# Analysis ofDs⁎D⁎K⁎ and Ds1D1K⁎ Vertices in Three-Point Sum Rules

Authors: M. Janbazi; R. Khosravi; E. Noori
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

In this study, the coupling constants ofDs⁎D⁎K⁎ and Ds1D1K⁎ vertices were determined within the three-point Quantum chromodynamics sum rules method with and without consideration of the SUf(3) symmetry. The coupling constants were calculated for off-shell charm and K⁎ cases. Considering the nonperturbative effect of the correlation function, as the most important contribution, the quark-quark, quark-gluon, and gluon-gluon condensate corrections were estimated and were compared with other predictive methods.


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## Body

## 1. Introduction

Considerable attention has been focused on the strong form factors and coupling constants of meson vertices in the context of quantum chromodynamics (QCD) since the last decade. In high energy physics, understanding the functional form of the strong form factors plays very important role in explaining of the meson interactions. Therefore, accurate determination of the strong form factors and coupling constants associated with the vertices involving mesons has attracted great interest in recent studies of the high energy physics.Quantum chromodynamics sum rules (QCDSR) formalism has been used extensively to study about the “exotic” mesons made of quark- gluon hybrid (qq¯g), tetraquark states (qq¯qq¯), molecular states of two ordinary mesons, glueballs [1], and vertices involving charmed mesons such as D∗D∗ρ [2, 3], D∗Dπ [2, 4], DDρ [5], D∗Dρ [6], DDJ/ψ [7], D∗DJ/ψ [8], D∗DsK, Ds∗DK, D0∗DsK, Ds0∗DK [9], D∗D∗P, D∗DV, DDV [10], D∗D∗π [11], DsD∗K, Ds∗DK [12], DDω [13], Ds0DK and D0DsK [14], Ds1D∗K, Ds1D∗K0∗ [15, 16], DsDsV, Ds∗Ds∗V [17, 18], and D1D∗π,D1D0π,D1D1π [19].In this study, 3-point sum rules (3PSR) method is used to calculate the strong form factors and coupling constants of theDs∗D∗K∗ and Ds1D1K∗ vertices. The 3PSR correlation function is investigated from the phenomenological and the theoretical points of view. Regarding the phenomenological (physical) approach, the representation can be expressed in terms of hadronic degrees of freedom which can be considered as responsible for the introduction of the form factors, decay constant, and masses. The theoretical (QCD) approach usually can be divided into two main contributions as perturbative and nonperturbative. In this approach, the quark-gluon language and Wilson operator product expansion (OPE) are usually used to evaluate the correlation function in terms of the QCD degrees of freedom such as quark condensate, gluon condensate, etc. Equating the two sides and applying the double Borel transformations with respect to the momentum of the initial and final states to suppress the contribution of the higher states, and continuum, the strong form factors can be estimated.The effective Lagrangian of the interaction for theDs∗D∗K∗ and Ds1D1K∗ vertices can be written as [20](1)LDs∗D∗K∗=igDs∗D∗K∗Ds∗μ∂μK∗νD¯ν∗-K∗ν∂μD¯ν∗+∂μDs∗νKν∗-Ds∗ν∂μKν∗D¯∗μ+K∗μDs∗ν∂μD¯ν∗-∂μDs∗νD¯ν∗,(2)LDs1D1K∗=igDs1D1K∗Ds1μ∂μK∗νD¯1ν-K∗ν∂μD¯1ν+∂μDs1νKν∗-Ds1ν∂μKν∗D¯1μ+K∗μDs1ν∂μD¯1ν-∂μDs1νD¯1ν,where gDs∗D∗K∗ and gDs1D1K∗ are the strong form factor. Using the introduced form of the Lagrangian, the elements related to the Ds∗D∗K∗ and Ds1D1K∗ vertices can be derived in terms of the strong form factor as(3)D∗p,εDs∗p′,ε′∣K∗q,ε′′=igDs∗D∗K∗q2×qα+p′αgμν-qμ+pμgνα+qνgαμ×εαpεμ′p′εν′′q,(4)D1p,εDs1p′,ε′∣K∗q,ε′′=igDs1D1K∗q2×qα+p′αgμν-qμ+pμgνα+qνgαμ×εαpεμ′p′εν′′q,where q=p-p′.The organization of the paper is as follows: In Section2, the quark-quark, quark-gluon, and gluon-gluon condensate contributions, considering the nonperturbative effects of the Borel transform scheme, are discussed in order to calculate the strong form factors of the Ds∗D∗K∗ and Ds1D1K∗ vertices in the framework of the 3PSR. The numerical analysis of the strong form factors estimation as well as the coupling constants, with and without consideration of the SUf(3) symmetry, is described in Section 3 and the conclusion is made in Section 4.


