Z. Phys. C - Particles and Fields 55, 479-483 (1992) Zeitschrift P a r t i d e s Nr Physik C

and FLaiLs �9 Springer-Verlag 1992

T-odd and CP-odd triple momentum correlations in the exclusive semi-leptonic charm meson decay D K* (Krt)lvl J.G. K6rner*, K. Schilcher, Y.L. Wu**

Institut ffir Physik, Johannes-Gutenberg Universitfit, Staudingerweg 7, Postfach 3980, W-6500 Mainz, Federal Republic of Germany

Received 8 March 1992

Abstract. We study T-odd and CP-odd triple momentum correlations in exclusive semi-leptonic charm meson decays D~K*(K~)Ivl . We define asymmetry ratios that measure the true CP-violating effects, and T-odd triple momentum correlation effects from unitarity contribu- tions. Possible new CP-violating contributions are par- ametrized in terms of an effective four-fermion Hamil- tonian where CP-violating effects should come from new non-standard sources. A detailed analysis of the left-right model and the Weinberg Higgs-boson model of CP-viola- tion is carried out.

Indirect and direct CP-violating effects have so far only been observed in the K ~ ~ system from the K L ~ branching ratio and a CP asymmetry in KL~7cI~ semilep- tonic decay [1]. These data, however, can be explained by a number of theoretical models. In order to gain a deeper understanding of the origin of CP-violation, it is necessary to investigate possible CP-violation effects in decay modes as well as in other flavour channels. Possible CP-violating effects in the B-meson sector have been at the focus of theoretical attention. In particular this concerned the study of partial rate asymmetries occuring in different exclusive channels [2]. Recently, a new type of CP-violat- ing signal in triple-product correlations were investigated [3-5].

In particular, the T-odd and CP-odd triple mo- mentum correlations in the exclusive semi-leptonic bot- tom meson decays B~D*(~D~)+l+vz were studied in detail by us [4]. We found that the asymmetry ratios that measure these T-odd triple momentum correlation effects would have to come from new non-standard model sour-

* Also at: Deutsches Elektronensynchroton (DESY), Notkestr. 85, 2000 Hamburg 52, FRG ** Supported by Bundesministerium ffir Forschung und Tech- nologie (BMFT), FRG

ces as there are no standard model contributions. Their magnitude was estimated to be (9(1%) in some extensions of the standard model. We also found that the strong interaction unitarity contributions are small or, in the case of a particular T-odd observable, absent. A similar dis- cussion can obviously be applied to the exclusive semi- leptonic charm meson decays D~K*(~Krr)+l+v~ , in which the magnitude of the CP-violating effects is ex- pected to be of the same order as in the B---> D* ( ~ D~) + l + vz decays. In the case of CP-violating partial rates the asymmetries in D-meson decays are much smaller than those in B-meson decays. For these reasons it is important and worthwhile to study the T-odd triple momentum correlations in the decay D ~ K * ( ~ K ~ ) + l+vl. In addition, the mode D~K*(~Krr )+l+vz has a large branching ratio. Also there is now new experi- mental information on the form factors that occur in this decay.

The analysis of the T-odd and CP-odd triple mo- mentum correlation effects proceeds in analogy to the case B~D*(~D~z)+l+qt treated in [4]. We begin by writing down a very general effective Hamiltonian including CP- violating effects:

G Heff = ~ Vcs [(1 + ZLL) qs 7u(1 -- 7S) qc s 7"( 1 -- 75) L2 ,/2

+ ZRLq~ 7,(1 + 75) qc/~1 7'( 1 - 75)L2

-]-~]LLqs(1 --75) qc/~1 (1 -- ];5) L2

+ r/RL Cis(1 + YS) qc s (1 --75) L2] , (1)

where G is the weak coupling constant and V~s is the KM matrix element. The qi and Li are quark and lepton fields, respectively. In writing down the effective Hamiltonian in (1) we have not included right-handed leptonic currents because any interference between the left- and right- handed leptonic section gives terms proportional to the left ucatrino mass which we neglect in this analysis.

The complex coupling coefficients 7~LL, •RL, ~/LL and qRL parametrize any new physics that would arise from vector and scalar boson exchange including CP-violating

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contributions. The standard model prediction is obtained from (1) by setting ZiL and qiL (i=L, R) to zero.

