content_image/430825.jpg content_image/430829.jpg content_image/430828.jpg content_image/430826.jpg content_image/430822.jpg content_image/430821.jpg content_image/430827.jpg content_image/430823.jpg content_image/430830.jpg content_image/430820.jpg content_image/430832.jpg content_image/430831.jpg content_image/430824.jpg
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 

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 

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 

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. 

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, 

<img src='content_image/430825.jpg'>

<img src='content_image/430829.jpg'>

<img src='content_image/430828.jpg'>

<img src='content_image/430826.jpg'>

<img src='content_image/430822.jpg'>

<img src='content_image/430821.jpg'>

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 2 

The result for a = 1 . 31 is 

<img src='content_image/430827.jpg'>

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 

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 

<img src='content_image/430823.jpg'>

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 

<img src='content_image/430830.jpg'>

<img src='content_image/430820.jpg'>

<img src='content_image/430832.jpg'>

<img src='content_image/430831.jpg'>

<img src='content_image/430824.jpg'>

## B. Nucleon 

