Off-shell single-top production at NLO matched to parton showers

Journal of High Energy Physics, Jun 2016

We study the hadroproduction of a W b pair in association with a light jet, focusing on the dominant t-channel contribution and including exactly at the matrix-element level all non-resonant and off-shell effects induced by the finite top-quark width. Our simulations are accurate to the next-to-leading order in QCD, and are matched to the Herwig6 and Pythia8 parton showers through the MC@NLO method. We present phenomenological results relevant to the 8 TeV LHC, and carry out a thorough comparison to the case of on-shell t-channel single-top production. We formulate our approach so that it can be applied to the general case of matrix elements that feature coloured intermediate resonances and are matched to parton showers.

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Off-shell single-top production at NLO matched to parton showers

Received: March -shell single-top production at NLO matched to parton showers R. Frederix 0 1 3 5 6 S. Frixione 0 1 3 6 A.S. Papanastasiou 0 1 3 4 6 S. Prestel 0 1 2 3 6 P. Torrielli 0 1 3 6 0 2575 Sand Hill Road, Menlo Park, CA 94025-7090 U.S.A 1 J.J. Thomson Avenue, CB3 0HE, Cambridge, U.K 2 SLAC National Accelerator Laboratory 3 Via Dodecaneso 33, I-16146, Genoa , Italy 4 Cavendish Laboratory, University of Cambridge 5 Physik Department T31, Technische Universitat Munchen 6 Via P. Giuria 1, I-10125, Turin , Italy We study the hadroproduction of a W b pair in association with a light jet, focusing on the dominant t-channel contribution and including exactly at the matrix-element level all non-resonant and o -shell e ects induced by the nite top-quark width. Our simulations are accurate to the next-to-leading order in QCD, and are matched to the Herwig6 and Pythia8 parton showers through the MC@NLO method. We present phenomenological results relevant to the 8 TeV LHC, and carry out a thorough comparison to the case of on-shell t-channel single-top production. We formulate our approach so that it can be applied to the general case of matrix elements that feature coloured intermediate resonances and are matched to parton showers. bINFN | Sezione di Genova - O 1 Introduction 3 Results 3.1 3.2 2 Matching setup, subtleties and technicalities 2.1 Integration of subtracted cross sections with resonances 2.1.1 Treatment of resonances in FKS 2.2 Matching to parton showers Single-top hadroproduction: process de nition and approximations Di erential distributions 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 Transverse momentum of reconstructed top quark, pT (W +; Jb) Transverse momentum of primary b-jet, pT (Jb) Invariant mass of reconstructed top quark, M (W +; Jb) Mass of primary b-jet, M (Jb) Relative transverse momentum of primary b-jet, pT;rel(Jb) Invariant mass of lepton+b-jet system, M (l+; Jb) 4 Conclusions and outlook A Technicalities on the treatment of resonances 1 3 by CDF and D0 [3{5] at the Tevatron, as well as by the ATLAS [6{8] and CMS [9{11] collaborations, at both the 7 TeV and 8 TeV LHC, with preliminary results [12, 13] available at 13 TeV as well. This channel has also been exploited by the ATLAS collaboration in the rst top-mass extraction from single-top events in ref. [14]. More recently, experimental evidence has been found [15{18] of s- and W t-channel production, which are characterised by cross sections smaller than that of the t channel. Single top production has been shown to be sensitive to anomalous tW b-couplings (see for example refs. [19, 20]), and e orts are being made by experiments [21, 22] to use the t-channel process to search for such features. Furthermore, with increased statistics at run II of the LHC, measurements of the top-quark decay products and of di erential quantities will be possible with vastly improved precision. With a view to matching the progress achieved on the experimental side, it is important to review, assess and improve the current theoretical predictions available for single-top { 1 { production. Top quarks are never observed as stable particles, but rather their production is inferred through a kinematic reconstruction of their decay products (jets, leptons and missing energy). Theoretical predictions, whenever possible, should therefore re ect this fact, namely they should deal with top decay products instead of stable top quarks as primary objects. This is particularly important for observables sensitive to the decay and o -shellness of the top, as well as for those sensitive to non-resonant contributions, which are completely missing in the stable-top approximation. Nonetheless, the current theoretical standard only partially ful ls this requirement. State-of-the-art predictions at the hadron level for this process are obtained through NLOmatching with parton showers (NLO+PS) both in the four- and ve- avour schemes [23{ 25] in the MC@NLO [26] and POWHEG [27, 28] approaches, assuming stable-top hard matrix elements. In such setups, the top-quark decay is performed with LO accuracy, and the o -shellness of the top propagator is introduced through a simple Breit-Wigner smearing, either by the PS itself, or at the matrix-element level (which allows one to correctly account for both production and decay spin correlations) by applying the method introduced in ref. [29]. At xed order (where the most advanced predictions for the stabletop t-channel cross section are the NNLO results of ref. [30]), alongside the NLO corrections to the production, NLO corrections to the decay of the top quark have been included in the narrow-width approximation (NWA) [31{34]. A systematic treatment of o -shell e ects for resonant top quarks was rst presented in refs. [35, 36], using an e ective-theory-inspired generalisation of the pole expansion. The NLO corrections to the t-channel process with full o -shell and non-resonant e ects have been computed in ref. [37] by adopting the complex-mass scheme (CMS) [38, 39]. Including NLO QCD corrections to the top decay has been shown to play a signi cant role, especially for observables such as transverse momentum of the b-jet or the invariant mass of the lepton+b-jet system [34, 40]. Additionally, treating the top quark as on-shell (as in the NWA) or o -shell can also lead to striking di erences in the NLO predictions of experimentally relevant observables [35{37], a prime example being the invariant mass of the reconstructed top (see also refs. [41{43] for similar features in tt production). In light of these observations at xed order, understanding to what extent these e ects survive the showering and hadronisation stages in a Monte Carlo (MC) is not only interesting from the theory point of view, but it also becomes crucial for improved predictions of the observables mentioned above. In particular, with predictions at NLO+PS accuracy and full o -shell e ects at the hard