# Techniques for the treatment of IR divergences in decay processes at NLO and application to the top-quark decay

## Abstract

We present the extension of two general algorithms for the treatment of infrared singularities arising in electroweak corrections to decay processes at next-to-leading order: the dipole subtraction formalism and the one-cutoff slicing method. The former is extended to the case of decay kinematics which has not been considered in the literature so far. The latter is generalised to production and decay processes with more than two charged particles, where new “surface” terms arise. Arbitrary patterns of massive and massless external particles are considered, including the treatment of infrared singularities in dimensional or mass regularisation. As an application of the two techniques we present the calculation of the next-to-leading order QCD and electroweak corrections to the top-quark decay width including all off-shell and decay effects of intermediate \({\mathrm {W}} \) bosons. The result, e.g., represents a building block of a future calculation of NLO electroweak effects to off-shell top-quark pair (\({\mathrm {W^+}} {\mathrm {W^-}} {\mathrm {b}} {\bar{\mathrm{b}}} \)) production. Moreover, this calculation can serve as the first step towards an event generator for top-quark decays at next-to-leading order accuracy, which can be used to attach top-quark decays to complicated many-particle top-quark processes, such as for \({\mathrm {t}} {\bar{\mathrm{t}}} +{\mathrm {H}} \) or \({\mathrm {t}} {\bar{\mathrm{t}}} +\text {jets}\).

## 1 Introduction

The calculation of radiative corrections to cross-section predictions at next-to-leading order (NLO) saw tremendous progress in the past decades leading to a high degree of automation of calculations. This advancement was also driven by the development of general techniques for the treatment of infrared (IR) singularities. Such divergences arise in the intermediate steps of the calculation involving massless states and cancel in the final result by virtue of the Kinoshita–Lee–Nauenberg (KLN) and factorisation theorems. The various methods that were proposed in the literature can be broadly classified into two types: the phase-space-slicing [1, 2, 3, 4, 5] and the subtraction formalisms [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Both approaches allow for the analytic cancellation of all IR singularities without jeopardising the flexibility of a fully differential numerical phase-space integration.

In the first part of the paper we consider the so-called *dipole subtraction formalism*, which is one of the most established subtraction methods. It was continuously extended since its original formulation in Ref. [12], which was restricted to the computation of NLO QCD radiative corrections involving only massless partons, to cover all possible cases that are relevant in scattering processes. In Ref. [16] the dipole formalism was developed for the calculation of NLO QED corrections where the emission of photons from both massless and massive fermions was considered. The extension of the original algorithm for NLO QCD calculations to the case with massive partons was worked out in Refs. [17, 18]. The treatment of non-collinear-safe observables in QED corrections was presented in Ref. [19] where the formalism was further extended to cover the various fermion–photon splittings not considered in Ref. [16].

For the case of decay processes, subtraction terms were constructed in Ref. [21] for the calculation of NLO QCD corrections to the radiative decay of the top-quark. These subtraction terms were based on the results of Ref. [22] and constructed to fit into the dipole subtraction formalism. Although the discussion in Ref. [21] was restricted to the case of the top-quark decay, the subtraction terms given there should in principle be applicable to the decay of any strongly interacting particle, however, with the restriction that only massless partons appear in the final state.

So far, no general algorithm exists for the calculation of NLO QED corrections to decay processes using the dipole subtraction formalism. In this paper we fill this gap by explicitly constructing such a process-independent subtraction term considering both massless and massive final-state fermions and work out its extension to the treatment of non-collinear-safe observables. The further extension of the formalism to treat massive partons in NLO QCD corrections to decay processes is then fully straightforward.

The *one-cutoff phase-space slicing* (OCS) method, considered in the second part of the paper, was initially introduced in Ref. [1] for the calculation of NLO QCD corrections to the multi-jet production cross section in \({\mathrm {e^+}} {\mathrm {e^-}} \) annihilation. As such, its application was limited to the case with only massless coloured partons in the final state. It was later extended to allow for incoming partons [2] and massive quarks [4]. The OCS method is distinguished by the feature that one single Lorentz-invariant cut is introduced to isolate the soft and/or collinear regions in the real-emission phase space. Another frequently used slicing method, which is not considered in detail in this paper, is the so-called two-cutoff slicing method [5, 23]. As a consequence of introducing two independent cuts on energies and angles, however, this approach is not manifestly Lorentz invariant.

In its original formulation, the OCS method is restricted to amplitudes that involve only two sources (particles) of IR singularities, a feature that is naturally supported by colour-ordered QCD amplitudes, but neither by sub-leading colour structures in QCD nor by IR singularities originating from *U*(1) gauge bosons, such as the photon in QED. Furthermore, no results employing mass regularisation are documented in the literature, although it is a common choice in the calculation of electroweak radiative corrections. In this paper we extend the OCS formalism in these two respects, i.e. presenting a formulation employing mass regularisation and dealing with an arbitrary number of soft or collinear singularities. To this end, we employ results from the aforementioned dipole subtraction formalism to obtain an approximation of the real-emission matrix elements in the singular regions together with the associated phase-space parametrisations suitable for the subsequent analytic integration. Furthermore, we discuss the subtleties that arise in the case of processes that involve arbitrary many sources of IR singularities.

At its design luminosity in Run 2, the LHC will be a top-quark factory, copiously producing top-quarks in pairs, as single particles, or in association with other particles. As the heaviest elementary particle known to date, the top-quark is a unique window to explore the mechanism of electroweak symmetry breaking and to test the Standard Model (SM) on the one hand, and to probe physics beyond the SM on the other. Apart from the top-quark mass, key observables in the top sector comprise the top-quark width and all accessible production cross sections, for which precise theoretical predictions are required. In these predictions the top-quark width \(\Gamma _{{\mathrm {t}}}\) plays a double role: as an observable in its own right and as an ingredient in cross-section predictions that take into account top-quark decays. In the latter case, \(\Gamma _{{\mathrm {t}}}\) enters the Breit–Wigner propagator of the top-quark resonance, so that it should be known to the same perturbative order as the calculated cross section.

