Semileptonic decays of pseudoscalar mesons to scalar f meson 0 7 T. M. Aliev ∗†, M. Savcı ‡ 0 0 Physics Department, Middle East Technical University, 06531 Ankara, Turkey 2 n a J Abstract 5 1 The transition form factors of D f ℓν, D f ℓν and B f ℓν decays are 1 calculated in 3–point QCD sum rulesm→eth0od, assu→min0g that f ius→a qu0ark–antiquark v 0 state withamixtureof strangeandlightquarks. Thebranchingratios ofthesedecays 8 0 are calculated in terms of the mixing angle. 1 1 0 7 0 / h p - p e h : v i X r a ∗e-mail: [email protected] †permanent address:Institute of Physics,Baku,Azerbaijan ‡e-mail: [email protected] 1 Introduction The inner structure of the scalar mesons in terms of quarks is still an open question in par- ticle physics and it is the subject of an intense and continuous theoretical and experimental investigations for establishing their their nature (for a review, see [1]). There are numerous scenarios for the classification of the scalar mesons. The established 0++ mesons are divided into two groups: 1) Near and above 1 GeV, and 2) in the region 1.3 GeV 1.5 GeV. The ÷ first group scalar mesons form an SU(3) nonet, which contains two isosinglets, an isotriplet and two strange isodoublets. Inthe quark model, the flavor structure of these scalar mesons would be σ > = cosθ n¯n > sinθ s¯s > , | | − | f > = cosθ s¯s > +sinθ n¯n > , 0 | | | 1 a0 = (u¯u d¯d) , a+ = ud¯, a− = d¯u , 0 √2 − 0 0 κ+ = s¯u , κ¯0 = d¯s , κ− = u¯s , (1) where n¯n >= (u¯u+d¯d)/√2, and θ is the mixing angle. Here we take into account the fact | ¯ that between isoscalars s¯s and u¯u + dd there is mixing, which follows from experiments. Indeed the observation 1 Γ(J/ψ f ω) Γ(J/ψ f φ) , 0 0 → ≃ 2 → indicates that the quark content of f (980) is not purely s¯s state, but should have non– 0 strange parts too [2]. Secondly, if f (980) is purely s¯s state, then f ππ should be 0 0 → OZI suppressed. But the decay width of f (980) is dominated by f ππ which leads to 0 0 → the conclusion that in f (980) there should be n¯n parts as well. Therefore f should be a 0 0 mixture of s¯s and n¯n, as is presented in Eq. (1). Analysis of the experimental data shows that the mixing angle θ lies in the range 250 < θ < 400 and 1400 < θ < 1650 [3]. Although there is another scenario where mesons below or about 1 GeV is described as a four–quark state (see for example [4]), in this work we restrict ourselves to considering the q¯q description for f (980) meson, but taking into account the mixing between s¯s and n¯n. In 0 the present work we study the semileptonic decays B+ f (980)ℓ+ν, D+ f (980)ℓ+ν → 0 d,s → 0 decays in order to get information about the quark content of f (980). 0 From theoretical point of view, investigation of the semileptonic decays is simpler com- pared to that of hadronic decays, because leptons do not participate in strong interactions. The experimental study of weak semileptonic decays of heavy flavored mesons is very im- portant for the more accurate determination of the Cabibbo-kobayashi-Maskawa (CKM) matrix elements, their leptonic decay constants, etc. The precise determination of the CKM matrix elements depends crucially on the possi- bility of controlling long distance interaction effects. So, in study of the exclusive semilep- tonic decays the main problem is calculation of the transition form factors, which involve the long distance QCD dynamics, belonging to the non–perturbative sector of QCD. For this reason, in calculation of the transition form factors some kind of non–perturbative 1 approach is needed. Among all non–perturbative approaches QCD sum rules method [5] is more powerful, since it is based on