# QED proper vertex Ward identity derived from global symmetry and Schwinger-Dyson Equations?

+ 1 like - 0 dislike
3784 views

In QED, according to Schwinger-Dyson equation, $$\left(\eta^{\mu\nu}(\partial ^2)-(1-\frac{1}{\xi})\partial^{\mu}\partial^{\nu}\right)\langle 0|\mathcal{T}A_{\nu}(x)...|0\rangle = e\,\langle 0|\mathcal{T}j^{\mu}(x)...|0\rangle + \text{contact terms}$$ And the term $\left(\eta^{\mu\nu}(\partial ^2)-(1-\frac{1}{\xi})\partial^{\mu}\partial^{\nu}\right)$ is just the inverse bare photon propagator, so if we put the photon on shell, then the l.h.s will yield the complete n-point Green function with the complete photon propagator removed and also multiplied by a factor $Z_3$, the vector field renormalization constant.
But the r.h.s gives $$\partial_{\mu}\, \langle 0|\mathcal{T}j^{\mu}(x)...|0\rangle = \text{contact terms}$$ which is the common complete (n-1)-point complete Green function.

So if we truncate all the n-1 external complete propagators, then we are left with the proper vertex Ward identity.

The problem is, now the constant $Z_3$ appeared.
But the well known Ward identity, e.g. $$p_\mu\Gamma^\mu_P(k,l)=H(p^2)[iS^{-1}(k)-iS^{-1}(l)]$$ doesn't contain $Z_3$.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.