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Further study of the effect of final-state phases on CP violation in the non-leptonic Bd0B̄d0 decays
Volume 261, number 3
PHYSICS LETTERS B
30 May 1991
Further study of the effect of final-state phases on CP violation inthe non-leptonic Bj-B, o -o decays Zhi-zhong Xing a,b and Dong-sheng Du
b
CCAST(~brld Laboratory), P.O. Box 8730, 100 080 Beijing, China b Institute ofttigh Energy Physics, Academia Sinica, P.O. Box 918(4), 100 039 Beijing, China
a
Received 7 December 1990; revised manuscript received 31 December 1990
We make a further study of the CP-violating asymmetry for the decays B° - D + D -, n+n -, and K+n -, which have both treelevel and penguin amplitudes. We confirm that the final-state phase shifts can lead to significant direct CP violation for these decays, and find that the absorptive part in the penguin amplitude can also lead to direct CP-violating asymmetry for each decay, which is about 5% in the standard model. The numerical predictions of the CP-violating asymmetries are given against final-state phase shifts, and they are quite different from those by Tanimoto et al. [ Phys. Rev. D 42 (1990) 252 ].
1. Introduction Because the B-mesons are heavy, there are many channels suitable for studying CP-violating phenomena [ 1,2 ]. To observe any C P violation, we require that there should be some interference between two amplitudes. This interference can occur either through B°-I] ° mixing or via final-state interactions, or by a c o m b i n a t i o n o f both effects. In this article, we only consider those non-leptonic neutral-B-meson decays which possess both the treelevel a m p l i t u d e and the loop ( p e n g u i n ) one. T a n i m o t o et al. [ 3 ] have i m p r o v e d G r o n a u ' s study o f this p r o b l e m [4] by investigating the effect o f the final-state phases on CP-violating a s y m m e t r y quantitatively. They drew the remarkable conclusion that the final-state phase shifts play an i m p o r t a n t role for the decays B ° ~ D + D - , n+n - and K + n - . However, there are a few errors in ref. [3 ]: ( i ) The K o b a y a h s i - M a s k a w a matrix factors in the quark subprocesses b-~eca, uud, and ~us, which should be Vc*bVcd, V*b Vud, V*b Vus respectively, were written as Vcb Vc*d, Vub V*ud, Vub V'us in ref. [3 ]. This will cause a reverse sign o f the CP-violating a s y m m e t r y Cr. (ii) The analytic formula o f the loop integral function Ii which were used in ref. [ 3 ] to calculate the penguin amplitudes, in fact, are not correct. This error makes the penguin a m p l i t u d e s considerably larger than the tree-level one, but it is screened in the calculations o f Cf because Cf is expressed as the ratio o f two quantities (see eqs. ( 1 ) and (2) ). (iii) The imaginary part o f the integral function li was neglected and expected to have only a small contribution to Cf in ref. [ 3 ]. In fact, the existence o f Im li can lead to a direct C P violation, thus, it has typical significance. Following G r o n a u and T a n i m o t o ' s idea, here we correct the errors in ref. [ 3 ], and make a further study o f the problem. We also calculate the CP-violating a s y m m e t r y for the case o f a non-vanishing imaginary c o m p o n e n t of L. We find that Im I, can affect the b e h a v i o r o f Cf considerably, and the final-state phases can lead to remarkable direct CP-violating effects for the typical processes B ° - D + D , n + n - , and K + n - . O u r numerical results are d r a m a t i c a l l y different from those in ref. [ 3 ].
