Bootstrapping QCD: the Lake, the Peninsula and the Kink

TL;DR

The paper investigates whether pion scattering amplitudes occupy a special region in the space of consistent four-dimensional S-matrices with $O(3)$ symmetry and no bound states. It develops a crossing-symmetric, unitarity-constrained bootstrap using an amplitude ansatz for $A(s|t,u)$ parameterized by coefficients $a_{nm}$ and $b_{nm}$ and mapped via $rho_z$, and interrogates low-energy observables such as $a_0^{(0)}$, $a_1^{(1)}$, $a_0^{(2)}$ together with Adler zeros $s_0$ and $s_2$ and resonance data. Numerical scans reveal lake-like, peninsula-like, and kink-like regions in the allowed parameter space; adding the presence of a $\rho$ resonance and inequalities on the scattering lengths constrains the boundary theories so that the effective ranges and resonance masses move toward experimental QCD values, with a prominent kink near the real world. These results show that the S-matrix bootstrap, when informed by chiral symmetry and resonance data, can reproduce qualitative hadronic structure and offers a principled route to infer chiral-zero positions and resonance content from fundamental constraints.

Abstract

We consider the S-matrix bootstrap of four dimensional scattering amplitudes with $O(3)$ symmetry and no bound-states. We explore the allowed space of scattering lengths which parametrize the interaction strength at threshold of the various scattering channels. Next we consider an application of this formalism to pion physics. A signature of pions is that they are derivatively coupled leading to (chiral) zeros in their scattering amplitudes. In this work we explore the multi-dimensional space of chiral zeros positions, scattering length values and resonance mass values. Interestingly, we encounter lakes, peninsulas and kinks depending on which sections of this intricate multi-dimensional space we consider. We discuss the remarkable location where QCD seems to lie in these plots, based on various experimental and theoretical expectations.

Figures (7)

• Figure 1: Exploration of the minimum values scattering lengths can take. The three surfaces here correspond to $N_{max}=12,14,16$ (orange, red, light-blue). The fact they are almost indistinguishable is the sign of their very good convergence. The QCD values (Table \ref{['tableExp']}) are represented by a dot here (the errors are smaller than the size of the dot) and are of course well within the allowed region where scattering lengths live.
• Figure 2: Schematic picture of the lake, panel (a), and of the peninsula determination, (b). In (a) the solid blue lines enclose the allowed region for the amplitude $\mathcal{T}_{0}^{(2)}$ when we impose the chiral zero condition at $s_0=1/2$ and the $\rho$ resonance. The same region is shown in panel (b) by dashed blue lines. The solid blue lines in (b) embraces the allowed region when the three scattering lengths are set to the experimental values within errors. In both panels we denote in red the region where we cannot fix $s_2$ and in green where we can.
• Figure 3: Pion Lake: the white region is the exclusion area in the plane $(s_0, s_2)$ when we fix the $\rho$ resonance. The black point corresponds to tree-level chiral perturbation theory \ref{['chitree']} which is now excluded. We show the shape of the lake for three different $N_{max}=12,13,14$ (blue to red): convergence is guaranteed by the three curves almost overlapping.
• Figure 4: Scattering lengths orbit around the lake. The colors in the inset and in the panel match in order to help following the orbit as we move around the lake. In the left inset we zoom in on the $A$-kink: the black dot corresponds to the tree-level chiral theory and the ellipsoid to the experimental values for QCD. All the curves shown are obtained at fixed $N_{max}=14$. The small non-smoothness of the $A$-$B$ arc is a numerical artefact (in any case it occurs in a region where $a_1^{(1)}$ is huge, far from the pion physics we are interested in).
• Figure 5: Pion peninsula: we fix the $\rho$ resonance and the inequalities given by the experimental intervals for the three scattering lengths. The colored region is the one allowed by unitarity and different colors correspond to different $N_{max}$ from 12 to 20 as we go from blue to red. The dashed contour encloses the pion lake.