The chirally rotated Schrödinger functional with Wilson fermions and automatic improvement

Abstract

A modified formulation of the Schrödinger 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 (χSF). On the lattice with Wilson fermions the relation is not exact, suggesting some interesting tests of universality. The main advantage of the χSF 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.

The chirally rotated Schrödinger functional: theoretical expectations and perturbative tests

“…The chirally rotated Schrödinger functional (χSF) [1,2] provides a new tool to address renormalization and O(a) improvement problems in lattice QCD and similar lattice gauge theories with Wilson type fermions. With an even number of massless fermion flavours it is formally related to the standard Schrödinger functional (SF) [3][4][5] by a non-singlet chiral field rotation.…”

Perturbative O$(a^2)$ effects in gradient flow couplings with SF and SF-open boundary conditions

“…To completely define the propagator for spatial indices we impose either SF boundary conditions, 9) or SF-open boundary conditions, 10) and analogously for the B-field at positive flow times. 1 We define e.g.…”

The gradient flow running coupling with twisted boundary conditions

“…Fermions induce additional boundary counterterms that one needs to compute to have O(a 2 ) scaling. Moreover the boundary conditions for the fermion fields typically break chiral symmetry, and therefore one also needs bulk improvement, although this last issue can be addressed by modifying the boundary conditions of the fermion fields [12]. On the other hand, the TPL scheme can not be used with an arbitrary number of fermions in the fundamental representation.…”

“…But when this scheme can be used, it guarantees a better scaling towards the continuum. O(a) improvement is automatic, even with Wilson fermions, provided that one works with massless quarks [12,13]. Nevertheless the observable used to define the coupling (a ratio of polyakov loops), tends to be more noisy than the SF coupling, especially in the non-perturbative domain.…”

“…Regardless of the representation but assuming that the matter fields are compatible with the twisted boundary conditions, O(a) improvement for massless Wilson quarks is automatically satisfied since the fields live on a torus, and the boundary conditions do not break chiral symmetry (see [12,13]). …”