%Title: Charged-Higgs effects in a new B -> D tau nu differential decay distribution
%Author: Ulrich Nierste, Stephanie Trine, and Susanne Westhoff
%Published: <EM> Phys. Rev. D </EM> <B> 78 </B> (2008) 015006. 
%arxiv/0801.4938
\documentclass[aps,prl,twocolumn]{revtex4}

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\newcommand{\BDTN}{B \to D \tau \nu_{\tau}}
\newcommand{\BDTNp}{B \to \bar{D} \tau^+ \nu_{\tau}}
\newcommand{\BDTNm}{\bar{B} \to D \tau^- \bar{\nu}_{\tau}}
\newcommand{\BDLN}{B \to D \ell \nu_\ell}
\newcommand{\BDLNp}{B \to \bar{D} \ell^+ \nu_\ell}
\newcommand{\BDLNm}{\bar{B} \to D \ell^- \bar{\nu}_\ell}
\newcommand{\BTN}{B \to \tau \nu_\tau}
\newcommand{\BTNp}{B^+ \to \tau^+ \nu_\tau}
\newcommand{\BTNm}{B^- \to \tau^- \bar{\nu}_\tau}

\newcommand{\Bbar}{\,\overline{\rm B}}
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\begin{document}
\title{\boldmath Charged--Higgs effects in a new $B\to D \tau \nu_{\tau}$ differential decay distribution \unboldmath}

\author{Ulrich Nierste} 
\author{St\'ephanie Trine} 
\author{Susanne Westhoff} 

\affiliation{
~\\ 
Institut f\"ur Theoretische Teilchenphysik\\
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
}




\begin{abstract}
We show that the decay mode $\BDTN$ is competitive with and complementary
to $\BTN$ in the search for charged--Higgs effects.    
Updating the relevant formfactors, we find that the differential distribution in the
 decay chain $\bar{B}\to D\bar{\nu}_{\tau}\tau^-[\to\pi^-\nu_{\tau}]$
 excellently discriminates between Standard--Model and charged--Higgs
 contributions.  By measuring the $D$ and $\pi^-$ energies and the angle
 between the $D$ and $\pi^-$ tracks one can determine 
 the effective charged--Higgs coupling including a 
possible CP--violating phase.
% the magnitude and phase of the effective charged--Higgs coupling $g_S$.
\end{abstract}

\maketitle

%\section{Introduction}    
\noindent
\textbf{1.~Introduction.}~
The $B$ factories BaBar and BELLE have
accumulated enough statistics to probe extensions of the Higgs sector of
the Standard Model. Notably, the decay $\BTNp$ allows us to place first
constraints on the parameters $\tan \beta$ and $M_{H^+}$ of the
two--Higgs--doublet model (2HDM) of type II \cite{btau}. 
Here $\tan\beta$ is the ratio of the two Higgs vacuum expectation values and $M_{H^+}$
is the mass of the physical charged Higgs boson $H^+$ in the model.
Since the couplings of $H^+$ to $b$'s and $\tau$'s grow with
$\tan\beta$, $\BTNp$ probes large values of $\tan\beta$.  At tree--level
the Higgs sector of the Minimal Supersymmetric Standard Model (MSSM)
coincides with the type--II 2HDM.
The coupling of $H^+$ to fermions can be modified by a 
factor of order one due to  $\tan\beta$--enhanced radiative 
corrections \cite{ar03,ltb},
yet this introduces only a few additional supersymmetric parameters
and the access to the Higgs sector is not obfuscated like in many other modes,
such as $b\to s\gamma$.
This explains the great theoretical
interest in the experimental ranges for ${\cal B} (\BTNp)$ \cite{bpexp}.

The decay $\BDTN$ provides an alternative route to charged--Higgs
effects \cite{t,itoh}. As we will show in the following,  
this mode is not only competitive with $B^+\to \tau^+\nu_\tau$, it also 
opens the door to a potential CP--violating phase in the Yukawa couplings
of the $H^+$ to $b$ and $\tau$. $\BDTN$ compares to $\BTN$ as follows:

\vspace{1mm}
i) ${\cal B} (\BDTN)$ exceeds ${\cal B} (\BTN)$
by roughly a factor of 50 in the Standard Model.

\vspace{1mm}
ii) $\BDTN$ involves the well--known element 
    $V_{cb}$ of the Cabibbo--Kobayashi--Maskawa (CKM) matrix. The 
    uncertainty on $|V_{ub}|$ entering $\BTN$ is much
    larger.

