Direct CP violation in charmed hadron decays via ρ-ω mixing

Guo, Xiao-Hui; Thomas, Anthony William

61(11):116009

© 2000 American Physical Society

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27

^th

March 2013

Direct CP violation in charmed hadron decays via ␳ - ␻ mixing

X.-H. Guo*

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide, SA 5005, Australia

and Institute of High Energy Physics, Academia Sinica, Beijing 100039, China A. W. Thomas^†

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide, SA 5005, Australia

共Received 12 July 1999; published 10 May 2000兲

We study the possibility of obtaining large direct C P violation in the charmed hadron decays D^⫹

→␳^⫹␳⁰(␻)→␳^⫹␲^⫹␲^⫺, D^⫹→␲^⫹␳⁰(␻)→␲^⫹␲^⫹␲^⫺, D⁰→␾␳⁰(␻)→␾␲^⫹␲^⫺, D⁰→␩␳⁰(␻)→␩␲^⫹␲^⫺, D⁰→␩⬘␳⁰(␻)→␩⬘␲^⫹␲^⫺, D⁰→␲⁰␳⁰(␻)→␲⁰␲^⫹␲^⫺, and ⌳c→p␳⁰(␻)→p␲^⫹␲^⫺ via ␳-␻ mixing. The analysis is carried out in the factorization approach. The C P violation parameter depends on the effective parameter Ncwhich is relevant to the hadronization dynamics of each decay channel and should be determined by experiment. It is found that for fixed N_cthe C P violation parameter reaches its maximum value when the invariant mass of the ␲^⫹␲^⫺ pair is in the vicinity of the ␻ resonance. For most of the parameter space explored the C P violating asymmetry is of order 10^⫺4. However, over a small range, 1.98⭐N_c⭐1.99 and 1.95⭐Nc⭐2.02, the asymmetries for D⁰→␲⁰␳⁰(␻)→␲⁰␲^⫹␲^⫺ and ⌳c→p␳⁰(␻)→p␲^⫹␲^⫺ 共respectively兲 can exceed 1%, at the cost of a small branching ratio. We also estimate the decay branching ratios for D⁰

→␲⁰␳⁰and⌳c→p␳⁰for these values of N_c, which should be tested by future experimental data.

PACS number共s兲: 11.30.Er, 12.39.⫺x, 13.20.Fc, 14.20.Lq I. INTRODUCTION

Although C P violation has been known in the neutral kaon system for more than three decades its dynamical origin still remains an open problem. In addition to the kaon sys- tem, the study of C P violation in heavy quark systems has been a subject of intense interest and is important in under- standing whether the standard model provides a correct de- scription of this phenomenon through the Cabibbo- Kobayashi-Maskawa 共CKM兲 matrix. Actually there have been many theoretical studies in the area of C P violation in b-flavored and charm systems and some experimental projects have been proposed关1兴.

Recent studies of direct C P violation in the B meson sys- tem关2兴have suggested that large C P-violating asymmetries should be observed in forthcoming experiments. However, in the charm sector, C P violation is usually predicted to be small. A rough estimate of C P violation in charmed systems gives an asymmetry parameter which is typically smaller than 10^⫺3 due to the suppression of the CKM matrix ele- ments关3兴. By introducing large final-state-interaction phases provided by nearby resonances, Buccella et al. predicted larger C P violation, namely, a few times 10^⫺3 关4兴. On the other hand, experimental measurements in some decay chan- nels are consistent with zero asymmetry关5兴.

Direct C P violation occurs through the interference of two amplitudes with different weak and strong phases. The weak phase difference is determined by the CKM matrix

elements and the strong phase is usually very uncertain. In Refs. 关6,7兴, the authors studied direct C P violation in had- ronic B decays through the interference of tree and penguin diagrams, where ␳^-␻ mixing was used to obtain a large strong phase 共as required for large C P violation兲. This mechanism was also applied to the hadronic decays of the heavy baryon, ⌳b, where even larger C P violation may be possible 关8兴. In the present paper we will investigate direct C P violation in the hadronic decays of charmed hadrons, involving the same mechanism, with the aim of finding chan- nels which may exhibit large C P asymmetry.

