fully probing the cabibbo-kobayashi-maskawa unitarity triangle at the Υ(4 s ...

PHYSICAL REVIEW D VOLUME 47, NUMBER 7 1 APRIL 1993

Fully probing the Cabibbo-Kobayashi-Maskawa unitarity triangle at the Y(4S) resonance

Dongsheng D u Institute of High Energy Physics, P.O. Box 918(4), Beijing 100039, People's Republic of China

Zhi-zhong Xing China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730,

Beijing 100080, People's Republic of China and Institute of High Energy Physics, P.O. Box 918(4), Beijing 100039, People's Republic of China*

(Received 18 August 1992; revised manuscript received 16 November 1992)

We analyze the joint decays of C = +1 B,OB pairs at a symmetric B factory. A few time-integrated measurements of decay rates and CP-violating asymmetries are proposed for B;,B ~ - + n - + n - - , ~ ~ n - ~ , y ? ~ ~ and B>-+n-'n-O both on the T(4S) resonance and just above it. We show that it is possible to extract three angles of the Cabibbo-Kobayashi-Maskawa unitarity triangle from such measurements unambiguously, even in the presence of significant final-state strong interactions and in the absence of exclusive B: versus B decay modes.

PACS numberb): 13.25. f m , 11.30.Er, 12.15.Ff, 14.40.J~

I. INTRODUCTION

Within the standard electroweak model, the three- family Cabibbo-Kobayashi-Maskawa (CKM) matrix V describes the admixtures of different quark flavors and offers a natural explanation of C P violation [I]. The unitarity of the 3 X 3 C K M matrix results in the relation

which corresponds to a unitarity triangle in the complex plane (see Fig. 1). The three angles of this triangle are usually defined by

and then

is a natural consequence of Eq. (1). The point is that one can make enough independent measurements of the sides and angles of the C K M unitarity triangle to check the va- lidity of the standard model. At present 1 VUd 1, I Vcd 1, and

1 Vcbl have been measured to good accuracy. Precise measurements of the ratios I Vub /Vcb / and I Vrd /Vcb 1 may determine the shape of the unitarity triangle and fix all the C K M parameters including the CP-violating phase. Conventionally three distinct classes of neutral B decays

a ailing address.

into CP eigenstates, e.g., ~: -7 i -+ . r r - , l / K S , and B P - ~ O K , , are suggested to probe the corresponding a, 8, and y via the CP-violating asymmetries in these modes [2-41. However, there are two obvious problems to such a program. First, the decays of B: and B: into r+r- can occur through both tree-level and QCD-loop-induced (penguin) diagrams so that their partial rate asymmetry suffers from final-state strong interactions and cannot be predicted reliably [5,6]. Neglecting penguin contributions to such decay modes is certainly a bad approximation, which may lead to significant deviation of the probed (process-dependent) angle from the geometrical (process-independent) one to a certain degree [7]. Second, B; and B: events have not been reconstructed and accumulated in current B-meson experiments; hence, observing their exclusive decay channels and possible CP-violating signals is still unfeasible. On the other hand, the expected rapid rate of BP-B; oscillation will make it very difficult to measure either time-dependent or time-integrated partial rate asymmetries in B: versus By decay modes [5].

To disentangle the effects of tree-level and penguin contributions for the final-state a f a - and TOTO, Gronau and London have analyzed the isospin relations among decay amplitudes of B: ,B:+T+T- ,T~~~-~ and B+-7i-+a0 [6]. On the condition that the time dependence of B: and B: decays into r f T - and rOrO has been measured, their analysis can be employed to extract the

FIG. 1. The CKM unitarity triangle in the complex plane.

47 2825 - @ 1993 The American Physical Society

2826 DONGSHENG DU AND ZHI-ZHONG XING 47

angle a, in most cases, without theoretical uncertainty. In contrast, Bigi has pointed out that a can be unambiguously probed from the CP-violating asymmetries in decay rate evolution for the related modes Bj-Do'* )KS versus BO, +Do' * 'K, or for B,O+DO( * 'K, versus BO, -DO( * )K,, if sufficient statistical events can be accumulated [5]. Re- cently three different approaches have been suggested for the extraction of y , which is the most problematic angle of the C K M unitarity triangle. Two of them are to measure CP-violating effects in six related B~-B; or Ab-Tb decay modes, e.g., B~-DOK(*),DOK'*) 7 D o C P K'* ' and . .

