I have a question about non-crossing approximation (NCA) stated in Condensed Matter Field Theory by Altland & Simons1. It is on page 200-203 for the 3rd edition (on page 223-227 for the 2nd edition). In short, I believe the expression for the first-order contribution of the self-energy operator in the book has two problems; one is quite important while the other might be unnecessary.

Below are the equation for the action (1) and for the self-energy (2) and the diagrammatic expression for each term; the first(second) term of the LHS of (2) for the first(second) diagram in the figure. $\phi = \{\phi^a\}$ is an N-component vector field with $a=1,...,N$. $G_{0,\textbf{p}}$ is the Fourier component of the free Green's function,
$G^{ab}_0(x-y)\equiv\langle\phi^a(x)\phi^b(y)\rangle_0$.

$(1) \quad S[\phi] \equiv \int d^dx \,(\frac{1}{2}\partial\phi\cdot\partial\phi
+\frac{r}{2}\phi\cdot\phi+\frac{g}{4N}(\phi\cdot\phi)^2)$

$(2) \quad\Sigma_{\textbf{p}}^{(1)}=-\delta^{ab}\frac{g}{L^d}(\frac{1}{N}\sum\limits_{\textbf{p}'}G_{0,\textbf{p}'}+\sum\limits_{\textbf{p}'}G_{0,\textbf{p-p}'})$

Here are the two points I doubt.

The $1/N$ coefficient should be in front of the second term of Eq(2) rather than the first one.

The Green's function in the second summation should be the same as the first one since $g$ does not carry any momentum.(Actually, I cannot understand why we should introduce such 'wavy-line' unless we adopt some auxiliary field[2] and write Eq(1) as
$$S[\phi, A] = \int d^dx \,
(\frac{1}{2}\partial\phi\cdot\partial\phi
+\frac{r}{2}\phi\cdot\phi
-\frac{N}{4g}A^2
+\frac{1}{2}A(\phi\cdot\phi)).$$ Integrating out for $A$ gives the same result with Eq(1). However, also in this case, the $A$ propagator does not carry any momentum due to the absence of a derivative term.)

To explain in more detail,

- For $\phi$ loop, it contains index summation which gives the factor $N$. For example, for the first order of the Green function we should evaluate the term like (with Einstein's summation)

$\langle\phi^a(x)\phi^b(y)\int d^dz \,\, \phi^c(z)\phi^c(z)\phi^d(z)\phi^c(z)\,\rangle_0$.

The first diagram represents the case when the contraction occurs for (a,c), (b,c) & (d,d); (d,d) contraction gives the $N$ factor. On the other hand the second diagram is the case for (a,c), (b,d) & (c,d). That is why I believe the expression is wrong.

- I know that
$\sum\limits_{\textbf{p}'}G_{0,\textbf{p}'}=\sum\limits_{\textbf{p}'}G_{0,\textbf{p-p}'}$ for this case and I think that it could be just pedagogical expression; to write similarly as the case for Coulomb interaction where bosonic field propagator carries momentum. Is it right to think in this way?

$1/N$ coefficient is important in NCA since it decides which term to ignore under the limit of $N\rightarrow \inf$. Considering the book's context and specified term such as "rainbow diagrams", I know I am wrong. But I cannot find what's the problem. Please help me.

1 Altland A, Simons BD. Condensed Matter Field Theory. 3rd ed.

[2] Hooft, G. 'T ., 2002. LARGE N, in: .. https://doi.org/10.1142/9789812776914_0001

This post imported from StackExchange Physics at 2024-07-19 15:15 (UTC), posted by SE-user Jinu.P