Searching for the squark flavor mixing in CP violations of Bs -> K+ K- and K0bar K0 decays
Atsushi Hayakawa, Yusuke Shimizu, Morimitsu Tanimoto, Kei Yamamoto
aa r X i v : . [ h e p - ph ] J a n Searching for the squark flavor mixingin CP violations of B s → K + K − and K ¯ K decays Atsushi Hayakawa , ∗ , Yusuke Shimizu , † ,Morimitsu Tanimoto , ‡ , and Kei Yamamoto , § Graduate School of Science and Technology, Niigata University,Niigata 950-2181, Japan Max-Planck-Institute f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Department of Physics, Niigata University, Niigata 950-2181, Japan
Abstract
We study CP violations in the B s → K + K − and B s → K K decays in order to find thecontribution of the supersymmetry, which comes from the gluino-squark mediated flavorchanging current. We obtain the allowed region of the squark flavor mixing parameters byputting the experimental data, the mass difference ∆ M B s , the CP violating phase φ s in B s → J/ψφ decay and the b → sγ branching ratio. In addition to these data, we take intoaccount the constraint from the asymmetry of B → K + π − because the B s → K + K − decay is related with the B → K + π − decay by replacing the spectator s with d . Underthese constraints, we predict the magnitudes of the CP violation in the B s → K + K − and B s → K K decays. The predicted region of the CP violation C K + K − is strongly cut fromthe direct CP violation of ¯ B → K − π + , therefore, the deviation from the SM predictionof C K + K − is not found. On the other hand, the CP violation S K + K − is possibly deviatedfrom the SM prediction considerably, in the region of 0 . ∼ .
5. Since the standard modelpredictions of C K ¯ K and S K ¯ K are very small, the squark contribution can be detectablein C K ¯ K and S K ¯ K . These magnitudes are expected in the region C K ¯ K = − . ∼ . S K ¯ K = − . ∼ .
3. More precise data of these CP violations provide us a crucialtest for the gluino-squark mediated flavor changing current. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected]
Introduction
Recently, there have been a lot of studies to search for new physics in the low energy flavorphysics such as B s decays. Actually, the LHCb collaboration has reported new data of theCP violations of the B s meson and the branching ratios of rare B s decays [1]-[12]. For manyyears, the CP violations in the K and B mesons have been successfully understood withinthe framework of the standard model (SM), so called Kobayashi-Maskawa (KM) model [13],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 indirectly discovered in the flavor changingneutral current (FCNC) of the B and B s decays at the LHCb experiment and the furthercoming experiment Belle II.The LHCb collaboration presented the data of the time dependent CP asymmetry inthe non-leptonic B s → J/ψφ decay [4, 11, 12], which is consistent with the SM prediction.Therefore, this observed value gives us a strong constraint of the new physics contribution tothe b → s transition. In addition to this result, the first measurement of time-dependent CPviolation in B s → K + K − decay has been reported at LHCb [14]. Some authors discussedthis process and the B s → K K one in order to search for new physics [15]-[20], becausethe penguin amplitudes dominate these decays. Especially, the SM prediction of the CPviolation of the B s → K K decay is very small, and so, the new physics contribution canbe detectable in the time dependent CP asymmetry.On the other hands, it is noticed that the B s → K + K − decay is related with the B → K + π − decay by replacing the spectator s with d . Thus, the B → K + π − decay associateswith the processes of B s → K + K − and B s → K K in order to search for the new physicsin the b → s penguin process. It is found that the recent experimental data of the directCP violation in B → K + π − decay is well agreement with the SM prediction with the QCDfactorization calculation [21, 22]. This process depends on the form factor F ( B → K ) andthe chiral enhancement factor (2 M K /m b m s ) in the framework of the QCD factorization. Theamplitudes of B s → K + K − and B s → K K decays also involve the common form factorand chiral enhancement factor under neglecting the difference of masses of the B and B s mesons.As the new physics, we examine the sensitivity of the effect of the supersymmetry (SUSY)in the CP violation of these B s decays. Although the SUSY is one of the most attractivecandidates for the new physics, the SUSY signals have not been observed yet. Since the lowerbounds of the superparticle masses increase gradually, the squark and the gluino masses aresupposed to