matrix-element level, it becomes possible to validate NLO+PS approaches where the underlying hard matrix elements are computed in the onshell-top approximation. It is also of great relevance to use these improved predictions to properly assess the systematics a ecting the extraction of the top mass when using, as is currently done, MCs within which the hard matrix elements do not include full NLO o -shell e ects. Recently, work has been performed in this direction in refs. [44, 45] within the POWHEG+Pythia8 framework, including NLO corrections in both production and decay, and considering tt production in the NWA and single-top t-channel production with full o -shell e ects, respectively. In this work we adopt the MC@NLO scheme, and study the NLO matching to parton showers of t-channel single-top hadroproduction with full o -shell and non-resonant e ects, { 2 { namely the t-channel contribution to the EW process pp ! W +bj, with j being a light jet, at the 8 TeV LHC. We match our computations to the Herwig6 [46, 47] and Pythia8 [48] parton showers. In this context, we discuss in general the subtleties that occur in NLO+PS simulations for processes with intermediate coloured resonances, and perform a thorough comparison to other available approximations of t-channel single-top cross section. In doing so, we present a study of hadron-level observables sensitive to top-decay radiative corrections and o -shell e ects. The shape of such observables is often a result of a sensitive interplay of a number of di erent phenomena, which we endeavour to disentangle and understand here. We perform our calculations in the framework of MadGraph5 aMC@NLO [49], The paper is structured as follows: in section 2 we describe the setup of the computation, and in particular the subtleties related to the phase-space parametrisation, integration, and MC@NLO-type matching of processes with intermediate coloured resonances; in section 3 we discuss some details of the various approximations to the complete W +bj process, and we present our results for a selected set of observables; in section 4 we draw our conclusions. 2 2.1 Matching setup, subtleties and technicalities Integration of subtracted cross sections with resonances Regardless of whether a computation that features an unstable intermediate particle (that henceforth we denote by , and assume to have mass m and width m ) is matched to parton showers, one problem which must be addressed is that of the e cient integration of the corresponding matrix elements. This is particularly non-trivial in the case where these matrix elements enter the real-emission contribution to an NLO cross section, owing to the necessity of IR-subtracting them. At the amplitude level, the unstable particle is represented by (at least) an s-channel propagator and thus the squared matrix elements will contain a term 1 (k2 m2 )2 + ( m )2 ; with k2 > 0 the virtuality of . Because of eq. (2.1), kinematic con gurations with k2 will be associated with large weights, and hence the corresponding unweighted events will be more likely to occur. The likelihood of this increases with decreasing , which is easy to understand also in view of the fact that the ! 0 limit of eq. (2.1) is proportional to the (2.1) Dirac delta function (k2 m2 ), that forces k2 = m2 exactly. An e cient matrix element integration therefore requires that the phase-space generation be biased towards k2 con gurations, a requirement which is independent of the perturbative order. At the LO (i.e. tree) level, this is not di cult to achieve. The most direct way is that of choosing k2 as one of the integration variables, so that the adaptive integration quickly knows where to throw most of the phase-space points. This is what is done in MadGraph5 aMC@NLO. While one would like to apply a similar strategy at the NLO and beyond, it is the IR subtractions relevant to the real-emission terms that prevent one from doing this in a straightforward manner. In contrast, all of the other non-subtracted contributions to an NLO cross section, such as the Born and the virtuals, can be dealt with in exactly the same way as the LO. In order to simplify the discussion of the relevant issues without loss of generality, let us assume that only one type of singularity is relevant (say, the soft singularities). Following FKS, we shall denote by the phase-space variable that in the limit ! 0 causes the matrix elements to be soft-singular, and by b all of the other (Bornlevel) phase-space variables. The typical structure of the integrated NLO cross section will thus be: Z dbd 1 where the redundant rst argument (k2 ) of the integrand & (& is equal to the matrix elements times phase-space factors) has been inserted explicitly only in view of its relevance to the present discussion. Because of the way eq. (2.2) is integrated (i.e. by choosing some b and for any given random number), its event and counterevent contributions ( rst and second term, respectively, under the integral sign in eq. (2.2)) will typically have very di erent weights, owing to eq. (2.1), unless the condition k2 (b; ) = k2 (b; 0) ; 8 ; b given is ful lled. Such a di erence in weights is responsible for a poorly-convergent integration. In principle, this is simply an e ciency problem, since the convergence in the largestatistics limit is guaranteed by the condition !0 lim k2 (b; ) = k2 (b; 0) ; which holds regardless of whether eq. (2.3) is true or not. In practice, however, the statistics one needs to accumulate rapidly grows with the inverse of , becoming in nite in the ! 0 limit. Indeed, it is instructive to consider the situation in the limiting case where eq. (2.1) is replaced by a Dirac delta. When this happens and the condition of eq. (2.3) is not ful lled, then for any given b either the event or the counterevent is non-null, but never both simultaneously (since it is either k2 (b; ) = m2 or k2 (b; 0) = m2 ). This implies that the phase space is partitioned into two disjoint regions, in which either only the event or only the counterevent contribution to the integrand of eq. (2.2) is non-null, which in turn renders the numerical integration impossible with nite statistics. Alternatively, and from a more physical viewpoint, in processes where is very small, for the majority of phase-space points the event is computed at the resonance peak, while the counterevents { 4 { are far away from the peak (or vice versa), even though the energy of the emitted parton ( in eq. (2.2)) might be small as well. Thus, a small-width resonance severely hampers the cancellation between event and counterevents. A viable solution stems from a re-mapping of the phase space: b ! (b) : There is ample freedom in choosing the speci c form of eq. (2.5), but nevertheless we can distinguish two classes of re-mappings. The members of the rst class ful ll the following condition: k2 (b; ) = k2 ( (b); 0) ; 8 ; (2.6) for