OCHA-PP-70 Decay constants and mixing parameters in a ¯ relativistic model for qQ system 6 9 9 1 n a J Mohammad R. Ahmadya, Roberto R. Mendel and James D. Talmanb 6 1 1 aDepartment of Physics, Ochanomizu University, 1-1 Otsuka 2, Bunkyo-ku, Tokyo 112, Japan v bDepartment of Applied Mathematics, The University of Western Ontario, London, Ontario, 1 7 Canada N6A 5B7 2 1 0 6 9 / h p - ABSTRACT p e h : v i X We extend our recent work, in which the Dirac equation with a “(asymp- r a totically free) Coulomb + (Lorentz scalar γ0σr) linear ” potential is used to ¯ obtain the light quark wavefunction for qQ mesons in the limit m → ∞, Q to estimate the decay constant f and the mixing parameter B of the pseu- P doscalar mesons. We compare our results for the evolution of f and B with P the meson mass M to the non-relativistic formulas for these quantities and P show that there is a significant correction in the subasymptotic region. For σ = 0.14 GeV−2 and Λ = 0.240 GeV we obtain: f = 0.371 , f = MS D Ds 0.442 , f = 0.301 , f = 0.368 GeV and B = 0.88 , B = 0.89 , B = B Bs D Ds B 0.95 , B = 0.96 , and B = 0.60. Bs K Recently, we presented our results for the form factor (Isgur-Wise function) which parametrizes the transition matrix elements of mesons containing a heavy quark [1]. In ourrelativistictreatmentofaheavy-light system, weassumedthat,withtheheavyquark fixedattheorigin,thelightquarkwavefunctionobeysaDiracequationwithaspherically symmetric potential. This potential, an asymptotically free Coulomb term plus a linear confining term, reflects the QCD interactions both at short and long distances. In this way, we obtained the shape and the slope at zero recoil of the form factor in the leading 1/m order by fitting our model parameters to the experimental results on the Q semileptonic B decays. In this paper, we extend our results by calculating the decay constants f and mixing P parameters B of the pseudoscalar qQ¯ mesons in our model. f parametrizes the matrix P element for the decay of a pseudoscalar meson through the axial-vector current A5: µ < 0|A5(x)|P(k) >= ik f e−ik·x , (1) µ µ P where k is the four-momentum of the meson. On the other hand, B is related to the matrix element for neutral-meson mixing which is conventionally written as 4 < P¯|(V −A5)2|P >= f2M B , (2) µ µ 3 P P where V is the vector current and M is the meson mass. From a phenomenological µ P point of view, knowledge of these matrix elements is necessary for extracting important quantities such asthequark mixing matrixandCPviolationwithin theStandardModel. Experimentally f = 132MeV and f = 167MeV are well-known. However, for heavy π K mesons only a few data with large uncertainties are available. Mark III sets an upper bound f < 290MeV [2] while WA75[3], CLEOII[4] and BES[5] find 225 ± 45 ± 20 ± D 41 , 344±37±52 and 430+150 ±40MeV for f respectively. −130 Ds There are various quark-model, QCD sum rules and lattice calculations of these parameters[6, 7, 8, 9]. A frequently used point of comparison is a scaling law for f that P is derived from a non-relativistic (NR) quark model: 1 fNR(m → ∞) ∝ , (3) P Q sMP where m is the mass of the heavy quark. The mixing parameter is identically 1 in any Q NR quark model: BNR(m ) = 1 . (4) Q In Refs. [8, 9], it is shown that the combination of a relativistic dynamics and an asymptotically free Coulomb interaction for qQ¯ system results in a significant deviation from the NR scaling law. Our approach here is complementary to Refs. [8, 9] in the 1 sense that we use a potential which is not only asymptotically free but also includes a linear confining term which determines the global behavior of the Dirac wavefunction. ∞ At the same time, we use a saturation value for the strong coupling α = α (r → ∞) s s therefore avoiding unphysical pair creation effects. We start with the time-independent Dirac equation: α~.