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\begin{document}
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\title{
\rightline  {\small}
\rightline  {\small{SINP-TNP/02-32}}
\rightline  {\small{DO-TH 02/19}}\vspace*{0.5cm}
\bf 
Can R-Parity violation explain
the apparent \boldmath $B\r\pi^+\pi^-$ anomaly ?}
\author{ 
{\large\bf Gautam Bhattacharyya}
\thanks{Electronic address: gb@theory.saha.ernet.in}\\
Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, 
India
\and
{\large\bf Amitava Datta}
\thanks{Electronic address: adatta@juphys.ernet.in}\\
Department of Physics, Jadavpur University, Kolkata 700032, India
\and 
{\large\bf Anirban Kundu}
\thanks{Electronic address: kundu@zylon.physik.uni-dortmund.de}
\thanks{On leave from Department of Physics, Jadavpur University, 
Kolkata 700032, India}\\
Universit\"at Dortmund, Institut f\"ur Physik, D-44221 Dortmund, Germany} 
\maketitle


\begin{abstract}
All predictions based on the standard model for the branching ratio
and CP asymmetries in the decay channels $B_d (\bar{B_d}) \r \pi^+
\pi^-$ in general rely on the predictive hard final state interaction
dominance hypothesis, yielding small strong phases. Such small values
are disfavoured by the current data, in particular by the large CP
asymmetries measured by the Belle collaboration, which seem to require
the relevant phase to be large, although experimental uncertainties do
not permit the final verdict. It is shown that in the presence of
R-parity violating supersymmetry a new tree level contribution to the
decay amplitude may lead to branching ratio and asymmetries compatible
with the data even if the strong phase happens to be small. Further
tests of this model via related $B$-decay modes and collider searches
are discussed.

\vskip 10pt \noindent
\texttt{PACS Nos:~11.30.Fs, 12.60.Jv, 13.25.Hw} \\
\texttt{Key Words:~R-parity violation, B-decays}

\end{abstract}

\newpage



It has long been known that the channel $B \r \pi^+ \pi^-$ would give us
the first estimate of the unitarity triangle (UT) angle $\alpha$ (we use
$B$ as a general shorthand for both $B_d$ and $\bar{B_d}$).
A large volume of data on this channel has recently been made available
by BaBar, Belle and CLEO collaborations.  The  branching
ratio (BR), averaged over $B_d$ and $\bar {B_d}$ decays,
has been measured by all the three groups. The world average is given by
\cite{branching}:
\be
{\rm BR} (B\r\pi^+\pi^-) =  (4.4\pm 0.9)\times 10^{-6}.
\ee


The CP-violating asymmetries have been measured by both BaBar and Belle
collaborations. The time dependent asymmetry $a^{dm}_{f}(t)$
 for any flavour non-specific decay $B_q \r f$, where $f$ is a
final state accessible to both $B_q$ and $\bar{B}_q$ mesons
and $q = d$ or $s$, is defined by
\bea
a^{dm}_{f}(t) &=& {\Gamma(B_q(t)\r f) - \Gamma(\bar{B_q}(t) \r f) \over
\Gamma(B_q(t)\r f) + \Gamma(\bar{B_q}(t) \r f)}\n\\
&=&a_{f}^d\cos\Delta mt+a_{f}^m\sin\Delta mt
\eea
where $\Delta m$ is the mass difference between the two mass eigenstates.
The data  for the $\pi^+\pi^-$ decay channel
are quoted  in terms of $S_{\pi\pi} = -  a_{\pi\pi}^m $
(the  asymmetry parameter corresponding to mixing induced CP
violation \cite{review})
and $C_{\pi\pi} = a_{\pi\pi}^d $
(the  asymmetry parameter corresponding to direct CP
violation \cite{review}).
The latter parameter is denoted by $A_{\pi\pi}$ in the notation of Belle,
where $C_{\pi\pi} = -A_{\pi\pi}$. The latest results are:
\bea
S_{\pi\pi} &=& 0.02\pm 0.34\pm 0.05   ~~({\rm BaBar}~ 
                                       ~\cite{babar}) ; ~~
           -1.21^{+0.38+0.16}_{-0.27-0.13} ~~ ({\rm Belle} 
                                            ~\cite{belle}) ; 
\nonumber\\
C_{\pi\pi} &=& -0.30\pm 0.25\pm 0.04  ~~({\rm BaBar}~ 
                                       ~\cite{babar}) ; ~~  
           -0.94^{+0.31}_{-0.25}\pm 0.09~~ ({\rm Belle} 
                                            ~\cite{belle}) .
\eea
In view of the model independent constraint $S_{\pi\pi}^2 +
C_{\pi\pi}^2 < $ 1.0, the large central values indicated by the Belle
data should not be taken at face values. Nevertheless this shows
for the first time that large (almost maximal) direct CP violating
asymmetry in this channel is a distinct possibility contrary to
earlier beliefs.

Now the crux of the matter is that {\em it is difficult to
simultaneously explain the BR($B\r\pi^+\pi^-$) and the Belle
CP-asymmetry data within the framework of the standard model (SM).}
We shall elaborate  this point in the following. 


