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Vertices of the vector mesons with the strange charmed mesons in QCD

R. Khosravi1,* and M. Janbazi2,

1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran2Physics Department, Shiraz University, Shiraz 71454, Iran

(Received 12 August 2012; published 3 January 2013)

We investigate the strong form factors and coupling constants of vertices containing the strange

charmed mesons Ds0, Ds, Ds , and Ds1 with the vector mesons and J=c in the framework of the three

point QCD sum rules. Taking into account the nonperturbative part contributions of the correlation

functions, the condensate terms of dimension 3, 4 and 5 related to the contributions of the quark-quark,

gluon-gluon, and quark-gluon condensate, respectively, are evaluated. The present work can give

considerable information about the hadronic processes involving the strange charmed mesons.

DOI: 10.1103/PhysRevD.87.016003 PACS numbers: 11.55.Hx, 12.38.Lg, 13.75.Lb, 14.40.Lb

I. INTRODUCTION

The strong form factors and coupling constants associ-ated with vertices involving mesons are important for theexplanation of hadronic processes in the strong interaction.They have received wide attention for the new research ofthe nature of the charmed pseudoscalar and axial vectormesons. The strong coupling constants among the charmedmeson such as gDDP, gDDP, gDDV , and gDDV , where Pand V stand for pseudoscalar and vector mesons, respec-tively, play an important role in understanding the finalstate interactions in QCD [1].

The QCD sum rules have been successfully applied to awide variety of problems in hadron physics [2] (for detailsabout this method, see Refs. [3,4]). Some possible verticesinvolving charmed mesons such as DD [5,6], DD [7],DD [8], DDJ=c [9], DDJ=c [10], DDsK, DsDK,D0DsK, Ds0DK [11], D

DP, DDV, DDV [12], DD[13], DDJ=c [14], DsDK, DsDK [15], and DD! [16]have been studied in the framework of the QCD sum rules.It is very important to know the precise functional form ofthe form factors in these vertices and even to know how thisform changes when one or the other (or both) mesons areoff shell [8].

In this work, we consider vertices of vector mesons withthe strange charmed mesons via the three point QCD sumrules (3PSR); i.e., we calculate the strong form factors andcoupling constants associated with the VDs0Ds0, VDsDs,VDsDs , and VDs1Ds1 vertices, where V can be or J=c .In the 3PSR theory, the calculation begins with a threepoint correlation function. The correlation function isinvestigated in two phenomenological and theoreticalsides. The strong form factors are estimated by equatingtwo sides and applying the double Borel transformationswith respect to the momentum of the initial and final statesto suppress the contribution of the higher states and con-tinuum. Calculating the theoretical part of the correlation

function consists of two perturbative and nonperturbativecontributions. The nonperturbative part of the correlationfunction is called the condensate contribution. The con-densate term of dimension 3 is related to the contributionof the quark-quark condensate, and dimension 4 and 5 areconnected to the gluon-gluon and gluon-quark condensate,respectively. The main points in the present work are thecalculation of the quark-quark, gluon-gluon, and gluon-quark corrections; these are the most important correctionsof the nonperturbative part of the correlation function inthe 3PSR method.This paper includes four sections. The calculation of the

sum rules for the strong form factors of the Ds0Ds0,DsDs, D

sD

s , and Ds1Ds1 vertices are presented in

Sec. II. With the necessary changes in the expressionsderived for the strong form factors of the vertices involvingthe meson, such as variations in the type of quarks, wecan easily apply the same calculations for the J=cDs0D

s0,

J=cDsDs, J=cDsD

s , and J=cDs1Ds1 vertices. In

Sec. III, the calculations of the quark-quark, gluon-quark,and gluon-gluon condensate contributions in the Boreltransform scheme are presented. In this section the strongform factors are derived. Section IV presents our numericalanalysis of the strong form factors as well as the couplingconstants.

