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.


Introduction
The CP-violation found in the K-and B-system [1][2][3] can be well explained in the standard model. It is well known that the electric dipole moment (EDM) of an elementary particle is a clear signal of CP-violation [4][5][6][7][8]. The Cabbibo-Kobayashi-Maskawa (CKM) phase is the only source of CP-violation in the SM, which has an ignorable effect on the EDM of the elementary particle. In the SM, even to two-loop order, the EDM of a fermion does not appear, and there are partial cancelations between the three-loop contributions [9][10][11][12]. If the 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 [13,14], it is unable to explain some phenomena. Physicists consider the SM to 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 [15][16][17]. There are also many new models beyond the SM, such as μνSSM [18,19]. Generally speaking, the new models introduce new CP-violating phases that a e-mail: zhaosm@hbu.edu.cn b e-mail: fengtf@hbu.edu.cn can affect the EDMs of fermions, B 0 -B 0 mixing etc. The EDMs of electron and neutron are strict constraints on the CP-violating phases [20][21][22][23]. In the models beyond SM, there are new CP-violating phases which can give large contributions to the electron and neutron EDMs [24][25][26]. To make the MSSM predictions of the electron and neutron EDMs under the experiment upper bounds, there are three possibilities [27,28]: (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, one obtains the minimal supersymmetric extension of the SM, which is the so-called BLMSSM [29,30]. The authors in Refs. [31][32][33] first proposed BLMSSM, where they studied some phenomena. At TeV scale, the local gauge symmetries of BLMSSM break 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 ) [34] in the BLMSSM, where some other processes [35][36][37] are also researched. Taking the CP-violating phases with nonzero values, the neutron EDM, the lepton EDM, and B 0 -B 0 mixing are researched in this model [38][39][40].
From the experimental neutron data, the bounding of the top EDM is analyzed [41]. Taking into account the precise measurements of the electron and neutron EDMs, the upper limits of heavy quark EDMs are also discussed [42]. The upper limits on the EDMs of heavy quarks are researched from e + e − annihilation [43]. In the CP-violating MSSM, the authors studied the c quark EDM including two-loop gluino contributions. There is also other work on the c quark EDM [44,45]. Considering the pre-existing work, the upper bounds of the 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 the charm quark and top quark in the framework of the CP-violating BLMSSM. At the low energy scale, the quark chromoelectric dipole moment (CEDM) may give important contributions to the quark EDM. So, we also study the quark CEDM with the renormalization group equations.
In Sect. 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 Sect. 3. In Sect. 4, we give the input parameters and calculate the numerical results. The last section is used to discuss the results and the allowed parameter space.
The SU (2) L doublets H u , H d obtain nonzero VEVs υ u , υ d , The SU (2) L singlets B , ϕ B and L , ϕ L obtain nonzero VEVs υ B , υ B and υ L , υ L , respectively, Therefore, the local gauge symmetry SU We write the superpotential of BLMSSM [34] with W MSSM representing the superpotential of the MSSM. The soft breaking terms L soft of the BLMSSM are collected here [29,30,34]: L MSSM soft are the soft breaking terms of MSSM.

Mass matrix
From the soft breaking terms and the scalar potential, we deduce the mass squared matrix for superfields X; we have 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 [38][39][40] −L mass with μ X denoting the mass ofX .
In the BLMSSM, there are the new baryon boson, the SU (2) L singlets B and ϕ B . Their superpartners are, respectively, λ B , ψ B and ψ ϕ B , and they mix together producing three baryon neutralinos. In the base (iλ B , ψ B , ψ ϕ B ), the mass mixing matrix M B N is obtained and diagonalized by the rotation matrix Z N B [46], represent the mass eigenstates of the baryon neutralinos.
The exotic quarks with charge 2/3 is written in terms of four-component Dirac spinors, whose mass matrix reads [34] −L mass Using the unitary transformations, the two mass eigenstates of exotic quarks with charge 2/3 are obtained by the rotation matrices U t and W t , The mass squared matrix for charge 2/3 exotic squarks M 2 t is obtained in our previous work [34]. For saving space, we do not show it here. ).

Needed couplings
To study the quark EDMs, the couplings between photon (gluon) and exotic quarks (exotic squarks) are necessary. We derive the couplings between photon (gluon) and exotic quarks, thus with F μ and G a μ representing the electromagnetic field and gluon field, respectively. T a (a = 1, . . . , 8) are the strong SU (3) gauge group generators. Similarly, the couplings between the 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 quark-exotic quark-X couplings are shown in the mass basis, From the superpotential W X , in the same way we can also obtain the Yukawa couplings of another type (quark-exotic squark-X ) [38][39][40]. We have Beyond the MSSM, there are couplings for the baryon neutralino, quarks, and squarks. They are deduced in our previous work [46], and can give new contributions to the quark EDMs. We have

