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20

The D → ρ semileptonic and radiative decays within the light-cone sum rules

Hai-Bing Fu,1, ∗ Long Zeng,1, † Rong Lu,1, ‡ Wei Cheng,2, § and Xing-Gang Wu3, ¶

1Institute of Particle Physics & Department of Physics,

Guizhou Minzu University, Guiyang 550025, P.R. China2State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing, 100190, P.R. China3Department of Physics, Chongqing University, Chongqing 401331, P.R. China

The measured branching ratio of the D meson semileptonic decay D → ρe+νe, which is based onthe 0.82 fb−1 CLEO data taken at the peak of ψ(3770) resonance, disagrees with the traditionalSVZ sum rules analysis by about three times. In the paper, we show that this discrepancy canbe eliminated by applying the QCD light-cone sum rules (LCSR) approach to calculate the D → ρtransition form factors A1,2(q

2) and V (q2). After extrapolating the LCSR predictions of these TFFsto whole q2-region, we obtain 1/|Vcd|

2 × Γ(D → ρeνe) = (55.45+13.34−9.41 ) × 10−15 GeV. Using the

CKM matrix element and the D0(D+) lifetime from the Particle Data Group, we obtain B(D0 →ρ−e+νe) = (1.749+0.421

−0.297 ± 0.006) × 10−3 and B(D+ → ρ0e+νe) = (2.217+0.534−0.376 ± 0.015) × 10−3,

which agree with the CLEO measurements within errors. We also calculate the branching ratiosof the two D meson radiative processes and obtain B(D0 → ρ0γ) = (1.744+0.598

−0.704) × 10−5 and

B(D+ → ρ+γ) = (5.034+0.939−0.958)× 10−5, which also agree with the Belle measurements within errors.

Thus we think the LCSR approach is applicable for dealing with the D meson decays.

PACS numbers: 12.38.-t, 12.38.Bx, 14.40.Aq

I. INTRODUCTION

The semileptonic decays of the heavy meson, whichcontains heavy c or b quark, are important for study-ing the weak and strong interactions and for studyingthe heavy-flavor physics. In the charm factories nowa-days, such as Belle, LHCb, BES and PANDA, the Dmeson semileptonic decays provide a good platform forthe precision test of standard model (SM) and for search-ing of new physics (NP) beyond the SM. For examples,the CLEO Collaboration present the first measurement ofthe branching fraction of D+ → ωe+νe [1, 2]. Lately, theCLEO Collaboration finished a more precise measure-ment on the D0 → ρ−e+νe and D+ → ρ0e+νe decaysbased on 0.82 fb−1 data taken at the peak of ψ(3770)resonance. The CLEO Collaboration gave the branchingfractions B(D0 → ρ−e+νe) = (1.77± 0.12± 0.10)× 10−3

and B(D+ → ρ0e+νe) = (2.17 ± 0.12+0.12−0.22) × 10−3 [3].

Recently, the BES-III Collaboration published their im-proved results, B(D0 → ρ−e+νe) = (1.445 ± 0.058 ±0.039)× 10−3 and B(D+ → ρ0e+νe) = (1.860± 0.070±0.061)× 10−3, by using a data sample corresponding toan integrated luminosity of 2.93 fb−1 at a center-of-massenergy of 3.773 GeV [4].

Meanwhile, the nonleptonic decays of the charm me-son such as D0 → ρ0γ and D+ → ρ+γ have also beena subject of high priority for many years and has re-

∗Electronic address: fuhb@cqu.edu.cn†Electronic address: zenglongz@outlook.com‡Electronic address: lvrong97@aliyun.com§Electronic address: chengwei@itp.ac.cn¶Electronic address: wuxg@cqu.edu.cn

sulted in a wealth of experimental data. In particu-lar, two measured observables by the Belle Collaborationare, B(D0 → ρ0γ) = (1.77 ± 0.30 ± 0.07) × 10−5 andACP (D

0 → ρ0γ) = 0.056 ± 0.152 ± 0.006 [5], where thecharge-parity (CP) asymmetry ACP is defined as

ACP (D → ργ) =Γ(D → ργ)− Γ(D → ργ)

Γ(D → ργ) + Γ(D → ργ). (1)

The D → ρ transition form factors (TFFs) are keycomponents of the D meson semileptonic decay D →ρeνe and the D meson radiative decay D → ργ. In or-der to achieve more precise predictions on the D mesonsemileptonic and radiative decays, it is important to havemore accurate D → ρ TFFs. Theoretically, the D → ρTFFs have been calculated under various approaches,such as the 3-point QCD sum rules (3PSR) [6], the heavyquark effective field theory (HQEFT) [7, 8], the relativis-tic harmonic oscillator potential model (RHOPM) [9],the quark model (QM) [10, 11], the light-front quarkmodel (LFQM) [12], the heavy meson and chiral sym-metries (HMχT) [13], and the Lattice QCD [14, 15].Most results are consistent with the CLEOmeasurementswithin errors; while the QCD sum rules (SR) leads tomuch smaller branching ratios, i.e. B(D0 → ρ−e+νe) =(0.5 ± 0.1) × 10−3 [6]. One may question the applica-bility of the QCD SR approach for those TFFs. Thereare large uncertainties for the 3PSR prediction, which ishowever based on the conventional Shifman-Vainshtein-Zakharov SR approach [16] and the approach itself hasmany defaults in dealing with such kind of TFFs [17].The QCD SR approach is applicable in both small and

intermediate q2-region, which provides a bridge betweenthe pQCD and lattice QCD predictions, and then a moreaccurate QCD SR prediction is helpful for a better un-derstanding of the TFFs. In the paper, as a new QCD

