Eﬀect of CP violation in the singlet-doublet dark …seminar/pdf_2017_kouki/...Eﬀect of CP violation in the singlet-doublet dark matter model Tomohiro Abe Institute for Advanced

Effect of CP violation in the singlet-doublet dark matter model

Tomohiro Abe Institute for Advanced Research Nagoya UniversityKobayashi-Maskawa Institute Nagoya University

!!!!!!

based on PLB 771 (2017) 125-130 (arXiv:1702.07236)

Seminar @ Osaka U. Tomohiro Abe (IAR, KMI Nagoya U) 1

https://arxiv.org/abs/1607.03706


Dark matter

[Planck Mission]

What we know ★ gravity is only the long range force for DM (no QED interaction) ★ short range force should be very weak (no QCD interaction) ★ no candidates in the SM (need new physics) ★ stable (or life time > age of the universe) (symmetry?) ★ energy density


WIMP dark matter

Features of WIMP (Weakly Interacting Massive Particle) • weakly interacting to the SM • mass = GeV ~ TeV • Freeze out scenario • correlation among various observables

DM

DM SM

SM

annihilation（thermal relic, indirect detection）

pair production（LHC)

scattering（

direct detection)


What we do in WIMP studies

calculate energy density • calculate annihilation cross section of DM, <σv> • substitute it into Boltzmann equation • solve it and get energy density • we can determine one parameter in the model

choose a model

dn

dt+ 3Hn = �h�vi

�n2 � n2

eq

�

Boltzmann suppression • E ~ mDM is dominant • DM is non-relativistic in annihilation process

h�vi /Z

|M12!34|2 e�E1/T e�E2/T d(phase space)


What we do in WIMP studies

calculate energy density • calculate annihilation cross section of DM, <σv> • substitute it into Boltzmann equation • solve it and get energy density • we can determine one parameter in the model

study direct detection • DM-nucleon scattering • compare with experiments

study other observables • indirect detection • LHC phenomenology • flavor physics • …

choose a model

DM DM

nuclear recoil • scintillation light • recoil energy <~ 100 keV

nuclear


)2WIMP mass (GeV/c10 210 310

)2SD

WIM

P-ne

utro

n cr

oss s

ectio

n (c

m-4210

-4110

-4010

-3910

-3810

-3710

-3610

-3510

-3410PandaX-II (this work)LUXXENON100CMSATLAS

=0.25DM=g

qg

=1.45DM=g

qg

=1DM=0.25, g

qg

[PandaX-II Collaboration, Phys. Rev. Lett. 118, 071301 (2017)][PandaX-II Collaboration, 1708.06917]

6

)2WIMP mass (GeV/c10 210 310 410

)2W

IMP-

nucl

eon

cros

s sec

tion

(cm

46−10

45−10

44−10

This work

LUX 2017

XENON1T 2017

PandaX-II 2016

FIG. 5: The 90% C.L. upper limits for the spinindependent WIMP-nucleon elastic cross sections fromthe combined PandaX-II Run 9 and Run 10 data (red),overlaid with that from PandaX-II 2016 [3] (blue), LUX2017 [2] (magenta), and XENON1T 2017 [4] (black).The green band represents the ±1� sensitivity band.

(No. 2016YFA0400301). We thank the support ofgrants from the O�ce of Science and Technology, Shang-hai Municipal Government (No. 11DZ2260700, No.16DZ2260200), and the support from the Key Labora-tory for Particle Physics, Astrophysics and Cosmology,Ministry of Education. This work is supported in partby the Chinese Academy of Sciences Center for Excel-lence in Particle Physics (CCEPP) and Hongwen Foun-dation in Hong Kong. We also would like to thank Dr.Xunhua Yuan and Chunfa Yao of China Iron and SteelResearch Institute Group, and Taiyuan Iron and Steel(Group) Co. LTD for crucial help on low backgroundstainless steel. Finally, we thank the following organiza-tions for indispensable logistics and other supports: theCJPL administration and the Yalong River HydropowerDevelopment Company Ltd.

⇤ Spokesperson: [email protected]

† Corresponding author: [email protected]‡ Corresponding author: [email protected]§ Corresponding author: [email protected]

[1] G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405,279 (2005), arXiv:hep-ph/0404175 [hep-ph].

