Renormalization Group, hidden symmetries and approximate Ward identities in the XYZ model, I
Using renormalization group methods, we study the Heisenberg-Ising XYZ chain in an external magnetic field directed as the z axis, in the case of small coupling J_3 in the z direction. We study the asymptotic behaviour of the spin space-time correlation function in the direction of the magnetic field and the singularities of its Fourier transform. The work is organized in two parts. In the present paper an expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. In the subsequent paper, by using hidden symmetries of the model, we show that this condition is indeed verified, if J_3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing.