Abstract
The cold and dense QCD equation of state (EoS) at high baryon chemical potential $\mu_B$ involves at order $\alpha^2_S$ an all-loop summation of the soft mode $m_E\sim \alpha_S^{1/2} \mu_B$ contributions. Recently, the complete soft contributions at order $\alpha^3_S$ were calculated, using the hard thermal loop (HTL) formalism. By identifying {\em massive} renormalization group (RG) properties within HTL, we resum to all orders $\alpha_S^p, p\ge 3$ the leading and next-to-leading logarithmic soft contributions. We obtain compact analytical expressions, that show visible deviations from the state-of-the art results, and noticeably reduced residual scale dependence. Our results should help to reduce uncertainties in extending the EoS in the intermediate $\mu_B$ regime, relevant in particular for the phenomenology of neutron stars.