arXiv subjects

Michael Buballa

Publications and source records attributed to Michael Buballa.

At least 19 recordsLinked to original sources

arXiv2026

A doubly critical point in the color-superconducting regime of the RG-consistent NJL model

Nambu--Jona-Lasinio (NJL) models of color-superconducting quark matter suffer from cutoff artifacts once temperature or quark chemical potential become comparable to the model cutoff. These artifacts can be removed employing a renormalization-group (RG) consistent regularization scheme. In this article we perform a systematic study of neutral beta-equilibrated quark matter within the three-flavor NJL model with RG consistent regularization. Varying the coupling constant in the scalar diquark channel, we map out the phase diagram in the plane of chemical potential and temperature, and identify the gapless domains. We particularly focus on the melting pattern of the color-flavor locked (CFL) phase. At larger couplings the CFL phase melts through a so-called $d$SC phase, as expected from leading-order Ginzburg--Landau analyses. Lowering the coupling, the phase structure becomes markedly richer: While at large densities the CFL phase still melts through a $dSC$ phase, we find a $uSC$ phase at lower chemical potential. These phases, $uSC$ and $dSC$ meet at a doubly critical point whose existence had been anticipated long ago but was never demonstrated explicitly in a model.

hep-ph↗

arXiv2026

Interplay between inhomogeneous chiral and crystalline color-superconducting phases in the two-flavor NJL model

We study the interplay between the chiral density wave (CDW) and the single-plane-wave Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase of color-superconducting matter in two-flavor quark matter at vanishing and non-vanishing temperature $T$, quark number chemical potential $\mu$ and isospin chemical potential $\delta\mu$. The analysis is performed within the two-flavor Nambu--Jona-Lasinio (NJL) model in the chiral limit, using a three-momentum cutoff scheme. Treating the CDW wave vector $\vec{q}$ and the LOFF pair momentum $\vec{q}\,'$ as independent variational parameters, we minimize the mean-field effective potential with respect to both amplitudes and both wave vectors, without constraining their relative orientation, and map out the $T$-$\mu$ and $\mu$-$\delta\mu$ phase diagrams for a range of diquark couplings $G_D$. Our central result is that $\vec{q}$ and $\vec{q}\,'$ are never simultaneously nonzero: inhomogeneous chiral and diquark condensates do not coexist across the entire parameter range.

hep-ph↗

arXiv2025

Neutrino absorption in two-flavor color-superconducting quark matter

We calculate the absorption mean free paths of electron and muon neutrinos in two-flavor color-superconducting (2SC) quark matter in the density and temperature range that is relevant to binary neutron star mergers. We model the strong interaction between quarks using a Nambu--Jona-Lasinio model, performing calculations self-consistently in the mean-field approximation. Since the 2SC gap is large we restrict our analysis to the contribution of unpaired quarks. We find that at low temperatures absorption by a down quark $\nu+d \to u+e^-/\mu^-$ is kinematically not allowed, so absorption by a strange quark $\nu+s \to u+e^-/\mu^-$ dominates the mean free path. As temperature or neutrino energy rises, the $d$ quark absorption channel becomes active, and the mean free path shrinks. We find that in equilibrated 2SC matter with an electron lepton fraction $Y_{L_e}=0.1$, the neutrinos form a degenerate gas with a mean free path of meters or less, independent of the temperature.

nucl-th↗

arXiv2025

Removing cutoff artifacts in the NJL model by a Renormalization Group consistent treatment

We summarize how a renormalization-group (RG)-consistent treatment removes well-known artifacts in NJL-model descriptions of color-superconducting quark matter. We introduce two RG-consistent schemes, "minimal" and "massless", and present analytic solutions for the diquark gap at $T=0$ and for the phase boundary $T_c(\mu)$ in symmetric massless matter, representing the high-density limit of the model. We compare the pairing gaps, phase diagram, and speed of sound with results obtained using conventional regularization.

hep-ph↗

arXiv2025

Renormalizing the Quark-Meson-Diquark Model

We present a comprehensive study of the two-flavor Quark--Meson--Diquark (QMD) model by comparing a renormalization approach with a renormalization-group (RG) consistent mean-field formulation based on the functional renormalization group (FRG). The renormalized QMD model allows analytical investigations of key quantities such as the zero-temperature diquark gap and the critical temperature for color superconductivity, ultimately reproducing the exact BCS relation in the high-density limit. We carry out the same analysis for different schemes of RG-consistent QMD models. We show that the RG-consistent approach yields a phase diagram and thermodynamic properties qualitatively similar to those of the renormalized model, provided both are embedded within a unified scheme that ensures consistent vacuum properties. In particular, both treatments recover the Stefan--Boltzmann limit at high densities. On the other hand, whether the BCS relation for the critical temperature is satisfied depends on the details of the RG-consistent setup. Our results highlight the relevance of renormalization and RG-consistent methods for accurately capturing the thermodynamics of QMD and related effective models with diquark degrees of freedom.

hep-ph↗

arXiv2024

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

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 (EoS) 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 $c_s^2 \sim 0.6$ and the CFL gap up to $\Delta\sim250\,$MeV. 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 $2.0 M_{\odot}$ 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 1.4 $M_\odot$. 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 ($\sim 1-3\, n_{\text{sat}}$).

hep-ph↗

arXiv2024

New Tool to Detect Inhomogeneous Chiral Symmetry Breaking

In this letter, 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.

hep-ph↗

arXiv2024

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

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 d-quark superconducting (dSC) phase within the melting pattern of the Color-Flavor Locked (CFL) phase at high chemical potentials, consistent with earlier Ginzburg-Landau analyses.

hep-ph↗

arXiv2024

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

hep-ph↗

arXiv2023

Towards a Stability Analysis of Inhomogeneous Phases in QCD

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 1+1 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 non-locality 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 analyse 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.

hep-ph↗

arXiv2022

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. I. The $O(N)$ model

The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows us to reliably treat the emergence of nonanalytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higher-dimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.

cond-mat.stat-mech↗

arXiv2021

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

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 $N_f = 2 + 1$ 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.

hep-ph↗

arXiv2021

Role of baryon resonances in the $π^-p \to ne^+e^-$ reaction within an effective-Lagrangian model

We present a study of the reaction $π^-p \to ne^+e^-$ for $\sqrt{s}=1.49\,\textrm{GeV}$, including non-resonant Born terms and contributions of the $N(1440)$, $N(1520)$, $N(1535)$ resonances ($R$), using an effective-Lagrangian model, which we extended by a phenomenological phase factor at the $RNρ$ vertex function. We give predictions for both the differential cross section $dσ/dm$ and the spin density matrix elements of the virtual photon that decays into the lepton pair. In the studied energy range, the cross section is dominated by the Born and $N(1520)$ contributions.

nucl-th↗

arXiv2021

Inhomogeneous phases in the quark-meson model with explicit chiral-symmetry breaking

We investigate the existence of inhomogeneous chiral phases in the quark-meson model with explicit chiral-symmetry breaking. We find that the inhomogeneous region shrinks with increasing pion masses but survives for the physical value of m_pi. The instability towards inhomogeneous matter occurs in the scalar channel, while pseudoscalar modes are disfavored.

hep-ph↗

arXiv2020

Regulator dependence of inhomogeneous phases in the 2+1-dimensional Gross-Neveu model

The phase diagram of the Gross-Neveu model in $2+1$ space-time dimensions at non-zero 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 $σ$ is non-uniform in space. To this end, we analyze the stability of the energetically favored homogeneous configuration $σ(\textbf{x}) = \barσ= \textrm{const}$ 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 $Λ$, we find a region in the phase diagram where an inhomogeneous order parameter is favored. This inhomogeneous region shrinks, however, when $a$ is decreased or $Λ$ is increased, and finally diappears for all non-zero 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.

hep-lat↗

arXiv2020

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

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 [1, 2]. 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 Refs. [1, 2]. 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.

hep-ph↗

arXiv2020

Inhomogeneous chiral condensates in three-flavor quark matter

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 allow 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 $m_s$ 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, critical and Lifshitz point split. For realistic strange-quark masses the effect is however very small and becomes sizeable only for small values of $m_s$.

hep-ph↗

arXiv2019

Dyson-Schwinger approach to baryon number fluctuations

We summarize our results on baryon number fluctuations at nonzero temperature and chemical potential. They are obtained from solutions of a coupled set of Dyson-Schwinger equations for the quark and gluon propagators of QCD in Landau gauge with $N_f=2+1$ quark flavors. In comparison with preliminary STAR data, our results are compatible with a critical endpoint at large chemical potential and a freeze-out line that bends below it.

hep-ph↗