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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 compute the collisional energy loss of an {energetic} massive fermion crossing a chiral plasma at finite temperature characterized by an imbalance between the populations of left-handed and right-handed fermions. We find a new contribution to the energy loss which is proportional to the helicity of the test fermion and depends on the amount of chiral imbalance in the plasma. We then compute the difference between the energy loss of a fermion with the two opposite helicities, to assess whether this could be used to quantify the chiral imbalance in the plasma. We find that the leading contribution to these helicity-dependent energy loss contributions comes from the exchange of hard photons (or gluons for QCD) with the medium constituents, and in some scenarios can become comparable to the leading-order result for a plasma without any chiral imbalance. We also evaluate the contribution arising from soft photon exchange, which is a subleading effect, and requires regularization. We illustrate how dimensional regularization is a well suited prescription to be applied to these energy loss computations., 23 pages, 1 figure. v2: expanded discussion, typos fixed. matches published version

We present compact expressions for the power corrections to the hard thermal loop (HTL) Lagrangian of QED in $d$ space dimensions. These are corrections of order $(L/T)^2$, valid for momenta $L \ll T$, where $T$ is the temperature. In the limit $d\to 3$ we achieve a consistent regularization of both infrared and ultraviolet divergences, which respects the gauge symmetry of the theory. Dimensional regularization also allows us to witness subtle cancellations of infrared divergences. We also discuss how to generalise our results in the presence of a chemical potential, so as to obtain the power corrections to the hard dense loop (HDL) Lagrangian., 12 pages, no figures

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 compute the damping rate of a fermion propagating in a chiral plasma when there is an imbalance between the densities of left- and right-handed fermions, after generalizing the hard thermal loop resummation techniques for these systems. In the ultradegenerate limit, for very high energies the damping rate of this external fermion approaches a constant value. Closer to the two Fermi surfaces, however, we find that the rate depends on both the energy and the chirality of the fermion, being higher for the predominant chirality. This comes out as a result of its scattering with the particles of the plasma, mediated by the exchange of Landau damped photons. In particular, we find that the chiral imbalance is responsible for a different propagation of the left and right circular polarised transverse modes of the photon, and that a chiral fermion interacts differently with these two transverse modes. We argue that spontaneous radiation of energetic fermions is kinematically forbidden, and discuss the time regime where our computation is valid., Comment: v2: extended version with more detailed analysis on dispersion relations and revised discussion on chiral plasma instability. Title modified. Version accepted for publication. v3: added missing piece in Eqs. (23)-(24), results and conclusions unchanged. 14 pages, 3 figures

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 formulate the on-shell effective field theory (OSEFT) in an arbitrary frame and study its reparametrization invariance, which ensures that it respects Lorentz symmetry. In this formulation, the OSEFT Lagrangian looks formally equivalent to the sum over lightlike velocities of soft collinear effective field theory in the Abelian limit, but differences remain in the scale of the gauge fields involved in the two effective theories. We then use the OSEFT Lagrangian expanded in inverse powers of the on-shell energy to derive how the classical transport equations for charged massless fermions are corrected by quantum effects, as derived from quantum field theory. We provide a formulation in a full covariant way and explain how the consistent form of the chiral anomaly equation can be recovered from our results. We also show how the side-jump transformation of the distribution function associated with massless charged fermions can be derived from the reparametrization invariance transformation rules of the OSEFT quantum fields. Finally, we discuss the differences in our results with respect to others found in the literature.

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 develop an innovative technique for studying inhomogeneous phases with a spontaneous broken symmetry. The method relies on the knowledge of the exact form of the free energy in the homogeneous phase and on a specific gradient expansion of the order parameter. We apply this method to quark matter at vanishing temperature and large chemical potential, which is expected to be relevant for astrophysical considerations. The method is remarkably reliable and fast as compared to performing the full numerical diagonalization of the quark Hamiltonian in momentum space, and is designed to improve the standard Ginzburg-Landau expansion close to the phase transition points. For definiteness we focus on inhomogeneous chiral symmetry breaking, accurately reproducing known results for 1D and 2D modulations and examining novel crystalline structures, as well. Consistently with previous results, we find that the energetically favored modulation is the so-called 1D real kink crystal. We propose a qualitative description of the pairing mechanism to motivate this result., 11 pages, 8 figures. v2 with minor changes. Accepted for publication on Physical Review D

I discuss some properties of spatially dependent chiral condensates, focusing on one-dimensional modulations. After briefly introducing a generic formalism for studying inhomogeneous solutions in dense quark matter, I consider a plane-wave and a sinusoidal ansatz, and discuss the relationship between their order parameters and free energies, as well as a comparison of their dispersion relations. I then introduce a real kink crystal, and discuss its relationship with the sinusoidal ansatz by performing a Fourier analysis of this kind of solution., Comment: Contribution to the proceedings of TNPI2016. 7 pages, 4 figures

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 study the thermodynamic properties of matter at vanishing temperature for non-extreme values of the isospin chemical potential and of the strange quark chemical potential. From the leading order pressure obtained by maximizing the static chiral Lagrangian density we derive a simple expression for the equation of state in the pion condensed phase and in the kaon condensed phase. We find an analytical expression for the maximum of the ratio between the energy density and the Stefan-Boltzmann energy density as well as for the isospin chemical potential at the peak both in good agreement with lattice simulations of quantum chromodynamics. We speculate on the location of the crossover from the Bose-Einstein condensate state to the Bardeen-Cooper-Schrieffer state by a simple analysis of the thermodynamic properties of the system. For $\mu_I \gtrsim 2 m_\pi$ the leading order chiral perturbation theory breaks down; as an example it underestimates the energy density of the system and leads to a wrong asymptotic behavior., Comment: 5 pages, 3 figures, minor changes. Accepted version as Rapid Communication on PRD

When the isospin chemical potential exceeds the pion mass, charged pions condense in the zero-momentum state forming a superfluid. Chiral perturbation theory provides a very powerful tool for studying this phase. However, the formalism that is usually employed in this context does not clarify various aspects of the condensation mechanism and makes the identification of the soft modes problematic. We re-examine the pion condensed phase using different approaches within the chiral perturbation theory framework. As a first step, we perform a low-density expansion of the chiral Lagrangian valid close to the onset of the Bose-Einstein condensation. We obtain an effective theory that can be mapped to a Gross-Pitaevskii Lagrangian in which, remarkably, all the coefficients depend on the isospin chemical potential. The low-density expansion becomes unreliable deep in the pion condensed phase. For this reason, we develop an alternative field expansion deriving a low-energy Lagrangian analog to that of quantum magnets. By integrating out the "radial" fluctuations we obtain a soft Lagrangian in terms of the Nambu-Goldstone bosons arising from the breaking of the pion number symmetry. Finally, we test the robustness of the second-order transition between the normal and the pion condensed phase when next-to-leading-order chiral corrections are included. We determine the range of parameters for turning the second-order phase transition into a first-order one, finding that the currently accepted values of these corrections are unlikely to change the order of the phase transition., Comment: 12 pages, 2 figures; improved Sec.4; added Appendix B

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

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

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