vertices in QCD sum rules

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1 Home Search Collections Journals About Contact us M IOPscience Analses of g D*sDK and g B*sBK vertices in QCD sum rules This article has been downloaded from IOPscience. Please scroll down to see the full text article. J. Phs.: Conf. Ser. 7 ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: The article was downloaded on 7/7/ at : Please note that terms and conditions appl.

2 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( Analses of g D s DK and g B s BK vertices in QCD sum rules J Y Süngü, H Sundu and K Azizi Department of Phsics, Kocaeli Universit, 8 Izmit, Turke Department of Phsics, Facult of Arts and Sciences, Doğuş Universit, Acıbadem-Kadıkö, 7 Istanbul, Turke jilmazkaa@kocaeli.edu.tr Abstract. The coupling constants g D s DK and g B s BK are calculated in the framework of QCD sum rules. We evaluate the correlation functions of these vertices considering both D(B and K mesons off-shell and obtain the results as g D s DK = (.89 ±. and g B s BK = (. ±.8.. Introduction A precise determination of the coupling constants is a vital task to get knowledge about cross sections and the nature and structure of the encountered particles. Heav-light pseudoscalar mesons strong coupling constants are of great importance especiall in evaluating charmonium cross sections. Experimentall, it is believed that in the production of the charmonium states like J/ψ and ψ from the B c or newl discovered charmonium X, Y and Z states b the BaBar and BELLE collaborations, there are intermediate two bod states containing D, D s, D and Ds mesons (for example, the kaon can annihilate the charmonium in a nuclear medium to give D and D s mesons, which deca to the final J/ψ and ψ states exchanging one or more virtual mesons. A similar stor would happen in decas of heav bottomonium. To exactl follow and analze the procedure in the experiment, we need to have knowledge about the coupling constants among the particles involved. In the literature, there have been series of works on coupling constants such as D Dπ [, ], DDρ [], DDJ/ψ [], D DJ/ψ [], D D π [6, 7], D D J/ψ [8], D s D K,DsDK [9, ], D D s K,D s DK[], DDω [], D D ρ [], D Dρ [], B s BK,B s B K [], DsDK (89, and BsBK (89 [] in the framework of QCD sum rules (QCDSR technique [6]. In this work, we calculate the DsDK and BsBK vertices using QCDSR (for details see [7]. These coupling constants belong to the low energ sector of QCD, which is far from perturbative regime. Therefore, for calculation of these coupling constants some nonperturbative methods are needed. QCDSR is one of the most promising and predictive one among all existing nonperturbing methods in studing the properties of hadrons. The paper is organized as follows. In section II, the model is shortl described. In this section, we calculate the correlation function when both the D(B and K mesons are off-shell. Then we obtain QCD sum rules for the strong coupling constants of the Ds D K and B s B K vertices. Finall in section III, we numericall analze the obtained strong coupling constant sum rules for the considered vertices. We will obtain the numerical values for each coupling constant when both the D(B and K states are off-shell. Then taking the average of the two Published under licence b Ltd

3 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( off-shell cases, we will obtain final numerical values for each coupling constant. In this section, we also compare our result on g D s DK with existing predictions in the literature [9].. QCD sum rules for the coupling constants The aim of this section is to calculate the coupling constants g D s DK and g B s BK which characterize the DsDK and BsBK decas, respectivel. We start b considering the threepoint correlation function. The three-point function associated to Ds DK(B sbk vertex for both D meson off-shell and K meson off-shell states is given respectivel b Π D(B µ (p,q = i d x d e ip x e iq T ( η K (x η D(B ( η D s(bs µ (, ( Π K µ (p,q = i d x d e ip x e iq T ( η D(B (x η K ( η D s (B s µ (. ( Here T is the time ordering product. Each meson interpolating field can be written in terms of the quark field operators as following: η K (x = s(xγ u(x, η D(B (x = u(xγ c(b(x, η D s (B s µ (x = s(xγ µ c(b(x. ( where u, s, c and b are the up, strange, charm and bottom quark field, respectivel. Each current has the same quantum numbers of the associated meson. According to the idea of the QCDSR, we should calculate this correlator both in terms of hadrons and in quark-gluon language, and then equate these representations. The first side, called phenomenological or phsical side, is obtained using hadronic degrees of freedom. The second, so called QCD or theoretical side is calculated using quark and gluon degrees of freedom b the help of the operator product expansion (OPE in deep Euclidean region. Firstl, let us focus on the calculation of the phsical side of the correlation function Eq.( for an off-shell D(B meson. The phsical part can be obtained b saturating Eq.( with the appropriate D(B, D s(b s and K states. After some straightforward calculation, we obtain: µ (p,p = ηk K(p η D(B D(B(q K(p D(B(q Ds (B s (p,ǫ D s (B s (p,ǫ ηd s (B s µ (q m D(B (p m Ds (B s (p m K Π D(B +... where... represents the contribution of the higher states and the continuum. The phenomenological side of the sum rule is defined in terms of meson masses, meson deca constants and coupling constants. We introduce the meson deca constants f K, f D s (B s and f D(B defined b the following matrix elements. η K K(p η D(B D(B(q = i m K f K m u + m s, = i m D(B f D(B m c(b + m u, Ds(B s(p,ǫ η D s (B s µ = m D s (Bs f D s (Bs ǫ µ, K(p D(B(q Ds(B s(p,ǫ = g D s DK(Bs BK (p q ǫ, ( (

