Asymmetric capture of Dirac dark matter by the Sun

Current problems with the solar model may be alleviated if a significant amount of dark matter from the galactic halo is captured in the Sun. We discuss the capture process in the case where the dark matter is a Dirac fermion and the background halo consists of equal amounts of dark matter and anti-dark matter. By considering the case where dark matter and anti-dark matter have different cross sections on solar nuclei as well as the case where the capture process is considered to be a Poisson process, we find that a significant asymmetry between the captured dark particles and anti-particles is possible even for an annihilation cross section in the range expected for thermal relic dark matter. Since the captured number of particles are competitive with asymmetric dark matter models in a large range of parameter space, one may expect solar physics to be altered by the capture of Dirac dark matter. It is thus possible that solutions to the solar composition problem may be searched for in these type of models.


I. INTRODUCTION
Over the past few decades, a large amount of evidence in support of the existence of dark matter (DM) has been assembled, see e.g., refs [1,2]. To describe the properties of DM has become one of the main issues not only in cosmology but also in particle physics, since the Standard Model leaves no room for its existence, which means extending it is necessary.
Some of the most studied particle candidates of DM are weakly interacting massive particles such as the lightest neutral supersymmetric particle, the neutralino (an in-depth review can be found in ref [3]).
If DM interacts with regular matter, it may scatter in astrophysical bodies such as the Sun and become gravitationally bound. Over time, this can lead to a large accumulation of DM, the effect of which used to be one of the proposed explanations of the solar neutrino problem [4]. This idea was later discarded in favour of neutrino flavour conversion by the neutral current phase of the SNO experiment [5]. Since the downward revision of heavy elements in the standard solar model [6], theoretical predictions and observations of helioseismology do not match and the Sun now faces the solar composition problem [7]. Again, a proposed solution is DM trapped in the Sun due to its interaction with regular matter. The effects of DM captured by the Sun has been extensively studied for various DM models [8][9][10][11][12][13][14] and also in other stars [15][16][17][18][19]. The idea is that DM particles collide with nuclei in the Sun, lose enough energy to become gravitationally bound and to eventually settle in the solar core after additional scattering. Once in the core, DM can scatter off of the thermal distribution of nuclei and gain some energy which is then lost by scattering in the outer regions of the Sun, the result of which is a lower core temperature [20]. This shift in temperature can change the solar neutrino fluxes that is observed by experiments, in particular the 8 B flux which varies as T 25 [21]. The implementation of DM being captured by the Sun would also affect helioseismology and may cure the solar composition problem. Annihilating DM can also provide a channel for indirect detection of DM as a flux of high energy neutrinos from these annihilations might be detectable on Earth. Such signals are being searched for by neutrino telescopes [22,23]. In the case where the DM has self interactions, DM particles already captured by the Sun provide a possibility of DM self-capture, which could lead to higher concentrations of DM inside the Sun. If DM annihilations produce high-energy neutrinos, this would increase the number of annihilations and thus the signal in neutrino telescopes could be enhanced [24]. Energy injection into the Sun by the DM annihilation products has been studied, but requires the halo density to be many orders of magnitude higher than the local density for the star to be affected [25].
Common for previously studied models in regard to its effect on the Sun is that there is always a single component in the dark matter halo. In this paper we will study the accumulation of DM in the Sun under the assumption that it is a Dirac fermion and that the DM halo is composed of equal amounts of DM and anti-DM particles. This opens up a possibility for an asymmetry in the captured number of DM and anti-DM to occur, either due to different capture cross sections for DM and anti-DM on solar nuclei or due to random fluctuations in the capture process.
The remainder of this paper is organised as follows: In section II, we present the framework that governs the amount of DM and anti-DM present in the Sun and discuss the behaviour of the DM to anti-DM asymmetry in the limiting cases. Then, in section III, we apply our framework to the case of the Sun and in section IV we present our results. Finally, in section V, we summarise our results and present our conclusions.

