- Email: [email protected]
Mixing-induced CP violation in the decay Bd→K0 K̄0 within the standard model
29 December1994 PHYSICS LETTERS B
I:.1~SEVIER
Physics Letters B 341 (1994) 205-212
Mixing-induced CP violation in the decay Ba standard model
K
within the
Robert Fleischer Technische Universitilt Miinchen. Physik Department. lnstitutfiir Theoretische Physik. D-85748 Garching, Germany
Received 15 September 1994 Editor: P.V. Landshoff
Abstract
Recently, flavour SU(3) symmetry of strong interactions has been combined with certain dynamical assumptions to derive triangle relations amonG B-meson decay-amplitudes. We show that these relations allow a prediction of the mixing-induced CP asymmetry ./~x-ina(Bd ----*K°R°). Contrary to statements made in several previous papers, this asymmetry should be non-vanishing in the Standard Model due to QCD-penguins with internal up- and charm-quark exchanges and could be as large as 0(30%). The branching ratio BR(Ba ~ K°K°) is expected to be of O(10-6). In the future, the results presented in this letter should allow interesting tests of the SU(3) triangle relations and of the Standard Model description of CP violation.
CP-violating asymmetries in neutral B-meson decays are of special interest for an experimental test of the Cabibbo-Kobayashi-Maskawa-model (CKMmodel) of CP violation [ 1 ]. In contrast to the situation arising in the charged B-meson system, where only direct CP violation is present, it is a characteristic feature of the neutral B-meson system that also mixinginduced CP violation, which is generated by the interference between B ° - B ° mixing and decay processes, may contribute significantly to the CP-violating asymmetries [ 2 ] - [ 8 ] . The major point of this letter is a prediction of the mixing-induced CP asymmetry of the decay Bd --~ K°K ° originating from the generic QCD-penguin process b ~ dgs. The main inputs are the SU(3) flavour symmetry of strong interactions and certain dynamical assumptions to be specified below. In the pre* Supported by the German Bundesministerium fiir Forschung und Technologieunder contract 06 TM 732.
vious literature, it has been claimed by several authors (see, e.g., Refs. [3,7,8]) that decays such as Bd ---, KsKs or Bd ---, K ° K ° (the CP asymmetries of both channels are equal) were useful modes to test the Standard Model, since it would predict zero CPviolating asymmetries due to a cancellation of weak decay- and mixing-phases. We point out that this statement is only correct, if the QCD-penguin amplitudes are dominated by internal top-quark exchanges. As we shall see, however, QCD-penguins with internal upand charm-quarks may also play an important role and could lead to rather large CP asymmetries of (9( 10-50) %. Therefore, non-vanishing CP-violating asymmetries measured in Bd ---, KsKs (or Bd ~ K ° K °) would not necessarily give hints to contributions from physics beyond the Standard Model as emphasized, e.g., in Ref. [8]. The importance of pure penguininduced b ---, d~s modes in respect of direct CP violation has been emphasized previously by G6rard and Hou [9].
0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSDI 0370-2693 (94)01287-3
206
R. Fleischer/ Physics Letters B 341 (1994) 205-212
If we consider a neutral Bq-decay (q E {d, s}) into a CP self-conjugate final state If), e.g., the transition Bd ---, K° g"°, the time-dependent and time-integrated CP asymmetries are given by
acp(t) = r(Bq°(t) ---'f ) -r(Bq°(t) ---'f ) r(BqO(t) - , f ) + r(Bq°(t) --, f ) Adir (B q "-+ f ) c o s ( a M q t )
= ,.."tCp
+ .mcmipx-ind(Bq---+ f ) s i n ( A M 0 t )
(I)
and fo°°dt[r(B°q(t) ~ f ) - r ( / ~ ° ( t ) - - , f ) ] acp - f o dt [F(BO(t) --, f ) + r ( B o ( t ) - - , f ) ] 1
[ dir
= 1 + X2 .~t~p (Bq -~ f ) + xq.mc~pX'ind(Bq ~ f ) ] ,
(2)
respectively. Here, AMq > 0 is the mass splitting of the physical Bq° - / ~ 0 mixing-eigenstates and xq =7"BqAMq denotes the so-called mixing-parameter. In Eqs. (1) and (2), we have separated the direct CPviolating contributions characterized by q
f ) -1 +
(3)
I q)l 2
from those describing mixing-induced CP violation which are proportional to .~x'ind(Bq ----+f)
=
2Im ~:~q) l +
(4)
I#:>I2.
