750 GeV diphoton resonance and electric dipole moments

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

750 GeV

diphoton

resonance

and

electric

dipole

moments

Kiwoon Choi

a

,

Sang

Hui Im

a

,

Hyungjin Kim

a

,

b

,

∗

,

Doh

Young Mo

a a_Center_for_Theoretical_Physics_of_the_Universe,_Institute_for_Basic_Science_(IBS),_Daejeon_34051,_Republic_of_Korea b_Department_of_Physics,_KAIST,_Daejeon_34141,_Republic_of_Korea

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received20June2016

Receivedinrevisedform20July2016 Accepted21July2016

Availableonline26July2016 Editor:J.Hisano

We examine theimplicationoftherecentlyobserved 750 GeVdiphotonexcessfortheelectricdipole moments ofthe neutronand electron.Iftheexcessisduetoaspinzeroresonancewhichcouplesto photons and gluonsthrough the loopsofmassive vector-likefermions, the resultingneutron electric dipole moment can becomparable to thepresent experimental boundif theCP-violating angle

α

in the underlyingnewphysicsisofO(10−1₎_._An_electron_EDM_comparable_to_the _present_bound_can_be achieved throughamixing betweenthe 750 GeVresonance and the Standard Model Higgsboson, if themixingangleitselfforanapproximatelypseudoscalarresonance,orthemixingangletimesthe CP-violatingangle

α

foranapproximatelyscalarresonance,isofO(10−3₎_._For_the_case_that_the_{750 GeV} resonancecorrespondstoacompositepseudo-Nambu–GoldstonebosonformedbyaQCD-likehypercolor dynamicsconﬁningatHC,theresultingneutronEDMcanbeestimatedwith

α

∼ (750 GeV/HC)2θHC, whereθHCisthehypercolorvacuumangle.

1. Introduction

RecentlytheATLASandCMScollaborationsreportedanexcess ofdiphotoneventsattheinvariantmassmγ γ

750 GeV withthe local signiﬁcance 3.6

σ

and 2.6

σ

, respectively [1,2]. The anal-ysis was updated later, yielding an increased local signiﬁcance, 3.9

σ

and3.4

σ

,respectively

[3,4]

.Ifthesignal persists,thiswill be an unforeseen discovery of new physics beyondthe Standard Model (SM). So one can ask now what would be the possible phenomenologyotherthanthediphotonexcess,whichmayresult fromthenewphysicstoexplainthe750 GeVdiphotonexcess.

With the presently available data,one simple scenario to ex-plain the diphoton excess is a SM-singlet spin zero resonance S which couples to massive vector-like fermions carryingnon-zero SM gauge charges[5–8]. Inthisscenario, the750 GeV resonance interacts with the SM sector dominantly through the SM gauge ﬁeldsandpossibly alsothroughthe Higgsboson. Insuch case, if thenewphysicssectorinvolvesaCP-violatinginteraction,the elec-tricdipolemoment(EDM)oftheneutronorelectronmayprovide themostsensitiveprobeofnewphysicsinthelowenergylimit.

*

Correspondingauthor.

E-mailaddresses:kchoi@ibs.re.kr(K. Choi),shim@ibs.re.kr(S.H. Im), hjkim06@kaist.ac.kr(H. Kim),modohyoung@ibs.re.kr(D.Y. Mo).

More explicitly, after integrating out the massive vector-like fermions,theeffectivelagrangianmayinclude

κ

s 2S F aμν_Fa μν

+

κ

p 2 S F aμν_F

_˜

a μν

+

dW 3 fabcF a μρF bρ ν

Fcμν

+ ...,

(1) where Fa

μν denotes the SM gauge ﬁeld strength and F

˜

μνa

=

1

2

μνρσ F

a_{μν is} _its _dual. _In _view _of _that _the _SM _weak interac-tions breakCPexplicitlythroughthecomplexYukawacouplings,1

itisquiteplausiblethattheunderlyingdynamicsof S generically breaks CP, which would result in nonzero value of the effective couplings

κ

s

κ

p

/

κ

2

s

+

κ

2p and dW.Asiswell known,inthe pres-enceofthoseCPviolatingcouplings,a nonzeroneutronorelectron EDM can be induced through the loops involving the SM gauge ﬁelds

[10–13]

.

In this paper, we examine the neutron and electron EDM in models for the 750 GeV resonance, in which the effective in-teractions (1) are generated by the loops of massive vector-like fermions.2 _We _ﬁnd _that _for _the _parameter _region _to _give _the

1 _Throughout_this_paper,_we_assume_the_CP_invariance_in_the_strong_interaction_is duetotheQCDaxionassociatedwithaPeccei–QuinnU(1)symmetry[9].

2 _A_similar_study_was_carried_out_right_after_the_discovery_of_the_Higgs_boson, consideringCP-oddcouplingsoftheHiggsboson[14,15].

http://dx.doi.org/10.1016/j.physletb.2016.07.056

(2)

diphoton cross section

σ

(

pp

→

γ γ

)

=

1

∼

10 fb, the neutron EDM can be comparableto the present experimental bound, e.g. dn

∼

a few

×

10−26e cm,iftheCP-violatingangle

α

inthe under-lying dynamicsis of

O(

10−1

₎

_,_where _{sin 2}

_α

_∼

_κ

s

κ

p

/

κ

2 s

+

κ

2p in termsoftheeffectivecouplingsin(1).Anelectron EDMnearthe presentbound canbe obtainedalsothrougha mixingbetween S andtheSM Higgsboson H .Weﬁndthat againfortheparameter regionof

σ

(

pp

→

γ γ

)

=

1

∼

10 fb, theelectronEDM isgivenby de

∼

6

×

10−26sin

ξ

S Hsin

α

e cm,where

ξ

S H isthe S

−

H mixing angle.3 OurresultontheneutronEDMcanbeappliedalsotothe models in which S corresponds to a composite pseudo-Nambu– GoldstonebosonformedbyaQCD-likehypercolordynamics which isconﬁningat

HC[16–19].Inthiscase,theCP-violatingorder pa-rameter

α

canbeidentiﬁedas

α

∼ θ

HCm2S

/

2HC,where

θ

HCdenotes thevacuumangleoftheunderlyingQCD-likehypercolordynamics. The organization of this paperis as follows. In section 2, we introduce a simple model for the 750 GeV resonance involving CPviolatinginteractions, andsummarizethe diphotonsignalrate given by the model. In section 3, we examine the neutron and electronEDMinthemodelofsection2,anddiscusstheconnection betweentheresultingEDMsandthediphotonsignalrate.Although wearefocusingonaspeciﬁcmodel,ourresultscanbeusedforan estimationofEDMsinmoregeneric modelsforthe750 GeV res-onance.In section 4, weapply ourresultto the casethat S isa compositepseudo-Nambu–Goldstonebosonformedby aQCD-like hypercolordynamics.Section5istheconclusion.

