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We extend the analysis of Higgs-regulated planar amplitudes of N=4supersymmetric Yang-Mills theory to four loops for the four-gluon amplitude andto two loops for the five-gluon amplitude. Our calculations are consistent witha proposed all-loop ansatz for planar MHV n-gluon amplitudes that is the analogof the BDS ansatz in dimensional regularization. In all cases considered, wehave verified that the IR-finite parts of the logarithm of the amplitudes havethe same dependence on kinematic variables as the corresponding functions indimensionally-regulated amplitudes (up to overall additive constants, which wedetermine). We also study various Regge limits of N=4 SYM planar n-gluonamplitudes. Euclidean Regge limits of Higgs-regulated n \geq 4 amplitudes yieldresults similar in form to those found using dimensional regularization, butwith different expressions for the gluon trajectory and Regge verticesresulting from the different regulator scheme. We also show that the Reggelimit of the four-gluon amplitude is dominated at next-to-leading-log order byvertical ladder diagrams together with the class of vertical ladder diagramswith a single H-shaped insertion.

More loops and legs in Higgs-regulated N=4 SYM amplitudes