CP Violation in Neutral K Meson Decays
Yukio IT0
Depar tment of Science E ducation
F acu l ty of E d uca tion F u kttshi m a O ni'1)er si ty , Fttku shi m a
We introduce the CP-violating parameters s,t and u with s2十t2十u2= 1, which have s = l and t = u= c it CP is conserved under CPT invariance. In order to analyse neutral K meson decays,the decay rates at,d the other quantities are expressed by these parameters. Unitarity relation,2(fm/r)= (t/su)and the other relations are also derived. Specially, it is shown that CP-violating para-
meter s and the quantities in K ・ > 2,t are described in terms of 1り,_ 1, e._, e , and e (phase shif t di ff erence in 7,1てscattering). Numerical calculations of e and
い10 1/1 1 _ l are illustrated in the case of - t ? su. .
St. Introduction
It is said that CPT is conserved in all interactions and that C,P and T are also conserved independently in strong and electro-magnetic interactions. But, in the weak interaction, this circumstance is different,namely,Po or T as well as P are not conserved in the weak interaction,but no evidence against the vio- lation of CPT. Ia this paper,we discuss on the PC(or T)violation which seems
to occur only in the decay of K mesons. Throughout this paper,we assume CPT invariance in all interactions.
First,we introduce the parameters which express the measures of CP vio- 1atio_n, in analogy with a,βand r 'mt -decays,1,amely,s,t and u,and discuss on the decay and mass matrix in neutral K meson decays in §2。 In S3;we get the so-called Be11-Steiaberger sum rule aad find the restriction on the CP vio- lating parameters. In §4,we take the case of 2,◆decay and discuss on the re- lations obtained from Wu-Yang relation. And,we show that the quantities re-lated to K→2- and CP violating parameters are expressible only by lり+_1, 8+_,
8oo and e. Examples of e and l?col/い,14._f in the case of - t ・::二;su are discussed
in §6. ,
32. Decay and mass matfix
According to the discussion of Wu and Yang'), the decay and mass matrix in neutral K meson decay is expressed as follows.
、 -P A
に
.
一一 M.1・
十 r2 Re A2・Im A2十y - 2 s ・u ・』m - t ・
2 (1 - u
A3十(Re A2)2 (Im A2)2十x
-2t ・f m 十s ・u ・r Mi = - - 2 (1 _ u2) 2s ・f m - t ・u ・r Mr - n ,- _2、一 with s2十t2十u2 = 1.
If Cp is conserved under CPT invariance,then,s=l and t=u=0 Usin.g these parameters,
2p ・q= 7・十21・f m
a:ad
(4)
q s 十i t
p -j -.十u ' (5)
the relations between quantities in (1) and the lifetimes, mass differen.ce of neutral K mesons are as follows,where relation (4) is derived from the diff er_ ence of eigenvahles of matrix (1),
2 u ・t ・dm 十s ・r (6) (7) (8) (9) 2 (1 _ u2)
Sci . Rep. Fukushima Univ., No. 20
= ? Ao Ao? 十? Af 2 A2.A?? 十一十i ?Mr_ j.M1 M,, ) (1)
7171:(I = 0) In:71(I = 2)
Using these p and q;Ks and KL are defined,respectively,as follows.
lKs > = ,1 121 ?/lp1 十lq 1 _ 22{P IKe > 十ql Ko> },
(2)
lKL> = 高/1P1 十 q 1{p lK°> - qlK°>}
Furthermore,it is convenientto introduce the following quantities related to CP violation in order to analyse the neutral K meson decay2).
2 Re (p* .q) 2 1m (p' ・q) l p l2_ l q l2
Ito :CP Vjolation in Neutral K Meson DecayS 3
(M 十M2) _ {Ao十(Re A2)2_ (Im A2)2十x }2- (2 Re A2・Im A2十y)2
= (」m )'- j- r2 00 where d m - m s - m L, r = r s - r L, X - ( R e 「 l 2) ・:,loop , .. y _ (Im r ',2) en ep, ..
Fo11owitlg relatio1,s are also derived.
p2 = ? ,(-I i? ;..・ (r 十21・4n ) 一 at) q2 = ? ? j ? ;・・ (r + 21・.4n ) or (1- u)・p2十(1 十u) ・q2 = s ・(r 十21・」m) 的 S3. Unitarity relation
We can construct the fo11owiag three summations.
Σta (Ks→a) 12 = 「'11十 2・・・j -1 u2)・・(4'Stu '4m 十(S2- t2)'「)n ( - ? ta (KL- n) l2= 「l,- 2-(1 1 -u-j)- (4 ・Stu '?m 十(S2- t2)'「) 04
・iu (s2- t2、)・4n
-? 一、一一一 一、一一一' - 1 - u? ? 、 '
_ st r }
a0
By def initions,1.h.s.of (13) and (14) are, just,rsand rL, respectively. Then,we can get the fo11owing relation between CP violating parameters.
2_4m. =r s ut 00
Further,we can get the fo1lowing relation from (15) using (16)_
4 Sci. Rep. Fukushima Univ_, No.20
with
1 ・ ・
Mj = mj- 2 ・1'rj (J= S,L)
This relation (17) is the so-called Bell-Steinberger sum rule3}or unitarity relation.
We can re-write the relations (6) (9) using (16) as follows.
2 Re A2・Im A2 十y - 0 a8;
A言十(Re A2)' - (Im A2)2 十x = ・21s ・r 的
1_s2
M'= - 2 s u '「 f的
Mr - 12 - ? 「 ?11
Relations (18) and (19) mean that Im 「,。_ 0 and Re f ',2= (1/2s) r.
Experimentally,absolute value of 1_h.s.of (16) is nearly equal to unity. Then,we get
- t s u u
using 」m < 0 from experiment4). The other relations are necessary to estimate the magnitudes of the CP violating parameters. In order to derive the relations, we will discuss on the relations to be determined by experiment in the following sections_
§4. 21n:decay
We consider the 27t decay of neutral K meson. From the definitions of Ks and Kt_, we can express the amplitudes of 2Tldecay of the neutral K meson in terms of CP violating parameters. From these amplitudes; we can.get the following decay rates.
R (K,.→2π) - Ao{(1 ・十Rｻ¥ 12)- (1 十R2- I') ・s - 2 ・RI ・t} ?
R (Ks →2π) - A3{(1 十R2十I2) 十(1 十R2- I2) ・s 十2 ・RI ・t} e4
R (K →2-) = 2 Ao (1+ R2十I2) 因
R (Ks→7,:°π°) ? (2/3)・Ac・(1- 2? 2 。R ・cos の ?
R (Ks- r,:?7t ) ? (4/3) ・ A3・(1 十 /2 ・R ・cos の ?
Ito:CP Violation in Netltral K Meson Decays 5
Re A。 Im A2. .
R - Ao- , I = Ae aad e - 0 1__。,_.- Il l=o;._
The amplitude ratioas of K →2-・are defined as
f __ a f '?- = ε. {a s,_,__ 1 十 I F .(R 十j _I .)}y2 .
x {1十_.1. ・F ・(R 十i ・_・I) }2 -' ? roe= a_I''o0 - _ ・ {a s,co 1一、/2 ・F ・(R 十j . I )}xo
x !1一、12 ・F・ (R 十i ・e・I) }1 -' ? where a ,__,_ = amplitude (Kj:→一'71:-),etc.. We can write the fo11owing
relations between -f1+_ and f oe.
(f oe十2 ・71+_) 十 /2 ・(ク+_ - 7,co) ・F ・(R 十i ・e ・I) = 3 ・g
(r+_ - ,loo) ・{1 - / 21 ・F・R - F2・(R'十I2) } SO
. 3 ?/2
- 1 2 'F 'I '(1 - f ._ 'fOe)
where
__ u - i 't 1 十 S and F _ exp (j .e)
Wu-Yang 1)take the fo11owing approximations for ,,+_ and roo instead of
(28) and (29). 1.l._ -- e 十一' e1, 1' o a - e - 2 g ' , where a' = (1/、/2)・i ・F ・I.
These amplitudes construct the so-ca11ed Wu-Yang triangle,namely,
・
f oe 十 2 1t +_ = 3 ε 13 ''
We examine the validity of this approximation through the fo11owing procedures.
Sci. Rep.Fukushima Univ No.20 1- 1, - l,7+ l,,=o ・J 2 ・I ・R ・ 〔 一f i s x
x {2 (u ・sin e 十t ・cos8) ・cos8 十 t }十 3 ・ I ・cos e 〕 ?/ 2 2 e?
and
1178011 - 1,7+_ i__o- 4・I ・R ・ 〔 jI S x
x {2 (u ・sin 8 十t ・cos の・cos 8- t }十 ?/2 ・I ・cos e〕 ?
where 、71,fRo(i = 十一or eO)corresponds to the one in Wu-Yang relation. For the special case.we take the fo11owing values.
u = - t_ = 1.18 x 10_35)
1十S 1十S
and
6)
8 - 500
The numerical results are summarized as fol1ows. (i) いf,._1R=:510-2/ l・り.,__lR=a -1.024 for i > 0 and 1.008 for I<0, where we use い,1.,._fRo=2.0x10-3. (ii)
1?COIR二:..,o-2/い'1oolRo for various l'7001 R_o are illustrated in Fig. 1.
Fig The ratio ,foe R_5.Io-2/ 7100,R o, where ,700 R__o means the value obtained from
neglecting ReA2 term in the definition of ηco. R = 5 x:10-2 1s taken from
Ito :CP Violation in Neutral K Meson Decays 7
As seen from the above discussions,the deviations of 117+_ ,,=5.,o-2/ 1リ+_ 1,=o and
7oolR__5.,o-2/ l?COIR=o from unity a-e at most 5 per-cent.
Under these circumstances, we summarize the results obtained from Wu- Yang triangle,namely; relations which neglect the reje ct Re A2 terms in リ+ー
a n d w 。8 .
い'col _ COS(6_トー一の
In_t_l - cos(e'.o 二6)
u _ co??_j:1 _ _{cos8・cos (eae一の 十 1 ・sin (e+_-goo)・sin8'1
1十S - 0) 3
二t = lyl+-1 {sing oes (8。。一の 十 1 sin (e,_- e。。) cos6 }
1十s cos(e - 8) 3 1-S 1十S I r+_1 -2 .2 / . 3
cos2 (e_l__一の 十 ( sin (8+_- coo) )2 -
3_ sin (e+_- coo) cos (8+_- 8) cos (90十coo一の }
sin (e+__e。。) COS (θ00一の
1 {1170012_ 1- 2 }- _1 ・I2十一 I・ { u . sin6 十 t_ cos0 }
3 2 1十S 1十S
1 . { nOel2 十2 . 1r+_ 2 }- 1二S 十 I2 00
3 1十S
If we take into account of Re A2 terms,the first relation,for example,should be read as follows.
一 = COS (θ+-一θ) {1+ R . Sj_;j _'Sin (0+.-- ?31_j..._} 的
l?+_1 cos (806一の ? 2 Cos _- e ) ・cos ( 一の
In (35) and (36),8+_ and θ00 are phases of ・11+_and 7100,respectively. Numerical discussions are summarized in S6.
§5. Semi-leptonie decay
In the case of semi-leptonic decay of the neutral K meson;we can express the decay rates and the decay ratios in terms of CP violating parameters and,moreover, we can show the method of finding these CP violating para_ meters;specially u,independent of s, t and the parameters of 」S/f Q= ±1.
8 Sci.Rop_ Fukushima Univ., No.20
36. Summary and discussions
We express mass difference,. lifetimes and the quantities in K →2- in terms of the CP violating parameters s, t and u in order to estimate the magnitudes of CP violation in the neutral K meson decay. Firstly, we show the restrictions on these CP violating parameters such as 2(f m/r)_ (t/su) and the unitarity relations. And,we show in (35) that the CP violating para- meters, r。el and Im A2/ Aoare expressible as functions of 1η.,._1, the phases of amplitude ratios in K→21, (6._ and 8.n) and the phase shift diff erence in
It7t Scatte「ing (の.
Recent experimental situations are summarized as follows: (1) 1,的1/ If _1 _,, 131'9:', (i i) 300 d".8._ ・d".80''' and (iii) 8 ? 50o6' ・ We i 11ustrate e and ;-y的li t・f ?_1 in Fig. 2 as f unction of go? (assuming Icu l ? 80「) and 61+_ in the case of - t -:? s u in (22).
c Oo
Fig 11 (phase shi f t dif ference in ・,一一scattering) and り8e/ ,1.,_ as f unctions of
ii_ and 11 in the case of - t '-s・u in (22). We neglect the points at 。./ y,.,_,= 1 f o r e ._ = 11。..
Since neither phases (e._, goo and の nor いfool / 1?+_f are given exactly by
experiment, we can not determine the magnitudes of CP violation in weak interaction for the present.
Ito:CP Vjolationin Neutral K Meson Decays 9
we neglect, in this paper, the eff ects from 3π and the othe「 modes; because jt seems that we cannot treat these decay modes without techniques of current algebra or some kinds of the approximations adding some Pa「amete「S (or form factors) which have no direct connectio1' with CP violation.
References
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3) i s _Bell and J Stejnberger , in Proceedings of the Oxf ord International Conte「once
on Elementary Particles, September ,1965.
4) E_.Fa1ssner, lj _Foeth, A.staude, K.Titte1, P.Darri ulat, K.Kieinnecht, C.Rubbia,
J sandwejss,M I Fer rero,and C . Grosse, Phys . Letters 30B, 204 (1969) : (mt,-mS) / 石= (0.555±0.020)・10-'a sce-1.
5) This stems f rom the e;xperjmental data on semi-leptonic decay :D.DO「f an et at ・,
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6) W D walker et at.,Phys.Rev.Letters 18,630 (1967) e = 530土200. 7) A H;Rosenferd et at.,Rev.Mod_Phys.40,77 (1968).
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Accele-rator Conference on K-Mesons,1967.
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