Abstract
We present a complete momentum-space prescription for the renormalisation of tensorial correlators in conformal field theories. Our discussion covers all 3-point functions of stress tensors and conserved currents in arbitrary spacetime dimensions. In dimensions three and four, we give explicit results for the renormalised correlators, the anomalous Ward identities they obey, and the conformal anomalies. For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly.
Highlights
The classic results for CFT 2- and 3-point functions are formulated in position space [1,2,3,4]
For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly
Away from the critical point, our toolkit is generally limited to perturbative methods such as Feynman diagrams, and such calculations are typically performed in momentum space
Summary
The classic results for CFT 2- and 3-point functions are formulated in position space [1,2,3,4]. We review the trace and transverse Ward identities, their role in decomposing the tensorial structure of correlators into scalar form factors, and the classification of conformal Ward identities into primary and secondary These identities can be solved in terms of triple-K integrals yielding the full momentum-space 3-point functions for generic values of the operator and spacetime dimensions. We give explicit results for the regulated form factors, valid in arbitrary spacetime dimensions, and carry out the full renormalisation procedure in dimensions three and four This enables a precise identification of the anomalies and the corresponding anomalous Ward identities satisfied by the renormalised correlators. Appendix D evaluates the counterterm contributions to the stress tensor 3-point function, and appendix E reviews the evanescent tensorial operators appearing in three spacetime dimensions.
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