\section{Introduction}

A key objective of particle physics research is the experimental
confirmation of a theoretically consistent description of elementary
particle masses and electroweak symmetry breaking.  The prevalent
formalism is the Higgs mechanism
\cite{Higgs:1964ia,Higgs:1964pj,Higgs:1966ev,Englert:1964et,Guralnik:1964eu},
which predicts the existence of one or more physical Higgs bosons.
Recently, a candidate Standard Model (SM) Higgs boson with
$M_H\approx 126$ GeV has been discovered \cite{Aad:2012tfa,Chatrchyan:2012ufa}.
If physics beyond the SM (BSM) is realized in nature,
additional, heavier Higgs bosons may be discovered at the
Large Hadron Collider (LHC).  A comprehensive analysis of Higgs boson searches
at the LHC can be found in
refs.~\cite{Dittmaier:2011ti,Dittmaier:2012vm,Heinemeyer:2013tqa}.

At the LHC and Tevatron, Higgs bosons are primarily produced in
gluon fusion \cite{Georgi:1977gs}.
Next-to-leading order (NLO) QCD corrections have been calculated in the heavy-top
limit \cite{Dawson:1990zj} and  with finite $t$ and $b$ mass effects \cite{Djouadi:1991tka,Graudenz:1992pv,Spira:1995rr}, and were found to be as large as
80--100\% at the LHC.
This motivated the calculation of next-to-next-to-leading
order (NNLO) QCD corrections \cite{Harlander:2002wh,Anastasiou:2002yz,Ravindran:2003um}
enhanced by soft-gluon resummation at next-to-next-to-leading logarithmic
level \cite{Catani:2003zt,deFlorian:2011xf} and beyond \cite{Moch:2005ky}.
At NNLO QCD, the residual scale uncertainty is of ${\cal O}(10\%)$
for inclusive observables \cite{Dittmaier:2011ti}.
In addition to higher-order QCD corrections, electroweak corrections
have been computed and found to be at the 1--5\%
level \cite{Djouadi:1994ge,Aglietti:2004nj,Degrassi:2004mx,Actis:2008ug,Anastasiou:2008tj}.

The $H\to WW\to \ell\bar{\nu}\bar{\ell}\nu$ decay mode\footnote{%
Charged leptons are denoted by $\ell$.}
in gluon fusion plays an important role in Higgs searches at the LHC
\cite{Dittmar:1996ss,Dittmaier:2012vm}, and fully differential
NNLO QCD \cite{Anastasiou:2007mz,Grazzini:2008tf,Anastasiou:2008ik} and
NLO electroweak \cite{Bredenstein:2006rh} corrections have
been calculated and studied for this process.\footnote{%
A comprehensive NLO QCD analysis of the irreducible $WW$+0,1 jet background
including squared quark-loop contributions has been performed in ref.\ \cite{Cascioli:2013gfa}.}

With inclusive NNLO signal
scale uncertainties of ${\cal O}(10\%)$, which can be further reduced by
experimental selection cuts, it is
important to study signal-background interference effects, because
they can be of similar size if invariant Higgs masses above the
weak-boson pair threshold contribute.  An accurate estimate of
the magnitude of signal-background interference effects allows
experimenters to decide if it has to be taken into
account or may be treated as an additional uncertainty.
We note that interference effects at the few percent level if neglected
in SM calculations could be wrongly identified as anomalous couplings.

Higgs-continuum interference in $gg\ (\to H)\to WW$ and $gg\ (\to H)\to ZZ$
has been studied for a light and heavy SM Higgs boson
in refs.\ \cite{Glover:1988fe,Glover:1988rg,Seymour:1995qg,Binoth:2006mf,Accomando:2007xc,Campbell:2011cu,Kauer:2012ma,Passarino:2012ri,Kauer:2012hd,Campanario:2012bh,Bonvini:2013jha,Caola:2013yja}.  Signal-background interference for
$gg\ (\to H)\to WW\to \ell\bar{\nu}\bar{\ell}'\nu'$ (different-flavour final state)
and Higgs masses between 120 GeV and 600 GeV has been analysed at LO in ref.\
\cite{Campbell:2011cu} as well as refs.\
\cite{Kauer:2012hd,Binoth:2006mf,Accomando:2007xc,Kauer:2012ma}, and
including (N)NLO corrections in soft-collinear approximation in ref.\
\cite{Bonvini:2013jha} for $M_H=600$ GeV.  Signal-background interference for
$gg\ (\to H)\to ZZ\to \ell\bar{\ell}\nu'\bar{\nu}'$ has been investigated in
ref.\ \cite{Kauer:2012hd} for $M_H=125$ GeV and $M_H=200$ GeV.

For the same-flavour final state
$\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ considered here,
in addition to Higgs-continuum interference one also has
interference between $WW$ and $ZZ$ intermediate states.
The $H\to WW/ZZ\to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$
decay including interference was calculated at LO and NLO in
ref.\ \cite{Bredenstein:2006rh,Dittmaier:2012vm,Heinemeyer:2013tqa}
and interference-induced deviations of 10\% for the shape of distributions
have been found.\footnote{%
We note that substantial interference effects between direct and indirect Higgs
boson decays to quarkonia have been identified in ref.\ \cite{Bodwin:2013gca}.}
For continuum production, the $WW/ZZ$ interference has been studied
in $q\bar{q}$ scattering at LO in ref.\ \cite{Melia:2011tj}.
No sizeable effects were found.
Here, we investigate both types of interference for the
gluon scattering subprocess at loop-induced LO.

The paper is organised as follows: In section \ref{sec:calculation},
calculational details are discussed.  $WW/ZZ$ and signal-background
interference is studied for the gluon-induced continuum background as well
as light and heavy Higgs boson signals with minimal and realistic experimental
selection cuts in section \ref{sec:results}.
Conclusions are given in section \ref{sec:conclusions}.

\section{Calculational details\label{sec:calculation}}

We implement the $gg\ (\to H) \to WW/ZZ\to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$
process (same-flavour final state) in the publicly available parton-level program
and event generator \textsf{gg2VV} \cite{gg2VV}, thus completing the implemention
of all $gg\ (\to H) \to 4$ leptons processes at loop-induced leading order (LO).

\subsection{Amplitude\label{sec:process}}

Figure \ref{fig:graphs} displays representative graphs for the Higgs signal
process (a,c) and the $gg$-initiated continuum background process (b,d).
For the same-flavour final state, $WW$ intermediate states (a,b) and
$ZZ$ intermediate states (c,d) contribute.  Note that the
amplitude contributions (a), (b), (c) and (d) interfere.
$\gamma^\ast\to \ell\bar{\ell}$ contributions are important for
Higgs (invariant) masses below the $Z$-pair threshold \cite{Binoth:2008pr}
and are therefore included.
In addition to box topologies, in principle also triangle topologies
contribute to the $gg$ continuum process (see figure \ref{fig:triangles}).
But, in the limit of vanishing lepton masses the triangle graphs do not contribute.\footnote{Note that the $gg\to Z^\ast$ triangle graphs do contribute for non-zero lepton masses, which was verified by explicit calculation.}
\begin{figure}[t]
\vspace{0.4cm}
\centering
\includegraphics[width=0.8\textwidth, clip=true]{figs/ggWWZZ_graphs.eps}\\[.0cm]
\caption{\label{fig:graphs}
Representative Feynman graphs for the LO amplitude of $gg\to WW \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ (Higgs signal (a) and continuum background (b) contributions)
and $gg\to ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ (Higgs signal (c) and continuum background (d) contributions).}
\end{figure}
\begin{figure}[t]
\vspace{0.4cm}
\centering
\includegraphics[height=2.6cm, clip=true]{figs/triangles1.eps}\hfil
\includegraphics[height=2.6cm, clip=true]{figs/triangles2.eps}
\caption{\label{fig:triangles}
Representative triangle graphs that formally contribute to $gg\to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$.}
\end{figure}

The implementation of the amplitudes for signal and background
processes is generated with FeynArts/FormCalc \cite{Hahn:2000kx,Hahn:1998yk}
and subsequently customised.  The Higgs amplitudes are implemented using
the complex-pole scheme described in ref.\ \cite{Goria:2011wa}.
The $gg$ continuum amplitude receives contributions from box graphs
that are affected by numerical instabilities when Gram determinants
approach zero.  In these critical phase space regions the amplitude
is evaluated in quadruple precision.  Residual instabilities are eliminated
by requiring that $p_{T,W}$ and $p_{T,Z}$ are larger than $1$ GeV.

\subsection{Phase space\label{sec:sampling}}

We use a modified version of the adaptive-importance-sampling-oriented phase space integration method employed in
ref.\ \cite{Melia:2011tj} for the same-flavour $ZZ\to \ell\bar{\ell}\ell\bar{\ell}$
decay mode.  Note that for a given final state, amplitudes with two distinct,
potentially resonant intermediate states related via permutation contribute to the
decay: $(Z\to \ell_1\bar{\ell}_1, Z\to \ell_2\bar{\ell}_2)$ and
$(Z\to \ell_1\bar{\ell}_2, Z\to \ell_2\bar{\ell}_1)$.  For the different-flavour
case, efficient importance sampling can be achieved by applying a mapping that
approximates the Breit-Wigner resonance as well as the $\gamma^\ast$-induced
divergence at zero virtuality when integrating over both (physically distinct)
$Z$ boson virtualities $s_i= p_i^2$, where $p_i$ is the 4-momentum of the
intermediate $Z$ boson state and $i=1,2$ is the flavour index.  To efficiently
sample the same-flavour final state, ref.\ \cite{Melia:2011tj} uses an
appropriately weighted sum of the two phase space integrals corresponding to the
two lepton permutations $(\bar{\ell}_1\leftrightarrow\bar{\ell}_2)$.  The
phase-space-configuration-dependent weight for each channel is chosen such that
not efficiently sampled resonant configurations are suppressed.  Proper
normalisation is achieved by overall
multiplication with the inverse sum of the weights.\footnote{As noted in
ref.\ \cite{Melia:2011tj}, in the $ZZ\to \ell\bar{\ell}\ell\bar{\ell}$ case it is
advisable to randomly permute the identical leptons if final state criteria
(e.g.\ selection cuts) are applied.}  When adapting the method to
$(Z\to \ell\bar{\ell}, Z\to \nu_\ell\bar{\nu}_\ell)$ and
$(W\to \ell\bar{\nu}_\ell, W\to \bar{\ell}\nu_\ell)$ an asymmetry is introduced,
and to preserve efficiency it is necessary to numerically balance both weight
factors.  This can be achieved by using
\begin{align}
f_{ZZ} &= \left\{\left[(s_{12}-M_W^2)^2 + (M_W\Gamma_W)^2\right] \times \left[(s_{34}-M_W^2)^2 + (M_W\Gamma_W)^2\right]\right\}^2\,,\\
f_{WW} &= \frac{s_{13}^3}{M_Z^6}\ \left[(s_{13}-M_Z^2)^2 + (M_Z\Gamma_Z)^2\right]^2 \times \frac{s_{24}^3}{M_Z^6}\ \left[(s_{24}-M_Z^2)^2 + (M_Z\Gamma_Z)^2\right]^2\,,
\end{align}
with 4-momentum indices $1=\ell$, $2=\bar{\nu}_\ell$, $3=\bar{\ell}$, $4=\nu_\ell$.
The integrated cross section is computed using normalised weight factors:
\begin{gather}
\sigma = \int_{WW} \frac{f_{WW}}{f_{WW}+f_{ZZ}}\ d\sigma + \int_{ZZ}  \frac{f_{ZZ}}{f_{WW}+f_{ZZ}}\ d\sigma\,,
\end{gather}
where the integral subscript indicates which phase space mapping is applied.

Adaptive Monte Carlo integration was carried out using the Dvegas
package \cite{Dvegas}, which was developed in the context of
refs.\ \cite{Kauer:2001sp,Kauer:2002sn}.
The correctness of the program was checked with cross section and squared
amplitude level comparisons with GoSam \cite{Cullen:2011ac}, MCFM \cite{Campbell:1999ah,Campbell:2011bn,Campbell:2011cu} and VBFNLO \cite{Arnold:2008rz,Arnold:2011wj}.

\section{Results\label{sec:results}}

In this section, we present cross sections
and measures for interference and off-shell effects for the
$gg\to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ process
in $pp$ collisions at $\sqrt{s}=8$ TeV.  The complete loop-induced
LO amplitude is included, i.e.\ Higgs signal and
continuum background contributions as well as $WW$ and $Z(Z,\gamma^\ast)$
intermediate states (collectively denoted by $VV$).
All results are given for a single lepton flavour combination.
No flavour summation is carried out for charged leptons or neutrinos.\footnote{%
Since our study focuses on interference effects, we do not include
final states with different neutrino flavours, which cannot be distinguished
experimentally.  Results for the corresponding different-flavour
final states have been presented in ref.\ \cite{Kauer:2012hd} and
can be computed with \textsf{gg2VV}.}
As input parameters, we use the specification of the
LHC Higgs Cross Section Working Group in App.~A of ref.\ \cite{Dittmaier:2011ti}
with NLO weak boson widths and $G_\mu$ scheme.
Finite top and bottom quark mass effects
are included.  Lepton masses are neglected.
The fixed-width prescription is used for weak boson propagators.
The complex-pole-scheme Higgs widths are calculated with
the program \textsf{HTO} (G.\ Passarino, unpublished).
We consider the Higgs masses 126 GeV, 200 GeV, 400 GeV, 600 GeV, 800 GeV and 1 TeV
with $\Gamma_H = 4.17116$ MeV, 1.42120 GeV, 26.5977 GeV, 103.933 GeV, 235.571 GeV
and 416.119 GeV, respectively.
The PDF set MSTW2008 NNLO \cite{Martin:2009iq} with 3-loop running for
$\alpha_s(\mu^2)$ and $\alpha_s(M_Z^2)=0.11707$ is used.
The CKM matrix is set to the unit matrix, which causes a
negligible error \cite{Kauer:2012ma}.
Unless otherwise noted, the renormalisation and factorisation scales are
fixed at $\mu_R=\mu_F=\mu_H= M_H/2$, which
yields a better perturbative convergence for the signal than
$\mu_R=\mu_F=M_H$ \cite{Anastasiou:2002yz}.
In section \ref{sec:lighthiggs}, the dynamic scale choice
$\mu_R=\mu_F=\mu_\text{offshell}= M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell}/2$
is employed for comparison.
This choice was proposed in ref.\ \cite{Caola:2013yja} for
the far-off-shell region, $M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell} > 2M_Z$,
which gives a significant contribution to the total cross section
\cite{Kauer:2012hd}. We note that at Higgs resonance, one has $\mu_H\approx \mu_\text{offshell}$.
When a full NLO calculation of the interference becomes available, the optimal
scale choice should be investigated in more detail.%
\footnote{(N)NLO corrections to signal-background interference in
$gg\ (\to H) \to WW\to\ell\bar{\nu}\bar{\ell}'\nu'$ have been calculated in
soft-collinear approximation in ref.\ \cite{Bonvini:2013jha}.}
When considering the continuum background only (section \ref{sec:ggcont}),
the fixed scale $\mu_R=\mu_F= (M_W+M_Z)/2$ is used.

