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Possible CPT violation and the K° decay
ANNALS
OF PHYSICS
64,
510-531 (1971)
Possible
CPT Violation
and the K” Decay*+
YOAV ACHIMAN Laboratoire
de Physique Thkorique, Bkt. 211, Facult.4 des Sciences, 91 Orsay, France Received June 19, 1970
The generalized expressions of the measurable CP violating parameters for the K system are given, without assuming CPTinvariance or the other usual assumptions such as: dQ = -AS, the approximate AZ = $ rule. The unitarity sum rule for the K” decays is also studied without using these assumptions. Some applications of these generalized expressions are discussed. We suggest a test for CPT invariance by testing an inequality between the charge asymmetries of KL ‘-f rev(&) and KL -+ v&R,). Using the upper bound on the e - p asymmetry for the K* decays, we show that any signitlcant difference between R, and R, indicates some CPT violation. Alternatively, even a small e - p asymmetry may lead to considerably different R,,,, , if CPTis violated. As another application we construct a CPT violating phenomenological model, in which there is no mass or lifetime difference between particles and their antiparticles even in the presence of CPT violation. We give also the “second-order” corrections to the phases of c and 8 in the case of CPT and T invariance, respectively. It is found that according to the available experimental results 4, should be considerably lower than 43”.
1. INTRODUCTION If CPT violation exists at all, it would probably be first observed in the K decays. The interactions of the K” complex are the most sensitive apparatus known to detect possible violations of CP, T or CPT. (Actually, CP violation has been observed, up to now, in this system only).l This is related to the fact that the weakly decaying particles are not K” and R”, but their mixtures which depend on the invariance properties of the system. The existing experimental situation in the K” decays is now consistent with CPT invariance [2], and the doubts raised recently [4] about it, have disappeared. However, one cannot yet exclude a violation of CPT, and it is worthwhile to study in detail this possibility in the K” system, due to the essential role that CPT conser* Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research. + Thisstudy was performed in partial fultiment of Ph.D. requirements at Tel-Aviv University. 1 For a recent experimental review see J. Steinberger [l, 21. For general remarks about the possibility of CPT violation see, for example, L. B. Okun [3].
510
CPT
VIOLATION
AND
K”
DECAY
511
vation plays in elementary particles physics. In this paper we aim to contribute to the study of the possibility of CPT nonconservation in the K" decays. To test CPT invariance one usually assumes it in order to obtain relations, which may be tested experimentally. We shall study here the formulation without assuming conservation of CPT. One of the advantages of such an approach is that it enables us to observe where it is most probably to detect CPT violation, if it does exist, In the following, we shall give the general expressions for the observed CP violating parameters, allowing a small nonconservation of CPT and without using the usual assumptions, such as OQ = AS or the flZ = 4 rule. The explicit general expressions, for the important contributions to the unitarity sum rule (u.s.r.) for the K" decays, will be also given. Using these, we shall calculate the “second-order” corrections to the phases of c and 6, in the cases of CPT and T invariance correspondingly. It will be found that when CPT is conserved the available experimental results require +C to be quite different from the frequently used value +t- = 43”, while if T is invariant the first-order term is a good approximation to bs = -47” & 180”. It will be also shown explicitly that when CPT or Tare conserved, the contributions to the u.s.r., other than due to K” --, 2rr, are practically pure imaginary or real, respectively. This may be useful in checking CPT or T invariance. However, a better way to test CPT invariance is pointed out in Section 6, as an application of the general formulation. This consists of looking for a difference between the charge-asymmetries in KL + rev(R,) and Kt + rrpv(R,). We show that even a very small e - ZLasymmetry may be enhanced considerably in the case of violation of CPT. in such a way as to lead to a substantial difference between R, and R, . (The existing experimental situation is not in contradiction with such a difference). On the other hand, CPT conservation in addition to the small upper-bound on the e - p asymmetry observed in the K-1: decays, leads only to a negligible difference between R, and R, In the last section we study, as another application, a phenomenological assumption that the scalar product between the wavefunctions of the short and long lived neutral kaons (Ks,L), is very small, i.e., i(K, 1KLjI
5 IOW.
This assumption requires CPT violation in order to explain the observed charge asymmetry in KL -+ nlv. Yet, in a special case of this model there is no mass or lifetime difference between particles and their antiparticles, regardless of the amount of CPT violation in the model. Using this example it is clear, therefore, that the upper-bound on the difference between quantities that should be equal due to CPT invariance, cannot serve as an absolute measure for CPT violation, as has been done in several papers.2 ’ See,for example,the review articleof Lee and Wu [S].
512
ACHIMAN
2. NOTATION
We use here the notation of Lee and Wu [5]. The K” complex is described using the equation
where A is in a 2 x 2 complex matrix Aij
= rij + iMij
(2)
and M and r are the mass matrix and decay matrix, respectively. In the Wigner-Weisskopf nonrelativistic perturbation approximation, obtains for A, the expression [513 (1,,,
= M;&,
+ (S 1 Hw 1S’) + c @ I Hw’ I n)(n I Hw I “) Mi. - M,” + ic n
+
one
...
(3)
=(SIM;.+H,
where H, and Hst are the Hamiltonians of the weak and strong interactions. The intermediate states 1n) are understood to be eigenstates of the strong interaction. The special choice of the many arbitrary phases in the K” system plays an important role in the discussion. We shall choose the phases of the states so that the representation is real, T I K”) = 1K”),
TIKO) = IR”)
(44
which satisfies TS = 1. (This does not, however, fix the arbitrary phase in the definition of T for the kaon field). In addition, we shall define CP so that CP 1K”) = 1R”),
CP I x0) = I K”),
W-4
satisfying (CP)2 = 1, and [CP, T] = 0. For the kaon field this commutativity restricts the phases of T and CP in such a way that their product is il. The special advantage of choosing a real representation is that an operator which commutes with T has real matrix elements, while an operator which anticommutes with T has pure imaginary matrix elements. In other words, an imaginary part of a K” decaying amplitude describes T violation (and CP violation if CPT is conserved).4 3 A careful and clear derivation of this expression may be found in the Appendix of Van-Hove’s, “Lectures on CP violation,” CERN, 1967. 4 Note, that we do not assume here the Wu and Yang [7] phase convention, i.e., that <2q1p0 1I& [ K”) is real, which is used in Lee and Wu [S].
CPT
VIOLATION
AND
K”
513
DECAY
One may show [5] that the observed eigenstates of A are, in this formulation 1 KS) N +z ((1 + E + 6) 1 K”) + (1 - E - 8) 1R”>), j KL) N +2 ((1 + E - 8) j K"; - (1 - E + 8) 1K”)j, where ~l_(K”/A(R”)-(RolA;Ko; rs
-
FL 4 2i(m,
-
m3
’ (6)
s~(K”~n~K”)-((R”:AIR”) rs - FL 4 2i(m,
-
md
.
ms,L andrs,, are the masses and decay rates of the observed
decaying short and one assumes that E and 6 are small parameters
long lived kaons. In the derivation
lSl<1,
lE/
and the approximate equality N denotes that second order in small quantities is neglected. Using (3) and (6), it follows that 6 = 0, E = 0, e=s=o,
and
if CPT is invariant, if T is invariant if CP is invariant.
C6’)
Also using (5), in our formulation
3.
THE
2~
DECAY
(7)
MODE
Let us denote A, = <277, I H,
I K”),
A, = (2Tr,{ Hwl R"),
where I 2x,) are the isospin I states of 27~. If CPT is conserved let A, = A,*,
@a)
and T invariance gives real
AI and A!.
Ob)
514
ACHIMAN
Using this notation one obtains, assuming small CP and CPT violation, following expressions for the measurable quantities
the
where SI are the s-wave 2~ phase-shifts, with isospin I and c.m. energy equal to the K” mass. Let us give these expressions in some special cases. If AZ = i is approximately valid, I and we neglect 0&4,/A,)
p 0
I
E). Then for AZ N 4,
7j+- = E - 6 + ; (1 _ St)
+ &
7700= E _ 8 + i (1 _ +)
- 5
(2
- At)
pa-80), (94
0
If CPT is conserved, due to (Ba) and (6’),
(9
- *) 0
ei(sO-aO)e 0
CPT
VIOLATION
AND
K”
DECAY
515
where i&-8,) ,
If in addition we neglect O((A,/A,) E),
A0 q+- ~16 + i Im Re A, ’ “’ (94 . ImAo ‘loo = ’ + ’ Re A,
2I ’ ’
These are the frequently used expressions (only that the phase convention of Wu and Yang is not assumed).5 In the case of T conservation we have to put E = 0 in (9) and A, will be real. The value of the Z = 4 violating term (AJA, + &/A,) ei@e+) is related to the parameter R = Rate(K, -+ 7~Ok’) (10) s Rate(K, -+ X+V-) * Explicitly, Rs=~
1 I-24’ I 1+At
I3
(10’)
where
A’ = -& ($ + 9, ei(wo). This gives the equation [I l] A’ = 4 -12R
(2 i- 2Rs - 3 aeeiE], S
i.e., for a fixed value of Rs , A’ is located on a circle. Experimentally Rs m $, (see Table I) hence, one may have also solutions for A’ that are not small. However, 5 The expressionsfor the q’s, when assuming the approximate I = 4 rule and using the Wu and Yang phase, i.e., the analog of (9a), is given in Lee and Wu [S]. The expressions when assuming CPT invariance only, in addition to the Wu and Yang phase, i.e., the analog of (9b), may be found in the book of P. K. Kabir [8]. The expressions in another representation may be found in the review of Bell and Steinberger [9]. See also Refs. [IO, 111.
