Dark matter, fine-tuning and $(g-2)_{\mu}$ in the pMSSM

In this paper we analyze spectra in the phenomenological supersymmetric Standard Model that simultaneously result in the right dark-matter relic density $\Omega_{\rm DM} h^2$, offer an explanation for the $(g-2)_{\mu}$ discrepancy $\Delta a_{\mu}$ and are minimally fine-tuned. We discuss the LHC phenomenology resulting from these spectra and the sensitivity of dark-matter direct detection experiments to these spectra. We find that the latter type of experiments with sensitivity to the spin-dependent dark-matter-nucleon scattering cross section $\sigma_{\rm SD,p}$ will probe all of our found solutions.


leon scatteri
g cross section σ SD,p will probe all of our found solutions.

Introduction

The Large Hadron Collider (LHC) has been searching for over a decade for signs of physics that originate from beyond-the-Standard-Model (BSM) scenarios, including searches for signals that originate from supersymmetric (SUSY) particle production.These high-energy searches are complemented by low-energy experiments such as dark-matter (DM) experiments, or experiments that search for small deviations in known Standard-Model (SM) processes from their SM prediction.In the former category, the XENON1T [1,2], PandaX-II [3,4] and PICO [5][6][7] experiments provide limits on the DM-nucleus scattering cross section, whereas the Planck collaboration provides a precise measurement of the DM relic abundance [8].In the latter category, the anomalous magnetic moment of the muon (g − 2) µ plays an important role.There is a longstanding discrepancy between the experimental result [9][10][11] and the SM prediction for the muon anomalous magnetic moment.The latter is composed of quantum-electrodynamic, weak, hadronic vacuu -polarization, and hadronic light-by-light contributions, and reads
a SM µ = (g − 2) µ 2 = 116 591 810(43) × 10 −11 ,(1)
where the value between parentheses represents the theoretical uncertainty.Th improved experimental results obt h the Brookhaven result [9][10][11] read
a exp µ = 116 592 061(41) × 10 −11 ,(2)
showing that the deviation is now
∆a µ = a exp µ − a SM µ = 251(59) × 10 −11 .(3)
An independent experiment with different techniques than those employed by the Fermilab experiment is being constructed at J-PARC [38,39].The Minimal Supersymmetric Standard Model (MSSM) with R-parity conservation predicts a DM candidate and can simultaneously provide an explanation for the (g − 2) µ discrepancy 1 .Furthermore, the MSSM provides a solution to the fine-tuning (FT) problem in the Higgs sector that any BSM model introduces, even after taking into account the constraints on colored sparticles originating from the LHC.It is clear that for a rich model such as the MSSM, the interplay between the various experimental results is of crucial importance.In this context, the interplay between the LHC limits and the (g − 2) µ discrepancy has been studied in e.g.Ref. [41][42][43][44][45][46][47][48].DM direct detection (DMDD) searches are complementary in regions of the MSSM parameter space where the LHC has little sensitivity, for example in compressed regions.Papers that explore the DM implications of spectra that explain the (g − 2) µ discrepancy include Refs.[47][48][49][50][51][52], where the relic density requirement is not always taken into account.Likelihood analyses or global fits, where all experimental data that constrain the MSSM parameter space are taken into account, have been performed in e.g.Ref. [52][53][54][55][56][57][58].The degree of FT in constrained models that explain t e (g − 2) µ discrepancy is studied in [59,60], whereas the role of FT in spectra with the right DM properties is studied in Ref. [61][62][63][64][65].

In this work we perform for the first time a study of the phenomenology of the MSSM that simultaneously accounts for the DM relic abundance and the observed discrepancy of (g − 2) µ , that includes all DMDD and LHC limits, and that constrains the model-parameter space to models that are minimally fine-tuned.Th paper is structured as follows.In Section 2 we introduce our notation, the muon anomalous magnetic moment, and the electroweak fine-tuning measure.

In Section 3

e explain the set-up of our analysis.In Section 4 we explore the
phenomenology of the viable spectra, and in Section 5 we present our conclusions.