2. AmodelfordiphotonexcesswithCPviolation

The750 GeVdiphoton excesscan be explainedmost straight-forwardlybyintroducingaSM-singletspinzeroresonanceS which couples to massive vector-like fermions to generatethe effective interactions (1) [7]. To be speciﬁc, here we consider a simple modelinvolvingNF Diracfermions

= (

1

,

2

,

...,

NF

)

carrying

acommoncharge undertheSM gaugegroup SU

(

3

)

c

×

SU

(

2

)

L

×

U

(

1

)

Y.ThenthemostgeneralrenormalizableinteractionsofS and

include

L

= ¯

i/D

− ¯

M

+

YsS

+

iYpSγ5

−

1 2m 2 SS2

−

AS HS

|

H

|

2

+ ...,

(2)

wherethemassmatrix M canbechosen toberealanddiagonal, whileYs,p arehermitianYukawacouplingmatrices.Here H isthe SM Higgsdoublet, andwe have chosen the ﬁeld basis forwhich S has avanishingvacuumexpectationvalueinthelimittoignore itsmixingwithH .Forsimplicity,inthefollowingweassumethat all fermion massesand the Yukawa couplings are approximately ﬂavor-universal,sotheycanbeparametrizedas

M

≈

m1NF×NF

,

Ys

≈

yScos

α

1NF×NF

,

Yp

≈

ySsin

α

1NF×NF

,

(3) where1NF×NF denotestheNF

×

NF unitmatrix.Notethatinthis

parametrizationsin 2

α

correspondstotheorderparameterforCP violation.Inthe following,wewilloftenuse

α

(orsin

α

) asaCP violatingorderparameter,althoughitshouldbe

α

−

π

/

2 (orcos

α

) foranapproximatelypseudoscalarS.

Undertheaboveassumptiononthemodelparameters,onecan computethe1PIamplitudesfortheproductionanddecayofS at theLHC,yielding[7]

3 _Note_that_if_ξ

S H correspondstoaCP-violatingmixingangle,thensinαinthis

expressionisnotaCP-violatingparameteranymore,andthereforeisaparameterof orderunity.

L

1PI

=

g2 3 16

π

2_m S S

c(₃s)Ga_μνGaμν

+

c(₃p)Ga_μν

Gaμν

+

g22 16

π

2_m S S

c(₂s)Wa_μνWaμν

+

c(₂p)W_μνa

Waμν

+

g21 16

π

2_m S S

c(₁s)BμνBμν

+

c(1p)Bμν

Bμν

,

(4) where c(_is)

=

NFyScos

α

Tr

(

Ti2

())

mS m A1/2

(

τ

)

2

,

c(_ip)

= −

NFySsin

α

Tr

(

Ti2

())

mS m f

(

τ

)

τ

,

(5)

with i

=

1

,

2

,

3 denoting the SM gauge groups U

(

1

)

Y, SU

(

2

)

L, SU

(

3

)

c, respectively, and

τ

≡

m2S

/

4m2. The loop functions

A1/2

(

τ

)

and f

(

τ

)

aregivenby

A1/2

(

τ

)

=

2

τ

+ (

τ

−

1

)

f

(

τ

)

]/

τ

2

,

f

(

τ

)

= −

1 2 1

0 dx1 xln

[

1

−

4x

(

1

−

x

)

τ

]

=

⎧

⎪

⎨

⎪

⎩

(

arcsin

√

τ

)

2

,

τ

≤

1

−

1 4

ln

1+1−τ−1 1−1−τ−1

−

iπ

2

,

τ

>

1

.

(6)

Note that with a nonzero value of the CP violating angle

α

, the 750 GeV resonance S couples to both Fa

μν Faμν and Faμν

Faμν . Thesetwocouplingsturnouttoincoherentlycontributetothe de-cayrateofS,sothattherelevantdecayratesaregivenby

γ γ

=

1 4

π

e2 16

π

2

2 mS

c_γ(s)

2

+

c_γ(p)

2

,

(7)

gg

=

8 4

π

g2₃ 16

π

2

2 mS

c(gs)

2

+

c(gp)

2

,

(8)

in the rest frame of S. The diphoton signal cross section atthe LHCcan beestimatedusingthe narrowwidthapproximation[7], yielding

σ

(

pp

→

S

→

γ γ

)

=

Cgg 1 s mS

S

γ γ mS

gg mS

,

(9)

wherethecoeﬃcient Cgg

=

2137 at

√

s

=

13 TeV,and

S denotes thetotaldecaywidthofS.Manipulatingthis,thedecayrateshould satisfythefollowingrelation,

γ γ mS

gg mS

=

2

.

17

×

10−9

S 1 GeV

_σ

_signal 8 fb

,

(10)

forthesignalcrosssection

σ

signal

≡

σ

(

pp

→

γ γ

)

=

1

∼

10 fb.Here we normalizethe totaldecayrateof S by

S

=

1 GeV,sinceit is a typicalvalue whenthere isan appreciablemixingbetweenthe singletscalar S andtheSMHiggsdoublet[20].

Plugging(5)and(7),

(8)

into(10),we obtainarelation which isusefulforanestimationoftheelectricdipolemomentsoverthe diphotonsignalregion:

m yS

=

96 GeV

×

QNF

2 3Tr

(

T 2 3

())

Tr

(

1

())

1/2

×

1 GeV

S

1/4

_{8 fb}

σ

signal

1/4 R

,

(11)

(3)

where R

(

α

,

τ

=

m2S

/

4m2

)

=

c2 α

(

A1/2

(

τ

)/

2

)

2

+

s2α

(

f

(

τ

)/

τ

)

2 c2₀_.₁

(

A1/2

(

1

/

4

)/

2

)

2

+

s2₀_.₁

(

4 f

(

1

/

4

))

2

1/2

=

O

(

1

).

Here Q and T3

()

denotetheelectromagneticandcolorcharge

of

,respectively, Tr

(

1

())

isthe dimensionof thegauge group representationof

,andsα

=

sin

α

andcα

=

cos

α

.Notethat R

representsthedependenceon

τ

=

m2S

/

4m2 and

α

,whichis

nor-malizedtothevalue at

τ

=

1

/

4 and

α

=

0

.

1.As R hasamild

dependenceon

τ

and

α

,therangeoftheparameterratiom

/

yS which would explain the diphoton excess can be easily read off fromtheaboverelation.

