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

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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).

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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

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  • 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)

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  • 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

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  • 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.The QCD sum rules for the strong form factors are

    obtained by equating two representations of the correlationfunction and applying the Borel transformations withrespect to the p2p2 ! M21 and p02p02 ! M21 on thephenomenological as well as the perturbative and non-perturbative parts of the correlation function in order tosuppress the contributions of the higher states and contin-uum.We obtain the equations for the strong form factors asfollows:

    gD0

    D0D0 q2 mq2 m2D0

    CD0m2D0ff2D0 m2D0 m2 q2e

    m2

    M21e

    m2

    D0M22

    142

    Z sD00

    mcms2ds0

    Z s0

    2m2s

    dsD0

    D0D0 s; s0; q2e s

    M21e

    s0M22 hssi

    M21M22

    CD0

    D0D0

    12

    ; (24)

    gD00

    D00D00 q2 2q2 m2D00

    mff2D00 3m2D00 m2 q2

    e

    m2

    M21e

    m2

    D00M22

    1

    42

    Z sD000mcms2

    ds0Z s

    0

    2m2s

    dsD00

    D00D00 s; s0; q2e s

    M21e

    s0M22 hssi

    M21M22

    CD00

    D00D00

    12

    ; (25)

    gD0D0 q2

    q2 m2CD0mm2D0ff2D0

    e

    m2

    D0M21 e

    m2

    D0M22

    142

    Z sD00

    mcms2ds0

    Z sD00

    mcms2ds

    D0D0 s; s0; q2e s

    M21e

    s0M22 iM21M22

    s

    G2C

    D0D0

    6

    ; (26)

    gD00D00 q2 2q2 m2

    mff2D00 4m2D00 q2

    e

    m2

    D00M21 e

    m2

    D00M22

    1

    42

    Z sD000

    mcms2ds0

    Z sD000

    mcms2dsD00D00 s; s0; q2e

    sM21e

    s0M22 iM21M22

    s

    G2C

    D00D00

    6

    ; (27)

    where s0 and sD0D000 are the continuum thresholds in the

    and D0D00 mesons, respectively.

    IV. NUMERICAL ANALYSIS

    In this section, we analyze the strong form factorsand coupling constants for the VDs0D

    s0, VDsDs, VD

    sD

    s ,

    andVDs1Ds1, (V , J=c ) vertices.We choose the valuesof meson masses as m 1:680 GeV, mJ=c3:097GeV,mD

    s0 2:317 GeV, mDs 1:968 GeV, mDs 2:112 GeV,

    mDs1 2:459 GeV [19], hssi 0:8 0:2hu ui, and

    hu ui hd di 0:240 0:010 GeV3 for which wechoose the value of the condensates at a fixed renormaliza-tion scale of about 1 GeV [20]. Also, the leptonic decayconstants used in the calculation of the QCD sum rule forthese vertices are presented in Table I. For a comprehensiveanalysis of the strong form factors and coupling constants,we use the following values of the quark masses mc and msin three sets presented in Table II.The expressions for the strong form factors in

    Eqs. (24)(27) contain also four auxiliary parameters:Borel mass parameters M1 and M2, and continuum

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  • thresholds s0 and sD0D000 . These are mathematical objects,

    so the physical quantities, i.e., strong form factors andcoupling constants, should be independent of them. Thevalues of the continuum thresholds are taken to be

    s0 m 2 and sD0D00

    0 mD0D00 2. We use0:4 GeV 1:0 GeV, where Q2 q2 [25]. Theworking regions for M21 and M

    22 are determined by requir-

    ing that the contributions of the higher states and contin-uum be effectively suppressed, and therefore it guaranteesthat the contributions of higher dimensional operators aresmall. In this work, we use the following relation betweenthe Borel masses M1 and M2 [8,9]:

    M21M22

    m2i

    m2o; (28)

    where mi and mo are the masses of the incoming andoutgoing meson, respectively. According to this relationbetween the M1 and M2 [Eq. (28)], we will have only oneindependent Borel mass parameter M. We found goodstability of the sum rule in the interval 14 GeV2 M2 22 GeV2 for all vertices. The dependence of the strongform factors gD

    s0D

    s0, gDsDs , gDsDs , and gDs1Ds1 on

    Borel mass parameters M2 in Q2 1 GeV2 consideringset II and 0:8 GeV is shown in Fig. 4.

    We calculated theQ2 dependence of the strong couplingform factors in the region where the sum rule is valid. Soto extend our results to the full region, we look for

    parametrization of the form factors in such a way that inthe validity region of 3PSR, this parametrization coincideswith the sum rules prediction. For off-shell charmed me-sons, our numerical calculations show that the sufficientparametrization of the form factors with respect to Q2 is(monopole fit function)

    gQ2 AQ2 B ; (29)

    and for the off-shell meson the strong form factors canbe fitted by the exponential fit function as given (Gaussianfit function)

    gQ2 AeQ2=B: (30)For different values of the three sets (Table II) and

    (0:4 GeV 1:0 GeV), we analyze the parameters Aand B for the Ds0Ds0, DsDs, DsDs , and Ds1Ds1vertices. The values of the parameters A and B are given inTable III. As Table III shows, the values of the parametersA and B in set II are increased by increasing the value.Therefore, the dependence of the strong form factors onQ2

    are changed. For example, Fig. 5 shows the variation ofthe strong form factors gD

    s0D

    s0for different amounts of.