The muon anomalous magnetic moment and fine-tuning in the pMSSM

Instead of exploring the full MSSM with 105 free parameters, we focus on the phenomenological MSSM (pMSSM) [66], which has 19 free parameters.In this phenomenologically motivated pMSSM one requires that the first and second generation squark and slepton masses are degenerate, that the trilinear couplings of the first and second generation sfermions are set to zero (leaving only those of the third generation, A t , A b and A τ ), and that no new sources of CP violation are introduced.In addition one assumes that all sfermion mass matrices are diagonal.
(= |M 1 |), M 2 (= |M 2 |) and M 3 (= |M 3 |).
As a result of electroweak symmetry breaking (EWSB), the gaugino and the higgsino interaction eigenstates mix into mass eigenstates, called neutralinos and charginos.The neutralinos, denoted by χ 0 i with i = 1, . . ., 4, are the neutral mass eigenstates of the bino, wino and higgsino interaction eigenstates.The neutralinos are ordered by increasing mass, with χ 0 1 the lightest neutralino.Given the constraints from DMDD experiments on sneutrino DM, we take the lightest ne s it our DM candidate.Depending on the exact values of M 1 , M 2 and |µ|, this lightest mass eigenstate can be mostly bino-like (if M 1 is smallest), wino-like (if M 2 is smallest) or higgsino-like (if |µ| is smallest).The amount of bino, wino and higgsino mixing of the lightest neutralino is given by N 11 , N 12 and N 2 13 + N 2  14 , where N ij are the entries of the matrix that diagonalizes the neutralino mass matrix.In the basis of ( B, W 0 , H 0 d , H 0 u ), this mass matrix is given by
M χ 0 =     M 1 0 −c β s θ W M Z s β s θ W M Z 0 M 2 c β c θ W M Z −s β c θ W M Z −c β s θ W M Z c β c θ W M Z 0 −µ s β s θ W M Z −s β c θ W M Z −µ 0     ,(4)
with s x ≡ sin x, c x ≡ cos x, and the ratio of the SM W -and Z-boson masses being denoted by
cos θ W = M W /M Z .
The charginos, denoted by χ ± i with i = 1, 2, are the charged mass eigenstates of the wino and higgsino interaction eigenstates, with χ ± 1 the lightest chargino.In the basis of ( W ± , H ± u/d ), their mass matrix at tree level reads
M χ ± = M 2 √ 2c β c θ W M two gaugino parameters are close in value.


Electroweak fine-tuning in the pMSSM

The EWSB conditions link M Z to the input parameters via the minimization of the scalar potential of the Higgs fields.The resulting equation at one loop is [67,68]
M 2 Z 2 = m 2 H d + Σ d d − (m 2 Hu + Σ u u ) tan 2 β tan 2 β − 1 − µ 2 ,(6)
where the two effective pot ections to the soft SUSY breaking Higgs masses (explicit expressions are shown in the appendix of Ref. [68]).In order to obtain the observed value of M Z = 91.2GeV, one needs some degr

of cancellation between the SUSY par
meters appearing in Eq. (6).If small relative changes in the SUSY parameters will result in a distinctly different value of M Z , the considered spectrum is said to b right value of M Z .FT measures aim to quantify this sensitivity of M Z to the SUSY input parameters.The electroweak (EW) FT measure [69,70] is an agnostic approach to the computation of fine-tuning.We take this approach because a generic broken minimal SUSY theory has two relevant energy scales: a high-scale one at which SUSY breaking takes place, and a low-scale one (M SUSY ) where the resulting SUSY particle spectrum is situated and the EWSB conditions must be satisfied.We do not know which and how many fundamental parameters exist for a possible high-scale theory.The EW FT measure does not take such underlying high-scale model assumptions into account for its computation.The EW FT measure (∆ EW ) parameterizes how sensitive M Z is to variations in each of the coefficients C i , which are evaluated at M Z .It is defined as
∆ EW ≡ max i C i M 2 Z /2 , (7)
where the C i are
C m H d = m 2 H d tan 2 β − 1 , C m Hu = −m 2 Hu tan 2 β tan 2 β − 1 , C µ = −µ 2 , C Σ d d = max(Σ d d ) tan 2 β − 1 , C Σ u u = − max(Σ u u ) tan 2 β tan 2 β − 1 .
The tadpole contributions Σ u u and Σ d d contain a sum of different contributions.These contributions are computed individually and the maximum contribution is used to compute the C Σ u u and C Σ d d coefficients.We will use an upper bound of ∆ EW < 100 (implying no worse than O(1%) fine-tuning on the mass of the Z-boson) to determine whe ters is fine-tuned ∆ EW = 100 is reached at |µ| 800 GeV [63,65,69,[71][72][73][74][75].The masses of the gluino, sbottom, stop and squarks are allowed to get large for models with low ∆ EW [64,76,77].Therefore, we assume that the masses of these sparticles are above 2.5 TeV (for the gluino), above 1.2 TeV (for the stops and bottoms) and above 2 TeV (for the squarks), such that they evade the ATLAS and CMS limits2 .


The muon anomalous magnetic moment

In the pMSSM, one-loop contributions to a µ arise from diagrams with a chargino-sneutrino or neutralino-smuon loop [78].The expressions for these one-loop corrections read [79]
δa χ 0 µ = m µ 16π 2 4 i=1 2 m=1 − m µ 12m 2 µm |n L im | 2 + |n R im | 2 F N 1 m 2 χ 0 i m 2 µm (8)+ m χ 0 i 3m 2 µm Re n L im n R im F N 2 m 2 χ 0 i m 2 µm , δa χ ± µ = m µ 16π 2 2 k=1 m µ 12m 2 νµ |c L k | 2 + |c R k | 2 F C 1 m 2 χ ± k m 2 νµ + 2m χ ± k 3m 2 νµ Re c L k c R k F C 2 m 2 χ ± k m 2 νµ ,(9)
with m µ the muon mass, m µm the first

r second smuon mass, m νµ the muon
neutrino mass, i, m and k the indices for the neutralinos, smuons and charginos and the couplings
n R im = √ 2g 1 N i1 X m2 + y µ N i3 X m1 , n L im = 1 √ 2 (g 2 N i2 + g 1 N i1 ) ile the unitary matrix X diagonalizes the smuon mass matrix M 2 µ , which reads for the pMSSM in the ( µ L , µ R ) basis
M 2 µ = m 2 L1 + s 2 θ W − 1 2 M 2 Z cos(2β) −m µ µ tan β −m µ µ tan β m 2 e R − s 2 θ W M 2 Z cos(2β) . (12)
The loop functions F N 1,2 and F C 1,2 can be found in Ref. [79].They are norma nic Barr-Zee diagrams are the source of the largest possible two-loop contribution.Here a Higgs boson and a photon connect to either a chargino or sfermion loop [80] 3 .As one can see in the expressions above, the chargino-sneutrino and neutralino-smuon contributions are controlled by M 1 , M 2 , tan β and µ (through m χ 0 i and m χ ± k ), as well as m L O(100) GeV) neutralinos/charginos and smuons/sneutrinos exist in the sparticle spectrum.The Barr-Zee diagrams are enhanced by large values of tan β, small values of m A and large Higgs-sfermion couplings.In general, the one-loop chargino-sneutrino contribution dominates over the neutralino-slepton contribution [79], unless there is a large smuon left-right mixing induced by a sizable value for µ [83].These latter spectra will however result in slightly higher FT values

which is a dire
t consequence of a higher value of |µ|.


Analysis setup

To create the SUSY spectra we use SoftSUSY 4.0 [84], the Higgs mass is calculated using Feyn-Higgs 2.14.2 [85][86][87][88][89], and SUSYHIT [90] is used to calculate the decay of the SUSY and Higgs particles.Vevacious [91][92][93] is used to check that the models have at least a meta-stable minimum state that has a lifetime that exceeds that of our universe and that this state is not color/charge breaking 4 .We use SUSY-AI [94] and SModelS [95][96][97][98][99] to determine the LHC exclusion of a model point.LHC cross sections for sparticle production at NLO accuracy are calculated using Prospino [100].HiggsBounds 5.1.1 is used to determine whether the SUSY models satisfy the LEP, Tevatron and LHC Higgs constraints [101][102][103][104][105][106][107][108].MicrOMEGAs 5.2.1 [109][110][111][112][113][114] is used to compute the DM relic density (Ω DM h 2 ), the present-day velocity-weighted annihilation cross section ( σv ) and the spin-dependent and spin-independent dark-matter -nucleon scattering cross sections (σ SD,p and σ SI,p ).For DM indirect detection we only consider the limit on σv stemming from the observation of gamma rays originating from dwarf galaxies, which we implement as a hard cut on each of the channels reported on the last page of Ref. [115].The current constraints on the dark-matter -nucleon scattering cross sections originating from various dark matter direct detection (DMDD) experiments are determined via MicrOMEGAs, while future projections of constraints are determined via DDCalc 2.0.0 [116].Flavor observables are computed with SuperIso 4.1 [117,118], while the muon anomalous magnetic moment and its theoretical uncertainty is determined with GM2Calc [81,[119][120][121].We use the Gaussian particle filter [122] to search the pMSSM parameter space for interesting areas.The lightest SM-like Higgs boson is required to be in the mass range of 122 GeV ≤ m h ≤ 128 GeV.Spectra that do not satisfy the LHC bounds on sparticle masses, branching fractions of B/D-meson decays, the DMDD, or DM indirect detection bounds are r moved.Our spectra are furthermore required to satisfy the LEP limits on the masses of the charginos, light sleptons and staus (m χ ± 1 > 103.5 GeV, m l ± > 90 GeV and m τ ± > 85 GeV) [123,124], and the constraints on the invisible and total width of the Z-boson (Γ Z,inv = 499.0± 1.5 MeV and Γ Z = 2.4952 ± 0.0023 GeV) [125].