Since we are considering direct C P violation, we have to consider hadronic matrix elements for both tree and penguin diagrams which are controlled by the effects of nonperturba- tive QCD and hence are uncertain. In our discussions we will use the factorization approximation so that one of the cur- rents in the nonleptonic decay Hamiltonian is factorized out and generates a meson. Thus the decay amplitude of the two body nonleptonic decay becomes the product of two matrix elements, one related to the decay constant of the factorized meson and the other to the weak transition matrix element between two hadrons. There have been some discussions of the plausibility of factorization关9,10兴, and this approach may be a good approximation in energetic decays. In some recent work corrections to the factorization approximation have also been considered by introducing some phenomenological nonfactorizable parameters which depend on the specific de- cay channels and should be determined by experimental data 关11–14兴.

The effective Hamiltonian for the ⌬S⫽1, weak, nonlep- tonic decays has been discussed in detail in Refs. 关15,16兴, where the Wilson coefficients for the tree and penguin op- erators were obtained to the next-to-leading order QCD and QED corrections by calculating the 10⫻10, two-loop,

*Email address: xhguo@physics.adelaide.edu.au

†Email address: athomas@physics.adelaide.edu.au

anomalous dimension matrix. The dependence of the Wilson coefficients on renormalization scheme, gauge and infra-red cutoff was also discussed. The formalism can be extended to the charmed hadron nonleptonic decays in a straightforward way.

The remainder of this paper is organized as follows. In Sec. II we calculate the six Wilson coefficients of tree and QCD penguin operators to the next-to-leading order QCD corrections by applying the results of Refs.关15,16兴. Then in Sec. III we give the formalism for the C P-violating asym- metry in charmed hadron nonleptonic decays and show nu- merical results. Finally, Sec. IV is reserved for a summary and some discussion.

II. THE EFFECTIVE HAMILTONIAN FOR NONLEPTONIC CHARMED HADRON DECAYS

In order to calculate direct C P violation in nonleptonic, charmed hadron decays we use the following effective weak Hamiltonian, which is Cabibbo first-forbidden, based on the operator product expansion:

H_⌬C⫽1⫽G_F

冑

²

冋

^q

兺

^{⫽d,s VuqVcq*共c1O1q⫹c2O2q兲}

⫺V_ubV_cb*_i

兺

_⫽⁶₃ ^ciOi

册

^{⫹H.c. 共1兲}

Here c_i(i⫽1, . . . ,6) are the Wilson coefficients and the op- erators O_i have the following expressions:

O₁^q⫽¯u_␣␥␮共1⫺␥5兲q_␤¯q_␤␥^␮共1⫺␥5兲c_␣, O₂^q⫽¯u␥␮共1⫺␥5兲qq¯␥^␮共1⫺␥5兲c, O₃⫽¯u␥␮共1⫺␥5兲c

兺

_q⬘ ^{¯q⬘␥␮共1⫺␥5兲q⬘,}

O₄⫽¯u_␣␥␮共1⫺␥5兲c_␤

兺

_q⬘ ^{¯q␤⬘␥␮共1⫺␥5兲q␣⬘,}

O₅⫽¯u␥␮共1⫺␥5兲c

兺

_q⬘ ^{¯q⬘␥␮共1⫹␥5兲q⬘,}

O₆⫽¯u_␣␥␮共1⫺␥5兲c_␤

兺

_q⬘ ^{¯q␤⬘␥␮共1⫹␥5兲q␣⬘, 共2兲}

where␣ ^and␤are color indices, and q⬘⫽u, d, s. In Eq.共2兲 O₁ and O₂ are the tree operators, while O₃⫺O₆ are QCD penguin operators. In the Hamiltonian we have omitted the operators associated with electroweak penguin diagrams.

The Wilson coefficients c_i(i⫽1, . . . ,6), are calculable in perturbation theory by using the renormalization group. The solution has the following form:

C共␮兲⫽U共␮^,mW兲C共m_W兲, 共3兲 where U(␮^,mW) describes the QCD evolution which sums the logarithms关␣sln(m_W²/␮²⁾兴ⁿ共leading-log approximation兲

and ␣s关␣sln(m_W²/␮²⁾兴ⁿ 共next-to-leading order兲. In Refs.

关15,16兴it was shown that U(m₁,m₂) can be written as U共m₁,m₂兲⫽

冉

^{1⫹␣s4共m␲1兲J}

冊

^{U0共m1,m2兲}

冉

^{1⫺␣s4共m␲2兲J}

冊

^,

共4兲 where U⁰(m₁,m₂) is the evolution matrix in the leading-log approximation and the matrix J summarizes the next-to- leading order corrections to this evolution.

