Eur. Phys. J. C (2013) 73:2519DOI 10.1140/epjc/s10052-013-2519-2

Regular Article - Theoretical Physics

Enhanced CP violation in D0 → ρ0(ω)ρ0(ω) → π+π−π+π−

Gang Lü1,a, Z.-H. Zhang2,b, X.-H. Guo3,c, Jun-Chao Lu1, Shi-Ming Yan1

1College of Science, Henan University of Technology, Zhengzhou 450001, China2School of Nuclear Science and Technology, University of South China, Hengyang, Hunan 421001, China3College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China

Received: 29 January 2013 / Revised: 29 June 2013 / Published online: 21 August 2013© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract In the factorization approach we study the directCP violation in D0 → ρ0(ω)ρ0(ω) → π+π−π+π− (withunpolarized ρ0(ω)) via the ρ − ω mixing mechanism. Wefind that the CP violation can be enhanced due to a largestrong phase difference when the masses of the π+π− pairsare in the vicinity of the ω resonance. The CP violation pa-rameter depends on the effective parameter Nc which in-cludes nonfactorizable effects and is determined by the ex-perimental data. Taking into account double ρ − ω mixingcontributions, we find the CP violating asymmetry is largerthan these in the cases where no ρ − ω mixing or singleρ −ω mixing is taken into account. We also discuss the pos-sibility to observe the predicted CP violation at BEPCII.

1 Introduction

CP violation remains an open problem in particle physicssince it was first observed in the neutral kaon system morethan four decades ago [1]. In the Standard Model (SM),weak complex phases in the Cabibbo–Kobayashi–Maskawa(CKM) matrix are responsible for CP violation [2, 3]. Inthe past few years more attention has been focused on thedecays of the B meson system both theoretically and ex-perimentally. Large CP violation has been observed in thedecays of B mesons [4]. However, in the charm sector, CPviolation is usually predicted to be small due to the sup-pression of the CKM matrix elements. Even so, CP viola-tion has been observed in the processes D± → K0

s π± andD0 → K+K− [4].

Direct CP violation occurs through the interference oftwo amplitudes with different weak phases and strongphases. The weak phase difference is directly determined

a e-mail: ganglv66@sina.comb e-mail: zhangzh@iopp.ccnu.edu.cnc e-mail: xhguo@bnu.edu.cn

by the CKM matrix elements, while the strong phase is usu-ally difficult to control. In order to obtain a large CP viola-tion signal, one needs to appeal to some phenomenologicalmechanism to get a large strong phase difference. ρ − ω

mixing has been used for this purpose in the past few years[5–13]. In this paper, we will investigate the CP violation forthe decay D0 → ρ0(ω)ρ0(ω) → π+π−π+π−. In charmedmeson decays, direct CP violation is small and difficult tobe detected in experiments. However, for the channel withtwo ρ(ω) mesons in the intermediate state, the decay pro-cess can get twice contribution from ρ–ω mixing. One canexpect that there should be a bigger CP violating asymmetrythan in the case with no or single ρ–ω mixing.

In our calculations, we have to deal with hadronic ma-trix elements for both tree and penguin operators in the ef-fective Hamiltonian, which are controlled by the effects ofnonperturbative QCD. The mass of the charm quark is about1.5 GeV, so it is difficult to apply the QCD factorizationapproach [14] to the D-meson decays, as 1/mc power cor-rections are much larger than those in the case of B-mesondecays. Hence in order to extract the strong phase differ-ence we will use the traditional factorization approach, inwhich one of the currents in the Hamiltonian is factorizedout and generates a meson. Thus, the decay amplitude be-comes the product of two matrix elements. Such factoriza-tion scheme was first argued to be plausible in energetic de-cays like bottom-hadron decays [15, 16], then was provedto be the leading order result in the framework of QCDfactorization when the radiative QCD corrections of orderO(αs(mb)) (mb is the b-quark mass) and O(1/mb) correc-tions are neglected [14]. Since the nonfactorizable contribu-tions are ignored in this factorization scheme, an effectiveparameter, Nc, is introduced in order to take into accountnonfactorizable contributions. In the present work, we willdetermine the value of Nc by comparing the theoretical re-sults with the experimental data for the decay branching ra-tios.