(−i▽~ )+V (r)+c +γ0(σr+m ) Ψ(~r) = ǫ Ψ(~r) , (5) c 0 q q h i where m is the light quark mass. V (r) is the asymptotically free Coulomb potential q c (tranforming as the zeroth component of a Lorentz four-vector): 4α (r) s V (r) = − (6) c 3 r where α (r) is obtained in the leading log approximation and is parametrized as follows s 2π α (r) = (7) s (11− 2NF)ln[δ + β] 3 r The parameter δ defines the “long distance” saturation value for α . We use δ = 2.0 s which corresponds to α (r = ∞) ≈ 1.0. The parameter β is related to the QCD scale s −1 Λ by β = (2.23Λ ) for N = 3 (we use N = 3 throughout this paper). We take MS MS F F Λ ≈ 0.240 GeV , which corresponds to β = 1.87 GeV−1 in most of the calculations. MS To describe the long distance behaviour, a linear term (γ σr) is introduced in the 0 potential which transforms as a Lorentz scalar. This form is favored based on many theoretical and phenomenological arguments [10, 11, 12]. For the parameter σ, we mainly use a value σ = 0.14 which is favored by a recent lattice estimate [13] and can also be extracted from the Regge slope using two-body generalization of Klein-Gordon equation [11, 12]. However, we also present results for σ = 0.25 and 0.18 GeV2, as the former value is compatible with the experimentally available 0.45 GeV 2P-1S splitting for D and D system and the latter is extracted from Regge slope data using string s model [14]. As mentioned in the discussion of our results in Ref. [1], the best fit of our model to the recent CLEO 1994 data analysis favors a smaller value for the parameter σ. The potential in the Dirac equation includes an additive constant c , which is clearly 0 subleading both in the short and long distance regimes. The only role of this constant is to define the absolute scale of the light quark energy ǫ which is identified with the q ”inertia” parameter often introduced in heavy quark effective theory: ǫ ≡ Λ¯ = lim (M −m ) . (8) q q mQ→∞ P Q 2 Therefore, only the difference Λ¯ −c ≈ ǫ −c can be extracted from Eq. (5). As we q 0 q 0 indicate later, in the heavy limit m → ∞, f and B are not very sensitive to Λ¯ (and Q P q in turn to c ). 0 The ground state solution to Eq.(5) has the form: χ(r) ψ(~r) = N , (9) −iσ.rˆφ(r) ! where N is a normalization constant. Figure 1 illustrates the functional behaviour of the normalized large component χn(= Nχ(r)) and small component φn(= Nφ(r)) of the wavefunction ψ(~r) for the cases where m = 0 (appropriate for q ≡ u,d) and q m = 0.175 GeV (for q ≡ s). We observe that χ(r) is finite and φ(r) → 0 as r → 0. q This, as noted in Refs. [8, 9], is due to using a running α (r) in the Coulomb poten- s tial (Eq.(6)) rather than a fixed α which would result a singular wavefunction at the origin. On the other hand, since our model incorporates the long distance behaviour of QCD interactions (through linear confining potential) and at the same time avoids a singularity in the Coulomb potential (by introducing a saturation value for α (r)), the s resulting wavefunction is physically meaningful for the whole range of r. This is our main improvement over previous works where a Dirac equation along with the leading- log Coulomb potential has been applied to heavy-light mesons [8, 9]. The functions χ(r), φ(r) and the normalization constant N are independent of m in our leading order (in Q 1/m ) approach where we assume that the heavy quark is fixed at the origin. How- Q ever, as it is illustrated in Figure 1, the SU(3) symmetry breaking strange quark mass F m = 0.175 GeV results in about 20% larger value for N (χ(r) is independent of m as s q r → 0). The normalization constant N depends on the global behaviour of the poten- tial, i.e. β and σ (see Eqs.