in the valence and sea quark sectors. As was discussed in detail in [@sachrajda2005; @bedaque2005] using chiral perturbation theory, the associated effects are expected to appear as finite-size effects exponentially small in spatial volume for channels which do not have final-state interactions such as three-point functions for form factor calculations. Since terms of such magnitude are also present in unitary theory with periodic valence and sea quarks, we ignore this issue in the present work.

The twisted boundary condition can be imposed on a periodic quark field configuration by the following transformation $$\label{eq:fund47}
\psi(x) \longrightarrow U(\theta,x) \psi(x) = e^{2\pi i \sum^{3}_{j=1}{\theta_j x_j/L}} \psi(x).$$ In practice we transfer the twist from the quark sector to the gluon sector by an ${\rm U(1)}$ transformation on the spatial gluon link fields given by $$\label{eq:fund48}
U_i(x) \rightarrow U^\theta_i(x)=e^{2\pi i \theta_i/L}U_i(x), \qquad i=1,2,3.$$ Thus valence quark propagators are solved with the periodic boundary condition but on the PACS-CS gauge configurations twisted by the U(1) transformation above.

In order to check that the term $\frac{2\pi \theta_j}{L}$ acts as true physical momentum, we carried out a test of the energy-momentum dispersion relation of the pion on some PACS-CS configurations. Of the two valence quarks inside the pion, 