II. ACCUMULATION AND THE ASYMMETRY
We model the total number of DM particles in the Sun N and the total number of anti-DM particles in the SunN using the coupled system of first order differential equationṡ Here, c andc are the capture rates on solar nuclei for DM and for anti-DM, respectively, while C andC will depend on the self-capture rates of DM and anti-DM and on the ejection rates of DM and anti-DM that is already trapped by incoming particles from the DM halo.
The DM-anti-DM annihilation rate is given by Γ. In principle, C andC also include evaporation effects, but it has been shown that this is negligible for DM particles with a mass 5 GeV [26,27] and we will neglect it in the following discussion.
We define the asymmetry ∆ as the difference between the DM and anti-DM numbers, ∆ = N −N . It evolves according to∆ where we have defined d = c −c and D = C −C. Assuming negligible amounts of both DM and anti-DM at the birth of the Sun, ∆(0) = 0, we find The absolute value of ∆ also represents the minimum amount of dark particles (DM or anti-DM) present in the Sun.
It is important to note here that there is a geometric limit for the self-capture rates.
When N andN are large enough, our approach is no longer valid. When DM and anti-DM is trapped inside the Sun, they will accumulate in the center roughly inside a sphere of radius r χ (derived in the next section). When σ χχ N +σ χχN is larger than the cross-sectional area of the sphere, πr 2 χ , every incoming DM particle will scatter off either trapped DM or anti-DM. When the distributions of trapped DM and anti-DM are identical, the fraction of collisions of DM on DM is f χχ = σ χχ N/(σ χχ N + σ χχN ) while the fraction of collisions on anti-DM is given by f χχ = σ χχN /(σ χχ N + σ χχN ). The same argument holds for anti-DM except the total cross section σ χχ N + σ χχN can not exceed πr 2 χ . In this case, the fraction of collisions on DM is given by f χχ with all N andN interchanged. Similarly, the fraction of collisions on anti-DM is given by f χχ , again with N andN interchanged. When this occurs, the equation for ∆ will fail since eqs. (1) and (2) must be corrected in order to take the geometric limit into account.
which has three interesting limits: When Dt is small, self-capture is negligible and the asymmetry will be proportional to the difference in the capture rates. When D < 0, the system eventually stabilizes, since the additional capture of anti-DM by already captured DM balances the difference in the capture rate on normal matter. For D > 0, DM captures itself at a larger rate than anti-DM.
Once this process becomes dominant, it leads to an exponential increase in the amount of DM captured in the Sun.

II.2. Stochastically induced difference
When the capture rates are equal (c =c), the amount of DM might at some point be larger than the amount of anti-DM simply due to random variations in the capture process which can be modelled by adding a white noise signal δ c to the capture rates The white noise is normalized such that the expected number of captured DM particles and its variation matches those of a Poisson distribution, i.e., n 2 − n 2 = n . We find that and hence s = c 0 . Using the same argument for the capture rate of anti-DM with a white noise signal δc, we find Since δ c and δc are independent, δ d has the properties The expectation value of the asymmetry ∆ is zero, which should be expected since the probability of having an over-abundance of DM to anti-DM must be the same as that of having an over-abundance of anti-DM due to symmetry. To estimate the typical magnitude of the asymmetry, we can study the standard deviation of the stochastic variable ∆, given by∆ = ∆ 2 . We find that Thus, the limiting behaviour is similar to the case for intrinsic different capture rates The major difference is that the short time limit |Dt| 1 gives an asymmetry that is expected to grow with the square root of t rather than linearly and that the coefficients are related to c 0 and D by a square root. For small Dt, we note that this result is precisely what would be expected from the difference between two Poisson distributions of expectation value c 0 t while an equilibrium or an exponential growth occur for strong self-capture.

