Decays of polarized top quarks to lepton, neutrino, and jets at NLO QCD

Open Access
Regular Article - Theoretical Physics

Abstract

We compute the differential and total rate of the semileptonic decay of polarized top quarks \(t\rightarrow \ell \nu _\ell + b~\mathrm{jet} + \mathrm{jet}\) at next-to-leading order (NLO) in the QCD coupling with an off-shell intermediate \(W\) boson. We present several normalized distributions, in particular those that reflect the \(t\)-spin analyzing powers of the lepton, the b-jet and the \(W^+\) boson at LO and NLO QCD.

Keywords

Large Hadron Collider Charged Lepton Semileptonic Decay Helicity Fraction Differential Decay Rate 

1 Introduction

The top quark, the heaviest known fundamental particle, is set apart from the lighter quarks by the fact that it is so short-lived that it does not hadronize. The top quark decays almost exclusively into a \(b\) quark and a \(W\) boson; other decay modes have so far not been observed.

Top-quark production and decay has been explored quite in detail at the Tevatron and especially at the Large Hadron Collider (LHC). So far almost all experimental results agree well with corresponding Standard Model (SM) predictions. (For recent overviews, see [1, 2, 3, 4].) On the theoretical side, significant recent progress1 includes the computation of the hadronic \(t\bar{t}\) production cross section at next-to-next-to-leading order (NNLO) in the QCD coupling \(\alpha _s\) [5, 6] and the calculation of the differential decay rate of \(t\rightarrow b\ell \nu _\ell \) at NNLO in perturbative QCD [7, 8].

Over the years, top-quark decay has been analyzed in detail within the SM. As to the total decay width \(\Gamma _t\), the order \(\alpha _s\) QCD corrections [9, 10], the order \(\alpha \) electroweak corrections [11, 12], and the order \(\alpha _s^2\) QCD corrections [13, 14] were calculated quite some time ago. The fractions of top-quark decay into \(W^+\) with helicity \(\lambda _W=0, \pm 1\) are also known to NNLO QCD [15], including the order \(\alpha \) electroweak corrections [16]. Differential distributions of semileptonic and non-leptonic decays of (un)polarized top quarks were determined to NLO in the gauge couplings [17, 18, 19, 20, 21, 22, 23, 24, 25], and \(b\)-quark fragmentation was analyzed in [26, 27, 28, 29, 30].

In this paper we compute the differential and total rate of polarized top quarks decaying into \(\ell \nu _\ell + b~\mathrm{jet} + \mathrm{jet}\) at NLO in the QCD coupling. The differential rate is of interest as a building block for predictions of top-quark production and decay at NLO QCD, for instance for \(t{\bar{t}} + \mathrm{jet}\) production [31, 32, 33], for single top-quark + \(\mathrm{jet}\) production at the LHC, or for \(t{\bar{t}} + \mathrm{jet}\) production at a future \(e^+e^-\) linear collider. In fact, this decay mode was already computed to NLO QCD by [33]. The results of this paper as regards \(t{\bar{t}} + \mathrm{jet}\) production at hadron colliders include also NLO jet radiation in top-quark decay. Distributions for this decay mode were not given separately in [33]. Therefore, we believe that it is useful to present, for possible applications to other processes, a separate detailed analysis of this top-quark decay mode.

The paper is organized as follows. In Sect. 2 we describe our computational setup. In Sect. 3 we present our results for the decay rate and for a number of distributions for (un)polarized top-quark decays. Section 4 contains a short summary. In the appendix we list the subtraction terms, for the Catani–Seymour subtraction formalism [34, 35] with extensions to the case of a colored massive initial state [33, 36, 37], which we use to handle the soft and collinear divergences that appear in the real radiation and NLO virtual correction matrix elements.