be at the TeV scale [23]. While, there are new sources of the CP violation in thelow energy flavor physics if the SM is extended to the SUSY model. The soft squark mass ma-trices contain the CP-violating phases, which contribute to the FCNC with the CP violation.Therefore, one expects the effect of the SUSY contribution in the CP-violating phenomena ofthe B s meson decays. We study the gluino-squark mediated flavor changing process, whichis the most important process of the SUSY contribution for the b → s transition [24]- [37].The gluino mass is expected to be larger than 1 . . O (10 − O (1) TeV. Then, the s → d transition mediated1y the first and second family squarks is suppressed by their heavy masses, and competingprocess is mediated by the second order contribution of the third family squark. In order toestimate the gluino-squark mediated FCNC for the B s meson decays, we work in the basisof the squark mass eigenstate. Then, the 6 × B and B s decays inthe QCD factorization. In section 3, we present the setup in our split-family scenario. Insection 4, we discuss the sensitivity of the gluino-squark mediated FCNC to the CP violationof the B → K + π − , B s → K + K − and, B s → K K decays. Section 5 is devoted to thesummary. Relevant formulations are presented in appendices A, B, and C. B decays in QCD factorization In this section, we present the formulation of the CP violation in B → K + π − , B s → K + K − ,and B s → K K decays within the framework of the QCD factorization [21, 22, 38, 39].First, we begin with the effective Hamiltonian for the ∆ B = 1 transition as H eff = 4 G F √ " X q ′ = u,c V q ′ b V ∗ q ′ q X i =1 , C i O ( q ′ ) i − V tb V ∗ tq X i =3 − , γ, G (cid:16) C i O i + e C i e O i (cid:17) , (1)where q = s, d . The local operators are given as O ( q ′ )1 = (¯ q α γ µ P L q ′ β )(¯ q ′ β γ µ P L b α ) , O ( q ′ )2 = (¯ q α γ µ P L q ′ α )(¯ q ′ β γ µ P L b β ) ,O = (¯ q α γ µ P L b α ) X Q ( ¯ Q β γ µ P L Q β ) , O = (¯ q α γ µ P L b β ) X Q ( ¯ Q β γ µ P L Q α ) ,O = (¯ q α γ µ P L b α ) X Q ( ¯ Q β γ µ P R Q β ) , O = (¯ q α γ µ P L b β ) X Q ( ¯ Q β γ µ P R Q α ) ,O = 32 (¯ q α γ µ P L b α ) X Q ( e Q ¯ Q β γ µ P R Q β ) , O = 32 (¯ q α γ µ P L b β ) X Q ( e Q ¯ Q β γ µ P R Q α ) ,O = 32 (¯ q α γ µ P L b α ) X Q ( e Q ¯ Q β γ µ P L Q β ) , O = 32 (¯ q α γ µ P L b β ) X Q ( e Q ¯ Q β γ µ P L Q α ) ,O γ = e π m b ¯ q α σ µν P R b α F µν , O G = g s π m b ¯ q α σ µν P R T aαβ b β G aµν , (2)where P R = (1 + γ ) / P L = (1 − γ ) /
2, and α , β are color indices, and Q is taken to be u, d, s, c quarks. Here, C i ’s and e C i ’s are the Wilson coefficients at the relevant mass scale,and e O i ’s are the operators by replacing L ( R ) with R ( L ) in O i . The e C i ’s are neglected in SM.We use the value of Wilson coefficients at µ = m b as follows: C = − . , C = 1 . , C = 0 . , C = − . ,C = 0 . , C = − . , C = − . / , C = 0 . / ,C = − . / , C = − . / , C G = − . , (3)2n the SM calculations [38].The hard scattering amplitude is given for the relevant decay modes as follows: T p = 4 G F √ X p = u,c V ∗ pq V pb h a p (¯ qγ µ Lu ) ⊗ (¯ uγ µ Lb ) + a p (¯ uγ µ Lu ) ⊗ (¯ qγ µ Lb ) + a p ( ¯ q ′ γ µ Lq ′ ) ⊗ (¯ qγ µ Lb )+ a p (¯ qγ µ Lq ′ ) ⊗ (¯ qγ µ Lb ) + a p ( ¯ q ′ γ µ Rq ′ ) ⊗ (¯ qγ µ Lb ) + a p ( − qRq ′ ) ⊗ ( ¯ q ′ Lb )+ a p e q ′ ( ¯ q ′ γ µ Rq ′ ) ⊗ (¯ qγ µ Lb ) + ( − a p e q ′ + a a )(¯ qRq ′ ) ⊗ ( ¯ q ′ Lb )+ a p e q ′ ( ¯ q ′ γ µ Lq ′ ) ⊗ (¯ qγ µ Lb ) + ( a p e q ′ + a p a )(¯ qγ µ Lq ′ ) ⊗ ( ¯ q ′ γ µ Lb ) i , (4)where the symbol ⊗ denotes h M M | j ⊗ j | B i ≡ h M | j | ih M | j | B i . The effective a pi ’s whichcontain next-to leading order (NLO) coefficients and O ( α s ) hard scattering corrections aregiven as, a c , = 0 , a ci = a ui ( i = 3 , , , , , , a, a ) , a u = C + C N + α s π C F N C F M ,a u = C + C N + α s π C F N C F M , a u = C + C N + α s π C F N C F M ,a p = C + C N + α s π C F N h C (cid:2) F M + G M ( s q ) + G M ( s b ) (cid:3) + C G M ( s q )+ ( C + C ) b X f = u G M ( s f ) + C G G M ,g i ,a u = C + C N + α s π C F N C ( − F M − ,a p = C + C N + α s π C F N h C G ′ M ( s p ) + C (cid:2) G ′ M ( s q ) + G ′ M ( s b ) (cid:3) + ( C + C ) b X f = u G ′ M ( s f ) + C G G ′ M ,g i ,a u = C + C N + α s π C F N C ( − F M − , a p = C + C N ,a p a = α s π C F N h ( C + C ) b X f = u e f G ′ M ( s f ) + C
32 [ e q G ′ M ( s q ) + e b G ′ M ( s b )] i ,a u = C + C N + α s π C F N C F M , a u = C + C N + α s π C F N C F M ,a p a = α s π C F N h ( C + C ) b X f = u e f G M ( s f ) + C
