J/ψ dissociation cross sections in a relativistic quark model

J

Õ

␺

dissociation cross sections in a relativistic quark model

Mikhail A. Ivanov

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

Juergen G. Ko¨rner

Institut fu¨r Physik, Johannes Gutenberg–Universita¨t, D-55099 Mainz, Germany

Pietro Santorelli

Dipartimento di Scienze Fisiche, Universita` di Napoli ‘‘Federico II,’’ Napoli, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Napoli, Italy

共Received 26 November 2003; published 27 July 2004兲

We calculate the amplitudes and the cross sections of the charm dissociation processes J/␺⫹␲ →DD¯ ,D*D¯ (D*D),D*D* within a relativistic constituent quark model. We consistently account for the contributions coming from both the box and triangle diagrams that contribute to the dissociation processes. The cross section is dominated by the D*D¯ and D*D*channels. When summing up the four channels we find a maximum total cross section of about 2.3 mb at

冑

s⬇4.1 GeV. We compare our results to the results of other

model calculations.

DOI: 10.1103/PhysRevD.70.014005 PACS number共s兲: 12.39.Ki, 13.75.Lb, 14.40.Lb

I. INTRODUCTION

The analysis of the J/␺ dissociation cross section is im-portant for understanding the suppression of J/␺ production observed in Pb-Pb collisions by the NA50 Collaboration at the CERN Super Proton Synchrotron 共SPS兲 关1兴. There are a number of theoretical calculations on the cc¯⫹light hadron cross sections共see, e.g., the review Ref. 关2兴兲. However, they give widely divergent results, which implies that one is still far away from a real understanding of the scattering mecha-nism. The nonrelativistic quark model has been applied in关3兴 and 关4–6兴 for the calculation of the cross sections for the dissociation processes cc¯⫹qq→¯ cq¯⫹qc¯. The calculated cross sections for the reactions J/␺⫹␲→DD¯ , D*D¯ , DD*,

D*D* have the following common features: they rise very fast from zero at threshold to a maximum value and finally fall off due the Gaussian form of the potential. The magni-tude of the maximum total cross section was found to be

⬇7 mb at

冑

s⬇4.1 GeV in 关3兴 and a somewhat smaller value

of ⬇1.4 mb at

冑

s⬇3.9 GeV in 关4–6兴.

Another approach to studying the charm dissociation pro-cess started with the model proposed by Matinian and Mu¨ller

关7兴. They assumed that the dissociation cross section is

domi-nated, in the t channel, by the D meson exchange. A gener-alization of this approach can be found in 关8–13兴 where an effective chiral SU共4兲 Lagrangian was employed. Such an approach seems to us quite problematic for the following reasons:共i兲 SU共4兲 is a badly broken symmetry, and 共ii兲 some of the couplings in the chiral SU共4兲 Lagrangian are un-known. Nevertheless, this is a relativistic approach that al-lows one to study the above processes in a systematic fash-ion. In this framework, the dissociation cross section of J/␺ by light hadrons is predicted to be, near the physical thresh-old, in the range 1–10 mb. Moreover, it is interesting to refer to the paper 关12兴, where the J/␺ dissociation by ␲ and ␳

mesons was examined in the meson exchange model 关12兴 and compared to the quark interchange model. The authors of this paper found that the meson exchange model could give predictions similar to those of the quark interchange models, not only for the magnitudes but also for the energy dependence of the low-energy dissociation cross sections of the J/␺ by␲ and␳ mesons.

It appears that the microscopic quark nature of hadrons is important in charm dissociation processes. The first step is to calculate the relevant form factors corresponding to the triple and quartic meson vertices in the kinematical region of the dissociation reaction. QCD sum rules have been used in Refs. 关14–17兴 to evaluate those form factors and to deter-mine the charm cross section. The cross section was found to be about 1 mb at

冑

s⬇4.1 GeV with a monotonic growth

when the energy is increased关17兴.

An approach based on the 1/N expansion in QCD com-bined with the Regge theory gives, for the total dissociation cross section, a value of a few millibarns near to

冑

s ⬇4 GeV 关18兴.

We also mention the work of Deandrea et al., where the strong couplings J/␺D*D*and J/␺D*D*␲were evaluated in the constituent quark model 关19兴. Finally, an extension of the finite-temperature Dyson-Schwinger equation approach to heavy mesons and its application to the reaction J/␺⫹␲ →D⫹D¯ was considered in关20兴.

We employ a relativistic quark model关21兴 to calculate the charm dissociation amplitudes and cross sections. This model is based on an effective Lagrangian which describes the coupling of hadrons H to their constituent quarks. The coupling strength is determined by the compositeness condi-tion ZH⫽0 关22兴 where ZHis the wave function

renormaliza-tion constant of the hadron H. One starts with an effective Lagrangian written down in terms of quark and hadron fields. Then, by using Feynman rules, the S-matrix elements describing the hadronic interactions are given in terms of a

set of quark diagrams. In particular, the compositeness con-dition enables one to avoid a double counting of the hadronic degrees of freedom. The approach is self-consistent and uni-versally applicable. All calculations of physical observables are straightforward. The model has only a small set of ad-justable parameters given by the values of the constituent quark masses and the scale parameters that define the size of the distribution of the constituent quarks inside a given had-ron. The values of all fit parameters are within the window of expectations.

The shape of the vertex functions and the quark propaga-tors can in principle be found from an analysis of the Bethe-Salpeter and Dyson-Schwinger equations as was done, e.g., in关23兴. In this paper, however, we choose a phenomenologi-cal approach where the vertex functions are modeled by a Gaussian form, the size parameter of which is determined by a fit to the leptonic and radiative decays of the lowest-lying light, charm, and bottom mesons. For the quark propagators we use a local representation.

We calculate the amplitudes and the cross sections of the charm dissociation processes

J/␺⫹␲→D⫹D¯ ,

J/␺⫹␲→D*⫹D¯_{共D*⫹D兲,} J/␺⫹␲→D*⫹D*.

These processes are described by both box and resonance diagrams which can be calculated straightforwardly in our approach. We compare our results with the results of other studies.

The layout of the paper is as follows: In Sec. II we briefly discuss our relativistic quark model. In Sec. III we outline the calculational technique of the arbitrary n-point one-loop diagrams with local propagators and Gaussian vertex func-tions. We give explicit result for the triangle and box dia-grams which are the building blocks of the charm dissocia-tion amplitudes. In Sec. IV we calculate the charm dissociation amplitudes and the total cross sections. In Sec. V we perform the numerical analysis and give our prediction for the cross sections.

II. THE MODEL

The coupling of a meson H to its constituent quarks q1

and q¯2 is determined by the Lagrangian

Lint Str_共x兲⫽g

HH共x兲

冕

dx1

冕

dx2FH共x,x1,x2兲q¯2共x2兲

⫻⌫H␭Hq1共x1兲⫹H.c. 共1兲

Here, ␭H and ⌫H are Gell-Mann and Dirac matrices which

describe the flavor and spin quantum numbers of the meson field H(x). The function F_H is related to the scalar part of the Bethe-Salpeter amplitude and characterizes the finite size of the meson. To satisfy translational invariance, the function

F_H has to satisfy the identity F_H(x⫹a,x₁⫹a,x₂⫹a)

⫽FH(x,x1,x2) for any four-vector a. In the following we use a

particular form for the vertex function,

FH共x,x1,x2兲⫽␦共x⫺c12 1

x1⫺c12 2

x2兲⌽H„共x1⫺x2兲2…, 共2兲

where ⌽H is the correlation function of two constituent

quarks with masses m1, m2 and ci j k_⫽m

k/(mi⫹mj).

The coupling constant gHin Eq.共1兲 is determined by the

so-calledcompositeness condition originally proposed in关22兴 and extensively used in 关24兴. The compositeness condition requires that the renormalization constant of the elementary meson field H(x) is set to zero,

ZH⫽1⫺ 3g_H2 4␲2⌸ ˜ H

⬘

共MH 2_{兲⫽0, 共3兲}

where ⌸˜_H

⬘

is the derivative of the meson mass operator. In order to clarify the physical meaning of this condition, we note that Z_H1/2 is also interpreted as the matrix element be-tween a physical particle state and the corresponding bare state. For ZH⫽0 it then follows that the physical state does

not contain the bare one and is described as a bound state. The interaction Lagrangian in Eq. 共1兲 and the corresponding free Lagrangian describe both the constituents 共quarks兲 and the physical particles共hadrons兲 which are bound states of the quarks. As a result of the interaction, the physical particle is dressed, i.e., its mass and wave function have to be renor-malized. The condition ZH⫽0 also effectively excludes the

constituent degrees of freedom from the physical space and thereby guarantees that a double counting of physical ob-servables is avoided. The constituent quarks exist in virtual states only. One of the corollaries of the compositeness con-dition is the absence of a direct interaction of the dressed charged particle with the electromagnetic field. Taking into account both the tree-level diagram and the diagrams with the self-energy insertions into the external legs yields a com-mon factor ZHwhich is equal to zero. We refer the interested

reader to our previous papers 关21,24,25兴 where these points are discussed in more detail.

