Nonperturbative Two-Point Correlators

• The paper presents an exact construction of two-point correlators using nonperturbative methods such as the form‐factor bootstrap and spectral representations.
• It demonstrates how the spectral decomposition of the fractional Laplacian reproduces RG-improved logarithmic scaling and validates asymptotic freedom at short distances.
• The study bridges the gap between ultraviolet power-law and logarithmic behavior and infrared exponential decay associated with the mass gap.

Nonperturbative completion of two-point correlators refers to the construction of exact, fully nonperturbative expressions for two-point Green’s functions in quantum field theory, encompassing all orders of perturbation theory and including genuinely nonperturbative effects such as strong-coupling dynamics, mass gaps, operator mixing, and resurgent or topological phenomena. In integrable models and certain gauge theories, these completions are achieved using specialized methods such as the form-factor bootstrap, functional techniques, Borel–Écalle resummation, lattice computations, and large‑N expansions. Nonperturbative completion is essential for validating the structure of quantum field theories, verifying asymptotic properties (e.g., asymptotic freedom), extracting physical observables beyond perturbation theory, and providing stringent benchmarks for any prospective definition of the full quantum theory.

1. Nonperturbative Form-Factor Bootstrap and Spectral Representation

In the $(1+1)$-dimensional $\mathrm{SU}(N) \times \mathrm{SU}(N)$ principal chiral sigma model (PCSM), nonperturbative completion of two-point correlators is realized by the form-factor bootstrap, exploiting integrability and the exact S-matrix (Orland, 2014). The method constructs the Euclidean two-point function

$$G(|x|,A) = N^{-1} \langle 0 | \operatorname{Tr}(\Phi(0)^{\dagger} \Phi(x)) | 0 \rangle,$$

for the multiplicatively renormalized field $\Phi(x)$ related to the bare field $U(x) \in \mathrm{SU}(N)$ by $\Phi(x) \sim Z^{-1/2} U(x)$.

The bootstrap approach determines matrix elements (“form factors”) of local operators between asymptotic states by solving Smirnov’s axioms using the known S-matrix. For the PCSM at large $N$, this enables the summation over all multi-particle intermediate states, yielding an explicit spectral representation of the correlator. This approach bypasses perturbative expansions and provides access to strong-coupling and mass-gap phenomena.

The correlator is represented in a compact form controlled by the spectrum of a fractional Laplacian operator: $$G(mR) = \text{const} \int_{-1}^1 du_1 \cdots \int_{-1}^1 du_{2l+1} \prod_{j=1}^{2l} \frac{L}{(u_j-u_{j+1})^2 + (m/L)^2},\quad L = \log(1/(mR)),$$ where the integration variables $u$ parametrize the spectral decomposition and the expression relates directly to the eigenvalues $A_n$ of $\Delta^{1/2} = \sqrt{-d^2/du^2}$ with Dirichlet boundary conditions.

2. Short-Distance Asymptotics and Asymptotic Freedom

A key achievement of the nonperturbative completion is the controlled ultraviolet asymptotics (short-distance behavior). In the PCSM, the two-point correlator exhibits the leading form

$$N^{-1} \langle \operatorname{Tr} (\Phi(0)^{\dagger}\Phi(x)) \rangle \simeq C_2 [\ln(1/(m|x|))]^2 + C_1 \ln(1/(m|x|)) + \text{const} + \cdots,$$

where $C_2$ and $C_1$ are computable in terms of the spectrum of $\Delta^{1/2}$, specifically $C_2 \propto \sum 1/A_n$. This leading behavior matches the prediction from perturbative renormalization group (RG) analysis in an asymptotically free model, confirming at the fully nonperturbative level the expected scaling properties.

The nonperturbative construction directly embodies RG-improved anomalous dimension scaling: $G(R, A) \sim C [\ln(RA)]^{1/B_1},$ with $B_1$ the one-loop coefficient, reproducing perturbative results in the ultraviolet and showing that the theory becomes effectively free at very short distances. The agreement reinforces the validity and completeness of the bootstrap method in encapsulating asymptotic freedom.

3. Spectral Approach: Fractional Laplacian and Poisson Kernel

The spectral representation of the correlator links its structure to the eigenvalues and eigenfunctions of the nonlocal operator $\Delta^{1/2} = \sqrt{-d^2/du^2}$, acting on $u \in(-1,1)$ with Dirichlet conditions. The expansion: $$Q_n(u):\ \Delta^{1/2} Q_n(u) = A_n Q_n(u),\quad Q_n(\pm1)=0,$$ determines the contribution of each “mode” to the short-distance expansion. The Poisson kernel formalism—expressed as

$$\langle u'|P(a)|u\rangle = \frac{a}{[(u'-u)^2 + a^2]^{3/2}},\quad P(a) = \exp(-a \Delta^{1/2}),$$

provides an explicit link between the Green’s function of the fractional Laplacian and the multi-point integration appearing in the correlator representation. This reveals the analytic structure (including nonperturbative contributions and decay rates) as controlled by spectral data rather than by Feynman diagrams or local field equations.

4. Comparison to Perturbative Results and Renormalization Group

The nonperturbative completion yields expressions that, when expanded in the appropriate limit (e.g., short distances), precisely reproduce the perturbative RG expectations. This matching is not limited to leading logarithms but persists at the subleading level, establishing the method as both exact and practically consistent with standard field-theoretic analyses.

One key result is that the summation over multi-particle contributions and the spectral expansion, which are inherently nonperturbative, yield the same logarithmic behavior (e.g., $[\ln(1/(m|x|))]^2$ at large $N$) as calculated diagrammatically by summing leading logarithms in perturbation theory. The matching strongly supports the idea that the nonperturbative solution incorporates all RG-improved perturbative data while also yielding finite, strong-coupling effects.

5. Mass Gap and Correlation Length

The large-distance (infrared) behavior is equally accessible in the nonperturbative formulation. The correlator decays exponentially at scales $|x| \gg 1/m$, where $m$ is the dynamically generated mass gap, a phenomenon that cannot be captured within any finite order of perturbation theory. The transition from power-law plus logarithmic scaling in the ultraviolet to exponential suppression in the infrared is realized in the spectral decomposition, with the correlation length set by the spectral gap of the fractional Laplacian and the physical mass gap of the sigma model.

6. Broader Implications and Structure

The nonperturbative completion paradigm—combining integrability, spectral theory of nonlocal operators, and exact resummation—provides a model for analyzing asymptotic freedom, operator mixing, and dynamical mass generation in other two-dimensional and higher-dimensional asymptotically free models. The interplay between functional analytic and operator-theoretic methods (e.g., spectral sums, Poisson kernels) and field-theoretic physics points towards deeper connections, such as the encoding of quantum dynamics in operator spectra and the use of nonlocal operators to characterize scaling regimes.

This formulation exemplifies nonperturbative completion: it yields a two-point function that unifies both perturbative RG-improved scaling at short distance and strong-coupling mass gap physics in the infrared, in exact analytic form. The resulting two-point correlator serves as a benchmark for other (non-integrable) asymptotically free theories and as a template for studying the full nonperturbative content of gauge and sigma models beyond the reach of perturbative methods (Orland, 2014).