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%% section 3 The Unitarity Triangle [slacpub7127003 in slacpub7127003: slacpub7127004]
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\section{\usemenu{slacpub7127::context::slacpub7127003}{The Unitarity Triangle}}\label{section::slacpub7127003}
In the Wolfenstein parametrization \cite{14}, the CKM matrix
is given by
%
\beq
V_{CKM}=\left(\begin{array}{ccc}
1\frac{1}{2}\lambda^2 & \lambda & A\lambda^3(\rhoi\eta)\\
\lambda & 1\frac{1}{2}\lambda^2 & A\lambda^2 \\
A\lambda^3(1\rhoi\eta) & A\lambda^2 & 1 \end{array}\right)
\eeq
%
We can best visualize the effects of the new contributions
on the CKM parameters by plotting the allowed regions in
the $\rho\eta$ plane.{\footnote{The parameter $\lambda$
corresponds to the Cabbibo angle,
and is extremely well measured in treelevel standard model
decays. Although the parameter $A$ occurs in all of the expressions
for neutral meson mixing, and we allow it to vary in our subsequent
fits, its best fit value is always close to that determined from the
CKM element $V_{cb}$ whose determination is again dominated by
treelevel standard model physics. Thus the effects of new physics are
dominantly felt by the parameters $\rho$ and $\eta$.}}
We will plot the usual three constraints
coming from the experimentally measured quantities
$V_{ub}/V_{cb}, ~\Delta m_{B_d}$, and $\epsilon$,
as well as the constraint from $Arg(M^{B_d}_{12})$ which will be
cleanly measured at the $B$ factories by the $CP$ asymmetry in the
decay $B_d \rightarrow \Psi K_S$.
Although the model satisfies the constraint
from $\Delta m_K$, we do not include it in the subsequent analysis
because of the large uncertainty in the standard
model prediction for this quantity due to long distance effects.
The curves to be plotted are determined by the following equations
%
\beq
1)~~~~~~~~~~~~~~~~~~
\frac{V_{ub}}{V_{cb}}=\lambda \sqrt{\rho^2+\eta^2}
\label{expt1}
\eeq
%
this determines a circle centered at the origin of the $\rho\eta$
plane. Since this quantity is determined by treelevel decays, it is
not affected by the presence of new physics.
%
\beq
2)~~~~~~~~~~~~~~~~~~
\Delta m_{Bd} = 2M_{12}^{B_d}
\eeq
%
which gives
%
\beq
(1\frac{c}{2A\lambda^2(a+b)}\rho)^2+\eta^2=
\frac{\Delta m_{B_d}}{2 K_{\bar b d} m_t^2 (a+b)}
\label{expt2}
\eeq
%
where $K_{\bar b d}$ and $a,~b,~c$
have been defined in Eqs.~(\docLink{slacpub7127002.tcx}[defK]{3},\docLink{slacpub7127002.tcx}[defabc]{8}). This once again
determines a circle on the $\rho\eta$ plane. The presence of new
physics has two effects here. The in phase supersymmetric contribution
given by $b$ in the denominator on the righthandside of
Eq.~(\docLink{slacpub7127003.tcx}[expt2]{14}) reduces the radius of the circle, and the out of
phase contribution proportional to $c$ displaces the center from $\rho
= 1$.
%
\beq
3)~~~~~~~~~~~~~~~~~~~
\epsilon = \frac{Im M_{12}^{K}}{\sqrt 2 \Delta m_K}
\eeq
%
which gives
%
\beq
\eta[(a+b)A^2\lambda^4(1\rho)c A\lambda^2(1\rho)\frac{c}{2}
A\lambda^2 + P'_0]=\frac{\Delta m_K \epsilon}{\sqrt 2 K_{\bar s
d} m_t^2 A^2\lambda^6}
\label{expt3}
\eeq
%
with
%
\beq
P'_0=\frac{m_c^2}{m_t^2}[\frac{\eta_{ct}}{\eta_{K}}f_3(y_c,y_t)
\frac{\eta_{cc}}{\eta_{K}}]
\eeq
%
This curve determines a hyperbola in the $\rho\eta$ plane, with the
new physics once again having two effects. The term proportional to
$b$ reduces the distance of the directrix from the origin,
and the term proportional to $c$ shifts the coordinates to which the
hyperbola is referred.
