Energy-Momentum tensor correlators in ϕ4 theory I: The spin-zero sector

Abstract

We revisit the construction of the renormalized trace Θ of the Energy-Momentum tensor in the four-dimensional $\lambda {\varphi }^4$ theory, using dimensional regularization in $d=4-\epsilon$ dimensions. We first construct several basic correlators such as $〈{\varphi }^2\varphi \varphi 〉$, $〈{\varphi }^4\varphi \varphi 〉$ to order ${\lambda }^2$ and from these the correlators $〈K_I\varphi \varphi 〉$ and $〈K_I{K}_J〉$ 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 $〈\Theta \varphi \varphi 〉$, we construct the operator Θ as a certain linear combination of the basis operators, using the requirements that Θ should vanish on the fixed point and that it should have zero anomalous dimension. Finally, we compute the 2-point function $〈\Theta \Theta 〉$ 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.