Table Of Content

B B∗K and B BK∗ vertices using QCD sum rules s s A. Cerqueira Jr., B. Osório Rodrigues Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, 20550-900, Rio de Janeiro, RJ, Brazil. M. E. Bracco Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, 5 1 Rod. Presidente Dutra Km 298, Pólo Industrial, 27537-000 , Resende, RJ, Brazil. 0 2 n M. Nielsen a J Instituto de F´ısica, Universidade de São Paulo, 2 1 C.P. 66318, 05389-970 São Paulo, SP, Brazil ] h Abstract p - p The form factors and the coupling constant of the B B∗K and B BK∗ vertices are calculated s s e h using the QCD sum rules method. Three point correlation functions are computed considering [ 1 both the heavy and light mesons off-shell in each vertex, from which, after an extrapolation of the v 6 QCDSR results at the pole of the off-shell mesons, we obtain the coupling constant of the vertex. 2 7 The form factors obtained have different behaviors but their simultaneous extrapolation reach the 2 0 same value of the coupling constant g = 8.41±1.23 and g = 3.3±0.5. We compare . BsB∗K BsBK∗ 1 0 our result with other theoretical estimates and compute the uncertainties of the method. 5 1 : v i X r a 1 I. INTRODUCTION In the recent years, many new charmonium and bottomonium states have been observed at the B-factories. As an example, in the bottomonium sector, the Belle Collaboration reported the observation of two charged narrow structures in the π±Υ(nS) (n = 1,2,3) and π±h (mP) (m = 1,2) mass spectra of the Υ(5S) → Υ(nS)π± and Υ(5S) → h (mP)π± b b decay processes [1]. These narrow structures were called Z (10610) and Z (10650). As b b pointed out by the Belle Collaboration, the proximity of the BB¯∗ and B∗B¯∗ thresholds and the Z (10610) and Z (10650) masses suggests that these states could be interpreted as b b weakly bound BB¯∗ and B∗B¯∗ states. In particular, using the one-boson exchange model and considering S-wave and D-wave mixing, the authors of Ref. [2] were able to explain both, Z (10610) and Z (10650), as BB¯∗ and B∗B¯∗ molecular states. The main ingredients b b in the one-boson exchange model are the effective Lagrangians, that describe the strong interactions between the heavy and light mesons. These Lagrangians are characterized by the strong coupling constants in the considered vertices which, in general, are not known. These heavy-heavy-light mesons coupling constants are fundamental objects, since they can provide essential information on the low energy behavior of the QCD. Depending on their numerical values, a particular molecular state may or may not be bound. Therefore, it is really important to have reliable ways to extract these values based on QCD calculations. However, such low-energy hadron interaction lie in a region which is very far away from the perturbative regime. Therefore, we need some non-perturbative approaches, such as the QCDsumrules(QCDSR)[3–5], tocalculatetheformfactorsandcouplingconstantsofthese vertices. There are already some QCDSR calculations for the heavy-heavy-light vertices like the B∗Bπ [6], B BK [7], B∗BK∗ [8], B∗B∗ρ [9], B BK∗, B∗BK [10] and B∗BK [11]. s0 s s 0 s 1 s In the charm sector, various vertices were evaluated with this approach and the results are systematized in [12]. Here we calculate the form factor and the coupling constant at the B B∗K and B BK∗ vertices in the framework of three-point QCDSR. More specifically, we s s evaluate the g (Q2) and g (Q2) form factors in three different ways, considering, BsB∗K BsBK∗ one by