JHEP10(2013)020

Published for SISSA by Springer

Received: June 5, 2013

Revised: August 13, 2013

Accepted: September 1, 2013

Published: October 2, 2013

Neutron electric dipole moment in CP violating

BLMSSM

Shu-Min Zhao,a Tai-Fu Feng,a Ben Yan,a Hai-Bin Zhang,a,b Yin-Jie Zhang,a

Biao Chena and Xue-Qian Lic

aDepartment of Physics, Hebei University,

Wusi East Road, No. 180, Baoding, 071002, ChinabDepartment of Physics, Dalian University of Technology,

Linggong Road No. 2, Dalian, 116024, ChinacDepartment of Physics, Nankai University,

Weijin Road No. 94, Tianjin, 300071, China

E-mail: zhaosm@hbu.edu.cn, fengtf@hbu.edu.cn, Yb118sdfz@163.com,

hibzhang@163.com, zhangyinjie@163.com, 1435566369@qq.com,

lixq@nankai.edu.cn

Abstract: Considering the CP violating phases, we analyze the neutron electric dipole

moment (EDM) in a CP violating supersymmetric extension of the standard model where

baryon and lepton numbers are local gauge symmetries(BLMSSM). The contributions from

the one loop diagrams and the Weinberg operators are taken into account. Adopting some

assumptions on the relevant parameter space, we give the numerical results analysis. The

numerical results for the neutron EDM can reach 1.05 × 10−25(e.cm), which is about the

experimental upper limit.

Keywords: Supersymmetric gauge theory, Extended Supersymmetry, CP violation

ArXiv ePrint: 1306.0664

c© SISSA 2013 doi:10.1007/JHEP10(2013)020

JHEP10(2013)020

Contents

1 Introduction 1

2 Some coupling in BLMSSM 2

3 The corrections to the neutron EDM in BLMSSM 5

3.1 The corrections to neutron EDM in MSSM 6

3.2 The corrections to the neutron EDM from superfields in BLMSSM 7

4 The numerical analysis 10

5 Summary 14

A The couplings between Higgs and exotic quarks 14

1 Introduction

Physicists have been interested in the minimal supersymmetric extension of the standard

model (MSSM) [1–5] for a long time. Considering the matter-antimatter asymmetry in

the universe, baryon number (B) should be broken. To explain the neutrino oscillation

experiment, neutrino should have tiny mass which can be induced by the seesaw mechanism

from the heavy majorana neutrinos. Then lepton number (L) is also expected to be broken.

A minimal supersymmetric extension of the SM with local gauged B and L(BLMSSM)

is more favorite. With the assumption that B and L are spontaneously broken gauge

symmetries around TeV scale, the author [6, 7] examines two extensions of the SM.

B and L are spontaneously broken near the weak scale in BLMSSM. The light neutri-

nos get mass from the seesaw mechanism, and proton decay is forbidden [6–9]. Therefore,

a large desert between the electroweak scale and grand unified scale is not necessary. This

is the main motivation for the BLMSSM. Many possible signals of the MSSM at the LHC

have been studied by the CMS [10] and ATLAS [11] experiments and they set very strong

bounds on the gluino and squarks masses with R-parity conservation. However, the pre-

dictions and bounds for the collider experiments should be changed with the broken B and

L symmetries [6, 7, 12].

Flavour violation in the quark and leptonic sectors is suppressed naturally, because

tree level flavor changing neutral currents involving the quarks (charged leptons) are for-

bidden by the gauge symmetries and particle content [8, 9]. We can look for lepton number

violation at the LHC, from the decays of right handed neutrinos [8, 9, 13]. The baryon

number violation in the decays of squarks and gauginos [14] can also be detected at the

LHC. For example, the baryon number violating decays of gluinos may give rise to channels

with multi-tops and multi-bottoms [8, 9].

– 1 –

JHEP10(2013)020

In BLMSSM, the mass and decays of the lightest CP-even Higgs are investigated, where

Yukawa couplings between Higgs doublets and exotic quarks are neglected [6–9]. Taking

into account the Yukawa couplings between Higgs and exotic quarks and ATLAS (CMS)

results for the neutral Higgs (mh0 ∼ 124 − 126 GeV) [15–17], the lightest CP-even Higgs

decays h0 → γγ, h0 → ZZ(WW ) are investigated in BLMSSM [18].

As is known to all, the magnetic dipole moments(MDMs) [19–24] and electric dipole

moments (EDMs) [25–28] of fermions open a wide window to find new physics beyond the

SM. The present experiment upper limit of the neutron EDM is 1.1× 10−25 e · cm [29, 30].

In SM, the CP phases of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements fully

induce very tiny neutron EDM, and it is too tiny to be detected by experiment in the near

future [31].

In MSSM, many new sources of the CP violation lead to large contributions to the

neutron EDM [32–37] at one loop level, and the theoretical prediction is larger than the

present experiment upper bound. To consistent with the experiment data, there are three

approaches. The first is fine tuning, making the CP phases sufficiently small, i.e. ≤ 10−2;

the second is that the supersymmetry spectra is very heavy at several TeV. Internal can-

cellation among the different contributions is the last one, and seems more reasonable [38].

The theoretical analysis of EDMs for the fermions at one-loop level is well studied in

MSSM. On other hand, authors obtain the corrections to neutron EDM from the two-loop

Barr-Zee type diagrams involving the Higgs bosons [39, 40]. For the neutron EDM, the two-

loop gluino corrections and gluonic Weinberg operator induced by two-loop gluino-squark

diagrams are carried out in the literature [41].

