Top-quark pair-production and decay at high precision

TL;DR

This paper provides a high-precision, fully-differential calculation of top-quark pair production and decay at the LHC, incorporating approximate and exact NNLO corrections, leading to improved agreement with experimental data.

Contribution

It introduces the first implementation of exact NNLO corrections to top-quark decay processes in a differential Monte Carlo framework.

Findings

NNLO corrections significantly affect cross section predictions

Improved agreement with ATLAS and CMS measurements

Enhanced accuracy in modeling top-quark events

Abstract

We present a fully-differential and high-precision calculation of top-quark pair-production and decay at the LHC, providing predictions for observables constructed from top-quark leptonic and $b$-flavoured jet final states. The calculation is implemented in a parton-level Monte Carlo and includes an approximation to the next-to-next-to leading order (NNLO) corrections to the production and, for the first time, the exact NNLO corrections to the decay subprocesses. The corrections beyond NLO are sizeable, and including them is crucial for an accurate description of the cross section constrained by experimental phase-space restrictions. We compare our predictions to published ATLAS and CMS measurements at the LHC, finding improved agreement compared with lower orders in the perturbative expansion.

Tables

Table 1. Table 1: Fiducial cross sections for a variety of LHC center-of-mass energies and setups. Theoretical predictions with uncertainties are tabulated at LO, NLO and N̂NLO as are the experimental measurements. The uncertainties on the measured cross sections have been obtained by summing the individual statistical, systematic, beam and luminosity uncertainties in quadrature. δ dec . subscript 𝛿 dec \delta_{\rm dec.} indicates the impact on the cross section of higher-order corrections to the top decay, see eq. ( 3 ). The Monte Carlo uncertainty on all theoretical predictions is better than 1‰.

ATLAS setup, $e^{\pm }{\mu }^{\mp }$ channel Aad et al. (2014)
energy	fiducial volume	LO [pb]	NLO [pb]	N̂NLO [pb]	${\delta }_{\mathrm{dec}.}$	ATLAS [pb]
7 TeV	$p_T\left(l^{\pm }\right)>25$ GeV,	${1.592}_{-26.0\%}^{+39.2\%}$	${2.007}_{-13.2\%}^{+11.9\%}$	${2.210}_{-6.0\%}^{+2.2\%}$	-0.3%	${2.305}_{-3.8\%}^{+3.8\%}$
7 TeV	$p_T\left(l^{\pm }\right)>30$ GeV,	${1.265}_{-26.1\%}^{+39.3\%}$	${1.585}_{-13.1\%}^{+11.8\%}$	${1.736}_{-6.0\%}^{+2.2\%}$	-0.8%	${1.817}_{-3.8\%}^{+3.8\%}$
8 TeV	$p_T\left(l^{\pm }\right)>25$ GeV,	${2.249}_{-25.5\%}^{+37.9\%}$	${2.855}_{-12.9\%}^{+11.9\%}$	${3.130}_{-6.0\%}^{+2.3\%}$	-0.3%	${3.036}_{-4.1\%}^{+4.1\%}$
8 TeV	$p_T\left(l^{\pm }\right)>30$ GeV,	${1.788}_{-25.5\%}^{+38.0\%}$	${2.256}_{-12.9\%}^{+11.7\%}$	${2.461}_{-6.1\%}^{+2.3\%}$	-0.7%	${2.380}_{-4.1\%}^{+4.1\%}$
CMS setup, $e^{\pm }{\mu }^{\mp },e^{+}{e}^{-},{\mu }^{+}{\mu }^{-}$ channel Khachatryan et al. (2016), 2 $b$-jets required (anti-$k_t$ algorithm Cacciari et al. (2008), $R=0.5$)
energy	fiducial volume	LO [pb]	NLO [pb]	N̂NLO [pb]	${\delta }_{\mathrm{dec}.}$	CMS [pb]
8 TeV	$p_T\left(l^{\pm }\right)>20$ GeV, ,	${3.780}_{-25.3\%}^{+37.4\%}$	${4.483}_{-11.5\%}^{+9.0\%}$	${4.874}_{-6.8\%}^{+2.5\%}$	-8.0%	${4.73}_{-4.7\%}^{+4.7\%}$
	$p_T\left(J_b\right)>30$ GeV,