For the sake of simplicity let us assume that only two interfering
amplitudes contribute to $\bar{B_d}\r\pi^+\pi^-$ and denote them by
$$
a_1\exp(i\phi_1)\exp(i\delta_1) \ \ \ {\rm and} \ \ \ 
a_2\exp(i\phi_2)\exp(i\delta_2),
$$
where $\phi_i$'s and $\delta_i$'s ($i=1,2$) are the weak and the 
strong phases respectively. 
We also use the notation \be
\Delta\delta = \delta_2-\delta_1; \ \ \ \Delta\phi = \phi_2-\phi_1
\ee
The observables $a_{CP}^{d}$ and $a^m_{CP}$ can be expressed in terms
of the above parameters. One obtains
\bea
a_{CP}^{d} & = &{1-|\lambda_{\pi\pi}|^2\over 1+|\lambda_{\pi\pi}|^2}
= {2a_1a_2\sin\Delta\phi\sin\Delta\delta\over a_1^2 + a_2^2 + 2a_1a_2
\cos\Delta\phi\cos\Delta\delta}, \\
%\eea
%\bea
{\rm and} ~~~~
a^m_{CP} &=& {2\ {\rm Im}\lambda_{\pi\pi}\over
1+|\lambda_{\pi\pi}|^2},
~~~{\rm where}~~~\\
%\eea
%\bea
%{\rm where} ~~~~
\lambda_{\pi\pi} &=& 
e^{-i\phi_M}{\bra \pi\pi|\bar{B_q}\ket\over \bra \pi\pi|B_q\ket}
= \eta_{CP} e^{i(-\phi_M+2\phi_1)}
{\left
(a_1^2+a_2^2e^{2i\Delta\phi} + 
2a_1a_2e^{i\Delta\phi}\cos\Delta \delta\right)
\over
\left (a_1^2+a_2^2+2a_1a_2\cos(\Delta\delta-\Delta\phi)\right) }.
\eea
Here  $\phi_M$ is the phase of the $B_d - \bar{B}_d$ mixing amplitude
(this may include  phases from the CKM elements as well as phases
from  new physics), and $\eta_{CP}$ is the CP eigenvalue ($+1$) for the
final state $\pi^+\pi^-$.

For the sake of completeness we also include the expressions for the
BR $(B\r\pi^+\pi^-)$: 
\bea {\rm BR} (B_d \r \pi^+ \pi^-) &\sim& a_1^2 + a_2^2
+ 2a_1a_2\cos(\Delta\delta -\Delta\phi) \nonumber\\ 
{\rm BR} (\bar{B}_d \r
\pi^+ \pi^- ) &\sim& a_1^2 + a_2^2 + 2a_1a_2 \cos(\Delta\delta+
\Delta\phi), 
\eea 
where the phase space factors have been
suppressed. When one averages over these two terms, one obtains the
expression in the denominator of Eq.~(5).


In the SM $a_1$ and $a_2$ are identified with the tree and the
top-mediated penguin amplitudes \cite{review} respectively, so that
$\phi_1=-\gamma$, $\phi_2 = \beta$ and $\phi_M = 2\beta$.  One expects
$a_2$ to be considerably suppressed with respect to $a_1$ due to the
standard loop suppression factors.  In addition the hard final state
interaction (HFSI) dominance hypothesis, which all calculations with
predictive power employ (see below) in one form or the other, implies
$|\sin\Delta\delta| \ll 1$. Thus the observable $a_{CP}^{d}$ appears
to be generically small.  Moreover, as we shall see below, the
measured BR in Eq.~(1), turns out to be smaller than the SM
prediction, the degree of discrepancy depending upon the method of
calculation.

This leads us to the problem: Explanation of the BR ($B \r
\pi^+\pi^-$) needs a strong destructive interference between the two
amplitudes, leading to $|\cos\Delta\delta|\sim 1$, whereas the large
asymmetry as measured by Belle requires $|\sin\Delta\delta|\sim 1$
(see Eq.~(5)), both of which cannot be reconciled at the same time.
As we shall see below, this inherent conflict shows up in most of the
analyses within the framework of the SM.  However, due to large error
bars in the current data and uncertainties in the theoretical
predictions, one cannot conclusively exclude the SM but it will be
under pressure if large asymmetries, as indicated by the the Belle
data and the central value of the BaBar data on $C_{\pi \pi}$, persist.

Now we shall comment very briefly on some detailed analysis in the
SM. The short distance part of the effective Hamiltonian governing
$B$-decays is well understood \cite{review}.  It consists of several
QCD corrected four-Fermi effective operators popularly known as tree
(or current-current) operators, QCD and electroweak penguin operators,
etc. The corresponding Wilson coefficients have been obtained to the
next-to-leading order (NLO) accuracy. We do not go into any
discussions on the uncertainties coming from the regularisation scheme
dependence, as well as the choice of the regularisation scale, and
other theoretical and experimental uncertainties like the magnitude of
the relevant CKM elements, the light quark masses, the decay form
factors, etc. They only add to the overall uncertainty in the
prediction.

