

 23 Oct 95

LMU-13/95

September, 1995

On Testing Unitarity of the Quark Mixing Matrix 1

Zhi-zhong XING 2 Sektion Physik, Theoretische Physik, Universit"at M"unchen,

Theresienstrasse 37, D-80333 M"unchen, Germany

Abstract Unitarity triangles and characteristic measurables of the 3 \Theta 3 Cabibbo-KobayashiMaskawa (CKM) matrix are discussed. Beyond the 3 \Theta 3 CKM scheme we make a rephasing-invariant generalization of the Gronau-Wyler-Dunietz approach to determine the weak phase shift in B\Sigma u ! DK\Sigma or B0d ! DK\Lambda 0 vs _B0d ! D _K\Lambda 0, which is only sensitive to the underlying new physics in D0 \Gamma _D0 mixing. We also show that the weak angle fl j arg(\Gamma V \Lambda ubV \Lambda cdVudVcb) is possible to be determined from the CP asymmetries of some Bd decays, even in a non-standard model with an extended quark sector. Brief comments are given on tests of unitarity of the 3 \Theta 3 CKM matrix.

1A talk given at the Conference on Production and Decay of Hyperons, Charm and Beauty Hadrons, Strasbourg, France, September 5 - 8, 1995 (to appear in the conference proceedings)

2Electronic address: Xing@hep.physik.uni-muenchen.de

1

Part I. Within the 3 \Theta 3 CKM Scheme In this part we explore some consequences of unitarity of the 3 \Theta 3 CKM matrix. An instructive discussion is given about the characteristic measurables of the CKM matrix and their relations with the unitarity triangles. We also take a look into the feature of the weak angle fl in exclusive B decays and CP asymmetries.

A. Unitarity Triangles and Parametrizations In the minimal standard electroweak model, the 3 \Theta 3 CKM matrix V provides a natural description of quark mixing and CP violation. Unitarity is the only but powerful constraint, imposed by the model itself, on V . This restriction can be expressed as two sets of orthogonalityplus-normalization conditions:X

ff=d;s;b (

V \Lambda iffVjff) = ffiij ; X

i=u;c;t (

V \Lambda iffVifi) = ffifffi ; (1)

where Latin subscripts run over the up-type quarks (u; c; t) and Greek ones over the down-type quarks (d; s; b). In the complex plane the six orthogonality relations given above correspond to six triangles (see Fig. 1), the so-called unitarity triangles.

By use of the unitarity conditions in Eq. (1), one is able to parametrize the CKM matrix in various ways. Several popular parametrizations are given in terms of three Euler angles and one CP -violating phase [1, 2]. A parametrization is also available in terms of four independent modulus of the CKM matrix elements [3], or four independent sides of the unitarity triangles, or four independent angles of the unitarity triangles [4]. To analyze data, the Wolfenstein parametrization is most convenient because it straightforwardly reflects the hierarchy of quark mixings [5]. However, the unitarity conditions in the original Wolfenstein form are satisfied only to the accuracy of O(*4), which is insufficient for a self-consistent description of all properties of the CKM matrix. A useful modified version can be given as [6]

V = 0BB@

1 \Gamma 12 *2 \Gamma 18*4 * A*3(ae \Gamma ij)\Gamma * h1 + 12A2*4(2ae \Gamma 1) + iA2*4ji 1 \Gamma 12*2 \Gamma 18 \Gamma 4A2 + 1\Delta *4 A*2

A*3(1 \Gamma ae \Gamma ij) \Gamma A*2 h1 + 12*2(2ae \Gamma 1) + i*2ji 1 \Gamma 12 A2*4

1CCA

; (2)

where unitarity is kept up to O(*6). This degree of accuracy is enough for V to confront all precise experimental data in the near future.

