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

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## 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 progress^{1} 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

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.

## 3 Results

**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

**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

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.

^{2}\(E_W=E_l+E_\nu \) 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.

The right sides of Figs. 4 and 5 display the distribution of the invariant mass \(M_{lb}\) of the lepton and the \(b\) jet^{3}. 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 \(-10\) %.

### 3.2 Top-spin analyzing power

**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

^{4}that the corresponding angular distributions for top antiquarks are given by

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.
- 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.
- 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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