- Email: [email protected]
CP violating asymmetries in charged D meson decays
Physics Letters B 302 (1993) 319-325 North-Holland
PHYSICS LETTERS B
CP violating asymmetries in charged D meson decays F. BucceUa a, M. Lusignoli b, G. Mangano a,c, G. Miele a,c A. Pugliese b and P. Santorelli a,c a Dipartimento di Scienze Fisiche, Universitgt di Napoli, 1-80125 Naples, Italy b Dipartimento di Fisica, Universith "La Sapienza", Roma, andINFN, Sezione di Roma L 1-00185 Rome, Italy c INFN, Sezione di Napoli, 1-80125 Naples, Italy
Received 22 December 1992
The CP violating asymmetries for Cabibbo suppressed charged D meson decays in the standard model are estimated in the factorized approximation, using the two-loop effective bamiltonian and a model for final state interactions previously tested for Cabibbo allowed D decays. No new parameters are added. The predictions are larger than expected and not too far from the experimental possibilities.
It is well known that C P violating effects show up in a decay process only if the decay amplitude is the sum of two different parts, whose phases are made of a weak (Cabibbo-Kobayashi-Maskawa) and a strong (final state interaction) contribution. The weak contributions to the phases change sign when going to the CP-conjugate process, while the strong ones do not. Let us denote a generic decay amplitude of this type by ~¢=A exp ( i ~ ) + B exp (i~2) ,
( 1)
and the corresponding C P conjugate amplitude by ~ = A * exp(i8~) + B * exp(i82).
(2)
The C P violating asymmetry in the decay rates will be therefore [ ~tl 2 _ 1~[2 ace-1~¢12+1~12-iAiZ
2~(AB*)sin(32--81) + iBi2+ 29~(AB.)cos(cSz_Ol) .
(3)
Both factors in the numerator of eq. (3) should be nonvanishing to have a nonzero effect. Moreover, to have a sizeable asymmetry the moduli of the two amplitudes A and B should not differ too much. While the phases of the weak amplitudes are expressed in terms of fundamental parameters of the theory (the angles and the C P violating phase in the C K M quark mixing matrix), the strong phases c~ are in general unknown for heavy flavour decays. For this reason the asymmetries in charged B (or D) decays are less studied and less under control, theoretically *~, than the asymmetries in mixing assisted channels in neutral B decays [ 1 ]. In a previous paper [2] we have presented a rather successful phenomenological analysis of Cabibbo-allowed D decays. In that analysis a specific model for final state interactions [ 3 ] was used, based on the assumption that the scattering phase-shifts are dominated by the nearest resonance. In this paper we exploit this model of final state interactions to estimate the C P violating asymmetries for two-body D decays in the standard model. These asymmetries have long been believed to be unmeasurably small. For Cabibbo allowed decays, this is a consequence of the smallness of the second interfering amplitude with different weak phase (B in ( 1 ) ), which can only be generated by mixing and doubly Cabibbo suppressed decay. In fact, asymmetries in D~/£Tr have #~ Although they are easier to measure experimentally, due to their self-taggingproperty. 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
319
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
been recently suggested [ 4 ] as a sensitive tool to observe possible "new physics" effects in CP violating amplitudes, taking advantage of the large strong phase differences. In the Cabibbo suppressed case the penguin diagram [ 5 ] contribution to the effective hamiltonian, although its coefficient is rather small, provides a nonnegligible second amplitude. We will therefore limit our calculations to Cabibbo suppressed decays. Moreover, only charged D will be considered. In fact, to describe final state interactions in D O decays that receive contributions from not yet observed isoscalar resonances, new undetermined parameters would be needed. As we will see, the asymmetries are small but larger than previously expected [ 6 ], and in some cases they are O (a few 10- 3). To observe at 3a level an asymmetry of 3 × 10- 3 in a channel having a decay branching ratio of 0.5%, a number of charged D mesons equal to 2 × 108 is required, with 100% efficiencies. Present projects of taucharm factories do not have a luminosity high enough to allow such a measurement, but it is not unconceivable that one could reach this sensitivity in a not too far future. The effective weak hamiltonian for Cabibbo suppressed nonleptonic decays of charmed particles is given by GF
H~ff= ~
GF
VudV*a(C, Qd +C2Qa2)+ ~
GF
6
VusVc*(flQSl + C 2 Q ~ ) - ~r~ Vut,Vc*b,=3ZC, Qi+h.c.
(4)
In eq. (4) the operators are [7]
Qd=a"~,,,( 1-75)dpdP~,u( 1-75)c,~, Q3 =a~Tu(1 - T s ) c , E #aTu(1 -ys)qa, q
Qa2=a"y~,( 1 - ys)d,~dP;,~'( 1- ys)c~, Q4 =a~Tu(1 -~,s)ca Z qPTu( 1-75)q~, q
Q~ =a~Yu(1-75)c, Z #PTU(l + Ts)qB, Q6 =a'~yu(1-ys)c~ ~ qSyu( l + 75)q,~ . q
(5)
q
The operator Q] (Q~) in eq. (4) is obtained from Qd (Q2a) with the substitution (d--,s). In eq. (5) ot and fl are colour indices and in the "penguin" operators q (4) is to be summed over all active flavours (u, d, s). We have evaluated the coefficients C~ using the two-loop anomalous dimension matrices recently calculated by Buras and collaborators [ 8 ]. The effect of next-to-leading corrections is particularly large for the coefficients of the penguin operators, as it has already been noted in ref. [ 8 ] in the case of AS= 1 decays. One problem that we have to face is the renormalization scheme dependence of the coefficients, that should be cancelled by a corresponding dependence in the matrix elements of the operators. Unfortunately we estimate the matrix elements of four-fermion operators using a specific model, and it is not evident to which scheme this corresponds ~2. We calculated the coefficients in both schemes (naive dimensional regularization and 't Hooft-Veltman scheme) for which anomalous dimension matrices have been given. We follow then the prescription of Buras et al., given in eq. (3.6) of ref. [ 8 ], to define "scheme independent" coefficients. Although not unique, this prescription is phenomenologically favoured, in that it reinforces the results of the leading order calculation and provides large penguin contributions, that are welcome in the AS= 1 case. It turns out that this prescription gives for the dominant penguin operator coefficient C6 a value that is about two and a half times larger than the leading order prediction C LL. We stress that the results for C6 in both schemes are also considerably larger than
C6LL. In ref. [2 ] we fitted the value of A4Ms, among other parameters, and obtained the result A4Ms ~ 200 MeV. We therefore adopted this value in the calculation of the coefficients C~. The results are reported in table 1. Given the effective hamiltonian, to obtain the decay amplitude
•rg~(D~jO = (Jqn~fflD),
(6)
we evaluate the matrix elements of the operators in the factorization approximation, neglecting moreover the #2 An analogousproblem for the dependence of the coefficientson the subtraction point,/z, alreadyappears at leading order. 320
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
Table 1 Coefficients in the effective hamiltonian, eq. (4), for A~s = 200 MeV and mb= 4.8 GeV, at a scale/t= m~= 1.5 GeV, in different renorrealization schemes. See text for further details.
