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

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.


I. INTRODUCTION
The CP violation found in the K-and B system [1] 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 violation [2]. 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 contributions [3]. 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 h 0 [4], 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 time [5]. There are also many new models beyond the SM, such as µνSSM [6]. Generally speaking, the new models introduce new CP-violating phases that can affect the EDMs of fermions, B 0 −B 0 mixing et al. The EDMs of electron and neutron are strict constraints on the CP-violating phases [7]. In the models beyond SM, there are new CP-violating phases which can give large contributions to electron and neutron EDMs [8]. To make the MSSM predictions of electron and neutron EDMs under the experiment upper bounds, there are three possibilities [9]: 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 B and L, people obtain the minimal supersymmetric extension of the SM, which is the so called BLMSSM [11]. The authors in Ref. [10] 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 h 0 → V V, V = (γ, Z, W ) [12] in the BLMSSM, where some other processes [13] are also researched. Taking the CP-violating phases with nonzero values, the neutron EDM, lepton EDM and B 0 −B 0 mixing are researched in this model [14].
From neutron experimental data, the bounding of top EDM is analyzed [15]. Taking into account the precise measurements of the electron and neutron EDMs, the upper limits of heavy quark EDMs are also discussed [16]. The upper limits on the EDMs of heavy quarks are researched from e + e − annihilation [17]. In the CP-violating MSSM, the authors study c quark EDM including two loop gluino contributions [18]. There are also other works on the c quark EDM [19]. Considering the pre-existing works, the upper bounds of EDMs for c and t are about d c < 5.0 × 10 −17 e.cm and d t < 3.06 × 10 −15 e.cm. 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.
We show the superpotential of BLMSSM [12] with W M SSM representing the superpotential of the MSSM. The soft breaking terms L sof t of the BLMSSM are collected here [11,12].
are the soft breaking terms of MSSM.

A. mass matrix
From the soft breaking terms and the scalar potential, we deduce the mass squared matrix for superfields X.
We diagonalize the mass squared matrix for the superfields X through the unitary transformation, ψ X and ψ X ′ are the superpartners of the scalar superfields X and X ′ . ψ X and ψ X ′ can composite four-component Dirac spinors, whose mass term are given out [14] −L mass with µ X denoting the mass ofX.
In the BLMSSM, there are the new baryon boson, the SU(2) L singlets Φ B and ϕ B . Their superpartners are respectively λ B , ψ Φ B and ψ ϕ B , and they mix together producing 3 baryon neutralinos. In the base (iλ B , ψ Φ B , ψ ϕ B ), the mass mixing matrix M BN is obtained and diagonalized by the rotation matrix Z N B [20].
represent the mass eigenstates of baryon neutralinos.
The exotic quarks with charged 2/3 is in four-component Dirac spinors, whose mass matrix reads as [12] − L mass Using the unitary transformations, the two mass eigenstates of exotic quarks with charged 2/3 are obtained by the rotation matrices U t and W t , The mass squared matrix for charged 2/3 exotic squarks M 2 t ′ is obtained in our previous work [12]. For saving space in the work, we do not show it here.

B. 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.
with F µ and G a µ representing electromagnetic field and gluon field respectively. T a (a = 1, · · · , 8) are the strong SU(3) gauge group generators. Similarly, the couplings between photon (gluon) and exotic squarks are also deduced From the superpotential W X , 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 W X .
The couplings of quark-exotic quark-X are shown in the mass basis, From the superpotential W X , in the same way we can also obtain another type Yukawa couplings (quark-exotic squark-X) [14].
Beyond the MSSM, there are couplings for baryon neutralino, quarks and squarks. They are deduced in our previous work [20], and can give new contributions to the quark EDMs.