Page 2 of 8 Eur. Phys. J. C (2013) 73:2519

The remainder of this paper is organized as follows. InSect. 2 we present the form of the effective Hamiltonianand the values of Wilson coefficients. In Sect. 3 we givethe formalism for the CP violation in D0 → ρ0(ω)ρ0(ω) →π+π−π+π− via ρ–ω mixing. In Sect. 4, we calculate thebranching ratios for D0 → ρ0ρ0 and present the rangeof Nc. A summary and discussion are included in the lastsection.

2 The effective Hamiltonian

With the operator product expansion, the effective weakHamiltonian for charmed hadron decays, which is singlyCabibbo-suppressed, is [17, 18]

H�C=1 = GF√2

[ ∑q=d,s

VuqV ∗cq

(c1O

q

1 + c2Oq

2

)

− VubV∗cb

6∑i=3

ciOi

]+ H.C., (1)

where GF represents Fermi constant, ci (i = 1, . . . ,6) arethe Wilson coefficients, Vuq , Vcq , Vub and Vcb are CKM ma-trix elements. The operators Oi have the following forms:

Oq

1 = uαγμ(1 − γ5)qβ qβγ μ(1 − γ5)cα,

Oq

2 = uγμ(1 − γ5)qqγ μ(1 − γ5)c,

O3 = uγμ(1 − γ5)c∑q ′

q ′γ μ(1 − γ5)q′,

O4 = uαγμ(1 − γ5)cβ

∑q ′

q ′βγ μ(1 − γ5)q

′α,

O5 = uγμ(1 − γ5)c∑q ′

q ′γ μ(1 + γ5)q′,

O6 = uαγμ(1 − γ5)cβ

∑q ′

q ′βγ μ(1 + γ5)q

′α,

(2)

where α and β are color indices, and q ′ = u,d or s quarks.In Eq. (2) O

q

1 and Oq

2 are tree operators, O3–O6 are QCDpenguin operators. We have omitted the operators associatedwith electroweak penguin diagrams.

From the renormalization group equation, one can calcu-late the Wilson coefficients ci (i = 1, . . . ,6) by perturbationtheory [17–20]. C(mW) can be obtained by matching thefull theory and the effective theory at the scale mW . One canfind the evolution matrix and the quark-threshold matchingmatrix are the same as those in b decays [17–20]. Hence,based on the flavor independence of strong interaction, wecan calculate the Wilson coefficients C(mc) at the scale mc .

The physical quantities should be renormalization schemeindependent. In this paper we use the scheme-independentWilson coefficients. Following Refs. [17–20] we obtain thefollowing scheme-independent Wilson coefficients for c-decays at the scale mc = 1.35 GeV:

c1 = −0.6941, c2 = 1.3777, c3 = 0.0652,

c4 = −0.0627, c5 = 0.0206, c6 = −0.1355.(3)

In obtaining Eq. (3) we have taken αs(mZ) = 0.118 which

leads to Λ(5)QCD = 0.226 GeV and Λ

(4)QCD = 0.329 GeV. To be

consistent, the matrix elements of the operators Oi shouldalso be renormalized to the one-loop order since we areworking to the next-to-leading order for the Wilson coeffi-cients. This results in effective Wilson coefficients, c′

i , whichsatisfy the constraint

ci(mc)⟨Oi(mc)

⟩ = c′i〈Oi〉tree, (4)

where 〈Oi(mc)〉 are the matrix elements, renormalized tothe one-loop order. The relations between c′

i and ci read[21, 22]

c′1 = c1, c′

2 = c2, c′3 = c3 − Ps/3,

c′4 = c4 + Ps, c′

5 = c5 − Ps/3, c′6 = c6 + Ps,

(5)

where

Ps = (αs(mc)/8π

)(10/9 + G

(m,mc, q

2))c2,

with

G(m,mc, q

2) = 4∫ 1

0dxx(1 − x) ln

m2 − x(1 − x)q2

m2c

.