\vspace{1mm}
iii) ${\cal B} (\BTN)$ is proportional to two
    powers of the $B$ decay constant $f_B$, which must be obtained with
    non--perturbative methods. Current lattice gauge theory computations
    are struggling with chiral logarithms and $f_B^2$ can only be
    determined with an uncertainty of 30\% or more \cite{latt07}.
    $\BDTN$ involves two formfactors, one of which can be
    measured in $\BDLN$ ($\ell=e,\mu$) decays \cite{b01,HFAG}. The
    other one is tightly constrained by Heavy Quark Effective
    Theory (HQET) \cite{n94,cn}, so that hadronic uncertainties 
    can be reduced to less than 10\% once the measurement of $\BDLN$ is improved.

\vspace{1mm}
iv) Unlike $\BTN$ the three--body decay 
    $\BDTN$ permits the study of decay distributions 
    which discriminate between $W^+$ and $H^+$ exchange \cite{gh,ks,miki02}. 

\vspace{1mm}
v) The Standard Model (SM) contribution to $\BTN$
    is helicity--suppressed, so that the sensitivity of ${\cal B}
    (\BTN)$ to $H^+$ is enhanced. For $\BDTN$ a similar effect only
    occurs near the kinematic endpoint, where the $D$ moves slowly
    in the $B$ rest frame \cite{gh}:
    While the transversely polarized $W^+$ contribution suffers
    from a P--wave suppression, the virtual $H^+$ recoils against the 
    $D$ meson in an unsuppressed S--wave.

\vspace{1mm}
Items iv) and v) strongly suggest to study differential decay
distributions in $\BDTN$. The $\tau$ in the final state
poses an experimental challenge, because it does not travel far enough
for a displaced vertex and its decay involves at least one more neutrino.
In particular, the $\tau$ polarization, known as a charged-Higgs
analyzer, is not directly accessible to experiment \cite{t}.
The most straightforward way to deal with this is to consider the full
decay chain down to the detectable particles stemming from the $\tau$. 
We have studied the decays $\tau^- \to \pi^- \nu_\tau$, $\tau^- \to
\rho^- \nu_\tau$, and $\tau^- \to \ell^-\bar{\nu}_\ell \nu_\tau$ and
assessed the sensitivity of the decay distributions to $H^+$ effects.
We find that the decay chain $\bar{B}\to D \bar{\nu}_{\tau}\tau^-[\to \pi^-
\nu_{\tau}]$ discriminates between $W^+$ and $H^+$ exchange in an
excellent way. In this Letter we present the results for the
  differential decay rate as a function of the $D$ and $\pi^-$ energies
  and the angle between the $D$ and $\pi^-$ three--momenta for this decay
chain. Our result greatly facilitates the determination of
the effective coupling $g_S$ governing $H^+$ exchange,
including a potential complex phase.  A
conventional analysis combining Monte Carlo simulations of $\bar{B}\to D
\bar{\nu}_{\tau} \tau^-$ and $\tau^- \to \pi^- \nu_\tau$ decays would be
very cumbersome, because the $\BDTN$ differential distributions strongly
depend on the a--priori unknown value of $g_S$.\\[2mm]
\boldmath
%\section{$B\rightarrow D$ Formfactors}
\textbf{2.~$B\rightarrow D$ formfactors.}~
\unboldmath
The effective hamiltonian describing $B\rightarrow(D)\tau\nu_{\tau}$
transitions mediated by $W^+$ or $H^+$ reads (with $q=u\,(c)$) 
\begin{equation}
\label{eq:1}
\begin{split}
H_{\textrm{eff}} & =\frac{G_{F}}{\sqrt{2}} V_{qb}\text{\thinspace}
\big\{\left[  \overline{q}\gamma^{\mu}(1-\gamma_{5})b\right]  \text{\thinspace}
\left[  \overline{\tau}\gamma_{\mu}(1-\gamma_{5})\nu_{\tau}\right] \\
& -\frac{\ov m_{b}m_{\tau}}{m_{B}^{2}}\text{\thinspace}\text{\thinspace}
\overline{q}\left[ g_{S} + g_P \gamma_5\right] b  \text{\thinspace}
\left[\overline{\tau}(1-\gamma_{5})\nu_{\tau}\right]\big\}\,+\,\mbox{h.c.}
\end{split}
\end{equation}
The effective coupling constant $g_P$ only enters the $\BTN$ decay, while $\BDTN$ is only sensitive to $g_S\,$.
The $B^+$ meson mass $m_B$ is introduced in Eq.~(\ref{eq:1}), so that 
${\cal B} (\BTN)$ vanishes for $g_P=1$.
The above operators as well as $\ov m_b$ are defined in the $\ov{\rm MS}$ scheme.
In the MSSM, which is our main focus, one has $g_S=g_P$.

