The energy identity for a sequence of Yang-Mills {\alpha}-connections

We prove that the Yang-Mills $\alpha$-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills $\alpha$-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as $\alpha \to 1$, a sequence of Yang-Mills $\alpha$-connections converges to a Yang-Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang-Mills $\alpha$-connections. As an application, we also prove an energy identity for the Yang-Mills flow at the maximal existence time.