We report on the progress in work on improving precision of the standard model theoretical predictions for the top quark pair production and Decay into six fermions at a linear collider. Two programs have been combined
into a single Monte Carlo program: eett6f, a MC program for e+e− ! 6f, and topfit, a program for electroweak radiative corrections to e+e− ! t¯t. The MC program is described and preliminary numerical results are shown

Biernacik, A.

Kołodziej, Karol

Lorca, A.

Riemann, T.

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Towards high precision predictions for top quark pair production and decay at a linear collider  Author: A. Biernacik, Karol Kołodziej, A. Lorca, T. Riemann  Citation style: Biernacik A., Kołodziej Karol, Lorca A., Riemann T. (2003). Towards high precision predictions for top quark pair production and decay at a linear collider. "Acta Physica Polonica B" (Vol. 34, no. 11 (2003), s. 5487-5495).   Vol. 34 (2003) ACTA PHYSICA POLONICA B No 11TOWARDS HIGH PRECISION PREDICTIONSFOR TOP QUARK PAIR PRODUCTIONAND DECAY AT A LINEAR COLLIDER∗ ∗∗Adam Biernacik, Karol KołodziejInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, PolandAlejandro Lorca and Tord RiemannDeutsches Elektronen-Synchrotron DESYPlatanenallee 6, D-15738 Zeuthen, Germany(Received October 31, 2003)We report on the progress in work on improving precision of the stan-dard model theoretical predictions for the top quark pair production and de-cay into six fermions at a linear collider. Two programs have been combinedinto a single Monte Carlo program: eett6f, a MC program for e+e− → 6f ,and topfit, a program for electroweak radiative corrections to e+e− → tt¯.The MC program is described and preliminary numerical results are shown.PACS numbers: 12.15.–y, 13.40.Ks, 14.65.Ha1. IntroductionIn order to disentangle possible new physics effects that may be revealedin the process of top quark pair productione+e− → tt¯ (1)at a linear collider from predictions of the standard model (SM), one needs toknow the latter with high precision, possibly at the level of a few per mill [1].∗ Presented at the XXVII International Conference of Theoretical Physics, “Matter tothe Deepest”, Ustroń, Poland, September 15–21, 2003.∗∗ Work supported in part by the Polish State Committee for Scientific Research (KBN)under contract No. 2 P03B 045 23, by European Commission’s 5-th Framework con-tract HPRN-CT-2000-00149 and by Sonderforschungsbereich/Transregio 9 of DFG’Computergestützte Theoretische Teilchenphysik’.(5487)5488 A. Biernacik et al.That high precision level requires taking into account radiative corrections.Because the top (and antitop) quark of reaction (1) decays into a b (and b¯)and fermion–antifermion pairs, we should study the 6 fermion reactions ofthe forme+e− → bf1f¯ ′1b¯f2f¯ ′2, (2)where f1, f′2 = νe, νµ, ντ , u, c and f′1, f2 = e−, µ−, τ−, d, s.Reactions (2) receive contributions from typically several hundred Feyn-man diagrams already at the tree level, e.g. the hadronic reaction e+e− →bud¯b¯du¯ in the unitary gauge, neglecting the Higgs boson couplings to ferm-ions lighter than a b-quark, gets contributions from 1484 Feynman diagrams.This is the main reason why the calculation of the full O(α) radiative cor-rections to any of reactions (2) seems not to be feasible at the moment.Therefore, in order to improve the precision of the lowest order predic-tions one should try to include