4 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( where ǫ are the polarization vectors associated with the D s(b s. Finall, using Eqs.(-(, the phsical side of the correlation function for an off-shell D(B meson can be written as: Π D(B µ (p,p = g D(B f D DsDK(B sbk (q s (Bs f D(B f K m K m D m Ds (B s (q m D(B (p m Ds (B s (p m K (m c(b + m u (m s + m u [( + m K q ] p µ p µ. (6 m D s Similarl, we get the final expression of the phsical side of the correlation function for an off-shell K meson as: Π K µ (p,p = gd K f D s DK(B s BK(q s (Bsf D(B f K m K m D m Ds(B s (q m D(B (p m Ds(B s (p m K (m c(b + m u (m s + m u [( + m D(B q m D s ] p µ p µ. (7 To calculate the coupling constant, we will choose the structure, p µ from both sides of the correlation functions. Now, we concentrate on the QCD side, the correlation function is calculated at deep Euclidean space, where p and p in terms of the operator product expansion. For this aim, each correlation function, Π i µ (p,p, where i stands for D(B or K, can be written in terms of perturbative and non-perturbative parts as: Π QCD = Π per + Π nonper, (8 where the perturbative part is defined in terms of double dispersion integral as: Π per = π ds ρ(s,s,q ds + subtraction terms, (9 (s p (s p where ρ(s,s,q is called spectral densit. In order to obtain the spectral densit, we need to calculate the bare loop diagram (a and (d in Fig.( for D(B and K off-shell, respectivel. We calculate these diagrams in terms of the usual Fenman integration technique b the help of the Cutkosk rules, i.e., b replacing the quark propagators with Dirac delta function: q m ( πiδ(q m. After some straightforward calculations, we obtain the spectral densities as following: ρ D(B (s,s,q N [ c = λ / (s,s,q (m u m s (q s (m c(b m s + m u (s m s q s ( m s m u + m c(b (m u m s m s q + m c(b (m sm u + q s + q (s q + m s m u (s + q + m c(b (m s m u (m s + q + s ] s (m c(b m c(bm s + m c(b m u + q, ( N c ρ K (s,s,q = λ / (s,s,q + s + [ ( (m c(b m u (q s m c(b (m c(b m u + m u ( m s m u q ( m c(b (m s m u + m s m u + m c(b ( m sm u q + m s (q s + q (s q m s m u (q + s + m c(b ( m s + m sm u + m u (q + s + m s (q + s s + ( m c(b m s + m s + m sm u + q s ], (

5 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( D[B] D[B] D[B] Ds[B s] (a K Ds[B s] (b K Ds[B s] (c K K K K Ds[B s] D[B] D (d s[b s] D[B] D (e s[b s] D[B] (f D[B] D[B] D[B] D[B] Ds[B s] K Ds[B s] K Ds[B s] K Ds[B s] K (g (h (i (j K K K K D s[b s] x D[B] Ds[B s] D[B] Ds[B s] D[B] Ds[B s] D[B] (k (l (m (n Figure. (a and (d: Bare loop diagram for the D(B and K off-shell, respectivel; (b and (c: Diagrams corresponding to quark condensate for the D(B off-shell; (e and (f: Diagrams corresponding to quark condensate for the K off-shell; (g, (h, (i, (j: Diagrams corresponding to gluon-quark condensate for the D(B off-shell; (k, (l, (m, (n: Diagrams corresponding to gluon-quark condensate for the K off-shell. for the Ds DK and B sbk vertex associated with the off-shell D and K meson, respectivel. Here λ(a,b,c = a + b + c ac bc ab and N c = is the color number. To calculate the nonperturbative contributions in QCD side, we consider the quark condensate diagrams presented in (b, (c, (e, (f, (g, (h, (i, (j, (k, (l, (m and (n parts of Fig. (. There is also a numericall negligible contribution from the heav quark condensates, which we will not take into account in this calculation. Therefore, for the nonperturbative part, we onl encounter contributions coming from light quark condensates. Contributions of the diagrams (c, (e, (f, (g, (i, (k, (l, (m and (n in Fig. ( are zero since appling double Borel transformation with respect to both of the variables p and p will kill them because onl one variable appears in the denominator in these cases. Hence, we calculate the diagrams (b, (h and (j in Fig. ( for the off-shell D(B meson. As a result, we obtain: Π D(B nonper { mu = ss rr + m m u r r + m m } u r r, ( for the off-shell D and B meson and Π K nonper =, ( for the off-shell K meson. Here r = p m c(b and r = p m u. Now, it is time to appl the double Borel transformations with respect to p (p M and p (p M to the phsical as well as the QCD sides and equate the coefficient of the