The corrections to the top-quark decay process \({\mathrm {t}} \rightarrow {\mathrm {b}} {\mathrm {W^+}} \) are known at NLO [24, 25, 26, 27] and next-to-next-to-leading order (NNLO) [28, 29, 30, 31, 32] in QCD and up to NLO for the electroweak (EW) corrections [33, 34, 35, 36, 37, 38]. However, all prior calculations either consider on-shell \({\mathrm {W}} \) bosons or only partially account for off-shell effects by performing an expansion about the intermediate W resonance. As an application of the techniques introduced in this work, we present the fully differential calculation of the QCD and EW corrections at NLO to the three-body decay of the top-quark, accounting for off-shell effects of the intermediate \({\mathrm {W}} \) boson within the complex-mass scheme [39, 40, 41] and retaining a finite bottom-quark mass. We consider both the hadronic and the semi-leptonic decay channels and further allow for non-collinear-safe photon emission (“bare leptons”) for the latter. The presented calculation of the NLO EW corrections to the decays \({\mathrm {t}} \rightarrow {\mathrm {b}} {\mathrm {W^+}} \rightarrow {\mathrm {b}} {\ell ^+} {\nu _\ell }/ {\mathrm {b}} {\bar{q}} {q} '\), thus, provides an ingredient to the (not yet existing) calculation of NLO EW corrections to the \({\mathrm {t}} {\bar{\mathrm{t}}} \) production process \({\mathrm {p}} {\mathrm {p}} \rightarrow {\mathrm {b}} {\bar{\mathrm{b}}} {\mathrm {W^+}} {\mathrm {W^-}} \rightarrow 6{f} \) including top-quark decays, similar to the respective NLO QCD corrections to the top-quark decay in the NLO QCD calculation to \({\mathrm {p}} {\mathrm {p}} \rightarrow {\mathrm {b}} {\bar{\mathrm{b}}} {\mathrm {W^+}} {\mathrm {W^-}} \rightarrow 6{f} \) [42, 43, 44, 45, 46].

On the experimental side, the first direct experimental determination of the top-quark decay width was performed by the experiments at the Tevatron collider [47, 48]. It, however, still suffered from large experimental uncertainties. Recently, the first measurement at the LHC was presented by the CMS experiment with the data collected at \(\sqrt{s}=8~\mathrm {TeV} \) [49], significantly improving on the previous measurements, with a precision at the \(10~\%\) level.

This paper is organised as follows: In Sect. 2 we briefly review the general structure of an NLO calculation to decay processes to set up the notation and conventions used throughout this work. Section 3 first outlines the dipole subtraction formalism in Sect. 3.1 and subsequently discusses its extension to deal with decay kinematics. Massive final-state fermions are treated in Sect. 3.2, and the special case of light fermions is covered in Sect. 3.3, where we also discuss the modifications for non-collinear-safe observables. The OCS method is presented in Sect. 4 with a general overview of the formalism in Sect. 4.1, explaining in particular the origin and treatment of so-called “surface terms” which arise in the case of processes with more than two sources of IR singularities. The various cases that arise from the different combinations of massless or massive particles in the initial or final states are covered in Sect. 4.2. The application of the two techniques to the top-quark decay at NLO is presented in Sect. 5 while the calculational details are summarised in Sect. 5.1. In Sects. 5.2 and 5.3 we present the numerical results for the corrections to the decay width and the differential distributions, respectively. The conclusions and an outlook are given in Sect. 6.

## 2 NLO corrections to decay processes – notation and conventions

*n*particles is given by

*n*-particle phase-space measure, where \(p_i\) and \(m_i\) denote the momentum and mass of the

*i*th final-state particle, respectively, and \(p_a\) the momentum of the decaying particle, with \(p_a^\mu =(m_a,\mathbf {0})\) in its centre-of-mass frame. Furthermore, we have used the shorthand notation \( \Phi _{n} \equiv \{p_1,\ldots ,p_n\}\) for the associated kinematics given by the set of momenta of the particles.

The calculation of higher-order corrections involving the emission of massless particles, in general, leads to the occurrence of IR singularities in both the virtual and the real corrections which cancel in the sum of Eq. (2.4) by virtue of the KLN theorem.^{1} Although the cancellation of IR singularities is well understood, in practise, the procedure is non-trivial, since the virtual and real corrections are defined on different phase spaces. Moreover, the divergences contained in the real corrections are implicitly hidden in the phase-space integration, whereas the singularities in the virtual corrections are made explicit using regulators when performing the integration over the loop momentum. To accomplish an analytic cancellation of all IR singularities therefore requires one to make the singularities of the real corrections explicit by means of regulators. On the other hand, it is desirable to retain the flexibility of a fully differential numerical approach for the phase-space integration. Both the slicing and the subtraction techniques provide a prescription to perform a fully differential calculation suitable for a numerical evaluation while having the full analytic control over the IR singularities and their cancellation. Both approaches exploit the factorisation properties of the real-emission amplitude in the various singular limits which we briefly review in the following for the case of photon radiation off fermions in order to set up our notation. For further details we refer to Ref. [16] which we follow closely. Throughout this work we regularise the IR singularities using *mass regularisation* where an infinitesimal photon mass \(m_{{\mathrm {\gamma }}}\) is introduced and the masses of the light fermions \(m_{{f}}\) are retained in the divergent logarithms. A prescription for the conversion of the results obtained using mass regularisation to the corresponding expressions using dimensional regularisation is given in the “Appendix”.

