Effective operators in tchannel single top production and decay
Abstract
The production of a single top quark in the tchannel and its subsequent decay is studied at NLO accuracy in QCD, augmented with the relevant dimension6 effective operators from the Standard Model Effective Theory. We examine various kinematic and angular distributions for protonproton collisions at the LHC at 13 TeV, in order to assess the sensitivity to these operators, both with and without the top quark narrow width approximation. Our results will be helpful when devising strategies to establish bounds on their coefficients, including the amount of CP violation of the weak dipole operator.
1 Introduction
Let us note here that more operators can contribute starting at \({\mathcal {O}}(1/\Lambda ^4)\) such as the operators involving right handed bottom quarks, e.g. the dipole operator of the bottom quark, whose contributions are suppressed by the bottom mass at \({\mathcal {O}}(1/\Lambda ^2)\). Fourfermion operators involving righthanded light quarks can also be relevant at \({\mathcal {O}}(1/\Lambda ^4)\) [10], but these are eliminated if one assumes Minimal Flavour Violation [11]. In general we assume the contribution from dimension8 operators to be sufficiently suppressed by their associated \(1/\Lambda ^4\) prefactor. We shall however use order \(1/\Lambda ^4\) contributions to the cross section arising from squared contributions of dimension6 ones to assess uncertainties. Finally flavour changing interactions can also contribute to single top production, but we do not consider this here. For a recent global analysis of topquark related flavour changing interactions in the effective operator framework see [12]. Global SMEFT constraints in the top sector are obtained in [13, 14].
This paper assesses the effect of the limited set of dimension6 operators on single top quark production in the tchannel (for brevity we show results for top production, but the same observations can be made in antitop production). We do so moreover at nexttoleading order (NLO) in QCD, including top quark decay to W and b, both in the narrow top width approximation (NWA), and by producing the Wb directly, including nonresonant contributions.
The paper is structured as follows. In the next section we discuss the necessary background to single top production in SMEFT. In Sect. 3 we present our results, highlighting the opportunities in constraining the dimension6 operators with present and future LHC data. In the final section we present our conclusions.
2 Single top production in SM extended to dimension6
To establish our context we recall here some basic aspects of single top production, and the associated charged current interaction. The leading order diagrams for tchannel single top production are shown in Fig. 1.
Any new physics altering the Wtb interaction can then be probed by studying single top production and decay. The SMEFT can parameterise deviations from the SM predictions and can be used to make quantitative predictions to be compared with experimental data.
An interesting feature of this production cross section is that each of the coefficients \(C^{(3)}_{\varphi Q}\), \(C^{(3)}_{qQ,rs}\) and \(\mathrm {Re\,} C_{tW}\) is associated with a specific angular dependence, enabling one to determine, or at least bound, the individual contributions experimentally.
The operator \(O^{(3)}_{\varphi Q}\) only modifies the magnitude of the Wtb interaction as shown in Eq. (2.2), but does not change the angular dependence of the SM prediction. By contrast, the operator \(O^{(3)}_{qQ,rs}\), with corresponding real coefficient \(C^{(3)}_{qQ,rs}\), represents a fourquark contact interaction and noticeably affects the angular distribution of the top quark production angle. Of course, Eq. (2.2) addresses only the dominant, lowest order parton process \(u+b\rightarrow d+t\). Other partonic processes also contribute but the different angular behaviour of the partonic cross section predicted by the different operators directly translates into different shapes of the top transverse momentum distribution. This is illustrated in Fig. 2, where the effect of \(C^{(3)}_{qQ,rs}\) on the top \(p_T\) distribution is clearly distinguishable. Finally, the contribution of \(\mathrm {Re\,}C_{tW}\) has a signature again different from the other two operators, but its effect is smaller and is better determined in the decay of top quarks than in their production [6].
The above discussion is somewhat simplified as it refers to the lowest order contributions in both QCD and the EFT expansion. Nexttoleading order (NLO) QCD corrections can be also relevant and can potentially modify the relative contributions of the operators. At NLO in QCD, the chromomagnetic dipole operator, \(O_{tG}\), contributes as discussed in [6] whilst additional operators contribute at \({\mathcal {O}}(1/\Lambda ^4)\). We omit these operator contributions in this work but in future work we intend to take them into consideration.^{2}
3 Numerical studies
To study the impact of the three operators on single top production we compute the corresponding contributions at LO and NLO matched to the parton shower (PS). The computation is performed within the MadGraph5_aMC@NLO (MG5_aMC) framework [22], and uses the NLO EFT implementation of Ref. [6]. While [6] produces results for stable top quarks, we will also consider the top quark decays. This can by achieved by either decaying the topquark in MadSpin [23] or by following the procedure of resonanceaware PS matching presented in [24], to produce a Wbj final state. By decaying the W boson in MadSpin, we retain spin information. Our setup is fully differential and allows us to assess the impact of NLO corrections as well as the impact of the operators entering either in the production or in the decay, or both, for any observable.