Phenomenology

We assume that the DM abundance is determined by thermal freeze-out and require that the lightest neutralino saturates Ω DM h 2 with the observed value of 0.12 [8] within 0.03 to allow for a theoretical uncertainty on the relic-density calculation.As shown above, the mass eigenstate of the DM particle is a mixture of bino, wino and higgsino interaction eigenstates.To obtain the correct relic density in the pMSSM with a pure state, one can either have a higgsino with a mass of m χ 0 1 800 Ge

or a wino with
m χ 0 1 2.5 TeV.Spectra that saturate the relic density with lower DM masses necessarily are predominantly bino-like, mixed with higgsino/wino components.Negligible higgsino/wino components are found in so-called funnel regions [126,127], i.e. regions where the mass of the DM particle is roughly half of the mass of the Z boson, SM-like Higgs boson or heavy Higgs boson.In such a scenario, the mass of the neutralino can even get below 100 GeV with M 1 < 100 GeV, and in particular the early-universe DM annihilation cross section is enhanced for m χ 0 1 m h /2 and M Z /2.Moreover, spectra with another particle close in mass to the LSP can satisfy the relic density constraint without having a large wino/higgsino component too, due to the co-annihilation mechanism [128].The case where the lightest neutralino is predominantly wino-like results in a fine-tuned spectrum: to obtain the right relic density M 2 2.5 TeV for a pure wino, so |µ| > 2.5 TeV in that scenario.Secondly, such high LSP neutralino masses do not give a large enough contribution to ∆a µ , since the other sparticle masses have to exceed this LSP mass.The pure-higgsino solutions do not allow for an explanation of ∆a µ [51] either.Therefore our solutions, as shown in Fig. 1, feature predominantly bino-like LSPs.Due to the combined ∆a µ constraint (requiring high tan β), DMDD limits and the FT requirement, the composition has a small higgsino component (< 20%) and a negligible wino component.The second-to-lightest neutralino and the lightest chargino are eith r wino-like, higgsino-like, or mixed wino-higgsino states.It might be surprising to see that spectra with bino-higgsino LSPs are allowed to have wino-like χ 0 2 / χ ± 1 .Such configurations can however be found in spectra for which M 1 , M 2 and |µ| are all of O(100) GeV with M 2 being smaller than |µ|, and that have moderate to large values of tan β (10 tan β 20).From Eq. ( 4) one may infer that for such spectra, little mixing can take place between the bino and wino.This results in negligible wino components of the LSP, whereas χ ± 1 and χ 0 2 can be predominantly wino-like.Moreover, decreasing |µ| for such models will not only result in a higher higgsino-component, but counter-intuitively also in a higher wino component of the LSP, while the wino component of χ ± 1 and χ 0 2 then decreases.Because of these higher wino/higgsino components of the LSP, such scenarios result in larger values of σ SI,p and σ SD,p .Therefore, decreasing |µ| for these scenarios is limited by the constraints imposed by the DMDD experiments.The spectra where χ ± 1 and χ 0 2 are predominantly higgsino-like are typically difficult to probe at the LHC due to low production cross sections compared to the pure wino χ ± 1 / χ 0 2 case.In Fig. 1 we show the spectra that survive all constraints and have ∆ EW < 100.Lower values for ∆ EW are generally found for lower DM masses.The mass of the DM particle does not exceed 500 GeV, which is a direct result of the combined requirements of having ∆ EW < 100 and a sufficiently high contribution to ∆a µ .The lowest-obtained value is ∆ EW = 12.3.From the right-hand side of Fig. 1, we can distinguish three different type of DM early-universe annihilation mechanisms: the funnel regions, the coannihilation regions and the bino-higgsino solution (indicated with b b and t t).As the LHC phenomenology of these three early-universe annihilation regimes can be quite different, we now discuss them one-by-one.


