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

We compute the differential and total rate of the semileptonic decay of polarized top-quarks $t\to \ell \nu_\ell + b{\rm jet} + {\rm jet}$ 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.


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 includes the computation of the hadronic tt production cross section at next-to-next-to-leading order (NNLO) in the QCD coupling α s [5,6] and the calculation of the differential decay rate of t → b ν 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 Γ t , the order α s QCD corrections [9,10], the order α electroweak corrections [11,12], and the order α 2 s QCD corrections [13,14] were calculated quite some time ago. The fractions of top-quark decay into W + with helicity λ W = 0, ±1 are also known to NNLO QCD [15], including the order α 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 ν + b jet + 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 tt+jet production [31][32][33], for single top-quark + jet production at the LHC, or for tt+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 on tt + 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 Sec. 2 we describe our computational set-up. In Sec. 3 we present our results for the decay rate and for a number of distributions for (un)polarized top-quark decays. Sec. 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 coloured 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.

Set-up of the computation
We consider the decay of polarized top quarks to leptons, a b-jet and an additional jet, 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 α 2 s , the differential decay rate of (1) is determined by the amplitudes of the following parton processes: and In the case of additional bb production in the real radiation process (3), we take into account only configurations where a bb pair is unresolved by the jet algorithm, i.e., we consider in (1) final states where the additional jet has zero b-flavour. To order α 2 s , the matrix element of (2) is the sum of the Born term |M B | 2 and the interference δM V of the Born and the 1-loop amplitude. The calculation of δM V , using dimensional regularisation, is standard. We expressed δM 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 δM 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 = (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 α s is defined in the MS scheme. The soft and collinear divergences that appear in the phase space integrals of the tree-level matrix elements of (3) and in δM 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 where E i , E j are the energies of partons i, j in the final state, θ 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 .

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 2 t = 1), differential distributions for the decay (1) of a 100 percent polarized ensemble of top quarks are of the form where O denotes some observable. In the fully differential case, the functions A and B (that 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 Sec. 3.1. In Sec. 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 the b-jet, and the W boson. For the numerial results given below, we use m t = 173.5 GeV, m W = 80.39 GeV and Γ W = 2.08 GeV. The QCD coupling for 5-flavour QCD is taken to be α s (m Z ) = 0.118. Its evolution to µ = m t and conversion to the 6-flavour MS coupling results in α s (m t ) = 0.108. Moreover, we use α(m t ) = 7.9 × 10 −3 and sin 2 θ W = 0.231 which yields the weak coupling g 2 W = 0.429. The normalized distributions given below do not depend on g 2 W because we work to lowest order in g 2 W .

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 Γ t→blν l + jet is shown at LO and NLO QCD, normalized to the leading order rate Γ t→blν l = 1.8698 · 10 −3 m t , for a renormalization scale µ = m t . It is clear that this ratio increases for decreasing Y . In the lower pane of this figure, the 'K factor' Γ NLO t→blν l + jet /Γ LO t→blν l + jet 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 ∼ 2.5 · 10 −3 they become negative. 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 Γ NLO t→blν l + jet , are expanded in powers of α s . Taking out a factor of α s both from the LO and NLO (differential) rate, we have 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 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 Fig. 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 energy 1 E W = E l + E ν may be compared with the case of the lowest-order on-shell decay t → bW where the (dimensionless) W energy is fixed,Ē W = m 2 W + k 2 W /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 belowĒ W , but approaches this value if the jet cut Y is decreased. The distributions on the right sides of Figs Radiation off the t and b leads to an upper bound on E b that is belowĒ 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 ∼ 10%. Figs. 8 and 9 show the distribution of cos θ bl , where θ 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 θ 2l , where θ 2l is the angle between + and the second jet. The distributions of cos θ 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 θ 2l is falling less steeply towards smaller angles θ 2l . The QCD corrections are markedly below 5% in most of the kinematic range. The distribution of cos θ * W l , where θ * W l 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 θ * W l the one-dimensional distribution has the well-known form with F L + F − + F + = 1. For events with an additional jet, one expects that for small jet cut Y the corresponding distribution tends towards the inclusive one. Performing a fit to the cos θ * W l 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) [15]) and are in agreement with recent results from ATLAS and CMS [43]. The plots on the right sides of Figs. 10 and 11 show the distribution of cos θ W b , where θ W b 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.