[2] D. S. Akerib et al. (LUX), Phys. Rev. Lett. 118, 021303(2017), arXiv:1608.07648 [astro-ph.CO].

[3] A. Tan et al. (PandaX-II), Phys. Rev. Lett. 117, 121303(2016), arXiv:1607.07400 [hep-ex].

[4] E. Aprile et al. (XENON), (2017), arXiv:1705.06655[astro-ph.CO].

[5] E. A. Bagnaschi et al., Eur. Phys. J. C75, 500 (2015),arXiv:1508.01173 [hep-ph].

[6] J. Liu, X. Chen, and X. Ji, Nature Phys. 13, 212 (2017).[7] K. J. Kang, J. P. Cheng, Y. H. Chen, Y. J. Li, M. B.

Shen, S. Y. Wu, and Q. Yue, Proceedings, 11th Inter-national Conference on Topics in astroparticle and un-derground physics in Memory of Julio Morales (TAUP2009): Rome, Italy, July 1-5, 2009, J. Phys. Conf. Ser.203, 012028 (2010).

[8] D. S. Akerib et al. (LUX), Phys. Rev. D93, 072009(2016), arXiv:1512.03133 [physics.ins-det].

[9] http://www.caen.it/servlet/checkCaenManualFile?Id=12364.

[10] Supplemental Material, see https://pandax.sjtu.edu.

cn/articles/2nd/supplemental.pdf, which includesRefs. [2], [3], [4] and [24].

[11] Q. Wu et al., (2017), arXiv:1707.02134 [physics.ins-det].[12] S. Agostinelli et al. (GEANT4), Nucl. Instrum. Meth.

A506, 250 (2003).[13] J. Allison et al., IEEE Trans. Nucl. Sci. 53, 270 (2006).[14] B. Lenardo, K. Kazkaz, M. Szydagis, and M. Tripathi,

IEEE Trans. Nucl. Sci. 62, 3387 (2015), arXiv:1412.4417[astro-ph.IM].

[15] C. H. Faham, V. M. Gehman, A. Currie, A. Dobi,P. Sorensen, and R. J. Gaitskell, JINST 10, P09010(2015), arXiv:1506.08748 [physics.ins-det].

[16] M. Xiao et al. (PandaX), Sci. China Phys. Mech. Astron.57, 2024 (2014), arXiv:1408.5114 [hep-ex].

[17] X. Xiao et al. (PandaX), Phys. Rev. D92, 052004 (2015),arXiv:1505.00771 [hep-ex].

[18] G. Cowan, K. Cranmer, E. Gross, and O. Vitells,Eur. Phys. J. C71, 1554 (2011), [Erratum: Eur. Phys.J.C73,2501(2013)], arXiv:1007.1727 [physics.data-an].

[19] E. Aprile et al. (XENON100), Phys. Rev. D84, 052003(2011), arXiv:1103.0303 [hep-ex].

[20] G. J. Feldman and R. D. Cousins, Phys. Rev. D57, 3873(1998), arXiv:physics/9711021 [physics.data-an].

[21] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, (2011),arXiv:1105.3166 [physics.data-an].

[22] A. L. Read, J. Phys. G28, 2693 (2002).[23] T. Junk, Nucl. Instrum. Meth. A434, 435 (1999),

arXiv:9902006 [hep-ex].[24] X. Ji, in TeV Particle Astrophysics 2017 (Columbus,

Ohio, US, 2017) https://tevpa2017.osu.edu/talks/

ji.pdf.

spin-independent spin-dependent

Direct detection


10 Direct Detection Program Roadmap 39

1 10 100 1000 10410!5010!4910!4810!4710!4610!4510!4410!4310!4210!4110!4010!39

10!1410!1310!1210!1110!1010!910!810!710!610!510!410!3

WIMP Mass !GeV"c2#

WIMP!nucleoncrosssection!cm2 #

WIMP!nucleoncrosssection!pb#

7BeNeutrinos

NEUTRINO C OHER ENT SCATTERING

NEUTRINO COHERENT SCATTERING

(Green&ovals)&Asymmetric&DM&&(Violet&oval)&Magne7c&DM&(Blue&oval)&Extra&dimensions&&(Red&circle)&SUSY&MSSM&&&&&&MSSM:&Pure&Higgsino&&&&&&&MSSM:&A&funnel&&&&&&MSSM:&BinoEstop&coannihila7on&&&&&&MSSM:&BinoEsquark&coannihila7on&&