The resonant process $gg\to H \to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$
(amplitude ${\cal M}_H$) and continuum process
$gg\to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$
(amplitude ${\cal M}_\text{cont}$) have the same initial and final states.
The amplitudes therefore interfere.  Furthermore,
amplitude contributions with $W^-W^+$ and $ZZ$
intermediate states interfere.   By choosing suitable lepton flavour
combinations, the $W^-W^+$ and $ZZ$ contributions can be separated.
Both are therefore gauge invariant.  We denote the amplitudes
corresponding to the graph sets (a), (b), (c) and (d) of
figure \ref{fig:graphs} with ${\cal M}_{H,WW}$, ${\cal M}_{\text{cont},WW}$,
${\cal M}_{H,ZZ}$ and ${\cal M}_{\text{cont},ZZ}$, respectively.
The interference is given by
\begin{equation}
I_{ij} = 2\,\mbox{Re}({\cal M}_i{\cal M}^\ast_j) =
|{\cal M}_i + {\cal M}_j|^2 - \rho_i - \rho_j \,,
\end{equation}
where $\rho_{i,j}= |{\cal M}_{i,j}|^2$.
As suggestive relative, symmetric, non-negative interference measure, we consider
\begin{equation}
R(i,j)=\frac{\sigma(\rho_i+\rho_j+I_{ij})}{\sigma(\rho_i+\rho_j)} \,,
\end{equation}
and apply it to Higgs signal and continuum background contributions
($i=H$, $j=\text{cont}$) as well as contributions from $WW$ and $ZZ$ intermediate
states ($i=WW$, $j=ZZ$):
\begin{align}
R_1 &= R(H,\text{cont}) \,, \\
R_3 &= R(WW,ZZ) \,. \label{eq:r3}
\end{align}
The contributions are described in section \ref{sec:process}.
When two interfering amplitude contributions are not viewed on an equal
footing, the interference as relative correction to the primary contribution
is a suggestive asymmetric measure:
\begin{equation}
\widetilde{R}(i,j)=\frac{\sigma(\rho_i+I_{ij})}{\sigma(\rho_i)} \,.
\end{equation}
Employing this definition, a Higgs-boson-search-inspired alternative to $R_1$ is
given by
\begin{equation}
R_2= \widetilde{R}(H,\text{cont}) \,.
\end{equation}
It measures the relative signal modification due to signal-background
interference, because the null hypothesis, i.e.\ the SM without Higgs,
is not sensitive to the interference.
In addition to $R_3$ given in eq.\ (\ref{eq:r3}), we also compute the $WW/ZZ$ interference as relative
correction to the cross section of the dominant intermediate weak boson state,
either $WW$ or $ZZ$ depending on the applied (search) selection cuts:
\begin{equation}
R_4 = \widetilde{R}(WW,ZZ)\qquad\text{and}\qquad R_5 = \widetilde{R}(ZZ,WW) \,,
\end{equation}
respectively.
Finally, for comparison the relative correction of the gluon-induced cross section
due to the subdominant intermediate weak boson state is given when interference is
neglected:
\begin{equation}
R_6 = \sigma(\rho_{ZZ})/\sigma(\rho_{WW})\qquad\text{and}\qquad
R_7 = \sigma(\rho_{WW})/\sigma(\rho_{ZZ}) \,,
\end{equation}
respectively.\footnote{%
Note that due to additional contributions from undetectable non-interfering
neutrino flavour combinations the phenomenological corrections are $3R_6$
and $R_7/3$.}

\subsection{Gluon-induced continuum background\label{sec:ggcont}}

First, we consider $WW/ZZ$ interference in the SM without Higgs boson.
Continuum weak-boson pair production in gluon scattering
\cite{Glover:1988fe,Glover:1988rg,Dicus:1987dj,%
Zecher:1994kb,Binoth:2005ua,Binoth:2006mf,Binoth:2008pr,Campbell:2011bn,%
Campbell:2011cu,Frederix:2011ss} formally contributes to
$pp\to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ at NNLO,\footnote{%
We note that electroweak corrections to $pp\to VV$ have been computed in
refs.\ \cite{Bierweiler:2012kw,Bierweiler:2013dja,Baglio:2013toa}
and threshold-resummed and approximate NNLO results for $pp\to WW$ in ref.\ \cite{Dawson:2013lya}.}
but is enhanced by the large gluon-gluon flux at the LHC (and is
further enhanced by Higgs search selection cuts \cite{Binoth:2005ua}).\footnote{%
We further note that cross sections for gluon-induced $VV+$jet production have
been calculated in refs.\
\cite{Melia:2012zg,Agrawal:2012df,Agrawal:2012as,Campanario:2012bh}.}
We note that $WW/ZZ$ interference in continuum production in quark-antiquark
scattering was calculated at LO in ref.\ \cite{Melia:2011tj} and found to be
negligible.
Here, results for the gluon-induced
continuum process $gg\to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ are presented
in table \ref{tab:ggcont}.
In addition to a minimal $M_{\ell\bar{\ell}} > 4$ GeV cut,
we also consider LHC standard cuts
for $WW$ production ($p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$,
$\sla{p}_T > 45$ GeV, $M_{\ell\bar{\ell}} > 12$ GeV,
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV) and $ZZ$ production
($p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$,
$|M_{\ell\bar{\ell}} - M_Z| < 15$ GeV) \cite{Aad:2013wqa,Chatrchyan:2013lba}.
The renormalisation and factorisation scales are set to
$(M_W+M_Z)/2$.
\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|l|ccc|cc|c|}
\cline{2-4}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$gg\ \to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{3}{|c}{} \\
\cline{5-7}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s}=8$ TeV} & \multicolumn{2}{c|}{interference} & \multicolumn{1}{c|}{correction} \\
\hline
cuts & $WW$ & $ZZ$ & $WW/ZZ$ & $R_3$ & $R_{4/5}$ & $R_{6/7}$ \\
\hline
minimal       & 18.780(5) & 2.4581(6) & 21.241(4) & 1.0001(3) & 1.0002(4) & 0.13089(5) \\
standard $WW$ & 6.243(2) & 0.03914(2) & 6.285(2) & 1.0003(4) & 1.0003(4) & 0.006270(3) \\
standard $ZZ$ & 2.3815(7) & 1.5267(4) & 3.9116(8) & 1.0009(3) & 1.0023(8) & 1.5599(6) \\
\hline
\end{tabular}}\\[.0cm]
\caption{\label{tab:ggcont}
Cross sections for $gg\ \to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions at $\sqrt{s}=8$ TeV for three sets of selection cuts calculated at loop-induced leading order.
The interference measures
$R_3=\sigma(|{\cal M}_{WW}+{\cal M}_{ZZ}|^2)/\sigma(|{\cal M}_{WW}|^2+|{\cal M}_{ZZ}|^2)$, $R_4=\sigma(|{\cal M}_{WW}|^2+2\,\mbox{Re}({\cal M}_{WW}{\cal M}^\ast_{ZZ}))/\sigma(|{\cal M}_{WW}|^2)$ and $R_5=\sigma(|{\cal M}_{ZZ}|^2+2\,\mbox{Re}({\cal M}_{WW}{\cal M}^\ast_{ZZ}))/\sigma(|{\cal M}_{ZZ}|^2)$ as well as the correction measures $R_6=\sigma(|{\cal M}_{ZZ}|^2)/\sigma(|{\cal M}_{WW}|^2)$ and $R_7=\sigma(|{\cal M}_{WW}|^2)/\sigma(|{\cal M}_{ZZ}|^2)$ are also displayed.  $R_{4/5} = R_4\ (R_5)$ and  $R_{6/7} = R_6\ (R_7)$ for minimal and standard $WW$ cuts (standard $ZZ$ cuts).
Minimal cuts: $M_{\ell\bar{\ell}} > 4$ GeV.
$WW$ standard cuts: $p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$,
$\sla{p}_T > 45$ GeV, $M_{\ell\bar{\ell}} > 12$ GeV and
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV.
$ZZ$ standard cuts: $p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$,
$|M_{\ell\bar{\ell}} - M_Z| < 15$ GeV.
$\gamma^\ast$ contributions are included in $ZZ$.
Cross sections are given for a single lepton flavour combination.
The integration error is displayed in brackets.
}
\end{table}
The results demonstrate that $WW/ZZ$ interference effects in gluon-fusion
continuum production are negligible.  Table \ref{tab:ggcont} also demonstrates
that the $ZZ$ correction to $WW$ production is of ${\cal O}(10\%)$ with minimal
selection cuts, but becomes negligible when standard $WW$ selection cuts are applied.
In contrast, the $WW$ correction to $ZZ$ production is of ${\cal O}(1)$
when standard $ZZ$ selection cuts are applied.

\subsection{Light Higgs boson signal\label{sec:lighthiggs}}

We now study signal-background and $WW/ZZ$ interference effects
in the Standard Model with a $126$ GeV Higgs boson motivated by
the recent discovery \cite{Aad:2012tfa,Chatrchyan:2012ufa}.
Results for minimal, standard and Higgs search cuts
are shown in tables \ref{tab:light_minimal}, \ref{tab:light_standard}
and \ref{tab:light_higgssearch}, respectively.
As above, a minimal $M_{\ell\bar{\ell}} > 4$ GeV cut is applied
to exclude the on-shell photon singularity.
In this section, we apply the following cuts as standard cuts for the
$WW \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ process
at the LHC:\footnote{The staggered $p_{T\ell}$ cut
accomodates the off-shell $W$ boson decay for $M_{WW}\approx 126$ GeV.
Leptons are ordered by decreasing $p_T$.}
$p_{T\ell,\text{1st}} > 25$ GeV, $p_{T\ell,\text{2nd}} > 15$ GeV, $|\eta_\ell| < 2.5$, $\sla{p}_T > 45$ GeV, $M_{\ell\bar{\ell}} > 12$ GeV and
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV.
As $H\to WW$ search cuts, we apply these standard cuts and
in addition $M_{\ell\bar{\ell}} <$ 50 GeV,
$\Delta\phi_{\ell\bar{\ell}} < 1.8$ and $0.75\,M_H < M_{T} < M_H$
\cite{Aad:2013wqa}.\footnote{Very similar
selection cuts are employed in ref.\ \cite{Chatrchyan:2013lba}.}
The cut on the transverse mass
\begin{gather}
\label{eq:MT1}
M_{T1}=\sqrt{(M_{T,\ell\bar{\ell}}+\sla{p}_{T})^2-({\bf{p}}_{T,\ell\bar{\ell}}+{\sla{\bf{p}}}_T)^2}\quad\text{with}\quad M_{T,\ell\bar{\ell}}=\sqrt{p_{T,\ell\bar{\ell}}^2+M_{\ell\bar{\ell}}^2}
\end{gather}
strongly reduces the contribution from
$M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell} > M_H$ \cite{Campbell:2011cu}.\footnote{
In the absence of additional final state particles,
the expression for $M_{T1}$ simplifies due to ${\sla{\bf{p}}}_T=-{\bf{p}}_{T,\ell\ell}$.
}
The renormalisation and factorisation scales are set to
$\mu_R=\mu_F= M_H/2$, except in table \ref{tab:light_minimal_dynscale}.

