Single top production at linear $e^-e^+$ colliders

We study single top production at linear lepton colliders with $\sqrt{s} = 0.5$, 1 and 3 TeV. A preliminary analysis shows that despite the large $t\bar t$ background at $\sqrt{s} = 0.5$ and 1 TeV it is possible to obtain a good sensitivity to the $tbW$ vertex; even more so at 3 TeV when single top becomes the dominant mode of production. Concerning the four dimension six operators involved, two of them simultaneously generate $ttZ$ couplings and their sensitivity decreases with energy. The opposite is true for the other two operators. Single top production at these machines is also useful to probe charged-current four fermion operators $e\nu bt$, some of which are related to $tbW$ operators through the equations of motion.


Introduction
Future linear lepton colliders such as the International Linear Collider (ILC) [1][2][3][4][5] and the Compact Linear Collider (CLIC) [6,7] have the top quark as one of their main areas of research. In particular, an extensive effort based on the dimension 6 operators [8] of the Standard Model Effective Field Theory is being developed for the top-quark physics program at these colliders as well as at the LHC [9]. For the LHC, significant limits have been obtained when the effective couplings enter in loop-level [10][11][12] as well as tree-level processes [13,14], and many experimental measurements can be found in the literature, for tt production, W -helicity in top quark decay, rare top decays, same-sign tops production, single-top, mono-top and multiple-top production [15].
In the context of top-quark production in e − e + colliders, so far most of the interest has been placed on tt and ttH production at the ILC for beam energies of √ s = 0.5 and 1 TeV [16], and very few studies have been done on single-top production [17][18][19][20][21]. It is now known that ttZ(γ) couplings will be far better probed in this machine than at the LHC [22,23]. However, with respect to the tbW coupling the LHC is already providing very strong limits through single-top production and W -helicity in top decays [24]. In contrast, tt production at the linear collider has very little sensitivity to tbW even for angular distributions of decay products [25]. To date, there is no study on the potential of single-top production in e − e + collisions to probe the effective tbW coupling. This is one of the goals of this paper: to find out what is the sensitivity to this coupling and how it compares with the potential of e + e − → tt as well as the LHC. Furthermore, we also study the sensitivity to four-fermion operators eνbt relevant to this process. We refer in this study to the basis of dimension-6, SU(3)×SU(2)×U(1) gauge-invariant operators provided in [8]. In tables 1 and 2 we show the flavor-diagonal operators in that basis that are relevant to top quark production at the ILC and CLIC. The notation used here is standard: τ I are the Pauli matrices, ϕ is the SM Higgs doublet withφ = iτ 2 ϕ * , q L is the third generation left-handed SU (2) doublet, t R and b R are the right-handed SU (2) singlets. The covariant derivative is defined as D µ ϕ = ∂ µ ϕ − ig/2τ I W I µ ϕ − ig /2B µ ϕ. As described in more detail below, we will follow the operator normalization used in [11,13].
ttZ + ttA ttZ + ttA + tbW tbW  Table 2: Four-fermion operators relevant for Top production at ILC. Indices 13 stand for first family leptons and third family quarks.
The basis operators generating couplings of the top quark to the gauge bosons are displayed in table 1. As shown there, there are three operators that only generate neutral current (NC) ttZ and ttγ vertices, and two that generate both charged-current (CC) and NC couplings. For these five operators the ILC tt process can indeed surpass the potential of the LHC [23,26]. The remaining two operators in table 1 generate solely CC tbW effective couplings that cannot be sensitively probed by top-pair production. In this paper we discuss whether the single-top production mode at the linear colliders would be able to give bounds for the two purely CC two operators similar to or more stringent than the LHC, and how the limits on the two mixed NC/CC operators from single-top compare to those from top-pair production.
We parenthetically point out here that, strictly speaking, the distinction between NC operators and CC operators that are sensitive to the top-pair and the single-top processes separately is not fully clear-cut. Indeed, it has been pointed out that off-shell effects in top pair production can indeed bring sensitivity to the tbW coupling, and in particular be used to measure the top-quark width with great accuracy [27,28], which argues in favor of the notion that the potential of the ILC and CLIC machines in studying top-quark physics will go beyond the context of on-shell tt production.
As has been pointed out in [29], a consistent analysis of top-gauge boson operators cannot exclude the effects of four-fermion operators. Indeed, the choice of dimension 6 basis top-gauge boson operators implies that other operators of the same type are deemed redundant because of the equations of motion [8]. These equations of motion involve four-fermion operators that are chosen to appear in the list of independent operators and, therefore, must be included in the analysis if it is to be mathematically consistent and model independent.
The basis operators generating four-fermion vertices involving the top quark are shown in table 2. As with the quark-gauge boson operators we focus on operators containing only third-generation quarks and, in the case of four-fermion operators, first-family leptons. As seen in table 2, there are four operators generating purely eett vertices that are related only to NC tt production. We will not consider them in this paper; a recent study on the ILC potential to probe them can be found in [30]. Another goal of this paper is to obtain the limits set by single-top production on the remaining four operators in the table, which generate CC-type eνtb couplings relevant to that process. This article is organized as follows. In section 2 we discuss in detail the SM process of single-top production and decay at an e − e + collider, as well as its reducible and irreducible backgrounds and the role of beam polarizations. In section 3 we review the flavor-diagonal effective operators relevant to single-top production and discuss the recent LHC results on effective top-gauge boson couplings and the projected sensitivity of top-pair production at the ILC to those couplings, which set the context against which single-top production at ILC and CLIC must be analyzed. In section 4 we obtain bounds on the effective couplings from the single-top total cross section, both at the individual-coupling level and for pairs of couplings, at √ s = 0.5, 1 and 3 TeV for certain ranges of experimental uncertainties at each energy. Finally, in section 5 we present our conclusions.