LHC phenomenology for the funnel regimes

We start with the funnel regions, of which there are two in our spectra5 .The first one centers around m χ 0 1 40 GeV, which is slightly less than M Z /2.This can be explained as follows.The velocities of the DM particles were much higher in the early universe than what they are in the present-day universe.This means that DM annihilations via s-channel Z exchanges could happen on-resonance in the early universe, whereas in the present-day universe these exchanges only happen off-resonance.This also explains the fact that the value for σv is allowed to get orders of magnitude smaller than the value that one usually expects for a thermal relic (around σv = 3 • 10 −26 cm 3 s −1 for a DM mass of 100 GeV).These models are characteri

d by small wino/higgsino components of th
LSP -otherwise the early-universe annihilation would be too efficient, resulting in a too-low value of Ω DM h 2 .The second funnel region is centered around m χ 0 1 60 GeV, slightly less than m h /2.These DM particles annihilated in the early universe predominantly via s-channel SM-like Higgs exchanges.No solutions are found for spectra with DM masses in-between the two funnel regions.Here, the wino/higgsino component necessarily needs to increase to satisfy the Ω DM h 2 requirement, and these spectra are excluded by DMDD experiments.The minimal value of ∆ EW for these spectra is 13.2.The two funnel regimes are characterized by light (m χ 0 1 < 100 GeV) bino-like LSPs.The χ ± 1 and χ 0 2 are degenerate in mass.They are wino mixtures for masses around 100−200 GeV, while they become higgsino-like for heavier χ ± 1 / χ 0 2 (up to m χ ± 1 / χ 0 2 500 GeV).The mass gap between χ 0
1 and χ 0 2 or χ ± 1 (∆(m χ 0 2 , m χ 0 1 ) or ∆(m χ ± 1 , m χ 0 1
)) is at least around 50 GeV, and exceeds 100 GeV for m χ ± 1 150 GeV (see Fig. 2, left panel).The masses of the sleptons are heavier than (at least) the masses of χ 0 2 and χ ± 1 .Therefore, three different sorts of decays for χ 0 2 can be identified:
1. χ 0 2 → h χ 0 1 when ∆(m χ 0 2 , m χ 0 1 ) > m h , 2. χ 0 2 → Z χ 0 1 when ∆(m χ 0 2 , m χ 0 1
) > M Z , and 3. off-shell decays when ∆(m χ 0 2 , m χ 0 1 ) < M Z .For χ ± 1 , there are only two sorts of decays:
1. χ ± 1 → W ± χ 0 1 when ∆(m χ ± 1 , m χ 0 1 ) > M W , and 2. off-shell decays when ∆(m χ ± 1 , m χ 0 1 ) < M W .
Searches for χ 0 2 χ ± 1 production with on-shell decays of χ 0 2 → Z χ 0 1 are most sensitive to our spectra [ drops when χ 0 2 can decay into the S air, whereas we deal with mixed wino-higgsino pairs.To recast their analysis, we show in the left panel of Fig. 3 the average cross section per 10 by 10 GeV bin for χ 0 2 χ ± 1 production.We find that our cross sections in the regime where M Z < ∆(m χ 0 2 , m χ 0 1 ) < m h do no not exceed the 95% confidence level (CL) limits of Ref. [130,131].The models with off-shell decays are slightly more constrained by the LHC experiments.Particularly Ref. [133] excludes some of our spectra in this regime that have m χ ± 1 up to 210 GeV and ∆(m χ 0 2 , m χ 0 1 ) < 55 GeV.


LHC phenomenology for the coannihilation regimes

The second regime is the coannihilation regime.It starts to open up at DM masses of roughly 75 GeV, as no charged sparticles (and therefore no coannihilation partners other than the sneutrino) can exist with masses below 85 GeV due to the LEP/LHC bounds.Three different types of coannihilation partners are identified: first-/second-generation sleptons, thirdgeneration sleptons, and charginos or

avier neutralinos.Interestingly, only with the he
p of slepton coannihilations the DM particle can have a mass between O(70 − 150) GeV and still give the right Ω DM h 2 .To obtain the right relic density in this regime without a sleptoncoannihilation partner, one generally needs high higgsino fractions, which increases the value of σ SI,p beyond the exclusion limit of the DMDD experiments.The lowest values of ∆ EW are found in the stau-coannihilation regime (∆ EW = 12.3), while the first-/second-generation slepton and The coannihilation regimes are all characterized by small mass differences between the LSP and its coannihilation partner(s).