Equations

$$d\sigma =d{\sigma }_{t\bar{t}}\times \frac{d{\Gamma }_{t\to b{l}^{+}{\nu }_l}}{{\Gamma }_t}\times \frac{d{\Gamma }_{\bar{t}\to \bar{b}{l}^{{}^{\prime }-}{\bar{\nu }}_{l^{\prime }}}}{{\Gamma }_t},$$

$$d{\sigma }_{t\bar{t}}$$

$$d{\Gamma }_{t(\bar{t})}$$

$$m_t$$

$$m_W$$

$$m_Z$$

$${\delta }_{dec.}=\text{NˆNLO}/{\text{NˆNLO}}_{\mathrm{LO}dec.}-1[\%],$$

Full text

††preprint: CAVENDISH-HEP-17-09

I Introduction

The presence of top quarks produced in collider experiments is always inferred via the top-quark decay products. The majority of data/theory comparisons are however, performed at the level of stable tops. Indeed, it is through such comparisons that the top quark sector of the Standard Model is being carefully scrutinized. High statistics have been collected at all three main center-of-mass energy runs at the LHC and many measurements of ‘top-quark’ inclusive cross sections and distributions are already down to the impressive level of a few per cent in uncertainty. On the theoretical side, there has been equally impressive progress with top-pair production Bäärnreuther et al. (2012); Czakon et al. (2013, 2016a, 2016b, 2016c); Abelof et al. (2015); Czakon et al. (2017a, b) and single top Brucherseifer et al. (2014); Berger et al. (2016) computed fully-differentially at next-to-next-to leading order (NNLO) in perturbative QCD (pQCD) for stable tops. The fixed-order predictions can often be supplemented with various resummations which stabilize the predictions against the effects of large logarithms, see for example refs. Kidonakis (2001); Kidonakis et al. (2001); Kidonakis and Vogt (2003); Banfi and Laenen (2005); Ahrens et al. (2010, 2011); Beneke et al. (2012); Cacciari et al. (2012); Ferroglia et al. (2012, 2014); Guzzi et al. (2015); Pecjak et al. (2016).

Despite this progress, to compare with stable-top predictions, experiments must, in general, do two things. Firstly, they must extrapolate their measurements from the detector fiducial volumes out to the full phase space, and secondly, they must translate their measurements of final states that they are sensitive to, back to some definition of top-quark ‘partons.’ Such extrapolation and unfolding corrections are typically derived from event generators that treat the top decay at leading order (LO), which can lead to inconsistencies when comparing with high-precision predictions. Moreover, the resulting systematic uncertainties can be difficult to estimate – the observed tensions Czakon et al. (2016d) between the ATLAS and CMS measurements of stable-top distributions may just hint at some unknown systematic errors in the above procedure.

To overcome these modelling uncertainties it is evident that experimental measurements of observables constructed directly from top-quark decay products in detector fiducial volumes are the quantities that should be compared with theoretical predictions. In fact, experiments have begun publishing such measurements Aad et al. (2014); Khachatryan et al. (2016) and importantly these often come with smaller systematic errors than measurements of the inclusive cross section Aad et al. (2014). To fully exploit these measurements it is crucial that theoretical predictions describe the decay products fully differentially, and that they are as accurate as possible, i.e. they include higher order perturbative corrections in both production and decay subprocesses.

Significant efforts have been made in this direction at next-to-leading order (NLO), both treating the top-quark propagator in the narrow-width approximation (NWA) Bernreuther et al. (2004); Melnikov and Schulze (2009); Campbell and Ellis (2015); Melnikov et al. (2011, 2012) as well as keeping the top quarks off their mass-shell Bevilacqua et al. (2011); Denner et al. (2011, 2012); Falgari et al. (2013); Heinrich et al. (2014); Frederix (2014); Cascioli et al. (2014); Falgari et al. (2010, 2011); Papanastasiou et al. (2013); Denner and Feger (2015); Bevilacqua et al. (2016a, b); Denner and Pellen (2016). Furthermore, frameworks have been recently developed to consistently match both sets of the above NLO calculations to parton showers Campbell et al. (2015); Ježo and Nason (2015); Frederix et al. (2016); Ježo et al. (2016). It is clear that these predictions come much closer to the quantities that are actually measured by experiments.

