000-001

# The Curious Early History of CKM Matrix
-miracles happen!-

Stephen Lars Olsen<EMAIL_ADDRESS>Institute for Basic Science, Daejeon
South Korea
CAU High Energy Physics Center, Chung-Ang University, Seoul South Korea

###### Abstract

The 1973 Kobayashi Maskawa paper proposed a compelling link between Cabibbo’s
flavor-mixing scheme and ${\mathcal{C}}$${\mathcal{P}}$violation but, since it
required the existence of six quarks at a time when the physics community was
happy with only three, it received zero attention. However, two years after
the paper appeared—at which time it had received a grand total of two
citations—the charmed quark was discovered and it finally got some notice and
acceptance. After this stumbling start, it subsequently emerged as the focal
point of an enormous amount of experimental and theoretical research activity.
In an invited talk at a KEK symposium to celebrate the 50th anniversary of the
KM paper, I reviewed some of the less well known circumstances that occurred
in the years preceding and following the paper’s appearance.

Some spoilers:

— Kobayashi and Maskawa (and a number of other Japanese physicists) were
convinced about the existence of the charmed quark nearly three years before
its “discovery” at Brookhaven and SLAC.

— The matrix provided in their seminal 1973 paper was mathematically
incorrect. Another version that was in common use for the following twelve
years was technically correct, but not really a rotation matrix.

— The CKM matrix ${\mathcal{C}}$${\mathcal{P}}$ phase was only measurable
because of the very specific hierarchy of the flavor mixing angles and meson
masses.

— Similarly, the neutrino mixing discovery, and the PMNS-matrix measurability
were only possible because of favorable values of the neutrino mass
differences and mixing angles.

In addition I include some speculations about what may be in store for the
future.

Quark flavor mixing, Neutrino flavor mixing, C${\mathcal{P}}$ violation

##  1 Introduction

The challenge of reviewing a subject that is fifty years old to a community of
experts is to find something to say that isn’t already well known to everyone
in the audience. However, this obvious truth didn’t occur to me when I was
invited by the organizers to speak at the KEK special symposium to celebrate
the fiftieth anniversary of the Kobayashi-Maskawa six-quark model. An
invitation that, in a reckless capitulation to my vanity, I immediately
accepted. Upon subsequent reflection, I realized my dilemma: there was
precious little that I could say about the hundreds of CKM-related published
Belle results—which I expect the organizers had in mind when they offered this
invitation—that wasn’t already very familiar to the symposium participants.
So, instead, I decided to exploit the one advantage I might have over most
other participants, and that was that I would be the oldest, or least one of
the oldest, person in attendance and reminisce about the early days of the KM
era, including some of its pre-history. So, with the forewarning that all
historical accounts suffer from mistakes and oversimplifications, and are
varnished to match the preconceptions and prejudices of the chronicler, here
goes:

##  2 Prehistory: Cabibbo flavor-mixing and the discovery of
${\mathcal{C}}$${\mathcal{P}}$ violation

The prehistory started sixty years ago during the 1963-64 academic year111This
happened to coincide with my first year as a graduate student at the
University of Wisconsin. when there were three major discoveries that all
played a major roles in the Kobayashi-Maskawa story: flavor-mixing, quarks,
and the observation of ${\mathcal{C}}$${\mathcal{P}}$ violation in
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ decays.

###  2.1 Cabibbo flavor mixing

In their classic paper that identified the $V$$-$$A$ coupling of the weak
interaction Feynman:1958ty , Feynman and Gell-Mann proposed that the weak
interaction was a current-current interaction where the hadron current has the
form

$J_{\mu}=g\big{[}\alpha\big{(}V_{\mu}^{\Delta{\mathcal{S}}=0}-A_{\mu}^{\Delta{\mathcal{S}}=0}\big{)}+\beta\big{(}V_{\mu}^{\Delta{\mathcal{S}}=1}-A_{\mu}^{\Delta{\mathcal{S}}=1}\big{)}\big{]},$
(1)

where $g$ is a coupling constant, $V^{\Delta{\mathcal{S}}=0}_{\mu}$ and
$A^{\Delta{\mathcal{S}}=0}_{\mu}$ are the vector and axial vector currents for
strangeness conserving processes and $V^{\Delta{\mathcal{S}}=1}_{\mu}$ and
$A^{\Delta{\mathcal{S}}=1}_{\mu}$ are corresponding currents for
$\Delta{\mathcal{S}}=\pm 1$ transitions. They also made two additional
conjectures. One was universality, the notion that the currents for the
$\Delta{\mathcal{S}}=0$ and $\Delta{\mathcal{S}}=\pm 1$ hadronic transitions
and the

$g_{W}\big{(}\bar{\nu}_{e}\gamma_{\mu}(1-\gamma_{5})e^{-}\big{)}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}g_{W}\big{(}\bar{\nu}_{\mu}\gamma_{\mu}(1-\gamma_{5})\mu^{-}\big{)}$
(2)

lepton currents all have a common coupling strength, i.e., $g=g_{W}$, and
$\alpha$ = $\beta$ = 1 in eqn. 1, where $g_{W}$ is related to the square root
of the Fermi constant $G_{F}$ by

$G_{F}=\frac{\sqrt{2}}{8}\bigg{(}\frac{g_{W}}{M_{W}}\bigg{)}^{2}.$ (3)

The other one was the so-called Conserved Vector Current (CVC) hypothesis that
says that the hadronic matrix elements for the vector component of the weak
interaction current are the same as those for the electromagnetic
interactions. This has the consequence that vector form-factors for weak
decays of hadrons at zero squared momentum-transfers are unity,
$f_{V}(q^{2}$=$0)$ = 1. These two conjectures translated into a prediction
that the coupling strength extracted from the vector-mediated semileptonic
process $K^{+}$$\rightarrow$$\pi^{0}e^{+}\nu_{e}$, i.e.,
$g_{V}^{\Delta{\mathcal{S}}=1}$ shown in Fig.1a) should be the same as $g_{W}$
in $\mu^{+}$$\rightarrow$$e^{+}\nu_{e}\bar{\nu}_{\mu}$.

Figure 1: a) (upper) The lowest order Feynman diagram for
$K^{+}$$\rightarrow$$\mu^{+}\nu$. b) The lowest-order Feynman diagram for
$\mu^{+}$$\rightarrow$$e^{+}{\nu}_{e}\bar{\nu}_{\mu}$.

In a paper that appeared in June 1963 Cabibbo:1963yz , Cabibbo pointed out
that Feynman-Gell-Mann universality conjecture failed miserably. His
comparison of experimental measurements of the partial width for the
$\Delta{\mathcal{S}}=1$ vector weak-interaction process
$K^{+}$$\rightarrow$$\pi^{0}\ell^{+}\nu$ Roe:1961zz  to the well known width
for muon decay found

$\frac{g_{V}^{\Delta{\mathcal{S}}=1}}{g_{W}}\approx 0.26$ (4)

and about a factor of four below expectations. He also found a similar
deviation from universality in the ratio of the axial-vector-mediated partial
decay widths
$\Gamma(K^{+}$$\rightarrow$$\mu^{+}\nu)$/$\Gamma(\pi^{+}$$\rightarrow$$\mu^{+}\nu)$:

$\frac{g_{A}^{\Delta{\mathcal{S}}=1}}{g_{A}^{\Delta{\mathcal{S}}=0}}\approx
0.26.$ (5)

(Although the axial-vector currents are not “protected” by CVC, corrections to
them were expected to be small GellMann:1960np , and certainly not large
enough to account for a factor of four.)

Cabibbo proposed modifying the Feynman-Gell-Mann $\alpha$ = $\beta$ = 1
conjecture to $\alpha^{2}$+$\beta^{2}$= 1, in which case

$g_{V}^{\Delta{\mathcal{S}}=1}=\beta g_{W}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\frac{g_{A}^{\Delta{\mathcal{S}}=1}}{g_{A}^{\Delta{\mathcal{S}}=0}}=\frac{\beta}{\alpha},$
(6)

where $\beta\approx 0.25$ could accommodate the abovementioned experimental
results. In his paper, Cabibbo proposed his eponymous angle $\theta_{C}$,
which he estimated to $\theta_{C}\approx 14.9^{\circ}$, as a convenient way to
express two parameters $\alpha$ and $\beta$ that were subject to the
constraint $\alpha^{2}$+$\beta^{2}$=1, and he didn’t mention anything about
rotations. The earliest experiments that addressed Cabibbo’s hypothesis
Eisele:1969gw  were focused on testing the validity of Cabibbo’s relation,
$\alpha^{2}$ \+ $\beta^{2}$ = 1.

The notion that this might represent a rotation didn’t become apparent until
the 1970 GIM paper Glashow:1970gm  that proposed the $c$-quark as a way to
suppress flavor-changing neutral currents. If one accepts the existence of two
quark doublets,

$\begin{pmatrix}u\\\ d\end{pmatrix}{\displaystyle
s}~{}\Longrightarrow~{}\begin{pmatrix}u\\\ d\end{pmatrix}\begin{pmatrix}c\\\
s\end{pmatrix},$ (7)

the Cabibbo $d$-$s$ mixed quark state $d^{\prime}$ = $d\cos\theta_{C}$ \+
$s\sin\theta_{C}$ is produced by the application of a 2x2 unitary rotation
matrix:

$\begin{pmatrix}d^{\prime}\\\
s^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\theta_{C}&\sin\theta_{C}\\\
-\sin\theta_{C}&\cos\theta_{C}\end{pmatrix}\begin{pmatrix}d\\\
s\end{pmatrix}=\begin{pmatrix}d\cos\theta_{C}+s\sin\theta_{C}\\\
-d\sin\theta_{C}+s\cos\theta_{C}\end{pmatrix},$ (8)

and has an orthogonal partner, $s$ =$-d\sin\theta_{C}$ \+ $s\cos\theta_{C}$.
In this formulation, it is apparent that Cabibbo’s form of weak universality
is the same as Feynman-Gell-Mann universality applied to the rotated
$d^{\prime}$ and $s^{\prime}$ quarks.222In addition to suppressing
$\Delta{\mathcal{S}}$ =$\pm 1$ weak interaction couplings relative to that for
muon decay by a factor of $\sin\theta_{C}$ = $0.2245$, Cabibbo’s weak
universality predicts that $\Delta{\mathcal{S}}$ = 0 couplings are suppressed
by a factor of $\cos\theta_{C}$ = $0.974$. In fact, nuclear physicists had
known since 1955 that the half-life for ${}^{14}{\rm
O}$$\rightarrow$${}^{14}{\rm N}\beta^{+}\nu$, a vector-mediated
$0^{+}$$\rightarrow$$0^{+}$ nuclear $\beta^{+}$ decay transition, was $\sim$3%
longer than the value that was predicted using the $g_{W}$ value determined
from muon decay Gerhart:1954zz ; Hendrie:1961zz . In 1960, three years before
Cabibbo’s paper, this discrepancy was noted in the introductory remarks of a
Nuovo Cimento article on the axial-vector current by Gell-Mann and Levy
GellMann:1960np , together with a footnote that suggested that this might be
because the unitarity condition might be, in fact, $\alpha^{2}$+$\beta^{2}$=
1, and not the $\alpha$ = $\beta$ = 1 condition that was conjectured in the
Feyman-Gell-Mann $V$-$A$ paper. The footnote includes a estimate on the mixing
that translates into $\theta$ $\approx$ 14∘, consistent with—and three years
before—Cabibbo’s estimate for $\theta_{C}$ based on $\Delta{\mathcal{S}}$ = 1
transitions. This may explain why K and M, but not C, were awarded the Nobel
prize in 2008.

###  2.2 Gell-Mann Zweig quarks

During this same year Gell-Mann GellMann:1964nj  and Zweig Zweig:1981pd
proposed the quark model in which hadrons were comprised of fractionally
charge fermionic constituents (Zweig called then “aces”). Gell-Mann’s paper
was published in January 1964; Zweig’s paper was never published.333The story
here is that the head of the CERN theory group in 1964, when Zweig was there
on a visiting appointment, thought Zweig’s proposed fractionally charged
particles was a crackpot idea and refused to provide him with the clerical and
drafting support that was needed to prepare a journal-worthy manuscript in the
pre-Latex & computer-graphics era. Gell-Mann won the 1969 Nobel physics prize
and by 1976, when the head of the theory group became the Director-General of
CERN, Zweig was doing biological research and no longer involved in particle
physics. With rotated quarks, the short-distance weak interaction hadronic
currents are the same as those for leptons:

$\displaystyle J_{\mu}^{q}$ $\displaystyle=$ $\displaystyle
g_{W}(\bar{u}\gamma_{\mu}(1-\gamma^{5})d^{\prime})~{}+~{}g_{W}(\bar{c}\gamma_{\mu}(1-\gamma_{5})s^{\prime})$
$\displaystyle=$ $\displaystyle
g_{W}\sum_{i,j}(\bar{u}_{i}\gamma_{\mu}(1-\gamma^{5})V_{ij}d_{j}),$

where $(u_{1},u_{2})$ = $(u,c)$ & $(d_{1},d_{2})$ = $(d,s)$, and $V_{ij}$ is
the eqn. 8 quark mixing matrix. The long distance quark-to-hadron processes
are described by form factors.

###  2.3 Discovery of ${\mathcal{C}}$${\mathcal{P}}$ violation

The Christenson, Cronin, Fitch and Turley discovery of the
${\mathcal{C}}$${\mathcal{P}}$ violating decay mode $K_{L}$$\rightarrow$
$\pi^{+}\pi^{-}$ was reported in the summer of 1964 Christenson:1964fg . This
was a relatively low priority experiment that was not aimed at investigating
${\mathcal{C}}$${\mathcal{P}}$ violation but, instead, was designed to
investigate some anomalies in coherent $K_{2}$$\rightarrow$$K_{1}$
regeneration measurements that had been reported during the previous year
Leipuner:1963zz . It failed to qualify for a spot in the main experimental
hall of the then, almost new, AGS synchrotron that was occupied by
spectrometers specialized for total cross section determinations, and $\pi$,
$K$, $\bar{p}$ and $\mu$-proton elastic scattering measurements. Instead, the
experimental apparatus was located in a relatively inaccessible area inside
the AGS magnet ring that the laboratory technical staff referred to as “Inner
Mongolia,”444In the 1960s diplomatic relations between the U.S. and China were
non-existent, and mainland China, including Inner Mongolia, was considered by
most Americans to be about as accessible as the far side of the Moon. in a
neutral particle line that was essentially a hole in the AGS shielding wall
that was pointed at a target located in the accelerator’s vacuum chamber, as
illustrated in Fig. 2a. The high flux of $\gamma$-rays emerging from the
target were attenuated by a 3.8 cm-thick lead block followed by a collimator
and a bending magnet that swept charged particles out of the beam aperture. A
double-arm spectrometer consisting of tracking spark chambers before and after
two vertically bending magnets measured the directions and momenta of charged
particles that were produced by $K_{L}$ meson decays that occurred in a 2
m-long decay volume that was a plastic bag filled with atmospheric pressure
helium—a low-budget approximation of a vacuum chamber—as shown in Fig. 2b.

