Strong Coupling Constant with Flavour Thresholds at Four Loops in the MS-bar Scheme

Abstract

We present in analytic form the matching conditions for the strong coupling constant alpha_s^(n_f)(mu) at the flavour thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient beta_3 of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavours n_f.

Figures (3)

• Figure 1: $\mu$ dependence of $\alpha_s^{(5)}(\mu)$ calculated from $\alpha_s^{(5)}(M_Z)=0.118$ using Eq. (\ref{['alp']}) at one (coarsely dotted), two (dashed), three (dot-dashed), and four (solid) loops. The densely dotted line represents the exact solution of Eq. (\ref{['rge']}) at four loops.
• Figure 2: Typical three-loop diagrams pertinent to $\Pi_G^h(q_G^2)$, $\Pi_c^h(q_c^2)$, and $\Gamma_\mu^h(q_c,q_{\bar{c}})$. Loopy, dashed, and solid lines represent gluons $G$, Faddeev-Popov ghosts $c$, and heavy quarks $h$, respectively.
• Figure 3: $\mu^{(5)}$ dependence of $\alpha_s^{(5)}(M_Z)$ calculated from $\alpha_s^{(4)}(M_\tau)=0.36$ and $M_b=4.7$ GeV using Eq. (\ref{['alp']}) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eq. (\ref{['oms']}) at the respective order.