A note on the total action of 4D Gauss–Bonnet theory

Abstract Recently, a novel four-dimensional Gauss–Bonnet theory has been suggested as a limiting case of the original D -dimensional theory with singular Gauss–Bonnet coupling constant $$\alpha \rightarrow \alpha /(D-4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mo>→</mml:mo><mml:mi>α</mml:mi><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> . The theory is proposed at the level of field equations. Here we analyse this theory at the level of action. We find that the on-shell action and surface terms split into parts, one of which does not scale like $$(D-4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> . The limiting $$D\rightarrow 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>→</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> procedure, therefore, gives unphysical divergences in the on-shell action and surface terms in four dimensions. We further highlight various issues related to the computation of counterterms in this theory.

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