%This is a submission to PRD (Rapid Communications).
%The title is "Shifting R_b with A_{FB}^b".
%The authors are Darwin Chang and Ernest Ma.
%One figure in postcript will be sent separately.

%Further correspondence should be sent to Ernest Ma.  Postal address: 
%Physics Department, University of California, Riverside, CA 92521, USA.

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\begin{flushright} UCRHEP-T220\\April 1998\
\end{flushright}
\vspace{0.5in}
\begin{center}
{\Large	\bf Shifting $R_b$ with $A^b_{FB}$\\}
\vspace{1.0in}
{\bf Darwin Chang\\}
\vspace{0.1in}
{\sl National Center for Theoretical Sciences and 
Department of Physics,\\} 
{\sl National Tsing-Hua University\\}
{\sl Hsinchu 30043, Taiwan, Republic of China\\}
\vspace{0.5in}
{\bf Ernest Ma\\}
\vspace{0.1in}
{\sl Department of Physics, University of California\\}
{\sl Riverside, California 92521, USA\\}
\vspace{1.0in}
\end{center}
\begin{abstract}\
Precision measurements at the $Z$ resonance agree well with the standard 
model.  However, there is still a hint of a discrepancy, not so much in $R_b$ 
by itself (which has received a great deal of attention in the past several 
years) but in the forward-backward asymmetry $A^b_{FB}$ together with $R_b$.  
The two are of course correlated.  We explore the possibilty that these and 
other effects are due to the mixing of $b_L$ and $b_R$ with one or more heavy 
quarks.
\end{abstract}

\newpage
\baselineskip 24pt

Ever since the $Z$ boson was produced as a resonance at the $e^- e^+$ 
collider LEP at CERN, precision measurements of electroweak parameters as 
well as $\alpha_S$ of quantum chromodynamics (QCD) became available.  
Certain deviations from the predictions of the standard model have been 
observed in the past, notably the excess in
\begin{equation}
R_b \equiv {\Gamma (Z \to b \bar b) \over \Gamma (Z \to {\rm hadrons})}.
\end{equation}
This had prompted a flood of theoretical speculations regarding the possible 
existence of new physics\cite{1}.  At present, however, the experimental 
data\cite{2} have settled down to a value of $R_b$ consistent with an 
excess of only 1.3$\sigma$ and it is certainly not an indication of new 
physics by itself.  On the other hand, the forward-backward asymmetry
$A^b_{FB}$ is now measured to be $-2.0\sigma$ away.  [This quantity used to 
be less accurately measured and it was always less than $\pm 1.0\sigma$ from 
the standard-model prediction.]  If one takes seriously the two measurements 
together, a possible discrepancy still remains.  In this paper we will 
explore how the mixing of $b_L$ and $b_R$ with one or more heavy quarks 
would explain the present data.

In the standard model, using $m_t = 175.6 \pm 5.5$ GeV and assuming 
$m_H = 300$ GeV, the overall best fit gives\cite{3} $\sin^2 \theta_W = 
0.23152$, for which $R_b = 0.21576$ and $A_{FB}^{0,b} = 0.10308$.  The 
experimental measurements are\cite{2,4}
\begin{eqnarray}
R_b &=& 0.2170 \pm 0.0009 ~~~ (+1.38\sigma), \\ 
A_{FB}^{0,b} &=& 0.0984 \pm 0.0024 ~~~ (-1.95\sigma),
\end{eqnarray}
where the number of $\sigma$'s is the ``pull" which is defined as the 
difference between measurement and fit in units of the measurement error.

Consider the couplings of the $b$ quark to the $Z$ boson:
\begin{equation}
{\cal L}_{int} = {g Z^\mu \over \cos \theta_W} \left( g_L \bar b_L \gamma_\mu 
b_L + g_R \bar b_R \gamma_\mu b_R \right),
\end{equation}
where the subscripts $L,R$ on $b$ refer to the left and right chiral 
projections $(1 \mp \gamma_5)/2$ respectively.  Hence $R_b \propto g_L^2 + 
g_R^2$, whereas $A_{FB}^{0,b} \propto (g_L^2 - g_R^2)/(g_L^2 + g_R^2)$. 
>From the data, it is clear that a larger $g_R^2$ is desirable because that 
would decrease $A_{FB}^{0,b}$ and increase $R_b$.  Previous attempts\cite{5} 
to increase $R_b$ mostly considered increasing $g_L^2$.  Two specific 
exceptions\cite{6,7} proposed to increase $g_R^2$ and we will discuss them 
in detail below.

