In view of current interest in the trapping of antihydrogen () atoms at very low temperatures, we have carried out a calculation of s-wave hydrogen–antihydrogen scattering at very low energies, using the Kohn variational method, taking into account rearrangement scattering into the three channels that contain positronium in its ground state and lie closest to threshold. We find that our values for the elastic cross section are in good agreement with the values obtained by Jonsell et al (Jonsell et al 2001 Phys. Rev. A 64 052712) using a distorted wave approximation. However, our values for the total rearrangement cross section are much larger than their values. In particular, the largest such cross section is for the N = 23 s-state of protonium and positronium in its ground state, a channel that was estimated to make a negligible contribution by Jonsell et al. As a consequence of our much larger values for the total rearrangement cross section, we predict that cooling of by cold H would be considerably less efficient than was found to be the case by Jonsell et al.

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A large group of experimentalists is currently working on the preparation of the ATHENA (ApparaTus for High precision Experiments on Neutral Antimatter), which is to be used at CERN to carry out experiments on antihydrogen to test the CPT invariance of Quantum Field Theory and also Einstein's Principle of Equivalence. It is intended to carry out these experiments by trapping at very low temperatures ( < 1 K ) in an inhomogeneous magnetic field. The size and extent of the collaboration on this project can be seen from ref. [14], which is entitled 'Antihydrogen production and precision experiments' and has 54 authors from 7 different countries.In this chapter, we consider some of the interesting theoretical problems that arise when in which a positron is bound to an antiproton, interacts with H, He andThe theoretical work that has been carried out for is described. This includes a very recent preliminary calculation that we have carried out using the Kohn variational method. Methods of extending this work to scattering by He and are discussed. These processes are the main cause of loss of trapped Thus there is considerable interest in obtaining cross sections for them, in order to determine annihilation rates under various experimental conditions. The aim is to make it possible for experimentalists to choose the conditions which maximise the lifetime of the 53

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