Near beta equilibrium, the flavor-changing weak rates of deconfined quark matter control strangeness and beta equilibration, bulk viscosity, and the damping of oscillations in neutron stars and mergers. Hoyos, Olzi and Rodriguez-Fernandez (JHEP 2024) derived a formula for these rates to leading order in the Fermi coupling, exact in the QCD coupling, in terms of spectral functions of left-handed flavor currents, and evaluated it in a simple bottom-up holographic model of massless, unpaired quark matter, finding a low-temperature rate suppressed by logarithmic factors of the temperature compared to the perturbative result. The open problem is to make this strong-coupling prediction quantitative for neutron-star applications, along independent improvement directions enumerated or motivated by the source, each a distinct advance: include nonzero quark masses (at least the strange quark) with their backreaction on the bulk geometry, which is expected to replace the near-horizon AdS2 throat by a hyperscaling-Lifshitz geometry and change the temperature scaling of the rate; treat paired (color-superconducting) quark phases; quantify model dependence by computing in refined holographic constructions (top-down D3-D7 or Sakai-Sugimoto, or V-QCD) including finite-Nc effects; assess the non-factorizable four-current corrections to the leading-order factorization, which originate from effective eight-fermion interactions and are expected to be relatively small even at finite NcN_c; and establish numerically the validity regime across neutron-star-relevant temperatures. No computation of the rate under these extensions exists in the literature, so the problem remains open.

Background

At the densities reached in the cores of neutron stars and in their binary mergers, cold QCD matter may be deconfined quark matter. Whether or not it is in beta equilibrium (the state in which the electrochemical potentials of the interacting species satisfy the weak-reaction balance conditions, for example μd=μs\mu_d=\mu_s for the strangeness-changing channel) is governed by flavor-changing weak processes, such as the nonleptonic strangeness-changing reaction u+d→u+su+d\to u+s and semileptonic channels involving electrons and neutrinos. The rate at which these weak processes restore equilibrium determines the equilibration timescale, the bulk viscosity of dense matter, the damping of stellar oscillations such as r-modes (inertial oscillation modes of rotating stars, restored by the Coriolis force), and features of gravitational-wave signals from mergers. Perturbative QCD calculations of these rates date back to Heiselberg and Madsen, with non-Fermi-liquid corrections (long-range gauge interactions producing logarithmic enhancements) computed by Schäfer and Schwenzer; however, at neutron-star densities the strong coupling is not small, and lattice QCD is inapplicable at finite baryon chemical potential, so the rate is genuinely non-perturbative.

Holography (gauge/gravity duality) addresses strong coupling by mapping the large-NcN_c (many-color) limit of a strongly coupled field theory onto classical gravity in one higher dimension. In 'bottom-up' holographic models (such as the hard-wall model of Erlich, Katz, Son and Stephanov, or the lattice-matched V-QCD construction) the dual geometry is chosen phenomenologically, while 'top-down' models such as D3-D7 brane intersections and the Witten-Sakai-Sugimoto model derive from string theory; all are reliable mainly at large NcN_c, where correlators of currents are computed from fluctuations of dual gauge fields on the gravity side.

Hoyos, Olzi and Rodriguez-Fernandez derived a field-theoretic formula, exact in the QCD coupling to leading order in the Fermi coupling GFG_F, expressing the weak equilibration rate of quark flavor densities near equilibrium in terms of the spectral functions (the imaginary parts of the retarded current-current correlators, which encode the dissipative response) of left-handed flavor currents; for the nonleptonic channel this uses a factorization of a four-current correlator into two two-point functions, exact at large NcN_c but an approximation otherwise. Evaluating these spectral functions in a simple bottom-up holographic model of massless, unpaired quark matter at chemical potential μq\mu_q and temperature T≪μqT\ll\mu_q, they found the strangeness-equilibration coefficient

λds≈GF2sin⁡2θCcos⁡2θCN2323π μq5T2(log⁡6μqπT)−3/2,\lambda_{ds}\approx G_{F}^{2}\sin^{2}\theta_{C}\cos^{2}\theta_{C}\frac{\mathcal{N}^{2}}{32}\sqrt{\frac{3}{\pi}}\,\mu_{q}^{5}T^{2}\left(\log\frac{6\mu_{q}}{\pi T}\right)^{-3/2},

with θC\theta_{C} the Cabibbo angle and N\mathcal{N} a UV matching constant. The near-horizon region of the extremal charged black hole is an AdS2AdS_2 throat (an emergent two-dimensional anti-de Sitter geometry associated by Faulkner, Liu, McGreevy and Vegh with marginal-Fermi-liquid behavior), and this IR geometry fixes the rate's logarithmic suppression — the opposite of the perturbative log enhancement. The analytic formula is valid for T/μq≲10−2T/\mu_q\lesssim10^{-2} in this model.

