DESY 03-004 January 2003 SUSY QCD corrections to the polarization and spin correlations of top quarks produced in e+e collisions − 3 0 0 2 n a J A.Brandenburg a, andM. Maniatisb ∗ 7 1 1 v a DESY-Theorie, D-22603Hamburg,Germany 2 b II. InsitutfürTheoretischePhysik,UniversitätHamburg, LuruperChaussee 149,D-22761 4 1 Hamburg, Germany 1 0 3 0 / h p - p e h Abstract: : v We compute the supersymmetricQCD corrections to the polarization and the spin correlations i X of top quarks produced abovethresholdin e+e collisions,taking into account arbitrary longi- − r a tudinalpolarizationoftheinitialbeams. Keywords: topquarks,supersymmetry,polarization,radiativecorrections supportedbyaHeisenbergfellowshipofD.F.G. ∗ 1 Introduction A future linear e+e collider will be an excellent tool to search for and investigate extensions − of the Standard Model (SM) of particle physics [1]. One particularly attractive extension of the SM is Supersymmetry (SUSY) [2], which solves several conceptual problems of the SM. Apartfromtheirdirectproduction,alsovirtualeffectsofSUSYparticlesmayleadtoobservable deviationsfromtheSMexpectations. Inparticular,topquarkpairproductionatalinearcollider maybeasensitiveprobeofsucheffects. Veryhighenergyscalesareinvolvedintheproduction and decay of top quarks. Moreover, since they decay very quickly, the spin of top quarks is not affected by hadronization effects and becomes an additional observable to probe top quark interactions. Atafuturelineare+e collider,theelectron(andpossiblyalsothepositron)beam − may haveasubstantiallongitudinalpolarization,which willbean asset to studytopquark spin phenomena. We therefore study in this paper the impact of virtual effects of SUSY particles on spin properties of tt¯pairs in e+e collisions. We restrict ourselves here to the SUSY QCD − sectoroftheMinimalSupersymmetricStandardModel(MSSM).SUSYQCDcorrectionstothe (spin-summed) differential cross section for e+e tt¯have already been studied quite some − → time ago [3], and we extend these results by keeping the full information on the tt¯spin state. The full MSSM corrections to the spin-summeddifferential cross section have been calculated in [4]. In section2 wedefinethespinobservablesthat wecalculateinthispaper and alsodiscusshow they can be measured. Section 3 gives analytic results for these observables, and section 4 contains numerical results for specific choices of the SUSY QCD parameters. In section 5 we present ourconclusions. 2 Spin observables We considerthereaction e+(p ,l )+e (p ,l ) (g ,Z ) t(k )+t¯(k )+X, (1) + + − ∗ ∗ t t¯ − − → → where l (l ) denotes the longitudinal polarization of the electron (positron) beam1. Within + − the Standard Model, spin effects of top quarks in reaction (1) have been analysed first in ref. [5]. QCD corrections to the production of top quark pairs, includingthe full information about their spins, can be found in refs. [6, 7]. Fully analytic results for the top quark polarization [8] and aspecific spincorrelation[9]to ordera arealso available. s The top quark polarization is defined as two times the expectation value of the top quark spin operatorS . TheoperatorS