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
315
Volume 261, number3
PHYSICSLETTERSB
30May 1991
2. Partial-decay-rate asymmetry The time-integrated CP-violating asymmetry is defined as [2,5 ] F(BOhys__,f) - - F ( g p-o h y s ---~[ ' ) C,-= F(B%y~--f) + F(Bphys -o ~t') ,
(1)
and in terms of the decay amplitudes, Cr is given as (2+z2) (IA 12- I/ll 2) +z2( IA2[ 2 - IA£12) - 2 z ( IA 12 I m 2 - 1312 Im 2-)
G= (2+z2)(IA[2+ I.dl2) +z2( IA212+ IA2-12)-2z( IAI 2 ImR+ IAI 2 lm2-) '
(2)
where A= (flBd°),
-8-=(fig°),
7,-
(fiB °
(l'l B° )
(fiB o
;~--qz, 2 - e 2 , P
q
and Am q F ' p
-
/M'~2- iFT2
(3)
The quark subprocesses 6-,eca, oua and fiug, and their CP-conjugate processes, occur via the tree-level and penguin diagrams. Therefore, (fl B° ) = (fl H, ree IBd° ) + ( f[ Hpenguin IB° ) =A~ exp (i6) --FA2,
(4)
where A~, A2 are the weak amplitudes and d is the strong phase difference between the tree-level amplitude and the penguin one. The effective weak hamiltonian of the tree-level diagram can be expressed as
Ht~ = 2x/2 Gv V*b Vjk [ C, ( qkk rUqjL ) ( q,L)',,bL ) + C2 (q, LYuqjL) (qaLp,,bL) ] , where i, j = u
(5)
or c, k = d or s, and V,b, Vjk are the K M matrix elements. At the scale ~t~rnb, the Q C D scale-
dependent coefficients Cl ~ 1.1, C 2 ~,
-
0.24 [ 6 ]. The effective penguin hamiltonian is given as [ 7 ]
X - ~(~LT~q~)(q~7~b~)+(F~LTuqg)(~7~b~)+
~¢(q]Lq~)(gl~b~)-2(q]kq~)(Ct~b~)
+h.c.,
(6)
where N is the number of colors, i runs on u, c and t, j is d or s, and the greek letters a, fl denote the color indices. The analytic form of the loop integral function I, is presented in section 3. Using the techniques of factorization approximation, 1/N expansion and Fierz transformation, which are used in ref. [ 3], we get the decay amplitudes in terms of one hadronic matrix element, (D+D-IBd°)=(D+D
IH,~e~lB°)+(D+D-IHpongui~lB ° )
+ 1 a~ = ~ 1 Gv{ *V~bVco(CI ~ C 2 ) e x p ( i ~ ) + ~ ( Z
+2
316
1-~
(md+m~)(rnb--m~)
VTbV,o ' , ) [ ( 1 - ~ )
(D+l-dT~,ysc[0)(D
1
le~blB°),
(7a)
Volume 261, number 3
PHYSICS LETTERSB
(
1 {.
30 May 1991
1 ) exp(i~)+~-nos(~ V~Y,dl,)[(,_1)~-~
(1) (md+m.)(mb--mu) ]}
+2
1-~
(7b)
1 o~s ,~ 1 Gv{V*bV,,s(C,+ ~C2)exp(ic~)+~(~V~V, sI,)[(l-~)
(K+n- IB°) =
(m~+m,~)(mb-mu) (K+l-.~7'.7'~u10) n-lab.,biB°).
+2 1 - ~
(7c)
The CP-conjugate decay amplitudes are obtained by replacing the KM matrix elements V~jwith V~, etc. in eq. (7). Note that the final-states D + D - and n + n - are CP eigenstates, but K+n - is not.
