effrit said:
I see the same basis everywhere and I can find the new physics corrections, but nowhere seems to have an actual number for the standard model value?
Aren't they all zero by construction?
The Wilson coefficients of an effective theory are calculated as the matching coefficients between the full (high energy) theory and the effective (low energy) theory at a given matching scale.
So to calculate the Wilson coefficients to a given order in perturbation theory you calculate all the relevant loop diagrams in the full theory (i.e. light degrees of freedom + heavy degrees of freedom) and in the effective theory, (i.e. only light degrees of freedom) and fix the Wilson coefficient by the condition that at some matching scale the results have to agree.
In case of SMEFT the standard model + higher dimensional operators of standard model fields ist the low energy effective theory, of an unknown high energy theory with some unknown heavy degrees of freedom (new heavy particles or whatever). So when you calculate the Wilson coefficients the pure standard model diagrams will exactly cancel out because they contribute both in the high energy theory above the matching scale and in the low energy theory below the matching scale.
Assuming there is really no new physics beyond the standard model, you would just match two identical theories (the standard model above and below the matching scale) onto each other, which leads to the Wilson coefficients of the higher dimensional operators all being zero.
What you seem to have in mind, that you shrink the standard model diagrams to a point interaction, would mean that you are integrating out heavy standard model particles. But that is not the point in SMEFT. You do that for example in Fermi theory of weak interaction when integrating out the W-boson, or when you introduce an effective gluon-Higgs coupling when integrating out the top loop, but in SMEFT the full standard model IS the low energy effective theory, and you integrate out only new physics.