Very Degenerate Higgsino Dark Matter

We present a study of the Very Degenerate Higgsino Dark Matter (DM), whose mass splitting between the lightest neutral and charged components is O1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}(1) $$\end{document} MeV, much smaller than radiative splitting of 355 MeV. The scenario is realized in the minimal supersymmetric standard model by small gaugino mixings. In contrast to the pure Higgsino DM with the radiative splitting only, various observable signatures with distinct features are induced. First of all, the very small mass splitting makes (a) sizable Sommerfeld enhancement and Ramsauer-Townsend (RT) suppression relevant to ∼1 TeV Higgsino DM, and (b) Sommerfeld-Ramsauer-Townsend effect saturate at lower velocities v/c ≲ 10−3. As a result, annihilation signals can be large enough to be observed from the galactic center and/or dwarf galaxies, while the relative signal sizes can vary depending on the locations of Sommerfeld peaks and RT dips. In addition, at collider experiments, stable chargino signatures can be searched for to probe the model in the future. DM direct detection signals, however, depend on the Wino mass; even no detectable signals can be induced if the Wino is heavier than about 10 TeV.


Introduction
The pure Higgsino (with the electroweak-radiative mass splitting ∆m = 355 MeV between its lightest neutral and charged components) is an attractive candidate of thermal dark matter (DM) for its mass around 1 TeV [1]. As null results at Large Hadron Collider (LHC) experiments push supersymmetry (SUSY) to TeV scale, such Higgsino as the lightest supersymmetric particle (LSP) has recently become an important target for future collider [2][3][4][5][6][7] and DM search experiments [5][6][7][8][9][10][11]. A priori, the Higgsino mass µ and gaugino masses M 1 , M 2 for the Bino and Wino are not related; thus, the pure Higgsino scenario with much heavier gauginos is possible and natural by considering two distinct Peccei-Quinn and R symmetric limits.
It is, however, difficult to test the pure Higgsino LSP up to 1-2 TeV at collider experiments (including future 100 TeV options) and dark matter detections. Standard collider searches of the pure Higgsino LSP based on jet plus missing energy become hard as final state visible particles become too soft to be well observed due to the small mass splitting [2][3][4]; but the splitting is still large enough for charged Higgino components to decay promptly at colliders so that disappearing track and stable chargino searches are not able to probe [2,12]. Furthermore, the purity of the Higgsino states suppresses DM direct detection signals. DM indirect detection signals are also not large enough because of relatively weak interactions and negligible Sommerfeld enhancements [8-11, 13, 14]. In contrast, the pure Wino DM with the radiative mass splitting of 164 MeV, another thermal DM candidate for its mass ∼ 3 TeV, provides several ways to test: monojet plus missing energy due to more efficient recoil and larger cross-section [2,4,15,16], disappearing track due to -1 -

Very Degenerate Higgsino DM
We discuss the SUSY parameter space of the Very Degenerate Higgsino DM, which involves the Higgsino mass parameter µ, the Bino and Wino masses M 1,2 , the ratio of the Higgs vacuum expectation values t β ≡ tan β = v u /v d , the weak mixing angle given by s W ≡ sin θ W ≈ 0.23, and the W gauge boson mass m W . We assume the limit |M 1 ±M 2 |, |M 2 ±µ|, |µ±M 1 | m W and |M 1 |, |M 2 | |µ|. We keep the signs of mass eigenvalues and make eigenvectors real. Later on, we will assume M 2 , µ > 0 and M 1 < 0 for the Very Degenerate Higgsino DM, but we will be agnostic about how such signs can be obtained. Meanwhile, all sfermions and heavy Higgs bosons are assumed to be very heavy and not relevant to our study.
Higgsino mass eigenvalues at tree-level are [20,21], where s 2β = sin 2β and so on. The subscripts S, A imply that the mass eigenstates are Which of χ 0 S or χ 0 A is the LSP depends on the relative sign of µ -2 -