In this letter we focus on the dominant top-quark production mode at the LHC, top-pair ($t\bar{t}$) production, and describe a calculation that goes beyond NLO in pQCD in both the production and decay stages in the NWA. The calculation includes an approximation of the NNLO corrections to the $t\bar{t}$ production subprocess and the exact NNLO corrections to the top and antitop decays. We present first results of our new calculation, which has been implemented in a parton-level Monte Carlo, making predictions for the $t\bar{t}$ process at the LHC in the di-lepton channel, fully differential in the final-state leptons, $b$-flavoured jets ($b$-jets) and missing energy. This represents a significant improvement to the current state-of-the-art at fixed order in perturbation theory and, as will be shown below, compares very favourably to published fiducial region measurements by the ATLAS and CMS experiments.

II Details of the Calculation

The full technical details of our calculation will be presented in a forthcoming publication, however we now briefly summarize the important building blocks. In the NWA the differential cross section for $t\bar{t}$ production and decay in a particular decay channel (e.g. the di-lepton channel) can be written schematically to all orders as

[TABLE]

where $d\sigma_{t\bar{t}}$, $d\Gamma_{t\to bl^{+}\nu_{l}}$ and $d\Gamma_{\bar{t}\to\bar{b}l^{{}^{\prime}-}\bar{\nu}_{l^{\prime}}}$ are the differential production cross section for a $t\bar{t}$ pair and the differential top- and antitop-quark decay widths (we write the latter as $d\Gamma_{t}$ and $d\Gamma_{\bar{t}}$ for brevity). $\Gamma_{t}$ is the total top-quark width. The ‘$\times$’ in eq. (1) indicates that production and decay are combined in a way that preserves spin-correlations. Each term in eq. (1) has a perturbative expansion in the strong coupling constant $\alpha_{s}$,

[TABLE]

In the expansion in $\alpha_{s}$ of eq. (1), we adhere to the convention of a strict expansion, namely, we do not include terms proportional to powers of $\alpha_{s}$ that are higher than the order that we work to Melnikov and Schulze (2009); Campbell and Ellis (2015). This means that at NLO (NNLO) perturbative contributions proportional to $\alpha_{s}^{4}$ ($\alpha_{s}^{5}$) or higher are not included. In this convention when integrating inclusively over the decay products of the top quarks the cross section for the production of a stable top pair is recovered (multiplied by the appropriate branching fractions for the $W$-boson decays). This feature constitutes a highly non-trivial check of the implementation of each contribution in the expansion.

In the present calculation we include the exact NNLO corrections (i.e. corrections up to $\alpha_{s}^{2}$) in the expansions of $d\Gamma_{t\,(\bar{t})}$ and $\Gamma_{t}$ and approximate-NNLO corrections to $d\sigma_{t\bar{t}}$ (the NLO corrections $d\sigma^{(1)}_{t\bar{t}}$ are included exactly). We denote our best predictions as N̂NLO (and not NNLO) since we make an approximation to $d\sigma^{(2)}_{t\bar{t}}$ (the exact NNLO corrections to the production where the spin information of the top quarks is kept explicit, required to construct eq. (1), are not currently known).

