Abstract
The three-loop master integrals for ladder-box diagrams with one massive leg are computed from an eighty-five by eighty-five system of differential equations, solved by means of Magnus exponential. The results of the considered box-type integrals, as well as of the tower of vertex- and bubble-type master integrals associated to subtopologies, are given as a Taylor series expansion in the dimensional regulator parameter epsilon = (4-d)/2. The coefficients of the series are expressed in terms of uniform weight combinations of multiple polylogarithms and transcendental constants up to weight six. The considered integrals enter the next-to-next-to-next-to-leading order virtual corrections to scattering processes like the three-jet production mediated by vector boson decay, V* -> jjj, as well as the Higgs plus one-jet production in gluon fusion, pp -> Hj.
Highlights
Magnus exponentialWe can use Magnus exponential [31, 38] to define a matrix B that implements a change of basis f → g, f ≡ Bg, Aσ ≡ B−1AσB − B−1∂σB,
Decomposition of the whole amplitude in terms of master integrals (MI’s)
The coefficients of the series are expressed in terms of uniform weight combinations of multiple polylogarithms and transcendental constants up to weight six
Summary
We can use Magnus exponential [31, 38] to define a matrix B that implements a change of basis f → g, f ≡ Bg, Aσ ≡ B−1AσB − B−1∂σB,. Such that the new basis of MI’s fulfills a canonical system of differential equations,. G is called canonical basis of MI’s. [31], in order to build the matrix B of (2.13), let us introduce Magnus exponential matrix [38, 39]. For a generic matrix M = M (t), this is defined as eΩ[M (t)] ,.
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