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 = + , b jet, W + . The corresponding normalized distribution has the a priori form 1 Γ where θ f = ∠(s t ,k f ). The coefficient κ f is the top-spin analyzing power of f and measures the degree of correlation. CP invariance implies 3 that the corresponding angular distributions for top antiquarks are given by The values of κ f can be extracted from the slope of the distributions (6) or from The results for κ 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 (6) for µ = m t . The uncertainties due to scale variations between m t /2 and 2m t are below 1%.
Y=0.01 Y=0.001 One may compare these t-spin analyzing powers with the corresponding ones of the dominant semileptonic decay modes t → b + ν . In the latter case one has κ NLO = 0.999 [18] and κ NLO b = −0.39 [19]. Moreover, in this inclusive case, κ NLO b = −κ NLO W . 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, κ b = −κ 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 + . This is due to the known fact that for t → + ν b (+jet), the amplitudes that correspond to the different polarization states of the intermediate W boson interfere constructively (destructively) when + is emitted in (opposite to) the direction of the top spin.

Summary
We have computed the differential and total rate of the semileptonic decay of polarized topquarks t → ν + b jet + 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 5% in most of the kinematic range. Near kinematic edges or significantly off the W resonance, the corrections can become ∼ 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.
Here, dφ 4 , dφ 5 and dφ (dip) are the 4-particle, 5-particle, and dipole phase-space measures, respectively, δM CT denotes, schematically, the dipole subtraction counterterms for the two real radiation processes (3), and denote jet functions. The quantity Y (ij),l is calculated from the momentum of the pseudo-jet that consists of partons i and j, cf. Sec. 2, whereas the quantityỸ (ij),l is calculated from the emitter and spectator momentak ij andk l , which are defined in terms of the 5-particle phase space φ 5 . The following formulae are given for conventional dimensional regularisation.

Un-integrated dipoles
The set of counterterms δM CT for the real radiation processes (3) can be constructed, using the emitter-spectator terminology of [34], with so-called final-final and final-initial dipoles. We denote the 4-momenta of the top-quark and of the b quark from the tW b vertex with k t and k b , and those of the two gluons or the q,q in (3) by k 1 , k 2 . The final-final dipoles required for (3) can be obtained from [34]: The indices λ i and ρ i transform according to the spinor and vector representations of the Lorentz group, i.e., they refer to the spin of the initial-state quark and gluon, respectively, in (10).
The final-initial dipoles for t → W bg 1 g 2 were constructed in [33,36] (using, in part, results of [37]). They contain the eikonal terms ∝ m 2 t k i · k j /(k t · k i ) 2 for canceling the soft singularities that arise from gluon radiation off the initial top-quark. The final-initial dipole for t → W bqq can be constructed analogously. Here In this case the vector Π ρ FI takes a more complicated form. For the sake of brevity we refer to eq. (20) of [33].

Integrated dipoles
For the analytical integration of the final-final dipoles over the respective subspaces we use a phase-space splitting of the form dφ 5 (k i , k j , k l , · · · ) = dφ 4 (k ij ,k l , · · · ) × dφ (dip.) ij,l (Y ij,l , Z ij,l ), such that one can, in the end, identify dφ 4 (k ij ,k l , · · · ) with the four-particle phase space of the Born matrix elements or of the virtual corrections. The momenta of the emitterk ij and the spectatork l are constructed according to [34] from the soft/collinear pair k i , k j and another parton momentum k l , whereas all remaining momenta are unaffected. In our case the emitter and spectator is either a b-quark or a gluon, i.e.k ij =k b/g andk l =k g/b . The phase space of the final-final dipoles can then be parameterized as The double index ij labels the soft/collinear pair and the index l refers to the momentum of the remaining final-state parton.
Integration of (10) over the dipole phase space yields Again dφ 4 (k ij ,R) can be identified with the phase space of the Born matrix elements or of the virtual corrections. Here, the phase-space mapping 5 → 4 affects, besides the soft/collinear pair, also all other final state momenta, denoted by R. This procedure, as well as the dipole phase-space parameterization, is adapted from [37], [33]. The boundary of the Y t ij integration is Integration of (11) over the dipole phase space yields hypergeometric functions 2 F 1 , which we expanded in powers of using the package HPL 2.0 [44]. We obtain where HereT b = (k t −k b ) 2 /m 2 t , andT g = (k t −k g ) 2 /m 2 t . Eq. (14) agrees with the result of [36], and eqs. (15) and (16) with those of [33].