III. DM AND ANTI-DM SELF-CAPTURE RATES
For a self-interacting model of DM, the total capture of DM in the Sun will be the sum of a capture rate due to interactions with solar nuclei, a term proportional to the already captured DM and a similar term proportional to the number of captured anti-DM particles.
The capture of anti-DM is completely analogous to DM capture although the rates may differ depending on the various scattering cross sections. The specific formulas to compute the capture rates and ejection rates are presented in the appendix but the complexity of the capture rates C andC requires some discussion. In what follows, we assume that the time it takes for a captured DM particle to fall into thermal equilibrium in the solar core is negligible.
Generally, the formula for the capture rates of DM by DM and anti-DM as well as the ejection rates are given by: The factor wΩ is the rate at which a particle with velocity w will scatter at radius r and contain information on the probability that the incoming particle is captured and whether the target particle is ejected or not.
The rate of capture of an incoming particle without ejecting the target particle is given by of the particle before it falls into the gravitational potential is u and the velocity at radius where v esc is the escape velocity. The radial distribution of the target particle that is hit is given by n(r) and is assumed to have the form where n 0 is the normalization constant. The function φ(r) is the gravitational potential energy at radius r from the core and is given by Here, M (r) is the total mass inside the sphere of radius r around the solar core; If the distribution of DM within the Sun is concentrated to the center where the density ρ(r) = ρ c and temperature T (r) = T c are approximately constant, then The normalization constant is n 0 = π − 3 2 r −3 χ N and r 2 χ = 3kT c /2πGρ c m χ . Defining (r) = n(r)/N , the distribution can be written n(r) = (r)N . For anti-DM, the radial distribution is taken to be the same except N →N .
The ejection rate of DM captured by in the Sun by collisions with DM or anti-DM from the halo is given by We must also take into account that while a dark matter particle is being ejected from the Sun, it is also likely that the particle from the halo is captured by the same process. The rate for this exchange occurring is C exch = C eject − C eject 2 , where C eject 2 is the rate of ejections in which both the incoming halo particle and the particle from the Sun are ejected, given by The possible cases depending on velocity for capture and ejection are shown schematically in Fig. 1 along with the velocity distribution of the standard halo model. As can be seen from this figure, the escape velocity will generally be so large that we expect that the self-capture will be dominant with some contribution from halo particles being captured while ejecting the target particle.
Summarizing this, the capture rates C andC relevant for the evolution of the dark matter and anti-dark matter numbers in the Sun can be written as Here, the single self-ejection events do not appear in C, as the net change in the DM number in the Sun is zero for these events, but the full ejection induced by the opposite species from the halo must be taken into account as the capture of the halo particle does not compensate for the ejected one. For the capture of DM on anti-DM, the relevant quantities are the capture without ejection and the ejection of the target particle while capturing the halo particle.
The annihilation rate is computed as [28] where σv is the thermally averaged annihilation cross section. As long as R r χ , the upper limit of the integral can be set to ∞ rather than R and the annihilation rate evaluates to Γ = σv

IV. RESULTS
In the following, we will make some explicit assumptions in order to estimate the effects described in the previous sections. The velocity distribution f (u) of the halo is assumed to be a standard Maxwell-Boltzmann distribution shifted to the solar frame moving through the halo at v = 220 km/s. It can be expressed as [4] f It is assumed that the DM and anti-DM components in the halo are identical and that the density of each are equal. They will then each have a density ρ χ = ρχ = 0.15 GeV cm −3 , which is equal to half the total local DM density of 0.3 GeV cm −3 [29,30]. The number density of DM and anti-DM is therefore n χ = ρ χ /m χ .
The dark matter is assumed to scatter with regular matter with velocity independent spinindependent (SI) and/or spin-dependent (SD) cross sections through effective operators.
For the case of SD capture in the Sun, we are interested in the bounds on the SD DMproton cross section. Limits on these cross sections have been set in various direct detection experiments [31][32][33][34][35][36]. In the DM mass range 10 − 1000 GeV, the limit on the SI cross section is σ SI 10 −44 cm 2 [31]. For smaller DM masses, the limits on the SI cross section weakens significantly. For a 5 GeV DM particle, σ SI 10 −40 cm 2 . The limits on the SD cross section in the mass range 10 − 1000 GeV is σ SD 10 −38 cm 2 [33]. For a 5 GeV particle, the bound is slightly reduced to σ SD 10 −37 cm 2 [36].
Limits on the self-interaction of DM comes from astrophysical sources. When galaxy clusters collide, drag forces acting on the gas while the DM passes through unhindered would produce an offset in the mass and gas distribution of the clusters, the size of which can be used to put upper limits on the self-interaction of DM. In [37], one such collision was analysed and and set an upper limit on the self-interacting cross section of σ χχ /m χ 2 · 10 −24 cm 2 /GeV. The relic abundance of DM has been precisely derived from WMAP [38] and Planck [39] experimental data. The thermally averaged annihilation cross section can be related to this relic abundance by solving the Boltzmann equation which is done in e.g., ref [3] and is here taken to be 3 · 10 −26 cm 3 /s. As a model of the Sun, the AGSS09 solar model [40] is chosen. It contains the mass and radial distribution of elements up to Ni. The solar age is taken to be t = 4.5 byrs.
The capture rate of a DM particle with mass m χ = 5 GeV is calculated to be at most 10 27 s −1 for a SD cross section of 10 −37 cm 2 and 2.9 · 10 25 s −1 for a SI cross section of 10 −40 cm 2 . For a DM particle of mass 10 GeV, the bounds push SI capture down by four orders of magnitude even though for a fixed SI cross section, the capture rate is only reduced at the percent level. Thus, the SD cross section allows for higher capture rates for all masses in the range 5-1000 GeV. Fig. 2 shows the values of C s , C exch and C eject 2 as a function of DM masses between 5 and 1000 GeV. While C exch is roughly 20 times smaller than C s , C eject 2 is almost 15 orders of magnitude lower. This is not surprising as the escape velocity is very large where the DM resides and particles with a large velocity in the halo are exponentially suppressed (cf. fig. 1). In the case of the Sun, C eject 2 may therefore be neglected, since C s and/or C exch will be completely dominant depending on the relative sizes of σ χχ and σ χχ . The capture rates can now be written as by using that C exch (σ) C eject (σ) and one finds that D takes the form Note that even if the scattering cross sections σ χχ and σ χχ are equal, D will be non-zero.
This is due to the fact that ejection of the more dominant species occurs at a larger rate. be no self-capture at all or the difference between σ χχ and σ χχ is such that D is small. On the other hand, the exponential growth is apparent when σ χχ is larger than σ χχ as to make D positive. Since ∆ is definitely smaller or equal to N , the geometric limit of self-capture has definitely been reached once σ χχ ∆ > πr 2 χ . In the case of fig. 3, a redefinition of C andC would have already been necessary for the two cases with D > 0. Heavier DM particles will breach the limit for smaller self-scattering cross sections since r χ decrease in size as m −1/2 χ .

