Modern hydrodynamic simulations of core-collapse supernovae and neutron-star mergers require knowledge not only of the equilibrium properties of strongly interacting matter, but also of the system’s response to perturbations, encoded in various transport coefficients. Using perturbative and holographic tools, we derive here an improved weak-coupling and a new strong-coupling result for the most important transport coefficient of unpaired quark matter, its bulk viscosity. These results are combined in a simple analytic pocket formula for the quantity that is rooted in perturbative quantum chromodynamics at high densities but takes into account nonperturbative holographic input at neutron-star densities, where the system is strongly coupled. This expression can be used in the modeling of unpaired quark matter at astrophysically relevant temperatures and densities.

South Korea

Mexico

Jesús Cruz Rojas

Germany

Tyler Gorda

Spain

Carlos Hoyos

Finland

Niko Jokela

Matti Järvinen

Norway

Aleksi Kurkela

Risto Paatelainen

Saga Säppi

Aleksi Vuorinen

Estimate for the Bulk Viscosity of Strongly Coupled Quark Matter Using Perturbative QCD and Holography

Deconfined quark matter at asymptotically high densities is weakly coupled, due to the asymptotic freedom of quantum chromodynamics. In this weak-coupling regime, bulk thermodynamic properties of quark matter, assuming a trivial ground state, are currently known to partial next-to-next-to-next-to-leading order. However, the ground state at high densities is expected to be a color superconductor, in which the excitation spectrum of (at least some) quarks exhibit a gap with a nonperturbative dependence on the strong coupling. In this work, we calculate the thermodynamic properties of color-superconducting quark matter at high densities and zero temperature at next-to-leading order (NLO) in the coupling in the presence of a finite gap. We work in the limit of two massless quark flavors, which corresponds to deconfined symmetric nuclear matter, and further assume that the gap is small compared to the quark chemical potential. In these limits, we find that the NLO corrections to the pressure and speed of sound are comparable in size to the leading-order effects of the gap, and further increase both quantities above their values for nonsuperconducting quark matter. We also provide a parametrization of the NLO speed of sound to guide phenomenology in the high-density region, and we furthermore comment on whether our findings should be expected to extend to the case of three-flavor quark matter of relevance to neutron stars.

Andreas Geißel

Jens Braun

Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order

The equation of state of QCD matter at high densities is relevant for neutron star structure and for neutron star mergers and has been a focus of recent work. We show how lattice QCD simulations, free of sign problems, can provide an upper bound on the pressure as a function of quark chemical potentials. We show that at large chemical potentials this bound should become quite sharp; the difference between the upper bound on the pressure P PQ  and the true pressure P is of order P PQ  − P =   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </math>  $$ {\alpha}_{\textrm{s}}^3 $$  P). The corrections arise from a single Feynman diagram; its calculation would render remaining corrections   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>4</mn> </msubsup> </math>  $$ {\alpha}_{\textrm{s}}^4 $$  P).

Guy Moore

Bounding the QCD Equation of State with the Lattice

Accurately understanding the equation of state (EOS) of high-density, zero-temperature quark matter plays an essential role in constraining the behavior of dense strongly interacting matter inside the cores of neutron stars. In this Letter, we study the weak-coupling expansion of the EOS of cold quark matter and derive the complete, gauge-invariant contributions from the long-wavelength, dynamically screened gluonic sector at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant <math><msub><mi>α</mi><mi>s</mi></msub></math>. This elevates the EOS result to the <math><mrow><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>ln</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow></math> level, leaving only one unknown constant from the unscreened sector at N3LO, and places it on par with its high-temperature counterpart from 2003.

Kaapo Seppänen

Equation of State of Cold Quark Matter to <math><mrow><mi>O</mi><mo>(</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>ln</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></mrow></math>

We compute the next-to-leading order (NLO) hard correction to the gluon self-energy tensor with arbitrary soft momenta in a hot and/or dense weakly coupled plasma in Quantum Chromodynamics. Our diagrammatic computations of the two-loop and power corrections are performed within the hard-thermal-loop (HTL) framework and in general covariant gauge, using the real-time formalism. We find that after renormalization our individual results are finite and gauge-dependent, and they reproduce previously computed results in Quantum Electrodynamics in the appropriate limit. Combining our results, we also recover a formerly known gauge-independent matching coefficient and associated screening mass in a specific kinematic limit. Our NLO results supersede leading-order HTL results from the 1980s and pave the way to an improved understanding of the bulk properties of deconfined matter, such as the equation of state.

