Electroweak decay of quark matter within dense astrophysical combustion flames

The transition from hadronic matter to quark matter may have a key role in several highly energetic astrophysical phenomena such as core collapse supernova explosions Benvenuto and Horvath (1989); Zha et al. (2020); Fischer et al. (2018), gamma ray bursts Lugones et al. (2002); Ouyed et al. (2002, 2020), binary neutron star (NS) mergers Bauswein et al. (2019); Most et al. (2019) and phase-transition-induced collapse of NSs Lin et al. (2006); Abdikamalov et al. (2009); Cheng et al. (2009). Although the potential relevance of this conversion has been recognised for decades, there are still several unresolved issues such the exact mechanism that triggers the transition Iida and Sato (1998); Olesen and Madsen (1994); Bombaci et al. (2004); Lugones (2016); Bombaci et al. (2009) and its propagation mode to the rest of the star Lugones et al. (1994); Herzog and Ropke (2011); Pagliara et al. (2013); Ouyed et al. (2018). This is of key importance to assess potentially observable astrophysical signatures.

According to theoretical models, the transition to quark matter in a compact star would begin with the nucleation of a small deconfined seed inside the stellar core when the density of hadronic matter goes beyond a critical density Olesen and Madsen (1994); Iida and Sato (1998); Lugones and Benvenuto (1998); Benvenuto and Lugones (1999); Bombaci et al. (2004); Lugones and Bombaci (2005); Bombaci et al. (2007, 2009); do Carmo and Lugones (2013); Lugones (2016). The formation of such seed would occur in two steps. First, a small lump of hadrons deconfines in a strong interaction time scale, leaving a small quark drop that is not in equilibrium under weak interactions. Then, weak interactions drive the system to chemical equilibrium in a weak interaction time scale. The energy released in such conversion can ignite hadronic matter in the neighborhood of the initial seed, creating a self sustained combustion process that may convert to quark matter the core of the star and even the whole star if quark matter is absolutely stable (see Lugones (2016) and references therein). During the conversion process, a combustion front (flame) separating the unburnt hadronic matter from the burnt quark matter travels outwards along the star Lugones et al. (1994, 2002); Keranen et al. (2004); Niebergal et al. (2010); Herzog and Ropke (2011); Fischer et al. (2011); Pagliara et al. (2013).

The structure of the flame is depicted in Fig. 1. The combustion front propagates to the right with a velocity that depends on the combustion mode, deflagration or detonation. In the fastest case, the velocity is of the order of $c/\sqrt{3}$, where $c$ is the speed of light Lugones et al. (1994). In such a case, at the flame front there is a region of thickness $l_{\mathrm{strong}}\sim\tau_{\mathrm{strong}}\times c/\sqrt{3}\approx 10^{-23}\mathrm{s}\times c/\sqrt{3}\sim 1\,\mathrm{fm}$ where hadronic matter deconfines. Behind it, there is a weak decay region of thickness $l_{\mathrm{weak}}\sim\tau_{\mathrm{weak}}\times c/\sqrt{3}\approx 10^{-8}\mathrm{s}\times c/\sqrt{3}\sim 1\,\mathrm{m}$ where quark matter approaches chemical equilibrium through weak interactions ($\tau_{\mathrm{weak}}\sim 10^{-8}\mathrm{s}$ is a rough estimate taken from Refs. Dai et al. (1995a, b, 1993); Anand et al. (1997) but its value is subject to density and temperature variations as we shall see later). If the propagation velocity is smaller, these thicknesses are proportionally smaller.

In this work, we focus on a fluid element that is initially located inside the deconfinement zone of the flame (see region 2 of Fig. 1). In this region, hadrons have just deconfined and matter is made up of quarks and leptons out of chemical equilibrium. The abundance of each quark and lepton species in region 2 is the same as in region 1, with the only difference that in region 1 the quarks are confined within hadrons and in region 2 they are deconfined. As time passes, the separation surface between regions 1 and 2 moves to the right in Fig. 1. In a reference frame in which the flame is at rest, we would see that our fluid element moves to the left in Fig. 1 across the entire region 3. Along this path quarks and leptons interact with each other through weak reactions so that they come closer and closer to chemical equilibrium. Finally, the fluid element enters region 4, just at the moment when it reaches full chemical equilibrium.