8BNeutrinos

Atmospheric and DSNB Neutrinos

CDMS II Ge (2009)

Xenon100 (2012)

CRESST

CoGeNT(2012)

CDMS Si(2013)

EDELWEISS (2011)

DAMA SIMPLE (2012)

ZEPLIN-III (2012)COUPP (2012)

SuperCDMS Soudan Low ThresholdSuperCDMS Soudan CDMS-lite

XENON 10 S2 (2013)CDMS-II Ge Low Threshold (2011)

SuperCDMS Soudan

Xenon1T

LZ

LUX

DarkSide G2

DarkSide 50

DEAP3600

PICO250-CF3I

PICO250-C3F8

SNOLAB

SuperCDMS

Figure 26. A compilation of WIMP-nucleon spin-independent cross section limits (solid curves), hintsfor WIMP signals (shaded closed contours) and projections (dot and dot-dashed curves) for US-led directdetection experiments that are expected to operate over the next decade. Also shown is an approximateband where coherent scattering of 8B solar neutrinos, atmospheric neutrinos and di↵use supernova neutrinoswith nuclei will begin to limit the sensitivity of direct detection experiments to WIMPs. Finally, a suite oftheoretical model predictions is indicated by the shaded regions, with model references included.

We believe that any proposed new direct detection experiment must demonstrate that it meets at least oneof the following two criteria:

• Provide at least an order of magnitude improvement in cross section sensitivity for some range ofWIMP masses and interaction types.

• Demonstrate the capability to confirm or deny an indication of a WIMP signal from another experiment.

The US has a clear leadership role in the field of direct dark matter detection experiments, with mostmajor collaborations having major involvement of US groups. In order to maintain this leadership role, andto reduce the risk inherent in pushing novel technologies to their limits, a variety of US-led direct search

Community Planning Study: Snowmass 2013

Snowmass [1310.8327]

Direct detection


good points • simple • predictable (one parameter can be determined from <σv> )

current status of WIMP

weak points • predict direct detection signals, but no signal yet • simple models can be easily excluded • many experiments are on going and planed • constraints is getting stronger and stronger

two choices (1) give up WIMP scenario (2) consider ideas to avoid the constraints from the direct detections

Seminar @ Osaka U. Tomohiro Abe (IAR, KMI Nagoya U)

In this talk

9

Discuss a model that • is in WIMP scenario (freeze out scenario) • can avoid the strong constraints from direct detection • is testable from experiments other than direct detections

keywords • fermion DM • scalar mediator • pseudo-scalar interaction


Plan

10

•brief review of WIMP scenario

• pseudo-scalar interaction

• singlet-doublet model

•Summary


Plan

11




•Summary


fermion DM with Pseudo-scalar coupling

If DM has a pseudo-scalar interaction, then

L � ig�5 a ψ = DM, a = mediator (scalar)

Suppression of the direct detection Annihilation cross section is not suppressed

We can avoid constraints from direct detection while keeping WIMP scenario

q

SM

DM

SM

DM

/ u(p)�5u(p) ' ~q · ~�

DM

DM

SM

SM

/ v(p)�5u(p) ' mDM

12

=X

s

Zd3p

(2⇡)3p

2Ep

�ap,s

us

(p)e�ipx + b†p,s

vs

(p)eipx�


How to realize pseudo-scalar interaction

pseudo-scalar interaction

DMi�5 DMh

h

DM

DM

SM

SM

CPV

CP violation in dark sector • CPV can generate pseudo-scalar int. • mediator scalar can be CP-even

( )Another way is to introduce CP-odd mediator • models require CPV in scalar sector • or models are complicated • σSI can be generated by loop effects

[Ghorbani (2015); Baek-Ko-Li (2017)]

[Berlin-Gori-Lin-Wang (2015)]

[Arcadi et.al. (1711.02110)]


Plan

14




•Summary


Singlet-doublet DM modelMahbubani, Senatore [hep-ph/0510064]D’Eramo [0705.4493] Enberg et.al. [arXiv:0706.0918]…

SU(3)c SU(2)L U(1)Y Z2

! 1 1 0 -1⌘ 1 2 1/2 -1⇠† 1 2 1/2 -1

Lint. = �M1

2!! �M2⇠⌘ � y!H†⌘ � y0⇠H! + (h.c.)

