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\title{The Superweak Theory 35 Years Later}

\author{L. Wolfenstein\thanks{Carnegie Mellon University, Pittsburgh,
PA 15213}}

\hyphenation{barr}

\begin{document}

\maketitle

\begin{abstract}
The origins, possible rationale and definition of the superweak theory are
reviewed.  The observation of direct CP violation in $K_L$ decay provides the
first significant evidence against the theory.  The much larger direct CP
violation expected in $B$ decays should definitively kill it. 
\end{abstract}
When CP violation was discovered in 1964, there existed a very successful
theory of weak interactions called the V--A theory.  This closely resembled the
original Fermi theory, but there were added the helicity projection operators
that led to maximal parity (P) violation and particle-antiparticle symmetry (C)
violation. However, this theory was CP invariant. 

Thus, two possibilities emerged.  Either one modified the V--A theory in some
way or one assumed there existed a much weaker interaction that led to CP
violation.  The second possibility existed because the observed effect could be
limited to CP violation in $K^0 - \bar{K}^0$ mixing causing a small admixture
of the CP-even state $K_1$ into $K_L$ 
\begin{equation}
|K_L\rangle = (|K_2\rangle + \epsilon |K_1\rangle) (1+|\epsilon|^2)^{-1},
\end{equation}
where $K_2$ is the CP-odd state.  Because $K^0 - \bar{K}^0$ mixing is a
second-order weak effect the CP-violating effect could be limited to a new
$\Delta S = 2$ interaction of the form 
\begin{equation}
G_{sdsd} (\bar{s} \mathcal{O} d)(\bar{s} \mathcal{O} d) + \mathrm{h.c.},
\label{wolf:eq:2}
\end{equation}
with $G_{sdsd}$ having a value of order $10^{-10}$ to $10^{-11}$ times the
standard weak interaction coupling $G_F$.  This came to be called the
\emph{superweak theory}. 

There are several consequences of the superweak theory:
\begin{enumerate}
\item The CP violation would be the same for different decay modes such as
$\pi^+ \pi^-$ and $\pi^0 \pi^0$ since it was determined entirely by the mixing
$\epsilon$.  In a standard notation, this says $\epsilon'$ vanishes. 
\item Assuming CPT invariance the phase of $\epsilon$ is given exactly by 
\[
\theta_\epsilon = \tan^{-1} (2\Delta M/\Delta\Gamma) \approx 43.5^\circ,
\]
where $\Delta M = M_L - M_S$ and $\Delta\Gamma = \Gamma_S - \Gamma_L \approx
\Gamma_S$. This is referred to as the superweak phase. \item It would be very
unlikely that CP violation would be discovered outside the $K^0$ system. 
\end{enumerate}
All these predictions appeared to be true for the next 25 years.

The superweak theory is really a large class of theories with the common
feature that their effect at low energies contains the term~(\ref{wolf:eq:2}). 
The original form I wrote in 1964~\cite{wolf:wolf1} had the standard $\Delta Q
= \Delta S$ current coupled very weakly to a postulated $\Delta Q = -\Delta S$
current. With the advent of the quark picture, this is no longer a possibility.
It is possible that the superweak character arises from the exchange of a
superheavy particle, from very weak coupling constants, or from higher-order
effects. 

In 1973, with the discovery of neutral currents, a new theory of weak
interactions came to be accepted, the Weinberg-Salam model.  As shown by
Glashow, Iliopoulos, and Maiani, this required a fourth quark.  This theory was
more elegant that the previous theory because it was based on a gauge principle
and was renormalizable. We all loved it, but it was CP-invariant, not by
design, but because of its basic form.  I'm not sure why people didn't worry
more about that, but perhaps it was because the one small effect could always
be explained by superweak. 

However, in 1973, a paper was published in the Progress of Theoretical
Physics~\cite{wolf:kobayashi}, a paper that, at first, few people noticed. It
pointed out in one paragraph that if there were six quarks instead of four (at
the time, in fact, only three were known), then CP violation could be
accommodated in the Weinberg-Salam model.  With the discovery of the $b$ quark
system, this became the standard model of CP violation.  In this model, the up
quarks ($u_j$) are connected to the down quarks ($d_i$) in a weak decay by 
\[
d_i \rightarrow V_{ji} u_j,
\]
where $V_{ji}$ is the $3\times 3$ CKM matrix.  The one significant phase in the
matrix is the origin of CP violation. 

