The one loop contributions to c(t) electric dipole moment in the CP violating BLMSSM

# The one loop contributions to c(t) electric dipole moment in the CP violating BLMSSM

Shu-Min Zhao111zhaosm@hbu.edu.cn, Tai-Fu Feng222fengtf@hbu.edu.cn, Zhong-Jun Yang, Hai-Bin Zhang, Xing-Xing Dong, Tao-Guo Department of Physics, Hebei University, Baoding 071002, China
School of Mathematics and Science, Hebei University of Geosciences, Shijiazhuang 050031, China
July 14, 2019
###### Abstract

In the CP violating supersymmetric extension of the standard model with local gauged baryon and lepton symmetries(BLMSSM), there are new CP violating sources which can give new contributions to the quark electric dipole moment (EDM). Considering the CP violating phases, we analyze the EDMs of the quarks c and t. We take into account the contributions from the one loop diagrams. The numerical results are analyzed with some assumptions on the relevant parameter space. The numerical results for the c and t EDMs can reach large values.

CP violating, electric dipole moment, local gauge symmetry
###### pacs:
11.30.Er, 12.60.Jv,14.80.Cp

## I Introduction

The CP violation found in the K- and B system CPVKB () can be well explained in the standard model. It is well known that, the electric dipole moment(EDM) of elementary particle is a clear sinal of CP violationcp1 (). The Cabbibo-Kobayashi-Maskawa(CKM) phase is the only source of CP violation in the SM, which has ignorable effect on the EDM of elementary particle. In the SM, even to two loop order, the EDM of a fermion does not appear, and there are partial cancelation between the three loop contributionsSMTwoLoop (). If EDM of an elementary fermion is detected, one can confirm there are new CP phases and physics beyond the SM.

Though SM has obtained large successes with the detection of the lightest CP-even Higgs Higgs (), it is unable to explain some phenomena. Physicists consider the SM should be a low energy effective theory of a large model. The minimal supersymmetric extension of the standard model(MSSM) is very favorite and people have been interested in it for a long timeMSSM (). There are also many new models beyond the SM, such as SSMmunuSSM (). Generally speaking, the new models introduce new CP-violating phases that can affect the EDMs of fermions, mixing et al. The EDMs of electron and neutron are strict constraints on the CP-violating phasesEDM (). In the models beyond SM, there are new CP-violating phases which can give large contributions to electron and neutron EDMsNEEDM (). To make the MSSM predictions of electron and neutron EDMs under the experiment upper bounds, there are three possibilitiesxiangxiao (): 1 the CP-violating phases are very small, 2 varies contributions cancel with each other in some special parameter spaces, 3 the supersymmetry particles are very heavy at several TeV order.

Taking into account the local gauged and , people obtain the minimal supersymmetric extension of the SM, which is the so called BLMSSMBLMSSM1 (). The authors in Ref.BLMSSM2 () first proposed BLMSSM, where they studied some phenomena. At TeV scale, the local gauge symmetries of BLMSSM breaks spontaneously. Therefore, in BLMSSM R-parity is violated and the asymmetry of matter-antimatter in the universe can be explained. We have studied the lightest CP-even Higgs mass and the decays weBLMSSM () in the BLMSSM, where some other processesweBLNCP () are also researched. Taking the CP-violating phases with nonzero values, the neutron EDM, lepton EDM and mixing are researched in this modelweBLCPV ().

From neutron experimental data, the bounding of top EDM is analyzedTopEDM (). Taking into account the precise measurements of the electron and neutron EDMs, the upper limits of heavy quark EDMs are also discussedHQEDM (). The upper limits on the EDMs of heavy quarks are researched from annihilationHQEEEDM (). In the CP-violating MSSM, the authors study c quark EDM including two loop gluino contributionsBFEDM (). There are also other works on the c quark EDMOCEDM (). Considering the pre-existing works, the upper bounds of EDMs for c and t are about and . In this work, we calculate the EDMs of charm quark and top quark in the framework of the CP-violating BLMSSM. At low energy scale, the quark chromoelectric dipole moment(CEDM) can give important contributions to the quark EDM. So, we also study the quark CEDM with the renormalization group equations.