3. The loop integral function I; The loop integral function I, in eqs. (6) and ( 7 ) is given as [ 8 ] Ii-
M~v + mU2 ~
~
M~v - m;
(L-J),
(8)
where 1
L= ~x(1-x)
ln[mZ-k2x(l-x)] &r,
(9a)
ln[M2(1-x)+mZx-k2x(1-x)] dx,
(9b)
0
J= ix(l-x) 0
where Mw and m, are the W boson mass and the ith type quark mass, respectively, and k is the momentum transfer carried by the gluon. The functions L and J are given according to the value of k 2 as follows:
( 2m2~k2 (r-l)= jlnkr+l J
L=U-~r 1+
L=U+½r(l+~r 2)arctg(1/r) L=U-~r l + ~ k S - ] l n \ r + l J=V+W+Fln
]
l+R)2 62
[
J=V+W-2F arctg
(k2<0),
(10a)
(0
(10b)
ignr
1+
(k2<~(Mw-m,) 2),
+arctg~-~-JJ
(lla)
(k2>(Mw-mi)2),
(llb)
where U=glnm2-5
2m~
3k2 ' V = ~ l n m 2 +
6sZ+6s--5 18
2M 2 3k 2 ,
(12) 317
Volume 261, number 3
PHYSICS
sM2w~ ( m 2 "~ ~-~.]ln ~ kMwJ ,
W = ( 2S3-~-3S2-- 1 " 12
LETTERS
( _2s2+ s - - 1 F=
12
B
30 M a y 1991
M2w) ~
n
(12 cont'd)
'
and
r=,~ll-4m2,/k2l,
s=
M2w- m , 2 k2 , R=~/l(l+s)2-4M2/k21.
(13)
We should emphasize that eq. ( 1 la) is not the same as eqs. (4d) and (4e) in ref. [8], i.e., the expression of li (eq. ( 8 ) ) is different from that used in refs. [3,8]. In this article, we only care for the timelike penguin amplitudes [ 3 ] and take the naive values of k 2 given by Gerard et al. [ 9 ] : 1 2 2 1 2 ~mb
(14)
4. Numerical analysis and conclusion
Now, we perform the numerical analysis of the CP-violating asymmetries for the typical decays mentioned above. The quark masses are taken as (mu, rod, ms, mc, rob) = (0.005, 0.01, 0.175, 1.5, 5.0) in units of GeV. The top-quark mass is a free parameter, and here we take m , = 100 GeV. The value of C~s is fixed as 0.23, and the 1/N=O limit is adopted [3]. We use the KM matrix parametrized by Chau and Keung [10], where we take s,,=0.046 and s_-/s>.=O.09following from the recent A R G U S and CLEO results [ 11 ]. The KM phase ~ = 150 ° is taken typically. We take for the B°a-BOa system z = 0.7 [ 11 ], and k = 3.5 GeV according to eq. (14). We calculate the changes of the loop integral function I, with m o m e n t u m k, and depict the results in fig. 1. These numerical results can be referred to in the calculations of the timelike penguin amplitudes. In table 1 we present the ratios of the penguin amplitude to the tree-level one for the decays BOa-,D+D -,
•""•"
O.
Imle
Relt
ImIu -1.
_
d
-2. 2.
R('lu
_
_1
3.
4.
5.
Fig. 1. The loop integral function li versus m o m e n t u m k for the t i m e l i k e penguin a m p l i t u d e s in the B-meson system.
k(GeV)
Table 1 Ratios of the penguin a m p l i t u d e to the tree-level one for the decays B ° ~ D + D - , n +rt - and K + n - . In case ( a ) , I m l i is neglected, a n d in case ( b ) , Im I, is retained. Q u a r k process
13~cca b~uud b~uus
318
Final state
D+D n+rtK+rt -
Case (a)
Case (b)
Apengui,/Atree
Aoenguin/Atree
0.135 0.216 3.45
0.144 0.231 3.43
Volume 261, number 3
"~
PHYSICS LETTERS B
] (a)
30 May 1991
.6
K+~-
(b) K+v -
.3
O.
I
_.3
~
-
-
-
-
-
o.
-
7+~ -
-,3
1
-.6
-.6 -180
\
°
-90
°
O °
90 °
180
°
Fig. 2. The CP-violating a s y m m e t ~ ' parameter Cr versus 6 for B ° ~ D + D (b) lm 1, is taken into account.
-180
°
-90 °
I 0 o
90 °
1.80 °
, n+n -, and K+n - in the case of (a) Im I, is neglected, and
n + n - , and K+n -. We find that in the B° -~K+n - decay, the penguin amplitude is larger than the tree-level one, so direct C P violation could be large.