Here q is the momentum transfer of the gluon in the penguindiagram and m is the mass of the quark in the loop of thepenguin diagram. G(m,mc, q

2) has the following explicitexpression [23]:

ReG = 2

3

(ln

m2

m2c

− 5

3− 4

m2

q2+

(1 + 2

m2

q2

)

×√

1 − 4m2

q2ln

1 +√

1 − 4m2

q2

1 −√

1 − 4m2

q2

), (6)

ImG = −2

3π

(1 + 2

m2

q2

)√1 − 4

m2

q2.

Based on simple arguments at the quark level, the valueof q2 is chosen to be in the range 0.3 < q2/m2

c < 0.5 [5, 6].From Eqs. (3), (5) and (6) we can obtain numerical valuesof c′

i . When q2/m2c = 0.3,

c′1 = −0.6941, c′

2 = 1.3777,

c′3 = 0.07226 + 0.01472i, c′

4 = −0.08388 − 0.04417i,

c′5 = 0.02766 + 0.01472i, c′

6 = −0.1567 − 0.04417i,

(7)

and when q2/m2c = 0.5,

c′1 = −0.6941, c′

2 = 1.3777,

c′3 = 0.06926 + 0.01483i, c′

4 = −0.07488 − 0.04448i,

c′5 = 0.02466 + 0.01483i, c′

6 = −0.1477 − 0.04448i.

(8)

Eur. Phys. J. C (2013) 73:2519 Page 3 of 8

The CKM matrix, which should be determined from ex-periments, can be expressed in terms of the Wolfenstein pa-rameters, A,λ,ρ and η [24, 25]:⎛⎝ 1 − 1

2λ2 λ Aλ3(ρ − iη)

−λ 1 − 12λ2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

⎞⎠ , (9)

where O(λ4) corrections are neglected. The latest values forthe parameters in the CKM matrix are [4]

λ = 0.2272 ± 0.0010, A = 0.818+0.007−0.017,

ρ = 0.221+0.064−0.028, η = 0.340+0.017

−0.045,(10)

where

ρ = ρ

(1 − λ2

2

), η = η

(1 − λ2

2

). (11)

From Eqs. (10), (11) we have

0.198 < ρ < 0.293, 0.302 < η < 0.366. (12)

3 CP violation in D0 → ρ0(ω)ρ0(ω) → π+π−π+π−

3.1 Formalism

The amplitude A (A) for the decay D0 → π+π−π+π−(D0 → π+π−π+π−) can be written as

A = ⟨π+π−π+π−∣∣HT

∣∣D0⟩+ ⟨

π+π−π+π−∣∣HP∣∣D0⟩, (13)

A = ⟨π+π−π+π−∣∣HT

∣∣D0⟩+ ⟨

π+π−π+π−∣∣HP∣∣D0⟩, (14)

with HT and HP being the Hamiltonian for the tree andpenguin operators, respectively.

We can define the relative magnitude and phases betweenthe tree and penguin operator contributions as follows:

A = ⟨π+π−π+π−∣∣HT

∣∣D0⟩[1 + rei(δ+φ)], (15)

A = ⟨π+π−π+π−∣∣HT

∣∣D0⟩[1 + rei(δ−φ)], (16)

where δ and φ are strong and weak phases, respectively. φ

arises from the CP-violating phase in the CKM matrix, andit is arg[VubV

∗cb/(VudV ∗

cd)] for the c → d transition. The pa-rameter r is the absolute value of the ratio of penguin andtree amplitudes:

r ≡∣∣∣∣ 〈π+π−π+π−|HP |D0〉〈π+π−π+π−|HT |D0〉

∣∣∣∣. (17)

The CP violating asymmetry, a, can be written as

a ≡ |A|2 − |A|2|A|2 + |A|2 = −2r sin δ sinφ

1 + 2r cos δ cosφ + r2. (18)

From (18), one can find the CP violation depends on theweak phase difference and the strong phase difference. Theweak phase is determined for a specific decay process.Hence, in order to obtain a large CP violation, we needsome mechanism to make sin δ large. It has been found thatρ–ω mixing (which was proposed based on vector mesondominance [26]) leads to a large strong phase difference[6–13]. Based on ρ–ω mixing and working to the first or-der of isospin violation, we have the following results:

⟨π+π−π+π−∣∣HT

∣∣D0⟩ = 2g2ρ

s2ρsω

Πρωtρω + g2ρ

s2ρ

tρρ, (19)

⟨π+π−π+π−∣∣HP

∣∣D0⟩ = 2g2ρ

s2ρsω

Πρωpρω + g2ρ

s2ρ

pρρ, (20)

where tρρ(pρρ) and tρω(pρω) are the tree (penguin) ampli-tudes for D0 → ρ0ρ0 and D0 → ρ0ω, respectively; gρ isthe coupling for ρ0 → π+π−; Πρω is the effective ρ–ω

mixing amplitude which also effectively includes the directcoupling ω → π+π−, and sV (V = ρ or ω) is proportionalto the inverse propagator of the vector meson V ,

sV = s − m2V + imV ΓV , (21)

with√

s being the invariant masses of the π+π− pairs. Forthe decay process of D0 → ρ0(ω)ρ0(ω) → π+π−π+π−,ρ–ω mixing contributes twice. Hence, there is a factor of2 in Eqs. (19), (20) compared with the case of single ρ–ω

mixing [7–12]. Furthermore, we have g2ρ and s2

ρ due to twoρ → ππ couplings and two ρ propagators. One can ignorethe s2

ω term which is of the second order of isospin violation.The direct coupling ω → π+π− has been effectively ab-

sorbed into Πρω which leads to the explicit s dependenceof Πρω [27, 28]. However, the s dependence of Πρω is neg-ligible in practice. We can make the expansion Πρω(s) =Πρω(m2

ω) + (s − mω)Π ′ρω(m2

ω). The ρ–ω mixing parame-ters were determined in the fit of Gardner and O’Connell[29]:

Re Πρω

(m2

ω

) = −3500 ± 300 MeV2,

Im Πρω

(m2

ω

) = −300 ± 300 MeV2, (22)

Π ′ρω

(m2

ω

) = 0.03 ± 0.04.

From Eqs. (13), (15), (19), (20) one has

reiδeiφ = 2Πρωpρω + sωpρρ

2Πρωtρω + sωtρρ

, (23)

where double ρ–ω mixing results in the factor of 2. Defining

pρω

tρρ

≡ r ′ei(δq+φ),tρω

tρρ

≡ αeiδα ,pρρ

pρω

≡ βeiδβ ,

(24)

where δα , δβ and δq are strong phases, one finds the follow-ing expression from Eqs. (23) and (24):

Page 4 of 8 Eur. Phys. J. C (2013) 73:2519

reiδ = r ′eiδq2Πρω + βeiδβ sω

2Πρωαeiδα + sω. (25)

In order to get the CP violating asymmetry in Eq. (18), sinφ

and cosφ are needed. The weak phase φ is fixed by theCKM matrix elements. In the Wolfenstein parametrization[24, 25], one has

sinφ = η√[ρ + A2λ4(ρ2 + η2)]2 + η2,

cosφ = − ρ + A2λ4(ρ2 + η2)√[ρ + A2λ4(ρ2 + η2)]2 + η2.

(26)

3.2 Calculational details

As mentioned in Introduction, we calculate the matrix ele-ments in the traditional framework of factorization. With theHamiltonian given in Eq. (1) one can present the matrix ele-ments for D0 → ρ0(ω)ρ0(ω). The decay amplitudes can bewritten as the product of matrix elements of the two currents.Omitting Dirac matrices and color labels, we define⟨ρ0

∣∣(dd)|0〉⟨ρ0∣∣(uc)

∣∣D0⟩ ≡ T , (27)

where (dd) and (uc) denote the V − A currents.The contribution of the tree-level operator Od

1 can bewritten as⟨ρ0ρ0

∣∣Od1

∣∣D0⟩ = 2⟨ρ0

∣∣(dd)|0〉⟨ρ0∣∣(uc)

∣∣D0⟩ = 2T . (28)