6 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( selected structure p µ from two representations. Finall, we get the following sum rules for the corresponding coupling constants: + g D(B D s DK(B s BK(q = [ s π ] BΠ D(B nonper s ds (m c(b +m s (q m D(B (m c(b + m u (m s + m u f D s (B s f D(B f K m D s (B s m K m D(B ( + m K q m D s (B s m Ds e (B s M m K e M (m s+m u ds ρ D(B (s,s,q θ[ (f D(B (s,s ]e s M e s M ( gd K (q m K s DK(B s BK(q = (m c(b + m u (m s + m u f D s (Bsf D(B f K m D s (Bsm K m D(B ( + m D(B q [ π s s ds (m c(b +m s m D s (B s (m c(b +m u ds ρ K (s,s,q θ[ (f K (s,s ]e s M e s m Ds e (B s m D(B M e M M ] ( for the off-shell D(B and K meson associated with the D sdk (B sbk vertex, respectivel. The integration regions for the perturbative part in Eqs.(-( are determined requiring that the arguments of the three δ functions coming from Cutkosk rule vanish simultaneousl. So, the phsical region of the s and s planes are described b the following non-equalities: f D(B (s,s = s (m s m u + s + (m c(b m s s( q + s + s λ / (m c(b,m s,sλ / (s,s,q (6 f K D(B (s,s = s ( m c(b + m u s + (m c(b m s + s( q + s + s λ / (m c(b,m s,sλ/ (s,s,q (7 for the D(B and K off-shell meson associated with the Ds DK(B sbk vertex, respectivel. These phsical regions are imposed b the limits on the integrals and step functions in the integrands of the sum rules. In order to subtract the contributions of the higher states and continuum, the quark-hadron dualit assumption is used, i.e., it is assumed that, ρ higherstates (s,s = ρ OPE (s,s θ(s s θ(s s (8 where s and s are the continuum thresholds. Note that, the double Borel transformation used in the calculations is written as: ˆB (p m m (p m n ( m+n Γ(m Γ(n e m /M e m /M (M m (M n. (9. Numerical analsis This section is devoted to the numerical analsis of the sum rules for the coupling constant. To obtain numerical values of the considered coupling constants, the following input parameters are used in calculations: m K = (.9677 ±.6 GeV, m D = (.868 ±. GeV, m D s = (. ±. GeV, m B = (.79 ±. GeV, m B s = (. ±. GeV

7 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( g (D D * s DK(Q = GeV g (D D * s DK(Q = GeV M (GeV M (GeV Figure. g D D sdk (Q = GeV M. The continuum thresholds, s = 6.8 GeV, s =.99 GeV and M = 8 GeV have been used. Figure. g D D sdk (Q = GeV M. The continuum thresholds, s = 6.8 GeV, s =.99 GeV and M = GeV have been used. [8], m c =. GeV, m b =.7 GeV, m s =. GeV [9, ], f K = 6 MeV [], f D s = (7 ± 6 MeV, f Bs = (9 ± 6 MeV [], f B = (9 ± MeV [], f D = ( ± ± 7 MeV [], ss =.8(. ±. GeV [], m = (.8 ±. GeV [6]. The sum rule contains the four auxiliar parameters, namel the continuum thresholds, s and s and Borel mass parameters, M and M. Since these parameters are not phsical quantities, our results should be independent of them. Therefore, the working regions for the Borel mass parameters M and M are determined requiring that both the contributions of the higher states and continuum are sufficientl suppressed and the contributions coming from higher dimensions are small. As a result, we can show that D off-shell stabilizes for 8GeV M GeV and GeV M GeV and K off-shell for 6GeV M GeV and GeV M GeV associated with the Ds DK vertex. Similarl, the regions, GeV M GeV and GeV M GeV for B off-shell, and 6GeV M GeV and GeV M GeV for K off-shell are obtained for the Bs BK vertex. The dependence of considered coupling constants on Borel parameters for different cases are shown in Figs.(-(9. From these figures, we see a good stabilit of the results with respect to the Borel mass parameters in the working regions. The continuum thresholds, s and s are not completel arbitrar but the are correlated to the energ of the first excited states with the same quantum numbers. Our numerical calculations show that in the regions (m i +. s (m i +.7 and (m f +. s (m f +.7, respectivel for the continuum thresholds s and s, our results have weak dependence on these parameters. Here, m i is the mass of initial particle and the m f stands for the mass of the final on-shell state. Now, using the working region for auxiliar parameters and other input parameters, we would like to discuss the behavior of the strong coupling constants in terms of q. In the case of off-shell D meson related to the DsDK vertex, our numerical result is described well b the following mono-polar fit parametrization g (D D s DK(Q = 8.76(GeV Q + 7.(GeV ( where Q = q. The coupling constants are defined as the values of the form factors at Q = m meson (see also [], where m meson is the mass of the off-shell meson. Using Q = m D 6

