Strong coupling constants of bottom and charmed mesons with scalar, pseudoscalar, and axial vector kaons

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(1)PHYSICAL REVIEW D 83, 114009 (2011). Strong coupling constants of bottom and charmed mesons with scalar, pseudoscalar, and axial vector kaons H. Sundu,1,* J. Y. Su¨ngu¨,1,† S. S¸ahin,1,‡ N. Yinelek,1,x and K. Azizi2,k 1. Department of Physics, Kocaeli University, 41380 Izmit, Turkey Physics Division, Faculty of Arts and Sciences, Dog˘us¸ University, Acbadem-Kadko¨y, 34722 Istanbul, Turkey (Received 4 March 2011; revised manuscript received 10 May 2011; published 2 June 2011). 2. The strong coupling constants, gDs DK0 , gBs BK0 , gDs DK , gBs BK , gDs DK1 and gBs BK1 , where K0 , K and K1 are scalar, pseudoscalar, and axial-vector kaon mesons, respectively, are calculated in the framework of three-point QCD sum rules. In particular, the correlation functions of the considered vertices when both BðDÞ and K0 ðKÞðK1 Þ mesons are off shell are evaluated. In the case of K1 , which is either K1 ð1270Þ or K1 ð1400Þ, the mixing between these two states are also taken into account. A comparison of the obtained result with the existing prediction on gDs DK as the only coupling constant among the considered vertices, previously calculated in the literature, is also made. DOI: 10.1103/PhysRevD.83.114009. PACS numbers: 11.55.Hx, 13.25.Ft, 13.25.Hw, 13.75.Lb. I. INTRODUCTION The strong coupling constants among the bottom and charmed mesons with light scalar, pseudoscalar, and axial strange mesons are the main ingredients in analysis of their strong interactions. More accurate determination of these coupling constants is needed to better understand the strong interactions among the participated mesons, construct the strong potentials among them, and obtain knowledge about the nature and structure of the encountered particles. Experimentally, it is believed that in the production of the charmonium states like J= c and c 0 from the Bc or newly discovered charmonium X, Y, and Z states by the BABAR and BELLE collaborations, there are intermediate two body states containing D, Ds , D and Ds mesons (for example, the kaon can annihilate the charmonium in a nuclear medium to give D and Ds mesons), which decay to the final J= c and c 0 states exchanging one or more virtual mesons. A similar story would happen in decays of heavy bottonium. To exactly follow and analyze the procedure in the experiment, we need to knowledge about the coupling constants among the particles involved. The strong coupling constants among mesons take place in low energies very far from the perturbative region, where the strong coupling constant between quarks and gluons is large and perturbation theory fails. Hence, in the hadronic scale, one should consult to some nonperturbative methods in QCD to describe nonperturbative phenomena. Among the nonperturbative methods, the QCD sum rules approach [1–4] is one of the most powerful, applicable, and attractive one as it is based on QCD Lagrangian and is free of a model dependent parameter. This approach has rendered many successful predictions such as its predictions. about the vector mesons [5–9]. The three-point correlation function has been widely used to calculate many parameters of hadrons (see for instance [10–13]). The QCD sum rules for some strong coupling constants were derived by means of the three-point functions in [14]. In the present work, we investigate various strong coupling constants among bottom (charmed)-bottom strange (charmed strange) mesons with scalar, pseudoscalar, and axial-vector kaons. Calculation of such coupling constants can help us in understanding the nature of the strong interaction among the participating particles. In the case of the scalar kaon, we consider the Bs B K0 and Ds D K0 vertices for both K0 ð800Þ and K0 ð1430Þ. Understanding the internal structure of the scalar mesons has been a striking issue in the last 30–40 years. Despite their investigation both theoretically and experimentally, most of their properties are not very clear yet. Detection and identification of the scalar mesons are difficult, experimentally, so the theoretical and phenomenological works can play a crucial role in this regard. In this work, we also calculate the coupling constants gBs BK and gDs DK for pseudoscalar K. The next aim in the present work is to consider the vertices Bs B K1 and Ds D K1 for both K1 ð1270Þ and K1 ð1400Þ axial states taking into account their mixture. Experimentally, the K1 ð1270Þ and K1 ð1400Þ are the mixtures of the strange members of two axial-vector SU(3) octets 3 P1 ðK1A Þ and 1 P1 ðK1B Þ. To avoid any confusion between the B meson and the sign B in the K1B , we will use the K1aðbÞ instead of K1AðBÞ in this article. The K1 ð1270; 1400Þ are related to the K1a;b states via [15,16] j K1 ð1270Þi ¼j K1a i sinþ j K1b i cos;. *hayriye.sundu@kocaeli.edu.tr † jyilmazkaya@kocaeli.edu.tr ‡ 095131004@kocaeli.edu.tr x neseyinelek@gmail.com k kazizi@dogus.edu.tr. 1550-7998= 2011=83(11)=114009(14). (1). j K1 ð1400Þi ¼j K1a i cos j K1b i sin; where the mixing angle takes the values in the interval 37 58 , 58 37 [15–19]. The sign. 114009-1. Ó 2011 American Physical Society.

