where the symbols Θ q i denote the quark parts of the grav- itational form factors of Eq. (49). A monopole fit has been undertaken, yielding m$_{σ}$ = 0 . 89(27)(9) MeV and A$_{22}$ (0) = − 0 . 076(5) ( cf. Table C.8 on p. 105 of Ref. [64]). By using several parameterizations it has been noted that the precise form of the function cannot be pinned down from the data unambiguously due to large uncertainties in A$_{22}$ ( t ). In the following we will use the low-energy theorem including infinitely many states, of the form with B ( x, y ) = Γ( x )Γ( y ) / Γ( x + y ) denoting the Euler Beta function. The function (68) fulfills the normaliza- tion condition For x ≫ y one has B ( x, y ) ∼ Γ( y ) x − $^{y}$, hence in the asymptotic region of M 2 − t ≫ ( b − 1) a we find which provides a relatively accurate fixing of A$_{22}$ at the origin and greatly helps the regression analysis. In terms of the form factors related to the Generalized Parton Distributions (GPD) of Ref. [63, 64], the quark contribution to the gravitational form factor reads The result for a = 1 . 31 is As we can see, the result is fully compatible with the monopole ( b = 2) and at present the large errors in the lattice data wash out any insight from the excited scalar spectrum, despite the large momenta $^{3}$. Thus, we restrict ourselves to a simple monopole fit Due to the multiplicative QCD evolution one has where the (valence) quark momentum fraction depends on the renormalization scale µ . Its leading-order pertur- bative evolution reads which yields χ $^{2}$/ DOF ∼ 2 . 4. Actually the large χ 2 is due to incompatible values of nearby points. In such a situation, in order to obtain reliable estimate of the model parameters, we rescale the errors by a factor of 1.5 to make them mutually compatible. Moreover, we enforce the low energy theorem, Eq. (62) as a constraint – a possibility not directly considered in Ref. [63, 64]. As a result we get χ $^{2}$/ DOF ≃ 1 (after the mentioned rescaling of the data errors) and the optimum values where the anomalous dimension is γ (0) 1 / (2 β$_{0}$ ) = 32 / 81 for N$_{F}$ = N$_{c}$ = 3. The QCD running coupling constant is equal to where we take Λ$_{QCD}$ = 226 MeV for N$_{c}$ = N$_{f}$ = 3. Phenomenologically, it is known from the Durham group analysis [66], based mainly on the E615 Drell- Yan data [67] and the model assumption that the sea quarks carry 10 − 20% fraction of the momentum, that 〈 x 〉 π u + d = 0 . 47(2) at the scale µ = 2 GeV. The analy- sis of the Dortmund group [68], based on the assump- tion that the momentum fraction carried by the valence quarks in the pion coincides with that in the nucleon, yields 〈 x 〉 π u + d = 0 . 4 at µ = 2 GeV. The lattice data at the lattice spacing a$_{lat}$ = 0 . 1fm as well as a recent chiral quark model calculation [69] support this view. In Fig. 3 we present the corresponding correlation ellipse. The gravitational form factor Θ$_{0}$( t ) at the optimum val- ues of the parameters (73) is presented in Fig. 4. The scalar component of the nucleon gravitational form factor is We recall that the large- N$_{c}$ analysis implies sums of monopoles in the form factors. The largest available mo- mentum transfer, t = − 4 GeV $^{2}$, obtained in Ref. [64], suggests that some information on the contribution of the excited states might be extracted. Therefore, following the approach already used for the electromagnetic pion form factor in Ref. [70] (see also [71]), we have attempted a Regge-like fit, 〈 N ( p $^{′}$) | Θ(0) | N ( p ) 〉 = ¯ u ( p $^{′}$) u ( p )Θ$_{N}$( q $^{2}$) , (74) Θ$_{π}$( t ) = tf$_{b}$ ( t ) , (67) α ( µ ) = 4 π β$_{0}$ log( µ $^{2}$/ Λ 2 $_{QCD}$) , (66) 〈 x 〉 π u + d = 0 . 52(2) m$_{σ}$ = 445(32) MeV . (73) Θ q $_{π}$( t ) = 〈 x 〉 π u + d t m 2 σ m 2 σ − t , (72) R = 〈 x 〉 $_{q}$( µ ) x $_{q}$( µ$_{0}$ ) = ( α ( µ ) α ( µ$_{0}$ ) ) γ (0) 1 / (2 β$_{0}$ ) , (65) Θ q $_{π}$( t, µ ) = 〈 x 〉 π $_{q}$( µ )Θ$_{π}$( t ) , (64) 〈 x 〉 π u + d = 0 . 52(3) , m$_{σ}$ = 495 +250 − 135 MeV , b = 2 . 24 +1 . 56 − 0 $_{55}$. (71) Θ q $_{π}$( t ) = − 4 tA q $_{22}$( t ) (63) A$_{22}$ (0) = − 1 4 A$_{20}$ (0) , (62) f$_{b}$ ( t ) ∼ Γ ( M 2 a/ 2 + b − 1 ) Γ ( M 2 a/ 2 ) ( a/ 2 M 2 − t ) b − 1 . (70) f$_{b}$ (0) = 1 . (69) f$_{b}$ ( t ) = B ( b − 1 , M $^{2}$− t a/ 2 ) B ( b − 1 , M 2 a/ $_{2}$) , (68) 9 B. Nucleon 3 We note that a similar fit [71] to the vector form factor using a = 1 . 2(1) GeV 2 yields m$_{ρ}$ = 775(15) MeV and b = 2 . 14(7), which can be distinguished from a simple monopole fit at the two-standard-deviation level.