NSF-PT-97-2





CASCADE MIXING AND THE CP-VIOLATING ANGLE BETA*



Boris Kayser

National Science Foundation
4201 Wilson Boulevard
Arlington, VA 22230 USA





Abstract

In the decay chain Bd + K + (l), neutral K mixing
follows on the heels of neutral B mixing. This "cascade mixing" leads
 17 Sep 1997 to an interference which probes cos 2, where is one of the three
CP-violating phase angles which characterize CP violation in the
Standard Model. Widely-discussed future B-system experiments will
determine trigonometric functions of these three phase angles,
leaving the underlying angles themselves discretely ambiguous. A
determination of cos 2 through cascade mixing would eliminate all
the discrete ambiguities entirely.




* This report is to appear in the Proceedings of the Moriond Workshop on Elec-
troweak Interactions and Unified Theories, held in Les Arcs, France, March
1997. It discusses work done in collaboration with L. Stodolsky. Some aspects of
this work are reported in .


Suppose a neutral B, which is a coherent superposition of mass eigenstates,
decays to a final state containing a neutral K, which is also such a superposition.
Then we can have K mixing following on the heels of B mixing. We refer to this as
"cascade mixing".1) As we shall see, cascade mixing could help us test the Stan-
dard Model of CP violation by providing important information which cannot be
obtained by the more commonly discussed future B- or K-system CP experiments.

--
Let us call the mass eigenstates of the Bd-Bd system BHeavy (BH) and BLight
(BL). These states have complex masses

M
B B
Heavy(Light) = MB +
()  i , (1)
2 2

where MB is their average mass, MB is their mass difference and is defined to be
positive, and B is their common width. Similarly, the mass eigenstates of the K0-
--
K0 system, KShort (KS) and KLong (KL), have complex masses

m
 K S(L)
S(L) = mK 
(+)  i , (2)
2 2

where mK is the average of the KS and KL masses, mK m(KL)  m(KS) is their
positive mass difference, and S,L are the KS,L widths.

Under special circumstances, the composition of a neutral meson in terms
of its mass-eigenstate components can be tuned. For example, consider the kaons
produced by a "regenerator". Incident on the regenerator is a pure KL beam. With
some amplitude r, the regenerator introduces into the beam a KS component.
That is, when a kaon emerges from the regenerator, it is in the state

|Kr |L + r|S . (3)

By changing the characteristics of the regenerator, one can change r and thereby
change the KS, KL composition of Kr.

Suppose, now, that we produce neutral kaons via the decay

Bd + K . (4)
After B


In this decay, the parent is produced as a pure Bd at time 0 in its rest frame,
and then decays into + K after a proper time B. It is straightforward to show
that the K created by this decay is born in the state |KFrom B given by


1


- M M
i B i B
B B
K 2 2

From B e + i tane KS

. (5)
M M

i B -i B
+ B B
e 2 + i tane 2 KL




Here,

*

arg- VcdVcb , (6)
*

V
tdVtb

where V is the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix.
-- --
Elements of V occur in the diagrams for Bd-Bd mixing, K0-K0 mixing, and BK
decay. As a result, the relative phase of these elements, which if nonvanishing
is CP-violating, occurs in the expression (5) for |KFrom B.

From Eq. (5), we see that by selecting events with a particular B, we can
tune the KS, KL composition of KFrom B, just as we can tune that of the Kr from a
regenerator. Both the relative phase and magnitude of the KS and KL components
of KFrom B vary with B. In particular, from the squares of the coefficients of KS
and KL in Eq. (5), we find that

Prob ( K ( )
S ) = 1- sin 2 sin MB B
Prob ( K 1 + sin 2 sin(M ) . (7)

L ) B B

Here, Prob (KS) is the probability of finding a KS in KFrom B, and similarly for
Prob (KL).

Existing data allow sin 2 to lie anywhere in the range 0.38 to 0.94.2) From
Eq. (7), we see that, so long as sin 2 is not very close to unity, Prob (KS) and Prob
(KL) are of the same order of magnitude.
As time passes after the decay Bd After + K produces a neutral kaon, the
B 
KS, KL composition of this kaon changes as a result of K-K mixing, and we no
longer have the composition given by Eq. (5). When the kaon itself decays, its KS
and KL pieces contribute coherently. The interference between their contributions
can reflect both the B mixing which took place before the B decayed, and the K
mixing which took place after it decayed. The B mixing plays a role because it
determines the state |KFrom B in which the kaon is born, and the subsequent K




2


mixing plays one because it determines how the KS, KL composition of the kaon
changes after it is born.

