Dispersion relations: foundations

TL;DR

Dispersion relations provide a causality-driven, analyticity-based framework for analyzing quantum-mechanical scattering and decays in the complex $s$-plane. The paper builds from basic complex-analysis tools to nonperturbative hadronic applications, illustrating how unitarity and crossing constrain amplitudes, and showing how form factors like $F_\pi^V(s)$ can be reconstructed via the Omnès method using phase shifts $\delta_1(s)$. A central contribution is the demonstration of resonance pole science through second-sheet continuations and universal pole positions across production and scattering, as well as the development of Roy equations and Khuri–Treiman formalisms for two- and three-body processes. Collectively, dispersion theory emerges as a powerful, model-independent toolkit for precise low-energy hadron physics, enabling robust extractions of resonance properties, radii, and phase shifts beyond perturbation theory.

Abstract

We give a pedagogical introduction to the founding ideas of dispersion relations in particle physics. Starting from elementary mechanical systems, we show how the physical principle of causality is closely related to the mathematical property of analyticity, and how both are implemented in quantum mechanical scattering theory. We present a personal selection of elementary applications such as the relation between hadronic production amplitudes or form factors to scattering, and the extraction of resonance properties on unphysical Riemann sheets. More advanced topics such as Roy equations for pion--pion scattering and dispersion relations for three-body decays are briefly touched upon.

Figures (4)

• Figure 1: Modulus (left) and phase (right) of the Green's function $G(\omega)$, with resonant behavior at $\omega \approx \pm \omega_0$.
• Figure 2: Integration in the complex $\omega$ plane along the contour $\Gamma$ (blue). Both poles $\omega_{1/2}$ (red crosses) lie in the lower half plane $\mathbb{I}_-(\omega)$, while the integration contour is closed in the upper half plane $\mathbb{I}_+(\omega)$ where the Green's function is analytic.
• Figure 3: Analytic structure of $S(E)$, with a right-hand cut (red zig-zag line) for $E>0$ and bound states (red crosses) for $E<0$. Extending the original contour of the Cauchy integral (blue dashed) towards infinity leads to the blue contour that wraps around both cut and poles.
• Figure 4: Unitarity relation for the pion vector form factor. The curly line stands for the electromagnetic current, the blue disk denotes the form factor itself, while the red one stands for the pion--pion scattering amplitude. The dotted line signifies the cut.