Dark matter interactions with muons in neutron stars

Neutron stars contain a significant number of stable muons due to the large chemical potential and degenerate electrons. This makes them the unique vessel to capture muonphilic dark matter, which does not interact with other astrophysical objects, including Earth and its direct-detection experiments. The infalling dark matter can heat up the neutron star both kinetically and via annihilations, which is potentially observable with future infrared telescopes. New physics models for muonphilic dark matter can easily be motivated by, and connected to, existing anomalies in the muon sector, e.g. the anomalous magnetic moment or LHCb's recent hints for lepton-flavor non-universality in $B\to K\mu^+\mu^-$ decays. We study the implications for a model with dark matter charged under a local $U(1)_{L_\mu-L_\tau}$.


INTRODUCTION
Among the many odd properties of neutron stars (NS), one of the weirdest is certainly the existence of stable hyperon and muon layers. Much like the neutrons, these usually unstable particles are stabilized by the degenerate Fermi-gas phase, which Pauli-blocks the would-be daughter particles of a decay. A typical old NS contains roughly 10 55 muons, compared to about 10 57 neutrons [1][2][3]. This lets NS interact with and potentially capture dark matter (DM) that couples mainly to muons, unique among astrophysical objects [4].
Assuming a large enough muon-DM cross section to capture a significant number of DM particles, the infalling DM unavoidably transfers heat to the NS, see Refs. [25][26][27][28][29]. This can increase the temperature of old NS from O(100 K) to O(2000 K), leading to an infrared blackbody spectrum that is in principle within range of future telescopes such as the James Webb Space Telescope, the Thirty Meter Telescope, or the European Extremely Large Telescope [25]. Further heating can occur through DM annihilations inside the NS.
In this letter we will discuss this capture of muonphilic DM in NS and the resulting heating, allowing for both DM and the mediator to be light to go beyond existing calculations. As a well-motivated example for muonphilic DM we consider an U (1) Lµ−Lτ model that can also ameliorate existing anomalies in the muon sector.

DARK MATTER IN NEUTRON STARS
The capture of DM particles by the NS depends on the macroscopic NS properties such as its mass M , radius R and temperature T , and also on the properties of its Fermi-degenerate medium, esp. chemical potential and lepton fractions. We consider a NS profile based on realistic two-and three-nucleon forces obtained in Refs. [1,2], where phenomenological fits are performed for equations of state of varying stiffness, neglecting the possibility of exotic quark phases. We present our results for the lowmass configuration of model BSK20 which corresponds to the NS values M = 1.52 M , R = 11.6 km, the core muon fraction Y µ = 2 × 10 −2 , and the muon chemical potential in the core µ µ = 65 MeV. We refer the reader to App. A of Ref. [4] for details regarding radial profiles of BSK20 and possible uncertainties on the DM capture rate.
Dark matter capture: The maximal capture rate, i.e. the rate at which all DM particles χ with mass m χ that intercept the NS are captured, is given by the geometric rate [4] C 5.6 × 10 25 s with DM energy density ρ χ around the NS. For DMmuon contact interactions this rate can be achieved for cross sections above πR 2 /N µ 5 × 10 −43 cm 2 . The geometric rate C is shown in Fig. 1 as a black dashed line fixing ρ χ = 0.4 GeV/cm 3 . We will consider interactions between DM particles χ and muons that are mediated by a potentially light gauge boson Z with vector coupling g q µ to muons and coupling g q χ to χ. The differential elastic scattering cross section in the non-relativistic limit is then  valid both for scalar and fermion DM.
is the recoil energy in the muon rest frame, and m red the reduced mass of the DM-muon system. The rate of accretion of DM particles from the halo into a NS by scattering off muons with density n µ then reads [30], in the zero-temperature limit, To approximate Pauli blocking we introduce the factor ζ(r) ≡ Min[1, δp(r)/p F (r)] following Refs. [4,31,32].
Here δp(r) √ 2m red v esc (r) is the typical momentum transfer in the muon rest frame and p F (r) = 2m µ µ µ (r) the Fermi momentum. f v is the DM velocity distribution in the NS frame and u χ the DM speed in the halo [4]. To capture a DM particle from the halo, the recoil energy should be between The Z -mass dependence of the capture rate is given by For m χ m µ > m Z the capture cross section and rate thus become independent of the mediator mass.
For a close-by NS we take the local DM density ρ χ = 0.4 GeV/cm 3 and present the DM capture rate on muons in Fig. 1. The colored dotted lines correspond to the capture rate obtained using Eq. (3), whereas the solid curves are more accurate rates which are numerically obtained by using the methods of Ref. [4], where possible non-trivial kinematics due to the degenerate nature of NS matter are taken into account. For m Z > m χ > m µ the rate C scales as m −1 χ ; for m χ p F /( √ 2v esc ) < m Z , Pauli blocking of the final state muon becomes efficient and the capture rate saturates and scales as m 2 χ . For m Z < m µ , Pauli blocking instead becomes important for m χ m Z . The full calculation of the capture rate differs from Eq. (3) only when Pauli blocking is relevant and suppresses C by a factor up to 35. Thermal effects in the capture rate are negligible for the range of DM masses of interest here. The main theoretical uncertainties in our calculation are the NS muon content and chemical potential, which should be at most off by a factor of two [4] (see also Ref. [29]). Neutron star heating: As shown in Refs. [25][26][27][28][29], the NS temperature increase due to DM capture is approximately As can be appreciated from Fig. 1, it is not difficult to saturate C ∼ C for much of the interesting parameter space. The SM-DM interaction underlying C is of course impossible to reconstruct from T kin , but the observation of NS with T < T kin in DM-rich environments would still allow to set limits on DM-µ interactions. Importantly, truly muonphilic DM can only be captured in NS, much like inelastic DM [25,27], whereas most other DM models also allow for capture in other objects, including Earth. Additional NS heating comes from DM annihilation into SM particles other than neutrinos, which arises for non-asymmetric DM. NS are expected to cool via neutrino emission through the modified Urca process for the first million years and through photon emission afterwards [33,34]. Once the rate of DM accretion equilibrates with DM annihilation the emissivity due to DM annihilation does not depend on temperature. This implies that once the temperature of the NS is sufficiently small, the power from DM annihilations that heat up the NS equals that of the photon emission, resulting in a constant temperature. For an old NS ten parsecs away, the maximal heating due to annihilations gives T ann 2480 K [ρ DM /(0.4 GeV/cm 3 )] 0.45 [25,34].
For asymmetric DM [35], which cannot self-annihilate by construction, it seems possible to accumulate enough DM inside the NS to form a black hole that destroys its host. The observation of very old NS in DM-rich regions would then allow us to put strong constraints on, e.g., the muon-DM cross section [4]. Fermionic DM is typically difficult to constrain in this way due to the additional pressure from the Pauli exclusion principle. Bosonic DM can in principle collapse much easier; however, as pointed out in Refs. [32,36], DM self-interactions play an important role and can hinder black hole formation even for bosonic DM, as discussed below.

