Chiral Symmetry Breaking

In summary, the vacuum expectation value of certain composite fields breaks a continuous symmetry of the Lagrangian, resulting in the emergence of Goldstone bosons. This can be seen in chiral QCD with light quarks, where the full chiral symmetry is broken down to a subgroup and fluctuations of the composite operator \bar{Q}^a_R Q^b_L are described by a field \Phi. These massless Goldstone bosons, such as pions and kaons, represent the low energy excitations of the system and can be given mass by including the effects of quark masses.
  • #1
Neitrino
137
0
Dear PF,

Could anyone help me to understand the following issue:

I don't understand the formation of Goldston bosons in chiral symmmetry breaking. Namely - suppose there is only fermion fields Lagrangian and due to some reasons the fermion field obteins VEV - <0|psi^bar psi|0> is non zero. And left/right chiral symmetry is reduced to its diagonal group.
I don't understant how Goldstone bosons are formed - I mean as folows"
In Scalar field case if we have multiple fields phi1, phi2, phi3... and when
this scalalar field develops some VEV - v, then for convience we can put only phi1 equals to this VEV -v, rest fields we can put zero...we will say this VEV of field phi1 is v, VEVs of rest field are zero: <0|phi1|0>=v,
<0|phi2|0>=0 etc. We say we have phi1 massive scalar field and rest Goldsone Bosons from Broken Symmetry.

So I don't understand what happens when suppose fermion fields develop VEV.

Thanks
George
 
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  • #2
The story goes as follows, the vacuum expectation value of certain composite fields break a continuous symmetry of the Lagrangian and Goldstone bosons emerge for each of the unbroken generators. I suspect this is what you have in mind anyway, so let me talk specifically about chiral QCD with the light quarks (u, d, s). The Lagrangian for the quark part of the theory is [tex] \mathcal{L} = \bar{Q}_R i \gamma^\mu D_\mu Q_R + \bar{Q}_L i \gamma^\mu D_\mu Q_L [/tex]. Clearly the whole thing is invariant under the chiral transformation [tex] SU(3)_L \times SU(3)_R [/tex] because we've left out the quark mass terms. Now, when you look at composite operators like [tex] \bar{Q}^a_R Q^b_L [/tex] (a,b are flavor indices) you find that these operators acquire a vacuum expectation. In this case the vev originates from the non-perturbative part of QCD. The full chiral symmetry is broken down to the subgroup [tex] SU(3)_V [/tex] just as you said, and the vacuum expectation value can be written like [tex] \langle \bar{Q}^a_R Q^b_L \rangle = v \delta^{a b} [/tex]. In order to describe fluctuations of the operator \bar{Q}^a_R Q^b_L one introduces another field [tex] \Phi [/tex] which is an [tex] SU(3) [/tex] matrix defined via [tex] \bar{Q}_R Q_L \rightarrow v \Phi [/tex]. Because it is a member of [tex] SU(3) [/tex] it can be written in terms the exponential of (i times) a traceless hermitian matrix. Such a hermitian matrix has 8 independent components, one for each of the broken generators, and each one corresponds to one of the massless Goldstone bosons. These fields are your pions, your kaons, and the eta, and they represent the low energy excitations of the system. That's how the Goldstone modes appear in the theory. They can be given mass by including the effects of quark masses to leading order, and then they are referred to as pseudo-Goldstone bosons because the chiral symmetry which would be spontaneuously broken by the vev is explicitly broken by the quark masses.

Hope this helps.
 
Last edited:
  • #3


Dear George,

Chiral symmetry breaking is a phenomenon in which the symmetry between left-handed and right-handed particles is broken, resulting in the formation of Goldstone bosons. This can occur in systems with fermion fields, where the fermion field obtains a non-zero vacuum expectation value (VEV). This breaks the chiral symmetry, reducing it to its diagonal group.

To understand how Goldstone bosons are formed in this scenario, it may be helpful to think of the fermion fields as analogous to the scalar fields in the example you mentioned. Just as the scalar fields develop a VEV and break the symmetry, the fermion fields also develop a VEV and break the chiral symmetry. This results in the formation of Goldstone bosons, which are the remnants of the broken symmetry.

In the case of scalar fields, we can choose one of the fields to have a non-zero VEV and the rest to be zero, resulting in a massive scalar field and Goldstone bosons. Similarly, in the case of fermion fields, the non-zero VEV of the fermion field results in a massive fermion and Goldstone bosons. These Goldstone bosons are important as they are associated with the spontaneous breaking of a symmetry and can have implications in various physical phenomena.

I hope this helps in understanding the formation of Goldstone bosons in chiral symmetry breaking.


 

1. What is chiral symmetry breaking?

Chiral symmetry breaking is a phenomenon in particle physics where the symmetrical behavior of a system is broken due to the presence of massless particles. This results in different behavior between left-handed and right-handed particles.

2. How does chiral symmetry breaking occur?

Chiral symmetry breaking occurs when the symmetrical potential of a system is distorted by the presence of particles with different masses. This leads to the formation of a non-zero vacuum expectation value, which breaks the chiral symmetry.

3. What is the significance of chiral symmetry breaking in physics?

Chiral symmetry breaking is important in understanding the behavior of particles and their interactions. It helps explain why certain particles have mass, and plays a role in the Standard Model of particle physics.

4. Can chiral symmetry breaking be observed in experiments?

Yes, chiral symmetry breaking has been observed in experiments such as high-energy collisions at particle accelerators. It is also used in theoretical models to make predictions about the behavior of particles.

5. Is chiral symmetry breaking related to the concept of handedness?

Yes, chiral symmetry breaking is related to the concept of handedness, or chirality, which describes the asymmetry between left and right orientations. In particle physics, this refers to the different behavior of left-handed and right-handed particles due to chiral symmetry breaking.

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