Study on contributions of hadronic loops to decays of

Study on contributions of hadronic loops to decays of J= ! vector� pseudoscalar mesons

Xiang Liu,* Xiao-Qiang Zeng,† and Xue-Qian Li‡

Department of physics, Nankai University, Tianjin 300071, China(Received 22 June 2006; published 5 October 2006)

In this work, we evaluate the contributions of the hadronic loops to the amplitudes of J= ! PV whereP and V denote light pseudoscalar and vector mesons, respectively. By fitting data of two well measuredchannels of J= ! PV, we obtain the contribution from the pure OZI process to the amplitude which isexpressed by a phenomenological quantity jGPV

S j, and a parameter � existing in the calculations of thecontribution of hadronic loops. In terms of � and jGPV

S j, we calculate the branching ratios of otherchannels and get results which are reasonably consistent with data. Our results show that the contributionsfrom the hadronic loops are of the same order of magnitude as that from the OZI processes and theinterference between the two contributions are destructive. The picture can be applied to study otherchannels such as PP or VV of decays of J= .

DOI: 10.1103/PhysRevD.74.074003 PACS numbers: 13.20.Gd

I. INTRODUCTION

By the commonly accepted point of view, narrowness ofthe J= resonance is interpreted by the OZI rule [1].Namely, J= dissolves into three gluons which wouldthen hadronize into measurable hadrons and these pro-cesses are OZI suppressed. In Fig. 1, the three-gluonprocess diagrams are shown and they are single-OZI-suppressed (SOZI) or double-OZI (DOZI) suppressed.Generally one can ignore the contributions from DOZI.

Because of complexity of the loop calculations andnonperturbative QCD effects which govern the hadroniza-tion, accurate estimate of the contributions from such OZI-suppressed processes to the decay rates is still missing.Therefore one cannot rule out other possible mechanismswhich also make substantial contributions to the decayrates of J= and other family members of charmonia.

Meanwhile, in the charm-tau energy regions, some phe-nomena cannot be understood in the regular theoreticalframework. The �� puzzle [2] is the most challengingone. Rosner recently suggested [3] that this puzzle maybe explained by the 2S-1D mixing. There exists an alter-native possibility which might explain the surprisinglysmall branching ratio of �3886� ! ��, i.e. the final stateinteraction (FSI) [4–6].

Of course the first challenge to theorists is to correctlyevaluate the widths of the exclusive J= nonleptonic de-cays. Besides the OZI-suppressed mechanism, there existsanother possible process which may also contribute to theamplitude. That is the contribution from the hadronicloops. In fact, except the phase space for various channels,the mechanism where three gluons are exchanged, shouldbe flavor-blind and universal for all the processes. It isanother reason to motivate us to seek for a new mechanism

because a simple analysis indicates that data of J= decaysdo not manifest the expected universality.

This picture looks similar to that for the final stateinteraction (FSI) [4,5], but essentially different. In theFSI picture, J= or �3770� decay into two real hadronsand the two real hadrons rescatter into another pair ofhadrons which have the same isospin structure, but differ-ent identities, via strong interaction. In that case, the twohadrons, no matter in the intermediate stage or in the finalstates, are real and on their mass shells, therefore the twolight hadrons cannot be charmed mesons such as D or D�

due to constraints of momentum-energy conservation i.e.MJ= < 2MD (or MD �MD�). One can derive the rescat-tering amplitude by calculating the absorptive (imaginary)part of the so-called triangle diagrams [4–6].

There exists a different contribution. Recently, Suzuki[7] indicates that the real part of the triangle where the twointermediate hadrons in the triangle can be off-shell virtualones, does also contribute. Thus to estimate the corre-sponding contributions, one needs to calculate the disper-sive (real) part of the triangle. The diagram correspondingto the hadronic loops is shown in Fig. 2 (there may be moresimilar diagrams which contribute to the processes (seeFig. 3)).

In fact, such a mechanism should also exist in all thedecay modes of J= and make sizable contributions. Inthis work, we are going to evaluate the contribution fromthe hadronic loops where the intermediate hadrons are off-mass-shell.

