content_image/394798.jpg content_image/394796.jpg content_image/394766.jpg content_image/394797.jpg content_image/394793.jpg content_image/394792.jpg content_image/394791.jpg content_image/394790.jpg content_image/394789.jpg content_image/394788.jpg content_image/394767.jpg content_image/394795.jpg content_image/394787.jpg content_image/394786.jpg content_image/394794.jpg
The gravitational form factors of the pion are defined via the matrix element of the energy-momentum tensor between the pion states, 

<img src='content_image/394798.jpg'>

where − 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 

where 

<img src='content_image/394796.jpg'>

Due to chiral symmetry constraints, this form factor sat- isfies [51, 55, 56], 

Two low energy theorems follow: 

These conditions can be achieved by assuming a deriva- tive coupling 

written as 

and thus 

<img src='content_image/394766.jpg'>

<img src='content_image/394797.jpg'>

<img src='content_image/394793.jpg'>

<img src='content_image/394792.jpg'>

<img src='content_image/394791.jpg'>

<img src='content_image/394790.jpg'>

<img src='content_image/394789.jpg'>

<img src='content_image/394788.jpg'>

## A. Pion 

<img src='content_image/394767.jpg'>

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 

<img src='content_image/394795.jpg'>

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 

<img src='content_image/394787.jpg'>

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 

<img src='content_image/394786.jpg'>

<img src='content_image/394794.jpg'>

