Anomalous Field Dimensions in Quantum Field Theory

• Anomalous field dimensions are deviations from canonical scaling that arise from quantum corrections, nonlocal operators, and holographic influences.
• They are computed using renormalization constants and RG flow, often involving operator mixing and nonperturbative methods in various models.
• These dimensions critically determine scaling behaviors and transport properties, influencing universality classes in both local and nonlocal settings.

Anomalous field dimensions arise when scaling dimensions of fields or composite operators deviate from their canonical, engineering dimensions due to interactions, geometry, or nonlocal effects. While often associated with quantum corrections (e.g., in conventional renormalization group flows), they also appear in nonlocal or nonstandard settings, including effective theories with fractional operators, infinite-flavor constructions, and boundary fields in holography. These anomalous dimensions play a central role in determining the scaling, universality classes, and critical properties of diverse quantum field systems.

1. Rigorous Frameworks for Anomalous Dimensions

In any renormalizable quantum field theory, the scaling (or "anomalous") dimension of an operator or field is defined by introducing a renormalization constant $Z_O(\mu)$ relating the bare operator $O_0$ and the renormalized operator $O_R(\mu)$:

$O_R(\mu) = Z_O(\mu) O_0,$

with anomalous dimension

$\gamma_O(\mu) = -\mu \frac{d}{d\mu} \ln Z_O(\mu).$

The full scaling dimension is $\Delta_O = d_\text{canonical} + \gamma_O$, where $d_\text{canonical}$ is the engineering (classical) dimension. In perturbation theory, $\gamma_O$ is generally expanded in powers of the coupling, but nonperturbative regimes (conformal field theories, large-$N$, bootstrap) admit alternative approaches (Giedt, 2015).

Operators of the same quantum numbers can mix, and the anomalous dimension becomes a matrix $\gamma_{ij}$ determined by the RG flow of the operator basis.

2. Mechanisms Yielding Anomalous Field Dimensions

Local QFTs and Renormalization

Canonical anomalous dimensions arise from quantum corrections due to loop diagrams. In asymptotically free gauge theories, such as QCD, the scaling dimension of the quark mass or bilinear operators is computed perturbatively and verified nonperturbatively on the lattice or in the conformal window (Luthe et al., 2016, Gracey et al., 2018).

Nonlocal and Fractional Operators

In certain effective, induced, or boundary actions, the propagation or dynamics is governed by nonlocal terms—most notably by fractional powers of the Laplacian or kinetic operator. For example, fractional Maxwell equations on a boundary take the form

$\Delta^\gamma A^t = 0,$

implying a nonstandard scaling $[A^t]=\gamma$ for the gauge potential—distinct from the canonical value even in the presence of a locally conserved current (Nave et al., 2017).

Holographic and Higher-Dimensional Constructions

In holography, bulk couplings (e.g., to a dilaton) can induce effective nonlocal actions on the boundary, leading to fractional scaling dimensions for fields and currents. The Caffarelli–Silvestre extension theorem and its generalization to $p$-forms provide a precise mathematical framework for how bulk equations reduce to fractional Laplacians on the boundary, directly determining the anomalous dimension (Nave et al., 2017).

Infinite-Band and Large-$N$ Models

In certain systems composed of an infinite (or very large) number of decoupled "bands" (flavors) with varying masses and charges, exact scale invariance emerges only in the $N\to\infty$ limit. Summing over an infinite spectrum produces an anomalous current dimension $\Phi$ determined by the power-law scaling of the density of states and the charge spectrum, modifying both thermodynamic exponents and correlator scaling (Karch, 2015).

3. The Caffarelli–Silvestre Mechanism and Boundary Fractional Dynamics

The Caffarelli–Silvestre (CS) extension theorem formalizes the relationship between an elliptic PDE in a higher-dimensional half-space and a fractional Laplacian on the boundary. For the scalar case, solutions $U(x, y)$ in $\mathbb{R}^{n+1}_+$ satisfy

$$\nabla\cdot(y^a \nabla U) = 0,\qquad U(x, 0) = f(x),$$

and the Dirichlet-to-Neumann map gives

$$-\lim_{y\to 0^+} y^a \partial_y U(x, y) = C_{n, a} (-\Delta_x)^\gamma f(x),$$

where $\gamma = (1 - a)/2$ and $(-\Delta)^\gamma$ denotes the fractional Laplacian (Nave et al., 2017).

