Signatures of bosonic squark decays in non-minimally flavour-violating supersymmetry
PPreprint typeset in JHEP style. - HYPER VERSION
DESY 10-060
Signatures of bosonic squark decays in non-minimally flavour-violating supersymmetry
Matthias Bruhnke ,a , Bj¨orn Herrmann , ,b , Werner Porod , ,c Institut f¨ur Theoretische Physik und Astrophysik,Universit¨at W¨urzburg, D-97074 W¨urzburg, Germany Deutsches Elektronen-Synchrotron (DESY), Theory group,Notkestraße 85, D-22603 Hamburg, Germany AHEP Group, Institut de F´ısica Corpuscular - C.S.I.C.,Universitat de Val`encia, E-46071 Val`encia, Spain a Email: [email protected] b Email: [email protected] c Email: [email protected]
Abstract:
We investigate couplings of squarks to gauge and Higgs-bosons withinthe framework of non-minimal flavour violation in the Minimal Supersymmetric Stan-dard Model. Introducing non-diagonal elements in the mass matrices of squarks, wefirst study their impact on the self-energies and physical mass eigenvalues of squarks.We then present an extensive analysis of bosonic squark decays for variations of theflavour-violating parameters around the two benchmark scenarios SPS1a’ and SPS1b.Signatures, that would be characteristic for a non-minimal flavour structure in thesquark sector, can be found in wide regions of the parameter space. a r X i v : . [ h e p - ph ] A ug . Introduction With the start of the Large Hadron Collider (LHC) first measurements at the TeV-scale will be realized in near future. Although the Standard Model (SM) of particlephysics has given most accurate predictions for a wide range of phenomena, there arestrong hints that it is not the ultimate theory, but should rather be thought of as alow-energy limit of a more fundamental framework. Among the numerous candidatesfor the latter, the Minimal Supersymmetric Standard Model (MSSM) is probably thebest-studied extension of the Standard Model. Postulating a superpartner with op-posite statistics for each SM particle, supersymmetry (SUSY) cures the hierarchyproblem by stabilising the Higgs mass, can lead to gauge-coupling unification, andincludes natural candidates for the dark matter observed in our Universe. Althoughit is clear that supersymmetry must be broken at the electroweak scale, there isno theoretical consensus about the exact breaking mechanism. One therefore intro-duces so-called soft SUSY breaking parameters which do not introduce quadraticdivergences. Thus, the stability of the ratio of the electroweak scale over a GUTscale or the Planck scale is maintained. It will be the task of LHC and other futurecolliders like the ILC to measure these parameters as precisely as possible [1] whichthen will give a clue on how SUSY is broken.One of the open questions related to the breaking mechanism concerns the flavourstructure of the MSSM. The hypothesis of minimal flavour violation (MFV) [2] as-sumes that it is the same as in the Standard Model, where all quark flavour-violatinginteractions are parametrised through the CKM-matrix. However, in principle newsources of flavour violation can appear within supersymmetric models, especially ifthey are embedded in larger frameworks such as grand unified theories [3, 4, 5]. Thisso-called non-minimal flavour violation (NMFV) allows for non-diagonal, i.e. flavour-violating, entries in the mass matrices at the weak scale, that cannot be related tothe CKM-matrix any more. These entries are conveniently taken as additional freeparameters and can imply a different phenomenology as compared to the case ofMFV. For a more detailed review on flavour violation in the MSSM see, e.g., ref. [6].In the last years, various publications have focused on the phenomenology of non-minimal flavour violation in the squark sector of the MSSM. In refs. [7] and [8], loop-induced effects on the decays of top-quarks or Higgs-bosons have been investigated,respectively. Various flavour-changing neutral current processes related to the top-quark have been discussed in refs. [9, 10]. The production of squarks and gauginosat hadron colliders has been studied in refs. [11, 12], and in refs. [13, 14, 15] possiblesignatures of gluino and squark decays have been discussed.The aim of the present paper is to provide a complementary study in the contextof bosonic decays of squarks. Therefore, we investigate the impact of new sources offlavour violation on the couplings of the squarks to gauge and Higgs-bosons. Suchflavour-violating couplings can induce new contributions to the squark mass matrices,1ither at the tree-level or at the one-loop level or both. These contributions influencedirectly the mass spectrum and the flavour decomposition of the different squarkmass eigenstates. Concerning the decays into vector bosons or Higgs-bosons, NMFVin the squark sector can allow for interesting signatures already at tree-level, thatwould not be possible within the framework of MFV. If squarks are discovered at theLHC and the relevant signals can be revealed in the decay chains, they would allowto exclude the hypothesis of MFV in the MSSM.This paper is organised as follows. In sec. 2, we review the parametrisation offlavour violation in the squark sector of the MSSM and the most relevant experimen-tal constraints. We then discuss in sec. 3 the impact of NMFV on the squark massspectrum at the one-loop level and the induced flavour-mixing terms. We finallystudy experimental signatures related to flavour-violating squark decays into finalstates containing gauge and Higgs-bosons in sec. 4. Conclusions are given in sec. 5.
2. Quark flavour violation in the MSSM
In supersymmetric models with the most general form of flavour mixing, the 6 × M q = (cid:32) M q,LL M q,LR M q,RL M q,RR (cid:33) , (2.1)for ˜ q = ˜ u, ˜ d , respectively. The 3 × M d,LL = ˆ m q + D ˜ d,LL + ˆ m d , M u,LL = V CKM ˆ m q V † CKM + D ˜ d,LL + ˆ m u , M d,RR = ˆ m d + D ˜ d,RR + ˆ m d , M u,RR = ˆ m u + D ˜ u,RR + ˆ m u , (2.2)where ˆ m ˜ q, ˜ d, ˜ u are the soft-breaking mass parameters of the squarks in the super-CKMbasis and ˆ m u,d denote the diagonal mass matrices of up- and down-type quarks. TheD-terms are D ˜ q,LL = cos 2 βm Z ( T q − e q sin θ W ) and D ˜ q,RR = e q sin θ W cos 2 βm Z .Here, T q and e q denote the isospin and electric charge of the (s)quarks, and θ W isthe weak mixing angle. Due to the SU (2) symmetry, the left-left blocks are relatedthrough the CKM-matrix V CKM . The off-diagonal blocks of eq. (2.1) are given by M d,RL = M † ˜ d,LR = v d √ T D − µ ∗ ˆ m d tan β, M u,RL = M † ˜ u,LR = v u √ T U − µ ∗ ˆ m u cot β, (2.3)where ˆ T D,U are the trilinear soft-breaking parameters in the super-CKM basis. Theparameters related to the Higgs sector are the ratio of the vacuum expectation valuesof the two Higgs doublets, tan β = v u /v d , and the higgsino mass parameter µ .2n order to diagonalize the mass matrices of eq. (2.1), two 6 × R ˜ u and R ˜ d are needed, defined such thatdiag( m q , . . . , m q ) = R ˜ q M q R † ˜ q and m ˜ q < · · · < m ˜ q . (2.4)For the sake of a dimensionless and scenario-independent description, non-minimalflavour violation (NMFV) in the squark sector is conveniently parametrised by theparameters δ IJij defined through( ˆ m q ) ij = δ LLij (cid:2) { ˆ m q } + Tr { ˆ m u } + Tr { ˆ m d } (cid:3) ( ˆ m d ) ij = δ d,RRij (cid:2) Tr { ˆ m q } + Tr { ˆ m d } (cid:3) , ( ˆ m u ) ij = δ u,RRij (cid:2) Tr { ˆ m q } + Tr { ˆ m u } (cid:3) , v d √ ( ˆ T D ) ij = δ d,RLij (cid:2) Tr { ˆ m q } + Tr { ˆ m d } (cid:3) , v d √ ( ˆ T ∗ D ) ij = δ d,LRij (cid:2) Tr { ˆ m q } + Tr { ˆ m d } (cid:3) , v u √ ( ˆ T U ) ij = δ u,RLij (cid:2) Tr { ˆ m q } + Tr { ˆ m u } (cid:3) , v u √ ( ˆ T ∗ U ) ij = δ u,LRij (cid:2) Tr { ˆ m q } + Tr { ˆ m u } (cid:3) , (2.5)for i, j = 1 , , i (cid:54) = j ). Note that with this definition one can account forpotential large left-right mixing effects as the traces correspond roughly to the sumof the squark masses squared at tree level.Many experimental measurements impose constraints on the parameter spaceof the MSSM. In the context of flavour transitions, the most relevant constraintscome from precision measurements of mixing and decays of K - and B -mesons, wherethe squarks enter at the same loop-level as the standard model contributions. Inparticular, very stringent constraints are imposed on generation mixing involving firstgeneration squarks [19, 20, 21]. We take them implicitly into account by consideringonly mixing between second and third generation squarks, which is least constrained.In particular, this means that we consider seven independent NMFV parameters, δ LL ≡ δ LL , δ RRu ≡ δ u,RR , δ RRd ≡ δ d,RR , δ RLu ≡ δ u,RL = (cid:16) δ u,LR (cid:17) ∗ δ LRu ≡ δ u,LR = (cid:0) δ u,RL (cid:1) ∗ , δ RLd ≡ δ d,RL = (cid:0) δ d,LR (cid:1) ∗ , δ LRd ≡ δ d,LR = (cid:0) δ d,RL (cid:1) ∗ , (2.6)where we omit for simplicity the generation indices. The so-defined parameters areassumed to be real, the influence of possible complex phases being beyond the scopeof this work.We then explicitly impose the constraints given in table 1 on the flavour mixingbetween second and third generation squarks. The experimental upper and lowerlimits on BR( B s → µ + µ − ) and m h are given at the 95% confidence level, while theerror intervals for the other observables are given at the 68% (1 σ ) confidence level.The calculation of the physical mass spectrum and the rotation matrices as wellas the observables shown in table 1 is done using SPheno 3.0 [22] . Furthermore, in An updated version including flavour effects can be obtained at ∼ porod/SPheno.html . × BR( b → sγ ) 3 . ± . ± .
09 [25] +0 . − . [26, 27]∆ M B s [ps − ] 17 . ± . ± .
07 [28] ± .
88 [29]10 × BR( ¯ B s → X s l + l − ) 1 . ± .
50 [30] ± .
11 [30]10 × BR( B s → µ + µ − ) ≤ . m h [GeV] ≥ . ± . Table 1:
Constraints on flavour violation in the squark sector, current experimental limitsand theoretical error estimates. order to perform scans over the parameter space in an efficient way on a computercluster, we make use of the
Mathematica package
MapCore described in ref. [23].The scenarios discussed in the following are in agreement with the the currentexperimental limits given above at 95% confidence level for wide ranges of the param-eters of eq. (2.6). Here, we also take into account the available theoretical error esti-mates given in table 1. A detailed study of the allowed ranges in the parameter spaceis, however, beyond the scope of this paper. For selected parameter configurations,we shall in sec. 4 indicate the allowed ranges of the NMFV-parameters δ IJq . Detailedstudies of constraints on the MSSM parameter space due to precision measurementsand low-energy observables can also be found in refs. [9, 10, 11, 12, 14, 15, 16, 17, 18].
