Metastable Pions in Dense Media

Metastable pions in dense media

Marcelo Loewe1,2,3, Alfredo Raya4, CristiánVillavicencio5 1Instituto de F´ısica, Pontificia Universidad Católica 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 Tecnológico de Valpara´ıso – CCTVAL, Universidad Técnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile 4Instituto de F´ısica y Matemáticas, Universidad Michoacana de San Nicolásde Hidalgo, Edificio C-3, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán,Mexico 5Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del B´ıo-B´ıo, Casilla 447, Chillán,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 = |k|, (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 conclusions 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 ∈ 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 scaling [30]. We adopt the latter scaling for fπ as well as the ¯ / + ¯ / − nucleon effective masses in this work. Lπ` = fπGF [ψν` Dπ (1−γ5)ψ`+ψ`Dπ (1−γ5)ψν` ], (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. Since there is L` = ψ`[i∂/ + µ`γ0 − m`]ψ` (4) isospin asymmetry, the propagation is different for π+ L = ψ¯ [i∂/ + µ γ ]ψ . (5) − ν` ν` ν` 0 ν` and π . For our purposes, we need to calculate the 3

and take the imaginary part of the self-energy. The general form for Γπ− is

Z 3 2 2 d k X −2πGF fπ n 2 Γ − = m [sE tE + q·k] π (2π)3 sE tE E π ` ν` s,t=±1 ` ν` π o − 2[sE` Eπ − p·q][tEν` Eπ + p·k]

FIG. 1. Feynman diagram representing the pion self-energy [1 − nF (sE` − µ`) − nF (tEν` + µν` )] from weak interactions in a thermal bath. δ (Eπ − µπ) − (sE` − µ`) − (tEν` + µν` ) . (15) retarded charged-pion propagator in momentum space1, A similar argument shows that including self-energy corrections generated through weak Γ + (µ , µ , µ ) = Γ − (−µ , −µ , −µ ). (16) coupling with leptons, namely, π π ` ν` π π ` ν`

ret i D ∓ (p) = , (10) π (p ± µ )2 − E2 − Π(±p) A. Pion decay rate at chemical equilibrium 0 π π p0→p0+i where Π(p) is the self-energy. We are interested in finding a configuration where pi- Neglecting the real part of the self-energy and expand- ons and leptons are in chemical equilibrium, which is ex- ing around the physical real pole p0 = Eπ − µπ, the pressed by the relation among chemical potentials negative-pion propagator can be written in the nonrela- tivistic Breit-Wigner form µπ = µ` − µν` . (17) ret i Dπ− (p) ≈ 2 2 , (11) Assuming the above relation to be valid, we can solve (p0 + µπ) − E + iEπΓ − π π exactly Eq. (15). We proceed in this form, but in Sec.V where the momentum dependent decay rate is defined as we explain the necessary conditions for chemical equilibrium. Summing up s and t and integrating in the neu- 1 trino momentum solid angle, considering mπ > m` we Γπ− = − Im Π(Eπ − µπ + i, p). (12) obtain Eπ

Z a`(Eπ +|p|) At zero temperature, this quantity is just the decay width mπ 1 − ¯ of the pion. Nevertheless, at finite temperature, the Γπ = Γπ` dEν` [1 − n` − nν¯` ] Eπ 2a`|p| meaning of this quantity should be interpreted differently a`(Eπ −|p|) (18) (see below). We proceed to calculate the leptonic contribution to where E` = Eπ − Eν` by energy conservation. The con- the pion self-energy. This process is shown diagrammat- stant a is defined as ically in Fig.1 and can be written as ` 2 2 Z d4k mπ − m` a` ≡ , (19) −iΠ(p) = tr V (p)S`(q)V (p)Sν (k) , (13) 2 (2π)4 ` 2mπ while where S` and Sν` are the time-ordered lepton and neutrino propagators, q = p + k as momentum conservation ¯ 1 2 2 2 2 requires, and the vertex obtained from the interaction Γπ` = fπGF mπm` a` (20) term in Eq. (1) is defined as π is the well-known pion decay width in vacuum for π− → V (p) = GF fπ(p/ + µπγ0)(1 − γ5). (14) ` +ν ¯`. Performing the traces over Dirac indices and summing Equation (18) shows explicitly the particle statistics over the Matsubara frequencies using the techniques de- averaged in the allowed range of energy for the emergent scribed in [40], we set the retarded condition p0 → p0 +i neutrinos, as well as the on-shell energy conservation. It is usually interpreted as the probability for inserting a lepton and an antineutrino into the system, minus the probability of the system to produce a lepton and an an- 1 A decay rate cannot be defined in the remote past; that is the reason why the retarded propagator is used. Usually there is tineutrino: (1 − n`)(1 − nν¯` ) − n` nν¯` = 1 − n` − nν¯` . In no difference at all in vacuum, but at finite temperature and other words, the direct decay rate minus the recombina- − density, the imaginary part of the retarded and the time-ordered tion process π ↔ ` +ν ¯`. So, this rate can be described self-energy produce different results. d R in terms of the direct rate Γπ− = (1 − n`)(1 − nν¯` ) and 4