Cl C2 6"3 (?4 C5 C6
Leading order
Naive dimensional regularization
't Hooft-Veltman
"Scheme independent"
--0.439 1.216 0.0073 -0.0185 0.0057 -0.0210
--0.303 1.139 0.0155 -0.0398 0.0099 -0.0447
--0.311 1.032 0.0126 -0.0291 0.0084 -0.0320
-0.514 1.270 0.0202 -0.0460 0.0127 -0.0541
colour-suppressed contributions, following ref. [ 9 ] and ref. [ 2 ]. We also neglect the contribution o f the d i m e n sion five o p e r a t o r
= mc~'~tr u~ ( 1 + 75 ) t~pc ~ G ~ ,
(7)
that is generated at the two-loop level. Its Wilson coefficient has not been evaluated in ref. [ 8 ], where the masses o f external quarks have been neglected. Its m a t r i x elements cannot be calculated in the factorized a p p r o x i m a t i o n a n d it would be anyhow suppressed by 1/Nc in the large Arc limit. As an illustration, we discuss the explicit example D + ~ K + / ( "*° - a decay channel that turns out to have a sizeable asymmetry. The expressions for the matrix elements o f the operators Q~ (i = 1..... 5 ) are easily obtained following ref. [ 2 ] a n d are (K+/(-.o I Qd iD + ) = ( K + / ( . o I Q] I D + ) = 0 . ( K + K * ° I Q2a ID + ) = --2Mr.(E*'pK) (m~ -Fred) WI, v, ( K + K * ° I Q~ ID + ) = 2 M K . ( e * ' p r ) f r 1
acs
2 2 , -- M r / M D s
(K+/('*° I Q 4 ID + > = ( K + ~ , * ° t Q d l D + > +
(K÷e*°LQ~I D + > ,
(K+K'*° I Q3 ID + > = (K+/('*° I Q51 D + > = 0 .
(8)
In eq. ( 8 ) , in a d d i t i o n to self-explanatory symbols for particle masses and m o m e n t a and for the K* polarization vector E, two quantities need a definition: acs= 0.8 is the axial charge and I'Fev parameterizes the annihilation contribution. Wev has been d e t e r m i n e d by a fit to the C a b i b b o allowed decay rates in ref. [ 2 ], with the result Wev= 0.53 + 0.13 ~3. The o p e r a t o r 06 must first be written in a Fierz-rearranged form: 06 = - 2 t ] a ( 1 - ys)caa~ ( 1 + Ys)q~ •
(9)
In the factorization a p p r o x i m a t i o n , its matrix elements are given in terms o f matrix elements o f scalar and pseudoscalar densities. These in turn m a y be related to the m a t r i x elements of the divergences o f vector and axial currents, a n d thereby to form factors that have been discussed in ref. [ 2 ]. The result is (K+/('*° 106 ID + ) =4Mz~.(~*'pr)
d
Wev-
(ms+mu)(mc+rns)
1-M~/M2J
.
(10)
We note that the penguin o p e r a t o r Q6 has, by far, the largest matrix element. Even if the coefficient C6 is #3 We will assume the central values for the parameters determined in ref. [2] in the following estimates. 321
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
considerably smaller than C2 o r C 1 its contribution would be dominant, were it not for the smallness of the CKM factor: I V,b V¢*b[ / I Vus Vc* [ -~ 10-3. In fact, the Q6 penguin operator gives the dominant contribution to the imaginarypart of the amplitude ~Jw, since all the CKM factors' imaginary parts are of the same order. Expressions analogous to (8) and (10) can be derived for the other two-body Cabibbo first forbidden decay channels o f D + and Ds+ . At this stage one would not have any CP violating asymmetry, since no strong phases from final state interactions have been included in the amplitudes ~¢w (D--,f). Rescattering, to be discussed in the next section, will provide the necessary nonzero strong phase-shifts. We make the assumption that the final state interactions are dominated by resonant contributions [ 3 ]. In the mass region of pseudoscalar charmed particles there is evidence, albeit not very strong [ I 0 ], for a J e = 0 + resonance K~ (with mass 1950 MeV, width 201 _+86 MeV and 52% branching ratio in Kn [ 11 ] ), a j e = O- K ( 1 8 3 0 ) (with F = 250 MeV and an observed decay to KO [12] ) and a j e = 0 - n(1770) with F = 310 MeV [ 13 ]. These resonances should dominate, respectively, the rescattering effects for Cabibbo forbidden decays of Ds+ -,PP, D + --,PV and D +--,PV, where P(V) indicates a pseudoscalar (vector) decay product. In ref. [2 ] we assumed the existence o f a j e = 0 + resonance ao, that should be relevant for Cabibbo forbidden D + ~ P P decays, with mass 1890 MeV and width ~200 MeV. We will not try to discuss the Cabibbo forbidden decay amplitudes for D Omesons; in this case several other unobserved isoscalar nonstrange resonances should determine the final state interactions and this would bring new unknown parameters in the calculations. The FSI effect modifies the amplitudes for Cabibbo suppressed D + and D + decays in the following way [ 3 ]: d(D~VhPk)=~(D~VhPk)+Chk[exp(i~8)--l]
~ Ch,k,~w(D-~Vh,Pk,)=dhkd27+Chkexp(i~8)d 8 •
(11)
h'k'
In ( 11 ) Chk (dhk) are the normalized (~C2k = 1 ) Couplings of a pseudoscalar (P) and a vector (V) octets to a pseudoscalar octet (27-plet). The resonance P i s present only in the octet case, and one has
F(P)
sin88 exp(iS8) = 2 ( m ~ - m D ) - i F ( P ) "
(12)
Similar expressions hold for d (D~PhPk). The CP violating asymmetry ofeq. (3) is given by 2 sin ~8 Chkdhk~("~127~¢8.) ace = c2k i ~¢a 12+ d~k I ~¢27 [ 2"t- 2Chkdhk~(5ff27~lS*)cos 88 "
( 13 )
To have a nonzero asymmetry the weak phases of d s and d 27 must be different. The phase difference is provided essentially by the penguin contributions, which dominate the imaginary parts of the amplitudes, and appear in d s only. On the other hand, it is true that in some channels there is no rescattering in our model and ace= O. This is the case for instance for the decay amplitude ~¢ (D +--, 7r+Tro), which only contains the term d 27. A particular treatment is needed for those decay channels where neutral kaons appear. In fact, these particles will be mostly detected through their decays into two pions, and these decays are by themselves affected by CP violation (in the K system). One has therefore to be able to disentangle the CP violating effects in D and K decays. Define the "experimental", time-integrated asymmetry for the D + decay to one charged and one neutral g meson, f T [ F ( D +--,K +K'°--,K +Tr+Tr- ) ( t) - F ( D - - - , K - K ° - - , K - T r +Tr- ) ( t) ]dt A ( T ) = f~[F(D+__,K+I(O__,K+Tr+n - ) (t) + F ( D - ~ K - K ° - - , K - n + T r - ) (t) ]dt'
(14)
and recall the usual definitions of parameters for the neutral kaon decays AM=MKL--Mxs,
322
F=½(Fs+FL) ,
A(KL ~+u - ) r/+_ = A(Ks__.Tr+~r_) = e + C .
