Dispersion relations
One method to compute the phaseshifts is to first compute the energy in a frame with non-zero total momentum and then compute the COM energy using the relativistic dispersion relation
\( W_{CM}^2 = W_{LAB}^2-\textbf{P}^2 \)
where \(\textbf{P}\) is the total momentum. The discretization of the lattice breaks Lorentz symmetry and as a result the relativistic dispersion relation is violated at finite lattice spacing. There a lattice dispersion relation expected describe the dispersion more accurately. The lattice dispersion relation for a single particle is
\( cosh (E(p)) = 1-cos(p) + cosh(m) \)
where \(m\) and \(p\) are the mass and momentum respectively. There for the case of two degenerate particles, the lattice dispersion is
\( cosh (E(p)/2) = 1-cos(p) + cosh(m) \)
where \(m\) and \(p\) are the mass and relative-momentum respectively. To test these relations, we compute the pion mass using twisted boundary conditions.
