New class of compact stars: Pion stars

We investigate the viability of a new type of compact star whose main constituent is a Bose-Einstein condensate of charged pions. Several different setups are considered, where a gas of charged leptons and neutrinos is also present. The pionic equation of state is obtained from lattice QCD simulations in the presence of an isospin chemical potential, and requires no modeling of the nuclear force. The gravitationally bound configurations of these systems are found by solving the Tolman-Oppenheimer-Volkov equations. We discuss weak decays within the pion condensed phase and elaborate on the generation mechanism of such objects.

potential is sufficiently small so that heavier charged hadrons are not excited. The strongest constraint is given by µ I < m K /2 ≈ 1.8 m π , where m K is the kaon mass, and is fulfilled in the following calculations.
The charge density n Q and the pion condensate σ π are obtained as expectation values involving the Euclidean path integral over the gluon and quark fields discretized on a space-time lattice. The positivity of the measure in the path integral 5 ensures that standard importance sampling methods are applicable. Since the spontaneous symmetry breaking associated to pion condensation does not occur in a finite volume, the simulations are performed by introducing a pionic source parameter λ that breaks the symmetry explicitly. 12 Physical results are obtained by extrapolating this auxiliary parameter to zero as discussed in Ref. 13 The details of our lattice setup are described in Methods.
The dependence of n Q on the isospin chemical potential is shown in the left panel of Fig. 1, clearly reflecting the phase transition to the pion condensed phase at µ I = m π /2. Due to effects from the finite volume and the small but nonzero temperature employed in our simulations, the density below µ I = m π /2 is not exactly zero. To approach the thermodynamic and T = 0 limits consistently, we employ χPT. In particular, we set the density to zero below m π /2 and fit the lattice data to the form predicted by χPT around the critical chemical potential (see Methods). Matching this fit to a spline interpolation of the lattice results at higher isospin chemical potentials gives the continuous curve shown in the left panel of Fig. 1. The resulting n Q (µ I ) curve is used to calculate the EoS, as shown in the right panel of Fig. 1. π + e π + µ π Figure 1: Left panel: Phase transition between the vacuum and the pion condensed phase, as exhibited by the charge density. The lattice data are fitted using χPT (yellow curve) and matched to a spline interpolation (blue curve). The error bars indicate the standard error of the mean of n Q . Right panel: Equation of state in the pion condensed phase in the QCD sector and for the electrically neutral systems also including leptons (either muons or electrons). The width of the curves represents the standard error of the mean and incorporates statistical uncertainties as well as the uncertainty in the lattice pion mass for the pion-lepton systems.
In the vacuum phase, charged pions decay weakly into leptons, with a characteristic lifetime of τ W ≈ 10 −8 s. However, the condensed phase carries nonzero electric charge that is stable against weak decays. To see this, consider the following argument: since the spontaneously broken symmetry group is part of the local gauge group of electromagnetism, the pion condensed phase is a superconductor, where the electric charge eigenstates π + and π − mix with each other. One linear combination ( π 1 ) plays the role of the Goldstone mode, while the other one ( π 2 ) is heavy, fulfilling m π2 > m π . 5 In the presence of dynamical photons the Goldstone mode disappears via the Higgs mechanism, 14 at the cost of a nonzero photon mass m γ ∝ e|σ π |. Thus, only π 2 can decay via weak interactions. However, if the temperature is sufficiently low (i.e. T m π2 ), this mode is not excited and no weak decay of electrically charged states can occur. An obvious analogue to this situation is the spontaneous symmetry breaking and the associated Higgs mechanism in the electroweak sector of the Standard Model. Below the electroweak scale the Higgs condensate is stable and carries nonzero weak hypercharge. The Higgs boson and the gauge bosons decay weakly but are irrelevant for low-temperature physics due to their large masses.
To ensure stability under electromagnetism, the pion condensate is neutralized by a gas of leptons. In the present approach we consider the leptons to be free relativistic particles. A systematic improvement over this assumption is possible by taking into account O(e 2 ) electromagnetic effects perturbatively, both in the electroweak sector and in lattice QCD simulations. The lepton density n l is controlled by a lepton chemical potential µ l , from which the leptonic contribution to the pressure p l and to the energy density l can be obtained, similarly to the QCD sector. We require local charge neutrality to hold, n Q + n l = 0, which uniquely determines the lepton chemical potential in terms of µ I (see Methods). We mention that a similar construction, assuming a first-order phase transition for pions, was discussed in Ref. 15 Using the resulting EoS, the mass M and radius R of pion stars can be computed by solving the Tolman-Oppenheimer-Volkov (TOV) equations, 16,17 which describe hydrostatic equilibrium in general relativity, assuming spherical symmetry. Further stability analyses are performed by requiring the star to be robust against density perturbations 18 and radial oscillations. 19 Our main result is the mass-radius relation for gravitationally stable pion stars, composed of a pion condensate and electrons, see the left panel of Fig. 2. The results for an electrically charged pure pion star (our preliminary results for this case were presented in Ref. 20 ) are also included in the figure. In addition, we considered a pion-muon system (muons decay weakly into electrons) and a mixture of electrons and muons in chemical equilibrium by setting their respective chemical potentials equal. We observed that the gravitationally stable configurations for the latter setup cannot maintain a muonic component and are thus identical to those for the pion-electron system. Fig. 2 reflects the R ∼ constant behavior for pure pion stars (with masses below 7 M ) -a telltale sign for an interaction-dominated EoS. The slope changes by the addition of leptons, scaling as M R 3 ∼ constant, similarly to stars made of fermions. Moreover, one can see that pion stars are considerably heavier and larger when compared to other branches of compact stars, attaining masses and radii up to M ∼ 250 M and R 30 000 km. This comparison is more strikingly illustrated in the right panel of Fig. 2, where neutron stars and white dwarfs, together with regular stars, are also included. 21  We note that about 10 −4 s after the Big Bang, the Universe consisted mainly of a pion gas with electrons, muons and neutrinos in weak equilibrium. It is therefore conceivable that a seed of charged pion condensed matter, neutralized by leptons, could have been created at that time. Assuming that a pion star can indeed be formed, we now consider its stability in free (cold) space. While the Bose-Einstein condensate in the bulk of the star behaves coherently and does not decay, the situation is different near the surface of the star, where n Q approaches zero and temperature effects become important. Provided that in this dilute environment, pions are well described by a free complex scalar field theory, the critical temperature for condensation can be found semianalytically to be T c (n Q ) ∼ 2π/m π [n Q /ζ(3/2)] 2/3 . Thus, for cosmic temperatures around T ≈ 2.7 K, the pion star boundary is marked by a critical charge density n Q ≈ 10 −19 fm −3 , corresponding to an energy density c . The nonzero pressure in this region results in an outward flow of pions subject to weak decays and a subsequent rearrangement of the star matter characterized by the speed of sound v s . The latter follows from the EoS as v 2 s = ∂p/∂ , giving v s ≈ 5 km/s at the star boundary. Assuming that the corresponding mass loss proceeds along the mass-radius relation M (R), it is described by the differential equation dM/dt = −4πR 2 (M )v s c . This results in typical lifetimes of the order of a million years for the heaviest stars. The recombination of pions, together with the capture and conversion of further charged particles into the condensate counteracts this process, making the above estimate a lower bound for the lifetime. Moreover, we stress that through v s the mass loss is very sensitive to the precise EoS for low density and might be subject to large corrections from perturbative QED effects that we neglected here. 25. Endrődi, G., "Magnetic structure of isospin-asymmetric QCD matter in neutron stars," Phys.