The analysis of $\BDTN$ requires the knowledge of the formfactors $F_V$
and $F_S$ which parametrize the vector and scalar current matrix
elements:
\begin{align}% 
  \langle D(p_D)|\bar{c}\gamma^{\mu}b|\bar{B}(p_B)\rangle \! =\,&
  F_V(q^2)\! 
  \left[p_B^{\mu}+p_D^{\mu}-m_B^2\frac{1-r^2}{q^2}q^{\mu}\right]\nn
  +\,& F_S(q^2)\,m_B^2\frac{1-r^2}{q^2}q^{\mu}\,,\nn \langle
  D(p_D)|\bar{c}b|\bar{B}(p_B)\rangle=\,& \frac{m_B^2\,(1-r^2)}{\ov
    m_b-\ov m_c}F_S(q^2)\,,\label{eq:2}
\end{align}
where $p_B$ and $p_D$ denote the meson four--momenta, $q=p_B-p_D$, and $r=m_D/m_B\,$.
It is convenient to introduce the normalized formfactors
$V_1\equiv F_V\,2\sqrt{r}/(1+r)$ and $S_1\equiv F_S(1+r)/(2\sqrt{r})$,
as well as the kinematic variable $w\equiv(1+r^2-q^2/m_B^2)/2r$.
%\begin{equation}
%\label{eq:3}
%w\equiv(1+r^2-q^2/m_B^2)/2r.
%end{equation}
In the limit of infinitely heavy quark masses $m_Q=m_b,\,m_c$
(which are properly infrared--subtracted pole masses),
both $V_1(w)$ and $S_1(w)$ reduce to the universal
Isgur--Wise function $\xi(w)$, normalized to $\xi(1)=1$\,.
At the kinematic endpoint $w=1$, corrections to this limit read
\begin{equation}
\label{eq:4}
V_1(1) =\eta_v-\frac{1-r}{1+r}\big(\delta_{\textrm{rad}}+\delta_{1/m_Q}\big)\,,\quad
S_1(1) =\eta_v\,,
\end{equation}
up to $\mathcal{O}(\alpha_s^2,1/m_Q^2)$.
Here $\eta_v$ denotes radiative corrections in the limit of equal
heavy meson masses, and $\delta_{\textrm{rad}}\,(\delta_{1/m_Q})$ are the
first order radiative $(1/m_Q)$ corrections to the function $\xi_-$
defined in \cite{n94}.  The $\delta_{1/m_Q}$ term depends on the
  subleading function $\xi_3(w=1)=\bar{\Lambda}\eta(w=1)$ and on the HQET parameter $\bar{\Lambda}$. We take $\bar{\Lambda}=0.5\pm
0.1\,\textrm{GeV}$, $\eta(1)=0.6\pm 0.2$ \cite{lnn}, $\eta_v$ and
$\delta_{\textrm{rad}}$ to $\mathcal{O}(\alpha_s)$ from Ref.~\cite{n92},
and add a $5\%$ error to the formfactors at $w=1$ to account for higher
order corrections. We obtain $V_1(1)=1.05\pm0.08$ and
$S_1(1)=1.02\pm0.05$.

The semileptonic decay into light leptons $\BDLN$ depends solely on the vector formfactor
$V_1(w)$. The measured quantity $|V_{cb}|V_1(w)$ was fitted
by the BELLE collaboration \cite{b01} to a two--parameter ansatz
$V_1(w,V_1(1),\rho_1^2)$ derived from dispersion relations
  and heavy quark spin symmetry \cite{cn}.  The fitted curve, however,
suffers from large statistical and systematic uncertainties near $w=1$.
We thus take $V_1(1)$ from HQET instead, use $|V_{cb}|=(4.17\pm0.07)\%$
from inclusive semileptonic $B$ decays \cite{pdg}, and only fix
the formfactor at large recoil $w=1.45$ from the data, including the dominant
systematic errors in a conservative way: $|V_{cb}|V_1(1.45)=(2.63\pm0.51)\%$.  The formfactor
over the whole kinematic range is then obtained using a two--parameter
description $F_{V}(w,a_0^{V},a_1^{V})$, 
which uses a conformal mapping $w\to z(w)$ resulting in an
  essentially linear dependence of $F_V$ on $z$ \cite{rh}. This
linearity in $z(w)$ is confirmed by the fact that fitting the $\BDLN$ data
with both $F_V$ parametrizations without further theoretical
constraints essentially gives the same result (see Fig.~\ref{ff}).
The sets of parameters corresponding to the minimal and maximal
  formfactors satisfying the HQET constraint at $w=1$ are displayed in
  Tab.~\ref{par} for both parametrizations $V_1(w,V_1(1),\rho_1^2)$ and
  $F_V(w,a_0^V,a_1^V)$.  They delimit the gray area in Fig.~\ref{ff}.
The HFAG \cite{HFAG} treats systematic errors in a different way and (including CLEO and ALEPH data)
finds smaller uncertainties at large recoil.