II. THE THREE POINT QCD SUM RULESMETHOD

We start with the correlation function in 3PSR to calcu-late the strong form factors associated with the Ds0Ds0,DsDs, D

sD

s , and Ds1Ds1 meson vertices when both

strange charmed and mesons can be off shell, in 3PSRwe start with the correlation function. For off-shellcharmed mesons, these correlation functions are given by

D0

p; p0 i2

Zd4xd4yeip0xpyh0jT fjD0 xjD0y0j ygj0i;

(1)*erezakhosravi@cc.iut.ac.irmehdijanbazi@yahoo.com

PHYSICAL REVIEW D 87, 016003 (2013)

1550-7998=2013=87(1)=016003(17) 016003-1 2013 American Physical Society

http://dx.doi.org/10.1103/PhysRevD.87.016003

D00

p; p0 i2

Zd4xd4yeip0xpyh0jT fjD00 xjD00y 0jygj0i;

(2)

where D0 stands for the scalar or pseudoscalar charmedmesons (Ds0, Ds), and D

00 stands for the vector or axialvector charmed mesons (Ds , Ds1). For the off-shell meson, these quantities are

p;p0 i2

Zd4xd4yeip0xpyh0jT fjD0 xj 0jD0yygj0i; (3)

p; p0 i2

Zd4xd4yeip0xpyh0jT fjD00 xj0jD00y ygj0i;

(4)

where jDs0 sUc, jDs s5c, jD

s

sc, jDs1 s5c, and j

ss are interpolating currents of Ds0,

Ds, Ds , Ds1, and mesons, respectively, and have

the same quantum numbers of the associative mesons.Also, T is the time ordering product, p and p0 are the

four-momentum of the initial and final mesons, respec-tively (see Fig. 1).Equations (1)(4) can be calculated in two different

ways: In the physical or phenomenological part, the rep-resentation is in terms of hadronic degrees of freedomwhich is responsible for the introduction of the form fac-tors, decay constants, and masses. In QCD or theoreticalrepresentation, we evaluate the correlation function inquark-gluon language and in terms of QCD degrees offreedom like quark-quark condensate, gluon-gluon con-densate, etc., with the help of the Wilson operator productexpansion (OPE).In order to calculate the phenomenological parts of the

correlation functions in Eqs. (1)(4), three complete sets ofintermediate states with the same quantum number should

be inserted in these equations. For and, we have

h0jjD0 jD0pih0jjD0 jD0p0ihD0pD0p0jq; 00ihq; 00jj j0i

p2 m2D0 p02 m2D0 q2 m2 higher and continuum states; (5)

h0jjD00 jD00p; ih0jjD00 jD00p0; 0ihD00p; D00p0; 0jq; 00ihq; 00jj j0i

p2 m2D00 p02 m2D00 q2 m2 higher and continuum states:

(6)

The same process can be easily used forD0

andD00. The following matrix elements are defined in the standard way

in terms of strong form factors as well as leptonic decay constants of the charmed and mesons as

hD0pD0p0jq; 00i gD0D0 q2p p000q;

hD00p; D00p0; 0jq; 00i igD00D00 q2p0 pg q p0g q pg00 q0D00 p0D00 p;

h0jjD0 jD0pi CD0mD0fD0 ;h0jjD00 jD00p; i mD00fD00p;h0jjjq; 00i mf00q; (7)

where q p0 p, gDD0D0 q2, and gVD00D00 q2 are the strong form factors,mi and fi, i D0,D00 and are the masses anddecay constants of mesons, , 0 and 00 are the polarization vector of the vector mesons. Also, CDs0 1 and CDs

mDsmsmc .

Using Eq. (7) in Eqs. (5) and (6), and after some calculations, we obtain

s s

c

s

s

c

D (D)D (D) D (D)

D (D)

(a) (b)

p p

q q

pp

FIG. 1. Perturbative diagrams for the off-shell (a) and theoff-shell D0D00 (b).

R. KHOSRAVI AND M. JANBAZI PHYSICAL REVIEW D 87, 016003 (2013)

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gD0D0 q2CD02mm2D0ff2D0

p2 m2D0 p02 m2D0 q2 m2p other structures higher and continuum states; (8)

gD00D00 q2mff

2D00 4m2D00 q2

2p2 m2D00 p02 m2D00 q2 m2gq other structures higher and continuum states;

(9)

D0

gD0D0D0 q2CD02m2D0ff2D0 m2D0 m2 q2mp2 m2p02 m2D0 q2 m2D0

p other structures higher and continuum states; (10)

D00

gD00D00D00 q2mff

2D00 3m2D00 m2 q2

2p2 m2p02 m2D00 q2 m2D00 gq other structures higher and continuum states:

(11)

