From data to the analytic S-matrix: A Bootstrap fit of the pion scattering amplitude

TL;DR

This work develops a UV-complete, analytic ππ scattering amplitude built from a multi-foliation rho-ansatz and constrained by unitarity, soft theorems, and spectrum zeros, then fits experimental and lattice data via a gradient-free PSO coupled to S-matrix Bootstrap. The resulting amplitude reproduces known resonances below 1.4 GeV, agrees with χPT at low energy, and makes nonperturbative predictions including a spin-3 state, a D2-related structure at high energy, and a genuine I=2 tetraquark around 2 GeV. The framework demonstrates how SDP-based Bootstrap and PSO can jointly yield a predictive, fully analytic scattering amplitude that encodes the spectrum and high-energy behavior, with broad potential extensions to more channels and observables. Overall, it provides a principled path from data to an analytic S-matrix consistent with fundamental constraints and QCD dynamics.

Abstract

We propose a novel strategy to fit experimental data using a UV complete amplitude ansatz satisfying the constraints of Analyticity, Crossing, and Unitarity. We focus on $ππ$ scattering combining both experimental and lattice data. The fit strategy requires using S-matrix Bootstrap methods and non-convex Particle Swarm Optimization techniques. Using this procedure, we numerically construct a full-fledged scattering amplitude that fits the data and contains the known QCD spectrum that couples to $ππ$ states below $1.4$ GeV. The amplitude constructed agrees below the two-particle threshold with the two-loop $χ$PT prediction. Moreover, we correctly predict the $D_2$ phase shift, the appearance of a spin three state, and the behavior of the high-energy total cross-section. Finally, we find a genuine tetraquark resonance around 2 GeV, which we argue might be detected by looking into the decays of B mesons.

Figures (14)

• Figure 1: The vertical side of the boxes corresponds to the uncertainty of the mass determination, the horizontal side to $\Gamma/2$, with $\Gamma$ the width (we do not show the error on the width here). Mass and width are in the same units. Gray refers to the experimental spectrum, red to our Bootstrap estimate. The particles marked by an asterisk, are still affected by numerical systematic as explained later in section \ref{['sec:predictions']}.
• Figure 2: High energy $S_2$ partial wave phase shift profile. In blue the best fit, in light blue all amplitudes with suboptimal $\chi^2$ (see section \ref{['sec:best_fit']}). We see a jump in the phase around $s\approx 200$ that signals the presence of a resonance. In the inset, the mass parameters are extracted from $m=\sqrt{s^*}$, where $s^*$ is the position of the zero $S_0^{(2)}(s^*)=0$ in the complex $s$-plane, and in units of $m_\pi=1$.
• Figure 3: The kink from Andrea in the scattering length space $\{a_0^{(0)}, a_1^{(1)}\}$. The allowed region is in blue. The point with error bars is the experimental determination, see also Table \ref{['tab:thresholds_predictions']}. The blue dot is the position of the kink as a result of our Bootstrap fit algorithm. On the left, we represent the target functional for $\theta=25\pi/18$.
• Figure 4: The four channels used to fit the pion amplitude. The points respectively in orange and green are experimental and lattice data. The thick blue curve is the best fit, the light blue cloud is given by all the curves with sub-optimal $\chi^2$. (Notation: $S_0$ stands for the $(I,\ell)=(0,0)$ channel, $P$ for the $(1,1)$, $S_2$ for the $(2,0)$, and $D_0$ for the $(0,2)$.
• Figure 5: Partial wave amplitudes $t_\ell^{(I)}$ in the real strip $0<s<4$. The black line is the Bootstrap prediction, dashed lines are different approximations from $\chi$PT Bijnens:1995yn.