Fig. 2 shows the CP-violating asymmetries Cf via the final-state phase shifts in the case of (a) Im I, is neglected, and (b) Im li is retained. We can see that the asymmetry parameters Cf depend remarkably on 6 for the decays B° ~ K + n -, and n + n , but rather mildly for B° ~ D + D -. The existence o f l m / , transforms the behavior of Cr for each process, and leads to the direct CP-violating asymmetry, which is about 5% in the case of the threegeneration model. Because we have corrected the errors in ref. [ 3 ], our numerical predictions on Cf in fig. 2 are not consistent with those in ref. [3]. We notice that although there is some similarity between Cf for B° D+D - presented in ref. [4] and the one shown in fig. 2a, they were calculated by a different loop integral function I,. Hence, this similarity is just an accident. According to the calculations above, we can conclude that the previous predictions of the CP-violating asymmetries by the standard model [5 ] need a suitable correction if the final-state phase shifts 6 are not zero. In summary, we have made a further study of the effect of final-state phase shifts on the CP-violating asymmetries for the decays B ° ~ D + D -, n + n - , and K+n -, by correcting the errors that appeared in ref. [3] and taking the contribution of the imaginary component of the loop integral function I, into account. We confirm that the existence of the penguin amplitude in addition to the tree-level amplitude can lead to direct CP violation via both the non-vanishing final-state phase shifts and the absorptive part in the penguin amplitude. Therefore, it is necessary to study the final-state interaction if we want to give more reliable predictions of the CP-violating asymmetries.
References [ 1 ] A R G U S Collab., H. Albrecht et al., Phys. Len. B 192 (1987) 245; CLEO Collab., M. Artuso et al., Phys. Rev. Lett. 62 (1989) 2233. [2] I.I. Bigi and A.I. Sanda, Nucl. Phys. B 193 ( 1981 ) 85; Phys. Rev. D 29 (1984) 1393; L. Wolfenstein, Nucl. Phys. B 246 (1984) 45; J.F. Donoghue, T. Nakada, E.A. Paschos and D. Wyler, Phys. Lett. B 195 (1987) 285; D.-S. Du and Z.-Y. Zhao, Phys. Rev. Lett. 59 (1987) 1072; I. Dunietz and R.G. Sachs, Phys. Rev. D 37 (1988) 3186; E.A. Paschos and U. Turke, Phys. Rep. 178 (1989) 145. [3] M. Tanimoto et al., Phys. Rev. D 42 (1990) 252. [4] M. Gronau, Phys. Rev. Lett. 63 (1989) 1451. [ 5 ] D. Du, 1. Dunietz and D. Wu, Phys. Rev. D 34 (1986) 3414. [6] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 29 (1985) 637; 34 (1987) 103.
319
Volume 261, number 3
PHYSICS LETTERS B
[ 7 ] A.I. Vainshtein, V.l. Zakharov and M.A. Shifman, Zh. Eksp. Teor. Fiz. 72 ( 1977 ) 1275; M.B. Gavela et al., Phys. Lett. B 154 (1985) 425; L.-L. Chau and H.-Y. Cheng, Phys. Rev. Lett. 53 (1984) 1037; 59 ( 1987 ) 958; Phys. Lett. B 165 ( 1985 ) 429. [8] M. Tanimoto, Phys. Lett. B 218 (1989) 481; Phys. Rev. Lett. 62 (1989) 2797; M. Tanimoto, T. Shinmoto, K. Hirayama and K. Senba, Phys. Rev. D 40 (1989) 2934. [9] J.M. Gerard and S.-W. Hou, Max-Planck report MPI-PAE/PTH 26/88 ( 1988 ). [ 10 ] L.L. Cbau and W.Y. Keung, Phys. Lett. B 53 (1984) 1804. [ 11 ] ARGUS Collab., Report No. DESY 89-152, 1989 (unpublished); CLEO Collab., R. Fulton et al., Phys. Rev. Lett. 64 (1990) 16.
