Knowledge of the pressure of cold and dense quark matter is known to significantly constrain the equation of state of neutron-star matter, a quantity playing a key role in deciphering the stars’ internal structure. In this Letter, we make important progress toward determining the last unknown contribution to the pressure at next-to-next-to-next-to-leading order in the strong coupling constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math>, available through the sum of all four-loop vacuum diagrams of dense quantum chromodynamics. We demonstrate the cancellation of both the covariant gauge parameter and all infrared divergences in the sum, showcasing the effective-field-theory paradigm in action. For the remaining infrared-finite four-loop integrals, we demonstrate the efficacy of the dense loop tree duality method—a novel numerical framework for multiloop calculations in thermal field theory. Together, our results show that completing the next-to-next-to-next-to-leading order pressure of cold quark matter is no longer merely possible but for the first time within reach.

Finland

Aapeli Kärkkäinen

Pablo Navarrete

Mika Nurmela

Risto Paatelainen

Kaapo Seppänen

Aleksi Vuorinen

Quark Matter at Four Loops: Hardships and How to Overcome Them

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

Niko Jokela

Matti Järvinen

Norway

Aleksi Kurkela

Saga Säppi

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

Understanding the properties and physical phase of the dense strongly interacting matter present in the cores of neutron stars or created in their binary mergers remains one of the most prominent open problems in nuclear astrophysics. While most microscopic analyses have historically relied on solvable phenomenological models of nuclear and quark matter, in recent years a model-independent approach utilizing only controlled ab-initio calculations and astrophysical observations has emerged as a viable alternative.In these lecture notes, I review recent progress in first-principles weak-coupling calculations within high-density quark matter, shedding light on its thermodynamic and transport properties. I cover the most important technical tools used in such calculations, introduce selected highlight results, and explain how this information can be used in phenomenological studies of neutron-star physics. The notes do not offer a self-consistent treatment of the topics covered, but rather aim at filling gaps in existing textbooks on thermal field theory and at connecting the dots in a story developed in several recent research articles, to which the interested reader is directed for further technical details.

Particle-theory Input for Neutron-star Physics

Both nonzero temperature and chemical potentials break the Lorentz symmetry present in vacuum quantum field theory by singling out the rest frame of the heat bath. This leads to complications in the application of thermal perturbation theory, including the appearance of novel infrared divergences in loop integrals and an apparent absence of four-dimensional integration-by-parts (IBP) identities, vital for high-order computations. Here, we propose a new strategy that enables the use of IBP techniques in the evaluation of Feynman integrals, in particular vacuum or bubble diagrams, in the limit of vanishing temperature T but nonzero chemical potentials μ. The central elements of the new setup include a contour representation for the temporal momentum integral, the use of a small but nonzero T as an IR regulator, and the systematic application of both temporal and spatial differential operators in the generation of linear relations among the loop integrals of interest. The relations we derive contain novel inhomogeneous terms featuring differentiated Fermi-Dirac distribution functions, which severely complicate calculations at nonzero temperature, but are shown to reduce to solvable lower-dimensional objects as T tends to zero. Pedagogical example computations are kept at the one- and two-loop levels, but the application of the new method to higher-order calculations is discussed in some detail.

Juuso Österman

Philipp Schicho

Integrating by parts at finite density

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.

Italy

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

The transport properties of dense QCD matter play a crucial role in the physics of neutron stars and their mergers but are notoriously difficult to study with traditional quantum field theory tools. Specializing to the case of unpaired quark matter in beta equilibrium, we approach the problem through the machinery of holography, in particular the V-QCD and D3-D7 models, and derive results for the electrical and thermal conductivities and the shear and bulk viscosities. In addition we compare the bulk to shear viscosity ratio to the speed of sound and find that it violates the so-called Buchel bound. Our results differ dramatically from earlier predictions of perturbative QCD, the root causes and implications of which we analyze in detail.

Sweden

Javier G. Subils

Javier Tarrío

Holographic approach to transport in dense QCD matter

High-order perturbative calculations for thermodynamic quantities in QCD are complicated by the physics of dynamical screening that affects the soft, long-wavelength modes of the system. Here, we provide details for the evaluation of this soft contribution to the 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>) pressure of high-density, zero-temperature quark matter (QM), complementing our accompanying Letter [T. Gorda et al., Phys. Rev. Lett. 127, 162003 (2021)]. Our calculation requires the determination of the pressure of the hard-thermal-loop effective theory to full two-loop order at zero temperature, which we go through in considerable detail. In addition to this, we comprehensively discuss the structure of the weak-coupling expansion of the QM pressure, and lay out a roadmap towards the evaluation of the contributions missing from a full <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math> result for this quantity.

Cold quark matter at <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math>: Soft contributions

Accurate knowledge of the thermodynamic properties of zero-temperature, high-density quark matter plays an integral role in attempts to constrain the behavior of the dense QCD matter found inside neutron-star cores, irrespective of the phase realized inside the stars. In this Letter, we consider the weak-coupling expansion of the dense QCD equation of state and compute the next-to-next-to-next-to-leading-order contribution arising from the non-Abelian interactions among long-wavelength, dynamically screened gluonic fields. Accounting for these interactions requires an all-loop resummation, which can be performed using hard-thermal-loop (HTL) kinematic approximations. Concretely, we perform a full two-loop computation using the HTL effective theory, valid for the long-wavelength, or soft, modes. We find that the soft sector is well behaved within cold quark matter, contrary to the case encountered at high temperatures, and find that the new contribution decreases the renormalization-scale dependence of the equation of state at high density.

Soft Interactions in Cold Quark Matter

Motivated by the possible presence of deconfined quark matter in neutron stars and their mergers and the important role of transport phenomena in these systems, we perform the first-ever systematic study of different viscosities and conductivities of dense quark matter using the gauge/gravity duality. Using the V-QCD model, we arrive at results that are in qualitative disagreement with the predictions of perturbation theory, which highlights the differing transport properties of the system at weak and strong coupling and calls for caution in the use of the perturbative results in neutron star applications.