The evolution matrices U⁰(m₁,m₂) and J can be obtained by calculating the appropriate one- and two-loop diagrams, respectively. The initial conditions C(m_W) are determined by matching the full theory and the effective theory at the scale m_W. At the scale m_c the Wilson coefficients are given by

C共m_c兲⫽U₄共m_c,m_b兲M共m_b兲U₅共m_b,m_W兲C共m_W兲, 共5兲 where U_f(m₁,m₂) is the evolution matrix from m₂ to m₁ with f active flavors and M (m_b) is the quark-threshold matching matrix at m_b. Since the strong interaction is inde- pendent of quark flavors, the matrices U₄(m_c,m_b), U₅(m_b,m_W), and M (m_b) are the same as those in b decays.

Hence, using the expressions for U⁰(m₁,m₂), J and M (m_b) given in Refs.关15,16兴, we can obtain C(m_c).

In general, the Wilson coefficients depend on the renor- malization scheme. The scheme-independent Wilson coeffi- cients C¯ (␮) are introduced by the following equation:

C¯共␮兲⫽

冉

^{1⫹4␣␲s RT}

冊

^{C共␮兲, 共6兲}

where R is the renormalization matrix associated with the four-quark operators O_i(i⫽1, . . . ,6) in Eq.共2兲, at the scale m_W. The scheme-independent Wilson coefficients have been used in the literature关4,17,18兴. However, since R depends on the infrared regulator 关15兴, C¯ (␮) also carries such a depen- dence. In the present paper we have chosen to use the scheme-independent Wilson coefficients.

From Eqs. 共5兲,共6兲 and the expressions for the matrices U_f(m₁,m₂), M (m_b), and R in Refs.关15,16兴 we obtain the following scheme-independent Wilson coefficients for c de- cays at the scale m_c⫽1.35 GeV:

¯c₁⫽⫺0.6941, ¯c₂⫽1.3777, ¯c₃⫽0.0652,

共7兲

¯c

4⫽⫺0.0627, ¯c

5⫽0.0206, ¯c

6⫽⫺0.1355.

In obtaining Eq. 共7兲 we have taken ␣s(m_Z)⫽0.118 which leads to ⌳QCD

(5) ⫽0.226 GeV and ⌳QCD

(4) ⫽0.329 GeV. To be consistent, the matrix elements of the operators O_i should also be renormalized to the one-loop order since we are working to the next-to-leading order for the Wilson coeffi- cients. This results in effective Wilson coefficients, c_i⬘^, which satisfy the constraint

c_i共m_c兲具^Oi共m_c兲典⫽c_i⬘具^Oi典^tree, 共8兲 where具^Oi(mc)典are the matrix elements, renormalized to the one-loop order. The relations between c_i⬘^{and c}iread关17,18兴

c₁⬘⫽¯c₁, c₂⬘⫽¯c₂, c₃⬘⫽¯c₃⫺P_s/3,

共9兲 c₄⬘⫽¯c₄⫹P_s, c₅⬘⫽¯c₅⫺P_s/3, c₆⬘⫽¯c₆⫹P_s, where

P_s⫽关␣s共m_c兲/8␲兴关10/9⫹G共m,m_c,q²兲兴¯c₂, with

G共m,m_c,q²兲⫽4

冕

0 1

dxx共1⫺x兲lnm²⫺x共1⫺x兲q² m_c² . Here q² is the momentum transfer of the gluon in the pen- guin diagram and m is the mass of the quark in the loop of the penguin diagram.¹ G(m,m_c,q²) has the following ex- plicit expression关19兴:

Re G⫽2

3

冋

^{lnmm2c2⫺53⫺4mq22⫹}

冉

^1⫹2mq22

冊

⫻

冑

^1⫺4mq²² ln

1⫹

冑

^1⫺4mq²² 1⫺

冑

^1⫺4mq²²

册

^,

Im G⫽⫺2

3␲

冉

^1⫹2mq22

冊 冑

^{1⫺4mq22. 共10兲}

Based on simple arguments at the quark level, the value of q² is chosen in the range 0.3⬍q²/m_c²⬍0.5 关6,7兴. From Eqs.共7兲,共9兲, and共10兲we can obtain numerical values of c_i⬘^. When q²/m_c²⫽0.3,

c₁⬘⫽⫺0.6941, c₂⬘⫽1.3777,

c₃⬘⫽0.07226⫹0.01472i, c₄⬘⫽⫺0.08388⫺0.04417i, c₅⬘⫽0.02766⫹0.01472i, c₆⬘⫽⫺0.1567⫺0.04417i,

共11兲 and when q²/m_c²⫽0.5,

c₁⬘⫽⫺0.6941, c₂⬘⫽1.3777,

c₃⬘⫽0.06926⫹0.01483i, c₄⬘⫽⫺0.07488⫺0.04448i,

c₅⬘⫽0.02466⫹0.01483i, c₆⬘⫽⫺0.1477⫺0.04448i.

