The tiny (g-2) muon wobble from small-μ supersymmetry

A new measurement of the muon anomalous magnetic moment, gμ− 2, has been reported by the Fermilab Muon g-2 collaboration and shows a 4.2 σ departure from the most precise and reliable calculation of this quantity in the Standard Model. Assuming that this discrepancy is due to new physics, we concentrate on a simple supersymmetric model that also provides a dark matter explanation in a previously unexplored region of supersymmetric parameter space. Such interesting region can realize a Bino-like dark matter candidate compatible with all current direct detection constraints for small to moderate values of the Higgsino mass parameter |μ|. This in turn would imply the existence of light additional Higgs bosons and Higgsino particles within reach of the high-luminosity LHC and future colliders. We provide benchmark scenarios that will be tested in the next generation of direct dark matter experiments and at the LHC.


Introduction
The Standard Model (SM) of particle physics has built its reputation on decades of measurements at experiments around the world that testify to its validity. With the discovery of the Higgs boson almost a decade ago [1,2] all SM particles have been observed and the mechanism that gives mass to the SM particles, with the possible exception of the neutrinos, has been established. Nonetheless, we know that physics beyond the SM (BSM) is required to explain the nature of dark matter (DM) and the source of the observed matterantimatter asymmetry. Furthermore, an understanding of some features of the SM such as the hierarchy of the fermion masses or the stability of the electroweak vacuum is lacking.
The direct discovery of new particles pointing towards new forces or new symmetries in nature will be the most striking and conclusive evidence of BSM physics. However, it may well be the case that BSM particles lie beyond our present experimental reach in mass and/or interaction strength, and that clues for new physics may first come from results for precision observables that depart from their SM expectations. With that in mind, since the discovery of the Higgs boson, we are straining our resources and capabilities to measure the properties of the Higgs boson to higher and higher accuracy, and flavor and electroweak physics experiments at the LHC and elsewhere are pursuing a complementary broad program of precision measurements. Breakthroughs in our understanding of what lies beyond the SM could occur at any time.
Recently, new results of measurements involving muons have been reported. The LHCb experiment has reported new values of the decay rate of B-mesons to a kaon and a pair of muons compared to the decay into a kaon and electrons [3], providing evidence at the 3 σ-level of the violation of lepton universality. This so-called R K anomaly joins the ranks of previously reported anomalies involving heavy-flavor quarks such as the bottom quark forward-backward asymmetry at LEP [4,5], and measurements of meson decays at the LHC and B-factories such as R K * [6][7][8] and R D ( * ) [9][10][11][12][13][14]. The Fermilab Muon (g-2) JHEP01(2022)025 experiment has just reported a new measurement of the anomalous magnetic moment of the muon, a µ ≡ (g µ − 2) /2. The SM prediction of a µ is known with the remarkable relative precision of 4×10 −8 , a SM µ = 116 591 810(43)×10 −11 . From the new Fermilab Muon (g-2) experiment, the measured value is a exp, FNAL µ = 116 592 040(54) × 10 −11 [36], which combined with the previous E821 result a exp, E821 µ = 116 592 089(63) × 10 −11 [37], yields a value a exp µ = 116 592 061(41) × 10 −11 . An important point when considering the tension between experimental results and the SM predictions are the current limitations on theoretical tools in computing the hadronic vacuum polarization (HVP) contribution to a SM µ , which is governed by the strong interaction and is particularly challenging to calculate from first principles. The most accurate result of the HVP contribution is based on a data-driven result, extracting its value from precise and reliable low-energy (e + e − → hadrons) cross section measurements via dispersion theory. Assuming no contribution from new physics to the low energy processes and conservatively accounting for experimental errors, this yields a value a HVP µ = 685.4(4.0) × 10 −10 [15,[20][21][22][23][24][25][26], implying an uncertainty of 0.6 % in this contribution. 1 The SM prediction for the anomalous magnetic moment of the muon and the measured value then differ by 4.2 σ, It is imperative to ask what these anomalies may imply for new physics. The most relevant questions that come to mind are: can the a µ and R K ( * ) anomalies be explained by the same BSM physics? Can they give guidance about the nature of DM? Are they related to cosmological discrepancies? How constrained are the possible solutions by other experimental searches? There are two broad classes of solutions to the (g µ − 2) anomaly that may be considered in the light of the above: • New relatively light particles with small couplings to muons, typically featuring particles with O(100) MeV masses and O(10 −3 ) couplings to muons. Examples of such models are new (light) scalars [43][44][45][46][47][48][49] and new (light) (Z ) vector bosons. A pseudoscalar particle may also lead to an explanation of ∆a µ , provided it couples not only to muons, but also to photons. The typical example are axion-like particles [50][51][52][53], although obtaining the proper ∆a µ requires a delicate interplay between the couplings to muons and to photons. 2 In the case of gauge bosons, a common explanation of both R K ( * ) and a µ is, however, strongly constrained by neutrino trident bounds on Z bosons coupled to muons [55][56][57]. 3 In addition, bounds from BaBar [60] and CMS [61] 1 The HVP contribution has recently been computed in lattice QCD, yielding a higher value of a HVP µ = 708.7(5.3) × 10 −10 [38]. Given the high complexity of this calculation, independent lattice calculations with commiserate precision are needed before confronting this result with the well tested data-driven one. We stress that if a larger value of the HVP contribution were confirmed, which would (partially) explain the (gµ − 2) anomaly, new physics contributions will be needed to bring theory and measurements of (e + e − → hadrons) in agreement [39][40][41][42].