32 [ e q G M ( s q ) + e b G M ( s b )] i , (5)where q = d, s q ′ = u, d, s f = u, d, s, c, b and C F = ( N − / (2 N ) with the number ofcolors N = 3. In Appendix A, we present the loop integral functions F M G M ,g , G M ( s q ), G ′ M ,g and G ′ M ( s q ), in which the internal quark mass enters as s f = m f /m b .3n this work, C i includes both SM contribution and squark-gluino one, such as C i = C SM i + C ˜ gi , where C SM i ’s are given in Ref. [40]. The Wilson coefficients of the gluino-squarkcontribution C ˜ g γ and C ˜ g G are presented in Appendix B. We should also take account of theSUSY contribution in e C i ’s( i = 3 − , γ , G ), which are derived by replacing L ( R ) with R ( L ) in C i . Then, C i ’s are replaced with C i − e C i in Eq.(5) for the decays B s → K + K − and B s → K K . The minus sign in front of e C i is due to the parity of the final states.By using these formula, we can write the decay amplitude for the ¯ B → K − π + , ¯ B s → K + K − and ¯ B s → K K decays, respectively, as follows:¯ A ( ¯ B → K − π + ) = G F √ if π ( M B − M K ) F B → K (0)(1 − λ | V cb | (cid:16) R CKM e − iγ (cid:2) a u + a u + R K ( a u + a u + a a ) + a u + a u a (cid:3) + (cid:2) a c + R K ( a c + a c ) + a c + a c a ] (cid:17) , (6)¯ A ( ¯ B s → K + K − ) = G F √ if K ( M B s − M K ) F B s → K (0)(1 − λ | V cb | (cid:16) R CKM e − iγ (cid:2) a u + a u + R K ( a u + a u + a a ) + a u + a u a (cid:3) + (cid:2) a c + R K ( a c + a c ) + a c + a c a ] (cid:17) , (7)¯ A ( ¯ B s → K ¯ K ) = G F √ if K ( M B s − M K ) F B s → K (0)(1 − λ | V cb | (cid:16) R CKM e − iγ (cid:2) a u + R K ( a u + a u + a a ) + a u + a u a (cid:3) + (cid:2) a c + R K ( a c + a c ) + a c + a c a ] (cid:17) , (8)where R CKM = λ − λ / (cid:12)(cid:12)(cid:12)(cid:12) V ub V cb (cid:12)(cid:12)(cid:12)(cid:12) , and f π ( K ) , F B ( B s ) → K (0) are decay constants and the form factors at q = 0, respectively.The CKM matrix elements V cb , V ud and V us are chosen to be real and γ is the phase of V ∗ ub ,and we take λ = V us = 0 . R K = 2 M K / (( m s + m ¯ d )( m b − m q )).Let us discuss the time dependent CP asymmetries of B s decaying into the final state f ,which are defined as [41] C f = 1 − | λ f | | λ f | , S f = 2Im λ f | λ f | , (9)where λ f = qp ¯ ρ , qp ≃ s M s ∗ M s , ¯ ρ ≡ ¯ A ( ¯ B s → f ) A ( B s → f ) . (10)In the B s → J/ψφ decay, we write λ J/ψφ in terms of phase factors as follow: λ J/ψφ ≡ e − iφ s . (11)4n the SM, the angle φ s is given as φ s = − β s , in which β s is one angle of the unitaritytriangle for B s . The SM predicts φ s as [42] φ s = − . ± . . (12)The recent experimental data of this phase is [4, 43] φ s = 0 . ± . ± . . (13)This value constrains the magnitude of the new physics, which contributes to M s in Eq.(10).For the gluino-squark contribution to M s , we present the formulation in Appendix C.The time dependent CP asymmetries of B s → K + K − and B s → K ¯ K are obtained bycalculating λ K + K − = e − iφ s ¯ A ( ¯ B s → K + K − ) A ( B s → K + K − ) , λ K ¯ K = e − iφ s ¯ A ( ¯ B s → K ¯ K ) A ( B s → K ¯ K ) . (14)The new physics contribution is often sensitive in the b → sγ decay. The branching ratioBR( b → sγ ) is given as [44]BR( b → sγ )BR( b → ce ¯ ν e ) = | V ∗ ts V tb | | V cb | απf ( z ) ( | C γ ( m b ) | + | ˜ C γ ( m b ) | ) , (15)where f ( z ) = 1 − z + 8 z − z − z ln z , z = m c,pole m b,pole . (16)Here C γ ( m b ) and ˜ C γ ( m b ) include both contributions from the SM and the new physics.The SM prediction including the next-to-next-to-leading order correction is given as [45]BR( b → sγ )(SM) = (3 . ± . × − , (17)on the other hand, the experimental data is obtained as [46]BR( b → sγ )(exp) = (3 . ± . × − . (18)By inputing this experimental value, the contribution of the gluino-squark mediated flavorchanging process, C γ and ˜ C γ , is constrained.In addition to the CP violating processes with ∆ B = 2 ,
1, the SUSY contribution is alsosensitive to the electric dipole moment [47], which is the the T violation of the flavor con-serving process. The experimental upper bound of the electric dipole moment of the neutronprovides us the upper-bound of the chromo-EDM(cEDM) of the strange quark [48]-[51]. ThecEDM of the strange quark d Cs is given in terms of the gluino-sbottom-quark interactions[37]. The upper bound of the cEDM of the strange quark is given by the experimental upperbound of the neutron EDM as [51], e | d Cs | < . × − ecm . (19)This bound constrains the SUSY flavor mixing angles and the phases in C G and ˜ C G .However, the experimental data of the direct CP violation in the B → K + π − decay givesa little bit stronger constraint for C G and ˜ C G in our framework. Therefore, we omit thediscussion about the cEDM in this work. 5 Setup of squark flavor mixing