共12兲 In calculating the matrix elements of the Hamiltonian共1兲, we can then simply use the effective Wilson coefficients in Eqs.

共11兲共12兲 to multiply the tree-level matrix elements of the operators O_i(i⫽1, . . . ,6).

III. CP VIOLATION IN CHARMED HADRON DECAYS A. Formalism for CP violation

in charmed hadron decays

The formalism for C P violation in B and ⌳b hadronic decays关6–8兴can be generalized to the case of charmed had- rons in a straightforward manner. Let H_c denote a charmed hadron which could be D^⫾, D⁰, or⌳c. The amplitude A for the decay H_c→f␲^⫹␲^⫺ ( f is a decay product兲is

A⫽具␲^⫹␲^⫺f兩H^T兩H_c典^⫹具␲^⫹␲^⫺f兩H^P兩H_c典^{, 共13兲} where H^T and H^P are the Hamiltonians for the tree and penguin operators, respectively.

The relative magnitude and phases of these two diagrams are defined as follows:

A⫽具␲^⫹␲^⫺f兩H^T兩H_c典关1⫹re^i␦e^i␾兴,

共14兲 A¯⫽具␲^⫹␲^⫺f¯兩H^T兩H¯_c典关1⫹re^i␦e^⫺i␾兴,

where␦ ^and␾ are strong and weak phases, respectively.␾ arises from the C P-violating phase in the CKM matrix, and it is arg关V_ubV_cb*/(V_uqV_cq*)兴 for the c→q transition (q⫽d or s). The parameter r is defined as

r⬅

冏

^具具␲^␲⫹⫹␲␲^⫺⫺ff兩H兩H^TP兩兩^HH^c_c^典典

冏

^{. 共15兲}

The C P-violating asymmetry, a, can be written as

a⬅兩A兩²⫺兩A¯兩²

兩A兩²⫹兩A¯兩²⫽ ⫺2r sin␦^sin␾

1⫹2r cos␦^cos␾⫹r². 共16兲 It can be seen from Eq. 共16兲 that both weak and strong phases are needed to produce C P violation. Since in r there is strong suppression from the ratio of the CKM matrix ele- ments, 关V_ubV_cb*/(V_uqV_cq*)兴, which is of the order 10^⫺3 关3兴 关for both q⫽d and q⫽s this suppression is 0.62⫻10^⫺3, see Eqs.共32兲and共43兲in Sec. III B兴, usually the C P violation in charmed hadron decays is predicted to be small.

The weak phase␾for a specific physical process is fixed.

In order to obtain possible large C P violation, we need some mechanism to produce either large sin␦or large r.␳^-␻^mix- ing has the dual advantages that the strong phase difference is large 共passing through 90° at the␻ ^resonance兲 and well known. In this scenario one has关7,8兴

具␲^⫹␲^⫺f兩H^T兩H_c典⫽ g_␳ s_␳s_␻⌸˜

␳␻t_␻⫹g_␳

s_␳t_␳, 共17兲

1m could be md or ms. However, the numerical values of c_i⬘ change by at most 4% when we change m from m_dto m_s. There- fore, we ignore this difference in our calculations, setting m⫽ms.

具␲^⫹␲^⫺f兩H^P兩H_c典^⫽_s^g␳

␳s_␻⌸˜

␳␻p_␻⫹g_␳

s_␳p_␳, 共18兲 where t_V (V⫽␳ ^or ␻) is the tree and p_V is the penguin amplitude for producing a vector meson V by H_c→f V; g_␳ is the coupling for␳⁰→␲^⫹␲^⫺; ⌸˜

␳␻ is the effective␳^-␻^mix- ing amplitude and s_V^⫺1 is from the propagator of V, s_V⫽s

⫺m_V²⫹im_V⌫V, with

冑

s being the invariant mass of the

␲^⫹␲^⫺ pair. The numerical values for the ␳^-␻ ^{mixing pa-} rameter are 关7,20,21兴 Re⌸˜

