Consider the Euclidean path integral for anharmonic oscillator:

Action:

\[ S_E[\phi] = \int d\tau \left( \frac{1}{2} \left( \frac{d\phi}{d\tau} \right)^2 + \frac{1}{2} m^2 \phi^2 + \frac{\lambda}{4!} \phi^4 \right) \]

Generating functional:

\[ Z[J] = \int \mathcal{D}\phi \exp\left( -S_E[\phi] + \int d\tau J(\tau) \phi(\tau) \right) \]

Time-ordered \(n\)-point function (without divided by \(Z[0]\)):

\[ \left. \frac{\delta^n Z[J]}{\delta J(\tau_1) \cdots \delta J(\tau_n)} \right|_{J=0} \]

Perturbative expansion for time-ordered correlators using functional derivative (without divided by \(Z[0]\)):

\[ \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{k! (4!)^k} \int d\tau_1' \cdots d\tau_k' \left. \frac{\delta^{n + 4k} Z[J]}{\delta J(\tau_1) \cdots \delta J(\tau_n) \delta J(\tau_1')^4 \cdots \delta J(\tau_k')^4} \right|_{J=0} \]

In this demo, each diagram is labeled by its order \(k\) (power of \(\lambda^k\)).