Physical interpretation of V and A couplings

In summary, the difference between V and A couplings lies in how the L and R projection operators commute through the coupling term and conjugate field term in a given Lagrangian. While purely vector or axial currents conserve parity, mixed terms involving a pseudoscalar violate parity. This can be shown mathematically by considering the eigenvalues of parity in the presence of a pseudoscalar.
  • #1
bomanfishwow
27
0
Hi,

I'm trying to remind myself of exactly what, physically, is the difference between V and A couplings. Now, a vector coupling is of the form [tex]\bar{\psi}\gamma^\mu\psi[/tex], and axial coupling of the form [tex]\bar{\psi}\gamma^\mu\gamma^5\psi[/tex]. Thinking in terms of a chiral fermion expanded as:

[tex]
f = \left[\left(\frac{1-\gamma^5}{2}\right) + \left(\frac{1+\gamma^5}{2}\right)\right]\psi
[/tex]

and where [tex]\bar{f} = \gamma^\dagger\gamma^0[/tex], I assume the difference between the V and A couplings has to do with how the L and R projection operators commute through either [tex]\gamma^\mu[/tex] or [tex]\gamma^\mu\gamma^5[/tex] from the 'coupling' term, and the [tex]\gamma^0[/tex] from the conjugate field term in a given Lagrangian.

However, as [tex][\gamma^5,\gamma^5] = 0[/tex], I don't see how a difference in the chiral treatment between V and A couplings can arise. Am I barking up completely the wrong tree? Any insight welcomed!
 
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  • #2
You can always redefine what you mean by "V" and "A", as purely vector or purely axial currents conserve parity. It's the mixed terms that generate parity violation.
 
  • #3
Vanadium 50 said:
You can always redefine what you mean by "V" and "A", as purely vector or purely axial currents conserve parity. It's the mixed terms that generate parity violation.

So, a mixed term is something like [tex]\bar{\psi}\gamma^\mu\left(1-\gamma^5\right)\psi[/tex]? How can it be shown that this term violates parity?

Cheers.
 
  • #4
Gamma 5 is a pseudoscalar. Adding a pseudoscalar to a scalar violates parity.
When used in beta decay, the V-A interaction leads to negative helicity for outgoing leptons. This violates parity.
 
  • #5
clem said:
Gamma 5 is a pseudoscalar. Adding a pseudoscalar to a scalar violates parity.

Sure, but I'm interested in showing why this is the case mathematically...
 
  • #6
Given [itex]P|\psi> = (-1)^n|\psi>[/itex]

Then [itex]P(a + b\gamma^5)|\psi> = aP|\psi> + bP \gamma^5|\psi> [/itex]

Which is [itex]\left( (-1)^na + (-1)^{n+1}b \right) |\psi>[/itex], so is only an eigenvalue of parity if [itex]a=0[/itex] or [itex]b=0[/itex].
 
  • #7
Ah yes, very elegant. Thanks!
 

1. What is the physical interpretation of V and A couplings?

The V and A couplings refer to the vector and axial vector couplings, respectively, in the Standard Model of particle physics. These couplings describe the strength of interaction between particles, specifically between fermions (particles with half-integer spin) and gauge bosons (force-carrying particles).

2. How do V and A couplings affect particle interactions?

V and A couplings affect the symmetry properties of particle interactions. While vector couplings preserve parity (mirror symmetry), axial vector couplings violate parity. This is because vector particles rotate in the same direction as their spin, while axial vector particles rotate in the opposite direction.

3. What is the significance of V and A couplings in the Standard Model?

The V and A couplings play a crucial role in the electroweak theory of the Standard Model, which describes the electromagnetic and weak nuclear forces. They also help explain certain phenomena, such as the universality of weak interactions and the matter-antimatter asymmetry in the universe.

4. How are V and A couplings measured?

V and A couplings can be measured through experiments that involve the production and decay of particles. For example, the ratio of the decay rates of two particles with different V and A couplings can be used to determine the strength of their couplings.

5. Can V and A couplings be modified or changed?

In the Standard Model, the V and A couplings are fixed values that cannot be modified or changed. However, there are theories beyond the Standard Model that suggest modifications to these couplings, which could potentially be observed through experiments.

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