On the derivation of an effective Higgs field

TL;DR

This paper explores how the Higgs mechanism can naturally emerge from the classical Lagrangian of a vector boson with longitudinal polarization, providing a criterion to determine boson mass based on field strengths, prior to quantization.

Contribution

It demonstrates a classical derivation of the Higgs field and mass criterion from the Yang-Mills Lagrangian without quantization, highlighting the role of longitudinal polarization.

Findings

Scalar Higgs field can arise automatically in the Lagrangian.

Massiveness of bosons can be inferred from field strength differences.

Method applies to both U(1) and SU(2) gauge theories.

Abstract

In one respect, the massive vector-boson shows its difference from a massless vector-boson by one more physical polarization, known as longitudinal polarization. In another respect, the quantized boson acquires its mass by Higgs mechanism. In this paper we study the effect of the longitudinal polarization in U(1) case by substituting it into the primary Yang-Mills Lagrangian $-\frac 14F_{\mu \nu}F^{\mu \nu}$. Under a hypothesis of strong transversal condition for free vector boson, it is found that in the Lagrangian the scalar field for the Higgs mechanism can automatically arise after we separate a part equivalent to the contribution of a massless boson. In addition, a criterion is obtained to infer whether the boson is massive or not: if $\mathbf{E}^2-\mathbf{B}^2\neq 0$, where $\mathbf{E}$ and $\mathbf{B}$ are field strengths, then it is massive. The analysis also pertains to SU(2) case. The method in this paper is performed before any quantizations.