J/ Ψ + · ·· J/ +Ψ · ··

)b()a(

FIG. 1. (a) and (b) correspond to the SOZI and DOZI pro-cesses for decays of J= ! two-light mesons.

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

PHYSICAL REVIEW D 74, 074003 (2006)

1550-7998=2006=74(7)=074003(8) 074003-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.74.074003

The physical picture is following: J= first dissolvesinto two virtual hadrons, then by exchanging an appropri-ate hadron (i.e it possesses appropriate charge, flavor, spinand isospin) they turn into two on-shell real light hadronswhich are to be seen by detector. The two intermediatehadrons are charmed hadrons, which contains flavor c or �c,thus the transition J= ! H1H2 where H1 and H2 are thetwo charmed hadrons, does not suffer from the OZI sup-pression. It is noted that the authors of [8] discuss apossible mechanism for the relativistic heavy ion collision,where J= absorbs a pion existing in the hot environmentto transit into two charmed mesons and the process mayinfluence the observed J= suppression at RHIC. Eventhough such process is different from that under discussionof this work, the pictures have similarities.

In the practical calculations, the coupling constant of theeffective vertex cannot be obtained by fitting data becausesuch a process cannot occur due to the limited phase space.Then certain reasonable symmetries are invoked to get thecoupling constants. We can argue that even though thecoupling constant itself is not accurately determined, thequalitative conclusion is not seriously influenced.

In contrast with the derivation of the absorptive part ofthe triangle, the loop integration seems to be ultraviolet-divergent. In fact, following the standard procedure, theeffective vertices are obtained from the chiral Lagrangian

where all the concerned hadrons are on-shell, i.e. are realones. To compensate the off-shell effects of the hadrons inthe triangle, one may introduce form factors at each vertex.The form factors are inversely proportional to (q2 ��2)where � is a phenomenological parameter [4] which isexpressed in terms of another phenomenological parameter� (see the following text for details). Because of existenceof the form factors the ultraviolet divergence disappears,namely � indeed plays a role similar to the cut-offs in thePauli-Villas renormalization scheme [9,10].

Picture is as follows, because, so far, we cannot derivethe contribution from the OZI-suppressed processes (i.e.via the three-gluon intermediate stage) to the total ampli-tude based on the quantum field theory yet, we denote suchcontribution by a phenomenological parameter jGPV

S j

which should be fixed by fitting data.Therefore, by the simple picture, there are only two

parameters, i.e. � and jGPVS j. Concretely, we suppose that

both the direct three-gluon process and hadronic loopscontribute to the amplitude altogether, by fitting data oftwo distinict channels of J= ! PV where P and V standfor pseudoscalar and vector mesons, respectively, we ob-tain the values of � and jGPV

S j simultaneously. Here wechoose to study processes J= ! PV, since there are moredata available. Of course when we derive jGPV

S j from data,we have to carefully take care of the phase space of finalstates. Applying the obtained � and jGPV

S j, we evaluate therates of other J= ! PV channels.

Indeed, our numerical results show that a destructiveinterference between the three-gluon process (OZI) and thehadronic loop can result in rates which are satisfactorilyconsistent with data. We will give a detailed discussion inthe last section.

This work is organized as follows, after the introduction,in Sec. II, we formulate the hadronic loop contribution for

J/ Ψ •

c̄

c

qq̄

FIG. 2. The hadronic loop diagram where the two hadronsinside the triangle are timelike, but virtual ones and the ex-changed hadron possessing appropriate quantum numbers arealso off-shell.

J/ ψ

J/ ψ J/ ψ J/ ψ

J/ ψJ/ ψD 0

D̄∗0

D 0

π 0

ρ0

D 0

D̄∗0

D∗0

π 0

ρ0

D 0

D̄ 0

D∗0

π 0

ρ0

(a) (b) (c)

D∗0

D̄∗0

D 0

π 0

ρ0

D∗0

D̄∗0

D∗0

π 0

ρ0

D∗0

D̄ 0

D∗0

π 0

ρ0

(d) (e) ( f )

FIG. 3. The Feynman diagrams which depict the decays of J= ! �0�0 through D�� hadronic loops.