## 2. The Strong Form Factor ofDs∗D∗K∗ and Ds1D1K∗ Vertices

To compute the strong form factor of theDs∗D∗K∗ and Ds1D1K∗ vertices via the 3PSR, we start with the correlation function. When the K∗ meson is off-shell, the correlation function can be written in the following form:(5)ΠμναK∗p,p′=i2∫d4xd4yeip′x-py0∣TjμDs∗xjνK∗†0jαD∗†y∣0,(6)ΠμναK∗p,p′=i2∫d4xd4yeip′x-py0∣TjμDs1xjνK∗†0jαD1†y∣0.For off-shell charm meson, the correlation function can be written as(7)ΠμανD∗p,p′=i2∫d4xd4yeip′x-py0∣TjμDs∗xjαD∗†0jνK∗†y∣0,(8)ΠμανD1p,p′=i2∫d4xd4yeip′x-py0∣TjμDs1xjαD1†0jνK∗†y∣0,where jμDs∗=c¯γμs,jαD∗=c¯γαu, jμDs1=c¯γμγ5s, jαD1=c¯γαγ5u, and jνK∗=u¯γνs are interpolating currents with the same quantum numbers of Ds∗,D∗, Ds1,D1, and K∗ mesons. As described in Figure 1, T, p, and p′ are time ordering product and four momentum instances of the initial and final mesons, respectively.Figure 1
Perturbative diagrams for off-shellK∗ (a) and off-shell D∗ (b).
(a) (b)Considering the OPE scheme in the phenomenological approach, the correlation functions (see (5) to (8)) can be written in terms of several tensor structures and their coefficients are found using the sum rules. It is clear from (3) and (4) that the form factor gDs∗D∗K∗ is used for the fourth Lorentz structure which can be extracted from the sum rules. We choose the Lorentz structure because of its fewer ambiguities in the 3PSR approach, i.e., less influence of higher dimension of the condensates and better stability as function of the Borel mass parameter [2]. For these reasons, the gμαqν structure is chosen which is assumed to better formulate the problem.In order to calculate the phenomenological part of the correlation functions in (5) to (8), three complete sets of intermediate states with the same quantum number as the currents jμDs∗, jαD∗, jμDs1, jαD1, and jνK∗ are selected. The matrix elements 〈0∣jμDs∗∣Ds∗(p,ε)〉, 〈0∣jαD∗∣D∗(p)〉, 〈0∣jμDs1∣Ds1(p,ε)〉, 0∣jαD1∣D1p, and 〈0∣jνK∗∣K∗(q,ε)〉 are defined as(9)0∣jμV∣Vp,ε=mVfVεμp,where mV and fV are the masses and decay constants of mesons V(Ds∗,D∗,Ds1,D1,K∗) and εμ is introduced as the polarization vector of the vector meson V(Ds∗,D∗,Ds1,D1,K∗).The phenomenological part of thegμαqν structure associated with the Ds∗D∗K∗ vertex for off-shell D∗ and K∗ mesons can be expressed as(10)ΠμναD∗=-gDs∗D∗K∗D∗q2mK∗fK∗fD∗fDs∗3mD∗2+mK∗2-q22mDs∗q2-mD∗2p2-mK∗2p′2-mDs∗2+⋯,ΠμναK∗=gDs∗D∗K∗K∗q2mK∗fK∗fD∗fDs∗3mDs∗2+mD∗2-q22mDs∗q2-mK∗2p2-mD∗2p′2-mDs∗2+⋯,The phenomenological part of thegμαqν structure associated with the Ds1D1K∗ vertex for off-shell D1 and K∗ mesons can be expressed as(11)ΠμναD1=-gDs1D1K∗D1q2mK∗fK∗fD1fDs13mD12+mK∗2-q22mDs1q2-mD12p2-mK∗2p′2-mDs12+⋯,ΠμναK∗=gDs1D1K∗K∗q2mK∗fK∗fD1fDs13mDs12+mD12-q22mDs1q2-mK∗2p2-mD∗2p′2-mDs12+⋯.Using the operator product expansion in Euclidean region and assumingp2,p′2→-∞, one can calculate the QCD side of the correlation function (see (5) to (8)) which contains perturbative and nonperturbative terms. Using the double dispersion relation for the coefficient of the Lorentz structure gμαqν appearing in