ToseetheoriginoftheCPviolatingangle

α

,onemayconsider aUV completion ofthemodel(2).Inregard tothis, anattractive possibility is that the model is embedded at some higher scales intoasupersymmetricmodelincludingasingletsuperﬁeld

φ

and NF ﬂavorsofvector-likechargedmattersuperﬁelds

ψ

+ψ

c [21,22]. Themostgeneralrenormalizablesuperpotential of

φ

and

ψ

+ ψ

c isgivenby W

= (

M

+

Y

φ)ψ ψ

c

+

1 2

μ

φ

2

₊

1 3

κ

φ

3

_,

₍₁₂₎

wherewithoutlossofgenerality M canbe chosentobe realand diagonal, det

(

Y

)

to be real, and

φ

to have a vanishing vacuum value in the limit to ignore the mixingwith the Higgsdoublets. Again, for simplicitylet us assume that the mass matrix M and theYukawacouplingmatrixY are approximatelyﬂavor-universal, andtherefore

M

≈

m1NF×NF

,

Y

≈

yS1NF×NF

.

(13)

Including the soft supersymmetry (SUSY) breaking terms, the scalarmasstermof

φ

isgivenby

|

μ

φ

|

2

+

m2φ

|φ|

2

₊

1 2

Bφ

μ

φ

2

+

h.c.

,

(14)

wheremφ isaSUSY breakingsoftscalarmass,while B isa

holo-morphicbilinearsoftparameter.Notethatinourprescription,both

μ

φ andBφ arecomplexingeneral.

Without relying on any ﬁne tuning other than the minimal one tokeep the SM Higgsto be light, onecan arrange theSUSY modelparameters to identifythe lighter masseigenstate of

φ

as the750 GeVresonanceS,andthefermioncomponentsof

ψ

+ ψ

c astheDirac fermion

to generatethe effectiveinteractions (4), while keeping all other SUSY particles heavy enough to be in multi-TeVscales.Thenourmodel(2)arisesasalowenergy effec-tive theoryatscales around TeVfromtheSUSY model(12),with thematchingcondition

1

√

2S

=

Re

(φ)

cos

α

+

Im

(φ)

sin

α

,

where tan 2

α

=

Im

(

Bφ

μ

φ

)

Re

(

Bφ

μ

φ

)

.

(15)

Another possibility, which is completely different but equally interesting, would be that S corresponds to a pseudo-Nambu– GoldstonebosonformedbyaQCD-likehypercolordynamicswhich conﬁnesatscalesnearTeV.Aswewillseeinsection4,theCP vi-olatingorderparameter

α

insuchmodelscanbeidentiﬁedas sin 2

α

∼

m2S

2_HCsin

θ

HC

,

(16)

Fig. 1. TheWeinberg’sthreegluoninteractiongeneratedasatwo-loopthreshold correction.Herethesmalldarksquarerepresentstheγ5-couplingofS tothe vector-likefermion.

where

HC isthescale ofspontaneouschiralsymmetry breaking bythehypercolordynamicsand

θ

HC isthehypercolorvacuum an-gle.

3. Electricdipolemoments

Inthissection,weestimatetheelectricdipolemoments(EDMs) inducedbythe750 GeVsectorintermsofthemodelintroducedin theprevioussection.Atenergyscalesbelowm andmS,theheavy fermions

andthesingletscalar S canbeintegratedout, while leavingtheirfootprintsintheeffectiveinteractionsamongtheSM gauge bosons and Higgs boson. Then those effective interactions eventuallygeneratethenucleonandelectronEDMsinthelow en-ergy limit through the loops involving the exchange of the SM gaugebosonsand/ortheHiggsboson.Inthisprocess,oneneedsto take intoaccounttherenormalizationgroup (RG)running, partic-ularlythoseduetotheQCDinteractions,fromtheinitialthreshold scalem

∼

mS downtothehadronicscale

QCD,aswellasthe in-termediate thresholdcorrectionsfromintegratingoutthemassive SMparticles.

To simplifythecalculation, we willignore theRGrunning ef-fectsduetotheQCD interactionsover thescalesfromm tothe

SM Higgsboson mass mH

=

125 GeV. In thisapproximation, the WilsonianeffectiveinteractionsatscalesjustbelowmH canbe de-termined by the leading order Feynman diagramsinvolving

,

S and the SM Higgs boson. We then take into account the subse-quentRGrunningduetotheQCDinteractions frommH to

QCD, while ignoring the threshold corrections due to the SM heavy quarks,toderivethelowenergyeffectivelagrangianatscalesjust above

QCD.

3.1. NeutronEDM

TheleadingcontributiontotheneutronEDMturnsouttocome from the Weinberg’s three gluon operator [10] generated by the diagramin

Fig. 1

.Inthepresenceofamixingbetweenthesinglet scalar S and the SM Higgs boson H ,the EDM andchromo EDM (CEDM)oflightquarksareinducedbytheBarr–Zeediagrams[11]

in

Fig. 2

,whichmayprovideapotentially importantcontribution totheneutronEDM.

Tobeconcrete,letustakeasimplemodelhavingNF vector-like Diracfermions

transformingunderSU

(

3

)

c

×

SU

(

2

)

L

×

U

(

1

)

Y as

= (

3

,

1

)

Y

,

(17) where Y denotes the U

(

1

)

Y hypercharge of

. As mentioned above, we take an approximation to ignore the RG running due tothe QCDinteractionsbetweenm andmH

=

125 GeV.Thenat scalesjustbelowmH,therelevantWilsonianeffectiveinteractions aredeterminedtobe

[10,23–26]

,

(4)

Fig. 2. The Barr–Zee diagrams for the EDM and chromo EDM (CEDM) of light fermions. The small cross denotes the S−H mixing.