In principle, the relevant decay amplitudes may be obtained by
computing the matrix elements of these operators for specific initial
and final states. However, here the long distance effects come into
play and there is no foolproof way of handling them. Even more
uncertain are the so called strong phases arising due to final state
interactions. All calculations which are fully predictive assume the
dominance of the so-called HFSI at the quark level. 
In the simplest approach the resulting
strong phases, usually small in magnitude, are perturbatively computed
from the absorptive part of the penguin diagram \cite{hfsi}.

Long distance effects may also arise through soft final state
interactions (SFI) at the hadron level.  These non-perturbative phases
can only be computed in specific models \cite{sfi}. The resulting
phases, which may  be small or large depending on the channel, are very
much model dependent.

There is no fully convincing argument which rigorously justify the
HFSI dominance hypothesis. There is of course the intuitive colour
transparency argument according to which the SFI's are subdominant
effects in decays which are not colour suppressed \cite{bjorken}.
Some justification of the colour transparency argument can be found in
the treatment of $B$ decays in the QCD factorisation approach
\cite{beneke}.  In this formulation, the basic assumption is that in
the heavy quark limit $\Lambda_{QCD} \ll m_b$ a nonleptonic $B$ decay
into two mesons is dominated by hard gluon exchange.  The decay
amplitude can be expressed in terms of a nonperturbative meson form
factor, the light-cone wavefunctions of the participating mesons and a
perturbatively calculable hard scattering kernel. While the first two
are always real, the decay amplitude develops an imaginary part
through the kernel.  In the heavy quark limit, the strong interaction
phases can be computed as expansions in $\alpha_s$ and hence are
small.  At least in the leading order in $\Lambda_{QCD}/m_b$, all long
distance effects can be absorbed in the meson form factor and the
light-cone amplitudes (though beyond leading order this simple picture
may break down). This prediction within the framework of perturbative
QCD more or less agrees with the branching ratio measurements,
particularly when both the final state mesons are light.

In view of the above uncertainties, both QCD and non-QCD, there is no
universally accepted theoretical prediction on the nonleptonic decay
modes in general.  Several methods involving varying degrees of
sophistication have been invoked.

The simplest method is the one based on conventional factorisation
\cite{ali}.  In this approach the relevant matrix elements are
computed in terms of phenomenological parameters like form factors,
the number of effective colours ($N_c$) varying between 2 and
infinity, quark masses etc.  It is further assumed that the final
state interactions can indeed be computed perturbatively as in
\cite{hfsi}.  The CKM parameters are varied over the entire range
currently allowed by indirect phenomenological fits to several
electroweak observables \cite{ckm}.  It was found that the
CP-asymmetries and branching ratios of some of the $B$-decay channels
(the so called type I processes) are remarkably stable with respect to
the variation in $N_c$.  These predictions were therefore marked as
robust \cite{ali,ali1}.  The decay $B \r \pi^+ \pi^- $ belongs to this
class.

When the predictions of \cite{ali} are confronted with recent data on
$B \r \pi^+ \pi^- $, several disagreements show up.  First, the
predicted BR for this mode, given by (9.0-15.0)$\times 10^{-6}$,
appears to be higher than the experimental data by about $5\sigma$,
even after all theoretical uncertainties are taken into account.  It
should, however, be emphasized that the branching ratios of many other
$B$ decay channels \cite{branching} are still in perfect agreement
with the predictions of \cite{ali} (except when the decay product
involves the $\eta$ or $\eta'$ mesons, which are outside the scope of
our discussions).

The predictions for the CP asymetries in the conventional
factorisation approach are \cite{ali1}: $S_{\pi\pi} = -0.35^{+
0.016}_{-0.022}$ , $C_{\pi\pi} = 0.07 ^{+ 0.036}_{-0.016}$, for $\rho$
= 0.12, $\eta$ = 0.34, $N_c$ = 3 and $k^2$ = $m_b^2 \pm 2$
GeV$^2$. Here $\rho$ and $\eta$ are well known Wolfenstein parameters
and $k^2$ is a measure of the degree of virtuality of the gluon.
Varying the Wolfenstein parameters over the allowed range, it can be
readily checked that the predictions are indeed incompatible with the
Belle data.


In the QCD factorization approach \cite{beneke} several observed
branching ratios 
including that of the $B \rightarrow \pi^+ \pi^- $ channel can be
explained (updated results can be found in Table 1 of
\cite{beneke1}). Unfortunately the same cannot be said about the
asymmetries. The predictions are clearly incompatible with the Belle
data (see figure 2 of \cite{beneke1}). 
%****************************************************
The BaBar data \cite{babar},
because of the large error bars involved, are inconclusive at the moment.
There is no  immediate demand to invoke physics beyond the SM but the
possibility of a future conflict with the SM remains open.


