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cp violation , initially observed  @xcite only in the @xmath7@xmath8 system , is one feature of the standard model ( sm ) that still defies clear theoretical understanding . the ckm picture , which describes _ all _ the _ observed _ cp violation in terms of a single phase in the quark - mixing matrix , has been vindicated by the recent measurements of @xmath9@xmath10 mixing at belle and babar  @xcite . cp violation is in fact one of the necessary ingredients for generating the observed excess of baryons over antibaryons in the universe @xcite . the amount of cp violation present in the quark sector is , however , too small to generate a baryon asymmetry of the observed level of @xmath11  @xcite . new sources of cp violation _ beyond _ the sm are therefore a necessity  @xcite . supersymmetry ( susy ) is arguably the most attractive extension of the sm , as it solves , for instance , the problem of the instability of the electroweak symmetry - breaking scale against radiative corrections . already the minimal supersymmetric standard model ( mssm ) @xcite provides possible new sources of cp violation through additional cp - violating phases , which can not be rotated away by simple field redefinitions . a large number of these phases , particularly those involving sparticles of the first and to a large extent of the second generation , are severely constrained by measurements of the electric dipole moments ( edms ) of the electron , muon , neutron as well as @xmath12hg and @xmath13tl . however , these constraints are model - dependent . it has been demonstrated @xcite that cancellations among different diagrams allow certain combinations of these phases to be large in a general mssm . furthermore , if the sfermions of the first two generations are sufficiently heavy , above the 1 tev range , the edm constraints on the phase of the higgsino mass parameter @xmath14 , in general constrained to @xmath15 , get weaker ; the sfermions of the third generation can still be light . non - vanishing phases of @xmath16 and/or the trilinear scalar couplings @xmath17 can induce explicit cp violation in the higgs sector via loop corrections @xcite . though these phases generate edms independently of the first two generations of sfermions , the edms are suppressed by the mass scale of the two heavy higgses @xcite . for a thorough discussion of the edms see @xcite and references therein . the above mentioned phases can also have a significant influence on the higgs production rates in the gluon fusion mode at the tevatron and the lhc @xcite . mssm cp phases can hence change the higgs phenomenology at colliders quite substantially . all this makes the mssm with cp - violating phases a very attractive proposition . it has therefore been the subject of many recent investigations , studying the implications of these phases on neutralino / chargino production and decay @xcite , on the third generation of sfermions @xcite , as well as the neutral @xcite and charged @xcite higgs sector . various cp - even and cp - odd ( t - odd ) observables , which can give information on these phases , have been identified . it is interesting to note that cp - even observables such as masses , branching ratios , cross sections , etc . , often afford more precise probes thanks to the larger magnitude of the effects . for direct evidence of cp violation , however , cp - odd / t - odd observables as discussed e.g.  in @xcite have to be measured . the latest study of the @xmath18 sector in @xcite demonstrates that it may be possible to determine the real and imaginary parts of @xmath19 to a precision of 23% from a fit of the mssm lagrange parameters to masses , cross sections and branching ratios at a future @xmath20 linear collider ( lc ) . this requires that both the @xmath21 , @xmath22 mass eigenstates can be produced at the lc and the branching ratios measured with high precision . in the @xmath23 sector @xcite the precision on @xmath24 is worse , around 1020% for low @xmath25 and about 37% for large @xmath25 .    in this paper , we show that the longitudinal polarization of fermions produced in sfermion decays , i.e.  @xmath26 and @xmath27 with @xmath28 a third generation ( s)quark or ( s)lepton , can also be used as a probe of cp phases . the fermion polarization can give complementary information to the decay branching ratios and will in particular be useful if the branching ratios can not be measured with high enough precision or if one decay channel dominates . the average polarization of fermions produced in sfermion decays carries information on the @xmath29@xmath30 mixing as well as on the gaugino  higgsino mixing @xcite . the polarizations that can be measured are those of top and tau ; both can be inferred from the decay lepton distributions . it is its large mass that causes the @xmath31 to decay before hadronization and thus the decay products can carry information about its polarization . for taus , also the energy distribution of the decay pions can be used . the polarization of the decay fermions has been used for studies of mssm parameter determination in the cp - conserving case in @xcite . for the cp - violating case , the phase dependence of the longitudinal fermion polarization has been mentioned in @xcite . we extend these studies by discussing in detail the sensitivity of the fermion polarization to cp - violating phases in the mssm . the paper is organized as follows : in section 2 , we summarize our notation for the description of the sfermion , neutralino and chargino systems in the mssm with cp violation . in section  3 , we discuss fermion polarization in sfermion decays to neutralinos , @xmath32 with @xmath33 . we present numerical results on the polarization as a function of different mssm parameters and discuss the sensitivity to cp - violating phases in the sfermion and neutralino sectors . in section 4 we perform an analogous analysis for @xmath34 decays . in section  5 we summarize the results and present our conclusions . ignoring intergenerational mixing , the sfermion mass matrices can be written as a series of @xmath35 matrices , each of which describes sfermions of a specific flavour : @xmath36 with @xmath37    { m_{\sf_{r}}}^2 & = & m^2_{\ti r }      + e_f\,m_z^2 \cos 2\b\,\sin^2\t_w + m_f^2 , \\[1 mm ]    a_f & = & a_f - \mu^*\ , \ { \cot\b , \tan\b \ }            = |a_f|\ , e^{i\phsf } \ , ,    \label{eq : aq}\end{aligned}\ ] ] for @xmath38up , down@xmath39-type sfermions ; @xmath40 , @xmath41 and @xmath42 are the mass , electric charge and the third component of the weak isospin of the partner fermion , respectively ; @xmath43 , @xmath44 and @xmath45 are soft susy - breaking parameters for each family , and @xmath16 is the higgsino mass parameter ; @xmath45 and @xmath16 can have complex phases : @xmath46 and @xmath47 . + according to eq . , @xmath48 is diagonalized by a unitary rotation matrix @xmath49 . the weak eigenstates @xmath50 and @xmath51 are thus related to their mass eigenstates @xmath52 and @xmath53 by @xmath54 with @xmath55 and @xmath56 the sfermion mixing angle and phase . since the off - diagonal element of @xmath48 is proportional to @xmath40 , this mixing is mostly relevant to the third generation , @xmath57 , on which we concentrate in the following . the mass eigenvalues are given by @xmath58 by convention , we choose @xmath52 to be the lighter mass eigenstate , @xmath59 . notice also that @xmath60 . for the mixing angle @xmath55 we choose @xmath61 which places @xmath55 in the 2nd quadrant of the unit circle . the @xmath62@xmath63 mixing is large if @xmath64 , with @xmath65 if @xmath66 and @xmath67 if @xmath68 . moreover , we see that the phase dependence of @xmath69 and @xmath49 is determined by @xmath70 . this dependence is strongest if @xmath71 . this issue will be discussed in more detail in the numerical analyses of sections  3 and 4 .      in the basis @xmath72 the neutralino mass matrix is : @xmath73    the gaugino mass parameters @xmath74 and the higgsino mass parameter @xmath16 can in principle all be complex . the phase of @xmath75 can be rotated away , which leaves us with two phases in this sector : @xmath76 , the phase of @xmath77 , and @xmath78 , the phase of @xmath16 . + the matrix of eq . is diagonalized by the unitary mixing matrix @xmath79 : @xmath80 where @xmath81 , @xmath82 , are the ( non - negative ) masses of the physical neutralino states . we choose the ordering @xmath83 . a concise discussion of the neutralino sector with complex phases can be found in @xcite . the chargino mass matrix is : @xmath84 it is diagonalized by the two unitary matrices @xmath85 and @xmath86 , @xmath87 where @xmath88 are the masses of the physical chargino states with @xmath89 . the sfermion interaction with neutralinos is ( @xmath91 ; @xmath82 ) @xmath92 where @xmath93 the @xmath94 and @xmath95 couplings are @xmath96 for stops and sbottoms , and @xmath97 for staus , with the yukawa couplings @xmath98 given by @xmath99    the gaugino interaction conserves the helicity of the sfermion while the higgsino interaction flips it . in the limit @xmath100 , the average polarization of the fermion coming from the @xmath101 decay can therefore be calculated as @xcite @xmath102    using eqs .  , and as well as @xmath103 , we obtain , for the @xmath104 decay ( omitting the overall factor @xmath105 and dropping the sfermion and neutralino indices for simplicity ) : @xmath106 \,.\end{aligned}\ ] ] we see that the phase dependence of @xmath107 is the largest for maximal sfermion mixing ( @xmath108 ) and if the neutralino has both sizeable gaugino and higgsino components . it is , moreover , enhanced if the yukawa coupling @xmath98 is large . furthermore , @xmath107 is sensitive to cp violation even if just one phase , in either the neutralino or the sfermion sector , is non - zero . in particular , if only @xmath45 and thus only the sfermion mixing matrix has a non - zero phase , the phase - dependent term becomes @xmath109 if , on the other hand , only @xmath76 is non - zero we get @xmath110\,\sin 2\t \,.\ ] ]    the polarization @xmath111 , eq . , depends only on couplings but not on masses . for the numerical analysis we therefore use @xmath77 , @xmath75 , @xmath16 , @xmath112 , @xmath55 and @xmath113 as input parameters , assuming @xmath114 to satisfy edm constraints more easily : assuming cancellations for the 1-loop contributions and the cp - odd higgs mass parameter @xmath115  gev , 1-loop and 2-loop contributions to the electron edm ( eedm ) , as well as their sum , stay below the experimental limit . we use the formulae of @xcite for the 2-loop contributions . in order not to vary too many parameters , we use , moreover , the gut relation @xmath116 and choose @xmath117 and @xmath118 ( large but not maximal mixing ) throughout this section . the free parameters in our analysis are thus @xmath75 , @xmath119 , and the phases @xmath76 , @xmath113 . before we present the numerical results , a comment is in order : cp violation in the neutralino sector is determined by the phases of @xmath77 and @xmath16 , while @xmath113 originates from relative phases of @xmath45 and @xmath16 . for stops the mixing is dominated by @xmath19 , while for sbottoms and staus it is dominated by @xmath120 ; quite generally we have @xmath121 unless @xmath122 , and @xmath123 unless @xmath124 . more precisely , @xmath125 for @xmath126 , any @xmath113 can be reached by an appropriate choice of @xmath127 , independent of @xmath128 . for @xmath129 , however , @xmath113 is restricted by @xmath128 . in the special case of @xmath130 and @xmath131 , @xmath132 . in the stop sector this is not a problem since @xmath133 can in general be easily achieved . for sbottoms and staus , choosing @xmath134 freely leads , however , to quite large @xmath135 , which may in some cases create problems with charge- or colour - breaking minima . ( 148,66 ) ( 0,0 ) ( 30,0 ) ( 104,0 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 ) ( 16,14 ) ( 90,14 )    ( 148,72 ) ( 0,0 ) ( 30,0 ) ( 104,0 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 ) ( 16,14 ) ( 110,14 )    shows the average tau polarization in @xmath136 decays as functions of @xmath75 and @xmath119 for @xmath117 , @xmath137 and various choices of @xmath76 and @xmath138 . the lower limits of @xmath75 and @xmath119 are given by the lep2 constraint of @xmath139  gev @xcite , which automatically takes care of all other lep constraints on the gaugino higgsino sector . as can be seen , @xmath140 is quite sensitive to cp phases for @xmath141 , that is if the @xmath142 has a sizeable higgsino component . analogously , shows the average top polarization in @xmath143 decays . we observe again a strong dependence on the cp phases if the neutralino has a sizeable higgsino component . unlike the case of @xmath140 , for @xmath144 the dependence is still significant when @xmath145 . we also note that some phase combinations lead to very similar polarizations , e.g.  @xmath146 and @xmath147 . at a future @xmath20 linear collider ( lc ) , one expects to be able to measure the tau polarization to about 35% and the top polarization to about 10% @xcite . we see from figs . [ fig : ptau_m2mu ] and [ fig : ptop_m2mu ] that the effects of cp - violating phases may well be visible in @xmath144 and/or @xmath140 , provided @xmath16 is not too large . we next choose specific values of @xmath75 and @xmath119 to discuss the phase dependences in more detail . a shows @xmath140 as a function of @xmath76 , for @xmath148  gev , @xmath149  gev and @xmath150 , @xmath151 , @xmath152 and @xmath153 . since for fixed @xmath75 and @xmath119 the @xmath142 mass changes with @xmath76 , we show in addition in b @xmath140 as a function of @xmath154 for various values of @xmath76 , with @xmath149  gev and @xmath75 adjusted such that @xmath155  gev . @xmath140 varies over a large