Supersymmetry and the Superconductor-Insulator Transition

a r X i v :0805.4481v 1 [h e p -t h ] 29 M a y 2008SUPERSYMMETRIC CHERN-SIMONS MODELS IN HARMONIC SU-PERSPACESB.M.ZupnikBogoliubov Laboratory of Theoretical Physics,JINR,Dubna,Moscow Region,141980,Russia;E-mail:zupnik@theor.jinr.ruAbstractWe review harmonic superspaces of the D =3,N =3and 4supersymmetries and gauge models in these superspaces.Superspaces of the D =3,N =5supersymmetry use harmonic coordinates of the SO (5)group.The superfield N =5actions describe the off-shell infinite-dimensional Chern-Simons supermultiplet.1IntroductionSupersymmetric extensions of the D =3Chern-Simons theory were discussed in [1]-[10].A superfield action of the D =3,N =1Chern-Simons theory can be interpreted as the superspace integral of the differential Chern-Simons superform dA +2iwhere i andˆk are two-component indices of the automorphism groups SU L(2)and SU R(2), respectively,αis the two-component index of the SL(2,R)group and m=0,1,2is the 3D vector index.The N=4supersymmetry transformations areδx m=−i(γm)αβ(ǫαjˆk θβjˆk−iǫβjˆkθαjˆk),(2.2)whereγm are the3Dγmatrices.The SU L(2)/U(1)harmonics u±i[11]can be used to construct the left analytic super-space[15]with the LA coordinatesζL=(x m L,θ+ˆkα).(2.3)The L-analytic prepotential V++(ζL,u)describes the left N=4vector multiplet A m,φˆkˆl ,λαiˆk,D ik.The D=3,N=4SY M action can be constructed in terms of this prepotential by anal-ogy with the D=4,N=2SY M action[14].Let us introduce the new notation for the left harmonics u±i=u(±1,0)iand the analogousnotation v(0,±1)ˆk for the right SL R(2)/U(1)harmonics.The biharmonic N=4superspaceuses the Grassmann coordinates[15]θ(±1,±1)α=u(±1,0)i v(0,±1)ˆkθiˆkα.(2.4)In this representation,we haveζL=(x m L,θ(1,±1)α),V++≡V(2,0),(2.5)D(1,±1)αV(2,0)=0,D(0,2)vV(2,0)=0.(2.6)The right analytic N=4coordinates areζR=(x m R,θ(±1,1)α),x m R=x m L−2i(γm)αβθ(−1,1)αθ(1,1)β.(2.7) The mirror R analytic prepotentialˆV(0,2)D(±1,1)ˆV(0,2)=0,D(2,0)uˆV(0,2)=0(2.8)describes the right N=4vector multipletˆA m,ˆφij,ˆλαiˆk ,ˆD ik,whereˆA m is the mirror vectorgaugefield using the independent gauge group.The right N=4SY M action is similar to the analogous left action.These multiplets can be formally connected by the map SU L(2)↔SU R(2).The N=4superfield Chern-Simons type(or BF-type)action for the gauge group U(1)×U(1)connects two mirror vector multipletsdud3x L dθ(−4,0)V(2,0)(ζL,u)D(1,1)αD(1,1)αˆV(0,−2),(2.9) where the right connection satisfies the equationD(0,2)v ˆV(0,−2)=D(0,−2)v˜V(0,2),D(2,0)uˆV(0,−2)=0.(2.10)The component form of this action was considered in[16,17].We can identify the left and right isospinor indices in the N=4spinor coordinatesθαjˆk→θαjk=θα(jk)+12D++αD++αV−−.(2.14)The action of the corresponding CS-theory can be constructed in the full or analytic N=3superspaces[8,9].3Harmonic superspaces for the group SO(5)The homogeneous space SO(5)/U(2)is parametrized by elements of the harmonic5×5 matrixU K a=(U+i a,U0a,U−ia)=(U+1a,U+2a,U0a,U−1a,U−2a),(3.1) where a=1,...5is the vector index of the group SO(5),i=1,2is the spinor index of the group SU(2),and U(1)-charges are denoted by symbols+,−,0.The basic relations for these harmonics areU+i a U+k a=U+i a U0a=0,U−ia U−ka =U−ia U0a=0,U+i a U−ka=δi k,U0a U0a=1,U+i a U−ib +U−ia U+ib+U0a U0b=δab.(3.2)We consider the SO(5)invariant harmonic derivatives with nonzero U(1)charges∂+i=U+i a ∂∂U−ia,∂+i U0a=U+i a,∂+i U−ka=−δi k U0a,∂++=U+ia∂∂U0a −U0a∂∂U+i a ,[∂−i,∂−k]=εki∂−−,∂−k∂−k=−∂−−,where some relations between these harmonic derivatives are defined.The U(1)neutral harmonic derivatives form the Lie algebra U(2)∂i k=U+i a ∂∂U−ia,[∂+i,∂−k]=−∂i k,(3.4)∂0≡∂k k=U+k a ∂∂U−ka,[∂++,∂−−]=∂0,∂i k U+l a=δl k U+i a,∂i k U−la =−δi l U−ka.(3.5)The operators∂+k,∂++,∂−k ,∂−−and∂i k satisfy the commutation relations of the Lie al-gebra SO(5).One defines an ordinary complex conjugation on these harmonicsU0a=U0a,(3.6) however,it is convenient to use a special conjugation in the harmonic space(U+i a)∼=U+i a,(U−ia)∼=U−ia,(U0a)∼=U0a.(3.7) All harmonics are real with respect to this conjugation.The full superspace of the D=3,N=5supersymmetry has the spinor CB coordi-natesθαa,(α=1,2;a=1,2,3,4,5)in addition to the coordinates x m of the three-dimensional Minkowski space.The group SL(2,R)×SO(5)acts on the spinor coordinates. The superconformal transformations of these coordinates are considered in Appendix.The SO(5)/U(2)harmonics allow us to construct projections of the spinor coordinates and the partial spinor derivativesθ+iα=U+i aθαa,θ0α=U0aθαa,θ−αi=U−iaθαa,(3.8)∂−iα=∂/∂θ+iα,∂0α=∂/∂θ0α,∂+iα=∂/∂θ−αi.The analytic coordinates(AB-representation)in the full harmonic superspace use these projections of10spinor coordinatesθ+iα,θ0α,θ−αi and the following representation of the vector coordinate:x m A ≡y m=x m+i(θ+kγmθ−k)=x m+i(θaγmθb)U+k a U−kb.