%Title: New Physics Effects in $B \to J/\Psi K$
%Author: Thomas Mannel
%Published: <EM> eConf  </EM> <B> C020516 </B> (2002) . 
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\boldmath
\Title{New Physics Effects in $B \to J/\Psi K$}
\unboldmath

\bigskip \bigskip

%+\addtocontents{toc}{{\it D. Reggiano}}
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\begin{raggedright}  

{\it Thomas Mannel\index{Mannel, T.}\\
Institut f\"ur Theoretische Teilchenphysik \\
University of Karlsruhe\\
D-76128 Karlsruhe, GERMANY}
\bigskip \bigskip
\end{raggedright}



\section{Introduction}
%
%
%
At present, we are at the beginning of the $B$-factory era in particle
physics, which will provide valuable insights into CP violation and
various tests of the CKM picture of this phenomenon.
Among the most interesting $B$-decay channels is the
``gold-plated'' mode $B_d\to J/\psi K_{\rm S}$ \cite{bisa}, which allows
the determination of the angle $\beta$ of the unitarity triangle of
the Cabibbo--Kobayashi--Maskawa (CKM) matrix. The present status
of these measurements is summarized in other contributions to this
conference \cite{FPCPBaBar,FPCPBelle}.

In this talk, which is based on a collaboration with
Robert Fleischer \cite{FM}, I consider the neutral mode
$B_d\to J/\psi K_{\rm S}$ together with its charged counterpart
$B^\pm\to J/\psi K^\pm$. Making use of the isospin symmetry of
strong interactions, we derive a model-independent parametrization
of the corresponding decay amplitudes.
After a careful analysis of the standard-model contributions, we consider
possible new-physics amplitudes and introduce 
a set of observables, which allows us a general analysis of new-physics
effects in the $B\to J/\psi K$ system.

There is a large variety of possible new physics scenarios. One of the most
popular one is a supersymmetric extension of the standard model, which has in
its most general form a very complicated flavour sector. 
Other models, like large extra dimension or left-right symmetric
models, have similar disadvantages, and a general feature of all these models
is an unacceptably large CP violation. It is fair to say that currently
there is no generally accepted way to model new physics effects in the
flavour sector.  

The standard model is the most general renormalizable theory with the
desired particle spectrum and interactions, and a 
generic parametrization of new-physics effects can in principle be
obtained by adding ``non-renormalizable'' dimension-6 operators ${\cal O}_i$
\cite{BW} 
\begin{equation}
{\cal L} = {\cal L}_{SM} +
           \frac{1}{\Lambda^2} \sum_i C_i {\cal O}_i
\end{equation}
where the $C_i$ are their coefficients and $\Lambda^2$ is
the scale of new physics.
The problem with this parametrization is that the number of possible
operators ${\cal O}_i$ is ${\cal O}(100)$, such that this parametrization
in its full generality is useless for practical applications. Still it
is useful to get an estimate of the size of a possible new physics
contribution.

Once we have added the new contributions given by the dimension-6 operators,
we proceed in the same way as before and consider a low-energy effective
theory by integrating out the top quark and the heavy gauge bosons. The result
of this step are four-fermion operators with couplings proportional to 
\begin{equation}
G_F = \frac{g^2}{M_W^2} = \frac{1}{<v>^2} 
\end{equation}
while contributions from new physics effects enter with coefficients of order
$1/\Lambda^2$. Furthermore, in the standard model we have in addition also
CKM factors, such that certain transitions in this model are small
and a possible new physics contribution could be observable even for
large scales $\Lambda$.

However, focussing on $\Delta B = \pm 2$ transition operators the number
of possible dimension-6 operators is not very large. Generically these
are given by 
\begin{equation}
{\cal L}_{new}^{\Delta B = 2} = \frac{1}{\Lambda^2} \sum C_i
\left[(\overline{b} \Gamma_i S_i d_1)(\overline{b} \Gamma_i S_i d_2)\right]
\end{equation}
where $\Gamma_i$ is an arbitrary Dirac matrix and $S_i$ indicates the color
structure ($S_i \times S_i = 1 \times 1$ or $T^a \times T^a$). Such a
contribution will  not only change the frequency of the $B - \overline{B}$
oscillations, but can also contribute to the mixing phase, which in the
standard model is simply $\sin (2 \beta)$.