    Our calculations show that the left and right diagrams inFig. 5 nearly cross each other at one point for differentvalues of, which means that the strong coupling constantcan be defined. In the next figure, we clearly show theintersection point of two diagrams of the strong formfactors in 0:8. With regard to the values of set IIand set III, the dependence of the strong form factors onQ2 for the Ds0Ds0, DsDs, DsDs , and Ds1Ds1 ver-tices is shown in Fig. 6. In this figure, the small circles andboxes correspond to the form factors via the 3PSR calcu-lations. As it is seen, the form factors and their fit functionscoincide well together.

    TABLE I. The leptonic decay constants in MeV.

    fJ=c f fDs0

    fDs fDs fDs1

    405 10 [14] 234 10 [21] 225 25 [22] 274 13 [23] 266 32 [22] 240 25 [24]

    TABLE II. Different values of the quark masses in GeV inthree sets.

    Set I Set II Set III

    mc 1.30 should [14] 1.26 [25] 1.47 [25]ms 0:142 1 GeV [26] 0.104 [19] 0.104 [19]

    14 15 16 17 18 19 20 21 22

    5

    10

    15

    20

    25

    M2(GeV2)

    g

    D D

    ( D

    D

    ) (

    Q2 =

    1GeV

    2 )

    offshell

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    Ds0* D

    s0*

    DsD

    s

    Ds*D

    s*

    Ds1

    Ds1

    14 15 16 17 18 19 20 21 22

    5

    10

    15

    20

    M2(GeV2)

    gD(D

    )

    D D

    (

    D

    D) (

    Q2 =

    1GeV

    2 )

    charm offshell

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    Ds0* D

    s0*

    DsD

    s

    Ds*D

    s*

    Ds1

    Ds1

    FIG. 4. The strong form factors gD0D0D00D00 as functions of the Borel mass parameterM2 with the values of set II for the off shellmeson (left) and the charmed off shell mesons (right).

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  • We define the coupling constant as the value of the strongcoupling form factor at Q2 m2m in Eqs. (24)(27),where mm is the mass of the off-shell meson. Consideringthe uncertainties in the values of the other input parameters,we obtain the values of the strong coupling constants indifferent values of the three sets shown in Table IV.

    Now, we will provide the same results for theJ=cDs0Ds0, J=cDsDs, J=cDsDs , and J=cDs1Ds1 ver-tices. With little change in the expressions presented inSecs. II and III, such as the change in the quark permuta-tions, we can easily find similar results in Eqs. (24)(27)for strong form factors of the new vertices as

    gD0

    J=cD0D0 q2 mJ=c q2 m2D0

    CD0m2D0fj=c f2D0 m2D0 m2J=c q2e

    m2J=c

    M21 e

    m2

    D0M22

    1

    42

    Z sD00

    mcms2ds0

    Z sJ=c

    0

    2m2c

    dsD0

    J=cD0D0 s; s0; q2e s

    M21

    s0M22 iM21M22

    s

    G2CJ=cJ=cD0D0

    6

    ; (31)

    where D0

    J=cD0D0 D0

    D0D0 js$c andCD0

    J=cD0D0 CD0D0 js$c. sJ=c0 is the continuum threshold in the J=c meson, and its valueis mJ=c 2 where 0:4 GeV 1:0 GeV,

    gD00

    J=cD00D00 q2 2q2 m2D00

    mJ=c fj=c f2D00 3m2D00 m2J=c q2

    e

    m2J=c

    M21 e

    m2

    D00M22

    1

    42

    Z sD000

    mcms2ds0

    Z sJ=c

    0

    2m2c

    dsD00

    J=cD00D00 s; s0; q2e s

    M21

    s0M22 iM21M22

    s

    G2CJ=c

    J=cD00D00

    6

    ; (32)

    TABLE III. Parameters appearing in the fit functions for the Ds0Ds0, DsDs, D

    sD

    s , and Ds1Ds1 vertices for various mc, ms,

    and , where 1 0:5 GeV, 2 0:6 GeV, 3 0:8 GeV, and 4 1:0 GeV.Set I Set II Set III

    Form factor A1 B1 A2 B2 A3 B3 A4 B4 A3 B3gD

    s0D

    s03.11 2.98 4.74 4.93 5.89 6.03 7.13 6.92 4.26 5.23

    gD

    s0

    Ds0D

    s0129.77 21.53 147.73 22.59 202.12 25.98 265.42 29.78 213.26 35.38

    gDsDs 1.47 2.72 2.76 4.82 3.14 5.28 3.87 6.68 2.83 5.02

    gDsDsDs

    67.11 19.69 87.48 21.67 117.56 24.7 152.17 28.13 166.62 35.94

    gDsDs 4.89 125.45 6.02 127.39 7.38 171.69 8.66 225.45 7.54 234.78

    gDsDsDs 90.20 21.39 132.178 23.23 202.75 31.39 290.73 36.72 162.15 25.45

    gDs1Ds1 12.83 315.78 13.08 317.13 13.82 478.09 14.94 641.45 14.31 868.31

    gDs1Ds1Ds1

    237.70 24.35 295.22 28.12 896.13 69.03 741.61 51.19 522.89 42.01

    6 4 2 0 2 4 6 80

    2

    4

    6

    8

    10

    12

    14

    16

    Q2(GeV

    2)

    g

    D* s0

    D* s0

    (Q

    2)

    offshell ms=0.104GeV

    mc=1.26GeV =0.4GeV

    =0.6GeV

    =0.8GeV

    =1GeV

    10 8 6 4 2 0 2 4 6 8 102

    4

    6

    8

    10

    12

    14

    Q2(GeV

    2)

    g

    Ds0*

    Ds0*

    (Q

    2)

    Ds0

    * offshell m

    s=0.104GeV

    mc=1.26GeV =0.4GeV

    =0.6GeV

    =0.8GeV

    =1GeV

    FIG. 5. The strong form factor gDs0D

    s0on Q2 for different values of for the off shell (left) and the Ds0 off shell (right).

