0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 GLYPH<Minus> t GLYPH<LBracket1> GeV $^{2}$GLYPH<RBracket1> G Θ GLYPH<LParen1> t GLYPH<RParen1> GLYPH<Minus> A$_{20}$ GLYPH<LParen1> 0 GLYPH<RParen1>GLYPH<Slash1> 4 1 2 3 4 GLYPH<Minus> t GLYPH<LBracket1> GeV $^{2}$GLYPH<RBracket1> GLYPH<Minus> 0.10 GLYPH<Minus> 0.05 0.05 0.10 A$_{22}$ GLYPH<LParen1> t GLYPH<RParen1> FIG. 5: (Color online) Spin-0 gravitational form factor of the nucleon, G$_{θ}$ ( t ), obtained from the lattice simulations of Refs. [73] at the pion masses m$_{π}$ = 352 and 356 MeV, together with the monopole fit with mass m$_{σ}$ = 624 MeV (solid line). The band indicates the uncertainty in m$_{σ}$ of 78 MeV obtained with the χ 2 method. FIG. 4: (Co l or o nline) Spin-0 gravitational form factor of the pion, A 22 ( t ) , fro m the lattice calculation of Refs. [63, 64] ex- trapolated t o th e physical pion mass (the points with the large error bars), toge ther with the monopole fit with parameters (73), indicated by the line. The band indicates the uncer- tainty in the m odel parameters. The low energy theorem fixes the value o f the A 22 form factor at t = 0. The lower data points sho w the data for − A 20 / 4. where u ( p ) an d u ( p $^{′}$) are nucleon Dirac spinors. In the meson-domina nce approximation it can be written as The nucleon m ass is given by Eq. (42). The quark c ontributions to the GPD’s of the nucleon have been deter mined on the lattice [72, 73] for masses as small as twice t he physical pion mass. The decomposition corresponding t o the energy-momentum tensor reads where σ µν = i $_{2}$[ γ $^{µ}$, γ$^{ν}$] (the Bjorken-Drell notation), the scalar functions are moments of the GPD’s, the momen- tum transfer is denoted as Δ = p ′ − p , and the average nucleon momentum is P = ( p ′ + p ) / 2. Taking the trace and applying the Gordon identity, 2 M$_{N}$ ¯ u ( p $^{′}$) γ $^{µ}$u ( p ) = ¯ u ( p $^{′}$)( iσ $^{µρ}$Δ$_{ρ}$ + 2 P $^{µ}$) u ( p ), as well as the Dirac equation, (/ p − M$_{N}$ ) u ( p ) = 0 and ¯ u ( p $^{′}$)( / p ′ − M$_{N}$ ) = 0, we obtain the following expression for the spin-0 gravitational form factor of the nucleon: The combination (77) can be directly constructed from the results of the LHPC Collaboration [73], where the data points for the A$_{20}$ , B$_{20}$ , and C$_{20}$ form factors are provided for six values the pion mass: m$_{π}$ = 757, 681, 595, 495, 356, and 352 MeV. The values of the corre- sponding nucleon masses are M$_{N}$ = 1565, 1489, 1379, 1292, 1216, and 1158 MeV. We add the statistical errors in quadrature. The result for the two lowest pion masses is shown in Fig. 5. We note that while the A$_{20}$ ( t ) function is determined to a better than 10% accuracy, the large errors in Θ q N originate from the less accurate B$_{20}$ ( t ) and Due to the multiplicative character of the QCD evolution for the conserved energy-momentum tensor operator, one has ∑ q Θ q $_{N}$( t ) = ∑ q 〈 x 〉 N q Θ$_{N}$( t ) ≡ 〈 x 〉 u + $_{d}$Θ$_{N}$ ( t ) . (78) Θ q $_{N}$( t ) = M$_{N}$ [ A q $_{20}$( t ) + t 4 M 2 M B q $_{20}$( t ) − 3 t M 2 N C q $_{20}$( t ) ] . (77) 〈 p $^{′}$| Θ q µν | p 〉 = ¯ u ( p $^{′}$) [ A q $_{20}$( t ) γ$_{µ}$P$_{ν}$ + γ$_{ν}$P$_{µ}$ 2 + B q $_{20}$( t ) i ( P$_{µ}$σ$_{νρ}$ + P$_{ν}$σ$_{µρ}$ )Δ ρ 4 M$_{N}$ + C q $_{20}$( t ) Δ$_{µ}$Δ$_{ν}$ − g$_{µν}$ Δ 2 M$_{N}$ ] u ( p ) , (76) Θ$_{N}$ ( q $^{2}$) = ∑ n g$_{nNN}$ f$_{n}$m 2 n m 2 n − q $_{2}$. (75) 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.5 1.0 1.5 2.0 m Σ GLYPH<LBracket1> GeV GLYPH<RBracket1> c FIG. 6: The Δ χ 2 = 2 . 3 and 4 . 6 contours, corresponding to the 68% and 90% confidence levels in the m$_{σ}$ − c plane. The fit is obtained from the nucleon spin-0 gravitational form factor obtained from the LHPC data [73]. See the text for details.