Functional Analysis and Optimization Methods in Hadron Physics

This book begins with a brief historical review of the early applications of standard dispersion relations in particle physics. It then presents the modern perspective within the Standard Model, emphasizing the relation of analyticity together with alternative tools applied to strong interactions, such as perturbative and lattice quantum chromodynamics (QCD), as well as chiral perturbation theory.The core of the book argues that, in order to improve the prediction of specific hadronic observables, it is often necessary to resort to methods of complex analysis more sophisticated than the simple Cauchy integral. Accordingly, a separate mathematical chapter is devoted to solving several functional analysis optimization problems. Their applications to physical amplitudes and form factors are discussed in the following chapters, which also demonstrate how to merge the analytic approach with statistical analysis tools.

Given its scope, the book offers a valuable guide for researchers working in precision hadronic physics, as well as graduate students who are new to the field.

Inhalt
PRELIMINARY: 1. Introduction (~ 10 pages) 1.1. The strong interactions before the Standard Model

1.2. Causality, unitarity and crossing symmetry

1.3. Standard dispersion relations 1.4. Pitfalls of analytic continuation

2. Modern approach to analyticity (~ 15 pages)

2.1. Oehme's argument for Quantum Chromodynamics (QCD)

2.2. Operator Product Expansion (OPE) and analytic continuation

2.3. Analyticity versus Chiral Perturbation Theory (ChPT) and lattice QCD

2.4. Phase and modulus constraints 2.5. Functional-analysis based approach

3. Optimization problems in complex analysis (~ 40 pages)

3.1. Extremal problems in L2 norm

3.2. Extremal problems in L norm

3.3. Generalizations: vector-valued and non-real analytic functions 3.4. Series optimization by conformal mappings

4. Optimization methods for precision predictions in hadron physics (~ 50 pages)

4.1. Unitarity bounds for weak hadronic form factors

4.2. Electromagnetic charge radius of the pion at high precision

4.3. Virtual Compton scattering on proton

4.4. Conformal mappings in perturbative QCD

4.5. Testing quark-hadron duality violation

5. Other applications of analyticity (~ 10 pages) 5.1. Rigorous results from axiomatic field theory

5.2. Detection of broad resonances

5.3. Coupled-channels and three-body decays

5.4. Divergent series and hyperasymptotics

5.5. Outlook