The Leptophilic Dark Matter in the Sun: the Minimum Testable Mass

# The Leptophilic Dark Matter in the Sun: the Minimum Testable Mass

Zheng-Liang Liang Institute of Applied Physics and Computational Mathematics
Beijing, 100088, China
Yi-Lei Tang School of Physics, KIAS,
85 Hoegiro, Seoul 02455, Republic of Korea
Zi-Qing Yang Big Data Research, IFLYTEK
Wangjiang Road West 666#, Hefei, 230088, China
###### Abstract

The physics of the solar dark matter (DM) that is captured and thermalises throught the DM-nucleon interaction has been extensively studied. In this work, we consider the leptophilic DM scenario where the DM particles interact exclusively with the electrons through the axial-vector coupling. We investigate relevant phenomenology in the Sun, including its capture, evaporation and thermalisation, and we calculate the equilibrium distribution using the Monte Carlo methods, rather than adopting a semi-analytic approximation. Based on the analysis, we then determine the minimum testable mass for which the DM-electron coupling strength can be probed via the neutrino observation. Compared to the case of the DM-nucleon interaction, it turns out that minimum detectable mass of the DM-electron interaction is roughly 1 GeV smaller, and a cross section about two orders of magnitude larger is required for the saturation of the annihilation signal.

1

## 1 Introduction

Several neutrino telescopes have been looking for the trace of the Weakly Interacting Massive Particles (WIMPs) 2011ApJ…742…78T (); Aartsen:2012kia (); Adrian-Martinez:2013ayv (); Avrorin:2014swy (), a generic kind of candidate for the Dark Matter (DM), from the Sun. This is based on the picture that the Galactic WIMPs collides with nuclei in the Sun as they pass by the solar neighbourhood, gradually sinking into the solar core after subsequent collisions, and end up annihilating into primary or secondary neutrinos that escape the environment of the dense plasma in the Sun, so to be observed by the terrestrial neutrino detectors.

The neutrino flux at the detector location is related to the solar DM annihilation through the following schematic relation:

 dΦνdEν=ΓA4πd2⊙dNνdEν, (1.1)

where is the Sun-Earth distance, and represent the neutrino differential flux at the Earth and the neutrino energy spectrum per DM annihilation event in the Sun, respectively. The total annihilation rate can be expressed in terms of the number of the trapped DM particles :

 ΓA = 12A⊙N2χ, (1.2)

where denotes twice the annihilation rate of a pair of DM particles. The evolution of the solar DM number is depicted with the following equation:

 dNχdt=C⊙−E⊙Nχ−A⊙N2χ, (1.3)

which involves the DM capture (evaporation) rate by scattering off atomic nuclei in the Sun, as well as the annihilation rate . Eq. (1.3) has an analytic solution

 (1.4)

with

 τe=(C⊙A⊙+E2⊙/4)−1/2 (1.5)

the time scale for the capture, evaporation and annihilation processes to equilibrate. Once the equilibrium is reached at the present day, , , with being the solar age, the annihilation output also reaches its maximum value. Depending on the ratio , or the DM mass , such equilibrium can be categorized into two different scenarios: (1) , that’s when the evaporation effect can be neglected and the equilibrium is between annihilation and solar capture, , , so we can either determine or constrain the strength of the DM-nucleon interaction from solar neutrino observation; (2) , under this circumstance evaporation overwhelms annihilation for the DM depletion, and the balance between evaporation and solar capture yields , which not only implies a heavy suppression of the neutrino flux, but also prevents us from drawing the coupling strength of the DM-nucleon interaction from the possible observed signals.

While relevant phenomenology associated with the DM-nucleon interaction have been studied extensively in literature, the tempting possibility that the DM particles couple exclusively to leptons, the so-called leptophilic scenario, has aroused wide interest in the community Bernabei:2007gr (); Fox:2008kb (); Kopp:2009et (); Feldstein:2010su (); Essig:2011nj (); Foot:2014xwa (); Lee:2015qva (); Roberts:2016xfw (). However, even for a broad range of leptophilic DM models, it turns out that the the effective DM-nucleon cross section arising from the loop-induced DM-quark interaction competes or overwhelms that of the DM-electron interaction Kopp:2009et (). A notable exception is that the DM particle interacts with electron through the axial-vector coupling, a case in which the loop-induced contribution vanishes.

