⾃然科学の歩き⽅

⾃然科学の歩き⽅

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Electric dipole moments and dark matter in a CP violating MSSM

Tomohiro Abe,^1,2 Naoya Omoto,³ Osamu Seto,^4,3 and Tetsuo Shindou⁵

1Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya, Aichi 464-8602, Japan

2Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho Chikusa-ku, Nagoya, Aichi 464-8602 Japan

3Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

4Institute for International Collaboration, Hokkaido University, Sapporo 060-0815, Japan

5Division of Liberal-Arts, Kogakuin University, Nakano-machi, Hachioji, Tokyo 192-0045, Japan

(Received 24 May 2018; published 30 October 2018)

We investigate electric dipole moments (EDMs) in a CP-violating minimal supersymmetric standard model with the binolike 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 ofCP-violating phases induce a difference in DM-nucleon scattering cross section by a factor.

DOI: 10.1103/PhysRevD.98.075029

I. 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 interpre- tation of muon anomalous magnetic moment is one example [1–3]. SUSY models contain new sources of CP violation and/or flavor violation, which potentially induce new CP or flavor violatingphenomena. The lightestSUSY 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

Oð1Þ CP violating phases and Oð100Þ GeV masses of SUSY particles [6–9]. The LHC results suggest that masses of many SUSY particles are larger than Oð10Þ TeV.¹ Therefore CP violating phases of order unity in the SUSY sector [10,11] seems still likely and worth investigating. For recent studies, see e.g., Refs. [12,13].

In the MSSM with R-parity, the lightest neutralino χ˜ 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 [14,15], XENON1T [16], and PandaX-II[17,18]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 signifi- cantly through SU(2) gauge interaction, or (ii) annihilation cross section of binolike 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, contri- bution to the EDMs, and collider signals have been precisely studied in Ref. [19].

In this paper, we focus on another case in the later class;

a binolike neutralino DM annihilates through heavy Higgs

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

1To be precise, a scenario with SUSY particles with masses of a few TeV is still allowed. The current limit on gluino mass is around 2 TeV and the squark masses can be smaller than 3 TeV in the degenerate case.

PHYSICAL REVIEW D 98, 075029 (2018)

2470-0010=2018=98(7)=075029(14) 075029-1 Published by the American Physical Society

雑誌情報，巻数等

タイトル

所属 著者

概要(アブストラクト) 受領日等

本文

Abe et al, PRD98, 075029より

MINI REVIEW published: 14 January 2019 doi: 10.3389/fphy.2018.00159

Frontiers in Physics | www.frontiersin.org 1 January 2019 | Volume 6 | Article 159

Edited by:

Stefano Moretti, University of Southampton, United Kingdom

Reviewed by:

Chandan Hati, UMR6533 Laboratoire de Physique de Clermont (LPC), France Frank Franz Deppisch, University College London, United Kingdom

*Correspondence:

Tetsuo Shindou [email protected]

Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received:10 October 2018 Accepted:20 December 2018 Published:14 January 2019

Citation:

Shindou T (2019) A UV Picture of a Loop Induced Neutrino Mass Model and Its Phenomenological Consequences. Front. Phys. 6:159.

doi: 10.3389/fphy.2018.00159

A UV Picture of a Loop Induced Neutrino Mass Model and Its

Phenomenological Consequences

Tetsuo Shindou*

Division of Liberal-Arts, Kogakuin University, Tokyo, Japan

In this article, we review several models where tiny neutrino masses are radiatively generated via loop diagrams. In such models, additional scalar fields are often introduced so that the Standard Model Higgs sector is extended.Such an extension results in a rich phenomenology of the model. We briefly discuss such a model and its UV completion to highlight some of its phenomenological consequences.

Keywords: neutrino mass, extended Higgs sector, UV theory, SUSY model, collider phenomenology

1. INTRODUCTION

Precise measurement of the Higgs boson property at the LHC experiments [1–6] suggests that the Standard Model (SM) provides quite a good explanation of the physics of elementary particles.