At first we thought that the small value of $\epsilon$ meant that the phase in
the CKM matrix was small.  The true picture came with the determination of the
relatively long $B$ lifetime.  It then became clear that the matrix had a
hierarchical structure.  It could be expanded in powers of the sine of the
Cabibbo angle ($\lambda = \sin \theta_0 = .22$) and the phase entered only at
order $\lambda^3$ in the elements connecting the first and third generation: 
\begin{eqnarray}
V_{ub} & = & A\lambda^3 (\rho - i\eta) \nonumber\\
V_{td} & = & A\lambda^3 (1-\rho - i\eta).
\label{wolf:eq:3}
\end{eqnarray}
The rate of $b\rightarrow u$ decays gives roughly
\[
(\rho^2 + \eta^2)^{1/2} \approx 0.4 \pm 0.15,
\]
while the value of $\epsilon$ requires $\eta$ greater than 0.2.  As a result,
the phases of Eq.~(\ref{wolf:eq:3}) are not small. 

Given the standard model, is there any motivation for thinking about superweak?
 The most compelling arguments are those given by Steve Barr~\cite{wolf:barr1}.
 One would like a more elegant way to introduce CP violation than by putting a
whole set of arbitrary phases in the Yukawa matrices.  Also, if one assumes CP
violation is allowed everywhere you can put it, then the standard model
contains an arbitrary parameter called $\theta_\mathrm{QCD}$, which is
restricted to values less than $10^{-9}$ by the neutron electric dipole moment.
A possible solution to these problems is to assume CP invariance is valid at
some higher mass scale, but it is broken softly, either spontaneously or in
terms of dimensions less than 4.  Then $\theta_\mathrm{QCD}$ is calculable and
in suitable theories is small enough.  Similarly, the parameter $\eta$ in the
CKM matrix must be calculated from the same origin and it may turn out
\cite{wolf:glashow} to be much too small to explain $\epsilon$. In that case,
the observed CP violation must be explained by other effective interactions
caused by the new physics at the higher mass scale, which may be effectively
superweak. 

The major feature of the weak Hamiltonian of the standard CKM theory that
distinguishes it from superweak is the presence of the CP-violating terms that
change flavor by one unit that are of the order $G_F A\lambda^3$.  Thus,
besides the CP violation in the mixing, we expect \emph{direct CP violation};
that is, CP violation in the decay amplitude.  However, direct CP violation for
the $K^0$ system given by the parameter $\epsilon'$ is very small and, as
discussed in these pages, the value of $\epsilon'$ in the CKM model is very
uncertain. 

A question arises as to the definition of a superweak model since there exist
many examples in the literature.  In general, a superweak model is defined by
the effective Lagrangian at the electroweak scale or below: 
\[
L_\mathrm{eff} = L_0 + L_\mathrm{SW}
\]
where $L_0$ is the standard model with $\eta$ much less than that required to
explain $\epsilon$ and $L_\mathrm{SW}$ contains various four-quark interaction
with coefficients $G_{ijkl}$ including the term~(\ref{wolf:eq:2}) needed to
explain $\epsilon$. 

A theory is superweak if all the $G_{ijkl}$ are much smaller than the usual
weak interactions in $L_0$.  An extreme superweak theory would be one in which
all $G_{ijkl}$ were as small as $G_{sdsd}$ and involved a change of flavor
$\Delta F$ of 2 units.  However, I would consider theories with larger values
of $G_{ijkl}$ as long as they were much smaller than those of the standard
model and allow for $\Delta F = 1$ as well as~2.  This raises the interesting
point of whether the observed value of $\epsilon'/\epsilon$ could possibly be
consistent with some theories that are essentially superweak.  The simplest
superweak models involve the tree-level exchange of a heavy boson.  The $\Delta
F = 1$ CP-violating term in $L_\mathrm{SW}$ then might have $G_{sddd}$ of order
$G_{sdsd}$ in which case $\epsilon'$ would be of order $10^{-9}$ to $10^{-10}$
or $\epsilon'/\epsilon \sim 10^{-6}$ to $10^{-7}$. However, it is also possible
that the exchange might involve a relatively light Higgs boson in a multi-Higgs
scenario with very small couplings.  In general, the contribution of the
neutral Higgs exchange to $\epsilon'/\epsilon$ in such models is expected to be
small, but it is possible that charged Higgs exchange could give
$\epsilon'/\epsilon$ as large as $10^{-3}$~\cite{wolf:wu}. 

There exists a large class of models that fit our definition of superweak
although they are called ``milliweak'' by Barr~\cite{wolf:barr1}. These are
models in which $\epsilon$ occurs only in a box diagram involving a new boson
$H$ and either a quark, a lepton, or a new fermion.  A large catalogue of such
models is given by Barr.  If $H$ is a charged diquark, $\epsilon'/\epsilon$ can
actually arise at tree level and be as large as $3\times 10^{-3}$; in contrast,
if $H$ is a leptoquark, $\epsilon'/\epsilon$ occurs only via a loop
(penguin-type) graph and is no larger than $10^{-7}$~\cite{wolf:barr2}.  In
another model~\cite{wolf:bowser}, $H$ is a charged scalar and the fermion is a
new ``quark'' with charge 4/3; here again $\epsilon'/\epsilon$ occurs at loop
level and is found to be of order $10^{-4}$.  It has been suggested
\cite{wolf:barbieri} that models that are effectively superweak can be
constructed with SUSY particles.  The possibility that $\epsilon'/\epsilon$
could be as large as $2\times 10^{-3}$ from loops with SUSY particles with
particle mixing has recently been noted~\cite{wolf:masiero}. 

