The determination of T- and CPT-violations for the (K0, K0 complex by K0K0 comparisons

ANNALS

OF PHYSICS

171, 463488

(1986)

The Determination of T- and CPT-Violations for the (K’, K”) Complex by R”/Ko Comparisons N. W. TANNER Depurtment

qf Nuclear

Physics.

Oxford

Uniwrsit~,

England

AND R. H. DALITZ Department

qf Theoretical Received

Physics,

O.rford

December

University,

England

13, 1985

Emphasis is given to the possibilities for new measurements concerning T-, CP-, and CPT-violation parameters through comparing neutral kaon decays following the at-rest production reactions ,Cp --f K” (or p) I(’ ?L*, a comparison made possible by the pure, intense, and well-defined antiproton beams of LEAR. Independent determinations may be made for parameters already measured, such as E and d. but also for the CPT-violation parameters (A; - A,)/(K” + A,), y (in K,,) and (x - .?) (the dQ = -As parameters for K/3 and R,, decay). These determinations will also allow a quantitive test of the Bell-Steinberger relationship. A brief review of the present evidence is also included. ‘i‘s 1986 Academic Press. Inc

1. INTR~OUCTI~N With the operation of LEAR, the antiproton storage ring at CERN, it has become possible to carry out accurate experiments making a direct comparison of kaons and anti-kaons. For neutral kaons, this can conveniently be done for an antiproton beam coming to rest and annihilating in hydrogen, by measurements on the charged kaon and charged pion for the following reactions of stopped antiprotons, p+p+K--

+n+ +P,

(l.la)

p+p+K+

+n-

(l.lb)

+p.

Taking the time of this annihilation to be t = 0, we shall denote the state of the neutral kaon at proper time t by II/(t) for the beam (l.la) which is ti at t = 0, and by q(t) for the beam ( 1.1b) which is P at t = 0. These reactions ( 1.1) are especially convenient for experiments concerning questions which involve a direct comparison 463 0003-4916/86

$7.50

CopyrIght 0 1986 by Academic Press, Inc. All ughts of reproduction in any form reserved

464

TANNER

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between &? and p beams, since the switch from Ko beam to F beam is equivalent to reversing the magnetic fields used for selection and measurement of the charged mesons. This new situation, together with the high quality of the antiproton beam available from LEAR, makes possible very accurate measurements of new types bearing on the questions of CP-, T-, and CPT-violation for the P-F’ complex, as Gabathuler and Pavlopoulos have also noted recently [ 11. Theoretical prejudice against CPT-violation is almost overwhelming. It has been proved [2-61 that any field theory which satisfies Lorentz invariance, hermiticity, local commutativity, and the spin-statistics theorem must be CPT-invariant, and these requirements are satisfied by all of the theories under investigation at present, including in particular quantum chromodynamics and the standard model of the electroweak interactions. If CPT-invariance is accepted, it follows that the interactions which violate CP-invariance must also violate T-invariance. On the other hand, no direct experimental evidence has been found to date which explicitly requires the failure of T-invariance, neither for the K0-p complex nor for any other weak decay process. Empirical evidence for T-violation in weak decay processes has been sought principally through measurements of muon polarization normal to the decay plane for the processes K + npv [7] and of spin-dependent correlations in polarized neutron decay [8], but the accuracy reached in these measurements is at the 1 % level at best, whereas the CP-violating interactions known for the Ko-F complex are only of order 0.1 % of the CP-conserving weak interactions. The argument usually given for T-violation in these interactions, based on the Bell-Steinberger relation (see Sect. 5), is really an argument for their CPT-invariance, their T-violation then being deduced from the empirical evidence of their CP-violation. Although this argument uses experimental data as input, it does rest on the validity of this theoretical relationship and the assumption that the decay processes observed are the only processes contributing to this relationship. Although there is no conflict at present between the absence of any empirical demonstration of T-violation and the expectation of T-violation on the basis of CP-violation with CPT-invariance, this is not really a satisfactory situation. There is therefore much interest in an experiment to demonstrate T-violation, which was proposed originally by Kabir [9] and has now been made possible by the new facility LEAR using the reactions (1.1 ), for it is highly desirable that the situation concerning T-violation for the CP-violating weak interactions should be established in a direct and unambiguous way. The question of CPT-invariance is ultimately experimental and cannot be left to the theoreticians. It is not impossible to construct CPT-violating field theories. An example of a free field theory which explicitly violates CPT-invariance has been discussed in detail by Oksak and Todorov [lo] and by Stoyanov and Todorov [ 1I]; the field operator 4(x) obeys local commutativity but it has an infinite number of components. The existence of this model CPT-violating theory does not actually violate the general theorem mentioned above, for the Wightman axioms [6] implicitly exclude field operators 4(x) with an infinite number of components. We do not wish to put forward any particular model for CPT-violation here but only

DETERMINATION OFT- AND CPT-VIOLATIONS

465

to point out that such a possibility is not completely excluded theoretically. The empirical evidence on CPT is very limited at present. It is concerned with the equality of masses and lifetimes, and the reversal of magnetic moments, between particle and antiparticle. The most accurate limits are those coming from the comparison of lifetimes for K', n’ and pL, particles, the former two being at the level of 0.1 %, the latter at 0.01 %; otherwise the evidence yields limits only at the level of several percent. For the (J?, I?) complex, there is the limit of order 0.1 % derived from the Bell-Steinberger relation, mentioned in the last paragraph and discussed in Sect. 5, but this is not a direct empirical check and there can be substantial uncertainties in the use of this theoretical relation. It is certainly not excluded that there could be CPT-violations of the same order of magnitude as the known CP-violations. Indeed, the present values of the phase angles 4 + ~ and do0 defined in Section 2 give the non-zero value (1.2)

where tan #sw = 2( M, - M,)/( Ts - f ,J, and so represent a 2-standard deviation discrepancy with CPT-invariance in the analysis of Barmin et al. [30], but this comes almost entirely from f&, which is very difficult to measure. The inequality of masses or of lifetimes between particle and antiparticle, requires not only CPT-violation, but also a violation of C-invariance [ 121. A good limit on CPT-violation, of order tO-3, is provided by the equality of the 1~~ lifetimes, these decays being ds = 0; this naturally directs our attention towards the mysterious CP-violation and the neutral kaons, since the properties of their mass matrix are particularly accessible to experiment and have a far greater sensitivity to CPT-violation than other particle properties which have been considered for such tests. The theoretical discussion of neutral kaon decays is, as far as CPT is concerned, often obscure and sometimes confused in the literature. The papers of Dass er al. [ 13, 141 and of Sachs [ 151 are particularly clear and to the point but the thrust of our discussion is in a different direction, being concerned with a new class of experiments. In this paper, we have endeavored to lay out the necessary algebraic relations ab initio, keeping to the notation of Cronin’s review article [ 163 but dropping, for simplicity, all terms quadratic in small parameters. In Section 2 and 3, we point out how the parameters involved in CP-, T-, and CPT-violation can be deduced from experimental data to come from the study of the leptonic decay events (1~) along the trajectories of I?’ and I? particles produced through the interaction processes (1.1) and we discuss Kabir’s experiment in Section 4. In Section 5, we discuss the Bell-Steinberger unitarity relation in the present context of CPT-violation. In Section 6, we collect together the data which exist, bearing on CPT-invariance, and endeavor to place an upper limit on the CPT-violation consistent with it.

