Dark matter in the Sun: scattering off electrons vs nucleons
The annihilation of dark matter (DM) particles accumulated in the Sun could produce a flux of neutrinos, which is potentially detectable with neutrino detectors/telescopes and the DM elastic scattering cross section can be constrained. Although the process of DM capture in astrophysical objects like the Sun is commonly assumed to be due to interactions only with nucleons, there are scenarios in which tree-level DM couplings to quarks are absent, and even if loop-induced interactions with nucleons are allowed, scatterings off electrons could be the dominant capture mechanism. We consider this possibility and study in detail all the ingredients necessary to compute the neutrino production rates from DM annihilations in the Sun (capture, annihilation and evaporation rates) for velocity-independent and isotropic, velocity-dependent and isotropic and momentum-dependent scattering cross sections for DM interactions with electrons and compare them with the results obtained for the case of interactions with nucleons. Moreover, we improve the usual calculations in a number of ways and provide analytical expressions in three appendices. Interestingly, we find that the evaporation mass in the case of interactions with electrons could be below the GeV range, depending on the high-velocity tail of the DM distribution in the Sun, which would open a new mass window for searching for this type of scenarios.
In memory of Haim Goldberg
1]Raghuveer Garani 2]and Sergio Palomares-Ruiz
Bethe Center for Theoretical Physics and Physikalisches Institut, Universität Bonn, Nußallee 12, D-53115 Bonn, Germany
\affiliationInstituto de Física Corpuscular (IFIC), CSIC-Universitat de València,
Apartado de Correos 22085, E-46071 València, Spain
Dark matter (DM) particles in the galactic halo could be brought into close orbits around the Sun after scattering off solar nuclei. Subsequent scatterings could finally capture those DM particles inside the Sun and thermalize them. It has been three decades since the effects of DM particles accumulated in the Sun were originally considered to solve the solar neutrino problem by modifying the energy transfer in the Sun [Steigman:1997vs, Spergel:1984re, Faulkner:1985rm]. However, these first papers did not attempt to explain the physical origin of the required DM concentration, i.e., how solar capture of galactic DM particles would proceed, which was studied for the first time in Ref. [Press:1985ug] (and later refined in Refs. [Gould:1987ju, Gould:1987ir]). Soon after those seminal works, it was realized that annihilations of DM particles accumulated in the Sun would give rise to a neutrino flux, potentially detectable at neutrino detectors [Silk:1985ax, Krauss:1985aaa, Freese:1985qw, Hagelin:1986gv, Gaisser:1986ha, Srednicki:1986vj, Griest:1986yu]. Since then, this is one of the existing strategies to indirectly detect DM, which is in turn complementary to DM direct searches, given that in both cases the signal would be proportional to the DM elastic scattering cross section. Indeed, numerous studies have evaluated the prospects of detection of the potential high-energy neutrino flux [Kamionkowski:1991nj, Bottino:1991dy, Halzen:1991kh, Gandhi:1993ce, Bottino:1994xp, Bergstrom:1996kp, Bergstrom:1998xh, Barger:2001ur, Bertin:2002ky, Hooper:2002gs, Bueno:2004dv, Cirelli:2005gh, Halzen:2005ar, Mena:2007ty, Lehnert:2007fv, Barger:2007xf, Barger:2007hj, Blennow:2007tw, Liu:2008kz, Hooper:2008cf, Wikstrom:2009kw, Nussinov:2009ft, Menon:2009qj, Buckley:2009kw, Zentner:2009is, Ellis:2009ka, Esmaili:2009ks, Ellis:2011af, Bell:2011sn, Kappl:2011kz, Agarwalla:2011yy, Chen:2011vda, Kundu:2011ek, Rott:2011fh, Das:2011yr, Kumar:2012uh, Bell:2012dk, Silverwood:2012tp, Blennow:2013pya, Arina:2013jya, Liang:2013dsa, Ibarra:2013eba, Albuquerque:2013xna, Baratella:2013fya, Guo:2013ypa, Ibarra:2014vya, Chen:2014oaa, Blumenthal:2014cwa, Catena:2015iea, Chen:2015uha, Belanger:2015hra, Heisig:2015ira, Danninger:2014xza, Blennow:2015hzp, Murase:2016nwx, Lopes:2016ezf, Baum:2016oow, Allahverdi:2016fvl] and of that of neutrinos in the (10-100) MeV range [Rott:2012qb, Bernal:2012qh, Rott:2015nma, Rott:2016mzs] using neutrino detectors/telescopes [Desai:2004pq, Desai:2007ra, Abbasi:2009uz, Abbasi:2009vg, Tanaka:2011uf, IceCube:2011aj, Scott:2012mq, Aartsen:2012kia, Adrian-Martinez:2013ayv, Avrorin:2014swy, Choi:2015ara, Aartsen:2016exj, Adrian-Martinez:2016ujo, Aartsen:2016zhm].