their charge-conjugated counterparts [8,9], or Ab +DOA,DOA,D;~A and their charge-conjugated counterparts [lo]. The feasibility of these two ways is of course dependent upon the accumulation sample size, which depends on the branching ratios and detection efficiencies (in particular, for reconstructing the neutral D mesons). The third approach, pointed out by Bigi, is to study the joint decay rates of Y(4S),,,,n,n,, + ( ~ d q ~ ~ ~ ~ B ~ , ~ ~ ~ ~ ) - f f2, where the final states f and f, are common to B; and B: decays. He gave a detailed discussion about this method in the approximation of neglecting penguin contributions to charmless decay modes such as B:,BO, - ~ + a - [5].

It is then natural to ask the question: Could the full extraction of a, p, and y be realized under a specific experimental circumstance with fewer B decay modes? This paper is aimed at giving an affirmative answer to this question and improving some of the previous studies of this problem. We are going to focus our attention on symmetric e + e - collisions on the Y(4S) resonance and just above it to produce C = f 1 B ~ , ~ ~ ~ ~ B O , , ~ ~ ~ ~ events, and to propose a few time-integrated measurements of joint decay rates and CP-violating asymmetries for B ~ - T + T - , ~ ~ ~ ~ , $ K ~ , B;+T+T~, and their charge- conjugated processes. With the help of isospin analysis we show that such measurements will be enough for us to probe a , P, and y unambiguously, even in the presence of significant final-state strong interactions and in the absence of B: versus B: decays into p o ~ s , e t ~ . I t is worthwhile to remark that our approach has obvious experimental advantages since the Y(4S) is one of the richest B-meson sources today, and measurements of the time-integrated partial rates and CP-violating asymmetries for B decays is proving feasible at the T(4S) threshold [11,12].

The remainder of this paper is organized as follows: in Sec. I1 we calculate the joint decay rates and CP-violating asymmetries for c = f 1 B:,,~,,BO,,,~,, pairs into CP eigenstates n + n - , n0n0, and $Ks, and discuss their experimental detectability at the Y(4S); Sec. I11 is devoted to extraction of three angles of the C K M unitarity triangle; a summary and conclusion are given in Sec. IV.

11. DECAYS OF ~dq,,,,B~,,,,,, PAIRS AT THE W4S)

Symmetric electron-positron collisions at the Y(4S) threshold offer some experimental advantages in studying b-flavor physics, where both C = - 1 and C = + 1 ~ d q ~ ~ ~ ~ ~ ~ , ~ ~ ~ ~ pairs can be produced [12,13]. O n the Y(4S) resonance the B's are produced in a two-body

B:B, or B~~,~,,BO,,~~,, final state with definite angular momentum and an accurately known total energy. The signal-to-background ratio is larger than that in any other B production process, and the mass uncertainty in ki- nematic reconstructions is quite small. The two C = - 1 neutral B mesons mix coherently until one of them decays; thus, the flavor can be tagged without dilution due to B,O-BO, oscillation. At a center-of-mass beam energy above ME +ME. but below 2MB *, the e + e - annihilation can produce BB* or B * B pairs which dominate the b6 final states. The B * (B* ) will decay radiatively to the B ( B ) , leaving B B ~ with the BZ in a C = + 1 state. To ver- ify that an observed neutral B decay mode comes from e +e - - B ~ ~ ~ ~ ~ ~ ; , * ~ ~ ~ ~ + rather than from e +e + B ~ , ~ , , B ~ , , ~ , , , it is unnecessary to detect the photon (with energy E y =MB*-M, =50 MeV) from

-0 ~ j , * p h , s -B2phys Y Or B;,*phphys-fBd,phys~. If One recon- structs the measured B mass using the beam energy con- straint (as if there were no photon), the reconstructed mass for either B in a BB* (or B *B ) event will fall E , /2 above MB. Since the rms mass resolution in a BB threshold experiment is typically 6m =2.5 MeV, there is no serious difficulty in identifying the C state of B~,~,,BO,,,~,, pairs [11-131.