Figure 2: a) The neutral $K_{L}$-beamline at the AGS that was used for the
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ search experiment. b) The two-arm
$\pi^{+}\pi^{-}$ spectrometer consisted of a helium-filled decay volume
followed by optical tracking spark chambers before and after momentum
analyzing magnets c) The distribution of events versus the cosine of the angle
between the direction of the two-track momentum sum and the beamline. The
upper, middle and lower panels are for events with two-track invariant masses
that are below, centered on, and above $m_{K_{L}}$, respectively.

Most of the detected events were due to ${\mathcal{C}}{\mathcal{P}}$-allowed
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}\pi^{0}$ decays and
$K_{L}\rightarrow\pi^{\pm}\ell^{\mp}\nu$ ($\ell$ = $e,\mu$) semileptonic
decays. In these decays, the $\pi^{0}$ or $\nu$ was not detected and, as a
result, the invariant mass of the two detected charged particles was not, in
general, equal to the $K_{L}$-meson mass ($m_{K_{L}}$ = $498$ MeV). For
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}\pi^{0}$ decays where the $\pi^{0}$ is
undetected, the $\pi^{+}\pi^{-}$ invariant mass is always below 363 MeV; for
$K_{L}$$\rightarrow$$\pi^{\pm}\ell^{\mp}\nu$, where the $\nu$ is missed and
the $\ell^{\mp}$ track is assigned a pion mass, the two charged track
invariant mass distribution ranges from 280 to 546 MeV, with no peak near
$m_{K_{L}}$. Although the energies of the decaying $K_{L}$ mesons were not
known, their three-momentum directions were confined to be within an rms
spread of $\pm 3.4$ mrad ($\pm 0.2^{\circ}$) around the beamline. A
consequence of the missed particle in the three-body decay channels was that
the vector sums of the two charge track’s momentum vectors did not usually
point along the well defined $K_{L}$ beamline.

Thus, the experimental signature for $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$
decays was a pair of oppositely charged tracks that, when assigned pion
masses, had an invariant mass that was within $\pm 5$ MeV of $m_{K_{L}}$ and
with a summed vector momentum that is directed along the $K_{L}$ beam
direction. Results for these two quantities are shown in Fig. 2c, where the
horizontal axis is the cosine of the angle between the $\vec{p}_{\pi^{+}}$ \+
$\vec{p}_{\pi^{-}}$ direction and the $K_{L}$ beamline, and the upper, central
and lower panels show the experimental distributions for $M(\pi^{+}\pi^{-})$
below, centered on, and above $m_{K_{L}}$, respectively. In the central panel
there is a pronounced peak totally contained within $\cos\theta$ $>$ $0.99996$
($\theta$ $<$ $9$ mrad), a feature that is absent in the distributions for
$M(\pi^{+}\pi^{-})$ below or above $m_{K_{L}}$ shown in the upper and lower
panels. The ratio of the branching fractions for
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ to the sum of all
(${\mathcal{C}}$${\mathcal{P}}$-conserving) decays to charged particles was
$(2.0\pm 0.4)$ $\times$ $10^{-3}$.

The signal peak in the central panel of Fig. 2c contained an excess of $45\pm
10$ events, but these were not all $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$
events. About ten of them were due to the coherent
$K_{L}$$\rightarrow$$K_{S}$$\rightarrow$$\pi^{+}\pi^{-}$ regeneration process
on the helium nuclei in the gas bag decay region, and were indistinguishable
from the $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ signal events. Nature was kind.
If the branching fraction had been much smaller or the regeneration cross
section were higher, the interpretation of the observed signal peak would have
been ambiguous. As mentioned above, this was a low-priority experiment. If it
had ended up by simply setting an upper limit on the
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ branching fraction, who knows when, if
ever, a follow-up experiment with higher sensitivity would have occurred.

##  3 The Kobayashi-Maskawa paper

The famous Kobayashi-Maskawa paper Kobayashi:1973fv  was written in mid-1972,
and published in the February 1973 issue of the Japanese journal Progress of
Theoretical Physics, where it was basically ignored; during the following two
and a half years, it received all of two citations. The paper’s title is
${\mathcal{C}}{\mathcal{P}}$ Violation in the Renormalizable Theory of Weak
Interaction, where the Renormalizable Theory of Weak Interaction is the term
they used for what we now call the Standard Model.

Simply put, a ${\mathcal{C}}$${\mathcal{P}}$ violation means that the
amplitudes for a processes that involve initial-state particles converting to
final-particles and its corresponding antiparticle equivalent are not the
same, e.g.,

${\mathcal{M}}(a\rightarrow
bc)=\bra{bc}H_{w}\ket{a}\neq\bar{\mathcal{M}}(\bar{a}\rightarrow\bar{b}\bar{c})=\bra{\bar{b}\bar{c}}H^{{\dagger}}_{w}\ket{\bar{a}}.$
(10)

But the hermiticity of the Hamiltonian requires that the squares of the
amplitudes are equal:

$|{\mathcal{M}}(a\rightarrow
bc)|^{2}=|\bra{bc}H_{w}\ket{a}|^{2}=|\bar{\mathcal{M}}(\bar{a}\rightarrow\bar{b}\bar{c})|^{2}=|\bra{\bar{b}\bar{c}}H^{{\dagger}}_{w}\ket{\bar{a}}|^{2},$
(11)

and the only way these two conditions can be satisfied is if ${\mathcal{M}}$
and $\bar{\mathcal{M}}$ differ by a phase, i.e.,

${\mathcal{M}}=|{\mathcal{M}}|e^{i\delta_{{\mathcal{C}}{\mathcal{P}}}}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\bar{\mathcal{M}}=|{\mathcal{M}}|e^{-i\delta_{{\mathcal{C}}{\mathcal{P}}}}.$
(12)

So, to incorporate ${\mathcal{C}}$${\mathcal{P}}$ violation into the Standard
Model, all you have to do is find a way to insert a complex phase in it
somewhere, which, at first glance, wouldn’t seem to be so difficult. However
this ${\mathcal{C}}$${\mathcal{P}}$-violating phase is special and, unlike all
other phases that show up quantum theories, and including it into the the
theory is not at all trivial.

There are countless phases that occur in quantum mechanics; both the
Schrödinger and Dirac equations have an imaginary coefficient and their
solutions are complex wave functions. But all of these phases, with, so far,
only one exception, have the same sign for particles and antiparticles. Only a
${\mathcal{C}}$${\mathcal{P}}$-violating phase has opposite signs for
particles and antiparticles.

The KM paper examines various possible ways that a complex
${\mathcal{C}}$${\mathcal{P}}$ phase might be incorporated into the Standard
Model. In the following I discuss the first five pages and the last page
separately.

###  3.1 The KM paper: pages 1$\rightarrow$5

In the first five pages, various possibilities were examined and the authors
concluded that “no realistic models of
${\mathcal{C}}$${\mathcal{P}}$-violation exist in the quartet scheme without
introducing any other new fields.” Here, by the “quartet scheme” they meant
the four-quark model that included the charmed quark. Note that this was
written in 1971, nearly three before the $c$-quark discovery in the “November
1974 revolution.” This, and their page-five conclusion that no realistic model
for ${\mathcal{C}}$${\mathcal{P}}$-violation exists with four quarks raise two
questions:

i) Why were Kobayashi and Maskawa so sure of the existence of the $c$-quark at
such an early data?

ii) Why can’t a ${\mathcal{C}}$${\mathcal{P}}$-violating phase be introduced
together with the Cabibbo angle into the eqn. 8 2$\times$2 quark-flavor mixing
matrix?

####  3.1.1 The discovery of charm: Japanese version

In 1970, a small team of experimenters in Japan led by Kiyoshi Niu, exposed a
stack of photographic emulsions to cosmic rays in a high altitude commercial
cargo airliner Niu:1971xu . Upon subsequent inspection they found a remarkable
event, shown in Fig. 3, in which an ultra-high energy (multi-TeV) cosmic ray
particle interaction produced four charged tracks and two very high energy,
closely spaced $\gamma$-rays that, when attributed to a
$\pi^{0}\rightarrow\gamma\gamma$ decay, had a total energy of $3.2\pm 0.4$
TeV. Two of the charged tracks, labeled B & C in the figure, have kinks within
$\sim$5 cm of the production point that are quite distinct in both the $X$ and
$Y$ projections shown in Figs. 3a & b, indicating that they decayed to charged
daughters (tracks B’ & C’). When the event is viewed along the flight
direction of track B (Fig. 3c), its daughter charged track (B’) and the high
energy $\pi^{0}$ are very close to being back-to-back. The transverse momentum
of the $\pi^{0}$ relative to the direction of track B was $627\pm 90$ MeV, and
much higher than was possible for the decay of any known particle at that
time. With the $\pi^{0}$ setting the energy scale and assumptions of two-body
decays at each kink: $\pi^{\pm}\pi^{0}$ for B$\rightarrow$B’ and $\pi^{0}p$
for C$\rightarrow$C’ where the secondary $\pi^{0}$ is missed, transverse
momentum balance was used to estimate the masses and lifetimes of B and C:

Assumed | Mass | lifetime
---|---|---
decay mode | (GeV) | (sec)
B$\rightarrow$$\pi^{+}\pi^{0}$ | 1.78 | 2.2$\times 10^{-14}$
C$\rightarrow$$(\pi^{0})p$ | 2.95 | 3.6$\times 10^{-14}$,

The estimated B mass and the proper time intervals are consistent with the GIM
estimates of $\sim$2 GeV for the charmed-quark mass (and in reasonable
agreement with what the now very well determined $D^{-}$ mass (1.869 GeV) and
$\Lambda_{c}$ mass (2.286 GeV)). The lifetimes were much shorter than that of
any known weakly decaying particle as well as the
${\mathcal{O}}(10^{-13\,}{\rm s}$) estimate that was given in the GIM paper.
But the latter fact is perhaps not too surprising since emulsion measurements
are biased towards shorter lifetimes. For these reasons, Nagoya theorist Shuzo
Ogawa interpreted Niu’s event as being the associated production of an
anticharmed meson charmed-baryon pair and their subsequent decays. Although
whether or not Niu’s event and Ogawa’s interpretation amounted to a Nobel-
prize-worthy claim of a discovery might be a subject of dispute, what matters
for our story here is that many people in the Japanese theoretical physics
community, especially those in Nagoya that included Kobayashi and Maskawa,
were convinced that the charmed quark had been discovered, and that four
quarks existed in nature, a scenario that they called the “quartet model.”

Figure 3: The a) $X$ and b) $Y$ projections of the Niu event. Here the tracks
labeled B and C have kinks at depth of 1.38 cm and 5.14 cm, respectively, that
are evident in both views. c) The same event viewed along track B, where the
direction of a high energy $\pi^{0}$, inferred from two detected
$\gamma$-rays, is very nearly opposite the direction of B’, the daughter track
that emerges from the track B kink. (The figures are taken from ref.
Niu:1971xu .)

####  3.1.2 A ${\mathcal{C}}$${\mathcal{P}}$ phase in the four-quark mixing
matrix?

In general, a 2$\times$2 matrix like that in eqn. 8 has four complex elements
that correspond to eight distinct real numbers. In the four-quark model,
flavor mixing is completely described by the single real number $\theta_{C}$,
why can’t one of the other seven numbers be used to specify
$\delta_{{\mathcal{C}}{\mathcal{P}}}$, a
${\mathcal{C}}$${\mathcal{P}}$-violating phase?

The flavor-mixing matrix describes a rotation and, thus, has to conserve
probability. This means it should be unitary: i.e. $\bm{VV^{\dagger}}$=
$\bm{\mathcal{I}}$, where $\bm{\mathcal{I}}$ is the identity matrix:

$\begin{pmatrix}V_{ud}&V_{us}\\\
V_{cd}&V_{cs}\end{pmatrix}\times\begin{pmatrix}V^{*}_{ud}&V^{*}_{cd}\\\
V^{*}_{us}&V^{*}_{cs}\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$
(13)

This corresponds to four relations

$\displaystyle|V_{ud}|^{2}+|V_{us}|^{2}=1~{}~{}~{}~{}~{}$ $\displaystyle{\rm
and}$ $\displaystyle~{}~{}~{}~{}~{}|V_{cd}|^{2}+|V_{cs}|^{2}=1$ (14)
$\displaystyle V_{ud}V^{*}_{cd}=-V_{us}V^{*}_{cs}~{}~{}~{}~{}~{}$
$\displaystyle{\rm and}$
$\displaystyle~{}~{}~{}~{}~{}V_{cd}V^{*}_{ud}=-V_{cs}V^{*}_{us},$ (15)

that reduces the number of independent parameters from eight to four.

In the weak interaction quark currents (eqn. 2.2), the total number of quarks
is conserved: $q_{j}$, which annihilates a $q_{j}$-quark, is always
accompanied by $\bar{q}_{i}$, which creates a $q_{i}$-quark. The theory has a
subtle property: if each quark field is multiplied by an arbitrary phase
factor,

$d_{j}\longrightarrow e^{i\phi_{j}}d_{j}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}\bar{u}_{i}\longrightarrow e^{-i\phi_{i}}\bar{u}_{i},$ (16)

and the interactions are modified by the same phases,

$V\longrightarrow\begin{pmatrix}e^{i\phi_{u}}&0\\\
0&e^{i\phi_{c}}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}\\\
V_{cd}&V_{cs}\end{pmatrix}\begin{pmatrix}e^{-i\phi_{d}}&0\\\
0&e^{-i\phi_{s}}\end{pmatrix},$ (17)

there is no net effect on on the $J_{\mu}^{q}$ current:

$\displaystyle(\bar{u}_{i}\gamma_{\mu}(1-\gamma_{5})V_{ij}d_{j})$
$\displaystyle\longrightarrow$
$\displaystyle(\bar{u}_{i}e^{-i\phi_{i}}\gamma_{\mu}(1-\gamma_{5})e^{i(\phi_{i}-\phi_{j})}V_{ij}e^{i\phi_{j}}d_{j})$
(18) $\displaystyle=$
$\displaystyle(\bar{u}_{i}\gamma_{\mu}(1-\gamma_{5})V_{ij}d_{j}).$

This process is called rephasing and the four phases can be expressed as three
independent phase differences plus one overall phase that has no effect on the
physics. Thus, of the the eight real numbers we started with, four are removed
by the unitarity constraint, and three can have any value with no net effect.
Thus there is only one number left to define the matrix, and that is needed
for the rotation angle $\theta_{C}$. There is no freedom to add a
${\mathcal{C}}$${\mathcal{P}}$ phase and is what led Kobayashi and Maskawa to
conclude that there was no way to incorporate ${\mathcal{C}}$${\mathcal{P}}$
violation into the four quark model.

###  3.2 The KM paper: page 6

In his 2008 Nobel prize lecture Maskawa:2009zza , Maskawa recalled that he and
Kobayashi had completed the work that was covered in pages 1-5 of their paper
and were disappointed with their failure to find any way to incorporate a
${\mathcal{C}}$${\mathcal{P}}$ phase into the four quark model, and were
reconciled to the unhappy likelihood that they would have to publish the
negative result. Then, one evening, while—as is customary in Japan—he was
taking his after-dinner bath, he mentally went through the calculation
described in the previous paragraph, except this time for a six-quark scenario
with a 3$\times$3 flavor-mixing rotation matrix in three dimensions. In this
case, there are 9 complex elements that are described by 18 real numbers. Of
these, 9 are constrained by the unitarity requirement, 5 are taken up by
rephasing, and 3 are needed to specify the 3-dimensional rotation,555In
general, for $N$ quark generations with an $N$$\times$$N$ mixing matrix, there
are $N^{2}$ elements characterized by $2N^{2}$ real numbers. Of these $N^{2}$
are used for unitarity, $2N$-$1$ for rephasing and $N(N$-$1)/2$ are needed to
define the rotation angles. The remaining degrees of freedom that can show up
as ${\mathcal{C}}$${\mathcal{P}}$ phases are $(N$-$1)(N$-$2)/2$, which
vanishes for $N$=$2$, but is $1$for $N$=$3$ (and would be $3$ if $N$=$4$). and
that left one number that remained available.

