NSF-PT-96-3
December 1996

CP VIOLATION, MIXING, AND QUANTUM MECHANICS

Boris Kayser
Physics Division, National Science Foundation
4201 Wilson Blvd., Arlington, VA 22230 USA


ABSTRACT

We discuss the quantum mechanics of B-factory experiments, and that of neutral K or
B decay. Predictions for the processes to be studied at the B factories can be made through
an approach based on amplitudes, rather than on wave functions. This approach avoids the
puzzles of the "collapse of the wave function." In the treatment of the decay of a neutral K
or B, the contributions of the different mass eigenstate components of the decaying particle
must be evaluated at exactly the same spacetime point. Otherwise, the frequency predicted
for the oscillation of the decay rate may be incorrect.

According to the Standard Model (SM), CP violation comes from complex phase factors in the
Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. In the B meson system, some of the
anticipated CP-violating asymmetries can cleanly probe the phases of various products of CKM el-
ements. Thus, these asymmetries can incisively test whether CKM phases are indeed the origin of
CP violation.

Most of the clean information on CKM phases will come from the decays of neutral B mesons.
 -- 
The neutral B meson Bd, a bd bound state, and its antiparticle Bd, a bd bound state, mix. As a re-
--
sult, the Bd Bd system has two mass eigenstates, BHeavy (BH) and BLight (BL), with complex
masses

 = + m
B -
H( L) mB (-) i . (1)
2 2

Here, mB is the average of the BH and BL masses, mB is their mass difference, and is the width
which to a very good approximation they have in common.1 Due to the mixing, a neutral B which
at some proper time = 0 is known to be a pure |Bd will not remain so. Rather, after a proper time
it will have evolved into a state |Bd(), given according to the SM by

 13 Feb 1997 -
i m -i
B

B 2 
d ( ) = e
(2)
*

 m V m
cos B tdVtb B
- i sin
2 Bd V* 2 Bd
tdVtb
--
Similarly, a neutral B which at = 0 was a pure |Bd will have evolved after proper time into a
-- --
state |Bd() which, like |Bd(), is a superposition of pure |Bd and pure |Bd. Note from Eq. (2)
--
that, until it decays, a |Bd() oscillates back and forth between being a pure |Bd and a pure |Bd.

Future, complementary experiments on CP violation in B decays will be carried out at dedi-
cated high-luminosity e+e colliders ("B factories") and at hadron facilities. At the B factories, the
B mesons will be produced via the process

e+e (4s) BB . (3)

Half the time, the B mesons yielded by this process will be neutral, and it is only this case with
which we shall be concerned.

To appear in the Proceedings of the 28th International Conference on High Energy Physics,
Warsaw, July 1996.
1


Let us consider the process (3) in the (4s) rest frame. Since the (4s) has intrinsic spin J = 1,
--
but B mesons are spinless, the B pair from (4s) BB is in a p wave. Owing to Bd Bd mixing,
--
each member of this pair oscillates back and forth between |Bd and |Bd [cf. Eq. (2)]. However,
owing to their common origin in the decay (4s) BB, the two B mesons in the pair are corre-
lated. In particular, they cannot both decay to the same final state f at the same time t in the (4s)
frame.2 For, if they did, then just after their decay we would have two identical J = 0 bosonic
systems--one from each of the B mesons--in an overall p wave. This would violate the rule that
two identical bosons cannot be in an antisymmetric state.

Consider, then, the situation in which the B mesons from (4s) BB decay at different
(4s)-frame times t1 and t2>t1. Suppose the first decay, at time t1, yields the final state f1. Then,
by the previous argument, we know that at time t1, the B which did not yet decay must be in a state
which cannot decay to f1. Thus, at time t1, this surviving B must be in the state |BNot f given by
1

B = -
Not f Bd f1 T Bd Bd f1 T Bd , (4)
1


since, obviously, f1|T|BNot f = 0. One says that the decay of the first B from (4s) BB at
1
time t1 "collapses the BB wave function," uniquely fixing the state of the remaining B at the same
time. With the state of the remaining B known, one can apply the Schrdinger equation for the Bd
--
 Bd system to evolve this state forward from time t1. One can then calculate the probability for this
surviving B to decay to a final state f2 at time t2. Putting everything together, one finds, for exam-
ple, that if f1 = lX, a semileptonic state, and f2 = Ks, then
-
Prob [One B l X at t1; Other B Ks at t2 ]
(5)
-(t +t )
e 2 1 1 - sin
{ ( ) [ ( - )]}
K sin mB t2 t1 .
s

Here, "Prob" stands for probability, and Ks is the phase of a certain product of CKM elements.
The expression (5), with t1 and t2 times in the (4s) rest frame, neglects the motion of the B
mesons. This is an excellent approximation, since in the (4s) frame each B has v/c 0.06.
However, it would be desirable to have a description of the sequence (4s) BB f1f2 which is
fully consistent, not only with quantum mechanics, but also with relativity. In addition, it would be
advantageous to have a description which avoids the collapse of the BB wave function. One would
then avoid the enigma of how, without any interaction between the two B mesons after they are
born and separate, the B which decays last can know what final state the one which decays first
produces, and when it does so.

