Sensitivity of the Squark Flavor Mixing to the CP Violation of K, B0 and Bs

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The flavor physics is on the new stage in the light of LHCb data. The LHCb collaboration has reported new data of the CP violation of the Bs meson and the branching ratios of 0 rare Bs decays [1]-[12]. For many years the CP violation in the K and B mesons has been successfully understood within the framework of the standard model (SM), so called Kobayashi-Maskawa (KM) model, where the source of the CP violation is the KM phase in the quark sector with three families. However, the new physics has been expected to be 0 indirectly discovered in the precise data of B and Bs meson decays at the LHCb experiment and the further coming experiment, Belle II. The supersymmetry (SUSY) is one of the most attractive candidates for the new physics. The SUSY signals have not been observed yet although the Higgs-like events have been confirmed [13]. Since the lower bounds of the superparticle masses increase gradually, the squark and the gluino masses are supposed to be at the TeV scale [14]. While, there are new sources of the CP violation if the SM is extended to the SUSY models. The soft squark mass matrices contain the CP-violating phases, which contribute to the flavor changing neutral current (FCNC) with the CP violation. Therefore, we expect the effect of the SUSY contribution in the CP-violating phenomena. However, the clear deviation from the SM prediction has not been observed yet in the LHCb experiment [1]-[12]. The LHCb collaboration presented the time dependent CP asymmetry in the non-leptonic Bs J/ψφ decay [11, 12, 4], which gives a constraint of the SUSY contribution on the b s transition.→ They have also reported the first measurement of the CP violating phase in→ the B φφ decay [2]. This decay process is occurred at the one-loop level in the SM, where s → the CP violating phase is very small. On the other hand, the gluino-squark mediated flavor changing process provides new CP violating phases. Thus, the CP asymmetry of B φφ is s → expected to be deviated considerably from the SM one. In this work, we discuss the sensitivity of the SUSY contribution to the CP asymmetry of Bs φφ and Bs φη′ by taking account of constraints from other experimental data of the→ CP violation.→ For these decay modes, the most important process of the SUSY contribution is the gluino-squark mediated flavor changing process [15]- [26]. This FCNC effect is constrained by the CP violations in B0 J/ψK and B J/ψφ decays. The CP violation of K meson, ǫ , also provides → S s → K a severe constraint to the gluino-squark mediated FCNC. In the SM, ǫK is proportional to sin(2β) which is derived from the time dependent CP asymmetry in B0 J/ψK decay [27]. → s The relation between ǫK and sin(2β) is examined by taking account of the gluino-squark mediated FCNC [28]. 0 0 0 0 0 0 The time dependent CP asymmetry of B φK , B η′K , and B K K¯ decays → S → → are also attractive ones to search for the gluino-squark mediated FCNC because the penguin amplitude dominates this process as well as Bs φφ. Furthermore, we discuss the FCNC → 0 with the CP violation in the semileptonic CP asymmetries of B and Bs mesons. In addition, it is remarked that the upper-bound of the chromo-EDM(cEDM) of the strange quark gives a severe constraint for the gluino-squark mediated b s transition → [29]-[32]. The lower bounds of the squark masses increase gradually. The gluino mass is expected to be larger than 1.3 TeV, and the squarks of the first and second families are also heavier

1 than 1.4 TeV [14]. Therefore, we take the split-family scenario, in which the ﬁrst and second family squarks are very heavy, (10) TeV, while the third family squark masses are at (1) TeV. Then, the s d transitionO mediated by the ﬁrst and second family squarks O → is naturally suppressed by their heavy masses, and competing process is mediated by the second order contribution of the third family squark. In order to estimate the gluino-squark 0 mediated FCNC for the K, B and Bs meson decays comprehensively, we work in the basis of the squark mass eigenstate. Then, the 6 6 mixing matrix among down-squarks and down-quarks is studied by input of the experimental× constraints. In section 2, we present the formulation of the gluino-squark mediated transition in our split-family scenario. In section 3, we discuss the gluino-squark mediated FCNC contribution to ǫK . In section 4, we discuss the sensitivity of the gluino-squark mediated FCNC to the 0 CP violation of the non-leptonic and the semi-leptonic decays of B and Bs mesons. Section 5 is devoted to the summary.

2 CP violation through squark ﬂavor mixing

2.1 Squark ﬂavor mixing Let us discuss the gluino-squark mediated ﬂavor changing process as the dominate SUSY contribution. We give the 6 6 squark mass matrix to be M (˜q =u, ˜ d˜) in the super-CKM × q˜ basis. In order to go to the diagonal basis of the squark mass matrix, we rotate Mq˜ as

2 (q) 2 (q) m˜ q˜dia =ΓG Mq˜ ΓG † , (1)

(q) where ΓG is the 6 6 unitary matrix, and we decompose it into the 3 6 matrices as (q) (q) (q)) T × × ΓG = (ΓGL, ΓGR ) in the following expressions. Then, the gluino-squark-quark interaction is given as

a a (q) (q) (˜gqq˜)= i√2g q∗(T )G (Γ ) L + (Γ ) R q + h.c. , (2) Lint − s i GL ij GR ij j q X{ } h i e e where Ga denotes the gluino field, and L and R are projection operators. This interaction leads to the gluino-squark mediated flavor changing process with ∆F = 2 and ∆F = 1 throughe the box and penguin diagrams. In our framework, the squarks of the first and second families are heavier than multi-TeV, on the other hand, the masses of the third family squarks, stop and sbottom, are around 1 TeV. Therefore, the first and second squark contribution is suppressed in the gluino-squark mediated flavor changing process by their heavy masses. The stop and sbottom interactions dominate the gluino-squark mediated flavor changing process. Then, the sbottom interaction (d) dominates ∆B = 2 and ∆B = 1 processes. We take a suitable parametrizations of ΓGL and

2 (d) ΓGR as follows [28]:

dL dL iφ 1 0 δ13 cθ 0 0 δ13 sθe (d) − Γ = 0 1 δdLc 0 0 δdLs eiφ , GL  23 θ − 23 θ  δdL∗ δdL∗ c 0 0 s eiφ − 13 − 23 θ − θ  

dR iφ dR 0 0 δ13 sθe− 1 0 δ13 cθ (d) dR iφ dR ΓGR = 0 0 δ23 sθe− 0 1 δ23 cθ , (3)  iφ dR dR  0 0 s e− δ ∗ δ ∗ c θ − 13 − 23 θ   ˜ dL dR where cθ = cos θ and sθ = sin θ, with the mixing angle θ in the bL,R sector and δj3 , δj3 are the couplings responsible for the flavor transitions. By using these rotation matrices, 0 we estimate the gluino-sbottom mediated flavor changing amplitudes in the K, B , and Bs meson decays. For the numerical analysis, we fix sbottom masses. The third family squarks can have substantial mixing between the left-handed squark and the right-handed one due to large