3. Impact of quark flavour violation on squark masses
The following study is based on the two reference scenarios SPS1a’ [32] and SPS1b[33] in the framework of minimal supergravity (mSUGRA) to fix the flavour diagonalparameters. The corresponding input parameters at the grand unification scale arethe universal scalar mass m = 70 (200) GeV, gaugino mass m / = 250 (400) GeV,and trilinear coupling A = −
300 (0) GeV for SPS1a’ (SPS1b). At the GUT scalewe take T f = A Y f where Y f is the corresponding Yukawa coupling. In contrasttan β = 10 (30) is given at the scale m Z . The higgsino mass parameter µ is positivefor both scenarios. For the input values of the standard model parameters we referthe reader to ref. [28]. The mass of the top quark has been set to m pole t = 172 . Q = 1TeV are obtained through two-loop renormalization group (RGE) running [24]. Thecalculation of the physical masses is done at the one-loop level, including leadingtwo-loop contributions to the Higgs masses. Here we have included flavour effects inthe renormalization group (RGE) running as well as in the calculation of the massesand mixing matrices.In table 2 we show the resulting physical masses and the corresponding flavourdecomposition of the up- and down-type squarks in the case of minimal flavour4PS1a’ Flavour content Mass˜ d
96% ˜ d L
4% ˜ s L d
96% ˜ s L
4% ˜ d L d d R d s R d
91% ˜ b R
9% ˜ b L d
91% ˜ b L
9% ˜ b R u
68% ˜ t L
32% ˜ t R u
99% ˜ u L
1% ˜ c L u
99% ˜ c L
1% ˜ u L u u R u c R u
68% ˜ t R
32% ˜ t L
367 SPS1b Flavour content Mass˜ d
98% ˜ d L
2% ˜ s L d
98% ˜ s L
2% ˜ d L d d R d s R d
73% ˜ b R
27% ˜ b L d
73% ˜ b L
27% ˜ b R u
99% ˜ c L
1% ˜ u L u
99% ˜ u L
1% ˜ c L u u R u c R u
76% ˜ t L
24% ˜ t R u
76% ˜ t R
24% ˜ t L Table 2:
Flavour content and masses of up- and down-type squarks for the referencescenario SPS1a’ (left) and SPS1b (right) assuming MFV. The masses are given in GeV. violation (MFV) for the two reference scenarios. For both points the left-right mixingis most important for third generation squarks. Due to the larger value of tan β inthe case of SPS1b the sbottom mixing is larger w.r.t. SPS1a’. The small mixingsinvolving the first and second generation are CKM-induced both in the RGE runningof the parameters as well as in corresponding entries of the one-loop mass matrices.Since here the CKM-matrix is the only source of flavour mixing, this is relevant forleft-handed squarks only.In the following, we focus mainly on the scenario SPS1a’ for a detailed discussionof variations around the MFV-case, and discuss additional effects induced due tothe larger value of tan β using SPS1b. We add at the electroweak scale additionalNMFV-parameters to the MFV-parameters obtained in the RGE running and studytheir effects on masses and mixing. As a first example, let us study the dependenceon δ RRd . This parameter induces a direct ˜ s R -˜ b R mixing and affects at tree-level thedown-sector only and at the one-loop level a tiny mixing is induced in the up-sectoras well. In fig. 1, we show the mass eigenvalues and the flavour decomposition ofselected down-type squarks. For δ RRd = 0, ˜ d is a pure ˜ s R state and almost mass-degenerate with ˜ d , as can be seen from table 2. Due to the small mass difference,a strong ˜ s R -˜ b R mixing is observed already for small values of δ RRd . This leads toseveral sharp level-crossings between the mass eigenstates when the flavour contentsof the corresponding squarks are exchanged. A rather smooth crossing occurs around δ RRd ≈ . b L content of ˜ d is taken over by ˜ d .A similar situation is present in the sector of up-type squarks for a variation ofthe parameter δ RRu . The only difference is the larger mass gap already in the MFV5 (cid:45) ∆ dRR G e V (cid:45) (cid:45) ∆ dRR (cid:160) R (cid:72) d (cid:142) , b (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , s (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , b (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) ∆ dRR (cid:160) R (cid:72) d (cid:142) , b (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , s (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , b (cid:142) L (cid:76) (cid:164) Figure 1:
Dependence of the masses (left) and the flavour decomposition (centre andright) of selected down-type squarks on the NMFV-parameter δ RRd based on the benchmarkscenario SPS1a’. The point δ RRd = 0 corresponds to the MFV-scenario of table 2. (cid:45) (cid:45) ∆ LL G e V (cid:45) (cid:45) ∆ LL (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) ∆ LL (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) Figure 2:
Dependence of the masses (left) and the flavour decomposition (centre andright) of selected up-type squarks on the NMFV-parameter δ LL based on the benchmarkscenario SPS1a’. The point δ LL = 0 corresponds to the MFV-scenario of table 2. case (see table 2). As a consequence, the exchange of flavour contents between theu-squarks is less pronounced.Due to SU (2) invariance a non-zero parameter δ LL induces both a ˜ c L -˜ t L and a ˜ s L -˜ b L mixing. In figure we show the squark mass eigenvalues and flavour decompositionsof selected up-type squarks as a function of δ LL while all other NMFV-parameters ofeq. (2.6) are kept to zero. Again, level-crossings and flavour exchanges are observedalready for moderate values of δ LL . Note that here the behaviour of the up-typesquarks is not symmetric w.r.t. the MFV-case, which is a consequence of the CKM-relation between the two left-left blocks of the mass matrices. In consequence, small˜ u L -˜ c L and ˜ u L -˜ t L mixing are induced in addition to ˜ c L -˜ t L . The fact that this asym-metry shows in the up-sector is due to the definition of the super-CKM basis, wherethe down-type squarks are diagonal.Let us now turn to flavour-violating entries in the off-diagonal blocks of thesquark mass matrices. Fig. 3 shows the dependence of up-type squark mass eigen-values and flavour contents as a function of δ RLd . Since this parameter affects notonly the mass matrix but primarily the squark-squark-Higgs coupling, we observesignificant differences to the previous cases. First, instead of an increasing masssplitting, the two lightest down-type squark masses decrease both with increasing δ RLd , implying that the trace of the mass matrix is not invariant under the influenceof the NMFV-parameter.Indeed, the parameter δ RLd induces squark self-energies stemming from the Higgs-loops shown in fig. 4, that are quadratic in the squark-squark-Higgs coupling ( ˆ T D ) .6 (cid:45) ∆ dRL G e V (cid:45) (cid:45) ∆ dRL (cid:160) R (cid:72) d (cid:142) , b (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , s (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , b (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) ∆ dRL (cid:160) R (cid:72) d (cid:142) , b (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , s (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) d (cid:142) , b (cid:142) L (cid:76) (cid:164) Figure 3:
Dependence of masses (left) and flavour decompositions (centre and right) ofselected down-type squarks on the NMFV-parameter δ RLd based on the benchmark scenarioSPS1a’. The point δ RLd = 0 corresponds to the MFV-scenario of table 2. ˜ s R ˜ b L h ˜ s R ˜ s R ˜ b L H ˜ s R ˜ s R ˜ b L A ˜ s R ˜ s R ˜ t L H − ˜ s R ˜ b L ˜ s R h ˜ b L ˜ b L ˜ s R H ˜ b L ˜ b L ˜ s R A ˜ b L ˜ t L ˜ s R H + ˜ t L Figure 4:
One-loop contributions from Higgs-bosons to the self-energies of right-handedstrange and left-handed bottom and top squarks.
In the limit where these contributions dominate we get from eqs. (B.2) and (B.3)∆ m (cid:39) − (cid:12)(cid:12)(cid:12) ( ˆ T D ) (cid:12)(cid:12)(cid:12) B ( p , m q , m H ) , (3.1)where we denote schematically m ˜ q and m H the masses of the squarks and Higgsbosons in the loop, respectively. The main contributions are due to due H , A and H + where those including a charged Higgs are about twice as large as the loopscontaining H or A . Assuming approximately equal masses for H , A , and H ± (seetable 2), the mass parameter of ˜ s R receives a two times larger corrections than themass parameter of ˜ s L . Note, that independent of the sign of ( ˆ T D ) this contributionswill reduce the trace of the down squark mass matrix compared to the MFV caseleading to the observed reduction of the two lightest states.The resulting decrease of the diagonal elements overcompensates the increase ofthe off-diagonal ones. In consequence, the strong mixing of the two lightest down-type squark states is reversed for δ RLd → ±
1. This “unmixing” effect is observed,e.g., in fig. 3 for | δ RLd | (cid:38) .
12. Due to the loop with a charged Higgs-boson (see fig.4), the parameter δ RLd also affects the sector of up-type squarks. In particular, themass parameter of ˜ t L receives a correction of similar size than its isospin partner ˜ b L .For this reason, also the two stop-states unmix for large | δ RLd | .The most interesting NMFV-parameters are those inducing off-diagonal entriesin the trilinear couplings of up-type squarks. In the mSUGRA scenarios under con-sideration here, the diagonal entry ( ˆ T U ) is relatively large due to large top Yukawa7 c R ˜ t L h , H , A ˜ t R ˜ c R ˜ b L H + ˜ t R Figure 5:
One-loop contributions to ˜ c R -˜ t R mixing that are linear in the parameter δ RLu . (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRL G e V (cid:68) m b (cid:142) L , b (cid:142) L (cid:68) m c (cid:142) R , c (cid:142) R (cid:68) m t (cid:142) L , t (cid:142) L m c (cid:142) R , t (cid:142) R m c (cid:142) R , t (cid:142) L (cid:45) ∆ uRL (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:45) ∆ uRL (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) Figure 6:
Dependence of up-type squark masses (left) and flavour decompositions ofselected down-type squarks (centre and right) on the NMFV-parameter δ RLu based on thebenchmark scenario SPS1a’. The point δ RLu = 0 corresponds to the MFV-scenario of table2. coupling. This does not only imply important corrections to the couplings as dis-cussed above, but can also lead to new flavour-mixing entries that are induced at theone-loop level.We first discuss the parameter δ RLu that induces the one-loop contributions to˜ c R -˜ t R mixing shown in fig. 5. Here, the contribution from the light Higgs-boson h is more important since in the up-squark sector in the decoupling limit the cou-plings to the heavy Higgs bosons are tan β -suppressed. Another difference w.r.t. thesdown-mixing parameters δ RL,LRd is that the loop-induced right-right mixing becomesrelevant because of the large value of ( ˆ T U ) which enters the loop via the combina-tion ( ˆ T U ) ( ˆ T U ) . The corrections to the mass parameters, however, are due to thelinear dependence much smaller than in the sdown-sector.In order to numerically illustrate this, we show in fig. 6 the mass parametersand selected entries of the mixing matrix as a function of the NMFV-parameter δ RLu . The graph also shows that both mixing elements depend linearly on δ RLu . Thequadratically dependent self-energies cannot challenge the left-right mixing, so that˜ u and ˜ u approach maximally mixed states consisting of ˜ c R and ˜ t L for δ RLu → ± u is a nearly pure ˜ t R state in this limit with a small ˜ c R admixture.Finally, let us study variations of parameter δ LRu inducing a direct mixing betweenthe gauge eigenstates ˜ c L and ˜ t R . At the one-loop level, this parameter inducescorrections to the mass parameter of ˜ s L , the isospin partner of the left-handed flavour˜ c L . On top of that, the graphs shown in fig. 7 lead to a sizeable mixing between ˜ c L and˜ t L and between ˜ s L and ˜ b L , the latter again due to the strong top Yukawa coupling.In contrast, the rather small charm Yukawa coupling does not allow for remarkableinfluence on the sector of sdown-squarks. This is also in contrast to the variation of δ RLu discussed above, where the opposite situation has been observed.8 c L ˜ t R h , H , A ˜ t L ˜ s L ˜ t R H − ˜ b L Figure 7:
One-loop contributions to ˜ c L -˜ t L and ˜ s L -˜ b L mixing that are linear in the param-eter δ LRu . (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR G e V (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR G e V (cid:68) m s (cid:142) L , s (cid:142) L (cid:68) m t (cid:142) R , t (cid:142) R (cid:68) m c (cid:142) L , c (cid:142) L m s (cid:142) L , b (cid:142) L m c (cid:142) L , t (cid:142) L m c (cid:142) L , t (cid:142) R (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , u (cid:142) L (cid:76) (cid:164) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR (cid:160) R (cid:72) u (cid:142) , t (cid:142) R (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , t (cid:142) L (cid:76) (cid:164) (cid:160) R (cid:72) u (cid:142) , c (cid:142) L (cid:76) (cid:164) Figure 8:
Dependence of physical squark masses (top left), one-loop contributions tosquark mass parameters (top centre) and flavour decompositions of selected up-type squarks(top right and bottom) on the NMFV-parameter δ LRu based on the benchmark scenarioSPS1a’. The point δ LRu = 0 corresponds to the MFV-scenario of table 2.