2

SR analysis, we take the improved version of QCD SRapproach, i.e. the light-cone sum rules (LCSR) [18–20],to recalculate the D → ρ TFFs. The LCSR avoids theproblems of the conventional SVZ SR by making a partialresummation of the operator product expansion (OPE)to all orders and reorganize the OPE expansion in termsof the twists of relevant operators rather than their di-mensions. The vacuum condensates of the SVZ SR arethen substituted by the light-mesons light-cone distri-bution amplitudes (LCDAs) of increasing twists. TheLCDA, which relates the matrix elements of the nonlo-cal light-ray operators sandwiched between the hadronicstate and the vacuum, has a direct physical significanceand provides the underlying links between the hadronicphenomena at small and large distances. Since its inven-tion, the LCSR approach has been widely adopted forstudying the heavy → light meson decays.The remaining parts of the paper are organized as fol-

lows. In Sec. II, we present the calculation technologyfor the D → ρ semileptonic and radiative decays, andtheir key components, e.g. the D → ρ TFFs, within theLCSR approach. In Sec. III, we present our numericalresults and discussions on the D → ρ TFFs, the D → ρsemileptonic decay widthes and branching ratios for twodifferent channels, and the branching ratio and the di-rect CP asymmetry of the D → ργ decay. Section IV isreserved for a summary.

II. CALCULATION TECHNOLOGY

A. The D → ρe+νe semileptonic decay

e+

W+

ρ0(ρ−)D+(D0)

νe

FIG. 1: Typical diagram for the D → ρe+νe semileptonicdecay.

We present the typical diagram for the D → ρe+νesemileptonic decay in Fig.1. The longitudinal and trans-verse helicity decay widths of D → ρe+νe can be ex-pressed in terms of three helicity amplitudes H0,±(q

2):

Γa =G2

F |KHV|2192π3m3

D

∫ q2max

m2e

q2√λ(q2)|Ha(q

2)|2, (2)

where a = +, −, 0, q2max = (mD −mρ)2, and λ(q2) is the

phase-space factor, which equals to (m2D +m2

ρ − q2)2 −4m2

Dm2ρ. The Fermi constant GF = 1.166×10−5 GeV−2.

The constant KHV parameterizes the quark flavor mix-ing relevant to a particular transition, which equals to|Vcd| for D0 → ρ−e+νe and |Vcd|/

√2 for D+ → ρ0e+νe.

Then the total decay width of D+/0 → ρ0/−e+νe can bewritten as

Γ = ΓL + ΓT, (3)

where ΓT = Γ+ + Γ− and ΓL = Γ0.Here we have adopted the helicity basis to express the

decay width. In the helicity basis, the TFF Ha(q2) corre-

sponds to a transition amplitude with definite spin-parityquantum number in the lepton pair center-of-mass frame.The transverse and longitudinal helicity TFF Ha(q

2) canbe calculated by relating them to the TFFs A1,2(q

2) andV (q2) via the following way

H±(q2) = (mD +mρ)A1(q

2)∓√λ(q2)

mD +mρV (q2) (4)

and

H0(q2) =

1

2mρ

√q2

{(m2

D −m2ρ − q2)(mD +mρ)A1(q

2)

− λ(q2)

mD +mρA2(q

2)

}. (5)

The TFFs A1,2(q2) and V (q2) can be defined by relating

them to the D → ρ matrix elements, e.g.

〈ρ(p, ǫ)|qγµc|D(p)〉 = ǫµναβ ǫ∗νpαpβ

2V (q2)

mD +mρ, (6)

〈ρ(p, ǫ)|qγµγ5c|D(p)〉 = iǫ∗µ(mD +mρ)A1(q2)

−i(ǫ∗ · p)(p+ p)µ1

mD +mρA2(q

2)

−i(ǫ∗ · p)(p− p)µ2mρ

q2[A3(q

2)−A0(q2)]. (7)

where ǫ is ρ-meson polarization vector and q is the four-momentum transfer between the two mesons.

B. The D → ργ decay

We give a mini-review of the Cabibbo-suppressed D →ργ decay in this subsection, which is an important chan-nel for testing the SM and for searching of new physicsbeyond the SM.The amplitude of D → ργ decay can be decomposed

into the following gauge invariant form,

A[D(p) → ρ(p, ǫ)γ(q, ǫ)] = ǫµναβqµǫ∗νpαǫ∗βAρ

PC

+i[(ǫ∗ · q)(ǫ · p)− (p · q)(ǫ∗ · ǫ∗)]AρPV, (8)

where ǫ and ǫ are polarization vectors of ρ and γ, respec-tively. Aρ

PC and AρPV are parity conserving and parity

violating amplitudes which can be calculated by using

3

the effective c→ uγ weak Lagrangian [21–24]. Then thedecay rate of D → ργ can be written as [25, 26]

Γ(D → ργ) =m3

D

32π

(1−

m2ρ

m2D

)3[|Aρ

PV|2 + |AρPC|2

]. (9)

Our remaining task is to calculate the amplitudesAρ

PV,PC, which contain both the long-distance contribu-tion and the short-distance contribution, e.g.

AρPV,PC = (Aρ

PV,PC)l.d. + (Aρ

PV,PC)s.d.. (10)

q

c

d, s d, s

γ

W u

q

FIG. 2: The typical VME diagram via the penguin-like c →uγ transition.

Firstly, the long-distance contributions for radiativedecays D → ργ can be treated as originating from thenonleptonic transition D → ρV0, followed by the con-version V0 → γ via the vector meson dominance (VME)mechanism [27], whose typical Feynman diagram via thepenguin-like c → uγ transition is shown in Fig.2. Moreprecisely, the long-distance amplitudes (Aρ

PV,PC)l.d. can

be divided into three parts, e.g.