Contributions to the cross section in pb for tchannel single top production at 13 TeV, as parameterised in Eq. (3.3). These values have been extracted from fitting Eq. (3.3), to a hundred computed cross sections with randomly chosen coupling strengths for the effective operators, both for LO and NLO separately. The statistical errors for each contribution in the table is below 1% except for the \(\sigma _{qQij,tW}\) term at NLO, which is at 1.1%. The righthandside column shows the Kfactor, which is defined for each row as the ratio of the NLO over the LO prediction. By the subscripts tW and itW we denote the contributions of the real and imaginary parts of \(C_{tW}\) respectively
Operator  LO  NLO  K  

\(\sigma \) [pb]  \(\frac{\sigma }{\sigma _{\text {SM}}}\) [%]  \(\sigma \) [pb]  \(\frac{\sigma }{\sigma _{\text {SM}}}\) [%]  
\(\sigma _{\text {SM}}\)  123  –  137  –  1.12 
\(\sigma _{qQ,rs^{(3)}}\)  \(\) 92.3  \(\) 75.3  \(\)102  \(\) 74.7  1.11 
\(\sigma _{\varphi Q^{(3)}}\)  14.6  11.9  16.3  11.9  1.12 
\(\sigma _{tW}\)  3.05  2.49  3.57  2.6  1.17 
\(\sigma _{itW}\)  –  –  –    – 
\(\sigma _{qQ,rs^{(3)},\,qQ,rs^{(3)}}\)  77.3  63.1  80.8  58.9  1.05 
\(\sigma _{\varphi Q^ {(3)},\varphi Q^ {(3)}}\)  0.434  0.354  0.485  0.354  1.12 
\(\sigma _{tW,tW}\)  0.758  0.619  1.03  0.752  1.36 
\(\sigma _{itW,itW}\)  0.761  0.616  1.03  0.752  1.35 
\(\sigma _{qQ,rs^{(3)},\,\varphi Q^ {(3)}}\)  \(\) 5.49  \(\)4.48  \(\) 6.08  \(\) 4.43  1.11 
\(\sigma _{qQ,rs^{(3)},\,tW}\)  \(\) 2.34  \(\) 1.91  \(\) 2.84  \(\) 2.07  1.22 
\(\sigma _{\varphi Q^ {(3)},tW}\)  0.182  0.148  0.212  0.155  1.17 
The benchmark choices for the coupling values of the effective operators, together with the corresponding tchannel single top cross section and the width of the top quark. The scale and PDF uncertainties of the cross sections are also shown
Operator  Coupling value  LO  NLO  

\(\sigma \)[pb] ±scale ±PDF  \(\Gamma _{\text {top}}\) [GeV]  \(\sigma \)[pb] ± scale ± PDF  \(\Gamma _{\text {top}}\) [GeV]  
\(\text {SM}\)  –  \(123^{+9.3\text {\%}}_{11.4\text {\%}} \pm 8.9\text {\%}\)  1.49  \(137^{+2.7\text {\%}}_{ 2.6\text {\%}} \pm {1.2\text {\%}}\)  1.36 
\(O^{(3)}_{qQ,rs}\)  \(\)0.4  \(172^{+8.7\text {\%}}_{ 10.8\text {\%}} \pm 8.9\text {\%}\)  1.49  \(190^{+2.4\text {\%}}_{ 1.8\text {\%}} \pm 1.1\text {\%}\)  1.35 
\(O_{\varphi Q}^{(3)}\)  1  \(137^{+9.3\text {\%}}_{11.4\text {\%}} \pm 8.9\text {\%}\)  1.67  \(154^{+2.3\text {\%}}_{ 2.3\text {\%}} \pm 1.2\text {\%}\)  1.52 
\(O_{tW}\) (Re)  2  \(132^{+9.3\text {\%}}_{11.4\text {\%}} \pm 8.8\text {\%}\)  1.83  \(148^{+2.3\text {\%}}_{2.5\text {\%}} \pm 1.2\text {\%}\)  1.68 
\(O_{tW}\) (Im)  1.75i  \(125^{+9.2\text {\%}}_{11.4\text {\%}} \pm 8.8\text {\%}\)  1.51  \(140^{+2.3\text {\%}}_{ 2.5\text {\%}} \pm 1.2\text {\%}\)  1.38 
3.1 Inclusive single top production
We start by computing the total single top production cross section for stable top quarks for the relevant operators at LO and NLO for the LHC, at 13 TeV. These results are also available in Ref. [6], but we reproduce them here in Table 1 for completeness. Our computation uses the fiveflavour number scheme.^{3} For these results the renormalisation and factorisation scales, \(\mu _{R}\) and \(\mu _F\) are both set to \(m_t = 172.5\) GeV. The NNPDF3.0 LO and NLO sets [27] are used for the LO and NLO predictions respectively and the only kinematic cuts are applied to the jets: \(p_T^j > 5\) GeV and \(\eta _j < 5\). To show the impact of the NLO corrections, Table 1 presents the Kfactors which are defined as the ratio \(\sigma _{\text {NLO}}/\sigma _{\text {LO}}\) for each contribution.
