<p>We determine parameters of the renormalization group-consistent three-flavor color-superconducting Nambu–Jona-Lasinio (NJL) model that are suited to investigate possible compact-star configurations. Our goal is to provide quark-matter equations of state (EOSs) that can be used for hadron-quark hybrid-star constructions. To that end, we mainly focus on the parameters of the quark-matter model. By varying the vector and diquark coupling constants, we analyze their impact on the EOS, the speed of sound, the maximum diquark gap, and the mass-radius relation. In almost all configurations, a stable color-flavor-locked (CFL) phase appears in the core of the maximum-mass configurations, typically spanning several kilometers in radius. In other cases, the star’s two-flavor color-superconducting (2SC) branch of the EOS becomes unstable before reaching the CFL transition density. At neutron-star densities, the speed of sound squared reaches up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>∼</mo><mn>0.6</mn></mrow></math> and the CFL gap up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><mo>∼</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. We argue that adding a hadronic EOS at lower densities by performing a Maxwell construction does not increase the maximum mass substantially. Thus we use the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2.0</mn><msub><mi>M</mi><mo stretchy="false">⊙</mo></msub></math> constraint to constrain the NJL model parameters that are suited for the construction of hybrid-star EOS. We construct three examples of the hybrid-star model, demonstrating that there is room for different color-superconducting compositions. The hybrid EOSs obtained in this way can have no 2SC matter or different ratios of 2SC and CFL quark matter in the core. We show that early hadron-quark transitions are possible that can modify the tidal deformability at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1.4</mn><msub><mi>M</mi><mo stretchy="false">⊙</mo></msub></math>. We find that these EOSs are consistent with the imposed constraints from astrophysics and perturbative QCD. They allow for different hybrid-star scenarios with a hadronic EOS that is soft at low to intermediate densities (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>1</mn><mi>–</mi><mn>3</mn><msub><mi>n</mi><mi>sat</mi></msub></math>).</p>

Germany

Hosein Gholami

Ishfaq Ahmad Rather

Marco Hofmann

Michael Buballa

Jürgen Schaffner-Bielich

Astrophysical constraints on color-superconducting phases in compact stars within the RG-consistent NJL model

<p>In this paper, we discuss a novel method to search for inhomogeneous chiral symmetry breaking in theories with fermions. The prime application we have in mind is QCD, but the method is also applicable for other theories, including solid state applications. It is based on an extension of the chiral susceptibility to inhomogeneous phases, and it works as a stability analysis. Our method allows us to determine when homogeneous solutions have the tendency to create spatially modulated condensates. As proof of principle, we apply this technique to a rainbow-ladder QCD model and find that its phase diagram contains an inhomogeneous region.</p>

Brazil

Theo F. Motta

Julian Bernhardt

Christian S. Fischer

New tool to detect inhomogeneous chiral-symmetry breaking

The Nambu–Jona-Lasinio (NJL) model and specifically its extension to color superconductivity (CSC) is a popular effective model for investigating dense quark matter. However, the reliability of its results is challenged by cutoff artifacts, which emerge if temperature or chemical potential are of the order of the cutoff energy scales. In this work, we generalize an idea from Braun et al. [SciPost Phys. 6, 056 (2019)], which is based on the requirement of renormalization-group (RG) consistency and has successfully been applied to the two-flavor quark-meson-diquark model, to the NJL model for electrically and color-neutral three-flavor color-superconducting quark matter. To this end, we analyze the medium divergences of the model and eliminate them by appropriate counterterms, introducing three different schemes. We show that the RG consistent treatment removes the cutoff artifacts of the conventional regularization and enables the investigation of CSC matter at higher densities by the model. Our studies reveal the emergence of a so-called <math><mi>d</mi></math>-quark superconducting (dSC) phase within the melting pattern of the color-flavor locked phase at high chemical potentials, consistent with earlier Ginzburg-Landau analyses.

Renormalization-group consistent treatment of color superconductivity in the NJL model

In this work we continue our efforts to study the existence of a phase with an inhomogeneous, i.e., spatially varying, chiral condensate in QCD. To this end we employ a previously established method of stability analysis of the two-particle irreducible effective action in a truncation that corresponds to a rainbow-ladder approximation of the quark-gluon interaction of QCD. If the analysis is restricted to homogeneous phases, the phase diagram features a first-order chiral transition in the lower-temperature regime. Performing the stability analysis along the lower-chemical-potential border of the corresponding spinodal region, we find that below a certain temperature the homogeneous chirally symmetric solution is unstable against inhomogeneous condensation. We argue that this instability may persist to chemical potentials above the homogeneous first-order phase boundary, in which case it signals the existence of an inhomogeneous ground state. Our methodology is also applicable for more sophisticated truncations of the QCD effective action.

Inhomogenuous instabilities at large chemical potential in a rainbow-ladder QCD model

The possible occurrence of crystalline or inhomogeneous phases in the QCD phase diagram at large chemical potential has been under investigation for over thirty years. Such phases are present in models of QCD such as the Gross-Neveu model in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions, Nambu–Jona-Lasinio (NJL) and quark meson models. Yet, no unambiguous confirmation exists from actual QCD. In this work, we propose a new approach for a stability analysis that is based on the two-particle irreducible effective action and compatible with full QCD calculations within the framework of functional methods. As a first test, we reproduce a known NJL model result within this framework. We then discuss the additional difficulties which arise in QCD due to the nonlocality of the quark self-energy and suggest a method to overcome them. As a proof of principle and as an illustration of the analysis, we consider the Wigner-Weyl solution of the quark Dyson-Schwinger equation (DSE) within a simple truncation of QCD in the chiral limit and analyze its stability against homogeneous chiral-symmetry breaking fluctuations. For temperatures above and below the tricritical point we find that the boundary of the instability region coincides well with the second-order phase boundary or the left spinodal, respectively, obtained from the direct solutions of the DSEs. Finally, we outline how this method can be generalized to study inhomogeneous fluctuations.