2 Top quark production at an e + e − collider.
To better understand why we have chosen the single top process defined below in (2) let us review the context of top pair and single top production in an e + e − collider. Unlike the LHC, tt production at the ILC is generated by the electroweak interaction and becomes an irreducible background for single top production. Single top production can be hard to distinguish from e + e − → tt particularly near the threshold region. This intermingling makes off-shell effects in top pair production sensitive to the tbW vertex. Therefore, it can be used to probe the tbW vertex and the top width [27,28]. At tree level σ(e + e − → tt) is given by just two diagrams (s-channel Z and γ). At √ s = 0.5 TeV the cross section is about 550 fb with an increase of about 15% when QCD corrections are included [31]. If we require one of the top quark lines to be about 20 GeV away from the resonance so as to obtain single top events, the contribution from these 2 diagrams yields: σ(e + e − → tt * → tbW − + tbW + ) 20fb. This does not mean that tt * is the main source of single top production. If we consider the tbW − final state, with no CKM mixing, we will find that there are a total of 7 diagrams, with only two of them corresponding to tt * , and that the cross section is actually σ(e + e − → tbW − + tbW + ) 50fb. At this level, one can ask what are the possible decay channels and the most interesting ones. We should bear in mind that the final states coincide with the well known tt decay channels. The dileptonic channel, with the 4.5% fraction of about 2.3 fb would yield about 2300 events with a luminosity of 1 ab −1 before cuts. This is actually a very rough estimate, let's consider specifically e + e − → bµ − ν µ be + ν e that with a 1.1% fraction we would expect to contribute with about 0.55fb. It turns out that this process in particular has 438 diagrams, indeed most of them with no t-lines. After imposing a cut on the invariant M bb to be away from the Z and Higgs boson resonances the cross section reduces to just about 0.2fb [20]. The dileptonic channel thus seems to yield rather poor statistics. Let us now consider the semileptonic mode, with final states − νbbjj or + νbbjj. Whether the lepton is an electron or a muon we now expect to have about a 2 × 7.2% fraction that is about 2 × 3.6 fb for each possibility = e, µ. However, if e ± is the lepton in the final state, one desirable feature arises: t-channel diagrams appear. In t-channel diagrams there are no ttZ(γ) vertices and the sensivility goes only to the tbW coupling. From [18] we find that the actual cross section for e + e − → tbe − ν e is about 3 fb where the invariant mass of the be − ν e system is at least 20GeV away from the top quark resonance. With a 67% fraction of the hadronic decay we then expect to have a total of 4 fb for the semileptonic mode with the electron. Kinematic cuts will still reduce this number significantly, as we shall see below, but yet enough cross section will survive that would yield good statistics. This is the final state of interest for this study: two b-jets, two light quarks, an electron or positron and its neutrino.

Semileptonic signal process and irreducible background
The set of Feynman diagrams for top-quark production and decay in e − e + collisions in the SM in the semileptonic channel is a subset of those for the six-fermion processes e − e + → q u q d bb e − ν e + q u q d bb e + ν e , with q u = u, c, q d = d, s.
The final states (1) can be reached through two different top production processes, one followed by hadronic top decay: and the other one followed by leptonic decay: The process (2a) has been studied in [17,18] at the top-production level (e − e + → tbe − ν e ). Here, we extend that study to include top decay and the process (2b). The Feynman diagrams for the process (2) are shown in figures 1-4 for the final states containing e − ν e . We set the electron mass m e = 0, thus decoupling the electron from the Higgs field. We take into account only Cabibbo mixing in our computations, since third-generation mixing can be safely neglected for our purposes. Thus, in (2) we have (q u , q d ) = (u, d), (u, s), (c, d), (c, s). With these considerations, the 6 topologies corresponding to t-channel vector boson exchange in figure 1 lead to 40 Feynman diagrams. Notice that those diagrams involve only hadronic top decay. s-channel vector boson exchange diagrams with one internal top line decaying hadronically are given by the 5 topologies in figure 2, corresponding to 32 diagrams. s-channel diagrams with one top decaying leptonically are given by the 5 topologies in figure 3, leading to 36 diagrams. Finally, figure 4 shows one topology, corresponding to 8 diagrams for s-channel vector boson exchange with two internal top lines, which contribute to single-top production when one top line is on its mass shell and the other one off shell. We have, then, a total of 116 diagrams for semileptonic single-top production and decay in the SM with Cabibbo mixing, in the e − ν e channel. If full CKM mixing is taken into account the number of diagrams doubles to 232, since the additional diagrams with third-generation mixing can be obtained from the ones without it by just exchanging the q d and b final-state lines in each diagram.   single top: m(beν) ∈ I t and m(bjj) ∈ I t or m(beν) ∈ I t and m(bjj) ∈ I t , top pair: m(beν) ∈ I t and m(bjj) ∈ I t , offshell t: m(beν) ∈ I t and m(bjj) ∈ I t , where I t is a mass interval around the top mass, I t = (132, 212) GeV, and m(beν), m(bjj) refer to the invariant mass of the three-particle sets in the final state that can originate from a top decay. The singletop region in (3) corresponds to the process we are interested in, or "signal" process. The top-pair region in (3) refers to the production of an on-shell top pair, whereas the line labeled "off-shell t" corresponds to the phase-space region where no on-shell top is produced. Figure 5 displays the dependence on √ s of the total cross sections for the processes (2) restricted to the regions (3), with minimal phase-space cuts (see (5) below). Also shown in the figure are the cross sections for the processes with µ ∓ final states, whose diagrams are given by those in figures 2-4 with the replacement e, ν e → µ, ν µ . The cross sections for the muonic final states are equal to those for the process (2) restricted to s-channel only, so that they illustrate the role of the s-channel in (2).
