Volume 242, number 1 PHYSICS LETTERS B 31 May 1990

T-ODD AND CP-ODD TRIPLE M O M E N T U M CORRELATIONS IN T H E EXCLUSIVE S E M I - L E P T O N I C B O T T O M M E S O N DECAY B ~ D*+ ~ + v~

J.G. KORNER, K. SCHILCHER and Y.L. W U t Institut f~r Physik, Yohannes-Gutenberg-Universitdt, Staudinger Weg 7, Postfach 3980. D-6500 Mainz, FRG

Received 17 February 1990

We study T-odd triple momentum correlations in exclusive semi,leptonic (S~ ~ bottom meson decays B--, D*( ~ Dn )+ ~ + v~. We define asymmetry ratios that measure these T-odd triple momentum correlation effects. We provide a careful discussion of possible unitarity contributions to the asymmetry ratios. The conclusion is that strong interaction unitarity contributions are small or, in the case of one particular T-odd observable, absent. CP-violating contributions to the asymmetry ratios would have to come from new non-standard sources as there are no standard model contributions. Possible new CP-violating contributions are parametrized in terms of an effective four-fermion hamiltonian. Their effect on the T-odd asymmetry ratios are computed in terms of specific quark models describing S~ exclusive heavy meson decays.

CP-violating effects have thus far been observed only in the K ° - K ° system. Despite the fact that they can be accommodated in the standard model of electroweak interactions via a phase angle in the three-generation Ko- bayashi-Maskawa matrix, a deep unders tanding of the origin of CP-violation is still missing.

It is generally believed that further insight into the underlying mechanism of CP-violation can be obtained from future high statistics experiments in the b-quark sector.

In this paper we study T-odd triple m o m e n t u m correlations in the exclusive semi-leptonic (S~) decay B ~ D*(--, Dn)~v~, which, in the absence of final state interaction effects, would signal true CP-violating effects. We enumerate the T-odd observables occurring in this process and define asymmetry ratios that measure them.

We carefully discuss possible sources of unitari ty contr ibutions to the T-odd observables. We find them to be small in general. In particular we find that one of our T-odd observables is not affected by strong interaction unitari ty effects, whereas another one is not affected by CP-violating contributions. Experimentally more de- manding is to take sums and differences of T-odd asymmetry ratios of SI~ B (b) and B (13) decays and thereby to establish the absence or presence of CP-violating a n d / o r uni tar i ty corrections.

In order to assess the sensitivity of the various T-odd asymmetry observables to various sources of CP-viola- tions, we write down a very general CP-violating four-fermion effective hamil tonian which includes vector- boson and scalar-boson exchanges with arbitrary chirality couplings. The T-odd asymmetries are computed in terms of the coupling constants of the effective hamil tonian using the helicity matching form factor model of ref. [ 1 ] (KS model) and the model of ref. [2] (BSW model) as input. The order of magnitude of the CP- violating effect is estimated in some extensions of the standard model and could be as big as 1%.

We begin by writing down the angular decay distr ibution of the S~ cascade decay B~D*(-- ,Dn)~9~ that is induced by an effective current -current interact ion with axial vector (A) and vector (V) currents.

Using e.g. the results of ref. [ 3 ] one obtains

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0370-2593/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland) t 19

Volume 242, number 1 PHYSICS LETTERS B 31 May 1990

dF(B--.-D*(--,-Dn)~')0 3G21Vb~ I2(qZ--m~)Zp .,..,.,,, ~ . dqedcosOdzdcos0* = ~ ( ~ DtLJ --,t)~)

m2 sineO) sine0*(lH+ le+ lH_12)+(sin2 0+ q~2 COS20) cosZ0*4[Ho12

- ( 1 - m ~ - ~ - ] sin20 cos 2zsin20* 2 Re(H+H*_) - ( - m2) -qT/s in 20 cos;/sin 20* Re(H+ + H _ )H~

+cos 0 sin20* 2( IH+ 12- IH_ ]2) - s i n 0 cos Z sin 20* 2 Re(H+ -H)H~ m~ m~ ~@COS 0 COS20 ~ + ~ - c o s Z 0 * 4 [ H t [ a + 8Re(H~H~)+-~ysinOcoszsin20*2Re(H+ +H )H* ( l a )

( ( - 1 - ~ - j sina0 sin 2Z sina0* 2 Im(H+H*_ ) - 1 - ~ - ] sin 20 s inz sin 20* I m ( H + - H _ ) H I

H* mg . - H _ ) H i lb) - s i n 0 s i n z s i n 2 0 * 2 I m ( H + + H _ ) o+ , ( ~ - sm 0 sin Z sin 20* 2 I m ( H +

where G is the weak coupling constant (G ~_ 1.02 m p 2 × 10 -5) and Vbc is the b--, c Kobayashi-Maskawa matrix element, q2 is the invariant momentum transfer squared and p is the momentum of the D* in the B rest system. 0 and Z are the polar and azimuthal angle of the lepton in the (~9~) CM system and 0* is the polar angle of the D relative to the D* in the D* rest system (see fig. l ). B ( D * ~ D ~) is the D*--, D~ branching ratio and m~ is the lepton mass.

The helicity form factors H~ ( g = t, +, 0 ) appearing m ( 1 ) are defined by the relevant helicity projections of the current-induced B ~ D* transition matrix elements. One has

H~ =g*~(a) <D* (2)IA~ + V~ IB>, (2)

where ~ (a) is the polarization vector associated with the current (or with the Woff-~h~,) [ 3 ]. ~r= 2 = +, 0 denote the transverse and longitudinal spin-1 components and a = t, 2 = 0 denotes the time- (or scalar-) component of

a} {I~) CM frame b) § rest frame

X / / / iX I-~ i~\---- - - [ -

• t 6 / . 1 t---'--~----~" w°ff-shell /

Y ~ I

c) D" rest frame

D x

~o ,Z R_°-- "~z D~

Fig. 1. Definition of angles O, 6and Z in (~9) CM-frame (a), 0* in D* rest flame (c). (b) shows B rest frame.

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the current transition. Note that the helicities in (2) have to balance since the decaying B meson is a spin-zero object.

In eq. (1) we have written down the angular decay distribution for B (b)-+D* (c)~v~. The corresponding decay distribution for I] (13)-+ I)*(e)l~v~ is obtained from eq. (1) by reversing the signs of the parity violating lepton side contributions, i.e. the terms proportional to ( IH+] 2 - IH_I2), R e ( H + - H _ ) H ~ and I m ( H +

H * + _ )Ho. In this case the helicity amplitudes in eq. ( 1 ) are defined by the corresponding projections of the B (b) + I)* (~) transitions, i.e. by H~ = ~* ~ (~r) ( IS)* (2) I A~ + V~ I B >-

We have separated the angular terms in eq. (1) into T-even ( l a ) and T-odd contributions ( lb ) . That the angular factors in (1 b) multiplying the imaginary parts of interference contributions represent T-odd correla- tions can be seen by rewriting them in terms of the polar angle Owhich measures the angle between the normal to the hadron plane (the hadron plane is defined by the decay products D and re) and the lepton's three momen- tum (see fig. l ). One has

sin 0 sin Z= cos cf20 oc p~. (PD* X P D ) ,

sin20 sin 2;~= 2 sin 0 cos Z sin 0 sin Z Ocp~'pD± p~" (PD* XPD ) ,

sin 20 sinz= 2 cos 0 sin 0 sin Z °ZP~'PD*P~" (PD* XPD) , (3)

where p~ is the three momentum of the lepton and PDL lies along the x-axis (see fig. 1 ). The scalar products P~'PD ± and p~'pD, are even under T whereas the pseudoscalar triple product p~- (PD* ×PD) is odd under T. This then ascertains the statement that the angular factors in ( 1 b) represent T-odd correlations.