516
ACHIMAN TABLE I Experimental Data
Quantity 103 1?+- I 103 j too I
4+400
Exp. value
Ref.
1.90 f 0.09
r291 [301 [3lI 1321 [331 [341 r351
3.6 i- 0.4 3.6 + 0.6 3.2 f 0.7 2.3 f 0.3 2.2 * 0.4 (-2 * 7)11*; t3 39.8” j, 6”* 17” + 31” 4.5 - lo-* 2 1 Re AZ/A,, I 2 8 . IO-2
PI WI WI
Quantity
Exp. value
_
108 Re lo3 R, Rex Imx R;’ 83 - 80
4.05 2.24 3.15 0.14 -0.13
f f f f &
2.16 2.33 -(SO0 -(30”
& 0.108 f 0.05 LIZ20”) & 10”)
AmlrS
1.4 0.36 0.3 0.05” 0.049’
0.469 * 0.0128
Ref. [361 [371 [3gl [391 I391 Il41 [401 [411 t421
PI
8 World average. b World average taking e, /L together.
using the fact that the AZ = 4 rule is approximately good for Kk, it is very improbable that there will be a large violation of AZ = 4 in KS + 27r. Note, that all these considerations are actually only qualitative because we have neglected the electromagnetic corrections, which may well be in the same order of magnitude as A’. In general, one can only say that [lo] A’ = A,, -+ O(10-3).
4. CHARGE ASYMMETRY
IN KL -+ dv
Let us denote: AS = -AQ
AS = AQ amplitudes
amplitudes
g, = A(K” --+ d-v), & = A@? 3 7rI%).
(12)
Et = a*,
(13)
fi ,fE , g, , and gl are all real.
(14)
fr = A(K” -+ d+v), jl = A(F 3 7TfI-q, If CPT is conserved .i=t=.fi*
and
and if T is invariant
517
CPT VIOLATION AND Kc DECAY
The quantity
vanishes if dS = LIQ.~ Assuming small CPT violation
and we neglect second order in these quantities. Assuming small CP and CPT violation, we have for the charge-asymmetry R = Rate& + r-1’~) - Rate(K, + ~~~f-5) 2 Rate(K, + n-l+v) + Rate(K, + n+l-C) ? 1 N Re-( 1) + / 1 - Xl 12 120 - I x1 I”> Re(E - 6) :;: .A 3) - Re [xl ($$ - --j$-j]/. -t /xz12Re ( F1
(17)
If dQ N dS holds, i.e., / x1 I < 1, one has & N 2 Re(e - S) + (Re +
-
(17’)
I).
5. EXPLICIT EXPRESSIONS FOR THE 2n AND nlv CONTRIBUTIONS TO THE U.S.R. AND SOME CONSEQUENCES
The unitarity sum rule (u.s.r.) is a way to express the conservation of probabilities in the K” decays.’ Explicitly t[(rs
+ r,) + 2ih
- m&KS
I KL,) = c (F j H,
I K,)*
(F I HFV I KL),
(18)
F
where l-‘,,, are the decay rates of Ks,L and ms,L are their masses. The summation CF is extended to all observed final states for the K” decays. and includes spin and isospin summation and phase space integration,
T - ,&, 1 dPF(274” W, - P). F 6 In general jE andg, shouldberegardedassuitableaverageson the variables[12]. 7 Theu.s.r.wasfirst derivedby Bell and Steinberger [9J. See also Ref. [ll]. In the literature one canfind differentformalderivationsof this sum rule. For the derivation and discussion of a generalized u.s.r., see L. P. Homitz
and J. P. Marchand
[13].
518
ACHIMAN
Using the expression (7) for
(I+i
F)
N 2 * 10-3, it is possible to give
(Re B - iIm6)
=CB,7jF, F
where Am=mL-mm,, B F = RateK
= E J--s ’
3 rs
(20)
and (21)
The u.s.r. is one of the most useful relations in the K" system. This is due to the fact that it is a relation between quantities that are measurable. The expressions on the rhs are on-mass-shell, in contradiction with the other expressions for E and 6 (6), which include off-mass-shell contributions. One more convenient fact about this sum is that only few terms contribute in practice. This is easily seen in the representation (18’), because BF are in general very small factors. A direct way to study this, is by using the Schwartz identity [9]
In experiment r? r,“’
z*Cg
hi 1
M 10-3,
rp ,( 5.5 * 10-4, --ji-
r,rp 5 10-4,
s
and rF#2?r,3a,nlv S,L
g
10-3.
But @+-)
= w?ocJ = lo-3,
hence I g I 22 1c3.
CPT
VIOLATION
AND
K”
519
DECAY
It is clear, therefore, that the contributions from F # 2x, 377, n/v may be neglected. The contribution from KssL ---f 377 may also be very probably neglected. This is because we saw that
Ic i < v’I’?l-p
< 2.34 . lO-4
3n
and one needs a very large CP violation in KS,L --f 3~ in order to have a considerable contribution to the sum. This is improbable, although the experimental situation with respect to this decay is yet very unclear. We shall, therefore discuss here in detail only xzrr and zPlv . Let us only note that Gourdin [ 151 showed that CPT invariance leads to (22a) and T invariance to
(22b) We shall calculate explicitly the expressions for the 2~ and r/v contributions to the u.s.r., in terms of the amplitudes, without using the assumptions of CPT invariance, or dQ = rlS, or rlZ N +. The calculation is especially important in the case of xVlv , where q,lV are not yet measurable. It will be shown that it is enough to know xL , in addition to the partial rates, in order to obtain a good estimate of this contribution. The available experimental data show an important contribution to the u.s.r. due to Ks,L --, XIV. In the case of KS,L -+ 2n, the explicit calculation enables us to obtain the contribution of flZ = fr breaking terms to the phases of E and 6. We do not neglect O(( A,/&, j E), and this gives us new consistency relations, in the special cases of CPT or T invariance. It is found that these relations are consistent with the existing experimental results. Finally, it will be shown that the contributions to the u.s.r., other than due to K S,L - 2rr, are practically imaginary or real in the special cases of CPT or T invariance correspondingly. This is useful especially when one uses the u.s.r. in order to check CPT or T invariance. Assuming small violation of CPT, one obtains by explicit calculation
-k 46 h
E - Re 8) Re x1 - 2i Im x1 + 2 j x1 12Re i s
- 2f Im [xz (-$595/64’2-14
- +)]I.
- $C) 1 (23)
520
ACHIMAN
This expression contains terms of O(E), however comparing it with (1 P-s) G,L - 10-3, it is clear that one can neglect, in the u.s.r., all terms of O(E), and we are finally left with* (239
We see that the KS,, -+ rrlv contribution is practically imaginary, and also, that it is proportional to Im xl and therefore vanishes practically if one assumes T invariance. This is an interesting property especially when noting that x3,, has also a similar behavior [15]. This is because due to the small value of (l/F,) rsu M lo-*, the terms (22a, b) are negligible. Hence, in these special cases, of CPT and T invariance, C3= is imaginary or real, respectively. Collecting contributions, we have for the u.s.r.
(l+i
$$)(ReE-iim6) S
N rsclrT
Rs) cv+- 4 13~~~) - i EE rs
2 Irn x lflX13
+ &
$.
(24)
In the special cases of CPT or T invariance:
(1 +iF)ReE2Am S
N r,(lrI
-
(
R,) (rl+- + G,bo) + &
i--2j$)ImSN S
rsclrI
[-rlePt*
As) b+-
f
* Im ’ + Im C] 1+ IXP 3m for CPT invariance; Rs~~oo)
(25)
+ $--S Re c,
397
for T invariance.
(26)
We see, therefore, that the contributions to &, other than the main term Czn , are either purely imaginary or real in the special cases of CPT or T invariance correspondingly.9 It can be shown, using this description, that Eq. (26), i.e., the assumption of T invariance, is inconsistant with experiment [19], therefore T is violated in the K” decays. 8 P. K. Kabir [16] has derived this expression, in the case of CPT invariance. B This may be shown using very general consideration and assumptions [17, 181. Here, however this fact is shown explicitly.
