The manifest symmetry of the all-loop-orders scalar effective potential of the spontaneously broken Abelian Higgs model derived from Ward-Takahashi identities

The weak-scale $U(1)_Y$ Abelian Higgs Model (AHM) is the spontaneous-symmetry-breaking (SSB) gauge theory of a complex scalar $phi$ and a vector $A^mu$. In Lorenz gauge, the SSB AHM has: a massless pseudo-scalar $pi$; a conserved (up to gauge-fixing terms) $U(1)_Y$ global physical current; no conserved $U(1)_Y$ charge; and a Goldstone Theorem, where $tilde pi$ in the Kibble representation becomes a true massless Nambu-Goldstone boson (NGB). Two towers of 1-soft-pion SSB global WTI govern the $phi$-sector. The 1$^{st}$ tower gives relations among connected off-shell Greenu0027s functions, and powerful constraints on the all-loop-orders $phi$-sector effective AHM Lagrangian. The 2$^{nd}$ tower, governing on-shell connected T-matrix elements, further severely constrains the physics. It guarantees IR finiteness for zero NGB mass; includes the Lee-Stora-Symanzik (LSS) Theorem, forcing the on-shell T-matrix element with only two external pseudoscalar legs to vanish; and is the source of an additional NGB global shift symmetry. Using the Greenu0027s functions and the LSS Theorem of the theory, we construct the manifestly symmetric quantum effective potential of the scalar sector of the Goldstone mode of the AHM, in the Lorenz gauge, to all orders in quantum loops. To do so, we solve the Ward Takahashi identities of the theory recursively, and express arbitrary $phi$-sector Greenu0027s functions in terms of those with no external scalar legs and an even number of pseudoscalar legs. The resulting expression for the effective potential is much easier to evaluate at a given order in quantum loops, since many fewer diagrams need be computed for a given number of external $phi$ legs. In the Kibble representation, the so-constructed effective scalar potential is gauge-independent.