(3.9)The analytic coordinates are real with respect to the special conjugation. The harmonic derivatives have the following form in AB:D+k=∂+k−i(θ+kγmθ0)∂m+θ+kα∂0α−θ0α∂+kα,D++=∂+++i(θ+kγmθ+k )∂m+θ+αk∂+kα,(3.10)D k l=∂k l+θ+kα∂−lα−θ−αl∂−kα.We use the commutation relations[D+k,D+l]=−εkl D++,D+k D+k=D++.(3.11) The AB spinor derivatives areD+iα=∂+iα,D−iα=−∂−iα−2iθ−βi∂αβ,D0α=∂0α+iθ0β∂αβ.(3.12) The coordinates of the analytic superspaceζ=(y m,θ+iα,θ0α,U K a)have the Grass-mann dimension6and dimension of the even space3+6.The functionsΦ(ζ)satisfy the Grassmann analyticity condition in this superspaceD+kαΦ=0.(3.13)In addition to this condition,the analytic superfields in the SO (5)/U (2)harmonic super-space possess also the U (2)-covariance.This subsidiary condition looks especially simple for the U (2)-scalar superfieldsD k l Λ(ζ)=0.(3.14)The integration measure in the analytic superspace dµ(−4)has dimension zerodµ(−4)=dUd 3x A (∂0α)2(∂−iα)4=dUd 3x A dθ(−4).(3.15)The SO (5)/U (1)×U (1)harmonics can be defined via the components of the real orthogonal 5×5matrix [20,21]U Ka = U (1,1)a ,U (1,−1)a ,U (0,0)a ,U (−1,1)a ,U (−1,−1)a (3.16)where a is the SO(5)vector index and the index K =1,2,...5corresponds to givencombinations of the U(1)×U(1)charges.We use the following harmonic derivatives∂(2,0)=U (1,1)b∂/∂U (−1,1)b−U (1,−1)b∂/∂U (−1,−1)b,∂(1,1)=U (1,1)b∂/∂U (0,0)b −U (0,0)b∂/∂U (−1,−1)b,∂(1,−1)=U (1,−1)b ∂/∂U (0,0)b−U (0,0)b∂/∂U (−1,1)b,∂(0,2)=U (1,1)b∂/∂U (1,−1)b−U (−1,1)b∂/∂U (−1,−1)b,∂(0,−2)=U (1,−1)b∂/∂U (1,1)b−U (−1,−1)b∂/∂U (−1,1)b.(3.17)We define the harmonic projections of the N =5Grassmann coordinatesθK α=θaαU K a =(θ(1,1)α,θ(1,−1)α,θ(0,0)α,θ(−1,1)α,θ(−1,−1)α).(3.18)The SO (5)/U (1)×U (1)analytic superspace contains only spinor coordinatesζ=(x m A ,θ(1,1)α,θ(1,−1)α,θ(0,0)α),(3.19)x m A=x m +iθ(1,1)γm θ(−1,−1)+iθ(1,−1)γm θ(−1,1),δǫx m A =−iǫ(0,0)γm θ(0,0)−2iǫ(−1,1)γm θ(1,−1)−2iǫ(−1,−1)γm θ(1,1),(3.20)where ǫKα=ǫαa U Ka are the harmonic projections of the supersymmetry parameters.General superfields in the analytic coordinates depend also on additional spinor coor-dinates θ(−1,1)αand θ(−1,−1)α.The harmonized partial spinor derivatives are∂(−1,−1)α=∂/∂θ(1,1)α,∂(−1,1)α=∂/∂θ(1,−1)α,∂(0,0)α=∂/∂θ(0,0)α,(3.21)∂(1,1)α=∂/∂θ(−1,−1)α,∂(1,−1)α=∂/∂θ(−1,1)α.We use the special conjugation ∼in the harmonic superspaceU (p,q )a =U (p,−q )a ,θ(p,q )α=θ(p,−q )α, x m A =x m A,(θ(p,q )αθ(s,r )β)∼=θ(s,−r )βθ(p,−q )α, f (x A )=¯f(x A ),(3.22)where ¯fis the ordinary complex conjugation.The analytic superspace is real with respect to the special conjugation.The analytic-superspace integral measure contains partial spinor derivatives(3.21)1dµ(−4,0)=−they are traceless and anti-HermitianΛ†=−Λ.We treat these prepotentials as connections in the covariant gauge derivatives∇+i=D+i+V+i,∇++=D+++V++,δΛV+i=D+iΛ+[Λ,V+i],δΛV++=D++Λ+[Λ,V++],(4.4) D+kαδΛV+k=D+kαδΛV++=0,D i jδΛV+k=δk jδΛV+i,D i jδΛV++=δi jδΛV++,where the infinitesimal gauge transformations of the gauge superfields are defined.These covariant derivatives commute with the spinor derivatives D+kαand preserve the CR-structure in the harmonic superspace.We can construct three analytic superfield strengths offthe mass shellF++=112πdµ(−4)Tr{V+j D++V+j+2V++D+j V+j+(V++)2+V++[V+j,V+j]},(4.5)where k is the coupling constant,and a choice of the numerical multiplier guarantees the correct normalization of the vector-field action.This action is invariant with respect to the infinitesimal gauge transformations of the prepotentials(4.4).The idea of construction of the superfield action in the harmonic SO(5)/U(2)was proposed in[18],although the detailed construction of the superfield Chern-Simons theory was not discussed in this work.The equivalent superfield action was considered in the framework of the alternative superfield formalism[20].The superconformal N=5invariance of this action was proven in[19].The action S1yields superfield equations of motion which mean triviality of the su-perfield strengths of the theoryF(+3) k =D++V+k−D+kV+++[V++,V+k]=0,F++=V++−D+k V+k −V+k V+k=0.