Fighting off field dependence in MSSM Higgs-mass corrections of order $\alpha_t\,\alpha_s$ and $\alpha_t^2$

The connection between gauge and Higgs sectors makes supersymmetric extensions of the Standard Model predictive frameworks for the derivation of Higgs masses. In this paper, we study the contamination of such predictions by field-renormalization constants, in the MSSM with two-loop gaugeless corrections of $\mathcal{O}{\left(\alpha_{t,b}\,\alpha_s,\,\alpha_{t,b}^2\right)}$ and full momentum dependence, and demonstrate how strict perturbative expansions allow to systematically neutralize the dependence on such unphysical objects. On the other hand, the popular procedure consisting in an iterative pole search remains explicitly dependent on field counterterms. We then analyze the magnitude of the intrinsic uncertainty that this feature implies for the iterative method, both in non-degenerate and near-degenerate regimes, and conclude that this strategy does not improve on the predictions of the more straightforward expansion. We also discuss several features related to the inclusion of the orders $\alpha_{t,b}\,\alpha_s$ and $\alpha_{t,b}^2$ in the so-called 'fixed-order' approach, such as the resummation of UV-logarithms for heavy supersymmetric spectra.