A. Joint decay rates

Let us calculate the joint decay rate of the B~,,~,,B;,,~,, pair at the Y(4S1, which depends on its charge conjugation state C. In the above two cases of B ~ , ~ ~ ~ ~ ~ ~ , ~ ~ ~ ~ prod_uction, the time-dependent wave function for a event in its rest frame can be de- scribed by

where K is the three-momentum vector of the B:,,~,, (or B ~ , p h , , ~ m e s o n , and C = f 1 are the charge parities of the B~, ,~ , ,B~, ,~ , , state. The proper time evolution of an ini- tially ( t = O ) pure B: or B; is given by

where

and p,q are the Pais-Treiman parameters for B~O-BO, mixing. To a high degree of accuracy, we obtain [2-41

!?I FULLY PROBING THE CABIBBO-KOBAYASHI-MASKAWA . . . 2827

Now we consider the case where one of the neutral B tonically to another final state f 2 at time t 2 . The joint mesons, with the momentum K , decays semileptonically decay rate for having such an event at the Y ( 4 S ) can be o r nonleptonically to a final state f , at (proper) time t l ; given as and the other, with the momentum -K, decays nonlep-

where A ( f , ) = A ( B:- f ) and x( f , ) r A (B: - f ) denote the corresponding amplitudes for B: and B; decays into f ( i = 1 , 2 ) , and

It is easy to check that for the C = - 1 case formula (8) is in agreement with that obtained in Ref. [ 5 ] . Here we give a more general expression including the C = + 1 case.

In view of experimental accessibility at the Y ( 4 S ) , we are interested in the time-independent decay rates of the B's. Integration of r( f , , t , ; f , , t 2 ) over both times t and t 2 leads to a simpler formula for the branching fraction of this joint decay ':

where x E Am / r is the B ~ - B ; mixing parameter, whose value has been measured in experiments [14].

B. Time-integrated observables for aa and $Ks

We are going to use the decay modes B++rk?1-0, B:,B; - n - + n - - , n - O ~ O , $KS to extract three angles of the CKM unitarity triangle at the Y ( 4 S ) . To succeed in this objective we need information on the magnitudes of decay amplitudes and CP-violating asymmetries of these processes. Instead of assuming that the time dependence of BZphys and B:,phys decays into n-+n--, n - O n - O , and $Ks has been determined somewhere [6,8,9], here we propose a few time-integrated measurements for such decay modes just at the Y ( 4 S ) . Explicitly we consider the following three types of joint decays of ~ j , p h y s ~ : phys pairs.

( 1 ) ( ~ ~ , , ~ ~ ~ ~ ~ , ~ ~ ~ ~ ) ~ + ( l ~ ~ ) ( T T T T ) , where the leptons I' can be used to tag the flavor of the initial B: and 8; mesons, and the n-n- (i.e., n-+n-- or * O n - O ) state is a CP eigenstate. In this case, A ( I - x + ) = Z ( ~ + X - ) = O , I A ( 1 + X - ) I = / A(1 -x+ ) I # o , and Ip(n-n-)l#l due to the penguin contributions to A (.nn-) and z(n-n-). As a result, one can obtain the branching fraction of this joint decay on the Y ( 4 S ) resonance:

Just above the Y ( 4 S ) resonance, the time-integrated partial rate asymmetry is given by [15]

In Eqs. ( 1 1 ) and (121, B ( I * X ' ; ~ n - ) ~ = - , , B ( 1 + x - 1 , and ~ 4 ( n - n - ) ~ = + , may be observed in the future experiments. Con- sequently the physical quantities B ( T T ) [or ( A ( ~ - T T ) ~ ~ ] , l p ( ~ a ) ( ~ , and I r n [ p ( ~ r v ) ] can be extracted from these observables to determine the CKM unitarity triangle.

(2 ) ( B ~ , ~ ~ ~ , B ~ , ~ ~ , ~ ) ~ + ( I * x ~ ) ($KS ), where $Ks is a C P eigenstate with Ip($Ks ) ) = 1 to a high degree of accuracy. To extract, the CP-violating term I m [ p ( $ K s ) ] , measuring the following time-integrated partial rate asymmetry is enough:

'It should be noted that, before integrating over t , and t 2 for T( f , , t , ; f 2 , t 2 ) ~ , we have to normalize the wave functions l B j S p h y s ( t ) ) and IBz, ,h, , ( t )) in Eq. (5) to unity.