(AρPV,PC)

l.d. = AIPV,PC +AII

PV,PC +AIIIPV,PC

with 8 schematic diagrams to be calculated, are the hy-brid model based on the heavy quark effective theoryand chiral perturbation theory. The rectangle in each di-agram of Fig.3 respects the weak transition due to theeffective Lagrangian which can be found in Ref.[27]. TheLagrangian contains a product of two left-handed quarkcurrents, which will be expressed in terms of the rele-vant hadronic degrees of freedom, D, D∗, P and V , asdescribed in Fig.3:

• The first partAIPC, which is represented by Fig.3(a)

and Fig.3(b), denotes the photon emitted from theD meson becomes D∗ meson and then weakly de-cays into ρ meson.

• The second part AIIPC, which is shown by Fig.3(c),

comes from the weak decay of D meson, whichfirstly decays into an off-shell light pseudoscalarand then decays into ργ.

• As shown by Fig.3(d) and Fig.3(e), there are alsolong-distance penguin-like contributions, which

contribute to both the parity-conserving ampli-tude AIII

PC and the parity-violating amplitude AIIIPV.

They will be vanished in the exact SU(3) flavourlimit.

• Fig.3(f) and Fig.3(g) are bremsstrahlung-like dia-grams, where the photon emission is due to thedirect coupling to charged initial D or final-stateρ-meson, both of which contribute to the parity vi-olating amplitude AI

PV.

• Fig.3(h) denotes the remaining parity violatingcontribution from other quark-level picture [27],whose amplitude is represented as AII

PV.

It has been found that Fig.3(c) has sizable contribu-tion, while Figs.3(d) and (e) have negligible contribu-tions which are less than 1%. To derive the ampli-

tudes AI,II,IIIPV,PC, we need to know the coupling strengthes

for the interaction of the light vector meson with D orD∗ and the heavy quark-photon interaction, which canbe characterized by the parameters λ and λ′, respec-tively. To do our numerical calculation, we shall adoptλ = 0.479 GeV−1 [27], and λ′ = 1/(6mc) which is pre-dicted by approximately relating it to the charm quarkmagnetic moment [28–31].Following the above discussions, the parity-conserving

amplitudes for D+ → ρ+γ are

AIPC(D

+ → ρ+γ) = 4a1G∣∣∣∣λ

′ − λgV

2√2

(fρmρ

− fω3mω

) ∣∣∣∣

×mρmD∗fDfρm2

D∗ −m2ρ

√mD∗

mD,

AIIPC(D

+ → ρ+γ) = 4a1Gm2

DfDfω3mω(m2

D −m2π)

|CV VΠ|,

AIIIPC(D

+ → ρ+γ) = a2G(−f2

ρ +1

3f2ω − V ∗

csVusV ∗cdVud

2

3f2φ

)

× |V (0)|mD +mρ

(11)

and the parity-violating amplitudes for D+ → ρ+γ are

AIPV(D

+ → ρ+γ) = 2a1GmρfρfDm2

D −m2ρ

,

AIIPV(D

+ → ρ+γ) = −a1Gfρ

m2D −m2

ρ

[fρ(mD +mρ)

×A1(m2ρ)− fω

mρ(mD +mω)

3mωADω

1 (m2ρ)

],

AIIIPV(D

+ → ρ+γ) = a2G(−f2

ρ +1

3f2ω − V ∗

csVusV ∗cdVud

2

3f2φ

)

× A1(0)

mD −mρ, (12)

where G = eGFVudV∗cd/

√2 with e =

√4παem, gV = 5.9

which can be fixed by the flavor symmetry, |CV VΠ| =

4

(b)

D∗

γ

ρD

(c)γ

ρ0

PρD

(d)

γ

D∗

ρ0

ρD

(h)ρ0

γ

DV

ρD

(g)γ

ρ0(f)

D

γ

D ρ

(e)

γ

ρ0

ρD

D∗

(a)ρ0

γ

ρD

ρ

FIG. 3: Skeleton diagrams for the long-distance amplitudes which contribute to the D → ργ decay. The rectangle denotes theweak transition which can be described with the help of the effective Lagrangian.

3g2V /(32π2) = 0.33 [23], a1 = 1.26±0.1 and a2 = −0.55±

0.1 [32], fρ,ω,φ are decay constants which can be fixed bythe leptonic decays of these mesons. ADω

1 (m2ρ) is the TFF

for D → ω [23].

The Cabibbo suppressed decay D0 → ρ0γ involvesthe contribution from the η − η′ mixing, whose parity-conserving amplitudes are

AIPC(D

0 → ρ0γ) = 4a2b0ρG∣∣∣∣λ

′ + λgV

2√2

(fρmρ

+fω3mω

) ∣∣∣∣

×mρmD∗fDfρm2

D∗ −m2ρ

√mD∗

mD,

AIIPC(D

0 → ρ0γ) = 4a2GBρm2DfD|CV V Π|,

AIIIPC(D

0 → ρ0γ) = a2G(−f2

ρ +1

3f2ω − V ∗

csVusV ∗cdVud

2

3f2φ

)

× V (0)

mD +mρ, (13)

where b0ρ = −1/√2 and the coefficients

Bρ =

3∑

i=1

BρPi

m2D −m2

Pi

, (14)

where Pi stands for the light mesons π, η, and η′,respectively, and Bρ

π = fω/(3√2mω), B

ρη = −c(c −√

2s)fρ/(√2mρ), B

ρη′ = −s(

√2c + s)fρ/(

√2mρ) with

s = sin θ and c = cos θ. Here θ is the η− η′ mixing angleand we set its value as θ = −20o, which is consistent withthe values derived in Refs.[33–36]. The parity-violating

amplitudes for D0 → ρ0γ are

AIPV(D

0 → ρ0γ) = 0,

AIIPV(D

0 → ρ0γ) = −a2Gfρ√

2(m2D −m2

ρ)

[fρ(mD +mρ)