Similarly, the scalar formfactor $F_{S}(w,a_0^{S},a_1^{S})$ is constrained by HQET at $w=1$,
while its value at large recoil is fixed from the relation $F_S(q^2=0)=F_V(q^2=0)$.
The resulting parameters are displayed in the third line of Tab.~\ref{par}.


\begin{figure}[htbp]
{\psfrag{ix}{\raisebox{0.05cm}{\hspace{-0.3cm}\scalebox{1}{$w$}}}
\psfrag{iy}{\raisebox{0cm}{\hspace{-0.5cm}\scalebox{1}{$|V_{cb}|V_1(w)$}}}
\psfrag{1.0}{\hspace{-0.05cm}$\scalebox{0.8}{1.0}$}\psfrag{1.1}{\hspace{-0.05cm}$\scalebox{0.8}{1.1}$}\psfrag{1.2}{\hspace{-0.05cm}$\scalebox{0.8}{1.2}$}\psfrag{1.3}{\hspace{-0.05cm}$\scalebox{0.8}{1.3}$}\psfrag{1.4}{\hspace{-0.05cm}$\scalebox{0.8}{1.4}$}
\psfrag{1.5}{\hspace{-0.05cm}$\scalebox{0.8}{1.5}$}\psfrag{1.6}{\hspace{-0.05cm}$\scalebox{0.8}{1.6}$}
\psfrag{0.02}{\hspace{0.02cm}$\scalebox{0.8}{0.02}$}\psfrag{0.03}{\hspace{0.02cm}$\scalebox{0.8}{0.03}$}\psfrag{0.04}{\hspace{0.02cm}$\scalebox{0.8}{0.04}$}\psfrag{0.05}{\hspace{0.02cm}$\scalebox{0.8}{0.05}$}\psfrag{0.06}{\hspace{0.02cm}$\scalebox{0.8}{0.06}$}\psfrag{0.07}{\hspace{0.02cm}$\scalebox{0.8}{0.07}$}
\begin{center}
\hspace{-0.2cm}
\scalebox{1.2}{\epsfig{file=ff2801.eps}}
\end{center}
}
\vspace{-0.8cm}
\caption{Vector formfactor $V_1(w)$. Dots: exp. data \cite{b01} with
  stat. errors only. Dashed: fit to parametrization in \cite{cn}. Plain:
  fit to linear parametrization in \cite{rh}. Gray band: formfactor with
  HQET constraint at $w=1$; systematic errors dominate at large recoil.}
\label{ff}
\end{figure}

\begin{table}[hbtp!]
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
Parameters & minimal $F(w)$ & maximal $F(w)$ & central $F(w)$\\
\hline
$\{V_1(1),\rho_1^2\}$ & $\{0.97,1.47\}$ & $\{1.13,1.06\}$ & $\{1.05,1.24\}$\\
$\{a_0^V,a_1^V\}[10^{-4}]$ & $\{2.30,-14\}$ & $\{3.01,-5\}$ & $\{2.66,-9\}$\\
$\{a_0^S,a_1^S\}[10^{-3}]$ & $\{3.96,-26\}$ & $\{5.05,-7\}$ & $\{4.52,-16\}$\\
\hline
\end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Parameters $\{V_1(1),\rho_1^2\}$ for $F_V$ \cite{cn} and $\{a_0^{V,S},a_1^{V,S}\}$ for $F_{V,S}$ (see
  \cite{rh}, $Q^2=0$, $\eta=2$, subthreshold poles:
  $m(1^-)=6.337,\,6.899,\,7.012\,\textrm{GeV}$ and
  $m(0^+)=6.700,\,7.108\,\textrm{GeV}$ \cite{poles}).}
\label{par}
\end{table}

