The two-dimensional sine-Gordon model is the Euclidean quantum field theory formally defined by reweighting the Gaussian free field with the renormalized interaction λ∫cos⁡(βφ)\lambda\int\cos(\beta\varphi); it is expected to exist as a non-Gaussian measure exactly in the subcritical regime β2<8π\beta^{2}<8\pi. Despite four decades of constructive work, no construction covers the whole regime without restricting the parameters: rigorous renormalization-group methods reach β2<8π\beta^{2}<8\pi only for small coupling λ\lambda (Dimock–Hurd) or in finite volume (Pelaič), the stochastic-quantization route stays below the first threshold 4π4\pi (Barashkov; Gubinelli–Hairer–Oh–Zine), and the recent forward–backward SDE route of Gubinelli–Meyer, pushed by Meyer (2026) to β2<67 8π\beta^{2}<\frac{6}{7}\,8\pi with a multipole Mayer expansion, stops exactly where its effective-force estimates become superlinear in the field (the large-field problem). The open problem, stated in Meyer's 2026 paper, is the full construction — for all λ∈R\lambda\in\mathbb{R} and β2∈[0,8π)\beta^{2}\in[0,8\pi), in the infinite volume R2\mathbb{R}^{2}, in both the massive and massless cases — including, on the FBSDE route, global-in-time existence for the weak FBSDE and uniform control of the Girsanov density M0,∞ρM^{\rho}_{0,\infty} as the volume cutoff is removed. The source also lists, among possible future extensions rather than as part of this open problem, the systematic extension of the order-by-order renormalization-flow analysis to (sub)critical dd-dimensional settings with field-dependent renormalizations.

Background

In constructive Euclidean field theory one seeks a probability measure on fields φ∈S′(R2)\varphi\in\mathcal S'(\mathbb R^{2}) (tempered distributions) whose correlation functions are the Schwinger functions of a quantum field theory. The sine-Gordon model is formally the Gibbs measure obtained by reweighting the Gaussian free field (GFF) — the centered Gaussian field with covariance (−Δ+m2)−1(-\Delta+m^{2})^{-1} — by a cosine interaction:

νsGρ(dφ)  ‘‘=    Ξ−1exp⁡(−∫R2(λcos⁡(βφ(x))−cρ)ρ(x) dx) μ(dφ),\nu^{\rho}_{\mathrm{sG}}(\mathrm d\varphi)\;``=\;\;\Xi^{-1}\exp\Big(-\int_{\mathbb R^{2}}\big(\lambda\cos(\beta\varphi(x))-c^{\rho}\big)\rho(x)\,\mathrm dx\Big)\,\mu(\mathrm d\varphi),

where ρ∈Cc∞(R2)\rho\in C_c^\infty(\mathbb R^{2}) is a smooth volume cutoff, λ∈R\lambda\in\mathbb R is the coupling, and cρc^{\rho} abbreviates the counterterms: because the GFF is a distribution, cos⁡(βφ(x))\cos(\beta\varphi(x)) is ill-defined and must be renormalized (normal ordered), with further charge-type renormalizations required as β2\beta^{2} grows. The model is massive if m>0m>0 and massless if m=0m=0. Its reach is unusually wide: it is equivalent to the massive Thirring fermionic model, represents the two-dimensional Coulomb gas and dimer models (classical statistical-mechanical systems whose height variables it describes), and in the massless case borders on Liouville theory (the conformal field theory of random two-dimensional surfaces).

