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Metastable Pions in Dense Media

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Metastable Pions in Dense Media Metastable pions in dense media Marcelo Loewe1;2;3, Alfredo Raya4, Cristi´anVillavicencio5 1Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago 22, Chile 2Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa 3Centro Cient´ıfico Tecnol´ogico de Valpara´ıso { CCTVAL, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile 4Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´asde Hidalgo, Edificio C-3, Ciudad Universitaria, C.P. 58040, Morelia, Michoac´an,Mexico 5Departamento de Ciencias B´asicas, Facultad de Ciencias, Universidad del B´ıo-B´ıo, Casilla 447, Chill´an,Chile. We study the leptonic decay of charged pions in a compact star environment. Considering leptons as a degenerated Fermi system, pions are tightly constrained to decay into these particles because their Fermi levels are occupied. Thus, pion decay is only possible through thermal fluctuations. Under these circumstances, pion lifetime is larger and hence can be considered to reach a metastable state. We explore restrictions under which such a metastability is possible. We also study conditions under which pions and leptons already in chemical equilibrium can reach simultaneously the thermal equilibrium, and obtain the neutrino emissivity from metastable pions. Scenarios that favor this metastable state are protoneutron stars. I. INTRODUCTION tron stars, where temperature reaches values higher than 1 MeV (∼ 1010 K). Improvements on the study of pions in The study of pions in a condensed state has widely hot and dense media in heavy ion collision experiments at moderate energies have been conducted, showing an im- been explored in different contexts and frameworks [1{ − + 17]. In particular, compact stars may provide a natural portant increase of the π /π ratio at high baryon den- scenario for such a state of matter. The possibility of sity [33{35]. Thus, an increase of the electron chemical generating pions in a condensed state in compact stars, potential is expected, particularly as density increases. however, is still matter of discussion. For the formation So, at least in protoneutron stars, an important asym- of a charged-pion condensate in a chemically equilibrated metric charged-pion production is expected. However, system, it is necessary that the electron chemical poten- due to the short pion decay time, it is reasonable to ig- tial reaches values up to the pion mass. Nevertheless, nore these pions in the EOS and only consider them as at high enough baryon density, the formation of hyper- a sudden source for neutrinos in the cooling process. ons is also possible and this process tends to reduce the The 99.99% of the charged pions decay into muons and electron chemical potential [18{21]. On the other hand, muonic neutrinos, and rarely into electrons and electronic at the inner core, the formation of kaon condensate is neutrinos as a subleading channel. Since muons and elec- favored in comparison with pion condensate due to a re- trons inside a compact star are degenerated, almost all duction of kaon mass [22{28]. Moreover, it is expected states are occupied up to Fermi levels and the only way that a pion-nucleon s-wave repulsive interaction increases for a negative pion to decay into leptons is through ther- the pion mass although the p-wave attractive potential mal fluctuations. In other words, there is not enough produces the opposite effect [21, 29]. All in all, there are phase space available for the byproduct of a charged- many dense nuclear effects affecting the pion mass and pion decay, which makes it an extremely slow process. leptonic decay constant and thus the range of values for This observation implies that pions are in a metastable these observables is broadened [30, 31]. Such consider- state [36]. Moreover, lepton decay rate may be induced arXiv:1610.05751v2 [hep-ph] 16 May 2017 ations make it unclear to distinguish whether the elec- through thermal fluctuations by considering scattering tron chemical potential may reach the pion mass, but with neutrinos from the thermal bath, generating as a definitely it can reach values close to the condensation result an increase of the number of π− in the thermal point. bath. When both the pionic and leptonic decay rates A vast portion of the existing literature regarding com- become of the same order, both types of particles reach pact stars does not incorporate pions in the equation of thermal equilibrium simultaneously [37{39]. state (EOS) because these studies are mainly based on In this article we explore the issue of charged-pion mean field models. In the absence of a pion condensate, metastability and conditions under which it would be dynamical pions do not play a significant role in such possible. We also discuss some