I have been studying the path integral approach to QFT and set myself the challenge of starting from a $\phi^3$ interaction Lagrangian and following the method through to completion.
I started with:
$$
\mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu \phi - \frac{m^2}{2}\phi^2 - \frac{\lambda}{3!}\phi^3
$$
By introducing a source term, $-J\phi$, I was then able to express the partition function in the following form:
$$
Z[J] = Z_{free} \exp{\biggl\{\frac{-i\lambda}{3!}\int \biggl(\prod_{i=1}^3 \frac{d^D k_i}{(2\pi)^D} ~i\frac{\delta~}{\delta J(k_i)} \biggr)(2\pi)^D \delta(k_1+k_2+k_3)\biggr\}} \\
\exp{\biggl\{\frac{-i}{2}\int \frac{d^D k}{(2\pi)^D} \frac{J(-k) J(k)}{k^2 - m^2} \biggr\}}
$$
So at this point I am quite happy, I have both the vertex contributions and the propagator. However, there is a numerical factor $Z_{free}$ given by:
$$
Z_{free} = \int[\mathcal{D}\phi] \exp \biggl\{ i \int \frac{d^D k}{(2\pi)^D} ~\phi(-k) \cdot \frac{1}{2}(k^2-m^2)\cdot \phi(k)\biggr\}
$$
After performing a Wick rotation this would look like a Gaussian integral, albeit a functional one. However, I am not sure how to proceed with calculating this term, the integral in the exponent is additionally troubling to me.
Any advice would be much appreciated, I am sure this follows from the generalisation of an $n$-dimensional Gaussian integral.
