The Frequency of Neutral Meson and Neutrino Oscillation

Home , Kaon

SLAC{PUB{7123

The Frequency of Neutral Meson and Neutrino

Oscillation

Boris Kayser

Stanford Linear Accelerator Center

Stanford University, Stanford, California 94309

and

Division of Physics, National Science Foundation

4201 Wilson Boulevard, Arlington, Virginia 22230

Abstract

Interference b etween the di erent mass eigenstate comp onents of a neutral K meson

causes its decay probability to oscillate with time. Related oscillations o ccur in the

decaychain ! KK ! f f where f are decaychannels, in neutral B decay,

1 2 1;2

in the chain 4s ! BB ! f f , and in massive neutrino propagation. Since the

1 2

mass eigenstates comprising a neutral K , a neutral B , or a neutrino have di erent

masses, they have di erent sp eeds at any given momentum.Thus, classically, they

b ecome separated in space and time. This circumstance can tempt one to evaluate

their contributions to the K or B decay, or to the neutrino interaction with a detector,

at di erent spacetime p oints. However, these quantum-mechanically interfering con-

tributions must always b e evaluated at precisely the same p oint. Evaluating them at

di erent p oints can lead to predicted oscillation frequencies double their true values.

Preliminary

Work partially supp orted by the Department of Energy, contract DE{AC03{76SF00515.

Permanent address.

The neutral K meson pro duced in a typical kaon exp eriment is a sup erp osition of

two mass eigenstates: the long-lived K , and the shorter-lived and very slightly lighter

K . When the K meson decays, its K and K comp onents contribute coherently.

S L S

The interference b etween their coherent contributions causes the probability of decay

into a given nal state to oscillate with time. The frequency of this oscillation is

m m m , the di erence b etween the K and K masses.

K L S L S

For a given momentum, the K and K comp onents of a kaon travel at di erent

L S

sp eeds, due to their di ering masses. Hence, classical ly, in the lab oratory frame of

reference they will arrive at the p oint x where the kaon decays at slightly di erent

L S

times, t and t . One might then b e tempted to evaluate the K and K contributions

L S

to the kaon decay amplitude at these classical arrival times. But then one would

be evaluating these two coherent contributions at di erent times. This would b e an

incorrect pro cedure. Following it can lead, as we shall see, to the erroneous conclusion

that the frequency of oscillation in the decay probability is not m , but 2m [1].

K K

The oscillation in the probability for decay of a single kaon has an analogue in the

decaychain

! K + K ;

! f ! f

1 2

where the kaons are neutral, and f are nal states of interest. There are further

1;2

analogues in single neutral B decay, in the decaychain 4s ! BB ! f f where

1 2

the B mesons are neutral, and in neutrino oscillation. In each of these situations, one

has a propagating particle or particles which is a coherent sup erp osition of several

mass eigenstates with di erent masses. In every case, one must evaluate the coherent

contributions of the di erent mass eigenstates to the amplitude for decay or detection

of the propagating particle at precisely the same spacetime p oint. Calculating these

contributions at di erent p oints and then adding them coherently is an erroneous

pro cedure whichinevery instance can yield an oscillation frequency double the true

value.

In this Letter, we will examine the oscillation in neutral meson decay, fo cusing 2

on the physics related to the oscillation frequency.We will see that a recent claim

that in ! KK ! f f this frequency is 2m , while in isolated single K decayit

1 2 K

is m , cannot b e correct, b ecause the B -meson analogue of this claim is decisively

contradicted by exp eriment. We will then turn to the analysis on which the claim is

based, and discover that for ! KK ! f f , this analysis entails the evaluation at

1 2

di erent spacetime p oints of the coherent K and K contributions to decayofa kaon.

L S

We will see explicitly that when the contributions of the di erent mass eigenstate

comp onents of a propagating particle to the decay or detection of this particle are

not calculated at the same spacetime p oint, the predicted oscillation frequency can

be twice its true value, not only in ! KK ! f f , but quite generally in b oth one-

1 2

and two-neutral-K and neutral-B pro cesses, and in neutrino oscillation.

