Cold quark matter at N3LO: soft contributions

High-order perturbative calculations for thermodynamic quantities in QCD are complicated by the physics of dynamical screening that affects the soft, long-wavelength modes of the system. Here, we provide details for the evaluation of this soft contribution to the next-to-next-to-next-to-leading order (N3LO) pressure of high-density, zero-temperature quark matter (QM), complementing our accompanying paper in arXiv:2103.05658. Our calculation requires the determination of the pressure of the hard-thermal-loop (HTL) effective theory to full two-loop order at zero temperature, which we go through in considerable detail. In addition to this, we comprehensively discuss the structure of the weak-coupling expansion of the QM pressure, and lay out a roadmap towards the evaluation of the contributions missing from a full N3LO result for this quantity.