All measurables of CP violation in the standard model are sensitively related to the angles of the unitarity triangles. It is instructive to express the nine inner angles in terms of the Wolfenstein

2

V\Lambda cbVtb LL

LL V\Lambda cdVtd

,,,,, ,,,,, ,, V\Lambda

csVts

1 2

3

[u]

V\Lambda csVcbX XXXXX

XXXXXX V\Lambda tsVtb

EE

E V

\Lambda usVub

2 9

6

[d]

V\Lambda tdVud BB

BBV \Lambda tsVus

\Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi V\Lambda tbVub

4 5

6

[c]

V\Lambda ubVud HHH

HHH

HHV\Lambda tbVtd

\Theta \Theta \Theta \Theta V

\Lambda cbVcd

5 7

1

[s]

V\Lambda udVcd CC

CV \Lambda ubVcb

(((((((( (((((((

V\Lambda usVcs

7 8

9

[t]

V\Lambda cdVcshh hhhhhhhh

hhhhV\Lambda udVus

DD

D V

\Lambda tdVts

8 3

4

[b]

Figure 1: The unitarity triangles of the 3 \Theta 3 CKM matrix in the complex plane. Each triangle is named in terms of the quark flavor that does not appear in its three sides. Note that the six triangles have the same area, and they only have nine different inner angles (versus eighteen different sides).

3

parameters. In lowest-order approximations, we obtain:

tan(6 1) ss tan(6 4) ss \Gamma tan(6 3) ss j1 \Gamma ae ; (3a)

tan(6 6) ss tan(6 7) ss \Gamma tan(6 9) ss jae ; (3b) and

tan(6 2) ss *2j ; tan(6 8) ss A2*4j ; tan(6 5) ss jae(ae \Gamma 1) + j2 : (3c)

It is clear that all the six triangles collapse into lines if j = 0. Conventionally one uses ff = 6 5, fi = 6 1, and fl = 6 7 to denote the three angles of unitarity triangle [s], which will be overdetermined at B-meson factories. Here it is worth emphasizing that the angles of the other unitarity triangles also have chances to be determined with the development of more precise experiments. We expect that the approximate relations given in Eq. (3) can be tested by various measurements of CP violation in the near future, either within or beyond the K-, D- and B-meson systems.

B. Characteristic Measurables The 3\Theta 3 CKM matrix has four independent characteristic measurables. The first one is a universal measure of CP violation in weak interactions of quarks, the so-called Jarlskog parameter J [7]:

J = fififiIm iViffVjfiV \Lambda ifiV \Lambda jffjfififi (i 6= j; ff 6= fi) : (4) It is straightforward to show that all the six unitarity triangles have the same area J =2, although their shapes are quite different (see Fig. 1 for illustration). If there is no CP violation, i.e., J = 0, then all six unitarity triangles collapse into lines. An interesting point is that J can be determined from three sides of each triangle, which are not directly related to any CP -violating signal.

The structure of the 3\Theta 3 CKM matrix is basically characterized by its two off-diagonal asymmetries [8]. They are denoted by A1 about the Vud \Gamma Vcs \Gamma Vtb axis [9] and A2 about the Vub \Gamma Vcs \Gamma Vtd axis (see Fig. 2 for illustration):

A1 = jVusj2 \Gamma jVcdj2 = jVcbj2 \Gamma jVtsj2 = jVtdj2 \Gamma jVubj2 ; (5a) A2 = jVusj2 \Gamma jVcbj2 = jVcdj2 \Gamma jVtsj2 = jVtbj2 \Gamma jVudj2 : (5b) The above relations are direct consequences of the normalization conditions given in Eq. (1). Note that the asymmetry parameters A1;2 are independent of each other, and they are independent of the CP -violating parameter J .

Note that the element Vcs sits at the centre of the CKM matrix and is independent of the offdiagonal asymmetries A1;2 (see Fig. 2). jVcsj is indeed the fourth characteristic measurable of the 3\Theta 3

4

_**^ud _**^us _**^ub _**^cd _**^cs _**^cb _**^td _**^ts _**^tb @@

@@ @@

@@ @@

@@I

@@ @@ @@

@@ @@

@@I

@@ @@ @@

@@ @@

@@IA

1

\Phi \Phi \Phi \Delta \Delta \Delta ff \Phi \Phi

\Phi \Phi \Phi

\Phi *

\Delta \Delta \Delta \Delta \Delta \Delta

\Phi \Phi \Phi \Delta

\Delta \Delta ff

(a)

_**^ud _**^us _**^ub _**^cd _**^cs _**^cb _**^td _**^ts _**^tb\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma `

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma `

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma

\Gamma \Gamma A 2

HHH A

AAUHH HHH HY

AA

AA

AA HHH A

AAU

(b) Figure 2: The off-diagonal asymmetries of the 3\Theta 3 CKM matrix: (a) A1 about the Vud \Gamma Vcs\Gamma Vtb axis; and (b) A2 about the Vub \Gamma Vcs \Gamma Vtd axis.

