Dispersion relations: foundations (2510.01962v1)

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.

• The paper demonstrates that causality enforces analyticity in response functions, establishing the basis for dispersion relations in scattering amplitudes.
• It details both unsubtracted and subtracted dispersion relations, including the treatment of resonance poles and the application of Feynman diagram techniques.
• The work applies these concepts to low-energy hadron interactions, such as pion form factors and three-body decays, highlighting implications for precision resonance analysis.

Foundations of Dispersion Relations in Particle Physics

Introduction

This chapter provides a comprehensive and pedagogical overview of the foundational principles underlying dispersion relations in particle physics, emphasizing their derivation from causality and analyticity, and their implementation in quantum mechanical and quantum field theoretical contexts. The discussion spans from elementary mechanical systems to advanced applications such as Roy equations for pion–pion scattering and dispersion relations for three-body decays. The treatment is informal, focusing on physical intuition and practical consequences rather than mathematical rigor.

Analyticity and Causality

The central thesis is the equivalence between the physical principle of causality and the mathematical property of analyticity. The analysis begins with the forced damped harmonic oscillator, demonstrating that the requirement of causality (no propagation of signals from future to past) enforces the analyticity of the response function in the upper half-plane of complex frequency. This connection is generalized: any linear, time-translation invariant, and causal system yields a Fourier transform (the causal transform) that is analytic in the upper half-plane. This result is foundational for the analytic structure of quantum mechanical amplitudes.

Analytic Structure of Scattering Amplitudes

Non-relativistic quantum mechanical scattering amplitudes are shown to be causal transforms, and thus analytic functions of energy. The $S$-matrix, defined via the ratio of outgoing to incoming wave coefficients, is analytic except for singularities on the real axis: branch cuts corresponding to physical thresholds and poles corresponding to bound states. The analytic continuation to complex energy reveals the existence of multiple Riemann sheets, with resonance poles residing on unphysical sheets. The Schwarz reflection principle relates the values of the $S$-matrix above and below the real axis.

Dispersion Relations and Subtractions

The analytic structure allows the derivation of dispersion relations, which relate the real and imaginary parts of amplitudes via Cauchy's theorem. For amplitudes that do not vanish sufficiently fast at infinity, subtracted dispersion relations are required, introducing subtraction constants that must be fixed by external input. The general form is:

$$\text{Re}\, S(E) = \frac{1}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{\text{Im}\, S(E')}{E' - E}\, dE' + \sum_j \frac{\text{Res}\, S(E_j)}{E_j-E}$$

with further subtractions as necessary for convergence.

Application to Feynman Diagrams

In relativistic quantum field theory, dispersion relations are applied to Feynman diagrams. The Cutkosky rules provide a practical method for computing the discontinuity (imaginary part) of loop integrals by replacing propagators with delta functions, corresponding to on-shell intermediate states. The imaginary part is then used in a subtracted dispersion relation to reconstruct the full amplitude. This approach isolates divergences in subtraction constants and provides a constructive tool for amplitude calculation.

Non-Perturbative Contexts: Pion Form Factor

The utility of dispersion relations is most pronounced in non-perturbative regimes, such as low-energy hadron interactions. The pion vector form factor $F_\pi^V(s)$ serves as a canonical example. Unitarity relates the discontinuity of the form factor to the $P$-wave $\pi\pi$ scattering amplitude, leading to Watson's theorem: the phase of the form factor matches the scattering phase shift below inelastic thresholds. The Omnès solution provides an explicit representation:

$$\Omega(s) = \exp\left\{ \frac{s}{\pi} \int_{4M_\pi^2}^\infty \frac{\delta_1(s')}{s'(s'-s)} ds' \right\}$$

This resums final-state interactions to all orders and yields sum rules for observables such as the pion charge radius, with numerical agreement to within a few percent of experimental values, the discrepancy attributed to inelastic effects.

Resonance Poles and Analytic Continuation

Resonance poles are rigorously defined as $S$-matrix poles on unphysical Riemann sheets. The analytic continuation of partial waves and form factors to these sheets is constructed, with the pole positions determined by zeros of the $S$-matrix on the physical sheet. The universality of pole positions across scattering and production amplitudes is established, and the extraction of residues provides direct access to coupling constants, e.g., $g_{\rho\pi\pi}$ and $g_{\rho\gamma}$ for the $\rho$ resonance.

Dispersion Relations for Scattering Processes

Scattering amplitudes depend on multiple Mandelstam variables, and their analytic structure is constrained by crossing symmetry. Fixed-$t$ and partial-wave dispersion relations are discussed, with the appearance of right-hand (unitarity) and left-hand (crossed-channel) cuts. Subtractions are generally required, with the Froissart–Martin bound ensuring convergence. The Roy equations provide a system of coupled integral equations for $\pi\pi$ phase shifts, incorporating crossing symmetry and subtractions, and serve as the basis for high-precision determinations of low-energy hadronic observables.

Three-Body Decays and Khuri–Treiman Equations

The extension to three-body decays introduces additional complexity due to the presence of multiple physical regions and intricate analytic structure. The amplitude is decomposed into single-variable functions via reconstruction theorems, and the unitarity relation for the $P$-wave leads to an inhomogeneous Omnès problem. The Khuri–Treiman formalism provides a solution via dispersive integration, with the inhomogeneity arising from crossed-channel dynamics. Numerical implementation requires careful treatment of analytic continuation, angular integration, and singularities at pseudothresholds.

Implications and Future Directions

The foundational principles of dispersion relations—causality, analyticity, and unitarity—provide powerful constraints and constructive tools for the analysis of quantum mechanical and quantum field theoretical amplitudes. Their application spans perturbative and non-perturbative regimes, enabling rigorous extraction of resonance properties, coupling constants, and low-energy observables. The formalism is central to modern amplitude analysis in hadron physics, with ongoing developments in multi-variable dispersion relations, three-body decays, and precision determinations of fundamental parameters.

Future directions include the refinement of numerical techniques for solving coupled integral equations, the extension to higher-point functions and multi-channel systems, and the integration with lattice QCD and effective field theory approaches. The interplay between analytic $S$-matrix theory and modern amplitude methods continues to be a fertile ground for theoretical and phenomenological advances.

Conclusion

Dispersion relations, rooted in the physical principles of causality and probability conservation, constitute a cornerstone of theoretical particle physics. Their rigorous analytic structure enables both consistency checks and predictive calculations across a wide range of phenomena, from elementary mechanical systems to complex hadronic interactions. The formalism provides a unifying framework for understanding the analytic properties of amplitudes, the extraction of resonance parameters, and the implementation of unitarity and crossing symmetry, with enduring relevance for both theoretical developments and experimental analyses.