Let us discuss the gluino-squark mediated flavor changing process as the dominant SUSYcontribution of the b → s transition. We give the 6 × M ˜ q (˜ q = ˜ u, ˜ d ) in the super-CKM basis. In order to go to the diagonal basis of the squark massmatrix, we rotate M ˜ q as ˜ m q, diagonal = Γ ( q ) G M q Γ ( q ) † G , (20)where Γ ( q ) G is the 6 × × ( q ) G = (Γ ( q ) GL , Γ ( q )) GR ) T in the following expressions. Then, the gluino-squark-quark interactionis given as L int (˜ gq ˜ q ) = − i √ g s X { q } e q ∗ i ( T a ) e G a h (Γ ( q ) GL ) ij L + (Γ ( q ) GR ) ij R i q j + h.c. , (21)where e G a denotes the gluino field, and L and R are projection operators. This interactionleads to the gluino-squark mediated flavor changing process with ∆ B = 2 and ∆ B = 1through the box and penguin diagrams.We take the split-family scenario, in which the first and second family squarks are veryheavy, O (10 − O (1) TeV. Therefore,the first and second squark contribution is suppressed in the gluino-squark mediated flavorchanging process by their heavy masses. In addition, we also assume the flavor symmetrysuch as U(2) [52] in order to suppress FCNC enough in the neutral K meson system [53]. Thestop and sbottom interactions dominate the gluino-squark mediated flavor changing process.Then, the sbottom interaction contributes ∆ B = 2 and ∆ B = 1 processes. We take a suitableparametrizations of Γ ( d ) GL and Γ ( d ) GR as follows [54]:Γ ( d ) GL = δ dL c θ − δ dL s θ e iφ δ dL c θ − δ dL s θ e iφ − δ dL ∗ − δ dL ∗ c θ − s θ e iφ , Γ ( d ) GR = δ dR s θ e − iφ δ dR c θ δ dR s θ e − iφ δ dR c θ s θ e − iφ − δ dR ∗ − δ dR ∗ c θ , (22)where c θ = cos θ and s θ = sin θ , with the mixing angle θ in the ˜ b L,R sector and δ dLj , δ dRj are the couplings responsible for the flavor transitions. The mixing angle θ comes from thetrilinear SUSY breaking terms. If this breaking is neglected, θ vanishes. In our work, wesuppose the large µ tan β , which leads to the non-negligible mixing angle θ in the ˜ b L − ˜ b R sector. By using these rotation matrices, we estimate the gluino-sbottom mediated flavorchanging amplitudes in the B s meson decay.For the numerical analysis, we fix sbottom masses. The third family squarks can havesubstantial mixing between the left-handed squark and the right-handed one due to the large6ukawa coupling, that is the large µ tan β . In our numerical calculation, we take the typicalmass eigenvalues m ˜ b and m ˜ b , and the gluino mass m ˜ g as follows: m ˜ b = 1 TeV , m ˜ b = 1 . , m ˜ g = 2 TeV , (23)where we take account of the present experimental bounds [23]. Once we fix mass eigenvalues m , m and µ tan β , we can estimate the mixing angle θ between the left-handed sbottomand the right-handed one [55]. Taking µ tan β = 20 −
50 TeV, we estimate θ in the rangeof 4 ◦ − ◦ , which is used in our numerical calculations. If we take µ tan β ≪
20 TeV, theleft-right mixing angle θ is much less than O (1 ◦ ). Then, the SUSY contribution in C G and C γ are tiny because the left-right mixing dominates C G and C γ . The smaller massdifference m ˜ b − m ˜ b gives the larger mixing angle θ . However, our results does not so changesince the SUSY contribution depends on the combination of θ and the mass deference assin 2 θ × ( m b − m b ) in our scheme.The relevant mixing angles are δ dL and δ dR for B s → K + K − and B s → K K decays.These mixing angles are complex, and then we take | δ dR | = | δ dL | , (24)for simplicity. On the other hand, the phases of δ dR and δ dL are free parameters, which areare constrained by experimental data.We comment on our assumption in Eq.(24). This one may be motivated from the SO (10)GUT model with SUSY apart from phases. In practice, this case of Eq.(24) give us thelargest SUSY contribution in our prediction because the SUSY one is symmetric for δ dR and δ dL in our framework. Therefore, our predicted region of the CP violations is not changedeven if this assumption is relaxed. We show predicted numerical results of the CP violation in our framework. Let us start withpresenting the SM prediction of the direct CP asymmetry of the B → K + π − process A ( ¯ B → K − π + ) = | ¯ A ( ¯ B → K − π + ) | − |A ( B → K + π − ) | | ¯ A ( ¯ B → K − π + ) | + |A ( B → K + π − ) | . (25)The predicted asymmetry depends on | V ub | and γ in the SM. We show it versus | V ub | inFigure 1(a), where the recent measurements of | V ub | and γ are taken as follows [56]: | V ub | = (3 . ± . × − , γ = (70 . ± . ◦ , (26)and other input parameters in our calculation are summarized in Table 1.As seen in Figure 1(a), the SM prediction completely agrees with the observed value − . ± .