CKM matrix [8]. In contrast, either 2 \Theta 2 or 4 \Theta 4 unitary matrix has not such a "central" element to characterize its geometrical structure. Of course, one can use J ; A1;2 and jVcsj2 to parametrize the whole CKM matrix V . In such a parametrization, the matrix elements jVusj, jVcdj, jVcbj and jVtsj are independent of the CP -violating parameter J , while jVudj, jVubj, jVtdj and jVtbj depend upon all the four parameters.

By use of the modified Wolfenstein parametrization in Eq. (2), J and A1;2 are given as [8]

J ss A2*6j ; A1 ss A2*6(1 \Gamma 2ae) ; A2 ss *2 i1 \Gamma A2*2j : (6) It is clear that A2 ?? A1 and A1 , J . Both A1 and A2 are independent of j, a parameter signifying CP violation. The allowed ranges of the Wolfenstein parameters have been analyzed by many authors with the help of current experimental data. Taking only the central values of *; A; ae and j into account [10], we find A2=A1 * 400, A1 , 10\Gamma 5 \Gamma 10\Gamma 4, and J , 10\Gamma 5.

From the direct measurements jVusj = 0:2205 \Sigma 0:0018 and jVcdj = 0:204 \Sigma 0:017 [1], we observe that the possibility of A1 ss 0 has not been completely ruled out. But one is convinced that A1 ? 0 should be of the dominant possibility. If this point is really true, then we can find a definite hierarchy for the nine CKM matrix elements:

jVtbj ? jVudj ? jVcsj ?? jVusj ? jVcdj

?? jVcbj ? jVtsj ?? jVtdj ? jVubj :

(7)

5

Since jVub=Vcbj = 0:08 \Sigma 0:02 [1], it is certain that all jViffj are nonzero. The above interesting result reflects our present understanding of the magnitudes of quark mixings.

Geometrically the nonvanishing A1 and A2 imply some differences in the six unitarity triangles. In general, these triangles have nine different inner angles and eighteen different sides, thus their shapes are different from one another (see Fig. 1). If A1 or A2 were vanishing, one would find three equivalence relations among the six triangles [8]:

A1 = 0 =) [u] ,= [d] ; [c] ,= [s] ; [t] ,= [b] (8a) with 6 1 = 6 6, 6 3 = 6 9, 6 4 = 6 7; and

A2 = 0 =) [u] ,= [b] ; [c] ,= [s] ; [t] ,= [d] (8b) with 6 1 = 6 4, 6 2 = 6 8, 6 6 = 6 7. In either case, the six unitarity triangles have six different inner angles and nine different sides. As a consequence, the CKM matrix can be parametrized by use of three independent quantities. In view of Eq. (6), there is no possibility for A2 = 0. The possibility ofA

1 = 0, which requires ae ss 0:5, is only allowed on the extreme margin of the existing data and should be absolutely ruled out in the near future. From the point of view that the quark mixing matrix

V is basically governed by the quark mass matrices MU and MD [11], we stress that the nonzero off-diagonal asymmetries of V would imply a kind of symmetry breaking appearing in MU;D. Thus it is suggestive that the study of specific patterns of MU;D (and their underlying dynamics) may start from the symmetry limit A1 = 0 (or J = 0) at a superheavy scale.