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and K M 2 : the χ 0 A is the LSP if the relative sign is positive, and vice versa. Expressing both possibilities, we write the LSP mass as Higgsino mass splitting at tree-level is then The physical mass splitting is ∆m = ∆m tree + ∆m loop , where the model-independent electroweak loop corrections give ∆m loop ≈ 355 MeV for the Higgsino [12].
Notably, the ∆m tree can be negative, so that the resulting physical mass splitting ∆m can be smaller than the ∆m loop . 1 From the above approximations, we find that one way to obtain negative ∆m tree is to satisfy the following conditions: • sign(µM 2 ) > 0 is required because only the first term in eq. (2.4) can be negative.
Assuming µ, M 2 > 0 from now on, we rewrite Thus, ∆m tree < 0 if the mass parameters satisfy • M 1 < 0 is preferred so that K < 1. We assume M 1 < 0.
• Small t β is preferred; t β 2 does not allow solutions for ∆m tree < 0 for the range of mass parameters considered.
We apply this set of approximate conditions to our full numerical calculation to narrow down the solution finding procedure.
In figure 1, we show one set of numerical solutions for ∆m tree < 0 for the range of µ ≤ 2 TeV and −2.5 ≥ M 1 ≥ −5 TeV with fixed benchmark parameters M 2 = 10 TeV and t β = 1.8. In most of the parameter space shown, ∆m is smaller than the radiative mass splitting of 355 MeV. Although the approximate equations above do not depend on µ, the full numerical solution does a bit. We will consider two benchmark cases of ∆m =2, 10 MeV in this parameter space throughout. Later, we will also comment on the case with smaller M 2 = 5 TeV. The solutions for ∆m =2, 10 MeV and our most discussions do not strongly depend on the value of M 2 , but direct detection signals do as will be discussed. The neutralino mass splitting, δm 0 ≡ |m 0 χ 2 | − |m 0 χ 1 |, is somewhat larger ∼ O(100) MeV, and it also does not strongly affect our discussion. 1 The negative ∆mtree has been used in exotic collider phenomenology of Higgsinos [22,23].
, and spin-independent direct detection rate σ SI (dotted; see section 4.1) are shown. We consider the two benchmark models along the ∆m = 2, 10 MeV contours throughout.

Indirect detection of annihilation signals
Non-perturbative effects in DM pair annihilation can lead to Sommerfeld enhancement [13,14] or Ramsauer-Townsend suppression [9,11]. The pure Higgsino DM with µ ∼ 1 TeV and ∆m ≈ 355 MeV does not experience large SRT effects. Only Higgsinos as heavy as ∼ 7 TeV can experience sizable effects, but they are too heavy to be relevant to collider experiments. On the other hand, the 1-3 TeV pure Wino DM with ∆m ≈ 164 MeV experiences much larger SRT effects with a resonance appearing at around 2.4 TeV [8-11, 13, 14, 17]. Since the SRT effects on the pure Wino DM saturate at relatively high velocities v/c ∼ 10 −2 , Wino annihilation cross-sections at various astronomical sites with different velocity dispersions are same.
We will discuss that the very small splitting of the Higgsino DM can make the relevant Higgsino mass scale down to ∼ 1 TeV and allow different annihilation cross-sections at various astronomical sites, postponing the saturation to lower velocities. Furthermore, there can appear not only Sommerfeld enhancements but also RT suppressions.