006 [43]. The predicted asymmetry is linear dependent on | V ub | . As far as | V ub | = (3 . − . × − , our prediction is successful. Our prediction is not sensitive to γ in the region of γ = (70 . ± . ◦ since sin γ is not so changed. More precise data of theasymmetry and | V ub | is crucial test of our SM prediction with the QCD factorization.7 s ( M Z ) = 0 . m s (2GeV) = 0 .
095 GeV [46] m c ( m c ) = 1 .
275 GeV [46] m b ( m b ) = 4 .
18 GeV [46] m t ( m t ) = 160 . S ) [46] M B s = 5 . M B s = (116 . ± . × − GeV [7] f B s = (233 ±
10) MeV [56] f π = (130 . ± .
4) MeV [46] f K = (156 . ± .
1) MeV [46] λ = 0 . | V cb | = (4 . ± . × − [56]Table 1: Input parameters in our calculation.We also present the CP averaged branching ratio versus the form factor F B → K (0) inFigure 1(b), in which the magnitude of the form factor is taken to be F B → K (0) = 0 . − . F B → K (0) =0 . − .
42. We omit figures of the | V ub | and γ dependences of the branching ratio becauseit is insensitive to | V ub | and γ . H a L - - - - - - È V ub È ‰ A C P S M H B ® K + Π - L ‰ H b L F B ® K H L B R S M H B ® K + Π - L ‰ Figure 1: Predictions of (a) the asymmetery versus | V ub | and (b) the branching ratio versusthe form factor F B → K (0) in the B → K + π − decay. The inside between dashed red linesdenotes the experimental allowed region at 90%C.L.The agreement between the SM prediction and the experimental data indicates that theSUSY contribution is constrained severely by the direct CP violation of ¯ B → K − π + . Wehave searched the allowed parameter region of δ dL ( dR )23 by scattering the magnitude of δ dL ( dR )23 and these phases in the region of 0 ∼ . − π ∼ π , respectively. These parameters areconstrained by the mass difference ∆ M B s , the CP violating phase φ s in B s → J/ψφ decayand the branching ratio of the b → sγ decay. In addition to these data, the asymmetry of A ( ¯ B → K − π + ) constrains the magnitude of δ dL ( dR )23 . We show the predicted asymmetry8ersus the magnitude of δ dL ( dR )23 in Figure 2, where its phase is taken in − π ∼ π . It is foundthat the SUSY contribution becomes important in the region of | δ dL ( dR )23 | ≥ . b → sγ decay versus the magnitudeof δ dL ( dR )23 in Figure 3. The significant contribution of the SUSY effect is also seen in theregion of | δ dL ( dR )23 | ≥ . - - - È ∆ H dR L È A C P H B ® K + Π - L ‰ Figure 2: The predicted A ( ¯ B → K − π + )versus | δ dL ( dR )23 | . The inside betweendashed red lines denotes the experimen-tal allowed region at 90%C.L. È ∆ H dR L È B R H b ® s Γ L ‰ Figure 3: The predicted branching ra-tio of b → sγ versus | δ dL ( dR )23 | . The insidebetween dashed red lines denotes the ex-perimental allowed region at 90%C.L.Let us show the allowed region on the plane of | δ dL ( R )23 | and those phases, taking accountof ∆ M B s , φ s in B s → J/ψφ decay, the branching ratio of b → sγ , and the asymmetry A ( ¯ B → K − π + ). The input experimental data are taken at 90 % C.L. We present the allowedregion of | δ dL ( R )23 | versus (arg δ dL + arg δ dR ) in Figure 4(a), and versus (arg δ dL − arg δ dR ) inFigure 4(b) with | δ dL | = | δ dR | , respectively. It is found that the squark flavor mixing isallowed in the region of | δ dL | ≤ .
02 for all region of the phase. If two phases arg δ dL andarg δ dR are tuned to suppress the imaginary part, | δ dL | is allowed up to 0 . B s → K + K − and B s → K K decaysunder the constraint of δ dL of Figure 4. We show the predicted regions among C K + K − , S K + K − , C K ¯ K and S K ¯ K in Figures 5(a)-5(d). As seen in Figure 5(a), the predicted regionof C K + K − is strongly cut by the constraint from the direct CP violation of ¯ B → K − π + .Therefore, the deviation from the SM prediction of C K + K − is not found. On the other hand, S K + K − is possibly deviated from the SM prediction considerably, that is expected to be in0 . ∼ .
