# $U(1)\otimes BRST$ symmetry, of on-shell T-matrix elements and (1-$\phi$-I) Green's functions, determines the vacuum state of the Abelian Higgs Model from symmetry alone: minimization of the scalar-sector effective potential is unnecessary

## TL;DR

This paper demonstrates that the vacuum state of the Abelian Higgs Model is uniquely determined by its symmetries alone, without the need for potential minimization, through the use of Ward identities and on-shell T-matrix symmetries.

## Contribution

It shows that the vacuum of the Abelian Higgs Model can be fixed solely by symmetry considerations, bypassing the traditional minimization of the effective potential.

## Key findings

- Symmetries lead to all-loop Ward identities in the model.
- T-matrix on-shell symmetry enforces the vanishing of tadpoles.
- Vacuum state is determined by symmetry without potential minimization.

## Abstract

The weak-scale $U(1)_{Y}$ Abelian Higgs Model (AHM) is the spontaneous-symmetry-breaking gauge theory of a complex scalar $\phi = \frac{1}{\sqrt{2}}(H + i \pi)$ and a vector $A_{\mu}$. Global $U(1)_{Y}\otimes BRST$ symmetry emerges: when it is realized that on-shell T-matrix elements enjoy an extra $U(1)_{Y}$ global symmetry beyond the Lagragian's BRST symmetry. The symmetries co-exist: $U(1)_{Y}$ generators $\delta_{U(1)_Y}$ commute with BRST generators s and $[\delta_{U(1)_Y},s]{\cal L} = 0$. Two towers of Ward Takahashi identities (WTI), which include all-loop-orders quantum corrections, emerge: a tower of relations among off-shell 1-$\phi$-I (but 1-$A_{\mu}$-Reducible) Green's functions; another tower of Adler-zero WTI for on-shell T-matrix elements. The T-matrix's LSS theorem forces tadpoles to automatically vanish (equivalently $m^{2}_{\pi} = 0$) by symmetry alone. We show that, when the full symmetries of Lorenz gauge AHM are enforced on the scalar-sector effective potential, the vacuum state of the theory is specified/decided by symmetry alone. We use recursive WTI relations among Green's functions to include opeators of dimension $\geq 1$. We express the fully renormalized scalar-sector effective potential in a form which shows explicitly that, for small scalar field values, the gauge-independent vacuum state of the theory $\langle H\rangle_{renormalized} = Z^{1/2}_{\phi}\langle H\rangle_{bare}$ is determined by $U(1)_{Y}\otimes BRST$ symmetry alone, without minimizing the effective potential.