A `novel' pure theory of Einstein-Gauss-Bonnet gravity in four-spacetime dimensions can be constructed by rescaling the Gauss-Bonnet coupling constant, seemingly eluding Lovelock's theorem. Recently, however, the well-posedness of this model has been called into question. Here we apply a `dimensional regularization' technique, first used by Mann and Ross to write down a $D\to2$ limit of general relativity, to the case of pure Einstein-Gauss-Bonnet gravity. The resulting four-dimensional action is a particular Horndeski theory of gravity matching the result found via a Kaluza-Klein reduction over a flat internal space. Some cosmological solutions of this four-dimensional theory are examined. We further adapt the technique to higher curvature Lovelock theories of gravity, as well as a low-energy effective string action with an $\alpha'$ correction. With respect to the $D\to4$ limit of the $\alpha'$-corrected string action, we find we must also rescale the dilaton to have a non-singular action in four dimensions. Interestingly, when the conformal rescaling $\Phi$ is interpreted as another dilaton, the regularized string action appears to be a special case of a covariant multi-Galileon theory of gravity.

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D→4 爱因斯坦-高斯-博内引力及其他

可以通过重新调整高斯-博内耦合常数来构建四时空维度的爱因斯坦-高斯-博内引力的“新颖”纯理论，这似乎避开了洛夫洛克定理。然而，最近，这个模型的适定性受到了质疑。在这里，我们将“维数正则化”技术应用到纯爱因斯坦-高斯-博内引力的情况下，曼恩和罗斯首先使用它来写下广义相对论的 $D\to2$ 极限。由此产生的四维作用是一种特殊的 Horndeski 重力理论，与通过在平坦内部空间上通过 Kaluza-Klein 归约发现的结果相匹配。检验了这个四维理论的一些宇宙学解。我们进一步将该技术应用于更高曲率的洛夫洛克引力理论，以及具有 $\alpha'$ 校正的低能量有效弦乐动作。关于 $\alpha'$-corrected string action 的 $D\to4$ 限制，我们发现我们还必须重新缩放膨胀以在四个维度上具有非奇异动作。有趣的是，当共形重标度 $\Phi$ 被解释为另一个膨胀时，正则化的弦作用似乎是协变多伽利略引力理论的一个特例。