Yukawa couplings. In our numerical calculation, we take the typical mass eigenvalues m˜b1 and m˜b2 , and the gluino mass mg˜ as follows:

m˜b1 = 1 TeV, m˜b2 =1.1 TeV, mg˜ = 2 TeV, (4) where we take account of the present experimental bounds [14]. Then, we can roughly estimate the mixing angle θ between the left-handed sbottom and the right-handed one by using RGE’s under the assumption of the universal mass of the GUT scale although it depends on the SUSY parameters in details [33]. Therefore, we scatter the left and right mixing θ in dL dL the range of 10◦ 35◦ in our numerical calculations. The mixing parameters δ and δ are − 13 23 complex, and will be constrained by the experimental data. For simplicity, we take

δdR = δdL , δdR = δdL , (5) | 13 | | 13 | | 23 | | 23 | dR dR on the other hand, the phases of δ23 and δ13 , and the phase φ are free parameters. Therefore, we have three mixing angles and ﬁve phases in the mixing matrices of Eq.(3), which are free parameters in our calculations.

2.2 CP violation in ∆B =2 and ∆B =1 processes Let us discuss the SUSY contribution in the ∆B = 2 process. The contribution of new q physics to the dispersive part M12 is parameterized as

q q,SM q,SUSY q,SM 2iσq M12 = M12 + M12 = M12 (1 + hqe ) , (q = d,s) (6)

q,SM q,SUSY where M12 and M12 are the SM and the SUSY contributions. The parameters hq and q,SUSY σq are given in terms of mixing parameters of Eq.(3). The M12 are given explicitly in Appendix A. By inputting experimental data of ǫK , ∆MB0 , ∆MBs , sin(2β), and sin(2βs), we constrain the magnitude hq and the phase σq.

3 q The indirect CP violation leads to the non-zero asymmetry asl in the semileptonic decays B µ−X(q = d,s) with ”wrong-sign” such as: q → + q q Γ(B¯ µ X) Γ(B µ−X) Γ Γ aq q → − q → Im 12 = | 12| sin φq , (7) sl ≡ Γ(B¯ µ+X)+Γ(B µ X) ≃ M q M q sl q → q → − 12 | 12| q ¯ where Γ12 is the absorptive part in the eﬀective Hamiltonian of the Bq-Bq system, where 0 Bd is denoted as the B meson in this paper. The SM contribution to the absorptive part Γq is dominated by tree-level decay b ccs¯ etc.. Therefore, we assume Γq = Γq,SM in our 12 → 12 12 calculation. In the SM, the CP phases are read [34],

sSM 3 dSM 2 φ = (3.84 1.05) 10− , φ = (7.50 2.44) 10− , (8) sl ± × sl − ± × which correspond to

sSM 5 dSM 4 a = (1.9 0.3) 10− , a = (4.1 0.6) 10− . (9) sl ± × sl − ± × The recent experimental data of these asymmetries are given as [8, 35]

s 2 d 3 a =( 0.24 0.54 0.33) 10− , a =( 0.3 2.1) 10− . (10) sl − ± ± × sl − ± × There are many interesting non-leptonic CP violating decays to search for new physics. The eﬀective Hamiltonian for the ∆B = 1 process is given as follows:

4GF (q′) H = V ′ V ∗′ C O V V ∗ C O + C O , (11) eff √ q b q q i i − tb tq i i i i 2 "q′=u,c i=1,2 i=3 6,7γ,8G # X X −X e e where q = s,d. The local operators are given as

′ ′ (q ) µ (q ) µ O1 = (¯qαγµPLqβ′ )(¯qβ′ γ PLbα),O2 = (¯qαγµPLqα′ )(¯qβ′ γ PLbβ), µ µ O3 = (¯qαγµPLbα) (Q¯βγ PLQβ),O4 = (¯qαγµPLbβ) (Q¯βγ PLQα), Q Q X X µ µ O5 = (¯qαγµPLbα) (Q¯βγ PRQβ),O6 = (¯qαγµPLbβ) (Q¯βγ PRQα), Q Q e X g X O = m q¯ σµν P b F ,O = s m q¯ σµν P T a b Ga , (12) 7γ 16π2 b α R α µν 8G 16π2 b α R αβ β µν where P = (1+ γ )/2, P = (1 γ )/2, and α, β are color indices, and Q is taken to be R 5 L − 5 u,d,s,c quarks. Here, Ci’s and Ci’s are the Wilson coeﬃcients at the relevant mass scale, and Oi’s are the operators by replacing L(R) with R(L) in Oi. In this paper, Ci includes SM g˜ SM both SM contribution and squark-gluinoe one, such as Ci = Ci + Ci , where Ci ’s are g˜ g˜ givene in Ref. [36]. The Wilson coeﬃcients of the gluino-squark contribution C7γ and C8G are g˜ g˜ presented in Appendix B, where it is remarked that the magnitudes of C7γ (mb) and C8G(mb) are reduced by the cancellation between the contributions of two sbottom ˜b1 and ˜b2.