CPT
VIOLATION
AND
K”
521
DECAY
The u.s.r. may be also used in order to compute the phases of E, 6. $C is known in the first approximation, we shall compute explicitly the second-order corrections. To do this, let us calculate the contribution due to KS,, -+ 2rr explicitly. We shall use two parameters R, and A’, which are related. However, the exact relation is not known, because of the unknown electromagnetic corrections (see discussion in Section 3), we shall use, therefore, for R, its directly measured value. The available estimate for d (in the case CPT is conserved) is [20] 8 . 10-Z 2 j fl j 2 4.5 . 10--z. Due to the fact that 1 E / = 0(10-3), O(EA) may be considerably higher than O(G), which is neglected in our calculations. Let us, therefore, neglect only terms of O(eA2).
In this approximation ?I+- = 6 - 6 + ; (1 - 2)
(1 - A’) + ~“(1 - A’), (27)
-
“loo CT+E - 13+ ; (1 - $)
0
(1 + 24’) -- 2d’(1 + 249,
where
Hence, (r]+-
im
R.y~o)
N
(1
+
R,)
[C -
+ (2Rs -
1) ;
8 +
i
(1 - -$)I
(1 - +‘)
+ (I - 2R,) E
- (1 + 4R,) A’,“.
0
This can be simplified,
using the fact that R, = 4 + O(A),
into the expression &
(T+- + R,q,,)
N E - 8 + ‘2 (1 - 2)
+ 2 [; (1 - 2RJ - A’] F”. (28)
The last term is neglected in the usual approximation, where one neglects O(EA). However, this term can in principle contribute more than x+ and obviously Cln , hence, we think that in the approximation where one takes into account the last contribution, it may not be justified to neglect O(A).
522
ACHIMAN
Using (28) we have a new expression for the u.s.r.,
(1+ iF)(Ree--ima) N E - 6 + ; (1 - 2)
. Fept.
2 Im
+ 2 [; (1 - 2Rs) - A(] E”
x
++
-zrsl+lX12
(29)
n
In the special cases of CPT and T invariance: 1+i--- 2Am J-s
Re E
_N E + i $$$.
+ 1/2 i [ ’ -32Rs
. FePt. 2 Im x
+&ImC,
-zTl+lxlz
- A] Im (2)
ei(‘*-*O)
for CPT invariance;
3n
CW
2Am 1 +iF)iIm6 -
-8 I;
(1 - $)
+kRex.
+ 5
[ l -32RS
_ 4(
;2o
$2) o ,&-80)
for T invariance.
(29’3
3n
It turns out, that in order that our approximation be valid, in these special cases, corresponding consistency conditions must hold. In the case of CPT invariance, taking the real parts on each side, one obtains Re E N Re E -
~‘2 Im (2)
Im I( ’ -32R”
- A) e"(8~-8~)/.
Hence, the following condition is needed 1 -
2Rs
Im
ee’b,-60)
3 or 1 - 2R,
3
v9
w cos(8, - 8,).
(304
CPT
VIOLATION
AND
K”
523
DECAY
This may be checked experimentally. Using, for example Re(A,/A,) = 8 . lo-” and R$ = 2.33, one finds 6, - 6, N_ 140”, which is consistent with the experimental available values (see Table I). In the case of T invariance, one obtains in the same way (for the imaginary parts) w cos(S, - So).
VW
Taking in each case, the other (imaginary or real) part, one can obtain information on the phases of E and 6: For CPT invariance, 2Am
2 Im x ltlX12
hReAA0 -- P-t. rs 0
rReE’VImE+ s + 9
(1 - 2Rs) Im (9)
- Re (2)
Im ($)
cos@, - 6,)
co: 2(6, - 6,).
Only (Im A,)/(Re A,) may be of O(1O-3), all other terms are lower by at least one order of magnitude. It is, therefore, convenient to define
In terms of <, one may write in the first approximation,
i.e., neglecting O(EA),
q+- = EI + E’, (33)
700 !x E^- 2E’.
The phase of E is due to (31),
- d/z ’ -32R” + Re (9)
Im ($)
Im (2) 0
cos(S, - 6,)
co: 2(6, - &,)I.
(34)
Usually, one takes the first-order term 2Am/r, only. Let us show that, according to the available experimental data, the second-order terms may contribute considerably, especially CIllV .
524
ACHIMAN
Using the experimental
values of Table I,
1 Pept. K-i-- r, which is obviously an important
2 Im x 1+lx12
N -(0.35
contribution
f 0.15),
to
The other terms can have a contribution of O(1O-2), and may be even a higher one, but the experimental situation, with respect to these terms, is not so clear. Hence, we may conclude, noting that Re Z = Re E > 0, thatlo
A similar analysis can be made for
Ls-++--$), 0
in the case of T invariance. In this case one has in the first approximation q+- -N 6 + El, (36)
qo(j N s - 2ER. Taking the real part of the u.s.r., we have $Irn$!=--Re8+2Re(
1-32R,
-,‘),“+kRe$.
The consistency condition says that the imaginary part of the Al = $ breaking terms vanish. 6, - 6, NN -45”, hence the real part is negligible as well as very probably zS,, . Also, Re 8 can be shown to be of 0(10-8), using (36) and the experimental data on the 7’s. In this case, therefore, the first approximation
lo Note however, that there are yet some doubts about the value of Im x, which is in contradiction with most of the CP violating theories. The theory of Sachs [21], that assumes CP to be violated in the AQ = -AS interactions, is one of the few theories in which Im x # 0. In this theory ds N 4++-N doO, and the above result is consistent with the available experimental values. (See Table I.)
CPT
VIOLATION
AND K”
DECAY
525
is a good one, in contrast to & when CPT is conserved. Equations (36) were used by C. S. Casella [22] to check T invariance, in an analogous way to the use of the Wu-Yang triangle [7].
6. R1 AND THE EXPERIMENTAL
TEST OF CPT INVARIANCE
As an example of the use of our CPT violating expressions, let us show on the one hand, using the observed upper-bound on the e - p nonuniversality, that a difference between R, and R, is practically an evidence for CPT violation; and on the other hand, that even a very small e - p nonuniversality may be enhanced, in the case of CPT violation, in such a way as to lead to a substantial difference between R, and R, .I1 The usual tests of CPT in the K” decays consist of checking the Wu-Yang triangle [7] or, what is better and almost equivalent, a direct use of the U.S.T. for the K” decays. In these methods one must know at least four of the parameters I rl+- i, I %I 1. 4+- y &JCIy and Rl , in addition to some information on the other modes of the K” decays. In contrast, the test proposed here may be realized in principle in one independent experiment, is more sensitive and it is more probable to observe CPT violation here, if it does exist.12 When CPT is invariant, &
=
2(1
-
I 1 -
1 XI x2
1”)
Re
I2
l ’
(37)
and if xU # xe , one may still obtain unequal values of R,,, . In order to study this possibility it is better to allow the ratio x1 to be not simply a constant but in general a function of the variables which specify the state of the nlv systems. This was done recently by L. M. Sehgal [12], assuming that the dQ = --dS amplitudes are small compared to OQ = dS ones. The result is that in this approximation one replaces x1 by a suitable average (x>~ on the variables. If one defines the asymmetry parameter R, - R, a = R, f R, i
(38)
in this case r: cpT = JMxh
- (xh).
(39)
I1 Note, that the existing experimental situation (see Table I) is not in contradiction with the existence. of such a difference. I2 The fact that a considerubfe difference between R,,and R, is evidence for CPT violation was mentioned by L. M. Sehgal [12]. Our idea was already pointed out in Refs. [23, 241, as well as in the first version of this paper.
526
ACHIMAN
The existing experimental resultP are not sufficient to observe a considerable difference between the weighted averages of Re(x>@ and Re(x), ; hence, taking into account the general world average (Table I) Re xear = 0.14 f 0.03, it is very improbable
that QcPr will be higher than N 0.4: Q CPT
d
0.4.
However, in order to reach the upper bound, in the CPT invariant case, one must assume a large violation of the e - p universality in these decays, which is in contradiction with the observed upper bound for Kt ,
f+(&$ = 1.01 f 0.05 f+(q) This is because &pr
(see Ref. [5]).
may be written &PT
=
Re
[l
-
Re$$].
w
Assuming that the e - ,u asymmetry of K; decays cannot beconsiderably than the observed upper bound (40), in the CP conserving interaction14
.f+KJ If+(K,o,)
higher
1 < 10-l. I
Experimentally15 I Re x2 I < lo-l, hence n ,-PT < lo-‘. Let us now show that even a small e - p nonuniversality consistent tiith the observed upper bound (40) may be enhanced in the CPT violating terms of Rl in such a way that it may lead to A w O(1 - 10). The CPT violating terms contain factors of the form
3l Rep-1 I Is For a discussion see Ref. 121. I4 We need only the real part of
and
-$
x1. I5 Sehgal assumes that second order in ,Q may be neglected.
