Perturbative Pions and the Effective Range Expansion

I discuss effective field theories for heavy bound systems, particularly bound systems involving two heavy quarks. The emphasis is on the relevant concepts and on interesting physical applications and results.

A new limit of pion form factor at very large $Q^2$ is obtained by using a pion wave function determined from an effective chiral field theory of mesons. It shows that when $Q^2>>(1.8GeV)^2$ the pion form factor reaches the asymptotic limit ${\alpha_s(Q^2)\over Q^2}$.

Analyses of the pion valence-quark distribution function (DF), ${u}^\pi(x;\zeta)$, which explicitly incorporate the behaviour of the pion wave function prescribed by quantum chromodynamics (QCD), predict ${u}^\pi(x\simeq 1;\zeta) \sim (1-x)^{\beta(\zeta)}$, $\beta(\zeta \gtrsim m_p)>2$, where $m_p$ is the proton mass. Nevertheless, more than forty years after the first experiment to collect data suitable for extracting the $x\simeq 1$ behaviour of ${u}^\pi$, the empirical sta...

The subject of effective interactions is introduced and applications in both quantum mechanics and quantum field theory are presented. In particular the use of chiral perturbation theory as an effective low energy description of QCD is developed and it is argued that such methods rally are effective in both the meson and baryonic sectors.

The reasons why a considerable effort is made to resolve the low energy structure of QCD are discussed. The effective field theory used for this purpose is illustrated with the recent progress made in the predictions for pipi scattering and in understanding the low energy properties of the theory in the large Nc limit.

We present a simple introduction to the techniques of effective field theory (EFT) and their application to QCD. For problems with more than one energy scale, the EFT approach is a useful alternative to more traditional model-building strategies. The most relevant such problem for this discussion is that of making contact between QCD and the hadronic phase of matter. As a simple example, an EFT calculation of the bound states of hydrogen within QED is sketched. A more signifi...

The quark-connected and the quark-disconnected Wick contractions contributing to the pion's scalar form factor are computed in the two and in the three flavour chiral effective theory at next-to-leading order. While the quark-disconnected contribution to the form factor itself turns out to be power-counting suppressed its contribution to the scalar radius is of the same order of magnitude as the one of the quark-connected contribution. This result underlines that neglecting q...

The current status of effective field theory (EFT) descriptions of nucleon-nucleon (NN) interactions is briefly reviewed, and a new formulation of EFT which treats pion interactions perturbatively is presented. This approach differs from the Kaplan-Savage-Wise (KSW) expansion in that the singular short distance part of the pion tensor interaction is summed to all orders.

The various dynamical scales below the pion mass involved in $\pi^{+}$ $\pi^{-}$ atoms are sequentially integrated out using non-relativistic effective field theory techniques. This allows us to systematically organise the corrections to the energy levels and decay width. We present our results in terms of a single unknown constant which may be obtained by matching to the Chiral Lagrangian with electromagnetic interactions at two loops.

Novel techniques are presented, which identify the chiral power-counting regime (PCR), and realize the existence of an intrinsic energy scale embedded in lattice QCD results that extend outside the PCR. The nucleon mass is considered as a benchmark for illustrating this new approach. Using finite-range regularization, an optimal regularization scale can be extracted from lattice simulation results by analyzing the renormalization of the low energy coefficients. The optimal sc...