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

    016003-8

  • 4 2 0 2 4 6 8 100

    2

    4

    6

    8

    10

    12

    Q2(GeV

    2)

    g

    Ds0*

    D s0* (

    Q2

    ) offshell Exponential fit

    Ds0

    * offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    4 2 0 2 4 6 8 10 12

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    Q2(GeV

    2)

    g D

    s0*D

    s0* (

    Q2

    )

    offshell Exponential fit

    Ds0

    * offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    6 4 2 0 2 4 6 8 10 120

    1

    2

    3

    4

    5

    6

    7

    8

    9

    Q2(GeV

    2)

    g D

    sD s

    (Q

    2)

    offshell Exponential fit

    Ds offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    6 4 2 0 2 4 6 8 100

    1

    2

    3

    4

    5

    6

    7

    8

    Q2(GeV

    2)

    g

    D sD s

    (Q

    2)

    offshell Exponential fit

    Ds offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    8 6 4 2 0 2 4 6 8 10 12

    5

    6

    7

    8

    9

    10

    Q2(GeV

    2)

    g

    D* sD

    * s

    (Q

    2)

    offshell Exponential fit

    D*

    s offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    8 6 4 2 0 2 4 6 8 10 124

    5

    6

    7

    8

    9

    10

    Q2(GeV

    2)

    g

    D* sD

    * s

    (Q

    2)

    offshell Exponential fit

    D*

    s offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    5 0 5 10 15

    8

    10

    12

    14

    16

    18

    Q2(GeV

    2)

    g

    D s1D s

    1

    (Q

    2)

    offshell Exponential fit

    Ds1

    offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    10 5 0 5 10 159

    10

    11

    12

    13

    14

    15

    16

    17

    18

    Q2(GeV

    2)

    g

    D s1D

    s1

    (Q

    2)

    offshell Exponential fit

    Ds1

    offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    FIG. 6. The strong form factors gD0D0 and gD00D00 on Q2 for different values of the mc and 0:8 for the off shell and the

    charmed off shell mesons. The small circles and boxes correspond to the form factors via the 3PSR calculations.

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

    016003-9

  • where D00

    J=cD00D00 D00

    D00D00 js$c and CD00

    J=cD00D00 CD00D00 js$c,

    gJ=cJ=cD0D0 q2

    q2 m2J=c CD0mJ=cm2D0fJ=c f2D0

    e

    m2

    D0M21 e

    m2

    D0M22

    142

    Z sD00mcms2

    ds0

    Z sD00mcms2

    dsJ=cJ=cD0D0 s; s0; q2e

    sM21

    s0M22 hssi

    M21M22

    CJ=cJ=cD0D0

    12

    ; (33)

    where J=cJ=cD0D0 D0D0 js$c and CJ=cJ=cD0D0 CD

    0D0D0 js$c,

    TABLE IV. The strong coupling constants gD0D0 and gD00D00 in GeV1 for various mc and ms.

    Set I Set II Set III

    Coupling constant ch-off-sh -off-sh ch-off-sh -off-sh ch-off-sh -off-sh

    gDs0D

    s08:03 1:30 8:10 1:20 9:20 1:67 9:00 1:72 7:11 1:29 7:31 1:40

    gDsDs 4:25 0:92 4:15 0:88 5:17 1:34 5:06 1:83 5:20 1:35 4:97 1:80gDsDs 5:33 0:21 4:89 0:74 7:28 1:74 6:78 1:63 7:72 1:86 7:64 1:84gDs1Ds1 12:99 0:64 12:95 0:81 14:18 2:25 13:41 1:88 14:54 2:31 14:35 2:01

    12 13 14 15 16 17 18 19 200

    2

    4

    6

    8

    10

    12

    14

    16

    18

    M2(GeV2)

    gJ/

    J/

    DD

    (J/

    DD

    ) (

    Q2 =

    1GeV

    2 )

    J/ offshell

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    J/Ds0* D

    s0*

    J/DsD

    s

    J/Ds*D

    s*

    J/Ds1

    Ds1

    12 13 14 15 16 17 18 19 202

    4

    6

    8

    10

    12

    14

    16

    18

    20

    22

    M2(GeV2)

    gD(D

    )J/

    D D

    (J/

    D

    D

    ) (

    Q2 =

    1GeV

    2 )

    charm offshell

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    J/Ds0* D

    s0*

    J/DsD

    s

    J/Ds*D

    s*

    J/Ds1

    Ds1

    FIG. 7. The strong form factors gJ=cD0D0J=cD00D00 as functions of the Borel mass parameter M2 with the values of set II for the J=coff shell (left) and the charmed off shell mesons (right).

    TABLE V. Parameters appearing in the fit functions for the gJ=cD0D0J=cD00D00 form factors for various mc, ms, and , where 1 0:5 GeV, 2 0:6 GeV, 3 0:8 GeV, and 4 1:0 GeV.