In BLMSSM, there are many new CP violation sources and new particles. Assuming

that the Yukawa couplings between the Higgs and exotic quarks cannot be ignored, we

study the neutron EDM from the one loop diagrams and the Weinberg operator contri-

butions in this work. After this introduction, we briefly summarize the main ingredients

of the BLMSSM, and show the needed couplings for exotic quarks in section 2. For the

neutron EDM, the one loop diagrams’ corrections and the Weinberg operator contribu-

tions are presented in section 3. In section 4, we show the numerical results and study the

relation between the neutron EDM and the BLMSSM parameters. Our conclusions are

summarized in section 5.

2 Some coupling in BLMSSM

Enlarging the SM, the author obtains the BLMSSM [6, 7] with the local gauge group of

SU(3)C⊗SU(2)L⊗U(1)Y ⊗U(1)B⊗U(1)L. In BLMSSM, to cancel L anomaly, there are the

exotic superfields of the new leptons L4 ∼ (1, 2, −1/2, 0, L4), Ec4 ∼ (1, 1, 1, 0, −L4),

N c4 ∼ (1, 1, 0, 0, −L4), L

c5 ∼ (1, 2, 1/2, 0, −(3 + L4)), E5 ∼ (1, 1, −1, 0, 3 + L4),

N5 ∼ (1, 1, 0, 0, 3 + L4). To break lepton number spontaneously, the superfields

ΦL ∼ (1, 1, 0, 0, −2) and ϕL ∼ (1, 1, 0, 0, 2) are introduced, and they acquire

nonzero vacuum expectation values (VEVs). Higgs mechanism has a solid stone because

of the detection of the lightest CP even Higgs at LHC [15–17].

– 2 –

JHEP10(2013)020

In the same way, the model includes the new quarks Q4 ∼ (3, 2, 1/6, B4, 0),

U c4 ∼ (3, 1, −2/3, −B4, 0), D

c4 ∼ (3, 1, 1/3, −B4, 0), Q

c5 ∼ (3, 2, −1/6, −(1 +B4), 0),

U5 ∼ (3, 1, 2/3, 1 + B4, 0), D5 ∼ (3, 1, −1/3, 1 + B4, 0), to cancel the B anomaly.

ΦB ∼ (1, 1, 0, 1, 0) and ϕB ∼ (1, 1, 0, −1, 0) are the ‘brand new’ Higgs super-

fields. With nonzero vacuum expectation values (VEVs) they break baryon number spon-

taneously. The exotic quarks are very heavy, because of the nonzero VEVs of ΦB and

ϕB. Therefore, the BLMSSM also introduces the superfields X ∼ (1, 1, 0, 2/3 + B4, 0),

X ′ ∼ (1, 1, 0, −(2/3+B4), 0) to make the unstable exotic quarks. The lightest superfields

X can be a candidate for dark matter.

In BLMSSM, the superpotential reads as follows

WBLMSSM = WMSSM +WB +WL +WX , (2.1)

with WMSSM representing the superpotential of the MSSM. The concrete forms of WB,WL

and WX are

WB = λQQ4Qc5ΦB + λU U

c4U5ϕB + λDD

c4D5ϕB + µBΦBϕB

+Yu4Q4HuU

c4 + Yd4Q4HdD

c4 + Yu5

Qc5HdU5 + Yd5Q

c5HuD5 ,

WL = Ye4L4HdEc4 + Yν4L4HuN

c4 + Ye5L

c5HuE5 + Yν5L

c5HdN5

+YνLHuNc + λNcN cN cϕL + µLΦLϕL ,

WX = λ1QQc5X + λ2U

cU5X′ + λ3D

cD5X′ + µXXX ′ . (2.2)

The soft breaking terms Lsoft of the BLMSSM can be found in our previous work [18].

To make the local gauge symmetry SU(2)L ⊗U(1)Y ⊗U(1)B ⊗U(1)L broken down to

the electromagnetic symmetry U(1)e, the SU(2)L singlets ΦB, ϕB, ΦL, ϕL and the SU(2)Ldoublets Hu, Hd should obtain nonzero VEVs υB, υB, υL, υL and υu, υd respectively.

Hu =

(

H+u

1√2

(

υu +H0u + iP 0

u

)

)

, Hd =

(

1√2

(

υd +H0d + iP 0

d

)

H−d

)

,

ΦB =1√2

(

υB +Φ0B + iP 0

B

)

, ϕB =1√2

(

υB + ϕ0B + iP

0B

)

,

ΦL =1√2

(

υL +Φ0L + iP 0

L

)

, ϕL =1√2

(

υL + ϕ0L + iP

0L

)

, (2.3)

In ref. [18], the mass matrixes of Higgs, exotic quarks and exotic scalar quarks are

obtained. In the mass basis, assuming CP conservation in exotic quark sector, we show

the couplings between the neutral Higgs and charged (2/3,−1/3) exotic quarks. Here we

investigate the CP violating effect and modify our previous results.

To save space, we define H0α(α = 1, 2, 3, . . . 8) = (h0, H0, A0, G0, h0B, H

0B, A

0B, G

0B).

LH0q′q′ =8∑

α=1

2∑

i,j=1

{

(NLH0

α)ijH

0αti+3PLtj+3 + (NR

H0α)ijH

0αti+3PRtj+3

+(KLH0

α)ijH

0αbi+3PLbj+3 + (KR

H0α)ijH

0αbi+3PRbj+3

}

, (2.4)

– 3 –

JHEP10(2013)020

where ti+3, bi+3(i = 1, 2) are the exotic quark fields, the coupling contents

(NLH0

α)ij , (NL

H0α)ij , (NL

H0α)ij , (NL

H0α)ij are collected in the appendix.

