The S-matrix Bootstrap II: Two Dimensional Amplitudes

TL;DR

This work establishes rigorous bounds on cubic couplings in 1+1D gapped QFTs using crossing symmetry and unitarity for fixed mass spectra. It develops a dual approach: a numerical dispersion-relation bootstrap and an analytic CDD-factor bootstrap, yielding maximally allowed residues g1^max that are saturated by integrable S-matrices. Concrete examples recover known integrable theories (e.g., sine-Gordon) and identify the magnetic point of the Scaling Ising model as a sharp bound saturation, with broader bounds matching independent conformal bootstrap results in an AdS box. The results bridge S-matrix, integrable, and conformal bootstrap methods, and point to future extensions to include heavier particles, inelastic channels, and higher-dimensional generalizations.

Abstract

We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 1 + 1 dimensions due to crossing symmetry and unitarity. In this way we establish rigorous bounds on the cubic couplings of a given theory with a fixed mass spectrum. In special cases we identify interesting integrable theories saturating these bounds. Our analytic bounds match precisely with numerical bounds obtained in a companion paper where we consider massive QFT in an AdS box and study boundary correlators using the technology of the conformal bootstrap.

Figures (15)

• Figure 1: The $2\rightarrow 2$ S-matrix element. Time runs vertically in this figure. In two dimensions energy-momentum conservation implies there is only one independent Mandelstam variable such that $S=S(s)$ with $\sqrt{s}$ the centre of mass energy.
• Figure 2: Analytic properties of the S-matrix element $S(s)$ for the scattering of the lightest particles of the theory. We have a cut starting at $s=4m^2$ corresponding to the two particle production threshold. As implied by (\ref{['crossing']}), we have another cut starting at $t=4m^2$ (or $s=0$) describing particle production in the $t$-channel process. The segment $s\in[0,4m^2]$ between the two particle cuts is where most of the action takes place for us. It is here that poles corresponding to fundamental particles or their bound-states can appear as in (\ref{['near pole of S']}). We distinguish $s$ and $t$ channel poles (solid and empty circles respectively) by the sign of their residues. When the external particles are not the lightest in the theory, we sometimes have more singularities such as further two particle cuts and/or Coleman-Thun poles.
• Figure 3: Approximation of an arbitrary density with a linear spline. The red dashed line represents some unknown $\rho(x)$ which we approximate with the grey spline passing through the points $(\rho_{n},x_{n})$. Explicitly we have $\rho(x) \approx \rho_{n}\frac{(x-x_{n+1})}{(x_{n}-x_{n+1})} + \rho_{n+1}\frac{(x-x_{n})}{(x_{n+1}-x_{n})}$ for $x\in[x_{n},x_{n+1}]$. We use this approximation up to some cutoff $x_{M}$ after which we assume the density decays as $\rho(x) \sim 1/x$. That is, we have $\rho(x) \approx \rho_{M} \, x_{M}/x$ for $x \ge x_{M}$ which allows us to explicitly integrate the tail from $x_{M}$ to $\infty$.
• Figure 4: Maximum cubic coupling $g_1^\text{max}$ between the two external particles of mass $m$ and the exchanged particle of mass $m_1$. Here we consider the simplest possible spectrum where a single particle of mass $m_1$ shows up in the elastic S-matrix element describing the scattering process of two mass $m$ particles. The red dots are the numerical results. The solid line is an analytic curved guessed above (\ref{['SGmatrix']}) and derived in the next section. The blue (white) region corresponds to allowed (excluded) QFT's for this simple spectrum.
• Figure 5: Result of numerics for (a) $m_1 = \sqrt{3}$ and (b) $m_1=1$. In both figures the green, orange and blue curves are $\text{Im}(S)$, $\text{Re}(S)$, $|S|$ respectively. Note that the blue curve is flat and equal to $1$. In other words, the S-matrix that maximizes $g_1$ saturates unitarity at all values of $s>4m^2$. The red dashed lines are real part, imaginary part and magnitude of the sine-Gordon S-matrix \ref{['SGmatrix']}. In figure (a) the numerical results match perfectly with \ref{['SGmatrix']}, while in figure (b) the numerics give precisely $(-1)$ times the sine-Gordon S-matrix as explained in the text.