Impact of the LZ Experiment on the DM Phenomenology and Naturalness in the MSSM

This paper uses approximate analytical formulas and numerical results with the bino-dominated dark matter (DM) as an example to analyze the impact of the LUX-ZEPLIN (LZ) experiment on the DM phenomenology and naturalness in the Minimal Supersymmet-ric Standard Model (MSSM). We conclude that the limitation of the latest LZ experiment worsens the naturalness of the MSSM, as the predictions of the Z -boson mass and DM relic density demonstrate, particularly in the regions where the correct DM relic density is obtained by the Z - or h -mediated resonant annihilations.


Introduction
Supersymmetric models of particle physics are renowned for providing an elegant solution to the daunting gauge hierarchy problem.The Minimal Supersymmetric Standard Model (MSSM), as the most economical supersymmetric expansion model, may provide a solid description of nature from the weak scale to energy scales associated with the grand unification [1], which also receives indirect experimental support from the measured strengths of weak-scale gauge couplings, measured value of the top quark mass, and discovery of an SM Higgs-like boson by ATLAS [2] and CMS [3] in 2012.However, this audacious extrapolation has suffered a string of serious setbacks because LHC and DM data have shown no signs of supersymmetric matter, which has led some physicists to question whether weakscale SUSY really exists or at least to concede that it suffers diversiform unattractive fine tunings [4].
In the MSSM, the Z-boson mass is as follows [5] where m 2 Hu and m 2 H d are the soft SUSY-breaking (not physical) Higgs mass terms; µ is the superpotential higgsino mass term; tan β ≡ v u /v d is the ratio of Higgs field vevs; Σ u u and Σ d d include various independent radiative corrections [6].Since the term ( m 2 is suppressed by tan 2 β − 1, for even moderate tan β values, Eq. (1.1) To naturally achieve m Z ≃ 91.2 GeV, −m 2 Hu , −µ 2 , and each contribution to −Σ u u should be comparable in magnitude to m 2 Z /2.The extent of the comparability can be quantified using the electroweak fine tuning parameter [7] 1 where C Hu = −m 2 Hu , C µ = −µ 2 , and C Σ u u = −Σ u u .A lower value of ∆ EW implies less fine tuning, and 1/∆ EW is the percentage of fine tuning, e.g., ∆ EW = 20 corresponds to ∆ −1 EW = 5% fine tuning among the terms that contribute to m 2 Z /2.Therefore, given the experimental lower bound, there is a general consensus that smaller values of |µ| are preferred in fine tuning issues.However, the current experiment limits impose a strong lower bound on |µ|.For example, in 2017, the analysis of a global fit for the MSSM, which considered various experimental constraints 2 showed that µ > 350 GeV was favored at a 1 Compared with the other two measures of EWFT (∆BG and ∆HS) as shown in Ref. [7], ∆EW is created from weak-scale SUSY parameters and consequently contains no information about any possible high-scale origin.Hence, ∆EW is advantageous because it is model-independent, where any model that yields the same weak-scale mass spectrum will generate the same value of ∆EW [8][9][10].Meanwhile, ∆EW < ∆BG ≲ ∆HS, i.e., ∆EW can be considered a lower bound on electroweak fine tuning [11].Any model with a large value of ∆EW is always fine tuned.
2 These experimental constraints include those from the DM relic density, PandaX-II (2017) results for the SI cross section [12], PICO results for the SD cross section [13], and searches for supersymmetric particles at the 13-TeV LHC with 36 fb −1 data (especially the CMS analysis of the electroweakino production ) [14].
95% confidence level (C.L.) [15].This value of µ can induce a tuning of approximately 3% to predict the Z-boson mass.The studies in Ref. [16] showed that the Xenon-1T direct search limits [17] imposed a strong lower bound on |µ|, particularly for µ > 0 or when the masses of the heavy Higgs bosons of the MSSM were near their current limit from LHC searches.Ref. [18] demonstrated that µ should be larger than approximately 500 GeV for M 1 < 0 and 630 GeV for M 1 > 100 GeV considering the recent measurement of the muon anomalous magnetic moment at Fermilab [74], first results of the LUX-ZEPLIN (LZ) experiment in the direct search for DM [20], and rapid progress of the LHC search for supersymmetry [14,21,22].In Ref. [23], systematic studies on DM in the hMSSM with a light gaugino/higgsino sector revealed that the stringent requirement of the conventional thermal paradigm as a mechanism to achieve the correct DM relic density had a significant impact on the viable parameter space, one of which is that the lower bounds of µ were elevated to approximately 500 GeV.These improved bounds imply a tuning of O(1%) to predict the Z-boson mass.
In this paper, using the approximate analytical formulas and numerical results, we analyzed the DM phenomenology and associated unnaturalness in the MSSM in detail under the latest LZ experimental limits.The rest of this paper is organized as follows.Section 2 briefly introduces the neutralino sections of the MSSM and demonstrates the DM scattering cross-sections with nucleons and annihilation for bino-like χ0 1 using the approximate analytical formulas.Section 3 briefly describes our scanning strategy and investigates the predictions for the surviving samples and properties of bino-dominated DM scenarios to understand the associated unnaturalness.Section 4 presents our conclusions.

