Show that the vacuum polarization is transverse

[leo.]

Homework Statement

Show that the vacuum polarization [itex]\Pi^{\mu\nu}_2(p)[/itex] in 1-loop is transverse. Decide whether you want to use Ward's identity and prove this to be true in all orders or only prove for 1-loop.

Homework Equations

Ward's identity [itex]q_\mu \mathcal{M}^{\mu}=0[/itex] which must hold where [itex]\mathcal{M}=\epsilon_\mu \mathcal{M}^\mu[/itex] is the amplitude for a process with an external photon with momentum [itex]q[/itex] and polarization [itex]\epsilon_\mu(q)[/itex].

Also the form of [itex]\Pi^{\mu\nu}_2(p)[/itex] guessed by means of Lorentz invariance by Schwartz in his book "Quantum Field Theory and the Standard Model":
[tex]\Pi^{\mu\nu}_2(p)=\Delta_1(p^2)g^{\mu\nu}+\Delta_2(p^2)p^\mu p^\nu[/tex].

The Attempt at a Solution

If I apply Ward's identity to [itex]\Pi^{\mu\nu}_2[/itex] using the form guessed by Lorentz invariance the result is easily obtained. Actually we have

[tex]p_\mu \Pi^{\mu\nu}_2(p)=\Delta_1(p^2) p^\nu+\Delta_2(p^2)p^2 p^\nu=0[/tex]

and this gives that [itex]\Delta_2(p^2)=-\frac{1}{p^2}\Delta_1(p^2)[/itex]. Since the photon isn't on-shell this four-momentum is not zero, and we are left with

[tex]\Pi^{\mu\nu}_2(p)=\left(g^{\mu\nu}-\dfrac{p^\mu p^\nu}{p^2}\right)\Pi_2(p)[/tex]

where I renamed [itex]\Pi_2(p)=\Delta_1(p)[/itex]. This was my initial guess. There are some problems however. The first one, is that this is redundant. If I assume Ward's identity to be valid to [itex]\Pi^{\mu\nu}_2[/itex] then I'm already obviously assuming transversality.

The second one is that I can't see why Ward's identity can be used in this case. As I've stated in the relevant equations, Ward's identity is applied to the amplitude for a process with an external photon. Here as the books explain, the vacuum polarization is just a part of a diagram, where the external photons cannot be considered on-shell.

More than that, in Peskin's book, he says Ward's identity is valid to [itex]\Pi^{\mu\nu}[/itex] where this last one is the "sum of all 1-particle-irreducible insertions", being [itex]\Pi^{\mu\nu}_2[/itex] the second order contribution. By his writing I understand Ward's identity should be valid just for that sum, not for the individual contributions.

So I'm quite confused on how Ward's identity must be used.

Also, how would be the solution without using it, just for one-loop as the problem states that can be done?

 

[mark726]

Thank you for your question about the vacuum polarization in 1-loop and whether it is transverse. You have made some good progress in your attempt at a solution, but I can see where you might have some confusion about the use of Ward's identity.

Firstly, I would like to clarify that Ward's identity is not a tool that can be used to prove transversality for all orders. It is a general symmetry relation that holds in quantum field theory, and it can be used to prove transversality in specific cases, such as in 1-loop calculations.

In your attempt at a solution, you correctly applied Ward's identity to the form of \Pi^{\mu\nu}_2(p) that was guessed by Lorentz invariance. However, as you pointed out, this is redundant because you are already assuming transversality in your use of Ward's identity. In order to use Ward's identity to prove transversality, you need to start with a form for \Pi^{\mu\nu}_2(p) that is not already transverse.

In Peskin's book, he states that Ward's identity is valid for the "sum of all 1-particle-irreducible insertions". This means that it is valid for the sum of all diagrams that contribute to the vacuum polarization at 1-loop. So, in order to use Ward's identity in this case, you would need to consider the full set of diagrams and their contributions to \Pi^{\mu\nu}_2(p), not just the second order contribution.

If you want to prove transversality for only the 1-loop contribution, you will need to use a different approach. One possible way to do this is to use the Feynman rules and evaluate the relevant diagrams to show that the transverse part of \Pi^{\mu\nu}_2(p) vanishes. This will require some algebraic manipulation and careful consideration of the kinematics involved.

I hope this helps clarify the use of Ward's identity in this problem. Good luck with your calculations!