- *,3l
fz
CPT VIOLATION
AND K"
527
DECAY
which obey Re h
1
- 1 1 5 lO-3
and
Suppose, for example, that
Wfe/fe*>- WfdC*) w 1o-2 Wf&*) + Wf&*) I and let us compute n caused by the term (Re(j=&*)
A tRe $
- I)
Re(f&*) - Wf,lf,*> - I) = Re(J&*) + Re(f,lf,*) - 2 = Wjklfe W&!L*)
*> - Re(f,lL*) +
W&L*)
f
-1
2
1_ Re(feiife*)
+
ReCLif,*~
1
’
hence
The same consideration are true for terms containing ((g,/gl*) - (I&*)). It is clear therefore that even a small e - p asymmetry may be enhanced by a factor of 0(103) in these terms and this may obviously infer a big h. Moreover, considering the upper bound (40), we deduced that ncPr < 10s2, if CPT is conserved.
7. A CPT VIOLATING
PHENOMENOLOGICAL
MODEL
It is frequently stated, that the fact that there was no difference observed between quantities which should be equal if CPT is invariant, is a proof of CPT conservation. One even calculates an upper bound on the amount of possible CPT violation, using the observed upper bound on the difference between particles and their antiparticles or their lifetimes .16 Let us show a counter example by studying a simple CPT violating model, based on a phenomenological assumption. It will be found in this model that it is possible that, regardless of the amount of CPT violation, there is no mass or lifetime difference between particles and their antiparticles. I0 See for example the review article of Lee and Wu [5].
ACHIMAN
528
If CPT is violated it will be convenient to replace it by other assumptions in order to reduce the number of parameters in the theory. Usually, one studies T invariance as an alternative [9], but this is very probably wrong experimentally [19,22]. The phenomenological assumption used here, is that the wavefunctions of KS and & are orthogonal. Explicitly, we shall assume
IWS I KL>I 25 1P4.
(41)
As a possible justification for this assumption, one may use the result of a special case of a nonlocal model for the K” decays [25]. It was also shown by L. A. Khalfin [26] that the orthogonality of KS and & must hold in any theory in which KS and K, are high poles of the Goldberger-Watson [29] type, without any relation to CP invariance.” The orthogonality assumption, when CPT is conserved, requires due to (7) Re E N 0. Hence, because of Eq. (17), in the case of R
2
N
2
invariance,
CPT
’ I Xl la Re I 1 - xz I2
e .
(42)
No charge asymmetry in KL + &v should be observed, in contradiction with experiments (see Table I). Our assumption (27) requires therefore CPT violation. However, the “mass difference” I@” ) M ) R”) - (K” 1M 1K”)l is 10s times smaller than mL - m, , the upper bound used for the calculation of the bound on the amount of CPT violations [5]. To show this, let us note that according to our assumption (41), Eq. (7) leads to
Therefore, using the expression for 6 of Eq. (6), one may see that (P(M(Ry--(K”IMIK”) 2h - md
6zRe6N
(43) -
(R”lrlP)-(K”lrlK”) rs
-
FL
I7 The orthogonality assumption has been used in several papers [28], but in addition to CPT This is obviously in contradiction with experiment (see discussion in the text). Our idea is discussed in Ref. [23]. invariance.
CPT
VIOLATION
AND
K”
DECAY
529
It will be shown later that in our model / 6 1 5 10-3,
W)
hence !;R”[MjR”)-(K”/M/K”)l
~lo-3~2(mL-wzs).
(45)
We see that if the nonconservation of CPT is restricted to the K” complex, the violation would practically not be observed through differences of masses or lifetimes between particles and their antiparticles. In order to prove (44), we use our assumption (41) and neglect all terms of 0(10-4) in the u.s.r. of Eq. (24). We find that j E - 6 + ; (1 - 211
5 Iv.
but 1 -
+
/ 5
10-3,
0
hence 161 5 10-Z
and
/ E I 2 10-S.
(44’)
The u.s.r. can be used also to obtain an estimate of the other experimental predictions of the orthogonality assumption. We shall take the approximation in which one neglects all the contributions to the u.s.r. other than Ks,L -+ 25r.ls In this approximation using (24) I T+- + &rloo I 5 1c4.
(46)
Experimentally, ) ?j+- / = (1.90 f 0.05) * 10-3, and c,h+- = 39.8” 9 6”, hence Re vO,,and Im T,,~, as well as Re q+- and lm v+- , are of 0(10-3), and the relation 700 = --2r),.-
(47)
is a good estimate for this model. Comparing the results with Table I, the experimental predictions of the model are not in contradiction with experiment, (up to one recent experiment for &J. la For discussionof this approximation seeSection 5.
530
ACHIMAN
The main point in the orthogonality model lies in the fact that its predictions do not depend explicitly on E or 6. One will obtain exactly the same predictions for the measurable quantities: T+-, voO, and RI , assuming E = 0 or 8 = 0 or even E = 6 = 0. The case 6 = 0 implies, using Eq. (43) that MRo - MKo = I’,. - I’,. = 0.
One sees therefore, as we claimed, that one may find special situations in which CPT violation in the K” system does not necessarily cause difference between the masses as lifetimes of particles and their antiparticles. Hence, obviously, an upper bound on these differences cannot be used to obtain an upper bound on the amount of possible CPT violation.
ACKNOWLEDGMENT I wish to thank Professor L. P. Horwitz and guidance during this research.
and Professor Y. Ne’eman for their encouragement
REFERENCES 1. J. STEINBERGER,K” decay and CP violation, CERN Report 70-1, 1970. 2. J. STEINBERGER,Proceeding of the Topical Conference on weak interactions, Geneva 1969, CERN Report 69-7, 1969. 3. L. B. OKUN, Comm. Nucl. Part. Phys. 2 (1968), 116. 4. S. BENNET, D. NYGREN, H. SAAL, J. STEINBERGER,AND J. SUNDERLAND, Phys. Rev. Letters 19 (1967), 5. 6.
7. 8. 9.
10. 11. 12. 13. 14.
15. 16. 17.
18.
997.
T. D. LEE AND C. S. Wu, Ann. Rev. Nucl. Sci. 17 (1967), 513. L. VAN-HOVE, Lectures at CERN academy training program 1967, CERN Report 67-27,1967. T. T. Wu AND C. N. YANG, Phys. Rev. Letters 13 (1964), 380. P. K. KABIR, “The C. P. Puzzle,” Academic Press, New York, 1968. J. S. BELL AND J. STEINBERGER,Proceedings of Oxford International Conference on Elementary Particles, 1965. G. CHARPAK AND M. GOURDIN, Notes of Lectures given at Madras, India, CERN Report 67-18, 1967. L. B. OKUN AND C. RUBBIA, Proceedings of the International Conference on Elementary Particles, Heidelberg, 1967. L. M. SEHGAL, Phys. Rev. Letters 21 (1968), 412. L. P. HORWITZ AND J. P. MARSHAND, Univ. of Denver preprint. N. BARASH-SCHMIDT, A. BARBARO-GALTIERY, L. R. PRINCE, A. H. ROSENFELD,P. S~D~VG, G. WOHL, M. Roos, AND G. CONFORTO, Rev. Mod. Phys. (1969). M. GOURD~N,Nuovo Cimento 53A (1968), 880. P. K. KABIR, Phys. Rev. Letters 21 (1968), 314. L. I. LAPWUS, Vsp. Fiz. Nauk. 98 (1968); Yaden F:7, 8 (1968). P. K. KABIR, Nature 220 (1968), 1310.
CPT
VIOLATION
AND
K”
DECAY
531
19. Y. ACHIMAN, Nuovo Cimento Letter 7 (1969), 301. 20. D. AUBERT, Proceedings of the Topical Conference on weak interactions, CERN, January 1969.
