1. Probing Flavor Structure in Supersymmetric Theories Shaaban Khalil Ain Shams University, Cairo, Egypt

2. I. Introduction  The recent results on the mixing-induced asymmetries of B → φ K and B → ή K are: Belle BaBar  These observations are considered as signals to new physics. →→  The direct CP violation in B 0 → K - π + and B - → K - π 0 :

3.  To accommodate the CP asymmetries of B decays, SUSY models with flavor non-universal soft breaking terms are favored.   The squark mixings are classified as i) LL and RR mixings given by (δ u,d LL ) ij and (δ u,d RR ) ij. ii) LR and RL mixings given by (δ u,d LR ) ij and (δ u,d RL ) ij.   Constraints are more stringent on the LR (RL) mass insertions than the LL (RR) mass insertions.   MIs between 1st & 2nd generations are severely constrained more than MIs between 1st or 2nd & 3rd generations.   This gives the hope that SUSY contributions to the B-system.

4. II. Squark Mixing: LL versus LR  The EDM constraints severely restrict the LL and RR mass insertions: (δ d LR ) 22eff ≈ (δ d LR ) 22 +(δ d LL ) 23 (δ d LR ) 33 (δ d RR ) 32 ≈ 10 -2 (δ d LL ) 23 (δ d RR ) 32 (for neg. (δ d LR ) 22 ). ≈ 10 -2 (δ d LL ) 23 (δ d RR ) 32 (for neg. (δ d LR ) 22 ).  From Hg EDM: Im(δ d LR ) 22 < 5.6 x 10 -6.  The SUSY contributions to the decay amplitudes B → φ K and ή K: R φ = -0.14 e -i0.1 (δ d LL ) 23 -127 e -i0.08 (δ d LR ) 23 +(L ↔ R), R ή = -0.07 e -i0.24 (δ d LL ) 23 - 64 (δ d LR ) 23 - (L ↔ R).

5.  The CP asymmetries S φK and S ήK can not be accommodated simultaneously by dominant (δ d LL ) 23 or (δ d RR ) 23. SφK and SήK  Combining the effects of LL and RR MIs, we can fit the exp. data of SφK and SήK.  However, the Hg EDM exceeds with many order of magnitudes its exp. Bound.

6.  Thus, SUSY models with dominant LR and RL may be the most favorite scenario.  However, it is difficult to arrange for (δ d LR ) 23 ≈ O(10 -2 ) whilst (δ d LR ) 12 remains small: From ΔM K and έ/ε: Re (δ d LR ) 12 < O(10 -4 ) & Im (δ d LR ) 12 < O(10 -5 ).

7.  The MI (δ d LR ) ij are given by (δ d LR ) ij ≈ [V d+ L. (Y d A d ). V d R ] ij.  The factorizable A-term is an example of a specific texture to satisfy this hierarchy between (δ d LR ) 12 and (δ d LR ) 23.  With intermediate/large tanβ, an effective (δ d LR ) 23 can be obtained: (δ d LR ) 23 eff = (δ d LR ) 23 + (δ d LL ) 23 (δ d LR ) 33  For negligible (δ d LR ) 23, we find

8.  Thus if (δ d LL ) 23 ≈ 10 -2, we get (δ d LR ) 23eff ≈ O(10 -2 – 10 -3 ).  These contributions are considered as LL (or RR).  The main effect is still due to the Wilson coefficient C 8g of the chromomagnetic operator.  Although (δ d LL ) 23 ≈ 10 -2 is not enough to explain the CP asymmetries of B-decays.  Still it can induce an effective LR mixing that accounts for these results.

9. Suggested supersymmetric flavor model  As an example, we consider the following SUSY model: M 1 =M 2 =M 3 = M 1/2 A u = A d = A 0 M U 2 = M D 2 = m 0 2 m H1 2 = m H2 2 = m 0 2  The masses of the squark doublets are given by tan β=15, m 0 =M 1/2 =A 0 =250 → a ≤5

10.  The Yukawa textures play an important rule in the CP and flavour supersymmetric results. Y u= 1/v sinβ diag(m u,m c,m t ) Y d =1/vsinβ V + CKM. diag(m d,m s,m b ). V CKM  Although, it is hierarchical texture, it lads to a good mixing between the second and third generations.  The LL down MIs are given by: (δ d LL ) ij =1/m q 2 [V L d+ (M d ) 2 LL V d L ] ij With a=5 → (δ d LL ) 23 ~ 0.08 e 0.4 I  Thus, one gets: (δ d LR ) 23eff ~ O (10 -2 -10 -3 ).  The corresponding single LR MI is negligible due to the degeneracy of the A-terms.

11. Contribution to S φK and S ήK  Gluino exchanges give the dominant contribution to the CP asymmetries: S φK and S ήK. S ήK  1 σ constraints on S ήK leads to a lower bound on a: a ≥ 3. SφK and SήK  With a large a, it is quite possible to account simultaneously for the experimental results SφK and SήK.

12. Contribution to B→ K π  The direct CP asymmetries of B→ K π are given by: A CP K - π + ~ 2 r T sin δ T sin(θ P + γ) + 2r EW C sin δ EW C sin(θ P -θ EW C ), A CP K - π 0 ~ 2 r T sin δ T sin(θ P + γ) - 2 r EW sin δ EW sin(θ P -θ EW ).  The parameters θ P, θ EW C, θ EW and δ T, δ EW, δ EW C are the CP violating and CP conserving phases respectively.  The parameter r T measures the relative size of the tree and QCD penguin contributions.  The parameter, r EW, r EW C measure the relative size of the electroweak and QCD contributions.

13. P e θ P = P sm (1+k e θ’ P ), r EW e δ EW e θ EW = r EW sm e δ EW (1+ l e θ’ EW ),r EW C e δ EW C e θ EW C =(r EW C ) sm e δ EW C (1+m e θ’ EW C ), r T e δ T =(r T e δ T ) sm /|1+ k e θ’ P |  k,l,m are given by (m g = m q =500, M 2 =200, μ =400): k e θ’ P = -0.0019 tan β (δ u LL ) 32 - 35.0 (δ d LR ) 23 +0.061 (δ u LR ) 32 l e θ’ EW = 0.0528 tan β (δ u LL ) 32 - 2.78 (δ d LR ) 23 +1.11(δ u LR ) 32 m e θ’ EW C =0.134 tan β (δ u LL ) 32 + 26.4 (δ d LR ) 23 +1.62 (δ u LR ) 32. a=5, m 0 =M 1/2 =A 0 =250 GeV → (δd LR ) 23 ~ 0.006xe -2.7i. Thus: K~0.2, l~ 0.009, m ~0.16 → r EW ~ 0.13, r c EW ~ 0.012, r T ~ 0.16 → A CP K - π + ~ -0.113 and A CP K - π 0 < A CP K - π +

14. Conclusions  We studied the possibility of probing SUSY flavor structure using K, B CP asymmetries constraints & EDM.  One possibility: large LR and/or RL sector.  Second possibility: large LL combined with a very small RR and also intermediate or large tan β.  Large LR requires a specific pattern for A terms.  Large LL seems quite natural and can be obtained by a non- universality between the squark masses.  As an example, a SUSY model with a non-universal left squark masses is considered.  one gets effective (δ d LR ) 23 that leads to a significant SUSY contributions to the CP asymmetries of B decays.