So, to begin here is the Hartree-Fock approximation in the book:

The part below tells us that in the Hartree approximation (mean field) only the first two diagrams contribute. Note that not only the loop but also the rightmost propagator is not bare in the Hartree term.

Let me now upload the relevant section of the book

(It looks longer than it is with this layout but they're not full pages. In any case. We have them there for reference)

Then rearraging the SE for the full propagator, an effective potential (screened potential) is define din $(3.76)$ and we can focus on figures now. Look at figure $3.22(b)$ they say it is the Hartree approximation but the rightmost propagator is not dressed (cf. my very first picture).

Note that in this figure $G(a,b, U)$ (dashed line) is what we have called $G(a,b)$ up to now, just that the $U$ dependence is explicitly written. On the other hand, I think that $G_0(a,b,U)$ is not $G_0(a,b)$ from above, since the latter would the the zero order term and it should not display $U$ dependence. I think this is an unfortunate notation for the propagator in the presence of the external potential and neglecting particle interactions.

I think that what we originally called $G_0(1,2)$ is with this notation $G(1,2,0)$ (which would be the zero order term). This view seems to be confirmed by the next figure (3.23), where it is expanded to first order in the external potential.

Finally, we have figure 3.24, where the full propagator (dashed) is written using all of the equations above. Note that the solid line, unlike the previous figure, is now $G(1,2,0)=G_0(1,2)$, not $G_0(1,2,U)$, which should mean that it contains also the "normal" tadpoles (the ones without an X) coming from particle interactions

That's pretty much it. From direct comparison of the final diagram and our equation we have an expression for the potential, but... The notation made the passage from figure 3.23 to 3.24 very nebulous and most of all, I don't see why we get away with not resumming (I'm talking about that rightmost propagator)

Probably this has something to do with the fact that we're seeking a first order response.

So, the real question is why figure 3.22(b)?

(Of course no obligations, I'll use this room as my archive anyways)

Hey. I'll have a look later.

@HerrFeinmann yes, from a first look I'd have interpreted $G_0(a,b,U)$ like this: non-interacting particles in the presence of an external potential $U$, and with that $G_0(a,b,U=0)$ is what previously has been called $G_0$. Is that also your understanding?

And yes...the notation is a bit... weird so to say :d

@TobiasFünke On the first question, yes. Regarding the second, I would say that $G(1,2,U=0)=G_0(1,2)$

ok. but $G_0(1,2)$ can mean two things, no? Either free particles or zeroth order in $U$

at least this is what I've understood from your comments

This is my understanding and the notation is very cumbersome (and understated)

(The first is just $(3.72)$)

Hello nI, I'll give you writing privilege

Done.

looks like a trainwreck to read

xD

@naturallyInconsistent Indeed, that's why I stated I don't expect you to read or answer. In any case this will be my archive. I think the book is more of a trainwreck that what I've written. This is the result of hours of interpretations :P

why do you use this book?

So, long story short. In my course this computation was sketched as an example and I needed to figure out some details. I checked the references and found it in this book. I don't like it either, but apparently it's the only one I can find it doing it this way. Now I'm at the point where I'm trying to figure out why this book does so. We've all been there, haven't we? :P

:d yes

@HerrFeinmann if all you are asking is about why the original diagram that should have dressed propagator in both the tadpole loop and also in the 1 prime 1 rightmost propagator, yet is not so in the new diagram, then, yes, it is obvious that this is due to restriction of consideration to linear response.

Diagrams like this one that dress the propagator independently from the external potential and all the other similar diagrams appearing in the resummation.

That's why I feel that one should do the resummation for the part bare in the external potential $U$ (so you can get $G(1,2, U=0)$ as in the description of fig. 3.24 and do what we are saying only for the diagrams that involve a $\times$. I hope I phrased it clearly.

Isnt that $U^2$ and above? It is quite annoying because, yes, you have to have all the correlation terms that are ignoring $U$, and that is already quite annoying to work with

But as far as that exposition goes, it seems to be wanting only the linear response, and when you expand both dressed propagators, you will end up with just that diagram being linear in U

Note that it can easily be an infinite resummation that still ends up being linear in U

I'm super tired, gonna sneeppuuu

It would be with the x's. Wait, I'll edit the pic

This is U^2. This book is a mess :P

@naturallyInconsistent Okie dokie. Thank you for your comments. I will keep thinking about it. Good night!

@HerrFeinmann this is wrong. The x-es are on the straight line where it joins the propagator. You'll see it matching the labels that appear in the integrals.

Indeed, all Feynman-like diagrams are meant to be integrals

shorthand for integrals

Unless I'm misunderstanding (sorry in such case), you can have X on loops, as in the picture above, fig 3.24

@HerrFeinmann oops, myow bad. Yes, in the notation that this paper is using, the x is being tadpole-ed out. It is just insane and internally inconsistent with how it is also drawing the x at the bottom on that exact same figure. I think the author is just drunk.

@naturallyInconsistent Ahahahahhaha well at least I got a good laugh out of it, thanks :D

This is basically my life as a student. Courses stealing from books that end up being inconsistent :P