� = Arg(M⇤1M

⇤2 yy

0)(|M1|, |M2|, |y|, |y0|,�)

There is a CP phase naturally • four complex parameters • three phases are unphysical, eliminated by the rotation of the three Weyl fermions

five parameters:

New particles (three Z2 odd Weyl fermions)

Mass terms and Yukawa interactions with the SM Higgs



Mass matrix for Z2 odd fermions

phase, and we explicitly write down the phase as M2e�i�. In this basis, all the parameters except

� are positive. After the Higgs field develops a vacuum expectation value (VEV), we find the

following mass terms

Lmass = �1

2

⇣! ⌘0 ⇠0

⌘

0

BBB@

M1vp2y vp

2y0

vp2y 0 M2e

�i�

vp2y0 M2e

�i� 0

1

CCCA

0

BBB@

!

⌘0

⇠0

1

CCCA�M2⇠

�⌘+ + (h.c.), (2.2)

where v is the VEV of the Higgs boson, v ' 246 GeV. We introduce �, ✓, and r for later

convenience,

y = � sin ✓, y0 = � cos ✓, r =y

y0= tan ✓. (2.3)

The mass of the charged Dirac fermion is M2. After we diagonalize the mass matrix, we obtain

three neutral Weyl fermions (�01,�

02,�

03) that are related to (!, ⌘0, ⇠0) by a unitary matrix V as

follows.0

BBB@

!

⌘0

⇠0

1

CCCA=

0

BBB@

V11 V12 V13

V21 V22 V23

V31 V32 V33

1

CCCA

0

BBB@

�01

�02

�03

1

CCCA, (2.4)

where �01 is the lightest neutral Z2-odd field and thus is the DM candidate in this model.

It is su�cient to study � in the range 0 � ⇡, although �⇡ < � ⇡ in general. Physics in

the negative � regime is related to physics in the positive � regime by the complex conjugate of

the mass matrix, and thus of the mixing matrix. If observables respect the CP symmetry, they do

not depend on the sign of �. All the processes for the relic abundance and the direct detection are

independent of the sign of � because they respect the CP symmetry. While the CP violating EDM

depends on the sign of �, the sign of � merely changes the sign of the EDM, so we focus only on

the positive � region.

The ratio of the two Yukawa couplings is important as we will see below. We denote it by

r = y/y0 and focus on 0 < r 1. Since both Yukawa couplings are positive, r is positive

definite. For r = 0, we can eliminate � by the redefinition of the fermion fields. We do not discuss

that situation because we aim to examine the CP violation e↵ect in the dark sector. Physics for

1 < r < 1 is equivalent to physics for 0 < r < 1 by renaming ⌘0 and ⇠0 as ⇠0 and ⌘0 respectively

as can be seen from the mass matrix. Therefore it is su�cient to discuss for 0 < r 1.

3

Mixing Majorana fermion with Dirac fermions (three neutral Majorana fermions)

one charged Dirac fermion

new particles are • 3 neutral Majorana fermions (← the lightest one is DM) • 1 charged Dirac fermions



Mixing matrix

gauge eigenstates

phase, and we explicitly write down the phase as M2e�i�. In this basis, all the parameters except

� are positive. After the Higgs field develops a vacuum expectation value (VEV), we find the

following mass terms

Lmass = �1

2

⇣! ⌘0 ⇠0

⌘

0

BBB@

M1vp2y vp

2y0

vp2y 0 M2e

�i�

vp2y0 M2e

�i� 0

1

CCCA

0

BBB@

!

⌘0

⇠0

1

CCCA�M2⇠

�⌘+ + (h.c.), (2.2)

where v is the VEV of the Higgs boson, v ' 246 GeV. We introduce �, ✓, and r for later

convenience,

y = � sin ✓, y0 = � cos ✓, r =y

y0= tan ✓. (2.3)

The mass of the charged Dirac fermion is M2. After we diagonalize the mass matrix, we obtain

three neutral Weyl fermions (�01,�

02,�

03) that are related to (!, ⌘0, ⇠0) by a unitary matrix V as

follows.0

BBB@

!