(2) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011). ambiguity for the mixing angle is correlated with the fact that one can add arbitrary phase to the j K1a i and j K1b i. Studies on B ! K1 ð1270Þ and ! K1 ð1270Þ lead to the following value for [20]: ¼ ð34 13Þ :. (i) correlation functions corresponding to the Ds ðBs Þ DðBÞ K0 vertex: for DðBÞ off shell: Z 0 DðBÞ ¼ i2 d4 xd4 yeip x eiqy h0jT ðK0 ðxÞDðBÞ ðyÞ. (2). In the present work, contributing the quark-quark and quark-gluon condensate diagrams as nonperturbative effects, we evaluate the corresponding correlation functions when both BðDÞ and K0 ðKÞðK1 Þ mesons are off shell. Note that recently, we have investigated the Ds DK ð892Þ, and Bs BK ð892Þ vertices for K being the vector meson in the framework of the three-point QCD sum rules in [21]. Moreover, the following coupling constants have been investigated via three-point and light cone QCD sum rules in the literature: D D [22,23], DD [24], DDJ= c [25], D DJ= c [26], D D [27,28], D D J= c [29], Ds D K, Ds DK [27,30], D0 Ds K, Ds0 DK [31], DD! [32], D D [33], D D [34], Bs0 BK, Bs1 B K [35], Bs0 BK [36], a0 0 , a0 0 0 [37], a0 Kþ K [38] and f0 Kþ K [38,39]. The outline of the paper is as follows. In Sec. II, we introduce the responsible correlation functions, and we obtain QCD sum rules for the strong coupling constant of the considered vertices. For each of the scalar, pseudoscalar, and axial kaon cases, we will calculate the correlation function when both the BðDÞ and K0 ðKÞðK1 Þ mesons are off shell. In the case of the K1 meson, first we will calculate the QCD sum rules for the vertices Bs B K1a and Ds D K1b , then using the relations in Eq. (1), we will acquire the QCD sum rules for the vertices Bs B K1 ð1270Þ and Ds D K1 ð1400Þ. In obtaining the sum rules for physical quantities, we will consider both light quark-light quark and light quark-gluon condensate diagrams as nonperturbative contributions. Finally, in Sec. III, we numerically analyze the obtained sum rules for the considered strong coupling constants. We will obtain the numerical values for each coupling constant when both the BðDÞ and K0 ðKÞðK1 Þ states are off shell. Then taking the average of the two off-shell cases, we will obtain final numerical values for each coupling constant. In this section, we also compare our result on gDs DK with existing prediction in the literature. II. QCD SUM RULES FOR THE STRONG COUPLING CONSTANTS. Ds ðBs Þy ð0ÞÞj0i; for K0 off shell: Z 0 K0 ¼ i2 d4 xd4 yeip x eiqy h0jT ðDðBÞ ðxÞK0 ðyÞ Ds ðBs Þy ð0ÞÞj0i; (ii) correlation functions corresponding to Ds ðBs Þ DðBÞ K vertex: for DðBÞ off shell: Z 0 DðBÞ ¼ i2 d4 xd4 yeip x eiqy h0jT ðK ðxÞDðBÞ ðyÞ . . Ds ðBs Þy ð0ÞÞj0i;. (4) the. (5). for K off shell: Z 0 2 d4 xd4 yeip x eiqy h0jT ðDðBÞ ðxÞK ðyÞ K ¼i . . Ds ðBs Þy ð0ÞÞj0i;. (iii) correlation functions corresponding to Ds ðBs Þ DðBÞ K1 vertex: for DðBÞ off shell: Z 0 2 DðBÞ ðyÞ 1 DðBÞ d4 xd4 yeip x eiqy h0jT ðK ¼ i ðxÞ . . Ds ðBs Þy ð0ÞÞj0i; 2 DðBÞ 0 ¼ i. Z. (6) the. (7). 0. DðBÞ 1 d4 xd4 yeip x eiqy h0jT ðK ðyÞ 0 ðxÞ . . Ds ðBs Þy ð0ÞÞj0i;. (8). for K1 off shell: Z 0 2 1 1 d4 xd4 yeip x eiqy h0jT ðDðBÞ ðxÞK K ¼ i ðyÞ . In this section, we obtain QCD sum rules for the strong coupling constants associated with the Ds ðBs Þ DðBÞ K0 , Ds ðBs Þ DðBÞ K, and Ds ðBs Þ DðBÞ K1 vertices. We start our discussion considering the sufficient correlation functions responsible for the corresponding strong transition involving each K0 , K and K1 mesons when both DðBÞ and K0 ðKÞðK1 Þ are off shell. The following three-point correlation functions describe the considered strong transitions:. (3). . Ds ðBs Þy ð0ÞÞj0i; 2 1 K 0 ¼ i. Z. (9). 0. 1 d4 xd4 yeip x eiqy h0jT ðDðBÞ ðxÞK 0 ðyÞ . . Ds ðBs Þy ð0ÞÞj0i;. (10). where T is the time ordering product, q is the momentum of the off-shell state, p0 is the momentum of the final. 114009-2.

(3) STRONG COUPLING CONSTANTS OF BOTTOM AND . . .. on-shell state. We will set the momentum of the initial state as p ¼ p0 þ q. In the vertex containing K1 meson, we have two correlation functions for both off-shell cases since this 1 meson couples into two interpolating currents K and K1 0 . We will define these couplings in terms of G-parity conserving and G-parity violating decay constants later. The interpolating currents, which produce the considered mesons from the vacuum with the same quantum numbers as the interpolating currents can be written in terms of the quark field operators as following form: . K0 ðxÞ ¼ sðxÞUuðxÞ; K ðxÞ ¼ sðxÞ5 uðxÞ; 1 K ðxÞ ¼ sðxÞ 5 uðxÞ;. (11). 1 0 K 0 ðxÞ ¼ sðxÞ 5 uðxÞ;. DðBÞ ðxÞ ¼ uðxÞ 5 cðbÞðxÞ; Ds ðBs Þ ðxÞ ¼ sðxÞ5 cðbÞðxÞ; . . s ðBs Þ D ðxÞ ¼ sðxÞ cðbÞðxÞ; . where U stands for unit matrix and u, s, c and b are the up, strange, charm, and bottom quark fields, respectively.. PHYSICAL REVIEW D 83, 114009 (2011). According to general philosophy of the QCD sum rules, we calculate the aforementioned correlation functions in two different representations. In physical or phenomenological representation, we calculate them in terms of hadronic parameters. In QCD or theoretical representation, we evaluate them in terms of QCD degrees of freedom like quark masses, quark condensates, etc. with the help of the operator product expansion, where the perturbative and nonperturbative contributions are separated. The QCD sum rules for strong coupling constants are obtained equating these two different representations through dispersion relation. To suppress contributions of the higher states and continuum, we will apply double Borel transformation with respect to the momentum squared of the initial and final on-shell states to both sides of the obtained sum rules. First, let us focus on the calculation of the physical sides of the aforesaid correlation functions, for example, when DðBÞ meson is off shell. Saturating the correlation functions with the complete sets of three participating particles and isolating the ground states and after some straightforward calculations, we obtain: (i) physical representation corresponding to the Ds ðBs Þ DðBÞ K0 vertex:. . DðBÞ. h0jK0 jK0 ðp0 Þih0jDðBÞ jDðBÞðqÞihK0 ðp0 ÞDðBÞðqÞjDs ðBs ÞðpÞihDs ðBs ÞðpÞjDs ðBs Þ j0i þ ...; ¼ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þðp02 m2K Þ. (12). 0. (ii) physical representation corresponding to the Ds ðBs Þ DðBÞ K vertex: Ds ðBs Þ h0jK jKðp0 Þih0jDðBÞ jDðBÞðqÞihKðp0 ÞDðBÞðqÞjDs ðBs Þðp; ÞihDs ðBs Þðp; Þj j0i DðBÞ þ ...; ¼ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þðp02 m2K Þ (iii) physical representation corresponding to the Ds ðBs Þ DðBÞ K1aðbÞ vertex: Ds ðBs Þ a 0 0 a 0 0 1 h0jDðBÞ jDðBÞðqÞihDs ðBs Þðp; Þj j0i h0jK jK1 ðp ; ÞihK1 ðp ; ÞDðBÞðqÞjDs ðBs Þðp; Þi DðBÞ ¼ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þ ðp02 m2Ka Þ 1 K1 b 0 0 b 0 0 h0j jK1 ðp ; ÞihK1 ðp ; ÞDðBÞðqÞjDs ðBs Þðp; Þi þ þ ...; ðp02 m2Kb Þ. (13). (14). 1. DðBÞ 0. ðB Þ K s s h0jDðBÞ jDðBÞðqÞihDs ðBs Þðp; ÞjD j0i h0j1 0 jK1a ðp0 ; 0 ÞihK1a ðp0 ; 0 ÞDðBÞðqÞjDs ðBs Þðp; Þi ¼ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þ ðp02 m2Ka Þ 1 K1 b 0 0 b 0 0 h0j0 jK1 ðp ; ÞihK1 ðp ; ÞDðBÞðqÞjDs ðBs Þðp; Þi þ ...; þ ðp02 m2Kb Þ. (15). 1. where . . .. represents the contributions of the higher states and continuum, and and 0 are the polarization vectors associated with the Ds and K1aðbÞ mesons, respectively. From the above equations it is clear that to proceed we need to define the following matrix elements in terms of decay constants as well as strong coupling constants:. 114009-3.