In the Perturbative QCD (PQCD) framework \cite{sanda,keum-sanda} of
the $B$ meson decaying into two light mesons, the hard process
dominates. The spectator quark, which is almost at rest, is required
to emit a hard gluon to receive sufficient kick to catch up a fast
moving quark involved in the weak decay to form a meson. In this
approach the final state interaction arising from soft gluon exchanges
between final state hadrons are negligible. The novel result is that
due to a sizable imaginary part in the factorized annihilation
contribution, one can generate a significant direct CP violation in
the $B\rightarrow \pi^+ \pi^-$ decay. PQCD generates a BR for the
above mode which is close to the experimental number (the two are
still incompatible at the $1\sigma$ level), but the direct CP
asymmetry (23$\pm 7\%$), though somewhat larger than the other
predictions, is not compatible with the Belle data even at the
$2\sigma$ level.

In the analysis of \cite{gronau} the magnitudes of the amplitudes are
taken from earlier phenomenological fits \cite{rosner}. These values
are consistent with the predictions of the QCD factorisation approach
\cite{beneke} and are typically larger than the corresponding results
of \cite{ali} by factors of 2.0 to 3.0. However, the strong phase
difference $\Delta\delta$ is fitted from the average of the Belle and
BaBar results.  It is then found that in order to explain the
asymmetry data the angle $\alpha$ of the unitarity triangle greater
than $\pi/2$ and $\Delta\delta$ in the vicinity of $-\pi/2$ are
preferred.  Such large magnitudes of $\Delta\delta$ are certainly
against the spirit of colour transparency and the QCD factorisation
approach \cite{beneke}. The BR data on the otherhand clearly prefer
values of $\Delta\delta$ close to zero or $\pi$ for reasons discussed
earlier. However, $|\Delta\delta|$ close to $\pi/2$, as preferred by
the asymmetry data, cannot be conclusively excluded in view of the
large error bars (see, {\em e.g.}, figure 4 of \cite{gronau}).

In a more recent work \cite{hou}, several branching ratios and CP
asymmetries in $B\r K\pi$ and $B\r \pi\pi$ channels were considered
(in view of the large error bars $S_{\pi\pi}$ and $C_{\pi\pi}$ were
not included in the fit). Using the amplitudes as given by the
conventional factorisation \cite{ali,ali1}, it was concluded that
large values of the strong phase difference are preferred by the data,
which is also against the spirit of the colour transparency argument.

In summary the following conclusions emerge. If there is indeed a
large direct CP violating asymmetry, as is indicated by the Belle
data, then no calculation from first principles based on the
hypothesis of HFSI dominance, can explain the data, be it the
conventional factorisation \cite{ali,ali1}, the QCD factorisation
\cite{beneke} or the PQCD approach \cite{sanda}.  In fact as long as
the large SFI's, apparently favoured by the data \cite{gronau,hou}
cannot be computed from first principles, no such theory has any
predictive power.

In view of this an alternative scenario seems to be quite appealing,
which we pursue in this letter. Does any kind of new physics, with
a relatively small strong phase compatible with the HFSI dominance
hypothesis \cite{bjorken,beneke}, describe the Belle asymmetry and the
BR ($B \r \pi^+ \pi^-$) data adequately? If so, what else can be said
about that new physics?

In this letter we shall show that this alternative can indeed be
achieved by supersymmetry with R-parity violation (RPV)
\cite{rpv_original,rpv}.  Possible impacts of RPV on $B$ decays has
been emphasized by many authors \cite{rpvcp,alak}. The main point is
that unlike most extensions of the SM, {\em RPV contributes to $B$
decay amplitudes at the tree level}.  Moreover, the current bounds
\cite{rpv} on sparticle masses and couplings leave open the possiblity
that such contributions can indeed be comparable to or even larger
than the SM amplitude. It may be recalled that the presence of two
interfering amplitudes of comparable magnitude is essential for a
large direct CP violating asymmetry (see Eq.~(5)).

It is well known that in order to avoid rapid proton decay one cannot
have both lepton number and baryon number violating RPV model, and we
shall work with a lepton number violating one. This leads to
slepton/sneutrino mediated $B$ decays.  Since the current lower bound
on the slepton mass \cite{rpv} is weaker than that on squark mass,
larger effects within the reach of current round of experiments are
more probable in this scenario. We start with the superpotential
\be
\label{w}
{\cal W}_{\lambda'} = \lambda'_{ijk} L_i Q_j D^c_k,
\ee
where $i, j, k = 1, 2, 3$ are quark and lepton generation indices;
$L$ and $Q$ are the SU(2)-doublet lepton and quark superfields and
$D^c$ is the SU(2)-singlet down-type quark
superfield respectively. For the process $B\rightarrow \pi^+\pi^-$,
the relevant four-Fermi operator is of the form
\be
    {\cal H}_{\lambda'} =
\frac{\lambda'_{i11}\lambda'^*_{i13}}{2m^2_{\tilde{e}_{L_i}}}
(\bar{u}\gamma^\mu P_L u)
                       (\bar{d}\gamma_\mu P_R b) + {\rm h.c.}
\ee
where $P_R(P_L)= (1+(-)\gamma_5)/2$. In the above formula $i$ 
is the generation index of the slepton.
The current bound on $\lambda'_{111}$ is too restrictive
($|\lambda'_{111}| <  3.5\times 10^{-4}$ \cite{rpv}), which rules
out the possibility that this coupling plays any significant role in 
$B$ decays.