%\section{Charged--Higgs Effects}
\noindent
\textbf{3.~Charged--Higgs effects.}~
The MSSM is a well--motivated new--physics scenario in which
charged scalar current interactions occur at tree--level.
Resumming the dominant $\tan\beta$-enhanced loop corrections to all
orders, the couplings $g_{S,P}$ in Eq.~(\ref{eq:1}) specify to \cite{yuk,itoh}
\begin{equation}
\label{eq:5}
g_{S}=g_{P}=\frac{m_{B}^{2}}{M_{H^+}^{2}}\, 
\frac{\tan^{2}\beta}{ (1+\widetilde \epsilon_0\,\tan\beta)
                      (1+\epsilon_{\tau}\,\tan\beta)}\,.
\end{equation}
This particular form holds in MSSM scenarios with Minimal Flavor 
Violation (MFV).
The loop factor $\widetilde \epsilon_0$ arises from the quark Yukawa sector
and depends on ratios of superparticle masses, resulting in a
  sizable non--decoupling effect $\widetilde
\epsilon_0\tan\beta={\cal O}(1)$ for $\tan\beta={\cal O}(50)$.  $\epsilon_{\tau}$
comprises the corresponding effect for the $\tau$ lepton. $\widetilde
\epsilon_0$ and $\epsilon_\tau$ can receive sizable complex phases from
the Higgsino mass parameter $\mu$, if first--generation sfermions are
sufficiently heavy to soften the impact of the bounds on electric dipole moments on
$\arg \mu$. Beyond MFV also phases from squark mass matrices will easily
render $g_S$ complex.
It is therefore mandatory to constrain -- and eventually measure -- both
magnitude and phase of $g_S$. The type--II 2HDM is recovered
setting $\widetilde \epsilon_0=\epsilon_{\tau}=0$.

The $\BDTN$ branching ratio has recently been measured by the BaBar collaboration \cite{babarbdt}:
\begin{equation}
\label{eq:6}
R^{\textrm{exp}}\equiv\frac{\mathcal{B}(\BDTN)}{\mathcal{B}(\BDLN)}=(41.6\pm 11.7\pm 5.2)\%\,.
\end{equation}
The normalization to $\mathcal{B}(\BDLN)$ reduces the dependence
on the vector formfactor $F_V$ and thus tames the main theoretical uncertainties.
In the presence of charged--Higgs contributions, the theoretical ratio is approximated to 1\% by
\begin{equation*}
R^{\textrm{th}}=\frac{ 1.126 + 0.037\, r_V + r_0^2\, ( 1.544 + 0.082\, r_S + N_{H^+} ) }{ 10 - 0.95\, r_V }\ ,
\end{equation*}
\vspace{-5mm}
\begin{equation}
\label{eq:7}
\begin{split}
N_{H^+}=&- r_{cb}\,Re[g_S]\,( 1.038 + 0.076\, r_S )\\ 
        &+ r_{cb}^2\,|g_S|^2\,( 0.186 + 0.017\, r_S ),
\end{split}
\end{equation}
with $r_V=(a_1^V/a_0^V)/(-3.4)$, $r_S=(a_1^S/a_0^S)/(-3.5)$, $r_0=(a_0^S/a_0^V)/17$, and $r_{cb}=0.8/(1-\ov m_c/\ov m_b)$.
The dependence on the slope parameters $a_1^{V,S}$ appears to be quite mild.
In Fig.~\ref{bdt} we compare $R^{\textrm{th}}$ to the one--sigma measurement for positive $g_S$.
We also display the theoretical prediction without experimental
uncertainties on the formfactors
at large recoil to emphasize the importance of a better experimental knowledge of the $\BDLN$ modes.
Taking $F_V$ from the HFAG \cite{HFAG}, the error band is close to the dark gray band.