IV.2. Symmetric capture and∆
The case of c =c implies a simple solution to the steady state of eqs. (1) and (2). When the capture rates are equal, the symmetry of the equations implies that N =N . Both the capture of DM and anti-DM will then be given bẏ In the steady state, the capture rate is equal to the annihilation rate so thatṄ = 0 which plugged into the above gives The total number of captured particles will then be 2N ∞ . With the same self-scattering cross section and setting σ χχ = 0, the stochastic asymmetry becomes 5 orders of magnitude larger than the expected number of particles which increase to N ∞ = 3.02 · 10 41 . Since∆ is defined as the standard deviation of ∆, we may in this case expect the asymmetry to be large. However, if the actual asymmetry was of this size, the geometric limit has kicked in and∆ is no longer given by eq. (12). Still, this is an indication that self-capture combined with a stochastically induced asymmetry has lead to a significant accumulation of DM in the Sun.

V. SUMMARY AND DISCUSSION
In this paper, we have considered the capture of DM in the Sun under the assumption that it is a self-interacting Dirac particle and that the galactic background density consists of equal amounts of DM and anti-DM. This opens up the possibility that a large difference in the captured amount of each type might occur so that, even though annihilation occurs, the total number of particles in the Sun may continue to grow. The initial asymmetry between the number of captured DM and anti-DM particles can occur either due to different scattering cross sections for DM and anti-DM on solar nuclei or due to stochastic fluctuations in the capture process. Any such asymmetry may then be amplified by self-capture or counteracted by capture of anti-DM by DM. The size of the asymmetry is independent of the annihilation rate and an analytical expression for its size was derived.
When the capture rates of DM and anti-DM are different (c =c), we have the case of asymmetric capture. If the capture rates of DM by DM is C and the capture rate of DM by anti-DM isC, we define the difference in these rates as D = C −C. When D < 0, the capture of anti-DM is more efficient than DM self-capture which implies that the asymmetry ∆ will at some point find an equilibrium. On the other hand, if D > 0, DM will capture itself more efficiently so that any initial asymmetry will grow exponentially. The asymmetry is found to become large enough to conclude that the geometric bound on the self-capture needs to be taken into account for a wide range of solar capture rates. This is due to an exponential dependence on the size of the DM on DM and DM on anti-DM capture rates when D > 1/t . This occurs for a 5 GeV DM particle with a capture rate of 10 25 s −1 on solar nuclei, corresponding to a spin-dependent cross section of 10 −39 cm 2 , for a D given by σ χχ = 2 · 10 −24 cm 2 and σ χχ = 0, where σ χχ is the DM self-scattering cross section and σ χχ is the scattering cross section for DM on anti-DM. When the capture rate of anti-DM by DM and the capture rate of DM by DM makes D negative, the asymmetry will always be smaller than in the case when there is no self-capture at all. In any case, the size of the asymmetry implies that the total amount of DM captured may be large, but that solving the system of (1) and (2) numerically while taking the geometric bound into account is necessary.
When the capture rates on solar nuclei are equal for DM and anti-DM (c =c), the stochastic asymmetry∆, which estimates the typical magnitude of the actual asymmetry induced by the stochastic variation of c andc, is always extremely small in comparison to the total number of particles in the Sun when the self-scattering cross sections σ χχ and σ χχ are such that D is small or negative. However, in the case when D is positive, the exponential dependence of D may bring the stochastic asymmetry to a size several orders of magnitude larger than the expected total number of trapped particles at steady-state with no asymmetry. However, this case is an extreme since the scattering cross section σ χχ is taken to be right around the upper bound while σ χχ = 0. Increasing σ χχ to half that of σ χχ , the stochastic asymmetry is reduced by over 10 orders of magnitude to a negligible level compared to the expected amount for symmetric capture. The window for the asymmetry induced by stochastic variations for the Sun is very small and requires σ χχ σ χχ so it may not be expected that the solar asymmetry is large. However, the capture rates increase proportionally to the background density of DM so that larger self-capture rates may be expected for stars in regions where the background density is larger. Even if the likelihood that the Sun has a negligible asymmetry since the background density is small, stars in such regions may have a D that is several orders of magnitude larger which would increase∆ significantly thus affecting the evolution of such stars.
In this work, we have neglected the fact that a stochastic capture rate on solar nuclei may imply a stochastic variation of the self-capture rates as well. One such scenario would occur if there are perturbations in the local background density. The investigation of this case is left for future work.

