quantity glueball q ¯ q meson m$_{n}$ 1 1 f$_{n}$ N$_{c}$ √$_{N}$$_{c}$ Γ$_{nππ}$ 1 /N 2 c 1 /N$_{c}$ g$_{nππ}$ 1 /N$_{c}$ 1 $_{/}$√$_{N}$$_{c}$ g$_{nNN}$ 1 √$_{N}$$_{c}$ 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.49 0.50 0.51 0.52 0.53 0.54 0.55 m Σ GLYPH<LParen1> GeV GLYPH<RParen1> GLYPH<Less> x GLYPH<Greater> u GLYPH<Plus> d 8 TABLE II: N$_{c}$ -scaling of various quantities. FIG. 3: The Δ χ 2 = 2 . 3 and 4 . 6 contours, corresponding to the 68% and 90% confidence levels, in the m$_{σ}$ − 〈 x 〉 u + d plane. The fit is obtained from the lattice data for the pion spin-0 gravitational form factor, as described in the text. A. Pion The gravitational form factors of the pion are defined via the matrix element of the energy-momentum tensor between the pion states, where P = $_{2}$( p + p ), q = p − p , and a, b are the isospin indices. The trace part of this form factor corresponds to the coupling of the spin-0, while the traceless part to spin-2 states. In the present analysis we are interested in the former. The spin-0 gravitational form factor is given by exactly as Eq. (37). Since according to Table II we have the scaling f$_{n}$g$_{nππ}$ ∼ N 0 $_{c}$, both mesons and glueballs may contribute to relation (57) on equal footing. At large momenta the form factor behaves as where Due to chiral symmetry constraints, this form factor sat- isfies [51, 55, 56], If we require the constant term to vanish, we get Clearly, cancellations are necessary and we need at least two contributing states to satisfy the sum rule (59). This high-energy relation plays a similar role in the scale- invariance considerations to the second Weinberg sum rule in the chiral framework. The only determination of the pion gravitational form factor to date is the lattice calculation of Brommel et al. [63, 64]. The data go up to t = − 4 GeV $^{2}$, however with substantial errors. The quark contribution to the energy momentum tensor, considered in [63, 64], is Two low energy theorems follow: These conditions can be achieved by assuming a deriva- tive coupling written as and thus from which a decomposition similar to Eq. (49) and the corresponding spin-0 component Eq. (51) follow. The data points are shown in Fig. 4. Following the standard notation for the moments of the pion GPD’s [65] we in- troduce A$_{20}$ ( t ) = 1 2 Θ q $_{1}$( t ) , A$_{22}$ ( t ) = − 1 2 Θ q $_{2}$( t ) , (61) Θ µν q ( x ) = 1 4 ¯ q ( x ) [ γ $^{µ}$i ↼ ⇀ D ν + γ ν i ↼ ⇀ D µ ] q ( x ) , (60) ∑ n f$_{n}$g$_{nππ}$ = 1 , (57) Θ$_{π}$( q $^{2}$) = ∑ n g$_{nππ}$f$_{n}$q $^{2}$m2 n m 2 n − q 2 , (56) 〈 π $^{a}$( p $^{′}$) | Φ$_{n}$ | π $^{b}$( p ) 〉 = δ ab g$_{nππ}$q 2 m 2 n − q $_{2}$, (55) Θ $_{π}$(0) = 1 . (54) Θ$_{π}$(0) = 0 , (53) ′ Θ$_{π}$( q $^{2}$) = q 2 + O ( p $^{4}$) . (52) ∑ n f$_{n}$g$_{nππ}$m 2 n = 0 . (59) Θ$_{π}$( q $^{2}$) → − ∑ n g$_{nππ}$m 2 $_{n}$f$_{n}$ − ∑ $_{n}$g$_{nππ}$ m 4 $_{n}$f$_{n}$ q 2 + . . . (58) Θ$_{π}$( q $^{2}$) = 3 2 q $^{2}$Θ$_{1}$ ( q $^{2}$) + 1 2 (4 m 2 π − q $^{2}$)Θ$_{2}$ ( q $^{2}$) . (51) 〈 π $^{b}$( p $^{′}$) | Θ(0) | π $^{a}$( p ) 〉 = δ $^{ab}$Θ$_{π}$ ( q $^{2}$) , (50) 〈 π $^{b}$( p $^{′}$) | Θ $^{µν}$(0) | π $^{a}$( p ) 〉 = 1 2 δ $^{ab}$[ ( g $^{µν}$q 2 − q $^{µ}$q$^{ν}$)Θ$_{1}$ ( q $^{2}$) + 4 P $^{µ}$P $^{ν}$Θ$_{2}$ ( q $^{2}$) ] , (49)