The diagonal mass parameters shown in fig. 8 are again quadratic in the NMFV-parameter δ LRu , while the tree-level ˜ c L -˜ t R mixing and the loop-induced ˜ s L -˜ b L mixingshow a linear dependence. The latter also holds asymptotically for the ˜ c L -˜ t L mixing,where the offset can be traced to the fact that the left-left block of the up-type squarkmass matrices is not diagonal in the super-CKM basis. This effect is also visible in aslight asymmetry in the flavour contents shown in fig. 8. The level-crossing between˜ u and ˜ u occurs, e.g., at δ LRu ≈ − . δ LRu ≈ . c L -˜ t R mixing. In consequence, we observe a rather “conventional”mass splitting tending towards a equipartition of the mixing flavours in the lightestand heaviest mass eigenstates ˜ u and ˜ u for δ LRu → ±
4. Quark flavour violating decays of squarks into bosons
Assuming MFV, the couplings of squarks to Z - and Higgs-bosons are in good ap-proximation diagonal in generation space. In consequence, only two squark flavoursof the same generation can be involved in the corresponding interactions. As a furtherconsequence, the decay of a given squark ˜ q i into a Z - or Higgs-boson can involve9PS1a’ Branching ratios W ± ˜ d → ˜ u W − d → ˜ u W − Z ˜ u → ˜ u Z h ˜ u → ˜ u h W ± ˜ d → ˜ u W − d → ˜ u W − Z ˜ u → ˜ u Z h ˜ u → ˜ u h Table 3:
Branching ratios of kinematically allowed decays of squarks into gauge and Higgsbosons for the reference scenarios SPS1a’ (left) and SPS1b (right) assuming MFV. only one further squark ˜ q j (if kinematically allowed). For our reference scenarios,this can be seen in table 3, where we show the branching ratios of the kinematicallyallowed decays of squarks into gauge and Higgs-bosons. Note that in both cases theheavier u -squark is mainly the heavier stop.In the same way, decays of a certain down- (up-)squark ˜ d i (˜ u i ) into W ± -bosonscan involve maximally two up- (down-)squarks ˜ d j (˜ u j ) and ˜ d k (˜ u k ) as final states.For our examples, we have the decays ˜ d → ˜ u W − and ˜ d → ˜ u W − . Note thatdue to the CKM-matrix further flavour-violating decays are in principal allowed,e.g. due to a ˜ s L → ˜ t L transition. These are, however, strongly suppressed w.r.t.to the generation-conserving channels mentioned above. The generation-conservingtransitions ˜ u → ˜ d , W + are kinematically forbidden in the SPS1a’ scenario.Let us now discuss which experimental signatures related to squark decays wouldbe able to challenge the hypothesis of MFV. If at least one of the NMFV-parametersin eq. (2.6) is non-zero, a further up-type squark mass eigenstate can obtain suffi-ciently large admixtures of ˜ t R or ˜ t L in order to open a new decay channel with a Z or h in the final state. In addition, the modified mass splitting alters the kinematicalconditions as discussed in sec. 3. In the same way, non-vanishing generation mixingcan allow for more than one (two) squark(s) decaying into the same final state squarkand a neutral (charged) boson. Observing such additional decays would be a clearhint towards a non-minimal flavour structure in the squark sector. Obviously, thesame reasoning also holds for the down-type squarks.To summarise, relevant NMFV-signatures for decays of squarks into final stateswith W ± , Z , and Higgs-bosons would be: • ˜ q i → ˜ q j Z or ˜ q i → ˜ q j h for a fixed value of i and at least two different valuesof j or for a fixed value of j and at least two different values of i , • ˜ q i → ˜ q (cid:48) j W ± or ˜ q i → ˜ q (cid:48) j H ± for a fixed value of i and at least three differentvalues of j or for a fixed value of j and at least three different values of i .In the following, we discuss our results for the scenarios SPS1a’ and SPS1b for thevariation of one or two NMFV-parameters.10 (cid:45) (cid:45) ∆ dRR BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:45) ∆ dRR G e V (cid:71) (cid:72) d (cid:142) (cid:174) Χ(cid:142) , b (cid:76) (cid:71) (cid:72) d (cid:142) (cid:174) Χ(cid:142) , s (cid:76) (cid:71) (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:71) tot (cid:72) d (cid:142) (cid:76) Figure 9:
Branching ratios of squarks decaying into W-bosons (left) and total and partialdecay widths of ˜ d (right) for variations of δ RRd around the reference scenario SPS1a’. Theshaded region indicates the experimentally disfavoured points.
As our main reference scenario for the following discussion we choose again thebenchmark point SPS1a’. For specific cases, we shall also comment on SPS1b. Theexperimental constraints of table 1 allow for rather large variations of the NMFV-parameter δ RRd . The strongest constraint is here imposed by the observable ∆ m B s ,leading to an allowed range of − . (cid:46) δ RRd (cid:46) .
69. The constraint from b → sγ is less stringent in this case. In the left panel of fig. 9, we show selected branchingratios of down-type squarks into W-bosons in dependence of this NMFV-parameter.The shaded regions correspond to the experimentally disfavoured parameter config-urations. In the case of MFV (i.e. δ RRd = 0), only the decays of ˜ d and ˜ d into ˜ u are possible (see table 3). However, already for small variations of δ RRd , the small ˜ b L content in ˜ d is sufficient to open the additional decay channel ˜ d → ˜ u W − with acomparable branching ratio.The right panel of fig. 9 shows that the rapid increase of the branching fractionis due to the increase of the partial width of this particular channel combined witha decrease of the decay ˜ d → ˜ χ s , which is dominant in the MFV-case. For smallvalues of δ RRd , the decrease of the coupling strength is compensated by the increasedmass splitting, so that the width Γ( ˜ d → ˜ u W − ) remains practically constant. ForSPS1a’ we observe thus a strong NMFV-signature with three important branchingratios around 30% related to decays into a W-boson for δ RRd (cid:46) ± .
1. For SPS1b, thechannels involving W-bosons are less important, reaching branching ratios from 3%to 13%.Since the parameter δ RRd has no influence on the sector of up-type squarks,no further decay of an up-squark into a Z - or Higgs-bosons can be induced. Inthe MFV-case (see table 3), the mass splitting is not sufficient to allow for such adecay for down-type squarks. For increasing flavour mixing, the decay of a down-type squark into a neutral bosons becomes possible allowed due to the increasedmass splitting. For δ RRd (cid:38) . . d → ˜ d Z ( ˜ d , → ˜ d Z )is kinematically allowed. For higher δ RRd , also decays into h would be possible.However, the necessary ˜ b L and ˜ s L contents are decreasing at the same time, so that11 ∆ uRR BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) (cid:45) ∆ uRR BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) Figure 10:
Branching ratios of squarks decaying into Z- (left) and Higgs-bosons (right)for variations of δ RRu around the reference scenario SPS1a’. The shaded region indicatesthe experimentally disfavoured points. the corresponding coupling is suppressed. For similar reasons, kinematically alloweddecays into h cannot be observed since the coupling is approximately zero due to anegative interference between the D- and F-term contributions.Variations of the parameter δ RRu only influence the sector of up-type squarks insuch a way that no NMFV-signatures with decays of sdown-squarks into W-bosonscan be induced. There are only the two possible modes given in table 3. Nevertheless,the corresponding branching ratios depend on the flavour-mixing parameter δ RRu dueto decreasing mass of ˜ u . At the same time, the mass of ˜ u increases, so that thedecay ˜ u → ˜ d W + opens for | δ RRu | (cid:38) .
15. However, this decay cannot by interpretedas a typical NMFV-signal within this context. The allowed range for this NMFV-parameter is − . (cid:46) δ RRu (cid:46) .
8, where the main constraints come from squark masslimits rather than the observables of table 1.Such signatures, however, can occur in the context of decays involving Z- orHiggs-bosons. In fig. 10 we show the corresponding branching ratios of up-typesquarks. While for δ RRu = 0, only one such decay is possible, additional channels openwith sizeable branching fractions already for moderate flavour mixing. In particular,there are three states containing ˜ t L , and the mass splitting creates sufficient phase-space to produce an on-shell Z- or Higgs-boson. As can be seen from fig. 10, basicallythe whole range of 0 . < | δ RRu | < . u and ˜ u decaying into ˜ u and Z or h . For | δ RRu | (cid:38) .
5, even asecond NMFV-signature shows up, namely the additional decays ˜ u → ˜ u Z ( h )reaching branching ratios of up to 0.02 (0.07).For comparison, we show in fig. 11 the corresponding branching ratios of up-type squarks for the reference scenario SPS1b. Generally, this point features lowerbranching ratios. It is interesting to note that there is even a small range around δ RRu ≈ . u and˜ u , where both states have a sizeable ˜ t L -content. The main difference w.r.t. SPS1a’lies in the initial mass splitting in the MFV-case, which is not sufficient to make ˜ t theheaviest up-type squark (see table 2). Instead the third generation mass eigenstates12 (cid:45) ∆ uRR BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) (cid:45) (cid:45) ∆ uRR BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) Figure 11:
Branching ratios of squarks decaying into Z- (left) and Higgs-bosons (right)for variations of δ RRu around the reference scenario SPS1b. The shaded region indicatesthe experimentally disfavoured points. are the two lightest ones, as is also the case for the down-type squarks. This leadsto a different structure of the flavour mixings induced by the NMFV-parameters.In contrast to the flavour mixing in the right-right sectors discussed above, theparameter δ LL is rather constrained due to the decay b → sγ , allowing only forthe narrow interval − . (cid:46) δ LL (cid:46) .
03. A second allowed window around δ LL ≈ .
8, where the large SUSY contributions to BR( b → sγ ) cancel, is disfavoured byboth ∆ m B s and ¯ B s → l + l − . Concerning NMFV-signatures, three decay channelsof down-type squarks into a W-boson and the lightest up-type squark are open for | δ LL | (cid:38) .
04. Further signal channels would open beyond the experimental exclusionlimit. In the same way, the two decay channels ˜ u → ˜ u Z and ˜ u → ˜ u Z leadto a weak NMFV-signature where the additional decays reach at most 10% at theboundary of the experimentally allowed range. Again, further decay modes wouldbe allowed outside the experimentally favoured interval for the parameter δ LL . Thebranching ratios of squark decays into Higgs-bosons are significantly smaller becausethe corresponding couplings get small for large left-right mixing.Apart from the effects on the squark mass eigenvalues discussed in sec. 3, varia-tions of the NMFV-parameter in the off-diagonal blocks of eq. (2.1) can also inducethe characteristic signatures in the decays of squarks. Flavour mixing in this sec-tor of down-type squarks is mainly constrained from the experimental limit on themeson-oscillation observable ∆ m B s and the branching ratio of b → sγ , which leaveonly the rather narrow interval of − . (cid:46) δ RLd (cid:46) .