Our analysis will concentrate on the time evolution of the thermodynamic properties of the fluid element described above using the Boltzmann equation and calculating the rates of all the relevant weak reactions. We will pay special attention to the thermodynamic conditions at which the phase conversion occurs in typical astrophysical conditions. For example, in an old NS, accretion from a companion or rotational slowdown may trigger the conversion at the stellar core. In this case, the conversion begins in a low temperature environment without trapped neutrinos. On the other hand, the conversion may occur in a proto NS formed immediately after the gravitational collapse of the core of a massive star or in the massive compact object that may form after a binary NS merging. Such objects have very high temperatures in their interiors (typically few tens of MeV) and a large amount of trapped neutrinos, i.e. their mean free path is much shorter than the size of the star. We will also analyse systematically the role of the initial hadronic composition, or equivalently, the quark and lepton concentrations in the deconfinement region of the flame.

The paper is organized as follows. In Sec. II we summarize the equations that describe the time evolution of the particle abundances and the temperature inside the flame. In Sec. III we describe the quark matter equation of state used inside the flame and the initial conditions assumed in the deconfinement region. In Sec. IV we summarize the reaction rates and the neutrino emissivities in hot and dense quark matter without making any approximation about the degeneracy of neutrinos (the expressions are derived in Appendix A). In Sec. V we present our results for cold deleptonized NSs and in Sec. VI for hot NSs with trapped neutrinos. In Sec. VII we present our conclusions.

The time evolution of quark matter composition inside the flame will be described by means of the Boltzmann equation. We will focus on a small fluid element within the flame whose size is much smaller than the flame thickness and we will neglect the effect of the gravitational external field. For simplicity, we will also neglect spatial gradients of the distribution function¹¹1The order of magnitude of the term with $v\partial/\partial x$ in the Boltzmann equation is $\sim v/L$, where $v$ is a typical fluid velocity and $L$ is the flame thickness. For fast combustions in neutron star matter we have $v\sim c_{s}\sim c$ (being $c_{s}$ the sound speed and $c$ the speed of light) and therefore $v/L\sim 10^{9}s^{-1}$. As we shall see later, this is essentially of the same order of the term $\partial/\partial t\sim\tau^{-1}$, being $\tau\sim 10^{-9}-10^{-8}\,\mathrm{s}$ the timescale for attaining chemical equilibrium. This means that the term with $v\partial/\partial x$ may be important in the case of fast combustions. However, for slow enough deflagrations $v$ is significantly smaller than $c$, and the term with $v\partial/\partial x$ can be neglected safely.. Matter in the flame is assumed to be composed by $u$, $d$, $s$ quarks, electrons, electron neutrinos (and their antiparticles) interacting among themselves through weak interactions. Therefore, the Boltzmann transport equation for each particle species in the system reads:

being

where $f_{i}$ is the distribution function of the particle species $i$, $W_{i,j,...}$ is the transition probability for the processes creating and destroying such particles, and ${\cal S}$ is a statistical blocking factor involving the Fermi-Dirac distribution functions of the ingoing and outgoing particles. In the case of a flame in a hot NS, the assumption of a Fermi-Dirac distribution for neutrinos is justified because they are trapped and their mean free path is very small. For a flame in a cold NS the situation may be different. Close to the deconfinement zone shown in Fig. 1, matter is essentially transparent to neutrinos, but as the fluid element approaches the rear side of the flame, the temperature increases significantly and neutrinos may get trapped. The neutrino-transparent and neutrino-trapped regimes are the simplified extremes of a continuum, which is realized at different regions of the flame. Between these extremes there is a semitransparent regime where the spectrum of neutrinos includes a low-energy population that escapes, a high-energy tail that is trapped, and an intermediate-energy range where the mean free path is of the order of the flame thickness. A full analysis of the neutrino spectrum in this case is complex and it is beyond the scope of the present work. For simplicity we will assume in this work that the neutrino distribution is always a Fermi-Dirac one and let a more detailed analysis for future work.