Dark Matter Section in the MSSM
In the MSSM, the neutralino mass matrix in the basis of Ψ 0 = (−i B0 , −i W 0 , H0 d , H0 u ) is [24]: where M 1 , M 2 , and µ are the soft SUSY-breaking mass parameters of the bino, wino, and higgsinos, respectively; m Z is the Z-boson mass; θ w is the Weinberg angle (c W ≡ cos θ W and s W ≡ sin θ W ); tan β ≡ s β /c β = v u /v d is the ratio of the vacuum expectation values for the two Higgs doublets (c β ≡ cos β and s β ≡ sin β) and GeV) 2 .Diagonalizing M neut with a 4 × 4 unitary matrix N yields the masses of the physical states χ0 i (ordered by mass) of four neutralinos: where m χ0 i is the root to the following eigenequation: 2) The eigenvector of m χ0 i is the column vector constituted by N ij (j = 1, 2, 3, 4), which is given by 3) The specific form of the normalization factor C i is: Then, the diagonalizing matrix is 13 + N 2 14 > 0.5), we call χ0 1 the bino-( wino-or higgsino-) dominant DM.The couplings of DM to the scalar Higgs states and Z-boson are included in the calculation of the DMnucleon cross sections and DM annihilation, which correspond to the Lagrangian [25,26]: The coefficients are: where h and H are two CP-even Higgs states predicted by the MSSM: the SM-like Higgs boson and non-SM doublet Higgs boson, respectively.Serving as a weakly interacting massive particle (WIMP), χ0 1 may be detected by measuring their spin-independent (SI) and spin-dependent (SD) scattering cross-sections after an elastic scattering of χ0 1 on a nucleus occurs.At the tree level, the contribution to the SD (SI) scattering cross-section in the heavy squark limit is dominated by the tchannel Z-boson (CP-even Higgs bosons h i ) exchange diagram.Therefore, the scattering cross-sections have the following form [18,27,28]: , (2.9) where N=p, n represents the proton and neutron, and [29,30].The form factors at zero momentum transfer are G , where f d, s) is the normalized light quark contribution to the nucleon mass, and f affects other heavy quark mass fractions in the nucleons [31,32].In this study, the default settings for f N q were used in the micrOMEGAs package [33], and they predicted F p u ≃ F n u ≃ 0.15 and F p d ≃ F n d ≃ 0.13.Hence, the DM-proton scattering and DM-neutron scattering had approximately equal SI cross-sections (i.e., σ SI χ0 In the pure bino limit (m χ0 1 ≈ M 1 and N 2 11 ≈ 1), the above formulas can be approximated as ) ≃ 0.14 [35] and tan β ≫ 1, we can conclude that (2.14) . (2.15) These two analytic formulas suggest that σ SD and σ SI are suppressed by µ 4 and µ 2 , respectively.Moreover, if ( ) and tan β have opposite signs, the contributions to σ SI from the light Higgs(h) and heavy Higgs(H) exchange channels destructively interfere with each other.σ SI will vanish for ( → 0, which is known as the "generalized blind spot" [28,35].For the convenience of subsequent descriptions, we defined A h = ( to represent the contributions from h and H to the SI cross-section, respectively. For DM that was produced via the standard thermal freeze-out, the relic density at freeze-out temperature T F ≡ m χ0 1 /x F (typically, x F ≃ 20) was approximately [28] Ω χ0 where ⟨σv⟩ x F corresponds to the effective (thermally averaged) annihilation cross-section.
In the MSSM, for m χ0 1 < 1 TeV, only the bino-dominated χ0 1 can predict the correct DM relic density.For the higgsino-or wino-dominated χ0 1 , the predicted relic density is much smaller than the observed DM relic density because they relatively strongly interacted with the Standard Model particles [37].In our scenario, considering bino-like χ0 1 as an example, we will discuss the fine tunings introduced by the DM sector in the MSSM under the current DM experimental limits.For bino-like χ0 1 , the contributions to ⟨σv⟩ x F are from two channels • The Z-or h-mediated resonant annihilation [38,39].The corresponding annihilation cross-sections are approximated to [40] ) , (2.17) ) Due to the hierarchy of Yukawa couplings, the contribution to the thermal crosssection from the ( χ0 1 → q q) annihilations will be dominated by bottom quarks for the lighter m χ0 1 .Here, m d is the down-type quark mass.As the expression shows, the measured relic density requires a high degeneracy between m Z(h) and 2m χ0 1 .We define degeneracy parameters ∆ Z | to quantize the fine tuning between m Z(h) and 2m χ0 1 , e.g., ∆ Z = 10 −3 implies 0.1% fine tuning among m Z and 2m χ0 1 .