i R d i 2 − the inverse rate Γπ− = n` nν¯` as Γπ = Γπ− − Γπ− . From Eq. (15), setting p = 0 and considering m` ≤ m˜ π, These rates are related to each other through the pion EDDR is

d 2 Γπ− (E −µ )/T a˜` = e π π , (21) Γ − = Γ¯π` [1 − nF (M` − µ`) − nF (Mν + µν )] , i π a2 ` ` Γπ− ` (29) result that can be veriﬁed from Eq. (18). This is an ex- where the mass terms tension of the same result obtained in [37, 40], including now chemical potential. M` ≡ |(1 − a˜`)m ˜ π|, (30) From Eq. (21) it can also be obtained Mν` ≡ |a˜` m˜ π|, (31)

d β(Eπ −µπ ) Γπ− = e nπ− Γπ− (22) are the rest energy of the terms in Eqs. (26) and (27), re- i spectively, but with the pion mass shifted with the chem- Γ − = n − Γ − . (23) π π π ical asymmetry term, and with the shifted constant Although this microscopic interpretation can be easily m˜ 2 − m2 understood (unlike the case of a fermion decaying into π ` a˜` ≡ 2 . (32) a boson and fermion, as we shortly see), it is better to 2m ˜ π adopt the macroscopic interpretation: Γ − is the rate π We observe once again in Eq. (29) the microscopic inter- at which the slightly out-of-equilibrium pion system can pretation in terms of availability of phase space to allow reach thermal equilibrium [37, 40], the resulting leptons to be created: (1 − n`)(1 − nν¯` ) − −t Γ n n = 1 − n − n . f(t) = n + c e π− , (24) ` ν¯` ` ν¯` π− π The condition needed to obtain Eq. (29) in this out- where f(t) is the nonequilibrium pion distribution func- of-equilibrium system can be expressed in terms of the neutrino chemical potential. The condition m` ≤ m˜ π tion and cπ some energy dependent function. From now c implies that µν ≤ µ , where onward we refer to Γπ− as the equilibrium deviation de- ` ν` cay rate (EDDR). µc ≡ (m − µ ) + (µ − m ). (33) ν` π π ` ` Integrating Eq. (18) with respect to neutrino momen- This critical quantity is positive if we consider the condi- tum, we get tions imposed in Sec.II: that charged pions are in normal phase (µ < m ) and charged leptons are degenerated " + ! π π −(E −µ`)/T ¯ mπ T 1 + e ` (µ` > m`). The condition in Eq. (33) is valid also for Γπ− = Γπ` 1 + ln − −(E −µ )/T Eπ 2a`|p| 1 + e ` ` negative values of the neutrino chemical potential, where + the thermal bath contains more antineutrinos than neu- −(E +µν )/T !# T 1 + e ν` ` trinos. + ln , −(E− +µ )/T 2a`|p| 1 + e ν` ν` (25) C. High neutrino density where the energy terms are deﬁned as It is interesting to see what happens if neutrino chem- ± c E = (1 − a`)Eπ ± a`|p|, (26) ical potential increases beyond µ . ` ν` ± Eν = a`(Eπ ± |p|), (27) ` • If 0 ≤ m˜ ≤ m , or equivalently 0 ≤ µ − µc ≤ π ` ν` ν` m`, the annihilation of pions induced by scattering with neutrinos from the thermal bath is favored: B. Pion EDDR in the rest frame − π + ν` ↔ `. The EDDR in this case is