2 Setup of the computation

We consider the decay of polarized top quarks to leptons, a \(b\)-jet, and an additional jet,
$$\begin{aligned} t\rightarrow W^{*+} + b~\mathrm{jet} + \mathrm{jet} \rightarrow {\ell }^+ \nu _\ell \, + b~\mathrm{jet} \, + \mathrm{jet}, \end{aligned}$$
(1)
at NLO QCD, for an off-shell intermediate \(W\) boson. The quarks and leptons in the final state are taken to be massless.
At NLO QCD, i.e., at order \(\alpha _s^2\), the differential decay rate of (1) is determined by the amplitudes of the following parton processes:
$$\begin{aligned} t\rightarrow {\ell }^+ \nu _\ell \, +b \, + g, \end{aligned}$$
(2)
and
$$\begin{aligned} t&\rightarrow {\ell }^+ \nu _\ell \, + b \, + g g, \quad t\rightarrow {\ell }^+ \nu _\ell \, + b \, + q{\bar{q}},\quad q=u,d,s,c,b.\nonumber \\ \end{aligned}$$
(3)
In the case of additional \(b\bar{b}\) production in the real radiation process (3), we take into account only configurations where a \(b \bar{b}\) pair is unresolved by the jet algorithm, i.e., we consider in (1) final states where the additional jet has zero \(b\)-flavor. To order \(\alpha _s^2\), the matrix element of (2) is the sum of the Born term \(|{\mathcal {M}}_B|^2\) and the interference \(\delta {\mathcal {M}}_V\) of the Born and the 1-loop amplitude. The calculation of \(\delta {\mathcal {M}}_V\), using dimensional regularization, is standard. We expressed \(\delta {\mathcal {M}}_V\), using Passarino–Veltman reduction [38], in terms of scalar one-loop integrals with up to four external legs. The scalar integrals that appear in \(\delta {\mathcal {M}}_V\) are known analytically in \(d\) space-time dimensions (cf., for instance, [39] and references therein). In particular, we extracted the ultraviolet (UV) and infrared (IR) poles in \(\epsilon =(4-d)/2\) that appear in several of these scalar integrals analytically. The processes (3) are described by tree-level matrix elements.

As to renormalization, the top-quark mass is defined in the on-shell scheme while the QCD coupling \(\alpha _s\) is defined in the \(\overline{\mathrm{MS}}\) scheme.

The soft and collinear divergences that appear in the phase-space integrals of the tree-level matrix elements of (3) and in \(\delta {\mathcal {M}}_V\) are handled with the dipole subtraction method [34, 35] and extensions that apply to the decay of a massive quark [33, 36, 37]. Details are given in the appendix.

We define jets by the Durham algorithm [40], i.e., we use the jet metric
$$\begin{aligned} Y_{ij} = 2 \frac{\min {\left( E_i^2,E_j^2\right) }}{m_t^2} \left( 1-\cos {\theta _{ij}}\right) , \end{aligned}$$
(4)
where \(E_i\), \(E_j\) are the energies of partons \(i,j\) in the final state, \(\theta _{ij}\) is the angle between them, and \(m_t\) denotes the mass of the top quark. We work in the rest frame of the \(t\) quark. The jet resolution parameter is denoted by \(Y\). An unresolved pair of final-state partons \(i,j\) is recombined by adding the four-momenta, \(k_{(ij)} = k_i + k_j\). The decay (2) contributes to (1) events with \(Y_{bg} > Y\). The real radiation processes (3) contribute to (1) events with one unresolved pair of partons \(i,j\), i.e., with \(Y_{ij} < Y\). The jet distance between the recombined pseudoparticle \((ij)\) and the remaining parton \(n\) must satisfy \(Y_{n(ij)} > Y\).

3 Results

As already mentioned, we work in the top-quark rest frame. If we denote the top-spin vector in this frame by s \(_t\) (where s \(_t^2=1\)), differential distributions for the decay (1) of a 100 % polarized ensemble of top quarks are of the form
$$\begin{aligned} \frac{\mathrm{d}\Gamma }{\mathrm{d}O} = A + \mathbf{B}\cdot \mathbf{s}_t, \end{aligned}$$
(5)
where \(O\) denotes some observable. In the fully differential case, the functions \(A\) and B (which transform as scalar and vector, respectively, under spatial rotations) depend on the independent kinematical variables of (1), and the vector B may be represented as a linear combination of terms proportional to the directions of the charged lepton and of the two jets in the final state.

Rotational invariance implies that a number of distributions hold both for polarized and unpolarized top quarks. This includes the distributions that will be presented in Sect. 3.1.

In Sect. 3.2 we consider distributions that are relevant for the decay of polarized top quarks, namely those that reflect the top-spin analyzing power of the charged lepton, the b-jet, and the \(W\) boson.

