Coupling the gauge bosons to the Higgs scalars

[QuantumDevil]

At page 52 of 4th chapter in "An Introduction to gauge theories & modern particle physics" by Leader & Predazzi one can find such statement:

"We must therefore rearrange (4.2.4) so that we can identify the field that multiplies$$\frac{1}{2}\left(1+\tau_{3}\right)$$ as gauge boson that remains massless i.e. as photon"

But I still don't understand why such field is needed and what it mean that "field multiplies (transformation mentioned above) as gauge massles boson". For me this statement sounds "slangish".

Next question: unbroken generator - what's that? And for what purpose one impose condition(4.2.8)

I will appreciate any kind of advice.
QuantumDevil.

 

[schieghoven]

I'm not familiar with this text, but perhaps the following will help:

It's not really about 'why such a field is needed'; the gauge fields follow from assuming some gauge symmetry in the Lagrangian, and in this case we're considering the Higgs scalar (2 complex components), which naturally admits SU(2)xU(1) symmetry. $$\frac{1}{2}\left(1+\tau_{3}\right)$$ is a 2-by-2 Hermitian matrix with complex entries, and if you exponentiate it you get a gauge transformation from SU(2)xU(1)... so it's called a generator. It's an unbroken generator, because the gauge transformations it generates do not change the Higgs' vacuum state. In more detail, we can write the Higgs vacuum state as
$$v \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$
with v a real number (the vacuum expectation value). Note that
$$\frac{1}{2}\left(1+\tau_{3}\right) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$
and hopefully it's clear that why the Higgs' vacuum state is unaffected under these gauge transformations. However, the other generators of SU(2)xU(1) do change the Higgs vacuum state... so these are called broken generators. The broken generators correspond to massive gauge bosons, while the unbroken generators are massless - this comes from the covariant derivative in the Higgs part of the Lagrangian, where only broken generators will couple to the Higgs' nonzero vacuum state.

Best wishes

Dave

 

[sir-pinski]

Coupling the gauge bosons to the Higgs scalars is an important concept in modern particle physics. It refers to the way in which the Higgs field interacts with the gauge bosons, which are the particles responsible for mediating the fundamental forces in the Standard Model.

In the Standard Model, the Higgs field is coupled to the gauge bosons through a process called spontaneous symmetry breaking. This is a key mechanism that allows the gauge bosons to acquire mass, while still preserving the gauge symmetry of the theory. The Higgs field is responsible for giving mass to the W and Z bosons, which are the carriers of the weak nuclear force. Without this coupling, the gauge bosons would remain massless, and the theory would not accurately describe the interactions we observe in nature.

In order to better understand this concept, we must first understand what is meant by a "field multiplying" the transformation mentioned in the quote. In this context, it refers to the mathematical operation of taking two quantities and multiplying them together. In the case of the Higgs field, it is multiplied by the transformation (1+τ3)/2. This transformation is a mathematical representation of the weak isospin, which is a fundamental symmetry of the Standard Model. By multiplying the Higgs field by this transformation, we are able to identify the field that gives mass to the W and Z bosons, and remains massless itself, as the photon.

The term "unbroken generator" refers to a specific symmetry generator in the Standard Model that remains unchanged after spontaneous symmetry breaking. This means that the corresponding particle, in this case the photon, remains massless. By imposing the condition (4.2.8), we are ensuring that this symmetry remains unbroken, and the photon remains massless.

In summary, coupling the gauge bosons to the Higgs field is essential for understanding the way in which particles acquire mass in the Standard Model. The "field multiplying" the transformation refers to the mathematical operation of multiplying the Higgs field by a symmetry generator. The unbroken generator is a symmetry that remains unchanged after spontaneous symmetry breaking, and imposing the condition (4.2.8) ensures that the corresponding particle remains massless. I hope this explanation helps clarify the concepts mentioned in the quote.