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Enhanced Neutrino Emissivities in Pseudoscalar-mediated Dark Matter Annihilation in Neutron Stars

We calculate neutrino emissivities from self-annihilating dark matter (DM) (χ) in the dense and hot stellar interior of a (proto)neutron star. Using a model where DM interacts with nucleons in the stellar core through a pseudoscalar boson (a) we find that the neutrino production rates from the dominant reaction channels or , with subsequent decay of the mediator , could locally match and even surpass those of the standard neutrinos from the modified nuclear URCA processes at early ages. We find that the emitting region can be localized in a tiny fraction of the star (less than a few percent of the core volume) and the process can last its entire lifetime for some cases under study. We discuss the possible consequences of our results for stellar cooling in light of existing DM constraints.

(Dated: January 8, 2018) We calculate neutrino emissivities for annihilating Dark Matter with pseudoscalar-mediated couplings to ordinary matter in dense stellar environments. We show that for Dark Matter models compatible with current direct detection limits and cosmological relic abundance, the neutrino production rates from channels χχ → νν or via pseudoscalar mediators χχ → aa with subsequent decay a → νν, can be locally much stronger than the modified URCA processes at early stages of star cooling. In light of these results we discuss the parameter ranges and thermodynamical stellar conditions that could give rise to inner temperature profiles warmer than those obtained for standard scenarios, having thus an impact on possible superfluid components in the inner core.
Dark Matter (DM) is an essential part of the standard cosmological model. Despite the tremendous amount of progress that has been made both on the theoretical and experimental fronts in the search for this missing matter, which we now know constitutes nearly 85% of the Universe's matter density, its true nature remains an open question [1]. The Standard Model (SM) of particle physics alone cannot explain the nature of this DM, suggesting that the model must be extended. Many theoretical model proposals have arised to try to explain the existing phenomenology. In this work, we consider the so-called Coy Dark Matter model [2] which belongs to the family of simplified models (see for instance [3][4][5]) including two new particles to the SM, i.e. a Dirac fermion DM candidate, χ, and a pseudoscalar mediator, a, with masses m χ and m a , respectively, connecting ordinary and dark sectors. The interaction lagrangian of the model reads where g χ is the DM-mediator coupling, g f corresponds to the couplings to the SM fermions, f , and g 0 is an overall scaling factor. From the usual schemes used for matter couplings when introducing Beyond-Standard-Modelmotivated physics we will restrict to the so-called flavouruniversal which sets g f = 1 for all SM fermions. Let us recall, however, that there are other schemes where a couples either to quarks or leptons exclusively, and with a flavour structure. Typically, in these models DM phenomenology is controlled by four parameters, m χ , m a , g χ , and g 0 g f . In the range m χ < m Higgs and m a < m χ , the relevant annihilation processes are χχ → ff and χχ → aa. For the first case the annihilation cross-section has a coupling strength σ a ∝ (g 0 g χ ) 2 σ 0 , while for the second σ a ∝ g 4 χ σ 0 as derived in [6]. Due to the pseudoscalar portal, the Coy DM model provides spin dependent interactions with nucleons (n) at tree level. Nevertheless, the n-DM interaction considered in direct searches is suppressed because it is momentum dependent (see [7][8][9] for details). Instead, the spin independent cross-section is not present at tree level but the effective interaction at one-loop can be constructed [10]. Estimations of both cross-sections in vacuum are given in [7]. In this work we will be interested, particularly, in the annihilation reactions χχ → ff and χχ → aa with subsequent decay a → ff and, further, we will explore possible astrophysical consequences in the neutrino fermionic channel as dense (compact) stars are efficient DM accretors. One of the key quantities that can dictate the internal stellar energetic balance is the local energy emissivity, Q E = dE dV dt , to be understood as the energy emitted/injected per unit volume per unit time, through a prescribed particle physics reaction. In order to be definite we will restrict our analysis to three different sets of flavour-universal parameters that are not in conflict with existing phenomenology of direct detection experiments [11] nor cosmological bounds [12]. The masses and couplings in sets A and B in Table I are mainly determined by DM relic abundance because the DM mass is in the region where direct detection experiments are less restrictive. On the contrary, the couplings in set C are constrained by LUX results [13] in spin independent and spin dependent cross-sections. In all cases, we estimate the parameters using MicroOmegas [14] and direct detection cross-sections at one-loop level [7,10].