C. Triangle [s] and Angle fl Among six unitarity triangles of the 3 \Theta 3 CKM matrix, triangle [s] is of most interest for studies of CP violation in B-meson systems. Its three inner angles are commonly denoted by Greek letters ff, fi and fl (see Fig. 3 for illustration):

ff = arg \Gamma V

\Lambda tbVtd

V \Lambda ubVud ! ; fi = arg \Gamma

V \Lambda cbVcd

V \Lambda tbVtd ! ; fl = arg \Gamma

V \Lambda ubVud

V \Lambda cbVcd ! : (9)

Of course,

ff + fi + fl = 1800 (10)

is a trivial consequence of the above definition, no matter whether the vectors V \Lambda ubVud, V \Lambda cbVcd and V \Lambda tbVtd form a closed triangle or not in the complex plane. This point can be seen more clearly in the following sections. Hence an experimental examination of the sum rule in Eq. (10) does not make much sense for testing unitarity of the 3 \Theta 3 CKM matrix. In the literature there are still some ambiguities or misleading remarks associated with this problem. It is therefore necessary to make a clarification.

6

---'' ''

''

''

''

''

''

''

''

''

''

''

''

''

''

''

''

''

///

@@ @@ @@ @@ @@ @@

@@ @@ @@ @@ @@ @@

@@ @@ @@ @@ @@ @@

IIIV \Lambda ubVud

V\Lambda cbVcd

V\Lambda tbVtd fl fi

ff

Figure 3: Unitarity triangle [s]. It is known that the angles ff and fi can directly enter the CP asymmetries of some Bd transitions, which are induced by the interplay of decay and B0d \Gamma _B0d mixing (see, e.g., Ref. [12]). The typical examples are B0d vs _B0d ! KS and ss+ss\Gamma , whose CP asymmetries are governed by fi and ff respectively. Taking into account the penguin contributions to Bd ! ss+ss\Gamma , one may extract ff from the correlative decay modes Bd ! ss+ss\Gamma ; ss0ss0 and B\Sigma d ! ss\Sigma ss0 with the help of isospin relations and time-dependent measurements [13].

Since the CKM factors V \Lambda ubVud and V \Lambda cbVcd come from the quark processes _b ! _uu _d and _b ! _cc _d respectively, their relative phase fl cannot be related to any CP -violating observable of Bu, Bd or Bs decays in an exact way. All the proposed approaches for extraction of fl are only able to probe the approximate magnitude of fl, and they may work to a good degree of accuracy only in the assumption of unitarity of the 3 \Theta 3 CKM matrix. For example, the CP asymmetry of Bs ! ae0KS is indeed governed by the phase of (V \Lambda tbV \Lambda udV \Lambda csVtsVubVcd) other than that of (\Gamma V \Lambda ubV \Lambda cdVudVcb), but the two phases are approximately identical if triangle [d] closes and its side jV \Lambda usVubj is vanishingly short [14]. In and only in weak B decays of the type

B+c \Gamma ! (ss+; ae+; a+1 ) + (D0; D\Lambda 0) ; B+c \Gamma ! (ss0; ae0; a01; !) + (D+; D\Lambda +) ; (11)

a pure fl can enter the decay rates. These decays occur through the spectator diagrams (_b ! _uu _d), the annihilation diagrams (_b ! _cc _d) and the penguin diagrams (_b ! _d). Due to the orthogonality condition performed in triangle [s], one of the three CKM factors in the penguin amplitudes (i.e., V \Lambda tbVtd) can be absorbed. Thus the overall amplitude of every decay mode in Eq. (11) contains two components, associated with V \Lambda ubVud and V \Lambda cbVcd respectively. For instance, the CP asymmetry in B+c ! ss+D0 vs B\Gamma c ! ss\Gamma _D0 is unambiguously proportional to sin fl. But this asymmetry involves large uncertainty from final-state strong interactions [15], hence in practice the above B\Sigma c transitions cannot be used to extract fl.