\begin{figure}[htbp]
{\psfrag{ix}{\raisebox{0.1cm}{$g_S$}}
\psfrag{iy}{\raisebox{-0.1cm}{\hspace{-0.2cm}$R$}}
\psfrag{X1}{$1\sigma$}
\psfrag{0.15}{}\psfrag{0.25}{}\psfrag{0.35}{}\psfrag{0.45}{}
\psfrag{0.0}{\hspace{0.06cm}$\scalebox{0.8}{0}$}\psfrag{0.5}{\hspace{-0.04cm}$\scalebox{0.8}{0.5}$}\psfrag{1.0}{\hspace{0.06cm}$\scalebox{0.8}{1}$}\psfrag{1.5}{\hspace{-0.04cm}$\scalebox{0.8}{1.5}$}\psfrag{2.0}{\hspace{0.06cm}$\scalebox{0.8}{2}$}\psfrag{2.5}{\hspace{-0.04cm}$\scalebox{0.8}{2.5}$}\psfrag{3.0}{\hspace{0.06cm}$\scalebox{0.8}{3}$}\psfrag{3.5}{\hspace{-0.04cm}$\scalebox{0.8}{3.5}$}
\psfrag{0.10}{\hspace{0.12cm}$\scalebox{0.8}{0.1}$}\psfrag{0.20}{\hspace{0.12cm}$\scalebox{0.8}{0.2}$}\psfrag{0.30}{\hspace{0.12cm}$\scalebox{0.8}{0.3}$}\psfrag{0.40}{\hspace{0.12cm}$\scalebox{0.8}{0.4}$}
\begin{center}
\hspace{-0.7cm}
\scalebox{1.2}{\epsfig{file=bdt2801.eps}}
\end{center}
}
\vspace{-0.8cm}
\caption{$R\equiv\mathcal{B}(\BDTN)/\mathcal{B}(\BDLN)$ as a function of $g_S$.
Light gray band: $R^{\textrm{exp}}$, see
  (\ref{eq:6}) \cite{babarbdt}. Gray band: $R^{\textrm{th}}$, see (\ref{eq:7}).
Dark gray band: $R^{\textrm{th}}$ without experimental systematic errors on the formfactors.
SM:
$R^{\textrm{th}}(g_S=0)=0.31\,{}^{+0.07}_{-0.05}\,(\textrm{gray})\,[\,\pm 0.02\,(\textrm{dark gray})]$.}
\label{bdt}
\end{figure}

The $\BDTN$ branching fraction is promising to discover -- or constrain --
charged--Higgs effects, but not to measure $g_S$ with good precision, as the dependence in Fig.~\ref{bdt} is too flat.
The differential distribution in the decay chain
$\bar{B}\to D \bar{\nu}_{\tau}\tau^-[\to\pi^-\nu_{\tau}]$ is better
suited for that purpose.
The experimentally accessible quantities are the energies $E_D$ and
$E_\pi$ of the $D$ and $\pi^-$ mesons, respectively, and the angle
$\theta$ between the three--momenta $\vec p_D$ and $\vec p_\pi$. We
define these quantities in the $B$ rest frame. We integrate over
  the phase space of the two unobserved
neutrinos in the final state. Our formulae contain the
full spin correlation between the production and decay of the $\tau$,
which is important to discriminate between SM and charged--Higgs contributions.
This approach further facilitates the rejection of
backgrounds from neutral particles escaping detection, as in $\bar{B}\to D
D^-[\to \pi^- \pi^0]$ with an undetected $\pi^0$: If the mass of the
undetected particle is $m$, this background can be suppressed by cuts
excluding the region around
\begin{align}
\cos\theta = \frac{(m_B-E_D-E_{\pi})^2-2(E_D^2-m_D^2)-m^2}{2(E_D^2-m_D^2)}\,. \label{eq:8}
\end{align}

