We consider the critical gravity theory with a scalar field in four dimensions. We find that this theory has the solution corresponding to the Minkowski background, in addition to the de Sitter (dS) and the anti-de Sitter (AdS) backgrounds, depending on whether the action includes the cosmological term or not. We should note that the Minkowski background is the solution which cannot be obtained in the model without a scalar field. Furthermore, when we choose the parameters to eliminate the scalar graviton, the additional scalar field is also simultaneously eliminated. At the critical point, we show that the Abbott-Deser (AD) mass of the Schwarzschild-de Sitter (SdS) black hole and the energy for the massless graviton vanish, whose situation is not changed from the model without a scalar field.

• Received 15 March 2012

DOI:https://doi.org/10.1103/PhysRevD.85.127503