With the effective Hamiltonian in (1), we can write down the angular distribution of the semileptonic cascade decay D ~ K * (~ Kn) lgl

dF (D ~ K * ) (--, Knlgt ) dq 2 dcos 0 dz dcos 0*

3Ge I V~I2(q z-mz~ )zp - 128(2n)4 m2 q2 B(K*~Kn)

�9 [{(( l+cos20)+~2sinZO)sineO * (IH+IZ+IH-I 2)

+(sin2 0+~2 cose O* )4lHol 2

m2~ O* - 1 - ~ - j sin 20 cos 2 )~ sin 2 2Re(H+ H*)

0, - 1 - ~ - j sin2 0 cos)~ sin 2 Re(H+ +H_) H*

+ cos 0 sin 20* 2(IB+ [ 2 - [H-[2)

- sin 0 cos Z sin2 0* 2Re(H+ - H_ )H~' 2 2

+ ~ cose0 * 41Htl 2 +~-~ cos0 cos20 * 8Re(HtH~)

2

+~2 ~. sin0 cosz sin 2 0* 2Re(H+ + H _ ) H *

+cos 20* 4IHI 2 (2a)

ml - s i n 0 cosz sin20* ~ 2Re(H+ +H_)H*

ml - c o s 0 cos 2 0* ~ 8Re(Ho H*)

} - c o s 20* ~ 8Re(H+H*)

+ - 1 - ~ j s i n 0 s i n 2 x s i n Z

. Im(H+-H_)(H+ +H-)*

O* - t - ~ j sin20 sin)~ sin 2 I m ( H + - H _ ) H *

--sin0 sinz sin20 * 2 I m ( H + - H )H~' 2

+~z ~ sin0 sinz sin 2 0* 2 I r a ( H + - H _ ) H *

ml 0* } ] (2b) x / ~ sin0 sinz sin e 2Im(H+ -H_)H* .

The last line of (2b) contains the contribution of a possible scalar exchange that would be generated by the last two lines of the effective Hamiltonian (1). Equation (2) holds for the decays D__(c)--*K*(s)l + vL. For the corresponding decays D(6)~K*(g)l-gt one needs to change the signs of the parity violating terms.

The invariant momentum transfer squared is denoted by q2 and p is the momentum of the K* in the D rest system. 0 and Z are the polar and azimuthal angles of the lepton in the (lgz) CM system, 0* is the angle between the K and the K* system in the K-n CM system (see Fig. 1). B(K*-*Kn) is the K*~Kn branching ratio and ml is the lepton's mass. The helicity form factors H~(a= t, +_, 0) and H are defined by

H~--g*"(a) (K*(2)IAu+ GID) (3a)

and

H = ( K * ( 2 ) I S+PID), (3b)

where V u, A,, S and P are the vector, axial vector, scalar and pseudoscalar currents, respectively. In our case, the currents take the form

V.=(1 + ZLL + )(RDG Tuqc,

A. = - (1 + )~LL - - ZRL)q~ 7u75 qc,

S =(?ILL 71- tlRL) [Is qc,

P=(tlRL--tlLL) q~5 qc, (4)

as can be read off from the effective Hamiltonian (1). ?,(a) is the polarization vector associated with the

currents, a = 2 = _+, 0 denote the transverse and longitu- dinal spin 1 components and a = t, 2 = 0 denotes the time- (or scalar) component of the current transition.

In the angular decay distribution (2), we have separate- ly written out the T-even parts in (2a) and the T-odd parts in (2b). The T-odd nature of contributions (2b) can be exhibited by rewriting them in terms of the triple mo- mentum products which are odd under T as discussed in [4].

We now turn to the discussion of triple momentum correlation measures by defining suitable asymmetry ra- tios as in I-4]. To this end we partition the full physical angular ranges into the following sub-domains:

(Z) I: 0_<Z<n/2,

II: n/2_<)~ < n,

III: n<_Z<~n,

IV: ~n<_Z<2n, (5a)

(0) A: O<O<n/2,

B: n/2_<0<n, (5b)

/A / K . / / z_ox,s

F i g . 1 .

