Gilberto Colangelo

These lectures aim to provide a basic introduction to dispersive methods and their modern applications to the phenomenology of the Standard Model at low en

We provide a nonperturbative determination of the scheme- and scale-independent low-energy constant ℓ 7 , appearing in the QCD effective chiral Lagrangian at next-to-leading order, by means of lattice QCD simulations with N f = 2 + 1 quark flavors. We adopt staggered fermions and extract ℓ 7 from the pion mass splitting by suitably generalizing the method introduced by Frezzotti [] for the Wilson discretization. Adopting 12 gauge ensembles with three different values of the pion mass, and four different values of the lattice spacing, we are able to achieve controlled extrapolations toward the continuum, infinite volume, and chiral limits. Our final result ℓ 7 × 10 3 = 2.79 ( 58 ) stat ( 19 ) syst = 2.79 ( 61 ) tot agrees with and substantially improves on previous determinations.

A reliable calculation of radiative corrections to τ → π π ν τ decays is an important prerequisite for using hadronic τ decays for a data-driven evaluation of the hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon, a μ HVP , LO [ π π , τ ] . In this Letter, we present an improved model-independent analysis of these radiative corrections, including, for the first time, effects beyond pointlike pions in the evaluation of the loop diagrams. These structure-dependent corrections, implemented via a dispersive representation of the pion form factor, lead to significant changes compared to previous calculations due to enhancements near the ρ ( 770 ) resonance. We also devise strategies for the matching to chiral perturbation theory and a stable implementation of the real corrections down to the two-pion threshold, which shows that some higher-order isospin-breaking corrections need to be kept due to a strong threshold enhancement. Finally, we perform dispersive fits to the currently available τ → π π ν τ spectra and discuss the consequences for isospin-breaking corrections in the evaluation of a μ HVP , LO [ π π , τ ] .

A bstract: This is the first of a series of papers devoted to a detailed analysis of isospin-breaking effects in the ππ scattering amplitude and the vector form factor of the pion. Isospin breaking originates from the mass difference between up and down quarks and from electromagnetic effects. The latter can be further split into effects due to the pion-mass difference and the remaining virtual and real photonic effects. In this paper, we derive the modifications to the Roy equations for ππ scattering due to the mass difference between the charged and the neutral pion. We solve the equations numerically after matching them to the chiral representation of the ππ scattering amplitude evaluated in the same approximation, which is also provided. Numerical results are presented and discussed in detail.

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.

A correction to this paper has been published: https://doi.org/10.1007/JHEP08(2022)295