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New method (arXiv:2104.11930) is applied to construct cubic interactions ofmassless spin 4 and real scalar fields. In contrast with the case of masslessspin 3 fields (arXiv:2208.05700, 2209.03678, 2210.02842) the procedure requiresto use an unconstrained formulation (arXiv:0702.161) for the Fronsdal theory.It is shown that in the unconstrained formulation there exists a four-parameterfamily of cubic interactions between massless spin 4 and real scalar fields,which contains four derivatives in vertices and is invariant with respect tothe original gauge transformation. These vertices contain cubic interactionsbetween auxiliary and scalar fields too. Eliminating all auxiliary fields fromthe obtained result using the equations of motion for the initial action givesa one-parameter family of cubic vertices for constrained massless spin 4 andreal scalar fields, Such cubic vertices are invariant with respect toconstrained gauge transformations of the Fronsdal theory.

Cubic vertices of interacting massless spin 4 and real scalar fields in unconstrained formulation