21. R. G. SACHS, Phys. Rev. Letters 13 (1964), 286. 22. R. S. CASSELA,Phys. Rev. Lerters 21(1968), 1128; 22 (1969), 554. 23. Y. ACHIMAN AND L. P. HORWITZ, Nlcovo Cimento 57A (1968), 863. 24. Y. ACHIMAN, Nuovo Cimento Letters 1 (1969), 608. 25. Y. ACHIMAN AND L. P. HORWITZ, NKOVO Cimento 62A (1969), 983. 26. L. A. KHALFIN, JETP Lett. 8 (1968), 65; see also L. I. Lapidus Dubna preprint P2-4082. 27. M. GOLDBERGER AND K. WATSON, Phys. Rev. 136B (1964), 1472. 28. S. H. PAUL, Y. TOMOZAWA, AND Y. P. YAO, Phys. Letters 19 (1965), 601; Phys. Rev. 142B (1966), 1041; V. S. MATHUR, NUOVO Cimento 44A (1966), 1268; T. BOWEN, Phys. Rev. Letters 16 (19661, I12; P. K. KABIR, Nature 213 (1967), 898. 29. J. W. CRONIN, in Proc. Int. Conf. High Energy Phys., 14th, Vienna, 1968. 30. R. J. CENCE, B. D. JONES, B. Z. PETERSON, V. J. STENGER, J. WILSON, D. J. CHENG, R. D. EAND, R. W. KENNEY, I. LINSCOTT, W. P. OLIVER, S. PARKER, AND C. REY of Ref. [29]. 31. J. M. GAILLARD, F. KREINEN, W. GOLBRAITH, A. HUSSRI, M. R. JANE, N. H. LIPMAN. G. MANNING, T. RATCLIFFE, H. FAISSNER, AND H. REITHLER of Ref. [29]. 32. J. CHOLLET, J. M. GAILLARD, M. R. JANE, T. J. RATCLIFFE, J. P. REPELLIN, K. R. SCHUBERT, AND B. WALFF, in Proc. Conf. Weak Int. CERN (1969). 33. M. BANNER, J. W. GRONIN, J. K. LIU, AND .I. E. PILCHER, Phys. Rev. Letters 21 (1969), 1107. 34. I. A. BUGADOV, D. C. CUNDY, G. MYATT, F. A. NEZRICK, G. H. TRILLING, W. VENUS, H. YOSHIKI, B. AUBERT, P. HEUSSE, I. LE DONG, J. P. LOWYS, D. MOFFERET, E. NAGY, C. PACALJD, L. BEHR, P. BEILLIERE, G. BOUTANG, M. SCHIFF, AND J. VAN DER VELDE, Phys. Letters 28B (1968), 215. 35. D. F. BARTLETT, R. K. CARNEGIE, V. L. FITCH, K. GOLJLIANOS, D. 1. HUTCHINSON, T. KAMAE, R. F. ROTH, J. S. Russ, AND W. VERNON, Phys. Rev. Letters 21 (1968), 558. 36. D. DORFAN, J. ENSTROM, D. RAYMOND, M. SCHWARTZ, S. WOJCICKI, D. H. MILLER, AND M. PACIOTTI, Phys. Rev. Letters 19 (1967), 987. 37. S. BENNET, D. NYGREN, H. SAAL, J. STEINBERGER, AND J. SUNDERLAND, Phys. Rev. Letters 19 (1967), 993. (See Ref. [l].) 38. T. KIRK, J. MARKS, D. NYGREN, J. PEOPLES, AND J. STEINBERGER, preliminary result reported. 39. C. RUBBIA, in “Proceedings of the Topical Conference on Weak Interaction,” Geneva 1969. 40. B. GOBBI, D. GREEN, W. HAKEL, R. MOFFET, AND J. ROSEN, in Proc. Int. Conf. High Energy
Phys., 14th, Vienna, 1968. 41. W. SELOVE, Ref. 40, abstract 801. 42. V. D. WALKER, Phys. Rev. Letters 18 (1967), (1968),
1613.
630; S. MAZATECK
et al, Phys. Rev. Letters 23
OF PHYSICS
64,
510-531 (1971)
Possible
CPT Violation
and the K” Decay*+
YOAV ACHIMAN Laboratoire
de Physique Thkorique, Bkt. 211, Facult.4 des Sciences, 91 Orsay, France Received June 19, 1970
The generalized expressions of the measurable CP violating parameters for the K system are given, without assuming CPTinvariance or the other usual assumptions such as: dQ = -AS, the approximate AZ = $ rule. The unitarity sum rule for the K” decays is also studied without using these assumptions. Some applications of these generalized expressions are discussed. We suggest a test for CPT invariance by testing an inequality between the charge asymmetries of KL ‘-f rev(&) and KL -+ v&R,). Using the upper bound on the e - p asymmetry for the K* decays, we show that any signitlcant difference between R, and R, indicates some CPT violation. Alternatively, even a small e - p asymmetry may lead to considerably different R,,,, , if CPTis violated. As another application we construct a CPT violating phenomenological model, in which there is no mass or lifetime difference between particles and their antiparticles even in the presence of CPT violation. We give also the “second-order” corrections to the phases of c and 8 in the case of CPT and T invariance, respectively. It is found that according to the available experimental results 4, should be considerably lower than 43”.
1. INTRODUCTION If CPT violation exists at all, it would probably be first observed in the K decays. The interactions of the K” complex are the most sensitive apparatus known to detect possible violations of CP, T or CPT. (Actually, CP violation has been observed, up to now, in this system only).l This is related to the fact that the weakly decaying particles are not K” and R”, but their mixtures which depend on the invariance properties of the system. The existing experimental situation in the K” decays is now consistent with CPT invariance [2], and the doubts raised recently [4] about it, have disappeared. However, one cannot yet exclude a violation of CPT, and it is worthwhile to study in detail this possibility in the K” system, due to the essential role that CPT conser* Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research. + Thisstudy was performed in partial fultiment of Ph.D. requirements at Tel-Aviv University. 1 For a recent experimental review see J. Steinberger [l, 21. For general remarks about the possibility of CPT violation see, for example, L. B. Okun [3].
510
CPT
VIOLATION
AND
K”
DECAY
511
vation plays in elementary particles physics. In this paper we aim to contribute to the study of the possibility of CPT nonconservation in the K" decays. To test CPT invariance one usually assumes it in order to obtain relations, which may be tested experimentally. We shall study here the formulation without assuming conservation of CPT. One of the advantages of such an approach is that it enables us to observe where it is most probably to detect CPT violation, if it does exist, In the following, we shall give the general expressions for the observed CP violating parameters, allowing a small nonconservation of CPT and without using the usual assumptions, such as OQ = AS or the flZ = 4 rule. The explicit general expressions, for the important contributions to the unitarity sum rule (u.s.r.) for the K" decays, will be also given. Using these, we shall calculate the “second-order” corrections to the phases of c and 6, in the cases of CPT and T invariance correspondingly. It will be found that when CPT is conserved the available experimental results require +C to be quite different from the frequently used value +t- = 43”, while if T is invariant the first-order term is a good approximation to bs = -47” & 180”. It will be also shown explicitly that when CPT or Tare conserved, the contributions to the u.s.r., other than due to K” --, 2rr, are practically pure imaginary or real, respectively. This may be useful in checking CPT or T invariance. However, a better way to test CPT invariance is pointed out in Section 6, as an application of the general formulation. This consists of looking for a difference between the charge-asymmetries in KL + rev(R,) and Kt + rrpv(R,). We show that even a very small e - ZLasymmetry may be enhanced considerably in the case of violation of CPT. in such a way as to lead to a substantial difference between R, and R, . (The existing experimental situation is not in contradiction with such a difference). On the other hand, CPT conservation in addition to the small upper-bound on the e - p asymmetry observed in the K-1: decays, leads only to a negligible difference between R, and R, In the last section we study, as another application, a phenomenological assumption that the scalar product between the wavefunctions of the short and long lived neutral kaons (Ks,L), is very small, i.e., i(K, 1KLjI
5 IOW.
This assumption requires CPT violation in order to explain the observed charge asymmetry in KL -+ nlv. Yet, in a special case of this model there is no mass or lifetime difference between particles and their antiparticles, regardless of the amount of CPT violation in the model. Using this example it is clear, therefore, that the upper-bound on the difference between quantities that should be equal due to CPT invariance, cannot serve as an absolute measure for CPT violation, as has been done in several papers.2 ’ See,for example,the review articleof Lee and Wu [S].
512
ACHIMAN
2. NOTATION
We use here the notation of Lee and Wu [5]. The K” complex is described using the equation
where A is in a 2 x 2 complex matrix Aij
= rij + iMij
(2)
and M and r are the mass matrix and decay matrix, respectively. In the Wigner-Weisskopf nonrelativistic perturbation approximation, obtains for A, the expression [513 (1,,,
= M;&,
+ (S 1 Hw 1S’) + c @ I Hw’ I n)(n I Hw I “) Mi. - M,” + ic n
+
one
...