SRT effects with very small mass splitting
We focus on today's DM annihilation cross-sections into W W, ZZ, γγ, Zγ channels. Thus, we do not consider co-annihilation channels. Pair annihilations with SRT effects can pro-

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ceed via various intermediate two-body states with the same charge Q = 0 and spin S = 0, 1 as those of the initial LSP pair, which are exchanged by photons and on/off-shell W, Z gauge bosons. We take into account all two-body states formed among Higgsino states; in addition, we add heavier gauginos if their masses are within 10 GeV of the Higgsino in order to accommodate non-zero effects from them, but this rarely happens in our study. We follow a general formalism developed for SUSY in ref. [24][25][26][27] to calculate absorptive Wilson coefficients and non-relativistic potentials between various two-body states, and we numerically solve resulting Schrödinger equations to obtain SRT effects.
We study two benchmark models with ∆m = 2, 10 MeV presented in figure 1. For the given µ ∈ {600, 2000} GeV (and other parameters as described), a unique solution for M 1 is found. As long as gaugino mixtures are small, the exact value of M 2 ( |M 1 |) does not matter much in annihilation signals. It is because leading contributions to annihilations and SRT effects already exist in the pure Higgsino model with vanishing gaugino mixings: for example, the direct annihilation χ 0 χ 0 → W W and the SRT effect χ 0 χ 0 → χ + 1 χ − can be mediated by the Higgsino-Higgsino-W interaction without need for any gaugino mixtures. Thus, we set M 2 = 10 TeV (and t β = 1.8) in this section.
In figure 2, we show contours of the annihilation cross-section into photon-line signals, σv for the benchmark models with ∆m = 2, 10 MeV and the usual pure Higgsino model with ∆m = 355 MeV for comparison. Similar features exist in photon-continuum signals from σv W W +ZZ ≡ σv W W + σv ZZ , and similar discussions apply.
Two types of enhancements are observed, most clearly from the ∆m = 2 MeV result. First, a series of threshold zero-energy resonances forms just below the excitation threshold of χ 0 χ 0 → χ + χ − with 1 2 µv 2 ∆m (blue-dashed line) [27][28][29], depicted as diagonal bands of enhancement. Photon exchanges between chargino pairs are responsible for the series of closely-located resonances, but not all of them are captured and shown in the figure; see ref. [27] for a demonstration of many closely-located threshold resonances. Well below the threshold, SRT effects are independent on the DM velocity as the W -boson exchange in χ 0 χ 0 → χ + χ − becomes governed by the W -mass rather than the DM momentum [13,14,27], depicted as vertical regions of enhancement. The SRT effect saturates at finite enhancement in the v → 0 limit because of the finite-ranged W -exchange Yukawa potential.
As ∆m increases, the excitation χ 0 χ 0 → χ + χ − becomes harder and the attractive potential becomes effectively shallower [14]. A heavier DM with a smaller Bohr radius can compensate this trend and can form zero-energy bound states. Thus, the larger ∆m, the heavier Higgsino Sommerfeld peaks. From µ ∼ 1.1 TeV for ∆m = 2 MeV, the Sommerfeld peak moves to a heavier µ ∼ 1.3 TeV for ∆m = 10 MeV and to much heavier µ ∼ 7 TeV for the pure Higgsino with ∆m = 355 MeV. Moreover, the threshold velocity becomes higher with the larger ∆m, making the SRT effects saturate at higher velocities. All such behaviors are clearly shown in figure 2.
Another remarkable is that RT dips are formed near Sommerfeld peaks [9,11,18,19] both near the excitation threshold and in the small-velocity saturation regime. RT dips are located at slightly heavier Higgsino masses and/or larger velocities. As ∆m increases, dips and peaks become more separated in µ and v.

