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Eur. Phys. J. C (2017) 77:102 DOI 10.1140/epjc/s10052-017-4627-x

Regular Article - Theoretical Physics

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

Shu-Min Zhao1,a, Tai-Fu Feng1,b, Zhong-Jun Yang1, Hai-Bin Zhang1, Xing-Xing Dong1,TaoGuo2 1 Department of Physics, Hebei University, Baoding 071002, China 2 School of Mathematics and Science, Hebei University of Geosciences, Shijiazhuang 050031, China

Received: 25 October 2016 / Accepted: 16 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract In the CP-violating supersymmetric extension of can affect the EDMs of , B0–B¯ 0 mixing etc. The the with local gauged and EDMs of and are strict constraints on the symmetries (BLMSSM), there are new CP-violating sources CP-violating phases [20–23]. In the models beyond SM, there which can give new contributions to the electric dipole are new CP-violating phases which can give large contribu- moment (EDM). Considering the CP-violating phases, we tions to the electron and neutron EDMs [24–26]. To make analyze the EDMs of the c and t. We take into account the MSSM predictions of the electron and neutron EDMs the contributions from the one-loop diagrams. The numerical under the experiment upper bounds, there are three possibil- results are analyzed with some assumptions on the relevant ities [27,28]: (1) the CP-violating phases are very small, (2) parameter space. The numerical results for the c and t EDMs varies contributions cancel with each other in some special can reach large values. parameter spaces, (3) the are very heavy at several TeV order. Taking into account the local gauged B and L, one obtains 1 Introduction the minimal supersymmetric extension of the SM, which is the so-called BLMSSM [29,30]. The authors in Refs. [31–33] The CP-violation found in the K- and B-system [1–3] can be first proposed BLMSSM, where they studied some phenom- well explained in the standard model. It is well known that the ena. At TeV scale, the local gauge symmetries of BLMSSM electric dipole moment (EDM) of an elementary is a break spontaneously. Therefore, in BLMSSM R-parity is vio- clear signal of CP-violation [4–8]. The Cabbibo–Kobayashi– lated and the asymmetry of –antimatter in the universe Maskawa (CKM) phase is the only source of CP-violation in can be explained. We have studied the lightest CP-even Higgs the SM, which has an ignorable effect on the EDM of the mass and the decays h0 → VV, V = (γ, Z, W) [34]in . In the SM, even to two-loop order, the the BLMSSM, where some other processes [35–37]arealso EDM of a does not appear, and there are partial can- researched. Taking the CP-violating phases with nonzero val- celations between the three-loop contributions [9–12]. If the ues, the neutron EDM, the lepton EDM, and B0–B¯ 0 mixing EDM of an elementary fermion is detected, one can confirm are researched in this model [38–40]. there are new CP phases and physics beyond the SM. From the experimental neutron data, the bounding of the Though SM has obtained large successes with the detec- top EDM is analyzed [41]. Taking into account the precise tion of the lightest CP-even Higgs h0 [13,14], it is unable measurements of the electron and neutron EDMs, the upper to explain some phenomena. Physicists consider the SM limits of heavy quark EDMs are also discussed [42]. The to be a low energy effective theory of a large model. The upper limits on the EDMs of heavy quarks are researched minimal supersymmetric extension of the standard model from e+e− annihilation [43]. In the CP-violating MSSM, the (MSSM) is very favorite and people have been interested in authors studied the c quark EDM including two-loop it for a long time [15–17]. There are also many new models contributions. There is also other work on the c quark EDM beyond the SM, such as μνSSM [18,19]. Generally speak- [44,45]. Considering the pre-existing work, the upper bounds −17 ing, the new models introduce new CP-violating phases that of the EDMs for c and t are about dc < 5.0×10 e cm and −15 dt < 3.06×10 e cm. In this work, we calculate the EDMs a e-mail: [email protected] of the and in the framework