one, each one of the mesons in the vertex to be off-shell. From these form factors, we extract the g and the g coupling constants. BsB∗K BsBK∗ 2 II. THE QCD SUM RULE FOR THE B B∗K AND B BK∗ VERTICES s s To perform the QCDSR calculation and obtain the form factors and coupling constants of the vertices B B∗K and B BK∗, we follow our previous works, as Ref. [12]. The starting s s point is the three-point correlation function given by : Π(Bs)(p,p(cid:48)) = (cid:90) d4xd4y(cid:104)0|T{jK(∗)(x)jBs†(y)jB(∗)†(0)}|0(cid:105)eip(cid:48)·xe−iq·y, (1) µ(ν) µ (ν) for the B meson off-shell, and: s Π(K)(p,p(cid:48)) = (cid:90) d4xd4y(cid:104)0|T{jB(∗)(x)jK(∗)†(y)jBs†(0)}|0(cid:105)eip(cid:48)·xe−iq·y, (2) (µ)ν (µ) ν for the K or K∗ meson off-shell. In Eqs. (1) and (2), q = p−p(cid:48) is the momentum of the off- shellmesonandpandp(cid:48) arethemomentumofotherones. Thecurrentsj(M) arethecurrents associated with each meson in the vertex and contain the quantum information about the state. The correlation functions in Eqs. (1) and (2) allow to obtain two different form factors corresponding to the same vertex. In this way, the vertex is tested by two different mesons, the heavier and the lighter ones in the corresponding vertex. The calculation of these two correlation functions allows to reduce the uncertainties of the evaluation of the coupling constant of the vertex [12]. Equations (1) and (2) contain different numbers of Lorentz structures, and for each structure, we can write a different sum rule. In principle all the structures would give the same result. However, due to different approximations each structure can lead to different results. Therefore, one has to choose the structures less sensitive to the different approximations. To obtain the sum rule, these functions are calculated in two different ways: using quarks degrees of freedom – the QCD side; and using hadronic degrees of freedom – the phenomenological side. In the QCD side, the correlators are evaluated using Wilson’s operator product expansion (OPE). The duality principle allows us to obtain an interval in which both repre- sentations are equivalent. Therefore, in this region, we can obtain the QCD sum rule from where the form factors are evaluated. To improve the matching between the two sides, we perform a Borel transformation to both QCD and phenomenological sides. 3 A. The QCD side The QCD side is obtained using the following meson currents for the B B∗K vertex: s jB∗(x) =¯bγ q, µ µ jBs(0) = i¯bγ s, 5 5 jK(y) = s¯γ γ q (3) ν ν 5 and the following ones for the B BK∗ vertex: s jB(x) = iq¯γ b, 5 5 jBs(0) = i¯bγ s, (4) 5 5 jK∗(y) = q¯γ s. µ µ Here, q, s and b are the light, strange and bottom quark fields respectively. Each one of these currents has the quantum numbers of the associated meson. In the case of K off-shell meson, we use, as usual in QCDSR, the pseudo scalar current for it, see refs. D∗Dπ [6], D∗DK [13] and B∗BK [11]. The general expression for the vertices has different structures, s s which can be written in terms of a double dispersion relation over the virtualities p2 and p(cid:48)2, holding Q2 = −q2 fixed: 1 (cid:90) ∞ (cid:90) ∞ ρ(s,u,Q2) Γ(p2,p(cid:48)2,Q2) = − ds du , (5) 4π2 (s−p2)(u−p(cid:48)2) smin umin where the spectral density ρ(s,u,Q2) can be obtained from the Cutkosky’s rules. The invariant amplitudes receive contributions from all terms in the OPE. In the case of formfactors,themaincontributionintheOPEistheperturbativeterm,whichisrepresented in Fig. 1, for the two cases that we are considering, the B (B ) and K(K∗) meson off-shell s s for the B B∗K (B BK∗) vertex. s s In order to obtain the form factor, we have to choose one of the different structures ap- pearing in Eqs.