Three-loop planar master integrals for heavy-to-light form factors

We calculate analytically the three-loop planar master integrals relevant for heavy-to-light form factors using the method of differential equations. After choosing a proper canonical basis, the boundary conditions are easy to be determined, and the solution of differential equations is greatly simplified. The results for seventy-one master integrals at general kinematics are all expressed in terms of harmonic polylogarithms.Comment: 18 pages, 2 figure

Role of the effective payoff function in evolutionary game dynamics

In most studies regarding evolutionary game dynamics, the effective payoff, a quantity that translates the payoff derived from game interactions into reproductive success, is usually assumed to be a specific function of the payoff. Meanwhile, the effect of different function forms of effective payoff on evolutionary dynamics is always left in the basket. With introducing a generalized mapping that the effective payoff of individuals is a non-negative function of two variables on selection intensity and payoff, we study how different effective payoff functions affect evolutionary dynamics in a symmetrical mutation-selection process. For standard two-strategy two-player games, we find that under weak selection the condition for one strategy to dominate the other depends not only on the classical {\sigma}-rule, but also on an extra constant that is determined by the form of the effective payoff function. By changing the sign of the constant, we can alter the direction of strategy selection. Taking the Moran process and pairwise comparison process as specific models in well-mixed populations, we find that different fitness or imitation mappings are equivalent under weak selection. Moreover, the sign of the extra constant determines the direction of one-third law and risk-dominance for sufficiently large populations. This work thus helps to elucidate how the effective payoff function as another fundamental ingredient of evolution affect evolutionary dynamics.Comment: This paper has been accepted to publish on EP

Influence of initial distributions on robust cooperation in evolutionary Prisoner's Dilemma

We study the evolutionary Prisoner's Dilemma game on scale-free networks for different initial distributions. We consider three types of initial distributions for cooperators and defectors: initially random distribution with different frequencies of defectors; intentional organization with defectors initially occupying the most connected nodes with different fractions of defectors; intentional assignment for cooperators occupying the most connected nodes with different proportions of defectors at the beginning. It is shown that initial configurations for cooperators and defectors can influence the stationary level of cooperation and the evolution speed of cooperation. Organizations with the vertices with highest connectivity representing individuals cooperators could exhibit the most robust cooperation and drive evolutionary process to converge fastest to the high steady cooperation in the three situations of initial distributions. Otherwise, we determine the critical initial frequencies of defectors above which the extinction of cooperators occurs for the respective initial distributions, and find that the presence of network loops and clusters for cooperators can favor the emergence of cooperation.Comment: Submitted to EP

Conical Defects, Black Holes and Higher Spin (Super-)Symmetry

We study the (super-)symmetries of classical solutions in the higher spin (super-)gravity in AdS$_3$. We show that the symmetries of the solutions are encoded in the holonomy around the spatial circle. When the spatial holonomies of the solutions are trivial, they preserve maximal symmetries of the theory, and are actually the smooth conical defects. We find all the smooth conical defects in the $sl(N), so(2N+1),sp(2N), so(2N), g_2$, as well as in $sl(N|N-1)$ and $osp(2N+1|2N)$ Chern-Simons gravity theories. In the bosonic higher spin cases, there are one-to-one correspondences between the smooth conical defects and the highest weight representations of Lie group. Furthermore we investigate the higher spin black holes in $osp(3|2)$ and $sl(3|2)$ higher spin (super-)gravity and find that they are only partially symmetric. In general, the black holes break all the supersymmetries, but in some cases they preserve part of the supersymmetries.Comment: 48 pages; more clarifications on conical defects in supersymmetric cas

Quasi parton distribution functions at NNLO: flavor non-diagonal quark contributions

We present a next-to-next-to-leading order (NNLO) calculation of the quasi parton distribution functions (Quasi-PDFs) in the large momentum effective theory (LaMET). We focus on the flavor non-diagonal quark-quark channel and demonstrate the LaMET factorization at the NNLO accuracy in the modified minimal subtraction scheme. The matching coefficient between the quasi-PDF and the light-cone PDF is derived. This provides a first step towards a complete NNLO analysis of quasi-PDFs and to better understand the nucleon structures from the first principle of QCD.Comment: 10 pages, 3 figures; v2: accepted for publication in Physical Review D as a rapid communicatio

Black holes in Truncated Higher Spin AdS3_3 Gravity

We study the higher spin black holes in a truncated version of higher spin gravity in $AdS_3$. This theory contains only finite number of even spins s=2,4,...,2N. We mainly focus on the simplest case, so-called (Type I and II) spin ${\tilde 4}$ gravity, which contains only spin 2 and spin 4 fields. This spin ${\tilde 4}$ gravity is as simple as spin 3 gravity, thus provides another example to test various ideas on higher spin gravity. We find that the asymptotical symmetry of this spin ${\tilde 4}$ gravity is a classical W(2,4)-symmetry. Moreover, we study the black hole solution with pure spin 4 hair and discuss its thermodynamics. One important feature of this black hole is that its entropy could be written in compact forms. Furthermore, we investigate a $G_2$ generated higher spin gravity. This higher spin gravity only contains spin 2 and spin 6 fields which makes it different from other kinds of higher spin gravity. We find the corresponding black hole with spin 6 hair, and discuss its thermodynamics analytically. It turns out that the black holes with spin 4 or spin 6 hair constructed in this paper are the only black holes with single higher spin hair, besides the spin 3 black hole found in arXiv:1103.4304.Comment: 23 pages;minor revision, references added; published versio