range depending on @xmath76 and @xmath154 ; if the neutralino mass parameters , @xmath25 and @xmath156 are known , @xmath140 can hence be used as a sensitive probe of these phases ( although additional information will be necessary to resolve ambiguities and actually determine the various phases ) . at a lc , the parameters of the neutralino / chargino sector and also sfermion masses and mixing angles can be determined very precisely , exploiting tunable beam energy and beam polarization @xcite . the actual precision depends of course on the specific scenario . to illustrate the influence of uncertainties in the knowledge of the model parameters , we take the case of @xmath148  gev , @xmath149  gev and vanishing phases as reference point and assume that the following precisions can be achieved : @xmath157 , @xmath158 , @xmath159 , and @xmath160 . varying the parameters within this range around the reference point leads to @xmath161 at @xmath150 , which is indicated as an error bar in b. ( the 35% uncertainty in the measurement of @xmath162 is comparatively negligible ) . we conclude that in our particular scenario , if no phase has been observed in the neutralino / chargino sector , a measurement of @xmath140 would be sensitive to @xmath163 . if @xmath25 can be measured to @xmath164 , this improves to @xmath165 and @xmath166 . according to , a measurement of a non - zero @xmath167 implies a lower limit on @xmath168 ; in our example where @xmath131 , @xmath169  gev ( 1  tev ) for @xmath170 ( @xmath171 ) . increasing the precision in @xmath172 , @xmath172 and @xmath173 from 0.5% to 0.1% barely improves these limits . ( 148,72 ) ( 0,0 ) ( 36,-1 ) ( 109,-1 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 )    ( 148,72 ) ( 0,0 ) ( 36,-1 ) ( 108,-1 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 )    we perform a similar analysis for @xmath144 , using @xmath174  gev and @xmath175  gev as reference point . the results are shown in a , b in analogy to a , b . again a high sensitivity to both @xmath76 and @xmath176 is observed . for the case of vanishing phases , we get @xmath177 . a variation of the parameters around the reference point as above ( with @xmath178 ) leads to a parametric uncertainty of @xmath179 . adding the experimental resolution @xmath180 in quadrature gives @xmath181 at @xmath182 , indicated as an error bar in b. we see that in this scenario @xmath144 would be sensitive to @xmath183 . if @xmath184 can be measured to @xmath185 this improves to @xmath186 ( @xmath187 ) and @xmath188 ; if @xmath77 , @xmath75 , @xmath119 can be measured to 0.1% and @xmath25 to @xmath189 , @xmath190 becomes negligible with respect to the experimental resolution of @xmath191 . since @xmath192 , a measurement of @xmath144 can be used to derive information on @xmath19 . in particular , if both mass eigenstates are known , @xmath19 is given by @xmath193 an analogous relation with @xmath194 holds for @xmath195 , although the precision on @xmath24 is in general much worse than on @xmath19 . in this context note that @xmath111 can also be useful to resolve the sign ambiguity in the @xmath196 determination from cross section measurements @xcite in the cp - conserving case . this corresponds to distinguishing the cases @xmath197 and @xmath198 . even though we have presented results of our analysis for @xmath199 , chosen in order to satisfy the edm constraints without having to appeal to cancellations , we have also investigated the case of a non - zero @xmath128 . we found that a non - zero @xmath128 shifts the curves in [fig : ptop_phases ] but does not cause a qualitative change of the results . ( 148,72 ) ( 0,0 ) ( 38,-1 ) ( 112,-1 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 )    last but not least we note that giving up the gut relation between @xmath200 and @xmath75 changes the picture completely , as the pattern of gaugino  higgsino mixing is strongly affected @xcite . this is illustrated in , where we plot @xmath140 and @xmath144 as functions of @xmath201 for @xmath202  gev , @xmath203  gev and the other parameters as in and [ fig : ptop_m2mu ] . a detailed study of the implications of non - universal gaugino masses will be presented elsewhere . + _ to sum up , _ both @xmath144 and @xmath140 can vary over a large range depending on @xmath76 and @xmath204 ( and also @xmath128 , though we did not discuss this case explicitly ) and may thus be used as sensitive probes of these phases . to this aim , however , the neutralino mass parameters , @xmath25 and the sfermion mixing angles need to be known . given the complexity of the problem , a combined fit of all available data