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

Aglietti, U.; Becchetti, Matteo; Bochicchio, Marco; Papinutto, Mauro; Scardino, Francesco

doi:10.48550/arxiv.2105.11262

Cited by 2 publications

(8 citation statements)

References 14 publications

(62 reference statements)

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“…Hence, to the leading large-N order, single-and double-trace operators only mix separately among themselves [20]. According to Eq.…”

Section: Dimension-8 Operators In Large-n Ym Theorymentioning

confidence: 99%

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

Becchetti¹,

Bochicchio

2022

Eur. Phys. J. C

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Recently, 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.

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“…Hence, to the leading large-N order, single-and double-trace operators only mix separately among themselves [20]. According to Eq.…”

Section: Dimension-8 Operators In Large-n Ym Theorymentioning

confidence: 99%

“…In this respect, a fundamental physics application of [1] and the present paper, in order to work out the UV asymptotics of the nonperturbative glueball one-loop effective action [19], consists in showing that twist-2 operators in SU(N ) YM theory belong to the case (I) [20].…”

Section: Introduction and Physics Motivationsmentioning

confidence: 99%

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

Becchetti¹,

Bochicchio

2022

Eur. Phys. J. C

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“…If the condition (14) is satisfied for all the eigenvalues of A 0 , then A(g) = − γ(g) β(g) may be set in the one-loop exact form (13) to all orders in perturbation theory for a choice of the operator basis.…”

Section: The Poincarè-dulac Theorem For Massless Qcd-like Theoriesmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Becchetti¹

2021

Preprint

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We review recent progress on operator mixing [1, 2] in the light of the theory of canonical forms for linear systems of differential equations and, in particular, of the Poincaré-Dulac theorem. We show that the matrix A• determines which different cases of operator mixing can occur, and we review their classification. We derive a sufficient condition for A(g) to be set in the one-loop exact form A(g) = γ 0 β 0 1 g . Finally, we discuss the consequences of the unitarity requirement in massless QCD-like theories, and we demonstrate that γ 0 is always diagonalizable if the theory is conformal invariant and unitary in its free limit at g = 0.

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

Cited by 2 publications

References 14 publications

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

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

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

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