O(a²) improvement of the overlap-Dirac operator

We construct an $O(a^2)$-improved overlap-Dirac operator by designing an improved overlap kernel, based on the Symanzik improvement program. Field rotation terms are also identified to improve off-shell amplitudes for both massless and massive fermions. We check the free dispersion relation and propagator, and show that improved results become to close to the continuum ones at low momentum region. We test the effect of improvement on the full-QCD gauge configuration and find that the relativistic dispersion relation is satisfied within a few percent error up to $m_qa \approx 0.5$.