Quark–Hadron Pasta Phases

• Quark–hadron pasta phases are spatially inhomogeneous structures that form during the first-order phase transition between hadronic and quark matter, featuring geometries like droplets, rods, slabs, tubes, and bubbles.
• They emerge from a detailed energy balance between surface tension and Coulomb interactions under Gibbs equilibrium, with finite-size and thermal effects critically shaping their properties.
• The presence of these pasta phases significantly affects the equation of state, neutrino transport, and observable phenomena in proto-neutron stars and supernova dynamics.

Quark–hadron pasta phases represent a class of spatially inhomogeneous structures that arise in dense astrophysical matter during a first-order phase transition between hadronic and quark matter—most notably in proto-neutron stars, neutron stars, and supernova environments. These structures result from the interplay of finite-size effects, mainly surface tension and Coulomb interactions, when the deconfinement transition is governed by multi-component Gibbs equilibrium and not the Maxwell construction. The term “pasta” refers to their diverse geometries—droplets (spheres), rods (cylinders), slabs (planes), tubes, and bubbles—analogous to various pasta shapes, and their presence or absence has major implications for the thermodynamics and evolution of compact objects.

1. Physical Principles and Gibbs Conditions

The emergence of the quark–hadron pasta phase is tightly connected to the nature of the first-order deconfinement transition in multi-component systems like stellar matter, where baryon number and electric charge conservation both apply. The equilibrium must fulfill the full set of Gibbs conditions:

• Pressure balance, incorporating surface tension σ at the interface: $P_H = P_Q - \frac{2\sigma}{R}$ where $P_H$ and $P_Q$ are the partial pressures in the hadronic and quark phases, and $R$ is a characteristic size (e.g., droplet radius).
• Chemical equilibrium, entailing coupled relations among chemical potentials for all relevant species. For example, in the presence of neutrinos ($\mu_{\nu_e}$) and hyperons ($\mu_\Lambda, \mu_{\Sigma^-}$), the system must satisfy:

$$\begin{align*} \mu_u + \mu_e - \mu_{\nu_e} &= \mu_d = \mu_s \ \mu_p + \mu_e - \mu_{\nu_e} &= \mu_n = \mu_\Lambda = \mu_u + 2\mu_d \ \mu_{\Sigma^-} + \mu_p &= 2\mu_n \end{align*}$$

• Thermal equilibrium, imposing $T_H = T_Q$ (equal temperatures), typically enforced by minimizing the Helmholtz free energy with all conserved densities and the temperature held fixed.

The global charge neutrality condition is imposed over the entire mixed-phase region (not locally in each domain), facilitating different charge distributions in each phase. Local deviations are energetically balanced by surface and Coulomb energy contributions.

2. Pasta Structures: Geometries and Energy Balance

Finite-size effects fundamentally shape the emergence of the pasta phase. The surface tension at the interface and the long-range Coulomb interaction drive the system toward non-uniform structures that minimize the total free energy. The sum of the surface energy ($E_S$) and the Coulomb energy ($E_C$) for a structure of size $R$ is typically written as:

$$E_S = \sigma S,\qquad E_C = \text{geometry-dependent term} \propto (q_1 - q_2)^2 R^2,$$

where $S$ is the interfacial area and $q_i$ are the charge densities in the two coexisting phases.

The competition between these energies yields optimal geometries for given parameters. The minimization condition for total structural energy (volume, surface, Coulomb, correlation) produces the well-known balance $E_S = 2E_C$ for the simplest geometry (e.g., an unscreened sphere), with further refinements necessary to include screening effects and nontrivial geometries.

Possible configurations, evolving with volume fraction $\chi$ of the minority phase, include:

Geometry	Dimensionality	Typical Density Range
Droplets	3	$\chi \ll 1$ (quark in hadron)
Rods	2	Intermediate
Slabs	1	Mid-transition ($\chi\sim0.5$)
Tubes	2	$\chi \gtrsim 0.5$
Bubbles	3	$\chi \to 1$ (hadron in quark)

Charge screening, chemical rearrangement (correlation energy), and surface curvature may significantly impact the actual sequence and the width of the pasta region.

3. Thermodynamic Consequences and Instabilities

The formation and persistence of pasta structures are closely regulated by thermal and lepton (neutrino) effects:

• Finite temperature introduces entropy contributions into the bulk free energies, shifting the minimum of the free energy per baryon and typically resulting in increased optimal structure sizes. Above $T \approx 60$ MeV, thermal fluctuations destabilize the pasta phase, often leading to a reversion to a Maxwell-like, locally neutral configuration.
• Neutrino trapping (characterized by lepton fraction $Y_{\nu_e}$ or $Y_l$) modifies chemical equilibrium and increases the electron fraction. Enhanced electron densities diminish the net charge in each phase, lowering both Coulomb and correlation energies. At high lepton number, the energy minimum for pasta structures disappears. This results in an amorphous, bicontinuous (non-crystalline) configuration where local charge neutrality is almost restored and the surface and Coulomb effects become negligible (Yasutake et al., 2012).
• Mechanical instability: At high temperatures or lepton fractions, for reasonable surface tension values, the optimal pasta size diverges (cell radius $R_W \to \infty$), and the transition sharply crosses over to a Maxwell-like regime which is energetically favored.

