode.
The two resolutions result in a roughly similar distribution in Fourier space.
There are no strong perturbations in amplitude caused by the lower resolution.
Neither is the pattern speed affected significantly.

\section[]{Conclusions}

{\sc Superbox} is a particle-mesh code where additional grids and sub-grids are
applied to regions of high particle density. This strategy turns out to be very efficient
since the code can run on any workstation or PC. Nevertheless, the code has its
limitations. For instance, stellar discs are poorly resolved in vertical
direction. We overcome this problem by introducing flattened grids. This is one
of the features of the new code {\sc Superbox-10} where, in addition, an individual
mass can be assigned to each particles.

We found that the computationally most intensive part of the code is the FFT.
We parallelised it using the library {\sc fftw}. This resulted in a
speed-up of 2 to 24, depending on grid size $N$ and the number of particles $n$.

We created a galaxy model with an exponential disc, a bulge with a Hernquist
profile and a dark matter halo, also with a Hernquist profile. The model was
realised using the program {\sc MaGaLie}.

The model was first simulated in isolation. These simulations show that flattening
the intermediate grid is an efficient means to reduce numerical heating in the simulation.

We also simulated the merging of the galaxy with small satellites in order to analyse the
proposed disc heating due to the interaction.
We find that satellites on prograde orbits with low eccentricity
and inclination heat the disc most efficiently. If the heat increase in
the solar neighbourhood were to be explained by that type of satellite mergers alone, a
rate between 0.2 and 1.7 mergers per Gyr would be
required. The detailed analysis of energy and angular momentum re-distribution shows that
most of the satellites energy and angular momentum is transfered to the dark matter halo.
This shows that the halo plays an important role even in 50:1 mergers. This confirms,
that simulations of such processes should represent the halo by live particles and not by
a fixed background potential. The presented pilot study of high-resolution simulations of
disc heating by merging satellite galaxies serves as a starting point for an extended
parameter study to quantify the heating rate in a cosmological context.

\section*{Acknowledgments}

We thank Markus Hartmann und Ingo Berentzen for
numerous and fruitful discussions.

This work was supported by Sonderforschungsbereich SFB 881 ``The Milky Way System''
(subproject A1) of the German Research Foundation (DFG).

The computer hardware used for the simulations was supported by project ``GRACE''
I/80 041-043 of the Volkswagen Foundation and by the Ministry of Science, Research
and the Arts of Baden-W\"urttemberg.