This construction is generalized to $p$-forms:

• For a gauge field $A$ propagating in the bulk with $d(y^a * dA) = 0$, the boundary field satisfies $(\Delta^\gamma A^t) = 0$, with $\gamma$ determined by the bulk-dilaton profile.
• The corresponding gauge symmetry is realized as $A^t \to A^t + d_\gamma \Lambda$ with $d_\gamma = \Delta^{(\gamma-1)/2} d$.
• The scaling relation $[A^t] = \gamma$ holds generically, except in the local ($a = -1$) limit.

Physically, the appearance of anomalous field dimensions via this mechanism reflects genuine nonlocality of the boundary theory, not RG-induced anomalous scaling as in local QFT.

4. Anomalous Field Dimensions for Currents and Gauge Fields

A key property of local QFTs is that Ward identities enforce protected scaling for conserved currents: $\gamma_J = 0$ for a strictly local, conserved current (Nave et al., 2017). However, in boundary or nonlocal settings:

• Fractional gauge theories: The current $J$ sourcing a fractional Maxwell field obeys $\Delta^\gamma A^t = J$, implying $[J] = d - 1 - \gamma$. Both the gauge potential and the current possess non-canonical, generically non-integer scaling dimensions depending on $\gamma$.
• Infinite-band or KK-summed models: The current scaling dimension is shifted by a hyperscaling-violating exponent $\Phi$, as in

$[J^i] = d + z - 1 + \Phi$

for spatial current, and $[J^0] = d - \Phi$ for charge density. The two-point correlators scale with an extra factor of $\Phi$:

$$\langle J^i(x) J^j(0) \rangle \sim \frac{1}{|x|^{2\Delta_J}}$$

where $\Delta_J = d + z - 1 + \Phi$ (Karch, 2015).

Such anomalous dimensions are not quantum corrections but emerge from the collective effect of infinitely many massive degrees of freedom or nonlocal integration kernels.

5. Consequences and Physical Implications

Nonlocal Theories and Ward Identities

In these settings, Ward identities no longer enforce conventional protection of scaling dimensions for boundary or generalized currents. Anomalous field dimensions control the scaling of response functions, such as conductivity or susceptibility, and dictate nontrivial power laws in correlation functions:

• Conductivity and charge transport inherited from a boundary fractional action scale in a nonuniversal way, controlled by the parameter $\gamma$ (Nave et al., 2017).
• In infinite-band systems, thermodynamic and transport exponents are shifted by $\Phi$, affecting, e.g., the scaling of the total free energy.

Dual CFT and Holographic Context

When viewed from the perspective of AdS/CFT, the presence of nonlocal boundary terms induced by bulk dynamics encodes emergent critical exponents inaccessible in purely local boundary actions. The scaling dimension of the dual operator in the boundary CFT is given by the asymptotic scaling behavior set by the bulk solution's normalizability and the effective boundary condition—often fractional due to the CS mechanism.

Signal in Observables

Observable consequences include:

• Fractional power-law responses in two-point correlators and linear response.
• Non-standard operator scaling in the spectrum of the boundary theory, with implications for the universality class.

A summary table of key scaling relations for boundary anomalous field dimensions:

Setting	Field	Anomalous Dimension	Controlling Parameter
Fractional Maxwell (holography)	$A^t$ (boundary gauge)	$[A^t]=\gamma$	$\gamma=\frac{1-a}{2}$
Boundary current	$J$	$[J]=d-1-\gamma$	$\gamma$ (nonlocality)
KK/infinite-band model	$J^i$ (current)	$\Delta_J = d + z - 1 + \Phi$	$\Phi=\tilde{y}z$

6. Broader Perspectives and Generalizations

Anomalous field dimensions are now recognized as a hallmark of nonlocality, nontrivial bulk/boundary interplay, or infinite-component systems. While the canonical setting is renormalization-induced scaling corrections in local field theories, the modern understanding encompasses:

• Induced nonlocal actions (fractional Laplacians, higher-derivative boundary conditions) via holography or effective field theory.
• General large-$N$ or infinite-band models where scaling is emergent only in the thermodynamic/infinite-$N$ limit, and anomalous exponents are "classical" in origin.
• Physical observables sensitive to anomalous field dimensions in, e.g., charge transport, correlation decay, response functions, and critical scaling.

Ward identities and standard field-theoretic theorems prohibiting anomalous dimensions for conserved currents are strictly valid only in local QFT; nonlocality and infinite sector summations evade these protections, leading to a much richer phenomenology of scaling and universality classes (Nave et al., 2017, Karch, 2015).