Eureka! With six-quarks there is room for a
${\mathcal{C}}$${\mathcal{P}}$-violating phase!

A sixth page was added to the manuscript that included the remarks:

> Next we consider a 6-plet model, another interesting model of
> ${\mathcal{C}}$${\mathcal{P}}$ violation,… with a 3$\times$3 instead of
> 2$\times$2 unitary matrix. In this case we cannot absorb all phases of
> matrix elements into the phase convention and can take, for example, the
> following expression:

$\begin{pmatrix}\cos\theta_{1}&-\sin\theta_{1}\cos\theta_{3}&-\sin\theta_{1}\sin\theta_{3}\\\
\sin\theta_{1}\cos\theta_{2}&\cos\theta_{1}\cos\theta_{2}\cos\theta_{3}-\sin\theta_{2}\sin\theta_{3}e^{i\delta}&\cos\theta_{1}\cos\theta_{2}\sin\theta_{3}+\sin\theta_{2}\cos\theta_{3}e^{i\delta}\\\
\sin\theta_{1}\sin\theta_{2}&\cos\theta_{1}\sin\theta_{2}\cos\theta_{3}+\cos\theta_{2}\sin\theta_{3}e^{i\delta}&\cos\theta_{1}\sin\theta_{2}\sin\theta_{3}-\cos\theta_{2}\sin\theta_{3}e^{i\delta}\end{pmatrix}.$
(19)

> Then we have ${\mathcal{C}}$${\mathcal{P}}$-violating effects … that appear
> only in the $\Delta{\mathcal{S}}$$\neq$0 non-leptonic processes and semi-
> leptonic decay of neutral strange mesons (we are not concerned with higher
> states with the new quantum number) …

(Here, $\theta_{1}$ is (approximately) the Cabibbo angle, $\theta_{2}$ is the
mixing angle between the 2nd\- & 3rd-generation quarks, $\theta_{3}$ mixes the
1st\- & 3rd-generations, and $\delta$ is the
${\mathcal{C}}$${\mathcal{P}}$-violating phase.) And that was it. Discussion
about the six-quark model was confined to one paragraph that, together with
the expression for the matrix, occupied only about half of the page.

Thus, Kobayashi and Maskawa discovered the way to incorporate a
${\mathcal{C}}$${\mathcal{P}}$ violating phase into the Standard Model, and
established a deep theoretical connection between quark-flavor mixing and
${\mathcal{C}}$${\mathcal{P}}$ violation, two subjects that had previously
been considered to be unrelated. But this came at a cost: you need to have
six-quarks. In 1971, thanks to Kiyoshi Niu, this was not such a big stretch
for Kobayashi and Maskawa, who were among the fortunate few who were already
convinced that there were (at least) four quarks. On the other hand, most of
the world-wide particle physics community outside of Japan was quite satisfied
with three quarks.

####  3.2.1 The first proposal for (and the naming of) charm

The distinction between electron- and muon-neutrinos had been established in
1962 Danby:1962nd , and, in what was at that time the beginnings of the
Standard Model, the electron and muon and their neutrinos occupied two weak-
isospin=1/2 spinors. In a 1964 paper, Bjorken and Glashow Bjorken:1964gz
discussed an expansion of $SU(3)$ to $SU(4)$ with the addition of another
strangeness-like flavor quantum number. They formulated their model in terms
of the Sakata model, the predecessor of the quark model that used the proton-
neutron isospin doublet and isoscalar Lambda as basic constituents, with the
Lambda replaced by a doublet that had a fourth baryonic constituent with a
non-zero value of the new quantum number. When reformulated in the context of
the quark model, which had just emerged at that time with three quarks, this
was equivalent to adding a fourth quark with matching patterns for the leptons
and quarks:

$\displaystyle{\rm leptons}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm quarks}$ (20)
$\displaystyle\begin{pmatrix}e^{-}\\\
\nu_{e}\end{pmatrix}~{}\begin{pmatrix}\mu^{-}\\\
\nu_{\mu}\end{pmatrix}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}\begin{pmatrix}u\\\
d\end{pmatrix}~{}\begin{pmatrix}\bm{c}\\\ s\end{pmatrix}.$

While their proposal was not very different from schemes that other authors
had proposed around that time (see, for example refs. Tarjanne:1963zza ;
Maki:1963ye ; Hara:1963gw ), Bjorken and Glashow gave the new flavor the
charismatic name “charm,” and that’s the one that stuck; the fourth quark was
known as the “charmed quark” (and not the grammatically awkward “charm quark”)
even before it was discovered. In retrospect, this four-quark scheme seems
like a pretty sound—almost obvious—idea;666In a passing comment in his
original paper on quarks GellMann:1964nj , Gell-Mann mentioned a four-quark
scheme that was “parallel with the leptons” as an interesting possibility. how
could anyone dismiss such a simple and sensible suggestion? Nevertheless, for
the next six years this idea didn’t go very far. By the time the GIM paper
Glashow:1970gm  appeared in 1970, the Bjorken-Glashow paper had received a
grand total of six citations. For some reason, there seem to have been a
special affinity in the physics community for the number three and an aversion
to the number four.777In east Asian cultures the number four is considered to
be very unlucky because the Chinese pronunciations of their words for “four”
and “death” are homophonous. As far as I know there is no such taboo in
Western cultures. Moreover, even the GIM paper that proposed a four-quark
scenario as a compelling explanation for the suppression of flavor-changing
neutral-currents, an important theoretical issue at that time, didn’t
experience a boom in citations until after the $J/\psi$ discovery, which
finally convinced the world-wide physics community that there were (at least)
four quarks.

In 1975, soon after the $J/\psi$ discovery, Maiani (the “M” in GIM), who, at
that time, was unaware of the KM result,888See footnote 2 in Maiani’s paper.
wrote an interesting paper Maiani:1975in  that contained all of the KM model
and then some. But, thanks to their three-year-long head start, it was
Kobayashi and Maskawa—and not Maiani—that got to go to Sweden in December
2008.

The six-quark era started in earnest in 1977, after the discovery of the
$\tau$-lepton Perl:1975bf  and the $b$-quark Herb:1977ek , and the KM model
was elevated to role of being the most likely mechanism for
${\mathcal{C}}$${\mathcal{P}}$ violation. Although the $b$-quark’s charge=2/3
partner, the $t$-quark, wasn’t discovered until 1995 Abe:1995hr ; D0:1995jca ,
there was very little doubt about its existence.999There were some papers on
“topless models,” Barger:1981vg ; Branco:1978mr ; Gorn:1980cj  including one
with the title: “Does bare-bottom rule out the topless E6 model?”
Sarkar:1983yx , but these did not get much attention. The only question was
its mass; based on the mass pattern of the known quarks, the general consensus
was that it was probably in the $\sim$25–35 GeV range Yamanaka:1981pa .

###  3.3 “Discovery” of the KM paper

As mentioned above, for over two years the KM paper remained completely
unnoticed outside of Japan (and only received limited attention inside Japan).
It was finally brought to the attention of the world-wide physics community in
a curious set of circumstances that are recounted here.

But first it should be noted that version of the quark-mixing matrix that
appeared in the KM paper and is reproduced above in eqn. 19 contains a pretty
obvious typographical (or transcription) error. Since the KM matrix nominally
describes a rotation, if all three mixing angles and the
${\mathcal{C}}$${\mathcal{P}}$ phase are set to zero, it should revert to the
identity matrix, i.e.,

$\bm{V}\xrightarrow{~{}(\theta_{1,2,3},\delta)\rightarrow
0~{}}\bm{\mathcal{I}}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$
(21)

However in the matrix that appears in the KM paper, the zero-angle limit for
the lower-right diagonal element $V_{tb}$, which should be
$V_{tb}$$\rightarrow$1, is incorrect:

$V^{\rm
KM}_{tb}=\cos\theta_{1}\sin\theta_{2}\sin\theta_{3}-\cos\theta_{2}\sin\theta_{3}e^{i\delta}\xrightarrow{~{}(\theta_{1,2,3},\delta)\rightarrow
0~{}}0.$ (22)

The first paper to establish that the KM model could account for all that was
known about ${\mathcal{C}}$${\mathcal{P}}$ violations at that time, was by
Pakvasa and Sugawara and published in the July 1976 issue of Physical Review D
Pakvasa:1975ti . In their paper, they pointed out that in the six-quark model,
a non-zero value of $\varepsilon$, the neutral $K$-meson mass-matrix
${\mathcal{C}}$${\mathcal{P}}$ violation parameter, of the correct magnitude
would be produced by the interference between the virtual $c$\- and $t$-quark
contributions to the $K^{0}$-$\bar{K}^{0}$ mixing box-diagram shown in Fig.
4a. They also pointed out that in the KM picture, the penguin
diagram101010This was before penguin diagrams were called penguin diagrams.
shown in Fig. 4b, that would mediate direct-${\mathcal{C}}$${\mathcal{P}}$
violating $K_{2}$$\rightarrow$$\pi^{+}\pi^{-}$ decays, i.e., the
$\varepsilon^{\prime}$ parameter, would be small and consistent with the then
existing experimental limits. Since PRD is a widely distributed physics
journal, this paper provided the first awareness of the KM paper to the
international particle physics community.

Figure 4: a) The $W$-exchange Standard Model box diagram for
$K^{0}$-$\bar{K}^{0}$ mixing (not shown is the one with heavy quark exchange).
In the KM six-quark model, the kaon’s mass-matrix
${\mathcal{C}}$${\mathcal{P}}$-violating parameter $\varepsilon$ is produced
by interference between the $c$\- and $t$-quark contributions. b) The penguin
diagram for direct-${\mathcal{C}}$${\mathcal{P}}$-violating
$K_{2}$$\rightarrow$$\pi\pi$ decays.

In their paper, Pakvasa and Sugawara made no mention of the typo in the KM
paper and included an expression for the matrix that they attributed to KM
but, in fact, was different. This paper also had a typo that mistakenly
identified Toshihide Maskawa as K. Maskawa in their citation to the KM paper.
Interestingly, many of the papers that immediately followed the Pakvasa-
Sugawara paper used the Pakvasa-Sugawara version of the CKM matrix with no
mention of the mistake, and with citations to the KM paper that had T. Maskawa
incorrectly listed as K. Maskawa (see, it e.g., refs. Lee:1976nx ;
Ellis:1977uk ; Weinberg:1977maa ; Altarelli:1977zq ). This recurrence of the
typo in the citations provided pretty clear evidence that the Pakvasa-Sugawara
PRD article was the source that researchers used for both the matrix and the
citation, and that the PTP paper itself was probably not very widely
read.111111But why didn’t Pakvasa and Sugawara alert their readers about the
problem with the matrix in the KM paper? Sugawara’s recollection is that when
he first learned about the KM paper at a University of Tokyo physics seminar,
he was impressed by their six-quark scheme and “reconstructed it in [his] own
way, without reading their paper carefully.” When he and Pakvasa subsequently
did their analysis and wrote up their results, they included Sugawara’s
version of the matrix, which was the KM version without the error, in their
paper. In fact, Pakvasa and Sugawara remained unaware of the KM paper’s typo.
According to Sugawara, “I never realized that the paper had this typo until it
was recently pointed out to me.”

But, in addition to the typo in their citation to the KM paper, the Pakvasa-
Sugawara version of the KM matrix had a problem of its own. In their paper,
the KM expression for $V_{tb}$, given above in eqn. 22, was replaced by

$V^{\rm
PS}_{tb}=\cos\theta_{1}\sin\theta_{2}\sin\theta_{3}-\cos\theta_{2}\cos\theta_{3}e^{i\delta},$
(23)

which doesn’t go to zero in the limit or zero mixing angles. But neither does
it go to 1, instead,

$V^{\rm PS}_{tb}\xrightarrow{~{}(\theta_{1,2,3},\delta)\rightarrow 0~{}}-1,$
(24)

and for $\delta=0$, has a negative determinant. Nevertheless, this version of
the matrix is unitary, which is the only essential requirement for a quark-
flavor mixing matrix, and this form was used in much (but not all) of the
literature until 1984, when the currently widely accepted Chau-Keung
parameterization was proposed and soon thereafter endorsed by the Particle
Data Group.

##  4 Reparameterizing the CKM matrix

A rotation in three dimensions can be accomplished by three successive
rotations: first by an angle $\theta_{1}$ around the $z$ axis that mixes $x$
and $y$ [$(x,y)$$\rightarrow$$(x^{\prime},y^{\prime})$)], then by $\theta_{2}$
about the new $x^{\prime}$ axis that mixes $y^{\prime}$ and $z$
[$(y^{\prime},z)$$\rightarrow$$(y^{\prime\prime},z^{\prime})$] and, finally,
by $\theta_{3}$ around the $y^{\prime\prime}$ axis that mixes $x^{\prime}$ and
$x^{\prime}$
[$(x^{\prime},z^{\prime})$$\rightarrow$$(x^{\prime\prime},z^{\prime\prime})$].
This is just one of of many ways that can be used to specify a given rotation.
For example, the order of the three rotations can be changed, and there is
freedom in the choice of the axes that are used to define the rotations. For
these different choices, the values of the individual rotation angles are
different, as are the expressions for each matrix element in terms of these
rotation angles. Ultimately, however, the numerical value of the magnitude of
each matrix element for any of these choices has to be the same, and
independent of the choice of the individual rotations. In addition, in the
case of the CKM matrix, which is complex, rephasing invariance provides five
independent arbitrary phase parameters that can be attached to the various
matrix elements to establish whatever phase convention may seem convenient.
The physics content is independent of these parameterization choices.

On what basis should a parameterization be selected? In answer to this, Haim
Harari suggested some criteria for what he would consider to be a “good”
parameterization. These included Harari:1986xf :

  * •

There should be a simple relation between the most directly measurable matrix
elements $V_{ij}$ and the quark mixing angles.

  * •

The matrix elements above the diagonal, which correspond to kinematically
allowed decay processes that are directly measurable, should have the simplest
possible expressions.

  * •

If possible, the ${\mathcal{C}}$${\mathcal{P}}$ violating phase should be
linked to only one angle, and preferably the sine of that angle.

During the years immediately following the wide recognition of the KM paper,
there was considerable effort aimed at finding a suitable parameterization.
This was aided by concurrent experimental measurements of the relative
magnitudes of the $V_{cb}$ and $V_{ub}$ matrix elements using $B$-mesons that
were produced via $e^{+}e^{-}$ annihilations at two colliders that existed at
that time: PEP at SLAC and CESR at Cornell.