In this work, we will investigate some interesting phenomena of the leptophilic DM trapped in the Sun. Specifically, we will explore the minimum testable mass through neutrino observation for the scenario where the DM particle couples exclusively to electron. The minimum detectable solar DM mass is determined by the parametric relations between capture, evaporation and annihilation, as has been extensively studied in the context of the DM-nucleon coupling scenario Gould:1987ju (); Gould:1989hm (); Gould:1991hx (); Busoni:2013kaa (); Catena:2015uha (); Vincent:2016dcp (); Liang:2016yjf (); Baum:2016oow (); Garani:2017jcj (); Busoni:2017mhe (); Widmark:2017yvd (). But considering that the medium of the ionised electrons is much softer than that of the nuclei (suppressed by a factor in terms of thermally averaged momentum, with the electron (nucleus) mass ()), the minimum detectable leptophilic DM mass is expected to be smaller accordingly, due to the less energetic collisions that would prevent the buildup of the solar DM through evaporation. While quantitative analyses on this issue have been discussed in ref. Garani:2017jcj (), where equilibrium distribution of leptophilic DM was described phenomenologically with a semi-analytic approximation, in this paper, we will pursue an accurate evaluation of the distribution with a Monte Carlo method adopted in ref. Gould:1987ju (); Liang:2016yjf (), in an effort to provide a more precise description of the leptophilic DM in the Sun.

This paper is organised as follows. In Sec. 2, we will take a brief review on the theoretical ground for the capture, evaporation, and annihilation of the leptophilic DM in the Sun, and put these formulas into numerical computation. Main results, along with relevant analyses and discussions, are provided in Sec. 3.

## 2 Distribution and evolution of solar DM

In this section we will discuss the distribution and evolution of the solar DM. An accurate description of the distribution of the captured DM is crucial for the evaluation of evaporation and annihilation rate, and together with capture rate, they determine the evolution of the solar DM population. We obtain the solar DM distribution by solving the Boltzmann equation in a numerical manner. Now we delve into the details.

### 2.1 capture of the dark matter by solar electrons

The buildup of the solar DM population begins with the capture of the Galactic DM particles. There is the possibility that the free-streaming DM particles will be gravitationally pulled inside the Sun and scattered by electrons therein to velocities lower than the local escape velocity, so to be captured. The standard procedure for evaluating the DM capture rate has been well established in the literature Gould:1987ir (); Gould:1987ww (); Gould:1991hx (). After a small modification to replace nuclei with electrons, the capture rate of the DM particle by solar electrons can be expressed as

 ˜C⊙ = (2.1)

where is the radius of the Sun, is the mass of the DM particle, and are the DM density in the solar neighborhood and the velocity distribution in the solar rest frame, respectively. In calculation, we use and model the velocity distribution as a Maxwellian form in the rest frame of the Galactic centre, with the dispersion velocity and a truncation at the Galactic escape velocity of . connects the velocity outside the solar influence sphere, , and the one accelerated by the gravitational pull at the radius , with the local escape velocity . The quantity represents the differential event rate of a DM particle with initial velocity down-scattered to final smaller one by solar electrons in unit volume as the following,

 R−e(w→v) = ne⟨dσχedv|w−ue|⟩ (2.2) = ne∫fe(ue)dσχedv|w−ue|d3ue,

where the differential cross section for the DM-electron system depends on their relative velocity , and denotes the average over the thermal velocity distribution of solar electrons. is the local electron number density. The Maxwellian distribution is written as

 fe(ue) = (√πu0)−3exp(−u2eu20), (2.3)

with . is the local temperature at radius . Hence the event rate eq. (2.2) can be further expressed explicitly in the following analytic form,

 R−e(w→v) = neσχe4η(η+)2vw{erf[(η−w−η+v)2u0,(η−w+η+v)2u0] (2.4) +exp[η(w2−v2)/u20]erf[(η+w−η−v)2u0,(η+w+η−v)2u0]},

where , and .

### 2.2 relaxation and distribution of the solar DM

In order to determine the distribution of the trapped DM, in this work we adopt the numerical method outlined in ref. Gould:1987ju (), which is in essence equivalent to solving the Boltzmann equation. The benefits are two-fold: first, this method is able to describe the high end of the velocity distribution of the solar DM, which is a prerequisite of an accurate evaluation of evaporation rate; second, this method can also provide a detailed description of the relaxation process of the newly captured DM particles. The second advantage is especially important because in contrast to the nucleus capture scenario, the marginally captured DM particles by solar electrons are prone to be ejected back to deep space before their distribution reach the final equilibrium. The leakage due to the evaporation over the relaxation process needs to be carefully quantified.

Here we take a brief introduction to the methodology. Our discussion is based on the assumption that the accumulated DM population does bring any significant impact on the solar structure, , the Sun as a heat reservoir is also modeled as the background for the residing DM particles. We keep track of a small portion of DM particles since their capture until they finally integrate into the rest already in equilibrium. The Boltzmann equation is linear due to the absence of the DM self-interaction, and can be further simplified as the following master equation if expressed with a convenient choice of parameters (total energy per mass) and (angular momentum per mass) Gould:1987ju ():

 df(E,L,t)dt = −f(E,L,t)∑E′,L′S(E,L;E′,L′)+∑E′,L′f(E′,L′,t)S(E′,L′;E,L) f(E,L,0) = fcap(E,L), (2.5)

where is distribution function at time , and represents the scattering matrix element for transition process . In practice the parameters and are nondimensionalised in terms of an energy reference value , and an angular momentum value . These values are constructed from a length unit, namely the solar radius , and a time unit , with the Newton’s constant and the solar mass .

As mentioned in ref. Liang:2016yjf (), in the optical thin limit and for a large enough time step , the probability for a collision in simulation is insensitive to the starting position in the periodic orbit defined by and , so these two parameters are sufficient for the representation of DM states bound to the Sun. The initial distribution is obtained by assuming the DM particles captured after first collision with electrons do not deviate much from their incident directions. Consequently, if is defined as the angle between the trajectory of scattered DM particle and radial direction from the centre of the Sun, the probability distribution scale proportionally with differential in each spherical layer , with . For illustration, we present the initial distribution for a GeV DM particle in the left panel of fig. 2.1.

It should be noted that compared with nucleus, the electron thermal momentum is suppressed by (with the nucleus mass), which suggests that the DM particle can only marginally fall into, and easily escape from the solar gravitational well. Therefore the evaporation effect is taken into account in eq. (2.5), at variance with the approach adopted in ref. Liang:2016yjf (), where only gravitationally bound states are involved in the simulation of relaxation process. This modification is necessary considering that the leakage due to evaporation may no longer be neglected over the relaxation process. Since all bound states are connected and the evaporated DM particles are not anticipated to be trapped again, in simulation we allocate one state to account for the escape state that is corresponding to the absorbing state in the context of Markov process, whereas all bound states are corresponding to the transient states. For long enough time, evaporation will deplete all the DM particles participating in simulation, but before that a steady normalised distribution among the survival DM particles is expected to be reached (for which we provide a proof in Appendix A). We evolve eq. (2.5) with the discrete time step until converges to this limiting distribution , and other physical details of the relaxation can be recorded at the same time. The equilibrium distribution is shown in the right panel of fig. 2.1. If the relaxation time scale is verified to be much smaller than the Sun age , the picture of instant thermalisation will keep unchanged except that an effective capture rate should be introduced as the original one , suppressed by the remaining proportion over the relaxation process. For illustration, the capture rate and the ratio between the two capture rates, , are presented in fig. 2.2, respectively.