In the above quoted reactions involving annihilating with dΦ = d 3 p1 ) the 4-body (12 → 34) phase space element and |M| 2 the spin-averaged squared matrix element of the reaction considered. The additional factor f (f 1 , f 2 , f 3 , f 4 ) accounts for the initial and final particle distribution functions contribution we will discuss below for each case. We will denote p 1 = (E 1 , p 1 ), p 2 = (E 2 , p 2 ) as the incoming 4-momenta while p 3 = (E 3 , p 3 ), p 4 = (E 4 , p 4 ) are the outgoing 4-momenta, respectively. The detailed associated Feynman diagrams are shown in Fig.(1).
Specifically, for the case of annihilation into fermionic pairs, Q ff E , we can write the general expression for the spin-averaged squared matrix element as where q 2 = s = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 is the transferred momentum, i.e. s, the Mandelstam variable. E q = | q| 2 + m 2 a is the transferred energy. In this case and f χ , f f are the local stellar distribution functions for DM and fermionic particles, respectively, containing density and temperature dependence we will discuss further below. Γ is the pseudoscalar particle decay width in the local medium through the reaction a → ff . It is obtained using the optical theorem as Γ = 1 E q Im Π( q) where Π( q) is the pseudoscalar polarizarion insertion given by [16,17] and the corresponding cut of the associated tadpole diagram involves the fermion propagator including a vacuum and matter contribution [16], G 0 (k) = For the DM annihilation into two pseudoscalars, Q aa E , instead, the average involved is performed over initial fermionic states and the squared matrix element reads, where 13 being θ ij the angle between p i and p j . From trivial integration of the Dirac delta function and one of the outgoing momenta, say p 4 in eq.(2), it is easy to see that meaningful real angular variables set kinematical limits over E 4 such that In In this case for the annihilation into pseudoscalars the factor f ( In order to see the physical relevance of the quantities under scrutiny, at this point we will assume our stellar scenario is that of a dense star, a Neutron Star (NS). Briefly, a NS is mostly constituted by nucleons ( 90% neutrons) forming a central core at a density in excess of nuclear saturation density, ρ 0 2.4×10 14 g/cm 3 . For the sake of our discussion we will consider baryonic density ρ b ∼ 2ρ 0 . NSs are born as hot lepton-rich objects with internal temperature T ∼ 20 MeV evolving into cold T ∼ 10 keV neutron-rich ones. An average NS has a radius R 12 km and mass M 1.5M being thus a star with large compactness ∼ GM/R, capable of accreting DM from an existing galactic distribution.
However, accretion of a dark component will proceed not only during the collapsed stage but during most of stellar lifetime at different capture rates, C χ . First, in the progenitor stages the progressively denser nuclear ash central core allows the build-up of a finite number density, n χ (r), over time. Later, when a compact object is formed the capture rate is enhanced, as it has been previously estimated, see for example [18,19]. In this way, assumed an equation of state for regular SM matter in the interior of the NS, at a given galactic location, and with a corresponding ambient DM density ρ χ , it is approximated by where ρ ambient is a factor dealing with the ratio of the scattering cross-section taken as reference when comparing experimental limits, to the geometrical value (σ 0 = mnR 2 M ∼ 10 −45 cm 2 ). It saturates to f χ ∼ 1 if σ s σ 0 . Following [2] we take the scattering cross-section as σ s ∼ where E r is the nuclear recoil energy, g n is the n-DM scattering coupling strength and µ is the n-DM reduced mass. For parameters in Table I used in this work f χ ∼ 1. Current limits for a DM candidate under the WIMP paradigm in direct detection searches remain at the level of σ s 10 −44÷−45 cm 2 in the m χ ∼ (0.1 − 100) GeV range.