×A1(m2ρ) + fω

mρ(mD +mω)

3mωADω

1 (m2ρ)

],

AIIIPV(D

0 → ρ0γ) = a2G(−f2

ρ +1

3f2ω − V ∗

csVusV ∗cdVud

2

3f2φ

)

× A1(0)

mD −mρ. (15)

Secondly, we need to deal with the short-distance con-tribution (Aρ

PC(PV))s.d.. There are ten operators which

enter into the weak interactions of the D → ρ decay [21–24]. It has been observed that due to the GIM suppres-sion and also the small magnitudes of the Wilson coef-ficients, only the operators Q7 and Q′

7 have sizable con-tributions to the D → ρ decay; Thus the short-distanceamplitudes (Aρ

PC(PV))s.d. have the following form [24, 26]

(AρPC(PV))

s.d. =

√4παeQuGFmc

2√2π2

(A7 +A′7)T (0), (16)

in which T (0) = T1(0) = T2(0) and the tensor TFFs

5

T1,2(q2) are defined by

〈ρ(p, ǫ)|uqνσµν(1 + γ5)c|D(p)〉

= −2iǫµναβ ǫ∗νpαpβT1(q

2)

+[(m2

D −m2ρ)ǫ

∗µ − (ǫ∗ · p)(p+ p)µ

]T2(q

2)

+(ǫ∗ · p)[(p− p)µ − (p+ p)µ

q2

m2D −m2

ρ

]T3(q

2).

(17)

Those two tensor TFFs T1,2(q2) can be calculated under

the LCSR approach. For the needed T1, by using theheavy quark effective field theory, it can be related tothe TFFs V and A1 via the following way [8]

T1(q2) =

m2D −m2

ρ + q2

2mD(mD +mρ)V (q2) +

mD +mρ

2mDA1(q

2).

Furthermore, the coefficient A(′)7 = C

(′)eff7 + · · · , where

the ellipse stands for the additional contribution fromwithin and outside the SM.As a final remark, for the Cabibbo suppressed decay

D0 → ρ0γ, in the limit where the strong phases of theamplitudes have a mild energy dependence, and assumingthat we can neglect the weak phase of the long-distanceamplitudes, and the CP asymmetry |ACP | is primarilysensitive to direct CP asymmetry |adirργ |, i.e.

|ACP | ≈ |adirργ | = 2|Im(Aρ

PC,PV)s.d.|

|(AρPC,PV)

l.d.| × | sin(∆φstrong)|.

(18)

C. The D → ρ TFFs

TABLE I: The ρ-meson LCDAs with different twist-structuresup to δ3 [38], where δ ≃ mρ/mc.

twist-2 twist-3 twist-4

δ0 φ⊥2;ρ

δ1 φ‖2;ρ φ⊥

3;ρ, ψ⊥3;ρ,Φ

‖3;ρ, Φ

⊥3;ρ

δ2 φ‖3;ρ, ψ

‖3;ρ,Φ

⊥3;ρ φ⊥

4;ρ, ψ⊥4;ρ,Ψ

⊥4;ρ, Ψ

⊥4;ρ

δ3 φ‖4;ρ, ψ

‖4;ρ

As mentioned in the Introduction, we shall adopt theLCSR approach to calculate the D → ρ TFFs. TheLCSR approach is based on the operator production ex-pansion near the light cone, and in different to the tra-ditional QCD SR approach which parameterizes all thenon-perturbative dynamics into vacuum condensates, theLCSR approach parameterizes those non-perturbativedynamics into LCDAs with increasing twists. Due tothe complex structures of the ρ-meson LCDAs, it is con-venient to arrange them by the parameter δ ≃ mρ/mc ∼

52% [37, 38]. A collection of the ρ-meson twist-2, twist-3and twist-4 LCDAs up to δ3-order are shown in Table I.

Up to twist-4 level, there are totally fifteen ρ-mesonLCDAs, all of which, especially the high-twist DAs, arefar from affirmation. As a tricky point of the LCSR ap-proach, one may choose proper current for the correla-tion function (correlator) so as to suppress less certainhigh-twist terms and improve the accuracy of the LCSRprediction [40–45]. For example, one can adopt a right-handed current to do the LCSR calculation, i.e., startingfrom the following chiral correlator

Πµ(p, q) = i

∫d4xeiq·x

× 〈ρ(p, ǫ)|T{q1(x)γµ(1− γ5)c(x), j

†D(0)

}|0〉,(19)

where the current j†D(x) = ic(x)(1+γ5)q2(x). This chiralcorrelator highlights the contributions from the chiral-

odd LCDAs φ⊥2;ρ at the δ0-order, φ‖3;ρ, ψ

‖3;ρ, Φ

⊥3;ρ, φ

⊥4;ρ,

ψ⊥4;ρ, Ψ

⊥4;ρ, Ψ

⊥4;ρ at the δ2-order; while all the contribu-

tions from the chiral-even ρ-meson LCDAs are negligible.

Following the standard LCSR procedures, we can ob-tain the LCSRs for the D → ρ TFFs A1,2(q

2) and V (q2)by using the above chiral correlator, which are similar tothe B → ρ TFFs. The B → ρ TFFs have calculated byvarious groups under the LCSR approach, and a recentwork is done by applying the vacuum-to-B-meson corre-lation function with an interpolating current for the lightvector meson [46]. The lengthy analytic expressions forthe D → ρ TFFs with the help of the present choice of achiral correlator can be obtained from Ref.[39] by replac-ing the B-meson inputs there as the present D mesonones. To short the length of the paper, we shall not listthose formulas here and the interesting reader may turnto Ref.[39] for detail.