⌘0

⇠0

1

CCCA=

0

BBB@

V11 V12 V13

V21 V22 V23

V31 V32 V33

1

CCCA

0

BBB@

�01

�02

�03

1

CCCA, (2.4)

where �01 is the lightest neutral Z2-odd field and thus is the DM candidate in this model.

It is su�cient to study � in the range 0 � ⇡, although �⇡ < � ⇡ in general. Physics in

the negative � regime is related to physics in the positive � regime by the complex conjugate of

the mass matrix, and thus of the mixing matrix. If observables respect the CP symmetry, they do

not depend on the sign of �. All the processes for the relic abundance and the direct detection are

independent of the sign of � because they respect the CP symmetry. While the CP violating EDM

depends on the sign of �, the sign of � merely changes the sign of the EDM, so we focus only on

the positive � region.

The ratio of the two Yukawa couplings is important as we will see below. We denote it by

r = y/y0 and focus on 0 < r 1. Since both Yukawa couplings are positive, r is positive

definite. For r = 0, we can eliminate � by the redefinition of the fermion fields. We do not discuss

that situation because we aim to examine the CP violation e↵ect in the dark sector. Physics for

1 < r < 1 is equivalent to physics for 0 < r < 1 by renaming ⌘0 and ⇠0 as ⇠0 and ⌘0 respectively

as can be seen from the mass matrix. Therefore it is su�cient to discuss for 0 < r 1.

3

mass eigenstates

2.1 couplings

The Z2-odd fermions couple to the Higgs boson and the gauge bosons. We can obtain the inter-

action terms by diagonalizing the mass matrix and going to the mass eigenbasis. Four component

notation is useful in calculations. The mass eigenstates of the charged and neutral Z2 odd particles

in four component notation are given by

+ =

0

@ ⌘+

(⇠�)†

1

A , j =

0

@ �0j

�0j†

1

A . (2.5)

The interaction terms including Z2-odd particles are2

Lint. ��X

j

+�µ(PLc

L�j

+ PRcR�j) jW

+µ

�X

j

j�µ(PL(c

L�j)⇤ + PR(c

R�j)⇤) +W

�µ

�✓

e

2sW cW(c2W � s2W )Zµ + eAµ

◆ +�

µ +

� 1

2

X

j,k

cZ�j�k j�µ�5 kZµ +

1

2

X

j,k

cpZ�j�k ji�

µ kZµ

� 1

2

X

j,k

ch�j�k j kh+

1

2

X

j,k

cph�j�k ji�5 kh, (2.6)

where

cL�j=

ep2sW

V2j , (2.7)

cR�j=

ep2sW

V ⇤3j , (2.8)

cZ�j�k =e

2sW cWRe(V ⇤

2jV2k � V ⇤3jV3k), (2.9)

cpZ�j�k=

e

2sW cWIm(V ⇤

2jV2k � V ⇤3jV3k), (2.10)

ch�j�k=

p2Re(yV1jV2k + y0V1jV3k), (2.11)

cph�j�k=

p2Im(yV1jV2k + y0V1jV3k). (2.12)

Among these terms, the following interactions are particularly important for our discussion.

�1

2cZ�1�1 1�

µ�5 1Zµ � 1

2ch�1�1 1 1h+

1

2cph�1�1

1i�5 1h. (2.13)

2We have checked these interaction terms are consistent with Ref. [23, 24] up to conventions of the fields and the

gauge couplings.

4

Important interaction terms

2.1 couplings





+ =

0

@ ⌘+

(⇠�)†

1

A , j =

0

@ �0j

�0j†

1

A . (2.5)


Lint. ��X

j

+�µ(PLc

L�j

+ PRcR�j) jW

+µ

�X

j

j�µ(PL(c

L�j)⇤ + PR(c

R�j)⇤) +W

�µ

�✓

e


◆ +�

µ +

� 1

2

X

j,k


1

2

X

j,k

cpZ�j�k ji�

µ kZµ

� 1

2

X

j,k

ch�j�k j kh+

1

2

X

j,k


where

cL�j=

ep2sW

V2j , (2.7)

cR�j=

ep2sW

V ⇤3j , (2.8)

cZ�j�k =e

2sW cWRe(V ⇤

2jV2k � V ⇤3jV3k), (2.9)

cpZ�j�k=

e

2sW cWIm(V ⇤

2jV2k � V ⇤3jV3k), (2.10)

ch�j�k=


cph�j�k=



�1

2cZ�1�1 1�

µ�5 1Zµ � 1

2ch�1�1 1 1h+

1

2cph�1�1

1i�5 1h. (2.13)


gauge couplings.

