Mapping the chiral-invariant-mass dependence of gravitational-wave spectra from first-order QCD phase transitions in the parity doublet model and breaking the order-parameter degeneracy

In the two-flavor parity doublet model at finite baryon chemical potential, the nucleon mass splits into a chiral-invariant part $m_0$ (which survives chiral symmetry restoration) and a part generated by spontaneous chiral symmetry breaking. A 2026 study computed the gravitational-wave (GW) spectra from the two first-order transitions of this model — the nuclear liquid–gas transition and the chiral transition — at a single representative value $m_0=800\ \mathrm{MeV}$ , showing the liquid–gas channel can fit pulsar-timing-array data while the chiral signal is suppressed by five orders of magnitude. What remains open is the systematic map of how the phase diagram, the nucleation parameters (transition strength $\alpha$ and inverse duration $\beta/H$ ), and the GW spectra vary across the phenomenologically viable $m_0$ range; whether GW observations, combined with neutron-star and heavy-ion constraints, can quantitatively constrain $m_0$ and thereby probe the origin of the nucleon mass; and whether such multi-messenger analyses can break the degeneracy whereby a detected spectrum characterized only by $\alpha$ and $\beta/H$ does not identify the microscopic order parameter (liquid–gas vs chiral vs deconfinement).

Background

The parity doublet model is an effective chiral model of dense baryonic matter in which the nucleon and its parity partner $N(1535)$ form a representation under chiral symmetry, so that the nucleon mass receives two contributions: a chiral-invariant mass $m_0$ that persists when chiral symmetry is restored, and a term proportional to the chiral condensate $\langle\bar q q\rangle$ that vanishes upon restoration. The parameter $m_0$ therefore quantifies the part of the nucleon mass not originating from spontaneous chiral symmetry breaking, making it a direct handle on the origin of hadron mass. At finite temperature $T$ and baryon chemical potential $\mu_B$ , the mean-field parity doublet phase diagram contains two first-order transition regions: a nuclear liquid–gas transition at relatively low chemical potential ( $\mu_B\sim 920\ \mathrm{MeV}$ ) and a chiral transition at higher chemical potential ( $\mu_B\sim 1635\ \mathrm{MeV}$ ). A first-order cosmological phase transition proceeds by bubble nucleation: the critical bubble profile is obtained by solving the bounce equation for the order parameter, and the nucleation rate is controlled by the three-dimensional Euclidean action $S_3/T$ , evaluated at the nucleation temperature. The transition strength $\alpha$ (the ratio of vacuum-energy release to the background radiation energy density) and the inverse transition duration $\beta/H$ (with $H$ the Hubble rate) then determine the stochastic GW background, produced chiefly by sound waves in the plasma and, to a lesser extent, by magnetohydrodynamic turbulence, with efficiency factors conventionally denoted $\kappa_v$ and $\kappa_{\rm tb}$ . Interest in QCD-epoch transitions was sharpened by the pulsar timing array (PTA) collaborations — NANOGrav, EPTA/InPTA, PPTA, and CPTA — reporting a nano-Hz stochastic GW background; a QCD-scale transition can reach these frequencies only if the early universe carried a large baryon asymmetry (pushing $\mu_B$ high) followed by entropy dilution, a scenario studied as 'QCD-induced little inflation'. In this framework, Gao, Shao, and Mao (arXiv:2604.00465) fixed $m_0=800\ \mathrm{MeV}$ as a representative value consistent with neutron-star observations, fitted the remaining model parameters to nuclear saturation properties, solved the bounce equation along both first-order lines, and computed the GW spectra: the liquid–gas transition yields $\alpha\sim\mathcal{O}(1)$ and $\beta/H\sim\mathcal{O}(10)$ – $\mathcal{O}(100)$ near the endpoint of the first-order line, with peak frequencies from the milli-Hz down to the nano-Hz band (fitting the NANOGrav 15-year spectrum near $\mu_B\approx 921.8\ \mathrm{MeV}$ ), while the chiral transition produces signals suppressed by roughly five orders of magnitude. Independently, $m_0$ is already constrained by neutron-star phenomenology: quark–hadron crossover equations of state (Minamikawa, Kojo, and Harada, 2021), tidal deformability from GW170817 in an extended parity doublet model (Yamazaki and Harada, 2019), and NICER (Neutron Star Interior Composition Explorer) radius measurements such as PSR J0614-3329 (Gao, Kong, and Ma, 2025) all bound $m_0$ , but no existing work connects these bounds to a cosmological GW spectrum. The authors of arXiv:2604.00465 point out that because varying $m_0$ shifts the location and strength of the first-order regions in the $T$ – $\mu_B$ plane, the GW signal carries an imprint of $m_0$ that future PTA and space-based interferometer (LISA, Taiji) observations could exploit — yet this dependence has never been mapped. They further emphasize the inverse-problem degeneracy: since the GW spectrum is primarily characterized by the two macroscopic parameters $\alpha$ and $\beta/H$ , a detection alone cannot reveal whether the underlying order parameter is that of a liquid–gas transition, a chiral transition, or a deconfinement transition, with only the signal amplitude offering a partial density diagnostic.