All these works have focused on DM-nucleon interactions. However, the possibility of DM particles having no direct couplings to quarks, but only to leptons, the so-called leptophilic scenarios, has been extensively considered in the literature; to alleviate the conflict of the DM interpretation [Bernabei:2007gr, Dedes:2009bk, Kopp:2009et, Feldstein:2010su, Chang:2014tea, Bell:2014tta, Foot:2014xwa, Roberts:2016xfw] between the signal observed at the DAMA experiment [Bernabei:2013xsa] and the null results of other direct searches [Felizardo:2011uw, Abe:2015eos, Agnese:2015nto, Amole:2016pye, Aprile:2016wwo, Tan:2016zwf, Akerib:2016vxi, Aprile:2016swn, Aprile:2017yea], for future strategies to search for sub-GeV DM particles with direct detection experiments [Essig:2011nj, Chen:2015pha, Lee:2015qva, Essig:2015cda], within the context of cosmic-ray anomalies in order to explain the positron, but not antiproton, excess [Fox:2008kb, Cao:2009yy, Bi:2009uj, Ibarra:2009bm, Cohen:2009fz, Cavasonza:2016qem] seen by different experiments [Adriani:2008zr, Chang:2008aa, Adriani:2010rc, FermiLAT:2011ab, Accardo:2014lma, Aguilar:2016kjl], to reduce the tension of the observed anomaly in the muon magnetic moment [Agrawal:2014ufa] or as potential signals in collider searches [DEramo:2017zqw].
Even in the case of tree-level DM couplings to electrons, in general, loop-induced DM-quark couplings are also present by coupling photons to virtual leptons [Kopp:2009et]. Therefore, DM would be captured in the Sun by both, interactions off solar electrons via tree-level processes and interactions off solar nuclei via loop processes. However, there are cases in which no loop-induced DM-quark contribution is present, such as axial vector couplings and thus, only DM capture by electrons is possible. Neutrino signals for leptophilic scenarios have been considered in Ref. [Kopp:2009et]. In that work, a constant (velocity-independent and isotropic) cross section was assumed to compute the solar capture rate of DM particles. However, DM-electron (and DM-nucleon) interactions could have a more complicated structure and non-trivial dependencies on the relative velocity () and the scattering angle () do appear for various operators [Fan:2010gt, Fitzpatrick:2012ix, Anand:2013yka, Hill:2013hoa, Gresham:2014vja, Panci:2014gga, Catena:2014epa, Gluscevic:2014vga, Catena:2014hla, Gluscevic:2015sqa, Dent:2015zpa, Catena:2015vpa, Kavanagh:2015jma, Catena:2015uha, Gazda:2016mrp]. Indeed, these possibilities, assuming couplings only to quarks, have been recently considered in the context of high-energy neutrino signals from the Sun [Kumar:2012uh, Guo:2013ypa, Liang:2013dsa], to reduce the tension between solar models and helioseismological data [Lopes:2013xua, Vincent:2013lua, Lopes:2014aoa, Vincent:2014jia, Vincent:2015gqa, Vincent:2016dcp], and to allow for a better compatibility among different results from direct searches [Masso:2009mu, Chang:2009yt, Chang:2010en, Barger:2010gv, Fitzpatrick:2010br, Foot:2011pi, Schwetz:2011xm, Farina:2011pw, Fornengo:2011sz, DelNobile:2012tx, Foot:2012cs, Fitzpatrick:2012ib, Catena:2014uqa, Barello:2014uda, Catena:2015uua, Catena:2016hoj, Rogers:2016jrx].
