Ultraviolet central charge of the periodic integrable U(N) amplitudes in the two-dimensional S-matrix bootstrap

The S-matrix bootstrap of two-dimensional massive theories with global $U(N)$ symmetry and no bound states contains three periodic integrable two-particle amplitudes — classes V, VI and VII of the Córdova–Rodrigues classification, class VII previously unknown — whose microscopic physical realization is not known. Their ultraviolet content has been probed only through the two-particle contribution $c_2(\theta_{\max})$ to Zamolodchikov's central-charge sum rule: classes V–VI exhibit a walking value $c\approx1$ for $N\gtrsim1$ and class VII a walking value $c\approx2$ for $N\gtrsim2$, while the full central-charge integral, which receives contributions from the form factors (matrix elements between multiparticle states) of arbitrarily many particles, is expected to diverge for these extremal amplitudes (those on the boundary of the allowed bootstrap region). The open problem is to determine the complete multiparticle ultraviolet central charge of these amplitudes — whether the full sum rule converges to a finite $c^{\rm UV}$, and to what value, for each class and each $N$ — and to resolve the discrepancy that the class VII walking value $c\approx2$ conflicts with the $c=1$ of the critical $PSU(N)$ (projective special unitary) loop model at $N=2$, the natural ultraviolet candidate: one must decide whether the mismatch reflects the difference between full $U(N)$ and projective $PSU(N)$ global symmetry, a liberation of fundamental excitations, or another mechanism, and thereby identify the ultraviolet theory that this amplitude describes.

Background

In two spacetime dimensions the S-matrix bootstrap asks which $2\to2$ scattering amplitudes of massive particles are consistent with analyticity, crossing symmetry and unitarity; the allowed region (the 'monolith') has a boundary whose special points are integrable theories — theories with no particle production, whose scattering factorizes into two-particle amplitudes satisfying the Yang–Baxter equation, the consistency condition ensuring that different orderings of sequential two-particle collisions give the same result. For a global $U(N)$ symmetry and no bound states, Córdova and Rodrigues (2025) mapped this space and classified its integrable solutions into classes I–VII, revising the 1978 classification of Berg–Karowski–Weisz–Kurak. Classes V–VII are periodic in the real rapidity $\theta$ (the kinematic variable with $s=4m^2\cosh^2(\theta/2)$, where $s$ is the center-of-mass energy) with period $2\pi/\mu$; in the singlet channel (the component of the amplitude acting on two-particle states in the trivial, one-dimensional $U(N)$ representation) $\cosh(\pi\mu_{\rm V})=N$, $e^{\pi\mu_{\rm VI}}=N$ and $\cosh(\pi\mu_{\rm VII})=N/2$. Class VII had not appeared in the literature before, and no microscopic lattice or continuum model is known whose S-matrix equals any of classes V–VII.

The ultraviolet content of a two-dimensional massive theory is quantified by Zamolodchikov's c-theorem, which yields a sum rule: the difference between the central charges (the coefficient $c$ measuring the number of effective degrees of freedom of a two-dimensional conformal field theory) of the conformal field theories (CFTs) at the two ends of a renormalization-group flow equals an integral over the spectral density of the trace of the stress tensor $\Theta$. For a gapped theory the infrared value vanishes, so the sum rule computes the ultraviolet central charge $c^{\rm UV}$ directly. Inserting a complete set of states decomposes the sum rule into $n$-particle contributions $c^{\rm UV}=\sum_n c_n$, each determined by the corresponding $n$-particle form factors of $\Theta$ — its matrix elements between the vacuum and $n$-particle states. In an integrable theory Watson's equation $F(s)=S(s)F^{*}(s)$, relating the two-particle form factor $F$ of $\Theta$ to the amplitude $S$, fixes $F$ from $S$ up to normalization, so only $c_2$ is directly accessible from two-body data; all higher terms require a genuine multiparticle form-factor construction.