The couplings between charged Higgs and exotic quarks are

LH+t′b′ =2∑

i,j=1

[

(NLH+)ijH

+ti+3PLbj+3 + (NRH+)ijH

+ti+3PRbj+3

]

+2∑

i,j=1

[

(NLG+)ijG

+ti+3PLbj+3 + (NRG+)ijG

+ti+3PRbj+3

]

+ h.c. (2.5)

with the coupling constants

(NLH+)ij = −

(

Yu4(W †

t )i2(Ub)1j + Yd5(W†t )i1(Ub)2j

)

cosβ,

(NRH+)ij =

(

Y ∗d4(U †

t )i1(Wb)2j + Y ∗u5(U †

t )i2(Wb)1j

)

sinβ,

(NLG+)ij = −

(

Yu4(W †

t )i2(Ub)1j + Yd5(W†t )i1(Ub)2j

)

sinβ,

(NRG+)ij = −

(

Y ∗d4(U †

t )i1(Wb)2j + Y ∗u5(U †

t )i2(Wb)1j

)

cosβ. (2.6)

Here, tanβ = vu/vd. Wt and Ut are the matrixes to diagonalize the mass matrix of the

up type exotic quarks. While, Wb and Ub are the matrixes to diagonalize the mass matrix

of the exotic quarks with charge −1/3. Ati+3, Adi+3

(i = 1, 2) are the same with those in

ref. [18]. We also derive the couplings between photon (gluon) and exotic quarks.

Lγq′q′ = −2e

3

2∑

i=1

ti+3γµti+3Fµ +

e

3

2∑

i=1

bi+3γµbi+3Fµ,

Lgq′q′ = −g3

2∑

i=1

ti+3Taγµti+3G

aµ − g3

2∑

i=1

bi+3Taγµbi+3G

aµ, (2.7)

where Fµ is electromagnetic field, Gaµ is gluon field, T a (a = 1, · · · , 8) denote the genera-

tors of the strong SU(3) gauge group.

The couplings of gluino, exotic quark and exotic scalar quark have relation with Wein-

berg operator, and are written in the following form

LΛt′U =√2g3T

a4∑

i=1

2∑

j=1

{

(NLU )ijU∗

i ΛGPLtj+3 + (NRU )ijU∗

i ΛGPRtj+3

}

+ h.c.

LΛb′D =√2g3T

a4∑

i=1

2∑

j=1

{

(NLD )ijD∗

i ΛGPLbj+3 + (NRD )ijD∗

i ΛGPRbj+3

}

+ h.c. (2.8)

where ΛG is gluino field. Ui, Di(i = 1, 2, 3, 4) are exotic scalar quarks with charge 2/3 and

−1/3 respectively. The concrete forms of the coupling constants are

(NLU )ij = U †

i4Ut

2j − U †i1U

t

1j , (NRU )ij = U †

i2Wt

2j − U †i3W

t

1j ,

(NLD )ij = D†

i4Ub2j −D†

i1Ub1j , (NR

D )ij = D†i2W

b2j −D†

i3Wb1j . (2.9)

– 4 –

JHEP10(2013)020

Using the scalar potential and the soft breaking terms, we write the mass squared

matrix for superfields X.

− LX = (X∗ X ′)

(

|µX |2 + SX −µ∗XB∗

X

−µXBX |µX |2 − SX

)(

X

X ′∗

)

, (2.10)

with SX =g2B

2 (23 +B4)(v2B − v2B). Adopting the unitary transformation,

(

X1

X2

)

= Z†X

(

X

X ′∗

)

, (2.11)

the mass squared matrix for the superfields X are diagonalized.

Z†X

(

|µX |2 + SX −µ∗XB∗

X

−µXBX |µX |2 − SX

)

ZX =

(

m2X1 0

0 m2X2

)

. (2.12)

In mass basis, we obtain the couplings of quark-exotic quark and the superfields X.

LXt′u =2∑

i,j=1

(

(NLt′ )ijXj ti+3PLu

I + (NRt′ )ijXj ti+3PRu

I)

+ h.c.

LXb′d =2∑

i,j=1

(

(NLb′ )ijXj bi+3PLd

I + (NRb′ )ijXj bi+3PRd

I)

+ h.c. (2.13)

where the coupling constants are

(NLt′ )ij = −λ1(W

†t )i1(ZX)1j , (NR

t′ )ij = −λ∗2(U

†t )i2(ZX)2j ,

(NLb′ )ij = λ1(W

†b )i1(ZX)1j , (NR

b′ )ij = −λ∗3(U

†b )i2(ZX)2j . (2.14)

3 The corrections to the neutron EDM in BLMSSM

The EDM df of the fermion can be obtained from the effective Lagrangian [41]

LEDM = − i

2dffσ

µνγ5fFµν , (3.1)

where Fµν represents the electromagnetic field strength, f is a fermion field. This effective

Lagrangian violates CP conservation obviously, and cannot be present at tree level in

the fundamental interactions. However in the CP violating electroweak theory, it can be

obtained at least at one loop level.

Because neutron is composed of quarks and gluons, the chromoelectric dipole moment

(CEDM) fT aσµνγ5fGaµν of quarks can also give contributions to the neutron EDM. Ga

µν

represents the gluon field strength. Weinberg has discovered a dimension-6 operator

fabcGaµρG

bρν Gc

λσǫµνλσ which violates CP conservation and it makes a very large contribution

to the neutron EDM [42, 43].

