Parity-violating CFT and the gravitational chiral anomaly

Abstract

We illustrate how the conformal Ward identities (CWI) and the gravitational chiral anomaly completely determine the structure of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">⟨</a:mo><a:mi>T</a:mi><a:mi>T</a:mi><a:msub><a:mi>J</a:mi><a:mn>5</a:mn></a:msub><a:mo stretchy="false">⟩</a:mo></a:math> (graviton-graviton-chiral gauge current) correlator in momentum space. This analysis extends our previous results on the anomaly vertices <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo stretchy="false">⟨</e:mo><e:mi>A</e:mi><e:mi>V</e:mi><e:mi>V</e:mi><e:mo stretchy="false">⟩</e:mo></e:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">⟨</i:mo><i:mi>A</i:mi><i:mi>A</i:mi><i:mi>A</i:mi><i:mo stretchy="false">⟩</i:mo></i:math>, as well as the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo stretchy="false">⟨</m:mo><m:mi>T</m:mi><m:mi>J</m:mi><m:mi>J</m:mi><m:mo stretchy="false">⟩</m:mo></m:math> parity-odd conformal anomaly vertex in general CFTs. The <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo stretchy="false">⟨</q:mo><q:mi>T</q:mi><q:mi>T</q:mi><q:msub><q:mi>J</q:mi><q:mn>5</q:mn></q:msub><q:mo stretchy="false">⟩</q:mo></q:math> plays a fundamental role in the analysis of the conformal backreaction in early Universe cosmology, affecting the particle content and the evolution of the primordial plasma. Our approach is nonperturbative and not Lagrangian based, requiring the inclusion of a single anomaly pole in the solution of the anomaly constraint. The pole and its residue, along with the CWIs, determine the entire correlator in all of its sectors (longitudinal/transverse), all of which are proportional to the same anomaly coefficient. The method does not rely on a specific expression of the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>C</u:mi><u:mi>P</u:mi></u:math>-odd anomalous current, which in free field theory can be represented either by a bilinear fermion current or by a gauge-dependent Chern-Simons current; it relies solely on the symmetry constraints. We compute the correlator perturbatively at one loop in free field theory and verify its exact agreement with the nonperturbative result. A comparison with the perturbative analysis confirms the presence of a sum rule satisfied by the correlator, similar to the parity-even <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo stretchy="false">⟨</w:mo><w:mi>T</w:mi><w:mi>J</w:mi><w:mi>J</w:mi><w:mo stretchy="false">⟩</w:mo></w:math> and the chiral <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mo stretchy="false">⟨</ab:mo><ab:mi>A</ab:mi><ab:mi>V</ab:mi><ab:mi>V</ab:mi><ab:mo stretchy="false">⟩</ab:mo></ab:math>. Published by the American Physical Society 2024

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