Quantization of Anomalous Dimensions

• Quantization of anomalous dimensions is a framework where quantum corrections yield discrete spectral values for scaling operators in quantum field theories.
• It uses conformal symmetry, bootstrap constraints, and operator mixing to define precise operator spectra across diverse models such as large-N CFTs and lattice field theories.
• Detailed studies in gauge, monopole operator, and AdS3/CFT2 contexts demonstrate how quantization reconciles perturbative and nonperturbative predictions.

Anomalous dimensions represent the quantum corrections to the scaling dimensions of operators in quantum and statistical field theories. Their quantization encapsulates the circumstances, mechanisms, and mathematical structures by which these dimensions assume discrete, quantized values or spectra under renormalization, operator mixing, and conformal symmetry constraints. The quantization of anomalous dimensions is central to the formulation and solution of conformal field theories (CFTs), the dynamics of gauge and string theories, and nonperturbative approaches such as lattice field theory.

1. Operator Anomalous Dimensions: Definition and Physical Context

An anomalous dimension $\gamma_\mathcal{O}$ of an operator $\mathcal{O}$ is the quantum correction to its canonical (engineering) scaling dimension, such that under an RG flow the renormalized operator satisfies

$$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$

where $\mu$ is the renormalization scale and $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$ in terms of the renormalization constant $Z_{\mathcal{O}}$ (Giedt, 2015). In theories with operator mixing, $\gamma$ becomes a matrix whose eigenvalues correspond to the scaling dimensions of "diagonalized" operators. The quantization problem is to determine the possible discrete and continuous spectra of these anomalous dimensions and the constraints that fix their allowed values.

2. Quantization via Conformal and Bootstrap Constraints

Conformal symmetry imposes powerful restrictions on the possible anomalous dimensions. In minimal CFTs and their deformations, the bootstrap program yields quantization through the requirement that interacting theory correlators match those of a free theory as the deformation is turned off ($\epsilon \to 0$). Specifically, the discrete set of upper-critical dimensions $d_m=2m/(m-2)$ arises from multiplet recombination: only when $d=d_m$ can certain primary and descendant fields recombine appropriately under perturbations (e.g., in $\mathcal{O}$0 scalar theories). This mechanism directly quantizes the allowed operator spectrum (Gliozzi, 2017).

Spin and charge quantum numbers further discretize possible scaling dimensions. Leading-order anomalous dimensions for spinning primaries and higher-spin currents are fixed by matching four- and five-point functions in both free and interacting theories, leaving no free parameters. The resulting anomalous dimensions reproduce all known perturbative data in $\mathcal{O}$1, illustrating how quantization emerges purely from conformal structure and OPE data (Gliozzi, 2017).

3. Quantization through Operator Mixing and Infinite-Dimensional Matrices

In gauge and fermionic CFTs, the operator basis (especially for four-fermion operators or "evanescent" operators in $\mathcal{O}$2) is infinite. The diagonalization problem thus involves infinite-dimensional anomalous dimension matrices with tridiagonal (three-term recurrence) structure. For instance, in $\mathcal{O}$3 QED or $\mathcal{O}$4 Gross-Neveu-Yukawa models, the one-loop anomalous dimension matrix leads to a three-term recurrence relation solved by continuous dual Hahn polynomials. The allowed spectrum is determined by demanding that two-point functions are normalizable (finite), which restricts spectral parameters (e.g., spectral parameter $\mathcal{O}$5 in QED) to quantized subsets: a continuous spectrum for general operators, and isolated discrete values when the hypergeometric series truncates. This yields a "quantization rule": the spectrum consists of a continuum constrained by normalizability, plus isolated physical (discrete) eigenvalues (Ji et al., 2018).

4. Explicit Computation in Specific Theories

Large-$\mathcal{O}$6 CFTs and Higher-Spin Currents

In the critical $\mathcal{O}$7 scalar and Gross-Neveu models in $\mathcal{O}$8 dimensions, anomalous dimensions can be computed exactly to all orders in $\mathcal{O}$9 by conformal perturbation theory. Starting from free theory and including double-trace or four-fermion deformations, one introduces auxiliary fields (e.g., $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$0) to generate the leading $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$1 corrections. Anomalous dimensions of spinning currents are extracted from divergent contributions to two-point functions; the final result is a closed form $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$2 valid for all $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$3 and $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$4, matched against both RG computations and AdS/CFT predictions (Hikida et al., 2016). The quantization here refers to the exact, nonperturbative determination of these anomalous dimensions in terms of discrete spin $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$5 for all integer $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$6.