Annihilations at GC and DG
We calculate annihilation cross-sections at GC and DG, main candidate sites for DM indirect detection. The GC is expected to support a huge DM density but also plenty of contaminations from baryons, whereas DG are very clean DM sources in spite of smaller DM density. In addition, velocity dispersions are an order of magnitude different, often further differentiating annihilation signals at DG and GC.
We convolute the annihilation cross-section calculated in the previous subsection with the Maxwell-Boltzmann velocity distributions for GC and DG [9,[30][31][32]  Resulting velocity-convoluted annihilation cross-sections at GC and DG are shown in figure 3. Sommerfeld enhancements and RT suppressions are both clearly observed near the 1 TeV Higgsino. Near Sommerfeld peaks and RT dips, annihilation cross-sections at GC and DG are different in general. The difference is larger for the ∆m = 2 MeV case because SRT effects saturate at lower velocities. Meanwhile, overall enhancements and suppressions are larger for the ∆m = 10 MeV case because peaks and dips are more separated in µ and v so that they lead to less cancellation in the velocity convolution. We also comment that GC cross-sections are not as sharp as DG ones in the figure because we had to average over very closely-separated peaks and dips appearing just below the excitation threshold (where the GC signal is most sensitive too), not all of which is well captured in our parameter scanning.
Another remarkable feature in figure 3 is that, owing to RT dips, the DG annihilation cross-section can be smaller than that of GC. It is a counter-example to the typical result that the DG annihilation cross-section is similar or larger because the DM velocity dispersion is smaller. The existence of RT dips is (accidentally) more clear in the photon-line signal than in the photon-continuum signal; as RT dips are produced from cancellations between various contributions (not necessarily related to resonances), their appearances and strengths can depend on annihilation channels.
However, the photon-line dip depth that will be observed at detectors is subject to the internal bremsstrahlung effect. Within detector resolutions of the photon energy, the photons radiated off the W W annihilation process can contribute to photon-line signals, and this extra contribution can smooth the RT-dips in figure 4. As shown in the 5plet and 7-plet DM cases studied in ref. [19], photon-line dips at some DM masses can disappear due to this extra contribution. For a better estimation of indirect detection, it is worthwhile to carry out a similar study for our doublet case; so our conclusions are subject to this uncertainty.
The peak heights shown in the figure may also be subject to uncertainties; our parameter scanning resolution very close to peak centers is limited, and perturbative cor- rections that may become important in this regime are not added. The perturbative corrections are most important when the unitarity is broken by unphysically enhanced cross-sections [33]. However, our annihilation cross-sections are well below the unitarity bound σv ≤ 4π/(µ 2 v) 10 −20 × 1 TeV µ 2 10 −2 v cm 3 /sec; and indeed, the regularizing velocity v c ∼ 10 −6 [33] is much smaller than our saturation velocity. Also, our scanning resolution is good enough just away from peak centers. Thus, we do not attempt to further improve the peak height calculation.
In figure 4, we finally overlay the latest constraints and some projection limits of indirect detections. Datasets presented include: HESS 2013 [34] and Fermi-LAT 2015 [35] for photon-lines from GC, MAGIC 2013 [36] for photon-lines from DG, Fermi-LAT+MAGIC combination [37] for photon-continuum from DG, and HESS 254h [38][39][40] for photoncontinuum from GC. Projection studies include: CTA 5h [11] for photon-lines from GC (see refs. [41,42] for similar results), CTA 500h [43,44] for photon-continuum from GC, and Fermi-LAT 15 years for photon-continuum from 16 DG [45] (see ref. [46] for CTA projections). Current and future DES constraints from DG photon-continuum [47] are similar or weaker than the results shown, so we do not show them. A full DM relic density is as-  sumed for all Higgsino masses to interpret these data as the constraints on the annihilation cross-sections.
Currently, Sommerfeld peaks in both ∆m = 2, 10 MeV models are constrained by DG searches. Also, GC searches constrain Sommerfeld peaks of the ∆m = 10 MeV case, while smaller peaks of the ∆m = 2 MeV are not yet constrained by GC searches. In the future, a large part of the Sommerfeld enhanced parameter space can be probed by CTA GC and Fermi DG searches. On the other hand, RT dips in photon-line signals are below future sensitivities although potential positive contributions from the internal bremsstrahlung can change this somewhat. RT dips in photon-continuum signals are less significant and close to CTA GC projections.
For reference, we also show as green bands the mass range where the thermal Higgsino DM with ∆m = 355 MeV can explain the full DM relic density. Although SRT effects on the Very Degenerate Higgsino model can alter the relic density somewhat, the pure Higgsino result is still a useful guide as SRT effects on relic density may not be so significant; not only nearby Sommerfeld peaks and RT dips may cancel each other during a thermal history, but also some co-annihilation channels may have opposite SRT effects (as for the pure Higgsino DM [27]) that can also nullify impacts on the relic density. Without dedicated relic density calculations, we are content with assuming a full DM relic density which may come, e.g., from a non-thermal origin, and in any case our signals can be scaled in proportion to a true relic density.

