Factorizable electroweak O(alpha) corrections for top quark pair production and decay at a linear e+e- collider

We calculate the standard model predictions for top quark pair production and decay into six fermions at a linear e+ e- collider. We include the factorizable electroweak O(alpha) corrections in the pole approximation and QED corrections due to the initial state radiation in the structure function approach. The effects of the radiative corrections on the predictions are illustrated by showing numerical results for two selected six-fermion reactions e+ e- ->b nu_mu mu+ bar{b} mu- bar{nu_mu} and e+ e- ->b nu_mu mu+ bar{b} d bar{u}.


INTRODUCTION
Precise measurements of top quark pair production e + e − → tt (1) at the threshold and in the continuum region will belong to the basic physics program of the future International Linear Collider (ILC) [1]. In order to fully profit from these high precision measurements one has to bring theoretical predictions to at least the same, or preferably better, precision, which obviously requires taking into account radiative corrections. The latter should be calculated not only for the on-shell production process (1). Due to their large widths the t-andt-quark of reaction (1) almost immediately decay into bW + andbW − , respectively, and the W -bosons subsequently into 2 fermions each, thus constituting six-fermion reactions of the form where f 1 , f ′ 2 = ν e , ν µ , ν τ , u, c and f ′ 1 , f 2 = e − , µ − , τ − , d, s. Typical lowest order Feynman diagrams of reaction (2) are shown in Fig. 1. As decays of the top and antitop take place before toponium resonances can form, the Standard Model (SM) predictions for reaction (1) can be obtained with the perturbative method. The QCD predictions for reaction (1) in the threshold region were obtained in [2] and then improved by calculation of the next-to-next-to-leading order QCD corrections [3], and by including the effects of initial state radiation and beamstrahlung [4]. The O(αα s ) [5,6,7] and O(αα 2 s ) [8] corrections to the subsequent top decay into a W boson and a b quark are also known. In the continuum above the threshold, the QCD predictions for reaction (1) are known to order α 2 s [9] and the electroweak (EW) corrections to one-loop order [10,11,12], including the hard bremsstrahlung corrections [13,11]. The QCD and EW corrections are large, typically of O(10%). Order α s [15] and α 2 s QCD, and EW corrections have been combined in [16]. Quite recently the EW radiative corrections to (1) have been recalculated with a program topfit [11,12] and thoroughly compared with results of other calculations, with hard bremsstrahlung [17] and without it [18]. Finally, the radiative corrections to W decays into fermion pairs, which have to be taken into account too, are also known [19,20,21].
At tree level, reactions (2) can be studied with a Monte Carlo (MC) program eett6f [22,23] or with any other MC program dedicated to the six fermion reactions, such as Sixphact [24], Sixfap [25], Lusifer [26], or with any of multi-purpose generators, such as Amegic [27], Grace [28]/Bases [29], Madgraph [30]/Madevent [31], Phegas [32]/Helac [33], or Whizard [34]/Comphep [35], Madgraph [30], or O'mega [36]. Thorough comparison of the lowest order predictions for several different channels of (2) obtained with Amegic++, eett6f, Lusifer, Phegas, Sixfap and Whizard have been performed in the framework of the Monte Carlo Generators group of the ECFA/DESY workshop [37]. A survey of SM cross sections of all six fermion reactions with up to four quarks in the limit of massless fermions (but the top quark), has been done in [26]. The latter contains also a fine tuned comparison of both the lowest order and lowest order plus ISR results, obtained in the structure function approach, between Lusifer and Whizard.
Concerning radiative corrections to the six-fermion reactions (2), the situation is less advanced. Already at the tree level, any of the reactions receives contributions from typically several hundred Feynman diagrams, e.g. in the unitary gauge, with neglect of the Higgs boson couplings to fermions lighter than the b quark, reactions e + e − → bν µ µ +b dū, e + e − → bν µ µ +b µ −ν µ , and e + e − → budbdū get contributions from 264, 452, and 1484 Feynman diagrams, respectively. Hence, the calculation of the full O(α) radiative corrections to any of reactions (2) seems not to be feasible at present. Therefore, in the present note we will make a step towards improving precision of the lowest order predictions for (2) by including leading radiative effects, such as initial state radiation (ISR) and factorizable EW radiative corrections to the process of the on-shell top quark pair production (1), to the decay of the t (t) into bW + (bW − ) and to the subsequent decays of the W -bosons. We will illustrate an effect of of these corrections by showing numerical results for the two selected six-fermion reactions and e + e − → bν µ µ +b dū.