The phase convention independent quantity ~(:q)contains essentiallyall the information needed to evaluate the CP-violating asymmetries. It is given by
[
~:;q) = exp - i O
:> A(BOq --* f ) ,
(5)
where A(/~ ° --~ f ) and A ( B ° --~ f ) are decay amplitudes and O Mi2 (q) = ~" + 2arg(VtqVtb) -- (bcp(Bq)
(6)
is the B ° - /~0 mixing phase which is a function of the complex phases of the CKM matrix [ 1]. The
phase ~bcp(Bq) arises from our freedom of choosing a CP phase-convention and is defined by the relation ( CP )IB °) = exp[ idpcp( Bq) ] IB°). In the convention independent expression (5), ~bce(Bq) is cancelled by the ratio of decay amplitudes. If a neutral Bq-meson decay into a final CP eigenstate is dominated by a single CKM-amplitude, .A~, vanishes and the uncertainties related to unknown hadronic matrix elements cancel in the mixinginduced asymmetry .Acm~ xi"d. In this very interesting case, .ACm~ x'ind is a theoretical clean measure of the angles appearing in the so-called unitarity triangle (for a recent phenomenological analysis of this triangle see, e.g., Ref. [ 10] ). An example of such a decay is the channel Bd --~ ¢Ks which should allow a very clean determination of the angle fl in the unitarity triangle from the CP asymmetry ./[cmipxind(Bd~ ~PKs) = - sin 2ft. On the other hand, if several amplitudes with both different CP-violating weak and CP-conserving strong phases contribute to a neutral Bq-decay, the hadronic uncertainties do not cancel and a theoretical clean prediction of the CP asymmetries (I) and (2) is a priori not possible. Furthermore, we expect significant direct CP violation. A decay in this category is, e.g., the pure penguin-induced mode Bd ---, K°/( ° [ I 1]. Here, the amplitudes with different weak and strong phases mentioned above arise from QCD-penguins with internal up-, charm- and top-quark exchanges. After this short introduction, let us now come to our main point. Using SU(3) flavour symmetry of strong interactions [ 12]-[ 16] and certain plausible dynamical assumptions (e.g., neglect of annihilation topologies), several triangle relations among B-meson decay amplitudes into 7rrr, 7rK and KK final states have been derived in a recent series of interesting publications [17]-[21]. In this letter, our main intention is to point out that these relations in combination with the recent results of Ref. [22] allow an interesting prediction of the mixing-induced CP-violating asymmetry .~px'ind(Bd ""+ g0/~0). To this end, let us use the same notation as in Refs. [17]-[22] and denote the amplitudes corresponding to b ~ d (b ---+ d) and b ---+ s (b --+ g) QCD-penguins generically by P (P) and P' (P~), respectively. Then, taking into account that (CP)IK°R °) = +IK°R °) and applying the Wolfenstein parametrization [23] of the CKM-matrix, we
207
R. Fleischer/ PhysicsLettersB 341 (1994)205-212 obtain ~(d) Ko~
Combining (7) and (8) gives the expression =
-- exp(--i2fl)/~ ~
(7)
In Refs. [ 1 7 ] - [ 2 1 ] , it has been assumed that the QCD-penguin amplitudes are dominated by internal top-quark exchanges. However, as has been pointed out in [22], sizable contributions may also arise from QCD-penguins with internal up- and charm-quarks. Including these additional penguin amplitudes, which will turn out to be essential for the CP-violating effects arising in the mode B d ~ gO~"0, we find [22] P--- = : P e x p [ i ( 2 f l + ~ b - ~ ) ] , P ,o?
(8)
where pe = 1,/R2
Rt v
- 2Rtlael
cos(# + Sap) + IAPI 2 (9)
and
tan~b --
IAPI s i n ( # + SAP) R, - IAPI cos(B + ~aP)"
(10)
The CP-conjugate quantities #e and ~ can be obtained easily from (9) and (10) by substituting/3 ----, - f t . In Eqs. (9) and (10), Ap describes the contributions of the QCD-penguins with internal u- and c-quarks and is defined by the ratio of strong penguin-amplitudes [221
Pc-P.
Ap _----IAPIexp(it$Ae) ~- p , - p . '
1 IVtdl a IVcbl
~
#e
= - - - exp [iQb - ~ ) ] . PP
(13)
If we consider only QCD-penguins with internal topquark exchanges corresponding to Ap = 0 [ 17 ] - [ 21 ], the weak decay- and mixing-phases cancel each other and we get
¢(d) ( A P = O ) = - I .
(14)
Consequently, the CP asymmetries ( 1 ) and (2) vanish in this approximation. In several previous papers (see, e.g., Ref. [8]), it has been claimed that this result would provide a test of the Standard Model and that observed non-vanishing CP asymmetries would indicate physics beyond the Standard Model. However, within the Standard Model, non-vanishing asymmetries .A~i~(Bd ---+ K° [(°) and ,A~-pX-ind(Bd ~ K° [(O) may arise from QCD-penguin contributions with internal u- and c-quarks. In order to illustrate this statement quantitatively, let us apply, as in Ref. [22], the perturbative approach of Bander, Silverman and Soni [24] - the so-called "BSS-mechanism" - to estimate AP. We will also investigate the expected magnitude of the "average" branching ratio BR(Bd --~ K°/(°) defined by BR(Bd --~ K°K °) O0
=-½/dtEr(B°(t) ---, KOgo) o
(11)
+ V(B°(t) ---, K°/~°)] (d) 2
whereas the CKM-factor
Rt =~
~:(d)
: !I e(1 r 0 +e 0 1 2 (12)
represents one side of the unitarity triangle. Present experimental data imply that Rt is of (.9( 1 ) [ 10]. Note that AP is affected strongly by hadronic uncertainties, in particular by unknown strong final state interaction phases. In the limit of degenerate u- and c-quark masses, Ap would vanish due to the GIM mechanism. However, since m, ~ 4.5 MeV, whereas m¢ "~ 1.3 GeV, this GIM cancellation is incomplete and in principle sizable effects arising from Ap could be expected [22].
) r(B° -~ K ° ~ ) r s , •
(15)
Here, F(B ° --, K°R°) denotes the transition rate of the decay Bd° ~ K°K° which can be obtained by performing the usual phase-space integrations. To simplify the discussion, we neglect the renormalization group evolution from/.t = O ( M w ) down to /x = O(mb) and take into account QCD renormalization effects only approximately through the substitution a , ~ as(/x) (for a discussion of QCDcorrections affecting Bd ---' /t°R °, see Ref. [1 1] ). Then, choosing Rt = 1 and various angles fl, we find the curves shown in Figs. 1-3 describing the dependencies of.Acmiex-ind(Ba ---+K°/(°), .A~(Bd --~ KOh"°)
R. Fleischer / Physics Letters B 341 (1994) 205-212
208
40
30 \
20
//~
~" ,~.
~, ~.
10
;~4¢ ............................................................................ ..-..-.-...-..-.5..?.?.5.5. 0
......... ;?:,'/
i!
2
..y""
, /. /
// /
-10
,. / ,/ ." / / ' // , /
-20
/
/
---___
p=45 ° p=30 °
....
p=20 °
............ p = 1 0 °
/
_ _
p_-0 °
/
-30
/ -40
o12
0.0
ol,
o16
o18
i.o
k2/mb 2
Fig. 1. The dependence o f ,A.~pX-ind(Ba --) K ° K ° ) on ~ / m 2 for
Rt =
I and various values o f the C K M - a n g l e / ~ .
60
50 i
....
i]__45 °
_ _ _
40 .
~.,. ,
\
30
.
.
.
