A knot or hitch which holds its form rigidly is also called a loop (Owen 1993). Formally, if is a topological space and, a loop is a continuous map such that and. Most simply, a loop is a closed curve whose initial and final points coincide in a fixed point known as the basepoint. The time to generate the corresponding symmetric tensors grows factorially with the rank of tensor integrals. The term 'loop' has a number of meanings in mathematics. Which utilizes Mathematica’s built-in function P e r m u t a t i o n s. These tensors are generated by a Package- X internal function (inside I n d e x A l g ‘), ⟶ μ … in the notation of Denner and Dittmaier ( 2006), containing products of external momentum four-vectors p μ i and the metric tensor g μ ν. Running Time: 5ms to 10s for integrals typically occurring in practical computations longer for higher rank tensor integrals. Unusual features: Includes rudimentary routines for tensor algebraic operations and for performing traces over Dirac gamma matrices. Restrictions: One-loop integrals are limited to those involving no more than three propagator factors. For start, test, incr, body executes start, then repeatedly evaluates body and incr until test fails to give True. Solution method: Passarino-Veltman reduction formula, Denner-Dittmaier reduction formulae, and two new reduction algorithms described in the manuscript. Nature of problem: Analytic calculation of one-loop integrals in relativistic quantum field theory for arbitrarily high-rank tensor integrals and any kinematic configuration of real-valued external invariants and internal masses. ![]() RAM required for execution: 10 MB, depending on size of computation ![]() Operating systems: Windows, Mac OS X, Linux (or any system supporting Mathematica 8.0 or higher) Programming language: Mathematica (Wolfram Languange) Licensing provisions: Standard CPC license, Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. The package is intended to be used both as a research tool and as an educational tool. Also included is a routine to compute traces of products of Dirac matrices, and a collection of projectors to facilitate the computation of fermion form factors at one-loop. Emphasis is on evaluation speed, on readability of results, and especially on user-friendliness. Output expressions can be readily evaluated numerically and manipulated symbolically with built-in Mathematica functions. Package- X computes arbitrarily high rank tensor integrals with up to three propagators, and gives compact expressions of UV divergent, IR divergent, and finite parts for any kinematic configuration involving real-valued external invariants and internal masses. Package- X, a Mathematica package for the analytic computation of one-loop integrals dimensionally regulated near 4 spacetime dimensions is described.
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