The response of the QCD vacuum to a constant external (electro)magnetic field is studied through the tensor polarization of the chiral condensate and the magnetic susceptibility at zero and at finite temperature. We determine these quantities using lattice configurations generated with the treelevel Symanzik improved gauge action and = 1 + 1 + 1 flavors of stout smeared staggered quarks with physical masses. We carry out the renormalization of the observables under study and perform the continuum limit both at > 0 and at = 0, using different lattice spacings. Finite size effects are studied by using various spatial lattice volumes. The magnetic susceptibilities reveal a spin-diamagnetic behavior; we obtain at zero temperature = −(2.08 ± 0.08) GeV −2 , = −(2.02 ± 0.09) GeV −2 and = −(3.4 ± 1.4) GeV −2 for the up, down and strange quarks, respectively, in the MS scheme at a renormalization scale of 2 GeV. We also find the polarization to change smoothly with the temperature in the confinement phase and then to drastically reduce around the transition region.* Corresponding author. gergely.endrodi@physik.uni-r.de 1 Employing physical quark masses in the simulation and extrapolating the results to the continuum limit, as was done in Refs. [5][6][7], proved to be essential. Studies where these ingredients are missing produce qualitatively different results, namely an increasing ( ) function [8,9]. A possible explanation for this discrepancy and a comparison to effective theories was given recently in Ref. [7].

We present preliminary results of the first lattice QCD calculation of the K → π matrix elements of the chromomagnetic operator O CM = gs σ µν G µν d, which appears in the effective Hamiltonian describing ∆S = 1 transitions in and beyond the Standard Model. Having dimension 5, the chromomagnetic operator is characterized by a rich pattern of mixing with operators of equal and lower dimensionality. The multiplicative renormalization factor as well as the mixing coefficients with the operators of equal dimension have been computed at one-loop in perturbation theory. The power divergent coefficients controlling the mixing with operators of lower dimension have been computed non-perturbatively, by imposing suitable subtraction conditions. The numerical simulations have been carried out using the gauge field configurations produced by the European Twisted Mass Collaboration with N f = 2 + 1 + 1 dynamical quarks at three values of the lattice spacing. Our preliminary result for the B-parameter of the chromomagnetic operator is B CMO = 0.29(11), which can be compared with the estimate B CMO ∼ 1 − 4 currently used in phenomenological analyses. K → π matrix elements of the chromagnetic operator on the lattice V. Lubicz

We present the results of the first lattice QCD calculation of the K → π matrix elements of the chromomagnetic operator O CM ¼ gsσ μν G μν d, which appears in the effective Hamiltonian describing ΔS ¼ 1 transitions in and beyond the standard model. Having dimension five, the chromomagnetic operator is characterized by a rich pattern of mixing with operators of equal and lower dimensionality. The multiplicative renormalization factor as well as the mixing coefficients with the operators of equal dimension have been computed at one loop in perturbation theory. The power divergent coefficients controlling the mixing with operators of lower dimension have been determined nonperturbatively, by imposing suitable subtraction conditions. The numerical simulations have been carried out using the gauge field configurations produced by the European Twisted Mass Collaboration with N f ¼ 2 þ 1 þ 1 dynamical quarks at three values of the lattice spacing. Our result for the B parameter of the chromomagnetic operator at the physical pion and kaon point is B Kπ CMO ¼ 0.273ð69Þ, while in the SU(3) chiral limit we obtain B CMO ¼ 0.076ð23Þ. Our findings are significantly smaller than the model-dependent estimate B CMO ∼ 1-4, currently used in phenomenological analyses, and improve the uncertainty on this important phenomenological quantity.

We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric N =1 QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use naïve discretization. The gauge group that we consider is SU (Nc), while the number of colors, Nc, the number of flavors, N f , and the gauge parameter, α, are left unspecified.We obtain analytic expressions for the renormalization factors of the coupling constant (Zg) and of the quark (Z ψ ), gluon (Zu), gluino (Z λ ), squark (ZA ± ), and ghost (Zc) fields on the lattice. We also compute the critical values of the gluino, quark and squark masses. Finally, we address the mixing which occurs among squark degrees of freedom beyond tree level: we calculate the corresponding mixing matrix which is necessary in order to disentangle the components of the squark field via an additional finite renormalization.

The determination of renormalization factors is of crucial importance in lattice QCD. They relate the observables obtained on the lattice to their measured counterparts in the continuum in a suitable renormalization scheme. Therefore, they have to be computed as precisely as possible. A widely used approach is the nonperturbative Rome-Southampton method. It requires, however, a careful treatment of lattice artifacts. In this paper we investigate a method to suppress these artifacts by subtracting one-loop contributions to renormalization factors calculated in lattice perturbation theory. We compare results obtained from a complete one-loop subtraction with those calculated for a subtraction of contributions proportional to the square of the lattice spacing.