A phenomenonological model of T violation in nonleptonic decays

.\NS.\LS

OF

PHYSICS:

A

The

Enrico

43,

25-71

(1967)

Phenomenonological in Nonleptonic

li’ermi The

Inslitccfe l~nirwsii~~

for

A~;rtclear

Model of Decays”?

Studies

of Phicago,

Chicago,

and

T Violation

fhe Department Illinois 606SY

of Ph?ysics,

A phellomenonological model of 7’ violation, which assllmes consta~lt Tviolating phases in an effective IIamiltonian describing nonleptonic interactions, is applied to all nonleptollic decays of hyperons and kaons in order to place constraints on the phases and to establish some connections between the Fviolating charact.eristirs of the variorls decays. It is fmlnd that within the framework of existing data, the model is consistent, brat that more diverse and accluate experiments are needed to determine the phases introdrlced in the model. In principle, by comparing all nonleptonic decays, the phases can he determined unicluely although they cannot, in general, be obtained by considering any one decay. The performance of certain experiments is slrggested. Accurate results from these will enable the phases to be determined if the mechanism of 7’ violation is the same, or closely related, in different decays. These include, among others, the following experiments which could throw light on whether or not T violation takes place via some dominant mechanism. (1) I)etermination of the asymmetry parameters in the variolls hyperon decays wollld enable 11s to distinguish between 2’ violation in the ( Al 1 = 15, “5 amplitudes. (2) A measurement of the charge asymmetry in the decay K?G + ~T+x-+’ is a direct test for 7’ violation in the 1 AI 1 = 32 amplitllde. (3) A comparison of either the T+ with the T- rate or the T’+ with the T’- rate is a test of T violation in the / AZ / = j.5 amplitude, and it is worth noting that,, of the two, the T’* comparison provides a much more sensitive test,. I. IXTRODUCTION

CP violation may occur in strong, electromagnetic or weak interactions. So far it has only definitely been observed in the weak decay (1) k’: -+ 2~. However, very little is laow~~ about the details of the CP violation. Assuming the validity of the CPT theorem, a violation of CP conservation implies a violation of T invariance. If T invariance is valid, then the amplitudes * This work supported in part i Submitt,ed t,o the 1)epartment of t.he requirements for the Ph.D. $ Present, address: Rutherford England.

by the U.S. Atomic Energy Commission. of Physics, University of Chicago, in partial degree. High Energy Laboratory, Chilton, Didcot, 25

fulfillment Berkshire,

26

KENNY

relevant t.0 some particular interaction are going to be relatively real, aside from phases which arise from final-state interactions. If we lift the restriction of 7’ invariance (and thus of CP conservation), then immediately we allow the various amplitudes to be relatively complex. If the phases between the amplitudes could be measured, then we would have a much clearer picture of the details of the CP violation. Unfortunately, as will be shown later, an analysis of the data for a particular decay process may be subject to an ambiguous interpretation in terms of phases between amplitudes even if the data are accurate and complete. One is then tempted to ask if, by comparing data on various processes, one may learn something about the details of T violation. Evidently, in order to do this we must assume some sort of universal Hamiltonian. Various models of T violation (.2), (5) in the weak interactions have been proposed within the framework of the current-current theory and SU( 3) symmetry (4). A disadvant’age of these models, however, is that one cannot compare the T violation in different nonleptonic processes since the matrix elements of the currentcurrent Hamiltonian between purely hadronic states cannot yet be reliably calculated. More recently several authors have proposed effective Hamiltonians for the nonleptonic decays which are quite different in structure from the usual currencurrent’ model. In one model (5) the effective Hamiltonian is taken to be proportional to the divergence of a current, which is a mixture of vector and axial vector and transforms under the octet representation of SU(3). In the other model (6), the effective Hamiltonian is taken to be a sum of scalar and pseudoscalar densities, each transforming under the octet representation of SU(3). Such models could be made to violate T invariance by the insertion of a simple constant phase difference between the scalar and pseudoscalar parts of the Hamiltonian. The important thing about such a phase would be its universality, i.e., the same T-violating phase would exist between X- and P-wave amplitudes in all of t’he hyperon decays and this phase would also exist between the K” + 2~ and Ku --+ 31r amplitudes thus leading to CP violation in the decay Kn” --$ 2a. Since the Hamiltonian transforms under the octet representation, it will satisfy the 1Al 1 = $2 rule for nonleptonic decays. Another model of nonleptonic decays which is of interest to us is that of Lee and Swift (7). They assumea pole model of weak interactions within the framework of SIi(3) symmetry. Under the assumptions of (1) CP invariance and (2) weak Hamiltonian transforms like Xc, they show that only P-wave amplitudes arise from the pole-dominance model of Feldman, Mathews, and Salam (8), whereas S-wave amplitudes may be described by a model which assumes K* pole dominance with wea.k interactions mediating the K* -+ ?rprocess. Thus since quite independent, pole diagrams lead to the S- and P-wave amplitudes, it

MODEL

OF

27

T VIOLATION

is clear that associating a constant phase with the K” - P vertex would again lead to a universal T-violating phase. In t,his model it is assumed that the effective weak Hamiltonian describing the t’wo-particle vertex transforms like X6 , so that the / AI / = f,i rule is satisfied. A natural extension of such models which contain a “universal phase” between X- and P-wave amplitudes would be a model in which not only would there be a universal phase between the parity-conserving and parity-violat,ing parts, but also between the parts of the effective Hamiltonian which transform like 1 AI 1 = ,I$, 1AI 1 = "i, .. . etc. Let us write a CP-conserving nonleptonic effective Hamiltonian as Hw = HI,? + Hw

+ HWZ + . . . (1)

i- Hi,? + Hi,, -I- H:,z + . . . , where Hn,2 (H:L,2) conserves (violates) space like a spinor of order sin. To allow explicity for the possibility t onian

+ H,,exp + fdf,

(-iq$) + H&

parity

and transforms

of T violation,

in isotopic

n-e rewrite

bhe Hamil-

+ ...

+ Hbf,exp

spin

(2) (ih’)

+ d/zexp

l-i&‘)

+ ... ,

= H + H’, where K,z

= (H,,z)+,

Hi,;2 = (&J+,

and we adopt the convention that H+ induces a AS = + 1 transition. The behavior of the various parts of the Hamiltonian under the symmetry operations T, P and C is shown in Table I. The phases 7P , 71~and qT are independent of n and are chosen so that the Hamiltonian commutes with the product of these three operations taken in any order as required by the CPT t,heorem. The conditions on the phases are 3-P = ?C??T = fl. In this model it is assumed that + is not a function of coordinates (or momentum transfer in momentum space) but it may vary as indicated by the subscript n and presence or absence of a prime. The convention has been adopted that phases are measured relative to Hii . T invariance is violated directly in nonleptonic decay if any of & , &’ are nonzero. By writing out the Hamiltonian in such an explicit form, we have classified

28

KENNY TABLE TR.\NSFORM.\TION

I

PROPERTIES OF THE VARIOUS P.IRTS OF THE HAMILTONIAN UNDER THE OPEHATIONS OF l’, P, AND C

implicitly some of the different ways T may be violated in nonleptonic decays. It must be emphasized, however, that the various c$,,, &’ are, in general, energydependent. Such would be the case, for example, if one expanded the T-violating current-current Hamiltonian of (3) in the form of Eq. (2) (in this case, of course, 1 A1 1 = 4; and only the phase +1 enters). However, we shall assume that &, , &’ are constant with respect to energy and independent of the decay involved. This model is not to be taken seriously, because of such a simple assumption, but we can hope that it serves to give scme idea of the order of magnitude of the correlations that may be expected between various T-violating effects. In this respect, the phases appearing as parameters may be considered as some sort of average of phases actually appearing in the fundamental interaction. To determine what sort of average would require a more specific model. In the following, we shall apply our phenomenonological model to hyperon and kaon nonleptonic decays in order to see what restrictions are placed by the data on the phasesC&, &‘. For the sake of simplicity and because of the lack of relevant data at the present time, we shall illustrate the model for five simple cases,viz. the caseswhere only one of 41 , C&, &‘, & , & is nonzero and all other phases are assumedto be zero. In general, how-ever, there is no reason to expect any of the phasesto be zero. In fact, given sufficiently accurate measurements of the relevant quantities, the various phases could be determined uniquely. II.

A-HYPERON

DECAY

In this decay, only H&t , H& , Hit, and Hit2 contribute. notation ATAI 21for the decay amplitude of a hyperon through of AI to a final state of isospin I. There is a certain ambiguity this notation since A,, and A,, could refer to A- or X-decay. Z-decay are discussed in different sections, this ambiguity difficulty. The reduced matrix elements for A-hyperon decay are

We introduce the an isospin change involved in using But since A- and should cause IIO

05 II Hh II A> = PII , (3)

.MODEL

OF

29

T VIOLATIOS

where P,, , Ps3 , S,; , Xaa are all real. We have neglected the final-stat’e-interaction phase shifts, t#he largest of which is associat’ed with the amplitude S1, and is about, 6”. This does not affect’ the main argument and seems reasonable in view of the fact t,hat the available data are sparse and inaccurat’e. For the same reason we shall neglect phase shift’s in the other h.vperon decays. In terms of the above amplitudes, the observed final-state amplitudes are (/IT- ( H / A) = - ( 33 j”“P 11exp (&j (p-

1H’ 1A) = - ( “3 )‘~“Sn

(ILTI’ 1H / ii)

=

(n,” 1H’ 1Li) =

(>i)l”P

llexp

(i&J

(1.2j1’2S11

+ (!$)“‘PK3

exp (z&j

= P-,

+ ( $d)“‘&

exp ($9’)

= S-,

(,2i)1’2Paaexp (&)

= P”,

+

(4)

+ (S~)“‘S,, exp (ids’) = So.

The measurable quantities for each channel are the tot’al rate the asymmetry parameter a

=

1 s

2 Re SP* 12 + 1pT2 ’

the T-violation paramet’er 2 Im SP” p = lAyI:!+ IpI”’ and 1s I2 - 1P I2 y = 18 12+ I p /i ’ which t’ells us whether the S- or P-wave amplitude is larger. Such measurements are possible for both channels. Thus one could determine I S 1, 1P I and the phase of S/P for both channels. There are a total of six experimental quantities which can be fitted by the seven parameters PII , P,, , Sn , S,, , 4, , +3 and ds’. Clearly, there canrlot be a unique fit so that A-decay alone does not provide enough informat’ion to determine the various T-violating phases in our model. But under our assumption of a universal Hamiltonian with constant +n and + n’ 1 it’ is possible to compare the const8raintsobtained from this decay Lvith constraints obtained from other decays. If we neglect t’erms quadrat’ic in the / A1 1 = ?.b amplikde, t’he branching ratio is given by r” 1 =-[1+3dS r2

PllP33 cos

6~ - +3j + slls,3 cos +3 SL + Cl

and the asymmetry parameter ratio is given by

(5)

30

KENNY

The data on the branching

ratio as summarized r’/r-

by Samios (9) give

= 0.506 f

0.020.

(7)

If time-reversal invariance is valid, this result, together with Eq. (5), tells us that the / AI 1 = 35 amplitude cannot be greater than 2 %. On t’he other hand, if time-reversal invariance is violated, we can only place an upper limit of 20%ontheIAII = x 9 amplitude, the upper limit being attained when the 1 AI 1 = M amplitude is 90” out of phase with the 1 AI 1 = 35 amplitude. If we consider the three examples mentioned in the introduction, we can place the following limits on the 1 AI 1 = 3s admixture: if

41 # 0,

if

43

#

0,

P33

833

&=43/=0,

then

p,

+1 = $3’ = 0,

then

%2%,

Fs2%; 11

11

(Sal g

5 20%;

(Sb)

< 20%. SC-

(SC)

11

if

43’ # 0,

& = $3 = 0,

then 2

11 833

,< 2%, 11

The data on the asymmetry parameter ratio (11) give cY”/a- = 1.10 f 0.27.

(9)

A comparison of this with Eq. 6 tells us that this result places much lesssevere restrictions on the ( AI 1 = y 2 amplitude than does the data on the branching ratio. In the following we shall assumethat the restrictions placed on t#heI AI 1 = jZ$jamplitude are those given in Eqs. (S). A useful measure of the T violation is the quantity /3/a = Im XP*/Re SP* = tan x,

(10)

where x is the phase angle between the S- and P-wave amplitudes. It has been determined by Cronin and Overseth (12) that x(h + PC)

= x- = (-15

f- 2O)O.