In this work, we present general results for the solar DM capture, annihilation and evaporation rates, as well as for the resulting neutrino fluxes from DM annihilations at production, for the cases of interactions with electrons with constant, -dependent and transfer momentum ()-dependent elastic scattering cross sections. All our results are compared to those obtained for the case of DM interactions with nucleons. We perform all computations taking into account thermal effects and study their importance. Moreover, we improve over the common calculation of the rates in a number of ways. We consistently compute the temperature in the regime of weak cross sections (Knudsen limit or optically thin regime) for each case including the effect of evaporation and the truncation of the DM velocity distribution, for which we also consider several cutoff velocities. We compute the minimum DM mass for which evaporation is not efficient enough to reduce the number of captured DM particles and find that, for the case of DM-electron scatterings, depending on the cutoff velocity, the minimum testable mass could be significantly smaller (below GeV) than the usually quoted evaporation mass in the case of DM-nucleon interactions. Finally, we compare the neutrino rates at production resulting from capture by electrons and nuclei. This is relevant to evaluate the importance of electron capture in leptophilic scenarios, which will be studied elsewhere [Garani:2017].
This paper is organized as follows. In Section 2 we describe different types of interactions we consider. In Section 3 we review the calculation of the capture rate and compare the results of capture by solar electrons and nuclei. In Section 4 we describe the velocity and radial distribution of DM particles in the Sun once equilibrium is attained and show the resulting temperature in the optically thin regime for the different types of interactions and targets (electrons and nuclei). With this at hand, we write down the expression for the annihilation rate. In Section 5 we review the calculation of the evaporation rate and illustrate our results. In Section 6 we compute the minimum testable mass below which evaporation is very effective for the different cases under study. In Section 7 we compare the neutrino rates at production obtained for capture by electrons and by nuclei for the different cross sections we consider. Finally, in Section 8 we summarize our findings and draw our conclusions. In three appendices we describe the calculation of the differential scattering rates (Appendix A), the calculation of the temperature in the optically thin regime (Appendix B) and the calculation of some quantities related to the propagation of DM particles in the Sun, mainly relevant in the conduction limit or optically thick regime (Appendix C).
2 Scattering cross sections
The scattering rates that govern the capture and evaporation rates of DM particles in the Sun scale with the scattering cross section in the Knudsen limit (optically thin regime). For the case of interactions off free electrons, the single-particle total (constant) cross section, which appears in the scattering rates, is simply given by . However, in the case of interactions off nuclei , depending on the type of interactions, either spin-dependent (SD) or spin-independent (SI), the total (constant) DM-nucleus cross sections, at zero momentum transfer, are given, in terms of the DM-proton and DM-neutron cross sections, by
where () is the reduced mass of the DM-nucleus (DM-proton/neutron) system, () and () are the SD and SI elastic scattering DM cross section off protons (neutrons), respectively, , and are the atomic number, the mass number and the total angular momentum of the nucleus , and and are the expectation values of the spins of protons and neutrons averaged over all nucleons, which we take
In the case of SD cross sections, the coupling with protons is the one which is mainly probed because almost all DM interactions are off hydrogen.
However, only in the case of constant cross sections, the scattering rate directly depends on the total cross section. For velocity-dependent and momentum-dependent cross sections, the differential cross section enters the calculation. In this work, in addition to the usual constant (velocity-independent and isotropic) cross section case, we also consider -dependent (isotropic) and -dependent cross sections, where and are the relative DM-target velocity and the transfer momentum, respectively. The differential cross sections (in the limit of zero transfer momentum
where is the center-of-mass scattering angle, and are a reference relative velocity and transfer momentum, and and are the DM and target masses, respectively. The mass-dependent term in Eq. (7) is included so that the total -dependent cross section is equal to the -dependent cross section when and is the same in both cases, i.e., . In this work, we use km/s. See Appendix A for further comments, definitions and for a description of how the differential cross sections enter the calculation of the differential scattering rates.
