We examine the properties of two-dimensional conformal field theories (CFTs) with vanishing central charge based on the extended Kac table for c_{(9, 6)} = 0 using a general ansatz for the stress energy tensor residing in a Jordan cell of rank two.

Within this set-up we will derive the operator product expansions (OPEs) and two-point functions of the stress energy tensor T(z) and its logarithmic partner field t(z) and illustrate this by a bosonic field realization.

We will show why our approach may be more promising than those chosen in the literature so far, including a discussion on properties of the augmented minimal model with vanishing central charge such as full conformal invariance of the vacuum as a state in an irreducible representation.

Furthermore we will present a more general solution of another solution to the c → 0 catastrophe based on a logarithmic CFT tensor model. As an example, we consider a tensor product of the well known c = −2 logarithmic CFT with a fourfold Ising model.

We give an overview of the possible configurations and various consequences for the two-point functions and the OPEs of the stress energy tensor T(z) and its logarithmic partner field t(z). We will motivate why, due to the full conformal invariance of the vacuum at c = 0, we should assume a Jordan cell for the identity, since t(z) is presumably a descendant of a new h = 0 field.