The interesting point about the $\Delta B = \pm 2$ contribution is that the
standard model contribution is very small. The only non-vanishing coefficient
in the standard model is the one for
$\Gamma_i \times \Gamma_i =
 \gamma_\mu (1-\gamma_5) \times \gamma^\mu (1-\gamma_5)$ and
$S_i \times S_i = 1 \times 1$. This coefficient is of the order
\begin{equation}
C_{SM} = \frac{G_{\rm F}}{\sqrt{2}} \left(\frac{G_{\rm F}
M_W^2}{\sqrt{128} \pi^2}\right)(V_{td} V_{tb}^\ast)^2
\end{equation}
which is small due to the CKM suppression by two powers of $|V_{td}|$.
Furthermore, the standard model diagramms are at least one-loop, in which
case one finds another suppression by a loop factor $1/(16 \pi^2)$.  
It has already been pointed out some time ago \cite{Nir} that this leads
to a significant sensitivity to new physics in the $\Delta B = \pm 2$ sector.
Furthermore, an ansatz of this type is the $B$-physics analogue of 
the superweak model invented in the context of kaon-CP violation. 

To get a rough idea of possible new physics effects in
$B - \overline{B}$ mixing, we assume the same coupling strenght
(including the loop factor) as in the
standard model and find 
\begin{equation}
C_{NP} = \frac{G_{\rm F}}{\sqrt{2}}\left(\frac{G_{\rm F}
M_W^2}{\sqrt{128} \pi^2}\right) \frac{M_W^2}{\Lambda^2} 
\end{equation}
where $\psi$ is an additional weak phase induced by new physics.
Adding this new piece to the standard-model contribution we get an
estimate for the phase $\phi_{\rm M}$ of $B - \overline{B}$ mixing
\begin{equation}
\tan \phi_{\rm M} =
\frac{\sin(2 \beta) + \varrho^2 \sin (2\psi)}
{\cos (2 \beta) + \varrho^2\cos(2\psi)},
\end{equation}
with
\begin{equation}
\varrho=\left(\frac{1}{\lambda^3 A R_t}\right)\left(\frac{M_W}{\Lambda}
\right) 
\end{equation}
where $\lambda$ is the Wolfenstein parameter and $R_t$ is the length of
side opposite to $\gamma$ of the unitarity triangle. Note that $\varrho$
could be ${\cal O} (1)$ due to the presence of the CKM factors,
yielding a significant deviation. In turn, 
this yields a sensitivity up to very high scale, roughly $\Lambda \sim 8$ TeV.

\boldmath
\section{Analysis of $B \to J/\Psi K$}
\unboldmath

The analysis of the $\Delta B = \pm 1$ transitions is much more involved
than the $\Delta B = \pm 2$ case, since the number of possible operators
is much larger. For that reason it is more convenient to study a particular
decay.

For the class of decays  $B \to J/\Psi K$ it is convenient to first perform
an isospin analysis. The initial as well as the final state carry $I = 1/2$,
consequently the effective interaction has either $I = 0$ or $I = 1$.
Taking into account the neutral and the charged decay modes we have
the following isospin doublets
\begin{equation}
\left(\begin{array}{c}
|+1/2\rangle\\
|-1/2\rangle\end{array}\right):
\quad
\underbrace{\left(\begin{array}{c} |B^+\rangle  \\
|B^0_d \rangle \end{array}\right),
\,
\left(\begin{array}{c} |\overline{B^0_d}\rangle  \\
-|B^- \rangle
\end{array}\right)}_{CP},
\,
\underbrace{\left(\begin{array}{c} |J/\psi K^+\rangle  \\
|J/\psi K^0 \rangle
\end{array}\right),
\,
\left(\begin{array}{c} |J/\psi\overline{K^0}\rangle  \\
-|J/\psi K^-  \rangle \end{array}\right)}_{CP}
\end{equation}
where we indicated their relation through the CP tranformation.

Purely from isospin we find the relations
\begin{eqnarray}
&& \langle J/\psi K^+|{\cal H}_{\rm eff}^{I=0}|B^+\rangle =
+\langle J/\psi K^0|{\cal H}_{\rm eff}^{I=0}|B^0_d\rangle \\
&& \langle J/\psi K^+|{\cal H}_{\rm eff}^{I=1}|B^+\rangle =
-\langle J/\psi K^0|{\cal H}_{\rm eff}^{I=1}|B^0_d\rangle
\end{eqnarray}
where ${\cal H}_{\rm eff}^I$ indicates the different isospin components
of the effective interaction. Note that these relations are completely
general and thus hold even in the presence of new physics.

