7.1 Operator-field matching The matching of operator dimensions to field solutions is complicated by the presence of a UV cutoff, Λ. Without it the matching is performed at r = ∞ : now the matching must be performed at r = Λ. This makes things more difficult for two reasons: These two points seem almost unsurmountable, but all is not lost. There are an infinite number of operators in QCD anyway, yet we have shown in chapters 5-6 that including just a handful gives us very close matches to experimental values. In addition, we do have some idea of what to expect from lattice QCD results from a subject called ‘perfect actions’ [146]. For the rest of this section, we first see the effect of imposing a hard UV cutoff, without worrying about the effects of anomalous dimensions or higher dimension operators. The results are very good. We then consider 98 2. The presence of large couplings for higher dimension operators before QCD can be matched to the perturbative gravity theory would be another signal of non-perturbative phenomena. Formally there are an infinite set of such couplings. They are called irrelevant perturbations (because they become small in the perturbative regime), and will ap- pear in the gravitational dual as deformations of the metric which grow at large r . 1. As we discussed in section 1.2.6, fields can have anomalous dimensions in the bulk which become small as r → ∞ , and so don’t require consideration when matching at r = ∞ . By introducing a finite cutoff r = Λ, we need to match not to the classical dimension, but to the anomalously-adjusted dimension of the fields.