Finally,
%
\beq
4)~~~~~~~~~~~~~~~~~~~~~~~~~
V_{td}=V_{td}e^{i\beta_{KM}}
\eeq
%
which leads to the straight line
%
\beq
\eta=(1\rho) \tan \beta_{KM}
\eeq
%
where $\beta_{KM}$ is determined from the expression
%
\beq
Arg(M_{12}^{B_d})=\tan^{1}[\frac{(a+b)\sin 2\beta_{KM}c'\sin\beta_{KM}}
{(a+b)\cos 2\beta_{KM}c'\cos\beta_{KM}}],
\label{expt4}
\eeq
%
with
%
\beq
c'=\frac{c}{A\lambda^2\sqrt{(1\rho)^2+\eta^2}}.
\eeq
%
Here the shift from the standard model expectation is entirely due to
the ``outofphase'' contribution proportional to $c$, which
tends to increase the phase of $B_d\bar
B_d$ mixing as compared to the standard model
(in Eqs.~(\docLink{slacpub7127003.tcx}[expt2]{14},~\docLink{slacpub7127003.tcx}[expt3]{16},~\docLink{slacpub7127003.tcx}[expt4]{20}), the standard
model limit can be recovered by setting $b = c = 0$. This corresponds
to the limit $\tilde m \rightarrow \infty$).
We plot the constraints from these curves
using the inputs from Table 1 in
Figs. 1. Our inputs are the same as those in \cite{15} except
for $V_{cb}$ where we use the value in \cite{8}, and
that we have been slightly less conservative in our estimates of the
uncertainties in $\sqrt {B_B}f_B$ and $B_K$.
%
\begin{table}
\begin{center}
\begin{tabular}{ll}
\hline
Parameter & Value \\
\hline
$V_{ub}/V_{cb}$ & $0.08 \pm 0.02$ \\
$\Delta m_{B_d}$ & $(0.306 \pm 0.0158) \times 10^{12}$ GeV \\
$\epsilon$ & $ (2.26 \pm 0.02) \times 10^{3}$ \\
$V_{cb}$ & $0.039 \pm 0.002$ \\
$\lambda$ & 0.2205 \\
$m_t$ & $170 \pm 10$ GeV \\
$m_c$ & $1.3$ GeV \\
$\sqrt{B_B}f_B$ & $180 \pm 30 MeV$ \\
$B_K$ & $0.8 \pm 0.2$ \\
$\eta_B$ & 0.55 \\
$\eta_K, ~\eta_{ct}, ~\eta_{cc}$ & 0.57, 0.47, 1.32 \\
$\tilde m, ~\tan\beta$ & 85 GeV, 1 \\
\hline
\end{tabular}
\end{center}
\caption {Input parameters for the $\rho\eta$ analysis
presented in Fig. 1 and Table 2.}
\end{table}
%
\begin{figure}
\epsfig{file=rhoeta.eps,height=6in,width=6in}
\caption {Constraints on $\rho$ and $\eta$ based on the
parameters of Table 1.
The small circles (dash) are from
$V_{ub}/V_{cb}$, the large circles (dot dash) are from
$\Delta m_{B_d}$, the hyperbolae (dot dot dash) from $\epsilon$ and the
straight lines (solid) from $Arg(M_{12}^{B_d})$.
(a) The standard model. (b) The correct supersymmetric analysis. (c)
The incorrect supersymmetric analysis.}
\end{figure}
%
Fig. 1(a) corresponds to the standard model case {\it i.e.}
no new physics. Fig.1(b) and Fig. 1(c) include
the supersymmetric contribution
with $\tilde m=85$ GeV, and $\tan\beta=1$. However Fig. 1(c) contains
the wrong analysis where we incorrectly assume that the supersymmetric
contribution is always in phase with the standard model one.