If the KS-KL interference when the kaon decays to some final state f is to be
significant, the KS and KL contributions to the decay must be comparable. Since,
as we have seen, the KS and KL components of the kaon wave function are of the
same order of magnitude, this means that the decay amplitudes A(KSf) and
A(KLf) must be comparable. In the table below, we show the ratio |A(KLf) /
A(KSf) | for the common final states.3) In the table, l is an e or .

A( K )
Final State f L f
A( K )
S f

0.002

0.007

l 1.00

+0 ~24

From this table, we see that it is in the semileptonic decays K l that KS-KL
interference will be appreciable.

Let us consider, then, the decay chain

Bd + K . (8)
After B
After K l+

Here, a Bd decays at proper time B after its birth, as measured in its rest frame,
and its daughter K decays at proper time K after its own birth, as measured in its
own rest frame. In the Standard Model (SM), the final state l+ can come only
--
from the K0, and not the K0, component of the K. (There is, of course, also the
--
alternative final state +l
 , which can come only from the K0 component. For
the moment, we consider only the final state l+.) The amplitude for the decay
chain (8) is given by4)

+
Amp B
[ ( )]
d
+ K
+ -l =
After B After K



= - -
A(B ) i i
M B ( ) N K
d is BM e A BM KN e A K N is K 0
( ) (9)
M = Heavy,

Light
N
=S,L
+
 A K 0 -l
( )





3


Here, A(Bd is BM) is the amplitude for the initial Bd to be, in particular, the B
mass eigenstate BM, exp(iMB) is the amplitude for this BM to propagate for a
proper time B, A(BMKN) is the amplitude for the BM to decay to a plus the K
mass eigenstate KN, exp(iNK) is the amplitude for this K mass eigenstate to
propagate for a proper time K, A(KN is K0) is the amplitude for the KN to be, in
particular, a K0 (the only component of a K that can decay to l+), and A(K0
l+) is the amplitude for a K0 to decay to l+. That the amplitude for a B of com-
plex mass M to propagate for a proper time B is exp(iMB)4) follows trivially
from Schrdinger's equation applied in the rest frame of the B. The amplitude for
a K to propagate follows similarly.

An amplitude similar to that of Eq. (9) describes the decay chain where the
--
parent B is born a Bd, and/or the K decay channel is +l
 .

It is interesting to compare the amplitude (9) for Bd + K + (l)
with the corresponding amplitude for the more familiar process Bd +KS. [At
the practical level, by "KS" one means here a neutral kaon which decays to
(which KL does only rarely) within roughly one KS lifetime.] We have


Amp Bd
+ KS
After
B =
= -
A(B ) iM B ( ) . (10)
d is BM e A BM KS
M = Heavy,

Light

Here, only one K mass eigenstate, and no nontrivial K propagation or K mixing, is
involved. While this amplitude sums only over B mass eigenstates, the amplitude
(9) for Bd + K + (l) sums over both B and K mass eigenstates, reflecting
the nontrivial role of both B and K mixing in the process it describes.

Inserting the SM values for A(Bd is BM) and A(BM KS) in Eq. (10), and
--
squaring this amplitude and its analogue for a parent B born as a Bd, we find the
famous result5)

(-)
-
B B B ( )
d
+ KS 1 - sin 2 sin M
( + ) B B
After

B e . (11)

Since KS is a CP eigenstate, these two decay rates are the rates for CP-mirror-
image processes. Thus, the difference between them violates CP invariance. As we
see, this difference measures sin 2. Moreover, it does so cleanly. That is, beyond
their dependence on the CP-violating angle , which one would like to determine
to test the SM of CP violation, the decay rates (11) depend only on parameters
which are already known, B and MB. There is no dependence on unknown or
uncertain quantities.


4


We now insert in Eq. (9) the SM values for A(Bd is BM), A(BM KN), and
A(KN is K0), neglecting the small violation of CP in the K system. Then, squaring
--
the amplitude of Eq. (9), and its counterparts for a parent B which is born as a Bd
and/or a daughter K which decays into +l
, we find that

(-)

B d
+ K
+ ml
( )
After B After K


- -
e B B e S K 1 - ( )
( + ) sin 2 sin MB B

-
+ e L K 1 + sin 2 sin(M )
( - ) B B

 - 1 . (12)
( + )
2e 2 S L K cos
[ (M )cos(m )
(m) B B K K

+ cos 2 sin(M ) ( )]
B B sin mK K




Here, at the head of the third line on the right-hand side, the pair of signs which
--
is not (is) in parentheses corresponds to an initial Bd (Bd). The upper sign of each
of these two pairs is for K l+, and the lower one for K +l
 .

