Finite-Coupling Model for Wilson Coefficients

• The paper introduces finite-coupling models that compute Wilson coefficients as nonperturbative functions of the coupling constant, bypassing traditional perturbative series.
• It leverages resummation techniques, integrability, and lattice matching to accurately evaluate operator product expansions and manage renormalon ambiguities.
• Applications include planar N=4 SYM amplitudes, weak effective Hamiltonians, and precision flavor physics, thereby enhancing UV/IR matching and phenomenological predictions.

A finite-coupling model for Wilson coefficients refers to theoretical approaches and explicit constructions in which the Wilson coefficients appearing in operator product expansions (OPEs) or effective actions are determined as nonperturbative functions of the coupling constants, rather than as power series truncations in the weak or strong coupling limits. This concept is essential in a variety of settings, including the resummation and duality structure of amplitudes in planar $\mathcal{N}=4$ super Yang-Mills (SYM), the interpretation of heavy flavor physics in the Standard Model, and the development of quantitative connections between ultraviolet (UV) and infrared (IR) physics in both perturbative and nonperturbative frameworks. Finite-coupling models for Wilson coefficients provide a bridge between integrability, resummation, bootstrap methods, and modern techniques in lattice gauge theory and effective field theory matching.

1. Definition and General Framework

A finite-coupling model for Wilson coefficients is an explicit, nonperturbative construction of the coefficients of local operators in OPEs or effective actions as analytic (or exactly resummed) functions of the coupling constant. Unlike standard weak- or strong-coupling expansions, such a model is valid for all values of the coupling and thus interpolates smoothly between the asymptotic regimes.

In practical terms, given a correlator or amplitude $\mathcal{A}(x)$, its short-distance expansion in the OPE reads

$$\mathcal{A}(x) \sim \sum_k C_k(x,\mu,g)\,\mathcal{O}_k(\mu),$$

where $C_k$ are Wilson coefficients, $\mathcal{O}_k$ are local operators, $\mu$ is the renormalization scale, and $g$ is the coupling. In a finite-coupling model, the $C_k$ are determined by nonperturbative methods, integrability, or via direct resummation, rather than by strict expansion in powers of $g$.

This framework is realized in several domains:

• Factorized OPEs for Wilson loops and scattering amplitudes in planar $\mathcal{N}=4$ SYM (Basso et al., 2015, Sever et al., 2021).
• Nonperturbative calculations in lattice field theory, where matching of Green’s functions at short versus long distances yields nonperturbative Wilson coefficients (Bruno et al., 2017, Tomii, 2019).
• Resummation and the cancellation of IR renormalon ambiguities by analytic methods, yielding cutoff-independent Wilson coefficients with well-defined power corrections (Mishima et al., 2016).
• Operator matching in the context of weak effective Hamiltonians, with RG evolution performed exactly or to highest known order (Boer et al., 2016).

2. Finite-Coupling OPE and Resummation in $\mathcal{N}=4$ SYM

A paradigmatic example arises in planar $\mathcal{N}=4$ SYM theory, where the operator product expansion for null polygonal Wilson loops (and via duality for scattering amplitudes) admits a manifestly nonperturbative, factorized representation.

The pentagon OPE (POPE) formulates the amplitude (or renormalized Wilson loop) as a sum over $n$-particle states (flux-tube excitations), with each term taking the form:

$$\mathcal{W} = \sum_{n} \frac{1}{S_n} \int \prod_i du_i\ \Pi_\mathrm{dyn} \cdot \Pi_\mathrm{FF} \cdot \Pi_\mathrm{mat},$$

where:

• $$\Pi_\mathrm{dyn} = \prod_i \mu(u_i) \exp[ - E(u_i)\tau + i p(u_i)\sigma + i m_i \phi ] \prod_{i<j} 1/|P(u_i|u_j)|^2$$ encodes the dynamical, kinematical, and dressing factor contributions.
• $\Pi_\mathrm{FF}$ is a form factor factor (encoding helicity and R-charge flow).
• $\Pi_\mathrm{mat}$ encodes the SU(4) R-symmetry structure via auxiliary rapidity integrals.

Here, the parameters $E(u), p(u), m_i, \mu(u)$, and $P(u|v)$ are known exactly in terms of the coupling $g=\sqrt{\lambda}/(4\pi)$, providing an all-$g$ result for each term (Basso et al., 2015). This POPE construction defines an all-coupling ("finite-coupling") model for the Wilson coefficients in the OPE expansion of Wilson loops. These exact expressions encode the scaling, mixing, and operator dressing for each sector and are valid in the full nonperturbative regime.

The collinear limit (large $\tau$) further isolates the dominant OPE contributions: higher-energy excitations are exponentially suppressed, exposing the leading Wilson coefficient contributions and enabling direct comparison to both perturbative calculations and integrability-based strong coupling expansions.

3. Nonperturbative and Lattice Approaches

Finite-coupling models are also realized via nonperturbative lattice matching, where the Wilson coefficients are extracted by matching matrix elements or Green’s functions computed in the full theory and in the effective theory at a common scale.

In weak effective Hamiltonian contexts, Green's functions with inserted operators are computed both in the full theory (including all dynamical fields, e.g., heavy $W$ bosons) and the corresponding effective theory. Imposing matching conditions at a defined scale yields nonperturbatively determined Wilson coefficients that include all-orders effects in the strong coupling:

$(G_F/\sqrt{2}) C_i(\mu) M_{ij}(\mu) = W_j(\mu)$

where $M_{ij}(\mu)$ is the matrix of projected Green’s functions in the EFT and $W_j(\mu)$ in the full theory (Bruno et al., 2017). Renormalization and mixing matrices are determined numerically or semi-analytically, and systematic errors are controlled by volume, momentum, and mass extrapolations. This approach provides a methodology to construct finite-coupling Wilson coefficients (i.e., those resumming all nonperturbative effects) for weak decays or hadron transitions.

