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We propose a systematic perturbative method for calculating the binding energy of threshold bound states-states which exist for arbitrary small coupling. The starting point is a (regularized) free theory. Explicit calculations are performed for quantum mechanics with arbitrary short-range potential in ID and various (1 4-l)-dimensional quantum field theories. We check the method by comparing the results with exact formulas available in solvable models.PACS numbers: 03.65. Ge, ll.10.St, 31.15.+q Threshold bound states.-The study of bound state spectra is one of the most interesting and difficult problems in quantum mechanics and especially in quantum field theory. The main difficulty is that the problem is essentially "nonperturbative." Other quantities, like scattering amplitudes, are usually calculated perturbatively for weak coupling. However, the study of even the simplest bound states, like the hydrogen atom, requires the use of special methods.In quantum mechanics in order to study a potential it is approximated by one of the solvable potentials, i.e., those for which the spectrum is known exactly. The deviations are then taken into account systematically in perturbation theory. In quantum field theory the available approximation schemes [1] are usually very complicated and by no means systematic. Even for weakly coupled theories like QED the calculation should be considered an "art" [1]. The standard wisdom in quantum mechanics as well as quantum field theory is that the perturbation theory around free theory is inapplicable for the study of bound states [2]. If all the interactions are considered a small perturbation, to any finite order in perturbation theory there are no poles in the scattering matrix. It is easy to understand the reason behind this conclusion by considering the bound state spectrum of a square-well potential in three dimensions. As the depth of the well decreases the bound states are "swallowed" one by one by the continuum. Beyond some critical depth there are no bound states left in the spectrum. Clearly, weak coupling perturbation theory has nothing to do with such bound states. There is, however, a class of bound states which exist for any value of the coupling constant no matter how small. An example of such "threshold" bound states is 2 the hydrogen atom E n = -^p. When the fine structure constant a -> 0, the spectrum becomes denser but no bound state is lost in the continuum. For such bound states the above conclusion is not warranted.Threshold bound states happen to be quite abundant in quantum mechanics and quantum field theory. They include the lowest bound state of essentially any shortrange potential in ID and 2D quantum mechanics and in 1+1 and 2+1 quantum field theories, all the bound states of QED, and even hadrons in QCD. Perturbation theory does "know" about such states. One encounters threshold or on-shell infrared singularities in scattering amplitudes when momenta approach the threshold.In this Letter we reconsider the use of weak coupling perturbation theory [3] f...

The anomalous magnetic moment of anyons is calculated to leading order in a 1/N expansion. It is shown that the gyromagnetic ratio g remains 2 to the leading order in 1/N . This result strongly supports that obtained in [2], namely that g=2 is in fact exact. 1