Strong-Coupling Constant with Flavor Thresholds at Five Loops in the   MS-bar Scheme
  We present in analytic form the matching conditions for the strong-coupling constant alpha_s^(n_f)(mu) at the flavor thresholds to four loops in the modified minimal-subtraction scheme. Taking into account the present knowledge on the coefficient beta_4 of the Callan-Symanzik beta function of quantum chromo-dynamics, we thus derive a five-loop formula for alpha_s^(n_f)(mu) together with appropriate relationships between the asymptotic scale parameters Lambda^(n_f) for different numbers of flavors n_f. 
B.A. Kniehl, A.V. Kotikov, A.I. Onishchenko, O.L. Veretin