From the way in which the various parts of the decay rates (12) depend on S
or L, it is obvious that the first line in the expression for these rates is the contri-
bution of the KS component of the daughter kaon, the second line is the contribu-
tion of the KL component, and the remaining lines are the KS-KL interference
term.

(--)
While the rates (12) for Bd + K + (l) are not as simple as the rates
(--)
for Bd + KS, they are every bit as clean. That is, they depend only on the angle
one would like to determine, and on the already-known parameters B, S, L,
MB, and mK.

To test the SM of CP violation, one would like to determine from B decays
the CP-violating angles , , and , where is defined by Eq. (6),

*

arg- VtdVtb , (13)
*

V
udVub
and
*

arg- VudVub . (14)
*

V
cdVcb



5


These three angles are the interior angles of what is commonly referred to as the
CKM unitarity trangle.6) To carry out the test of the SM, one would like to deter-
mine , , and themselves, and not just trigonometric functions of them such as
sin 2, which leave the underlying angles discretely ambiguous. Unfortunately,
(--)
the rates for Bd + KS, the B decays where CP violation will probably be first
(--)
sought, measure only sin 2 (see Eq. (11)). In contrast, the rates for Bd + K
+ (l) are sensitive to both sin 2 and cos 2 (see Eq. (12)). Thus, once sin 2
(--) (--)
(hence |cos 2|) has been determined from Bd + KS, the rates for Bd + K
+ (l) could be used to determine the sign of cos 2. This additional informa-
tion could do much more than partially eliminate the discrete ambiguity in .
(--) (--)
Indeed, if, as hoped, sin 2 will have been found from Bd + KS, sin 2 from Bd
, and cos 2 from B DK, a determination of Sign(cos 2) would then suf-
fice to eliminate all discrete ambiguities from , , and .7)

(--)
To extract Sign(cos 2) from the decay rates for Bd + K + (l), one
would measure them as functions of B and K, and then compare them with Eqs.
(12), taking the known values of B, S, L, MB, mK, and sin 2 as inputs. The
ease with which the extraction could be performed would depend, of course, on
(--)
the event rate. Now, the popular decays Bd + KS will be detected principally
through the + decay mode of KS. Thus, we normalize the decay rate (12) rela-
tive to (11) by taking into account the branching ratios for kaon decay to ml and
to +. Then, integrating over K, we find that if, for example, cos 2 = sin 2,
then the cos 2 term in [Bd + K + (l+)] contributes an event rate 1/600
that contributed by the sin 2 term in [Bd + KS + (+)]. Hopefully, such
(--)
an event rate makes the study of the sign of the cos 2 term in Bd + K + (l)
feasible at hadron facilities, although it may not be feasible at e+e B factories. It
(--)
should be noted that the total number of decays of the type Bd + K + (l)
(--)
will actually be comparable to the number of the type Bd + KS + (+).
However, while the latter decays will be concentrated at values of K <
~ 1/S, the
former decays will be spread out over the much larger region K <
~ 1/L. The KS-KL
(--)
interference term in Bd + K + (l), where cos 2 appears, is only signifi-
cant when K <
~ 2/S, so that the KS component of the kaon has not already decayed
(--)
away. Thus, only the small fraction of all Bd + K + (l) decays which
(--)
have K <
~ 2/S are usable. This is how the event rate from the cos 2 term in [Bd
+ K + (l)] comes to be much smaller than that from the sin 2 term in
(--)
[Bd + KS + (+)].

(--)
To determine the decay times B and K in an event of the type Bd + K
+ (l), one would measure the B and K pathlengths and energies. The pres-
ence of an undetectable neutrino in the final state to which the K decays does not




6


make it impossible to determine the K energy. Indeed, despite the neutrino, one
would have a four-constraint fit to the kinematics of the entire decay chain.8)

Why does Bd + KS probe only sin 2, while Bd + K + (l)
probes both sin 2 and cos 2? To answer this question, let us consider Fig. 1. As

BH +KL

(--)
B
d +(l)


BL +KS

(--)
Figure 1. The paths through which the decay chain Bd +K +(l) can proceed.