For  $i=2$ or 3, the bound on the product $\lambda'_{i11}\lambda'^*_{i13}$
is rather modest ($|\lambda'_{i11}  \lambda'_{i13}|<  3.6 \times 10^{-3}$)
\cite{mixing}. 
Following the standard practice we shall assume that the RPV 
couplings are hierarchical {\em i.e.}, only one combination of the 
couplings is numerically significant.

The matrix element of the RPV operator for $B\r\pi^+\pi^-$ is given,
using conventional factorisation \cite{ali}, by
\be
\bra \pi^+\pi^-|{\cal H}_{\lambda'}|\bar{B_d}\ket =
-{1\over 4} \frac{\lambda'_{i11}\lambda'^*_{i13}}{m^2_{\tilde{e}_{L_i}}}
{m_\pi^2\over (m_d+m_u)(m_b-m_u)} f_\pi F_0^{B\r\pi}(m_\pi^2)
(m_B^2-m_\pi^2)
\ee
where $f_\pi$ is the pion decay constant and $F_0$ is the BSW formfactor.
We consider the use of conventional factorisation to be the most
conservative approach in this context since the SM predictions within
this framework maximally disagree with the data. 


Using this and the matrix elements of the SM operators \`a la the
conventional factorisation \cite{ali}, one can now calculate the BR
and CP asymmetries. In order to obtain an intuitive feeling of the
numerical results, it will be useful to give a close look at
Eqs.~(5-8), with the SM tree and RPV as the two interfering
amplitudes. However, for actual numerical results the SM penguin
amplitudes have been included.

A comment on the $B$-$\bar{B}$ mixing phase $\phi_M$ in Eq.~(7) is now
in order.  In the SM this phase is computed from the box diagram, and
$\phi_M$ turns out to be 2$\beta$, where $\beta$ is one of the angles
of the unitarity triangle. When R-parity is violated some additional
box diagrams contribute to the mixing amplitude \cite{mixing}. The
additional contribution is governed by the same combination
($\lambda'_{i11}\lambda'^*_{i13}$) of RPV couplings which is at the
focus of attention of this paper.  Consistency, therefore, demands
that this contribution be included. For simplicity, we do not take
into consideration the contributions coming from the R-parity
conserving sector of supersymmetry. 

This may have profound consequences for the extraction of the angle
$\beta$ of the unitarity triangle.  If we assume, as usual,
hierarchical RPV couplings then only a particular combination of
$\lambda^{\prime}$ couplings is numerically significant, which may
directly influence only a few related $B$ decay channels, where the
underlying quark-level process is either $b\r u\bar{u}d$ or $b\r
d\bar{d} d$.  However, RPV can universally affect all the mixing
induced asymmetries through $\phi_M$. For example, the mixing induced
asymmetries in $B \r J/\psi K_S$ and $B \r \phi K_S$ happen to be
proportional to sin 2$\beta$ in the SM.  In the presence of RPV, the
apparent $\beta$ ($\beta_{eff}$) extracted from these asymmetries will
not be equal to the SM $\beta$. If the other two angles of the
unitarity triangle ($\alpha$ and $\gamma$) are measured through decay
channels not affected by RPV, the measured values of the three angles
may not add up to $\pi$. It is noted that even in the presence of RPV
the same asymmetry should be measured in both the decay channels
mentioned above. Thus the recently reported discrepancy in $\beta$
measured in the above two channels \cite{phi} cannot be explained by
our choice of RPV couplings. For such an explanation direct tree level
RPV contribution to one of the above channels through another
combination of couplings will be needed \cite{alak}.


We now turn to the numerical results. The RPV model introduces four
extra parameters compared to the SM: (a) the left slepton and
sneutrino masses, which are equal up to the SU(2) breaking D-terms, (b)
the magnitude of the product $\lambda'_{i11}\lambda'^*_{i13}$ (which
according to our convention can be either positive or negative), (c)
the phase of this product, hereafter called $\phi$ or the weak RPV
phase, which can have any value between 0 and $\pi$ to maintain
consistency with the sign convention in (b), and (d) the strong phase
between the SM tree and the RPV amplitudes varying between 0 and
$2\pi$. We fix the slepton mass at 100 GeV which is consistent with
the current bound coming from direct searches \cite{aleph}.  The
magnitude of the product coupling and the remaining two parameters are
randomly varied within their allowed ranges or bounds.  The most
interesting point is that in spite of so many apparently free
parameters, a very restrictive pattern will emerge at the end of the
analysis.