actsonthetensorproductofthet andt¯spinspacesandisgivenby s t t S = 1l, where the first (second) factor in the tensor product refers to the t (t¯) spin space. t 2 ⊗ s (The spin operator of the top antiquark is defined by S = 1l .) The expectation value is t¯ ⊗ 2 1Foraright-handedelectron(positron),l =+1. ∓ 1 taken with respect to the spin degrees of freedom of the tt¯sample described by a spin density matrix R,i.e. Tr[RS ] t P =2 S =2 . (2) t t h i TrR For details on the definition and computation of R, see e.g. [6]. The polarization of the top antiquarkP isdefinedbyreplacingS byS in(2). FortopquarkpairsproducedbyCPinvariant t¯ t t¯ interactions,wehaveP =P . Thespincorrelationsbetweent andt¯can becalculatedby using t¯ t thematrix Tr RS S t,i t¯,j C =4 S S =4 . (3) ij t,i t¯,j h i TrR (cid:2) (cid:3) Using arbitrary spin quantization axes aˆ and bˆ for the t and t¯ spins, the spin correlation with respect totheseaxesisgivenby aˆC bˆ (P aˆ)(P bˆ) c(aˆ,bˆ)= i ij j− t· t¯· . (4) 1 (P aˆ)2 1 (P bˆ)2 t t¯ − · − · q q Thedirectionsaˆ,bˆ canbechosenarbitrarily. Differentchoiceswillyielddifferentvaluesforthe spin correlation c(aˆ,bˆ). The spin properties of the top quarks and antiquarks can be measured by analysing the angular distributions of the t and t¯ decay products. For example, if both t and t¯decay semileptonically,t bℓ+n , t¯ b¯ℓ n¯ , the following double differential lepton ℓ ′− ℓ → → ′ angulardistributionissensitivetothett¯spinstate: 1 d2s 1 = (1+B cosq +B cosq Ccosq cosq ), (5) s dcosq +dcosq 4 1 + 2 −− + − − with s being the cross section for the channel under consideration. In Eq. (5) q (q ) denotes + theanglebetweenthedirectionofflightoftheleptonℓ+(ℓ )inthet (t¯)restframean−dthecho- ′− sen spin quantization axis aˆ (bˆ). The coefficients B andC are related to the mean (averaged 1,2 over the scattering angle)t (t¯) polarization and spin correlation projected onto the directions aˆ andbˆ. Usingthedoublepoleapproximation[10]forthet andt¯propagators,oneobtainsforthe so-called factorizablecontributions[11, 12] B = k P aˆ, 1 + t · B = k P bˆ, 2 t¯ − − · C = k k aˆC bˆ , (6) + i ij j − where theoverlineindicatestheaverageoverthescattering angle,e.g. 1 dyTr[RS aˆ] P aˆ =2 1 t· , (7) t· R− 1 dyTrR 1 − etc., where y is the cosine of the top quarkRscattering angle. In (6), k is the spin analysing power of thecharged lepton ℓ . At leading order, k =+1. QCD corre±ctions to this result are ± at the per mill level [13]. SUSY QCD corrections t±o the spin analysing power k are exactly ± zero [14]. 2 3 Analytic results We now turn towards the calculation of the SUSY QCD corrections to the polarization and spin correlations of top quark pairs produced in e+e collisions. These corrections directly − determine the SUSY QCD corrections to the double lepton distribution (5) within the double pole approximation, since the corrections to the LO result k = +1 are exactly zero and the ± non-factorizablecontributionsduetoSUSY particlesalso vanishwithinthatapproximation. Theamplitudeforreaction (1)includingSUSY QCD correctionsmay bewrittenas follows: 4pa iTfi = i s c (s)v¯(p+) gVeg µ−geAg µg 5 u(p−)HZµ−v¯(p+)g