␳␻(m_␻²)⫽⫺3500⫾300 MeV², Im⌸˜

␳␻(m_␻²)⫽⫺300⫾300 MeV². The direct coupling ␻

→␲^⫹␲^⫺ is effectively absorbed into⌸˜

␳␻ 关21兴. Defining

p_␻

t_␳ ⬅r⬘^{ei(␦q⫹␾), t}_t^␻

␳⬅␣^ei␦␣, _p^p␳

␻⬅␤^ei␦␤, 共19兲 where␦␣,␦␤, and␦qare strong phases, one has the follow- ing expression for r and␦^,

re^i␦⫽r⬘^{ei␦q⌸˜␳␻⫹␤e}

i␦_␤s_␻ s_␻⫹⌸˜

␳␻␣^{ei␦␣. 共20兲} It will be shown that in the factorization approach, for all the decay processes H_c→f␲^⫹␲^⫺we are considering,␣^ei␦␣ is real 共see Sec. III B for details兲. Therefore, we let

␣^ei␦␣⫽g, 共21兲 where g is a real parameter. Letting

␤^ei␦␤⫽b⫹ci, r⬘^ei␦q⫽d⫹ei, 共22兲 and using Eq.共20兲, we obtain the following result when

冑

^s

⬃m_␻:

re^i␦⫽ C⫹Di

共s⫺m_␻²⫹g Re⌸˜

␳␻兲²⫹共g Im⌸˜

␳␻⫹m_␻⌫_␻兲², 共23兲 where

C⫽共s⫺m_␻²⫹g Re⌸˜

␳␻兲兵^d关Re⌸˜␳␻⫹b共s⫺m_␻²兲⫺cm_␻⌫_␻兴

⫺e关Im⌸˜

␳␻⫹bm_␻⌫_␻⫹c共s⫺m_␻²兲兴其

⫹共g Im⌸˜

␳␻⫹m_␻⌫_␻兲兵^e关Re⌸˜␳␻⫹b共s⫺m_␻²兲⫺cm_␻⌫_␻兴

⫹d关Im⌸˜

␳␻⫹bm_␻⌫_␻⫹c共s⫺m_␻²兲兴其^, D⫽共s⫺m_␻²⫹g Re⌸˜

␳␻兲兵^e关Re⌸˜

␳␻⫹b共s⫺m_␻²兲⫺cm_␻⌫_␻兴

⫹d关Im⌸˜

␳␻⫹bm_␻⌫_␻⫹c共s⫺m_␻²兲兴其

⫺共g Im⌸˜

␳␻⫹m_␻⌫_␻兲兵^d关Re⌸˜

␳␻⫹b共s⫺m_␻²兲

⫺cm_␻⌫_␻兴⫺e关Im⌸˜

␳␻⫹bm_␻⌫_␻⫹c共s⫺m_␻²兲兴其^{. 共24兲}

The weak phase comes from 关V_ubV_cb*/(V_uqV_cq*)兴. If the operators O₁^d,O₂^d contribute to the decay processes we have

sin␾兩d⫽ ␩

冑

关␳⫹A²␭⁴共␳²⫹␩²兲兴²⫹␩^2,

共25兲 cos␾兩d⫽⫺ ␳⫹A²␭⁴共␳²⫹␩²兲

冑

关␳⫹A²␭⁴共␳²⫹␩²兲兴²⫹␩^2, while if O₁^s and O₂^s contribute, we have

sin␾兩s⫽⫺ ␩

冑

␳²⫹␩^2,

共26兲 cos␾兩s⫽ ␳

冑

␳²⫹␩^2,

where we have used the Wolfenstein parametrization关22兴for the CKM matrix elements. In order to obtain r sin␦^{, r cos}␦^, and r we need to calculate␣^ei␦␣,␤^{ei␦␤, and r}⬘^ei␦q. This will be done in the next subsection.

B. CP violation in H_c\f␲^¿␲^À

In the following we will calculate the C P-violating asym- metries in H_c→f␲^⫹␲^⫺. In the factorization approximation

␳⁰⁽␻) is generated by one current which has the proper quantum numbers in the Hamiltonian in Eq. 共1兲. In the fol- lowing we will consider the decay processes D^⫹

→␳^⫹␳⁰⁽␻⁾→␳^⫹␲^⫹␲^{⫺, D⫹}→␲^⫹␳⁰⁽␻⁾→␲^⫹␲^⫹␲^{⫺, D0}

→␾␳⁰⁽␻⁾→␾␲^⫹␲^{⫺, D0}→␩␳⁰⁽␻⁾→␩␲^⫹␲^{⫺, D0}

→␩⬘␳⁰⁽␻⁾→␩⬘␲^⫹␲^{⫺, D0}→␲⁰␳⁰⁽␻⁾→␲⁰␲^⫹␲^{⫺, and}

⌳c→p␳⁰⁽␻⁾→p␲^⫹␲^⫺, individually.