of the b e-mail: [email protected] CP-violating BLMSSM. At the low energy scale, the quark 123 102 Page 2 of 9 Eur. Phys. J. C (2017) 77:102 chromoelectric dipole moment (CEDM) may give important Therefore, the local gauge symmetry SU(2)L ⊗ U(1)Y ⊗ contributions to the quark EDM. So, we also study the quark U(1)B ⊗ U(1)L breaks down to the electromagnetic sym- CEDM with the renormalization group equations. metry U(1)e. In Sect. 2, we briefly introduce the BLMSSM and show the We write the superpotential of BLMSSM [34] needed mass matrices and couplings, after this introduction. The EDMs (CEDMs) of c and t are researched in Sect. 3.In WBLMSSM = WMSSM + WB + WL + WX , Sect. 4, we give the input parameters and calculate the numer- W = λ Qˆ Qˆ cˆ + λ Uˆ cUˆ ϕˆ + λ Dˆ c Dˆ ϕˆ ical results. The last section is used to discuss the results and B Q 4 5 B U 4 5 B D 4 5 B +μ ˆ ϕˆ + ˆ ˆ ˆ c + ˆ ˆ ˆ c the allowed parameter space. B B B Yu4 Q4 HuU4 Yd4 Q4 Hd D4 + ˆ c ˆ ˆ + ˆ c ˆ ˆ , Yu5 Q5 HdU5 Yd5 Q5 Hu D5 2 The BLMSSM W = ˆ ˆ ˆ c + ˆ ˆ ˆ c + ˆ c ˆ ˆ L Ye4 L4 Hd E4 Yν4 L4 Hu N4 Ye5 L5 Hu E5 ˆ c ˆ ˆ ˆ ˆ ˆ c ˆ c ˆ c ˆ + ν + ν + λ c ϕˆ + μ  ϕˆ , Considering the local gauge symmetries of B(L) and enlarg- Y 5 L5 Hd N5 Y L Hu N N N N L L L L ( ) ⊗ ( ) ⊗ W = λ ˆ ˆ c ˆ + λ ˆ c ˆ ˆ  + λ ˆ c ˆ ˆ  + μ ˆ ˆ , ing the local gauge group of the SM to SU 3 C SU 2 L X 1 QQ5 X 2U U5 X 3 D D5 X X X X ( ) ⊗ ( ) ⊗ ( ) U 1 Y U 1 B U 1 L one can obtain the BLMSSM (3) model [29,30]. In the BLMSSM, there are the exotic super- ˆ ∼ ( , , / , , ) fields including the new quarks Q4 3 2 1 6 B4 0 , W ˆ c ∼ (¯, , − / , − , ) ˆ c ∼ (¯, , / , − , ) with MSSM representing the superpotential of the MSSM. U4 3 1 2 3 B4 0 , D4 3 1 1 3 B4 0 , L ˆ c ∼ (¯, , − / , −( + ), ) ˆ ∼ ( , , / , + The soft breaking terms soft of the BLMSSM are collected Q5 3 2 1 6 1 B4 0 , U5 3 1 2 3 1 ˆ here [29,30,34]: B4, 0), D5 ∼ (3, 1, −1/3, 1 + B4, 0), and the new lep- tons Lˆ ∼ (1, 2, −1/2, 0, L ), Eˆ c ∼ (1, 1, 1, 0, −L ), 4 4 4 4 L = LMSSM − (m2 ) N˜ c∗ N˜ c − m2 Q˜ † Q˜ ˆ c ∼ ( , , , , − ) ˆ c ∼ ( , , / , , −( + )) soft soft N˜ c IJ I J Q˜ 4 4 N4 1 1 0 0 L4 , L5 1 2 1 2 0 3 L4 , 4 ˆ ˆ 2 ˜ c∗ ˜ c 2 ˜ c∗ ˜ c E5 ∼ (1, 1, −1, 0, 3 + L4), N5 ∼ (1, 1, 0, 0, 3 + L4) to −m U U − m D D U˜ 4 4 D˜ 4 4 cancel the B and L anomalies. 4 4 2 ˜ c† ˜ c 2 ˜ ∗ ˜ 2 ˜ ∗ ˜ h0 −m ˜ Q Q − m ˜ U U5 − m ˜ D D5 With the detection of the lightest CP-even Higgs at Q5 5 5 U5 5 D5 5 LHC [13,14], the is very convincing for 2 ˜ † ˜ 2 ˜ c∗ ˜ c −m ˜ L L4 − mν˜ N N , and BLMSSM is based on the Higgs mech- L4 4 4 4 4 ˆ ( , , , , − ), ∗ anism. The introduced Higgs superfields L 1 1 0 0 2 −m2 E˜ c E˜ c − m2 L˜ c† L˜ c ˆ e˜4 4 4 L˜ 5 5 ϕˆL (1, 1, 0, 0, 2) and B(1, 1, 0, 1, 0), ϕˆB (1, 1, 0, −1, 0) 5 2 ˜ ∗ ˜ 2 ˜ ∗ ˜ 2 ∗ break the lepton number and baryon number spontaneously. −m N N − m E E − m   ν˜5 5 5 e˜5 5 5 B B B These Higgs superfields acquire nonzero vacuum expecta- − 2 ϕ∗ ϕ − 2 ∗  − 2 ϕ∗ ϕ mϕ B B m L L mϕ L L tion values (VEVs) and provide masses to the exotic  B L L and exotic quarks. To make the heavy exotic quarks unstable − m λ λ + m λ λ + h.c. ˆ ( , , , / + , ) ˆ ( , , , −( / + B B B L L L the superfields X 1 1 0 2 3 B4 0 , X 1 1 0 2 3  B4), 0) are introduced in the BLMSSM. + ˜ ˜ c + ˜ ˜ c Au4 Yu4 Q4 HuU4 Ad4 Yd4 Q4 Hd D4 The SU(2)L doublets Hu, Hd obtain nonzero VEVs υ ,υ + ˜ c ˜ + ˜ c ˜ u d , Au5 Yu5 Q5 HdU5 Ad5 Yd5 Q5 Hu D5   ˜ ˜ c ˜ c ˜ + +ABQλQ Q4 Q B + ABU λU U U5ϕB H 5 4  =  u  , Hu √1 υ + 0 + 0 + λ ˜ c ˜ ϕ + μ  ϕ + . . u Hu iPu ABD D D4 D5 B BB B B B h c 2     √1 0 0 ˜ ˜ c ˜ ˜ c υd + H + iP + A Y L H E + Aν Yν L H N = 2 d d . e4 e4 4 d 4 4 4 4 u 4 Hd − (1) Hd + ˜ c ˜ + ˜ c ˜ Ae5 Ye5 L5 Hu E5 Aν5 Yν5 L5 Hd N5 ˜ ˜ c ˜ c ˜ c +A Yν LH N + A c λ c N N ϕ The SU(2)L singlets B,ϕB and L ,ϕL obtain nonzero N u N N L υ , υ υ , υ VEVs B B and L L , respectively, +BL μL L ϕL + h.c.      1 1 + λ ˜ ˜ c + λ ˜ c ˜   = √ υ +0 + 0 ,ϕ = √ υ +ϕ0 + 0 . A1 1 QQ5 X A2 2U U5 X B B B iPB B B B i P B  2 2 ˜ c ˜       +A3λ3 D D5 X + BX μX XX + h.c. . (4)  = √1 υ +0 + 0 ,ϕ = √1 υ +ϕ0 + 0 . L L L iPL L L L i P L 2 2 LMSSM (2) soft are the soft breaking terms of MSSM. 123 Eur. Phys. J. C (2017) 77:102 Page 3 of 9 102

