Renormalization group, hidden symmetries and approximate ward identities in the XYZ model

Using renormalization group methods, we study the Heisenberg-Ising XYZ chai n in an external magnetic field directed as the z axis, in the case of smal l coupling J(3) in the z direction. In particular, we focus our attention o n the asymptotic behaviour of the spin correlation function in the directio n of the magnetic field and the singularities of its Fourier transform. An expansion for the ground state energy and the effective potential is der ived, which is convergent if the running coupling constants are small enoug h. Moreover, by using hidden symmetries of the model, we show that this con dition is indeed verified, if J(3) is small enough, and we derive an expans ion for the spin correlation function. We also prove, by means of an approx imate Ward identity, that a critical index, related with the asymptotic beh aviour of the correlation function, is exactly vanishing, together with oth er properties, so obtaining a rather detailed description of the XYZ correl ation function.