What we wish to describe is a decay chain of the form

(4s) B + B , (6)

f (t , x )
2 2 2

f (t , x )
1 1 1

where f1 is the final state into which one of the B mesons decays, (t1, x1) is the spacetime point at

which the decay occurs, and similarly for f2 and (t2, x2). One can describe this chain in a way
which takes relativity fully into account, and avoids the collapse of the wave function, by simply
calculating directly, without going through a BB wave function, the amplitude for the entire
chain.3,4 To this end, it is convenient to work in the neutral B mass eigenstate basis. The amplitude
for the chain (6) has two terms. The first of these represents a process in which the B which prop-

agates from the (4s) decay point to (t1,x1) and decays to f1 is a BH, while the one which propa-


2



gates to (t2, x2) and decays to f2 is a BL. The second term represents a process in which the roles
of BH and BL are interchanged. Since the BHBL mass difference is tiny, these two processes are
experimentally indistinguishable, so their amplitudes must be added coherently. Due to the anti-
symmetry of the (4s) BB amplitude, the two B mesons in (6) cannot both be BH or BL.

The amplitude AHL for the process where the B which decays to f1 (f2) is a BH (BL) is given
by

AHL = A(BH to 1; BL to 2) eiH1 eiL2 
 A(BHf1) A(BLf2) . (7)

Here, A(BH to 1; BL to 2) is the amplitude for an (4s) to decay to a BH moving towards (t1, x1)

and a BL moving towards (t2, x2). (This amplitude is antisymmetric under BH BL.) The factor
exp(iH1) is the amplitude for the BH to propagate from the spacetime point where it is born to

the point (t1, x1) where it decays. In this factor, 1 is the proper time which elapses in the BH rest
frame during the propagation. Similarly, exp(iL2) is the amplitude for the BL to propagate to

(t2, x2). [That the amplitude for a particle of mass  to propagate for a proper time is exp(i)
follows trivially from Schrdinger's equation applied in the rest frame of the particle.] Finally,
A(BHf1) is the amplitude for BH to decay to f1, and similarly for A(BLf2). If the various A's
on the right-hand side of Eq. (7) are Lorentz invariant, AHL is Lorentz invariant.

Adding to AHL the amplitude for the process in which the roles of BH and BL are interchanged,
and using the antisymmetry of A(BH to 1; BL to 2), we find that the complete amplitude A for the
decay chain (6) is simply

A eiH1 eiL2 A(BHf1) A(BLf2)
 eiL1 eiH2 A(BLf1) A(BHf2) . (8)

If, for example, we apply this general expression to the case where f1 = lX and f2 = KS, we
find that

Prob [One B lX at 1; Other B KS at 2]
e(2+1) {1  (sin KS) sin[mB(21)]} . (9)
This result agrees with the one of Eq.(5) found by collapsing the BB wave function, except that the
times t1,2 in the rest frame of the (4s) have been replaced by the proper times 1,2 in the rest
frames of the B mesons.5 This is a negligible correction for (4s) BB, but a (2-3)% effect for
KK, a process involving almost identical physics to be studied at the factory DANE.
Of course, the incorporation of small relativistic effects is not the main benefit of the amplitude
approach. A more important benefit is the ability to derive theoretical expressions such as (9)
which describe B-factory experiments without having to invoke the "collapse of the wave function"
or to puzzle over its riddles.


As our discussion of B-factory experiments illustrates, the behavior of a neutral B meson in-
volves some very interesting quantum mechanical effects. Indeed, this is true of a neutral K, D, B,
or, assuming neutrino mass, a neutrino. Any of these particles is a superposition of mass eigen-
states with different masses. These mass eigenstate components contribute coherently when the
particle decays or interacts. Interference between these coherent contributions causes the probabil-
ity for the decay or interaction to oscillate with the time or distance that the particle travels. Let us
consider this oscillation.6




3


The oscillatory behavior of a particle which is a superposition of several mass eigenstates is
nicely illustrated by the propagation of a kaon produced by a KS regenerator. Incident on the re-
generator is a pure KL beam. With amplitude r, the regenerator introduces into the beam a KS
component. Thus, a kaon emerging from the regenerator is in the state

|Kr = |KL + r |KS . (10)

Let us call the spacetime point where the kaon leaves the regenerator (0, 0). Suppose this kaon then
propagates to a spacetime point (t, x), where it is observed to decay into +. Since its |KL and
|KS components propagate differently, the kaon is no longer in the state |Kr of Eq. (10) when it
arrives at (t, x), but in a different state we shall call |Kr(t, x). The probability [Kr(t, x) +]
for this kaon to decay to + at the point (t, x) is given by

Kr(t, x) +-
[ ]=
(11)
- 2
N
A(K i N +-
r is KN )e A(KN ) .
N = S, L


Here, A(Kr is KN) is the amplitude for the original Kr to be the mass eigenstate KN, exp(iNN)
is the amplitude for this mass eigenstate to propagate from the regenerator to the decay point (t, x),
and A(KN +) is the amplitude for KN to decay to +. In the propagation amplitude exp
(iNN), N mN iN/2 is the complex mass of KN, and N is the proper time which elapses in
the KN rest frame during its propagation. By including a superscript N on N, we are allowing for
the possibility that N may depend on whether the KN is a KS or KL.

The precise meaning of the proper time N is somewhat subtle. Since understanding its mean-
ing is crucial to the correct treatment of particles which are mixtures of mass eigenstates, let us try
to clarify what N is.3,6

The key point is that the interfering KS and KL contributions to the decay of our kaon, like the
interfering components of any spacetime-dependent wave, must be evaluated at precisely the same
spacetime point (t, x).3,6 To be sure, for a given momentum p, the KS and KL components of the
kaon move at different speeds, because they have different masses. Thus, classically , these com-
ponents, which were both born at the spacetime point (0, 0), cannot both arrive at the decay point x
at the same time t. However, quantum mechanically, the propagating kaon is described by a wave
packet, with some central momentum p. This wave packet has KS and KL pieces. Since the KS is
slightly lighter than the KL, the center of the KS piece of the wave packet moves faster than that of
the KL piece. Thus, the centers of the KS and KL pieces become displaced relative to each other.
Nevertheless, at the time t that the kaon decays, the KS and KL pieces of its wave packet overlap.
It is the contributions from these two overlapping pieces at the common point x where the kaon
decays that are to be added coherently. Thus, in Eq. (11), N is simply the time in the KN rest
frame of the decay event (t, x). That is,

N = 1 [E ]
N ( p)t - px , (12)
mN

where p is the central momentum of the wave packet, and E 2
N(p) = (p2 + mN )1/2.

From Eq. (12), it is easily shown that through first order in mK mL  mS, S = L. Thus,
through first order in mK, one may compute the relative phase between the two interfering ampli-
tudes in Eq. (11) taking S and L to have a common value . Experimentally, this is given in
terms of the meaured distance x travelled by a kaon before decay and the measured momentum p of
the kaon by


4


m
= x K . (13)
p

1
Here, mK (m
2 L + mS) is the average kaon mass.

[In complete analogy with the kaon proper times, the B meson proper times in Eqs. (7) and (8)
are, through first order in mB, independent of whether a BH or a BL is propagating. For this
reason, we have not distinguished between the proper times for these two cases.]

With S and L taken to have a common value in Eq. (11), this expression yields the familiar
result7
- 2 -
[K S + L +
r (t, x) +- ] r 2 e +- e
(14)
+ - 1( + )
2 r +- e 2 S L cos(m + -
K r +- ) .

Here, + A(KL+) / A(KS+), and r and + are, respectively, the phases of r and
+. The oscillatory last term on the right-hand side of Eq. (14) arises from the interference be-
tween the KS and KL contributions to the decay. Note that the frequency of oscillation of this term
with is mK. Our knowledge of mK (and similarly mB) stems solely from measurements of
this frequency and that of related oscillations.

