Positivity and Geometric Function Theory Constraints on Pion Scattering

TL;DR

This work builds a rigorous link between geometric function theory and nonperturbative scattering with $O(N)$ symmetry by formulating three fully crossing-symmetric dispersion relations in a $z$-variable framework. Through Robertson representations, the authors show the associated amplitudes are typically real, enabling Bieberbach-Rogosinski bounds on Taylor coefficients and new locality (sum-rule) constraints. They derive positivity conditions from partial-wave unitarity and Gegenbauer positivity, and translate these into concrete bounds on both abstract expansion coefficients $\mathcal{W}_{p q}^{(k)}$ and physical amplitude coefficients $\mathcal{C}_{p,q}$, with explicit checks against known theories like two-loop chiral perturbation theory and Lovelace-Shapiro models. Crucially, the bounds are largely independent of $N$, supporting a universal geometric-function-theory perspective on pion scattering. The framework sets the stage for cross-pollination with CFT Mellin amplitudes, crossing-antisymmetric bases, and S-matrix bootstrap analyses, highlighting the practical potential of these mathematical structures in constraining high-energy scattering data.

Abstract

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with $O(N)$ global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the $z$-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for $O(N)$ model in $z$-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the $O(N)$ model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, $π^+π^-\to π^0π^0$) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

Figures (3)

• Figure 1: The light blue region is the theory space allowed by positivity and typically real-ness. Some of the known theories has been indicated. Green region is the $O(3)$ Lovelace-Shapiro model \ref{['LS_model_full']} with $\beta'=\frac{1}{m^2_\rho}(1-\beta_0)$ and $\beta_0$ varies from $0.465<\beta_0<0.489$ (see ABAS). The red dot is the the $O(3)$ Lovelace-Shapiro model \ref{['LS_model_full']} with $\beta_0=\beta'=1/2$. The blue dot is the 2-loop chiral perturbation theory with parameters are taken from experimental values values. The orange regions are obtained from the S-matrix bootstrap amplitude for the upper river boundaries ABPHASABAS
• Figure 2: The light blue region is the theory space allowed by the constraints in the table \ref{['tab:Cpqbounds']}. Some of the known theories are indicated. Green region is the $O(3)$ Lovelace-Shapiro model \ref{['LS_model_full']} with $\beta'=\frac{1}{m^2_\rho}(1-\beta_0)$ and $\beta_0$ varies from $0.465<\beta_0<0.489$ (see ABAS). The red dot is the the $O(3)$ Lovelace-Shapiro model \ref{['LS_model_full']} with $\beta_0=\beta'=1/2$. The blue dot is the 2-loop chiral perturbation theory with parameters are taken from experimental values values. The orange regions are obtained from the S-matrix bootstrap amplitude for the upper river boundaries ABPHASABAS.
• Figure 3: We calculate $\mathcal{W}^{(k)}_{n-m,m}$ for $m>n$, using formula \ref{['eq:invWk']} and \ref{['aell_LS']}. The $\mathcal{W}^{(k)}_{n-m,m}$ for $m>n$ should vanish. We have truncated the $k$-sum to $k_{\max}=40$ and $\ell$-sum to $L_{\max}$. We see that as we increase $L_{\max}$ the $\mathcal{W}_{-1,3}^{(0)}, \mathcal{W}_{-1,3}^{(0)}, \mathcal{W}_{-1,2}^{(2)}$ go toward zero.