466

TANNERANDDALITZ

2. PARAMETERS OF THE (&?,p)

COMPLEX

AND ITS DECAY

For an arbitrary neutral kaon state, (Q IK?) +a IF)}, determined by the Schrodinger equation [ 161

its time-dependence

--$ (3=(iM+iT)(!J

where M and r are 2 x 2 hermitian

is

(2.1)

matrices generally written as (2.2)

where the suffixes 1 and 2 refer to the Ko and ir?’ states, respectively, and t is the proper time. The stationary states IF) and [Z?) are defined as orthogonal eigenfunctions of the strong and electromagnetic interactions, which are assumed to be CP-, T-, and CPT-invariant. The eigenvectors of the matrix (iM+ ir) are the exponentially decaying states IK,) and (KL) with eigenvalues (a)

(b)

ys=iMs++rs,

yL=iML+fT,,

(2.3 1

where MS and ML are the masses, and rs and rL the decay rates, of the K, and K, states, respectively (see Appendix A). In general, to first order in the small parameters E and d defined below, these eigenvectors are given by (2.4a)

(Ks)=~(l+~+d)llYO)+(l-c-d)i~)}/~, IKL)=

{(I +&-A)

lK?)-(1

-&+A)

IF)}/&

(2.4b)

The presence of E and A in these expressions represents failures of CP-invariance: E corresponds to the case where CPT-invariance holds and T is violated, A to the case where CPT is violated and T-invariance holds. In terms of the mass matrix (2.1), they are given by (cf. Appendix A) E= C-2 Im Ml2 + i Im r,,)/My, A = (i(M,,

-J&J

(2.5a)

- Y,)),

+ (Cl - L)/WWs

- Y~)L

(2.5b)

where ys and yL are defined by Eqs. (2.3). We note that A is linearly related (with empirically known coefficients) with the mass and decay rate-differences between K? and p, the differences characteristic of CPT-violation. The CP-violation parameters q measured by experiment on neutral kaon decay to rc+V and rc”rcoare defined by ‘I+- = (7c+c71-lTlK,)/(n+n-I yloo= (TC~TC~I T IKL)/(nonol

T/K,), T [KS).

(2.6a) (2.6b)

DETERMINATION

467

OF T- AND CPT-VIOLATIONS

These parameters and the pL charge asymmetry parameter 6,( cc) given by (3.8) below are the only explicit evidence we have at present for the failure of CP-invariance. In general, there will also be CP-violation and CPT-violation in the K-+ 2n amplitudes themselves. These amplitudes have two factors, the first being the amplitude calculated for (nrr) standing waves, the second arising from the (rrrc) scattering in the final state. Thus, we have (a)

(b)

A(Ko -+ (rm),) = A,e’“‘,

A(p

--) (rtrc),) = .&e’“‘,

(2.7)

where 6, denotes the s-wave rcx scattering phase for isospin I at the energy of the kaon mass. If CPT-invariance holds, then 2, = A:, and, since the phase of the I@ state is still free and the J? state has the opposite phase, it is possible to choose this phase to make 2, = A, real. With CPT-violation 2, #A,, but the j?’ phase may still be chosen to give A, and 2, the same phase, so that X,/A, = real [ 16, 171. In this case, it is convenient to choose the real number A,, given by i, = (A, - A,)/(A, + A,)

(2.8)

as a measure of the CPT-violation for the I= 0 decay amplitude. The corresponding measure for the I= 2 decay amplitude is AZ= (A, - A,)/(A, + A,), which is a complex number. The usual analysis [ 17, 1S] then leads to the following expressions for v: 9 + _ = Eg+ E’, (2.9a) yIoo

where E,, and

E’

=

Eg -

2C’,

(2.9b)

are given by (2.10a)

Eo=~-A-&,

E1-Ao-Ao Ao+Ao

~(A,-A,)+(dz-~*)exp(i(d,-B J 2(4+A, ) + (-A2+Az)exp(i(&-6,)’

0)

(2.10b)

keeping only the leading terms in E, A, &,, and AZ. The amplitudes (A, + 2,) and (A, + J2) are, of course, insensitive to any T- and CPT-violating interactions and the ratio z = (A, + A2)/(A0 + A,) is real, insofar as the dominant weak interactions are CP-, T-, and CPT-invariant. It is well known that z is a small quantity, with value (4.5 + 0.1) x lo-*, from the ratio of the K’ -+ n&z0 decay rate and the A?, -+ rr~ total decay rate, analyzed in terms of the standard model of electroweak interactions, which allows only AZ= $ and AI = 5 transitions for non-leptonic K-meson decay. The deviation of the branching ratio (n +n. - )/(7c”rco)= (2.185 t 0.025) for Ks -+ n7t from the value 2 given by the Al= + transition alone is in quantitative accord with this small value for z, which measures the strength of 595!171:2-16

468

TANNER

AND

DALITZ

the AZ= 3 interactions relative to those for AZ= 4. In any event, this smallness of : allows us to expand the denominator of (2.10b) and so to reach the result El= -- 1 (A,-A,)--(A,-A,) &+A, ,/z

exp(i(d, - 6,)).

(2.11)

The relative magnitudes of (A, - AJ and (A, - A,) are not at all known, a priori. In the limit of CPT-invariance, where we have 2, = A:, and neglecting z, Eq. (2.11) reduces to the usual expression [16, 171 ImA,

ELI &

The combination given by

.-.

Ao

exp(i(b, - 6,)).

of q + _ and g,, appropriate

(2.12)

to Z= 0 does not depend on g’, and is

(21+ - + VW)/3 = 80 (2.13)

=&-A-&.

Since ;io is real, the imaginary

part of this equation gives us

Im E- Im d = Im( 2q+ _ + v],)/3,

(2.14)

as a consequence of the phase convention chosen above, viz. A,/A, = real. From the real part of Eq. (2.13), we see that the new CPT-violation parameter 1, is related to the other parameters for the Ko - Z?” complex by the equation A0 = Re E- Re A - Re(21+

+ ~,,)/3,

(2.15)

The PDG values for v + _ and 11~ at present are [S] (a) (b)

(g, ) = (2.274 f 0.022) x 10-3, I’looj = (2.33 f 0.08) x 10-3,

qS+- =(44.6+ 1.2)” q& = (54 * 5)“.

However, the ratio Iv+ _ l/l~& has recently been measured, with considerable precision, in two experiments, one at Brookhaven National Laboratory [ 191 and the other at Fermilab [20], with the following results: Yale-BNL

group:

Chicago-Scalay

group:

(‘1++~/~~~(=1+3(+0.0017+0.0072+0.0043), 1y1+_ )/ lqool = 1 + 3( - 0.0046 + 0.0052 + 0.0024)

the second error being the estimate of the range of systematic uncertainty in each experiment. These results will affect the PDG values for In + _ ( and Irooj, bringing their value for JqooJdown to (2.295 _+O.OSl) x 10e3, close to (q+ -1. An assessment of these experiments and of two other experiments now in progress has recently been published by Gollin [21]. Our only concern here is to note the limits these

DETERMINATION

469

OF T- AND CPT-VIOLATIONS

measurements place on the amplitude E’. Since the amplitudes of r~+ _ and qoo are so nearly equal, and the phase difference 6# = (Qloo- 4, _ f = (9.4 + 5)” is not large, the vector from qoO to q + _ on their Argand plot is essentially perpendicular to .Q and we have the firm estimate Im(&‘/&J

= (n/180)(64)/3

Given that E’/E~ is small, the (amplitude)’ l(9+-/rloo)12~

= (0.055 f 0.030).

ratio gives (including

the PDG value)

1 + 6 Re(e’/&,) + 21(Re(&‘/&,))* - 3(Im(E’/e,))‘+

..