Once the DM particles are effectively captured by the star, a DM particle population N χ resides inside. Its number will depend on the capture and annihilation rate C χ , C a [22]. Note that in the range of masses we consider, evaporation effects [20] (as well as decay [21]) do not substantially modify the DM population as the kinetic to gravitational potential ratio remains small. As a function of time t, the two processes compete to yield a population N χ (t) = C χ /C a tanh(t/τ + γ(N χ,0 )) where γ(N χ,0 ) = tanh −1 ( C a /C χ N χ,0 ) and τ = 1/ C χ C a . For a typical progenitor of a compact star, N χ,0 ∼ 10 39 ( mχ 1 GeV ) [22], assuming that local densities for average NS galactic distances peak around ∼ 2 kpc where ρ χ ∼ 10 2 ρ ambient χ,0 . For times t τ ∼ 10 3÷4 yr the equilibrium sets N χ (t) ∼ C χ /C a .
As thermalization times for DM particles are much smaller than dynamical cooling times [23], inside the star the DM particle number density takes the form n χ (r) = n 0, χ e − mχ k B T Φ(r) , with n 0, χ the central value.
is the gravitational potential. Finally n χ (r) = n 0, χ e −(r/r th ) 2 with a thermal radius r th = 3k B T 2πGρnmχ . Normalization requires R 0 n χ (r)dV = N χ at a given time. Possible limiting values of N χ could arise from the fact that a fermionic Chandrasekhar mass could be achieved [24]. This possibility is safely not fulfilled as long as N χ (t) < N Ch , where N Ch ∼ ( M Pl /m χ ) 3 ∼ 1.8 × 10 54 (1 TeV/m χ ) 3 with M Pl the Planck mass.
Given the fact that inside the star the small DM fraction provides a Fermi energy E F,χ k B T , we can approximate its distribution function by a classical Maxwell-Boltzmann type 2mχ T . Note that the phase space factor f (f 1 , f 2 , f 3 , f 4 ) in eq. (2) will introduce DM density and T dependence into the calculation as a thermalized DM distribution exists inside the NS core. As for the outgoing fermions the medium density effects, will generally arise from the phase space blocking factors and collective effects [25].
We now discuss one interesting astrophysical consequence of this scenario by restricting the final fermion states to be under the form of neutrinos.
These weakly interacting SM fermions play a key role in releasing energy from energetic systems. For example, the end of a very massive star involves the gravitational collapse of the central core. From standard physics most of its gravitational binding energy is emitted in neutrinos (and antineutrinos) of the three families. A very efficient cooling scenario emerges in the first ∼ 10 5 yr. Standard processes such as the URCA cooling or the modified URCA (MURCA) cooling [26,27] can release neutrinos with associated emissivities respectively. Typical energetic scales are given by the conversion factor 1 MeV ∼ 10 10 K. R is a control function of order unity we will discuss below. We must keep in mind the fact that these neutrinos cool off the star as they leave, having scattered a few times with ordinary nucleon matter [28,29] after a first rapid trapping stage. These standard processes effectively release energy from the baryonic system, having a genuinely different effect in the energetic balance when compared to average energy injection from stellar DM annihilation processes [30].
In Fig.(2) left panel the logarithm (base 10) of the energy emissivity for the process χχ → νν is shown as a function of temperature for three sets of DM parameters in Table I (dashed, dotted and dash-dotted lines). The standard physics cooling is depicted here by the MURCA emissivity (solid line) for the sake of comparison. Although this latter is not the only process contributing to the effective cooling, it sets an upper limit to standard emissivities in the scenario depicted. We do not include URCA emissivities either since central densities must be higher to provide the ∼ 11% protons fraction required [31] to sustain the fast reaction. Further, we assume neutrinos do not get trapped after being produced and therefore their Fermi-Dirac distributions fulfill f ν ∼ 0. We can see that around T ∼ 0.1 MeV local central emissivities log 10 (Q E ) ∼ 21 ÷ 22 are balanced by the heating processes. At the thermodynamical conditions selected we obtain that typical values for the involved decay length are of order of ∼ 10 2 fm making it a negligible contribution to the neutrino transport. In Fig.(2) right panel, the logarithm (base 10) is shown for the reaction χχ → aa with subsequent decay a → νν. In this case the neutrino emissivity is largely enhanced with respect to the direct production of neutrinos χχ → νν. reactions, however.