µu(p−)Hgµ , (8) (cid:26) (cid:27) (cid:0) (cid:1) where ge = 1 +2sin2J , and ge = 1, with J denoting the weak mixing angle. The functionVc is−giv2enby W A −2 W 1 s c (s)= , (9) 4sin2J cos2J s m2 W W − Z where m standsfor themass of theZ boson. We neglect theZ width,since wework at lowest Z order in the electroweak coupling and the c.m. energy is far above m . The hadronic currents Z haveaformfactordecompositionas follows: µ (k k ) HZµ,g =u¯(kt) VZ,g g µ−AZ,g g µg 5+SZ,g t2−m t¯ v(kt¯) (10) t (cid:20) (cid:21) with VZ,g = VZ0,g +VZ1,g , AZ,g = A0Z,g +A1Z,g . (11) In (11), Vg0 =Qt, where Qt denotes the electric charge of the top quark in units of e=√4pa , A0g =0,andVZ0=gVt = 12−43sin2J W,A0Z =gtA= 12 arethetreelevelvector-andtheaxial-vector couplingsofthetop quarkto theZ boson. The one-loop SUSY QCD contributions to the different form factors are denoted byV1 , A1 g ,Z g ,Z andSg ,Z. Scalarandpseudoscalarcouplingsproportionalto(kt+kt¯)µand(kt+kt¯)µg 5havebeen neglected in (8), since they induce contributions proportional to the electron mass. In addition CP violating formfactors proportional to (k k )µg are possible in SUSY QCD through a t t¯ 5 − complex phase in the squark mass matrices. In [15] it has been shown that the dependence of the cross section on these phases is weak and that CP odd asymmetries are typically of the order of 10 3. We therefore set these phases to zero in the following. To make this paper self- − contained we list the form factors Vg1,Z, A1g ,Z and Sg ,Z in the appendix. We have performed an analyticcomparisontothecorrespondingresults in[3]and foundcompleteagreement. We define f = Q +c (ge +ge)(gt gt ), LL(LR) t V A V A − ± f = Q +c (ge ge)(gt gt ), (12) RR(RL) t V A V A − − ∓ 3 and P =1 l l (l l ). (13) + + ± − − ± −− Theelectroweak couplingsthat entertheBorn resultsarethen givenby 1 g = P (f + f )2 P (f + f )2 , V±V 8 + RR RL ± − LL LR g = 1(cid:2)P (f f )2 P (f f )2(cid:3), ±AA 8 + RR− RL ± − LL− LR g = 1(cid:2)P (f2 f2 ) P (f2 f2 ) .(cid:3) (14) V±A 8 + RR− RL ± − LL− LR (cid:2) (cid:3) Likewise,defining gLL(LR) = c (gVe +geA)(VZ1 A1Z) (Vg1 A1g ) ± − ± gRR(RL) = c (gVe gVe)(VZ1 A1Z) (Vg1 A1g ), − ∓ − ∓ sL(R) = c (gVe geA)SZ Sg ,. (15) ± − theSUSY QCD contributionsmaybewrittenin termsofthefollowingquantities: 1 h = [P (f + f )(g +g ) P (f + f )(g +g )], V±V 8 + RR RL RR RL ± − LL LR LL LR 1 h = [P (f f )(g g ) P (f f )(g g )], ±AA 8 + RR− RL RR− RL ± − LL− LR LL− LR 1 Re h = Re [P (f g f g ) P (f g f g )], V±A 8 + RR RR− RL RL ± − LL LL− LR LR 1 Im h = Im [P (f g f g ) P (f g f g )], V±A 8 + RL RR− RR RL ± − LR LL− LL LR 1 s = [P (f + f )s P (f + f )s ], V± 4 + RR RL R± − LL LR L 1 s = [P (f f )s P (f f )s ]. (16) ±A −4 + RR− RL R± − LL− LR L It is convenient to write the results in terms of the electron and top quark directions pˆ and kˆ definedinthec.m. system,thecosineofthescatteringangley=pˆ kˆ,thescaledtopquarkmass · r =2m /√s,and thetopquark velocityb =√1 r2. t − Thedifferentialcross sectionincludingtheSUSY QCD correctionsreads: ds ds 0 ds 1 3N b = + =s C 2 b 2(1 y2) g+ +2Re h+ dy dy dy pt 8 − − VV VV (cid:26) (cid:2) (cid:3)(cid:0) (cid:1) + b 2(1+y2) g+ +2Re