5. The precise measurement of S K + K − is important to search for the SUSY effect.As seen in Figure 5(b), the SM predictions of C K ¯ K and S K ¯ K are very small since wehave ¯ A ( ¯ B s → K ¯ K ) A ( B s → K ¯ K ) ≃ V tb V ∗ ts V ∗ tb V ts , qp ≃ V ∗ tb V ts V tb V ∗ ts , λ K ¯ K ≃ , (27)where the CKM matrix elements canceled out each other in λ K ¯ K . Since the SUSY contribu-tion violates this cancellation, we expect the observation of the CP violation for both C K ¯ K a L -Π-Π (cid:144) Π (cid:144) Π È ∆ H dR L È a r g @ ∆ d L + ∆ d R D H b L -Π-Π (cid:144) Π (cid:144) Π È ∆ H dR L È a r g @ ∆ d L - ∆ d R D Figure 4: The allowed region of | δ dL ( R )23 | versus (a) the sum of two phases and (b) the differenceof two phases.and S K ¯ K in the B s → K ¯ K decay. These predicted magnitudes are roughly proportionalto each other in the region C K ¯ K = − . ∼ .
06 and S K ¯ K = − . ∼ . C K ¯ K and C K + K − in Figures 5(c), and between S K ¯ K and S K + K − in Figures 5(d), respectively. While the predicted value of C K + K − is restrictedaround 0 . C K ¯ K is expected in the region of − . ∼ .
06. On the other hand, S K ¯ K isroughly proportional to S K + K − , which gives us a crucial test for the SUSY contribution. In order to search for the gluino-squark mediated flavor changing effect, we have studiedthe CP violations in the B s → K + K − and B s → K K processes, in which the b → s transition penguin amplitudes dominate the decays. We have searched for the allowed regionof the flavor mixing δ dL , by putting the experimental data the mass difference ∆ M B s , the CPviolating phase φ s in B s → J/ψφ decay and the b → sγ branching ratio. In addition to thesedata, we have taken into account the constraint from the asymmetry of B → K + π − becausethe B s → K + K − decay is related with the B → K + π − decay by replacing the spectator s with d . We have obtained the constraint of | δ dL | ≤ . B s → K + K − and B s → K K decays. The predicted region of the CP violation C K + K − is strongly cut by theconstraint from the direct CP violation of ¯ B → K − π + , which is well agreement with theSM prediction with the QCD factorization calculation, Therefore, the deviation from the SMprediction of C K + K − is not expected. On the other hand, S K + K − is possibly deviated fromthe SM prediction considerably, in the region of 0 . ∼ .
5. Since the SM predictions of C K ¯ K and S K ¯ K are tiny, the SUSY contribution is expected to be detectable in C K ¯ K and S K ¯ K .These expected magnitudes are in the region C K ¯ K = − . ∼ .
06 and S K ¯ K = − . ∼ . a L - - C K + K - S K + K - H b L - - - - C K K S K K H c L - - - C K K C K + K - H d L - - - S K K S K + K - Figure 5: The predicted CP violations of (a) C K + K − − S K + K − , (b) C K ¯ K − S K ¯ K , (c) C K ¯ K − C K + K − , and (d) S K ¯ K − S K + K − . The inside between dashed red lines denotes theexperimental allowed region at 90%C.L., and yellow regions denote the SM predictions. Acknowledgment
Y.S. is supported by JSPS Postdoctoral Fellowships for Research Abroad, No.20130600.This work is also supported by JSPS Grand-in-Aid for Scientific Research, 21340055 and24654062, 25-5222, respectively.
AppendixA Loop integral in penguins
The loop integrals in Eq.(5) are given as follows [38, 39]:11 M = −
12 ln µm b −
