A LIGHT DILATON IN GAUGE THEORIES?

We study gauge theories having a slowly varying running coupling and chlral symmetry breaking. It has been suggested that such theories contain a hght dflaton, a pseudo-Goldstone boson assocmted with approxtmate scale mvanance. We calculate ~ts mass by studying the scahng properties of the effective action describing chiral symmetry breaking. We also consider the effect of a physical high energy cutoffas motivated m a techmcolor context. Our results indicate that a hght dllaton is unhkely We wish to study asymptotically free gauge theories containing no exphcit dimensionful parameters and having a small r-function over the momentum range relevant for chiral symmetry breaking. Below we shall exhibit theories of this type for which we can trust the lowest order perturbative r-function throughout this momentum range. But if the r-function is small then the theory possesses an approximate scale invariance, and chiral symmetry breaking would imply a spontaneous breaking of this scale symmetry. This makes necessary a pseudo-Goldstone boson associated with broken approximate scale mvariance, a pseudo-dilaton (PD) [ 1,2]. The nonzero r-function is the only explicit source of scale symmetry breaking, and the PD mass is related to ~t in some way. We might guess that the mass squared is proportional to the value of the r-function at the scale of chiral symmetry breaking [ 3]. Others [ 4] suggest that the mass is given by the confinement scale A, the inverse size ofglueball states. The theories we study will have A less than the chiral symmetry breaking scale A~,. The PD will appear in the effective theory immediately below A~,. In this paper we will focus on short &stanc