(11)

In terms of the quantities introduced in Eq. (3), tmx-=E=

a!-

-tan4

1

l + d2 53 sin (41 - 43) sin 24~ Pll [

1

++e&

1

0

)

(12)

tan x0 = $ P33 =-

tan 41 [ 1 - 2 4

p 11

sin

(41

-

43)

sin 241

43 _

2 4

1

&I &cYiizg ’ SF

SS

where we have neglected terms quadratic in the 1AI 1 = N amplitude.

(12’)

MODEL

OF

31

T VIOLATION

We now consider t’he three simple examples of T violation introduction: (a i 41 # 0, q& = 43’ = 0.

From Eqs. (la),

(12’),

ttan X- M t’an x0 ,k: -tan

mentioned

in the

and (S), we have

& ;

(13)

i.e., x- ,z x0 ,% -41 . Thus, from the experimental

result [Eq. (ll)]

(13)

we have

$1 = (15 f 20)“.

(14)

It, is clear that’ this hypothesis could be tested by more accurate measurements of the angle x for both decay modes. The prediction of the hypothesis is that, x- and x0 should be approximately equal. (h) & # 0, ~$1= & = 0.

From Eqs. (12) and (12’1, we have

1 P33sin

t,an x- = ~ 42 p,,

&

=

-

1 tan x”. 2

-

If T violnt~ion occurs t’hrough the / A1 1 = 3p ampWide, it is clear that the overall phase x bet’ween the S- and P-wave amplitudes is “naturally” small even if 43 is large. 12rom Eq. ( 11 ), &

(16)

$E sin & = 0.3 f 0.4.

This is consistent with the resbrictions placed on PBd in Eq. (8) without placing any limitation on the phase & . This hypothesis could be tested by a measurement of the phase x for both decay modes. The prediction is that X- and x0 should be opposit’e in sign and connected by the relation given in Eq.( 15). ( c) &’ # 0, $1 = 43 = 0.

From Eqs. (12) and (12’)

we have

- 1 &a

tan x- = _dS x,, sin ~$31= - 1 tan x0. Once a,gain the phase x is “naturally”

- 3

(17)

small. From Eq. (11 j,

1 833 sin &’ = 0.3 f 0.4. Sl,

This is consistent with the restrictions placed on S,, in Eqs. (8) without placing any limitation on the phase +3’. In summary we can say that hypothesis (a) and hypotheses (b) and (c) represent extremes such that (a) x- M x0 # 0, (b) or (c) 2 t,an x- = -tan

xc; x-, x0 # 0.

32

KENNY

A measurement of the angles x- and x0 would be expected to lie somewhere between these extreme predictions and hence would lead to some inference about the relative importance of the different phases. However this measurement would not enable us to distinguish between (b3 and &‘. This behavior is to be expected since, as we stated earlier, in general it will not be possible to make a unique solution for all the amplitudes and phases involved in this decay. III.

Z:-HYPERON

DECAY

Our discussion of X-decay is very+similar to t,hat of A-decay in the previous section. In this decay H& , HiI , H1/2 , and H:T2 contribute. The reduced matrix elements are

(19)

where plz , P,? , fin and Xs2 are all real. served final-state amplitudes are

t’erms of these amplitudes,

In

the ob-

(ha- 1H ( E-) = PI2 exp ($1) + !4 P,, exp (z&s), (AT- 1H' / Z-) = ~‘5% + ?5 S, exp ($al), (AT' 1H 1Z") = &

[PIZ exp ($1) - Paz exp (i&)1,

(ha0 I H' 1Z") = &

[Sn

If we neglect terms quadratic is given by

rF=2 and the asymmetry

According

[

1+3

PI2

parameter

- 832 exp (i+)31)1.

in the I AI ( = 35 amplitude, p32

cos

(20)

(41

-

43)

+ Xl2 -______XL + PT2

832

the branching cm

43

1

rat.io (21)

ratio is given by

to Samios (9), the corresponding r-/l?’

experimental

data are

= 1.66 zt 0.23,

and a-/cY” = 1.23 f 0.58. It is clear that, even in the absence of

T violation, the present data are consistent

~MODEL

OF

T VIOLATION

33

a-it’h a 10% admixture of ] AI / = g$ If T invariance is violated, the allowed admixture could be much larger as in the caseof A-decay. The data on the T-violation parameter x from Rosenfeld et al. (IO) are x- = (17 f

x0 = ( -8 f

ls)“,

(23)

62)O.

Note that our definition of t’he T-violating phase has the opposite sign from that of Ref. (10). Note also that x-(x”) here is the S-P relative phase for Z-( 2”) decay whereas in the previous section it was the corresponding phase for A + p?r-(A --+ MO). In terms of the quantities introduced in Eq. (19)) tan x- =

-tan

+1

1

tan x0 =-tan91

P32

sin

PI,

[

(+I

-

43)

_

sin 24,

S I sin & SlZ Gz&

2

1

1+2~sini~l-~3)+2~~ill~3

[

PI,

sin 2&

As12 Gz&

(24)

’

1 ’

(24’)

where we have neglected terms quadratic in the 1AI 1 = a$ amplitude. We consider the samethree simple examples of T violation. (a) 41 # 0; +3 = +3’ = 0.

now

From Eq. (24) we have

tan x- = tan x0 = -tan 4, .

(25)

From the experimental result [Eq. (23)], x- and x0 are equal within the rather large errors. Also the value of & obtained from a comparison of Eqs. (23) and (24) is consistent with that obtained in the previous section [see Eq. (14)]. Of course, in view of the large errors involved, such a comparison does not mean very much. Again this hypothesis could be tested by more accurate measurements of the angle x for both decay modes. (b) 43 # 0; 41 = I$; = 0. -2 tanx-

From Eq. (24) we have = ( P32/P12)sin 43 = tan x0.

(26)

Again the overall phase x between the S- and P-wave amplitudes is “naturally” small becauseof the smallnessof t,he ] AI ] = ,3/i amplitude relative to the 1Al 1 = ,1$amplitude. The data in Eq. (23) are consistent with a 10 % or larger admixture of 1AI ] = s and a phase C#B~ as large as we please. Within the large errors, the relation -2 tan x- = tan x” (26’) is satisfied. (c) 43’ # 0; 41 = ~$3= 0.

From Eq. (24) we have

2 tan X- = (Ssz/Siz) sin I#J~’= -tan x0. We may make the same comments here as we did above in hypothesis (b).

(27)

34

KENNY

It is clear that the Z-decay is very similar to the h-decay insofar as it is capable of giving us information on possible T violation. In fact we refer to the summary at the end of the section on A-decay in order to point out how more accurat’e measurements of x would enable us to distinguish between the various hypotheses. IV. Z-HYPERON

DECAY

In this decay H&t , Hip , Hip , H:f, , Hit2 , and H:fi contribute. The reduced P-wave matrix elements are

(28)

where PII , PI3 , Pzl , Pg3and Ps3 are all real and we have neglected final-state interactions. The reduced S-wave matrix elements are defined similarly. In terms of these quantities, the observed final-state P-wave amplitudes are (pr” 1H 1z+) = s

3

[P

l1

+ P .exp ($3) + (315)“’ P53exp ($5)

= P1” exp (id (n.rr+ 1H ( 2’) = %[--2Pl1

+ P30exp (i43) + P50exp ($5) , + P131exp (til)

+ ?&[P3~ -

2(?3)““P,,]

(29)

.exp (ti3) + (>f5J1”P53 exp (&) = PI+ exp (GM + P3+ exp (%3) + P5+ exp ($51, (m- 1H j 2-) = P 13 exp (til)

+

(%i)1’2P33

exp (i&)

+

(f/i5)li2PS3

-exp (f&J = PI- exp ($1) + Pa- exp ($3) + P5- exp ($5). There are similar expressions for the observed final-state S-wave amplitudes. The 1AI 1 = 4; amplitudes P1pand X1’ satisfy equations of the form &‘A;

+ A,+ = Al-.

(30)

This relation is independent of T invariance and is usually exhibited in the form of a vector diagram in the S-P plane (IS). It is found experimentally (14) that not only do these vector amplitudes form a closed triangle according to Eq. (30) (and thus satisfy the / AI 1 = 46 rule within experimental error) but also that the triangle is approximately right-angled and isosceleswith one side along the

MODEL

OF

T VIOLATION

:25

P-axis and one side along the X-axis. This fact may be expressed by writming I‘+ hN I‘“=

algebraically

r-

and

(31) ff + ,e ff- 2’ 0,

0 a%

-1

[see Refs. (9) and (14) for more exact numbers], where I” and c/’ refer to the decay in which rp is emitted (p = +, -, 0). Without a knowledge of y+ or y-, it was not possible to know whether Z+ --+ na+ was almost pure P-wave and Z- -+ n6 was almost pure S-wave or vice-versa. However, recent, experiments (15) indicate the former to be the case. This is also consistent with t,heoret,ical predictions made within the framework of SU( 3) and CP invariance. Lee ( 16) and others have predicted under these assumptions that 2+ + nT+ is pure Pwave. Consequently, from the Gell-Mann-Rosenfeld diagram, Z- -+ nF must be almost pure X-wave. We conclude from this that the 1A1 1 = ,1,L;component of the S- and P-wave amplitudes is suppressedin the decays Z+ + ~LS+and S- + na-, respectively. However, we shall assume that, the j AI / = !,Q component remains unaffected in these decays. We shall neglect the 1Al / = 5,samplitudes. Because there are 110 data on the parameter @in Z-decay and becauseat present it is difficult to make an estimate of the possible / AI 1 = 5~ admixture in the amplitude and finally becausethe large number of paramet’ers involved [seeEq. (28 )] make it impossible to make simple quantitat’ive comparisons between t,he various decay modes, we shall give a qualitative discussion of T ViOhttkJn in S-decay. There should be no difficulty in following the argument’ since the problem is basically the same as t#hat discussedin t’he previous t#wosections. The discussion differs here, however, in that it deals with the parameter p rather t.han P/CYsince O(is close to zero for two of t,he decays. Let us again consider our t,hree simple examples of T-violatJion. (a ) Suppose that 41 # 0, but 43 = 43’ = 0. Then to second order in small quantities p+ = p- = 0

(32)

assuming sin +1 is small. We shall see that this assumption is justified when we consider 8+ ----fpr”. From the definition of cy (see Sec. II), it is clear t’hat, I cos #q / 2 / aa j = 0.960 f 0.067, [see (14)] which t’ells us only that

/ 91 I < 25”.

