I hope that such a post is appropriate in this subreddit.
I have just found this video on youtube and looked/skipped through some of the parts. The explanations and recaps seem extremely well made.
I wanted to share this with everyone else, as this video, naturally, does not have a lot of views.
Hi everyone,
recently I made a post about EFT's in general and I thank you guys very much for answering my questions! I would now have another fundamental question regarding Soft collinear effective theory (SCET). We can restrict ourselves to the scalar phi^3 SCET toy theory. I am using arXiv:1410.1892 as one of my primary sources for study.
My understanding of the construction of the EFT so far is:
Split up the field into collinear, anti-collinear and soft contributions (the heavy contributions are integrated out)
Plug them into the general phi^3 theory and only leave those which are allowed by momentum conservation
Multipole expand the soft fields
Having done that, one finds out that the theory is non-renormalized in any order in perturbation theory as the resulting integrals are always scaleless. So the resulting operators do not have any Wilson coefficients (meaning they are equal to 1).
So here is somewhat where the confusion for me starts. In other effective theories, the approach in my lectures was always to construct the most general Lagrangian for the EFT which respects the symmetries, up to some order in the small parameter. The higher dimensional operators have Wilson coefficients in front of them, which can be determined through matching calculations, where we try to use the respective operators in a given S-matrix element or correlation function.
In the SCET on the other hand we don't seem to know the higher dimensional operators at first. So one now looks at a "hard external" current in the full and then effective theory and finds out that the Wilson coefficients are apparently functions of the collinear and anti-collinear momenta.
And here is the main point I don't really understand, although it should be really basic: We now expect the corresponding position space operator in the EFT to correspond to the Fourier transform of the momentum dependent Wilson coefficient (so we replace all momenta by derivatives) from which it follows that the full operators are non-local.
I don't understand the following: When determining Wilson coefficients, we are computing Feynman diagrams, which give us Lorentz invariant scalars and all of the EFT's I have looked at so far, the Wilson coefficients were simple scalars. Here, the Wilson coefficient is a function of the external momenta, ok, but why should this correspond to the "momentum space Wilson coefficient"? Isn't it just simply the value of the Wilson coefficient, as again, Feynman diagrams don't give us "momentum" or "position" space numbers, but simply Lorentz scalars?
I would highly appreciate any help.
Hi everyone,
I am currently participating in an introductory course on EFT's where we discuss their general ideas and applications (Euler-Heisenberg, Weak EFT, HQET, SCET, xPT). I have a very fundamental question regarding the integration of heavy particles.
Say we have a toy scalar theory consisting of a heavy and a light field. Then one can imagine, that to get the EFT Lagrangian, one integrates out the heavy field, expands the resulting action in the small ratio E/M (if M is the mass of the heavy particle) and as such gets new local operators in the EFT (I know that this is in some sense an outdated point of view, but it shall suffice).
Then on the other hand, we can (without eplicitly calculating the path integral) construct the most general Lagrangian out of the light d.o.f (in our case the light field) which obeys the symmetries of the UV complete Lagrangian, up to some order in our expansion. This will give us our EFT Lagrangian including the Wilson coefficients for the higher order operators. After performing the path integral above we can then compare both Lagrangians to determine the Wilson coefficients.
What I do not quite understand is why this procedure only allows us to determine the tree level Wilson coefficients. How, using the path integral, can I determine say the 1-loop Wilson coefficients?
I know that the path integral approach is not necessary for a matching calculation and we can instead calculate correlation functions or S-matrix elements in the UV-complete and the EFT separately and then compare results to get the Wilson coefficients. But I would like to understand where in the path integral approach we have "neglected" the higher order corrections.
I would really appreciate any answers as always!