15. The experimentally al-lowed ranges for the parameter δ LRd are even more narrow. Here, the constraintfrom b → sγ only allows for − . (cid:46) δ LRd (cid:46) . δ RLd and δ LRd in detail. Inparticular, the latter does not lead to NMFV-signatures for our reference scenar-ios SPS1a’ and SPS1b. For SPS1a’ and variations of δ RLd , sizeable branching ratiosBR( ˜ d → ˜ u W + ) ≈ BR( ˜ d → ˜ u W + ) ≈ BR( ˜ d → ˜ u W + ) ≈ −
30% can be ob-served around δ RLd ≈ − .
016 and δ RLd ≈ . b → sγ and ∆ M B s . Additional decay channels involving Z-or Higgs-bosons are not opened within this range. The same qualitative picture is13 ∆ uRL BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) (cid:45) (cid:45) ∆ uRL BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) Figure 12:
Branching ratios of squarks decaying into Z- (left) and Higgs-bosons (right)for variations of δ RLu around the reference scenario SPS1a’. The shaded region indicatesthe experimentally disfavoured points. obtained for SPS1b.Again, the most interesting parameters are the ones leading to NMFV-signatureswith neutral bosons. They allow for rather large experimentally allowed ranges − . (cid:46) δ RLu (cid:46) .
39 and − . (cid:46) δ LRu (cid:46) .
10. Interestingly, the limit on δ RLu comesnot from the low-energy observables, but from the experimental limit on the massof the lightest Higgs-boson. The latter becomes lighter for increasing flavour mixingdue to the corrections from squark-loops. For δ LRu , the decay b → sγ remains themost stringent constraint.Let us start the discussion with the parameter δ LRu inducing a ˜ c R -˜ t L mixing. Here,additional decays into W-bosons cannot be achieved due to the fact that, except ˜ d and ˜ d , no down-type squarks obtain a ˜ b L admixture, which can lead to decays into˜ u . In the case of MFV, the decay ˜ u → ˜ u Z is the only channel involving a Z-boson. For already rather small | δ RLu | (cid:38) .
03, a second channel ˜ u → ˜ u Z with thesame final state opens with a branching ratio of up to about 10% as shown in fig.12. The reason therefore is the non-zero ˜ t L content in ˜ u (see fig. 6). Its decreasefor | δ RLu | (cid:38) .
25 is compensated by the increasing phase space so that the branchingratio remains nearly constant. The branching ratio of the initial decay ˜ u → ˜ u Z increases with the mass splitting. The corresponding coupling only changes withina range of 10%.In fig. 12, we show the decays of squarks into light Higgs bosons as a functionof the NMFV-parameter δ RLu for the reference scenario SPS1a’. Here, the initialdecay ˜ u → ˜ u h decreases with increasing flavour mixing. The relevant coupling isdominated by a left-right component, but the ˜ t R content of ˜ u and ˜ u decreases (seefig. 6). For | δ RLu | (cid:38) .
3, the new channel ˜ u → ˜ u h opens leading to a characteristicNMFV-signature. However, the decay ˜ u → ˜ u h falls rapidly below 1%, so thatonly the rather narrow interval − . ≤ δ RLu ≤ .
18 allows for a realistic signature.In the case of SPS1b, variations of the parameter δ RLu qualitatively show thesame signature. However, the different mass splitting and the associated smoothlevel-crossing between ˜ u and ˜ u at δ RLu ≈ ± . u , , can decay into the same final state ˜ u Z . This14 (cid:45) (cid:45) ∆ uRL BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) Figure 13:
Branching ratios of squarks decaying into Z-bosons for variations of δ RLu aroundthe reference scenario SPS1b. The shaded region indicates the experimentally disfavouredpoints. (cid:45) (cid:45) (cid:45) (cid:45) ∆ uLR BR (cid:72) u (cid:142) (cid:174) d (cid:142) , W (cid:43) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:45) (cid:45) (cid:45) ∆ uLR BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) Figure 14:
Branching ratios of squarks decaying into W-bosons for variations of δ LRu around the reference scenarios SPS1a’ (left) and SPS1b (right). The shaded region indicatesthe experimentally disfavoured points. behaviour is analogous to the case of δ RRu discussed above. The numerical results forthe branching ratios of these decays are shown in fig. 13.Finally, let us discuss the squark decays as a function of the parameter δ LRu . Forthis parameter, it is again possible to observe a NMFV-signature from decays into W-bosons. In fig. 14, we show examples of numerical branching ratios with the lightestup-type squark and a W-boson in the final state for our two reference scenarios.While around δ LRu ≈ d → ˜ u W + opens for δ LRu (cid:46) − . δ LRu (cid:38) . s L -˜ b L mixing discussed in sec. 3. Due to the mass splitting, the threerelevant branching ratios increase with the flavour-mixing parameter.For SPS1b, the branching ratios of the three signal decays ˜ d , , → ˜ u W + are withbranching ratios of about 30–60% rather sizeable. For δ LRu (cid:38) − .
35, even a fourthchannel due to the decay of ˜ d opens up. However, the corresponding branchingfraction in the experimentally allowed range remains below 1.6%.The above results for SPS1a’ and SPS1b can be summarized as follows: • The NMFV-parameters in the sdown-sector only induce signatures related toW-bosons. 15
The parameters in the sup-sector induce mainly signatures related to Z- andHiggs-bosons. The latter have mostly smaller branching ratios due to thesuppressed phase space. • The parameter δ LL acts on both the sup- and sdown-sectors and can thereforeinduce three W-channels as well as two Z- or Higgs-channels. However, thisparameter is heavily constraint from experimental data. • The parameters δ RRu and δ RRd are rather unconstrained and allow for large masssplitting leading to large branching ratios for decays into vector bosons.These features are to some extend a consequence of the original mSUGRA boundaryconditions. However, the basic structure is the same as in other breaking schemeslike GMSB or AMSB. One would need some non-universal boundary condition atthe high scale to depart from this feature. One possibility are, e.g., extra D -termcontributions occurring from the breaking of a higher rank group to the SM group.For example, if SO (10) or E (6) get broken to the SM, there are D -terms contributingdifferently to the left- and right squarks [34].Finally, selected signatures for the variation of one single NMFV-parameter forthe reference scenarios SPS1a’ and SPS1b are collected in tables 4 and 5, respectively.They have been chosen by the requirement to provide potentially large signals. The signatures related to non-minimal flavour violation in the squark sector discussedabove can be extended or amplified if more than one of the parameters defined in eq.(2.6) is non-zero. In particular, the relevant branching ratios can be increased andadditional NMFV-signatures can appear. Here, we limit ourselves to the simultane-ous variation of two NMFV-parameters, leaving more involved variations for lateranalyses. We also discuss only the combinations of parameters that lead to newsignatures w.r.t. the variations discussed in sec. 4.1.Since the above analysis has shown that the most interesting parameters are δ LL and δ LRu , we start our discussion with their simultaneous variation. A secondinteresting combination of parameters is δ LRu together with δ RRu . The latter is experi-mentally rather unconstrained and leads to sizeable NMFV-signatures. In particular,it is the only parameter inducing a signature with two decays of the same initial statesquark into different final state squarks and a light Higgs boson (see tables 4 and 5).Since δ RRu does not act on the the sector of down-type squarks, no signatures withW-bosons can be realized in the case of the single-parameter variation. However,taking a second NMFV-parameter to be non-zero can relax this limitation and opennew signals.In fig. 15 we show the experimentally allowed ranges at the 95% confidencelevel in the δ LL - δ LRu and δ RRu - δ LRu -planes around our reference scenario SPS1a’. The16PS1a’ δ RRd = ± . δ RRu = ± . d → ˜ u W − ) = 30%BR( ˜ d → ˜ u W − ) = 34%BR( ˜ d → ˜ u W − ) = 25% BR(˜ u → ˜ u Z ) = 32%BR(˜ u → ˜ u Z ) = 9.8%BR(˜ u → ˜ u Z ) = 2.4%BR(˜ u → ˜ u h ) = 18%BR(˜ u → ˜ u h ) = 2.8%BR(˜ u → ˜ u h ) = 5.7% δ LL = − . δ LRu = − . d → ˜ u W − ) = 29%BR( ˜ d → ˜ u W − ) = 33%BR( ˜ d → ˜ u W − ) = 3.2%BR(˜ u → ˜ u Z ) = 5.4%BR(˜ u → ˜ u Z ) = 30% BR( ˜ d → ˜ u W − ) = 42%BR( ˜ d → ˜ u W − ) = 49%BR( ˜ d → ˜ u W − ) = 18%BR(˜ u → ˜ u Z ) = 2.4%BR(˜ u → ˜ u Z ) = 39% δ RLu = ± . δ RLd = − . u → ˜ u Z ) = 7.4%BR(˜ u → ˜ u Z ) = 37%BR(˜ u → ˜ u h ) = 12%BR(˜ u → ˜ u h ) = 2.8% BR( ˜ d → ˜ u W − ) = 30.0%BR( ˜ d → ˜ u W − ) = 27.0%BR( ˜ d → ˜ u W − ) = 34.0% Table 4:
Branching ratios of squark decays leading to typical NMFV-signatures for se-lected parameter points beyond MFV based on the benchmark scenario SPS1a’. decisive constraint is here again the inclusive decay b → sγ , leading to a rather smallinterval for the left-left mixing parameter δ LL . The limits on the Higgs-mass and on∆ M B s are secondary. Non-zero values of δ LRu lead to small additional contributions toBR( b → sγ ) due to the loop-induced ˜ s L -˜ b L mixing, so that the different contributionscancel in certain regions of the δ LL - δ LRu plane. The experimentally allowed regionis therefore slightly twisted w.r.t. the axes. The applied constraints allow for alarge interval for the parameter δ RRu , so that a rather wide concave favoured region isobserved in the δ RRu - δ LRu -plane. For the second reference scenario SPS1b, the situationis qualitatively the same, with the exception that the vacuum stability [35] excludescertain regions in the δ RRu - δ LRu -plane that are allowed concerning the constraints oftable 1.Let us now turn to the decays of squarks into W-bosons. Fig. 16 shows thatup to four decay channels can involve the same final state ˜ u W − . The single graphsshow the corresponding branching ratios in the δ LL - δ LRu plane. The main new feature17PS1b δ RRd = ± . δ RRu = ± . d → ˜ u W − ) = 5.0%BR( ˜ d → ˜ u W − ) = 14%BR( ˜ d → ˜ u W − ) = 6.8% BR(˜ u → ˜ u Z ) = 12%BR(˜ u → ˜ u Z ) = 3.9%BR(˜ u → ˜ u Z ) = 1.7%BR(˜ u → ˜ u h ) = 8.5%BR(˜ u → ˜ u h ) = 1.3%BR(˜ u → ˜ u h ) = 3.5% δ LL = − . δ LRu = − . d → ˜ u W − ) = 4.6%BR( ˜ d → ˜ u W − ) = 12%BR( ˜ d → ˜ u W − ) = 1.9%BR(˜ u → ˜ u Z ) = 8.2%BR(˜ u → ˜ u Z ) = 3.5%BR(˜ u → ˜ u h ) = 2.4%BR(˜ u → ˜ u h ) = 1.2% BR( ˜ d → ˜ u W − ) = 7.1%BR( ˜ d → ˜ u W − ) = 16%BR( ˜ d → ˜ u W − ) = 7.4%BR(˜ u → ˜ u Z ) = 8.0%BR(˜ u → ˜ u Z ) = 9.8%BR(˜ u → ˜ u h ) = 1.7%BR(˜ u → ˜ u h ) = 4.8% δ RLu = ± . u → ˜ u Z ) = 5.2%BR(˜ u → ˜ u Z ) = 8.0%BR(˜ u → ˜ u Z ) = 8.1% Table 5:
Branching ratios of squark decays leading to typical NMFV-signatures for se-lected parameter points beyond MFV based on the benchmark scenario SPS1b. compared to the aspects discussed in sec. 4.1 is that there is a wide experimentallyallowed range where the channels ˜ d → ˜ u W − and ˜ d → ˜ u W − both are open. Notethat this signature with up to four W-bosons is possible in wide regions of parameterspace.As discussed in the previous section, NMFV-signatures due to decays into Z-bosons are present over wide ranges of the analysed parameter space. However, thereare only rather small regions where branching ratios for three different u -squarks into˜ u Z exceed five percent.Concerning decays into light Higgs-bosons, the reference scenario SPS1a’ doesnot lead to new features if one allows for two non-zero NMFV-parameters. Althoughthe couplings can become rather important, the phase space does not allow for ad-ditional branching ratios larger than 1%. However, new decay channels involvingHiggs-bosons open in the case of SPS1b, as can be seen in fig. 17. Here, we have˜ u ≈ ˜ c L if all NMFV-parameters are zero, and the two lightest squarks are the stops.18 (cid:45) (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R B s (cid:174) Μ (cid:43) Μ (cid:45) (cid:68) m B s B s (cid:174) X s (cid:123) (cid:43) (cid:123) (cid:45) m h b (cid:174) s Γ (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R B s (cid:174) Μ (cid:43) Μ (cid:45) B s (cid:174) X s (cid:123) (cid:43) (cid:123) (cid:45) (cid:68) m B s m h b (cid:174) s Γ Figure 15:
Experimental constraints for simultaneous variation of two NMFV-parametersaround the reference scenario SPS1a’. The legend indicates the constraints in order of theexcluded regions starting from the allowed white region. (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) d (cid:142) (cid:174) u (cid:142) , W (cid:45) (cid:76) Figure 16:
Dependence of the branching ratios of selected down-type squarks into thelightest up-type squark and a W-boson on the NMFV-parameters δ LL and δ LRu aroundthe reference scenario SPS1a’. The shaded region indicates the experimentally disfavouredpoints.
