FERMILAB-CONF-96/429-T
NSF-PT-96-2
November 1996



CP VIOLATION IN THE K AND B SYSTEMS





Boris Kayser


Fermi National Accelerator Laboratory
Batavia, IL 60510 USA
and
National Science Foundation*
4201 Wilson Boulevard
Arlington, VA 22230 USA





Abstract

Although CP violation was discovered more than thirty years
ago, its origin is still unknown. In these lectures, we describe the CP-
violating effects which have been seen in K decays, and explain how
CP violation can be caused by the Standard Model weak interaction.
 5 Feb 1997 The hypothesis that this interaction is indeed the origin of CP viola-
tion will be incisively tested by future experiments on B and K decays.
We explain what quantities these experiments will try to determine,
and how they will be able to determine them in a theoretically clean
way. To clarify the physics of the K system, we give a phase-conven-
tion-free description of CP violation in this system. We conclude by
briefly exploring whether electric dipole moments actually violate CP
even if CPT invariance is not assumed.





To appear in the Proceedings of the Summer School in High Energy Physics and Cosmology, held
at the International Center for Theoretical Physics, Trieste, June-July 1995.

* Permanent Address


1. Preamble

Being lovers of symmetry, we are tempted to expect that the behavior of a
physical system will not change at all if we replace every particle in it by its anti-
particle. However, in some processes nature violates this expected matter-anti-
matter symmetry. Particularly interesting is the violation of invariance under
CP, the combined action of charge conjugation C and parity P. Before we turn to
CP, let us first briefly consider the simplest operation which replaces a particle by
its antiparticle, namely C.


The effect of C on a particle f(p ,) of momentum p and helicity is given by


Cf(p ,) = C f (p ,) , (1.1)


where f is the antiparticle of f, and C is a phase factor. Note that C does not alter
a particle's momentum or helicity.

It has long been known that some processes are not invariant under C.
Consider, for example, the decay . Invariance under C would require that
the muons produced in + + + and - - + have identical helicity. But it
is found that actually they have opposite helicity: the + in + decay is always left-
handed, while the - in - decay is always right-handed. Thus,  is not
invariant under C.

A somewhat more subtle operation which replaces a particle by its antipar-
ticle is CP. The effect of CP on f(
p ,) is given by


CPf(p ,) = CP f ( p ,) , (1.2)

Here, the momentum and helicity reversals are due to the action of P, and CP is a
phase factor.

From Eq. (1.2), the CP-mirror image of the decay +  + + , where the
LH
subscript LH reminds us that the + has left-handed helicity, is the decay -
  + , in which the  has right-handed helicity. These two decays are the ones
RH
actually observed, and they have equal rates. Thus, , while not invariant
under C, is invariant under CP.

One might wonder whether perhaps all processes, even those which are not
invariant under C, are nevertheless invariant under CP. The decays of the neu-
tral K mesons have taught us that this is not the case. Let us turn, then, to the
phenomenology of the neutral kaon system.



1


2. CP Violation in the Neutral K System

Let us consider neutral kaons at rest. We choose our phase conventions in
this Section so that


CPK0 = +--
K0 . (2.1)

Simple field theory then implies that


CP--
K0 = +K0 . (2.2)

In Section 5, we shall free ourselves of this phase convention, and adopt a conven-
tion-free formalism.

--
While K0 and K0 have opposite strangeness, the weak interactions do not
--
conserve strangeness, and so they mix K0 and K0. The time evolution of a neutral
kaon is then described by a two-component Schrdinger equation of the form

a(t) a(t)
i = M . (2.3)
a a
t (t)
(t)


Here, a(t) is the amplitude for us to have a K0 at time t, anda(t) is the amplitude
--
for us to have a K0. The quantity M is a 2x2 matrix,


K0|M|K0 K0|M|K0
M M

M = 11 12
, (2.4)
M M
K0|M|K0 K0|M|K0 21 22


known as the neutral K mass matrix, which serves as the effective Hamiltonian
for a neutral kaon at rest. Since a kaon disappears with time through its decay, M
is non-Hermitean.

We shall assume that the world is CPT invariant. It is readily shown that
this invariance implies that


M11 K0MK0 = --
K0M--
K0 M22 . (2.5)

-- 
Writing M11 M  i/2, and M22 M  i/2, we see that the real part of Eq. (2.5), M
--
= M , is an example of the well-known CPT requirement that a particle and its anti-

particle have the same mass. The imaginary part, = , is an example of the re-
quirement that they have the same total width or, equivalently, the same lifetime.




2


Assume for the moment that CP invariance holds. Then our effective
Hamiltonian obeys (CP)M (CP) = M, and we have


M12 K0M--
K0 = K0(CP)M (CP)--
K0

--
= (CP)K0M(CP)K0 = --
K0MK0 M21 . (2.6)

Thus, from Eqs. (2.5) and (2.6), M has the form

X Y
M = . (2.7)
Y X


The eigenstates of M--the mass eigenstates of the neutral K system--are then
1 1
1 1
and - . That is, they are the CP eigenstates
2 1
2 1


K = 1
1 K0 + K0
2 (2.8)

and


K = 1
2 K0 - K0
2 . (2.9)

Note that K1 and K2 have opposite CP parity:

CPK1(2) = +
() K1(2) . (2.10)

Now, experimentally, the two mass eigenstates of the neutral K system are
the K-short KS, with a short lifetime S = (0.8926  0.0012) x 1010 sec, and the K-
long KL, with a much longer lifetime L = (5.17  0.04) x 108 sec. Essentially all KS
decays are to + or 00. It is easy to show that both of these final states have CP
= +1. Thus, if CP invariance and the associated CP conservation law hold, KS
must be K1. Then KL must be K2. But then, since K2 has CP = 1, the decays KL
+ and KL 00 are forbidden. Nevertheless, these decays do occur.1 Thus, CP
is violated in neutral K decays.

To be sure, the observed violation is small. The amplitudes for the CP-violat-
ing decay KL + and the CP-conserving one KS + are in the ratio2

+- T KL = -
(2.285  0.019)  10 3 . (2.11)
+- T KS




3


Similarly,2


00 T KL = -
(2.275  0.019)  10 3 . (2.12)
00 T KS


Nevertheless, this violation of CP is nonvanishing. Like C, CP is a symmetry
which is not always respected.

In addition to KL , other CP-violating effects have been seen in the
neutral kaon system. One of these is found in the semileptonic decays KL l,
where l is an e or a . When CP invariance holds, the KL is a CP eigenstate. Then
the CP-mirror image of the decay KL l+ is KL +l
. To be sure, CP
reverses the momenta and helicities of all the outgoing particles, but that is irrel-
evant when we integrate over these variables to get the full rate for decay into the
particles under consideration. Thus, when CP invariance holds, we require that
(KL l+ ) = (KL +l
). However, it is found experimentally that2

+ -
K
( )- K
( )
L -l L +l

" + "

= 3.27  0.12 x 103 . (2.13)

Further observed CP-violating effects will be discussed in Section 5.

3. The Origin of CP Violation in the Standard Model

All CP-violating effects observed to date have been seen in the decays of neu-
tral kaons. These decays are known to be due to the weak interaction. Therefore, it
is natural to speculate that CP violation may well be an effect of the weak inter-
action. This is the possibility that we shall emphasize here.

The weak interaction is very successfully described by the so-called
Standard Model (SM). In the SM, the weak interaction is carried by the charged
weak boson W, and the neutral weak boson Z. These bosons couple to the leptons
and to the three generations, or families, of quarks:

Generation: 1 2 3

Charge = 2/3 u c t
Charge = 1/3 d s b

Mass



4


As indicated, the quarks in the third generation are the heaviest ones, those in the
second generation are lighter, and those in the first generation are lighter still. In
a typical Feynman diagram for a hadronic weak decay, a relatively heavy quark
decays to lighter ones via W exchange. This is illustrated in Fig. 1, in which a KS
decays via its --
K0 ( =s 
d) component into +. The diagram of Fig. 1 entails the
W-mediated quark decay s u
ud.


* u
Vud 
W d
V
s us u
K +
s

d

Figure 1. One of the diagrams for KS +-.

According to the SM, the coupling of the W boson to the quarks is given by
the Hamiltonian

H   *
W = g W V
i + g W V
i  . (2.14)
2 i L L i L L
= 2
u,c,t
i=d,s,b i

15
Here, g is a real overall coupling strength, W is the W boson field, L is
2
the left-handed projection of the quark field and similarly for iL, and the numer-
ical coefficients Vi are the elements of the Cabibbo-Kobayashi-Maskawa (CKM)
quark mixing matrix

Vud Vus Vub
V = V
cd Vcs Vcb
. (2.15)

Vtd Vts Vtb

Note from Eq. (2.14) that any of the three negatively-charged quarks i can turn into
any of the three positively-charged ones by emitting a W, the amplitude for its
doing so being proportional to Vi. Thus, the off-diagonal elements of V describe
transitions in which a quark in one family turns into a quark in another family.
Hence, these elements mix the families, earning for V the name "quark mixing
matrix".




5


The SM coupling of the Z boson to the quarks is described by the
Hamiltonian

H = g  [{ ( )- ( ) ] ( ) } . (2.16)
Z Z I3 qL Q q sin2 W qL qL - Q q sin2 W qR qR
cosW q=u,c,t,
d,s,b

Here, W is the Weinberg angle, I3(qL) is the weak isospin of the left-handed
(1+5)
quark qL, Q(q) is the electric charge of q, and qR q is the right-handed
2
projection of the quark field q.

g 
Under CP, the term W V
i L  iL in the W-quark interaction HW trans-
2
forms as

(  
CP) g W V ( )-1 [ ( )*( )] g
i L  iL CP = * W
( ) i W ViiL L . (2.17)
2 2

Here, (W), etc. are phases, and we may choose [*(W)()*(i)] = 1. Then

H  
W = g W V i + g W V*ii 
2 i L L L L
,i 2 ,i . (2.18)

g  
W V  + g W V*i i
CP iiL L L L
2 2
,i ,i

We see that HW is CP-invariant if, and only if, V is real, or can be made real by
changing the phase conventions for the quark fields.

