Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Salcedo, L. A. M.; Melo, J. P. B. C. de; Hadjmichef, D.; Frederico, T. Weak Decay Constant of Pseudoscalar Mesons in a QCD-Inspired Model Brazilian Journal of Physics, vol. 34, núm. 1A, march, 2004, pp. 297-299 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46400132 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 297 Weak Decay Constant of Pseudoscalar Mesons in a QCD-Inspired Model L. A. M. Salcedoa, J.P.B.C. de Melo b, D. Hadjmichefc, and T. Fredericoa aDep. de F´ısica, Instituto Tecnologico´ de Aeronautica,´ Centro Tecnico´ Aeroespacial, 12.228-900 Sao˜ Jose´ dos Campos, Sao˜ Paulo, Brazil bInstituto de F´ısica Teorica,´ Universidade Estadual Paulista, 01405-900, Sao˜ Paulo, Brazil c Instituto de F´ısica e Matematica,´ Universidade Federal de Pelotas, 96010-900, Campus Universitario´ Pelotas, Rio Grande do Sul, Brazil Received on 15 August, 2003. We show that a linear scaling between the weak decay constants of pseudoscalars and the vector meson masses is supported by the available experimental data. The decay constants scale as fm=f¼ = MV =M½ (fm decay constant and MV vector meson ground state mass). This simple form is justified within a renormalized light- front QCD-inspired model for quark-antiquark bound states. 1 Introduction proximation [1]. In a simplified form [2, 3], the effective mass operator equation is written as: Effective theories applied to describe hadrons, which are in- " # spired by Quantum Chromodynamics [1, 2, 3], can be useful ~k2 + m2 ~k2 + m2 M 2 Ã (x;~k ) = ? 1 + ? 2 Ã (x;~k ) in indicating direct correlations between different hadronic m m ? x 1 ¡ x m ? properties. In this way, it is possible to pin down the relevant Z µ ¶ dependence of the observables with some physical scales 4m1m2 ® ¡ dx0d~k0 »(x; x0) ¡ ¸ Ã (x0;~k0 ); (1) that otherwise would have no simple reason to show a di- ? 3¼2 Q2 m ? rect relation, besides being properties of the same underly- ing theory. For example, if one points out a systematic de- where the phase space factor is pendence of a hadron observable with its mass even in a phe- nomenological model, this fact may be regarded as an useful θ(x0)θ(1 ¡ x0) »(x; x0) = p ; guide for presenting results obtained in Lattice QCD. In fact, x(1 ¡ x)x0(1 ¡ x0) systematic correlations between different meson properties with mass scales were found from the solution of Dyson- and Ãm is the projection of the light-front wave-function Schwinger equations [4]. in the quark-antiquark Fock-state component. The mean 0 2 0 2 One intriguing aspect is the dependence of the weak square momentum transfer ((k1 ¡k1) +(k2 ¡k2) )=2 gives 2 0 decay constant of the pseudoscalar meson with its mass. ¡Q (ki and ki are the quark four-momenta). The coupling For light mesons up to D, the weak decay constant tends constant ® defines the strength of the Coulomb-like poten- to increase with the mass, while numerical simulations of tial and ¸ is the bare coupling constant of the Dirac-delta 2 quenched lattice-QCD indicate that fD > fB [5], which is hyperfine interaction. The energy transfer in Q is left out. still maintained with two flavor sea quarks [5, 6]. General Confinement comes through the binding of the constituents arguments, within Dyson-Schwinger formalism for QCD in in the meson, which in practice keeps the quarks inside the the heavy quark limit, says that thep weak decay constant mesons. should be inversely proportional to Mm [7] (Mm is the The mass operator equation (1) needs to be regularized pseudoscalar meson mass). Effective QCD inspired mod- and renormalized in order to give physical results, such de- els valid for low energy scales can also be called to help to velopment has been performed in Ref. [8]. In that work, it investigate this subtle point. In these models [2, 3, 8], the was obtained the renormalized form of the equation for the interaction is flavor independent, while the masses of con- bound state mass, which is i) invariant under renormaliza- stituent quarks can be changed, which naturally implies in tion group transformations, ii) the physical input is given by correlations between observables and masses. the pion mass and radius, and iii) no regularization parame- Our aim here, is to investigate the pseudoscalar weak ter. decay constant within a QCD inspired model [8]. The mass In the work of Ref. [8], the quark mass