= 1 + 6( -0.0036 + 0.0088). Adopting

the value just obtained for Im(E’/E,),

we reach the estimate

Re(&‘/e,) = -0.0021 2 0.0090. These values are negligible in relation to the effects considered in this paper and we shall generally neglect E’/E~ in this paper. From the above numbers and Eqs. (2.9), it follows that the Argand vector E’ is given by E’ = (rl+ - - rloov3.

Since q + _ and qoo are so close in magnitude, the direction of E’ is perpendicular to the bisector of the angle between ye+~ and qoO and therefore makes angle - (41.7k2.5)” with the real axis, on the basis of the measured phases for q+ _ and qoo. If CPT invariance were to hold, E’ would be given by (2.12), which has phase (90 + 6, - 6,)’ = (37 + S)‘, where 6, denotes the s-wave 7~7~scattering phase shift for isospin I. The discrepancy between these angles would be a substantial violation of CPT-invariance if the value of &, were firm; in fact, as boo comes close to (b+ _, the direction of E’ deduced from the Argand vectors 9 + _ and qoo becomes increasingly ill-defined and this discrepancy less and less significant. In the Introduction, we referred to another discrepancy with CPT-invariance, namely Eq. (1.2). From the unitarity relation, discussed below in Section 5 and given by (5.1 l), CPT-invariance would imply that, to an accuracy of order 10e3, the phase of E is &.w = arctan{2(M, - M,)/(T, - TL)} = (43.6 + 0.4)“. Given that E’/E is small, the expressions (2.9) imply that the phase of E should be given by (24 + _ + &,,)/3, irrespective of CPT-invariance. With CPT-invariance and the unitarity relation, we can identify these two phases for E, which leads us to Eq. (1.2) with zero on the right-hand side. At present, this large difference between the measured values for IP+ _ and &, gives rise to the largest deviation from CPT-invariance deducible from the ti - p data available today, and it continually bedevils any conclusions which may be drawn at present concerning the question of CPT-invariance. Since the eigenstates KS and K, have time-dependence exp( -ys t) and exp( - yLt), respectively, the Eqs. (2.4) for time t may be inverted to give

470

TANNER

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DALITZ

expressions for the state $(I), which is pure K?’ at t = 0, and for the state t,?(t), which is pure p at t=O, with the result JZIl/(t)=(l-E+d)lKs)e-~‘s’+(l-&--d)(KL)e~”I~’

(2.16a)

~~(t)=(l+E-~)]Ks)e-~s’-(l+&+d)~K~)e~’~.’.

(2.16b)

The distribution of rrrr decays along the neutral kaon trajectories is obtained using the xn decay amplitudes (2.7) for Ko and p and squaring. Apart from a common factor 1(rcz] T /Ks)12/2, these distributions have the form Ae-rS’

+ 2CeCr’cos(t6M)

+ 2Se-rrsin(t6M)

+DeprLf,

(2.17)

where F= (rs + fJ2, 6M= (M, -M,), and the coefficients have the following forms (retaining terms no higher than linear in E and A), A=(lT(2Ree-2Red)},

(2.18a)

C= + I(lT2Re&)Re?,,+2ImdIm?,,),

(2.18b)

S= f_j(l~2RRe&)Im?,,g2Imd.ReI?,,),

(2.18~)

D={lT(2Rer+2Red)]

(2.18d)

(v,~(‘,

the upper signs to be taken for the case Ii/(t), the lower signs for the case g(t). We note that the time-dependence of rtcnevents along a trajectory of a particle initially Ko differs from the time-dependence along a trajectory of a particle initially in the CP-mirror state $?, because the KS-K, interference is constructive in one case, and destructive in the other, as is illustrated in Fig. 1. We note that this qualitative difference between p and p beams holds whether or not there is T-violation (case of E# 0) or CPT-violation (case of d #O), or both, because E and A appear in these expressions (2.18) only as a small correction. It is, of course, from the measurement of the coefficients C and S of such interference terms that our knowledge of the phase of qnn has been derived, for both rr+n- and norto modes. We note that, when both p and p initial beams are available, the comparison between the expressions (2.17) for each of them has some sensitivity to the parameters e and A. For example, far downstream where f$ l/r,, the ratio of the rate of rcz events for the two beams is given to first order in E and A, by R,,(m)/a,,(m)

= 1 -4(Re

E +

Re A),

(2.19)

where the notation R,,(t) is used for the rc7t rate observed for the Ko beam IC/(t), R,,(t) for the p beam $(t). It is unrealistic to propose this ratio (2.19) for measurement because of the very low rate of rc1t events in the time region $1/f, owing to the smallness of /?,,I ‘. As a second example, the rate of rrn7tevents at t = 0 is given by (A + 2C+ D), leading to the ratio R,,(O)/8,,(0)

= 1 - 4(Re E- Re A - Re q%,),

(2.20)

DETERMINATION

OF

i

i ‘I

,o.’

:\ :I

:\ :’ :\ .\ :\ :I

01 Ti L ,.

471

T- AND CPT-VIOLATIONS

lo-

:I :\ :.\ ..,I .I 1.

2 0"

10-'

&\

1o-6

K0

I '.. / 1 '.

;:;=.-

, 5

10

I I/ “1 15

. ..-.II.-.-.-

,

1

1

20

25

30

FIG. 1. The time dependences for Z+Rdecays along the path of a neutral kaon beam initially and along the path of the CP mirror situation. that of a neutral kaon beam initially p, illustrating failure of CP invariance.

A”’ the

which involves a different combination of these parameters, but this also is not readily accessible to direct measurement. In practice, it is the K, -+ rc1t region, t z l/r,, which it is most convenient to observe. It would be practicable, for example, to include all events up to some proper time t,,, such that l/r, < t,,, < l/r,. Carrying out the integrations of (2.13) and retaining only terms linear in the small parameters, we have

&m(t)i sd””L(t) dt = 1 + (4(Re A - Re E) + 4(TRe qnn + SMIm

-4 kA2 RW +d U,d~

q,,) r,/(F’+

(6M)‘)

(2.21)

where terms of relative order of magnitude exp( - Ts &ax) have been neglected in the first term within the curly brackets, of relative order of magnitude exp( -fSfmax/2) in the second term, and of relative order of magnitude (rLtmax)/2 is the third term. Of course, for any reasonable choice of t,,,, the coefficient so 1~,,12 ~Sblax is a rather small number, being only 2.5 x 10e3 for t,,, = l/r,,

472

TANNERANDDALITZ

that this third term is negligible. We have given this expression (2.21) in order to show that, although it is primarily the phase of qnn which is determined from the observed time dependence (2.17), the availability of a precise comparison between the rcrcdistributions for the Ko and K(’ beams does allow the possibility of measuring the quantity (Re d - Re 8) in the same experiment. It is of interest to note that for the rr+z- mode, we have from (2.9a) and (2.10a) Red-Res=Res’-Rev+-

-A,,,

(2.22)

so that, since /Re E’( < lop4 and Re q + ~ is well determined, this measurement would give directly the CPT-violating contribution & to the I=0 amplitudes (K, R)O -+ 7c7c.