The approximation to the exact NNLO corrections in production builds on the work presented in ref. Broggio et al. (2014). The starting point for this approximation is a factorization formula derived in the Soft-Collinear-Effective-Theory (SCET) framework Bauer et al. (2002, 2001); Beneke et al. (2002). It was shown in ref. Ahrens et al. (2010) that in the soft-gluon limit $z=M_{t\bar{t}}/\hat{s}=(p_{t}+p_{\bar{t}})^{2}/\hat{s}\to 1$ (‘Pair Invariant Mass’ kinematics) the cross section for $t\bar{t}$ production can be written as a convolution of a hard function, containing the effects of virtual corrections, and a soft function, which contains the effects of emissions of soft gluons, together with standard parton distribution functions (PDFs). The factorised structure makes it possible to resum large logarithms of $(1-z)$ and the subsequent expansion to fixed order of the resummed cross section provides an approximation to the exact NNLO. In ref. Broggio et al. (2014) this approach was generalised beyond the stable-top approximation and the spin-correlated LO decay of the top-quarks was attached to the approximate-NNLO production kernels. For our central prediction we take the kernels of refs. Broggio et al. (2014) and additionally include terms subleading in $(1-z)$, that arise from the soft-expansion of the Altarelli-Parisi splitting functions Kramer et al. (1998); Catani et al. (2001), which are known to bring improvements to the NNLO approximation Kramer et al. (1998); Catani et al. (2001); Broggio et al. (2016); Muselli et al. (2015). Furthermore, we explicitly use the freedom to include different subleading effects to help construct a reliable estimate of the theoretical uncertainty. Specifically, to make this estimate we take the envelope of scale variation together with variations (switching on and off) of subleading corrections of different origin – from the splitting functions Kramer et al. (1998); Catani et al. (2001) and from phase space Broggio et al. (2016).

The quality of the NNLO approximation in the production can ultimately be assessed by comparing to the exact NNLO cross sections for stable tops Czakon et al. (2013, 2016a). We find excellent agreement with the NNLO inclusive cross section, for example, for LHC 8 TeV, $m_{t}=173.3$ GeV, $\mu_{F}=\mu_{R}\in[0.5,1.0,2.0]m_{t}$ and using MSTW2008 pdfs Martin et al. (2009) we have $\sigma({\rm NNLO}_{\rm approx.})=239.4^{+\phantom{1}5.7}_{-14.0}$ pb, whilst the exact NNLO cross section computed with top++ Czakon and Mitov (2014) is $\sigma({\rm NNLO})=239.2^{+\phantom{1}9.2}_{-14.8}$ pb (with equally good agreement for LHC 7 and 13 TeV). Furthermore, as displayed in figure 1 for the average transverse momentum of the top and antitop, we also find very good agreement with the exact results at the differential level (similar agreement is also found for the invariant mass of the top-quark pair and the average rapidity of the top and antitop). These validation checks for stable top quarks provide us with confidence when using the approximate-NNLO kernels for the case when the top quarks are decayed.

The NNLO corrections to the top quark decay are also calculated retaining full spin-correlations between production and decay. Using the SCET-inspired phase-space slicing method presented in ref. Gao et al. (2013), a small cutoff on the invariant mass ($m_{j}$) of all QCD partons from the top quark decay is introduced to split the phase space in the computation of $d\Gamma^{(2)}_{t\,(\bar{t})}$ into resolved and unresolved regions. The resolved region receives contributions from the NLO corrections to the process of top decay plus an additional jet, and can be dealt with straightforwardly. The contribution in the unresolved region can be factorized and calculated using SCET, up to power corrections in $m_{j}^{2}/m_{t}^{2}$ Gao et al. (2013). The sum of resolved and unresolved contributions then converges to the exact NNLO correction when the cutoff is sufficiently small. In practice we find that a cutoff of $10^{-5}$ on $m_{j}^{2}/m_{t}^{2}$ is sufficient to ensure the remaining power corrections are negligible for all kinematic distributions considered. We note that the NNLO decay was also computed in ref. Brucherseifer et al. (2013).

Finally, the required NLO$\times$NLO production-decay ($\sim d\sigma_{t\bar{t}}^{(1)}\times d\Gamma^{(1)}_{t}\times d\Gamma^{(0)}_{\bar{t}}$) and decay-decay corrections ($\sim d\sigma_{t\bar{t}}^{(0)}\times d\Gamma^{(1)}_{t}\times d\Gamma^{(1)}_{\bar{t}}$) have also been computed. Since production and decay subprocesses can be treated separately in the NWA, as far as their singularity structure is concerned, standard NLO techniques Catani and Seymour (1997); Catani et al. (2002); Frixione et al. (1996) can be adapted to deal with IR-singularities in these contributions.

