We determine the non-perturbatively renormalized axial current for O($a$) improved lattice QCD with Wilson quarks. Our strategy is based on the chirally rotated Schrodinger functional and can be generalized to other finite (ratios of) renormalization
constants which are traditionally obtained by imposing continuum chiral Ward identities as normalization conditions. Compared to the latter we achieve an error reduction up to one order of magnitude. Our results have already enabled the setting of the scale for the $N_{rm f}=2+1$ CLS ensembles [1] and are thus an essential ingredient for the recent $alpha_s$ determination by the ALPHA collaboration [2]. In this paper we shortly review the strategy and present our results for both $N_{rm f}=2$ and $N_{rm f}=3$ lattice QCD, where we match the $beta$-values of the CLS gauge configurations. In addition to the axial current renormalization, we also present precise results for the renormalized local vector current.
The gradient flow provides a new class of renormalized observables which can be measured with high precision in lattice simulations. In principle this allows for many interesting applications to renormalization and improvement problems. In practice,
however, such applications are made difficult by the rather large cutoff effects found in many gradient flow observables. At lowest order of perturbation theory we here study the leading cutoff effects in a finite volume gradient flow coupling with SF and SF-open boundary conditions. We confirm that O($a^2$) Symanzik improvement is achieved at tree-level, provided the action, observable and the flow are O($a^2$) improved. O($a^2$) effects from the time boundaries are found to be absent at this order, both with SF and SF-open boundary conditions. For the calculation we have used a convenient representation of the free gauge field propagator at finite flow times which follows from a recently proposed set-up by Luscher and renders lattice perturbation theory more practical at finite flow time and with SF, open, SF-open or open-SF boundary conditions.
Precision tests of QCD perturbation theory are not readily available from experimental data. The main reasons are systematic uncertainties due to the confinement of quarks and gluons, as well as kinematical constraints which limit the accessible ener
gy scales. We here show how continuum extrapolated lattice data may overcome such problems and provide excellent probes of renormalized perturbation theory. This work corresponds to an essential step in the ALPHA collaborations project to determine the $Lambda$-parameter in 3-flavour QCD. I explain the basic techniques used in the high energy regime, namely the use of mass-independent renormalization schemes for the QCD coupling constant in a finite Euclidean space time volume. When combined with finite size techniques this allows one to iteratively step up the energy scale by factors of 2, thereby quickly covering two orders of magnitude in scale. We may then compare perturbation theory (with $beta$-functions available up to 3-loop order) to our non-perturbative data for a 1-parameter family of running couplings. We conclude that a target precision of 3 percent for the $Lambda$-parameter requires non-perturbative data up to scales where $alpha_sapprox 0.1$, whereas the apparent precision obtained from applying perturbation theory around $alpha_s approx 0.2$ can be misleading. This should be taken as a general warning to practitioners of QCD perturbation theory.
The chirally rotated Schrodinger functional ($chi$SF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schrodinger functional (SF), with different lattice symmetries and a common continuum limit expected from u
niversality. 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.
Symanzik showed that quantum field theory can be formulated on a space with boundaries by including suitable surface interactions in the action to implement boundary conditions. We show that to all orders in perturbation theory all the divergences in
duced by these surface interactions can be absorbed by a renormalization of their coefficients.
We introduce a new algorithm which we call the {Rational Hybrid Monte Carlo} Algorithm (RHMC). This method uses a rational approximation to the fermionic kernel together with a noisy Kennedy-Kuti acceptance step to give an efficient algorithm with no molecular dynamics integration step-size errors.
