Gauge Theory Bootstrap: Pion amplitudes and low energy parameters

TL;DR

This paper applies the Gauge Theory Bootstrap method to compute pion scattering phase shifts and low energy parameters, successfully connecting low-energy effective theories with high-energy QCD data, and identifying key resonances.

Contribution

It introduces a novel bootstrap approach that constrains the pion S-matrix by matching low-energy amplitudes with high-energy QCD data, providing new insights into strongly coupled gauge theories.

Findings

Computed pion phase shifts up to 2 GeV for all partial waves with ℓ≤3.

Calculated low energy χPT coefficients 1,2,4,6.

Identified resonances 90, f2(1270), 90(1450).

Abstract

Following the Gauge Theory Bootstrap method proposed in our previous work [arXiv:2309.12402], we compute pion scattering phase shifts for all partial waves with angular momentum $\ell\le 3$ up to 2 GeV and calculate the low energy $\chi$PT coefficients $\bar{\ell}_{1,2,4,6}$. The method looks for the most general S-matrix that matches at low energy the tree level amplitudes of the non-linear sigma model and at high energy, QCD sum rules and form factors. This is a theoretical/numerical calculation that uses as only data the pion mass $m_\pi$, pion decay constant $f_\pi$ and the QCD parameters $N_c=3$, $N_f=2$, $m_q$ and $\alpha_s$. All results are in reasonable agreement with experiment. In particular, we find the $\rho(770)$, $f_2(1270)$ and $\rho(1450)$ resonances. The interplay between the UV gauge theory and low energy pion physics is an example of a general situation where we know the microscopic theory as well as the effective theory of long wavelength fluctuations but we want to solve the strongly coupled dynamics at intermediate energies. The bootstrap builds a bridge between the low and high energy by determining the consistent S-matrix that matches both and provides, in this case, a new direction to understand the strongly coupled physics of gauge theories.