III Phenomenology

We now apply the calculation outlined above to LHC phenomenology and in particular, to the $t\bar{t}$ process in the di-lepton channel. We first focus our attention on the fiducial cross sections measured by the ATLAS experiment for the $e^{\pm}\mu^{\mp}$ channel at 7 and 8 TeV Aad et al. (2014), and the CMS experiment in the full di-lepton channel ($e^{\pm}\mu^{\mp}$, $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$) at 8 TeV Khachatryan et al. (2016). For simplicity we compare to measurements where the indirect decays $W\to\tau\to e(\mu)$ are considered as backgrounds. The corresponding definitions of the fiducial volumes for each experiment, constructed through cuts on final-state leptons (and $b$-jets, $J_{b}$), can be found in table 1.

The inputs for our theoretical predictions are set to

[TABLE]

where we note that $\Gamma_{t}^{(0)}$ is a function of $m_{t}$, $M_{W}$ and $G_{F}$. We use fixed factorization and renormalization scales 111In future work we intend to explore the effects of dynamical scales, which are known to improve the perturbative stability of the tails of distributions. $\mu=\mu_{F}=\mu_{R}\in[0.5,1.0,2.0]m_{t}$ and vary the scale in the NLO and NNLO corrections to the top width $\Gamma^{(1,2)}_{t}(\mu)$ for consistency. The theoretical uncertainty bands are obtained by taking the envelope of the predictions for each scale. For the approximate-NNLO corrections in production, as stated earlier, we additionally take the envelope of predictions computed with different subleading terms in $(1-z)$. We use LO, NLO, NNLO MMHT2014 PDFs Harland-Lang et al. (2015) with the corresponding value for $\alpha_{s}(M_{Z})$ for our LO, NLO and N̂NLO predictions. In the results presented here we treat the $W$-bosons in the NWA.

Our theoretical predictions of the cross sections for the corresponding fiducial setups of ATLAS and CMS are tabulated in table 1. The ATLAS and CMS measurements are also shown in the same table. The errorbars on the experimental data have been obtained by summing the published individual uncertainties (statistical, systematic, luminosity and beam) in quadrature. One observes that for each setup shown, there is a reduction in the uncertainty bands with increasing perturbative order, with the N̂NLO bands being roughly half the size of the NLO bands. Additionally the corrections to the cross section going from LO to NLO and from NLO to N̂NLO are also reduced, indicating an improved perturbative convergence. The corrections beyond NLO are significant, around 9-10%, underlining that such corrections are crucial for an accurate description of fiducial regions 222We point out that large NNLO corrections in fiducial regions were also found in single top-quark production Berger et al. (2016).. This statement is strengthened when a comparison to the experimental measurements is made: for each setup considered, table 1 reveals an improvement in the agreement between measurement and data. More precisely, use of the N̂NLO prediction brings the difference between the central values of theory and measurement to within 3-4%.

An important aspect to quantify is the size of the contributions involving corrections to the top decay. This can be done by studying the % difference between the N̂NLO prediction, which includes higher-order corrections to the decay, and the prediction N̂NLO${}_{\rm LO\,dec.}$ that just includes the LO decay ($d\Gamma^{(0)}_{t\,(\bar{t})}$),

[TABLE]

also found in table 1. For the ATLAS setup, whose definition of the fiducial volume involves no cuts on the $b$-jets, we find $\delta_{\rm dec.}<1\%$, i.e. the contributions from higher orders in the decay are very small. However, this is not the case for the CMS setup, where the constraint $p_{T}(J_{b})>30$ GeV is in place, and where the prediction that treats the top decay only at LO is 8% larger than the prediction that consistently includes corrections in the decay. Coupled with the comparison to the precise experimental measurements, these findings point to two important conclusions. Firstly, NNLO corrections in general are vital to describe fiducial-region cross sections accurately. Secondly, corrections to the production subprocess alone do not uniformly give a good description of the measurements – higher-order corrections to the decay must be included to see an improved agreement for all setups considered.