###  4.1 Experimental information about quark transitions

The PEP project was initially conceived as a two ring proton-electron-positron
collider with an electron-positron ring that could support $E_{\rm cm}$= 30
GeV $e^{+}e^{-}$ collisions primarily to search for the top-quark (if its mass
was less than 15 GeV), and a second 150 GeV proton ring that could support
$e^{\pm}p$ collisions with $E_{\rm cm}$ $\approx$ 100 GeV for measurements of
deep inelastic scattering at high energies and $Q^{2}$ values. The CESR
collider was conceived during 1974 as a follow-up to the Cornell fixed-target
$e^{-}p$ scattering program, and proposed to the U.S. National Science
Foundation in 1975, soon after the $J/\psi$ discovery, as an $E_{\rm
cm}$$\approx$ 16 GeV $e^{+}e^{-}$ collider primarily aimed at studies of
charmed particles. The PEP and CESR projects were both well underway when the
$b$-quark was discovered in 1977. Fortuitously, the initial CESR design energy
could comfortably operate in the $E_{\rm cm}$= 9$-$11 GeV range, and cover the
three narrow $\Upsilon(1S,2S,3S)$ resonances and the threshold region for
$e^{+}e^{-}\rightarrow B\bar{B}$ meson pair production, that was expected to
be around $E_{\rm cm}$= $10.5$ GeV.121212The CESR facility spent its life
operating at $E_{\rm cm}$$\sim$10–11 GeV.

The two projects started running in 1979 and soon thereafter provided
convincing experimental evidence for a strong hierarchy among the weak
interaction mixing angles for the three quark generations. It was already well
established that transitions within the first quark generation, e.g.,
$u$$\rightarrow$$d$ and within the second generation ($c$$\rightarrow$$s$)
were strongly favored over transitions between the first and second
generations ($s$$\rightarrow$$u$ and $c$$\rightarrow$$d$), i.e., the Cabibbo
angle. Experiments at PEP found that transitions between the second and third
generation ($c$$\rightarrow$$b$) are more suppressed those between the first
and second generations, and the CESR experiment determined that transitions
between the first and third generations ($u$$\rightarrow$$b$) are the least
favored of all.

1st$\leftrightarrow$ 2nd generation:

The suppression of strangeness-changing decays that was noted by Cabibbo in
1963, led to the realization that the relative strengths of the
$u$$\rightarrow$$d$ and the strangeness-changing $u$$\rightarrow$$s$
transitions are modulated by factors of $\cos\theta_{C}$ and $\sin\theta_{C}$,
respectively, where $\theta_{C}$ = 13∘ is the Cabibbo angle. Thus the diagonal
element $V_{ud}$ = $0.974$ is nearly unity and almost five times larger that
its adjacent entry $V_{us}$ = 0.225.

2nd$\leftrightarrow$ 3rd generation:

In 1983, the MAC experiment at the PEP collider used $b$-flavored hadrons
(mainly $B^{\pm}$ and $B^{0}$ mesons) produced via the
$e^{+}e^{-}$$\rightarrow$$b\bar{b}$ annihilation process (about 1/10th of the
total annihilation cross section) to determine the $b$-quark lifetime by
measuring the impact parameters of charged leptons from
$B$$\rightarrow$$X\mu\nu$ and $Xe\nu$ inclusive semileptonic decays, where $X$
is a hadronic system Fernandez:1983az . (The definition of the impact
parameter is indicated in Fig. 5a.) This was a difficult measurement: the
parent $B$-meson’s energy and direction were not precisely known on an event-
by-event basis; there was contamination from semileptonic decays of charmed
mesons; and the mean value of the impact parameter that was eventually
determined ($\approx$170 $\mu$m) was substantially smaller than the
experimental resolution ($\sim$600 $\mu$m) as well as the horizontal size of
the beam-beam interaction region ($\sim$400 $\mu$m). Nonetheless, the measured
impact parameter distributions for muons and electrons shown in the upper and
lower panels of Fig. 5b, respectively, both had excesses at positive values
and these translated into a $b$-quark lifetime of $\tau_{b}$ = $1.8\pm 0.7$
ps. Since this was about a factor of four times longer than the lifetime of
the much lighter $c$-quark, it was a big surprise at that time. As discussed
below, $b$$\rightarrow$$c$ transitions are the $b$-quark’s dominant decay
mechanism, and the measured lifetime could be used to determine a value for
$|V_{cb}|$ using the expression

$\Gamma_{b}=1/\tau_{b}=|V_{cb}|^{2}\frac{G_{F}^{2}m_{b}^{5}c^{4}}{192\pi^{3}\hbar^{7}}(2_{\rm
quarks}\times 3_{\rm colors}+3_{\rm leptons}),$ (25)

which is the (textbook) expression for the muon lifetime tailored to the
$b$-quark mass, modified to include the effect of $|V_{cb}|$ on the coupling
strength, and multiplied by the number of accessible final states: two quark
flavors, three quark colors, and three types of leptons, as illustrated in
Fig. 5c. The result was $|V_{cb}|$ $\approx$ 0.04, and about a factor of five
smaller than $|V_{us}|$, a difference that is similar to (but not exactly the
same as) the factor of five Cabbibo suppression131313The PDG 2020 Zyla:2020zbs
value for the $B$ meson lifetime is $\tau_{b}$= $1.519\pm 0.004$ ps and that
for the $V_{cb}$ matrix element is $|V_{bc}|$ = $0.0410\pm 0.0014$, which are
both within the error ranges of the MAC measurements. between $V_{us}$ and
$V_{ud}$. The MAC results were confirmed by the MarkII and DELCO experiments
at PEP Lockyer:1983ev ; DELCO:1984jng  and the TASSO experiment TASSO:1984ult
at PETRA, an $E_{\rm cm}$=$40$ GeV $e^{+}e^{-}$ collider at the DESY
laboratory in Hamburg.

Figure 5: a) A sketch of the projections of tracks from the semileptonic decay
of a $B$ meson onto a plane perpendicular to the $e^{+}e^{-}$ beamline,
together with indications of the size of the beam-beam interaction region, and
the definition of the impact parameter, i.e., the distance of closest approach
of the charged lepton track to the $e^{+}e^{-}$ interaction point. The unknown
$B$ meson’s direction was assigned with reasonably good accuracy to be along
the event’s thrust axis. b) Impact parameter distributions for muons (upper)
and electrons (lower) (from the MAC experiment Fernandez:1983az ). Here each
entry is weighted by the inverse square of its measurement error. c) A quark-
line diagram for $b$-quark decays. The subscript $\alpha$ indicates the quark
color.

1st$\leftrightarrow$ 3rd generation:

The CLEO experiment studied semileptonic $B$$\rightarrow$$X\ell\nu$ decays of
$B$ mesons that were produced at $E_{\rm cm}$ = $10.58$ GeV, the peak141414The
$\Upsilon(4S)$ is the 3rd radial excitation of the $\Upsilon(1S)$, the lowest-
lying $J^{PC}$= $1^{--}$ “bottomonium” ($b\bar{b}$) meson, and the first that
is above the $B\bar{B}$ “open bottom” threshold. At its peak, the cross
section for $e^{+}e^{-}$$\rightarrow$$B\bar{B}$ is about 1 nb. of the
$\Upsilon(4S)$$\rightarrow$$B\bar{B}$ resonance CLEO:1984uti . Since this
energy is only 20 MeV above the $B\bar{B}$ threshold, the $B$ mesons are
produced very nearly at rest (the boost factor is $\gamma\beta$ = $0.062$) and
the energy of a decay lepton ($\ell$ = $e,\mu$) in the laboratory frame is
very nearly equal to what it is in the $B$ meson rest frame. In
$b$$\rightarrow$ $c\ell\nu$-mediated decays, the minimum mass of the hadronic
system is $M^{\rm min}_{X}$= $m_{D}$ = $1.86$ GeV, and this translates into a
maximum lepton momentum of $p_{c\ell\nu}^{\rm max}$ = $2.31$ GeV/$c$; in
$b$$\rightarrow$ $u\ell\nu$-mediated decays, the mass of the hadronic system
can be as light as $M^{\rm min}_{X}$= $m_{\pi}$, and the end-point momentum is
$p_{u\ell\nu}^{\rm max}$ = $2.60$ GeV/$c$. Thus, measurements of the lepton
momentum spectra in the end-point region can be used to determine the relative
strengths of the $b$$\rightarrow$$c$ and $b$$\rightarrow$$u$ transitions and
extract the value of $|V_{ub}|^{2}/|V_{bc}|^{2}$. Figure 6 shows the measured
momentum distributions for electrons (upper) and muons (lower) together with
expectations for $b$$\rightarrow$ $c\ell\nu$ (dashed curves) and
$b$$\rightarrow$ $u\ell\nu$ (dotted curves). There are no events in the
2.31$<$$p_{\rm lepton}$$<$2.60 GeV/$c$ range that could be unambiguously
attributed to $b$$\rightarrow$$u\ell\nu$ decays and the shapes of the spectra
are consistent with expectations for $\sim$100% $b$$\rightarrow$ $c\ell\nu$
with no significant contribution from $b$$\rightarrow$ $u\ell\nu$. From these
data, the CLEO group established a 90% CL upper limit151515The PDG 2020
Zyla:2020zbs  value is $|V_{ub}|/|V_{cb}|$ = $0.093\pm 0.004$. of
$|V_{ub}|/|V_{cb}|$$<$0.14.

Figure 6: Lepton momentum spectra for $B$$\rightarrow$$Xe\nu$ (upper) and
$B$$\rightarrow$$X\mu\nu$ (lower) decays. The dashed (dotted) curves are
expectations for $b$$\rightarrow$$c\ell\nu$ ($b$$\rightarrow$$u\ell\nu$). The
dash-dot curves shows the expectation for secondary leptons from semileptonic
decays of $D$ decays produced via $B$$\rightarrow$$D\ell\nu$ and of $\tau$
leptons from $B\rightarrow X\tau\nu$. (From the CLEO experiment CLEO:1984uti
.)

q

###  4.2 The Chau-Keung and Wolfenstein parameterizations

Since the early parameterizations fail on all three of the Harari criteria, a
better one was needed. Of a number of proposed replacementsMaiani:1975in ;
Chau:1984fp ; Wolfenstein:1983yz ; Gronau:1984dn ; Harari:1986xf , two
continue to be widely used today: one by Chau and Keung Chau:1984fp  and the
other by Wolfenstein Wolfenstein:1983yz .

####  4.2.1 Chau-Keung parameterization

The parameterization proposed by Chau and Keung was specifically motivated by
the occurrence of the ${\mathcal{C}}$${\mathcal{P}}$ violating phase in the
$V_{tb}=\cos\theta_{1}\cos\theta_{2}\cos_{3}-\cos\theta_{2}\cos\theta_{3}e^{i\delta}$
term in the Pakvasa-Sugawara version of the original KM parameterization that
seemed to suggest that there is a large ${\mathcal{C}}$${\mathcal{P}}$
violation that is confined to the ($t,b$) quark sector. This was in sharp
contrast to the results of detailed calculations of measurable
${\mathcal{C}}$${\mathcal{P}}$ violating effects that invariably resulted in
small numbers that involved factors of
$\sin\theta_{2}\sin\theta_{3}e^{i\delta}$.

(For clarity, in the following we follow the more transparent PDG notation
that uses $\theta_{ij}$ to denote the mixing angle around an axis that is
perpendicular to the $(i,j)$ plane. The translation between $\theta_{1,2,3}$
and $\theta_{ij}$ is: $\theta_{1}$ = $\theta_{12}$, $\theta_{2}$ =
$\theta_{23}$, and $\theta_{3}$ = $\theta_{13}$. In addition we abbreviate
$\cos\theta_{ij}$ by $c_{ij}$ and $\sin\theta_{ij}$ by $s_{ij}$.)

The Chau-Keung parameterization, which is exactly unitary, is given by the
(right-to-left) sequence of two-dimensional rotations:

$\displaystyle V_{\rm CKM}$ $\displaystyle=$
$\displaystyle\overbrace{\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix}}^{\theta_{23}~{}{\rm
about}~{}d^{\prime\prime}}\overbrace{\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta}\\\
0&1&0\\\ -s_{13}e^{i\delta~{}}&0&c_{13}\end{pmatrix}}^{\theta_{13}~{}{\rm
about}~{}s^{\prime}~{}({\rm
incl.}~{}\delta)}\overbrace{\begin{pmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{pmatrix}}^{\theta_{12}~{}{\rm about}~{}b}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}~{}~{}&~{}~{}c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}~{}~{}&~{}~{}s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}~{}~{}&~{}~{}-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}~{}~{}&~{}~{}c_{23}c_{13}\end{pmatrix},$

where it is apparent that this version passes the requirement that
${\bm{V}}$$\rightarrow$$\bm{\mathcal{I}}$ when $\theta_{ij}$$\rightarrow$ 0\.
Here Harari’s three criteria are satisfied: the
$\theta_{12},\theta_{23},\theta_{13}$ mixing angles correspond to
experimentally distinct $u$$\leftrightarrow$$s$, $c$$\leftrightarrow$$b$, &
$u$$\leftrightarrow$$b$ transitions, respectively; the
${\mathcal{C}}$${\mathcal{P}}$ phase factor is always multiplied by a factor
containing $s_{13}\,(=\,|V_{ub}|)$; and the three above-diagonal terms have
simple forms. Moreover, in this parameterization, $\theta_{12}$, $\theta_{23}$
and $\theta_{13}$ are all in the first quadrant, which means that $s_{ij}$ and
$c_{ij}$ are all positive, the ${\mathcal{C}}$${\mathcal{P}}$ phase $\delta$
is also positive and in the range $0\leq\delta<2\pi$, and the matrix reduces
to the $2\times 2$ Cabibbo matrix for $\theta_{23}$ = $\theta_{13}$ = 0. This
has been the PDG parameterization of choice since 1986
ParticleDataGroup:1986kuw  and, according to recent measurements Zyla:2020zbs
,

$\displaystyle\theta_{12}=13.09^{\circ}\pm
0.03^{\circ}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}\theta_{23}=2.32^{\circ}\pm 0.04^{\circ}$ (27)
$\displaystyle\theta_{13}=0.207^{\circ}\pm 0.007^{\circ}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}\delta=68.53^{\circ}\pm 0.51^{\circ}.$

####  4.2.2 Wolfenstein parameterization

The Wolfenstein parameterization is an approximation that employs a polynomial
expansion in terms of $\lambda$ $\equiv$ $\sin\theta_{C}$= 0.2265 that
reflects the hierarchical character of the CKM matrix. With an accuracy up to
${\mathcal{O}}(\lambda^{3})$ it has the form:

$V_{\rm
CKM}=\begin{pmatrix}1-{\textstyle\frac{1}{2}}\lambda^{2}&\lambda&A\lambda^{3}({\rho}-i{\eta})\\\
-\lambda&1-{\textstyle\frac{1}{2}}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}(1-{\rho}-i{\eta})&-A\lambda^{2}&1\end{pmatrix}+{\mathcal{O}}(\lambda^{4}),$
(28)

where the parameter $A$$\approx$ 0.8 accounts for the fact that the Cabbibo-
like suppression between $V_{cb}$ and $V_{cd}$ is about 20% more severe than
that between $V_{cd}$ and $V_{ud}$. In this parameterization, all of the
${\mathcal{C}}$${\mathcal{P}}$ violation resides in the single parameter
${\eta}$ that is confined to the $V_{cd}$ and $V_{ub}$ corners of the matrix
where it is multiplied by $\lambda^{3}A$, a small number.