In the standard model,
\begin{equation}
\left( g_L^2 \right)_{SM} = \left( -{1 \over 2} + {1 \over 3} \sin^2 \theta_W 
\right)^2 = 0.17878, ~~~ \left( g_R^2 \right)_{SM} = \left( {1 \over 3} 
\sin^2 \theta_W \right)^2 = 0.00596.
\end{equation}
In Fig.~1 we plot $R_b$ versus $A_{FB}^{0,b}$ as a function of $g_R^2 (g_L^2)$ 
with $g_L^2 (g_R^2)$ fixed at its standard-model value.  The experimental 
range is also displayed.  For $g_L^2$ fixed at $(g_L^2)_{SM}$, the best 
fit is \begin{equation}
g_R^2 = 0.00736
\end{equation}
for which $R_b = 0.2174$ and $A_{FB}^{0,b} = 0.1015$ are obtained.  If we 
let both $g_L^2$ and $g_R^2$ be free parameters, then we get the central 
values of $R_b$ and $A_{FB}^{0,b}$ with
\begin{equation}
g_L^2 = 0.17586, ~~~ g_R^2 = 0.00994.
\end{equation}

We now discuss how the above two cases, {\it i.e.} Eqs.~(6) and (7), may be 
obtained.  In Ref.~[6], a vector doublet of quarks with the conventional 
charges, {\it i.e.} 2/3 and $-1/3$, is added.  We call this Model (A) with 
$(Q_1, Q_2)_{L,R} \sim (3,2,1/6)$ under $SU(3)_C \times SU(2)_L \times 
U(1)_Y$.  Since $(Q_1, Q_2)_L$ transforms in the same way as the known 
quark doublets, we define it precisely as the one that forms an invariant 
mass with $(Q_1, Q_2)_R$.  Hence the mass matrix linking $(\bar b_L, 
\bar Q_{2L})$ with $(b_R, Q_{2R})$ is given by
\begin{equation}
{\cal M}_{b,Q_2} = \left( \begin{array} {c@{\quad}c} m_b & 0 \\ m_{Qb} & 
M \end{array} \right),
\end{equation}
which shows that $b_R$-$Q_{2R}$ mixing is dominant, and that $b_L$-$Q_{2L}$ 
mixing is suppressed by $m_b/m_Q$ and is thus negligible\cite{1}.  We now have
\begin{equation}
g_R^2 = \left[ {1 \over 3} \sin^2 \theta_W \cos^2 \theta_2 + \left( 
-{1 \over 2} + {1 \over 3} \sin^2 \theta_W \right) \sin^2 \theta_2 
\right]^2 = \left[ {1 \over 3} \sin^2 \theta_W - {1 \over 2} \sin^2 \theta_2 
\right]^2.
\end{equation}
In order to increase $g_R^2$ from its standard-model value, it is clear that 
$\sin^2 \theta_2$ must be greater than $(4/3) \sin^2 \theta_W$.  Hence a 
rather large mixing with $Q_2$ is required in this model.  Numerically, to 
obtain Eq.~(6), we need
\begin{equation}
\sin^2 \theta_2 = 0.3260.
\end{equation}

In Ref.~[7], a vector doublet of quarks with the unconventional charges 
$-1/3$ and $-4/3$ is added.  We call this Model (B) with $(Q_3, Q_4)_{L,R} 
\sim (3,2,-5/6)$.  The $b$-$Q_3$ mass matrix is of the same form as Eq.~(8) 
because there cannot be a $\bar b_L Q_{3R}$ term for lack of a Higgs triplet. 
In this case,
\begin{equation}
g_R^2 = \left[ {1 \over 2} \sin^2 \theta_W \cos^2 \theta_3 + \left( 
{1 \over 2} + {1 \over 3} \sin^2 \theta_W \right) \sin^2 \theta_3 \right]^2 
= \left[ {1 \over 3} \sin^2 \theta_W + {1 \over 2} \sin^2 \theta_3 \right]^2.
\end{equation}
Now we need only a small mixing to obtain Eq.~(6), namely
\begin{equation}
\sin^2 \theta_3 = 0.0173.
\end{equation}

