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\begin{titlepage}

\begin{flushright}
{\bf TP-USl-02/01 \\ January 2002 }

\end{flushright}


\vspace{0.5 cm}
\begin{center}{\bf\Large Transverse spin effects }\end{center}
\begin{center}{\bf\Large in $H/A\to \tau^+ \tau^-;~ 
\tau^\pm \to \nu X^\pm$,  }\end{center}
\begin{center}{\bf\Large Monte Carlo approach $^{\dag}$}\end{center}
\vspace{0.9 cm}
\begin{center}
  {\large\bf   Z. W\c{a}s$^{a,b}$} ~{\large \bf and}~ {\large\bf   M. Worek$^{c}$ }
\vspace{0.3 cm}
\\
{\em $^a$Institute of Nuclear Physics\\
         Kawiory 26a, 30-055 Cracow, Poland.}\\
{\em $^b$CERN, Theory Division, CH-1211 Geneva 23, Switzerland.}\\
{\em $^c$ Institute of Physics, University of Silesia\\ Uniwersytecka 4, 
40-007 Katowice, Poland.}\\
\vspace{0.3 cm}
e-mail: {\tt Zbigniew.Was@cern.ch}\\
e-mail: {\tt Malgorzata.Worek@phys.us.edu.pl}
\vspace{1mm}

\end{center}

\vspace{1mm}
\begin{abstract}
The  transverse spin effects may be helpful to distinguish between 
scalar $(J^{PC}=0^{++})$ or pseudoscalar $(J^{PC}=0^{-+})$
nature of the spin zero (Higgs) particle once discovered in future accelerator
experiments. 
The correlations can manifest themselves
{\it e.g.} in distribution of acollinearity angle of  $ X^\pm $ in decay chain
$H/A\to \tau^+ \tau^-;~ \tau^\pm \to \nu X^\pm $. This delicate measurement 
will require however reconstruction of the Higgs boson rest-frame. Then,
questions of the combined detector-theoretical effects
may be critical to establish reliability of the method. 
Appropriate Monte Carlo program
is essential.

In the present paper we  extend the standard universal interface, of the
{\tt TAUOLA} $\tau$-lepton decay library, to include the complete spin effects
for
$\tau$ leptons originating  from the spin zero particle.  
As usual, the  interface is expected to work with any
Monte Carlo generator providing Higgs boson production, and subsequent decay 
into pair of $\tau$ leptons. 


Some example of numerical results and cross checks of the program, 
will be also given.


\end{abstract}
\begin{center}
%{\it To be submitted to Computer Physics Communications.}
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\vspace{1mm}
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\footnoterule
\noindent
{\footnotesize
\begin{itemize}
\item[${\dag}$]
This work is partly supported by
the Polish State Committee for Scientific Research 
(KBN) grants Nos 
5P03B12420, 5P03B09320, 5P03B10121, 
and the European Commission 5-th framework contract 
HPRN-CT-2000-00149.
\item[${\;}$] Home page: {\tt http://wasm.home.cern.ch/wasm/}
\end{itemize}
}

\end{titlepage}

One of the main goals for future high energy experiments is to measure
properties of the Standard Model (${\cal SM}$) Higgs sector.
Proton-Proton Colliders, such as Tevatron \cite{Tevatron}  or LHC
 \cite{LHC1,LHC2}
are expected to discover the Higgs boson,
if ${\cal SM}$ or one of its ${\cal MSSM}$ extensions  is true.
%see {\it e.g.} \cite{Spira:1997qz}. 
However, the comprehensive precise measurements  of all Higgs boson 
 properties are  expected to be  left for future experiments
on high energy $e^+e^-$ linear colliders such as   JLC \cite{Abe:2001gc},
NLC \cite{:2001ve}, or TESLA \cite{TESLA}. 

One of the important measurement, just after establishing that the newly 
discovered 
particle has indeed spin zero is to check  if it is a scalar or pseudoscalar. 
Depending on the mass of the (to be) discovered Higgs boson, 
different observables can give access to this information.
Already long
time ago, see  {\it e.g.} \cite{Kramer:1994jn}, it was argued that
exploring transverse spin correlations in 
the Higgs boson decay  
$H/A \to \tau^+ \tau^-; \tau^\pm \to \nu_{\tau} X^\pm$ may be,
in some cases, helpful.
Interest in such process is recently
revived in context of proposals for Linear Colliders.
%  \cite{Ronan,Peskin}. 
The measurement may be however involved, as it require reconstruction of  
acollinearity angle between  $\tau^+$  $\tau^-$ decay 
products {\it in H/A rest-frame}.
The distribution in this angle is sensitive to the transverse 
$\tau^+ \tau^-$ spin correlations, 
which are different for the scalar and pseudoscalar.
Precise enough reconstruction of the {\it  H/A rest-frame} may turn out to
 be a challenge.
Many effects, theoretical ({\it e.g.} QED bremsstrahlung), or experimental 
(beamstrahlung, not sufficient
hermeticity of the detector, angular/energy resolution {\it etc.}) may 
invalidate the method.
 It is generally expected  
that the Monte Carlo method is the only way to estimate whether such 
measurement can be
realized in practice, and which performance features  
of the future detectors may turn out to be crucial.