Formulation
Using the effective Lagrangian [47] method, one obtains the fermion EDM d f from with F μν representing the electromagnetic field strength and f denoting a fermion field. It is obvious that this effective Lagrangian is CP-violating. In the fundamental interactions, this CP-violating Lagrangian cannot 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) f T a σ μν γ 5 f G a μν of the quark can also give contribution to the quark EDM. G a μν denotes the gluon field strength.
To describe the CP-violating operators obtained from the loop diagrams, the effective method is convenient. The coefficients of the quark EDM and CEDM at the matching scale μ should be evolved down to the quark mass scale with the renormalization group equations. At the 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 with representing the energy scale where the Wilson coefficients C i ( ) are evaluated. In our previous work [38][39][40], we have studied the neutron EDM in the CP-violating BLMSSM, where the contributions from baryon neutralino-squark andX -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 CPviolating 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 selfenergy diagrams according to the virtual particles, thus: (1) gluino-squark, (2) neutralino-squark, (3) chargino-squark, (4) X-exotic quark, (5) baryon neutralino-squark, (6)Xexotic squark.
From the quark self-energy diagrams, one obtains the triangle diagrams needed by attaching a photon or gluon on Fig. 1 In the BLMSSM, one-loop self-energy diagrams are collected here, and the corresponding triangle diagrams are obtained from them by attaching a photon or a gluon in all possible ways 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. 1, 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 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 [27,28]  We show the gluino-squark corrections to the quark EDMs and CEDMs, thus with α s = g 2 3 /(4π). ZŨ is the matrix for the up-type squarks, with the definition Z † U m 2Ũ ZŨ = diag(m 2 q 1 , . . . , m 2 q 6 ). The concrete form of the loop function C(r ) is [27,28] To check the functions A(r ), B(r ) and C(r ) in Refs. [27,28], we calculate the one-loop triangle diagrams using the effective Lagrangian method. In the calculation, we use the approximation 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 the t quark. The ratio m 2 t 1 TeV 2 ∼ 0.03 is small enough to use the approximation formula.

For the diagram where the photon is just attached on the internal charged fermions, our result corresponding to A(r ) is the function a[F, S], a[F, S]
with the definition F = When the photon is just emitted from the internal charged scalars, our result for With the same approach as that of a[F, S], b[F, S] turns into the form C(r ) is obtained from the diagrams where the photon is attached on both the internal charged fermions and the charged scalars. Therefore, C(r ) is the linear combination of A(r ) and B(r ), From the above discussion, the results in Refs. [27,28] are the same as 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 neutralinosquark diagrams are also obtained 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 contributions Eqs. (18), (20), and (28).
At one-loop level, there are three corrections of new type beyond MSSM. The corrections from the virtual X and exotic up-type quark have been deduced in [38][39][40], 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 Eqs. (6) and (10). The one-loop baryon neutralino and up-type squark contributions read 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 uptype squark, respectively. Using the renormalization group equations [48,49], we evolve the coefficients of the quark EDM and CEDM at matching scale μ down to the quark(c, t) mass scale, 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 [50]

The numerical results
Here, the results are studied numerically. We take into account not only the experimental constraints from Higgs and neutrino, but also our previous work on this model. From ATLAS collaboration, mg ≥ 1460 GeV is the updated bound on the gluino mass [51]. The parameters are assumed to be

c quark EDM
For the c quark EDM, we use the following parameters: tan β = 10, μ = 800 GeV, mg = 1600 GeV, The baryon neutralino and squarks can give contributions to the c quark EDM, which is relevant to the parameters g B and m B . m B , representing baryon gaugino masses, can have a 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, = 1, 2, 3), we study the c quark EDM versus g B . If we do not mention the other CPviolating phases, it indicates the other CP-violating phases are zero. In Fig. 2

t quark EDM
To calculate the t quark EDM numerically, we use the parameters The quark-gluino-squark coupling corrections to the 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 mixing matrices of squarks and exotic squarks have a relation with tan β. The absolute value of the gluino mass obviously also influences the results from Eq. (20). Using the parameters θ 3 = −0.5π, μ = 800 GeV, tan  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 B t = 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 iθ μ GeV, the numerical results for the t quark EDM are plotted, respectively, by the dotted line, solid line, and dashed line. The results are all decreasing functions of Mus and at the order of 10 −18 e cm as Mus < 3500 GeV. The absolute value of μ also influences the results, and the extent is small (Fig. 6).

Discussion
In the CP-violating BLMSSM, there are new CP-violating phases θ X , θ μ B , θ m B beyond MSSM. For the c quark EDM, we consider the conditions θ X = 0, θ μ B = 0 and θ m B = 0, respectively. The results show θ X can give large contributions, even reach the experimental upper bound (5 × 10 −17 e cm) for the c EDM. The effects produced from θ m B and θ μ B are at the order of 10 −21 e cm, which are much smaller than those from θ X . For the t quark EDM the CP-violating phases θ 3 , θ μ , and θ X are studied. Under the two conditions θ X = 0 and θ 3 = 0 , we find d t to be at the order of 10 −17 e cm. Especially for nonzero θ 3 with mg near its lower bound, the t quark EDM can reach 10 −16 e cm and even higher. They are both larger than the results for θ μ = 0.
In BLMSSM, at one-loop level there are contributions of three types [(1) the virtual X and exotic up-type quark, (2) baryon neutralino and up-type squark, (3) exotic up-type squark andX ] to the quark EDM beyond MSSM. For the contributions beyond MSSM, to obtain large d c and d t the CP-violating phase θ X should be nonzero in our parameter space. With only θ X = 0, the nonzero contributions come from Eqs. (23) and (25). In Fig. 4 13) and (14). The coupling constants λ 1 and λ 2 are important parameters. In our numerical calculation, we adopt the case that the values of λ 1 (λ 2 ) for c quark are the same as 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 the neutron EDM, we find that θ 3 and θ X are important CPviolating phases. tan β, Y u 4 , Y u 5 , mg, m 2 Q , m 2 U , λ 1 , λ 2 , V B t are also important. On the whole, our numerical results are sufficiently large so as to be detected in the future. The work can confine the parameter space in this model and has significance for the relevant experiments for the c and t quark EDMs.