Direct detection
The spin-independent direct detection (SIDD) signal of the nearly degenerate Higgsino DM depends on the mass splitting between the neutral states δm 0 and the amount of the gaugino mixture. The neutral mass gap δm 0 should be larger than O(0.1) MeV, otherwise its inelastic scattering mediated by the Z exchange should have been already observed [9]. For the sufficiently large δm 0 as in our study (see figure 1), the elastic scattering rate is controlled by gaugino mixtures (via Higgsino-gaugino-Higgs coupling), that is, the signal vanishes in the pure Higgsino limit. Therefore, we consider two benchmark values of M 2 = 10 and 5 TeV in this subsection, representing the cases with relatively small and large gaugino mixings and SIDD signals. For each M 2 benchmark, the value of M 1 is fixed (as a function of other parameters) to obtain the desired ∆m = 2, 10 MeV, and thus SIDD rates are determined.
The SIDD cross-section is approximately given by [48] σ SI 8 × 10 −47 g hχχ 0.01 where the sign ∓ implies the sign(-K ) and we assume the Higgs alignment limit. We obtain σ SI = (3 ∼ 5) × 10 −48 , (4 ∼ 9) × 10 −47 cm 2 for the M 2 = 10, 5 TeV with the range spanned by µ = 600 ∼ 1500 GeV (see figure 1 for M 2 = 10 TeV result). The dependence on the ∆m (indirectly via Bino mixtures) is not significant for ∆m 10 MeV. The former range of σ SI with M 2 = 10 TeV is close to the coherent neutrino scattering background floor so that searches will be difficult in the near future, while the latter range with M 2 = 5 TeV is expected to be probed at future experiments such as DarkSide-G2 [49,50] and LZ [49,51].
Although indirect detection signals are sizable for both M 2 benchmark values, the absence or existence of detectable SIDD signals still depends on the Wino mixture (hence, the Wino mass), and either is not a necessary consequence of the Very Degenerate Higgsino DM. Meanwhile, more interesting direct detection signals of our model can be produced by the formation of a DM-nucleus bound state through the inelastic scattering of χ 0 N Z → χ − N Z+1 [52][53][54]. The latest analysis adopting a semi-classical calculation in the Fermi gas model of nuclei [54] showed that neutrinoless double-beta decay experiments like EXO-200 and Kamland-Zen are able to provide a unique and strong sensitivity to the model parameter space with ∆m smaller than the chargino-nucleus binding energy ∼ 20 MeV. Further progresses in understanding nuclear model dependences of the nuclear transition element and/or improving experimental sensitivies will be crutial to test our model.