4. Implementation in Astrophysical Equations of State

The global impact of the pasta phase on the equation of state (EOS) is intricate:

• The EOS in the mixed-phase region is not a simple superposition of hadron and quark EOS; it incorporates finite-size and Coulomb contributions, minimizing the full free energy:

$$F = \int_{V_H} d^3 r\, \mathcal{A}_H + \int_{V_Q} d^3 r\, \mathcal{A}_Q + F_e + F_{\nu_e} + E_C + E_S.$$

Here, $\mathcal{A}_{H/Q}$ denotes the free energy density for hadronic/quark matter, $F_{e/\nu_e}$ for leptonic components, and $E_C$, $E_S$ are as above.

• Gibbs vs. Maxwell construction: When the mixed phase is stable and pasta structures are realized, the EOS interpolates between the 'soft' Gibbs and the 'hard' Maxwell limits. If temperature or lepton fraction is high, the system is well described by the Maxwell construction (constant pressure plateau). At lower $T$ and $Y_{\nu_e}$, a continuous mixed phase region (with varying pressure) emerges, possibly featuring a structural sequence of pasta geometries.
• The presence of a pasta phase suppresses the onset of hyperons in the hadronic phase because the positively charged hadronic domain disfavors negatively charged hyperons (e.g., $\Sigma^-$). This counters the softening of the EOS that the emergence of hyperons would otherwise cause in locally charge-neutral models (Tatsumi et al., 2011).
• In the case of protoneutron stars evolving after core collapse, the existence and properties of the pasta phase can strongly influence neutrino transport, the speed and nature of deleptonization and cooling, and the global stability against gravitational collapse.

5. Effects on Proto-Neutron Star Evolution and Observational Signatures

The sequence of pasta phase emergence and disappearance orchestrates the thermodynamical and transport properties in supernova and proto-neutron star evolution:

• Early post-bounce phase (high $T$, high $Y_{\nu_e}$): The mixed phase is amorphous, local charge neutrality prevails, and the classic crystalline pasta phase is mechanically unstable. The EOS does not exhibit a density discontinuity and is well described by the Maxwell construction (Yasutake et al., 2012).
• Intermediate cooling stage: As temperature and $Y_{\nu_e}$ decrease (deleptonization and cooling), the fine balance between surface and Coulomb energies can sustain pasta phases. These phases can critically alter mechanical properties (shear modulus, viscosity) and transport (thermal/electrical conductivity), with consequences for pulsar glitches, supernova neutrino signals, and gravitational wave emission.
• Later cooling (neutrino-transparent) phase: If temperature and lepton fraction drop below critical values, the system may transition to a locally charge-neutral (Maxwell) regime with no mixed-phase region.

The pasta phase’s impact on the supernova explosion mechanism, neutron-star cooling, and gravitational-wave signal spectrum (e.g., supporting or dampening certain stellar oscillation modes) is significant, particularly due to its effect on density discontinuities and elasticity (Yasutake et al., 2010).

6. Quantitative Characterization and Limiting Factors

The existence and extent of the quark–hadron pasta phase are regulated by several key physical quantities:

• Surface tension $\sigma$: For moderate $\sigma$ (e.g., $\lesssim 40$ MeV·fm$^{-2}$), the mixed phase with pasta structures can extend over a wide density range. For high $\sigma$ ($\gtrsim 65$ MeV·fm$^{-2}$), the region is suppressed—system reverts to the Maxwell construction (Tatsumi et al., 2011).
• Correlation and rearrangement (charge screening) energy: Critical for the stabilization of finer geometries, and for the shift of optimum sizes as $T$ and $Y_{\nu_e}$ vary.
• Finite temperature and neutrino storage: Both decrease the width of the mixed-phase region and can eventually disrupt the regularity of pasta structures, resulting in an amorphous or Maxwell-like regime.
• Hyperon content: Suppression within positively charged hadronic domains can yield a stiffer EOS at high density, compatible with supporting larger neutron star masses.

The quantification of these limits and the calculation of the resulting EOS require detailed minimization of the total free energy—including bulk, surface, Coulomb, and stabilizing/destabilizing factors—subject to phase equilibrium conditions.

7. Summary Table: Key Energy Contributions in Pasta Phase Modeling

Contribution	Expression	Relevance
Surface Energy	$E_S = \sigma S$	Favors larger structures
Coulomb Energy	$E_C \propto (q_1 - q_2)^2 R^2$	Penalizes charge separation
Correlation	$E_{\mathrm{corr}}$ (from charge screening)	Alters size and stability
Thermal	$-T S$ (entropy term in free energy)	Tends to destabilize pasta at high $T$
Neutrino Trapping	Modifies chemical equilibrium, lowers charges	Tends to destabilize pasta at high $Y_{\nu_e}$

8. Implications for Observations and Astrophysical Modeling

The existence and nature of quark–hadron pasta phases influence the mass–radius relation, maximum mass, stability of compact stars, and their transport and mechanical properties. Their presence or instability against melting due to temperature or lepton fraction are crucial for simulation codes modeling supernova dynamics, proto-neutron star cooling, and late-stage pulsar behavior. Density discontinuities, existence of amorphous phases, or suppression of elastic crystalline lattices have direct consequences for the gravitational wave signal spectrum and the interpretation of observed neutron star properties.

In essence, the quark–hadron pasta phases are an inherent feature of the mixed-phase region arising from a first-order deconfinement transition in multi-component, charge-neutral compact star matter, persistently shaped and often destabilized by coupling among finite-size effects, temperature, and lepton content. Their quantitative characterization is vital for the construction of realistic equations of state and the predictive modeling of neutron star evolution and associated astrophysical phenomena (Yasutake et al., 2010, Tatsumi et al., 2011, Yasutake et al., 2012, Yasutake et al., 2012).