We briefly discuss the introduction of the electromagnetic field into the nonlocal Lagrangian in Eq. 共1兲 in a gauge in-variant manner. This can be accomplished by using the path exponential关26兴

Lint

Str⫹em_共x兲⫽g

HH共x兲

冕

dx1

冕

dx2FH共x,x1,x2兲

⫻q¯2共x2兲eieq2I(x2,x, P)⌫_H␭_He⫺ieq1I(x1,x, P)q₁共x₁兲,

共4兲 where I共xi,x, P兲⫽

冕

x x_i dz_␮A␮共z兲 共5兲

and where z␮ is a coordinate point on the path P. At first sight it appears that the results depend on the path P which connects the end points x and x1 in the path integral in Eq.

共5兲. However, we need to know only derivatives of such

integrals for the perturbative calculations here. Therefore, we use a formalism关26兴 which is based on the path-independent definition of the derivative of I(x,y , P):

lim

dx␮→0 dx␮ ⳵

⳵x␮I共x,y,P兲⫽ lim_dx␮_→0关I共x⫹dx,y,P⬘兲

⫺I共x,y,P兲兴, 共6兲

where the path P

⬘

is obtained from P by shifting the end point x by dx. The use of the definition in Eq.共6兲 leads to the key rule

⳵

⳵x␮I共x,y,P兲⫽A␮共x兲, 共7兲

which in turn states that the derivative of the path integral

I(x, y , P) does not depend on the path P originally used in

the definition. This allows one to construct the perturbation theory in a consistent way and guarantees the implementa-tion of charge conservaimplementa-tion and the Ward identities.

As an example, we consider the transition J/␺→␥ which is now described by two diagrams in Fig. 1:共a兲 the standard ‘‘bubble’’ and 共b兲 the ‘‘tadpole’’ ones. The total matrix ele-ment has a manifestly gauge invariant form

M_J/␮␯_␺共p兲⫽共g␮␯p2⫺p␮p␯兲MJ/␺共p2兲, 共8兲 where MJ/␺共p2兲⫽ec 3gJ/␺ 4␲2 1 p2

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ 1 d␣兵ya⌽˜J/␺共za兲 ⫺yb⌽˜J/␺共zb兲其, ya⫽

冉

1⫹ t 2

冊

mc 2_⫹p 2 4 1⫹3t/2⫺2t3␣共1⫺␣兲 共1⫹t兲2 , za⫽t关mc 2_{⫺␣共1⫺␣兲p}2_兴⫺ t 1⫹t

冉

1 2⫺␣

冊

2 p2, yb⫽

冉

1⫹ t 2

冊

冋

mc 2_⫺ ␣ 2 共1⫹t兲2 p2 4

册

, z_b⫽tm_c2⫺

冉

1⫺ ␣ 1⫹t

冊

␣p2 4 .

For the pseudoscalar and vector mesons treated in this paper the derivatives of the mass operators are written as

⌸˜ P

⬘

共p2_兲⫽ 1 2 p2p ␣ d d p␣

冕

d4k 4␲2i⌽ ˜ P 2 共⫺k2_兲 ⫻tr关␥5_S 1共k”⫹c12 1 p” 兲␥5S2共k”⫺c12 2 p ” 兲兴, ⌸˜ V

⬘

共p2_兲⫽1 3

冋

g ␮␯_⫺p␮p␯ p2

册

1 2 p2p ␣ d d p␣

冕

d4k 4␲2i⌽ ˜ V 2_共⫺k2_兲 ⫻tr关␥␯_S 1共k”⫹c21 1 _{p” 兲␥}␮_S 2共k”⫺c12 2 _{p” 兲兴. 共9兲}

The leptonic decay constants f_P and f_V are FIG. 2. The strong triangle and box diagrams. FIG. 1. The diagrams corresponding to the J/␺→␥ transition:

3gP 4␲2

冕

d4k 4␲2i⌽ ˜_P共⫺k2_兲tr关␥␮_␥5_S 1共k”⫹c12 1 _{p” 兲␥}5_S 2共k”⫺c12 2_{p” 兲兴} ⫽ fPp␮, 3gV 4␲2

冕

d4k 4␲2i⌽ ˜_V共⫺k2_兲tr关␥␮_S 1共k”⫹c12 1 _{p” 兲␥•⑀} VS2共k”⫺c12 2 _{p” 兲兴} ⫽mVfV⑀V␮.

We use free fermion propagators for the valence quarks

Si共k” 兲⫽

1

mi⫺k” 共10兲

with an effective constituent quark mass mi. As discussed in 关21,24兴 we assume for the meson mass MHthat

M_H⬍m₁⫹m₂ 共11兲

in order to avoid the appearance of imaginary parts in the physical amplitudes. This holds true for the light pseudo-scalar mesons but is no longer true for the light vector me-sons. We shall therefore employ identical masses for the pseudoscalar mesons and the vector mesons in our matrix element calculations but use physical masses in the phase space calculation. This is quite a reliable approximation for the heavy vector mesons, e.g., D*and B*, where the hyper-fine splitting between the D* and D and the B* and B, respectively, is quite small.

The shape of the vertex functions and the quark propaga-tors can in principle be found from an analysis of the Bethe-Salpeter and Dyson-Schwinger equations as was done, e.g., in关23,27兴. In this paper, however, we choose a phenomeno-logical approach where the vertex functions are modeled by a Gaussian form, the size parameter of which is determined by a fit to the leptonic and radiative decays of the lowest-lying light, charm, and bottom mesons 共see Table I兲. Our previous studies of phenomena involving the low-lying had-rons have shown that this approximation is successful and reliable关21,24兴. We employ a Gaussian for the vertex func-tion of the form⌽˜H(kE

2

)⬟exp(⫺k_E2/⌳_H2), where kE is a

Eu-clidean momentum. The size parameters⌳_H2 are determined

by a fit to experimental data, when available, or to lattice results for the leptonic decay constants f_P and f_V where P

⫽␲,D,B and V⫽J/␺,⌼. Here we improve the fit by using the MINUIT code in a least-squares fit. The values of the fit parameters are displayed in Table II. The quality of the fit may be assessed from the entries in Table I.

III. STRONG TRIANGLE AND BOX DIAGRAMS

Transition matrix elements involving composite hadrons are specified in the model by the appropriate quark diagram. Here, we give explicit expressions for the integrals corre-sponding to the strong triangle and box diagrams shown in Fig. 2.

First, we will make the transformations which are com-mon for all one-loop diagrams with local propagators and Gaussian vertex functions. The Feynman integral corsponding to a one-loop diagram with n propagators and, re-spectively, n-vertex functions may be written in Minkowskii space as In共p1, . . . , pn兲⫽

冕

d4_k 4␲2itr

冋

i

兿

_⫽1 n ⌽˜_{i共⫺共k⫹vi⫹n兲}2_兲 ⫻⌫iSi共k”⫹v”i兲

册

共12兲

where the vectors viare linear combinations of the external

momenta p_ito be specified later on, k is the loop momentum, and⌫iare Dirac matrices for the i meson关cf. Eq. 共1兲兴. The

external momenta are all chosen as ingoing such that one has

兺i⫽1 n _p

i⫽0.

The propagators can be written as

TABLE I. The physical quantities used in least-squares fitting our parameters. The values are taken from the Particle Data Group关28兴 or from lattice simulations 关29兴. The value of fB_c is our average of QCD sum

rules calculations关30兴. All numbers are given in MeV except for g_␲0_␥␥.

This model Expt. or lattice This model Expt. or lattice

f_␲ 130.7 130.7⫾0.1⫾0.36 g_␲0_␥␥ 0.272 GeV⫺1 0.273 GeV⫺1 fK 159.8 159.8⫾1.4⫾0.44 fJ/␺ 405 405⫾17 fD 211 203⫾14 226⫾15 fB 182 173⫾23 198⫾30 fD_s 244 230⫾14 250⫾30 fBs 209 200⫾20 230⫾30 fB_c 360 360 f⌼ 710 710⫾37

TABLE II. The fitted values of the model parameters.

