QCD Coupling from a Nonperturbative Determination of the Three-Flavor $\mathrm{\Lambda }$ Parameter

We present a lattice determination of the $\mathrm{\Lambda }$ parameter in three-flavor QCD and the strong coupling at the $Z$ pole mass. Computing the nonperturbative running of the coupling in the range from 0.2 to 70 GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain ${\mathrm{\Lambda }}_{\overline{\mathrm{MS}}}^{(3)}=341(12)\mathrm{MeV}$. The nonperturbative running up to very high energies guarantees that systematic effects associated with perturbation theory are well under control. Using the four-loop prediction for ${\mathrm{\Lambda }}_{\overline{\mathrm{MS}}}^{(5)}/{\mathrm{\Lambda }}_{\overline{\mathrm{MS}}}^{(3)}$ yields ${\alpha }_{\overline{\mathrm{MS}}}^{(5)}(m_Z)=0.11852(84)$.

Figure 1

The shaded area shows the $a>0.04\mathrm{fm}$ region of large volume results which dominate the present PDG and FLAG estimates [7, 17] of ${\alpha }_{\overline{\mathrm{MS}}}(\mu )$. The boundary of the shaded area is determined by ${\alpha }_{\overline{\mathrm{MS}}}(\mu )$ evaluated at two-loop order for $N_f=3$ and ${\mathrm{\Lambda }}_{\overline{\mathrm{MS}}}^{(3)}=341\mathrm{MeV}$. The $x$ coordinate is then obtained by setting $a=0.04\mathrm{fm}$, the smallest value of lattice spacings reached in the large volume simulations entering [7, 17]. The data points on the left are finite-size scaling computations [8, 13, 14].

Figure 2

Running couplings of $N_f=3$ QCD from integrating the nonperturbative $\beta$ functions in the SF and GF schemes [13, 14]. They are matched nonperturbatively by defining ${\overline{g}}_{\mathrm{SF}}^2({\mu }_0)=2.012$ and computing ${\overline{g}}_{\mathrm{GF}}^2({\mu }_0/2)=2.6723(64)$.

Figure 3

Dependence of $f_{\pi \mathrm{K}}$ at ${\varphi }_4=1.11$ on the pion mass through ${\varphi }_2=m_{\pi }^2/{\mu }_{\mathrm{ref}}^2$ [31]. We normalized to $f_{\pi \mathrm{K}}^{\mathrm{sym}}$ at the symmetric point $m_u=m_d=m_s$. The ratio follows the parameter-free prediction of NLO chiral PT.

Figure 4

Continuum extrapolations described in the text. Extrapolated values are shown in proximity of $a=0$.