(3)
=(SIM;.+H,
where H, and Hst are the Hamiltonians of the weak and strong interactions. The intermediate states 1n) are understood to be eigenstates of the strong interaction. The special choice of the many arbitrary phases in the K” system plays an important role in the discussion. We shall choose the phases of the states so that the representation is real, T I K”) = 1K”),
TIKO) = IR”)
(44
which satisfies TS = 1. (This does not, however, fix the arbitrary phase in the definition of T for the kaon field). In addition, we shall define CP so that CP 1K”) = 1R”),
CP I x0) = I K”),
W-4
satisfying (CP)2 = 1, and [CP, T] = 0. For the kaon field this commutativity restricts the phases of T and CP in such a way that their product is il. The special advantage of choosing a real representation is that an operator which commutes with T has real matrix elements, while an operator which anticommutes with T has pure imaginary matrix elements. In other words, an imaginary part of a K” decaying amplitude describes T violation (and CP violation if CPT is conserved).4 3 A careful and clear derivation of this expression may be found in the Appendix of Van-Hove’s, “Lectures on CP violation,” CERN, 1967. 4 Note, that we do not assume here the Wu and Yang [7] phase convention, i.e., that <2q1p0 1I& [ K”) is real, which is used in Lee and Wu [S].
CPT
VIOLATION
AND
K”
513
DECAY
One may show [5] that the observed eigenstates of A are, in this formulation 1 KS) N +z ((1 + E + 6) 1 K”) + (1 - E - 8) 1R”>), j KL) N +2 ((1 + E - 8) j K"; - (1 - E + 8) 1K”)j, where ~l_(K”/A(R”)-(RolA;Ko; rs
-
FL 4 2i(m,
-
m3
’ (6)
s~(K”~n~K”)-((R”:AIR”) rs - FL 4 2i(m,
-
md
.
ms,L andrs,, are the masses and decay rates of the observed
decaying short and one assumes that E and 6 are small parameters
long lived kaons. In the derivation
lSl<1,
lE/
and the approximate equality N denotes that second order in small quantities is neglected. Using (3) and (6), it follows that 6 = 0, E = 0, e=s=o,
and
if CPT is invariant, if T is invariant if CP is invariant.
C6’)
Also using (5), in our formulation
3.
THE
2~
DECAY
(7)
MODE
Let us denote A, = <277, I H,
I K”),
A, = (2Tr,{ Hwl R"),
where I 2x,) are the isospin I states of 27~. If CPT is conserved let A, = A,*,
@a)
and T invariance gives real
AI and A!.
Ob)
514
ACHIMAN
Using this notation one obtains, assuming small CP and CPT violation, following expressions for the measurable quantities
the
where SI are the s-wave 2~ phase-shifts, with isospin I and c.m. energy equal to the K” mass. Let us give these expressions in some special cases. If AZ = i is approximately valid, I and we neglect 0&4,/A,)
p 0
I
E). Then for AZ N 4,
7j+- = E - 6 + ; (1 _ St)
+ &
7700= E _ 8 + i (1 _ +)
- 5
(2
- At)
pa-80), (94
0
If CPT is conserved, due to (Ba) and (6’),
(9
- *) 0
ei(sO-aO)e 0
CPT
VIOLATION
AND
K”
DECAY
515
where i&-8,) ,
If in addition we neglect O((A,/A,) E),
A0 q+- ~16 + i Im Re A, ’ “’ (94 . ImAo ‘loo = ’ + ’ Re A,
2I ’ ’
These are the frequently used expressions (only that the phase convention of Wu and Yang is not assumed).5 In the case of T conservation we have to put E = 0 in (9) and A, will be real. The value of the Z = 4 violating term (AJA, + &/A,) ei@e+) is related to the parameter R = Rate(K, -+ 7~Ok’) (10) s Rate(K, -+ X+V-) * Explicitly, Rs=~
1 I-24’ I 1+At
I3
(10’)
where
A’ = -& ($ + 9, ei(wo). This gives the equation [I l] A’ = 4 -12R
(2 i- 2Rs - 3 aeeiE], S
i.e., for a fixed value of Rs , A’ is located on a circle. Experimentally Rs m $, (see Table I) hence, one may have also solutions for A’ that are not small. However, 5 The expressionsfor the q’s, when assuming the approximate I = 4 rule and using the Wu and Yang phase, i.e., the analog of (9a), is given in Lee and Wu [S]. The expressions when assuming CPT invariance only, in addition to the Wu and Yang phase, i.e., the analog of (9b), may be found in the book of P. K. Kabir [8]. The expressions in another representation may be found in the review of Bell and Steinberger [9]. See also Refs. [IO, 111.
516
ACHIMAN TABLE I Experimental Data
Quantity 103 1?+- I 103 j too I
4+400
Exp. value
Ref.
1.90 f 0.09
r291 [301 [3lI 1321 [331 [341 r351
3.6 i- 0.4 3.6 + 0.6 3.2 f 0.7 2.3 f 0.3 2.2 * 0.4 (-2 * 7)11*; t3 39.8” j, 6”* 17” + 31” 4.5 - lo-* 2 1 Re AZ/A,, I 2 8 . IO-2
PI WI WI
Quantity
Exp. value
_
108 Re lo3 R, Rex Imx R;’ 83 - 80
4.05 2.24 3.15 0.14 -0.13
f f f f &
2.16 2.33 -(SO0 -(30”
& 0.108 f 0.05 LIZ20”) & 10”)
AmlrS
1.4 0.36 0.3 0.05” 0.049’
0.469 * 0.0128
Ref. [361 [371 [3gl [391 I391 Il41 [401 [411 t421
PI
8 World average. b World average taking e, /L together.
using the fact that the AZ = 4 rule is approximately good for Kk, it is very improbable that there will be a large violation of AZ = 4 in KS + 27r. Note, that all these considerations are actually only qualitative because we have neglected the electromagnetic corrections, which may well be in the same order of magnitude as A’. In general, one can only say that [lo] A’ = A,, -+ O(10-3).
4. CHARGE ASYMMETRY
IN KL -+ dv
Let us denote: AS = -AQ
AS = AQ amplitudes
amplitudes
g, = A(K” --+ d-v), & = A@? 3 7rI%).
(12)
Et = a*,
(13)
fi ,fE , g, , and gl are all real.
(14)
fr = A(K” -+ d+v), jl = A(F 3 7TfI-q, If CPT is conserved .i=t=.fi*
and
and if T is invariant
517
CPT VIOLATION AND Kc DECAY
The quantity
vanishes if dS = LIQ.~ Assuming small CPT violation
and we neglect second order in these quantities. Assuming small CP and CPT violation, we have for the charge-asymmetry R = Rate& + r-1’~) - Rate(K, + ~~~f-5) 2 Rate(K, + n-l+v) + Rate(K, + n+l-C) ? 1 N Re-( 1) + / 1 - Xl 12 120 - I x1 I”> Re(E - 6) :;: .A 3) - Re [xl ($$ - --j$-j]/. -t /xz12Re ( F1
(17)
If dQ N dS holds, i.e., / x1 I < 1, one has & N 2 Re(e - S) + (Re +
-
(17’)
I).
5. EXPLICIT EXPRESSIONS FOR THE 2n AND nlv CONTRIBUTIONS TO THE U.S.R. AND SOME CONSEQUENCES
The unitarity sum rule (u.s.r.) is a way to express the conservation of probabilities in the K” decays.’ Explicitly t[(rs
+ r,) + 2ih
- m&KS
I KL,) = c (F j H,
I K,)*
(F I HFV I KL),
(18)
F
where l-‘,,, are the decay rates of Ks,L and ms,L are their masses. The summation CF is extended to all observed final states for the K” decays. and includes spin and isospin summation and phase space integration,
T - ,&, 1 dPF(274” W, - P). F 6 In general jE andg, shouldberegardedassuitableaverageson the variables[12]. 7 Theu.s.r.wasfirst derivedby Bell and Steinberger [9J. See also Ref. [ll]. In the literature one canfind differentformalderivationsof this sum rule. For the derivation and discussion of a generalized u.s.r., see L. P. Homitz
and J. P. Marchand
[13].
518
ACHIMAN
Using the expression (7) for
(I+i
F)
N 2 * 10-3, it is possible to give
(Re B - iIm6)
=CB,7jF, F
where Am=mL-mm,, B F = RateK
= E J--s ’
3 rs
(20)
and (21)
The u.s.r. is one of the most useful relations in the K" system. This is due to the fact that it is a relation between quantities that are measurable. The expressions on the rhs are on-mass-shell, in contradiction with the other expressions for E and 6 (6), which include off-mass-shell contributions. One more convenient fact about this sum is that only few terms contribute in practice. This is easily seen in the representation (18’), because BF are in general very small factors. A direct way to study this, is by using the Schwartz identity [9]
In experiment r? r,“’
z*Cg
hi 1
M 10-3,
rp ,( 5.5 * 10-4, --ji-
r,rp 5 10-4,
s
and rF#2?r,3a,nlv S,L
g
10-3.
But @+-)
= w?ocJ = lo-3,
hence I g I 22 1c3.