(15)
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
It is also convenient to define [ t 4 ] f(T)-
2 (AM)Z+F2
× { (/-~t/+ _ + AM ~t/+ _ ) [ 1 - exp ( - FT) cos AMT] + (AM 9~t/+_ -/-~t/+ _ ) exp ( - F T ) sin AMT}, g ( T ) = FL[ 1 - e x p ( - F s T )
FL Fsf( T) ] + I s [ 1- e x p ( - F L T )
] l t/+_ I2'
Y~-ace +2~R¢.
(16)
The resulting asymmetry is zi(T) =
7-g(T) 1 -Tg(T)
(17)
"
It is instructive to consider the limits of this expression for short ( << rs) and long ( >> rs) time T: lim d (T) = ace,
lim ,4 (T) ... a c e - 29~.
T~0
T~oo
(18)
Eq. ( 18 ) shows that to perform a measurement of the D + decay asymmetry in these channels will be essentially hopeless if the asymmetry acp turns out to be considerably less than 291~, the asymmetry in KL semileptonic decays. For Ds decays, the equation analogous to (14) would be As(T) = f ~ [ F ( D ~ --,n+K°--,n+n+n - ) (t) - F ( D j - ~ n-/('°--, ~z- it + n - ) (t) ]dt f ~ [ F ( D + ~ n + K ° ~ n + n + n - ) (t) + F ( D 7 ~ n - I ( ° - - , n - n + ~ z - ) (t) ]dt"
(19)
The resulting expression can be obtained from (16) and (17) by changing the signs of the terms containing E and r/+ _. Therefore lim As (T) ~- ace + 29~e.
(20)
T ~ oo
The decay amplitudes for Cabibbo-suppressed charm decays (and especially their CP violating asymmetries) depend on all the entries in the first two rows of the quark mixing matrix. In the Wolfenstein parametrization [ 15 ], extended to keep terms up to O (25) in the imaginary parts, the matrix is written as follows: f 1 -- ½22 V = ~ --2(l+A224pei6) k A 2 3 [ 1 - ( 1 - ½ 2 2 ) p e i~]
2
1--122 - A 2 2 ( l + 2 2 p e ia)
h23p e - i ~ A22 ) . 1
(21)
In (21), 2=0.220 is the Cabibbo angle, we have chosen Vcb=A2Z=O.046 and allowed different values for p, p = 0.50 _+0.14, and for the CP violating phase ~. Following the analysis of ref. [ 16 ] and assuming for the top quark mass the "favourite" value ( 140 GeV), we considered both positive (0.47-0.81) and negative ( - 0 . 9 8 - 0 . 8 0 ) values for cos g, corresponding to the two possible shapes of the unitarity triangle compatible with present data on B - B mixing and the value of the ~ parameter. The positive cos 6 values correspond to larger CP violating effects in K decays ( ( / E and charged K decay asymmetries), in B decays and also in D decays. We have evaluated the branching ratios for decay into twelve different two-particle final states, and the corresponding CP violating asymmetries, both for D + and for D + . The results are reported in tables 2 and 3, respectively. We note that the predicted branching ratios are in reasonable agreement with the existing data [ 10 ], listed in the second columns, which have not been used in fitting the parameters. The variations in the theoretical predictions with p and cos 6 are less than 10 -4 and therefore neglected in the third column. 323
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
Table 2 Branching ratios and CP-violating decay asymmetries for D + Cabibbo forbidden decays. Experimental data and 90% CL upper bounds taken from ref. [ 10 ]. Decay channel
D+~n+Tt ° D+~n+~ I D+~zt+q ' D+~K+z("° D+-.*p°r~ + D+--,p+~t ° D+~p+rl D+~p+tf D +--+to~ + D+~¢~
+
D+--*K*+I("° D+~I~*°K+
103 X ace
102 × BR experiment
theory
cos J > 0
cos J < 0
<0.53 0.66_+0.22 <0.8 0.73_+0.18 <0.12
0.21 0.78 0.10 1.11 0.20 0.39 0.12 0.07 0.07
- 1 . 5 _+0.4 0.04+0.01 1.0 _+0.3 - 2 . 3 +0.6 2.9 +0.8 -
- 0 . 7 +0.4 0.02_+0.01 0.5 +0.3 - 1 . 2 _+0.6 1.5 +0.8 -
< 1.0 < 1.4 <0.6 0.60_+0.08
0.36
0.47-+0.09
1.85 0.36
-
-
-
-
- 0 . 9 -+0.3 2.8 +0.8
- 0 . 5 _+0.3 1.4 +0.7
Table 3 Branching ratios and CP-violating decay asymmetries for D~+ Cabibbo forbidden decays. Decay channel
experiment
D+-.-,K+n ° D+~K+rl D+~K+rl ' D+--,K°Tt + D~.-.p+K ° D~+---,p°K+ D~coK + D+ ~ K + D~+~K*+n ° D~-.,K*+q D+~K*+tf D~+~K*°~ +
103)
10 2× BR
<0.6 a)
<0.2 b)
theory
cos J > 0
cos J < 0
0.12 0.004 0.80 0.48 1.59 0.15 0.03 0.18 0.03 0.05 0.01 0.23
- 0 . 5 _+0.2 2.3 ___0.7 1.0 _+0.3 - 1 . 5 +0.5 0.6 +0.2 1.0 _+0.3 - 5 . 5 +1.6 0.14+0.04 - 5 . 8 +1.7 - 3 . 5 +1.0 -8.1 +2.3 - 2 . 6 _+0.8
- 0 . 2 +0.t 1.2 _+0.6 0.5 +0.3 - 0 . 8 +0.4 0.3 +0.2 0.5 -+0.3 - 2 . 7 _+1.4 0.07_+0.04 - 2 . 9 _+1.5 - 1 . 7 +0.9 -4.1 _+2.1 - 1 . 3 +0.7
Experimental 90% CL upper bounds are taken from ref. [ 10 ]. b) Experimental 90% CL upper bounds from ref. [ 17 ]. ")
W e also n o t e t h a t v e r y r e c e n t l y a d o u b l y C a b i b b o - s u p p r e s s e d d e c a y r a t e h a s b e e n m e a s u r e d f o r t h e first t i m e [ 17 ], w i t h t h e result
F ~ c ~~rt+ K + )) exp = ( J~- o•+3.2 --2. 6 + 0 . 7 ) ) < 1 0 -2 F (( D D ++--'¢
(22)
W i t h t h e p a r a m e t e r s d e t e r m i n e d i n ref. [2 ] o u r m o d e l p r e d i c t s
F(D+ofbK+)F(D+oO n + ) t h = 3 . 2 X 10 - 2 .
