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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- 

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 

where we have chosen the inverse of the half-width squared as the weight. The result of this procedure, called Fit II, is 

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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. 

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. 

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]. 

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. 

## III. FORMALISM FOR THE EFFECTIVE SCALAR FIELDS 

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. 

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