L

eff

(

mH

)

= −

dW

(

mH

)

6 fabc

μνρσ_Ga ρσGbμλGcνλ

−

i 2

q

dq

(

mH

)

eqσ

¯

μν

γ

5qFμν

+ ˜

dq

(

mH

)

g3qσ

¯

μν

γ

5T3aqGaμν

,

(18) with dq

(

mH

)

=

4NF e2

(

4

π

)

4 mq v

6Y2 yS m sαsξcξ

×

Qq

+

t2_wQq

−

T_q3_L 2c2 w

g

m2 m2 H

−

g

m2 m2 S

,

˜

dq

(

mH

)

=

4NF g2₃

(

4

π

)

4 mq v

yS m sαsξcξ

×

g

m2 m2_H

−

g

m2 m2_S

,

dW

(

mH

)

= −

NF g3 3

(

4

π

)

4 y2_S m2cαsα

s2_ξh

m2 m2_H

+

c2_ξh

m2 m2_S

,

(19) whereq

=

u

,

d

,

s standsforthelightquarkspecies,sα

=

sin

α

,sξ

=

sin

ξ

S H forthe S

−

H mixing angle

ξ

S H, v

=

246 GeV is the SM Higgsvacuumvalue,cw

=

cos

θ

w,tw

=

tan

θ

w fortheweakmixing angle

θ

w,andtheloopfunctionsg andh aregivenby4

g

(

z

)

≡

z 2 1

0 dx 1 x

(

1

−

x

)

−

zln x

(

1

−

x

)

z

,

h

(

z

)

≡

z2 1

0 dx 1

0 dy x 3_y3

₍

₁

₋

_x

₎

[

zx

(

1

−

xy

)

+ (

1

−

x

)(

1

−

y

)

]

2

.

(20) Letusrecallthattheparameterratiom

/

yS hasaspeciﬁc connec-tionwiththediphotoncrosssection

σ

(

pp

→

γ γ

)

,whichisgiven by(11).ThisallowsustoestimatetheexpectedsizeoftheEDMs intermsofafewmodelparameterssuchas

α

and

ξ

S H.

In order to estimate the resulting neutron EDM, we should bringtheeffectiveinteractions(18)downtotheQCDscalethrough theRGevolution.Forthis, it isconvenientto redeﬁnethe coeﬃ-cientsas C1

(

μ

)

=

dq

(

μ

)

mqQq

,

C2

(

μ

)

=

˜

dq

(

μ

)

mq

,

C3

(

μ

)

=

dW

(

μ

)

g3

,

(21)

4 _It_is_useful_to_note_the_{asymptotic behavior}_of_the_loop_functions:_h (z1)

z ln(1/z),h(z1)1/4,andg(z1)1+ (ln z)/2.

whicharesatisfyingtheRGequation

[27,28]

:

μ

∂

C

∂

μ

=

g2₃

16

π

2

γ

C

,

(22)

withtheanomalousdimensionmatrix

γ

≡

⎛

⎝

γ

0e

γ

eqq

γ

0Gq 0 0

γ

G

⎞

⎠

=

⎛

⎝

8C0F 16C8CF

−

F4Nc 2N0c 0 0 Nc

+

2nf

+ β

0

⎞

⎠ ,

(23)

whereC

= (

C1

,

C2

,

C3

)

T,Nc

=

3 isthe numberofcolor, CF

=

4

/

3 isaquadraticCasimir,nf isthenumberofactivelightquarks,and

β

0

= (

33

−

2nf

)/

3 istheone-loopbetafunctioncoeﬃcient.Solving thisRGequations,oneﬁnds[27]

C1

(

μ

)

=

η

κeC1

(

mH

)

+

γ

qe

γ

e

−

γ

q

(

η

κe

−

_η

κq

₎

_C 2

(

mH

)

+

γ

Gq

γ

qe

η

κe

(

γ

q

−

γ

e

)(

γ

G

−

γ

e

)

+

γ

Gq

γ

qe

η

κq

(

γ

e

−

γ

q

)(

γ

G

−

γ

q

)

+

γ

Gq

γ

qe

η

κG

(

γ

e

−

γ

G

)(

γ

q

−

γ

G

)

C3

(

mH

),

C2

(

μ

)

=

η

κqC2

(

mH

)

+

γ

Gq

γ

q

−

γ

G

η

κq

−

_η

κG

_C 3

(

mH

),

C3

(

μ

)

=

η

κGC3

(

mH

),

(24)

where

η

≡

g₃2

(

mH

)/

g23

(

μ

)

and

κ

x

=

γ

x

/(

2

β

0

)

.Theanalytic expres-sions for Ci

(

μ

∼

QCD

)

in terms of Ci

(

mH

)

are complicated ex-cept C3,howeverfortunatelyitturnsoutthatthedominant contri-butiontotheneutronEDMcomesfromC3

(

μ

∼

QCD

)

.From(24), weobtain dW

(

μ

)

=

g3

(

mc

)

g3

(

μ

)

g3

(

mb

)

g3

(

mc

)

33 25

_g₃

₍

_m_H

₎

g3

(

mb

)

39 23 dW

(

mH

).

(25)

Itcanbeshownnumericallythatdq

(

μ

)

andd

˜

q

(

μ

)

alsogeta sim-ilar amountof suppressionby the RGevolutioncompared tothe highscalevaluesatmH.

Now one can relate the Wilsonian coeﬃcients dW

(

μ

),

dq

(

μ

)

andd

˜

q

(

μ

)

at

μ

∼

QCDtotheneutronEDM:

−

i

2dnnσ

¯

μν

_γ

5nFμν

,

(26)

whichisthemostambiguousstep.Forthis,onecantaketwo ap-proaches, theNaive Dimensional Analysis (NDA)[29] or theQCD sum rule [30–32], essentially yielding similar results. As for the neutronEDMestimatedbytheNDA,oneﬁnds

dn

/

e

=

O

(

dq

(

μ

))

+

O

(˜

dq

(

μ

)/

√

(5)

where the corresponding scale

μ

is chosen to be the one with g3

(

μ

)

4

π

/

√

6 [10]. Onthe other hand,applying the QCD sum rulefortheneutronEDMdqn fromthe(C)EDMoflightquarks,one ﬁndsamoreconcreteresult5_[32]_:

dqn

/

e

−

0

.

2du

(

μ

)

+

0

.

78dd

(

μ

)

+

0

.

29d

˜

u

(

μ

)

+

0

.

59d

˜

d

(

μ

),

(28) for

μ

=

1 GeV. Asforthe neutronEDM dW

n fromthe Weinberg’s threegluonoperatorintheQCDsumruleapproach,onesimilarly ﬁnds[33]

|

dW_n

/

e

| =

1

.

0+₋1₀._.0₅

×

20 MeV

× |

dW

(

μ

)

|

(29) for

μ

=

1 GeV.Wecannowmakeacomparisonbetweenthe neu-tronEDMd_nW originatingfromdW

(

μ

)

andtheotherpartdnq origi-natingfromdq

(

μ

)

andd

˜

q

(

μ

)

.WithintheQCDsumruleapproach, weﬁndnumerically

dqn

/

dnW

3 sin

ξ

S H

+

0

.

07

.

(30) This implies that the neutron EDM is dominated by the contri-bution from the Weinberg’s three gluon operator for the S

−

H mixingangle

ξ

S H

0

.