Let us recall the standard results for neutral meson decay. The probability

[K ! f at ] for an isolated neutral K to decay to a nal state f at prop er time

, if at time = 0 this K was a pure K ,isgiven by [2]

0 2 +

S L S L

[K ! f at ] / e + j j e +2j je cosm : 1

f f K f

Here, and are, resp ectively, the widths of the K and K , and

S L S L

hf jT jK i

: 2 j je

f f

hf jT jK i

The oscillatory last term in Eq. 1 comes from the interference b etween the K and

K contributions to the decay. Next, supp ose wehave a neutral kaon pair pro duced

in a p wave via the pro cess ! KK. The joint probability [One K ! f at ;

1 1

Other K ! f at ] for one memb er of this pair to decay to the nal state f at

2 2 1

prop er time after its birth in the decay, while the other decays to nal state f

1 2

at time , is given by [3, 4 , 5]

[One K ! f at ; Other K ! f at ]

1 1 2 2

i i

1 2

L S

/ e e AK ! f AK ! f 3

L 1 S 2

i i

2 1

L S

: AK ! f AK ! f e e

S 1 L 2

Here, m are the complex masses of K , and AK ! f , etc.,

L;S L;S L;S L;S L 1

are decay amplitudes. The interference term in the probability 3 has the oscillatory 3

time dep endence

+ +

2 1

S L

cos[m + ] ; 4 e

K 2 1 0

where is time-indep endent. Note that the frequency of this oscillation, m ,is

0 K

the same as that of the oscillatory term in the decay rate 1 for a single kaon in

isolation.

The standard result for the probability[B ! f at ] for an isolated neutral

nonstrange B to decay to a nal state f at prop er time ,ifat = 0 this B was a

pure B , is Eq. 1 with the K -meson quantities replaced by their B -meson coun-

terparts [6]. The frequency of the oscillation is nowm mB mB , the

B H L

di erence b etween the masses of the heavier mass eigenstate B and the lighter

mass eigenstate B of the B B system.

For B mesons, the counterpart to a K pair pro duced in a p wave via ! KK

is a B pair pro duced in a p wave via 4s ! BB [7]. The standard result for the

joint probability [One B ! f at ; Other B ! f at ] for the decays of the

1 1 2 2

memb ers of this pair is Eq. 3 with K and K replaced by the B mass eigenstates,

L S

and and by their complex masses [8, 9 , 6, 5, 10 ]. The interference term 4 in

L S

this joint probabilitynow has frequency m . AsintheK system, this frequency is

the same as in the decay of a single B in isolation.

Sto dolsky and I have presented an approach to the treatment of a decaychain

suchas ! KK ! f f or 4s ! BB ! f f in which the amplitude for

1 2 1 2

the entire chain is calculated directly [5, 10 ]. Unlike the more traditional treatment

[3, 4 , 8 , 9 ] of suchachain, our metho d do es not entail the intro duction of a wave

function, or state vector, for the KK system. Consequently, this metho d avoids the

somewhat mysterious \collapse of the KK wave function," in which the decay of one

K ,or K K , content of the other K K at a certain instant determines the K

S L

at the same instant [3]. The amplitude approach also has the advantage of manifest

Lorentz covariance. Applying this approach, one readily repro duces the standard

expression 3 for the decay probability, without having to invoke the collapse of the

wave function. Obtaining 3 by the amplitude metho ds makes it evident that the

times in this decay rate are prop er times in the K rest frames, not times in the 4

rest frame. In the \collapse" approach, this p oint is not so clear, and one must guess

astutely.

In a Comment [11] on Ref. [5], and in other work [12] to which this Comment calls

attention, Srivastava, Widom, and Sassaroli SWS argue that in ! KK ! f f ,

1 2

the frequency of the oscillation in the decay rate is not m , as in 4, but 2m .

K K

Thus, these authors disagree with the early analyses of this pro cess, and with the

analysis in Ref. [5]. However, they agree that in the decay of an isolated single

neutral kaon that is not part of a kaon pair pro duced via ! KK, the oscillation

frequency is indeed m , as in the standard result, Eq. 1. Hence, they predict

that once the oscillation frequency in ! KK ! f f is measured, it will prove

1 2

to b e twice the already-determined oscillation frequency in the decay of an isolated

single kaon.