In the paper, we will present an algorithm for generating decays of 
$\tau^\pm$ leptons produced
in $H/A \to \tau^+ \tau^-$ including full spin correlations. We will 
show  some distributions
to check correct functioning of the program as well. Finally, distributions
with smearings emulating  detection-like effects will be given. 

We  start from  an algorithm presented
in Refs. \cite{Pierzchala:2001gc,Golonka:2000iu}, where
$ \tau$ lepton decay is combined,
 with the part of the event for $\tau$  production provided by any (host) 
Monte Carlo program. 
The only requirement is that generated  events are stored in {\tt HEPEVT}
common block \cite{PDG:1998}. 
The kinematical information
on the momenta of all particles forming an event is  enough to calculate
spin state of the $ \tau$, at least at the level of the approximation used. 
The  {\tt TAUOLA} library \cite{Jadach:1990mz,Jezabek:1991qp,Jadach:1993hs}
 is used 
to decay  $\tau$ lepton(s) of the fixed helicity state. The
 {\tt PHOTOS} \cite{Barberio:1990ms,Barberio:1994qi} Monte Carlo program 
is used for generating radiative corrections in the decay.



Since Higgs boson spin is zero,
the correlations {\it do not} depend at all, on the mechanism of  Higgs
 boson production.
This facilitated our work a lot. Technical difficulties related to the 
choice of
$\tau^+$ and $\tau^-$ spin quantization frames, present in the case  
of  $e^+e^- \to  Z/\gamma \to \tau^+  \tau^-$ \cite{gps:1998,jadach-was:1984}
(bremsstrahlung effects included or not), were not present.
Some modifications  were nonetheless necessary for introduction of full spin
correlations into the algorithm. 
Let us list changes introduced with respect to  algorithm of 
Ref. \cite{Pierzchala:2001gc}:
\begin{enumerate}
\item
  The {\tt TRALOR} routine \cite{Jadach:1990mz}  defining relation 
of $\tau^\pm$ spin quantization frames and laboratory frame, had to be 
replaced. New version assures
that in case of Higgs boson decay, the quantization frames for spin states of 
$ \tau^+$ and  $\tau^-$ are properly oriented with respect to each other. 
In fact they are 
simply connected 
 by the single  boost along the $z$-axis (no rotation). Boosts along the 
same axis
connect the $\tau^\pm$ rest-frames  with 
the Higgs boson rest frame. The Higgs boson 
rest frame is connected with the Laboratory frame in the way as explained 
in \cite{Pierzchala:2001gc}.
\item
   The density matrix was taken from Ref. \cite{Kramer:1994jn} and defined 
for the quantization 
   frames as specified in previous point. 
   Case of purely scalar or pseudoscalar Higgs boson is implemented only.
   Any further extension is straightforward.
\item
    Generation of the decays is implemented in subroutine ({\tt  SPINHIGGS}) 
    following the method explained  in Ref.  
    \cite{Jadach:1990mz} and already  previously used in 
{\tt  KORALB} \cite{jadach-was:1984}.
\item 
   We  assume, that in the Higgs boson decay, production generator, does 
not include
   bremsstrahlung.  Instead,  
{\tt PHOTOS} \cite{Barberio:1990ms,Barberio:1994qi}
   can be used for that purpose, once generation of $\tau^\pm$  
decays is completed.
\item
   More complete inclusion of  bremsstrahlung corrections, would require 
   substantial re-write and extension of the  program to the solution 
   as in Ref. \cite{kkcpc:1999} or a similar one.
\end{enumerate}


Once we have explained the main principles of the generation algorithm, 
let us turn to the discussion 
of numerical results. We will take  Higgs boson of $120$ $GeV$ as an example
and we will look for the difference if it is scalar or pseudoscalar.
In the first group of our plots thick line will denote predictions for  
scalar Higgs boson and 
thin line for the pseudoscalar one. 
For simplicity, and to better visualize spin effects, 
we  take single   $\tau^\pm \to \nu \pi^\pm$ decay mode only. 