For the numerical results given below, we use \(m_t= 173.5\) GeV, \(m_W =80.39\) GeV and \(\Gamma _W = 2.08\) GeV. The QCD coupling for 5-flavor QCD is taken to be \(\alpha _s(m_Z)=0.118\). Its evolution to \(\mu =m_t\) and conversion to the 6-flavor \(\overline{\mathrm{MS}}\) coupling results in \(\alpha _s(m_t)=0.108\). Moreover, we use \(\alpha (m_t)=7.9\times 10^{-3}\) and \(\sin ^2\theta _W=0.231\) which yields the weak coupling \(g_W^2=0.429\). The normalized distributions given below do not depend on \(g_W^2\) because we work to lowest order in \(g_W^2\).

3.1 Distributions for (un)polarized top-quark decay

First, we compute the decay rate of (1) as a function of the jet resolution parameter \(Y\). In Fig. 1 the ratio of \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}\) and the leading order rate \(\Gamma ^{LO}_{t \rightarrow b \, \bar{l}\nu _l}=0.162\) GeV is shown at LO and NLO QCD, for a renormalization scale \(\mu =m_t\). It is clear that this ratio increases for decreasing \(Y\). In the lower pane of this figure, the ‘K factor’ \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}^\mathrm{NLO}/\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}^\mathrm{LO}\) is displayed. One sees that in a large range of \(Y\), the QCD corrections are positive and at most of order 8 %, while for \(Y\) below \(\sim 2.5\times 10^{-3}\) they become negative.
Fig. 1

Upper pane decay rate \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}\) (LO and NLO) normalized to \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l}\) (LO) as a function of the jet resolution parameter \(Y\) for \(\mu =m_t\). Lower pane ratio of \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}^\mathrm{NLO}/\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}^\mathrm{LO}\) as a function of \(Y\). The solid line corresponds to \(\mu =m_t\), the shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

In the remainder of this section we compute normalized decay distributions, both at LO and NLO QCD for two values of the jet resolution parameter, \(Y=0.01\) and \(Y=0.001\). The NLO decay distributions, which are normalized to the NLO decay rate \(\Gamma _{t \rightarrow b \, \bar{l}\nu _l \, + \, \mathrm{jet}}^\mathrm{NLO}\), are expanded in powers of \(\alpha _s\). Taking out a factor of \(\alpha _s\) both from the LO and NLO (differential) rate, we have
$$\begin{aligned} \frac{{\mathrm{d}\Gamma }^\mathrm{NLO}}{{\Gamma }^\mathrm{NLO}}&= \frac{\mathrm{d}\Gamma _{0} + \alpha _s \mathrm{d}\Gamma _{1} + {\mathcal {O}}(\alpha _s^2)}{\Gamma _{0} + \alpha _s \Gamma _{1} + {\mathcal {O}}(\alpha _s^2)}\nonumber \\&= \frac{\mathrm{d}\Gamma _{0}}{\Gamma _{0}}\left( 1-\alpha _s\frac{\Gamma _{1}}{\Gamma _{0}}\right) + \alpha _s\frac{\mathrm{d}\Gamma _1}{\Gamma _0} + {\mathcal {O}}(\alpha _s^2). \end{aligned}$$
(6)

In the following we rescale all dimensionful variables with \(m_t\). That is, in the following, the energies \(E_W\), \(E_l\), \(E_b\), and \(E_2\) of the \(W\) boson, the charged lepton, \(b\)-jet, and the second jet with zero \(b\)-flavor, respectively, and the \(W\) and \(\ell b\)-jet invariant masses \(M_W\), \(M_{lb}\) denote dimensionless variables.

The invariant mass distribution and the energy distribution of the off-shell \(W\) boson are displayed in Figs. 2 and 3 for \(Y=0.01\) and \(Y=0.001\), respectively. The QCD corrections to the invariant mass of the \(W\) boson are very small. The distribution of the \(W\) energy2 \(E_W=E_l+E_\nu \) may be compared with the case of the lowest-order on-shell decay \(t\rightarrow b W\) where the (dimensionless) \(W\) energy is fixed, \({\bar{E}}_W =\sqrt{m_W^2+\mathbf{k}_W^2}/m_t =0.61.\) In the case of additional jet radiation and allowing the \(W\) boson to be off-shell, one expects therefore that the maximum of the distribution of \(E_W\) is below \({\bar{E}}_W\), but it approaches this value if the jet cut \(Y\) is decreased. The distributions on the right sides of Figs. 2 and 3 show this behavior. The QCD corrections are small at and in the near vicinity of the maximum of the distribution, whereas they can become rather large if the \(W\) boson is significantly off-shell.
Fig. 2