We find that for the single top process the squared terms and interference between the operators, i.e. the \({\mathcal {O}}(1/\Lambda ^4)\) terms, are suppressed for coefficients of \({\mathcal {O}}(1)\) for the \(O_{tW}\) and \(O_{\varphi Q^{3}}\) operators but are not negligible for the 4fermion operator. Taking its coefficient to be of order one we find a large cancellation between the interference and squared contributions. We also observe that Kfactors vary considerably between the various operators, and can be quite different from the SM contribution. This underlines the importance of including genuine NLO corrections in predictions, since a universal Kfactor does not summarise the table. In the table we also include the contribution of the imaginary part of the coefficient \(C_{tW}\), which only enters squared at \({\mathcal {O}}(1/\Lambda ^4)\) as it does not interfere with the SM or the other operators. We will discuss this contribution in detail in Sect. 3.5.
Total crosssection results give a good first indication on the impact of the operators on the single top production process, but more information can be extracted by considering differential distributions. To demonstrate the effect of the operators on the differential distributions we select a set of benchmark scenarios. The benchmark coupling values that will be used throughout the paper are presented in Table 2. We follow the EFT analyses of Refs. [6, 28] to ensure that our coupling values fall within the current limits. The effects on the inclusive cross section and the top width are also given for both LO and NLO. The predicted deviations from the SM predictions lie within the uncertainty of recent single top measurements: \(\sigma = 156 \pm 35\) pb and \(0.6 \le \Gamma _{\text {top}} \le 2.5\) GeV [29, 30, 31, 32]. In the table we also include the scale uncertainties obtained by varying the central renormalisation and factorisation scale by a factor of two up and down, and the PDF uncertainties. We note the significant decrease in the scale and PDF uncertainties going from LO to NLO, a wellknown feature of NLO computations. At NLO the combined uncertainty is only of the order of 3%, in agreement with previous results [22]. We shall therefore refrain from showing uncertainty bands in our differential distributions, even though these can be straightforwardly computed with our setup.
In the distributions we see that the 4fermion operator in particular has an effect on the shapes in both the transverse momentum and rapidity distributions, leading to harder and more central tops. The impact of the other two operators on the distribution shapes is milder. It can also be observed that the shape difference between LO and NLO has its largest effect in the sensitive region of these distributions, highlighting again the importance of NLO predictions for experimental analyses of this process.
3.2 Single top production and decay
To study the process in more detail and extract maximal information on the impact of the operators, we should consider the distributions of the top decay products. This requires studying the full process of \(p p \rightarrow b \ell \nu j\), where we have assumed that the top quark decays leptonically. In such a computation several difficulties arise compared to that for the inclusive \(pp \rightarrow tj\) computation.

The full matrix element up to the leptons \((b\nu l j)\) in MG5_aMC (fullchain).

Wbj production in MG5_aMC and decay the W in MadSpin (halfchain).

Single top production (tj) in MG5_aMC and decay the top and W in MadSpin (nochain).