Now, the B -meson analogue of the SWS argument for kaons would predict that in

4s ! BB ! f f , the oscillation frequency in the decay rate is twice as large as

1 2

in the decay of an isolated single neutral B .However, this prediction is contradicted

by exp eriment. To see this, consider rst the decay of an isolated B , or, equivalently,

a B which is part of a complicated many-b o dy state pro duced in a Z decayora

high-energy p p collision. Supp ose that through tagging this B is known to b e a pure

0 0

B at prop er time =0. Owing to B B mixing, at subsequent times the B

will b e a B , B sup erp osition. Supp ose that at time >0, the B decays into a

nal state f which can come only from its B comp onent, or into the CP-conjugate

state f , which, of course, can come only from its B comp onent. The probabilities

[B ! f at ] for these two decays are given by[6]

[B ! f at ] / e [1 cos! ] : 5

B 1

Here, ! is the oscillation frequency, and all authors, including SWS, agree that

! =m . The quantity is the decay width that B and B unlike K and

1 B H L L

K have in common. The results 5, with ! =m ,may b e obtained from the

S 1 B

K -meson formula 1 by substituting for and , and m for m , and then

S L B K

using the easily demonstrated fact that the B -meson analogue of the parameter of

Eq. 2 is +1 for the nal state f , and 1 for f . Analysis of the decay of neutral B

B B 5

mesons pro duced in Z decays at LEP, and in pp collisions at the Tevatron, in terms

of Eqs. 5 has yielded the value [13]

! =0:458 0:020ps : 6

This analysis has included direct observation of the oscillation with time. The prop er

time of a B decay is determined by measuring the distance x which the B has

traveled and the B momentum p. Then

= x ; 7

where m is the average neutral B mass, and do es not distinguish b etween B and

B H

B .

Consider, next, the chains 4s ! BB ! f g , where the nal states f and

B B B

g can come only from a B , and g only from a B .For these chains, all agree that

B B

[One B ! f at ; Other B ! g at ]

B 1 B 2

2 1

1 / e + cos ! ; 8

2 2 1

except for a disagreement concerning the value of the oscillation frequency ! . All

early treatments [9, 6] and the more recent amplitude approach [5, 10 ] predict that

! =m . Indeed, Eqs. 8 with ! =m follow trivially from the B -meson

2 B 2 B

analogue of the standard K -meson formula 3 and the usual expressions for B and

B .However, by following the SWS approach, as describ ed B in terms of B and

in Ref. [12 ], we nd that they would predict that ! =2m . This is a B -meson

2 B

analogue of the oscillation frequency doubling they claim to b e presentin!KK.

Apart from the value of ! , their approach yields precisely the same decay probabil-

ities, Eqs. 8, as found by others. In the SWS treatment, these probabilities are

initially expressed in terms of the B pathlengths x and momenta p, but they coincide

with Eqs. 8, with ! =2m , once we use Eq. 7 to express them in terms of the

2 B

exp erimentally measured prop er times.

The ARGUS and CLEO collab orations have studied the time integrals over the

decay rates 8. These groups rep ort the ratio

f g N f g +N

B B B B

; 9 r =

Nf g +Nf g

B B B B 6

where

Z Z

1 1

N f g = d d [One B ! f at ; Other B ! g at ] 10

B B 1 2 B 1 B 2

0 0

is the total number of events in which one B decays to f and the other to g , and

B B

similarly for the other quantities. The state f is chosen to b e D ` , and g to

B B

b e the inclusive state X` . In employing the measured r to learn ab out ! , one

makes use of the fact that the omitted prop ortionality constants in the two relations

8 are equal. This equality holds so long as CP violation in the decay amplitudes

0 0

A may b e neglected. Since in the Standard Mo del semileptonic B B ! g

decay is heavily dominated by a single tree-level diagram, which is unlikely to have

signi cant comp etition from b eyond the Standard Mo del, this should b e an excellent

approximation. Similarly, so should the CP relations N f g =Nf g and

B B B B

f g = Nf g . Thus, integrating Eqs. 8, we conclude that N

B B B B

r = ; 11

where ! =. From the combined ARGUS-CLEO result [14],

r =0:179 0:039 ; 12

we then learn that [14]

=0:66 0:09 : 13

Now, the common B B lifetime,1=, has b een found to b e [15]