1,whichmightbe requiredtobeconsistent withtheHiggsprecisiondata

[20,34,35]

.6

With the above observation, plugging (11), (19) and (25)

into(29),weobtainthefollowingexpressionfortheexpected neu-tron EDMoverthe750 GeVsignalregion:

dn

/

e

3

×

10−25cm

×

cαsα NFY2

S 1 GeV

_σ

_signal 8 fb

×

Rn

,

(31) where Rn

=

h

(

4

τ

)

h

(

1

)

_c2 0.1

(

A1/2

(

1

/

4

)/

2

)

2

+

s20.1

(

4 f

(

1

/

4

))

2 c2 α

(

A1/2

(

τ

)/

2

)

2

+

s2α

(

f

(

τ

)/

τ

)

2

1/2

=

O

(

1

).

Here Rn represents the dependence on the loop functions A1/2, f andh deﬁnedin(6)and(20),whichisnormalizedtothevalue at

τ

=

m2S

/

4m2

=

1

/

4 and

α

=

0

.

1. With this result, one can

easily see that the neutron EDM from the 750 GeV sector satu-ratesthecurrentexperimentalupperbound

∼

3

×

10−26_{e cm}_[37] fortheparameterregionwithsin 2

α

/

NFY2

∼

0

.

1.Inaddition,we

notethatdespiteoftheoreticaluncertainties, a recent experimen-talboundonmercuryEDMcouldgiveafactortwostrongerbound ontheneutron EDM,

|

dn

|

<

1

.

6

×

10−26e cm[38],leadingtoa fac-tortwo stronger constraint on sin 2

α

/

NFY2. In

Fig. 3

,we depict

the resultingneutron EDM as afunction of CP violating angle

α

forthe model parameters which give the diphoton cross section

σ

(

pp

→

γ γ

))

=

1

∼

10 fb.Thegrayregionabovethesolid lineis excludedby[37],whilethelightgrayregionabovethedashedline isexcludedbyHgEDM[38].

3.2. ElectronEDM

In the presence ofthe S

−

H mixing, a sizable electron EDM can arise from the Barr–Zee diagram in Fig. 2. In case of the

5 _We_are_using_“the_modiﬁed_QCD_sum_rule”_obtained_by_assuming_the_Peccei– QuinnmechanismtodynamicallycanceltheQCDvacuumangle.

6 _If_one_uses_the_NDA_rule_or_the_chiral_perturbation_theory_[36]_,_the_resulting neutronEDMinducedbythe(C)EDMofthestrangequarkcanbecomparableto thecontributionfromtheWeinberg’sthreegluonoperatorforthe S−H mixing angleξS H∼0.1.

Fig. 3. TheneutronelectricdipolemomentasafunctionoftheCPviolatingangle αforthemodelparameterstogiveσsignal=1–10 fb.Forthisplot,wechoosethe totaldecaywidthofS asS=1 GeV,thenumberofDiracfermionsasNF=1,

themassandU(1)Y hypercharge ofasm=750 GeV andY=1.(For inter-pretationofthereferencestocolorinthisﬁgure,thereaderisreferredtotheweb versionofthisarticle.)

model with NF ﬂavors of

= (

3

,

1

)

Y, we obtain the electron

EDM

−

ie

2de

(

μ

)

eσ

¯

μν

_γ

5e Fμν

,

(32)

withthecoeﬃcient[25,26] de

= −

24NF e2

(

4

π

)

4 me v

Y2 yS m sαsξcξ

1

+

t2_w

−

1 4c2w

×

g

m2 m2_H

−

g

m2 m2_S

,

(33)

wheretheloopfunctiong

(

z

)

isgivenin(20)andtheother param-etersaredeﬁnedassameasin(19).Applyingtherelation(11)for theaboveresult,weﬁnd

de

= [−

5

.

9

×

10−26cm

]

×

sαsξcξY

S 1 GeV

1/4

σ

signal 8 fb

1/4

×

Re

,

(34) where Re

=

g

(

m2_S

/

4

τ

m2H

)

−

g

(

1

/

4

τ

)

g

(

m2_S

/

m2_H

)

−

g

(

1

)

×

c2₀_.₁

(

A1/2

(

1

/

4

)/

2

)

2

+

s2₀_.₁

(

4 f

(

1

/

4

))

2 c2 α

(

A1/2

(

τ

)/

2

)

2

+

s2α

(

f

(

τ

)/

τ

)

2

1/2

=

O

(

1

)

for

τ

=

m2S

/

4m2.TheaboveresultshowstheelectronEDM

asso-ciatedwiththeS

−

H mixingcansaturatethecurrentexperimental upperlimit8

.

7

×

10−29_cm_[39] _when_sin

_α

_sin

_ξ

S H

=

O(

10−3

)

.In

Fig. 4,wedepicttheelectronEDMoverthe750 GeVsignalregion forthetwodifferentvaluesoftheS

−

H mixingangle:

ξ

S H

=

10−1 and10−2_.

The electron EDM is sensitive to sin

α

sin

ξ

H S. On the other hand, theneutron EDM issensitive to the sin 2

α

. Thisallows us toderivecombinedconstraintsonthetwoangleparameters

α

and

ξ

H S.In

Fig. 5

,wepresent theboundson

(

α

,

ξ

H S

)

fromtheelectron andneutronEDMs.

If the vector-like fermions

carry a nonzero SU

(

2

)

L charge, therecanbeanonzeroelectronEDMeveninthelimit

ξ

S H

=

0.For

(6)

Fig. 4. Theelectron electric dipolemoment for the minimal model with = (3,1)Y, m=750 GeV,ξS H= (10−

1_,₁₀−2₎_,_Y

=1, S=1 GeV,andσsignal= 1–10 fb.

Fig. 5. Constraintsonthe angleparameters(α,ξH S)fromEDMs.Thegreen

dot-dashedlinerepresentstheneutronEDM,whilethebluedashedlineisthe elec-tronEDM. HereweuseS=1 GeV,σsignal=8 fb,m=750 GeV,Y=1.The blueshadedregionisexcludedbythecurrentexperimentalresultontheelectron EDM[39],whilethegreenshadedregionisexcludedbythecurrentexperimental resultsontheneutronEDM[37,38].TheredshadedregionisexcludedbytheHiggs bosonpropertiesmeasuredbytheLargeHadronCollider[20,34,35].(For interpre-tationofthereferencestocolorinthisﬁgurelegend,thereaderisreferredtothe webversionofthisarticle.)

instance,inthemodelwith NF ﬂavorsof

= (

3

,

2

)

Y,a CP-odd

threeW -bosonoperatoroftheform

˜

dW 3

i jkW i μρW jρ ν

Wkμν

canbe generatedby the loopsof

. Following

[12,13]

,7 _we _ﬁnd

theresultingelectronEDMisgivenby

7 _The_authors_in_[13]_noticed_that_the _result_is_{scheme-dependent.}_This_means thatthepreciseresultdependsonthedependenceofd˜W ontheexternalW -boson

de

−

NF 4 g4₂

(

16

π

2

₎

3me y2_S m2cαsα

s2_ξh

m2_H m2

+

c2_ξh

m2_S m2

[−

6

.

