Anomalous Dimensions: Double-Trace Operators

• The study explains that anomalous dimensions quantify quantum corrections to double-trace operators by interpreting them as binding energies in AdS space.
• The Hamiltonian approach bypasses complex OPE analyses by directly applying quantum-mechanical perturbation theory to compute energy shifts.
• The work shows that both positive and negative anomalous dimensions can emerge, influencing operator mixing, RG flows, and supersymmetric model building.

Anomalous dimensions of double-trace operators quantify quantum corrections to the scaling dimensions of composite operators, typically of schematic form $[\mathcal{O}_1 \mathcal{O}_2]_{n,\ell}$, built as bilinears in single-trace primaries in conformal field theories (CFTs). These corrections are central to understanding operator mixing, unitarity, RG flows, and spectra of weakly and strongly coupled CFTs, especially those with AdS (gravity) duals. This article provides a technical overview of fundamental results, calculational methods, and their implications across several paradigmatic settings.

1. Fundamental Definition and Physical Context

Double-trace operators are composite primaries composed of two (possibly distinct) single-trace operators, with spacetime derivatives and spin indices symmetrized as needed. Their classical (engineering) dimension is the sum of the constituent operators' dimensions plus derivatives: for spin-$\ell$,

$$\Delta^{(0)}_{n,\ell} = \Delta_1 + \Delta_2 + 2n + \ell.$$

Quantum corrections shift this by an anomalous dimension,

$$\Delta_{n,\ell} = \Delta^{(0)}_{n,\ell} + \gamma_{n,\ell},$$

where $\gamma_{n,\ell}$ is a key observable. In holographic CFTs at large $N$, anomalous dimensions originate from interactions—encoded on the gravity side as binding energies of two-particle AdS states—while in perturbative gauge theories they arise from nonplanar diagrams and operator mixing.

Anomalous dimensions of double-trace operators, especially non-chiral (non-protected) ones, are pivotal for both conceptual considerations (e.g., spectrum stability, unitarity, OPE crossing) and phenomenological applications (e.g., sequestering mechanisms for supersymmetry breaking).

2. Direct Hamiltonian Approach: Anomalous Dimensions as Binding Energies

A key result (Fitzpatrick et al., 2011) is the translation of double-trace anomalous dimension calculations in superconformal field theories (SCFTs) with weakly-coupled AdS duals into a Hamiltonian language. Rather than extracting anomalous dimensions from four-point correlation functions and their conformal block decompositions—a technically formidable route involving complicated AdS propagators and OPE analysis—the spectrum of operator dimensions is directly tied to the quantum-mechanical binding energy of the corresponding AdS two-particle state.

For a scalar $\phi$ of dimension $\Delta_0$ and AdS mass $m^2 = \Delta_0(\Delta_0-4)$, the unperturbed two-particle energy is $2\Delta_0$. Upon including interactions (suppressed by the gravitational coupling $k \sim 1/N$), one expands the Hamiltonian as

$$H = H_\text{free} + k H_\text{exchange} + k^2 H_\text{contact} + \dots$$

First-order perturbation theory then gives

$$E_{\phi\phi^\dagger} = 2\Delta_0 + k^2 \langle\phi\phi^\dagger|H_\text{eff}|\phi\phi^\dagger\rangle + O(k^4),$$

with the anomalous dimension identified as

$$\gamma_{\phi\phi^\dagger} = \Delta_{\phi\phi^\dagger} - 2\Delta_0 = V_\text{bound},$$

where $H_\text{eff}$ encodes effective contact terms plus graviphoton and graviton exchanges, constructed by integrating out the relevant dynamical fields using their classical equations of motion.

The explicit formula is

$$\gamma_{\phi\phi^\dagger} = \int d^4x\,\sqrt{-g}\, \langle\phi(x)\phi^\dagger(x) | V_\text{eff}[\phi,\phi^\dagger] | \phi(x)\phi^\dagger(x) \rangle,$$

with $V_\text{eff}$ a sum of quartic contact interactions, photon-exchange, and graviton-exchange contributions. For ground state wavefunctions (e.g., $$\phi_0(x) = \mathcal{N}_0 (e^{it}\cos p)^{\Delta_0}$$), the integral is tractable.

Key points and implications:

• This method provides direct physical intuition: the anomalous dimension is the AdS binding energy.
• For chiral double-trace operators, supersymmetry ensures the anomalous dimension must vanish; this yields a nontrivial constraint among interaction couplings.
• With appropriate couplings, non-chiral double-trace operators can acquire either positive or negative anomalous dimensions, depending on details of the AdS model and bulk parameters—an important distinction for phenomenological applications.

3. Comparison to Four-point Function/OPE Methods

In standard CFT approaches, anomalous dimensions are extracted from the structure of four-point functions: $$F_{O_1O_2}(u,v) = \sum_O C_{O_1O_2O} \, G(\Delta_O,\ell_O;u,v),$$ where logarithmic terms ($\log u$) in channel expansions encode anomalous dimensions. This path requires computation of Witten diagrams (contact and exchange) in AdS and careful expansion in conformal blocks, a technically demanding enterprise.