3. THE LEPTONIC DECAYS The dominant K, decay modes are K, --f lrcv, where I denotes e’ or p+, n the corresponding pion 7cT, and v the corresponding neutrino, their total rate being about 66 % of rL. The transition amplitudes for the individual decay processes may be written as follows:

(l+v,z-I

TlK(‘) =f,,

(3.la)

(l+v,n-1

T@?‘)=g,=x,f,,

(3.lb)

(l-V1n+l

T @)*

=&=.&f,,

(3.lc)

(I-V,n+I

T IF)*

=fi

(3.ld)

In these expressions, CPT-invariance has not been assumed. The dQ = As rule, where s denotes strangeness, would impose the restrictions Ko + I- rc+V/ and &? + I+z-v,, that is, x!=X,= 0. Although the standard model for electroweak interactions leads us to expect this rule to be valid to order 10-14, the experimental evidence only places limits of order ~0.1 on the magnitudes of each of the AQ = -As amplitudes x, and .fI, so we have retained them. With CPT-invariance, and with neglect of final state (Coulomb) interactions [35], we would have (a) fl=f,,

Here we shall leave open the possibility and write, (a) f,=F,(l

-v/L

(b)

“, = x,.

(3.2)

of CP- and CPT-violations (b) A=F,(l

+Y,),

in K13decays (3.3)

noting that F, and y, will be complex numbers in general, and that y,= 0 for CPT-invariance and no final state interactions. We should remark here that there are two matrix elements f, and g, required for a full specification of the K,3 decay

DETERMINATIONOF

473

T- AND CPT-VIOLATIONS

amplitude. The structure discussed here for f, will hold equally explicit inclusion of both is necessary in the present context only properties in the (Zvn) final states are to be correlated with the time the neutral kaon beam. In order to use the amplitudes (3.1), it is now convenient wavefunctions (2.16) in terms of $? and p states,

for g, and the if polarization development of to rewrite the

2~(t)={(1+2A)~KO)+(1-2&3~~)$e~“~’ (3.4a)

+((1-24)JKO)-(1-2~)1J?)}e-~L’, 2$(t)={(1+2~)IJ?)+(1-2A)IR)je~~s’ -{(1+2~)(KO)-(1+2A)\&?))e-~L’.

(3.4b)

The leptonic decay rates along the particle path are then given by the following expressions, apart from a factor lF,i*/4,

For KL decays, where t 9 T; ‘, these expressions simplify. Dropping a common factor exp( -r,t), we then have, to first order in the small quantities E, A, y,, x,, and -u,, (a)

R,+(co)=l-2Re(2A+x,+y,),

(b)

R,+(co)=l+2Re(2~-x,-~~,); (3.6)

(a)

R,-(a)=

1 -2Re(2&+.?-4’0,

(b)

8,-(a)=

1 +2Re(2A-.?,+y,). (3.7)

Owing to CP-violation, the rates for the two charge states in K, --f l’v,n + decay are different. This charge asymmetry is well measured [8], the parameter h,(m) being (3.8) This lepton asymmetry must be the same as that for $ or $, since it is a characteristic of the KL meson and does not depend on how the K, beam is formed. We therefore have for t $ r,- ’ ,

6,(aJ)=

R,+(a)-R,-(m) ~,+(4+&-w=

R,+(m)-R,&(co) R,+(CO)+R,-(co)

474

TANNER

AND

DALITZ

the latter equality being verified, to first order in small quantities, by insertion of the expressions (3.6) and (3.7). The explicit expression for 6,( co) is S,( co) = 2 Re(s - A) - Re(2y, + .Y!- X,), In the limiting case of AQ = As and CPT-invariance tion, this reduces to

(3.10)

for the (K, K)’ -+ Iv,n interac-

6,( co) = 2 Re(s - A).

(3.11)

We note that this involves the quantity (E-A) whose imaginary part is given empirically by Eq. (2.14). Taken together, these values specify (E - A) in the complex plane but they do not separate E from A; that this should be so is evident from the structure of the K, wavefunction (2.4b). In any event, there may also be contributions to 6,( co) from the parameters ~1,, xl, and X, coming from the K,, interactions themselves. In the experiments envisaged here, there are other KJ3 ratios which can be measured far downstream, for t 9 l/r,, and which are potentially more helpful. For example, using (3.6) and (3.7), we find the equalities R,+(ccl)-R,+(cu)=R,-(co)-R,-(cr:)=R,(cu)-R,(cu) R,+(co)+R,+(oo) R,-(m)+R,-(cc) R,(co)+R,(m)

(3.12)

where RI = R,+ + R,- and i?, = &+ + R,-, their common value (denoted by y,( co) being given completely, to first order in small parameters, by y,(m)=2Re(~+A).

(3.13

Since it does not involve y,, x,, or X,, this would be a particularly interesting ratio to measure. It may seen puzzling that different ratio measurements in the K, region of the beam can yield different parameter combinations, viz. (3.10) and (3.13), while the KL wavefunction involves only the combination (E - A). The explanation can be seen by referring to Eqs. (3.4), in each of which the coefficient of the term exp(-y,t) does not agree with KL wavefunction (2.2b) but differs from it by terms of order (E or A). In (3.4a), this coefficient: is (1 -E -A) times the K, wavefunction, whereas in (3.4b) it is - (1 + e + A) times the K, wavefunction. In other words, the particles represented by $(t) and G(t) have different intensities in the K, region t $ l/r,, the difference between their intensities being 4 Re(s + A) times their mean intensity. It follows that the same ratio (difference)/(mean), for the yields far downstream for the particles initially p/p, will hold for all K, decay modes, including x+~~rr’, ~~~~~~~ and yy decays, for example. Hence we can drop the suffix on yr( ocl) in Eq. (3.13) and write this ratio generally, for any decay mode A 2y(a3)=(R~cc)-R,-(co))/((R~co)+RXcc,))/2)=4Re(&+A). To measure this ratio requires that the relative intensity

(3.14) (at t = 0) of the particles

DETERMINATION

OF T- AND CPT-VIOLATIONS

415

initially K” and initially p be known to a high degree of accuracy, and this appears to be quite possible using the jip annihilation reaction (1.1) to produce these particles. Other K,, ratios can be investigated in the K, region, but they determine only linear combinations of 6,( co) and y,( oo), and so involve the combination Re(2y, + x, - X/). We shall mention two of these:

%(a)=

Ri-(co)-R,+(co)

= 4 Re A + Re( 2y, + xI - X,),

(3.15)

=4Re~-Re(2y,+x,-2,).

(3.16)

R,-(co)+R,+(cxl)

At this point, it is appropriate to emphasize that there do not exist accurate measurements of K,, rates normalized to the kaon production rate, i.e., no absolute branching fraction measurements. The K,, “branching fractions” quoted in the literature [22-251 are actually K, + X/V rates normalized to K, + z+?I- rates in the same experiment. If we define “the K, -+ rt +n- rate” as the integral for the IZ+X ~ process taken between t = 0 and t,,, such that r< ’ < t,,, G T; ‘, as was adopted above in Section 2, we can consider the asymmetry in these “branching fractions” for p and P sources. For example, the quantity Y,( co ) corresponding to y,(co) defined by Eq. (3.12) is given by (3.17a) =4ReA+2Ts(~Re?+-+6MIm?+~)/(~2+(6~)2)+o(lrl+_~’r,t,,,), (3.17b) where I,, is the integral (3.18) of the ~c+x- rate R,,(t) for the P’ beam, I,, being that of R,,(t) for the i?’ beam. The relation (3.17b) gives Re A directly in terms of empirical quantities but there do not yet exist empirical values for Y, sufficiently accurate to be useful for this purpose. More generally, for arbitrary t, the distribution of (K, g)‘-+ Iv,n events is given by the form (2.17) with appropriate values for the coefficients A, C, S, and D. To first order in the small quantities E, A, y, X, and X, and apart from an overall coefficient IF12/4, these values are tabulated here for the four cases of interest; see Table I. Common factors have been taken out and listed at the left of the table. When products are specified, terms of second order or more are to be omitted; the expressions are valid only to first order. We note from (3.1) that the weak decays

476

TANNER

AND

TABLE The Coefficients

Decay mode

Common factor

I

in the Expressions

A

C

1+4Red+2Rex, ti(O+l’ (1 - 51). I +2Rex, $(t) + l+ (1 - 2-v,). ( 1 + 4 Re E). ( 1 + 2~1,). ( 1 - 4 Re E). 1 + 2 Re .Y, l-4Red+2Ret, (1 + %,I. Note. Ae-rs’+2Ce-ncos(t6M)+2Se-~sin(16M)+ dence of the KR -+ Ivn decay events along initially P (case Qft)).

DALITZ

a neutral

+I -1 -1 +l

kaon

(2.17)

S -4Imd-2Imx, 2 Im x, -2 Im MU, +4Imd+2Im.f, Decrl’ beam

D I-4Red-2Rex, 1 - 2 Re x, 1 - 2 Re .?, 1+4Red-2Re.f,

represents the proper-time depenwhich is initially Ko (case $(t)) or

leading to a definite final state, l+v,rc- or I-V,rc+, involve the same amplitude factor (f, or f,) and therefore the same factor, I( 1 - y[)(’ z (1 - 2 Re yI) or I( 1 + y,)12 z (1 + 2 Re y,); this is the reason why Im ,vI does not appear, to first order, so that we have been able to treat y, as real, in our discussion. We shall suppose that the coefficients A, C, S, and D can be obtained from the data on these leptonic decay modes as a function of proper time t for particles initially K” and initially p. This is possible, at least in principle, by use of a maximum likelihood method on the basis of the expressions (2.17) and the experimental efficiencies. First we discuss the situation for analysis when the selection rule AQ = As is imposed, namely xl= X,= 0. We note that the square matrix on Table I then depends only on A; the common factors depend only on Re E and yI. We make the following remarks: (i) The ratio of II/ -+ I+ for large t (i.e., term D) to $ -+ I+ for small t (i.e., term A) is given by D($-P)/(A(tj+l+)=l-8ReA. (3.19) The value of the corresponding ratio D( $ --* I- )/A ( I$ -+ I- ) is (1 + 8 Re A ). These comparisons allow two measurements for Re A and their consistency provides a check on the absence of .Y and X. Taking them together allows the estimate (o(Ji-I’)/D(~~l-))/(A(llljIf)/A(~~I-))=l-

16ReA,

(3.20)

showing a very considerable sensitivity to Re A. (ii) The coefticients S for $(t) + I+ and I+?(C)-P I- are proportional to Im A. The ratio S/A is 14 Im A for the cases t)(t) + I+ and $(t) + I-, respectively. The change of sign provides a check and there is a strong sensitivity to Im A. (iii) After the determination of Re A, the value of yc can be determined from the ratio between t,+(t) + I+ and $((t) + I-. This determination may be based on all events, or on all events within a specified proper time span.

DETERMINATION OFT- AND CPT-VIOLATIONS

477

(iv) After the determination of Re A and y,, the value of Re E can be determined from the ratio between $(t) -+ I+ and 6(t) --) I-, the relevant factor being (1+4Res)/(l-4Re~)=(l+8Rec). The experimental situation for the determination of the values of A, C, S, D for each (Iv,rr) process is more favorable than for the corresponding coefficients for the 7crrprocesses, because all four coefficients are of the same order of magnitude. We now consider the full matrix of Table I, including the terms x, and .?,, thus with four additional parameters. It is clear that Re x, is determined uniquely from D/A for $-I’, and Im x, from S/A for $ -+ I+. Similarly, Re ZI and Im .%! are determined uniquely from D/A and S/A, respectively, for $ + I-. When these are known, then Re A and Im A are determined twice over, as before; yc can then be determined from the comparison of $ -+ I’ and $ -+ I-, and Re E from the comparison of $-I+ and $ -+ I-. The last parameter required to give a complete description of the situation concerning CP-, T-, and CPT-invariance for the weak interactions relevant to the (K, R)’ system is Im E; this is given by the phase convention 2,/A, = real and is related with Im A and experimental data on qnn according to Eq. (2.14). Thus, we can conclude that the study of the time-dependence of K,3 decays along J? and J? particle trajectories allows the possibility of an independent evaluation of all of these parameters.

4. KABIR'S DIRECT TEST FOR T-VIOLATION

The ratio p,(t) defined by (4.1)

and used in Eq. (3.16) for the case t -+ cc, is of particular interest in view of its connection with Kabir’s proposal [9] for an empirical and explicit demonstration of the failure of time-reversal invariance for the (K, IQ0 system. Kabir considered the following intensities at time I for beams initially L? or Ko: P-,k?:

PKx(f)=

l(P,

tl tint

IP, O)j2,

(4.2a)

P-+P:

PRK(f) = I (P,

tl ciN’

IF?, O>l’.

(4.2b)

and pointed out the importance characterize by the ratio

of measuring

d(f) = (PKR(f) - PRK(t))I(PKR(f)

their difference, which we may + P&f)).

(4.3 1

He argued that a non-zero &(t) would represent a deviation from reciprocity, a property guaranteed by time-reversal invariance, and that this deviation would

478

TANNER

AND

DALITZ

therefore imply a fundamental failure of time-reversal invariance for the weak decay interactions. The last step in his argument may be demonstrated briefly as follows. The operation of time-reversal gives the identity [26] (ZPle-‘H’J

Ii+) = (TIP1 = (jq

e +ifhtj Tp)* g-i-

= (T&-l

e-iHTl ITF)

p-0)

(4.4a) (4.4b)

where T is the time-reversal operator and HT is the time-reverse of H. In view of the identity (4.4b), the rates (4.2) are necessarily equal when HT. = H, i.e., when the Hamiltonian is time-reversal invariant. If the rates (4.2) were found empirically not to be equal, i.e., if d(t) were non-zero, Kabir would conclude that H,# H, i.e., that time-reversal invariance must be violated. The question is how to measure d(t). With CPT-invariance and the AQ = As rule valid, so that yr = 0 and xI = X, = 0, observations of K,, decays along the beam can achieve this purpose; the observation of an I+ would identify a ti decay and the observation of an l- would identify a p decay, and these two decay interacLOW ENERGYANTI-PROTON (@RING AT CERN.

t 20MeVP

1 HYDROGEN TARGET 4 PP 4 Exactly equal

IDENTIFIES RQ

1 KO*RO

t RO+KQ

I

NET: K"-i?