However, there still are several unsolved problems in the SM. For example, there is no dark matter (DM) candidate, no successful baryogenesis scenario works, gauge hierarchy problems should be solved by some additional mechanism, and so on. An origin of tiny neutrino mass has been one of such problems for more than two decades. The neutrino oscillation data [7–12] requires that there are tiny mass squared differences among three neutrino mass eigenvalues, and the absolute value of the neutrino masses have quite a severe upper bound of mν ! ^O(0.1) eV [13,14].

In many models, the tiny neutrino masses are originated from the dimension five operator (H · ¯ℓ^c)(H · ℓ) [15] after the electroweak symmetry breaking. The question is how to provide the suppressed coefficient of the operator. There are essentially three possibilities to get such a suppression factor naturally. One idea is using a suppression by a mass scale. Since the operator is dimension five, the coefficient is suppressed by some mass scale. If such a mass scale is significantly larger than the electroweak scale, the coefficient of the dimension five operator gets a strong suppression. The necessary mass scaleM in this case is naively estimated by the relation⟨H⟩²/M ∼ mν, so thatmν ∼ 0.1 eV suggestsM ∼ 10¹⁵ GeV. The most famous mechanism of this possibility is so-called type I seesaw model [16–20], where heavy right handed neutrinos (RHNs) are introduced to the SM and the dimension five operator is suppressed by this heavy mass scale after decoupling of the RHNs. The second mechanism is that the smallness of the coefficient is naturally explained as a result of slightly broken symmetry. This idea is realized e.g., in inverse seesaw mechanism [21,22].

The third possibility is that the operator is generated through quantum loop effect [23–34]. In this case, the suppression comes from the loop factor. For example, in a one-loop model, the coefficient gets a suppression factor of 1/(4π)² in addition to a suppression by a particle mass in the loop. In Figure 1, examples of relevant diagrams for neutrino masses are shown in several models. A recent comprehensive review on the third possibility can be found, for example, in Cai et al. [35].

Comparing to the first cases (e.g., type-I seesaw mechanism), one can find that the mass scale of new particles should be much lower in the second cases. In a case that the

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本文

T.Shindou, Frontier in Physics, 6, 159より

本⽂の構成

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実際にこのレポート・論⽂で⾏った調査，実験，

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結果のまとめと総括，今後の展望，残った課題等

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Electric dipole moments and dark matter in a CP violating MSSM

Tomohiro Abe,^1,2 Naoya Omoto,³ Osamu Seto,^4,3 and Tetsuo Shindou⁵

1Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya, Aichi 464-8602, Japan

2Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho Chikusa-ku, Nagoya, Aichi 464-8602 Japan

3Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

4Institute for International Collaboration, Hokkaido University, Sapporo 060-0815, Japan

5Division of Liberal-Arts, Kogakuin University, Nakano-machi, Hachioji, Tokyo 192-0045, Japan

(Received 24 May 2018; published 30 October 2018)

We investigate electric dipole moments (EDMs) in a CP-violating minimal supersymmetric standard model with the binolike 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 thatCP-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 ofCP-violating phases induce a difference in DM-nucleon scattering cross section by a factor.

DOI:10.1103/PhysRevD.98.075029

I. 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 interpre- tation of muon anomalous magnetic moment is one example [1–3]. SUSY models contain new sources of CP violation and/or flavor violation, which potentially induce newCPor flavor violatingphenomena. The lightestSUSY 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 ofCPviolation. Parameters in the MSSM generally pose severalCPviolating phases. It used to be regarded that the null experimental EDM results confront the MSSM with

Oð1Þ CP violating phases and Oð100Þ GeV masses of SUSY particles[6–9]. The LHC results suggest that masses of many SUSY particles are larger than Oð10Þ TeV.¹ ThereforeCP violating phases of order unity in the SUSY sector[10,11]seems still likely and worth investigating. For recent studies, see e.g., Refs. [12,13].