The analogue of Eq. (2.17) for the terms in the Z-quark interaction HZ states
that each of these terms transforms back into itself under CP. Thus, HZ is neces-
sarily CP-invariant.

We conclude that in the SM weak interaction, CP violation can arise only if
some of the numbers Vi are complex. How their complexity can produce physi-
cal CP-violating effects will be explained shortly.

3.1. The CKM Matrix

The SM requires that the CKM quark mixing matrix be unitary. Apart from
this unitarity, the matrix is not predicted, so its elements must be determined
experimentally.





6


How many independent parameters are needed to determine fully the CKM
matrix V? In answering this question, we must bear in mind that some of the
complex phases which V may contain are not physically meaningful. To see this,
note from Eq. (2.14) that, apart from irrelevant factors, Vi is just the amplitude
HWi for the quark transition i through W emission. Thus, if the arbi-
trary relative phase of the i and quarks is changed, the phase of Vi will change
correspondingly. Redefining the down-type quark i by i eii multiplies the i
column of V by ei. Similarly, phase redefining the up-type quark multiplies the
row of V by a phase factor. Hence, without changing the physics, we may mul-
tiply any column or row of V by a phase factor, or carry out any number of such
operations. We may use these operations to remove from V five phases corre-
sponding to the five relative phases of the six quarks, leaving five of the elements
of V real. We may do this, for example, by multiplying each of the columns of V by
a phase factor chosen to make its bottom element real, and then multiplying each
of the top two rows by a phase factor chosen to make its rightmost element real.

Mindful of this freedom to remove at least some phases from V, let us now
suppose that there are, not just three doublet quark families, but N of them, so
that V becomes an NxN matrix. How many parameters are necessary to deter-
mine it completely? Before constraints are imposed, 2N2 real numbers are
required to fully specify the N2 complex elements of V. But the unitarity of V
demands that the sum of the absolute squares of the elements in any of its
columns be unity--a demand that imposes N constraints. Furthermore, unitarity
demands that any two of the columns of V be orthogonal. Now, there are N(N1)/2
pairs of columns, and the equation expressing the orthogonality of any pair has
both a real and an imaginary part. Thus, orthogonality of columns imposes N(N
1) constraints. Hence, the most general NxN unitary matrix depends on 2N2  N
 N(N1) = N2 real parameters. Now, in the N quark families there are 2N
quarks, with 2N1 relative phases. Thus, 2N1 phases in V are not physically
meaningful, and may be removed by phase redefinitions of the quarks or, equiva-
lently, by multiplying columns and rows of V by phase factors. Hence, the number
of physically meaningful independent real parameters in V is N2  (2N1) = (N1)2.

One possible choice for these parameters is mixing angles (parameters
which would be present even if V were real) and phases. To calculate the number
of mixing angles on which V depends, imagine that it is real. It is then an ortho-
gonal (i.e., a rotation) matrix. It contains N2 real elements, subject to N con-
straints expressing the requirement that each of its columns be a vector of unit
length, and N(N1)/2 constraints expressing the requirement that any pair of its
columns be orthogonal. Thus, V depends on N2  N  N(N1)/2 = N(N1)/2 mix-
ing angles.


7


In summary, the complex NxN quark mixing matrix depends on (N1)2
parameters. If we take these to be mixing angles and phases, N(N1)/2 of them
are mixing angles, so that (N-1)2  N(N1)/2 = (N1)(N2)/2 of them are phases.
Note from this result that there are no physically significant phases in the mixing
matrix unless N 3.3 Had there been fewer than three quark families, it would
have been impossible for the weak interaction, as described in the SM, to violate
CP.

(It is instructive and easy to explicitly construct the most general unitary
quark mixing matrix for the case N = 2, and show that all phases can be removed
from this matrix by multiplying its rows and columns by phase factors. Since this
leaves the matrix real, it cannot violate CP.)

Although there are in reality (at least) three quark families, the fact that
the quark mixing matrix cannot violate CP in a world with only two families has
an important consequence. Namely, it implies that CP violation in K decays will
be small, as observed, even if the complex phases in the true 3x3 mixing matrix
are large. Furthermore, it tells us where we must look if we wish to see large CP-
violating effects.4 To see why it implies that CP violation in K decays will be small,
note that, as illustrated in Fig. 1, the dominant diagrams for K decays involve only
quarks from the first two families. The quarks t and b are not involved. Thus, in
first approximation, K decays "do not know" that there are not just two, but three
quark generations. In this approximation, K decay processes do not contain
enough physics to violate CP. Now, when one goes beyond the first approximation,
one finds that K decays do involve the quarks of the third generation in several
ways, so that these reactions can (and do!) violate CP. However, because one must
go beyond the leading approximation before the third generation quarks come into
the picture, CP violation in K decays is small.

From this discussion, it is clear that if we wish to see large CP-violating
effects coming from the CKM matrix, we must look for them in processes which
involve, even in leading approximation, quarks from all three generations. To this
end, new facilities are being built and new experiments are being developed
which will study CP violation in the decays of the B or beauty mesons. The B
mesons and their quark content are
 
B+ = [bu ] B = [bu]
 -- 
Bd = [bd] Bd = [bd]
 -- 
Bs = [bs] Bs = [bs]
 -- 
Bc = [bc] Bc = [bc] .



8


In a typical B decay, the heavy b orb quark in the B--a quark of the third genera-
tion--decays down to lighter quarks in the first and/or second generations. Often,
all three generations are involved. Thus, CP violation can be large.

As we shall see shortly, the CP-violating effects in B decays can also yield
clean information on the phases in the CKM matrix. Thus, the study of these
effects will be a very good test of whether these phases are indeed the origin of CP
violation.

4. CP Violation in the B System

The effects to be sought in the B system are CP-violating inequalities
between the rates for CP-mirror-image decays. When CP invariance holds, the
amplitude fTi for the decay of any initial state i into any final one f obeys

fTi = CP[f]TCP[i] . (4.1)

Thus, for example, any inequality between the rates for the CP-mirror-image
decays B+ f and B f, wheref CP[f] is the CP-mirror image of the final
state f, is a violation of CP invariance. It is violations of this general sort, which
are B-system analogues of the K-system asymmetry of Eq. (2.13), which will be
sought.

As we noted earlier, CPT invariance, which we assume to hold exactly,
requires that any unstable particle and its antiparticle have the same total width.
Thus, if there is some final state f for which, in violation of CP, [B+ f] > [B
f], then there must be some other final state (or states) f for which [B+ f] <
[B f]. Otherwise, the CPT constraint that total [B+] = total[B] could not be
satisfied.

The complex phases in the CKM matrix, like complex phases anywhere in
quantum mechanics, lead to physical consequences only through interferences
between amplitudes. In particular, it is through interferences that the CKM
phases produce CP violation. How they do this is nicely illustrated by the compari-
son between the CP-mirror-image processes B+ f and B f. Suppose that the
weak decay B+ f receives contributions from two Feynman diagrams. Each of
these diagrams is proportional, like the diagram of Fig. 1, to some product of
CKM elements. Thus, the amplitude a for the first diagram has the form

f
a = MeiCKM eiS , (4.2)





9


where M is the magnitude of a, f
CKM is the phase of the product of CKM elements
to which the diagram is proportional, and S is a phase arising from strong in-
teraction effects such as final-state rescattering. Similarly, the amplitude a for
the second diagram has the form

f
a = Mei CKM ei S , (4.3)

where M' a, f
CKM is the phase of the product of CKM elements to which the
second diagram is proportional, and S is the strong-interaction phase of this
diagram. The rate for B+ f is then

f f 2
+
B f
[ ]= MeiCKMeiS +Mei CKMei S , (4.4)
= 2 + ( )
M2 + M 2MM cos + S

where f
CKM  f
CKM is the relative CKM phase of the two amplitudes, and S
S  S is their relative strong-interaction phase.

Now, the diagrams for the CP-mirror-image decay B f are, of course,
the same as those for B+ f, except that every quark (antiquark) is replaced by its
antiquark (quark). From HW, Eq. (2.14), we see that, owing to this replacement,
every CKM element appearing in a diagram for B+ f is replaced by its complex
conjugate in the corresponding diagram for B f. However, apart from CKM
phases, the SM weak interaction of Eqs. (2.14) and (2.16) is completely CP invari-
ant, as is the SM strong interaction. Thus, apart from the reversal of its CKM
phase, the amplitude of a diagram does not change at all when we go from B+ f
to B f. Hence, the rate for B f is

f f 2
- - -
B f
[ ]= Me i i
CKM eiS + Me CKM ei S . (4.5)
= 2 + ( )
M2 + M 2MM cos - + S

Comparing Eqs. (4.4) and (4.5), we see that when CKM phases are present,
the two interfering amplitudes can have a different relative phase in B f than
they do in B+ f. As a result, [B f] and [B+ f] can differ, in violation of
CP.

To test the SM of CP violation, one would like not only to observe a CP-violat-
ing inequality between [B+ f] and [B f], but also to determine the CKM
phase . Of course, [B+ f] and [B f] are only two measurable quantities,
and as we see from Eqs. (4.4) and (4.5), they depend on four parameters: M, M, ,


10


and S. Thus, by themselves, they cannot determine . Consequently, in general,
a clean test of the SM of CP violation is not possible in decays of charged B mesons.
(To be sure, in the exceptional cases where M and M can be determined indepen-
dently of [B+ f] and [B f], the measurement of these two decay rates de-
termines sin2, up to a two-fold ambiguity, and so does provide a test of the SM.5)

4.1. Decays of Neutral B Mesons

In decays of neutral B mesons, a clean test of the SM of CP violation is
-- --
possible. To see why, let us discuss the Bd  Bd system; the Bs  Bs system behaves
similarly.

