1 2 3 4 n 250 300 350 400 F$_{Ρ}$ 1 GLYPH<Slash1> $^{2}$GLYPH<Slash1> MeV GLYPH<CapLambda>GLYPH<Equal>GLYPH<Infinity> GLYPH<CapLambda>GLYPH<Equal> 1940 MeV 776 MeV GLYPH<CapLambda>GLYPH<Equal> 582 MeV 1 2 3 4 n 250 300 350 400 F$_{Ρ}$ 1 GLYPH<Slash1> $^{2}$GLYPH<Slash1> MeV GLYPH<CapLambda>GLYPH<Equal> 272 MeV GLYPH<CapLambda>GLYPH<Equal> 388 MeV GLYPH<CapLambda>GLYPH<Equal> 194 MeV Figure 7.2: The decay constant F 1 / 2 ρ for different ρ excitations, plotted for varying UV cutoffs 7.4 Anomalous Dimensions at the Cutoff Scale There remains the possibility that the classical dimension of any operator is not respected at the quantum level. Here we investigate possible effects to the operator ¯ qγ $^{µ}$q ∗ . Classically is has dimension three, and this is encoded in the gravity dual by fixing the UV boundary condition as In general though we should write and numerically investigate the preferred value of ω . The result is that for all values of the cutoff, from ∞ down to our tuned value of 194 MeV, the preferred value of ω is two, so it appears that the classical value is conserved at the quantum level. However, this is a rather naive analysis: if one evaluates the derivative of the flows f$_{n}$ ( r ) at finite values of r in the case of the model with an infinite cutoff, the derivatives differ for each value of n . Hence we should be $^{∗}$There is a theorem [148] that says that conserved currents are forbidden from ac- quiring anomalous dimensions. However this theorem is only true for theories that do not contain vector mesons with the same quantum numbers as the conserved current, hence the theorem is not applicable here 102 f$_{n}$ ( r ) ∼ $_{r}$− $^{ω}$, $_{f}$′ $_{n}$( r ) ∼ − $_{ωr}$− ω − 1 (7.6) f$_{n}$ ( r ) ∼ $_{r}$− $^{2}$, $_{f}$′ $_{n}$( r ) ∼ − 2 $_{r}$− 3 (7.5)