h+ +4b y g+ +2Re h+ 2b 2(1 y2)Res+ , (17) AA AA VA VA − − V (cid:27) (cid:0) (cid:1) (cid:0) (cid:1) 4 where 4pa 2 s = , (18) pt 3s and ds 0/dy is obtained by setting h+ =h+ =h+ = s+ =0. We further introduce a vector VV AA VA V perpendicular to k in the production plane k = pˆ ykˆ and a vector normal to this plane, ⊥ n=pˆ kˆ. The topquark polarization includingtheS−USY QCD corrections is equal to thetop × antiquark polarizationand reads: P = P0+P1 t t t 3N b = s C b (1+y2) g +2Reh +y g +2Re h +b 2y g +2Re h kˆ pt 4 V−A V−A V−V V−V −AA −AA (cid:26) (cid:2) (cid:0) (cid:1) (cid:0)b 2 (cid:1) (cid:0) (cid:1)(cid:3) + r b y g +2Reh +g +2Re h Res b yRe s k V−A V−A V−V V−V − r2 V−− −A ⊥ (cid:20) (cid:21) (cid:0) b 2 (cid:1) ds(cid:0)0 −1 ds 1 (cid:1)ds 0 −1 + 2b rImh+ + yIm s+ b Ims+ n P0 . (19) VA r V − A dy − t dy dy (cid:20) (cid:21) (cid:27)(cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) ForthematrixC defined in(4)wefind ij 1 ds 1 ds 0 −1 C = C0 +C1 = d 1+ ij ij ij 3 ij dy dy " # (cid:18) (cid:19) + s 3NCb ds 0 −1 g+ +2Reh+ b 2 g+ +2Re h+ k k 1d (1 y2) pt 4 dy VV VV − AA AA i⊥ ⊥j −3 ij − (cid:18) (cid:19) (cid:26) (cid:20) (cid:21) (cid:2) (cid:0) (cid:1)(cid:3) + y2+b 2(1 y2) g+ +2Re h+ +b 2y2 g+ +2Re h+ +2b y g+ +2Re h+ − VV VV AA AA VA VA h(cid:0) (cid:1)(cid:0) 1 (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) + 2b 2(1 y2)Res+ kˆ kˆ d − V i j−3 ij (cid:20) (cid:21) i b 2 + r y g+ +2Re h+ +b g+ +2Re h+ yRe s+ b Res+ k kˆ +k kˆ VV VV VA VA − r2 V − A i⊥ j ⊥j i (cid:20) (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) b (cid:0) (cid:1) h i + 2b Imh k n +k n +b 2yrIm h + Im s b yIm s kˆ n +kˆ n V−A i⊥ j ⊥j i V−A r V−− −A i j j i (cid:20) (cid:21) h i ds 1 ds 0 −1 (cid:0) (cid:1) (cid:2) (cid:3) C0 . (20) − ij dy dy (cid:18) (cid:19) (cid:27) TheBornresultsP0 andC0 areobtainedfrom(19)and(20)bysettingh =h =h =s = t ij V±A V−V −AA V± s =ds 1/dy=0. ±A For fully polarized electrons (or positrons)a so-called ‘optimal spin basis’ can be constructed. This is an axis dˆ with respect to which thet andt¯spins are 100% correlated at the tree level in theStandardModelforanyvelocityandscatteringangle[16]. Thisaxisdˆ isthesolutionofthe equation dˆC0dˆ =1. (21) i ij j 5 Onegets dˆ =xkˆ + 1 x2kˆ , (22) ⊥ − with x [ 1,1] only if either P = 0 or Pp= 0. For P = 0, which can be realized with + + left-han∈ded−electrons (l = 1), onefinds − − − f (b +y)+ f (y b ) LL LR x= − . (23) − (1+yb )2f2 +(1 yb )2f2 +2(y2b 2+1 2b 2)f f 1/2 LL LR LL LR − − For right-handed (cid:2)electrons, the optimal basis is obtained by replacing fLL(cid:3) fRR, fLR fRL in Eq. (23). Note that at threshold dˆb →0pˆ, i.e. the optimal basis at threshol→d is defined→by the direction of the beam, while in the h−ig→h-energy limit dˆb →1kˆ, i.e. the optimal basis coincides with the helicity basis. By analytically evaluating dˆC1d−ˆ→we find that the virtual SUSY QCD i ij j corrections tothett¯spincorrelationsin theoptimalbasisareexactlyzero. 