With the help of the OPE in the Euclidean region, where p2, p02 ! 1, we calculate the QCD side of the correlationfunctions containing perturbative and nonperturbative parts. For this aim, the correlation functions for the D0D0 and theD00D00 vertices are written, respectively, as follows:

D0

p2; p02; q2 D0per D0nonperp ; D00

p2; p02; q2 D00per D00nonpergq ; (12)

where denotes other structures and higher states.First, we calculate the perturbative part whose diagrams are shown in Fig. 1.Using the double dispersion relation for each coefficient of the Lorentz structures p and gq appearing in the

correlation functions [Eq. (12)], we get

Mperp2; p02; q2 142

Zds

Zds0

Ms; s0; q2s p2s0 p02 subtraction terms; (13)

where Ms; s0; q2 is the spectral density, and M stands for off-shell mesons D0, D00, and . We calculate the spectraldensities in terms of the usual Feynman integrals with the help of the Cutkosky rules, where the quark propagators arereplaced by Dirac delta functions 1

p2m2 ! 2ip2 m2.

(i) For the p structure related to the D0D0 vertex:

D0

D0D0 Nc20I0 2I0msms mc 2B14msmc 4m2s u 20;D0D0 Nc20I0 4I0mcms mc 2B14msmc 2m2c m2s u;

where 1 for D0 Ds0 and 1 for D0 Ds(ii) For the gq structure related to the D

00D00 vertex:

D00

D00D00 Nc4A 8C1 C2 2B1 B2 I0 0 2m2sB1 B2 2I0msmc 2B1 B2 I0m2s uB1 B2;

D00D00 Nc4A 8C1 C2 2B1 B2 I0 0 2m2sB1 B2

4I0msmc 2B1 B2 2I0m2c uB1 B2;

where 1 forD00 Ds and 1 forD00 Ds1. The explicit expressions of the coefficients in the spectral densitiesentering the sum rules are given as

VERTICES OF THE VECTOR MESONS WITH THE . . . PHYSICAL REVIEW D 87, 016003 (2013)

016003-3

I0s; s0; q2 14

12s; s0; q2 ; a; b; c a

2 b2 c2 2ac 2bc 2ac; sm23 m21;

0 s0 m23 m22; u s s0 q2;B1 I0

s; s0; q2 2s0 0u; B2 I0

s; s0; q2 2s0 u;

A I02s; s0; q2 4ss

0m23 s02 s02 u2m23 u0;

C1 I022s; s0; q2 8s

02m23s 2s0m23u2 4um230ss0 u3m230 2s023 3s0u20

202ss0 02u2 us3;C2 I0

22s; s0; q2 8s2m23

0s0 2s203 4um23ss0 220ss0 3us02 2sm23u2

s0u3 u3m23 20u2;

where Nc 3 is the color factor. It should be noted that inthese coefficients, m1 m2 ms, m3 mc for D0D0 ,D00D00 , and m1 m3 ms, m2 mc for D

0D0D0 ,

D00D00D00

(see Fig. 1).

III. CONDENSATE CONTRIBUTIONS

In this section, the nonperturbative part contributions ofthe correlation function are discussed [Eq. (12)]. In QCD,the three point correlation function can be evaluated by theOPE in the deep Euclidean region. For this aim, we expandthe time ordered products of currents contained in the threepoint correlation function in terms of a series of localoperators with increasing dimension. Taking into accountthe vacuum expectation value of OPE, the expansion of thecorrelation function in terms of local operators is written asfollows:

perp2; p02; q2 C0;nonperp2; p02; q2 C3h qqi C4hGaGai

C5h qTaGaqi C6h qq q0qi ; (14)

where Ci are the Wilson coefficients,Ga is the gluon field

strength tensor, and and 0 are the matrices appearing inthe calculations. In Eq. (14), C0 is related to the contribu-tion of the perturbative part of the correlation function, andthe rest of the terms are related to the nonperturbativecontributions of it. The perturbative part contribution ofthe correlation function was discussed before. For thecalculation of the nonperturbative contributions (conden-sate terms), we consider these points:

(a) The condensate terms of dimension 3, 4, and 5 arerelated to the contributions of the quark-quark,gluon-gluon, and quark-gluon condensate, respec-tively, and are more important than the other termsin OPE.

(b) In 3PSR, when the light quark is a spectator, thegluon-gluon condensate contributions can be easilyignored (see Fig. 2) [17].