2.1 Mass matrix The mass squared matrix for charge 2/3 exotic squarks M2 t˜ is obtained in our previous work [34]. For saving space, 2 From the soft breaking terms and the scalar potential, we M Z ˜ we do not show it here. t˜ is diagonalized by t through deduce the mass squared matrix for superfields X;wehave † 2 2 2 2 2 the formula Z M Z ˜ = diag(m , m , m , m ).   t˜ t˜ t U˜ U˜ U˜ U˜ 2 ∗ ∗ 1 2 3 4 ∗  |μ | + S −μ B X − L = ( ) X X X X , X X X −μ |μ |2 − ∗ X BX X SX X 2.2 Needed couplings  2 gB 2 2 2 SX = + B (v −¯v ). (5) 2 3 4 B B To study the quark EDMs, the couplings between () and exotic quarks (exotic squarks) are necessary. We diagonalize the mass squared matrix for the superfields We derive the couplings between photon (gluon) and exotic X through the unitary transformation,   quarks, thus X1 † X = Z ∗ , 2 2 X 2e μ a μ a X2 X L   =− ¯ γ μ− ¯ γ ,   γ(g)q q ti+3 ti+3 F g3 ti+3 T ti+3 Gμ ∗ ∗ 2 3 |μ |2 + S −μ B m 0 i=1 i=1 Z † X X X X Z = X1 . X −μ B |μ |2 − S X 0 m2 (11) X X X X X2 (6) a with Fμ and Gμ representing the electromagnetic field and a ( = ,..., ) ψX and ψX are the of the scalar superfields X gluon field, respectively. T a 1 8 are the strong  ( ) and X . ψX and ψX can composite four-component Dirac SU 3 gauge group generators. Similarly, the couplings spinors, whose mass term are given out [38–40] between the photon (gluon) and exotic squarks are also  deduced ¯ ψ −Lmass = μ X˜ X˜ , X˜ = X , (7) ˜ X ψ¯  4 X X 2 ∗ μ L   =− δ U˜ ∂˜ U˜ γ(g)q˜ q˜ e jβ Fμ j i β ˜ 3 with μX denoting the mass of X. j,β=1 In the BLMSSM, there are the new baryon , the 4 ( )  ϕ − a δ a U˜ ∗ ∂˜ μU˜ . SU 2 L singlets B and B. Their superpartners are, respec- g3T jβ Gμ j i β (12) λ ψ ψ ,β= tively, B, B and ϕB , and they mix together producing j 1 three baryon . In the base (iλ ,ψ ,ψϕ ),the B B B From the superpotential W , one can find there are interac- mass mixing matrix M is obtained and diagonalized by X BN tions at tree level for quark, exotic quark and X. The needed the rotation matrix Z [46], NB Yukawa interactions can be deduced from the superpotential ⎛ ⎞   2m −v g v¯ g W . The quark–exotic quark–X couplings are shown in the B B B B B k0 X M = ⎝ −v g 0 −μ ⎠ ,χ0 = Bi , mass basis, BN B B B Bi k¯0 v¯ g −μ 0 Bi B B B 2  1i 0 2i 0 3i 0 L ¯ I R iλ = Z k ,ψ = Z k ,ψϕ = Z k . (8) L  = (N ) X t + P u +(N ) B NB Bi B NB Bi B NB Bi Xt u t ij j i 3 L t ij i, j=1 χ 0 (i = 1, 2, 3) represent the mass eigenstates of the baryon  Bi × ¯ I + . . neutralinos. X j ti+3 PR u h c The exotic quarks with charge 2/3 is written in terms of † (N L ) =−λ (W ) (Z ) , four-component Dirac spinors, whose mass matrix reads [34] t ij 1 t i1 X 1 j ∗ (N R) =−λ (U †) (Z ) . (13)    t ij 2 t i2 X 2 j √1 λ v − √1 v  Q B Yu5 d t −Lmass = (¯ ¯ ) 2 2 4L . From the superpotential W , in the same way we can also t t4R t5R √1 √1  (9) X − Yu vu λuv¯B t 2 4 2 5L obtain the Yukawa couplings of another type (quark–exotic squark–X˜ )[38–40]. We have Using the unitary transformations, the two mass eigen- 4   states of exotic quarks with charge 2/3 are obtained by the ∗ ˜ ˜ L ˜ ˜ =− u¯ λ (Z ˜ ) P + λ (Z ˜ ) P XU . (14) rotation matrices Ut and Wt , u¯ XU 1 t 3i L 2 t 4i R i     i=1   t4L = † t4L , t4R = † t4R , Ut  Wt  Beyond the MSSM, there are couplings for the baryon t5L t t5R t  5L  5R , quarks, and squarks. They are deduced in our √1 √1 λQvB − Yu vd † 2 2 5 = ( , ). previous work [46], and can give new contributions to the Wt √1 √1 Ut diag mt4 mt5 (10) − Yu vu λuv¯B 2 4 2 quark EDMs. We have 123 102 Page 4 of 9 Eur. Phys. J. C (2017) 77:102 √ 3 6 L(χ 0 ˜) = 2 χ¯ Bqq gB B0 3 i I,i=1 j=1 1i Ij∗ 1i∗ (I +3) j∗ I ˜ ∗ ×(Z Z PL − Z Z PR)u U + h.c. (15) NB U˜ NB U˜ j