Numerical Results and Theoretical Analysis
We used the MultiNest algorithm [47] with n live = 100003 to comprehensively scan the following parameter space: where tan β was defined at the electroweak scale, and the others were defined at the renormalization scale Q = 1 TeV.To obtain the SM-like Higgs boson mass (m h ≈ 125 GeV), the soft trilinear coefficients A t and A b were assumed to be equal and freely change to adjust the Higgs mass spectrum.The masses of the second-generation sleptons (M μL , M μR ) were used as free parameters to explain the muon g-2 anomaly and predict the measured DM relic abundance by co-annihilation with sleptons.Other SUSY parameters of sleptons were fixed at 3 TeV to reduce the number of free parameters.The release of τ may change the e/µ signals of this study and relax the LHC restrictions [48,49].Other unimportant parameters were also fixed at 3 TeV, including the gluino mass M 3 and three generations of squarks except A t and A b .
During the scan, some experimental constraints were imposed by constructing the following corresponding likelihood function to guide the process as follows: where • L h and L h,extra are the likelihood functions for the consistency of h's properties with the LHC Higgs data at the 95% C.L. [53] and collider searches for extra Higgs bosons [57].The two restrictions were implemented by the programs HiggsSignal 2.6.2 [50][51][52][53] and HiggsBounds 5.10.2 [54][55][56][57] , respectively.
• L B is the likelihood function for the measured branching ratio of the B → X s γ and B s → µ + µ − .These ratios were calculated by the formulae in Refs.[58,59] and should be consistent with their experimental measurements at the 2σ level [60].
• L V ac is the likelihood function for the vacuum stability of the scalar potential, which consists of the Higgs fields and the last two generations of the slepton fields.This condition was implemented by the code Vevacious [71,72].
• L aµ is the likelihood function of the muon g − 2 anomaly given by where ∆a µ ≡ a Exp µ − a SM µ is the difference between experimental central value of a µ and its SM prediction, and δa µ is the total uncertainties in determining ∆a µ [73][74][75].
To probe into the impact of the LZ experiment on the DM phenomenology and naturalness of the complete parameter space in the MSSM, we did not consider the constraints from the LHC search for SUSY.Similar to the studies in Ref. [18], the restrictions from the LHC experiment require that the lower bounds of µ become approximately 400 GeV, and the upper bounds of M 1 become approximately 570 GeV, which severely compresses the surviving space of the MSSM.For example, compared with the restriction from the PandaX-4T experiment on the SI scattering cross-section, the lower bounds of µ improve by approximately 100 GeV.As a result, the regions with the Z-and h-mediated resonant annihilations are more unnatural at predicting the correct relic density so that they are commonly missed in the scans by the MultiNest algorithm.
The acquired samples were refined using the following criteria: the observed DM relic abundance within ±10% of the measured central value is Ωh 2 = 0.12 ( i.e., 0.108 ≤ Ωh 2 ≤ 0.132) [36], and N 2 11 > 0.5 to guarantee that the LSP is a bino-like neutralino.Then, we projected the refined samples on the corresponding planes of Fig. 1, Fig. 2 and Fig. 3 .
In Fig. 1, we projected the surviving samples on the m χ0 1 −σ SI only related to µ and will be suppressed by a large µ, which is consistent with the analysis based on Eq. (2.14), i.e., σ SD χ0 The current LZ experiment constraint on σ SD χ0 1 −n requires that µ is greater than 370 GeV.However, the distribution of the σ SI  To find the combination that will make σ SI χ0 1 −p satisfy the latest experimental limit, as presented in Fig. 2, we projected the surviving samples onto the A h −σ SI  m H . Fig. 2 shows that compared with the contributions from H (A H ) and v 2 /µ 2 to the SI cross-section, the contribution of h (A h ) played a dominant role.
In Fig. 3, we projected the samples onto the µ − ∆ EW plane and σ SD According to DM annihilation mechanisms, we divided the refined samples into four categories to discuss in detail.
1. Type-I samples: m χ0 1 ≈ − 1 2 m Z , 5 < tan β < 58, 337 GeV < µ < 462 GeV,100 GeV < M 2 < 1206 GeV.χ0 1 is mainly annihilated to d d by exchanging a resonant Z-boson in the s-channel to obtain its measured relic density.According to Eq. (2.17), the corresponding annihilation cross-section in this area is approximately From Fig. 1, σ SI χ0 1 −p can also satisfy the direct detection constraints with small µ values, but σ SD χ0 1 −n cannot.As discussed above, σ SI χ0 1 −p was proportional to (A h + A H ) and suppressed by µ 2 , so it even vanishes under the arranged limits of M 1 , µ, tan β, and m H . Fig. 2 shows that |A h | < 0.1, which corresponds to M 1 /µ and tan β having opposite signs, e.g., suppresses the SI cross section.However, σ SD  Briefly, the characteristics of this area are similar to the cases with 100 GeV < m χ0 1 < 550 GeV.Moreover, these regions of the parameter space will shrink or disappear with further improvement of sensitivity in future experiments.3. When the LZ experiment was considered, the lower bounds of the higgsino mass µ and ∆ EW were elevated, and µ should exceed 600 GeV, 370 GeV, 340 GeV, 440 GeV and 580 GeV for the cases of and M 1 ≳ 100 GeV, respectively.In particular, µ was more tightly limited for the case of M 1 > 0 than for M 1 < 0, when |M 1 | was fixed.The reason is that, under the approximation m χ0 1 ≃ M 1 , a negative M 1 can lead to the cancelation between different contributions to the SI DM-nucleon scattering cross-section in Eq. (2.15).These improved bounds of µ imply a tuning of O(1%) to predict the Z-boson mass and simultaneously worsen the naturalness of the Z-or h-mediated resonant annihilations to achieve the measured DM relic density.When considering the restrictions from the LHC experiment, the lower bounds of µ improved by approximately 100 GeV, and the upper bound of M 1 was approximately 570 GeV [18], so the surviving region greatly decreased that will be exacerbated when future DM DD experiments fail to detect the signs of DM, which implies larger unnaturalness of the MSSM.