The first type of coannihilation is that of first-/second-generation sleptons ( l ± 1 ).The compression between m l ± 1 and m χ 0 1 is increased for higher LSP masses such that the right Ω DM h 2 can still be obtained.Spectra with ∆(m χ 0 2 , m χ 0 1 ) > M Z are under strong constraints from searches for χ 0 2 χ ± 1 → l llν l (see e.g.[132]).We explicitly remove those points from our spectra, leaving only models with ∆(m χ 0 2 , m χ 0 1 ) < M Z .The χ ± 1 and χ 0 2 sparticles are typically higgsino-like with a small wino component, and have masses between 180 and 500 GeV.The second coannihilation regime is characterized by low τ ± 1 masses.The masses of χ ± 1 / χ 0 2 can still be as light as 105 GeV in this regime, where they are predominantly wino-like.The higgsino component of these particles increases when their masses increase, up to m χ ± 1 / χ 0 2 500 GeV.Alt ough we have a large production cross section for the wino-like χ ± 1 / χ 0 2 pair, these models are not constrained by the LHC experiments due to the presence of the light staus.The staus are often lighter than χ ± 1 and χ 0 2 , and the searches for τ ± 1 -mediated decays of χ + 1 χ − 1 / χ ± 1 χ 0 2 production have no sensitivity when ∆(m χ 0 1 , m τ ±


1

) < 100 GeV [134, 135].The latter holds for our spectra in the second coannihilation regime, since the mass differences between the LSP and τ ± 1 are between 5 − 50 GeV in that case.Additionally, relatively few LHC searches for low-mass τ ± particles exist.Small τ + τ − production cross sections and low signal acceptances make these searches difficult, so the experiments have no constraining power in the compressed regime [136,137].A dedicated low mass τ ± search without an assumed mass degeneracy between τ ± 1 and τ ± 2 would be interesting to probe the sensitivity of the LHC to these scenarios.The last coannihilation regime has a χ ± 1 or χ 0 2 that is close in mass to the LSP.Note that although the slepton masses in these regions can be O(200) GeV, the results from the l + R,L l − R,L searches with l ± = e ± , µ ± or τ ± (e.g.[137-139]) are not directly applicable here, as often one or more of the chargino/heavier neutralino states is lighter than the sleptons.Therefore, the slepton will not decay with a 100% branching ratio to χ 0 1 l ± , although this is assumed in the above-mentioned searches.Instead, in this regime, only the χ ± 1 χ 0 2 searches are of relevance, similar to the case in the funnel region discussed above.Interestingly, although the mass compression for the slepton coannihilation regimes needs to increase to obtain the right relic density for higher DM masses, for the gaugino-coannihilation regime it instead needs to decrease.The mass compression between the LSP and wino-higgsino like χ ± 1 / χ 0 2 sparticles is generally around 15-20 GeV, and Ref.

[133] excludes our solutions with m χ ± 1 up to 140 − 180 GeV.


LHC phenomenology for the bino-higgsino LSP

The last regime we identify consists of bino-higgsino LSPs and is labeled with b b and t t.These early-universe annihilation channels are mediated by either s-channel Z or h/H exchanges.The t t annihilation channel opens up when m χ 0 1 becomes larger than the mass of the top quark m t , as then the invariant mass of the two LSPs is enough to create a t t pair6 .For the Z-exchange channel this annihilation becomes favored over the annihilati n into a lighter fermion pair, since any Z-mediated annihilatio

