Proposition: First-order display for a general compact gauge parameter

Let $\varphi$ be $\kappa$-independent and let $T^{\mu \nu }[\varphi ]={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi -{\eta }^{\mu \nu }{\mathfrak{L}}_{\eta }[\varphi ]$. Then

Proof. The ${\kappa }^1$ coefficient of (eq:varied) is

Since ${\partial }_{\mu }((1-g){\xi }^{\mu }{\mathfrak{L}}_{\eta })=-({\partial }_{\mu }g){\xi }^{\mu }{\mathfrak{L}}_{\eta }+(1-g)({\partial }_{\mu }{\xi }^{\mu }){\mathfrak{L}}_{\eta }+(1-g){\xi }^{\mu }{\partial }_{\mu }{\mathfrak{L}}_{\eta }$ integrates to zero, the expression equals $\int (1-g)({\partial }^{\mu }{\xi }^{\nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -({\partial }_{\mu }{\xi }^{\mu }){\mathfrak{L}}_{\eta })=\int (1-g)T^{\mu \nu }{\partial }_{\mu }{\xi }_{\nu }$. Integration by parts and ${\partial }_{\mu }{T}^{\mu \nu }=(\square +m^2)\varphi {\partial }^{\nu }\varphi$ give the second form.