Long-range self-interacting dark matter in the Sun

We investigate the implications of the long-rang self-interaction on both the self-capture and the annihilation of the self-interacting dark matter (SIDM) trapped in the Sun. Our discussion is based on a specific SIDM model in which DM particles self-interact via a light scalar mediator, or Yukawa potential, in the context of quantum mechanics. Within this framework, we calculate the self-capture rate across a broad region of parameter space. While the self-capture rate can be obtained separately in the Born regime with perturbative method, and the classical limits with the Rutherford formula, our calculation covers the gap between in a non-perturbative fashion. Besides, the phenomelogy of both the Sommerfeld-enhanced s- and p-wave annihilation of the solar SIDM is also involved in our discussion. Moreover, by combining the analysis of the Super-Kamiokande (SK) data and the observed DM relic density, we constrain the nuclear capture rate of the DM particles in the presence of the dark Yukawa potential. The consequence of the long-range dark force on probing the solar SIDM turns out to be significant if the force-carrier is much lighter than the DM particle, and a quantitative analysis is provided.


I. INTRODUCTION
a large self-capture rate.
Following the literature [11,35,42] in which authors managed to discuss the DM selfscattering beyond the perturbative level, we similarly give a quantitative description of the self-capture rate for the Majorana DM particles. However, instead of adopting the transfer cross section as a proxy quantity to describe the self-scattering, we calculate the self-capture rate with long-range interaction in a straightforward manner, considering that the kinetic requirement for self-capture automatically regulates the forward-scattering divergence and hence makes the self-capture rate a well-defined quantity.
In addition, in order to explore the phenomenology of the solar SIDM from a comprehensive perspective, our discussion includes the Sommerfeld-enhanced annihilation of the solar SIDM as well. We illustrate that while the DM particles predominantly annihilate to light mediators through the p-wave channel, a small but non-vanishing quota of annihilation proceeding through the s-wave channel in the thermal freeze-out may become dominant over its p-wave counterpart within the Sun, and produce signals at the terrestrial neutrino telescopes. For such case we use the neutrino telescope Super-Kamiokande [57] to find out whether the Sommerfeld effect can help constrain the nuclear capture rate C n in a nontrivial manner. These constraints can be translated into upper limits on the DM-nucleon cross section for other SIDM models. It is found that in the approach of the Coulomb limit of the long-range self-interaction, the constraints from the Super-Kamiokande on nuclear capture rate can be significantly tightened.
This paper is organized as follows. In Sec. II we take an introduction to the model setup for our study. In Sec. III we introduce the related background about the solar DM, including its self-capture and evolution within the Sun. We discuss the implications of the light mediator on the self-capture and annihilation of the solar DM particles, as well as on relic density in Sec IV. We further investigate its impact on the neutrino telescope Super-Kamiokande in Sec. V, and conclusions are drawn in Sec. VI.

II. MODEL SETUP
We are interested in the the dark sector that gives rise to the Yukawa potential of the DM self-interaction. As a familiar prototype, Yukawa potential between DM particles provides us with a representative description and a concrete example of the long-range selfinteraction [10,12,42]. The self-interacting Yukawa potential * is written as which describes in non-relativistic limit the following interactions of the dark sector [42]: L χφ = g χχ γ µ χφ µ vector mediator g χχ χφ scalar mediator , (2) * Strictly speaking, Yukawa potential narrowly refers to the attractive interaction in Eq. (1). However, for the purpose of convenience, here we use it to refer to both the attractive and repulsive interactions. where m φ is the mass of the mediator particle φ, and the dark fine-structure constant α χ is connected to the coupling g χ through α χ = g 2 χ / (4π). For a vector force-carrier, the interaction between DM could be attractive (χχ scattering) or repulsive (χχ orχχ scattering), while the force can only be attractive (−) between DM for the scalar mediator scenario. Since in this paper we are interested in spin-1/2 identical DM, i.e., Majorana DM particles, the interaction between DM and the vector mediator in Eq. (2) is absent, and an extra s-channel process should be included in the calculation of relevant scattering amplitude for a scalar mediator φ, which is shown in Fig. 1. While the second and the third diagrams in Fig. 1 correspond to the long-range Yukawa interaction in non-relativistic limit, the first s-channel process can be described non-relativistically with a contact interaction, which is subject to the Sommerfeld enhancement that will be reviewed in detail below. However, due to its highly suppressed cross section σ χχ ∼ α 2 χ v 3 rel /m 2 χ , contribution of the s-channel process can be safely ignored compared with that of the Yukawa potential, even if the Sommerfeld enhancement is involved. So only the Yukawa potential is relevant for the DM scattering.