the first principles of QCD. About the most recent status of QCD sum rules, the interested readers are advised to consult [6]. Semileptonic decays D K¯0eν¯ [7], D+ K(K0∗)e+ν [8], D πeν¯ [9], D ρeν¯ e e e e → → → → [10], B D(D∗)ℓν¯ [11] and D φℓν¯ [12] are all studied in the framework of 3–point ℓ ℓ → → QCD sum rules method. In this work we study the semileptonic B f (980)ℓ+ν and D f (980)ℓ+ν u 0 ℓ s(d) 0 ℓ → → decays in the 3–point QCD sum rules method. The paper is organized as follows: In section 2, we derive the sum rules for the form factors, responsible for pseudoscalar to scalar meson transition. Section 3 is devoted to the numerical analysis of the transition form factors and discussion and contains our conclusions. 2 Pseudoscalar–scalar meson transition form factors from QCD sum rules Pseudoscalar–scalar transition form factors are defined via the matrix element of the weak current sandwiched between initial and final meson states S(p′) q¯ γ (1 γ )q P(p) , 1 µ 5 2 h | − | i where q and q are the relevant quarks, P and S are the pseudoscalar and scalar me- 1 2 son states, respectively. It follows from parity conservation in strong interaction that only axial part of weak current gives non–zero contribution to this matrix element, and imposing Lorentz invariance, it can be written in terms of the form factors as follows: S(p′) q¯ γ (1 γ )q P(p) = i f (p+p′) +f q , (2) 1 µ 5 2 + µ − µ h | − | i − h i where q = p p . µ 1 2 − For evaluation of these form factors in the QCD sum rule, we consider the following the 3–point correlation function Π (p2,p′2,q2) = d4xd4yei(p′y−px) 0 T J (y)JA(0)J (x) 0 , (3) µ − S µ P Z D (cid:12) n o(cid:12) E (cid:12) (cid:12) where, J = q¯ q , JA = q¯ γ γ q and J = q¯ γ q(cid:12) are the interpolating(cid:12) currents of scalar S 2 2 µ 2 µ 5 1 P 1 5 2 and pseudoscalar mesons, and weak axial currents, respectively. It should be noted here that, q = u, q = u and q = b for the B f (980) transition; and q = s(d), q = s(d) 3 2 1 u 0 3 2 → and q = c for the D f (980) transition, respectively. 1 s(d) 0 → The decomposition of the correlation function (3) into the Lorentz structures, obviously, has the form Π = Π (p+p′) +Π (p p′) . (4) µ + µ − µ − For the amplitudes Π and Π , we have the following dispersion relation + − 1 ρ (s,s′,Q2)dsds′ Π (p2,p′2,Q2) = ± +subtraction terms , (5) ± −(2π)2 (s p2)(s′ p′2) Z − − where ρ is the corresponding spectral density and Q2 = q2 > 0. According to QCD sum ± − rules approach, the correlation function is calculated by the operator product expansion 2 (OPE) at large Euclidean momenta p2 and p′2 on one side, and on the other side it is calculated by inserting a complete set of intermediate states having the same quantum numbers with the currents J and J . S P The phenomenological part of (3) is obtained by saturating correlator it with the lowest pseudoscalar (in our case B , D or D mesons) and scalar f (980) mesons, yielding u s 0 0 J S(p′) S(p′) JA(0) P(p) P(p) J (x) 0 Π = h | S| i µ h | P | i +excited states . (6) µ D(m2 (cid:12) p′2)(m(cid:12) 2 pE2) S −(cid:12)(cid:12) (cid:12)(cid:12) P − The matrix elements in Eq. (6) are defined as 0 J S(p′) = λ , S S h | | i m2f P J 0 = i P P , (7) P h | | i − m +m 1 2 where f and f are the leptonic decay constants of scalar and pseudoscalar mesons, and S P m and m are being their masses, respectively. Note that, leptonic decay constant f in S P S Eq. (7) is scale dependent for which we choose the scale to be µ = 1 GeV2, and m for B f ℓν , b u 0 → m = 1   mc for Ds f0ℓν , D f0ℓν , → →  m for B f ℓν , D f ℓν , u u 0 0 → → m = 2   ms for Ds f0ℓν , →  Using Eqs. (2), (4), (6) and (7), for the invariant