A Short Introduction to Non - Relativistic Effective Field Theories

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. 1 NON-RELATIVISTIC BOUND SYSTEMS In nature there are many particle bound systems for which the relative velocity v of the particle in the system is small. Typical examples in the domain of electromagnetic interactions are positronium (e+e−) and muonium (e−μ+), their counterpart in the strong interaction domain are heavy quarkonia (tt̄, bb̄, bc̄, cc̄) and somehow in the middle lie systems like hydrogen, hydrogenoid atoms, pionium (π+π−). All such systems are non-relativistic bound systems. At the beginning of this paper, I will use the example of positronium to show what are the typical (technical) problems inherent to a non-relativistic bound state calculation, even in a pure perturbative situation and I will relate such problems to the existence of several physical scales. Then, I will discuss the further complications that arise in bound state calculations inside a strongly coupled theory like QCD. For the rest of the paper, I will introduce non-relativistic Effective Field Theories (EFT) and I will explain how EFT greatly simplify the bound state calculations, both technically and conceptually, allowing us to obtain systematically interesting and new physical results. 2 AN EXAMPLE of bound state dynamics: POSITRONIUM To understand what are the peculiarities of the bound state interaction, let us first consider positronium. Here, the interaction causing the binding is the electromagnetic interaction: QED fully describes this system and the coupling constant is the fine structure constant α, which is small and completely under control in the region of physical interest. The positronium energy levels are given by the poles of the four-point positron-electron Green function G = 〈0|ψ̄1ψ2ψ̄2ψ1|0〉, which in turn is given as a formal series in terms of Feynman diagrams and thus in terms of α. An integral equation can be written, called Bethe-Salpeter equation, whose solution is the four-point Green function G and whose kernel K is the subset of two-particle irreducible diagrams. Appropriate techniques allow us to write down formally the energy levels as a series of contributions involving kernel insertions on the zeroth-order Green function averaged on the zeroth order wave functions (for a general review see [1], for an explicit calculation see e.g. [2]). There is an important difference between the calculation of a scattering amplitude and the calculation of bound state wave functions and energy levels. Both are given in terms of Green Invited Talk given at the XXIII International Workshop on the Fundamental Problems of High Energy Physics, Protvino (Russia), June 2000. functions projected on initial and final states, but in the first case the initial and final states are on shell, i.e. the wave functions describing initial and final particles are free, while in the second case the initial and final wave functions are the bound state ones and thus bear a dependence on α. This last fact produces that from the expansion of the energy levels it is not trivial to select the set of diagrams contributing at any given order in α. In fact, being the wave function α-dependent, the number of vertices in a diagram do not allow to trace back the order in α of the contribution of the diagram. Precisely, it happens that: the contribution of each graph is a series in α; the leading order in α does not follow from the number of vertices in the graphs. For example, diagrams differing only for the number of ladder photons that they contain, contribute all at the same leading order in α to the energy levels, plus subleading contributions. Therefore, it is necessary to resum an infinite series of diagrams to complete an order in α in the energy levels. In other words, the bound state calculation are ’non-perturbative’ in the binding interaction. Moreover, the contribution of a diagram depends strongly on the gauge. In particular, spurious terms can be generated and canceled only by subsequent contributions. In fact, while in a scattering calculation gauge invariance is manifest term by term at any order of the expansion in α, in a bound state calculation one has to look for particular subsets of diagrams and prove that they are gauge invariant. This turns out to be quite difficult in practice. The same fact can be explained in the following alternative way. Let us consider the calculation of the perturbative corrections to the electron anomalous magnetic moment. This is a typical scattering amplitude calculation and in the integrals related to the actual evaluation of the Feynman diagrams we have only one relevant physical scale: the mass m of the electron. On the other hand, let us consider the calculation of the positronium energy levels. This is a typical bound state calculation and in the integrals related to the actual evaluation of the Feynman diagrams we have three relevant physical scales: the mass m of the electron, the relative momentum p ∼ mv and the bound state energy E ∼ mv2. For positronium v ∼ α ≪ 1 and thus such scales are quite different and get entangled in the calculations causing the Feynman diagrams to contribute in a nontrivial way to the perturbative expansion in α, as explained above [1]. This is what happens in the QED calculation of the bound state energy levels. Alternatively, one can take explicitly advantage of the non-relativistic nature of the system and start directly from the non-relativistic reduction of the QED bound state problem. In this case the zeroth-order problem is the Schrödinger equation