In Section 2, we briefly introduce the BLMSSM and show the needed mass matrices and couplings, after this introduction. The EDMs(CEDMs) of c and t are researched in Section 3. In Section 4, we give out the input parameters and calculate the numerical results. The last Section is used to discuss the results and the allowed parameter space.

## Ii The BLMSSM

Considering the local gauge symmetries of B(L) and enlarging the local gauge group of the SM to one can obtain the BLMSSM modelBLMSSM1 (). In the BLMSSM, there are the exotic superfields including the new quarks , , , , , , and the new leptons , , , , , to cancel the and anomalies.

With the detection of the lightest CP even Higgs at LHCHiggs (), Higgs mechanism is very convincing for particle physics, and BLMSSM is based on the Higgs mechanism. The introduced Higgs superfields and break lepton number and baryon number spontaneously. These Higgs superfields acquire nonzero vacuum expectation values (VEVs) and provide masses to the exotic leptons and exotic quarks. To make the heavy exotic quarks unstable the superfields , are introduced in the BLMSSM.

The doublets obtain nonzero VEVs ,

 Hu=⎛⎝H+u1√2(υu+H0u+iP0u)⎞⎠,    Hd=⎛⎝1√2(υd+H0d+iP0d)H−d⎞⎠. (1)

The singlets and obtain nonzero VEVs and respectively,

 ΦB=1√2(υB+Φ0B+iP0B),         φB=1√2(¯¯¯υB+φ0B+i¯¯¯¯P0B). ΦL=1√2(υL+Φ0L+iP0L),          φL=1√2(¯¯¯υL+φ0L+i¯¯¯¯P0L). (2)

Therefore, the local gauge symmetry breaks down to the electromagnetic symmetry .

We show the superpotential of BLMSSM weBLMSSM ()

 WBLMSSM=WMSSM+WB+WL+WX, WB=λQ^Q4^Qc5^ΦB+λU^Uc4^U5^φB+λD^Dc4^D5^φB+μB^ΦB^φB +Yu4^Q4^Hu^Uc4+Yd4^Q4^Hd^Dc4+Yu5^Qc5^Hd^U5+Yd5^Qc5^Hu^D5, WL=Ye4^L4^Hd^Ec4+Yν4^L4^Hu^Nc4+Ye5^Lc5^Hu^E5+Yν5^Lc5^Hd^N5 +Yν^L^Hu^Nc+λNc^Nc^Nc^φL+μL^ΦL^φL, WX=λ1^Q^Qc5^X+λ2^Uc^U5^X′+λ3^Dc^D5^X′+μX^X^X′, (3)

with representing the superpotential of the MSSM. The soft breaking terms of the BLMSSM are collected hereBLMSSM1 (); weBLMSSM ().

are the soft breaking terms of MSSM.

### ii.1 mass matrix

From the soft breaking terms and the scalar potential, we deduce the mass squared matrix for superfields X.

 −LX=(X∗   X′)(|μX|2+SX−μ∗XB∗X−μXBX|μX|2−SX)(XX′∗), SX=g2B2(23+B4)(v2B−¯v2B). (5)

We diagonalize the mass squared matrix for the superfields X through the unitary transformation,

 (X1X2)=Z†X(XX′∗),      Z†X(|μX|2+SX−μ∗XB∗X−μXBX|μX|2−SX)ZX=(m2X100m2X2). (6)

and are the superpartners of the scalar superfields and . and can composite four-component Dirac spinors, whose mass term are given outweBLCPV ()

 −Lmass~X=μX¯~X~X,      ~X=(ψX¯ψX′), (7)

with denoting the mass of .

In the BLMSSM, there are the new baryon boson, the singlets and . Their superpartners are respectively , and , and they mix together producing 3 baryon neutralinos. In the base , the mass mixing matrix is obtained and diagonalized by the rotation matrix DCPC ().