Although we want to explore phenomenologically the impact of the DM long-range selfinteraction as far as possible, it is unrealistic for our discussion to be entirely modelindependent. At the very least, the phenomenological consequence of the irreducible annihilation process χχ → φφ has to be taken into consideration (see Fig. 2). So, we study a model that not only gives rise to significant Sommerfeld effects within the Sun, but also cover as wide a range of phenomenology as possible, namely, the self-capture and both the s-and p-wave annihilation of the solar SIDM.
To facilitate the novelty brought by the long range self-interaction, we assume that the irreducible p-wave process (shown in Fig. 2) dominates the annihilation during the thermal freeze-out. The reason for this assumption is two-fold: 1) As opposed to the s-wavedominated case, the p-wave-dominated annihilation avoids severe constraint on the coupling α χ imposed by the DM relic density, so that the Sommerfeld effect remains significant [12,58].
2) The p-wave-dominated annihilation within the Sun is subject to such significant velocitysuppression that the total annihilation rate Γ A will be enhanced remarkably (see Eq. (9) and (18) in the following section), which may further favor the indirect detection of the solar DM.
On the other hand, given the dark sector of the Majorana DM χ and the scalar mediator φ, it is natural to assume that they couple to the SM sector through the Higgs portal [59][60][61][62][63][64][65]. In addition, extra four-fermion interactions are introduced to account for the DM s-wave annihilation into the SM particles. Integrating all these considerations, we consider the following interaction as an example: where sin θ describes the mixing between φ and Higgs boson, and y f = m f /v EW is the Yukawa coupling of the SM fermion f , with the SM fermion mass m f and the Higgs vacuum expectation value v EW ≈ 246 GeV. G f is the coupling strength between the DM and the SM fermion f . The coupling of φ and right-handed sterile neutrino N is introduced in Ref. [66] to save the light mediator φ from the Big Bang Nucleosynthesis (BBN) constraints. To avoid the overclosure problem, overproduction of entropy and changes to the light element abundances during the BBN, φ is required to decay before the BBN (t ∼ 1 s) [67], which corresponds to the bound [66] y N 6 × 10 −12 100 MeV m φ

A. Dark matter in the Sun
The accumulation of the DM in the sun is initiated by DM-nuclei collisions that cost the passing-by Galactic DM particles sufficient kinetic energy to escape from the gravitational pull of the Sun. In this process, subsequent collisions tend to further lower the kinetic energy of the DM rather than eject them back into the deep space as long as the DM mass is heavier than a few GeV, hence the number of the trapped DM grows until their capture and annihilation reach an equilibrium. If the DM self-interaction is taken into account, the evolution of the solar DM number N χ is depicted by the following equation where C n is the DM capture rate by (scattering off) nuclei in the Sun, C a is twice the annihilation rate of a pair of DM particles, E n and E s represent respectively the evaporation rate by nuclei and by the captured DM particles, and C s is the capture rate by the trapped DM particles, a key physical quantity of our concern in this paper. It should be noted that in the range of the DM mass of our interest (m χ ≥ 10 GeV) it is justified for us to omit the the evaporation rate E n in Eq. (5), since it is irrelevant for a DM mass larger than 5 GeV [68][69][70]. * Besides, we also omit the the self-evaporation rate E s because C a ≫ E s , which will be demonstrated in detail in Appendix B. Thus Eq. (5) is reduced to which has an analytic solution with ξ = C n · C a + C 2 s /4.
If tanh (t ⊙ · ξ) ≃ 1 in Eq. (7) with solar age t ⊙ = 4.5 × 10 9 yr , the capture and annihilation of the DM reach an equilibrium today. Then the annihilation rate Γ A for a steady number of DM within the Sun can be written as which is reduced to the trivial case where the dark matter self-interaction is absent, i.e., when Γ A = C n /2 if C 2 s /(C n C a ) ≪ 1 is satisfied.