The EDDR in the rest frame can be obtained easily 2 a˜` Γ − = Γ¯ [n (M − µ ) − n (M − µ )] . (34) even considering the system out of chemical equilibrium. π π` a2 F ν` ν` F ` ` Let us define a shifted pion mass as ` Since n − n = n (1 − n ) − n (1 − n ), the pre- m˜ ≡ m − µ + µ − µ . (28) ν` ` ν` ` ` ν` π π π ` ν` vious result can be interpreted as the probability of finding a neutrino in the thermal bath from which the pion will scatter off, minus the inverse process.

2 Here we adopt the notation of Ref. [37]. However, the notation • If −m` ≤ m˜ π ≤ 0, or equivalently m` ≤ µν` − where Γ> = Γd and Γ< = Γi is more standard [40]. µc ≤ 2m , it is more favorable that pions scatter ν` ` 5

off from antileptons generating neutrinos: π− +`¯↔ condensation critical temperature. With this definition ν. From energy conservation, the EDDR is we say then that the particle is metastable if Γ0 = 0. 2 Let us analyze the case where the system is in chemical a˜` − ¯ equilibrium. From Eq. (25), we have Γπ = Γπ` 2 [nF (Mν` − µν` ) − nF (M` + µ`)] . (35) a` " + mπ µ` − E + Notice that n`¯ − nν` = n`¯(1 − nν` ) − nν` (1 − n`¯), ¯ ` Γ0 = Γπ` 1 + θ(µ` − E` ) which gives the probability of finding an antilepton Eπ 2a`|p|

in the thermal bath from which the pion scatters − # oﬀ leaving a neutrino, minus the probability of the µ` − E` − − θ(µ` − E` ) . (38) inverse process. 2a`|p| • Finally, ifm ˜ ≤ −m , or equivalently 2m ≤ µ − π ` ` ν` We can immediately see that if µ > E+, the zero tem- µc , a π− is totally annihilated by an antilepton and ` ` ν` perature part of the pion decay rate vanishes. This condi- a neutrino, disappearing in the thermal bath. The tion depends on the pion energy and the lepton chemical opposite reaction means that the thermal bath pro- potential. Therefore, in order to get Γ = 0, the pion duces those particles, π− +`¯+ν ↔ (thermal bath). 0 energy must be less than a certain critical energy Thus, EDDR in this case is

2 Eπ < Ec ≡ (1 + b`) µ` − b` qF , (39) a˜` Γ − = Γ¯ [n (M + µ ) + n (M − µ ) − 1] . π π` a2 F ` ` F ν` ν` ` or, in terms of the pion momentum, (36) Writing n¯ + nν − 1 = n¯ nν − (1 − n¯)(1 − nν ), ` ` ` ` ` ` |p| < pc ≡ (1 + b`) qF − b` µ`, (40) we interpret this rate as the probability to ﬁnd an antilepton and a neutrino in the thermal bath mi- with nus the probability to introduce an antilepton and 2 2 a neutrino into the thermal bath. Backreaction mπ − m` b` ≡ 2 , (41) means the thermal bath can spontaneously gener- 2m` ate a pion, an antilepton and a neutrino. and where All the above-mentioned EDDRs are positive. These q reactions are possible because we are out of chemical 2 2 qF = µ − m (42) equilibrium in a high neutrino density environment. The ` ` meaning of those processes is explained in [37] at zero is the lepton Fermi momentum. From the critical values chemical potentials. However, we recall that the mi- exposed, one can see that the leptonic chemical potential croscopical description is not well understood, especially is restricted to be higher than a certain critical value when dealing with the fermion EDDR. The advantage of calculating the imaginary part of the semileptonic self- µ` > µc ≡ (1 − a`)mπ. (43) energy instead of using the Fermi golden rule is because in the latter, thermal probability factors must be added At chemical equilibrium, considering muons as the result- by hand. ing leptons with mπ = 139.5 MeV, we have the critical value µc ≈ 109.74 MeV.

IV. METASTABILITY CONDITIONS If we analyze the hypothetical system with π+, metastability can be reached if the neutrino chemical po-

For metastability, we understand a special situation tential is high enough so that µνµ − µµ ≥ a`mπ. This where the decay rate of a certain particle becomes much was noticed in [41] but at zero temperature, where pions smaller (a bigger lifetime) than the usual case, in analogy should condense. with what happens in nuclear isomerism. In our case, this special situation is produced by the environment that inhibits the particle decay and is related to the Pauli V. PION-LEPTON THERMAL EQUILIBRIUM blocking effect. Only thermal fluctuations are capable of overcoming this situation. In order to describe this The system we are interested in analyzing is in chem- scenario, let us separate the direct decay rate into the ical equilibrium, but not in thermal equilibrium, with zero temperature and thermal fluctuation parts metastable pions decaying slowly. The EDDR in a d thermalized system must be seen, as it is explained in Γ − = Γ + δΓ (37) π 0 T Ref. [37], in terms of the time needed for a system to where the first term is defined as Γ0 ≡ limT →0 Γπ− and reach thermal equilibrium [see Eq. (24)]. The decay of pi- the second term is the thermal fluctuation. The tem- ons and the inverse process consider, statistically speak- perature for our analysis must be higher than the pion ing, pions as real test particles and “virtual” leptons and 6

Now the procedure is the same as in Sec.III by setting q0 → q0 + i in the retarded propagator. The EDDR for each lepton helicity is

1 Γ± = − Im Π±(E` − µ` + i, q). (50) E` In order to simplify the analysis, we calculate the average FIG. 2. Feynman diagram representing the lepton self-energy of the EDDRs for the different helicities. Defining Γ` = from weak interactions in a thermal bath. (Γ+ + Γ−)/2, after Matsubara summation, the average lepton EDDR is neutrinos from the heath bath. We also have to consider 3 Z b`(E`+|q|) mπ m` 1 another process involving the same participants [38, 39] Γ = Γ¯ dE [n − + n ] ` π` 2m E 2b |q| ν` π ν¯` if we want to explore the system as a whole: real test lep- ` ` ` b`(E`−|q|) tons recombined with antineutrinos from the heat bath (51) increasing the pion population. The latter process oc- where E = E + E by energy conservation. curs very slowly since it is a purely thermal fluctuating π ` ν` phenomenon. To compare how slow metastable pions The lepton EDDR is related to a direct process, where can decay, we explore under which conditions pions and an external lepton scatters with a thermalized antineu- leptons reach thermal equilibrium simultaneously. This trino, giving rise to a thermalized pion ` +ν ¯` → π. obviously happens if both EDDR are of the same magni- The corresponding inverse process can be thought as a thermalized pion decaying into a thermalized antineu- tude: Γ` ∼ Γπ− . trino and a observable lepton π → ` +ν ¯. In terms d i R − We need then to find the lepton EDDR assuming the of rates, Γ` = Γ` + Γ` = (nπ + nν¯` ), where the di- d R − system is in chemical equilibrium, i.e., µπ = µ` − µν . rect rate is Γ` = (1 + nπ )nν¯` and the inverse rate is ` i R − The dressed lepton propagator is Γ` = nπ (1−nν¯` ). Here, the direct and the inverse processes have to be added in opposition to the case of boson i Sdr(q) = , (44) decay. This is a general feature of decaying fermions as ` Q/ − m − Σ was pointed in [37]. The direct and inverse rates are related as where Q = (q0 + µ`, q) and Σ(q) is the lepton self-energy generated by weak coupling, shown in the Feynman dia- Γd ` = e(E`−µ`)/T , (52) gram in Fig.2, which corresponds to i Γ` Z d4p −iΣ(q) = − V (p)S (k)V (p)D − (p) , (45) and also from this result, these rates can be written as (2π)4 ν` π Γd = e(E`−µ`)/T n Γ , (53) with p = q − k by virtue of energy conservation and V (p) ` ` ` i as already defined in Eq. (14). The general structure of Γ` = n`Γ` . (54) the self-energy in this process is Σ = A/(1 − γ5), where The lepton EDDR is interpreted as the rate at which Z d4k P 2K − 2K·PP the slightly out-of-equilibrium lepton system can reach A (q) = G2 f 2 µ µ . (46) µ F π 4 2 2 2 the thermal equilibrium: (2π) (P − mπ)K

