Abstract—We reexamine the work of Rashdan et al., who considered a chiral model for the nucleon based on the linear sigma model with scalar-isoscalar scalar-isovector mesons coupled to quarks. The dependence of the axial-vector coupling constant g A and the pion nucleon coupling constant g NN on the quark masses and sigma masses have been investigated in the frame work of the extended linear sigma model. In this work we calculate both of g A and g NN to investigate the effect of the quark masses on the g A in the framework of the extended linear sigma model, which is proposed by Rashdan et al. and compare it with the free Skyrmion model, extended Skyrmion model and finally with Birse and Banerjee model. The field equations have been solved in the mean-field approximation by Goldflam and Wilets. Our study shows a better fitting to the experimental data compared with the existing models. Index Terms—Extended linear sigma model, axial vector coupling constant, quark mass and mean field approximation. I. INTRODUCTION The axial-vector coupling constant g A is important to understand Quantum Chromodynamics (QCD). In recent years there has been a growing interest in studying A g . A lot of groups have made significant progress towards understanding A g using several models (see Cloet et al. [1] and Ali et al. [2]). We study the extended linear sigma model as one of these models to describe the interactions of quarks and  meson in a mean field approximation which has the hedgehog property. A similar model has been considered by Kalbermann and Eisenberg [3], Birse and Banerjee [4], while the higher order of the mesonic inteactions in the linear sigma model was considered by Sahu and Ohnishi [5], [6] and M.Rashdan et al. [7]-[9], who used the mean field approximation to get a better description of the g A . In our study [10], we used the coherent pair approximation to study the g A . Few solutions for the lagrangian of chiral linear soliton models applied to the nucleon have already been suggested. The mean-field equations are a straightforward extension of the finding by Goldflam and Wilets [11]. In this work, we consider a model based on the idea of strong QCD forces. The aim is to investigate the effect of the quark masses on the A g in the framework of the extended linear sigma model, which is proposed by Rashdan et al. [7] with prameters like the pion decay constant 91.9 =  f MeV and the pion mass 138.04 =  m MeV fixed in similar way as Struber and Rischke [12]. The paper is organized as follows; first, the explaination of extended linear sigma model in Section II, the numerical results and the discussion in Section III , and finally, the conclusion presented in Section IV. II. THE EXTENDED LINEAR SIGMA MODEL The extended linear sigma model is described in details in [7]. We describe the interactions of quarks with   mesons and pions by Birse and Banerjee [4]. The Lagrangian density is,     , ) ( 4 = , 2 2 2 2 2 2 2          f m U      (1) It is clear that this potential also satisfies chiral symmetry. Applying the PCAC we get, , 2 = 2 2 2 4 2      f m f  (2) and . 8 3 = 6 2 2 2     f m m  (3) Now, we expand the extremum, with the shifted field defined as , =    f '  (4) inserting equ.(4) into equ.(1), we obtain                                      . . 2 1 = 5  ig g f g i r L ' ' '   , ,   ' U  (5) with       . 4 = , 2 2 2 2 2 2 2 2 2             f m f m f U ' ' '              (6) The time-independent fields  r '  and  r   are to satisfy Determination of the Axial-Vector Coupling Constant from the Extended Linear Sigma Model Tarek Sayed Taha Ali International Journal of Applied Physics and Mathematics, Vol. 4, No. 3, May 2014 184 DOI: 10.7763/IJAPM.2014.V4.280 Manuscript received January 10, 2014; revised April 4, 2014. Tarek Sayed Taha Ali is with Faculty of Science, UAE University, AL-AIN, U.A.E. (e-mail: t.ali@uaeu.ac.ae).