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[14,15], XENON1T[16], and PandaX-II[17,18]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 signifi- cantly through SU(2) gauge interaction, or (ii) annihilation cross section of binolike 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, contri- bution to the EDMs, and collider signals have been precisely studied in Ref. [19].

In this paper, we focus on another case in the later class;

a binolike neutralino DM annihilates through heavy Higgs Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

1To be precise, a scenario with SUSY particles with masses of a few TeV is still allowed. The current limit on gluino mass is around 2 TeV and the squark masses can be smaller than 3 TeV in the degenerate case.

PHYSICAL REVIEW D 98, 075029 (2018)

2470-0010=2018=98(7)=075029(14)背景知識やこの研究に⾄るまでの流れ075029-1 Published by the American Physical Society

この論⽂であつかう問題

「なぜこの論⽂を書くのか」を説明 論⽂の構成

boson resonance [20–24].² In this scenario where heavy Higgs boson resonance 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 binolike neutralino contains small Higgsino component, the neutra- lino can be searched through Higgs bosons exchange by spin- independent scattering off nucleus [26]. Masses of stops would be around 10 TeV in order to reproduce the measured SM-like Higgs boson mass (m_h ¼125GeV)[27,28]. Then, all the other SUSY particle masses and parameters except for the bino mass, the Higgsino mass parameterμ,Bμ, and stop masses can be much larger thanOð1ÞTeV. 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 inCPviolating 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 byCPviolation in neutralino DM sector with taking account ofCP-violating phase effects into the thermal DM abundance[29–34].

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, e.g., Refs. [29,32,35,36]. The magnitude of scattering off cross section between DM and nucleon is affected by the CP violating phases [32,37–42]. 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.II, we define a benchmark scenario for studying phenomenology in our DM scenario. In Sec.III, we show the results of our analysis on several EDM measurements and the spin-independent cross section. Summary and conclusion are presented in Sec.IV.

II. 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 Ref. [43]

W ¼ϵ_ab½ðy_eÞijH^a₁L^b_iE¯_jþ ðy_dÞijH^a₁Q^b_iD¯_j

þ ðyuÞijH^a₂Q^b_iU¯j−μH^a₁H^b₂&; ð2:1Þ and

Lsoft¼−M1

2 B˜B˜ −M2

2 W˜^αW˜^α−M3

2 G˜^AG˜^A

−m²_H1H^'_1aH^a₁þm²_H2H^'_2aH^a₂

−q˜^'_iLaðM²_q˜Þijq˜^a_jL−l˜^'_iLaðM²_l˜Þijl˜^a_jL

−u˜iRðM²_u˜Þiju˜^'_jR−d˜iRðM²_d˜Þijd˜^'jR−e˜iRðM²_e˜Þije˜^'_jR

−ϵab½ðTeÞijH^a₁l˜^b_iLe˜jRþ ðTdÞijH^a₁q˜^b_iLd˜jR

þ ðT_uÞijH^a₂q˜^b_iLu˜_jRþm²₃H^a₁H^b₂þH:c:&; ð2:2Þ respectively. The convention of the epsilon tensor is ϵ₁₂¼−ϵ₂₁¼1. Here, we note that gaugino mass parameter for binoM1, winoM2, and gluinoM3are in general complex.

In the following, we focus on the Yukawa couplings of the third generation quarks and leptons, so that we usey_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 M²_q˜i¼ ðM²_q˜Þii, M²_l˜

i ¼ ðM²_l˜Þii, M²_u˜i¼ ðM²_u˜Þii, M²_d˜

i ¼ ðM²_d˜Þii, andM²_e˜i¼ ðM²_e˜Þii. For the trilinear couplings, Aparameters defined byðTuÞ33¼A_τyt,ðTdÞ33¼A_τyb, and ðTeÞ33¼A_τy_τare used.

