Operator mixing in massless QCD-like theories and Poincare'-Dulac theorem

Abstract

Recently, a geometric approach to operator mixing in massless QCD-like theories -that involves canonical forms, obtained by means of gauge transformations, based on the Poincare'-Dulac theorem for the linear system that defines the renormalized mixing matrix in the coordinate representation Z(x, µ) -has been advocated in [1]. In particular, it has been determined under which conditions a renormalization scheme exists where the linear system -and correspondingly Z(x, µ) -may be set in a diagonal canonical form that is one-loop exact to all perturbative orders -the nonresonant diagonalizable case -according to the Poincare'-Dulac theorem. Following the aforementioned approach, in the present paper we work out the most general canonical form for the linear system above and its solution Z(x, µ). Accordingly, we point out that a new phenomenon occurs, which is hardly discussed in the literature: If the matrix γ 0 β 0 , with γ(g) = γ 0 g 2 + • • • the matrix of the anomalous dimensions and β(g) = −β 0 g 3 + • • • the beta function, either is nondiagonalizable or is diagonalizable but -according to the Poincare'-Dulac theorem -a resonant condition for its eigenvalues and the system holds, Z(x, µ) is nondiagonalizable. If γ 0 β 0 is nondiagonalizable, Z(x, µ) is modified with respect to the nonresonant diagonalizable case by a matrix factor that contains asymptotically in the UV sums of powers of logs of the running coupling, i.e., powers of loglogs of the coordinates. We observe that this is the closest analog for asymptotically free theories of logCFTs. Yet, we argue that in the gauge-invariant (Hermitian) sector of a massless QCD-like theory, which to the order of g 2 (µ) should be conformal and unitary, γ 0 β 0 should always be diagonalizable, because otherwise a nonunitary logCFT would arise to the order of g 2 (µ). Nevertheless, even if the eigenvalues of γ 0 β 0 are diagonalizable, the associated linear system may be resonant, thus realizing the second alternative above. However, we demonstrate that in the latter case the corresponding leading UV asymptotics of Z(x, µ), despite its nondiagonalizability, coincides with the one of the nonresonant diagonalizable case.

Operator mixing, UV asymptotics of nonplanar/planar $2$-point correlators, and nonperturbative large-$N$ expansion of QCD-like theories

“…Besides, we work out (section 3) the close interplay between our lowest-order computation and the actual nonperturbative asymptotics of 2-point correlators in the case of operator mixing according to [2,9,10].…”

“…We summarize briefly the results about operator mixing in [2,9,10], both to the lowest orders and to all orders, that are relevant for the present paper.…”

“…) is a dimensionless RG-invariant matrix function -which is symmetric, as the operators are bosonic and either P T even or odd, and real, as the operators are Hermitian -of the running coupling only [10], and:…”

A differential-geometry approach to operator mixing in massless QCD-like theories and Poincaré-Dulac theorem

“…This is the case (I) of the classification introduced in [1]. The remaining cases, (II),(III) and (IV), of the aforementioned classification, where such a reduction is not actually possible, have been discussed in [2], which also contains physical applications based on [14,17,34,35].…”

“…The approach to operator mixing based on the Poincaré-Dulac theorem allows us to study in the most general case the UV asymptotics of the renormalized mixing matrix Z(x, µ) [1,2], which enters a number of applications of the renormalization group (RG), ranging from the deep inelastic scattering [15] in QCD to the evalutation of the ratio ε ε [16][17][18] for the possible implications of new physics, and to the constraints [14,[19][20][21][22][23][24] to the eventual nonperturbative solution of the large-N limit [25][26][27][28] of massless QCD-like theories.…”