XIANG LIU, XIAO-QIANG ZENG, AND XUE-QIAN LI PHYSICAL REVIEW D 74, 074003 (2006)

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J= ! PV. In Sec. III, we present our numerical resultsalong with all the input parameters. Finally, Sec. IV isdevoted to discussion and conclusion.

II. FORMULATION

The effective Lagrangian for J= decaying into lightpseudoscalar and vector mesons is written as [11]

L � �GPVS � GPV

H �� Tr�@�Py@�V��

� GPVD �� Tr�@�P

y�Tr�@�V��; (1)

where GPVS , GPV

D and GPVH respectively denote the effective

couplings corresponding to single OZI diagram [Fig. 1(a)],double-OZI diagram [Fig. 1(b)] and hadronic loop diagram(Fig. 2). In this work, we neglect the DOZI contribution. Pand V respectively denote the nonet pseudoscalar and thenonet vector meson matrices

P �

�0��2p �

x��x�0�0��

2p �� K�

�� 0��2p �

x��x�0�0��

2p K0

K� �K0 y�� y�0�0

0BBB@1CCCA;

(2)

V �

�0��2p � !��

2p �� K��

�� 0��2p � !��

2p K�0

K�� K�0

0BB@

1CCA: (3)

In the above expression, x�;�0 and y�;�0 describe the �-�0smixing, and

x� � y�0 �cos�

��2p

sin��3p ;

x�0 � �y� �sin�

��2p

cos��3p ;

(4)

where � �19:1 [12] is the mixing angle of � and �0.Taking J= ! �0�0 as an example, we illustrate the

calculations of the contributions from hadronic loops.The Feynman diagrams describing the contributions of

the hadronic loops to J= ! �0�0 are depicted in Fig. 3.In the hadronic loops of Figs. 3(a)–3(f) only D��0 and�D��0 are involved, but of course they can be simply re-

placed by D�� to compose new diagrams.In the former works [13,14], the effective Lagrangians,

which are related to our calculation, are constructed basedon the chiral symmetry and heavy quark symmetry as

L � igJ= DD�Di@$

�Djy�� gJ= DD��@�Di@�D

�jy� � igJ= D�D� �D��

i �@�D�jy� �� D�j�y�@�D�

i��

� �D��i @$

�D�jy� ��

�1

2gD�D�P"��D

��i @

�Pij@$�

D��yj � igD�DP�D

i@�PijD�jy� �D�i

�@�PijD

jy�

� igDDVDyi @$

�Dj�V��ij � 2fD�DV"��@

�V��ij�Dyi @$�

D��j �D��yi @

$�Dj� � igD�D�VD

��yi @$

�D�j� �V��ij

� 4ifD�D�VD�yi��@

�V� � @�V��ijD�j�

�; (5)

where D and D� are, respectively, pseudoscalar and vectorheavy mesons, i.e. D�� D0��; �D��; �D�s ��. Theactual values of the coupling constants will be given in thefollowing section.

With these Lagrangians, we write out the decay ampli-tudes of J= ! �0�0. For Fig. 3(a),

Ma �A�0�0

Z d4q

�2��4�igJ= DD�"��

�J= �ip

�2��ip

�1 ��

� �igDD��p1 � q� � ��gD�D��ip4��

�i

p21 �m

2D

i��g��

p22 �m

2D�

i

q2 �m2D

F 2�mD; q2�: (6)

For saving the space, we collect the concrete expressionsof the amplitudes corresponding to Fig. 3(b)–3(f) in theAppendix.