the correlation function (see (3) and (4)), we get(12)ΠperMp2,p′2,q2=-14π2∫ds∫ds′ρMs,s′,q2s-p2s′-p′2+subtractionterms,where ρM(s,s′,q2) is spectral density and M stands for off-shell charm and K∗ mesons. The Cutkoskys rule allows us to obtain the spectral densities of the correlation function for the Lorentz structure appearing in the correlation function. As shown in Figure 1, the leading contribution comes from the perturbative term. As a result, the spectral densities are obtained in the case of the double discontinuity in (12) for the vertices; see Appendix A.In order to consider the nonperturbative part of the correlation functions for the case of spectator light quark (for off-shell charm meson), we proceed to calculate the nonperturbative contributions in the QCD approach which contain the quark-quark and quark-gluon condensates [23]. Figure 2 describes the important quark-quark and quark-gluon condensates from the nonperturbative contribution of the off-shell charm mesons [23].Figure 2
Nonperturbative diagrams for the off-shellD∗ meson.In the 3PSR frame work, when the heavy quark is a spectator (for off-shellK∗ meson), the gluon-gluon contribution can be considered. Figure 3 shows related diagrams of the gluon-gluon condensate. More details about the nonperturbative contributions CDs∗D∗K∗D∗ and CDs1D1K∗D1(sum contributions of quark-quark and quark-gluon condensates) and CDs∗D∗K∗K∗ and CDs1D1K∗K∗(for gluon-gluon condensates) corresponding to Figures 2 and 3 are given in Appendix B, respectively.Figure 3
Nonperturbative diagrams for the off-shellK∗ meson.Considering the perturbative and nonperturbative parts of the correlation function in order to suppress the contributions of the higher states, the strong form factors can be calculated in the phenomenological side by equating the two representations of the correlation function and applying the Borel transformations with respect top2(p2→M12) and p′2(p′2→M22). The equations for the strong form factors gDs∗D∗K∗K∗ and gDs∗D∗K∗D∗ are obtained as(13)gDs∗D∗K∗K∗q2=2mDs∗q2-mK∗2mK∗fK∗fD∗fDs∗3mDs∗2+mD∗2-q2emD∗2/M12emDs∗2/M22-14π2∫mc+ms2s0Ds∗ds′∫mu+mc2s0D∗dsρDs∗D∗K∗K∗s,s′,q2e-s/M12e-s′/M22-iM12M22αsπG2×CDs∗D∗K∗K∗,gDs∗D∗K∗D∗q2=2mDs∗q2-mD∗2mK∗fK∗fD∗fDs∗3mD∗2+mK∗2-q2emK∗2/M12emDs∗2/M22-14π2∫mc+ms2s0Ds∗ds′∫mu+ms2s0K∗dsρDs∗D∗K∗D∗s,s′,q2e-s/M12e-s′/M22+M12M22ss¯×CDs∗D∗K∗D∗.The equations describing the strong form factorsgDs1D1K∗K∗ and gDs1D1K∗D1 can be written as(14)gDs1D1K∗K∗q2=2mDs1q2-mK∗2mK∗fK∗fD1fDs13mDs12+mD12-q2emD12/M12emDs12/M22-14π2∫mc+ms2s0Ds1ds′∫mu+mc2s0D1dsρDs1D1K∗K∗s,s′,q2e-s/M12e-s′/M22-iM12M22αsπG2×CDs1D1K∗K∗,gDs1D1K∗D1q2=2mDs1q2-mD12mK∗fK∗fD1fDs13mD12+mK∗2-q2emK∗2/M12emDs12/M22-14π2∫mc+ms2s0Ds1ds′∫mu+ms2s0K∗dsρDs1D1K∗D1s,s′,q2e-s/M12e-s′/M22+M12M22ss¯×CDs1D1K∗D1.where the quantities s0Ds∗, s0D∗s0Ds1, s0D1, and s0K∗ are introduced as the continuum thresholds in Ds∗, D∗Ds1, D1, and K∗ mesons, respectively, and ρDs∗D∗K∗K∗, ρDs∗D∗K∗D∗, CDs∗D∗K∗K∗, CDs∗D∗K∗D∗, ρDs1D1K∗K∗, ρDs1D1K∗D1, CDs1D1K∗K∗, and CDs1D1K∗D1 are defined in Appendices A and B.


