QED Renormalization: Feynman Diagrams with Counterterms on External Legs

[paweld]

I have a question concerning renormalization of QED. I don't know if
Feynman diagrams with counterterms on ecternal legs are allowed.
Normally to find S matrix amputated green function is necesary and to find
it one don't take into account all propagators on external legs - they
are canceled according to LSZ formula.

 

[meopemuk]

If you follow the general recipe, then you must draw loops in external legs. For example, there should be diagrams with photon half-circles attached to electron external lines. This diagrams are represented by divergent integrals and should be "renormalized" by adding counterterms to the original Hamiltonian. The counterterms lead to the appearance of new diagrams. For example, there will be graphs in which counterterms are inserted in the electron external legs. They are sometimes depicted by placing a cross on the electron line.

The important fact is that diagrams with crosses (counterterms) *exactly* cancel similar diagrams with loops. This cancellation is true only if both the cross and the loop are placed in an external line, where energy-momentum is "on the mass shell". For this reason, one can decide to ignore both loops and counterterms in external legs.

The cancellation between loops and counterterms is not perfect for internal lines (where the energy-momentum if "off-shell"). So, after adding infinite loop contribution and -infinite counterterm contribution you are left with a small finite residual term. These residual terms are responsible for "radiative corrections" to scattering amplitudes.

Eugene.

 

[TriTertButoxy]

According to the LSZ formula, upon computing the renormalized 1PI (or amputated, or truncated, or proper) diagrams, you have to multiply the result by factors of the square-root of the residue of the poles associated with the external particles. To get these residues, you must do a separate self-energy correction, and find the relevant pole and residue. This separate exercise requires renormalization as well.

 

[paweld]

Thanks.
Could you give me a reference to book or paper where all rules how to
draw QED diagrams with counterterms and write formal formula for
amplitude are given.

 

[meopemuk]

Thanks.
Could you give me a reference to book or paper where all rules how to
draw QED diagrams with counterterms and write formal formula for
amplitude are given.

There are many QFT textbooks where all the rules are laid out. My favorite books are Schweber, Bjorken&Drell, and, of course, Weinberg. It is not a bad idea to read the original Feynman's papers. However, in my opinion there is not a single textbook explaining QFT and renormalization in a satisfactory manner. If you want to get a full logical picture of what is going on there (not just learn the rules by heart) you'll need to study and compare several textbooks and do a lot of homework.

Eugene.

 

[paweld]

Could you explain me what extra terms in the formula for amplitude appear from
the counterterm [tex] i (Z_2-1) \bar{\psi} {\not\partial} \psi [/tex] in QED
lagrangian. This term is indicated as a crossed line in Feynamnn diagram. Is this just
[tex] i (Z_2 -1) {\not p} [/tex] or this with two propagators:
[tex] i \frac{{\not p}+m}{p^2-m^2+i\epsilon} (- i (Z_2 -1) \displaystyle{\not p})
i\frac{{\not p}+m}{p^2-m^2+i\epsilon}[/tex]?

 

[Avodyne]

Thanks.
Could you give me a reference to book or paper where all rules how to
draw QED diagrams with counterterms and write formal formula for
amplitude are given.

Srednicki. Draft copy available for free download from his webpage.

 

[Avodyne]

Could you explain me what extra terms in the formula for amplitude appear from
the counterterm [tex] i (Z_2-1) \bar{\psi} {\not\partial} \psi [/tex] in QED
lagrangian. This term is indicated as a crossed line in Feynamnn diagram. Is this just
[tex] i (Z_2 -1) {\not p} [/tex] or this with two propagators:
[tex] i \frac{{\not p}+m}{p^2-m^2+i\epsilon} (- i (Z_2 -1) \displaystyle{\not p})
i\frac{{\not p}+m}{p^2-m^2+i\epsilon}[/tex]?

The latter.