H L

=1:57 0:05 ps : 14

Combining this lifetime with the value of , Eq. 13, we nd that

! =0:42 0:06 ps : 15

This frequency is in excellent agreement with the frequency ! , Eq. 6, found in

single B decay. Had ! b een twice ! , as exp ected by SWS, the measured ! , Eq. 6,

2 1 1

would have predicted that ! =0:920:04ps ,invery strong disagreement with the

measured ! value, 0:42 0:06ps .Thus, strictly on the basis of exp erimental data,

2 7

we can conclude that the oscillation frequency in 4s ! BB ! f f is the same as

1 2

in single B decay, rather than twice as great. Since the physics of ! KK ! f f

1 2

is virtually identical to that of 4s ! BB ! f f ,we can also conclude that the

1 2

oscillation frequency in ! KK ! f f is the same as in single K decay.

1 2

From the standp oint of the \collapse" approach, it would have b een very surprising

if ! had di ered from ! . In this approach, the decay of one B in a B pair pro duced

2 1

via 4s ! BB xes the state of the remaining B at the same instant. Subsequently,

this remaining B oscillates b etween B and B states. Since this B is now alone, it

would b e surprising if the frequency of this oscillation were di erent than in the case

of a single B which is alone from the very b eginning [16 ].

Let us now compare the SWS treatment with the amplitude approach of Refs.

[5] and [10], and identify the features which lead SWS to exp ect that the oscillation

frequency in ! KK ! f f is 2m ,twice that in single K decay, while from the

1 2 K

amplitude approachwe exp ect b oth frequencies to b e m . In treating ! KK,

SWS intro duce a state vector for the KK system, and use eld-theoretic propagators

to describ e the propagation of the kaons. In the amplitude approach, the intro duction

of a KK wave function or state vector is carefully avoided, and eld theory is not

used. Complete decaychains suchas ! KK ! f f , including the kaon decay

1 2

amplitudes, which are omitted by SWS, are treated. Nevertheless, it app ears from

Ref. [12 ] that SWS would agree with the authors of the amplitude approach that the

oscillation frequency in ! KK ! f f is m , and not 2m , if the former only

1 2 K K

treated prop er times as do the latter.

The treatment of prop er times in the amplitude approach is most easily explained

by discussing the decay of a single kaon, b orn as a pure K ,into a nal state f .For the

amplitude for this pro cess, A, the amplitude approach yields the readily understo o d

result

0 i

AK is K e A = AK ! f : 16

N N

N =L;S

0 0

Here, the pure K is a sup erp osition of the mass eigenstates K and K , and AK

L S

is K is the amplitude for it to b e, in particular, the mass eigenstate K . The

N N

factor expi is the amplitude for this K to propagate for the prop er time

N N 8

interval that elapses b etween its birth and decay [17]. Finally, AK ! f is the

amplitude for the K to decayinto the state f . In principle, the prop er time that

elapses b etween the kaon birth and decaymay dep end on whether the kaon is a K or

a K ,sowe denote it by , with N = L or S . As explained in Ref. [5], the meaning

of this prop er time is somewhat subtle. We picture the kaon as b eing describ ed bya

wave packet, with some central momentum p [18]. Supp ose that the kaon is b orn at

the spacetime p oint 0,0, and decays at the p ointt; x. Now, the K and K have

L S

di erent masses, so, for a given momentum p, it is not p ossible classical ly for b oth the

K and K comp onents of the kaon, b orn at the p oint 0,0, to arrive at the lo cation

L S

x at the same time t.However, it is p ossible for the K and K comp onents of the

L S

kaon wave packet to overlap at the p ointt; x, with the center of the K comp onent

displaced relative to that of the K comp onent. These overlapping comp onents of

the wave packet are two amplitudes, corresp onding to the two terms in Eq. 16,

for the kaon to b e at the spacetime p ointt; x. It is the interference b etween these

amplitudes which leads to the oscillation in the decay probability. In calculating this

interference, wemust, of course, evaluate all phase factors at the common p ointt; x

