https://doi.org/10.1140/epjs/s11734-026-02136-x
Review
A brief introduction to dispersive methods
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland
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Received:
29
September
2025
Accepted:
12
January
2026
Published online:
3
June
2026
Abstract
These lectures aim to provide a basic introduction to dispersive methods and their modern applications to the phenomenology of the Standard Model at low energy. This approach exploits analyticity properties of Green functions and scattering amplitude, often combined with unitarity constraints. To find a logically coherent set of topics in this vast subject, I start with the two-point Green’s function, show that this needs the three-point function as input, which in turn needs the four-point function. The sequence stops here, just like these lectures, because the four-point function is related only to itself (if one ignores inelastic effects). I will discuss these dispersion relations both in the case of toy models, simple scalar theories, as well as in the phenomenologically relevant case of QCD. The two-point function of the electromagnetic current in QCD plays a role in the evaluation of the hadronic vacuum polarization contribution to the 
of leptons. The most important contribution to this two-point function is due to the two-pion intermediate state. To evaluate this, one needs the electromagnetic form factor of the pion as input. The dispersion relation for the latter takes the form of an Omnès problem and the solution is given by the Omnès function, which can be expressed in terms of the phase-shifts for the 
scattering amplitude. The dispersion relation for the scattering amplitude takes the form of the so-called Roy equations. In this case, we encounter for the first time a left-hand cut and see that this is constrained by crossing symmetry. If one takes into account also the nonlinear constraints given by unitarity one ends up with the Roy equations, which take the form of coupled, non-linear integral equations. A brief discussion of their numerical solutions and a few selected applications concludes the lectures.
© The Author(s) 2026
modified publication 2026
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