(5), (6) and (7)). We mainly use β = 1.87 (corresponding to Λ = 0.240 GeV) and σ = 0.14 GeV−1 (see the discussion following Eq. (7)), however, MS we do vary these parameters to investigate the sensitivity of our results. We now proceed to compute the decay constant f and mixing parameter B in our P model. The decay constant is given by the overlap integral [15] 12 f2(m ) = N2 d3~r|Ψ (~r)|2χ2(r) , (10) P Q M Q P Z where 12 is a color-spin-flavour coefficient and the quantum-mechanical fluctuations in the position of the heavy quark are assumed to be described by a non-relativistic wavefunction Ψ (~r). Considering that Ψ (~r) is “spread“ over a distance r = O(1/m ), Q Q Q Q one can reduce (10) to the expression [9]: 12 f (m ) = Nχ(r ) . (11) P Q Q sMP 3 At this point, we need to make a definite connection between r and the heavy quark Q mass m . It is assumed that r = κ/m , where κ ≥ 1 is expected on physical grounds. Q Q Q We obtain a value for κ by extrapolating the decay constant formula (11) to K-meson system where f = 167MeV is known experimentally. This extrapolation, even though K of uncertain reliability due to finite m effects, has been frequently made in purely NR Q models. For mconstituent ≈ M = 0.495 GeV, we obtain κ = 1.67 for Λ = 0.240 and s K MS κ = 1.73 for Λ = 0.200. In comparing our results for the decay constant with the MS NR scaling law, Eq. (3), we also include a finite renormalization factor suggested by Voloshin and Shifman [16] and Politzer and Wise [17]. In Figure 2, comparison is made between the ”improved” NR scaling law γ 1 1 fNR(M ) ∝ , (12) P P sMP α˜s(MP)! and our relativistic results γ 1 κ 1 f (M ) ∝ χ , (13) P P sMP (cid:18)MP(cid:19) α˜s(MP)! where in the latter we have made the approximation M ≈ m in the expression for r . P Q Q One can justify this approximation in view of the uncertainty in the determination of r from the extrapolation to K-meson system. Because of this assumption, r does not Q Q depend on the inertia parameter Λ¯ and therefore is also independent of the constant q c . Also in Eq. (13), γ = 2/9 for N = 3 and α˜ (M ) = 2π/9ln(M /Λ ) in the 0 F s P P MS renormalization correction factor. In the Figure, the above functions are normalized to their value at a large mass scale (∼ 60Λ ) where the physics (of m → ∞) is MS Q well understood. Our results are presented for m = 0 and m = 0.175 GeV. The q q main feature that distinguishes the relativistic graph from the NR scaling law is the maximum at around D meson mass, as noted in Refs. [8, 9]. We would like to emphasize again, that our improved model is free of unphysical behaviour for the whole range of r. Therefore, here we rigorously confirm that the maximum in the scaling law is physical and is due entirely to the relativistic dynamics of the light quark at the short distances. The position of this maximum is around 7.0Λ for m = 0 (for Λ = MS q MS peak 0.240 GeV , M ≈ 1.68 GeV which is somewhat below M ≈ 1.87 GeV) and around P D peak 8.1Λ for m = 0.175 GeV (resulting M ≈ 1.94 GeV which is roughly the same as MS q Ps M ≈ 1.96 GeV). From Figure 2, we also notice that, roughly speaking, from the B Ds meson mass scale onward, SU(3) is a good symmetry as far as the scaling of the decay F constant is concerned. The mixing parameter B can also be written as an overlap integral [18] 12 B(m ) = N2 d3~r|Ψ (~r)|2 χ2(r)−φ2(r) , (14) Q f2M Q P P Z h i 4 which will be approximated by 2 φ(r ) Q B(m ) = 1− , (15) Q "χ(rQ)# by the same argument that followed Eq.