Monopole Operators and Large-Charge Asymptotics

For monopole operators in 2+1-dimensional CFTs (QED$$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$7, QED$$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$8-Gross-Neveu, and related models), quantization emerges in two ways. First, state-operator correspondence and saddle-point methods yield the full $$\mu\frac{d}{d\mu}\mathcal{O}(\mu) = -\gamma_\mathcal{O}(\mu)\mathcal{O}(\mu),$$9 expansion of scaling dimensions, with both leading and sub-leading quantized contributions for fixed monopole charge $\mu$0 (discrete). Second, for large monopole charge, the expansion

$\mu$1

shows that the constant term $\mu$2 is universal and independent of $\mu$3, matching predictions from nonperturbative CFT arguments and illustrating quantization through universal large-charge expansions (Dupuis et al., 2021).

5. Lattice Field Theory and Spectroscopic Extraction

On the lattice, quantization of anomalous dimensions is operationalized by discretization and nonperturbative measurement of RG flows and operator mixing. Lattice analogues of continuum operators are constructed, renormalization constants $\mu$4 are extracted from scaling of correlators or step-scaling of renormalization factors, and anomalous dimensions are then determined. Operator mixing on the lattice, when represented as a matrix, is diagonalized or treated via generalized eigenvalue problems. The extraction process is inherently quantized, as fitting procedures select discrete eigenvalues corresponding to scaling operators of the continuum limit. Results for the anomalous mass dimension $\mu$5 in various near-conformal gauge theories demonstrate the practical quantization in numerical spectra, with $\mu$6 in all studied cases (Giedt, 2015).

6. Quantum Wilson Lines and Ambiguities in Quantization

In AdS$\mu$7/CFT$\mu$8, the quantization of anomalous dimensions is realized through quantum expectation values of open SL(2,$\mu$9) Wilson lines in Chern-Simons theory coupled to a point particle of spin $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$0 (Besken et al., 2017). The scaling dimension $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$1 is obtained via a loop expansion in $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$2, with each order introducing a further quantized correction. While the one-loop (order $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$3) result matches exactly the CFT prediction, at two loops ($$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$4) renormalization-scheme ambiguities appear due to the lack of a canonical local counterterm and UV subtraction ambiguity. The result is that while exponentiation into a pure power-law is preserved, the discrete coefficient of $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$5 at two loops (corresponding to $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$6) is scheme-dependent and not uniquely quantized until further physical constraints are imposed. This highlights that quantization in this context requires not just diagonalization or computation, but a consistent choice of renormalization prescription consistent with conformal invariance and the Virasoro algebra (Besken et al., 2017).

7. Mathematical Mechanisms and Physical Implications

Across all these contexts, quantization of anomalous dimensions arises due to:

• Operator algebra structure and representation theory (discrete $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$7, $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$8, $$\gamma_\mathcal{O}(\mu) = -\mu d\ln Z_{\mathcal{O}}/d\mu$$9, etc.).
• Conformality and bootstrap constraints (multiplet recombination, crossing, OPE matching).
• Normalizability conditions for two-point functions, enforcing spectral parameter quantization in infinite-matrix diagonalization.
• Scheme-dependence and renormalization ambiguities, especially at higher loops, where physical consistency (algebra closure, BRST invariance) must dictate the unique prescription.
• Universality at large quantum numbers, where quantized constants in large-$Z_{\mathcal{O}}$0 expansions link bulk and boundary CFT data.

These mechanisms jointly reveal that the quantization of anomalous dimensions is a structural property of quantum field theory, reflecting both mathematical consistency and physical universality.

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References:

• Anomalous dimensions from quantum Wilson lines (Besken et al., 2017)
• Anomalous dimensions of higher spin currents in large N CFTs (Hikida et al., 2016)
• Anomalous dimensions on the lattice (Giedt, 2015)
• Anomalous dimensions of monopole operators at the transitions between Dirac and topological spin liquids (Dupuis et al., 2021)
• On operator mixing in fermionic CFTs in non-integer dimensions (Ji et al., 2018)
• Anomalous dimensions of spinning operators from conformal symmetry (Gliozzi, 2017)