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

Abstract

We revisit the construction of the renormalized trace $\Theta$ of the Energy-Momentum tensor in the four-dimensional $\lambda {\varphi }^4$ theory,using dimensional regularization in $d=4-\text{ve}$ 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 $\Theta$ 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 $⟨\Theta \Theta ⟩$ 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.

Comments: 84 pages, clarification comments added, typos corrected