-- --
The key feature of the Bd  Bd system is the fact that the Bd and Bd mix. In
the SM, they do so largely as a result of the box diagram in Fig. 2. The phase of the
--
mixing amplitude A(Bd Bd) is then


arg A
[ (B )]= *
( )2
m
d Bd arg VtdVtb . (4.6)
-2CKM
--
We shall refer to m
CKM (where m stands for mixing) as the Bd  Bd mixing phase.


Vtb
* t V
td
b d
Bd W W Bd

d b
Vtd t Vtb
*

--
Figure 2. The SM box diagram for Bd  Bd mixing.

--
Time evolution in the Bd  Bd system is described by a two-component
--
Schrdinger equation, just like the one for neutral kaons, Eq. (2.3). The Bd  Bd
Schrdinger equation involves a mass matrix,


M = X M 12
, (4.7)
M

21 X
-- --
which is the Bd  Bd counterpart to the K mass matrix of Eq. (2.4). In the Bd  Bd
mass matrix, the diagonal elements X are equal because of CPT, and the off-diag-
onal element M21 comes from the box diagram of Fig. 2. The remaining off-diago-



11


nal element, M 12, comes from a similar box diagram in which every quark
(antiquark) has been replaced by its antiquark (quark). As in the case of charged B
decays, this means that every CKM element has been replaced by its complex con-
jugate, but there have been no other changes. Since the box diagram of Fig. 2 has
no strong phase (owing to the fact that the B meson is far belowtt threshold), we
see that

M12 = M21(V V*) = M *
21 . (4.8)

--
Let us call the mass eigenstates of the Bd  Bd system BHeavy (BH) and
BLight (BL). From Eq. 4.7, the complex masses of these mass eigenstates--the
eigenvalues of M-- are


= M M - i
H(L) X + m
(-) 12 21 H(L) H(L) . (4.9)
2

Here, mH(L) are the masses of BH(L), respectively, and H(L) are their widths.
Note that since M12M21 is real and positive, so that H and L have the same
imaginary part, the widths of BH and BL are equal:

H = L . (4.10)

(To a very good approximation, this equality holds even if the SM diagram of Fig. 2
--
is not a good approximation to A(Bd Bd). This is simply because, unlike KS and
KL, neither B mass eigenstate has a special decay mode which is an appreciable
fraction of its decays and which is unavailable to the other mass eigenstate. Thus,
BH and BL have approximately equal widths.)

From Eqs. (4.7) and (4.9), the mass eigenstates BH(L) are given by

- m
B = 1 + 2iCKM
H(L) B
d e B
(-) d . (4.11)
2

Here and hereafter we assume that M21 does come from the SM diagram of Fig. 2.

--
Owing to the Bd --Bd mixing, a neutral B at rest which at time t = 0 is a
pure Bd will not remain that way. Rather, in time t it will evolve into a state
Bd(t) which is a coherent superposition of Bd and --
Bd. From Eqs. (4.9), (4.11),
and Schrdinger's equation, it is straightforward to show that

-
i m-i
t - m
B 2 - 2iCKM
{ }
d (t) = e c Bd ie s Bd . (4.12)



12


Here,

mH + mL
m (4.13)
2

is the average BH, BL mass,

m mH  mL (4.14)

is the BH  BL mass difference, and


c cos (mt), s sin (mt) . (4.15)
2 2

Note from Eq. (4.12) that, before it decays into some final state, a neutral B meson
which at time t = 0 is a pureBd oscillates between being a Bd and a --
Bd. This
oscillation has been observed,6 and it is found that

m
= 0
.66  0.09 ARGUS & CLEO7

0.72  0.04 LEP6,8


Thus, before a typical B decays, it undergoes a non-negligible fraction of one oscil-
lation.

--
Suppose, now, that f is a final state into which both a pure Bd and a pure Bd
can decay. Examples of such a final state are +, D0Ks, +, and Ks. Let f(t)
(Bd(t) f) be the time-dependent probability for the time-evolved particle Bd(t),
which at t = 0 was a pure Bd, to decay into f. From the wave function for Bd(t), Eq.
(4.12), f(t) is given by

2
2 = -t - -2i mCKM
f (t) = f T Bd (t) e c f T Bd ie s f T Bd . (4.16)


Let us now assume that the decay amplitudes fTBd and fT--
Bd are each
dominated by a single Feynman diagram. Then

f
=
f T Bd MeiCKM eiS , (4.17)

where M is the magnitude of the diagram which dominates fTBd, f
CKM is the
phase of the product of CKM elements to which this diagram is proportional, and
S is the strong interaction phase of the diagram. Similarly,

- f
f T B = Me iCKM eiS , (4.18)
d


13


-- 
where M ,  f
CKM, andS are respectively the magnitude, CKM phase, and strong
phase of the diagram which dominates fT--
Bd. From Eqs. (4.16)-(4.18), we then
have9

--
f(t) = et {c2M2 + s2--
M 2  2csMM sin ( + s)} , (4.19)

where

m + f + f (4.20)
2CKM CKM CKM

is the relative CKM phase of the two interfering amplitudes in Eq. (4.16), and

S = S S (4.21)

is their relative strong phase.

--
The CP-mirror image of the decay Bd(t) f is the process Bd(t) f, where
-- --
Bd(t) is the time-evolved particle which at time t = 0 is a pure Bd. As before, when
we go from a process to its CP-mirror image, the CKM phases reverse, but noth-
ing else changes. Thus, from the expression (4.19) for f(t), we may infer that the
 --
probability f(t) (Bd(t) f) is given by9


 --
f (t) = et {c2M2 + s2--
M 2  2csMM sin ( + s)} . (4.22)

--
Now, since Bd(t) f and Bd(t) f are CP-conjugate reactions, CP invari-

ance would require that 
f(t) = f (t). Comparing Eqs. (4.19) and (4.22), we see that
when 0, this requirement is violated. Note that, as always, the CKM phase
produces this CP violation through an interference; in this case the interference
--
between the two terms in Eq. (4.16), or between their analogues in Bd(t) f.
Physically, the first term in Eq. (4.16) corresponds to a Bd remaining a Bd and
decaying directly into f. The second term corresponds to a Bd evolving, through
--
mixing, into a Bd, which then decays into f.

Recalling that and m are already known, it is trivial to see from Eqs.

(4.19) and (4.22) that measurements of the functions 
f(t) and f (t) will determine
--
M, M , s+ sin ( + s) and s sin ( + s). Once s+ and s are known, one can
find sin2, up to a two-fold ambiguity, by using


sin2 = 1 1 - s 2 2
+s-  (1 - s+ )(1 - s- )
2 . (4.23)





14


Note that, apart from the discrete ambiguity, this expression gives a theoretically
clean value for sin2. This value does not depend on any unknown or difficult-to-
calculate parameters. This value can be compared directly to the prediction from
the CKM matrix to test cleanly whether phases in this matrix are indeed the
source of CP violation.

As we have seen, the CKM phase which is probed in a given decay, Bd(t)
f, is the relative CKM phase of the two interfering terms in Eq. (4.16). That is,
recalling Eq. (4.6),

A(B f )
= d
CKM Phase , (4.24)
A(B ) ( )
d B A B
d d
f

where "A" denotes an amplitude. As an example, in Bd(t) +, we expect A(Bd
+ --
) to be dominated by the diagram in Fig. 3. Similarly, we expect Bd + to
be dominated by the diagram in Fig. 4.

u
Vud +

d
 V *
b ub u



Bd
d

Figure 3. The diagram which dominates Bd +.


* u
Vud 
d
V
b ub u
-- +
Bd

d

--
Figure 4. The diagram which dominates Bd +.

The mixing amplitude A(Bd --
Bd) is dominated by the diagram in Fig. 2. Thus, in
Bd(t) +,




15


*
= V V
arg ud ub
2
( *
V V . (4.25)
td tb ) *
V V
ub ud
= 2 [
arg * *
V V V V
ud ub tb td ]


In a similar way, one may easily find what CKM phase is probed by any particu-
lar decay. Note that since each of the amplitudes in Eq. (4.24) is always propor-
tional to some product of CKM elements (assuming each amplitude is dominated
by one diagram), is always the phase of some product and quotient, or equiva-
lently of some product, of CKM elements.

The neutral B decay rates, and the extraction of a CKM phase from them,
become particularly simple when the final state f is a CP eigenstate. Examples of
such a final state are + and (neglecting CP violation in the kaon system) Ks.

When f is a CP eigenstate, we have f CPf = ff, where f is the CP parity of
 --
f. Then fT--
Bd = ffT--
Bd. Now, Bd is the CP conjugate of Bd, andf is the

CP conjugate of f, so fT--
Bd is the CP conjugate of fTBd. As before, CP-con-
jugate amplitudes have opposite CKM phase but are otherwise identical. Thus,
from Eq. (4.17), when f is a CP eigenstate,

- f
f T B = iCKM eiS
d f Me . (4.26)

Using this relation and Eq. (4.17) in Eq. (4.16), we find that

f(t) = M2et{1  f sin sin(m t)} , (4.27)

where , the relative CKM phase of the two interfering terms, is now given by

= 2 m f
( )
CKM + . (4.28)
CKM

-- 
For the CP-mirror-image decay, Bd(t) f, the decay rate f(t) must be the same as
f(t) except for a reversal of the CKM phase. That is,


f(t) = M2et{1 + f sin sin(m t)} . (4.29)

Now, m is known, as is the CP parity f of any particular final state f of interest.

