- #1

JD_PM

- 1,131

- 158

- Homework Statement
- Given the generating functional (repeated indices in the same term are assumed to be summed over)

\begin{equation}

Z[w] := \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 w_i (x) \theta_j - \frac 1 2 w_i (x) w_i (x) \right) \tag{1}

\end{equation}

We define

\begin{equation*}

w_i(x) := x_i + \frac 1 2 g_{ijk} x_j x_k, \quad \text{where} \quad g_{ijk} = g_{ikj} = g_{kij}

\end{equation*}

Where ##g_{ijk}## is assumed to be small.

a) Find the Feynman rules.

b) Find the one- and two-loop contributions to ##Z[g]##

- Relevant Equations
- N/A

To approach the problem I first studied section 1.3 and, more importantly, 1.4 of Osborn's notes.

We first need to compute ##\partial_j \omega_i (x)## and ##\omega_i (x)\omega_i (x)##

\begin{equation*}

\partial_j \omega_i (x) = \delta_{ij} + \underbrace{\partial_j (g_{ilm})}_{=0}x_l x_m + g_{ijk}x_k

\end{equation*}

\begin{align*}

\omega_i \omega_i &= \left( x_i + \frac 1 2 g_{ijk} x_j x_k\right) \left(x_i + \frac 1 2 g_{ilm} x_l x_m \right) \\

&= x_i x_i + \frac 1 4 g_{ijk} g_{ilm} x_j x_k x_l x_m + g_{ijk}x_i x_j x_k

\end{align*}

Plugging the given definition of ##\omega_i (x)## into ##(1)## yields

\begin{align*}

Z[w] &= \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 w_i (x) \theta_j - \frac 1 2 w_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 x_i -\frac 1 2 g_{ijk} x_j x_k - \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m \Big) \\

&= \int \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp(-\bar{\theta}_i \delta_{ij} \theta_j ) \frac{1}{(2 \pi)^{n/2}} \int d^n x \exp\Big( -\frac 1 2 \vec x \cdot \vec x \\

&+ \vec b \cdot \vec x - V(x) \Big)

\end{align*}

Where

\begin{equation*}

b_k := g_{ijk} \theta_i \bar{\theta}_j, \quad \text{noticing} \quad \{ \theta, \bar{\theta}\} = 0

\end{equation*}

\begin{equation*}

V(x) = + \frac 1 2 g_{ijk} x_j x_k + \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m

\end{equation*}

Next we work out the potential. Based on 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

\end{equation*}

\begin{equation*}

V_{ij}(x) = 3 g_{ijk} x_k + \frac 1 2 \left( g_{ijk} g_{klm} x_l x_m + 2g_{ikl} g_{jkm} x_l x_m \right)

\end{equation*}

\begin{equation*}

V_{ijk}(x) = 3 g_{ijk} + g_{ijk} g_{klm} x_m + 2g_{ikl} g_{jkm} 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*}

At this point I should have all necessary tools to write down the Feynman rules and proceed with the diagrammatic interpretation. However I am struggling to see how I can proceed, in an analogous way Osborn did in section 1.4

Any guidance is much appreciated.

Thank you

We first need to compute ##\partial_j \omega_i (x)## and ##\omega_i (x)\omega_i (x)##

\begin{equation*}

\partial_j \omega_i (x) = \delta_{ij} + \underbrace{\partial_j (g_{ilm})}_{=0}x_l x_m + g_{ijk}x_k

\end{equation*}

\begin{align*}

\omega_i \omega_i &= \left( x_i + \frac 1 2 g_{ijk} x_j x_k\right) \left(x_i + \frac 1 2 g_{ilm} x_l x_m \right) \\

&= x_i x_i + \frac 1 4 g_{ijk} g_{ilm} x_j x_k x_l x_m + g_{ijk}x_i x_j x_k

\end{align*}

Plugging the given definition of ##\omega_i (x)## into ##(1)## yields

\begin{align*}

Z[w] &= \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 w_i (x) \theta_j - \frac 1 2 w_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 x_i -\frac 1 2 g_{ijk} x_j x_k - \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m \Big) \\

&= \int \prod_{i=1}^n d \bar{\theta}_i d \theta_i \exp(-\bar{\theta}_i \delta_{ij} \theta_j ) \frac{1}{(2 \pi)^{n/2}} \int d^n x \exp\Big( -\frac 1 2 \vec x \cdot \vec x \\

&+ \vec b \cdot \vec x - V(x) \Big)

\end{align*}

Where

\begin{equation*}

b_k := g_{ijk} \theta_i \bar{\theta}_j, \quad \text{noticing} \quad \{ \theta, \bar{\theta}\} = 0

\end{equation*}

\begin{equation*}

V(x) = + \frac 1 2 g_{ijk} x_j x_k + \frac 1 8 g_{ijk} g_{ilm} x_j x_k x_l x_m

\end{equation*}

Next we work out the potential. Based on 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

\end{equation*}

\begin{equation*}

V_{ij}(x) = 3 g_{ijk} x_k + \frac 1 2 \left( g_{ijk} g_{klm} x_l x_m + 2g_{ikl} g_{jkm} x_l x_m \right)

\end{equation*}

\begin{equation*}

V_{ijk}(x) = 3 g_{ijk} + g_{ijk} g_{klm} x_m + 2g_{ikl} g_{jkm} 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*}

At this point I should have all necessary tools to write down the Feynman rules and proceed with the diagrammatic interpretation. However I am struggling to see how I can proceed, in an analogous way Osborn did in section 1.4

Any guidance is much appreciated.

Thank you