共1兲 D^⫹→␳^⫹V(V⫽␳^{0 or} ␻). First we consider D^⫹

→␳^⫹␳⁰⁽␻). After factorization, the contribution to t_␳^␳⫹ 共the superscript denotes the decay product f in H_c→f␲^⫹␲^⫺) from the tree level operator O₁^d is

具␳^⫹␳⁰兩O₁^d兩D^⫹典^⫽具␳⁰兩共¯ dd 兲兩0典具␳^⫹兩共¯ cu 兲兩D^⫹典^⬅T1, 共27兲 where (d¯ d) and (u¯c) denote the V-A currents. If we ignore isospin violating effects, then the matrix element of O₂^dis the same as that of O₁^d. After adding the contributions from Fierz transformation of O₁^d and O₂^d we have

t_␳^␳⫹⫽共c₁⬘⫹c₂⬘兲共1⫹1/N_c兲T₁, 共28兲 where we have omitted the CKM matrix elements in the expression of t_␳^␳⫹. Since in Eq. 共28兲we have neglected the color-octet contribution, which is nonfactorizable and diffi- cult to calculate, N_cshould be treated as an effective param- eter which depends on the hadronization dynamics of differ- ent decay channels. In the same way we find that t_␻^␳⫹⫽

⫺t_␳^␳⫹, so that, from Eq.共19兲, we have

共␣^ei␦␣兲^␳⫹⫽⫺1. 共29兲 The penguin operator contributions, p_␳^␳⫹ and p_␻^␳⫹, can be evaluated in the same way with the aid of the Fierz identities.

From Eq.共19兲we have

共␤^ei␦␤兲^␳⫹⫽0 共30兲 and

共r⬘^ei␦q兲^␳⫹⫽2

共c₃⬘⫹c₄⬘兲

冉

^1⫹N1c

冊

^{⫹c5⬘⫹N1cc6⬘}

共c₁⬘⫹c₂⬘兲

冉

^1⫹N1c

冊 冏

^VVubudVVcb*cd*

冏

^,

共31兲 where

冏

^VVubudVVcb*cd*

冏

^{⫽1A⫺2␭␭24/2}

冑

共1⫹A²␭^␳42␳^⫹兲^2␩⫹²A⁴␭⁸␩^{2. 共32兲} 共2兲 ⌳c→pV. Next we consider⌳c→p␳⁰⁽␻). Defining

具␳^0p兩O₁^d兩⌳c典⫽具␳⁰兩共¯ dd 兲兩0典具^p兩共¯ cu 兲兩⌳c典⬅T₂, 共33兲 we have

t_␳^p⫽

冉

^{c1⬘⫹N1cc2⬘}

冊

^{T2. 共34兲}

After evaluating t_␻^p and the penguin diagram contributions we obtain the following results:

共␣^ei␦␣兲^p⫽⫺1, 共35兲

共␤^ei␦␤兲^p⫽

c₄⬘⫹ 1 N_cc₃⬘

冉

^2⫹N1c

冊

^c3⬘⫹

冉

^1⫹N2c

冊

^c4⬘⫹2

冉

^{c5⬘⫹N1cc6⬘}

冊

^,

共36兲

共r⬘^ei␦q兲^p⫽

冉

^2⫹N1c

冊

^c3⬘⫹

冉

^1⫹N2c

冊

^c4⬘⫹2

冉

^{c5⬘⫹N1cc6⬘}

冊

c₁⬘⫹ 1 N_cc₂⬘

⫻

冏

^VVubudVVcb*cd*

冏

^{. 共37兲}

共3兲 D⁰→␾V. For the decay channel D⁰→␾␳⁰⁽␻⁾

→␾␲^⫹␲^⫺the operators O₁^s and O₂^s contribute to the decay matrix elements. If we define