Soft gluon self-energy at finite temperature and density: hard NLO corrections in general covariant gauge

The equation of state of neutron-star cores can be constrained by requiring a consistent connection to the perturbative Quantum Chromodynamics (QCD) calculations at high densities. The constraining power of the QCD input depends on uncertainties from missing higher-order terms, the choice of the unphysical renormalization scale, and the reference density where QCD calculations are performed. Within a Bayesian approach, we discuss the convergence of the perturbative QCD series, quantify its uncertainties at high densities, and present a framework to systematically propagate the uncertainties down to neutron-star densities. We find that the effect of the QCD input on the neutron-star inference is insensitive to the various unphysical choices made in the uncertainty estimation.

Oleg Komoltsev

Aleksas Mazeliauskas

Bayesian uncertainty quantification of perturbative QCD input to the neutron-star equation of state

We present the first complete calculation of the soft photon self-energy to the next-to-leading order in a hot and/or dense ultrarelativistic plasma in QED. The calculation is performed within the real-time formalism utilizing dimensional regularization in <math><mn>4</mn><mo>−</mo><mn>2</mn><mi>ϵ</mi></math> dimensions, while the result is reported including explicit <math><mi>O</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo></math> terms in the zero-temperature limit. This information is required to extend the perturbative calculation of the pressure of cold and dense QED matter to partial next-to-next-to-next-to-leading order in a weak-coupling expansion, reported in a companion paper. These results pave the way for a similar future calculation in quantum chromodynamics.

Juuso Österman

Italy

Philipp Schicho

Soft photon propagation in a hot and dense medium to next-to-leading order

We determine the pressure of a cold and dense electron gas to a nearly complete next-to-next-to-next-to-leading order (<math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math>) in the fine-structure constant <math><msub><mi>α</mi><mi>e</mi></msub></math>, utilizing a new result for the two-loop photon self-energy from a companion paper. Our result contains all infrared-sensitive contributions to the pressure at this order, including the coefficient of the <math><mi>O</mi><mo>(</mo><msubsup><mi>α</mi><mi>e</mi><mn>3</mn></msubsup><mi>ln</mi><msub><mi>α</mi><mi>e</mi></msub><mo>)</mo></math> term, and leaves only a single coefficient associated with the contributions of unresummed hard momenta undetermined. Moreover, we explicitly demonstrate the complete cancellation of infrared divergences according to the effective-field-theory paradigm by determining part of the hard contributions at this order. Our calculation provides the first improvement to a 45-year-old milestone result and demonstrates the feasibility of the corresponding <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math> calculation for cold and dense quark matter.

Degenerate fermionic matter at <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math>: Quantum electrodynamics

At zero temperature and finite chemical potential, <math><mi>d</mi></math>-dimensional loop integrals with complex-valued integrands in the imaginary-time formalism yield results dependent on the integration order. We observe this even with the simplest one-loop dimensionally regularized integrals. Computing such integrals by evaluating the spatial <math><msup><mi>d</mi><mi>d</mi></msup><mi>p</mi></math> integral before the temporal <math><mi>d</mi><msub><mi>p</mi><mn>0</mn></msub></math> integral yields results consistent with those obtained at small but nonvanishing temperatures. Computing the temporal integral first by applying the residue theorem to the integrand yields a different answer. The same holds for general complexified propagators. In this work we aim to understand the theoretical background behind this difference, in order to fully enable the powerful techniques of residue calculus in applications. We cast the difference into the form of a derivative term related to Dirac deltas, and further demonstrate how the difference originates from the zero-temperature limit of the Fermi–Dirac occupation functions treated as complex-valued functions. We also discuss a generalization to propagators raised to noninteger powers.

Augmenting the residue theorem with boundary terms in finite-density calculations

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths (formation lengths) of two consecutive splitting processes overlap, avoiding soft-emission approximations. Previous work made a “nearly-complete” calculation of the effect of overlapping formation times on gluonic splittings such as g → gg → ggg (with simplifying assumptions such as an infinite QCD medium and the large-N c limit). In this paper, we extend those previous rate calculations from nearly-complete to complete by including processes involving the exchange of longitudinally-polarized gluons. In the context of Lightcone Pertubation Theory, used earlier for the “nearly-complete” calculation, such exchanges are instantaneous in lightcone time and have their own diagrammatic representation.