4 g˜ g˜ The Wilson coeﬃcients of C7γ (mb) and C8G(mb) at the mb scale are given at the leading order of QCD as follows [36]:

g˜ g˜ 8 g˜ C (mb)= ζC (mg˜)+ (η ζ)C (mg˜), 7γ 7γ 3 − 8G (13) g˜ g˜ C8G(mb)= ηC8G(mg˜),

where 16 16 14 14 α (m ) 21 α (m ) 23 α (m ) 21 α (m ) 23 ζ = s g˜ s t , η = s g˜ s t . (14) α (m ) α (m ) α (m ) α (m ) s t s b s t s b 0 Let us discuss the time dependent CP asymmetries of B and Bs decaying into the ﬁnal state f, which are deﬁned as [37] 2Imλ S = f , (15) f 1+ λ 2 | f | where q 0 q q M ∗ A¯(B¯ f) λ = ρ¯ , 12 , ρ¯ q → . (16) f p p ≃ M q ≡ A(B0 f) s 12 q → q Here M12(q = s,d) include the SUSY contribution in addition to the SM one. In the B0 J/ψK and B J/ψφ decays, we write λ and λ in terms of phase → S s → J/ψKS J/ψφ factors, respectively: iφd iφs λ e− , λ e− . (17) J/ψKS ≡ − J/ψφ ≡ In the SM, the angle φd is given as φd =2β, in which β is one angle of the unitarity triangle with respect to B0. On the other hand, φ is given as φ = 2β , in which β is one angle of s s − s s the unitarity triangle for Bs. Once φd is input, the SM predicts φs as [38]

φ = 0.0363 0.0017 . (18) s − ± The recent experimental data of these phases are [4, 39]

sin φ =0.679 0.020 , φ =0.07 0.09 0.01 , (19) d ± s ± ± in which the contribution of the gluino-squark-quark interaction is expected to be found because of

φ =2β + arg(1 + h e2iσd ) , φ = 2β + arg(1 + h e2iσs ) , (20) d d s − s s where β(βs) is given in terms of the CKM matrix elements. These experimental values also constrain the mixing parameters in Eq.(3). Let us consider the contribution from the gluino-sbottom-quark interaction in the non- 0 0 leptonic decays of the B meson. Since the B J/ψKS process occurs at the tree level in → d the SM, the CP asymmetry in this process mainly originates from M12. The CP asymmetries 0 0 0 d of the penguin dominated decays B φKS and B η′K also come from M12 in the 0→ 0 → 0 0 SM. Then, the CP asymmetries of B J/ψK , B φK , and B η′K decays are → S → S →

5 expected to be the same magnitude within 10%. On the other hand, if the gluino-sbottom- quark interaction contributes to the decay at the one-loop level, its magnitude could be 0 0 0 comparable to the SM penguin one in B φKS and B η′K decays, but the effect of → 0 → the gluino-sbottom-quark interaction is tiny in the B J/ψKS decay because this process is at the tree level in the SM. Therefore, there is a possibility→ to find the SUSY contribution by observing the different CP asymmetries among those processes [40, 41].

The time dependent CP asymmetry SJ/ψKS has been precisely measured. We take the data of these time dependent CP asymmetries in HFAG [39], which are

+0.11 ′ 0 SJ/ψKS =0.679 0.020 , SφKS =0.74 0.13 , Sη K =0.59 0.07 . (21) ± − ± These values may be regarded to be same within the experimental error-bar. Thus, the experimental values are consistent with the prediction of the SM. In other words, these data dL(dR) severely may constrain the flavor mixing parameter δ23 . Recently, LHCb reported the first flavor-tagged measurement of the time-dependent CP- violating asymmetry in the Bs decay [2]. In this decay process, the CP-violating weak phase arises due to the CP violation in the interference between B B¯ mixing and the b sss¯ s − s → gluonic penguin decay amplitude. The CP-violating phase φs is measured to be in the interval

φ = [ 2.46, 0.76] rad , (22) s − − at 68%C.L. [2]. We expect that the precise data will be presented in the near future.

2.3 The b s transition → 0 0 0 The CP asymmetries S for B φK and B η′K are given in terms of λ in Eq. (16): f → S → f CSM O + Cg˜ O + Cg˜ O i h ii i h ii i h ii iφ i=3 6,7γ,8G ′ 0 d − λφKS, η K = e− X , (23) SM g˜ e g˜ e − C ∗ O + C ∗ O + C ∗ O i h ii i h ii i h ii i=3 6,7γ,8G −X e e 0 0 0 where Oi is the abbreviation of f Oi B . It is noticed φKS Oi B = φKS Oi B h 0 i 0 0 0 h | | i h | | i h | | i and η′K Oi B = η′K Oi B , because these ﬁnal states have diﬀerent parities [41, 40]. h | | i −h | | i g˜ Since the dominant term comes from the gluon penguin C8G, the decay amplitudes of f =e φKS 0 and f = η′K are given as follows:e

A¯(B¯0 φK ) C (m )+ C˜ (m ), → S ∝ 8G b 8G b 0 0 A¯(B¯ η′K¯ ) C (m ) C˜ (m ). (24) → ∝ 8G b − 8G b

Since C˜8G(mb) is suppressed compared to C8G(mb) in the SM, the magnitudes of the time 0 dependent CP asymmetries Sf (f = J/ψφ, φKS, η′K ) are almost same in the SM prediction. However, the squark ﬂavor mixing gives the unsuppressed C˜8G(mb), then, the CP asymmetries in those decays are expected to be deviated among them. Therefore, those experimental data give us the tight constraint for C8G(mb) and C˜8G(mb).

6 We have also λ for B φφ and B φη′ as follow: f s → s → CSM O + Cg˜ O + Cg˜ O i h ii i h ii i h ii iφs i=3 6,7γ,8G λφφ,φη′ = e− −X , (25) SM g˜ g˜ C ∗ O + C ∗ O + Ce ∗ eO i h ii i h ii i h ii i=3 6,7γ,8G −X e e with φφ O B = φφ O B and φη′ O B = φη′ O B . The decay amplitudes of h | i| si −h | i| si h | i| si h | i| si f = φφ and f = φη′ are given as follows: e e A¯(B¯ φφ) C (m ) C˜ (m ), s → ∝ 8G b − 8G b A¯(B¯ φη′) C (m )+ C˜ (m ). (26) s → ∝ 8G b 8G b Since C O and C˜ O˜ dominate these amplitudes, our numerical results are insensi- 8Gh 8Gi 8Gh 8Gi tive to the hadronic matrix elements. In order to obtain precise results, we also take account of the small contributions from other Wilson coeﬃcients Ci (i =3, 4, 5, 6) and C˜i (i =3, 4, 5, 6) in our calculations. We estimate each hadronic matrix element by using the factorization relations in Ref. [42]:

1 1 O = O = 1+ O , O = O , h 3i h 4i N h 5i h 6i N h 5i c c

αs(mb) 2mb 1 O8G = O4 + O6 ( O3 + O5 ) , (27) h i 8π − q2 ! h i h i − Nc h i h i h i 2 2 where q = 6.3 GeV and Nc =p 3 is the number of colors. One may worry about the reli- abilityh ofi these naive factorization relations. However, this approximation has been justiﬁed numerically in the relevant b s transition as seen in the calculation of PQCD [43]. → 2.4 The b d transition → 0 0 0 The time dependent CP asymmetry S 0 ¯ 0 in the B K K¯ decay is also the interesting K K → one to search for the new physics since there is no tree process of the SM in the B0 K0K¯ 0 decay [44, 45]. The amplitude A¯(B¯0 K0K¯ 0) is given in Ref. [44], in which the→ QCD factorization is taken for the hadronic matrix→ elements [46] 1, as

0 0 0 4GF q q A¯(B¯ K K¯ ) V V ∗ [a (m )+ r a (m )] X. (28) → ≃ √ qb qd 4 b χ 6 b 2 q=u,c X Here X is the factorized matrix element (See Ref. [44].) as

X = if F (m2 )(m2 m2 ), (29) − K 0 K B − K 1Improved analyses with SU(3) ﬂavor symmetry were presented in Refs. [47, 48, 49].

7 2 where fK and F0(mK) denote the decay coupling constant of the K meson and the form 2 factor, respectively, and rχ = 2mK /((mb ms)(ms + md)) denotes the chiral enhancement q − factor. The coeﬃcients ai ’s are given as [44, 46] ˜ q ˜ (C3 C3) αs(mb) CF ˜ a4(mb)=(C4 C4)+ − + (C3 C3)[FK + GK (sd)+ GK(sb)] − Nc 4π Nc " − b

+ C2GK(sq)+ (C4 C˜4)+(C6 C˜6) GK(sf )+(C8G C˜8G)GK,g , − − − # h i Xf=u ˜ q ˜ (C5 C5) αs(mb) CF ˜ a6(mb)=(C6 C6)+ − + (C3 C3)[GK′ (sd)+ GK′ (sb)] − Nc 4π Nc " − b ˜ ˜ ˜ + C2GK′ (sq)+ (C4 C4)+(C6 C6) GK′ (sf )+(C8G C8G)GK,g′ , (30) − − − # h i Xf=u where q takes u and c quarks, C =(N 2 1)/(2N ), and the loop functions F , G , G , F c − c K K K,g GK′ , and GK,g′ are given in Refs. [44, 46]. The internal quark mass in the penguin diagrams enters as s = m2 /m2. 2 The minus sign in front of C˜ (i =3 6, 8G) comes from the parity f f b i − of the ﬁnal state. The CP asymmetry SK0K¯ 0 is given in terms of λK0K¯ 0 : ¯ ¯0 0 ¯ 0 iφd A(B K K ) λ 0 ¯ 0 = e− → . (31) K K − A(B0 K0K¯ 0) → 2.5 Chromo EDM of strange quark In addition to the CP violating processes with ∆B =2, 1, we should discuss the T violation of ﬂavor conserving process, that is the electric dipole moment. The T violation is expected to be observed in the electric dipole moment of the neutron and the electron. The experimental upper bound of the electric dipole moment of the neutron provides us the upper-bound of the chromo-EDM(cEDM) of the strange quark [29]-[32]. C The cEDM of the strange quark ds is given in terms of the gluino-sbottom-quark interactions as seen in Appendix C. The upper bound of the cEDM of the strange quark is given by the experimental upper bound of the neutron EDM as [32], C 25 e d < 0.5 10− ecm. (32) | s | × dL(dR) This bound severely constrains phases of the mixing parameters δ23 of Eq.(3).

3 Tension between ǫK and sin2β

We start our numerical discussion by looking at the ǫK parameter, which is given in the following theoretical formula K iφǫ Im(M12 ) ImA0 1 2∆MK ǫ = e sin φ + ξ , ξ = , φ = tan− , (33) K ǫ ∆M ReA ǫ ∆Γ K 0 K 2 g˜ ∗ ∗ q˜ The Ci in Eq. (30) should be replaced with [(VtbVtq )/(VqbVqd)]Ci in Appendix B.

8 K with A0 being the isospin zero amplitude in K ππ decays. Here, M12 is the dispersive 0 ¯0 → part of the K K mixing, ∆MK is the mass difference in the neutral K meson. An effect of suppression factor− κ which indicates effects of ξ = 0 and φ < π/4, was given by Buras ǫ 6 ǫ and Guadagnoli [27] as: κ =0.92 0.02 . (34) ǫ ± K In the SM, the dispersive part M12 is given as follows,

M 12 = K K¯ K h |H∆F =2| i 4 G 2 = F M 2 Bˆ F 2 M η λ2E(x )+ η λ2E(x )+2η λ λ E(x , x ) , (35) −3 4π W K K K cc c c tt t t ct c t c t where λc = VcsVcd∗ , λt = VtsVtd∗ , and E(x)’s are the one-loop functions [50]. Then, we obtain ǫSM in terms of the Wolfenstein parameters λ, ρ and η as follows: | K | ǫSM = κ C Bˆ V 2λ2η¯ V 2(1 ρ¯)η E(x ) η E(x )+ η E(x , x ) | K | ǫ ǫ K| cb| | cb| − tt t − cc c ct c t 1 = κ C Bˆ V 2λ2 V 2R2 sin(2β)η E(x )+ R sin β( η E(x )+ η E(x , x )) , ǫ ǫ K| cb| 2| cb| t tt t t − cc c ct c t (36) where

2 2 2 GF FKmK MW Cǫ = , (37) 2 6√2π ∆MK and 1 1 ρ¯ = ρ 1 λ2 , η¯ = η 1 λ2 . (38) − 2 − 2 In Eq.(36), we use the relation:

R sin β =η, ¯ R cos β =1 ρ,¯ (39) t t − where Rt is

exp 1 Vtd 1 FBs √Bs MBs ∆MB0 Rt = | | = exp . (40) λ Vts λ F √B MB0 ∆M | | B s s Bs SM As seen in Eq.(36), ǫK is given in terms of sin(2β) because there is only one CP violating phase in the SM. | | If we take into account the gluino-sbottom-quark interaction, ǫK is modiﬁed as

