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


    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


    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


    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.


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


    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


    p; p0 i2

    Zd4xd4yeip0xpyh0jT fjD0 xjD0y0j ygj0i;


    PHYSICAL REVIEW D 87, 016003 (2013)

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

  • D00

    p; p0 i2

    Zd4xd4yeip0xpyh0jT fjD00 xjD00y 0jygj0i;


    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;


    where jDs0 sUc, jDs s5c, jD


    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:


    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





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

    D (D)

    (a) (b)

    p p

    q q


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



  • 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;



    gD0D0D0 q2CD02m2D0ff2D0 m2D0 m2 q2mp2 m2p02 m2D0 q2 m2D0

    p other structures higher and continuum states; (10)


    gD00D00D00 q2mff

    2D00 3m2D00 m2 q2

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


    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:


    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



    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:


    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:


    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



  • 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 ,


    (see Fig. 1).


    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


    s s s ss

    c cc



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

    c c



    s ss



    s s ss

    ss s

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



  • 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



    p2 m2sm 1







    p02 m2cn 1






    where M21 and M22 are the Borel parameters, respectively,

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


    nonper hssiM21M222CD


    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:;b;c

    Z d4k24

    k12...nk2m2capk2m2sbp0 k2m2sc



    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.,


    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 ;


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

    Bp2M21Bp02M22p2mp02nI12...n M21mM22ndm



    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



  • U0a;bZ 10dyyM21M22aybexp






    B1 1M22M


    m2sM21 M222 M22M21Q2;

    B0 1M21M


    m2s m2cM21 M22; B1 m2c




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

    nonper is


    6; (23)

    where the explicit expressions for CD0D0 and C

    D00D00 are

    given in Appendix B.Th...