The fate of pion condensation in quark matter: from the chiral limit to the physical pion mass
H. Abuki, R. Anglani, R. Gatto, M. Pellicoro, M. Ruggieri
备注：
6 pages, 4 figures, RevTex4, (v2) title changed, Fig. 1 replaced, fourth section enlarged, references added, version accepted for publication in Phys. Rev. D, (v3) typos corrected, published version
We study aspects of the pion condensation in two-flavor neutral quark matter using the Nambu--Jona-Lasinio model of QCD at finite density. We investigate the role of electric charge neutrality, and explicit symmetry breaking via quark mass, both of which control the onset of the charged pion $(π^c)$ condensation. We show that the equality between the electric chemical potential and the in-medium pion mass, $μ_{e}=M_{π^-}$, as a threshold, persists even for a composite pion system in the medium, provided the transition to the pion condensed phase is of the second order. Moreover, we find that the pion condensate in neutral quark matter is extremely fragile to the symmetry breaking effect via a current quark mass $m$, and is ruled out for $m$ larger than the order of 10 keV.
The formation of charged pion condensate under parallel electromagnetic fields is studied within the two-flavor Nambu--Jona-Lasinio model. The technique of Schwinger proper time method is extended to explore the quantity locating in the off-diagonal flavor space, i.e., charged pion. We obtain the associated effective potential as a function of the strength of the electromagnetic fields and find out that it contains a sextic term which possibly induce weakly first order phase transition. Dependence of pion condensation on model parameters is investigated.
The phase diagram of dense quark matter with chiral imbalance is considered with the conditions of electric neutrality and $β$-equilibrium. It has been shown recently that chiral imbalance can generate charged pion condensation in dense quark matter, so it was interesting to verify that this phenomenon takes place in realistic physical scenarios such as electrically neutral matter in $β$-equilibrium, because a window of pion condensation at dense quark matter phase diagram (without chiral imbalance) predicted earlier was closed by the consideration of these conditions at the physical current q…
The indications of a possible pion condensation at the LHC are summarized. The condensation is predicted by the non-equilibrium hadronization model for 2.76 TeV Pb+Pb collisions at the LHC. The model solves the proton/pion puzzle and reproduces the low $p_T$ enhancement of the pion spectra, as well as the spectra of protons and antiprotons, charged kaons, $K^0_S$, $K^*(892)^0$ and $ϕ(1020)$. The obtained parameters allow to estimate the amount of pion condensate on the level of 5\% from the total number of pions at the LHC. The condensate is located at $p_T<250$ MeV.
The Bose-Einstein condensation and the liquid-gas first order phase transition are studied in the interacting pion matter. Two phenomenological models are used: the mean-field model and the hybrid model. Free model parameters are fixed by fitting the lattice QCD data on the pion Bose condensate density at zero temperature. In spite of some minor differences the two model demonstrate an identical qualitative and very close quantitative behavior for the thermodynamic functions and electric charge fluctuations. A peculiar property of the considered models is an intersection of the Bose-Einstein c…
S-wave pion-nucleon interactions in the linear sigma model, and in Manohar-Georgi and Gasser-Sainio-Svarc models with finite number of terms in Lagrangians, as well as in a general phenomenological approach are reviewed. Subtleties associated with the current algebra theorems and field redefinitions are discussed. In the first and third models most likely the s-wave pion condensation in the isospin-symmetric matter does not occur at least up to high densities, whereas within the second model it may appear already at moderate densities. In the phenomenological approach two parameterizations of …
We analyze the phase diagram of a three-flavor Nambu-Jona-Lasinio model at finite temperature $T$ and chemical potentials $μ_u, μ_d, μ_s$. We study the competition of pion and kaon condensation and we propose a physical situation in which kaon condensation could be led only by light quark finite densities.
We analyse in detail the possibility of Bose-Einstein condensation of pions produced in heavy-ion collisions at the beam energy $\sqrt{s_{\rm NN}}$ = 2.76 TeV. Our approach is based on the chemical non-equilibrium thermal model of hadron production which has been generalised to include separately the contribution from the local zero-momentum state. In order to study both the hadronic multiplicities and the transverse-momentum spectra, we use the Cracow freeze-out model which parameterises the flow and space-time geometry of the system at freeze-out in a very economic way. Our analysis indicate…
We investigate the possibility of charged pion condensation in the presence of parallel rotation and magnetic field within the Nambu--Jona-Lasinio model with quarks as the fundamental degrees of freedom. Previous study based on non-interacting Klein-Gordon theory for pions showed that the charged pions will undergo Bose-Einstein condensation under this circumstance [Y. Liu and I. Zahed, Phys. Rev. Lett. ${\bf 120}$, 032001 (2018)]. In this work, we take into account the internal quark structures of charged pions self-consistently through quark polarization loops in an interacting theory, i.e.,…
The behavior of pions is studied in systems where their normal leptonic decay is forbidden. When thermal fluctuations are present, a low decay rate is generated, and as a consequence of lepton recombination, the amount of pions remains almost unaltered. Compact stars conditions are favorable for the formation of such intermediate state of charged pions: near condensation and almost stable, leading to a continuum source of anti-neutrinos. In particular, protoneutron stars could be an scenario where this state of matter is relevant.
Particle number fluctuations are studied in the ideal pion gas approaching Bose-Einstein condensation. Two different cases are considered: Bose condensation of pions at large charge densities $ρ_Q$ and Bose condensation at large total densities of pions $ρ_π$. Calculations are done in grand canonical, canonical and microcanonical ensembles. At high collision energy, in the samples of events with a fixed number of all pions, $N_π$, one may observe a prominent signal. When $N_π$ increases the scaled variances for particle number fluctuations of both neutral and charged pions increase dramaticall…
We present microcanonical ensemble calculations of particle number fluctuations in the ideal pion gas approaching Bose-Einstein condensation. In the samples of events with a fixed number of all pions, $N_π$, one may observe a prominent signal. When $N_π$ increases the scaled variances for particle number fluctuations of both neutral and charged pions increase dramatically in the vicinity of the Bose-Einstein condensation line. As an example, the estimates are presented for $p+p$ collisions at the beam energy of 70 GeV.
Using a simple kinetic model we study whether a pion condensate can be formed during the hadronization of the quark-gluon plasma in ultrarelativistic heavy-ion collisions. If hadronization proceeds fast and the pion effective mass is close to zero, some pions condense into the zero momentum state. On the other hand, for the pion effective mass equal to the vacuum one or larger no onset of the Bose-Einstein condensation is observed. The constraints on the condensation process coming from the requirement of the entropy increase are discussed in detail.
We analyze various aspects of pion and kaon condensation in the framework of chiral perturbation theory. Considering a system at vanishing temperature and varying the isospin chemical potential and the strange quark chemical potential we reproduce known results about the phase transition to the pion condensation phase and to the kaon condensation phase. However, we obtain mesonic mixings and masses in the condensed phases that are in disagreement with the results reported in previous works. Our findings are obtained both by a theory group analysis and by direct calculation by means of the same…
We studied thermal evolution of isolated neutron stars (NSs) including the pion condensation core, with an emphasis on the stiffness of equation of state (EOS). Many temperature observations can be explained by the minimal cooling scenario which excludes the fast neutrino cooling process. However, several NSs are cold enough to require it. The most crucial problem for NS cooling theory is whether the nucleon direct Urca (DU) process is open. The DU process is forbidden if the nucleon symmetry energy is significantly low. Hence, another fast cooling process is required in such an EOS. As the ca…
Dilepton Production as a Probe of Pion Condensation in Hot and Dense QCD Matter
University of Wrocław（弗罗茨瓦夫大学）；West University of Timişoara（蒂米什瓦拉西大学）；Aliah University（阿利亚大学）；Polish Academy of Sciences PAN（波兰科学院）；Hiroshima University（广岛大学）
We show that the combined effects of a rotation plus a magnetic field can cause charged pion condensation. We suggest that this phenomenon may yield to observable effects in current heavy ion collisions at collider energies, where large magnetism and rotations are expected in off-central collisions.