Thus, the CP-violating asymmetry between f(t) and f(t),

f (t) - f (t) = ( )
f sin sin mt , (4.30)
" + "



16


cleanly determines the CKM phase quantity sin.10

It should be emphasized that the ability to cleanly extract CKM phase
information from decay rates does depend on the assumption that fTBd and
fT--
Bd are each dominated by one Feynman diagram. When fTBd or fT--
Bd
involves several competing diagrams with different CKM phases, the rate for Bd(t)
f involves several interferences, rather than just one, and no longer cleanly
determines any one relative CKM phase of two amplitudes. Fortunately, in at
least some of the decay modes of greatest interest, there are strong reasons for
believing that one diagram does dominate.11

4.2. Future Experiments

In Section 3, it was argued that CP-violating effects in B decay can be large.
We now see, for example, from Eq. (4.30) for the asymmetry in decay to a CP
eigenstate, that these effects can indeed be large. If the CKM phase quantity sin
in the asymmetry (4.30) is O (1), then obviously the asymmetry itself is O (1).
However, it will take a large sample of B mesons to observe even a large CP-violat-
ing asymmetry. The reason is that each of the asymmetries on which the experi-
mental search will focus occurs in the decay to some specific final state, or CP-
conjugate pair of final states, and the branching ratio for B decay to any of the
final states of interest is rather small. Thus, a lot of B mesons will be required
before a CP-asymmetry in some particular decay mode can be seen.

As an example, consider the CP eigenstate final state f = Ks. If the decay
rate [Bd(t) Ks] is measured by observing N events, the measurement has a
--
statistical error of order N. Similarly for [Bd(t) Ks]. Thus, if the asymmetry

[B ]-[ ]
d (t) Ks Bd(t) Ks (4.31)
" + "

is, for example, of order 0.1, we must have N << (0.1)N in order to measure it with
any accuracy. Hence, we require N >
~ 103. Now, typically a is detected via its
decay to + or e+e. Since only 12% of particles decay in this way, we need ~104
Bd(t) Ks events in order to detect 103 of them. Furthermore, BR(Bd(t) Ks)
4x104.2 Thus, to detect 103 BdKs decays, we need ~108 Bd mesons. For other
typical decay modes of interest, the number of Bd mesons required is similar.
However, the total number of Bd mesons recorded to date at CESR, for example, is
only ~5x106.12 To produce and study enough B mesons to measure CP-violating
asymmetries in the B system, future experiments are being planned for hadron
facilities, and special high-luminosity e+ e colliders ("B factories") are being built



17


at SLAC and KEK. The experiments to be done at the hadron facilities and the B
factories will complement each other nicely.

--
To experimentally compare the rate for Bd(t) f with that for Bd(t) f (or,
--
when f is a CP eigenstate, that for Bd(t) f), we must, of course, be able to distin-
--
guish a Bd(t) from a Bd(t). That is, we must be able to tag the B as having been a
--
pure Bd, or a pure Bd, at some specific time t = 0. Several methods for doing this
are being considered. Let us briefly review them.

At the B factories, B mesons will be produced in pairs via the reaction

--
e+e (4s) BdBd . (4.32)

Since the (4s) [the upsilon(4s)] has intrinsic spin S = 1, and B mesons are spin-
less, the B pair created in this reaction will be in a p wave. Now, after it is pro-
duced, each B meson in the pair will evolve, thanks to mixing, into a coherent
--
mixture of pure Bd and pure Bd. However, at no time can one have two identical
bosons in an antisymmetric state such as a p wave. Thus, if at some time which
we shall call t = 0, one of the B mesons in the pair decays in a fashion which re-
--
veals that at the instant of decay it is, say, a Bd, then, at the same instant, the
other B meson in the pair must be a Bd. That is, the decay of the one B at t = 0 tags
the remaining B as a Bd(t). This type of tagging is an interesting modern applica-
tion of the quantum mechanical correlation first discussed by Einstein, Podolsky,
and Rosen (the EPR effect).

What kind of neutral B decay will reveal that at the instant of decay the par-
--
ent was a Bd? An example of such a decay is semileptonic decay, the diagrams for
which are shown in Fig. 5. From these diagrams, we see that a positively-charged
--
lepton l+ can come only from a Bd, and a negatively charged one l only from a Bd.
Thus, the charge of the lepton tells us whether, at the instant of decay, the parent
--
was a Bd or a Bd.

A typical B factory experiment might study the decay chain

e+ + e (4s) B + B , (4.33)

f
 CP
l +
( ) + ...


where l(+) + ... is a semileptonic final state and fCP is a CP eigenstate. Let us
consider this chain in the (4s) rest frame. In this frame the B mesons are quite
nonrelativistic, so we may, for the moment, neglect their motion, and take B-rest-
frame proper times and (4s)-frame times to be indistinguishable. Changing the


18


l+


W
 
b c

Bd
d


l



W
b c
--
Bd

d

Figure 5. The diagrams for semileptonic neutral B decay.
The symbol l denotes a charged lepton.

notation, let us now call the time of the decay (4s) B + B, t = 0; the time of the

decay B l(+) + X, tl; and the time of the decay B fCP, tCP. The probability that
one B will decay semileptonically at time tl is proportional to exp[tl]. The proba-
bility that the other B will live at least until time tl is proportional to a second fac-
tor of exp[tl]. If the B undergoing the semileptonic decay yields an l (l+), then
--
at time tl the other B must be a pure Bd (Bd). Thus, the probability that this B will
decay to fCP at time tCP is given by Eq. (4.27) [Eq. (4.29)] with f taken to be fCP. Most
importantly, in applying Eq. (4.27) or (4.29), we must take the time variable, which
as we recall represents the time of the decay to the CP eigenstate relative to the
--
time when the parent was known to be a pure Bd or Bd , to be tCP  tl. Combining
all factors, we have for the joint probability of the two B decays in (4.33)

-

Probability One B (+)
l + X at time tl; Other B fCP at time tCP



- - - - -
e t t (t t )
le le CP l 1
(+) [ ( )]
f sin sin m t . (4.34)
CP CP - tl


= - + -
e (tCP t )
l 1
(+) [ ( )]
f sin sin m t
CP CP - tl




19


Although it is not obvious from what has been said, this result is true even if tCP is
earlier than tl.

To take the (so far neglected) motion of the B mesons in the (4s) rest frame
and all the requirements of relativity into account, we may replace the treatment
above by one in which we do not speak of the semileptonic decay of one B as deter-
--
mining the Bd or Bd nature of the other B. Rather, we simply calculate directly the
amplitude for the entire decay chain (4.33).13 This amplitude approach also has
the advantage of avoiding a puzzling question raised by the treatment based on the
EPR effect: How does the second B know the charge of the lepton produced in the
decay of the first B, and how does it know when that decay occurred? For the joint
probability of the two B decays in (4.33), the amplitude approach yields precisely
the same result, Eq. (4.34), as the EPR approach, provided that the times in that
result are taken to be proper times in the B rest frames, rather than times in the
(4s) rest frame. The time tl must be taken to be the proper time elapsed in the
frame of the semileptonically decaying B between its birth and decay, and
similarly for tCP.

Suppose one does an experiment in which there is not enough resolution to
measure the decay times tl and tCP, so one simply measures the time integral
over the joint decay probability (4.34). The contribution to this time integral of the
term in (4.34) proportional to sin, the quantity one would like to determine,
vanishes. This is because



- +
dt e (tCP t )
l sin[m(t ] = 0 (4.35)

dtl
0 0 CP CP - tl)

by the antisymmetry of the integrand under tl tCP. Thus, to determine sin
with neutral B mesons at a B factory, one must be able to measure the B decay
times, at least to some extent. To measure the decay time of a B, one would deter-
mine the pathlength it covers before decay and its energy. Now, in every e+ e
collider built so far, the e+ and e beams have equal and opposite momenta, so that
in the reaction e+ e (4s) BB, the (4s) is at rest in the laboratory frame.
Thus, at these colliders, one would be trying to determine the B pathlength in the
(4s) rest frame. However, as already mentioned, in this frame the B mesons are
quite nonrelativistic. In fact, they are so slow ( 0.06) that, before decaying in
1.6x1012 sec,8 a typical B covers only ~30m. Pathlengths this short cannot be
measured. To make the B pathlengths long enough to be measurable, the SLAC
and KEK B factories will be asymmetric colliders. That is, in each of them the
positron beam will have a different energy from the electron beam. As a result,
the (4s) formed in the e+ e collision will be moving in the laboratory, and will
transmit its motion to its daughter B mesons. The asymmetry between the beam


20


energies will be sufficient to lead to B mesons which typically will travel ~200m
before decaying. Such a distance is large enough to be measured.

Another method for tagging, which may prove useful at hadron facilities, is
based on the expectation that some fraction of the neutral B mesons made at those
facilities will be created via the production and decay of a B**. By B** we mean an
excited B meson heavy enough to decay to B + . Such mesons are expected as p-
wave quark-antiquark bound states, and are observed at LEP.8 Now, as Fig. 6
+  --
makes clear, a B** decays to Bd+, but a B** to Bd.

--
B B
d d

b b 
+ d 
B** B** d

d  d
u u
+ 

+  --
Figure 6. The diagrams for B** B +
d and B** Bd .

Suppose, then, that in some event one finds a neutral B and a charged which
are close to each other in phase space and whose momenta are such that the
invariant mass of the B system is the known mass of a B**. Then, neglecting
background and assuming that the B system came from a B**, if the charge of
the is positive (negative), we can conclude that, at the moment of its production
--
in B** B, the neutral B was a pure Bd (Bd ).14 Results from LEP8 suggest that
the fraction of B mesons made via a B** may be appreciable at hadron facilities, so
this method of tagging may be quite helpful.

4.3. What There is to Measure

As we have seen, the CKM phase which is probed by CP violation in any B
decay is the phase of some product of CKM elements. What, then, are the inde-
pendent phases of all possible products of CKM elements? That is, what is there to
measure?