any given b. This can be exploited by rewriting eq. (2.2) as follows: Z dbd 1 In other words, eq. (2.5) is used for changing the integration variables of the counterevent (see appendix A for a few technical details on this procedure). Thanks to this, the virtuality of is the same in the event and counterevent of eq. (2.7), hence solving the original problem. Conversely, the re-mappings that belong to the second class ful ll the condition: k2 ( (b); ) = k2 (b; 0) ; 8 ; for any given b. One thus changes the integration variables of the event contribution, whence the analogous of eq. (2.7) reads: Z dbd ; which again solves the problem. In summary, there are three possible ways out. The rst, which we call a type-I solution, is that of choosing a phase-space parametrisation such that eq. (2.3) is ful lled (this is essentially what has been done in ref. [45]). The second (called type-IIa solution) entails a re-mapping of the phase-space variables relevant to the counterevent, so that eq. (2.6) is ful lled. Finally, with a type-IIb solution the re-mapping acts on the variables relevant to the event, so that eq. (2.8) is ful lled. While approaches of type I are simpler than those of type II, they are not necessarily more convenient in the context of NLO or NLO+PS computations, where the primary concern is that of nding a phase-space parametrisation which is ideally suited to IR subtractions (and, in the case of NLO+PS, to MC matching). The latter requirement might render eq. (2.3) (and thus the direct application of the method of ref. [45]) di cult to achieve. It also implies that it is hard to nd a solution to the problem which can be applied to a generic IR-subtraction formalism. It is much more convenient to work in the context of a speci c subtraction scheme, and for this reason in what follows we shall concentrate on FKS and on its implementation in { 5 { (2.5) (2.7) (2.8) (2.9) the relevant information can be found in the original papers [50, 51] and in ref. [52]. The latter work deals speci cally with the issues relevant to automation, and hence to the MadGraph5 aMC@NLO implementation. There are two possible situations, depicted in gure 1: the case where the FKS pair | identi ed in what follows by the indices i (the FKS parton) and j (its sister) | is not directly connected to the tree1 that stems from the resonance (left panel); and the case where the FKS pair is part of the tree whose root is (right panel). We also have to keep in mind that, at variance with the simpli ed treatment presented in section 2.1, in QCD there are both soft and collinear singularities. However, one of the key properties of the phase-space parametrisations relevant to FKS in MadGraph5 aMC@NLO is that for a given real-emission resolved con guration, the reduced (i.e. Born-like) con gurations associated with the soft and collinear limits are identical to each other. We observe that this is a su cient condition for a type-IIb approach to work (since the re-mapping of eq. (2.8) requires that the r.h.s. of that equation be unique for a given b). Conversely, type-IIa solutions might be implemented in any case, however with possibly di erent re-mappings associated with soft and collinear con gurations. The situation depicted in the left panel of gure 1 can occur with either initial-state or nal-state singularities, and the phase-space parametrisation in MadGraph5 aMC@NLO o ers a type-I solution in this case. This is because for both types of singularities all of the nal-state momenta relevant to a given event (excluding i and j in the case of a nal-state singularities) are related to those of the associated counterevents by means of boosts. Since neither i nor j contribute to the invariant mass of , this implies that eq. (2.3) is ful lled, and therefore one can choose k2 as an integration variable. Let us now turn to the situation depicted in the right panel of gure 1, which occurs solely in the case of nal-state singularities. Relevant cases are for example that of a Z branching, with ( ; ; ) = (Z; q; q) and (i; j) = (g; q), or that of a top-quark branching, with ( ; ; ) = (t; b; W ) and (i; j) = (g; b). In the current version of MadGraph5 aMC@NLO, the phase-space parametrisation adopted is that of section 5.2 of ref. [28], and its generalisation to the case of a massive FKS sister. Such a parametrisation 1Note that this is a sensible de nition, because is an s-channel, and hence it is the root of a tree that can be separated from the rest of the diagram by a single cut. { 6 { HJEP06(21)7 does not obey eq. (2.3), and we have therefore considered type-II approaches. In order to keep the present discussion at a non-technical level, we limit ourselves here to saying that for the sake of this work, and for future versions of MadGraph5 aMC@NLO, we have implemented a type-IIb solution. However, further details are provided in appendix A. order at which it is carried out. In order to simplify the following discussion we shall always refer to the process we study in this paper; however, it should be clear that our considerations and procedure are valid in general. At the parton level, simulations for o -shell non-resonant single-top hadroproduction are based on processes of the type xy ! W bq(+z), rather than on their on-shell analogues xy ! tq(+z). Therefore, there is no physical way (nor formal necessity) of agging a speci c subset of generated events as stemming from top-quark contributions. Nonetheless, despite the formal categorising of events as containing or not containing an intermediate top being unphysical, the description of higher-order contributions induced through parton showering might be very di erent in the two cases. MC event generators typically handle the showering from a coloured resonance and from its decay products in a factorised fashion: emissions from the resonance are treated rst and are in competition with all other sources of radiation, and emissions o the resonance decay products are added in a second, separate step. This choice is physically motivated by the NWA, which dictates a factorisation into production and decay subprocesses, as well as a suppression of the interference of radiation in production and in decay by t=mt 1. In the presence of a top-quark resonance, the showers from its decay products will usually be forced to preserve the reconstructed top invariant mass mtrec (to be precisely de ned below). On the other hand, no such constraint is applied if the information on the intermediate top quark is absent. Such a disparity may lead to very di erent shower evolutions2 even when starting from exactly the same nal-state kinematics. For certain observables, especially those related to the invariant mass of the W b-jet system, W bq samples for which the top quark is not written in any of the events may thus produce results in visible disagreement with analogous on-shell t(! W b)q samples, even in