Abstract
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by continuous scale invariance, which is broken into discrete scale invariance when three or more particles come into play. This symmetry induces a universal behavior for different systems that is independent of the details of the underlying interaction and rooted in the smallness of the ratio ℓ/ a B ≪ 1, where the length a B is associated with the binding energy of the two-body system [Formula: see text], and ℓ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the onset of system-specific nonuniversal effects. In this review, we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which a B → ∞, and we show that nuclear systems belong to this class of universality.
Highlights
Bound systems define a class of universality; the particles stay most of the time outside the interaction range, and many of their properties can be explained in terms of the probability of being inside the classically forbidden region
The 4He2 molecule has an extremely low binding energy, E2 ≈ 1 mK, several orders of magnitude smaller than the typical interaction energy [2], 2/mrv2dW ≈ 1.5 K, given in terms of its van der Waals length, rvdW = 5.08 a0. Nuclear physics is another example; the deuteron binding energy is E2 = 2.22456 MeV, which is much smaller than the typical nuclear energy, 2/m 2 ≈ 20 MeV
Two-body systems manifest continuous scale invariance (CSI), depicted in Figure 1, which is broken into discrete scale invariance (DSI) in three-body systems
Summary
The dynamics of two-body systems inside the universal window is highly independent of the details of their mutual interaction. Though the zero-range case was used many times as a first approximation to describe systems inside the universal window, we proceed differently and start our description from the ERE (Equation 2) relating the three parameters that determine the low-energy dynamics of the system. It can be cast in the following compact form: rea = 2rBaB, where we have introduced the length rB = a − aB—which, together with the energy length aB, completely determines the S-matrix of systems that have one bound state [60, 61]. To study the dynamics of the systems inside the universal window, we make use of a two-parameter short-range potential and consider this potential a minimal low-energy representation of the two-particle interaction fixed by two low-energy data points, aB (or a) and rB (or re)
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