 MBN=⎛⎜⎝2mB−vBgB¯vBgB−vBgB0−μB¯vBgB−μB0⎞⎟⎠,     χ0Bi=(k0Bi¯k0Bi), iλB=Z1iNBk0Bi,      ψΦB=Z2iNBk0Bi,      ψφB=Z3iNBk0Bi. (8)

represent the mass eigenstates of baryon neutralinos.

The exotic quarks with charged 2/3 is in four-component Dirac spinors, whose mass matrix reads asweBLMSSM ()

 −Lmasst′=(¯t′4R   ¯t′5R)⎛⎜⎝1√2λQvB−1√2Yu5vd−1√2Yu4vu1√2λu¯vB⎞⎟⎠(t′4Lt′5L), (9)

Using the unitary transformations, the two mass eigenstates of exotic quarks with charged 2/3 are obtained by the rotation matrices and ,

 (t4Lt5L)=U†t(t′4Lt′5L),        (t4Rt5R)=W†t(t′4Rt′5R), W†t⎛⎜⎝1√2λQvB−1√2Yu5vd−1√2Yu4vu1√2λu¯vB⎞⎟⎠Ut=diag(mt4,mt5). (10)

The mass squared matrix for charged 2/3 exotic squarks is obtained in our previous workweBLMSSM (). For saving space in the work, we do not show it here. is diagonalized by through the formula .

### ii.2 needed couplings

To study quark EDMs, the couplings between photon (gluon) and exotic quarks(exotic squarks) are necessary. We derive the couplings between photon (gluon) and exotic quarks.

 Lγ(g)q′q′=−2e32∑i=1¯ti+3γμti+3Fμ−g32∑i=1¯ti+3Taγμti+3Gaμ, (11)

with and representing electromagnetic field and gluon field respectively. are the strong gauge group generators. Similarly, the couplings between photon (gluon) and exotic squarks are also deduced

 Lγ(g)~q′~q′=−23e4∑j,β=1δjβFμ~U∗ji~∂μ~Uβ−g3Ta4∑j,β=1δjβGaμ~U∗ji~∂μ~Uβ. (12)

From the superpotential , one can find there are interactions at tree level for quark, exotic quark and X. The needed Yukawa interactions can be deduced from the superpotential . The couplings of quark-exotic quark-X are shown in the mass basis,

 LXt′u=2∑i,j=1((NLt′)ijXj¯ti+3PLuI+(NRt′)ijXj¯ti+3PRuI)+h.c. (NLt′)ij=−λ1(W†t)i1(ZX)1j,        (NRt′)ij=−λ∗2(U†t)i2(ZX)2j. (13)

From the superpotential , in the same way we can also obtain another type Yukawa couplings (quark-exotic squark-)weBLCPV ().

 L¯u~X~U=−4∑i=1¯u(λ1(Z~t′)∗3iPL+λ2(Z~t′)4iPR)~X~Ui. (14)

Beyond the MSSM, there are couplings for baryon neutralino, quarks and squarks. They are deduced in our previous workDCPC (), and can give new contributions to the quark EDMs.

 L(χ0Bq~q)=3∑I,i=16∑j=1√23gB¯χB0i(Z1iNBZIj∗~UPL−Z1i∗NBZ(I+3)j∗~UPR)uI~U∗j+H.c. (15)

## Iii Formulation

Using the effective LagrangianneuEDM () method, one obtains the fermion EDM from

 LEDM=−i2df¯¯¯fσμνγ5fFμν, (16)

with representing the electromagnetic field strength, denoting a fermion field. It is obviously that this effective Lagrangian is CP-violating. In the fundamental interactions, this CP-violating Lagrangian can not be obtained at tree level. Considering the CP-violating electroweak theory, one can get this effective Lagrangian from the loop diagrams. The chromoelectric dipole moment (CEDM) of quark can also give contribution to the quark EDM. denotes the gluon field strength.

