Automatic O(a) improvement for twisted-mass QCD

We present a condition for automatic O(a) improvement in twisted mass lattice QCD, using symmetries of the Symanzik effective theory. If the continuum part of the Symanzik effective theory is invariant under a particular transformation, named T1 in this report, scaling violations of all quantities invariant under T1 transformation are even in the lattice spacing a. On the other hand, quantities non-invariant under T1 vanish in the continuum limit with odd powers in a. We prove this statement even for the massive case without using the equation of motion. We also consider a few different criteria for the T1 invariant condition in lattice theories and discuss ambiguities of the lattice condition for O(a) improvement.