    Set I Set II Set III

    Form factor A1 B1 A2 B2 A3 B3 A4 B4 A3 B3gJ=cJ=cD

    s0D

    s03.73 14.31 5.62 31.07 7.20 48.83 8.02 46.03 4.64 46.73

    gD

    s0

    J=cDs0D

    s0174.95 29.88 179.92 30.10 221.17 30.94 254.72 32.16 129.89 29.18

    gJ=cJ=cDsDs 1.73 253.20 1.96 187.02 2.34 242.44 2.96 251.02 1.86 194.32

    gDsJ=cDsDs

    160.21 89.05 176.22 92.22 211.61 89.91 284.62 89.32 186.54 101.63

    gJ=cJ=cDsDs 2.14 23.59 2.79 25.54 3.67 27.76 4.48 31.72 2.87 26.91

    gDsJ=cDsDs 499.86 166.02 847.23 199.97 1452.91 282.26 2341.12 403.88 868.24 254.32

    gJ=cJ=cDs1Ds1 5.98 29.48 7.46 32.88 8.66 46.79 9.58 60.37 6.93 38.69

    gDs1J=cDs1Ds1

    538.94 70.62 724.31 83.10 1033.08 107.17 1489.78 138.33 658.84 86.47

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

    016003-10

  • 2 0 2 4 6 8 105.5

    6

    6.5

    7

    7.5

    8

    8.5

    Q2(GeV

    2)

    gJ/

    D

    s0*D

    s0* (

    Q2)

    J/ offshell Exponential fit

    Ds0

    * offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    5 0 5 103

    3.5

    4

    4.5

    5

    5.5

    6

    Q2(GeV

    2)

    gJ/

    Ds0

    *D

    s0* (

    Q2

    )

    J/ offshell Exponential fit

    Ds0

    * offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    2 0 2 4 6 8 10 122.15

    2.2

    2.25

    2.3

    2.35

    2.4

    2.45

    Q2(GeV

    2)

    gJ/

    DsD

    s

    (Q

    2)

    J/ offshell Exponential fit

    Ds offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    6 4 2 0 2 4 6 8 10 12

    1.7

    1.75

    1.8

    1.85

    1.9

    1.95

    2

    Q2(GeV

    2)

    gJ/

    DsD s

    (Q

    2)

    J/ offshell Exponential fit

    Ds offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    15 10 5 0 5 10 152

    3

    4

    5

    6

    7

    Q2(GeV

    2)

    gJ/

    D* sD

    * s

    (Q

    2)

    J/ offshell Exponential fit

    D*

    s offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    10 5 0 5 10 151.5

    2

    2.5

    3

    3.5

    4

    4.5

    Q2(GeV

    2)

    gJ/

    D* sD

    * s

    (Q

    2)

    J/ offshell Exponential fit

    D*

    s offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    10 5 0 5 106

    7

    8

    9

    10

    11

    12

    13

    Q2(GeV

    2)

    gJ/

    Ds1

    D s1

    (Q

    2)

    J/ offshell Exponential fit

    Ds1

    offshell Monopolar fit

    mc=1.26GeV

    ms=0.104GeV

    =0.8GeV

    15 10 5 0 5 10 15

    5

    6

    7

    8

    9

    10

    11

    Q2(GeV

    2)

    gJ/

    Ds1

    D s1

    (Q

    2)

    J/ offshell Exponential fit

    Ds1

    offshell Monopolar fit

    mc=1.47GeV

    ms=0.104GeV

    =0.8GeV

    FIG. 8. The strong form factors gJ=cD0D0 and gJ=cD00D00 onQ2 for different values of themc and 0:8 for the J=c off shell and the

    charmed off shell mesons. The small circles and boxes correspond to the form factors via the 3PSR calculations.

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

    016003-11

  • gJ=cJ=cD00D00 q2 2q2 m2J=c

    mJ=c fJ=c f2D00 4m2D00 q2

    e

    m2

    D00M21 e

    m2

    D00M22

    1

    42

    Z sD000

    mcms2ds0

    Z sD00

    0

    mcms2dsJ=cJ=cD00D00 s; s0; q2e

    sM21

    s0M22 hssi

    M21M22

    CJ=cJ=cD00D00

    12

    ; (34)

    where J=cJ=cD00D00 D00D00 js$c and CJ=cJ=cD00D00

    CD00

    D00D00 js$c.The dependence of the strong form factors gJ=cD

    s0D

    s0,

    gJ=cDsDs , gJ=cDsDs , and gJ=cDs1Ds1 on the Borel mass pa-

    rameters M2 for the values of set II, 0:8 GeV andQ2 1 GeV2 are shown in Fig. 7.

    For all these vertices, good stability occurs in the inter-val 12 GeV2 M2 20 GeV2.

    The dependence of the above strong coupling formfactors in Eqs. (31)(34) on Q2 to the full physical regionis estimated using Eqs. (29) and (30) for the off-shellcharmed mesons and the off-shell J=c , respectively. Thevalues of the parameters A and B for the gJ=cD

    s0D

    s0,

    gJ=cDsDs , gJ=cDsDs , and gJ=cDs1Ds1 form factors in different

    amounts of the mc, ms, and are given in Table V. Thedependence of the strong form factors on Q2 for the valuesof set I and set II and 0:8 is shown in Fig. 8. As beforein these figures, the small circles and boxes correspond tothe form factors via 3PSR calculations.Considering the errors of the input parameters,

    the strong coupling constant values of the J=cD0D0J=cD00D00 vertices for three sets are shown in Table VI.Our numerical analysis shows that the contribution of

    the nonperturbative part containing the quark-quarkand quark-gluon diagrams is about 17%, the gluon-gluoncontribution is about 6% of the total, and the main con-tribution comes from the perturbative part of the strongform factors.The errors are estimated by the variation of the Borel

    parameter M2, the variation of the continuum thresholds

    s0 , sJ=c0 , and s

    D0D000 , the leptonic decay constants, and

    uncertainties in the values of the other input parameters.The main uncertainty comes from the continuum thresh-olds and the decay constants, which is25% of the central

    TABLE VI. The strong coupling constants gJ=cD0D0 and gJ=cD00D00 in GeV1 for various mc and ms.

