The chirally rotated Schr\"odinger functional: theoretical expectations and perturbative tests

The chirally rotated Schr\"odinger functional ($\chi$SF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schr\"odinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O($a$) improvement to be operational in the $\chi$SF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the $\chi$SF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O($a$) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the $\chi$SF framework and prepares the ground for non-perturbative applications.