3 Formulation

Using the effective Lagrangian [47] method, one obtains the fermion EDM d f from

i μν L =− d f f σ γ fFμν, (16) EDM 2 5 Fig. 1 In the BLMSSM, one-loop self-energy diagrams are collected here, and the corresponding triangle diagrams are obtained from them with Fμν representing the electromagnetic field strength and by attaching a photon or a gluon in all possible ways 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. the internal lines in all possible ways. After the calculation, Considering the CP-violating electroweak theory, one can we obtain the effective Lagrangian contributing to the quark get this effective Lagrangian from the loop diagrams. The EDMs and CEDMs. The BLMSSM is larger than MSSM and a μν a chromoelectric dipole moment (CEDM) fT σ γ5 fGμν includes the MSSM contributions. In Fig. 1, we plot all the of the quark can also give contribution to the quark EDM. one-loop self-energy diagrams of the up-type quark. a Gμν denotes the gluon field strength. In this section, we show the one-loop corrections to the To describe the CP-violating operators obtained from the quark EDMs (CEDMs). The one-loop chargino–squark con- loop diagrams, the effective method is convenient. The coef- tributions are α ficients of the quark EDM and CEDM at the matching scale γ I e † d ± (u ) = V V μ χ π 2 UD DU should be evolved down to the quark mass scale with the k 4 sW renormalization group equations. At the matching scale, we 6 2     mχ± D D † k can obtain the effective Lagrangian with the CP-violating × ( ) , ( ) Im AC k i BC i,k 2 operators. The effective Lagrangian containing operators m ˜ ⎡i k ⎛ ⎞ ⎛ ⎞D⎤i relating with the quark EDM and CEDM are 2 2 m ± m ± 1 χ χ × ⎣− B ⎝ k ⎠ + A ⎝ k ⎠⎦ , 4 2 2 3 m ˜ m ˜ Leff = C ()O (), Di Di i i α i g ( I ) = g3 † μν μν dχ± u 2 VUDVDU O = qσ P qF , O = qσ P qF , k 4πs 1 L μν 2 R μν W ⎛ ⎞ a μν a a μν a O = σ , O = σ , 6 2   2 3 qT PL qGμν 4 qT PR qGμν (17)   mχ± mχ± D D † k ⎝ k ⎠ × Im (A ) , (B ) B ,  C k i C i,k 2 2 with representing the energy scale where the Wilson coef- m ˜ m ˜ i k Di Di ficients Ci () are evaluated. m I ( D) = √ u ( )Ji( )2k, In our previous work [38–40], we have studied the neu- AC k,i Z D˜ Z+ 2m sβ tron EDM in the CP-violating BLMSSM, where the contri- W m I ˜ ( D) = √ d ( )(J+3)i ( )2k − ( )Ji( )1k. butions from baryon neutralino–squark and X–exotic squark BC k,i Z D˜ Z− Z D˜ Z− are neglected, because they are all small in the used param- 2mW cβ eter space. Here we take into account all the contributions (18) at one-loop level to study the c and t EDMs. In the CP- 2 Here α = e /(4π),sW = sin θW , cW = cos θW , θW is violating BLMSSM, the one-loop corrections to the quark the Weinberg angle, V is the CKM matrix. We define the EDMs and CEDMs can be divided into six types according one-loop functions A(r) and B(r) as [27,28] to the quark self-energy diagrams. We divide the quark self- − energy diagrams according to the virtual particles, thus: (1) A(r) =[2(1 − r)2] 1[3 − r + 2lnr/(1 − r)], − gluino–squark, (2) neutralino–squark, (3) chargino–squark, B(r) =[2(r − 1)2] 1[1 + r + 2r ln r/(1 − r)]. (19) (4) X-exotic quark, (5) baryon neutralino–squark, (6) X˜ – m ˜ (i = 1 ...6) are the squark masses and m (k = exotic squark. Di χ0 k From the quark self-energy diagrams, one obtains the tri- 1, 2, 3, 4) denote the eigenvalues of the neutralino mass angle diagrams needed by attaching a photon or gluon on matrix. Z ˜ is the rotation matrix used to diagonalize the Di 