The irreducible background to single-top production consists of all processes (1) not proceeding through the on-shell production of a single top. We distinguis three contributions to the irreducible background: (i ) top-pair production, arising from the diagrams in figure 4 with both top lines on shell, (ii ) off-shell top processes, comprising the diagrams in figures 1-4 with all internal top lines off shell, and (iii ) no-top processes originating from all Feynman diagrams for (1) not containing any top quark propagator. It is clear by definition that there can be no interference between the processes (i ) and (ii ), and the interference between (i ) and (iii ) turns out to be suppressed, as discussed below. Therefore, it is appropriate to adopt the convention to refer to the contributions (ii ) and (iii ) together as irreducible background, and to (i ) as a separate top-pair production background.
For the process (1), with Cabibbo mixing and for the e − ν e channel, there are 2064 diagrams without internal t lines (1808 with 6 electroweak vertices and 256 with 4 electroweak vertices and 2 strong ones). As we discuss in more detail below, at ILC/CLIC energies, 90% of the irreducible background cross section stems from the W H and W Z associated production processes and their charge conjugates. The process (4a) involves 96 Feynman diagrams and (4b) 320, for a total of 416 diagrams. In the computation of the irreducible background described below, however, we take into account the full process (1).

Phase-space cuts and event selection
We compute the tree-level cross sections for single-top production and decay, and for the background processes, with the matrix-element Monte Carlo program MadGraph5 aMC@NLO (henceforth MG5) version 2.3 [32]. In all cases we set m t = 172 GeV, The masses of the lighter quarks, e and µ are set to vanish, and the Higgs vacuum-expectation value v = 246.22 GeV. Furthermore, we take into account Cabibbo mixing with θ c = 0.228. For event analysis we use Root version 5.34 [33]. In order to make the cross section well defined, and to improve the signal-to-background ratio, we apply several phase-space cuts discussed in detail in what follows. We impose minimal centrality and isolation cuts in the form where η(e) refers to the pseudorapidity of the final-state electron or positron, η(j) to that of the jets, and ∆R ch. to the distance in the η-φ plane between any pair of charged particles. We assume that the central detector system covers the central region |η| < 3-3.5 and the forward detectors the region 3 < |η| < 4, as is expected to be the case at the ILC/CLIC [5,6]. As shown in figure 6, the cut on η(e) is substantially more restrictive at the CLIC energy than at ILC's. This is a consequence of the fact that at √ s = 3 TeV the process (2) occurs mostly through the t-channel vector boson exchange diagrams of figure 1, while at the ILC the s-channel diagrams of figures 2-4 dominate, as shown in figure 5. The cut on ∆R = ∆η 2 + ∆ϕ 2 in (5) is an isolation cut setting the minimal distance between any two charged particles in the final state. The reducible background to the process (2), which is studied in more detail below in section 2.4, consists of final states with four jets, an electron/positron and E ⊥ , with the number of b-jets N b = 2. Such final states contain pairs of massless partons, light quarks or gluons, which lead to infrared singularities. In order to avoid those singularities we require the final-state jets to satisfy the condition where m(j, j ) refers to the mass of any pair of partons in the final state. As shown in  Figure 6: Differential cross sections for the e ∓ pseudorapidity for the signal process (2) with the cuts A 0 , (5), and C 0 , (6), normalized to total cross section (see table 3).
energies, and it cuts about 10% of the signal at the ILC and 30% at the highest CLIC energy. In figure 7 we display several mass differential cross sections computed with the cuts (5) , resp.) are seen to be dominated by the Z, h peaks and W peak, respectively, as expected from the main irreducible-background processes (4).
For instance, at √ s = 3 TeV the total cross section for (4) with the cuts A 0 and C 0 is found to be 6.57 fb, amounting to 92% of that of the total irreducible backgroung, 7.15 fb, as given in table 3.
To suppress the reducible background we require the final state to contain exactly two b-tagged jets: We assume the b-jet tagging efficiency to be η b = 80%, the mistagging probabilities to be p c = 10% for c-jets and p q = 1% for the lighter jets. These are realistic values, consistent with the efficiencies currently achieved by LHC detectors [34]. In our analysis we simulate b tagging by relabeling b quarks as lighter ones with 20% probability and by relabeling as b quarks c quarks with 10% probability and lighter partons with 1% probability. Thus, cut (7) results in the signal and irreducible background cross sections being reduced to about 61% of their value. The cut C 1 plays an important role in rejecting the reducible background, as discussed in detail in the following section.
Most of the rejection of the irreducible background, and of the reducible background events remaining after cut (7), is achieved by requiring that each event must contain the decay products of exactly one on-shell top quark, as described by the "single top" line in (3). We denote the four jets in the final state by J 0,...,3 , with J 0,1 the two b-tagged jets and J 2,3 the two light jets, the latter ones in no particular order. In events with a hadronically decaying top, we denote the promptly produced b jet as J 0 and the b jet from top decay as J 1 , and for leptonic top decays we denote the prompt b jet as J 1 and the b jet  Table 3: Effect of the phase-space cuts (5)-(10) on the total cross section for the process (2) and its subprocesses (3). The b-tagging efficiency and mistagging probabilities involved in C 1 are given in the text under equation (7).
from top decay as J 0 . Thus, with this convention and with the notation introduced in (3), the phase space cut for single-top events can be written as where the first parenthesis corresponds to leptonically decaying and the second to hadronically decaying top quarks. In figure 7 (c), the differential cross section for the invariant mass m(b, j, j) for the signal process (solid line) corresponds to m(J 1 , J 2 , J 3 ), and in 7 (d), m(b, e, ν) corresponds to m(J 0 , e, ν). For the irreducible background, the distributions of m(b, j, j) and m(b, e, ν) obtained with each one of the two b quarks in the final state are essentially identical. The irreducible background contains a substantial combinatorial component satisfying (8), as is apparent from figures 7 (c) where the distribution of m(b, q, q) is seen to contain a significant number of events under the top mass peak. This yields a sizable irreducible background cross section even after the cut C 2 has been applied, as shown in table 3. Since most of the irreducible background consists of processes (4) in which the bb pair comes from Z or h decay, we are led to introduce the cut to further suppress the remaining irreducible background. Finally, we require that the two light jets be the decay products of an on-shell W boson (cf. figures [1][2][3][4] and that the final state contains substantial E T , C 4 : 60 GeV < m(J 2 , J 3 ) < 100 GeV, and E T > 10 GeV. TeV and with the cuts A 0 , (5), and C 0 , (6). All differential cross sections are normalized to total cross section: 3.58 fb for the signal and 3.13 fb for the background processes.