Numerically, we observe that the leading-twist termsare dominant for the chiral LCSRs of the TFFs, agreeingwell with the usual δ-power counting rule. Thus, thoseTFFs shall provide us a useful platform for testing theproperties of the leading-twist φ⊥2;ρ via comparisons withthe data or predictions from other theoretical approaches,such as those of Refs.[47–50].

More over, it is convenient to define two ratios over thethree TFFs A1,2(q

2) and V (q2),

rV =V (0)

A1(0)and r2 =

A2(0)

A1(0). (20)

Those two ratios could suppress the theoretical errors foreach TFF within the LCSR approach, and also suppressthe differences of the predictions on the TFFs from vari-ous approaches. Thus a better comparison with the datacan be achieved.

6

III. NUMERICAL RESULTS AND DISCUSSION

A. Distribiton amplitudes and TFFs

To do the numerical calculation, we take the decayconstant f⊥

ρ = 0.165(9)GeV [47]. The ρ and D mesonmasses are taken as mρ = 0.775 GeV and mD = 1.864GeV [33]. The Cabibbo-Kobayashi-Maskawa matrix ele-ment |Vcd| = 0.2252±0.0007 [33], and the D meson decayconstant fD shall be determined by using the QCD sumrules approach [51]. We adopt the WH model [52] as theρ-meson transverse leading twist wavefunction, whose ra-dial part is from the BHL-prescription [53] and the spin-space wavefunction χh1h2

ρ (x,k⊥) is from Wigner-Meloshrotation. And then, after integrating out the transversemoment dependence, we obtain the ρ-meson LCDA

φ⊥2;ρ(x, µ) =A⊥

2;ρ

√3xxmq

8π3

2 f⊥ρ b

⊥2;ρ

[1 + B⊥2;ρC

3

2

2 (ξ)]

×[Erf

(b⊥2;ρ

õ2 +m2

q

xx

)− Erf

(b⊥2;ρ

√m2

q

xx

)],

(21)

where f⊥ρ = f⊥

ρ /√3 and the error function, Erf(x) =

2∫ x

0 e−t2dt/

√π. The constitute light-quark mass is taken

asmq ≃ 300 MeV. The two parameters A⊥2;ρ and b⊥2;ρ can

be fixed by the normalization condition and the average

value of the squared transverse momentum 〈k2⊥〉

1/22;ρ =

0.37 ± 0.02 GeV [39]. The parameter B⊥2;ρ can be fixed

by using the second Gegenbauer moment, i.e. a⊥2;ρ(µ0 =1 GeV) = 0.14(6) [47].Using those input values, we obtain

A⊥,C2;ρ = 23.808, b⊥,C

2;ρ = 0.572, B⊥,C2;ρ = 0.100; (22)

A⊥,U2;ρ = 22.679, b⊥,U

2;ρ = 0.555, B⊥,U2;ρ = 0.151; (23)

A⊥,D2;ρ = 25.212, b⊥,D

2;ρ = 0.595, B⊥,D2;ρ = 0.050, (24)

where C, U and D stand for center, upper and lowervalues, respectively.We present the ρ meson transverse twist-2 LCDA

φ⊥2;ρ(x, µ0 = 1 GeV) in Fig. 4. As a comparison, theBall and Braun (BB) model [47], the Linear potentialmodel (PM) and the harmonic oscillator potential model(HOPM) [48], the AdS/QCD model [49], and the asymp-totic model have also been presented. Fig. 4 shows thatthe shape of φ⊥2;ρ is still unfixed, which varies from a sin-gle peaked behavior to a double peaked behavior. Theφ⊥2;ρ shape is primarily controlled by the magnitude of

B⊥2;ρ or equivalently the second Gegenbauer moment a⊥2;ρ.

Thus the WH model provides a convenient way to fix thebehavior of φ⊥2;ρ via comparing with the data.To set the Borel windows for the LCSRs of the D → ρ

TFFs, we adopt the following criteria

(i) We require the continuum contribution to be lessthan 30% of the total LCSR.

¨

¨

¨

¨

¨

¨

¨

¨

¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨

¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨

¨

¨

¨

¨

¨

¨

¨

¨

WH (This work)

BB

Linear PM

HOPM

AdS/QCD

Asymptotic¨

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

x

Φ2;Ρ¦

Hx,1

GeVL

FIG. 4: The ρ-meson leading-twist LCDA φ⊥2;ρ(x,µ0 =

1 GeV) predicted from the WH model. As a comparison,the BB prediction [47], the Linear PM and HOPM [48], theAdS/QCD prediction [49], and the asymptotic form have alsobeen presented.

(ii) We require all the high-twist LCDAs’ contributionsto be less than 15% of the total LCSR.

(iii) The derivatives of LCSRs for TFFs with respect to(−1/M2) give three LCSRs for the D meson massmD. We require the predicted D meson mass to befulfilled in comparing with the experiment one, e.g.|mth

D −mexpD |/mexp

D less than 0.1%.

We take the continuum thresholds for D → ρ TFFsA1,2(q

2) and V (q2) as s0(A1) = 6.1(3) GeV2, s0(A2) =

7.1(3) GeV2 and s0(V ) = 6.6(3) GeV2, which are closeto the squared mass of the D meson’s first excited stateD1(2420). Numerically, we observe that the TFFs changeslightly with s0, thus the uncertainties caused by differentchoices s0 are small 1.Following those criteria, the determined Borel windows

are M2 = 4.5(3)GeV2 for A1, M2 = 6.2(3)GeV2 for A2,

and M2 = 5.0(3)GeV2 for V , respectively. More explic-itly, we present the Borel windows for those TFFs at thelarge recoil point q2 = 0 in Fig. 5, which shows that theTFFs change slightly within the Borel windows, beingconsistent with conventionally adopted qualitative crite-rion that the TFF should be flat within the Borel window(since the physical observable should not depend on thisartificial parameter). Contributions for different LCDAsfor those TFFs are presented in Table II. It shows thatthe δ0-order twist-2 LCDA φ⊥2;ρ provides dominate con-

tribution, while the contributions from the δ2-order high-twist LCDAs are small.Table III shows the D → ρ TFFs at the large re-

coil region q2 0 GeV2, where the uncertainties are

1 Such a small s0 dependence also plays a role to suppress the un-

wanted scalar contribution due to the choice of chiral correlator.

