Abstract
Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schrödinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale 1/L_0approx 4 GeV through bar{g}^2(L_0) =2.012, we quote L_0 Lambda ^{N_mathrm{f}=3}_{{overline{mathrm{MS}}}} =0.0791(21). This error is dominated by statistics; in particular, the remnant perturbative uncertainty is negligible and very well controlled, by connecting to infinite renormalization scale from different scales 2^n/L_0 for n=0,1,ldots ,5. An intermediate step in this connection may involve any member of a one-parameter family of SF couplings. This provides an excellent opportunity for tests of perturbation theory some of which have been published in a letter (ALPHA collaboration, M. Dalla Brida et al. in Phys Rev Lett 117(18):182001, 2016). The results indicate that for our target precision of 3 per cent in L_0 Lambda ^{N_mathrm{f}=3}_{{overline{mathrm{MS}}}}, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of alpha _s=bar{g}^2/(4pi ). In the present work we reach this precision by studying scales that vary by a factor 2^5= 32, reaching down to alpha _sapprox 0.1. We here provide the details of our analysis and an extended discussion.
Highlights
The Standard Model seems to describe all high energy physics experiments carried out to date, in some cases with extraordinary accuracy
Using numerical simulations and finite volume step-scaling techniques, we have studied a family of Schrödinger functional (SF) couplings, parameterized by ν, over a range of scales corresponding to energies of 4–128 GeV, differing by a scale factor 32
This, together with an unprecedented control of statistical and systematic errors represents a luxury which we have exploited to test the accuracy of perturbation theory
Summary
The Standard Model seems to describe all high energy physics experiments carried out to date, in some cases with extraordinary accuracy (cf. [2] for the most recent PDG review). As part of the project to determine αs(m Z ) from low energy hadronic input in 3-flavour QCD [1,12,13], our collaboration has applied these techniques to a 1-parameter family of finite volume couplings in Schrödinger functional (SF) schemes, for which the 3-loop β-function is known [14,15,16,17] We have measured these couplings in numerical simulations and for a range of lattice sizes with unprecedented precision. The QCD 3-loop β-function is currently known in the case of infinite space-time volume [30], and there is progress for the case of a finite volume with SF boundary conditions [29] using numerical stochastic perturbation theory [31,32,33] These results seem to point to a 3-loop β-function coefficient which is significantly larger than in the MS- and SF-schemes. Further studies are required and one should re-assess the situation once more perturbative information becomes available
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