Quark Energy ⌳H Energy ⌳H Energy

masses 共GeV兲 共GeV兲 共GeV兲

mu⫽md 0.223 ⌳␲ 1.074 ⌳B_c 1.959

ms 0.356 ⌳K 1.514 ⌳J/_␺ 2.622

mc 1.707 ⌳D⫽⌳D_s 1.844 ⌳⌼ 3.965

S_i共k”⫹v”_i兲⫽共m_i⫹k”⫹v”_i兲•

冕

0 ⬁ d␤_ie⫺␤i[mi 2 ⫺(k⫹vi)2]_. 共13兲

For the vertex functions one writes

⌽˜_{i共⫺共k⫹vi}

⫹n兲2兲⫽e␤i⫹n•(k⫹vi⫹n)

2

, i⫽1, . . . ,n, 共14兲

where the parameters ␤i⫹n⫽si⫽1/⌳i

2

are the size param-eters. One can then easily perform the integration over k:

冕

d4k ␲2_iexp

再

兺

_i⫽1 2n ␤i共k⫹vi兲2

冎

⫽ 1 ␤2exp

再

1 ␤ 1_{⭐i⬍ j⭐2n}

兺

␤i␤j共vi⫺vj兲 2

冎

_{. 共15兲}

Here, ␤⫽兺_i2n_⫽1␤i. The numerator mi⫹k”⫹v”i can be

re-placed by a differential operator in the following manner:

冕

d4k ␲2_ii

兿

_⫽1 n ⌫i共mi⫹k”⫹v”i兲e␤k 2_⫹2kr ⫽

兿

i⫽1 n ⌫i

冉

mi⫹v”i⫹ 1 2⳵”r

冊

• 1 ␤2e ⫺r2_/␤ ⫽ 1 ␤2e ⫺r2_/␤

兿

i⫽1 n ⌫i

冉

mi⫹v”i⫺ 1 ␤r”⫹ 1 2⳵”r

冊

. 共16兲 We thus have In共p1, . . . , pn兲⫽

兿

i⫽1 n

冕

0 ⬁ d␤i ␤2 exp

再

⫺

兺

_i⫽1 n ␤imi 2 ⫹1_␤

兺

1⭐i⬍ j⭐2n ␤i␤j共vi⫺vj兲 2

冎

⫻1 4tr

冋

兿

i⫽1 n ⌫i

冉

mi⫹v”i⫺ 1 ␤r”⫹ 1 2⳵”r

冊

册

, 共17兲

where r⫽兺_i2n_⫽1␤ivi. Finally, we effect some further

transfor-mations on the integration variables to get the integral into a form suitable for numerical evaluation. We use the formula

兿

i⫽1 n

冕

0 ⬁ d␤if共␤1, . . . ,␤n兲 ⫽

冕

0 ⬁ dttn⫺1

兿

i⫽1 n

冕

0 ⬁ d␣_i␦

冉

1⫺

兺

i⫽1 n ␣i

冊

f共t␣1, . . . ,t␣n兲;

and we scale the t variable by t→wt with w⫽兺_in_⫽1␤i⫹n ⫽兺i⫽1

n _s

i and introduce the new variable ␤˜i⫽␤i/w(i⫽n ⫹1, . . . ,2n). We have In共p1, . . . , pn兲 ⫽

冕

0 ⬁ dt t n⫺1 共1⫹t兲2

兿

_i⫽1 n

冕

0 ⬁ d␣i␦

冉

1⫺

兺

i_⫽1 n ␣i

冊

•e⫺wz ⫻w 2 4 tr

冋

兿

i⫽1 n ⌫i

冉

mi⫹v”i⫺ 1 1⫹tr”˜⫹ 1 2w⳵”˜r

冊

册

, 共18兲 where r ˜_⫽t

兺

i_⫽1 n ␣ivi⫹

兺

i_⫽n⫹1 2n ␤ ˜_ivi.

The z form in the exponential function is written as

z⫽tzloc⫺ t 1⫹tz1⫺ 1 1⫹tz2, 共19兲 zloc⫽

兺

i⫽1 n ␣imi 2_⫺

兺

1⭐i⬍ j⭐n ␣i␣j ai j, z₁⫽

兺

i⫽1 n ␣i

兺

j⫽n⫹1 2n ␤ ˜ jai j⫺

兺

1⭐i⬍ j⭐n ␣i␣j a_{i j}, z2⫽

兺

n⫹1⭐i⬍ j⭐2n ␤ ˜ i␤˜jai j.

The matrix ai j⫽(vi⫺vj)2 depends on the invariant

kine-matical variables. Explicit expressions for this matrix for the triangle and box diagrams are given in Tables III and IV, respectively. We will introduce the variable v⫽t/(1⫹t)(0

⭐v⭐1) in Eq. 共18兲 in what follows.

TABLE III. The matrix ai j⫽(vi⫺vj)2 for the triangle diagram. Here,v1⫽0, v2⫽⫺p2, v3⫽p1, v4⫽c13 1 p1, v5⫽⫺c12 1 p2, v6⫽p1 ⫹c23 3 p3. a12⫽p2 2 a13⫽p1 2 a23⫽p3 2 a14⫽(c13 1 )2p1 2 a15⫽(c12 1 )2p2 2 a16⫽c23 2 p1 2_⫺c 23 2 c23 3 p3 2_⫹c 23 3 p2 2 a24⫽c13 3 p2 2 ⫺c13 1 c13 3 p1 2 ⫹c13 1 p3 2 a25⫽(c12 2 )2_p 2 2 a26⫽(c23 2 )2_p 3 2 a34⫽(c13 3 )2_p 1 2 a35⫽c12 2 p₁2⫺c₁₂1c₁₂2p2₂⫹c₁₂1p₃2 a36⫽(c23 3 )2_p 3 2 a45⫽c13 1 (c₁₃1⫺c₁₂1) p₁2 a46⫽c13 3 (c₁₃3⫺c₂₃3) p₁2 a56⫽c12 2 (c₁₂2⫺c₂₃2) p₂2 ⫹c12 1 (c12 1_⫺c 13 1 ) p2 2_⫹c 12 1 c13 1 p3 2 _⫹c 23 3 (c23 3_⫺c 13 3 ) p2 2_⫹c 13 3 c23 3 p3 2 _⫹c 23 2 (c23 2_⫺c 12 2 ) p3 2_⫹c 12 2 c23 2 p1 2

Some further remarks are appropriate. One can see that the existence of the loop integral Eq. 共18兲 is defined by the local␣ form zloc. Even if we apply the constraints Eq.共11兲

it does not guarantee that this form is always positive. For instance, in the simplest triangle diagram with p₁2⫽p₂2⫽p₃2

⬅p2 _{and m}

1⫽m2⫽m3⫽m the form zloc⫽0 if p2⫽3m2.

Such singularities in the Feynman diagrams with local propagators are called anomalous thresholds共see the discus-sion in Ref.关31兴 and other references therein兲. The situation is much more complicated in the case of the box diagrams. Obviously, we cannot consider the annihilation processes when the energy is large enough to go beyond the normal thresholds corresponding to quark production. However, in the case of the dissociation processes considered here, the local form z_locis always positive, and we are able to make self-consistent and reliable predictions for physical observ-ables.

IV. THE JÕ␺ DISSOCIATION AMPLITUDES AND CROSS SECTIONS

In our approach the dissociation processes J/␺⫹␲⫹ →D⫹D0_{, D*}⫹_⫹D0_{, and D*}⫹_⫹D*0 _{are described by the}

diagrams in Fig. 3.

Our momentum labeling is defined by

J/␺共p1兲⫹␲⫹共p2兲→D3⫹共q1兲⫹D4 0_共q 2兲, 共20兲 where D₃⫹⫽D⫹ or D*⫹, D₄0⫽D0 or D*0_{, p} 1 2_⫽m H1 2 ⬅mJ/␺ 2 , p₂2⫽m_H22 ⬅m_␲2, q2₁⫽m_H32 ⬅m_D2⫹(mD*⫹ 2 ), q₂2⫽m_H42 ⬅m2 D0_(m2_D*0_).