In the following we shall also be concerned with current-current interactions involving scalar (S) and pseu- doscalar (P) currents. The hadronic pseudoscalar current induces T-odd contributions through interference with the standard model vector current contribution. Noting that the time component polarization vector e. (t) is proportional to the four momentum transfer qu [ e~(t) = q u / x / ~ ] one can obtain the relevant angular distri- bution from eq. ( 1 ) by the substitution Ht~Hprn~/v /q 7. Hp is the pseudoscalar helicity form factor defined by

H e = ( D * ( 2 = O ) I P I B ) , (4)

and P is the pseudoscalar current ~l. Returning to the decay distribution eq. ( 1 ) one notes that the coefficients of the T-odd angular factors in ( 1 )

can be projected out by defining suitable asymmetry ratios. To this end we partition the angular phase space into the following sectors:

(Z) I: O~<Z< ½re, II: ½7c~<Z<~r, III: zc~<Z<3Zc, IV: 37r~<X<2zc, (5a)

(0) A : 0 ~ 0 < ~ z c , B:lzE~<0<zc, (5b)

(0") A:0~<0*<½~r, B*: ½ ~ 0 * < z ~ . (5c)

We then define the following asymmetry ratios:

dF/dq2 ] i_ii+in_lv A VTAT = 2

d F / d q [I+II+m+xv (6a)

1 ( -- 1 + m 2 / q 2) l m H . H * _ A(q 2) = 7~ d F / d q 2 ' (6b)

~1 T-odd triple momentum correlations resulting from the interference of a psendoscalar and vector current contribution have also been discussed in ref. [4]. We disagree with the final formulas (36) and (37) written down in ref. [4].

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2 d l ' / d q ] I + I I - I I I - I V : A - B;A*--B*

A VTAL = di . /dq2 I+II+III+IV;A+B;A*+B*

2 ( - - 1 2 2 1 + m ~ / q )~ I m ( H + - H _ ) H " d A ( q 2) 7c d l ' / dq 2 '

d F / d q 2 [ i + i i _ l l i _ IV:A*_B* ALR = 2

dF/dq [x+n+ni+W;A*+U*

=AATAL +A vxAs +A vTp ,

where

3 ½ I ra(H+ + H _ ) H " d A ( q 2) AATAL - - 2 d I ' / d q 2 '

3 (rn~/q 2) ½ I ra(H+ - H _ ) H * A ( q 2) AVTAS - 2 d l ' / d q 2 '

3 ( r n ~ / v / ~ ) ~ I m ( H + - H _ ) H ~ , A ( q 2) A VTe = 2 dF /dq 2 '

and where

31 May 1990

(7a)

(7b)

(8a)

(8b)

(9)

(lo)

(11)

GZl Vbc]z(q 2 - A(q2)= 12(2zc)3M2q~)2PB(D*--~Dx). (12)

The denominator of the asymmetry ratios is given by the angle integrated partial rate d n (B-oD* ( ~ Dn)~v~)/ dq 2

dq--~=A(q2) l + ~ q 2 ( [H+IZ+IH_I2+LHolZ)+3~q21H~I 2 . (13)

The asymmetry ratios ( 6 ) - ( 11 ) are defined for any q Z-value. For fixed q2 the factor A (q2) in the numerator and the denominator cancels. When integrating over q2 (numerator and denominator separately! ) one obtains corresponding integrated asymmetry values. In this case A (q2) can no longer be cancelled, but the branching ratio B (D* ~ Drc) still cancels. Angular variables whose domain are not specified in (6) - ( 11 ) are understood to be integrated over their physical ranges.