The answer to this question grows out of the fact that, in the SM, the CKM
matrix must be unitary. The requirement of unitarity demands, among other
things, that any pair of columns of the CKM matrix be orthogonal, and similarly
for any pair of rows. Thus, we have the six orthogonality constraints





21


ds VudV*us + VcdV*cs + VtdV*ts = 0
5

sb VusV*ub + VcsV*cb + VtsV*tb = 0
4 2 2

db VudV*ub + VcdV*cb + VtdV*tb = 0
3 3 3 (4.36)

u c VudV*cd + VusV*cs + VubV*cb = 0
5

ct VcdV*td + VcsV*ts + VcbV*tb = 0
4 2 2

ut VudV*td + VusV*ts + VubV*tb = 0
3 3 3

To the left of each constraint is indicated the pair of columns, or of rows, whose
orthogonality is expressed by that constraint. Under each term in each constraint
is given the rough empirical size of that term, expressed as a power of the Cabibbo
angle = 0.22. Each term in any of the constraints may be pictured as a vector in
the complex plane. The constraint then states that its three terms form the sides
of a closed triangle, called a "unitarity triangle",15 in the complex plane. The six
unitarity triangles corresponding to the constraints of Eqs. (4.36) are shown,
somewhat schematically, in Fig. 7. As we see from Eqs. (4.36), in two of the trian-
gles, the three sides are of comparable size, so that the interior angles can all be
large. However, in each of the remaining triangles, one of the sides is much
shorter than the other two, and the angle opposite this short side must be small.

Any angle in one of the unitarity triangles is, of course, (apart from an
extra ) just the relative phase of the two adjacent sides. Thus, this angle is the
phase of a product of CKM elements. Furthermore, the product concerned will
always be one whose phase is convention-independent. For example, the relative
phase of the two sides adjacent to the angle in the db triangle is arg
(VtdV*tbV*udVub). Now, this phase is invariant under phase redefinition of the t
quark, since this redefinition causes equal and opposite phase changes in Vtd and
V*tb. Similarly, is invariant under phase redefinition of the u, d, or b quark.
Thus, the angles in the unitarity triangles do not depend on phase conventions.





22


ds uc
V * *
cdVcs V
V * udVcd
tdVts V *
V ubVcb
udVus
*
' V *
usVcs





sb ct
V *
tsVtb V *
csVts
V *
cdVtd
V *
* usVub
VcsVcb V *
cbVtb





db ut

V *
udVub V * * *
tdVtb VudVtd VubVtb

V * *
cdVcb VusVts


Figure 7. The unitarity triangles. To the left of each triangle is indicated the pair of
columns, or of rows, whose orthogonality this triangle expresses. The significance of
the angles labeled , , , , and is explained in the text.

Now, it can be shown that if is the phase of any phase-convention-independent
product of CKM elements (that is, if is the CKM phase probed in some experi-
ment on CP violation), then16

= n + n + n + n . (4.37)

Here, , , , and are the four unitarity triangle angles identified in Fig. 7, and
n, n, n, and n are integers. From Eq. (4.37), we see that, presuming , , ,
and are independent, these four angles may be taken to be the independent



23


phases of all possible (convention-independent) products of CKM elements. The
CKM phase probed by any CP experiment is a simple linear combination of
these four angles. The future experiments on CP violation in the B system may be
thought of as, in part, an attempt to determine these four angles.

It can be proved that, once they are known, , , , and completely de-
termine the entire CKM matrix.16 Since, as is well known, it takes four indepen-
dent parameters to determine this matrix, it follows that , , , and must
indeed be independent, as we just assumed. Furthermore, since , , , and do
completely determine the full CKM matrix, CP experiments in the B system are
not merely measurements of angles in the unitarity triangles, but, in principle at
least, probes of the entire content of the CKM matrix.17

From the magnitudes of the terms in the "ds" orthogonality constraint of
Eqs. (4.36), we see that the angle in the ds unitarity triangle is at most of order
5/ , or 2x103 radians. Thus, in a B decay where the CKM phase which is
probed is , the CP violation would be very small. As a result, it may not be possi-
ble to measure . However, plans are being developed, and facilities being con-
structed, to measure the three remaining independent angles, , , and .

Wolfenstein has introduced a very good (~3%) approximation18 to the CKM
matrix V which is based on the empirical observation that, as far as the magni-
tudes of its elements are concerned, V has approximately the form

1 3

V ~ 1 2 . (4.38)
3 2 1


The implications of the magnitudes summarized here for the unitarity constraints
(4.36) have already been indicated beneath them. In Wolfenstein's approximation,
in effect one neglects the small term in the ds constraint of Eqs. (4.36) relative to
the larger terms, and does the same in the sb constraint. The ds and sb unitarity
triangles then each collapse to two antiparallel lines of equal length, and the an-
gles and vanish (cf. Fig. 7). Of the four independent unitarity-triangle angles
originally present, only the angles and , in the db triangle, remain. These
angles, and the dependent angle =   in the same triangle, are in any case
the angles on which the early CP experiments on the B system will concentrate,
since they are the angles which may be large and which, therefore, may produce
large CP-violating asymmetries. Consequently, in the literature, attention has
been focused on the db triangle.



24


The program to test the SM of CP violation through experiments on B
decays may be summarized as follows:

1. Measure the four independent angles of the unitarity triangles. If the smallest
angle, , is beyond reach, at least measure , , and . Focus first on and ,
since these angles may both be large.

2. To see whether the SM provides a consistent picture of CP-violating phenom-
ena, or leads to inconsistencies which point to physics beyond the SM, overcon-
strain the system as much as possible. To do so--

a. Measure, if possible, CP asymmetries in different decay modes which, if
the SM of CP violation is correct, all yield the same angle (, for example).
See whether these asymmetries actually yield the same numerical result.

b. Measure independently the angles , , and in the db triangle, and see
whether these angles actually add up to .

c . Measure the lengths of the sides of the db triangle (via experiments on non-
CP-violating effects such as decay rates and neutral B mixing). See whether
the interior angles implied by the measured lengths agree with those in-
ferred directly from CP-violating asymmetries.

Table 1. Decay modes and the CKM phase angle which they probe. In the final
state K*0, the K*0 is required to decay as shown. Similarly for the final state
( )
D0K+; gCP is a CP eigenstate, such as + or K+K. References are given in the
last column.



Decay Mode Ref.

B +
d(t) +, +, a1  2 11, 9, 19

Bd(t) Ks, K*0 2 10, 20

Ks 0

B +
s(t) Ds K + 2 - 21
( )
B+ D0K+ - 5
gCP

Bs(t) 2 22, 23



25


In Table 1 are listed some decay modes which (in combination with their
CP conjugates) are potential probes of the independent angles , , and , and the
dependent angle . In this table, Bs(t), in analogy with Bd(t), is the time-evolved
state which at time t = 0 was a pure Bs. The B+ decay listed is one of the excep-
tional charged B decays from which clean CKM phase information can be ex-
tracted.5 Note that, neglecting and relative to , the decays B +
s(t) Ds K and
( )
B+ D0K+ (gCP)K+ both yield the latter angle.

5. Testing the SM of CP Violation in the K System

The future tests of the SM of CP violation will include experiments on the
neutral K system, where CP violation was discovered. Before discussing these
experiments, we shall introduce a phase-convention-independent description of
CP violation in this system. Such a description has several advantages. First, it
clarifies the meaning of the phases which have been experimentally observed.
Secondly, it makes possible a useful test for errors in theoretical calculations.
Namely, if one computes the theoretical prediction for an experimental observable
using nothing but convention-independent variables, then it is easy to check by in-
spection that the prediction is convention-independent, as it must always be. If it
is not convention-independent, then one has made a mistake.

With the convention-independent description of CP violation in hand, we
shall discuss past experiments on the kaon analogues of the time-dependent Bd(t)
decays we considered in Section 4.1. Finally, we shall turn to future kaon experi-
ments.

5.1 Convention-Free Description of CP Violation

The existence of different phase conventions arises from the freedom to
redefine any quantum state by multiplying it by a phase factor. To develop a
phase-convention-free formalism, we must express every quantity of interest in
terms of variables that are manifestly invariant under such phase redefinitions of
the states.

--
When the phases of the states K0 and K0, and in particular their relative
phase, are arbitrary, we have

--
CP K0 = K0 , (5.1)

where is a phase factor. Elementary field theory then implies that


CP --
K0 = *K0 . (5.2)


26


--
Thus, within the neutral K system, in the K0, K0 basis, the operator CP is the
matrix


CP = 0 *

. (5.3)
0


From this matrix, we see that within the neutral K system,

(CP)1 = CP = CP , (5.4)

and

(CP)2 = I , (5.5)

where I is the identity matrix. From this last relation, it follows that the neutral
kaon CP eigenstates, K1,2, are given by


K = ei1(2)
1(2) K0 + CP K0
(-) , (5.6)
2

with

CPK1(2) = + K
(  ) 1(2) . (5.7)

In Eqs. (5.6), the overall phases 1,2 are arbitrary. However, when, as in either of
Eqs. (5.6), a state is expressed as a coherent superposition of several components,
the relative phases of the components had better not be arbitrary, because the con-
tributions from these components can interfere, with physical consequences,
when the state decays. To make this non-arbitrariness manifest in each of Eqs.
(5.6), we have written both components on the right-hand side in terms of the
same state, K0. It is then obvious that no arbitrary relative phase is involved.
(An operator, such as the CP operator in Eq. (5.6), does not introduce arbitrary
phases. These come only from states, or from the matrix elements of operators
between states.)

Let us now turn to the neutral K mass matrix M of Eq. (2.4). The diagonal
elements of this matrix are convention-free, since the arbitrary phase of the state
--
K0 obviously cancels out of M11 K0MK0 and that of K0 cancels out of M22
-- --
K0MK0. Thus, the CPT constraint that M11 = M22 X holds in any convention.