In order to carry out the numerical analysis we need some more inputs
like quark masses, form factors and the relevant CKM elements.  We use
$m_u = 4.2$ MeV, $m_d = 7.6$ MeV, $m_b = 4.88$ GeV, pion decay
constant $f_\pi = 132$ MeV, and the decay formfactor in the BSW model
$F_0^{B\r\pi} (m_\pi^2) = 0.39$ GeV.  Our philosophy is to take the
strong phase difference between the tree and the penguin amplitudes to
be small as required by the HFSI hypothesis.  For the purpose of
illustration, we have randomly varied this phase difference in the
interval $-1^\circ$ and $23^\circ$ --- a range motivated by the QCD
factorisation approach \cite{beneke1}. The CKM parameters whose values
are not precisely known have been varied randomly within the range
allowed by the CKM fit \cite{ckm}. In particular $V_{td}$ is allowed
to lie in the range between 0.0030 and 0.0096. Arguably such ranges
may change in the presence of RPV, since the $B_d$-$\bar{B_d}$ mixing
amplitude and the resulting mass difference of $B_d$ meson mass
eigenstates ($\Delta m_{B_d}$), an important ingredient of the CKM
fit, are affected for reasons discussed above. In order to compensate
for the restricted inputs we have not constrained the weak phase
$\gamma$ within the SM range, but varied it randomly in the entire
allowed range 0 to $\pi$. The other important input parameter $\sin
(2\beta)$ has been varied between 0.25 and 1.0 though we present our
results for the benchmark value $\sin(2\beta) = 0.79$.  We have
checked that none of our results, apart from the allowed range of RPV
weak phase, depends sensitively on the choice of the angle $\beta$, and
thus this analysis holds for some other slightly different CKM fits
too (see, {\em e.g.}, \cite{buras}).

As stated above the effective Hamiltonian in Eq.\ (10) also leads to a
pair of new box amplitudes for $B$-$\bar{B}$ mixing. The first kind
has two $\lambda'$ vertices, two SU(2) gauge couplings, and two
$u$-type quarks, one slepton and one $W$ inside the box. The second
type has four $\lambda'$ vertices and involves only sleptons and $u$
quarks inside the box. Neglecting the SM box completely, and taking
the product coupling to be real, the authors in \cite{mixing} found
the conservative bound $|\lambda'_{i11}\lambda'_{i13}| \leq 3.6\times
10^{-3}$. We, on the other hand, take into account the SM box and the
possible phase of the product coupling which is varied over the range
already given.  This, as discussed above, modifies the phase $\phi_M$
from its SM value of $2\beta$ to $2\beta_{eff}$.  We now impose the
constraint that $\sin(2\beta_{eff})$ should satisfy the observed
CP-asymmetry in the $B\rightarrow J/\psi K_S$ channel (i.e.,
$\beta_{eff}$, which is a combination of $\beta$, RPV weak phase
$\phi$, and the box amplitudes, should satisfy $0.69 \leq
\sin(2\beta_{eff})\leq 0.89$).

%-----------------------------------------
\begin{figure}[htbp]
\vspace{-10pt}%3in
\centerline{\hspace{-3.3mm}
\rotatebox{-90}{\epsfxsize=6cm\epsfbox{fig_1a.ps}}
\hspace{-0.1cm}
\rotatebox{-90}{\epsfxsize=6cm\epsfbox{fig_1b.ps}}}
\vspace*{3mm}
\centerline{\hspace{-0.5cm} (a) \hspace{7.0cm} (b)}
\hspace{3.3cm}\caption[]{\small Allowed values of strong phase difference
$\Delta\delta$ between RPV and SM tree amplitudes.}
\protect\label{fig1}
\end{figure} 
%------------------------------------



We now list all the constraints imposed in our study of the allowed
space of the RPV parameters: (i) $\Delta m_{B_d}$ \cite{branching},
(ii) CP asymmetry from the decay $ B\r J/\psi K_S$, (iii) BR
($B\r\pi^+\pi^-$) as in Eq.~(1), and (iv) the asymmetries $C_{\pi\pi}$
and $S_{\pi\pi}$ as in Eq.~(3).  Since data from BaBar and Belle are
incompatible at the level of more than $1\sigma$, it may not be a good
idea to take their averages.  Instead, we consider the Belle CP
asymmetry data at 90\% confidence interval -- this range has a
significant overlap with the BaBar direct CP asymmetry data.  In
addition, we also impose the model independent constraint
$S_{\pi\pi}^2+C_{\pi\pi}^2<1$.  We shall later comment on the
implication of BaBar data on RPV models. There is no obvious
inconsistency in the branching ratio measurements reported by all the
three collaborations. Therefore we require the branching ratio
constraint to be satisfied within 1$\sigma$.

%-----------------------------------------
\begin{figure}[htbp]
\vspace{-10pt}%3in
\centerline{\hspace{-3.3mm}
\rotatebox{-90}{\epsfxsize=8cm\epsfbox{fig_2.ps}}}
\hspace{3.3cm}\caption[]{\small Possible values of the RPV weak phase 
($\phi$). For more details, see text.}  \protect\label{fig2}
\end{figure}
%------------------------------------




The random variation of the parameters as discussed above leads to the
scatter plots displayed in figures 1 to 5. The following salient
features are to be noted.
