Some questions in QFT (EM vertex, Ward identity, etc.)

[Wrichik Basu]

I am reading the A First Book of Quantum Field Theory. I have reached the chapter of renormalization, where the authors describe how the infinities of the self-energy diagrams can be corrected. They have also discussed later how the infrared and ultraviolet divergences are corrected. Just before this chapter, they have discussed the general EM vertex, $\Gamma$ and the EM form factors.

After reading all these, two things are not getting clear to me. The first one is trivial, but the second one is more important. I should have written what the authors have said, but that is too long, so I have attached some scanned pages of the book.

The main Feynman diagram that will be used is this one:

Fig. 1​

Here are the questions.

1. The authors have used the terms "2-point function", "3-point function" and "2-point amplitude" in a number of places. What do these "points" mean? For example, if you refer to the attached pdf, on page 2, the authors write, "Let us first draw the total 2-point amplitude as the sum of the tree-level and the rest." Sometime later, they write, "Let us first evaluate the 1-loop corrections to the 2-point function for the fermions, i.e., the self-energy of the fermions." What does 2-point mean here?​

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2. Consider Fig. 1. The vertex function for the above diagram is:​

Eqn. 1​

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where $eQ$ is the charge of the particle, and the superscript on $\Gamma$ denotes the number of loops.​

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The authors have said, "Let us first draw the total 2-point amplitude as the sum of the tree-level and the rest." And they have given the following diagram:​

Fig. 2​

The undecorated line on the RHS is the tree level diagram.​

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Next, consider this diagram (from the attached pdf):​

Fig. 3
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The contribution of this diagram can be written as: $$-i\Sigma^{(1)} (p) \ = \ (-ieQ)^2 \int \frac{d^4 k}{(2\pi)^4} \gamma_\mu \dfrac{i}{\not\! p + \not\! k - m} \ \gamma_\nu \ i D^{\mu \nu} (k), \quad \text{...Eqn. 2}$$ where $iD^{\mu\nu}$ is the photon propagator.​

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After some manipulations, what comes out is, $$q^\mu \Gamma_\mu ^{(1)} (p,p') \ = \ Q \left[ \left(-\Sigma^{(1)}(p)\right) - \left(-\Sigma^{(1)}(p')\right) \right] \quad \text{...Eqn. 3}$$ Then they wrote, "Adding this (Eqn. 3) to the tree level relation gives Eq. (12.13) up to 1-loop contributions." (I have highlighted this line in the pdf.) This is Eq. 12.13, the Ward identity: $$q^\mu \Gamma_\mu (p, \ p - q) \ = \ Q \left[S_F^{-1}(p) - S_F^{-1} (p - q)\right]$$ where $S_F$ is the fermionic propagator.​

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I just couldn't understand what they are saying above. What is "tree-level relation"? This one? $$q^\mu \Gamma_\mu ^{(0)} (p, \ p - q) \ = \ Q \left[S_F^{-1}(p) - S_F^{-1} (p - q)\right]^{(0)}$$ What does this mean in terms of Feynman diagrams?​

 

[HomogenousCow]

1. The authors have used the terms "2-point function", "3-point function" and "2-point amplitude" in a number of places. What do these "points" mean? For example, if you refer to the attached pdf, on page 2, the authors write, "Let us first draw the total 2-point amplitude as the sum of the tree-level and the rest." Sometime later, they write, "Let us first evaluate the 1-loop corrections to the 2-point function for the fermions, i.e., the self-energy of the fermions." What does 2-point mean here?

An "n-point function" is a time-ordered vacuum expectation value of n-number of field operators at specified spacetime positions. For example the fermion "2-point function" is $\langle \Omega |T \psi (x) \bar\psi (y) | \Omega \rangle$, where $|\Omega\rangle$ is the interacting vaccum. In the free theory this is just the spinor Feynman propagator $S_F(x-y)$. A "3-point function" in QED might be something like $\langle \Omega |T \psi (x_1) \bar\psi (x_2) A^\mu (x_3)| \Omega \rangle$, from which you may calculate corrections to the vertex factor.