(4.6)These classical superfield equations have pure gauge solutions for the prepotentials onlyV+k=e−ΛD+k eΛ,V++=e−ΛD++eΛ,(4.7) whereΛis an arbitrary analytic superfield.The transformation of the sixth supersymmetry can be defined on the analytic N=5 superfieldsδ6V++=ǫα6D0αV++,δ6V+k=ǫα6D0αV+k,(4.8)δ6D+k V+l=ǫα6D0αD+k V+l,δ6D++V+l=ǫα6D0αD++V+l,(4.9) whereǫα6are the corresponding odd parameters.This transformation preserves the Grass-mann analyticity and U(2)-covariance{D0α,D+kβ}=0,[D k l,D0α]=0,[D+k,D0α]=D+kα,[D++,D0α]=0.(4.10)The action S1is invariant with respect to this sixth supersymmetryδ6S1= dµ(−4)ǫα6D0αL(+4)=0.(4.11)In the SO(5)/U(1)×U(1)harmonic superspace,we can introduce the D=3,N=5 analytic matrix gauge prepotentials corresponding to thefive harmonic derivativesV(p,q)(ζ,U)=[V(1,1),V(1,−1),V(2,0),V(0,±2)],(V(1,1))†=−V(1,−1),(V(2,0))†=V(2,0),V(0,−2)=[V(0,2)]†,(4.12) where the Hermitian conjugation†includes∼conjugation of matrix elements and trans-position.We shall consider the restricted gauge supergroup using the supersymmetry-preserving harmonic(H)analyticity constraints on the gauge superfield parametersH1:D(0,±2)Λ=0.(4.13) These constrains yield additional reality conditions for the component gauge parameters.We use the harmonic-analyticity constraints on the gauge prepotentialsH2:V(0,±2)=0,D(0,−2)V(1,1)=V(1,−1),D(0,2)V(1,1)=0(4.14) and the conjugated constraints combined with relations(4.12).The superfield CS action can be constructed in terms of these H-constrained gauge superfields[21]2ikS=−V(2,0)V(2,0)}.(4.15)2Note,that the similar harmonic superspace based on the USp(4)/U(1)×U(1)harmon-ics was used in[22]for the harmonic interpretation of the D=4,N=4super Yang-Mills constraints.This work was partially supported by the grants RFBR06-02-16684,DFG436RUS 113/669-3,INTAS05-10000008-7928and by the Heisenberg-Landau programme.References[1]W.Siegel,Nucl.Phys.B156(1979)135.[2]J.Schonfeld,Nucl.Phys.B185(1981)157.[3]B.M.Zupnik,D.G.Pak,Teor.Mat.Fiz.77(1988)97;Eng.transl.:Theor.Math.Phys.77(1988)1070.[4]B.M.Zupnik,D.G.Pak,Class.Quant.Grav.6(1989)723.[5]B.M.Zupnik,Teor.Mat.Fiz.89(1991)253,Eng.transl.:Theor.Math.Phys.89(1991)1191.[6]B.M.Zupnik,Phys.Lett.B254(1991)127.[7]H.Nishino,S.J.Gates,Int.J.Mod.Phys.8(1993)3371.[8]B.M.Zupnik,D.V.Khetselius,Yad.Fiz.47(1988)1147;Eng.transl.:Sov.J.Nucl.Phys.47(1988)730.[9]B.M.Zupnik,Springer Lect.Notes in Phys.524(1998)116;hep-th/9804167.[10]J.H.Schwarz,Jour.High.Ener.Phys.0411(2004)078,hep-th/0411077.[11]A.Galperin,E.Ivanov,S.Kalitzin,V.Ogievetsky,E.Sokatchev,Class.Quant.Grav.1(1984)469.[12]A.Galperin,E.Ivanov,S.Kalitzin,V.Ogievetsky,E.Sokatchev,Class.Quant.Grav.2(1985)155.[13]A.Galperin,E.Ivanov,V.Ogievetsky,E.Sokatchev,Harmonic superspace,Cam-bridge University Press,Cambridge,2001.[14]B.M.Zupnik,Phys.Lett.B183(1987)175.[15]B.M.Zupnik,Nucl.Phys.B554(1999)365,Erratum:Nucl.Phys.B644(2002)405E;hep-th/9902038.[16]R.Brooks,S.J.Gates,Nucl.Phys.B432(1994)205,hep-th/09407147.[17]A.Kapustin,M.Strassler,JHEP04(1999)021,hep-th/9902033.[18]P.S.Howe,M.I.Leeming,Clas.Quant.Grav.11(1994)2843,hep-th/9402038.[19]B.M.Zupnik,Chern-Simons theory in SO(5)/U(2)harmonic superspace,arXiv0802.0801(hep-th).[20]B.M.Zupnik,Phys.Lett.B660(2008)254,arXiv0711.4680(hep-th).[21]B.M.Zupnik,Gauge model in D=3,N=5harmonic superspace,arXiv0708.3951(hep-th).[22]I.L.Buchbinder,O.Lechtenfeld,I.B.Samsonov,N=4superparticle and super Yang-Mills theory in USp(4)harmonic superspace,arxiv:0804.3063(hep-th).。

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a r X i v :h e p -p h /0002115v 3 5 J u n 2000UWThPh-2000-06HEPHY-PUB 729/00FTUV/00-10IFIC/00-10hep-ph/0002115LC-TH-2000-031Phenomenology ofStops,Sbottoms,τ-Sneutrinos,and Stausat an e +e −Linear ColliderA.Bartl,1H.Eberl,2S.Kraml,2W.Majerotto,2W.Porod 31Institut f¨u r Theoretische Physik,Universit¨a t Wien,A–1090Vienna,Austria2Inst.f.Hochenergiephysik,¨Osterr.Akademie d.Wissenschaften,A–1050Vienna,Austria3Inst.de F´ısica Corpuscular (IFIC),CSIC,E–46071Val`e ncia,SpainAbstractWe discuss production and decays of stops,sbottoms,τ-sneutrinos,and staus in e +e −annihila-tion in the energy range√s =0.5−1.5TeV,wherethese states are expected to be pair produced.Moreover,at an e +e −Linear Collider with this energy and an integrated luminosity of about 500fb −1it will be possible to measure masses,cross sections and decay branching ratios with high precision [6].This will allow us to obtain information on the fundamental soft SUSY breaking parameters.Therefore,it is necessary to investigate how this information can be extracted from the experimental data,and how precisely these parameters can be determined.In this way it will be possible to test our theoretical ideas about the underlying SUSY breaking