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In section 6.4 of Weinberg QFT, the book says on page 286:

It is important to also consider Feynman diagrams "off the mass shell", for which the external line energies like the energies associated with internal lines are free variables, unrealted to any three-momenta. For one thing, these arise as parts of larger Feynman diagrams; for instance, a loop appearing as an insertion in some internal line of a diagram could be regarded as a Feynman diagram with two external lines, both off the mass shell."

My question is: what does it mean Feynman diagrams "off the mass shell". Do we simply use the usual S matrix Feynman diagram, but we do not assume incoming and outgoing particles to be on the mass shell, or do we actually replace the on shell external lines with propogators? Because in the first possible definition, the off shell diagram would not be a part of a larger Feynman diagram. However, later the book says

Of course, once we calculate the contribution of a given Feynman diagram off the mass shell, it is easy to calculate the associated S matrix elements by going to the mass shell, taking the four momentum $p^\mu$ flowing along the line into the diagram to have $p^0=\sqrt{\boldsymbol{p}^2+m^2}$ ...

This paragraph seems to imply the first definition is what Weinberg means.

For my 2nd question, on page 287, the book goes on to say

Feynman graphs with lines off the mass shell are just a special case of a wider generalization of the Feynman rules that takes into account the effects of various possible external fields."

My question is: how does existence of external fields causing the Feynman diagram to be off shell? The incoming particles and outgoing particles of an S matrix are always on shell by definition.

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  • $\begingroup$ The external lines do not need to carry propagators, because these smaller diagrams are part of larger diagrams where they will be connected via propagators to the rest of the larger diagram. $\endgroup$ Dec 16, 2023 at 4:21

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  1. We are apparently here only considering Feynman diagrams in momentum space, i.e. all external legs carry a 4-momentum, and the diagrams obey total momentum conservation.

  2. An on-shell (off-shell) diagram means that the 4-momenta of the external legs satisfy (do not satisfy) the mass-shell condition, respectively.

  3. Note that a diagram may exclude or include their external legs, although an on-shell diagram always excludes their legs, i.e. is amputated. Also diagrams are not necessarily realized as $S$-matrix elements. However, check out the LSZ reduction formulas.

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  • $\begingroup$ Does the off shell diagram contains factors like $1/(p^2-m^2+i\epsilon)$ in the external lines? $\endgroup$ Oct 13, 2022 at 21:21
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    $\begingroup$ If and only if the external lines are included in the parts that belong to the diagram. $\endgroup$
    – Qmechanic
    Oct 13, 2022 at 21:35

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