Using the Fierz transformation, the contribution of Od2 is

(1/Nc)T . Hence we have

tρρ = 2

(c′

1 + 1

Nc

c′2

)T , (29)

where we have neglected the nonfactorizable color-octetcontribution which is difficult to evaluate. Therefore, Nc

should be treated as an effective parameter and may deviatefrom the naive value 3 [7–9, 18]. Since the hadronizationinformation is included in Nc, the value of Nc may be dif-ferent for different decay channels. Furthermore, since thecolor-octet contribution associated with each operator in theHamiltonian (1) can vary, the effective Nc in the Fierz trans-formation for each operator may be different. In general,nonfactorizable effects can be absorbed into the effective pa-rameter aj (j = 1, . . . ,6) after the Fierz transformation

a2i = c′2i + c′

2i−1

(Nc)2i

,

a2i−1 = c′2i−1 + c′

2i

(Nc)2i−1(i = 1,2,3),

(30)

where

1

(Nc)j= 1

3+ ξj (j = 1, . . . ,6), (31)

with ξj representing the nonfactorizable effects, which maybe different for each operator. However, since we do not

have enough information about the operator dependence ofξj , we assume ξj is universal for each operator [18, 30, 31]and hence for each operator we use the same effective Nc[=(Nc)j ].

In the same way we find that tρω = 0. This leads to

αeiδα = 0, (32)

from Eq. (24).In a similar way, we can evaluate the penguin operator

contributions pρρ and pρω with the aid of the Fierz identi-ties. From Eq. (24) we have

βeiδβ = a4

a3 + a4 + a5, (33)

r ′eiδq = −a3 − a4 − a5

a1×

∣∣∣∣VubV∗cb

VudV ∗cd

∣∣∣∣, (34)

where∣∣∣∣VubV∗cb

VudV ∗cd

∣∣∣∣ = A2λ4

1 − λ2/2

√ρ2 + η2

(1 + A2λ4ρ)2 + A4λ8η2. (35)

3.3 Numerical results

The CP violation parameter, a, depends on q2, Nc, and theCKM matrix elements. As mentioned above, Nc includesnonfactorizable effects and should be determined by exper-imental data. We extract the range as 1.12 ≤ Nc ≤ 1.28,4.47 ≤ Nc ≤ 4.91, 6.31 ≤ Nc ≤ 8.10 and 1.11 ≤ Nc ≤ 1.32,4.45 ≤ Nc ≤ 4.95, 6.33 ≤ Nc ≤ 8.13 for q2/m2

c = 0.3and q2/m2

c = 0.5, respectively. The allowed range will bediscussed in detail in Sect. 4. From most previous stud-ies, it seems that Nc is usually less than the value 3 [6–9, 32]. Hence, for the process of D0 → ρ0(ω)ρ0(ω) →π+π−π+π−, we take the ranges of Nc to be 1.12 ≤Nc ≤ 1.28 and 1.11 ≤ Nc ≤ 1.32 when q2/m2

c = 0.3 andq2/m2

c = 0.5, respectively. In fact, when Nc increases to avalue larger than 3, it can be found that the CP violation ismainly caused by effective Wilson coefficients, which arecomplex numbers. The CKM matrix elements, which relateto ρ, η, λ and A, are given by Eq. (10).

From the numerical results, we find that for a fixedNc there is a maximum CP violating parameter value,amax, when the invariant masses of the π+π− pairs arein the vicinity of the ω resonance. For example, whenq2/m2

c = 0.3 and Nc = 1.2, in the case of (ρmax, ηmax) and(ρmin, ηmin), the maximum CP violating parameter variesfrom around −3.4 × 10−4 to −4.3 × 10−4. This is shownin Fig. 1. However, with the same parameters as the above,we find the CP violating parameter is only −5.0 × 10−5 inthe case of no ρ–ω mixing. Similarly, when q2/m2

c = 0.5and Nc = 1.2, one can also see that the CP violating param-eter is much enhanced by ρ–ω mixing from Fig. 1.