(0") A*: 0 < 0 * < n / 2 ,

B*: 7~/2_<0" <m (5c)

We then define the following asymmetry ratios:

dF/dq 2 h-n +m-w AvT A~ = d ~ ~ ~ (6a)

( - 1 + m{/q 2) Im H+ H* A(q 2) - dF/dq2 (6b)

dF/dq 2 I I + I I - I n - I V ; A - B ; A * -B* A V r A L - - 2 = (7a)

d F / d q [1 +11 +Ill + IV;A+B;A* +B*

2 ( - 1 +m 2" t /q 2"11m(H+ )~ - H - ) H ~ A ( q 2 ) - n dF/dq 2 (7b)

dF/dq 2 II+n-m-W;A*-B* ALR - - 2 (8a)

dF/dq Ii+n+ni+w;a*+B*

= A A T A L + AVTAs-F A v r e , (8b)

where 3 � 8 9 )H~A(q 2)

Aa~'AL= 2 dF/dq 2 ' (9)

3 m2/q 2 �89 Im (H + -- H_ )H* A (q2) AvT As= 2 dF / dq 2 ' (10)

3 m2/q 2 �89 Im(H+ - H _ )H* A(q 2) Av~e = 2 dF/dq 2 ' (11)

and where

G2l Vbcl2 (q2--m2)2p . . . . ,__,.. , A(q 2) = ~ ~ ( ~ ~z) . (12)

The denominator of the asymmetry ratios is given by the angle integrated partial rate d F ( D ~ K * ( ~ K n ) lv~)/dq 2 which reads

dF=A(q2) 1+ (IH 12+IH-IN+IHol 2) dqZ +

+ 3~ql~ ]Htl2 + ,HI2]. (13)

The asymmetry ratios (6)-(11) are defined for any q2-value. For fixed q2 the factor A(q 2) 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(q 2) can no longer be cancelledl but the branching ratio B(K *-~Kn) still cancels. Angular variables whose domain are not specified in (6)-(11) are understood to be inte- grated 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

AV~-AT: transverse vector-transverse axial vector in- terference

AV~-A~: transverse vector-longitudinal axial vector interference

AATA~: transverse axial vector longitudinal axial vector interference

AViAn: transverse vector-scalar axial vector interfer- ence

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Avre: transverse vector- pseudoscalar interference (14)

where Vv denotes the (necessarily!) transverse vector cur- rent contribution, Ar and AL 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.

We emphasize that T-odd correlations can occur in general even if all interactions in the decay process con- serve T. The sources of such T-odd contributions are unitarity corrections (or final state interaction effects). The situation in B-~D* (-~Dn)+I+ vt decay has been analyzed in 1-43 in terms of the (Dr 0 channel. This analysis can be carried over to the case D-~K*(-~Kn)+l+vz. The main conclusions are:

i) unitarity effects from the interference of different partial waves in Av~.ATvanish due to the orthogonality properties of the Legendre polynomials. ii) the lowest partial waves contributing to the trans- verse-longitudinal asymmetry ratios AVTA,_ and ALR are d-wave- s-wave interferences and thus appear to be negli- gible. The Coulombic unitarity corrections are expected to be small and down by at least the fine structure con- stant c~.

A more detailed discussion of the unitarity effects can be found in [4] and will not be repeated here.

As already indicated in [4] one has the possibility to distinguish unitarity effects and truly CP-violating effects by measuring sums and differences of the T-odd asymmet- ries of s.1. D(c) and /)(6) decays. This can be seen by exhibiting the phases of the helicity amplitudes. One has

H + - H _ = ]Hvl e ia veie/' v

H+ + H_ = IH A] ei6Te i ~

Ho = I Ho I eiaLe i~L

H, = IHtl eiase i~S

H = I HI e i~'e i~"

lq + - H_ = - IHv l e iave- i~

/~+ -4- /L = IHAI ei6Te -ia~,

Ho = [Hol eiaLe - i~ ,

/4,--In, I ei~se -ia's,

H = IH[ eiaPe -i~p, (15)

where Hi and /4~ denote the helicity amplitudes of D(c) and/5(6) meson decays, respectively. The phases 6i denote the strong interaction phases (or unitarity phases) which arise from different amplitudes, for example, isospin am- plitudes. The phases (hi denote the CP-violating phases as would e.g. arise from the weak Hamiltonian (1). With the effective Hamiltonian (11) one can see that

(I) L ---~ t~ S = tJ) T ~- ~I) A , (16)

with

I m (ZRL - - ~(LL)

tgq)a= 1 +Re(ZRL--ZLL)' (17)

I m ( ,~RL "]- ~LL )

tg q~V = 1 + Re (ZRL -]'~LL)' (18)

Im (rigL-- riLL)

tg ~ - Re (riRL-- rILL)" (19)