was changed to operator equation for the valence component of the meson allow the study of mesons with one light antiquark plus a light-front wave function, described as a bound system of a strange, charm or bottom quark. The masses of the con- constituent quark and antiquark of masses m1 and m2, was stituent quarks were within the range of 300 up to 5000 derived in the effective one-gluon-exchange interaction ap- MeV. A mass of 384 MeV was obtained for the up and 298 L.A.M. Salcedo et al. down quarks from the rho meson mass, which in the model where is weakly bound. The Dirac-delta interaction comes from an effective hyperfine interaction which splits the pseudo- ~k2 + m2 ~k2 + m2 scalar and vector meson states. In the singlet channel the 2 ? 1 ? 2 M0 = + ; (4) hyperfine interaction is attractive, which is not valid for spin x 1 ¡ x one mesons. In the model, the Dirac-delta interaction mock up short-range physics which are brought by the empirical in the frame in which the meson has zero transverse mo- value of the pion mass, and from that a reasonable descrip- 02 2 ~ mentum. (M 0 is obtained from M0 by substitution of k? tion of the binding energies of the constituent quarks form- ~ 0 0 by k? and x by x .) The overall normalization of the qq ing the pseudoscalar mesons was found [8]. The model, Fock-component of the meson wave-function (3) is G . without the Coulomb like interaction, was also able to de- m scribe the binding energies of the ground state of spin 1/2 In this first calculation of the decay constant within this baryons containing two light quarks and a heavy one [9]. model, we are going to assume the dominance of the asymp- Within the effective model of Eq.(1), the low-lying vec- totic form of the meson wave function and simply use tor mesons are weakly bound systems of constituent quarks while the pseudo-scalars are more strongly bound [8]. This ~ 1 Gm allows to calculate the masses of constituent quarks directly Ãm(x; k?) = p : (5) x(1 ¡ x) M 2 ¡ M 2 from the masses of the ground state of vector mesons [9]: m 0 1 m = M = 384 MeV ; u 2 ½ To obtain the pseudoscalar decay constants, we follow 1 Ref. [11]. To construct the observables in terms of the me- ms = MK¤ ¡ M½ = 508 MeV ; son wave function, one has to account for the coupling of 2 1 the spin of the quarks, which is described by an effective mc = MD¤ ¡ M½ = 1623 MeV ; Lagrangian density with a pseudo-scalar coupling between 2 the quark (q (~x) and q (~x)) and meson (© (~x)) fields [11] 1 1 2 m m = M ¤ ¡ M = 4941 MeV ; (2) b B 2 ½ L (~x) = ¡iG © (~x) q (~x)γ5q (~x) + h:c: ; (6) where it is used the values of 768 MeV, 892 MeV, 2007 MeV eff m m 1 2 and 5325 MeV for the ½, K¤, D¤ and B¤ masses, respec- tively [10]. the coupling constant is Gm. From the effective Lagrangian Here, we use the effective model to predict a physical above one can derive meson observables and write them in property directly related to the wave-function of the ground terms of the light-front asymptotic wave function, Eq. (5). state of the pseudo scalar mesons. We calculate the weak To achieve this goal, it is necessary to eliminate the rela- + + + + + decay constants (fm) of K , D , Ds , for which exper- tive x -time (x = t + z) between the constituents in the imental values are known [10]. Besides the constituent physical amplitude, which then allows to write the meson quark masses from Eq.(2) and the pion mass, our calcu- observable in terms of the wave function [11]. lation needs as input the pion weak decay constant, f¼ = 92:4 § 0:07 § 0:25 MeV [10]. The eigenfunction of the in- teracting mass squared operator from Eq.(1), for large trans- verse momentum, behaves as the asymptotic wave-function, 3 Results for the weak decay constant 2 which decreases slowly as 1=~p?. Therefore, in the calcu- lation of the weak decay constants it is necessary to reg- of pseudoscalar mesons ulate the logarithmic divergence in the transverse momen- tum integration and take care of the cut-off dependence to The pseudoscalar meson weak decay constant is calculated be able to give an unique answer. One has to consider that from the matrix element of the axial current A¹(0), between the pion decay constant provides the short-range informa- the vacuum state j0i and the meson state jqmi with four mo- tion contained in the pion wave function, which we suppose mentum qm [10]: to be the same for all pseudo-scalars.