ACKNOWLEDGMENTS
This work was supported by the Göran Gustafsson Foundation.
Appendix A: Capture rates of DM The capture of dark matter in celestial bodies is a standard calculation, first done by Press and Spergel [4], later improved and corrected by Gould [41]. Given the velocity distribution of halo dark matter, f (u), in the frame of the Sun where u is the velocity very far away where the gravitational potential of the Sun is negligible. The capture rate is given by where wΩ is the rate at which a particle with velocity w at radius r will scatter and lose enough energy to be captured.

Solar element capture
For the SI cross section, Gould found that while for the SD cross section n i (r) is the radial distribution of element i in the Sun. The mass of and scattering cross section on element i is m i and σ i , respectively. The mass of the DM particle is m χ , µ = mχ m i and µ + = 1+µ 2 . The SI scattering cross section scale as where A i is the number of nucleons in element i and µ i the reduced mass of element i and the DM. σ p is the proton scattering cross section and µ p the reduced mass of the DM and the proton. The more complex formula for SI scattering is due to the form factor which takes the nuclear structure of the target into account for larger energy transfers given by and M i is the mass of nuclei i. For hydrogen, this form factor is set to unity. The total capture rate for a SI capture rate is then the sum of the capture rate by each individual element. For SD capture, hydrogen is the only element of importance since there is no A 2 enhancement of the cross sections on other elements and the fraction of elements with spin is completely negligible compared to hydrogen.

Self-capture
If DM has a non-zero scattering cross section on other DM and anti-DM particles, it may also be captured by colliding with other DM and anti-DM particles. A derivation of self-capture is given in ref. [24] and we review the result and add three cases for which the target particle is ejected. For self-capture, Ω is broken down to Ω = σn(r)wP cap .
Here, σ is the DM self-scattering cross section, n(r) the radial distribution of already captured DM and P cap the probability that the particle is captured in a collision while not giving the target particle enough energy to escape the Sun. The projectile and target particles are gravitationally unbound when their kinetic energy is greater than m χ v 2 esc /2. This means that, for capture of a particle without ejecting the target, the energy transfer ∆E must be in the interval The energy transfer distribution is assumed uniform on the interval which gives Ω(r, w) as and the self-capture rate is given by When the transferred energy in a collision involving DM and anti-DM is greater than m χ v 2 esc /2, the particle that is hit will be gravitationally unbound and escape the Sun. This ejection rate is calculated using the same formula as self-capture but with a different Ω. We can divide ejection into two regions, one in which u < v esc and one in which u > v esc .
If u < v esc and the incoming particle is trapped after a collision, the target particle may or may not be trapped. If the target is trapped, we have the case of self-capture that is described above. If the target particle is ejected, the transferred energy is in the range and wΩ is given by If u > v esc , entrapment of the incoming particle will always result in the ejection of the target particle. In the case that the target particle is still trapped, ∆E falls in the interval The factor wΩ is then wΩ = σn(r)v 2 esc Θ(u − v esc ).
However, if ∆E falls in the interval v 2 esc w 2 < ∆E E < u 2 w 2 (A15) The target particle will be ejected and the incoming particle will still have a velocity that is larger than the escape velocity and thus also escape. For this case, wΩ is found to be wΩ = σn(r)(u 2 − v 2 esc )Θ(u − v esc ).