3 Capture of dark matter by the Sun
DM particles from the galactic halo could get eventually captured by the Sun if, after scattering off solar targets (nuclei and electrons), they lose energy so that their resulting velocity is lower than the Sun’s escape velocity at a distance from the center of the Sun, . The capture rate of DM particles with mass for weak cross sections, for which the probability of interaction is small, is (to good approximation) given by
where the sum is over all possible targets. In this work we consider electrons and 29 nuclei as targets and use their density and temperature distributions as determined within the standard solar model [Asplund:2009fu, Serenelli:2011py] (see Ref. [Vinyoles:2016djt] for a recent update). The factor (and the analogous ) is the differential scattering rate at which a DM particle with velocity scatters off a target with mass to a final velocity (). They are explicitly given in Appendix A for constant, -dependent and -dependent cross sections.
The nuclear form factor for nucleus is , which we approximate as the one corresponding to a Gaussian nuclear density distribution with root-mean-square radius (i.e., equal to that of a uniform sphere of radius ), i.e.,
For SI interactions [Eder:1968],
and given that the nuclear density distribution is different from the spin distribution [Engel:1991wq], for SD interactions [Belanger:2008sj],
For electrons and hydrogen, .
A few comments are in order. Note that a more realistic Woods-Saxon nuclear density distribution (for SI interactions) results in a form factor which is very similar to Eq. (9) for relatively low values [Engel:1989ix, Engel:1992bf]. Moreover, Eq. (8) is strictly correct if target nuclei are assumed to be at rest (for electrons and hydrogen, it is always correct as ). In that case: . Otherwise, up-scatterings with a final velocity below the escape velocity must also be considered (a term with the factor) and the nuclear form factor cannot be factored out, but has to be included in the calculation of the differential scattering rates and . However, the former correction is negligible and, given the current uncertainties, the fact that we are not using more accurate nuclear response functions [Anand:2013yka, Catena:2015uha, Gazda:2016mrp] and that in the end factoring out the nuclear form factor represents at most an overall (much smaller in the constant case) reduction with respect to the results from Eq. (8) for the case of SI interactions only, we do not refine the calculation further and consider the form factor as computed in the zero-temperature limit (but not the differential scattering rates), so that it can be factored out in Eq. (8) and the analytical expressions in Appendix A can be used.
The local DM density is given by GeV/cm, is the Sun radius and is the halo velocity distribution seen by an observer moving at speed , the speed of the Sun with respect to the DM rest frame,
where is the DM velocity at infinity in the Sun’s rest frame, is the angle between the DM and the solar system velocities and is the DM velocity distribution in the galactic rest frame, which is assumed to be a Maxwell-Boltzmann distribution (the so-called standard halo model) and thus,
with , the square of the DM velocity at a distance from the center of the Sun. We take the values km/s for the velocity of the Sun with respect to the DM rest frame and thus, for the velocity dispersion. Actually, does not extend beyond the local galactic escape velocity, km/s at 90% confidence level [Piffl:2013mla]. However, this represents a correction on the capture rate below the percent level [Choi:2013eda], much smaller than the very same form of the velocity distribution [Kundu:2011ek, Danninger:2014xza, Choi:2013eda]. Finally, note that we are assuming the Sun to be in free space, but the presence of the planets (mainly Jupiter) could affect the solar capture rate
On the other hand, Eq. (8) is only valid for weak scattering cross sections, such that the probability of interaction is very small: the capture rate cannot grow indefinitely with the cross section. The saturation value for the capture rate is set by the geometrical cross section of the Sun (when the probability of interaction and capture is equal to one) [Bottino:2002pd, Bernal:2012qh],
where is the average speed in the DM rest frame and the factor takes into account the suppression due to the motion of the Sun (),
For the chosen values of and , . Finally, in order to allow for a smooth transition between these two regimes, we estimate the capture rate as [Bernal:2012qh]
In the left panels of Fig. 1, we show the capture rates as a function of the DM mass for the case of DM-electron interactions (solid red curves), DM-nucleon SD interactions (dashed green curves) and DM-nucleon SI interactions (dot-dashed blue curves), for constant cross sections with (top panels), -dependent cross sections with (middle panels) and for cross sections with dependence with (bottom panels). In each panel, we also indicate the geometric limit (dashed black curve), Eq. (14). We stress again that even for leptophilic DM models, in general, interactions with nucleons are possible via loop processes, so the capture rates by nuclei are relevant and need to be considered.