(4) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011) K0. h0j jK0 ðp0 Þi ¼ mK0 fK0 ; h0jK jKðp0 Þi ¼ h0jDðBÞ jDðBÞðqÞi ¼ h0jDs ðBs Þ jDs ðBs ÞðqÞi ¼ . m2K fK ; mu þ ms m2DðBÞ fDðBÞ mcðbÞ þ mu. ;. (16). m2Ds ðBs Þ fDs ðBs Þ mcðbÞ þ ms. ;. . Ds ðBs Þ j0i ¼ mDs ðBs Þ fDs ðBs Þ ; hDs ðBs Þðp; Þj. hK0 ðp0 ÞDðBÞðqÞjDs ðBs ÞðpÞi ¼ gDs DK0 ðBs BK0 Þ p p0 ; hKðp0 ÞDðBÞðqÞjDs ðBs Þðp; Þi ¼ gDs DKðBs BKÞ ðp0 qÞ ; hK1aðbÞ ðp0 ; 0 ÞDðBÞðqÞjDs ðBs Þðp; Þi ¼ gDs DKaðbÞ ðBs BKaðbÞ Þ fm2Ds ðBs Þ ð 0 : Þ þ ð 0 :pÞð :p0 Þg; 1. 1. and. a 0 0 0 1 h0jK jK1 ðp ; Þi ¼ mK1a fK1a ;. 9 > =. b 0 0 0 0 0 0 > 1 ; h0jK 0 jK1 ðp ; Þi ¼ fK b? ð p0 0 p Þ 1. a 0 0 1 h0jK 0 jK1 ðp ; Þi. ¼. ?;K a fK1a a0 1 ð 0 p00. 9 00 p0 Þ; > =. k;K1b 0. . b 0 0 1 h0jK jK1 ðp ; Þi ¼ mK b fK b? a0 1 1. > ;. where fK0 , fK , fDðBÞ , fDs ðBs Þ and fDs ðBs Þ are leptonic decay constants of the K0 , K, DðBÞ, Ds ðBs Þ, and Ds ðBs Þ mesons, respectively. The fK1a and fKb? are decay constants encoun1 tered to the calculations from both definitions of the G-parity conserving and violating matrix elements for the axial K1a and K1b astates (for b more details see ?;K k;K [15,20,40,41]). The a0 1 and a0 1 are zeroth order Gegenbauer moments. In the above equations, the gDs DK0 ðBs BK0 Þ , gDs DKðBs BKÞ , and gDs DKaðbÞ ðBs BKaðbÞ Þ are strong 1 1 coupling constants, which we are going to obtain QCD sum rules for them in this section. Using Eqs. (12)–(16), and summing over the polarization vectors using the. 0 0 ¼ g þ. p0 p0 ; m2 aðbÞ K1. ¼ g þ. p p ; m2Ds ðBs Þ (18). the final physical representations of the correlation functions for each vertices in the case of DðBÞ off shell is obtained as. 114009-4. G-parity conserving definitions; (17) G-parity violating definitions;. (i) Ds ðBs Þ DðBÞ K0 vertex: 2 DðBÞ ¼ gDðBÞ Ds DK ðBs BK Þ ðq Þ 0. . 0. fDðBÞ m2DðBÞ fDs ðBs Þ m2D ðB Þ s s mcðbÞ þmu mcðbÞ þms m2DðBÞ Þðp02 m2K Þðp2 m2Ds ðBs Þ Þ 0. fK0 mK0 2ðq2 . ðm2Ds ðBs Þ þ m2K q2 Þ þ . . . :; 0. (19). (ii) Ds ðBs Þ DðBÞ K vertex: DðBÞ 2 DðBÞ ¼ gD DKðB BKÞ ðq Þ s s 2 f m2 fDðBÞ mDðBÞ mcðbÞ þmu 2 ðq m2DðBÞ Þðp02 m2K Þðp2 m2Ds ðBs Þ Þ. fDs ðBs Þ mDs ðBs Þ msKþmKu. . m2 q2 1þ K 2 p 2p0 þ ...:; mDs. (20).

(5) STRONG COUPLING CONSTANTS OF BOTTOM AND . . .. (iii). Ds ðBs Þ DðBÞ . DðBÞ K1 vertex: mK1a fK1a 2 ¼ gDðBÞ ðq Þ a a Ds DK1 ðBs BK1 Þ ðp02 m2Ka Þ 1. ðq2 Þ þ gDðBÞ D DKb ðB BK b Þ 1. s. 1. s. k;K1b . mK1b fKb? a0 1. ðp02 m2Kb Þ 1. . fDðBÞ m2DðBÞ mcðbÞ þmu fDs ðBs Þ mDs ðBs Þ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þ. fm2Ds ðBs Þ g þ other structuresg þ . . . ;. PHYSICAL REVIEW D 83, 114009 (2011). sides of the correlation functions corresponding to the 1 vertices containing the K and K1 with current K (K1 K1 with current 0 ), respectively. In a similar way, one can easily obtain the physical representations of the correlation functions associated with the K0 ðKÞðK1 Þ off shell. Now, we calculate the QCD or theoretical sides of the considered correlation functions. The QCD representations of the correlation functions are obtained in the deep Euclidean region, where p2 ! 1 and p02 ! 1 via operator product expansion. To this aim, each correlation function in QCD side is written in terms of the perturbative and nonperturbative parts as. (21) . i ¼ iper þ inonper ;. DðBÞ 2 gDðBÞ 0 ¼ Ds DKa ðBs BKa Þ ðq Þ 1. 1. fK1a a0. ðq2 Þ þ gDðBÞ D DKb ðB BK b Þ s. 1. s. 1. where i stands for DðBÞ or K0 ðKÞðK1 Þ and the perturbative parts are defined in terms of a double dispersion integral as follows:. ðp02 m2Ka Þ 1. . fK b ? 1. ðp02 m2Kb Þ 1. . (23). ?;K1a. fDðBÞ m2DðBÞ mcðbÞ þmu fDs ðBs Þ mDs ðBs Þ ðq2 m2DðBÞ Þðp2 m2Ds ðBs Þ Þ. iper ¼ . 1 Z 0Z i ðs; s0 ; q2 Þ ds ds 2 4 ðs p2 Þðs0 p02 Þ. þ subtraction terms;. (24). fm2Ds ðBs Þ ðg p00 g0 p0 Þ þ other structuresg þ . . . ;. (22). where to calculate the coupling constants, we will choose the structures p and g ðg p00 g0 p0 Þ from both. where i ðs; s0 ; q2 Þ are called spectral densities. In order to obtain the spectral density, we need to calculate the bare loop diagrams (a) and (d) in Fig. 1 for DðBÞ and K0 ðKÞðK1 Þ off shell, respectively. We calculate these diagrams in. FIG. 1. Diagrams considered in the calculations. The first and third line diagrams refer to the BðDÞ off shell, and the second and fourth line diagrams show the case when K0 ðKÞðK1 Þ is off shell.. 114009-5.

(6) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011). terms of the usual Feynman integrals with the help of the Cutkosky rules, where the quark propagators are replaced by 1 2 2 Dirac delta function, i.e., q2 m 2 ! ð2iÞ ðq m Þ. As a result, the spectral densities are obtained as follows: (i) Ds ðBs Þ DðBÞ K0 vertex: (i) DðBÞ off shell: N (25) DðBÞ ðs;s0 ;q2 Þ ¼ 1=2 c 0 2 fms ðmu ðms þ mu Þ q2 Þ smu mcðbÞ ððms þ mu Þ2 s0 mcðbÞ ðms þ mu ÞÞg; 2

(7) ðs;s ;q Þ (ii) K0 off shell: . K0 ðs;s0 ;q2 Þ ¼. Nc fms s0 þ mcðbÞ ððms þ mu Þ2 q2 Þ m2cðbÞ ðms þ mu Þ þ mu ðs ms ðms þ mu ÞÞg; 1=2 2

(8) ðs;s0 ;q2 Þ. (26). (ii) Ds ðBs Þ DðBÞ K vertex: (i) DðBÞ off shell: N DðBÞ ðs; s0 ; q2 Þ ¼ 3=2 c 0 2 ½ðmu ms Þðq2 sÞðmcðbÞ m2s þ mu ðs m2s q2 ÞÞ s0 ðm3s mu þ 2m3cðbÞ ðmu ms Þ

(9) ðs; s ; q Þ 2m2s q2 þ m2cðbÞ ð2ms mu þ q2 sÞ þ q2 ðs q2 Þ þ ms mu ðs þ q2 Þ þ mcðbÞ ðms mu Þðm2s þ q2 þ sÞÞ 2. s0 ðm2cðbÞ mcðbÞ ms þ mcðbÞ mu þ q2 Þ;. (27). (ii) K off shell: K ðs; s0 ; q2 Þ ¼. Nc ½ðmcðbÞ 3=2

(10) ðs; s0 ; q2 Þ. mu Þðq2 sÞðm2cðbÞ ðmcðbÞ mu Þ þ mu ðms mu q2 þ sÞÞ þ ðm3cðbÞ ðms mu Þ. þ 2m3s mu þ m2cðbÞ ðms mu 2q2 Þ þ m2s ðq2 sÞ þ q2 ðs q2 Þ ms mu ðq2 þ sÞ 2. þ mcðbÞ ð2m3s þ 2m2s mu þ mu ðq2 þ sÞ þ ms ðq2 þ sÞÞÞs0 þ ðmcðbÞ ms þ m2s þ ms mu þ q2 Þs0 ;. (28). (iii) Ds ðBs Þ DðBÞ K1 vertex: (i) DðBÞ off shell: DðBÞ ðs; s0 ; q2 Þ ¼ 2Nc I0 ðs; s0 ; q2 Þ½2m3s þ 2m2s mu 2mcðbÞ ms ðms þ mu Þ þ ms ðs þ s0 q2 Þ þ 4AðmcðbÞ mu Þ 1 þ Bð2mu s þ mcðbÞ ðq2 s s0 Þ þ ms ðq2 þ 3s þ s0 ÞÞ þ Cð2mcðbÞ s0 þ mu ðq2 þ s þ s0 Þ þ ms ðq2 þ s þ 3s0 ÞÞ;. (29). DðBÞ ðs; s0 ; q2 Þ ¼ 4Nc I0 ðs; s0 ; q2 Þ½2A þ ðmcðbÞ ms m2s Þ ðB þ 2GÞs þ Hðq2 s s0 Þ þ CðmcðbÞ ðms þ mu Þ 2 ms ðms þ mu Þ þ q2 s s0 Þ;. (30). (ii) K1 off shell: 0 2 0 2 2 0 1 K 1 ðs; s ; q Þ ¼ 2Nc I0 ðs; s ; q Þ½2mcðbÞ ðmcðbÞ þ ms mu mcðbÞ ðms þ mu Þ þ s Þ 4DðmcðbÞ mu Þ þ Eð2mcðbÞ ms mu Þðq2 s s0 Þ þ 2Fs0 ð2mcðbÞ þ ms þ mu Þ; 0 2 0 2 2 1 K 2 ðs; s ; q Þ ¼ 4Nc I0 ðs; s ; q Þ½2D þ mcðbÞ mcðbÞ ms Es FðmcðbÞ ðmcðbÞ ms mu Þ þ ms mu Þ;. K1 1 where 1 and 2 correspond to the currents K and 0 , respectively, and. 114009-6. (31). (32).

(11) STRONG COUPLING CONSTANTS OF BOTTOM AND . . .. PHYSICAL REVIEW D 83, 114009 (2011). 1 fm4 q2 þ m4cðbÞ s0 þ q2 ss0 þ m2cðbÞ ðm2s ðq2 þ s s0 Þ þ s0 ðq2 s þ s0 ÞÞ m2s q2 ðq2 þ s þ s0 Þg; 2 s 1 B ¼ fm2s ðq2 s þ s0 Þ s0 ð2m2cðbÞ q2 s þ s0 Þg; 1 C ¼ fm2cðbÞ ðq2 þ s þ s0 Þ þ sðq2 s þ s0 Þ m2s ðq2 s þ s0 Þg; 1 D ¼ fm4cðbÞ q2 þ m4s s0 þ q2 ss0 m2cðbÞ q2 ðq2 þ s þ s0 Þ þ m2s ðm2cðbÞ ðq2 þ s s0 Þ þ s0 ðq2 s þ s0 ÞÞg; 2 1 E ¼ fs0 ð2m2s q2 s þ s0 Þ m2cðbÞ ðq2 s þ s0 Þg; 1 fðm2s þ m2cðbÞ þ sÞðq2 þ s þ s0 Þ 2sðm2cðbÞ þ s0 Þg; F¼

(12) ðs;s0 ;q2 Þ 1 G ¼ 2 f3m4cðbÞ s0 ðq2 s s0 Þ þ m4s ð2q4 ðs s0 Þ2 q2 ðs þ s0 ÞÞ ss0 ð2q4 þ ðs s0 Þ2 þ q2 ðs þ s0 ÞÞ þ m2s ðq6 q4 ðs þ s0 Þ þ ðs s0 Þ2 ðs þ s0 Þ q2 ðs2 6ss0 þ s02 ÞÞ 2m2cðbÞ ðs0 ðq4 2s2 A¼. (33). þ q2 ðs 2s0 Þ þ ss0 þ s02 Þ þ m2s ðq4 þ s2 þ ss0 2s02 þ q2 ð2s þ s0 ÞÞÞg; H¼. 1 2 4 fs ðq 2q2 ðs 2s0 Þ þ ðs s0 Þ2 Þ þ m4s ðq4 þ 2q2 ð2s s0 Þ þ ðs s0 Þ2 Þ 2 þ m4cðbÞ ðq4 þ s2 þ 4ss0 þ s02 2q2 ðs þ s0 ÞÞ 2m2s sð2q4 þ ðs s0 Þ2 þ q2 ðs þ s0 ÞÞ 2m2cðbÞ ðm2s ðq4 2s2 þ q2 ðs s0 Þ þ ss0 þ s02 Þ þ sðq4 þ s2 þ ss0 2s02 þ q2 ð2s þ s0 ÞÞÞg;. and I0 ðs; s0 ; q2 Þ ¼. 1 4

(13) 1=2 ðs; s0 ; q2 Þ. ¼ q4 þ ðs s0 Þ2 2q2 ðs þ s0 Þ.

(14) ða; b; cÞ ¼ a2 þ b2 þ c2 2ac 2bc 2ab: (34). In the spectral densities, Nc ¼ 3 is the color number, and we have kept terms linear in mu . Now, we proceed to calculate the nonperturbative contributions in QCD side. We consider the quark-quark and quarkgluon condensate diagrams presented as (b), (c), (e), (f), (g), (h), (i), (j), (k), (l), (m), and (n) in Fig. 1. Contributions of the diagrams (c), (e), (f), (g), (i), (k), (l), (m), and (n) in Fig. 1 are zero since applying double Borel transformation with respect 2 to the both variables p2 and p0 will kill their contributions because of only one variable appearing in the denominator in these cases. Hence, we consider contributions of only diagrams (b), (h), and (j) in Fig. 1. As a result, we obtain: (i) Ds ðBs Þ DðBÞ K0 vertex: (i) DðBÞ off shell: 2 2 2 2 2 hssi 2mcðbÞ mu mcðbÞ þ q 1 1 m0 ð4mcðbÞ mu mcðbÞ þ q Þ m20 m20 þ DðBÞ nonper ¼ 2 r r0 rr0 4rr0 4r2 4r2 r0 m20 ðm2cðbÞ 4mcðbÞ mu q2 Þ m20 m20 ; (35) þ þ þ 4rr02 4r2 4rr0 (ii) K0 off shell:. K. 0 nonper ¼ 0;. (ii) Ds ðBs Þ DðBÞ K vertex: (i) DðBÞ off shell:. m m2 m m2 m DðBÞ ssi u0 þ 02 u0 þ 0 02u ; nonper ¼ h rr 4r r 2rr. (ii) K off shell:. 114009-7. (36). (37).

(15) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011). K nonper ¼ 0;. (38). (iii) Ds ðBs Þ DðBÞ K1 vertex: (i) DðBÞ off shell: 2 m2 q2 m2cðbÞ q2 q2 þ 2mu mcðbÞ m2cðbÞ m 2 1 1 1 1 cðbÞ DðBÞ nonper1 ¼ hssi 0 þ þ þ þ ; þ rr0 r2 r02 2r r0 24 2rr0 r2 r0 rr02 mcðbÞ m20 mcðbÞ DðBÞ ¼ h s si ; nonper2 rr0 24rr02. (39). (40). (ii) K1 off shell: 1 K nonper1 ¼ 0;. (41). 1 K nonper2 ¼ 0;. (42). K1 1 where r ¼ p2 m2cðbÞ and r0 ¼ p02 m2u . The nonper1 and nonper2 correspond to also the currents K and 0 , respectively. In this step, we equate the physical side in the case of K0 and the coefficients of the selected structures in physical sides of K and K1 to the corresponding QCD sides and obtain QCD sum rules for the considered strong coupling constants. To suppress the contributions of the higher states and continuum, we also apply the double Borel transformation 2 2 2 with respect to the variables p2 ðp2 ! M2 Þ and p0 ðp0 ! M0 Þ. Finally, we get the following sum rules for the considered coupling constants: (i) Ds ðBs Þ DðBÞ K0 vertex: (i) DðBÞ off shell: 2 02 2ðq2 m2DðBÞ ÞðmcðbÞ þ mu ÞðmcðbÞ þ ms Þ 2 2 m =M 2Þ ¼ emDs ðBs Þ =M e K0 ðq gDðBÞ Ds DK0 ðBs BK0 Þ 2 2 2 2 2 mDs ðBs Þ mDðBÞ mK0 fDs ðBs Þ fDðBÞ fK0 ðmDs ðBs Þ þ mK q Þ 0 Z s00 1 Z s0 2 0 02 DðBÞ ^ nonper 2 ds ds0 DðBÞ ðs;s0 ;q2 Þ½1 ðfDðBÞ ðs;s0 ÞÞ2 es=M es =M þ B ; 4 ðmcðbÞ þms Þ2 ðms þmu Þ2. (43) (ii) K0 off shell: K gD0s DK ðBs BK Þ ðq2 Þ 0 0. ¼. 2ðq2 m2K ÞðmcðbÞ þ mu ÞðmcðbÞ þ ms Þ. m2. =M2 m2. =M02. 0 e Ds ðBs Þ e Ds ðBs Þ m2Ds ðBs Þ m2DðBÞ mK0 fDs ðBs Þ fDðBÞ fK0 ðm2Ds ðBs Þ þ m2DðBÞ q2 Þ Z s00 1 Z s0 2 s0 =M 02 K0 K0 0 0 2 0 2 s=M 2 ds ds ðs; s ; q Þ½1 ðf ðs; s ÞÞ e e ; (44) 4 ðmcðbÞ þms Þ2 ðmcðbÞ þmu Þ2. (ii) Ds ðBs Þ DðBÞ K vertex: (i) DðBÞ off shell: ðq2 m2DðBÞ ÞðmcðbÞ þ mu Þðms þ mu Þ m2 =M2 m2 =M02 2 Ds ðBs Þ gDðBÞ ðq Þ ¼ e K 2 q2 e Ds DKðBs BKÞ m 2 2 K fDs ðBs Þ fDðBÞ fK mDs ðBs Þ mK mDðBÞ ð1 þ m2 Þ . Ds ðBs Þ. Z s00 1 Z s0 2 0 02 2 ds ds0 DðBÞ ðs; s0 ; q2 Þ½1 ðfDðBÞ ðs; s0 ÞÞ2 es=M es =M 2 2 4 ðmcðbÞ þms Þ ðms þmu Þ ^ DðBÞ þ B nonper. 114009-8. (45).