123 Eur. Phys. J. C (2017) 77:102 Page 5 of 9 102 mass squared matrix for the down-type squark. Z− and Z+ and obtain are the rotation matrices used to obtain the mass eigenstates r 2 − 4r + 2log(r) + 3 of the charginos. a[r, 1]=− = A(r). (24) 2(r − 1)3 We show the gluino–squark corrections to the quark EDMs and CEDMs, thus When the photon is just emitted from the internal charged scalars, our result for B(r) is 6    γ I 4 (I +3)i Ii∗ −iθ 2 2 d (u ) =− eα Im (Z ˜ ) (Z ˜ ) e 3 NP mS g˜ π s U U b[F, S]= dk4 9 = π 2 ( 2 − 2 )( 2 − 2 )3 ⎛i 1 ⎞ i k m F k mS 2 2 − ( ) + ( ) − 2 |m ˜ | |m ˜ | F 2FSlog F 2FSlog S S × g B ⎝ g ⎠ , = . (25) 2 2 2(F − S)3 m ˜ m ˜ Ui Ui With the same approach as that of a[F, S], b[F, S] turns into α 6   g I g3 s (I +3)i Ii∗ −iθ the form d (u ) = Im (Z ˜ ) (Z ˜ ) e 3 g˜ π U U 4 = r 2 − 2r log(r) − 1 i 1⎛ ⎞ b[r, 1]= = B(r). (26) 2(r − 1)3 |m ˜ | |m ˜ |2 × g C ⎝ g ⎠ , 2 2 (20) C(r) is obtained from the diagrams where the photon m ˜ m ˜ Ui Ui is attached on both the internal charged fermions and the 2 charged scalars. Therefore, C(r) is the linear combination of with αs = g /(4π). Z ˜ is the matrix for the up-type squarks, 3 U A(r) and B(r), Z† 2 Z = ( 2 ,..., 2 ) with the definition ˜ m ˜ U˜ diag mq˜ mq˜ .The U U C( ) 1 6 1 concrete form of the loop function r is [27,28] C(r) = B(r) − 3A(r). (27) 3 C( ) =[ ( − )2]−1[ − − ( − ) /( − )]. r 6 r 1 10r 26 2r 18 ln r r 1 From the above discussion, the results in Refs. [27,28]arethe (21) same as our results. In our calculation, we do not ignore the mass of the external fermion. So, it is clear that the analytical To check the functions A(r), B(r) and C(r) in Refs. [27, expressions for the quark EDM in this work are practicable 28], we calculate the one-loop triangle diagrams using the for both c and t. effective Lagrangian method. In the calculation, we use the Similarly, the contributions from the one-loop neutralino– approximation squark diagrams are also obtained · + 2 ( · )2 1 2k p p 4 k p 6 4 = 1 − + , γ eα (k + p)2 − m2 k2 − m2 (k2 − m2)2 ( I ) = dχ0 u 2 2 k 12πs c (22) W W i=1 k=1 ⎛ ⎞   2 mχ0 mχ0 with k representing the loop integral momentum and p rep- † k ⎝ k ⎠ ×Im (AN )k,i (BN ) , B , resenting the external momentum. It is reasonable because i k m2 m2 U˜ U˜ the internal particles are at the order of TeV, and the exter- i i α 6 4 nal quark is lighter than TeV, even for the t quark. The ratio g ( I ) = g3 m2 dχ0 u 2 2 t ∼ 0.03 is small enough to use the approximation for- k 8πs c 1TeV2 W W i=1 k=1 ⎛ ⎞ mula. 2  m 0 m 0 † χ χ For the diagram where the photon is just attached on the × ( ) , ( ) k B ⎝ k ⎠ , Im AN k i BN i,k 2 2 internal charged fermions, our result corresponding to A(r) m ˜ m ˜ Ui Ui is the function a[F, S], 4 (I +3)i 1k  (A ) , =− s (Z ˜ ) (Z ) 2 2 N k i W U N [ , ]= NP 4 k 3 a F S dk m I c iπ 2 ( 2 − 2 )3( 2 − 2 ) + u W ( )Ii( )4k, k m F k mS ZU˜ Z N mW sβ F2 + 2S2 log(F) − 4FS + 3S2 − 2S2 log(S) =− , Ii sW 1k∗ 2k∗ ( − )3 (BN )k,i = (Z ˜ ) ( (Z N ) + cW (Z N ) ) 2 F S U 3 (23) mu I cW (I +3)i 4k∗ + (Z ˜ ) (Z N ) . (28) β U m2 m2 mW s = F = S λ with the definition F 2 and S 2 . NP represents NP NP Z N is the mixing matrix to get the eigenvalues mχ0 (k = the energy scale of the new physics. In order to compare with , , , ) k m2 1 2 3 4 of neutralino mass matrix. In the MSSM, there are A( ) λ2 = 2 , → , → F = the function r ,weuse NP mS S 1 F 2 r also the contributions Eqs. (18), (20), and (28). mS 123 102 Page 6 of 9 Eur. Phys. J. C (2017) 77:102

