  We generalize the electromagnetic duality between a massless, canonical
scalar field and a 2-form gauge field in 4-dimensional spacetime to
scalar-tensor theories. We derive the action of 2-form gauge field that is dual
to two kinds of scalar-tensor theories: shift symmetric K-essence theory and
the shift symmetric Horndeski theory up to quadratic in scalar field. The
former case, the dual 2-form has a nonlinear kinetic term. The latter case, the
dual 2-form has non-trivial interactions with gravity through Einstein tensor.
In both case, the duality relation is modified from usual case, that is, the
dual 2-form field is not simply given by the Hodge dual of the gradient of the
scalar field.