The Hamiltonian approach (Fitzpatrick et al., 2011) circumvents this complexity, sidestepping explicit four-point function calculations and integral manipulations in curved backgrounds. The key advantage is the reduction of the problem to quantum-mechanical perturbation theory, with the spectrum and wavefunctions in global AdS coordinates used directly.

4. Sign Structure and Applications of Anomalous Dimensions

A novel outcome is that the sign of $\gamma_{\phi\phi^\dagger}$ is not fixed: it may be positive or negative depending on the competition between different interaction contributions and the value of bulk parameters (e.g., the parameter $A$ characterizing the model). For large $A$, graviton and photon exchanges yield attractive forces (negative binding energy), while for smaller $A$, curvature effects can reverse this, yielding positive anomalous dimensions.

Physical consequences:

• Positive $\gamma_{\phi\phi^\dagger}$: Operator dimensions increase under renormalization group flow. This may be leveraged in supersymmetric model building—for instance, to suppress dangerous higher-dimensional (SUSY-breaking) operators, facilitating sequestering.
• Negative $\gamma_{\phi\phi^\dagger}$: Common in theories where operator mixing leads to attractive binding, relevant for double-scaling limits and nonplanar corrections in, e.g., $\mathcal{N}=4$ SYM at finite $N$ (Kimura et al., 2015).
• The Hamiltonian formalism shows both scenarios are consistent with unitarity and crossing symmetry, though only models with proper UV completions (e.g., consistent supergravity or string embeddings) represent true SCFTs.

5. Explicit Formulas and Calculation Steps

The core algorithm consists of:

1. Specifying the spectrum and wavefunctions for the free theory in AdS:

$$E_{n,\ell} = \Delta_0 + 2n + \ell, \quad m^2 = \Delta_0(\Delta_0-4).$$

1. Constructing the two-particle state $|\phi\phi^\dagger\rangle$, with energy $2\Delta_0$ at leading order.
2. Expanding the interaction Hamiltonian and integrating out intermediate fields (e.g., photon and graviton) to construct $H_\text{eff}$.
3. Using first-order perturbation theory to compute the energy shift:

$$\gamma_{\phi\phi^\dagger} = \langle \phi\phi^\dagger | V_\text{eff} | \phi\phi^\dagger \rangle = \int d^4x\,\sqrt{-g}\, V_\text{eff}[\phi, \phi^\dagger].$$

1. Verifying vanishing anomalous dimensions for chiral operators as a supersymmetric consistency check.

The method generalizes to higher-point states and spinning operators by suitable adaptation of the state construction.

6. Implications for Model Building and UV Completion

The ability to engineer both positive and negative anomalous dimensions for double-trace operators is of interest in constructing superconformal field theories with desirable phenomenological properties, such as suppression of higher-dimensional operators to ensure stability against SUSY-breaking deformations.

A crucial open issue is UV completion: while effective AdS$_5$ supergravity models can realize these properties at low energy, full string embeddings (e.g., IIB on Sasaki–Einstein manifolds) are typically required to guarantee the existence of non-chiral operators with positive anomalous dimensions within a fully consistent SCFT framework.

7. Summary Table: Key Aspects of the Hamiltonian Approach

Step	Description	Output
State construction	$|\phi\phi^\dagger\rangle$ via AdS modes	Free energy $2\Delta_0$
Hamiltonian expansion	$$H = H_\text{free} + k H_\text{exch} + k^2 H_\text{cont}$$	Perturbative structure
Effective Hamiltonian	Integrate out dynamic fields (photon, graviton)	$H_\text{eff}$ at $O(k^2)$
Perturbation theory	$$\Delta E = \langle \text{state}| H_\text{eff} |\text{state}\rangle$$	Anomalous dimension $\gamma$
Consistency for chiral states	$V_\text{eff} \to 0$ by SUSY	$\gamma=0$, operator protected
Sign of binding energy	Controlled by model details (bulk parameters, couplings)	$\gamma$ positive or negative

• Hamiltonian approach and positive anomalous dimensions in effective AdS$_5$ SCFTs: (Fitzpatrick et al., 2011)
• Nonplanar, large $N$ analysis and negative anomalous dimensions: (Kimura et al., 2015)
• Bootstrap and holography for universal anomalous dimensions: (Kaviraj et al., 2015)
• Operator mixing, unitarity, and spectrum degeneracies: (Aprile et al., 2018, Kulaxizi et al., 2018, Shyani, 2019)

Conclusion

The study of anomalous dimensions of double-trace operators has evolved towards powerful and physically transparent methodologies—most notably the Hamiltonian approach, which links anomalous dimensions directly to AdS binding energies. This shift not only simplifies practical computations but deepens insight into RG flows, operator mixing, and UV completion. The sign and magnitude of anomalous dimensions serve as diagnostic tools for the structure and stability of strongly coupled CFTs and their holographic duals, with direct applications to model building and the analysis of supersymmetry breaking mechanisms.