For the $O(N)$ periodic Yang–Baxter amplitude — to which the singlet amplitudes of the $U(N)$ classes are formally identical — Gorbenko and Zan (2020) computed $c_2(\theta_{\max})$ and found a long walking plateau — a wide range of cutoff rapidities over which $c_2$ stays nearly constant — near the real part of the central charge of the complex CFT pair that the flow passes between for $N>2$ (the two real $O(N)$ fixed points merge at $N=2$ and leave the real axis), with the central charge blowing up beyond the plateau; Córdova–Correia–Georgoudis–Vuignier (2024) later derived divergence criteria showing that the full sum rule diverges on most of the $O(N)$ bootstrap boundary, including the periodic point. For the $U(N)$ classes, Córdova and Rodrigues computed only $c_2(\theta_{\max})$, the two-particle contribution to the sum-rule integral cut off at rapidity $\theta_{\max}$: classes V–VI walk around $c\approx1$ for $N\gtrsim1$ (expected, since class V is the $O(2N)$ periodic solution reducing at $N=1$ to the $O(2)$ sigma model with $c=1$), while class VII walks around $c\approx2$ for $N\gtrsim2$. A natural ultraviolet candidate for class VII is the critical oriented loop model of Roux–Jacobsen–Ribault–Saleur (2025), conjectured to share the $O(N)$ phase diagram with merging fixed points at $N=2$; however, that model's symmetry is in fact $PSU(N)$ — the projective special unitary group, $SU(N)$ modulo its center — and it is built from a spin chain alternating fundamental and antifundamental sites, so only meson-like composites exist; its central charge at $N=2$ is $c=1$, not $2$. In neighboring massive models — the $CP^{N-1}$ sigma model (a model whose target space is complex projective space) coupled to a minimally coupled Dirac fermion with quartic interaction (Basso–Rej 2013) — the fundamental excitations are 'liberated' and the ultraviolet central charge is $2N-1$, one unit above that of $CP^{N-1}$, illustrating one candidate mechanism for such a shift.

Problem Statement

Determine the full ultraviolet central charge of the periodic integrable $U(N)$-symmetric amplitudes (classes V–VII of the Córdova–Rodrigues classification) and reconcile it with candidate ultraviolet theories. Concretely: (i) compute the contributions of all $n$-particle form factors of the stress-tensor trace $\Theta$ to Zamolodchikov's c-sum rule for each of classes V–VII as a function of $N$ — going beyond the two-particle term $c_2(\theta_{\max})$ — and establish whether the complete sum rule converges to a finite $c^{\rm UV}$ or diverges, and if finite, to what value; (ii) resolve the class VII discrepancy: explain why the two-particle walking value is $c\approx2$ for $N\gtrsim2$ while the critical $PSU(N)$ loop model, the natural ultraviolet candidate, has $c=1$ at $N=2$ — deciding whether the difference comes from the distinction between full $U(N)$ and projective $PSU(N)$ global symmetry, from a liberation of fundamental (deconfined) excitations analogous to the $CP^{N-1}$ plus massless-fermion models where the central charge shifts by one unit, or from another mechanism — and thereby identify the ultraviolet theory (a $U(N)$-symmetric loop model, or a pair of complex conformal field theories connected by a walking flow) that the class VII amplitude describes. A deliverable is resolved only to the stated depth: (i) requires the explicit multiparticle form-factor construction — the $n$-particle contributions of $\Theta$ to the c-sum rule for each class and each $N$, evaluated beyond the two-particle term $c_2(\theta_{\max})$ — and is not resolved by a convergence-or-divergence verdict carried over from the two-particle singlet amplitude, while (ii) requires identifying the ultraviolet theory together with the mechanism that fixes the class VII walking value; the two deliverables are jointly required, and neither alone resolves the task. The same multiparticle question for the generic, non-integrable extremal amplitudes saturating the $U(N)$ bootstrap bounds — for which the two-particle amplitude alone does not determine the higher form factors — is part of the unresolved central-charge status of this bootstrap space; the present problem targets the integrable periodic classes, where the two-particle amplitude is fixed and the multiparticle form-factor construction is well-posed. The common objective uniting these deliverables is to establish the ultraviolet central-charge content of the periodic $U(N)$ extremal amplitudes and the ultraviolet theory they point to: the multiparticle computation and the discrepancy resolution jointly serve this objective, the former fixing the number and the latter fixing its physical interpretation.

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