at least the leading radiative effects, suchas initial state radiation (ISR) and factorizable radiative corrections to theprocess of the on-shell top quark pair production (1) and to the decay of thetop and antitop quarks.In the present lecture, we report on the progress of the SM theoreticalpredictions for the top quark pair production and decay into six fermions ata linear collider, paying special attention to the precision improvement.We will show a sample of results on reactions (2) including some higherorder effects including the ISR correction obtained with eett6f [2], a MonteCarlo program for reactions e+e− → 6 fermions relevant for the top produc-tion. Further, we introduce the one-loop electroweak corrections to reaction(1) obtained with topfit [3], a numerical package for calculating electroweakradiative corrections to the on-shell pair production of a massive fermion–antifermion pair. The two programs have been combined into a single MonteCarlo program.2. Calculating ISR with eett6fThe ISR correction has been implemented recently into eett6f [4] in theleading logarithmic (LL) approximation utilizing the structure function ap-proach. In this approach, the corrected differential cross section dσLL(p1, p2)of any reaction (2) readsdσBorn+ISR (p1, p2) =1∫0dx11∫0dx2 ΓLLee(x1, Q2)ΓLLee(x2, Q2)×dσBorn (x1p1, x2p2) , (3)Towards High Precision Predictions for Top Quark Pair Production . . . 5489where p1 (p2) is the four momentum of a positron (electron), x1 (x2) is thefraction of the initial momentum of the positron (electron) that remainsafter emission of a collinear photon and dσBorn(x1p1, x2p2) is the lowestorder cross section calculated at the reduced four momenta of the positronand electron. The structure function ΓLLee(x,Q2)is given by Eq. (67) of [5],with ‘BETA’ choice for non-leading terms. The splitting scale Q2, which isnot fixed in the LL approximation is chosen to be equal s = (p1 + p2)2.The ISR corrected cross sections, σBorn+ISR, of different channels of (2)obtained with the current version of eett6f are compared with the resultsof LUSIFER [6] in Table I [4], where also the corresponding lowest ordertotal cross sections σBorn are compared. The reader is referred to [7] for acomparison of the lowest order predictions with other existing MC programs.The input parameters are the same as in [6] and cuts are given byθ(l, e±) > 5◦ , θ(q, e±) > 5◦ , θ(l, l′) > 5◦ , θ(l, q) > 5◦ ,El > 10 GeV , Eq > 10 GeV , m(q, q′) > 10 GeV , (4)where θ(i, j) is the angle between particles i and j in the laboratory system,q and l denote a quark and a final state charged lepton, respectively, andm(q, q′) is the invariant mass of a qq′ quark pair.TABLE IComparison of the cross sections of different channels of (2) at√s = 500 GeV ofLUSIFER [6] and eett6f [2, 4]. The input parameters and cuts, see Eqs. (4) arethe same as in [6]. All cross sections are in fb. The number in parenthesis showthe uncertainty of the last decimals.LUSIFER eett6fe+e− → σBorn σBorn+ISR σBorn σBorn+ISRb νe e+e−ν¯e b¯ 5.8530(68) 5.6465(70) 5.8622(63) 5.6441(67)b νe e+µ−ν¯µ b¯ 5.8188(45) 5.6042(38) 5.8189(37) 5.6075(59)b νµ µ+µ−ν¯µ b¯ 5.8091(49) 5.5887(36) 5.8065(33) 5.5929(54)b νµ µ+τ−ν¯τ b¯ 5.7998(36) 5.5840(40) 5.7992(32) 5.5844(33)b νµ µ+d u¯ b¯ 17.171(24) 16.561(24) 17.213(23) 16.569(17)without QCD: 17.095(11) 16.454(10) 17.106(15) 16.457(16)b νe e+d u¯ b¯ 17.276(45) 16.577(21) 17.301(26) 16.741(43)without