Electric dipole moments and dark matter in a CP violating MSSM

We investigate electric dipole moments (EDMs) in a CP-violating minimal supersymmetric standard model with the Bino-like neutralino dark matter (DM) annihilating through the heavy Higgs funnel. Motivated by the current experimental results, in particular, the measured mass of the standard model-like Higgs boson, we consider a mass spectrum with stop masses of about 10 TeV. For the other sfermions, we consider masses of about 100 TeV. We show that CP-violating phases of the order of ten degrees in gaugino and Higgsino mass parameters are consistent with the current bound by EDMs of the electron, the neutron, and the mercury. They are within the reach of future experiments. We also show that effects of CP-violating phases induce a difference in DM-nucleon scattering cross section by a factor.


INTRODUCTION
Supersymmetry is an attractive candidate of physics beyond the Standard Model (SM), although current results from LHC experiments indicate that supersymmetric (SUSY) particles are heavier than have been expected. Attractive aspects come from the fact that, for instance, the gauge coupling unification is realized in the minimal SUSY SM (MSSM), the gauge hierarchy problem is improved, and elementary scalar fields such as Higgs fields are introduced in a theoretically natural way. Moreover, SUSY models may provide additional interesting consequences. SUSY interpretation of muon anomalous magnetic moment is one example [1][2][3]. SUSY models contain new sources of CP violation and/or flavor violation, which potentially induce new CP or flavor violating phenomena. The lightest SUSY particle (LSP) is stable and hence a good candidate for dark matter (DM) in our Universe, if the R-parity is unbroken [4,5].
Electric dipole moments (EDMs) of the neutron and other heavy atoms are prime physical quantities for probing sources of CP violation. Parameters in the MSSM generally pose several CP violating phases. It used to be regarded that the null experimental EDM results confront the MSSM with O(1) CP violating phases and O(100) GeV masses of SUSY particles [6][7][8][9]. The LHC results indicate that masses of many SUSY particles are larger than O(10) TeV. Therefore CP violating phases of order unity in the SUSY sector [10,11] seems still likely and worth investigating.
In the MSSM with R-parity, the lightest neutralinoχ is a candidate of the weakly interacting mass particle (WIMP) DM. While LHC experiments as well as direct detection experiments of DM, such as LUX [12,13], XENON1T [14], and PandaX-II [15,16] are constraining large parameter space of the MSSM, there are still viable scenarios reproducing thermal relic abundance of the DM consistently. Appropriate magnitude of annihilation cross section of neutralino in the early Universe is realized if (i) neutralinos annihilate significantly through SU(2) gauge interaction, or (ii) annihilation cross section of Bino-like neutralino is enhanced with a particular mass spectrum of other associated particles.
Higgsino-like neutralino DM with the mass of about 1 TeV is an example in the former class.
Phenomenology in this scenario such as the direct detection of DM, contribution to the EDMs, and collider signals have been precisely studied in Ref. [17].
In this paper, we focus on another case in the later class; a Bino-like neutralino DM annihilates through heavy Higgs boson resonance [18][19][20][21][22] 1 . In this scenario where heavy Higgs boson resonance 1 For a study of CP violation in stau coannihilation scenario, see e.g., Ref. [23] in the neutralino DM annihilation is utilized, masses of the heavy Higgs boson are about twice of the mass of the neutralino DM. Since the Bino-like neutralino contains small Higgsino component, the neutralino can be searched through Higgs bosons exchange by spin-independent scattering off nucleus [24]. Masses of stops would be around 10 TeV in order to reproduce the measured SMlike Higgs boson mass (m h = 125 GeV) [25,26]. Then, all the other SUSY particle masses and parameters except for the Bino mass, the Higgsino mass parameter µ, Bµ, and stop masses can be much larger than O(1) TeV. With such SUSY particle mass spectrum, most of SUSY contributions to the low energy phenomena can be decoupled as the irrelevant SUSY particles are heavier, SUSY contributions to the EDMs in CP violating models can still be significantly large nevertheless. The main goal of this article is, by decoupling the other particles, to estimate the magnitude of EDMs induced by CP violation in neutralino DM sector with taking account of CP-violating phase effects into the thermal DM abundance [27][28][29][30][31][32].
We examine the electron EDM, the nucleon EDM, and the mercury EDM, as well as the DM-nucleon scattering cross section on the parameter space, where appropriate thermal DM relic abundance is reproduced, for order unity CP-violating phases of gaugino mass parameters and the µ parameter. For non-vanishing CP phase of µ and A parameters, see, for example, Refs. [27,30,33,34]. The magnitude of scattering off cross section between DM and nucleon is affected by the CP violating phases [30,[35][36][37][38][39][40]. We also study the dependence of spin-independent cross section of the DM in our scenario and find that the effect changes by a factor. We show that wide parameter regions in our scenario are now unconstrained yet, but will be explored by future experiments.
This article is organized as follows. In Sec. 2, we define a benchmark scenario for studying phenomenology in our DM scenario. In Sec. 3, we show the results of our analysis on several EDM measurements and the spin-independent cross section. Summary and conclusion are presented in Sec. 4.

SETUP OF THE SCENARIO
In this section, we briefly review the MSSM Lagrangian, and we describe the parameter setup for our analysis. The superpotential and the soft SUSY breaking terms in the MSSM are given by [41] and respectively. The convention of the epsilon tensor is 12 = − 21 = 1. Here, we note that gaugino mass parameter for Bino M 1 , Wino M 2 , and gluino M 3 are in general complex. In the following, we focus on the Yukawa couplings of the third generation quarks and leptons, so that we use y t , y b , and y τ for the Yukawa couplings of top, bottom, and tau, respectively. Neglecting the flavor mixing in the soft SUSY breaking terms, we take flavor diagonal soft scalar masses as M 2 For the trilinear couplings, A parameters defined by (T u ) 33 = A τ y t , (T d ) 33 = A τ y b , and (T e ) 33 = A τ y τ are used.
In the MSSM, the mass of the SM-like Higgs boson is expressed with some SUSY breaking parameters. In our analysis, we take tan β := H 2 / H 1 = 30 and we fix the stop mass parameters as Mq 3 = 7 TeV, Mt := Mũ 3 = 7 TeV and A t = 10 TeV, then the measured SM-like Higgs boson mass m h 125 GeV can easily reproduced [25,26]. The other SUSY particles are relevant to neither the mass of the SM-like Higgs boson nor the DM relic density. We may assume that those are much heavier than stop so that those are decoupled from low energy observables. We here take masses of the other sfermions as 100 TeV and the Wino and gluino masses to be 10 TeV. Higgsino mass |µ| should be of the order of TeV. We leave M 1 as a free parameter and solve |µ| from the measured dark matter energy density. We summarize the parameter set in our analysis 4 as follows: The other A-terms are zero. With the above parameter set, besides the CKM phase and the CP phases in the sfermion mass matrices, five parameters, µ, gaugino masses M i and A t , may have There is a rephasing degree of freedom in the MSSM. Thus, all the physical quantities are described by the following combinations, Without loss of generality, we can take the basis of CP phases as φ M 3 = 0. In addition, we take φ At = 0 to concentrate on the CP violation in the neutralino sector as well as, technically speaking, to keep m h 125 GeV avoiding complicated parameter dependence of the SM-like Higgs boson mass. Hence, in the rest of this paper, we scan the following four parameters, (2.10)