The asymmetry ratios appearing in ( 4 ) - ( 9 ) can be seen to arise from the interference of the following cur- rent-current contributions:

A vTa~: transverse vector-transverse axial vector interference,

A VrAL: transverse vector-longitudinal axial vector interference,

AA~AL: transverse axial vector-longitudinal axial vector interference,

A VTAs: transverse vector-scalar axial vector interference,

A vTce: transverse vector-pseudoscalar interference, (14)

where Vx denotes the (necessarily!) transverse vector current contribution, AT andAL the axial vector transverse and longitudinal spin-1 contributions, respectively, As the axial vector spin-zero contribution and P is the pseudo- scalar current contribution.

From eq. (8) it is clear that the last three contributions of (14) cannot be kept apart by an angular measure- ment alone but require a separation via their different dependences on the lepton mass, cf. eqs. (9) - ( 11 ).

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It is well known that there are in general two distinct sources that can contribute to T-odd observables. These are unitarity corrections (or final state interaction effects) and truly CP-violating contributions (see e.g. ref. [ 5 ] ). The "bare" helicity amplitudes describing the current-induced transition from the B to a stable, elemen- tary D* are real in the physical region of the S~ decay process and thus do not generate any contribution to the

T-odd observables. Let us begin by discussing possible unitarity corrections. First note that unitarity effects leading to the imagi-

nary parts of the W and D* propagators (Breit-Wigner effects) do not contribute to T-odd observables since they contribute equal phases to each helicity amplitude. Also there are no unitarity corrections from O(as ) gluon exchange contributions to the WQCl vertex because there are no cuts in the physical region of the S£ decay process as already mentioned above.

Another possible source of unitarity corrections are (D~) final state interaction effects f rom the interference of differenl partial waves and /o r A / = 0 and A I = 1 contributions. Qualitatively one expects these to be quite small. First note that the D* is quite narrow (FD, < 2 MeV). Relative to the dominant p-wave resonance contri- bution the nonresonant background is suppressed. The available phase space in the resonance region is so small that one expects only appreciable s- and p-wave contributions. The basic weak b ~ c transition is A / = 0 with possible suppressed A / = 1 contributions from electromagnetism feeding an already suppressed background. Taking these facts together one finds no strong interaction unltarity contributions to the T-odd asymmetries ( 6 ) - ( 11 ) the approximation that only I = ½ s- and p-waves are considered in the (Drc) system.

In order to be more specific we write down partial wave expansions for the currem induced B-,Drt helicity transition amplitudes/~o which are defined by

H~ = ~-u(a) ( D n l A , + gu I B ) . (15)

Compared to the helicity amplitudes H~ defined in (2) the (Dn) system is no longer required to be a p-wave. The general helicity ampli tudes/ to may then be expanded in partial waves [ 6 ]. One has

/~* ~ [ ]~/12 exp ( i~)/2) -'t- ~3/2 . . . . +l exp(ifi 3/2) ]P~(cos 0*) , l= 1

[ho l exp(l& )_,,o l exp(ifi3/2)]p~(cosO*) , (16)

where

d P~(cos 0") = d cos 0* Pl(cos 0") . (17)

The ff112(3/2) are the (real!) partial wave amplitudes in the I = ½ (3) scattering state, and the ~)/2(3/2) are their associated phase factors.

The general cos 0* distribution corresponding to eq. ( 1 ) may be obtained from eq. ( 1 ) by the substitution H+ ~ / 1 + and cos O*Ho ~Ho. For the p-wave case one of course recovers eq. ( 1 }.

In the following we ~shall'neglect the doubly suppressed electromagnetism-induced ~ r = 1 transitions and set all partial wave amplitudes/73f 2 to zero. In this approximation one has unitarity effects only from the interfer- ence of different partial waves. Since the transverse helicity ampl i tudes /7 , start with a p-wave the lowest con- tribution to the asymmetry ratio A V~A~ in eq. (6) would be from p-wave-d-wave interference. However, with the cos 0* integration prescribed in (6), this interference drops out exactly. In fact, one can show that, due to the orthogonality properties of the Legendre polynomials, any interference contribution to AvxAx vanishes in that integration range [ 6 ]. The lowest partial waves contributing to the transverse-longitudinal asymmetry ratios A VxA~ and ALR are d-wave-s-wave interference and thus appear to be negligible. In particular, the lowest partial wave that could interfere with the dominant p-wave contribution would be f-wave with its severe phase- space suppression.