N N

where the interference o ccurs. Thus, in the factor expi , the prop er time

is the kaon-rest-frame time which corresp onds to the decay p ointt; x. Hence, from

the Lorentz transformation, wehave for the mass eigenstate K with momentum p

2 2 1=2

and corresp onding energy E p=p +m ,

= E ptpx : 17

To rst order in m , this relation is

m m

K K

0 LS 0

t = ; 18

2m E p

K K

2 2 1=2

where m m + m =2 is the average neutral kaon mass, E p p + m

K L S K

is the corresp onding energy at momentum p, and [E pt px]=m is the

K K

L S 0

value of or for vanishing m .Now, and t are, of course, related by time

dilation: =m =E pt.Thus, from Eq. 18 we see that to rst order in m ,

K K K

L S 0

= = , and we shall refer to this prop er time simply as .To rst order in

m , the relative phase of the factors expi in the twointerfering terms in

K N 9

Eq. 16 is just

m m =m : 19

L S K

Hence, the oscillation with induced by the interference has frequency m . Using

0 0

the well-known fact that AK is K =AK is K , the amplitude of Eq. 16

L S

immediately yields the decay rate of Eq. 1.

In practice, the observer-frame time of the decay, t, is not measured. The lo cation

of the decay, x, and the momentum of the kaon, p, are measured, and t is then inferred

from x and p using the relation t =E p=px. The prop er time ofthe

decay is then found from t using the time-dilation relation =m =E pt. That

K K

is,

= x : 20

Equation 7 already gave the B -exp eriment analogue of this relation.

Wehave just seen that in K decay and similarly in B decay, the variation of

the prop er time of a given decay p ointt; x from one contributing mass eigenstate

to another is completely negligible. However, in the propagation of a neutrino, the

prop er time of a given neutrino interaction p ointt; xvaries very imp ortantly from

one mass eigenstate to another [5]. Thus, whether the prop er times asso ciated with

di erent mass eigenstates are equal or di erent dep ends on which problem one is

treating.

For the decaychain ! KK ! f f , the amplitude approach yields the time-

1 2

dep endent probability of Eq. 3, in which the rst term on the right-hand side is

the amplitude for a K to decayinto the nal state f at prop er time while a K

L 1 1 S

decays into the state f at prop er time , and the second term is the amplitude for

2 2

the pro cess in which the roles of K and K are interchanged. The time dep endence

L S

in Eq. 3 arises from the factors expi , with N = L or S and j = 1 or 2. Just

N j

as in the case of single kaon decay, to rst order in m the prop er time in eachof

K j

these factors do es not dep end on whether the factor is for propagation of a K or K .

L S

Accordingly,wehave written this prop er time simply as , not . Supp ose that,

in the rest frame, the kaons created in ! KK are observed to travel distances

x and x b efore decaying. Then, if p is the -frame momentum of either kaon, the

1 2 10

prop er times of the two decays are given by

= x ; j =1;2 ; 21

j j

just as in Eq. 20 for a single K decay.To rst order in m , the phase of the

factor expi is just m .Thus, to this order, the time-dep endent part of

N j N j

the relative phase of the twointerfering terms in Eq. 3 is

m m =m : 22

L S 1 2 K 1 2

Hence, the oscillation with prop er time induced by the interference has frequency

m |identical to the frequency in single kaon decay.

In the SWS approach, the kaon pair created in ! KK is describ ed by a state

vector j i of the form [12]

L S S L

i i i i

L S S L

2 1 2 1

jK K i : 23 e jK K ie e j i e

S L L S

To explain the notation, let us supp ose, as b efore, that in the rest frame the two

kaons are observed to travel distances x and x b efore decaying. In the rst term in

1 2

j i, jK K i is a state in whichitisaK which has traveled to x and a K to x ,

L S L 1 S 2

and in the second term the roles of K and K are interchanged. The prop er times

L S

N = L or S and j = 1 or 2 of the decays are not the prop er times of Eq. 21.

Rather, SWS state [11, 12] that if the kaon which travels the distance x is the mass

eigenstate K , the prop er time of its decay, to b e used in Eq. 23, is given by

= x : 24

Here, p is, as b efore, the -frame momentum of either memb er of the kaon pair.