In the amplitudes, A�0�0 � 1=2. F �mi; q2�s representthe form factors which compensate the off-shell effects ofthe mesons at the vertices and are written as [15]

F �mi; q2� �

�2 �m2i

�2 � q2 (7)

where � is a phenomenological parameter. It is obviousthat as q2 ! 0 it becomes a number and if � mi, it turnsto be unity. Whereas, as q2 ! 1, the form factor ap-proaches to zero. In that situation, the distance becomesvery small, so that the inner structure would manifest itselfand the whole picture of hadron interaction is no longervalid. At that region the form factor turns to zero and playsa role to cut off the end effect. The concrete expression of� is represented as [16]

��mi� � mi � ��QCD; (8)

STUDY ON CONTRIBUTIONS OF HADRONIC LOOPS TO . . . PHYSICAL REVIEW D 74, 074003 (2006)

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where mi denotes the mass of exchanged meson. �QCD �

220 MeV. � is a phenomenological parameter. In fact, inliterature, some other forms for the form factors are sug-gested [17], all of them depend on phenomenologicalparameters which are obtained by fitting data, thereforein the calculations all the forms are somehow equivalenteven though some of them may provide a better convergentbehavior for the triangle loop integration. That is verysimilar to the case of the pauli-Villas renormalizationscheme [9,10].

Taking Fig. 3(a) as an example, we carry out theFeynman parameter integration for (6) and get

M�a� �A�0�0gJ= DD�gDD�gD�D�Z 1

0dx

�Z 1�x

0dy�

2

�4��2log

��mD;mD� ;��mD��

��mD;mD� ; mD�

�

��2 �m2

D�y

8�2��mD;mD� ; mD�

��2"��

�J= �

��p�3p

�4 �;

(9)

where

��a; b; c� � m23�1� x� y�

2 � 2�p3 � p4��1� x� y�x

�m24x

2 � �m23 � a

2��1� x� y�

� �m24 � b

2�x� yc2;

m3 (m4), p3 (p4) are the masses and four- momenta of �0

(�0), respectively. Because of the form factors, the ultra-violet divergence does not exist as expected.

We use the same method to deal with the amplitudescorresponding to Fig. 3(b)–3(f) and also put them in theappendix.

To obtain the amplitudes corresponding to hadronicloops which contain D��, one only needs to replace theparameters corresponding to D��0 and �D��0 by that toD�� and D�� in the above six expressions (9)–(17)and (A1)–(A10). Thus contribution from the hadronicloops to the total amplitude can be eventually expressed as

M�0�0

H � �Ma �Mb �Mc �Md �Me �Mf�

� �Ma �Mb �Mc �Md

�Me �Mf�jmD��0!m

D��

� G�0�0

H ��"��J= �

��p�3p

�4 ; (10)

where G�0�0

H �� denotes the effective coupling for J= !

�0�0 induced by the hadronic loops where G�0�0

H �� is afunction of variable �.

Similar to J= ! �0�0, we can also obtain contribu-tions of hadronic loops to J= ! K��K� � c:c:,K�0 �K0 �

c:c:, !��0�, ��

0�. It is noticed that the hadronic loops forprocesses J= ! K��K� � c:c:, K�0 �K0 � c:c:, ��

0� in-

cludeD��s mesons and in the calculations, the coefficientA�0�0 is replaced by AK�K � 1 (or A!��

0 � � x��0 �=2,

A��0 � � y��0 �=2).

We have obtained seven quantities which contain bothcontributions of the OZI process and the hadronic loop

G �0�0� GPV

S ��0�0� G�0�0

H ��; (11)

G K��K� � GPVS �K

��K� �GK��K�H ��; (12)

G !� � GPVS �!� � G!�

D �G!�H ��; (13)

G !�0 � GPVS �!�

0� G!�0

D � G!�0

H ��; (14)

G � � GPVS �� G�

D �G�H ��; (15)

G �0 � GPVS ��

0� G�0

D � G�0

H ��; (16)

where the contributions from the DOZI processes are

ignored, i.e. one can set G!��0 �

D � G��0 �

D � 0. Geometryfactors �PV satisfy the following relation:

��0�0

S :�K��K�

S :�!�S :�!�0

S :��S :��0

S � 1:1:x�:x�0 :y�:y�0 :

(17)

The SU(3) geometry factors �PV is obtained by taking thetrace Tr�PV� as J= is an SU(3) singlet, and in fact, it isjust the SU(3) C-G coefficient for the concerned channel.

Thus using Eqs. (11) and (12), the parameter � isdetermined. With it, we are able to calculate the valuesof GPV

H .