(23)
T h i s r a t i o agrees w i t h i n o n e s t a n d a r d d e v i a t i o n w i t h t h e e x p e r i m e n t a l result ( 2 2 ) , h o w e v e r it s h o u l d b e n o t e d 324
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
t h a t the p r e d i c t e d i n d i v i d u a l rates are s m a l l e r t h a n the d a t a b y a s o m e w h a t larger a m o u n t , see table 2. T h e errors i n the a s y m m e t r i e s r e p o r t e d in the f o u r t h a n d fifth c o l u m n s have b e e n e s t i m a t e d b y semi-dispersion in the n u m b e r s o b t a i n e d v a r y i n g one at a t i m e p a n d cos 8 in the ranges previously m e n t i o n e d . T h e asymm e t r i e s for a n o b t u s e 8 (fifth c o l u m n ) are generally smaller b y a factor two. T h e r e are s o m e reasons to prefer the acute-angled s o l u t i o n for the u n i t a r i t y triangle, though. Lattice Q C D calculations [ 18 ], as well as recent e v a l u a t i o n s with Q C D s u m rules [ 19 ], f a v o u r a r a t h e r large value for the B m e s o n decay c o n s t a n t fa. This, together with the large top q u a r k mass i n d i c a t e d by the electroweak data analyses [20], w o u l d select a n acutea n g l e d u n i t a r i t y triangle. H o p i n g that n a t u r e is k i n d to us a n d cos 8 > 0, o u r results predict the largest a s y m m e t r i e s ( ~ 10 - 2 ) for some Ds rare decay channels. C o n s i d e r i n g h o w e v e r that D, are also difficult to procedure, we c o n s i d e r the D + decay c h a n n e l s h a v i n g the highest a s y m m e t r i e s ( ~ 3 × 1 0 - 3 ) as m o r e p r o m i s i n g to observe C P v i o l a t i n g effects. T h e best c a n d i d a t e is p r o b a b l y the decay D + - , K + / ( *°. Here the final state c o n t a i n s three charged particles -~ o f the times, a n d the decay b r a n c h i n g ratio has b e e n already m e a s u r e d to be h a l f a percent. These are the n u m bers we used in the d i s c u s s i o n at the e n d o f the i n t r o d u c t i o n . A t a u - c h a r m factory at the ~ / ' ( 3 7 7 0 ) with a n integrated l u m i n o s i t y Ae= 1040 c m - 2 w o u l d p r o d u c e a b o u t 10 s charged D m e s o n s a n d therefore ~ 1 . 5 × 105 decay e v e n t s D + ~ K + F , * ° - - - , K + K - r c + a n d as m a n y D - ~ K - K * ° - - - , K - K + n - . W i t h o u t i n c l u d i n g the f u r t h e r r e d u c t i o n d u e to actual e x p e r i m e n t a l efficiencies, this n u m b e r o f e v e n t s w o u l d i m p l y for the a s y m m e t r y a statistical accuracy o f 1.7 × 10 -3. As we already said, the a s s u m e d l u m i n o s i t y is n o t e n o u g h to establish the effect, b u t at the s a m e t i m e n o t too far f r o m this goal.
References [ 1 ] A. Carter and A.I. Sanda, Phys. Rev. Lett. 45 (1980) 952; Phys. Rev. D 23 ( 1981 ) 1567; I.I. Bigi and A.I. Sanda, Nucl. Phys. B 193 ( 1981 ) 85. [2] F. Buccella, M. Lusignoli, G. Miele and A. Pugliese, Z. Phys. C 55 (1992) 243. [3] F. Buccella, M. Forte, G. Miele and G. Ricciardi, Z. Phys, C 48 (1990) 47. [4] A. Le Yaouanc, L. Oliver and J.-C. Raynal, Phys. Lett. B 292 (1992) 353. [ 5 ] M. Shifman, A. Vainshtein and V. Zakharov, Sov. Phys. JETP 45 ( 1977 ) 670. [6] L.-L. Chau and H.-Y. Cheng, Phys. Rev. Lett. 53 (1984) 1037. [ 7 ] F.J. Gilman and M.B. Wise, Phys. Rev. D 20 ( 1979 ) 2392. [ 8] A.J. Buras, M. Jamin, M.E. Lautenbacher and P.E. Weisz, Nuel. Phys. B 370 (1992) 69; B 375 (1992 ) 501 (A). [ 9 ] M. Bauer and B. Stech, Phys. Lett. B 152 ( 1985 ) 380; M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 ( 1987 ) 103. [ 10 ] Particle Data Group, K. Hikasa et al., Review of panicle properties, Phys. Rev. D 45 ( 1992 ) Part II. [ I 1 ] D. Aston et al., Nucl. Phys. B 296 ( 1988 ) 493. [ 12 ] T. Armstrong et al., Nucl. Phys. B 221 ( 1983 ) 1. [ 13] G. Bellini et al., Phys. Rev. Lett. 48 (1982) 1697. [ 14] G. Amelino-Camelia, F. Buccella, G. D'Ambrosio, A. Gallo, G. Mangano and M. Miragliuolo, Z. Phys. C 55 (1992) 63. [ 15] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 16] M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, Nucl. Phys. B 369 (1992) 139. [ 17 ] J.C. Anjos et al., Phys. Rev. Lett. 69 ( 1992 ) 2892. [ 18] C.R. AUton et al., Nucl. Phys. B 349 ( 1991 ) 598; Nucl. Phys. B (Proc. Suppl.) 20 ( 1991 ) 504; C. Alexandrou et al., Phys. Lett. B 256 ( 1991 ) 60; A. Abada et al., Nucl. Phys. B 376 ( 1992 ) 172; C. Bernard, C. Heard, J. Labrenz and A. Soni, Nucl. Phys. B (Proc. Suppl. ) 26 (1992) 384. [ 19 ] S. Narison, QCD spectral sum rules, World Scientific Lecture Notes in Physics, Vol. 26 (World Scientific, Singapore, 1989 ); M. Neubert, Phys. Rev. D 45 (1992) 2451. [ 20 ] J. Ellis and G.L. Fogli, Phys. Lett. B 249 (1990) 543; P. Langacker and M.-X. Luo, Phys. Rev. D 44 ( 1991 ) 1591; A. Borrelli, L. Maiani and R. Sisto, Phys. Lett. B 244 (1990) 117.
325
PHYSICS LETTERS B
CP violating asymmetries in charged D meson decays F. BucceUa a, M. Lusignoli b, G. Mangano a,c, G. Miele a,c A. Pugliese b and P. Santorelli a,c a Dipartimento di Scienze Fisiche, Universitgt di Napoli, 1-80125 Naples, Italy b Dipartimento di Fisica, Universith "La Sapienza", Roma, andINFN, Sezione di Roma L 1-00185 Rome, Italy c INFN, Sezione di Napoli, 1-80125 Naples, Italy
Received 22 December 1992
The CP violating asymmetries for Cabibbo suppressed charged D meson decays in the standard model are estimated in the factorized approximation, using the two-loop effective bamiltonian and a model for final state interactions previously tested for Cabibbo allowed D decays. No new parameters are added. The predictions are larger than expected and not too far from the experimental possibilities.
It is well known that C P violating effects show up in a decay process only if the decay amplitude is the sum of two different parts, whose phases are made of a weak (Cabibbo-Kobayashi-Maskawa) and a strong (final state interaction) contribution. The weak contributions to the phases change sign when going to the CP-conjugate process, while the strong ones do not. Let us denote a generic decay amplitude of this type by ~¢=A exp ( i ~ ) + B exp (i~2) ,
( 1)
and the corresponding C P conjugate amplitude by ~ = A * exp(i8~) + B * exp(i82).
(2)
The C P violating asymmetry in the decay rates will be therefore [ ~tl 2 _ 1~[2 ace-1~¢12+1~12-iAiZ
2~(AB*)sin(32--81) + iBi2+ 29~(AB.)cos(cSz_Ol) .
(3)
Both factors in the numerator of eq. (3) should be nonvanishing to have a nonzero effect. Moreover, to have a sizeable asymmetry the moduli of the two amplitudes A and B should not differ too much. While the phases of the weak amplitudes are expressed in terms of fundamental parameters of the theory (the angles and the C P violating phase in the C K M quark mixing matrix), the strong phases c~ are in general unknown for heavy flavour decays. For this reason the asymmetries in charged B (or D) decays are less studied and less under control, theoretically *~, than the asymmetries in mixing assisted channels in neutral B decays [ 1 ]. In a previous paper [2] we have presented a rather successful phenomenological analysis of Cabibbo-allowed D decays. In that analysis a specific model for final state interactions [ 3 ] was used, based on the assumption that the scattering phase-shifts are dominated by the nearest resonance. In this paper we exploit this model of final state interactions to estimate the C P violating asymmetries for two-body D decays in the standard model. These asymmetries have long been believed to be unmeasurably small. For Cabibbo allowed decays, this is a consequence of the smallness of the second interfering amplitude with different weak phase (B in ( 1 ) ), which can only be generated by mixing and doubly Cabibbo suppressed decay. In fact, asymmetries in D~/£Tr have #~ Although they are easier to measure experimentally, due to their self-taggingproperty. 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
319
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
been recently suggested [ 4 ] as a sensitive tool to observe possible "new physics" effects in CP violating amplitudes, taking advantage of the large strong phase differences. In the Cabibbo suppressed case the penguin diagram [ 5 ] contribution to the effective hamiltonian, although its coefficient is rather small, provides a nonnegligible second amplitude. We will therefore limit our calculations to Cabibbo suppressed decays. Moreover, only charged D will be considered. In fact, to describe final state interactions in D O decays that receive contributions from not yet observed isoscalar resonances, new undetermined parameters would be needed. As we will see, the asymmetries are small but larger than previously expected [ 6 ], and in some cases they are O (a few 10- 3). To observe at 3a level an asymmetry of 3 × 10- 3 in a channel having a decay branching ratio of 0.5%, a number of charged D mesons equal to 2 × 108 is required, with 100% efficiencies. Present projects of taucharm factories do not have a luminosity high enough to allow such a measurement, but it is not unconceivable that one could reach this sensitivity in a not too far future. The effective weak hamiltonian for Cabibbo suppressed nonleptonic decays of charmed particles is given by GF
H~ff= ~
GF
VudV*a(C, Qd +C2Qa2)+ ~
GF
6
VusVc*(flQSl + C 2 Q ~ ) - ~r~ Vut,Vc*b,=3ZC, Qi+h.c.
(4)
In eq. (4) the operators are [7]
Qd=a"~,,,( 1-75)dpdP~,u( 1-75)c,~, Q3 =a~Tu(1 - T s ) c , E #aTu(1 -ys)qa, q
Qa2=a"y~,( 1 - ys)d,~dP;,~'( 1- ys)c~, Q4 =a~Tu(1 -~,s)ca Z qPTu( 1-75)q~, q
Q~ =a~Yu(1-75)c, Z #PTU(l + Ts)qB, Q6 =a'~yu(1-ys)c~ ~ qSyu( l + 75)q,~ . q
(5)
q
The operator Q] (Q~) in eq. (4) is obtained from Qd (Q2a) with the substitution (d--,s). In eq. (5) ot and fl are colour indices and in the "penguin" operators q (4) is to be summed over all active flavours (u, d, s). We have evaluated the coefficients C~ using the two-loop anomalous dimension matrices recently calculated by Buras and collaborators [ 8 ]. The effect of next-to-leading corrections is particularly large for the coefficients of the penguin operators, as it has already been noted in ref. [ 8 ] in the case of AS= 1 decays. One problem that we have to face is the renormalization scheme dependence of the coefficients, that should be cancelled by a corresponding dependence in the matrix elements of the operators. Unfortunately we estimate the matrix elements of four-fermion operators using a specific model, and it is not evident to which scheme this corresponds ~2. We calculated the coefficients in both schemes (naive dimensional regularization and 't Hooft-Veltman scheme) for which anomalous dimension matrices have been given. We follow then the prescription of Buras et al., given in eq. (3.6) of ref. [ 8 ], to define "scheme independent" coefficients. Although not unique, this prescription is phenomenologically favoured, in that it reinforces the results of the leading order calculation and provides large penguin contributions, that are welcome in the AS= 1 case. It turns out that this prescription gives for the dominant penguin operator coefficient C6 a value that is about two and a half times larger than the leading order prediction C LL. We stress that the results for C6 in both schemes are also considerably larger than
C6LL. In ref. [2 ] we fitted the value of A4Ms, among other parameters, and obtained the result A4Ms ~ 200 MeV. We therefore adopted this value in the calculation of the coefficients C~. The results are reported in table 1. Given the effective hamiltonian, to obtain the decay amplitude
•rg~(D~jO = (Jqn~fflD),
(6)
we evaluate the matrix elements of the operators in the factorization approximation, neglecting moreover the #2 An analogousproblem for the dependence of the coefficientson the subtraction point,/z, alreadyappears at leading order. 320
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
Table 1 Coefficients in the effective hamiltonian, eq. (4), for A~s = 200 MeV and mb= 4.8 GeV, at a scale/t= m~= 1.5 GeV, in different renorrealization schemes. See text for further details.