*soft-photon limit*(\(k\rightarrow 0\)) can be described by the well-known eikonal approximation where the squared real-emission amplitude, summed over the photon polarisations \(\lambda _{{\mathrm {\gamma }}}\), can be written as follows:

*e*is the elementary charge. Here the summation over \({f},{f} '\) extends over all charged external particles (of any spin), and the reduced matrix element \(\mathcal {M} _0\) is evaluated with the set of momenta that is obtained by omitting the momentum

*k*of the soft photon. We have further introduced the sign factors \(\sigma _{{f}}=\pm 1\) describing the charge flow, which are defined as \(\sigma _{{f}}=+1\) for incoming particles and outgoing anti-particles and \(\sigma _{{f}}=-1\) for incoming anti-particles and outgoing particles. Overall charge conservation can then be expressed as

*i*which might change to any other spin value \(\tau \kappa _i\). If

*i*is a spin-\(\frac{1}{2}\) fermion, the spin can only potentially change sign, i.e. there is just a sum over \(\tau =\pm \). In this case, the functions \(g_{i,\tau }^\text {(out)}\) are given by

Note that Eq. (2.8a) and (2.8b) does not describe particles with a spin other than \(\frac{1}{2}\). However, any subtraction formalism based on this asymptotics may still be used for particles with another spin (such as a W boson) as long those particles are heavy enough to avoid collinear singularities, because the asymptotic behaviour (2.5) in the soft-photon limit is spin independent.

## 3 The dipole subtraction formalism

As was pointed out in the introduction, so far, no general formalism exists yet in the literature for the calculation of NLO QED corrections to decay processes within the dipole subtraction formalism. We present the construction of such a general subtraction term in the following and begin in Sect. 3.1 with a brief overview of the formalism and of our approach for its generalisation to decay processes. The explicit form of the corresponding radiator function and its integrated counterpart is given in Sect. 3.2 for the case of a massive final-state fermion. Section 3.3 considers the case of light fermions and the extension to non-collinear-safe observables.

### 3.1 Overview of the method

*dipoles*\(\left|{\mathcal {M} _{\text {sub} , {f} {f} ' }}\right|^2\) according to

*emitter*and

*spectator*particles, respectively. The dipole terms are written as a product of the auxiliary functions \(g^\text {(sub)}_{{f} {f} '} \) and the squared matrix element \(\left|{\mathcal {M} _0}\right|^2\), where the former represent universal functions that encapsulate the singularity structure and can be integrated analytically over the singular photon phase space once and for all. The associated squared matrix element \(\left|{\mathcal {M} _0}\right|^2\) is evaluated on the reduced

*n*-particle phase space \( \widetilde{\Phi }_{n ,{f} {f} ' } \) that remains after isolating the singular one-particle subspace \([\mathrm {d}k]\) of the photon. These

*dipole mappings*are required to reproduce the appropriate kinematic configuration in the singular regions and further respect exact momentum conservation as well as all mass-shell conditions.

A plain extension in the spirit of the dipole subtraction formalism for the decay process would construct two new subtraction terms: a dipole where the decaying particle *a* is the emitter, \(g^\text {(sub)}_{ai}\), and second where *a* acts as the spectator \(g^\text {(sub)}_{ia}\). However, since there are no collinear singularities associated with the initial-state particle *a*, we combine the two subtraction terms to a single term called \(d^\text {(sub)}_{ia} = g^\text {(sub)}_{ia} + g^\text {(sub)}_{ai} \) which is illustrated in Fig. 1 in terms of the two schematic diagrams that would be associated with \(g^\text {(sub)}_{ia}\) and \(g^\text {(sub)}_{ai}\).

*i*,

*j*extends over all charged particles in the final state,

*a*denotes the decaying particle, and \(g^\text {(sub)}_{ij}\) are the known subtraction functions of the dipoles with a final-state emitter–spectator pair given in Ref. [16].

### 3.2 Subtraction function for massive final-state fermions

In the following we present the construction of the subtraction function \(d^\text {(sub)}_{ia}\) for the decay of a particle *a* of mass \(m_a\) with a massive final-state fermion *i* of mass \(m_i\). The special case of light fermions can be obtained by a systematic expansion in \(m_i\) and will be discussed in the next section together with the extension to non-collinear-safe observables.

*a*fixed. This constraint can be maintained by the deformation of the momenta \(k_l\) of all particles that are not involved in the respective dipole term via a Lorentz transformation. Such a transformation is necessary in order to compensate the recoil induced by the radiated photon so that four-momentum conservation and all mass-shell conditions are retained. To this end, we define the auxiliary momentum \(P_{ia}\) of the “recoiling system”, see Fig. 1,

*j*of Ref. [16], where particle

*i*plays the role of the emitter and the recoiling system the role of the spectator of mass \(m_j^2=P_{ia}^2=\widetilde{P}_{ia}^2\). In detail, the following substitutions have to be made in order to transfer the quantities of Sect. 4.1 of Ref. [16] to the kinematics considered here:

### 3.3 Light fermions and non-collinear-safe observables

The corresponding result where the IR singularities are regularised using dimensional regularisation can be obtained from Eq. (3.23) by performing a transition between the regularisation schemes using the results of Ref. [52] which is briefly summarised in the “Appendix”.

*non-collinear-safe*observables, photons arbitrarily close to an outgoing charged particle are not treated inclusively, but it is assumed that such a configuration can be experimentally resolved. This is, for instance, the case for muons in the final state, which are observed in the muon chambers further out in the detector, whereas the photons are detected independently in the electromagnetic calorimeter. Such an observable definition is sensitive to the details of the collinear \({f} \rightarrow {f} +{\mathrm {\gamma }} \) splitting. The subtraction formalism can be extended to cover this case by exploiting the information on the dipole variable \(z_{ia}\) which corresponds to the momentum fraction \(z_i\) of Eq. (2.9) carried away by the emitter fermion in the collinear limit. This is accomplished by treating the