structures we get f m2 λ f Π = P P S ± . (8) ± −m +m (m2 p′2)(m2 p2) 1 2 S − P − From QCD side, the correlation function can be calculated with the help of the OPE at short distance, and in this work we will consider operators up to dimension six. The theoretical part of the correlator for the B D (2317)ℓν is calculated in [13], and in the s → s0 present work, for the theoretical part of the corresponding sum rules, we will use the results of this work. For the spectral densities we have N Π = c (∆′ +∆)(1+A+B)+(m2 +2m m +Q2)(A+B) , (9) + 4λ1/2(s,s′,Q2) 1 1 2 h i N Π = c ∆′ +∆+m2 +2m m +Q2 +2m m (A B) − 4λ1/2(s,s′,Q2) 1 1 2 1 2 − h(cid:16) (cid:17) + ∆′ ∆ 2m m , (10) 1 2 − − i 3 where N = 3, ∆ = s m2, ∆′ = s′ m2, and c − 1 − 2 1 A = (s+s′ +Q2)∆′ +2s′∆ , λ(s,s′,Q2) − h i 1 B = (s+s′ +Q2)∆′ +2s∆′ . λ(s,s′,Q2) − h i Forthe decays under consideration, m is m (m ) orm , and therefore, to take into account 2 u d s SU(3)–violatingeffects, hereandinallfollowingcalculationswewillretaintermsthatlinear with m , while neglecting the terms higher order in m . 2 2 For power corrections (PC)we get 1 m m 1 m2 2 ΠPC = q¯ q 1 − 2 + m q¯ q 1 + 2h 2 2i rr′ 4 2h 2 2i r2r′ − rr′! 1 3m2(m m ) 2(m 2m ) 2(2m m ) m2q¯ q 1 1 − 2 + 1 − 2 + 1 − 2 − 12 0 2 2i" r3r′ rr′2 r2r′ m (2m2 +m m +2Q2) 2m (m2 +Q2) + 1 1 1 2 − 2 1 r2r′2 # 4 12m3(m m ) 8m m (m2 +Q2) 56m m + πα q¯ q 2 1 1 − 2 + 1 2 1 + 1 2 81 sh 2 2i "− r4r′ r2r′3 rr′3 4m2(2m2 +m m +2Q2) 8m m (m2 +Q2) 8m (8m 7m ) 1 1 1 2 − 1 2 1 1 1 − 2 − r3r′2 − r3r′ 48 48 4(5m2 20m m 2Q2) + + 1 − 1 2 − rr′2 r2r′ − r2r′2 # 1 m2(m2 +Q2) 5m2 +4Q2 6m4 10m2 + m2m q¯ q 2 1 1 + 1 + 1 + 1 , (11) 9 0 2h 2 2i "− r3r′2 r2r′2 r4r′ r3r′ # 1 m +m 1 m ΠPC = q¯ q 1 2 + m m q¯ q 1 − −2h 2 2i rr′ 4 1 2h 2 2i − r2r′! 1 3m2(m +m ) 2(m +3m ) 2(3m +m ) + m2q¯ q 1 1 2 + 1 2 + 1 2 12 0 2 2i" r3r′ rr′2 r2r′ m (2m2 +m m +2Q2)+2m (m2 +Q2) + 1 1 1 2 2 1 r2r′2 # 1 12m3(m +m ) 8m m (m2 +Q2) 56m m + πα q¯ q 2 1 1 2 1 2 1 1 2 81 sh 2 2i " r4r′ − r2r′3 − rr′3 4m2(2m2 +m m +2Q2)+8m m (m2 +Q2) 8m (9m +7m ) + 1 1 1 2 1 2 1 + 1 1 2 r3r′2 r3r′ 28m2 8 8 + 1 + r2r′2 rr′2 − r2r′# 1 m2(m2 +Q2) m2 6m4 4 4 24m2 + m2m q¯ q 2 1 1 1 1 + 1 , (12) 9 0 2h 2 2i " r3r′2 − r2r′2 − r4r′ rr′2 − r2r′ − r3r′ # 4 where r = p2 m2 and r′ = p′2. Note that the D f (980)ℓ+ν and D f (980)ℓ+ν − 1 s → 0 ℓ → 0 ℓ decays which are considered in [14] differ from our results in three aspects: Our result on spectral density is two times smaller compared to that given in [14]. • Since it is known that the main contribution to the sum rules comes from the spectral density, it is indispensable that our results on the form factors differ from those predicted in [14]. In[14], partofthosediagramswhich areproportionaltom arenottakeninto account s • (in our case they correspond to the terms proportional to m m2 q¯ q ). 2 0h 2 2i Sum rules for the form factor f are totally absent in [14], which could be essential − • for the B f (980)τν decay. u 0 τ → Contribution of higher states in the physical part of the sum rules aretaken into account with the help of the hadron–quark duality, i.e., corresponding spectral density for higher states is equal to the perturbative spectral density for s and s′ starting from s > s and 0 0 0 s′ > s′, where s and s′ ar the continuum thresholds in the corresponding channels. 