2 A similar mechanism applies for (ge − 2) in the case of the QCD axion; see, for instance, ref. [54]. 3 There are also bounds from Coherent ν-Nucleus Scattering (CEνNS), although these are not yet competitive with the bounds from neutrino trident processes [58,59].

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rule out the values of the new gauge (or scalar) coupling which could explain the observed value of a µ for m Z (m φ ) ≥ 2m µ 210 MeV. Explanations of the flavor anomalies require the gauge boson to decay into a pair of muons, preventing simultaneous explanations of R K ( * ) and a µ .
• New heavy fermions [62,63] or scalars (possibly accompanied by additional new particles, like in the supersymmetric extensions of the SM discussed in detail below), as well as leptoquark particles [64][65][66][67], with larger couplings to muons. Leptoquarks provide one of the most interesting solutions to ∆a µ , since they can lead to a simultaneous explanation of the R K anomaly. Note however that a careful choice of the leptoquark couplings is necessary to avoid flavor-changing neutral currents, which is perhaps the least attractive feature of such scenarios.
In section 2, we shall discuss a supersymmetric solution in the simplest supersymmetric model at hand, the Minimal Supersymmetric Standard Model (MSSM). We focus on a region of the parameter space of the MSSM where the (g µ − 2) anomaly can be realized simultaneously with a viable DM candidate. We show that in the region of moderate |µ| and moderate-to-large values of tan β, a Bino-like DM candidate can be realized in the proximity of blind spots (that require µ × M 1 < 0) for spin-independent direct detection (SIDD) experiments [68]. In this way, our MSSM scenario explores a different region of parameter space than the one considered in the study of refs. [69,70], which considers regions of large µ as a way to accommodate current SIDD bounds. We summarize and conclude in section 3. In appendix A, we give details about the LHC constraints on these scenarios.
2 Tiny (g µ − 2) muon wobble with small |µ| in the MSSM Supersymmetric extensions of the SM remain among the most compelling BSM scenarios [71][72][73], not least because the stability of the Higgs mass parameter under quantum corrections can be ensured. In minimal supersymmetric extensions of the SM, the SMlike Higgs is naturally light [74][75][76][77][78][79][80][81][82][83][84] and the corrections to electroweak precision as well as flavor observables tend to be small, leading to good agreement with observations. Supersymmetric extensions can also lead to gauge coupling unification and provide a natural DM candidate, namely the lightest neutralino.
In this section, we propose simultaneous (g µ − 2) and DM solutions in the Minimal Supersymmetric Standard Model (MSSM) [71][72][73] which have not been explored before. Related recent (but prior to the publication of the Fermilab Muon (g-2) result) studies can, for example, be found in refs. [69,70,[85][86][87][88][89][90]. One crucial difference between our study and the very recent work in refs. [69,70] is that the spin-independent direct detection (SIDD) cross section is suppressed not by decoupling the Higgsino and heavy Higgs contributions, but by a partial cancellation between the amplitudes mediated by the two neutral CP-even Higgs boson mass eigenstates. This cancellation requires opposite signs of the Higgsino and the Bino mass parameters, (µ × M 1 ) < 0 [68]. Demonstrating that one can explain the a µ anomaly in this region of parameter space is non-trivial, as this combination of the Higgsino and Bino mass parameters renders the contribution of the neutralino-smuon loop JHEP01(2022)025 to a µ negative, while the experimentally observed value is larger than the SM prediction. Explaining the experimental measurement is only possible if the chargino-sneutrino contribution to a µ is positive and has larger absolute magnitude than the neutralino-smuon contribution, and if the values of the individual contributions are such that the observed anomaly, ∆a µ = (20 − 30) × 10 −10 , can be explained. Moreover, this can only be achieved for moderate (absolute) values of the Higgsino mass parameter |µ| 500 GeV, and values of the heavy Higgs boson masses than are not far away from the current experimental limit coming from direct searches.