Conclusion
In this study, the LZ experiment recently released its first results in the direct search for DM, where the sensitivities to the SI and SD cross-sections of DM-nucleon scattering reached approximately 6.0 × 10 −48 cm 2 and 1.0 × 10 −42 cm 2 , respectively, for the DM mass of approximately 30 GeV [20].These unprecedented precision values strongly limit the DM coupling to SM particles that are determined by the SUSY parameters.Considering this strong experimental limitation, the DM phenomenology and unnaturalness related to DM physics in the MSSM were discussed in detail using approximate analytical formulas.It was found that under the limit of the latest dark matter experiment, the unnaturalness associated with DM in the MSSM is embodied in the large higgsino mass µ elevated by the latest LZ experiment requirement.For example, after considering the LZ experiment, the lower bounds of the higgsino mass µ and ∆ EW were elevated, and µ had to be larger than 600 GeV, 370 GeV, 340 GeV, 440 GeV and 580 GeV for the cases of M 1 < −400 GeV, −400 GeV ≲ M 1 ≲ −100 GeV, M 1 ≃ −m Z /2, M 1 ≃ m h /2, and M 1 ≳ 100 GeV, respectively.In particular, µ was more tightly limited for the case of M 1 > 0 than for M 1 < 0, when |M 1 | was fixed.These improved bounds of µ implied a tuning of O(1%) to predict the Z-boson mass and simultaneously worsen the naturalness of the Z-and h-mediated resonant annihilations to achieve the correct DM relic density.If the LHC experiment is included, the surviving region greatly decreases which could be exacerbated if future DM DD experiments fail to detect the signs of DM, thus implying larger unnaturalness of the MSSM.