IDENTIFIES @

1 k+K,

1 W, K,AMPLlTUDE, DECAYS

-

,K,AMPLlTUDE DECAYS

+

E%Ei3;ED

IDENTIFIES KO

+KO

DIFFERENT BYQ65%

FIG. 2. Low energy anti-proton (p) ring at CERN: The chain of reactions and decays which relates a difference in the rates for K-eand K+e+ events to a test of detailed balance, @ + i? vs $? -+ p, and therefore to a test of time reversal invariance. This test of T-invariance is valid only if there is no CPT-violation in K. decay.

DETERMINATION

OFT-

AND CPT-VIOLATIONS

479

tions have the same magnitude when CPT-invariance holds. In this case, the quantity fl,( t) is identical with JZZ(t), so that observation of any non-zero value for /3,(t) would imply P,& t) # PRK( t) and therefore a violation of time-reversal invariance. The links in this chain of processes and argumentation are laid out graphically in Fig. 2. This argument of Kabir has great generality, being independent of what nonzero value is observed, for its validity depends only on the property (4.4) and not on other details of the theory. At the present time, assuming CPT-invariance and the absence of dQ = -ds interactions, we do know something further about the parameters of the (&?, p) complex, namely the value of Re E. In this context, Kabir predicted a definite value for LX!(~), namely 4 Re E, with value about 0.65 %, and the observable asymmetry P!(t) will then have the same value. When t is large, t 9 ‘ss, the basic reasons for the final lepton charge asymmetry are that (i), as remarked in Section 2, the normalization of the final K, beam depends on whether it came from initial Ko or p particle, and (ii) there is a charge asymmetry in the decay process K, --t rrfv itself. More generally, the relation between (3,(t) and ..d(t) is modified by both the CPT-violating and the LIQ = -ds components in the K,, decay interaction. This is already apparent in the expression (3.16) for p,(cc ), appropriate to the K, regime of large t. Even if dQ = -ds transitions are excluded, as expected, there remains the contribution of Re(y,) to the non-zero value of fl[(r). Since this CPT-violating interaction affects the relation between the K” -+ rrfv and I? -+ rc/$ amplitudes, it necessarily involves C-violation, so that its CP-violating component need not involve T-violation. It follows that the mere observance of a non-zero P[(t) no longer necessarily implies the presence of a T-violating interaction, since the value observed may result from a CPT-violating interaction. This modification destroys the qualitative character of this criterion for the existence of T-violating interactions, and puts it on the plane of quantitative measurement. If the value obtained for fi,( co) turns out not to equal the value 4 Re E predicted by Kabir, the difference must be due to contributions from CPT-violating interactions and its measurement may then be regarded as a quantitative demonstration of their existence. For finite t, the explicit expression for fl,( t) can be obtained from Eq. (2.17) and the entries in Table I. as follows:

a,(t) = 4 Re E- 2 Re y, - Re( x, - Z,) + Im(x, + .fI).

sinh( tr) cosh( tr) - cos(t 6M)

sin( t 6M) cosh( tr) - cos( t 6M)’

In the limit t -+ co, this expression reduces to (3.16). For the case of CTP-invariance, x, = Xl and y, = 0, p/(t) takes the form P,(t)=4Res+2Imx,.

sin( f 6M) cosh( tr) - cos( t 6M)’

480

TANNER

AND

DALITZ

where the coefficient of Im xI becomes small for tr, 2 5, in which domain P,(t) has the value 4 Re E x 6.5 x 10-3, independent of 1. The leptonic decay mode is of course not the only means available to distinguish a Ko component from a X0 component of a neutral kaon beam and so to measure d(t). The &? component has strong absorptive interactions with nucleons, e.g., iP+p4+71+,

(4.7)

whereas the Ko component does not, its most striking being the charge exchange process

interaction

p+p-m+K+.

with nucleons (4.8 1

In principle, these two reactions could be used to sample the Ko and F components of the beam at a given proper time, and so lead to a demonstration of the existence of T-violation free from the ambiguity introduced by the possibility of CPT-violating interactions. The practical difficulty is that the cross sections for these reactions are rather poorly known as functions of momentum and reaction angle. Also, these cross sections are only a small fraction of the Pp and Pp total cross sections, so that they are not efficient indicators of strangeness, in contrast with the K,, decay interactions for p and p, which have amplitudes of similar magnitude for E?’ and p states, and account for more than two-thirds of the KoL decays. It is apparent that the determination of /3,(t) is an important measurement, which can throw light on the supposed violation of time-reversal invariance and the possible violation of CPT-invariance and which can also lead to an improvement in the experimental status of the dQ=ds rule.

5. THE BELL-STEINBERGER

RELATION

A further relation involving E and d comes from the unitarity Steinberger [27]. This stems from the equation

relation of Bell and

(5.1)

where If) is a final state normalized to include the density of states and other numerical factors. This equation states that the rate at which neutral kaons disappear equals rate at which products appear. Inserting into this equation the wavefunction

DETERMINATION

481

OF T- AND CPT-VIOLATIONS

and picking out the L-S cross terms at t = 0 leads to the relation (idM+n

~KsIW=~,(fl

TlKs)*

(fl

TIK,)3

where the sum is over all physical decay channels f common namely zrc, rczrc, ‘~lev, rcpv, etc. From (2.4), we have

(5.3)

to both K, and K,,

(K,J K,)=2ReE-2iImA.

(5.4)

The zrc terms of the sum on the right of (5.3) are expressible in terms of the nrrn defined in Section 2 and the widths rs(rcrc), in the usual way. The contributions of the rrrcrt channels are similarly expressible in terms of observable quantities ye+ -0 and qooo defined by I?+.-~= (TC+X-TC~(T IKs)/(r+n-noI

(5.5a)

T IKL),

(5.5b)

qooo= (T-C~TC~TC~\ T ~Ks)/(~ono~o~ T IK,),

and the partial widths r,(rr+n-n’) tonic modes, (I+v,n-1

T (KS)* (Z+v,n-1

may similarly

and f,(zOrcOnO) [28,29].

T IKL) + (I-C,z+l

T (K,)*

The sum over the lep-

(l-V,n+J

T IK,)

(5.6)

be written @,,-

I(l+v,n-I

TIKL)I’+@,n+

I(/-V,n+l

TlK,)l’

=~L(~v~){(1+~,(~))yI~+“,~+(1--,(~))~T-,,+}/2.

(5.7)

where 6,( co) was defined in Eq. (3.8) and the q,va are defined by )Il+“,n- = (l+v,n-(

T(K,)/(l+v,~--1

T IK,)

(5.8a)

q,-v,n+=(I-Vl~+f

T\K,)/(IpV,n+l

TIK,).

(5.8b)

Using the eigenfunctions (2.4) and the decay amplitudes (3.1), these ylcyncan be readily expressed in terms of E, A, y,, .x1, and Z,, the result being, to first order in these parameters, v/+v,n- =(l

(5.9a)

+24+2x,),

~]/-~,~t = -( 1 - 24 + 2X:). Insertion

(5.9b)

of these expressions into (5.7) leads to the leptonic term ~L(hr)(b,( co) + 2A* +x:

- XJ.