5

×

10−31cm

] ×

cαsα NFQ2

S 1 GeV

_σ

_signal 8 fb

× ˜

Re

,

(35) where

˜

Re

=

c_ξ2h

(

4

τ

)

+

s2ξh

(

m2H

/

m2

)

c2₀_.₁h

(

1

)

+

s₀2_.₁h

(

m2_H

/

m2_S

)

×

c2₀_.₁

(

A1/2

(

1

/

4

)/

2

)

2

+

s2₀_.₁

(

4 f

(

1

/

4

))

2 c2 α

(

A1/2

(

τ

)/

2

)

2

+

s2α

(

f

(

τ

)/

τ

)

2

1/2

=

O

(

1

).

Forsin 2

α

0

.

1,whichmightberequiredtosatisfytheboundon theneutronEDM,theresultingelectronEDMisaboutthreeorders ofmagnitudesmallerthan thecurrentbound,thereforetoosmall tobeobservableinaforeseeablefuture.

4. Compositepseudo-Nambu–Goldstoneresonance

In the previous section, we discussed the neutron and elec-tron EDM in models where the 750 GeVresonance is identiﬁed as an elementary spin zero ﬁeld (at least at scales around TeV) which couples to vector-like fermions to generate the effective couplingsto explainthediphoton excess

σ

(

pp

→

γ γ

)

∼

5 fb. On the other hand,it has beenpointed out that in most cases this scheme confronts with a strong coupling regime at scales not far above the TeV scale [40–42]. Inregard to this, an interesting possibility is that S corresponds to a composite pseudo-Nambu– Goldstone (PNG) boson of the spontaneously broken chiral sym-metry ofa newQCD-like hypercolor dynamics whichconﬁnes at

HC

=

O(

1

)

TeV[16–18].Asiswellknown,suchmodelsinvolvea unique sourceofCPviolation, thehypercolor vacuumangle

θ

HC,8 which can yield a nonzero neutron or electron EDM in the low energylimit[16].

To proceed, we consider a speciﬁc example, the model dis-cussedin[17],involvingahypercolorgauge group SU

(

N

)

HC with chargedDiracfermions

(ψ,

χ

)

whichtransformunder SU

(

N

)

HC

×

SU

(

3

)

c

×

SU

(

2

)

L

×

U

(

1

)

Y as

ψ

= (

N

,

3

,

1

)

Yψ

,

χ

= (

N

,

1

,

1

)

Yχ

,

(36) where Yψ,χ denotethe U

(

1

)

Y hypercharge.At scalesabove

H C, thelagrangianofthehypercolorcolorsectorisgivenby

L

HC

= −

1 4g2 HC HaμνHa_μν

−

θ

HC 32

π

2H aμν

_Ha μν

+ ¯ψ

i/D

ψ

+ ¯

χi/

Dχ

− ¯ψ

mψ

ψ

− ¯

χm

χ

,

(37) where Haμν denotes the SU

(

N

)

HC gauge ﬁeld strength,

Haμν is its dual,andthefermionmassesmψ,χ arechosen tobe realand

γ

5-free. For a discussion of the low energy consequence of the CP-violating vacuumangle

θ

HC, itis convenient to make a chiral rotationoffermionﬁeldstorotateaway

θ

HC intothephaseofthe fermionmassmatrix,whichresultsin

M

=

Mθ

≡

diag

mψeixψθHC

,

mψeixψθHC

,

mψeixψθHC

,

mχeixχθHC

,

(38)

momenta.Herewesimplyusetheresultfromthedimensionalregularizationfor thepurposeofestimationoftheelectronEDM.

8 _A_{phenomenological}_study_of_the_hypercolor_vacuum_angle_on_diphoton_excess has beencarriedoutinRef.[43].

(7)

where 3xψ

+

xχ

=

1

.

Formψ,χ

HC,themodelisinvariantunderan approximate chiralsymmetry SU

(

4

)

L

×

SU

(

4

)

R whichisspontaneouslybroken downtothediagonalSU

(

4

)

V bythefermionbilinearcondensates:

| ¯ψ

L

ψ

R

| | ¯

χ

L

χ

R

|

N

16

π

2

3

HC

.

(39)

Thecorresponding pseudo-Nambu–Goldstone(PNG)boson canbe described by an SU

(

4

)

-valued ﬁeld U

=

exp

(

2i

/

f

)

whose low energydynamicsisgovernedby

L

eff

=

1 4f 2_tr

_D μU DμU†

+

μ

3tr

MθU†

+

h.c.

+

L

WZW

+

L

CPV

...,

(40)

wherethenaivedimensionalanalysissuggests

f2

N 16

π

2

2 H

,

μ

3

N 16

π

2

3 H

,

(41)

and

L

WZW and

L

CPV denote the Wess–Zumino–Wittenterm and the additionalCP-violating term, respectively. For a discussion of CPviolationdueto

θ

HC

=

0,itisconvenienttochoosethefermion massmatrixMθ as

−

i 2

Mθ

−

Mθ†

=

mθ14×4

,

(42)

for which the PNG boson has a vanishing vacuum expectation value.ThentheCPviolationdueto

θ

HC

=

0 isparametrizedsimply by mθ

≡

mψsin xψ

θ

HC

!

=

m_χsin x_χ

θ

HC

!

3xψ

+

xχ

=

1

!

,

(43) whichmanifestlyshowsthat CPisrestoredif

θ

HC oranyofmψ,χ is vanishing. In the limit

|θ

HC

|

1, this order parameter for CP violationhasasimpleexpression:

mθ

θ

HC tr

M−_¯θ1 H=0

=

mψmχ 3mχ

+

mψ

θ

HC

.