In the MSSM, the mass of the SM-like Higgs boson is expressed with some SUSY breaking parameters. In our analysis, we take tanβ≔hH₂i=hH1i¼30and we fix the stop mass parameters asMq˜₃ ¼7TeV,M˜t≔Mu˜₃¼7TeV andAt¼10TeV, then the measured SM-like Higgs boson mass mh≃125 GeV can easily reproduced [27,28]. 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. In this article, we focus on the binolike DM with the Higgs funnel scenario, where heavy Higgs boson mass is close to twice the mass of the DM so that the binolike neutralino rapidly annihilate through the heavy Higgs bosons resonance and has left with the appropriate cosmic abundance for DM. Since masses of heavier neutral Higgs bosons,m_H andm_A, are close to the charged Higgs boson massm_H( in the MSSM, we fixm_H( to be twice of bino mass parameterM1to realize resonant annihilation by the heavy Higgs bosons. In addition, theχ-˜ χ-Higgs boson˜ coupling depends on non-vanishing Higgsino component in the neutralino. Thus, both the bino mass jM1j and the Higgsino massjμjshould be of the order of TeV. We leave M₁ as a free parameter and solve jμj from the measured dark matter energy density. We summarize the parameter set in our analysis as follows:

jM₂j¼jM₃j¼10TeV; ð2:3Þ

2For a study ofCPviolation in stau coannihilation scenario, see, e.g., Ref. [25].

ABE, OMOTO, SETO, and SHINDOU PHYS. REV. D 98,075029 (2018)

075029-2

Abe et al, PRD98, 075029より

本⽂の構成

boson resonance [20 – 24]. ² In this scenario where heavy Higgs boson resonance 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 binolike neutralino contains small Higgsino component, the neutra- lino can be searched through Higgs bosons exchange by spin- independent scattering off nucleus [26]. Masses of stops would be around 10 TeV in order to reproduce the measured SM-like Higgs boson mass (m _h ¼ 125 GeV) [27,28]. 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 ð 1 Þ 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 [29 – 34].

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, e.g., Refs. [29,32,35,36]. The magnitude of scattering off cross section between DM and nucleon is affected by the CP violating phases [32,37 – 42]. 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. II, we define a benchmark scenario for studying phenomenology in our DM scenario. In Sec. III, we show the results of our analysis on several EDM measurements and the spin-independent cross section. Summary and conclusion are presented in Sec. IV .

II. 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 Ref. [43]

W ¼ ϵ _ab ½ð y _e Þ ij H ^a ₁ L ^b _i E ¯ _j þ ð y _d Þ ij H ^a ₁ Q ^b _i D ¯ _j

þ ð y _u Þ ij H ^a ₂ Q ^b _i U ¯ _j − μH ^a ₁ H ^b ₂ & ; ð 2:1 Þ

and

L _soft ¼ − M ₁

2 B ˜ B ˜ − M ₂

2 W ˜ ^α W ˜ ^α − M ₃

2 G ˜ ^A G ˜ ^A

− m ² _H

1

H ^' _1a H ^a ₁ þ m ² _H

2

H ^' _2a H ^a ₂

− q ˜ ^' _iLa ð M ² _{q ˜} Þ ij q ˜ ^a _jL − l ˜ ^' _i

_a ð M ² _{l ˜} Þ ij l ˜ ^a _jL

− u ˜ _iR ð M ² _{u ˜} Þ ij u ˜ ^' _jR − d ˜ _iR ð M ² _{d ˜} Þ ij d ˜ ^' _jR − e ˜ _iR ð M ² _{e ˜} Þ ij e ˜ ^' _jR

− ϵ _ab ½ð T _e Þ ij H ^a ₁ l ˜ ^b _iL e ˜ _jR þ ð T _d Þ ij H ^a ₁ q ˜ ^b _iL d ˜ _jR

þ ð T _u Þ ij H ^a ₂ q ˜ ^b _iL u ˜ _jR þ m ² ₃ H ^a ₁ H ^b ₂ þ H:c: & ; ð 2:2 Þ respectively. The convention of the epsilon tensor is ϵ ₁₂ ¼ − ϵ ₂₁ ¼ 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 _{q ˜}