Cut C 4 provides some further suppression of the irreducible background without significantly affecting the signal. The cumulative effect of the phase-space cuts (5)-(10) on signal and background is shown in table 3. The lines of this table labeled C 4 give the cross sections for the signal and irreducible background, including the effect of b-tagging efficiencies and all phase-space cuts. From those results, and assuming an integrated luminosity L = 1 ab −1 , we estimate the statistical uncertainty in the signal cross section σ sgn to be 4.0%, 2.4%, 3.1% at √ s = 0.5, 1, 3 TeV, respectively. The irreducible background is seen to amount to 4.1%, 3.3%, 4.4% of σ sgn , respectively, at the same three energies. Table 3 shows that the top-pair production background is fully rejected by the cut C 2 , but that is true only when measurement uncertainties are not taken into account. If we allow for the effect on the cut C 2 of measurement uncertainties in m(J 1 , J 2 , J 3 ) and m(J 0 , e, ν), however, a fraction of top-pair events would pass that cut. For reasonably small measurement uncertainties we expect that fraction of events to be a relatively small fraction of the pair-production cross section. In the case of CLIC, in which the tt cross section is already very small compared to that of single-top, this effect is expected to be of secondorder. At the ILC, however, top-pair production is substantially larger than single-top, so that even a small fraction of these events can become a large background. In this section we quantify the top-pair background to single-top production taking into account the top-mass reconstruction uncertainty. A goal of the ILC and CLIC detectors is to achieve high enough jet energy-and mass-resolution to be able to separate the W and Z peaks in the dijet mass spectrum [5,6]. As shown in figure 2.6 of [6], for that purpose the mass resolution σ m /m must be better than 5%, with good separation of the W and Z peaks obtained for σ m /m = 2.5%. Motivated by that observation, we assume that the top mass will be reconstructed from three jets at the ILC/CLIC with a relative uncertainty in the range 5-7.5%. In order to obtain a quantitative estimate of the effect of measurement uncertainties on the tt background, we randomly smear the reconstructed masses m(J 1 , J 2 , J 3 ) and m(J 0 , e, ν) before applying the cut C 2 to each top-pair production event. We assume those masses to be independently normal-distributed with standard-deviation parameter σ m = ε m m, where ε m is the assumed relative uncertainty. For simplicity we assume the same value of ε m for both m(J 1 , J 2 , J 3 ) and m(J 0 , e, ν). We carry out this randomized analysis of the tt event sample a few thousand times to obtain a statistical sample of the tt cross section after cuts A 0 , C 0,...,4 . The resulting σ tt distribution is strongly asymmetric with a long tail to the right, so we characterize it by the interval ( σ tt , σ tt + ∆σ tt ) .

Top-pair background
At √ s = 3 TeV, if we assume the top mass to be reconstructed with uncertainty ε m = 5%, the tt background turns out to be 0.48%-0.50% of the single-top cross section with all cuts, A 0 , C 0,...,4 , as given in table 3. For ε m = 7.5%, we find the tt background to be 0.51%-0.79% of the single-top cross section. As expected, that background turns out to represent a small uncertainty at CLIC energy. At √ s = 1 TeV, for ε m = 5%, the tt background result is 1.6%-3.3% of the single-top cross section. For ε m = 7.5%, we get 3.5%-16.4%. We see that for ε m less than 7.5% the tt background is limited to about 15% of the single-top cross section, and for lower values near ε m 5% that background can be somewhat less than 5%.
At √ s = 0.5 TeV, for ε m = 5%, the tt background is 8.6%-17% of the single-top cross section. If ε m = 7.5%, we get 16.5%-50.5% for the tt background. These results are also not unexpected, since the tt cross section is large at the ILC at 0.5 TeV and can potentially swamp the single-top process. They also suggest, however, that the top-pair background can be limited to the range 10-30% for ε m less than 7%, and to the range 10-20% if ε m is close to 5%.

Reducible background
The reducible background to the single-top production process (2) is given by processes of the form e − e + → j 1 j 2 j 3 j 4 e − ν e + j 1 j 2 j 3 j 4 e + ν e , with j = u, d, c, s, b, g or their antiparticles. The number of b/b jets in (11) can be N b = 0, 1, 3 (with N b = 2 corresponding to the signal and irreducible background processes (1), and N b = 4 being forbidden by electric charge conservation). Since the probability to mistag a final state (11) with N b = 2 as one with N b = 2 as in (1) depends on N b and on the number N c of c/c quarks, we have to consider separately the cases with different values of N b , N c . For our computation of the reducible background we adopt two-generation Cabibbo mixing, since the effects of third-generation mixing on cross sections are numerically inconsequential. This implies in particular that only processes (11) with N b = 0 and without internal t lines are possible. Indeed, explicit computation shows that the final states (11) with N b = 1, 3, which can only occur through third-generation mixing, lead to cross sections of O(10 −2 fb) at most even before the restrictive cuts C 1...4 in equations (7)-(10) are applied. For the same reason, equally negligible cross sections are obtained for diagrams with N b = 0 with one or more internal t lines.