mechanism.Phenomenological studies on SUSY particle searches at the LHC have shown that the detection ofthe scalar top quark may be very difficult due to the overwhelming background from t ¯tproduction [7,8,9,10,11].This is in particular true for m ˜t 1<∼250GeV [7].In principle,such a light stop could be discovered at the Tevatron.The actual mass reach,however,strongly depends on the luminosity,decay modes,and the available phase–space [12,13].Thus an e +e −Linear Collider with√s =500GeV and 1TeV.We give numerical results for the production cross sectionstaking into account polarization of both the e −and e +beams.In particular,we show that by using polarized beams it will be possible to determine the fundamental SUSY parameters with higher precision than without polarization.Moreover,we discuss the decays of these particles.The production crosssections as well as the decay rates of the sfermions show a distinct dependence on the ˜f L –˜f R mixingangles.Squarks (sleptons)can decay into quarks (leptons)plus neutralinos or charginos.Squarks may also decay into gluinos.In addition,if the splitting between the different sfermion mass eigenstates is large enough,transitions between these states by emmission of weak vector bosons or Higgs bosons are possible.These decay modes can be important for the higher mass eigenstates,and lead to complicated cascade decays.In the case of the lighter stop,however,all these tree–level two–body decays may bekinematically forbidden.Then the ˜t1has more complicated higher–order decays [14,15,16].The framework of our calculation is the Minimal Supersymmetric Standard Model (MSSM)[1]whichcontains the Standard Model (SM)particles plus the sleptons ˜ℓ±,sneutrinos ˜νℓ,squarks ˜q ,gluinos ˜g ,twopairs of charginos ˜χ±i ,i =1,2,four neutralinos,˜χ0i ,i =1,...,4,and five Higgs bosons,h 0,H 0,A 0,H ±[17].In Section 2we shortly review the basic features of left–right mixing of squarks and sleptons of the 3rd generation,and present formulae and numerical results for the production cross sections with polarized e −and e +beams.In Section 3we discuss the decays of these particles and present numerical results for their branching ratios.In Section 4we give an estimate of the errors to be expected for the fundamental soft SUSY–breaking parameters of the stop mixing matrix.In Section 5we compare the situation concerning stop,sbottom,and stau searches at LHC and Tevatron with that at an e +e −Linear Collider.Section 6contains a short summary.2Production Cross SectionsLeft–right mixing of the sfermions is described by the symmetric 2×2mass matrices which in the (˜f L ,˜f R )basis (f =t,b,τ)read [2,17]M 2˜f =M 2˜f La f m f a f m f M 2˜f R.(1)The diagonal elements of the sfermion mass matrices areM 2˜f L=M 2˜F +m 2Z cos 2β(T 3f −e f sin 2ΘW )+m 2f ,(2)M 2˜f R=M 2˜F ′+e f m 2Z cos 2βsin 2θW +m 2f(3)where m f ,e f and T 3f are the mass,charge and third component of weak isospin of the fermion f ,and θW is the Weinberg angle.Moreover,M ˜F =M ˜Q for ˜f L =˜t L ,˜b L ,M ˜F =M ˜L for ˜f L =˜τL ,and M ˜F ′=M ˜U ,M ˜D ,M ˜E for ˜f R =˜t R ,˜b R ,˜τR ,respectively.M ˜Q ,M ˜U ,M ˜D ,M ˜L ,and M ˜E are soft SUSY–breaking mass parameters of the third generation sfermion system.The off–diagonal elements of the sfermion mass matrices arem t a t=m t (A t −µcot β),(4)m b a b =m b (A b −µtan β),(5)m τa τ=m τ(A τ−µtan β)(6)for stop,sbottom,and stau,respectively.A t ,A b ,A τare soft SUSY–breaking trilinear scalar couplingparameters.Evidently,in the stop sector there can be strong ˜tL -˜t R mixing due to the large top quark mass.In the case of sbottoms and staus the ˜f L −˜f R mixing effects are also non-negligible if tan β>∼10.We assume that all parameters are real.Then the mass matrices can be diagonalized by 2×2orthogonalmatrices.The mass eigenvalues for the sfermions ˜f=˜t ,˜b,˜τare m 2˜f 1,2=1(M 2˜f L−M 2˜fR)2+4m 2f a 2f ),(7)and the mass eigenstates are˜f 1=˜f L cos θ˜f +˜f R sin θ˜f ,(8)˜f 2=˜f R cos θ˜f −˜f L sin θ˜f,(9)where ˜t 1,˜b 1,˜τ1denote the lighter eigenstates.The sfermion mixing angle is given bycos θ˜f =−a f m f(M 2˜f L−m 2˜f 1)2+a 2f m 2f,sin θ˜f =M 2˜f L−m 2˜f1(M 2˜f L−m 2˜f 1)2+a 2f m 2f.(10)The ˜ντappears only in the left–state.Its mass ism 2˜ντ=M 2˜L+1s 4e 2f δij(1−P −P +)−e f c ij δij16s 4W c 4W(v 2e +a 2e)(1−P −P +)−2v e a e (P −−P +) D ZZ,(12)where P −and P +denote the degree of polarization of the e −and e +beams,with the conventionP ±=−1,0,+1for left–polarized,unpolarized,right–polarized e ±beams,respectively.