The parameter regime has a rigid structure. The cosine interaction is relevant exactly for β2<8π\beta^{2}<8\pi; for β2≥8π\beta^{2}\ge8\pi the renormalized partition function diverges (the supercritical regime, related to critical Gaussian multiplicative chaos, the critical case of the random measures obtained by exponentiating a log-correlated Gaussian field), so 8π8\pi is the natural completion boundary. Within [0,8π)[0,8\pi) lie infinitely many thresholds βℓ2=ℓ−1ℓ 8π\beta^{2}_{\ell}=\frac{\ell-1}{\ell}\,8\pi, ℓ∈N\ell\in\mathbb N — 4π4\pi, 16π/316\pi/3, 6π6\pi, …\ldots, 678π\frac{6}{7}8\pi, …\ldots — at each of which a higher harmonic cos⁡(ℓβφ)\cos(\ell\beta\varphi) becomes relevant and one more renormalization is needed. Below the first threshold 4π4\pi a single counterterm suffices; above it the theory is genuinely non-perturbative in the renormalization. For β2≥4π\beta^{2}\ge4\pi the resulting measure is mutually singular with respect to the GFF (Gubinelli–Meyer, 2024), so the object to be constructed is far from Gaussian. For the parameter ranges already constructed, structural results such as log-Sobolev inequalities (bounds on the relative entropy of the measure by its Dirichlet energy; Bauerschmidt–Bodineau, 2021) and tail bounds and couplings (Bauerschmidt–Hofstetter, 2022) are available, indicating what a complete construction should support.

No existing construction covers the full regime without restricting a parameter. (i) Renormalization-group route: Benfatto–Gallavotti–Nicoló (1982) and successors proved ultra-violet stability via bounds on the partition function in the 'regions of collapse'; Dimock–Hurd (2000) constructed the massive model for the full range β2<8π\beta^{2}<8\pi in infinite volume but only for small coupling λ≪1\lambda\ll1; Pelaič (2025) reached arbitrary λ\lambda in the full subcritical range but only in finite volume (also addressing an issue in the Dimock–Hurd analysis). (ii) Stochastic-quantization route: Barashkov (2022) and Gubinelli–Hairer–Oh–Zine (2025) constructed the measure below the first threshold β2<4π\beta^{2}<4\pi, including infinite volume; for the associated Langevin dynamics, local well-posedness holds in the full subcritical regime (Chandra–Hairer–Shen, 2018, after Hairer–Shen, 2016) but global well-posedness only up to 6π6\pi (Bringmann–Cao, 2026). (iii) Massless and higher-dimensional models: Vihko (2026) treats the massless log-correlated model in any dimension d≥1d\ge1 non-perturbatively for all coupling, but for β2<(d+1) 2π\beta^{2}<(d+1)\,2\pi when d≥2d\ge2 (for d=2d=2 this is β2<6π\beta^{2}<6\pi) and primarily at the level of charge and gradient correlation functions; Bauerschmidt–Mason–Webb (2025) treat the massless model at the free-fermion point β2=4π\beta^{2}=4\pi.

The source paper (Meyer, 2026) pushes the stochastic-control/FBSDE route — the line to which this question belongs — to the current frontier. The route interpolates the covariance, realizing the GFF as the terminal value of a Brownian martingale WtW_t with ⟨W⟩t=Gt\langle W\rangle_t=G_t for a scale interpolation GtG_t from 00 to (−Δ+m2)−1(-\Delta+m^{2})^{-1}, and represents the measure through the SDE Xt=∫0tG˙sFs(Xs) ds+WtX_t=\int_0^t\dot G_s F_s(X_s)\,\mathrm ds+W_t, where Ft=−∇VtF_t=-\nabla V_t is the force of a scale interpolation VtV_t of the renormalized interaction (an effective potential), together with a Girsanov-type density M0,∞ρM^{\rho}_{0,\infty} (a normalized stochastic exponential) equivalently characterized by a weak stochastic control problem, i.e. a weak forward–backward SDE (FBSDE — a system coupling a forward Itô diffusion with a backward equation carrying the terminal condition). Two requirements compete: the drift G˙sFs\dot G_sF_s must grow at most sublinearly in the field for the SDE to be solvable globally in scale time, and the interpolation must track the renormalization flow equation sufficiently accurately. Meyer's Theorem 1.1 constructs, in the full subcritical regime, effective potentials with integrable error — but with force bounds growing like (1+D)KF(β2)(1+D)^{K_F(\beta^{2})} in the field size DD, where KF(β2)→∞K_F(\beta^{2})\to\infty as β2→8π\beta^{2}\to8\pi. This superlinear field-dependence — the large-field problem — first bites at the seventh threshold β72=67 8π\beta^{2}_7=\frac{6}{7}\,8\pi, which is exactly where the resulting non-perturbative construction of the measure (Theorem 1.2, valid for all λ∈R\lambda\in\mathbb R in finite volume) stops. The order-by-order perturbative analysis behind Theorem 1.1 is itself a new systematic account of the sine-Gordon renormalization flow, based on a modified ('multipole') version of the Mayer expansion, the classical cluster-expansion technique of statistical mechanics here adapted to clusters carrying higher charge multipoles.