of two Majorana fermions is helicity suppres
ed at tree level [140].This is explained as follows.The two identical LSPs form a Majorana pair.Such a pair is even under the operation of charge-conjugation C = (−1) L+S with S the total spin and L the total orbital angular momentum, so L and S must either both be even, or both be odd.Taking the limit of zero velocity, as the present-day velocity of DM particles is non-relativistic, we may assume L = 0 and even S.The final-state fermion pair can have a total spin of S = 1 or S = 0, but only the latter is allowed for the Majorana-pair annihilation in the non-relativistic limit.For a Dirac-field pair, an S = 0 configuration is obtained if the fermion and anti-fermion are from different Weyl spinors: a left-and right-handed one.In the SM, a coupling with this combination only arises (at tree level) by a mass insertion.Therefore, the transition amplitude is proportional to the mass of the final-state fermions, and a decay to a heavier pair of fermions is generally preferred.In spectra where tan β is large we also see the heavy-Higgs-mediated decays to b b, as the bottom-Yukawa coupling is enhanced.As can be seen in Fig. 2, in the regime of m χ 0 1 m t , the masses of χ ± 1 and χ 0 2 are relatively close to that of the LSP, so due to the coannihilation mechanism these spectra tend to show slightly lower values of σv than naively would be expected.The minimal value of ∆ EW is around 14.2 for these models.The χ 0 2 and χ ± 1 are predominantly higgsino-like with masses from 180 to 500 GeV.Due to their small production cross section, the LHC searches do not have exclusion power in this regime.


Dark-matter direct detection experiments

In the previous subsections we discussed the phenomenology of the viable spectra at the LHC.We now comment on the sensitivity of DMDD experiments.The resulting values for σ SI,p and σ SD,p may be seen in Fig. 4.While the value of σ SI,p varies by over 7 orders of magnitude, σ SD,p is relatively constrained.Moreover, we observe that σ SD,p is directly correlated with ∆ EW : lower values of σ SD,p result in higher values of ∆ EW .This is due to the fact that the LSP in our spectra is always bino-like with a small higgsin

component.The value of σ SD,p decreases w
th smaller higgsino fractions in the LSP, while ∆ EW increases since |µ| needs to increase (for a given fixed LSP mass).For this reason, future DMDD experiments that probe σ SD,p will be sensitive to all our solutions, irrespective of the masses and compositions of the rest of the sparticle spectrum.In Fig. 4, we indicate the projected limit of the PICO-40L and the PICO-500 experiments [141].We observe that the latter one is sensitive to all of our solutions with ∆ EW < 62.The LUX-ZEPLIN experiment [142] (whose projected limit is not shown in Fig. 4) will exclude all of our solutions with ∆ EW < 100.


Conclusion

In this paper we have analyzed the spectra in the pMSSM that are minimally fine-tuned, result in the right Ω DM h 2 and simultaneously offer an explanation for ∆a µ .In terms of DM phenomenology, we have distinguished three interesting branches of solutions: the funnel regimes, three types of coannihilation regimes, and the generic bino-higgsino solution.All these solutions have in common that the LSP is predominantly bino-like with a small higgsino component.Masses of the DM particle range between 39 − 495 GeV.We discussed the phenomenology at the LHC for each of the regimes.The first and second regime are relatively more constrained by χ 0 2 χ ± 1 searches at the LHC (in particular by the one presented in Ref.

[133]) than the last re

me, which i
due to the lower wino-components and higher masses of the χ 0 2 / χ ± 1 sparticles that is typical in the last regime.On the other hand, in particular when the coannihilation partner of the LSP is a light stau, the LHC searches show little to no sensitivity to our found solutions.A dedicated low-mass τ ± search without an assumed mass degeneracy between τ ± 1 and τ ± 2 would be interesting to probe the sensitivity of the LHC to these scenarios.The requirement of satisfying ∆a µ excludes models with a higher-mass higgsino of around 600 GeV as the LSP, which means that the value of σ SD,p is directly linked to ∆ EW .Therefore, DMDD experiments that probe σ SD,p will ultimately be sensitive to all of our found solutions Figure 1 :
1
Figure 1: The mass of the DM particle (m χ 0 1) vs the velocity-weighted annihilation cross section ( σv ).The value of ∆ EW is shown as a color code on the left, where the points are ordered such that spectra with lower values of ∆ EW lie on top of those with higher values of ∆ EW .On the right we show the dominant early-universe annihilation process that contributes to the value of Ω DM h 2 .In both plots, we only show points that satisfy all experimental constraints, and have 133 × 10 −11 < ∆a µ < 369 × 10 −11 , allowing for a 2σ uncertainty.


Figure 2 :
2
Figure 2: The mass difference between the DM particle and the lightest chargino (left), lightest smuon (middle) and lightest stau (right) versus the mass of the heavier particle.The color