∆a µ and direct dark matter detection constraints
The MSSM contributions to a µ have been discussed extensively in the literature, see, for example, refs. [89,[91][92][93][94][95][96][97]. The most important contributions arise via chargino-sneutrino and neutralino-smuon loops, approximately described by [89] where M 2 is the Wino mass parameter and m f are the scalar particle f masses, with the loop functions see refs. [94,97] for the full (one-loop) expressions. It is interesting to note that these two contributions can be of the same order of magnitude: the chargino-sneutrino contribution is proportional to Higgsino-Wino mixing which can be sizeable, but suppressed by the smallness of the Higgsino-sneutrino-muon coupling which is proportional to the muon Yukawa coupling, ∝ m µ tan β/v, with the SM Higgs vacuum expectation value v. The neutralino-smuon contribution, on the other hand, arises via muon-smuon-neutralino vertices which are proportional to the gauge couplings, but is suppressed by the small smuon left-right mixing, ∝ m µ (µ tan . Regarding corrections beyond oneloop [98,99], the most relevant contribution is associated with corrections to the muon Yukawa coupling, ∆ µ . These corrections become relevant at large values of µ tan β and can be re-summed at all orders of perturbation theory [100]. While these corrections lead to small modifications of a µ , they do not change the overall dependence of ∆a µ on the masses of the supersymmetric particles. From eqs. (2.1)-(2.2) we can observe that the signs of the MSSM contributions to a µ depend sensitively on the relative signs of the gaugino masses M 1 and M 2 and the

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Higgsino mass parameter µ. As emphasized before, a DM candidate compatible with the current null-results from direct detection experiments can be realized for |µ| 500 GeV if M 1 and µ have opposite signs. For this combinations of signs, the contribution from the neutralino-smuon loop to a µ will be negative, a χ 0 − µ µ < 0. Since the measured value of a µ is larger than the SM prediction by ∆a µ 25 × 10 −10 , we require the chargino-sneutrino contribution to be positive and larger than the neutralino-smuon contribution. This can be realized if M 2 has the same sign as µ and if |M 2 | is of similar size as |µ| and the soft smuon masses. In the regime of moderate or large values of tan β, and assuming all weakly interacting sparticles have masses of the same order, m, one obtains approximately (2.5) The factor 1.3 reduces to values closer to 1 if M 1 and M 2 have opposite signs. This implies that for values of tan β 10, sparticles with masses m ∼ 200 GeV can lead to an explanation of the observed ∆a µ anomaly, while for tan β = 60, the characteristic scale of the weakly interacting sparticle masses may be as large as m ∼ 500 GeV. The range of tan β and of sparticle masses consistent with the observed ∆a µ has implications on the DM properties. We will concentrate on DM candidates with masses comparable to the weak scale, such that the thermal DM relic density reproduces the observed value. In the MSSM, DM candidates in this mass range can be realized if the lightest supersymmetric particle is an almost-pure Bino, m χ |M 1 |.
For the moderate-to-large values of tan β required to explain the (g µ − 2) anomaly, the SIDD amplitude for the scattering of DM with nuclei (N ) is proportional to where m h and m H are the masses of the SM-like and the new heavy neutral Higgs boson, respectively. We see that the SIDD amplitude depends in a crucial way on the sizes and signs of M 1 and µ. There are two options to lower the SIDD amplitude: for large values of |µ|, the Higgsino components of the DM candidate become small and the SIDD amplitude is suppressed. Alternatively, the light and heavy CP-even Higgs contributions (first and second terms inside the brackets in eq. (2.6)) may interfere destructively, leading to a suppression of the SIDD amplitude. The latter option is particularly interesting since it allows |µ| to remain of the order of the electroweak scale; see, for example ref. [101] for a recent discussion of naturalness and the connection with direct detection bounds. Regarding the first term in eq. (2.6), if M 1 −µ sin 2β, the contributions of the Higgsino-up and the Higgsino-down admixtures to the (χχh) interaction cancel. The second term is the contribution to the (χN → χN ) amplitude arising from the t-channel exchange of the non-SM-like heavy Higgs boson H. The generalized blind spot condition for the SIDD cross section of a Bino-like DM candidate is then [68] 2 (M 1 + µ sin 2β) If the condition in eq. (2.7) is satisfied, the amplitudes mediated by h and by H exchange interfere destructively, suppressing the SIDD cross section; a property that also holds at the one-loop level [102]. In general, even if one is not in the proximity of the blind spot solution, if the neutralino is mostly Bino-like, for a given value of |µ| and M 1 , the cross section is suppressed (enhanced) if µ and M 1 have opposite (the same) sign. 4 The mass of the heavy Higgs boson plays an important role in the blind-spot cancellation. In the presence of light electroweakinos, the current LHC bounds on m H coming from searches for heavy Higgs bosons decaying into tau-leptons [103-106] can be approximated by For values of m H close to this bound, the SIDD amplitude is proportional to To exemplify the relevance of the relative sign and size of µ and M 1 , consider M SI p for tan β = 16. As a reference value for the SIDD amplitude, let us set µ −M 1 . Keeping M 1 fixed, but increasing the value of |µ| to µ −2M 1 , the value of M SI p becomes a factor of ≈ 1/6 smaller. Let us compare this to the situation for which µ and M 1 have the same sign. First, we can note that for µ = M 1 , the SIDD amplitude is almost a factor 2 larger than for µ = −M 1 . Furthermore, in order to obtain a reduction of M SI p by a factor of 1/6, one would have to raise the value of |µ| from µ ∼ M 1 to µ ∼ 4M 1 . This exemplifies that obtaining SIDD cross sections compatible with experimental limits either requires (µM 1 ) < 0 (blind spot solution) or, to compensate for a positive sign of this product, one must sufficiently enhance the ratio µ/M 1 (large-µ solution).
The spin dependent (SD) interactions are instead dominated by Z-exchange, and can only be suppressed by lowering the Higgsino component of the lightest neutralino. At moderate or large values of tan β, the amplitude for SD interactions is proportional to [87] M SD ∝ v µ 2 cos 2β . neither has the sign nor the magnitude to account for the (g µ − 2) anomaly. If anything, depending on the sign of M 1 × M 2 , it will partially cancel the Wino contribution to ∆a µ . For smaller values of |µ|, an explanation for the (g µ −2) anomaly requires a Wino-mediated contribution enabled by µ × M 2 > 0 and moderate values of M 2 .
For the LHC constraints indicated in figures 1 and 2 (|M 1 | 240 GeV, shown with the purple dash-dotted line) we have assumed that the lightest stau is the next-to-lightest supersymmetric particle, with a mass such that the proper relic density is obtained by coannihilation of the lightest stau with the lightest neutralino. In such a case, the Wino-like chargino and neutralino have sizable branching ratios into staus, increasing the stau production rate. In order to estimate the LHC limits, we use a recent analysis [111] searching for tau-lepton final states, which assumed that the mass gap between the lightest chargino and neutralino is 50 GeV and the lightest stau mass lies in the middle of the lightest chargino and neutralino masses, which is close to the situation found under our assumptions. This shows that the LHC is already putting strong constraints on the realization of this scenario. Note that we chose the Wino-(M 2 ) and the first and second generation slepton (M 1,2 L , M 1,2 R ) mass parameters to be approximately degenerate ( GeV) such that current LHC limits for direct slepton searches are avoided for slepton masses above ∼ 200 GeV [112-123].

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Additional constraints from LHC searches with charged leptons in the final states can arise from production of the Higgsino-like neutralino and chargino states. These states decay into gauge and Higgs bosons and lighter charginos and neutralinos. If µ is large compared to M 2 , Higgsino production and their decays can lead to relevant signals at the LHC despite Higgsinos having much smaller production cross section that Winos, because for such a choice of parameters, the Higgsino-decays lead to final states with much harder leptons than the leptons arising from Wino or slepton production. We have checked a number of example points from the preferred regions of figures 1 and 2 (i.e., where all direct detection constraints are satisfied, where |M 1 | 240 GeV, avoiding the LHC constrain indicated in the figures, and where the ∆a µ contribution explains the observed value) using checkmate2 to check that no additional LHC constraints arise from slepton and electroweakino (including Higgsino) production.

Benchmark points explaining ∆a µ , dark matter and avoiding LHC constraints
For a Bino-like DM candidate with mass in the few-hundred GeV range, the observed relic density can be realized via thermal production through different mechanisms, such as coannihilation with sleptons or charginos [128][129][130][131][132][133], t-channel annihilation via light left-right mixed staus [134] or smuons [135], or resonant s-channel annihilation [131,132]. In table 1 we present a few benchmark scenarios which simultaneously accommodate the (g µ − 2) anomaly and a viable DM candidate. All of them are consistent with the observed relic density, the observed value of ∆a µ , and satisfy LHC constraints as well as constraints from direct detection. For all benchmark points, we set the parameters in the squark and gluino sectors such that experimental bounds are satisfied and that the observed mass of the SM-like Higgs boson is reproduced. In general, the supersymmetric partners of the colorcharged particles must have masses of the order of a few TeV to satisfy current experimental bounds (see, for instance, refs. [136][137][138][139]). Note that in some of our benchmark scenarios, the hierarchy between the gluino and the weak gaugino masses is larger than the hierarchy induced by the running of the gaugino masses from (approximately) universal values at the Grand Unification scale. While a Grand Unified Theory (GUT) is theoretically attractive, we do not know if any GUT is realized in nature. The symmetries of the low energy theory do not impose any constraint on the hierarchy between the gluino and the weak gaugino masses. Furthermore, even in GUT models, higher order operators at the GUT scale can lead to departures from universal gaugino masses [140]. A somewhat related point is that the small-|µ| region we are interested in here tends to require particular choices for the soft supersymmetry breaking masses of the Higgs doublets. As is well known, the large stop masses required to reproduce a 125 GeV SM-like Higgs boson result in large radiative corrections to m 2 Hu . Starting from universal soft parameters m 2 Hu = m 2 H d at high energy scales, m 2 Hu is driven to large negative values at the electroweak scale, while m 2 H d receives much smaller radiative corrections. The correct electroweak symmetry breaking (and, in particular, the correct mass of the electroweak gauge bosons) is then achieved for −m 2 Hu ≈ |µ| 2 m 2 H d at the electroweak scale. We leave a dedicated investigation of the required values of the soft parameters to achieve the correct electroweak symmetry breaking pattern JHEP01(2022)025 for future work, however, let us note that (radiative) electroweak symmetry breaking as well as the correct mass of the electroweak gauge bosons are straightforward to achieve regardless of the value of |µ| if one allows for different values of the soft supersymmetry breaking parameters m 2 Hu and m 2 H d at high energy scales. The constraints from Higgsino and Wino pair production depend on a careful consideration of the production cross sections and decay branching ratios [141,142]. Here, we consider a compressed spectrum, for which the electroweakino and slepton constraints are weakened. The results for the spectrum, ∆a µ , the relic density, as well as the SI and SD cross sections have been obtained with Micromegas 5.2.7.a [143][144][145]. We use SUSY-HIT 1.5 [146] to compute branching ratios relevant for checking the electroweakino and slepton constraints. One problem in the analysis of the LHC limits is that, in many cases, signals can be obtained from the chain decay of many different electroweak particles, and therefore it is difficult to directly apply the bounds from LHC analyses which are typically presented in terms of simplified models. In order to solve this problem, we use checkmate2 [147][148][149][150], that uses Monte Carlo event generation to compare all production and decay channels for the neutralinos, charginos and sleptons with the current LHC analyses . Although most of the relevant LHC analyses have been included in checkmate2, a few of the most recent analyses are not yet implemented in this code. In these cases, we check the compatibility of our points by using conservative estimates of the particle contributions to the different search signals, as explained in appendix A.
The scenarios presented below correspond to different origins of the observed DM relic density and should serve as a guidance for experimental probes of the supersymmetric explanation of the muon (g − 2).
• BMSM: a DM production scenario closely related to the relatively low masses of the muon (neutrino) superpartners required to address the a µ -anomaly is co-annihilation of the lightest neutralino with the light slepton states. The benchmark BMSM gives a representation of this possibility, where we set the masses of the tau-lepton superpartners to be larger than those of the first and second generation sleptons. Since multiple production channels contribute to final states containing leptons at the LHC, current searches strongly constrain the presence of light electroweak interacting particles in this scenario. In order to be compatible with ∆a µ , DM phenomenology and LHC searches, BMSM features the largest values of |µ| and tan β of the benchmark points presented in this article.
• BMST: a similar solution to BMSM is associated with the co-annihilation of a light stau with the lightest neutralino. For universal soft slepton masses, this happens naturally at large values of tan β, where the lightest stau is pushed to masses lower than those of the sneutrinos. BMST gives a representative example of this possibility.