the narrow-width limit. The reconstructed top-quark mass is itself a prime example of this issue. This situation is disturbing, since it is ultimately due to arbitrary choices made in MC modelling. That is, despite being physically motivated, the decision of whether or not to write the resonance in the event record (which is based on the NWA that breaks down precisely in the o -shell region which one aims to investigate), and the constraint that the resonance mass is kept constant if written in the event record, are both choices that are non-parametric in nature. These can therefore easily o set the increase in precision attained by computing 2The verb \may" is emphasised here. In fact, the type of direct and indirect e ects that could be seen in the nal state crucially depend on the details of the speci c MC implementation. We shall give explicit examples in section 3, for the cases of Pythia8 and Herwig6. { 7 { higher-order corrections. At the same time, this implies that systematic studies of these aspects of MC modelling can sensibly be carried out also at the LO. Pending thorough comparisons with data, at the theoretical level one can assume the on-shell-top limit of the showered results to be a sensible benchmark. Hence, the discussion above suggests that the explicit presence of the intermediate top in a W bq sample is a desirable feature under certain conditions. The most straightforward example of the latter is that kinematic con gurations for which mtrec is \su ciently close" to the pole top mass mt should include an intermediate top quark at the level of hard events to be given in input to the shower. Evidence for this feature comes from the xed-order results, where it can be shown that near top-quark resonances, the dominant contributions to the cross section arise from Feynman diagrams involving intermediate top quarks [35{37]. It would appear consistent that the leading topologies at xed-order should also be the dominant ones after parton-showering, and the presence of intermediate top quarks in W bq samples would allow for this. The strategy we have adopted in MadGraph5 aMC@NLO works essentially in the same way at the LO (where it was already the default [60]) and at the NLO. It is completely general, and is not restricted to the W bj process we are considering in this paper. Moreover, it is already used, in a simpli ed format, for resonances that are not charged under QCD. The method relies on the adaptive multi-channel integration, in which integration channels are roughly in one-to-one correspondence with Born-level Feynman diagrams.3 When considering a single integration channel, the corresponding diagram has a well-de ned structure, possibly with intermediate s-channel contributions. We use this information to decide whether any of these intermediate resonances is written in the hard-event record. In particular, for each of such resonances: a.) If the diagram that de nes the integration channel does not feature the intermediate resonance in an s-channel con guration, then that resonance is not written in the hard-event le. b.) If the diagram contains a resonance in an s-channel con guration, then we distinguish various cases depending on the FKS sector which is presently integrated, as follows. b.1) If the diagram and the FKS sector are such that the grandmother of the realemission radiation can be identi ed with that resonance (this is the situation depicted in the right panel of gure 1) the resonance is written on the hardevent record, and a type-IIb re-mapping is used, as outlined in section 2.1 and appendix A. b.2) The procedure of item b.1) is always carried out in the relevant FKS sector, except in the two following cases: either the MC one matches to does not conserve the reconstructed resonance mass when o -decay emissions occur;4 or a kinematic con guration is generated such that the re-mapping cannot be performed 3Even in the NLO mode of MadGraph5 aMC@NLO, the integration channels correspond to the underlying Born diagrams only | see section 6.3 of ref. [52], with f = 1. 4In this work this happens with Pythia8 in the global-recoil scheme.. { 8 { the hard event. the condition (see appendix A). In these cases, no information on the resonance appears in b.3) For all of the other FKS con gurations or at the LO, we use a dimensionless number, xcut, to determine whether the resonance will be part of the hardevent record.5 In detail, if and only if the given kinematic con guration satis es mrec m < xcut (2.10) will the resonance be part of the hard-event record. The quantity mrec denotes the reconstructed mass of the resonance , and is de ned as the invariant mass of the four-momentum sum of all the decay products of the resonance, as determined by the diagram used in the multi-channel integration (and the FKS sector in the case of real-emission-like events); xcut is a free parameter. What is done here is not motivated by considerations of numerical stability, as in the xed-order NLO (fNLO henceforth [49]) computation (stability is merely a by-product in the present case), but rather by the fact that it is in keeping with the procedures adopted internally by the MCs. This implies that the construction of the MC counterterms relevant to the MC@NLO matching is modi ed in order to take the above information into account. In particular, the features described in items a.) and b.) are applied to the MC counterterms as well, in order to precisely mimic the kinematic constraints imposed by the PS at O( sb+1). It should thus be clear that the only new ingredient w.r.t. the current implementation in MadGraph5 aMC@NLO is one relevant to emissions o the resonance decay products, which, being a phase-space re-mapping, can be trivially automated. A couple of comments on the procedures adopted in MCs are in order here. Firstly, for emissions that do not involve s-channel resonances (left panel of gure 1, that corresponds to item b.3) above) MCs can easily, and do in the cases considered in MadGraph5 aMC@NLO, conserve mrec, regardless of whether the resonance is written on the event record. Therefore, the dependence of physical observables on the parameter xcut is not induced by the conservation (or the lack thereof) of the reconstructed resonance mass, but by the di erent behaviours of the showers driven by the presence of the resonance in the event record | we shall see explicit examples of this in section 3. Secondly, and as far as emissions o the resonance decay products are concerned (right panel of gure 1), in the case of Herwig6, mrec is conserved, whilst in Pythia8 it is not (owing to global recoil). In the MadGraph5 aMC@NLO matching to Pythia8, the global-recoil strategy is adopted in the latter, which allows one to simplify the MC counterterm de nition in the former. In particular, the presence of resonances is ignored, since radiation from the decay products is generated identically