SM g˜ ǫK = ǫK + ǫK. (41)

g˜ Here, ǫK is given by the imaginary part of the gluino-sbottom box diagram, which is presented in Appendix A. The magnitude of ǫg˜ is proportional to the product δdL(dR) δdL(dR) because K | 13 × 23 | 9 the ﬁrst and second families are decoupled in the gluino-sbottom box diagrams. We should also modify Rt as follows:

exp 1 F √B M ∆M 0 C R = Bs s Bs B s , (42) t λ √ M ∆M exp C FB B r B s Bs r d where

2iσq Cq =1+ hqe , (q = d, s). (43)

Now, we remark the non-perturbative parameter BˆK in eq.(36). Recently, the error of this parameter shrank dramatically in the lattice calculations. The most updated value is presented as [51, 52] Bˆ =0.73 0.03 . (44) K ± SM By inputting this value, we can calculate ǫK for the ﬁxed sin(2β). In other words, we can | | 0 test numerically the overlap region of among ǫK , ∆MB /∆MBs and sin(2β) in the unitarity triangle of the SM. | | We obtain the relation between sin(2β) and ǫSM/Bˆ , which is shown with the ex- | K K| 0.0050 perimental allowed region with 90% C.L. 0.0045 in Figure 1. It is noticed that the consis- 0.0040

È tency between the SM prediction and the K

` 0.0035 SM B ˆ experimental data in sin(2β) and ǫK /BK

K | | SM 0.0030

Ε is marginal. This fact was pointed out by È 0.0025 Buras and Guadagnoli [27], and called as 0.0020 the tension between ǫK and sin(2β). This 0.0015 | | 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 situation may indicate the new physics. We sinH2ΒL will show that this tension is understood by taking account of the SUSY box diagram Figure 1: The predicted region on sin 2β- through the gluino-sbottom-quark interac- ˆ ǫK /BK plane in SM. Solid and dotted lines de- tion, which also predicts the deviation from | | 0 note the experimental best ﬁt and bounds with the SM in the CP violations of B φKS, 0 0 → 0 90% C.L. B η′K , Bs φφ, Bs φη′, B 0 →0 0 → → → K K¯ , B µ−X, and Bs µ−X decays. → →

4 Numerical Results

Let us present the numerical results. In order to constrain the gluino-sbottom-quark mixing parameters, we input the experimental data of the CP violations, ǫK , φd, and φs. The experimental upper bound of cEDM of the strange quark is also put. In addition to these experimental data of the CP violations and T violation, we take account of the observed values ∆M 0 , ∆M , the CKM mixing V , and the branching ratio of b sγ. The input B Bs | ub| →

10 0.005 0.0055 0.004 0.0050 K ` 0.003 0.0045 B È 0.0040 ub K SM

0.002 V È Ε 0.0035 È 0.001 0.0030 0.0025 0.000 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.0020 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 sinH2ΒL sinH2ΒL

Figure 2: Predicted region on ǫSM /Bˆ - | K | K Figure 3: The predicted Vub versus sin(2β) plane. Vertical and horizontal sin(2β). Horizontal dashed| lines| de- dashed lines denote the experimental al- note the experimental allowed region with lowed region with 90%C.L. Vertical and 90%C.L. of V . Horizontal solid lines | ub| horizontal solid lines denote observed cen- denote the observed central values. tral values. parameters in our calculation are summarized in Table 1. SM ˆ At ﬁrst, we present the allowed region on the plane of ǫK /BK and sin(2β) in Figure 2, where SM components in Eqs. (20) and (41) are only shown.| | The present experimental data of sin φ in Eq. (19) allows the range of sin(2β)=0.57 0.88, where β is one angle d − of the unitarity triangle. Once we take account of the contribution of the gluino-sbottom- quark interaction, the allowed regions of ǫ and sin φ converge within the experimental | K | d error-bars. SM SM 0 When sin 2β and ∆MB /∆MBs are ﬁxed, the Vub is predicted. In Figure 3, we show the relation between sin(2β) and V , where the outside| | the experimental error-bar of V | ub| | ub| are cut. In the present measurement of Vub , there is 2.6 σ discrepancy in the exclusive and inclusive decays as follows [51]: | |

3 3 V = (3.28 0.30) 10− (exclusive), V = (4.40 0.31) 10− (inclusive), (45) | ub| ± × | ub| ± × 3 although the average value is Vub = (3.82 0.56) 10− . The precise observation of Vub leads to the determination of sin(2| |β). ± × | | The allowed region of the mixing parameters δdL(dR) and δdL(dR) are shown in Figure 4, | 13 | | 23 | where we input the experimental data of the CP violations, ǫK , φd, and φs. The experimental upper bound of cEDM of the strange quark is also input. We also take account of the observed 0 values ∆MB , ∆MBs , the CKM mixing Vub , and the branching ratio of b sγ. As seen in Figure 4, we obtain the allowed| | region of →

δdL(dR) =0 0.01, δdL(dR) =0 0.04. (46) | 13 | ∼ | 23 | ∼ By using these values, we discuss the sensitivity of the SUSY contribution to the CP violation 0 of the B and Bs decays.

11 αs(MZ )=0.1184 [35] mc(mc)=1.275 GeV [35] mt(mc)=1.275 GeV (MS¯ ) [35]

MBs =5.36677(24) GeV [35] 13 ∆Ms = (116.942 0.1564) 10− GeV [7] ± × 13 ∆Md = (3.337 0.033) 10− GeV [35] f = (233 10)± MeV [51]× Bs ± f /f 0 =1.200 0.02 [51] Bs B ± ξs =1.21(6) [27] λ =0.2255(7) [35] 2 V = (4.12 0.11) 10− [51] | cb| ± × ηcc =1.43(23) [27] ηct =0.47(4) [27] ηtt =0.5765(65) [27] f = (156.1 1.1) MeV [35] K ± κǫ =0.92(2) [27]

Table 1: Input parameters in our calculation.