The computation, however, is deliberately idealized, and the authors list what must be solved before it becomes a realistic prediction: the quark masses were set to zero, whereas the strange quark in neutron-star matter is massive, and a nonzero mass turns on a bulk scalar field whose backreaction can replace the AdS2AdS_2 throat with a hyperscaling-Lifshitz geometry (a scale-covariant but non-relativistic IR geometry characterized by a dynamical exponent), changing the power of TT in the rate; quark pairing was not considered, although color superconductivity (Cooper pairing of quarks, e.g. the color-flavor-locked (CFL) or two-flavor color-superconducting (2SC) phases) is expected at these densities and modifies the low-energy excitations through a condensate with non-trivial flavor structure; and the model itself is a simple bottom-up construction at Nc→∞N_c\to\infty, while realistic neutron-star modeling uses D3-D7, Sakai-Sugimoto or V-QCD constructions whose rate predictions (and finite-NcN_c corrections, including non-factorizable four-current contributions beyond the leading large-NcN_c factorization) have not been computed. In addition, the physical temperature range relevant for neutron-star interiors and mergers extends beyond the regime in which the source validated its analytic formula, requiring more extensive numerical evaluation than performed.

Problem Statement

Compute the weak flavor-equilibration rate of cold, dense, strongly coupled quark matter — for instance the strangeness-equilibration coefficient λds\lambda_{ds} governing the relaxation of quark flavor densities near beta equilibrium — to leading order in the Fermi coupling and non-perturbatively in the strong coupling, under neutron-star-relevant conditions, going beyond the massless, unpaired, single-model computation of Hoyos, Olzi and Rodriguez-Fernandez along the independent improvement directions enumerated or motivated by the source, each of which constitutes a distinct advance toward a quantitative neutron-star prediction:

  1. Including nonzero quark masses (minimally the strange-quark mass) — noted by the source as 'perhaps the most significant' improvement — together with their backreaction on the bulk scalar and metric, thereby determining how the infrared geometry (the AdS2AdS_2 throat versus a hyperscaling-Lifshitz geometry) and consequently the low-temperature scaling and momentum dependence of the current spectral functions and of the rate are modified.
  2. Extending the computation to paired (color-superconducting) phases of quark matter, whose flavor-structured condensates change the low-energy excitations and require the corresponding scalar-field profiles in the holographic dual.
  3. Quantifying model dependence: repeating the computation in at least one refined holographic construction (top-down D3-D7 or Sakai-Sugimoto, or V-QCD), estimating finite-NcN_c corrections, and assessing the non-factorizable four-current contributions neglected in the leading-order large-NcN_c factorization, which originate from effective eight-fermion interactions and are expected to be relatively small even at finite NcN_c.
  4. Establishing numerically the regime of validity of the resulting rate formulas over the temperatures and chemical potentials relevant for neutron-star interiors and mergers, beyond the T/μq≲10−2T/\mu_q\lesssim10^{-2} range validated for the massless model.

These directions serve one objective: a weak equilibration rate with a controlled uncertainty budget — from quark masses, pairing, model choice, and factorization — that can be used as a quantitative microphysical input for bulk-viscosity and equilibration modeling of compact stars; each direction is an independent advance toward that objective rather than a jointly required component of it.

References

  1. Carlos Hoyos, Andrea Olzi, David Rodriguez-Fernandez, 'Weak rates in strongly coupled cold quark matter', JHEP 12 (2024) 058, arXiv:2407.21643, DOI:10.1007/JHEP12(2024)058, https://arxiv.org/abs/2407.21643
  2. Thomas Apostolidis, Matti Järvinen, Elias Kiritsis, Francesco Nitti, Andrea Olzi, Edwan Préau, 'Flavour current correlators and the non-Abelian hydrodynamic approximation: the charged sector', 2026, arXiv:2607.20991, https://arxiv.org/abs/2607.20991
  3. Matti Järvinen, Toshali Mitra, 'Quark flavors in hot and dense holographic QCD: setup and comparison to data', JHEP 03 (2026) 210, arXiv:2507.08087, DOI:10.1007/JHEP03(2026)210, https://arxiv.org/abs/2507.08087
  4. Carlos Hoyos, Niko Jokela, Andrea Olzi, 'Subtleties of non-Abelian D-brane actions and their effect on holographic heavy-light meson spectra', JHEP 04 (2026) 127, arXiv:2511.17409, DOI:10.1007/JHEP04(2026)127, https://arxiv.org/abs/2511.17409
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