7

FIG. 5: The determined Borel windows for the TFFs at thelarge recoil point, A1,2(0) and V (0).

TABLE II: Contributions from the LCDAs with various twiststructures for the D → ρ TFFs A1,2(0) and V (0), in whichonly the main non-zero contributions are listed.

A1(0) A2(0) V (0)

φ⊥2;ρ 0.539 1.067 0.989

ψ‖3;ρ 0.089 −0.187 /

φ⊥4;ρ −0.011 −0.157 −0.174

IL −0.016 −0.091 /

H3 −0.021 −0.170 /

Φ⊥3;ρ / 0.0003 /

Total 0.580 0.468 0.815

squared averages of all the mentioned error sources forthe LCSR approach. As a comparison, we also presentthe predictions from various approaches in Table III,i.e. from the CLEO collaboration [3], the 3PSR [6],the HQEFT [7, 8], RHOPM [9], the QM [10, 11], theLFQM [12], the HMχT [13], and the Lattice QCD pre-dictions [14, 15], respectively.

Table III shows that the TFFs under many approachesare consistent with each other within reasonable errors.To show the relative importance of various TFFs withindifferent approaches more clearly, we present a compari-son of the ratios r2 and rV in Fig. 6. The LCSR uncer-

tainties for the TFFs are(+13%−12%

)for rV and

(+15%−15%

)for

r2, which are much smaller than the previous 3PSR pre-dictions (which are ±45% and ±48% [6], respectively).Thus, by using the LCSR approach, more accurate QCDSR predictions can be obtained.

The physically allowable range for the TFFs is 0 ≤q2 ≤ (mD − mρ)

2 = 1.18GeV2. Theoretically, the LC-SRs for the D → ρ TFFs are applicable in low and in-termediate q2-regions, e.g. q2 ∈ [0, 0.8] GeV2. We canextrapolate them to whole q2-regions via a rapidly con-

verging series over the z(t)-expansion [54, 55]

Fi(q2) = Pi(q

2)∑

k=0,1,2

aik[z(q2)− z(0)]k, (25)

where

z(t) =

√t+ − t−√

t+ − t0√t+ − t+

√t+ − t0

(26)

with t± = (mD ± mρ)2, t0 = t+(1 −

√1− t−/t+), and

the Fi are three TFFs A1,2 and V , respectively. ThePi(q

2) = (1 − q2/m2R,i)

−1 is a simple pole correspondingto the first resonance in the spectrum. The appropriateresonance masses are given in Table IV. The parametersaik can be fixed by requiring ∆ < 0.1%, and the resultsare put in Table V. Here the parameter ∆ is introducedto measure the quality of extrapolation,

∆ =

∑t

∣∣Fi(t)− F fiti (t)

∣∣∑

t |Fi(t)|× 100, (27)

where t ∈ [0, 140 , · · · , 4040 ]× 0.8GeV2.

The extrapolated TFFs in whole q2-region are pre-sented in Fig. 7, where the shaded bands are uncertain-ties from various input parameters. As a comparison,we also give the results from various approaches, whichare from the CLEO Collaboration [3], the QM [11], theLFQM [12], the HLχPT [13], and the Lattice predic-tions [56], respectively. The CLEO Collaboration onlyissues the TFFs at large recoil region, and the presentCLEO curves are fitted ones from the large energy chiral-quark model [57].

B. The semi-leptonic decay D → ρeνe

By using the extrapolated D → ρ TFFs, we calculatethe total decay width 1/|Vcd|2 × Γ, the ratio of longitu-

TABLE III: The D → ρ TFFs A1,2(q2) and V (q2) at the

large recoil region q2 ≃ 0. The errors are squared averagesof all the mentioned error sources. As a comparison, we alsopresent the prediction from various methods.

A1(0) A2(0) V (0)

This work 0.580+0.065−0.050 0.468+0.052

−0.053 0.815+0.070−0.051

CLEO2013 [3] 0.56(1)+0.02−0.03 0.47(6)(4) 0.84(9)+0.05

−0.06

3PSR [6] 0.5(2) 0.4(1) 1.0(2)

HQETF-I [7] 0.57(8) 0.52(7) 0.72(10)

HQETF-II [8] 0.599+0.035−0.030 0.372+0.026

−0.031 0.801+0.044−0.036

RHOPM [9] 0.78 0.92 1.23

QM-I [10] 0.59 0.23 1.34

QM-II [11] 0.59 0.49 0.90

LFQM [12] 0.60(1) 0.47(0) 0.88(3)

HMχT [13] 0.61 0.31 1.05

Lattice [14] 0.45(4) 0.02(26) 0.78(12)

Lattice [15] 0.65(15)+0.24−0.23 0.59(31)+0.28

−0.25 1.07(49)(35)

8

FIG. 6: Comparison of theoretical predictions for the ratios r2 and rV within various approaches. The CLEO measurementsare presented as a comparison.

FIG. 7: The extrapolated D → ρ TFFs A1,2(q2) and V (q2), in which the shaded bands are squared average of those from the

mentioned error sources. As a comparison, we also present the central value of the QM [11], the LFQM [12], the HLχPT [13],the Lattice QCD predictions [56], and CLEO measurments [3] in those figures.

TABLE IV: Resonance masses of quantum number JP as in-dicated necessary for the parameterisation of D → ρ TFFsA1,2 and V .