Upper panes normalized distribution of the invariant mass \(M_W\) of the \(W\) boson (left) and of the \(W\) energy \(E_W\) (right) for \(Y=0.01\) and \(\mu =m_t\). Lower panes ratio of the NLO and corresponding LO distribution. The shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

Fig. 3

Same as Fig. 2, but for a jet resolution parameter \(Y=0.001\)

The left sides of Figs. 4 and 5 show the distribution of the energy \(E_l\) of the charged lepton. For decreasing jet cut the distribution moves toward the lepton-energy distribution of the inclusive semileptonic decay which, at tree level and for a massless \(b\) quark, has its maximum at \(E_l=0.25\).
Fig. 4

Upper panes normalized distribution of the energy \(E_l\) of the charged lepton (left) and of the invariant mass \(M_{lb}\) of the \(b\) jet and the charged lepton (right) for \(Y=0.01\) and \(\mu =m_t\). Lower panes ratio of the NLO and corresponding LO distribution. The shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

Fig. 5

Same as Fig. 4, but for \(Y=0.001\)

The right sides of Figs. 4 and 5 display the distribution of the invariant mass \(M_{lb}\) of the lepton and the \(b\) jet3. In the case of the LO decay \(t\rightarrow b\ell \nu _\ell \) and an on-shell intermediate \(W\) boson, \(M_{lb}\) has a sharp upper bound, which, in terms of our dimensionless variables, is given by \(M_{lb}^\mathrm{max}=\sqrt{1-m_W^2/m_t^2}\). In the case of (2), (3), where gluons or \(q\bar{q}\) are radiated, the invariant mass \(M_{lb}\) cannot exceed the LO kinematic boundary, as long as the \(W\) boson is kept on-shell. The distance between the maximum of the \(M_{lb}\) distribution and \(M_{lb}^\mathrm{max}\) is expected to decrease with decreasing jet cut \(Y.\) An off-shell \(W\) boson leads to a tail of the \(M_{lb}\) distribution beyond \(M_{lb}^\mathrm{max}\). All of these features arise in the results shown on the right sides of Figs. 4 and 5. In the vicinity of \(M_{lb}^\mathrm{max}=0.89\) the QCD corrections are about \(-10\) %.

The distribution of the b-jet energy \(E_b\) and of the energy \(E_2\) of the second jet is displayed in Figs. 6 and 7. In the case of the LO decay \(t\rightarrow b W\), the energy of the massless \(b\) quark is fixed to be \({\bar{E}}_b =(1-m_W^2/m_t^2)/2 =0.39\). Radiation off the \(t\) and \(b\) leads to an upper bound on \(E_b\) that is below \({\bar{E}}_b\) for \(Y>0\). An off-shell \(W\) boson can, however, lead to some events with \(E_b\) above this value. The average energy \(E_2\) of the second jet is smaller than that of the \(b\) jet. These features are exhibited by the results shown in Figs. 6 and 7. Near the kinematic edges the QCD corrections can become \(\sim \)10 %.
Fig. 6

Upper panes normalized distribution of b-jet energy \(E_b\) (left) and of the energy \(E_2\) of the second jet (right) for \(Y=0.01\) and \(\mu =m_t\). Lower panes ratio of the NLO and corresponding LO distribution. The shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

Fig. 7

Same as Fig. 6, but for \(Y=0.001\)

Figures 8 and 9 show the distribution of \(\cos {\theta _{bl}}\), where \(\theta _{bl}\) is the angle between the directions of flight of the charged lepton and the b-jet in the \(t\) rest frame, and of \(\cos {\theta _{2l}}\), where \(\theta _{2l}\) is the angle between \(\ell ^+\) and the second jet. The distributions of \(\cos {\theta _{bl}}\) are qualitatively similar to the corresponding distributions in the case of inclusive semileptonic top-decay; for most of the events the charged lepton and the \(b\) jet are almost back-to-back. As expected, the distribution of \(\cos {\theta _{2l}}\) is falling less steeply toward smaller angles \(\theta _{2l}\). The QCD corrections are markedly below 5 % in most of the kinematic range.
Fig. 8