(33)

36

KENNY

Thus sin ~$1is certainly small, as assumed. This limit placed on 41 is compatible with the A- and Z-decay data. If this hypothesis were correct, we would expect, the value of 0’ to be comparable in magnitude to the values of p for A- and ,“decays. However, we would expect the values of p+ and p- to be suppressed relative to these by about the same ptor that the 1AI 1 = 52, S-wave amplitude is suppressedin the decay S+ -+nr . (b) Suppose that 43 # 0, but +L = & = 0. Since, in this casep” is proportional t’o the interference between Sf and P2” (p = +, -, 0), we wdd expect the values of /3” and 6 to be comparable in magnitude to the values of p for A- and Z- decays. However, we would expect the value of of to be suppressed relative to these by the same factor that the 1AI / = pi, X-wave amplitude is suppressed in the decay Z+ --+ na + . (c) Suppose that 43’ # 0, but 41 = ds = 0. Since, in this case, 0” is proportional t’o t’he interference between PIP and 8,’ (p = +, -, 0), we would expect the values of /3” and @ to be comparable in magnitude to the values of p for Aand Z-decays. However, we would expect the value of @-to be suppressedby the usual factor. In summary we can say that, although the Z-decays represent a much more complex problem in terms of parameters necessary for its description than either of A- or Z-decay, it enables us to make a quite clear-cut distinction between the various hypotheses. Briefly, the results of t’he discussionof t,he various hypotheses were (a) P+, P- << P”, (b) Of cc P”, P-, Cc) P- << PO,p+. Thus, accurate measurements of fl+, p-, and p” would enable us to say which, if any, of the above hypotheses is the dominant mechanism for T violat,ion or which combinations were important. Since both A- and E-decay make a distinction between hypothesis (a) and hypotheses (b) and (c) [but not between (b) and (c)l, it is clear that t’he most valuable contribution that measurements of p+, @-and 0” in Z-decay can make is to give some information about the relative importance of hypotheses (b) and (c) ; i.e., informat.ion about the relative importance of the phases4a and +3/. We have also pointed out above [see Eq. (33)] that an upper limit on the phase 4, can be obtained by measurement of (Y’. Consequently, a more accurate measurement of (Y’ would lead to a more definite upper limit on the phase41than that given in Eq. (83). V. K ---t 2~ lIECAY

and HL, contribute. We denote the relevant In this decay, only H:~z , I&, reduced matrix elements (17) by (I /I Hh

II K) = 6, exp (iS11,

(n

=

1, 3, 5),

(34)

MODEL

OF

T VIOLATION

37

where b,, is real, I is the isotopic spin of the final 2~ state, 6* is the S-wave 2~ scattering phase shift for the pions in a state of isotopic spin I with total center of mass energy equal to the K mass. In terms of t,he above quantities, the observed final-skte amplitudes are (0, 0 1H,

/ K”)

= T”“bl

exp (&)

= Pa exp (i& 1,

(2, 0 j H,

1 K”)

= T”‘[b 3 exp (-~4~‘)

+

b, exp (-&‘)I

exp

(i&)

= & exp (i&j, (2, 1 1H,

1Kf)

= (3A)“‘[b3

(35)

exp (-i&‘)

-

(;3)b5 exp (-&‘.)I

exp (2~)

= 0, exp (IS,). The Kf decay rate into g+r” is: r(K+

+ a+~“)

and the following I&I

= $1 I bl exp ( - &’ ) - ?,3bs cxp ( - &’ ) 1’

parameters

characterize

(36)

K” decay (i8) :

= 1(a+x-IHwfK,“)/,

I a? j = I (T”,” j H w I K$

I, (37)

v+- = a:?/ay! qoo = a:%‘/& We have introduced

= (~‘6

I H w ( K,“)/(T+/

= (,“ir’ 1H w I K,“)/(T”T”

I H w I KF), / H M,1Klu).

I k’t)

and ) K;) nhich are defined by (19)

1 Kl”)

= (1 + / I’ Iy)--li’( 1K”) + I’ I k:“) ),

/ &“)

= (1 + I i‘ 1’)-1”( / h’“) - I’ 1 r;“)).

The phase of I k:‘) is defined b> 1I?“) = CP I K”). In terms of the paramet’ers PO and p2 [Eq. (%?)I, ai’lL’ = (1 +

1 ~l’)P”‘([(2$)1’2&

exp i(&

-

So)]

f ~[(?~)‘~‘/3~ + ($$)“‘&* exp i(& - So)]], ( 1,‘L) = (1 + / )’ 12)-1’2{[(?$)1’Eflo - (,Z)““& exp j(& a00

-

&)j

f ~[(Ji)(3~

-

(,2$)‘“&*

+

(>3)“‘&

(38)

exp i(& - SO)]},

where the upper sign refers to a”’ the lower to a”‘. We have six measurable quantities from K” decay to determine

PO, ,& and T

35

KENNY

assuming a knowledge we may know

of the scattering

phase shift difference

(& - &,). Hence

po = 22”%, ) P2 = 2-1’2[b3 exp ( -G$Q’) + b5 exp (-z&l)], / 8~ I = (%>“‘[b3

exp (-i&‘)

-

(39)

3,$bs exp ( -&‘)].

It is impossible from an analysis of K 3 2?r decay to determine the parameters bl , b0 , bg , &I, and C#Q’uniquely. A similar situation arose in the individual hyperon decays. We shall consider two simple cases of T violation. (a) Suppose that &’ = ~$5’ = 0, but & # 0; in other words, T-violating phases occur only in the parity-conserving parts of the Hamiltonian. In this case the various p [Eq. (35)] become po = 2-l’%

)

p2 = 2P2( bp + bb), p1 = (gy(b3

(40)

- %bb).

We thus have the simple relations amp (ICI’----) ?r + ?r)/amp

(K” -+n

amp (Kzo ---f r + 7r)/amp

(K” + ST+ 7r) = (1 - r),/(l

+ r) = (1 + r)l(l

+ 1r 1y, + ) r ]2)1’2,

(41)

for either decay mode ~+a- or TOT’. Thus 9+- = 1700= (1 - r)l(l

+ r).

(42)

In this case the Kz” --f 21r decay, which exhibits CP violation, occurs because of the contribution to the self-energy (and hence to r) of the K” ---f 3r mode which we have assumed violates CP. A measurement of either q+- or r]oowould determine Y in magnitude and phase. r depends on the T-violating phases 4,, which enter into the parity-conserving K + 3a decay, through the 3~ contribution to the self-energy. However, since this dependence involves off-mass-shell effects, which we cannot calculate because of divergence difficulties, we cannot estimate the dependence of r on the T-violating phases and thus place any limits on the phases & . (b) Suppose that ~$31# 0, #Q = $JS= 0. Assume that ) AI ] = >s does not contribute. In this case, the various /3 [Eq. (35)] become po = 22”%

)

P2 = 2-1’2b3 exp ( -&‘), 81 = (,$i)-“‘b3

exp (-i&‘).

(43)

MODEL

Then we have, following v+- Z .I;[6 + i&

OF

Wu and Yang (18))

Im (p2/p0) exp i(& - &I)] = ?Z[E -

700

M $z[e - 224s

39

!I’ VIOLATION

idZ(bJbl)

sin &’ exp i(6, - So)], (44)

Im (&/PO) exp i(& - So)] = j,$[c + i2&(b3/b1)

sin &’ exp i(6, - &I)],

where 6=1--r. A knowledge of q+- and r]oo in magnitude and phase would determine c and bl sin &‘. To separat,e b3 and sin &‘, we need to use t’he branching ratio formula (18) r (K1” * 27r”) ly2-Q + 27r)

1 - 2&ReFcos 1-

242;

1

1

1

(& - So)

COY 43’ COB (62 -

1

(45)

60) .

We also have (46) thus (20) r&O+ w) =~[l*4(~)I:~c~,s~~cos(6~-60~]. r(K1o -+ 2rj Unfortunately,

60 and & are not well-known

(47)

at, present. Recent experiments

6. = 48” (modulo r) ($1)

give (48)

at a dipion mass of 500 RleV, and 35’ < 6. (modulo r)

< 55” (92)

at a dipion mass of 400-500 MeV, and the & phase shift is small at the same energies (22). In addition, current theoretical speculation (23) about the behavior of 6. close to threshold is not conclusive. However, if we suppose for illustrative purposes that (60 - 62) = ,1&r f na, we find that for fixed +31, it’ is possible to have a branching

(49) ratio of either

r ( KIO + 2$) = 0.333 + 0.030 cos 43’ ) r(k’? ---f 2a)

40

KENNY

or = 0.333 - 0.030 cos c#J3’. From the data summarized

by Trilling

1b&l

(24), in deriving

(50)

Eq. (50)) we have used

j = 0.045 f 0.001,

(51)

where the relation of 1bs/b, 1 to the two pion decay rates of K+ and Kl” is given by Eq. 46. On the other hand, if we admit complete uncertainty in the phase shifts, then the branching ratio may lie anywhere in the interval

+ 0.333 - 0.042 1 cos 43' 1 < lyK1° 27r") < 0.333 + 0.042 1 cos 43' I . l-(KlO--t 27r)

(52)

[As noted by Dalitz (do), there is a slight correction to be made for phase space differences between Kt ---f ?T+T- and Kf + lr”ro such that 0.333 above should be replaced by 0.337.1 On comparing this with the experimental result (24)

r(K1'--,2?r') I'(K1° -+2r)

= 0.309 f 0.022)

(53)

it is clear that it is impossible to estimate cos &‘, even if we assume a knowledge of the phase shift difference, e.g., (60 - 62) = y&r f nr,

(49)

as above, we would have

f0.030

cos 43' = 0.027 f 0.022,

(54)

which places no limits on &‘. However, it has been pointed out by Truong (25) that the phase $31 must be small if there is no I AI 1 = 95 cont,ribution because, under this assumption,

1 63/b 1 = 0.045 f 0.001,

(51)

but q+- E X[E - iz/Z(b&) and the experimental

sin c&’ exp i(&

value (1)

/ ?j+- 1 - 2 x 10-3, means that we must have sin &’ More exactly,

Truong

0.1.

(56)

(25) concludes that 43’

5 2".

(56’)

MODEL

OF

T VIOLATIOri

41

This small phase is consistent with the branching ratio data given above. The conclusion is not altered if one admits nonzero T-violating phases in the parit#yconserving part of the Hamiltonian, i.e., +n # 0. Such phases would only affect the term E on the right-hand side of Eq. (44) and hence t’he general argument is not changed. It should be noted t#hat our conclusion here is different from a more recent’ paper by Truong (26) in which he assumes (a) cos (& - 60) = 1, (b) I’(&’ ---f 2rU)/I’(Ki0 (cj I b,/bl 1 = 0.045 f

+ 277) = 0.335 f 0.014 [see (97)], 0.001 [i.e., no 1 AI ] = 5;; see Eq. (51)]

and concludes from an analysis similar to the above that, &’ is large, which contradict’s the small value of &’ implied by the smallness of ] q+- / [see Eq. (56)]. He concludes from this that, / A1 I = 5’2 is present, and that there is a large phase between I A1 ] = 1,~ and I A1 ( > 3;~ in K --+ 2n decay. Our conclusion is different because (a) experimental and theoretical evidence does not, in our OphkJn, seem to permit a good estimate of (& - &,), (b) we have used an average value for the KI” + 2~ branching ratio from a compilation by Trilling (24) which contains the branching ratio quoted by Truong, but is quite different from it. Lee and Wu (98) also conclude from an analysis of the present dat#a that, the relative phase shift (62 - 60) as determined from the branching ratio dat#a is con sistent with the direct measurement of (62 - So) assuming that 43’ = 0. In summary, there seemsto be no reason for invoking I AI ] = 5:; in order to explain t’he K -+ 2a data. If n-e exclude such amplitudes, K -+ 2~ data constrain &’ to be lessthan 2”. We note finally a result peculiar to this example,

which was pointed out by Truong (.2;), (26’). This result follows from the ohservation that E is small compared with 11in this case. Kow E # 0 arises from CP-nonconservation effects in the self-energy matrix of the P? - h” system. In this example, the lowestorder contribut’ion to c is proportional to (Re p2.1rn fl,)/ po”2 (b,)’ sin &‘/bl’, whereas the second term in the expansion [Eq. (4411 of 7 is proportional to Im &/fio = b3 sin &‘/bl . Consequently, the contribution of 6 to 7 will be of order bl/b, - 5 % compared with t’he second term and may be neglected, yielding Eq. (57). In particular, the relation

2/11+-l= 1’700/,

(57’)

$2

KENNY

may be tested experimentally. If this does not hold, then we may conclude that there is a significant amount of T violation in some other mode. We can also see from Eq. (57) that t,he phase of f may be related to the phase-shift difference (& - 60). However, since neither the phase of 17nor (Se - 60) are reliably known at the present Cme, the validit’y of this phase relat’ionship cannot be decided. VI. K 4 3~ DECAY A.

LINEAR

~SXPANHION

OF

AMPLITUDES

It is found that a linear expansion of the amplitudes in terms of the pion energies (%!9), (30) gives an adequate account of the experimental data. We base our notation on a paper by Barton et al. (50) and expand the amplitude for the decay, KP + KP + ~2’ + my, as M(apf)

= E(aPy)

[

1 + a(c&)

=$

+ c(c&)

WL2 1 (58)

%Z

.

Here p denotes t’he charge of the kaon; or, fi, y, the charges of the pions; m the average pion mass; and si = (K - kJ*, IZisi = 3so = mK2 + ml2 + m2’ + ms2. K, ki are the four-momenta of the K-meson and ith pion, respectively, and mK , mi their masses.The labels 3 and y are reserved for the unlike pion in 7 +-0 and 7’ decay (i.e., 6 in 7 and rr+ in 7’) and for r” in K” -+lrn-lr. We differ from Barton et al. in expanding the amplitude so that u(a@y) and c(c&) are dimensionless. In addition we have included a term proportional to ( s2- s,) which leads to an asymmetry between K+ and ?r in K” decay and which can survive for Kzo decay since we are not assuming CP invariance. Rewriting the amplitude in terms of the Dalitz variables

y=Q--Ts Q

and

x = 4% -I,

Q

where

Q= mK - ml - m2 - m3, Ti = kinetic energy of ith pion,

(59)

MODEL OF

T

43

VIOLATIOK

In deriving this, we have noted the differences between the K+-p massesand the T+-~O massesin order t80det#ermineQ. In the kinematics, however, we have assumedthe equality of t,hesemassesand the validity of nonrelaCivistic mechanics. We shall continue to make use of these assumptions. B. IZATES AND DIFFERENTIAL

SPECTRA

To obtain the rate, we integrate 1M(&) 1”over phase space. For simplicity, let us drop the charge assignment (a&) and denote

(61) so that (60’) The differential spectra are given by

(@La)

= 2 z / E I'[ I + 2 Re ~‘y + 1O’ l”.y’ + f I E’ I”( 1 - yt)] (1 - 2/?)l’*, and do _=(1X

p a

(l-G)"2

s-(l-,211/2

= ,;,E,f

I Al I2 dy

1

(62b)

1 + i I g’ I”( 1 - x2) + 2 Re e/z + I t’ 12n2(1 - J?)I’*.

The total rate is found by integrating either of these: (631 where p is a phase-spacefactor proportional to Q’ in the nonrelativistic approximation. (1 - x2)1’2 and (1 - y’)“* are also phase-spacefactors. Terms in the square brackets t,hat are different from 1 arise from nonsymmetric contributions to the final three-pion state. The basis of the linear approximation is that O’ and 6’ are small, since the3 represent those contributions from final states that’ are suppressedby angu!ar momentum barriers at such low- energies.This is borne out by experimental data which we shall quote later. Further, we shall give reasonsfor supposing

44

IiENNY

In this light we can rewrite

the differential

clw -= dY

2 i ( E I’[1 + 2 Re a’~](1

dw -=

2;;,*[

spectra - y’)l”,

1

1 + a / C’ I’(1 - x”) + 2 Re E’X (1 - ?)l”.

dX

(64b)

The first of these spectra, Ey. (64a), represents the usual linear variation of the density of the Dalitz plot, with respect to the unlike pion energy in Kf decay, or the ?Y’ energy in KC + a+a-a’. The second of these spectra, Eq. (64b), applies only to K” + x+F7r”, and represems a dependence of the density of the Dalitz plot with respect to the g+-renergy difference. We may integrate Eq. (64b) to obtain a right-left asymmetry of the Dalitz plot. If N+(K) is the number of decays in which the T+( K-) energy exceeds the x-( r’) energy, t’hen N+ - NN, + N-

s = -%i

1 Re t’ 1 + % j u’/*

Thus, although we should-for consistency-include the term ya ] a’ I’( 1 - 2’) in the differential spectrum [Eq. (64b)] since C’ and 1 a’ I2 are of the same order it makes very little difference if we consider only the right,-left asymmetry of the Dalitz plot. Similarly, we may approximate the total rate by I‘ = pR 2, p ) R I’,

(‘3’3)

since the quadratic terms contribute on the order of 1 %. It should be noted when comparing experimental data on rates that the quantity R should be used rather than r since R is independent of the phase-space differences between the various K -+ 3~ decays. c.

CLASSIFICATION

OF

ihPLITUDES

In general, the final 3a K+(p) decay. However, amplitude on the variables states can contribute to K

USING

ISOTOPIC

SPIN

states may be classified into 6(7) isospin states for under the requirement of linear dependence of the s, , Barton et al. (30) point out that ouly four of the -+ 3a. Using their notation, t#hesestates are

MODEL

where the notation ( I, I,(AY) li , t,hat is ckher independent, on them (X = I,). H,, , H,, , H& and II?/2 duced matrix elements using (l(S)

OF

T VIOLA’IYON

45

) indicates a state of tot8al isospin I, Z component, of the variables si( S = S) or linearly dependent, contrihut~c to these final stnt*es. WC define t,he rethe not,ation of Barton et al. (JO),

11H6pi 1) 12) = X,, exp [iS(l, S)]

(n = 1, 31,

1)H,,,

(n = 1,:31.

II b) = p,t exp [iS(l, Ii,] (2(L) !I H;,, 11!z) = vn exp [i6(2, I,,] (3(S) 11HI/r /I 1?) = 7),,exp [iS(3, S)] (l(L)

(,i8)

(n = 3, 5), (71= 5, 'i),

where A,,, AL,, , v,, and v,, are real. exp [i6( 1, s)] is nn clement of the diagonnlized S-matrix for %-zr scattering for a particular isospin stat,e and moment,um cmlfiguration. Since the t,otal energy is very close t’o threshold, we do not, expect t,hat there will be an appreciable int,eraction in the final state between piOJls which are in states of relative orbital angular momentum different, from zero. So we take 6(1, I,) = 6(2, I,) = 0. The relevant amplitudes arc then (1, O(S) ( Hw

/ K") = +T [XI exp ( -i&)

+ As exp ( - &)I

. exp [iS(l, S)] = a°C1, S) exp [iS(l, X)], (1, l(S) / Hn

/ K+) = [XI exp ( --$I)

- 3,s X3exp ( - &)I

exp [i6(1, S)] = a+(l, S) exp [iS(l, S)], (1,0(L) 1Hw 1K”) = $5 [~1 (1, l(L) 1H,

exp

C-i&)

/ K+) = [WIexp (-i41)

+

cl3

exp (--&)I

= a"(l,LJ,

- !,2 LGexp (-z’I#J~)] = a+(l, L), (69)

4

@,O(L) I Hw I K”) = * (2, l(L) 1Hw 1K+) =

[v3exp C.-i&

+ IQ exp (-&)]

= a”(~, L),

(3.i)“*[v3 exp (- z$~) - ?& exp (-i&l

(3,0(S) I Hw I K”) = 5

[75 exp ( -i+s)

= a+(~, L),

+ 77exp ( -i&7)]

. exp [i6(X, AS)] = a”(& S) exp [i6(3, S)l, (3, l(s) 1Hw 1K+> = (~$)1’2[~~exp ( -i+b)

- !~TJ~exp ( --i&1

. exp [i6(3, S)l = a+(3,

S) exp [i6(3, S)].

By projecting out the various charge st,ates, we find that t,he amplitudes for K+

43

KENNY

decay are JI(++-)

= (& ~

and the amplitudes

a+(l, S) exp [2X(1, S)]

for K” decay are

We can obtain the amplitudes for both 3~ decay modes of the K- from Eq. 70 by application of the CPT theorem: A[*(--++)

= -(&

a+(l, 8) exp[--i6(l,SIl

. exp [---i&(3, S)] + &

+ &

[a+(2, L) -

a+(3, S) a+(l, L)] y, (70’)

M*(oo-)

= -(& . exp [--iS(3,

The amplitudes -a*c+-oj

[a+(2, L) +

for the 3a decay modes of 2

a+(l,

are obtained

L)I

7.

similarly:

= . exp [--is(3,

-~*(ooo)

S)] - &

~ = -(l&2

S>l + -&

~“(1, L) ‘y

u”(l, X) exp

[---iS(l,

(71,)

- f ~‘(2, L) ‘9, S)]

u”(3, S) exp [--iS(3,

S)],

MODEL

OF

47

T VIOLATIOX

where we have used 1K-j

= c 1K+),

/ 2)

= CP / K”).

We can determine the rates and differential spectra for the various decays [see Eqs. (64a), (64h) ( (66 )] by rewriting t’he various amplitudes in the form

It is clear - that, t’he partial rates and differential spectra may differ for K+( K”) and K-( K”) decay into charge-conjugst’e channels (31 ). This \\-ould be a consequence of CP violation and final-state interactions. Since Kl” and Kz”, rather t’han A? and 2, arc the relevant, neutral particles (see previous section for definition of K 1’ and K2”), we introduce t,he amplitudes

and EI(&y),

nfl(apr,

= (1 + ) r J2)-“2[M(aPr)

+ ldT(c43~~],

AI,

= (1 + j ?‘/‘)-“‘$lfkx~~)

- riG(a/ly,],

u;(c&y),

Afi(apy)

~~(a$y) are defined from

= E’JaPr)

[

1 + Ui(&)

sy

+ Ci(&)

where i = 1, 2 and (~$7) is (+ -0)

or (000) as before.

D.

3

RATES

AND

SPECTI~A

WHEN

If we suppose that there is K+, K- decay we have EC++--)

172)

I # IN)

= ___ (&

I = 3 component

a+(l, S) exp (2)

222

1 ,

(7.3)

in the final stat,e, then for

= -2E(OO+

1, (74)

E*(--+)

= (&

a+(l, S) exp (--i6)

where 6 = 6(1, 8). Thus as a consequence of t’he linear approximation component in the final state, we have EC++-)

= R(-

-+I,

R(OO+)

= -2E*(OO-),

and neglect,ing the I = .? = R(oo--);

i.e., a consequence of these special assumptions is that the partial and K- into charge-conjugate channels are the same. Further, we have a(++-)

= 4R(oo+),

(7.5) rat#es of K+

(76)

48

KEhwT

which

may be compared

l&h

the experimental

R(++-)/4R(oo+)

result

(24)

= 1.03 f 0.04.

(77) The close agreement between the theoretical prediction [Eq. (76)] and the experimental result [Eq. (77)] is the best available evidence for neglecting I = 3 in the final state, although in principle, it could be due to an accidental cancellation of 1 AI ] = 45 and 1 AI 1 = x amplitudes. From Eq. 69 it may be seen that, if / AI ] = $2 and 1AI 1 = 56 cancel in Kf decay, they must add in K" decay. Such a possibility may be checked by examining the K:, K: decay rates. When there is no I = 3 contribution to the final states, ure have, setting r = 1 in Eqs. (72) and (73), a(+-01

= im

E2( +-0)

= li4

4

Im a”(;, S) exp (is)

(75) Re ~“(1, S) exp (is)

We shall examine the justification Ri(OO0) tvhere the factor 46 is introduced

= - i I&(000).

of setting 1’ = 1 later in this section. Since = 3,s’ I Ei(OO0)

I*,

because the pions are identical,

?nR,(+-0) ivhich may be compared

= - i I&(000),

= Ri(OOO),

with the experimental

result

1v-ehave i = 1, 2.

(79)

(24)

R*(OOOj = 1.07 f 0.12 ?$'Rz(+-0)

(SO)

Although the experimental error is large, the result again supports the hypothesis that there is no I = 3 in the final state. There are no data on K1° -+ 3?r decay relevant to this question. We shall assume henceforth1 that there is no I = 3 in the final state. The terms in the spectra of K+ and K- decay representing a linear variation of Dalitz-plot density u-ith respect to the odd pion energy are characterized by (5)liZ a+& L) - a+(2 L) Reu(++-) = - -Re -A exp (--is) , a+(l, S)

1

1 See,

Re a(OO+)

= (5)l”

Re

a+(l, L) + a+(27 h) exp (-is) a+(L 8

Re o(OO-)

= (5)l’* Re

a+(& L) + u+(2, L) exp (is) u+(l, 8)

however,

Sec. VII.

1 .

MODEL

OF

T VIOLATION

49

Thus, under our assumptions, the partial rates for k’+ and K- decay into charge conjugate channels are the same [Eq (75)], but the differential spectra may differ, provided CP is not conserved and final-state interactions are present. The terms in the specka of KF and K,’ decay representing a linear variation of t,he Dnlitz plot densit’y with respect to the TOenergy are characterized by

(82)

The terms in the spectra of KIo and K: decay representing a linear variation of the Dalitz-plot density with respect to the =+-VT- energy difference are charact’erized by

The available experimental . 2. 3 various expressions is

information

(24) with which we must compare t’hese

Ii(+-O)2R(+ 00)

= osy f .

007 . ’

u( ++

= -0.093

f 0.011,

a(OOf)

=

0.25

f 0.02,

u(f-0)

=

0.24

f 0.02,

-)

(84)

and Q( + -0) is consistent with zero (32). Within the fairly large experimental errors, t’hese results are consistent with CP conservation and the 1AI / = f,i rule which predict (SO) R( + -0) -2fJ++-) 6,(+-o)

= 2R(+oo), = u(OO+)

= u(+-O),

(85)

= 0.

2 It shotdd be noted that Trilling’s summary of K-decay data [see ref. (24)] combines results on 7+ and r decay to get an average value of u(++). To test CP conservation, an independent, analysis of the T+ and T- data would be needed together with greater accuracy of measurement. 3 Note that our sign convention for ~(a@?) is opposite from Trilling [Ref. (24)] because of the difference in definition. See Eq. (38).

IiENNY

50

However, we want to explore the constraints placed by the existing data on violations of CP conservation and the 1 AI 1 = ,I 2 rule. Since we have excluded I = 3 in the final state, we shall consider only ( AI ( = 92, ,3;. In the following, we shall consider our three simple examples of T violation under the following assumptions. (i) Transitions to the I = 1 symmetric state are more favored than transitions to the 1 = 1, 2 nonsymmetric states becauseof angular momentum barriers which are important at such low energies; and (ii) 1AI 1 = $$ transitions are more favored than 1AI 1 = 34 transitions. In terms of the reduced matrix element’s[Eq. (68)], this gives the order-of-magnitude ordering XI >>

E. SIMPLE

EXAMPLES

x3 ,Pl

>>

/.4 , v3 .

OF T VIOLATION

(a) Suppose that r#~’# 0, but 41 = $Q = 0. In this case I* # 1 would arise only from K” -+ 2a contributions to the self-energy, and the B?, KO + 3~ amplitudes would be real aside from phasesarising from final-state interactions. Similarly to the previous section, since we cannot relate the deviation of r from 1 to the T-violating phase &’ because of divergence difficulties in the calculation of the self-energy, we are unable to place limits on &’ by a consideration of k? + z&r. (b) Suppose that 41 # 0, but &’ = & = 0. Then we have for Kf, K- decay

EC++-) = &l,, [ XI exp

C--i&)

1

- i X3 exp (is) = --2E(00+)

Reu(++--)

- (5Y = 2

Re

Rea(--+)

- (5y = 2

Re pl exp ( - &) - $&.Q- (34)“‘,, XI exp (-&) - j&

Re a(OO+) = (5)“‘Re Re a(OO-)

= (5)“‘Re

~11

exp

(--i41)

-

Y&3

X1 exp (-&)

~1 ev

PI

exp

-

(34)““~

- 1 iA,

,

1 1

exp (--is) exp (i6)

,

, (86)

( -$I) - 92~ + (%$‘2~3 cup I .+) . Xl exp (-i&) - pgb (-i4d

-

Xl exp (-z&)

S;W

+

(~~Y”v~

exp

cisj

- 4iX,

To the leading order, the differential spectra are in agreement with the ] AI 1 = $i, CP conservation prediction [see Eq. (85)]. However, there are small differences which arise from interference between the I AI j = x amplitudes and the IAIl = $4 amplitudes. In particular, differences between the Dalitz plots for decay of K+ and K- into charge conjugate final states arise from interference between the / AI 1 = 9; amplitude and the CP-violating part of Dhe ] AI I = Ji

MODEL

amplitude.

51

VIOLBTION

For example,

Re a(OO+) Re u(OO+) For K,‘,

T

OF

- Re a(OO-) + Re a(OO-) K2’ decay n-e have

&

&(+-0)

= -ih2Xl

sin $1 exp (is),

E2(f-0)

= (l;)l,2 (XI cos 41 + X3) exp (CS), r cos 6,

Re ul( + -0)

= (5)"2

Re IQ+-0)

= ( 5)“’ r z”,z ,“: z r cos b, 1 3

Re Q( + -0)

1

= ( 5)“2 L x1

&

(8s)

Re ~2(+ -0)

sin 6, SlIl

41

= 0.

From Eqs. (86) and (88)) we have, bo the leading order, -2 Re

UC++-)

=

Re a(OO+)

= Re a*(+-0)

= (5)l'" p&

cos 6, (89)

which means the 1AI 1 = pi, CP conservation prediction [Eq. (85)] is satisfied t,o the lowest order. We also have, to leading order, Rz(+-O)

=

cos’

&

+

;

(2

cos

c$l

+

cm3

zlz(+OO) a(+-o) Rz(+--O)

= tar12 o1

4d,

(90) (91)

From Eqs. (84) and (90) we have j&j

= (20 f 6)”

if

XS = 0,

(92s)

while if 1Xx/Xi ) 5 0.1, then I dl / 5 40”,

(92b)

which is closely related to the result of M. Gaillard (33). Eq. (91)) together with the available data on KF + 3a (34)) cannot place any restriction on & . Thus, for a range of values of C#Q , and provided the I AI I = g amplitude is small compared wit,h the 1AI I = $5 amplitude, the deviation of the rat’io of

52

KENNY

the rates from the 1AI / = 32, CP conservation prediction [Eq. (85)] is small. This, together with the approximate equality of Eq. (89) above, suggeststhat our ordering of the small quantities was reasonable. Violation of -2 Rc a(++-)

= lie u(OO+)

[seeEq. (S:i)]

implies an admixture of I = 2 in the final state. From Eq. (86), Reu(OO+)

+ 2Re,(++-)

M (15)“2(~3/X1) cos (41 - 6), = 0.06 f

(93)

0.04

from the experimental results [Eq. (84)]. Th’is is also consistent with our ordering of the small quantities; in fact, this is consistent with zero admixture of I = 2. However, an admixture of 1 = 2 in the final state of 1% or 2 % compared with the symmetric I = 1 cannot be excluded. Such an admixture is necessary if a charge asymmetry is to be observed in the final state for Kg0 decay. In t’his particular example, however, the charge asymmetry vanishes for Ks” decay as may be seen from Eq. (88), even if there is a nonzero admixture of I = 2 in the final state and CP is not conserved. This is because we have assumed all of the CP violation occurs in the / AI 1 = $5 transition. (c) Suppose t’hat & # 0, but +3’ = til = 0. Theu we have for Kf, K- decay

EC++-1 = $& Rea(+i--)

X1 - k X3exp (-$3)

= -2

1

exp (i6) = -2E(OO+),

(5y

. Re ~1 - ?w3 exp C-W - (:s,i)112y3 cxp ( -i43) cup (-is) XI - ?$v exp (-i&) [ Reu(--+)

=

. ~~

I>

-q

Pl - %~a exp C-f&) - (3,&)“2~, exp (-&) [ XI - ! $X3exp (- &)

exp (is)

(94)

11

Re u(OO+) = (5)“’

Re a(OO-)

= (s)112 . Re

PI

[

-

$2~3exp (--i&3) + (3$)1’p~3exp (-&) XI - f $B exp (- &)

exp (ia)

1 .

MODEL

OF

53

T VIOLAWON

Again, to the leading order, the differential spectra are in agreement with the prediction. Differences between the Dalitz plots 1AI 1 = >‘L, CP conservation for decay of K+ and K- int,o charge conjugate final states arise from interference between the 1AI ( = ,L,i amplitude and the CP-violating part) of the 1 AZ 1 = 8z amplitude. Ii’or example, 1te aCoo+)

-

Re (TOM+)

+ Re dOO-1

Kc u(OO- 1

(9.5)

E’or Kl”, Kzo decay we have E1( +-0)

Re al( + -0)

= - (l~),,2 iX3 sin 43

(is),

= (5)1’2 f cos 6, .I

z/S lie tli+-0)

= (5)l” F cot d

&

= -(5)“2

Re c2(+--0)

exp

sin 6,

43

” Xl

+

sin

X3

(‘OS

43

sin 6.

+a

From Eqs. (94) and (96)) we have, to the leading order, -2 Re U( ++-)

= Re a(OO+)

= Re c?( + -0)

= (.i)“’ j&

cos 6, (97)

which means that the 1AI I = ?2, CP conservation prediction [Eq. (83)] is satisfied to the lowest order. We also have J&+---O) 2R(oo+)

>:

J&(+---0)

~

R2(+-0)

1

i

3x3

co?

Xl

( h3

2 qi*12

0 x,

+3 .

1 ’

~3,

(9s) (99)

‘~

It is clear that for all values of (63, R2( + -o~/m~oo+)

E 1,

in agreement with the I AI 1 = 52, CP conservation prediction [Ey. (85)] provided only I X3/X1( << 1. In other words, the effect of a large T-violating phase could be suppressedby a small / Ai ( = :j2 amplitude. This would also give R1( + -0)/h%

+ -0)

<< 1,

54

KENNT

for all values of & . This is not inconsistent with the present experimental situation (34). Violation of -2 Re U( ++

-)

= Re a(OO+),

implies an admixture of I = 2 in the final state. From Eq. (94), Re a(OO+) + 2 Re U( ++

-)

NN (lrj)l”(

Y&)

cos (& + 6)

= 0.06 f 0.04

(100)

from the experimental results [Eq. (84)]. In this case, it is possible to have a charge asymmetry in the Kl -+ 3~ decay. Let us supposefor the purposes of illustration that (15)““( Q/X1) cos (43 + 6) = 0.05,

(101a)

and further that T is maximally violated; i.e., & = 35~. Then -( 15)“7 v&1) sin 6 = 0.05.

(101b)

The charge asymmetry is given by

N+ - N8 m”T”““2/3c(+-o), N+ f N- = 3?r m2 = 3.6%,

(65) (101’)

from Eqs. (96) and (1Olb). It is amusing to note that, in this case of maximal T violation, we did not need to know 6 in order to calculate the charge asymmetry. In general, however, the asymmetry would be lessthan 3.6 %, assuming that the I = 2 admixture is fixed at -1% in the sense of Eq. (101a). The exact asymmetry in this case could not be calculated without a knowledge of 6. It should be noted that, if we assume&’ # 0 in examples (b) and (c) above, the arguments are essentially unchanged since 43’ # 0 will only affect the value of r. The effect of r different from 1 will be discussedlater. F. K10 ---f 3n DECAY It is not possible to observe K: directly in the same way that K: can be observed directly as a pure beam. Instead one must look at the interference between K: and Kl at short times after the production of a K” or KO. This would result in a deviation from a pure exponential distribution of the 3a decay mode. At the present time, there is no positive evidence for such a deviation (S4), although the statistics are limited. Since it is difficult to observe time variations in the rate, we would like to suggest rather that the average rate of decay into the three-pion mode be measured

MOl)EI,

OF

F),n

2’ VIOLATION

over t,he first few K1’ lifetimes and compared with the pure K21)rate into the same mode. For more detailed results, one could compare the Dalitz plot averaged over tfhc first few KY,’ lifet’imes wit,h t’he Dalitz plot obtained from t,he pure K,” beam. If a K” is produced at rest at time t = 0, t’he state vector for the particle at, time t aft,er production is (35) K”(t)

= (1 + / I’ I’)-“‘[KF

exp ( -+iA,t

- inlIt) + K20 exp ( -J$A,t

-

(102 ) imd)]j

1

where X, , nl; are the decay rate and mass of Kio; KIO and Kzo are defined in Sec. V. Thus the amplitude for decay into t,he t’hree-pion mode is given bl AI(t)

= ( 1 + 1 9‘ I’)-“‘[Al,

exp ( -)9&t

- in~t) + A/, exp ( -J

where Mi = A!,( + -0) j A/(t)

$X2t

-

( 103) im,tj],

and is defined in Eq. (72). The decay rate is

1’ = (1 + I r I’)-‘[

I .Jfl /‘exp

+ I M, I’exp (-U)

(-id)

+ 2 Re MJl~*

exp (iAmt)

exp --$2(X1 + X?)t],

where Am = m2 - ml , Firstly let us compare the rates at short and long times from production. drop the charge assignment for convenience. From Eq. (A16), R(h)

= (In~E’)2exp .exp -!,;(A, sz (Imfi’)‘exp

+ (ReB’)2exp

(-Ah)

+

X2)tl

(-Altl)

.exp ( -$$Altl)

( 104’1

We

- 2ImE’ReE’

(-k,t,j

sin (Aw&) + (Rek’)’

(,105 1

- 2ImB’ReE”

sin ( Amtl), R(tz) CY (Re ,%“J2 exp ( -h2t2) c

(Re JC’)’

(106)

= h&(+-O), xvhere fl = O(Q), and I3’ exp (is) Then

= E = E( +-0);

i-1 << t2 <<

72

,

see Eq. (All).

R(h) R(b) -R(t2) ‘? !‘2R(+ -0) exp (-Ml)

(107)

+ 1 - 2 . exp (--$/2X&)

sin (A&l).