320
30 May 1991
PHYSICS LETTERS B
30 May 1991
Further study of the effect of final-state phases on CP violation inthe non-leptonic Bj-B, o -o decays Zhi-zhong Xing a,b and Dong-sheng Du
b
CCAST(~brld Laboratory), P.O. Box 8730, 100 080 Beijing, China b Institute ofttigh Energy Physics, Academia Sinica, P.O. Box 918(4), 100 039 Beijing, China
a
Received 7 December 1990; revised manuscript received 31 December 1990
We make a further study of the CP-violating asymmetry for the decays B° - D + D -, n+n -, and K+n -, which have both treelevel and penguin amplitudes. We confirm that the final-state phase shifts can lead to significant direct CP violation for these decays, and find that the absorptive part in the penguin amplitude can also lead to direct CP-violating asymmetry for each decay, which is about 5% in the standard model. The numerical predictions of the CP-violating asymmetries are given against final-state phase shifts, and they are quite different from those by Tanimoto et al. [ Phys. Rev. D 42 (1990) 252 ].
1. Introduction Because the B-mesons are heavy, there are many channels suitable for studying CP-violating phenomena [ 1,2 ]. To observe any C P violation, we require that there should be some interference between two amplitudes. This interference can occur either through B°-I] ° mixing or via final-state interactions, or by a c o m b i n a t i o n o f both effects. In this article, we only consider those non-leptonic neutral-B-meson decays which possess both the treelevel a m p l i t u d e and the loop ( p e n g u i n ) one. T a n i m o t o et al. [ 3 ] have i m p r o v e d G r o n a u ' s study o f this p r o b l e m [4] by investigating the effect o f the final-state phases on CP-violating a s y m m e t r y quantitatively. They drew the remarkable conclusion that the final-state phase shifts play an i m p o r t a n t role for the decays B ° ~ D + D - , n+n - and K + n - . However, there are a few errors in ref. [3 ]: ( i ) The K o b a y a h s i - M a s k a w a matrix factors in the quark subprocesses b-~eca, uud, and ~us, which should be Vc*bVcd, V*b Vud, V*b Vus respectively, were written as Vcb Vc*d, Vub V*ud, Vub V'us in ref. [3 ]. This will cause a reverse sign o f the CP-violating a s y m m e t r y Cr. (ii) The analytic formula o f the loop integral function Ii which were used in ref. [ 3 ] to calculate the penguin amplitudes, in fact, are not correct. This error makes the penguin a m p l i t u d e s considerably larger than the tree-level one, but it is screened in the calculations o f Cf because Cf is expressed as the ratio o f two quantities (see eqs. ( 1 ) and (2) ). (iii) The imaginary part o f the integral function li was neglected and expected to have only a small contribution to Cf in ref. [ 3 ]. In fact, the existence o f Im li can lead to a direct C P violation, thus, it has typical significance. Following G r o n a u and T a n i m o t o ' s idea, here we correct the errors in ref. [ 3 ], and make a further study o f the problem. We also calculate the CP-violating a s y m m e t r y for the case o f a non-vanishing imaginary c o m p o n e n t of L. We find that Im I, can affect the b e h a v i o r o f Cf considerably, and the final-state phases can lead to remarkable direct CP-violating effects for the typical processes B ° - D + D , n + n - , and K + n - . O u r numerical results are d r a m a t i c a l l y different from those in ref. [ 3 ].
