Resonance M [MeV] Γ [MeV] n M (Fit I) M (Fit II) f 0 (600) 400 − 1200 500 − 1000 0 530 556 f 0 (980) 980(10) 70(30) 1 976 983 f 0 (1370) 1350(150) 400(100) 2 1275 1274 f 0 (1500) 1505(6) 109(7) 3 1516 1510 f 0 (1710) 1724(7) 137(8) 4 1724 1714 f 0 (2020) 1992(16) 442(60) 5 1909 1896 f 0 (2100) 2103(8) 209(19) 6 2078 2062 f 0 (2200) 2189(13) 238(50) 7 2234 2215 f 0 (2330) 2321(30) 223(30) 8 2380 2359 TABLE I: PDG values of resonance parameters [36], com- pared to the fits to the radial Regge spectrum of the 0 ++ scalar-isoscalar states, M 2 n = 1 $_{2}$an + m 2 $_{σ}$. In fits I and II the uncertainties are taken as the error in the mass, or as the one half of the resonance width, respectively. In both cases the lowest n = 0 state, corresponding to the f 0 (600) resonance, is excluded from the fit. FIG. 2: The Δ χ 2 = 2 . 3 and 4 . 6 contours, corresponding to the 68% and 90% confidence levels in the m$_{σ}$ − a plane, for Fit I (solid lines) and Fit II (dashed lined). The optimum values are indicated with the dots, filled for Fit I, and open for Fit II. 0.3 0.4 0.5 0.6 0.7 0.8 1.2 1.3 1.4 1.5 m Σ GLYPH<LBracket1> GeV GLYPH<RBracket1> a GLYPH<LBracket1> GeV 2 GLYPH<RBracket1> As is well known [18, 19], at large N$_{c}$ mesons and glueballs are stable states with fixed mass, M$_{S,n}$ ∼ N 0 $_{c}$, and small decay widths Γ$_{S,n}$ ∼ 1 /N$_{c}$ and ∼ 1 /N 2 $_{c}$, re- spectively. Moreover, a general feature of the observed mesonic spectrum is that while the masses grow with n , the widths stay constant. Therefore, one would naively expect the large- N$_{c}$ approximation to work better for the higher masses of the excited states. One might incorpo- rate the widths by using a particular model for the spec- tral density, such as the Breit-Wigner parametrization. Rather than using such a parametrization for the finite width, it may be better to take the widths themselves as an uncertainty on the value of the mass up to O (1 /N$_{c}$ ). This is clearly an upper bound on the uncertainty of the resonance position. Actually, the PDG resonance val- In this Section we review the basic formalism used in the following parts of the paper. A particular attention is paid to the N$_{c}$ -scaling of coupling constants of the meson and glueball scalar-isoscalar states. In Sect. IV we will provide a mounting evidence on the coupling of the lightest state to the gravitational form factor of both the pion and the nucleon. This is relevant, as it shows that the σ state should be explicitly consid- ered in the evaluation of the correlation functions, where it plays an essential dynamical role. Although our fitting strategy was rather crude, the re- sulting Regge-like spectrum of the scalar-isoscalar states turns out to be very reasonable, given the simplicity of the mass formula of Eq. (2). The crucial aspect of the foregoing analysis is that the f$_{0}$ (600) resonance corre- sponds to the lightest state in the radially-excited spec- trum of the Regge-like scalar-isoscalar states. This is an important point, since scalar glueballs [42, 43] and scalar-isoscalar mesons [44, 45] have been studied within the AdS/CFT framework, however neglecting the possi- ble role of the lightest scalar in any of these approaches. The soft-wall version has been reviewed in Ref. [46, 47]. We remark that the large- N$_{c}$ -motivated analysis of the NN-scattering in the $^{1}$S$_{0}$ -channel [27, 41] yields a quite similar value for the lowest scalar mass, m$_{σ}$ = 500(20) MeV. The correlation plot for the a − m$_{σ}$ parameters is pre- sented in Fig. 2 for fits I (solid lines) and II (dashed lines). We notice the compatibility of the two methods. where we have chosen the inverse of the half-width squared as the weight. The result of this procedure, called Fit II, is ues are often quoted as the Breit-Wigner values, which, depending on the assumed background, changes from process to process and in general leads to some model dependence. However, a rigorous description of reso- nances as quantum mechanical states requires to deter- mine them as poles of the meson-meson scattering ampli- tudes in the second Riemann sheet. Then their position in the complex plane becomes model-independent. For the lowest 0 ++ state the shift between the Breit-Wigner and the pole values is rather large, m$_{σ}$ = 800 MeV vs. m$_{σ}$ = 444 MeV, so the difference is compatible with the corresponding half-width. In view of this discussion we fit Eq. (2) by minimizing 4 χ 2 = ∑ n ( M$_{f,n}$ − M$_{S}$ ( n ) Γ$_{f,n}$ / 2 ) 2 , (8) a = 1 . 31(12) GeV $^{2}$, m$_{σ}$ = 556(127) MeV , (9) χ $^{2}$/ DOF = 0 . 12 (Fit II) . III. FORMALISM FOR THE EFFECTIVE SCALAR FIELDS