For both parameters δ LL and δ LRu , the channel ˜ u → ˜ u h opens with branchingratios of up to almost 15%. The branching fraction of the original decay channel˜ u → ˜ u h (see table 3) decreases at the same time.Finally, let us discuss NMFV-signatures related to a simultaneous variation of δ RRu and δ LRu . For our scenario based on SPS1a’, the branching ratios of up-type19 (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) (cid:45) (cid:45) (cid:45) ∆ LL ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) Figure 17:
Dependence of the branching ratios of the down-type squarks ˜ u and ˜ u intothe lightest up-type squark ˜ u and a Higgs-boson h on the NMFV-parameters δ LL and δ LRu around the reference scenario SPS1b. The shaded region indicates the experimentallydisfavoured points. (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76)
00 0 00 0 0 0000 00.010.01 0.010.01 0.010.010.02 0.02 0.02 0.020.020.030.03 0.030.030.04 0.04 (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , Z (cid:76) Figure 18:
Dependence of the branching ratios of selected up-type squarks into Z-bosonson the NMFV-parameters δ RRu and δ LRu around the reference scenario SPS1a’. The shadedregion indicates the experimentally disfavoured points. squarks into Z- and Higgs-bosons are shown in figs. 18 and 19, respectively. A largepart of the two-dimensional parameter space allows for three decay channels into thesame final state ˜ u Z . The branching ratio BR(˜ u → ˜ u Z ) increases mainly alongthe δ RRu direction, while it is almost independent of δ LRu . This is explained by thefact that for MFV, ˜ u is a pure ˜ c R state and no ˜ t L content is introduced for variationsof δ LRu . For the decay of ˜ u , the opposite is observed. This channel remains closedalong the δ RRu -axis, since this parameter mixes neither a ˜ t L content into ˜ u nor a ˜ c L content into ˜ u . This only happens for | δ LRu | >
0. However, due to the destructiveinterference of the ˜ c L and ˜ t L parts of the coupling, this branching ratio cannot exceed3% if only δ LRu is non-zero. If both parameters are non-zero, the branching fractioncan be as large as about 4%. Therefore, while the decays ˜ u → ˜ u Z and ˜ u → ˜ u Z are sizable over most of the parameter space considered, the other two are small andlarge statistics will be necessary to measure them.We observe a similar picture in the context of decays into Higgs-bosons. Again,20 (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76)
00 0 0 000 00.010.01 0.010.010.02 0.02 0.02 0.020.030.03 0.03 0.030.04 0.040.05 0.050.06 0.060.07 0.07 0.07 (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) ∆ uRR ∆ u L R BR (cid:72) u (cid:142) (cid:174) u (cid:142) , h (cid:76) Figure 19:
Dependence of the branching ratios of selected up-type squarks into Higgs-bosons on the NMFV-parameters δ RRu and δ LRu around the reference scenario SPS1a’. Theshaded region indicates the experimentally disfavoured points. the branching fractions are generally smaller than for the Z-boson due to scalar phasespace. The decay ˜ u → ˜ u h can only open for non-zero values of δ RRu , while thebranching ratio of ˜ u → ˜ u h decreases for | δ RRu | → | δ LRu | → u into ˜ u is only allowed for large | δ RRu | (cid:38) . δ RRu only, the branching fraction BR(˜ u → ˜ u h ) canhere reach values of 7%. This shows again, that combinations of NMFV-parameterscan emphasise certain signatures.Looking at decays into W-bosons, the channel ˜ d → ˜ u W − can open beside thetwo original (see table 3) ones, ˜ d → ˜ u W − and ˜ d → ˜ u W − . The branching fractionof the additional channel reaches about 20% within the experimentally allowed regionof the NMFV-parameters. This is possible due to the loop-induced ˜ s L -˜ b L mixing. Thebranching fractions depend therefore only very weakly on the parameter δ RRu , as theseare left-left transitions and the impact is only via the phase space, and thus are notshown separately as the main information is already contained in fig. 14.Last but not least, we mention that in case of SPS1b regions in parameter spaceare found where ˜ u decays into three different final state squarks ˜ d , , and a W + .This becomes possible only if both NMFV-parameters δ RRu and δ LRu are non-zero.In order to give a compact overview over the various signals, we summariseselected parameter configurations for the two reference scenarios and related NMFV-signatures in table 6.
5. Conclusion
In summary, we have discussed which signatures related to bosonic decays of squarkscould be able to challenge the hypothesis of minimal flavour violation in supersym-metry. Therefore, we have investigated in detail the effect of non-minimally flavour-violating couplings of squarks to gauge- and Higgs-bosons within the MSSM. Starting21PS1a’ δ LL = − . δ LRu = 0 . δ RRu = 0 . δ LRu = − . d → ˜ u W − ) = 24%BR( ˜ d → ˜ u W − ) = 14%BR( ˜ d → ˜ u W − ) = 10%BR( ˜ d → ˜ u W − ) = 30%BR(˜ u → ˜ u Z ) = 2.5%BR(˜ u → ˜ u Z ) = 34% BR(˜ u → ˜ u Z ) = 33%BR(˜ u → ˜ u Z ) = 19%BR(˜ u → ˜ u Z ) = 19%BR(˜ u → ˜ u Z ) = 1.7%BR(˜ u → ˜ u h ) = 19%BR(˜ u → ˜ u h ) = 6.7%BR(˜ u → ˜ u h ) = 3.7%BR(˜ u → ˜ u h ) = 4.3%SPS1b δ LL = 0 . δ LRu = − . δ RRu = 0 . δ LRu = − . d → ˜ u W − ) = 5.7%BR( ˜ d → ˜ u W − ) = 13%BR( ˜ d → ˜ u W − ) = 5.3%BR( ˜ d → ˜ u W − ) = 7.1%BR(˜ u → ˜ u Z ) = 5.8%BR(˜ u → ˜ u Z ) = 12%BR(˜ u → ˜ u h ) = 1.1%BR(˜ u → ˜ u h ) = 5.7% BR(˜ u → ˜ u Z ) = 21%BR(˜ u → ˜ u Z ) = 190%BR(˜ u → ˜ u Z ) = 5.2%BR(˜ u → ˜ u Z ) = 3.4%BR(˜ u → ˜ u Z ) = 1.6%BR(˜ u → ˜ u h ) = 16%BR(˜ u → ˜ u h ) = 19%BR(˜ u → ˜ u h ) = 1.4%BR(˜ u → ˜ u h ) = 4.8%BR(˜ u → ˜ u h ) = 2.0% Table 6:
Branching ratios of squark decays leading to NMFV-signatures for selectedparameter points with two non-zero NMFV-parameters. from the benchmark scenarios SPS1a’ and SPS1b we have introduced off-diagonal,i.e. flavour violating, parameters in the mass matrices of up- and down-type squarks.After discussing their contributions to squark self-energy at the tree- or the one-loop level, we have studied in detail their implications for the decay of squarks intogauge- and Higgs-bosons. In particular, we have shown that each of the parametersinducing a flavour mixing between second and third generation squarks has a specialcharacteristic, which is independent of the exact reference scenario. Combinationof two parameters leads to a superposition of the associated effects on the massspectrum and the decay signatures. 22ypical signatures of non-minimally flavour-violating couplings can be observedin wide ranges of the analyzed parameter space: they involve either multiple decaymodes of one single squark state into final states with gauge or Higgs bosons. Thesecond possibility is that more than one squarks decay into the same final statecontaining a squark and a Z/Higgs-boson or that more than two squarks decay intothe same final state containing a squark and W-boson or the charged Higgs boson.Clearly disentangling these final states is experimentally challenging and furtherdetailed Monte Carlo studies will be necessary to demonstrate the feasibility of thisidea.We note that the detailed results presented here depend on the SUSY pointchosen, e.g., the dominance of final states containing W -bosons ( Z - and neutral Higgsbosons) in case of down-type squarks (up-type) squarks changes as sign of NMFV.Other SUSY breaking schemes and/or additional D -term contributions stemmingfrom the breaking of larger rank groups to the SM gauge group might change thehierarchy between the soft SUSY breaking parameters leading to additional decaymodes and/or suppressing the modes discussed here due to different kinematics.However, in general one gets sizable NMFV decay branching ratios except for theregions where all squark masses parameters squared are nearly degenerate and at thesame time all left-right mixing entries are small.As final result, if it will be possible to observe squarks at the LHC and to re-construct their decay channels, the observation of the discussed signal would excludethe hypothesis of minimal flavour violation. Then, the purely bosonic decay modesdiscussed in this paper would deliver complementary information w.r.t. fermionicdecays into charginos, neutralinos, or gluinos, which will be helpful for reconstruct-ing the couplings and mass parameters of supersymmetric partners of the StandardModel particles. Acknowledgments
We thank T. Hurth for useful discussions and F. Staub for cross checking the formulaegiven in Appendix B. This work has been supported by the DFG, project numberPO-1337/1-1. B.H. acknowledges support from the Hamburg Excellence Cluster“Connecting particles to the cosmos” and W.P. from the Alexander von HumboldtFoundation and the Spanish grant FPA2008-00319/FPA.