Problem Statement

Complete the constructive program for the subcritical sine-Gordon Euclidean field theory, as posed at the close of Meyer's FBSDE construction paper. The primary objective — the 'full construction' — is:

(1) For every β2∈[0,8π)\beta^{2}\in[0,8\pi) and every coupling λ∈R\lambda\in\mathbb R, construct the renormalized sine-Gordon measure non-perturbatively, beyond the current FBSDE restriction β2<67 8π\beta^{2}<\frac{6}{7}\,8\pi. On the FBSDE route this requires overcoming the large-field problem: the available effective-force estimates grow like (1+D)KF(β2)(1+D)^{K_F(\beta^{2})} in the field size DD with KF(β2)→∞K_F(\beta^{2})\to\infty as β2→8π\beta^{2}\to8\pi, and the route needs either a better understanding of the large-field problem, global-in-time existence for the weak FBSDE, or improved (sublinear) estimates on the effective force replacing the superlinear bounds.

(2) Remove the volume cutoff: construct the model in infinite volume R2\mathbb R^{2}. On the FBSDE route the effective-force control carries over, but the Girsanov density M0,∞ρM^{\rho}_{0,\infty} must be controlled uniformly as the cutoff ρ↑1\rho\uparrow1, which is not currently known.

(3) Treat both the massive (m>0m>0) and the massless (m=0m=0) models.

Items (1)–(3) together amount to: construct the sine-Gordon EQFT for all values of λ∈R\lambda\in\mathbb R, β2∈[0,8π)\beta^{2}\in[0,8\pi), in the infinite volume R2\mathbb R^{2}, massive and massless. A solution should identify the limiting measure (or, equivalently, its Schwinger functions) with the standard properties — normalization, Euclidean invariance, reflection positivity (positivity under reflection in affine hyperplanes, the axiom guaranteeing a unitary quantum theory after analytic continuation), and the correct renormalized short-distance behavior; the absolute continuity with respect to the GFF below 4π4\pi and the mutual singularity above 4π4\pi should emerge from the construction rather than be assumed.

Beyond this open problem, the source mentions — as a possible future extension rather than as a required deliverable of the program — the systematic extension of the order-by-order perturbative analysis of the renormalization flow equation (developed in the source for the two-dimensional model) to (sub)critical dd-dimensional settings with field-dependent renormalizations, i.e. to log-correlated cosine-interaction models in general dimension, where the current state of the art covers restricted parameter ranges and smeared correlation functions.