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|ccc|cc|}
\cline{2-4}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s}=8$ TeV,} & \multicolumn{2}{|c}{} \\
\cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$M_H=126$ GeV, min.\ cuts} & \multicolumn{2}{c|}{interference} \\
\hline
$VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ \\
\hline
$WW$ & 17.593(4) & 20.978(5) & 36.480(8) & 0.9458(3) & 0.8811(6) \\
$ZZ$ & 0.9308(3) & 2.7543(7) & 3.436(2) & 0.9324(5) & 0.732(2) \\
$WW/ZZ$ & 17.729(3) & 23.742(5) & 39.125(8) & 0.9434(3) & 0.8677(6) \\
\hline
$R_3$ & 0.9571(3) & 1.0004(3) & 0.9802(3) &  \multicolumn{2}{|c}{}\\
$R_4$ & 0.9548(3) & 1.0004(4) & 0.9783(3) &  \multicolumn{2}{|c}{}\\
$R_6$ & 0.05291(2) & 0.13129(5) & 0.09419(5) &  \multicolumn{2}{|c}{}\\
\cline{1-4}
\end{tabular}}\\[.0cm]
\caption{\label{tab:light_minimal}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV.
The off-shell Higgs cross section at $M_H=126$ GeV, the gluon-induced continuum cross section and
the sum including interference are given.
The signal-background interference measures
$R_1=\sigma(|{\cal M}_H + {\cal M}_\text{cont}|^2)/\sigma(|{\cal M}_H|^2 + |{\cal M}_\text{cont}|^2)$
and
$R_2=\sigma(|{\cal M}_H|^2+2\,\mbox{Re}({\cal M}_H{\cal M}^\ast_\text{cont}))/\sigma(|{\cal M}_H|^2)$
as well as several $WW/ZZ$-related interference and correction measures
(see table \protect\ref{tab:ggcont})
are also displayed.
A minimal $M_{\ell\bar{\ell}} > 4$ GeV cut is applied.
Other details as in table \protect\ref{tab:ggcont}.
}
\end{table}

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|ccc|cc|}
\cline{2-4}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s}=8$ TeV,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$M_H=126$ GeV, min.\ cuts,} & \multicolumn{2}{|c}{} \\
\cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\mu_R=\mu_F= M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell}/2$} & \multicolumn{2}{c|}{interference} \\
\hline
$VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ \\
\hline
$WW$ & 17.318(4) & 16.925(4) & 32.803(8) & 0.9580(3) & 0.9169(6) \\
$ZZ$ & 0.8822(2) & 2.1553(6) & 2.872(1) & 0.9455(4) & 0.813(2) \\
$WW/ZZ$ & 17.402(3) & 19.084(4) & 34.884(7) & 0.9561(3) & 0.9079(5) \\
\hline
$R_3$ & 0.9562(3) & 1.0002(3) & 0.9778(3) &  \multicolumn{2}{|c}{}\\
$R_4$ & 0.9540(3) & 1.0002(4) & 0.9759(4) &  \multicolumn{2}{|c}{}\\
$R_6$ & 0.05094(2) & 0.12735(5) & 0.08756(4) &  \multicolumn{2}{|c}{}\\
\cline{1-4}
\end{tabular}}\\[.0cm]
\caption{\label{tab:light_minimal_dynscale}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV and $M_H=126$ GeV.
The dynamic scale
$\mu_R=\mu_F= M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell}/2$
is employed.
Details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

For the Higgs signal cross section, we find negative $WW/ZZ$ interference of
approximately $5\%$, whereas no significant $WW/ZZ$ interference occurs
for the continuum background in agreement with our findings in
section \ref{sec:ggcont}.  This difference can be traced back to the fact that
for $H\to VV$ with $M_V < M_H < 2M_V$ the most likely configuration features
one  weak boson ($W$ as well as $Z$) that is (far) off-shell, whereas for the
continuum background no such kinematic constraint exists.

Signal-background interference effects for minimal, standard and Higgs search cuts
are compatible with what would be expected on the basis of different-flavour final
state results \cite{Campbell:2011cu,Kauer:2012hd}.  With minimal and standard cuts,
the signal-background interference of the subdominant $ZZ$ contribution
when measured with $R_2$ is significantly larger than for the dominant $WW$
contribution.  Due to the suppression of the $ZZ$ contribution (high with standard
and search cuts), the induced change of the overall signal-background interference
is, however, at most at the one-percentage-point level.    As seen in table
\ref{tab:light_higgssearch}, the application of a $M_{T1} < M_H$ cut suppresses
the interference effects even stronger for the $ZZ$ contribution than for the
$WW$ contribution.

Results for $\mu_R=\mu_F= M_{\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell}/2$ and minimal cut are displayed in table \ref{tab:light_minimal_dynscale}.  Signal-background interference effects are smaller compared to $\mu_R=\mu_F= M_H/2$, because the dynamic scale reduces
the contribution of the region with strong interference, due to the smaller strong coupling at higher $VV$ invariant mass.  $R_2$ decreases by 3--8 percentage points
relative to table \ref{tab:light_minimal}.  Changes in the $WW/ZZ$
interference measures are at the sub-percentage point level.

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|ccc|cc|}
\cline{2-4}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s}=8$ TeV,} & \multicolumn{2}{|c}{} \\
\cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$M_H=126$ GeV, std.\ cuts} & \multicolumn{2}{c|}{interference} \\
\hline
$VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ \\
\hline
$WW$ & 3.606(2) & 8.084(3) & 10.442(3) & 0.8932(3) & 0.654(1) \\
$ZZ$ & 0.009752(6) & 0.05099(3) & 0.05387(3) & 0.8868(7) & 0.295(5) \\
$WW/ZZ$ & 3.545(3) & 8.17(2) & 10.46(2) & 0.893(3) & 0.646(8) \\
\hline
$R_3$ & 0.9804(8) & 1.004(3) & 0.996(2) &  \multicolumn{2}{|c}{}\\
$R_4$ & 0.9803(8) & 1.004(3) & 0.996(2) &  \multicolumn{2}{|c}{}\\
$R_6$ & 0.002704(2) & 0.006308(5) & 0.005159(4) &  \multicolumn{2}{|c}{}\\
\cline{1-4}
\end{tabular}}\\[.0cm]
\caption{\label{tab:light_standard}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV and $M_H=126$ GeV.
$WW$ standard cuts for $M_{WW}\approx 126$ GeV are applied:
$p_{T\ell,\text{1st}} > 25$ GeV, $p_{T\ell,\text{2nd}} > 15$ GeV, $|\eta_\ell| < 2.5$, $\sla{p}_T > 45$ GeV, $M_{\ell\bar{\ell}} > 12$ GeV and
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV.
Other details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|ccc|cc|}
\cline{2-4}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s}=8$ TeV,} & \multicolumn{2}{|c}{} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{$M_H=126$ GeV,} & \multicolumn{2}{|c}{} \\
\cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{3}{|c|}{Higgs search cuts} & \multicolumn{2}{c|}{interference} \\
\hline
$VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ \\
\hline
$WW$ & 2.9303(7) & 0.7836(4) & 3.6649(8) & 0.9868(4) & 0.9833(4) \\
$ZZ$ & 0.004658(3) & 0.002851(2) & 0.007494(3) & 0.9979(6) & 0.9966(9) \\
$WW/ZZ$ & 2.8758(7) & 0.7864(4) & 3.6131(8) & 0.9866(3) & 0.9829(4) \\
\hline
$R_3$ & 0.9799(4) & 0.9999(8) & 0.9839(3) &  \multicolumn{2}{|c}{}\\
$R_4$ & 0.9798(4) & 0.9999(8) & 0.9838(3) &  \multicolumn{2}{|c}{}\\
$R_6$ & 0.0015898(9) & 0.003638(3) & 0.002045(1) &  \multicolumn{2}{|c}{}\\
\cline{1-4}
\end{tabular}}\\[.0cm]
\caption{\label{tab:light_higgssearch}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV and $M_H=126$ GeV.
Higgs search cuts are applied, i.e.\ $WW$ standard cuts (as in
table \protect\ref{tab:light_standard}) and $M_{\ell\bar{\ell}} <$ 50 GeV,
$\Delta\phi_{\ell\bar{\ell}} < 1.8$ and $0.75\,M_H < M_{T1} < M_H$.
The transverse mass $M_{T1}$
is defined in eq.\ (\ref{eq:MT1}) in the main text.
Other details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

\subsection{Heavy Higgs boson signal\label{sec:heavyhiggs}}

We now study the signal-background and $WW/ZZ$ interference in the
same-flavour decay mode for a SM-like Higgs boson with mass between 200 GeV
and 1 TeV.  A priori, the search for a SM Higgs boson in the intermediate and
heavy mass range is a crucial task.  It continues without premature conclusion
about the exact nature of the discovered boson at 126 GeV.
More specifically, one can search for a heavier SM Higgs boson with the advantage
that assumptions about the realized SM extension are not required.  Given that the
discovered 126 GeV boson appears to have all characteristics of the SM Higgs boson,
an alternative, slightly better motivated approach is to search for a second,
heavier Higgs boson.  A first
proposal for a framework for the interpretation of the continuing LHC Higgs
searches at masses other than 126 GeV has been set out in ref.\
\cite{Heinemeyer:2013tqa}.  These searches are being conducted in
a model-independent way by suitably rescaling SM predictions in order to
preserve unitarity cancellations at high energies as well as by considering
specific BSM benchmark models.
It is straightforward to apply the rescaling procedure
described in ref.\ \cite{Heinemeyer:2013tqa} to the results
presented in this section.  We note that the program \textsf{gg2VV} can be used
to study Higgs-continuum interference and off-shell effects for BSM scenarios
with a SM-like Higgs boson with rescaled $Hgg$, $HWW$ and
$HZZ$ couplings.  The analysis of BSM benchmark models is beyond
the scope of this paper.

In this context, the question arises if Higgs-Higgs interference effects
are small.  The SM-like Higgs boson at 126 GeV appears to have the expected
tiny width of approx.\ 4 MeV.\footnote{Bounding $\Gamma_H$ at $M_H\approx 126$ GeV
with LHC and Tevatron data has been studied in refs.\ \cite{Barger:2012hv,Dobrescu:2012td,Ellis:2013lra,Djouadi:2013qya,CMS:yva,Martin:2012xc,Cheung:2013kla,Dixon:2013haa,Belanger:2013xza,Caola:2013yja}.}
The overlap of the extremely narrow Breit-Wigner lineshape with the lineshape of a
heavier Higgs boson with experimentally discriminable mass is insignificant.
Higgs-Higgs interference effects are thus expected to be
negligible provided that off-shell effects \cite{Kauer:2012hd,Kauer:2013cga}
are suppressed by the search selection cuts.\footnote{%
We note that the role of interference effects in analysing the tensor structure
of the $HZZ$ coupling has been studied in ref.\ \cite{Chen:2013waa}.}

Both approaches, a heavy SM or BSM Higgs boson search at the LHC (denoted by
``SM'' and ``BSM'' below), have been
pursued by ATLAS and CMS for Higgs masses up to 1 TeV.
In the following, we quote the experimental exclusion limits for a heavy Higgs
boson with SM properties.
The $H\to ZZ\to \ell\bar{\ell}\ell\bar{\ell}$ (same or different flavour) and
$H\to ZZ\to \ell\bar{\ell}\nu\bar{\nu}$ channels
yield the strongest limits.\footnote{%
All exclusion limits are given at 95\% confidence level.}
The four-charged-lepton channel has been studied by ATLAS (SM)
\cite{ATLAS:2013nma}
and CMS (SM)
\cite{CMS:xwa}.
An exclusion region of 130--827 GeV is given in the latter work.
Secondly, the $H\to ZZ\to \ell\bar{\ell}\nu\bar{\nu}$ channel has been
studied by CMS (SM and BSM) \cite{CMS-PAS-HIG-13-014}
and ATLAS (SM, $M_H < 600$ GeV, 7 TeV data)
\cite{Aad:2012ora}.
CMS and ATLAS exclude a mass in the ranges 248--930 GeV
and 319--558 GeV, respectively.
Furthermore, the $H\to WW\to \ell\bar{\nu}\bar{\ell}\nu$ channel has been analysed
by ATLAS (SM) \cite{ATLAS-CONF-2013-067} and CMS (SM)
\cite{CMS:bxa},
excluding the mass ranges 260--642 GeV and 128-600 GeV, respectively.
The semileptonic decay modes $H\to WW\to \ell\nu jj$
\cite{Aad:2012me,CMS-PAS-HIG-13-008}
and $H\to ZZ\to \ell\bar{\ell} jj$
\cite{Aad:2012oxa}
have also been studied by ATLAS (SM) and CMS (SM and BSM).
A combined analysis of $H\to ZZ$ and $H\to WW$ channels in the
mass range 145 GeV to 1 TeV has been carried out by CMS \cite{Chatrchyan:2013yoa}
and an update is in preparation \cite{JianWangTalk}.

\subsubsection{Minimal cuts}

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|c|ccc|cc|c|}
\cline{3-5}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{3}{|c}{} \\
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s} = 8$ TeV,} & \multicolumn{3}{|c}{} \\
\cline{6-7}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{minimal cut} & \multicolumn{2}{c|}{interference} & \multicolumn{1}{|c}{}\\
\hline
\multicolumn{1}{|c|}{$M_H$ [GeV]} & $VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ & $H$/cont \\
\hline
\multirow{7}{*}{200} & $WW$ & 23.662(7) & 17.781(5) & 41.677(9) & 1.0057(3) & 1.0099(5) & 1.3307(6) \\
 & $ZZ$ & 3.125(1) & 2.327(2) & 5.565(2) & 1.0208(4) & 1.0363(7) & 1.3431(8) \\
 & $WW/ZZ$ & 26.781(7) & 20.114(5) & 47.242(9) & 1.0074(3) & 1.0129(5) & 1.3315(5) \\
\cline{2-8}
 & $R_3$ & 0.9998(4) & 1.0003(4) & 1.0000(3) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 0.9998(4) & 1.0003(4) & 1.0000(3) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.13208(6) & 0.13086(8) & 0.13354(5) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{400} & $WW$ & 8.548(3) & 14.078(4) & 22.405(5) & 0.9902(3) & 0.9742(8) & 0.6072(3) \\
 & $ZZ$ & 1.5053(7) & 1.8298(7) & 3.318(1) & 0.9948(4) & 0.9886(9) & 0.8226(5) \\
 & $WW/ZZ$ & 10.053(3) & 15.912(4) & 25.727(5) & 0.9908(3) & 0.9764(7) & 0.6318(3) \\
\cline{2-8}
 & $R_3$ & 1.0000(5) & 1.0002(4) & 1.0002(3) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0000(5) & 1.0003(4) & 1.0002(3) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.1761(1) & 0.12998(6) & 0.14809(5) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{600} & $WW$ & 1.430(3) & 12.354(4) & 14.142(5) & 1.0260(5) & 1.250(5) & 0.1157(2) \\
 & $ZZ$ & 0.2600(4) & 1.6014(7) & 1.9016(8) & 1.0216(6) & 1.154(5) & 0.1624(3) \\
 & $WW/ZZ$ & 1.690(3) & 13.959(4) & 16.047(5) & 1.0255(4) & 1.236(4) & 0.1211(2) \\
\cline{2-8}
 & $R_3$ & 1.000(2) & 1.0003(4) & 1.0002(4) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.000(3) & 1.0003(5) & 1.0003(5) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.1819(4) & 0.12963(7) & 0.13447(7) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{800} & $WW$ & 0.2857(7) & 11.286(3) & 11.806(3) & 1.0202(4) & 1.82(2) & 0.02531(6) \\
 & $ZZ$ & 0.0527(2) & 1.4602(6) & 1.5425(6) & 1.0195(6) & 1.56(2) & 0.03611(8) \\
 & $WW/ZZ$ & 0.3384(7) & 12.750(3) & 13.352(3) & 1.0201(4) & 1.78(2) & 0.02654(6) \\
\cline{2-8}
 & $R_3$ & 1.000(3) & 1.0003(4) & 1.0003(4) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.000(4) & 1.0003(4) & 1.0003(4) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.1845(6) & 0.12938(6) & 0.13065(6) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{1000} & $WW$ & 0.0759(4) & 10.547(3) & 10.760(3) & 1.0129(4) & 2.81(6) & 0.00720(3) \\
 & $ZZ$ & 0.01409(4) & 1.3616(6) & 1.3939(6) & 1.0132(6) & 2.29(6) & 0.01034(3) \\
 & $WW/ZZ$ & 0.0900(4) & 11.911(3) & 12.157(3) & 1.0129(4) & 2.73(5) & 0.00755(3) \\
\cline{2-8}
 & $R_3$ & 1.000(5) & 1.0003(4) & 1.0003(4) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.000(6) & 1.0003(4) & 1.0003(4) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.1856(9) & 0.12910(7) & 0.12955(7) &  \multicolumn{3}{|c}{}\\
\cline{1-5}
\end{tabular}}\\[.0cm]
\caption{\label{tab:heavy_minimal}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV with a Higgs boson mass in the range 200 GeV to 1 TeV.
A minimal $M_{\ell\bar{\ell}} > 4$ GeV cut is applied.
Other details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

As in section \ref{sec:lighthiggs},
the renormalisation and factorisation scales are set to
$\mu_R=\mu_F= M_H/2$.
Again, we first give results for
a minimal $M_{\ell\bar{\ell}} > 4$ GeV cut.  To cover the heavy Higgs
mass range of interest, results are shown in table \ref{tab:heavy_minimal}
for a Higgs mass of 200 GeV, 400 GeV, 600 GeV, 800 GeV and 1 TeV.