B. Self-capture and annihilation of the self-interacting dark matter The formula for the self-capture rate can be obtained after a minor modification to the procedures in Ref. [69,72] as the following: where ρ χ = 0.3 GeV is the DM local density in our solar neighborhood, m χ manifestly the mass of the DM. f (u) is the velocity distribution of DM in the Sun's rest frame, which is expressed as where |v ⊙ | = 220 km · s −1 is the velocity of the Sun, u being the DM velocity at infinity where the solar gravitational effects is negligible, v 0 = 220 km · s −1 the DM velocity dispersion, and N(v esc ) is the normalization constant dependent on the Galactic escape velocity v esc = 544 km · s −1 . w = v 2 esc (r) + u 2 is the velocity of the incident DM particle, with v esc (r) the solar escape velocity at radius r. The DM distribution within the Sun can be described with a characteristic temperature T χ as follows * where G is the Newton's constant and M (r) represents the solar mass contained within radius r. T χ is used to be determined self-consistently through the following equation: in which the right hand side represents the solar temperature at the mean value of DM radius We use the Standard Sun Model (SSM) GS98 [73] to obtain a fitting function of R χ as follows with solar radius R ⊙ . The effective cross section for self-capture, σ sc (w), is dependent on the velocity of the incoming DM particle w, if we approximate the target DM trapped within the Sun to be at rest compared with the accelerated Galactic ones. On the other hand, for the mass range concerned in this paper (m χ ≥ 10 GeV), we further assume that the captured DM particles are located at the center of the Sun (see Eq. (15)), thus the self-capture rate can be simplified as where However, here we note that instead of growing infinitely with the accretion of the captured DM particles, C s N χ saturates to a maximum value set by the geometric limit of the resident DM, above which C s N χ is replaced by On the other hand, the annihilation rate of the solar DM can be expressed as where σ ann v rel ⊙ is the thermal average of the DM annihilation cross section times relative velocity v rel within the Sun, and the effective volume occupied by the trapped DM particles can also be described with a fitting function as follows:

IV. IMPLICATIONS OF THE LONG-RANGE SELF-INTERACTING DM
In this section, we study the implications of the long-range self-interaction on the selfcapture and annihilation within the Sun. Due to the multiple exchanges of the light forcecarrier between DM particles, the relevant calculations have to be carried out beyond the perturbative approach. The quantum mechanical description of the two-body scattering is proved to be an effective framework to do so, which is connected to the self-capture and annihilation in a manner that the asymptotic wave-function at infinity describes the final state of the scattering, and the wave-function at origin partly determines the probability of the annihilation. Now we delve into the details.

A. Calculation of the self-capture rete
We calculate the self-capture cross section σ sc (w) in Eq. (10) in the center-of-mass (CM) frame, in which the self-capture differential cross section for a pair of identical DM can be expressed as (after averaging over the initial spins and summing over the final spins ) where S represents the total spin of the DM-pair, the factor 1/2 accounting for the symmetry of the out-going DM particles, and the elastic scattering amplitude f (θ) is related to the wave function in position space via where the radial partial-wave amplitude A ℓ (r) has the following asymptotic solution for a large r: with j ℓ (n ℓ ) as the spherical Bessel (Neumann) function, and δ ℓ the phase shift for a partial wave ℓ . Thus f (θ) can be further expressed as with p = m χ w/2. As for the scattering angle θ, to ensure a net capture it is required to with where v J = 18.5 km · s −1 is the solar escape velocity at the radius of Jupiter's orbit. As a conservative estimate [74], v J is introduced not only to take into account the gravitational effects of the Jupiter from a more realistic three-body perspective, but to regulate the divergence encountered in the calculation of C s for the long-range self-interaction scenario.
We follow the same procedure as proposed in Ref. [42] to numerically calculate the phase shift δ ℓ and then the self-capture rate C s for specific parameters m χ , m φ , and coupling strength α χ . The value of C s is scanned in Fig. 3 over the the parameter space of the mass ratio z = m χ /m φ and the self-interaction strength α χ , for the Majorana DM with mass m χ = 10 GeV. While the Born regime occupies the lower left corner of the figure, the Sommerfeld effect becomes increasingly significant in the Coulomb limit (z → ∞), where a large partial wave number (ℓ max 1000) is required in calculation to ensure the convergence of C s . It is worth mentioning that given z and α χ fixed, C s will be suppressed for larger DM masses through a m −3 χ dependence, which is obvious from Eq. (16, 20, and 23).

