Gauge invariance for a class of tree diagrams in the standard model

For gauge theory, the matrix element for any physical process is independent of the gauge used. Since this is a formal statement and examples are known where gauge invariance is violated, for any specific process this gauge invariance needs to be checked by explicit calculation. In this paper, gauge invariance is found to hold for a large non-trivial class of processes described by tree diagrams in the standard model-tree diagrams with two external $W$ bosons and any number of external Higgs bosons. This verification of gauge invariance is quite complicated, and is based on a direct study of the difference between different gauges through induction on the number of external Higgs bosons.