The reducible bakground (11) contains a large number of subprocesses whose detailed description is not needed for our purposes. With only two-generation mixing taken into account, it involves 15632 Feynman diagrams with e − ν e final states. However, some general features of this background, with cuts A 0 , C 0 , are easily understood. With those cuts the gluon final states q u q d gg are a minority fraction of the cross section. The final states q u q d qq which dominate the cross section originate mostly from W Z associated production, as seen from figure 8, in a similar way as the irreducible background. ) and e − e + → udgge − ν e ((c) and (d)), with the cuts A 0 , (5), and C 0 , (6), normalized to total cross section.
We compute the reducible-background cross section by applying the acceptance cuts A 0 , (5), to particles with electric or color charge (i.e., to charged leptons, quarks and gluons). The cut C 0 , (6), is also required to make the cross section infrared finite. Simulation of light-parton mistagging is carried out, as discussed in the text below (7), by relabeling c quarks as b ones with probability p c = 0.1 and lighter partons with probability p = 0.01. The resulting mistagging probabilities for processes (11) with N b = 0 are given by 0 b 0 c : with q = 1 − p, q c = 1 − p c . The mistagged final states, containing two "fake" b quarks, are then required to pass the cuts C 2...4 , as defined in (8)- (10), in the same way as the signal and irreducible background. The effects of cuts A 0 , C 0,1 on the total cross sections is shown in table 4. Notice that, at each energy, the cross sections corresponding to cut C 1 can be obtained by multiplying the results for the cuts A 0 and C 0 by the corresponding probabilities in (12), up to a small numerical uncertainty. As seen from the table, those cross sections are at most O(10 −2 fb). After cuts C 2...4 are applied, the resulting reducible-background cross sections are at most O(10 −3 fb) and therefore negligible.  Table 4: Effect of the phase-space cut (7) on the total cross section for the process (11). The b-tagging efficiency and mistagging probabilities involved in C 1 are given in the text under equation (7).

Beam polarization
The ILC baseline design supposes a polarization of the electron beam of at least 80%, and 30% for the positron beam [4]. For CLIC, the baseline design assumes an electron beam polarization of 80% and an unpolarized positron beam [6]. In both accelerators a later upgrade is foreseen that would increase the positron beam polarization to 60% [4,6]. In the study of single-top production beam polarization may lead to a reduction of the measurement uncertainties, either by increasing the signal cross section, therefore reducing the statistical uncertainty, or by suppressing important backgrounds. At √ s = 3 TeV the cross-section uncertainties are dominated by the irreducible background and the statistical uncertainty (about 4.4% and 3.1%, respectively, as noted at the end of section 2.2). The toppair production background uncertainty is subdominant, as discussed in section 2.3. The longitudinal beam polarization combinations that cause the signal cross section to increase or decrease have the same effect on the irreducible background, though not necessarily in the same amount. With longitudinal polarizations (P e − , P e + ) = (−90%, 60%) both the signal and irreducible background increase, leading to a statistical uncertainty of 2% and an irreducible background of 6.2%, thus worsening the overall uncertainty. For (P e − , P e + ) = (−90%, −60%) we get a statistical uncertainty of 3.2% and an irreducible background of 4.7%, and for (P e − , P e + ) = (90%, 60%) we get 4.4% and 4.5%, respectively; these cases show little change in the uncertainties with respect to the unpolarized case. Finally, for (P e − , P e + ) = (90%, −60%) the statistical uncertainty grows to 8% while the irreducible background decreases to 3.3%, for an overall uncertainty considerably worse than the unpolarized result. At this energy, therefore, we find that even the highest possible beam polarizations do not lead to a significant reduction in the cross section uncertainty with respect to the unpolarized case. At √ s = 1 TeV, assuming an integrated luminosity L = 1 ab −1 , we found in section 2.2 a statistical uncertainty of 2.4% and an irreducible background of 3.3%. As discussed in section 2.3, top-pair production may be significantly larger than the irreducible background. Assuming the highest possible beam polarizations, we find that (P e − , P e + ) = (−90%, 60%) increases the signal cross section σ sgnl by a factor 2.72 relative to the unpolarized case, thus reducing the statistical uncertainty to 1.5%. The irreducible and top-pair production backgrounds increase by a factor of 2 and 2.24, respectively, so that σ irr /σ sgnl decreases by 25% and σ tt /σ sgnl by 20% with respect to the unpolarized case. The polarizations (P e − , P e + ) = (−90%, −60%) also reduce the ratio σ tt /σ sgnl , but by decreasing all cross sections. The signal cross section σ sgnl decreases by a factor 0.79, thus slightly increasing the statistical uncertainty to 2.7%. The ratio σ irr /σ sgnl remains unchanged, and σ tt /σ sgnl decreases by 25% with respect to the unpolarized case. At √ s = 0.5 TeV from section 2.2 we get a statistical uncertainty of 4% and an irreducible background of 3.3% relative to the signal cross section σ sgnl . From section 2.3 it is apparent that at this energy the tt production background strongly dominates the cross section uncertainty. The beam polarizations (P e − , P e + ) = (−90%, 60%) increases σ sgnl by a factor of 2.7 and σ tt by 2.2, thus leading to a decrease in σ tt /σ sgnl of 20%. The polarizations (P e − , P e + ) = (−90%, −60%) decrease σ sgnl by a factor 0.7 and σ tt by 0.6, thus reducing σ tt /σ sgnl by 20%. Therefore, at ILC energies we expect the highest beam polarizations to lead to a moderate reduction of single-top cross section uncertainties by 20-25%. Lower polarizations would yield correspondingly smaller uncertainty reductions.