    Set I Set II Set III

    Coupling constant ch-off-sh J=c -off-sh ch-off-sh J=c -off-sh ch-off-sh J=c -off-sh

    gJ=cDs0D

    s07:14 1:50 7:32 1:20 7:82 1:97 8:13 1:93 5:45 1:37 5:70 1:35

    gJ=cDsDs 1:88 0:73 1:80 0:65 2:33 1:01 2:26 0:81 1:91 0:83 1:95 0:70gJ=cDsDs 3:09 0:52 3:21 0:61 4:69 1:34 4:59 1:58 3:47 0:99 4:09 1:41gJ=cDs1Ds1 8:35 0:91 8:29 0:97 9:91 1:35 10:19 1:30 8:19 1:12 8:89 1:14

    TABLE VII. Parameters appearing in the fit functions for theD0D0D00D00 and J=cD0D0J=cD00D00 form factors inSUf3 symmetry with mc 1:26 GeV and 0:8 GeV.Form factor A B Form factor A B

    gDs0D

    s06.15 5.67 g

    J=cJ=cD

    s0D

    s07.39 54.43

    gD

    s0

    Ds0D

    s0214.64 26.40 g

    Ds0

    J=cDs0D

    s0237.60 31.73

    gDsDs 2.13 4.63 gJ=cJ=cDsDs

    2.01 213.63

    gDsDsDs

    118.08 32.74 gDsJ=cDsDs

    204.69 103.05

    gDsDs 7.19 254.34 gJ=cJ=cDsDs 3.47 29.19

    gDs

    DsDs 191.06 29.08 gDsJ=cDsDs 2570.27 522.58

    gDs1Ds1 15.14 370.05 gJ=cJ=cDs1Ds1

    9.13 43.75

    gDs1Ds1Ds1

    650.85 48.39 gDs1J=cDs1Ds1

    1149.21 113.47

    TABLE VIII. The strong coupling constants gD0D0 , gD00D00 , gJ=cD0D0 , and gJ=cD00D00 in GeV1 in SUf3 symmetry with mc

    1:26 GeV.

    Coupling constant ch-off-sh -off-sh Coupling constant ch-off-sh J=c -off-sh

    gDs0D

    s010:21 1:81 10:12 1:79 gJ=cD

    s0D

    s09:02 2:15 8:81 2:01

    gDsDs 4:09 1:15 3:91 1:03 gJ=cDsDs 2:07 0:91 2:10 0:92gDsDs 7:76 1:79 7:27 1:61 gJ=cDsDs 4:96 1:42 4:82 1:38gDs1Ds1 15:37 2:51 15:26 2:47 gJ=cDs1Ds1 10:70 2:42 11:37 2:55

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

    016003-12

  • value, while the other uncertainties are small, constitutinga few percent.

    So far, the strong coupling constant values were inves-tigated via the SUf3 symmetry breaking, and the mass ofthe s quark was considered in the expressions of thecondensate terms and spectral densities. Now, we want toanalyze the strong coupling constants considering theSUf3 symmetry. For the stated purpose, the mass of thes quark is ignored in all equations, i.e., ms ! mu 0 GeVand hssi ! hu ui.

    In view of the SUf3 symmetry, the values of theparameters A and B for the D0D0D00D00 andJ=cD0D0J=cD00D00 vertices inmc 1:26 GeV and 0:8 GeV are given in Table VII. Also considering theSUf3 symmetry, we obtain the values of the strong cou-pling constants in mc 1:26 GeV shown in Table VIII.

    Finally, we would like to compare our results with thevalues predicted by other methods. It should be remindedthat with the SUf3 symmetry consideration, it is possibleto compare the coupling constant value of gJ=cDsDs with

    gJ=cDD . Taking an average of the two values of the strong

    form factor gJ=cDsDs for the off-shell Ds and the off-shell

    J=c , we obtain

    gj=cDsDs 4:89 1:40 GeV1: (35)

    We compare our result for gJ=cDsDs in Eq. (35) with those

    of other calculations for gJ=cDD in Table IX. Our value is

    smaller than the values obtained using all the approachessuch as 3PSR [14,28], the quark model (QM) [27], thevector meson dominance approach (VMD) [29], and theconstituent quark-meson model (CQM) [30], but it is ingood agreement with the QM calculation.

    In summary, considering the contributions of the quark-

    quark, quark-gluon, and gluon-gluon condensate correc-

    tions, we estimate the strong form factors for the D0D0,D00D00, J=cD0D0, and J=cD00D00 (D0 Ds0, Ds andD00 Ds , Ds1) vertices within 3PSR. The dependence ofthe strong form factors on the transferred momentum

    square Q2 was plotted. We also evaluated the couplingconstants of these vertices. Detection of these strong

    form factors and the coupling constants and their compari-

    son with the phenomenological models like QCD sum

    rules could give useful information about strong interac-

    tions of the strange charmed mesons.

    ACKNOWLEDGMENTS

    Partial support of the Isfahan University of TechnologyResearch Council is appreciated.

    APPENDIX A

    In this Appendix, the explicit expressions of the coefficients of the quark-quark and quark-gluon condensate of thestrong form factors for the vertices D0D0 and D00D00 with application of the double Borel transformations are given.