482

We then find that the T-odd asymmetry ratios can be written as

AVrA,=IAvTA, I sin(6vi+ ~), i= T, L, S, (20)

AArAL = [AArAL I sin 6rL, (21)

Av~p = I Av~el sin (6ve + qs') (22)

for D(c) semileptonic decay, and

AVTA, = --IAvra, I sin(6~i-- cb), i= T, L, S, (23)

AA~AL = --IAA~a=I sin (6TL), (24)

Av,p = -]Av~el sin (6re - 4~') (25)

for/)(g) semileptonic decay, where

6i j=6i-6 j i , j= V, T, L, S, P,

cb = 4~v - q~A,

4~' = ~v -- 4)p. (26)

The additional signs in the front of each of the equations in (23)-(36) come from the parity violation on either the lepton side or the meson side.

Note that the T-odd asymmetry ratio AArALcan only arise from unitarity effects.

Taking sums and differences of the D(c) and /5(0) asymmetry ratios we obtain

cP avrA, = AvrAi + AVTAi = I AvTAi I COS (6Vi) s i n 4 , (27)

U AVTA.- -AvrA,=IAvr&I s i n ( 6 v i ) C O S ~ , (28) aVTAi = , ,

with i= T, L, S and

av~pCP = Av~p + AvTe = [ A v~P[ co s (3 ~p) sin 4~', (29)

aW~e = Av~e- Av~e = [Av~p[ sin (bye) cos ~b', (30)

where cP av~j ( j = A r , Am, As, P) are a clean signal of CP violation whereas aW~j depends only on unitarity effects.

The CP violation phases 45 and 4)' are related to the complex coupling coefficients )~iL and rhm ( i=L ,R) through (17)-(19) and the definition (26). The magnitude of the coupling coefficients Z~L and thL are presumably small, since the present phenomenology of the C-system does not provide any evidence for large deviations from the standard model predictions. We therefore include only T-odd linear effects in the effective CP-violating coupling coefficients.

We thus have

sin �9 ~ 2Im 7~ ~L (31)

and

sin ~b' ~ Im (r/*z- r/~z) �9 (32)

In order to get a quantitative handle on the amplitude of the T-odd asymmetries [Avail ( j = A r , AL, As, P) we need explicit expressions for the current induced D ~ K * form factors including their qZ-dependence. Unfortunately present theoretical predictions of the form factors for D ~ K * decay are not well settled [6].

We therefore turn to an analysis of the present data in terms of the contributing form factors. Such an analysis was performed in [7], where a consistent set of form factors fitting experimental data [8] was found. There is, of course, still a big uncertainty due to the experimental

errors. Nevertheless, we shall take the form factor values of [7] as a basis of our numerical calculation. For the qZ = 0 values of the relevant four form factors contributing t o O - + K * e v e w e u s e

V(0) = 0.94,

A 1 (0) = 0.47,

A2(0) =0.03,

A0(0) =0.71. (33)

The qe-dependence is assumed to be given by monopole dominance as in [9].

Our numerical results for the IAvTj[ ( j=AT, AL, As, P) are displayed in Table 1.

As Table 1 shows the T-odd asymmetry measure AvTATobtains the largest contribution from sin 4) = I m )~*L in all two cases l=e, #. This is quite fortunate since the T-odd measure AvTA,has no contributions from CP con- serving strong interaction unitarity corrections as argued above (i.e. 6vT~O). One notes the different dependence of the asymmetry measures on the lepton mass m,. The strongest dependence occurs for AvTaswhich is quadrati-

cally dependent on the scaled lepton mass ml/,~q 7 in the differential qZ-rate (see (10)). Av~p is only linearly depen- dent on m l / x / ~ (see (11)) and AVT a ~and AvTALare almost lepton mass independent in their dominating term.

Let us now turn to some specific models that can generate CP-violating interactions. We shall consider the left-right symmetric model (LRM) [10] and the Higgs- boson model (HBM) (for reviews of the HBM see [11]), which both are attractive extensions of the standard model.

In the left-right symmetric model we have

ZLL ~ O, ZLR = ~LR ~"

ZRR~O~RRP, ZRL=~RLr (34) with

2 2 p = mL/mR, (35)

where r is the mixing angle between "left" and "right" bosons and aLR, C~RL and ~RR are phase factors associated with the right-handed couplings which, in general, are non-vanishing even for only two families of fermions, mL and mR are the masses of the left-handed and right-handed gauge bosons, respectively.