In contrast to the light Higgs case, we find that $WW/ZZ$ interference for
the Higgs signal is negligible for all considered heavy
Higgs masses.  This further supports our conclusion in section \ref{sec:lighthiggs},
which is that non-negligible $WW/ZZ$ interference effects occur only if
at least one
weak boson of the pair is dominantly off-shell due to kinematic constraints.
This assertion is also confirmed by the results given in sections
\ref{sec:heavyhiggszz} and \ref{sec:heavyhiggsww},
where heavy Higgs search selection cuts are applied.
With minimal cut the $ZZ$ correction is of ${\cal O}(10$--$20\%)$
and effects a change of the signal-background interference at the few
percentage-point level.  For Higgs masses beyond 700 GeV, signal-over-background
ratios are at the percent and per mil level, and
signal-background interference ($R_2-1$) is of ${\cal O}(1)$.

\subsubsection{$H\to ZZ$ search cuts\label{sec:heavyhiggszz}}

While the four-charged-lepton channel has the highest sensitivity in the
$H\to ZZ$ search for masses below 500 GeV, the
$H\to ZZ\to \ell\bar{\ell}\nu\bar{\nu}$ channel considered here dominates above
500 GeV \cite{JianWangTalk}.
As above, we consider the Higgs mass range 200 GeV--1 TeV and note that
for the lower end of this range the $H\to WW\to \ell\bar{\ell}\nu\bar{\nu}$
process can contribute as much as 70\% to the signal after selection cuts
\cite{Aad:2012ora}.  For our calculations, we adopt the analysis strategy
of the recent study presented in ref.\ \cite{CMS-PAS-HIG-13-014}.
In more detail, as $H\to ZZ$ search cuts we apply the basic selection cuts
$\ p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$ and
$|M_{\ell\bar{\ell}} - M_Z| < 15$ GeV.
In addition, to select the signal contribution with $M_{ZZ}\approx M_H$
Higgs-mass-dependent bounds are imposed on
the transverse mass $M_{T2}$ and $\sla{p}_T$.\footnote{%
In our calculation, $p_{T,\ell\bar{\ell}}$ is constrained by the
$\sla{p}_T$ cut.}
$M_{T2}$ is given by
\begin{gather}
\label{eq:MT2}
M_{T2}=\sqrt{ \left(M_{T,\ell\bar{\ell}}+\sla{M}_{T}\right)^2-({\bf{p}}_{T,\ell\bar{\ell}}+{\sla{\bf{p}}}_T)^2 }\quad\text{with}\quad
\sla{M}_{T}=\sqrt{\sla{p}_{T}^2+M_{\ell\bar{\ell}}^2} \,.
\end{gather}
$M_{T,\ell\bar{\ell}}$ is defined in eq.\ (\ref{eq:MT1}).\footnote{%
In the absence of additional final state particles:
$M_{T2}=2M_{T,\ell\bar{\ell}}=2\sla{M}_{T}$.}
The employed $M_{T2}$ and $\sla{p}_T$ bounds are given in table \ref{tab:zzcuts}.
Note that an upper bound is imposed on $M_{T2}$ for all Higgs masses
in order to suppress increasingly large interference effects at high invariant
masses.
\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|l|ccccc|}
\hline
$M_H$ [GeV] &                   200 & 400 & 600 & 800 & 1000 \\
\hline
$M_{T2}$ lower bound [GeV] &    180 & 300 & 420 & 500 & 550 \\
$M_{T2}$ upper bound [GeV] &    220 & 450 & 700  & 900 & 1100 \\
$\sla{p}_T$ lower bound [GeV] &  20 &  90 & 110 & 150 & 170 \\
\hline
\end{tabular}}\\[.0cm]
\caption{\label{tab:zzcuts}
Higgs-mass-dependent $H\to ZZ$ search cuts.
Bounding cuts are applied to the transverse
mass $M_{T2}$ defined in eq.\ (\ref{eq:MT2}) in the main text
and the missing transverse momentum $\sla{p}_T$.
}
\end{table}

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|c|ccc|cc|c|}
\cline{3-5}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\ell}\nu_\ell\bar{\nu}_\ell$,} & \multicolumn{3}{|c}{} \\
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s} = 8$ TeV,} & \multicolumn{3}{|c}{} \\
\cline{6-7}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$H\to ZZ$ cuts} & \multicolumn{2}{c|}{interference} & \multicolumn{1}{|c}{} \\
\hline
\multicolumn{1}{|c|}{$M_H$ [GeV]} & $VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ & $H$/cont \\
\hline
\multirow{7}{*}{200} & $ZZ$ & 1.7507(5) & 0.6224(3) & 2.5153(6) & 1.0600(4) & 1.0813(5) & 2.813(2) \\
 & $WW$ & 2.2529(9) & 1.0064(6) & 3.338(2) & 1.0243(5) & 1.0351(7) & 2.239(2) \\
 & $ZZ/WW$ & 3.997(1) & 1.6302(6) & 5.848(2) & 1.0392(3) & 1.0552(5) & 2.452(2) \\
\cline{2-8}
 & $R_3$ & 0.9983(4) & 1.0009(6) & 0.9990(3) &  \multicolumn{3}{|c}{}\\
 & $R_5$ & 0.9961(8) & 1.002(2) & 0.9976(7) &  \multicolumn{3}{|c}{}\\
 & $R_7$ & 1.2869(6) & 1.617(2) & 1.3272(6) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{400} & $ZZ$ & 0.8919(3) & 0.07674(7) & 0.9508(3) & 0.9816(5) & 0.9800(5) & 11.62(1) \\
 & $WW$ & 0.02340(2) & 0.01400(3) & 0.03909(4) & 1.045(2) & 1.073(2) & 1.672(4) \\
 & $ZZ/WW$ & 0.9154(3) & 0.09074(7) & 0.9901(3) & 0.9840(4) & 0.9824(5) & 10.088(9) \\
\cline{2-8}
 & $R_3$ & 1.0001(5) & 1.000(2) & 1.0001(5) &  \multicolumn{3}{|c}{}\\
 & $R_5$ & 1.0001(5) & 1.000(2) & 1.0001(5) &  \multicolumn{3}{|c}{}\\
 & $R_7$ & 0.02623(2) & 0.1824(4) & 0.04111(4) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{600} & $ZZ$ & 0.15889(9) & 0.02187(1) & 0.18949(9) & 1.0483(7) & 1.0549(8) & 7.263(5) \\
 & $WW/10^{-4}$ & $0.232(2)$ & 1.57(3) & 1.98(4) & 1.10(3) & 1.8(2) & 0.148(3) \\
 & $ZZ/WW$ & 0.15891(9) & 0.02204(1) & 0.18970(9) & 1.0484(7) & 1.0551(8) & 7.211(5) \\
\cline{2-8}
 & $R_3$ & 1.0000(8) & 1.0002(7) & 1.0001(7) &  \multicolumn{3}{|c}{}\\
 & $R_5$ & 1.0000(8) & 1.0002(7) & 1.0001(7) &  \multicolumn{3}{|c}{}\\
 & $R_7$ & 0.000146(2) & 0.0072(2) & 0.00105(2) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{800} & $ZZ$ & 0.03191(2) & 0.010964(4) & 0.05113(2) & 1.1928(6) & 1.2590(8) & 2.910(2) \\
 & $WW/10^{-5}$ & 0.069(3) & 2.91(9) & 3.1(1) & 1.04(5) & 3(2) & 0.024(2) \\
 & $ZZ/WW$ & 0.03191(2) & 0.010994(4) & 0.05117(2) & 1.1927(6) & 1.2591(8) & 2.902(2) \\
\cline{2-8}
 & $R_3$ & 1.0000(7) & 1.0001(6) & 1.0000(5) &  \multicolumn{3}{|c}{}\\
 & $R_5$ & 1.0000(7) & 1.0001(6) & 1.0000(5) &  \multicolumn{3}{|c}{}\\
 & $R_7$ & $2.2(1)\!\cdot\! 10^{-5}$ & 0.00265(9) & 0.00061(2) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{1000} & $ZZ$ & 0.008415(3) & 0.007268(2) & 0.020719(6) & 1.3211(5) & 1.5984(9) & 1.1578(5) \\
 & $WW/10^{-5}$ & 0.0086(4) & 1.22(5) & 1.25(5) & 1.02(6) & 4(8) & 0.0071(5) \\
 & $ZZ/WW$ & 0.008416(3) & 0.007281(2) & 0.020733(6) & 1.3208(5) & 1.5984(9) & 1.1558(5) \\
\cline{2-8}
 & $R_3$ & 1.0000(5) & 1.0001(4) & 1.0000(4) &  \multicolumn{3}{|c}{}\\
 & $R_5$ & 1.0000(5) & 1.0001(4) & 1.0000(4) &  \multicolumn{3}{|c}{}\\
 & $R_7$ & 0.00001(1) & 0.00168(7) & 0.00060(3) &  \multicolumn{3}{|c}{}\\
\cline{1-5}
\end{tabular}}\\[.0cm]
\caption{\label{tab:heavy_HtoZZ}
Cross sections and interference measures for
$gg\ (\to H)\to ZZ/WW \to \ell\bar{\ell}\nu_\ell\bar{\nu}_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV with a Higgs boson mass in the range 200 GeV to 1 TeV.
$H\to ZZ$ search cuts are applied:
$p_{T\ell} > 20$ GeV, $|\eta_\ell| < 2.5$, $|M_{\ell\bar{\ell}} - M_Z| < 15$ GeV
and the $M_H$-dependent cuts displayed in table \protect\ref{tab:zzcuts}.
Other details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

Applying these $H\to ZZ$ search cuts, we obtain the results shown
in table \ref{tab:heavy_HtoZZ}, which demonstrate the absence of
$WW/ZZ$ interference for the Higgs signal and continuum background.
Signal-background interference ($R_2-1$) ranges from $-2\%$ to
$+60\%$ and is not affected by the $WW$ correction for $M_H\gtrsim 400$ GeV.