4

2.1 couplings





+ =

0

@ ⌘+

(⇠�)†

1

A , j =

0

@ �0j

�0j†

1

A . (2.5)


Lint. ��X

j

+�µ(PLc

L�j

+ PRcR�j) jW

+µ

�X

j

j�µ(PL(c

L�j)⇤ + PR(c

R�j)⇤) +W

�µ

�✓

e


◆ +�

µ +

� 1

2

X

j,k


1

2

X

j,k

cpZ�j�k ji�

µ kZµ

� 1

2

X

j,k

ch�j�k j kh+

1

2

X

j,k


where

cL�j=

ep2sW

V2j , (2.7)

cR�j=

ep2sW

V ⇤3j , (2.8)

cZ�j�k =e

2sW cWRe(V ⇤

2jV2k � V ⇤3jV3k), (2.9)

cpZ�j�k=

e

2sW cWIm(V ⇤

2jV2k � V ⇤3jV3k), (2.10)

ch�j�k=


cph�j�k=



�1

2cZ�1�1 1�

µ�5 1Zµ � 1

2ch�1�1 1 1h+

1

2cph�1�1

1i�5 1h. (2.13)


gauge couplings.

4

DM

2.1 couplings





+ =

0

@ ⌘+

(⇠�)†

1

A , j =

0

@ �0j

�0j†

1

A . (2.5)


Lint. ��X

j

+�µ(PLc

L�j

+ PRcR�j) jW

+µ

�X

j

j�µ(PL(c

L�j)⇤ + PR(c

R�j)⇤) +W

�µ

�✓

e


◆ +�

µ +

� 1

2

X

j,k


1

2

X

j,k

cpZ�j�k ji�

µ kZµ

� 1

2

X

j,k

ch�j�k j kh+

1

2

X

j,k


where

cL�j=

ep2sW

V2j , (2.7)

cR�j=

ep2sW

V ⇤3j , (2.8)

cZ�j�k =e

2sW cWRe(V ⇤

2jV2k � V ⇤3jV3k), (2.9)

cpZ�j�k=

e

2sW cWIm(V ⇤

2jV2k � V ⇤3jV3k), (2.10)

ch�j�k=


cph�j�k=



�1

2cZ�1�1 1�

µ�5 1Zµ � 1

2ch�1�1 1 1h+

1

2cph�1�1

1i�5 1h. (2.13)


gauge couplings.

4

2.1 couplings





+ =

0

@ ⌘+

(⇠�)†

1

A , j =

0

@ �0j

�0j†

1

A . (2.5)


Lint. ��X

j

+�µ(PLc

L�j

+ PRcR�j) jW

+µ

�X

j

j�µ(PL(c

L�j)⇤ + PR(c

R�j)⇤) +W

�µ

�✓

e


◆ +�

µ +

� 1

2

X

j,k


1

2

X

j,k

cpZ�j�k ji�

µ kZµ

� 1

2

X

j,k

ch�j�k j kh+

1

2

X

j,k


where

cL�j=

ep2sW

V2j , (2.7)

cR�j=

ep2sW

V ⇤3j , (2.8)

cZ�j�k =e

2sW cWRe(V ⇤

2jV2k � V ⇤3jV3k), (2.9)

cpZ�j�k=

e

2sW cWIm(V ⇤

2jV2k � V ⇤3jV3k), (2.10)

ch�j�k=


cph�j�k=



�1

2cZ�1�1 1�

µ�5 1Zµ � 1

2ch�1�1 1 1h+

1

2cph�1�1

1i�5 1h. (2.13)


gauge couplings.

