Convergence of the form-factor expansion of correlation functions in the 1+1-dimensional sinh-Gordon model beyond the space-like vacuum two-point function, and the sharp large-N asymptotics of its summands

The bootstrap form-factor program expresses every correlation function of the 1+1-dimensional sinh-Gordon quantum field theory as an explicit series of multiple integrals over rapidities. Kozlowski (Inventiones mathematicae, 2023) proved convergence of the first and only case settled so far — the vacuum two-point function of space-like separated operators whose form-factor amplitudes satisfy a polynomial-exponential growth bound — together with a large-N upper bound $\exp[-c\,N^2/(\ln N)^3]$ on the N-th summand. Everything beyond this case is open: time-like separated two-point functions, $k\ge3$ point functions (whose closed $\boldsymbol{r}$-truncated representations were only recently constructed, the full correlators being conjectural sums), matrix elements between non-vacuum states, the generalization to the sine-Gordon model (whose exponential-field form factors were constructed in 2026), and the sharp large-N asymptotics of the summands, expected to follow a two-scale $-N^2(\ln N)^p$ expansion. The problem is to establish rigorous convergence in these settings and determine the true asymptotics. This would complete the construction of the sinh-Gordon correlation functions and discharge the convergence hypothesis under which the bootstrap expressions have been shown to satisfy all Wightman axioms — i.e. yield a rigorously constructed interacting massive quantum field theory with exact correlation functions.

Background

The sinh-Gordon model is the massive quantum field theory in 1+1 space-time dimensions of a single Hermitian scalar field with Lagrangian density $\mathcal{L}=\frac{1}{2}\partial_\mu\varphi\,\partial^\mu\varphi-\frac{m^2}{g^2}\cosh(g\varphi)$. It is integrable: particle production is absent, scattering is elastic and diagonal, and the exact two-body S-matrix is $S(\beta)=\tanh(\beta/2-i\pi b)/\tanh(\beta/2+i\pi b)$ with $b=\frac{1}{2}\frac{g^2}{8\pi+g^2}\in(0,\frac{1}{2})$, where the rapidity $\beta$ parametrizes on-shell momenta as $(p^0,p^1)=(m\cosh\beta,m\sinh\beta)$. Since $S$ has no poles in the physical strip $0<\mathrm{Im}\,\beta<\pi$, the spectrum consists of a single massive particle and no bound states.

Form factors are the matrix elements $\langle 0|\mathcal{O}(0)|\beta_1,\dots,\beta_N\rangle$ of local operators between the vacuum and $N$-particle asymptotic states. In the bootstrap program for integrable quantum field theories (Zamolodchikov; Smirnov), these matrix elements are characterized — and in favorable cases computed in closed form — by a system of functional equations (unitarity, crossing symmetry, factorized fusion at annihilation poles) depending only on $S$. Inserting a resolution of the identity over asymptotic states then expresses every correlation function as a series of $N$-fold multiple integrals over rapidities: the form-factor expansion. For the vacuum two-point function of local operators $\mathcal{O}_1,\mathcal{O}_2$ at space-like separation, parametrized by $r=m\sqrt{x^2-t^2}>0$, the series takes the form $U(r)=\sum_{N\ge0}U_N(r)$, where $U_N(r)$ is an $N$-fold integral over $\mathbb{R}^N$ whose integrand is built from the two-body weight $e^{w(\beta_a-\beta_b)}$ with $w=\ln(F(\beta)F(-\beta))$ logarithmic at the origin ($F$ being the minimal two-particle form factor, the elementary building block solving the two-body form-factor equation), the confining one-body factor $e^{-r\cosh\beta_a}$, and combinatorial kernels $K_N^{(\mathcal{O})}$ encoding the form factors of $\mathcal{O}$, known explicitly for the exponential operators $:\!e^{\gamma\varphi}\!:$ and their descendants.