B. Sommerfeld enhancement of the annihilation rate
Apart from the self-capture, the light force-carrier brings about significant consequences on the annihilation of DM as well, which have been widely studied in determining the thermal relic density and in calculating signals in indirect DM searches. To gauge the Sommerfeld effect in annihilation we invoke the Sommerfeld factor as follows [75,76] S ℓ = non−perturbative partial wave cross section tree − level partial wave cross section where ℓ stands for the ℓ-th partial wave. If one skips somewhat formal theoretical discussions, the implication of Sommerfeld enhancement on annihilation is encoded in the following heuristic relation between the non-perturbative M ℓ ladder and "bare" amplitude M ℓ 0 at treelevel [75,76]: where φ (k) = k ψ + p + (−1) S −k ψ + p /2 is the familiar scattering wave function for identical DM-pair in the reduced system, p = m χ v rel /2 being the initial momentum in the CM frame, with relative velocity v rel . It is noted that for identical fermion pairs L + S must be even, thus we have φ (k) = k ψ + p . After Fourier transformation and by repeated use of addition theorem the Sommerfeld factor for partial wave ℓ can be written in terms of the wave function in position space which can be calculated numerically along with δ ℓ . Therefore, after taking into consideration the Sommerfeld enhancement, we express the annihilation rate for the trapped DM particles as follows where S is the Sommerfeld factor dependent on v rel , the footnote "tree" indicating that the cross section is calculated at the tree-level, and the Maxwellian velocity distribution is written as with u 0 = 2T χ /m χ . Note that in Eq. (30) we neglect the cut-off at the solar escape velocity, which is equivalent to the approximation v esc → ∞.
Since we are only interested in the case where other p-wave processes are suppressed by couplings y f sin θ or y N in Eq. (3), the total thermally averaged cross section can be written as where included are the thermal cross sections for the irreducible p-wave process and its s-wave counterparts arising from the four-fermion interaction with the color factor c f = 3 (1) for quarks (leptons).

C. Constraints from the relic density
In Sec.IV A we calculate the self-capture rate C s with free parameters α χ and z = m χ /m φ . However, the coupling strength α χ is not an independent parameter as it can be determined by the observational DM relic density with specific m χ and m φ . For the given m χ , m φ and α χ the evolution of the relic density of DM is described by the following Boltzmann equation [77,78] dn χ dt + 3Hn χ = − σ ann v rel n 2 χ − n eq 2 where n χ and n eq χ are the number density of the DM and its counterpart in equilibrium with the background particles, respectively, H the Hubble constant and σ ann v rel is the thermally averaged cross section introduced in Eq. (31). In practice by changing variables such that Y = n χ /s Y eq = n eq χ /s with s the entropy density and x = m χ /T , the above Boltzmann equation can be rewritten as follows [77,78] where g * s and g * are the effective degrees of freedom for entropy and energy density, respectively, and the Planck mass m P = 1.22 × 10 19 GeV. We follow Ref. [58,79] to numerically solve Eq. (35) so as to give the coupling α χ determined by the relic density. In Fig. 5 shown is the value of α χ as a function of m χ and z = m χ /m φ , with a specific value η = 10 −2 introduced in Eq. (31). One can see from Fig. 5 that owing to the high velocity at the chemical decoupling stage, α χ is insensitive to the Sommerfeld effect. Here we note that the effect of the kinetic decoupling is not included in our calculation, because it is insignificant for the case of p-wave-dominated annihilation when the temperature of kinetic decoupling T kd is much less than that of the chemical decoupling T f , even if the Sommerfeld enhancement is taken into consideration [42,58,80]. This requirement is in coincidence with the onset of the Sommerfeld effect: a large z = m χ /m φ ≫ 1 guarantees that T f ≃ m χ /25 ≫ m φ ∼ T φ kd , where the kinetic decoupling temperature for the χ-φ interaction T φ kd provides a maximal value for T kd [12]. Given η = 0.01, in Fig. 6 we use the relic density to calculate the thermally averaged cross section for the s-wave (σv rel ) 0 S 0 ⊙ and p-wave annihilation (σ φφ v rel ) tree S 1 ⊙ within the Sun, respectively. We note that smaller η will not bring remarkable changes to Fig. 6 because the relic density turns out to remain constant as the s-wave component gets smaller (η < 0.01) [58].