In the case of constant cross sections (top-left panel), for high DM masses, capture by nuclei is several orders of magnitude (up to two for SD and four for SI) larger than capture by electrons. The differences decrease for lower masses and capture by electrons is comparable or larger for GeV, which can be relevant if the DM velocity distribution has a cutoff at (see below). The results for -dependent and -dependent cross sections are similar to each other (for the normalizations used in this work). Unlike for constant cross sections, in these cases, at high masses the capture rate by electrons is a factor of a few larger than capture by nuclei via SD interactions and the differences with respect to the SI capture rate decrease, being of three orders of magnitude. These results can be understood from the even more important impact of thermal effects for these cross sections as compared to the constant case and can have important consequences in some models [Garani:2017]. Moreover, the SD capture rate is much smaller than the SI case (up to four orders of magnitude). Overall, the capture rate via -dependent and -dependent cross sections is a factor of about four, three and two orders of magnitude larger than the case with constant cross sections (assuming the same for all cases) for capture by electrons, SI and SD interactions off nucleons, respectively.
In the right panels of Fig. 1, we illustrate the impact of thermal effects on the capture rates. These effects are driven by two competing factors: the ratio of the solar temperature to the DM escape energy, , where is the most probable speed of the targets at position , and the ratio of the targets thermal speed to the escape velocity. Whereas a larger average kinetic energy of the DM particles suppresses capture, thermal effects enlarge the range of velocities contributing to it. For the same cases of the left panels, we show the ratio of the capture rates obtained using the thermal distribution of the target particles with respect to the capture rates obtained in the limit , i.e., when the targets are at rest.
In the case of velocity-independent and isotropic cross sections, as discussed in Ref. [Kopp:2009et] and as can be seen in the top-right panel, thermal effects represent an order of magnitude correction if the target particles are electrons. However, the correction in the case of interactions with nucleons is very small for GeV. These differences can be explained by the larger thermal speed of electrons as compared to that for nuclei by a factor . In the case of -dependent and -dependent cross sections, thermal effects on the capture rates by electrons are very important and represent an increase of three orders of magnitude in the range of masses we show. This can be understood from the extra factors in the differential scattering rates and (see Appendix A). For these cross sections, even in the case of capture by nucleons, thermal effects cannot always be neglected. For GeV, for DM-nucleon interactions the increase in the capture rate is of a few tens of percent for both, -dependent and -dependent cross sections, although for SI interactions the correction is negligible for -dependent cross sections. On the other hand, for GeV, for all cases, thermal effects suppress the capture rates contrary to the results at higher masses, given that and . This explains the dip in the ratios for the case of DM-nucleon interactions at GeV.
Finally, as mentioned above, we have checked that the correct calculation of the capture rate by nucleons (mainly for SI interactions), i.e., including the form factor in the factors, only represents a decrease of with respect to the results shown here.
4 Dark matter distribution and annihilation rate in the Sun
After DM particles are trapped inside the Sun, successive scatterings with the target material (nuclei and electrons), which is in local thermodynamic equilibrium (LTE), would thermalize them at a temperature . Therefore, the velocity distributions of target and DM particles can be assumed to have a Maxwell-Boltzmann form,
where and are the solar temperature and the thermal DM velocity at a distance from the center of the Sun, respectively. Whereas in the case of large scattering cross sections (conduction limit or optically thick regime), DM particles would also be in LTE with the solar medium, i.e., , in the case of weak cross sections (Knudsen limit or optically thin regime), the DM distribution could be approximated as being isothermal, i.e., with a single temperature, .