    CD

    s0

    Ds0D

    s0

    3msm2cM21 3m2smcM21 2m20msM21 3msq2M22 3m2smcM22 3msm20M22 m20mcM22 3msm2cM22

    3m3sm

    2cM

    21

    M22 3

    2

    m20m3cM

    21

    M22 3

    2

    m20msm2cM

    21

    M22 3m

    3cm

    2sM

    21

    M22 3m

    5sM

    22

    M21 7

    4

    m20m2smcM

    22

    M21 3m

    4smcM

    22

    M21

    74

    m20m3sM

    22

    M21 3m5s 3m20m2smc 3m3sm2c 3m3sq2 3m3cm2s m20msq2 2m20mcq2 3mcq2m2s

    3m4smc m20m3s 2m20m3c e

    m2s

    M21e

    m2cM22 ;

    CDsDsDs

    3m20mcM213m3sM21

    7

    2m20msM

    213m2smcM213msm2cM213msq2M223msm2cM226m3sM22

    5

    2m20mcM

    22

    32

    m20msm2cM

    21

    M223m

    3sm

    2cM

    21

    M223m

    3cm

    2sM

    21

    M2232

    m20m3cM

    21

    M223m

    5sM

    22

    M213m

    4smcM

    22

    M2132

    m20m2smcM

    22

    M2132

    m20m3sM

    22

    M21

    12m20mcq

    232m20m

    2smc32m

    20msm

    2c12m

    20msq

    212m20m

    3s12m

    20m

    3c

    e

    m2s

    M21e

    m2cM22 ;

    TABLE IX. Values of the strong coupling constant using different approaches: 3PSR [14], QM [27], 3PSR [28], VMD [29], andCQM [30].

    Coupling constant Ours Reference [14] Reference [27] Reference [28] Reference [29] Reference [30]

    gJ=cDD 4:89 1:40 6:2 0:9 4.9 5:8 0:8 7.6 8:0 0:5

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  • CDsDsDs

    3msM

    21M

    22 3m3sM21 9m2smcM21 6msm2cM21 2m20mcM21 m20msM21 3msq2M21 3m3sM22

    3msm20M22 3m2smcM22 3m3sm

    2cM

    21

    M22 3

    2

    m20m3cM

    21

    M22 3

    2

    m20msm2cM

    21

    M22 3m

    3cm

    2sM

    21

    M22 3m

    4smcM

    22

    M21

    3m5sM

    22

    M21 3

    2

    m2smcm20M

    22

    M21 3

    2

    m20m3sM

    22

    M21 3m5s 2m20m3s m20m3c 3m3sq2 3m3cm2s 3m4smc 3m3sm2c

    2m20msq2 m20mcq2 3m20msm2c 3mcq2m2s e

    m2s

    M21e

    m2cM22 ;

    CDs1Ds1Ds1

    3msM

    21M

    22 3m3sM21 9m2smcM21 6msm2cM21 2m20mcM21 m20msM21 3msq2M21 3m3sM22

    3msm20M22 3m2smcM22 3m3sm

    2cM

    21

    M22 3

    2

    m20m3cM

    21

    M22 3

    2

    m20msm2cM

    21

    M22 3m

    3cm

    2sM

    21

    M22 3m

    4smcM

    22

    M21

    3m5sM

    22

    M21 3

    2

    m2smcm20M

    22

    M21 3

    2

    m20m3sM

    22

    M21 3m5s 2m20m3s m20m3c 3m3sq2 3m3cm2s 3m4smc 3m3sm2c

    2m20msq2 m20mcq2 3m20msm2c 3mcq2m2s e

    m2s

    M21e

    m2cM22 :

    APPENDIX B

    In this Appendix, the coefficients of the gluon condensate contributions of the strong form factors for the D0D0 andD00D00 vertices in the Borel transform scheme are presented.

    CDs0D

    s0 I23; 2; 2m6c I13; 2; 2m6c I13; 2; 2m4cm2s I23; 2; 2m4cm2s I23; 2; 2m3cm3s

    I13; 2; 2m3cm3s 3I03; 2; 1m4c I13; 2; 1m4c 3I02; 2; 2m4c I03; 1; 2m4c I23; 2; 1m4c 2I12; 3; 1m3cms 3I14; 1; 1m3cms 4I02; 3; 1m3cms 3I24; 1; 1m3cms 2I12; 2; 2m3cms 2I22; 2; 2m3cms I1;02 3; 2; 2m3cms I1;01 3; 2; 2m3cms 2I22; 3; 1m3cms I23; 2; 1m3cms I13; 2; 1m3cms I0;11 3; 2; 2m2cm2s I0;12 3; 2; 2m2cm2s 3I03; 2; 1m2cm2s 2I13; 2; 1mcm3s 2I23; 2; 1mcm3s 2I11; 2; 2m2c 3=2I11; 3; 1m2c 3I1;00 3; 2; 1m2c 3I0;11 3; 1; 2m2c 3I1;01 3; 2; 1m2c I0;10 3; 2; 1m2c 6I01; 3; 1m2c 2I21; 2; 2m2c 3I0;12 3; 1; 2m2c I0;20 3; 2; 2m2c 3I1;02 3; 2; 1m2c 2I01; 2; 2m2c 3I0;10 4; 1; 1m2c 5I1;12 3; 2; 2m2c 5I1;11 3; 2; 2m2c 3=2I21; 3; 1m2c 3I0;12 3; 1; 2mcms I1;12 3; 2; 2mcms 2I1;01 3; 2; 1mcms 4I12; 2; 1mcms I1;11 3; 2; 2mcms 2I1;02 3; 2; 1mcms 3I0;11 3; 1; 2mcms 2I11; 2; 2mcms 4I22; 2; 1mcms 2I21; 2; 2mcms 4I01; 3; 1mcms 2I0;10 2; 2; 2m2s I01; 2; 2m2s I0;20 3; 2; 2m2s 2I0;10 2; 2; 1 2I11; 2; 1 2I1;10 2; 2; 2 2I21; 2; 1 2I12; 1; 1 2I22; 1; 1 I1;11 2; 2; 2 I1;00 2; 2; 1 3I02; 1; 1 I0;10 1; 2; 2 I1;12 2; 2; 2 I1;20 3; 2; 2 6I0;10 1; 3; 1 3I0;10 2; 1; 2;