Integrating Eq. (1) over the momentum of particle $i$, we have

where $n_{B}$ is the baryon number density, $n_{i}$ is the particle number density, $Y_{i}\equiv n_{i}/n_{B}$ is the abundance of particle $i$, and $\Gamma_{P}$ is the rate of decay or production of the particle $i$ due to the process $P$ given by:

For the present problem, the transition probability $W$ has the form (see Colvero and Lugones (2014) and references therein):

where $\delta^{4}$ is a Dirac delta function for the conservation of four-momentum that will be specified later, $P_{i}=(E_{i},\mathbf{p}_{i})$ is the four-momentum of any quark or lepton and $\left\langle|\mathcal{M}|^{2}\right\rangle$ denotes the squared matrix element summed over final spins and averaged over the initial spins. In Table 1 we list all the processes under consideration and the corresponding expressions for $\left\langle|\mathcal{M}|^{2}\right\rangle$.

The time evolution of the temperature can be obtained by means of the first law of thermodynamics. Since baryon number is a conserved quantity it is convenient to write the first law on a per baryon basis. Let $\epsilon$ be the total energy density and $s$ the entropy per baryon. Then, we have:

We will neglect the expansion of the fluid, i.e. the volume per baryon $v$ will be assumed to be essentially unchanged; $dv=d(1/n_{B})=0$. We will also assume that the energy per baryon $\epsilon/n_{B}$ changes due to neutrinos that leave the system, i.e. $d(\frac{\epsilon}{n_{B}})/dt=\varepsilon_{\nu}$, where $\varepsilon_{\nu}$ is the neutrino emissivity per baryon due to the weak processes (see Sec. IV for more details). Thus, the first law of thermodynamics reads:

The latter equation can ve rewritten as:

Using

and defining

we arrive to the equation that governs the time evolution of the temperature:

Providing appropriate initial conditions (see Sec. III.2), Eqs. (3) and (11) allow us to describe the time evolution of a fluid element within the flame since deconfinement until the time at which full chemical equilibrium is attained.

Once dense hadronic matter deconfines into quark matter, the initial state doesn’t have in general the quark abundances that guarantee chemical equilibrium. Thus, weak interaction processes will take place and will drive the composition to an equilibrium configuration. Just after deconfinement of hadronic matter, the following processes may occur:

Depending on the astrophysical environment at which deconfinement is initiated, some of these processes may have a different relevance. For example, in a cold and deleptonized NS, neutrinos are free to escape from the system and neutrino captures in processes III and IV do not happen, but as temperature increases, these processes are more relevant and should be considered. On the other hand, in just born PNSs or in hot compact stars potentially formed in a binary merger, the temperatures are very high (typically some tens of MeV) and there is a large amount of trapped neutrinos, in the sense that they have a mean free path much shorter than the size of the star.

Reaction rates and neutrino emissivities have been calculated in previous works for two different approximate cases Anand et al. (1997): (1) cold deleptonized matter, where quarks and electrons are degenerate, and (2) hot neutrino rich matter, where quarks, electrons and neutrinos were treated as degenerate. In this section, we generalize these results and obtain the reaction rates and neutrino emissivities assuming degenerate quarks and electrons, but without making any assumption about the degeneracy state of neutrinos. This approximation is implemented only in the squared matrix element $\left\langle|\mathcal{M}|^{2}\right\rangle$ by replacing the particle momenta and energies by the corresponding Fermi momenta and chemical potentials. The approximation is not used in the Fermi blocking factors, nor in the delta function. We show below only the relevant results and present a detailed derivation in Appendix A. The error due to this assumption is small, as estimated in Appendix B where we compare the approximate and the exact rates Madsen (1993) for the nonleptonic reaction, which is the dominant one.