0 Equating the two representations for the invariant structures Π , and applying double ± Borel transformation on the variables p2 and p′2 (p2 M2, p′2 M′2) in order to suppress → → the higher states and continuum contributions, we get the following sum rules for the form factors f and f : + − m +m 1 f (q2) = 1 2 em2P/M2em2S/M′2 dsds′ρ (s,s′,Q2)e−s/M2−s′/M′2 ± − fPm2P λS (Z ± + BM2BM′2ΠP±C) . (13) The double Borel transformation for the quantity 1/rnr′m is defined as: 1 (M2)n−1(M′2)m−1 BM2BM′2rnr′m = (−1)n+m Γ(n) Γ(m) e−m21/M2 . (14) Theintegrationregionfortheperturbativecontributionisdeterminedfromthefollowing inequalities: 2ss′ +(m2 s)(s+s′ +Q2) 1 1 − 1 . (15) − ≤ λ1/2(s,s′,Q2)(m2 s) ≤ 1 − In the calculation of the widths of the considered decays, it is necessary to know the q2 dependence of the form factors in the whole physical region m2 q2 q2 . ℓ ≤ ≤ max 3 Numerical analysis In this section we present our results for the form factors f (q2) and f (q2) for the decays + − under consideration. The main input parameters for the sum rules are the Borelparameters 5 M2 and M′2, continuum thresholds s and s′. The values of other parameters needed are: 0 0 m = (4.7 0.1) GeV [6], m = 1.4 GeV, m = 0.15 GeV, u¯u = (0.243)3 GeV3, b c s µ=1 GeV ± h i| − s¯s = 0.8 u¯u [15]. The values of the leptonic decay constants of B , D and D u s h i × h i mesons are determined from the analysis of the corresponding two–point correlators: f = Bu (0.14 0.01GeV [16], f = (0.22 0.02 GeV [17]andf = (0.17 0.02GeV [6,16,17]. For ± Ds ± D ± the continuum thresholds we take the values sBu = (33 2) GeV2, sDs = (7.7 1.1) GeV2, 0 ± 0 ± sD = (6 0.2) GeV2 and s′ = 1.6 GeV2 which is determined from 2–point sum rules 0 ± 0 analysis [6,14,16,18]. The Borel parameters M2 and M′2 are the auxiliary parameters and therefore the phys- ical quantities should be independent of them. For this reason we need to find the working regions of M2 and M′2 where form factors are practically independent of them. In obtaining the working regions of M2 and M′2 the following two conditions should be satisfied: The continuum contribution should be small, and, • power corrections should be convergent. • Our numerical analysis shows that, both conditions are satisfied in the region 10 GeV2 ≤ M2 20 GeV2 for B f ℓν¯ , 4 GeV2 M2 8 GeV2 for D (D) f ℓν¯ , and u 0 ℓ s 0 ℓ ≤ → ≤ ≤ → 1.2 GeV2 M′2 2 GeV2 for all channels. ≤ ≤ Varying the input parameters s , s′, f , f , f and f in the respective regions as 0 0 0 Ds B D mentioned in the text, we get the following results for the form factors at q2 = 0 fBu(0) = 1.7(0.25 0.02) , + ± fBu(0) = 1.7(0.24 0.03) , − − ± fD(0) = 1.7(0.32 0.03) , + ± fDs(0) = 1.7(0.27 0.02) , (16) + ± The multiplying factor 1.7 corresponds to the case for λ = 0.19 GeV2, and without this S factor λ = 0.35 GeV2 [3]. For a comparison we present the results of the form factor f S + for the B f (980) transition coming from the covariant light front dynamics [19] and u 0 → dispersion relation approach [20], as well as the result for the B π transition [21]. u → fBu→f0(0) = 0.27 [19] , + fBu→f0(0) = 0.09 [20] , + fBu→π(0) = 0.25 [21] . (17) + From a comparison of Eqs. (16) and (17) we see that, our prediction on f for the B + u → f (980) transitions quite close to the prediction of the light front dynamics and that of 0 B π transition when λ = 0.35, and approximately three times larger compared to that u S → of thedispersion relationapproach. These close results ofthe formfactorf for theB π + u → and B f transitions could indicate of the that λ should have the value λ = 0.35, u 0 S S → which is obtained in [3] by taking (α ) corrections into account. s O Note that we present the form factor f only for the B f τν¯ decay, because this − u 0 τ → form factor can give considerable contribution to this decay. 