I just couldn't understand what they are saying above. What is "tree-level relation"? This one?

The tree level relation is simply $-ieQ\gamma^\mu$. What he's saying is you dot this with $q$ and then rewrite it to something like $$-ieQq_\mu \gamma^\mu = -ieQ(q_\mu + p_\mu - p_\mu)\gamma^\mu -ieQ(m-m), $$ then when you add this to equation 3 it works out. Mind you those $S_F$ terms in Eq. 12.13 aren't the free propagators but rather the effective QED ones:

$$S(\not\! p) = \int \frac{d^4p}{(2\pi)^4}e^{-ip(x-y)}\langle \Omega |T \psi (x) \bar\psi (y) | \Omega \rangle = \frac{i}{\not\! p - m +\Sigma(\not\! p)},$$

from this you can see that $S^{-1}(\not\! p) = -i(\not\! p - m +\Sigma(\not\! p))$. My signs and placement of $i$s might not agree with your textbook by the way, so interpret them as you may.

 

[Wrichik Basu]

@HomogenousCow Thanks, that was a helpful post. But tell me something: How is the Ward-Takahashi identity useful? What I mean to say is, I can easily figure out the vertex function of the loops by looking at the Feynman diagrams. Why do I need the Ward identity in that case? In the specific example that I have shown, the authors have first written a tree-level relation and then added a correction term to it from Fig. 3 using Ward identity. Was all this at all necessary (other than the purpose of demonstration that the Ward identity is valid for 1-loop contributions)?

Also, I know how to put the vertex function into the Lagrangian (the vertex function was defined from the $\langle j^\mu\rangle A_\mu$ interaction term of the lagrangian, so the steps can be retraced.) But how is this vertex function related to the Feynman amplitude of the whole diagram? In other words, consider Fig. 1, the vertex function for which is given by Eq. 1. Now, what will be the Feynman amplitude for the whole diagram? Do I just multiply the vertex function along with the input from the external fermion and photon lines?

 

[HomogenousCow]

Thanks, that was a helpful post. But tell me something: How is the Ward-Takahashi identity useful? What I mean to say is, I can easily figure out the vertex function of the loops by looking at the Feynman diagrams. Why do I need the Ward identity in that case? In the specific example that I have shown, the authors have first written a tree-level relation and then added a correction term to it from Fig. 3 using Ward identity. Was all this at all necessary (other than the purpose of demonstration that the Ward identity is valid for 1-loop contributions)?

The key is that the Ward-Takahashi identity can be proven non-perturbatively through the path integral formation very easily. This means it is true order by order and to all orders (well, assuming non-perturbative QED really exists), so it allows you prove things quite generally such as the equality of certain renormalization counter-terms to all orders.

Also, I know how to put the vertex function into the Lagrangian (the vertex function was defined from the ⟨jμ⟩Aμ⟨jμ⟩Aμ\langle j^\mu\rangle A_\mu interaction term of the lagrangian, so the steps can be retraced.) But how is this vertex function related to the Feynman amplitude of the whole diagram? In other words, consider Fig. 1, the vertex function for which is given by Eq. 1. Now, what will be the Feynman amplitude for the whole diagram? Do I just multiply the vertex function along with the input from the external fermion and photon lines?

Yeah so I was wondering about this myself earlier today, in fact I made a thread about it but no one has yet to reply. I looked it up later myself and thankfully found a good explanation in Schwartz's text and some lecture PDFs online (http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf).