*n*-particle kinematics of the dipole phase space \( \widetilde{\Phi }_{n ,ia } \) as an \((n+1)\)-particle event with momenta \(p_i\rightarrow z_{ia}\tilde{p}_i\) and \(k\rightarrow (1-z_{ia})\tilde{p}_i\). For further details we refer to Ref. [19] where this extension was introduced. Owing to the explicit dependence of the observables on the dipole variable, the integration over \(z_{ia}\equiv z\) cannot be performed analytically, but is left open for a numerical evaluation. Employing a plus prescription, the singular endpoint part can be isolated so that the extension to the non-collinear-safe case amounts to an additional term which vanishes for inclusive observables,

*g*(

*x*). The function \(\bar{\mathcal {D}}^\text {(sub)}_{ia}\) is obtained after carrying out the analytical integration over the dipole variable \(y_{ia}\) and explicitly reads

## 4 The one-cutoff phase-space slicing method

The main idea of the phase-space slicing approach is the isolation of resolved hard emissions from the unresolved soft and/or collinear regions in the real-emission phase space upon introducing a technical cut parameter. The hard-emission contributions are free of singularities, so that the corresponding phase-space integration can be performed numerically using Monte Carlo methods. The contributions from the unresolved regions encapsulate the IR singularities of the real corrections, and the integrations over the subspaces containing the singularities are performed analytically, exploiting the universal factorisation properties of amplitudes in the soft or collinear limits. This makes the IR singularities that were implicitly contained in the real-emission phase space explicit by means of regulators and allows for an analytic cancellation of all IR singularities. Various approaches have been proposed for the introduction of the technical cut. The OCS method considered in this work imposes a single cut \(2p_i p_j>\Delta s\) on the invariant scalar products of the momenta \(p_i,p_j\) of two particles that can cause a soft singularity (\(p_i\rightarrow 0\) or \(p_j\rightarrow 0\)) or a collinear singularity (\(p_i p_j\rightarrow 0\)). The method, thus, isolates soft and collinear singularities simultaneously and results in a formalism that is manifestly Lorentz invariant. The technical cut parameter \(\Delta s\) should be chosen small enough to suppress all effects of order \(\mathcal{O}(\Delta s)\) to a numerically irrelevant level. Despite the drawback of introducing an arbitrary resolution parameter, such as \(\Delta s\), slicing methods offer the possibility to widely suppress negative weights and to avoid unbounded weights in Monte Carlo integrations, which considerably simplifies a later extension of the Monte Carlo integrator to a Monte Carlo event generator with unweighted events.

Although the OCS method has been extensively studied in NLO QCD calculations involving massless [1, 2] or massive [4, 23, 53] cases, there is so far no exhaustive survey of results covering all relevant configurations of massive particles in production and decay processes. The OCS method is particularly suited to NLO QCD calculations in leading-colour approximation where the corrections are decomposed into colour-ordered contributions that naturally involve only singularities associated with the two neighbouring hard radiators of the unresolved parton in the colour ordering. If more than two collinear singularities appear in one contribution, as for instance in the sub-leading colour contributions, the application of OCS becomes subtle. In this case, process-specific treatments are required, such as reconstructing the IR singularities from the leading-colour building blocks as described in Refs. [1, 2].

*m*cannot be obtained upon simply taking the limit \(m\rightarrow 0\) in the OCS method, since the hierarchy \(\Delta s\ll m^2\) is used in the integration over the unresolved regions for a radiating particle of mass

*m*. In order to isolate the mass singularity regulated by a small mass

*m*(which defines a mass regularisation), however, the inverted hierarchy \(m^2\ll \Delta s\) is required. In the following we describe OCS for QED radiation covering all relevant mass configurations (with large and small masses). The transfer to QCD calculations is straightforward and merely requires the reintroduction of colour charges known from the massless QCD case.

^{2}

### 4.1 Overview of the method

*k*are the momenta of a charged fermion

*f*and the photon, respectively. The definition (4.2) holds irrespective whether the fermion is massless or massive. In order to formally partition the phase space into hard and soft–collinear regions, we introduce shorthands for the \(\theta \)-functions

*hard*region we require \(s_{ f\gamma }>\Delta s \) for all charged fermions

*f*of the process, so that the corresponding contribution to the real correction reads

*soft–collinear*region is given by

The integrals \(I_{\mathrm {S},ff'}\) and \(I_{\mathrm {C},ff'}\) are confined to the soft and collinear regions, respectively, and involve only the technical cuts on the invariants \(s_{f\gamma }\) and \(s_{f'\gamma }\) of the emitter and spectator fermions. For this reason, the integrations in \(I_{\mathrm {S},ff'}\) and \(I_{\mathrm {C},ff'}\) over the singular subspaces can be performed analytically for small \(\Delta s\), as outlined below. On the other hand, the quasi-soft integrals \(I_{\mathrm {QS},ff'}\) involve cuts on all charged particles, rendering a process-independent analytical treatment impossible; nevertheless their numerical evaluation is straightforward, as outlined at the end of this section. Note that the quasi-soft integrals \(I_{\mathrm {QS},ff'}\) do not occur if only two sources of collinear singularities exist in the bremsstrahlung phase space. It is the occurrence of these subtle terms that requires process-specific and non-generic manipulations of the formalism as described in Refs. [1, 2] for the sub-leading colour contributions.

