A specific integration with Grassmann variables

Question

I have recently read (for example, here) that this relation below is true

$$ \int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}), $$ where $Pf(\mathbf{A})$ is the Pfaffian of an even dimensional skew-(or anti-)symmetric matrix $\mathbf{A}$ and $\{ z_i \}$ are Grassmann variables.

Could anyone point out if there is a similar formula for the form below:

$$ \int dz \: \big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^k, $$ where $k > 0$ is an integer?

My guess is yes since

$$ e^{\mathbf{A}} = \sum_{j=0}^\infty \frac{\mathbf{A}^j}{j!}, $$ but I do not have enough experience with these types of integrals.

Thanks in advance!

Answer 1

Let's denote the number of Grassmann variables $z_i $ by $N$, i.e., $i=1,\ldots ,N$. Then, in your first equation, the result on the right-hand side is exactly generated by the term $$ \int dz \: \frac{1}{(N/2)!}\big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^{(N/2)} $$ from the exponential series on the left-hand side, since only the monomial containing each $z_i $ precisely once yields a non-zero result upon integration. Therefore the result for your desired integral is zero for $k\neq N/2$, and $(N/2)!\ Pf(\mathbf{A})$ for $k=N/2 $.