## Principle of Gauge Symmetry Breaking

The main objective of this post is to introduce the following principle of gauge symmetry breaking, based on the principle of interaction dynamics (PID), first postulated in [3]:

Principle of Gauge Symmetry Breaking. The gauge symmetry holds true only for the Lagrangian actions for the electromagnetic, week and strong interactions, and it will be violated in the field equations of these interactions.

Inspired by the great vision of Einstein, we adopt the view that the laws of Nature are dictated by a few basic universal symmetries and fundamental first principles. As we explained in the post: Dark Matter and Dark Energy: a Property of Gravity, the Einstein-Hilbert action is fully dictated by the symmetry of general relativity: the law of gravity is covariant under general coordinate transformations. The gravitational field equations are then uniquely determined by the principle of interaction dynamics (PID). Gauge symmetry dictates the Lagrangian actions for the electromagnetic, the weak and the strong interactions. The field equations of these interactions are then uniquely by PID as well.

Physically, symmetries are displayed in two levels in the laws of Nature:

1. invariance of the Lagrangian action, and
2. the covariance of the variation equations of the action.

The symmetries for the principle of general relativity, the Lorentz invariance and the principle of representation invariance (PRI) are universally true for both the actions and the variation equations. Basically, these symmetries require that the validity of laws of Nature be independent of the coordinate systems expressing them.

However, the physical implication of the gauge symmetry is different in the following sense:

• The gauge invariance of the Lagrangian action says that the energy contributions of the interaction fields caused by the ${N}$ fermions carrying the same charge ${g}$ are indistinguishable. Here ${g}$ plays also the role of the coupling constant of ${SU(N)}$ gauge theory.
• If the variation field equations were gauge covariant, then the particles involved in the interaction are indistinguishable. Physically this is in general not the case. In other words, the gauge field equations must break the gauge symmetry. This is the phenomenon, often called spontaneous symmetry breaking.

The physical implication discussed here clearly suggests us to postulate the principle of gauge symmetry breaking for interactions described by an ${SU(N)}$ gauge theory, stated in the beginning of this post.

To be more specific, let us recall the ${SU(N)}$ the gauge transformations:

$\displaystyle (\widetilde \Psi, \widetilde G^a_\mu \tau_a, \widetilde m) = \left(\Omega \Psi, G^a_\mu \Omega \tau_a \Omega^{-1} + \frac{i}{g}(\partial_\mu \Omega) \Omega^{-1}, \Omega m \Omega^{-1}\right), \ \ \ \forall \Omega=e^{i \theta^k(x) \tau_k} \in SU(N).$

Then the Yang-Mills action is dictated by the gauge invariance under the gauge transformations (see the previous post: Yang-Mills Theory and Principle of Representation Invariance (PRI)):

$\displaystyle L_{YM}=\int_{M^4}\left[-\frac{1}{4}{\mathcal G}_{ab}g^{\mu\alpha}g^{\nu\beta}F^a_{\mu\nu}F^b_{\alpha\beta} +\bar{\Psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\Psi\right]dx. \ \ \ \ \ (1)$

By PID, the corresponding field equations are given by

$\displaystyle {\mathcal G}_{ab} \left[ \partial^{\nu}F^b_{\nu\mu} - g \lambda^{b}_{cd} g^{\alpha \beta}F^c_{\alpha\mu}G^d_{\beta}\right] - g \bar{\Psi} \gamma_{\mu}\tau_a \Psi = (\partial_\mu + \alpha_b G^b_\mu) \phi_a, \ \ \ \ \ (2)$

together with the Dirac equations

$\displaystyle (i\gamma^{\mu} (\partial_{\mu} + igG^a_{\mu}\tau_a)- m)\Psi =0. \ \ \ \ \ (3)$

It is then clear that the Yang-Mills action (1) is gauge invariant. However, the term

$\displaystyle \alpha_b G^b_\mu \phi_a$

on the right-hand side of the field equations (2) is not covariant under the gauge transformations, and consequently breaks the gauge symmetry. This spontaneous gauge symmetry breaking represents the distinction of the ${N}$ particles involved.

We remark that the PID induced mass generation mechanism follows from the Nambu & Jona-Lasinio mechanism proposed in the early 60s. The mass generation of the ${W^\pm}$ and ${Z^0}$ particles using the Higgs fields, discovered by Robert Brout & Francois Englert, Higgs, and Gerald Guralnik, Richard Hagen & Tom Kibble, follows also from the Nambu & Jona-Lasinio mass generation mechanism. We refer interested readers to the original papers of Nambu and Jona-Lasinio as well as the Nambu’s Nobel lecture, delivered by Jona-Lasinio for the original ideas for the mass generation mechanism.

The key advantage for PID induced spontaneous symmetry breaking and mass generation mechanism addressed here (see also the original papers [3, 6]) is that it is based only on a first principle, PID,  offering much richer physical insights to the nature of Higgs fields and mass generation mechanism. More shall be explored in our future posts on this issue.

Tian Ma & Shouhong Wang