*k*, we can simplify the evaluation even further. For the relevant contribution we can factorise the \((n+1)\)-particle phase space into the corresponding

*n*-particle phase space for all particles but the photon and the 1-particle photon phase space, leading to the alternative form

### 4.2 Integration over the singular regions

#### 4.2.1 Final-state emitter and final-state spectator

*i*and a final-state spectator

*j*(illustrated in Fig. 3) the full integral over the soft–collinear regions is given by

*i*become collinear, this photon recombination ensures that all

*i*-collinear configurations integrated over in \(G_{ij}(P_{ij}^2,\Delta s)\) are treated inclusively, i.e. either the full contribution represented by \(G_{ij}(P_{ij}^2,\Delta s)\) contributes or it is completely excluded by cuts. This even holds true for contributions in \(G_{ij}(P_{ij}^2,\Delta s)\) where the photon is collinear to the spectator

*j*, since the photon has to be soft in this case, i.e. \(s_{j\gamma }=\mathcal{O}(\Delta s)\). In non-collinear-safe observables, however, the photon can be separated from collinear fermions, which is for instance the case for muons detected in the muon chambers. In this case the differential information on the energy flow inside the

*i*-collinear cone should be kept in the phase-space integral (4.20). This energy flow in the soft–collinear region is controlled by the variable \(z_{ij}\), where it is effectively given by \(z_i=p_i^0/(p_i^0+k^0)\). In this situation, the integration over \(z_{ij}\) in Eq. (4.20) should be included in the numerical phase-space integration, so that instead of the full integral \(G_{ij}(P_{ij}^2,\Delta s)\) the following function of \(z=z_{ij}\) appears:

*z*-integration,

*(i) Massive emitter and massive spectator*

*z*values in the soft–collinear region are of the order \(1-\mathcal{O}(\Delta y)\). The integral \(G_{ij}(P_{ij}^2,\Delta s)\), thus, only receives contributions from a tiny neighbourhood of the point \((y_{ij},z_{ij})=(0,1)\), i.e. it does not involve an integration over the momentum flow in the collinear regions. In this case non-collinear-safe observable can be calculated using \(G_{ij}(P_{ij}^2,\Delta s)\) without the need to introduce \(\bar{\mathcal {G}}_{ij}(P_{ij}^2,\Delta s,z)\).

*(ii)* *Massive emitter i and light spectator j*

*(iii)* *Light emitter i and massive spectator j*

*i*-collinear singularities, which are both isolated by the cut parameter \(\Delta s \) in the integration over the hard photon emission which is based on matrix elements with \(m_i=0\).

*(iv)*

*Light emitter i and light spectator j*

#### 4.2.2 Final-state emitter and initial-state spectator, and vice versa

*i*and an initial-state spectator

*a*, and of an initial-state emitter

*a*and a final-state spectator

*i*(depicted on the left- and on the right-hand sides of Fig. 4, respectively), again the variables and abbreviations of Ref. [16] are used.

^{3}Here we use

*i*and

*a*. The soft and collinear regions are characterised as follows:

*(i) Massive final-state emitter i and massive initial-state spectator a, and vice versa*

*i*and initial-state spectator

*a*(or vice versa) are massive, we have to respect the following hierarchy of parameters:

*(ii) Light particle i and massive particle a*

*i*is light and particle

*a*is heavy, we have to consider the hierarchy

*(iii) Light particle a and massive particle i*

*a*is light and particle

*i*is heavy, we have to consider the hierarchy

*(iv) Light particles*

*a*

*and*

*i*

*a*and

*i*being light, so that the mass hierarchy is

#### 4.2.3 Initial-state emitter and spectator

*a*and spectator

*b*in the initial state with masses \(m_a\) and \(m_b\). The corresponding structural diagram is shown in Fig. 5. Using again the kinematical variables introduced in Ref. [16],

*(i) Massive initial-state emitter a and spectator b*

*a*and initial-state spectator

*b*are massive, we have to respect the following hierarchy of parameters:

*(ii) Massive emitter a and light spectator b*

*(iii) Light emitter a and massive spectator b*

*a*is light and the spectator

*b*is heavy, we have to consider the hierarchy

*(iv) Light emitter a and spectator b*

*a*and

*b*being light, so that the mass hierarchy is

## 5 Application to the top-quark decay

As an application of the techniques described in Sects. 3 and 4, in the following we present the calculation of the QCD and EW radiative corrections to the top-quark decay at NLO. First, we summarise our calculational setup in Sect. 5.1. Then, we discuss the different contributions to the top-quark decay width in Sect. 5.2. Finally, in Sect. 5.3 we present differential distributions in the top-quark rest frame for a set of kinematic observables.

### 5.1 Calculational setup and input parameters

*complex-mass scheme*[39, 40, 41], a method that takes into account the effects of the instability and the off-shellness of the unstable particle in a gauge-invariant way, providing a consistent procedure at the one-loop level. The main idea of the complex-mass scheme is to consider the squared boson masses as complex quantities, defined through the gauge-invariant pole mass \(M_V\) and width \(\Gamma _V\),

Contributions to the decay width of the top-quark at NLO in \(\alpha _\mathrm {s} \) and \(\alpha \), divided into the semi-leptonic and hadronic decay channels. Results using the narrow-width approximation (NWA) are given for comparison

\(\Gamma _{{\mathrm {t}}}^\text {LO} (\mathrm {GeV})\) | \(\Gamma _{{\mathrm {t}}}^\text {NLO} (\mathrm {GeV})\) | \(\delta ^{\alpha _\mathrm {s}} (\%)\) | \(\delta ^\alpha (\%)\) | ||
---|---|---|---|---|---|

Semi-leptonic | Off-shell | 0.161065 (1) | 0.148109 (2) | \(-9.38\) | \(+1.34\) |

NWA | 0.163634 | 0.150504 | \(-9.38\) | \(+1.35\) | |

Hadronic | Off-shell | 0.483194 (1) | 0.46242 (3) | \(-5.59\) | \(+1.29\) |

NWA | 0.490902 | 0.46987 | \(-5.59\) | \(+1.31\) | |

Total | Off-shell | 1.449582 (4) | 1.36918 (6) | \( -6.85\) | \(+1.30\) |

NWA | 1.472707 | 1.39126 | \(-6.85\) | \(+1.32\) |

#### 5.1.1 Details on the calculations

The implementation employing the slicing method of Sect. 4 was performed using matrix elements that were calculated using the Weyl–van-der-Waerden spinor formalism (as formulated in Ref. [61]) for the tree-level amplitudes and in-house Mathematica routines for the virtual one-loop corrections. Optimised phase-space parametrisations, based on Refs. [39, 62], were chosen for each of the pronounced structures in the integrand induced by collinear poles or Breit–Wigner resonances. The numerical integration was performed using the Vegas algorithm [63, 64].

