Perturbative Pions and the Effective Range Expansion

In these notes we present an introductory review on various topics about low energy pion physics (some kaon physics is discussed as well). Among these, we include the uses of analyticity and unitarity to describe partial wave amplitudes (for which we give accurate and economical parametrizations) and form factors; (forward) dispersion relations; and the use of the Froissart--Gribov representation to evaluate accurately the low energy parameters (scattering lengths and effecti...

Various aspects of the application of Effective Field Theory (EFT) to the Nucleon-Nucleon (NN) interaction are considered. We look for contributions beyond One Pion Exchange which are predicted by Chiral Symmetry. Using the formalism of the Wilson Renormalisation Group (RG) we review power counting in a simplified EFT containing only nucleons. For weak scattering at low energy, we find a natural expansion of the scattering matrix around the unique trivial fixed point of the...

We investigate p-wave pion production in nucleon-nucleon collisions up to next-to-next-to-leading order in chiral effective field theory. In particular, we show that it is possible to describe simultaneously the p-wave amplitudes in the $pn\to pp\pi^-$, $pp\to pn\pi^+$, $pp\to d\pi^+$ channels by adjusting a single low-energy constant accompanying the short-range operator which is available at this order. This study provides a non-trivial test of the applicability of chiral e...

After a brief introduction to chiral perturbation theory, the effective field theory of the standard model at low energies, two recent applications are reviewed: elastic pion-pion scattering to two-loop accuracy and the complete renormalized pion-nucleon Lagrangian to $O(p^3)$ in the chiral expansion.

The inverse amplitude method is analysed to two-loop order in the chiral expansion in the case of $\pi\pi$ scattering and the pion form factors. The analysis is mainly restricted to the elastic approximation but the possible extension to the inelastic case is also discussed in some detail. It is shown how the two-loop approach improves the inverse amplitude method applied to one-loop order in the chiral expansion. For both $\pi\pi$ scattering and the pion form factors, it is ...

We discuss an alternative scheme for including effective range corrections in pionless effective field theory. The standard approach treats range terms as perturbative insertions in the T -matrix. In a finite volume this scheme can lead to singular behavior close to the unperturbed energies. We consider an alternative scheme that resums the effective range but expands the spurious pole of the T -matrix created by this resummation. We test this alternative expansion for severa...

An explicit expression of the generating functional of two-flavor low-energy QCD with external sources in the presence of non-vanishing nucleon densities has been derived recently [1]. Within this approach we derive power counting rules for the calculation of in-medium pion properties. We develop the so-called standard rules for residual nucleon energies of the order of the pion mass and a modified scheme (non-standard counting) for vanishing residual nucleon energies. We als...

Chiral perturbation theory is the effective field theory of the standard model. In this talk, I discuss some applications of this framework to the pion--nucleon system. These are chiral corrections to the S--wave pion--nucleon scattering lengths, the reaction $\pi N \to \pi \pi N$ at threshold and low--energy theorems in $\pi^0$ photoproduction.

A brief introduction to the subject of chiral perturbation theory ($\chi$pt) is presented, including a discussion of effective field theory and applications of $\chi$pt in the arena of purely mesonic interactions as well as in the $\pi N$ sector.

We study the two-pion exchange potential at next-to-next-to-leading order in chiral effective field theory. We propose a new cut-off scheme for the pion loop integrals based on spectral function regularization. We show that this method allows for a consistent implementation of constraints from pion-nucleon scattering. It leads to a much improved description of the partial waves with angular momentum l \geq 2 as compared to the calculation utilizing dimensional regularization.