4

couplings

4-component notation


Three important couplings

DM

DM

h

DM

DM

Z

= �i�ch�1�1 � i�5cPh�1�1

�

= i�µ�5cZ�1�1

• contributes to direct detection (σSI)• can be zero by parameter choice (blind spot)

• contributes to direct detection (σSD)• becomes zero if y = y’

all the fermions we introduced in the above are odd. The lightest neutral Z2-odd fermion is a DM

candidate.

The mass and Yukawa interaction terms for the Z2 odd particles are given by

Lint. =− M1

2ωω −M2e

−iφξη − yωH†η − y′ξHω + (h.c.). (2.1)

All the parameters have phases in general but we can eliminate three of them after the redifinition

of fermion fields. We work in the base where only the Dirac mass term has a phase and we explicitly

write down the phase as M2e−iφ. In this base, all the parameter except φ are positive. After the

Higgs field develops a vacuum expectation value (VEV), we find the following mass terms

Lmass = −1

2

(ω η0 ξ0

)

⎛

⎜⎜⎜⎝

M1v√2y v√

2y′

v√2y 0 M2e−iφ

v√2y′ M2e−iφ 0

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎝

ω

η0

ξ0

⎞

⎟⎟⎟⎠−M2ξ

−η+ + (h.c.), (2.2)

where v is the Higgs boson VEV, v ≃ 246 GeV. We introduce λ, θ, and r for later conveience as

y = λ sin θ, y′ = λ cos θ, r =y

y′= tan θ. (2.3)

We find that the mass of the charged Dirac fermion is M2. After we diagonalize the mass matrix,

we obtain three neutral Weyl fermions (χ01,χ

02,χ

03) that are related by a unitary matrix V as follows.

⎛

⎜⎜⎜⎝

ω

η0

ξ0

⎞

⎟⎟⎟⎠=

⎛

⎜⎜⎜⎝

V11 V12 V13

V21 V22 V23

V31 V32 V33

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎝

χ01

χ02

χ03

⎞

⎟⎟⎟⎠, (2.4)

where χ01 is the lightest neutral Z2-odd fields and thus is the DM candidate in this model.

It is sufficient to study the range of φ as 0 ≤ φ ≤ π, although −π < φ ≤ π in general. Physics

in the negative φ regime is related to physics in the positive φ regime by the complex conjugate of

the mass matrix and thus of the mixing matrix. If observables respect the CP symmetry, they do

not depend on the sign of φ. All the processes for the relic abundance and the cross sections of the

dark matter to neucleons are independent of the sign of φ because they respect the CP symmetry.

The EDM is an observable to measure CP violation and thus depends on the sign of φ. However,

the effect of changing the sign of φ is just to change the sign of EDM. Therefore we only focus on

the positive φ region.

The ratio of the two Yukawa coupling is important as we will see below. We denote it by

r = y/y′ and focus on 0 < r ≤ 1. Since both of two Yukawa couplings are positive, r is positive

definite. For r = 0, we can erase φ by the redefinition of the fermion feilds. We do not discuss

3

• does not contribute to direct detection• becomes zero if no CPV phase in dark sector

Cohen, Kearney, Pierce and Tucker-Smith [1109.2604]Cheung, Hall, Pinner and Ruderman [1211.4873]Cheung and Sanford [1311.5896]

We can avoid the direct detection constraints while keeping WIMP scenario if cZχχ = chχχ = 0 and chχχP ≠ 0 .


Another effect of CP violation in this model

prediction of the electric dipole moment (EDM) ★ EDM is predicted via Barr-Zee type diagram

complementary to direct detection ★ In large CPV case, σSI is small, EDM is large ★ In small CPV case, σSI is large, EDM is small ★ EDM helps to test this model in the region where direct

detection signal is not expected

Figure 1: Barr-Zee type contributions to the EDM. The Z2-odd charged and neutral fermions run in the

triangle part.

0, we find

0 =m4DM �m2

DM

✓M2

2 +v2�2

2�M1M2 sin 2✓ cos�

◆�M2

2 sin 2✓

✓M1M2 cos�� v2

2�2 sin 2✓

◆.

