Explicit construction of the energy-momentum tensor in the large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> limit

We construct the energy-momentum tensor of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi mathvariant="normal">O</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>N</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> linear sigma model explicitly in the large <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>N</f:mi> </f:math> limit using the exact renormalization group (ERG) formalism. The energy-momentum tensor is obtained as a cutoff dependent functional of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>N</h:mi> </h:math> scalar field variables. Our guiding principles behind the construction are twofold: first the energy-momentum tensor must satisfy the Ward identity for translation and rotation invariance, and second the energy-momentum tensor must satisfy a variant of the exact renormalization group equation. In the limit that the momentum cutoff goes to zero, our energy-momentum tensor gives the one-particle irreducible (1PI) effective action with the insertion of a single energy-momentum tensor operator. We verify that the energy-momentum tensor constructed satisfies the expected trace formula, and that the trace vanishes at the Wilson-Fisher critical point.