A Determination of the Lambda Parameter from Full Lattice QCD

TL;DR

The paper delivers a lattice QCD determination of the MSbar Lambda parameter for $n_f=0$ and $n_f=2$, employing $O(a)$-improved Wilson fermions, nonperturbative scale setting via $r_0$, and a boosted coupling $g_{\Box}$ to connect to $\alpha_s^{\overline{MS}}$. By exploring multiple lattice-to-MSbar conversion strategies and applying Padé-improved beta functions, the authors obtain $\Lambda^{\overline{MS}}_0 = 259(1)(20)$ MeV and $\Lambda^{\overline{MS}}_2 = 261(17)(26)$ MeV, with an extrapolated $\Lambda^{\overline{MS}}_3 = 260(12)(26)$ MeV. Matching across flavours and higher-loop analysis yields $\alpha_s^{\overline{MS}}(m_Z) = 0.112(1)(2)$ when evolved to five flavours. The results, while consistent with previous lattice determinations and broadly agreeing with phenomenology, tend to sit slightly below the experimental central value, highlighting ongoing systematic uncertainties primarily from scale setting and continuum extrapolation. Overall, the study demonstrates reliable continuum and chiral limit control and provides a careful pathway to precise $\alpha_s$ determinations from lattice QCD with dynamical quarks.

Abstract

We present a determination of the QCD parameter Lambda in the quenched approximation (n_f=0) and for two flavours (n_f=2) of light dynamical quarks. The calculations are performed on the lattice using O(a) improved Wilson fermions and include taking the continuum limit. We find Lambda_{n_f=0} = 259(1)(20) MeV and Lambda_{n_f=2} = 261(17)(26) MeV}, using r_0 = 0.467 fm to set the scale. Extrapolating our results to five flavours, we obtain for the running coupling constant at the mass of the Z boson alpha_s(m_Z) = 0.112(1)(2). All numbers refer to the MSbar scheme.

Figures (9)

• Figure 1: $\alpha_s^{\hbox{\tiny $\overline{MS}$}}(\mu)$ versus $\mu/\Lambda^{\hbox{\tiny $\overline{MS}$}}$ for $n_f=0$ (left picture) and $n_f=2$ (right picture), using successively more and more coefficients of the $\beta$ function.
• Figure 2: $F^{\hbox{\tiny $\overline{MS}$}}(g_{\hbox{\tiny $\overline{MS}$}})$ for $g_{\hbox{\tiny $\overline{MS}$}}^2 = 2$ versus $\beta$ function coefficient number $n$. The $n_f=0$ values are filled circles, while the $n_f=2$ values are filled squares. The $[1/1]$ Padé approximations are given as open symbols.
• Figure 3: The quenched $r_0\Lambda^{\hbox{\tiny $\overline{MS}$}}$ points versus $(a/r_0)^2$, together with a linear extrapolation to the continuum limit for method IIP. The filled circles are used for the extrapolation. The star represents the extrapolated value.
• Figure 4: The plaquette $P$ (filled symbols) plotted against the bare quark mass $am_q$ for $\beta=5.20$ (lower curve) until $\beta = 5.40$ (upper curve). The fits use eq. (\ref{['plaq_fit_eq']}), giving the extrapolated values in the chiral limit (open symbols).
• Figure 5: The force parameter $r_0/a$ plotted against $am_q$. The same notation as in Fig. \ref{['fig_amq_plaq_nf2_quad']} is used.