\subsubsection{$H\to WW$ search cuts\label{sec:heavyhiggsww}}

We adopt the $H\to WW$ search cuts of ref.\ \cite{CMS:bxa}.\footnote{In ref.\ \cite{ATLAS-CONF-2013-067} the same-lepton-flavour final states have not been included.}
To simulate detector coverage, select $WW\to \ell\bar{\nu}\bar{\ell}\nu$
and suppress
reducible backgrounds the following cuts are applied:
$|\eta_\ell| < 2.5$, $\sla{p}_T > 20$ GeV, $M_{\ell\bar{\ell}} > 12$ GeV,
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV and
$p_{T,\ell\bar{\ell}} > 45$ GeV.
To select the $H\to WW$ signal, following ref.\ \cite{CMS:bxa}, we
define the transverse mass
\begin{gather}
\label{eq:MT3}
M_{T3}= \sqrt{2\,p_{T,\ell\bar{\ell}}\:\sla{p}_T(1-\cos\Delta\phi_{\ell\bar{\ell},\text{miss}})}\,,
\end{gather}
where $\Delta\phi_{\ell\bar{\ell},\text{miss}}$ is the angle between
${\bf{p}}_{T,\ell\bar{\ell}}$ and ${\sla{\bf{p}}}_T$,
and impose the cuts\ \,
$p_{T\ell,\text{max}} > 0.2\,M_H$,
$p_{T\ell,\text{min}} > 25$ GeV,
$M_{\ell\bar{\ell}} < 90$ GeV for $M_H=200$ GeV and
$M_{\ell\bar{\ell}} < M_H - 100$ GeV for $M_H\geq 400$ GeV,
$\Delta\phi_{\ell\bar{\ell}} < 100^\circ$ for $M_H=200$ GeV and
$\Delta\phi_{\ell\bar{\ell}} < 175^\circ$ for $M_H\geq 400$ GeV,
and 120 GeV $< M_{T3} < M_H$.\footnote{In the absence of additional final
state particles: $M_{T3}=2\,p_{T,\ell\bar{\ell}}=2\,\sla{p}_T$.}

\begin{table}[t]
\vspace{0.4cm}
\renewcommand{\arraystretch}{1.2}
\centering
{\footnotesize
\begin{tabular}{|c|c|ccc|cc|c|}
\cline{3-5}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$gg\ (\to H)\to VV \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$,} & \multicolumn{3}{|c}{} \\
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$\sigma$ [fb], $pp$, $\sqrt{s} = 8$ TeV,} & \multicolumn{3}{|c}{} \\
\cline{6-7}
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$H\to WW$ cuts} & \multicolumn{2}{c|}{interference} & \multicolumn{1}{|c}{} \\
\hline
\multicolumn{1}{|c|}{$M_H$ [GeV]} & $VV$ & $H$ & cont & $|H$+cont$|^2$ & $R_1$ & $R_2$ & $H$/cont \\
\hline
\multirow{7}{*}{200} & $WW$ & 3.686(2) & 1.5060(8) & 5.614(2) & 1.0812(4) & 1.1144(6) & 2.448(2) \\
 & $ZZ$ & 0.008678(7) & 0.003983(3) & 0.012141(8) & 0.9589(8) & 0.940(2) & 2.179(3) \\
 & $WW/ZZ$ & 3.701(2) & 1.5100(8) & 5.633(2) & 1.0809(4) & 1.1139(6) & 2.451(2) \\
\cline{2-8}
 & $R_3$ & 1.0017(5) & 1.0000(7) & 1.0012(4) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0017(5) & 1.0000(7) & 1.0012(4) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.002354(2) & 0.002645(3) & 0.002163(2) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{400} & $WW$ & 2.5009(9) & 0.6197(3) & 2.9806(9) & 0.9551(4) & 0.9440(5) & 4.036(3) \\
 & $ZZ$ & 0.03929(3) & 0.00983(2) & 0.04896(4) & 0.997(1) & 0.996(2) & 3.995(9) \\
 & $WW/ZZ$ & 2.5401(9) & 0.6298(4) & 3.0297(9) & 0.9558(4) & 0.9448(5) & 4.033(3) \\
\cline{2-8}
 & $R_3$ & 1.0000(5) & 1.0004(7) & 1.0000(5) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0000(5) & 1.0004(7) & 1.0000(5) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.01571(2) & 0.01587(4) & 0.01643(2) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{600} & $WW$ & 0.5260(3) & 0.2659(2) & 0.8386(4) & 1.0590(6) & 1.0888(9) & 1.978(2) \\
 & $ZZ$ & 0.006632(5) & 0.00347(2) & 0.01099(2) & 1.087(3) & 1.133(4) & 1.909(8) \\
 & $WW/ZZ$ & 0.5326(3) & 0.2694(2) & 0.8496(4) & 1.0593(6) & 1.0893(9) & 1.977(2) \\
\cline{2-8}
 & $R_3$ & 1.0000(7) & 1.0002(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0000(7) & 1.0002(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.01261(2) & 0.01306(6) & 0.01310(3) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{800} & $WW$ & 0.10346(5) & 0.10562(5) & 0.23894(9) & 1.1428(6) & 1.289(2) & 0.9795(7) \\
 & $ZZ$ & 0.0012318(8) & 0.001252(9) & 0.00291(1) & 1.173(6) & 1.35(2) & 0.984(7) \\
 & $WW/ZZ$ & 0.10469(5) & 0.10689(5) & 0.24185(9) & 1.1431(6) & 1.289(2) & 0.9794(7) \\
\cline{2-8}
 & $R_3$ & 1.0000(7) & 1.0001(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0000(7) & 1.0001(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.01191(1) & 0.01186(9) & 0.01219(4) &  \multicolumn{3}{|c}{}\\
\hline
\multirow{7}{*}{1000} & $WW$ & 0.02426(2) & 0.04357(2) & 0.08049(3) & 1.1867(6) & 1.522(2) & 0.5568(4) \\
 & $ZZ$ & 0.0002958(2) & 0.000512(4) & 0.000986(4) & 1.220(7) & 1.60(2) & 0.577(4) \\
 & $WW/ZZ$ & 0.02455(2) & 0.04408(2) & 0.08148(3) & 1.1871(6) & 1.523(2) & 0.5570(4) \\
\cline{2-8}
 & $R_3$ & 1.0000(7) & 1.0001(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_4$ & 1.0000(7) & 1.0001(7) & 1.0000(6) &  \multicolumn{3}{|c}{}\\
 & $R_6$ & 0.01219(1) & 0.01176(8) & 0.01225(5) &  \multicolumn{3}{|c}{}\\
\cline{1-5}
\end{tabular}}\\[.0cm]
\caption{\label{tab:heavy_HtoWW}
Cross sections and interference measures for
$gg\ (\to H)\to WW/ZZ \to \ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ in $pp$ collisions
at $\sqrt{s}=8$ TeV with a Higgs boson mass in the range 200 GeV to 1 TeV.
$H\to WW$ search cuts are applied:
$p_{T\ell,\text{max}} > 0.2\,M_H$,
$p_{T\ell,\text{min}} > 25$ GeV, $|\eta_\ell| < 2.5$,
$\sla{p}_T > 20$ GeV, $p_{T,\ell\bar{\ell}} > 45$ GeV,
$M_{\ell\bar{\ell}} > 12$ GeV,
$|M_{\ell\bar{\ell}} - M_Z| > 15$ GeV,
$M_{\ell\bar{\ell}} < 90$ GeV for $M_H=200$ GeV and
$M_{\ell\bar{\ell}} < M_H - 100$ GeV for $M_H\geq 400$ GeV,
$\Delta\phi_{\ell\bar{\ell}} < 100^\circ$ for $M_H=200$ GeV and
$\Delta\phi_{\ell\bar{\ell}} < 175^\circ$ for $M_H\geq 400$ GeV,
120 GeV $< M_{T3} < M_H$.
The transverse mass $M_{T3}$
is defined in eq.\ (\ref{eq:MT3}) in the main text.
Other details as in table \protect\ref{tab:light_minimal}.
}
\end{table}

With these $H\to WW$ search cuts, we obtain the results shown
in table \ref{tab:heavy_HtoWW}.  The cuts reduce the $ZZ$ correction
to the one-percent level, and the signal-background interference
is dominated by the $WW$ contribution.

\section{Conclusions\label{sec:conclusions}}

We studied $WW/ZZ$ interference for Higgs signal and continuum background as well
as signal-background interference
for same-flavour $\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$ final
states produced in gluon-gluon scattering at the LHC for light and heavy
Higgs masses with minimal and realistic experimental selection cuts.
For the signal cross section, we find
$WW/ZZ$ interference effects of ${\cal O}(5\%)$ at $M_H=126$ GeV.
For $M_H\ge 200$ GeV, we find that $WW/ZZ$ interference is negligible.
For the $gg$ continuum background, we also find that $WW/ZZ$ interference
is negligible.
As general rule, we conclude that non-negligible $WW/ZZ$ interference
effects occur only if at least one weak boson of the pair is dominantly
off-shell due to kinematic constraints.
Our results demonstrate that in cases where $WW/ZZ$ interference can be
neglected it is nevertheless important to take into account the subdominant
weak boson pair as its contribution is of ${\cal O}(10$--$20\%)$ before
search selection cuts are applied.  Considering $WW$ intermediate
states in $H\to ZZ$ searches is crucial due to the intrinsically larger
continuum cross section.
The subdominant weak boson pair contribution induces a
correction to the signal-background interference, which is
at the few percentage point level except when selection cuts
strongly suppress the subdominant contribution.
To mitigate the signal reduction in heavy Higgs searches due to increasingly large
negative signal-background interference for invariant masses above the Higgs
mass, we employed an upper bound cut on the transverse mass also for
$M_H \gtrsim 600$ GeV.
The kinematical dependence of signal-background and $WW/ZZ$ interference
for $gg\ \to H \to WW/ZZ\to\ell\bar{\nu}_\ell\bar{\ell}\nu_\ell$
can be calculated and simulated with the public parton-level
event generator \textsf{gg2VV}.

\acknowledgments

I would like to thank the conveners and
members of the LHC Higgs Cross Section Working Group,
notably S.\ Bolognesi, D.\ de Florian, S.\ Diglio, C.\ Mariotti,
G.\ Passarino and R.\ Tanaka, for stimulating
and informative discussions,
G.\ Heinrich, M.\ Rauch and M.\ Rodgers for useful comparisons,
and the Centre for Particle Physics at Royal Holloway, University of London
and the Institute for Theoretical
Particle Physics and Cosmology at RWTH Aachen University for access to
their computing facilities.
Financial support from the Higher Education Funding Council for England,
the Science and Technology Facilities Council and the Institute for
Particle Physics Phenomenology, Durham, is gratefully acknowledged.


\section{Introduction}
This paper introduces a new \emph{model-based} approach to
constructing Cylindrical Algebraic Decompositions (CADs).  The
model-based approach, building on \cite{JovanovicdeMoura:12}
and \cite{Brown:2013}, has some very nice properties
(described later in the paper) that make it appealing.
However, prior work has not applied it to constructing CADs.
Jovanovic and de Moura's work \cite{JovanovicdeMoura:12}, which
introduced the approach, uses it to determine the satisfiability of
Tarski formulas.  In some sense, their approach can be seen as
building a CAD-like decomposition.  However, what is constructed is
an unstructured list of cells, which makes it unsuitable for some
of what CADs are used for.  Moreover, the method is not obviously
parallelizable, and it doesn't take as strong advantage of the
``model-based approach'' as is possible. \cite{Brown:2013} shows
how to make stronger use of the ``model'' during the construction
of a single open cell.  This paper continues in one of the directions
outlined in that paper, using the strong model-based approach to
construct not just a single cylindrical cell, but a whole
decomposition of real space into cylindrical cells.

A particularly
exciting aspect of this new model-based approach is that while each
cell in the
decomposition is cylindrical, those cells need not by cylindrically
arranged with respect to one another.  This frees us to construct
more general decompositions than CADs, thereby representing
semi-algebraic sets with fewer cells.  To make use of this freedom, we
introduce a new generalization of CAD, the
\emph{Open Non-uniform Cylindrical Algebraic Decomposition} (Open NuCAD),
and an algorithm TI-Open-NuCAD that efficiently constructs an Open
NuCAD from an input formula.  As demonstrated by an example
computation that is worked out in detail in this paper, the
flexibility of NuCADs allow sets to be represented using fewer cells
than with a CAD.

\section{Non-uniform Cylindrical Algebraic Decomposition}
In this section we define Non-uniform Cylindrical Algebraic
Decomposition.  We assume the reader is already familiar with
the usual CAD notions --- like delineability, level of a polynomial,
etc.  Note that $\lambda$ denotes the empty string in what follows,
$||$ indicates concatenation,
and $\pi_k(\cdot)$ denotes projection down onto $\mathbb{R}^k$.
This paper deals with open cylindrical cells which, except in the
trivial case of a single cell, cannot truly decompose $\mathbb{R}^n$.
Instead, we say that a set of open regions defines a
\emph{weak decomposition} of $\mathbb{R}^n$ if the regions are
pairwise disjoint, and the union of their closures contains $\mathbb{R}^n$.
We here provide a definition of an open cylindrical cell.  This is
entirely in keeping with the usual definition of a cell in the CAD
literature.

\begin{definition}
\label{definition:NuCAD}
An \emph{Open Cylindrical Cell} is a subset of $\mathbb{R}^n$
is a set of the form
$$
\{ (\alpha_1,\ldots,\alpha_n) \in B\times\mathbb{R} |
f(\alpha_1,\ldots,\alpha_{n-1}) < \alpha_n < g(\alpha_1,\ldots,\alpha_{n-1})\}
$$
or
$$
\{ (\alpha_1,\ldots,\alpha_n) \in B\times\mathbb{R} |
f(\alpha_1,\ldots,\alpha_{n-1}) < \alpha_n\}
$$
or
$$
\{ (\alpha_1,\ldots,\alpha_n) \in B\times\mathbb{R} |
\alpha_n < g(\alpha_1,\ldots,\alpha_{n-1})\}
$$
where $B$ is an open cylindrical cell in $\mathbb{R}^{n-1}$ and
the graphs of $f$ and $g$ over $B$ are disjoint sections of
polynomials,
and $()$ is considered an open cylindrical cell in $\mathbb{R}^0$.
\end{definition}

Next we define Open Non-uniform Cylindrical Algebraic Decomposition
(Open NuCAD), which relaxes the requirements of the usual CAD.  In
particular, it is possible to have two cells whose projections onto
a lower dimension are neither equal nor disjoint.  In other words,
while each individual cell is cylindrical, distinct cells are not
necessarily organized into cylinders.

\begin{definition}
  \label{definition:OpenNuCAD}
  An \emph{Open Non-uniform Cylindrical Algebraic Decomposition}
  (Open NuCAD) of $\mathbb{R}^n$
  is a collection $C$ of open cylindrical cells, each of
  which is labelled with a unique string of the form
  $([0-9]+(L,U,X))*$.
  The relation
  $$
  E = \{(C_1,C_2) | \mbox{$C_1$ and $C_2$ are cells with labels
    $lab_1$ and $lab_2$
    satisfying $lab_2 = lab_1([0-9]+(L,U,X))$}\}$$
    defines a graph on the cells.
  \begin{enumerate}
    \item the graph $(C,E)$ is a tree, rooted at cell $\mathbb{R}^n$,
      with label $\lambda$ (the empty string),
    \item the children of cell $C_0$ with label  $lab_0$ have labels
      taken from the set
      $$\{lab_01L,\ldots,lab_0nL,lab_01U,\ldots,lab_0nU,lab_0nX\}$$
      and if $C_0$ has children, then one of them is labelled
      $lab_0nX$,
    \item if cell $C_2$ is the child of $C_1$ with label
      $lab_1nX$, then $C_2 \subseteq  C_1$ and for each $i \in \{1,\ldots,n\}$,
      in the cylinder over over $\pi_{i-1}(C_2)$
      the section that defines the lower (resp. upper)
      boundary of $C_2$ in $x_i$ is either identical to or disjoint
      from the section that defines the lower (resp. upper)
      boundary of $C_1$ in $x_i$
    \item
      \label{point:LandU}
      if cell $C_X$ is the child of $C_0$ with label
      $lab_0nX$,
      then
      \begin{equation}
        \label{equation:ExprLandU}
      (\pi_{i-1}(C_X) \times \mathbb{R}) \cap \pi_i(C_0) - \overline{\pi_i(C_X)}
      \end{equation}
      consists of zero one or two open cells: the region with
      $i$-coordinates below
      $\overline{\pi_i(C_X)}$ if it is non-empty, which is denoted
      $B_L$,
      and the region with $i$-coordinates above
      $\overline{\pi_i(C_X)}$ if it is non-empty, which is denoted
      $B_U$.
      There is a cell with label $lab_0iL$ if and only if
      $B_L$ is non-empty and, if it exists, that cell is
      $(B_L \times \mathbb{R}^{n-i}) \cap C_0$.
      There is a cell with label $lab_0iU$ if and only if
      $B_U$ is non-empty and, if it exists, that cell is
      $(B_U \times \mathbb{R}^{n-i}) \cap C_0$.
  \end{enumerate}
\end{definition}

Next we prove that NuCADs really do define decompositions of
$\mathbb{R}^n$ or, more properly, Open NuCADs define weak
decompositions of $\mathbb{R}^n$.

\begin{theorem}
If cell $C_0$ is a non-leaf node in the graph $(C,E)$,
its children form a weak decomposition of $C_0$.
\end{theorem}
\noindent{\sc Proof.}\enspace
What needs to be proved is that there is no open subset of $C_0$
having empty intersection with all of the children of $C_0$.
Let $S$ be an open, connected subset of $C_0$.  Let $i$ be the maximum
element of $\{1,\ldots,n+1\}$ such that $\pi_{i-1}(S) \subseteq \pi_{i-1}(C_X)$.
If $i = n+1$, then $S$ is contained in $C_X$, the child that, by
definition, must exist.  So the theorem holds in this case.

