Erik Panzer (panzer@mathematik.hu-berlin.de), May 2013.On the analytic computation of massless propagators in dimensional regularization,Nuclear Physics B 874 (2) (2013) 567--593. doi:10.1016/j.nuclphysb.2013.05.025. arXiv:1305.2161v2.This worksheet shows how to use the functions in "massless_propagators" to extract the epsilon-expansions of three- and four-loop massless propagators, stored in "massless_propagators_periods".These files can be obtained from http://www.math.hu-berlin.de/~panzer/QyRBUTVtYXNzbGVzc19wcm9wYWdhdG9yczYiISIiJSFHJSFHInitially, results are in the G-scheme, i.e. the prefactor G0 is taken out per loop such that the one-loop-master integral in D=4-2*epsilon is just 1/epsilon:LUknc2VyaWVzRyUqcHJvdGVjdGVkRzYkLUkiR0c2IjYkIiIiRiovSShlcHNpbG9uR0YoIiIhTo change the scheme, set the per-loop factor:QyU+SStsb29wRmFjdG9yRzYiIiIiISIiLUknc2VyaWVzRyUqcHJvdGVjdGVkRzYlLUkiR0dGJTYkRiZGJi9JKGVwc2lsb25HRiUiIiEiIiM=Back to the G-scheme:QyU+SStsb29wRmFjdG9yRzYiKiRJI0cwR0YlISIiRigtSSdzZXJpZXNHJSpwcm90ZWN0ZWRHNiQtSSJHR0YlNiQiIiJGMC9JKGVwc2lsb25HRiUiIiE=Graphs are named as in the article. The argument is the list [nu_1,...,nu_E] of nu_e for each edge (edges are numbered as depicted in the figures in the article). The propagator of e is taken to power 1 + nu_e*epsilon.So for propagators raised to power one, the non-planar three-loop propagator has the expansion:LUkiTkc2IjYjNyoiIiFGJ0YnRidGJ0YnRidGJw==Homogeneous weight (transcendentality) can be achieved through suitable rational prefactors, as in the results in the article:LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJ3Nlcmllc0dGJTYkKiYtSSJOR0YnNiM3KiIiIUYxRjFGMUYxRjFGMUYxIiIiLCZGMkYySShlcHNpbG9uR0YnISIjRjUvRjRGMQ==With epsilon-shifts of the propagator powers:LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJ3Nlcmllc0dGJTYkKiYtSSJOR0YnNiM3KiIiIkYxRjFGMUYxRjFGMUYxRjEsJkYxRjFJKGVwc2lsb25HRichIiNGNC9GMyIiIQ==The polynomial for arbitrary insertions:LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJ3Nlcmllc0dGJTYlKiYtSSJOR0YnNiM3Iy1JJHNlcUdGJTYkJkkjbnVHRic2I0kiZUdGJy9GNzsiIiIiIilGOiwmRjpGOkkoZXBzaWxvbkdGJyEiI0Y+L0Y9IiIhIiIk