A misleading Wilsonian fixed point

We exhibit here, for a scalar theory, an apparently non-trivial Wilsonian fixed point, which surprisingly describes a free theory. This modest note is an observation which can be of interest in the framework of functional methods in Quantum Field Theory. The search for Wilsonian fixed points is important in Quantum Field Theory, as these represent configurations independent of the energy scale of the process under study, and hence for which quantum fluctuations are controlled. We exhibit here, for a scalar theory, an exact solution of a renormalisation group equation, leading to an IR fixed point configuration, which is not analytical in the dynamical field. Although this configuration seems to describe an interacting theory, its non-analiticity prevents from defining a loop expansion. We show then that the path integral quantization of an UV configuration similar to the latter fixed point can be recovered by the quantization of a free theory, by implementing a functional change of variable in the partition function. The non-analytic and apparently non-trivial fixed point is therefore just another representation of a free theory, which explains why it satisfies an exact renormalization equation. The Jacobian generated by the functional change of variable is essential for the equivalence between the would-be non-trivial fixed point and a free theory. More generally, the definition of the integration measure can play an important role in the functional integral, especially for gauge theories defined on curved space times [1]. Also, the Jacobian arising from a Weyl transformation of the world sheet metric, in the framework of non-critical strings, it responsible for the generation of the Liouville action [2]. Finally, it has been shown in the framework of scalar electrodymanics, that the parametrization of matter using polar coordinates, in the field space, also leads to a non-trivial quantization [3]. Wegner Houghton exact renormalization equation. Consider a scalar field Φ = φ + φ̃, where the IR field φ has non-vanishing Fourier components for momenta smaller than some value k, and the UV field φ̃ has non-vanishing Fourier components for momenta p such that k < |p| ≤ Λ, where Λ is a fixed cut off. The bare Euclidean action we consider is, in D dimensions, SΛ[Φ] = ∫ dx { 1 2 ZΛ(Φ)∂μΦ∂ Φ+ UΛ(Φ) }