Momentum cutoffs free field theory

[thoughtgaze]

for a given diagram in some interacting theory that needs a momentum cutoff

shouldn't the same momentum cutoff be used for diagrams that don't need a momentum cutoff for convergence

for example, phi3 theory has a self energy diagram that diverges, so if one imposed a momentum cutoff there, shouldn't there also be one for the tree level diagram for consistency?

Shouldn't it also be imposed on the free part of the lagrangian too?

 

[The_Duck]

Well, remember that in the end we always take the limit as the momentum cutoff goes to infinity. For divergent diagrams this limit is nontrivial and defining and evaluating this limit is the whole point of renormalization. But for convergent diagrams this limit is trivial and is equal to what you would get if you just evaluated them without a momentum cutoff.