For comparison against the above two vectorial models, we consider also the 
addition 
of one mirror family of heavy fermions.  The heavy quarks here are 
right-handed doublets and left-handed singlets.  We call this Model (C) with 
$(Q_5, Q_6)_R \sim (3,2,1/6)$, $Q_{5L} \sim (3,1,2/3)$, and $Q_{6L} \sim 
(3,1,-1/3)$.  The $b$-$Q_6$ mass matrix is then
\begin{equation}
{\cal M}_{b,Q_6} = \left( \begin{array} {c@{\quad}c} m_b & m_{bQ} \\ m_{Qb} & 
m_Q \end{array} \right),
\end{equation}
which allows both $b_R$-$Q_{6R}$ and $b_L$-$Q_{6L}$ mixings, so that Eq.~(7) 
may be satisfied.  However, it is a somewhat unnatural solution because $m_b$ 
and $m_Q$ come from the vacuum expectation value of the Higgs doublet, whereas 
$m_{bQ}$ and $m_{Qb}$ are invariant mass terms.  It is thus difficult to 
understand why the latter two masses are not much greater.  Using
\begin{equation}
g_L^2 = \left[ \left( -{1 \over 2} + {1 \over 3} \sin^2 \theta_W \right) 
\cos^2 \theta_{6L} + {1 \over 3} \sin^2 \theta_W \sin^2 \theta_{6L} \right]^2 
= \left[ -{1 \over 2} + {1 \over 3} \sin^2 \theta_W + {1 \over 2} \sin^2 
\theta_{6L} \right]^2,
\end{equation}
and Eq.~(9) with $\theta_2$ replaced by $\theta_{6R}$ to fit Eq.~(7), we find
\begin{equation}
\sin^2 \theta_{6L} = 0.0189, ~~~ \sin^2 \theta_{6R} = 0.3537.
\end{equation}

In Model (A) and Model (C), large mixing of $b_R$ with a heavy quark is 
required, as shown in Eqs.~(10) and (15) respectively.  This has important 
implications on the electroweak oblique parameters $S, T, U$ or $\epsilon_1, 
\epsilon_2, \epsilon_3$.  In Model (A), assuming that $Q_1$ does not mix with 
$t$, $c$, or $u$, we have the following physical doublets:
\begin{equation}
\left( \begin{array} {c} Q_1 \\ Q_2 \end{array} \right)_L, ~~~ \left( 
\begin{array} {c} Q_1 \\ Q_2 \cos \theta_2 - b \sin \theta_2 \end{array} 
\right)_R,
\end{equation}
which would contribute to $T$ or $\epsilon_1$.  In the above, the masses of 
$Q_1$ and $Q_2$ are related by $m_1 = m_2 \cos \theta_2$, assuming that 
$m_b << m_1, m_2$.  Let $x \equiv \sin^2 \theta_2$, then we find 
\begin{equation}
\Delta \epsilon_1 = {3 \alpha \over 16 \pi \sin^2 \theta_W} {m_2^2 \over 
M_W^2} F(x),
\end{equation}
where
\begin{equation}
F(x) = -2(1-x)(2-x) \left[ 1 + {\ln (1-x) \over x} \right] -2x + 3x^2.
\end{equation}
Note that $F(0) = 0$ and $F(1) = 1$ as expected.  Also, $F(x) > 0$ 
for $0 < x < 1$.  Taking $x = 0.3260$ as in Eq.~(10), we get $F_1(x) = 0.141$. 
Let us choose $m_1 = 200$ GeV so that the decay $Q_1 \to b + W$ would not be 
a significant contribution to the top signal at the Tevatron.  In that case, 
$m_2 = 244$ GeV and $\Delta \epsilon_1 = 2.6 \times 10^{-3}$ which would 
take this model far away\cite{3} from the data.  Since 
our purpose is to find out if mixing with heavy quarks would improve the 
overall agreement with data, this numerical result tells us that Model (A) 
as it stands is not the answer.

