A note on the definition of gravitational energy for quadratic curvature gravity via topological regularization

Within the framework of four-dimensional quadratic curvature gravities in the appearance of a negative cosmological constant,a definition for the gravitational energy of solutions with anti-de Sitter(AdS)asymptotics was put forward by Giribet et al.[Phys.Rev.D 98044046(2018)].This was achieved by adding proper topological invariant terms to the gravity action to render the variation problem well-posed.We prove that the definition via the procedure of topological regularization can be covered by our previous work[Int.J.Mod.Phys.A 352050102(2020)]in four dimensions.Motivated by this,we further generalize the results to generic diffeomorphism invariant theories of gravity in arbitrary even dimensions.

Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to HERA data

We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution equation with fixed and running coupling constants in the saturation region. The analytic solution of the S-matrix shows that the exp(-O(η^2))rapidity dependence of the solution with the fixed coupling constant is replaced by the exp(-O(η^3/2))dependence in the smallest dipole running coupling case, as opposed to obeying the law found in our previous publication, where all the solutions of the next-to-leading order evolution equations comply with exp(-O(η))rapidity dependence once the QCD coupling is switched from the fixed coupling to the smallest dipole running coupling prescription. This finding indicates that the corrections of the sub-leading double logarithms in the Sudakov suppressed evolution equation are significant, which compensate for a part of the evolution decrease of the dipole amplitude introduced by the running coupling effect. To test the analytic findings, we calculate the numerical solutions of the Sudakov suppressed evolution equation, and the numerical results confirm the analytic outcomes. Moreover, we use the numerical solutions of the evolution equationto fit the HERA data. This demonstrates that the Sudakov suppressed evolution equation can achieve a good quality fit to the data.