7

Gronau and Wyler have developed an approach to extract fl from the decay modes B\Sigma u ! D01(2)K\Sigma , where D01(2) = \Theta D0 + (\Gamma ) _D0\Lambda =p2 denotes a CP even (odd) state [16]. Following the same idea, Dunietz carried out an analysis of B0d ! D01(2)K\Lambda 0 vs _B0d ! D01(2) _K\Lambda 0 [17]. Note that the relevant weak phase shift in these transitions is indeed

fl0 = arg V

\Lambda ubVcs

V \Lambda cbVus \Delta

VusV \Lambda cs V \Lambda usVcs ! = arg

V \Lambda ubVus

V \Lambda cbVcs ! (12)

other than fl. In the above formula we have taken into account the weak phases from D0 \Gamma _D0 mixing, otherwise, fl0 is not rephasing-invariant. However, fl0 ss fl is a very good approximation within the 3 \Theta 3 CKM scheme. This point can be clearly seen as follows. From the orthogonality relation shown by triangle [t], we find that V \Lambda udVcd ss \Gamma V \Lambda usVcs holds up to the accuracy of O(*5). Hence one obtains fl ss arg(V \Lambda ubV \Lambda csVcbVus), equal to fl0 in Eq. (12). It should be noted that fl0 ss fl will not be valid if unitarity of the 3 \Theta 3 CKM matrix is violated. In this case, an apparent weak phase from new physics may enter fl0 through D0 \Gamma _D0 mixing.

Part II. Beyond the 3 \Theta 3 CKM Scheme In this part we first make a rephasing-invariant generalization of the Gronau-Wyler-Dunietz approach to determine a weak phase beyond the 3 \Theta 3 CKM scheme. In either the model of four quark families or that of Z-mediated flavor changing neutral currents (FCNC's), we show that fl is possible to be determined from CP asymmetries of some Bd decays with the help of Eq. (10), although both ff and fi may be significantly contaminated by new physics. Finally we comment briefly on tests of unitarity of the 3 \Theta 3 CKM matrix.

D. Determination of a Weak Phase Before the 3 \Theta 3 CKM mechanism passes stringent tests, it is useful to develop some modelindependent approaches for extraction of the weak phases from specific B-meson decays. Such ideas rely on the fact that there is no significant effect of new physics on the direct decay of b quark via the tree-level W -mediated diagrams [12]. Violation of the unitarity conditions in Eq. (1) mainly manifests itself in B0 \Gamma _B0 mixing (or D0 \Gamma _D0 mixing) and loop-induced penguin channels. To illustrate, we make a generalization of the Gronau-Wyler-Dunietz approach to determine a weak phase from the decay modes B\Sigma u ! DK\Sigma or from B0d ! DK\Lambda 0 vs _B0d ! D _K\Lambda 0. This weak phase can be denoted, generally and rephasing-invariantly, as

' = arg V

\Lambda ubVcs

V \Lambda cbVus \Delta

qD pD ! ; (13)

8

_b

d B0d

_u

d c

_s K\Lambda 0

D0 =) DL;H* _

oe

oe -

oe

(a)

_b

d B0d

_c

d u

_s K\Lambda 0

_D0 =) DL;H* _

oe

oe -

oe

(b) Figure 4: The dominant tree-level W -mediated diagrams for B0d ! DLK\Lambda 0 or DH K\Lambda 0. where the complex parameters qD and pD connect D0 and _D0 to their mass eigenstates through

jDLi = pDjD0i + qDj _D0ij

DHi = pDjD0i \Gamma qDj _D0i (14)

(with jpDj2 + jqDj2 = 1). As pointed out by Blaylock, Seiden and Nir [18], jqD=pDj ss 1 is a very reasonable approximation in all reasonable models of D0 \Gamma _D0 mixing with new physics. This implies that CP violation in the D0 \Gamma _D0 mass matrix is negligible. In the minimal standard model, qD=pD = (VusV \Lambda cs)=(V \Lambda usVcs), ' turns out to be fl0 as given by Eq. (12). For some non-standard models like those listed in Ref. [18], however, qD=pD may introduce a significant phase shift into '.