phenomenological con- models because they immediately decay and mostly be- sequences of such a state. For this purpose, this article have as an interaction mediating particle [19]. is organized as follows: In Sec. II we present the initial If a pion production mechanism exists, the pion density considerations for the system we are dealing with. In number is negligible at low temperature by exponential Sec. III we calculate the pion decay rate in dense lep- suppression. However, this is not the case in protoneu- ton medium and explore the case when there is chemical 2 equilibrium between pions and leptons, and nonchemical We define the particle four-momentum as p for pions, equilibrium with pions in rest frame. In Sec.IV we show q for leptons and k for neutrinos. The energy of pions, the conditions for metastability and we define the criti- charged leptons and neutrinos is defined as cal parameters to reach such a state. We calculate the q lepton decay rate in Sec.V and the conditions in param- E = pp2 + m2 ;E = q2 + m2;E = jkj; (6) eter space where pion-lepton thermal equilibrium can be π π ` ` ν` reached simultaneously. The neutrino emissivity through respectively. Also, we define the corresponding particle pion decay is calculated in Sec.VI and compared with the number density distributions as URCA process emissivity. Finally we present our conclu- sions and future perspectives in Sec. VII. nπ− = nB(Eπ − µπ); (7) n` = nF (E` − µ`); (8) II. INITIAL CONSIDERATIONS nν¯` = nF (Eν` + µν` ); (9) x=T −1 x=T −1 Metastability of negative charged pions in a where nB(x) = (e − 1) and nF (x) = (e + 1) degenerate-muon and -electron environment is due represent the Bose-Einstein and Fermi-Dirac distribu- to Pauli blocking, which suppresses the meson decay tions, respectively. into leptons. This condition is achieved by most of the existing neutron star models. In this section we sum- Our treatment includes thermal effects, which we con- marize the general considerations adopted, the models sider within the Matsubara formalism. For this purpose, to be used, and the notation employed throughout the we change the four-momenta zeroth components l to the article. 0 respective Matsubara frequencies at a given temperature • Charged-pion number is conserved in normal phase, T according to p0 = i2npπT for pions, q0 = i(2nq + 1)πT for leptons, and k0 = i(2nk + 1)πT for neutrinos, for namely, µπ ≡ µπ− = −µπ+ , and µπ < mπ. np; nq; nk 2 Z. Correspondingly, the momentum inte- R P • Charged leptons are degenerated, which means that grals change to dl0 = i2πT n for bosons and fermions. we consider all lepton chemical potentials bigger than their respective masses, µ` > m`, where ` As we mentioned, nuclear interactions affect consid- stands for muon (µ) or electron (e). Masses con- erably pion dynamics. In particular, it is not well un- sidered here are mµ = 105:6 MeV, me = 0:5 MeV, derstood whether the repulsive s wave or the attrac- and neutrinos (ν) are considered massless. tive p wave dominates [33], so we consider the effective pion mass undetermined. However, it can be observed • The nuclear medium modifies hadronic parameters. in hadronic models that medium effects are incorporated For simplicity, and because we attempt to describe through the pion self-energy and the field renormalization weak interaction effects only, we do not specify par- strength [31, 32]. It is always possible to expand these ticular values for m and f since we do not know π π quantities at low momentum and the result is a freelike a priori the influence of dense nuclear matter on particle propagator with modified mass, and pion veloc- these parameters. However, we always consider ity [32]. Here, for simplicity, we consider only the pion m > m . π ` mass modification. In the same way, the pion decay constant can be ob- Weak interactions between pions and leptons are de- tained expanding at low momentum of the axial current scribed by the effective Fermi model using PCAC [31, 32] or by adopting the Brown-Rho scal- ing [30]. We adopt the latter scaling for fπ as well as the ¯ = + ¯ = − nucleon effective masses in this work. Lπ` = fπGF [ ν` Dπ (1−γ5) `+ `Dπ (1−γ5) ν` ]; (1) where the derivative ∓ ∓ III. IN-MEDIUM PION DECAY RATE Dαπ = (@α ∓ iµπδα0)π (2) includes the charged-pion chemical potential. The value In order to find the conditions for metastability, let us −5 −2 used for the Fermi constant is GF = 1:17 × 10 GeV . first examine in detail the in-medium pion decay rate, The free Lagrangians for pions, leptons, and neutrinos which we derive using the optical theorem. The same are, respectively, results are obtained from the Fermi golden rule. The charged-pion propagator must be understood as + α − 2 + − Lπ = Dαπ D π − mππ π ; (3) the creation of a pion with a given charge an the annihi- ¯ lation of a pion with the opposite charge.
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