dγ ( ) = . dγ (), dg( ) = . dg(). At one-loop level, there are three corrections of new type q χ 1 53 q q χ 3 4 q (32) beyond MSSM. The corrections from the virtual X and exotic up-type quark have been deduced in [38–40], The quark CEDMs can contribute to the quark EDMs at low energy scale. Therefore, they must be taken into account in 2 γ eλ λ mt + the numerical calculation and the formula is [50] d (u I ) = 1 2 i 3 X j 2 2 24π m γ e i, j=1 X j d = d + dg.   c c π c (33)   m2 4 ∗ ∗ ti+3 ×Im (Wt )1i (Z X ) (Ut ) (Z X )2 j A , 1 j 2i m2 X j 2 4 The numerical results g λ λ mt + dg (u I ) = 3 1 2 i 3 X j 16π 2 m2 , = X j Here, the results are studied numerically. We take into i j 1     m2 account not only the experimental constraints from Higgs ∗ ∗ ti+3 ×Im (Wt )1i (Z X ) (Ut ) (Z X )2 j A , and , but also our previous work on this model. From 1 j 2i m2 X j ATLAScollaboration, mg˜ ≥ 1460 GeV is the updated bound (29) on the gluino mass [51]. The parameters are assumed to be ( = , ) mti+3 and m Xi i 1 2 are mass eigenvalues of the exotic m1 = m2 = ABQ = ABU = 1TeV, up-type quarks and X superfields. Wt , Ut and Z X are the 2 2 BX = 500 GeV, m = δij TeV ,(i, j = 1, 2, 3), mixing matrices defined in Eqs. (6) and (10). D = =  =  = , The one-loop baryon neutralino and up-type squark con- Au Ad Au Ad 500 GeV = = . ,λ= λ = . , tributions read Yd4 Yd5 0 7Yb Q u 0 5 m2 = m2 = m2 = m2 = 1TeV2, 2 2 6 Q˜ Q˜ U˜ U˜ γ I egB 4 5 4 5 dχ (u ) =− B 108π 2 3 = = B = , Au = Au = 500 GeV. (34) i 1 j 1 ⎛ ⎞ 4 2 4 5   m2 ( + ) ∗ mχi χi ×Im (Z 1i )2 Z I 3 j Z Ij B B ⎝ B ⎠ , NB U˜ U˜ 2 2 4.1 c quark EDM m ˜ m ˜ U j U j 2 2 6 For the c quark EDM, we use the following parameters: g I g3gB dχ (u ) =− B π 2 72 = = tan β = 10,μ= 800 GeV, m ˜ = 1600 GeV, i 1 j 1 ⎛ ⎞ g 2 β = ,v = , = = . .   m tan B 2 Bt 3TeV Yu4 Yu 0 7Yt (35) ( + ) ∗ mχi χi 5 ×Im (Z 1i )2 Z I 3 j Z Ij B B ⎝ B ⎠ . NB U˜ U˜ 2 2 m ˜ m ˜ The baryon neutralino and squarks can give contributions U j U j to the c quark EDM, which is relevant to the parameters g (30) B and m B. m B, representing baryon masses, can have a ( = , , ) θ mχi i 1 2 3 are the eigenvalues of the baryon neutralino nonzero CP-violating phase m B .Bothm B and gB influence B masses. the baryon neutralino masses. Furthermore, gB is the cou- The exotic up-type squark and X˜ can also contribute to pling constant for the quark–squark–baryon neutralino. So, θ =− . π, λ = λ = . ,μ = μ = , the c(t) EDM and CEDM with m B 0 5 1 2 0 1 X B 3TeV 2 = 2 = δ 2 ( , = , , ),   m Q mU ijTeV for i j 1 2 3 we study the c λ λ 2   m2 γ e 1 2 3i∗ 4i∗ m X˜ X˜ quark EDM versus gB. If we do not mention the other CP- d ˜ = Im (Z ˜ ) (Z ˜ ) B , X 24π 2 t t m2 m2 violating phases, it indicates the other CP-violating phases i=1 t˜ + t˜ + i 3  i 3  are zero. In Fig. 2, the numerical results corresponding to   2 λ λ 2 m iθ g g3 1 2 3i∗ 4i∗ m X˜ X˜ m = (1, 2, 3) × e m B TeV are plotted by the solid line, d = Im (Z ˜ ) (Z ˜ ) B , B X˜ π 2 t t 2 2 16 m ˜ m ˜ dotted line, and dashed line, respectively. On the whole, the i=1 ti+3 ti+3 (31) three lines are all increasing functions of gB. The dotted line and the dashed line vary slightly. The solid line quickly gets ˜ m ˜ m ˜ X g with X and ti+3 denoting the masses of and exotic up- larger with increasing B. These three lines also imply the type squark, respectively. results are suppressed by large |m B|. The obtained numeri- θ Using the renormalization group equations [48,49], we cal results from the nonzero CP-violating phase m B are at evolve the coefficients of the quark EDM and CEDM at the order of 10−21 e cm, four orders smaller than the upper matching scale μ down to the quark(c, t) mass scale, bound of the c quark EDM. 123 Eur. Phys. J. C (2017) 77:102 Page 7 of 9 102