The effective method is convenient to describe the CP violation operators at loop

level. At matching scale, we can integrate out the heavy particles in the loop and obtain

– 5 –

JHEP10(2013)020

q χ−k q

Qi

q q

qi

g, χ0k

Figure 1. The MSSM one loop self energy diagrams from which the corresponding triangle dia-

grams are obtained by attaching a photon or a gluon in all possible ways.

the resulting effective Lagrangian with a full set of CP violation operators. Here, we just

show the effective Lagrangian containing operators relevant to the neutron EDM.

Leff =5∑

i

Ci(Λ)Oi(Λ) ,

O1 = qσµνPLqFµν , O2 = qσµνPRqFµν , O3 = qT aσµνPLqGaµν ,

O4 = qT aσµνPRqGaµν , O5 = −1

6fabcG

aµρG

bρν Gc

λσǫµνλσ . (3.2)

where Λ is the energy scale, at which the Wilson coefficients Ci(Λ) are evaluated.

3.1 The corrections to neutron EDM in MSSM

In SUSY, the one-loop supersymmetric corrections to the quark EDMs and CEDMs are

obtained [41]. It can be divided into three types according to the quark self-energy dia-

grams. The internal line particles are chargino and squark (neutralino and squark, gluino

and squark). The needed triangle diagrams are obtained by attaching a photon or gluon

on the internal lines of the quark selfenergy diagrams in all possible ways. Calculating

these diagrams one can obtain the effective Lagrangian contributing to the quark EDMs

and CEDMs. Before we show the BLMSSM corrections, we present the one loop MSSM

contributions to the quark EDMs and CEDMs, which can be obtained in the previous

works. The needed one loop self energy diagrams are shown in figure 1.

The one-loop neutralino-squark corrections to the quark EDMs and CEDMs are

dγχ0k(1)

= eqeα

16πs2wc2w

2∑

i=1

4∑

k=1

Im(

(AqN )k,i(B

qN )†i,k

)mχ0k

m2qi

B

(m2χ0k

m2qi

)

,

dgχ0k(1)

=g3αs

64πs2wc2w

2∑

i=1

4∑

k=1

Im(

(AqN )k,i(B

qN )†i,k

)mχ0k

m2qi

B

(m2χ0k

m2qi

)

, (3.3)

Here eu = 2/3, ed = −1/3, α = e2/(4π), αs = g23/(4π), sW = sin θW , cW = cos θW ,

θW is the Weinberg angle. mqi (i = 1, 2) are the scalar quarks masses and mχ0k(k =

1, 2, 3, 4) denote the eigenvalues of neutralino mass matrix. We define the one loop

function B(r) as B(r) = [2(r − 1)2]−1[1 + r + 2r ln r/(r − 1)]. The concrete forms of the

coupling constants (AuN )k,i, (B

uN )k,i, (A

dN )k,i, (B

dN )k,i can be found in ref. [41].

The contributions from the one loop gluino-squark diagrams are written as

dγg(1) = − 2

3πeqeαs

2∑

i=1

Im(

(Zq)2,i(Z†q )i,1e

−iθ3) |mg|m2

qi

B

( |mg|2m2

qi

)

,

– 6 –

JHEP10(2013)020

g t t

Figure 2. The MSSM two loop diagram contributing to the Weinberg operator by attaching three

gluons in all possible ways.

dgg(1) =

g3αs

4π

2∑

i=1

Im(

(Zq)2,i(Z†q )i,1e

−iθ3) |mg|m2

qi

C

( |mg|2m2

qi

)

, (3.4)

with θ3 denoting the phase of the gluino mass mg and the loop function C(r) = [6(r −1)2]−1[10r − 26− (2r − 18) ln r/(r − 1)]. Zq are the mixing matrices of the squarks, which

diagonalize the mass squared matrixes of squarks Z†qm

2qZq = diag(m2

q1, m2

q2).

We give the chargino-squark contributions here

dγχ±

k(1)

=eα

4πs2wV †qQVQq

∑

i,k

Im(

(AQC)k,i(B

QC )

†i,k

)mχ±

k

m2Qi

×[

eQB

(m2χ±

k

m2Qi

)

+ (eq − eQ)A

(m2χ±

k

m2Qi

)]

,

dgχ±

k(1)

=g3α

4πs2wV †qQVQq

∑

i,k

Im(

(AQC)k,i(B

QC )

†i,k

)mχ±

k

m2Qi

B

(m2χ±

k

m2Qi

)

, (3.5)

where mχ±

k(k = 1, 2) denote the chargino masses, V is the CKM matrix, and the concrete

form of A(r) is A(r) = [2(1 − r)2]−1[3 − r + 2 ln r/(1 − r)]. One can obtain the concrete

forms of (AdC)k,i, (B

dC)k,i, (A

uC)k,i, (B

uC)k,i in the literature [41].

In MSSM, the two-loop “gluino-squark” diagram shown in figure 2 gives contributions

to the gluonic Weinberg operator, whose Wilson coefficient is [41–43]

CSUSY5 = −3αsmt

( g34π

)3Im(

(Zt)2,2(Zt)†2,1

)m2t1−m2

t2

|mg|5H

( m2t1

|mg|2,m2

t2

|mg|2,

m2t

|mg|2)

(3.6)

Here, the H function can be found in [42, 43]. Zt is the matrix to diagonalize the mass

squared matrix of stop. Up to now, in MSSM the dominant contributions to the neutron

EDM are complete.