18 + f I M + f II M ,f I M = Z dx g ( x ) φ M ( x ) , g ( x ) = 3 1 − x − x ln x − iπ,f II M = 4 π N f M f B f B → M + (0) M B Z dz φ B ( z ) z Z dx φ M ( x ) x Z dy φ M ( y ) y ,G M ,g = − Z dx x φ M ( x ) ,G M ( s q ) = 23 −
43 ln µm b + 4 Z dxφ M ( x ) Z du u ¯ u ln[ s q − u ¯ u ¯ x − iǫ ] ,G ′ M ,g = − Z dx φ M ( x ) = − ,G ′ M ( s q ) = 13 − ln µm b + 3 Z dxφ M ( x ) Z du u ¯ u ln[ s q − u ¯ u ¯ x − iǫ ] , (28)where ¯ x = 1 − x and ¯ u = 1 − u . The internal quark mass in the penguin diagrams enters as s f = m f /m b . The functions φ ( x ) and φ ( x ) are meson’s leading-twist distribution amplitudeand twist-3 distribution amplitude, respectively. For π and K mesons, we use well knownform [58, 59]: φ π,K ( x ) = 6 x (1 − x ) , φ π,K ( x ) = 1 . (29)For the B meson, we use [60, 61, 62] φ B ( x ) = N B x (1 − x ) exp (cid:20) − M B x ω B (cid:21) , (30)where ω B = 0 . . GeV for the B and B s mesons, respectively, and N B is thenormalization constant to make R dxφ B ( x ) = 1 . B Squark contribution in ∆ B = 1 process The Wilson coefficients for the gluino contribution in Eq.(1) are written as [63] C ˜ g γ ( m ˜ g ) = 83 √ α s π G F V tb V ∗ tq × " (cid:0) Γ ( d ) GL (cid:1) ∗ k m d (cid:26)(cid:0) Γ ( d ) GL (cid:1) (cid:18) − F ( x g ) (cid:19) + m ˜ g m b (cid:0) Γ ( d ) GR (cid:1) (cid:18) − F ( x g ) (cid:19)(cid:27) + (cid:0) Γ ( d ) GL (cid:1) ∗ k m d (cid:26)(cid:0) Γ ( d ) GL (cid:1) (cid:18) − F ( x g ) (cid:19) + m ˜ g m b (cid:0) Γ ( d ) GR (cid:1) (cid:18) − F ( x g ) (cid:19)(cid:27) , (31)12 ˜ g G ( m ˜ g ) = 83 √ α s π G F V tb V ∗ tq " (cid:0) Γ ( d ) GL (cid:1) ∗ k m d (cid:26)(cid:0) Γ ( d ) GL (cid:1) (cid:18) − F ( x g ) − F ( x g ) (cid:19) + m ˜ g m b (cid:0) Γ ( d ) GR (cid:1) (cid:18) − F ( x g ) − F ( x g ) (cid:19)(cid:27) + (cid:0) Γ ( d ) GL (cid:1) ∗ k m d (cid:26)(cid:0) Γ ( d ) GL (cid:1) (cid:18) − F ( x g ) − F ( x g ) (cid:19) + m ˜ g m b (cid:0) Γ ( d ) GR (cid:1) (cid:18) − F ( x g ) − F ( x g ) (cid:19)(cid:27) , (32)where k = 2 , b → q ( q = s, d ) transitions, respectively. The loop functions F i ( x I ˜ g ) are given as F ( x I ˜ g ) = x I ˜ g log x I ˜ g x I ˜ g − + ( x I ˜ g ) − x I ˜ g − x I ˜ g − ,F ( x I ˜ g ) = − ( x I ˜ g ) log x I ˜ g x I ˜ g − + 2( x I ˜ g ) + 5 x I ˜ g − x I ˜ g − ,F ( x I ˜ g ) = log x I ˜ g ( x I ˜ g − + x I ˜ g − x I ˜ g − ,F ( x I ˜ g ) = − x I ˜ g log x I ˜ g ( x I ˜ g − + x I ˜ g + 12( x I ˜ g − = 12 g ( x I ˜ g , x I ˜ g ) , (33)with x I ˜ g = m g /m d I ( I = 3 , e C ˜ gi ( m ˜ g )’s are obtained by replacing L ( R ) with R ( L ) in C ˜ gi ( m ˜ g )’s.The Wilson coefficients of C ˜ g γ ( m b ) and C ˜ g G ( m b ) at the m b scale are given at the leadingorder of QCD as follows [40]: C ˜ g γ ( m b ) = ζ C ˜ g γ ( m ˜ g ) + 83 ( η − ζ ) C ˜ g G ( m ˜ g ) ,C ˜ g G ( m b ) = ηC ˜ g G ( m ˜ g ) , (34)where ζ = (cid:18) α s ( m ˜ g ) α s ( m t ) (cid:19) (cid:18) α s ( m t ) α s ( m b ) (cid:19) , η = (cid:18) α s ( m ˜ g ) α s ( m t ) (cid:19) (cid:18) α s ( m t ) α s ( m b ) (cid:19) . (35) C Squark contribution in ∆ B = 2 process The ∆ B = 2 effective Lagrangian from the gluino-sbottom-quark interaction is given as L ∆ F =2eff = −
