Convergence of the form-factor series for the multipoint correlation functions of the 1+1-dimensional sinh-Gordon model

The integrable bootstrap program represents each vacuum $k$-point correlation function of the massive $1+1$-dimensional sinh-Gordon quantum field theory as a series of multiple integrals whose terms are labelled by the numbers of intermediate particles running between the insertion points. Closed, rigorous integral representations exist for every truncated (finite-particle-content) partial sum, and it has been proven -- conditionally on the convergence of the full series -- that the resulting multipoint functions satisfy all the Wightman axioms. The convergence itself, however, is established only for the two-point function with mutually space-like separated supports. This problem asks to establish, or refute, the conjectured absolute convergence and tempered-distribution property of these series for all $k\ge 2$ and all Schwartz test functions, including time-like configurations (Conjecture 4.10 of Kozlowski--Simon 2025 and its general form, Conjecture 2.5 of Kozlowski--Simon 2025). A positive resolution would make the conditional Wightman-axiomatics theorem unconditional and complete the first construction of an interacting relativistic quantum field theory from its exact factorized two-particle $S$-matrix alone. As of September 2026 no proof or refutation of this conjecture is known, and the convergence problem remains open.

Background

The sinh-Gordon model is the massive scalar quantum field theory in $1+1$ dimensions whose interaction potential is a cosh potential, $\mathcal{L}[\varphi]=(\partial_t\varphi)^2-(\partial_x\varphi)^2-\frac{m^2}{g^2}\cosh(g\varphi)$ up to inessential normalization conventions. It is integrable: it carries infinitely many commuting conserved charges, so scattering is elastic and factorizes into two-particle processes. Relativistic two-particle momenta are parameterized by rapidities $\beta$ (energy $m\cosh\beta$, momentum $m\sinh\beta$), and the exact two-particle $S$-matrix is the scalar function $S(\beta)=\frac{\sinh\beta-i\sin(2\pi b)}{\sinh\beta+i\sin(2\pi b)}$ with $b=\frac{g^2}{2(8\pi+g^2)}\in(0,\frac{1}{2})$ a reparameterization of the coupling (Zamolodchikov and Zamolodchikov, 1979); it has no poles on the physical strip $0<\operatorname{Im}\beta<\pi$, reflecting the absence of bound states.

The integrable bootstrap program (Smirnov, 1992) reverses the logic of quantization: starting only from the exact $S$-matrix and the Fock Hilbert space $\mathcal{H}=\bigoplus_{n\ge0}L^2(\mathbb{R}^n_>)$ built on ordered rapidity configurations, one reconstructs the local quantum fields as operator-valued kernels. Their building blocks are the form factors, the matrix elements $F^{(O)}_n(\beta_1,\dots,\beta_n)$ of a local operator $O$ between the vacuum and $n$-particle states, fixed by a Riemann--Hilbert-type system of equations encoding unitarity, crossing, locality and Lorentz covariance of the scattering. Correlation functions -- vacuum expectation values of products of fields -- then arise as series over the numbers of intermediate particles running between the insertion points, each term being a multiple integral whose dimension grows with the particle number; these are the rigorous counterparts of the form-factor expansions used as exact computational tools throughout integrable-QFT physics.

For a $k$-point function, the $r$-truncated partial sums -- in which the numbers of particles summed over between consecutive insertion points are bounded componentwise by a multi-index $r\in\mathbb{N}^{k-1}$, corresponding to correlators of regularized field operators that connect only a finite portion of the Fock space -- can be constructed on fully rigorous grounds. Kozlowski and Simon (arXiv:2502.03894) derived the first closed, explicit integral representations for all such $r$-truncated $k$-point functions of the sinh-Gordon model, as smooth functions on the space-like Weyl chambers: the connected regions of configuration space in which the points $x_1,\dots,x_k$ are pairwise space-like separated, $(x_a-x_b)^2<0$, with a fixed ordering (the regime corresponding to Euclidean separations). Before that work no closed result of any kind existed for $k\ge3$: earlier literature (e.g. Babujian--Karowski--Tsvelik, 2017, and earlier partial computations in the Ising, Potts and $O(n)$ models) had only evaluated the first one or two terms of the series. The full, per se $k$-point functions are obtained by summing the truncated ones over $r\in\mathbb{N}^{k-1}$, and precisely this convergence is unproven: Conjecture 4.10 of arXiv:2502.03894 asserts absolute convergence in the space-like regime, while the analogous statement for arbitrary Schwartz test functions $G\in\mathcal{S}(\mathbb{R}^{1,1})^k$ (rapidly decreasing smooth functions) -- which includes time-like separated points $(x_a-x_b)^2>0$, for which the relevant series already differ at $k=2$ (Kozlowski, ICM 2022) -- is stated as Conjecture 2.5 in the follow-up work. Convergence has been proven only for the two-point function with mutually space-like supports (Kozlowski, Inventiones mathematicae 2023), a roughly hundred-page analysis.

The stakes are sharp because of the Wightman axioms: the standard conditions under which a system of correlation functions defines a genuine quantum field theory with a Hilbert space, a unique vacuum and Poincare covariance (temperedness, covariance, the spectral condition, local commutativity at space-like separation, positive definiteness, and clustering). Partial aspects, notably local commutativity at the heuristic level of operator products, were discussed much earlier (Smirnov, 1992; Babujian and Karowski, 2004). Kozlowski and Simon (arXiv:2511.10800) proved that, if the convergence conjecture holds, the bootstrap-built multipoint functions of the sinh-Gordon model satisfy all of the Wightman axioms -- the first time the complete Wightman program has been carried out for an interacting integrable quantum field theory, albeit conditionally; the conditional Wightman-axioms step was thereby settled in November 2025. Establishing the well-definiteness of the series is therefore the single missing step in constructing an interacting relativistic quantum field theory starting solely from its exact factorized $S$-matrix.