Let us take the decay modes B0d ! DK\Lambda 0 and _B0d ! D _K\Lambda 0 for example. Since B0d ! D0K0, B0d ! _D0K0 and their CP -conjugate counterparts occur only through the tree-level W -mediated quark diagrams (see Fig. 4 for illustration), we parametrize the transition amplitudes of Bd ! DL(H)K\Lambda as follows:

A(B0d ! DL(H)K\Lambda 0) = p\Lambda D (V \Lambda ubVcs) Aa eiffia + (\Gamma ) q\Lambda D (V \Lambda cbVus) Ab eiffib ;

A( _B0d ! DL(H) _K\Lambda 0) = p\Lambda D (VcbV \Lambda us) Ab eiffib + (\Gamma ) q\Lambda D (VubV \Lambda cs) Aa eiffia ; (15)

where Aa and Ab are real (positive) hadronic matrix elements, and ffia and ffib are the corresponding strong phases. Unlike ref. [17], here one cannot use a simple triangular relation to describe the above decay amplitudes. Some specific measurements are possible to establish the following (dimensionless) decay-rate asymmetry:

\Delta ij j jA(B

0 d ! DiK\Lambda 0)j2 \Gamma jA( _B0d ! Di _K\Lambda 0)j2j

A(B0d ! D0K\Lambda 0)j jA(B0d ! _D0K\Lambda 0)j (16)

with i; j = L or H. Denoting the strong phase difference ffib \Gamma ffia j ffi and using the reasonable approximation jqD=pDj ss 1, we explicitly obtain

\Delta LL = 2 sin ' sin ffi ; \Delta HH = \Gamma 2 sin ' sin ffi ; \Delta LH = 2 cos ' cos ffi ; \Delta HL = \Gamma 2 cos ' cos ffi : (17)

In experiments, the relations \Delta LL = \Gamma \Delta HH and \Delta LH = \Gamma \Delta HL can be well examined. Note that only the asymmetries \Delta LL and \Delta HH represent CP violation, and they vanish if the weak phase shift ' vanishes.

9

Obviously Eq. (17) can be used to extract '. If the CP asymmetries \Delta LL and \Delta HH were substantially suppressed due to the smallness of ffi, then \Delta LH = \Gamma \Delta HL ss 2 cos ' would be a good approximation. In general, we have`

\Delta LL sin ''

2

+ ` \Delta LHcos ' '

2

= ` \Delta LLsin ffi '

2

+ `\Delta LHcos ffi '

2

= 4 : (18)

Note that the angle ' (or ffi) extracted from the above equation has a few ambiguities in its size and sign. This kind of ambiguities can be removed by studying a set of exclusive decay modes Bd ! (D0; _D0; DL; DH) + X0, where X0 is any mode with flavor content (_sd) or (s _d), as long as its net strangeness can be unambiguously deduced [17]. All such processes have a common weak phase shift ', but their strong phase shifts ffi should be different from one another.

In a similar way, one can make a rephasing-invariant generalization of Gronau and Wyler's work in Ref. [16], so as to extract the weak phase shift ' from the processes B\Sigma u ! DL(H)K\Sigma , etc [19].

E. fl in two Models with an Extended Quark Sector A violation of unitarity of the 3 \Theta 3 CKM matrix implies

\Omega j V \Lambda ubVud + V \Lambda cbVcd + V \Lambda tbVtd 6= 0 ; (19) i.e., the three sides of triangle [s] do not close. In this case, V \Lambda ubVud, V \Lambda cbVcd, V \Lambda tbVtd and \Gamma \Omega form a quadrangle in the complex plane (see Fig. 5 for illustration). Note that the angle sum rule in Eq. (10) remains valid, although the magnitudes of ff, fi and fl all have been contaminated by new physics. Beyond the minimal standard model, here we consider two basic approaches to extend the quark sector [12, 20], which allow breaking of the unitarity constraints in Eq. (1):

(1) In the standard model with four quark families, quark mixing is described by a 4 \Theta 4 unitary matrix. Thus \Omega = \Gamma V \Lambda t0bVt0d, where t

0 denotes the fourth up-type quark. The t0 quark can contribute

to B0d \Gamma _B0d mixing via box diagrams, proportional to (Vt0bV \Lambda t0d)2.

(2) In the model with an iso-singlet down-type quark, \Omega = Udb, where Udb is a non-diagonal coupling of the Z gauge boson. There may be tree-level Z-mediated FCNC's contributing to direct b decays, but they are negligibly small in comparison with the tree-level W -mediated channels. However, B0d \Gamma _B0d mixing is possible to be significantly modified by tree-level Z-mediated diagrams, proportional to (Udb)2.

