arXiv : A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD

Dalla Brida, Mattia; Sommer, Rainer; Fritzsch, Patrick; Ramos, Alberto; Korzec, Tomasz; Sint, Stefan

doi:10.1140/epjc/s10052-018-5838-5

Article
Report number	arXiv:1803.10230 ; CERN-TH-2018-060 ; DESY 18-044 ; WUB/18-01 ; DESY-18-044 ; WUB-18-01
Title	A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD
Author(s)	Dalla Brida, Mattia (INFN, Milan Bicocca ; Milan Bicocca U.) ; Fritzsch, Patrick (CERN) ; Korzec, Tomasz (Wuppertal U.) ; Ramos, Alberto (Hamilton Math. Inst., Dublin) ; Sint, Stefan (Hamilton Math. Inst., Dublin) ; Sommer, Rainer (NIC, Zeuthen ; Humboldt U., Berlin)
Collaboration	ALPHA Collaboration
Publication	2018-05-10
Imprint	2018-03-27
Number of pages	40
Note	* Temporary entry * 40 pages, 11 figures
In:	Eur. Phys. J. C 78 (2018) 372
DOI	10.1140/epjc/s10052-018-5838-5
Subject category	hep-ph ; Particle Physics - Phenomenology ; hep-lat ; Particle Physics - Lattice
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 Schr\"odinger 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 $1/L_0\approx 4$ GeV through $\bar{g}^2(L_0) =2.012$, we quote $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}} =0.0791(21)$. 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 [1]. The results indicate that for our target precision of 3 per cent in $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}}$, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of $\alpha_s=\bar{g}^2/(4\pi)$. In the present work we reach this precision by studying scales that vary by a factor $2^5= 32$, reaching down to $\alpha_s\approx 0.1$. We here provide the details of our analysis and an extended discussion.
Copyright/License	publication: (License: CC-BY-4.0) preprint: (License: arXiv nonexclusive-distrib 1.0)

$The step scaling function $\sigma(u)$, a discrete version of the $\beta$-function, defined in \eq{eq:sigdef}. The combination shown here yields directly the lowest order cofficient, $b_0$ of the $\beta$-function as $(\sigma(u)-u)/u^2 = 2 b_0\ln\,2 +\rmO(u)$. The dashed lines show the perturbative 2-loop behavior. The purple 1-sigma band shows our result (fit~C in table~\ref{tab:Sigfits}). Data points for $\nf=0,2,3,4$ are taken from the literature \cite{Capitani:1998mq,DellaMorte:2004bc,Aoki:2009tf,Tekin:2010mm}.$ $The step scaling function $\sigma(u)$, a discrete version of the $\beta$-function, defined in \eq{eq:sigdef}. The combination shown here yields directly the lowest order cofficient, $b_0$ of the $\beta$-function as $(\sigma(u)-u)/u^2 = 2 b_0\ln\,2 +\rmO(u)$. The dashed lines show the perturbative 2-loop behavior. The purple 1-sigma band shows our result (fit~C in table~\ref{tab:Sigfits}). Data points for $\nf=0,2,3,4$ are taken from the literature \cite{Capitani:1998mq,DellaMorte:2004bc,Aoki:2009tf,Tekin:2010mm}.$ $Coefficients $c^\nu_1(s)$ and $c^\nu_2(s)$ for two different SF$_\nu$ schemes. Left $\nu=0$, right $\nu=-0.5$. The values $s^\star$ defined by the condition $c_1(s^\star)=0$ are approximately $s^\star\approx 3$ (left) and $s^\star\approx 5$ (right).$ Show more plots

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