Cl C2 6"3 (?4 C5 C6
Leading order
Naive dimensional regularization
't Hooft-Veltman
"Scheme independent"
--0.439 1.216 0.0073 -0.0185 0.0057 -0.0210
--0.303 1.139 0.0155 -0.0398 0.0099 -0.0447
--0.311 1.032 0.0126 -0.0291 0.0084 -0.0320
-0.514 1.270 0.0202 -0.0460 0.0127 -0.0541
colour-suppressed contributions, following ref. [ 9 ] and ref. [ 2 ]. We also neglect the contribution o f the d i m e n sion five o p e r a t o r
= mc~'~tr u~ ( 1 + 75 ) t~pc ~ G ~ ,
(7)
that is generated at the two-loop level. Its Wilson coefficient has not been evaluated in ref. [ 8 ], where the masses o f external quarks have been neglected. Its m a t r i x elements cannot be calculated in the factorized a p p r o x i m a t i o n a n d it would be anyhow suppressed by 1/Nc in the large Arc limit. As an illustration, we discuss the explicit example D + ~ K + / ( "*° - a decay channel that turns out to have a sizeable asymmetry. The expressions for the matrix elements o f the operators Q~ (i = 1..... 5 ) are easily obtained following ref. [ 2 ] a n d are (K+/(-.o I Qd iD + ) = ( K + / ( . o I Q] I D + ) = 0 . ( K + K * ° I Q2a ID + ) = --2Mr.(E*'pK) (m~ -Fred) WI, v, ( K + K * ° I Q~ ID + ) = 2 M K . ( e * ' p r ) f r 1
acs
2 2 , -- M r / M D s
(K+/('*° I Q 4 ID + > = ( K + ~ , * ° t Q d l D + > +
(K÷e*°LQ~I D + > ,
(K+K'*° I Q3 ID + > = (K+/('*° I Q51 D + > = 0 .
(8)
In eq. ( 8 ) , in a d d i t i o n to self-explanatory symbols for particle masses and m o m e n t a and for the K* polarization vector E, two quantities need a definition: acs= 0.8 is the axial charge and I'Fev parameterizes the annihilation contribution. Wev has been d e t e r m i n e d by a fit to the C a b i b b o allowed decay rates in ref. [ 2 ], with the result Wev= 0.53 + 0.13 ~3. The o p e r a t o r 06 must first be written in a Fierz-rearranged form: 06 = - 2 t ] a ( 1 - ys)caa~ ( 1 + Ys)q~ •
(9)
In the factorization a p p r o x i m a t i o n , its matrix elements are given in terms o f matrix elements o f scalar and pseudoscalar densities. These in turn m a y be related to the m a t r i x elements of the divergences o f vector and axial currents, a n d thereby to form factors that have been discussed in ref. [ 2 ]. The result is (K+/('*° 106 ID + ) =4Mz~.(~*'pr)
d
Wev-
(ms+mu)(mc+rns)
1-M~/M2J
.
(10)
We note that the penguin o p e r a t o r Q6 has, by far, the largest matrix element. Even if the coefficient C6 is #3 We will assume the central values for the parameters determined in ref. [2] in the following estimates. 321
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
considerably smaller than C2 o r C 1 its contribution would be dominant, were it not for the smallness of the CKM factor: I V,b V¢*b[ / I Vus Vc* [ -~ 10-3. In fact, the Q6 penguin operator gives the dominant contribution to the imaginarypart of the amplitude ~Jw, since all the CKM factors' imaginary parts are of the same order. Expressions analogous to (8) and (10) can be derived for the other two-body Cabibbo first forbidden decay channels o f D + and Ds+ . At this stage one would not have any CP violating asymmetry, since no strong phases from final state interactions have been included in the amplitudes ~¢w (D--,f). Rescattering, to be discussed in the next section, will provide the necessary nonzero strong phase-shifts. We make the assumption that the final state interactions are dominated by resonant contributions [ 3 ]. In the mass region of pseudoscalar charmed particles there is evidence, albeit not very strong [ I 0 ], for a J e = 0 + resonance K~ (with mass 1950 MeV, width 201 _+86 MeV and 52% branching ratio in Kn [ 11 ] ), a j e = O- K ( 1 8 3 0 ) (with F = 250 MeV and an observed decay to KO [12] ) and a j e = 0 - n(1770) with F = 310 MeV [ 13 ]. These resonances should dominate, respectively, the rescattering effects for Cabibbo forbidden decays of Ds+ -,PP, D + --,PV and D +--,PV, where P(V) indicates a pseudoscalar (vector) decay product. In ref. [2 ] we assumed the existence o f a j e = 0 + resonance ao, that should be relevant for Cabibbo forbidden D + ~ P P decays, with mass 1890 MeV and width ~200 MeV. We will not try to discuss the Cabibbo forbidden decay amplitudes for D Omesons; in this case several other unobserved isoscalar nonstrange resonances should determine the final state interactions and this would bring new unknown parameters in the calculations. The FSI effect modifies the amplitudes for Cabibbo suppressed D + and D + decays in the following way [ 3 ]: d(D~VhPk)=~(D~VhPk)+Chk[exp(i~8)--l]
~ Ch,k,~w(D-~Vh,Pk,)=dhkd27+Chkexp(i~8)d 8 •
(11)
h'k'
In ( 11 ) Chk (dhk) are the normalized (~C2k = 1 ) Couplings of a pseudoscalar (P) and a vector (V) octets to a pseudoscalar octet (27-plet). The resonance P i s present only in the octet case, and one has
F(P)
sin88 exp(iS8) = 2 ( m ~ - m D ) - i F ( P ) "
(12)
Similar expressions hold for d (D~PhPk). The CP violating asymmetry ofeq. (3) is given by 2 sin ~8 Chkdhk~("~127~¢8.) ace = c2k i ~¢a 12+ d~k I ~¢27 [ 2"t- 2Chkdhk~(5ff27~lS*)cos 88 "
( 13 )
To have a nonzero asymmetry the weak phases of d s and d 27 must be different. The phase difference is provided essentially by the penguin contributions, which dominate the imaginary parts of the amplitudes, and appear in d s only. On the other hand, it is true that in some channels there is no rescattering in our model and ace= O. This is the case for instance for the decay amplitude ~¢ (D +--, 7r+Tro), which only contains the term d 27. A particular treatment is needed for those decay channels where neutral kaons appear. In fact, these particles will be mostly detected through their decays into two pions, and these decays are by themselves affected by CP violation (in the K system). One has therefore to be able to disentangle the CP violating effects in D and K decays. Define the "experimental", time-integrated asymmetry for the D + decay to one charged and one neutral g meson, f T [ F ( D +--,K +K'°--,K +Tr+Tr- ) ( t) - F ( D - - - , K - K ° - - , K - T r +Tr- ) ( t) ]dt A ( T ) = f~[F(D+__,K+I(O__,K+Tr+n - ) (t) + F ( D - ~ K - K ° - - , K - n + T r - ) (t) ]dt'
(14)
and recall the usual definitions of parameters for the neutral kaon decays AM=MKL--Mxs,
322
F=½(Fs+FL) ,
A(KL ~+u - ) r/+_ = A(Ks__.Tr+~r_) = e + C .