In Model (C), the physical doublets are
\begin{equation}
\left( \begin{array} {c} Q_5 \\ Q_6 \cos \theta_{6R} - b \sin \theta_{6R} 
\end{array} \right)_R, ~~~ \left( \begin{array} {c} t \\ b \cos \theta_{6L} 
- Q_6 \sin \theta_{6L} \end{array} \right)_L,
\end{equation}
but since $\theta_{6L}$ is small, it can be neglected, and $m_5$ is unrelated 
to $m_6$.  We now find
\begin{equation}
\Delta \epsilon_1 = {3 \alpha \over 16 \pi \sin^2 \theta_W} {1 \over M_W^2} 
\left[ m_5^2 + m_6^2 \cos^4 \theta_{6R} - {2 m_5^2 m_6^2 \cos^2 \theta_{6R} 
\over m_5^2 - m_6^2} \ln {m_5^2 \over m_6^2} \right].
\end{equation}
Hence we can fine-tune $m_5^2/m_6^2$ to make $\Delta \epsilon_1$ small.  For 
example, if we let $m_5 = 200$ GeV, then the above expression is 
minimized with $m_6 = 273$ GeV for which $\Delta \epsilon_1 = 0.52 \times 
10^{-3}$.  This much smaller shift is acceptable.  On the other hand, unlike 
Models (A) and (B) where the heavy quarks are doublets in both left and right 
chiralities, Model (C) has $Q_{5L}$ and $Q_{6L}$ as singlets, hence the shift 
in $S$ or $\epsilon_3$ becomes nonnegligible.  Let $x \equiv \sin^2 
\theta_{6R}$ and assume $M_Z << m_5, m_6$, then we find
\begin{equation}
\Delta \epsilon_3 = {\alpha \over 24 \pi \sin^2 \theta_W} \left[ 3 - 
8 x + 5 x^2 - \ln {m_5^2 \over m_6^2} - x (2-3x) 
\ln {m_b^2 \over m_6^2} \right] = 1.8 \times 10^{-3}
\end{equation}
for $x = 0.3537$, $m_5 = 200$ GeV, and $m_6 = 273$ GeV.  This shift 
would already take this model far away\cite{3} from the data, not to mention 
that there is also the leptonic contribution of $0.44 \times 10^{-3}$.  
Hence Model (C) is also not the answer.

