OPE Coefficients in QFT/CFT

• OPE coefficients are fundamental distributions that describe the short-distance behavior of local operator products in quantum and conformal field theories.
• They are computed using recursive flow equations and renormalization techniques, ensuring UV-finite and accurate results across perturbative and nonperturbative models.
• Their algebraic structure underpins bootstrap constraints, selection rules, and geometric formulations, impacting high-energy, condensed matter, and celestial holographic theories.

Operator Product Expansion (OPE) coefficients are the central data that control the short-distance (or collinear, or coincident-operator) behavior of quantum fields in quantum field theory (QFT) and conformal field theory (CFT). These coefficients determine how products of local operators decompose into sums over other operators, anchoring the algebraic structure underlying both general QFTs and special classes such as critical statistical models, gauge theories, and celestial holographic dualities. Their precise determination, classification, and interrelations are now foundational to computational and conceptual advances across high-energy theory, condensed matter, and mathematical physics.

1. Definition and Theoretical Framework

The operator product expansion asserts that products of local operators possess an asymptotic expansion in the limit where their insertion points approach coincidence. For operators $\mathcal{O}_i(x)$ and $\mathcal{O}_j(y)$, the OPE takes the form

$$\mathcal{O}_i(x)\,\mathcal{O}_j(y)\sim\sum_k C_{ij}^{\;\;k}(x,y)\,\mathcal{O}_k(y)$$

where the symbol “$\sim$” denotes an asymptotic expansion as $|x-y|\to0$. The coefficient $C_{ij}^{\;k}(x,y)$ is a distribution encoding all singularities and finite behavior as $x\to y$, constrained by the field content, spacetime symmetry, and the renormalization prescription (Hollands et al., 2023).

For composite operators in curved spacetime, $C_{ij}^k(x,y;z)$ are locally and covariantly constructed (e.g., using geodesic distance and curvature tensors), and their scaling degree is bounded by the engineering and anomalous dimensions of $\mathcal{O}_k$. OPE coefficients provide well-defined, state-independent local data, in contrast to vacuum expectation values, which generally exhibit non-analyticity across theory parameters (0805.3419).

In conformal field theory, the OPE reduces to

$$\mathcal{O}_i(x)\,\mathcal{O}_j(0) = \sum_k f_{ij}{}^k\,\frac{\mathcal{O}_k(0)}{|x|^{\Delta_i+\Delta_j-\Delta_k}}$$

with $f_{ij}{}^k$ (structure constants) and all tensorial and differential structure determined by conformal invariance (Hollands et al., 2023).

2. Computation, Recursion, and Renormalization Flow

OPE coefficients can be constructed recursively via flow equations or “action principles,” both in flat and curved spaces, and for interacting as well as free theories. For perturbatively renormalizable theories, such as $\varphi^4$ in $d=4$, the recursion for the $g$-derivative of any coefficient is of the form (Holland et al., 2014, Holland et al., 2015, Fröb, 2020, Hollands, 2017): $$\partial_g\,\mathcal{C}_{A_1\cdots A_N}^B(x_1,\dots,x_N) = - \text{(vertex insertion)} + \text{(spectator subtractions)} + \text{(target subtractions)}$$ The explicit formulae, e.g.

$$\partial_g\,\mathcal{C}_{A_1\cdots A_N}^B = -\frac{1}{4!}\int d^4y\, \bigl[\mathcal{C}_{\varphi^4\,A_1\cdots A_N}^B - \sum \mathcal{C}_{\varphi^4\,A_i}^C\,\mathcal{C}_{A_1\dots C \dots}^B - \sum \mathcal{C}_{A_1\cdots A_N}^C\, \mathcal{C}_{\varphi^4\,C}^B \bigr]$$

yield, upon term-by-term order expansion, absolutely finite quantities requiring no additional UV counterterms (Holland et al., 2014). The approach generalizes to curved backgrounds, to arbitrary many composite insertions, and (with additional care) to gauge theories (Fröb et al., 2016).

Renormalization group (RG) flow for OPE coefficients is made manifest: under coupling deformations, the RG equations are differential equations for the coefficients and their associated field redefinitions and scaling dimensions (Hollands, 2017). In the conformally-invariant case, this reduces to: $$A_i\,\frac{d \Delta_i}{dg} = D_i\,\lambda_{V i i}\,, \quad A_i\,\frac{d\lambda_{ijk}}{dg} = \sum_m T_{ijk}^m \lambda_{V j m}\lambda_{i m k} + \text{cyc.}$$ where all constants are set by the (spinning) conformal block algebra.

3. Algebraic Structure: Associativity, Bootstrap, and Cohomology

The associativity (OPE "bootstrap") condition, which is a higher-dimensional analogue of the Jacobi identity, imposes nontrivial quadratic relations among OPE coefficients. Specifically, for three operators,

$$\mathcal{C}_{A_1A_2A_3}^B(x_1,x_2,x_3) = \sum_E \mathcal{C}_{A_1A_2}^E(x_1,x_2)\,\mathcal{C}_{EA_3}^B(x_2,x_3)$$

whenever $|x_1 - x_2| \ll |x_2 - x_3|$ (Holland et al., 2012, Holland et al., 2015). This recursive structure ensures that, in conjunction with convergence and analyticity (away from diagonal configurations), the entire set of $n$-point OPE coefficients is uniquely determined by the $2$- and $3$-point data ("coherence theorem" for vertex algebras) (Holland et al., 2015).