(16) STRONG COUPLING CONSTANTS OF BOTTOM AND . . .. PHYSICAL REVIEW D 83, 114009 (2011). (ii) K off shell: 2 gK Ds DKðBs BKÞ ðq Þ ¼. ðq2 m2K ÞðmcðbÞ þ mu Þðms þ mu Þ fDs ðBs Þ fDðBÞ fK mDs ðBs Þ m2K m2DðBÞ ð1 þ. m2 . m2DðBÞ q2 m2 Ds ðBs Þ. Ds ðBs Þ. e. =M2 m2 =M02 DðBÞ. e. Þ. Z s00 1 Z s0 2 0 02 2 ds ds0 K ðs; s0 ; q2 Þ½1 ðfK ðs; s0 ÞÞ2 es=M es =M ; 4 ðmcðbÞ þms Þ2 ðmcðbÞ þmu Þ2. (46). (iii) Ds ðBs Þ DðBÞ K1 vertex: (i) DðBÞ off shell: 2 02 m2Ka =M02 k;K1b mKb =M DðBÞ 2 2 1 1 a a b gDðBÞ ðq Þm f e þ g ðq Þm f a e b? K1 K1 K1 K 0 Ds DKa ðBs BK a Þ D DKb ðB BK b Þ 1. ¼. 1. 1. s. 1. 1. s. m2DðBÞ Þ 1 m2 =M2 Ds ðBs Þ e 2 m2DðBÞ 4 fDs ðBs Þ fDðBÞ mcðbÞ þmu m3Ds ðBs Þ ðq2. Z s0 ðmcðbÞ þms Þ2. ds. Z s00 ðms þmu Þ2. ds0 DðBÞ ðs; s0 ; q2 Þ 1. 2 0 02 DðBÞ ^ nonper1 ½1 ðfDðBÞ ðs; s0 ÞÞ2 es=M es =M þ B ; . 2 02 ?;K1a mKa =M 2 1 aa ðq Þf e gDðBÞ a a K 0 Ds DK1 ðBs BK1 Þ 1. ¼. ðq2 m2DðBÞ Þ. m2 . m2DðBÞ. fDs ðBs Þ fDðBÞ mcðbÞ þmu m3Ds ðBs Þ ½1 ðf. DðBÞ. þ. 0. e. =M Ds ðBs Þ. m2 b =M02 K 2 1 gDðBÞ ðq Þf b? e b b K Ds DK1 ðBs BK1 Þ 1 2. . s=M2 s0 =M02. 2. ðs; s ÞÞ e. (47). e. . . Z s00 1 Z s0 ds ds0 DðBÞ ðs; s0 ; q2 Þ 2 2 2 4 ðmcðbÞ þms Þ2 ðms þmu Þ. DðBÞ ^ þ Bnonper2 ;. (48). (ii) K1 off shell: k;K b a mK1b fKb? a0 1 a fK a m b K K1 K 1 gDs DKa ðBs BKa Þ ðq2 Þ 2 1 21 þ gD1 DKb ðB BKb Þ ðq2 Þ s s 1 1 1 1 ðq mKa Þ ðq2 m2Kb Þ 1 1 Z s0 Z s00 1 1 m2 =M2 m2 =M02 0 2 1 Ds ðBs Þ DðBÞ e e ds ds0 K ¼ 1 ðs; s ; q Þ 2 m2DðBÞ 2 2 4 ðm þm Þ ðm þm Þ 3 cðbÞ s cðbÞ u fDs ðBs Þ fDðBÞ mcðbÞ þmu mDs ðBs Þ 2 0 02 1 ^ K ½1 ðfK1 ðs; s0 ÞÞ2 es=M es =M þ B nonper1 ; . Ka gD1s DKa ðBs BKa Þ ðq2 Þ 1 1. ?;K1a. fK1a a0. ðq2 m2Ka Þ. 1. s. s. 1. 1. ¼. 1 m2. 3 fDs ðBs Þ fDðBÞ mcðbÞDðBÞ þmu mDs ðBs Þ. ½1 . m2 . e. Ds ðB sÞ. =M2 m2 =M02 DðBÞ. 2 0 02 ðfK1 ðs; s0 ÞÞ2 es=M es =M. e. . . fKb?. Kb. þ gD1 DKb ðB BKb Þ ðq2 Þ. (49). 1. ðq2 m2Kb Þ 1. Z s00 1 Z s0 0 2 1 ds ds0 K 2 2 ðs; s ; q Þ 4 ðmcðbÞ þms Þ2 ðmcðbÞ þmu Þ2. K1 ^ þ Bnonper2 ;. (50). ^ nonper represents the double Borel transformation of the nonperturbative part in each case, s0 and s00 are the where the B continuum thresholds and the functions fi ðs; s0 Þ inside the step functions are determined requiring that the arguments of the three functions coming from the Cutkosky rule vanish simultaneously. As a result, we find: (i) DðBÞ off shell:. 114009-9.

(17) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011). fDðBÞ ðs; s0 Þ ¼. (ii) K0 ðKÞðK1 Þ off shell: K ðKÞðK1 Þ. f1 0. ðs; s0 Þ ¼. 2sðm2s m2u þ s0 Þ þ ðm2cðbÞ m2s sÞðq2 þ s þ s0 Þ

(18) 1=2 ðm2cðbÞ ; m2s ; sÞ

(19) 1=2 ðs; s0 ; q2 Þ. ;. 2sðm2cðbÞ þ m2u s0 Þ þ ðm2cðbÞ m2s þ sÞðq2 þ s þ s0 Þ

(20) 1=2 ðm2cðbÞ ; m2s ; sÞ

(21) 1=2 ðs; s0 ; q2 Þ. (51). :. (52). Here, we should stress that the physical regions are imposed by the limits on the integrals and step functions in the integrands in the sum rules expressions. In order to subtract the contributions of the higher states and continuum, the quarkhadron duality assumption in the following form is used: higherstates ðs; s0 Þ ¼ OPE ðs; s0 Þðs s0 Þðs0 s00 Þ: (53) The double Borel transformation used in calculations is also defined in the following way: B^. 1 1 1 1 m2 =M2 m2 =M02 1 e 1 e 2 ! ð1Þmþn : 2 m 02 2 n 2 m1 ðmÞ ðnÞ ðp m1 Þ ðp m2 Þ ðM Þ ðM02 Þn1 2. (54). At the end of this section, we would like to mention that using Eqs. (1) and (16) the couplings to K1 ð1270Þ and K1 ð1400Þ are obtained in terms of the couplings to the K1aðbÞ as gDs DK1 ð1270ÞðBs BK1 ð1270ÞÞ ¼ gDs DK1a ðBs BK1a Þ sin þ gDs DK1b ðBs BK1b Þ cos (55) a a gDs DK1 ð1400ÞðBs BK1 ð1400ÞÞ ¼ gDs DK1 ðBs BK1 Þ cos gDs DK1b ðBs BK1b Þ sin III. NUMERICAL ANALYSIS In the present section, we numerically analyze the expressions of QCD sum rules obtained for the considered strong coupling constants. Some input parameters used in the calculations are mK ¼ ð493:677 0:016Þ MeV, mK0 ð800Þ ¼ ð672 40Þ MeV, mK0 ð1430Þ ¼ ð1425 50Þ MeV, mK1 ð1270Þ ¼ ð1272 7Þ MeV, mK1 ð1400Þ ¼ ð1403 7Þ MeV, mD ¼ ð1:8648 0:000 14Þ GeV, mB ¼ ð5:2792 0:0003Þ GeV, mDs ¼ ð1:96847 0:00033Þ MeV, mBs ¼ ð5:36630:0006Þ MeV, mBs ¼ ð5:4154 mDs ¼ ð2:1123 0:0005Þ GeV, 0:0014Þ GeV [42], mc ¼ 1:3 GeV, mb ¼ 4:7 GeV, ms ¼ 0:14 GeV [43], fK ¼ 160 MeV [44], fK0 ð800Þ ð1 GeVÞ ¼ ð340 20Þ MeV, fK0 ð1430Þ ð1 GeVÞ ¼ ð445 50Þ MeV fBs ¼ ð229 [45], fDs ¼ ð272 16020 Þ MeV, 31 2016 Þ MeV [46], fB ¼ 190 13 MeV [47], fD ¼ ð202 41 17Þ MeV [48], fDs ¼ ð286 44 ¼ 41Þ MeV [49], fBs ¼ 196 MeV [50], hssi ¼ 0:8huui 0:8ð0:24 0:01Þ3 GeV3 [43], m20 ¼ ð0:8 0:2Þ GeV2 [51], mK1a ¼ 1:31 GeV, mK1b ¼ 1:34 GeV, fK1a ð1 GeVÞ ¼ k;K1b. 0:25 GeV, fKb? ð1 GeVÞ ¼ 0:19 GeV, a0 1. ?;K A a0 1 ð1. ð1 GeVÞ ¼. 0:19 0:07 and GeVÞ ¼ 0:27þ0:03 0:17 [15,20,40]. The sum rule for strong coupling constants also contains four auxiliary parameters, namely, the continuum thresholds s0 and s00 related to the initial and final channels, respectively, as well as Borel mass parameters M2. and M02 . These quantities are mathematical objects, so according to the general criteria and standard procedure in QCD sum rules, our physical results should be insensitive to them. Therefore, we shall look for working regions of these quantities at which the dependence of coupling constants on these auxiliary parameters are weak. The working regions for the Borel mass parameters M2 and 2 M0 are determined demanding that both the contributions of the higher states and continuum are sufficiently suppressed and the contributions coming from the higher dimensional operators are small. Our calculations lead to the following working regions common for all cases: (i) DðÞ s DK0 ðKÞðK1 Þ vertex: (i) D off shell: 8 GeV2 M2 25 GeV2 and 2 5 GeV2 M0 15 GeV2 , (ii) K0 ðKÞðK1 Þ off shell: 6 GeV2 M2 15 GeV2 and 2 4 GeV2 M0 12 GeV2 , (ii) BðÞ s BK0 ðKÞðK1 Þ vertex: (i) B off shell: 14 GeV2 M2 30 GeV2 and 2 5 GeV2 M0 20 GeV2 , (ii) K0 ðKÞK1 off shell: 6 GeV2 M2 20 GeV2 and 2 5 GeV2 M0 15 GeV2 . The continuum thresholds, s0 and s00 are not completely arbitrary but they are correlated to the energy of the first excited states with the same quantum numbers as the considered interpolating currents. Our numerical calculations show that in the regions ðmi þ 0:3Þ2 s0 ðmi þ 0:7Þ2. 114009-10.