The Mandelstam variables are defined in the standard form s⫽共p₁⫹p₂兲2_⫽共q 1⫹q2兲2, t⫽共p1⫺q1兲2⫽共p2⫺q2兲2, u⫽共p1⫺q2兲2⫽共p2⫺q1兲2, where s⫹t⫹u⫽m_H12 ⫹m_H22 ⫹m_H32 ⫹m_H42 .

Cross sections are calculated by using the formula

␴共s兲⫽₁₉₂1_␲s 1 p_1,cm2

冕

t_⫺

t_⫹

dt兩M共s,t兲兩2, 共21兲 where M (s,t) is an invariant amplitude and

t_⫾⫽共E1,cm⫺E3,cm兲2⫺共p1,cm⫿q1,cm兲2, E1,cm⫽ s⫹mH1 2 _⫺m H2 2 2

冑

s , E3,cm⫽ s⫹mH3 2 _⫺m H4 2 2

冑

s ,

TABLE IV. The matrix ai j⫽(vi⫺vj)2 for the box diagram. Here, v1⫽0, v2⫽⫺p2, v3⫽⫺p2⫺p3, v4⫽p1, v5⫽c14 1 p1, v6 ⫽⫺c12 1 p2, v7⫽⫺p2⫺c23 2 p3, v8⫽p1⫹c34 4 p4. a12⫽p2 2 a13⫽(p2⫹p3)2 a14⫽p1 2 a23⫽p3 2 a24⫽(p1⫹p2) 2 a34⫽p4 2 a15⫽(c14 1 )2_p 1 2 a16⫽(c12 1 )2_p 2 2 a17⫽c23 3 p2 2 ⫺c23 2 c23 3 p3 2 ⫹c23 2 ( p2⫹p3)2 a18⫽c34 3 p₁2⫺c₃₄3c₃₄4p₄2 a25⫽c14 4 p₂2⫺c₁₄1c₁₄4p₁2 a26⫽(c12 2 )2_p 2 2 ⫹c34 4 ( p1⫹p4)2 ⫹c14 1 ( p1⫹p2)2 a27⫽(c23 2 )2p3 2 a28⫽c34 4 p3 2_⫺c 34 3 c34 4 p4 2 a35⫽c14 1 p4 2_⫺c 14 1 c14 4 p1 2 ⫹c34 3 ( p3⫹p4)2 ⫹c14 4 ( p1⫹p4)2 a36⫽c12 1 p3 2 ⫺c12 1 c12 2 p2 2 ⫹c12 2 ( p2⫹p3)2 a37⫽(c23 3 )2_p 3 2 a38⫽(c34 3 )2_p 4 2 a45⫽(c14 4 )2_p 1 2 a46⫽c12 2 p₁2⫺c₁₂1c₁₂2p₂2 a47⫽c23 2 p₄2⫺c₂₃2c₂₃3p₃2 ⫹c12 1 ( p1⫹p2)2 ⫹c23 3 ( p3⫹p4)2 a48⫽(c34 4 )2p4 2

FIG. 3. The Feynman diagrams describing the charm dissocia-tion processes J/␺⫹␲⫹→D⫹⫹D0, D*⫹⫹D0_{, D*}⫹_⫹D*0

p1,cm⫽ ␭1/2_共s,m H1 2 _,m H2 2 _兲 2

冑

s , q1,cm⫽ ␭1/2_共s,m H3 2 ,m_H42 兲 2

冑

s .

The reaction threshold is equal to s₀⫽(m_H3⫹m_H4)2_{. Note}

that Eq. 共21兲 contains the statistical factor 1/3 which comes from averaging over the initial state J/␺ polarizations.

The dissociation processes are described by both the box and the resonance diagrams as shown in Fig. 3. The reso-nance diagrams depend explicitly only on the t or u variables whereas the box diagrams are functions of s and t.

In the following three subsections we write explicitly the amplitudes for the processes J/␺⫹␲⫹→D⫹⫹D0, D*⫹

⫹D0_{, D*⫹⫹D*}0_{in terms of form factors; all the analytical}

expressions for them are reported.

A. The channel JÕ␺¿␲¿\D¿¿D0

The invariant matrix element is written as1

M共VPPP兲⫽M_䊐共VPPP兲⫹Mres共VPPP兲, 共22兲 where M_䊐共VPPP兲⫽⑀_␮ 1共p1兲•␧ ␮1p1p2q2_F VPPP共s,t兲, 共23兲 Mres共VPPP兲⫽⑀␮₁共p1兲•兵⫺M_⌬a␣ 共PVP兲DD* ␣␤_共p 2⫺q2兲 ⫻M_⌬a␮1␤_{共VVP兲⫺M} ⌬b ␣ _共PPV兲D D_* ␣␤_共p 2⫺q1兲 ⫻M_⌬,b␮1␤_共VPV兲其_, M_⌬a␣ 共PVP兲⫽p₂␣F_PVP(1a)共t兲, M_⌬a␮1␤共VVP兲⫽␧␮1␤p1( p2⫺q2)_F VVP (a) _共t兲, M_⌬b␣ 共PPV兲⫽p₂␣F_PPV(1b)共u兲, M_⌬b␮1␤_{共VPV兲⫽␧}␮₁␤p₁( p₂⫺q₁)_F VPV (b) _共u兲.

The common minus sign is due to the extra quark loop in the resonance diagrams as compared to the box diagram. This extra sign is very important, as will be seen later on, since the extra sign leads to a constructive interference of reso-nance and box graph contributions. The loop momenta direc-tions come from explicit calculation of the S-matrix elements and are shown in Fig. 3. The vector (D*) and pseudoscalar

共D兲 meson propagators are given by

D_D * ␣␤_共p兲⫽⫺g␣␤⫹p␣p␤/mD_* 2 m_D * 2 ⫺p2 , i 2_D D共p兲⫽ ⫺1 m_D2⫺p2.

For the form factors one obtains

F_VPPP共s,t兲⫽g_J/␺g_D2g_␲

冕

d␴_䊐•exp关⫺w•z_䊐共s,t兲兴 •w2_关m c共1⫹d2兲⫺mqd2兴, F_PVP(1a)共t兲⫽g_␲gD_*gD

冕

d␴⌬•exp关⫺wz⌬共t兲兴 •兵w关⫺m_qm_c⫺m_q2b₁⫹m_␲2b₁2共b₁⫹b₂⫺1兲 ⫺mD 2 b₁2b2⫹tb2共b1b2⫹b1 2_⫺1⫹b 2兲兴 ⫺共1⫺v兲共1⫹3b1兲其, F_VVP(a) 共t兲⫽gJ/␺gD_*gD

冕

d␴⌬•exp关⫺wz⌬共t兲兴 •兵w关mc共b2⫺1兲⫺mqb2兴其, FPPV (1b)_共u兲⫽g ␲gD*gD

冕

d␴⌬•exp关⫺wz⌬共u兲兴 •兵w关mqmc⫹mq 2 共1⫺b1⫺b2兲 ⫹m_␲2_共⫺2b 1b2⫹b1b2 2_⫹b 1⫹2b1 2_b 2⫺2b1 2 ⫹b1 3_兲⫹m D 2_共⫺2b 1b2⫹2b1b2 2_⫹b 1 2_b 2⫹b2 ⫺2b2 2_⫹b 2 3_兲⫹u共2b 1b2⫺b1b2 2_⫺b 1 2 b2兲兴 ⫹4⫹3b1v⫺3b1⫹3b2v⫺3b2⫺4v其,

F_VPV(b) 共u兲⫽gJ/␺gDgD_*

冕

d␴⌬•exp关⫺wz⌬共u兲兴 •兵w关⫺mc⫹b2共mc⫺mq兲兴其.

The integration measures are defined by

冕

d␴_䊐⫽ 3 4␲2

冕

0 1 dv

冉

v 1⫺v

冊

3

冕

d4␣␦

冉

1⫺

兺

i⫽1 4 ␣i

冊

,

冕

d␴_⌬⫽ 3 4␲2

冕

0 1 dv

冉

v 1⫺v

冊

2

冕

d3␣␦

冉

1⫺

兺

i_⫽1 3 ␣i

冊

. 共24兲

The coefficients bi共triangle diagrams兲 and di共box diagrams兲

are given by b₁⫽v␣₃⫹共1⫺v兲共s˜₁c₁₃1⫹s˜₃c₂₃2 兲, b2⫽v␣2⫹共1⫺v兲共s˜2c12 1_⫹s˜ 3c23 3 _兲, d1⫽v␣4⫹共1⫺v兲共s˜1c14 1_⫹s˜ 4c34 3 _兲, d2⫽⫺v共␣2⫹␣3兲⫺共1⫺v兲共s˜2c12 1 _⫹s˜ 3⫹s˜4c34 4 _兲,

1_{Note that the symbol ␧}␣␤pq_⬅␧␣␤␮␯_p

␮q␯, where ␧␣␤␮␯ is the

Levi-Civita` tensor. Moreover, the order of indices V and P of the form factors should be read looking at the Fig. 3 and following the arrows backward in the loops.

d₃⫽⫺v␣₃⫺共1⫺v兲共s˜₃c2₂₃⫹s˜₃⫹s˜₄c₃₄4 兲.

