1. T. A. Minelli, A. Pascolini andC. Villi:Nuovo Cimento A,104, 1589 (1991); hereafter this paper will be referred to as [A].
2. See note (61) in [A].
3. See fig. 23 in [A].
4. Apart from the achievement of our specific aim, this Lagrangian is a sort of theoretical bonanza in QCD studies inspired by the claimed analogy of QCD with dual superconductivity theories. In this case the confinement is expected to be explained in terms of a (nontrivial) chromo-magnetic condensate. To this end a bosonic Lagrangian density describing the chromo-electric field will be coupled to one simulating an Ising (chromo)-magnet. For a general reference seeI. D. Lawrie:A Unified Grand Tour of Theoretical Physics (Adam Hilger, Bristol, 1990); for models of confinements inspired to dual superconductivity seeW. C. Haxton andL. Heller:Phys. Rev. D,22, 1198 (1980);J. W. Alcock, M. J. Burfitt andW. N. Cottingham:Nucl. Phys. B,226, 299 (1983);H. Lehman andT. S. Wu:Nucl. Phys. B,237, 205 (1984);J. S. Ball andA. Caticha:Phys. Rev. D,37, 524 (1988).
5. The numerical solution of eq. (2.5) has been performed choosing as initial conditionsM 0(0), =0,M′ (0)=M 0 ′ * =M ∞ M, where σ is given by (3.4) in order to relate the present calculations to the Padua model, and expressing parameterb by means of (2.12) in terms ofa andc, which remain the only free parameters. In order to be sure of our numerical results, and avoid attributing mathematical significance to purely numerical effects, we have used three different methods of integration,i.e. a Runge-Kutta Merson one, one based on a variable-order, variable-step Adams technique and the Backward Differentiation Formulae (for a description of these methods see, for example,G. Hall andJ. M. Watt (Editors):Modern Numerical Methods for Ordinary Differential Equations (Clarendon Press, Oxford, 1976)), obtaining in all cases the same results. As a function of parametera, keepingc fixed, eq. (2.5) possesses solutions which diverge with increasingr and oscillating ones: between these two classes of solutions it is possible to find one which remains constant up to a value ofr which can be made larger and larger by fine-tuning the value of parametera; this process is finally limited by the computer precision. These discriminating values ofa decrease very rapidly as a functions ofc, and in general a better agreement of the numerical solution with function (2.11) may be obtained for smaller values ofc.