(5.10)

We note that this expression does not depend explicitly on E or y,, when rL (IvK) is taken from experiment. Bringing the 2n, 371, pvn, and evn terms together leads to our result,

482

TANNER AND DALITZ

+ (6,(m)

+2A*

+x,* -2,)

r,(nev)

+ (h,(cc)

+ 2A* +x,*-iii)

rL(qu).

(5.11) The Kn terms of this expression differ from those given in [ 16,301 since those authors assume CPT-invariance for the & interactions. The 7crrterms dominate the sum in (5.11) by a factor of order (f,/T,) and quite a good approximation is obtained by using them alone. Using also the good empirical approximations that (ML - M,) z +f s 9 f r, q + ~ z qoo, and Arg q + z &.w = arctan{ 2( M, - M,)/( rs - r,) > z 7c/4, we obtain the well-known estimate Res-iirn

Az(l/$)

Iq++l

[l +0(10p3)},

(5.12)

from which we conclude that Im A is very small (of order 10p6), while Re Ez 1~+ ~ I/;‘“- From (2.14) we conclude also that Im Ez Im q + , in the same approximation. On the basis of the notion that CPT-violation might be the result of non-Hermitian components in the effective weak interactions, Kenny and Sachs [31] have questioned the validity of the unitarity relation (5.11). We have based our derivation on Eq. (5.1), which expresses only the conservation of probability; it is not identical with the unitarity relation used by them although the two relations become identical when CPT-invariance is imposed. However, any theory of CPT-violation, in order to be acceptable, must lead to positive probabilities and to probability conservation as we have assumed in Eq. (5.1). In this context, we wish to point out that, following the determination of the parameters E, A, x/, and Z, as envisaged in Section 3 above, the Bell-Steinberger relation (5.11) can be subjected to direct experimental test, since all of the quantities we have specified in Eq. (5.11) will be available from independent measurements.

6. CURRENT EXPERIMENTAL LIMITS ON CPT-INVARIANCE

In this section, apart from several side remarks, we focus attention on the case where T-, CP- and CPT-violations occur only in the mass matrix, i.e., where only E and A are non-zero. Physically, the parameter A is closely related with the mass and decay rate differences of p and p. Using the good approximation (M, - M,) z (rs - r,)/2 and the formula (A.8), we have

(a)

Ml1 - Mz* z2(Im ML-MS

A-Re

A),

(b)

rll - r22 z2(ImA+ReA), rs-rL

which are indeed direct measures of the CPT-odd magnitudes characteristic of the weak interactions.

interactions

relative

(6.1) to the

DETERMINATION

OFT-

AND

CPT-VIOLATIONS

483

When we neglect CPT-failure for the K,, and K,3 decay interactions and allow only, LIQ = As interactions for K13, the lepton asymmetry is 6,( co) = Re(s - A), and the term & in Eq. (2.13) is zero. When this is taken together with the BellSteinberger expression (5.11) and the phase convention (2.14), there are sufficient equations to determine the two complex numbers E and d and this is the path which has generally been followed in the literature [16]. However, it is logically unjustifiable to assume CPT-invariance in some aspects of the problem, viz. ,l., = y, = xI - X: = 0, while questioning the validity of CPT-invariance elsewhere. When the decay interactions violate CPT-invariance (and the LIQ = As rule), 6,( cc ) receives an additional unknown contribution - Re(2?, + X, - .f,), and it is no longer possible to follow this path; one further relation involving Re d is needed. With existing data, there are two possibilities for a further J?/p comparison: (A) (PI &) -+ rc+7c- decay in the K, region. There are a number of measurements of the reaction pp -+ Ko (or 130) KT + pions, from hydrogen bubble chamber photographs which were scanned for K, --, Z+X- decay events and which distinguished between J? and p production [32-341. Assuming C-invariance for the strong interaction production process, these measurements give essentially the quantity (2.21). Combining all of the data from these experiments there were 3220 K” events and 3208 F’ events; thus, this ratio deviates from unity by ( + 3.7 + 25) x 10p3. The second term in the curly brackets of (2.21) has a magnitude of about 4 $ Iq + _ / = 12.8 x 10 m3, leading to the estimate Red-ReE=(-2.316)x

10-j

(6.2)

consistent with zero, but also consistent wih the value (- 1.65 + 0.06) x 10e3 given by (3.11) and the empirical value of 6,( ~3) (eq. (3.8)). In passing, it is worth mentioning that, using Eq. (3.10), we can estimate the value Re(2y,+x,-.%,)=(-0.5)12)~10~~,

(6.3)

which gives at least an upper limit on the magnitude of this quantity. (B) (K, f?)O -+ lvn branching ratios. The tabulated branching ratios [S] are hydrogen bubble chamber measurements of K, + rclv rates normalized to KS -+ 71+rt Cho et ul. [22] and Burgun et al. [23] studied Ko decay following the reaction K +p + Ppn + ; the former obtained the value (l.OO+O.OS) x 1O-3 for R,( i;o)/I,, on the basis of 215 G3 events, the latter ( 1.ll +_0.06) x 10-j on the basis of about 1000e3 events. Webber et al. [24] and Mann et al. [25] studied F decay following the reaction K-p + pn; the former obtained the value (1.13&0.11)x 10e3 for R,(c0),/1=,, on the basis of 178 eJ events, the latter (0.73 kO.15) x 1O-3 on the basis of 116 events, the difference between these two values being almost three standard deviations. All of these values were derived with the assumption x=X=0, and it is clear that they can only be regarded as very preliminary values, obtained from very early work and with poor statistics; we quote them only for the purpose of illustration. Together they give the estimate ?V4:171,‘2-17

484

TANNER AND DALITZ

Y,(co)~(50+40)x 1op3. Inserting it into the relation (3.17b) and using 4 J”i (q+ _ I as the value for the second term within its curly brackets leads to the rough estimate Red=(+9+10)xlO-3. (6.4)

If we were to adopt this value, and to estimate Re(x-.F) = (0 + 30) x lop3 from the values found for x and X in the literature [8], the analysis of the BellSteinberger unitarity relation (5.11) would lead to much the same values as those reported by Baldo-Ceolin et al. [29], namely Res=(1.62+0.05)x10~3,

Im.s=(1.58fO.O9)x Imd=(-0.11+0.10)x

With the direct empirical of Re E the value

10e3.