All three options show good agreement, as shown in Fig. 4. We verified this to be the case for other observables as well. Given the level of agreement we find at LO between the Wbj and \(l\nu b j\) distributions we will follow the halfchain method for our NLO results, i.e. we produce Wbj and decay the W in MadSpin, employing the relatively narrow Wwidth. A similar agreement is expected to hold at NLO, in particular as the leptonic decay of the W is not sensitive to higher order QCD corrections.
3.3 Treatment of top quark width and impact of multiple operator insertions
 (i)
The width of the top enters in the production of the Wbj final state. The effective operators affect the numerical value of this width, which has to be computed accordingly. We examine the modifications of the width value and its impact on the validity of the narrow width approximation for the top decay.
 (ii)
By considering the Wbj production matrix elements, the effective operators can now enter both in top production and in top decay. Allowing more insertions in the amplitude generates higher order terms in \(1/\Lambda ^2\). These higherorder terms are expected to be suppressed but we will check this explicitly. Studying the Wbj final state moreover implies that configurations without top quarks contribute. The dimension6 operators can affect also these irreducible backgrounds, hence their contributions should be included and their impact studied.
In order to examine whether the conclusions reached so far apply to differential distributions as well we show in Fig. 10 the top polarisation angle, defined in (3.4), obtained for two different values of the coefficient for one and two EFT insertions. The left pane shows both EFT options for \(C_{tW}=1\). One can observe that the two distributions coincide within statistical errors. The right pane shows the case of \(C_{tW}=6\), here the impact of higher order terms are important and these cannot be described by a global normalisation factor as shown in the ratio inset. This indicates that higher order effects in the EFT can be nonnegligible. Therefore, for consistency with the production cross section and to avoid missing large higher order effects, all distributions in the rest of this paper have been obtained by generating Wbj allowing up to two EFT insertions, with the top width computed as a function of the coupling.
We note here that we validated our leadingorder results with the ones discussed in [19] for the topquark polarisation (P), analysing power (\(a_i\)) and lepton angular distributions. We performed a detailed comparison by allowing all possible insertions of the operators and matching all parameters of the computation with the one implemented in the generator Protos [18], and found perfect agreement.
3.4 Results at NLO
Having studied the various effects at LO we proceed by computing the Wbj cross section at NLO in QCD in the presence of the dimension6 operators. The W boson is decayed leptonically through MadSpin, and Pythia8 [35] is used for parton showering and hadronisation. Since we also generate the irreducible backgrounds, a loose invariant mass cut is imposed on the Wb system, centered on the top mass \(100\, \mathrm {GeV}< M_{Wb\mathrm {jet}} < 250\, \mathrm {GeV}\) [24]. Jet clustering is done using fastjet [36] and the anti\(k_t\) algorithm [37], with the jet radius parameter set to 0.4. All other generator settings and kinematic cuts are the same as in Sect. 3.1.
Since the spin axis of the top is known [15] a rich set of angular observables showing spin correlations, can be exploited. Below we will elaborate on the definitions of the angles involved. In general, based on the choice of reference frame, it is possible to probe the production and decay vertex of the single top separately. In any frame, a new set of coordinates can be defined based on the spin axis of the top. These additional coordinate axes provide the ability to construct other angles that contain spin information. For brevity, only the angular distributions that show the most sensitivity to the effective operators will be presented in this section.
We investigate the distributions of the angles between the directions of the top quark decay products and these new directions. The angle of the charged lepton with respect to the three axes defined above is affected most by the polarisation of the top [38].
We also mention here that we examined event samples where the operators were only allowed to enter in the production of the top quark. Here it was observed that for the W helicity angles, Eqs. 3.11 and 3.12, no deviation for the SM was observed. This validates that these angles probe the decay vertex only.
3.5 CPviolation in single top
In this subsection we study possible CPviolating effects in single top production. In the SM CP violation is too small for baryogenesis, which motivates the search for new sources of CPviolation. Within the EFT, the coefficient of the \({\mathcal {O}}_{tW}\) operator can have an imaginary part, leading to a new CPviolating interaction. Here we study how large this effect could be and identify observables sensitive to it.
As discussed in [19], the polarisation angle \(\cos {\theta ^y_{\ell }}\) defined in Eq. 3.10 shows a sensitivity to the phase of \({\mathcal {O}}_{tW}\) coefficient. This can indeed be observed in Fig. 16, where an asymmetry is clearly visible, for the imaginary part of the coefficient. The SM, charged current, fourfermion operator and real part of the dipole operator show no asymmetry in this distribution.