12 [ C V LL O V LL + C V RR O V RR ] − X i =1 h C ( i ) SLL O ( i ) SLL + C ( i ) SRR O ( i ) SRR + C ( i ) SLR O ( i ) SLR i , (36)13hen, the P - ¯ P mixing, M , is written as M = − m P h P |L ∆ F =2eff | ¯ P i . (37)The hadronic matrix elements are given in terms of the non-perturbative parameters B i as: h P |O V LL | ¯ P i = 23 m P f P B , h P |O V RR | ¯ P i = h P |O V LL | ¯ P i , h P |O (1) SLL | ¯ P i = − m P f P R P B , h P |O (1) SRR | ¯ P i = h P |O (1) SLL | ¯ P i , h P |O (2) SLL | ¯ P i = 112 m P f P R P B , h P |O (2) SRR | ¯ P i = h P |O (2) SLL | ¯ P i , h P |O (1) SLR | ¯ P i = 12 m P f P R P B , h P |O (2) SLR | ¯ P i = 16 m P f P R P B , (38)where R P = (cid:18) m P m Q + m q (cid:19) , (39)with ( P, Q, q ) = ( B d , b, d ) , ( B s , b, s ).The Wilson coefficients for the gluino contribution in Eq. (36) are written as [63] C V LL ( m ˜ g ) = α s m g X I,J =1 ( λ ( d ) GLL ) ijI ( λ ( d ) GLL ) ijJ (cid:20) g ( x ˜ gI , x ˜ gJ ) + 29 g ( x ˜ gI , x ˜ gJ ) (cid:21) ,C V RR ( m ˜ g ) = C V LL ( m ˜ g )( L ↔ R ) ,C (1) SRR ( m ˜ g ) = α s m g X I,J =1 ( λ ( d ) GLR ) ijI ( λ ( d ) GLR ) ijJ g ( x ˜ gI , x ˜ gJ ) ,C (1) SLL ( m ˜ g ) = C (1) SRR ( m ˜ g )( L ↔ R ) ,C (2) SRR ( m ˜ g ) = α s m g X I,J =1 ( λ ( d ) GLR ) ijI ( λ ( d ) GLR ) ijJ (cid:18) − (cid:19) g ( x ˜ gI , x ˜ gJ ) ,C (2) SLL ( m ˜ g ) = C (2) SRR ( m ˜ g )( L ↔ R ) ,C (1) SLR ( m ˜ g ) = α s m g X I,J =1 ( ( λ ( d ) GLR ) ijI ( λ ( d ) GRL ) ijJ (cid:18) − (cid:19) g ( x ˜ gI , x ˜ gJ )+ ( λ ( d ) GLL ) ijI ( λ ( d ) GRR ) ijJ (cid:20) g ( x ˜ gI , x ˜ gJ ) − g ( x ˜ gI , x ˜ gJ ) (cid:21) ) ,C (2) SLR ( m ˜ g ) = α s m g X I,J =1 ( ( λ ( d ) GLR ) ijI ( λ ( d ) GRL ) ijJ (cid:18) − (cid:19) g ( x ˜ gI , x ˜ gJ )+ ( λ ( d ) GLL ) ijI ( λ ( d ) GRR ) ijJ (cid:20) g ( x ˜ gI , x ˜ gJ ) + 109 g ( x ˜ gI , x ˜ gJ ) (cid:21) ) , (40)14here ( λ ( d ) GLL ) ijK = (Γ ( d ) † GL ) Ki (Γ ( d ) GL ) jK , ( λ ( d ) GRR ) ijK = (Γ ( d ) † GR ) Ki (Γ ( d ) GR ) jK , ( λ ( d ) GLR ) ijK = (Γ ( d ) † GL ) Ki (Γ ( d ) GR ) jK , ( λ ( d ) GRL ) ijK = (Γ ( d ) † GR ) Ki (Γ ( d ) GL ) jK . (41)Here we take ( i, j ) = (1 , , (2 ,
3) which correspond to the B and B s mesons, respectively.The loop functions are given as follows: • If x ˜ gI = x ˜ gJ ( x ˜ gI,J = m d I,J /m g ), g ( x ˜ gI , x ˜ gJ ) = 1 x ˜ gI − x ˜ gJ x ˜ gI log x ˜ gI ( x ˜ gI − − x ˜ gI − − x ˜ gJ log x ˜ gJ ( x ˜ gJ − + 1 x ˜ gJ − ! ,g ( x ˜ gI , x ˜ gJ ) = 1 x ˜ gI − x ˜ gJ ( x ˜ gI ) log x ˜ gI ( x ˜ gI − − x ˜ gI − − ( x ˜ gJ ) log x ˜ gJ ( x ˜ gJ − + 1 x ˜ gJ − ! . (42) • If x ˜ gI = x ˜ gJ , g ( x ˜ gI , x ˜ gI ) = − ( x ˜ gI + 1) log x ˜ gI ( x ˜ gI − + 2( x ˜ gI − ,g ( x ˜ gI , x ˜ gI ) = − x ˜ gI log x ˜ gI ( x ˜ gI − + x ˜ gI + 1( x ˜ gI − . (43)In this paper, we take ( I, J ) = (3 , , (3 , , (6 , , (6 , C V LL ( m b ) = η BV LL C V LL ( m ˜ g ) , C V RR ( m b ) = η BV RR C V LL ( m ˜ g ) , C (1) SLL ( m b ) C (2) SLL ( m b ) ! = C (1) SLL ( m ˜ g ) C (2) SLL ( m ˜ g ) ! X − LL η BLL X LL , C (1) SRR ( m b ) C (2) SRR ( m b ) ! = C (1) SRR ( m ˜ g ) C (2) SRR ( m ˜ g ) ! X − RR η BRR X RR , C (1) SLR ( m b ) C (2) SLR ( m b ) ! = C (1) SLR ( m ˜ g ) C (2) SLR ( m ˜ g ) ! X − LR η BLR X LR , (44)where η BV LL = η BV RR = (cid:18) α s ( m ˜ g ) α s ( m t ) (cid:19) (cid:18) α s ( m t ) α s ( m b ) (cid:19) ,η BLL = η BRR = S LL η d LL b ˜ g η d LL b ˜ g ! S − LL , η BLR = S LR η d LR b ˜ g η d LR b ˜ g ! S − LR ,η b ˜ g = (cid:18) α s ( m ˜ g ) α s ( m t ) (cid:19) (cid:18) α s ( m t ) α s ( m b ) (cid:19) , (45)15 LL = 23 (1 − √ , d LL = 23 (1 + √ , d LR = − , d LR = 2 ,S LL = (cid:18) √ −√ (cid:19) , S LR = (cid:18) − (cid:19) ,X LL = X RR = (cid:18) (cid:19) , X LR = (cid:18) −
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