3.2 The corrections to the neutron EDM from superfields in BLMSSM

In BLMSSM, the quark, exotic quark and superfields X have a coupling at tree level in

eq. (2.13) to avoid stability for the exotic quarks. Therefore, there are one loop triangle

– 7 –

JHEP10(2013)020

X

q qq′

Figure 3. The one loop selfenergy diagram from which the corresponding triangle diagrams are

obtained by attaching a photon or a gluon in all possible ways.

g Q q′ q′ H0i q′

b′ H+ t′ q′ X q

Figure 4. The BLMSSM two loop diagrams contributing to the Weinberg operator by attaching

three gluons in all possible ways.

diagrams obtained by attaching a photon or a gluon in all possible ways on the self energy

diagram in figure 3. Then we get the quark EDMs and CEDMs.

dγXj= − eqe

64π2

2∑

i,j=1

mqi+3

m2Xj

Im(

(NLq′ )ij(NR

q′ )†ji

)

A

(

m2qi+3

m2Xj

)

,

dgXj= − g3

64π2

2∑

i,j=1

mqi+3

m2Xj

Im(

(NLq′ )ij(NR

q′ )†ji

)

A

(

m2qi+3

m2Xj

)

, (3.7)

where q = t(b), q′ = t′(b′), et′ = 2/3, eb′ = −1/3, mXj(j = 1, 2) denote the masses

of the superfields X, mti+3(i = 1, 2) represent the masses of the up type exotic quarks,

mbi+3(i = 1, 2) are the masses of the down type exotic quarks.

The Weinberg operator comes from the two loop gluon diagrams. In this model, there

are four exotic quarks and eight exotic scalar quarks, which leads to a lot of diagrams

contributing to the Wilson coefficient C5(Λ) of the Weinberg operator. These two loop

diagrams are drawn in figure 4. The two loop “gluino-exotic quark-exotic scalar quark”

diagrams have corrections to the Wilson coefficient of the purely gluonic Weinberg operator,

– 8 –

JHEP10(2013)020

and the results are presented here.

CΛt′U5 = 3αs

(

g34π

)3 2∑

i=1

mti+3

[

Im(

(NLU )†i2(NR

U )2i

)m2U1

−m2U2

|mg|5H

( m2U1

|mg|2,m2

U2

|mg|2,m2

ti+3

|mg|2)

+Im(

(NLU )†i4(NR

U )4i

)m2U3

−m2U4

|mg|5H

( m2U3

|mg|2,m2

U4

|mg|2,m2

ti+3

|mg|2)]

,

CΛb′D5 = 3αs

( g34π

)32∑

i=1

mbi+3

[

Im(

(NLD )†i2(NR

D )2i

)m2D1

−m2D2

|mg|5H

( m2D1

|mg|2,m2

D2

|mg|2,m2

bi+3

|mg|2)

+Im(

(NLD )†i4(NR

D )4i

)m2D3

−m2D4

|mg|5H

( m2D3

|mg|2,m2

D4

|mg|2,m2

bi+3

|mg|2)]

. (3.8)

Here, mUi(i = 1, 2, 3, 4) denote the masses of the up type exotic scalar quarks, and mDi

(i =

1, 2, 3, 4) are the masses of the down type exotic scalar quarks.

From the couplings of neutral Higgs and exotic quarks in eq. (2.4), we obtain their

contributions to the Wilson coefficient C5(Λ).

CH0t′t′

5 = 2g33

(

1

4π

)4 8∑

α=1

[ 2∑

i=1

Im(

(NRH0

α)ii(NL

H0α)†ii

) 1

m2ti+3

h(mti+3,mH0

α)

+Im(

(NRH0

α)12(NL

H0α)†21

) 1

mt4mt5

h′(mt4 ,mt5 ,mH0α)

]

,

CH0b′b′

5 = 2g33

(

1

4π

)4 8∑

α=1

[ 2∑

i=1

Im(

(KRH0

α)ii(KL

H0α)†ii

) 1

m2bi+3

h(mbi+3,mH0

α)

+Im(

(KRH0

α)12(KL

H0α)†21

) 1

mb4mb5

h′(mb4 ,mb5 ,mH0α)

]

, (3.9)

where the functions h(x, y) and h′(x, y, z) are given in [42, 43]. In the same way, the

contributions from charged Higgs and exotic quarks are written in the following form.

CH+t′b′

5 = 2g33

(

1

4π

)4 2∑

i,j=1

1

mti+3mbj+3

[

Im(

(NRH+)ij(NL

H+)†ji

)

h′(mti+3,mbj+3

,mH+)

+Im(

(NRG+)ij(NL

G+)†ji

)

h′(mti+3,mbj+3

,mG+)

]

. (3.10)

The coupling of quark-exotic quark-superfield X in eq. (2.13) can not only give the

corrections to the neutron EDM at one loop level, but also contribute to the Wilson

coefficient C5(Λ) at two loop level.