The constraints on p and ~ from data were found to be

p < 10- 2 for ~ arbitrary,

r x 10 -2 for p~0 , (36)

from the measurement of #-decay [12]. Further

p_<6• -2, ~ < 4 x 1 0 -3, (37)

Table 1. Values of a symmet ry measure ampl i tudes [AvrAr[, IAvrAr[, [Av~A~ [ and IAvTpI defined in (6)-(11)

tAvTATI IAv~-AL] IAv~-As] IAvTPI

(e) 14 .5 • -2 7 .9x 10 -2 3 .75 • -7 5.65 • 10 - s (#) 13 .4 • -2 7.02 x 10 2 1.09 x 10-2 1 . 0 7 •

483

from the analysis of K o 3 n with some additional assum- ptions about hadronic matrix elements [13].

Relevant to our discussion is the parameter r As a possible range for ~ we consider

~ < 4 x 10-2-4 x 10 .3 . (38)

In terms of the CP-violating phase factor sin (b defined in (18), (19) and (26) this implies

sinqO<(10 1 10 -2 ) I m p e l . (39)

Using the results of Table 1 we see that the T-odd asym- metries could be of order (10- 2 - 1 0 - 3) Im c~L in the L RM. Considering the fact that Im c~L is proportional to the difference in the phase fators between the L- and R-KM matrix and therefore essentially unrestricted (i.e. C(1)) the LR-model in its present form allows for asymmetry ratios of order 10-2-10 .3

We now consider the Higgs boson model (HBM). In the HBM the scalar coupling constants t/is appearing in (1) are given by [3, 11]

(mlms~

,=1 /

qzL = ei~* ( ~ } (40) i= 1 \ m m ]

with the following combinations of Higgs-boson mass- matrix mixing angles:

H H H H S1 r S1 $3

0 ~ 1 = C H ~ 2 = c H ,

H H H H H i5~ H H ~l=(cl szc3-c2s3 e )~sis2,

= ( C l S2S3 -]-C2C 3 e ) / $ 1 s 2 , ~2 H H H H H i~H H H

= (Cl C2 C 3 + S~Sg ,~i~,-,~/o.oH l l H H H ~ I I ' ) l ~2 ,

H H H H H f12 =(c1 c2s3 - s : c3 eiO')/s~d]. (41)

The c~ = cos 0~ and s/~ = sin 0~ (i = 1, 2, 3) are the mixing angles of the Higgs sector. 6u is the CP-violating phase angle in the HBM. The two Higgs masses of the HBM are denoted by MR, (i= 1, 2).

In this case we find

sin ' * * = Im(qRL-- ~/LL)

_c~s~ sin6n (rncm, mcm,~( m~c~ ~ n n - - 225- 1 .~-~n2 �9 (42) cl s2 \ m 2, mn: / rncSl c2 /

The T-odd signal is seen to vanish in any of the limits s~=0 , z/2, 6 u = 0 and mu,=mng. At present the para- meter space of the HBM is not sufficiently restricted to warrant a detailed discussion of (42).

It should be noted that the asymmetry Av~p has the same structure as Av~As, which means that one can not distinguish the LRM from the HBM solely by studying these correlations. It is in addition necessary to measure the asymmetry A VT A r or A VrAL to determine the coupling constants sin 4, and sin 4". In this case both asymmetries AviAn, and Av~p can be extracted.

We see that in the LRM

I AvTA TI >IAv~-ALI >> I AvTasl �9 (43)

We have calculated T-odd and CP-odd triple mo- mentum correlations in the s.1. D-decay D--,K*(KTz)+ l+v~. Rather than employing specific models we have based our calculation on the most general effective inter- action in a gauge theory framework. It is left to experi- ment to determine the respective coupling constants or to provide upper limits to them. Various models of weak interaction can then be checked against these parameters. True T- or CP-violating effects often compete with final state interactions in T-odd amplitudes. We have found an experimental asymmetry observable (AVTAr) where these effects can be separated. An observed nonzero value of this observable AeTATWOuld represent a clear signature of CP-violation if Coulombic unitarity corrections can be neglected. In the standard model the T-violating effects considered here vanish- - therefore any observation would be an indication of new physics. The order of magnitude of the CP-violating effects has been estimated in some extensions of the standard model and found to be (9(1%). Such effects should be measurable in the very high statistics experiments envisaged in the z-charm factories of the future.

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