0.020

0.015 È L dR H 0.010 dL 13 ∆ È 0.005

0.000 0.00 0.01 0.02 0.03 0.04 0.05 dL HdRL È∆23 È

Figure 4: Allowed region of the mixing parameters, δdL(dR) and δdL(dR) . | 13 | | 23 |

′ 0 Let us discuss the time dependent CP asymmetries SφKS and Sη K . The SM leads to S (SM) S (SM) = S ′ 0 (SM), while the present data of these time dependent CP J/ψKS ≃ φKS η K asymmetries are given in Eq. (21). We predict the deviation from the SM in Figure 5 for the two cases, where the constraint of the cEDM of the strange quark is imposed or is not imposed. It is clearly seen that the cEDM of the strange quark reduces the deviation from the SM. Thus, it is very diﬃcult to observe the gluino-sbottom-quark contribution in these non-leptonic decays. In Figure 6, we show the prediction of the time dependent CP asymmetries Sφφ and Sφη′ , where the constraint of the cEDM is imposed. We use the experimental result of SJ/ψφ for the phase φs, which is given in Eq. (19), in our calculations. We denote the small pink region as the SM value S (SM) = 0.0363 0.0017 [38] in the ﬁgure. It is found that J/ψφ − ±

12 SJ/ψφ is almost proportional to Sφφ. If the ∆B = 1 SUSY contribution is seizable, these asymmetries should be different each other as seen in Eq.(26). That is, the gluino interaction induced ∆B = 1 contribution is very small. On the other hand, the gluino induced ∆B =2 contribution (SUSY box diagrams) could be detectable as seen in Eqs.(20) and (25). This g˜ situation is understandable because the magnitude of C8G(mb) is reduced by the cancellation ˜ ˜ between the contributions of two sbottom b1 and b2 as seen in Appendix B. In conclusion, we predict 0.1 . Sφφ . 0.2 and 0.1 . Sφη′ . 0.2, respectively. Since the phase φs has still large experimental− error bar, our− prediction will be improved if the precise experimental data of SJ/ψφ will be given in the near future at LHCb. In order to see this situation clearly, we show the SJ/ψφ dependence for the predicted Sφφ in Figure 7. LHCb reported the first flavor-tagged measurement of the time-dependent CP-violating asymmetry in Bs φφ decay [2]. The CP-violating phase is measured to be in the interval φ = [ 2.46, 0.76]→ rad as seen in Eq.(22). The precision of the CP violating phase mea- s − − surement is dominated by the statistical uncertainty and is expected to improve with larger LHCb data in the near future.

0.80 0.80 0.75 without cEDM 0.75 with cEDM 0.70 0.70 0 0 K K 0.65 0.65 ' ' Η Η

S 0.60 0.60 S 0.55 0.55 0.50 0.50 0.45 0.45 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 S S ΦKS ΦKS

′ 0 Figure 5: The predicted time dependent CP asymmetries on SφKS –Sη K plane without/with ′ 0 the constraint of cEDM of the strange quark. The SM prediction SJ/ψKS SφKS = Sη K is plotted by the pink slant lines. The experimental data with error-bar is plotted≃ by the red solid lines at 90% C.L.

In Figure 8 , we show the prediction of the time dependent CP asymmetry SK0K¯ 0 de- dL(dR) pending on δ . The predicted region is 0.4 S 0 ¯ 0 0.3, on the other hand, one | 13 | − ≤ K K ≤ predicts S 0 ¯ 0 (SM) 0.06 in the SM [44]. The present experimental data are given as K K ≃ S 0 ¯ 0 (exp) = 0.8 0.5 [35]. Since the SM predicted value is tiny, we have a chance to K K − ± observe the SUSY contribution by the precise experimental data in the near future. d s At last, we present the prediction of the indirect CP violation asl and asl in Figure 9. d s The predicted region is asl = 0.0017 0.002 and asl = 0.001 0.001, on the other dSM − ∼ 4 sSM − ∼ 5 hand, the SM gives asl = (4.1 0.6) 10− and asl = (1.9 0.3) 10− as shown in Eq.(9). The experimental data− still± have× large error-bars [8, 35]. ±The precise× measurement d of the semi-leptonic asymmetry asl at Belle II will provide us an interesting test of the SUSY contribution.

13 0.4 0.4

0.2 0.2 ΦΦ 0.0 ΦΦ 0.0 S S

-0.2 -0.2

-0.4 -0.4 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.10-0.05 0.00 0.05 0.10 0.15 0.20 0.25

SΦΗ' SJΨΦ

Figure 6: The predicted time dependent Figure 7: The predicted Sφφ versus SJ/ψφ CP asymmetries on Sφη′ –Sφφ plane. The plane, where SJ/ψφ is plotted within the small pink region denotes the SM predic- experimental error at 90% C.L. The small tion. pink region denotes the SM prediction.

1.0 0.010

0.5 0.005

0.000 s sl KK 0.0 a S -0.005 -0.5 -0.010

-1.0 -0.015 0.000 0.005 0.010 0.015 0.020 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 dL HdRL ad È∆13 È sl

Figure 8: The predicted time dependent Figure 9: Predicted semi-leptonic CP dL(dR) d s CP asymmetry S 0 ¯ 0 versus δ . asymmetries asl and asl. The red solid K K | 13 | The red solid and red dotted lines denote and red dotted lines denote the best ﬁt the best ﬁt value and the experimental value and the experimental bounds with bound with 90% C.L., respectively. 90% C.L., respectively.

5 Summary

We have discussed the sensitivity of the gluino-sbottom-quark interaction to the CP violating 0 phenomena of the K, B and Bs mesons. We take the split-family scenario, which is the consistent with the LHC data. In this scenario, the ﬁrst and second family squarks are very heavy, (10) TeV, on the other hand, the third family squark masses are at (1) TeV. Then, theO s d transition is mediated by the second order contribution of the thirdO family → sbottom. We have used mg˜ = 2 TeV, m˜b1 = 1 TeV, and m˜b2 =1.1 TeV. In order to constrain the gluino-sbottom-quark mixing parameters, we input the experimental data of the CP violations, ǫK , φd, and φs. The experimental upper bound of the cEDM of the strange quark 0 is also input. In addition, we take account of the observed values ∆MB , ∆MBs , the CKM mixing V , and the branching ratio of b sγ. | ub| →