If $i \leq n$, then we have $\pi_{i-1}(S) \subseteq \pi_{i-1}(C_X)$, but
$\pi_{i}(S) \nsubseteq \pi_{i}(C_X)$.  Consider the key expression (\ref{equation:ExprLandU})
from Point~\ref{point:LandU} of Definition~\ref{definition:OpenNuCAD} with
regards to $i$:
\[
\underbrace{
(\overbrace{\pi_{i-1}(C_X)}^{\pi_{i-1}(S)\subseteq} \times \mathbb{R})
\cap \pi_i(C_0)}_{\pi_{i}(S)\subseteq} -
\underbrace{\overline{\pi_i(C_X)}}_{\pi_{i}(S)\nsubseteq}
\]
This shows that one or both of the regions $B_L$ and $B_U$ from Point~\ref{point:LandU}
have non-empty intersection with $\pi_i(S)$, and thus is/are
non-empty.  Suppose $B_L \cap \pi_i(S) \neq \emptyset$ (the case for
$B_U$ is entirely analogous, and so will not be given explicitly).
Since $B_L$ is non-empty, by definition
$C_0$ has a child with label $lab_0iL$ that is $(B_L \times \mathbb{R}^{n-i}) \cap C_0$.
Since $S\subseteq C_0$ and $\pi_i(S) \subseteq B_L$, we have
$$\left((B_L \times \mathbb{R}^{n-i}) \cap C_0\right) \cap S \neq \emptyset,$$
which proves the theorem.
\qed

\begin{corollary}
The leaf cells of an Open NuCAD comprise a weak decomposition of $\mathbb{R}^n$.
\end{corollary}

\section{Algorithms}

We will follow the OpenCell data structure definition
provided in \cite{Brown:2013}, with the following
additions:
\begin{enumerate}
\item each cell carries a sample point $\alpha$ with it
\item each cell has an associated set $P$ of irreducible
  polynomials that are known to be sign-invariant
  (which implies order-invariant, since these are open cells)
  within the cell.
\item each cell has an associated label $lab$ of the form
  $([0-9]+(L,U,X))*$.
\end{enumerate}

We assume the existence of a procedure OC-Merge-Set
that is analogous to the procedure O-P-Merge defined in
\cite{Brown:2013}, except that instead of merging a
single polynomial $P$ with a given OneCell $C$, it merges
a set $Q$ of polynomials with a given OneCell $C$.  This
could be realized by simply applying O-P-Merge iteratively,
or via a divide-and-conquer approach as alluded to in the
final section of \cite{Brown:2013}.  We will assume that
this procedure manipulates OneCell data structures with the
augmentations described above.  The label $lab$ and point
$\alpha$ for the refined cell returned by OC-Merge-Set is
simply inherited from the input OneCell $C$, and the
associated set of polynomials is the super-set of $P \cup Q$
(where $P$ is the set associated with $C$) defined by the
projection factors computed during the refinement process
--- all of which are known to be sign-invariant in the refined
OneCell.

\textbf{Algorithm: Split}\\
\textbf{Input:}
OpenCell $D$ (with point $\alpha \in \mathbb{R}^n$,
projection factor set $P$, and label $lab$), and Formula $F$\\
\textbf{Output:}
queue of OpenCells that is either empty (in which case $F$ is
truth-invariant in $D$), or whose elements comprise a valid set
of children for $D$ according to Definition~\ref{definition:NuCAD}
(in which case $F$ is truth-invariant in the cell with label $labnX$).
\begin{enumerate}
\item choose $Q \subset \mathbb{Z}[x_1,\ldots,x_n]$ such that
  $Q \cap P = \emptyset$ and the sign-invariance of the elements of $P \cup Q$
  within a connected region containing $\alpha$ implies the
  truth-invariance of $F$; if $Q = \emptyset$ return an empty queue

\item $D' = \mbox{OC-Merge-Set}(D,\alpha,Q)$

\item if $D' = (\textsc{Fail},f)$ then \emph{/$\ast$ perturb $\alpha$ $\ast$/}
  \begin{enumerate}
  \item $L = \{f\}$, $i = \mbox{ level of $f$}$
  \item while at least one element of $L$ is nullified at
    $(\alpha_1,\ldots,\alpha_{i-1})$ do
    \begin{enumerate}
      \item $L = \bigcup_{g \in L} \factors{\ldcf{x_i}{g}}$
      \item $i = i - 1$
    \end{enumerate}
  \item $\zeta = \mbox{max}\{ \beta\in \mathbb{R}\ |\ \beta < \alpha_i \mbox{ and }
    g(\alpha_1,\ldots,\alpha_{i-1},\beta) = 0 \mbox{ for some $g\in L$}\}$
  \item choose $\gamma_i \in \left(\max(\zeta,D[i].L),\alpha_i\right)$
  \item for $j$ from $i+1$ to $n$ do
    \begin{enumerate}
      \item choose $\gamma_j$ so that
        \[
        \begin{array}{c}
          root(D[j].l(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
          D[j].L.j,x_j) < \gamma_j \mbox{ and}\\
          \gamma_j <
          root(D[j].u(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
          D[j].U.j,x_j)
        \end{array}
        \]
      \end{enumerate}
  \item set $\alpha =
    (\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_n)$,
    adjusting data-structure $D$ accordingly
  \item goto Step~2
  \end{enumerate}

\item enqueue $D',\alpha,P',lab'$ on output queue, where $P'$ is
  produced by the merge process, and $lab' = lab||nX$

\item for $i$ from 1 to $n$ do \emph{/$\ast$ split $D$ based on $D'$ $\ast$/}

      \begin{enumerate}
      \item
        if $D'[i].l \neq D[i].l$ then \emph{/$\ast$ lower bound at
          level $i$ changes $\ast$/}
        \begin{enumerate}
        \item $D'_{iL} = D'[1],\ldots,D'[i-1],
          (D[i].l,D[i].L,D'[i].l,D'[i].L)
          ,D[i+1],\ldots,D[n]$
        \item for $j$ from $i$ to $n$, choose $\gamma_j$ so that
          \[
          \begin{array}{c}
            root(D'_{iL}[j].l(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
            D'_{iL}[j].L.j,x_j) < \gamma_j \mbox{ and}\\
            \gamma_j <
            root(D'_{iL}[j].u(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
            D'_{iL}[j].U.j,x_j)
          \end{array}
          \]
        \item $\alpha'_{iL} = (\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_n)$
          \item $P'_{iL} = P \cup (P' \cap
            \mathbb{R}[x_1,\ldots,x_{i-1}])$, where $P$ and $P'$ are
            the sign-invariant polynomial sets for $D$ and $D'$, respectively.
            {\small Note: we might sometimes deduce that there are
              other polynomials that are sign-invariant in $D'_{iL}$.
            This could be quite worthwhile!}
        \item $lab_{iL} = lab||iL$, where $lab$ is the label for $D$
        \item enqueue new cell $D'_{iL},\alpha'_{iL},P'_{iL},lab_{iL}$
          in output queue
        \end{enumerate}

      \item
        if $D'[i].u \neq D[i].u$ then \emph{/$\ast$ upper bound at
          level $i$ changes $\ast$/}
        \begin{enumerate}
        \item $D'_{iU} = D'[1],\ldots,D'[i-1],
          (D[i]'.u,D'[i].U,D[i].u,D[i].U)
          ,D[i+1],\ldots,D[n]$
        \item for $j$ from $i$ to $n$, choose $\gamma_j$ so that
          \[
          \begin{array}{c}
            root(D'_{iU}[j].l(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
            D'_{iU}[j].L.j,x_j) < \gamma_j \mbox{ and}\\
            \gamma_j <
            root(D'_{iU}[j].u(\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_{j-1},x_j),
            D'_{iU}[j].U.j,x_j)
          \end{array}
          \]
          \item $\alpha'_{iU} = (\alpha_1,\ldots,\alpha_{i-1},\gamma_i,\ldots,\gamma_n)$
          \item $P'_{iU} = P \cup (P' \cap
            \mathbb{R}[x_1,\ldots,x_{i-1}])$, where $P$ and $P'$ are
            the sign-invariant polynomial sets for $D$ and $D'$, respectively.
            {\small Note: we might sometimes deduce that there are
              other polynomials that are sign-invariant in $D'_{iL}$.
            This could be quite worthwhile!}
        \item $lab_{iU} = lab||iU$, where $lab$ is the label for $D$
        \item enqueue new cell $D'_{iU},\alpha'_{iU},P'_{iU},lab_{iU}$
          in output queue
        \end{enumerate}
      \end{enumerate}
    \item return output queue
\end{enumerate}

\textbf{Algorithm: TI-Open-NuCAD}\\
\textbf{Input:} Formula $F$ in variables $x_1,\ldots, x_n$\\
\textbf{Output:}
Open-NuCAD $C$ in the leaf cells of which $F$ is truth invariant
\begin{enumerate}
  \item $C = \{ \}$
  \item let $Q$ be an empty queue
  \item enqueue in $Q$ and add to $C$ the OneCell representing $\mathbb{R}^n$, with point
    $\alpha$ chosen arbitrarily, $P = \{ \}$, and label $lab =
    \lambda$.
  \item while $Q$ is not empty
    \begin{enumerate}
      \item dequeue $D$ from $Q$ \emph{/$\ast$ need not actually
          follow FIFO $\ast$/}
      \item if the label of $D$ ends in $X$, continue to next iteration
      \item $Q' = \mbox{Split}(D,F)$
      \item for each $D'$ in $Q'$ do
        \begin{enumerate}
        \item add $D'$ to $Q$
        \item add $D'$ to $C$
        \end{enumerate}
    \end{enumerate}
  \item return $C$
\end{enumerate}

Note that no one method for choosing $Q$ in Step 1 of the algorithm
Split is specified.  There are different ways to do this, and which
one is employed may well affect practical performance quite a bit and
will warrant future investigation.  One point we will make, however,
is that $\alpha$ plays a role in making this choice.  For example,
suppose $F = f_1 > 0 \wedge f_2 > 0 \wedge \cdots \wedge f_r > 0$, and
suppose $F$ is $\textsc{False}$ at $\alpha$.
To choose $Q$ we need only find one $f_i \notin P$ that is negative at
$\alpha$. If there are multiple such $f_i$'s, we could choose among
them in several different ways.  We could take the lowest level $f_i$.
We could prefer low-degree $f_i$'s.  If all potential $f_i$'s are of
level $n$, we could substitute $(\alpha_1,\ldots,\alpha_{n-1})$ into
all of them, examine the CAD of $\mathbb{R}^1$ that results, and choose
the $f_i$ based on that information.