III. NUMERICAL RESULTS

Because of the flavor SU(3) symmetry, it is reasonable toset gJ= Ds

�Ds� gJ= D �D. Based on the naive vector domi-

nance model and using the data of the branching ratio ofJ= ! e�e�, the authors of Ref. [18] determinedg2J= D �D=�4�� 5. As a consequence of the spin symmetry

in the HQET, gJ= Ds�D�s and gJ= D�s �D�s satisfy the relations:

gJ= Ds�D�s � gJ= D �Ds

=mDsand gJ= D�s �D�s � gJ= Ds

�Ds[19].

Other coupling constants related to J= ! PV include[16]:

gD�D�� gD�D��mDmD�p �

2gf�; gDDV � gD�D�V �

�gV��2p ;

fD�DV �fD�D�VmD�

��gV��

2p ; gD�DsK �

��mDs

mD

sgD�D�;

gD�sDK �

��mD�s

mD�

sgD�D�; gV �

m�

f�;

where f� � 132 MeV, gV; � and � are parameters in theeffective chiral Lagrangian that describes the interaction of


074003-4

heavy mesons with low-momentum light vector mesons[14]. Following Ref. [20], we take g � 0:59, � � 0:9 and� � 0:56 GeV�1 [21]. Meanwhile, the coupling constantsgD��s D��s �K�� (fD��s D��s �K��) are related to gD��D��(fD��D��) and one can expect the relations in the limit ofSU(3) symmetry [16]

gD��s D��s �mD��s

mD��gD��D��; fD��s D��s �

mD��s

mD��fD��D��;

gD��D��s K� �

��mD��s

mD��

sgD��D��;

fD��D��s K� �

��mD��s

mD��

sfD��D��:

Using Eqs. (11) and (12), we have determined � andGPVS as

� � 0:13; jGPVS j � 4:51:

Namely, in our scheme, the hadronic loop contributionto J= ! PV is theoretically calculated whereas the OZIcontribution is obtained by fitting the data of the twochannels. The obtained jGPV

S j � 4:51 is supposed to beuniversal for all channels of J= ! PV based on SU(3)symmetry. With Eqs. (13)–(16), we predict the values ofGPV for all channels of J= ! PV. The obtained resultsare listed in Table I.

IV. CONCLUSION AND DISCUSSION

From above discussions and numerical results, one hasthe following observations.

(1) The contribution from the hadronic loops is sizableand has the same order of magnitude as that from theSOZI process.

(2) To get reasonable results which are consistent withdata, the interference between the SOZI contribu-tion and that from hadronic loops is destructive.

(3) We derive the parameter � and the universal jGPVS j

by fitting data of the first two distinct modes of J= and present in the first two columns of Table I. In ourpicture, there are only two free parameters. Thevalues of GPV which we predict in terms of � andGPVH are reasonably consistent with the experimental

data.

In this work, we rely on two key assumptions. The first isthat the value jGPV

S j is universal for all channels of J= !PV due to the flavor-blindness of gluon coupling to quarks.This treatment is similar to that made [8] for study process�� J= ! D�� D�� in a hot RHIC environment.

In this work, we reparametrize � in terms of� followingthe ansatz suggested by the authors of Ref. [16]. It seemsthat one may gain some information about the parameter �from the FSI calculations [4,6], however, it is not really thecase. Even though in both calculations, one deals with thetriangle diagrams which are composed of only hadrons, thesituations are different. For FSI, one calculates the absorp-tive part where only the exchanged meson is off-shell,whereas for the calculation of the dispersive part, all thethree mesons in the triangle are off-shell. Form factors arephenomenologically introduced to compensate the off-shell effects at each vertex, thus the form factors shouldnot be the same for calculating the dispersive and absorp-tive parts, i.e. even though they have the same expressions,the involved free parameters may be different. It is notedthat in the FSI case, the value of � in the form factor isexpected to be of order unity [16], but in the case ofhadronic loop the value of � is obviously smaller. In ourstrategy, we do not priori set its value, but let data deter-mine it.