Finally, there are coulombic unitarity corrections from one photon exchange between the charged lepton and

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the D* or D, g (particular picture), or the c-quark or the spectator quark (constituent picture). These contri - butions are down by at least the fine structure constant or. In the particle picture the coulombic unitarity correc- tions could be minimized by investigating S~ B decays with neutral final state particles only as e.g. B + ~ D *° ( ~ D ° ~ ° ) ~ % In the quark picture one expects the first order coulombic unitarity corrections to vanish in the limit mc/mb = 0 [ 7 ] which indicates that the coulombic unitarity effects may be smaller than the naive O (o~) estimate e2. The one photon contribution certainly deserves further theoretical study once the experimental sensitivity on the T-odd asymmetry measures has reached the level of O (1%). Note also that the coulombic unitarity corrections alter the angular structure as predicted in eq. ( 1 ) which was derived from the assumption that one has a factorizing current-current structure L.H. These coulombic unitarity corrections would, however, necessarily again lead to the characteristic T-odd triple momentum correlation structure p~. (PD* ×PD)"

One possibility to establish the presence or absence of unitarity effects and /o r CP-violating effects would be to measure the sums and differences o f the T-odd asymmetries of S~ B (b) and 1] (13) decays. For the purpose o f the pursuant discussion we adopt a generic notation introduced by Valencia in ref. [ 9 ] for a discussion of cor- responding issues in nonleptonic B-decays. Let a and c be two amplitudes describing S~ B-decays with relative phase (~+ q0, where the phase g be generated from unitarity effects and ~ from CP-violations. For the T-odd asymmetries in B (b) decays one finds

A B ~ I m ac*~ lacl sin(O+ q~) . (18)

The corresponding asymmetry for SJ~ B (b) decays described by the amplitudes d and ( reads

A~ ~ I m a(*~ - lac[ sin(c~- q 0 , (19)

where we have used CPT-invariance to rewrite the RHS of ( 19 ). Taking sums and differences of the B (b) and B (b) reactions one obtains

AB+AB~ lacl cos~s in qb, AB-Ar~~ lacl sinc~cos qb. (20,21)

If there are no unitarity effects (c~= 0) one has AB =AN, whereas AB = --AB in the absence of CP-violations (q~= 0 ). I f there are no unitarity effects and no CP-violations (~= qb= 0 ) t h e n the asymmetries vanish as ascertained before.

Further insights into the presence and absence of unitarity effects and /o r CP-violations can o f course be gained by comparing rates of B- and B-decays. As this point is discussed in detail in the literature (see e.g. ref. [4] ) we shall not pursue this point any further.

Let us now turn our discussion to possible CP-violating contributions to the T-odd asymmetries. In order to get a quantitative handle on these we write down the most general effective four-fermion hamiltonian including vector (axial vector) boson and scalar (pseudoscalar) boson exchange terms with either left or right chiral couplings to fermions ~3. One has

G nef f= ~ Vcb [qc ( 1 +ZLL) 7u( 1 --yS)qb/~l ~ ( 1 -- ps)L2 -bZRLqcyu( 1 + ~5)qb/~l y~( 1 "y s )L2

+ r/rLq¢( 1 --7s)qb/2~ ( 1 -- 7s)L2 + t/RLq~ ( 1 + yS)qb/2, ( 1 -- 75)L2 ] , (22)

where the qt and L~ are quark and lepton fields, respectively. In writing down the effective hamiltonian in eq. (22) we have not included right-handed leptonic currents because any interference between the left- and right- handed leptonic sectors gives terms proportional to the light neutrino mass which we neglect in this analysis.