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
315
Volume 261, number3
PHYSICSLETTERSB
30May 1991
2. Partial-decay-rate asymmetry The time-integrated CP-violating asymmetry is defined as [2,5 ] F(BOhys__,f) - - F ( g p-o h y s ---~[ ' ) C,-= F(B%y~--f) + F(Bphys -o ~t') ,
(1)
and in terms of the decay amplitudes, Cr is given as (2+z2) (IA 12- I/ll 2) +z2( IA2[ 2 - IA£12) - 2 z ( IA 12 I m 2 - 1312 Im 2-)
G= (2+z2)(IA[2+ I.dl2) +z2( IA212+ IA2-12)-2z( IAI 2 ImR+ IAI 2 lm2-) '
(2)
where A= (flBd°),
-8-=(fig°),
7,-
(fiB °
(l'l B° )
(fiB o
;~--qz, 2 - e 2 , P
q
and Am q F ' p
-
/M'~2- iFT2
(3)
The quark subprocesses 6-,eca, oua and fiug, and their CP-conjugate processes, occur via the tree-level and penguin diagrams. Therefore, (fl B° ) = (fl H, ree IBd° ) + ( f[ Hpenguin IB° ) =A~ exp (i6) --FA2,
(4)
where A~, A2 are the weak amplitudes and d is the strong phase difference between the tree-level amplitude and the penguin one. The effective weak hamiltonian of the tree-level diagram can be expressed as
Ht~ = 2x/2 Gv V*b Vjk [ C, ( qkk rUqjL ) ( q,L)',,bL ) + C2 (q, LYuqjL) (qaLp,,bL) ] , where i, j = u
(5)
or c, k = d or s, and V,b, Vjk are the K M matrix elements. At the scale ~t~rnb, the Q C D scale-
dependent coefficients Cl ~ 1.1, C 2 ~,
-
0.24 [ 6 ]. The effective penguin hamiltonian is given as [ 7 ]
X - ~(~LT~q~)(q~7~b~)+(F~LTuqg)(~7~b~)+
~¢(q]Lq~)(gl~b~)-2(q]kq~)(Ct~b~)
+h.c.,
(6)
where N is the number of colors, i runs on u, c and t, j is d or s, and the greek letters a, fl denote the color indices. The analytic form of the loop integral function I, is presented in section 3. Using the techniques of factorization approximation, 1/N expansion and Fierz transformation, which are used in ref. [ 3], we get the decay amplitudes in terms of one hadronic matrix element, (D+D-IBd°)=(D+D
IH,~e~lB°)+(D+D-IHpongui~lB ° )
+ 1 a~ = ~ 1 Gv{ *V~bVco(CI ~ C 2 ) e x p ( i ~ ) + ~ ( Z
+2
316
1-~
(md+m~)(rnb--m~)
VTbV,o ' , ) [ ( 1 - ~ )
(D+l-dT~,ysc[0)(D
1
le~blB°),
(7a)
Volume 261, number 3
PHYSICS LETTERSB
(
1 {.
30 May 1991
1 ) exp(i~)+~-nos(~ V~Y,dl,)[(,_1)~-~
(1) (md+m.)(mb--mu) ]}
+2
1-~
(7b)
1 o~s ,~ 1 Gv{V*bV,,s(C,+ ~C2)exp(ic~)+~(~V~V, sI,)[(l-~)
(K+n- IB°) =
(m~+m,~)(mb-mu) (K+l-.~7'.7'~u10) n-lab.,biB°).
+2 1 - ~
(7c)
The CP-conjugate decay amplitudes are obtained by replacing the KM matrix elements V~jwith V~, etc. in eq. (7). Note that the final-states D + D - and n + n - are CP eigenstates, but K+n - is not.