(15)
Volume 302, number 2,3
PHYSICSLETTERSB
25 March 1993
It is also convenient to define [ t 4 ] f(T)-
2 (AM)Z+F2
× { (/-~t/+ _ + AM ~t/+ _ ) [ 1 - exp ( - FT) cos AMT] + (AM 9~t/+_ -/-~t/+ _ ) exp ( - F T ) sin AMT}, g ( T ) = FL[ 1 - e x p ( - F s T )
FL Fsf( T) ] + I s [ 1- e x p ( - F L T )
] l t/+_ I2'
Y~-ace +2~R¢.
(16)
The resulting asymmetry is zi(T) =
7-g(T) 1 -Tg(T)
(17)
"
It is instructive to consider the limits of this expression for short ( << rs) and long ( >> rs) time T: lim d (T) = ace,
lim ,4 (T) ... a c e - 29~.
T~0
T~oo
(18)
Eq. ( 18 ) shows that to perform a measurement of the D + decay asymmetry in these channels will be essentially hopeless if the asymmetry acp turns out to be considerably less than 291~, the asymmetry in KL semileptonic decays. For Ds decays, the equation analogous to (14) would be As(T) = f ~ [ F ( D ~ --,n+K°--,n+n+n - ) (t) - F ( D j - ~ n-/('°--, ~z- it + n - ) (t) ]dt f ~ [ F ( D + ~ n + K ° ~ n + n + n - ) (t) + F ( D 7 ~ n - I ( ° - - , n - n + ~ z - ) (t) ]dt"
(19)
The resulting expression can be obtained from (16) and (17) by changing the signs of the terms containing E and r/+ _. Therefore lim As (T) ~- ace + 29~e.
(20)
T ~ oo
The decay amplitudes for Cabibbo-suppressed charm decays (and especially their CP violating asymmetries) depend on all the entries in the first two rows of the quark mixing matrix. In the Wolfenstein parametrization [ 15 ], extended to keep terms up to O (25) in the imaginary parts, the matrix is written as follows: f 1 -- ½22 V = ~ --2(l+A224pei6) k A 2 3 [ 1 - ( 1 - ½ 2 2 ) p e i~]
2
1--122 - A 2 2 ( l + 2 2 p e ia)
h23p e - i ~ A22 ) . 1
(21)
In (21), 2=0.220 is the Cabibbo angle, we have chosen Vcb=A2Z=O.046 and allowed different values for p, p = 0.50 _+0.14, and for the CP violating phase ~. Following the analysis of ref. [ 16 ] and assuming for the top quark mass the "favourite" value ( 140 GeV), we considered both positive (0.47-0.81) and negative ( - 0 . 9 8 - 0 . 8 0 ) values for cos g, corresponding to the two possible shapes of the unitarity triangle compatible with present data on B - B mixing and the value of the ~ parameter. The positive cos 6 values correspond to larger CP violating effects in K decays ( ( / E and charged K decay asymmetries), in B decays and also in D decays. We have evaluated the branching ratios for decay into twelve different two-particle final states, and the corresponding CP violating asymmetries, both for D + and for D + . The results are reported in tables 2 and 3, respectively. We note that the predicted branching ratios are in reasonable agreement with the existing data [ 10 ], listed in the second columns, which have not been used in fitting the parameters. The variations in the theoretical predictions with p and cos 6 are less than 10 -4 and therefore neglected in the third column. 323
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
Table 2 Branching ratios and CP-violating decay asymmetries for D + Cabibbo forbidden decays. Experimental data and 90% CL upper bounds taken from ref. [ 10 ]. Decay channel
D+~n+Tt ° D+~n+~ I D+~zt+q ' D+~K+z("° D+-.*p°r~ + D+--,p+~t ° D+~p+rl D+~p+tf D +--+to~ + D+~¢~
+
D+--*K*+I("° D+~I~*°K+
103 X ace
102 × BR experiment
theory
cos J > 0
cos J < 0
<0.53 0.66_+0.22 <0.8 0.73_+0.18 <0.12
0.21 0.78 0.10 1.11 0.20 0.39 0.12 0.07 0.07
- 1 . 5 _+0.4 0.04+0.01 1.0 _+0.3 - 2 . 3 +0.6 2.9 +0.8 -
- 0 . 7 +0.4 0.02_+0.01 0.5 +0.3 - 1 . 2 _+0.6 1.5 +0.8 -
< 1.0 < 1.4 <0.6 0.60_+0.08
0.36
0.47-+0.09
1.85 0.36
-
-
-
-
- 0 . 9 -+0.3 2.8 +0.8
- 0 . 5 _+0.3 1.4 +0.7
Table 3 Branching ratios and CP-violating decay asymmetries for D~+ Cabibbo forbidden decays. Decay channel
experiment
D+-.-,K+n ° D+~K+rl D+~K+rl ' D+--,K°Tt + D~.-.p+K ° D~+---,p°K+ D~coK + D+ ~ K + D~+~K*+n ° D~-.,K*+q D+~K*+tf D~+~K*°~ +
103)
10 2× BR
<0.6 a)
<0.2 b)
theory
cos J > 0
cos J < 0
0.12 0.004 0.80 0.48 1.59 0.15 0.03 0.18 0.03 0.05 0.01 0.23
- 0 . 5 _+0.2 2.3 ___0.7 1.0 _+0.3 - 1 . 5 +0.5 0.6 +0.2 1.0 _+0.3 - 5 . 5 +1.6 0.14+0.04 - 5 . 8 +1.7 - 3 . 5 +1.0 -8.1 +2.3 - 2 . 6 _+0.8
- 0 . 2 +0.t 1.2 _+0.6 0.5 +0.3 - 0 . 8 +0.4 0.3 +0.2 0.5 -+0.3 - 2 . 7 _+1.4 0.07_+0.04 - 2 . 9 _+1.5 - 1 . 7 +0.9 -4.1 _+2.1 - 1 . 3 +0.7
Experimental 90% CL upper bounds are taken from ref. [ 10 ]. b) Experimental 90% CL upper bounds from ref. [ 17 ]. ")
W e also n o t e t h a t v e r y r e c e n t l y a d o u b l y C a b i b b o - s u p p r e s s e d d e c a y r a t e h a s b e e n m e a s u r e d f o r t h e first t i m e [ 17 ], w i t h t h e result
F ~ c ~~rt+ K + )) exp = ( J~- o•+3.2 --2. 6 + 0 . 7 ) ) < 1 0 -2 F (( D D ++--'¢
(22)
W i t h t h e p a r a m e t e r s d e t e r m i n e d i n ref. [2 ] o u r m o d e l p r e d i c t s
F(D+ofbK+)F(D+oO n + ) t h = 3 . 2 X 10 - 2 .