For both cases, the presence of \Omega induces a new weak phase \Phi to the B0d \Gamma _B0d mixing parameter of the minimal standard model:

qB pB =

V \Lambda tbVtd VtbV \Lambda td =)

qB pB =

V \Lambda tbVtd VtbV \Lambda td e

i\Phi ; (20)

10

---\Delta \Delta \Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta \Delta

\Delta

\Delta ff\Delta ff\Delta ff

AA AA AA AA AA AA AA A

AA AA AA AA AA AA AA A

AA AA AA AA AA AA AA A

AKAKAK ,,,,,

,, ,,,,,

,, ,,,,,

,,999qqqqqqqqqq

qqqqqqq

q

qqqqqqq qqqqqqq q

V\Lambda ubVud

V\Lambda cbVcd

V\Lambda tbVtd \Gamma \Omega

fl fi

ff

Figure 5: The unitarity quadrangle in the complex plane. where

\Phi = arg "1 + 2 V

\Lambda t

0bVt0d

V \Lambda tbVtd

E(t; t

0)

E(t; t) +

(V \Lambda t0bVt0d)2

(V \Lambda tbVtd)2

E(t

0; t0)

E(t; t) # (21a) for the model of four quark families; and

\Phi = arg "1 + U

2 db

(V \Lambda tbVtd)2

4ss sin2 `w

ffeE(t; t) # (21b)

for the model of Z-mediated FCNC's. In the above equations, E(i; j) denotes the box-diagram function with internal i and j quarks [21], ffe and `w are the standard electroweak parameters.

Let us show how \Phi enters the CP asymmetries induced by the interplay of decay and B0d \Gamma _B0d mixing. This kind of CP -violating signals can be described by the following rephasing-invariant quantity:

,f = Im " qBp

B

A( _B0d ! f ) A(B0d ! f ) # : (22)

For Bd ! KS in the minimal standard model, the penguin contribution is expected to be negligibly small and K0 \Gamma _K0 mixing in the final state leads to an additional weak phase (V \Lambda cdVcs)=(VcdV \Lambda cs). These two arguments may also be valid for either of the above two models with an extended quark sector [12, 22]. Therefore, one can obtain

,KS = Im "\Gamma V

\Lambda tbVtd

VtbV \Lambda td e

i\Phi \Delta V \Lambda csVcb

VcsV \Lambda cb \Delta

V \Lambda cdVcs VcdV \Lambda cs # = sin(2fi \Gamma \Phi ) : (23)

11

For Bd ! ss+ss\Gamma , we have

,ss+ss\Gamma = Im " V

\Lambda tbVtd

VtbV \Lambda td e

i\Phi \Delta V \Lambda udVub

VudV \Lambda ub # = sin(2ff + \Phi ) ; (24)

if we neglect the penguin amplitudes. Uncertainties arising from the penguin contribution can be eliminated by use of the isospin relations among Bd ! ss+ss\Gamma ; ss0ss0 and B\Sigma u ! ss\Sigma ss0 [13]. Although either V \Lambda t0bVt0d or Udb may contaminate the penguin amplitudes, the above phase combination (2ff + \Phi ) can still be extracted from an isospin analysis. The reason is that the I = 2 amplitude of B ! ssss only contains the tree-level quark diagrams with V \Lambda udVub or VudV \Lambda ub, and all other weak and strong phases can be absorbed into a set of complex parameters which are determinable from the isospin triangles.

From Eqs. (23) and (24) we find

arcsin (,KS ) + arcsin (,ss+ss\Gamma ) = 2 (ff + fi) = 3600 \Gamma 2fl ; (25) irrelevant to the weak phase \Phi . Thus fl could be extracted from the CP asymmetries of Bd ! KS and ss+ss\Gamma , even though B0d \Gamma _B0d mixing is contaminated due to the presence of new physics.

F. On Testing Unitarity of the 3 \Theta 3 CKM Matrix Finally we comment briefly on tests of unitarity of the 3 \Theta 3 CKM matrix. From current experimental constraints on various non-standard electroweak models, we know that no new physics can significantly affect direct decays of b quark. In addition to D0 \Gamma _D0 mixing, B0 \Gamma _B0 mixing and loop-induced penguin transitions are two possible places to accommodate new physics beyond the minimal standard model. Thus the CP asymmetries of B decays, induced either by the interplay of decay and B0 \Gamma _B0 mixing or by penguin diagrams, could be contaminated by new physics. This leads to some difficulties for us to determine a specific weak phase cleanly and to test the unitarity conditions in Eq. (1) meaningfully.