14 By using the non-perturbative parameter BˆK = 0.73 0.03, which is the most updated value in the lattice calculations, it is clearly presented that± the consistency between the SM prediction and the experimental data is marginal on the sin(2β) ǫSM plane. This tension −| K | has been solved by taking account of the gluino-sbottom-quark interaction. dL(dR) The allowed region of the mixing parameters are obtained as δ13 = 0 0.01 and dL(dR) | | ∼ δ23 = 0 0.04. By using these values, the deviations from the SM prediction are | | ∼ 0 0 estimated in the CP violation of B and Bs decays. The CP asymmetries of the B φKS 0 0 → and B η′K decays are found to be tiny due to the cEDM constraint of the strange quark. → On the other hand, the CP asymmetries of the B φφ and B φη′ decays could be s → s → largely deviated from the SM predictions such as 0.1 . S . 0.2 and 0.1 . S ′ . 0.2. It − φφ − φη is remarked that the time dependent CP asymmetry Sφφ is almost proportional to Sφη′ . That is, the gluino-sbottom interaction induced ∆B = 1 transition is very small, but the ∆B =2 transition could be detectable. Since the phase φs has still large experimental error-bar, our prediction will be improved if the precise experimental data of SJ/ψφ will be given in the near future at the LHCb experiment. We also predict the time dependent CP asymmetry S 0 ¯ 0 as 0.4 S 0 ¯ 0 0.3 while K K − ≤ K K ≤ one predicts SK0K¯ 0 (SM) 0.06 in the SM. More precise data will test the SUSY contribution ≃ d s d in the near future. The semi-leptonic CP asymmetries asl and asl are predicted as asl = s dSM 4 0.0017 0.002 and asl = 0.001 0.001, while the SM predicts asl = (4.1 0.6) 10− − sSM∼ − 5 ∼ − d± × and asl = (1.9 0.3) 10− . We expect the precise measurement of the asl at Belle II, which will provide± us interesting× tests of the squark ﬂavor mixing.

Acknowledgment This work is supported by JSPS Grand-in-Aid for Scientiﬁc Research, 21340055 and 24654062, 25-5222, respectively.

Appendix

A Squark contribution in ∆F =2 process

The ∆F = 2 eﬀective Lagrangian from the gluino-sbottom-quark interaction is given as 1 ∆F =2 = [C O + C O ] Leﬀ − 2 V LL V LL V RR V RR 1 2 C(i) O(i) + C(i) O(i) + C(i) O(i) , (47) − 2 SLL SLL SRR SRR SLR SLR i=1 X h i 0 0 then, the P -P¯ mixing, M12, is written as

1 0 ∆F =2 ¯0 M12 = P eﬀ P . (48) −2mP h |L | i

2 6 αs (d) ij (d) ij 11 g˜ g˜ 2 g˜ g˜ CV LL(mg˜)= 2 (λGLL)I (λGLL)J g2[1](xI , xJ )+ g1[1](xI , xJ ) , mg˜ 18 9 I,JX=1 C (m )= C (m )(L R), V RR g˜ V LL g˜ ↔ 2 6 (1) αs (d) ij (d) ij 17 g˜ g˜ CSRR(mg˜)= 2 (λGLR)I (λGLR)J g1[1](xI , xJ ), mg˜ 9 I,JX=1 C(1) (m )= C(1) (m )(L R), SLL g˜ SRR g˜ ↔ 2 6 (2) αs (d) ij (d) ij 1 g˜ g˜ CSRR(mg˜)= 2 (λGLR)I (λGLR)J g1[1](xI , xJ ), mg˜ −3 I,JX=1 C(2) (m )= C(2) (m )(L R), SLL g˜ SRR g˜ ↔ α2 6 11 C(1) (m )= s (λ(d) )ij(λ(d) )ij g (xg˜, xg˜ ) SLR g˜ m2 GLR I GRL J − 9 2[1] I J g˜ I,J=1 ( X 14 2 +(λ(d) )ij(λ(d) )ij g (xg˜, xg˜ ) g (xg˜, xg˜ ) , GLL I GRR J 3 1[1] I J − 3 2[1] I J ) α2 6 5 C(2) (m )= s (λ(d) )ij(λ(d) )ij g (xg˜, xg˜ ) SLR g˜ m2 GLR I GRL J −3 2[1] I J g˜ I,J=1 ( X 2 10 +(λ(d) )ij(λ(d) )ij g (xg˜, xg˜ )+ g (xg˜, xg˜ ) , (51) GLL I GRR J 9 1[1] I J 9 2[1] I J ) where (d) ij (d) K (d) j (d) ij (d) K (d) j (λGLL)K = (ΓGL†)i (ΓGL)K , (λGRR)K = (ΓGR†)i (ΓGR)K , (d) ij (d) K (d) j (d) ij (d) K (d) j (λGLR)K = (ΓGL†)i (ΓGR)K , (λGRL)K = (ΓGR†)i (ΓGL)K . (52)

16 0 0 Here we take (i, j) = (1, 3), (2, 3), (1, 2) which correspond to B , Bs, and K mesons, respectively. The loop functions are given as follows:

g˜ g˜ g˜ 2 2 If xI = xJ (xI,J = m ˜ /mg˜), • 6 dI,J

g˜ g˜ g˜ g˜ g˜ g˜ 1 xI log xI 1 xJ log xJ 1 g1[1](x , x )= + , I J g˜ g˜ g˜ 2 g˜ g˜ 2 g˜ xI xJ (xI 1) − xI 1 − (xJ 1) xJ 1! − − − − − g˜ 2 g˜ g˜ 2 g˜ g˜ g˜ 1 (xI ) log xI 1 (xJ ) log xJ 1 g2[1](x , x )= + . (53) I J g˜ g˜ g˜ 2 g˜ g˜ 2 g˜ xI xJ (xI 1) − xI 1 − (xJ 1) xJ 1! − − − − −

If xg˜ = xg˜ , • I J g˜ g˜ g˜ g˜ (xI + 1) log xI 2 g1[1](x , x )= + , I I g˜ 3 g˜ 2 − (xI 1) (xI 1) g˜ − g˜ g˜ − g˜ g˜ 2xI log xI xI +1 g2[1](x , x )= + . (54) I I − (xg˜ 1)3 (xg˜ 1)2 I − I − In this paper, we take (I,J)=(3, 3), (3, 6), (6, 3), (6, 6), because we assume the split-family. The eﬀective Wilson coeﬃcients are given at the leading order of QCD as follows:

B(K) B(K) CV LL(mb(Λ = 2 GeV)) =ηV LL CV LL(mg˜), CV RR(mb(Λ = 2 GeV)) = ηV RR CV LL(mg˜), (1) (1) CSLL(mb(Λ = 2 GeV)) CSLL(mg˜) 1 B(K) (2) = (2) XLL− ηLL XLL, CSLL(mb(Λ = 2 GeV))! CSLL(mg˜)! (1) (1) CSRR(mb(Λ = 2 GeV)) CSRR(mg˜) 1 B(K) (2) = (2) XRR− ηRR XRR, CSRR(mb(Λ = 2 GeV))! CSRR(mg˜)! (1) (1) CSLR(mb(Λ = 2 GeV)) CSLR(mg˜) 1 B(K) (2) = (2) XLR− ηLR XLR, (55) CSLR(mb(Λ = 2 GeV))! CSLR(mg˜)! where

