I'm considering an "anisotropic" Hamiltonian of the form $$\beta H = \int d^n r_{||} d^{d-n} r_{\bot} \frac{K}{2} (\nabla_{||} m)^2 + \frac{L}{2} (\nabla^2_\bot m)^2 + \frac{t}{2}m^2 - hm$$
which in momentum space reads $$\beta H = \frac{1}{(2\pi)^d} \int_{|q_{||}|<\Lambda_{||}} d^n q_{||} \int_{|q_\bot|<\Lambda_\bot} d^{d-n} q_\bot \left[ \frac{K}{2}q_{||}^2 + \frac{L}{2} q_\bot^4 + \frac{t}{2}\right] |m(q_{||},q_\bot|^2$$
First, are my integration limits correct? I wonder, because if I'd imagine that the cutoffs for the different directions were the same, then shouldn't I be able to write the two integrals as an integral over all momenta $|q| < \Lambda$ for some $\Lambda$?
Second, when I do a renormalization group treatment, the idea is to integrate out the higher momenta and I'm not sure how to split the field in order to do so. I can imagine that I will get four integrals: One where both $q_{||}$ and $q_{\bot}$ are small, one where both are large, and two where one is small and the other large. Which of these can I ignore as an additive constant and which do I keep? Only the small-small one, or do the small-large ones also affect the renormalization?