Volume 220, number 1,2 PHYSICS LETTERS B 30 March 1989 HIGHER MASS SCALES AND MASS HIERARCHIES Thomas APPELQUIST, T. TAKEUCHI Centerfor Theoretical Physics, Department of Physics, Yale University,New Haven, CT06511, USA Martin EINHORN Department of Physics, University of Michigan, Ann Arbor, M148109, USA and L.C.R. WIJEWARDHANA Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA Received 15 February 1989 The impact of new interactions at very high energies on the spontaneous breaking of chiral symmetry in gauge field theories is studied. Described at low energies by higher dimension operators, these interactions can enhance the chiral condensate and even the Goldstone boson decay constant relative to the confinement scale. This can lead to important consequences in technicolor theories, including realistically large quark and lepton masses and the possibility of new physics at energies below the weak scale. In addition, small differences in strength of the four-fermion interactions, due say to isospin or flavor breaking, can be greatly magnified in the resulting quark and lepton mass spectrum. In this letter, we summarize a study of the effect of new, high energy interactions on the dynamical breaking of chiral symmetry in asymptotically free gauge field theories, in particular technicolor theo- ries. We will show that these interactions, described at low energies by higher dimension operators, can have an important effect on the mass scales that emerge from the breaking. In asymptotically free theories, the running cou- pling oz(p) is strong at momenta on the order of the confinement scales Ac. It then decreases like 1/In(p) until it reaches some smaller value a(A ), where A is the cutoff beyond which the theory can no longer be used in isolation. The value of or(A) is determined by the unknown high energy physics that lurks be- yond A. Dynamical chiral symmetry breaking, if it is driven purely by the gauge interaction, can only take place when o~ (p) exceeds some critical value otc [ 1 ], expected to be reasonably strong ~l. Thus if ct (A) << otc, the breaking must take place at scales well below the cutoff and perhaps on the order of the con- finement scale Ac. In QCD this is precisely what hap- pens, and technicolor theories are expected to behave in a similar way. The Goldstone boson decay con- stants for each case are f~ = 93 MeV and F= O (250 GeV). In either case, the new physics beyond A can be de- scribed at low energies in terms of effective higher dimensions operators. The dominant contribution at low energies should arise from dimension six, four- fermion operators. In technicolor theories, these op- erators (ETC interactions) [3 ] ~2, treated as pertur- bations to the dominant technicolor gauge interac- tions, give rise to quark and lepton masses, pseudo- Goldstone boson masses, and typically to flavor- ~ There is some evidence, however, that the ladder expansion may be reasonably convergent for ot= O (ctc); see ref. [ 2 ]. ~2 For a recent review of technicolor see ref. [4]. 223