8 6

5 6 4 e.cm e.cm

21 4

3 17 10 10 c

2 c d d 2 1

0 0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 gB Lam

θ = Fig. 2 The one-loop corrections to the c EDM versus gB with m B Fig. 4 The one-loop corrections to the c EDM versus Lam, while the −0.5π,thesolid line, dotted line,anddashed line corresponding to solid line, dotted line and dashed line denoting the results with μX = iθm i(0.5π,0.3π,0.1π) m B = (1, 2, 3) × e B TeV, respectively e TeV, respectively

7 results versus Lam are denoted, respectively, by the solid line, 6 dotted line, and dashed line. The three lines are the increas- −17 5 ing functions of Lam and the results are around 10 ecm. As Lam > 0.8, the solid line even exceeds the upper bound e.cm 4 . × −17 21 5 0 10 e cm. From Figs. 2, 3, and 4, one can see that

10 θ θ 3 the effects from X are much larger than the effects from μB c

d θ 2 and m B .

1 4.2 t quark EDM 0 1500 2000 2500 3000 3500 To calculate the t quark EDM numerically, we use the param- mB GeV eters θ = Fig. 3 The one-loop corrections to the c EDM versus mB with μB 1 . π solid line dotted line dashed line g = gB = , m B = 1TeV,μB = 3TeV. (36) 0 5 ,the , and corresponding to B 3 (1/3, 1/5, 1/10), respectively The quark–gluino–squark coupling corrections to the t quark EDM are shown in Eq. (20). tan β is important, because it θμ μ Here, we discuss the effects from the new phase B of B; influences the masses of chargino, neutralino, squark, and it may influence the baryon neutralino masses. Based on the so on. The mixing matrices of squarks and exotic squarks iθμ μ = × B (θμ = . π), λ = supposition B 2 e TeV B 0 5 1 have a relation with tan β. The absolute value of the gluino λ = . ,μ = 2 = δ 2, 2 = 2 0 5 X 3TeV,m Q 4 ij TeV mU mass obviously also influences the results from Eq. (20). δ 2 ( , = , , ) 2 ij TeV i j 1 2 3 , the results versus m B are shown Using the parameters θ3 =−0.5π, μ = 800 GeV, tan βB = as the solid line for g = 1 . The solid line decreases , = = . ,λ = λ = . ,μ = , = B 3 2 Yu4 Yu5 0 7Yt 1 2 0 5 X 1TeV VBt quickly in the region 1000 GeV < m < 1300 GeV. When , 2 = δ 2 2, 2 = δ 2 B 3300 GeV m Q ij1500 GeV mU ijTeV with m B > 1300 GeV, the extent of the change of the solid line is (i, j = 1, 2, 3),forthet quark EDM we plot the results small. The dotted line and the dashed line, respectively, rep- versus mg˜ .InFig.5, the solid line (tan β = 5), dotted line = 1 = 1 resent the results for gB 5 and gB 10 , and they are both (tan β = 10) and dashed line (tan β = 15) are all decreasing slowly decreasing functions of m B. Generally speaking, the functions of m ˜ . During the m ˜ region (1500–2000) GeV, − g g results are around 10 21 e cm, at the same order as of Fig. 2. the results of the three lines shrink quickly. Near the point λ1 and λ2 are important parameters for the couplings of m ˜ = 1480 GeV, the theoretical predictions are of the order ˜ g quark–exotic quark–X and quark–exotic squark–X. There- of 10−16 e cm and even reach 10−15 e cm. On the other hand, fore, the numerical results may obviously be influenced by the extent of the influence from tan β is not large. the varying λ1 and λ2. For simplicity, we suppose λ1 = λ2 = The effects to the t quark EDM from the μ parameter are , = 1 , = ,μ = , 2 = 2 = Lam gB 3 m B 1TeV B 3TeV m Q mU also of interest. μ is included in the mass matrices of chargino δ 2 ( , = , , ) 2 2 ij TeV for i j 1 2 3 . With the nonzero CP-violating and neutralino. On the other hand m Q and mu can affect the iθ phase θX = (0.5π, 0.3π, 0.1π) and μX = e X TeV, the masses and mixings of the up-type squark. For simplification 123 102 Page 8 of 9 Eur. Phys. J. C (2017) 77:102