\section{An Example Open NuCAD Construction}
Consider the input formula
$F = [ 16 y - 16 x^2 - 8 x - 1 > 0 \wedge x^2 + y^2 - 1 > 0 ]$.
We will follow the execution Algorithm TI-Open-NuCAD on this input.
In the interest of space, we will name the polynomials that will
appear in the computation up front:
$$
f_1 = 16 y - 16 x^2 - 8 x - 1 , f_2 = x^2 + y^2 - 1,
f_3 = 256 x^4 + 256 x^3 + 352 x^2 + 16 x - 255, f_4 = x+1, f_5 = x-1
$$
\pagebreak
\begin{enumerate}
\item Cell $C_0 = ([\ ],lab=\lambda, \alpha=(0,0), P=\{\})$ consisting of $\mathbb{R}^2$ enqueued on $Q$
\raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C0.pdf}}}

\item Split($C_0$): $F(\alpha)=\textsc{False}$, choose $Q = \{f_1\}$, enqueue
  the following
  cells
  \begin{itemize}[leftmargin=0in]
  \item $C_1 = ([y<root(f_1,1,y)],lab=2X,\alpha=(0,0),P={f_1})$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C1.pdf}}}
  \item $C_2 = ([y>root(f_1,1,y)],lab=2U,\alpha=(0,1/2),P={f_1})$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C2.pdf}}}
  \end{itemize}
  \item $C_1$'s label ends in $X$, so it is not processed further
  \item Split($C_2$): $F(\alpha)=\textsc{False}$, choose $Q = \{f_2\}$, enqueue
  the following
  cells
   \begin{itemize}[leftmargin=0in]
   \item $C_3 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge y < root(f_2,2,y)\wedge \\
         x > root(f_3,1,x) \wedge x < root(f_3,2,x)
         \end{array}
       \right],
lab=2U2X,\alpha=(0,\frac{1}{2}),P=\{f_1,f_2,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C3.pdf}}}
   \item $C_4 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge x < root(f_3,1,x)
         \end{array}
       \right],
lab=2U1L,\alpha=(-\frac{3}{2},2),P=\{f_1,f_3\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C4.pdf}}}
   \item $C_5 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge x > root(f_3,2,x)
         \end{array}
       \right],
lab=2U1U,\alpha=(\frac{3}{2},4),P=\{f_1,f_3\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C5.pdf}}}
   \item $C_6 = \left(\left[
       \begin{array}{c}
         y > root(f_2,2,y)\wedge \\
         x > root(f_3,1,x) \wedge x < root(f_3,2,x)
         \end{array}
       \right],
lab=2U2U,\alpha=(0,2),P=\{f_1,f_2^*,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C6.pdf}}}
  \end{itemize}
\item $C_3$'s label ends in $X$, so it is not processed further
\item Split($C_4$): $F(\alpha)=\textsc{True}$, choose $Q = \{f_2\}$, enqueue
  the following
  cells
   \begin{itemize}[leftmargin=0in]
   \item $C_7 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge x < -1
         \end{array}
       \right],
lab=2U1L2X,\alpha=(-\frac{3}{2},2),P=\{f_1,f_2,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C7.pdf}}}
   \item $C_8 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge \\
         x > -1 \wedge x < root(f_3,1,x)
         \end{array}
       \right],
lab=2U1L1U,\alpha=(-\frac{15}{16},2),P=\{f_1,f_2,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C8.pdf}}}
  \end{itemize}
\item Split($C_5$): $F(\alpha)=\textsc{True}$, choose $Q = \{f_2\}$, enqueue
  the following
  cells
   \begin{itemize}[leftmargin=0in]
   \item $C_9 = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge x > 1
         \end{array}
       \right],
lab=2U1U2X,\alpha=(\frac{3}{2},4),P=\{f_1,f_2,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C9.pdf}}}
   \item $C_{10} = \left(\left[
       \begin{array}{c}
         y>root(f_1,1,y)\wedge \\
         x > root(f_3,2,x)\wedge x < 1
         \end{array}
       \right],
lab=2U1U1L,\alpha=(\frac{15}{16},2),P=\{f_1,f_2,f_3,f_4,f_5\}\right)$ \raisebox{-.4\height}{\scalebox{0.1}{\includegraphics{ex2/ex2C10.pdf}}}
  \end{itemize}
\item all remaining cells in $Q$ either have labels that end in $X$
  or, when the call to Split is made, are not split further.
\end{enumerate}
\begin{figure}
{\scalebox{0.8}{\includegraphics{ex2/tree.pdf}}}
\caption{Depicted here is the Open NuCAD graph structure produced by
  the example run of TI-Open-NuCAD for input formula $F = [ 16 y - 16
  x^2 - 8 x - 1 > 0 \wedge x^2 + y^2 - 1 > 0 ]$.  The leaf nodes are a
  weak decomposition of $\mathbb{R}^2$ into open cylindrical cells in
  which $F$ is truth-invariant.  Also shown (circled) is the
  truth-invariant CAD for $F$.}
\label{figure:tree}
\end{figure}
\pagebreak
Figure~\ref{figure:tree} shows the NuCAD tree resulting from the above
execution of the TI-Open-NuCAD algorithm.
There are seven leaf nodes, which mean $\mathbb{R}^2$ has been
decomposed into seven cells.  The standard truth-invariant CAD for
input formula $F$ (shown circled in Figure~\ref{figure:tree})
contains 16 open cells in $\mathbb{R}^2$.  The Open NuCAD fails to be
an Open CAD because the projections onto $\mathbb{R}^1$ of
the cell $2X$ and any other leaf cell are neither disjoint nor
identical.

The primary purpose of this example is to illustrate the basic
functioning of TI-Open-NuCAD, and to illustrate the Open NuCAD data
structure.  Hopefully it has been successful in this.  There are
two important limitations to this example, though.  First of all,
Step~3, which deals with ``fail'' results returned by the
OC-Merge-Set operation, is not illustrated.  Secondly, and more
importantly, because this example only involves two variables there
is no opportunity to illustrate the reduction in the number and size
of projection factor sets that we expect to accompany the model-based
approach to CAD construction.

\section{The correctness of TI-Open-NuCAD}
In this section we sketch a proof of the correctness of
TI-Open-NuCAD.  In fact, TI-Open-NuCAD clearly
meets its specification provided that Split meets its specification,
and that termination can be proved.  First we prove a lemma that is
key to showing the termination of TI-Open-NuCAD.

For Open OneCell $D$ we
denote the set of polynomials whose sections define the boundaries
of $D$ by $\mbox{bpolys}(D)$ (note that they will be irreducible).  For
Tarski formula $F$ we denote the
set of irreducible factors of polynomials appearing on the
left-hand-side of the atomic
formulas of $F$ when they are normalized to be of the form
$f\ \sigma \ 0$ by $factors(F)$.

\begin{lemma}
\label{lemma:factors}
Suppose the call $\mbox{Split}(D,F)$ produces a non-empty queue $Q'$.
Let $H$ be the closure under the Open McCallum projection of
$\mbox{bpolys}(D) \cup factors(F)$.
For each cell $C \in Q'$, $\mbox{bpolys}(C) \subseteq H$.
\end{lemma}
\noindent{\sc Proof.}\enspace
First we note that if Step~2 produces $D' = (\textsc{Fail},f)$ then although
the sample point $\alpha$ and some of the algebraic numbers in the
data-structure may change, the defining formula for $D'$ and,
therefore, the elements of $\mbox{bpolys}(D)$ remain the same.
Next we note that if Step~2 produces a cell $D'$ (i.e. does not
produce $\textsc{Fail}$) then the specification of the O-P-Merge algorithm
from \cite{Brown:2013}, and by extension the OC-Merge-Set algorithm
called in Step~2, guarantees that $\mbox{bpolys}(D')$ is a subset
of the closure under the Open McCallum projection of
$\mbox{bpolys}(D) \cup Q$.  Since $Q \subseteq factors(F)$, we have
$\mbox{bpolys}(D') \subseteq H$.
For any cell $C$ enqueued on the output queue, at each level $i$,
the boundaries of $C$ are sections of polynomials from the set
$\{D[i].l,D'[i].l,D'[i].u,D[i].u\}$, which is a subset of $H$.
\qed

\begin{lemma}
\label{lemma:split}
The Algorithm $\mbox{Split}(D,F)$ terminates and meets its
specification.
\end{lemma}
\noindent{\sc Proof.}\enspace
As long as Step~3 only produces new values for point $\alpha$ that
are in the cell defined by $D$ and Step~2 eventually produces a non-$\textsc{Fail}$
result, $\mbox{Split}(D,F)$ clearly meets it specification.
Moreover, if the body of Step~3 is executed and $\alpha$ is in the
cell defined by $D$ (which is certainly true initially), then the new
value of $\alpha$ is also in the cell defined by $D$.  This is clear
because $\gamma_i$ is chosen from the interval
$(\mbox{max}(\zeta,D[i].L),\alpha_i) \subset (D[i].L,D[i].U)$,
and for $j \in \{i+1,\ldots,n\}$, $\gamma_j$ is chosen specifically
to satisfy the defining formula
$$root(D[j].l,D[j].L.j,x_j) < x_j < root(D[j].u,D[j].U.j,x_j).$$
What remains to be proven is termination, which boils down to showing
that the call to OC-Merge-Set in Step~2 eventually returns a
non-$\textsc{Fail}$ result.  If we were assured that OC-Merge-Set would
produce the same projection factors for the perturbed $\alpha$ as for
the original, this would be clear.  Unfortunately, we cannot be sure
of that.  Thus, we require a more subtle argument.  First, we note
that each perturbation leaves the $x_k$th coordinate unchanged for all
$k < i$, reduces the $i$th coordinates $\alpha_i$ so that it changes
from a root of $g(\alpha_1,\ldots,\alpha_{i-1},x_i)$ to something
slightly smaller (Step 3c), and potentially changes the remaining
coordinates.

Suppose Split does not terminate.  Then there is an infinite sequence
of $\alpha$ values and associated $f$'s satisfying $f(\alpha) = 0$.
Note that all the polynomials $f$ as well as all the elements of the
set $L$ constructed from $F$ come from the closure under the
McCallum projection of $\mbox{bpolys}(D) \cup \mbox{factors}(F)$,
which we'll denote $P_{MC}$. Let
$\alpha^{(0)}, \alpha^{(1)},\ldots$ be the infinite sequence of values
for $\alpha$ as the process progresses, let
$f^{(0)}, f^{(1)},\ldots$ be the infinite sequence of associated $f$'s
and $L^{(0)}, L^{(1)},\ldots$ and $i^{(0)}, i^{(1)},\ldots$
be the associated values for $L$ and $i$ arrived at by Step 3b.
We note that for any $k$, the elements $L^{(k)}$ all divide
$A(i^{(k)},\alpha^{(k)})$, where
$$
A(m,\rho) = \prod_{g \in P_{MC} |level(g) = m \wedge
  g(\rho_1,\ldots,\rho_{m-1},x_m) \neq 0}g.$$
We also note that the polynomial set $\{A(m,\rho)|m\in\{1,\ldots,n\} \wedge \rho
\in \mathbb{R}^n\}$ is finite.
So, for each $k$ we have that that $i^{(k)}$th coordinate of
$\alpha^{(k)}$ is a zero of some $g \in L^{(k)}$ that is not
nullified at $(\alpha_1^{(k)},\ldots,\alpha_{i^{(k)}-1}^{(k)})$
and thus is a zero of $A(i^{(k)},\alpha^{(k)})$.

We will show that for each level $r$, there is a value $k$ after which
the $r$th coordinate of $\alpha^{(k)}$ never changes.  We proceed by
induction on $r$.

Consider the case $r=1$.  Consider the subsequence $k_1,k_2,\ldots$
of all indices $k$ for
which $i^{(k)} = 1$.  For each $k_j$ in this subsequence,
$\alpha_1^{(k_j)}$ is a zero of $A(1,\alpha^{(k_j)})$.
Moreover, the new
value of $\alpha_1$ is smaller than the previous value and, since the
value of the 1st component of $\alpha$ is otherwise never changed,
$\alpha_1^{(k)}$
is strictly decreasing over the subsequence $k_1, k_2, \ldots$.  Since
$A(1,\beta)$ is the same for any $\beta\in\mathbb{R}^n$, and it has
finitely many roots, there are only finitely many elements of the
subsequence. In particular, there is a largest index $k^*$ in the
subsequence ($k^*$ can be taken as zero if the subsequence is empty),
and $\alpha_1$ is constant over all indices greater than $k^*$.

Suppose $r > 1$.  Assume, by induction, that the result holds for all
smaller values of $r$.  Then there is an index $k'$ such that for all
$k > k'$ the first $r-1$ components of $\alpha^{(k)}$ are constant.
So, for all $k > k'$, the $r$th component of $\alpha^{(k)}$ is
non-increasing.
Consider the subsequence $k_1,k_2,\ldots$
of all indices $k > k'$ for
which $i^{(k)} = r$.
Note that because the $r$th component of $\alpha$ is reduced at each step for
which $i^{(k)} = r$,
the sequence of values $\alpha_r^{(k_1)},\alpha_r^{(k_2)},\ldots$
is strictly decreasing.
For each $k_j$ in the subsequence,
$\alpha_1^{(k_j)}$ is a zero of $A(r,\alpha^{(k_j)})$.
Since there are only finitely many polynomials $A(r,\beta)$,
where $\beta \in
(\alpha_1^{(k'+1)},\ldots,\alpha_{r-1}^{(k'+1)})\times\mathbb{R}^{n-r+1}$,
each having only finitely many roots, there are only finitely many
elements in the subsequence.  In particular, there is a largest index
$k^*$ in the
subsequence ($k^*$ can be taken as $k'$ if the subsequence is empty),
and $\alpha_r$ is constant over all indices larger than $k^*$.