To describe the CP-violating operators obtained from loop diagrams, the effective method is convenient. The coefficients of the quark EDM and CEDM at the matching scale should be evolved down to the quark mass scale with the renormalization group equations. At matching scale, we can obtain the effective Lagrangian with the CP-violating operators. The effective Lagrangian containing operators relating with the quark EDM and CEDM are

 Leff=4∑iCi(Λ)Oi(Λ), O1=¯¯¯qσμνPLqFμν,          O2=¯¯¯qσμνPRqFμν, O3=¯¯¯qTaσμνPLqGaμν,      O4=¯¯¯qTaσμνPRqGaμν. (17)

with representing the energy scale, where the Wilson coefficients are evaluated.

In our previous workweBLCPV (), we have studied the neutron EDM in the CP-violating BLMSSM, where the contributions from baryon neutralino-squark and -exotic squark are neglected, because they are all small in the used parameter space. Here we take into account all the contributions at one loop level to study the c and t EDMs. In the CP-violating BLMSSM, the one-loop corrections to the quark EDMs and CEDMs can be divided into six types according to the quark self-energy diagrams. We divide the quark self-energy diagrams according to the virtual particles as: 1 gluino-squark, 2 neutralino-squark, 3 chargino-squark, 4 X-exotic quark, 5 baryon neutralino-squark, 6 -exotic squark.

From the quark selfenergy diagrams, one obtains the needed triangle diagrams by attaching a photon or gluon on the internal lines in all possible ways. After the calculation, we obtain the effective Lagrangian contributing to the quark EDMs and CEDMs. The BLMSSM is larger than MSSM and includes the MSSM contributions. In Fig.(6), we plot all the one loop self energy diagrams of the up-type quark.

In this section, we show the one loop corrections to the quark EDMs (CEDMs). The one loop chargino-squark contributions are

Here , is the Weinberg angle, is the CKM matrix. We define the one loop functions and asxiangxiao ()

 A(r)=[2(1−r)2]−1[3−r+2lnr/(1−r)], B(r)=[2(r−1)2]−1[1+r+2rlnr/(1−r)]. (19)

are the squarks masses and denote the eigenvalues of neutralino mass matrix. is the rotation matrix to diagonalize the mass squared matrix for the down type squark. and are the rotation matrices to obtain the mass eigenstates of charginos.

We show the gluino-squark corrections to the quark EDMs and CEDMs

 dγ~g(uI)=−49πeαs6∑i=1Im((Z~U)(I+3)i(Z~U)Ii∗e−iθ3)|m~g|m2~UiB(|m~g|2m2~Ui), dg~g(uI)=g3αs4π6∑i=1Im((Z~U)(I+3)i(Z~U)Ii∗e−iθ3)|m~g|m2~UiC(|m~g|2m2~Ui), (20)

with . is the matrix for the up type squarks, with the definition . The concrete form of the loop function isxiangxiao ()

 C(r)=[6(r−1)2]−1[10r−26−(2r−18)lnr/(r−1)]. (21)

To check the functions and in the Ref.xiangxiao (), we calculate the one loop triangle diagrams using the effective Lagrangian method. In the calculation, we use the approximation

 1(k+p)2−m2=1−2k⋅p+p2k2−m2+4(k⋅p)2(k2−m2)2, (22)

with k representing the loop integral momentum and p representing the external momentum. It is reasonable because the internal particles are at the order of TeV, and the external quark is lighter than TeV, even for t quark. The ratio is small enough to use the approximation formula.

For the diagram that the photon is just attached on the internal charged Fermions, our result corresponding to is the function

 a[F,S]=Λ2NPiπ2∫dk4k2(k2−m2F)3(k2−m2S) =−F2+2S2log(F)−4FS+3S2−2S2log(S)2(F−S)3,

with the definition and . represents the the energy scale of the new physics. In order to compare with the function , we use and obtain

 a[r,1]=−r2−4r+2log(r)+32(r−1)3=A(r). (24)

When the photon is just emitted from the internal charged scalars, our result for is

 b[F,S]=Λ2NPiπ2∫