The values of z_⌬, z_䊐, and w are defined by the relevant diagrams共size parameters and quark and hadron masses兲.

B. The channel JÕ␺¿␲¿\D*¿¿D0

The invariant matrix element can be written as

M共VPPV兲⫽M_䊐共VPPV兲⫹Mres共VPPV兲, 共25兲 with M_䊐共VPPV兲⫽⑀_␮ 1共p1兲⑀␮2共q1兲•关p2 ␮1_p 1 ␮2_F VPPV (1) _共s,t兲 ⫹q2 ␮1_p 1 ␮2_F VPPV (2) _{共s,t兲⫹p} 2 ␮1_p 2 ␮2_F VPPV (3) _共s,t兲 ⫹q2 ␮1_p 2 ␮2_F VPPV (4) _{共s,t兲⫹g␮1␮2} F_VPPV(5) 共s,t兲兴, Mres共VPPV兲⫽⑀␮₁共p1兲⑀␮₂共q1兲关⫺M_⌬a␣ 共PVP兲DD_* ␣␤ ⫻共p2⫺q2兲M_⌬a ␮1␤␮2共VVV兲 ⫺M_⌬b␣␮2_共PVV兲D D_* ␣␤_共p 2⫺q1兲M_⌬,b ␮1␤_共VPV兲 ⫹M_⌬c␮2_共PVP兲D D共p2⫺q1兲M_⌬,c ␮1_{共VPP兲兴.}

The expressions for the form factors are given in the Appen-dix.

C. The channel JÕ␺¿␲¿\D*¿¿D*0

The amplitude for the process J/␺⫹␲⫹→D*⫹⫹D*0

can be written as M共VVPV兲⫽M_䊐共VVPV兲⫹M_res共VVPV兲, 共26兲 where M_䊐共VVPV兲⫽⑀_␮ 1共p1兲⑀␮2共q2兲⑀␮3共q1兲关g ␮2␮3␧p1p2q2␮1_F VVPV (1) _{共s,t兲⫹g}␮₁␮_3␧p₁p₂q₂␮_2F VVPV (2) _共s,t兲 ⫹g␮1␮2␧p1p2q2␮3_F VVPV (3) _{共s,t兲⫹p} 1 ␮3␧p₁p₂␮₁␮_2F VVPV (4) _{共s,t兲⫹p} 1 ␮2␧p₁p₂␮₁␮_3F VVPV (5) _共s,t兲 ⫹p2 ␮3_␧p₁p₂␮₁␮_2F VVPV (6) _{共s,t兲⫹p} 2 ␮2_␧p₁p₂␮₁␮_3F VVPV (7) _{共s,t兲⫹p} 2 ␮1_␧p₁p₂␮₂␮_3F VVPV (8) _{共s,t兲⫹q} 2 ␮1_␧p₁p₂␮₂␮_3F VVPV (9) _共s,t兲 ⫹p2 ␮2␧p₁q₂␮₁␮_3F VVPV (10) 共s,t兲⫹p2 ␮3␧p₁q₂␮₁␮_2F VVPV (11) 共s,t兲⫹p1 ␮3␧p₂q₂␮₁␮_2F VVPV (12) 共s,t兲⫹p1 ␮2␧p₂q₂␮₁␮_3F VVPV (13) 共s,t兲 ⫹p₂␮1␧p₂q₂␮₂␮_3F VVPV (14) _{共s,t兲⫹q} 2 ␮1␧p₂q₂␮₂␮_3F VVPV (15) _{共s,t兲⫹␧}p₁␮₁␮₂␮_3F VVPV (16) _{共s,t兲⫹␧}p₂␮₁␮₂␮_3F VVPV (17) _{共s,t兲兴.}

FIG. 4. Dependence of F_␲D(1a)_*D(t) 共solid line兲 and F_J/(a)_␺D*D(t) 共dashed line兲 on the invariant variable t, which is the D* momen-tum squared.

FIG. 5. Comparison of our共solid line兲 form factors FDJ/␺D (1a)

(t) and F_J/(a)_␺D*D(t) with those共dashed line兲 obtained in 关35兴.

When taking into account parity invariance one knows from helicity counting that there are only 14 independent amplitudes in the共VVPV兲 case. The above set of 17 ampli-tudes are in fact not independent. They can be reduced to a set of 14 independent amplitudes by making use of the Schouten identity 共see, e.g., 关32兴兲

g␮␮1␧␮2␮3␮4␮5⫹cycl共␮

1,␮2,␮3,␮4,␮5兲⫽0.

We have made use of the Schouten identity as an additional check on our numerical calculations.

Moreover, the resonance term M_res(VVPV) and all the structures involved are reported in the following expressions:

Mres共VVPV兲⫽⑀␮₁共p1兲⑀␮₂共q2兲⑀␮₃共q1兲关⫺M_⌬a␣␮2共PVV兲 ⫻DD* ␣␤_共p 2⫺q2兲M_⌬a␮1␤␮3共VVV兲 ⫺M_⌬b␣␮3_共PVV兲D D_* ␣␤_共p 2⫺q1兲 ⫻M_⌬,b␮1␤␮2共VVV兲⫹M ⌬c ␮2共PPV兲 ⫻DD共p2⫺q2兲M_⌬,c␮1␮3共VPV兲 ⫹M_⌬,d␮3_共PVP兲D D共p2⫺q1兲M_⌬,d ␮1␮2_{共VVP兲兴,} M_⌬a␣␮2共PVV兲⫽␧p₂( p₂⫺q₂)␮₂␣_F PVV (a) _共t兲,

M_⌬a␮1␤␮3共VVV兲⫽共p 2⫺q2兲␮1p1␤p1 ␮3_F VVV (1a)_共t兲 ⫹共p2⫺q2兲␮1共p2⫺q2兲␤p1 ␮3_F VVV (2a)_共t兲 ⫹g␮1␮3_p 1 ␤_F VVV (3a)_共t兲⫹g␮₁␮₃ ⫻共p2⫺q2兲␤FVVV (4a)_{共t兲⫹g␮1␤} p₁␮3_F VVV (5a)_共t兲 ⫹g␮3␤共p 2⫺q2兲␮1FVVV (6a)_共t兲, M_⌬b␣␮3_{共PVV兲⫽␧}p₂( p₂⫺q₁)␮₃␣_F PVV (b) 共u兲, M_⌬b␮1␤␮2_{共VVV兲⫽共p} 2⫺q1兲␮1p1␤p1 ␮2_F VVV (1b) 共u兲 ⫹共p2⫺q1兲␮1共p2⫺q1兲␤p1 ␮2_F VVV (2b) 共u兲 ⫹g␮1␮2_p 1 ␤_F VVV (3b)_共u兲⫹g␮₁␮_2共p 2⫺q1兲␤ ⫻FVVV (4b)_{共u兲⫹g␮1␤} p₁␮2_F VVV (5b)_共u兲 ⫹g␮2␤共p 2⫺q1兲␮1FVVV (6b)_共u兲, M_⌬c␮2共PPV兲⫽p 2 ␮2_F PPV (c) _共t兲, M_⌬,c␮1␮3_{共VPV兲⫽␧}p₁q₁␮₁␮_3F VPV (c) 共t兲,

FIG. 7. Dependence of the form factor FJ/_␺D¯␲D(s,t) on s at

t⫽0.

FIG. 8. Contributions of the box 共dotted curve兲 and resonance 共dashed curve兲 diagrams to total 共solid curve兲 cross section for the process J/␺⫹␲⫹→D⫹⫹D0_.

FIG. 9. The cross sections of the processes J/␺⫹␲⫹→D⫹⫹D0_{, J/␺⫹␲}⫹_→D*⫹_⫹D0_{, and J/␺⫹␲}⫹_→D*⫹_⫹D*0_{. All three curves} are shown together in the last plot.