4 Numerical results In this section we present numerical results for the SUSY QCD corrections to the top quark polarization and tt¯ spin correlations. We also include a discussion of the corrections to the differential crosssectionand compareourresultstotheliterature. Wetakeintoaccounttheeffectsofmixingofthechiralcomponentsofthetopsquark. Thestop mass matrixcan beexpressedinterms ofMSSMparameters as follows: M2 +m2+m2(1 Q s2 )cos2b m (A µcotb ) M 2 = Q˜ t Z 2− t W t t− , (24) t˜ m (A µcotb ) M2 +m2+m2Q s2 cos2b t t− U˜ t Z t W ! where M , M are the soft SUSY-breaking parameters for the squark doublet q˜ (q = t,b) Q˜ U˜ L and the top squark singlet t˜ , respectively. Further, A is the stop soft SUSY-breaking trilinear R t coupling,andµistheSUSY-preservingbilinearHiggscoupling. TheratioofthetwoHiggsvac- uumexpectationvaluesisgivenbytanb ,andweusetheabbreviations =sinq . Thesquared W W physical masses of the stops are the eigenvalues of the above matrix. In order to simplify the discussion,weset tanb =1forall followingresults. Further, weassumethatthesbottommass matrix is diagonal with degenerate mass eigenvalues, M 2 = diag(m2,m2). Neglecting m in b˜ b˜ b˜ b the sbottom mass matrix this leads to M =m , and the stop mass matrix simplifies under the Q˜ b˜ aboveassumptionsto m2+m2 m M M 2 = b˜ t t LR , (25) t˜ m M M2 +m2 (cid:18) t LR U˜ t (cid:19) with M =A µ. Thestopmasseigenstatesare obtainedfrom thechiral statesby arotation: LR t − t˜ cosq sinq t˜ 1 = t˜ t˜ L . (26) t˜ sinq cosq t˜ 2 t˜ t˜ R (cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19) 6 2 1 0 [%] −1 mg˜=250GeV 10 mg˜=150GeV ss −2 −3 −4 (√s=500GeV,mb˜=100GeV) −5 −200 −100 0 100 200 MLR[GeV] Figure 1: Relative correction to the total cross section as a function of the mixing parameter M forafixedmixingangleq =p /4. Shownisthecorrectionfortwodifferentgluinomasses LR q˜ m =150 GeV andm =250 GeV. g˜ g˜ Maximal mixing (q = p and M =0) corresponds to M2 =m2. The latter relation will also t˜ 4 LR 6 U˜ b˜ beassumedfor M =0,leading tothefollowingstopmasseigenvalues2: LR m = m2+m2 m M . (27) t˜1,2 b˜ t ± t LR q Notethatweuseherethesamesetofassumptionsonthesquarkmassmatricesaswedidinour study of the SUSY QCD corrections in the decay of polarized top quarks [14]. Further we use sin2q =0.2236, a = 0.11, and we set the top mass to m = 174 GeV and the sbottom mass W s t that enters Eq.(27)tom =100 GeV. b˜ Fig. 1 showstherelativeSUSY QCD correction s 1/s 0 to thetotal cross sectionfor e+e tt¯ − → with unpolarized beams at √s = 500 GeV as a function of the mixing parameter M , where LR s 0 and s 1 are obtained from Eq. (17) by integrating over y. Shown are the relative corrections for twodifferent gluinomasses,namelym =150 GeV and m =250 GeV. Foralarge mixing g˜ g˜ parameter M and a small gluino mass of m = 150 GeV we find a large negativecorrection. LR g˜ Thecorrectiondecreases asthegluinomassincreases. AmixingparameterofM =200GeV LR corresponds to a light stop mass of m = 74 GeV, which is above the current experimental t˜ 2 lowerlimit[17]. Withourchoiceofthemasses, weare faraway from thethresholdsingularity at m =m +m ,where amoresophisticatedcalculationisnecessary. t g˜ t˜ Fig. 2 shows the differential cross section ds /dy, again for two different gluino masses m = g˜ 150 GeV and m = 250GeV, and forthecases of’no mixing’(M =0)and ’mixing’(M = g˜ LR LR 200 GeV and q = p /4), again at √s = 500 GeV. We have compared our results for s and q˜ ds /dywith[3]andfoundagreementwiththeirFig.3(nomixingcase),whilewedisagreewith the resultsdepicted in Fig. 4 (s and forward-backward asymmetrywithstop mixing). Wehave also compared ourresultsincludingthemixingwith[4,18]and find completeagreement. 