The eigenvalues of M--the complex masses of the mass eigenstates KS and
KL--are



27


S(L) = X +

M
(  ) 12M21 . (5.8)

We shall prove shortly that, as the notation implies, the eigenvalue X +

M12M21
(X 

M12M21) corresponds to the KS (KL). Being physically observable, these
eigenvalues cannot depend on conventions. As we have just seen, X is indeed con-
-- --
vention-free, and M12M21 = K0MK0K0MK0 clearly does not depend on the
phase of any state either.

The eigenstates belonging to the eigenvalues S(L) are, respectively,


K = eiS(L)
S(L) K0 + CP K0
(-) . (5.9)

1 + 2

Here, S(L) are arbitrary phases, and

12
K0 (CP)M K0

. (5.10)
K0 M (CP) K0


The arbitrary phase of the state K0 obviously cancels out of, so this quantity is
convention-free. Hence, so too is the relative phase of the two terms on the right-
hand side of Eqs. (5.9).

In terms of the CP eigenstates, the mass eigenstates KS(L) of Eqs. (5.9) are

1 +
K = + ~
[ ]
S(L) eiS(L) K2(1) . (5.11)
2 1 + 2
( ) ~K1(2)


Here,

~ -
K e i1(2) K
1(2) 1(2) , (5.12)

and

1 1
2 2
K0 K K0 M (CP) K0 - K0 (CP)M K0
1 K1 KL = 1 - = . (5.13)
1 1
K0 K 1 + 2 2
2 K2 KL K0 M (CP) K0 + K0 (CP)M K0






28


Note that is convention-free, and that, from Eqs. (5.12) and (5.6), the same is true
of the relative phase of ~
K1 and ~
K2. Thus, the relative phase of the two terms on
the right-hand side of Eqs. (5.11) is independent of conventions.

When the neutral kaon mass matrix M is CP-invariant, we have (CP)1M(CP)
= M, so that M(CP) = (CP)M, and consequently vanishes. Thus, is a convention-
free measure of CP violation in the neutral K mass matrix.

As we noted earlier, CP violation in the neutral K system is small. From
the fact that the amplitude for KL is much smaller than that for KS
[see Eqs. (2.11) and (2.12)], and the fact that CP() = +1, we know that it is KS
which is close to being a CP-even eigenstate of CP, and KL which is close to being
a CP-odd one. From Eq. (5.13), we see that when CP-noninvariance of M is
small, is small. Thus, it is clear from Eq. (5.11) that the mass eigenstates we
have labeled "KS" and "KL" are indeed respectively the KShort and KLong.
Hence, the corresponding eigenvalues, "S" and "L" of Eq. (5.8), are indeed re-
spectively the complex masses of KShort and KLong.

In studying the decays of neutral kaons to a final state f, it will be useful to
have the convention-free parameter


f T K
L KL K0
f . (5.14)
f T K
S KS K0

When f is a CP eigenstate with even CP parity,
--f would vanish in the absence of
CP violation, and serves as a convention-free measure of this violation.

In the literature, discussions of CP violation in the kaon system are almost
always carried out within specific phase conventions. Almost universally, these
discussions adopt the convention that S = L = 0 in Eqs. (5.9) for the states KS(L).
They also adopt the independent convention that 1 = 2 = 0 in Eqs. (5.6) for
K(1(2). Finally, they choose the additional convention that = +1 in the CP rela-
tion (5.1), as we did in Section 2. Alternatively, they choose = 1.

In the literature, neutral kaon decay to the final state f is commonly
described in terms of the parameter


f T KL
f , (5.15)
f T KS




29


especially when f is a 2 state. We note that the phase of f depends on the conven-
tions for the phases of KL and KS. Now, from Eqs. (5.9), we see that in the con-
vention where S = L, KLK0 / KSK0 = 1. Thus, in this convention,


--f = f . (5.16)

That is, our
--f is a convention-free analogue of the traditional parameter f, and
the two agree in the most commonly used convention for the phases of KL and
KS.

The violation of CP in the neutral K mass matrix M is traditionally de-
scribed in terms of the convention-dependent parameter , which may be defined
by


K1 KL . (5.17)
K2 KL

When M is CP-invariant, KL has no CP-even (i.e., K1) component, so vanishes.
From Eqs. (5.13) and (5.6),

= ei(2  1) . (5.18)

Thus, is a convention-free analogue of , and in the popular phase convention
where 2 = 1 = 0, the two agree.24

5.2. Some Existing Observations of CP Violation in the K System

In Section 2, we already mentioned two CP-violating effects which have
been seen in neutral kaon decay. The first of these is the decay of KL, which in the
absence of CP violation would have CP = 1, to , which has CP = +1. Since
KLK0 / KSK0 is just a phase factor [see Eqs. (5.9)], we see from Eqs. (2.11) and
(2.12) that the magnitudes of -- --
+
-- + and oo
-- 00 are both approximately
2.28 x 103, and, within errors, are equal. The second CP-violating effect we men-
tioned is the charge asymmetry of Eq. (2.13).

There is a third observed CP-violating effect, closely related to the decay KL
, and to the non-exponential decays of Bd(t) mesons to CP eigenstates de-
scribed by Eq. (4.27). This effect is found in the decay K0(t) f of a time-evolved
neutral K, which at time t = 0 was a pure K0, into the final state f = + or f =
00. Now, the KN (N = S or L) mass eigenstate component of a K0 evolves in
time t into KNexp(iNt). From this fact and Eqs. (5.9) and (5.14), it is trivial to




30


show that the time-dependent probability for the decay K0(t) f, (K0(t) f), is
given by

- 2 -
K0(t) f
( )e St + Lt +
f e

+ - 1 +
2 (
2 S L )t cos(m ) . (5.19)
f e Kt - f

Here, we have written the complex mass N of KN as mN  iN/2, where mN is
the mass of KN and N is its width. The mass difference mK is defined as mL 
m -- --
S, and f is the phase of f. Note from Eq. (5.19) that because both the KS and KL
components of a K0(t) can decay into (in violation of CP), the rate for K0(t)
receives a contribution from the decay of the KS component, another from that of
the KL component, and a third from an interference term.

A fourth observed CP-violating effect, very similar to the one found in K0(t)
f, is seen in the decay of neutral kaons produced by a regenerator. The regener-
ator is a slab of material on which is incident a pure KL beam--a neutral K beam
from which the KS component has long since decayed away. The regenerator
recreates a KS component in this beam. It is able to do so because a KL is a coher-
--
ent superposition of K0 and K0, and the amplitudes for the latter two particles to
scatter in a material medium differ. Thus, what emerges from the medium will
--
be a different K0K0 superposition from the one which was incident. That is, the
emerging kaon beam will contain a KS component. In particular, if a kaon enters
the regenerator as a pure KL, it will emerge in the state Kr given by

Kr = KLKLRTKLR + KSKSRTKLR . (5.20)

Here, R stands for the regenerator, so that KL(S)RTKLR is the amplitude for
the regenerator to emit a KL (KS) when a KL is incident. Now, after a time t in the
rest frame of the kaon Kr, its KN (N = L or S) mass eigenstate component will
have evolved into exp(iNt)KN. Thus, the Kr will have evolved into the state
Kr (t) given by

Kr(t) = eiLtKLKLRTKLR + eiStKSKSRTKLR . (5.21)

Omitting an irrelevant overall constant, the amplitude for this time-evolved kaon
to decay to the final state f, fTKr(t), is just

fTK --
r(t) f eiLt + reiSt . (5.22)

Here,r is the convention-free KS regeneration amplitude defined by



31


KL K0
r KSR T KLR . (5.23)
KLR T KLR K
S K0

From Eq. (5.22), the probability (Kr(t) f) for a neutral kaon to decay to a final
state f at a proper time t after emerging from a regenerator is given by

(K --
r(t) f) 
r 2eSt + f2eLt +
+ 2
r
--fe(S+L)t/2 cos(mKt +r f) . (5.24)

Here,r is the phase of 
r . If f is a state (hence CP-even), only the first term in
Eq. (5.24) would be present were it not for CP violation.

Through experimental studies of KS  KL interference terms such as those
in (K0(t) f), Eq. (5.19), and (Kr(t) f), Eq. (5.24), we have learned that25

mK = (3.4894  0.0073) eV , (5.25)

that25

-- --
+ arg (+) = (43.56  0.56) , (5.26)

and that2

-- --
oo arg (oo) = (43.5  1.0) . (5.27)

In the literature, the numbers quoted in Eqs. (5.26) and (5.27) are referred to, re-
spectively, as "the phase of +" and "the phase of oo". In the most popular phase
convention, in which --
f = f, these numbers do have this significance. However,
they do not have this meaning in general, since, as we have noticed, the phase of
f, Eq. (5.15), depends on conventions. The convention-free quantities whose
phases, in any convention, have the values quoted in Eqs. (5.26) and (5.27) are,
respectively, -- --
+ and oo.


5.3. Indirect and Direct CP Violation

There are two ways in which CP can be violated in neutral K decay. First, it
can be violated as a consequence of the CP-noninvariance of the neutral K mass
matrix, which causes the mass eigenstates KS and KL to deviate slightly from
being pure CP eigenstates. When the KL, while dominantly the CP-odd state K2,
contains a small admixture of the CP-even state K1, as in Eq. (5.11), it can decay to
the CP-even state + through its K1 component. It can do this even if the actual
K decay amplitudes conserve CP, so that +TK2 = 0.


32


The violation of CP stemming from the fact that KS and KL are not CP
eigenstates is called "indirect CP violation".

The other way in which CP can be violated is through the decay amplitudes
themselves. Examples of possible CP violations in K decay amplitudes would be a
nonvanishing value of the CP-changing decay amplitude +TK2, or a non-
--
vanishing value of the difference l+TK0  +l
TK0 between the ampli-
tudes for two CP-mirror-image processes.

The violation of CP in decay amplitudes themselves is called "direct CP
violation".