The phase structure of the Nambu -- Jona-Lasinio model at zero temperature and in the presence of baryon- and isospin chemical potentials is investigated. It is shown that in the chiral limit and for a wide range of model parameters there exist two different phases with pion condensation. In the first, ordinary phase, quarks are gapped particles. In the second, gapless pion condensation phase, there is no energy cost for creating only $u$- or both $u$ and $d$ quarks, and the density of baryons is nonzero.
We revisit the suggestion that charged rho-mesons undergo Bose-Einstein condensation in isospin-rich nuclear matter. Using a simple version of the Nambu-Jona-Lasinio (NJL) model, we conclude that rho-meson condensation is either avoided or postponed to isospin chemical potentials much higher than the rho-meson mass as a consequence of the repulsive interaction with the preformed pion condensate. In order to support our numerical results, we work out a linear sigma model for pions and rho-mesons, showing that the two models lead to similar patterns of medium dependence of meson masses. As a byp…
Pion condensation is studied at one--loop level and nonzero baryochemical potential in the framework of two flavor constituent quark model using the one--loop level optimized perturbation theory for the resummation of the perturbative series. A Landau type of analysis is presented for the investigation of the phase boundary between the pion condensed/non-condensed phases. The statement that the condensation starts at $\muI = m_π$ is slightly modified by one--loop corrections. The second order critical surface is determined and analysed in the $\muI-\muB-T$ space. The $\muI$ dependence of the o…
QCD at finite isospin chemical potential mu_I is studied. This theory has no fermion sign problem and can be simulated on the lattice using present-day techniques. We solve this theory analytically in two limits: low mu_I where chiral perturbation theory is applicable, and asymptotically high mu_I where perturbative QCD is at work. At low isospin density the ground state is a superfluid pion condensate. At very high density it is a Fermi liquid with Cooper pairing. The pairs carry the same quantum numbers as the pions. Motivated by this observation, we put forward a conjecture that the transit…
We show that the superconducting 2SC phase at high density and normal chiraly broken quark phase at low density is separated by the mixed non-uniform phase along the baryon density line.
The general features of quark and pion condensation in dense quark matter with flavor asymmetry have been considered at finite temperature in the presence of a chromomagnetic background field modelling the gluon condensate. In particular, pion condensation in the case of a constant abelian chromomagnetic field and zero temperature has been studied both analytically and numerically. Under the influence of the chromomagnetic background field the effective potential of the system is found to have a global minimum for a finite pion condensate even for small values of the effective quark coupling c…
The possibility of the pion condensation phenomenon in cold and electrically neutral dense baryonic matter is investigated in $β$-equilibrium. For simplicity, the consideration is performed in the framework of a NJL model with two quark flavors at zero current quark mass and for rather small values of the baryon chemical potential, where the diquark condensation might be ignored. Two sets of model parameters are used. For the first one, the pion condensed phase with finite baryon density is realized. In this phase both electrons and the pion condensate take part in the neutralization of the qu…
We have studied the phenomenology of pion condensation in 2-flavor neutral quark matter at finite density with Nambu--Jona-Lasinio (NJL) model of QCD. We have discussed the role of the bare quark mass $m$ and the electric chemical potential $μ_{e}$ in controlling the condensation. The central result of this work is that the onset for $π$-condensed phase occurs when $|μ_{e}|$ reaches the value of the in-medium pion mass $M_π$ provided the transition is of the second order, even for a composite pion system in the medium. Finally, we have shown that the condensation is extremely fragile with resp…
Recently it was pointed out that coherent or condensated states of pions may account for the explanation of the Centauro events observed in cosmic ray showers. We argue that an occurrence of condensed pions requires that the system evolves far out of thermal equilibrium. Besides an unusual charge ratio distribution we show that such a produced state also would lead to strong isospin correlations.