A. Couplings
In the following, we give a compilation of the couplings of squarks to vector- andHiggs-bosons taking into account the most general squark mixing as described in Sec.2. All couplings are given in both the super-CKM basis, ˜ q (s) , and the mass eigenbasisof the squarks, ˜ q (m) . 23 .1 Squark-squark-vector couplings Since gluons and photons are gauge bosons of the unbroken symmetry SU(3) × U(1) em and couple in equal manner to left- and right-handed squarks, their couplings tothe physical mass eigenstates are not influenced by the presence of non-minimallyflavour-violating terms. The latter only affect the couplings to Z- and W-bosons. Inthe super-CKM basis and using the same notation as in sec. 2, the relevant termsare given by the Lagrangian L ˜ q ˜ q (cid:48) V = − i g cos θ W Z µ (cid:20) ˜ q ∗ ( s ) Li (cid:16) T ˜ q − e ˜ q sin θ W (cid:17) ↔ ∂ µ ˜ q ( s ) Li − ˜ q ∗ ( s ) Ri e ˜ q sin θ W ↔ ∂ µ ˜ q ( s ) Ri (cid:21) − i √ g (cid:20) W + µ ˜ u ∗ ( s ) Li ( V CKM ) ij ↔ ∂ µ ˜ d ( s ) Lj + W − µ ˜ d ∗ ( s ) Li ( V † CKM ) ij ↔ ∂ µ ˜ u ( s ) Lj (cid:21) − ie A µ (cid:20) ˜ q ∗ ( s ) Li e ˜ q ↔ ∂ µ ˜ q ( s ) Li + ˜ q ∗ ( s ) Ri e ˜ q ↔ ∂ µ ˜ q ( s ) Ri (cid:21) . (A.1)These terms are almost identical to the quark-quark-vector couplings with the dif-ference that the squark fields have to be transformed to the mass eigenbasis ˜ q (m) according to ˜ q ( s ) Li = (cid:88) t =1 ( R † ˜ q ) it ˜ q (m) t , ˜ q ( s ) Ri = (cid:88) t =1 ( R † ˜ q ) ( i +3) t ˜ q (m) t . (A.2)The resulting coupling terms in the mass basis are given by L ˜ q ˜ q (cid:48) V = − i g cos θ W (cid:104) T ˜ q ( R ˜ q ) si ( R † ˜ q ) it − e ˜ q sin θ W δ st (cid:105) Z µ ˜ q *(m) s ↔ ∂ µ ˜ q (m) t − iee ˜ q A µ ˜ q ∗ ( m ) s ↔ ∂ µ ˜ q (m) t δ st − i g √ R ˜ u ) si ( V CKM ) ij ( R † ˜ d ) jt W + µ ˜ u ∗ ( m ) s ↔ ∂ µ ˜ d (m) t − i g √ R ˜ d ) si ( V † CKM ) ij ( R † ˜ u ) jt W − µ ˜ d ∗ ( m ) s ↔ ∂ µ ˜ u (m) t . (A.3)Here, in addition to the CKM-matrix, the rotation matrices R ˜ q enter the couplingsexplicitly, leading to NMFV-effects, e.g., in squark decays. While for couplings withW-bosons, both rotation matrices are present, in couplings to Z-bosons, only prod-ucts of two entries of the same rotation matrix appear. In consequence, the cou-plings to W-bosons are affected by all NMFV-parameters defined in sec. 2, while fora given squark decay into a Z-boson only at most four of them are relevant. Notethat, although the matrices R ˜ q are unitary, the product ( R ˜ q ) si ( R † ˜ q ) it (cid:54) = δ st , sinceno summation over the right-handed indices is performed. The term including thephoton field A µ is included only for completeness.24 .2 Squark-squark-Higgs couplings In the super-CKM basis, the Lagrangian including the coupling of squarks to thelighter neutral Higgs boson, h , is given by L ˜ u ˜ uh = − g m W h (cid:34) ˜ u ∗ ( s ) Li ˜ u (s) Lj (cid:16) m W sin( α + β )(1 − tan θ W ) δ ij + 2 cos α sin β m u,i δ ij (cid:17) + ˜ u ∗ ( s ) Ri ˜ u (s) Rj (cid:16) + m W sin( α + β ) tan θ W δ ij + 2 cos α sin β m u,i δ ij (cid:17) + (cid:104) ˜ u ∗ ( s ) Ri ˜ u (s) Lj (cid:16) + µ ∗ sin α sin β m u,i δ ij + cos α sin β v u √ T U ) ij (cid:17) + h . c . (cid:105)(cid:35) (A.4)and L ˜ d ˜ dh = g m W h (cid:34) ˜ d ∗ ( s ) Li ˜ d (s) Lj (cid:16) m W sin( α + β )(1 + tan θ W ) δ ij + 2 sin α cos β m d,i δ ij (cid:17) + ˜ d ∗ ( s ) i ˜ d (s) R j (cid:16) m W sin( α + β ) tan θ W δ ij + 2 sin α cos β m d,i δ ij (cid:17) + (cid:104) ˜ d ∗ ( s ) Ri ˜ d (s) L j (cid:16) µ ∗ cos α cos β m d,i δ ij + sin α cos β v d √ T D ) ij (cid:17) + h . c . (cid:105)(cid:35) (A.5)for up- and down-type squarks, respectively. The terms including tan θ W stem fromthe D-terms of the scalar potential and are flavour-universal. Expressions with quarkmasses m u,d are the Yukawa- and F-terms, and the trilinear couplings are explicitbreaking terms that couple left-handed to right-handed squarks. Transformation intothe mass basis of the squark fields, the above expressions become L ˜ u ˜ uh = − g m W ˜ u ∗ ( m ) s ˜ u (m) t h (cid:20) m W sin( α + β ) (cid:104) (1 − tan θ W )( R ˜ u ) si ( R † ˜ u ) it + tan θ W ( R ˜ u ) s ( i +3) ( R † ˜ u ) ( i +3) t (cid:105) + 2 cos α sin β (cid:104) ( R ˜ u ) si m u,i ( R † ˜ u ) it + ( R ˜ u ) s ( i +3) m u,i ( R † ˜ u ) i +3 t (cid:105) + sin α sin β (cid:104) µ ∗ ( R ˜ u ) s ( i +3) m u,i ( R † ˜ u ) it + µ ( R ˜ u ) si m u,i ( R † ˜ u ) ( i +3) t (cid:105) + cos α sin β v u √ (cid:104) ( R ˜ u ) s ( i +3) ( ˆ T U ) ij ( R † ˜ u ) jt + ( R ˜ u ) si ( ˆ T † U ) ij ( R † ˜ u ) ( j +3) t (cid:105)(cid:21) (A.6)25nd L ˜ d ˜ dh = g m W ˜ d ∗ ( m ) ˜ d (m) t h (cid:20) m W sin( α + β ) (cid:104) (1 + tan θ W ) ( R ˜ d ) si ( R † ˜ d ) it + tan θ W ( R ˜ d ) s ( i +3) ( R † ˜ d ) ( i +3) t (cid:105) + 2 sin α cos β (cid:104) ( R ˜ d ) si m d,i ( R † ˜ d ) it + ( R ˜ d ) s ( i +3) m d,i ( R † ˜ d ) ( i +3) t (cid:105) + cos α cos β (cid:104) µ ∗ ( R ˜ d ) s ( i +3) m d,i ( R † ˜ d ) it + µ ( R ˜ d ) si m d,i ( R † ˜ d ) ( i +3) t (cid:105) + sin α cos β v d √ (cid:104) ( R ˜ d ) s ( i +3) ( ˆ T D ) ij ( R † ˜ d ) jt + ( R ˜ d ) si ( ˆ T † D ) ij ( R † ˜ d ) ( j +3) t (cid:105)(cid:21) . (A.7)The couplings of squarks to the heavier neutral Higgs-boson H are obtained byreplacing h → H , sin α → cos α , and cos α → − sin α .The pseudoscalar Higgs-boson A is a mixture of the imaginary parts of theneutral components of the two doublets. Therefore it is anti-hermitian and the realdiagonal contributions to the couplings vanish. The structure of the remaining termsis rather simple, the corresponding Lagrangian in the super-CKM basis is given by L ˜ q ˜ qA = − i g m W A (cid:104) ˜ d ∗ ( s ) Ri ˜ d (s) Lj (cid:16) µ ∗ m d,i δ ij + tan β v d √ T D ) ij (cid:17) +˜ u ∗ ( s ) Ri ˜ u (s) Lj (cid:16) µ ∗ m u,i δ ij + cot β v u √ T U ) ij (cid:17) + h . c . (cid:105) . (A.8)Transformation into the mass eigenbasis leads to L ˜ d ˜ dA = − i g m W ˜ d ∗ ( m ) s ˜ d (m) t A (cid:104) µ ∗ ( R ˜ d ) s ( i +3) m d,i ( R † ˜ d ) it + tan β v d √ R ˜ d ) s ( i +3) ( ˆ T D ) ij ( R † ˜ d ) jt + h . c . (cid:105) (A.9)and L ˜ u ˜ uA = − i g m W ˜ u ∗ (m) s ˜ u (m) t A (cid:104) µ ∗ ( R ˜ u ) s ( i +3) m u i ( R † ˜ u ) it + cot β v u √ R ˜ u ) s ( i +3) ( ˆ T U ) ij ( R † ˜ u ) jt + h . c . (cid:105) . (A.10)Finally, the couplings of charged Higgs-bosons to squarks are given in super-26KM basis by L ˜ q ˜ qH ± = g √ m W H − (cid:34) ˜ d ∗ ( s ) Li ˜ u (s) Lj (cid:16) − m W sin 2 β ( V † CKM ) ij + tan β m d,i ( V † CKM ) ij + cot β ( V † CKM ) ij m u,j (cid:17) + ˜ d ∗ ( s ) Ri ˜ u (s) Lj (cid:16) µ ∗ m d,i δ ij + tan β v d √ T D ) ij (cid:17) + ˜ d ∗ ( s ) Li ˜ u (s) Rj (cid:16) µm u,i δ ij + cot β v u √ T † U ) ij (cid:17) + ˜ d ∗ ( s ) Ri ˜ u (s) Rj (tan β + cot β ) m d,i ( V † CKM ) ij m u j (cid:35) + h . c ., (A.11)leading to the following expression in the squark mass basis L ˜ q ˜ qH ± = g √ m W ˜ d ∗ ( m ) s ˜ u (m) t H − (cid:34) − m W sin 2 β ( R ˜ d ) si ( V † CKM ) ij ( R † ˜ u ) jt + tan β ( R ˜ d ) si m d,i ( V † CKM ) ij ( R † ˜ u ) jt + µ ∗ ( R ˜ d ) s ( i +3) m d,i ( R † ˜ u ) it + tan β v d √ R ˜ d ) s ( i +3) ( ˆ T D ) ij ( R † ˜ u ) jt + cot β ( R ˜ d ) si ( V † CKM ) ij m u,j ( R † ˜ u ) jt + µ ( R ˜ d ) si m u,i ( R † ˜ u ) ( i +3) t + cot β v u √ R ˜ d ) si ( ˆ T † U ) ij ( R † ˜ u ) ( j +3) t + (tan β + cot β )( R ˜ d ) s ( i +3) m d,i ( V † CKM ) ij m u,j ( R † ˜ u ) ( j +3) t (cid:35) + h . c .. (A.12) B. One-loop mass matrices