CXq′q5 = 2g33

(

1

4π

)4 2∑

i,j=1

[

Im(

(NRt′ )ij(NL

t′ )†ji

) 1

mti+3mt

h′(mti+3,mt,mXj

)

+Im(

(NRb′ )ij(NL

b′ )†ji

) 1

mbi+3mb

h′(mbi+3,mb,mXj

)

]

. (3.11)

The results obtained at the matching scale Λ should be transformed down to the

chirality breaking scale Λχ [44, 45]. So the renormalization group equations (RGEs) for

– 9 –

JHEP10(2013)020

the Wilson coefficients of the Weinberg operator and the quark EDMs, CEDMs should be

solved. The relations between the results at two deferent scales Λ and Λχ are presented here.

dγq (Λχ) = 1.53dγq (Λ), dgq(Λχ) = 3.4dgq(Λ), C5(Λχ) = 3.4C5(Λ) , (3.12)

There are three type contributions to the quark EDM, 1 the electroweak contribution dγq , 2

the CDEM contribution dgq , 3 the Weinberg operator contribution. Each type contribution

can be calculated numerically. At a low scale, the quark EDM can be obtained from dγq , dgq

and C5(Λχ) by the following formula [46].

dq = dγq +e

4πdgq +

eΛχ

4πC5(Λχ) . (3.13)

From the quark model, the EDM of neutron is derived from u quark’s EDM du and d

quark’s EDM dd with the following expression

dn =1

3(4dd − du). (3.14)

4 The numerical analysis

The lightest neutral CP even Higgs is discovered by LHC, with a mass mh0 ≃ 125.9 GeV,

which gives the most stringent constraint on the parameter space. Furthermore, the ATLAS

and CMS experiments announce the Higgs production and decay into diphoton channel are

larger than SM predictions with a factor 1.2 ∼ 2. The current values of the ratios are [15–

17, 47]: Rγγ = 1.6 ± 0.4 , RV V ∗ = 0.94 ± 0.40 , (V = Z, W ). CMS collaboration declare

that it is impossible to find a SM Higgs with mass from 127.5 to 600GeV. Considering all

the above constraints, we obtain a parameter space in our recent work [18].

In this work to obtain the numerical results, the relevant parameters in SM are shown

here.

αs(mZ) = 0.118, α(mZ) = 1/128, s2W(mZ) = 0.23,

mW = 80.4 GeV, mt = 174.2 GeV, mb = 4.2 GeV,

mu = 2.3× 10−3 GeV, md = 4.8× 10−3 GeV, (4.1)

To simplify the numerical discussion, we suppose At, Ab, Aν4 , Ae4 , Aν5 , Aν5 , Au4, Au5

, Ad4 ,

Ad5 , ABQ, ABU , ABD and Yu4, Yu5

, Yd4 , Yd5 are all real parameters.

The other parameters are adopted [18].

tanβ = tanβB = tanβL = 2, B4 = L4 =3

2,

mQ3= mU3

= mD3= 1 TeV, mZB

= mZL= 1 TeV,

mU4= mD4

= mQ5= mU5

= mD5= 1 TeV, mQ4

= 790 GeV,

mL4= mν4 = mE4

= mL5= mν5 = mE5

= 1 TeV ,

Aν4 = Ae4 = Aν5 = Aν5 = Au4= Au5

= Ad4 = Ad5 = 550 GeV ,

υBt=√

υ2B + υ2B = 3 TeV , υLt=√

υ2L + υ2L = 3 TeV ,

– 10 –

JHEP10(2013)020

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

Θ3�Π

d n´

1025He

.cmL

Figure 5. As BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, |µX | = 2400 GeV, θX = 0 and mQ1= mU1

=

mD1= MQ, the solid-line and dashed-line represent dn varying with θ3, for MQ = 1000GeV and

1200GeV respectively.

ABQ = ABU = ABD = −Ab = −At = 1 TeV, m1 = TeV,

Yu4= 0.76Yt, Yd4 = 0.7Yb, Yu5

= 0.7Yb, Yd5 = 0.13Yt,

m2 = 750 GeV , µB = 500 GeV, Λ = 1 TeV, Λχ = 1 GeV,

λQ = λu = λd = 0.5, |mg| = 300 GeV, BB = 1 TeV,

mν4 = mν5 = 90 GeV, me4 = me5 = 100 GeV, mµ = −800 GeV. (4.2)

The CP violating phase θ3 of gluino affects the neutron EDM obviously. Supposing

BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, |µX | = 2400 GeV and the CP violating phase of

µX is zero (θX = 0), we plot the neutron EDM dn varying with the phase θ3 in figure 5,

with the condition mQ1= mU1

= mD1= MQ. When MQ = 1000 GeV and θ3 = ±π

2 , the

absolute value of the neutron EDM dn reaches the biggest value 1.04×10−25 (e.cm), which

is very close to the present experiment upper limit. For the same value of θ3, the absolute

value of dn with MQ = 1200 GeV is smaller than that with MQ = 1000 GeV.

Then one finds MQ affects the theoretical results strongly. Taking BX =

300 GeV, λ1 = λ2 = λ3 = 0.1, |µX | = 2400 GeV, θX = 0, θ3 = π2 (

π4 ), and mQ1

= mU1=

mD1= MQ, the neutron EDM dn varying with MQ is plotted in figure 6. Obviously, dn

becomes small quickly, with the enlarging MQ in the region from 1000 GeV to 1500 GeV.

When MQ = 2000 GeV, dn turns very small, whose value is around 5× 10−28 (e.cm).

The superfields X can give contributions to the neutron EDM dn at one-loop level, so

the parameters having relation with the mass squared matrix of superfields X and the cou-

plings of superfields X-exotic quark-quark can affect the theoretical predictions strongly.