10 10

8 1

e.cm 6 e.cm 17

0.1 17 10 10 t 4 d t d 0.01 2

1600 1800 2000 2200 2400 0 mg GeV 0.0 0.2 0.4 0.6 0.8 1.0 Yu45

Fig. 5 t m ˜ θ = The one-loop corrections to the EDM versus g with 3 μ = −0.5π,thesolid line, dotted line,anddashed line corresponding to Fig. 7 The one-loop corrections to the t EDM versus Yu45 with X iθX tan β = (5, 10, 15) e TeV, the solid line, dotted line,anddashed line corresponding to the results with θX = (0.5π, 0.2π, 0.05π)

7

6 Yu45 and as Yu45 > 0.8 the numerical results get quickly larger. At the point Yu45 = 0.7, the solid line and dotted 5 line are larger than 1 × 10−17 e cm. The three lines are at e.cm 4 the order of 10−18 e cm, when Yu45 is small. For the c and t 18

10 3 quark EDMs, in our parameter space the CP-violating phases t

d θ and θX are important and can provide large contributions. 2 3

1

0 2000 3000 4000 5000 5 Discussion Mus GeV In the CP-violating BLMSSM, there are new CP-violating Fig. 6 The one-loop corrections to the t EDM versus Mus with θμ = phases θX ,θμ ,θm beyond MSSM. For the c quark EDM, −0.5π,thedotted line, solid line,anddashed line corresponding to the B B θ θ = ,θμ = θ = results with μ = (1500, 2500, 3500)ei μ GeV we consider the conditions X 0 B 0 and m B 0, respectively. The results show θX can give large contribu- tions, even reach the experimental upper bound (5 × 10−17 e of the numerical discussion, we adopt the relation m2 = θ θ Q cm) for the c EDM. The effects produced from m B and μB 2 = 2 β = , = −21 mU Mus and use the parameters tan 10 mg˜ are at the order of 10 e cm, which are much smaller than , β = . , = , = = θ 1600 GeV tan B 1 5 VBt 3600 GeV Yu4 Yu5 those from X .Forthet quark EDM the CP-violating phases 0.7Yt ,λ1 = λ2 = 0.1,μX = 1TeV.Asθμ =−0.5π θ3,θμ, and θX are studied. Under the two conditions θX = 0 iθμ −17 and μ = (1500, 2500, 3500)e GeV, the numerical results and θ3 = 0 , we find dt to be at the order of 10 ecm. for the t quark EDM are plotted, respectively, by the dotted Especially for nonzero θ3 with mg˜ near its lower bound, the line, solid line, and dashed line. The results are all decreas- t quark EDM can reach 10−16 e cm and even higher. They − ing functions of Mus and at the order of 10 18 ecmas are both larger than the results for θμ = 0. Mus < 3500 GeV. The absolute value of μ also influences In BLMSSM, at one-loop level there are contributions the results, and the extent is small (Fig. 6). of three types [(1) the virtual X and exotic up-type quark, The exotic squarks and exotic quarks are in connection (2) baryon neutralino and up-type squark, (3) exotic up-type ˜ with Yu4 and Yu5 , and the related contributions are shown squark and X] to the quark EDM beyond MSSM. For the in Eqs. (29, 31). As discussed in a previous subsection, contributions beyond MSSM, to obtain large dc and dt the nonzero θX can give large contributions. In Fig. 7 we plot CP-violating phase θX should be nonzero in our parameter the solid line, dotted line, and dashed line versus Yu45 with space. With only θX = 0, the nonzero contributions come = = ∗ ,θ = ( . π, . π, . π) ∼ −17 Yu4 Yu5 Yu45 Yt X 0 5 0 2 0 05 from Eqs. (23) and (25). In Fig. 4 dc 10 e cm and in iθ −17 and μX = 1eX TeV. The other parameters used are Fig. 7 dt ∼ 10 e cm, the EDMs dc and dt are of the same tan β = 10,μ= 800 GeV, mg˜ = 1600 GeV, tan βB = order of magnitude. In the other work, the EDMs dc and dt , = ,λ = λ = . , 2 = 2 = δ 2 2 VBt 3TeV 1 2 0 5 m Q mU ijTeV should be of different order. What is the reason in this work? with (i, j = 1, 2, 3). They are all increasing functions of The reason can be found from the couplings in Eqs. (13) 123 Eur. Phys. J. C (2017) 77:102 Page 9 of 9 102 and (14). 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