(6.5)

estimate given by (6.2) we would obtain from the value Red=(-0.7f6.0)x10p3,

consistent the limits assuming (6.4) and

10e3,

(6.6)

with (6.4) but not well determined. With these values for Im d and Re d, which can be placed on the mass and decay rate differences of Ko and I?, unitarity but not CPT invariance for K,, decay (so that we can only use (6.6)), and which represent limits on CPT-odd weak interactions, are

Ml, M --MM22 <3x10-“, L

S

rllr --= r22 <3x10-‘, s

(6.7)

L

at the level of two standard deviations. A better limit on Re A can be obtained by from 6,( cc ) assuming CPT-invariance for K,, decay [ 161 but even so the measured value for q&, gives a value for A differing from zero by two standard deviations [30]. The values obtained for the real and imaginary parts of E and A all depend on the valid application of the complicated unitarity relation (5.11). Without invoking it, nothing firm can be said at present about their imaginary parts and their real parts have only upper limits of order 2 x 10--2. It is the substance of this paper that p/F comparisons, especially for the K,, decays, could determine the real and imaginary parts of E and A, and the values of other CPT-violation parameters, independent of this unitarity relation. 7. CONCLUSION

In the above, we have discussed the possibility of determining the CP-violation parameter E and the CPT-violation parameters A, A,, y, x, and 2 from observations of K + lvrt events for neutral kaons initially K? or F at production. We have shown that these parameters can all be determined, at least in principle, from K,, observations alone. The new features giving rise to this possibility are that LEAR has made available clean, intense anti-proton beams with good momentum resolution, and that we can now switch from observations for p to observations

DETERMINATION

OF T- AND CPT-VIOLATIONS

485

for p simply by magnet current reversals, which means that those systematic errors in the comparison of Ko with p in one setting, which arise from mechanical asymmetries in the setup, are readily averaged out without identification. An experiment of the kind envisaged here can now be carried out as LEAR. The most recent analysis of existing data is that of Barmin et al. [30]. In order to find sufficient input information, they had to make use of Bell-Steinberger unitarity relation. There are some uncertainties with this relation, arising from the possibility of undetected decay modes for neutral kaons and from possible modifications to it due to CPT-violation and violations of the AQ = As rule. The results we have obtained following their analysis but not using 6,( m ) are E= ((1.62_+0.05)+i(1.58f0.09)$

x 10 -3

The parameter most poorly determined is Re A and the limit it gives on CPT-invariance in J?’ decay is only 6 13 x 10e3 of the weak interaction, which is substantially larger than the known failure of CP-invariance, /q + .~ ( = 2.23 x 10 p3. There is still no explicit empirical demonstration of T-violation for the weak interactions. Although this is implied by the CPT-theorem, that CP-violation must imply T-violation in so far as CPT-invariance holds, this situation is not really satisfactory and we badly need to see some direct evidence of T-violation in some domain of the weak interactions. We have discussed the experiment proposed for this end by Kabir [9]. In its most accessible form, where the KO and p components are sampied through their K,3 decays, it is sensitive also to CPT-violation, so that the experiment becomes a quantitative, rather than a qualitative, test for T-violation. The possibility of failure of CPT-invariance raises difficult questions. We have not attempted to put forward any particular theory of CPT-violation, but have only remarked that it is not inconceivable and that CPT is the only discrete symmetry which has not yet been shown to fail at some level. Some authors have raised the possibility whether it might not fail because of a failure of hermiticity at this level. As a unitarity relation, the BelllSteinberger relation would then be seriously affected. It is therefore rather desirable that the CPT-violation parameters should be determined without appeal to this relation. At present, Re E and Re A can be determined, but only the combination Im(s - A). From the experiments discussed here, it hould be possible to determine all of these parameters, and then to use them to test the Bell-Steinberger relation, using also the existing data on K + ~71 and 7crrrt decay processes. The present theories of CP-violation are rather unsatisfactory. They are based on spontaneous symmetry breaking of the standard model of electroweak interactions, involving parameters which we cannot predict from that model, Even so, it has proved extremely difficult to tit a value for F’/E as small as the limit now determined empirically. It may well be that CP-violation does not arise from within the standard model. Immense experimental effort has been devoted to the determination of

486

TANNER ANDDALITZ

further CP-violation phenomena and their parameters. Although the parameters already known to be non-zero have become thereby more accurately determined, no new phenomena have become established and only upper limits have been determined for the possible parameters. The thesis of this paper is that improvements in our knowledge of T-and CPT-invariance are most likely to be obtained, at this stage in the subject, from a precise comparison between p and i?’ decays free of systematic errors. This is hardly possible using sources such as K-p + J?n and 71 p + PA, but it does appear possible that jip annihilation at rest provides a symmetric p/K? source which will enable much progress to be made in this fundamental research. Nofe added in prooJ As pointed out to us by Professor R. G. Sachs, Aharony 1361 has shown that the measurement of both fl and p intensities downstream, from initial production of K” and p separately, allows a clean separation between CPT-violating and T-violating effects. However, this work does not include the effects of CPT-violation on the (Ko, F?) decay amplitudes themselves. GaliC [37] has recently demonstrated that the difference I(&, - 4, _ )I cannot exceed 0.4” if there is no CPT-violation and the present data on other (@. p) parameters are correct. We also wish to acknowledge helpful correspondence with Professor P. K. Kabir and Professor R. L. Jaffe on the topics of this paper.

APPENDIX A THE MASS MATRIX AND EIGENSTATES FORTHE (J?,p)

COMPLEX

We write the mass matrix in the following form

where the suffices (1,2) refer to (J?‘, p),

respectively, I@ and i= are diagonal,

and

a=;of,,-&J-f (r,,-l-22)

(A.2a)

a = Re Ml2 - (i/2) Re f I*

(A.2b)

~=iImM,,+(1/2)ImTIz.

(A.2c)

If c1= b = 0, the final matrix in (A.1 ) has eigenvalues + a and the corresponding eigenstates K, have definite CP values w = f 1, being written as (a)

K, = (K- R)lfi,

(b)

K- = (K+K)/$.

(A.3)

where we have adopted charge conjugation operator C such that CK= R and CR= K, and parity P= - 1 for K and g mesons. Using base states K+ , the final matrix in (A.1 ) takes the form +a ( a-p

a+-a -a )

K KY

(A.4)

DETERMINATION

OF

487

T- AND CPT-VIOLATIONS

The element G arises from CP-conserving weak interaction terms such as P + arc -+ KY’ and generates the bulk of the mass difference 6M = M, - M,. As the form (A.4) displays, the terms CIand p connect K, and K_ states and are therefore CP-violating terms. If T-invariance holds, M,, and r12 are real; only T-odd weak and superweak interactions contribute to fl. If CPT-invariance holds, M,, = MZ2 and rll = rZ2 hold valid; only CPT-odd weak interactions contribute to a. Retaining all terms in (A.l) for the present, the eigenvalues of the last matrix are A, = 7&iqFTF).

(A.5)

We denote the eigenstates by (a+ Ko + a + K?). For eigenvalue L + , we have

to first order in the small parameters M and p. For eigenvalue R_ , we have

By comparison

with the forms (2.2), we see that (a)

P/o = 2~

(b)

a/o = 24.

(A.81

Note that c( and fl occur in (AS) only to second order. To this accuracy, therefore, we have V+ -A_)=

-2~=(M,-M,)+~(T,-T,),

(A.9)

where rL Jr, z &, so that (r, - f,) z 2r, to the accuracy appropriate although we shall not use this approximation here. For the CPT-even, CP-odd interaction, we thus have the expression - Im M,, + i Im r,J2 to the accuracy specified above. For the CPT-odd, CP-even interaction, A =x/(20)

here,

(A.lO)

we have, to the same accuracy,

=-*1 i(M,, - MU) + (rll - rd/2 2 (r,-r,)/2+i(M,-M,)

.

(A.1 I)

We note that this parameter is zero if the diagonal elements of the mass matrix are equal, irrespective of the nature of the off-diagonal elements. A superweak interaction having only As = +2 matrix elements therefore cannot contribute to the CPTviolating effects discussed in this paper.

488

TANNER AND DALITZ REFERENCES

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