Thus, we have proven that there is an index $k'$  such that
for all $k > k'$, all coordinates of $\alpha^{(k)}$ are constant.
This is a contradiction, since executing Step 3 changes $\alpha$,
which means that our assumption that there is an input for which Split
does not terminate is invalid.
This completes our proof of the termination and correctness of Split.
\qed

\begin{theorem}
Algorithm TI-Open-NuCAD terminates, and meets its specification.
\end{theorem}
\noindent{\sc Proof.}\enspace
Lemma~\ref{lemma:split} shows that Split terminates and is correct.
Lemma~\ref{lemma:factors} shows that the boundary polynomials for
the cells returned by Split are elements of the closure
under the Open McCallum projection of $\mbox{factors}(F)$.
Thus for any cell $D'$ returned by
Split, and any cell $C$ from the
CAD produced by the Open McCallum projection for $F$, either
$C \cap D' = \emptyset$ or $C \subseteq D'$.  This means that
for each each cell $D$ enqueued on $Q$, we can imagine associating
with $D$ the
set of cells from the
CAD produced by the Open McCallum projection for $F$ that are
contained in $D$ --- we call this set $M_D$.
Note that $M_D$ is never empty.
Recall that when a cell with label ending in $X$ is dequeued from $Q$,
no call to
Split is made.
Define $X_Q$ to be the set of cells in $Q$ with label ending in $X$.
Consider the quantity
\begin{equation}
\label{equation:cq}
c_Q = |X_Q| + \sum_{E\in Q - X_Q}2|M_E|^2.
\end{equation}
We will show
 that at each iteration of the loop in Step 4 of
TI-Open-NuCAD the quantity $c_Q$ is reduced.
Every iteration, a cell $D$
is dequeued from $Q$ and
one of the following occurs:
\begin{enumerate}
\item no new cells are enqueued --- in which case one of the terms
on the right-hand side of (\ref{equation:cq}) gets smaller and the
other term is unchanged,
\item a single cell whose label ends in $X$ is enqueued --- in which
  case $|X_Q|$ increases by one, but
  $\sum_{E\in Q - X_Q}2|M_E|^2$ is reduced by $2|M_D|^2 > 1$
,
\item more than one cell is enqueued --- in which case
  the $|X_Q|$ term is increased by one, but in the sum
  the term $2|M_D|^2$ is replaced by
  $2|M_{D_1}|^2 + 2|M_{D_2}|^2 + \cdots + 2|M_{D_t}|^2$ where
  $|M_D| = |M_{D_1}| + |M_{D_2}| + \cdots + |M_{D_t}|$, $t \geq 2$.
  So the net change is
$$
1 + 2|M_{D_1}|^2 + 2|M_{D_2}|^2 + \cdots + 2|M_{D_t}|^2 - 2|M_D|^2 < 0.
$$
\end{enumerate}
Thus, termination is proven and, as noted
previously, correctness is then easily verified.
\qed

\section{Advantages of the model-based approach}
Further work is required to either produce an implementation of these
algorithms and provide a systematic empirical comparison between them
and the usual Open CAD construction algorithm, or to provide an
analytical comparison.  Moreover, in as much as an Open NuCAD is less
structured than an Open CAD, it cannot necessarily be used for the
same purposes.  So yet more work is required to understand the
applications and limitations of this new variant of CAD.  Given these
points, it is worth listing some of the reasons why the model-based
approach and Open NuCADs are important and worth developing.
\begin{enumerate}
\item The model-based approach produces smaller projection-factor sets
  and larger sign-invariant cells.  This point is demonstrated in
  \cite{Brown:2013}, and further experiments showing this were
  presented in the ISSAC 2013 talk accompanying that paper.  This is
  perhaps the most important reason to pursue this new approach,
  because the reduction in the number of projection factors and the
  increase in cell size is substantial.  For a single cell,
  experiments point to exponentially smaller projection factor sets
  and exponentially larger cells.
\item NuCADs allow for truth-invariant decompositions for an input
  formula using fewer cells than CADs.  The example in this paper
  demonstrates this point, although certainly more analysis, either
  empirical or analytical, is required to understand how substantial
  the difference between NuCADs and CADs really is.
\item Model-based construction of NuCADs is incremental.  After one
  loop iteration, which requires a small amount of time and space relative to
  even just the projection step for CAD construction, the new approach
  produces a cell in which the input formula is truth-invariant.  This
  is in marked contrast with traditional CAD construction, for which
  the entire projection must be computed before even the first cell is
  constructed.
\item Model-based construction of NuCADs is naturally parallelizable.
  Splitting of one cell in the queue $Q$ is completely independent of
  splitting other cells, so all the splitting can be done in
  parallel.  In fact, nodes could keep their own queues of cells to
  split, and would only need to communicate when one node ran out of
  cells to split and had to steal some from another's queue.  The one
  kind of information that one would probably want nodes to share
  would be the results of particularly expensive resultant and
  discriminant computations and the accompanying factorizations.
\end{enumerate}

Perhaps the most interesting of all, however, is that none of the
proofs of the doubly-exponential worst-case running time of CAD apply
to NuCADs.
\cite{BrownDavenport:07,DavenportHeintz:88,Weispfenning:88}
all deduce the doubly-exponential worst-case performance of CAD
from its connection to quantifier elimination --- in particular,
quantifier elimination for formulas with many quantifier alternations.
NuCADs, however, do not directly allow for quantifier elimination, at
least not for formulas with quantifier alternations, so they are not
subject to that argument.  This leaves open the intriguing possibility
that the model-based approach and NuCADs may provide a CAD-style
algorithm for satisfiability, existential quantifier elimination, or
even full quantifier elimination with an asymptotic complexity
competitive with modern QE algorithms, but with the kind of practical
utility that has made CAD attractive for smaller problems.

\bibliographystyle{plain}


\section{Introduction}

In order to show that the work of Dwivedi and Srivastava [2] (hereinafter DS)
can be investigated even analytically,  Chandra [1] used the condition $\omega
\nu > > v_A^2$ for the case of $\eta = 0$ and $\omega \eta < < v_A^2$ for the
case of $\nu = 0$. His also proved that the results of DS are wrong. Dwivedi and
 Srivastava [3] should not misguide the scientific
community by wrongly interpreting these conditions, as the numerical
calculations can also prove that the results of DS are wrong. Moreover, as a
consequence of the wrong approach of DS, some numerical results obtained by
Chandra [1]  are not physically acceptable. In the present work we have shown
numerically that the results of DS are wrong.

\section{Formulation}

The dispersion relation derived by Pek\"unl\"u et al. [4] for the MHD
equations can be expressed as [1, 2, 4]
\begin{eqnarray}
\nu \eta k^4 + [v_A^2  - i \omega (\nu + \eta)] k^2 - \omega^2 =  0
\label{eq0}
\end{eqnarray}

\noindent
Consideration of an atmosphere with (i) $\nu = 0$ (no viscosity) or (ii) with
$\eta = 0$ (no magnetic diffusivity) may not be a physically acceptable
situation. However, DS discussed these two cases numerically, where they got
wrong results.  Let us now investigate these
 two cases numerically. For either of the cases, the first term $\nu \eta k^4$
is zero and we have
\begin{eqnarray}
\Big[v_A^2  - i \omega (\nu + \eta)\Big] k^2 - \omega^2 =  0 \label{eq1}
\end{eqnarray}

\noindent
It is matter of surprise that Dwivedi and Srivastava [3] are asking for
solving equation (ref{eq0}) numerically.
For real values of $\omega$, the wave number $k$ can be assumed to have complex
 values. For $k = k_r + i k_i$, we can derive the values of $k_r$ and $k_i$
for the said two cases as the following:

\vspace{2mm}

(i) There is viscosity only . That is, there is no magnetic diffusivity
($\eta = 0$).  For this case, equation (\ref{eq1}) gives [1, 2]
\begin{eqnarray}
k_r^4 - P \ k_r^2 - Q^2 = 0 \hspace{1.5cm} \mbox{and} \hspace{1.5cm} k_r k_i =
Q  \label{eq2}
\end{eqnarray}

\noindent
where
\begin{eqnarray}
P = \frac{\omega^2 v_A^2}{v_A^4 + \omega^2 \nu^2}  \hspace{1.5cm} \mbox{and}
\hspace{1.5cm} Q = \frac{\omega^3 \nu}{2(v_A^4 + \omega^2 \nu^2)}
\nonumber
\end{eqnarray}

\noindent
Form equation (\ref{eq2}), we get
\begin{eqnarray}
k_r^2 = \frac{\omega^2 v_A^2}{2 (v_A^4 + \omega^2 \nu^2)} \Big[1 + \sqrt{
1 + (\omega \nu/v_A^2)^2}\Big] \label{eq3}
\end{eqnarray}

(ii) There is magnetic diffusivity only. That is, there is no viscosity
($\nu = 0$). For this case, equation (\ref{eq1}) gives [1, 2]
\begin{eqnarray}
k_r^4 - P'k_r^2 - Q'^2 = 0 \hspace{1.5cm} \mbox{and} \hspace{1.5cm} k_r k_i = Q'
 \label{eq4}
\end{eqnarray}

\noindent
where
\begin{eqnarray}
P' = \frac{\omega^2 v_A^2}{v_A^4 + \omega^2 \eta^2}  \hspace{1.5cm} \mbox{and}
 \hspace{1.5cm} Q' = \frac{\omega^3 \eta}{2(v_A^4 + \omega^2 \eta^2)}
\nonumber
\end{eqnarray}

\noindent
Form equation (\ref{eq4}), we get
\begin{eqnarray}
k_r^2 = \frac{\omega^2 \ v_A^2}{2 (v_A^4 + \omega^2 \eta^2)} \Big[1 + \sqrt{
1 + (\omega \eta/v_A^2)^2}\Big] \label{eq5}
\end{eqnarray}

\section{Results}

In order to get numerical values for $k_r$ and $k_i$, for the two cases, we
require values of physical parameters. The values of physical parameters are
provided by Chandra [1] and some of them are given in Table 1. In order to find
out the source of error in the work of DS, we asked them several times to
provide us the values of physical parameters used in their work, but they never
showed the courtesy to reply. For $\tau = 10^{-4}$ s, the
calculations are made and the results are given in Table 1.
The damping length scale $D$ and the wavelength $\lambda$ of the wave are
\begin{eqnarray}
D = \frac{2 \pi}{k_i} \hspace{1.5cm} \mbox{and} \hspace{1.5cm} \lambda =
\frac{2 \pi}{k_r} \nonumber
\end{eqnarray}

\noindent
The results for the case $\eta = 0$ show that the damping length scale and the
wavelength are of the same magnitude, which is of the order of $10^4$ m. For the
 case $\nu = 0$, the damping length scale comes out of the order of 10$^{11}$ m.This physically unacceptable result is obviously a consequence of the wrong
 approach of DS. Comparison of the present results with those of DS show that
their results are wrong. Even the nature of the variation of $D$ is quite
different.

\vspace{6mm}

\begin{tabular}{cccc|ll|ll}
\multicolumn{8}{l}{Table 1: Values of $k_r$ and $k_i$ for the two cases.}\\
\hline
 & & & & \multicolumn{2}{c}{$\nu =0$} & \multicolumn{2}{|c}{$\eta =0$}  \\
\cline{5-6}
\cline{7-8}
$R^a$ & $\nu^b$ & $\eta^c$ & $v_A^d$ & $k_r$ & $k_i$ & $k_r$ & $k_i$ \\
\hline
 1.05 &   0.20 &   3.79 &   3.25 &   1.94E-02 &   2.19E-10 &   1.26E-03 &   1.25E-03\\
 1.07 &   0.48 &   2.45 &   3.26 &   1.93E-02 &   1.40E-10 &   8.09E-04 &   8.06E-04\\
 1.09 &   0.93 &   1.80 &   3.26 &   1.93E-02 &   1.03E-10 &   5.82E-04 &   5.81E-04\\
 1.11 &   1.56 &   1.44 &   3.26 &   1.93E-02 &   8.23E-11 &   4.48E-04 &   4.48E-04\\
 1.13 &   2.40 &   1.21 &   3.24 &   1.94E-02 &   7.01E-11 &   3.62E-04 &   3.62E-04\\
 1.15 &   3.44 &   1.06 &   3.23 &   1.95E-02 &   6.24E-11 &   3.02E-04 &   3.02E-04\\
 1.17 &   4.65 &   0.96 &   3.21 &   1.96E-02 &   5.77E-11 &   2.60E-04 &   2.60E-04\\
 1.19 &   6.01 &   0.89 &   3.18 &   1.97E-02 &   5.48E-11 &   2.29E-04 &   2.29E-04\\
 1.21 &   7.46 &   0.85 &   3.16 &   1.99E-02 &   5.34E-11 &   2.05E-04 &   2.05E-04\\
 1.23 &   8.92 &   0.83 &   3.13 &   2.00E-02 &   5.30E-11 &   1.88E-04 &   1.88E-04\\
 1.25 &  10.31 &   0.82 &   3.11 &   2.02E-02 &   5.35E-11 &   1.75E-04 &   1.75E-04\\
 1.27 &  11.50 &   0.83 &   3.10 &   2.03E-02 &   5.46E-11 &   1.65E-04 &   1.65E-04\\
 1.29 &  12.40 &   0.85 &   3.10 &   2.03E-02 &   5.63E-11 &   1.59E-04 &   1.59E-04\\
 1.31 &  12.88 &   0.89 &   3.11 &   2.02E-02 &   5.86E-11 &   1.56E-04 &   1.56E-04\\
 1.33 &  12.86 &   0.96 &   3.14 &   2.00E-02 &   6.12E-11 &   1.56E-04 &   1.56E-04\\
 1.35 &  12.26 &   1.06 &   3.20 &   1.97E-02 &   6.43E-11 &   1.60E-04 &   1.60E-04\\
\hline
\multicolumn{8}{l}{\small $^a$Distance from the center of sun in $R_\odot$}\\
\multicolumn{8}{l}{\small $^b$Coefficient of viscosity in 10$^{11}$ m$^2$
s$^{-1}$;}\\
\multicolumn{8}{l}{\small $^c$Magnetic diffusivity in m$^2$ s$^{-1}$}\\
\multicolumn{8}{l}{\small $^d$Alfv\'{e}n velocity in 10$^6$ m s$^{-1}$}\\
\end{tabular}

\vspace{6mm}

\section*{Acknowledgments}
We are thankful to the Department of Science \& Technology (DST), New Delhi and
the Indian Space Research Organization (ISRO), Bangalore for financial support
in the form of research projects.

\vspace{6mm}

\noindent
{\large\bf  References}
\begin{description}

\item{} [1] Suresh Chandra. Comment on propagation and dissipation of
Alfv\'{e}n waves in  coronal holes. Open Astron J. 2009; {\bf 2}: 16 - 18

\item{} [2] Dwivedi B N and Srivastava A K. On the propagation and dissipation
of Alfv\'{e}n waves in coronal holes. Solar Phys. 2006; {\bf 237}: 143 - 152

\item{} [3] Dwivedi B N and Srivastava A K. Comment on ``Comment on propagation
 and dissipation of Alfv\'{e}n waves in  coronal holes" by Suresh Chandra. Open
 Astron J. 2009; {\bf 2}: 72 - 73 (DS).

\item{} [4] Pek\"unl\"u E R, Bozkurt Z, Afsar M, Soydugan E and Soydugan F.
 Alfv\'{e}n waves in the inner polar coronal hole. Mon. Notices Roy. Astron.
Soc. 2002; {\bf 336}: 1195 - 2000.

\end{description}

\end{document}