Fi JP mR,i/GeV

V 1− 2.007

A1, A2 1+ 2.427

dinal and transverse decay width ΓL/ΓT for the D → ρsemileptonic decays, and the ratio of positive and nega-tive decay width Γ+/Γ−. The results are presented in Ta-ble VI, where the results under various approaches havealso been presented. Table VI shows that our LCSR pre-dictions for 1/|Vcd|2×Γ, ΓL/ΓT and Γ+/Γ− are consistentwith other approaches within errors, only the value of1/|Vcd|2 ×Γ is quite larger than the 3PSR prediction [6].As a further step, we calculate the branching ratios

for the two D → ρ semileptonic decays. One is the D0-type decay via the process D0 → ρ−e+νe with the life-

TABLE V: The fitted parameters ai1,2 for theD → ρ TFFs Fi,in which all the LCSR parameters are set to be their centralvalues. ∆ is the measure of the quality of extrapolation.

A1 A2 V

ai1 0.711 −1.149 −0.797

ai2 29.23 15.108 10.370

∆ 0.05% 0.04% 0.01%

time τ(D0) = 0.410± 0.002 ps, and the other one is theD+-type decay via the process D+ → ρ0e+νe with thelifetime τ(D+) = 1.040 ± 0.007 ps [33]. The results aregiven in Table VII, where the first uncertainty is squaredaverage of the mentioned error sources, and the seconduncertainty is from the experimental errors for the life-time. As a comparison, we also list the branching ratiosderived from various approaches in Table VII. It indicatesthat a smaller 1/|Vcd|2 × Γ predicted by 3PSR leads to

9

TABLE VI: Total decay width 1/|Vcd|2 × Γ, the ratio of lon-

gitudinal and transverse decay width ΓL/ΓT , and the ratio ofpositive and negative decay width Γ+/Γ−.

1/|Vcd|2 × Γ ΓL/ΓT Γ+/Γ−

This paper 55.45+13.34−9.41 1.18+0.14

−0.13 0.22+0.04−0.03

3PSR [6] 15.80 ± 4.61 1.31± 0.11 0.24± 0.03

HQEFT [7] 71± 14 1.17± 0.09 0.29± 0.13

RHOPM[9] 90.83 0.91 0.19

QM [10] 88.86 1.33 0.11

Lattice[14] 54.63 ± 12.51 1.86± 0.56 0.16

Lattice[15] 71.75 1.10 0.18

TABLE VII: The branching ratios of the semileptonic decaysD0 → ρ−e+νe and D+ → ρ0e+νe (in unit: 10−3). As acomparison, we also present the results of CLEO Collabora-tion [2, 3], 3PSR [6], HQEFT [7], NWA [58] with HQEFT [8]and LFQM [12], FK [13] and ISGW2 [59].

Decay Mode D0 → ρ−e+νe D+ → ρ0e+νe

This paper 1.749+0.421−0.297 ± 0.006 2.217+0.534

−0.376 ± 0.015

CLEO2005 [2] 1.94± 0.39 ± 0.13 2.1± 0.4± 0.1

CLEO2013 [3] 1.77± 0.12 ± 0.10 2.17 ± 0.12+0.12−0.22

3PSR [6] 0.5± 0.1 –

HQEFT [7] 1.4± 0.3 –

NWA [58]+HQEFT [8] 1.67± 0.27 2.16 ± 0.36

NWA [58]+LFQM [12] 1.73± 0.07 2.24 ± 0.09

FK [13] 2.0 2.5

ISGW2 [59] 1.0 1.3

a smaller branching ratio. This explains why the previ-ous SR prediction is inconsistent with other approaches.However, by using the LCSR approach, we observe thata more reasonable and accurate SR prediction can beachieved. The LCSR predictions for the branching ratiosof the two D → ρ semileptonic decays also show betteragreement with the CLEO measurements.

C. The radiative decay D → ργ

After inputting the D → ρ TFFs into the parity con-serving and parity violating effective couplings Aρ

PV,PC,we get the branching ratio for the two D meson radiativeprocesses D0 → ρ0γ and D+ → ρ+γ. The results aregiven in Table VIII, where the uncertainties are squaredaverage of the theoretical and experimental error sources.As a comparison, we also listed the branching ratiosderived from various approaches. It indicates that theLCSR predictions for the branching ratios for radiativedecay D0 → ρ0γ shows a better agreement with the BelleCollaboration result, which is larger than other theoret-ical predictions.

TABLE VIII: The branching ratios for D → ργ decays (inunit: 10−5). As a comparison, we also present other theoret-ical predictions.

B(D0 → ρ0γ) B(D+ → ρ+γ)

This paper (SM) 1.744+0.598−0.704 5.034+0.939

−0.958

QCD SM [60] 0.38 0.46

Hybrid [26] 0.606 ± 0.565 1.174 ± 1.157

FS [23] 0.55± 0.45 3.35 ± 2.95

Pole Diagrams and VMD [21] 0.3± 0.2 4± 2

Belle Collaboration [5] 1.77± 0.31 −

For the direct CP asymmetry of the D → ργ decay, werecall that the maximal value of Eq.(18) can be reachedin the limit of maximal constructive interference (namelyof ±π/2 strong phase difference) of the amplitudes withdifferent weak phases. This way we get the upper limitof our predictions for the direct CP asymmetry.

The Wilson coefficient C(′)eff7 mainly comes from the

hard spectator interaction and weak annihilation contri-butions within the framework of SM, which results inCSM

7 (mc) ∈ [−0.00949+ 0.0014i,−0.00019+ 0.002i] [26].By taking input parameters into Eq.(18), we obtain theSM prediction for the CP asymmetry of D0 → ρ0γ, e.g.

ASMCP =

[1.329(±0.234)C7

(±0.089)mc

(+0.094−0.073

)F.F.(

+0.295−0.204

)a2

(+0.052−0.049

)D.C.