References

  1. Sarah-Jean Meyer. An FBSDE Construction of the Sine-Gordon EQFT for β2<67 8π\beta^{2}<\frac{6}{7}\,8\pi and Perturbative Renormalization in the Full Subcritical Regime. arXiv:2607.20632 (2026). DOI 10.48550/arXiv.2607.20632. https://arxiv.org/abs/2607.20632
  2. Massimiliano Gubinelli and Sarah-Jean Meyer. The FBSDE approach to sine-Gordon up to 6π6\pi. arXiv:2401.13648 (2024). DOI 10.48550/arXiv.2401.13648. https://arxiv.org/abs/2401.13648
  3. Jonathan Dimock and T. R. Hurd. Sine-Gordon Revisited. Annales Henri Poincaré 1(3):499–541 (2000). DOI 10.1007/s000230050005. https://doi.org/10.1007/s000230050005
  4. Jaka Pelaič. The subcritical finite-volume massive sine-Gordon model. arXiv:2508.13778 (2025). DOI 10.48550/arXiv.2508.13778. https://arxiv.org/abs/2508.13778
  5. Massimiliano Gubinelli, Martin Hairer, Tadahiro Oh, and Younes Zine. A simple construction of the sine-Gordon model via stochastic quantization. Journal of the London Mathematical Society 112(1):e70214 (2025). DOI 10.1112/jlms.70214. https://arxiv.org/abs/2412.16404
  6. Bjoern Bringmann and Sky Cao. Global well-posedness of the dynamical sine-Gordon model up to 6π6\pi. The Annals of Probability 54(3):1564–1607 (2026). DOI 10.1214/25-AOP1797. https://arxiv.org/abs/2410.15493
  7. Ajay Chandra, Martin Hairer, and Hao Shen. The dynamical sine-Gordon model in the full subcritical regime. arXiv:1808.02594 (2018). DOI 10.48550/arXiv.1808.02594. https://arxiv.org/abs/1808.02594
  8. Nikolay Barashkov. A stochastic control approach to Sine Gordon EQFT. arXiv:2203.06626 (2022). DOI 10.48550/arXiv.2203.06626. https://arxiv.org/abs/2203.06626
  9. Nikolay Barashkov and Massimiliano Gubinelli. On the variational method for Euclidean quantum fields in infinite volume. Probability and Mathematical Physics 4(4):761–801 (2023). DOI 10.2140/pmp.2023.4.761. https://doi.org/10.2140/pmp.2023.4.761
  10. Roland Bauerschmidt and Thierry Bodineau. Log-Sobolev Inequality for the Continuum Sine-Gordon Model. Communications on Pure and Applied Mathematics 74(10):2064–2113 (2021). DOI 10.1002/cpa.21926. https://doi.org/10.1002/cpa.21926
  11. Roland Bauerschmidt and Michael Hofstetter. Maximum and coupling of the sine-Gordon field. The Annals of Probability 50(2):455–508 (2022). DOI 10.1214/21-AOP1537. https://doi.org/10.1214/21-AOP1537
  12. G. Benfatto, G. Gallavotti, and F. Nicoló. On the massive sine-Gordon equation in the first few regions of collapse. Communications in Mathematical Physics 83(3):387–410 (1982). DOI 10.1007/BF01213609. https://doi.org/10.1007/BF01213609
  13. Sami Vihko. The massless sine-Gordon model with logarithmic correlations in arbitrary dimension. arXiv:2606.06657 (2026). DOI 10.48550/arXiv.2606.06657. https://arxiv.org/abs/2606.06657
  14. Roland Bauerschmidt, Scott Mason, and Christian Webb. Twisted Dirac operators and fractional correlations of the massless sine-Gordon model at the free fermion point. arXiv:2508.14806 (2025). DOI 10.48550/arXiv.2508.14806. https://arxiv.org/abs/2508.14806
  15. Martin Hairer and Hao Shen. The Dynamical Sine-Gordon Model. Communications in Mathematical Physics 341:933–989 (2016). DOI 10.1007/s00220-015-2525-3. https://doi.org/10.1007/s00220-015-2525-3
  16. Shanjian Tang and Rundong Xu. A quadratic BSDE approach to normalization for the finite volume 2D sine-Gordon model in the finite ultraviolet regime. arXiv:2501.12172 (2025). DOI 10.48550/arXiv.2501.12172. https://arxiv.org/abs/2501.12172