8 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( g (K D * s DK(Q = GeV g (K D * s DK(Q = GeV 6 9 M (GeV 6 8 M (GeV Figure. g K D sdk (Q = GeV M. The continuum thresholds, s = 6.8 GeV, s =.97 GeV and M = 7 GeV have been used. Figure. g K D sdk (Q = GeV M. The continuum thresholds, s = 6.8 GeV, s =.9 GeV and M = GeV have been used. g (B B * s BK(Q = GeV.. g (B B * s BK(Q = GeV M (GeV.. M (GeV Figure 6. g B B sbk (Q = GeV M. The continuum thresholds, s =.99 GeV, s =.99 GeV and M = GeV have been used. Figure 7. g B B sbk (Q = GeV M. The continuum thresholds, s =.99 GeV, s =.99 GeV and M = GeV have been used. in Eq.(, gd D s DK =.79 ±. is obtained. The result for an off-shell K meson can be well fitted b the exponential parametrization: g (K Q Ds DK(Q =. e7.(gev.88. ( Using Q = m K in Eq.(, gk Ds DK =.99 ±.6 is obtained. Taking the average of two above obtained values, finall we get the value of the g D s DK coupling constant as: g D s DK(Q = (.89 ±.. ( This result is consistent with the result obtained in [9]as g D s DK =.8 ±.. 7

9 nd International Conference on Particle Phsics Journal of Phsics: Conference Series 7 ( g (K B * s BK(Q = GeV g (K B * s BK(Q = GeV M (GeV 6 9 M (GeV Figure 8. g K B sbk (Q = GeV M. The continuum thresholds, s =.99 GeV, s =.9 GeV and M = GeV have been used. Figure 9. g K B sbk (Q = GeV M. The continuum thresholds, s =.99 GeV, s =.9 GeV and M = 7 GeV have been used. Similarl, for B sbk vertex, our result for B off-shell is better extrapolated b the exponential fit parametrization, g (B and for K off-shell case, the parametrization is Q Bs BK(Q =.66 e.(gev +. ( g (K Q Bs BK(Q =.9 e.(gev. ( Using Q = m B in Eq.(, the coupling constant is obtained as gb BsBK =. ±.. Also gb K sbk =.6 ±. is obtained at Q = m K in Eq.(. Taking the average of these results, we get the following result g B s BK(Q = (. ±.8. ( The errors in the results are due to the uncertainties in determination of the working regions for the auxiliar parameters as well as the errors in the input parameters.. Acknowledgments The authors thank E. Veli Veliev for his useful discussions. This work has been supported partl b the Scientific and Technological Research Council of Turke (TUBITAK under the research project T8. References [] Navarra F S, Nielsen M, Bracco M E, Chiapparini M E and Schat C L Phs. Lett. B 89 9 [] Navarra F S, Nielsen M, Bracco M E Phs. Rev. D 6 7 [] Bracco M E, Chiapparini M, Lozea A, Navarra F S and Nielsen M Phs. Lett. B [] Matheus R D, Navarra F S, Nielsen M and da Silva R R Phs. Lett. B 6 [] Matheus R D, Navarra F S, Nielsen M and da Silva R R Int. J. Mod. Phs. E [6] Wang Z G 7 Nucl. Phs. A 796 6; 7 Eur. Phs. J. C ; 7 Phs. Rev. D 7 7 8

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