The one-loop mass matrizes are given by the equations below and have been cross-checked using the package SARAH [36, 37]. The mass eigenvalues squared are ob-tained by taking the real part of the poles of the propagator matrixDet (cid:2) p k − M q ( p k ) (cid:3) = 0 , m q k = Re( p k ) , k = 1 , . . . , (cid:0) M q ( p k ) (cid:1) ij = (cid:0) M q (cid:1) ij − (cid:0) Π( p k ) (cid:1) ij (B.2)In this expression, M q is the tree-level mass matrix given in eq. (2.1) where all entriescontain running DR parameters at a common scale Q . Π( Q ) contains the squarkself-energy functions evaluated at the scale Q .27e obtain (cid:0) Π( p ) (cid:1) ij = (cid:88) n =1 (cid:20) F (cid:16) p , m q n , (cid:17) Γ ∗ ˆ˜ q j ,G, ˜ q n Γ ˆ˜ q i ,G, ˜ q n + F (cid:16) p , m q n , (cid:17) Γ ∗ ˆ˜ q j ,γ, ˜ q n Γ ˆ˜ q i ,γ, ˜ q n + F (cid:16) p , m q (cid:48) n , m W (cid:17) Γ ∗ ˆ˜ q j ,W + , ˜ q (cid:48) n Γ ˆ˜ q i ,W + , ˜ q (cid:48) n + F (cid:16) p , m q n , m Z (cid:17) Γ ∗ ˆ˜ q j ,Z, ˜ q n Γ ˆ˜ q i ,Z, ˜ q n +2 A (cid:16) m Z (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j ,Z,Z + 4 A (cid:16) m W (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j ,W + ,W − − (cid:88) l =1 (cid:20) A (cid:16) m h l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j ,h l ,h l + 12 A (cid:16) m A l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j ,A l ,A l + A (cid:16) m H + l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j ,H − l ,H + l (cid:21) + (cid:88) l =1 6 (cid:88) n =1 (cid:20) B (cid:16) p , m A l , m q n (cid:17) Γ ∗ ˆ˜ q j ,A l , ˜ q n Γ ˆ˜ q i ,A l , ˜ q n + B (cid:16) p , m h l , m q n (cid:17) Γ ∗ ˆ˜ q j ,h l , ˜ q n Γ ˆ˜ q i ,h l , ˜ q n + B (cid:16) p , m H + l , m u n (cid:17) Γ ∗ ˆ˜ q j ,H + l , ˜ q (cid:48) n Γ ˆ˜ q i ,H + l , ˜ q (cid:48) n (cid:21) − (cid:88) l =1 (cid:20) A (cid:16) m d l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j , ˜ d ∗ l , ˜ d l + A (cid:16) m u l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j , ˜ u ∗ l , ˜ u l + A (cid:16) m e l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j , ˜ e ∗ l , ˜ e l (cid:21) − (cid:88) l =1 A (cid:16) m ν l (cid:17) Γ ˆ˜ q ∗ i , ˆ˜ q j , ˜ ν ∗ l , ˜ ν l + 43 (cid:88) l =1 (cid:20) G (cid:16) p , m q l , m g (cid:17)(cid:16) Γ L ∗ ˆ˜ q j ,q l , ˜ g Γ L ˆ˜ q i ,q l , ˜ g + Γ ∗ R ˆ˜ q j ,q l , ˜ g Γ R ˆ˜ q i ,q l , ˜ g (cid:17) − m ˜ g m q l B (cid:16) p , m q l , m g (cid:17)(cid:16) Γ L ∗ ˆ˜ q j ,q l , ˜ g Γ R ˆ˜ q i ,q l , ˜ g + Γ R ∗ ˆ˜ q j ,q l , ˜ g Γ L ˆ˜ q i ,q l , ˜ g (cid:17)(cid:21) + (cid:88) l =1 3 (cid:88) n =1 (cid:20) G (cid:16) p , m χ + l , m q (cid:48) n (cid:17)(cid:16) Γ L ∗ ˆ˜ q j , ˜ χ + l ,q (cid:48) n Γ L ˆ˜ q i , ˜ χ + l ,q (cid:48) n + Γ R ∗ ˆ˜ q j , ˜ χ + l ,q (cid:48) n Γ R ˆ˜ q i , ˜ χ + l ,q (cid:48) n (cid:17) − m ˜ χ + l m q (cid:48) n B (cid:16) p , m χ + l , m q (cid:48) n (cid:17)(cid:16) Γ L ∗ ˆ˜ q j , ˜ χ + l ,q (cid:48) n Γ R ˆ˜ q i , ˜ χ + l ,q (cid:48) n + Γ R ∗ ˆ˜ q j , ˜ χ + l ,q (cid:48) n Γ L ˆ˜ q i , ˜ χ + l ,q (cid:48) n (cid:17)(cid:21) + (cid:88) l =1 3 (cid:88) n =1 (cid:20) G (cid:16) p , m χ l , m q n (cid:17)(cid:16) Γ L ∗ ˆ˜ q j , ˜ χ l ,q n Γ L ˆ˜ q i , ˜ χ l ,q n + Γ R ∗ ˆ˜ q j , ˜ χ l ,q n Γ R ˆ˜ q i , ˜ χ l ,q n (cid:17) − m ˜ χ l m q n B (cid:16) p , m χ l , m q n (cid:17)(cid:16) Γ L ∗ ˆ˜ q j , ˜ χ l ,q n Γ R ˆ˜ q i , ˜ χ l ,q n + Γ R ∗ ˆ˜ q j , ˜ χ l ,q n Γ L ˆ˜ q i , ˜ χ l ,q n (cid:17)(cid:21) , (B.3)where the notation ˆ˜ q i indicates that the corresponding squark is in the electroweakeigenbasis. The following couplings are independent of the squark type:Γ ˆ˜ q j ,g, ˜ q n = g ( R ˜ q ) jn , (B.4)Γ ˆ˜ q j ,γ, ˜ q n = e ˜ q e ( R ˜ q ) jn , (B.5)Γ ˆ˜ q j ,Z, ˜ q n = g cos θ W (cid:104) T ˜ q Θ(4 − j ) − e ˜ q sin θ W (cid:105) ( R ˜ q ) jn , (B.6)28 ˆ˜ q ∗ i , ˆ˜ q j ,Z,Z = 2 g cos θ W (cid:16) T ˜ q Θ(4 − i ) − e q sin θ W (cid:17) δ ij , (B.7)Γ ˆ˜ q ∗ i , ˆ˜ q j ,W + ,W − = g δ ij Θ(4 − i ) , (B.8)Γ ˆ˜ q ∗ i , ˆ˜ q j ,H − l ,H + l = (cid:104)(cid:0) D (cid:48) ˜ q − Y d,i Θ(4 − i ) (cid:1) ( R H + l ) − (cid:0) D (cid:48) ˜ q + Y u,i − Θ( i − (cid:1) ( R H + l ) (cid:105) δ ij , (B.9)Γ ˆ˜ q ∗ i , ˆ˜ q j , ˜ ν ∗ l , ˜ ν l = − D ˜ q δ ij , (B.10)Γ L ˆ˜ q j ,q l , ˜ g , = √ g δ i,l e iϕ ˜3 , (B.11)Γ R ˆ˜ q i ,q l , ˜ g = √ g δ i,l +3 e − iϕ ˜3 . (B.12)For the down-squarks we haveΓ ˆ˜ d j ,W + , ˜ u n = g √ δ jk ( V † CKM ) kl ( R † ˜ u ) ln , (B.13)Γ ˆ˜ d i ,h l , ˜ d n = g m W (cid:20) m W ( R h l sin β − R h l cos β ) (cid:104) (1 + tan θ W ) ( R † ˜ d ) in Θ(4 − i )+ tan θ W ( R † ˜ d ) in Θ( i − (cid:105) − R h l cos β (cid:104) ( R † ˜ d ) in m d,i Θ(4 − i ) + ( R † ˜ d ) in m d,i − Θ( i − (cid:105) + R h l cos β (cid:104) µ ∗ ( R † ˜ d ) in m d,i Θ(4 − i ) + µ ( R † ˜ d ) in m d,i − Θ( i − (cid:105) − R h l cos β v d √ (cid:104) ( R † ˜ d ) jn ( ˆ T D ) ji Θ(4 − i ) + ( R † ˜ d ) jn ( ˆ T † D ) j ( i − Θ( i − (cid:105)(cid:21) , (B.14)Γ ˆ˜ d i ,A l , ˜ d n = R A l √ µ ∗ ( R † ˜ d ) in Y d,i Θ(4 − i ) − R A l ( R † ˜ d ) jn ( ˆ T D ) j ( i − Θ( i − , (B.15)Γ ˆ˜ d i ,H + l , ˜ u n = g √ m W (cid:34) − g √ m W sin 2 β Θ(4 − i )( V † CKM ) ij ( R † ˜ u ) jn + tan β Θ(4 − i ) m d,i ( V † CKM ) ij ( R † ˜ u ) jn + µ ∗ Θ( i − m d,i ( R † ˜ u ) in + tan β v d √ i − T D ) ij ( R † ˜ u ) jn + cot β Θ(4 − i )( V † CKM ) ij m u,j ( R † ˜ u ) jn + µ Θ(4 − i ) m u,i ( R † ˜ u ) ( i +3) n + cot β v u √ − i )( ˆ T † U ) ij ( R † ˜ u ) ( j +3) n + (tan β + cot β )Θ( i − m d,i ( V † CKM ) ij m u,j ( R † ˜ u ) ( j +3) t (cid:35) (B.16)29 ˆ˜ d ∗ i , ˆ˜ d j ,h l ,h l = (cid:20) − (cid:18) D ˜ q + 12 Y d,i Θ(4 − i ) + 12 Y d,i − Θ( i − (cid:19) ( R h l ) + D ˜ q ( R h l ) (cid:21) δ ij , (B.17)Γ ˆ˜ d ∗ i , ˆ˜ d j ,A l ,A l = (cid:20) − (cid:18) D ˜ q + 12 Y d,i Θ(4 − i ) + 12 Y d,i − Θ( i − (cid:19) ( R A l ) + D ˜ q ( R A l ) (cid:21) δ ij , (B.18)Γ ˆ˜ d ∗ i , ˆ˜ d j , ˜ e ∗ l , ˜ e l = − g (cid:20) Θ(4 − i ) (cid:18) −
112 tan θ W (cid:19) (cid:88) k =1 ( R † ˜ l ) lk ( R ˜ l ) kl − Θ( i −
3) 16 tan θ W (cid:88) k =1 (cid:16) ( R † ˜ l ) lk ( R ˜ l ) kl − R † ˜ l ) l ( k +3) ( R ˜ l ) ( k +3) l (cid:17) +Θ(4 − i ) 16 tan θ W (cid:88) k =4 ( R † ˜ l ) lk ( R ˜ l ) kl (cid:21) δ ij − (cid:32) (cid:88) k =1 ( R † ˜ l ) lk Y l,k ( R ˜ l ) ( k +3) l (cid:33) Y d,i δ i ( j +3) − (cid:32) (cid:88) k =1 ( R † ˜ l ) lk Y l,k ( R ˜ l ) ( k +3) l (cid:33) ∗ Y d,i δ ( i +3) j (B.19)Γ L ˆ˜ d i , ˜ χ + l ,u n = − g ( V † CKM ) in V l + Y u,n ( V † CKM ) n ( i − V l (B.20)Γ R ˆ˜ d i , ˜ χ + l ,u n = ( V † CKM ) ni Y d,i U l (B.21)Γ L ˆ˜ d j , ˜ χ l ,d n = −√ g (cid:18) − N l −