The density of the Bose-Einstein condensate for non-interacting pions in a cubic box, at given temperature and (average) total pion density, is calculaated for three sets of boundary conditions. The densities are much larger than predicted from thermodynamics and depend significantly on the choice of the boundary conditions, even for volumes as large as $6.4\cdot 10$fm$^3$.
In this work pion and kaon condensation in the framework of chiral perturbation theory is studied. I consider a system at vanishing temperature with nonzero isospin chemical potential and strangeness chemical potential; meson masses and mixing in the normal phase, the pion condensation phase and the kaon condensation phase are described. There are differences with previous works, but the results presented here are supported by both theory group analysis and by direct calculations. Some pion decay channels in the normal and the pion condensation phases are studied, finding a nonmonotonic behavi…
We study the QCD vacuum structure under the influence of an electromagnetic field with a nonzero second Lorentz invariant $I_2=\vec{E}\cdot{\vec B}$. We show that the presence of $I_2$ can induce neutral pion ($π^0$) condensation in the QCD vacuum through the electromagnetic triangle anomaly. Within the frameworks of chiral perturbation theory at leading small-momenta expansion as well as the Nambu--Jona-Lasinio model at leading $1/N_c$ expansion, we quantify the dependence of the $π^0$ condensate on $I_2$. The stability of the $π^0$-condensed vacuum against the Schwinger charged pair producti…
We study pion condensation at zero temperature in a hard-wall holographic model of hadrons with isospin chemical potential. We find that the transition from the hadronic phase to the pion condensate phase is first order except in a certain limit of model parameters. Our analysis suggests that immediately across the phase boundary the condensate acts as a stiff medium approaching the Zel'dovich limit of equal energy density and pressure.
The influence of the HBT effect on the momentum spectra of independently produced pions is studied using the method developed earlier for discussion of multiplicity distributions. It is shown that in this case all the spectra and multiparticle correlation functions are expressible in terms of one function of two momenta. It is also shown that at the critical point all pions are attracted into one quantum state and thus form a Bose-Einstein condensate.
In this paper, we study O(2N)-symmetric $ϕ^4$-theory at finite temperature and density using the 2PI-1/N expansion. As specific examples, we consider pion condensation at finite isospin chemical potential and kaon condensation at finite chemical potential for hyper charge and isospin charge. We calculate the phase diagrams and the quasiparticle masses for pions and kaons in the large-N limit. It is shown that the effective potential and the gap equation can be renormalized by using local counterterms for the coupling constant and mass parameter, which are independent of temperature and chemica…
After the first predictions in seventies that in dense nuclear matter, for example, in neutron stars there could be pion condensation phenomenon, it was shown that at least s-wave pion condensation is unlikely to occur in such a medium in different approaches including NJL model consideration with neutrality and $β$-equilibrium condition. Then it has been found a condition that can promote this phenomenon in dense baryonic matter, it is chiral imbalance, so now it is interesting if this generation can survive rather strict requirement of electric neutrality and $β$-equilibrium. Here in this pa…
We argue that using an equilibrated gas of neutrinos it is possible to probe the phase diagram of QCD for finite isospin and small baryon chemical potentials. We discuss this region of the phase diagram in detail and demonstrate that for large enough neutrino densities a Bose-Einstein condensate of positively charged pions arises. Moreover, we show that for nonzero neutrino density the degeneracy in the lifetimes and masses of the charged pions is lifted.