M_⌬,d␮3 _{共PVP兲⫽p} 2 ␮3_F PVP (d) 共u兲, M_⌬,d␮1␮2_{共VVP兲⫽␧}p₁q₂␮₁␮_2F VVP (d) _共u兲.

All the expressions for the form factors are reported in the Appendix.

Note that the J/␺ dissociation amplitudes are not equal to zero when contracted with the four-momentum of the J/␺ except in Eq. 共22兲 which involves the Levi-Civita` tensor. In our approach we consider the J/␺ and other vector mesons as bound states of constituent quarks and not as gauge fields.

V. NUMERICAL RESULTS AND DISCUSSION

Some comments about and comparisons of the t depen-dence of the form factors are in order. The behavior of

F_PVP(1a)(t)⫽F_␲D(1a)_*D(t) 共in the literature, for t⫽m_D2_*, it is called gD_*D␲) and FVVP

(a)

(t)⫽F_J/(a)_␺D*D(t) in the kinematical region is shown in Figs. 4 and 5. In order to be able to compare with other calculations we quote the value of

F_␲D *D (1a) (m_D * 2

)(⬅gD_*D␲), which is equal to 22. This value is

about 2␴ larger than the recent experimental results from CLEO, gD*D␲⫽17.9⫾0.3⫾1.9 关33兴. A very small value for gD_*D␲ was predicted by the light cone QCD sum rules

approach关34兴. For the F_J/␺D

*␲

(a)

(t) form factor, we cannot go on the mass shell due to the presence of an anomalous threshold.

The dependence of FJ/␺D␲¯ D(s,t) on t for different values

of

冑

s is shown in Fig. 6. One can see that the t behavior is

rather flat. Moreover, the dependence of F_{J/␺D␲¯ D}(s,t) on

冑

s

at t⫽0 is shown in Fig. 7.

In Fig. 8 we separately plot the contributions coming from the box and resonance diagrams for the process J/␺⫹␲⫹ →D⫹_⫹D0_{. We display the dependence of the cross sections}

on the variable

冑

s for each channel in Fig. 9. The total cross

section is a sum over all channels,

␴tot共s兲⫽␴D⫹D0共s兲⫹␴D*⫹D0共s兲⫹␴D⫹D*0共s兲 ⫹␴D_*⫹D_*0共s兲, 共27兲

which are plotted in Fig. 9 Note that␴D⫹D_*0⫽␴_D*⫹D0. We

plot ␴tot(s) as a function of

冑

s in Fig. 10. One can see that

the maximum is about 2.3 mb at

冑

s⬇4.1 GeV. This is close

to the result obtained in 关4–6兴.

ACKNOWLEDGMENT

We thank David Blaschke and Craig Roberts for useful discussions. M.A.I. gratefully acknowledges the hospitality and support of Mainz University during a visit in which some of this work was conducted. This work was supported in part by a DFG Grant, the Russian Fund of Basic Research Grant No. 01-02-17200, and the Heisenberg-Landau Program.

APPENDIX: EXPRESSIONS FOR FORM FACTORS

Here we report the analytical expressions for the form factors involved in the calculation of the amplitudes

J/␺⫹␲⫹→D*⫹⫹D0 _{and J/␺⫹␲}⫹_→D*⫹_⫹D*0_, respectively. 1. JÕ␺¿␲¿\D*¿¿D0 F_VPPV(i) 共s,t兲⫽gJ/␺gDgD_*g␲

冕

d␴_䊐•exp关⫺w•z_䊐共s,t兲兴•NVPPV (i) _, NVPPV (1) _⫽⫺m qmcw2⫹mq 2 w2共2d1d3⫺2d2d3⫺d3兲⫹mc 2 w2共⫺1⫹d1⫺d2兲⫹mJ/␺ 2 w2共⫺4d1d2d3⫺2d1d2d3 2_⫺2d 1d2⫺2d1d2 2 d3 ⫺d1d2 2_⫺2d 1d3⫺d1d3 2_⫺d 1⫹4d1 2_d 2d3⫹2d1 2_d 2⫹4d1 2_d 3⫹2d1 2_d 3 2_⫹2d 1 2_⫺2d 1 3_d 3⫺d1 3_兲⫹m D 2_w2_共⫺6d 1d2d3⫹2d1d2d3 2 ⫺2d1d2⫺4d1d2 2_d 3⫺2d1d2 2_⫺2d 1d3⫹2d1d3 2_⫹2d 1 2_d 2d3⫹d1 2_d 2⫹2d1 2_d 3⫹2d2d3⫺3d2d3 2_⫹d 2⫹4d2 2_d 3⫺2d2 2_d 3 2_⫹2d 2 2 ⫹2d2 3_d 3⫹d2 3_⫺d 3 2_兲⫹m D 2_w2_共⫺2d 1d3 3_⫹d 1 2_d 3⫹2d1 2_d 3 2_⫺2d 2d3⫺2d2d3 2_⫹2d 2d3 3_⫺d 2 2_d 3⫺2d2 2_d 3 2_⫺d 3⫺d3 2_⫹d 3 3_兲 ⫹uw2_共3d 1d2d3⫹2d1d2⫹2d1d2 2_d 3⫹d1d2 2_⫹d 1d3⫺2d1 2_d 2d3⫺d1 2_d 2⫺2d1 2_d 3兲⫹sw2共d1d2d3⫹2d1d2d3 2_⫹d 1d3 ⫹d1d3 2_⫺d 1 2_d 3⫺2d1 2_d 3 2_兲⫹tw2_共⫺d 1d2d3⫺2d1d2d3 2_⫺d 1d3⫺2d1d3 2_⫹2d 2d3⫹3d2d3 2_⫹d 2 2_d 3⫹2d2 2_d 3 2_⫹d 3⫹d3 2_兲 ⫹w共⫺3⫺8vd1d3⫺3vd1⫹8vd2d3⫹3vd2⫹5vd3⫹3v⫹8d1d3⫹3d1⫺8d2d3⫺3d2⫺5d3兲,

FIG. 10. The total cross section共dotted line兲 together with the contributions already plotted in Fig. 9.