16 tan θ W N l (cid:19) δ in − Y d,n N l δ i ( n +3) (B.22)Γ R ˆ˜ d i , ˜ χ l ,d n = √ e ˜ u g tan θ W N ∗ l δ i ( n +3) − Y d,n N ∗ l δ in (B.23)Γ ˆ˜ d ∗ i , ˆ˜ d j , ˜ d ∗ l , ˜ d l = 43 g ( R ˜ d ) li ( R † ˜ d ) jl (cid:20) Θ( i − j −
3) + Θ(4 − i )Θ(4 − j ) − Θ( i − − j ) − Θ(4 − i )Θ( j − (cid:21) − g (cid:34)(cid:20) (cid:0) θ W (cid:1) ( R ˜ d ) li ( R † ˜ d ) jl + 3 (cid:88) k =1 (cid:18) (cid:0) θ W (cid:1) ( R ˜ d ) lk ( R † ˜ d ) kl +2 tan θ W ( R ˜ d ) l ( k +3) ( R † ˜ d ) ( k +3) l (cid:19) δ ij (cid:21) Θ(4 − i )Θ(4 − j )+2 tan θ W (cid:20) ( R ˜ u ) li ( R † ˜ u ) jl +3 (cid:88) k =1 (cid:16) ( R ˜ d ) lk ( R † ˜ d ) kl + 2( R ˜ d ) l ( k +3) ( R † ˜ d ) ( k +3) l (cid:17) δ ij (cid:21) Θ( i − j − θ W ( R ˜ d ) li ( R † ˜ d ) jl (Θ(4 − i )Θ( j −
3) + Θ( i − − j )) (cid:35) Y d,i δ i ( j − Θ(4 − i )Θ( j − (cid:88) k =1 Y d,k ( R ˜ d ) lk ( R † ˜ d ) ( k +3) l − Y d,j δ ( i − j Θ(4 − j )Θ( i − (cid:88) k =1 Y d,k ( R ˜ d ) lk ( R † ˜ d ) ( k +3) l − Y u,i Y d,j ( R ˜ d ) l ( i +3) ( R † ˜ d ) ( j +3) l Θ(4 − j )Θ(4 − i ) − Y d, ( i − Y d, ( j − ( R ˜ d ) l ( i − ( R † ˜ d ) ( j − l Θ( j − i −
3) (B.24)Γ ˆ˜ d ∗ i , ˆ˜ d j , ˜ u ∗ l , ˜ u l = − g (cid:20) (cid:18) ( V CKM ) il ( V † CKM ) lj Θ(4 − l ) − δ ij + 112 tan θ W δ ij (cid:19) × Θ(4 − j )Θ(4 − i ) (cid:88) k =1 ( R ˜ u ) lk ( R † ˜ u ) kl + 16 tan θ W δ ij Θ( j − i − (cid:88) k =1 ( R ˜ u ) lk ( R † ˜ u ) kl −
13 tan θ W δ ij Θ(4 − j )Θ(4 − i ) (cid:88) k =4 ( R ˜ u ) lk ( R † ˜ u ) kl −
23 tan θ W δ ij Θ( j − i − (cid:88) k =4 ( R ˜ u ) lk ( R † ˜ u ) kl (cid:21) − (cid:88) k =1 (cid:16) ( R † ˜ u ) lk Y u,k ( R ˜ u ) ( k +3) l (cid:17) Y d,i δ i ( j +3) − (cid:88) k =1 (cid:16) ( R † ˜ u ) lk Y u,k ( R ˜ u ) ( k +3) l (cid:17) ∗ Y d,i δ ( i +3) j (B.25)whereas for the up-squarks we haveΓ ˆ˜ u j ,W + , ˜ d n = g √ δ jk ( V CKM ) kl ( R † ˜ d ) ln , (B.26)Γ ˆ˜ u ∗ i , ˆ˜ u j ,h l ,h l = (cid:20) − D ˜ q ( R h l ) + (cid:18) D ˜ q − Y u,i Θ(4 − i ) − Y u,i − Θ( i − (cid:19) ( R h l ) (cid:21) δ ij , (B.27)Γ ˆ˜ u ∗ i , ˆ˜ u j ,A l ,A l = (cid:20) − D ˜ q ( R A l ) + (cid:18) D ˜ q − Y u,i Θ(4 − i ) − Y u,i − Θ( i − (cid:19) ( R A l ) (cid:21) δ ij , (B.28)31 ˆ˜ u i ,h l , ˜ u n = − g m W (cid:20) m W ( R h l sin β − R h l cos β ) (cid:104) (1 − tan θ W )( R † ˜ u ) in Θ(4 − i )+ tan θ W ( R † ˜ u ) in Θ( i − (cid:105) + 2 R h l sin β (cid:104) ( R † ˜ u ) in m u,i Θ(4 − i ) + ( R † ˜ u ) in m u,i − Θ( i − (cid:105) − R h l sin β (cid:104) µ ∗ ( R † ˜ u ) in m u,i Θ(4 − i ) + µ ( R † ˜ u ) in m u,i Θ( i − (cid:105) + R h l sin β v u √ (cid:104) ( R † ˜ u ) jn ( ˆ T U ) ji Θ(4 − i ) + ( R † ˜ u ) jn ( ˆ T † U ) j ( i − Θ( i − (cid:105)(cid:21) , (B.29)Γ ˆ˜ u i ,A l , ˜ u n = − R A l √ µ ∗ ( R † ˜ u ) in Y u i Θ(4 − i ) + R A l ( R † ˜ u ) jn ( ˆ T U ) ji Θ( i −
3) (B.30)Γ ˆ˜ u i ,H + l , ˜ d n = g √ m W (cid:34) − m W sin 2 β Θ(4 − i )( V CKM ) ij ( R † ˜ d ) jn + tan β Θ(4 − i )( V CKM ) ij m d,j ( R † ˜ d ) jn + µ Θ(4 − i ) m d,i ( R † ˜ d ) ( i +3) n + tan β v d √ − i )( ˆ T † D ) ij ( R † ˜ d ) ( j +3) n + cot β Θ(4 − i ) m u,i ( V CKM ) ij ( R † ˜ d ) jn + µ ∗ Θ( i − m u,i ( R † ˜ d ) in + cot β v u √ i − T U ) ij ( R † ˜ d ) jn + (tan β + cot β )Θ( i − m u,i ( V CKM ) ij m d,j ( R † ˜ d ) ( j +3) t (cid:35) (B.31)Γ ˆ˜ u ∗ i , ˆ˜ u j , ˜ e ∗ l , ˜ e l = − g (cid:20) − Θ(4 − i ) (cid:18)
112 tan θ W + 14 (cid:19) (cid:88) k =1 ( R † ˜ l ) lk ( R ˜ l ) kl +Θ( i −
3) 13 tan θ W (cid:88) k =1 (cid:16) ( R † ˜ l ) lk ( R ˜ l ) kl − R † ˜ l ) l ( k +3) ( R ˜ l ) ( k +3) l (cid:17) +Θ(4 − i ) 16 tan θ W (cid:88) k =4 ( R † ˜ l ) lk ( R ˜ l ) kl (cid:21) δ ij (B.32)Γ L ˆ˜ u i , ˜ χ + l ,d n = − g ( V CKM ) in U l + ( V CKM ) n ( i − Y d, ( i − U l (B.33)Γ R ˆ˜ d i , ˜ χ + l ,d n = Y u,n ( V CKM ) ni V l (B.34)Γ L ˆ˜ u j , ˜ χ l ,u n = −√ g (cid:18) N l −
16 tan θ W N l (cid:19) δ in − Y u,n N l δ i ( n +3) (B.35)Γ R ˆ˜ u i , ˜ χ l ,u n = √ e ˜ u g tan θ W N ∗ l δ i ( n +3) − Y u,n N ∗ l δ in (B.36)32 ˆ˜ u ∗ i , ˆ˜ u j , ˜ d ∗ l , ˜ d l = − g (cid:20) (cid:18) ( V † CKM ) li ( V CKM ) jl Θ(4 − l ) − δ ij + 112 tan θ W δ ij (cid:19) × Θ(4 − j )Θ(4 − i ) (cid:88) k =1 ( R ˜ d ) lk ( R † ˜ d ) kl −
13 tan θ W δ ij Θ( j − i − (cid:88) k =1 ( R ˜ d ) lk ( R † ˜ d ) kl + 16 tan θ W δ ij Θ(4 − j )Θ(4 − i ) (cid:88) k =4 ( R ˜ d ) lk ( R † ˜ d ) kl −
23 tan θ W δ ij Θ( j − i − (cid:88) k =4 ( R ˜ d ) lk ( R † ˜ d ) kl (cid:21) − (cid:88) k =1 (cid:16) ( R † ˜ d ) lk Y d,k ( R ˜ d ) ( k +3) l (cid:17) Y u,i δ i ( j +3) − (cid:88) k =1 (cid:16) ( R † ˜ d ) lk Y d,k ( R ˜ d ) ( k +3) l (cid:17) ∗ Y u,i δ ( i +3) j (B.37)Γ ˆ˜ u ∗ i , ˆ˜ u j , ˜ u ∗ l , ˜ u l = 43 g ( R ˜ u ) li ( R † ˜ u ) jl (cid:20) Θ( i − j −
3) + Θ(4 − i )Θ(4 − j ) − Θ( i − − j ) − Θ(4 − i )Θ( j − (cid:21) − g (cid:34)(cid:20) (cid:0) θ W (cid:1) ( R ˜ u ) li ( R † ˜ u ) jl + 3 (cid:88) k =1 (cid:18) (cid:0) θ W (cid:1) ( R ˜ u ) lk ( R † ˜ u ) kl − θ W ( R ˜ u ) l ( k +3) ( R † ˜ u ) ( k +3) l (cid:19) δ ij (cid:21) Θ(4 − i )Θ(4 − j )+4 tan θ W (cid:20) R ˜ u ) li ( R † ˜ u ) jl +3 (cid:88) k =1 (cid:16) − ( R ˜ u ) lk ( R † ˜ u ) kl + 4( R ˜ u ) l ( k +3) ( R † ˜ u ) ( k +3) l (cid:17) δ ij (cid:21) Θ( i − j − − θ W ( R ˜ u ) li ( R † ˜ u ) jl (Θ(4 − i )Θ( j −
3) + Θ( i − − j )) (cid:35) − Y u,i δ i ( j − Θ(4 − i )Θ( j − (cid:88) k =1 Y u,k ( R ˜ u ) lk ( R † ˜ u ) ( k +3) l − Y u,j δ ( i − j Θ(4 − j )Θ( i − (cid:88) k =1 Y u,k ( R ˜ u ) lk ( R † ˜ u ) ( k +3) l − Y u,i Y u,j ( R ˜ u ) l ( i +3) ( R † ˜ u ) ( j +3) l Θ(4 − j )Θ(4 − i ) − Y u, ( i − Y u, ( j − ( R ˜ u ) l ( i − ( R † ˜ u ) ( j − l Θ( j − i −
3) (B.38)33ithΘ( i ) = (cid:26) , i > , i ≤ D ˜ q = g θ W (cid:104)(cid:16) T ˜ q − e ˜ q sin θ W (cid:17) Θ(4 − i ) + e ˜ q sin θ W Θ( i − (cid:105) , (B.40) D (cid:48) ˜ q = g θ W (cid:104)(cid:16) T ˜ q cos 2 θ W + e ˜ q sin θ W (cid:17) Θ(4 − i ) − e ˜ q sin θ W Θ( i − (cid:105) , (B.41) h l = ( h , H ) , (B.42) R h = (cid:18) − sin α cos α cos α sin α (cid:19) , (B.43) A l = ( G , A ) , (B.44) R A = (cid:18) cos β sin β − sin β cos β (cid:19) , (B.45) H + l = ( G + , H + ) , (B.46) R H + = R A . (B.47) U, V are the chargino mixing matrices, N is the neutralino mixing matrix, Y d , Y u and Y l are the usual fermion Yukawa couplings. We are summing over repeated indicesand usually it is assumed that in case of 3 × A , B , F and G can be found inappendix B of ref. [38], where also the tree-level masses for charginos, neutralinos,and the Higgs-bosons are given. The slepton and sneutrino masses including flavourviolation can be found, e.g., in refs. [39, 40]. References [1] G. Weiglein et al. [LHC/LC Study Group], Phys. Rept. (2006) 47 [arXiv:hep-ph/0410364].[2] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B (2002)155 [arXiv:hep-ph/0207036].[3] J. F. Donoghue, H. P. Nilles and D. Wyler, Phys. Lett. B (1983) 55.[4] L. J. Hall, V. A. Kostelecky and S. Raby, Nucl. Phys. B (1986) 415.[5] F. Gabbiani, A. Masiero, Nucl. Phys.
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