Evaluation of a Double Integral with Exponentials and Two Generalized Laguerre Polynomials and Its Associated Moments

We analytically evaluate in terms of finite sums, the following definite improper integral, which appears in the quantum field theoretical calculation involving elementary charged particles having charge $\pm e$ in presence of a constant external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$ $$ I_{abln}^{\mu_1\dots\mu_r}(\mathbf{q}_\perp,|eB|) = \int\!\!\dfrac{d^2k_\perp}{(2\pi)^2}k_\perp^{\mu_1} \dots k_\perp^{\mu_r} \exp \bigg(-\frac{\mathbf{k}_\perp^2+\mathbf{p}_\perp^2}{|eB|}\bigg) L^a_l \bigg(\frac{2\mathbf{k}_\perp^2}{|eB|}\bigg) L^b_n \bigg(\frac{2\mathbf{p}_\perp^2}{|eB|}\bigg)\bigg|_{\mathbf{p}_\perp=(\mathbf{q}_\perp+\mathbf{k}_\perp)} $$ where $\mu_i \in \{1,2\}$, $l\ge 0$, $n\ge 0$ and $L_l^a(x)$ are the generalized Laguerre polynomials. We also provide explicit expressions of the result up to the fourth moment i.e. $r=4$, which are sufficient for most of the field theoretical calculations.