• BMW: the lightest neutralino may co-annihilate with the lightest chargino. The benchmark BMW represents such a possibility. Although the mechanisms controlling the relic density are different for the different benchmark points, they share many characteristics. They feature masses of weakly interacting sparticles masses lower than about 500 GeV and values of tan β of the order of a few 10's, leading to values of ∆a µ in the desired range. Apart from BMH, which we will discuss further below, all benchmark points in table 1  The observed relic density for a Bino-like DM candidate may also be obtained via resonant s-channel annihilation via the heavy Higgs boson A and H. However, for the values of tan β necessary to enhance ∆a µ , LHC bounds on the heavy Higgs bosons become very strong, implying a heavy spectrum. Using the bound on m H provided in eq. (2.8), the approximate expression for ∆a µ in eq. (2.5), and assuming that all the weakly interacting sparticles have masses close to m H /2, the maximal value for ∆a µ that may be obtained is which is a factor of a few smaller than the observed anomaly. Therefore, we shall not discuss this particular solution further. are preferred. Note that, for BMH, we chose the sleptons and the Winos to be heavy to avoid the bounds from the LHC. Hence, obtaining the proper value of ∆a µ requires relatively large values of tan β.

Future prospects
The benchmark points presented above are compatible with current experimental limits, but will be tested in the near future in several ways.
First, all four benchmark points will be probed by the next generation of direct detection experiments: the SIDD cross sections of all four benchmark points are within the projected sensitivities of the LZ and XENONnT experiments [215,216]. More generally, for µ × M 1 < 0, and for fixed values of M 1 , µ and tan β, the smallest possible value of the SIDD cross section is associated with the smallest allowed value of the heavy Higgs mass, see eq. (2.6). For masses 200 GeV |M 1 | 500 GeV, a hierarchy 1 |µ/M 1 | 3, and tan β 20, compatible with collider physics, muon (g − 2), and Dark Matter relic density constraints, the smallest possible SIDD cross section is (see eq. (2.9)) (2.13) The LZ and XENONnT experiments will probe cross sections as small as σ SI p ∼ O 10 −12 pb for |M 1 | ∼ 40 GeV, growing to σ SI p ∼ O 10 −11 pb for |M 1 | ∼ 500 GeV, implying full coverage of this representative region of parameters.
Furthermore, the spin-dependent WIMP-neutron cross sections can be probed by LZ and XENONnT, while the next generation of the PICO experiment will probe the spindependent WIMP-proton cross sections [217]. From eq. (2.10) we can see that the spindependent WIMP-nucleon cross sections are with a mild dependence on M 1 . The future sensitivities of LZ/XENONnT on σ SI n move from a few times 10 −7 pb for |M 1 | ∼ 100 GeV to ∼ 10 −6 pb for |M 1 | ∼ 500 GeV, while PICO-500 will probe σ SD p ∼ 10 −6 pb for |M 1 | ∼ 100 GeV and σ SD p ∼ 5 × 10 −6 pb for |M 1 | ∼ 500 GeV. Hence, these experiments will probe the region of parameter space where |µ| 500 GeV. In particular, LZ, XENONnT and PICO-500 will probe the spin-dependent cross sections of the benchmark points BMST, BMW, BMH, while BMSM has spin-dependent interactions smaller than the projected sensitivities of these experiments.
Second, for all benchmark points with M 1 × µ < 0 (BMSM, BMST, and BMW), the SIDD cross section is suppressed below current experimental limits due to the destructive interference between the amplitudes mediated by the SM-like and the heavy Higgs bosons discussed above. For this suppression to be effective, the masses of the non-SM-like Higgs bosons must be low enough to within the reach (see, for example, ref. [218]) of future runs of the LHC: the high-luminosity LHC will be sensitive to Higgs bosons with masses of about a factor 1.5 larger than current exclusion limits (keeping all other parameters, in particular tan β, fixed). From the expression of the SIDD cross section, eq. (2.6) we see that

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increasing m H → 1.5 m H corresponds to a factor 2-3 increase of the SIDD cross section. Such SIDD cross sections would be in conflict with current experimental constraints, or conversely, values of the heavy Higgs mass allowed by current direct detection bounds will be efficiently probed by the high-luminosity LHC. For BMSH, on the other hand, the SIDD cross section is suppressed by a large hierarchy between the Higgsino and Bino mass parameters, |µ| |M 1 |. Such "large |µ|" solutions to suppressing the SIDD cross sections allow for heavy Higgs masses beyond the projected reach of the high-luminosity LHC.
Last but not least, our benchmark scenarios are also testable in searches for electroweakly interacting particles at future runs of the LHC, see, for example, refs. [219,220]. We note that some of these projections have already been surpassed by innovative searches with current LHC data, like those presented in ref. [221], further bolstering the prospects of probing our benchmark points and similar scenarios in the upcoming runs of the LHC. The extrapolation of these conclusions to the whole region of parameters analyzed in this article should be the object of an independent dedicated study, that we plan to perform but is beyond the scope of the current article. Let us also emphasize that future lepton colliders play an important role to probe sleptons and charginos, especially for (semi-)compressed spectra, see refs. [222][223][224][225][226][227][228][229][230][231][232][233][234].