\begin{enumerate}
\item
%*******************************************************************
Figures (1a) and (1b) show that there are only two favoured regions
for $\Delta \delta$, the strong phase difference between the SM tree
and the RPV amplitude. They are near zero (which is equivalent to
2$\pi$) and $\pi$. Values of $\delta$ near $\pi/2$ are disfavoured. To
understand this result intuitively, let us neglect the SM penguin
amplitude temporarily and identify the amplitudes $a_1$ and $a_2$ in
Eq. (8) with the SM and RPV tree amplitudes respectively. Since the
experimentally observed branching ratio is significantly below the SM
prediction, what one requires is a strong destructive inteference
between the two amplitudes. From the expression of the average BR (the
denominator of Eq.\ (5)) it is obvious that this observable will be
smaller than $a_1^2$ provided $2 |a_1| > a_2$ and sign ($a_1 a_2 \cos
\Delta \delta$ cos $\Delta \phi$) is negative. However, $a_2$ can not
be too small with respect to $a_1$ since that would supress the
magnitude of this reduction via destructive interference as well as
$C_{\pi\pi}$. This automatically restricts the magnitude of the
product of the cosines. It should be emphasized that this fine balance
between $a_1$ and $a_2$ can be maintained only because within the
current experimental constraints the RPV amplitude at tree level can
be as large as or even larger than the SM amplitude. Moreover, to make
the destructive interference effective $|\cos \Delta \delta| \sim 1$
and $|\cos \Delta \phi| \sim 1$ are preferred. These choices, however,
are incompatible with a large $C_{\pi\pi}$ and this leads to
intermediate values of these parameters. Another important factor that
pushes $\Delta \delta$ close to zero or $\pi$ is the large value of
$S_{\pi\pi}$.  It is indeed gratifying to note that the favoured
values of $\Delta \delta$ are consistent with the colour transparency
argument which endows a small strong phase to a colour allowed decay.

%-----------------------------------------
\begin{figure}[htbp]
\vspace{-10pt}%3in
\centerline{\hspace{-3.3mm}
\rotatebox{-90}{\epsfxsize=8cm\epsfbox{fig_3.ps}}}
\hspace{3.3cm}\caption[]{\small Correlation between the RPV weak phase
($\phi$) and the UT angle $\gamma$.}
\protect\label{fig3}
\end{figure}
%------------------------------------



\item One also notes from figures (1a) and (1b) that $C_{\pi\pi}$ up to
$-0.66$ and $S_{\pi\pi}$ up to $-0.65$ can be accommodated. Thus, 
even in the presence of RPV the asymmetries cannot be arbitrarily large
unless future data indicate a significant upward shift in BR ($B \r
\pi^+\pi^-$).

\item The RPV weak phase ($\phi$) can only have a limited range, as is
evident from figure (2). This figure shows allowed bands near zero and
$\pi$, but one should interpret this figure with some care, since the
value of $\phi$ is quite sensitive to the choice of $\beta$. The
allowed range of $\phi$ is controlled by the mismatch between
$\sin(2\beta)$ and $\sin(2\beta_{eff})$.


\item The possible values of the UT angle $\gamma$ are not totally
arbitrary but its possible range is correlated to the range of $\phi$.
In fact, $\Delta \phi$ ($= \phi+\gamma$ in our convention) is a
crucial parameter for both branching ratio and CP asymmetries. One
should remember that in the presence of RPV interaction $B \r
\pi^+\pi^-$ ceases to be a good channel for $\alpha$ determination.

\item We obtain a new bound on the product coupling (see figure 4)
\begin{equation}
|\lambda'_{i11}\lambda'_{i13}| \leq 2.5\times 10^{-3},
\end{equation}
which is a marginal improvement over its existing bound of $0.0036$.
However, our bound is more general since we take into account the
possibility of destructive interference between the SM and the RPV box
amplitudes. This bound is more or less stable against the
variation of $\beta$.

%-----------------------------------------
\begin{figure}[htbp]
\vspace{-10pt}%3in
\centerline{\hspace{-3.3mm}
\rotatebox{-90}{\epsfxsize=8cm\epsfbox{fig_4.ps}}}
\hspace{3.3cm}\caption[]{\small Allowed range of $|\lambda'_{i11}
{\lambda'}^*_{i13}|$.}
\protect\label{fig4}
\end{figure}
%------------------------------------


\item Figures (5a) and (5b) show the interesting correlations between
the weak and the strong phase differences. While the branching ratio
data prefer both cosines to be near $+1$ or $-1$, the Belle CP
asymmetry data require the sines to be large. The allowed region is
therefore the one that appears as a compromise between these two.


%-----------------------------------------
\begin{figure}[htbp]
\vspace{-10pt}%3in
\centerline{\hspace{-3.3mm}
\rotatebox{-90}{\epsfxsize=6cm\epsfbox{fig_5a.ps}}
\hspace{-0.1cm}
\rotatebox{-90}{\epsfxsize=6cm\epsfbox{fig_5b.ps}}}
\vspace*{3mm}
\centerline{\hspace{-0.5cm} (a) \hspace{7.0cm} (b)}
\hspace{3.3cm}\caption[]{\small Correlation between cosines and sines of
weak and strong phase differences.}
\protect\label{fig5}
\end{figure}
%------------------------------------


\item The width of the so-called bands of solutions is essentially a
reflection of the uncertainties in the input parameters; once they are
narrowed down, one or two unique solutions may emerge. It is to be
checked whether they are compatible with the data from other related
decay channels.