In order to focus on the effects of the imaginary part of \(C_{tW}\), Fig. 17 shows results for a range of coupling values that are within the current global limits [28]. It is interesting to see that this observable is sensitive to both the size and to the sign of the coupling for \(\mathrm {Im}\,{\mathcal {O}}_{tW}\). We note here that we additionally studied the asymmetry suggested in [3], but found this to be less sensitive to \(\text {Im}C_{tW}\) than \(\cos {\theta ^y_{\ell }}\).
4 Conclusions
Single top production provides an excellent opportunity of probing top quark couplings. The SMEFT is a framework which allows us to parametrise deviations from the SM couplings in a consistent and modelindependent way. Predictions in the SMEFT can be systematically improved by computing higherorder corrections. In this work we computed for the first time single top production and decay at NLO in QCD, in the presence of dimension6 operators.
We studied the impact of these QCD corrections, both at the inclusive and differential level, and found that NLO effects affect both the total rates and the differential distributions in a nontrivial way, with different operator contributions receiving different Kfactors. NLO effects can be large and are therefore needed to reliably predict the impact of the dimension6 operators. We computed all relevant contributions at \({\mathcal {O}}(1/\Lambda ^2)\) (and some \({\mathcal {O}}(1/\Lambda ^4)\) terms), and examined their relative importance.
We then included also the decay of the top, examining the validity of the NWA and the impact of the top width in computing results for the Wbj final state. We find that the impact of the dimension6 operators on the top width needs to be taken correctly into account to ensure that the Wbj and tj cross sections are consistent. We then computed top production and decay at NLO matched to the parton shower using the resonanceaware matching within MG5_aMC, including offshell and interference effects. We obtained NLO distributions for both the top and its decay products for the SM and a series of benchmarks with nonzero operator coefficients. We find that the weak dipole and fourfermion operators can lead to harder tails in the distributions.
In order to fully exploit the power of spin correlations, we explored a series of angular observables that can be used to probe new physics couplings in either the production or decay of the top. These include the socalled polarisation angle and W helicity fractions. We find these angular distributions to be sensitive to different operators. The sensitivity becomes weaker when we apply cuts on the top decay products, but can still be probed by defining the corresponding asymmetries. Finally we considered CPviolating effects coming from the imaginary part of the dipole operator coefficient and studied an angular distribution that can be used to identify such an interaction.
Our study is an example of using an accurate and realistic simulation framework to compute deviations from the SM within SMEFT for a limited number of operators. Our results can be used in combination with the experimental results to obtain reliable constraints on the operator coefficients as part of the ongoing effort of EFT interpretations of LHC topquark measurements [41].
Footnotes
 1.
Only electronic W decays are used in this study. Similar results are expected for top quark events with a muon in the final state. Events where the W is decaying to a \(\tau \), or hadronically, are experimentally more difficult to isolate.
 2.
We remark that a slightly different approach [17, 18, 19, 20, 21], not using operators but anomalous couplings, has also been used in the literature. The connection between the operator coefficients to the anomalous couplings is discussed in [4]. Here all types of Lorentzinvariant interaction structures that involve the W boson and the top quark are allowed, including those that the Standard Model does not allow. An advantage of the present approach is the limited number of parameters, a restriction following from symmetry requirements.
 3.
We note here that a comparison between the fourflavour and fiveflavour number schemes for the SM tchannel single top production can be found in the literature [25, 26]. Differences between the two schemes are reduced at NLO, but can be up to 10% at the LHC, depending on the observable. A detailed comparison of the two schemes is beyond the scope of this work, but we expect our conclusions regarding the distribution shapes and Kfactors of the dim6 contributions in comparison to the SM to hold also in the fourflavour scheme.
 4.
This top is selected based on its particle ID (i.e. in this example it is not reconstructed from its decay products), and therefore stable. We note that this is the case only for Sect. 3.1. In the following sections of the paper, the top is reconstructed from its decay products.
 5.
For a fit of Wtb anomalous couplings using the Whelicity fractions and single top crosssection measurements, see Ref. [39].
Notes
Acknowledgements
We would like to thank Fabio Maltoni, Jordy de Vries and Cen Zhang for discussions, and Rikkert Frederix, Andrew Papanastasiou and Paolo Torrielli for valuable technical assistance. This work was supported by the Netherlands Organisation for Scientific Research (NWO). EV is supported by a Marie SkłodowskaCurie Individual Fellowship of the European Commission’s Horizon 2020 Programme under contract number 704187.
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