The scalar-only second coefficient

Following X-A, with $\mathcal{V}[H]\psi ={\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \beta }[H]{\partial }_{\beta }\psi )$ and $\mathcal{W}[H,k]\psi ={\partial }_{\alpha }({\sigma }_{\epsilon }({a_1}^{\alpha \beta }[k]+{a_2}^{\alpha \beta }[H]){\partial }_{\beta }\psi )$,

Here $\mathcal{W}[H,k]x^{\mu }={\partial }_{\alpha }({\sigma }_{\epsilon }({a_1}^{\alpha \mu }[k]+{a_2}^{\alpha \mu }[H]))$. Equation (eq:Asc) defines a functional of every smooth $\varphi$; for $P_m\varphi =0$ it is X-A's on-shell second coefficient with fixed free input, and it is the object in (eq:XAinv). It is quadratic in $(H,k)$ jointly, with $k$ entering linearly, and linear in $\varphi$. All five terms are retained; none is omitted.