We obtain the differential distribution
\begin{align}
  & \frac{d\Gamma(\bar{B}\to D\bar{\nu}_{\tau}\tau^-
    [\to\pi^-\nu_{\tau}])}{dE_D\,dE_{\pi}\,d\cos\theta}=G_F^4f_{\pi}^2|V_{ud}|^2|V_{cb}|^2\tau_{\tau}\label{eq:9}\\
  &
  \times\left[C_W(F_V,F_S)-C_{WH}(F_V,F_S)\,Re[g_S]+C_H(F_S)|g_S|^2\right]\nonumber
\end{align}
with formfactor--dependent functions of $E_D$, $E_{\pi}$, and
$\cos\theta$ for the SM ($C_W$), interference
($C_{WH}$), and Higgs ($C_H$) contributions, given 
as follows for vanishing $m_\pi$ (this approximation, 
which is good to 1\%, is not used in our numerical analysis),
\begin{equation}
\begin{split}
C_W& =\kappa \frac{m_{\tau}^4}{2}\frac{l^2}{p_{\pi}\cdot
    l} \, 
  \bigg\{ P^2 (b-1) + (P \cdot l)^2 \frac{2b}{l^2} + \\
  & \Bigg[
   \frac{l^2 (P\cdot p_\pi)^2 }{ (p_{\pi}\cdot l)^2} 
   - \frac{2 (P\cdot l)(P\cdot p_{\pi}) 
         }{p_{\pi}\cdot l} \Bigg] (3 b - 1)
  \bigg\}\, , \\
C_{WH} & = 2\kappa \, m_{\tau}^4  \, 
\frac{(1-r^2)F_S}{1-\ov m_c/\ov m_b} \, b \, 
  \bigg[ P \cdot l -\frac{l^2 P\cdot p_\pi }{p_{\pi}\cdot l} \bigg]\, , \\
C_H &=\kappa\,m_{\tau}^6 \frac{(1-r^2)^2
    F_S^2}{(1-\ov m_c/\ov m_b)^2}
  \Big(1-\frac{m_{\tau}^2}{2\,p_{\pi}\cdot l} \Big)\, , \\
\end{split} \label{defc}
\end{equation}
where $\ov m_c$ and $\ov m_b$ must be evaluated at the same scale so that 
$\ov m_c/\ov m_b=0.20\pm 0.02$ \cite{run}, and   
\begin{eqnarray}
P \!\!& =&\! F_V (p_B+p_D) - 
    (F_V-F_S)\frac{m_B^2(1-r^2)}{q^2} (p_B-p_D) \,, \nn
\kappa \!\!& =&\!
\frac{E_{\pi}\sqrt{E_D^2-m_D^2}}{
             128\,\pi^4\,m_B m_{\tau}}\,,
\quad b = \frac{m_\tau^2}{p_\pi \cdot l} 
      \Big( 1 - \frac{m_{\tau}^2}{2\,p_{\pi}\cdot l} \Big)\,, \nn
l \!\!& =&\! p_B-p_D-p_\pi \,,\quad 
   q^2 = (p_B-p_D)^2.\label{defpkb}
\end{eqnarray}
The dot products appearing in \eqsand{defc}{defpkb} are related to the energies,
momenta, and the angle $\theta$ measured in the $B$ rest frame as
$p_B\!\cdot\! l=m_B (m_B-E_D-E_\pi)$,
$p_D\!\cdot\! l = E_D (m_B -E_D-E_\pi) +
|\vec p_D|^2+|\vec p_D| E_\pi \cos\theta$, 
 $p_\pi\!\cdot\! l = E_\pi (m_B-E_D) 
+ |\vec p_D| E_\pi \cos\theta$, and $p_B\!\cdot\! p_D = m_B E_D$.
 Further $\tau_{\tau}=(290.6\pm 1.0)\times 10^{-15}s$ is the $\tau$
lepton lifetime, $f_{\pi}=(130.7\pm 0.1\pm 0.36)\,\textrm{MeV}$ the
pion decay constant, and the CKM matrix elements are $|V_{ud}|=0.97377\pm
0.00027$ and $|V_{cb}|=(41.7\pm 0.7)\times 10^{-3}$, the latter being
well determined from inclusive semileptonic $B$ decays \cite{pdg}.
Remarkably, one can probe a CP--violating phase of $g_S$ by exploiting the
shape of the distribution in Eq.~(\ref{eq:9}), which is not possible from the branching
fraction of either $\BDTN$ or $\BTN$.

\begin{figure*}[t]
\begin{minipage}{0.5\linewidth}
{\psfrag{ix}{$\cos\theta$}
\psfrag{iy}{\hspace{-2.2cm}$d\Gamma(B^0\to D^+\pi^-\bar{\nu}\nu)\
  [10^{-16}\textrm{GeV}]$}
\psfrag{X1}{$E_D=2\,\textrm{GeV}$, $E_{\pi}=1\,\textrm{GeV}$}
\psfrag{X2}{$g_S=0$ (SM)}
\psfrag{X3}{$g_S=1+i$}
\psfrag{X4}{$g_S=2$}
\begin{center}
\hspace{-1.5cm}
\scalebox{0.8}{\epsfig{file=bdp22801.eps}}
\end{center}
}
\end{minipage}
\begin{minipage}{0.45\linewidth}
{\psfrag{ix}{$\cos\theta$}
\psfrag{iy}{\hspace{-2.2cm}$d\Gamma(B^0\to D^+\pi^-\bar{\nu}\nu)\
  [10^{-16}\textrm{GeV}]$}
\psfrag{X1}{$E_D=2\,\textrm{GeV}$, $E_{\pi}=1\,\textrm{GeV}$}
\psfrag{X2}{$g_S=0$ (SM)}
\psfrag{X3}{$g_S=0.5$}
\begin{center}
\hspace{0cm}
\scalebox{0.8}{\epsfig{file=bdp052901-2.eps}}
\end{center}
}
\end{minipage}
\vspace{-0.4cm}
\caption{
$\bar{B}^0\to D^+\bar{\nu}_{\tau}\tau^-[\to\pi^-\nu_{\tau}]$ angular distribution
for $E_D=2\gev$ and $E_{\pi}=1\gev$.
Left: $g_S=0,1+i,2$. Right: $g_S=0,0.5$ (dark gray: without uncertainties in $F_V(w)$ and $V_{cb}$,
errors from $S_1(1)$ and $\ov m_c/\ov m_b$).
\label{bdp}
}
\end{figure*}