---.__ao o
p-_20 °
............ J]=10 ° ~ ---i~=0o
\ \
--.
\\
\ \
20
\
10
-100. 0
0.2
i
i
0 4
0 6
0.8
.0
k2/n~ 2
Fig. 2. The dependence of J~--p(Bd --) K°K O) on k2/tn~ for Rt = i and various values of the CKM-angle ,8.
R. Fleischer / Physics Letters B 341 (1994) 205-212
209
50
/ ,4
40 /"
\
/
\
/
30
\
/
/
/^
/
~
/ ~"
E
20
/"
/ /
\ \
/*/ ."
•
8=30*
....
~20" ~
- -
I~=0"
\ X
~
13=45°
---
............ ,.=_ I~ t n *
\
\
/* /
_._
\
X
"~.
10 ......
-1%.0
- . . . . ...
.
i
I
I
i
0.2
0.4
0.6
0.8
1.0
k=/rr~2 Fig. 3. The dependence o f the rime-integrated CP asymmetry acp o f the decay Ba --~ K ° K ° on the C K M - a n g l e / 3 .
and of the corresponding time-integrated asymmetry acp, respectively, on the momentum transfer k 2 of the gluons appearing in the usual QCD-penguin diagrams. Applying, in addition to the BSS-mechanism [24], the factorization approximation and the form factors presented by Bauer, Stech and Wirbel [25] to estimate the relevant hadronic matrix elements, we obtain the curves for BR(Bd ---* K°/f °) depicted in Fig. 4. The details of the calculation of AP within the perturbative BSS-mechanism can be found in Ref. [ 22], whereas the evaluation of the branching ratio BR(Bd --* K°/~"°) has been outlined in Ref. [ 11 ]. In drawing Figs. 1-4, we have taken into account that the present range of IV~b/V~bl implies/~ < 45 ° [ 10]. Looking at these figures and choosing k 2 to lie within the "physical" range 1
k2
< m2 <
~/m~for Rt =
] and various values o f
(xd = 0 . 7 . )
1
(16)
which follows from simple kinematical considerations at the quark level, we expect rather large asymmetries of the order (10-50)%, which are quite promising from the experimental point of view, and BR(Bd ---* K°/(O) = (9(10-6). We are aware of the fact that the
numerical estimates given here are very rough. They illustrate, however, the expected orders of magnitude. After this short quantitative illustration, let us now turn to the prediction of the mixing-induced CP asymmetry . m ~ p x ' i n d ( B d ~ g 0 j ~ 0 ) . Using two rates that determine IPI and IP'[, e.g., those of the modes B + ---* K+/~ ° and B + ---*1r+K°, respectively, and the triangle relations [ 19]-[21 ] A ( B ° ---* zr+ ~r- ) -t- V/2A ( B ° ---* 7r° ~r°) = v ~ A ( B + ---, zt+zr O)
(T -t- P ) 4- (C - P ) = (T 4- C)
(17)
and A ( B o --, 7 r - K + ) /ru + v / 2 A ( B ° ---, ~.°K°) / r u = v ~ A ( B + --, rr+zrO) ( T 4- P ' / r u ) 4- ( C - P ' / r u ) = ( T 4- C ) ,
(18)
where ru = Vus/Vud and T ( C ) refers to the "tree" ("colour-suppressed") amplitude of the decay B + ---, 7r+'n"°, the relative angle O between P and P ' can be measured. In contrast to the assertions made in [ 19]-
210
R. Fleischer / Physics Letters B 341 (1994) 205-212
1.1
r
1.0
o.9
N
0"50.0
i
~x
~ O°
I,'~
............ ~ 1 0 °
1"5\ ~ ,y~\
.... ___
1~20 ° i~ao o
/"
0[2
0[4
016
018
1.0
k2/mb2 Fig. 4. The dependence of the "average" branching ratio BR(Bd -+ K°/~0) defined by Eq. (15) on of the CKM-angle/3. (~'~a = 1.6 ps, ]Vcbl= 0.04, A(4.~) = 0.3 GeV). MS
[21 ], a9 is not equal to the CKM-angle/3, but receives some hadronic corrections [22]:
which yields
.A~(Ba -~ K°~)
0=/3+0 -0'.
(20)
If we consider in addition the corresponding CPconjugate processes, the angle = -/3 + 6
-
~b'
IPI2 - Ip12 IPI 2 + iP12
(24)
and ,A~x'ind(Bd ~ K°I~-'0 )
IAPI sin 6ap
tan ~b' = 1 -- ~ ' / ~ c--~s6rip "
=
(19)
Here, ~p' is a pure strong phase that is given by [22]
k2/m~ for Rt = I and various values
(21)
= 21PIIP I sin(2/3 + O - O)
(25)
IPl = + IPI 2 Note that Eq. (24) implies that ~ ( B d ---* K°KO) should be equal to the CP-violating asymmetry
can be determined as well. Note that 6 t = ~1, since no weak phases are present in (20). Consequently, combining (19) and (21) appropriately, we find
acp(B :l: --+ K:LKo)
2 / 3 + 0 - z 9 = 6 -~P
The expression (25) for the mixing-induced CP asymmetry .Z~x-ind(Ba ~ K°/~°) is interesting in two respects: i) If 2/3 is known, the triangle relations (17) and (18) (and those of the corresponding CPconjugate processes) allow a prediction of the
(22)
and ~KORO ~(a) can be expressed as ~:(a) xo~-
IPl exp [_i(2/3 + ~ _ O)] IPI
(23)
=
F(B + ~ K+/~°) - F ( B - ~ K - K °) F(B + --~ K+K°) + F ( B - ~ K - K ° ) "
(26)
R. Fleischer / Physics Letters B 341 (1994) 205-212