DARK MATTER CHARGED UNDER Lµ − Lτ
As a well-motivated example for muonphilic DM we consider an extension of the SM by the anomaly-free gauge group U (1) Lµ−Lτ [13][14][15][16]. Its gauge boson Z does not couple to first-generation particles and is thus only weakly constrained, as shown in Fig. 2. It is arguably the simplest explanation of the (g − 2) µ anomaly if g ∼ 5 × 10 −4 and M Z ∼ 10-100 MeV [20], fully testable with currently running experiments. For masses around 10 MeV this Z could furthermore ameliorate the observed tension in the Hubble parameter H 0 by contributing slightly to N eff [37], and could also affect the high-energy neutrino flux measured in IceCube [38,39]. In a different region of parameter space, a heavy L µ − L τ gauge boson could resolve the persistent anomalies in b → sµ + µ − transitions, as long as additional Z µ couplings to bγ µ P L s are generated, e.g. via vector-like fermions [21][22][23][24]. This requires m b m Z in order to obtain the desired contact-operator coefficient C µ 9 −0.95 [40] and m Z /g 1.1 TeV to evade the strong constraints from B s -B s mixing (at 2σ) [41]. In the following we will focus on two benchmark values for L µ − L τ that satisfy the existing constraints from Fig. 2 and could play a role either for (g − 2) µ and H 0 (m Z = 10 MeV, g = 5 × 10 −4 ) or b → sµ + µ − (m Z = 100 GeV, g = 0.1).
DM charged under U (1) Lµ−Lτ has been discussed for some time, typically as a WIMP with thermal abundance [54,55]. An appropriately chosen L µ − L τ charge q χ can render a new particle χ stable with muonphilic Z connection to the SM -the DM interactions with tauons being practically irrelevant. For most values of q χ the full Lagrangian actually has an additional global symmetry U (1) χ that corresponds to conserved DM number. In analogy to the observed baryon number asymmetry one could assume that the cosmological history also led to a DM asymmetry [35] which results in a density Ω asym proportional to the asymmetry. Unless the Z couplings are tiny, DM χ will also be produced thermally, leading to a symmetric DM component Ω sym . We will consider both options here but remain agnostic about the potential DM asymmetry. As a concrete example we will discuss scalar DM, the fermionic case is similar.
the annihilation for m χ > m Z . The orange regions indicate that the DM capture rate is maximized, C C * , so a nearby NS would be kinetically heated to 1700 K. Notice that our asymmetric scalar DM unavoidably has repulsive self-interactions mediated by the Z . As a result, even asymmetric scalar DM will not lead to black hole formation inside the NS [32,36], unless the DM masses are far above the values considered here.
In the light-mediator case these repulsive selfinteractions can also have an impact on structure formation, which has been argued to prefer DM-DM transfer cross sections σ T /m χ ∼ 1 cm 2 /g on dwarf-galaxy scales (DM velocities v dwarf ∼ 10 km/s) [57]. We show this contour in Fig. 3 using the formulae of Refs. [58][59][60] for σ T (χχ → χχ). Cross sections of similar size do not seem to be favored on cluster scales, although the issue is currently far from settled. Suppressing σ T /m χ on cluster scales (v cluster > 2000 km/s) requires a velocitydependent cross section [61], which arises in our model for m Z /m χ v cluster /c, as can be seen in Fig. 3 (left) [see also Ref. [43]]. The DM mass region 1-10 GeV in the light Z case is thus potentially preferred due to smallscale structure formation, while the region above the blue dashed line in Fig. 3 is disfavored by observations such as the bullet cluster [62].
For symmetric DM there is the possibility of additional NS heating through the annihilations χχ → µ + µ − , τ + τ − , Z Z , followed by Z decay to charged leptons. For the light Z case of Fig. 3 2)µ (left) and one for b → sµ + µ − (right). The black line denotes the WIMP scenario where the thermal symmetric abundance Ωsym matches the observed DM abundance Ω obs ; above is the region of asymmetric DM. In the orange parameter space we expect kinetic heating of old nearby NS to 1700 K by DM capture on muons; in the red region additional annihilation increases the NS temperature to 2500 K. The blue line shows the DM self-interaction transfer cross section σT (χχ → χχ)/mχ = 1 cm 2 /g that can resolve small-scale structure issues [61], esp. if suppressed on cluster scales as here for mχ GeV (green ellipse).