Upper panes normalized distribution of \(\cos {\theta _{bl}}\), where \(\theta _{bl}\) is the angle between the directions of flight of the charged lepton and the b-jet in the \(t\) rest frame (left), and of \(\cos {\theta _{2l}}\), where \(\theta _{2l}\) is the angle between \(\ell ^+\) and the second jet. The jet resolution parameter is chosen to be \(Y=0.01\) and \(\mu =m_t\). Lower panes ratio of the NLO and corresponding LO distribution. The shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

Fig. 9

Same as Fig. 8, but for \(Y=0.001\)

The distribution of \(\cos \theta ^*_{Wl}\), where \(\theta ^*_{Wl}\) is the angle between the \(W^+\) direction in the t rest frame and the lepton direction in the \(W^+\) rest frame, is presented in the plots on the left side of Figs. 10 and 11. This distribution has been used ever since at the Tevatron and the LHC for measuring the \(W\)-boson helicity fractions in inclusive semileptonic top-decay. With \(x=\cos \theta ^*_{Wl}\) the one-dimensional distribution has the well-known form
$$\begin{aligned} \Gamma ^{-1}\frac{\mathrm{d}\Gamma }{\mathrm{d}x}&= \frac{3}{4}F_L\left( 1-x^2\right) \nonumber \\&+ \frac{3}{8}F_-\left( 1-x\right) ^2 + \frac{3}{8}F_+\left( 1+x\right) ^2, \end{aligned}$$
with \(F_L + F_- + F_+= 1\). For events with an additional jet, one expects that for small jet cut \(Y\) the corresponding distribution tends toward the inclusive one. Performing a fit to the \(\cos \theta ^*_{Wl}\) distributions of Figs. 10 and 11 (where we take into account that our NLO distributions are not exactly normalized to one, due to the expansion (6)), we obtain \(F_L^\mathrm{NLO}= 0.668\) and \(F_-^\mathrm{NLO}=0.321\) for \(Y=0.01\), and \(F_L^\mathrm{NLO}= 0.689\) and \(F_-^\mathrm{NLO}=0.308\) for \(Y=0.001\). The size of the QCD corrections is \(\lesssim \)1 %. For \(Y=0.001\) the helicity fractions agree very well with the corresponding inclusive ones at NLO QCD (cf., for instance, [15]) and are in agreement with recent results from ATLAS and CMS [44].
The plots on the right sides of Figs. 10 and 11 show the distribution of \(\cos {\theta _{Wb}}\), where \(\theta _{Wb}\) is the angle between the \(W\) and the b-jet directions in the \(t\) rest frame. As in the inclusive case this distribution peaks when the \(W\) boson and the \(b\) jet are back-to-back.
Fig. 10

Upper panes normalized distribution of \(\cos \theta ^*_{Wl}\), where \(\theta ^*_{Wl}\) is the angle between the \(W^+\) direction in the t rest frame and the lepton direction in the \(W^+\) rest frame (left). The right plot shows the normalized distribution of \(\cos {\theta _{Wb}}\), where \(\theta _{Wb}\) is the angle between the \(W\) and the b-jet directions in the \(t\) rest frame. The jet resolution parameter is \(Y=0.01\) and \(\mu =m_t\). Lower panes ratio of the NLO and corresponding LO distribution. The shaded band results from scale variations between \(m_t/2\) and \(2 m_t\)

Fig. 11

Same as Fig. 10, but for \(Y=0.001\)

3.2 Top-spin analyzing power

Finally we consider, for the decay (1) of a 100 % polarized top-quark ensemble, the angular correlation of the top-spin vector s \(_t\) and the direction of flight of a final-state particle or jet \(f\) in the top rest frame, where \(f=\ell ^+,~b~\mathrm{jet},~W^+\). The corresponding normalized distribution has the a priori form
$$\begin{aligned} \frac{1}{\Gamma }\frac{\mathrm{d}\Gamma }{\mathrm{d}\cos {\theta _{f}}} = \frac{1}{2} \left( 1 + \kappa _f \cos {\theta _{f}} \right) , \end{aligned}$$
(7)
where \(\theta _{f} =\angle ({\mathbf {s}}_t, {\hat{\mathbf {k}}}_f)\). The coefficient \(\kappa _f\) is the top-spin analyzing power of \(f\) and measures the degree of correlation. CP invariance implies4 that the corresponding angular distributions for top antiquarks are given by
$$\begin{aligned} \frac{1}{\bar{\Gamma }}\frac{\mathrm{d}\bar{\Gamma }}{\mathrm{d}\cos {\theta _{\bar{f}}}} = \frac{1}{2} \left( 1 - \kappa _f \cos {\theta _{\bar{f}}} \right) . \end{aligned}$$
(8)
The values of \(\kappa _f\) can be extracted from the slope of the distributions (7) or from
$$\begin{aligned} \langle \cos \theta _f \rangle = \frac{1}{\Gamma }\int _0^\pi \mathrm{d}\cos {\theta _{f}} \, \left( \cos {\theta _{f}} \, \frac{\mathrm{d}\Gamma }{\mathrm{d}\cos {\theta _{f}} } \right) = \frac{\kappa _f}{3}. \end{aligned}$$
(9)
The results for \(\kappa _f\) at LO and NLO QCD are listed in Table 1 for two values of the jet resolution parameter \(Y\).
Table 1