2828 DONGSHENG DU AND ZHI-ZHONG XING - 47

On the other hand, the magnitude of the decay amplitude A ($Ks) can be obtained by observing B ($Ks) on the Y(4S) resonance:

(3) (B:,,,,,B~,~~,,)~=-~+~$K~)(T+~-), where the final states $Ks and .rr+a- are both CP eigenstates with the opposite CP parities. In this case, lp($Ks ) / = 1 but lp(7-r'~- ) #I . On the Y(4S) resonance, the corresponding branching fraction of this joint decay is given by

It can be seen later on that the term Re[p($Ks ) p ( a + ~ - ) * ] is associated with the CKM angle y, while the term Re[p($Ks)p(a+a- ) ] is relevant to the other two angles a and 0. On condition that measurements (1) and (2) have been realized [i.e., I A ($KS )I2,

1 A (.rr+.rrP ) 12, and I p ( .rr+.rrF ) / have been measured] and the angles a and 0 have been determined, one can employ Eq. (15) to extract the third angle y, which is conventionally suggested to be probed in the decay modes BP,B;-~OK~, etc. [2-41.

It should be noted that B: and B; decays into $Ks and a + a are both flavor-nonspecific processes. In experiments one may sum over all the $Ks and a+a- events - originating from the ( B ~ , ~ ~ ~ ~ B ~ , ~ ~ ~ ~ ) ~ = - state:

in order to extract y. In this case, flavor tagging is unnecessary.

C. Experimental detectability

To probe three angles of the CKM unitarity triangle, we have proposed a few time-integrated measurements of decay rates and CP-violating asymmetries for B ; , B ~ -a+a- , aOaO, $KS and B;-a*ao at the Y(4S). The detector requirements for symmetric e +e - colliders just above the Y(4S) are well understood, because this is the energy range in which the CLEO and ARGUS experiments have been successful in reconstructing B mesons [14,16]. In practice, we need good mass resolution to separate reconstructed C = + 1 BB events (coming from radiative decays of B * or B * from those C = - 1 events coming from direct BB production. On the other hand, good vertex detection would be a very useful tool for re- ducing backgrounds by separating D decay products from B decay products. According to our present knowledge of e + e - colliders, there are no serious problems in technology to meet these requirements [12,13]. Although the symmetric option can avoid the problem of the asymmetric collider, i.e., the uncertain effect of energy asymmetry on luminosity and the necessity of accurate

track measurements close to the interaction point, it suffers a loss of a factor of about 6 in data rate and in non-b& background rejection compared with the Y(4S) resonance. To produce a large sample of C = + 1 B decays for probing CP violation, however, a symmetric e +e - collider can be operated at an energy just above the Y(4S) threshold with either higher luminosity (Lpk- cm-2 s -1 ) or a large running time in compensation for

the loss of resonant enhancement of the cross section [131.

To give one a quantitative feeling of the proposed measurements, we make a rough estimate of the needed number of the B's. The semileptonic modes B ~ - + D * ( ~ o ~ o ) + ~ - v ~ and B ~ - D * ( 2 0 1 0 ) ~ 1 - ~ ~ can be used to tag the decay modes ~ ~ - a + a - , a ~ . r r ~ , $ ~ ~ and B:-a+aO, respectively. Current experimental data give B (D*+1 -V1 )-(4.9*0.8)%, B (D*Ol -v,) =(4.6+1.0)%, and B ( $ K ~ ) - - ~ . O X ~ O - ~ [17], while model-dependent calculations show that B ( T + T - ) - ~ . o x B ( a O a O ) - 1 . 0 ~ l o 6 , and B ( n - + a 0 ) - 6 . 0 ~ 10K6 [18]. Using Eqs. ( l l ) , (14), and (15), the order of branching fractions of these joint decay modes is estimated to be

where we take x ~ 0 . 7 [14] and Ip(.rr.rr) 1 -- 1 for simplicity. Assuming the composite detection efficiency for each mode to be loo%, the C = - 1 BB events needed to measure these joint decay rates should be at least on the order of lo5- lo9 [corresponding to the five modes in Eq. (17)]. Using current values of the CKM matrix elements [17], Ac=+,(.rr+.rr-), ~ ~ = + ~ ( . r r ~ . r r ~ ) , and are estimated to be on the order of 0.1 to 0.5. To discern these partial rate asymmetries at the 30 significance level with only statistical errors, we find that the needed num-

47 - FULLY PROBING THE CABIBBO-KOBAYASHI-MASKAWA , . . 2829

ber of C = + 1 BB events are on the order of lo9, lo1', and lo7, respectively. In fact, one has to consider the efficiencies of tagging, detection and reconstruction as well as the cost of not having the resonant enhancement of the Y(4S). These will make our estimates of the number of BB events, required for measuring joint decay rates and CP-violating asymmetries, multiplied by 10 to 100. So many events are only possible to be accumulated in symmetric e + e - colliders with higher luminosity (LPk> ~ m - ~ s - l ) and a large running time. We ex- pect that this can be realized within the reach of second- round experiments at a symmetric B factory.