(10). The non-relativistic scaling law for mixing parameter (Eq. (4)) is recovered for m → ∞ (i.e. r → 0) as φ(r → 0) = 0 (see Fig. Q Q 1). Figure 3 illustrates the evolution of the mixing parameter B with the meson mass M (as in the case of the decay constant, r = κ/m and we use the approximation P Q Q M ≈ m ) for m = 0 and m = 0.175 GeV. We observe that in the subasymptotic P Q q q region, deviation from the NR scaling law is 5% and 12% for the B and D meson mass scale respectively. On the other hand, the SU(3) symmetry breaking effects in the F < < scaling, even for M ∼ M ∼ M , is negligible. D P B In Table 1, we present our numerical predictions for the decay constant and mixing parameter of D, D , B, B mesons plus an estimate of B , the mixing parameter for the s s K K-meson. To test the sensitivity of these predictions, we vary the model parameters β and σ. For example, changing Λ from 0.240 GeV (β = 1.87 GeV−1) to 0.200 GeV MS (β = 2.24 GeV−1), results in a very small change (ranging from 1% to 4%) in f and P B. The ratios f /f = 1.22 and f /f = 1.18 (for σ = 0.14 GeV−1) are independent Bs B Ds D of Λ . On the other hand, the sensitivity to the linear potential coefficient is much MS more significant. Besides σ = 0.14 GeV−1, the decay constants and mixing parameters are also estimated for σ = 0.18, 0.25 GeV−1 (see the paragraph on the linear potential following Eq. (7)). An increase of 30% to 40% in the decay constants is observed for increasing σ from 0.14 to 0.25 GeV−1. However, the mixing parameters are far less sensitive to this model parameter. The results obtained for f all exceed the upper bound of 0.290 of Ref. 2. However, D the finite renormalization factor is uncertain at the lower mass of the D meson; if this factor, which is 1.27, is not included, f is 0.292 for σ = 0.14. The results may therefore D not be incompatible with Ref. 2 at the smaller value of σ. A comparison between our predictions (Table 1) and other theoretical results [6] also favors a smaller value for σ. Even though our results are larger than other theoretical predictions, for σ = 0.14 GeV−1 and taking into account the uncertainty factor 1.27 (from the finite renormalization factor below D-meson mass scale) there are agreements with some lattice and potential model estimates. However, QCD sum rules predictions are generally smaller than ours. Our estimated B and B is on the low side of a recent B Bs lattice prediction by UKQCD[7]. In conclusion, we used a relativistic model with a phenomenological potential that accounts for QCD interactions at all length scales, to estimate the decay constant f P and mixing parameter B of heavy-light mesons. The evolution of f and B with the P 5 meson mass significantly deviates from the NR scaling law in the phenomenologically < < interesting subasymptotic region M ∼ M ∼ M . D P B Acknowledgement The authors would like to thank H. Trottier for useful discussions. Work of M. A. is supported by the Japanese society for the Promotion of Science. 6 References [1] M.R. Ahmady, R.R. Mendel and J.D. Talman, Phys. Rev. D52 (1995) 254. [2] J. Adler, et al (MARK III Collaboration), Phys. Rev. Lett. 60 (1988) 1375. [3] S. Aoki, et al (WA75 Collaboration), Prog. Theor. Phys. 89 (1993) 131. [4] D. Acosta, et al (CLEO II Collaboration), Phys. Rev. D49 (1994) 5690. [5] J.Z. Bai, et al (BES Collaboration), SLAC preprint SLAC-PUB-95-6746. [6] See Table IV of the review by J.D. 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[18] R.E. Schrock and S.B. Treiman, Phys. Rev. D19 (1979) 2148; P. Colic et al, Nucl. Phys. B221 (1983) 141. 7 Table Captions Table 1: The estimated decay constants (in GeV) and mixing parameters for various choices of the model parameters. 8 Figure Captions Figure 1: The normalized large χn = Nχ(r) and small φn = Nφ(r) component of the wavefunction for q ≡ u,d (m = 0) and q ≡ s (m = 0.175 GeV). q q Figure 2: The evolution of the decay constant with the meson mass (m = 0 and q m = 0.175 GeV) compared with the NR scaling law. q Figure 3: The scaling of the mixing parameter B with the meson mass. 9