## 3. Numerical Analysis

In order to numerically estimate the strong form factors and coupling constants of the verticesDs∗D∗K∗ and Ds1D1K∗, the values of the quark and meson masses are chosen as ms=0.14±0.01 GeV, mK∗=0.89 GeV, mDs∗=2.11 GeV,mDs1=2.46 GeV, and mD1=2.42 GeV [24]. Moreover, the leptonic decay constants of the vertices are fK∗=220±5 [24], fDs∗=314±19 [25], fD∗=242±12 [25], fDs1=225±20 [26], and fD1=219±11 [27]There are four auxiliary parameters containing the Borel mass parametersM1 and M2 and continuum thresholds s0K∗, s0D∗(D1), and s0Ds∗(Ds1) in ((13) and (14)). The strong form factors and coupling constants are physical quantities which are independent of the mass parameters and continuum thresholds. However, the continuum thresholds are not completely arbitrary and can be related to the energy of the first exited state. The values of the continuum thresholds are taken to be s0K∗=(mK∗+δ)2, s0D∗(D1)=(mD∗D1+δ′)2, and s0Ds∗(Ds1)=(mDs∗Ds1+δ′)2 where 0.50 GeV ≤δ≤0.90 GeV and 0.30 GeV ≤δ′≤0.70 GeV [2–4].Our results should be almost insensitive to the intervals of the Borel parameters. In this work, the Borel masses are related asM12/M22=mK∗2+mc2/mDs∗(Ds1)2 and M12=M22 for off-shell charm mesons and K∗, respectively [5, 6]. The form factors for the Ds∗D∗K∗ and Ds1D1K∗ vertices with respect to the Borel parameters M12 are shown in Figure 4. It is found from the figure that the stability of the form factors, as function of Borel parameters, is good in the region of 13GeV2<M12<18GeV2 for off-shell K∗ and charm mesons. We get M12=15GeV2 and calculate the strong form factors gDs∗D∗K∗ in some points of Q2 via the 3PSR formalism.Figure 4
The strong form factorsgDs∗D∗K∗ (left) and gDs1D1K∗ (right) as functions of the Borel mass parameter M12 for off-shell charm and K∗ mesons.To extract the coupling constants from the form factors, it is needed to extend theQ2 dependency of the strong form factors to the ranges that the sum rule results are not valid. Therefore, we fitted two sets of points (boxes and circles) imposing the condition that the two resulting parameterizations lead to the same result for Q2=-mm2, where mm is the mass of the off-shell mesons. This procedure is sufficient to reduce the uncertainties. It is found that the sum rule predictions of the form factors in ((13) and (14)) are well fitted to the function (15)gQ2=Ae-Q2/B. The values of the parameters A and B are given in Table 1.Table 1
Parameters appearing in the fit functions for theDs∗D∗K∗ and Ds1D1K∗ vertices for various mc and (δ,δ′), where (δ1,δ1′)=(0.50,0.30),(δ2,δ2′)=(0.70,0.50), and (δ3,δ3′)=(0.90,0.70) GeV.
Set I      Set II Form factor A ( δ 1 , δ 1 ′ ) B ( δ 1 , δ 1 ′ ) A ( δ 2 , δ 2 ′ ) B ( δ 2 , δ 2 ′ ) A ( δ 3 , δ 3 ′ ) B ( δ 3 , δ 3 ′ ) A ( δ 2 , δ 2 ′ ) B ( δ 2 , δ 2 ′ ) g D s ∗ D ∗ K ∗ K ∗ ( Q 2 ) 4.04 183.10 4.95 197.49 5.93 215.62 4.43 266.52 g D s ∗ D ∗ K ∗ D ∗ ( Q 2 ) 3.67 61.46 4.42 52.33 5.56 47.96 4.18 57.95 g D s 1 D 1 K ∗ K ∗ ( Q 2 ) 4.19 15.24 4.39 21.18 4.68 29.21 4.22 7.69 g D s 1 D 1 K ∗ D 1 ( Q 2 ) 2.62 13.27 3.13 19.47 4.02 25.32 3.03 14.87Variations of the strong form factorsgDs∗D∗K∗K∗ and gDs∗D∗K∗D∗ for Ds∗D∗K∗ vertex and gDs1D1K∗K∗ and gs1D1K∗D1 for Ds1D1K∗ vertex with respect to the Q2 parameter are shown in Figure 5. The boxes and circles show the results of the numerical evaluation of the form factors via the 3PSR. It is clear from the figure that the form factors are in good agreement with the fitted function.Figure 5