(1) and (2). As commented above, different structures can lead to different results. Therefore, one has to choose the structure less sensitive to higher dimension con- densates, that provide a better stability as a function of the Borel mass, and that have a larger pole contribution, when compared with the continuum contribution. This is considered a “good” structure. If there is more than one “good” structure, the others can also be considered to estimate the uncertainties of the method. 4 (cid:54)qα (cid:54)qα B (B ) K(K∗) s s b (cid:0)(cid:18)y(cid:64) q s (cid:0)y(cid:64)(cid:73) q (cid:0) (cid:64) (cid:0) (cid:64) p p(cid:48) p p(cid:48) µ (cid:45)(cid:0)(cid:27)0 x(cid:64)(cid:82) (cid:45)ν µ (cid:45)(cid:0)(cid:9)0 x(cid:45)(cid:64) (cid:45)ν B∗(B) s K(K∗) B (B) b B∗(B ) s s FIG. 1: Perturbative diagrams for the B (B ) off-shell meson (left) and for K(K∗) off-shell meson s s (right), for B B∗K vertex (B BK∗ vertex). s s For B B∗K form factor and in the case B off-shell meson, we choose the p(cid:48) p(cid:48) structure, s s µ ν because it satisfies the criteria above. For K off-shell meson, we can work with both p and ν p(cid:48) structures In this case, we are going to work with the p(cid:48) structure while the other one ν ν will be used for the estimate of the uncertainties. The corresponding perturbative spectral densities, which enter in Eq. (5), are : 3 ρ(Bs)(s,u,Q2) = √ [(2m −2m )E −2m B], (6) b s b 2π λ for the p(cid:48) p(cid:48) structure of the B off-shell case, and µ ν s 3 (cid:104) (cid:105) ρ(K)(s,u,Q2) = − √ A(p·p(cid:48) −2k ·p−m m +m2)+2π(m2 −k ·p(cid:48)) , (7) b s b b 2π λ forthep structureintheK off-shellcase. Inbothcases, thequarkcondensatecontributions ν are neglected after the Borel transform. For B BK∗ vertex, we use the p structure for B off-shell meson and for K∗ off-shell s µ s meson, we have p and p(cid:48) structures, both giving excellent sum rules. Again we show the µ µ resultsforonestructureandtheotherisusedtoestimatetheuncertainties. Theperturbative contribution to the spectral density, when the B meson is off-shell is : s 3 ρ(Bs)(s,u,Q2) = √ [A(p·p(cid:48) −m m −2p·k)−p(cid:48) ·k] (8) b s 2π λ for the p structure. In this case, the quark condensate, (cid:104)qq¯(cid:105), contribution to the same µ structure is: m (cid:104)qq¯(cid:105) Π(cid:104)qq¯(cid:105) = s . (9) (p2 −m2)(p(cid:48)2 −m2) b s For K∗ off-shell case, the spectral density, for both structures, is given by: ρ(K∗)(s,u,Q2) = − 3√ {p (cid:104)A(p·p(cid:48) −m m −m2)+m2 −k ·p(cid:48) −m m )(cid:105) µ b s b b b s 2π λ 5 (cid:104) (cid:105) +p(cid:48) B(p·p(cid:48) +m m −m2)−k ·p+m2 }. (10) µ b s b b In Eqs. (6) to (10), we have defined λ = λ(s,u,t) = s2+t2+u2−2st−2su−2tu, s = p2, u = p(cid:48)2, t = −Q2 and A, B and E are functions of (s,u,t), given by:  →−  k |k | p(cid:48) |(cid:126)k| A = 2π√0 − →− cosθ√0 ; B = 2π cosθ ; (11) s |p(cid:48)| s |p(cid:126)(cid:48)| 2 (cid:126) π|k| (cid:16) (cid:17) E = − 3cosθ−1 , (12) 2 |p(cid:126)(cid:48)| where 2 2p(cid:48)k −u−m2 −ηm2 |(cid:126)k| = k 2 −m2; cosθ = − 0 0 i b; 0 i 2|p(cid:126)(cid:48)||(cid:126)k| s+u−t λ s+m2 −(cid:15) m2 p(cid:48) = √ ; |p(cid:126)(cid:48)|2 = ; k = i√ b; (13) 0 2 s 4s 0 2 s Finally, the OPE side, is calculated using Eq. (5) with the limits in the integration given by: s = (m + m )2 and