Generalized Dimensional Analysis

I describe a version of so-called naive dimensional analysis, a rule for estimating the sizes of terms in an effective theory below the scale of chiral symmetry breaking induced by a strong gauge interaction. The rule is simpler and more general than the original, which it includes as a special case. I also give a simple qualitative interpretation of the rule. Research supported in part by the National Science Foundation under Grant #PHY-8714654. Research supported in part by the Texas National Research Laboratory Commission, under Grant #RGFY9206. In dealing with effective field theories describing physics of mesons below a symmetry breaking scale in a strongly interacting theory, it is important to have a tool for estimating the coefficients of nonrenormalizable interactions. Naive dimensional analysis (NDA) [1] was proposed as such a tool. It works pretty well in QCD. However, a better instrument is needed for theories in which the number of colors and flavors may be very different from what they are in QCD. In this very brief note, I describe one. I will give a rule for such dimensional estimates that is both simpler and more general than the original. This simplicity and generality is obtained by introducing an additional parameter, the ratio of the Goldstone boson decay constant to the mass of the lightest non-Goldstone bound-states. I will also give an extremely simple, qualitative argument to interpret the rule. I will consider only strongly interacting theories that are “QCD-like” — with fermions transforming only under the simplest representation of the gauge group, in order to avoid the additional complications of chiral fermions and of dependence on ratios of Casimir operators. For example, I don’t want to think about “tumbling” [2] because it makes my head hurt. [3] The low energy physics described by these QCD-like theories is the physics of the light pseudo-Goldstone mesons. The effective field theory is only useful at energies small compared to the scale at which other bound-states appear. At the end, I speculate on the dependence of the extra parameter on the color and flavor structure of the theory and discuss possible generalizations of the trivial idea described here. The discussion in this paper is very simple. I do not pretend that it is very deep. It has probably been stated in only slightly different form by others. Nevertheless, I think that the very simplicity of the statement is a virtue. It strips NDA down to its barest essentials. I think that this is useful in trying to determine the form the dimensional analysis will take in more interesting effective theories. In order to be able to keep track of things like the number of colors and flavors in the various strong groups, I will distinguish the Goldstone boson decay constant, f , from the typical mass of the low-lying (non-Goldstone) bound states, Λ. In QCD, f = fπ is the Goldstone boson decay constant and Λ ≈ 1GeV (or the ρ mass — take your pick) is the typical mass of the light but non-Goldstone bound states. The simple rule to assign a dimensional coefficient of the right size to any term in the effective