The complex coupling coefficients ZLL, ZRL, qLL and ?~RL parametrize any new physics that would arise from

~2 For kaon decays, the effect of coulombic unitarity corrections has been estimated to occur at O ( 10- 6) [ 8 ]. ~3 We do not include a possible tensor exchange coupling of the type cru~oaU" in the present analysis. Although such a term Would have

the same dimension as the above coupling terms we are not aware of any candidate theory that would produce such a coupling term. Note also that such a tensor-tensor contribution has a more involved angular structure than the one given in eq. ( 1 ) .

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vector and scalar boson exchange including CP-violat ing contributions. The s tandard model predic t ion is ob-

ta ined from eq. (22) by setting ZiL and t/iL ( i = L , R) to zero. The magni tude o f the coupling coefficients Z~L and q~'L are presumably small since the present phenomenology

of the b-system does not provide any evidence for large deviat ions f rom the s tandard model predictions. We therefore include only T-odd effects linear (and not quadra t ic ) in the effective CP-violating couplings Z~L and t/~L in the following. Hence, in our approach the T-odd asymmet ry rat ios A VTAT, AVTAL and ALR defined in eqs. ( 6 ) - ( 11 ) are sensit ive to the CP-violat ing phases of the coupling coefficients ZRL and (t/RL--t/LL) through their

interference with the s tandard model ampli tudes. In order to es t imate the magni tude o f CP-violat ions included by H~fl- eq. (22) in S~ B--, D* decays one needs

explici t expressions for the current induced B - , D * form factors including their qZ-dependence. For tunate ly there are a number o f models available for these [ 1,2,10,11 ] which do not differ significantly in their predic t ions and are in agreement with the present S~ B ~ D* exper imental decay data. As two representat ive models we take

the models ofref . [ 1 ] (KS) and ref. [2] (BSW). Our numerica l results for the T-odd asymmet ry values defined in eqs. ( 6 ) - ( 11 ) are d isplayed in table 1. They

are p ropor t iona l to the imaginary parts of the effective couplings ZRL and (t/RL--t/LL) given by

av=2 ImZ~L, avp=Im(tl~L--q[L) t~-3)

ImzLL does not contr ibute te the T-odd asymmetr ies because of its lef t- lef t structure which is identical to the s tandard model structure. Similar ly there is no contr ibut ion to the (axial v e c t o r ) - ( a x i a l vector) T-odd term I m ( H + + H _ )H~ because of the lack of spatial interference.

Fo r the pseudoscalar form factor He contr ibut ing to the asymmet ry A vTe we have used the divergence relat ion qUA u (mb "4-m~)P between the axial vector current A u and pseudoscalar current P to relate it to the axial vector form factors given in ref. [ 2 ] (m b =-4.9 GeV, mo = 1.7 GeV) . The same relat ion is used to derive the pseudosca- lar form factor at q2 = 0 in the KS model [ 1 ], however, with ( m b 4- me) replaced by ( ms + mD.) and a monopole form factor behaviour .

As table 1 shows the T-odd asymmet ry measure A V~AT obtains the largest contr ibut ion from av= 2 ImZ~L in all three cases ~= e, g, z. This is quite for tunate since the T-odd measure AVTAT has no contr ibut ions from CP- conserving strong interact ion uni tar i ty corrections as argued above. One notes the different dependence of the asymmet ry measures on the lepton mass m~. The strongest dependence occurs for A VTAs which is quadrat ica l ly dependent on the scaled lepton mass m ~ / x ~ in the differential qZ-rate [see eq. (10) ]. Av~p is only linearly dependent on m ~ / x / ~ [see eq. (11 )] and A v ~ and AVrAL a r e lepton mass independent in their domina t ing term. The predic t ions of the KS and BSW models do not differ much from one another, where the largest differ- ences occur for the asymmet ry A V.AT and for all asymmetry measures in the r-sector.