Whether such series converge is the central analytic gap of the bootstrap program: without convergence the exact-correlation formulas remain formal. For decades the only rigorous convergence proof concerned a specific two-point function in the Lee-Yang model, obtained by Smirnov in an unpublished note via model-specific bounds (the Lee–Yang model is the simplest member of the family of 1+1-dimensional integrable quantum field theories, with a single unstable particle). Kozlowski (2023) established the first general result (Theorem 1.1): for operators whose form-factor amplitudes satisfy the growth bound $|p_N^{(\mathcal{O})}(\beta_N|\ell_N)|\le C_1^N\prod_{a=1}^N e^{C_2|\beta_a|^s}$ with $N$-independent constants $C_1,C_2,s>0$ — a class comprising the exponentials of the field — the series converges uniformly in $r\in[r_0,+\infty)$ for every $r_0>0$, and the $N$-th summand obeys

$$|U_N(r)|\le\exp\Big[-\frac{3\pi^4 b\hat b}{4}\cdot\frac{N^2}{(\ln N)^3}\Big(1+O\big(\tfrac{1}{\ln N}\big)\Big)\Big],\qquad\hat b=\tfrac12-b.$$

The remainder is not uniform as $r\to0^+$, where power-law ultraviolet behaviour is expected. The method is indirect: $U_N(r)$ is dominated by a random-matrix-type $N$-fold partition function whose large-$N$ behaviour is governed by the infimum of an $N$-dependent variational (energy) functional over probability measures on $\mathbb{R}$; the unique minimizer (the equilibrium measure) has a density supported on a single interval, characterized by a singular integral equation of truncated Wiener–Hopf type (an integral equation on a finite interval involving the Hilbert transform), which is solved through a $2\times2$ matrix Riemann–Hilbert problem (a boundary-value problem for piecewise-analytic matrix functions) analyzed in the large-$N$ limit by the Deift–Zhou non-linear steepest-descent method — a technique that reduces the problem, through a sequence of explicit factorizations of the jump matrix, to a form whose asymptotics can be read off directly. The author stresses that the two scales $N$ and $\ln N$ are intrinsic — the one-body confining potential $r\cosh\beta$ carries no prefactor of $N$ and balances the $N^2$ two-body repulsion only after a $\ln N$ rescaling of the integration variables — and expects the true asymptotics to take the two-scale form (eq. (1.13))

$$\ln U_N(r)\simeq -N^2(\ln N)^p\Big\{Q_{0,0}+\sum_{(s,\ell)\neq(0,0)}\frac{Q_{s,\ell}}{N^{\ell}(\ln N)^s}\Big\},$$

with the exponent $p$ and the coefficients $Q_{s,\ell}$ left undetermined, while the proven upper bound is not known to be sharp.

The surrounding picture has since advanced on the structural side. Kozlowski–Simon (2025) constructed the first closed, rigorous representations of the $\boldsymbol{r}$-truncated $k$-point functions ($\boldsymbol{r}\in\mathbb{N}^{k-1}$ a multi-index truncation; each truncated correlator is a finite sum of well-defined multiple integrals), so that the per se $k$-point functions are now explicit conjectural sums over $\boldsymbol{r}$. Kozlowski–Simon (2025) also proved that, if these series converge, the resulting multipoint functions satisfy all the Wightman axioms (temperedness, Poincaré covariance, spectral condition, locality, positivity and clustering — the standard axiomatic properties characterizing the correlation functions of a quantum field theory): the rigorous construction of the sinh-Gordon model as a Wightman quantum field theory is thereby reduced exactly to the convergence problem. For the sine-Gordon model — the Lagrangian with $\cos(g\varphi)$ in place of $\cosh(g\varphi)$, with topological soliton sectors, non-diagonal soliton–antisoliton scattering and an infinite tower of breather bound states — Kozlowski–Simon (2026) constructed the form factors of the field exponentials in all Fock sectors, providing the raw material for the same convergence question there. Relatedly, the large-$N$ analysis of the Lukyanov integral representation of the finite-volume vacuum expectation value of $e^{\alpha\varphi}$ has been initiated through the equilibrium measure of the associated $N$-dependent energy functional.

What remains unresolved is the convergence itself in every setting beyond the one solved case — time-like separation, $k\ge3$ points, non-vacuum matrix elements, the sine-Gordon model — together with the sharp large-$N$ asymptotics of the summands.

