Crossover Equation of State

• Crossover EOS is a theoretical framework that continuously interpolates between hadronic and quark-gluon matter using an analytic switching function to capture a smooth QCD transition.
• It combines relativistic mean-field models for hadronic matter and perturbative QCD for the quark sector, ensuring thermodynamic consistency and adherence to causal limits.
• The model supports realistic neutron star structures by reconciling heavy-mass observations with lattice QCD and nuclear data without invoking sharp phase boundaries.

A crossover equation of state (crossover EOS) is a phenomenological or theoretical construction that provides a continuous, thermodynamically consistent interpolation between qualitatively distinct phases of matter—typically hadronic matter and deconfined quark-gluon matter—across regimes where QCD predicts a rapid but smooth transition (crossover) rather than a true first-order phase transition. These EOSs are critical for modeling compact astrophysical objects (e.g., neutron stars), cosmological phase transitions, and heavy-ion collision matter, as crossover behavior is supported by both lattice QCD at low baryon chemical potential and by phenomenology in multiple physical systems.

1. Basic Principles and Mathematical Formulation

Crossover EOSs model a smooth transition between two limiting equations of state, generally denoted as a hadronic component $P_h$ and a quark/gluon component $P_q$. The transition is governed by a differentiable switching or interpolation function. For cold neutron star matter at zero temperature, as in (Kapusta et al., 2021), the pressure as a function of baryon chemical potential $\mu$ is typically written: $$P(\mu) = S(\mu)\, P_q(\mu) + [1 - S(\mu)]\, P_h(\mu)$$ where $S(\mu)$ (with $0 \leq S(\mu) \leq 1$) must satisfy:

• $S(\mu) \to 0$ as $\mu \to 0$ (hadronic regime)
• $S(\mu) \to 1$ as $\mu \to \infty$ (quark regime)
• $S(\mu)$ is infinitely differentiable (analytic) to avoid spurious phase transitions.

A commonly used form is: $$S(\mu) = \exp\left[-\left(\frac{\mu_0}{\mu}\right)^4\right]$$ where $\mu_0$ controls the onset and width of the crossover; for instance, values in the range $\mu_0 \sim 1.6-2.0$ GeV are adopted to accommodate neutron star observations near $2\,M_\odot$ (Kapusta et al., 2021).

The energy density is then obtained via first law thermodynamics: $$n(\mu) = \frac{dP}{d\mu},\quad \varepsilon(\mu) = \mu\, n(\mu) - P(\mu)$$ The adiabatic sound speed is

$$c_s^2(\mu) = \frac{dP}{d\varepsilon} = \frac{dP/d\mu}{d\varepsilon/d\mu} = \frac{dP/d\mu}{\mu\, d^2P/d\mu^2}$$

These constructions guarantee thermodynamic consistency and a physically motivated transition without introducing nonphysical artifacts.

2. Physical Implementation: Input Models and Interpolants

Hadronic EOS $P_h(\mu)$

For neutron star applications, $P_h(\mu)$ is often calculated using relativistic mean-field models calibrated to nuclear matter saturation properties: saturation density $n_0$, binding energy $B$, effective Landau mass $m_L^*$, incompressibility $K$, and symmetry energy $S$. For instance, the model in (Kapusta et al., 2021) fixes parameters to $n_0=0.153\,\mathrm{fm}^{-3}$, $B=16.3\,\mathrm{MeV}$, $m_L^*/m_N=0.83$, $K=250\,\mathrm{MeV}$, and $S=32.5\,\mathrm{MeV}$.

Quark EOS $P_q(\mu)$

The high-density sector relies on perturbative QCD calculations, often truncated to next-to-leading or next-to-next-to-leading order in $\alpha_s$, the strong coupling, with running computed to three loops: $$P_q(\mu) = \frac{N_f}{4\pi^2}\left(\frac{\mu}{3}\right)^4 \left[1 - \frac{2}{\pi}\alpha_s(\mu) + O(\alpha_s^2)\right]$$ where $N_f=3$ for massless quark flavors. The renormalization scale is chosen to match lattice QCD at moderate $\mu$.

Interpolation/Switching Function

The form $S(\mu) = \exp\left[-(\mu_0/\mu)^4\right]$ ensures infinite differentiability. The single parameter $\mu_0$ is tuned to nuclear and astrophysical constraints.

Possible generalizations include two-sided tanh windows (Li et al., 2018): $$P(\mu) = P_H(\mu)\, f_-(\mu) + P_Q(\mu)\, f_+(\mu),\quad f_\pm(\mu) = \frac{1}{2}[1 \pm \tanh((\mu - \tilde\mu)/\Gamma)]$$ where $\tilde\mu$ and $\Gamma$ are central position and width, respectively.

3. Thermodynamic and Causal Consistency

Explicit parameter choices are driven by the requirement that:

• $dP/d\varepsilon > 0$ (convexity and stability),
• $c_s^2 \leq 1$ (causality).