Note that we have included a cutoff in the DM velocity distribution, , which in general depends on the position (a valid assumption for circular orbits) and it is usually assumed to be equal to the escape velocity at a distance from the center of the Sun, , but we also consider another possibility, . The last choice is motivated by the fact that the bulk of evaporation and annihilation takes place in the solar core and for DM particles only passing through the core such a cutoff is a reasonable approximation to the actual distribution function [Gould:1987ju]. As apparent from the comparison of the results of this approximation with those of Refs. [Gould:1987ju, Liang:2016yjf], the actual non-thermal distribution (obtained by solving the collisional Boltzmann equation numerically) cannot be accurately mimicked by the approximate radial and truncated velocity distributions assumed in this work (and in most works in the literature). This has already been noted long ago, as the distribution function is locally non-isotropic with radial orbits always dominating and the local temperature in the Knudsen limit is not uniform [Gould:1989ez].
As we will assume, in the case of weak cross sections, after DM particles are captured by the Sun, they would thermalize non-locally by multiple interactions, with a single isothermal (iso) distribution. In this limit (Knudsen limit), their radial distribution can be written as [Spergel:1984re, Faulkner:1985rm, Griest:1986yu]
which corresponds to an isothermal sphere following the law of atmospheres, with a radial dependence set by the gravitational potential , with the gravitational constant and the solar mass at radius , and where is the total population of DM particles at a given time .
A relatively simple semi-analytical method to treat the problem in the Knudsen limit was proposed in Ref. [Spergel:1984re]. By assuming a Maxwell-Boltzmann velocity distribution for the DM and target particles, one can obtain a solution to the isothermal assumption by requiring the DM distribution to satisfy its first energy moment and solving for . By imposing that there is no net heat transferred between the two gases, the equation to be solved reads [Spergel:1984re]
is the energy transfer per unit volume and time, with being the energy transfer per collision averaged over the scattering angle and (see Eq. (19)) and being the radial distributions of DM particles and targets , respectively. As we mentioned above, this approximation relies on the assumption of a uniform and locally isotropic Maxwell-Boltzmann distribution for the DM particles, conditions which do not hold in a realistic situation [Gould:1987ju, Gould:1989ez, Liang:2016yjf]. Indeed, the above approximation overestimates the efficiency of energy transfer by a factor of a few, which depends on the DM and target mass ratio [Gilliland:1986, Nauenberg:1986em, Gould:1989ez]. Baring in mind the approximated nature of this approach, which is the usual one followed in the literature, we also compute the DM distribution function in the Knudsen limit in this way. However, we implement two semi-analytical corrections. First, we perform the calculation with a cutoff in the DM velocity distribution, in order to be consistent with the inputs used for the computation of the annihilation and evaporation rates. Second, we also include the energy flow in the form of evaporated DM particles that escape the Sun, which is relevant for DM masses of a few GeV and below, so that the final equation we solve is
where is defined in Appendix B. Indeed, when there is a velocity cutoff, in the case of interactions with electrons, unless this correction is included, wrong solutions are found for GeV and GeV and there are no solutions for GeV and GeV, for and , respectively. All the relevant expressions for different types of cross sections (velocity-independent and isotropic, velocity-dependent and isotropic and momentum-dependent) and with a generic cutoff in the DM velocity distribution are provided in Appendix B.
In Fig. 2 we show the results for the temperature as a function of the DM mass in the one-zone model or isothermal approximation for electrons (red curves), nucleons with SD (green curves) and SI (blue curves) interactions, for the case of no cutoff, , (solid curves), (dashed curves) and (dotted curves). We show the temperatures for constant (top panel), -dependent (bottom-left panel) and -dependent (bottom-right panel) cross sections. For GeV, the temperatures for the three velocity distributions are practically equal, i.e., the cutoff has no effect. This can be understood by the fact that the larger the mass the lower the typical velocities of the DM particles and thus, the high-velocity tail of the distribution is less important. Notice also that all results converge in the large mass limit. For GeV, the lower the cutoff velocity, the lower the temperature, but the differences are never larger than 10% for all shown cases. For these low masses, the temperature in the case of interactions off electrons is slightly larger than that obtained when DM interacts with nucleons, being relatively larger for -dependent and -dependent cross sections. Notice also that, in the case of interactions with nucleons, the temperatures for the three cross section dependences are very similar. In the case of thermalization with electrons, constant cross sections result in a bit lower temperatures than the other two cases (up to ), the differences getting reduced for low-cutoff velocities. Whereas for the no-cutoff case, for which the correction due to evaporation is negligible, the temperatures for -dependent and -dependent cross sections (as defined in this work) are exactly equal (see Appendix B), for the case with cutoff, -dependent cross sections result in slightly larger temperatures. Overall, the differences in the temperatures for the cases under consideration are small for the relevant range of masses and have a small impact on the final neutrino fluxes.