CALCULATIONAL SCHEME
We calculate the ISR and the factorizable SM corrections for the reaction where the particle momenta and helicities have been indicated in the parentheses, according to the following formula: where x 1 p 1 (x 2 p 2 ) is the four momentum of the positron (electron) after emission of a collinear photon. The structure function Γ LL ee (x, Q 2 ) is given by Eq. (67) of [38], with 'BETA' choice for non-leading terms. The splitting scale Q 2 , which is not fixed in the LL approximation is chosen to be s = (p 1 + p 2 ) 2 . By dσ Born+FEWC we denote the cross section including the factorizable EW O(α) corrections where M Born is the matrix element of reaction (5) obtained with the complete set of the lowest order Feynman diagrams, M tt and δM tt,FEW is, respectively, the lowest order amplitude of the 'signal' Feynman diagram of Fig. 1a and the corresponding factorizable EW O(α) correction, both in the pole approximation. The overlines in (7) denote, as usual, an initial state particle spin average and a sum over final state particle polarizations, and dΦ 6f is the Lorentz invariant six-particle phase space element. The basic phase space parametrizations which are used in the program are given by Eqs. (7)-(9) of [22]. The corrections that we take into account in δM tt,FEW are illustrated diagramatically in Fig. 2. In the pole approximation, the polarized lowest order amplitude M tt and the one-loop correction δM tt,FEW of Eq. (7) can be expressed analytically as follows: where the lowest order t andt decay amplitudes and the corresponding one-loop corrections read In (8)(9)(10)(11)(12)(13), σ t , σt and λ W + , λ W − denote polarizations of the intermediate top quarks and W bosons which are treated as on-shell particles, except for keeping their actual off-shell momenta in the denominators D t (p) and D W (p) of their propagators The fixed widths Γ t and Γ W of (15) are calculated in the program for a given set of initial parameters. They are set to their SM lowest order values, Γ W , for the Born cross sections, or they include radiative corrections of the same kind as those included in the numerators of (9), (12) and (13) for the radiatively corrected cross sections. While explaining further the notation of Eqs. (8-13) we will suppress the polarization indices. M e + e − →tt and δM e + e − →tt are the lowest order and the EW one-loop amplitudes of the on-shell top quark pair production process (1). They can be decomposed in a basis composed of the following invariant amplitudes The projected four momenta k t , kt of the on-shell top-and antitop-quark of (16), as well as the four momenta k W + , k W − of the on-shell W -bosons and the four momenta k 3 , . . . , k 8 of the decay fermions, which are used later, have been obtained from the four momenta of the final state fermions p 3 , . . . , p 8 of reaction (2) with the projection procedure described in Appendix A.
In terms of invariant amplitudes (16), the lowest order amplitude of (1) reads where the 4 Born form factors F ab 1B are given by In (18), e W is the effective electric charge, e W = √ 4πα W , with the Z-boson propagator is contained in the factor and we have used the following conventions for couplings of the electron and top quark to a photon and Z-boson We have introduced a constant Z-boson width Γ Z in (20), in a similar way as Γ t and Γ W have been introduced in (15), although the Z-boson propagator in the e + e − annihilation channel never becomes resonant in the CMS energy range above the tt-pair production threshold. Generally speaking, the constant width Γ of an unstable particle is introduced into the lowest order matrix elements by replacing its mass with the complex mass parameter in the corresponding propagator, both in the s-and t-channel one, while keeping the electroweak mixing parameter sin 2 θ W of (19) real. This approach is usually referred to in the literature as the fixed width scheme (FWS). The approach, in which m 2 W and m 2 Z are replaced with their complex counterparts according to (22) also in sin 2 θ W of (19) is on the other hand referred to as the complex mass scheme [39]. The latter has the advantage that it preserves Ward identities. Let us note, that in Eqs. (8)(9)(10)(11)(12)(13), substitution (22) with the six independent form factors:F ab 1 , a, b = 1, 5,F 11 3 andF 51 3 which are calculated numerically with a program topfit [11,12] that is tailored to a subroutine of a new version of eett6f. Note that a factor i has been omitted on the left hand side of (16) compared to [11]. Keeping it would result in an extra minus sign on the right hand side of (8) and (9), as we neglect the i factor in every vertex and propagator and consequently the resulting common +i factor for every Feynman diagram in the present work. The flags in topfit switch off all photonic corrections there, including the running of the electromagnetic coupling. This means that only the genuine weak corrections will contribute.
