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The Kapustin-Witten equations on R^4 are equations for a pair of connectionon the product principle SU(2) bundle and 1-form with values in the product Liealgebra bundle. The 1-form is the Higgs field. A dichotomy is proved to theeffect that either the averaged norm of the Higgs field on large radius spheresgrows faster than a power of the radius, or its 1-form components everywherepairwise commute.

Growth of the Higgs field for solutions to the Kapustin-Witten equations on R^4