Effective operators for single-top production at e − e + colliders
The Lagrangian for the single-top production processes (2) is of the form where O denotes dimension 6 effective operators, Λ is the new-physics scale, and the ellipsis refers to higher-dimensional operators. It will be convenient in what follows to express our results in terms of the modified dimensionless couplings where v is the Higgs-field vacuum expectation value. At tree level the coupling constants C O are independent of the scale Λ. We denote complex couplings as Throughout this paper we use the dimension 6 effective operators from the operator basis given in [8]. However, we adopt the sign convention in the covariant derivatives and the operator normalization defined in [11], where a factor y t is attached to an operator for each Higgs field it contains, and a factor g (g ) for each W µν (B µν ) field-strength tensor. We are interested in those operators that can contribute to single top production at an e + e − collider. There are operators with flavor changing couplings, but in this study we will not consider them.

Operators that generate tbW couplings.
There are four dimension 6 operators in the basis [8] that give rise to effective tbW couplings: O ϕq so as to eliminate the bbZ neutral current term. Expanding these operators in physical fields we obtain: The relation between the couplings C O in (14) and the usual δV L , V R , g L and g R tbW form factors is: There have been many studies that have set bounds on the coefficients of these operators based mostly on LHC ( √ s = 7, 8 TeV) single top production and W -helicity fractions in top decay [24,[35][36][37][38][39][40]. A recent LHC combined extraction of |f LV V tb | has been presented at 1σ level [35]: where f LV V tb ≡ V L in our notation. From this we can set −0.16 < C (−)33 ϕq < 0.20 at 2σ (or 95%) level, if we assume V tb = 1. Concerning the other couplings, with the constraint V L = 1, CMS has reported a global analysis based on two and three-dimensional fit scenarios, from which they have obtained the most stringent 95% C.L. bounds to date [24]: This limits can be converted to bounds on the effective couplings C O through (16), which are therefore the current LHC bounds on the effective dimension-6 couplings C O . What could be the improvement of these limits in the LHC top physics program, assuming no BSM physics is found? With integrated luminosities of up to 20 fb −1 already obtained, the statistical uncertainties are subdominant relative to the systematic and other uncertainties. We could then expect that even the HL-LHC phase will not necessarily yield an order-of-magnitude improvement over the limits (18), (19). For instance, in [23] the 95%C.L. limits |C The ILC tt production process has the potential of improving the C (−)33 ϕq and C 33 uW bounds by an order of magnitude or more [22,23,41]. Specifically, as reported in [22], the ILC at √ s = 500 GeV based on tt cross section and forward-backward (FB) asymmetry measurements, and assuming an experimental uncertainty of 5% in the cross section and 2% in the FB asymmetry, would give the single-coupling bounds, In [23], for the same energy an experimental uncertainty of 1% is assumed, which yields correspondingly tighter individual-coupling direct bounds, as shown in figure 6 of that reference. Notice that (21) assumes also longitudinal beam polarizations (P e − , P e + ) = (−80%, +30%) [23]. To bear in mind, there are other two operators with ttZ couplings that also contribute to tt production. When the simultaneous contributions of these operators are considered, the marginalised limits are relaxed. This is indeed so with the marginalised limits from the multivariate analysis carried out in [23], which are reported there to be larger than the individual ones (21) by a factor 17. For the purposes of the present study, however, we take the single-coupling bounds (21) and (20) as benchmarks of the projected sensitivity of tt production at the ILC.

Diagonal four fermion operators
The use of effective Lagrangians in the top quark physics program aims ultimately to constrain simultaneusly all the non-redundant operators (at a certain level, like dimension 6) based on all the available experimental measurements. For instance, a recent global fit has been presented in [38] where 4 topgauge boson and 5 four-fermion operator coefficients were constrained using both tt as well as single top production measurements from the LHC and the Tevatron. Not surprisingly, the bounds obtained by considering effects from one operator at a time tend to greatly relax when other operators are also taken into account [38]. There is indeed a great effort to perform global fit studies as is found in the literature [37]. Besides the motivation for making an analysis complete, the goal of considering all the operators is for consistency. Top-gauge boson operators with derivatives on fermion fields [42] do not appear in the basis of [8] because equations of motion relate them to the ones considered here. These equations involve four-fermion terms that are part of the non-redundant operators. For instance, a tbW coupling is generated by an operator O qW = q L γ µ τ I D ν q L W I µν that should be considered in a CC interaction of the top quark. Bearing in mind the general tbW vertex generated by the operators in (15), we can in fact isolate the nonredundant contribution by this operator [29]. However, it is convenient to implement this effect with the four-fermion operator O (3) lq that generates a e L γ µ ν L t L γ µ b L interaction instead [29]: In this sense, an analysis of top-gauge boson couplings should be considered complete and consistent only if it includes four-fermion operators.
There are 8 diagonal four-fermion operators involving two first-generation leptons and two thirdgeneration quarks. Four of them are associated to CC couplings, and the other four to only NC couplings.
The CC operators that contribute to single top production are: where we have used the abbreviated notation O 13 for O 1133 .
To date, there are no reported limits based on LHC nor Fermilab top production and/or decay processes for these operators. In [11] it has been pointed out that distribution-based measurements like the W -helicity fractions could in principle be used to this end as the leptonic decay is used to analyse the W -polarization. However, the experimental requirement that m ν be close to M W severely reduces the sensitivity of the fractions to the four fermion operators [11].

Effective couplings in single-top production and decay
For the computation of the cross section with anomalous effective vertices we use MG5, as described in section 2.2. For computational purposes we set the scale Λ = 10 TeV in (13), (14). The effective operators (15), (23) were implemented in MG5 by means of the program FeynRules version 2.0 [43].