(2.15)

Using Eqs. (2.14) and (2.15), we can obtain two relations. For example, we can solve for mDM and

�,

mDM =

0

B@M2

1M22 sin

2 2✓ sin2 �

M21 +M2

2 sin2 2✓ + 2M1M2 sin 2✓ cos�

1

CA

1/2

, (2.16)

� =

s2(M2

2 �m2DM)(m2

DM +M1M2 sin 2✓ cos�)

v2(M22 sin

2 2✓ �m2DM)

. (2.17)

This is the blind spot condition where ch�1�1 = 0. Eq. (2.16) is given in Ref. [23]

For � = 0, there is no blind spot because it requires mDM = 0. For � = ⇡, mDM is non-zero if

the denominator of Eq. (2.16) is zero and we find the following blind spot condition for � = ⇡,

mDM =M1 = M2 sin 2✓. (2.18)

This is the same as the blind spot condition given in Refs. [22, 24].3

2.3 EDM

This model predicts a new contribution to the EDM because of the new source of CP violation

�. The electron EDM is an important complement to direct detection experiments as we will see

in Sec. 3. We summarize the formulae here. The leading contribution comes from the Barr-Zee

type diagram shown in Fig. 1. The electron EDM is defined through

3Note that mDM = M2 sin 2✓ = �M2 cos⇡ sin 2✓, and sin 2✓ in Refs. [22, 24] corresponds to cos� sin 2✓ in our

notation.

6

×CPV�± �±

�0j

` `


setup • simple • CP phase naturally arises

important points of this model

three important couplings • different features to direct detection • We need at least one coupling

to realize WIMP scenario • We can avoid the direct detection constraints while keeping

WIMP scenario if cZχχ = chχχ = 0 and chχχP ≠ 0 .

ch�1�1 cPh�1�1cZ�1�1

h�vi Yes Yes Yes

�SI Yes No No

�SD No No Yes

SU(3)c SU(2)L U(1)Y Z2

! 1 1 0 -1⌘ 1 2 1/2 -1⇠† 1 2 1/2 -1

testable • EDM is complimentary to direct detection

Numerical Results


Result 1 : all couplings exist case (1/2)chχχ ≠ 0, cZχχ ≠ 0, chχχP ≠ 0 (M2 = 1000GeV, y/y’ = 0.5)

pink : excluded by σSIorange : excluded by σSDcontours: electron EDM

electron EDM (de) ★ current bound : |de| < 8.7 x 10-29 e cm

★ future prospect: |de| < O(10-30) e cm

[prospects for LZ is given in 1310.8327]

ACME experiment [1310.7534]

[1208.4507, 1502.04317, …]



0.03

0.03

0.03

0.03

0.01

0.01

0.010.001

0.001

0.01

Result 1 : all couplings exist case (2/2)chχχ values


Result 2 : chχχ = 0 casechχχ = 0, cZχχ ≠ 0, chχχP ≠ 0 (y/y’ = 0.5)



★ future prospect: |de| < O(10-30) e cmACME experiment [1310.7534]

[1208.4507, 1502.04317, …]


Result 3 : only pseudo-scalar coupling casechχχ = 0, cZχχ = 0, chχχP ≠ 0 (y/y’ = 1)

1.µ 10-30

1.µ 10-30

1.µ 10-29

1.µ 10-29

2.µ 10-29

2.µ 10-29

3.µ 10-29

200 400 600 800 10000.5

0.6

0.7

0.8

0.9

1.0

mDM@GeVD

fêp

Hblind spot, r=1L

In the gray region, the dark matter density cannot be the measured value.



★ future prospect: |de| < O(10-30) e cmACME experiment [1310.7534]

[1208.4507, 1502.04317, …]


Results

This model • is compatible with the current constraints

• can avoid the constraints from the direct detection

• can keep the WIMP scenario

• is testable thanks to the correlation between

“the direct detections“ and “EDM”

Summary


Summary

Effect of CP violation ★ generate pseudo-scalar interaction ★ avoid strong constraints from direct detection ★ keep DM annihilation cross section for WIMP scenario

CPV in singlet-doublet DM model ★ CP violation naturally arises ★ constraints from direct detection is avoidable ★ prediction of EDM > 10-30 [e cm], within future prospect ★ direct detection and EDM complementary to each other

Documents

Eﬀect of CP violation in the singlet-doublet dark …seminar/pdf_2017_kouki/...Eﬀect of CP violation in the singlet-doublet dark matter model Tomohiro Abe Institute for Advanced