Adding flavour to the S-matrix bootstrap

TL;DR

This work probes the space of 1+1D gapped QFTs with O(N) symmetry via the S-matrix bootstrap, focusing on two-particle scattering and the interplay of unitarity and crossing to extremize couplings. By combining analytic large-N insights with detailed numerical exploration, it recovers the integrable O(N) GN and NLSM S-matrices and the sine-Gordon kink S-matrix, while uncovering a family of nonintegrable deformations that exhibit an intricate spectrum of resonances and virtual states, including exotic S-matrices with no particle production and lacking Yang–Baxter factorization. The authors develop explicit large-N optimal S-matrix constructions and finite-N analytic ansätze, revealing how the singlet channel can dominate and how zeros/poles propagate across Riemann sheets under crossing. They discuss the physical relevance of these exotic solutions and propose future work to connect them to more realistic theories with particle production and to broaden the bootstrap program to multi-particle sectors and form factors. Overall, the paper advances understanding of the landscape of O(N) S-matrices, highlighting the role of cusps and the rich analytic structure that bounds and shapes quantum field theories in 1+1 dimensions.

Abstract

We explore the S-matrices of gapped, unitary, Lorentz invariant quantum field theories with a global O($N$) symmetry in 1+1 dimensions. We extremize various cubic and quartic couplings in the two-to-two scattering amplitudes of vector particles. Saturating these bounds, we encounter known integrable models with O($N$) symmetry such as the O($N$) Gross-Neveu and non-linear sigma models and the scattering of kinks in the sine-Gordon model. We also considered more general mass spectra for which we move away from the integrable realm. In this regime we find (numerically, through a large N analysis and sometimes even analytically) that the S-matrices saturating the various coupling bounds have an extremely rich structure exhibiting infinite resonances and virtual states in the various kinematical sheets. They are rather exotic in that they admit no particle production yet they do not obey Yang-Baxter equations. We discuss their physical (ir)relevance and speculate, based on some preliminary numerics, that they might be close to more realistic realistic theories with particle production.

Figures (17)

• Figure 1: Mapping between $s$ and $\theta$ variables. The cuts corresponding to the two particle thresholds with branch points at $s=0$ and $s=4m^2$ get opened in the $\theta$ plane. These cuts are square roots, so that going twice around one of the branch points leaves you back where you began. This is exemplified in the red path for the branch point $s=4m^2$ (or the regular point $\theta=0$). The thick lines in black represent possible inelastic thresholds. Unitarity is imposed for physical values of the center of mass energy $s\geq4m^2$ ($\theta>0$) represented by the green dashed line. The physical sheet(strip) in $s$($\theta$) is highlighted in grey. The bound state poles are located in the window $s\in0,4m^2$ ($\theta\in0,i\pi$) (blue line). Finally, the points in yellow are related by crossing which acts as a reflection around $s=2m^2$ ($\theta=i\pi/2$). Notice that in general there are infinitely many sheets.
• Figure 2: Analytic structure of the non-linear sigma model S-matrix in the $\theta$ plane corresponding to the 'minimal' integrable solution. The physical strip is highlighted in grey. The bullets $\color{grey} \bullet$ represent simple poles and the crosses $\color{grey} \boldsymbol\times$ simple zeros. There is a zero ${\color{blueblue}\boldsymbol\times}$ in the physical strip in the symmetric component at the position $\theta=i\lambda_\text{GN}=2\pi/(N-2)$. Integrability implies that unitarity is saturated, i.e. $S_\text{rep}(\theta)S_\text{rep}(-\theta)=1$, so that in a given representation a pole at $\theta$ has an associated zero at $-\theta$. Crossing mixes the different representations relating points at $\theta$ and $i\pi-\theta$ (see discussion around \ref{['eq2']}).
• Figure 3: Analytic structure of the Gross-Neveu S-matrix in the $\theta$ plane. The conventions are the same as in figure \ref{['fig_nlsmtheta']}. The dark blue poles ${\color{blueblue}\bullet}$ indicate the bound state poles in the singlet and anti-symmetric representations at $\theta=i\lambda_\text{GN}=2\pi/(N-2)$. The lighter blue poles ${\color{lightblue}\bullet}$ depict the t-channel poles at $\theta=i \pi-i\lambda_\text{GN}$ . As explained in the main text, the Gross-Neveu and non-linear sigma model S-matrices are related through a CDD factor which introduces the bound state poles in the singlet and anti-symmetric representations and their corresponding t-channel poles in all representations.
• Figure 4: Numerical results for the Gross-Neveu spectra $m_\text{GN}=2\cos$πN-2$$ maximizing the anti-symmetric coupling $g_\text{anti}$ in the $\theta=i\beta$ line. Notice that the singlet (left) and anti-symmetric (center) S-matrices have two poles: one for each bound state at $\theta=i\lambda_\text{GN}=i2\pi/(N-2)$ and another one imposed by the crossing equations at $\theta=i\pi-i\lambda_\text{GN}$. The symmetric channel (right) has only the latter pole. The three curves in each representation correspond to different values of N: 7 (blue), 11 (orange) and 15 (green). The bound state pole approaches $\theta=0$ ($s=4m^2$) as we increase the parameter $N$. The numerical results are in perfect agreement with analytic solutions \ref{['SmatrixGN']} plotted in dashed lines. The results were obtained with $n_\text{grid}=70$.
• Figure 5: ( a) Analytic structure of the Gross-Neveu anti-symmetric S-matrix in the Mandelstam plane. There is a zero at $s=0$ (in green), a bound state pole at $s=m_\text{GN}^2$ (in dark blue) and a t-channel pole in lighter blue. The green, blue and red dashed lines are respectively the regions below the left cut, between $s\in[0,4m^2]$ and above the right hand cut where we impose unitarity. ( b) Numerical results for the corresponding regions in the $\theta=\alpha+i\beta$ plane wit $N=7$ ($m_\text{GN}=2\cos(\pi/5)$) and $n_\text{grid}=75$. The top panel in green depicts the zero at $\theta=i\pi$ ($s=0$). The middle panel in blue shows the bound state and t-channel poles. In the red bottom panel we see that unitarity is saturated (solid curve representing the absolute value of the function).