Z. Phys. C - Particles and Fields t4, 179-186 (1982) Zeitschriff P a r t i c l e s fl3r Physik C and Fields Springer-Verlag1982 The Effect of Quark Masses on the Running of the QCD Coupling Constant Carl Rosenzweig Physics Department, SyracuseUniversity, Syracuse,NY 13210, USA Received 5 October 1981; in revised form 30 March 1982 Abstract. The presence of thresholds for finite quark masses alters the behavior of the QCD coupling constant. While the alterations are small they can have significant influence on the determination of the scale parameter A. Part of the discrepancy be- tween recent high Q2 determinations of A and older low Q2 determinations can be attributed to charm and bottom thresholds. We present simple, analytic expressions for the mass dependence of ~(Q2) and exploit these to discuss A determinations, scale breaking and the phenomenon of temporary free- dom when more than 17 quark flavors exist. I. Introduction Quantum Chromodynamics has become the para- digm of a theory for the strong interactions [1]. Although it is more readily supported by general theoretical arguments than precise experimental tests it has become the standard language for describing large classes of experiments. As physicists strive for more detailed experimental confirmation of the theory it behooves them to consider small effects which have heretofore been neglected. One such ef- fect is the finiteness of quark masses. In the first round of comparison between theory and experi- ment it was justified to neglect quark masses and use the theory of massless quarks. Now we are con- cerned about such matters as precise predictions of proton lifetimes and second or higher order cor- rections to scaling violations. It is inconsistent to neglect quark mass when we are considering other corrections of similar magnitude. We have reached a stage where a complete, systematic inclusion of quark mass effects in QCD is necessary. To date, the only process where this has been attempted is e + e- annihilation into hadrons [2]. While I do not pres- ent a complete analysis here, a first step in this direction is taken. The influence of quark masses on the evolution of the QCD coupling constant is treat- ed via a simple approximation. The major phenome- nological consequences arise in scaling violation in deep inelastic scattering. The effects are small but are of magnitude comparable to second order per- turbation contributions. In Sect. II we establish approximations which lead to a simple analytic expression for ~(Q2) includ- ing quark threshold up to second order in 1/lnQ 2. The analytic expression is attractive since we can readily exploit it to produce the qualitative features of the threshold effects. The appropriate infinite mass and zero mass limits are manifest. Section III applies this detailed knowledge of ~(Q2) to discuss the qualitative and quantitative changes induced by thresholds on scaling violation phenomena in deep inelastic scattering. Section IV is a pedagogical aside on the phenomena of temporary freedom, which oc- curs when 17 or more massive quarks are present. It is an explicit example of the Applequist-Carrazone theorem [3] and illustrates how it is possible to have infrared slavery without having asymptotic freedom. This discussion may resolve an old para- dox in theories based on expansions in 1/NF, the number of quark flavors. An appendix presents the second order scale violating predictions of QCD in- cluding threshold effects. II. The Quark Mass Dependence of the Running Coupling Constant The running coupling constant in QCD is defined by t Q2 ~ , M 2' g (m2) --2 i ~Q2 -/3 g ,Q2~ (1) 0170-9739/82/0014/0179/$01.60