]× 10−2. (28)

=(1.329+0.402

−0.335

)× 10−2. (29)

In the first line, the separate uncertainties are caused bythe errors of the quantities C7, mc, TFFs (F.F.), a2 anddecay constant (D.C.), respectively, and in the secondline, all the errors are added up in quadrature. The cen-tral value of the SM prediction is smaller than the Belleresult, Aexp

CP = 0.056±0.152±0.006 [5] by almost 3 times.Because of the large statistical error for the present Bellemeasurements, the SM prediction roughly agrees withthe Belle result within errors.

One may hope that the possible discrepancy can beaccommodated by a well-motivated extension of the SM.To quantify the size of the Wilson coefficients, one cannormalize the effective Hamiltonian within new-physicscontributions as

Heff−NP =GF√2

∑

i

CiQi + h.c., (30)

where the complete list of potentially relevant opera-tors can be found in Ref. [25]. The Wilson coefficients

δC(′)7,8(M) are generically related beyond SMmodels, with

M denotes the matching scale. The initial conditions ofthe four operators are assuming M > mt, taking intoaccount the renormalization group evolution of the oper-

10

ators at the leading log level, leads to

C(′)7 (mc) = η[ηC

(′)7 (M) + 8(η − 1)C

(′)8 (M)], (31)

C(′)8 (mc) = ηC

(′)8 (M), (32)

where the coefficients η and η can be found in Ref.[61].A non-vanishing value for ∆ACP = ACP(K

+K−) −ACP(π

+π−) has been observed by LHCb and CDF col-laborations [62, 63], which could be used to restrict thenew physics contribution from the operator Q8 by us-ing the relationship ∆ACP ≈ −1.8|Im[CNP

8 (mc)] [64].Considering the world average value ∆Aexp

CP = −(0.67 ±0.16)% [65], one can get |Im[CNP

8 (mc)]| ≈ 0.4 × 10−2.Furthermore the renormalization group evolution implies|Im[CNP

7 (mc)]| ≈ |Im[CNP8 (mc)]| if the initial scale M is

set to be around 1 TeV, which, for instance, could be hap-pened for super-symmetry theory. Taking into accountthe uncertainties in determining |Im[CNP

8 (mc)]| and theuncertainties from the initial conditions of |CNP

7 (M)|, onecan obtain a conservative range (0.2 − 0.8) × 10−2 for|Im[CNP

7 (mc)]| [24]. Thus we obtain a prediction for theCP asymmetry of D0 → ρ0γ,

ANPCP =

[3.907(±2.344)C7

(±0.260)mc

(+0.276−0.215

)F.F.(

+0.868−0.601

)a2

(+0.153−0.145

)D.C.

]× 10−2 (33)

=(3.907+2.533

−2.448

)× 10−2. (34)

In the second line, all the errors are added up in quadra-ture. This value agrees with the Belle Collaboration re-sult within errors.

IV. SUMMARY

In the paper, we have investigated the D → ρ TFFswithin the LCSR approach. As shown by Table IIIand Fig.6, more accurate QCD SR predictions for theTFFs A1,2(q

2) and V (q2) can be achieved by applyingthe LCSR approach other than the 3PSR approach. Tocompare with the CLEO measurements, the LCSR ap-proach can give reasonable explanations for the D → ρTFFs. The pQCD factorization approach is applicable inlarge recoil region q2 0 and the lattice QCD approachis applicable in very large q2-region, thus the extrapola-tion of the results under those two approaches shall be

strongly model dependent. The LCSR prediction pro-vides a bridge between the pQCD and lattice QCD pre-dictions, since it is applicable for a wider range, i.e. inboth small and intermediate q2-region.

After extrapolating the D → ρ TFFs to whole q2-region, the LCSR predictions for the branching ratiosof the two D meson semileptonic decays are B(D0 →ρ−e+νe) = (1.749+0.421

−0.297 ± 0.006) × 10−3 and B(D+ →ρ0e+νe) = (2.217+0.534

−0.376 ± 0.015) × 10−3, respectively,which agree with the CLEO measurements within errors.And as shown by Table VIII, the branching ratios of theD meson radiative decay D0 → ρ0γ and D+ → ρ+γ alsoshow good agreement with the experimental results incomparing with other theoretical predictions.

Based on the short and long distance parity conservingand parity violating amplitudes, i.e. (Aρ

PV/PC)s.d./l.d.,

and in the limit of maximal constructive interference ofthe amplitudes, we get the SM prediction for the CPasymmetry of D0 → ρ0γ, ASM

CP = (1.329+0.402−0.335) × 10−2,

which has large discrepancy with the Belle Collabora-tion data. By taking |Im[CNP

7 (mc)]| = (0.2− 0.8)× 10−2

for the NP Wilson coefficient, we obtain the NP predic-tion, ANP

CP = (3.907+2.533−2.448)× 10−2, which agrees with the

Belle Collaboration result within errors. Since the short-distance amplitude is relatively much smaller than thelong-distance amplitude, the NP effect induced by theoperator Q8 shall give negligible effect (less than 0.1%)to the SM predictions of the branching ratios. Thuswe think that the LCSR approach can provide a self-consistent way to deal with the D meson decays, and theD meson involved processes could be adopted for testingNP beyond the SM.

Acknowledgments: This work was supportedin part by the National Science Foundation ofChina under Grant No.11881240255, No.11765007 andNo.11625520, No.11847301, the Project of GuizhouProvincial Department of Science and Technology un-der Grant No.KY[2017]1089 and No.KY[2019]1171, theChina Postdoctoral Science Foundation under GrantNo.2019TQ0329, the Project of Guizhou Minzu Univer-sity under Grant No.GZMU[2019]YB19.

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