6 6 21 23 B B αs(mg˜) αs(mt) ηV LL = ηV RR = , αs(mt) αs(mb) 1 1 dLL dLR B B ηbg˜ 0 1 B ηbg˜ 0 1 η = η = SLL 2 S− , η = SLR 2 S− , LL RR dLL LL LR dLR LR 0 ηbg˜ ! 0 ηbg˜ ! 1 3 14 46 αs(mg˜) αs(mt) ηbg˜ = , αs(mt) αs(mb) 6 6 6 α (m ) 21 α (m ) 23 α (m ) 25 ηK = ηK = s g˜ s t s b , V LL V RR α (m ) α (m ) α (Λ = 2 GeV) s t s b s

17 1 1 dLL dLR K K ηΛ˜g 0 1 K ηΛ˜g 0 1 η = η = SLL 2 S− , η = SLR 2 S− , LL RR dLL LL LR dLR LR 0 ηΛ˜g ! 0 ηΛ˜g ! 1 3 3 α (m ) 14 α (m ) 46 α (m ) 50 η = s g˜ s t s b , Λ˜g α (m ) α (m ) α (Λ = 2 GeV) s t s b s 2 2 d1 = (1 √241), d2 = (1 + √241), d1 = 16, d2 =2, LL 3 − LL 3 LR − LR 16+√241 16 √241 2 1 60 −60 SLL = , SLR = − , 1 1 3 0 1 0 0 2 X = X = , X = . LL RR 4 8 LR 1− 0 (56)

For the parameters B(d)(i =2 5) of B mesons, we use values in [54] as follows: i − (Bd) (Bd) B2 (mb)=0.79(2)(4), B3 (mb)=0.92(2)(4), (Bd) +5 (Bd) +20 B4 (mb)=1.15(3)( 7), B5 (mb)=1.72(4)( 6 ), − − (Bs) (Bs) B2 (mb)=0.80(1)(4), B3 (mb)=0.93(3)(8), (Bs) +5 (Bs) +21 B4 (mb)=1.16(2)( 7), B5 (mb)=1.75(3)( 6 ) . (57) − − ˆ(d) ˆ(s) On the other hand, we use the most updated values for B1 and B1 as [51, 52] Bˆ(Bs) =1.33 0.06 , Bˆ(Bs)/Bˆ(Bd) =1.05 0.07 . (58) 1 ± 1 1 ± For the paremeters BK (i =2 5), we use following values [55], i − B(K)(2GeV) = 0.66 0.04, B(K)(2GeV) = 1.05 0.12, 2 ± 3 ± (59) B(K)(2GeV) = 1.03 0.06, B(K)(2GeV) = 0.73 0.10, 4 ± 5 ± (K) and we take recent value of Eq.(44) for deriving B1 (2GeV).

B Squark contribution in ∆F =1 process

The Wilson coeﬃcients for the gluino contribution in Eq.(11) are written as [53]

g˜ 8 √2αsπ C7γ (mg˜)= 3 2GF VtbVtq∗ (d) ∗ ΓGL k3 (d) 1 3 mg˜ (d) 1 3 2 ΓGL 33 F2(xg˜) + ΓGR 33 F4(xg˜) × m −3 mb −3 " d˜3 (d) ∗ ΓGL k6 (d) 1 6 mg˜ (d) 1 6 + 2 ΓGL 36 F2(xg˜) + ΓGR 36 F4(xg˜) , (60) m −3 mb −3 d˜6 # 18 (d) ∗ g˜ 8 √2αsπ ΓGL k3 (d) 9 3 1 3 C8G(mg˜)= 2 ΓGL 33 F1(xg˜) F2(xg˜) 3 2GF VtbV ∗ m −8 − 8 tq " d˜3 m 9 1 + g˜ Γ(d) F (x3) F (x3) m GR 33 −8 3 g˜ − 8 4 g˜ b (d) Γ ∗ 9 1 + GL k6 Γ(d) F (x6) F (x6) m2 GL 36 −8 1 g˜ − 8 2 g˜ d˜6 m 9 1 + g˜ Γ(d) F (x6) F (x6) , (61) m GR 36 −8 3 g˜ − 8 4 g˜ b # where k = 2, 1 correspond to b q (q = s,d) transitions, respectively. The loop functions I → Fi(xg˜) are given as

xI log xI (xI )2 5xI 2 F (xI )= g˜ g˜ + g˜ − g˜ − , 1 g˜ 2(xI 1)4 12(xI 1)3 g˜ − g˜ − (xI )2 log xI 2(xI )2 +5xI 1 F (xI )= g˜ g˜ + g˜ g˜ − , 2 g˜ − 2(xI 1)4 12(xI 1)3 g˜ − g˜ − log xI xI 3 F (xI )= g˜ + g˜ − , 3 g˜ (xI 1)3 2(xI 1)2 g˜ − g˜ − xI log xI xI +1 1 F (xI )= g˜ g˜ + g˜ = g (xI , xI ) , (62) 4 g˜ −(xI 1)3 2(xI 1)2 2 2[1] g˜ g˜ g˜ − g˜ − with xI = m2/m2 (I =3, 6). g˜ g˜ d˜I

C cEDM

The cEDM of the strange quark from gluino contribution is given by [53]

dC = 2 4πα (m )Im[Ag22], (63) s − s g˜ s q where

α (m ) 1 1 Ag22 = s g˜ m (λ(d) )22 + m (λ(d) )22 9F (x3)+ F (x3) s − 4π 3 2m2 s GLL 3 s GRR 3 1 g˜ 2 g˜ " ˜3 d (d) 22 3 3 + mg˜(λGLR)3 9F3(xg˜)+ F4(xg˜) 1 + m (λ(d) )22 + m (λ(d) )22 9F (x6)+ F (x6) + m (λ(d) )22 9F (x6)+ F (x6) . 2m2 s GLL 6 s GRR 6 1 g˜ 2 g˜ g˜ GLR 6 3 g˜ 4 g˜ ˜6 # d (64)

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