Since L π ∼ N$_{c}$ and f$_{π}$ ∼ √$_{N}$$_{c}$$_{, it follows (under the}$ assumption that all states are contributing in the large- N$_{c}$ limit and no cancellations occur) that g$_{nππ}$f$_{n}$ ∼ N 0 $_{c}$, or, due to Eq. (20), Lagrangean (41) fulfills also the standard Goldberger- Treiman relation, g$_{A}$M$_{N}$ = f$_{π}$g$_{πNN}$ . In addition, it is scale-invariant, such that as a consequence that D µ is the Noether current generating dilatations we have the identity [58–61] Upon subtraction of the vacuum part and using that Φ$_{n}$ = f$_{n}$ + ϕ$_{n}$ it follows that A construction of the chiral Lagrangean in the linear re- alization is presented in Appendix B. The scalar fields Φ$_{n}$ are invariant under chiral trans- formations. 2 Lagrangean (35) is nothing else but a gen- eralization of the dilaton Lagrangean of Refs. [58, 60] for the case of many dilatons. We treat here all scalars on equal footing, as suggested by the fact that they all fit into the same radial Regge trajectory. The tree-level decay width of the scalar n into two pions is given by the expression in agreement with (42). In other words, the pole mass and the energy-momentum mass are the same, as re- quired by consistency. The contribution of the scalar-isoscalar degrees of free- dom to the πN scattering is given by the formula Because of the scaling (38) which vanishes at the threshold q = 0, in agreement with the Weinberg-Tomozawa result. According to Eqs. (38) and (40), the meson contribution scales as N 0 c and the glueball contribution as 1 /N$_{c}$ , hence is relatively sup- pressed. which displays the weaker coupling of the glueballs to the pion compared to the mesons. The Lagrangean describing the interaction of the nu- cleon with the scalar-isoscalar particles is Finally, the nucleon-nucleon central potential stem- ming from the exchange of the J PC = 0 ++ isoscalar particles reads in the large- N$_{c}$ limit with u 2 = U 5 = e iγ$_{5}$⃗τ · π/f π . The nucleon field N trans- forms non-linearly, but we can return to the linear re- alization by undoing the chiral transformation, Ψ = uN , which effectively replaces the nucleon mass M$_{N}$ by the combination ∑ $_{n}$g$_{nNN}$ Φ$_{n}$. This yields the general- ized Goldberger-Treiman relation for the scalar-isoscalar channel, Passing to the coordinate space we obtain the standard Yukawa potential, The large- N$_{c}$ counting rules (40) yield V$_{C}$ ∼ N$_{c}$ from the meson exchange, while the glueball exchange pro- duces a subleading contribution ∼ N 0 $_{c}$. Other tenso- rial components of the NN potential scale as ∼ N$_{c}$ or ∼ 1 /N$_{c}$ [29, 30]. The N$_{c}$ scaling of various quantities is collected for convenience in Table II. derived by Carruthers in the early seventies [62]. Again, under the assumption that all states contribute to (42) and no cancellations occur, we find In this section we use the lattice data for the gravita- tional form factors of the pion and nucleon to place fur- ther bounds on the mass of the σ meson. Our analysis is based on the fact that the spin-0 channel is dominated by the scalar-isoscalar mesons. IV. GRAVITATIONAL FORM FACTORS 2 This is unlike the standard assignment of the linear sigma-model where one takes ( σ, ⃗ π ) as chiral partners in the (1 / 2 , 1 / 2) rep- resentation of the chiral SU (2)$_{R}$ ⊗ SU (2)$_{L}$ group. See also Ap- pendix B. g$_{nNN}$ ∼ N 0 c (glueball) , g$_{nNN}$ √$_{N}$$_{c}$ (meson) . (43) M$_{N}$ = ∑ n g$_{nNN}$f$_{n}$, (42) V$_{C}$ ( r ) = − ∑ n g 2 nNN 4 π e − m n r r (48) V$_{C}$ ( | q | ) = ∑ n g 2 nNN q 2 + m 2 n . (47) L N = ¯ N ( i / ∂ − ∑ n g$_{nNN}$ Φ$_{n}$ − g$_{A}$ f$_{π}$ u $^{†}$γ$_{5}$ / ∂u ) N, (41) Γ$_{nππ}$ ∼ 1 /N 2 c (glueball) , Γ$_{nππ}$ 1 /N$_{c}$ (meson) , (40) Γ(Φ$_{n}$ → ππ ) ≡ Γ$_{nππ}$ = (39) 3 g 2 $_{nππ}$m 2 n 32 π [ 1 − 2 m 2 π m 2 n ] 2 √ m 2 n − 4 m 2 π ∼ 3 g 2 $_{nππ}$m 3 n 32 π . T ab πN = δ ab ¯ u ( p $^{′}$) u ( p ) ∑ n q $^{2}$g$_{nππ}$ g$_{nNN}$ m 2 n − q 2 , (46) 〈 N | Θ(0) | N 〉 = − ∑ n m 2 $_{n}$f$_{n}$ 〈 N | ϕ$_{n}$ | N 〉 = n f$_{n}$g$_{nNN}$, (45) ∂ $^{µ}$D$_{µ}$ = Θ = ∑ n [4 V (Φ$_{n}$) − Φ$_{n}$ V $^{′}$(Φ$_{n}$ )] = − ∑ n m 2 n 4 f 2 n Φ 4 $_{n}$. (44) g$_{nππ}$ ∼ 1 /N$_{c}$ (glueball) , g$_{nππ}$ 1 $_{/}$√$_{N}$$_{c}$ (meson) . (38) 7