The strong form factorsgDs∗D∗K∗ (left) and gDs1D1K∗ (right) as functions of Q2 for off-shell charm and K∗ mesons.So-called harder is used. In the present analysis we find that the form factor is harder when the lighter meson is off-shell. This is in line with the results of our previous work [17], whereas this is in contrast with other previous calculations quoted by the authors [28, 29].The value of the strong form factors atQ2=-mm2 is defined as coupling constant. Calculation results of the coupling constant of the vertices Ds∗D∗K∗ and Ds1D1K∗ are summarized in Table 2. It should be noted that the coupling constants gDs∗D∗K∗ and gDs1D1K∗ are in the unit of GeV-1.Table 2
The coupling constant of the verticesDs∗D∗K∗ and Ds1D1K∗.
g Set I  Set II off-shell charmed off-shellK∗ off-shell charmed off-shellK∗ g D s ∗ D ∗ K ∗ 4.77 ± 0.63 4.96 ± 0.64 4.48 ± 0.58 4.45 ± 0.58 g D s 1 D 1 K ∗ 4.22 ± 0.55 4.56 ± 0.59 4.48 ± 0.58 4.67 ± 0.62In order to estimate the error of the calculated parameters, variations of the Borel parameter, continuum thresholds, and leptonic decay constants, as the most significant reasons of the uncertainties, are considered.To investigate the value of the strong coupling constant via theSUf(3) symmetry, the mass of the s quark is ignored in all equations. Calculated parameters A and B for the gDs∗D∗K∗ and gDs1D1K∗ vertices, considering (δ,δ′)=(0.70,0.50) GeV, are given in Table 3.Table 3
Parameters appearing in the fit functions for thegDs∗D∗K∗ and gDs1D1K∗ form factors in SUf(3) symmetry with mc=1.26 GeV and (δ,δ′)=(0.70,0.50) GeV.
Form factor A B Form factor A B g D s ∗ D ∗ K ∗ K ∗ ( Q 2 ) 5.01 218.91 g D s 1 D 1 K ∗ K ∗ ( Q 2 ) 4.56 20.63 g D s ∗ D ∗ K ∗ D ∗ ( Q 2 ) 4.54 52.08 g D s 1 D 1 K ∗ D 1 ( Q 2 ) 3.66 20.89Estimated coupling constants of the verticesDs∗D∗K∗, and Ds1D1K∗, considering the SUf(3) symmetry, are summarized in Table 4. The comparisons of the coupling constants gDs∗D∗K∗ with gD∗D∗ρ, considering other methods described in [2, 3], are given in Table 5. It is found that the results of the calculated parameters are in reasonable agreement with that of [2, 3] and a factor of two orders of magnitude larger in comparison with [21, 22].Table 4
The coupling constant of the verticesDs∗D∗K∗ and Ds1D1K∗, in SUf(3) symmetry.
g off-shell charmed off-shellK∗ g off-shell charmed off-shellK∗ g D s ∗ D ∗ K ∗ 4.88 ± 0.64 5.03 ± 0.65 g D s 1 D 1 K ∗ 4.85 ± 0.63 4.75 ± 0.62Table 5
Values of the strong coupling constant reporting different reference of the coupling constantgD∗D∗ρ [2, 3, 21, 22].
g Ours Reference [2, 3] Reference [21, 22] g D s ∗ D ∗ K ∗ 4.95±0.64 6.60±0.30 2.52


## 4. Conclusion

Strong form factors and coupling constants ofDs∗D∗K∗ and Ds1D1K∗ vertices were calculated in the frame work of 3-point sum rules of quantum chromodynamics with and without consideration of the SUf(3) symmetry. Considering nonperturbative contributions of the correlation functions, the quark-quark, quark-gluon, and gluon-gluon condensate corrections were estimated as the most effective terms. It was found from the numerical results that the obtained coupling constants are in good agreement with the other prediction methods described in [2, 3].



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