As our conclusion, it is obvious that the contributionfrom the hadronic loop to J= ! two-light mesons shouldexist. The question is how large it is compared to that of theSOZI processes. Namely, if only the SOZI process isresponsible for the decays of J= , it is hard to explainthe data and as the contribution from the hadronic loopsbeing added, one can see that the data are well accommo-dated within a tolerable range. In our scenario, there areonly two parameters in the whole scenario and they aredetermined by fitting data of two channels. With these twoparameters, we evaluate the branching ratios of the otherchannels of J= ! PV and the theoretical predictions areconsistent with data. It indicates that the ansatz is reason-able and the whole picture makes sense. Indeed, there isonly one channel, i.e. J= ! !�, for which the theoreticalprediction on the branching ratio is about 0.6 of the ex-perimental data. Even though there exists such a discrep-ancy, for a rough estimate the consistency is not bad at alland we would like to urge our experimental colleagues toredo the measurement to get more accurate results. On theother side, so far the SOZI processes have not well calcu-

TABLE I. The first two modes are well measured, we obtain � and jGPVS j � 4:51 by fitting them.

Decay mode �0�0 K��K� � c:c: � �0 !� !�0

BR� 10�3 (Experiment) [22] 4:2 0:5 5:0 0:4 0:65 0:07 0:33 0:04 1:58 0:16 0:167 0:025GPV (10�3 GeV�1) 2:08 0:25 1:65 0:26 0:89 0:096 0:71 0:086 1:27 0:13 0:46 0:069GPVH (10�3 GeV�1) (Theory) 6.59 6.16 3.55 5.13 5.46 4.07

GPV (10�3 GeV�1) (Theory) 2.08 (fitting) 1.65 (fitting) 0.92 0.62 0.95 0.40BR� 10�3 (Theory) 4.2 (fitting) 5.0 (fitting) 0.70 0.25 0.86 0.14


074003-5

lated in the framework of quantum field theory yet, we alsoneed to complete such a derivation and detect the valuewhich we obtained by fitting data and assuming a univer-sality for all modes. To be more accurate, however, furthertheoretical and experimental efforts may be needed. Thismechanism should exist not only for J= exclusive decays,but also for other members of charmonia. However, forhigher excited states of chamonia, the channels with finalstates composed of charmed hadrons are open, and thesituation becomes more complicated. This picture alsoapplies to the � decays and can be tested by the futureexperimental data.

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienceFoundation of China (NNSFC).

APPENDIX

One obtains the decay amplitudes of J= ! �0�0 cor-responding to Fig. 3(b)–3(f).

For Fig. 3(b),

Mb �A�0�0

Z d4q

�2��4�igJ= DD�"��

�J= �ip

�2��ip

�1 ��2ifD�D�" ��ip 3 ��

��ip�1 � iq

��igD�D��"��!�ip�4��ip

�2 ��

�i

p21 �m

2D

i��g��

p22 �m

2D�

i��g�!�

q2 �m2D�

F 2�mD� ; q2�: (A1)

For Fig. 3(c),

M c �A�0�0

Z d4q

�2��4�igJ= DD�p1 � p2� � �J= ��2ifD�D�" ��ip 3 ��

��ip�1 � iq

��gD�D��ip�4 ��

i

p21 �m

2D

�i

p22 �m

2D

i��g��

q2 �m2D�

F 2�mD� ; q2�: (A2)

For Fig. 3(d),

Md �A�0�0

Z d4q

�2��4fgJ= D�D� �ip�1�

�J= � ip

�2 �

�J= � i�p2 � p1� � �J= g��g�2ifD�D�"��!�ip�3 ��

��ip�1 � iq

��

� �gD�D��ip�4 ��

i��g!��

p21 �m

2D�

i��g��

p22 �m

2D�

i

q2 �m2D

F 2�mD; q2�: (A3)

For Fig. 3(e),

M e �A�0�0

Z d4q

�2��4fgJ= D�D� �ip

�1�

�J= � ip

�2 �

�J= � i�p2 � p1� � �J= g

��gf�gD�D��ip1 � iq� � ��g��

� 4fD�D��ip�3 �� ip

�3 �

��g�igD�D��"��!�ip

�4��ip

�2 ��

i��g��

p21 �m

2D�

i��g��

p22 �m

2D�

i��g!��

q2 �m2D�

F 2�mD� ; q2�: (A4)

