Determination of the Lambda parameter from full lattice QCD

We present a determination of the QCD parameter $\Lambda$ in the quenched approximation ($n_f=0$) and for two flavors ($n_f=2$) 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 ${\Lambda }_{n_f=0}^{\overline{MS}}=259(1)(19)\mathrm{MeV}$ and ${\Lambda }_{n_f=2}^{\overline{MS}}=261(17)(26)\mathrm{MeV}$, using $r_0=0.467\mathrm{fm}$ to set the scale. Extrapolating our results to five flavors, we obtain for the running coupling constant at the mass of the $Z$ boson ${\alpha }_s^{\overline{MS}}(m_Z)=0.112(1)(2)$.

Figure 1

${\alpha }_s^{\overline{MS}}(\mu )$ versus $\mu /{\Lambda }^{\overline{MS}}$ for $n_f=0$ (left picture) and $n_f=2$ (right picture), using successively more and more coefficients of the $\beta$ function.

Figure 2

$F^{\overline{MS}}(g_{\overline{MS}})$ for $g_{\overline{MS}}^2=2$ 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 $r_0{\Lambda }^{\overline{MS}}$ 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 $a{m}_q$ for $\beta =5.20$ (lower curve) until $\beta =5.40$ (upper curve). The fits use Eq. (41), giving the extrapolated values in the chiral limit (open symbols).

Figure 5

The force parameter $r_0/a$ plotted against $a{m}_q$. The same notation as in Fig. 4 is used.

Figure 6

The unquenched $r_0{\Lambda }^{\overline{MS}}$ points (filled circles) versus $(a/r_0{)}^2$, together with a linear extrapolation to the continuum limit for methods IIP and IIIP. Stars represent the extrapolated values.

Figure 7

The ratio ${\Lambda }_3^{\overline{MS}}/{\Lambda }_2^{\overline{MS}}$ against the ratio ${\Lambda }_2^{\overline{MS}}/{\Lambda }_0^{\overline{MS}}$ from 1-loop (dotted line), 2-loop (dashed line) and 3-loop (solid line) matching.

Figure 8

Values of ${\Lambda }^{\overline{MS}}$ versus number of quark flavors $n_f$. The filled circles are our $n_f=0$, 2 results, and the open circle is our extrapolated value. The inner error bars give the statistical errors, while the outer error bars give the total errors. The square is from a 3-loop analysis of the nonsinglet structure functions [35]. The triangles are taken from [36]. The open triangles are evaluated using the 4-loop expansion of ${\alpha }_s$ and 3-loop matching at the quark thresholds. The entries at $n_f=3$ and 4 have been displaced horizontally.

Figure 9

Comparison of ${\alpha }_s^{\overline{MS}}(m_Z)$ from this work (solid circle) with other lattice results [40, 41, 42, 2, 43, 4, 44, 21] (from top to bottom). The circles are from Wilson fermions and the squares from staggered fermions. The dashed line indicates the mean phenomenological value [36].