We present an unquenched N_f=2 lattice computation of the B _K parameter which controls K0-K̄0 oscillations. A partially quenched setup is employed with two maximally twisted dynamical (sea) light Wilson quarks, and valence quarks of both the maximally twisted and the Osterwalder-Seiler variety. Suitable combinations of these two kinds of valence quarks lead to a lattice definition of the B_K parameter which is both multiplicatively renormalizable and O(a) improved. Employing the nonperturbative RI-MOM scheme, in the continuum limit and at the physical value of the pion mass we get BKRGI=0.729±0.030, a number well in line with the existing quenched and unquenched determinations.

In this work we calculate the corrections to the amputated Green's functions of four-fermion operators, in 1-loop lattice perturbation theory. One of the novel aspects of our calculations is that they are carried out to second order in the lattice spacing, O(a2). We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and a family of Symanzik improved actions for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients (Plaquette, Tree-level Symanzik, Iwasaki, TILW and DBW2 action). While our Green's function calculations regard any pointlike four-fermion operators which do not mix with lower dimension ones, we pay particular attention to ΔF=2 operators, both parity conserving and parity violating (F stands for flavor: S, C, B). By appropriately projecting those bare Green's functions we compute the perturbative renormalization constants for a complete basis of four-fermion operators and we study their mixing pattern. For some of the actions considered here, even O(a0) results did not exist in the literature to date. The correction terms which we calculate (along with our previous O(a2) calculation of Z _Ψ) are essential ingredients for minimizing the lattice artifacts which are present in nonperturbative evaluations of renormalization constants with the RI^′-MOM method. Our perturbative results, for the matrix elements of ΔF=2 operators and for the corresponding renormalization matrices, depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible to the reader, we have included them in the distribution package of this paper, as an ASCII file named: 4-fermi.m; the file is best perused as Mathematica input. The main results of this work have been applied to improve nonperturbative estimates of the B_K-parameter in N _F=2 twisted mass lattice QCD

Perturbative and nonperturbative results on the renormalization constants of the fermion field and the twist-2 fermion bilinears are presented with emphasis on the nonperturbative evaluation of the one-derivative twist-2 vector and axial-vector operators. Nonperturbative results are obtained using the twisted mass Wilson fermion formulation employing two degenerate dynamical quarks and the tree-level Symanzik improved gluon action. The simulations have been performed for pion masses in the range of about 450-260 MeV and at three values of the lattice spacing a corresponding to β=3.9, 4.05, 4.20. Subtraction of O(a2) terms is carried out by performing the perturbative evaluation of these operators at 1-loop and up to O(a2). The renormalization conditions are defined in the RI⊃′-MOM scheme, for both perturbative and nonperturbative results. The renormalization factors, obtained for different values of the renormalization scale, are evolved perturbatively to a reference scale set by the inverse of the lattice spacing. In addition, they are translated to MS̄ at 2 GeV using 3-loop perturbative results for the conversion factors

We present results for the renormalization constants of bilinear quark operators obtained by using the tree-level Symanzik improved gauge action and the N _f = 2 twisted mass fermion action at maximal twist, which guarantees automatic O(a)-improvement. Our results are also relevant for the corresponding standard (un-twisted) Wilson fermionic action since the two actions only differ, in the massless limit, by a chiral rotation of the quark fields. The scale-independent renormalization constants Z _V, Z _A and the ratio Z _P /Z _S have been computed using the RI-MOMapproach, as well as other alternative methods. For Z _A and Z _P /Z _S, the latter are based on both standard twisted mass and Osterwalder-Seiler fermions, while for Z _V a Ward Identity has been used. The quark field renormalization constant Z _q and the scale dependent renormalization constants Z _S, Z _P and Z _T are determined in the RI-MOM scheme. Leading discretization effects of O(g ²a ²), evaluated in one-loop perturbation theory, are explicitly subtracted from the RI-MOM estimates

We calculate corrections to the fermion propagator and to the Green's functions of all fermion bilinear operators of the form , to one-loop in perturbation theory. We employ the Wilson/clover action for fermions and the Symanzik improved action for gluons. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, (a ²). Consequently, they have addressed a number of new issues, most notably the appearance of loop integrands with strong IR divergences (convergent only beyond 6 dimensions). Such integrands are not present in (a ¹) improvement calculations; there, IR divergent terms are seen to have the same structure as in the (a ⁰) case, by virtue of parity under integration, and they can thus be handled by well-known techniques. We explain how to correctly extract the full (a ²) dependence; in fact, our method is generalizable to any order in a. The (a ²) corrections to the quark propagator and Green's functions computed in this paper are useful to improve the nonperturbative RI-MOM determination of renormalization constants for quark bilinear operators. Our results depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible to the reader, we have included them in the distribution package of this paper, as an ASCII file named: Oa2results.m; the file is best perused as Mathematica input