The scattering matrix element is determined with Monte Carlo approach. Specifically, a large number of DM random walk samples are generated and tallied in the fixed time step , which is required to be long enough to ensure that the test particle receive substantial transfer momentum from solar electrons. The periodic radial trajectory of the bound DM particle between successive collisions is numerically integrated with the Standard Sun Model (SSM) GS98 Serenelli:2009yc () inside the Sun (), and is matched to analytic Keplerian orbit beyond the solar radius (if any). Thus the -th collision location and time can be determined with the random renewal collision probability via

 Pic = 1−exp[−∫ti+1tiλ(τ)dτ], (2.6)

where

 λ = ne⟨σχe(|w−ue|)|w−ue|⟩ (2.7) =

is dependent on the temporal parameter once the DM trajectory is determined with the method mentioned above. By generating further random numbers that help pick out the colliding solar electron’s velocity, and the scattering angle in the centre-of-mass (CM) frame, we then determine the outgoing state of the scattered DM particle after a coordinate transformation back to the solar reference.

### 2.3 evaporation and annihilation

Given the distribution, both evaporation and annihilation rate of the bound DM particle can be determined. The theoretical expression of the evaporation rate differs with the capture rate only in the way that the distribution of the incident DM particles is replaced by the normalised distribution of the DM particles trapped in the Sun, , and an up-scatter event rate with is introduced to account for the evaporation rather than . Thus the evaporation rate is expressed as

 E⊙ = ∫R⊙0dr∫f⊙(r,w)dw∫+∞vescR+e(w→v)dv, (2.8)

where

 R+e(w→v) = neσχe4η(η+)2vw{erf[(η+v−η−w)2u0,(η+v+η−w)2u0] (2.9) +exp[η(w2−v2)/u20]erf[(η−v−η+w)2u0,(η−v+η+w)2u0]}.

Besides, the evaporation rate can also be determined from simulation straightforwardly. Specifically speaking, the evaporation rate can also be constructed by collecting all the inflow probability into the escape state in each time step, at any instant time during the evolution of solar DM.

Evaporation rates obtained from these two approaches are found quite consistent in our study. For illustration in the left panel of fig. 2.3 shown are the relevant evaporation rates for a benchmark cross section and the DM mass ranging from to  GeV, with the blue solid line representing the evaporation rate drawn from the simulation, and the red dashed line corresponding to the calculated one. In the right panel of fig. 2.3, we present the simulated evolution of the evaporation rate for a 2 GeV DM. In order to justify the assumption of the instant thermalisation, the time scale is expressed in terms of the solar age . Once the evaporation rate reaches its equilibrium, the time scale of thermalisation is determined.

On the other hand, the annihilation coefficient is expressed in terms of the thermal cross section and the effective occupied volume of the solar DM, , as the following:

 A⊙ ≡ ⟨σv⟩⊙Veff, (2.10)

where the effective volume can be approximated from the simulated equilibrium distribution as the following function in the mass range from to  GeV:

 Veff = (2.11)

## 3 Minimum testable mass of the leptophilic DM

Based on the above numerical efforts on the capture, evaporation and annihilation of the leptophilic DM in the Sun, now we are ready to explore the parameter space where the solar neutrino observational approach is effective for the detection.

In analysis, we adopt the criterion or equivalently for the assumption that the neutrino flux reaches its full strength. On the other hand, in order to specify the parameter region for the annihilation- and evaporation-dominated scenarios, we set the criteria as and , respectively, where the canonical -wave thermal annihilation cross section is adopted in definition (see Eq. (2.10)).

The relevant parameter regions are presented in fig. 3.1. The quantitative analysis enables us to draw clear boundaries among different signal topologies. For instance, for a DM-electron cross section , the assumption of the equilibrium between capture and annihilation is only valid for a DM particle heavier than  GeV, while for a DM mass smaller than  GeV, one can no longer extract the coupling strength of the DM-electron interaction from the observed neutrino flux, because the number of DM particles turns independent of cross section . Moreover, if the cross section is smaller roughly than , the equilibrium among capture, evaporation and annihilation has not yet been reached at the present day. As a consequence, the signal flux is suppressed and the unsaturated number of the solar DM particles needs to be specified to determined or constrain the coupling strength Albuquerque:2013xna ().