Diagrams with one, two and three effective vertices entering the amplitude for (2), contribute to it at O(Λ −n ) with n = 2, 4 and 6, respectively. In fact, once the top propagator dependence on effective couplings through the top decay width is taken into account, the scattering amplitude is given as a power series of Λ −2 . We remark that diagrams with two effective vertices must be kept in the amplitude since, through their interference with SM diagrams, they make contributions to the cross section of the same order, O(Λ −4 ), as the square of diagrams with only one effective vertex. We have actually taken into account the contributions from diagrams with three effective vertices in our calculation as well as the dependence of the top decay width on the effective couplings, but we have explicitly verified in all cases that the contribution to the cross section from terms of order higher than O(Λ −4 ) is actually negligible Figure 10: All Feynman diagrams with one anomalous four-particle vertex from the operators O 33 uW , O 33 dW in (15). Figure 11: Sample Feynman diagrams with two and three anomalous vertices from the operators (15).
for values of the effective couplings within the bounds given below. (We remark here, parenthetically, that the contributions to the cross section at order 1/Λ 4 from dimension 8 operators interfering with the SM are currently unknown and constitute an inherent uncertainty of the EFT analysis at dimension 6.)

Methodology and assumptions
In order to obtain bounds on the effective couplings, we consider the ratio of the cross section σ eff ({C O }) obtained from the Lagrangian (13) at tree level to the SM cross section σ SM = σ eff ({0}) where {C O } is the set of anomalous coupling constants. For a given relative experimental uncertainty ε exp , the region of allowed values for the effective couplings {C O } is determined at the 1σ level by the inequalities We obtain allowed intervals on the effective couplings taken to be non-zero one at a time by parameterizing the ratio (24) as where the ellipsis refers to higher powers of C O . Similarly, we consider also allowed two-coupling regions for pairs of effective couplings by parameterizing (24) as Figure 12: Sample Feynman diagrams with one and two four-fermion vertices from the operators (23).  (26) and (27) are determined from an extensive set of MG5 simulations to which (26) and (27) are fitted. Once those parameters are known, (25) yields the desired one-or two-dimensional limits on the effective couplings being considered. The consistency condition that the contribution to the cross section from terms of O Λ −6 and higher in (13) be negligibly small entails on the parameterizations (26), (27) the requirement that the terms of O(C 3 ) and higher must be correspondingly negligible within the allowed region determined by (25). We check this consistency condition in all cases considered below.
In order to obtain bounds on the effective couplings through (25), below we assume ε exp to take values within certain intervals. We motivate those assumed ranges for ε exp by estimating the uncertainties in the signal cross section in our Monte Carlo simulations, through the addition in quadrature of the statistical uncertainty and the irreducible background given at the end of section 2.2, and the uncertainties arising from the tt background as discussed in section 2.3. This leads us to assume that at √ s = 3 TeV ε exp is in the range 5%-9%, and at √ s = 1 TeV within 5%-15%. In the case of √ s = 0.5 TeV we assume ε exp to lie in the interval 10%-20%, but present also some results at 30%. As discussed in more detail below, at all three energies the individual-coupling limits we obtain can be extrapolated to values of ε exp moderately lower or higher than the ranges we assume. Furthermore, as discussed in section 2.5, beam polarization may contribute to reduce background uncertainties, which may help reach the lower end of the assumed uncertainty intervals, especially at √ s = 0.5 TeV.

Results
The single-coupling bounds obtained from (25), are reported in  When the interference of diagrams containing one effective vertex from the operator O with those from the SM vanishes, or is suppressed by a small mass parameter, the linear term in (24) At that energy the signal cross section is most sensitive to C (−)33 ϕq and C 33 uW r , for which the bounds in (28) are close to the current limits set by CMS as quoted above in (19). Thus, in order to improve on the current CMS bounds on those couplings, the single-top cross section should be measured by the ILC at √ s = 0.5 TeV with an uncertainty ε exp < 30%. We point out also that, at that energy and with an unrealistically low uncertainty ε exp = 5%, we would obtain the limits −0.0093 < C (−)33 ϕq < 0.035. Comparison with the bounds at the same uncertainty in table 5, leads to the conclusion that the sensitivity to C (−)33 ϕq for a fixed relative uncertainty is smaller at CLIC than at the ILC. The single-coupling bounds in table 6 show both a large sensitivity to F couplings relative to that of C's, and a strong enhancement of that sensitivity with increasing energy, as expected of four-fermion interactions. As also seen from the table, the single-top cross section is most sensitive to F which still are rather strong. At each energy tables 5 and 6 give lower and upper bounds for each coupling, for three different values of ε exp . As is easy to check, any of those three values results from linear interpolation of the other two, within about 5%. This shows, heuristically, that linear interpolation is valid and can be used to find bounds corresponding to other values of ε exp within the range given in the table. Linear extrapolation can also be used to obtain bounds for ε exp moderately smaller than the lowest value used in the table, or moderately larger than the highest one. An illustration of this is provided by the bounds at √ s = 0.5 TeV and ε exp = 30% given above in (28), (resp., (29)) which agree with an extrapolation from table 5 (resp., 6) within at most 5% deviation. The allowed regions for pairs of effective couplings involving gauge bosons are displayed in figure  14, where the current LHC bounds from (19) are also displayed for reference. The allowed regions for pairs of effective four-fermion couplings are displayed in figure 15. As can be seen in figures 14, 15, with the exception of the couplings in figures 14(d) and 15(d),(e),(f), the cross section at the ILC at either √ s = 0.5 or 1 TeV does not determine a small, simply connected neighborhood of the origin but rather an extended toroidal band. However, the intersection of those two regions does provide a simply connected vicinity of the SM. Those ILC-allowed regions are further constrained by the bounds imposed by CLIC, as shown in the figures. The term containing the coefficient c in (27), corresponding to the interference of two anomalous amplitudes, leads to a rotation of the symmetry axes of the allowed region relative to the coordinate axes. Interference effects between the amplitudes proportional to C 33 ϕud r and C 33 dW r are apparent in figure 14(d), sizeable at √ s = 0.5 TeV and significantly weaker at higher energies. Smaller, but still noticeable interference between amplitudes proportional to C

Final remarks
Top quark physics is an essential part of the future ILC and CLIC collider programs. While so far most of the interest has focused on tt production and its great sensitivity to NC ttZ(A) couplings, in this paper we discuss whether the single-top mode would provide any useful information. In this context, it is important to stress the fact that while at the ILC single-top is subdominant to top-pair production, which then becomes a strong background, at CLIC single-top is the dominant top production mode.