−t Γ` Here, K = (k0 + µν` , k) and P = (p0 + µπ, p). Using f(t) = n` + c`e (55) the general structure of the self-energy, and considering that, because of spatial symmetry, A = qˆ|A|, the dressed where f(t) is the nonequilibrium lepton distribution func- lepton propagator can be written as tion and c` some energy dependent function. dr P+ P− Integrating out the neutrino momentum, we obtain the S` = (Q/ + m − Σ) 2 2 + 2 2 , Q − m` − Π+ Q − m` − Π− general result (47) " + ! where the helicity projectors are 3 −(E −µπ )/T ¯ mπ m` T 1 − e π Γ` = Γπ` ln − , 1 2m` E` 2b`|q| 1 − e−(Eπ −µπ )/T P± = (1 ± γ0γ·qˆγ5) , (48) + 2 −(E˜ +µν )/T !# 1 + e ν` ` − ln , and the mass corrections are −(E˜− +µ )/T 1 + e ν` ν` (56) Π±(q) = 2Q · A ± (Q0A·qˆ − |q|A0). (49) 7 where 1.00

± Eπ = (1 + b`)E` ± b`|q| , (57) 0.98 ± E˜ = b`Eπ ± b`|q|. (58) ν` 0.96 π /m

μ 0.94

Now, we need to ﬁnd a window in the parameter space μ where pions and leptons reach thermal equilibrium sat- 0.92 isfying Γ` ∼ Γπ− . The parameters involved are the neutrino chemical potential, the lepton chemical potential, the temperature, the pion energy and the lepton energy. 0.90 To reduce the number of parameters we consider pions 2 4 6 8 10 in the rest frame, where the decay rate is given in Eq. T[MeV] (29), withm ˜ π → mπ. In the same way, near the Fermi surface, ﬂuctuations of leptons in the degenerated environment are produced. So we take the lepton energy as 0.98 the Fermi energy in the decay rate, which leads to 0.96 3 mπ m` T 0.94 Γ` = Γ¯π` 2m` µ` 2b` qF π 0.92 /m

e sinh[(b`µ` + µν` )/T ] + sinh[b`qF /T ] × ln . (59) μ 0.90 sinh[(b`µ` + µν` )/T ] − sinh[b`qF /T ] 0.88 For simplicity we consider µν` = 0, which means that all real neutrinos escape from the star once created. This is a 0.86 usual approximation in neutron stars, but not necessarily valid for protoneutron stars [22, 42–44]. With such con- 2 4 6 8 10 siderations, the EDDRs depend only on temperature and T[MeV] lepton chemical potential. The regions where Γπ− ∼ Γ` are plotted in Fig.3, describing the temperature and lep- FIG. 3. Chemical potential and temperature values where ton chemical potential conditions for reaching thermal pion-lepton chemical equilibrium is favorable. The band equilibrium simultaneously. This is considered for pions widths correspond to a half order of magnitude difference be- decaying into muons (upper panel) as well as pions decay- tween the widths: 0.5 < Γπ− /Γ` < 1.5. The colors and lines refer to mπ = 140 MeV (solid blue), mπ = 115 MeV (dashed ing into electrons (lower panel) for different values of mπ. red) and mπ = 200 MeV (dot-dashed green). For decay rates such that 0.5 < Γπ− /Γ` < 1.5, we can see in Fig.3 that there is a wide region where simultaneous equilibrium is possible, especially at high temperature, that governs such a process is the emissivity (energy loss and always for values of the lepton chemical potential by neutrino emission per unit of time), the Urca pro- greater than the critical chemical potential for metasta- cess being the most efficient one [49]. In this section we bility, defined in Eq. (43). If pions are condensed they calculate the neutrino emissivity due to pion decay in a still can decay [15]; therefore, at finite temperature, such pion-lepton equilibrium regime. a kind of equilibrium analysis must be considered in the The neutrino emissivity generated by leptonic decay of condensed case. pions is defined as the transition probability matrix mul- Notice that in compact stars we have leptonic equilib- tiplied by: the ejected neutrino energy, the probability rium: µµ − µνµ = µe − µνe , and also beta equilibrium, − nπ of finding a pion in the thermal bath, and the proba- µn−µp = µe−µνe . Therefore, if there is pion-lepton equi- − bility 1 − n` of finding a hole below the Fermi level. All librium, then µn−µp = µπ. The processes n ↔ p+π are this integrated in phase space. Therefore always present since pions strongly interact with nuclear medium. Z ¯ ¯ ¯ X 2 π = dp dq dk |M| k0 nB(p0) [1 − nF (q0)] spin VI. NEUTRINO EMISSION (2π)4δ4(p − q − k), (60)