(E.g.,P −=−0.9means that 90%of the electrons are left–polarized and the rest is unpolarized.)v e =4s 2W −1,a e =−1are the vector and axial–vector couplings of the electron to the Z ,s 2W ≡sin 2θW ,c 2W ≡cos 2θW ,and c ijis the Z ˜f i ˜f j coupling (up to a factor 1/cos θW )c ij = T 3f cos 2θ˜f −e f s 2W−12T 3f sin 2θ˜f T 3f sin 2θ˜f −e f s 2W .(13)Furthermore,in Eq.(12)√(s −m 2Z )2+Γ2Z m 2Z,D γZ =s (s −m 2Z )θ˜t =0.98,θ˜b =1.17,and θ˜τ=0.82,respectively.There is a destructive interference between the γand Z –exchange contributions that leads to characteristic minima of the cross sections at specific values of the mixing angles θ˜f ,which according to Eq.(12)depend on √s dependence of the stop and sbottom pair production cross sections for m ˜t 1=220GeV,m ˜t 2=450GeV,cos θ˜t =−0.66,m ˜b 1=284GeV,m ˜b 2=345GeV,cos θ˜b =0.84,and m ˜g =555GeV.Here we have included supersymmetric QCD (i.e.gluon and gluino)corrections [21,22]and initial state radiation (ISR)[23].1The latter typically changes the cross section by ∼15%.The relative importance of the gluon and gluino corrections can be seen in Figs.1c,d where we plot∆σ/σ0for ˜t 1¯˜t 2and ˜b 1¯˜b 1production for the parameters of Fig.1a,b.In addition we also show the leading electroweak corrections in order of Yukawa couplings squared [24]for M =200GeV,µ=800GeV,m A =300GeV,and tan β=4.Let us discuss these corrections in more detail:The standard QCD correction [21](due to virtual gluon exchange and real gluon emission)is proportional to the tree–level cross section:σ=σ0(1+4αss =500GeV in (a)and√s =500GeV for two values of cos θ˜t :cos θ˜t =0.4in (a)andcos θ˜t =0.66in (b).The white windows show the range of polarization of the TESLA design [27].As40060080010001200140001020304050600800100012001400-100102030Figure 1:(a,b)Total cross sections for e +e −→˜ti ¯˜t j ,˜b i ¯˜b j as a function of √Figure 2:Total cross sections of stau and sneutrino pair production as a function of√one can see,one can significantly increase the cross section by using the maximally possible e−and e+ polarization.Here note that the(additional)positron polarization leads to an effective polarization[28] ofP eff=P−−P+σL+σR(16)whereσL:=σ(−|P−|,|P+|)andσR:=σ(|P−|,−|P+|).This observable is sensitive to the amount of mixing of the produced sfermions while kinematical effects only enter at loop level.In Fig.5we show A LR for e+e−→˜t i¯˜t i(i=1,2)as a function of cosθ˜t for90%polarized electrons and unpolarized as well as60%polarized positrons;√s=800GeV we obtainσ(˜t1˜t2)=7.9fb,9.1fb,and14.1fb for (P−,P+)=(0,0),(−0.9,0),and(−0.9,0.6),respectively.The left–right asymmetry,however,hardly varies with cosθ˜t:A LR(˜t1˜t2)≃0.14(0.15)for|P−|=0.9and|P+|=0(0.6),0<|cosθ˜t|<1,and the other parameters as above.We next discuss sbottom production using the scenario of Fig.1b,i.e.m˜b1=284GeV and m˜b2=345GeV.In this case,all three combinations˜b1¯˜b1,˜b1˜b2(≡˜b1¯˜b2+˜b2¯˜b1),and˜b2¯˜b2can be produced at√s=500GeV for unpolarized as well as for polarized e−beams(P+=0).The usefulness of beam polarization to(i)increase the cosθ˜τdependence and(ii)enhance/reduce˜τ1¯˜τ1relative to˜τ2¯˜τ2production is obvious.The left–right asymmetry of˜τi¯˜τi production for the parameters of Fig.9is shown in Fig.10. Here note that,in contrast to˜t and˜b production,A LR(˜τi¯˜τi)is almost zero for maximally mixed staus.-1.-0.50.0.5 1.020*********120140160-1.-0.50.0.5 1.51015202530Figure 3:cos θ˜t dependence of stop pair production cross sections for m ˜t 1=200GeV,m ˜t 2=420GeV,m ˜g =555GeV;√s =1TeV in (b);the label “L”(“R”)denotes P −=−0.9(0.9)with the dashed lines for P +=0,and the dotted lines for |P +|=0.6[sign(P +)=−sign(P −)];the full lineslabeled “U”are for unpolarized beams (P −=P +=0).----Figure 4:Dependence of σ(e +e −→˜t 1¯˜t 1)on degree of electron and positron polarization for √s =1TeV,m ˜t 1=200GeV,m ˜t2=420GeV,and m ˜g =555GeV;the solid lines are for 90%polarized electrons and unpolarized positrons,the dashed lines are for 90%polarized electrons and 60%polarized positrons.-1.-0.50.0.5 1.0510152025-1.-0.50.0.5 1.010203040-1.-0.50.0.5 1.246810Figure 6:cos θ˜b dependence of sbottom pair production cross sections for m ˜b 1=284GeV,m ˜b 2=345GeV,m ˜g =555GeV,√s =800GeV,m ˜b 1=284GeV,m ˜b 2=345GeV,and m ˜g=555GeV.----Figure 8:Dependence of (a)σ(e +e −→˜b 1¯˜b 1)and (b)σ(e +e −→˜b 2¯˜b 2)(in fb)on the degree of electron and positron polarization for√-1.-0.50.0.5 1.20406080-1.-0.50.0.5 1.20406080-1.-0.50.0.5 1.20406080Figure 9:cos θ˜τdependence of stau pair production cross sections for m ˜τ1=156GeV,m ˜τ2=180GeV,√s =500GeV,m ˜τ1=156GeV,m ˜τ2=180GeV,and cos θ˜τ=0.77.