From Eq. (18), one can see that the CP violating param-eter is dependent on sin δ and r . In our calculation, for the

Eur. Phys. J. C (2013) 73:2519 Page 5 of 8

Fig. 1 The CP violating asymmetry, a, as a function of√

s forq2/m2

c = 0.3(0.5). The solid (dashed) line corresponds to minimum(maximum) CKM matrix elements when Nc = 1.2 and q2/m2

c = 0.3;The dotted (dot-dashed) line corresponds to minimum (maximum)CKM matrix elements when Nc = 1.2 and q2/m2

c = 0.5

Fig. 2 sin δ as a function of√

s for q2/m2c = 0.3(0.5) when Nc = 1.2.

The solid (dashed) line corresponds to q2/m2c = 0.3(0.5)

fixed ranges of Nc, it is found that the ρ–ω mixing mech-anism produces a large sin δ which is independent of theCKM matrix elements. The plot of sin δ as function of

√s

is shown in Fig. 2. One can see the ρ–ω mixing mecha-nism leads to the strong phase, δ, reaching 90◦ (sin δ = 1)at the ω resonance for the process D0 → ρ0(ω)ρ0(ω) →π+π−π+π−. From Fig. 3, it can be seen that r increasesrapidly when the invariant masses of the π+π− pairs are inthe vicinity of the ω resonance.

For the process of D0 → ρ0(ω)ρ0(ω) → π+π−π+π−the existence of double ρ–ω mixing leads to larger CP vi-olation than that in the case of single ρ–ω mixing. This ismainly because double ρ–ω mixing makes sin δ bigger thanthat in the case of single ρ–ω mixing. The involvement ofdouble ρ–ω mixing may also change the value of r . How-ever, as found from our detailed analysis the effect of the

Fig. 3 r as a function of√

s for q2/m2c = 0.3(0.5). The solid (dashed)

line corresponds to minimum (maximum) CKM matrix elements whenNc = 1.2 and q2/m2

c = 0.3; The dotted (dot-dashed) line correspondsto minimum (maximum) CKM matrix elements when Nc = 1.2 andq2/m2

c = 0.5

change of r on a is small compared with the change of sin δ

due to the contribution of double ρ–ω mixing. In our cal-culations there are some uncertainties due to Nc, q2 and theCKM matrix elements. We find that these uncertainties donot change our qualitative conclusions.

4 Branching ratios for D0 → ρ0ρ0

4.1 Formalism

The matrix element for D → V (where V denotes a vectormeson) can be decomposed as follows [33, 34]:

〈V |Jμ|D〉= 2

mD + mV

εμνρσ ε∗νpρDpσ

V V(k2)

+ i

{ε∗μ(mD + mV )A1

(k2) − ε∗ · k

mD + mV

× (pD + pV )μA2(k2) − ε∗ · k

k22mV kμA3

(k2)}

+ iε∗ · kk2

2mV kμA0(k2), (36)

where Jμ is the weak current (Jμ = qγ μ(1 − γ5)c with q =u,d, s), pD(mD) and pV (mV ) are the momenta (masses)of D and V , respectively, k = pD − pV for D → V tran-sition and εμ is the polarization vector of V , Ai (i =0,1,2,3) are the weak form factors which satisfy A3(0) =A0(0), and A3(k

2) = [(mD + mV )/2mV ]A1(k2) − [(mD −

mV )/2mV ]A2(k2).

The decay amplitude for D → V1V2 can be written asthe form A(D → V1V2) = αX(DV1,V2), where X(DV1,V2) de-notes the factorized amplitude with the form 〈V2|(q2q3)|0〉〈V1|(q1c)|D〉, then the decay rate is given by [32]

Page 6 of 8 Eur. Phys. J. C (2013) 73:2519

Γ (D → V1V2)

= pc

8πm2D

∣∣α(mD + m1)m2fV2ADV11 (m2)

∣∣2H, (37)

where α is related to the CKM matrix elements and Wil-son coefficients, fV2 is the decay constant of V2, pc is thec.m. momentum of the decay particles, mD and m1(m2) arethe masses of the D meson and the vector meson V1(V2),respectively, and

H = (a − bx)2 + 2(1 + c2y2), (38)

where

a = m2D − m2

1 − m22

2m1m2, b = 2m2

Dp2c

m1m2(mD + m1)2,

c = 2mDpc

(mD + m1)2,

x = ADV12 (m2

2)

ADV11 (m2

2), y = V DV1(m2

2)

ADV11 (m2

2),

pc =√

[m2D − (m1 + m2)2][m2

D − (m1 − m2)2]2mD

.

