Chiral symmetry and quark condensate

In summary, the conversation discusses chiral symmetry in QCD and the requirement for a non-zero vacuum expectation value for spontaneous symmetry breaking to occur. The question is posed whether it can be proven that if chiral symmetry is an exact symmetry of QCD, then the vacuum expectation value of the chiral condensate must be zero. The answer is yes, as zero is the only value that is invariant under chiral symmetry. This can be shown explicitly by picking a one-parameter transformation and using the u,d parameterization.
  • #1
Einj
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I'm studying chiral symmetry in QCD. I understand that in order for a spontaneous symmetry breaking to occur, there must be some state with a vacuum expectation value different from zero. My question is: can someone prove that is the chiral symmetry is an exact symmetry of the QCD then necessarily:

$$\langle 0 | \bar{\psi}\psi |0\rangle = 0$$

In understand that this has to be derived from the invariance of the vacuum but I can't prove it explicitely.

Thanks
 
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  • #2
Einj said:
I'm studying chiral symmetry in QCD. I understand that in order for a spontaneous symmetry breaking to occur, there must be some state with a vacuum expectation value different from zero. My question is: can someone prove that is the chiral symmetry is an exact symmetry of the QCD then necessarily:

$$\langle 0 | \bar{\psi}\psi |0\rangle = 0$$

In understand that this has to be derived from the invariance of the vacuum but I can't prove it explicitely.

Thanks

It's actually easy to see that [itex]\langle 0 | \bar{\psi}\psi |0\rangle[/itex] is not invariant under chiral symmetry. For QCD with one flavor,

[tex]\psi = \begin{pmatrix} u \\ d \end{pmatrix},[/tex]

where we can consider the up and down quarks [itex]u,d[/itex] as Dirac spinors. Then we can write the chiral symmetry as

[tex]\psi \rightarrow \exp\left[i\gamma^5 \left( \vec{\theta}\cdot\vec{\tau}\right) \right] \psi,[/tex]

where the [itex]\vec{\tau}[/itex] are the generators of [itex]SU(2)[/itex] flavor transformations.

Some algebra will show that the kinetic term [itex] \bar{\psi} \gamma^\mu\partial_\mu \psi[/itex] is chiral invariant, but [itex] \bar{\psi} \psi[/itex] is not, because the [itex]\gamma^5[/itex] in the chiral transformation anticommutes with the factor of [itex]\gamma^0[/itex] in the Dirac conjugate.
 
  • #3
Ok, I knew that. Actually my question was a little bit different (or maybe it's the same thing but I can't see it :tongue2:). I was asking: IF the theory is invariant under chiral symmetry (i.e. the vacuum state is invariant) how can I show that necessarily [itex]\langle \bar{\psi}\psi\rangle=0[/itex]??
 
  • #4
Einj said:
Ok, I knew that. Actually my question was a little bit different (or maybe it's the same thing but I can't see it :tongue2:). I was asking: IF the theory is invariant under chiral symmetry (i.e. the vacuum state is invariant) how can I show that necessarily [itex]\langle \bar{\psi}\psi\rangle=0[/itex]??

Zero is the only value that is invariant under the chiral symmetry. You can be as explicit as you like by picking a one-parameter transformation and using the [itex]u,d[/itex] parameterization. Show that, if [itex]\rho = \langle \bar{\psi}\psi\rangle_0[/itex], then [itex]\delta\rho \neq 0[/itex] unless [itex]\rho =0[/itex].
 

1. What is chiral symmetry?

Chiral symmetry is a fundamental concept in particle physics that describes the symmetrical behavior of particles with respect to their chirality, or handedness. In other words, it refers to the behavior of particles that are mirror images of each other.

2. How does chiral symmetry relate to quark condensates?

Quark condensates are a manifestation of chiral symmetry breaking, where the chiral symmetry of the underlying theory is not preserved. This results in the formation of a non-zero vacuum expectation value for the quark fields, known as the quark condensate.

3. What is the significance of chiral symmetry and quark condensates in the Standard Model?

Chiral symmetry and quark condensates play a crucial role in the Standard Model of particle physics, as they are responsible for the masses of the elementary particles. Without chiral symmetry breaking and the resulting quark condensates, the theory would not be able to explain the observed masses of particles.

4. Can chiral symmetry and quark condensates be experimentally observed?

While chiral symmetry itself cannot be directly observed, its breaking through the formation of quark condensates can be indirectly observed through experiments such as deep inelastic scattering and lattice QCD simulations.

5. Are there any current research developments related to chiral symmetry and quark condensates?

Yes, there is ongoing research in this field, particularly in understanding the role of chiral symmetry breaking in the formation of hadrons and the study of chiral phase transitions. There is also research being done on the effects of chiral symmetry restoration in extreme conditions, such as in the early universe or in high-energy collisions.

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