CP-violating asymmetry .A~p mix-ind(Bd --, K°~°). In principle, the angle fl can be determined from the SU(3) triangle relations as well [ 19]-[21 ]. However, irrespectively of SU(3)-breaking effects and certain neglected diagrams (annihilation topologies, etc.), the QCD-penguin contributions with internal u- and c-quarks preclude a clean determination of fl by using the branching ratios only (see Eqs. (19) and (21)). As has been pointed out in Ref. [22], this difficulty can be overcome by measuring in addition the ratio Xd/X.~ of B ° - / ~ 0 to Bs° - / ~ 0 mixings to obtain the CKM-parameter Rt. The probably cleanest way of determining 2fl is the measurement of the mixing-induced CP asymmetry Z CP mix'ind(~ Bd ~ ~PKs) An interesting possibility to extract sin 2/3 from the two branching ratios BR(K + --~ ¢r+v~) and BR(KL ~ ¢r°v~) has been proposed recently by Buchalla and Buras [26]. ii) If one d e t e r m i n e s ¢4Cm~x-ind(Ba ~ gO/~"0) from measurements of the corresponding CPviolating asymmetries and 0, O from the twotriangle constructions outlined in Refs. [ 19][21 ], the hadronic uncertainties arising in (19) and (21) from QCD-penguins with internal u- and c-quarks affecting the extraction of the CKM-angle fl can be eliminated and another clean determination of,8 is possible (up to corrections related to SU(3)-breaking and certain neglected diagrams). In summary, we have shown that the CP-violating asymmetries "cpAdir(Bd --~ K°/~°) and "-'cpAmix-ind(Bd ----+ K°/~"°) are generated within the framework of the Standard Model by QCD-penguins with internal up'and charm-quark exchanges and are, thus, interesting quantities to obtain experimental insights into the physics of these contributions. Estimates obtained by applying the perturbative approach of Bander, Silverman and Soni give rather promising asymmetries of the order (10-50) % depending strongly on the angle fl and branching ratios at the 10 - 6 level. Therefore, measured non-vanishing CP asymmetries would not necessarily imply physics beyond the Standard Model as claimed in several previous papers. While the direct CP-violating asymmetry ,A~,(Bd ---+K°K °) should be equal to the one arising in the charged B-decay B ~ --+ K+/~, the mixing-
211
induced CP asymmetry ,ACm~x-ind(Bd ---+ gOj~0) can be predicted by using triangle relations among B-meson decay-amplitudes which follow from the SU(3) flavour symmetry of strong interactions and certain dynamical assumptions. These assumptions consist, e.g., of neglecting annihilation-like topologies. Uncertainties related to SU(3)-breaking effects have been discussed in [ 17]-[22]. Additional corrections to the triangle relations could also arise from electroweak penguins which may, in the presence of a heavy top-quark, lead to sizable contributions to the penguin sectors of B-decays into final states containing mesons with CP-self-conjugate quark contents [27][29]. Therefore, experimental tests of the validity of the triangle relations (17) and (18) are desirable. In the future, when it will hopefully be possible to measure CP-violating asymmetries arising in the decay Bd ---, K°K °, the results presented in this letter should provide such a test of the SU(3) triangle relations and, moreover, of our understanding of CP violation. It is a great pleasure to thank Andrzej Buras for constant encouragement, useful discussions and a careful reading of the manuscript. References [ 1] M. Kobayashi and T. Maskawa, Progr. Theor. Phys. 49 (1973) 652. 121 I. Dunietz and J. Rosner, Phys. Rev. D 34 (1986) 1404. 13] D. London and R.D. Peccei, Phys. Lett. B 223 (1989) 257. [4] M. Gronau, Phys. Rev. Lett. 63 (1989) 1451. [5] I.I. Bigi, V.A. Khoze, N.G. Uraltsev and A.I. Sanda, in CPViolation, edited by C. Jadskog (World Scientific, Singapore, 1989). 161 B. Grinstein, Phys. Letl. B 229 (1989) 280. 171 Y. Nir, SLAC report No. SLAC-PUB-5874 (1992). [81 H.R. Quinn, Nucl. Phys. B (Proc. Suppl.) A 37 (1994) 21. [91 J.-M. G6rard and W.-S. Hou, Phys. Rev. D 43 ( 1991 ) 2902; Phys. Lett. B 253 (1991) 478. [10] A.J. Buras, M.E. Lautenbacher and G. Ostermaier0 Phys. Rev. D 50 (1994) 3433. I l l l R. Fleischer, Z. Phys. C 58 (1993) 483. [121 D. Zeppenfeld, Z. Phys. C 8 (1981) 77. 113] M. Savage and M.B. Wise, Phys. Rev. D 39 (1989) 3346; D 40 (1989) 3127 (E). [14] L.L. Chau et al., Phys. Rev. D 43 (1991) 2176. I 151 J. Silva and L. Wolfenstein, Phys. Rev. D 49 (1994) R 115 I. [161 T. Hayashi, M. Matsuda and M. Tanimoto, Phys. Lett. B 323 (1994) 78.
212
R. Fleischer / Physics Letters B 341 (1994) 205-212
[ 17] M. Gronau, J.L. Rosner and D. London, Phys. Rev. Lett. 73 (1994) 21. [18] M. Gronau, O.E Hem~mdez, D. London and J.L. Rosner, Phys. Rev. D 50 (1994) 4529. [19] O.E Hem/mdez, D. London, M. Gronau and J.L. Rosner, Phys. Lett. B 333 (1994) 500. [201 O.E Hem~ndez and D. London, Universit6 de M o n ~ preprint UdeM-LPN-TH-94-198, hep-ph/9406418 (1994). [211 D. London, Universit6 de Montr6al preprint UdeM-LPN-TH94-199, hep-ph/9406412 (1994). [22] A.J. Buras and R. Fleischer, Technical University Munich preprint TUM-T31-69/94, hep-ph/9409244 (1994).
[23] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [24] M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43 (1979) 242. [251 M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103. 1261 G. Buchalla and A.J. Bur'as, Phys. Lett. B 333 (1994) 221. [271 I. Dunietz, Phys. Lett. B 316 (1993) 561. [281 R. Fleischer, Z. Phys. C 62 (1994) 81; Phys. Lett. B 321 (1994) 259; B 332 (1994) 419. [291 N.G. Deshpande and X.-G. He, Phys. Lett. B 336 (1994) 471; University of Oregon preprint OITS-553, hepph/9408404 (1994).