Let us go back to Model (A) and try to reduce $\Delta \epsilon_1$ of Eq.~(17) 
by allowing $Q_1$ to mix with $t$.  In that case, we have
\begin{equation}
\left( \begin{array} {c} Q_1 \cos \theta_{1R} - t \sin \theta_{1R} \\ 
Q_2 \cos \theta_{2R} - b \sin \theta_{2R} \end{array} \right)_R, 
\left( \begin{array} {c} Q_1 \cos \theta_{1L} - t \sin \theta_{1L} \\ 
Q_2 \end{array} \right)_L, \left( \begin{array} {c} t \cos \theta_{1L} + 
Q_1 \sin \theta_{1L} \\ b \end{array} \right)_L,
\end{equation}
where the masses of $Q_1$, $Q_2$, and $t$ are related by
\begin{equation}
m_1 = M (\cos \theta_{1L} / \cos \theta_{1R}), ~~~ m_2 = M/ \cos \theta_{2R}, 
~~~ m_t = M (\sin \theta_{1L} / \sin \theta_{1R}),
\end{equation}
where $M$ is defined as in Eq.~(8). 
After a straightforward calculation, we find
\begin{eqnarray}
\Delta \epsilon_1 &=& {3 \alpha \over 8 \pi \sin^2 \theta_W} {1 \over M_W^2} 
\left[ -4 M^2 + {1 \over 2} m_1^2 (1 + c_{1R}^4) + {1 \over 2} m_2^2 (1 + 
c_{2R}^4) + {1 \over 2} m_t^2 s_{1R}^4 \right. \nonumber \\  && ~~~~~~~~~~
~~~~~ \left.+ A_1 m_1^2 \ln m_1^2 + A_2 m_2^2 \ln m_2^2 + A_t m_t^2 \ln m_t^2 
\right],
\end{eqnarray}
where $c_{1R} \equiv \cos \theta_{1R}$, $s_{1R} \equiv \sin \theta_{1R}$, 
{\it etc.}, and
\begin{equation}
A_1 = -c_{1R}^2 s_{2R}^2 - s_{1L}^2 + s_{1R}^4 + (4 c_{1R}^2 - c_{1R}^2 
c_{2R}^2 - c_{1L}^2) {m_1^2 \over m_1^2 - m_2^2} + (c_{1R}^2 s_{1R}^2) 
{m_1^2 \over m_1^2 - m_t^2},
\end{equation}
\begin{equation}
A_2 = s_{2R}^2 + (-4 c_{1L}^2 c_{2R}^2 + c_{1R}^2 c_{2R}^2 + c_{1L}^2) {m_2^2 
\over m_1^2 - m_2^2} + (4 s_{1L}^2 c_{2R}^2 - s_{1R}^2 c_{2R}^2 - s_{1L}^2) 
{m_2^2 \over m_2^2 - m_t^2},
\end{equation}
\begin{equation}
A_t = c_{1R}^4 - s_{1R}^2 s_{2R}^2 - c_{1L}^2 - (c_{1R}^2 s_{1R}^2) {m_t^2 
\over m_1^2 - m_t^2} + (-4 s_{1R}^2 + s_{1R}^2 c_{2R}^2 + s_{1L}^2) {m_t^2 
\over m_2^2 - m_t^2}.
\end{equation}
Note that $A_1 m_1^2 + A_2 m_2^2 + A_t m_t^2 = 0$ as expected.  We now let 
$m_1 = 200$ GeV and using $m_t = 175.6$ GeV, we have
\begin{equation}
{s_{1L}^2 \over c_{1L}^2} = \left( {175.6 \over 200} \right)^2 {s_{1R}^2 
\over c_{1R}^2}.
\end{equation}
For a given value of $c_{1R}^2$, we then fix $c_{1L}^2$ and hence $M$ (= $m_1 
c_{1R}/c_{1L}$) as well as $m_2$ (= $M/c_{2R}$), assuming of course that 
$c_{2R}^2 = 0.6740$ from Eq.~(10).  We vary $c_{1R}^2$ and compute the 
right-hand side of Eq.~(24) numerically.  We find that it is in fact a 
monotonically increasing function of decreasing $c_{1R}^2$.  Hence the 
value obtained earlier for $\Delta \epsilon_1$ assuming no $Q_1$ mixing 
({\it i.e.} $c_{1R}^2 = c_{1L}^2 = 1$), which was already too far away from 
the experimental data, cannot be reduced, and Model (A) is not saved by 
additional mixings.

Now that both Models (A) and (C) are eliminated by the precision electroweak 
measurements, we focus our remaining discussion on Model (B).  The additional 
heavy quarks $Q_3$ and $Q_4$ have charges $-1/3$ and $-4/3$ respectively. 
The physical doublets are
\begin{equation}
\left( \begin{array} {c} Q_3 \\ Q_4 \end{array} \right)_L, ~~~ \left( 
\begin{array} {c} Q_3 \cos \theta_3 - b \sin \theta_3 \\ Q_4 \end{array} 
\right)_R,
\end{equation}
whereas
\begin{equation}
b_R \cos \theta_3 + Q_{3R} \sin \theta_3
\end{equation}
is a singlet.  Hence $Q_4$ decays into $b + W^-$ and would have been 
observed in the top-quark search at the Tevatron if its mass is below 
200 GeV or so.  In fact, the Tevatron top-quark events cannot tell $b$ 
from $\bar b$, hence it is even conceivable that $Q_4$ was actually 
discovered instead of $t$.  However, $Q_3$ would also have been produced, 
since $m_3 = m_4/\cos \theta_3 \simeq 1.01 m_4$, and it would decay into 
either $b + Z$ or $b + H$.  The nonobservation of the $b + Z$ mode at CDF 
requires the $b + H$ mode to dominate.  However, since LEP data 
already require the Higgs scalar mass to be greater than about 90 GeV, 
the $b + H$ mode cannot dominate and this exotic possibility is ruled 
out. We conclude that $Q_3$ and $Q_4$ are hitherto undiscovered 
and must be heavier than about 200 GeV.  Note that this is beyond the 
reach of LEP for producing $\bar b Q_3 + b \bar Q_3$.