For Fig. 3(f),

Mf �A�0�0

Z d4q

�2��4�igJ= DD�"��

�J= �ip

�2��ip

�1 ��f�gD�D��ip1 � iq� � ��g

�� 4fD�D��ip�3�

�� ip

�3�

��g

� �gD�D��ip 4 ��i��g��

p21 �m

2D�

i

p22 �m

2D

i��g� �

q2 �m2D�

F 2�mD� ; q2�: (A5)

By further calculations, we obtain


074003-6

M�b� �A�0�0gJ= DD�2fDD��gD�D��Z 1

0dxZ 1�x

0dy��

��2 �m2D� �y

16�2�2�mD;mD� ;��mD� ��

1

16�2��mD;mD� ; mD� �

�1

16�2��mD;mD� ;��mD� ��

��2xy2�p3 � p4�

2 � 2x2y�p3 � p4�2 � 2xy�p3 � p4�

2 � 2xy2m23m

24

� 2x2ym23m

24 � 2xym2

3m24

��

�2

�4��2log

��mD;mD� ;��mD� ��

��mD;mD� ; mD� �

��

��2 �m2D� �y

8�2��mD;mD� ; mD� �

�

� ��6y� 2x� 1��p3 � p4� � 2m23 � 2m2

3x� 2m23y� 2y�p3 � p4��

�"��

�J= �

��p�3p

�4 ; (A6)

M �c� �A�0�0gJ= DDfDD��gDD��Z 1

0dxZ 1�x

0dy�

2

�4��2log

��mD;mD;��mD� ��

��mD;mD;mD� �

��

��2 �m2D� �y

8�2��mD;mD;mD� �

��8"��

�J= �

��p

�3p

�4 �; (A7)

M�d� �A�0�0gJ= D�D�fDD��gD�D�Z 1

0dxZ 1�x

0dy��

��2 �m2D�y

16�2�2�mD� ; mD� ;��mD��

1

16�2��mD� ; mD� ; mD�

�1

16�2��mD� ; mD� ;��mD��

��4x2m2

4 � 4x2�p3 � p4� � 4xy�p3 � p4��

� 12�

2

�4��2log

��mD� ; mD� ;��mD��

��mD� ; mD� ; mD�

��

��2 �m2D�y

8�2��mD� ; mD� ; mD�

��"��

�J= �

��p�3p

�4 ; (A8)

M�e� �A�0�0gJ= D�D�gD�D��Z 1

0dxZ 1�x

0dy��

��2 �m2D� �y

16�2�2�mD� ; mD� ;��mD� ��

1

16�2��mD� ; mD� ; mD� �

�1

16�2��mD� ; mD� ;��mD� ��

�fD�D��4x

2m23 � 4y2m2

3 � 4m23 � 8xm2

3 � 8ym23 � 8xym2

3 � 4x2�p3 � p4�

� 4�p3 � p4� � 8x�p3 � p4� � 4y�p3 � p4� � 4xy�p3 � p4��

�

�2

�4��2log

��mD� ; mD� ;��mD� ��

��mD� ; mD� ; mD� �

��

��2 �m2D� �y

8�2��mD� ; mD� ; mD� �

��16fD�D�� 2gD�D��

�"��

�J= �

��p�3p

�4 ;

(A9)

M �f� �A�0�0gJ= D�DfD�D��gDD��Z 1

0dxZ 1�x

0dy��

��2 �m2D� �y

16�2�2�mD� ; mD;��mD� ��

1

16�2��mD� ; mD;mD� �

�1

16�2��mD� ; mD;��mD� ��

��4y�p3 � p4��

�"��

�J= �

��p�3p

�4 : (A10)

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[21] In Ref. [14] the authors also gave the values of � and gwhich are somewhat different from the updated value of �and g presented in [20]. In this work, we use the updatedvalues. Since we obtain other parameters by fitting data,the final results are not sensitive to the values of � and g.

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Study on contributions of hadronic loops to decays of