56

IiENNY

If we make a time average over the first few K? lifetimes using Am = =t$Q,, then

R(t) [ %Rr(+-0) if the averaging interval

R(t) [

?iRz(+-0) -

1

= 1+0.32(&;~TO.5O(I&J,

(108a)

is 0 5 t 5 ~~~ (see Appendix)

1

= 1 + 0.16 (g;)’

and

F 0.34 (s;),

(108b)

if the averaging interval is 0 5 t 5 2~7~ . It is clear that if CP is violated, this ratio can differ from 1. If we assume that 41 # 0, but 43 = $31 = 0 [as in example (b) of Sec. VI.E], then from Eq. 88, Im E”/Re

E’ no - tan 41.

(109)

If we take ] I$~ ( = 20” [see Eq. (92a)], then, from Eq. (108a), the ratio of the rates is 0.86 or 1.22, depending on the sign of +I~ , and assuming that the averaging interval is 0 5 t 5 rri . If we extend the interval to 0 5 t 5 27~~) then the ratio of the rates from Eq. (108b) is 0.89 or 1.13 depending on the sign of +i . Clearly, it is important to restrict the time interval of averaging to be as small as possible, otherwise T-violating effects will be suppressed. If we assume that & # 0, but & = 431 = 0 [as in example (c) of Sec. VI.E], then from Eq. (96), Im E’ ReE’

-X3 sin 43 Xl *

(110)

If 1 X3/X1 ) = fit, then, even if 43 = $4~, the deviation of the ratio of the rates from 1 is only about 5 % for a time average over the interval 0 5 t 6 a71 . The deviation from 1 will be even smaller for a time average over the interval 0 5 t 5 27r71. If one looks at the variation of density of the Dalitz plot with x0 energy, then a comparison of the time-averaged plot with the pure Kzo plot does not yield much information. In the appendix we give an expression [Eq. (A15)] from which one may calculate the time-dependent or time-average differential spectra. The existence of the CP-violating K1° + 3s decay affects the spectra to about the same order as it affects the rate. This effect decreases with time in the same way as the effect on the rate decreases with time. Finally we note an unusual possibility with regard to the charge asymmetry of the time-averaged Dalitz plot. Even if CP is conserved and there is no final-state interaction, it is possible to have an asymmetry between ?r+ and F. From the

MODEL

appendix R(t)

OF

57

T VIOLATION

we see that, under these assumptions, = E* I”[exp

(-Xzt)

+ 26/x cos And exp (-‘*

.Y E* 1’ [l + 2 c’x cos A& exp ( -$&t)] -1

t>] clr,

(111)

CZX,

where

and we have dropped the charge assignment (+ -0) for convenience. e’ is defined in Eq. (61). From this we see that the time-dependent charge asymmetry is

N, - Ns - E’ ros Arrzt exp (-?&t). N, -I- N- = - 3r Thus there is a time-dependent there is no CP violation and only on the existence of I = 2 result of interference between states, decreases rapidly with 0 5 t 5 ml, then (N+

- N-)/(N+

(112)

charge asymmetry which survives even when no final-state interaction. Its existence depends in the final state. However this effect, which is the K,’ -+ I = 2 and K,” -+ I = 1 symmetric, final increasing time. If we average the effect over

+ N-)

= -0.32~

= -O.l4c’,

if we average the effect over 0 5 t 6 2~7~ . Remembering

(113a, b)

that,

e’ = -&l?$!k,(+-o, ,* ,% _ d;? Tl ,%TInax ) In2

from Eqs. (Bl), (69), (71) and using (15)“*( Q/XI) = 0.05 from Eq. (101a) as an estimate of the I = 2 admixture (assuming all T-violating phases41 = C#S~ = C#Q’= 0 and that there is no final-state interaction), we have -v+ - N- = 14y N++N’ ” = 0.6%)

0 5 t 5 Trl

(114a)

0 5 t, 5 27r71.

(114b)

Thus, assuming an I = 2 admixture of about 1% compared with the I = 1 symmetric amplitude [see Eq. (lola)], it w-ould be possible to obtain a 13 %

charge asymmetry of the Dalitz plot if decay events were restricted to be less than ~7~ from the origin. If we assume more generally that CP is violated and that final-state interactions are present, then the charge asymmetry arises from both CP-conserving and CP-nonconserving effects. In this case it would be difficult to separate these additive effects without making an analysis of how the Dalitz plot changes with time. We can make only the two following unambiguous statements about a possible charge asymmetry. (i) If a charge asymmetry is observed within a few K,’ lifetimes of the production of the l?, then I = 2 is present in the final state.4 (ii) If a charge asymmetry is observed with a pure KZo beam, then I = 2 is present in the final state, and CP is not conserved. G. CONSEQUENCES

OF 1' #

1

So far we have made the approximation that r = 1. This cannot be true if CP is violated. In an analysis of K1’ --+ 2a and KZO---f B+T- experimental data, Glashow and Weinberg conclude (33) (115)

[ 1 - 11’I21 < 555. If 1‘ = 1 - E, this means -2 Re E + ( e j2 < $d5. From this equation, we conclude that Re c is small

I Re EI <

450,

(116a)

but, if Ewere pure imaginary, we can say only that I Im c I < 36

(116b)

which means that present data do not place a very severe restriction on the deviation of r from 1. [Note, however, that the smallnessof I v+- j suggeststhat ( c 1 is probably ~-10~~; seeEq. (44), Sec. V.] We are interested in the consequencesof 1’ # 1 on the foregoing arguments. We concluded that a comparison of R(+ -0) and R(OO+) did not lead to definite conclusions because 1AI 1 = ?,s and CP-violating effects cannot be separated in this context. It is clear that, if T is taken different from 1, interpreta4 Note that Z = 0 could, in principle, give rise to a charge asymmetry, but we have excluded Z = 0 in the final state because of our assumption of linear amplitudes. Z = 0 would give rise to a term of third order in the variables si .

MOI~EL

OF

7’ VIOLATION

59

tion of such a comparison becomes even more ambiguous. However, it was COD eluded that a definite test of CP violation in K -+ 37r would be provided by the charge asymmetry in Ks” + 7r+?r-7r”. If we suppose that K” --) 3a is CP conserving, but r’ # 1 because CP is violated in other decay channels, it is important to know whether a charge asymmetry can arise in Kzo --$ 3~ which is of the same order of magnitude as that expected when CP is violated directly in k” + 3~. If the two were of t#he same order of magnitude, then observation of a charge asymmetry would not, be an unambiguous t#est of direct CP violation in K -+ 3~. Following our earlier notation, and setting 4, = +3 = 0, #Q’ # 0, n-e have from Eqs. (,71), (71’), IlI(+--o)

= (&

. -L fi

(XI + X3) exp (ia) + & (117) . $

( Pl

+

113)

s~+;-~++

so that from Eq. (72), (11s) Thus 4

jRe4+-O)(

< fi (119)

from the inequality in Eq. (116b). Assuming a 1% admixture of I = 2 in the final state [see Eq. (lola)], we conclude that the above inequality for Re E~(+ -0) implies that the charge asymmetry due to 1’ # 1 would be less than 45 %. Such an asymmetry is an upper limit which is not likely to be attained for the following reason: We have chosen an extreme case in order to maximize / c 1; i.e., Eis taken to be pure imaginary. If 1Re e 1z 1Im c 1,then the inequality for I c / could be made much stronger; i.e.,

60

KENNY

In fact 1 E j is probably even smaller than this because of the smallness of 1 v+- ) (see comment above). However, we conclude that, until 1 - Y = E is known, we cannot exclude the possibility that’ 1’ # 1 could produce a detectable charge asymmetry in K: -+ ?r+a-$ although CP is conserved in K” + 3~. If the asymmetry turns out to be larger than >d %, then CP must be directly violated in this decay. It must be remembered that this discussion is based on the supposition that there is an I = 2 admixture of about 1% in the final state. We summarize our examination of the KTo + 3~ data thus (a). If 41 # 0, but C& = +3/ = 0, then comparison of the K+ and K: decay rates tells us that (92a)

1411 = (20f6)“, if there is

no

1 AI 1 = Fi admixt#ure and I 41 I s 40”,

(92b)

if we allow a 10 % admixture of / AI I = “5. There is no charge asymmetry in the decay Kfo + 3~ because the ( A1 I = B.2 amplitude is taken to be purely real. (b). If & # 0, but 4, = +3’ = 0, then comparison of the Kf and K: decay rates does not enable a limit to be placed on the phase +3 . Assuming that +3 = 3;~ and an I = 2 admixture of about 1 %, we calculated that’ the charge asymmetry in the decay KZO + 3~ was (N,

- N-),‘(N+

+ N-)

= 3.6%.

(101’)

In conjunction wit’h an experiment to determine the charge asymmetry, it would be important to make a measurement of the I = 2 admixture since the charge asymmetry is proportional to such an admixture [see Eqs. (65) and (83)]. Such an estimate could be obtained from accurate measurements of Re U( + + - ) and Re u(OO+ ) [see Eq. (loo)]. It may be noted that both examples predict differences bet’ween the odd pion spectra for both K+ and K- decay [see Eqs. (87) and (95)]. Because of our lack of knowledge of the various amplitudes involved in the difference, me cannot predict quantitatively what the difference might be. However, since t’he examples predict differences proportional to sin & and sin & , respectively, one could at present allow a greater difference between the spectra in the latter example because & is unrestricted whereas Eqs. (92a, b) suggest 141 1 < 40” in the respective examples. At present there is no evidence for the KF -+ 3a decay. It is suggested here that the average rate of decay of a K” beam into three pions be measured over the first few lifetimes and compared with the rate of decay of a KIo beam. Any differences between the two could be taken as evidence of Klo -+ 3~ decay and

MODEL

OF

(il

T VIOLATIOX

consequent~ly CP violation (since the rate K1’ --j CP-conserving channels I = 0, 2 is negligible to the orders of magnitude considered here). In the first example, it was noted that a 20% discrepancy in the rate could exist in the time interval 0 5 t 5 ~7~ assuming / $1 1 = 20” but t’hat in the second example we could expect only a .i’% discrepancy in t’he rate averaged over the same interval even if +3 = 12~. Thus such an experiment would be useful in detecting CP-violat#ion if the phase 41 were nonzero and reasonably large, say 20”. VII.

~31 >

‘i

TRANSITIONS

Very recently, measurements of the KS + 2~’ rat’e have been made (37)) (38 ) which indicate bhat a 1 AI 1 2 “5 amplitude conkibutes significantly to the CPnonconserving decay K,” + T + T. In particular, Cronin et al. (38) find I 700 !/I ?J-+- 1 = 2.5 f

0.3,

(120)

which is in agreement with t’he prediction of the example in which only +a’ is nonzero, viz. Eq. (.57’). Since bhis experimental result, has emphasized the importance of T violation inthe I AI / > $2 part of the weal<-interaction Hamiltonian, we shall extend our investigation t’o cover the possibility that there is T violation in the I AI 1 = 2x? part’ of t’he Hamiltonian. In particular, we shall suppose that, 4s and 45’ are t,he ordy nonzero phases and t,hat there are no amplitudes involved which have 1AI I > 52. We shall consider only h’s0+ 2a, 3~ and Z-decay. From Eqs. (3.5J and (M), wc have to lowest order q+- = >i[t - iv5 70~

= J,$e + 2i&

C&/b,) sin q&’exp i(& - 60)], CD&) sin #5)exp i(& - 60)].

(121)

We may give a similar argument here to that given for Eq. (57’) to just.ify the negl.ect of 6. Thus, as before, the contlribution t,o 4 from E will be of order 5 $$ and we have a relation similar t#oEq. (57’), -2v+-

c .$y)o :2 id2

(6,/b, ) sin &’ exp i( 6? - 60).

(122j

We may make t,he same comment,shere about, t’he relation betiveen I q+- I and 1~0~1 and the relation of the phase of 7 to the phase shift difference t,hat \vere made above in Sec. V. for the case&’ # 0. The experimental dat’a place no restriction on +5’. It could be as large as ?,& in lvhich case 1b, l/j bl I 2 3 x lo-",

(123)

which follows from the magnitude of 7, I 11) - 2 X 10Y3.If this is the case,then t’he / AI / = ~~ amplitude is significantly smaller than the / AI 1 = ?.bamplitude

62

KENNY

which is characterized

by [ b3 l/l bl 1 = 0.045 f 0.001.