This parameterization is very convenient and is widely used, but in a somewhat
modified form that was suggested by Buras and colleagues Buras:1994ec . In the
Buras version, the Wolfenstein $\lambda$, $A$, $\rho$ and $\eta$ parameters
are redefined in terms of the Chao-Keung angles to be exactly

$\displaystyle\lambda$ $\displaystyle\equiv$ $\displaystyle
s_{12}=\frac{|V_{us}|}{\sqrt{|V_{ud}|^{2}+|V_{us}|^{2}}}$ (29) $\displaystyle
A\lambda^{2}$ $\displaystyle\equiv$ $\displaystyle
s_{23}=\lambda\frac{|V_{cb}|}{|V_{us}|}$ (30) $\displaystyle
A\lambda^{3}(\rho-i\eta)$ $\displaystyle\equiv$ $\displaystyle
s_{13}e^{i\delta}=V^{*}_{ub}.$ (31)

With these parameter definitions, $V_{ub}$ is the same as in the Chau and
Keung parameterizations, and the higher corrections to $V_{us}$ and $V_{cb}$
start at ${\mathcal{O}}(\lambda^{7})$ and ${\mathcal{O}}(\lambda^{8})$,
respectively. In addition Buras et al. defined new parameters $\bar{\rho}$ and
$\bar{\eta}$ as

$\bar{\rho}+i\bar{\eta}=-\frac{V_{ud}V^{*}_{ub}}{V_{cd}V^{*}_{cb}},$ (32)

in which case

$\bar{\rho}=\rho\big{(}1-\frac{\lambda^{2}}{2}\big{)}+{\mathcal{O}}(\lambda^{4})~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\bar{\eta}=\eta\big{(}1-\frac{\lambda^{2}}{2}\big{)}+{\mathcal{O}}(\lambda^{4})$
(33)

and

$V_{td}=A\lambda^{3}(1-\bar{\rho}-i\bar{\eta})+{\mathcal{O}}(\lambda^{7}).$
(34)

Although the distinction between Wolfenstein’s $\rho,\ \eta$ and (the nearly
equal) $\bar{\rho},\ \bar{\eta}$ parameters may seem to be confusing and
unnecessary, the latter parameters are preferred and are more generally used
(for reasons that are discussed in detail in ref. Charles:2004jd ). The PDG
review Zyla:2020zbs  only provides values for the four (redefined)
“Wolfenstein parameters” that, in 2020, were

$\displaystyle\lambda=0.22650\pm 0.00048~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}A=0.790^{+0.017}_{-0.012}$ (35)
$\displaystyle\bar{\rho}=0.141^{+0.016}_{-0.017}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}\bar{\eta}=0.357\pm 0.011.$

The eqn. 28 form of the matrix is carefully tuned for
${\mathcal{C}}$${\mathcal{P}}$ violations in the $b$-quark sector that is
produced by the imaginary parts of $V_{ub}$ and $V_{td}$, and, since all the
matrix elements in the second row and column, i.e., the ones that involve the
strange and charmed quarks, are real, it is not applicable to descriptions of
${\mathcal{C}}$${\mathcal{P}}$ violation the $c$\- and $s$-quark sectors. For
this, Wolfenstein provided a version of the matrix that is expanded to include
${\mathcal{C}}$${\mathcal{P}}$-violating terms of ${\mathcal{O}}(\lambda^{4})$
in $V_{ts}$ and ${\mathcal{O}}(\lambda^{5})$ in $V_{cb}$:

$V_{\rm
CKM}=\begin{pmatrix}1-{\textstyle\frac{1}{2}}\lambda^{2}&\lambda&A\lambda^{3}(\bar{\rho}-i\bar{\eta}))\\\
-\lambda(1+iA^{2}\lambda^{4}\bar{\eta})&1-{\textstyle\frac{1}{2}}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\bar{\rho}-i\bar{\eta})&-A\lambda^{2}(1+i\lambda^{2}\bar{\eta})&1\end{pmatrix}+{\mathcal{O}}(\lambda^{6}).$
(36)

An expression involving the same parameters that is exactly unitary is given
by Kobayashi in ref. Kobayashi:1994ps

##  5 ${\mathcal{C}}$${\mathcal{P}}$ violation in $b$-quark decays?

In the KM model, ${\mathcal{C}}$${\mathcal{P}}$ violation in neutral $K$-meson
decays, other than the $\varepsilon$ mass-matrix parameter, are mainly
produced by complex phases in the upper-left 2$\times$2 corner of the KM
matrix161616$V_{ub}$ can contribute to kaon decays via penguin diagrams, but
these contributions are suppressed by similar ${\mathcal{O}}(A\lambda^{4})$
factors. that, in Wolfenstein’s ${\mathcal{O}}(\lambda^{5})$ parameterization,
(eqn. 36) is the confined to the phase of $V_{cd}$, and is tiny:

$\arg(V_{cd})=A^{2}\lambda^{4}\bar{\eta}=5.9\times 10^{-4}\approx
0.03^{\circ}.$ (37)

In contrast, the CKM matrix element for charmless decays of $B$-mesons that
proceed via $b$ $\rightarrow$ $u$ transitions is $V_{ub}$, with a
${\mathcal{C}}$${\mathcal{P}}$ phase $\delta$ that (we now know) is $\delta$
$\approx$ $70^{\circ}$. If the kaon’s
direct-${\mathcal{C}}$${\mathcal{P}}$-parameter $\varepsilon^{\prime}$, caused
by a tiny, fraction of a degree phase, can be measured in the 20th century,
the observation of ${\mathcal{C}}$${\mathcal{P}}$ violations produced by a 70∘
phase in $B$-meson decay in the 21st century should be easy.

Wrong! There are a number of important differences between the neutral kaon
and $B$-meson systems that make the types of measurements that were used to
discover and elucidate the properties of ${\mathcal{C}}$${\mathcal{P}}$
violations in the kaon system inapplicable to the $B$-meson system.

$B$-mesons have a huge number of different decay channels.

In contrast to the $K$-meson system, where 99.98% of $K_{S}$ decays are to
either $\pi^{+}\pi^{-}$ or $\pi^{0}\pi^{0}$ final states, and 99.7% of $K_{L}$
decays are to either $\pi\ell\nu$ or $\pi\pi\pi$ final states, $B$-mesons have
hundreds of different decay modes almost all of which have, at best, fraction
of a percent level branching ratios.

The $B^{0}$-$\bar{B}^{0}$ mass eigenstates have very short, and nearly equal
lifetimes.

In the $K$-meson system, the $K_{S}$ and $K_{L}$ mass eigenstates have large
lifetime differences (0.1 ns vs. 52 ns, respectively), and an essentially pure
$K_{L}$ beam can be achieved by simply making a neutral beam line that is
longer than several $K_{S}$ proper decay lengths. In comparison, the
equivalent $B_{H}$ and $B_{L}$ mass eigenstates have very nearly equal
lifetimes of a mere 1.5 ps ($c\tau_{B}$ = 0.45 mm), and there is no
possibility for making a beamline of ${\mathcal{C}}$${\mathcal{P}}$-tagged
$B$-mesons, much less one that distinguishes between the two different
${\mathcal{C}}$${\mathcal{P}}$ values.

$B$-meson decays to final states that are eigenstates of
${\mathcal{C}}$${\mathcal{P}}$ are infrequent.

$K_{S}$ mesons decay almost exclusively to ${\mathcal{C}}$${\mathcal{P}}$-even
$\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ eigenstates; $K_{L}$ decays to
${\mathcal{C}}$${\mathcal{P}}$-odd $\pi^{+}\pi^{-}\pi^{0}$ and
$\pi^{0}\pi^{0}\pi^{0}$ eigenstates occur with branching fractions of 19.5%
and 12.5%, respectively. In contrast, the most prominent
${\mathcal{C}}$${\mathcal{P}}$-eigenstate decay mode for neutral $B$-mesons is
$B^{0}$$\rightarrow$$J/\psi K_{S}$, with a meager 0.045% branching fraction.

Moreover, as noted above, $V_{cb}$, which has no
${\mathcal{C}}$${\mathcal{P}}$-violating phase, has a magnitude that is an
order of magnitude larger than $V_{ub}$ and, thus, branching fractions for $b$
$\rightarrow$ $u$ mediated “non-charmed” decays of $B$ mesons are strongly
suppressed. As a result the prospects for finding and studying
${\mathcal{C}}$${\mathcal{P}}$ violations in the $B$-meson system looked
pretty hopeless.

###  5.1 Prospects for testing the KM ${\mathcal{C}}$${\mathcal{P}}$
mechanism: pre 1980

Sometime around 1979-80, Abraham Pais who, along with Murray Gell-Mann was
responsible for many of the fundamental theoretical discoveries in the early
days of flavor physics, discussed the prospects for
${\mathcal{C}}$${\mathcal{P}}$ measurements with charmed and beauty mesons in
a seminar at Rockefeller University in New York City, where he had this to say
about ${\mathcal{C}}$${\mathcal{P}}$ violations with heavy quarks Sanda:2009zz
:

> “ There is good news and bad news. The good news is that
> ${\mathcal{C}}$${\mathcal{P}}$ violation in a heavy meson system is quite
> similar to that of the $K$-meson system. The bad news is that there is
> little distinction like the $K_{S}$-$K_{L}$ mass eigenstates. For heavy
> meson systems, both lifetimes are short.”

In the audience was a young theorist Ichiro (Tony) Sanda, who recalls thinking
at that time Sanda:2009zz :

> “‘${\mathcal{C}}$${\mathcal{P}}$ violation in a heavy meson system is quite
> similar to that of the K meson system?’—How could anything as interesting as
> ${\mathcal{C}}$${\mathcal{P}}$ violation be so uninteresting.”

and he resolved to find a way to prove that Pais was wrong.

###  5.2 Tony Sanda’s great idea

At this same time, I was one of the founding members of the CLEO experiment
that was located on the Cornell University campus in upstate New York, about a
two-hour drive from New York City.171717At that time I was at the University
of Rochester, a two-hour drive in the opposite direction. The CESR
$e^{+}e^{-}$ collider was in its infancy and had a maximum instantaneous
luminosity of ${\mathcal{L}}$$\sim$ $5\times 10^{30}$cm-2s-1. We had just
discovered the $\Upsilon(4S)$ resonance CLEO:1980tem  and while running at
$E_{\rm cm}$ = 10.58 GeV, its peak energy, we could collect about 30
$\Upsilon(4S)$$\rightarrow$$B\bar{B}$ events/day (see Fig. 7a). This was, at
that time, the most prolific source of $B$ mesons in the world and we were
anxious to make good use of them. To this end, my Cornell colleagues invited
Sanda for some seminars. During his first seminar, the best strategy that
Sanda had to offer was a vague plan to search for a $\ell^{+}\ell^{+}$ vs.
$\ell^{-}\ell^{-}$ asymmetry in events of the type

$e^{+}e^{-}\rightarrow B^{0}\bar{B}^{0}\rightarrow\ell^{\pm}\ell^{\pm}+{\rm
anything},$ (38)

with the faint hope that somehow a measurable ${\mathcal{C}}$${\mathcal{P}}$
violation would show up. But this was very much like the frequently performed
$K^{0}(\tau)$$\rightarrow$$\pi^{-}\ell^{+}\nu$ vs.
$K^{0}(\tau)$$\rightarrow$$\pi^{+}\ell^{-}\bar{\nu}$ asymmetry measurements,
but without any of the above-listed advantages that make the neutral kaon
system so special. The experimenters in the audience, who were all hyped up to
do great and wondrous things with the $B$ mesons that they had worked so hard
to produce, were noticeably disappointed. When Sanda got back to New York
City, he felt under strong pressure to come up with something that was new and
unique to $B$ mesons.

Figure 7: a) CLEO’s 1980 measurements of $\sigma(e^{+}e^{-}$$\rightarrow$${\rm
hadrons})$ in the $\Upsilon(3S)~{}\&~{}\Upsilon(4S)$ resonance region (from
ref. CLEO:1980tem ). b) Tree diagrams for $B^{0}$$\rightarrow$$J/\psi K^{0}$;
$K^{0}$$\rightarrow$$K_{S}$ (or $K_{L}$) (upper) and for
$\bar{B}^{0}$$\rightarrow$$J/\psi\bar{K}^{0}$;
$\bar{K}^{0}$$\rightarrow$$K_{S}$ (or $K_{L}$) after it is produced via
$B^{0}$$\rightarrow$$\bar{B}^{0}$ mixing (lower). The $\bar{B}^{0}$ picks up a
common phase $\delta_{\rm com}=\Delta M\tau$ from the mixing and a
${\mathcal{C}}$${\mathcal{P}}$ violating phase
$\phi_{{\mathcal{C}}{\mathcal{P}}}$=$-2\beta$ from the two $V_{td}$ vertices
in the mixing box.

A few months later, in his second seminar at Cornell, he did just that. He
proposed a scheme that he developed in collaboration with Ashton Carter
Carter:1980hr ; Carter:1980tk  for using interference between the
$B^{0}$$\rightarrow$$K_{S}J/\psi$ &
$B^{0}$$\rightarrow$$\bar{B}^{0}$$\rightarrow$$K_{S}J/\psi$ decay amplitudes
that eventually became the primary motivation for the BaBar and Belle
asymmetric $B$-factory experiments and was the basis for Kobayashi and
Maskawa’s Nobel prize.

The idea, which is illustrated in Fig. 7 b, was very elegant. You start with a
flavor-tagged $B^{0}$ (or $\bar{B}^{0}$—here I use a tagged $B^{0}$ to
illustrate the idea) that can directly decay via the
$B^{0}$$\rightarrow$$K^{0}J/\psi$ diagram show in the top panel, or decay via
the indirect route where it first mixes into a $\bar{B}^{0}$ that then decays
via $\bar{B}^{0}$$\rightarrow$$\bar{K}^{0}J/\psi$. But experiments don’t
distinguish between $K^{0}$ or $\bar{K}^{0}$ decays, instead they measure
$K_{S}$ and $K_{L}$ decays. Thus, in events where a $K_{S}$ is detected, the
direct and indirect decay routes access identical final states and interfere.
(Likewise for events where a $K_{L}$is detected, except here the interference
term has an opposite sign.)