Classification of OPE deformations aligns with Hochschild-type cohomology of the underlying associative algebra; first-order deformations reside in cohomology $H^2$, with higher-order obstructions in $H^3$. The recursive associativity equations ensure that any allowed deformation propagates to the full $n$-point sector (Holland et al., 2015, Holland et al., 2012).

4. Concrete Computations and Applications

OPE coefficients are explicitly calculated in numerous contexts:

• Perturbative QFT: For $\varphi^4_4$, recursive formulas produce the standard one-loop and higher corrections for e.g., $$(\varphi\varphi)\sim \mathbf{1} + \varphi^2 + \varphi^3 + \ldots$$ with exact cancellation of divergences (Holland et al., 2014).
• Lattice Spin Systems and Integrable Models: In the XXX spin chain, coefficients extracted via fermionic bases display binomial and polynomial dependence with system size, reflecting underlying fusion rules. Explicit expansion coefficients are constructed for up to 11 lattice sites, suggesting general patterns for all $N$ (Francesco et al., 2017).
• Critical CFTs: In the 3D Ising model, fuzzy-sphere regularization enables extraction of OPE coefficients that agree with the conformal bootstrap and MC methods to within $\sim2\%$ (Hu et al., 2023).
• Nonperturbative FRG: The BMW approximation to the functional RG flow enables computation of OPE coefficients such as $c_{112}$ in O$(N)$ models in arbitrary $2 \leq d \leq 4$ at $\sim1$–$3\%$ accuracy compared to conformal bootstrap and MC, with full interpolation across the critical regime (Rose et al., 2021).
• Gauge Theories and QCD: In light-cone OPE for QCD, singlet and non-singlet Wilson coefficients are extracted in terms of composite operator matrix elements, enabling operator-based determination of mixing and anomalous dimensions (Kisselev, 2011).
• Nuclear EFT: Wilson coefficients in the OPE expansion for non-local nucleonic bilinears, matched to pionless EFT, systematize short-distance factorization of momentum distributions in nuclei (Yu et al., 31 Dec 2024).
• Celestial CFT and Gravity: The leading OPE coefficients for gluon and graviton primaries in the celestial basis are determined exactly by soft theorems and are expressed in terms of Euler beta functions. The soft limits correspond precisely to the insertion of conformally soft asymptotic-symmetry currents (Pate et al., 2019, Himwich et al., 14 May 2025).

Table: Example OPE Coefficient Results in the 3D Ising Model (Hu et al., 2023)

Triplet $(\phi_i,\phi_j,\phi_k)$	Spin	$f_{ijk}$ (Fuzzy)	$f_{ijk}$ (Bootstrap)
$(\sigma,\sigma,\epsilon)$	0	1.0539(18)	1.0519
$(\epsilon,\epsilon,\epsilon)$	0	1.5441(23)	1.5324
$(\sigma,\sigma,T)$	2	0.3248(35)	0.3261
$(\epsilon,\epsilon,T)$	2	0.8951(35)	0.8892

5. Extensions: Geometry, Topology, and Derived Coefficients

In QFTs on curved spacetime, OPE coefficients are locally and covariantly built from geometric invariants (e.g., Van Vleck determinant, Synge's world function), and their singular structure encodes the short-distance behavior and recursion with respect to local couplings, even when no global symmetries exist (0805.3419, Fröb, 2020).

The modern mathematical framework recognizes that OPE coefficients themselves are often not just scalar- or tensor-valued analytic functions, but elements of sheaf cohomology, especially in theories with holomorphic, topological, or mixed topological-holomorphic structure (e.g., in higher-dimensional chiral or "raviolo" theories) (Alfonsi et al., 7 Feb 2025). The precise ghost-number assignments and the cohomological degree of these "derived" coefficients are controlled by the global topology of the excised space $$\mathbb{R}^m\times\mathbb{C}^n\times\mathbb{R}^p\backslash\{0\}$$ and the structure sheaf, with ghost number equaling the cohomological degree.

6. Constraints, Selection Rules, and Vanishing OPE Coefficients

Symmetry constraints, both continuous (conformal, Lorentz, supersymmetry) and discrete, enforce selection rules on which OPE coefficients may be nonzero. In superconformal theories, character multiplication and index analysis can determine vanishing of OPE coefficients for certain short multiplet products, based on the absence of required representations in the OPE channel. In the $(A_1,A_{2n})$ Argyres-Douglas SCFT, for example, vanishing of a specific OPE coefficient is precisely linked to the central charge matching that of the AD theory (Agarwal et al., 2018).

In celestial CFTs, conformal soft theorems and shadow transforms further relate and sometimes constrain sets of OPE coefficients through integral identities and Beta function factors (Himwich et al., 14 May 2025).

7. Physical and Mathematical Implications

OPE coefficients encode the essential local observable data in quantum field theory. Their (associative) algebraic structure underlies vertex operator algebras in 2d, the conformal and non-conformal bootstrap, classification of critical points, and the universal features of both Lorentzian and Euclidean QFTs in arbitrary dimensions.

The recursive, flow-equation-based, and geometric methods developed in the last two decades have provided rigorous, UV-finite, and fully general tools for computing and classifying OPE coefficients at all orders in perturbation theory, as well as nonperturbatively in special settings. The interplay between their local, analytic properties and the global, nonperturbative data in vacuum or thermal states anchors their key role in the modern structural understanding of quantum field theory across disciplines.