2. The mass of the lowest 0 ++ state (the σ ) can also be extracted from the analysis of gravitational form factors of the pion and nucleon, available from the lattice calculations. Extracted masses are com- patible with the benchmark determinations within the expected accuracy of the large- N$_{c}$ framework, showing the consistency of the approach and the proximity of the large- N$_{c}$ limit to the real world in the scalar-isoscalar sector. 3. The constraints of chiral symmetry and the high- energy behavior of the energy-momentum correla- tor imposes a set of sum rules involving various cou- pling constants. Saturation of these sum rules with just two states suggest that f$_{0}$ (600) is a meson and f$_{0}$ (980) is a glueball. Such an assignment, specific to the simple two-state model, results in natural N$_{c}$ -scaling of the ratios of the coupling constants of these states. 4. Correlation functions of the energy-momentum tensor are saturated by (infinitely many) scalar- isoscalar states. The analysis of the Operator Product Expansion shows that there appears the dimension-2 condensate, whose vanishing requires a fine tuning of parameters. Then, in the simplest Regge model the resulting mass of the lowest glue- ball state is ∼ 800 MeV. We thank J. Prades for clarifications and S. Nari- son for remarks. This work is partially supported by the Polish Ministry of Science and Higher Education (grants N202 034 32/0918 and N202 249235), Span- ish DGI and FEDER funds (grant FIS2008-01143/FIS), Junta de Andaluc´ıa (grant FQM225-05), the EU Inte- grated Infrastructure Initiative Hadron Physics Project (contract RII3-CT-2004-506078). In the large N$_{c}$ limit only tree-level diagrams survive, however, with infinitely many resonances. Thus, the infi- nite sums may need to be regularized. In the dimensional regularization the coupling of the resonance to the cur- rent acquires an additional dimension f$_{n}$ → f$_{n}$µ ϵ with ϵ = d − 4. By choosing as a natural scale µ = m$_{n}$ , one gets f 2 n → f 2 $_{n}$m 2 ϵ n , which means This is completely equivalent to formally expanding the two-point correlator at large Q 2 and reinterpreting the coefficients within the zeta-function regularization [84]. In our case the mass formula is m 2 n = an + m 2 $_{0}$, hence one has to consider the sums of the form which is the generalized Riemann zeta function. Then, we arrive at the explicit formulas is the scale behaviour of the latter, a different vacuum energy density (factor of two difference), and modified couplings containing more than one scalar. The σππ ver- tex is the same on shell , thus yields the same decay width of the σ into two pions. Note the resemblance to the bosonic part of the dilated chiral quark model of Ref. [110]. The main difference between potential (B5) and the standard Mexican hat potential, Imposing the condition f$_{σ}$ = f$_{π}$ removes the A term. Note, however, that the identity f$_{π}$ = f$_{σ}$ is not a conse- quence of any symmetry or physical requirement. With this condition the Lagrangean reads Note that σ and ⃗ π are chiral partners. where in the present case one gets an extension of the linear sigma model with the so-called A -term [109], allowed in the linear realization of the chiral symmetry: Introducing the Cartesian decomposition For only one scalar-isoscalar state, Lagrangean (35) with condition (37) reads Appendix B: The linear Lagrangean for one scalar 16 L Φ ,π = f 2 π 4 f 2 σ Φ $^{2}$〈 ∂ $^{µ}$U $^{†}$∂$_{µ}$ U $^{†}$〉 + 1 2 ∂ $^{µ}$Φ ∂$_{µ}$ Φ − V (Φ) . (B1) Σ = f$_{π}$ f$_{σ}$ Φ U = σ + i⃗ τ · ⃗ π, (B2) L = 1 2 [ ∂ $^{µ}$σ∂$_{µ}$ σ + ∂ $^{µ}$⃗ π · ∂$_{µ}$⃗ π ] + A 2 2 [ σ∂ $^{µ}$σ + ⃗ π · ∂$_{µ}$⃗ π ] 2 − V ( σ 2 + ⃗ π $^{2}$) , (B3) A $^{2}$σ2 vac = ( f 2 σ f 2 π − 1 ) . (B4) L = 1 2 [ ∂ $^{µ}$σ∂$_{µ}$ σ + ∂ $^{µ}$⃗ π · ∂$_{µ}$⃗ π ] − m 2 σ 8 f 2 π ( σ 2 + ⃗ π $^{2}$) 2 [ log ( σ 2 + ⃗ π 2 f 2 π ) − 1 2 ] . (B5) V ( σ, ⃗ π ) = m 2 σ 8 f 2 π ( σ 2 + ⃗ π 2 − f 2 π ) 2 , (B6) ∑ n f 2 $_{n}$m 2 k n ≡ lim s → k ∑ n f 2 $_{n}$m 2 s n . (A1) ∑ n ( n + α ) 2 k ≡ lim s → k ∑ n ( n + α ) 2 s = ζ ( − 2 s, α ) , (A2) ∞ ∑ n =0 ( n + α ) 0 = 1 2 − α, (A3) ∞ n =0 ( n + α ) 1 = 1 2 − α 2 + α − 1 6 , (A4) Appendix A: The ζ -regularization