The reaction rate for the decay process $d\rightarrow u+e^{-}+\bar{\nu}_{e}$ is

where $c_{1}=3G_{F}^{2}\cos^{2}\theta_{C}/(2\pi^{5})$ and the angular integral $I$ given by Eq. (47) (see also Ref. Wadhwa (1995)).

The rate $\Gamma_{\text{II}}$ for the process $s\rightarrow u+e^{-}+\bar{\nu}_{e}$ can be obtained replacing $\mu_{d}$ by $\mu_{s}$ and $\cos^{2}\theta_{C}$ by $\sin^{2}\theta_{C}$ in the latter expression.

For the process $u+e^{-}\leftrightarrow d+\nu_{e}$ we find

for the direct process (electron capture by $u$ quarks) and $\Gamma_{\text{III}}^{rev}=e^{-\xi_{d}}\Gamma_{\text{III}}^{dir}$ for the reverse process (neutrino absorption by $d$ quarks) where $\xi_{d}=(\mu_{u}+\mu_{e}-\mu_{d}-\mu_{\nu_{e}})/T$. The angular integral $J$ is given in Eq. (54).

The rate $\Gamma_{\text{IV}}^{dir}$ for $u+e^{-}\rightarrow s+\nu_{e}$ can be obtained replacing $\mu_{d}$ by $\mu_{s}$ and $\cos^{2}\theta_{C}$ by $\sin^{2}\theta_{C}$ in the expression for $\Gamma_{\text{III}}^{dir}$. For the reverse process $s+\nu_{e}\rightarrow u+e^{-}$ we have $\Gamma_{\text{IV}}^{rev}=e^{-\xi_{s}}\Gamma_{\text{IV}}^{dir}$ where $\xi_{s}=(\mu_{u}+\mu_{e}-\mu_{s}-\mu_{\nu_{e}})/T$.

Finally, for $u_{1}+d\rightarrow u_{2}+s$ we have

where $c_{2}=9G_{F}^{2}\sin^{2}\theta_{C}\cos^{2}\theta_{C}/(2\pi^{5})$. The rate for the reverse process $u+s\rightarrow u+d$ is given by $\Gamma_{\text{V}}^{rev}=e^{-(\mu_{d}-\mu_{s})/T}\Gamma_{\text{V}}^{dir}$.

The neutrino emissivity rate per baryon is given below for all the relevant processes. For $d\rightarrow u+e^{-}+\bar{\nu}_{e}$ we have

The emissivity $\varepsilon_{\text{II}}$ for $s\rightarrow u+e^{-}+\bar{\nu}_{e}$ can be obtained replacing $\mu_{d}$ by $\mu_{s}$ and $\cos\theta_{C}$ by $\sin\theta_{C}$ in the previous expression. For $u+e^{-}\rightarrow d+\nu_{e}$ we find

Similarly, the emissivity $\varepsilon_{\text{IV}}$ for $u+e^{-}\rightarrow s+\nu_{e}$ is obtained replacing $\mu_{d}$ by $\mu_{s}$ and $\cos\theta_{C}$ by $\sin\theta_{C}$ in the previous formula.

The total neutrino and antineutrino emissivities in cold deleptonized matter are $\varepsilon_{\nu}=\varepsilon_{\text{III}}+\varepsilon_{\text{IV}}$ and $\bar{\varepsilon}_{\bar{\nu}}=\varepsilon_{\text{I}}+\varepsilon_{\text{II}}$. For hot neutrino-rich matter, the total neutrino emissivity is $\varepsilon_{\nu}=\varepsilon_{\text{III}}(1-e^{-\xi_{d}})+\varepsilon_{\text{IV}}(1-e^{-\xi_{s}})$.