(c) When the heavy quark is a spectator, the quark-quark condensate contributions are suppressed bythe inverse of the heavy quark mass and can besafely omitted (see Fig. 3) [17].

(d) The quark condensate contributions of the lightquark, which is a nonspectator, are zero after apply-ing the double Borel transformation with respect toboth variables p2 and p02, because only one variableappears in the denominator.

Therefore, only three important diagrams of dimension3 and 5 remain from the nonperturbative part contributionswhen the charmed mesons are off shell. These diagramsnamed quark-quark and quark-gluon condensate are

ss

s s s ss

c cc

s

s

FIG. 2. Nonperturbative diagrams for the off-shell charmedmesons in theDs0Ds0, DsDs, DsDs , and Ds1Ds1 vertices.

c c

cc

ss

s ss

c

c

s s ss

ss s

FIG. 3. Nonperturbative diagrams for the off-shell meson inconsidering vertices.

R. KHOSRAVI AND M. JANBAZI PHYSICAL REVIEW D 87, 016003 (2013)

016003-4

shown in Fig. 2. When is an off-shell meson, the impor-tant diagrams are the six diagrams of dimension 3. Figure 3shows these diagrams related to gluon-gluon condensate.

After some straightforward calculations and applyingthe double Borel transformations with respect to thep2p2 ! M21 and p02p02 ! M21 as

Bp2M21

1

p2 m2sm 1

m

mem2s

M21

M21m;

Bp02M22

1

p02 m2cn 1

n

nem2c

M22

M22n;

(15)

where M21 and M22 are the Borel parameters, respectively,

the following results are obtained for the quark-quark andquark-gluon contributions (Fig. 2):

D0D00

nonper hssiM21M222CD

0D00

12: (16)

The explicit expressions for CD0

D0D0 and CD00D00D00 associated

with the D0D0 and D00D00 vertices are given inAppendix A.

To obtain the gluon condensate contributions (Fig. 3),we will follow the same procedure as stated in Ref. [18].The calculations of the gluon condensate diagrams areperformed in the Fock-Schwinger fixed-point gaugexAa 0, where Aa is the gluon field. In the calculationof these diagrams, the following type of integrals appears:

I12...na;b;c

Z d4k24

k12...nk2m2capk2m2sbp0 k2m2sc

;

(17)

where k is the momentum of the spectator quark c. Theseintegrals can be calculated using the Schwinger represen-tation for the Euclidean propagator, i.e.,

1

p2 m2 1

Z 10

dn1ep2m2: (18)

After the Borel transformation using

Bp2M2ep2 1=M2 ; (19)we obtain

I0a; b; c 1abc

162abc M212abM222acU0a b c 4; 1 c b;

Ia; b; c I1a; b; cp I2a; b; cp0;Ia; b; c I3a; b; cpp I4a; b; cpp0 p0p I5a; b; cp0p0 I6a; b; cg;Ip; p0 I7a; b; cgp gp gp I8a; b; cgp0 gp0 gp0 ;

(20)

where I12...n in the above equations represents the double Borel transformed form of integrals as

Bp2M21Bp02M22p2mp02nI12...n M21mM22ndm

dM21mdn

dM22nM21mM22nI12...n;

also Il (l 1; . . . ; 8) are defined as

I ka; b; c i 1abc1

162abc M211abkM224ackU0a b c 5; 1 c b;

Ima; b; c i 1abc1

162abc M21ab1mM227acmU0a b c 5; 1 c b;

I6a; b; c i 1abc1

322abc M213abM223acU0a b c 6; 2 c b;

Ina; b; c i 1abc

322abc M214abnM2211acnU0a b c 7; 2 c b;

where k 1, 2, m 3, 4, 5 and n 7, 8. We can define the function U0; as

VERTICES OF THE VECTOR MESONS WITH THE . . . PHYSICAL REVIEW D 87, 016003 (2013)

016003-5

U0a;bZ 10dyyM21M22aybexp

B1

yB0B1y

;

(21)

where

B1 1M22M

21

m2sM21 M222 M22M21Q2;

B0 1M21M

22

m2s m2cM21 M22; B1 m2c

M21M22

:

(22)

After straightforward but lengthy calculations, weget the following results for the gluon condensatecontributions:

nonper is

G2C

6; (23)

where the explicit expressions for CD0D0 and C

D00D00 are

given in Appendix B.Th...

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