I:.1~SEVIER
Physics Letters B 341 (1994) 205-212
Mixing-induced CP violation in the decay Ba standard model
K
within the
Robert Fleischer Technische Universitilt Miinchen. Physik Department. lnstitutfiir Theoretische Physik. D-85748 Garching, Germany
Received 15 September 1994 Editor: P.V. Landshoff
Abstract
Recently, flavour SU(3) symmetry of strong interactions has been combined with certain dynamical assumptions to derive triangle relations amonG B-meson decay-amplitudes. We show that these relations allow a prediction of the mixing-induced CP asymmetry ./~x-ina(Bd ----*K°R°). Contrary to statements made in several previous papers, this asymmetry should be non-vanishing in the Standard Model due to QCD-penguins with internal up- and charm-quark exchanges and could be as large as 0(30%). The branching ratio BR(Ba ~ K°K°) is expected to be of O(10-6). In the future, the results presented in this letter should allow interesting tests of the SU(3) triangle relations and of the Standard Model description of CP violation.
CP-violating asymmetries in neutral B-meson decays are of special interest for an experimental test of the Cabibbo-Kobayashi-Maskawa-model (CKMmodel) of CP violation [ 1 ]. In contrast to the situation arising in the charged B-meson system, where only direct CP violation is present, it is a characteristic feature of the neutral B-meson system that also mixinginduced CP violation, which is generated by the interference between B ° - B ° mixing and decay processes, may contribute significantly to the CP-violating asymmetries [ 2 ] - [ 8 ] . The major point of this letter is a prediction of the mixing-induced CP asymmetry of the decay Bd --~ K°K ° originating from the generic QCD-penguin process b ~ dgs. The main inputs are the SU(3) flavour symmetry of strong interactions and certain dynamical assumptions to be specified below. In the pre* Supported by the German Bundesministerium fiir Forschung und Technologieunder contract 06 TM 732.
vious literature, it has been claimed by several authors (see, e.g., Refs. [3,7,8]) that decays such as Bd ---, KsKs or Bd ---, K ° K ° (the CP asymmetries of both channels are equal) were useful modes to test the Standard Model, since it would predict zero CPviolating asymmetries due to a cancellation of weak decay- and mixing-phases. We point out that this statement is only correct, if the QCD-penguin amplitudes are dominated by internal top-quark exchanges. As we shall see, however, QCD-penguins with internal upand charm-quarks may also play an important role and could lead to rather large CP asymmetries of (9( 10-50) %. Therefore, non-vanishing CP-violating asymmetries measured in Bd ---, KsKs (or Bd ~ K ° K °) would not necessarily give hints to contributions from physics beyond the Standard Model as emphasized, e.g., in Ref. [8]. The importance of pure penguininduced b ---, d~s modes in respect of direct CP violation has been emphasized previously by G6rard and Hou [9].
0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSDI 0370-2693 (94)01287-3
206
R. Fleischer/ Physics Letters B 341 (1994) 205-212
If we consider a neutral Bq-decay (q E {d, s}) into a CP self-conjugate final state If), e.g., the transition Bd ---, K° g"°, the time-dependent and time-integrated CP asymmetries are given by
acp(t) = r(Bq°(t) ---'f ) -r(Bq°(t) ---'f ) r(BqO(t) - , f ) + r(Bq°(t) --, f ) Adir (B q "-+ f ) c o s ( a M q t )
= ,.."tCp
+ .mcmipx-ind(Bq---+ f ) s i n ( A M 0 t )
(I)
and fo°°dt[r(B°q(t) ~ f ) - r ( / ~ ° ( t ) - - , f ) ] acp - f o dt [F(BO(t) --, f ) + r ( B o ( t ) - - , f ) ] 1
[ dir
= 1 + X2 .~t~p (Bq -~ f ) + xq.mc~pX'ind(Bq ~ f ) ] ,
(2)
respectively. Here, AMq > 0 is the mass splitting of the physical Bq° - / ~ 0 mixing-eigenstates and xq =7"BqAMq denotes the so-called mixing-parameter. In Eqs. (1) and (2), we have separated the direct CPviolating contributions characterized by q
f ) -1 +
(3)
I q)l 2
from those describing mixing-induced CP violation which are proportional to .~x'ind(Bq ----+f)
=
2Im ~:~q) l +
(4)
I#:>I2.