111. PROBING THE CKM UNITARITY TRIANGLE

As we have mentioned before, a precise measurement of I Vub I and / Vtd may determine the shape of the C K M unitarity triangle and fix all the C K M parameters including the CP-violating phase. At the Y(4S) resonance, one can reliably extract the ratio Vub /Vcb I from the semileptonic decay modes B:-n--l+vi versus B:-D-I+V~ or B:-p-l+vi versus B:-D *-I+vl . Before the top quark is discovered, an indirect extraction of I V t d , e.g., from the B:-B; mixing parameter x, suffers from some theoretical uncertainties in evaluating hadronic matrix elements [4,5]. However, we can independently probe the three angles a , P, and y, and combine all the information on the sides and angles to construct the C K M unitarity triangle unambiguously.

A. Extraction of a

Following the analysis given in Ref. [6], we first relate the decay amplitudes of B:,B;-n-+n--,n-On-' and B;-T*T' in terms of their isospin components. Be- cause of Bose statistics the final state n-+ n-- (or n - ' ~ ' ) has both the I = 2 and I = 0 components, and the latter con- tains both the tree-level and penguin contributions. On the other hand, the final state T+T' (or n--ro) solely has the I = 2 component from the tree-level transition. Therefore the corresponding overall decay amplitudes for these six modes can be given by [6]

where A, ( 7, ) and A. ( To are the isospin amplitudes for a B (B) decaying into a n-n- pair with I = 2 and I =0, respectively. The charge-conjugated amplitudes ;? obtained from the amplitudes A by simply changing the sign of the C K M phases while the relevant strong phases remain the same. Because B'-n-'n-' occur only through the tree-level diagrams, we have

I A ( T + T ~ ) I = I ~ ( T - ~ ~ ) or 1 ~ ~ 1 = 1 2 ~ 1 , a n d t h e n

where 62 is the I = 2 final-state-interaction phase. For B:,B;+T+T-, the CP-violating term from B:-B; mix-

ing in Eq. (12) can be expressed in terms of isospin components as

where

with the ratios of isospin amplitudes r E A o / A 2 and - - - r = A, / A2. In a similar manner, we obtain

where

On condition that I A (n-=)I2 and I x ( n - ~ ) ~ [or I p ( n - ~ r ) ~ ] have been measured at the 'Y(4S), it is shown in the Ap- pendix that r (7) can be determined up to a twofold ambiguity in the sign of its phase. Consequently the magnitude of R ( R ) is determined uniquely but its phase O (a) can obtain four different values denoted by +a1 and ir02 (*GI and +a2). In general, Eqs. (20) and (22) determine s in(2a) unambiguously. The only exceptions are the very special cases in which some of the ambiguities of Eq. (20) overlap with those of Eq. (22). This happens when either O 1 or O2 equals one of the four phases fa, and + a 2 . In these cases, s in(2a) retains a twofold ambiguity [ 6 ] . This ambiguity, if it exists, may be eliminated by combining all the obtained information on the sides and angles of the C K M unitarity triangle.

B. Extraction of f i

Compared with the angles a and y , P is easier to be probed, e.g., in the B: versus B; decays into C P eigenstate $Ks. For such processes in which the final-state particles have nonzero strangeness, the KO-KO mixing parameter [3]

should appear in their overall decay amplitudes because B:+KO, B; +KO, and interference is possible only due to KO-R0 mixing. If the partial decay rate asymmetry A ($Ks ), = + has been measured just above the Y(4S) resonance, one can use Eq. (1 3) to extract /3 because2

2There is an extra minus sign in p($Ks since $Ks is a CP-odd state.

DONGSHENG DU AND ZHI-ZHONG XING

Unlike those charmless exclusive channels such as B:+K +np and B:+n+.ir-, whose one-gluon-exchange penguin amplitudes may be comparable with their CKM-suppressed tree-level amplitudes, the decay mode B:-$K, gets the penguin contribution at least through a two-gluon-emission diagram (the so-called hair-pin diagram) [19]. Thus neglecting small penguin contributions for such charmed processes is a quite safe approximation, at least at the 0 (a:) level.

