Unraveling Trace Anomaly of Supradense Matter via Neutron Star Compactness Scaling

Abstract: The trace anomaly $\Delta\equiv 1/3-P/\varepsilon=1/3-\phi$ quantifies the possibly broken conformal symmetry in supradense matter under pressure $P$ at energy density $\varepsilon$. Perturbative QCD (pQCD) predicts a vanishing $\Delta$ at extremely high energy or baryon densities when the conformal symmetry is realized but its behavior at intermediate densities reachable in neutron stars (NSs) are still very uncertain. The extraction of $\Delta$ from NS observations strongly depends on the employed model for nuclear Equation of State (EOS). Using the IPAD-TOV method based on an Intrinsic and Perturbatively Analysis of the Dimensionless (IPAD) Tolman-Oppenheimer-Volkoff (TOV) equations that are further verified numerically by using $10^5$ EOSs generated randomly with a meta-model in a very broad EOS parameter space constrained by terrestrial nuclear experiments and astrophysical observations, here we first show that the compactness $\xi\equiv GM_{\rm{NS}}/Rc^2\equiv M_{\rm{NS}}/R$ of a NS with mass $M_{\rm{NS}}$ and radius $R$ scales very accurately with $\overline{\Pi}{\rm{c}}\equiv\Pi{\rm{c}}\cdot(1+18\x/25)\equiv\x/(1+3\x^2+4\x)\cdot(1+18\rm{X}/25)$ where $\x\equiv\phi_{\rm{c}}= P_{\rm{c}}/\varepsilon_{\rm{c}}$ is the ratio of pressure over energy density at NS centers. The scaling of NS compactness thus enables one to readily read off the central trace anomaly $\Delta_{\rm{c}}=1/3-\x$ directly from the observational data of either the mass-radius or red-shift measurements. We then demonstrate indeed that the available NS data themselves from recent X-ray and gravitational wave observations can determine model-insensitively the trace anomaly as a function of energy density in NS cores, providing a stringent test of existing NS models and a clear guidance in a new direction for further understanding the nature and EOS of supradense matter.