In addition, recall that our discussions on the distribution and evolution of the solar DM rely on the assumption of the instant thermalisation, which means the time scale of the thermalisation should be much smaller than that of the solar age , so we also plot the contour line of the ratio in fig. 3.1 for reference purpose. The instant thermalisation turns out to be a reliable assumption for the parameter region of our interest.

In above investigation on the parameter space for the leptophilic DM detection, the cross section is capped at , which corresponds to a mean free path at the centre of the Sun. As coupling strength increases, collisions between DM particles and electrons begin to be frequent in processes such as capture, evaporation and energy transfer, and hence the optically thin approximation will no longer be valid for the description of the solar DM. Due to the multiple collisions, evaporation will be suppressed and the minimum testable DM mass begin to decrease accordingly Garani:2017jcj (). In that regime, the Monte Carlo approach adopted in this study will break down, since short DM free path requires an extra parameter for the description of the solar DM distribution, as mentioned in Sec. 2.2, and a full consideration of the Botlzmann equation is required, which is beyond the scope of this work.

###### Acknowledgements.
YLT is supported by National Research Foundation of Korea (NRF) Research Grant NRF- 2015R1A2A1A05001869, and the Korea Research Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2017H1D3A1A01014127).

## Appendix A relative probability distribution of the transient states

In this appendix we will prove that for the given amount of the captured DM particles, while suffer loss from evaporation, their relative probabilities will eventually evolve to a steady distribution. Towards this end, we consider a Markov process with finite discrete states, which is governed by the equation

 dpi(t)dt = ∑jWijpj(t), (A.1)

or in a compact form

 dp(t)dt = Wp(t), (A.2)

where is the transition matrix that describes the gains and losses between the (n+1)-state probabilities . Based on the following two properties of :

 Wij≥0, fori≠j; ∑iWij=0, foreachj, (A.3)

it is well known that regardless of the initial distribution , the probabilities of the n+1 states have a steady distribution in the long-time limit tagkey2007x (). On the other hand, eq. (A.2) has the following apparent solution,

 p(t) = eWtp(0). (A.4)

While the transition matrix may not be diagonalisable, there exists an invertible matrix almost doing the job such that , where the Jordan normal form is expressed as

 J = ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝J00⋯00J1⋯0⋮⋮⋱⋮00⋯Jm⎞⎟ ⎟ ⎟ ⎟ ⎟⎠, (A.5)

with the -dimensional Jordan block

 Jμ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λμ10⋯00λμ1⋯0⋮⋮⋱⋯000⋯λμ100⋯0λμ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (A.6)

and the sum over all dimensionalities of the blocks equals that of the Jordan matrix, ,

 ∑μdμ = n+1. (A.7)

Thus eq. (A.4) can be further expressed as

 p(t) = (A.8) = SeJtS−1p(0) = S⎛⎜ ⎜ ⎜ ⎜ ⎜⎝eJ0t0⋯00eJ1t⋯0⋮⋮⋱⋮00⋯eJmt⎞⎟ ⎟ ⎟ ⎟ ⎟⎠S−1p(0) = S⎛⎜ ⎜ ⎜ ⎜ ⎜⎝eλ0t⋅eY0t0⋯00eλ1t⋅eY1t⋯0⋮⋮⋱⋮00⋯eλmt⋅eYmt⎞⎟ ⎟ ⎟ ⎟ ⎟⎠S−1p(0),

where

 eYμt = exp[Jμt−diag(λμ,λμ⋯λμ)t] (A.9) = exp⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010⋯0001⋯⋮⋮⋮⋱⋯000⋯0100⋯00⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⋅t⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1tt22!⋯tdμ−1(dμ−1)!01t⋯tdμ−2(dμ−2)!⋮⋮⋱⋯⋮00⋯1t00⋯01⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

That is to say, the probability distribution can be expressed as a linear combination of the terms , which proves our conclusion that the transient states still have a stationary relative distribution in the long-time limit if their probabilities are renormalised to a finite number, and the thermalisation time scale can be described with the second largest real part of the set .

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