We conclude that while it is reasonable to expect $\epsilon'/\epsilon$ to be
small in superweak models, one cannot rule out models that are essentially
superweak and still have $\epsilon'/\epsilon$ of the order $2\times 10^{-3}$ as
observed. The reason is that $\epsilon'$ is only $5\times 10^{-6}$ which
corresponds to $G_{sdqq}$ of order $10^{-5}$ to $10^{-6}$. 

In fact, one might say that if $L_\mathrm{eff}$ is defined below the $b$-mass
scale, then the standard model becomes practically superweak: $L_0$ is then the
CP-conserving four-quark model and $L_\mathrm{SW}$ represents the effective
four-fermion interactions due to virtual effects of $t$ quarks. 

The most definitive way to distinguish the standard model from superweak is in
$B$ physics.  The first observation of CP violation should come from the
asymmetry in the decay $B^0 (\bar{B}^0)\rightarrow J/\psi K_S$.  Within the
standard model this measures $\sin 2\beta$, where $\beta$ is the phase of $1 -
\rho + i \eta$.  The standard model predicts a large positive value for $\sin
2\beta$ between 0.3 and 0.9.  This could be blamed entirely on CP violation  in
$B - \bar B$ mixing and is thus analagous to the $\epsilon$ parameter of the
$K$ system.  It could be explained by a superweak interaction with the coupling
$G_{bdbd}$ of order $10^{-7}$ times $G_F$.  The final, completely definitive
death of any superweak theory will come from the observation of direct CP
violation in the $B$ system. 

This arises from the phase $\gamma$ in $b\rightarrow u$ decays (where $\gamma$
is the phase of $\rho + i\eta$).  To be more accurate, one is comparing the
phase of $b\rightarrow u$ decays to that of $b\rightarrow c$ (such as
$B\rightarrow J/\psi K_S$). Evidence for such direct CP violation would be
given by the difference between the asymmetry parameter in a decay such as $B^0
(\bar{B}^0)\rightarrow \pi^+ \pi^-$ from that of $J/\psi K_S$.  This can be
considered \cite{wolf:wolf2} the $\epsilon'$ experiment for the $B$ system.  In
the tree approximation, this is given by $\sin 2 (\beta + \gamma)$, which is
expected to be very different from $\sin 2\beta$.  Thus, $\epsilon'$ for the
$B$ system is of order unity in contrast to the $K$ system where it is $5\times
10^{-6}$.  It should be emphasized that ``penguin pollution'' does not affect
the Yes-No question as to the existence of direct CP violation; all that is
required is a significant difference in the two asymmetries. 

It may well be that $B\rightarrow \pi\pi$ will not be the first place to find
direct CP violation.  It may be found in the difference in partial rates of
$B^+$ and $B^-$ decays if final state interactions are large enough.  In
thinking about $B$ physics, it is important to recognize the importance of
finding evidence for direct CP violation even if the extraction of $(\beta,
\gamma)$ or $(\rho,\eta)$ is not directly possible from the data. 

In retrospect, it is very surprising that it has taken over 30 years to kill
the superweak theory as the origin of CP violation.  It will still take a long
time to establish whether the CKM phase is the only source of CP violation or
whether there are important effects from physics beyond the standard model. 

I am indebted to discussions with Darwin Chang and Steve Barr.  This research
was supported, in part, by the Department of Energy under Contract
DE-FG02-91-ER-40682. 

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\bibitem{wolf:kobayashi} M.~Kobayshi and T.~Maskawa, Prog.~Theor.~Phys.\ \textbf{49}, 652 (1973).
\bibitem{wolf:barr1} S.~M.~Barr, Phys.~Rev.~D \textbf{34}, 1567 (1986).
\bibitem{wolf:glashow} See, for example, S.~L.~Glashow and H.~Georgi,
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\bibitem{wolf:bowser} D.~Bowser-Chao, D.~Chang, and W.~Y.~Keung,
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\bibitem{wolf:barbieri} R.~Barbieri \textit{et al.}, Phys.~Lett.~B
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\bibitem{wolf:masiero} A.~Masiero and H.~Murayama, Berkeley preprint
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\bibitem{wolf:wolf2} L.~Wolfenstein, Nucl.~Phys.~B \textbf{246}, 45 (1984).
\end{thebibliography}

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