The possible impact of the radial extent of an emitting inner region through these exotic processes should be correlated to the ratio ξ = ( √ 2r th /R b ), signaling the fraction where 95% of the aproximately Gaussian distribution of DM particles can be found, versus R b ∼ 9 km, radial extent of the boundary or limit of the core-to-crust region. Since the crust region has a tiny mass we will not consider this refinement [32] here. For the models analyzed in this work this ranges from ξ ∈ [0.03, 0.42] at T ∼ 1 MeV to ξ ∈ [0.01, 0.14] at T ∼ 0.1 MeV. As thoroughly studied [33] the effect of enhanced emissivities can have an impact on internal temperatures and matter phases depending critically on them, such as superfluid phases. In this sense, recent works [34,35] quote that the rapid cooling of the Cas A may be an indication of the existence of global neutron and proton superfluidity in the core. In these Bardeen-Cooper-Schrieffer (BCS) phases the critical temperature T c is related to the gap energy at zero temperature, ∆(T = 0) = 1.75k B T c , so that for T < T c there is a quenching of quantities such as emissivities [36] or heat capacities. This behavior can be described by the introduction of families of control functions R depending on the explicit process [37] and usually described through a Boltzmann-like factor ∼ e −∆/(k B T ) .
Focusing now in a fixed stellar evolutionary time, t, the radial (internal) heat equation r < R b can be written as 2 where κ is the thermal conductivity due to baryons and leptons (electrons) [38] κ = κ b + κ e . Q ν is the standard MURCA emissivity due to all processes releasing energy from the system arising from the nucleons, Q ν ∼ Q MURCA E . The DM heating term, H(r), has the radial dependence induced by the DM density so that H(r) = H 0 e −(r/r0) 2 with H 0 = Q aa E n 2 0,χ and r 0 = r th (T )/ √ 2. Although a detailed solution of the full evolution equation is out of the scope of this work one can nevertheless get a glimpse of the radial T -profile at different fixed evolutionary times solving eq.(7) assuming a flat initial profile to see how it is distorted from the presence of internal cooling and heating terms. We emphasize that only a full dynamical simulation will be able to obtain how the cooling mechanisms adjust inside the star as a temporal sequence.
For example, fixing a value T (r) ≡T for r ∈ [0, R b ] the central value T 0 is related to it through the approximate solution T (r) = T 0 + α r 2 6 − β  Table I in the rangeT ∼ (1 − 3) × 10 9 K. AtT = 10 9 K ∼ 0.1 MeV one can verify that the net effect of models B (β ∼ 10 5 K/cm 2 ) and C (10 7 K/cm 2 ), respectively, is heating the central volume while model A (β ∼ 10 −2 K/cm 2 ) provides net cooling, instead. We argue that this change in inner core temperatures with respect to standard cooling can thus affect the onset of superfluid ( 3 P 2 -3 F 2 neutron) phases at typical temperatures T c ∼ 5 × 10 9 K as local T (r) must be smaller than the T c for a finite gap energy. The star will self-adjust, however, in a consisting way that is to be dynamically determined to see at what extent the subsequent thermal evolution is affected in the stellar volume.
We have calculated the energy emissivity in DM annihilation into neutrinos from reaction channels χχ → νν or via pseudoscalar mediators χχ → aa, this latter with subsequent decay a → νν. We conclude that in the inner stellar regions where most of the DM population distribution exists the emissivity into neutrinos can be enhanced orders of magnitude with respect to the MURCA standard neutrino processes for parameter sets respecting constraints of direct detection limits and cosmological bounds. This, in turn, may lead to warmer inner temperature stellar profiles that can modify the onset of superfluid ( 3 P 2 -3 F 2 neutron) phases at typical temperatures T c ∼ 5 × 10 9 K. Although some other heating processes have been quoted in the literature (rotochemical heating [39] or hot blobs located at different depths in the crust in young NS [40]) the qualitative picture arising from the DM annihilation process is different, as the warmer inner pit can produce thermal instabilities in the inner core.