Let us now turn to some specific models that can generate CP-violat ing interactions. We shall consider the lef t - r ight symmetr ic model ( L R M ) [ 12 ] and the Higgs-boson model ( H B M ) (for reviews of the HBM see ref. [ 13 ] ), which are both at t ract ive extensions of the s tandard model.

Table 1 Values of asymmetry measures AVTAT, AreAL, AVTAS and AvTe defined in eqs. (6)-( 11 ) in terms of phase factors av=2 ImZ~L and avp--Im(tl[L--q[L) forKS [1 ] andBSW [2] models.

A VTAT A VTAL A Vras A VTP

(el KS 8.50× lO-2av 6.38;< lO-2av - 1.30× 10-8av BSW 7.89× lO-2av 6.26× lO-2av - 1.31 × 10-Say

(g) KS 8.46X lO-2av 6.36× 10-2av 4.06× lO-4av BSW 7.87× lO-2av 6.14N lO-2av --4.06× 10-4av

(z) KS 3.6 × lO-2av 2.14× lO-2av 2.30× 10-2a~ BSW 3.19× lO-2av 1.94× lO-2av -2.04× 10-2av

(e+g+z) KS 7.54X lO-2av 5.90× lO-2av 0.28N lO-2av BSW 7.35X 10-2av 5.72× lO-2av -0.25× lO-2av

0.72>( lO-Savv 0.74;4 lO-Save 1.49× lO-Save 1.53× lO-Save 1.16× 10-Zave 1.07 × 10-2ave 0.13× lO-2ave 0.12× lO-2avp

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In the lef t - r ight symmetr ic model we have

,~LL ~ 0 , ~ L R = O d L R ~ , , ~ R R ~ a R P . , 0 , ~ R L = a R L ~ , (24)

with

P=mzL/m~, (25)

where ~is the mixing angle between " lef t" and "r ight" bosons and OLLR , OLRL and O~RR are phase factors associated with the r ight-handed couplings which, in general, are non-vanishing even for only two families of fermions, m E and mR are the masses of the lef t -handed and r ight-handed gauge bosons, respectively.

The constraints on p and ~ f rom data were found to be

p~<lO -2 for ~ arbi trary, ~ 4 . 1 X l O -2 f o r p ~ O , (26)

f rom the measurement o f g-decay [ 14]. Fur ther

P~<6XIO -2 , ~ 4 X 1 0 -3 , (27)

from the analysis o f K--.3= with some addi t iona l assumptions about hadronic matr ix elements [ 15 ]. Relevant to our discussion is the parameter ~. As a possible range for { we consider

~ < 4 X 1 0 - 2 ~ 4 X 10 -3 • (28)

In terms o f the CP-violating phase factor a v d e f i n e d in eq. ( 15 ) this implies

av<~ ( 1 0 - 1 ~ 10 -2 ) Im O~r . (29)

Using the results of table 1 we see that the T-odd asymmetr ies could be of order ( 1 0 - 2 ~ 10 -3 ) Im OZ~L in the LRM. Considering the fact that Im ot ~L is p ropor t iona l to the difference in the phase factors between the L- and R-KM matr ix and therefore essentially unrestr icted (i.e. O( 1 ) ) the LR model in its present form allows for asymmet ry rat ios of order 10- 2 ~ 10- 3.

We now consider the Higgs boson model ( H B M ) . In the HBM the scalar coupling constants qo appearing in eq. ( 2 2 ) a r e given by [4,13]

r/RL= fl, a T \ V j , m n , rKL= a , a * , (30) i=1 i=1 \ mH, /

with the following combina t ions of Higgs-boson mass-matr ix mixing angles:

snc H SHS~ .H~HoH H . e x p ( i g H ) n H H n H Cl S2 S3 +C2 C3 exp(i0H) t-I o2 t, 3 - - C 2 S 3 o q - cH , o~2=n c-~-~ ' a l = s~s~ , d 2 = s~sp '

H H H H H Cl C2 C 3 -t-S 2 S 3 exp (igH) t- I~H~Ht~2 °3oH --°2°H'H~3 exp (iOta) i l l= sH@ , f12= S~C~ (31}

The c~ = cos 0~ and sH = sin O H ( i = 1, 2, 3 ) are the mixing angles of the Higgs sector• OH is the CP-violat ing phase angle in the HBM. The two Higgs masses of the HBM are denoted by mH, ( i = 1, 2) .

