Forces within hadrons on the light front

In this work, we find the light front densities for momentum and forces, including pressure and shear forces, within hadrons. This is achieved by deriving relativistically correct expressions relating these densities to the gravitational form factors $A(t)$ and $D(t)$ associated with the energy momentum tensor. The derivation begins from the fundamental definition of density in a quantum field theory, namely the expectation value of a local operator within a spatially localized state. We find that it is necessary to use the light front formalism to define a density that corresponds to the internal hadron structure. When using the instant form formalism, it is impossible to remove the spatial extent of the hadron wave function from any density, and—even within instant form dynamics—one does not obtain a Breit frame Fourier transform for a properly defined density. Within the front formalism, we derive new expressions for various mechanical properties of hadrons, including the mechanical radius, as well as for stability conditions. The multipole ansatz for the form factors is used as an example to illustrate all of these findings.

Figure 1

Light front transverse densities associated with the EMT, for several multipole orders. Each density is weighted by $2\pi x_{\perp }$. The $P^{+}$ density is divided by $P^{+}$, while the pressure and shear are multiplied by $P^{+}$, to remove the $P^{+}$ dependence.

Figure 2

Light front transverse densities associated with the pure stress tensor using the dipole ($n=2$) ansatz. Each density is weighted by $2\pi x_{\perp }$ and multiplied by $P^{+}$ to remove the $P^{+}$ dependence.