In order to fix normalization we give the formula for the EW one-loop corrected cross section dσ e + e − → tt of the on-shell top production (1) where the matrix elements M e + e − →tt and δM e + e − →tt are given by (17) and (23) and dΦ 2f is the Lorentz invariant two-particle phase space element with p t being the momentum and Ω t the solid angle of the t-quark.
The t-andt-quark decay amplitudes M t→bW + and Mt →bW − , and the corresponding one-loop corrections δM t→bW + and δMt →bW − can be decomposed in terms of the invariant amplitudes where P σ = (1+σγ 5 )/2, σ = ±1, are the chirality projectors and we have used real polarization vectors for W bosons. The decomposition reads In (27), g W f f is the SM W boson coupling to fermions which, similarly to the Born form factors of (18), is defined in terms of the effective electric charge e W t,i are calculated numerically with a newly written dedicated subroutine that reproduces results of [6,7]. The one-loop form factors of the antitop decay are then obtained assuming CP conservation which lead to the following relations Note that the imaginary parts of the form factors do not contribute at the one-loop order.
Similarly the W + -and W − -boson decay amplitudes M W + →f 1f ′ 1 and M W − →f 2f ′ 2 , and the corresponding one-loop corrections δM W + →f 1f ′ 1 and δM W − →f 2f ′ 2 are given by with the invariant amplitudes and the EW one-loop form factors of the W -boson decays F (σ) W ± ,i being calculated numerically, this time with a new subroutine that reproduces results of [21,7] for the EW corrected Wboson width. Again, the imaginary parts of the form factors do not contribute at the one-loop order.
The calculation of the EW factorizable corrections to reaction (2) Fig. 1(a) stop to dominate the cross section and the association of the reduced phase space point, at which the EW factorizable O(α) corrections depicted in Fig. 2 are calculated, with the phase space point of the full six particle phase space of (2) may lead to unnecessary distortion of the off resonance background contributions. Therefore in the following we will impose kinematical cuts on the quantities (34) which describe the relative departures of the invariant masses of (32) and (33) from m t and m W , respectively.

NUMERICAL RESULTS
In this section, we will illustrate the effect of the factorizable EW O(α) corrections described in Section 2 on the SM predictions for six fermion reactions relevant for detection of the top quark pair production and decay at the ILC (2) by showing results for total cross sections of its two specific channels (3)  Assuming a value of the Higgs boson mass, the W boson mass and the Z boson width are determined with ZFITTER [40], while the SM Higgs boson width is calculated with HDECAY [41]. We obtain the following values of these parameters for m H = 120 GeV: The actual values of the Z and Higgs boson widths are not very relevant in the context of the top quark pair production as they enter the calculation through the off resonance background contributions. The EW corrected top quark and W boson widths, which on the other hand play an essential role for the calculation, are calculated with a newly written dedicated subroutine that reproduces results of [21,7]. We obtain the following values for them for the parameters specified in (35)(36)(37) Γ W = 2.03777 GeV, Γ t = 1.67432GeV.
We have neglected the QCD correction to the widths Γ W and Γ t , as no QCD corrections have been included in the one-loop corrections to the tt-pair production process. The EW corrected widths of (39) are used in the calculation of the cross sections that include the EW factorizable corrections. For the calculation of the lowest order cross sections of (3) and (4) the corresponding lowest order SM values of the top quark and W -boson widths are used.