2Notethatbyfixingq t˜= p4 thelightstopcanbeeithert˜1 ort˜2 dependingonthesignofMLR. 7 2 no mixing 1 0 %] −1 mixing [ 1dy/0dy/ −2 sdsd −3 −4 mg˜=150GeV mg˜=250GeV mixing −5 −6 −1 −0.5 0 0.5 1 y Figure 2: Relative correction to the differential cross section ds /dy at √s=500 GeV for the cases ofno mixing(M =0)andmixing(q =p /4and M =200GeV). LR q˜ LR We nowturntowardsthediscussionoftheSUSY QCD correctionsto thett¯spinproperties. In Fig. 3 we investigate the expectation value of the top spin operator as a function of the centre-of-massenergy. Wehavecomputedtheaverageprojectedpolarizationdefined inEq.(7) for three choices of the quantization axis aˆ, namely for aˆ = kˆ (flight direction of the top), for aˆ =pˆ (electronbeamdirection),andforaˆ =nˆ (normaltotheeventplane). Thesequantitiesare showninthreedifferentplots,wherethincurvescorrespondtothetreelevelresultsandthethick curves are therelativecorrections in percent. The corrections are shown forthe case of mixing (q =p /4andM =200GeV)andagluinomassofm =150GeV.Forthepolarizationsofthe q˜ LR g˜ initialbeamswechoosel =0andconsiderthethreecasesl = 1,0,+1. Theprojectionof + − − thetop quark polarizationontonˆ vanishesat treelevel,and thusweonlyshowthecontribution from SUSY QCD absorptive parts in percent. In all cases SUSY QCD effects change the tree levelresultsby lessthan1% and vanishat threshold. InFig.4weshowtheaveragedspincorrelationsaˆC bˆ forthechoicesaˆ =bˆ =kˆ (helicitycor- i ij j relation),aˆ =bˆ =pˆ (beamlinecorrelation),andaˆ =pˆ, bˆ =kˆ,forthesamechoiceofparameters as in Fig. 3. Again the SUSY QCD correction are tiny. Fig. 5 shows the correlations for the choicesaˆ =kˆ, bˆ =nˆ andaˆ =pˆ, bˆ =nˆ. Thefirstofthesetwochoicesofspinquantizationaxes leads to SUSY QCD effects slightlylarger than1% around c.m. energies of700GeV and fora fully polarizedelectron beam. 5 Conclusions In this paper we have derived analytic expressions for the SUSY QCD corrections to the po- larization and spin correlations of tt¯ pairs produced in e+e annihilation with longitudinally − 8 0.7 1 0.6 k) c hi 0.5 (t 0.5 %] 0.4 l =0 ˆ0Pk[/t0.3 (thin) ll −−−==−+11 ˆ1Pkt0.2 ˆ0Pkt0 0.1 0 −0.5 −0.1 −0.2 −1 400 500 600 700 800 900 1000 √s[GeV] 1 k) 0.6 c hi (t 0.5 0.5 %] l =0 0ˆPp[/t0.4 (thin) ll −−−==−+11 1ˆPpt0.3 0ˆPpt0 0.2 0.1 −0.5 0 −0.1 −1 400 500 600 700 800 900 1000 √s[GeV] 0.6 0.5 0.4 %] [ 0.3 ˆn Pt 0.2 0.1 l =0 l −=+1 l −= 1 0 − − −0.1 −0.2 400 500 600 700 800 900 1000 √s[GeV] Figure 3: Average projected top quark polarization P aˆ defined in (7) for the choices aˆ = kˆ t (top), aˆ =pˆ (middle), aˆ =nˆ (bottom) as a function of the centre-of-mass energy. In each plot weshowthetreelevelresults(thinlines)andtherelativecorrectionsinpercent(thicklines)for unpolarized positronsand thethreecases l = 1,0,+1. − − 9