Let us start with  non-observable distributions 
defined in the Higgs boson rest-frame, which can nonetheless serve as 
convenient technical tests of our program. 
Fig.~\ref{rysunek1}  presents  distribution in the 
polar angle  ($\phi^*$) denoting relative
acoplanarity  of $\pi^\pm$ momenta around $\tau^\pm$ momentum axis.
The  distribution is indeed, as it should be \cite{Kramer:1994jn}, 
proportional to  \newline
$\sim 1 \mp {\pi^2 \over 16} \cos \phi^*$
respectively for scalar and pseudoscalar Higgs.
In Fig.~\ref{rysunek2} we plot the  distribution of $\pi^+ \pi^-$ 
acollinearity angle ($\delta^*$).
The difference between scalar and pseudoscalar Higgs is clearly visible 
especially
for acollinearities close to $\pi$ (see Fig.~\ref{rysunek3}).
If not washed out by the detector and/or bremsstrahlung effects this 
would offer
very interesting possibility. 

Let us now turn to the distributions defined for the combined process 
of decay and production 
of the Higgs boson. For the production we  have taken the process
$e^+e^- \to Z H$; $Z \to \mu^+\mu^-; H \to \tau^+ \tau^-$ (only scalar $H$  
can be produced
in this process), at Center-of-Mass-System energy of $350$ $GeV$. Monte 
Carlo program {\tt PYTHIA 6.1}
\cite{Pythia} was used, and effects due to initial state bremsstrahlung were
taken into account. 
The thick line will denote now, and in all further plots, the case when all 
spin effects are included. 
Thin line will denote the  case when longitudinal spin correlations 
are included only.
The difference between the two lines visualizes the size of the transverse 
spin effects.
If we  compared predictions for scalar and pseudoscalar, the difference would 
be 
roughly factor of two larger. However, ambiguities due to generally distinct
 production mechanisms would make 
the picture more involved  and less suitable for our discussion.

As we can see in Fig.~\ref{rysunek5}, the $\pi^+ \pi^-$ acollinearity angle 
($\delta$) distribution 
in laboratory frame looks quite different than in the Higgs boson rest-frame, 
the two cases of different spin treatments are practically indistinguishable, 
distribution is 
not peaked at $\delta \sim \pi$ at all.
  
If information on the beam energies and energies of all other
observed particles (high $p_T$ initial state bremsstrahlung 
photons, decay products of $Z$ {\it etc.}) are taken into considerations
the Higgs rest frame  can be reconstructed. 
We may  define the 'reconstructed' Higgs boson  momentum as the difference of 
sum of beam energies
and momenta of all visible particles, that is decay products of $Z$ and all 
radiative photons of 
$| \cos\theta| < 0.98$.
We will mimic detector and beamstrahlung  effects in a very crude way, 
assuming 
flat  spread over the range of  $\pm$ $2$ $GeV$ with respect to the 
reconstructed 
transverse momentum of the 
Higgs boson  and over the range of $\pm$ $5$ $GeV$ for the longitudinal 
component. 
As we can see  ( Figs.~\ref{rysunek6} and ~\ref{rysunek7}) in 
distribution of  acollinearity 
angle ($\delta^\bullet$) defined  
in reconstructed Higgs boson rest-frame, the effects due to transverse spin 
effects 
are  barely, but still visible. 

In our example study, we have neglected detector uncertainty for the 
measurement of directions and 
energies 
of charged $\pi^\pm$.
We have not discussed  at all,  
the effects due to final state bremsstrahlung, even though we  have 
installed the possibility in our program. 
We think, bremsstrahlung and detector effects need to be discussed together. 
Some work was already started on studying detector effects, see \cite{Bower} 
for details.
With the Monte Carlo 
tool like the present one, it will be  possible to extend this study.
 More realistic assumptions 
on detector functioning can be made, and all $\tau$ decay modes can be used.
Also, properties of the other $\tau$ decay modes can be used to improve 
sensitivity. 
The methods fruitful for {\it e.g.} measurement of $\tau$ polarization at 
LEP 1 and for study of 
CP parity are known since long 
\cite{Harton:1995dj,Nelson:1995vt}, and 
may become useful in our case as well.

Question of precision required by experiment for the ultimate Monte Carlo 
solution and
question of the uncertainty of our solution, will need to be addressed
at a certain point. Definitely before  interpretation of the real data of 
Linear Collider experiments. 
This is however out of the scope of the present paper.

