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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., Comment: 9 pages, 4 figures; v2: reference and minor typos corrected; v3: references added, comment on fluctuation effects, version accepted for publication at Phys. Lett. B

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 $m_s$., Comment: 13 pages, 4 figures. v2: comments added, matches published version. Added plots data

We construct analytic (3+1)-dimensional inhomogeneous and topologically non-trivial pion systems using chiral perturbation theory. We discuss the effect of isospin asymmetry with vanishing electromagnetic interactions as well as some particular configurations with non-vanishing electromagnetic interactions. The inhomogeneous configurations of the pion fields are characterized by a non-vanishing topological charge that can be identified with baryons surrounded by a cloud of pions. This system supports a topologically protected persistent superflow. When the electromagnetic field is turned on the superflow corresponds to an electromagnetic supercurrent., Comment: v2: extended discussion and added references; matches published version on Physical Review D

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., Comment: 14 pages, 5 figures, contains ancillary files with plot data; v2: figures and typos corrected, corresponds to published version in the EPJ ST Special Issue: Strong Correlations in Dense Matter Physics

Several model calculations of the QCD phase structure at nonzero temperature and density suggest that in certain regions of the phase diagram inhomogeneous condensates are favored over homogeneous ones. In particular, in a two-flavor NJL model in the chiral limit the would-be first-order boundary associated with the chiral phase transition is entirely covered by an inhomogeneous phase, characterized by a spatially modulated chiral condensate, whose tip coincides exactly with the chiral critical point. In this contribution we discuss how this result is altered by the inclusion of nonzero quark masses and strange quarks in the model., 1+10 pages, contribution to the proceedings of XIII Quark Confinement and the Hadron Spectrum, Maynooth University, Ireland, 2018, accepted for publication on PoS(Confinement2018)202

We show that the on-shell effective theory (OSEFT) is the quantum field theory counterpart of a Foldy-Wouthuysen diagonalization of relativistic quantum mechanics for massless fermions. Thus, it is free of the Zitterbewegung oscillations that would yield an ill-defined meaning to the semiclassical transport approach at short distances if derived from the pure Dirac picture. We present a detailed derivation of the collision terms in the chiral kinetic theory using the OSEFT. Collision integrals are derived up to order 1 / E , where E is the energy of an on-shell fermion. At this order, the collision terms depend on the spin tensor of the fermion, and in the presence of chiral imbalance, it describes how a massless fermion of a given helicity interacts differently with the transverse photons of different circular polarization. In order to back up our results, we check that they allow us to reproduce the fermion decay rate in an ultradegenerate plasma with a chiral imbalance computed directly from QED., J. M. T.-R. acknowledges the hospitality of the Institut de Ciencies de l’Espai (CSIC) and Universitat de Barcelona, where part of this work was carried out. We have been supported by the Ministerio de Ciencia e Innovación (Spain) under Projects No. FPA2016-81114-P and No. FPA2016-76005-C2-1-P, as well as by Project No. 2017-SGR-929 (Catalonia). J. M. T.-R. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Projects No. 411563442 (Hot Heavy Mesons) and No. 315477589—TRR 211 (Strong-interacting matter under extreme conditions). This work was also supported by the COST Action CA15213 THOR.

We update a previous study of the effects of vector interactions in the Nambu--Jona-Lasinio model on the formation of inhomogeneous chiral symmetry breaking condensates. In particular, by properly considering a spatially modulated vector mean-field associated with the quark number density of the system we show that, as the value of the vector coupling increases, a chiral density wave modulation can become thermodynamically favored over a real sinusoidal modulation. This behavior is found both via a Ginzburg-Landau analysis close to the Lifshitz point, as well as with a full numerical diagonalization of the mean-field Dirac Hamiltonian at vanishing temperature., 17 pages, 3 figures

We extend a previous work on the study of heavy charmonia and bottomonia in a deconfined quark-gluon plasma by considering the B c family of mesons. With the introduction of this bound state of a charm and a beauty quark, we investigate at finite temperature the behavior of the quarkonium, in an energy region between the ψ and the Y states.

We revisit the renormalization prescription for the quark-meson model in an extended mean-field approximation, where vacuum quark fluctuations are included. At a given cutoff scale the model parameters are fixed by fitting vacuum quantities, typically including the sigma-meson mass $m_\sigma$ and the pion decay constant $f_\pi$. In most publications the latter is identified with the expectation value of the sigma field, while for $m_\sigma$ the curvature mass is taken. When quark loops are included, this prescription is however inconsistent, and the correct identification involves the renormalized pion decay constant and the sigma pole mass. In the present article we investigate the influence of the parameter-fixing scheme on the phase structure of the model at finite temperature and chemical potential. Despite large differences between the model parameters in the two schemes, we find that in homogeneous matter the effect on the phase diagram is relatively small. For inhomogeneous phases, on the other hand, the choice of the proper renormalization prescription is crucial. In particular, we show that if renormalization effects on the pion decay constant are not considered, the model does not even present a well-defined renormalized limit when the cutoff is sent to infinity., Comment: 13 pages, 4 figures

We investigate the effects of isospin asymmetry on the competition between color-superconductivity and inhomogeneous chiral symmetry breaking in dense two-flavor quark matter using an extended Nambu--Jona-Lasinio model. We confirm the appearance of a coexistence window where chiral symmetry is inhomogeneously broken and a nonzero spatially homogeneous diquark gap is present, consistently with previous works, and show that such a phase survives at nonzero isospin chemical potentials. We also discuss how the model phase structure becomes modified as large isospin asymmetries are considered., Comment: 14 pages and 7 figures. Contribution to the proceedings of Modern Physics of Compact Stars 2015-Yerevan

We investigate the influence of spatially inhomogeneous chiral symmetry-breaking condensates in a magnetic field background on the equation of state for compact stellar objects. After building a hybrid star composed of nuclear and quark matter using the Maxwell construction, we find, by solving the Tolman-Oppenheimer-Volkoff equations for stellar equilibrium, that our equation of state supports stars with masses around 2 $M_\odot$ for values of the magnetic field that are in accordance with those inferred from magnetar data. The inclusion of a weak vector interaction term in the quark part allows one to reach 2 solar masses for relatively small central magnetic fields, making this composition a viable possibility for describing the internal degrees of freedom of this class of astrophysical objects., Added new graphs, references and physical discussions. This version to appear at Phys. Rev. D

An increasing number of model results suggests that chiral symmetry is broken inhomogeneously in a certain window at intermediate densities in the QCD phase diagram. This could have significant effects on the properties of compact stars, possibly leading to new astrophysical signatures. In this contribution we discuss this idea by reviewing recent results on inhomogeneous chiral symmetry breaking under an astrophysics-oriented perspective. After introducing two commonly studied spatial modulations of the chiral condensate, the chiral density wave and the real kink crystal, we focus on their properties and their effect on the equation of state of quark matter. We also describe how these crystalline phases are affected by different elements which are required for a realistic description of a compact star, such as charge neutrality, the presence of magnetic fields, vector interactions and the interplay with color-superconductivity. Finally, we discuss possible signatures of inhomogeneous chiral symmetry breaking in the core of compact stars, considering the cases of mass-radius relations and neutrino emissivity explicitly., Invited contribution to the EPJA Topical Issue "Exotic Matter in Neutron Stars". v2: extended discussion on strange quarks, updated mass-radius section, other small changes; matches published version. 16 pages, 14 figures

The chiral condensate, which is constant in vacuum, may become spatially modulated at moderately high densities where in the traditional picture of the QCD phase diagram a first-order chiral phase transition occurs. We review the current status of this idea, which originally dates back to Migdal's pion condensation, but recently received new momentum through studies on the nature of the chiral critical point and by the conjecture of a quarkyonic-matter phase. We discuss how these nonuniform phases emerge in generalized Ginzburg-Landau analyses as well as in specific calculations, both within effective models and in Dyson-Schwinger or large-$N_c$ approaches to QCD. Questions about the most favored shape of the modulations and its dimension, and about the effects of nonzero isospin chemical potential, strange quarks, color superconductivity, and external magnetic fields on these inhomogeneous phases will be addressed as well., 82 pages, 37 figures, review article submitted to Prog. Part. Nucl. Phys.; v2: accepted version

Inhomogeneous chiral-symmetry breaking phases at non-vanishing chemical potential and temperature are studied within a two-flavor quark-meson model in the chiral limit. The analysis is performed beyond the standard mean-field approximation by taking into account the Dirac-sea contributions of the quarks. Compared with the case where the Dirac sea is neglected, we find that the inhomogeneous phase shrinks, but in general does not disappear. It is shown within a Ginzburg-Landau analysis that the Lifshitz point of the inhomogeneous phase coincides with the tricritical point if the ratio between sigma-meson and constituent quark mass in vacuum is chosen to be $m_\sigma/M = 2$, corresponding to the fixed mass ratio in the Nambu--Jona-Lasinio model. In the present model, however, this ratio can be varied, offering the possibility to separate the two points. This is confirmed by our numerical calculations, which demonstrate a strong sensitivity of the size of the inhomogeneous phase on $m_\sigma$. Finally, we uncover a general instability of the model with respect to large wave numbers of the chiral modulations, which calls for further improvements beyond the present approximation., Comment: 21 pages, 12 figures. v2: extended discussions, to be published in PRD

The existence and the properties of self-bound quark matter in the NJL model at zero temperature are investigated in mean-field approximation, focusing on inhomogeneous structures with one-dimensional spatial modulations. It is found that the most stable homogeneous solutions which have previously been interpreted as schematic quark droplets are unstable against formation of a one-dimensional soliton-antisoliton lattice. The solitons repel each other, so that the minimal energy per quark is realized in the single-soliton limit. The properties of the solitons and their interactions are discussed in detail, and the effect of vector interactions is estimated. The results may be relevant for the dynamics of expanding quark matter., Comment: 21 pages, 6 figures; v2: modified title, discussion of soliton-soliton interactions corrected; version accepted for publication

We investigate the role of the isoscalar vector interaction and the dynamics of the Polyakov loop on inhomogeneous phases in the phase diagram of the two-flavor Nambu-Jona--Lasinio (NJL) model. Thereby we concentrate on phases with a one-dimensional modulation, explicitly domain-wall solitons and, for comparison, the chiral spiral. While the inclusion of the Polyakov loop merely leads to quantitative changes compared to the original NJL model, the presence of a repulsive vector-channel interaction has significant qualitative effects: Whereas for homogeneous phases the first-order phase transition gets weakened and eventually turns into a second-order transition or a cross-over, the domain of inhomogeneous phases is less affected. In particular the location of the Lifshitz point in terms of temperature and density is not modified. Consequently, the critical point disappears from the phase diagram and only a Lifshitz point (showing a different critical behavior) remains. As a result, susceptibilities remain finite., 28 pages, 10 figures

We present some recent developments in our study of inhomogeneous chiral symmetry breaking phases in the Nambu--Jona-Lasinio model. First, we investigate different kinds of one- and two-dimensional spatial modulations of the chiral condensate within the inhomogeneous "island" and compare their free energies. Next, we employ the Polyakov-loop extended version of the model to study the effects of varying the number of colors on the inhomogeneous region. Finally, we discuss the properties of an inhomogeneous "continent" which appears in our model at higher chemical potentials, and analyze its origin.

In the past few years, a growing number of compelling arguments, backed up by model calculations, pointed out that the intermediate-density region of the QCD phase diagram may be characterized by the formation of inhomogeneous condensates which spontaneously break some of the spatial symmetries of the theory. In the following we provide a brief recapitulation of these arguments and describe some recent results in a 3+1-dimensional QCD-inspired NJL model with quark-hole condensation in the form of a plane wave in the scalar and tensor channels. This model exhibits particular features in close analogy to its 1+1-dimensional counterpart, most notably an asymmetric spectral density and the arising of an anomalous contribution to the free energy., Comment: Presented at ICHEP-2014, Valencia, Spain