\item 
While fitting the Belle data, we have witnessed above a tussle between
the requirement of large cosines by the branching ratio measurement
and large sines by the CP asymmetry measurements. This is considerably
eased when we fit the BaBar data which is compatible with
large as well as small  values of CP
asymmetries -- the range at 90\% CL are: $S_{\pi\pi}$ between $0.6$
and $-0.6$, and $C_{\pi\pi}$ between $0.13$ and $-0.73$.  We have, in
fact, observed that the entire range of $S_{\pi\pi}$ and $C_{\pi\pi}$
observed by BaBar can be accommodated in our scenario. The upper bound
on the $\lambda'$ product coupling turns out to be $2.2\times
10^{-3}$, which is more or less equal to that found from the Belle
data.  Finally, the small strong phase solutions are indeed preserved
while fitting the BaBar data.
\end{enumerate}


So far we have focussed our attention on the quark level process
$b\rightarrow u\bar{u} d$ and studied its impact in $B \r \pi^+ \pi^-$
decay. It is now time to wonder what would be the impact of the
related quark level process $b\rightarrow d\bar{d} d$? Both operators
contributes to $B \r \pi^0 \pi^0$ and to understand the nature of the
SM and RPV contributions to this process it is important to recall
that the quark composition of $\pi^0$ is the antisymmetric combination
$(u\bar u - d\bar d)/\surd{2}$.  In the SM, while $b\rightarrow
u\bar{u} d$ corresponds to a colour suppressed tree diagram,
$b\rightarrow d\bar{d} d$ can proceed only through penguin
graphs. Because $u\bar u$ and $d\bar d$ combine antisymmetrically
inside $\pi^0$ the QCD penguin diagrams exactly cancel leaving the
colour suppressed tree and the subdominant electroweak penguin graphs
contributing to $B \r \pi^0 \pi^0$. The predicted BR is, therefore,
rather small and well within the current experimental upper bound
\cite{branching}.  What happens in the RPV scenario?  At the quark
level there are two tree diagrams, with $\lambda'_{i11}$ and
$\lambda'_{i13}$ at the two vertices, the final state being $u\bar{u}
d$ in one case and $d\bar{d} d$ in the other. While the former
proceeds through the exchange of a virtual slepton, for the latter the
propagator is a sneutrino. If we consider the slepton and the
sneutrino of a given generation to be exactly degenerate, the quark
composition of $\pi^0$ again ensures a strong cancellation of the two
amplitudes. So a significant nonvanishing RPV contribution would only
arise when the degeneracy between the slepton and the sneutrino mass
is lifted (possibly owing to a small D-term contribution). But in any
case the RPV contribution is hugely suppressed and the present
experimental upper limit on the branching ratio can easily accommodate
the combined SM plus RPV contributions. Similar arguments can be
advanced for the channel $B^+\rightarrow \pi^+\pi^0$ as well.
However, an interesting test of the scenario presented in this paper
may be possible in a few years when we will have sufficient data on
the $B\rightarrow\rho\pi$ channel (recall, $\rho$ has a symmetric
combination of $u\bar u$ and $d\bar d$).


The RPV scenario that we have considered in this paper can be directly
tested at colliders. The associated light sleptons/sneutrinos can be
produced at the Tevatron and at LHC via resonant production
\cite{dreiner} providing a useful cross-check of this scenario.  The
$\lambda'_{i11}$ couplings (in particular, $\lambda'_{211}$) give rise
to a distinct collider signature in the form of like-sign dilepton
signals. Such final states have low SM and R-parity conserving
supersymmetry background. The dominant production mechanism is a
$\lambda'$ induced resonant charged slepton production at tree level
at hadron colliders. This is followed by a R-parity conserving gauge
decay of the charged slepton into a neutralino and a charged
lepton. The neutralino can then decay via the crossed process to give
rise to a second charged lepton, which due to the majorana nature of
the neutralino can have the same charge as the hard lepton produced in
the slepton decay.  The study of ref. \cite{dreiner} shows that for a
value of $\lambda'_{211} = 0.05$, which is perfectly compatible with
our bound on the product $\lambda'_{211} \lambda'^*_{213}$, a smuon
mass of about 310 GeV would be visible above the backgrounds with 2
fb${}^{-1}$ integrated luminosity at the Tevatron Run II, while for
the same coupling a resonant smuon can be observed with a mass of 750
GeV at LHC with 10 fb${}^{-1}$ integrated luminosity. If RPV indeed
plays a role in $B$ decays, as discussed in this paper, then the smuon
mass is likely to be in the range 100-200 GeV. It is, therefore,
reasonable to expect smuon signals at the upgraded Tevatron.
\vspace*{1cm}


AD and AK have been supported by the BRNS grant 2000/37/10/BRNS of DAE,
Govt.\ of India. AK has also been supported by
the grant F.10-14/2001 (SR-I) of UGC, India, and
by the fellowship of the Alexander von Humboldt Foundation.



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\end{document}