Comparing Eqs. 23 and 3, we conclude that if the SWS expression for j i had

involved the same prop er times as do es the amplitude on the right-hand side of

Eq. 3, SWS would have found the same oscillation frequency in ! KK ! f f

1 2

as we did using the amplitude metho d; namely,m .However, if one uses the SWS

prop er times of Eq. 24, then the time-dep endent part of the relative phase of the

two terms in the SWS j i is

x x x x

1 2 1 2

2 2

m m = 2m m : 25

K K

L S

p p 11

In terms of the prop er times of Eq. 21 which will b e inferred from measurements,

this relative phase is just

2m : 26

K 1 2

Thus, SWS predict that in the oscillation as a function of exp erimentally determined

prop er times, the frequency is 2m . We see that it is their treatment of prop er

time that has led to the spurious factor of two in this prediction.

What is wrong with the SWS prop er time of Eq. 24? To answer this puzzle, let

us consider the reaction ! KK in the rest frame, which will b e the detector frame

at the factory DANE. SWS assume that the pathlength x of a kaon pro duced in

! KK is determined by measurement, and is indep endent of whether the kaon is

a K or a K . Similarly, they assume that the momentum p carried by the kaon is

L S

xed by the kinematics of ! KK, and is also indep endent of whether the kaon is

a K or a K . They then take the prop er time of the kaon decay to b e the quantity

L S

of Eq. 24, which do es dep end on whether the kaon is a K or a K .Now, the

L S

Lorentz transformation implies that the prop er time of the decay, , the -frame

time of this decay, t , and the measured -frame lo cation of the decay, x , are related

N N

m = E p t px ; 27

N N j

j j

2 2 1=2 N

where E p p + m .For the of Eq. 24, this relation implies that

N j

E p

t = x : 28

We recognize that this t is just the classical arrival time of the mass eigenstate

K at the decay p oint x . In particular, for SWS, the detector-frame time t of

N j

the decay dep ends on whether the decaying kaon is a K or a K . As a result,

L S

when the amplitude for ! KK ! f f is calculated from the SWS state vector

1 2

j i of Eq. 23, the contributions to this amplitude from the two terms in j i are

evaluated for di erent detector-frame times. The contribution from the rst term

which corresp onds to a K decaying at x and a K at x isevaluated for the pair

L 1 S 2

L S

of spacetime decay p oints t ;x , t ;x . But the contribution from the second term

1 2

;x , in which the roles of K and K are reversed is evaluated for the p oints t

1 L S

t ;x . The two contributions are then added coherently.

2 12

As noted at the b eginning of this Letter, such coherent adding of an amplitude

for decays at one pair of spacetime p oints to that for decays at a di erent pair of

p oints is an incorrect pro cedure. This is true even if, as here, the di erence b etween

the pairs is to o small to b e resolved exp erimentally. Note, for example, that one

cannot reliably calculate the intensityofa two-comp onent electromagnetic waveby

evaluating the amplitudes for the two comp onents at slightly di erent times and then

adding the results coherently. If, as in kaon decay, the two comp onents havevery

rapid time dep endence, such a pro cedure would yield completely incorrect results.

Similarly, quantum mechanical amplitudes to b e added coherently must corresp ond

to the same nal states and precisely the same spacetime p oints.

SWS describ e an isolated single kaon by a state vector [12] with a K and a K

L S

term, each of which has the same time dep endence as the corresp onding term in the

decay amplitude of Eq. 16. Thus, we see from Eq. 24 that, were SWS to interpret

prop er times in single kaon decay as they do in ! KK ! f f , the relative phase

1 2

of the K and K contributions to their single kaon decay amplitude would b e

L S

2 2

m m = 2m : 29

L S

Here, x and p are, resp ectively, the measured pathlength and momentum of the kaon,

and is the measured prop er time of its decay, inferred from x and p using Eq. 20.

We see that SWS would then predict that the oscillation frequency in single kaon

decayis2m , just as they do for ! KK ! f f . The reason that they actually

K 1 2

predict a frequency of m for single kaon decay is that, for this case, they assume

the K and K comp onents of the kaon to have, not a common momentum, but a

L S

common speed [12]. Thus, these comp onents cover the measured kaon pathlength x

in the same time, and, since their time dilation factors are equal, in the same prop er

time as well. The relative phase of the K and K pieces of the state vector is then

L S

just m , so that the oscillation frequency is m . While this agrees with the

K K

standard result, we cannot understand the basis for taking the K and K to have

L S

equal sp eeds. In the kaon wave packet, these comp onents have equal momenta, and

the true explanation for the frequency m is as given earlier.