3. The loop integral function I; The loop integral function I, in eqs. (6) and ( 7 ) is given as [ 8 ] Ii-
M~v + mU2 ~
~
M~v - m;
(L-J),
(8)
where 1
L= ~x(1-x)
ln[mZ-k2x(l-x)] &r,
(9a)
ln[M2(1-x)+mZx-k2x(1-x)] dx,
(9b)
0
J= ix(l-x) 0
where Mw and m, are the W boson mass and the ith type quark mass, respectively, and k is the momentum transfer carried by the gluon. The functions L and J are given according to the value of k 2 as follows:
( 2m2~k2 (r-l)= jlnkr+l J
L=U-~r 1+
L=U+½r(l+~r 2)arctg(1/r) L=U-~r l + ~ k S - ] l n \ r + l J=V+W+Fln
]
l+R)2 62
[
J=V+W-2F arctg
(k2<0),
(10a)
(0
(10b)
ignr
1+
(k2<~(Mw-m,) 2),
+arctg~-~-JJ
(lla)
(k2>(Mw-mi)2),
(llb)
where U=glnm2-5
2m~
3k2 ' V = ~ l n m 2 +
6sZ+6s--5 18
2M 2 3k 2 ,
(12) 317
Volume 261, number 3
PHYSICS
sM2w~ ( m 2 "~ ~-~.]ln ~ kMwJ ,
W = ( 2S3-~-3S2-- 1 " 12
LETTERS
( _2s2+ s - - 1 F=
12
B
30 M a y 1991
M2w) ~
n
(12 cont'd)
'
and
r=,~ll-4m2,/k2l,
s=
M2w- m , 2 k2 , R=~/l(l+s)2-4M2/k21.
(13)
We should emphasize that eq. ( 1 la) is not the same as eqs. (4d) and (4e) in ref. [8], i.e., the expression of li (eq. ( 8 ) ) is different from that used in refs. [3,8]. In this article, we only care for the timelike penguin amplitudes [ 3 ] and take the naive values of k 2 given by Gerard et al. [ 9 ] : 1 2 2 1 2 ~mb
(14)
4. Numerical analysis and conclusion
Now, we perform the numerical analysis of the CP-violating asymmetries for the typical decays mentioned above. The quark masses are taken as (mu, rod, ms, mc, rob) = (0.005, 0.01, 0.175, 1.5, 5.0) in units of GeV. The top-quark mass is a free parameter, and here we take m , = 100 GeV. The value of C~s is fixed as 0.23, and the 1/N=O limit is adopted [3]. We use the KM matrix parametrized by Chau and Keung [10], where we take s,,=0.046 and s_-/s>.=O.09following from the recent A R G U S and CLEO results [ 11 ]. The KM phase ~ = 150 ° is taken typically. We take for the B°a-BOa system z = 0.7 [ 11 ], and k = 3.5 GeV according to eq. (14). We calculate the changes of the loop integral function I, with m o m e n t u m k, and depict the results in fig. 1. These numerical results can be referred to in the calculations of the timelike penguin amplitudes. In table 1 we present the ratios of the penguin amplitude to the tree-level one for the decays BOa-,D+D -,
•""•"
O.
Imle
Relt
ImIu -1.
_
d
-2. 2.
R('lu
_
_1
3.
4.
5.
Fig. 1. The loop integral function li versus m o m e n t u m k for the t i m e l i k e penguin a m p l i t u d e s in the B-meson system.
k(GeV)
Table 1 Ratios of the penguin a m p l i t u d e to the tree-level one for the decays B ° ~ D + D - , n +rt - and K + n - . In case ( a ) , I m l i is neglected, a n d in case ( b ) , Im I, is retained. Q u a r k process
13~cca b~uud b~uus
318
Final state
D+D n+rtK+rt -
Case (a)
Case (b)
Apengui,/Atree
Aoenguin/Atree
0.135 0.216 3.45
0.144 0.231 3.43
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] (a)
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.6
K+~-
(b) K+v -
.3
O.
I
_.3
~
-
-
-
-
-
o.
-
7+~ -
-,3
1
-.6
-.6 -180
\
°
-90
°
O °
90 °
180
°
Fig. 2. The CP-violating a s y m m e t ~ ' parameter Cr versus 6 for B ° ~ D + D (b) lm 1, is taken into account.
-180
°
-90 °
I 0 o
90 °
1.80 °
, n+n -, and K+n - in the case of (a) Im I, is neglected, and
n + n - , and K+n -. We find that in the B° -~K+n - decay, the penguin amplitude is larger than the tree-level one, so direct C P violation could be large.