In all of these figures we include the error from $m_t$ only in its
effect on the leading coefficients of Eqs.~(\docLink{slacpub7127002.tcx}[massbd]{5},~\docLink{slacpub7127002.tcx}[massk]{7}),
ignoring its effect on the terms in the square brackets. We also
ignore the effects of the error in $V_{cb}$.
The straight lines in the figures
corresponding to $Arg(M_{12}^{B_d})$ are obtained in the
following way: we determine the point
%(marked by the cross on the plots)
in the overlap region of the
other three curves that give us the largest values for
the phase $\beta_{KM}$. This is then plugged into Eq.~(\docLink{slacpub7127003.tcx}[expt4]{20}),
and we include an error of $\pm 0.059$ in the determination of
$\sin(Arg(M_{12}^{B_d}))$ as quoted in \cite{16}.
Comparing Fig. 1(a) with Figs. 1(b) and 1(c), we notice
that although there
is a large overlap between the allowed regions for the Standard Model
and for the supersymmetric case, it could be possible that
the supersymmetric contributions, as discussed above shift, $\rho$ and
$\eta$ into a region excluded by the Standard Model. This
possibility which has been noticed in Refs.~\cite{5,8}
is not very interesting from the point of view of the
$B$factory CP violation
experiments. This is because supersymmetric particles in the mass
range we are cosidering would already be detected at the high energy
colliders like LEP2 before the $B$factories turn on, thus we would know
already that the Standard Model analysis is incorrect. More
interesting is the fact that the line denoting $Arg(M^{B_d}_{12})$
lies outside the allowed region for some values of the phase
$\beta_{KM}$, a possibility that can never occur in the constrained
version of the MSSM.
Thus, if we incorrectly interpreted this phase as measured by the CP
asymmetry in $B_d \rightarrow \Psi K_S$ as the CKM phase $\beta_{KM}$
(as in Fig. 1(c)),
we would come to the false conclusion that the unitarity triangle does
not close, and that there
are additional sources of CP violation in the theory besides the
complex Yukawa couplings between the quark and Higgs fields.
The obvious way to check for this supposed deviation from unitarity
is by the clean measurement of phases
that the CP violation experiments at the Bfactories allow.
The CP violating rate asymmetries in the decays
$B_d \rightarrow \pi\pi$,
$B_d \rightarrow \Psi K_S$, $B_s \rightarrow \rho K_S$ measure the
quantities $\sin2\alpha'$, $\sin2\beta'$, and $\sin2\gamma'$ where
%
\beq
2\alpha'=Arg(M^{B_d}_{12})+2\gamma,~~
2\beta'=Arg(M^{B_d}_{12}),~~
2\gamma'=Arg(M^{B_s}_{12})+2\gamma.
\eeq
%
and $\gamma$ is the phase of $V_{ub}$. In the standard model,
$Arg(M^{B_d}_{12})=2\beta_{KM}$ and $Arg(M^{B_s}_{12})=0$ so the
measured quantities
reduce to $\sin2\alpha$, $\sin2\beta_{KM}$, and $\sin2\gamma$ (after
making the replacement $\beta_{KM}+\gamma = \pi  \alpha$) where
$\alpha$, $\beta_{KM}$ and $\gamma$ are the angles of the ``unitarity''
triangle.
Thus if the phases of $B_d\bar B_d$ or $B_s
\bar B_s$ mixing are affected by new physics, the three measured
angles $\alpha'$, $\beta'$ and $\gamma'$ will not correspond to the
angles of the unitarity triangle, and
in general will not add up to $180^{\circ}$.
This test, however, does not work in
this case, because although the phase of $B_d\bar B_d$ mixing is
affected by the supersymmetric contribution,
that of $B_s \bar B_s$ mixing isn't [Eq. (\docLink{slacpub7127002.tcx}[massbs]{6})].
Thus, if we repeat the
above analysis making the replacement $\beta'+\gamma=\pi\alpha'$, we
will still obtain a triangle that closes, with angles
$\alpha\delta,~\beta_{KM}+\delta$ and $\gamma$, where $\delta$ is the
amount by which the phase of $B_d\bar B_d$ mixing is shifted from the
standard model value.
This possibility that the angles measured by the above CP
violating experiments would still add up to $180^{\circ}$ if the phase
of $B_d\bar B_d$ mixing is changed, but that of $B_s \bar B_s$
mixing isn't was pointed out in ~\cite{17}.
An alternative method to measure $\gamma$ is in the $CP$ violating
decay $B_d \rightarrow D^0_{CP}K^{\ast}$ \cite{18}.
However, since this measurement is a result of interfering treelevel
amplitudes, it is not affected by the new physics, and we would
still measure the true angle $\gamma$.
Thus, as in the case above, we would mistakenly
interpret the three angles obtained as summing to $180^{\circ}$.
Another interesting manifestation of this new phase in the
$B_d\bar B_d$ mixing matrix could be in the existence of $CP$
violating asymmetries in decays where the standard model predicts
none. A simple example of this is the penguin mediated decay $B_d
\rightarrow K_SK_S$ where the phase of the top mediated penguin
exactly cancels the phase $2\beta_{KM}$ of $B_d\bar B_d$ mixing
in the
standard model. In the model we are considering, this cancellation
would not be exact because of the new phase in $B_d\bar B_d$ mixing
matrix, and there could be observable $CP$ asymmetries in the
decay. It has recently been observed, however, that subdominant
penguins mediated by up and charm quarks could contribute to $CP$
violation in this channel \cite{19}. We have checked that this
contribution is not only comparable in magnitude to the one due to the
new mixing phase, but is also uncertain in sign. Thus the observation
(or non observation) of $CP$ violation in this decay could not
distinuish this model from either the standard model or the
constrained MSSM.
The considerations of the previous paragraphs show us that only phase
information is not enough to tell us that we are wrong in assuming that
supersymmetric contributions do not modify the phase of neutral $B$ meson
mixing. In order to detect this, we need to combine the phase
information from the $CP$ violating experiments with
independent information on
magnitudes (and phases) of the CKM matrix elements available in the
quantities $V_{ub}/V_{cb}$, $\Delta M_{B_d}$ and $\epsilon$
discussed earlier.
To this end we do a $\chi^2$ analysis for the central values of $\rho$
and $\eta$ using the quantities listed in Table 1. Our experimental
inputs are $V_{ub}/V_{cb}$, $\Delta M_{B_d}$, $\epsilon$,
$V_{cb}$, $m_t$, and ``projected values'' for $\sin 2\beta'$ and
$\sin 2\alpha'$, while allowing $\rho$, $\eta$, $A$ and $m_t$ to vary.
We display our results in Table 2, where analyses I and II correspond
to two different choices for the inputs $\sin 2\beta'$ and $\sin
2\alpha'$.
In both analyses we first do the $\chi^2$ minimization without any input
for $\sin 2\beta'$ and $\sin 2\alpha'$, to obtain central values and
errors on $\rho$ and $\eta$ (these would correspond to the allowed
regions of Figs. 1 without including the constraints from the straight
lines representing $Arg(M_{12}^{B_d}$)).
In analysis I, we then include as inputs,
$\sin 2\beta'$ and $\sin 2\alpha'$ calculated using these central
values, and repeat the $\chi^2$ minimization
to obtain a new minimum $\chi^2$ and central values for $\rho$ and
$\eta$. These are the values displayed
in Table 2. Analysis II follows the same procedure, except that
the inputs $\sin 2\beta'$ and $\sin 2\alpha'$ are calculated
using values of $\rho$ and $\eta$ that are one standard deviation
above the central values obtained in the first part of the procedure
(the central values and errors obtained here would correspond to the
allowed regions of Figs. 1 where we hsave included all the constraints
including those from $Arg(M_{12}^{B_d}$)).
In both the analyses we include an experimental error on
$\sin 2\beta'$ of $\pm 0.059$ and on $\sin 2\alpha'$ of $\pm 0.085$
which are the errors quoted by the BABAR colloboration in
\cite{16}.{\footnote{Although the determination of $\sin 2\beta'$ is
not affected by ``penguin pollution'' in this model, the determination
of $\sin 2\alpha'$ could be affected by outofphase supersymmetric
penguins in addition to the standard model ones. We assume that these
effects could be accounted for by an isospin analysis
\cite{20}.}}
The three cases in Table 2 correspond to those of
Fig. 1 {\it i.e.} case (a) corresponds to the standard model where
there is no new physics, in case (b) we correctly include the
supersymmetric contribution, whereas in case (c) we include the
supersymmetric contributions but neglect the outofphase part.
We have checked that the results of our $\chi^2$ analysis for the
standard model agree with those of \cite{15} for similar choices of
inputs.
%
\begin{table}
\begin{center}
\begin{tabular}{llll}
\hline
& Analysis & $(\rho,\eta)$ & $\chi^2_{\rm min}$ \\
\hline
I & a) Standard Model & $(0.04\pm 0.03,0.35\pm 0.04)$ & 0.013 \\
& b) Correct Susy & $(0.12\pm 0.03,0.34\pm 0.03)$ & 0.0051\\
& c) Incorrect Susy & $(0.13 \pm 0.03,0.36 \pm 0.03)$ & 0.88 \\
\hline
II& a) Standard Model & $(0.17\pm 0.04,0.42\pm 0.03)$ & 2.0 \\
& b) Correct Susy & $(0.23\pm 0.05,0.39\pm 0.03)$ & 1.9\\
& c) Incorrect Susy & $(0.25\pm 0.05,0.42\pm 0.03)$ & 3.8\\
\hline
\end{tabular}
\end{center}
\caption {Results of the $\chi^2$ analysis for $\rho$ and
$\eta$ based on the inputs of Table 1.}
\end{table}
%
We see that the results presented in Table 2 corroborate the visual
information of Fig. 1. Firstly the central values for $\rho$ and
$\eta$ for the supersymmetric case are indeed different from those for
the standard model. In particular more positive values for $\rho$ are
preferred by the supersymmetric case. Secondly, it is only if the
actual values for $\rho$ and $\eta$ were to lie near their current $1
\sigma$ upper bounds, as in analysis II,
that the $B$ factory experiments would be
sensitive to the new phase in $M_{12}^{B_d}$. This is signalled by the
large value of the minimum $\chi^2$ for the incorrect analysis in II
where we assume that there are no new phases in $M_{12}^{B_d}$
(since we have seven experimental inputs and four variables, we consider
$\chi^2 < 3$ indicative of a good fit).
This is as in Fig. 1(c) where
we can see that for $\rho$ and $\eta$ near their central
values, the area predicted by the incorrect analysis would lie within
the allowed region, whereas with $\rho$ and $\eta$ close to
their $1\sigma$ upper bounds, the area predicted by the incorrect
analysis clearly lies outside the allowed region. Thus, it seems that
even with the precise phase information provided by the $B$ factory
experiments, we would still have to be lucky in order to be able to
notice any deviation from the usual expectations of no new phases in
neutral meson mixing. However,
the insensitivity of this analysis to the new phase is
mostly due to the large errors in the experimental and theoretical
inputs into the analysis. Since we expect most of these to decrease
before the $B$ factory data analyses begin, we redo the analysis of
Table 2 using the same central values for the inputs, but with the
improved errors expected in the future.
We display our new inputs in Table 3, and the results of our $\chi^2$
analysis in Table 4. We base our estimates for the improved errors on
the inputs on ~\cite{21}. Figs. 2 display the
same constraints as Figs. 1,
but are plotted using the reduced errors of Table 3.