In this case we f ind

~ a o H . n [ m~ mc m mb) • - m£mc'~ b2 t.3 o3 - * * sm 6H [ ~ - - - + sin OH aVp--Im(t/RL--t/LL) -- ~1 ~3 ~3

°I~'H2"HoHt" 2 o 2 \ m m m ~ J ClHS H \ m 2 l m 2 2

H H . m b m ~ f C3S3 Sln0H( ~ mbm~'~ ~k mccHZ "~ - - . . - S T - 1 - -

cHI S H \ m i l l m H 2 ,] m b s H c H 2 , ] " (32)

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Volume 242, number 1 PHYSICS LETTERS B 31 May 1990

T h e T-odd signal is seen to van i sh in any o f the l imi t s s3 ~ = 0, ½7~, 3H = 0 and mnl = mH2- At p resen t the p a r a m e t e r

space o f the H B M is no t suff ic ient ly res t r ic ted to war ran t a de ta i l ed quan t i t a t i ve d iscuss ion o f e q . (32) .

I t should be n o t e d tha t the a s y m m e t r y Av~e has the same s t ructure as AVTAS, which m e a n s tha t one can no t

d i s t inguish the L R M f r o m the H B M solely by s tudying these corre la t ions . I t is in add i t i on necessary to measu re

the a s y m m e t r y A V~Ax or A V~AL to d e t e r m i n e the coupl ing cons tan t s a v and a w, then b o t h a symmet r i e s A vxAs and

A vzP can be extracted.

We see tha t in the L R M

]AVTAT I 7> IAvTAL I >> IAVTAs I

for exclus ive s emi l ep ton i c B-decay in the channels B--, D* ( D re) ege and B ~ D* ( D u) g9 w bu t for the channe l

B - ~ D * ( D u ) w ~ one has

IAvxAT ~ 1.5IAVTAsl, IAvTAs ] > IAv'rAL

We h a v e ca lcu la ted T-odd t r ip le m o m e n t u m corre la t ions in the S~ B-decay B ~ D * (Drc) + ~ + v. Ra the r t han

e m p l o y i n g specif ic mode l s we based our ca lcu la t ion on the m o s t genera l e f fec t ive in t e rac t ion in a gauge theory

f r amework . It is left to expe r imen t to d e t e r m i n e the respec t ive coupl ing cons tants or to p r o v i d e uppe r l imi t s to

them. Var ious m o d e l s o f w e a k in t e r ac t ion can than be checked against these paramete rs . T r u e T- or CP-v io la t ing

effects of ten c o m p e t e wi th f inal s tate in te rac t ions in T-odd ampl i tudes . We h a v e found an expe r imen t a l asym-

me t ry observab le (AvTax) where these effects can be separa ted , An obse rved nonze ro va lue o f this observab le

AvTAT w o u l d represen t a c lear s ignature o f CP-v io l a t i on i f c o u l o m b i c un i t a r i ty cor rec t ions can be neglected. In

the s t andard m o d e l the T-v io la t ing effects cons ide red here vanish , the re fore any obse rva t ion w o u l d be an indi-

ca t ion o f new physics. T h e o rde r o f m a g n i t u d e o f the CP-v io la t ing effects has been e s t ima ted in some ex tens ions

o f the s t anda rd m o d e l and found to be O (1%) . Such effects should be measurab le in the very high stat is t ics

expe r imen t s env i saged in the B-factor ies o f the future.

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