to any other hard-process external partons, which means that the same evolution kernels, phase-space boundaries and kinematics are applied for similar evolution steps. Thus, all nal-state particles compensate for the momentum shifts necessary to give non-null virtuality to the radiating parton. In the case of top decays, this 5In principle, the use of xcut can be envisaged in item b.1) as well. We have refrained from doing so, for the reasons explained further below in the main text. { 9 { ultimately implies that the invariant mass of the W b system (before an emission) and the invariant mass of the W bg system (after an emission) are not identical. For this reason, as already mentioned in general in item b.) above, in the case of Pythia8 we do not apply a type-IIb phase-space re-mapping. We have further explicitly checked that, at the LO, di erences between the global and local (which is mtrec-preserving) recoil schemes are negligible6 for all di erential distributions studied in the context of this paper (a sample of which will be presented in section 3). We remark that one can envisage the possibility of introducing an xcut dependence for emissions o the decay products as well (i.e. of using the condition of eq. (2.10) to decide whether or not to perform a re-mapping). We have presently refrained from doing so for a couple of reasons. Firstly, it is technically more complicated at the matrix-element level. Secondly, this e ort would not really be justi ed in view of the fact that the Breit-Wigner function is steeply falling, and thus even for relatively small values of xcut one is actually quite close to the asymptotic case xcut ! 1. Because of this, in what follows Herwig6based predictions have been obtained by setting xcut = 35 (our default, with 35 being our arbitrary choice of a very large value, consistently with the Breit-Wigner lineshape), and compared to those obtained with xcut = 0 (in which case, we disallow the type-IIb re-mapping in the phase space; this option will not be made available in the public version of MadGraph5 aMC@NLO). On the other hand, with Pythia8 these precautions are not necessary, and several values of xcut will be considered. While suitably writing the top quark in the hard events addresses the on-shell-limit issue, it poses another problem in the context of the MC@NLO matching. Despite the top quark not being an external particle in o -shell W bq production, the factorised-shower structure described above allows MCs to radiate gluons both o the W b system and o the top. The latter radiation may spoil the formal NLO accuracy of the computation. In fact, since gluon emission from an intermediate resonance is not IR-singular, in the context of the MC@NLO approach it is not associated with an MC counterterm, whence a potential double counting with the radiation of MC origin mentioned before. Two solutions to this double-counting issue are possible: either a nite extra MC counterterm is added to the MC@NLO short-distance cross section in order to match at the NLO level the e ects of the radiation o the top quark, or this type of radiation is directly disallowed in the PS. For simplicity, and without loss of formal accuracy, we have chosen the latter alternative,7 but have nevertheless veri ed (by switching it on and o , which at least at the LO is fully consistent) that the impact of such top-quark radiation is negligible for all of the observables considered in this paper. The issue of double counting is speci c to the NLO, and therefore no special measures regarding radiation from intermediate top quarks need to be taken at the LO; in our LO simulations, top quarks have been allowed to radiate. 6This is still not a proof that the same feature would hold at the NLO as well, but it is a strong indication that this is indeed the case. Note, furthermore, that in all the cases we have considered where Pythia8 and Herwig6 have signi cant shape di erences, these are not due to the choice of recoil scheme in the former MC, but rather to the intrinsic di erent shower evolutions when these are (or not) \resonance-aware". 7We are grateful to Bryan Webber for providing us with a version of Herwig6 that disallows radiation o intermediate top quarks. 3.1 considered was8 Single-top hadroproduction: process de nition and approximations The starting point of the present work is the fNLO calculation for the t-channel singletop cross section in the ve- avour scheme presented in ref. [37], in which the reaction p p ! W + Jb Jlight + X: (3.1) As in typical single-top searches, at least two jets are present in the nal state we study: a b-jet, Jb, de ned as a jet containing the outgoing b-quark from the hard interaction (for this reason, we call this jet the primary b-jet), and an additional light jet, Jlight, that does not necessarily contain bottom quarks. The assumption of a third-generation diagonal CKM matrix (Vtb = 1) is made in order to have a self-consistent de nition of the t-channel process [37].9 We remark that a consistent treatment of nite-width and non-resonant e ects for top-quark production at NLO can be achieved through the use of the complexmass scheme [38, 39]. This is a renormalisation scheme that introduces the top-width parameter t as part of a complex top-quark mass at the Lagrangian level. Examples of recent applications in NLO calculations with full o -shell e ects are the NLO results for toppair [41, 61{65] and t-channel single-top [37] production. In MadGraph5 aMC@NLO, the generation of the process of eq. (3.1) is obtained by issuing the following commands (see ref. [49] for details on the syntax, in particular table 14 at page 118 there): ./bin/mg5 aMC MG5 aMC> import model loop sm-no b mass MG5 aMC> set complex mass scheme True MG5 aMC> define p = p b b~; define j = p MG5 aMC> generate p p > w+ b j $$ w+ w- z a QED=3 QCD=0 [QCD] MG5 aMC> output; launch As was discussed in ref. [37], a consistent de nition of the process of eq. (3.1) in the veavour scheme requires a kinematic cut on the primary b-jet transverse momentum, which is not necessary when o -shell e ects are neglected, i.e. when the top quark is taken to be stable. This constraint, which is relatively straightforward to impose when working at xed order, becomes more complicated in the presence of parton showers, since it may become far from obvious which of the B-hadrons in the nal state descends from the hardinteraction b-quark. In order to directly compare our present results to previous work at fNLO [37], and to ensure that the conclusions presented below at the hadron level are in as close analogy as possible with those of ref. [37], we have chosen to exploit MC truth.10 This 8We emphasise that even if in ref. [37] and in this paper we simulate W + (i.e. top) production, the case of W (i.e. antitop) production is fully identical. 