------s =500GeV,m ˜τ1=156GeV,m ˜τ2=180GeV,cos θ˜τ=0.77,and m ˜ντ=148GeV.Finally,the dependence of σ(e +e −→˜τi ¯˜τi ),for cos θ˜τ=0.77,and σ(e +e −→˜ντ¯˜ντ)on both the electron and positron polarizations is shown in Fig.11.Notice that one could again substantially increase the cross sections by going beyond 60%e +polarization.3DecaysOwing to the influence of the Yukawa terms and the left–right mixing,the decay patterns of stops,sbottoms,τ-sneutrinos,and staus are in general more complicated than those of the sfermions of the first two generations.As for the sfermions of the first and second generation,there are the decays into neutralinos or charginos (i,j =1,2;k =1,...4):˜t i →t ˜χ0k ,b ˜χ+j ,˜b i →b ˜χ0k ,t ˜χ−j ,(17)˜τi →τ˜χ0k ,ντ˜χ−j ,˜ντ→ντ˜χ0k ,τ˜χ+j .(18)Stops and sbottoms may also decay into gluinos,˜t i →t ˜g ,˜b i →b ˜g (19)and if these decays are kinematically allowed,they are important.If the mass differences |m ˜t i −m ˜b j |and/or |m ˜τi −m ˜ντ|are large enough the transitions ˜t i →˜b j +W +(H +)or ˜b i →˜tj +W −(H −)(20)as well as˜τi →˜ντ+W −(H −)or ˜ντ→˜τi +W +(H +)(21)can occur.Moreover,in case of strong ˜f L −˜f Rmixing the splitting between the two mass eigenstates may be so large that heavier sfermion can decay into the lighter one:˜t2→˜t 1+Z 0(h 0,H 0,A 0),˜b 2→˜b 1+Z 0(h 0,H 0,A 0),˜τ2→˜τ1+Z 0(h 0,H 0,A 0).(22)The SUSY–QCD corrections to the squark decays of Eqs.(17)to (22)have been calculated in [30].TheYukawa coupling corrections to the decay ˜b i →t ˜χ−j have been discussed in [31].All these corrections will be important for precision measurements.The decays of the lighter stop can be still more complicated:If m ˜t 1<m ˜χ01+m t and m ˜t 1<m ˜χ+1+m b the three–body decays [15]˜t 1→W +b ˜χ01,H +b ˜χ01,b ˜l +i νl ,b ˜νl l +(23)can compete with the loop–decay [14]˜t 1→c ˜χ01,2.(24)If also m ˜t 1<m ˜χ01+m b +m W etc.,then four–body decays [16]˜t 1→b f ¯f ′˜χ01(25)have to be taken into account.We have studied numerically the widths and branching ratios of the various sfermion decay modes.In the calculation of the stop and sbottom decay widths we have included the SUSY–QCD correctionsaccording to [30].In Fig.12we show the decay width of ˜t 1→b ˜χ+1as a function of cos θ˜tfor m ˜t 1=200GeV,m ˜t 2=420GeV,tan β=4,M =180GeV,µ=360GeV,(gaugino–like ˜χ+1),and M =360GeV,-1.-0.50.0.5 1.0.0.10.20.30.40.5Figure 12:Decay width of ˜t 1as a function of cos θ˜t for m ˜t 1=200GeV,m ˜t 2=420GeV,tan β=4,{M,µ}={180,360}GeV in (a),and {M,µ}={360,180}GeV in (b);the dashed lines show the results at tree level,the full lines those at O (αs ).µ=180GeV (higgsino–like ˜χ+1).If Γ(˜t 1)<∼200MeV ˜t 1may hadronize before decaying [32].In Fig.12this is the case for a gaugino–like ˜χ+1as well as for a higgsino–like one if cos θ˜t <∼−0.5(or cos θ˜t >∼0.9).In case that all tree–level two–body decay modes are forbidden for ˜t1,higher order decays are impor-tant for its phenomenology.In the following we study examples where three–body decay modes,Eq.(23),are the dominant ones.For fixing the parameters we choose the following procedure:in addition to tan βand µwe use m ˜t 1and cos θ˜t as input parameters in the stop sector.For the sbottom (stau)sector we fix M ˜Q ,M ˜D and A b (M E ,M L ,A τ)as input parameters.(We use this mixed set of parameters in order to avoid unnaturally large values for A b and A τ.)Moreover,we assume for simplicity that the soft SUSY breaking parameters are equal for all generations.Note that,because of SU(2)invariance M ˜Q appears in both the stop and sbottom mass matrices,see Eqs.(1)–(3).The mass of the heavier stop can thus be calculated from the above set of input parameters as:m 2˜t2=2M 2˜Q +2m 2Z cos 2β 13sin 2θW +2m 2t −m 2˜t 1(1+cos 2θ˜t )tan βM m ˜χ01m ˜χ+2500223M Qm ˜b 23701503420.98M EA τm ˜τ1cos θ˜τ190200m ˜e R220195181Table 1:Parameters and physical quantities used in Fig.13and 14.All masses are given in GeV.size as tan βis small.BR(˜t 1→c ˜χ01)is of order 10−4independent of cos θ˜t and therefore negligible.Near cos θ˜t =−0.3the decay into b W +˜χ01has a branching ratio of ∼100%.Here the ˜t 1b ˜χ+1coupling vanishesleading to the reduction of the decays into sleptons.In Fig.13b the branching ratios for the decays into the different sleptons are shown.As tan βis small the sleptons couple mainly to the gaugino components of ˜χ+1.Therefore,the branching ratios of decays into staus,which are strongly mixed,are reduced.However,the sum of both branching ratios is nearly the same as BR(˜t 1→b νe ˜e +L ).The decays into sneutrinos