A complementary position-space approach matches Wilson coefficients between three- and four-flavor theories through long-distance ratios of lattice two-point functions, with spherical averaging used to restore discretization symmetry and enable continuum extrapolation (Tomii, 2019).

4. Finite-Coupling Models from Analytic Resummation and Renormalon Analysis

Wilson coefficients are also subject to ambiguities from infrared renormalons in perturbation theory, which manifest as uncertainties of order $\Lambda_\mathrm{QCD}^n/Q^{2n}$ in truncated series. A finite-coupling definition of the Wilson coefficient subtracts (or bypasses) such ambiguities by constructing the cutoff-independent ultraviolet part, $X_\mathrm{UV}$, via analytic methods (e.g., contour deformations in the Borel plane) (Mishima et al., 2016).

For a physical observable $X(Q^2)$ with a perturbative integral representation, one writes:

$$X_\mathrm{UV}(Q^2) = X_0(Q^2) + \sum_{0<n<u_{\mathrm{IR}}} \frac{4\pi c_n}{\beta_0} \left( e^{5/3} \Lambda_\mathrm{QCD}^2/Q^2 \right)^n,$$

with $X_0(Q^2)$ an explicit integral along a contour avoiding IR renormalon poles. The power corrections identified here are true UV-induced corrections (as shown via integration-by-regions), independent of ambiguity, and systematically included in the finite-coupling Wilson coefficient. Scheme dependence is controlled by analyticity properties, and the "massive gluon" scheme is uniquely optimal in this context.

This approach, when applied to observables such as the Adler function or static QCD potential, yields Wilson coefficients that are both stable and analytic across the physical range of $Q^2$, enabling robust matching between OPE and nonperturbative lattice or phenomenological results.

5. Relation to Amplitudes, Operator Duality, and Resummation

Finite-coupling Wilson coefficients are central in settings where observables can be re-expressed as sums over composite excitations, and their OPE structure is governed by integrability, bootstrap, or duality. In $\mathcal{N}=4$ SYM, the amplitude/Wilson loop duality allows for resummation of planar scattering amplitudes via the same POPE expansion as Wilson loops.

For instance, the exponential factors $\exp[-E(u)\tau + i p(u)\sigma + i m\phi]$ directly encode the scaling expected from the OPE, while measure and dressing factors provide the analog of Wilson coefficient dressing at finite coupling (Basso et al., 2015).

The universality of critical structures, such as the cusp anomalous dimension, is established at strong coupling via AdS/CFT, where the cusp contribution to the regularized minimal area is

$$A_\epsilon = \frac{l}{\epsilon} + \sum_i \Gamma_\mathrm{cusp}(\theta_i) \log \epsilon + A_\mathrm{ren},$$

with $\Gamma_\mathrm{cusp}(\theta_i)$ depending only on geometric data (cusp angle), showing that the finite-coupling Wilson coefficient in this nonperturbative regime is universal and independent of local curvature (Dorn, 2015).

6. Phenomenological and Lattice Applications

A robust finite-coupling model for Wilson coefficients underpins precision flavor physics, deep-inelastic scattering, and SM/BSM matching:

• Weak Hamiltonian coefficients for $c\to u \ell^+\ell^-$ or $b\to s$ transitions incorporate two-step RG running and matching matrices with all leading and subleading logs, producing Wilson coefficients that robustly interpolate between high ($M_W$) and low ($m_c$ or $m_b$) scales (Boer et al., 2016).
• In flavor physics, optimized angular observables in rare $B$ decays allow form-factor-independent determination of Wilson coefficients directly from data, effectively isolating finite-coupling effects due to the independence of kinematically defined crossing points (Kindra et al., 2017).
• Lattice studies make possible a nonperturbative matching of Wilson coefficients in effective theories by measuring ratios of two-point functions and employing spherical averaging to control discretization uncertainties; this improves accuracy in predicting observables such as $K\to\pi\pi$ transitions near charm threshold (Tomii, 2019).

7. Outlook and Extensions

Finite-coupling models for Wilson coefficients continue to be an active area of research, with developments in

• Integrable bootstrap and all-coupling OPE programs for amplitudes, correlators, and form factors in 4D gauge theories (Sever et al., 2021).
• Nonperturbative and lattice approaches to UV/IR matching in effective field theory, including new methods for renormalon subtraction, and robust numerical schemes for extracting improvement and normalization coefficients in lattice QCD (Divitiis et al., 2019).
• Automated computation of Wilson coefficients in BSM scenarios via symbolic calculation packages, ensuring all finite-coupling and gauge dependencies are captured for high-precision phenomenology (Bakshi et al., 2018, Uhlrich, 2022).
• Theoretical advances in analytic resummation, scheme optimization, and the exploration of universality properties (as with the cusp anomalous dimension) for constructing interpolating models between weak and strong coupling.

In all these contexts, the finite-coupling model for Wilson coefficients serves as a critical theoretical bridge, capturing the all-order, nonperturbative imprint of UV dynamics on IR observables and enabling more reliable confrontations with experimental data and nonperturbative numerical simulations.