seems to be the most convenient method for the extraction of the mssm parameters . the sfermion interaction with charginos is ( @xmath206 ) @xmath207 where @xmath208 ( @xmath209 ) stands for up - type ( s)quark and ( s)neutrinos , and @xmath210 ( @xmath211 ) stands for down - type ( s)quark and charged ( s)leptons . the couplings @xmath212 and @xmath213 are @xmath214 for stops and sbottoms and @xmath215 for staus and sneutrinos . analogous to the decay into a neutralino , eq . , the average polarization of the fermion coming from the @xmath216 decay is given by @xmath217    since only top and tau polarizations are measurable , we only discuss @xmath218 and @xmath219 decays . the latter case is especially simple because @xmath220 depends only on the parameters of the chargino sector : @xmath221 a measurement of @xmath220 may hence be useful to supplement the chargino parameter determination . however , the dependence of @xmath222 on @xmath199 turns out to be very small , the effects being in general well below 1% ( i.e. @xmath223 ) . only for the decay into the heavier chargino , the effect of a non - zero phase is severely restricted by the non - observation of the eedm . ] may be sizeable . as an example , shows the differences in @xmath224 between @xmath131 and @xmath225 in the @xmath226 plane for @xmath227  gev . @xmath228 can go up to @xmath229 . however , it requires quite heavy sneutrinos for this decay to be kinematically allowed . moreover , the measurement of @xmath224 will be diluted by @xmath230 decays . +    ( 70,74 ) ( 0,0 ) ( 27,-3 ) ( -2,71 ) ( 7,60 ) ( 36,60 ) ( 22,50 ) ( 24,38 ) ( 28,26.5 ) ( 34,18 )    let us now turn to top polarization in @xmath218 decays . for @xmath231 decays , we have @xmath232 \cos\varphi         + { { \cal i}m\ , } [ u_{j1}^ { } u_{j2}^ { * } ] \sin\varphi ) \ , . \label{eq : sbot1}\end{aligned}\ ] ] for @xmath233 decays , the corresponding expression is given by the rhs of with @xmath234 , @xmath235 interchanged , and a change in sign of the term @xmath236 . we see that the phase dependence of @xmath237 is proportional to @xmath238 and the amount of gaugino  higgsino mixing of the charginos ; it will therefore be largest for @xmath239 , @xmath240 and large @xmath112 . again , there is a non - zero effect even if there is just one phase in either the sbottom or chargino sector . note , however , that the only cp phase in the chargino sector is @xmath128 , which also enters the sfermion mass matrices . complex @xmath85 and @xmath86 hence imply @xmath241 . more precisely , @xmath242 for medium and large @xmath112 , and thus @xmath243 unless @xmath244 ; see eq .   and the related discussion . for the sake of a general discussion of the phase dependence of @xmath237 ( and since @xmath245 is still a free parameter ) , we nevertheless use @xmath199 and @xmath246 as independent input parameters .    shows the average top polarization in @xmath247 decays as a function of @xmath119 for @xmath248  gev , @xmath249 and 30 , and various combinations of @xmath199 and @xmath246 . here we have fixed @xmath250 , since from renormalization - group running one expects @xmath251 . as in the previous section , we find large effects from cp - violating phases if the @xmath252 has a sizeable higgsino component ; as expected , these effects are enhanced for large @xmath25 . the results stay the same if both @xmath199 and @xmath246 change their signs . moreover , @xmath253 . if @xmath199 and @xmath246 have the same sign , the difference in @xmath237 from the case of vanishing phases is larger than if they have opposite signs . in particular , we find @xmath254 over large regions of the parameter space . with an experimental resolution of the top polarization of about 10% this implies that in many cases @xmath255 can not be distinguished from @xmath256 by measurement of @xmath237 . furthermore , the value of @xmath237 is quite sensitive to the running @xmath257 quark mass , which enters the bottom yukawa coupling of eq .   and is subject to possibly large susy loop corrections . for the lines in we have used @xmath258  gev the grey bands show the range of @xmath237 when @xmath259 is varied between 2.5 and 4.5 gev . as can be seen , the uncertainty in @xmath259  more precisely in @xmath260  tends to wash out small effects of cp - violating phases , specially in the case of large @xmath25 . ( 148,72 ) ( 0,0 ) ( 32,-1 ) ( 104,-1 ) ( -2,37 ) ( 72,37 ) ( -2,64 ) ( 72,64 ) ( 30,67 ) ( 102,67 )    ( 148,72 ) ( 5,3.3 ) ( 81,2 ) ( 33,-1 ) ( 112,-1 ) ( -4,37 ) ( 73,37 ) ( -4,63 ) ( 73,63 ) ( 14,68 ) ( 92,68 )    in order to see what information can be extracted from a @xmath237 measurement , we pick two values of @xmath119 from a ; namely @xmath261  gev and @xmath262  gev , and show the phase dependences at these points in . a shows @xmath237 as a function of @xmath246 , for @xmath261  gev , @xmath249 , @xmath263 , and various values of @xmath128 . @xmath75 is chosen such that @xmath264  gev ( i.e.  @xmath174  gev for @xmath131 ) . the range obtained by varying @xmath259 within 2.54.5  gev is shown as grey bands for two of the curves , for @xmath131 and @xmath265 . we estimate the effect of an imperfect knowledge of the model parameters in the same way as in the previous section . for @xmath266 gev , @xmath267  gev , @xmath268 , @xmath269 and @xmath270 , we get @xmath271 at @xmath272 . varying in addition @xmath2734.5 gev gives @xmath274 . adding a 10% measurement error on @xmath237 in quadrature , we end up with @xmath275 ( @xmath276 ) without ( with ) the @xmath259 effect . these are shown as error bars in a. we see that the case of @xmath277 can not be distinguished from @xmath278 in this scenario . however , @xmath237 turns out to be quite a sensitive probe of @xmath279 , i.e.  the deviation from the ` natural ' alignment @xmath277 . in the example of a , @xmath280 @xmath281 can be resolved if @xmath260 is ( not ) known precisely , quite independently of @xmath199 . observing such a @xmath282 also implies a bound on @xmath283 of @xmath284 @xmath285  gev . if the precision on @xmath75 and @xmath119 is 0.1% and @xmath164 , we get @xmath286 at @xmath272 , so that the error is dominated by the experimental uncertainty . however , the resultant improvement in the sensitivity is limited to @xmath287 and @xmath288  gev . b shows @xmath237 as a function of @xmath246 , for @xmath262  gev , @xmath289  gev , and the other parameters as above ( @xmath174  gev at @xmath131 ) . the effect of an uncertainty in @xmath260 is again shown as grey bands for @xmath131 and @xmath265 . estimating the parametric uncertainty in the same way as above , but with @xmath290  gev , we get @xmath291 at @xmath272 . varying in addition @xmath2734.5 gev gives @xmath292 . adding a 10% measurement error on @xmath237 in quadrature , we end up with @xmath293 ( @xmath294 ) without ( with ) the effect of @xmath259 , shown as error bars in b. in a three - dimensional plot , @xmath237 has a bell - like shape in the @xmath128@xmath246 plane , with contours of constant @xmath237 being ellipses in this plane . if @xmath199 is not known , a measurement of @xmath237 may therefore be useful to put limits on @xmath128 and @xmath246 , but not on @xmath282 , which restricts @xmath283 . in our case study , we have assumed that @xmath295 is known . in this case , a measurement of @xmath296 , for instance , would restrict @xmath297 at @xmath298 , while a measurement of @xmath299 would disfavour @xmath300 as well as @xmath301 . the latter would also allow a constraint on @xmath282 . as mentioned above , a lower limit on @xmath282 implies a lower limit on @xmath283 . an upper limit on @xmath302 can be used to set an upper limit on @xmath283 as a function of @xmath303 : @xmath304 . note , however , that this becomes unbounded for @xmath305 . we have also investigated the case of large @xmath25 ( @xmath306 ) . it reveals a @xmath246 dependence similar to that of a , with almost no dependence on @xmath199 and the @xmath307 dependence accordingly more pronounced . we encounter , however , a large parametric uncertainty , which practically washes out the sensitivity to @xmath302 . + _ to sum up , _ tau polarization in @xmath219 decays depends only little on @xmath199 . @xmath220 is hence not a promising quantity to study cp phases , but may be useful for ( consistency ) tests of the gaugino  higgsino mixing . top polarization in @xmath218 decays , on the other hand , can be useful to probe @xmath199 , @xmath246 and/or @xmath308 in some regions of the parameter space . the measurement of @xmath237 , revealing phases or being consistent with vanishing phases , may also constrain @xmath283 . we have discussed the influence of cp - violating phases on the fermion polarization in sfermion decays to neutralinos or charginos , @xmath309 and @xmath310 ( @xmath206 ; @xmath82 ; @xmath311 ) . this polarization is considered as a useful tool for the mssm parameter determination @xcite .    