OBSERVABLES
As we mentioned in the previous section, we choose |µ| to achieve the correct DM relic density.
We use micrOMEGAs 4.3.5 [42] with CPsuperH2.3 [43] in calculations of dark matter thermal relic density. With the correct DM relic abundance, we calculate the electron EDM, the neutron EDM, and the mercury EDM. The electron and mercury EDMs give strong constraints on the parameter space as we will see later. We also discuss the scattering cross section for the direct detection experiments.
where the convention of the epsilon tensor is 0123 = +1. We calculate d u , d d , d e , d C u , and d C d by using CPsuperH2.3 [43] implemented in micrOMEGAs 4.3.5 [42]. We use the formulae given in Ref. [44] and couplings calculated by CPsuperH2.3 to evaluate ω. 2 These EDMs and the Wilson coefficient are evaluated at the electroweak scale µ W = m t . The neutron EDM and the mercury EDM have to be evaluated at the hadronic scale (µ H 1 GeV). The renormalization group evolution from the electroweak scale to the hadronic scale is taken into account [45]. At the leading order of QCD, we find 3 Here the unit of the EDMs and of the cEDMs are GeV −1 , and the unit of ω is GeV −2 . In the evaluation, we used the following values, We estimate the neutron EDM and the Mercury EDM as (3.11) Here, we used the results given in Refs. [46,47] for d n and the results given in Refs. [33,34,48] for d Hg . We found the contribution from the Weinberg operator is much smaller than the contributions from cEDMs. Thus, we ignore the contribution from the Weinberg operator in the following analysis.
Upper bounds on the electron EDM [49], the mercury EDM [50], and the neutron EDM [51] are |d n | <2.9 × 10 −26 e · cm (90% C.L.). (3.14) We note that the constraint from the thallium EDM [52] is in practice equivalent with that to the electron EDM [53][54][55][56]. Prospects for the electron and neutron EDMs are [57,58], (3.15) |d n | =2.5 × 10 −29 e · cm [59]. 7 with We discuss the φ µ and φ M 2 dependence of the EDMs. The left panels in Fig. 3 shows the electron EDM, the mercury EDM, and the neutron EDM with φ M 1 = 0. The shaded regions are already excluded by the current upper bound on the EDMs. We find the combination of the    EDM and the neutron EDM shown in Eqs. (3.15) and (3.16). The neutron and the mercury EDMs are sensitive to φ µ , and also weakly depend on φ M 2 . On the other hand, the electron EDM is sensitive to φ M 2 + φ µ , and weakly depend on φ M 1 . Therefore, we can determine the three CP phases by using the correlation among the EDMs measured in future experiments.
We next discuss the decoupling property for larger M 2 . Figure 5 shows the EDMs for M 2 = 100 TeV and 200 TeV. Comparing with the right panels in Fig. 3, we find that larger M 2 drastically reduce the values of the EDMs. In particular, the electron EDM is sensitive to the M 2 choice but the mercury EDM is not. This property can be understood as follows. The dominant contribution to the electron EDM is the H 0 i -γ-γ Barr-Zee diagram, where Wino loop gives the main contribution. On the other hand, the dominant contribution to the mercury EDM which originates from the quark EDMs is H 0 i -g-g Barr-Zee diagram, and the diagram is independent of Wino contribution. In Figs. 6 and 7, we show how each contribution to the electron EDM and to the down quark cEDM is decoupled for a larger value of M 2 .

DM-nucleon scattering cross section
Since we are working in the Higgs funnel scenario, the DM candidate couples to neutral scalar bosons. The DM candidate and nucleon interact with each other through these couplings. The couplings are rather small because of the funnel scenario. However, the couplings lead to a significant size of the spin-independent cross section which is within future prospects of the DM direct detection experiments.
There is also a Z-exchange diagram that generates spin-dependent cross section. This coupling depends on the mixing between Bino-Higgsino and Bino-Wino in the Bino-like DM scenario. The mixings are suppressed by the soft breaking neutralino mass parameters. We find that this coupling is so small that the resultant spin-dependent cross section σ SD = O(10 −8 ) pb is smaller than the prospect [60]. In the following, we focus on the spin-independent cross section. Figure 8 shows the φ M 2 and φ µ dependence of σ SI where its parameter choice is the same as in Fig. 3. Figure 9 shows the φ M 1 and φ µ dependence of σ SI with the same parameter choice as in Fig. 4. We find that the spin-independent cross section is smaller than the current upper bound [13,14,16] in all the region of the parameter space but within the prospects of the DARWIN [60], the DarkSide-20k [61], and the LZ [62]. We also find that the scattering cross section depends on φ M 1 + φ µ , and the φ M 2 dependence is not important. Since |M 2 | is much larger than |M 1 | and |µ| in our analysis, the sector related to dark matter physics is approximately the Bino-Higgsino system, and thus the scattering cross section weakly depends on φ M 2 . In the Bino-Higgsino system, there is only one physical CP phase. This is the reason why the scattering cross section depends on one combination of the CP phases, φ M 1 + φ µ .

SUMMARY
In this paper, we have estimated EDMs of the electron, the neutron, and the mercury as well as the DM-nucleon scattering cross section and shown the present constraints and prospects in the Once a few non-vanishing EDMs will be measured, it is possible to estimate individual CP phases.
We also have calculated the dependence of spin-independent cross section of the DM in our scenario. In fact, the non-vanishing CP violation effects change the cross section just by a factor.
The predicted scattering cross section with a nucleon is within the sensitivity of future experiments.