¼ ð M ² _{q ˜} Þ ii , M ² _{l ˜}

i

¼ ð M ² _{l ˜} Þ ii , M ² _{u ˜}

¼ ð M ² _{u ˜} Þ ii , M ² _{d ˜}

i

¼ ð M ² _{d ˜} Þ ii , and M ² _{e ˜}

¼ ð M ² _{e ˜} Þ ii . For the trilinear couplings, A parameters defined by ð T _u Þ 33 ¼ A _τ y _t , ð T _d Þ 33 ¼ A _τ y _b , and ð T _e Þ 33 ¼ 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 H ₂ i = h H ₁ i ¼ 30 and we fix the stop mass parameters as M _{q ˜}

¼ 7 TeV, M _{˜ t} ≔ M _{u ˜}

¼ 7 TeV and A _t ¼ 10 TeV, then the measured SM-like Higgs boson mass m _h ≃ 125 GeV can easily reproduced [27,28]. 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. In this article, we focus on the binolike DM with the Higgs funnel scenario, where heavy Higgs boson mass is close to twice the mass of the DM so that the binolike neutralino rapidly annihilate through the heavy Higgs bosons resonance and has left with the appropriate cosmic abundance for DM. Since masses of heavier neutral Higgs bosons, m _H and m _A , are close to the charged Higgs boson mass m _H

in the MSSM, we fix m _H

to be twice of bino mass parameter M ₁ to realize resonant annihilation by the heavy Higgs bosons. In addition, the χ ˜ - χ ˜ -Higgs boson coupling depends on non-vanishing Higgsino component in the neutralino. Thus, both the bino mass j M ₁ j and the Higgsino mass j μ j should be of the order of TeV. We leave M ₁ as a free parameter and solve j μ j from the measured dark matter energy density. We summarize the parameter set in our analysis as follows:

j M ₂ j ¼ j M ₃ j ¼ 10 TeV; ð 2:3 Þ

2

For a study of CP violation in stau coannihilation scenario, see, e.g., Ref. [25].

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1. 序論

2. セットアップ・⽅法 3. 解析

4. 結果やまとめ

boson resonance [20–24].² In this scenario where heavy Higgs boson resonance 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 binolike neutralino contains small Higgsino component, the neutra- lino can be searched through Higgs bosons exchange by spin- independent scattering off nucleus [26]. Masses of stops would be around 10 TeV in order to reproduce the measured SM-like Higgs boson mass (m_h ¼ 125 GeV) [27,28]. 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 thanOð1Þ 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 byCP violation in neutralino DM sector with taking account of CP-violating phase effects into the thermal DM abundance [29–34].

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, e.g., Refs. [29,32,35,36]. The magnitude of scattering off cross section between DM and nucleon is affected by the CP violating phases [32,37–42]. 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. II, we define a benchmark scenario for studying phenomenology in our DM scenario. In Sec. III, we show the results of our analysis on several EDM measurements and the spin-independent cross section. Summary and conclusion are presented in Sec. IV.

II. 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 Ref. [43]

W ¼ ϵ_ab½ðy_eÞijH^a₁L^b_i E¯_j þ ðy_dÞijH^a₁Q^b_i D¯ _j

þ ðy_u ÞijH^a₂Q^b_i U¯ _j − μH^a₁H^b₂&; ð2:1Þ

and

L_soft ¼ −M₁

2 B˜B˜ − M₂

2 W˜ ^αW˜ ^α − M₃

2 G˜ ^AG˜ ^A

− m²_H

1H^'_1aH^a₁ þ m²_H

2H^'_2aH^a₂

− q˜^'_iLaðM²_q˜Þijq˜^a_jL − l˜^'_iLaðM²_l˜Þijl˜^a_jL

− u˜_iRðM²_u˜Þiju˜^'_jR − d˜_iRðM²_d˜Þijd˜^'_jR − e˜_iRðM²_e˜Þije˜^'_jR

− ϵ_ab½ðT_eÞijH^a₁l˜^b_iLe˜_jR þ ðT_dÞijH^a₁q˜^b_iLd˜_jR

þ ðT_u ÞijH^a₂q˜^b_iLu˜_jR þ m²₃H^a₁H^b₂ þ H:c:&; ð2:2Þ respectively. The convention of the epsilon tensor is ϵ₁₂ ¼ −ϵ₂₁ ¼ 1. Here, we note that gaugino mass parameter for binoM₁, winoM₂, and gluinoM₃are in general complex.

In the following, we focus on the Yukawa couplings of the third generation quarks and leptons, so that we usey_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 M²_q˜i ¼ ðM²_q˜Þii, M²_l˜

i ¼ ðM²_l˜Þii, M²_u˜i ¼ ðM²_u˜Þii, M²_d˜

i ¼ ðM²_d˜Þii, andM²_e˜i ¼ ðM²_e˜Þii. For the trilinear couplings, A parameters defined by ðT_u Þ33 ¼ A_τy_t, ðT_dÞ33 ¼ A_τy_b, and ðT_eÞ33 ¼ A_τy_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 tanβ ≔ hH₂i=hH₁i ¼ 30 and we fix the stop mass parameters asM_q˜3 ¼ 7 TeV,M˜t ≔ M_u˜3 ¼ 7 TeV and A_t ¼ 10 TeV, then the measured SM-like Higgs boson mass m_h ≃125 GeV can easily reproduced [27,28]. 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. In this article, we focus on the binolike DM with the Higgs funnel scenario, where heavy Higgs boson mass is close to twice the mass of the DM so that the binolike neutralino rapidly annihilate through the heavy Higgs bosons resonance and has left with the appropriate cosmic abundance for DM. Since masses of heavier neutral Higgs bosons, m_H and m_A, are close to the charged Higgs boson mass m_H( in the MSSM, we fix m_H( to be twice of bino mass parameter M₁ to realize resonant annihilation by the heavy Higgs bosons. In addition, the χ˜-χ˜-Higgs boson coupling depends on non-vanishing Higgsino component in the neutralino. Thus, both the bino mass jM₁j and the Higgsino mass jμj should be of the order of TeV. We leave M₁ as a free parameter and solve jμj from the measured dark matter energy density. We summarize the parameter set in our analysis as follows:

jM₂j ¼ jM₃j ¼ 10 TeV; ð2:3Þ

2For a study of CP violation in stau coannihilation scenario, see, e.g., Ref. [25].

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M_q˜1;2 ¼ M_u˜1;2 ¼ Md˜1;2;3 ¼ Ml˜1;2;3 ¼ M_e˜1;2;3 ¼ 100 TeV;

ð2:4Þ M_q˜3 ¼ M_˜t ¼ 7 TeV; ð2:5Þ A_t ¼ 10 TeV; ð2:6Þ m_H$ ¼ 2M₁; ð2:7Þ

tanβ ¼ 30: ð2:8Þ

The other A-terms are zero. With the above parameter set, besides the Cabibbo-Kobayashi-Maskawa (CKM) phase and the CP phases in the sfermion mass matrices, five param- eters, μ, gaugino masses M_i and A_t, may have CP phases ðϕ_μ;ϕ_M1;ϕ_M2;ϕ_M3;ϕ_AtÞ, respectively. Here, each phases of a quantity X are defined by X ¼ jXje^iϕX.

There is a rephasing degree of freedom in the MSSM.