(39)

ADV11 , A

DV12 and V DV1 in Eq. (39) are the form factors as-

sociated with D → V1 transition.The decay amplitudes for D0 → ρ0ρ0, D0 → ρ0ω are

A(D0 → ρ0ρ0) = α1X

(Dρ0,ρ0), (40)

A(D0 → ρ0ω

) = α2X(Dρ0,ω), (41)

where

α1 = GF√2

[−2a1VudV ∗cd − 2a4VubV

∗cb

], (42)

α2 = GF√2

(2a3 + 2a4 + 2a5)VubV∗cb. (43)

In our case we should take into account the ρ–ω mix-ing contribution for branching ratios since we are workingto the first order of isospin violation. Then, we obtain thebranching ratio for D0 → ρ0ρ0:

BR(D0 → ρ0ρ0)

= pc

8πm2DΓD0

∣∣∣∣(

α1 + α22Πρω

(sρ − m2ω) + imωΓω

)

× (mD + mρ0)mρ0fρ0A1(m2

ρ0

)∣∣∣∣2

H. (44)

4.2 Form factor models

The form factors A1(k2), A2(k

2) and V (k2) depend on theinner structure of hadrons and are hence model dependent.We adopt the following form factors obtained in several phe-nomenological models:

Model 1 (2), (3) [33–36]:

V(k2

) = V (0)

1 − k2/(m21−)

, A1(k2) = A1(0)

1 − k2/(m21+)

,

A2(k2

) = A2(0)

1 − k2/(m21+)

,

(45)

where the form factor at k2 = 0 in different models areV (0) = 1.225(0.90)(0.735), A1(0) = 0.775(0.59)(0.590),and A2(0) = 0.923(0.49)(0.528), respectively. m1− =2.01 GeV, m1+ = 2.42 GeV.

4.3 Numerical results

As mentioned before, Nc includes the nonfactorizable ef-fects, which is difficult to deal with at present. Therefore,we treat Nc as an effective parameter to be determined byexperimental data. Currently, the latest experimental data forthe branching ratio of D → ρ0ρ0 [4] is

BR(D → ρ0ρ0) = (1.82 ± 0.13) × 10−3. (46)

Hence, we can determine the range of Nc by comparing thetheoretical value of the branching ratio with the experimen-tal data for the decay channel of D0 → ρ0ρ0. We calculatethe branching ratios for D0 → ρ0ρ0 with the formula givenin Eq. (44) in three models for the weak form factors whichare mentioned in the previous subsection. In addition, thebranching ratio also depends on the CKM matrix elementswhich are parameterized by λ, A, ρ and η, with their exper-imental values being given in Eqs. (10), (12). In the allowedranges for the parameters λ, A, ρ and η, we can get the rangeof Nc.

In the process of calculation, we find the branching ra-tio is not sensitive to the value of the CKM matrix param-eters. Therefore, we show the result in Fig. 4 for centralvalues of the CKM matrix parameters when q2/m2

c = 0.3.One can find the experimental data constrain the value of

Fig. 4 The branching ratio for D0 → ρ0ρ0 as a function of Nc forq2/m2

c = 0.3. The solid (dashed) (dotted) line correspond to Model 1(Model 2) (Model 3)

Eur. Phys. J. C (2013) 73:2519 Page 7 of 8

Table 1 The range of Nc for allthe models and the maximumrange of Nc

q2/m2b = 0.3 q2/m2

b = 0.5

Model 1 (1.21,1.28) (4.47,4.91) (1.20,1.29) (4.45,4.95)

Model 2 (1.12,1.19) (6.71,8.10) (1.11,1.32) (6.75,8.13)