The phase convention independent quantity ~(:q)contains essentiallyall the information needed to evaluate the CP-violating asymmetries. It is given by
[
~:;q) = exp - i O
:> A(BOq --* f ) ,
(5)
where A(/~ ° --~ f ) and A ( B ° --~ f ) are decay amplitudes and O Mi2 (q) = ~" + 2arg(VtqVtb) -- (bcp(Bq)
(6)
is the B ° - /~0 mixing phase which is a function of the complex phases of the CKM matrix [ 1]. The
phase ~bcp(Bq) arises from our freedom of choosing a CP phase-convention and is defined by the relation ( CP )IB °) = exp[ idpcp( Bq) ] IB°). In the convention independent expression (5), ~bce(Bq) is cancelled by the ratio of decay amplitudes. If a neutral Bq-meson decay into a final CP eigenstate is dominated by a single CKM-amplitude, .A~, vanishes and the uncertainties related to unknown hadronic matrix elements cancel in the mixinginduced asymmetry .Acm~ xi"d. In this very interesting case, .ACm~ x'ind is a theoretical clean measure of the angles appearing in the so-called unitarity triangle (for a recent phenomenological analysis of this triangle see, e.g., Ref. [ 10] ). An example of such a decay is the channel Bd --~ ¢Ks which should allow a very clean determination of the angle fl in the unitarity triangle from the CP asymmetry ./[cmipxind(Bd~ ~PKs) = - sin 2ft. On the other hand, if several amplitudes with both different CP-violating weak and CP-conserving strong phases contribute to a neutral Bq-decay, the hadronic uncertainties do not cancel and a theoretical clean prediction of the CP asymmetries (I) and (2) is a priori not possible. Furthermore, we expect significant direct CP violation. A decay in this category is, e.g., the pure penguin-induced mode Bd ---, K°/( ° [ I 1]. Here, the amplitudes with different weak and strong phases mentioned above arise from QCD-penguins with internal up-, charm- and top-quark exchanges. After this short introduction, let us now come to our main point. Using SU(3) flavour symmetry of strong interactions [ 12]-[ 16] and certain plausible dynamical assumptions (e.g., neglect of annihilation topologies), several triangle relations among B-meson decay amplitudes into 7rrr, 7rK and KK final states have been derived in a recent series of interesting publications [17]-[21]. In this letter, our main intention is to point out that these relations in combination with the recent results of Ref. [22] allow an interesting prediction of the mixing-induced CP-violating asymmetry .~px'ind(Bd ""+ g0/~0). To this end, let us use the same notation as in Refs. [17]-[22] and denote the amplitudes corresponding to b ~ d (b ---+ d) and b ---+ s (b --+ g) QCD-penguins generically by P (P) and P' (P~), respectively. Then, taking into account that (CP)IK°R °) = +IK°R °) and applying the Wolfenstein parametrization [23] of the CKM-matrix, we
207
R. Fleischer/ PhysicsLettersB 341 (1994)205-212 obtain ~(d) Ko~
Combining (7) and (8) gives the expression =
-- exp(--i2fl)/~ ~
(7)
In Refs. [ 1 7 ] - [ 2 1 ] , it has been assumed that the QCD-penguin amplitudes are dominated by internal top-quark exchanges. However, as has been pointed out in [22], sizable contributions may also arise from QCD-penguins with internal up- and charm-quarks. Including these additional penguin amplitudes, which will turn out to be essential for the CP-violating effects arising in the mode B d ~ gO~"0, we find [22] P--- = : P e x p [ i ( 2 f l + ~ b - ~ ) ] , P ,o?
(8)
where pe = 1,/R2
Rt v
- 2Rtlael
cos(# + Sap) + IAPI 2 (9)
and
tan~b --
IAPI s i n ( # + SAP) R, - IAPI cos(B + ~aP)"
(10)
The CP-conjugate quantities #e and ~ can be obtained easily from (9) and (10) by substituting/3 ----, - f t . In Eqs. (9) and (10), Ap describes the contributions of the QCD-penguins with internal u- and c-quarks and is defined by the ratio of strong penguin-amplitudes [221
Pc-P.
Ap _----IAPIexp(it$Ae) ~- p , - p . '
1 IVtdl a IVcbl
~
#e
= - - - exp [iQb - ~ ) ] . PP
(13)
If we consider only QCD-penguins with internal topquark exchanges corresponding to Ap = 0 [ 17 ] - [ 21 ], the weak decay- and mixing-phases cancel each other and we get
¢(d) ( A P = O ) = - I .
(14)
Consequently, the CP asymmetries ( 1 ) and (2) vanish in this approximation. In several previous papers (see, e.g., Ref. [8]), it has been claimed that this result would provide a test of the Standard Model and that observed non-vanishing CP asymmetries would indicate physics beyond the Standard Model. However, within the Standard Model, non-vanishing asymmetries .A~i~(Bd ---+ K° [(°) and ,A~-pX-ind(Bd ~ K° [(O) may arise from QCD-penguin contributions with internal u- and c-quarks. In order to illustrate this statement quantitatively, let us apply, as in Ref. [22], the perturbative approach of Bander, Silverman and Soni [24] - the so-called "BSS-mechanism" - to estimate AP. We will also investigate the expected magnitude of the "average" branching ratio BR(Bd --~ K°/(°) defined by BR(Bd --~ K°K °) O0
=-½/dtEr(B°(t) ---, KOgo) o
(11)
+ V(B°(t) ---, K°/~°)] (d) 2
whereas the CKM-factor
Rt =~
~:(d)
: !I e(1 r 0 +e 0 1 2 (12)
represents one side of the unitarity triangle. Present experimental data imply that Rt is of (.9( 1 ) [ 10]. Note that AP is affected strongly by hadronic uncertainties, in particular by unknown strong final state interaction phases. In the limit of degenerate u- and c-quark masses, Ap would vanish due to the GIM mechanism. However, since m, ~ 4.5 MeV, whereas m¢ "~ 1.3 GeV, this GIM cancellation is incomplete and in principle sizable effects arising from Ap could be expected [22].
) r(B° -~ K ° ~ ) r s , •
(15)
Here, F(B ° --, K°R°) denotes the transition rate of the decay Bd° ~ K°K° which can be obtained by performing the usual phase-space integrations. To simplify the discussion, we neglect the renormalization group evolution from/.t = O ( M w ) down to /x = O(mb) and take into account QCD renormalization effects only approximately through the substitution a , ~ as(/x) (for a discussion of QCDcorrections affecting Bd ---' /t°R °, see Ref. [1 1] ). Then, choosing Rt = 1 and various angles fl, we find the curves shown in Figs. 1-3 describing the dependencies of.Acmiex-ind(Ba ---+K°/(°), .A~(Bd --~ KOh"°)
R. Fleischer / Physics Letters B 341 (1994) 205-212
208
40
30 \
20
//~
~" ,~.
~, ~.
10
;~4¢ ............................................................................ ..-..-.-...-..-.5..?.?.5.5. 0
......... ;?:,'/
i!
2
..y""
, /. /
// /
-10
,. / ,/ ." / / ' // , /
-20
/
/
---___
p=45 ° p=30 °
....
p=20 °
............ p = 1 0 °
/
_ _
p_-0 °
/
-30
/ -40
o12
0.0
ol,
o16
o18
i.o
k2/mb 2
Fig. 1. The dependence o f ,A.~pX-ind(Ba --) K ° K ° ) on ~ / m 2 for
Rt =
I and various values o f the C K M - a n g l e / ~ .
60
50 i
....
i]__45 °
_ _ _
40 .
~.,. ,
\
30
.
.
.
---.__ao o
p-_20 °
............ J]=10 ° ~ ---i~=0o
\ \
--.