u = t − m2 for B off-shell and s = m2 − m2 and min b s min b s min b s u = t+m2−m2 for K off-shell, for B B∗K vertex. And s = (m )2 and u = t+m2 min b s s min b min b for B and K∗ off-shell for the B BK∗ vertex. s s B. The phenomenological side The three-point functions from Eqs. (1) and (2), when written in terms of hadron masses, decay constants and form factors, give the phenomenological side of the sum rule. 1. For the B B∗K vertex: s The meson decay constants f , f and f are defined by the following matrix elements: K Bs B∗ (cid:104)0|jK|K(p)(cid:105) = if p , (14) ν K ν (cid:104)0|jB∗|B∗(p(cid:48))(cid:105) = m f (cid:15)∗(p(cid:48)), (15) µ B∗ B∗ µ and m2 (cid:104)0|jBs|B (p)(cid:105) = Bs f , (16) 5 s m +m Bs b s 6 and the vertex function is defined by: (cid:104)K(q)|B∗(−p(cid:48))B (p)(cid:105) = −g(K) (cid:15)µ∗(p(cid:48))(2p−p(cid:48)) , (17) s BsB∗K µ which is extracted from the effective Lagrangian [14]: L = ig (cid:104)B∗µ(B¯ ∂ K −∂ B¯ K)−B¯∗µ(B ∂ K¯ −∂ B K¯)(cid:105). , (18) BsB∗K BsB∗K s µ µ s s µ µ s Saturating the correlation function with B , B∗ and K intermediate states we arrive at s −f f f m m2 g(Bs) (q2) Π(Bs) = K Bs B B∗ Bs BsB∗K µν (m +m )(p(cid:48)2 −m2 )(q2−m2 )(p2 −m2 ) s b K Bs B∗ (cid:34) (cid:32) (m2 −q2)(cid:33) (cid:35) × p p(cid:48) 1− K −2p(cid:48) p(cid:48) +“continuum”, (19) µ ν m2 µ ν B∗ for an off-shell B . Using the matrix element of K meson equal to s m2 (cid:104)0|jK|K(q)(cid:105) = Kf , (20) 5 m K s we arrive at an expression for an off-shell K: −f f f m m2 m2 g(K) (q2) Π(K) = B∗ K Bs B∗ Bs K BsB∗K µ (m +m )m (p2 −m2 )(p(cid:48)2 +m2 )(q2 −m2 ) b s s B B∗ K s (cid:34) (cid:32) (m2 −q2)(cid:33)(cid:35) −2p +p(cid:48) 1+ Bs +“continuum”. (21) µ µ m2 B∗ 2. For the B BK∗ vertex: s In this case, the effective Lagrangian is [14]: L = ig (cid:104)K∗µ(B∂ B¯ −B¯ ∂ B)+K¯∗µ(B ∂ B¯ −B¯∂ B )(cid:105) , (22) BsBK∗ BsBK∗ µ s s µ s µ µ s from where we can extract the vertex element, which is given by: (cid:104)K∗(q)|B (−p(cid:48))B(p)(cid:105) = ig(K∗) (cid:15)∗µ(q)(p+p(cid:48)) , (23) s BsBK∗ µ and the matrix elements which introduce the meson decay constants f , f and f are: K∗ Bs B (cid:104)0|jK∗|K∗(p)(cid:105) = f (cid:15)∗(q)m , (24) µ K∗ µ K∗ m2 (cid:104)B(p)|jB|0(cid:105) = f B , (25) 5 B m2 b 7 and m2 (cid:104)0|jBs|B (p(cid:48))(cid:105) = f Bs , (26) 5 s Bsm +m b s After some algebra we arrive at the following expression: f f f m m2 m2 Π(Bs) = −i K∗ B Bs K∗ Bs B µ (m2 +m m )(p2 −m2 )(p(cid:48)2 −m2 )(q2 −m2 ) b s b B K∗ Bs (cid:34) (cid:32) m2 −m2 (cid:33)(cid:35) ×g(Bs) (q2) −2p +p(cid:48) 1− B Bs +“continuum”, (27) BsBK∗ µ µ m2 K∗ when B is off-shell. s For K∗ off-shell we arrive at: f f f m m2 m2 Π(K∗)(p,p(cid:48),q) = − B K∗ Bs K∗ Bs B µ (m m +m2)(p2 −m2 )(p(cid:48)2 −m2 )(q2 −m2 ) s b b B Bs K∗ (cid:34) (cid:32) (m2 −m2 )(cid:33) ×g(K∗) (q2) p 1− B Bs BsBK∗ µ m2 K∗ (cid:32) (m2 −m2 )(cid:33)(cid:35) +p(cid:48) 1− Bs B +“continuum” (28) µ m2 K∗ III. THE SUM RULE The sum rule is obtained after performing a double Borel transform (BB), P2 = −p2 → M2 and P(cid:48)2 = −p(cid:48)2 → M(cid:48)2, to both the phenomenological and OPE sides: (cid:104) (cid:105) (cid:104) (cid:105) BB ΓOPE(I) (M,M(cid:48)) = BB Γphen(I) (M,M(cid:48)), (29) µ µ where M and M(cid:48) are the Borel masses and I is the off-shell meson. In order to eliminate the continuum contribution in the phenomenological side, instead of doing the integrals in Eq. (5) up to ∞, we do the integrals up to the continuum threshold parameters s and u . The threshold