Suppose that f+ is a CP-even final state. Suppose further that there is no
direct CP violation. Then f+T ~
K2 = 0. Thus, from Eqs. (5.14), (5.11), and (5.9),

f
= + T ~
K1 =
f . (5.28)
+ f
+ T ~
K1

That is, when there is no direct CP violation, the parameters --
f for different CP-
+
even final states f+ are all equal. In particular, they are all equal to. Now, Eq.
(5.13) clearly implies that


K0 M
[ ,CP] K0
= . (5.29)
2
K0 M (CP) K0
[ ]12 + K0(CP)M K0
[ ]12


This expression makes it particularly obvious that vanishes when M is CP
invariant. Since is small, the two terms in the denominator D of Eq. (5.29) are
approximately equal. Thus, from Eqs. (5.3) and (5.8),


D 4

M12M21 = 2 (S  L) . (5.30)

The numerator N of Eq. (5.29), being convention-independent, may be evaluated in
the convention where the of the CP relation (5.1) is unity. In this phase convention,

N = M12  M21 . (5.31)

Now, it can be shown that in the difference M12  M21, the dispersive part of the
matrix element dominates strongly over the absorptive part.26 Furthermore, the
dispersive part of M12 is real and equal to that of M21, except for CKM elements in
the former which are replaced by their complex conjugates in the latter. Thus, N


33


= M 12  M21 is pure imaginary. If, in particular, arg N = /2, then, from Eq.
(5.30),

-
arg = tan 1 2 mK
= (43.46  0.08)o . (5.32)

S - L

In the absence of direct CP violation, this angle (or, for arg N = +/2, this angle
plus ) is the predicted phase of -- --
+ and of o o. Comparing Eq. (5.32) with Eqs.
(5.26) and (5.27), we see that the agreement is superb. We note that in obtaining
arg, we used Eqs. (5.8) for the eigenvalues of M. These equations assume the CPT
constraint M11 = M22 X. Thus, the agreement between the phase we calculated
for and the measured phases of -- --
+ and oo is a test of CPT invariance.


All confirmed CP-violating effects observed to date can be explained in
terms of indirect CP violation alone. For example, as we have already remarked,
the measured magnitudes of -- --
+ and oo are compatible with equality, as required
when there is no direct CP violation. (We shall return to this point.) In addition,
the measured value of the charge asymmetry , Eq. (2.13), is compatible with the
hypothesis that this asymmetry arises purely from indirect CP violation. This
hypothesis is expected to be a very good one, since, as illustrated in Fig. 8, in the
SM there is only one diagram for the decay KL l+, and, similarly, only one
for KL +l
. The violation of CP arises from phase factors, and these phase
factors never produce physical effects unless there is an interference between
amplitudes proportional to them. When a decay involves only one diagram, hence
only one amplitude, there can be no interference. Therefore, the decay amplitude
cannot violate CP. That is, there can be no "direct" CP violation.

l+ l


s
 u s u
K K
L - L +

d d
(a) (b)

Figure 8. (a) The sole SM diagram for KL l+. (b) The sole SM diagram for KL
+l
. Note that KL l+ proceeds only through the K0(
sd) component of the KL,
-- 
while KL +l
proceeds only through the K0(sd) component.

To see that the value of is compatible with the absence of direct CP viola-
tion, we note from Eqs. (5.9) and (5.13) that




34


--
KL (1 +)K0  (1 )K0 . (5.33)

Recalling (see Fig. 8) that KL l+ and KL +l
occur only through the K0
--
and K0 components of the KL, respectively, we have

l+TKL (1 +) l+TK0 (5.34)


and

--
+l
TKL (1 ) +l
TK0 . (5.35)

--
If there is no direct CP violation, then l+TK0 and +l
TK0, being
decay amplitudes for CP-mirror-image processes, have equal magnitude. Then

+ -
(K
L -l ) - (KL +l )
" + "

= 1 + 2 - 1 - 2 (5.36)
" + "
2e ,



where we have used the fact that
2 1. Now, absent direct CP violation,  =
--
+ = 2.28 x 103. Thus, if direct CP violation is also absent from , then, from
Eq. (5.36), cannot exceed 2 (2.28 x 103) = 4.56 x 103. The measured value of
quoted in Eq. (2.13) satisfies this constraint easily.

While there is as yet no firm evidence for direct CP violation, a great effort
has been made to find such evidence by showing experimentally that in K ,
-- --
oo + , in violation of Eq. (5.28). However, so far, this challenging effort has
been inconclusive. The reported experimental results are

2
-4
NA31 Experiment 27
"e = (23  6.5)  10
"= 1 1- oo . (5.37)
6 +- (7.4  5.2  2.9)  10-4 E731 Experiment 28



(In the second of these results, the first error is statistical and the second system-
atic.) Plainly, more needs to be done to clarify the situation. More sensitive exper-
iments which will try to establish that
-- --
oo + are planned for both Fermilab and
CERN. In addition, at the coming factory DANE, an effort will be made to es-
tablish the existence of direct CP violation by following the ingenious suggestion29
to study the decay chain




35


K + K . (5.38)


0 0
+

To see that the probability of this chain depends on whether there is direct CP vio-
lation, consider the special case where the two kaons decay simultaneously in the
rest frame. Since the has S = 1, the primary decay KK leaves the kaons in
a p wave. As a result, these two kaons cannot decay simultaneously to the same
final state.30 For, if they did, then just after their decay, we would have two identi-
cal spinless bosonic systems (one from each of the kaons) in an overall p wave, in
violation of the rule that one cannot have two identical bosons in an antisymmet-
ric state. Thus, if at some time t one of the kaons decays to +, then at this time,
--
the other kaon must be that linear combination of K0 and K0 which cannot decay
to +. Now, in the absence of direct CP violation, we have TK2 = 0. Then
--
the linear combination of K0 and K0 which cannot decay to + is simply K2.
However, (in the absence of direct CP violation) K2 cannot decay to 00 either.
Thus, when there is no direct CP violation, the two kaon decays in the decay
sequence (5.38) cannot occur simultaneously.

Of course, the experiment to study the decay chain (5.38) will not restrict
itself to events in which the two kaons decay simultaneously. However, by consid-
ering this special case, we have seen that the experiment will be sensitive to
whether direct CP violation is present or not.

If, as the SM states, CP violation is due to complex phases in the CKM ma-
trix, then direct CP violation is indeed expected to occur, both in K and B decays,
apart from exceptions such as K (  )
L +
l . In particular, barring an accident,
in K the direct CP violation TK -- --
2 0 does indeed occur. Then + oo,
or equivalently, the parameter "e ('/)" of Eq. (5.37) is nonvanishing. However,
calculating the precise SM prediction for e ('/) is very challenging. From exist-
ing calculations, one predicts only that31

2 x 104 < e ('/) < 13 x 104 . (5.39)

Nevertheless, for e ('/) to vanish, or to be much smaller than 104, seems un-
likely. Thus, it is very interesting to search, with a sensitivity at the level of 104, for
a nonvanishing value of this directly-CP-violating quantity. Establishing a nonvan-
ishing value at this level would not only serve as a test, at least qualitative, of the
SM picture of CP violation, but would also discriminate against the models which
ascribe CP violation to a so-called "superweak interaction"32 lying beyond the SM.
In general, superweak models of CP violation predict that e ('/) << 104. 32,33


36


5.4. The Rare Decay KL 0 

Measurement of the branching ratio for the so far unobserved rare decay
KL 0 would provide a clean test of the SM of CP violation, complementing the
tests to come from B decays.

The system 0 can be in either a CP = +1 or a CP = 1 state. However,
neglecting neutrino mass, when this system is produced by SM interactions in KL
decay, it will be in a pure CP = +1 state. But in the absence of CP violation, CP(KL)
= 1. Thus, the decay KL 0 violates CP.

To see why the SM interactions yield a purely CP-even final state in KL
0, we note that the CP of the final state is given by

CP(0) = CP(0) CP() (1)L , (5.40)

where CP() is the CP of the  pair, and L is the orbital angular momentum of
the 0 relative to this pair in the KL rest frame. Since the KL is spinless, L = J(),
where J() is the total angular momentum of the pair. Now, when neutrino
mass is neglected, a neutrino produced by SM interactions will be left-handed,
and an antineutrino right-handed. Thus, in the effective Hamiltonian Heff for KL
0, in the rest frame of the  pair, the only operator which can create this
pair is L
L, where L is the left-handed projection of the neutrino field. (Other
operators bilinear in the neutrino field would create a neutrino pair with the
wrong helicities. For example, the scalar operator  would create a and  of
like, rather than opposite, helicity.) Now, the  pair created by the action of L
L
on the vacuum will have CP() = +1, since L
L is even under CP. In addition,
this pair will have J() = 1, since L
L is a spatial three-vector operator. Thus,
since CP(0) = 1, Eq. (5.40) yields CP(0) = +1.

In the SM, KL 0 comes from the diagrams in Fig. 9, plus the related
--
diagrams in which the decay goes through the K0, rather than the K0, component
of the K *
L. Notice that all the diagrams in Fig. 9 are proportional to VtsVtd. Thus,
--
their K0 analogues, in which every quark has been replaced by its antiquark, are
proportional to V *
tsVtd. Now, from Eqs. (5.9), (5.10), and (5.1),

--
KL
M12K0  M21K0 . (5.41)
--
At the quark level, the K0 K0 mixing amplitude M12 arises from diagrams such
as, for example, the one in Fig. 10, proportional to (V *
csVcd)2. Suppose this diagram





37


 


Z Z
 
t t W W
  
s
 W d s
 t d
KL 0 KL 0
d d d d




l

W W
 
s
 t d
KL 0
d d


Figure 9. The SM diagrams for KL 0 through the K0(sd) component of the KL.

c
s d

K0 W W K0

d s
c
--
Figure 10. A diagram for K0 K0 mixing.

dominates M *
12, so that its CP-conjugate, proportional to (VcsVcd)2, dominates
--
M 21. Then, from Eq. (5.41), the diagrams of Fig. 9, and their K0 analogues, we
have for the KL 0 amplitude

0TK * * * *
L VcsVcdVtsVtd  VcsVcdVtsVtd . (5.42)

That is,

0TK * *
L m(VcsVcdVtsVtd) J . (5.43)





38


From Fig. 7, it is easy to see that the phase-convention-independent quantity J is
just twice the area of the ds unitarity triangle. What is less obvious is that all six
unitarity triangles have the same area, so that J, which is known as the Jarlskog
invariant, is twice the area of any of them.34 Since the unitarity triangles can
have nonzero area only if the CKM matrix contains CP-violating complex phase
factors, J is a convention-free measure of CP violation in the CKM matrix.