CPT
VIOLATION
AND
K”
519
DECAY
It is clear, therefore, that the contributions from F # 2x, 377, n/v may be neglected. The contribution from KssL ---f 377 may also be very probably neglected. This is because we saw that
Ic i < v’I’?l-p
< 2.34 . lO-4
3n
and one needs a very large CP violation in KS,L --f 3~ in order to have a considerable contribution to the sum. This is improbable, although the experimental situation with respect to this decay is yet very unclear. We shall, therefore discuss here in detail only xzrr and zPlv . Let us only note that Gourdin [ 151 showed that CPT invariance leads to (22a) and T invariance to
(22b) We shall calculate explicitly the expressions for the 2~ and r/v contributions to the u.s.r., in terms of the amplitudes, without using the assumptions of CPT invariance, or dQ = rlS, or rlZ N +. The calculation is especially important in the case of xVlv , where q,lV are not yet measurable. It will be shown that it is enough to know xL , in addition to the partial rates, in order to obtain a good estimate of this contribution. The available experimental data show an important contribution to the u.s.r. due to Ks,L --, XIV. In the case of KS,L -+ 2n, the explicit calculation enables us to obtain the contribution of flZ = fr breaking terms to the phases of E and 6. We do not neglect O(( A,/&, j E), and this gives us new consistency relations, in the special cases of CPT or T invariance. It is found that these relations are consistent with the existing experimental results. Finally, it will be shown that the contributions to the u.s.r., other than due to K S,L - 2rr, are practically imaginary or real in the special cases of CPT or T invariance correspondingly. This is useful especially when one uses the u.s.r. in order to check CPT or T invariance. Assuming small violation of CPT, one obtains by explicit calculation
-k 46 h
E - Re 8) Re x1 - 2i Im x1 + 2 j x1 12Re i s
- 2f Im [xz (-$595/64’2-14
- +)]I.
- $C) 1 (23)
520
ACHIMAN
This expression contains terms of O(E), however comparing it with (1 P-s) G,L - 10-3, it is clear that one can neglect, in the u.s.r., all terms of O(E), and we are finally left with* (239
We see that the KS,, -+ rrlv contribution is practically imaginary, and also, that it is proportional to Im xl and therefore vanishes practically if one assumes T invariance. This is an interesting property especially when noting that x3,, has also a similar behavior [15]. This is because due to the small value of (l/F,) rsu M lo-*, the terms (22a, b) are negligible. Hence, in these special cases, of CPT and T invariance, C3= is imaginary or real, respectively. Collecting contributions, we have for the u.s.r.
(l+i
$$)(ReE-iim6) S
N rsclrT
Rs) cv+- 4 13~~~) - i EE rs
2 Irn x lflX13
+ &
$.
(24)
In the special cases of CPT or T invariance:
(1 +iF)ReE2Am S
N r,(lrI
-
(
R,) (rl+- + G,bo) + &
i--2j$)ImSN S
rsclrI
[-rlePt*
As) b+-
f
* Im ’ + Im C] 1+ IXP 3m for CPT invariance; Rs~~oo)
(25)
+ $--S Re c,
397
for T invariance.
(26)
We see, therefore, that the contributions to &, other than the main term Czn , are either purely imaginary or real in the special cases of CPT or T invariance correspondingly.9 It can be shown, using this description, that Eq. (26), i.e., the assumption of T invariance, is inconsistant with experiment [19], therefore T is violated in the K” decays. 8 P. K. Kabir [16] has derived this expression, in the case of CPT invariance. B This may be shown using very general consideration and assumptions [17, 181. Here, however this fact is shown explicitly.
CPT
VIOLATION
AND
K”
521
DECAY
The u.s.r. may be also used in order to compute the phases of E, 6. $C is known in the first approximation, we shall compute explicitly the second-order corrections. To do this, let us calculate the contribution due to KS,, -+ 2rr explicitly. We shall use two parameters R, and A’, which are related. However, the exact relation is not known, because of the unknown electromagnetic corrections (see discussion in Section 3), we shall use, therefore, for R, its directly measured value. The available estimate for d (in the case CPT is conserved) is [20] 8 . 10-Z 2 j fl j 2 4.5 . 10--z. Due to the fact that 1 E / = 0(10-3), O(EA) may be considerably higher than O(G), which is neglected in our calculations. Let us, therefore, neglect only terms of O(eA2).
In this approximation ?I+- = 6 - 6 + ; (1 - 2)
(1 - A’) + ~“(1 - A’), (27)
-
“loo CT+E - 13+ ; (1 - $)
0
(1 + 24’) -- 2d’(1 + 249,
where
Hence, (r]+-
im
R.y~o)
N
(1
+
R,)
[C -
+ (2Rs -
1) ;
8 +
i
(1 - -$)I
(1 - +‘)
+ (I - 2R,) E
- (1 + 4R,) A’,“.
0
This can be simplified,
using the fact that R, = 4 + O(A),
into the expression &
(T+- + R,q,,)
N E - 8 + ‘2 (1 - 2)
+ 2 [; (1 - 2RJ - A’] F”. (28)
The last term is neglected in the usual approximation, where one neglects O(EA). However, this term can in principle contribute more than x+ and obviously Cln , hence, we think that in the approximation where one takes into account the last contribution, it may not be justified to neglect O(A).
522
ACHIMAN
Using (28) we have a new expression for the u.s.r.,
(1+ iF)(Ree--ima) N E - 6 + ; (1 - 2)
. Fept.
2 Im
+ 2 [; (1 - 2Rs) - A(] E”
x
++
-zrsl+lX12
(29)
n
In the special cases of CPT and T invariance: 1+i--- 2Am J-s
Re E
_N E + i $$$.
+ 1/2 i [ ’ -32Rs
. FePt. 2 Im x
+&ImC,
-zTl+lxlz
- A] Im (2)
ei(‘*-*O)
for CPT invariance;
3n
CW
2Am 1 +iF)iIm6 -
-8 I;
(1 - $)
+kRex.
+ 5
[ l -32RS
_ 4(
;2o
$2) o ,&-80)
for T invariance.
(29’3
3n
It turns out, that in order that our approximation be valid, in these special cases, corresponding consistency conditions must hold. In the case of CPT invariance, taking the real parts on each side, one obtains Re E N Re E -
~‘2 Im (2)
Im I( ’ -32R”
- A) e"(8~-8~)/.
Hence, the following condition is needed 1 -
2Rs
Im
ee’b,-60)
3 or 1 - 2R,
3
v9
w cos(8, - 8,).
(304
CPT
VIOLATION
AND
K”
523
DECAY
This may be checked experimentally. Using, for example Re(A,/A,) = 8 . lo-” and R$ = 2.33, one finds 6, - 6, N_ 140”, which is consistent with the experimental available values (see Table I). In the case of T invariance, one obtains in the same way (for the imaginary parts) w cos(S, - So).
VW
Taking in each case, the other (imaginary or real) part, one can obtain information on the phases of E and 6: For CPT invariance, 2Am
2 Im x ltlX12
hReAA0 -- P-t. rs 0
rReE’VImE+ s + 9
(1 - 2Rs) Im (9)
- Re (2)
Im ($)
cos@, - 6,)
co: 2(6, - 6,).
Only (Im A,)/(Re A,) may be of O(1O-3), all other terms are lower by at least one order of magnitude. It is, therefore, convenient to define
In terms of <, one may write in the first approximation,
i.e., neglecting O(EA),
q+- = EI + E’, (33)
700 !x E^- 2E’.
The phase of E is due to (31),
- d/z ’ -32R” + Re (9)
Im ($)
Im (2) 0
cos(S, - 6,)
co: 2(6, - &,)I.
(34)
Usually, one takes the first-order term 2Am/r, only. Let us show that, according to the available experimental data, the second-order terms may contribute considerably, especially CIllV .
524
ACHIMAN
Using the experimental
values of Table I,
1 Pept. K-i-- r, which is obviously an important
2 Im x 1+lx12
N -(0.35
contribution
f 0.15),
to
The other terms can have a contribution of O(1O-2), and may be even a higher one, but the experimental situation, with respect to these terms, is not so clear. Hence, we may conclude, noting that Re Z = Re E > 0, thatlo
A similar analysis can be made for
Ls-++--$), 0
in the case of T invariance. In this case one has in the first approximation q+- -N 6 + El, (36)
qo(j N s - 2ER. Taking the real part of the u.s.r., we have $Irn$!=--Re8+2Re(
1-32R,
-,‘),“+kRe$.
The consistency condition says that the imaginary part of the Al = $ breaking terms vanish. 6, - 6, NN -45”, hence the real part is negligible as well as very probably zS,, . Also, Re 8 can be shown to be of 0(10-8), using (36) and the experimental data on the 7’s. In this case, therefore, the first approximation
lo Note however, that there are yet some doubts about the value of Im x, which is in contradiction with most of the CP violating theories. The theory of Sachs [21], that assumes CP to be violated in the AQ = -AS interactions, is one of the few theories in which Im x # 0. In this theory ds N 4++-N doO, and the above result is consistent with the available experimental values. (See Table I.)
CPT
VIOLATION
AND K”
DECAY
525
is a good one, in contrast to & when CPT is conserved. Equations (36) were used by C. S. Casella [22] to check T invariance, in an analogous way to the use of the Wu-Yang triangle [7].
6. R1 AND THE EXPERIMENTAL
TEST OF CPT INVARIANCE
As an example of the use of our CPT violating expressions, let us show on the one hand, using the observed upper-bound on the e - p nonuniversality, that a difference between R, and R, is practically an evidence for CPT violation; and on the other hand, that even a very small e - p nonuniversality may be enhanced, in the case of CPT violation, in such a way as to lead to a substantial difference between R, and R, .I1 The usual tests of CPT in the K” decays consist of checking the Wu-Yang triangle [7] or, what is better and almost equivalent, a direct use of the U.S.T. for the K” decays. In these methods one must know at least four of the parameters I rl+- i, I %I 1. 4+- y &JCIy and Rl , in addition to some information on the other modes of the K” decays. In contrast, the test proposed here may be realized in principle in one independent experiment, is more sensitive and it is more probable to observe CPT violation here, if it does exist.12 When CPT is invariant, &
=
2(1
-
I 1 -
1 XI x2
1”)
Re
I2
l ’
(37)
and if xU # xe , one may still obtain unequal values of R,,, . In order to study this possibility it is better to allow the ratio x1 to be not simply a constant but in general a function of the variables which specify the state of the nlv systems. This was done recently by L. M. Sehgal [12], assuming that the dQ = --dS amplitudes are small compared to OQ = dS ones. The result is that in this approximation one replaces x1 by a suitable average (x>~ on the variables. If one defines the asymmetry parameter R, - R, a = R, f R, i
(38)
in this case r: cpT = JMxh
- (xh).
(39)
I1 Note, that the existing experimental situation (see Table I) is not in contradiction with the existence. of such a difference. I2 The fact that a considerubfe difference between R,,and R, is evidence for CPT violation was mentioned by L. M. Sehgal [12]. Our idea was already pointed out in Refs. [23, 241, as well as in the first version of this paper.
526
ACHIMAN
The existing experimental resultP are not sufficient to observe a considerable difference between the weighted averages of Re(x>@ and Re(x), ; hence, taking into account the general world average (Table I) Re xear = 0.14 f 0.03, it is very improbable
that QcPr will be higher than N 0.4: Q CPT
d
0.4.
However, in order to reach the upper bound, in the CPT invariant case, one must assume a large violation of the e - p universality in these decays, which is in contradiction with the observed upper bound for Kt ,
f+(&$ = 1.01 f 0.05 f+(q) This is because &pr
(see Ref. [5]).
may be written &PT
=
Re
[l
-
Re$$].
w
Assuming that the e - ,u asymmetry of K; decays cannot beconsiderably than the observed upper bound (40), in the CP conserving interaction14
.f+KJ If+(K,o,)
higher
1 < 10-l. I
Experimentally15 I Re x2 I < lo-l, hence n ,-PT < lo-‘. Let us now show that even a small e - p nonuniversality consistent tiith the observed upper bound (40) may be enhanced in the CPT violating terms of Rl in such a way that it may lead to A w O(1 - 10). The CPT violating terms contain factors of the form
3l Rep-1 I Is For a discussion see Ref. 121. I4 We need only the real part of
and
-$
x1. I5 Sehgal assumes that second order in ,Q may be neglected.
- *,3l
fz
CPT VIOLATION
AND K"
527
DECAY
which obey Re h
1
- 1 1 5 lO-3
and
Suppose, for example, that
Wfe/fe*>- WfdC*) w 1o-2 Wf&*) + Wf&*) I and let us compute n caused by the term (Re(j=&*)
A tRe $
- I)
Re(f&*) - Wf,lf,*> - I) = Re(J&*) + Re(f,lf,*) - 2 = Wjklfe W&!L*)
*> - Re(f,lL*) +
W&L*)
f
-1
2
1_ Re(feiife*)
+
ReCLif,*~
1
’
hence
The same consideration are true for terms containing ((g,/gl*) - (I&*)). It is clear therefore that even a small e - p asymmetry may be enhanced by a factor of 0(103) in these terms and this may obviously infer a big h. Moreover, considering the upper bound (40), we deduced that ncPr < 10s2, if CPT is conserved.
7. A CPT VIOLATING
PHENOMENOLOGICAL
MODEL
It is frequently stated, that the fact that there was no difference observed between quantities which should be equal if CPT is invariant, is a proof of CPT conservation. One even calculates an upper bound on the amount of possible CPT violation, using the observed upper bound on the difference between particles and their antiparticles or their lifetimes .16 Let us show a counter example by studying a simple CPT violating model, based on a phenomenological assumption. It will be found in this model that it is possible that, regardless of the amount of CPT violation, there is no mass or lifetime difference between particles and their antiparticles. I0 See for example the review article of Lee and Wu [5].
ACHIMAN
528
If CPT is violated it will be convenient to replace it by other assumptions in order to reduce the number of parameters in the theory. Usually, one studies T invariance as an alternative [9], but this is very probably wrong experimentally [19,22]. The phenomenological assumption used here, is that the wavefunctions of KS and & are orthogonal. Explicitly, we shall assume
IWS I KL>I 25 1P4.
(41)
As a possible justification for this assumption, one may use the result of a special case of a nonlocal model for the K” decays [25]. It was also shown by L. A. Khalfin [26] that the orthogonality of KS and & must hold in any theory in which KS and K, are high poles of the Goldberger-Watson [29] type, without any relation to CP invariance.” The orthogonality assumption, when CPT is conserved, requires due to (7) Re E N 0. Hence, because of Eq. (17), in the case of R
2
N
2
invariance,
CPT
’ I Xl la Re I 1 - xz I2
e .
(42)
No charge asymmetry in KL + &v should be observed, in contradiction with experiments (see Table I). Our assumption (27) requires therefore CPT violation. However, the “mass difference” I@” ) M ) R”) - (K” 1M 1K”)l is 10s times smaller than mL - m, , the upper bound used for the calculation of the bound on the amount of CPT violations [5]. To show this, let us note that according to our assumption (41), Eq. (7) leads to
Therefore, using the expression for 6 of Eq. (6), one may see that (P(M(Ry--(K”IMIK”) 2h - md
6zRe6N
(43) -
(R”lrlP)-(K”lrlK”) rs
-
FL
I7 The orthogonality assumption has been used in several papers [28], but in addition to CPT This is obviously in contradiction with experiment (see discussion in the text). Our idea is discussed in Ref. [23]. invariance.
CPT
VIOLATION
AND
K”
DECAY
529
It will be shown later that in our model / 6 1 5 10-3,
W)
hence !;R”[MjR”)-(K”/M/K”)l
~lo-3~2(mL-wzs).
(45)
We see that if the nonconservation of CPT is restricted to the K” complex, the violation would practically not be observed through differences of masses or lifetimes between particles and their antiparticles. In order to prove (44), we use our assumption (41) and neglect all terms of 0(10-4) in the u.s.r. of Eq. (24). We find that j E - 6 + ; (1 - 211
5 Iv.
but 1 -
+
/ 5
10-3,
0
hence 161 5 10-Z
and
/ E I 2 10-S.
(44’)
The u.s.r. can be used also to obtain an estimate of the other experimental predictions of the orthogonality assumption. We shall take the approximation in which one neglects all the contributions to the u.s.r. other than Ks,L -+ 25r.ls In this approximation using (24) I T+- + &rloo I 5 1c4.
(46)
Experimentally, ) ?j+- / = (1.90 f 0.05) * 10-3, and c,h+- = 39.8” 9 6”, hence Re vO,,and Im T,,~, as well as Re q+- and lm v+- , are of 0(10-3), and the relation 700 = --2r),.-
(47)
is a good estimate for this model. Comparing the results with Table I, the experimental predictions of the model are not in contradiction with experiment, (up to one recent experiment for &J. la For discussionof this approximation seeSection 5.
530
ACHIMAN
The main point in the orthogonality model lies in the fact that its predictions do not depend explicitly on E or 6. One will obtain exactly the same predictions for the measurable quantities: T+-, voO, and RI , assuming E = 0 or 8 = 0 or even E = 6 = 0. The case 6 = 0 implies, using Eq. (43) that MRo - MKo = I’,. - I’,. = 0.
One sees therefore, as we claimed, that one may find special situations in which CPT violation in the K” system does not necessarily cause difference between the masses as lifetimes of particles and their antiparticles. Hence, obviously, an upper bound on these differences cannot be used to obtain an upper bound on the amount of possible CPT violation.
ACKNOWLEDGMENT I wish to thank Professor L. P. Horwitz and guidance during this research.
and Professor Y. Ne’eman for their encouragement
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