(44)

The PNG bosons of SU

(

4

)

L

×

SU

(

4

)

R

/

SU

(

4

)

V include a unique SM-singletcomponent S whichcan be identiﬁedasthe750 GeV resonance:

=

1

2

√

6diag

(

S

,

S

,

S

,

−

3S

)

+ · · · ,

(45)

whereU

=

exp

(

2i

/

f

)

,andtheellipsis denotesthe SU

(

3

)

c octet andtripletPNGbosonswhichareheavierthan S.ThentheWess– Zumino–Wittentermgivesrisetothefollowingeffectivecouplings between S and the SM gauge bosons, which would explain the diphotonexcess:

L

WZW

= −

N 16

π

2 S f

1 2

√

6g 2 3Gaμν

Gaμν

+

√

6 2 g1 2

_Y2 ψ

−

Yχ2

Bμν

+ · · · .

(46)

Withmθ

=

0,accordingtotheNDA,theunderlyinghypercolor

dy-namics generatesthe following CPviolating effective interactions renormalizedat

HC:

L

CPV

=

N 16

π

2 mθ

H S f

×

cGg23GaμνGaμν

+

cBg2

Y2_ψ

−

Y_χ2

BμνBμν

Fig. 6. Theexpectedneutronelectric dipolemomentasafunctionofthe hyper-colorvacuumangleθHCinmodelsforacompositePNG750 GeVresonance.Forthe analysis,weassumemψ=mχ,andamixingbetweenPNGbosonS andtheHiggs bosonwhichallowustotakeS=1 GeV asabenchmarkvalue.Wealsochoose

N=3,Yψ=1/3 andYχ=1.

+

N 16

π

2 mθ

H

κ

G

2_H g3₃fabc 3 G a μρGbνρ

Gcμν

+ · · · ,

(47) wherecG

,

cB and

κ

G arealloforderunity.

Itisnowstraightforwardtouseourpreviousresultstofindthe nucleon and electron EDM induced by the above effective inter-actions. Bymatching the coefficientsof the relevant interactions withthesimplemodelpresentedinsection2,wefindthe follow-ingcorrespondence: yS m

∼

N 2 f

,

sin 2

α

∼

m2_S

2_HCsin

θ

HC

,

(48)

wherewehaveusedtherelation

m2_S

= (

750 GeV

)

2

(

mψ

+

3mχ

)

μ

3

f2

(

mψ

+

3mχ

)

HC

.

(49)

Since theratio m

/

yS shouldbe around 100 GeVto explain the 750 GeV diphoton excess, it turns out that f

∼

N

×

50 GeV and

HC

4

π

f

/

√

N

∼

√

N TeV,implyingthatroughlysin 2

α

isa few factor smallerthan

θ

HC. In

Fig. 6

,we depict theneutron EDM in the minimal modelfor a composite PNG 750 GeVresonance for theparameterregiontogive

σ

(

pp

→

γ γ

)

=

1

∼

10 fb.

The minimal model of [17] can be generalized or modiﬁed to include a hypercolored fermion carrying a nonzero SU

(

2

)

L charge [16],e.g.

χ

can transformas

(

N

,

1

,

2

)

Yχ under SU

(

N

)

H

×

SU

(

3

)

c

×

SU

(

2

)

L

×

U

(

1

)

Y. Then the hypercolor dynamic with nonzero

θ

HCcangeneratethefollowingCP-oddthreeW -boson op-erator:

L

CPV

=

N 16

π

2 mθ

H

κ

W

2_H g₂3

i jk 3 W i μρWνjρ

Wkμν

,

(50) where

κ

W

=

O(

1

)

accordingtotheNDArule.Applyingour previ-ous result(35)undertherelation(48),weﬁndtheelectronEDM resulting fromtheabove three W -bosonoperator istoo smallto beobservableevenwhen

θ

HC

=

O(

1

)

.

Finally let us note that a composite PNG boson S can have a mixing withthe SM Higgs boson if the underlying hypercolor modelincludesahigherdimensionaloperatoroftheform:

1

ψ

|

H

|

2

¯ψ

L

ψ

R

+

1

χ

|

H

|

2

χ

¯

L

χ

R

+

h.c.

,

(51)

(8)

where

ψ,χ are complex in general. For instance, this form of dim

−

5 operatorscanbegeneratedbyanexchangeofheavyscalar ﬁeld

σ

whichhasthecouplings

L

σ

= −

1 2m 2 σ

σ

2

+

Aσ

σ

|

H

|

2

+

λ

ψ

σ

¯ψ

L

ψ

R

+ λ

χ

σ

χ

¯

L

χ

R

+

h.c.

!

+ ...,

(52) yielding 1

ψ

=

Aσ

λ

ψ m2 σ

,

1

χ

=

Aσ

λ

χ m2 σ

.

Theresulting S

−

H mixingangleisestimatedas

ξ

S H

∼

v

2_HC 4

πm

2 S Im

(

ψ,χ

)

|

ψ,χ

|

2

,

(53)

where v

=

246 GeV is the vacuum value of the SM Higgs dou-blet H . One can apply this mixing angle for our previous result

(34)toestimatetheresultingelectronEDM.NotethathereS isan approximatelypseudoscalarboson,andtherefore

ξ

S H corresponds toaCP-violatingmixingangle,whilesin

α

isCP-conservingandof orderunity.OnethenﬁndsthecurrentboundontheelectronEDM implies

HC

|

ψ,χ

|

Im

(

ψ,χ

)

|

ψ,χ

|

O

(

10−2

).

(54) 5. Conclusion

Therecentlyannounceddiphotonexcessat750 GeVintheRun II ATLAS andCMSdatamayturn out tobe the ﬁrstdiscovery of newphysicsbeyondtheStandardModelatcolliderexperiments.In thispaper,weexaminedtheimplicationofthe750 GeVdiphoton excessfortheEDMofneutronandelectroninmodelsinwhichthe diphotonexcessisduetoa spinzeroresonance S whichcouples to photons and gluons through the loops of massive vector-like fermions.We found that a neutron EDM comparableto the cur-rentexperimentalboundcanbeobtainediftheCPviolatingorder parametersin 2

α

intheunderlyingnewphysicsisof

O(

10−1

₎

_._An electronEDMnearthepresentboundcan beobtainedalsowhen sin

ξ

S H

×

sin

α

=

O(

10−3

)

,where

ξ

S H isthemixinganglebetween S and the SM Higgs boson. For the case that S corresponds to a pseudo-Nambu–Goldstone boson of a QCD-like hypercolor dy-namics,one can use thecorrespondence sin 2

α

∼

m2_Ssin

θ

HC

/

2_HC toestimatetheresultingEDMs,where

HC isthescaleof sponta-neouschiralsymmetry breaking by thehypercolor dynamics and

θ

HC isthehypercolor vacuumangle.In viewofthat anucleonor electronEDMnearthecurrentboundcanbeobtainedovera natu-ralparameterregionofthemodel,futureprecision measurements ofthenucleonorelectronEDMarehighlymotivated.

