Goldstone Theorem: Calculate Massless Gauge Bosons & Scalars

[sureman]

goldstone theorem help!

Homework Statement

Using goldstone theorem how can I calculate number of massless gauge bosons and massless scalars?
where G1 = SO(2) ∼ U(1) is the global symemtry, and G2 = SU (4) ∼ SO(6) is the gauge symmetry, and G1 ×G2 is completely broken.

Homework Equations

dimensions of SU(n)= n^2-1
dimensions of SO(n)=(n(n-1))/2

The Attempt at a Solution

dimensions of G1 and G2
G1= 1
G2=15

so I know we have n-m massless scalars and I think G2=m and G1xG2=n

so m=15 n=16

therefore

I think we have 1 gauge boson from n-m

and 15 massless scalars from n?

is this close to being right?

 

[mark726]

Thank you for your question. The Goldstone theorem is a powerful tool in understanding the behavior of symmetries in particle physics. In order to calculate the number of massless gauge bosons and massless scalars in your scenario, we will need to use the Goldstone theorem in conjunction with the dimensions of the symmetry groups G1 and G2.

First, let's review the Goldstone theorem. This theorem states that for every broken symmetry, there will be a corresponding massless particle, called a Goldstone boson. In other words, the number of Goldstone bosons is equal to the number of broken symmetries.

In your case, G1×G2 is completely broken, which means that both G1 and G2 are broken. This means that there will be a total of two Goldstone bosons, one for each broken symmetry.

Next, we need to determine the dimensions of G1 and G2. As you correctly stated, the dimension of G1 is 1. However, the dimension of G2 is not 15, but rather 15^2-1=224. This is because the dimension of SU(4) is (4^2-1)=15^2-1=224.

Now, let's use the Goldstone theorem to determine the number of massless gauge bosons and massless scalars. Since there are two broken symmetries, there will be two massless gauge bosons and two massless scalars. This means that there will be a total of 4 massless particles in your scenario.

I hope this helps clarify the application of the Goldstone theorem in your problem. If you have any further questions, please feel free to ask.
Scientist