(left) this is irrelevant since the
annihilation into charged fermions is p-wave suppressed and the s-wave channel χχ → Z Z only generates neutrinos; this also eliminates indirect-detection constraints [63]. For heavier Z , the s-wave χχ → Z Z → 4 could lead to indirect-detection signatures, but is currently only relevant for m Z < m χ 100 GeV [64], i.e. not for Fig. 3 (right). χχ → Z Z → 4 still heats the NS though, since the Z decays inside the NS with BR(Z → µμ + ττ ) 2/3. Accordingly, the m χ > 100 GeV WIMP parameter space in Fig. 3 (right) leads to nearby NS temperatures up to 2480 K [25].
So far we have ignored additional interactions that arise from kinetic mixing or the Higgs portal, which will induce DM couplings to non-muonic matter and are highly constrained by direct-detection experiments [24,[65][66][67]. For example, the one-loop scattering of DM via Z on the protons inside a nucleus N takes the form [6,24] keeping both the tree-level contribution from a Lagrangian term 2 Z µν A µν [68,69] and the finite one-loop contribution from muon and tauon loops [65,69]. For heavy DM and = 0, XENON1T [70] seemingly excludes the entire parameter space of Fig. 3 (right) using Eq. (7), while CRESST-III [71] excludes the entire m χ 1 GeV parameter space of Fig. 3 (left). However, there is no reason to ignore the tree-level kinetic mixing angle and it is reasonable to expect other new particles in the model to contribute further to γ-Z mixing. As a result, the full kinetic mixing should be treated as a free parameter, constrained by direct-detection experiments. The observation of old cold NS can on the other hand set a constraint that is independent of the kinetic mixing angle and thus perfectly complementary to Earth's direct-detection experiments. This holds even more true for other muonphilic mediators, e.g. pseudoscalars [29].

CONCLUSION
The muon g − 2 experiment at Fermilab (E989) is expected to confirm or dispute the longstanding magneticmoment anomaly by the end of 2019 [72], while Belle II will probe the L µ − L τ solution to said anomaly. Together with LHCb, Belle II is also expected to scrutinize the B → K ( * ) µ + µ − anomalies that hint at lepton non-universality. With these tantalizing and soon to be reevaluated hints for new physics in the muon sector it behooves us to consider muonphilic DM, which only has highly suppressed couplings to first-generation particles. Using the curious fact that NS contain a large population of stable muons we can expect limits on DM-muon interactions from the DM capture on NS. Indeed, the infalling DM will heat old NS up to O(2000 K) due to elastic scattering and potentially annihilation, which is in reach of future infrared telescopes. For light mediator masses, relevant for (g − 2) µ , muonphilic DM can also have self-interactions that resolve structure-formation issues. Overall we have shown that muonphilic DM is not as elusive as naively expected, both in terms of motivation and in terms of signatures.