The direct amplitude has no ${\mathcal{C}}$${\mathcal{P}}$ phase (at least not
at leading order), but the indirect amplitude has an extra factor of
$V_{td}^{2}$ (not $|V_{td}|^{2}$!) and so, a ${\mathcal{C}}$${\mathcal{P}}$
phase of $-2\beta$. For tagged $\bar{B}^{0}$ decays, the CKM factor is
$V_{td}^{*2}$ and the ${\mathcal{C}}$${\mathcal{P}}$ phase is $+2\beta$. Thus,
the $\bar{B}^{0}(\tau)$$\rightarrow$$f_{{\mathcal{C}}{\mathcal{P}}}$ vs.
$B^{0}(\tau)$$\rightarrow$$f_{{\mathcal{C}}{\mathcal{P}}}$ time-dependent
asymmetry, where $f_{{\mathcal{C}}{\mathcal{P}}}$ is any
${\mathcal{C}}$${\mathcal{P}}$-eigenstate is

${\mathcal{A}}^{{\mathcal{C}}{\mathcal{P}}}_{B\rightarrow
f_{{\mathcal{C}}{\mathcal{P}}}}(\tau)=\frac{\bar{\Gamma}_{\bar{B}^{0}\rightarrow
f_{{\mathcal{C}}{\mathcal{P}}}}(\tau)-\Gamma_{B^{0}\rightarrow
f_{{\mathcal{C}}{\mathcal{P}}}}(\tau)}{\bar{\Gamma}_{\bar{B}^{0}\rightarrow
f_{{\mathcal{C}}{\mathcal{P}}}}(\tau)+\Gamma_{B^{0}\rightarrow
f_{{\mathcal{C}}{\mathcal{P}}}}(\tau)}=-\xi_{{\mathcal{C}}{\mathcal{P}}}\sin(2\beta)\sin(\Delta
M_{B}\tau),$ (39)

where $\xi_{{\mathcal{C}}{\mathcal{P}}}$ (=$-$1 for $K_{S}J/\psi$ and +1 for
$K_{L}J/\psi$) is the ${\mathcal{C}}$${\mathcal{P}}$ eigenvalue of
$f_{{\mathcal{C}}{\mathcal{P}}}$, $\Delta M_{B}$ = $M_{H}$-$M_{L}$ is the mass
difference between the neutral $B_{H}$ and $B_{L}$ mass eigenstates (i.e., the
$B^{0}$-$\bar{B}^{0}$ mixing frequency), and $\tau$ is the proper time between
the $B^{0}$$\rightarrow$$KJ/\psi$ ($B_{{\mathcal{C}}{\mathcal{P}}}$) decay and
the flavor-specific decay of the accompanying $\bar{B}^{0}$ meson ($B_{\rm
tag}$), whose decay products are used to tag the $B^{0}$ meson’s flavor. Note
that in $e^{+}e^{-}$ colliders, $\tau$ can be positive (when the
$B_{{\mathcal{C}}{\mathcal{P}}}$ decay occurs after the $B_{\rm tag}$ decay),
or negative (when the $B_{{\mathcal{C}}{\mathcal{P}}}$ decay occurs first),
and the time-integrated asymmetry is zero.

####  5.2.1 Great idea! but is it practical?

The idea was new, and the mechanism was unique to $B$ mesons, but there were
many pieces that had to fall into just the right places for Sanda’s proposal
to have any chance of being practical. Since at that time there was no
experimental information about the $b$-quark-related CKM elements or
$B^{0}$-$\bar{B}^{0}$ mixing, there was no way to form any opinion about the
prospects for their favorability.

Tens of millions of tagged $B$$\rightarrow$$f_{{\mathcal{C}}{\mathcal{P}}}$
decays would be required:

In 1980, the world’s best source of $B$-mesons was CESR, with a production
rate of $\sim$30 $B\bar{B}$ events/day, of which only half were the desired
$B^{0}\bar{B}^{0}$ pairs. Sanda’s golden mode was
$B^{0}(\bar{B}^{0})$$\rightarrow$$K_{S}J/\psi$, which he estimated to have an
${\mathcal{O}}(10^{-3})$ branching fraction, and this implied that the
fractional probability of usable events would be

${\mathcal{F}}_{K_{S}J/\psi}<\underbrace{{\cal B}(B^{0}\rightarrow
K_{S}J/\psi)}_{\sim 10^{-3}}\underbrace{{\cal
B}(K_{S}\rightarrow\pi^{+}\pi^{-}){\cal B}(J/\psi\rightarrow\ell\ell)}_{\sim
10^{-1}}\underbrace{(\epsilon_{\rm trk})^{4}\epsilon^{\rm tag}_{\rm
eff}}_{\sim 10^{-1}}\approx 10^{-5},$ (40)

where $\epsilon_{\rm trk}$ is the efficiency for charged track detection that,
even in a nearly perfect detector, cannot be much higher than $\epsilon_{\rm
trk}$ $\approx$ 0.85, and $\epsilon^{\rm tag}_{\rm eff}$ $\approx$ 0.3 is an
estimate of the maximum possible effective ${\mathcal{B}}$-flavor tagging
efficiency. Thus, an ${\mathcal{A}}^{{\mathcal{C}}{\mathcal{P}}}_{{KJ/\psi}}$
measurement with even modest precision would require $\sim$30 M $B\bar{B}$
events (and a million days of operation at 1980 state of the art collider and
detector performance levels).

$B^{0}$-$\bar{B}^{0}$ mixing had to be substantial, $|V_{cb}|$ had to be
small, and $|V_{ub}|$ even smaller:

An essential part of the Sanda-Carter scheme is that the fraction of $B^{0}$
mesons that oscillate into a $\bar{B}^{0}$ before they decay has to be
reasonably large. This meant that $\Delta M_{B}$ would have to be similar in
magnitude to $\Gamma_{B}$ = $1/\tau_{B}$, which was then known to be
$\Gamma_{B}$$\approx$ $4.4\times 10^{-10}$ MeV. In the 1980s, when the
$t$-quark mass was (almost universally) expected to be $m_{t}$ $\sim$ 35 GeV,
calculations Buras:1984pq  found $\Delta M_{B}$ $\approx$ $1.2\times 10^{-10}$
MeV. If this were the case, only $\sim$6% of the tagged $B^{0}$-mesons would
oscillate into a $\bar{B}^{0}$ before decaying. (After three lifetimes, the
$\sin\Delta M_{B}\tau$ factor in eqn. 39 would have barely reached 0.5.) In
addition the $B$ lifetime had to be relatively long: i.e., $|V_{cb}|$$<$0.1,
and $|V_{ub}|$$<$$|V_{cb}|$.

The time sequence between the tag- and $f_{{\mathcal{C}}{\mathcal{P}}}$-decay
has to be distinguished:

The asymmetry in eqn. 39 has opposite signs for negative and positive values
of $\tau$, which makes it essential to distinguish between events in which the
$B_{{\mathcal{C}}{\mathcal{P}}}$ decays occurred first from those when it
decayed last. The $B$ mesons that are produced in
$\Upsilon(4S)$$\rightarrow$$B\bar{B}$ decays have c.m. momenta $p_{B}$ = 327
MeV/$c$, corresponding to $\gamma\beta$ = $0.062$ and have a mean decay
distance of $\beta\gamma c\tau_{B}$ = 28$\mu$m, which is unmeasurably small in
a c.m. $e^{+}e^{-}$ collider environment. For a collider operating at the
${\Upsilon}(4S)$, it would be impossible to distinguish the time sequence of
the $B_{{\mathcal{C}}{\mathcal{P}}}$ and $B_{\rm tag}$ decays.

Since the existence of six-quarks was pretty well established, the KM-
mechanism provided a compelling and almost obvious mechanism for explaining
the existence of ${\mathcal{C}}$${\mathcal{P}}$ violation. However the
prospects or a conclusive experimental test of this idea seemed hopeless. The
fortuitous set of circumstances that made studies of
${\mathcal{C}}$${\mathcal{P}}$ violation in the neutral kaon system possible
seemed unlikely to be repeated.

###  5.3 Three miracles

Nevertheless, in spite of these obstacles, Sanda maintained a nearly mystical
belief that “Mother Nature has gone out of Her way to show us
${\mathcal{C}}$${\mathcal{P}}$ violation, and She will also show us the way to
the fundamental theory” Sanda:2009zz , and forcefully advocated an aggressive
program of experimental investigations of ${\mathcal{C}}$${\mathcal{P}}$
violation in the decays of $B$ mesons. However, for the reasons itemized
above, Sanda’s advocacy was initially met with considerable skepticism from
his colleagues in both the theoretical and experimental physics communities.

And then three miracles occurred:

Miracle 1: ${\bm{B^{0}}}$-${\bm{\bar{B}^{0}}}$ mixing was discovered at DESY:

The most exciting event in flavor physics during the 1980s was the 1987
discovery of a large signal for $B^{0}$-$\bar{B}^{0}$ mixing by the ARGUS
experiment at DESY ARGUS:1987xtv . The strength of the mixing was clear
evidence the that top-quark mass, now known to be 173 GeV, was nearly an order
of magnitude larger than expected, which was shocking news to almost everyone.
This discovery, coupled with the 1.5 ps $B$-meson life-time measurements from
PEP and PETRA that translated into $|V_{cb}|$$\approx$ 0.04, and the
suppression of $b$$\rightarrow$$u$ relative to $b$$\rightarrow$$c$ transitions
meant that $|V_{ub}|$ was about a factor of ten smaller than $|V_{cb}|$. These
measurements confirmed Sanda’s strong belief that Mother Nature would indeed
help us “find the way to the fundamental theory.”

Miracle 2: Three-order-of-magnitude improvement in ${\bm{e^{+}e^{-}}}$
collider luminosity:

Advances in the understanding and modeling of beam dynamics, the use of
separate magnet rings that enabled multibunch collisions, and major advances
in RF feedback systems provided the huge increases in the
$e^{+}e^{-}$$\rightarrow$$\Upsilon(4S)$ production rate that were required by
the experiment Funakoshi:2014yfa ; Seeman:2009zza .

Miracle 3: The invention of asymmetric $e^{+}e^{-}$ colliders and innovations
in detector technology:

Pierre Oddone realized that an $e^{+}e^{-}$ collider operating at the
$\Upsilon(4S)$ resonance with a modest (i.e., factor of $\sim$2) difference
between the $e^{+}$ and $e^{-}$ beam energies would produce boosted $B$-mesons
with ${\mathcal{O}}(100\,\mu{\rm m})$ separation distances between the $B_{\rm
tag}$ and $B_{{\mathcal{C}}{\mathcal{P}}}$ vertices Oddone:1987up . This idea,
coupled with the concurrent development of high resolution silicon-strip
vertex detectors that could measure such small displacements in a collider
environment Litke:1987xy ; Batignani:1990jv ; BaBar:1999vnr ; Belle:2000mhy ,
offered a realistic solution to the decay-time-sequence determination problem.
Parallel improvements in detector performance levels in areas of particle
identification DIRC:1999mex ; Iijima:2000uv , and $\gamma$-ray & $K_{L}$
detection BaBar:2001yhh ; Belle:2000cnh ; Belle:2000jde  advanced the state of
the art levels of detection efficiencies and ${\mathcal{B}}$-flavor-tagging
quality.

##  6 First measurements of the KM angle $\beta$

At leading order, measurements of the eqn. 39 Carter-Sanda asymmetry
determines $\sin 2\beta$, where $\beta$ =
$\tan^{-1}\big{(}\bar{\eta}/(1-\bar{\rho})\big{)}$ and $\bar{\eta}$ &
$\bar{\rho}$ are the modified Wolfenstein parameters described in Section
4.2.2. Prior to the summer of 2001, the best measurements of $\sin 2\beta$
were from CDF CDF:1999ijp  ($0.79\pm 0.44$), BaBar BaBar:2001ags  ($0.34\pm
0.21$) and Belle Belle:2001qdd  ($0.58\pm 0.34$). The BaBar result was based
on a sample of 23 M ${\Upsilon}(4S)$$\rightarrow$$B\bar{B}$ events and Belle,
which was struggling with electron cloud effects in the KEKB positron ring
Wang:2002ds , had a smaller data sample of 11 M
${\Upsilon}(4S)$$\rightarrow$$B\bar{B}$ events. Each of the three measurements
were about $1.5\sigma$ from zero, and their weighted average, $0.46\pm 0.17$
indicated a non-zero ${\mathcal{C}}$${\mathcal{P}}$ violation at the
$\sim$2.5$\sigma$ level. The situation was tantalizing, but not conclusive.

This changed in August 2001 when, in back-to-back articles in Physical Review
Letters, BaBar, now with a sample of 32 M $B\bar{B}$ pairs, reported
BaBar:2001pki

$\sin 2\beta=0.59\pm 0.14\pm 0.05~{}~{}~{}~{}{\rm BaBar~{}(2001)},$ (41)

and Belle, with a 31 M $B\bar{B}$ pair data sample, reported Belle:2001zzw

$\sin 2\beta=0.99\pm 0.14\pm 0.06~{}~{}~{}~{}{\rm Belle~{}(2001)}.$ (42)

The BaBar data sample contained 803 $B_{{\mathcal{C}}{\mathcal{P}}}$ event
candidates with a signal purity of about 80%. The top three panels in Fig. 8a
show the BaBar experiment’s measured time distributions for
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ $B_{{\mathcal{C}}{\mathcal{P}}}$
decays. The uppermost plot shows the number of $B^{0}$ tags where it is
evident that there are more events with $\tau$$>$0 than with $\tau$$<$0, while
the $\bar{B}^{0}$ tags in the panel beneath it display an opposite pattern.
The third panel shows the bin-by-bin asymmetry where there is a clear
indication of the sine-like behavior that is expected for a
${\mathcal{C}}$${\mathcal{P}}$ violation as given in eqn. 39. The three lowest
panels show the corresponding results for $B\rightarrow$$K_{L}$$J/\psi$ decays
with $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$, where the $\tau$-dependent
asymmetries have opposite signs, again as expected.

Figure 8: a) BaBar results: the top three plots show the proper time
distributions and fit results for $B^{0}$-tags, $\bar{B}^{0}$-tags and their
asymmetry for $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$
$B_{{\mathcal{C}}{\mathcal{P}}}$ decays. The bottom three plots show the same
distributions for $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$
$B_{{\mathcal{C}}{\mathcal{P}}}$ decays. The shaded areas indicate the
background levels. (From ref. BaBar:2001pki ). b) Belle results: the top three
plots on the left show the time-dependent asymmetry and fit results for: (from
top down) the combined ($\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ events minus
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$ events) samples; the
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ (mostly $K_{S}J/\psi$) events; and the
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=+1 ($K_{L}J/\psi$) events. The bottom plot
shows results for non-${\mathcal{C}}$${\mathcal{P}}$-eigenstate decay modes
where no asymmetry is expected. The open circles and dashed curve on the right
show background-subtracted time distributions for
$q\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ events and the fit results. The solid
circles and curve show the same quantities for
$q\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$ events. (From ref. Belle:2001zzw ).

Figure 8b shows the results from the Belle experiment Belle:2001zzw  that used
747 $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ $B$-decay candidates (mostly
$B$$\rightarrow$$K_{S}J/\psi$) with 92% purity and 569
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$ $B$$\rightarrow$$K_{L}J/\psi$ decay
candidates with 61% purity. The top plot on the left side of the panel shows
the proper-time distribution for the $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$
modes minus that for $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$ modes. The 2nd
and 3rd panels from the top show the $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$
and $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$ modes separately, where the
different ${\mathcal{C}}$${\mathcal{P}}$ modes have opposite-sign asymmetries
as expected. The curves show the results of fits to the data. The bottom panel
shows the asymmetry for a large sample of self-tagged
(non-${\mathcal{C}}$${\mathcal{P}}$ eigenstate) $B$ decays
($B^{0}$$\rightarrow$$D^{(*)-}\pi^{+}$, $D^{*-}\rho^{+}$,
$K^{*0}(K^{+}\pi^{-})J/\psi$, and $D^{*-}\ell^{+}\nu$), where a non-zero
asymmetry would have to be due to instrumental effects; the fit to this sample
returned an asymmetry amplitude of $0.05\pm 0.04$.

The open circles in the plot on the right side of Fig. 8b show the time
distribution for $\bar{B}^{0}$-tags ($q$=1) in
$\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$ $B_{{\mathcal{C}}{\mathcal{P}}}$
decays plus $B^{0}$-tags ($q$=-1) in $\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$
decays (i.e., $q\xi_{{\mathcal{C}}{\mathcal{P}}}$=$-1$), with the fit results
shown as a dashed curve. The black dots and solid curve show the sum of the
opposite combinations ($q\xi_{{\mathcal{C}}{\mathcal{P}}}$=$+1$) and their fit
result.

