Evaluating Feynman integrals by the hypergeometry

The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear partial differential equations satisfied by the corresponding scalar integrals. Taking examples of the one-loop $B_{_0}$ and massless $C_{_0}$ functions, as well as the scalar integrals of two-loop vacuum and sunset diagrams, we verify our expressions coinciding with the well-known results of literatures. Based on the multiple hypergeometric functions of independent kinematic variables, the systems of homogeneous linear partial differential equations satisfied by the mentioned scalar integrals are established. Using the calculus of variations, one recognizes the system of linear partial differential equations as stationary conditions of a functional under some given restrictions, which is the cornerstone to perform the continuation of the scalar integrals to whole kinematic domains numerically with the finite element methods. In principle this method can be used to evaluate the scalar integrals of any Feynman diagrams.Comment: 39 pages, including 2 ps figure

Cooperative Label-Free Moving Target Fencing for Second-Order Multi-Agent Systems with Rigid Formation

This paper proposes a label-free controller for a second-order multi-agent system to cooperatively fence a moving target of variational velocity into a convex hull formed by the agents whereas maintaining a rigid formation. Therein, no label is predetermined for a specified agent. To attain a rigid formation with guaranteed collision avoidance, each controller consists of two terms: a dynamic regulator with an internal model to drive agents towards the moving target merely by position information feedback, and a repulsive force between each pair of adjacent agents. Significantly, sufficient conditions are derived to guarantee the asymptotic stability of the closed-loop systems governed by the proposed fencing controller. Rigorous analysis is provided to eliminate the strong nonlinear couplings induced by the label-free property. Finally, the effectiveness of the controller is substantiated by numerical simulations

Theory of magnetoelectric photocurrent generated by direct interband transitions in semiconductor quantum well

A linearly polarized light normally incident on a semiconductor quantum well with spin-orbit coupling may generate pure spin current via direct interband optical transition. An electric photocurrent can be extracted from the pure spin current when an in-plane magnetic field is applied, which has been recently observed in the InGaAs/InAlAs quantum well [Dai et al., Phys. Rev. Lett. 104, 246601 (2010)]. Here we present a theoretical study of this magnetoelectric photocurrent effect associated with the interband transition. By employing the density matrix formalism, we show that the photoexcited carrier density has an anisotropic distribution in k space, strongly dependent on the orientation of the electron wavevector and the polarization of the light. This anisotropy provides an intuitive picture of the observed dependence of the photocurrent on the magnetic field and the polarization of the light. We also show that the ratio of the pure spin photocurrent to the magnetoelectric photocurrent is approximately equal to the ratio of the kinetic energy to the Zeeman energy, which enables us to estimate the magnitude of the pure spin photocurrent. The photocurrent density calculated with the help of an anisotropic Rashba model and the Kohn-Luttinger model can produce all three terms in the fitting formula for measured current, with comparable order of magnitude, but discrepancies are still present and further investigation is needed.Comment: 13 pages, 9 figures, 2 table

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller's transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.Comment: 47 pages, 2 figures. arXiv admin note: text overlap with arXiv:2209.1519