We study the asymmetric nuclear matter using a holographic QCD model by introducing a baryonic charge in the infrared boundary. We first show that, in the normal hadron phase, the predicted values of the symmetry energy and it's slope parameter are comparable with the empirical values. We find that the phase transition from the normal phase to the pion condensation phase is delayed compared with the pure mesonic matter: The critical chemical potential is larger than the pion mass which is obtained for the pure mesonic matter. We also show that, in the pion condensation phase, the pion contribu…
We consider the Bose-Einstein condensation (BEC) in a relativistic pion gas. The thermodynamic limit when the system volume $V$ goes to infinity as well as the role of finite size effects are studied. At $V\to \infty$ the scaled variance for particle number fluctuations, $ω=< ΔN^2>/< N>$, converges to finite values in the normal phase above the BEC temperature, $T>T_C$. It diverges as $ω\propto V^{1/3}$ at the BEC line $T=T_C$, and $ω\propto V$ at $T<T_C$ in a phase with the BE condensate. Possible experimental signals of the pion BEC in finite systems created in high energy proton-proton coll…
We study the influence of the isospin asymmetry on the phase structure of strongly interacting quark matter near the critical point (CP) using a Ginzburg-Landau approach. The effect is found to be drastic, not only bringing about the shift of the location of the CP, but resulting in a rich phase structure in the vicinity of the CP. In particular, new tricritical and triple points emerge as soon as the isospin density becomes finite. Moreover, we find the CP being washed out from the phase diagram due to the stabilization of a homogeneous charged pion condensate when the isospin chemical potent…
Finite isospin chemical potential $μ_I$ and temperature $T$ have been introduced in the framework of soft-wall AdS/QCD model. By self-consistently solve the equation of motion, we obtain the phase boundary of pion condensation phase, across which the system undergoes a phase transition between pion condensation phase and normal phase. Comparing the free energy of solutions with and without pion condensation, we find that the phase transition is of first order type both at large $μ_I$ and small $μ_I$. Qualitatively, the behavior at large $μ_I$ is in agreement with the lattice simulation in Phys…
We use the Polyakov-loop extended two-flavor quark-meson model as a low-energy effective model for QCD to study the phase diagram in the $μ_I$--$T$ plane where $μ_I$ is the isospin chemical potential. In particular, we focus on the Bose condensation of charged pions. At $T=0$, the onset of pion condensation is at $μ_I={1\over2}m_π$ in accordance with exact results. The phase transition to a Bose-condensed phase is of second order for all values of $μ_I$ and in the $O(2)$ universality class. The chiral critical line joins the critical line for pion condensation at a point whose position depends…
The properties of the two-flavored Gross-Neveu model in the (1+1)-dimensional spacetime are investigated in the presence of the isospin $μ_I$ as well as quark number $μ$ chemical potentials both at zero and nonzero temperatures. The consideration is performed in the limit $N_c\to\infty$, i.e. in the case with an infinite number of colored quarks. In the plane of parameters $μ_I,μ$ a rather rich phase structure is found, which contains phases with and without pion condensation. Moreover, we have found a great variety of one-quark excitations of these phases, including gapless and gapped quasipa…
Signatures from pion condensation and lepton flavor asymmetries in the cosmological gravitational wave background
Instituto de Física, Universidade de São Paulo（圣保罗大学物理研究所）；Instituto de Física, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学物理研究所）；Institut für Theoretische Physik, J. W. Goethe Universität（歌德大学理论物理研究所）
We study pion condensation and the phase structure in a two-flavor Nambu-Jona-Lasinio model in the presence of baryon chemical potential $μ$ and isospin chemical potential $μ_I$at zero and finite temperature. There is a competition between the chiral condensate and a Bose-Einstein condensate of charged pions. In the chiral limit, the chiral condensate vanishes for any finite value of the isospin chemical potential, while there is a charged pion condensate that depends on the chemical potentials and the temperature. At the physical point, the chiral condensate is always nonzero, while the charg…
The quark-meson model is often used as an effective low-energy model for QCD to study the chiral transition at finite temperature $T$, baryon chemical potential $μ_B$, and isospin chemical potential $μ_I$. The parameters of the model are determined by matching the meson and quark masses, as well as the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. In this paper, we study the possibility of different phases at zero temperature. In particular, we investigate the competition between an inhomogeneous chiral condensate and a pion condensat…