6 In estimating the width of P f (980)ℓν¯ decay, we need to know the q2 dependence of 0 ℓ → the form factors f (q2) and f (q2) in the whole kinematical region m2 q2 (m m )2. + − ℓ ≤ ≤ P − f0 The q2 dependence of the form factors can be calculated from QCD sum rules (see [8,9]). Unfortunately QCD sum rule cannot reliably predict q2 dependence of the form factors in the full kinematical region. The QCD sum rules can reliably predict q2 dependence of the form factors in the region approximately 1 GeV2 below the perturbative cut. In order to extend the dependence of the form factors on q2 to the full kinematical region, we look such a parametrization of the form factors where it coincides with the sum rules prediction of in the above–mentioned region. Our numerical calculations shows that the best parametrization of the form factors with respect to q2 are as follows: f (0) f (q2) = P , (18) P 1 a qˆ+b qˆ2 c qˆ3 +d qˆ4 P P P P − − where P = B , D , D and qˆ= q2/m2. The values of the parameters f (0), a , b , c and u s P P P P P d at λ = 0.19 GeV2, are given in table 1. P S f (0) f (0) a b c d + − D 1.7 0.27 0.87 0.17 0.37 1.46 s × − D 1.7 0.32 0.89 0.40 0.18 1.00 × − − B 1.7 0.25 0.48 0.30 0.47 0.99 u × − − − B 1.7 0.24 0.41 0.42 0.95 1.55 u − × − − − Table 1: Form factors for the D f ℓν¯ , D f ℓν¯ and B f ℓν¯ decays in a four– s 0 ℓ 0 ℓ u 0 ℓ → → → parameter fit. The dependence of the form factors f and f (for B f τν¯ decay) are given in Figs. + − u 0 τ → (1)–(4). Using the parametrization of Eq. (2), for the P f ℓν¯ differential decay width, we 0 ℓ → get dΓ A q2 m2 2 = G2 V 2λ1/2(m2,m2 ,q2) − ℓ dq2 192π3m3 | ij| P f0 q2 ! P (2q2 +m2) 2 ℓ f (q2) (2m2 +2m2 q2)+2(m2 m2 )Re[f (q2)f∗(q2)] × (− 2 + P f0 − P − f0 + − h(cid:12) (cid:12) + f (q2) 2q2 + (q(cid:12)(cid:12)2 +2m2ℓ(cid:12)(cid:12)) f (q2) 2(m2 m2 )2 − q2 + P − f0 (cid:12) (cid:12) i h(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12)2(m2 (cid:12) m2 )q2Re[f (q2)f∗((cid:12)q2)]+(cid:12)f (q2) 2q4 , (19) P − f0 + − − ) (cid:12) (cid:12) i (cid:12) (cid:12) where (cid:12) (cid:12) cos2θ for D f ℓν¯ and , s 0 ℓ → A =   sin2θ for D f ℓν¯ and B f ℓν¯ , 0 ℓ u 0 ℓ 2 → →  7 V = 4.31 10−3 for B f ℓν¯ , ub u 0 ℓ | | × →  Vij =  |Vcs| = 0.96 for Ds → f0ℓν¯ℓ , V = 0.23 for D f ℓν¯ , Taking into accountth|ecqd2| dependence of the form→fac0torℓs f and f and performing + − integration over q2 and using the lifetimes of B , D and D mesons, we get the following u s values for the branching ratios when λ = 0.19 GeV2: S sin2θ (B f τν¯ ) = (7.17 10−5) , u 0 τ B → 2 × × sin2θ (B f µν¯ ) = (2.1 10−4) , u 0 µ B → 2 × × sin2θ (B f eν¯ ) = (2.1 10−4) , u 0 e B → 2 × × (D f µν¯ ) = cos2θ (3.85 10−3) , s 0 µ B → × × (D f eν¯ ) = cos2θ (4.07 10−3) , s 0 e B → × × sin2θ (D f µν¯ ) = (4.98 10−4) , 0 µ B → 2 × × sin2θ (D f eν¯ ) = (5.29 10−4) . (20) 0 e B → 2 × × We see from (20) that the ratios of the widths (D f ℓν¯ ) 0 ℓ R = B → , 1 (D f ℓν¯ ) s 0 ℓ B → (B f ℓν¯ ) u 0 ℓ R = B → , 2 (D f ℓν¯ ) s 0 ℓ B → aredirectlyrelatedwiththemixing angleθ. Ontheotherhand, asfarastheflavorstructure of f (980), as is given in Eq. (1), is considered, the ratio 0 (B f ℓν¯ ) u 0 ℓ R = B → , 3 (D f ℓν¯ ) 0 ℓ B → isindependentofthemixingangleθ. Therefore,experimentalmeasurementofthebranching ratios of B f ℓν¯ , D f ℓν¯ and D f ℓν¯ decays can give direct information about u 0 ℓ s 0 ℓ 0 ℓ → → → the mixing angle θ, as well as, about the flavor structure of f (980) meson. 0 In conclusion, we study the semileptonic decay of pseudoscalar mesons to the scalar f (980) meson. The transition form factors are calculated using 3–point QCD sum rule 0 analysis and then we estimate the corresponding branching ratios. 8 References [1] S. Spanier and N. A. T¨ornqvist, ”Note on scalar mesons”, Phys. Lett. B 592, 1 (2004); S. Godfrey and J. Napolitano, Rev. Mod. 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