With the assumption in the parameter space BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, θ3 = 0

and mQ1= mU1

= mD1= 1000 GeV, we investigate the relation between dn and the CP

violating phase θX numerically in figure 7. The shapes of the diagrams in figure 7 are very

similar as those in figure 5. The biggest absolute value of dn is obtained with θX = π2 and

– 11 –

JHEP10(2013)020

1000 1200 1400 1600 1800 20000.0

0.2

0.4

0.6

0.8

1.0

MQ�GeV

d n´

1025He

.cmL

Figure 6. As BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, |µX | = 2400 GeV, θX = 0 and mQ1=

mU1= mD1

= MQ, the solid-line and dashed-line represent dn varying with MQ, for θ3 = π2and

π4respectively.

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

ΘX �Π

d n´

1025He

.cmL

Figure 7. As BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, θ3 = 0 and mQ1= mU1

= mD1= 1000 GeV,

the solid-line and dashed-line represent dn varying with θX , for |µX | = 2400 GeV and |µX | =3000 GeV respectively.

|µX | = 2400 GeV. When |µX | = 3000 GeV, the absolute value of dn is smaller than that

with |µX | = 2400 GeV.

In figure 8, we can easily find the neutron EDM turns smaller and smaller, as the ab-

solute value of µX becomes larger and larger. The reason should be that some imaginary

parts come from the non-diagonal elements in the diagonalizing matrix ZX . Large diago-

nalized elements in the mass squared matrix of the superfields X lead to small non-diagonal

elements in ZX . Therefore, we obtain the small theoretical prediction on the neutron EDM.

Choosing λ1 = λ2 = λ3 = 0.1, θ3 = 0, θX = π2 , mQ1

= mU1= mD1

= 1000 GeV, we

plot dn varying with the parameter BX with |µX | = 2400(3000) GeV in figure 9. From

– 12 –

JHEP10(2013)020

2500 3000 3500 40000.0

0.2

0.4

0.6

0.8

1.0

ÈΜX È�GeV

d n´

1025He

.cmL

Figure 8. As BX = 300 GeV, λ1 = λ2 = λ3 = 0.1, θ3 = 0 and mQ1= mU1

= mD1= 1000 GeV,

the solid-line and dashed-line represent dn varying with |µX |, for θX = π2and π

4respectively.

50 100 150 200 250 3000.0

0.2

0.4

0.6

0.8

1.0

BX �GeV

d n´

1025He

.cmL

Figure 9. As λ1 = λ2 = λ3 = 0.1, θ3 = 0, θX = π2

and mQ1= mU1

= mD1= 1000 GeV,

the solid-line and dashed-line represent dn varying with BX , for |µX | = 2400 GeV and 3000GeV

respectively.

figure 9, the neutron EDM dn is almost the linear increasing function of BX . Because BX

is the non-diagonal element in the mass squared matrix of the superfields X, large BX gives

big contribution to imaginary elements in the matrix ZX .

From the Lagrangian of quark-exotic quark and the superfields X in eqs. (2.13)

and (2.14), one finds λ1, λ2, λ3 are important parameters, and they affect those coupling

strongly. So, we analyse how λ1, λ2, λ3 influence the neutron EDM with the hypothesis

in parameter space as BX = 300 GeV, θ3 = 0, θX = π2 , mQ1

= mU1= mD1

= 1000 GeV

and λ1 = λ2 = λ3 = Lam. The numerical results are plotted in the figure 10 for

µX = 2400(3000) GeV. When Lam is very small, neutron EDM dn is almost zero. The

diagrams in figure 10 is symmetrical for the axis Lam = 0.

– 13 –

JHEP10(2013)020

-0.10 -0.05 0.00 0.05 0.100.0

0.2

0.4

0.6

0.8

1.0

Lam

d n´

1025He

.cmL

Figure 10. As BX = 300 GeV, θ3 = 0, θX = π2, mQ1

= mU1= mD1

= 1000 GeV and λ1 = λ2 =

λ3 = Lam, the solid-line and dashed-line represent dn varying with Lam, for |µX | = 2400GeV and

3000GeV respectively.

5 Summary

We study the neutron EDM in the framework of the BLMSSM. The constraints from the

experiment data on the lightest CP-even Higgs mass (mh0 = 125.9 GeV) and Higgs decay

into diphoton (ZZ∗, WW ∗) are all considered. Furthermore the confine on the heaviest

CP-even Higgs and CP-odd Higgs mass is also taking into account (mA0 ,mH0) > 600 GeV.

We analyse how the neutron EDM depends on the related parameters. If the new physics

mass at TeV range, the present experiment upper bound on the neutron EDM gives a

serious constraint on the CP violating phase and the BLMSSM parameters. Our numerical

results show that in some parameter space, the neutron EDM dn can reach it’s present

experimental upper bound 1.1× 10−25(e.cm).

Acknowledgments

The work has been supported by the National Natural Science Foundation of China

(NNSFC) with Grant No. 11275036, No. 11047002, the Fund of Natural Science Foun-

dation of Hebei Province (A2011201118, A2013201277), the open project of State Key

Laboratory of Chinese Academy of Sciences Institute of theoretical physics (Y3KF311CJ1)

and Natural Science Fund of Hebei University with Grant No. 2011JQ05, No. 2012-242.

