Note that here we can define ${\varphi }^{\prime }(x^{\prime })=\varphi (x^{\prime })+\delta \varphi (x)=\varphi (x)+{\zeta }^{\mu }{\partial }_{\mu }\varphi (x)+\delta \varphi (x)$ and hence $\delta \varphi (x)=-{\zeta }^{\mu }{\partial }_{\mu }\varphi (x)$, and no additional terms are generated; alternatively we can do an active transformation $\varphi (x)\to \varphi (x)+{\zeta }^{\mu }{\partial }_{\mu }\varphi (x)$ and additional terms are generated but vanish by integration by parts and use of equations of motion.

This is only an estimate as we do not have the two-loop contribution to ${\beta }_1$. These corrections are not required in the two-loop estimate of the trilinear, etc., corrections to the Higgs potential below since we input $m_h^2$ in that case.

This dual derivation of the conserved current is fundamental to any gauge theory, and is analogous to the fact that the electromagnetic current can be obtained by locally varying the vector potential in the Dirac action, $\delta A_{\mu }$, or by varying the phase of the electron wave function, $\delta \psi =i\theta (x)\psi$. Doing both at the same time with $\delta A_{\mu }={\partial }_{\mu }\theta$ is just a gauge transformation, under which the Dirac action is invariant.

The reason for introducing the source term is to remove all the linear cross terms, $\propto \widehat{\varphi }$, arising from the shift. In this perturbative approach there remain $O({\hslash }^{3/2})$ terms, $\sim \delta m^2{\varphi }_c\widehat{\varphi }$. These we ignore since we are working to $O(\hslash )$. We then add back a term $+J{\varphi }_c$ which cancels the $-J{\varphi }_c$ arising from the shift. The general formalism of the Legendre transformed potential is given in Ref. [16].