The cooling process in compact stars is mainly pro- where the phase space measure for pions is duced by neutrino emission and provides information d4p dp¯ = θ(p + µ )δ4((p + µ )2 − E2) , (61) about the existence of nontrivial hadronic matter states. (2π)3 0 π 0 π π In particular, neutrino emission through pion decay has been studied in diﬀerent models [45–48] The quantity and an equivalent expression for the leptons. The proba- 8

] 2 π / ϵ

urca 0

[ ϵ 14

FIG. 4. Integration limits of the lepton energy as a func- 10 tion of pion energy. The critical chemical potential where 12 metastability starts is indicated, as well as the pion energy Log ± 10 values where E` = µ`. 8 bility amplitude for pions going into leptons and antineu- 6 trinos is deﬁned as 4 Z 4 − 4 4 2 h` ν¯| d x Lπ` |π i = iM(2π) δ (p − q − k). (62) 0 0 2 4 6 8 10 At chemical equilibrium, the neutrino emissivity from pi- T MeV ons is then [ ]

¯ 2 Z ∞ FIG. 5. Comparison between emissivities produced by the Γπ` mπT URCA process and pion decaying into muons (upper panel) π = 2 dEπ nB(Eπ − µπ) 2π a` mπ and electrons (lower panel). In both cases we use the values µ = 0.95m . f = 0.8 × 93.4 MeV and the pion masses m Eπ − E` ` π π π ln 1 + e(E`−µ`)/T =115 (dashed red), 140 (solid blue), and 200 MeV (dot-dashed T green). E+ ` (E`−µ`)/T −Li2 −e , (63) E− ` At temperature much higher than 20 MeV, the star turns to be opaque to neutrinos [43]. Neutrino total where E± was defined in Eq. (57) and plotted in Fig.4 ` mean free path is inversely proportional to the proba- as a function of pion energy. bility of absorption and scattering with the participants. A comparison between the neutrino emissivity due to Because the metastable state of pions emerges by the pion decay and by the URCA process is shown in Fig.5 difficulty for these particles to decay into degenerated for pion decaying into muons (upper panel) and electrons leptons (lepton Fermi levels are almost filled), one can (lower panel). straightforwardly infer that neutrino absorption by pi- URCA emissivity [49] is given by ons is highly suppressed in degenerated lepton environ- − 457π ments (π + ν` → `). On the other hand, such a = G2 |V |2(1 + 3g2 )m∗ m∗m∗T 6Θ , metastable state enlarges the amount of pions in the envi- URCA 10080 F ud A n p e npe (64) ronment, and thus, the possibility of pion-neutrino scat- − − ∗ ∗ tering (π +ν` → π +ν`) increases. It is not clear which where gA = 1.2 is the axial coupling, mn ≈ mp ≈ 0.8 × 940 MeV are the nucleon effective masses (in the effect dominates. The detailed calculation must be done ∗ to see the overall effect. Brown-Rho scaling) and me ≈ µe is the electron effective mass. The function Θnpe = θ(pF p + pF e − pF n) is the triangular condition between the Fermi momenta of the proton, electron, and neutron, which we consider here A. Low temperature approximation as satisfied. We see that neutrino emission tends to be ∼ 10−2 times Now we explore how the emissivity behaves in the low lower than the URCA process at high temperature for temperature approximation. The scale here is the pion pions decaying into muons, and ∼ 10−5 in the case when mass, so T ∼ 10 MeV is small enough. To expand the pions decay into electrons. Notice that for a small pion exponentials in Eq. (63) we need a negative exponent. ± mass, the muonic neutrino emissivity is similar to the The lepton energy integration limits E` (Eπ) are plotted URCA emissivity at T ∼ 1 MeV. in Fig.4, where all the terms in the plot were already 9