A The couplings between Higgs and exotic quarks

The coupling constants (NLH0

α)(α = 1, 2 · · · 8) for neutral Higgs and up type exotic quarks

are expressed as

(NLH0

1

)ij =1√2

[

Yu4(W †

t )i2(Ut)1j cosα+ Yu5(W †

t )i1(Ut)2j sinα]

,

– 14 –

JHEP10(2013)020

(NRH0

1

)ij =1√2

[

Y ∗u4(U †

t )i1(Wt)2j cosα+ Y ∗u5(U †

t )i2(Wt)1j sinα]

,

(NLH0

2

)ij =1√2

[

Yu4(W †

t )i2(Ut)1j sinα− Yu5(W †

t )i1(Ut)2j cosα]

,

(NRH0

2

)ij =1√2[Y ∗

u4(U †

t )i1(Wt)2j sinα− Y ∗u5(U †

t )i2(Wt)1j cosα]

,

(NLH0

3

)ij =i√2

[

Yu4(W †

t )i2(Ut)1j cosβ + Yu5(W †

t )i1(Ut)2j sinβ]

,

(NRH0

3

)ij = − i√2

[

Y ∗u4(U †

t )i1(Wt)2j cosβ + Y ∗u5(U †

t )i2(Wt)1j sinβ]

,

(NLH0

4

)ij =i√2

[

Yu4(W †

t )i2(Ut)1j sinβ − Yu5(W †

t )i1(Ut)2j cosβ]

,

(NRH0

4

)ij = − i√2

[

Y ∗u4(U †

t )i1(Wt)2j sinβ − Y ∗u5(U †

t )i2(Wt)1j cosβ]

,

(NLH0

5

)ij = − 1√2

[

λu(W†t )i2(Ut)2j cosαB − λQ(W

†t )i1(Ut)1j sinαB

]

,

(NRH0

5

)ij = − 1√2

[

λ∗u(U

†t )i2(Wt)2j cosαB − λ∗

Q(U†t )i2(Wt)2j sinαB

]

,

(NLH0

6

)ij = − 1√2[λu(W

†t )i2(Ut)2j sinαB + λQ(W

†t )i1(Ut)1j cosαB

]

,

(NRH0

6

)ij = − 1√2[λ∗

u(U†t )i2(Wt)2j sinαB + λ∗

Q(U†t )i2(Wt)2j cosαB

]

,

(NLH0

7

)ij = − i√2

[

λu(W†t )i2(Ut)2j cosβB − λQ(W

†t )i1(Ut)1j sinβB

]

,

(NRH0

7

)ij =i√2

[

λ∗u(U

†t )i2(Wt)2j cosβB − λ∗

Q(U†t )i2(Wt)2j sinβB

]

,

(NLH0

8

)ij = − i√2

[

λu(W†t )i2(Ut)2j sinβB + λQ(W

†t )i1(Ut)1j cosβB

]

,

(NRH0

8

)ij =i√2

[

λ∗u(U

†t )i2(Wt)2j sinβB + λ∗

Q(U†t )i2(Wt)2j cosβB

]

. (A.1)

We show the coupling constants (NLH0

α)(α = 1, 2 · · · 8) for neutral Higgs and down type

exotic quarks here.

(KLH0

1

)ij =1√2

[

Yd4(W†b )i2(Ub)1j sinα− Yd5(W

†b )i1(Ub)2j cosα

]

,

(KRH0

1

)ij =1√2[Y ∗

d4(U †

b )i1(Wb)2j sinα− Y ∗d5(U †

b )i2(Wb)1j cosα]

,

(KLH0

2

)ij = − 1√2

[

Yd4(W†b )i2(Ub)1j cosα+ Yd5(W

†b )i1(Ub)2j sinα

]

,

(KRH0

2

)ij = − 1√2

[

Y ∗d4(U †

b )i1(Wb)2j cosα+ Y ∗d5(U †

b )i2(Wb)1j sinα]

,

(KLH0

3

)ij =i√2

[

Yd4(W†b )i2(Ub)1j sinβ − Yd5(W

†b )i1(Ub)2j cosβ

]

,

(KRH0

3

)ij = − i√2

[

Y ∗d4(U †

b )i1(Wb)2j sinβ − Y ∗d5(U †

b )i2(Wb)1j cosβ]

,

– 15 –

JHEP10(2013)020

(KLH0

4

)ij = − i√2

[

Yd4(W†b )i2(Ub)1j cosβ + Yd5(W

†b )i1(Ub)2j sinβ

]

,

(KRH0

4

)ij =i√2

[

Y ∗d4(U †

b )i1(Wb)2j cosβ + Y ∗d5(U †

b )i2(Wb)1j sinβ]

,

(KLH0

5

)ij = − 1√2

[

λd(W†b )i2(Ub)2j cosαB + λQ(W

†b )i1(Ub)1j sinαB

]

,

(KRH0

5

)ij = − 1√2

[

λ∗d(U

†b )i2(Wb)2j cosαB + λ∗

Q(U†b )i2(Wb)2j sinαB

]

,

(KLH0

6

)ij = − 1√2

[

λd(W†b )i2(Ub)2j sinαB + λQ(W

†b )i1(Ub)1j cosαB

]

,

(KRH0

6

)ij = − 1√2

[

λ∗d(U

†b )i2(Wb)2j sinαB + λ∗

Q(U†b )i2(Wb)2j cosαB

]

,

(KLH0

7

)ij = − i√2

[

λd(W†b )i2(Ub)2j cosβB + λQ(W

†b )i1(Ub)1j sinβB

]

,

(KRH0

7

)ij =i√2

[

λ∗d(U

†b )i2(Wb)2j cosβB + λ∗

Q(U†b )i2(Wb)2j sinβB

]

,

(KLH0

8

)ij = − i√2

[

λd(W†b )i2(Ub)2j sinβB + λQ(W

†b )i1(Ub)1j cosβB

]

,

(KRH0

8

)ij =i√2

[

λ∗d(U

†b )i2(Wb)2j sinβB + λ∗

Q(U†b )i2(Wb)2j cosβB

]

. (A.2)

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