On the other hand, in the case of large scattering cross sections (conduction limit or optically thick regime), DM particles thermalize locally, i.e., , and the DM radial distribution can be approximated as [Nauenberg:1986em, Gould:1989hm]
where is set by the normalization . For an admixture of targets, a good approximation for the dimensionless thermal diffusivity is represented by the weighted mean of the single-target solutions [Gould:1989hm, Gould:1990],
where is the thermal diffusivity for a single target and it is tabulated as a function of in Ref. [Gould:1989hm] for constant cross sections and in Ref. [Vincent:2013lua] for velocity-dependent and momentum-dependent cross sections. The total mean free path of DM particles in the solar medium is defined as , where is the partial mean free path for DM interactions at a distance from the center of the Sun with a thermal averaged scattering cross section off targets with density . This thermal average is performed over the DM and target velocity distributions and is given by
The expressions for this thermal average for different types of cross sections (constant, velocity-dependent and momentum-dependent) are given in Appendix C.1.
The transition from one regime to the other is indicated by the so-called Knudsen number,
where is the approximate scale height of the DM distribution, with the density at the solar center. The Knudsen limit corresponds to . Note that the definition of the Knudsen number should in principle be a function of the position in the Sun. Nevertheless, given that most of the DM would be concentrated in the center of the Sun, this is sufficient for our purposes and a more accurate definition is beyond the scope of this paper.
Although the actual solution of the problem can only be obtained by solving the collisional Boltzmann equation, however, an approximate solution can be considered by interpolating between the optically thin () and the optically thick () regimes. In order to do so, we follow Refs. [Bottino:2002pd, Scott:2008ns], which motivated by the results of Ref. [Gould:1989hm], approximated the DM radial and velocity distribution as
where is the value of the Knudsen number for which DM particles transport energy most efficiently [Gould:1989hm]. This value was obtained by assuming a spherical harmonic oscillator potential and keeping the mean free path as a constant throughout the entire star, which is also the reason why we used the position-independent definition in Eq. (26). Note that a given , which marks the transition from one regime to the other, corresponds to different values of for different types of cross sections [Vincent:2015gqa] and targets.
Once the DM distribution is known, we can compute the annihilation rate , defined as
where we have used
in the denominator and where is the DM annihilation cross section times the relative velocity of the two DM particles, . In general, , but in this work, our default case is that of an -wave annihilation cross section corresponding to a thermal DM candidate, i.e., , where denotes thermal average over the two DM velocity distributions. In such a case, Eq. (29) simplifies as
Note that for -wave annihilations, for -dependent and -dependent cross sections equilibrium would be attained for smaller values of than for the constant case, but we will not discuss this possibility here.
5 Evaporation rate of dark matter from the Sun
In general, for sufficiently small DM masses, below a few GeV, interactions with the targets of the solar medium would bring most of the DM particles to velocities above the escape velocity , so that they can evaporate from the Sun. The evaporation rate is given by
where the factor accounts for the suppression of the fraction of DM particles that, even after acquiring a velocity larger than the escape velocity, would actually escape from the Sun due to further interactions in their way out, and can be written as [Gould:1990]
where is the optical depth at radius . The factors and , which take into account that DM particles travel in non-radial trajectories and that multiple scatterings are possible, are described in Appendix C.2. Although the result for the factors in is based on a calculation for a velocity-independent and isotropic cross section [Gould:1990], lacking a better estimate, we also use it for the other cases under study. In the optically thin regime, the suppression factor is nearly one, but we always included it in the calculations.
