How to construct a diagrammatic interpretation out of ##Z[J]##

[JD_PM]

TL;DR Summary

I am aimed at understanding how to construct a diagrammatic interpretation (and then derive the Feynman rules) out of a generating functional $Z[J]$fds

I am aimed at understanding how to derive the Feynman rules out of a generating functional $Z[J]$, which depends on the set of coordinates $x=(x_1,...,x_n)^T \in \Bbb R^n$ and Grassmann variables $\bar{\theta}, \theta$

\begin{equation}
Z[J] := \frac{1}{(2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp \left( -\bar{\theta}_i \partial_j J_i (x) \theta_j - \frac 1 2 J_i (x) J_i (x) \right) \tag{1}
\end{equation}As an example let us set a source $J_i(x) := x_i + \frac 1 2 g_{ijk} x_j x_k$ where $g_{ijk} = g_{ikj} = g_{kij}$. $Z[J]$ then takes the form

\begin{align*}
Z[J] &= \frac{1}{(2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp \left( -\bar{\theta}_i \partial_j J_i (x) \theta_j - \frac 1 2 J_i (x) w_i (x) \right) \\
&= \frac{1}{(2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp \Big( -\bar{\theta}_i \delta_{ij} \theta_j - g_{ijk} \bar{\theta}_i \theta_j x_k \\
&- \frac 1 2 x_i \delta_{ij} x_j -\frac 1 2 g_{ijk} x_i x_j x_k - \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m \Big) \\
&= \frac{1}{(2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp \Big( -\bar{\theta}_i \delta_{ij} \theta_j - \frac 1 2 x_i \delta_{ij} x_j\Big) \times \\
&\times \exp\left( - V(\theta, \bar{\theta}, x) \right) \\
&= \frac{1}{(2 \pi)^{n/2}} \exp\left( V\left(\frac{\partial}{\partial \bar{\theta}}\frac{\partial}{\partial \theta}\frac{\partial}{\partial x} \right)\right)\times \int d^n x \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp \Big( -\bar{\theta}_i \delta_{ij} \theta_j - \frac 1 2 x_i \delta_{ij} x_j\Big)
\end{align*}

Where

\begin{equation*}
V(\theta, \bar{\theta}, x) = + \frac 1 2 g_{ijk} x_i x_j x_k + \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m + g_{ijk} \bar{\theta}_i \theta_j x_k
\end{equation*}

I interpret $\frac 1 2 g_{ijk} x_i x_j x_k$, $\frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m$ and $g_{ijk} \bar{\theta}_i \theta_j x_k$ as a $\phi^3$ term, a $\phi^4$ term and a Yukawa term respectively.

Next we evaluate the derivatives of the potentials (Osborn's (1.132)). I get

\begin{equation*}
V_i(x) = \frac 3 2 g_{ijk} x_j x_k + \frac 1 2 g_{ijk} g_{jlm} x_k x_l x_m + g_{ijk}\bar{\theta}_j \theta_k - g_{ijk}\bar{\theta}_j x_k + g_{ijk}\theta_j x_k
\end{equation*}
\begin{equation*}
V_{ij}(x) = 3 g_{ijk} x_k + \frac 1 2 g_{ijk} g_{klm} x_l x_m + g_{ikl} g_{jkm} x_l x_m + 2g_{ijk}\theta_k - 2g_{ijk}\bar{\theta}_k
\end{equation*}
\begin{equation*}
V_{ijk}(x) = 3 g_{ijk} + g_{ijl} g_{klm} x_m + 2g_{ikl} g_{jlm} x_m
\end{equation*}
\begin{equation*}
V_{ijkl}(x) = g_{ijm}g_{klm} + 2g_{ikm} g_{jlm} x_m = 3g_{ijm}g_{klm}
\end{equation*}

Evaluating each potential term at $x=0$ yields $V_i(x)\Big|_{x=0} = g_{ijk}\bar{\theta}_j \theta_k$, $V_{ij}(x)\Big|_{x=0} = 2g_{ijk}\theta_k - 2g_{ijk}\bar{\theta}_k$, $V_{ijk}(x)\Big|_{x=0} = V_{ijk}(x)$ and $V_{ijkl}(x)\Big|_{x=0} = V_{ijkl}(x)$

OK my question at this point is: how to translate the above results into a diagrammatic interpretation? (so that we can write down the Feynman rules). I tried to follow Osborn but I do not see how he wrote the vacuum diagrams.

I already posted a similar thread here. However, I would like to gain deeper understanding and not only focus on solving the problem.

I followed Osborn's 1.4 section approach because is the way we have been taught in class.

However, while checking bibliography (mainly Mandl & Shaw [chapters 12, 13], QFT for the gifted amateur and Hendrik van Hees notes [section 4.7]), I noticed they all work in terms of Green's functions. Is this the standard way to approach this kind of problem?

It is perfectly OK to me to switch to Green's function language; at the end of the day, once I understand the main idea, I should be able to switch approach at will.

Thank you!

 

[AndyW]

Hello there,

First of all, great job on working through the problem and deriving the potentials! To answer your question, the diagrammatic interpretation of these potentials is as follows:

1. $V_i(x)$: This potential corresponds to a vertex with three external legs (one for each $x_i$) and one internal leg (corresponding to the $\bar{\theta}_j \theta_k$ term). This is a $\phi^3$ vertex, as you correctly identified.

2. $V_{ij}(x)$: This potential corresponds to a vertex with four external legs (two for each $x_i$ and $x_j$) and two internal legs (corresponding to the $\theta_k$ and $\bar{\theta}_k$ terms). This is a $\phi^4$ vertex.

3. $V_{ijk}(x)$: This potential corresponds to a vertex with five external legs (three for $x_i$, $x_j$, and $x_k$) and three internal legs (corresponding to the $\theta_k$ and $\bar{\theta}_k$ terms). This is a Yukawa vertex.

4. $V_{ijkl}(x)$: This potential corresponds to a vertex with six external legs (four for $x_i$, $x_j$, $x_k$, and $x_l$) and four internal legs (corresponding to the $\theta_k$ and $\bar{\theta}_k$ terms). This is a quartic Yukawa vertex.

As for your question about using Green's functions, it is indeed a common approach in QFT. In fact, Green's functions are closely related to Feynman diagrams, as they can be used to calculate the amplitudes associated with each diagram. So, if you would like to switch to Green's function language, it should not be a problem. However, it may require some additional knowledge about how to calculate Green's functions and their relation to Feynman diagrams.

I hope this helps! Let me know if you have any other questions or need further clarification. Keep up the good work on your studies!