Fig. 2 shows the CP-violating asymmetries Cf via the final-state phase shifts in the case of (a) Im I, is neglected, and (b) Im li is retained. We can see that the asymmetry parameters Cf depend remarkably on 6 for the decays B° ~ K + n -, and n + n , but rather mildly for B° ~ D + D -. The existence o f l m / , transforms the behavior of Cr for each process, and leads to the direct CP-violating asymmetry, which is about 5% in the case of the threegeneration model. Because we have corrected the errors in ref. [ 3 ], our numerical predictions on Cf in fig. 2 are not consistent with those in ref. [3]. We notice that although there is some similarity between Cf for B° D+D - presented in ref. [4] and the one shown in fig. 2a, they were calculated by a different loop integral function I,. Hence, this similarity is just an accident. According to the calculations above, we can conclude that the previous predictions of the CP-violating asymmetries by the standard model [5 ] need a suitable correction if the final-state phase shifts 6 are not zero. In summary, we have made a further study of the effect of final-state phase shifts on the CP-violating asymmetries for the decays B ° ~ D + D -, n + n - , and K+n -, by correcting the errors that appeared in ref. [3] and taking the contribution of the imaginary component of the loop integral function I, into account. We confirm that the existence of the penguin amplitude in addition to the tree-level amplitude can lead to direct CP violation via both the non-vanishing final-state phase shifts and the absorptive part in the penguin amplitude. Therefore, it is necessary to study the final-state interaction if we want to give more reliable predictions of the CP-violating asymmetries.
References [ 1 ] A R G U S Collab., H. Albrecht et al., Phys. Len. B 192 (1987) 245; CLEO Collab., M. Artuso et al., Phys. Rev. Lett. 62 (1989) 2233. [2] I.I. Bigi and A.I. Sanda, Nucl. Phys. B 193 ( 1981 ) 85; Phys. Rev. D 29 (1984) 1393; L. Wolfenstein, Nucl. Phys. B 246 (1984) 45; J.F. Donoghue, T. Nakada, E.A. Paschos and D. Wyler, Phys. Lett. B 195 (1987) 285; D.-S. Du and Z.-Y. Zhao, Phys. Rev. Lett. 59 (1987) 1072; I. Dunietz and R.G. Sachs, Phys. Rev. D 37 (1988) 3186; E.A. Paschos and U. Turke, Phys. Rep. 178 (1989) 145. [3] M. Tanimoto et al., Phys. Rev. D 42 (1990) 252. [4] M. Gronau, Phys. Rev. Lett. 63 (1989) 1451. [ 5 ] D. Du, 1. Dunietz and D. Wu, Phys. Rev. D 34 (1986) 3414. [6] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 29 (1985) 637; 34 (1987) 103.
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[ 7 ] A.I. Vainshtein, V.l. Zakharov and M.A. Shifman, Zh. Eksp. Teor. Fiz. 72 ( 1977 ) 1275; M.B. Gavela et al., Phys. Lett. B 154 (1985) 425; L.-L. Chau and H.-Y. Cheng, Phys. Rev. Lett. 53 (1984) 1037; 59 ( 1987 ) 958; Phys. Lett. B 165 ( 1985 ) 429. [8] M. Tanimoto, Phys. Lett. B 218 (1989) 481; Phys. Rev. Lett. 62 (1989) 2797; M. Tanimoto, T. Shinmoto, K. Hirayama and K. Senba, Phys. Rev. D 40 (1989) 2934. [9] J.M. Gerard and S.-W. Hou, Max-Planck report MPI-PAE/PTH 26/88 ( 1988 ). [ 10 ] L.L. Cbau and W.Y. Keung, Phys. Lett. B 53 (1984) 1804. [ 11 ] ARGUS Collab., Report No. DESY 89-152, 1989 (unpublished); CLEO Collab., R. Fulton et al., Phys. Rev. Lett. 64 (1990) 16.
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