What if, instead of evaluating the KS and KL contributions to the kaon decay at a common de-
cay point x and a common decay time t, we mistakenly evaluate them at the (unequal!) KS and KL
classical arrival times at the point x? The classical arrival time of KN at x is tN = x[EN(p)/p], and
the corresponding proper time is N = x(mN/p). If we use classical arrival times, then in Eq. (11)
this value of N replaces the quantity of Eq. (13). The relative phase of the KS and KL terms in
Eq. (11) is then

x
m 2 2
LL  mSS = (mL  mS ) = 2(m
p K) , (15)

where is the proper time of the decay determined experimentally via Eq. (13). But then the KS 
KL interference term in the predicted decay rate oscillates with at a frequency of 2(mK), rather
than the correct frequency, mK. That is, evaluating the contributions of the different mass eigen-
states at their classical times of arrival at the decay point yields an oscillation frequency double the
true value.

In summary, the contributions of the different mass eigenstate components of a neutral K, D,
or B to the decay of this particle must be evaluated at exactly the same spacetime point. Evaluating
them this way shows that the frequency with which the decay probability oscillates is always m,
the difference between the masses of the mass eigenstates. This is true both for an isolated neutral
K, D, or B, and for one which is part of a pair created in KK or (4s) BB. Several recent
descriptions of neutral particle propagation which differ from our own also conclude that the oscil-
lation frequency is m.8

Nevertheless, there is one analysis which does not agree.9 This analysis finds that, while the
probability for decay of an isolated neutral K oscillates like cos (mK), the probability for
KK f1f2, the -factory analogue of the decay chain (6), oscillates like cos [2(mK)(2  1)].
Thus, this analysis predicts that when the world's first factory is turned on, the oscillation fre-
quency will be found to be twice what we and most others expect.

Now, inspection of the treatment of Ref. 9 shows that in dealing with KK f1f2,, it
takes the KS and KL components of each kaon to have a common momentum, and evaluates their


5


contributions to the kaon decay at their classical arrival times at the decay point. As we have seen,
such a procedure gives an oscillation frequency of 2(mK), while the correct value is mK. Thus,
the frequency found in Ref. 9 for KK f1f2 is not correct. In particular, the prediction of
this reference that the oscillation frequency in KK f1f2 is twice that in decay of an isolated
K is not correct. This becomes especially clear when one notes that the treatment of Ref. 9, applied
to B mesons, would predict that the oscillation frequency in (4s) BB f1f2, 2, is twice the
frequency in decay of an isolated B, 1. This prediction is already experimentally contradicted.
Experiments at LEP and the Tevatron find that10

1 = (0.458  0.020) ps1 , (16)

while measurements at ARGUS and CLEO teach us that11

2 = (0.42  0.06) ps1 . (17)

Clearly, 2 is not twice 1, but equal to 1.

To sum up, the quantum mechanics of the B factory experiments can be simply understood,
without invoking the "collapse of the wave function," by calculating amplitudes directly. The
quantum mechanics of a propagating neutral K, D, B, or neutrino requires that the contributions of
the different mass eigenstate components of this particle to its decay or interaction always be eval-
uated at precisely the same spacetime point.



References

1I. Bigi, V. Khoze, N. Uraltsev, and A. Sanda, in CP Violation, ed. C. Jarlskog (World
Scientific, Singapore, 1989) p. 175.

2H. Lipkin, Phys. Rev. 176, 1715 (1968).

3B. Kayser and L. Stodolsky, Phys. Lett. B359, 343 (1995).

4B. Kayser, in `95 Electroweak Interactions and Unified Theories (Proceedings of the XXXth
Rencontres de Moriond), ed. J. Trn Thanh Vn (Editions Frontieres, Gif-sur-Yvette, 1995) p. 373.

5An astute guesser could have guessed that the effect of taking the relativistic motion of the B
mesons into account would be to replace t1,2 by 1,2.
6The discussion which follows is based on B. Kayser, SLAC-PUB-7123, to be published.

7See, for example, K.Kleinknecht, in CP Violation, ed. C. Jarlskog (World Scientific, Singapore,
1989) p. 41.

8Y. Grossman and H. Lipkin, Weizmann Inst. of Science preprint WIS-96/23; H. Lipkin, Phys.
Lett. B348, 604 (1995); J. Lowe et al., Phys. Lett. B384, 288 (1996) (This paper, like ours,
emphasizes that the contributions of different mass eigenstates must be evaluated at the same
spacetime point.). See also M. Nieto, e-print .

9Y. Srivastava, A. Widom, E. Sassaroli, Z. Phys. C66, 601 (1995).

10S. L. Wu, in Proceedings of the 17th International Symposium on Lepton-Photon Interactions,
eds. Z. Zhi-Peng and C. He-Sheng (World Scientific, Singapore, 1996) p. 273.

11H. Schrder, in B Decays (Revised 2nd Edition), ed. S. Stone (World Scientific, Singapore,
1994) p. 449.





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