This procedure has been carried out for the Einstein–Yang–Mills system  and for massless scalar electrodynamics . Both systems have a single length scale 1/e (in geometric units c = G = 1), where e is the gauge coupling constant. All values of e can be said to form one universality class of field equations  represented by e = 0. This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state (EOS) that do not affect the limit of high density . A simple example is that any scalar field potential becomes dynamically irrelevant compared to the kinetic energy in a self-similar solution , so that all scalar fields with potentials are in the universality class of the free massless scalar field. Surprisingly, even two different models like the SU(2) Yang–Mills and SU(2) Skyrme models in spherical symmetry are members of the same universality class .
If there are several scales , , etc. present in the problem, a possible approach is to set the arbitrary scale in Equation (29) equal to one of them, say , and define the dimensionless constants from the others. The scope of the universality classes depends on where the appear in the field equations. If a particular appears in the field equations only in positive integer powers, the corresponding appears only multiplied by , and will be irrelevant in the scaling limit. All values of this therefore belong to the same universality class. From the example above, adding a quartic self-interaction to the massive scalar field gives rise to the dimensionless number but its value is an irrelevant (in the language of renormalisation group theory) parameter.
Contrary to the statement in , we conjecture that massive scalar electrodynamics, for any values of e and m, is in the universality class of the massless uncharged scalar field in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the k of the perfect fluid, the of the 2-dimensional sigma model or, probably, a conformal coupling of the scalar field  (the numerical evidence is weak but a dependence should be expected).
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