Wrichik Basu
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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 selfenergy 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.
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 "2point function", "3point function" and "2point 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 2point amplitude as the sum of the treelevel and the rest." Sometime later, they write, "Let us first evaluate the 1loop corrections to the 2point function for the fermions, i.e., the selfenergy of the fermions." What does 2point mean here?
2. Consider Fig. 1. The vertex function for the above diagram is:
Eqn. 1
where ##eQ## is the charge of the particle, and the superscript on ##\Gamma## denotes the number of loops.
The authors have said, "Let us first draw the total 2point amplitude as the sum of the treelevel and the rest." And they have given the following diagram:
Fig. 2
The undecorated line on the RHS is the tree level diagram.
Next, consider this diagram (from the attached pdf):
Fig. 3
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.
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 1loop 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.
I just couldn't understand what they are saying above. What is "treelevel 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?
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