We use the Polyakov-loop extended two-flavor quark-meson model as a low-energy effective model for QCD to study 1) the possibility of inhomogeneous chiral condensates and its competition with a homogeneous pion condensate in the $μ$--$μ_I$ plane at $T=0$ and 2) the phase diagram in the $μ_I$--$T$ plane. In the $μ$--$μ_I$ plane, we find that an inhomogeneous chiral condensate only exists for pion masses lower that 37.1 MeV and does not coexist with a homogeneous pion condensate. In the $μ_I$--$T$ plane, we find that the phase transition to a Bose-condensed phase is of second order for all value…
We investigate some aspects of 2-flavor QCD with $m_u\not= m_d$ at low-energy, using the leading order chiral perturbation theory including anomaly effects. While nothing special happens at $m_u=0$ for the fixed $m_d\not=0$, the neutral pion mass becomes zero at two critical values of $m_u$, between which the neutral pion field condenses, leading to a spontaneously CP broken phase, the so-called Dashen phase. We also show that the "topological susceptibility" in the ChPT diverges at these two critical points. We briefly discuss a possibility that $m_u=0$ can be defined by the vanishing the "to…
In the framework of an extended Nambu--Jona-Lasinio model we are studying pion condensation in quark matter with an asymmetric isospin composition in a gravitational field of the static Einstein universe at finite temperature and chemical potential. This particular choice of the gravitational field configuration enables us to investigate phase transitions of the system with exact consideration of the role of this field in the formation of quark and pion condensates and to point out its influence on the phase portraits. We demonstrate the effect of oscillations of the thermodynamic quantities a…
The phase structure of dense quark matter in the two color case has been investigated with nonzero baryon $μ_B$, isospin $μ_I$ and chiral isospin $μ_{I5}$ chemical potentials. %in the framework of Nambu--Jona-Lasinio model with quark-antiquark and quark-quark interaction channels. It has been shown in the mean-field approximation that there exist three dualities, one of them between phases with spontaneous chiral symmetry breaking and condensation of charged pions, found in the three color case. The other two dual symmetries between the phase with condensation of charged pions and the phase wi…
We derive the pseudoscalar condensate induced by anomaly and vorticity from the Wigner function for massive fermions in homogeneous electromagnetic fields. It has an anomaly term and a force-vorticity coupling term. As a mass effect, the pseudoscalar condensate is linearly proportional to the fermion mass in small mass expansion. By a generalization to two-flavor and three-flavor cases, the neutral pion and eta meson condensates are calculated from the Wigner function and have anomaly parts as well as force-vorticity parts, in which the anomaly part of the neutral pion condensate is consistent…
Pion stars consisting of Bose-Einstein condensed charged pions have recently been proposed as a new class of compact stars. We use the two-particle irreducible effective action to leading order in the $1/N$-expansion to describe charged and neutrals pions as well as the sigma particle. Tuning the parameters in the Lagrangian correctly, the onset of Bose-Einstein condesation of charged pions is exactly at $μ_I=m_π$, where $μ_I$ is the isospin chemical potential. We calculate the pressure, energy density, and equation of state, which are used as input to the Tolman-Oppenheimer-Volkov equations. …
We summarize main points of our recent work where we studied the pion condensation for non-zero isospin chemical potential within a holographic QCD model. We confirmed that the second order phase transition with the mean field exponent occurs when the isospin chemical potential $μ_I$ exceed the pion mass, which is consistent with the result obtained by the chiral effective Lagrangian at $O(p^2)$. We find that a deviation for large $μ_I$ region can be understood as a higher order effects in the chiral effective Lagrangian. Our result shows that the chiral condensate defined by $\tildeσ \equiv \…
The properties of two-flavored massive Nambu--Jona-Lasinio model in (1+1)-dimensional spacetime are investigated in the presence of isospin and quark number chemical potentials. The consideration is performed in the large-$N_c$ limit, where $N_c$ is the number of colored quarks. It is shown in the framework of this model that charged pion condensation phenomenon of dense quark/hadron isotopically asymmetric matter is rather a spatially inhomogeneous than a homogeneous one.