Summary and conclusions
A wide range of possible extensions of the Standard Model (SM) can lead to an explanation of the value of ∆a µ measured at the Fermilab and Brookhaven experiments. While arguably the simplest explanation is the addition of a scalar particle, one can also rely on new gauge bosons, vector-like fermions or leptoquark models. The leptoquark (or R-parity violating supersymmetry) solution seems to be interesting since it can accommodate not only the values of ∆a µ , but can also lead to an explanation of the flavor anomalies, although at the prize of a delicate choice of the couplings of the leptoquarks.
In this work, we explore a solution based on the (R-parity conserving) Minimal Supersymmetric extension of the SM, in which, although one cannot address the flavor anomalies, one can find solutions leading to a compelling DM explanation. In particular, we discuss the conditions that are required to be consistent with the observed ∆a µ , existing direct dark matter (DM) detection constraints, and the bounds from the LHC on new Higgs bosons and supersymmetric particles. We look for solution in which direct DM detection constraints are fulfilled by a partial cancellation of the light and heavy CP-even Higgs mediated contributions which significantly differ from previous studies relying on very heavy Higgs and Higgsino particles. This cancellation requires negative values of µ×M 1 . Since the observed value of a µ is larger than the SM prediction, the Bino contribution to a µ , which is proportional to µ × M 1 , must be subdominant. This can only be realized for small-to-moderate values of |µ|. We present corresponding benchmark scenarios associated with different DM production mechanisms to achieve the observed relic density, including co-annihilation with sleptons, or resonant s-channel annihilation mediated by the SM-like Higgs or Z bosons.
The corresponding spectra have a number of interesting consequences: 1) The relatively small values of the Higgsino mass parameter lead to a more natural model, in terms of the

A LHC constraints from chargino and slepton searches
In this appendix, we discuss the constraints from chargino and slepton searches on our benchmark points presented in table 1. The most severe chargino constraints tend to stem from production of the lightest chargino ( χ ± 1 ) and the next-to-lightest neutralino ( χ 0 2 ) at the LHC, pp → χ ± 1 χ 0 2 . Note that for all of our benchmark points, the lightest neutralino is Bino-like, while χ 0 2 is Wino-like (for BMSM, BMST and BMW) or Higgsino-like (for BMH) depending on the hierarchy of |µ| and |M 2 |. Hence, χ 0 2 and χ ± 1 will typically be mass degenerate. All the benchmark points presented in this article fulfill the current LHC constraints  implemented in checkmate2 [147][148][149][150]. We also check compatibility with very recent LHC searches which are not yet implemented in checkmate2 by using conservative estimates of the particle contribution to these search channels.
In order to gain a physical intuition of how the benchmark points avoid the LHC constraints, we provide a brief discussion of their properties. We have computed the χ 0 2 + χ ± 1 production cross section at the 13 TeV LHC with MadGraph5_v3.1.1 [148], finding σ(pp → χ 0 2 + χ ± 1 ) = 0.08 pb . Comparing this to the upper limit from ref.
[123] σ(pp → W Z + 2 χ 0 1 ) 0.6 pb at these masses (this search is not yet implemented in checkmate2; we have taken the limit from the supplementary material of ref. [123] accessible via HEPdata or the CERN Document Server), we see that this

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benchmark point is not constrained by this search even before taking into account that the χ 0 2 and χ ± 1 decay branching ratios into gauge bosons are small in this scenario. For this benchmark point, however, the Wino-and Higgsino-like neutralinos and charginos can undergo cascade decays involving the light sleptons, giving rise to potentially detectable signatures in searches for charged leptons and missing energy at the LHC. Due to BMSM's mass spectrum, the production of the Wino-like χ 0 2 and χ ± 1 gives rise to relatively soft leptons. The most sensitive search corresponding to this final state currently implemented in checkmate2 is ref. [196], for which we find a signal strength of r ∼ 0.8. = m χ ± 1 = 296 GeV. Both the next-to-lightest neutralino and the lightest chargino decay into staus for this benchmark point, BR( χ 0 2 → τ ± 1 + τ ∓ ) = BR( χ ± 1 → τ ± 1 + ν τ ) = 100 % from our SUSY-HIT results. The staus in turn decay into tau-leptons, BR( τ ± → τ ± + χ 0 1 ) = 100 %, leading to tau-leptons + missing transverse energy final states from χ 0 2 and χ ± 1 production at the LHC. Although the corresponding searches are quite challenging, studies with initial state radiation jets in a compressed region with chargino-neutralino mass gap m χ ± . We have arranged the spectrum of BMST such that this bound is approximately applicable, and accordingly, we chose the masses of the Wino-like next-to-lightest neutralino and the lightest chargino to be larger than 290 GeV. Regarding the slepton searches, the selectrons and smuons have masses m ± = 323 GeV for BMST. Hence, m ± − m χ 0 1 = 68 GeV, which is below the mass gaps excluded by current LHC searches [122]. We note that out of the searches implemented in checkmate2, BMST has the largest signal strength (r ∼ 0.3) for the search in ref. [197]. 2 + χ ± 1 production cross section, we find σ(pp → χ 0 2 + χ ± 1 ) = 0.26 pb. While this point is not constrained by any of the analyses included in checkmate2, it may be constrained by the recent bounds coming from the multi-lepton final state analyses in ref.