Let us assume that the hadron-quark combustion process occurs in a cold NS. Due to the energy released by weak reactions, quark matter within the flame becomes hot in a very short timescale. On the other hand, during flame propagation we do not expect a significant heating of the hadronic matter ahead the combustion front because the flame propagates at a very high speed Lugones et al. (2002); Niebergal et al. (2010) and there isn’t enough time for heat conduction to occur. Moreover, if the flame is supersonic there is no heat conduction at all to the hadronic layers during flame propagation. Thus, the evolution of quark matter abundances towards chemical equilibrium is governed by

together with Eq. (11) for the time evolution of the temperature. Electric charge neutrality and baryon number conservation (Eqs. (17) and (18)) can be used to relate $s$ quark and electron abundances with $u$ and $d$ quark abundances:

Integrating the above equations numerically, we obtain the time evolution of the particle abundances and the temperature, as well as the neutrino emissivity as the system approaches to equilibrium.

Figs. 2 and 3 show that after quark deconfinement there is a significant increase in the temperature $T$ and in the strange quark abundance $Y_{s}$ in a timescale of $\sim 10^{-9}\,\mathrm{s}$ due to weak interaction decays that drive quark matter to chemical equilibrium. In this process, the abundances of quarks $u$, $d$ and electrons decrease. The increase in $T$ strongly depends on the initial strangeness of hadronic matter: for vanishing initial strangeness the temperature increases considerably more than for large initial strangeness (see Table 3). In fact, for transitions at $n_{B}=0.32\,\mathrm{fm}^{-3}$ (see Fig. 2), $T$ goes from 1 MeV to about 40 MeV for $\eta$=0 (vanishing initial strangeness), and to about 20 MeV for $\eta$=0.4 (large initial strangeness). This occurs because for $\eta=0.4$ just deconfined matter is closer to chemical equilibrium than matter without strangeness. As a result, there is more energy release when $\eta=0$ and the final temperatures attained for $\eta=0$ are larger than for $\eta=0.4$. Transitions at higher densities present a more drastic temperature rise, as can be seen in Fig. 3 for $n_{B}=0.96\,\mathrm{fm}^{-3}$ ($T$ goes from 1 MeV to about 60 MeV for $\eta$=0 and to about 30 MeV for $\eta$=0.4). When the strong interaction is turned on, temperature increments are further enhanced compared with the zero strong coupling constant case. In fact, we find that for $\alpha_{c}$ = 0.47 the final temperature is about 10% larger than for $\alpha_{c}=0$, for both $n_{B}=0.32\,\mathrm{fm}^{-3}$ and $n_{B}=0.96\,\mathrm{fm}^{-3}$.

The neutrino and antineutrino energy loss per baryon can be seen in Fig. 4. The largest emissivities are attained during the first nanosecond after deconfinent. Thereafter, they decline by several orders of magnitude in a timescale of $\sim 10^{-8}-10^{-7}\,\mathrm{s}$. The largest values of the emissivities per baryon range between $10^{9}-10^{12}\,\mathrm{MeV/s}$ for neutrinos and $10^{6}-10^{9}\,\mathrm{MeV/s}$ for antineutrinos. Notice that the emissivity follows the same trend as the temperature: it increases with $\alpha_{c}$, decreases with $\eta$, and increases with the baryon number density.

The relevance of the different weak interaction processes in a cold NS is analyzed in Fig. 5. In all cases, the nonleptonic process $u+d\rightarrow u+s$ dominates the rate until matter reaches chemical equilibrium.

The electron capture reactions have a smaller contribution to the rate, but they are the most important processes that emit neutrinos. However, once matter reaches equilibrium, their contribution to the total rate decays steeply. Notice that the contribution of the decay of $s$ and $d$ quarks to the total rate is always negligible. After $10^{-9}-10^{-8}$ s, chemical equilibrium is maintained essentially by the two nonleptonic processes $u+d\leftrightarrow u+s$.

