S-matrix bootstrap for resonances

TL;DR

The paper develops a bound on the coupling of asymptotic states to unstable resonances within the 1+1D S-matrix bootstrap framework, using zeros in the physical strip and a factorized S-matrix representation. It derives an explicit inequality relating resonance couplings to a dispersive exponential, and demonstrates its perturbative interpretation via a simple resonance model. A numerical optimization scheme confirms the analytical bound and provides a practical route toward generalization to higher dimensions, including a symmetric s-t channel ansatz that can extend to 3+1D. Overall, the work links resonance width and mass-gap constraints to bootstrap-consistent S-matrix data, offering both analytical insight and a scalable computational method.

Abstract

We study the $2\rightarrow2$ $S$-matrix element of a generic, gapped and Lorentz invariant QFT in $d=1+1$ space time dimensions. We derive an analytical bound on the coupling of the asymptotic states to unstable particles (a.k.a. resonances) and its physical implications. This is achieved by exploiting the connection between the S-matrix phase-shift and the roots of the S-matrix in the physical sheet. We also develop a numerical framework to recover the analytical bound as a solution to a numerical optimization problem. This later approach can be generalized to $d=3+1$ spacetime dimensions.

Figures (6)

• Figure 1: Illustration of the conformal map in Eq. (\ref{['conf1']}). The complex $s$-plane, the left plot, is mapped into the complex $\theta$-strip $\mathop{\mathrm{Im}}\nolimits \theta \in (0,\pi)$, right plot. We have also depicted the mapping of a dashed curve, a dotted curve and a gray grid.
• Figure 2: Left: Section of the complex plane of the phase of Eq. (\ref{['scatex']}), the lines $\delta(i\pi)\sim\delta(-i\pi)$ should be indentified. Right: phase-shift of Eq. (\ref{['scatex']}) localized around the position of the branch points generated by the zeros and poles of $S_\text{ex}(\theta)$.
• Figure 3: In the left plot, maximal coupling to the lowest mass resonance at $\theta_1=x+i\pi/7$ for an $S$-matrix with a single resonance (solid black), and an $S$-matrix with a second resonance at $\theta_2=6+i\pi/9$ (dashed light gray) and $\theta_2=4+i\pi/9$ (dotted darker gray). On the right, minimal value of $\text{Re}\theta_2$ as a function of the coupling to the $\theta_1$ resonance for an $S$-matrix with two resonances.
• Figure 4: Illustration of the conformal map in Eq. (\ref{['map']}) with $\beta=i\pi$.
• Figure 5: In the left plot solid black lines depict the imaginary part, real part and absolute value of $S_\text{th}(e^{i\phi})$. Superimposed we show the imaginary part (dashed white), the real part (doted white) and absolute value (solid white) of $S_\text{ans}(e^{i\phi})$. In the lower left plot, sharing the same horizontal axis, we show the real part (dotted) and the imaginary part (dashed) of $S_\text{th}(e^{i\phi})-S_\text{ans}(e^{i\phi})$. Finally the plot to the right is a comparison of the phase of $S_\text{ans}$ (white) and $S_\text{th}$ (black) in the $\theta$ strip.