In (Kapusta et al., 2021), the sound speed profile is nonmonotonic:

• At low $\mu$, the hadronic EOS can yield $c_s^2 \to 1$ due to strong vector repulsion,
• In the crossover region, $c_s^2$ peaks at values below unity,
• At large $\mu$, the quark EOS forces $c_s^2 \to 1/3$ from below (with logarithmic corrections).

This structure avoids the mechanical instability and extremely soft mixed-phase EOS typical of first-order constructions, eliminating early collapse to black holes.

4. Neutron Star Structure and Astrophysical Constraints

Neutron star properties are obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equations: $$\frac{dm}{dr} = 4\pi r^2 \varepsilon(r),\qquad \frac{dP}{dr} = -G \frac{[\varepsilon(r) + P(r)][m(r) + 4\pi r^3 P(r)]}{r[r - 2Gm(r)]}$$ with boundary conditions $m(0)=0$, $P(R)=0$. The input EOS is implemented as $P(\varepsilon)$ or via $(P(\mu),\varepsilon(\mu))$.

Findings in (Kapusta et al., 2021):

• For $\mu_0 \simeq 1.8\,\mathrm{GeV}$, maximum mass $M_\mathrm{max} \approx 2.0\,M_\odot$,
• Increasing $\mu_0$ to $2.0\,\mathrm{GeV}$ yields $M_\mathrm{max} \approx 2.2\,M_\odot$,
• Radii for maximum-mass configurations are $R \sim 11-13$ km and slightly decrease with increasing $\mu_0$,
• Central energy densities reach $\varepsilon_c \sim 0.8-1.2\,\mathrm{GeV/fm}^3$,
• The local quark pressure fraction at the centers of the heaviest stars is $S(\mu_c)$, with only $\sim 1-10\%$ due to quark matter,
• The EOS supports massive stars without a sharp phase boundary, in line with modern observations of heavy neutron stars (PSR J0348+0432, PSR J0740+6620).

5. Theoretical Significance and Distinctiveness

The crossover EOS realizes the key QCD prediction (from lattice) that at low or zero baryon chemical potential, the transition from hadrons to quarks and gluons is not a first-order phase transition, but a rapid, yet analytic crossover. Employing a crossover construction allows:

• Stiffness sufficient for 2$-$2.2 $M_\odot$ neutron stars without the EOS collapse seen in models with a first-order transition.
• Quark matter can coexist with hadrons on a subdominant basis in the core, without destabilizing the star.
• The absence of non-analyticities permits straightforward TOV integration and credible extrapolation to high densities within the tested parameter range.

This approach is supported by additional models (Li et al., 2018, Baym et al., 2019) wherein smooth interpolants in $\mu$ or density are used, with parameters tuned to laboratory nuclear data, lattice QCD, and astrophysical constraints (mass, radii, tidal deformability). The fraction of mixed/hybrid matter and the detailed location of the crossover are thus rendered predictive and testable.

6. Practical Implementation and Model Variants

Parameter Tuning and Model Selection

$$S(\mu) = \exp\left[-\left(\frac{\mu_0}{\mu}\right)^4\right]$$

with $\mu_0 \simeq 1.6-2.0$ GeV for cold neutron stars, as in (Kapusta et al., 2021). Hadronic sector couplings are fixed by nuclear experimental data; in the quark sector, the strong coupling $\alpha_s(\mu)$ is determined by 3-loop running, with the renormalization scale selected to match lattice QCD thermodynamics.

Key trade-off: increasing $\mu_0$ pushes the onset of quark pressure to higher density, leading to a stiffer EOS and higher $M_\mathrm{max}$, but reduces quark fraction in observed stars. Numerical integration of the structure equations proceeds as usual, with the reconstructed EOS guaranteeing stability and causality within the fit parameter range.

Generalization

Analogous interpolation schemes (e.g., tanh-based windows, polynomial Hermite interpolants) and similar hadron–quark model mixtures appear throughout the literature, both for dense cold matter and for finite-temperature QCD matter (including cosmological and heavy ion collision contexts), provided thermodynamic consistency is maintained and the switching function remains analytic.

7. Astrophysical and Observational Implications

Crossover EOSs provide the framework to resolve the long-standing problem of reconciling heavy neutron stars ($M_\mathrm{max} \sim 2\,M_\odot$), their observed radii ($R_{1.4} \sim 11-13\,\mathrm{km}$), and modern nuclear/lattice data with the microphysical expectation of deconfined quark matter at high densities. The smoothness of the transition ensures that even as quark matter contributes at high central densities, the EOS retains enough stiffness for observed neutron stars, and the pressure contribution from quarks remains modest—typically $\lesssim 10\%$ in the most massive stable stars with current constraints (Kapusta et al., 2021). This model avoids the rapid softening and destabilization encountered in first-order mixed-phase constructions and provides an explicit, implementable link between nuclear microphysics, QCD, and macroscopic astrophysical observations.