A second independent calculation was performed using the subtraction method described in Sect. 3. Here, the matrix elements were obtained with the combined packages FeynArts [65] and FormCalc [66, 67]. An independent implementation of routines for the phase-space generation was employed. As in the calculation based on the slicing approach, the numerical integration was performed with Vegas.

Both implementations resort to the Collier library [68], which is mainly based on the results of Refs. [69, 70] for the numerical evaluation of loop integrals and supports both dimensional and mass regularisation for the treatment of IR singularities.

### 5.2 NLO corrections to the top-quark decay width

In Table 1 we collect the numerical results on the various contributions to the top-quark decay width, providing separate results on the semi-leptonic and hadronic decay channels.

^{4}The full NLO prediction

^{5}\(\Gamma ^{\text {NLO}}_{{\mathrm {t}}}=1.3693(2)\mathrm {GeV} \) for the total width (see Table 1) can be compared to the most recent CMS measurement [49] of

As discussed in Sect. 4, the slicing approach introduces a cut parameter \(\Delta s \) in order to isolate the singular region of the real-emission phase space. The dependence on this cut parameter cancels between the individual parts of the calculation in a non-trivial manner when combined to the final result, constituting a powerful check on the correctness of the calculation. Although this implies a certain degree of arbitrariness in the choice of the specific value for the cut parameter, in practice, sensible values for \(\Delta s \) are constrained to a certain range. This interval of validity is limited by the approximation used in the analytic integration on the upper, and by the stability of the numerical evaluation on the lower end. It is therefore necessary to perform a scan over the cut parameter to identify the region where both restrictions are simultaneously fulfilled and where the result can be trusted. Figure 6 illustrates such a variation of \(\Delta s \) covering three orders of magnitude, clearly depicting the on-set of the aforementioned breakdown of the calculation. Furthermore, we observe good agreement between the results obtained by the two techniques which overlap within their Monte Carlo errors where the \(\Delta s \) variation develops a plateau. The OCS results presented in the following are obtained for a specific value of \(\Delta s \) where a point in the middle of the plateau was chosen after performing such a scan. This procedure is repeated for each decay mode (semi-leptonic and hadronic) and each type of correction (QCD and EW).

At this point, a comment on the relative performance of the two methods is in order: Although the OCSM has the advantage of avoiding negative weights to a large extent, the additional variation of the slicing cut that is required to locate the plateau constitutes an additional computational overhead. Furthermore, we require a higher degree of optimisation in the integrator to sample the regions of the real-emission phase space close to the technical cut to high accuracy, which is then cancelled against the large contribution that arises from the analytically integrated soft–collinear contributions. These are the well-known drawbacks of slicing approaches compared to local subtraction methods such as the dipole formalism which generally display better numerical convergence and stability. More specifically, in order to reach a comparable Monte Carlo error between the two implementations, the dipole subtraction formalism roughly requires an order of magnitude less statistics in the integration compared to the OCSM method.

### 5.3 Differential distributions in the top-quark rest frame

In the following, we present differential distributions for the decay of the top-quark in its rest frame. After describing the event reconstruction in Sect. 5.3.1, we present the numerical results for the semi-leptonic and hadronic decay channels in Sects. 5.3.2 and 5.3.3, respectively. The differential distributions and their corrections are shown as plots that are subdivided into three frames with the following conventions: The upper panel depicts the absolute distributions, while the bottom two frames show the relative EW (middle panel) and QCD (bottom panel) corrections.

#### 5.3.1 Event reconstruction

It is not always possible to fully distinguish the particles of the process in the soft and collinear configurations. In our calculation, we try to be as close as possible to an experimental situation and distinguish the following scenarios:

*(i)* *Electroweak corrections to the semi-leptonic decay “Dressed leptons”:* The EW corrections contain IR singularities originating from soft photons and configurations where the photon becomes collinear to the final-state charged lepton. For sufficiently inclusive observables these singularities cancel in the final result. To this end, we apply the *photon-recombination procedure* described in Ref. [71] which attributes the momentum of a collinear photon to the charged lepton when close to it, i.e., when \(\Delta R({\ell },\,{\mathrm {\gamma }})=\sqrt{\Delta y({\ell },\,{\mathrm {\gamma }})^2 + \Delta \phi ({\ell },\,{\mathrm {\gamma }})^2} < 0.1\), where \(\Delta y\,(\Delta \phi )\) is the rapidity (azimuthal angle) difference between two objects. Such a recombination procedure is mandatory for a realistic experimental description in the case of electrons. *“Bare muons”:* In the case of muons, a collinear muon–photon pair can be experimentally resolved, so that no photon recombination is required. This results in non-collinear-safe observables, which are treated following the procedures described in Sects. 3.3 and 4.2.

*(ii)* *QCD corrections to the semi-leptonic and hadronic decays*

In contrast to the semi-leptonic case, where we assume that the bottom quark could be directly identified with a bottom jet, in the hadronic case we are confronted with the situation that experimentally indistinguishable partons are present in the final state. It is therefore necessary to define an IR-safe observable which groups soft and/or collinear partons into jets. In our calculation we adopt the inclusive generalised anti-\(k_\mathrm {T}\) algorithm [72, 73] for two values of the clustering parameter, \(R=0.6\) and \(R=1.0\). The first value corresponds to the ATLAS default choice [74], the second one will be motivated in Sect. 5.3.3. We assume a bottom-tagging efficiency of \(100~\%\) and identify the jet containing the \({\mathrm {b}} \) quark as the \({\mathrm {b}} \)-tagged jet. In the following, we refer to the jet candidate with the highest transverse momentum^{6} as the leading jet.

*(iii)* *Electroweak corrections to the hadronic decay*

When computing the EW corrections to the hadronic decay, we do not distinguish the photon from the other QCD partons and apply the same jet algorithm as described in the previous paragraph.