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  • CDsDs I13; 2; 2m5cms I23; 2; 2m5cms I13; 2; 2m3cm3s I23; 2; 2m3cm3s I0;11 3; 2; 2m4c 3I03; 2; 1m4c

    3I14; 1; 1m4c 3I12; 2; 2m4c 3I24; 1; 1m4c I23; 2; 1m4c 3I22; 2; 2m4c I0;12 3; 2; 2m4c 3I02; 2; 2m4c I13; 2; 1m4c I1;00 3; 2; 2m4c 2I22; 2; 2m3cms 3I03; 2; 1m3cms 3I24; 1; 1m3cms I23; 2; 1m3cms 3I14; 1; 1m3cms 2I12; 2; 2m3cms I13; 2; 1m3cms 6I21; 4; 1m2cm2s 2I13; 2; 1m2cm2s 2I23; 2; 1m2cm2s 6I11; 4; 1m2cm2s 3I03; 2; 1m2cm2s 2I12; 3; 1mcm3s I0;11 3; 2; 2mcm3s I0;12 3; 2; 2mcm3s 2I22; 3; 1mcm3s I02; 1; 2m2c I0;20 3; 2; 2m2c 2I0;10 2; 2; 2m2c 3I0;11 3; 1; 2m2c 5I1;12 3; 2; 2m2c 3I1;00 3; 2; 1m2c 5I1;11 3; 2; 2m2c 3I0;12 3; 1; 2m2c I03; 1; 1m2c I1;12 3; 2; 2mcms 3I11; 3; 1mcms 3I21; 3; 1mcms 2I1;02 2; 3; 1mcms 2I1;01 2; 3; 1mcms I0;10 3; 2; 1mcms 4I12; 2; 1mcms 4I22; 2; 1mcms I1;11 3; 2; 2mcms I02; 2; 1mcms I0;11 2; 2; 2m2s I0;12 2; 2; 2m2s 2I0;10 2; 2; 2m2s I12; 2; 1m2s I02; 2; 1m2s I22; 2; 1m2s 2I0;20 3; 1; 2 I0;10 1; 2; 2 I02; 1; 1 2I11; 2; 1 2I21; 1; 2 I1;00 1; 2; 2 3I1;10 3; 2; 1 2I11; 1; 2 I0;10 2; 1; 2 2I21; 2; 1 I1;20 3; 2; 2 2I0;10 3; 1; 1 I01; 2; 1;

    CDsDs I03; 2; 2m6c I03; 2; 2m5cms I23; 2; 2m4cm2s I13; 2; 2m4cm2s I03; 2; 2m3cm3s I13; 2; 2m2cm4s I23; 2; 2m2cm4s 3I04; 1; 1m4c 3I02; 2; 2m4c 2I63; 2; 2m4c 2I12; 3; 1m3cms I0;12 3; 2; 2m3cms 2I02; 2; 2m3cms I1;00 3; 2; 2m3cms 3I04; 1; 1m3cms 2I22; 2; 2m3cms 2I22; 3; 1m3cms 2I02; 3; 1m3cms 2I12; 2; 2m3cms I0;11 3; 2; 2m3cms 2I1;01 3; 2; 2m2cm2s 2I63; 2; 2m2cm2s 2I1;02 3; 2; 2m2cm2s 2I22; 3; 1mcm3s I0;10 3; 2; 2mcm3s 6I01; 4; 1mcm3s I02; 2; 2mcm3s 2I12; 3; 1mcm3s I12; 2; 2m4s I0;12 3; 2; 2m4s I0;11 3; 2; 2m4s I22; 2; 2m4s I02; 1; 2m2c 8I83; 1; 2m2c I13; 1; 1m2c I1;00 2; 2; 2m2c 2I0;16 3; 2; 2m2c 4I1;08 3; 2; 2m2c 4I63; 2; 1m2c 4I82; 2; 2m2c 2I1;06 3; 2; 2m2c I23; 1; 1m2c 8I73; 1; 2m2c 4I72; 2; 2m2c 4I1;07 3; 2; 2m2c 6I64; 1; 1m2c 6I63; 1; 2m2c 8I62; 3; 1mcms 2I01; 2; 2mcms I0;11 2; 2; 2mcms 2I11; 2; 2mcms I0;10 2; 2; 2mcms I1;12 3; 2; 2mcms I22; 2; 1mcms 2I0;10 2; 3; 1mcms 2I21; 2; 2mcms I0;12 2; 2; 2mcms I12; 2; 1mcms I1;11 3; 2; 2mcms 9=2I11; 3; 1m2s I1;10 3; 2; 2m2s 12I61; 4; 1m2s 24I71; 4; 1m2s 6I1;01 1; 4; 1m2s 2I1;00 3; 2; 1m2s 6I1;02 1; 4; 1m2s I0;10 2; 2; 2m2s 9=2I21; 3; 1m2s 4I63; 2; 1m2s 2I0;16 3; 2; 2m2s 2I62; 2; 2m2s 4I73; 2; 1m2s 2I1;01 2; 2; 2m2s I03; 1; 1m2s 24I81; 4; 1m2s 2I1;02 2; 2; 2m2s 4I83; 2; 1m2s I2;12 3; 2; 2 4I1;08 2; 2; 2 2I1;06 2; 2; 2 I2;01 3; 2; 1 2I1;02 1; 2; 2 4I1;06 3; 2; 1 I2;11 3; 2; 2 4I1;07 2; 2; 2 2I1;10 2; 2; 2 2I1;16 3; 2; 2 8I72; 2; 1 4I1;18 3; 2; 2 I0;20 3; 1; 2 8I82; 2; 1 3I61; 3; 1 I2;02 3; 2; 1 4I62; 2; 1 8I62; 1; 2 12I73; 1; 1 2I0;10 1; 2; 2 3I1;00 2; 2; 1 I1;00 2; 1; 2 2I0;16 2; 2; 2 6I71; 3; 1 6I81; 3; 1 4I1;17 3; 2; 2 4I63; 1; 1 4I0;17 2; 2; 2 4I0;18 2; 2; 2 2I0;10 2; 2; 1 2I1;01 1; 2; 2 12I83; 1; 1 2I0;16 3; 1; 2;