We have so far assumed that $Q_3$ mixes only with $b$, but of course it 
could also 
mix with $s$ and $d$.  In that case, the state $b$ in Eqs.~(29) and (30) 
should be considered as a linear combination of $b$, $s$, and $d$, but 
dominated by $b$.  As a result, there would be flavor-nondiagonal 
couplings of the $Z$ to $d \bar s + s \bar d$, $d \bar b + b \bar d$, and 
$s \bar b + b \bar s$, and in addition, the charged-current mixing matrix 
mediated by $W$ would lose its unitarity.  These couplings are 
presumably very small, but they could affect the standard-model 
phenomenology regarding the $K^0 - \bar K^0$, $B^0 - \bar B^0$, and 
$B_s^0 - \bar B_s^0$ systems.  
%In other words, although the 
%charged-current quark mixing matrix, which involves only left-handed 
%fields, is the same in Model (B) as in the Standard Model, 
Furthermore, the indirect 
effect of flavor-nondiagonal neutral currents in the right-handed sector 
could result in the failure of the Standard Model to describe all data, 
especially the precision measurements to be obtained with the upcoming 
$B$ factories, at KEK and at SLAC.
For example, the exotic contributions to the decay $B \rightarrow X_s 
\gamma$ in vector quark models [including our Model (B)] have 
been analyzed recently\cite{8}.  The dominant extra contribution is 
from the violation of unitarity in the charged-current mixing matrix.  
It gives rise to nontrivial constraints on the off-diagonal mass 
matrix elements $b$-$Q_3$ and $s$-$Q_3$.  Note that whereas the Standard 
Model predicts a branching fraction of $b \rightarrow s \gamma$, 
including the next-to-leading-order correction, which is still allowed by the 
experimental data, future reduction in the experimental error with 
the same central value may be a potential signal for new vector 
quarks.

In conclusion, there may still be a hint of new physics in the current 
precision measurements of $R_b$ and $A_{FB}^b$.  If it is due to the mixing 
of $b$ with heavy quarks, the only viable model\cite{7} is to add a heavy 
vector doublet of quarks with the unconventional charges $-1/3$ and $-4/3$. 
Two other models are eliminated because they require large mixings, which in 
turn generate large shifts in $\epsilon_1$ and $\epsilon_3$, and are thus 
in disagreement with present precision data.

\vspace{0.3in}

\begin{center} {ACKNOWLEDGEMENT}
\end{center}

This work was supported in part by the U.~S.~Department of Energy under 
Grant No.~DE-FG03-94ER40837 and by a grant from the National Science 
Council of R.O.C.  We thank G. P. Yeh and C.-H. V. Chang for discussions. 
DC thanks the Physics Department, U.C. Riverside for hospitality 
during a visit when this work was initiated.


\newpage
\bibliographystyle{unsrt}
\begin{thebibliography}{99}
\bibitem{1} For a review, see for example P. Bamert, C. P. Burgess, J. M. 
Cline, D. London, and E. Nardi, Phys. Rev. {\bf D54}, 4275 (1996).
\bibitem{2} The LEP Collaborations {\it et al.}, CERN-PPE/97-154 (Dec 97).
\bibitem{3} G. Altarelli, R. Barbieri, and F. Caravaglios,  
(Dec 97).
\bibitem{4} D. Ward, in Proc. of the EPS High Energy Physics Meeting, 
Jerusalem, Israel (Aug 97).
\bibitem{5} See for example E. Ma, Phys. Rev. {\bf D53}, R2276 (1996) and 
many other references found in Ref.~[1].
\bibitem{6} T. Yoshikawa, Prog. Theor. Phys. {\bf 96}, 269 (1996).
\bibitem{7} C.-H. V. Chang, D. Chang, and W.-Y. Keung, Phys. Rev. {\bf D54}, 
7051 (1996).
\bibitem{8} C.-H. V. Chang, D. Chang, and W.-Y. Keung, "Vector Quark 
Model and B Meson Radiative Decay", unpublished.


\end{thebibliography}

\begin{center} {FIGURE CAPTION}
\end{center}

\noindent Fig.~1.  Plot of $R_b$ versus $A_{FB}^{0,b}$.  The solid lines 
are the experimental ranges.  The dashed (dotted) line is obtained by 
varying $g_L^2$ ($g_R^2$) holding $g_R^2$ ($g_L^2$) fixed in the Standard 
Model.
\end{document}