(1) First of all, the normalization relations of unitarity in Eq. (1) can be well checked with the help of more precise data on jViffj (i = u; c and ff = d; s; b) and on jVtbj. Among these seven matrix elements, it is urgent to minimize the experimental (and theoretical) errors associated with the values of jVubj, jVcdj, jVcsj and jVcbj. A determination of jVtbj will be available from the top-quark lifetime. At least, one can check the following three conditions:

jVudj2 + jVusj2 + jVubj2 = 1 (?) ;j

Vcdj2 + jVcsj2 + jVcbj2 = 1 (?) ;j

Vubj2 + jVcbj2 + jVtbj2 = 1 (?) :

(26)

A clean extraction of jVtdj and jVtsj from direct production or decays of the top quark will be very difficult in experiments. Although these two elements can in principle be determined from B0d \Gamma _B0d

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and B0s \Gamma _B0s mixings respectively, this approach itself could be affected by unknown new physics. To test the validity of Eq. (26) up to O(*5) or O(*6), of course, much effort is needed to make.

(2) Within the 3 \Theta 3 CKM scheme, the smallness of A1 is governed by *6 and it is insensitive to the exact values of other Wolfenstein parameters. This implies that jVusj \Gamma jVcdj , *5 ! 10\Gamma 3 is a reliable constraint that can now be obtained from unitarity. However, the difference between the experimental central values of jVusj and jVcdj is about 0.0165, significantly larger than the above unitarity restriction. Since the relative errors associated with jVusj and jVcdj are about 0:8% and 8:3% respectively [1], we expect that more precise measurements should enhance the existing value of jVcdj and lead it to approach jVusj closely. Within the accuracy of 0:1% for both elements, a clear deviation of jVcdj from jVusj would imply unitarity breaking in the 3 \Theta 3 CKM matrix. From Eqs. (2) and (7), we notice that within the 3 \Theta 3 CKM scheme jVudj is larger than jVcsj and their difference is of the order *4 , 10\Gamma 3. The current experimental data give jVudj = 0:9744 \Sigma 0:0010 and jVcsj = 1:01 \Sigma 0:18 [1]. Of course, the precision associated with jVcsj is very unsatisfactory and need be improved in the forthcoming experiments [23]. Within the accuracy of 1%, the measured value of jVcsj should be indistinguishable from that of jVudj, as required by unitarity.

(3) In comparison with ff and fi, fl could play an interesting role in testing unitarity of the 3\Theta 3 CKM matrix. Among six unitarity triangles in Fig. 1, only triangle [t] is determinable from measurements of the six matrix elements in the first two rows of V . If its three sides can be constrained up to the accuracy of O(*5), then the angle fl (i.e., 6 7) are calculable through

fl = arccos jV

\Lambda udVcdj2 + jV \Lambda ubVcbj2 \Gamma jV \Lambda usVcsj2

2 jV \Lambda udVcdj jV \Lambda ubVcbj ! : (27)

We have observed in the preceding sections that ' = fl0 ss fl for the minimal standard model. Beyond the 3 \Theta 3 CKM scheme, fl is expected to be extracted from CP asymmetries in some B-meson decays, as illustrated in Eqs. (23 - 25). In this case, we may generally have ' 6= fl. Thus a comparison between the values of the relevant weak phases (e.g., fl, fl0, 6 7 and ') obtained from different approaches should be able to check the orthogonality conditions in Eq. (1).

I would like to thank H. Fritzsch for his warm hospitality and constant encouragement. My gratitude goes also to C. Jin, D.M. Kaplan, H. Simma and D.D. Wu for useful discussions. I owe a great debt to A. Fridman and J.P. Engel for their financial support, so that I was able to participate in this nice workshop in this nice city. My research was supported by the Alexander von Humboldt Foundation of Germany.

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