#### 5.3.2 Semi-leptonic decay of the top-quark

They comprise differential distributions in the energy of the charged lepton (\(E_{{\ell }}\)) and of the bottom quark (\(E_{{\mathrm {b}}}\)), the invariant mass of the bottom quark and the charged lepton (\(M_{{\mathrm {b}} {\ell }}\)), and the cosine of the angle between the directions of the lepton and the bottom quark (\(\cos {\theta _{{\mathrm {b}} {\ell }}}\)).

For the remaining distributions, i.e. in \(M_{{\mathrm {b}} {\ell }}\) and \(E_{{\ell }}\), we observe EW corrections that are larger in magnitude for “bare muons”, stemming from the large mass-singular logarithm \(\alpha \ln m_\mu \) and reflecting the non-collinear-safe nature of the corresponding observable. The emitted photon in the \({\ell } \rightarrow {\ell } \,{\mathrm {\gamma }} \) splitting removes some energy from the charged lepton. Contrary to the “bare muon” case, the “dressed lepton” effectively can reabsorb some photons in its redefinition when the photon-recombination procedure is applied. Hence, the EW correction to \(E_{{\ell }}\), always monotonically decreasing, are steeper for the “bare muon” than for the “dressed lepton”, as can be seen in the bottom left plot of Fig. 7. For similar reasons, this effect is present also in the distribution in \(M_{{\mathrm {b}} {\ell }}\), as shown in the top left plot of Fig. 7. Here, the EW corrections decrease from \(+\)6 to \(-\)4 % for “dressed leptons”, while they drop from \(+\)8 to \(-\)10 % when “bare muons” are considered in the top-quark decay. The QCD corrections to this distribution are decreasing down to \(-\)45 % for large invariant masses.

#### 5.3.3 Hadronic decay of the top-quark

Here, we present the results for the QCD and EW corrections to the top-quark hadronic decay width. In our setup, the corrections to the two processes \({\mathrm {t}} \rightarrow {\mathrm {b}} \, {\mathrm {u}} {\mathrm {\bar{d}}} \) and \({\mathrm {t}} \rightarrow {\mathrm {b}} \, {\mathrm {c}} {\mathrm {\bar{s}}} \) coincide and are given in Table 1 for the integrated decay width.

In the top-left plot of Fig. 8 we show the distribution in the invariant mass of the \(\mathrm {b}\)-jet and the leading jet.

In the upper frame we notice that the QCD corrections evaluated for \(R=0.6\) turn the distribution negative at \(M_{{\mathrm {b}} j}\sim m_{{\mathrm {t}}}\). This is of course an unphysical effect: For a too low value of *R*, as in this case, the recombination between the real and the virtual parts is not sufficiently inclusive, so that the IR divergence leaves a trace as \(\alpha _s\ln R\) corrections. This problem can be solved by enlarging the cone, e.g. to \(R=1\), as shown in the same plot. With a wider cone, there is indeed a larger probability for the particles to be merged. In the case where all the final-state particles (excluding the \(\mathrm {b}\) quark) are combined into a single jet, the invariant mass \(M_{{\mathrm {b}} j}\) reproduces the top-quark mass value. This explains the appearance of a peak at \(M_{{\mathrm {b}} j}=m_{{\mathrm {t}}}\). This obviously happens more frequently for a \(1\rightarrow 3\) kinematics (as the LO) rather than at NLO. At NLO, the extra gluon can populate \(M_{{\mathrm {b}} j}\) regions less probable at LO. This in turns explains the huge corrections in the lower frame of Fig. 8 (top left): The magnitude of the NLO correction, shown in the middle frame, gets very large due to the strong suppression of the LO in the normalisation. The EW correction is much smaller than the QCD correction, but is similar in shape. The curve for \(R=0.6\) does not suffer from the same issue as the QCD counterpart. The \(R=1\) case is nonetheless shown for comparison.

The distributions obtained choosing \(R=0.6\) or \(R=1.0\) are compared also for the other observables. The main difference is that the relative corrections induced by the choice of the larger radius are milder than those obtained when using \(R=0.6\), as expected from the inclusiveness argument. This effect can be clearly seen in the bottom-left and bottom-right plots in Fig. 8, where the NLO corrections to the energy of the leading jet and of the \(\mathrm {b}\)-jet are presented, respectively. The energy of the leading jet receives positive QCD corrections due to gluon radiation, which turn negative above the energy peak at \(\sim \)65 \(\mathrm {GeV}\). As a consequence, the shape of the energy distribution at NLO is shifted to the left, more evidently for \(R=0.6\). In the low-energy tails the relative corrections are enhanced owing the normalisation to the small LO prediction in this region. The QCD and EW corrections to the distribution in the energy \(E_{\mathrm {b}} \) of the \(\mathrm {b}\)-jet, displayed in Fig. 8 (bottom right), again show the distinctive radiative tails as already discussed for the semi-leptonic top decay in the previous section.

The different choice of the jet-algorithm parameter affects also the cosine of the angle between the \(\mathrm {b}\)-jet and the leading jet, as displayed in the top-right plot in Fig. 8. Once again, a larger jet radius implies a larger probability to merge all final-state partons other than the \(\mathrm {b}\) quark into a single jet. The latter will then be back-to-back to the \(\mathrm {b}\)-jet since in this calculation the top-quark is considered at rest. This explains the peak appearing in the bin close to \(\cos \theta _{{\mathrm {b}} j}=-1\) when \(R=1\) is employed. As in the previous case, the shape of the EW corrections match closely the QCD ones, however, are smaller in size by an order of magnitude. The strong rise of the QCD corrections toward larger values of \(\cos \theta _{{\mathrm {b}} \mathrm {j}}\) again originates from the fact that gluon radiation populates the jet phase space more uniformly than at LO, where there is a strong tendency that the b quark and the hardest jet are back-to-back.