V. CONSEQUENCES ON THE NEUTRINO TELESCOPE
The signal of the solar DM is associated with the observation of primary and secondary high energy neutrinos, the annihilation products of the DM accumulated in the center of the Sun. Among other goals, many terrestrial neutrino detection projects such as Ice-Cube [81], Super-Kamiokande [82], Baikal Neutrino Project [83] and ANTARES [84] are dedicated to such observation, and next generation telescopes including PINGU [85] and Hyper-Kamiokande [86] are also in proposal. In general the neutrino differential flux at the detector location can be schematically described as where Γ A is the annihilation rate introduced in Eq. (9), d ⊙ the Sun-Earth distance, and dN ν /dE ν denotes the energy spectrum of neutrino ν e,µ,τ andν e,µ,τ per DM annihilation event, after folding the effects of annihilation branch ratio, hadronization of quarks and neutrino oscillation, which are usually simulated with DarkSUSY [87] and WimpSim [88]. For the purpose of illustration the neutrino flux is connected to the readout at the detector terminal with the following relation: where N µ is the number of the muon events that arise from the charge and neutral current interaction between the incident neutrino and medium surrounding the detector, the effects of which is encoded in the effective area A ν , E th the threshold energy of the detector, ∆Ω = 2π (1 − cos Ψ) being the solid angle surrounding the Sun with angular resolution Ψ, and t exp is the exposure time. If no excess of muon over the background is observed, an upper limit on the annihilation rate Γ A can be obtained. Now we use the recent release of the Super-Kamiokande (SK) analysis [57] to constrain the strength of the DM-nucleon interaction in the presence of the Sommerfeld effect, in terms of the upper limits on the nuclear capture rate C n that depends on the specific DM-nucleon interaction model. One can translate these constraints back on the couplings of the DMnucleon interaction with the nuclear form factors given by Ref. [89,90], once the interaction is specified. For the purpose of simple illustration we assume a leptophilic four-fermion interaction in Eq. (3), so that the incoming DM particle with velocity w interacts with the target nucleus through the Higgs portal. The relevant differential cross section is expressed as where q is the transferred momentum of the incident DM particle, and A is the atomic number of the target nucleus. The nucleon mass is approximated as the protonic one m p , and the coefficient f N = 0.351 arises from the scalar bilinear for the underlying quarks. F 2 N (q 2 ) = [3 j 1 (q R 1 ) / (q R 1 )] 2 e −q 2 s 2 is the nuclear form factor, where j 1 (x) = sin (x) /x 2 − cos (x) /x is the Bessel spherical function of the first kind, and the parameter R 1 = R 2 0 − 5s 2 with R 0 ⋍ 1.23A 1/3 fm, and s ⋍ 1 fm [91].
Current DM direct detections provide a stringent upper bound on the mixing angle such that sin θ 10 −8 [66]. Mixing angle at such scale is 2 to 3 orders of magnitude smaller for the light force-carrier φ to decay before t ∼ 1 s, jeopardizing the success of the BBN, so φ is assumed to decay quickly to the sterile neutrino N instead [66], as explained below Eq. (3). As a consequence we regard the s-wave annihilation as the only source relevant for the DM indirect detection. In Fig. 7 we present the upper limits at 90% confidence level (C. L) on C n imposed by SK with various z = m χ /m φ = 10 2 , 10 3 , and 10 4 , and η = 10 −2 , 10 −4 , 10 −5 and 10 −6 , respectively. As is discussed similarly in Ref. [57], two extreme benchmark scenarios in which DM exclusively annihilate via hard (τ + τ − ) and soft bb swave channel are taken into consideration over a DM mass range from 10 to 100 GeV. The number of the trapped solar DM particles N χ , or equivalently, the nuclear capture rate C n is constrained by the upper limit on the annihilation rate Γ up with the following relation: where the branch ratio of the s-wave annihilation which is responsible for the neutrino signal can be expressed as The constraints on C n are affected mainly by three factors: Γ up , z and η. As a rough estimate, Eq. (9) can be used to explain the features of the constraints in Fig. 7: when the Sommerfeld effect is insignificant (z = 10 2 , 10 3 ) so that Γ up ≫ C 2 s / (2C a ), upper limits are sensitive to η through the dependence on BR s ; the sharp increases of the sensitivity on C n in Fig. 7 imply that Γ up ∼ O (C 2 s BR s / (2C a )) when the Sommerfeld effect turns remarkable (z = 10 4 ), which is of our interest in this study. For reference purpose in Fig. 7 we also present the nuclear capture rate (in the black dash-dot line) derived from the DM-nucleus cross section in Eq. (38), for which the values of the φ-Higgs mixing angle sin θ = 10 −8 and z = 10 4 are adopted. Parameters at such scales are comparable to the sensitivity of direct detection discussed in Ref. [66]. Although the direct detection approaches are proved to be efficient in exploring the spin-independent nucleon-DM interaction, the solar neutrino detection may impose even more stringent constraints on that as the relevant parameter space is significantly squeezed by the Sommerfeld effect. Such phenomenon occurs at z = 10 4 for the present SK sensitivity, as is shown in Fig. 7.