%
\begin{table}
\begin{center}
\begin{tabular}{ll}
\hline
Parameter & Value \\
\hline
$V_{ub}/V_{cb}$ & $0.08 \pm 0.01$ \\
$\Delta m_{B_d}$ & $(0.306 \pm 0.0158) \times 10^{12}$ GeV \\
$\epsilon$ & $ (2.26 \pm 0.02) \times 10^{3}$ \\
$V_{cb}$ & $0.039 \pm 0.001$ \\
$\lambda$ & 0.2205 \\
$m_t$ & $170 \pm 5$ GeV \\
$m_c$ & $1.3$ GeV \\
$\sqrt{B_B}f_B$ & $180 \pm 10 MeV$ \\
$B_K$ & $0.8 \pm 0.05$ \\
$\eta_B$ & 0.55 \\
$\eta_K, ~\eta_{ct}, ~\eta_{cc}$ & 0.57, 0.47, 1.32 \\
$\tilde m, ~\tan\beta$ & 85 GeV, 1 \\
\hline
\end{tabular}
\caption {Inputs for the $\rho\eta$ analysis presented in
Fig. 2 and Table 4. The central values are the same as those of Table
1, however the errors reflect our expectations for experimental and
theoretical improvements in estimating these quantities. }
\end{center}
\end{table}
%
%
\begin{table}
\begin{center}
\begin{tabular}{llll}
\hline
& Analysis & $(\rho,\eta)$ & $\chi^2_{\rm min}$ \\
\hline
I & a) Standard Model & $(0.05\pm 0.03,0.35\pm 0.02)$ & 0.07 \\
& b) Correct Susy & $(0.12\pm 0.02,0.34\pm 0.02)$ & 0.03\\
& c) Incorrect Susy & $(0.12 \pm 0.02,0.34 \pm 0.02)$ & 4.1 \\
\hline
II& a) Standard Model & $(0.03\pm 0.02,0.37\pm 0.02)$ & 1.1 \\
& b) Correct Susy & $(0.16\pm 0.02,0.35\pm 0.02)$ & 1.1\\
& c) Incorrect Susy & $(0.16\pm 0.03,0.36\pm 0.02)$ & 6.0\\
\hline
\end{tabular}
\caption {Results of the $\chi^2$ analysis for $\rho$ and
$\eta$ based on the inputs of Table 3.}
\end{center}
\end{table}
%
\begin{figure}
\epsfig{file=rhoeta2.eps,height=6in,width=6in}
\caption {Constraints on $\rho$ and $\eta$ based on the
reduced errors on the input parameters we expect in the future (Table
3). The small circles (dash) are from
$V_{ub}/V_{cb}$, the large circles (dot dash) are from
$\Delta m_{B_d}$, the hyperbolae (dot dot dash) from $\epsilon$ and the
straight lines (solid) from $Arg(M_{12}^{B_d})$.
(a) The standard model. (b) The correct supersymmetric
analysis. (c) The incorrect supersymmetric analysis.}
\end{figure}
Table 4 (as well as Fig. 2) contains what we believe to be an accurate
representation of
the physics results obtained at the $B$ factories if the scenario
outlined in this paper were to hold, {\it i.e.}, the existence of low
energy supersymmetry with small $\tan\beta$, light righthanded
uptype squarks and no new CP violating phases. Once again we notice
that the central value for $\rho$ is more positive and clearly
different from what would be the standard model value. Here however,
in contrast with the results of Table 2, we see that incorrectly
assuming that $Arg(M_{12}^{B_d})$ is not affected by the new physics
yields a poor fit over most of the allowed region for $\rho$ and
$\eta$ (cases I(c) and II(c)).
Interestingly though, this deviation from the standard model
is not due to the existence of new $CP$ violating phases in the
theory as one would naively infer, but simply due to the fact that
the mixing patterns of the
squarks could be different from those of the quarks, resulting in the
CKM phase showing up in physical quantities in combinations different
from those in the standard model. It is exciting to know that the
experiments at the proposed $B$ factories are sensitive to this
possibility. Although we have based our analysis on one particular
choice for $\tan\beta$ and the mass of the lightest squark, the
explicit formulas presented make generalizations to other values
trivial. In particular, the effects we discuss become larger for
smaller squark mass and $\tan\beta$, and are reduced in the opposite
limit (as long as $\tan\beta \la 30$, after which this analysis no
longer holds).
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