parameters are defined as s = (m +∆ )2 and u = 0 0 0 i i 0 (m +∆ )2, where ∆ and ∆ are usually taken as 0.5 MeV, and m and m are the masses o o i o i o of the incoming and outgoing mesons respectively. In Eqs. (19), (21), (27), (28), g(I) (Q2) and g(I) (Q2) are the form factors when the BsB∗K BsBK∗ I meson is off-shell. As in our previous works, we define the coupling constant as the value of the form factor, g(I)(Q2), at Q2 = −m2, where m is the mass of the off-shell meson. I I 8 IV. RESULTS AND DISCUSSION Table I shows the value of the hadronic parameters used in the present calculation. We have used the experimental value for f of Ref. [15], for f and f from Ref. [16] and for K B∗ Bs f of Ref. [17]. K∗ TABLE I: Hadronic parameters used. K K∗ B B∗ B s m (GeV) 0.49 0.89±5 5.40 5.20 5.28 f (MeV) 160±1.4 220±5 208±10 250±10 191±0.87 ±44 ±39 ±29 We neglect the light quark mass (m = 0.0MeV). The strange and bottom quark masses q weretakenfromtheParticleDataGroup(PDG)andhavethevaluesm = 104+26−34MeV s and m = 4.20 + 0.17 − 0.07GeV respectively. In the next subsections, we show the two b form factors used to extract the coupling constants of each vertex. A. B B∗K vertex s 1,00 10 Pole s= u=0.5 Continuum s= u=0.6 al0,75 8 s= u=0.7 ot of t bution in % 0,50 2 g(Q)BsB*K 46 ntri o C0,25 2 0,00 0 5 10 15 20 5 10 15 20 M2(GeV2) M2(GeV2) (a) (b) FIG. 2: (a) The pole-continuum contributions for the B B∗K sum rule for the B off-shell and (b) s s g(Bs) (Q2 = 1GeV2) stability for different values of the continuum thresholds. BsB∗K In the case of a B off-shell meson, we work with the p(cid:48)p(cid:48) structure. In Fig. 2(a) we s ν µ show the contribution of the pole versus the continuum contribution for the sum rule and in Fig. 2(b) the stability of the form factor as a function o Borel mass, for Q2 = 1 GeV2 and 9 three different values for the continuum thresholds. We use the usual relation between the Borel masses M(cid:48) and M [12]: M(cid:48)2 = m2K M2. mBs∗−m2b FromFig.2(b)weclearlyseeawindowofstabilityforM2 ≥ 10 GeV2, andfromFig.2(a), we can see that the pole contribution is bigger than the continuum contribution for M2 ≤ 19GeV2. Therefore, there is a Borel window where the sum rule can be used to extract the form factor. 1,00 Pole s= u=0.5 Continuum s= u=0.6 12 s= u=0.7 al0,75 ot of t ution in % 0,50 2 g(Q)BsB*K 8 b ntri Co0,25 4 0,00 0 0 10 20 30 0 10 20 30 2 2 2 2 M(GeV) M(GeV) (a) (b) FIG. 3: a) The pole and continuum contributions for B B∗K sum rule with the K off-shell and b) s the g(K) (Q2 = 2 GeV2,M2) stability for different values of the continuum thresholds BsB∗K In the case of a K meson off-shell, we have two structures in Eq. (21) that can be used: p(cid:48) and p . Both structures give good sum rule results, that means a good pole- µ µ continuum contribution and good stability. We show in Fig. 3(a) and (b), repectively, the pole-continuum contribution and the stability only for the p structure. For the p(cid:48) , we ν µ obtain a very similar result and we use it to evaluate the uncertainties. From Fig. 3, we find a Borel window 10GeV2 ≤ M2 ≤ 14GeV2 where the sum rule can be used to extract he form factor. In Fig. 6 we show the QCDSR results for these two form factors, represented by squares and triangles for the cases of the B and K off-shell respectively. s B. B BK∗ vertex s For the case of the B meson off-shell, we use the structure p . In Fig. 4(a) and (b) we s µ showthepole-continuumcontributioncontributionandtheBorelmassstabilityrespectively. 10