9The inclusion of the s-channel contribution would require a re nement to the de nition of the b-jet in order make the process well-de ned. Moreover, note that the requirement that the production be EW implies that no Born-level channel features a gluon in the nal state, regardless of the value of Vtb. 10In both the Herwig6 and Pythia8 showers, the mother (JMOHEP) and daughter (JDAHEP) arrays in the event record are used to perform this identi cation. In the Herwig6 analysis we also make use of the information on the space-time vertices (VHEP) where particles are produced. pT (Jb) > 25 GeV pT (Jlight) > 25 GeV j (Jb)j < 4:5 j (Jlight)j < 4:5 140 GeV < M (W +; Jb) < 200 GeV kt jet algorithm [66, 67], with Rjet = 0:5 enables the tagging of the primary B-hadron, identi ed as the B-hadron originating from the outgoing b-quark from the hard interaction. We tag the primary b-jet as the jet that contains the primary B-hadron. In addition to the pT cut on the primary b-jet, we impose other cuts at the analysis level, summarised in table 1, which we adopt throughout our simulations. They are identical to those of ref. [37], and thus allow for a direct comparison with that paper. We limit ourselves here to reminding the reader that this setup has been chosen so as to avoid arti cial enhancements of non-resonant contributions, so that a meaningful comparison with the on-shell single-top simulations can be made. Given that observables sensitive to the leptonic decays of the W boson are also of interest in single-top analyses (e.g. the invariant mass of the lepton+b-jet system, M (l+; Jb), or the transverse mass of the reconstructed top quark, MT (l+; l; Jb)), we consider a leptonically decaying W +. For events passed to the PS, these decays are carried out by the showers themselves, whereas at xed order we simply decay the W + isotropically in its rest frame (which is also what the MCs do). We note that in this way production spin correlations for the leptons are not included. While these may be phenomenologically important, they are not relevant for the assessment of the o -shell and non-resonant e ects we are presently interested in, and therefore do not warrant the more involved process de nition that would be necessary when generating directly leptonic matrix elements.11 In view of the process de nition and of the cuts in table 1, one expects its on-shell analogue to constitute a reasonable approximation. Indeed, as it has been shown both in single top [35{37] and in tt [41{43] production, the NWA does an excellent job in approximating the fully-o -shell results for many distributions. However, since it does fail to capture the dominant e ects in regions of phase space that are sensitive to the top o shellness and to non-resonant contributions, misuse of the NWA may therefore introduce errors that vastly exceed the nave estimate of O( t=mt), which ultimately may have a bearing on experimental procedures such as top tagging and top-mass extractions. The presence of potentially large e ects of this kind will be studied in the following through a systematic comparison of the W +bj results with their on-shell counterparts, which we generically denote by tj. In tj production the top is a stable external particle, hence its radiation in the shower is consistently matched at the level of the MC@NLO short-distance cross sections. Thus, in this case, showering from the top must be allowed in order to avoid double counting: this is the usual procedure [23, 25]. The inclusion 11Having said that, the generation of the full process p p > e+ ve b j, i.e. the process with spincorrelated W -boson decays, is possible with the current version of MadGraph5 aMC@NLO. of NLO corrections to tj production, while highly desirable, is largely incomplete from the phenomenology viewpoint, since it does not improve the description of the top decays (which is left to the MC), and does not include any non-resonant contributions. This situation is addressed in part by the use of the procedure of ref. [29], automated in MadGraph5 aMC@NLO in the module MadSpin [68], which allows one to include both production and decay spin correlations, and to give a rough description of o -shell e ects through a simple Breit-Wigner smearing. For this reason, in the following we shall always use tj predictions in conjunction with MadSpin (which we denote by tj+MS). Although an improvement w.r.t. the \bare" tj results, these still do not contain NLO corrections to top decays, and fully ignore the non-resonant contributions to the W +bj nal state. In summary, by systematically comparing fNLO, tj+MS NLO+PS, and W +bj NLO+PS results, as well as their LO counterparts, we shall be able to assess the impact of a variety of mechanisms, since the above simulations are characterised by an increasing degree of complexity, owing to the inclusion of parton showers, of NLO corrections to decays, and of o -shell and non-resonant contributions. 3.2 Di erential distributions In this section we present our predictions for several observables, obtained with the different computational schemes discussed in section 3.1. All of these have been derived by setting the input parameters that enter the hard matrix elements as shown in table 2. The width value labelled by \NLO" in table 2 is adopted in the context of the W +bj NLO+PS and fNLO calculations, whereas that labelled by \LO" is used in all the other cases. Theory uncertainties are estimated by varying the renormalisation and factorisation scales independently in the range [ 0=2; 2 0]. These variations are performed automatically by MadGraph5 aMC@NLO in the course of a single run, thanks to the re-weighting technique introduced in ref. [69].12 We point out that, for certain observables, the LO scale dependence may be pathologically small, since the Born cross section does not contain any s factor. We have refrained from reporting the uncertainties associated with PDF errors, chie y in view of the fact that they a ect equally the W +bj and tj production processes. We have run the two MCs by adopting the respective default parameters, except for the PDFs, which have been taken equal to those used in the short-distance computations. The simulation of the underlying events is turned o , and in order to simplify the analysis B hadrons are imposed to be stable. In the following, to each observable we associate a gure that contains two main panels, one for Herwig6 and one for Pythia8, each accompanied by three insets. The main panels display four curves: W +bj results at NLO+PS and LO+PS with xcut = 35 (solid and dashed blue, respectively), at fNLO (dashed green with full diamonds), and tj+MS results at NLO+PS (solid red with full circles). MadSpin decays are characterised by a user-de ned parameter, BWcut, that sets the allowed range (i.e. the distance, in width units, from the resonance pole mass) for the invariant mass of the system composed of the resonance decay products. We choose BWcut = 35 as our default, and indicate this explicitly 12This implies that the scale dependence of the top width is neglected. HJEP06(21)7 Z = 2:4952 GeV GF = 1:6639 mt = 173:2 GeV tLO = 1:5017 GeV 10 5 GeV 2 mW = 80:3980 GeV W = 0 GeV e by appending the value of BWcut to the label `MS' in the plots (e.g. `MS35' indicates MadSpin results with BWcut = 35, and so on). We emphasise that the parameters xcut and BWcut are technically di erent, even though they are both associated with a distance from the resonance pole mass. MadSpin simulations feature a top quark in all of their LesHouches events and therefore are independent of xcut (more precisely, they are characterised by xcut = 1) but depend on BWcut. Conversely, simulations of the full W +bj process do not require MS decays, thus these events are strictly BWcut-independent, but do carry a dependence on xcut. Still, xcut and BWcut have a similar meaning from a physics viewpoint, because they parametrise, in di erent contexts, e ects related to top-quark o -shellness. This is the reason why we have chosen their default values to be identical. Similarly, BWcut = 0:1 is the analogue13 of xcut = 0. The rst (upper) inset in each gure contains ratios of the various perturbative approximations to the full W +bj process: LO+PS/NLO+PS (solid blue), fLO/fNLO (dashed green), Herwig6/Pythia8 at NLO+PS and at LO+PS (solid and dashed brown on the right, respectively), in addition to bands indicating the scale variations (N)LO( )= (N)LO( 0) (Herwig6 only, LO in yellow, NLO in grey on the left). The second (middle) inset contains the ratio, with respect to NLO+PS W +bj with xcut = 35, of NLO+PS tj+MS with BWcut = 35 and with BWcut = 0:1 (solid red and dashed magenta, respectively) and of fNLO (dashed green). Finally, the third (lower) inset displays the ratio, with respect to NLO+PS W +bj with xcut = 35, of NLO+PS W +bj with xcut = 0, 1, and 5 (solid cyan, dashed red, and solid green, respectively, with the latter two values adopted only for Pythia8, as explained in section 2.2). In order not to further complicate the discussion of the di erent e ects that play a role in the results presented below, for each gure our analysis is structured as follows. After a brief overview of the observable examined and the important features of the xedorder results (including the e ects of NLO corrections, sensitivity to o -shellness, and non-resonant contributions), we address in turn: a.) the e ect of NLO corrections on the matched results for the full W +bj process ( rst inset, solid blue curve); 13In MadSpin one cannot set BWcut strictly equal to zero, and thus we have used 0:1 instead. Given the values of the top mass and width, this di erence is fully irrelevant. b.) the e ect of parton showering with respect to the xed-order results for the full W +bj process ( rst and second insets); c.) the di erences between results for the full W +bj process, showered with Herwig6 and Pythia8 ( rst inset, dashed and solid brown curves on the right-hand panels); d.) the quality of the stable-top+MadSpin approximations to the full result (second inset, solid red and dashed magenta curves); e.) the sensitivity of the results to the arbitrary xcut parameter (third inset). Transverse momentum of reconstructed top quark, pT (W +; Jb) The rst observable we examine is pT (W +; Jb), the transverse momentum of the reconstructed top quark (de ned as the system composed of the W + and the primary b-jet), shown in gure 2. This observable is inclusive in the invariant mass of the reconstructed top and is thus barely sensitive to o -shell e ects [37]. Resonant/non-resonant interferences and pure non-resonant e ects also do not play a major role. By comparing the fNLO results to MCFM [31] (t-channel single top in the NWA, with NLO corrections in both production and decay; not shown here) it can be deduced that the trend of the spectrum becoming harder at high pT (W +; Jb) at fNLO w.r.t. fLO is a direct consequence of the corrections to production, but is however also enhanced by those to the decay. a.) The inclusion of NLO corrections to the matched simulation mirrors the hardening of the spectrum observed when including NLO corrections at a xed order. b.) The dashed green curve in the second inset indicates that PS e ects are on the whole not large (below 10% in all bins except the rst bin for Pythia8), and do not signi cantly alter the shape of the xed-order results. c.) The agreement between Herwig6 and Pythia8 improves at NLO (the solid brown curve on the rst right-hand inset is systematically closer to unity than the dashed brown curve). However, it should be pointed out that the agreement is already good at LO (especially shape-wise). d.) The second inset reveals that there is a general trend of the tj+MS spectra becoming softer compared to those of the NLO+PS W +bj results. This e ect follows from the same reasoning as the softer behaviour of the fLO or the LO+PS spectra w.r.t. their NLO counterparts, but is of much smaller size since the MadSpin results do include radiative corrections to the production subprocess, while they lack those to the top decay. Overall, the results for NLO+PS tj+MS are in agreement with the full NLO+PS W +bj ones to better than 10% for both showers, indicating that not only corrections to the decay but also non-resonant contributions are small in this case. Moreover, the invariance of the MadSpin results under variation of the BWcut parameter con rms that this distribution is relatively insensitive to o -shell e ects. e.) The third inset reveals that pT (W +; Jb) is largely stable against the choice of xcut, with only xcut = 0 displaying any visible e ects. The latter are however smaller than (Pythia8) or comparable to (Herwig6) the NLO scale uncertainty illustrated by the band in the upper left inset. 3.2.2 Transverse momentum of primary b-jet, pT (Jb) Figure 3 shows the transverse momentum of the primary b-jet, pT (Jb). This observable is less inclusive than pT (W +; Jb) over the top decay products, and therefore NLO corrections to the decay are expected to play a more important role. This is indeed the case since the non-trivial shape at low pT of the di erential K-factor at xed order is driven by the NLO corrections to the top decay (this has again been cross-checked with MCFM). These corrections also result in a harder pT (Jb)-tail at NLO w.r.t. LO. The feature at small pT in the fLO/fNLO ratio can be attributed to the kinematical fact that real radiation o the b-quark carries energy away from the b-jet, thus softening the NLO spectrum; such a leakage occurs less often when moving towards large pT 's, where the jets tend to be more collimated. a.) The di erential K-factors for the showered results at large values of pT (Jb) display the same features as the xed-order results (NLO+PS distributions are harder than LO+PS ones). However, the kinematic suppression in