C. Extraction of y

Instead of considering B: versus By decays into p°Ks, here we use the joint decay mode 0 -0 (Bd,physBd,phys)C= - -($KS ) (n inp to extract the third angle y on the Y(4S) resonance. On condition that a and 0

have been determined, we find that those two CP-violating terms in Eq. (15) can be expressed, in terms of the isospin amplitudes of B -nn, as

With the help of Eqs. (3) and (26), Eq. (15) is then transformed into

where y is the only unknown quantity if measurements (1)-(3) have been realized at the Y(4S). Neglecting the penguin contribution to B:,B: -n+.rrp, i.e., I R / = 1 and O =0, Eq. (27) is simplified as

This approximate result is in agreement with that obtained by Bigi [5] except for a different coefficient from wave-function normalization.

We may also consider the joint decay mode ( B ~ , , ~ ~ ~ ~ ~ , ~ ~ ~ , ) ~ = - 1-(D +D - )(n+.rrp) for probing y on the Y(4S) resonance. However, B:-D +D - can occur through the one-gluon-exchange penguin diagram so that it is not as good as B:+$K,. In fact, the four decay modes B: -a+nO, B:-T+T-, nO.rrO, $Ks and their charge-conjugated processes are enough for us to extract three angles of the CKM unitarity triangle at the Y(4S). As we have seen, this model-independent treatment does not (in principle) suffer from the presence of final-state strong interactions.

IV. SUMMARY AND CONCLUSION

We have proposed a few time-integrated measurements of joint decay rates and CP-violating asymmetries for B:,@ -n-+.rr-,nO.rrO, $KS and B,'-ninO both on the Y(4S) resonance and just above it. The experimental detectability for these processes was discussed. By means of the aforementioned measurements, we have shown that it is possible to probe three angles of the CKM uni- - tarity triangle unambiguously, even in the presence of significant final-state strong interactions and in the absence of B: versus B: decay modes. Combining all the

ments of the b-flavored mesonic or baryonic decays. Compared with those previous studies, our approach needs fewer B-meson decay modes and can be realized at one specific experimental situation, the Y(4S). No doubt, the Y(4S) is one of the richest B-meson sources today, and the relevant techniques are being developed by exper- imenters to enhance both C = - 1 and C = + 1 BB event number [12]. It is evident that a B-factory capable of luminosity cmp2 s and center-of-mass energy around 10 GeV will produce a wealth of information on weak decays and CP violation. We hope that our analysis can be tested within the reach of second-round experiments at a symmetric B factory running on the Y(4S) and just above it.

ACKNOWLEDGMENTS

One of the authors (Z.-Z.X.) would like to thank Inter- national Centre for Theoretical Physics (ICTP) for its hospitality and financial support. He is also grateful to A. H. Chan, F. Hussain, and D. S. Lin for their interest- ing discussions at ICTP. This work was supported in part by the National Natural Science Foundation of China.

APPENDIX

information on the sides and angles, one can construct On condition that the magnitudes of the decay ampli- the CKM unitarity triangle reliably. tudes for B:,E:-n+.rr-,n0n0 and B$+n ' . i ro have been

Up to the present, any attempts at probing the CKM measured at the Y(4S) or somewhere else, we can define unitarity triangle are strongly dependent upon measure- the observables

47 FULLY PROBING THE CABIBBO-KOBAYASHI-MASKAWA . . .

A + T ~ - /;?(r+rTT-)I2 r and f a r e given by a = a r I A (r+r0)/' ' I Z ( r - r0 )12 '

( A l ) I r = d 3 ( a + b ) - 2 ,

b = 1 ~ ( n ~ r ~ ) ' - A ( r O r O ) ' b = 6= karccos I 6 b - 3 a - 2 I A (r+r0)12 ' 1 ;?(r-r0)12 ' 4 d 3 ( a + b ) - 2 1 '

By means of Eqs. (18) and ( A l ) , we obtain two coupled equations for the ratios of isospin amplitudes rand E

and

Accordingly the corresponding magnitudes and phases of

and

Clearly r (F ) can be determined up to a twofold ambiguity in the sign of its phase. Unlike the analysis made in Ref. [ 6 ] , here extracting r and T' needs no geometrical con- siderations of isospin triangles for ~ ~ - t r + r - , r ~ r ~ , B,: -r+rO, and their charge-conjugated processes.

-

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fully probing the cabibbo-kobayashi-maskawa unitarity triangle at the Υ(4 s ...

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