The chirally rotated Schr\"odinger functional with Wilson fermions and automatic O(a) improvement

A modified formulation of the Schr\"odinger functional (SF) is proposed. In the continuum it is related to the standard SF by a non-singlet chiral field rotation and therefore referred to as the chirally rotated SF ($\chi$SF). On the lattice with Wilson fermions the relation is not exact, suggesting some interesting tests of universality. The main advantage of the $\chi$SF consists in its compatibility with the mechanism of automatic O($a$) improvement. In this paper the basic set-up is introduced and discussed. Chirally rotated SF boundary conditions are implemented on the lattice using an orbifold construction. The lattice symmetries imply a list of counterterms, which determine how the action and the basic fermionic two-point functions are renormalised and O($a$) improved. As with the standard SF, a logarithmically divergent boundary counterterm leads to a multiplicative renormalisation of the fermionic boundary fields. In addition, a finite dimension 3 boundary counterterm must be tuned in order to preserve the chirally rotated boundary conditions in the interacting theory. Once this is achieved, O($a$) effects originating from the bulk action or from insertions of composite operators in the bulk can be avoided by the mechanism of automatic O($a$) improvement. The remaining O($a$) effects arise from the boundaries and can be cancelled by tuning a couple of O($a$) boundary counterterms. The general results are illustrated in the free theory where the Sheikholeslami-Wohlert term is shown to affect correlation functions only at O($a^2$), irrespective of its coefficient.