It is also noted that when the s-wave component is sufficiently small that its cross section drops below the p-wave counterpart, constraints will turn loose because that's when BR s begins to be suppressed by a small η. To get some sense, in Fig. 8 we plot BR s as a function of DM mass m χ for z = 10 3 and z = 10 4 . It is evident that the dips in BR s in the left panel are attributed to the p-wave resonance encountered along the cross section at z = 10 3 in Fig. 6, and hence are responsible for the corresponding bumps in sensitivity on C n in Fig. 7 for z = 10 3 .

VI. CONCLUSIONS
In this work we investigate an example of the solar DM that self-interact via light force carriers, specifically, Majorana DM particles scattering with long-range self-interaction and annihilating dominantly through p-wave channel. To optimize and cover the relevant phenomenology of the solar SIDM as comprehensively as possible, we study a scenario where a small but non-vanishing quota of DM annihilation proceeded through the s-wave channel in the thermal decoupling epoch, so that the self-capture, both s-and p-wave annihilation are all involved in our consideration. Instead of evaluating relevant DM self-capture and annihilation processes at tree level, we calculate them in a non-perturbative way by solving the Schrödinger equation with Yukawa potential. We find that the Sommerfeld effect can be significant for DM self-capture if coupling α χ and mass ratio z are regarded as free parameters. Besides, the impact of the light mediator on the thermal freeze-out that is assumed to produce the the observed DM relic density is also taken into consideration, and is used to determine the self-interaction coupling α χ for the given masses of DM particle and mediator.
Moreover we also explore the consequence of the Sommerfeld effect on terrestrial neutrino telescopes which is utilized to probe signals from the solar DM. Since the light mediators produced from the p-wave DM annihilation do not produce any observable signal at the terrestrial neutrino telescope, the neutrino signals originate exclusively from the s-wave component, which may become close to or even dominant over its p-wave counterpart within the Sun, due to the Sommerfeld enhancement. Therefore we combine the sensitivity of SK and the observed DM relic density to constrain the nuclear capture rate, or equivalently, the strength of DM-nucleon interaction. Due to SK's high sensitivity, the constraint on the DM-nucleon interaction can be remarkably tightened in the low DM mass region, where the Sommerfeld effect on the DM self-capture rate has not yet been outstripped by the m −3 χ suppression.
It is also noted that while our discussion is based on a generic class of models in which the fermion DM particles self-interact via a light scalar mediator, the consequence of this force-carrier in astrophysics and cosmology is another interesting topic that is beyond the scope of this work (For a most recent study, see Ref. [92]). Besides, as a tentative study on the phenomenology of the solar SIDM, our discussion can be further extended to other SIDM models, not confined to the case of scalar mediator or Majorana DM.
Simple as it is, this work has captured the main features of the solar SIDM. To put it in a nutshell, the interplay between the long-range dark force and the solar neutrino detection is encoded in the following expression up to a possible correction due to the self-capture saturation, where σv rel swave ( σv rel pwave ) is the s-wave (p-wave) thermal cross section, and BR signal is the branch ratio for the signal annihilation channel. Qualitatively speaking, a model with a combination of enhanced C s and velocity-suppressed σv rel pwave may maximize the novel effects of the SIDM, as long as such enhancement has not been offset by an extremely small BR signal . For instance, in the model where φ decays to the active neutrinos before the BBN [66], BR signal will be significantly boosted if such decay proceeds before φ reaches the Earth. In this case, even for a smaller z the neutrino detector SK will lead in the sensitivity on C n .