We have carried out a preliminary parton level analysis of the semi-leptonic six fermion final state bbe − ν e q u q d (and its charge conjugate) in the context of single-top production and have found that at the three energies considered, √ s = 0.5, 1 and 3 TeV, the signal cross sections are about 1-2 fb (see table 3), including phase-space cuts and b-tagging efficiencies, enough to keep the statistical uncertainties under 5%. A detailed discussion of backgrounds is given in section 2.
We have obtained individual limits on tbW vertices in the context of the SU (2) × U (1)-gauge invariant effective dimension-6 operators [8], as shown in table 5 and figure 14. In order to discuss our results for the four top-gauge boson couplings generating CC vertices, it is convenient to separate them in two pairs. In the first pair we have the operators O (−)33 ϕq and O 33 uW , which have in common that they both generate interference with the SM, simultaneously generate an additional NC ttZ/A coupling and contribute substantially to s-channel diagrams that appear in both, single-top and top-pair production. With increasing energy both the effects of interference and the s-channel contribution decrease, and so does the sensitivity to these couplings. To compare our results to those from the LHC given in (19) and to the projections for tt production at the ILC, (20) and (21), we take the inverse of the length of the interval determined by the single-coupling bounds in those equations and in table 5 as a measure of the sensitivity. For the coupling C (−)33 ϕq the sensitivity obtained at √ s = 0.5 TeV if we assume an experimental uncertainty of 10% is four times larger than at the LHC, (19), and about the same as that of tt production at the same energy, with the experimental uncertainties for tt assumed in (20), but three times smaller if the uncertainties are those assumed in (21). Larger assumed experimental uncertainties lead, of course, to correspondingly smaller sensitivities. As discussed in relation to (28), for an experimental uncertainty of about 30%, the sensitivity to C TeV if we assume an experimental uncertainty of 5%. In that case the sensitivity is about eight times larger than the current LHC result, almost twice as large as the ILC tt sensitivity in (20), and slightly smaller than in (21). Thus, for the coupling C (−)33 ϕq we expect the sensitivity of single-top production at linear e − e + colliders to significantly improve the current one at the LHC and the projected one at the HL-LHC, and to be competitive with the sensitivity of tt production at the ILC.
For the coupling C 33 uW r , at √ s = 0.5 TeV and with an experimental uncertainty of 10%, the sensitivity from table 5 is three times larger than that from the current LHC result (19), though almost three times smaller than the ILC tt production result (20), and about 15 times smaller than (21). The largest sensitivity to this coupling in table 5 occurs at √ s = 1 TeV for an assumed experimental uncertainty of 5%, which is five times larger than the current LHC result (19), slightly smaller than the ILC tt sensitivity (20) and an order of magnitude smaller than (21). In this case we conclude that the sensitivity to C 33 uW r of single-top production at the ILC will improve on the current LHC sensitivity by a factor 3-5, but will be significantly worse than that of tt production at the ILC.
The operators O 33 ϕud and O 33 dW do not appreciably contribute to tt production at the ILC, and lead to very modest interference with the SM amplitudes. They contribute to single-top production in e − e + collisions mostly through t-channel diagrams, with the contribution from s-channel diagrams remaining essentially a SM input that is important at lower energies but becomes very small for energies higher than √ s = 2 TeV. In this respect, we notice that O 33 ϕud does not generate NC interactions, and that O 33 dW only generates, besides tbW , a bbZ coupling with very little effect on s-channel diagrams. As a result, the sensitivity to these operators increases with energy.
The sensitivity to C 33 ϕudr at √ s = 0.5 TeV with an assumed experimental uncertainty of 10%, as given in table 5, is less than half that of the current LHC limits (19). At √ s = 1 TeV with an assumed uncertainty of 5% is equal to the sensitivity of the current LHC results (19), and at √ s = 3 TeV with uncertainty 5% is about 60% larger than the current LHC sensitivity. For this coupling, therefore, at most a slight improvement over the current LHC sensitivity can be expected from CLIC, and none from the ILC.
The sensitivity to C 33 dW r at √ s = 0.5 TeV with an assumed experimental uncertainty of 10%, as given in table 5, is about half that of the current LHC limits (19). At √ s = 1 TeV with an assumed uncertainty of 5% it is twice as large as the current LHC sensitivity, and 2.5 times as large at √ s = 3 TeV with uncertainty 5%. Thus, the sensitivity to C 33 dW r at the highest energy e − e + collisions increases by a factor of 2-2.5 with respect to the current LHC sensitivity.
These conclusions depend, of course, on the fact that at each one of the three energies discussed here, we have assumed the lowest experimental uncertainties considered in table 5. With the information contained in that table, however, these conclusions can be adapted to other experimental uncertainty assumptions.
Limits on the four-fermion operators from (23) are given in table 6 and in figures 15 and 16. Those CC four-fermion couplings cannot be observed at hadron colliders, or in NC processes such as tt production at e − e + colliders. Therefore, single-top production at those colliders is the only possibility to bound them. Since the sensitivity to four-fermion interactions increases very rapidly with energy, for these couplings CLIC is clearly the best option. The sensitivity at the 1 TeV ILC is still substantial, however, being about half that for CLIC, as shown in table 6.   qr four-fermion coupling, excluded at 68% C.L. by a measurement of the total cross section for process (2). Color codes as in the previous figure.