(22) STRONG COUPLING CONSTANTS OF BOTTOM AND . . .. PHYSICAL REVIEW D 83, 114009 (2011). 2 2 2 02 2 FIG. 2 (color online). Left: gðDÞ Ds DK0 ð800Þ ðQ ¼ 1 GeV Þ as a function of the Borel mass M with M ¼ 10 GeV . Right: ðDÞ 2 2 02 2 2 gDs DK ð800Þ ðQ ¼ 1 GeV Þ as a function of the Borel mass M with M ¼ 17 GeV . The continuum thresholds s0 ¼ 6:09 GeV2 0 and s00 ¼ 1:37 GeV2 have been used.. and ðmf þ 0:3Þ2 s00 ðmf þ 0:7Þ2 , respectively, for the continuum thresholds s and s0 , our results have weak dependence on these parameters. Here, mi is the mass of initial particle and the mf stands for the mass of the final on-shell state. For instance consider the gD Ds DK0 coupling constant at which the D meson is off shell. This coupling constant describe the strong transition Ds ! DK0 , and for this case mi ¼ mDs and mf ¼ mK0 . As an example, we present the dependence of strong ðDÞ coupling constant gD on Borel mass parameters at s DK0 ð800Þ 2 2 2 Q ¼ 1 GeV , where Q ¼ q2 in Fig. 2. This figure demonstrates a good stability of the results with respect to the variations of Borel mass parameters in their working regions. Now, we proceed to find the Q2 behavior of the considered strong coupling constants using the working regions for auxiliary parameters. First, we consider the scalar kaon case for both K0 ð800Þ and K0 ð1430Þ. The strong coupling constant in this case obeys from the following Boltzmann function: gðQ2 Þ ¼ A1 þ. A2 2. x0 1 þ exp½Q x . ½GeV1 :. In the case of pseudoscalar kaon and D off shell, the strong coupling constant is well described by the following monopolar fit parametrization: 2 gðDÞ Ds DK ðQ Þ ¼. (57). The value of coupling constant obtained at Q2 ¼ m2meson is presented in Table IV. The result for strong coupling constant of pseudoscalar case and an off-shell K meson can be well fitted by the exponential parametrization 2 ðQ =7:25 ðGeV ÞÞ gðKÞ 0:88; Ds DK ðQ Þ ¼ 3:55e 2. 2. (58). where using Q2 ¼ m2K , we obtain the result as also presented in Table IV. We also depict the final result for this case taking the average of two above obtained values. TABLE I. Parameters appearing in the fit function of the coupling constants for the Ds DK0 ð800Þ, Ds DK0 ð1430Þ, Bs BK0 ð800Þ, and Bs BK0 ð1430Þ vertices. A1 and A2 are in GeV1 units, while x0 and x are in the units of GeV2 .. (56). The values of the parameters A1 , A2 , x0 and x for the considered coupling constant form factors are given in Table I. The coupling constants are defined as the values of the form factors at Q2 ¼ m2meson , where mmeson is the mass of off-shell meson. The results of the coupling constants obtained using Q2 ¼ m2meson are given in Tables II and III. The final result for each coupling constant is obtained taking the average of the coupling constants obtained from two different off-shell cases. The errors in the numerical results are due to the uncertainties in determination of the working regions for the auxiliary parameters as well as the errors in the input parameters.. 8:76 ðGeV2 Þ ; Q2 þ 7:12 ðGeV2 Þ. A1. A2. x0. x. 3.468. 2:741. 8.067. 4.995. 0:024. 0.772. 5.723. 1.257. ðDÞ gD ðQ2 Þ s DK ð1430Þ. 4.712. 3:818. 24.863. 10.985. ðK ð1430ÞÞ gDs0DK ð1430Þ ðQ2 Þ 0. 0:022. 0.772. 4.729. 1.637. gBðBÞs BK ð800Þ ðQ2 Þ. 4.151. 1:932. 13.842. 12.149. ðK0 ð800ÞÞ gBs BK ðQ2 Þ 0 ð800Þ. 0:017. 0.547. 5.431. 1.121. 2.055. 0:207. 11.239. 5.084. 0:004. 0.255. 4.819. 1.146. ðDÞ gD ðQ2 Þ s DK0 ð800Þ ðK ð800ÞÞ gDs0DK ð800Þ ðQ2 Þ 0 0. 0. gBðBÞs BK ð1430Þ ðQ2 Þ 0 ðK0 ð1430ÞÞ gBs BK ð1430Þ ðQ2 Þ 0. 114009-11.