In the standard model we can write the amplitudes for the decays as
\begin{eqnarray} \label{SMPsiK}
&& A(B^+\to J/\psi K^+)=\frac{G_{\rm
F}}{\sqrt{2}}\left[V_{cs}V_{cb}^\ast\left\{
A_c^{(0)}-A_c^{(1)}\right\}+V_{us}V_{ub}^\ast\left\{A_u^{(0)}-
A_u^{(1)}\right\}\right]
\\
&& A(B^0_d\to J/\psi K^0)=\frac{G_{\rm F}}{\sqrt{2}}\left[V_{cs}V_{cb}^\ast
\left\{A_c^{(0)}+A_c^{(1)}\right\}+V_{us} V_{ub}^\ast
\left\{A_u^{(0)}+A_u^{(1)}\right\}\right],
\end{eqnarray}
where
\begin{eqnarray}
&& A_c^{(0)}=A_{\rm CC}^c-A_{\rm QCD}^{\rm pen}-A_{\rm EW}^{(0)},\quad
A_c^{(1)}=-A_{\rm EW}^{(1)} \\
&& A_u^{(0)}=A_{\rm CC}^{u (0)}-A_{\rm QCD}^{\rm pen}-A_{\rm EW}^{(0)},\quad
A_u^{(1)}=A_{\rm CC}^{u (1)}-A_{\rm EW}^{(1)}
\end{eqnarray}
where $CC$ indicates the contribution from the current-current operators,
$QCD-pen$ the one from the QCD penguin operators and $EW$ the one from the
electroweak penguins, while the superscript index denotes the isospin of
the contribution.

Relations (\ref{SMPsiK}) yield an idea about the sizes of the various terms.
The explicit CKM factors show a suppression of the terms carrying a weak
phase by a factor $\lambda^2$. Furthermore, all these contributions correspond
to penguin topologies and hence one expects a further, dynamical suppression.
It is commonly believed that this is of the order of the Wolfenstein parameter
$\lambda$, and we shall asusme for the sake of ``power counting'' that this
suppression factor is just $\lambda$. Due to the fact that the electroweak
penguin contributions are also strongly suppressed by the dynamics and
CKM factors, we get the well-known standard-model result
\begin{equation}
A(B^+\to J/\psi K^+) = A(B^0_d\to J/\psi K^0) =
\frac{G_{\rm F}}{\sqrt{2}}\left(1-\frac{\lambda^2}{2}\right)\lambda^2 A\,
A_c^{(0)} + {\cal O} (\lambda^3)
\end{equation}

A possible new physics contribution can enter this decay mode in
various ways. Firstly, there could be an additional weak phase from
new physics in the $\Delta B = \pm 2$ which has been discussed above.
Secondly, there could be new contributions in both
the $I=0$ and $I=1$ pieces of the effective Hamiltonian. These contributions
are parametrized as
\begin{eqnarray} \label{NPE}
A(B^+\to J/\psi K^+) &=& A_{\rm SM}^{(0)}
\left[1+\sum_k r_0^{(k)}e^{i\delta_0^{(k)}}e^{i\varphi_0^{(k)}}-
\sum_j r_1^{(j)}e^{i\delta_1^{(j)}}e^{i\varphi_1^{(j)}} \right] \\
A(B^0_d\to J/\psi K^0)&=&A_{\rm SM}^{(0)}
\left[1+ \sum_k r_0^{(k)}e^{i\delta_0^{(k)}}e^{i\varphi_0^{(k)}}+
\sum_j r_1^{(j)}e^{i\delta_1^{(j)}}e^{i\varphi_1^{(j)}} \right]
\end{eqnarray}
where $A_{\rm SM}^{(0)}$ is the standard model amplitude, 
$r_I^{(l)}$ is the Modulus, $\delta_I^{(l)}$ is the strong and
$\varphi_I^{(l)}$ the weak phase of the $l^{th}$  contribution with
isospin $I$. Note that according to the above discussion we have 
$ r_I^{(l)} \sim {\cal O} (M_W^2 / \Lambda^2) $