具␳⁰␾兩O₁^s兩D⁰典⫽具␾兩共¯ss 兲兩0典具␳⁰兩共¯ cu 兲兩D⁰典⬅T₃, 共38兲 we have

t_␳^␾⫽

冉

^{c1⬘⫹N1cc2⬘}

冊

^{T3, 共39兲}

and

共␣^ei␦␣兲^␾⫽1, 共40兲 共␤^ei␦␤兲^␾⫽1, 共41兲

共r⬘^ei␦q兲^␾⫽⫺

c₃⬘⫹ 1

N_cc₄⬘⫹c₅⬘⫹ 1 N_cc₆⬘ c₁⬘⫹ 1

N_cc₂⬘

冏

^{VVubusVVcb*cs*}

冏

^{, 共42兲}

where

冏

^VVubusVVcb*cs*

冏

^{⫽A2␭14⫺}

冑

^{␭␳22/2⫹␩2. 共43兲}

共4兲 D⁰→␩⁽␩⬘)V. For the decay channels D⁰→␩␳⁰⁽␻⁾

→␩␲^⫹␲^{⫺and D0}→␩⬘␳⁰⁽␻⁾→␩⬘␲^⫹␲^⫺, things become a little complicated. It is known that ␩ ^and␩⬘ have both u¯ u

⫹¯ d and s¯s components. The decay constants, fd _␩^u_(␩⬘) and f_␩^s_(␩⬘), defined as

具^0兩¯u␥␮␥5u兩␩共␩⬘兲典^{⫽i f}_␩^u(␩⬘⁾p_␮,

共44兲 具⁰兩¯s␥_␮␥5s兩␩共␩⬘兲典⫽i f_␩^s_(␩⬘)p_␮,

are different. After straightforward derivations we have 共␣^ei␦␣兲^␩(␩⬘)⫽1, 共45兲 共␤^ei␦␤兲^␩(␩⬘)⫽1, 共46兲

共r⬘^ei␦q兲^␩(␩⬘)⫽⫺2 f_␩(␩

⬘⁾

u ⫹f_␩(␩

⬘⁾

s

f_␩(␩

⬘⁾

u ⫺f_␩(␩

⬘⁾

s

⫻ c₃⬘⫹ 1

N_cc₄⬘⫺c₅⬘⫺ 1 N_cc₆⬘ c₁⬘⫹ 1

N_cc₂⬘

冏

^{VVubusVVcb*cs*}

冏

^.

共47兲 In the derivations of Eqs. 共45兲–共47兲 we have made the ap- proximation that V_ubV_cb*/V_udV_cd*_⫽⫺V_ubV_cb*/V_usV_cs*. It is noted that the minus signs associated with c₅⬘ ^{and c}6⬘ ^{in Eq.}

共47兲arise because␩⁽␩⬘) are pseudoscalar mesons. Since the imaginary part of c₃⬘^(c4⬘) is the same as that of c₅⬘^(c6⬘^),␦q is zero. This leads to the strong phase, ␦, being zero, in com- bination with Eqs.共45兲,共46兲.

The decay constants f_␩(␩

⬘⁾

u and f_␩(␩

⬘⁾

s were calculated phenomenologically in Ref. 关23兴, based on the assumption that the decay constants in the quark flavor basis follow the pattern of particle state mixing. It was found that

f_␩^u⫽78 MeV, f_␩^s⫽⫺112 MeV, f_␩^u_⬘⫽63 MeV, 共48兲 f_␩^s_⬘⫽137 MeV.

共5兲 D→␲V. For the decay process D^⫹→␲^⫹␳⁰⁽␻⁾

→␲^⫹␲^⫹␲^⫺, two kinds of matrix element products are in- volved after factorization, i.e.,

具␳⁰共␻兲兩共¯ dd 兲兩0典具␲^⫹兩共¯ cu 兲兩D^⫹典 and

具␲^⫹兩共¯ du 兲兩0典具␳⁰共␻兲兩共d¯ c兲兩D^⫹典^.

These two quantities cannot be related to each other by sym- metry. Therefore, we have to evaluate them in some phe- nomenological quark models and hence more uncertainties are involved. Similarly for D⁰→␲⁰␳⁰⁽␻⁾→␲⁰␲^⫹␲^{⫺ we} have to evaluate 具␳⁰⁽␻⁾兩(d¯ d)兩0典具␲⁰兩(u¯ c)兩D⁰典 ^and 具␲⁰兩(d¯ d)兩0典具␳⁰⁽␻⁾兩(u¯ c)兩D⁰典 separately.