--
shown there, a particle born as a pure Bd or a pure Bd has a BH and a BL compo-
nent. Either of these mass eigenstate components can decay to either + KL or
+ KS. Subsequently, either the KL or KS can decay to l. Thus, as illustrated in
(-- )
Fig. 1, there are four paths through which the parent Bd can produce the final
state + (l). Now, in the limit where CP is conserved and vanishes, the
intermediate states in Fig. 1, BH, BL, KL, and KS, are all CP eigenstates. In
particular, CP(BH) = CP(KS) = , while CP(BL) = CP(KL) = +.9) Thus, the decays
BH KL and BL KS, represented by dashed lines in Fig. 1, connect states
which in the CP-conserving limit are of opposite CP parity. Consequently, the
amplitudes for these decays must vanish as CP violation (hence ) goes to zero,
and one finds by explicit calculation that they are proportional to sin . In
contrast, the decays BH KS and BL KL, represented by solid lines in Fig. 1,
connect states which in the CP-conserving limit are of the same CP parity. Thus,
the amplitudes for these decays are expected to survive as CP violation goes to
zero, and one finds that they are proportional to cos . Now, from Fig. 1, we see
(--)
that the decays Bd + KS involve only two paths, one through BH KS, and
one through BL KS. It is the interference between the amplitudes for these two
(--)
paths that leads to CP violation in Bd + KS. Since A(BH KS) cos , while
A(BL KS) sin , this interference is proportional to cos sin , or sin 2.
(--) (--)
This is why Bd + KS probes only sin 2. In contrast, in the decay chain Bd
+ K + (l), there are the four paths shown in Fig. 1, and all of them inter-
fere. Since A(BH KS) and A(BL KL) are both cos , the interference
between them is proportional to cos2. Similarly, the interference between A(BH
KL) and A(BL KS) is proportional to sin2. Obviously, a suitable linear



7


(--)
combination of cos2 and sin2 will yield cos 2. This is why Bd + K +
(l) probes cos 2.10)

(--)
In conclusion, in Bd + K + (l), the very interesting phenomenon
of cascade mixing11) makes possible the determination of cos 2. In combination
with other measurements which probably would precede it, a measurement of the
sign of cos 2 would eliminate all the discrete ambiguities in the CP-violating
phase angles , , and of the Standard Model.12)


Acknowledgments

This paper was written at CERN and the Max-Planck-Institut fr Physik.
The author warmly thanks Guido Altarelli for the excellent hospitality of CERN,
and Leo Stodolsky for that of the Max-Planck-Institut and for a fruitful and
enjoyable collaboration.





References:

1. Earlier work on K mixing following B mixing is reported in Y. Azimov, Phys.
Rev. D 42, 3705 (1990).

2. A. Ali and D. London, Nucl. Phys. Proc. Suppl. 54A, 297 (1997).

3. M. Barnett et al. (the Particle Data Group), Phys. Rev. D 54, 1 (1996).

4. B. Kayser and L. Stodolsky, Phys. Lett. B359, 343 (1995).

5. See, for example, I. Bigi, V. Khoze, A. Sanda, and N. Uraltsev, in C P
Violation, edited by C. Jarlskog (World Scientific, Singapore, 1989) p. 175.

6. For a discussion of how the SM of CP violation may be tested in B decays, see,
for example, Y. Nir and H. Quinn, in B Decays, edited by S. Stone (World
Scientific, Singapore, 1992) p. 362, and B. Kayser, in Trieste HEP Cosmology
1995, p. 432 (and .

7. B. Kayser and D. London, in preparation.

8. We thank R. Raja for a clear explanation of the relevant kinematics.

9. For the B mass eigenstates, this assumes as usual that the hadronic matrix
--
element for Bd-B d mixing has the sign which follows from the vacuum
insertion approximation. This sign is given in I. Dunietz, Phys. Rev. D 52,
3048 (1995). If this sign is incorrect, the CP parities and roles of BH and BL are
interchanged, but the present argument is otherwise unaltered. We thank I.
 
Dunietz for a helpful conversation about the signs of B-B and K-K mixing. For




8


a discussion of the role of these signs in determining CP asymmetries, see Y.
Grossman, B. Kayser, and Y. Nir, in preparation.

10. An alternative approach to probing cos 2 is described in L. Oliver, talk given
at the Babar Workshop, Princeton, March 1997, and in J. Charles, A. Le
Yaouanc et al., in preparation.

(--)
11. It has been suggested that cascade mixing in Bs + K + (l or )
might facilitate the determination of the mass difference between the mass
--
eigenstates of the Bs-Bs system. See Y. Azimov and I. Dunietz, Phys. Lett.
B395, 334 (1997).

12. The Moriond talk on which this written version is based included a discussion
of the oscillation frequency in neutral particle mixing. This topic is treated by
B. Kayser, in ICHEP 96, p.1135, and in B. Kayser, Stanford Linear Accelerator
Center preprint SLAC-PUB-7123, so it has been omitted here.





9