are preferred by kinematics.The decay ˜t 1→b W +˜χ01is dominated by top quark exchange,followed by chargino contributions.In many cases the interference term between t and ˜χ+1,2is more important than the pure ˜χ+1,2exchange.Moreover,we have found that the contribution from sbottom exchange is in general negligible.In Fig.14we show the branching ratios of ˜t 1decays as a function of tan βfor cos θ˜t =0.6and the other parameters as above.For small tan βthe decay ˜t 1→b W +˜χ01is the most important one.The branchingratios for the decays into sleptons are reduced in the range tan β<∼5because the gaugino component of ˜χ+1decreases and its mass increases.For tan β>∼10the decays into the b τE/final state become more important because of the growing τYukawa coupling and because of kinematics (m ˜τ1decreases with increasing tan β).Here ˜t 1→b ντ˜τ1gives the most important contribution as can be seen in Fig.14b.Even for large tan βthe decay into c ˜χ01is always suppressed.From the requirement that no two–body decays be allowed at tree level follows that m ˜χ+1>m ˜t 1−m b .Therefore,one expects an increase of BR(˜t 1→b W +˜χ01)if m ˜t 1increases,because the decay into b W +˜χ01is dominated by top–quark exchange whereas for the decays into sleptons the ˜χ+1contribution is the dominating one.In general BR(˜t 1→b W +˜χ01)is larger than 80%if m ˜t 1>∼350GeV [15].If the three–body decay modes are kinematically forbidden (or suppressed)four–body decays ˜t 1→b f ¯f ˜χ01come into play.Depending on the MSSM parameter region,these decays can also dominate overthe decay into c ˜χ01.For a discussion,see [16].We now turn to the decays of ˜t 2.Here the bosonic decays of Eqs.20and 22can play an important rˆo le as demonstrated in Figs.15and 16.In Fig.15we show the cos θ˜t dependence of BR(˜t 2)for m ˜t 1=200GeV,m ˜t 2=420GeV,M =180GeV,µ=360GeV,tan β=4,M ˜D =1.1M ˜Q ,A b =−300GeV,and m A =200GeV.As can be seen,the decays into bosons can have branching ratios of several ten percent.The branching ratio of the decay into the gaugino–like ˜χ+1is large if ˜t 2has a rather strong ˜t L component.The decay ˜t 2→˜t 1Z is preferred by strong mixing.The decays into ˜b i W +only occur for |cos θ˜t |>∼0.5because the sbottom masses are related to the stop parameters by our choice M ˜D =1.1M ˜Q .Notice thatBR(˜t 2→˜b i W +)goes to zero for |cos θ˜t |=1as in this case ˜t2=˜t R .We have chosen m A such that decays into all MSSM Higgs bosons be possible.These decays introduce a more complicated dependence on the mixing angle,Eq.(10),because A t =(m 2˜t1−m 2˜t 2)/(2m t )+µcot βdirectly enters the stop–HiggsFigure 13:Branching ratios for ˜t 1decays as a function of cos θ˜t form ˜t 1=220GeV,tan β=3,µ=500GeV,and M =240GeV.The other parameters are given in Table 1.The curves in a)correspond tothe transitions:◦˜t 1→b W +˜χ01,△˜t 1→c ˜χ01,˜t 1→b e +˜νe ,and •˜t 1→b τ+˜ντ.Figure 14:Branching ratios for ˜t 1decays as a function of tan βfor m ˜t 1=220GeV,µ=500GeV,cos θ˜t =0.25and M =240GeV.The other parameters are given in Table 1.The curves in a)correspondto the transitions:◦˜t 1→b W +˜χ01,△˜t 1→c ˜χ01,˜t 1→b e +˜νe ,and •˜t 1→b τ+˜ντ.In gray area m h 0<90GeV.couplings.Here notice also the dependence on the sign of cos θ˜t .Figure 16shows the branching ratios of ˜t 2decays as a function of m ˜t 2for cos θ˜t =−0.66and the other parameters as in Fig.15.Again we compare the fermionic and bosonic decay modes.While for a rather light ˜t 2the decay into b ˜χ+1is the most important one,with increasing mass difference m ˜t 2−m ˜t 1the decays into bosons,especially ˜t2→˜t 1Z ,become dominant.-1.-0.50.0.5 1.01020304050-1.-0.50.0.5 1.1020304050-1-0.500.5100.20.40.60.81-1-0.500.510.20.40.60.81Figure 17:Branching ratios of ˜τ1(a)and ˜τ2(b)decays as afunction of cos θ˜τfor m ˜τ1=156GeV,m ˜τ2=180GeV,M =120GeV,µ=300GeV,and tan β=4.0.0130(b)-----Figure 19:(a)Error bands and 68%CL error ellipse for determining m ˜t 1and cos θ˜t from cross section measurements;the dashed lines are for L =100fb −1and the full lines for L =500fb −1.(b)Error bands for the determination of cos θ˜t from A LR .In both plots m˜t1=200GeV,cos θ˜t=−0.66,√s =800GeV also ˜t2can be produced:σ(˜t 1¯˜t 2+c.c.)