\\
\ \
20
\
10
-100. 0
0.2
i
i
0 4
0 6
0.8
.0
k2/n~ 2
Fig. 2. The dependence of J~--p(Bd --) K°K O) on k2/tn~ for Rt = i and various values of the CKM-angle ,8.
R. Fleischer / Physics Letters B 341 (1994) 205-212
209
50
/ ,4
40 /"
\
/
\
/
30
\
/
/
/^
/
~
/ ~"
E
20
/"
/ /
\ \
/*/ ."
•
8=30*
....
~20" ~
- -
I~=0"
\ X
~
13=45°
---
............ ,.=_ I~ t n *
\
\
/* /
_._
\
X
"~.
10 ......
-1%.0
- . . . . ...
.
i
I
I
i
0.2
0.4
0.6
0.8
1.0
k=/rr~2 Fig. 3. The dependence o f the rime-integrated CP asymmetry acp o f the decay Ba --~ K ° K ° on the C K M - a n g l e / 3 .
and of the corresponding time-integrated asymmetry acp, respectively, on the momentum transfer k 2 of the gluons appearing in the usual QCD-penguin diagrams. Applying, in addition to the BSS-mechanism [24], the factorization approximation and the form factors presented by Bauer, Stech and Wirbel [25] to estimate the relevant hadronic matrix elements, we obtain the curves for BR(Bd ---* K°/f °) depicted in Fig. 4. The details of the calculation of AP within the perturbative BSS-mechanism can be found in Ref. [ 22], whereas the evaluation of the branching ratio BR(Bd --* K°/~"°) has been outlined in Ref. [ 11 ]. In drawing Figs. 1-4, we have taken into account that the present range of IV~b/V~bl implies/~ < 45 ° [ 10]. Looking at these figures and choosing k 2 to lie within the "physical" range 1
k2
< m2 <
~/m~for Rt =
] and various values o f
(xd = 0 . 7 . )
1
(16)
which follows from simple kinematical considerations at the quark level, we expect rather large asymmetries of the order (10-50)%, which are quite promising from the experimental point of view, and BR(Bd ---* K°/(O) = (9(10-6). We are aware of the fact that the
numerical estimates given here are very rough. They illustrate, however, the expected orders of magnitude. After this short quantitative illustration, let us now turn to the prediction of the mixing-induced CP asymmetry . m ~ p x ' i n d ( B d ~ g 0 j ~ 0 ) . Using two rates that determine IPI and IP'[, e.g., those of the modes B + ---* K+/~ ° and B + ---*1r+K°, respectively, and the triangle relations [ 19]-[21 ] A ( B ° ---* zr+ ~r- ) -t- V/2A ( B ° ---* 7r° ~r°) = v ~ A ( B + ---, zt+zr O)
(T -t- P ) 4- (C - P ) = (T 4- C)
(17)
and A ( B o --, 7 r - K + ) /ru + v / 2 A ( B ° ---, ~.°K°) / r u = v ~ A ( B + --, rr+zrO) ( T 4- P ' / r u ) 4- ( C - P ' / r u ) = ( T 4- C ) ,
(18)
where ru = Vus/Vud and T ( C ) refers to the "tree" ("colour-suppressed") amplitude of the decay B + ---, 7r+'n"°, the relative angle O between P and P ' can be measured. In contrast to the assertions made in [ 19]-
210
R. Fleischer / Physics Letters B 341 (1994) 205-212
1.1
r
1.0
o.9
N
0"50.0
i
~x
~ O°
I,'~
............ ~ 1 0 °
1"5\ ~ ,y~\
.... ___
1~20 ° i~ao o
/"
0[2
0[4
016
018
1.0
k2/mb2 Fig. 4. The dependence of the "average" branching ratio BR(Bd -+ K°/~0) defined by Eq. (15) on of the CKM-angle/3. (~'~a = 1.6 ps, ]Vcbl= 0.04, A(4.~) = 0.3 GeV). MS
[21 ], a9 is not equal to the CKM-angle/3, but receives some hadronic corrections [22]:
which yields
.A~(Ba -~ K°~)
0=/3+0 -0'.
(20)
If we consider in addition the corresponding CPconjugate processes, the angle = -/3 + 6
-
~b'
IPI2 - Ip12 IPI 2 + iP12
(24)
and ,A~x'ind(Bd ~ K°I~-'0 )
IAPI sin 6ap
tan ~b' = 1 -- ~ ' / ~ c--~s6rip "
=
(19)
Here, ~p' is a pure strong phase that is given by [22]
k2/m~ for Rt = I and various values
(21)
= 21PIIP I sin(2/3 + O - O)
(25)
IPl = + IPI 2 Note that Eq. (24) implies that ~ ( B d ---* K°KO) should be equal to the CP-violating asymmetry
can be determined as well. Note that 6 t = ~1, since no weak phases are present in (20). Consequently, combining (19) and (21) appropriately, we find
acp(B :l: --+ K:LKo)
2 / 3 + 0 - z 9 = 6 -~P
The expression (25) for the mixing-induced CP asymmetry .Z~x-ind(Ba ~ K°/~°) is interesting in two respects: i) If 2/3 is known, the triangle relations (17) and (18) (and those of the corresponding CPconjugate processes) allow a prediction of the
(22)
and ~KORO ~(a) can be expressed as ~:(a) xo~-
IPl exp [_i(2/3 + ~ _ O)] IPI
(23)
=
F(B + ~ K+/~°) - F ( B - ~ K - K °) F(B + --~ K+K°) + F ( B - ~ K - K ° ) "
(26)
R. Fleischer / Physics Letters B 341 (1994) 205-212