Collider searches
With the very small mass splitting, the charged Higgsino can be long-lived at LHC experiments. If it decays outside or the outer part of LHC detectors, stable chargino searches can apply, that is, characteristic ionization patterns of traversing massive charged particles can be identified. If it decays in the middle of detectors, disappearing charged track searches can apply as soft charged decay products are not efficiently reconstructed.
For ∆m much smaller than the pion mass, the dominant chargino decay mode is χ + → e + ν e χ 0 [12,[55][56][57]: with the function P (x) given in ref. [12]. For ∆m ∼ O(1 − 10) MeV, the decay length is very long, cτ ∼ 10 7 − 10 12 m (equivalently τ ∼ 10 −1 − 10 4 sec), so that almost all charginos traverse LHC detectors and thus only stable chargino searches apply. By reinterpreting the CMS 8 TeV constraints on the stable charged pure Wino [58], we obtain the constraint µ 400 − 600 GeV for ∆m much smaller than the pion mass. The uncertainty range quoted is partly owing to the lack of our knowledge of r min , the minimum decay length of the chargino for the CMS stable chargino search to be applied; it is needed because CMS considered the range of charged Wino decay length cτ = O(0.1 − 10) m where only a fraction of charged Winos traverse detectors and become stable charginos. From the CMS acceptance curve in ref. [58], we choose to vary r min = cτ min 1.5 − 6 m (τ min = 5 − 20 ns) to obtain the constraint and uncertainty.
We conclude that the ∼ 1 TeV Very Degenerate Higgsino DM is currently allowed, but future LHC searches of stable charginos will better constrain the model.

Cosmological constraints
The long-lived charged Higgsino can be cosmologically dangerous. The above quoted lifetime in our model τ ∼ 10 −1 − 10 4 sec could endanger the standard bing-bang nucleosynthesis (BBN) prediction. Although the chargino decay releases only soft leptons not directly affecting BBN, its metastable existence can form a bound state with a helium and can catalyze the 6 Li production. The lifetime limit τ 5000 sec of such a metastable charged particle [59] constrains the Higgsino mass splitting to be ∆m 1.2 MeV. 2 The (∆m) 5 dependence of the decay width in eq. (4.3) makes the BBN constraints quickly irrelevant to larger ∆m cases that we focus on.
As the enhancement is saturated at modestly small velocity, early-universe constraints from the era with the very small DM velocity such as recombination and DM protohalo formation are not strong. For example, σv W W 10 −24 cm 3 /sec is generally safe from such considerations (see, e.g., refs. [60][61][62]), so that the model is not constrained possibly except for very small parameter spaces close to Sommerfeld peaks.

JHEP01(2017)009 5 Summary and discussions
We have studied the Very Degenerate Higgsino DM model with O(1) MeV mass splitting, which is realized by small gaugino mixings and leads to dramatic non-perturbative effects. Owing to the very small mass splitting, SRT peaks and dips are present at around the 1 TeV Higgsino mass, and the velocity saturation of SRT effects is postponed to lower velocities v/c ∼ 10 −3 . As a result, indirect detection signals of ∼ 1 TeV Higgsino DM can be significantly Sommerfeld-enhanced (to be constrained already or observable in the near future) or even RT-suppressed. Annihilation cross-sections at GC and DG are different in general: either of them can be larger than the other depending on the locations of Sommerfeld peaks and RT dips. But our conclusions are subject to unaccounted internal bremsstrahlung effects which can smooth RT dips. Further studies are required to check that our results are robust. Meanwhile, other observable signature is also induced in stable chargino collider searches, which can probe the 1 TeV scale in the future. However, the rates of direct detection signals depend on the M 2 value (the smaller M 2 , the larger signal) so that M 2 ∼ 5(10) TeV can(not) produce detectable signals. The potentially unusual aspects of indirect detection signals discussed in this paper are well featured by the two benchmark models of ∆m = 2 and 10 MeV and shall be well taken into account in future searches and interpretations in terms of Higgsino DM models.
The Very Degenerate Higgsino DM also provides an example where "slight" gaugino mixings can have unexpectedly big impacts on the observation prospects of the Higgsino DM. The mixing is slight in the sense that the direct detection, whose leading contribution is induced by gaugino mixings, can still be small (for heavy enough Winos). But the phenomenology is unexpectedly interesting because such small mixings are usually thought not to affect the indirect detection signal, as the signal is already sizable in the zeromixing limit. In all, nearly pure Higgsino DM can have vastly different phenomena and discovery prospects from the pure Higgsino DM, and we hope that more complete studies can be followed.