Finally, we calculate the total energy released by each baryon of quark matter in the form of neutrinos $({\cal E}_{\nu_{e}})$ and antineutrinos $({\cal E}_{\bar{\nu}_{e}})$. Both, ${\cal E}_{\nu_{e}}$ and ${\cal E}_{\bar{\nu}_{e}}$, can be obtained by integrating the corresponding emissivities with time, since the initial time of hadron deconfinement (at region 2 in Fig. 1) until the moment when chemical equilibrium is attained (when the baryon enters region 4 in Fig. 1). Our results are shown in Table 3 and show that each baryon releases $\sim 6-60\,\mathrm{MeV}$ in neutrinos and $\sim 0.03-0.5\,\mathrm{MeV}$ in antineutrinos depending on the initial strange quark abundance and the initial baryon number density. For a typical NS with $10^{58}$ baryons, we can estimate an order of magnitude of the energy released if the whole star were converted into quark matter. This value is just a rough estimate because e.g. we don’t consider that the density of matter changes along the star. Anyway, it gives a hint of the “chemical” energy released by the hadron-quark conversion and suggests that it could be sufficient to power a gamma ray burst. Notice that this estimate doesn’t take into account the additional amount of (gravitational) energy that would be released by the rearrangement of the stellar configuration after the hadron-quark conversion.

We assume now that the hadron-quark combustion process occurs in a hot NS, such as in a proto NS born in the aftermath of a core collapse supernova or in the compact object formed after a merging event in a binary system. Due to the energy released by weak reactions, quark matter just behind the combustion front becomes even hotter in $\sim 1$ nanosecond. Additionally, hadron matter ahead the front is already hot. As a consequence, neutrinos are expected to be trapped in quark matter and, therefore, the reverse reactions ($\nu$-captures) III and IV are allowed. On the other hand, we have shown that the contribution of the decay of $s$ and $d$ quarks to the total rate is always negligible. Thus, the equations for the evolution of the quark abundances are now:

As before, electric charge neutrality and baryon number conservation give:

For the case of a phase transition taking place in a hot leptonized NS, there is an additional constraint coming from lepton number conservation because of the fact that during the short duration of the phase transition, neutrinos are trapped. As a consequence, the lepton abundance verifies:

The evolution of temperature is obtained from Eq. (11).

Our results are shown in Figs. 6–13. To fix the initial values of the particle number densities we have considered $\xi=1.4$, $\eta=0$ and $0.4$, and $\kappa$= 0.01 where the latter determines the initial value of the neutrino number density (see Table 2). Notice that now we are using two different initial temperatures, $T_{i}=20$ and $40\,\mathrm{MeV}$. The results for the temperature and the abundances of $u$, $d$, $s$ quarks have some similarities with the cold NS case studied before. In particular, it is valid the same analysis about the effect of the strong coupling constant and the presence of finite strangeness in quark matter. However, the difference between the final and the initial temperatures ($\Delta T\equiv T_{f}-T_{i}$) is smaller than in the NS case (see Figs. 6, 7, 8, 9 and Table 4).

The most significant differences with respect to the case of cold NS matter are related to the evolution of the electron and neutrino abundances, as shown in the right panel of Figs. 6, 7, 8, and 9. Due to neutrino trapping, the lepton abundance $Y_{L}$ is assumed to be constant during the transition. The electron abundance decreases as in the case of cold NSs, but now it doesn’t go to zero. The neutrino abundance increases up to about 0.1 neutrinos per baryon for $\eta$= 0. We also find that if quark matter has an initial strangeness $\eta$= 0.4, the final abundance of neutrinos tends to be smaller ($\sim 0.02-0.03$ neutrinos per baryon).