The matrix elements for D→X and D→X* (X and X* denote pseudoscalar and vector mesons, respectively兲can be decomposed as 关24兴

具^X兩J␮兩D典^⫽

冉

^{pD⫹pX⫺mD2k⫺2mX2k}

冊

_␮^F1共k2兲

⫹m_D²⫺m_X²

k² k_␮F₀共k²兲, 共49兲

具^X*兩J_␮兩D典⫽ 2

m_D⫹m_X*⑀_␮␯␳␴⑀*^␯p_D^␳p_X^␴_*V共k²兲

⫹i

冋

^{⑀␮*共mD⫹mX*兲A1共k2兲⫺mD⑀⫹•kmX}*

⫻共p_D⫹p_X*兲␮A₂共k²兲⫺⑀•k

k² 2m_X*k_␮A₃共k²兲

册

⫹i⑀•k

k² 2m_X*k_␮A₀共k²兲, 共50兲 where J_␮ is the weak current, k⫽p_D⫺p_X(X*) and⑀_␮ ^{is the} polarization vector of X*. The form factors satisfy the rela- tions F₁(0)⫽F₀(0), A₃(0)⫽A₀(0) and A₃(k²)⫽关(m_D

⫹m_X*)/2m_X*兴A₁(k²)⫺关(m_D⫺m_X*)/2m_X*兴A₂(k²).

Using the decomposition in Eqs. 共49兲,共50兲, we have for D^⫹→␲^⫹␳⁰⁽␻^),

t_␳^␲⫹⫽⫺

冑

^2mD兩pជ_␳兩

冋冉

^{c1⬘⫹N1cc2⬘}

冊

^{f␳F1共m␳2兲}

⫹

冉

^{c2⬘⫹N1cc1⬘}

冊

^{f␲A0共m␲2兲}

册

^{, 共51兲}

where f_␳ and f_␲ are the decay constants of the ␳ ^and ␲^, respectively, and pជ_␳ is the three momentum of the␳^.

It can be shown that t_␻^␲⫹⫽⫺t_␳^␲⫹. After calculating the penguin operator contributions, we have

共␣^ei␦␣兲^␲⫹⫽⫺1, 共52兲

共␤^ei␦␤兲^␲⫹⫽

关f_␳F₁共m_␳²兲⫺f_␲A₀共m_␲²兲兴

冉

^{c4⬘⫹N1cc3⬘}

冊

^{⫺共m2mc⫹␲2mfd␲兲共Am0共um⫹␲2m兲d兲}

冉

^{c6⬘⫹N1cc5⬘}

冊

x , 共53兲

共r⬘^ei␦q兲^␲⫹⫽ x

冋

^{f␳F1共m␳2兲⫹N1cf␲A0共m␲2兲}

册

^c1⬘⫹

冋

^{N1cf␳F1共m␳2兲⫹f␲A0共m␲2兲}

册

^c2⬘

冏

^VVubudVVcb*cd*

冏

^{, 共54兲}

where x is defined as

x⫽

冋

^{2 f␳F1共m␳2兲⫹f␳F1共m␳2兲⫹Ncf␲A0共m␲2兲}

册

^c3⬘⫹

冋

^{2 f␳FN1c共m␳2兲⫹f␳F1共m␳2兲⫹f␲A0共m␲2兲}

册

^c4⬘

⫹2

冋

^{f␳F1共m␳2兲⫺Nc共mmc␲2⫹fm␲Ad兲共0共mmu␲2⫹兲md兲}

册

^c5⬘⫹2

冋

^{f␳FN1共cm␳2兲⫺共mcm⫹␲2mf␲dA兲共0m共mu⫹␲2兲md兲}

册

^{c6⬘. 共55兲}

We can consider the process D⁰→␲⁰␳⁰⁽␻⁾→␲⁰␲^⫹␲^⫺in the same way. We find

共␣^ei␦␣兲^␲0⫽⫺f_␳F₁共m_␳²兲⫺f_␲A₀共m_␲²兲

f_␳F₁共m_␳²兲⫹f_␲A₀共m_␲²兲, 共56兲

共␤^ei␦␤兲^␲0⫽

关f_␳F₁共m_␳²兲⫹f_␲A₀共m_␲²兲兴

冉

^{c4⬘⫹N1cc3⬘}

冊

^{⫺共m2mc⫹␲2mfd␲兲共Am0共um⫹␲2m兲d兲}

冉

^{c6⬘⫹N1cc5⬘}

冊

x , 共57兲

共r⬘^ei␦q兲^␲0⫽ x

关f_␳F₁共m_␳²兲⫹f_␲A₀共m