Understanding the Higgs Mechanism

[touqra]

Let me take an example.
Suppose we have a Lagrangian, of a scalar field coupled to itself and a massless gauge boson field. Next, we expand the Lagrangian about the vacuum expectation value of the scalar field, and it turns out that, in the end, the gauge boson is massive.
My question is, this massive boson has a mass in the vicinity of the scalar field's VEV. But if we expand the scalar around some other values, the mass of the gauge boson changes too. And we can also expand the Lagrangian around some scalar value, such that the gauge boson remains massless...

 

[RafaMarce]

I think "yes", we have to choose the point where all fields vanish to do the expansion of lagrangean... There is a possible vacuum state in this point. But this is a trivial solution, i think is the only place where we have massless gauge boson, since that the symmetry is broken.

 

[sir-pinski]

The Higgs mechanism is a fundamental concept in particle physics that explains how particles acquire mass. In the example you have provided, the Lagrangian describes the interaction between a scalar field and a gauge boson field. The gauge boson is initially massless, but after expanding the Lagrangian about the vacuum expectation value (VEV) of the scalar field, it becomes massive.

This arises from the spontaneous breaking of a symmetry in the Lagrangian, known as the Higgs field. The Higgs field permeates all of space and interacts with other particles, giving them mass. The Higgs mechanism works by creating a field that permeates space, and as particles move through this field, they experience resistance and therefore acquire mass.

In your example, when the scalar field is expanded around a different VEV, the mass of the gauge boson changes because the strength of its interaction with the Higgs field is different. This shows that the Higgs mechanism is intimately linked to the VEV of the scalar field, and the mass of particles is dependent on the strength of their interaction with the Higgs field.

Furthermore, by expanding the Lagrangian around a different scalar value, it is possible to keep the gauge boson massless. This is because the Higgs field only interacts with particles that have a non-zero mass, so if the scalar field is expanded in such a way that the gauge boson remains massless, it means that the Higgs field is not interacting with it.

In summary, the Higgs mechanism is a crucial concept in understanding how particles acquire mass and how this mass is dependent on the VEV of the scalar field. It is a fundamental part of the Standard Model of particle physics and has been confirmed through experimental evidence, making it an essential tool for understanding the fundamental building blocks of our universe.