The massless sine-Gordon model with logarithmic correlations in arbitrary dimension

Abstract: In this article, we study the sine-Gordon model with logarithmic correlations in arbitrary dimension $d \ge 1$. The model is defined as a Euclidean field theory whose interaction term in the classical action is [ S_{\mathrm{int}}(\varphi) := 2z \int_Λ \cos(\sqrtβ\,\varphi)\,\dx, ] where $\varphi$ is a logarithmically correlated Gaussian field on a compact domain $Λ\subset \mathbb{R}^d$ that coincides with the Gaussian free field when $d=2$, and where $z \in \mathbb{R}$ and $β>0$. We treat both the massive and massless cases, with the main emphasis on the massless regime. We construct the field without imposing boundary conditions, together with its charge and gradient correlation functions, and we show that the partition function is renormalizable. The results are non-perturbative, holding for all $z \in \mathbb{R}$. If $d=1$, our results are valid for the full subcritical range $β\in (0,4π)$ and if $d\geq 2$, the results are valid for $β\in (0,(d+1)2π)$. Our analysis is carried out directly in the continuum: we perform renormalization via a scale decomposition and control the partition function by controlling the renormalized potential that arises in this procedure, using the iterated Mayer expansion of Brydges and Kennedy~\cite{BrKe87a}.

• The paper rigorously constructs the massless sine-Gordon model using nonperturbative methods and an iterated Mayer expansion.
• It establishes sharp renormalizability criteria by precisely controlling charge cumulants and collapse thresholds across different dimensions.
• The work provides analytic control over both massive and massless limits, paving the way for applications in quantum field theory and statistical mechanics.

Detailed Analysis of the Massless Sine-Gordon Model With Logarithmic Correlations in Arbitrary Dimension

Introduction and Motivation

The paper "The massless sine-Gordon model with logarithmic correlations in arbitrary dimension" (2606.06657) presents a rigorous nonperturbative construction and analysis of the sine-Gordon (sG) model defined via a logarithmically correlated Gaussian field as reference, in arbitrary dimension $d \geq 1$. This model is of particular significance as it merges paradigms from both Euclidean quantum field theory (QFT) and statistical mechanics, in particular the study of Coulomb/log-gases and their associated critical phenomena. The focus of the work is on the massless field theory, formulated without boundary conditions, and the main innovation is the generalization of techniques to higher dimensions, producing sharp dimensional thresholds for renormalizability and analyticity of the correlation structures.

Model Definition and Scope

The model is defined by the formal measure:

$$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$

where $z \in \mathbb{R}$ and $\beta > 0$ are coupling and inverse-temperature parameters, $\Lambda \subset \mathbb{R}^d$ is compact, and $\varphi$ is a centered Gaussian field with logarithmic covariance. In $d=2$, this is the usual Gaussian free field (GFF), but in $d \geq 3$, a fractional Laplacian structure is imposed to retain logarithmic correlations; specifically, the covariance is generated by the Green’s function of $(-\Delta + m^2)^{d/2}$, yielding a kernel with a genuine logarithmic singularity in the massless limit.

The construction addresses both the massive and massless field cases. The main technical results are established directly in the continuum (as opposed to lattice approximations), and crucially, the field is constructed without specifying boundary conditions.

The analysis is valid for all $z \in \mathbb{R}$ nonperturbatively, for $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$0 in a maximal 'subcritical' range that varies sharply with dimension: for $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$1, $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$2, and for $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$3, $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$4. These regimes mark the precise bounds where nontrivial continuum constructions are possible using the authors’ methods and are shown to be optimal (i.e., beyond these, the model ceases to be renormalizable, see below).

Main Technical Results

Renormalizability and Analyticity

A hierarchical scale-decomposition of the field is performed, along the lines of the renormalization group (RG), with renormalization via explicit counterterms constructed recursively via an iterated Mayer expansion (following the approach of Brydges and Kennedy). The key theorems are:

• Renormalizability of Partition Function: For the specified $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$5 and $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$6, after implementing a (finite) number of multiplicative counterterms (which grow with $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$7), the partition function has a finite nontrivial continuum and massless limit. The entire partition function is analytic in $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$8 (even function, entire) and positive for real $$Z^{-1} \exp\left(2z \int_\Lambda \cos(\sqrt{\beta}\,\varphi)\right) \nu(d\varphi),$$9.
• Correlation Functions: Mixed charge and gradient (derivative) truncated correlation functions (in both massive and massless limits) exist and are analytic in $z \in \mathbb{R}$0. The allowed number of charge insertions in the correlation functions is precisely controlled by the renormalization structure, and for sufficiently separated supports, two-point and higher-point functions are locally integrable up to the dimensionally dependent collapse threshold.
• Field Construction: The continuum Euclidean probability law for the field is constructed as a measure on suitable function/distribution spaces, with the partition function acting as the characteristic function. For test functions with zero total charge, the model admits a canonical construction as a probability measure modulo constants (on $z \in \mathbb{R}$1 in test function space).