0 defined except Ec = (1 + b`)µ` + b`qF . The procedure in degenerated lepton matter and modified hadronic pa- for the expansion is to separate the pion energy integral rameters due to dense nuclear matter. We found that, at ± in the ranges where E` − µ is negative expanding the certain values of the pion chemical potential, pions drift exponential in the logarithm and the polylogarithm. In into a metastable state and their decay is generated only ± the integration ranges where E` − µ is positive, the log through thermal fluctuations. This scenario is generated and polylog can be expanded after the use of the inversion by the fact that all leptonic energy states are occupied formulas up to the Fermi level, so the pion has no phase space to decay. In this way, the pion EDDR is calculated at fi- ln(1 + ex) − ln(1 + e−x) − x = 0, (65) nite temperature and chemical potentials. The particular x2 π2 case of pion EDDR at chemical equilibrium with leptons, Li (−ex) + Li (−e−x) + + = 0. (66) as well as the case out of equilibrium in rest frame was 2 2 2 6 studied. The Bose-Einstein term is expanded also up to the lead- The pion momentum is restricted for momentum below ing order. Keeping the leading terms, the integration a certain critical momentum, and the allowed values of in pion energy can be performed exactly from mπ to the chemical potential must be lower than the pion mass 0 0 and higher than a critical chemical potential. These crit- Ec or Ec. From Ec and Ec to infinity, the integrands are expanded around the lowest value due to exponential ical parameters are obtained by studying the behavior suppression. As a result, the emissivity is quite differ- of the direct pion decay rate at zero temperature. The ent from the one expected in a simple boson gas, where thermal equilibrium was also studied in terms of equiva- ∼ T 3/2e−(mπ −µπ )/T [45–48]. lent pion EDDR and lepton EDDR. The considered values of temperature and chemical potential for rest frame When pc T , which implies that lepton chemical potential should be bigger than the critical chemical poten- pions and leptons in the Fermi surface are reasonable. tial, the emissivity is proportional to T 2, namely Finally, we calculate the neutrino emissivity generated from metastable pion decay in chemical equilibrium, and T 2 compare with the URCA emissivity. The order of mag- ≈ Γ¯ m4 g(µ ) e−(Ec−µπ )/T , (67) nitude turns closer for T ∼ 4 MeV and is considerably π π` π ` 2πm π smaller than the usual boson gas emissivity. with We conclude that this state of matter is more favorable in protoneutron stars where temperature reaches values higher than 1 MeV. This scenario therefore can have sig- 2qF 2a`mπ 2pc qF − pc g(µ`) = + + . (68) nificant repercussions only for a short time. In cold com- b m p a m q + p ` π c ` π F c pact stars, metastable pions can also be founded, but are For p T , which means µ ∼ µ , the emissivity is much less abundant. c ` c The assumptions we consider can be reproduced for other kind of particles. In particular, the case of kaons ( 2 3/2) 2 T T could be a better example in compact stars since their π ≈ 2(1 − a` − a` ) + a` 2πmπ 2πmπ mass reduces considerably due to dense nuclear environment and the possibility to get metastable kaons in cold ¯ 4 −(Ec−µπ )/T × Γπ` mπ e . (69) dense matter in principle is enhanced. These approximations fairly fit the full expression and explicitly show that the neutrino emission from chemically ACKNOWLEDGMENTS equilibrated pions is much smaller than it was expected. This work has been supported by FONDECYT (Chile) Grant No. 1130056, No. 1150471, No. 1150847 and No. VII. CONCLUSIONS 1170107, ConicytPIA/Basal (Chile) Grant No. FB0821, CIC-UMSNH (México) Grant No. 4.22, and Consejo In this work we have studied the effects of dense lepton Nacional de Ciencia y Tecnologa (México)Grant No. matter over the decay rate of charged pions. We use con- 256494. CV acknowledges the group F´ısica de Altas En- ditions like in the neutron star environment, consisting erg´ıas at UBB.

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