Problem Statement

Establish the well-definiteness of the bootstrap construction of the multipoint correlation functions of the infinite-volume sinh-Gordon quantum field theory, or refute it. The two parts form one scientific objective -- absolute convergence of the form-factor series in all separation regimes -- and both must be addressed by a complete answer.

(1) Space-like case (Conjecture 4.10 of Kozlowski--Simon, arXiv:2502.03894). Prove that for every integer $k\ge2$ and any smooth, compactly supported test functions $g_1,\dots,g_k$ on $\mathbb{R}^{1,1}$ whose supports are mutually space-like separated and ordered within a single space-like Weyl chamber, the series over $r\in\mathbb{N}^{k-1}$ of $r$-truncated $k$-point functions converges absolutely and represents the $k$-point function $\langle f_{\mathrm{vac}},\,O_1[g_1]\cdots O_k[g_k]\,f_{\mathrm{vac}}\rangle$ by a smooth function $W(x_1,\dots,x_k)$, where the $O_a$ are the field operators built by the bootstrap construction.

(2) General case (Conjecture 2.5 of Kozlowski--Simon, arXiv:2511.10800). Prove that for every $k\ge2$ and every Schwartz test function $G\in\mathcal{S}(\mathbb{R}^{1,1})^k$ -- including configurations with time-like separated insertion points, for which the series differ from the space-like ones already at $k=2$ and no closed multipoint representations exist -- the form-factor series $\sum_{n}I^{(n)}_{\alpha^k}[G]$ converges absolutely and defines a tempered distribution; here the summands are the multiple integrals generated by the bootstrap construction (products of $S$-matrices and form factors integrated over intermediate rapidities) and the operator labels $\alpha^k$ range over the solutions of the form-factor equations.

A proof of (2) implies, via the conditional theorem of arXiv:2511.10800, that the bootstrap-built multipoint functions satisfy all the Wightman axioms unconditionally, completing the construction of the sinh-Gordon model as a Wightman quantum field theory from its two-particle $S$-matrix alone. A rigorous refutation -- an explicit coupling, integer $k$, and test functions for which the series fails to converge absolutely or fails to define a tempered distribution -- is equally admissible. The objective is completed by proving both parts, or by exhibiting a single divergence that falsifies the general claim of part (2), which already contains part (1) as its special case -- whether that divergence occurs in the space-like or the time-like regime. Answers covering a proper sub-class (for example only space-like supports, only $k=3$, only the two-point time-like case, or convergent closed representations on time-like Weyl chambers) are valid contributions but must state their scope precisely and do not by themselves complete the objective.

References

1. K. K. Kozlowski and A. Simon, 'On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory', arXiv:2502.03894 (2025). https://arxiv.org/abs/2502.03894
2. K. K. Kozlowski and A. Simon, 'Wightman axiomatics of the bootstrap construction of the 1+1 dimensional Sinh-Gordon model', arXiv:2511.10800 (2025). https://arxiv.org/abs/2511.10800
3. K. K. Kozlowski, 'On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions', Inventiones mathematicae 233 (2023) 725-827. DOI: 10.1007/s00222-023-01192-7. arXiv:2007.01740. https://doi.org/10.1007/s00222-023-01192-7
4. K. K. Kozlowski, 'Bootstrap approach to 1+1-dimensional integrable quantum field theories: the case of the sinh-Gordon model', Proceedings of the International Congress of Mathematicians (ICM 2022), Vol. 5, pp. 4096-4118. DOI: 10.4171/ICM2022/96. arXiv:2112.14274. https://doi.org/10.4171/ICM2022/96
5. K. K. Kozlowski and A. Simon, 'Form factors of field exponentials in the 1+1 dimensional Sine-Gordon model', arXiv:2608.14365 (2026). https://arxiv.org/abs/2608.14365
6. A. Simon, 'On correlation functions of the 1+1 dimensional integrable quantum field theories', PhD thesis, Ecole normale superieure de Lyon (2026). HAL: tel-05719674. https://theses.hal.science/tel-05719674
7. H. M. Babujian, M. Karowski and A. M. Tsvelik, 'Multipoint Green's functions in 1+1 dimensional integrable quantum field theories', Nuclear Physics B 917 (2017) 122-153. DOI: 10.1016/j.nuclphysb.2017.02.002. https://doi.org/10.1016/j.nuclphysb.2017.02.002
8. A. B. Zamolodchikov and Al. B. Zamolodchikov, 'Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models', Annals of Physics 120 (1979) 253-291. DOI: 10.1016/0003-4916(79)90391-9. https://doi.org/10.1016/0003-4916(79)90391-9
9. F. A. Smirnov, 'Form Factors in Completely Integrable Models of Quantum Field Theory', Advanced Series in Mathematical Physics Vol. 14, World Scientific, Singapore (1992). DOI: 10.1142/1115. https://doi.org/10.1142/1115
10. H. Babujian and M. Karowski, 'Towards the construction of Wightman functions of integrable quantum field theories', International Journal of Modern Physics A 19 (2004) 34-49. DOI: 10.1142/S0217751X04020294. https://doi.org/10.1142/S0217751X04020294