You can have a look at the above link but I'll just summarize it for you. So in order to be more instructive let's look at something more physical like Compton scattering $\gamma e^- \rightarrow \gamma e^-$. In order to calculate this amplitude from first principles (or near enough) we need to consider the four-point function $$G^{(4)} = \langle \Omega | T A^\mu (x_1) A^\nu (x_2) \psi (x_3) \bar \psi (x_4) | \Omega \rangle.$$ Now the Fourier transform of this expansion will look something like $$\tilde{G^{(4)}} = (2\pi)^4 \delta^4 (\sum p_f - \sum p_i) \Delta^{\mu\alpha}(p_1)\Delta^{\nu\beta}(p_2)S(\not\!p_3) i \mathcal{M}_{\alpha\beta} S(\not\!p_4) ,$$ with $\Delta$ as the effective photon propagators, $S$ the effective fermion propagator, and $i \mathcal{M}$ the interesting part of the amplitude. Now when we feed this thing into the LSZ, it's going to cut off the effective propagators by applying the differential operator found in the free part of the equations of motion to the Green's function. Relevantly to us the photon propagators get cut off by an operation which looks like

$$ \int d^4 x e^{-ipx} (\partial^2 g_{\mu\alpha} - \partial_\mu \partial_{\alpha}) \langle \Omega| T A^\alpha (x)...|\Omega\rangle \sim i \mathcal{M}_\mu, $$ where the analogous operations have been done to the other fields on the LHS. Now using the Schwinger-Dyson equation, which is the QFT version of the Euler-Lagrange equations, the LHS turns into $$\int d^4 x e^{-ipx} (\partial^2 g_{\mu\alpha} - \partial_\mu \partial_{\alpha}) \langle \Omega| T A^\alpha (x)...|\Omega\rangle \sim \int d^4 x e^{-ipx} \langle \Omega| T j^\mu (x)...|\Omega\rangle $$ up to some contact terms which are not relevant here. Now if we consider $$ip_\mu \mathcal{M}^\mu = p_\mu\int d^4 x e^{-ipx} \langle \Omega| T j^\mu (x)...|\Omega\rangle \sim \int d^4 x e^{-ipx} \partial_\mu \langle \Omega| T j^\mu (x)...|\Omega\rangle.$$ The RHS here is exactly what we have in the Ward-Takahashi identity, and it equals certain contact terms which are not relevant here since they don't have the correct poles to survive the LSZ, thus $$ip_\mu \mathcal{M}^\mu = 0.$$

 

[Wrichik Basu]

The key is that the Ward-Takahashi identity can be proven non-perturbatively through the path integral formation very easily. This means it is true order by order and to all orders (well, assuming non-perturbative QED really exists), so it allows you prove things quite generally such as the equality of certain renormalization counter-terms to all orders.

Path integral formalism is something that I have not yet studied. I started it before QFT from Feynman's book but realized some insights into QFT will be useful before I start reading it.

And thanks for the rest of the explanation. I had to do some reading because my book didn't use the term "LSZ reduction formula" anywhere. The method the book proposed (before introducing the Feynman amplitude) was to literally write out each and every term of the S-matrix and then contract fields using Wick contraction. I think I have to read a more rigorous book (like Schwartz or Peskin & Schroeder) before proceeding further.

 

[vanhees71]

The Ward-Takahashi identies in gauge theories are crucial for their consistence. They precisely reflect that gauge invariance of observable quantities (in vacuum QFT these are the S-matrix elements for physical processes and thus cross sections and decay rates). First of all they guarantee that naively Dyson-renormalizable gauge theories (like the Standard Model of elementary particles physics) are renormalizable to all orders in perturbation theory. This is most easily proven in the socalled background-field gauge, where also in the non-Abelian gauge-group case you don't need the more complicated Slavnov-Taylor identities and BRST symmetry as in the standard proof but you can do with Ward-Takahashi identities as in the Abelian case (QED). One crucial point in the poof is that in QED the four-photon proper vertex function is finite, i.e., you do not need to introduce non-renormalizable counter terms for it.

For a detailed discussion of renormalization and symmetries, also for local gauge symmetries, see

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf