The gradient flow coupling from numerical stochastic perturbation theory

Perturbative calculations of gradient flow observables are technically challenging. Current results are limited to a few quantities and, in general, to low perturbative orders. Numerical stochastic perturbation theory is a potentially powerful tool that may be applied in this context. Precise results using these techniques, however, require control over both statistical and systematic uncertainties. In this contribution, we discuss some recent algorithmic developments that lead to a substantial reduction of the cost of the computations. The matching of the ${\overline{\rm MS}}$ coupling with the gradient flow coupling in a finite box with Schr\"odinger functional boundary conditions is considered for illustration.Comment: Talk given at the 34th annual International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK; LaTeX source, 7 pages, 2 figure

SMD-based numerical stochastic perturbation theory

The viability of a variant of numerical stochastic perturbation theory, where the Langevin equation is replaced by the SMD algorithm, is examined. In particular, the convergence of the process to a unique stationary state is rigorously established and the use of higher-order symplectic integration schemes is shown to be highly profitable in this context. For illustration, the gradient-flow coupling in finite volume with Schr\"odinger functional boundary conditions is computed to two-loop (i.e. NNL) order in the SU(3) gauge theory. The scaling behaviour of the algorithm turns out to be rather favourable in this case, which allows the computations to be driven close to the continuum limit.Comment: 35 pages, 4 figures; v2: corrected typos, coincides with published versio

Study of nanogranite inclusions in garnets from the Ulten zone (Austroalpine domain, Eastern Alps)

The study of melt inclusions (MI) in high-grade metamorphic rocks represents a novel tool to characterize anatectic melt and partial melting P-T conditions. In this work, high-grade migmatitic gneisses from the Ulten-Nonsberg Zone (UZ) have been investigated, in order to characterize MI hosted in peritectic garnet. UZ rocks belong to the Austroalpine domain of the Eastern Alps and represent a pre-Alpine basement unit that underwent a polymetamorphic evolution marked by high-pressure metamorphism and migmatization processes during the Variscan orogeny, related to continental collision between Laurussia and Gondwana in the late Carboniferous (about 330 Ma). Previous geothermobarometric studies suggest P-T conditions of the migmatization event of P<1.0 GPa and T<900°C and P-T conditions for the peak metamorphism of 1-2 GPa and 600-900°C (with the wide range depending upon the assumption of the H2O activity of the fluid phase during metamorphic evolution; Godard et al., 1996). Other studies by Braga et al. (2007) and Braga and Massonne (2008), based on thermobarometry on garnet-kyanite gneisses, suggest a P climax (w1.2 GPa) during progressive heating and a thermal peak between 700-750°C at about 1 GPa, during the first stages of the exhumation process. For this study, after the sampling activity, 67 thin sections have been analyzed. Optical microscope and SEM-EDS observations allowed to identify the best samples with MI-bearing garnets. MI appear as a polycrystalline aggregate of Qtz +- Bt +- Kfs +- Pl +- Chl, with a maimum size of about 20 μm, and are therefore dened nanogranites or nanogranitoids. The presence of MI in the core of peritectic garnets allowed to conrm their primary origin and thus to consider the EMP composition of garnet core in order to perform the phase equilibria modeling. Phase equilibria modeling was performed with the Perple X software (Connolly, 2009), in order to estimate probable P-T conditions of entrapment of MI in peritectic garnets. The mineral assemblage of sample N3L (the one selected for phase equilibria modeling) is Qtz + Kfs + Wm + Pl + Grt + Ky, and Rt as accessory mineral. Bulk rock composition of sample NB5, very similar to N3L, was used and the chemical system Na2O-CaO-MnO-K2O-Fe2O3-MgO-Al2O3-SiO2-H2O-TiO2 (NCMnKFMASHT) was selected. A phase diagram section was obtained for P=0.5-2.0 GPa and T=600-950°C and garnet isopleths (for almandine, grossular, pyrope and spessartine components) have been calculated within the same P-T space, in order to further constrain the stability field. Garnet core isopleths of sample N3L did not match the mineralogical association of the rock in the pseudosection; therefore, the core composition of garnet from a rock very similar to NB5 was used (sample NONS17, from Braga et al., 2007). These data allowed to constrain the stability field of rock NONS17 and rock NB5, corresponding to a soprasolidus field with P-T conditions ranging from 1-1.4 GPa and 800-845°C, characterized by the mineralogical association Grt + Bt + Wm + Pl + Kfs + Qtz + Rt + melt. Therefore, the investigated MI clearly represent droplets of the anatectic melt produced during the prograde phase of the host migmatites.ope

Precision Determination of αs\alpha_s from Lattice QCD

We present an overview of the recent lattice determination of the QCD coupling $\alpha_s$ by the ALPHA Collaboration. The computation is based on the non-perturbative determination of the $\Lambda$-parameter of $N_{\rm f}=3$ QCD, and the perturbative matching of the $N_{\rm f}=3$ and $N_{\rm f}=5$ theories. The final result: $\alpha_s(m_Z)=0.11852(84)$, reaches sub-percent accuracy.Comment: 14 pages, 4 figures. Contribution from the "Selected Papers from the 7th International Conference on New Frontiers in Physics (ICNFP 2018)

Investigation of new methods for numerical stochastic perturbation theory in φ4 theory

Numerical stochastic perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice field theory. The standard algorithms based on the Langevin equation, however, suffer from several limitations which in practice restrict the potential of this technique. In this work we investigate some alternative methods which could in principle improve on the standard approach. In particular, we present a study of the recently proposed Instantaneous Stochastic Perturbation Theory, as well as a formulation of numerical stochastic perturbation theory based on Generalized Hybrid Molecular Dynamics algorithms. The viability of these methods is investigated in $\varphi^4$ theory.Comment: 45 pages, 12 figures. Added new section on cost comparison with Langevin NSPT. Matches published versio