A spurious factor of two has app eared, not only in the SWS analysis of ! KK ! 13

f f , but also in other discussions of neutral meson or neutrino oscillation. In every

1 2

case, this spurious factor can b e traced to the mistake of taking the spacetime p oint

of meson decay or neutrino detection to b e di erent for di erent mass eigenstates,

rather than b eing de ned by the exp eriment and common to all the mass eigenstates.

To see this, let us rst lo ok brie y at neutrino oscillation. A neutrino of de nite

avor that is, a ,a ,ora is a sup erp osition of mass eigenstates .For a

e m

2 2 1=2

given momentum p, the eigenstate has an energy E p=p +M . Assuming

m m

that the neutrino masses are small, M p, E p p + M =2p. In the time t that

m m

elapses b etween the birth of the neutrino and its detection, the comp onent of the

2 2

exp[ipt1 + M =2p ]. neutrino state vector acquires a phase factor exp[iE pt]

In practice, t is not measured. Rather, it is the pathlength x traversed by the neutrino

b efore its detection which is measured. Since the neutrino is highly relativistic, one

x.Thus, the relative phase of the and comp onents may then infer that t

m m

2 2

of the neutrino state vector is M M x=2p.From this relative phase, the usual

m m

expressions for neutrino oscillation follow.

Supp ose, now, that we do not consider the spacetime p oint of detection, t; x, to

b e a xed p oint common to all the mass eigenstate comp onents of the neutrino state

vector. Supp ose that, instead, we make the mistakeofevaluating the di erent mass

eigenstate comp onents at their di ering classical times of arrival at the measured

detection lo cation x. That is, wenowevaluate the comp onent of the state vector

at time t = xE p=p. This comp onent then has the phase factor

m 2 2 2

exp[iE pt ] = exp[iE px=p] = exp[ipx1 + M =p ] :

m m

2 2

The relative phase of the and comp onents is nowM M x=p,twice as

m m

big as b efore. Corresp ondingly, the oscillation frequency is twice its true value. Note

that the source of the spurious factor of two is the incorrect assumption that the

times of a given detection can b e taken to b e di erent for di erent comp onents of a

single neutrino state.

Wehave already seen that a spurious factor of two will arise in single kaon decay

if one takes the prop er time of the decay to b e given by Eq. 24, so that in the

exp erimental observer's frame, the K and K comp onents of the kaon decayat

L S 14

di erent times. We also observe from the discussion by Lipkin [1] that the spurious

factor of two in the frequency arises precisely when the di erent mass eigenstate

comp onents of the meson are taken to decay at di erent times, and is absent if these

comp onents are taken to decay at the same spacetime p oint.

In summary, the oscillation frequency in ! KK ! f f , and that in single K

1 2

decay, are b oth m . Similarly, the frequencies in 4s ! BB ! f f and single

K 1 2

B decay are b oth m . That the latter two frequencies are equal is an exp erimental

fact. That they are equal to m , and their K -meson analogues to m , follows

B K

from quantum mechanics.

In treating neutral meson or neutrino oscillation, it is imp ortanttotake the dif-

ferent mass eigenstate comp onents of the oscillating particle to decay or b e detected

at precisely the same spacetime p oint. Otherwise, a spurious factor of two in the

oscillation frequency can result.

Acknowledgments

The analysis rep orted here has grown out of a very enjoyable ongoing collab oration

with Leo Sto dolsky.Iwould like to thank Professor Sto dolsky for his many insights,

and in particular for his very useful comments on the topic of the present pap er. I am

grateful to the Stanford Linear Accelerator Center for its hospitality during part of

this work. I wish to thank Helen Quinn for a numb er of particularly helpful discussions

of neutral meson oscillation, Richard Blankenb ecler for several useful conversations,

and Sharon Jensen for her excellenttyping of the manuscript. 15

References

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Singap ore, 1989 p. 41.

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[14] H. Schroder, BDecays Revised Second Edition, ed. S. Stone World Scienti c,

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22 1963 239. 17