Abstract
We explore the validity of using nonrelativistic wave functions for lightly-bound states in relativistic situations. This is shown to be acceptable for scattering problems, but not necessarily for those involving a decay. In the latter case the wave function would need to have special properties. These do not occur for a deuteron-like wave function but do occur for a positronium-like system, the crucial features being that positronium is bound by zero mass exchange.
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Much of the material in this section is similar to what has previously been described by P.V. Landshoff, J.C. Polkinghorne: Phys. Rev.D18, 153 (1978) or I. Schmidt, R. Blankenbecler: Phys. Rev.D16, 1318 (1977)
In more old-fashioned language, this result is equivalent to saying that atq 2=0 the form factor is dominated by its triangle singularity represented by Fig. 1: See R.J. Eden et al.:The analytic S-matrix, Sect. 2.3. Cambridge: Cambridge University Press 1966
The vertex functionV does have an anomalous threshold, but this is harmless (see [2])
For a recent review of such processes, with references to earlier work, see W. Celemaster: 1980 Madison Conference talk, preprint ANL-HEP-CP-80-60
Because of the zero mass of the photon, the branch points in Fig. 3 now coalesce with the poles giving a cut starting atk 22 =m 2 and a cut startingk 21 =m 2. Neark 22 =m 2 the cut can be parametrised by (k 22 −m 2)β−1 where β=αv 2 (see [6]). In the region of interest the velocity ν∼α and so the discontinuity across the cut is strongly peaked near the beginning and the dominant contribution comes from setting β=0 and approximating the cut by the original pole
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Donnachie, A., Horgan, R.R. & Landshoff, P.V. Nonrelativistic wave functions in relativistic physics. Z. Phys. C - Particles and Fields 10, 71–76 (1981). https://doi.org/10.1007/BF01545785
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DOI: https://doi.org/10.1007/BF01545785