The BaBar and Belle results excluded a zero value for $\sin 2\beta$ by
$4\sigma$, and $6\sigma$, respectively, and their combined significance
established conclusively that ${\mathcal{C}}$${\mathcal{P}}$ symmetry is
violated in the $B$-$\bar{B}$ mixing process as as predicted by the KM six-
quark model. Makoto Kobayashi and Toshihide Maskawa shared half of the 2008
Nobel Prize in Physics “for the discovery of the origin of the broken symmetry
which predicts the existence of at least three families of quarks in nature.”
The Nobel committee’s remarks that accompanied their announcement included the
following:

> “[Kobayashi and Maskawa] explained broken symmetry within the framework of
> the Standard Model, but required that Model be extended to three families of
> quarks. These predicted, hypothetical new quarks have recently appeared in
> physics experiments. As late as 2001, the two particle detectors BaBar at
> Stanford, USA and Belle at Tsukuba, Japan, both detected broken symmetries
> independently of each other. The results were exactly as Kobayashi and
> Maskawa had predicted almost three decades earlier.”

During the two decades following the Belle and BaBar reports, there have been
many hundreds of measurements of non-zero ${\mathcal{C}}$${\mathcal{P}}$
violating symmetries in $B$ meson decays, mostly by BaBar and Belle, which
continued operating until 2008 and 2010, respectively. This program is being
continued by the LHCb Belyaev:2021cyr , an experiment specialized for heavy
flavor physics at the CERN Large Hadron Collider that began operating in 2010,
and Belle II Belle-II:2010dht , an upgraded version of Belle at KEK that began
operating in 2018. As is discussed in more detail in other contributions to
this symposium, all of these hundreds of measured
${\mathcal{C}}$${\mathcal{P}}$ violations are well explained as being due to
the effects of the single KM ${\mathcal{C}}$${\mathcal{P}}$-phase angle
$\delta$.

The KM model has been a remarkable success.

##  7 A few comments on mixing and ${\mathcal{C}}$${\mathcal{P}}$ violation in
the neutrino sector

As mentioned above, the two-doublet nature of the of the leptons was
identified in 1961 Danby:1962nd , a decade before it was established for
quarks. The notion of neutrino-mixing was first proposed four years earlier by
Pontecorvo Pontecorvo:1957qd  in 1957 and the PMNS neutrino mixing matrix was
suggested Maki, Nakagawa and Sakata Maki:1962mu  in 1962, a year before
Cabibbo’s paper appeared. When the $\tau$-lepton was discovered in 1975, the
six-lepton picture was established and the PMNS matrix for neutrinos expanded
to the same 3$\times$3 structure as the CKM matrix for quarks. If neutrinos
are Dirac particles, the mathematics of the neutrino-flavor mixing matrix are
the same as the KM matrix with three mixing angles $\theta_{ij}$ and one
${\mathcal{C}}$${\mathcal{P}}$ violating phase
$\delta_{{\mathcal{C}}{\mathcal{P}}}$, the so-called the “Dirac” phase. If
neutrinos are Majorana particles, lepton number is not conserved and the
matrix’s number of degrees of freedom increases by two and there are two
additional ${\mathcal{C}}$${\mathcal{P}}$-violating “Majorana” phases
Schechter:1980gr  that have no measurable effects on neutrino mixing
experiments (which makes them hard to measure). The commonly used
parameterization of the PMNS matrix that doesn’t include the Majorana phases
uses same mixing angle definitions as the eqn. 4.2.1) Chau-Keung version of
the CKM matrix, but with the sequence of rotations reversed, i.e.,

$\begin{pmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{pmatrix}=\stackrel{{\scriptstyle\rm
Atmospheric}}{{\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix}}}\stackrel{{\scriptstyle\rm
Reactor~{}experiments}}{{\begin{pmatrix}c_{13}&~{}0~{}&e^{-i\delta_{{\mathcal{C}}{\mathcal{P}}}}s_{13}\\\
0&~{}1~{}&0\\\
-e^{-i\delta_{{\mathcal{C}}{\mathcal{P}}}}s_{13}&~{}0~{}&c_{13}\end{pmatrix}}}\stackrel{{\scriptstyle\rm``Solar"}}{{\begin{pmatrix}c_{12}&s_{12}&~{}0~{}\\\
-s_{12}&c_{12}&~{}0~{}\\\ 0&&~{}1~{}\end{pmatrix}}}\begin{pmatrix}\nu_{1}\\\
\nu_{2}\\\ \nu_{3}\end{pmatrix},$ (43)

where $\nu_{1},\nu_{2},\nu_{3}$ denote the three neutrino mass eigenstates and
$\theta_{12}$, $\theta_{23}$, and $\theta_{13}$ are the “solar,”
“atmospheric,” and “reactor” neutrino mixing angles. The explicit form of the
matrix is:181818${\mathcal{C}}$${\mathcal{P}}$-violating Majorana phases
$\psi_{1}$ & $\psi_{2}$ can be included by $U^{\rm Maj}_{\rm PMNS}$ = $U^{\rm
Dirac}_{\rm PMNS}{\mathbb{P}}$, where ${\mathbb{P}}$ =
$\begin{pmatrix}e^{i\psi_{1}}&0&0\\\ 0&e^{i\psi_{2}}&0\\\ 0&0&1\end{pmatrix}$.

$\displaystyle U^{\rm Dirac}_{\rm PMNS}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}~{}U_{e1}&~{}U_{e2}&~{}U_{e3}\\\ ~{}U_{\mu
1}&~{}U_{\mu 2}&~{}U_{\mu 3}\\\ ~{}U_{\tau 1}&~{}U_{\tau 2}&~{}U_{\tau
3}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}~{}c_{12}c_{13}&~{}s_{12}c_{13}&~{}s_{13}e^{-i\delta_{{\mathcal{C}}{\mathcal{P}}}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{{\mathcal{C}}{\mathcal{P}}}}&~{}c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{{\mathcal{C}}{\mathcal{P}}}}&~{}s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{{\mathcal{C}}{\mathcal{P}}}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{{\mathcal{C}}{\mathcal{P}}}}&~{}c_{23}c_{13}\end{pmatrix},$

and the PDG 2020 Zyla:2020zbs  world averages for the three rotation angles
are:

$\displaystyle\sin^{2}\theta_{12}$ $\displaystyle=$ $\displaystyle 0.307\pm
0.013~{}~{}~{}~{}\Rightarrow\theta_{12}=33.6^{\circ}\pm 0.8^{\circ}$ (45)
$\displaystyle\sin^{2}\theta_{23}$ $\displaystyle=$ $\displaystyle 0.545\pm
0.021~{}~{}~{}~{}\Rightarrow\theta_{23}=47.6^{\circ}\pm 1.2^{\circ}$
$\displaystyle\sin^{2}\theta_{13}$ $\displaystyle=$ $\displaystyle 0.0218\pm
0.007~{}~{}\Rightarrow\theta_{13}=8.48^{\circ}\pm 0.14^{\circ}.$

Although mixing in the quark- and lepton-sectors have the same mathematical
structure, there are important differences in their practical applications,
including:

Figure 9: a) The area of the squares illustrates the magnitudes of the CKM
(left) and PMNS (right) matrix elements. b) A not-to-scale illustration of the
normal and (left) and inverted (right) neutrino mass hierarchy. The flavor
content of each of the $\nu_{1},\nu_{2},\nu_{3}$ mass eigenstate is indicated
by different shades of gray.

Strikingly different hierarchies: The three PMNS mixing angles listed above
differ in values by at most a factor of five, in contrast with the the CKM-
matrix where the corresponding mixing angles are smaller and span a two-order-
of-magnitude range in magnitudes:

$\theta^{\rm CKM}_{12}\approx 13.0^{\circ}~{}~{}~{}~{}\theta^{\rm
CKM}_{23}\approx 2.4^{\circ}~{}~{}~{}~{}\theta^{\rm CKM}_{13}\approx
0.20^{\circ}.$ (46)

Differences between the two matrices are illustrated in Fig. 9a, where the
areas of the squares are proportional to the magnitudes of the matrix
elements. Here Nature has been helpful again; if the PMNS matrix had a
hierarchy that was similar to that of the CKM matrix, neutrino oscillations
and neutrino masses would likely not have been discovered.

They operate in opposite “directions:” In the CKM case, the quark mass-
eigenstates are well known and the matrix is used to determine the flavor
states. For the PMNS case, the neutrino flavor states are well known and the
matrix is used to determine the mass eigenstates. Oscillation experiments have
determined the mass-difference hierarchies shown in Fig. 9b, where
Zyla:2020zbs

${\rm atmos:}~{}\Delta m^{2}_{32}=\pm(2.44\pm 0.03)\times
10^{-3}{\rm(eV)^{2}}~{}~{}~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}~{}~{}{\rm
solar:}~{}\Delta m^{2}_{21}=(7.53\pm 0.18)\times 10^{-5}{\rm(eV)^{2}}.$

The still unknown sign of $\Delta m^{2}_{32}$ (= $m^{2}_{3}$-$m^{2}_{2}$)
leaves two possible hierarchies as shown in Fig. 9b.

Quarks decay, neutrinos (probably) don’t: CKM-related measurements are almost
entirely based on measurements of decay processes with a formalism for
oscillations and ${\mathcal{C}}$${\mathcal{P}}$ violations is done in the
hadron’s restframe. In contrast case, PMNS-related measurement always involve
production processes that produce pure flavor states and detection experiments
located at some baseline distance that determine how the neutrino flavor-
contents changed during their propagation to the detector. Since the
neutrino’s restframe is ill defined,191919If the lightest neutrino has zero
mass it doesn’t have a restframe. the formalism is usually done in the
laboratory frame and expressed in terms of $E_{\nu}$\- and $\Delta
m^{2}$-dependent oscillation lengths, i.e.,

$l(E_{\nu})\equiv\lambda/2\pi=E_{\nu}/1.27\Delta m^{2},$ (47)

where the factor of 1.27 is specific to ($l,E_{\nu},\Delta m^{2}$) units of
(m, MeV, eV2), or (km, GeV, eV2). For the atmospheric and solar mass
differences given above, these are

$l^{\rm atm}(E_{\nu})\approx 320~{}{\rm(km/GeV)}\times
E_{\nu}~{}~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}l^{\rm sol}(E_{\nu})\approx
10,500~{}{\rm(m/MeV)}\times E_{\nu},$

where the units km/GeV and m/MeV are equivalent (and interchangeable), with
km/GeV usually attached to atmospheric and m/MeV to solar for historical
reasons.

The 320 km atmospheric length is long, but not impossibly long. It is nearly
the same as the 295 km distance between J-PARC and Super Kamiokande (and the
soon-to-be-completed HyperK detector), and corresponds to a 90∘ phase-change-
induced oscillation maximum for 600 MeV muon-neutrinos that can be copiously
produced by the J-PARC synchrotron. The 10.5 km solar length corresponds to an
oscillation maximum for $E_{\nu}$= 3 MeV reactor anti-electron-neutrinos at a
baseline of 50 km, which is the baseline for the JUNO reactor neutrino
experiment that will soon be operating in China. The HyperK Hyper-
Kamiokande:2018ofw  and JUNO JUNO:2015sjr  experiments will certainly not be
easy, but they will be done. If the neutrino oscillation lengths were factors
of two or more longer, both experiments would be much more difficult, if not
impossible.

##  8 Conclusions

The related subjects of ${\mathcal{C}}$${\mathcal{P}}$-violation and quark &
neutrino flavor mixing provide peeks into some of Nature’s most intimate
secrets. When Fitch and collaborators measured the $\sim$0.2% branching
fraction for $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ in 1963, they were seeing
the influence of the $t$-quark that wasn’t discovered until thirty years later
(see Fig. 4). Thanks to Kobayashi and Maskawa, the existence and most of the
properties of the $t$-quark (other than its mass) were pretty well understood
more than twenty years before it was discovered.

###  8.1 Nature has been kind

(Einstein famously said “The Lord God is subtle, but not malicious.”)

A common thread that characterizes this entire story is that we have been able
to probe these subjects at considerable depth, even though it didn’t a priori
have to be that way. The 0.2% $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ branching
fraction is as large as it is because of the phase-space suppression of the
partial decay width for the ${\mathcal{C}}$${\mathcal{P}}$-allowed
$K_{L}$$\rightarrow$$\,3\pi$ mode that has a $Q$-value of only 83 MeV. This is
a consequence of the relative masses of the $K$\- and $\pi$-mesons: if the
kaon mass were higher and/or the pion mass were lighter, the
$K_{L}$$\rightarrow$ $3\pi$ partial width would be larger, and the
$K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ branching fraction would be reduced. It
wouldn’t take very large changes in these masses to push the $\pi^{+}\pi^{-}$
branching fraction down to a value that was below the Fitch experiment’s
sensitivity level. Unlike parity violation, which is a huge effect that is
difficult to miss, the particle physics consequences of
${\mathcal{C}}$${\mathcal{P}}$-violations in processes other than $K_{L}$
decays have only been seen in elaborate, highly focused experiments that,
absent the $K_{L}$ measurements, very likely wouldn’t have occurred.

As described above, the KM phase was only measurable because Nature’s
parameters are aligned in a way that meet the stringent requirements that were
first identified in the Carter-Sanda papers, including a strong
$|V_{us}|$$>$$|V_{cb}|$$>$$|V_{ub}|$ hierarchy and a large enough $t$-quark
mass to produce a $B$-$\bar{B}$ mixing rate that is close to the $B$-meson
lifetime, but not so large that the mixing rate was much faster than the
lifetime. As a result, the phase could be measured, as could all three of the
CKM matrix’s mixing angles.

Remarkably, the same thread extends into the neutrino sector in which the PMNS
matrix has a very different, almost flat hierarchy that facilitated the
discovery of neutrino oscillations. This hierarchy has also enabled precise
measurements of all three of its mixing angles and will likely allow for a
determination of its Dirac phase in the not-so-distant future. Moreover, and
as mentioned above, Nature’s choices of the differences between the
$\nu_{1},\nu_{2},\nu_{3}$ eigenstate masses have made these measurements
realizable.

###  8.2 What’s next?… ${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$?

Although ${\mathcal{C}}$${\mathcal{P}}$ violations only show up as small,
subtle effects in particle physics experiments, the influence of
${\mathcal{C}}$${\mathcal{P}}$ violations on the evolution of the Universe is
glaringly obvious Sakharov:1967dj . However, the
${\mathcal{C}}$${\mathcal{P}}$ violations that we measure in $K$-, $B$\- and
(recently LHCb:2019hro ) $D$-meson decays that are produced by the KM
mechanism for quarks and, likely, leptons, cannot nearly account for the
matter-antimatter asymmetry of the Universe (see, e.g., ref. Canetti:2012zc ).
There must be other sources of C${\mathcal{P}}$ violation that have yet to be
discovered, and these are the primary motivations for the LHCb and Belle II
experiments. With the KM phase, Nature has given us a glimpse of
${\mathcal{C}}$${\mathcal{P}}$ violation, but not the whole story.