Toward a stability analysis of inhomogeneous phases in QCD

We study the phase structure of the two-flavor Nambu–Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [M. Sadzikowski, Phys. Lett. B 553, 45 (2003); M. Sadzikowski, Phys. Lett. B 642, 238 (2006)]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in the presence of a homogeneous 2SC condensate, if the dispersion relations in the absence of color superconductivity are known. In our calculations, we employ two different Ansätze for the inhomogeneous chiral condensate, the CDW as well as the real-kink crystal (RKC). Depending on the value of the diquark coupling we find a region of the phase diagram where the inhomogeneous chiral and the 2SC condensates coexist, confirming results of M. Sadzikowski [Phys. Lett. B 553, 45 (2003); 642, 238 (2006)]. Decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase. On the other hand, increasing the diquark coupling leads to a larger 2SC phase, while the inhomogeneous chiral and the coexistence phases become smaller. In agreement with previous studies the RKC Ansatz is energetically preferred over the CDW Ansatz. Both Ansätze lead to a qualitatively similar phase diagram, however, the coexistence phase is smaller for the RKC Ansatz.

Phillip Lakaschus

Dirk H. Rischke

Competition of inhomogeneous chiral phases and two-flavor color superconductivity in the NJL model

The phase diagram of the Gross-Neveu model in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> space-time dimensions at nonzero temperature and chemical potential is studied in the limit of infinitely many flavors, focusing on the possible existence of inhomogeneous phases, where the order parameter <math><mi>σ</mi></math> is nonuniform in space. To this end, we analyze the stability of the energetically favored homogeneous configuration <math><mi>σ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mover><mi>σ</mi><mo>¯</mo></mover><mo>=</mo><mtext>const</mtext></math> with respect to small inhomogeneous fluctuations, employing lattice field theory with two different lattice discretizations as well as a continuum approach with Pauli-Villars regularization. Within lattice field theory, we also perform a full minimization of the effective action, allowing for arbitrary 1-dimensional modulations of the order parameter. For all methods special attention is paid to the role of cutoff effects. For one of the two lattice discretizations, no inhomogeneous phase was found. For the other lattice discretization and within the continuum approach with a finite Pauli-Villars cutoff parameter <math><mi>Λ</mi></math>, we find a region in the phase diagram where an inhomogeneous order parameter is favored. This inhomogeneous region shrinks, however, when the lattice spacing is decreased or <math><mi>Λ</mi></math> is increased, and finally disappears for all nonzero temperatures when the cutoff is removed completely. For vanishing temperature, we find hints for a degeneracy of homogeneous and inhomogeneous solutions, in agreement with earlier findings.

Lennart Kurth

Marc Wagner

Marc Winstel

Regulator dependence of inhomogeneous phases in the (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional Gross-Neveu model

We investigate the effect of strange-quark degrees of freedom on the formation of inhomogeneous chiral condensates in a three-flavor Nambu–Jona-Lasinio model in mean-field approximation. A Ginzburg-Landau study complemented by a stability analysis allows us to determine in a general way the location of the critical and Lifshitz points, together with the phase boundary where the (partially) chirally restored phase becomes unstable against developing inhomogeneities, without resorting to specific assumptions on the shape of the chiral condensate. We discuss the resulting phase structure and study the influence of the bare strange-quark mass <math><msub><mi>m</mi><mi>s</mi></msub></math> and the axial anomaly on the size and location of the inhomogeneous phase compared to the first-order transition associated with homogeneous matter. We find that, as a consequence of the axial anomaly, the critical and the Lifshitz point split. For realistic strange-quark masses, the effect is, however, very small and becomes sizeable only for small values of <math><msub><mi>m</mi><mi>s</mi></msub></math>.

Spain

Stefano Carignano

Inhomogeneous chiral condensates in three-flavor quark matter

We present results for fluctuations of the baryon number for QCD at nonzero temperature and chemical potential. These are extracted from solutions to a coupled set of truncated Dyson-Schwinger equations for the quark and gluon propagators of Landau-gauge QCD with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> quark flavors, that has been studied previously. We discuss the changes of fluctuations and ratios thereof up to fourth order for several temperatures and baryon chemical potential up to and beyond the critical endpoint. In the context of preliminary STAR data for the skewness and kurtosis ratios, the results are compatible with the scenario of a critical endpoint at large chemical potential and slightly offset from the freeze-out line. We also discuss the caveats involved in this comparison.

Philipp Isserstedt

Pascal J. Gunkel

Baryon number fluctuations in the QCD phase diagram from Dyson-Schwinger equations

The effect of explicit chiral-symmetry breaking on inhomogeneous chiral phases is studied within a Nambu–Jona-Lasinio model with nonzero current quark mass. Generalizing an earlier result obtained in the chiral limit, we show within a Ginzburg-Landau analysis that the critical endpoint of the first-order chiral phase boundary between two homogeneous phases gets replaced by a “pseudo-Lifshitz point” when the possibility of inhomogeneous order parameters is considered. Performing a stability analysis we also show that the unstable mode along the phase boundary is in the scalar but not in the pseudoscalar channel, suggesting that modulations which contain pseudoscalar condensates, like a generalized dual chiral density wave, are disfavored against purely scalar ones. Numerically we find that the inhomogeneous phase shrinks as one moves away from the chiral limit, but survives even at significantly large values of the current quark mass.