(51)

If 45’ were different from ?+r, then the estimate (123) may be taken only as a lower limit for the j AI 1 = $5 amplitude. It is clear that the Kto + 2~ decay does not enable a clear distinction to be made between the possibilities of T violation either in the ] AI = 35 amplitude or in the ] AI 1 = 4/i amplitude. Thus we turn to the decay K + 3a. K -+ 37r We had supposed earlier that there was no I = 3 amplitude in the final state. However, since we are now allowing for the possibility of a T violation in the IAl/ = ,5$amplitude, we must consider the effect of an I = 3 amplitude in the final state. We assume,as mentioned above, that there is no 1AI ] = x amplitude. From Eqs. (63), (69)-(71), we have toleading order

R(++-) = [l + si I d++-)I21 4R(oo+) [l + x I J(c@+)121 . [ 1 + 5 (;J’

;I: cos {6(1, S) - 6(3,X)

+

4J51]1

(124) Rz(OO0)

%Rd+-0)

= [l + x I :Y+-o)12] .[l-5(?)li2;

cos 46 cos {6(1, S) - s(3, S)]

1 .

If we assumethat the 1AI ] = yi amplitude is 3 X lop3 [see Eq. (123)], then the I = 3 contribution to the right-hand side is of order 1%. On the other hand, Trilling’s data summary (24) indicates that the effect of the asymmetry is of the order of 3% in the comparison of the K: decay rates. Therefore, careful measurements of the rates and asymmetries are needed to determine the magnitude of the I AI ] = 35 contribution. The experimental data mentioned in Eqs. (77) and (80) do not exclude a I AI ] = 34 admixture as large as 5 %, which is an order of magnitude greater than that suggested by Eq. (123). A test for CP conservation in the I = 3 amplitude is obtained by comparing rates into charge conjugate states. These rates would be equal if CP were conserved. We have

R(++-)

112

;1’,sin

45

sin [6(1, S) - 6(3, S)],

R(---+I

R(OO+ > WC@--)

112

F1 sin 45 sin [6(1, S) - 6(3, AS)].

(12.5)

63

MODEL OF T VIOLATION

In t,his case, it is not important to consider the effects of the nonsymmetric states, since t#hedifference between, e.g., c’( + + - j and g’( - - +) does not contribute to lowest, order as we shall seeshortly. It is interesting to note that the T’* comparison is much more sensitive to the existence of a CP-nonconserving 1AI 1 = ,“5 amplitude than is the T* comparison. Finally we consider the structure of the Dalite plot. A comparison of the struct’ures of the T* and 7’* gives

Re [UC++-) Re [UC++-)

- d---+)1 + a(--+)]

1. 6 sin & tan 6( 1, S) = -1/gL 2 q5sin L28(1,S) - 6(3, 811 sin us +yzs

Re [a(OO+) - u(OO- )] -Iie [u(OO+) + u(OO-)]

2 =

c’os6(1, S)

, (126)

” sin 45 tan 6(1, S)

ZL

_ iL?

46 XI

sin B(l,

Sj -

6(3, S)] sin ~5

cos 6(1, S)

Since both v5 and ~1 are suppressedby comparable angular momentum barriers, it seemsreasonable to assumethat b5/p1and qs/X1 are of the same order. Thus the differences between the Dalitz plots of charge-conjugate decays will be quite small, being of the order of 1% if we assumethe 1AI 1 = “i amplitude is given by Eq. (123). This may be contrasted with the example of T violation in the I Al I = 3’; amplitude which is discussedin Sec. VIE, where the differences between the plots could be as large as 10 %. l”lJr the decay &” + B+P-~‘, t’his example predicts a charge asymmetry N+ - NN++ N-=

,Y WYT,,,,, -1,,?~~~sin~isins(l,s). -3?r 1

(127)

This asymmetry is expected to be of the order of 0.1% if we assumethe I AI / = ,9i amplitude is given by Eq. (123) and is thus an order of magnitude smaller than the asymmetry which would occur if the T violation occurred in the 1AI 1 = :,a amplitude alone. Z-DECAY

The only place where a ( AI 1 = $i transition can occur in hyperon decay is in t,he S-decay. It is difficult to extract information about the behavior of the various amplitudes under CP becausethe parameter /3is very small in two of the decays (X+ -+ nr+ and 8- --+ ,na- have s- and p-wave amplitudes suppressed, respectively), while in the remaining Z-decay ( Z+ + pro), the size of the I AI I = 5/z amplkude is expected to produce a phase between the s- and p-waves which is small compared with that arising from final-state interactions.

64

KENNY

It is possible to learn something from a careful comparison of t,he 2+-decays, however. Although, in general, there are 10, possibly complex, amplitudes involved in X-decay [see Eq. (29)], we may relate the Z+-decays by only four parameters; viz., the amplitudes for Z&decay into pion-nucleon, I = 3.5, j%i for both s-wave and p-wave. Apart from the final-state-interaction phase shifts which are known, the I = 36 amplitude will be complex for both s- and p-waves. Hence, in total, there are six parameters describing the XX+-decay. In principle, a knowledge of the rate and the various asymmetry parameters enable one to determine these six parameters. In particular, we may determine the magnitude and phase of the I = !d amplitude for the s- and p-waves. Using the notation of Eq. (29), for the p-wave this is 2 112 3 p13 (128) Q p33 + 15 P63 exp ($s),

2/2

4

0

where we have subtracted the phase due to final-state interactions which is known. The phase of this amplitude is given by 43 arctan

dri

P53 PI3

-

sin 46 a-\/ZP,,

*

It was pointed out in Sec. IV. that, from Z- --f n?r- decay, we know that PI3 is suppressed. Assuming that neither of the 1AI 1 = ,“i, yi amplitudes are suppressed,the phase of this I = $5 amplitude [Eq. (128)] could be of the order of 5” or more if C#J~ were as large as 35~. We can repeat the argument for s-wave decay, but since Sj3 is not suppressed, the phase of the I = K amplitude [Eq. (128)] will be correspondingly smaller, We may note here that, in the 1AI 1 > 35 CP-violating model of Truong (26), the phase of the I = ?d P-wave amplitude [Eq. (128)] could be quite large since, in this model, P33and P5, may be t,aken as pure imaginary. In general, we cannot learn anything from Z- -+ nF decay about CP violation. To understand this, we note that 10 amplitudes (possibly complex) are required to describe Z-decay, but only nine quantities can be measured [viz., I S I, I P I andphase (S/P) for each decay]. Thus, in general, not all amplitudes can be determined. In conclusion, available data place no restriction on the phases&, or 45’. In particular, it would seem to be rather difficult to learn much about the phase 45’ because its effect on K2’ -+ 27r decay is similar to the effect of +3’ # 0, and because of its small effect in Z-decay. However, more accurate information about the K10 branching ratio [seeEq. (47)] should be able to throw some light on the phase 43’. The existence of a nonzero &, does lead to some unique consequences in the K + 3~ decay which could not arise simply from T violation in the / AI 1 = 35 amplitude.

MODEL

OF

VIII.

65

T VIOLATION

CONCLUSION

We have applied our phenomenological model of T-violation to all hyperon nonleptonic decays and both kaon nonleptonic decays. Because of the lack of accurate experimental data, it is not possible to draw precise conclusions at the present. However, by setting all phases except one equal t#o zero, we were able t,o test our model for three simple cases. (a) ~$1# 0 but +3 = +a’ = 0. From the hyperon decays and KZO---f 3a decay, we were able to deduce A ---) pa-:+1

= (15 f

2O)O,

= (-17

f

= (8 f

62)“,

$- +ha-:f#ll z” -+hr0:g51 z+--,p7r”:

I&/

IS)“,

5225”,

Kg + 3~ : 1.C#~1 = (20 f 6)O I 41

I 5

if there is no 1 A1

j =

3i

if there is a 10 % admixture

‘w

and of

I A1 1 = 95. These numbers are consistent wit#h each other. They are consistent but’ +1 could be anywhere in the range -30”

with @I = 0,

< 41 < 30”,

because of the large experimental errors. (b) 93 # 0 but 41 = ds’ = 0. The hyperon and kaon decay dat)a place no limit on C#Q . (c) &’ # 0 but +l = $3 = 0. The hyperon decay data place no limit on 431, but, from K: + 2n we deduced that

If we allow all phases 4, , 43 , and C#Q’to be nonzero, our conclusions above are not altered. At present the experimental and theoretical situation is unsatisfactory. In order to see how the constraints upon the phases could be improved, we summarize here briefly the conclusions of the various sections with respect to experiments which might be expected to help clarify the situation: (1) A more accurate knowledge of /3/a for the decays -0 A -p* A --f mr”, z- ---f AT--, z *Aa0 1 would give information M and in the 1 AI j 43

, 43’.

=

on the relative importance of T violation in the I A1 / = ,34 channels, i.e., cm the relative importance of 4, and

66

KENNY

(2) An accurate

knowledge 0

z.z+ --f Pa 7

of @for the decays z+ -+na + , 2- --f na-

should enable us to distinguish between T violation in the P- and S-wave parts of the 1 AI 1 = 35 channel. Since we know &’ is small ( 52”) from Kzo + 2~ data, we expect that /3+ will be extremely small here and thus an accurate measurement of /?+ should provide a useful consistency check on the phase #3’. If on the other hand 43 is large, p- may be quite substantial; e.g., ->io. (3) A comparison of 1q+- ( and 1qoo1in the decay KZo-+ 27rwill enable us to seeif 43’ is small but nonzero (-2’) or if it is consistent with zero. (4) A measurement of the charge asymmetry in the Dalitz plot of Kzo + ?r+a-g’ is a direct check on the phase+s . Since such an asymmetry is proportional to the I = 2 admixture in the final state, it is important to have an estimate of such an admixture. This could be obtained from a comparison of the Dalitz plots in r- and r/-decay. Assuming that the 1AI 1 = 58 amplitudes are --4/;,~ of the 1AI 1 = 45 amplitudes, the available data place no restriction on the phases& and &‘. In particular, it would seem to be rather difficult to learn much about the phase ~$5’since it has an effect on the KZobranching ratio similar to that of the phase +3’. However, more accurate information about the K,’ branching ratio, which can give information about the phase+3’ [seeEq. (47)], should help to distinguish between these effects. The existence of such a nonzero-phase &,’ will have very little effect on the Z-decay and none on the other hyperon decays. The phase+5 # 0 will lead to a charge asymmetry in the decay K: + r+rT-ro. Such an asymmetry is expected to be considerably smaller than that produced by & # 0 because of our assumption that the 1Al 1 = 45 amplitude is ->{o of the 1AI j = 56 amplitude. However; the existence of & # 0 may lead to one substantial effect. (5) A comparison of the T*, 7’+ rates is a test for & # 0. In particular, the 7‘* rate comparison is a rather more sensitive test than is the r* rate comparison. In addition, the existence of &, # 0 will have a small effect on Bf decay as can be seen from the discussionof Sec. VII. Such experiments would enable a check for consistency to be made on this model. Because of the large numbers of amplitudes involved in all of these decays (especially in Z-decay and K + 3~ decay), it would take a substantial effort to determine the relevant amplitudes and phaseswith enough accuracy to check our model for constant phases.However, since this model is not to be taken seriously in the sense that, in general, one would not expect T-violating phases to be energy-independent, a completely detailed comparison of the model with experiment may not be useful. Nevertheless, an analysis of the various decays would be possible along the lines we have suggested and would be useful in this wider context.

MODEL

OF

T VIOLATION

67

Such an analysis may reveal, for example, that one of the phases tends to provide t,he main source of T violation, hut the value of this phase would vary from decay to decay because it is averaged in some fashion depending on the details of the underlying correct theory. This would occur, for example, if the correct theory were given hy the current-current picture with a phase inserted between the vector and axial vector strangeness-chainging currents (5). Such a phase could not he determined directly in nonleptonic decays, hut it w-uuld he manifested as a phase between the S- and P-wave amplitudes which was energydependent. The existence of such an underlying structure to the T violation could he tested and analyzed in the manner we have suggested. Since the work for this paper was completed, a relat’ed discussion of K -+ 3a decay has appeared (39 ) . ACKNOWLEDGMENTS

The aut,hor is grateful to l)r. Robert G. Sachs to him for continual guidance and stimulation. (:. Venturi for useful discussions.

for suggesting The author

this analysis and is indebt,ed would also like to thank Dr.

