Abstract:
In order to study the running coupling in four-flavour QCD, we review the set-up of the Schr\"odinger functional (SF) with staggered quarks. Staggered quarks require lattices which, in the usual counting, have even spatial lattice extent $L/a$ while the time extent $T/a$ must be odd. Setting $T=L$ is therefore only possible up to ${\rm O}(a)$, which introduces different cutoff effects already in the pure gauge theory. We re-define the SF such as to cope with this situation and determine the corresponding classical background field. A perturbative calculation yields the coefficient of the pure gauge ${\rm O}(a)$ boundary counterterm to one-loop order.

Abstract:
We present renormalization factors for the bilinear operators obtained using the non-perturbative renormalization method (NPR) in the RI-MOM scheme with improved staggered fermions on the MILC asqtad lattices ($N_f = 2+1$). We use the MILC coarse ensembles with $20^3 \times 64$ geometry and $am_{\ell}/am_s = 0.01/0.05$. We obtain the wave function renormalization factor $Z_q$ from the conserved vector current and the mass renormalization factor $Z_m$ from the scalar bilinear operator. We also present preliminary results of renormalization factors for other bilinear operators.

Abstract:
Improved staggered quark actions are designed to suppress flavour changing strong interactions. We discuss the perturbation theory for this type of actions and show the improvements to reduce the quark mass renormalisation compared to naive staggered quarks. The renormalisations are of similar size as for Wilson quarks.

Abstract:
We present preliminary results of matching factors of the four-fermion operators relevant to $B_K$, which are obtained using the non-perturbative renormalization (NPR) method in the RI-MOM scheme with HYP-smeared improved staggered fermions. We use the MILC asqtad coarse ($a \cong 0.12\,$fm) ensembles with $20^3 \times 64$ geometry and $am_{\ell}/am_s = 0.01/0.05$. We compare NPR results with those of one-loop perturbative matching.

Abstract:
We have simulated lattice QCD directly in the chiral limit of zero quark mass by adding an additional, irrelevant 4-fermion interaction to the standard action. Using lattices having temporal extent of six and spatial extents of twelve and eighteen, we find that the theory with 2 massless staggered quark flavors has a second order finite temperature phase transition. The critical exponents $\beta_{mag}$, $\delta$ and $\nu$ are measured and favour tricritical behaviour over that expected by universality arguments. The pion screening mass is consistent with zero below the transition, but is degenerate with the nonzero $\sigma(f_0)$ mass above the transition, indicating the restoration of chiral symmetry.

Abstract:
We present results from a systematic perturbative investigation of taste-changing in improved staggered quarks. We show one-loop taste-changing interactions can be removed perturbatively by an effective four-quark term and calculate the necessary coefficients.

Abstract:
We present O(g^4) calculations of both planar and non-planar Wilson loops for various actions in the presence of sea quarks. In particular, the plaquette, the static potential and the static self energy are calculated to this order for massive Wilson, Sheikholeslami-Wohlert and Kogut-Susskind fermions, including the mass and n_f dependence. The results can be used to obtain alpha_{MS} and m_b(m_b) from lattice simulations. We compare our perturbative calculations to simulation data of the static potential and report excellent qualitative agreement with boosted perturbation theory predictions for distances r<1/GeV. We are also able to resolve differences in the running of the coupling between n_f=2 and n_f=0 static potentials. We compute perturbative estimates of the ``beta-shifts'' of QCD with sea quarks, relative to the quenched theory, which we find to agree within 10 % with non-perturbative simulations. This is done by matching the respective static potentials at large distances. The prospects of determining the QCD running coupling from low energy hadron phenomenology in the near future are assessed. We obtain the result $\Lambda^{(2)}_{\bar{MS}}r_0=0.69(15)$ for the two flavour QCD Lambda-parameter from presently available lattice data where $r_0^{-1}\approx 400$ MeV and estimate $\alpha_{\bar{MS}}^{(5)}(m_Z)=0.1133(59)$.

Abstract:
Recently the four-loop perturbative QCD contributions to the relations between pole and running masses of charm, bottom and top quarks were evaluated in the $\rm{\overline{MS}}$-scheme. In this work the flavour-dependence of the $\mathcal{O}(\alpha_s^4)$ correction to this relation is obtained with the help of the least squares method. The numerical inaccuracies of the two concrete terms in the flavour dependent $\mathcal{O}(\alpha_s^4)$ correction are presented. Our results are in agreement with the recently estimated similar numbers, which however do not contain the exact definition of theoretical uncertainties. It is found that in the case of the $c$-quark mass the asymptotic structure of the corresponding perturbative series is starting to manifest itself from the third perturbative QCD correction. At the fourth order the numerical value of the $\mathcal{O}(\alpha_s^4)$-contribution is significantly larger than all previously known terms, including the leading order one. In the case of $b$-quark mass the $\mathcal{O}(\alpha_s^4)$ correction is comparable with the $\mathcal{O}(\alpha_s^3)$ contribution. The numerical effect of the fourth-order corrections to the pole top-quark mass is estimated. The necessity of decreasing its presented theoretical uncertainty is emphasized.

Abstract:
The improved staggered quark action has enabled breakthrough calculations of QCD with three light dynamical quarks. The largest remaining $a^2$ scaling violations come from "taste-changing" interactions, which are significantly reduced over naive quarks because of carefully chosen smearing of the gauge links. Here we examine further improvements to the quark action and show that additional smearing of the links can reduce the taste-changing errors by another factor of two, whilst retaining a fully $a^2$-improved action. The taste-changing errors can be understood within a perturbative calculation of the four-quark operators that appear at one-loop. We also discuss unquenching techniques with such a highly improved action.

Abstract:
We consider the flavour non-singlet Reggeon within the context of perturbative QCD. This consists of ladders built out of ``reggeized'' quarks. We propose a method for the numerical solution of the integro-differential equation for the amplitude describing the exchange of such a Reggeon. The solution is known to have a sharp rise at low values of Bjorken-x when applied to non-singlet quantities in deep-inelastic scattering. We show that when the running of the coupling is taken into account this sharp rise is further enhanced, although the Q^2 dependence is suppressed by the introduction of the running coupling. We also investigate the effects of simulating non-perturbative physics by introducing a constituent mass for the soft quarks and an effective mass for the soft gluons exchanged in the t-channel.