NVPPV (2) _⫽⫹m q 2 w2共⫺2d1d2⫹d2⫹2d2 2_兲⫹m J/␺ 2 w2共d1d2d3⫹2d1d2 2 d3⫹3d1d2 2_⫹2d 1d2 3_⫺2d 1 2 d2d3⫺3d1 2 d2⫺4d1 2 d2 2_⫹2d 1 3 d2兲 ⫹mD 2_w2_共⫺2d 1d2d3⫹2d1d2⫺2d1d2 2_d 3⫹7d1d2 2_⫹4d 1d2 3_⫺2d 1 2_d 2⫺2d1 2_d 2 2_⫹d 2d3⫺d2⫹3d2 2_d 3⫺4d2 2_⫹2d 2 3_d 3⫺5d2 3 ⫺2d2 4_兲⫹m ␲ 2_w2_共⫺d 1d2d3⫹2d1d2d3 2_⫺2d 1d2⫺2d1 2_d 2d3⫹3d2d3⫺d2d3 2_⫹3d 2 2_d 3⫺2d2 2_d 3 2_⫹2d 2 3_d 3⫹d3兲 ⫹uw2_共⫺d 1d2⫺3d1d2 2_⫺2d 1d2 3_⫹2d 1 2_d 2⫹2d1 2_d 2 2_兲⫹sw2_共⫺d 1d2d3⫹d1d2⫺2d1d2 2_d 3⫹2d1 2_d 2d3兲⫹tw 2_共2d 1d2d3 ⫹d1d2⫹2d1d2 2_d 3⫺d2d3⫺3d2 2_d 3⫺2d2 3_d 3兲⫹w共1⫹8vd1d2⫹2vd1⫺7vd2⫺8vd2 2_⫺v⫺8d 1d2⫺2d1⫹7d2⫹8d2 2_兲, N_VPPV(3) ⫽⫹mqmcw2共2d3兲⫹mq 2 w2共⫺2d2d3⫹2d3 2_兲⫹m c 2 w2共⫺1⫺d2兲⫹mJ/␺ 2 w2共⫺d1d2d3⫺d1d2⫺2d1d2 2 d3⫺d1d2 2_⫹d 1d3 ⫹4d1d3 2_⫹2d 1d3 3_⫺d 1⫹2d1 2 d2d3⫹d1 2 d2⫺2d1 2 d3⫺2d1 2 d₃2⫹d₁2兲⫹m2_Dw2共2d1d2d3 2_⫺d 1d2⫺2d1d2 2 d3⫺d1d2 2_⫹d 1d3 ⫹2d1d3 2_⫺d 2d3⫺4d2d3 2_⫹2d 2d3 3_⫹d 2 2 d3⫺4d2 2 d₃2⫹d₂2⫹2d₂3d3⫹d2 3_⫹2d 3 3_兲⫹m ␲ 2 w2共⫺d1d2d3⫺2d1d2d3 2_⫹2d 1d3 2 ⫹2d1d3 3_⫺d 2d3⫹d2d3 2_⫹4d 2d3 3_⫺d 2 2 d3⫺2d2 2 d₃2⫹d₃2⫺2d4₃兲⫹uw2共⫺2d1d2d3 2_⫹d 1d2⫹2d1d2 2 d3⫹d1d2 2_⫺d 1d3 ⫺2d1d3 2_兲⫹sw2_共d 1d2d3⫹2d1d2d3 2_⫺2d 1d3 2_⫺2d 1d3 3_兲⫹tw2_共d 2d3⫺2d2d3 3_⫹d 2 2 d3⫹2d2 2 d₃2⫺2d₃2⫺2d₃3兲⫹w共⫺1 ⫹8vd2d3⫹3vd2⫺4vd3⫺8vd3 2_⫹v⫺8d 2d3⫺3d2⫹4d3⫹8d3 2_兲, N_VPPV(4) ⫽⫹mqmcw2共⫺2d2兲⫹mq 2_w2_共⫺2d 2d3⫹2d2 2_兲⫹m c 2_w2_共1⫹d 2兲⫹mJ/␺ 2 _w2_共⫺3d 1d2d3⫺2d1d2d3 2_⫺d 1d2⫹d1d2 2_⫹2d 1d2 3 ⫹d1d3⫹d1⫹2d1 2_d 2d3⫹d1 2_d 2⫺2d1 2_d 2 2_⫺d 1 2_兲⫹m D 2_w2_共⫺2d 1d2d3⫺2d1d2 2_d 3⫹d1d2 2_⫹2d 1d2 3_⫹d 2d3⫺2d2d3 2_⫹5d 2 2_d 3 ⫺2d2 2_d 3 2_⫺d 2 2_⫹4d 2 3_d 3⫺3d2 3_⫺2d 2 4_兲⫹m ␲ 2_w2_共⫺d 1d2d3⫺2d1d2d3 2_⫺d 1d2⫹2d1d2 2_d 3⫹d1d3⫹d2d3⫺d2d3 2_⫹2d 2d3 3 ⫹d2 2 d3⫺4d2 2 d₃2⫹2d₂3d3⫺d3 2_兲⫹uw2_共2d 1d2d3⫹2d1d2 2 d3⫺d1d2 2_⫺2d 1d2 3_兲⫹sw2_共d 1d2d3⫹2d1d2d3 2_⫹d 1d2 ⫺2d1d2 2 d3⫺d1d3兲⫹tw2共d2d3⫹2d2d3 2_⫺d 2 2 d3⫹2d2 2 d₃2⫺2d₂3d3兲⫹w共1⫹8vd2d3⫺vd2⫺8vd2 2_⫹2vd 3⫺v⫺8d2d3 ⫹d2⫹8d2 2_⫺2d 3兲, N_VPPV(5) ⫽⫹mqmcmJ/␺ 2 _w2_{共⫺1/2兲⫹m} qmcm_␲2w2共⫺1/2兲⫹mqmcsw2共1/2兲⫹mq 2_m c 2_w2_共1兲⫹m q 2_m J/␺ 2 _w2_共d 1d2⫹d1d3⫹d1⫺d1 2 ⫺1/2d2⫺1/2d3兲⫹mq 2_m D 2_w2_共d 1d2⫹d2d3⫺1/2d2⫺d2 2_兲⫹m q 2_m ␲ 2_w2_共d 1d3⫹d2d3⫺1/2d3⫺d3 2_兲⫹m q 2_uw2_共⫺d 1d2 ⫹1/2d2兲⫹mq 2_sw2_共⫺d 1d3⫹1/2d3兲⫹mq 2_tw2_共⫺d 2d3兲⫹mq 2_{w共1⫺v兲⫹m} c 2_m J/␺ 2 _w2_共d 1d2⫹d1d3⫹3/2d1⫺d1 2_兲 ⫹mc 2 m_D2w2共⫺1/2⫹d1d2⫹d1⫹d2d3⫺3/2d2⫺d2 2_⫹d 3兲⫹mc 2 m_␲2w2共1/2⫹d1d3⫹1/2d1⫹d2d3⫹1/2d2⫺d3 2 兲⫹mc 2 uw2 ⫻共⫺d1d2⫺d1兲⫹mc 2 sw2共⫺d1d3⫺1/2d1兲⫹mc 2 tw2共⫺1/2⫺d2d3⫺1/2d2⫺d3兲⫹mc 2 w共2⫺2v兲⫹m_J/2_␺m_D2w2共d1d2d3 ⫹2d1d2d3 2_⫺4d 1d2⫺5d1d2 2_⫺2d 1d2 3_⫹d 1d3 2_⫺1/2d 1⫹11/2d1 2 d2⫹4d1 2 d₂2⫹3/2d₁2⫺2d₁3d2⫺d1 3_⫺1/2d 2d3 2_⫹d 2⫹3/2d2 2 ⫹1/2d2 3_⫺1/2d 3 2_兲⫹m J/␺ 2 m_␲2w2共2d1d2d3⫹2d1d2 2 d3⫹1/2d1d2 2_⫺7/2d 1d3 2_⫺2d 1d3 3_⫹1/2d 1⫹7/2d1 2 d3⫹4d1 2 d₃2⫺2d₁3d3 ⫺1/2d1 3_⫺d 2d3⫺1/2d2 2 d3⫺d3⫹1/2d3 3_兲⫹m J/␺ 2 uw2共d1d2d3⫹3/2d1d2⫹d1d2 2_⫹1/2d 1d3⫺2d1 2 d2d3⫺4d1 2 d2⫺2d1 2 d₂2 ⫺d1 2 d3⫺d1 2_⫹2d 1 3 d2⫹d1 3_兲⫹m J/␺ 2 sw2共d1d2d3⫹1/2d1d2⫹3/2d1d3⫹d1d3 2_⫺2d 1 2 d2d3⫺1/2d1 2 d2⫺7/2d1 2 d3⫺2d1 2 d₃2 ⫺1/2d1 2_⫹2d 1 3 d3⫹1/2d1 3_兲⫹m J/␺ 2 tw2共⫺4d1d2d3⫺2d1d2d3 2_⫺1/2d 1d2⫺2d1d2 2 d3⫺1/2d1d2 2_⫺3/2d 1d3⫺d1d3 2_⫺1/2d 1 ⫹2d1 2 d2d3⫹1/2d1 2 d2⫹d1 2 d3⫹1/2d1 2_⫹d 2d3⫹1/2d2d3 2_⫹1/2d 2 2 d3⫹1/2d3⫹1/2d3 2_兲⫹m J/␺ 2 w共⫺3/2⫺5vd1d2⫺5vd1d3 ⫺13/2vd1⫹5vd1 2_⫹3/2vd 2⫹3/2vd3⫹3/2v⫹5d1d2⫹5d1d3⫹13/2d1⫺5d1 2_⫺3/2d 2⫺3/2d3兲⫹mJ/␺ 4 w2共⫺d1d2d3 ⫺d1d2⫺1/2d1d2 2_⫺d 1d3⫺1/2d1d3 2_⫹2d 1 2_d 2d3⫹3d1 2_d 2⫹d1 2_d 2 2_⫹3d 1 2_d 3⫹d1 2_d 3 2_⫹3/2d 1 2_⫺2d 1 3_d 2⫺2d1 3_d 3⫺5/2d1 3_⫹d 1 4_兲 ⫹mD 2_m ␲ 2_w2_共⫺d 1d2d3⫺d1d3⫹2d1 2_d 2d3⫹1/2d1 2_d 2⫹d1 2_d 3⫹5/2d2d3 2_⫺2d 2d3 3_⫹1/2d 2⫺d2 2_d 3⫹4d2 2_d 3 2_⫺2d 2 3_d 3⫺1/2d2 3 ⫺1/2d3⫺d3 3_兲⫹m D 2_uw2_共⫺2d 1d2d3⫹3/2d1d2⫺2d1d2 2_d 3⫹7/2d1d2 2_⫹2d 1d2 3_⫺2d 1 2_d 2⫺2d1 2_d 2 2_⫹1/2d 2d3⫺1/2d2 ⫹1/2d2 2_d 3⫺d2 2_⫺1/2d 2 3_兲⫹m D 2_sw2_共2d 1d2d3⫺2d1d2d3 2_⫹1/2d 1d2⫹2d1d2 2_d 3⫹1/2d1d2 2_⫹d 1d3⫺d1d3 2_⫺2d 1 2_d 2d3

