Quarks effective masses and the pions

[jacopo23]

Hi everybody :)

A lagrangian of the u and d quarks where they don't have a mass, shares chiral SU(2) and U(1) symmetries.
In the vacuum, we write
$<0| \bar{Q} Q | 0 > = \eta$
where Q is the douplet of u and d quarks. In such a way we break chiral symmetries.
The current associated with the three axial broken symmetries creates three goldstone bosons, the pions, and in the approximation where d and u are massless, the pions are massless too.

Now I just don't understand the following point: breaking the symmetry, we give a mass to u and d quarks, as Peskin says in pag. 669.
How can these quarks and their relative antiquarks form the pions, which we said to be massless?

 

[eljose79]

Hi there,

It's great to see someone discussing the chiral symmetries and the mass of quarks. Let me try to explain the point you are struggling with.

Firstly, it is important to understand that the mass of quarks is not the only factor that determines the mass of particles. In the Standard Model, the Higgs mechanism is responsible for giving mass to particles through the Higgs field. This is also the case for pions, which acquire mass through the Higgs mechanism.

Now, in the context of chiral symmetry breaking, when we break the chiral symmetries by giving mass to the u and d quarks, it doesn't necessarily mean that the pions will also acquire mass. This is because the chiral symmetries are still present in the Lagrangian, and the pions are the Nambu-Goldstone bosons associated with the broken symmetries. This means that the pions are still massless in the Lagrangian, even though the u and d quarks have acquired mass.

To understand this concept better, think of a ball rolling down a hill. When the ball reaches the bottom, it has kinetic energy and is in motion. Now, if we were to give the ball a small push, it would start rolling back and forth on the hill, but its motion would still be governed by the same underlying dynamics. Similarly, in the context of chiral symmetry breaking, even though the u and d quarks have acquired mass, the underlying dynamics of the pions are still governed by the same chiral symmetries.

I hope this helps to clarify the point you were struggling with. Keep exploring and asking questions, that's how we make progress in science!