(23)
T h i s r a t i o agrees w i t h i n o n e s t a n d a r d d e v i a t i o n w i t h t h e e x p e r i m e n t a l result ( 2 2 ) , h o w e v e r it s h o u l d b e n o t e d 324
Volume 302, number 2,3
PHYSICS LETTERS B
25 March 1993
t h a t the p r e d i c t e d i n d i v i d u a l rates are s m a l l e r t h a n the d a t a b y a s o m e w h a t larger a m o u n t , see table 2. T h e errors i n the a s y m m e t r i e s r e p o r t e d in the f o u r t h a n d fifth c o l u m n s have b e e n e s t i m a t e d b y semi-dispersion in the n u m b e r s o b t a i n e d v a r y i n g one at a t i m e p a n d cos 8 in the ranges previously m e n t i o n e d . T h e asymm e t r i e s for a n o b t u s e 8 (fifth c o l u m n ) are generally smaller b y a factor two. T h e r e are s o m e reasons to prefer the acute-angled s o l u t i o n for the u n i t a r i t y triangle, though. Lattice Q C D calculations [ 18 ], as well as recent e v a l u a t i o n s with Q C D s u m rules [ 19 ], f a v o u r a r a t h e r large value for the B m e s o n decay c o n s t a n t fa. This, together with the large top q u a r k mass i n d i c a t e d by the electroweak data analyses [20], w o u l d select a n acutea n g l e d u n i t a r i t y triangle. H o p i n g that n a t u r e is k i n d to us a n d cos 8 > 0, o u r results predict the largest a s y m m e t r i e s ( ~ 10 - 2 ) for some Ds rare decay channels. C o n s i d e r i n g h o w e v e r that D, are also difficult to procedure, we c o n s i d e r the D + decay c h a n n e l s h a v i n g the highest a s y m m e t r i e s ( ~ 3 × 1 0 - 3 ) as m o r e p r o m i s i n g to observe C P v i o l a t i n g effects. T h e best c a n d i d a t e is p r o b a b l y the decay D + - , K + / ( *°. Here the final state c o n t a i n s three charged particles -~ o f the times, a n d the decay b r a n c h i n g ratio has b e e n already m e a s u r e d to be h a l f a percent. These are the n u m bers we used in the d i s c u s s i o n at the e n d o f the i n t r o d u c t i o n . A t a u - c h a r m factory at the ~ / ' ( 3 7 7 0 ) with a n integrated l u m i n o s i t y Ae= 1040 c m - 2 w o u l d p r o d u c e a b o u t 10 s charged D m e s o n s a n d therefore ~ 1 . 5 × 105 decay e v e n t s D + ~ K + F , * ° - - - , K + K - r c + a n d as m a n y D - ~ K - K * ° - - - , K - K + n - . W i t h o u t i n c l u d i n g the f u r t h e r r e d u c t i o n d u e to actual e x p e r i m e n t a l efficiencies, this n u m b e r o f e v e n t s w o u l d i m p l y for the a s y m m e t r y a statistical accuracy o f 1.7 × 10 -3. As we already said, the a s s u m e d l u m i n o s i t y is n o t e n o u g h to establish the effect, b u t at the s a m e t i m e n o t too far f r o m this goal.
References [ 1 ] A. Carter and A.I. Sanda, Phys. Rev. Lett. 45 (1980) 952; Phys. Rev. D 23 ( 1981 ) 1567; I.I. Bigi and A.I. Sanda, Nucl. Phys. B 193 ( 1981 ) 85. [2] F. Buccella, M. Lusignoli, G. Miele and A. Pugliese, Z. Phys. C 55 (1992) 243. [3] F. Buccella, M. Forte, G. Miele and G. Ricciardi, Z. Phys, C 48 (1990) 47. [4] A. Le Yaouanc, L. Oliver and J.-C. Raynal, Phys. Lett. B 292 (1992) 353. [ 5 ] M. Shifman, A. Vainshtein and V. Zakharov, Sov. Phys. JETP 45 ( 1977 ) 670. [6] L.-L. Chau and H.-Y. Cheng, Phys. Rev. Lett. 53 (1984) 1037. [ 7 ] F.J. Gilman and M.B. Wise, Phys. Rev. D 20 ( 1979 ) 2392. [ 8] A.J. Buras, M. Jamin, M.E. Lautenbacher and P.E. Weisz, Nuel. Phys. B 370 (1992) 69; B 375 (1992 ) 501 (A). [ 9 ] M. Bauer and B. Stech, Phys. Lett. B 152 ( 1985 ) 380; M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 ( 1987 ) 103. [ 10 ] Particle Data Group, K. Hikasa et al., Review of panicle properties, Phys. Rev. D 45 ( 1992 ) Part II. [ I 1 ] D. Aston et al., Nucl. Phys. B 296 ( 1988 ) 493. [ 12 ] T. Armstrong et al., Nucl. Phys. B 221 ( 1983 ) 1. [ 13] G. Bellini et al., Phys. Rev. Lett. 48 (1982) 1697. [ 14] G. Amelino-Camelia, F. Buccella, G. D'Ambrosio, A. Gallo, G. Mangano and M. Miragliuolo, Z. Phys. C 55 (1992) 63. [ 15] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 16] M. Lusignoli, L. Maiani, G. Martinelli and L. Reina, Nucl. Phys. B 369 (1992) 139. [ 17 ] J.C. Anjos et al., Phys. Rev. Lett. 69 ( 1992 ) 2892. [ 18] C.R. AUton et al., Nucl. Phys. B 349 ( 1991 ) 598; Nucl. Phys. B (Proc. Suppl.) 20 ( 1991 ) 504; C. Alexandrou et al., Phys. Lett. B 256 ( 1991 ) 60; A. Abada et al., Nucl. Phys. B 376 ( 1992 ) 172; C. Bernard, C. Heard, J. Labrenz and A. Soni, Nucl. Phys. B (Proc. Suppl. ) 26 (1992) 384. [ 19 ] S. Narison, QCD spectral sum rules, World Scientific Lecture Notes in Physics, Vol. 26 (World Scientific, Singapore, 1989 ); M. Neubert, Phys. Rev. D 45 (1992) 2451. [ 20 ] J. Ellis and G.L. Fogli, Phys. Lett. B 249 (1990) 543; P. Langacker and M.-X. Luo, Phys. Rev. D 44 ( 1991 ) 1591; A. Borrelli, L. Maiani and R. Sisto, Phys. Lett. B 244 (1990) 117.
325