Thus, all the physical quantities are described by the following combinations,

argðM_iM^%_jÞ; argðM_iA^%_tÞ; argðμM_iÞ;

argðμA_tÞ; ði; j ¼ 1;2;3Þ: ð2:9Þ Without loss of generality, we can take the basis ofCPphases as ϕ_M3 ¼ 0. In addition, we take ϕ_At ¼ 0 to concentrate on the CP violation in the neutralino sector as well as, technically speaking, to keepm_h ≃125 GeV avoiding com- plicated parameter dependence of the SM-like Higgs boson mass. In general, the nonzero value of ϕ_At significantly contributes to the predictions of the EDMs. However, in our parameter set given in Eqs. (2.3)–(2.8), we find the con- tribution from ϕ_At is negligible because the mass splitting between two stops is small. Therefore, we here set ϕ_At ¼ 0, and we scan the following four parameters,

ðjM₁j;ϕ_μ;ϕ_M1;ϕ_M2Þ: ð2:10Þ

III. OBSERVABLES

As we mentioned in the previous section, we choose jμj to achieve the correct DM relic density as ΩDMh² ¼ 0.1198 $ 0.0015 [44]. We use micrOMEGAs 4.3.5 [45] with

CPsuperH2.3 [46] in calculations of dark matter thermal relic density and the Higgs mass. In our benchmark point, the Higgs mass is almost fixed to be 125 GeV. There is small fluctuation of order of 0.1 GeV by scattering the param- eters. On the other hand, the calculation of the Higgs mass has theoretical uncertainty of order of a few GeV. So we consider that our benchmark points are consistent with measurements of the Higgs mass at the LHC.

With the correct DM relic abundance and the correct Higgs mass, 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.

A. EDM

The EDMs of fermions (d_f), the EDM of electron (d_e), the chromo EDM (cEDM) of quarks (d^C_q), and the Wilson coefficient of the Weinberg operator (ω) are defined by

L ⊃ −d_f i

2fσ¯ ^μνγ₅fF_μν − g_sd^C_q i

2qσ¯ ^μνγ₅qG_μν

− ω1

6f^abcG^a_μνG^bν_ρG^c_αβϵ^ρμαβ; ð3:1Þ where the convention of the epsilon tensor is ϵ⁰¹²³ ¼ þ1.

We calculate d_u, d_d, d_e, d^C_u , and d^C_d by using CPsuperH2.3

[46] implemented in micrOMEGAs 4.3.5 [45]. We use the formulae given in Ref. [47] and couplings calculated by

CPsuperH2.3 to evaluate ω.³ These EDMs and the Wilson coefficient are evaluated at the electroweak scale μ_W ¼ m_t. The neutron EDM and the mercury EDM have to be evaluated at the hadronic scale (μ_H ≃ 1 GeV). The renorm- alization group evolution from the electroweak scale to the hadronic scale is taken into account [48]. At the leading order of QCD, we find⁴

d_u

e ðμ_HÞ ¼ 0.35d_u

e ðμ_WÞ −0.17g_sðμ_WÞd^C_u ðμ_WÞ

−ð9.24874× 10⁻⁵ GeVÞωðμ_WÞ; ð3:2Þ d^C_u ðμ_HÞ ¼ 0.34g_sðμ_WÞd^C_u ðμ_WÞ þ ð0.00031 GeVÞωðμ_WÞ;

ð3:3Þ d^e_d

e ðμ_HÞ ¼ 0.40d_u

e ðμ_WÞ þ 0.098g_sðμ_WÞd^C_dðμ_WÞ

þ ð0.00010 GeVÞωðμ_WÞ; ð3:4Þ d^C_d ðμ_HÞ ¼ 0.38g_sðμ_WÞd^C_dðμ_WÞ þ ð0.00070 GeVÞωðμ_WÞ;

ð3:5Þ ωðμ_HÞ ¼ 0.39ωðμ_WÞ: ð3:6Þ 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,

3CPsuperH2.3 also calculate the Wilson coefficient of the Weinberg operator, but it returns very unstable numbers during the scanning the parameter space because of the loss of significant digits.

4The definition of the cEDM in the ^CPsuperH2.3 is different from ours, d^C_qjCPSUPERH2.3 ¼ g_sd^C_q.

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本論

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Abe et al, PRD98, 075029より