Chiral gauge theory and C-symmetry

In summary, the statement that follows is confusing because it is not clear what equality is being claimed. The equality is between the three-point functions ##\langle A(x) A(y) A(z) \rangle##, which is valid under the C operation, but not under any other operation.
  • #1
mkgsec
5
0
Hi, I have a question in Srednicki's QFT textbook.

In p.460 section 75(about Chiral gauge theory), it says

"In spinor electrodynamics, the fact that the vector potential is odd under charge conjugation implies that the sum of these diagrams(exact 3photon vertex at one-loop) must vanish."

That's good, because it's just the Furry's theorem for odd number of external photons. But I find the subsequent statement confusing.

"For the present case of a single Weyl field(coupled to U(1) field), there is no charge conjugation symmetry, and so we must evaluate these diagrams."

But shouldn't the transformation rule of [itex]A^\mu (x)[/itex] under C,P or T be universal regardless of specific theory?
 
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  • #2
In spinor electrodynamics, the Lagrangian is *invariant* under the C operation, which among other things takes ##A \to -A##. One consequence of this is that there should be an equality between the following three-point functions:

##\langle A(x) A(y) A(z) \rangle = \langle(-A(x))(-A(y))(-A(z))\rangle = -\langle A(x) A(y) A(z) \rangle##

The only way this equality can be satisfied is if ##\langle A(x) A(y) A(z) \rangle## vanishes. This is Furry's theorem. This result relies crucially on the fact that the C operation, under which ##A \to -A## and also ##\psi## transforms nontrivially, is a *symmetry of the Lagrangian*. That's why we have the equality above.

In the chiral case, you can still define a transformation under which ##A \to -A## but it is no longer a symmetry of the Lagrangian, no matter what transformation rule you choose for ##\psi##. Therefore you cannot make the same argument and Furry's theorem does not go through.
 
  • #3
But doesn't Furry's theorem still hold even if the langrangian has no C-symmetry? I thought it is valid as long as the vacuum is invariant under C. What am I missing here?
 
  • #4
Why do you think the vacuum is invariant under C?
 
  • #5
Well... Maybe because the vacuum contains no particle? Are the invariance of the vacuum and invariance of lagrangian related to each other?
 
  • #6
mkgsec said:
Are the invariance of the vacuum and invariance of lagrangian related to each other?

Right. If your theory has a symmetry and you have a unique vacuum state, then the vacuum is invariant under that symmetry. If you have some operation that is not a symmetry of your theory, there's no reason to expect the vacuum to be invariant under that symmetry.
 
  • #7
It's very clear now! I always assumed that the vacuum is invariant under any operation. Thank you, it was very helpful :)
 

1. What is chiral gauge theory?

Chiral gauge theory is a type of quantum field theory that describes the interactions between elementary particles. It takes into account the chiral (handedness) symmetry of particles, which refers to their spin and direction of motion.

2. What is C-symmetry in chiral gauge theory?

C-symmetry, also known as charge conjugation symmetry, is a fundamental symmetry in particle physics that describes the transformation of a particle into its antiparticle. In chiral gauge theory, C-symmetry is used to ensure that the equations describing particle interactions are consistent.

3. Why is chiral gauge theory important?

Chiral gauge theory is important because it provides a framework for understanding the behavior of fundamental particles and their interactions. It has been successfully used to make predictions and explain experimental results in areas such as high-energy physics and cosmology.

4. What are the key principles of chiral gauge theory?

The key principles of chiral gauge theory include the concept of gauge symmetry, which describes how particles interact with each other through the exchange of force-carrying particles, and the chiral symmetry mentioned earlier, which takes into account the handedness of particles. Another important principle is the use of renormalization, which allows for the removal of infinities that arise in the calculations of physical quantities.

5. How does C-symmetry relate to other symmetries in physics?

C-symmetry is one of several fundamental symmetries in physics, including P-symmetry (parity) and T-symmetry (time reversal). In chiral gauge theory, these symmetries are combined and described by the CPT theorem, which states that the combined operation of charge conjugation, parity, and time reversal must remain unchanged for any physical process.

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