Dimensional Scaling, Collapse Points, and Renormalization Structure

The dimensional bounds for $z \in \mathbb{R}$2 arise naturally from the analytic structure of the charge cumulants and the RG flow. The paper demonstrates that in dimension $z \in \mathbb{R}$3, the first collapse (divergence) point for the charge cumulant scaling—i.e., where two-point functions become nonintegrable—emerges already at $z \in \mathbb{R}$4, lower than the corresponding multiple-collapse points conjectured for log-gases. The number of counterterms and the recursive structure of the expansions is dictated by the collapse structure, matching the known super-renormalizable/renormalizable/trivial RG regimes for the two-dimensional model.

Key Technical Innovations

• Uniform Control in Arbitrary Dimension: Sharp combinatorial and analytic estimates are established for the iterated Mayer expansions and the associated (scale-dependent) cumulant functions, allowing for recursive bounds that propagate through the induction regardless of dimension.
• Onsager-type Inequality: A version of the Onsager inequality (to control the divergence of Coulomb gas energies for neutral charge configurations) is established and forms a backbone for the uniformity of the high-dimensional construction.
• Analytic Control Beyond $z \in \mathbb{R}$5: The approach bypasses the use of functional Polchinski RG equations and instead relies on explicit expansions and inductive estimates. This enables a more robust and transparent argument, and, importantly, one that generalizes seamlessly from $z \in \mathbb{R}$6 and $z \in \mathbb{R}$7 to arbitrary $z \in \mathbb{R}$8.

Numerical and Analytic Bounds

A variety of explicit bounds are derived for the scale-dependent charge cumulants and their derivatives with respect to the mass. These yield:

• Uniform (in UV cutoff $z \in \mathbb{R}$9 and mass $\beta > 0$0) integrability properties for all terms in the renormalized partition function and correlation functions.
• Explicit geometric convergence rates for the Mayer expansion in terms of parameters such as $\beta > 0$1, $\beta > 0$2 (scale parameter), and domain size.
• Uniform existence of limits for correlation functions in the full subcritical domain, with explicit analytic continuation domains in the coupling $\beta > 0$3.

In particular, the authors demonstrate that the necessity of additional counterterms arises precisely as $\beta > 0$4 approaches and crosses integer multiples of $\beta > 0$5, with each new 'collapse point' necessitating a further layer in the expansion, but with a strict upper bound at $\beta > 0$6 for integrability.

Contrasts With Previous Work

Whereas prior rigorous constructions (e.g., [BaWe24a], [LaRhVa23a], [Fr76a]) were restricted to lower dimensions or to small coupling, this paper achieves:

• Full nonperturbative construction for all $\beta > 0$7 in any dimension $\beta > 0$8 up to the sharp subcritical $\beta > 0$9-bound,
• Uniform analytic control over both massive and massless field limits,
• Canonical construction of mixed derivative and charge correlation functions,
• Identification and proof (in arbitrary $\Lambda \subset \mathbb{R}^d$0) that the first RG transition/collapse point scales linearly with $\Lambda \subset \mathbb{R}^d$1 (i.e., as $\Lambda \subset \mathbb{R}^d$2).

The approach does not rely on boundary conditions and is not limited to lattice regularizations.

Implications and Outlook

Theoretical Implications

• Universality and Dimensional Thresholds: The sharp scaling of divergence points with dimension reinforces universality conjectures for log-gas and sG models, and the continuity with Coulomb/Yukawa gas phase transitions.
• Bosonization and Dualities: The construction preserves functional structures necessary for bosonization correspondences, notably the celebrated Coleman duality at special $\Lambda \subset \mathbb{R}^d$3.
• Tools for Statistical and Field-Theoretic Physics: The flexibility to handle arbitrary $\Lambda \subset \mathbb{R}^d$4 may open the way for rigorous study of high-dimensional disordered systems, log-gases, and related models of 2D turbulence and Liouville quantum gravity.

Practical Implications

• Modeling of Log-Gas Dynamics: The results provide a direct Euclidean field-theoretic framework for investigating critical log-gas behavior in any dimension, relevant for problems in statistical mechanics, plasma physics, and complex systems.
• Numerical Simulations and Scaling Prediction: Explicit bounds and criteria for integrability at each step facilitate numerical approximations and simulations in complex geometries.

Further Developments

The general methodology appears suitably robust to be extended to the full range $\Lambda \subset \mathbb{R}^d$5 in $\Lambda \subset \mathbb{R}^d$6, provided refined estimates for cumulant bounds at the Kosterlitz-Thouless point are available. The authors suggest that further combinatorial and analytic advances (in particular, handling cancellations in tree expansions) could allow treatment of higher critical regimes and possibly rigorous construction of models obeying full conformal invariance, with direct implications for integrable field theories, Liouville CFT, and complex/imaginary chaos.

The formalism paves the way for further studies in the continuum limit of lattice field theories, universality in random geometry, and deep connections with integrable probability and random matrix theory.

Conclusion

This work provides a mathematically rigorous and nonperturbative foundation for the massless sine-Gordon model with logarithmic correlations in any dimension. Through explicit scale decomposition, recursive renormalization, and detailed analytic control, the paper establishes the existence, renormalizability, and analyticity of both the partition function and truncated correlation functions up to the sharp collapse thresholds determined by the dimension. The results both close previous gaps in the constructive theory of sine-Gordon-type models and open pathways to new developments in both rigorous quantum field theory and statistical mechanics.