Appendix A: Distribution of the self-interacting dark matter in the sun
The distribution of the DM particles trapped in the Sun has been well studied in the literature [93][94][95]. While in Ref. [93] authors adopts a Maxwellian velocity distribution , * to describe the non-thermal equilibrium between the DM particles and solar nuclei, the studies in Ref. [94,95] indicate a deviation from the Maxwellian form, in a manner that the actual velocity distribution is suppressed at the tail and tends to be anisotropic at large radius. Such deviation can be attributed to the fact that the energetic collisions that send the DM particles into the high orbits occur predominantly near the hotter core of the Sun. Thus one expects a lower angular momentum distribution, or equivalently a larger radial-velocity component (v r /v) for the high-energy orbits. This is illustrated in Fig. 2 of Ref. [95], where the normalized average kinetic energy of the DM particles (2/3) E k /T χ decreases toward the outer layers of the Sun, and the average square of the radial velocity component (v r /v) 2 deviates from 1/3 at radii larger than around 0.2 R ⊙ , indicating an anisotropic nature of the velocity distribution beyond that radius. Nevertheless, it should be stressed that while such correction to the thermal distribution is necessary for the evaluation of the energy transport and evaporation in the Sun that rely sensitively on the high end of the velocity distribution [94,95], the phenomenology of interest in this study such as self-capture and annihilation can still be described with the bulk of the Boltzmann form to a good approximation.
To illustrate this point, we first investigate the solar DM distribution by following the "Brownian motion" method pioneered by the author of Ref. [94]. A total number of 10 5 successive collisions between the 10 GeV trial DM particle and solar elements are simulated. In the left panel of Fig. 9 we present the quantity (2/3) E k /T χ and (v r /v) 2 from the solar center up to 0.1 R ⊙ , a radius sufficient to envelope almost all the DM particles trapped in the Sun (see Eq. (15)). The curves would be horizontal (at 1 and 1/3 respectively) for a thermal distribution. We compare the simulated velocity distribution with the Maxwellian form in the right panel, where the velocity variable v is nondimensionalized in terms of the velocity unit (GM ⊙ /R ⊙ ) 1/2 ≈ 436 km · s −1 . It is evident that the Maxwellian distribution with the characteristic temperature T χ indeed provides a good description of the realistic solar DM distribution for the mass of interest. T χ = 0.967 T ⊙ (0) is determined from Eq. (13). For larger DM masses, one can approximate T χ as the solar center temperature T ⊙ (0).
In addition, the Boltzmann form is expected to remain an accurate description for the solar SIDM because it naturally describes the thermal distribution of the SIDM particles gathering in the solar core, while the effective temperature T χ accounts for the non-thermal equilibrium between the SIDM particles and the solar nuclei to a good approximation. Therefore we still adopt the Boltzmann distribution in the evaluation of self-capture and annihilation in the Sun. * By demand of satisfying the Boltzmann equation, the Maxwellian velocity distribution in turn leads to the Boltzmann distribution f χ (r, v) ∝ exp [− (V (r) + E k ) /T χ ], and hence the density distribution n χ (r) ∝ exp [−V (r) /T χ ], with V (r) the gravitational potential of the DM particle at radius r. where we make approximation |v 1 − v 2 | ≃ |v 1 | = v 1 for the case v 1 > v esc ≫ v 2 , in which the contribution to evaporation rate doesn't suffer the Maxwellian tail suppression of f χ (v 2 ), and σ sc (v) is no other than the self-capture cross section with incident velocity v. On the other hand, the annihilation rate in a unit volume is written as where σ ann (v) is the annihilation cross section with relative velocity v. Given that compared with Eq. (B4) the integral over subspace´v >vesc d 3 v f χ (v) in Eq. (B3) so overwhelmingly suppresses ǫ s in the DM mass region of our interest that we can safely ignore the contribution from dark matter self-interaction in this study. Finally it is worth mentioning that while our discussion is based on the assumption of a large mean free path for the SIDM particles, the self-evaporation will be heavily suppressed if the optical depth is much smaller than the DM radius. Because in such case, the would-be evaporation events will end up being blocked by multiple collisions, which further favors our conclusion.