CP-violating asymmetry .A~p mix-ind(Bd --, K°~°). In principle, the angle fl can be determined from the SU(3) triangle relations as well [ 19]-[21 ]. However, irrespectively of SU(3)-breaking effects and certain neglected diagrams (annihilation topologies, etc.), the QCD-penguin contributions with internal u- and c-quarks preclude a clean determination of fl by using the branching ratios only (see Eqs. (19) and (21)). As has been pointed out in Ref. [22], this difficulty can be overcome by measuring in addition the ratio Xd/X.~ of B ° - / ~ 0 to Bs° - / ~ 0 mixings to obtain the CKM-parameter Rt. The probably cleanest way of determining 2fl is the measurement of the mixing-induced CP asymmetry Z CP mix'ind(~ Bd ~ ~PKs) An interesting possibility to extract sin 2/3 from the two branching ratios BR(K + --~ ¢r+v~) and BR(KL ~ ¢r°v~) has been proposed recently by Buchalla and Buras [26]. ii) If one d e t e r m i n e s ¢4Cm~x-ind(Ba ~ gO/~"0) from measurements of the corresponding CPviolating asymmetries and 0, O from the twotriangle constructions outlined in Refs. [ 19][21 ], the hadronic uncertainties arising in (19) and (21) from QCD-penguins with internal u- and c-quarks affecting the extraction of the CKM-angle fl can be eliminated and another clean determination of,8 is possible (up to corrections related to SU(3)-breaking and certain neglected diagrams). In summary, we have shown that the CP-violating asymmetries "cpAdir(Bd --~ K°/~°) and "-'cpAmix-ind(Bd ----+ K°/~"°) are generated within the framework of the Standard Model by QCD-penguins with internal up'and charm-quark exchanges and are, thus, interesting quantities to obtain experimental insights into the physics of these contributions. Estimates obtained by applying the perturbative approach of Bander, Silverman and Soni give rather promising asymmetries of the order (10-50) % depending strongly on the angle fl and branching ratios at the 10 - 6 level. Therefore, measured non-vanishing CP asymmetries would not necessarily imply physics beyond the Standard Model as claimed in several previous papers. While the direct CP-violating asymmetry ,A~,(Bd ---+K°K °) should be equal to the one arising in the charged B-decay B ~ --+ K+/~, the mixing-
211
induced CP asymmetry ,ACm~x-ind(Bd ---+ gOj~0) can be predicted by using triangle relations among B-meson decay-amplitudes which follow from the SU(3) flavour symmetry of strong interactions and certain dynamical assumptions. These assumptions consist, e.g., of neglecting annihilation-like topologies. Uncertainties related to SU(3)-breaking effects have been discussed in [ 17]-[22]. Additional corrections to the triangle relations could also arise from electroweak penguins which may, in the presence of a heavy top-quark, lead to sizable contributions to the penguin sectors of B-decays into final states containing mesons with CP-self-conjugate quark contents [27][29]. Therefore, experimental tests of the validity of the triangle relations (17) and (18) are desirable. In the future, when it will hopefully be possible to measure CP-violating asymmetries arising in the decay Bd ---, K°K °, the results presented in this letter should provide such a test of the SU(3) triangle relations and, moreover, of our understanding of CP violation. It is a great pleasure to thank Andrzej Buras for constant encouragement, useful discussions and a careful reading of the manuscript. References [ 1] M. Kobayashi and T. Maskawa, Progr. Theor. Phys. 49 (1973) 652. 121 I. Dunietz and J. Rosner, Phys. Rev. D 34 (1986) 1404. 13] D. London and R.D. Peccei, Phys. Lett. B 223 (1989) 257. [4] M. Gronau, Phys. Rev. Lett. 63 (1989) 1451. [5] I.I. Bigi, V.A. Khoze, N.G. Uraltsev and A.I. Sanda, in CPViolation, edited by C. Jadskog (World Scientific, Singapore, 1989). 161 B. Grinstein, Phys. Letl. B 229 (1989) 280. 171 Y. Nir, SLAC report No. SLAC-PUB-5874 (1992). [81 H.R. Quinn, Nucl. Phys. B (Proc. Suppl.) A 37 (1994) 21. [91 J.-M. G6rard and W.-S. Hou, Phys. Rev. D 43 ( 1991 ) 2902; Phys. Lett. B 253 (1991) 478. [10] A.J. Buras, M.E. Lautenbacher and G. Ostermaier0 Phys. Rev. D 50 (1994) 3433. I l l l R. Fleischer, Z. Phys. C 58 (1993) 483. [121 D. Zeppenfeld, Z. Phys. C 8 (1981) 77. 113] M. Savage and M.B. Wise, Phys. Rev. D 39 (1989) 3346; D 40 (1989) 3127 (E). [14] L.L. Chau et al., Phys. Rev. D 43 (1991) 2176. I 151 J. Silva and L. Wolfenstein, Phys. Rev. D 49 (1994) R 115 I. [161 T. Hayashi, M. Matsuda and M. Tanimoto, Phys. Lett. B 323 (1994) 78.
212
R. Fleischer / Physics Letters B 341 (1994) 205-212
[ 17] M. Gronau, J.L. Rosner and D. London, Phys. Rev. Lett. 73 (1994) 21. [18] M. Gronau, O.E Hem~mdez, D. London and J.L. Rosner, Phys. Rev. D 50 (1994) 4529. [19] O.E Hem/mdez, D. London, M. Gronau and J.L. Rosner, Phys. Lett. B 333 (1994) 500. [201 O.E Hem~ndez and D. London, Universit6 de M o n ~ preprint UdeM-LPN-TH-94-198, hep-ph/9406418 (1994). [211 D. London, Universit6 de Montr6al preprint UdeM-LPN-TH94-199, hep-ph/9406412 (1994). [22] A.J. Buras and R. Fleischer, Technical University Munich preprint TUM-T31-69/94, hep-ph/9409244 (1994).
[23] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [24] M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43 (1979) 242. [251 M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103. 1261 G. Buchalla and A.J. Bur'as, Phys. Lett. B 333 (1994) 221. [271 I. Dunietz, Phys. Lett. B 316 (1993) 561. [281 R. Fleischer, Z. Phys. C 62 (1994) 81; Phys. Lett. B 321 (1994) 259; B 332 (1994) 419. [291 N.G. Deshpande and X.-G. He, Phys. Lett. B 336 (1994) 471; University of Oregon preprint OITS-553, hepph/9408404 (1994).