\hspace{1 mm}
\vspace{1.5 cm}
\vskip 3 mm
\centerline{\large \bf Acknowledgements}
\vskip 3 mm
Inspiring atmosphere at LC workshops in Cracow, Tokyo and Chicago was 
essential for this work to be performed.
Authors are specially grateful to M. Ronan and M. Peskin for initial 
discussions.
 Useful discussion with G. Bower  is also acknowledged.

%*******************************************************************
\begin{figure}[!ht]
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\put(130, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek1.ps,width=120mm,
height=140mm}}}
\end{picture}
\caption
{\it  The $\pi^+ \pi^-$ acoplanarity  distribution (angle $\phi^*$)  
in the  Higgs boson rest frame.   Thick line denotes the case 
of the scalar Higgs boson and 
thin line the pseudoscalar one.}
\label{rysunek1}
\end{figure}
%*************************************************************************  
%*******************************************************************
\begin{figure}[!ht]
%\centering
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\put(130, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek2.ps,width=120mm,
height=140mm}}}
\end{picture}
\caption
{\it The  $\pi^+ \pi^-$ acollinearity distribution  (angle $\delta^*$)  
in the  Higgs boson rest frame. Full angular range $0 <\delta^* < \pi $ is shown.
Thick line denotes the case 
of the scalar Higgs boson and 
thin line the pseudoscalar one. }
\label{rysunek2}
\end{figure}
%*************************************************************************
%*******************************************************************
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%\centering
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%\put( 0,0){\framebox( 1600,800){ }}
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\put(-200, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek3.ps,width=120mm,height=140mm}}}
\put(500, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek4.ps,width=120mm,height=140mm}}}
\end{picture}
\caption
{\it The  $\pi^+ \pi^-$ acollinearity distribution  (angle $\delta^*$)  
in the  Higgs boson rest frame.
Parts of the distribution close to the end of spectrum; $\delta^* \sim \pi$ are shown.
Thick line denotes the case 
of the scalar Higgs boson and 
thin line the pseudoscalar one.}
\label{rysunek3}
\end{figure}
%*************************************************************************

%*******************************************************************
\begin{figure}[!ht]
%\centering
\setlength{\unitlength}{0.1mm}
\begin{picture}(1600,800)
%\put( 0,0){\framebox( 1600,800){ }}
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\put(130,-400){\makebox(0,0)[lb]{\epsfig{file=rysunek5.ps,width=120mm,height=140mm}}}
\end{picture}
\caption
{\it  The  $\pi^+ \pi^-$ acollinearity distribution  (angle $\delta$)  
in the laboratory frame. Full angular range $0 <\delta< \pi $ is shown.
Thick line denotes the case when all spin effects are included in the decay 
of the scalar Higgs boson and for 
thin line only longitudinal spin correlations are included. The two lines are nearly 
indistinguishable.}
\label{rysunek5}
\end{figure}
%*************************************************************************

  
%*******************************************************************
\begin{figure}[!ht]
%\centering
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%\put( 0,0){\framebox( 1600,800){ }}
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\put(130, -405){\makebox(0,0)[lb]{\epsfig{file=rysunek6.ps,width=120mm,
height=140mm}}}
\end{picture}
\caption
{\it The  $\pi^+ \pi^-$ acollinearity distribution  (angle $\delta^\bullet$)  
in the scalar Higgs boson reconstructed rest-frame. Full angular range $0 <\delta^\bullet < \pi $ is shown.
Thick line denotes the case when all spin effects are included  and for 
thin line only longitudinal spin correlations are taken.
 }
\label{rysunek6}
\end{figure}
%************************************************************************* 

%*******************************************************************
\begin{figure}[!ht]
%\centering
\setlength{\unitlength}{0.1mm}
\begin{picture}(1600,800)
%\put( 0,0){\framebox( 1600,800){ }}
\put( 375,750){\makebox(0,0)[b]{\large }}
\put(1225,750){\makebox(0,0)[b]{\large }}
\put(-200, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek7.ps,width=120mm,height=140mm}}}
\put( 500, -400){\makebox(0,0)[lb]{\epsfig{file=rysunek8.ps,width=120mm,height=140mm}}}
\end{picture}
\caption
{\it The  $\pi^+ \pi^-$ acollinearity distribution  (angle $\delta^\bullet$)  
in the scalar Higgs boson reconstructed rest-frame. 
Parts of the distribution close to the end of spectrum; $\delta^\bullet \sim \pi$ are shown.
Thick line denotes the case when all spin effects are included  and for 
thin line only longitudinal spin correlations are taken.
 }
\label{rysunek7}
\end{figure}
%*************************************************************************


\newpage
\hspace{1 mm}
\vspace{12 cm}


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