Results for the total cross sections of reactions (3) and (4) at three different centre of mass (CMS) energies in the presence of the following cuts on quantities δ t , δt, δ W + , δ W − , defined in (34), are shown in Table 1. The second column shows the Born cross sections calculated with the complete set of the lowest order Feynman diagrams. The third column shows the Born 'signal' cross section, i.e. the cross section obtained with the two lowest order signal diagrams of Fig. 1a only. We see that imposing the invariant mass cuts (40) efficiently reduces the off resonance background, which becomes quite sizeable if the cuts are not imposed [23,42]. The fourth and fifth columns show the cross sections including the ISR and factorizable EW corrections separately and the sixth column shows the results including both the ISR and EW factorizable corrections. Note that the cross sections of (4) are almost exactly 3 times larger than the cross sections of (3), in agreement with the naive counting of the colour degrees of freedom. This is because the neutral current off resonance background contributions that make reaction (3) differ from (4) are almost completely suppressed in the presence of cuts (40).
How the radiative corrections for the six fermion reactions (2) depend on the CMS energy is illustrated in Fig. 3, where, on the left hand side, we plot the total cross sections of reaction (4) as a function of the CMS energy, both in the lowest order and including different classes of corrections. The dashed-dotted line shows the Born cross section, the dotted line is the cross section including the ISR correction, the dashed line shows an effect of the factorizable EW correction while the solid line shows an effect of the combined ISR and factorizable EW correction. The plots on the right hand side of Fig. 3 show the corresponding relative corrections δ cor. = σ Born+cor. − σ Born σ Born , cor. = FEW, ISR, ISR + FEW.
The dashed line shows the relative factorizable EW correction. The correction is small and positive a few GeV above the tt-pair production threshold, but already about 20 GeV above the threshold it becomes negative and it falls down logarithmically towards more and more negative values, due to large logarithmic terms ∼ [ln (m 2 W /s)] 2 and ∼ ln (m 2 W /s), reaching 20%  (3) and (4) in fb at three different CMS energies in the presence of cuts (40). The numbers in parenthesis show the uncertainty of the last decimals.

SUMMARY AND OUTLOOK
We have calculated the SM predictions for top quark pair production and decay into six fermions at a linear e + e − collider. We have included the factorizable EW O(α) corrections in the pole approximation and QED corrections due to the initial state radiation in the structure function approach into SM predictions for the top quark pair production and decay into six fermions at the ILC. We have illustrated an effect of the radiative corrections on the predictions by showing numerical results for two selected six-fermion reactions (3) and (4). The ISR and factorizable EW radiative corrections are sizeable and therefore should be included in the analysis of future precision data on the top quark pair production and decay from the ILC.
In order to obtain a complete EW next to leading order result for six fermion reactions (2) in the pole approximation one should include the nonfactorizable virtual photonic corrections corresponding to an exchange of a virtual photon between the electrically charged lines of the signal diagrams of Fig. 1(a) which has not been included in the shaded blobs of Fig. 2  For example, an exchange of a photon between the initial state electron and any of the final state fermions or intermediate W bosons, or between the b andt quark, or its decay products should be taken into account. This would allow for inclusion of the real photon emission from the external legs in an exclusive way. Taking into account the QCD coreections would also be higly desirable. k t = Table A: A comparison of two randomly selected sets of the four momenta p i , i = 3, ..., 8, p 3 + p 4 + p 5 , p 6 + p 7 + p 8 , p 4 + p 5 , p 7 + p 8 and their projections k i , i = 3, ..., 8 , k t , kt , k W + , k W − , respectively. Quantities δ t = m 345 /m t − 1, δt = m 678 /m t − 1, δ W + = m 45 /m W − 1 and δ W − = m 78 /m W − 1 describe relative departures of the corresponding final state particle subsystems from a mass-shell of the t,t, W + and W − , respectively. δ t = 0.03%, δt = 0.19%, δ t = 0.06%, δt = 0.17%, GeV δ W + = 0.26%, δ W − = 0.85% δ W + = 0.78%, δ W − = 3.23%