## 6 Conclusions

In this paper we have described extensions of two techniques for the treatment of infrared singularities in the computation of electroweak corrections at next-to-leading order: In Sect. 3 we have supplemented the construction of universal subtraction terms within the dipole subtraction formalism to cover decay kinematics. The one-cutoff slicing method described in Sect. 4 was derived on the basis of the dipole subtraction formalism and organises the singular contributions in terms of pairs of fermions. These results represent the first ones employing mass regularisation and further allow for an arbitrary number of charged particles, a case that has not been discussed in detail in the literature so far. When dealing with more than three charged particles in the process, the appearance of new “surface” terms has been emphasised. For both methods we have covered all possible cases of massless and massive fermions in the final state and further worked out the respective extensions for the treatment of non-collinear-safe observables. The extension of the described techniques to the calculation of NLO QCD corrections is straightforward.

Both methods have been applied to compute the NLO QCD and electroweak corrections to the top-quark decay width. In this computation, we have considered the \({\mathrm {b}} \)-quark as massive and evaluated the correction to the top-quark three-body decays fully accounting for the off-shellness of the intermediate \({\mathrm {W}} \) boson within the complex-mass scheme. Both the hadronic and the semi-leptonic decay channels are considered, further accounting for the case of non-collinear-safe observables in the latter. For the semi-leptonic decay, the NLO QCD corrections to the partial decay width amount to \({\sim }-\)9.4 and \({\sim }+\)1.3 % for the electroweak corrections. In the hadronic decay channel, the NLO QCD correction are more moderate with \({\sim }-\)5.6 %, however, still much larger than the EW correction which amount to \({\sim }+\)1.3 %. While off-shell effects of the W boson modify the integrated decay widths by about \(1.5~\%\), the relative NLO QCD and EW corrections are modified only by few per mille by W off-shell effects. We have further presented differential distributions in the top-quark rest frame for different kinematic observables.

The codes developed in the context of this work can serve as a first step towards an unweighted Monte Carlo generator for the decay of the top-quark at NLO, which can easily be linked to any top-quark production process in order to simulate the decay at NLO accuracy. This is particularly important for processes with many particles in the final state when all intermediate particles are decayed. For instance, \({\mathrm {t}} \overline{{\mathrm {t}}}+V(V)\) (\(V={\mathrm {\gamma }}, {\mathrm {Z}}, {\mathrm {W^\pm }} \)), \({\mathrm {t}} \overline{{\mathrm {t}}}{\mathrm {H}} \), \({\mathrm {t}} \overline{{\mathrm {t}}}{\mathrm {b}} \overline{{\mathrm {b}}}\), and \({\mathrm {t}} \overline{{\mathrm {t}}}{\mathrm {t}} \overline{{\mathrm {t}}}\), among others. Moreover, the top-quark decay width including both NLO QCD and EW corrections and W-boson off-shell effects comprises an ingredient in NLO predictions for the production of single top-quarks or \({\mathrm {t}} \bar{\mathrm {t}} \) pairs including top-quark decays and off-shell effects, such as the process \({\mathrm {p}} {\mathrm {p}} \rightarrow {\mathrm {b}} {\bar{\mathrm{b}}} {\mathrm {W^+}} {\mathrm {W^-}} \rightarrow 6{f} \), for which NLO QCD corrections are known, but not yet the EW corrections.

## Footnotes

- 1.
In case of processes with identified hadrons in the initial and/or final state, collinear singularities arise for which the inclusiveness assumptions of the KLN theorem are not fulfilled. These universal singularities can be absorbed into a redefinition of the corresponding (NLO) structure functions given by the parton distribution functions and the fragmentation functions, respectively.

- 2.
In this paper, we consider only photon bremsstrahlung in detail, which up to colour factors literally translates into gluon radiation off massless or massive quarks in QCD (with splittings \(q\rightarrow q^*{\mathrm {g}} \), \(q^*\rightarrow q{\mathrm {g}} \) and likewise for \(\bar{q}\)). Other QCD splittings, such as \({\mathrm {g}} \rightarrow {\mathrm {g}} ^*{\mathrm {g}} \), \({\mathrm {g}} \rightarrow q^*q\), etc., either are covered by the known results for massless quarks or do not need any special treatment for massive quarks, because no soft or collinear singularity exists in those cases.

- 3.
For \(P_{ia}^2>0\) and \(0 < \sqrt{P_{ia}^2} < m_a-m_i\), the lower limit has to obey \(x_0 > \hat{x} = \frac{-\bar{P}_{ia}^2}{2m_a(m_a-\sqrt{P_{ia}^2})}\), see Ref. [16].

- 4.
A readjustment of the prefactor of the result given in Ref. [24] according to \(G_\mu \rightarrow G_\mu \cdot \mathfrak {R}(s_\mathrm {w} ^2) / s_\mathrm {w} ^2\) is required in order to be consistent with the complex-mass scheme employed in our calculation.

- 5.
The total width is calculated using lepton universality, i.e. the \(\tau \) mass \(m_\tau \) is neglected, since those effects are of \(\mathcal{O}(m_\tau ^2/M_{\mathrm {W}} ^2)\).

- 6.
Note that no direction is distinguished in the top-quark rest frame and we have fixed an arbitrary reference axis to compute the transverse component.

## Notes

### Acknowledgments

L.B. thanks Patrick Motylinski for some technical help in the phase-space integration. L.B. received support from the Theorie-LHC France initiative of the CNRS/IN2P3 and by the French ANR 12 JS05 002 01 BATS@LHC. S.D., L.B. and L.O. were supported by the Deutsche Forschungsgemeinschaft through the Research Training Group grant GRK 1102 *Physics at Hadron Accelerators*. S.D., A.H. and L.O. received support from the German Research Foundation (DFG) via Grant DI 784/2-1. Moreover, A.H. is supported via the ERC Advanced Grant MC@NNLO (340983).

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