Operator mixing in massless QCD-like theories and Poincarè–Dulac theorem

Abstract

AbstractRecently, a differential-geometric approach to operator mixing in massless QCD-like theories – that involves canonical forms, obtained by means of gauge transformations, based on the Poincarè–Dulac theorem for the system of linear differential equations that defines the renormalized mixing matrix in the coordinate representation $$Z(x,\mu )$$ Z ( x , μ ) – has been proposed in [1]. Specifically, it has been determined under which conditions a renormalization scheme exists where $$Z(x,\mu )$$ Z ( x , μ ) may be set in a diagonal canonical form that is one-loop exact to all perturbative orders – the nonresonant diagonalizable $$\frac{\gamma _{0}}{\beta _{0}}$$ γ 0 β 0 case (I) –. Moreover, the remaining cases, (II), (III) and (IV), of operator mixing, where such diagonalization is not possible, have also been classified in [1]. Accordingly, if $$\frac{\gamma _{0}}{\beta _{0}}$$ γ 0 β 0 , with $$\gamma (g)=\gamma _{0} g^{2}+\cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the matrix of the anomalous dimensions and $$\beta (g)=-\beta _{0} g^{3} + \cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function, either is diagonalizable but a resonant condition for its eigenvalues and the system holds (II) or is nondiagonalizable and nonresonant (III), or is nondiagonalizable and resonant (IV), $$Z(x,\mu )$$ Z ( x , μ ) is nondiagonalizable. In the cases (II), (III) and (IV), we demonstrate that its canonical form may be factorized into the exponential of a linear combination of upper triangular nilpotent constant matrices with coefficients that asymptotically in the UV are logs of the running coupling, i.e., asymptotically loglogs of the coordinates, and a diagonal matrix as in the nonresonant diagonalizable case (I). Hence, its ultraviolet asymptotics differs intrinsically from the case (I) and, for asymptotically free theories, this is the closest analog of logCFTs. We also work out actual physics realizations of the cases (I) and (II), while we argue that the cases (III) and (IV) are ruled out by a unitarity argument in the gauge-invariant sector.