III. FORMULATION
Using the effective Lagrangian [21] method, one obtains the fermion EDM d f from with F µν representing the electromagnetic field strength, f denoting a fermion field. It is obviously that this effective Lagrangian is CP-violating. In the fundamental interactions, with Λ representing the energy scale, where the Wilson coefficients C i (Λ) are evaluated.
In our previous work [14], In this section, we show the one loop corrections to the quark EDMs (CEDMs). The one loop chargino-squark contributions are Here α = e 2 /(4π), s W = sin θ W , c W = cos θ W , θ W is the Weinberg angle, V is the CKM matrix. We define the one loop functions A(r) and B(r) as [9] A We show the gluino-squark corrections to the quark EDMs and CEDMs with α s = g 2 3 /(4π). ZŨ is the matrix for the up type squarks, with the definition Z † U m 2 U ZŨ = diag(m 2 q 1 , . . . , m 2 q 6 ). The concrete form of the loop function C(r) is [9] C(r) = [6(r − 1 To check the functions A(r), B(r) and C(r) in the Ref. [9], we calculate the one loop triangle diagrams using the effective Lagrangian method. In the calculation, we use the 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 with the definition F = When the photon is just emitted from the internal charged scalars, our result for B(r) is and B(r), From the above discussion, the results in Ref. [9] are the same with our results. In our calculation, we do not ignore the mass of the external fermion. So, it is clear that the analytical expressions for the quark EDM in this work are practicable for both c and t.
Similarly, the contributions from the one loop neutralino-squark diagrams are also ob- Z N is the mixing matrix to get the eigenvalues m χ 0 k (k = 1, 2, 3, 4) of neutralino mass matrix. In the MSSM, there are also the front three type contributions Eqs. (18)(20)(28).
At one loop level, there are three new type corrections beyond MSSM. The corrections from the virtual X and exotic up-type quark has been deduced in the work [14] d γ X j (u I ) = m t i+3 and m X i (i=1,2) are mass eigenvalues of the exotic up type quarks and X superfields.
W t , U t and Z X are the mixing matrices defined in the Eqs.(6)(10).
The one loop baryon neutralino and up-type squark contributions read as ,2,3) are the eigenvalues of baryon neutralino masses. The exotic up-type squark andX can also contribute to the c(t) EDM and CEDM with mX and mt i+3 denoting the masses ofX and exotic up-type squark respectively.
Using the renormalization group equations [22], we evolve the coefficients of the quark EDM and CEDM at matching scale µ down to the quark(c, t) mass scale [23] The quark CEDMs can contribute to the quark EDMs at low energy scale. Therefore, they must be taken into account in the numerical calculation and the formula is [24]

IV. THE NUMERICAL RESULTS
Here, the results are studied numerically. We take into account not only the experiment constraints from Higgs and neutrino, but also our previous works in this model. From ATLAS collaboration, mg ≥ 1460 GeV is the updated bound on the gluino mass [25]. The parameters are supposed as

A. c quark EDM
For the c quark EDM, we use the following parameters as tan β = 10, µ = 800 GeV, mg = 1600 GeV, The baryon neutralino and squarks can give contributions to c quark EDM, which is relevant to the parameters g B and m B . m B representing baryon gaugino masses, can have nonzero CP-violating phase θ m B . Both m B and g B influence the baryon neutralino masses.
Furthermore, g B is the coupling constant for the quark-squark-baryon neutralino. So, with  Figs.(2, 3, 4), one can find the effects from θ X are much larger than the effects from θ µ B and θ m B .

B. t quark EDM
To calculate the t quark EDM numerically, we use the parameters as The quark-gluino-squark coupling corrections to t quark EDM are shown in Eq. (20). tan β is important, because it influences the masses of chargino, neutralino, squark and so on. The effects to the t quark EDM from the µ parameter are also of interest. µ is included in the mass matrices of chargino and neutralino. On the other hand m 2 Q and m 2 u can affect the masses and mixings of the up-type squark. For simplification of the numerical discussion, we adopt the relation m 2 Q = m 2 U = Mus 2 and use the parameters tan β = 10, mg = 1600 GeV, tan β B = 1.5, V Bt = 3600 GeV, Y u 4 = Y u 5 = 0.7Y t , λ 1 = λ 2 = 0.1, µ X = 1 TeV.
As θ µ = −0.5π and µ = (1500, 2500, 3500)e iθµ GeV, the numerical results for t quark  In the other works, the EDMs d c and d t should be of different order. What is the reason in this work? The reason can be found from the couplings in Eqs. (13,14). The coupling constants λ 1 and λ 2 are important parameters. In our numerical calculation, we adopt that the values of λ 1 (λ 2 ) for c quark are same with the values of λ 1 (λ 2 ) for t quark. That is to say, for the up type quark generation 2 and generation 3, the adopted values of λ 1 (λ 2 ) are the same.
From these numerical results and our previous work on neutron EDM, we find θ 3 and θ X are important CP-violating phases. tan β, Y u 4 , Y u 5 , mg, m 2 Q , m 2 U , λ 1 , λ 2 , V Bt are also important. In the whole, our numerical results are large to be detected in the future. The work can confine the parameter space in this model and possess meaning to the relevant experiments for c and t quark EDMs.