In addition to the mixing diagram of Fig. 10, one expects significant contri-
butions from other processes where the quark (antiquark) line involves an inter-
mediate u (u
 ).35 These contributions to M *
12 are proportional to (VusVud)2, rather
than (V * *
csV c d)2, and their counterparts in M 21 are proportional to (Vu sVud)2.
However, in view of the relative sizes of the terms in the ds unitarity constraint of
Eqs. (4.36), this constraint implies that, apart from a minus sign, V *
usVud has the
same phase as V *
csVcd to within a few milliradians. As a result, even when the u-
quark contributions to M12 and M21 are included, the CP-violating phase probed by
K * *
L 0
 is still that of VcsVcdVtsVtd, and we still have 0TKL J.

The decay KL 0 is not only CP violating, but is strongly dominated by
direct CP violation.36 One way to see this is to estimate
--
o . To this end, we note
that once the heavy W and Z boson degrees of freedom in the diagrams of Fig. 9
are integrated out, the effective Hamiltonian Heff for KL 0 is given by31

H * 
eff = G VtsVtd (s
 d) (L L) + h.c. . (5.44)

Here, G is a constant which is fairly well determined once the mass of the top
quark is given. For KS 0, the appropriate effective Hamiltonian represents
both the diagrams of Fig. 9 (with KL KS) and similar diagrams witht replaced
by c, and so is more complicated than the Heff of Eq. (5.44). However, we will get
the right order of magnitude for the KS 0 amplitude if we neglect the charm
contribution, and take Heff to be given by Eq. (5.44) for both KL 0 and KS
0.37 Then from the definition (5.14) for --f, Eq. (5.41) for KL and its analogue
for K --
S, and Eq. (5.44) for Heff, we have for o
 the estimate


M * 0 * 0
12VtsVtd s d K0 - M 21VtsVtd d s K0 L 0
L
0 ~ . (5.45)
" + " "

Now, as we have already noted, the CKM phase of
M *
12 is that of VcsVcd. Thus,
we may write

M *
12 = r12VcsVcd , (5.46)




39


where r12 has no CKM phase. Similarly, we may write

M *
21 = r21VcsVcd , (5.47)

where r21 has no CKM phase. Let us now go to a phase convention in which V is
real when CP is conserved. In such a convention, we must have


r 0 0
12 s d K0 = r21 d s K0 , (5.48)


since --
o  must vanish when CP is conserved. Hence, from Eq. (5.45), in our
chosen convention, when CP is not conserved, we must have


im V V* V* V
( )
cs cd ts td
0 ~ e V V* *
( ) . (5.49)
cdVtsV
cs td

Since both sides of this relation are phase-convention independent, this estimate
holds in any convention.

Combined with our knowledge of the CKM matrix,38 the relation (5.49)
yields the estimate 0.1 --
~
<o ~< 1. Even though this estimate was obtained
neglecting the charm-exchange contribution to KS 0, we may safely con-
clude that -- --
o  is much larger than the corresponding K parameters +
and
--
oo, both of which are ~2x103. However, from Eq. (5.28) we know that when
direct CP violation is absent, the parameters
--f for different CP-even final states
are all equal. Thus, we conclude that if the SM description of K 0 is correct,
direct CP violation is present in neutral K decays.

From Eqs. (5.14) and (5.11), we have

A
-- 2 +A1
o  = , (5.50)
A1 +A2

~
where A2 is the CP-violating decay amplitude 0TK2 , and A1 is the CP-
~
conserving one 0T K -- --
1 . As we have seen,
+ 2x103, but o
is much larger than this. Thus, from Eq. (5.50), A2 >> A1. That is, in
0TKL, which from Eq. (5.11) is proportional to A2 +A1, the directly CP
violating term A2 dominates over the indirectly CP violating oneA1.39

To calculate (KL 0), one must evaluate the matrix element




40


0H * 
effK0 = G VtsVtd 0s
 dK0 L L0 (5.51)

of the effective Hamiltonian of Eq. (5.44), and the simply-related matrix element
--
0HeffK0. In Eq. (5.51), the leptonic matrix element is, of course, trivial, and
the hadronic matrix element 0 s
 dK0 is related by isospin40 to 0s
uK+, a matrix element which has been measured by determining the rate for K+
0e+. For fixed input parameters (the top quark mass, for example), the theo-
retical uncertainty in BR(KL 0) is only 1.4%.31 Thus, measurement of this
branching ratio could yield a rather accurate value of the Jarlskog invariant J--
information about the CKM matrix which must be consistent with that from other
sources.

To be sure, the measurement would not be easy. Given our present knowl-
edge of the relevant CKM elements, BR(KL 0) = (1-5)x1011 in the SM.
Observing such a rare decay will be a challenge. Adding to the challenge will be
the fact that BR(KL 00) / BR(KL 0) ~ 108, so that one must take steps to
ensure that any decay identified as KL 0 is not really the much more likely
KL 00, with one of the pions having escaped detection. Nevertheless, it is to be
hoped that BR(KL 0) will indeed be measured.

6. Do Electric Dipole Moments Violate CP?

Outside of the neutral meson systems, CP violation has been sought by
trying to show that a spin one-half fermion, such as the neutron or electron, has
an electric dipole moment (EDM). It is trivial to prove, as we shall shortly, that
such an EDM would violate invariance under time reversal T. If one then
assumes that the world is invariant under CPT, it follows that an EDM would
violate CP. We would like to close the present article by asking whether one can
prove directly that an EDM violates CP without invoking CPT invariance.

Let us first prove that if a quantum system with definite angular momen-

tum s has an EDM, then T invariance is violated. By the rotational properties of

the system, the EDM, El
, must point along the vector s . That is,


El
= gEl s , (6.1)

where gEl is a constant. Imagine, now, that the system is in a static external

electric field E. The interaction energy E due to the EDM is then


E = El
 E = gEl s E . (6.2)





41



Now, it is obvious that under time reversal, E E, and s s . Thus, E  E.
That is, if the interaction energy of the world includes a term stemming from an
EDM, then this energy is not invariant under T.

If nature is described by a local, relativistically-invariant quantum field
theory (such as the SM and its extensions), then it is invariant under CPT.41
Thus, it is relatively safe to assume that CPT invariance does hold, so that an
EDM violates not only T but also CP.

But what if CPT does not hold? Can we prove that an EDM still violates CP?
To try to do so, we recall that, apart from a constant, the EDM of a spin one-half
fermion f, El(f), is just the q2 = 0 value of the form factor E(q2) in the "electric
dipole term"

i E(q2)  u
 2  q 5 u1 (6.3)

in the general decomposition of the amplitude for + f f, the absorption of a pho-
ton by f. Here, q is the momentum carried by the photon,  is the photon polariza-
tion, and u1 and u2 are, respectively, the initial and final Dirac spinors for f. To
explore the CP properties of the electric dipole coupling, (6.3), let us go to the cross
channel, where we have the process f + f. In this channel, the amplitude (6.3)
becomes

i E(q2)  u
 f  q 5 vf , (6.4)

where v is the Dirac spinor for an antifermion. We are now in a region where q2
(2mf)2, where mf is the mass of f, while in considering + f f we were in a re-
gion where q2 0. Now, one can show that if f + f proceeds through the cou-
pling (6.4), then theff pair will be produced in a state which in the nonrelativistic
limit is 1P1. Furthermore, the CP of anff pair in a state with orbital angular
momentum L and total spin S is just (1)S+1. Thus, the electric dipole coupling
(6.4) leaves theff pair in a state with CP = 1. But a photon has CP = +1. Thus,
when f + f proceeds through the electric dipole coupling (6.4), CP is not
conserved.

We have obtained this result for f + f without invoking CPT. Have we
thereby proved that an EDM violates CP even when CPT does not hold? Actually,
we have not! Recall that the EDM of f is proportional to the value of E(q2) at q2 = 0.
To relate E(q2=0) to E(q2) in the positive q2 region which corresponds to f + f,
we must invoke "crossing". Crossing tells us that the analyticity properties of E(q2)
are such that if this form factor is nonzero in the region q2 0, which corresponds
to + f f, then it is also nonzero in the region q2 (2mf)2, which corresponds to


42


f + f. Thus, from our previous argument, if El(f) 0, then f + f violates CP.
However, this author does not know of any way to prove that crossing holds with-
out assuming that nature is described by a local, relativistically-invariant quan-
tum field theory. And, if we assume that nature is described by such a theory,
then CPT invariance holds! Thus, we have not succeeded in proving that an EDM
violates CP even when CPT does not hold. Indeed, we still do not know whether an
EDM necessarily violates CP under such circumstances.

7. Conclusion

It is particularly natural to hypothesize that CP violation is an effect of the
SM weak interaction, which means that it comes from phases in the quark mix-
ing matrix. During the next two decades, this hypothesis will be cleanly tested
through elegant experiments on B and K decays. Regardless of whether these
experiments confirm or disprove the hypothesis, their results will be exciting
news.

Acknowledgements

It is a pleasure to thank G. Buchalla, I. Dunietz, B. Holstein, and B.
Winstein for helpful conversations, and to thank the Fermi National Accelerator
Laboratory for excellent hospitality during the time that these lectures were
prepared.





43


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