APPENDIX

j, setting 11’ 1’ =

From Eq. (104 / A/it)

/* E fg 1 nr, 12exp c--d)

1 we

+ 1 df2 12exp ( -ht)

+ 2Re ill,M,* We are interested

have

exp (iA&)

exp - >,h( X1 + XZjt].

in the average value over time T, i.e.,

/ A!r(t)l”= ; lT / Ai%)(”czt. If 2’ << TV , then we may rewrite

Eq. (A.l)

I AT(t) I2 h” ,I$j[ 1AJI I* exp ( --t/n)

For t,hc purpose of calculation,

+ 1M2 I* exp ( iAmt ) exp ( - t/2r,j.

7 1 T o exp (-t/n) J

1 exp ( - f/2~) T -0I’

COY (And)

(A.3)

we shall assume

j Anz 1 = X1/2 = l/27,

T

(A.2)

as

+ 2Re MJil,*

Then

(A.11

clt = ;[I

.

- exp C-I’/71)],

(A.4) CA.5j

cl1

=- 1 T exp ( - 1/2~~) (‘0s ( t/2T1) cZt 7’ s0 = ; [I - exp (-T/271)

{ cos (T/271) - sin (T/271)}],

CA.6)

68

KENNY

esp ( -t/271)

sin (Amt)

= l LT I‘oT exp = 13

[I -

dt ( -t/271)

sin (t/271) dt

exp (-T/~T~)(

(A.7)

cos (T/271)

+ sin (T/271) )I,

where the upper sign holds for Am > 0, the lower sign for Am < 0, Thus, if the averaging interval is 0 5 2 5 7~~ , then 1 M(t)

I* cz 45[0.32 ( MI 1’ + 1M2 1’ + 0.78 Re MJM,*

=F 0.50 Im M1M2*],

and m

(A.8) N” >$[O.lS ( Ml I2 + ( M, (* + 0.34 Re M,M,*

if the averaging interval is 0 5 t 5 2~7, . From Eq. (71), the amplitude for K” -+ $CK’

MC+-0) = (&*

~‘(1, S) exp (is)

=F 0.34 Im M1M2*], is

+ &o”(l,

L) s3z2 So

(A9)

+ ; a0(2, L) y, where we have set the I = 3 contribution equal to zero. In terms of the Dalitz variables z and y, we may rewrite this as in Eq. (60’), (A.lO)

M = JY[l + (T’y + El%]. We rewrite

this further

as M = E’ exp (is)

+ sy + ex,

(A.ll)

where E’ exp (is)

= .G,

s = EC’,

e = Ee’.

We have explicitly separated out the phase due to the final-state Comparing this with Eq. (AJ), we see that

e = --&

mKTmax o”(2, L) m2 .

interaction.

MODEL

The amplitude

69

T VIOLBTION

for KO + a+CrrO is given by Xf* = -E’

For li:,

OF

exp ( --is)

K: decay, we have the amplit’udes

illI E s5 (Ill + AT) = d

[i Im E’ exp (i6) + iy Im s + x Re s], (A.13)

= 45

[Re E’ exp (ia) + y Re s + ,ix Im s].

For a K” produced at rest we have the time-dependent 1M(t)

I2 M 55 [ 1 dl, I2 exp (--hi)

n’eglecting

terms quadratic

rate

exp (iAM)

exp --x(X1

- 2ImE’ReE’exp

+ (Re E’)’ -pi(X,

(A.11

+ X,)t].

in s and e, we have from Eqs. (A.13)

exp ( --AIt)

I2 = (ImE’)”

(A.14)

+ I Al2 I2 exp (--Ml

+ 2Re M,M,*

1N(t)

(A.12)

- sy + ex.

and (A.14)

exp ( -&f)

+ X2)2sin (Amt)

+ 2y[cos 6 Im E’ Im s exp ( -U) + cos6ReE’Resexp - Im (E’s*)

(-X,t)

sin 6 cos (AM)

exp -,l4(X,

- Im (R’S) cos 6 sin (Amt) + 2x[ -sin

+ Re (E’e)

(-X,t)

cos 6 cos (AM) sin 8 sin (AM)

exp -4$(X, exp -t-i;(X,

We could t,hen define a time-dependent R(t)

= (ImE’)’

exp (---It)

+ Xz)t + Xz)t].

rat’e [cf. Eq. (SS)]

+ (Re E’)’

- 21mE’ReE’exp Similarly we could define time-dependent (64a) and (64b). RECEIVED:

(A.15)

+ X,)t]

6 Im E’ Re e exp ( --~lt)

+ sin6ReE’Imeexp + Re (E’e*)

exp -:$(X1

+ X2)t

exp (-&t) -x(X, differential

+

(A.16)

Xz)t sin (AM).

spectra

analogous to Eqs.

December 15, 1966 REFERENCES

1.

J. H.

J. W. CKONIN, V. L. FITCH, -INI) R. TCRL.~Y, Phys. Rev. A. AB~SHIAN, R. J. ADAMS, D. W. CARPEXTER, G. I’. FISHER, .IND J. H. SMITH, ibid. 13,243 (1964).

CHRISTENSON,

138 (1964); NEFKENS,

Letters B.

&‘I.

13, K.

70

ICENNY

2. N. C6BmB0, Phys. Lefters 12, 137 (1964). S. S. L. GLASHO~, Phfys. Rev. Leffers 14, 35 (1964);

B. D’EsP.\GNAT AND M. K. GAILLARD, “Proceedings of t,he Second Coral Gables Conference.” Freeman. San Francisco and London, 1965; A. M~KALES, R. NUNEZ-LAGOS, AND M. SOLER, NILOVO Cimento 38, 1607 (1965). 4. N. CABIBBO, Phys. Rev. Leffers 10, 531 (1963). 5. K. NISHIJIMA AND J. L. STANK, Phys. Rev. 146, llG1 (1966). 6. M. K. GAILL.~RD, Phys. Letters 20, 533 (1966). 7. B. W. LEE AND A. R. SWIFT, Phys. Rev. 136, B228 (1964). 8. G. FELDMAN, P. T. MATHEWS, AND A. SALAM, Phys. Rev. 121, 302 (1961). 9. N. P. SAMIOS, Argonne National Laboratory Report No. ANL-7130,1965 (unpublished). 10. A. II. ROSENFELD, A. B.\RB~~Ho-G.\LTIERI, W. H. BARKAS,~'IERRE L. BASTIEN, J. KIRZ, AND M. Roos, Rev. Mod. Phys. 37, 633 (1965). 11. B. CORK, L. KEHTH. W. A. WENZEL, J. W. CRONIN,.\ND R. L. COOL, Phys. Rev. 120, 1000 (1960). 12. J. W. CRONIN AND 0. E. OVERSETH, Phys. Rev. 129, 1795 (1963). IS. M. GELL-M.ANN .\ND A. H. ROSENFELD, dwn. Rev. Nucl. Sci. 7, 454 (1957). 14. R. 0. BANGERTER, A. BARBARO-GALTIERI, J. I'. BERGE,J. J. MURRAY, F. T. SOLMITZ, M. L. STEVENSON, AND R. I). TRIPP, Phys. Rev. Leflers 17,495 (1966). See also Ref. (9). 15. 1). CLINE ANII J. ROBINWN, in t’he Proceedings of the Thirteenth International Conference on High Energy Physics, Berkeley, 1966 (unpublished); D. BERLEY, S. HERTZBACH, 11. KOFLER, S. YAMAMOTO, W. HEINTZELMAN, M. SCHIFF, J. THOMPSON, AND W. WILLIS, Phys. Rev. Leffers, 17, 1071 (1966). 16. B. W. LEE, Phys. Rev. Letfers 12, 83 (1964). 17. 11. H. 1).4LITZ, Proc. Phys. Sot. (London), A69, 527 (1956). 18. T. T. Wrr AND C. N. YANG, Phys. Rev. Letfers 13, 501 (1964). 19. R. G. SACHS, Ann. Phys. (N.Y.) 22, 239 (1963). 20. R. H. DALITZ, paper presented at the Brookhaven Conference on Weak Interactions, 1963 (unpublished). 21. L.W. JONES, D.O. CBLD~ELL, B. ZACHAROV,~~. HARTING,E.BLEULER, W. C. MIDDELKOOP, AND B. ELSNER, Phys. Letters 21, 590 (1966). 22. L. 1). JACOBS .\ND W. SELOVE, Phys. Rev. Letters 16, 669 (1966). 23. See, e.g., G. F. CHEW, Phys. Rev. Leffers 16, 60 (1966); L. F. COOK, ibid. 17, 212 (1966). 24. G. H. TRILLING, Argonne National Laboratory Report No. ANL-7130, 1965 (unpublished)-or University of California Radiation Laboratory Report No. UCRL-16473, 1965 (unpublished). 25. TRIN N. TRUONG, Phys. Rev. Letters 13,358a (1964). 26. TRAN N. TRUONG, Phys. Rev. Letters 17, 153 (1966). 27. J. L. BROWN, J. A. KADYK,G. H. TRILLING, B. P. RoE,L). SINCL.~IR.~ND J. C. VAN DER VELDE, Phys. Rev. 130, 769 (1963). The value obtained here for the branching ratio r(K1O --f 2nO)/r(K10 + 2~) is included in the compilation by Trilling [Ref. (@)I. 28. T. I>. LEE;\ND C. S,Wu,.4nn. Rev.NucZ. Sci. 16,471 (1966). 29. S. WEINBERG, Phys. Rev. Letters 4, 87 (1960). SO. G. BARTON, C. KACSER, AND S. P. ROSEN, Phys. Rev. 130,783 (1963). 51. N. BYERS, H. MCDOWELL, AND C. N. YANG, in Proceedings of the Seminar on High International Atomic Energy Agency, Energy Physics and Elementary Particles.” Vienna, Austria, 1965. 39. IT. W. K. HOPKINS, T. C. BACON, AND F. 1~. EISLER, Argonne National Laboratory Report No. ANL-7130, 1965 (unpublished).

MOI)EL

33. 34.

OF

7’ VIOLATION

‘71

M. IX. G.\ILL.\KD,N~LOUO C'in~enlo 35, 1225 11965). L. BEHR, V. BRISSON, P. PETIIN, C. P.\sc.\cn, B. AUBERI', E. BELLOTTI, A. PCLILI, hl. BALDO-CEOLIN, Ii:. C.\Lrhf.wI, S. CI.\MPOLILLO, H. HTTZIT.\, AND A. HCONZA, Argonne PIJational Laboratory Report No. AXL-7130, 1965 (lmpublished) ; J. A. AKDEltSON, F. R. CR.I\YFORD I',. L. (GOLDEN. 11. SERN, T. 0. BINFOKU, .\NI) 1'. G. LINU,

Phys. ReD. Letfers 14, 475 (1965); ibid., 15 615(E) (1965); ibid. 16, 968(E) (1966). On the basis of 18events, the latter conclnde t.hat- E,(+-O)/E,(+-0) = .r + ii/, where z = O.l~~:~, ~1 = 0.G & 0.9 for ul? > PII . It, is to be noted that. we cannot, ,lse their second solution for .C and !, since t,his assumes 1 Al / = fi ride. 35. 8. B. TREIM:IN .IND 11. (:. S.\CHS, Ph!/s. Rel). 103, 15-15 (1956). 36. l;j. L. GLASHOLV .\ND S. WEINBERG, Whys. Rec. Letlers 14, 835 (1965). $7. J.-M GAILL.IILD, F. GIEKE~~;, W. G\LBIS.\I.TH, A. Iluss~, &I. 1:. J.CGE, N. II. LIIW.\X, G. M.INNING, T. K\TCLIFFI, I'. D.\Y, A. G. P.\ILH.\M, B. T. P_\TNE, ,4. C. SHEIDVOOI), H. FAISSNER, AND H. REITHLER, Phys. ffw. Lellera 18, 20 (1967). 38. J. W. CRONIN, I'. F. KTNZ, W. S. RISK, .\ND 1'. C. WHEELER, f’hys. lieu. LetleTs 18, 25 (1967). 39. B. BARRETT .IND TRAN N. TRITONC:, Phyx. Kev. Lellers 17, 880 (1966).