We show the differential decay distribution including
charged--Higgs effects in comparison with the SM for the meson energies
$E_D=2\,\textrm{GeV}$ and $E_{\pi}=1\,\textrm{GeV}$, so that the
whole range of $\cos\theta$ is kinematically accessible. In this
particular region of phase space the SM rate is
strongly suppressed for $\cos\theta=-1$. For a large scalar coupling
$g_S=2$ (Fig.~\ref{bdp}, left), the Higgs contribution dominates the rate at this
point (dark gray band), such that we can clearly distinguish it from
the SM (light gray band).
The experimental information from $\mathcal{B}(\BTN)$ constrains $|1-g_P|$. For real
$g_P$ this permits a range near $g_P=0$ and another range around
$g_P=2$. In the MSSM situation with $g_P=g_S$, the case $g_S=2$
therefore is in agreement with $\BTN$, but can be
confirmed or ruled out by measuring our distribution. The discrimination potential for the phase
of $g_S$ shows up in the light gray band: It corresponds to a complex
$g_S=1+i$, which yields the same
$\mathcal{B}(\BTN)$ as $g_S=0,2$.
The $\BDTN$ branching ratio alone may also help to distinguish between these solutions,
depending on the future experimental value of $\mathcal{B}(\BDTN)$, see Fig.~\ref{bdt}.
For general $g_S$ values, a fit to the differential distribution in
Eq.~(\ref{eq:9}) would excellently
quantify charged--Higgs effects, especially once better experimental information is available,
as we illustrate in Fig.~\ref{bdp} (right--hand side) for $g_S=0.5$ and fixed $D$ and $\pi^-$ energies.\\[2mm]
%\section{Conclusions}
\textbf{4.~Conclusions.}~
We have studied charged--Higgs effects in a differential distribution of the
decay chain $\bar{B}\to D\bar{\nu}_{\tau}\tau^-[\to\pi^-\nu_{\tau}]$, 
which has the following advantages over the branching fractions
$\mathcal{B}(\BTN)$ and $\mathcal{B}(\BDTN)$:

\vspace{1mm} 
i) The Higgs coupling constant $g_S$ can be determined from 
  the \emph{shape} of the distribution in sensitive phase space regions.
  This analysis should be possible with current $B$ factory data.

\vspace{1mm}
ii) The dependence on both $|g_S|$ and $\real [g_S]$ allows to
  quantify a possible CP--violating phase. 
  Since our decay distribution is a 
  CP--conserving quantity, the phase of $g_S$ is determined with a  
  two--fold ambiguity. In the MSSM such a phase stems from 
  the $\mu$ parameter or the soft breaking terms and enters through the 
  $\tan\beta$--enhanced loop factors 
  $\widetilde \epsilon_0$,\,$\epsilon_{\tau}$. Collider studies of this
  phase involve smaller loop factors and require the measurements of 
  tiny asymmetries \cite{ccp}.

\vspace{0.8mm} 
The main uncertainties stem from the formfactors. One can gain a
much better accuracy with better data on the vector formfactor
$F_V$. The recent $\BDLN$ measurement by BaBar
\cite{babarbdl} furnishes promising data for a new fit.

%% In the context of the MSSM, our distribution provides a further
%% stringent $B$ physics constraint to be included in the global
%% analysis of the $\tan\beta-M_{H^+}$ plane. The general character of
%% our result facilitates the confirmation or exclusion of different
%% MSSM scenarios with large $\tan\beta$.
Within the MSSM, one will be able to place new constraints 
on the $\tan\beta-M_{H^+}$ plane, once our results are confronted with 
actual data from the $B$ factories. If $\tan\beta/M_{H^+}$ is indeed
large, there is a fair chance to reveal charged--Higgs effects ahead of
the LHC.
\vspace{3mm}\\
\emph{Acknowledgments}: The authors acknowledge helpful discussions
  with Richard Hill and Michael Feindt.  This work is supported in part
  by the DFG grant No.~NI 1105/1--1, by the DFG--SFB/TR9,
and by the EU Contract No.~MRTN-CT-2006-035482, \lq\lq FLAVIAnet''.
\vspace{-1mm}

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\end{thebibliography}

\end{document}