Finally, we make a comparison to differential CMS measurements Khachatryan et al. (2015) in the di-lepton channel. In figure 2 we present absolute distributions for the average lepton pseudo-rapidity, $\eta(l)_{\rm ave.}$ and the transverse momentum of the lepton-pair, $p_{T}(l^{+},l^{-})$, and $b$-jet pair, $p_{T}(J_{b},J_{\bar{b}})$, normalised to the N̂NLO prediction. We have chosen to rescale the published normalised data by the fiducial cross section found in ref. Khachatryan et al. (2016) in order to make the differences between theoretical predictions at different orders more visible. Since there are no published uncertainties for the absolute distributions, we show the experimental points with two errorbars – the smaller errorbars are those of the normalized cross section whilst the larger ones are those of the normalized cross section added in quadrature with the uncertainty of the fiducial cross section used for rescaling. Overall, there is again good agreement between the measurements and the N̂NLO predictions – the latter agreeing with the former within uncertainties in all bins. The N̂NLO brings an improvement in the agreement not only in the overall normalization, but also in the shape of each distribution for the bulk of the region of phase-space measured. In the last bin of the $p_{T}(l^{+},l^{-})$ and $p_{T}(J_{b},J_{\bar{b}})$ distributions the agreement becomes less good, however, in these regions both theoretical and experimental uncertainty bands become large.

IV Conclusions and Outlook

In this letter we have presented high-precision results for the fully-differential production and decay of a top-quark pair in fiducial regions at the LHC. Our results are based on the NWA and are accurate at approximate-NNLO in the production subprocess and exact-NNLO in the decay subprocess. The approximation we use in the production does an excellent job at approximating the exact NNLO for stable tops, giving us confidence in the results we present for decayed top quarks.

We have shown that, in general, the NNLO corrections are significant. Moreover, it is vital to include corrections to the decay as well as to the production subprocess for an accurate description of observables constructed from top-quark decay products. The importance of going beyond NLO is clearly seen when comparing theoretical predictions to available ATLAS and CMS fiducial cross section measurements. For different center-of-mass energies and setups we consistently find that the agreement between theory and measurement improves when the N̂NLO predictions are used. Additionally, we see an overall improvement in the agreement, in normalization as well as in shape (for the bulk of the ranges considered) when comparing to distributions constructed from lepton and $b$-jet final states published by CMS.

We envision that the calculation presented in this letter will open up a number of exciting possibilities for the study of top quarks. Given the impressively small experimental uncertainties on the measurements of fiducial cross sections, it would be particularly interesting and timely to use these measurements and exploit this new calculation to perform an extraction of $\alpha_{s}$ and $m_{t}^{\rm pole}$. This would bypass the need to extrapolate measurements to the full phase space and the modelling back to top-quark partons, that affect extractions from the inclusive stable-top cross section. With this calculation at hand it will also be possible to quantify the impact that the exact NNLO top-quark decay corrections have in methods of $m_{t}$-extraction sensitive to the decay (see for example refs. Frixione and Mitov (2014); Agashe et al. (2016)).

Data/theory comparisons of top-quark production, at the level of stable tops, have brought numerous impactful applications, for example, constraining the high-$x$ region of the gluon PDF Czakon et al. (2016d). We advocate that moving towards such applications, but working with observables at the level of the decay products of top quarks, as we have done here, will maximize the impact that current and future top-quark measurements will have both within and beyond the area of top-quark physics.

Acknowledgements We thank A. Broggio and A. Signer for early collaboration and many insightful discussions regarding the NNLOapprox. predictions. We also thank E. Laenen for helpful discussions regarding subleading power terms, D. Heymes for clarifications regarding the exact NNLO distributions for $t\bar{t}$ and C. Diez Pardos for making us aware of the fiducial numbers in ref. Khachatryan et al. (2016). Finally, we thank A. Signer for his comments on a draft of this letter. The work of JG is supported by the national 1000-talent plan of China. The work of AP is supported by the UK Science and Technology Facilities Council [grant ST/L002760/1].