Problem Statement

Complete the rigorous convergence theory of the form-factor expansion of the 1+1-dimensional sinh-Gordon model: extend the sole settled case — the vacuum two-point function at space-like separation, for operators whose form-factor amplitudes obey the growth bound $|p_N^{(\mathcal{O})}(\beta_N|\ell_N)|\le C_1^N\prod_a e^{C_2|\beta_a|^s}$ — to the correlation functions of the theory at their natural generality, and determine the sharp large-$N$ asymptotics of the summands. All components below are instances of one question — does the bootstrap form-factor series converge, and at what rate — and they jointly support one objective, the well-definedness of the bootstrap construction of the sinh-Gordon quantum field theory:

1. (Time-like regime) Prove convergence of the form-factor series for time-like separated two-point functions, obtained from the space-like representation by deforming every rapidity integration contour to a curve $\beta_a\in\gamma(\mathbb{R})$ with $\gamma(u)=u+i\vartheta(u)$, $0<\varepsilon<\pi/2$, and $\vartheta(u)\to-\mathrm{sgn}(t)\mathrm{sgn}(u)\,\varepsilon$ as $|u|\to\infty$; specify in what sense the resulting functions exist (uniformly away from the light cone, or as tempered distributions on appropriate test functions).

2. (Multipoint functions) Prove, for every $k\ge3$, convergence of the series over $\boldsymbol{r}\in\mathbb{N}^{k-1}$ of the $\boldsymbol{r}$-truncated $k$-point functions constructed by Kozlowski–Simon, so that each $k$-point Wightman function exists as a tempered distribution — i.e. establish the well-definiteness conjecture under which the bootstrap expressions have been shown to satisfy the Wightman axioms.

3. (Operator and state generality) Extend convergence beyond vacuum-to-vacuum matrix elements: to matrix elements between general asymptotic multi-particle states and to local operators outside the growth-bound class, so far as needed for products and commutators of bootstrap-constructed local operators to be well-defined — completing the causality/locality proof for the model.

4. (Sharp asymptotics) Determine the true large-$N$ behaviour of the summand $U_N(r)$: prove or disprove the two-scale form $\ln U_N(r)\simeq -N^2(\ln N)^p\{Q_{0,0}+\sum_{(s,\ell)\neq(0,0)}Q_{s,\ell}/(N^{\ell}(\ln N)^s)\}$, identify the exponent $p$ and the leading constant $Q_{0,0}$ when the form holds, and decide whether the $\exp[-c\,N^2/(\ln N)^3]$ decay of the known upper bound is sharp.

5. (Model generalization) Extend the convergence proof to more complex massive integrable quantum field theories, the sine-Gordon model being the benchmark case, now that the form factors of its field exponentials are available in all Fock sectors. The sine-Gordon model is the concrete target of this component — a rigorous convergence result for the sine-Gordon form-factor series completes it — and any further theories the argument covers should be stated explicitly.

An answer resolves the problem as stated by establishing rigorous convergence (with the natural notion of summation for each setting) in these directions together with the sharp asymptotics of the summands. Partial resolutions that settle some components completely while stating precisely what remains open constitute genuine progress but do not close the problem as stated.

References

1. Karol 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
2. Karol K. Kozlowski, 'Bootstrap approach to 1+1 dimensional integrable quantum field theories: the case of the Sinh-Gordon model', arXiv:2112.14274 (2021); contribution to the Proceedings of the International Congress of Mathematicians (ICM 2022). https://arxiv.org/abs/2112.14274
3. Karol K. Kozlowski, Alex 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
4. Karol K. Kozlowski, Alex 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
5. Karol K. Kozlowski, Alex 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. Charlie Dworaczek Guera, Karol K. Kozlowski, 'On the equilibrium measure for the Lukyanov integral', Journal of Mathematical Physics 66 (2025), 082301. DOI: 10.1063/5.0252653; arXiv:2406.00145. https://doi.org/10.1063/5.0252653
7. Sergei Lukyanov, 'Finite temperature expectation values of local fields in the sinh-Gordon model', Nuclear Physics B 612 (2001), 391–412. DOI: 10.1016/S0550-3213(01)00365-0. https://doi.org/10.1016/S0550-3213(01)00365-0
8. Wojciech Dybalski, 'Exact Schwinger functions for a class of bounded interactions in d≥2', Communications in Mathematical Physics 406 (2025), article 211. DOI: 10.1007/s00220-025-05391-6; arXiv:2502.07546. https://doi.org/10.1007/s00220-025-05391-6
9. András László, Zsigmond Tarcsay, 'On the running and the UV limit of Wilsonian renormalization group flows', Classical and Quantum Gravity 41 (2024), 125009. DOI: 10.1088/1361-6382/ad4a1a; arXiv:2303.03740. https://doi.org/10.1088/1361-6382/ad4a1a