QCD: 17.187(21) 16.511(12) 17.149(16) 16.522(17)5490 A. Biernacik et al.The results of both programs agree nicely, basically within one standarddeviation, except for e+e− → b νe e+d u¯ b¯, where the deviation is bigger,amounting to about 3 standard deviations. The discrepancy should be clar-ified by fine tuned comparisons. Differential cross sections obtained witheett6f that are plotted in Fig. 1 agree with those presented in [6] withinaccuracy of the plots.ps = 500 GeVBornBorn + ISRe+e ! budb (with cuts)fbGeVddmbudmbud(GeV)178177176175174173172171170876543210ps = 500 GeVBornBorn + ISRe+e ! budb (with cuts)(fb)dd cos budcos bud10.50-0.5-1181614121086420Fig. 1. Invariant mass (left) and angular (right) distributions of a b¯du¯-quark tripleat√s = 500 GeV with cuts.3. Calculating one-loop electroweak corrections with topfittopfit [3] is a numerical package for calculating the complete one-loopcorrections to top-pair production (1) at a linear e+e− collider in the con-tinuum energy region. The corrections are represented in terms of six in-dependent form factors, which is suitable for implementation in the MonteCarlo generator. We recall the normalization of the differential cross sectiondσ(e+e− → tt¯)d cos ϑ=βt32pi s∑conf[|MB|2 + 2Re (M∗BδM)] , (5)where ϑ is the antitop production angle with respect to the initial e+ beam,βt =(1− 4m2t /s)1/2and the sum stands for the spin and color configurationof external fermions. There are four nonvanishing form factors F ab1Bin BornapproximationMB =∑a,b={1,5}F ab1B M1,ab, (6)while the one-loop correction δM to the amplitude can be expressed interms of six independent form factors: Fˆ ab1 , a, b = {1, 5}, plus additionallyTowards High Precision Predictions for Top Quark Pair Production . . . 5491Fˆ 113 and Fˆ513 :δM =∑a,b={1,5}Fˆ ab1 M1,ab + Fˆ 113 M3,11 + Fˆ 513 M3,51 . (7)The invariant amplitudes in Eqs. (6) and (7) are given byiM1, ab = [v¯(p1) γµ ga u(p2) ] [u¯(p3) γµ gb v(p4) ] , a, b = {1, 5},iM3,11 = [v¯(p1) 1lu(p2) ] [u¯(p3) p/2 1l v(p4) ] ,iM3,51 = [v¯(p1) p/3 γ5 u(p2) ] [u¯(p3) 1l v(p4) ] . (8)The normalization of the form factors is chosen such that the pure pho-tonic Born amplitude becomes F 11,γ1B= e2QeQt/s; (see [3] for the details onthe analytic expressions).The resulting corrected differential cross section formula can then bewritten in terms of the computed form factors, the top quark color factor(ct) and the kinematical Mandelstam variables s, t and u (s+ t+u = 2m2t ):dσd cos ϑ=βtctsα2pis2Re{4sm2t(F¯ 111 F¯11∗1B − F¯ 151 F¯ 15∗1B + F¯511 F¯51∗1B − F¯ 551 F¯ 55∗1B)+2(u2 − t2 − 2m2t (u− t))(F¯ 111 F¯55∗1B + F¯551 F¯11∗1B + F¯511 F¯15∗1B + F¯151 F¯51∗1B)+2(u2 + t2 + 2m2t (s −m2t ))∑a,b={1,5}F¯ ab1 F¯ab∗1B+4(ut −m4t )(F¯ 113 F¯11∗1B + F¯513 F¯51∗1B)}, (9)where the dimensionless barred form factors are defined for convenience asF¯ ab∗1B ≡se2F ab∗1B ,F¯ ab1 ≡se2(12F ab1B + Fˆab1),F¯ ab3 ≡se2mtFˆab3 . (10)topfit is rather flexible and includes also real photonic bremsstrahlung.In order to fit the needs of the MC program described here, some specificvalues of flags have been chosen: IWEAK = 1 (inclusion of pure weak correc-tions), IQED = IQEDAA = 0. The latter two settings switch off the calculationof photonic corrections due to bremsstrahlung (IQED, with the related vir-tual corrections) and of the fermionic vacuum polarization effects (IQEDAA).The sample corrections were computed using the same input parameters asin [8]. In Table II, form factors are given for√s = 500 GeV and cos ϑ = 0.7,and in Table III some differential cross section values for√s = 500 GeV.5492 A. Biernacik et al.TABLE IIBorn and weak one-loop form factors for e+e−→tt¯ at√s=500GeV and cosϑ = 0.7.Born F.F. (GeV−2) Weak one-loop F.F. (GeV−2)Re Im Re ImFˆ 111 −2.50926472×10−7 0 1.08874859×10−8 −2.55015622×10−9Fˆ 151 1.56200966×10−8 0 −9.75979648×10−9 −9.02716758×10−9Fˆ 511 5.62400131×10−8 0 −7.00614109×10−9 −6.39336661×10−9Fˆ 551 −1.37479846×10−7 0 −1.11566191×10−9 7.86314816×10−9mtFˆ113 0 0 8.14431322×10−10 −8.44590997×10−10mtFˆ513 0 0 −9.09997715×10−10 4.88604910×10−10TABLE IIIDifferential cross sections for e+e− → tt¯ at √s = 500 GeV.cosϑ dσd cosϑ∣∣∣Born(pb) dσd cosϑ∣∣∣Born+Weak(pb)−0.9 0.1088391941 0.1011751626−0.7 0.1218770828 0.1136440560−0.5 0.1422750694 0.13326689880.0 0.2254704640 0.21156406460.5 0.3546664703 0.33006331430.7 0.4192250441 0.38839654730.9 0.4911437158 0.45281854184. Numerical resultsIn the present section, we will show the results of implementation ofthe one-loop electroweak form factors for on-shell top quark pair productionobtained with topfit in the MC program eett6f. Amplitudes (8) havebeen implemented in eett6f with the helicity amplitude method of [9, 10].The actual numerical values of the form factors F ab1B, Fˆ ab1 , a, b = {1, 5}, Fˆ 113and Fˆ 513 are computed by topfit and then transferred to eett6f. Unfortu-nately, the direct calculation of the one-loop corrections slows down the MCcomputation by more than a factor 1000. In practice, this problem has beensolved applying the following strategy: the values of the one-loop form fac-tors are computed for a given c.m.s. energy for a grid of several hundred, or afew thousand different values of cosine of the antitop scattering angle, cos ϑ.Towards High Precision Predictions for Top Quark Pair Production . . . 5493These fixed values of the form factors are stored in the computer memoryand then used for computation of the form factors at any intermediate valueof cos ϑ by means of a linear interpolation between the neighboring fixedvalues. The interpolation routine has been tested and shown to be veryprecise and efficient.As a first step we calculate the differential cross section of reaction (2) inthe narrow width approximation for the top and antitop, which is obtainedby multiplying the one-loop corrected cross section (5) with the correspond-ing branching fractions leading to a specific final state of (2)dσ(e+e− → tt¯→ bf1f¯ ′1b¯f2f¯ ′2)= dσ(e+e− → tt¯)×dΓ(t→ bf1f¯ ′1)dΓ(t¯→ b¯f2f¯ ′2)Γt2. (11)This approach is gauge invariant and can be used as the reference forany other way of implementation of the one-loop electroweak corrections inthe MC generator for a simulation of reactions (2) to O(α).Typical results for one specific channel of reaction (2) in the narrow topwidth approximation of Eq. (11) are shown in Figs. 2 and 3. The differentialcross sections of reaction e+e− → bνµµ+b¯du¯ at√s = 500 GeV without cutsare plotted on the left hand side of Fig. 2. Both the lowest order (dottedline) and one-loop corrected (solid line) differential cross sections in the b-quark energy are shown. The same differential cross sections with cuts ofconditions (4) are plotted on the right hand side of Fig. 2. The correspondingdifferential cross sections in the energy of µ+ are plotted in Fig. 3.ps = 500 GeVBorn + weakBorne+e ! tt! b+bdufbGeVddEbEb(GeV)2502001501005000.20.150.10.050ps = 500 GeV(with cuts)Born + weakBorne+e ! tt! b+bduEb(GeV)2502001501005000.20.150.10.050Fig. 2. Energy distributions of a b-quark at√s = 500 GeV without cuts and withcuts.5494 A. Biernacik et al.ps = 500 GeVBorn + weakBorne+e ! tt! b+bdufbGeVddE+E+ (GeV)2502001501005000.30.250.20.150.10.050(with cuts)ps = 500 GeVBorn + weakBorne+e ! tt! b+bduE+ (GeV)2502001501005000.30.250.20.150.10.050Fig. 3. Energy distributions of a µ+ at√s = 500 GeV without cuts and with cuts.We see that the quite substantial effect of the one-loop electroweak cor-rection on differential cross sections of e+e− → bνµµ+b¯du¯ at√s = 500 GeVis reduced by the cuts (4). This kind of reduction is of course not desirable ifone wants to test the consistency of the SM at the quantum level. However,it might be helpful in performing relatively fast numerical simulations ofdifferent channels of reaction (2) with the use of the MC generators workingonly with the lowest order amplitudes.5. Summary and outlookWe have reported on the progress in our work on improving precisionof SM theoretical predictions for the top quark pair production and decayinto six fermions at a linear collider. Apart from including the ISR effectsin eett6f [4], we have created a MC program that combines two programseett6f and topfit. The program allows for including one-loop electroweakcorrections to the top quark pair production in reactions (2) in the narrowwidth approximation for the top and antitop. The description of these cor-rections to the on-shell top pair production has to be complemented by thecorresponding ones to the decays of the top quarks [11], in order to completethe double pole approximation. This is in preparation [12].Towards High Precision Predictions for Top Quark Pair Production . . . 5495REFERENCES[1] R.-D. Heuer, D. Miller, F. Richard, P. M. Zerwas (eds.), TESLA Technical De-sign Report, Part III: Physics at an e+e− Linear Collider, hep-ph/0106315;T. Abe et al. (American Linear Collider Working Group Collaboration),hep-ex/0106055-058; K. Abe et al. (ACFA Linear Collider Working GroupCollaboration), hep-ph/0109166.[2] K. Kołodziej, Eur. Phys. J. C23, 471 (2002); Comput. Phys. Commun. 151,339 (2003); A. Biernacik, K. Kołodziej, Nucl. Phys. Proc. Suppl. 116, 33(2003).[3] J. Fleischer, A. Leike, T. Riemann, A. Werthenbach, Eur. Phys. J. C31,37 (2003), hep-ph/0302259; J. Fleischer, A. Leike, A. Lorca, T. Riemann,A. Werthenbach, Fortran Program topfit v.0.93 (10 Jun 2003); available athttp://www-zeuthen.desy.de/˜riemann or /˜alorca.[4] K. Kołodziej, in preparation.[5] W. Beenakker et al., in G. Altarelli, T. Sjöstrand, F. Zwirner (eds.), Physicsat LEP2 (Report CERN 96-01, Geneva, 1996), Vol. 1, p. 79, hep-ph/9602351.[6] S. Dittmaier, M. Roth, Nucl. Phys. B642, 307 (2002).[7] S. Dittmaier, hep-ph/0308079.[8] T. Hahn, W. Hollik, A. Lorca, T. Riemann, A. Werthenbach,hep-ph/0307132.[9] K. Kołodziej, M. Zrałek, Phys. Rev. D43, 3619 (1991).[10] F. Jegerlehner, K. Kołodziej, Eur. Phys. J. C12, 77 (2000).[11] A. Denner, T. Sack, Nucl. Phys. B358, 46 (1991).[12] K. Kołodziej et al., in preparation.

Towards high precision predictions for top quark pair production and decay at a linear collider

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Towards high precision predictions for top quark pair production and decay at a linear collider

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