⫺1/2d1 2 d2⫺d1 2 d3⫺1/2d2d3⫹1/2d2d3 2_⫺1/2d 2⫺1/2d2 2 d3⫺1/2d2 2_⫹1/2d 3 2_兲⫹m D 2 tw2共⫺2d1d2d3⫺1/2d1d2⫺2d1d2 2 d3 ⫺1/2d1d2 2_⫹1/2d 2d3⫺2d2d3 2_⫹5/2d 2 2_d 3⫺2d2 2_d 3 2_⫹1/2d 2 2_⫹2d 2 3_d 3⫹1/2d2 3_兲⫹m D 2_{w共⫺1⫺5vd} 1d2⫺2vd1⫺5vd2d3 ⫹4vd2⫹5vd2 2_⫺2vd 3⫹v⫹5d1d2⫹2d1⫹5d2d3⫺4d2⫺5d2 2_⫹2d 3兲⫹mD 4_w2_共2d 1d2d3⫺d1d2⫹2d1d2 2_d 3⫺3d1d2 2 ⫺2d1d2 3_⫹d 1 2_d 2⫹d1 2_d 2 2_⫺d 2d3⫹d2d3 2_⫹1/2d 2⫺3d2 2_d 3⫹d2 2_d 3 2_⫹3/2d 2 2_⫺2d 2 3_d 3⫹2d2 3_⫹d 2 4_兲⫹m ␲ 2_uw2_共2d 1d2d3 2 ⫺2d1d2 2_d 3⫺1/2d1d2 2_⫹1/2d 1d3⫹d1d3 2_⫺2d 1 2_d 2d3⫺1/2d1 2_d 2⫺d1 2_d 3⫹1/2d2d3⫺1/2d2d3 2_⫹1/2d 2 2_d 3⫹1/2d3兲⫹m_␲2sw2 ⫻共⫺d1d2d3⫺2d1d2d3 2_⫹1/2d 1d3⫹3/2d1d3 2_⫹2d 1d3 3_⫺d 1 2_d 3⫺2d1 2_d 3 2_⫹1/2d 2d3⫹1/2d2d3 2_⫹1/2d 3⫺1/2d3 3_兲⫹m ␲ 2_tw2 ⫻共⫺d1d2d3⫺2d1d2d3 2_⫺1/2d 1d3⫺d1d3 2_⫺1/2d 2d3⫹2d2d3 3_⫺d 2 2_d 3⫺2d2 2_d 3 2_⫹1/2d 3⫹d3 2_⫹d 3 3_兲⫹m ␲ 2_w共⫺1/2 ⫺5vd1d3⫺3/2vd1⫺5vd2d3⫺3/2vd2⫹5/2vd3⫹5vd3 2_⫹1/2v⫹5d 1d3⫹3/2d1⫹5d2d3⫹3/2d2⫺5/2d3⫺5d3 2 兲 ⫹m_␲4 w2共d1d2d3⫹2d1d2d3 2_⫺d 1d3 2_⫺2d 1d3 3_⫹1/2d 1 2 d3⫹d1 2 d₃2⫺d2d3 2_⫺2d 2d3 3_⫹1/2d 2 2 d3⫹d2 2 d₃2⫺1/2d3⫺1/2d3 2_⫹1/2d 3 3 ⫹d3 4_兲⫹usw2_共⫺d 1d2d3⫺1/2d1d2⫺1/2d1d3⫹2d1 2 d2d3⫹1/2d1 2 d2⫹d1 2 d3兲⫹utw2共2d1d2d3⫹1/2d1d2⫹2d1d2 2 d3 ⫹1/2d1d2 2_⫺1/2d 2d3⫺1/2d2 2 d3兲⫹uw共1/2⫹5vd1d2⫹2vd1⫺3/2vd2⫺1/2v⫺5d1d2⫺2d1⫹3/2d2兲 ⫹u2_w2_共⫺1/2d 1d2⫺1/2d1d2 2_⫹d 1 2 d2⫹d1 2 d₂2兲⫹stw2共d1d2d3⫹2d1d2d3 2_⫹1/2d 1d3⫹d1d3 2_⫺1/2d 2d3⫺1/2d2d3 2_⫺1/2d 3 ⫺1/2d3 2_{兲⫹sw共1⫹5vd} 1d3⫹3/2vd1⫺3/2vd3⫺v⫺5d1d3⫺3/2d1⫹3/2d3兲⫹s2w2共⫺1/2d1d3⫺1/2d1d3 2_⫹1/2d 1 2 d3 ⫹d1 2 d₃2兲⫹tw共⫺1/2⫹5vd2d3⫹3/2vd2⫹2vd3⫹1/2v⫺5d2d3⫺3/2d2⫺2d3兲⫹t2w2共1/2d2d3⫹d2d3 2_⫹1/2d 2 2 d3 ⫹d2 2 d3 2_{兲⫹3⫺6v⫹3v}2_; F_PVP(1a)共t兲⫽g_␲gD_*gD

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d␴_⌬•exp关⫺wz_⌬共t兲兴•关wmqmc共⫺1兲⫹wmq 2_共⫺b 1兲⫹wm_␲2共b1 2_b 2⫺b1 2_⫹b 1 3_兲⫹wt共b 1b2 2_⫹b 1 2_b 2⫺b2⫹b2 2_兲 ⫹wmD 2_共⫺b 1 2_b 2兲⫹v共1⫹3b1兲⫺1⫺3b1兴, F_PVP(2a)共t兲⫽gJ/␺gD_*gD

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d␴⌬•exp关⫺wz⌬共t兲兴•关wmq 2_共1⫺b 2兲⫹wm_␲ 2_共⫺3b 1b2⫹b1b2 2_⫹b 1⫹b1 2 b2⫺b1 2_⫹b 2⫺b2 2_兲 ⫹wt共⫺b1b2⫹b1b2 2_⫹b 2⫺2b2 2_⫹b 2 3_兲⫹wm D 2_共b 1b2⫺b1b2 2_⫺b 2⫹b2 2_{兲⫹v共⫺2⫹3b} 2兲⫹2⫺3b2兴,

F_PVV(b) 共u兲⫽g_␲g_D2_*

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d␴_⌬•exp关⫺wz_⌬共u兲兴•兵w关mq共b2⫺1兲⫺mcb2兴其,

F_VPV(b) 共u兲⫽g_J/␺g_Dg_D*

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d␴_⌬•exp关⫺wz_⌬共u兲兴•兵w关m_c共b₂⫺1兲⫺m_qb₂兴其,

F_PVP(c) 共u兲⫽g_␲gD_*gD

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d␴_⌬•exp关⫺wz_⌬共u兲兴•兵w关共⫺mcmq⫺mq

2_b 1兲⫹m_␲2b1 2_共b 2⫺1⫹b1兲⫹mD 2_b 2共b1b2⫹b1 2_⫺1⫹b 2兲⫺ub1 2_b 2兴 ⫺共1⫺v兲共1⫹3b1兲其, F_VPP(c) 共u兲⫽gJ/␺gD 2

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d␴_⌬•exp关⫺wz_⌬共u兲兴•兵w关⫺2mcmqb2⫹mc 2_共⫺1⫹b 2兲⫹mJ/␺ 2 b1共b2 2_⫺1⫹b 1b2⫹b1兲⫹mD 2 b₂2共b1⫺1⫹b2兲 ⫺ub1b2 2_{兴⫺共1⫺v兲共1⫹3b} 2兲其. 2. JÕ␺¿␲¿\D*¿¿D*0 F_VVPV(i) 共s,t兲⫽g_J/␺g_D * 2 g_␲

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d␴_䊐•exp关⫺w•z_䊐共s,t兲兴•N_VVPV(i) , N_VVPV(1) ⫽⫺w2mqd2,