(23) SUNDU et al.. PHYSICAL REVIEW D 83, 114009 (2011) TABLE II.. Value of the g. Ds DK0 ð800;1430Þ. gDs DK0 ð800Þ. coupling constant in GeV1 unit.. Q2 ¼ m2D. Q2 ¼ m2K ð800Þ. 0:97 0:02. 0:74 0:05. 0:85 0:08. Q2 ¼ m2D. Q2 ¼ m2K ð1430Þ. Average. 1:16 0:12. 0:49 0:07. 0:83 0:09. gDs DK0 ð1430Þ. Average. 0. 0. TABLE III. Value of the gBs BK0 ð800;1430Þ coupling constant in GeV1 unit.. gBs BK0 ð800Þ. Q2 ¼ m2B. Q2 ¼ m2K ð800Þ. Average. 2:28 0:18. 0:53 0:09. 1:14 0:21. Q2 ¼ m2B. Q2 ¼ m2K ð1430Þ. Average. 1:85 0:53. 0:25 0:04. 1:05 0:32. gBs BK0 ð1430Þ. TABLE IV.. 0. 0. Value of the gDs DK coupling constant.. gDs DK (Present work) gDs DK ([30]). Q2 ¼ m2D. Q2 ¼ m2K. Average. 2:79 0:24 2.72. 2:99 0:26 2.87. 2:89 0:25 2:84 0:31. Table IV also shows the predictions of [30] on gDs DK as the only existing previously calculated coupling constant among the considered vertices. Comparing our results with that of [30], we see a good consistency between two predictions. Similarly, for Bs BK vertex, our result for the pseudoscalar kaon and B off shell is better extrapolated by the exponential fit parametrization 2 ðQ gðBÞ Bs BK ðQ Þ ¼ 0:66e. 2 =23:34. ðGeV2 ÞÞ. þ 0:23;. (59). and in the K off-shell case by the parametrization 2 ðQ =4:02 ðGeV ÞÞ 1:03: gðKÞ Bs BK ðQ Þ ¼ 4:39e 2. 2. gBs BK. TABLE VI. Parameters appearing in the fit function of the coupling constants for the Ds DK1 ð1270Þ, Ds DK1 ð1400Þ, Bs BK1 ð1270Þ, and Bs BK1 ð1400Þ vertices. A1 and A2 are in GeV1 units, while x0 and x are in the units of GeV2 . A2. x0. x. 5.062. 2:337. 1.182. 1.531. 73.848. 87:162. 118.101. 74.590. 2 1 ð1270ÞÞ gðK Ds DK1 ð1270Þ ðQ Þ. 0.137. 1:507. 6.951. 1.845. 2 1 ð1400ÞÞ gðK Ds DK1 ð1400Þ ðQ Þ. 0:106. 1.234. 6.843. 1.847. gBðBÞs BK1 ð1270Þ ðQ2 Þ. 0.764. 0.412. 11.343. 4.708. gBðBÞs BK1 ð1400Þ ðQ2 Þ. 2.463. 2:178. 38.732. 18.980. 0.047. 23681:595 31:416. 2.914. 0:021. (60). Using the same procedure as above, we find the values depicted in Table V. In the case of axial-vector kaon, the strong coupling constant obey also the same Boltzmann function as the scalar case. The values of the parameters A1 , A2 , x0 and x for coupling constants in this case are given in Table VI. TABLE V.. The same procedure as in the scalar and pseudoscalar cases leads to the numerical results for the corresponding coupling constants as presented in Tables VII and VIII. In summary, the strong coupling constants, gBs BK0 , gDs DK0 , gBs BK , gDs DK , gBs BK1 , and gDs DK1 , have been. Value of the gBs BK coupling constant.. A1 ðDÞ 2 gD DK ð1270Þ ðQ Þ s 1 ðDÞ 2 gD DK ð1400Þ ðQ Þ s 1. Q2 ¼ m2B. Q2 ¼ m2K. Average. 2 1 ð1270ÞÞ gðK Bs BK1 ð1270Þ ðQ Þ. 2:40 0:22. 3:62 0:34. 3:01 0:28. 2 1 ð1400ÞÞ gðK Bs BK1 ð1400Þ ðQ Þ. 114009-12. 0.282. 3.080. 1.233.

(24) STRONG COUPLING CONSTANTS OF BOTTOM AND . . . TABLE VII. Values of the g. Ds DK1 ð1270Þ. gDs DK1 ð1270Þ. gDs DK1 ð1400Þ. TABLE VIII.. gBs BK1 ð1270Þ. gBs BK1 ð1400Þ. PHYSICAL REVIEW D 83, 114009 (2011) and g. Ds DK1 ð1400Þ. coupling constants in GeV1 .. Q2 ¼ m2D. Q2 ¼ m2K1 ð1270Þ. Average. 2:83 0:09. 1:36 0:14. 2:09 0:82. Q2 ¼ m2D. Q2 ¼ m2K1 ð1400Þ. Average. 0:97 0:15. 1:12 0:54. 1:04 0:78. Values of the gBs BK1 ð1270Þ and gBs BK1 ð1400Þ coupling constants in GeV1 . Q2 ¼ m2B. Q2 ¼ m2K1 ð1270Þ. Average. 1:18 0:07. 0:81 0:45. 1:99 0:11. Q2 ¼ m2B. Q2 ¼ m2K1 ð1400Þ. Average. 0:35 0:05. 0:26 0:04. 0:30 0:05. calculated in the framework of three-point QCD sum rules. The correlation functions of the considered vertices when both BðDÞ and K0 ðKÞðK1 Þ mesons are off shell are evaluated. The final numerical values have been obtained taking the average of the numerical values obtained from both off-shell cases. In the case of the axial vector K1 , which is either K1 ð1270Þ or K1 ð1400Þ, the mixing between these two states have also been taken into account. A comparison of the obtained result on Ds DK as the only previously. calculated coupling constant among the considered strong coupling constants has also been made.. [1] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B147, 385 (1979). [2] M. A. Shifman, A. I. Vainstein, and V. I. Zakharov, Nucl. Phys. B147, 448 (1979). [3] L. J. Reinders, H. Rubinstein, and S. Yazaki, Phys. Rep. 127, 1 (1985). [4] S. Narison, Lect. Notes Phys. 26, 1 (1990). [5] S. Leupold and U. Mosel, 8th International Conference on the Structure of Baryons (Baryons 98), Bonn, Germany, 1998, edited by D. W. Menze and B. Metsch (World Scientific, Singapore, 1999) p. 117. [6] S. Leupold and U. Mosel, Prog. Part. Nucl. Phys. 42, 221 (1999). [7] S. Leupold, W. Peters, and U. Mosel, Nucl. Phys. A628, 311 (1998). [8] F. Klingl, N. Kaiser, and W. Weise, Nucl. Phys. A624, 527 (1997). [9] T. Hatsuda and S. H. Lee, Phys. Rev. C 46, R34 (1992). [10] P. Colangelo and A. Khodjamirian, in At the Frontier of Particle Physics, edited by M. Shifman (World Scientific, Singapore, 2001), Vol. 3, p. 1495. [11] A. Yu. Khodjamirian, Phys. Lett. 90B, 460 (1980). [12] L. J. Reinders, H. R. Rubinstein, and S. Yazaki, Phys. Lett. 113B 411 (1982).. [13] A. I. Vainshtein, M. B. Voloshin, V. I. Zakharov, and M. A. Shifman, Sov. J. Nucl. Phys. 28, 237 (1978). [14] L. J. Reinders, H. Rubinstein, and S. Yazaki, Nucl. Phys. B213, 109 (1983). [15] J. P. Lee, Phys. Rev. D 74, 074001 (2006); H. Hatanaka and K.-C. Yang, Phys. Rev. D 77, 094023 (2008). [16] M. Suzuki, Phys. Rev. D 47, 1252 (1993). [17] H. Y. Cheng and C. K. Chua, Phys. Rev. D 69, 094007 (2004). [18] L. Burakovsky and J. T. Goldman, Phys. Rev. D 57, 2879 (1998). [19] H. Y. Cheng, Phys. Rev. D 67, 094007 (2003). [20] H. Hatanaka and K. C. Yang, Phys. Rev. D 78, 074007 (2008). [21] K. Azizi and H. Sundu, J. Phys. G 38, 045005 (2011). [22] F. S. Navarra, M. Nielsen, M. E. Bracco, M. Chiapparini, and C. L. Schat, Phys. Lett. B 489, 319 (2000). [23] F. S. Navarra, M. Nielsen, and M. E. Bracco, Phys. Rev. D 65, 037502 (2002). [24] M. E. Bracco, M. Chiapparini, A. Lozea, F. S.Navarra, and M. Nielsen, Phys. Lett. B 521, 1 (2001). [25] R. D. Matheus, F. S. Navarra, M. Nielsen, and R. R. da Silva, Phys. Lett. B 541, 265 (2002).. ACKNOWLEDGMENTS The authors thank E. Veli Veliev for his useful discussions. This work has been supported partly by the Scientific and Technological Research Council of Turkey ¨ BITAK) under the research project 110T284. (TU. 114009-13.

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