Energy-Momentum tensor correlators in $\phi^4$ theory I: The spin-zero sector

We revisit the construction of the renormalized trace $\Theta$ of the Energy-Momentum tensor in the four-dimensional $\lambda\phi^4$ theory,using dimensional regularization in $d=4-\ve$ dimensions. We first construct several basic correlators such as $\braket{\phi^2 \phi\phi}$, $\braket{\phi^4 \phi \phi}$ to order $\lambda^2$ and from these the correlators $\braket{K_I \phi \phi}$ and $\braket{K_I K_J}$ with $K_I$ the basis of dimension $d$ operators. We then match the limit of their expressions on the Wilson-Fisher fixed point to the corresponding expressions obtained in Conformal Field Theory. Then, using the 3-point function $\braket{\Theta\phi\phi}$, we construct the operator $\Theta$ as a certain linear combination of the basis operators, using the requirements that $\Theta$ should vanish on the fixed point and that it should have zero anomalous dimension. Finally, we compute the 2-point function $\braket{\Theta\Theta}$ and we show that it obeys an eigenvalue equation that gives additional information about the internal structure of the Energy-Momentum tensor operator to what is already contained in its Callan-Symanzik equation.