Note that, to keep it general, we should have considered a term with corresponding to down-scatterings to velocities above the escape velocity and hence, the limits for the and integrals would be and , respectively. Moreover, a priori, the nuclear form factor for the case of interactions off nuclei must be included too. Whereas in the term, the nuclear form factor can be factored out by computing it in the zero-temperature limit, in that limit, the contribution from the term is exactly zero. However, for non-zero temperatures, the form factor has to be included in the calculation of the term, so an analogous simplification to the one for the term cannot be made. Nevertheless, if the DM velocity distribution has a cutoff at , the term is absent and, for the case of the Sun, the nuclear form factors can be approximated to one (at these velocities ). In such cases, which are the ones we consider in this work, the evaporation rate is given by the usual expression, Eq. (32).
In Fig. 3 we show the evaporation rates for constant (top panels), -dependent (middle panels) and -dependent (bottom panels) cross sections, in the case of DM-electron (red curves), DM-nucleon SD (green curves) and DM-nucleon SI (blue curves) interactions. We consider the same cases depicted in Fig. 1, i.e., cross sections in the optically thin regime (left panels), but also show the results for large cross sections in the optically thick regime (right panels). In all the panels we show the results of a DM velocity distribution with a cutoff at (solid lines) and (dashed lines). The usual exponential fall off at large masses, due to the dependence of the evaporation rate per unit volume on , is clearly visible. Besides this, there are a number of other features worth noticing.
In the case of interactions with electrons, the effect of the modification of the high-velocity tail of the DM distribution is striking, with a huge impact for masses above (0.1) GeV, for which evaporation is very suppressed in the case of . On the other hand, in the case of interactions with nucleons, the evaporation rates are only moderately modified when a different cutoff in the DM velocity distribution is considered. This can be understood by the mass scales involved in the problem. Electrons, being light compared to the DM particles, carry little momentum, so that in the case of DM-electron interactions only DM particles with velocities close to the escape velocity are susceptible of gaining enough energy to escape after one interaction. In practice, the differential scattering rate for interactions with electrons peaks at DM velocities close to the escape velocity, whereas for the case of scattering off nucleons it has a broader shape. Therefore, a maximum velocity smaller than the escape velocity significantly suppresses evaporation in the former case. This has important consequences on the available parameter space, as we will discuss below.
Moreover, whereas in the Knudsen limit the evaporation rate scales linearly with the scattering cross section, for large cross sections, the suppression of evaporation results in a slower-than-linear increase with the cross section. Therefore, although in the optically thin regime the evaporation rate for SI interactions is larger than for SD interactions due to the coherent enhancement of the cross section in the first case, this very same enhancement implies a shorter mean free path and hence, a larger suppression of the evaporation rate for SI cross sections in the optically thick regime. The relative suppression of the rate in the case of interactions with electrons is similar to the case of DM-nucleon SD interactions. We also note that the behavior of the evaporation rates in the case of -dependent cross sections is very similar to that of -dependent cross sections. We find that, in these two cases, the evaporation rates corresponding to DM interactions with electrons are larger than for scatterings off nucleons. In addition, although in general low masses enter the optically thick regime for smaller cross sections (see Eq. (26)), for -dependent and -dependent cross sections this effect is more pronounced (see Eqs. (C.1) and (100)).
6 Evaporation mass
Once all the ingredients are computed, we are interested in knowing what is the minimum DM mass which is testable,
The solution of this equation, computed at the present time ( Gyr), is given by [Gaisser:1986ha, Griest:1986yu]
where is the equilibration time scale in the absence of evaporation and . For the usual value assumed for the thermal annihilation cross section, , and for , and for constant cross sections (much smaller for the other cases), equilibrium is reached (, ). In the limit when evaporation is important, , , and the number of accumulated DM particles decreases exponentially with decreasing mass (in the optically thin regime). In the limit when evaporation is negligible, , , and the number of accumulated DM particles decreases with increasing mass (as for large masses). Given these considerations, one can define the minimum testable mass or evaporation mass
In the limit when equilibrium has been reached, i.e., , it can be written as