Model 3 (1.13,1.19) (6.31,7.52) (1.12,1.19) (6.33,7.55)

Maximum range (1.12,1.28) (4.47,4.91) (6.31,8.10) (1.11,1.32) (4.45,4.95) (6.33,8.13)

Nc in two regions. One can see that the branching ratio isdependent on different models in Fig. 4. For a complete de-scription, we take the maximum range for Nc as shown inTable 1. In previous articles [5–13], the effective parame-ter Nc is determined to be below the value 3 for other de-cay channels. Considering this, we take the ranges of Nc as1.12 ≤ Nc ≤ 1.28 and 1.11 ≤ Nc ≤ 1.32 for q2/m2

c = 0.3and q2/m2

c = 0.5, respectively. One can see that the range isinsensitive to q2/m2

c .

5 Summary and discussion

We have studied the CP violation in the decay D0 →ρ0(ω)ρ0(ω) → π+π−π+π− due to the contribution ofρ–ω mixing. Since the CP violation is small in the pro-cesses of charmed meson decays, it is not easy to mea-sure in experiments. It is found that ρ–ω mixing can causea large strong phase difference so that large CP violationcan be obtained at the ω resonance. For the process D0 →ρ0(ω)ρ0(ω) → π+π−π+π−, there are two ρ–ω mixingcontributions which lead to even larger strong phase differ-ence than that in the case of single ρ–ω mixing. As a result,it is found that the CP violation could reach −3.4 × 10−4

which is one order larger than that in the case of no ρ–ω

mixing.We have worked in the traditional factorization approach

in which an effective parameter Nc is introduced to take intoaccount nonfactorizable contributions. We have determinedthe range of Nc by comparing the theoretical values and theexperimental data for the branching ratios for D0 → ρ0ρ0.

For the decay processes of bottom mesons, the tradi-tional factorization approach is the leading order of QCDfactorization scheme with αs(mb) and 1/mb corrections be-ing neglected. The QCD factorization scheme suffers fromendpoint singularities which are not well controlled. Forcharmed meson decays, the situation is more complex. Themass of charm quark is about 1.5 GeV which is smallermuch than the b quark mass. One may expect more uncer-tainties comparing with b quark systems. Currently, moreand more experimental data have been accumulated for CPviolation in decays of D mesons [4]. The magnitude of CPviolation which is of order 10−3 may be measured accu-rately in experiments, such as the CP violating asymmetry−(0.21 ± 0.17) % for the process of D0 → K+K− [4].

The Beijing Electron Positron Collider (BEPC) is fo-cused on the research of charm physics. BEPC was suc-cessfully upgraded to BEPCII in 2009. At the beam energyregion 1.89 GeV, the luminosity reaches 1032 cm−2 s−1,which is about 30 times of that of BEPC. The large pro-duction rate for charm quarks at BEPCII is crucial to studyCP violation in charmed meson decays. In order to observethe CP violation in D0 → ρ0(ω)ρ0(ω) → π+π−π+π− wehave predicted, the number of D0D0 needed is [37–39]:

ND0D0 ∼ 1

BRa2

(1 − a2)( 9

BRa2

(1 − a2)) (47)

for 1σ (3σ) signature, where BR is the branching ratio forD → ρ0ρ0. The branching ratio for D → ρ0ρ0 is of order10−3. The CP violation can be of order 10−4 via double ρ–ω

mixing in the determined range of Nc. Hence, the requirednumbers of D0D0 pairs are 107(108) for 1σ (3σ) signatureto observe the enhanced CP violation. The design report ofBEPCII presents that 2.5 × 107 D0D0 events can be col-lected from BESIII detector per year [40]. So it is possibleto observe the predicted CP violation.

Acknowledgements This work was supported by National NaturalScience Foundation of China (Project Numbers 11147003, 11047166,11175020, 10975018, 11275025 and 11147197), the Special Grants(Project Number 2009BS028) for Ph.D. from Henan University ofTechnology, Plan For Scientific Innovation Talent of Henan Universityof Technology (Project Number 2012CXRC17) and the FundamentalResearch Funds for the Central Universities in China.

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