Problem Statement

Within the two-flavor parity doublet model at finite temperature and baryon chemical potential, carry out the program that arXiv:2604.00465 identifies as future work, whose unifying objective is to connect the macroscopic gravitational-wave parameters of first-order QCD phase transitions to the microscopic physics of the QCD phase diagram — specifically the chiral invariant mass $m_0$ and the identity of the transition's order parameter. Concretely: (1) systematically map, over the phenomenologically viable range of $m_0$ fixed by nuclear saturation properties and neutron-star constraints, the $T$ – $\mu_B$ phase diagram, the nucleation parameters $\alpha$ and $\beta/H$ along both first-order lines (obtained from the bounce equation and the Euclidean action $S_3/T$ ), and the resulting gravitational-wave spectra, including peak frequencies and amplitudes for the liquid–gas and chiral channels; (2) quantify whether the mapped spectra for different $m_0$ values are distinguishable by current and planned detectors (pulsar timing arrays, LISA, Taiji), and what constraints on $m_0$ a gravitational-wave detection or non-detection would imply when combined with existing neutron-star (masses, radii, tidal deformabilities) and heavy-ion constraints from pion-production measurements in intermediate-energy Sn+Sn collisions (Zhou et al., 2025); and (3) assess quantitatively whether such joint multi-messenger analyses can break the degeneracy whereby a detected spectrum characterized only by $\alpha$ and $\beta/H$ does not uniquely identify the microscopic order parameter — liquid–gas, chiral, or deconfinement — including whether any feature of the signal (amplitude, spectral shape) or combination of probes provides a genuine discriminant. A complete resolution requires parts (1)–(3) jointly, except that the negative outcomes named immediately below each resolve the question on their own. A conclusion that the liquid–gas channel's spectrum is insufficiently $m_0$ -sensitive for discrimination, or that the degeneracy cannot be broken by the stated probes, would itself constitute a resolution of the question as stated.

References

Bikai Gao, Jingdong Shao, Hong Mao, Gravitational wave spectrum from first-order QCD phase transitions based on a parity doublet model, Phys. Rev. D 114, 023026 (2026), doi:10.1103/v1r6-gp2l, arXiv:2604.00465, https://arxiv.org/abs/2604.00465
Jingdong Shao, Hong Mao, Mei Huang, Transition rate and gravitational wave spectrum from first-order QCD phase transitions, Phys. Rev. D 111, 023052 (2025), arXiv:2410.06780, https://arxiv.org/abs/2410.06780
Hui-wen Zheng, Fei Gao, Ligong Bian, Si-xue Qin, Yu-xin Liu, Quantitative analysis of the gravitational wave spectrum sourced from a first-order chiral phase transition of QCD, Phys. Rev. D 111, L021303 (2025), doi:10.1103/PhysRevD.111.L021303, arXiv:2407.03795, https://arxiv.org/abs/2407.03795
Takuya Minamikawa, Toru Kojo, Masayasu Harada, Quark-hadron crossover equations of state for neutron stars: constraining the chiral invariant mass in a parity doublet model, Phys. Rev. C 103, 045205 (2021), doi:10.1103/PhysRevC.103.045205, arXiv:2011.13684, https://arxiv.org/abs/2011.13684
Takahiro Yamazaki, Masayasu Harada, Constraint to chiral invariant masses of nucleons from GW170817 in an extended parity doublet model, Phys. Rev. C 100, 025205 (2019), doi:10.1103/PhysRevC.100.025205, arXiv:1901.02167, https://arxiv.org/abs/1901.02167
Bikai Gao, Yuk-Kei Kong, Yong-Liang Ma, Origin of nucleon mass in the light of PSR J0614-3329 with quark-hadron crossover, Phys. Rev. D 112, 083041 (2025), doi:10.1103/xwkv-s8lg, arXiv:2509.03008, https://arxiv.org/abs/2509.03008
Aleksandr Chatrchyan, M. C. David Marsh, Charalampos Nikolis, Gravitational waves from a dilaton-induced, first-order QCD phase transition, Phys. Rev. Lett. 136, 041005 (2026), doi:10.1103/2v2f-1jvz, arXiv:2507.01191, https://arxiv.org/abs/2507.01191
Joshua Cesca, Archil Kobakhidze, QCD and electroweak phase transitions with hidden scale invariance: implications for primordial black holes, quark-lepton nuggets and gravitational waves, arXiv:2603.18406 (2026), https://arxiv.org/abs/2603.18406
Tae Hyun Jung, Seyong Kim, Jong-Wan Lee, Chang Sub Shin, Hee Beom Yang, Revisiting QCD-induced little inflation with chiral density wave state and its implications on pulsar timing array gravitational-wave signals, Phys. Rev. D 114, 023554 (2026), doi:10.1103/px47-qkxm, arXiv:2603.29772, https://arxiv.org/abs/2603.29772
Axel Brandenburg, Emma Clarke, Yutong He, Tina Kahniashvili, Can we observe the QCD phase transition-generated gravitational waves through pulsar timing arrays?, Phys. Rev. D 104, 043513 (2021), doi:10.1103/PhysRevD.104.043513, arXiv:2102.12428, https://arxiv.org/abs/2102.12428
Roman Pasechnik, António P. Morais, Probing cosmological phase transitions with SKAO through nano-Hz gravitational-wave portal, arXiv:2608.20897 (2026), https://arxiv.org/abs/2608.20897
Jia Zhou, Kyungil Kim, Sangyong Jeon, Jun Xu, Youngman Kim, Pion Production in an Extended Parity Doublet Model, Symmetry 17, 2155 (2025), doi:10.3390/sym17122155, https://doi.org/10.3390/sym17122155