Neutron Stars Narrow the Limits of Color Superconductivity ✦
At asymptotically high densities, QCD predicts a transition of quark matter into a color-flavor-locked (CFL) superconducting phase, but away from the asymptotics, the gap magnitude is undetermined. Multimessenger observations of neutron stars are combined with perturbative QCD and chiral effective theory in a Bayesian analysis using a neural network parameterization of the equation of state. Matching with QCD at a baryon chemical potential of 2.6 GeV, including next-to-leading order corrections for the CFL contribution, yields a gap Δ*_CFL = 34^{+32}_{-28} MeV and an upper limit of 66 MeV (95% credible) — twice as precise and at the lower end of model predictions. For the first time, a constraint on the high-order pressure constant pQCD: c0 = -21^{+9}_{-8} is obtained. It is shown that color-superconducting pairing contributes only a small amount to the pressure, deepening the connection between observable stellar parameters and the phase diagram of strong interactions.
Context
Quantum chromodynamics predicts that at extremely high densities — like those achieved in the cores of massive neutron stars — quark matter transitions into a state of color superconductivity, particularly the color-flavor locked (CFL) phase. However, the magnitude of the order parameter for this transition, which determines the binding energy of quark Cooper pairs, remained highly uncertain far from the asymptotic regime. Back in the 1930s, Fritz Zwicky predicted the existence of neutron stars as supernova remnants, and later Jocelyn Bell Burnell discovered pulsars, turning them into cosmic laboratories. Today, multi-messenger astronomy, including observations of gravitational waves from mergers and X-ray photometry with LIGO and NICER, allows us to peer into the heart of these objects and test fundamental theories.
Methods
To solve the inverse problem — reconstructing the equation of state (EOS) of dense matter from observational data — the authors developed a hybrid approach that combines quantum field calculations of QCD with machine learning. In the density regime typical of neutron stars, the speed of sound was modeled via a Gaussian process, allowing uncertainties from chiral effective field theory to be incorporated. For intermediate densities up to matching with perturbative QCD at a chemical potential of 2.6 GeV, a compact neural network with imposed boundary conditions was used. Bayesian inference was performed using nested sampling, simultaneously varying all EOS parameters, QCD constants, and color superconductivity. Additionally, data on maximum masses of pulsars, tidal deformability from GW170817 (detected by LIGO), and mass-radius measurements from NICER were included.
Results
The analysis showed that with 95% probability, the CFL order parameter at μB = 2.6 GeV does not exceed 66 MeV, and its most likely value is around 34 MeV. This is twice as tight as previous model-independent constraints and lies at the lower boundary of most microscopic predictions (50–150 MeV). Moreover, for the first time, a data-constrained value of the fourth-order constant of perturbative QCD was obtained: c0 = −21.2 +8.8 –7.5. The reconstructed equation of state shows a peak in the squared speed of sound above the conformal limit (c2s > 1/3) at densities of 3–5 times nuclear density and subsequent softening near the maximum mass. Interestingly, the maximum mass of a neutron star was found to be 2.16 M☉, and the radius for a canonical 1.4 M☉ mass is 11.5 km, consistent with multi-messenger estimates for GW170817.
Implications
These constraints imply that color superconductivity contributes only a small correction to the pressure of dense matter at densities relevant for neutron stars. This simplifies the connection between astrophysical data and the QCD phase diagram, making predictions more robust. The small order parameter also suggests that the transition to the CFL phase likely occurs at higher densities than previously thought and does not significantly affect the macroscopic properties of stars.
Future development
Further progress depends on accumulating precision data from NICER, the future eXTP mission, and next-generation gravitational wave detectors. Combining these observations with complete calculations of the four-loop contribution to the QCD pressure will either tighten existing constraints or detect a non-zero color superconductivity contribution if it exists at the edge of current sensitivity.
Impact
The results will impact several related fields: nuclear astrophysics, high-density QCD physics, and the interpretation of gravitational wave observations. In particular, refining the equation of state will help better model neutron star mergers and the evolution of remnants.
Next steps
The next steps include improving radius measurements of neutron stars with NICER and searching for ever more massive pulsars, which will allow testing the equation of state at extreme densities.
Key open problems
This research is directly connected to one of the central unsolved problems in physics — determining the phase diagram of quantum chromodynamics and the nature of dense matter at a few times nuclear density. Constraining color superconductivity also sheds light on cooling and transport mechanisms in compact objects.
🎯 The name color-flavor locking (CFL) reflects a remarkable property of the phase: in it, the color charges of quarks and their flavors (types) become 'locked' such that a rotation in color space simultaneously rotates flavor space, much like a dance where the partners cannot move independently.
Key numbers
- CFL order parameter upper limit: 66 MeV
- peak value: 34 MeV
- maximum neutron star mass: 2.16 M☉
- radius for 1.4 M☉: 11.5 km
- constant c0: −21.2