The net neutrino energy loss per baryon is shown in Figs. 10 and 11 for the processes $u+e^{-}\leftrightarrow d+{\nu_{e}}$ and $u+e^{-}\leftrightarrow s+{\nu_{e}}$. The emissivity is high during $\sim 10^{-9}\,\mathrm{s}$ and is followed by a steep decline to a value several orders of magnitude smaller in a timescale of $\sim 10^{-8}\,\mathrm{s}$. The maximum value of the emissivity per baryon is between $10^{9}\,\mathrm{MeV}/\mathrm{s}$ and $10^{12}\,\mathrm{MeV}/\mathrm{s}$ as for cold NSs. Initially, most neutrinos are emitted as a consequence of the $u+e^{-}\leftrightarrow d+{\nu_{e}}$ process, but around $t\approx 10^{-10}$ s the emissivity of this process falls significantly because the direct and reverse processes attain equilibrium around this time (see the point in Figs. 12 and 13 at which $\Gamma_{\text{III}}^{rev}$ and $\Gamma_{\text{III}}^{dir}$ become equal). On the other hand, the emissivity of the process $u+e^{-}\leftrightarrow s+{\nu_{e}}$ remains active for a longer time because processes involving $s$ quarks attain equilibrium later (see curves for the processes $\mathrm{IV}$ and $\mathrm{V}$ in Figs. 12 and 13). In fact, for $t\gtrsim 10^{-10}$ s, the neutrino emission is dominated by the process $u+e^{-}\leftrightarrow s+{\nu_{e}}$.

In Table 4 we show the total energy per baryon released by quark matter in the form of neutrinos, which has been obtained through time integration of the total neutrino emissivity. The total energy per baryon is in the range ${\cal E}_{\nu_{e}}=2-44\,\mathrm{MeV}$ for different initial conditions. As for cold NSs, we estimate the order of magnitude of the energy released if the whole star is converted into quark matter. For a typical star with $10^{58}$ baryons we find ${\cal E}_{total}\approx 0.4-7\times 10^{53}\,\mathrm{erg}$. We emphasize that neutrinos produced within the flame stay trapped in the hot quark matter core of the star until it cools and deleptonizes. As a consequence, they are released in a diffusion timescale which is of the order of tens of seconds Pons et al. (2001).

Finally, in Figs. 12 and 13 we show the reaction rates for all the processes in a hot NS. The nonleptonic process $u+d\rightarrow u+s$ is dominant most of the time in most cases but the process $u+e^{-}\leftrightarrow d+{\nu_{e}}$ has a significant contribution in the scenario of low initial strangeness, and becomes dominant near chemical equilibrium.

In this work, we have performed a detailed analysis of a flame that converts hadronic matter into quark matter in a compact star (see Fig. 1). We focused on a small portion of just deconfined quark matter which is initially at the deconfinement region of the flame and followed its evolution as it approaches equilibrium by means of weak interactions. For quark matter, we employed the MIT bag model at finite temperature including the effect of the finite mass of strange quarks and QCD corrections to the first order in the coupling constant $\alpha_{c}$ (see Sec. III.1).

The time evolution of quark matter was described by means of the Boltzmann equation (Sec. II) using the reaction rates of all the relevant weak interaction processes (Sec. IV). These reaction rates have already been calculated in the literature (see e.g. Dai et al. (1993, 1995b, 1995a); Madsen (1993); Anand et al. (1997); Lai et al. (2008a, b)) but we have introduced some improvements here. For example, in Refs. Dai et al. (1993, 1995b, 1995a); Anand et al. (1997); Lai et al. (2008a, b) neutrinos in hot NSs are taken as completely degenerate and those in cold NSs as completely non-degenerate. Although this is a reasonable approximation we have generalized the results, i.e. we treated the neutrinos without making any assumption about their degeneracy. Also, we have treated the equation of state in more detail. In some previous works (e.g. Refs. Dai et al. (1993, 1995b, 1995a); Anand et al. (1997); Lai et al. (2008a, b)) the equation of state was considered at $T=0$ even for matter at finite temperature. Again, this is a plausible approximation when quarks are degenerate, but we have generalized this issue by considering the full expressions at finite temperature.