40717 - 2 Response to Brodsky and Lu ’ s Letter “ On the Self - Consistency of Scale - Setting Methods ”

The claim that the Principle of Minimal Sensitivity is “disfavored” because it does not satisfy certain “self-consistency requirements” is meaningless, and shows a basic misunderstanding of the renormalization-scheme-dependence problem. Brodsky and Lu (BL) [1] have criticized the “Principle of Minimal Sensitivity” (PMS) optimization method [2], claiming that it fails to satisfy certain “reflexivity”, “symmetry”, and “transitivity” conditions. These conditions involve the series expansion of one couplant (a = αs/π) in terms of another: a1(μ1) = a2(μ2)(1 + r12(μ1, μ2)a2(μ2) + . . .). (1) BL apply PMS to this series as though the left-hand side were a physical quantity. This is a basic error: couplants (and Green’s functions, anomalous dimensions, etc.) are not physical quantities and are not renormalization-group (RG) invariant. PMS relies on the knowledge that the quantity in question is RG invariant – or would be, were it not for the truncation of the perturbation series. It makes no sense to apply PMS to “optimize” a1(μ1) because it is not RG invariant. Moreover, there is no point in trying to optimize things that are not physical quantities; they are ill-defined (being intrinsically scheme dependent), and they cannot be compared to experiment, anyhow. Equations like (1) contain no information about physics. Thus, BL’s conditions are meaningless. [It is quite amusing, though, to note that if optimization of (1) and “self-consistency” arguments were meaningful, then they would favour PMS over other schemes. Suppose we take a second-order approximation to (1): a (2) 1 ≡ a (2) 2 (1 + r12a (2) 2 ), (2) where a (2) 2 , by definition, satisfies the second-order truncated β-function: μ2 ∂a (2) 2 ∂μ2 = −b(a (2) 2 ) (1 + ca (2) 2 ). (3) Now, consider the μ1 dependence of a (2) 1 : μ1 ∂a (2) 1 ∂μ1 = μ1 ∂r12 ∂μ1 (a (2) 2 ) 2 = −b(a (2) 2 ) . (4) Substituting for a (2) 2 in terms of a (2) 1 from (2) (reversed) yields: μ1 ∂a (2) 1 ∂μ1 = −b [ a (2) 1 (1− r12a (2) 1 + . . .) ]2 (5) = −b(a (2) 1 ) (1− 2r12a (2) 1 + . . .). (6) But, for “self-consistency” in BL’s sense, we would want a (2) 1 to satisfy the second-order β-function equation (i.e., Eq. (3) with ‘2’ subscripts replaced by ‘1’ subscripts), at least up to higher-order corrections. In FAC, where the “optimization” gives r12(FAC) = 0,