But what about ${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$? The
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ theorem Schwinger:1951xk  states
that any quantum field theory that is Lorentz invariant, has local point-like
interaction vertices, and is Hermitian (i.e., conserves probability) is
invariant under the combined operations of ${\mathcal{C}}$, ${\mathcal{P}}$
and ${\mathcal{T}}$. Since the three QFTs that make up the Standard Model—QED,
QCD, and Electroweak theory—all satisfy these criteria,
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ symmetry has been elevated to a
kind of hallowed status in particle physics. But the non-renormalizability of
quantm gravity calls into question the validity of the locality assumption
Weinberg:1980kq , and suggests that at some scale,
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ has to be violated. Strictly
speaking, this violation only has to occur at impossibly high energies near
the ${\mathcal{O}}(10^{19}\,{\rm GeV})$ Planck scale, but maybe the same
thread of Nature that gives us a taste of ${\mathcal{C}}$${\mathcal{P}}$
violations at energy scales well below the scale needed to explain the baryon
asymmetry of the Universe, will give us a hint of
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ violation at scale below the one
that’s needed to rescue quantum gravity. In any case, since it is a
fundamental feature of the Standard Model,
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ invariance should be routinely
challenged at the highest feasible experimental sensitivities.

To date the most stringent test of the
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ prediction that particle-
antiparticle masses are equal comes from kaon physics202020The
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ test in kaon physics involves a
comparison the phase of the $\eta_{+-}$ ${\mathcal{C}}$${\mathcal{P}}$
violation parameter in $K_{L}$$\rightarrow$$\pi^{+}\pi^{-}$ decays with the
“superweak phase” $\phi_{\rm
SW}$$\equiv$$\tan^{-1}\big{(}2(M_{K_{L}}$$-$$M_{K_{S}})/(\Gamma_{K_{S}}$$-$$\Gamma_{K_{L}})\big{)}$.
Schwingenheuer:1995uf ; Apostolakis:1999zw  and sets a 90% C.L. limit on the
$K^{0}$-$\bar{K}^{0}$ mass difference of

$|M_{\bar{K}^{0}}-M_{K^{0}}|<5\times 10^{-19}~{}{\rm GeV},$ (48)

which is seven orders-of-magnitude more stringent than that for
$m_{\bar{e}}$$-$$m_{e}$ and nine orders-of-magnitude more stringent than the
$m_{\bar{p}}$$-$$m_{p}$ limit. This high sensitivity is because of the magic
of the virtual processes in the Fig. 4a box diagram and the unique properties
of the neutral kaons.

The kaon result is based on experiments done nearly thirty years ago with data
samples containing tens of millions of $K$$\rightarrow$$\pi^{+}\pi^{-}$
decays. One of the reasons these measurements have not been updated since then
is that technologies for improved sources of flavor-tagged neutral kaons have
not been pursued. However, if the above-described three order of magnitude
improvements in $e^{+}e^{-}$ collider luminosity that were developed for the
$B$-factories would be applied to a dedicated collider operating at $J/\psi$
mass peak, multi-billion-event/year samples of flavor-tagged neutral kaons,
produced via $J/\psi$$\rightarrow$$K^{\mp}\pi^{\pm}K^{0}(\bar{K}^{0})$ decays,
would be available to support ${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$
tests with more than an order-of-magnitude improved sensitivity. More modest
improvements in ${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ sensitivity
would be provided by new colliders in the $\tau$-charm energy range that are
being proposed in China Achasov:2023gey  and Russia Barnyakov:2020vob , if
they spend sufficient time operating at the $J/\psi$ peak.

Maybe during the next sixty years,
${\mathcal{C}}$${\mathcal{P}}$${\mathcal{T}}$ violation studies will prove to
be as interesting and provocative as ${\mathcal{C}}$${\mathcal{P}}$ studies
have been during the past sixty years.

## Acknowledgment

I thank the organizers of KM50 for inviting me to participate in this
interesting symposium, the editors of PTEP for inviting this manuscript, and
this paper’s referee, whomever he or she may be, for many important
corrections and helpful suggestions. This work was supported in part by the
National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335.

## References

  * (1) R. P. Feynman and Murray Gell-Mann, Phys. Rev., 109, 193–198 (1958).
  * (2) Nicola Cabibbo, Phys. Rev. Lett., 10, 531–533 (1963).
  * (3) Byron P. Roe, Daniel Sinclair, John L. Brown, Donald A. Glaser, John A. Kadyk, and George H. Trilling, Phys. Rev. Lett., 7, 346–348 (1961).
  * (4) Murray Gell-Mann and M Levy, Nuovo Cim., 16, 705 (1960).
  * (5) F. Eisele, R. Engelmann, H. Filthuth, W. Foehlisch, V. Hepp, E. Leitner, W. Presser, H. Schneider, and Gunter Zech, Z. Phys., 225, 383–392 (1969).
  * (6) S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D, 2, 1285–1292 (1970).
  * (7) J. B. Gerhart, Phys. Rev., 95, 288–289 (1954).
  * (8) D. L. Hendrie and J. B. Gerhart, Phys. Rev., 121, 846–849 (1961).
  * (9) Murray Gell-Mann, Phys. Lett., 8, 214–215 (1964).
  * (10) G. Zweig,  An SU(3) model for strong interaction symmetry and its breaking. Version 1,  CERN-TH-401, unpublished (1 1964).
  * (11) J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Lett., 13, 138–140 (1964).
  * (12) L. B. Leipuner, W. Chinowsky, R. Crittenden, R. Adair, B. Musgrave, and F. T. Shively, Phys. Rev., 132, 2285–2290 (1963).
  * (13) Makoto Kobayashi and Toshihide Maskawa, Prog. Theor. Phys., 49, 652–657 (1973).
  * (14) Kiyoshi Niu, Eiko Mikumo, and Yasuko Maeda, Prog. Theor. Phys., 46, 1644–1646 (1971).
  * (15) Toshihide Maskawa, Rev.Mod. Phys., 81, 1027 (2009).
  * (16) G. Danby, J. M. Gaillard, Konstantin A. Goulianos, L. M. Lederman, Nari B. Mistry, M. Schwartz, and J. Steinberger, Phys. Rev. Lett., 9, 36–44 (1962).
  * (17) J. D. Bjorken and S. L. Glashow, Phys. Lett., 11, 255–257 (1964).
  * (18) Pekka Tarjanne and Vigdor L. Teplitz, Phys. Rev. Lett., 11, 447–448 (1963).
  * (19) Ziro Maki, Prog. Theor. Phys., 31, 331 (1964).
  * (20) Yasuo Hara, Phys. Rev., 134, B701–B704 (1964).
  * (21) L. Maiani, Phys. Lett. B, 62, 183–186 (1976).
  * (22) Martin L. Perl et al., Phys. Rev. Lett., 35, 1489–1492 (1975).
  * (23) S. W. Herb et al., Phys. Rev. Lett., 39, 252–255 (1977).
  * (24) F. Abe et al., Phys. Rev. Lett., 74, 2626–2631 (1995), hep-ex/9503002.
  * (25) S. Abachi et al., Phys. Rev. Lett., 74, 2632–2637 (1995), hep-ex/9503003.
  * (26) Vernon D. Barger, Wai-Yee Keung, and R. J. N. Phillips, Phys. Rev. D, 24, 1328, The journal editors revised the title in the published version (1981).
  * (27) G. C. Branco and Hans Peter Nilles, Nucl. Phys. B, 151, 529–535 (1979).
  * (28) M. Gorn and E. A. Paschos, Phys. Lett. B, 101, 255–258 (1981).
  * (29) Utpal Sarkar and Amitava Raychaudhuri,  DOES BARE BOTTOM RULE OUT THE TOPLESS E6 MODEL? (4 1983).
  * (30) Y. Yamanaka, H. Sugawara, and S. Pakvasa, Phys. Rev. D, 25, 1895, [Erratum: Phys.Rev.D 29, 2135 (1984)] (1982).
  * (31) Sandip Pakvasa and Hirotaka Sugawara, Phys. Rev. D, 14, 305 (1976).
  * (32) Benjamin W. Lee, Phys. Rev. D, 15, 3394 (1977).
  * (33) John R. Ellis, M. K. Gaillard, Dimitri V. Nanopoulos, and S. Rudaz, Nucl. Phys. B, 131, 285, [Erratum: Nucl.Phys.B 132, 541 (1978)] (1977).
  * (34) Steven Weinberg, Experientia Suppl., 31, 339–352 (1977).
  * (35) Guido Altarelli, L. Baulieu, N. Cabibbo, L. Maiani, and R. Petronzio, Nucl. Phys. B, 125, 285, [Erratum: Nucl.Phys.B 130, 516 (1977)] (1977).
  * (36) Haim Harari and Miriam Leurer, Phys. Lett. B, 181, 123–128 (1986).
  * (37) E. Fernandez et al., Phys. Rev. Lett., 51, 1022 (1983).
  * (38) P. A. Zyla et al., PTEP, 2020(8), 083C01 (2020).
  * (39) N. Lockyer et al., Phys. Rev. Lett., 51, 1316 (1983).
  * (40) D. E. Klem et al., Phys. Rev. Lett., 53, 1873 (1984).
  * (41) M. Althoff et al., Phys. Lett. B, 149, 524–532 (1984).
  * (42) A. Chen et al., Phys. Rev. Lett., 52, 1084 (1984).
  * (43) Ling-Lie Chau and Wai-Yee Keung, Phys. Rev. Lett., 53, 1802 (1984).
  * (44) Lincoln Wolfenstein, Phys. Rev. Lett., 51, 1945 (1983).
  * (45) M. Gronau and J. Schechter, Phys. Rev. D, 31, 1668–1675 (1985).
  * (46) M. Aguilar-Benitez et al., Phys. Lett. B, 170, 1–350 (1986).
  * (47) Andrzej J. Buras, Markus E. Lautenbacher, and Gaby Ostermaier, Phys. Rev. D, 50, 3433–3446 (1994), hep-ph/9403384.
  * (48) J. Charles, Andreas Hocker, H. Lacker, S. Laplace, F. R. Le Diberder, J. Malcles, J. Ocariz, M. Pivk, and L. Roos, Eur. Phys. J. C, 41(1), 1–131, [updated results and plots available at: http://ckmfitter.in2p3.fr] (2005), hep-ph/0406184.
  * (49) M. Kobayashi, Prog. Theor. Phys., 92, 287–288 (1994).
  * (50) A. I. Sanda, Prog. Theor. Phys., 122, 103–123 (2009).
  * (51) D. Andrews et al., Phys. Rev. Lett., 45, 219 (1980).
  * (52) Ashton B. Carter and A. I. Sanda, Phys. Rev. Lett., 45, 952 (1980).
  * (53) Ashton B. Carter and A. I. Sanda, Phys. Rev. D, 23, 1567 (1981).
  * (54) A. J. Buras, W. Slominski, and H. Steger, Nucl. Phys. B, 245, 369–398 (1984).
  * (55) H. Albrecht et al., Phys. Lett. B, 192, 245–252 (1987).
  * (56) Y. Funakoshi,  Operational experience with crab cavities at KEKB,  In ICFA Mini-Workshop on Beam-Beam Effects in Hadron Colliders, pages 27–36 (2014), arXiv:1410.4036.
  * (57) J. Seeman et al., ICFA Beam Dyn. Newslett., 48, 72–87 (2009).
  * (58) P. Oddone,  DETECTOR CONSIDERATIONS,  In Donald H. Stork, editor, Proceedings, Conference on Linear Collider B-B Factory Conceptual Design, Los Angeles, California, 26-30 Jan. 1987, pages 423–446, Los Angeles, Calif (1987). World Scientific.
  * (59) A. Litke et al., Nucl. Instrum. Meth. A, 265, 93–98 (1988).
  * (60) G. Batignani et al., Nucl. Phys. B Proc. Suppl., 23, 291–296 (1991).
  * (61) C. Bozzi et al., Nucl. Instrum. Meth. A, 447, 15–25 (2000).
  * (62) G. Alimonti et al., Nucl. Instrum. Meth. A, 453, 71–77 (2000).
  * (63) I. Adam et al., Nucl. Instrum. Meth. A, 433, 121–127 (1999).
  * (64) T. Iijima et al., Nucl. Instrum. Meth. A, 453, 321–325 (2000).
  * (65) Bernard Aubert et al., Nucl. Instrum. Meth. A, 479, 1–116 (2002), hep-ex/0105044.
  * (66) A. Abashian et al., Nucl. Instrum. Meth. A, 479, 117–232 (2002).
  * (67) A. Abashian et al., Nucl. Instrum. Meth. A, 449, 112–124 (2000).
  * (68) T. Affolder et al., Phys. Rev. D, 61, 072005 (2000), hep-ex/9909003.
  * (69) Bernard Aubert et al., Phys. Rev. Lett., 86, 2515–2522 (2001), hep-ex/0102030.
  * (70) A. Abashian et al., Phys. Rev. Lett., 86, 2509–2514 (2001), hep-ex/0102018.
  * (71) L. F. Wang, H. Fukuma, K. Ohmi, S. Kurokawa, K. Oide, and F. Zimmermann, Phys. Rev. ST Accel. Beams, 5, 124402 (2002).
  * (72) Bernard Aubert et al., Phys. Rev. Lett., 87, 091801 (2001), hep-ex/0107013.
  * (73) Kazuo Abe et al., Phys. Rev. Lett., 87, 091802 (2001), hep-ex/0107061.
  * (74) I. Belyaev, G. Carboni, N. Harnew, C. Matteuzzi, and F. Teubert, Eur. Phys. J. H, 46(1), 3 (2021), arXiv:2101.05331.
  * (75) T. Abe et al.,  Belle II Technical Design Report (11 2010), arXiv:1011.0352.
  * (76) B. Pontecorvo, Zh. Eksp. Teor. Fiz., 34, 247 (1957).
  * (77) Ziro Maki, Masami Nakagawa, and Shoichi Sakata, Prog. Theor. Phys., 28, 870–880 (1962).
  * (78) J. Schechter and J. W. F. Valle, Phys. Rev. D, 22, 2227 (1980).
  * (79) K. Abe et al.,  Hyper-Kamiokande Design Report (5 2018), arXiv:1805.04163.
  * (80) Zelimir Djurcic et al.,  JUNO Conceptual Design Report (8 2015), arXiv:1508.07166.
  * (81) A. D. Sakharov, J. Exptl. Theoret. Phys., 5, 24–27 (1967).
  * (82) Roel Aaij et al., Phys. Rev. Lett., 122(21), 211803 (2019), arXiv:1903.08726.
  * (83) Laurent Canetti, Marco Drewes, and Mikhail Shaposhnikov, New J. Phys., 14, 095012 (2012), arXiv:1204.4186.
  * (84) Julian S. Schwinger, Phys. Rev., 82, 914–927 (1951).
  * (85) Steven Weinberg and Edward Witten, Phys. Lett. B, 96, 59–62 (1980).
  * (86) B. Schwingenheuer et al., Phys. Rev. Lett., 74, 4376–4379 (1995).
  * (87) A. Apostolakis et al., Phys. Lett. B, 458, 545–552 (1999).
  * (88) M. Achasov et al.,  STCF Conceptual Design Report: Volume I - Physics & Detector (3 2023), arXiv:2303.15790.
  * (89) A. Yu. Barnyakov, J. Phys. Conf. Ser., 1561(1), 012004 (2020).