Phase transition and nuclear symmetry energy from neutron star observations: Constraints in light of PSR J0614-3329

The possible occurrence of a first-order hadron-quark phase transition in neutron-star interiors remains an open question. Whether such a transition can be directly tested with improved observations is a key challenge. Here, we incorporate the latest constraints, especially a new radius measurement by the neutron star interior composition explorer for PSR J0614-3329, into a nonparametric Gaussian process equation of state framework that explicitly includes a first-order transition. We find a Bayes factor of $B\ensuremath{\approx}2.3$ when comparing models with and without an explicit phase transition, marginally favoring its presence. At 68% credibility, the transition onset density ${n}_{\mathrm{PT}}$ is either below $2{n}_{s}$ (corresponding to masses $\ensuremath{\lesssim}1{M}_{\ensuremath{\bigodot}}$, with density jump $\mathrm{\ensuremath{\Delta}}n\ensuremath{\sim}0.5{n}_{s}$) or, more prominently, above $4{n}_{s}$ (near the central density of the heaviest neutron star, with $\mathrm{\ensuremath{\Delta}}n\ensuremath{\sim}3{n}_{s}$), where ${n}_{s}$ represents the nuclear saturation density. In addition, by using symmetry-energy expansion at low densities ($<1.1{n}_{s}$), we infer a slope parameter $L={40.2}_{\ensuremath{-}14.3}^{+19.3}\text{ }\text{ }\mathrm{MeV}$, in good agreement with nuclear-experiment values. Intriguingly, $L$ correlates positively with the radius difference between $1.4{M}_{\ensuremath{\bigodot}}$ and $2.0{M}_{\ensuremath{\bigodot}}$ stars.