Universal Equation for Efimov States.pdf

a r X i v : c o n d - m a t / 0 2 0 1 2 8 1 v 3 1 3 M a r 2 0 0 3 Universal Equation for Efimov States Eric Braaten,? H.-W. Hammer,? and M. Kusunoki? Department of Physics, The Ohio State University, Columbus, OH 43210, USA (Dated: November 22, 2002) Abstract Efimov states are a sequence of shallow 3-body bound states that arise when the 2-body scattering length is large. Efimov showed that the binding energies of these states can be calculated in terms of the scattering length and a 3-body parameter by solving a transcendental equation involving a universal function of one variable. We calculate this universal function using effective field theory and use it to describe the three-body system of 4He atoms. We also extend Efimov’s theory to include the effects of deep 2-body bound states, which give widths to the Efimov states. PACS numbers: 03.65.Ge, 36.40.-c, 31.15.Ja, 21.45.+v Keywords: Efimov states, universality, 4He, effective field theory ?Electronic address: braaten@mps.ohio-state.edu ?Electronic address: hammer@itkp.uni-bonn.de; Present address: Helmholtz-Institut fu?r Strahlen- und Kernphysik (Abt. Theorie), Universita?t Bonn, 53115 Bonn, Germany ?Electronic address: masa@mps.ohio-state.edu 1 The interactions of nonrelativistic particles (such as atoms) with short-range interactions at extremely low energies are determined primarily by their S-wave scattering length a. If |a| is much larger than the characteristic range l of their interaction, low-energy atoms exhibit universal properties that are insensitive to the details of the interaction potential. In the 2-body sector, the universal properties are simple and familiar. The differential cross section for two identical bosons with relative wave number k ? 1/l and mass m is dσ/d? = 4a2/(1 + k2a2). If a 0, there is also a shallow 2-body bound state (the dimer) with binding energy B2 = h? 2/ma2. In the 3-body sector, there are also universal properties that were first deduced by Efimov [1]. The most r