In order to test for these additional contributions, we look at the
observables in the $B \to j/\Psi K$ system. Aside from the information
gathered from the measurement of the time-dependent asymmetry in the
neutral modes 
\begin{equation}
a_{\rm CP} (t) = {\cal A}_{\rm CP}^{{\rm dir}}\,\cos(\Delta M_d t)+
{\cal A}_{\rm CP}^{\rm mix}\,\sin(\Delta M_d t)
\end{equation}
we suggest to also measure the direct CP asymmetry in the charged modes
\begin{equation}
{\cal A}_{\rm CP}^{(+)} = \frac{|A(B^+\to J/\psi K^+)|^2-
                                |A(B^-\to J/\psi K^-)|^2}
                               {|A(B^+\to J/\psi K^+)|^2+
                                |A(B^-\to J/\psi K^-)|^2}
\end{equation}
as well as the difference of the CP averaged rates of the charged and the
neutral modes ( $\langle ... \rangle$ denotes the CP average)
\begin{equation}
B = \frac{\langle|A(B_d\to J/\psi K)|^2\rangle-
          \langle|A(B^\pm\to J/\psi K^\pm)|^2\rangle}
         {\langle|A(B_d\to J/\psi K)|^2\rangle+
          \langle|A(B^\pm\to J/\psi K^\pm)|^2\rangle}
\end{equation}

Given the parametrization (\ref{NPE}) of new-physics effects, we suggest
a slightly different set of observables which is
\begin{equation}
S = \frac{1}{2}\left[
{\cal A}_{\rm CP}^{{\rm dir}}+{\cal A}_{\rm CP}^{(+)}
\right],\quad
D = \frac{1}{2}\left[
{\cal A}_{\rm CP}^{{\rm dir}}-{\cal A}_{\rm CP}^{(+)}
\right]
\end{equation}
and $B$ as given before.

In terms of the parameters introdued in (\ref{NPE}) we see that $S$
measures the $I=0$ new physics contribution
\begin{equation}
S= -2 \left[\sum_k r_0^{(k)}\sin\delta_0^{(k)}\sin\varphi_0^{(k)}\right]
\left[1-2\sum_l r_0^{(l)}\cos\delta_0^{(l)}\cos\varphi_0^{(l)}\right] \, , 
\end{equation}
$B$ is sensitive to a CP conserving $I=1$ new-physics contribution,
while $D$ is sensitive to a CP violating new physics contribution
\begin{equation}
B=+ 2 \sum_j r_1^{(j)}\cos\delta_1^{(j)}\cos\varphi_1^{(j)} 
\quad 
D=- 2 \sum_j r_1^{(j)}\sin\delta_1^{(j)}\sin\varphi_1^{(j)}
\end{equation}

However, one has to keep in mind the limitations of this approach.
The size of the standard-model contributions which are under poor
theoretical control is ${\cal O}(\lambda^3)$ which means that a
possible new physics contributions has to be larger than this.



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%Patient &  Initial level($\mu$g/cc) &  w. Magnet &  
%w. Magnet and Sound \\ \hline
% Guglielmo B.  &   0.12     &     0.10      &     0.001  \\
% Ferrando di N. &  0.15     &     0.11      &  $< 0.0005$ \\ \hline
%\end{tabular}
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\bigskip
This work was supported by the DFG Forschergruppe ``Quantenfeldtheorie,
Computeralgebra und Monte Carlo Simulationen'' and by the German Ministry
for Education and Research (BMBF).

\begin{thebibliography}{99}
%
%%
%%  bibliographic items can be constructed using the LaTeX format in SPIRES:
%%    see    http://www.slac.stanford.edu/spires/hep/latex.html
%%  SPIRES will also supply the CITATION line information; please include it.
%%
\bibitem{bisa}
A.B. Carter and A.I. Sanda,
{\it Phys.\ Rev.\ Lett.}~{\bf 45} (1980) 952;
%%CITATION = PRLTA,45,952;%%
{\it Phys.\ Rev.}~{\bf D23} (1981) 1567; 
%%CITATION = PHRVA,D23,1567;%%
I.I. Bigi and A.I. Sanda,
{\it Nucl.\ Phys.}~{\bf B193} (1981) 85.
%%CITATION = NUPHA,B193,85;%%
%
\bibitem{FPCPBaBar} S. Rahatlou (BaBar Collaboration), talk at this conference.
%
\bibitem{FPCPBelle} M. Hazumi (Belle Collaboration), talk at this conference.
%
\bibitem{FM} R.~Fleischer and T.~Mannel,
Phys.\ Lett.\ B {\bf 506}, 311 (2001).
%%CITATION = HEP-PH 0101276;%%
%
\bibitem{BW}W. Buchm\"uller and D. Wyler, {\it Nucl.\ Phys.}~{\bf B268}
(1986) 621.
%%CITATION = NUPHA,B268,621;%%
%
\bibitem{Nir} A review is given in:
Y.~Grossman, Y.~Nir and R.~Rattazzi,
Adv.\ Ser.\ Direct.\ High Energy Phys.\  {\bf 15}, 755 (1998)
%%CITATION = HEP-PH 9701231;%%
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\end{document}