Top-spin analyzing powers extracted from the normalized distributions (7) for \(\mu =m_t\). The uncertainties due to scale variations between \(m_t/2\) and \(2 m_t\) are below 1 %

 

\(Y = 0.01\)

\(Y = 0.001\)

\(\kappa _\ell ^\mathrm{LO} \)

0.981

0.993

\(\kappa _\ell ^\mathrm{NLO} \)

0.983

0.996

\(\kappa _W^\mathrm{LO} \)

0.359

0.387

\(\kappa _W^\mathrm{NLO} \)

0.351

0.381

\(\kappa _b^\mathrm{LO} \)

-0.326

-0.368

\(\kappa _b^\mathrm{NLO} \)

-0.319

-0.364

One may compare these \(t\)-spin analyzing powers with the corresponding ones of the dominant semileptonic decay modes \(t\rightarrow b \ell ^+ \nu _\ell .\) In the latter case one has \(\kappa _\ell ^\mathrm{NLO}=0.999\) [18] and \(\kappa _b^\mathrm{NLO} = -0.39\) [19]. Moreover, in this inclusive case, \(\kappa _b^\mathrm{NLO} = -\kappa _W^\mathrm{NLO}\). The charged lepton is the best top-spin analyzer in the semileptonic decays both without and with an additional jet. This is due to the V-A structure of the charged weak current and angular momentum conservation. If an additional jet is produced in top-quark decay, \(\kappa _b = - \kappa _W\) no longer holds, of course, cf. Table 1. In semileptonic \(t\) decays both without and with an additional jet the \(t\)-spin analyzing power of the \(W\) boson is weaker than that of its daughter lepton \(\ell ^+\). This is due to the known fact that for \(t\rightarrow \ell ^+ \nu _\ell b~(+\mathrm{jet})\), the amplitudes that correspond to the different polarization states of the intermediate \(W\) boson interfere constructively (destructively) when \(\ell ^+\) is emitted in (opposite to) the direction of the top spin.

4 Summary

We have computed the differential and total rate of the semileptonic decay of polarized top quarks \(t\rightarrow \ell \nu _\ell + b~\mathrm{jet} + \mathrm{jet}\) at next-to-leading order QCD. We have defined the jets by the Durham algorithm, and we have presented a number of distributions for two different values of the jet resolution parameter. The QCD corrections to the leading-order distributions are \(\lesssim \)5 % in most of the kinematic range. Near kinematic edges or significantly off the \(W\) resonance, the corrections can become \(\sim \)10 %. Our results should be useful as a building block for future analyses of top-quark production and decay in hadron and in \(e^+e^-\) collisions.

Footnotes

  1. 1.

    Other recent results include [42, 43].

  2. 2.

    Here, we tacitly assume that the neutrino energy and momentum can be reconstructed in an experiment, which is usually possible only with ambiguities.

  3. 3.

    The distribution of \(M_{lb}\) in inclusive hadronic \(t\bar{t}\) production and decay was first analyzed at NLO QCD in [41] and was proposed as a tool to measure the top-quark mass. Cf. also [42].

  4. 4.

    The effect of the non-zero Kobayashi–Maskawa phase, which would show up only if higher-order weak corrections are taken into account, is completely negligible in these decays.

Notes

Acknowledgments

The work of W.B. was supported by BMBF and that of C.M. by Deutsche Forschungsgemeinschaft through Graduiertenkolleg GRK 1675.

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Authors and Affiliations

  1. 1.Institut für Theoretische PhysikRWTH Aachen UniversityAachenGermany
  2. 2.Department of Radiation OncologyNetherlands Cancer InstituteAmsterdamThe Netherlands

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