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  • CDs1Ds1 I23; 2; 2m5cms I13; 2; 2m5cms I1;01 3; 2; 2m4c 4I83; 2; 2m4c 2I63; 2; 2m4c 4I73; 2; 2m4c

    3I04; 1; 1m4c I1;02 3; 2; 2m4c 2I03; 2; 1m3cms I1;00 3; 2; 2m3cms 3I04; 1; 1m3cms 2I02; 2; 2m3cms 2I23; 2; 1m2cm2s 3I14; 1; 1m2cm2s I22; 2; 2m2cm2s I12; 2; 2m2cm2s 2I13; 2; 1m2cm2s 2I63; 2; 2m2cm2s 3I24; 1; 1m2cm2s I23; 2; 1mcm3s I13; 2; 1mcm3s I0;10 3; 2; 2mcm3s 6I01; 4; 1mcm3s I0;11 3; 2; 2mcm3s I0;12 3; 2; 2mcm3s I0;12 3; 2; 2m4s I03; 2; 1m4s I0;11 3; 2; 2m4s I13; 2; 1m4s 3I01; 4; 1m4s I23; 2; 1m4s 8I83; 2; 1m2c I2;02 3; 2; 2m2c 8I83; 1; 2m2c I0;10 2; 2; 2m2c 4I0;18 3; 2; 2m2c 4I0;17 3; 2; 2m2c 2I1;06 3; 2; 2m2c 4I63; 2; 1m2c I2;01 3; 2; 2m2c 8I73; 2; 1m2c 6I64; 1; 1m2c 2I0;16 3; 2; 2m2c 4I01; 2; 2m2c 6I63; 1; 2m2c 8I73; 1; 2m2c I11; 3; 1mcms I02; 1; 2mcms I21; 3; 1mcms I12; 1; 2mcms I1;02 2; 2; 2mcms 2I1;02 2; 3; 1mcms 2I0;10 3; 1; 2mcms 2I1;01 2; 3; 1mcms I01; 3; 1mcms 8I62; 3; 1mcms I1;12 3; 2; 2mcms I1;01 2; 2; 2mcms I1;11 3; 2; 2mcms I22; 1; 2mcms 12I61; 4; 1m2s 6I1;02 1; 4; 1m2s 4I83; 2; 1m2s I13; 1; 1m2s 6I1;01 1; 4; 1m2s 4I0;18 3; 2; 2m2s I11; 2; 2m2s I1;10 3; 2; 2m2s 2I0;16 3; 2; 2m2s I03; 1; 1m2s I21; 2; 2m2s 3I0;10 1; 4; 1m2s 4I63; 2; 1m2s I0;10 2; 2; 2m2s 4I0;17 3; 2; 2m2s I23; 1; 1m2s 2I62; 2; 2m2s 3I1;00 1; 4; 1m2s 4I73; 2; 1m2s 2I0;10 2; 2; 1 12I83; 1; 1 4I0;17 2; 2; 2 4I0;18 2; 2; 2 I1;00 3; 1; 1 2I0;16 3; 1; 2 4I1;17 3; 2; 2 4I1;18 3; 2; 2 2I0;16 2; 2; 2 8I72; 1; 2 8I62; 1; 2 4I1;07 2; 2; 2 2I12; 1; 1 I1;12 2; 2; 2 I1;11 2; 2; 2 2I22; 1; 1 8I72; 2; 1 8I82; 2; 1 4I63; 1; 1 4I1;08 2; 2; 2 4I62; 2; 1 3I02; 1; 1 4I1;06 3; 2; 1 12I73; 1; 1 2I1;06 2; 2; 2 2I1;16 3; 2; 2 3I0;11 2; 1; 2 3I0;12 2; 1; 2 8I82; 1; 2 2I1;01 2; 2; 1 2I1;02 2; 2; 1 I0;10 3; 1; 1 3I61; 3; 1 2I0;10 1; 2; 2 2I1;10 2; 2; 2;

    where

    I m;nl a; b; c M21mM22ndm

    dM21mdn

    dM22nM21mM22nIla; b; c:

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