Lattice setup
In the presence of the isospin chemical potential µ I and the auxiliary pionic source parameter λ, the fermion matrices for the light and strange quarks read where / D is the Dirac operator, τ a denote the Pauli matrices acting in flavor space and m ud and m s are the light and strange quark masses, respectively. We discretize M ud and M s using the rooted staggered formulation, so that the Euclidean path integral over the gauge field A µ becomes where S g is the gluonic part of the QCD action, for which we use the tree level-improved Symanzik discretization.
For the fermion matrices we employ stout smearing of the gauge fields. The determinants of M ud and of M s are positive, 5 allowing for a probabilistic interpretation and standard Monte-Carlo algorithms. The quark masses are tuned to their physical values so that, in particular, m π ≈ 135 MeV in the vacuum. The error of the pion mass used in the simulations originates primarily from the uncertainty of the lattice scale and amounts to 2%. Further details of our lattice action and our simulation algorithm are given in Refs. [23][24][25] The isospin density and the pion condensate are obtained as derivatives of the partition function, where V 4 = V /T is the four-dimensional volume of the system that includes the spatial volume V = (N s a) 3 and the temperature T = (N t a) −1 in terms of the lattice spacing a and the lattice geometry N 3 s × N t . The extrapolation of the isospin density to λ = 0 is performed using the singular value representation of the massive Dirac operator. 13 Here we perform simulations on a 24 3 × 32 lattice ensemble with a lattice spacing of a ≈ 0.29 fm, a wide range of chemical potentials 0 < µ I /m π ≤ 1 and three pionic source parameters 0.17 ≤ λ/m ud ≤ 0.88. The systematic uncertainties originating from lattice artefacts and from neglecting O(e 2 ) electromagnetic effects will be investigated in a future publication. The volume of our system is around 7 fm 3 , sufficiently large so that finite size effects are under control. The temperature is well below the relevant QCD scales so that it well approximates T = 0.

Equation of state and the TOV equations
In χPT the isospin density reads, 5 where f π is the chiral limit of the pion decay constant, which is the only free parameter for the χPT fit depicted in the left panel of Fig. 1. For free relativistic leptons, the density is where m l is the lepton mass. The pionic pressure and energy density is calculated from n I (µ I ) at zero temperature via and very similarly for the leptons, using n l (µ l ). The impact of a temperature of T ≈ 2.7 K on the pionic and leptonic EoS is of the order of e −mπ/T and e −m l /T , respectively and can be safely neglected. After requiring local charge neutrality n l = n Q = n I /2, the pion-lepton system is unambiguously characterized by the lepton chemical potential µ l . The total pressure p and energy density enter the TOV equations, 16,17 which can be rewritten in terms of the chemical potentials as dµ l dr = −Gµ l M + 4πr 3 p r 2 − 2rGM 1 + 2 µ I µ l 1 + 4 n l n I where G is Newton's constant, the primes denote derivatives with respect to the corresponding chemical potentials and we used natural units c = = 1. Eq. (7) is integrated numerically up to the star boundary r = R, where the pressure vanishes and the total mass M = M (R) is attained. The points of the mass-radius curves in Fig. 2 correspond to different values of the central chemical potential. The data that support the findings of this study are available from the authors on request.