=8.75fb for P −=−0.9and P +=0.If this cross section can be measured with a precision of 6%this leads to m ˜t 2=420±8.9GeV (again,we took into account a theoretical uncertainity of 1%).2With tan βand µknown from other measurements this then allows one to determine the soft SUSY breaking parameters of the stop sector.Assuming tan β=4±0.4leads to M ˜Q =298±8GeV andM ˜U =264±7GeV for L =500fb−1.In addition,assuming µ=800±80GeV we get A t =587±35(or −187±35)GeV.The ambiguity in A t exists because the sign of cos θ˜t can hardly be determined from cross section measurements.This may,however,be possible from measuring decay branching ratios or the stop–Higgs couplings.A different method to determine the sfermion mass is to use kinematical distributions.This was studied in [37]for squarks of the 1st and 2nd generation.It was shown that,by fitting the distribution of the minimum kinematically allowed squark mass,it is possible to determine m ˜qwith high precision.To be precise,[37]concluded that at√2Herenote that ˜t 1¯˜t 1is produced at √s =500GeV.One can thusimprove the errors on m ˜t 1,m ˜t 2,and cos θ˜t by combining the information obtained at different energies.However,this is beyond the scope of this study.of <∼0.5%using just 20fb −1of data (assuming that all squarks decay via ˜q →q ˜χ01and m ˜χ01is known).The influence of radiative effects on this method has been studied in [38].Taking into account initial state radiation of photons and gluon radiation in the production and decay processes it turned out that a mass of m ˜q =300GeV could be determined with an accuracy of <∼1%with 50fb −1of data.(Thisresult will still be affected by the error on the assumed m ˜χ01,hadronization effects,and systematic errors.)Although the analysis of [37,38]was performed for squarks of the 1st and 2nd generation,the method is also applicable to the 3rd generation.For the determination of the mixing angle,one can also make use of the left–right asymmetry A LR ,Eq.(16).This quantity is of special interest because kinematic effects and uncertainities in experimental efficiencies largely drop out.At √10σ(pp →t ¯t )for m ˜t 1∼m t and σ(pp →˜t 1¯˜t 1)∼1L =20fb −1.Similar results have been obtained for ˜t1three–body decays into sleptons.The authors of [13]also studied the search for light sbottoms at the Tevatron Run II concentrating on the decay ˜b 1→b ˜χ01within mSUGRA.They conclude that with 2fb −1of data the reach is m ˜b 1<∼200(155)GeV for m ˜χ01≃70(100)GeV.With 20fb −1one is sensitive to m ˜b 1<∼260(200)GeV for m ˜χ01≃70(100)GeV.Moreover,their analysis requires a mass difference of m ˜b 1−m ˜χ01>∼30GeV.The higher reach compared to ˜t 1→c ˜χ01is due to the higher tagging efficiency of b ’s.Similarly,also at the LHC the searchfor sbottoms is,in general,expected to be easier than that for stops.There are,however,cases where the analysis is very difficult,see e.g.[8].The search for staus crucially depends on the possibility of τidentification.At hadron colliders,˜τ’s are produced directly via the Drell–Yan process mediated by γ,Z or W exchange in the s –channel.They can also be produced in decays of charginos or neutralinos originating from squark and gluino cascade decays e.g.,˜q →q ′˜χ±j with ˜χ±j →˜τ±i ντor ˜χ±j →˜νττ±.At Tevatron energies,W pair production is the dominant background,while t ¯tevents,with the b jets being too soft to be detected,are the main background at the LHC.SUSY background mainly comes from ˜χ±˜χ∓production followed by leptonic decays.The Drell–Yan production has a low cross section,and it is practically impossible to extract the signal from the SM background (SUSY background is less important).The situation is different if chargino and neutralino decays into staus have a large branching ratio.As pointed out in [41,42,44]this is the case for large tan βwhere the tau Yukawa coupling becomes important.In [42,43,11]the decays ˜χ±1→˜τ1ντand ˜χ02→˜τ1τ(˜τ1→τ˜χ01)with the τ’s decaying hadronically have been studied.In [44,45],the dilepton mass spectrum of final states with e +e −/µ+µ−/e ±µ∓+E miss T +jets has been used to identify ˜τ1in the decay chain ˜χ02→˜τ1τ→˜χ01τ+τ−with τ→e (µ)+νe (µ)+ντ.It turned out that ˜τ1with m ˜τ1<∼350GeV ought to be discovered at the LHC if m ˜τ1<m ˜χ02and tan β>∼10.From this one can conclude that there exist MSSM parameter regions for which (light)sfermions of the 3rd generation may escape detection at both the Tevatron and the LHC.This is in particular the case for ˜t 1if m ˜t 1<∼250GeV,and for ˜τ1if tan β<∼10(or m ˜τ1>m ˜χ02).In these cases,an e +e −Linear Collider would not only allow for precision measurements but even serve as a discovery machine.6SummaryIn this contribution we discussed the phenomenology of stops,sbottoms,τ–sneutrinos,and staus at an e +e −Linear Collider with√s =500GeV.Also the detection of ˜τ’s is possible atthe LHC only in 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