χ0 1 −
p values is relatively complex.Although large µ can suppress σ SI χ0 1 −p , σ SI χ0 1 −p can also be small for small µ.Based on Eq. (2.15), σ SI χ0 1 −p should be related to a combination of M 1 , µ, tan β, and m H .

Figure 2 .
Figure 2. Projection of the refined samples onto the A h − σ SI χ0 1 −p plane, where the colors indicate the contributions from H to the SI cross-section A H , and onto the m χ0 1 − A h plane, where the colors indicate the value of the higgsino mass µ.

χ0 1 −
p plane, where the colors indicate the contributions from H to the SI cross-section A H , and onto the m χ0 1 − A h plane, where the colors indicate the value of the higgsino mass µ.The contribution of the heavy Higgs (H) to the SI cross-section for most samples was suppressed below 0.14 by

Figure 3 .
Figure 3. Projection of the refined samples onto the µ − ∆ EW plane and σ SD χ0 1 −n − σ SI χ0 1 −p plane with the colors indicating the values of ∆ EW .

χ0 1 −
n − σ SI χ0 1 −p plane with the colors indicating the values of EWFT (∆ EW ).Fig. 3 shows that the samples were fine-tuned with large ∆ EW > 70(∆ −1 EW = 1.4%EWFT).According to the above discussion, the reason is that the latest LZ experiment requires µ > 370 GeV.However, low ∆ EW solutions are only possible for low values of µ [7, 8].As shown in the second figure of Fig. 3, ∆ EW of the samples allowed by the LZ experiment on the SI scattering cross-section may be relatively low due to the existence of the "generalized blind spot" .Because the values of µ 2 /(m 2 Z /2) are the lower bounds of ∆ EW , we will use µ 2 /(m 2 Z /2) to measure ∆ EW hereafter.

µ 4 .
According to Eq.(2.14), the LZ direct detection constraint on large for large µ.The latest LZ experiment also increased µ to above 600 GeV, and the right relic density is obtained when |∆ M 2 | < 0.07 or |∆ m l | < 0.07.

Table 1 .
Values of µ and ∆ EW for four types of samples after and before considering the LZ experiment.×and√ indicate that the corresponding experimental limitations cannot and can be satisfied, respectively.Table1summarizes the above discussion of different samples.∆ EW is the values presented in Fig.