[123], which is not yet implemented in checkmate2. Note that the dominant production mechanism of charged lepton final states from the charginos and neutralinos in BMS is via tau-leptons. In
[123], however, the limits are obtained assuming decays of the charginos and neutralinos into gauge bosons and the lightest neutralinos and hence the limits are not directly applicable to this case. In order to make a conservative comparison to the upper limit σ(pp → W Z +2 χ 0 1 ) 0.9 pb [123] at these masses, we can note that, including the dominant contribution coming from τ lepton decays, the total leptonic branching ratio from χ 0 2 + χ ± 1 production 5 for this benchmark points is 4.5 %, while ref.
Regarding direct slepton searches, the lightest charged sleptons for this benchmark point are the staus, m τ 1 = 305 GeV, followed by the selectrons and smuons with m ± = 353 GeV. Such mass gaps, m ± −m χ 0 1 = 82 GeV, are not constrained by current LHC bounds even under the assumption of BR( ± → ± + χ 0 1 ) = 100 % for all four of the left-and righthanded selectron and smuon states, with stau bounds being even weaker. For this benchmark point, the left-handed charged sleptons decay preferentially into charginos, BR( ± L → ν + χ ± 1 ) = 53 %, and have sizeable branching ratios into the next-to-lightest (Wino-like) neutralino, BR( ± L → ± + χ 0 2 ) = 28 %. The reduced decay branching ratios into the lightest neutralino implies softer spectra of visible decay products at the LHC and hence even weaker bounds. Moreover, due to the compressed chargino and neutralino spectrum, no relevant additional constraints emerge from the decay of the sleptons into the Wino-like states.
We note that out of the searches implemented in checkmate2, BMW has the largest signal strength (r ∼ 0.4) for the search in ref. [197]. are Higgsino-like, leading to a relatively small χ 0 2 + χ ± 1 production cross section of σ(pp → χ 0 2 + χ ± 1 ) = 0.013 pb at the 13 TeV LHC. This is significantly below the upper limit from ref.
[123] at theses masses, σ(pp → W Z + 2 χ 0 1 ) 0.02 pb, even before taking the branching ratios of χ 0 2 and χ ± 1 into account [BR( χ 0 2 → χ 0 1 + h) = 62 % and BR( χ 0 3 → χ 0 1 + h) = 34 % for this point]. The masses of the Wino-like state are m χ 0 4 = m χ ± 2 = 745 GeV, beyond the current limit on these states [121,190,235], even before accounting for the decay patterns of the heavy Winos. The Wino-like states dominantly decay into the intermediate Higgsino-like states, Thus, production of Wino-like states at the LHC will mostly lead to cascade decays with softer visible final states than if the Wino-like states would directly decay into the lightest neutralino, χ 0 4 / χ ± 2 → χ 0 1 + W ± /Z/h, complicating experimental searches. Furthermore, let us stress that the bounds on Wino production presented by the experimental collaborations assume that the squarks are decoupled, and accordingly ignore the important t-channel squark-mediated contributions to the Wino JHEP01(2022)025 production cross section which can lower the cross section by order one factors depending on the exact squark masses [142]. These arguments apply to very recent searches for Winos in hadronic final states [221,236] that would rule out this scenario in the absence of cascade decays and the t-channel squark contributions. Nonetheless, these impressive searches clearly show the potential of the experimental collaborations to test the regions of parameters represented by our scenarios in future runs of the LHC. Regarding the slepton searches, the lightest charged sleptons for this benchmark point are the staus, m τ 1 = 710 GeV, and the selectrons and smuons have masses m ± = 751 GeV, beyond the reach of current LHC searches [122]. We note that out of the searches implemented in checkmate2, BMH has the largest signal strength (r ∼ 0.4) for the search in ref. [197].
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