in decays into charginos , the polarization depends on the phase of @xmath16 . since this dependence is weak in the case of @xmath219 , @xmath220 does not provide a promising probe of cp phases ( on the other hand , exactly this feature can make @xmath220 useful for consistency tests of gaugino  higgsino mixing ) . in @xmath312 decays , the dependence on @xmath199 can be rather large ; in addition , also the phase of the sbottom - mixing matrix plays a role . if @xmath313 , @xmath314 . we found that this case can be difficult to distinguish from the cp - conserving case by measuring @xmath315 . if , however , a deviation from @xmath316 is observed , these phases can be constrained and also limits on @xmath245 can be derived . the decays @xmath317 and @xmath318 provide a more effective probe of cp violation because an additional phase , the phase of @xmath77 , contributes . we found that cp phases can have a significant effect on the top and tau polarizations , especially if the involved neutralino has a sizeable higgsino component . if the parameters of the neutralino sector can be measured precisely , e.g.  in @xmath20 annihilation with polarized beams , @xmath319 and @xmath320 can be useful for the determination of cp phases . in particular , since @xmath192 unless @xmath119 is very large , a measurement of @xmath144 can give information on @xmath19 . in this respect it is important to note that ( for fixed masses ) the sfermion production cross sections do not depend on cp phases . in the sfermion sector , these can be manifest in branching ratios as discussed in @xcite , polarization of the decay fermions as discussed in this paper , and cp - odd asymmetries . branching ratios are in general rather difficult to measure with high precision . the information that can be gained from branching ratios is also limited if one decay channel dominates , e.g. @xmath136 in case of a light stau . this makes the polarization of the decay fermions a very interesting possibility to explore cp phases . last but not least we note that the computations in this paper , leading to effects of a few percent , have been performed at tree level . the influence of radiative corrections @xcite can be of comparable size and will therefore have to be taken into account for precision analyses . a measurement of the cp phases in the sfermion/@xmath321 sector will also complement cp studies of the higgs sector @xcite , since in the mssm higgs - sector cp violation is generated through quantum corrections @xcite . last but not least we emphazise that , since the effects can be large , the possibility of cp violation should be taken into account in precision susy parameter analyses , especially in a general analysis project as envisaged in @xcite . t.g . acknowledges the financial support of the institut fr hochenergiephysik in vienna and the cern theory division . r.g . wishes to thank the cern theory division and lapth for hospitality and financial support , where this work was completed . thanks the centre for high energy physics in bangalore for hospitality and financial support . rg wishes to acknowledge the partial support of the department of science and technology , india , under project number sp / s2/k-01/2000-ii . the sm constants used in the numerical analysis are : @xmath322          a.  d.  sakharov , pisma zh . teor .  fiz . * 5 * ( 1967 ) 32 , jetp lett . 6 , 24 ( 1967 ) . for a recent summary , see a.  d.  dolgov , hep - ph/0211260 . c.  l.  bennett _ et al . _ , astrophys . j.  suppl .   * 148 * , 1 ( 2003 ) [ astro - ph/0302207 ] . for a review , see e.g. m.  dine and a.  kusenko , rev . phys .   * 76 * , 1 ( 2004 ) [ hep - ph/0303065 ] . see , e.g.h . e.  haber and g.  l.  kane , phys . rept .   * 117 * ( 1985 ) 75 . t.  ibrahim and p.  nath , phys . b * 418 * , 98 ( 1998 ) [ hep - ph/9707409 ] ; 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the longitudinal polarization of fermions ( tops and taus ) produced in sfermion decays to neutralinos or charginos can be a useful tool for the determination of susy parameters . 
 we discuss this fermion polarization in the context of the mssm with complex parameters . 
 we show that the dependence on cp - violating phases can be large and that the fermion polarization may hence be used as a sensitive probe of cp phases in the mssm . 
 cern - ph - th/2004 - 086 + hephy - pub 790/04 + iisc - chep/7/04 + fi2004 - 15 + hep - ph/0405167    * fermion polarization in sfermion decays as + a probe of cp phases in the mssm * + thomas  gajdosik@xmath0 , rohini  m.  godbole@xmath1 , sabine  kraml@xmath2 + _ 
 @xmath3  institute of physics , vilnius lt-2600 , lithuania + @xmath4  centre for high energy physics , indian institute of science , bangalore 560012 , india + @xmath5  inst . 
 f.  hochenergiephysik , sterr . 
 akademie d.  wissenschaften , 1050 vienna , austria + @xmath6  department of physics , cern , theory division , 1211 geneva 23 , switzerland + _