Abstract
Effective actions, containing the logarithm of a functional Dirac determinant, appear in 1/N expansions of fermionic theories (N being the number of flavours). We introduce a method to find symmetric solutions of the corresponding non-linear and non-local saddle-point equations. This method consists in using the scattering data of the rotationally symmetric Dirac equation in two dimensions with the angular mometum as a spectral parameter. We apply the method to fermionic theories with scalar and pseudoscalar quartic couplings. The effective action that generates the 1/N expansion admits a closed form in terms of the scattering data only in the particular case when the model is integrable (Gross-Neveu and Chiral Gross-Neveu model). No instanton solutions are present in these two particular cases. This fact, together with the exact results for theS-matrix and form factors, suggests that the 1/N expansion could be convergent. In the general case, the quantum model has an additional dimensionless parameterg R·g R→±∞ gives the Chiral Gross-Neveu model. Wheng R>0, tachyons appear. Forg R→0−, andg R→−∞, generically complex-action instantons exists, indicating a possibly Borel-summable 1/N expansion.
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