Acknowledgements

Thiswork was supported by IBS under the projectcode, IBS-R018-D1.

References

[1] ATLASCollaboration,ATLAS-CONF-2015-081,2015. [2] CMSCollaboration,CMS-PAS-EXO-15-004,2015.

[3] M.Delmastro,DiphotonsearchesinATLAS,Talkat51stRencontresdeMoriond EW2016,March17,2016.

[4] P.Musella,SearchforhighmassdiphotonresonancesatCMS,Talkat51st Ren-contresdeMoriondEW2016,March17,2016.

[5]S.Knapen,T.Melia,M.Papucci,K.Zurek,Phys.Rev.D93 (7)(2016)075020, arXiv:1512.04928[hep-ph].

[6]D. Buttazzo, A. Greljo, D. Marzocca, Eur. Phys. J. C 76 (3) (2016) 116, arXiv:1512.04929[hep-ph].

[7]R.Franceschini,etal.,J. HighEnergyPhys.1603(2016)144,arXiv:1512.04933 [hep-ph].

[8] S. Di Chiara, L. Marzola, M. Raidal, Phys. Rev. D 93 (9) (2016) 095018, http://dx.doi.org/10.1103/PhysRevD.93.095018,arXiv:1512.04939[hep-ph]. [9]J.E.Kim,G.Carosi,Rev.Mod.Phys.82(2010)557,arXiv:0807.3125[hep-ph]. [10]S.Weinberg,Phys.Rev.Lett.63(1989)2333.

[11]S.M.Barr,A.Zee,Phys.Rev.Lett.65(1990)21;Phys.Rev.Lett.65(1990)2920 (Erratum).

[12]W.J.Marciano,A.Queijeiro,Phys.Rev.D33(1986)3449.

[13]F.Boudjema, K.Hagiwara,C. Hamzaoui,K.Numata,Phys. Rev.D43 (1991) 2223.

[14]D. McKeen, M. Pospelov, A. Ritz, Phys. Rev. D 86 (2012) 113004, arXiv:1208.4597[hep-ph].

[15]R. Harnik, J. Kopp, J. Zupan, J. High Energy Phys. 1303 (2013) 026, arXiv:1209.1397[hep-ph].

[16]K.Harigaya,Y.Nomura,Phys.Lett.B754(2016)151,arXiv:1512.04850 [hep-ph];J. HighEnergyPhys.1603(2016)091,arXiv:1602.01092[hep-ph]. [17]Y. Nakai, R. Sato, K. Tobioka, Phys. Rev. Lett. 116 (15) (2016) 151802,

arXiv:1512.04924[hep-ph].

[18]M.Redi,A.Strumia,A.Tesi,E.Vigiani,arXiv:1602.07297[hep-ph]. [19]D.Buttazzo,A.Greljo,G.Isidori,D.Marzocca,arXiv:1604.03940[hep-ph]. [20]A. Falkowski, O.Slone, T. Volansky,J. High Energy Phys. 1602(2016) 152,

[21]L.J. Hall, K. Harigaya, Y. Nomura, J. High Energy Phys. 1603 (2016) 017, arXiv:1512.07904[hep-ph].

[22]H.P.Nilles,M.W.Winkler,arXiv:1604.03598[hep-ph]. [23]D.A.Dicus,Phys.Rev.D41(1990)999.

[24]T.Abe,J.Hisano,T.Kitahara,K.Tobioka,J. HighEnergyPhys.1401(2014)106, arXiv:1311.4704[hep-ph];J. HighEnergyPhys.1604(2016)161(Erratum). [25]M.Jung,A.Pich,J. HighEnergyPhys.1404(2014)076,arXiv:1308.6283

[hep-ph].

[26]W.Dekens,J.deVries,J.Bsaisou,W.Bernreuther,C.Hanhart,U.G.Meißner, A.Nogga,A.Wirzba,J. HighEnergyPhys.1407(2014)069,arXiv:1404.6082 [hep-ph].

[27]G.Degrassi, E.Franco,S.Marchetti,L.Silvestrini, J. HighEnergyPhys. 0511 (2005)044,arXiv:hep-ph/0510137.

[28]J.Hisano,K.Tsumura,M.J.S.Yang,Phys.Lett.B713(2012)473,arXiv:1205.2212 [hep-ph].

[29]A.Manohar,H.Georgi,Nucl.Phys.234(1984)189;

H.Georgi,WeakInteractionsandModernParticleTheory,Benjamin/Cummings, MenloPark,1984;

H.Georgi,L.Randall,Nucl.Phys.276(1986)241.

[30]M.Pospelov,A.Ritz,Phys.Rev.D63(2001)073015,arXiv:hep-ph/0010037. [31]J. Hisano,J.Y. Lee, N. Nagata,Y. Shimizu, Phys. Rev.D 85 (2012) 114044,

[32]J.Hisano,D.Kobayashi,W.Kuramoto,T.Kuwahara,J. HighEnergyPhys.1511 (2015)085,arXiv:1507.05836[hep-ph].

[33]D.A.Demir,M.Pospelov,A.Ritz,Phys.Rev.D67(2003)015007, arXiv:hep-ph/0208257.

[34]L.Berthier,J.M.Cline,W.Shepherd,M.Trott,J. HighEnergyPhys.1604(2016) 084,arXiv:1512.06799[hep-ph].

[35]K.Cheung,P.Ko,J.S.Lee,J.Park,P.Y.Tseng,arXiv:1512.07853[hep-ph]. [36]K. Fuyuto, J. Hisano, N. Nagata, Phys. Rev. D 87 (5) (2013) 054018,

[37]C.A.Baker,etal.,Phys.Rev.Lett.97(2006)131801,arXiv:hep-ex/0602020. [38] B.Graner,Y.Chen,E.G.Lindahl,B.R.Heckel,Phys.Rev.Lett.116 (16)(2016)

161601, http://dx.doi.org/10.1103/PhysRevLett.116.161601, arXiv:1601.04339 [physics.atom-ph].

[39]J.Baron,etal.,ACMECollaboration,Science343(2014)269,arXiv:1310.7534 [physics.atom-ph].

[40]M.Son,A.Urbano,arXiv:1512.08307[hep-ph].

[41]J.Gu,Z.Liu,Phys.Rev.D93 (7)(2016)075006,arXiv:1512.07624[hep-ph]. [42]P.S.B.Dev,R.N. Mohapatra,Y.Zhang, J. HighEnergyPhys. 1602(2016)186,

[43]P.Draper,D.McKeen,J. HighEnergyPhys.1604(2016)127,arXiv:1602.03604 [hep-ph].