fLO/fNLO at low pT (Jb), driven by the fLO shape, does not carry over to the showered case. This is due to the fact that the shower, already at the LO, accounts for multiple emissions from the nal-state b quark, hence the radiation leakage outside the b-jet, induced by real corrections in the xed-order case, has a much milder impact at the showered level. b.) The dashed green curve in the second inset indicates that the shower e ects for Herwig6 at NLO are small, i.e. that NLO+PS is very close to fNLO, with a distribution only marginally harder ( 5% at low and high pT (Jb), respectively). With e ects of around 10%, Pythia8 departs more from the fNLO result. c.) From the brown dashed and solid curves in the upper right inset, we conclude that there are only mild shape di erences between the Herwig6 and Pythia8 predictions; Pythia8 tends to be slightly harder than Herwig6. The ratio of the two MC predictions displays a more regular behaviour (i.e. in a larger pT range) at the NLO than at the LO. This is most likely due to the fact that the impact of the matrixelement normalisation constraint is more important in the former than in the latter case. Such a pattern is similar to that observed in the case of pT (W +; Jb) but is of slightly bigger size here, which is consistent with the fact that the present observable is more dependent on MC modelling than the transverse momentum of the pseudo top. d.) The trend of the NLO+PS tj+MS curves (solid red, second inset) closely follows that of the LO+PS W +bj predictions, namely they are softer than the NLO+PS W +bj benchmarks. The e ect is similar to that observed in pT (W +; Jb) but is somewhat more pronounced here. Given that at fNLO in the NWA it is the corrections to the decay that induce the dominant features of the fLO/fNLO ratio, and that it is precisely these corrections that are missing in the MadSpin results, this is a strong indication that corrections to the decay subprocess are important for this observable. The independence of the MadSpin result on the BWcut parameter indicates that o shell e ects are essentially irrelevant for this observable | a feature that can also be seen at xed order. e.) As for the case of pT (W +; Jb), the present observable is relatively insensitive to the value of xcut. We only observe a marked e ect for xcut = 0, with di erences to the xcut = 35 result of up to 20% in the hard tails, for both showers. 3.2.3 Invariant mass of reconstructed top quark, M (W +; Jb) The reconstructed top quark mass, M (W +; Jb), displayed in gure 4, is an important observable used to tag top quarks and to help separate the single-top signal from its backgrounds. It may also be used in various ways to extract the top mass from data. At xed order, the real-radiation corrections to both production and decay are important and are the dominant contributions to the shape of the fLO/fNLO ratio. The region above the peak is sensitive to radiation from the production subprocess, whereas the region below the peak is sensitive to radiation from the top decay products. Additionally, treating the top quark as o -shell is vital to sensibly describe this distribution at xed order, with the predictions using the NWA failing to capture most of its features; see ref. [37] for more details. a.) The e ect of the NLO corrections on the LO+PS curve is to skew the distribution towards the right. The agreement in shape between LO+PS and NLO+PS in the low-mass region, i.e. the region sensitive to radiation from the nal-state b-quark, is satisfactory for both showers (and slightly better for Herwig6). This is an indication that for this observable, and in this phase-space region, hard radiation originating from the top-quark decay products is well approximated by the parton showers. The harder spectra at NLO+PS, particularly visible from the large-mass slope of the Herwig6 result, stem from hard radiation in the production subprocess being clustered into Jb by the jet algorithm. The Pythia8 high-mass tail does not show as strong a trend as Herwig6, likely pointing to more (or harder) production radiation in the LO results compared to the Herwig6 shower. The fact that this behaviour is mostly driven by the LO predictions can be inferred from the two brown histograms in the upper right-hand inset | see item 3 below. b.) The e ect of parton showering with respect to xed-order results is very signi cant over the full range considered, and exceeds 50% in the bins near the peak. Radiation by both showers smears and attens the sharply-peaked xed-order distribution. This smearing results from the combined e ect of ISR and FSR enhancing the high-mass tail when clustered into the b-jet, and of b-quark FSR enhancing the low-mass tail when leaking out of the b-jet. c.) The dashed and solid brown curves in the rst right-hand inset indicate that, both at LO and NLO, the Pythia8 distributions are atter overall than the corresponding Herwig6 ones, with e ects as large as 20% and 40% at NLO and LO, respectively. Despite the remaining visible di erences, there is a substantial improvement in the consistency of the two showers at NLO, compatible with the increased formal accuracy of the simulation. Di erences between the two showers are to some extent expected | the smearing cannot be attributed to one single factor, but rather is a combination of various sources that vary between the showers: di erent s(mZ ) or QCD choices, di erent showering models (interleaved ISR/FSR in Pythia8 and sequential ISR/FSR in Herwig6) and di erent hadronisation models. d.) The results for tj+MS are, for large values of BWcut, in good agreement (20% or better in the case BWcut = 35) with the NLO+PS W +bj distributions. This is particularly true for Pythia8, especially above the peak, while in the case of Herwig6 the tj+MS result displays a softer behaviour over the full mass range considered. As for the observables considered previously, for both showers the tj+MS/W +bj ratio has a similar pattern, though milder (i.e. it is closer to one), as that of the W +bj LO+PS/NLO+PS ratio. The large discrepancies between the BWcut = 35 and BWcut = 0:1 results, which are signi cant close to the peak (and, to a lesser extent, above it), indicate the importance of including o -shell e ects for a good description of this observable in that region. On the other hand, all BWcut choices appear to be in good mutual agreement below the peak. This is likely due to o -shell e ects being subdominant in this region w.r.t. corrections to the top decay products. e.) M (W +; Jb) exhibits a sensitivity to xcut qualitatively similar to that seen for pT (Jb) and pT (W +; Jb), namely there is a very small dependence for xcut 1, while the extreme choice xcut = 0 gives some visible shape distortions.14 For both MCs the xcut = 0 results are atter, more markedly so with Pythia8, which has also a mild tendency to skew the distribution rightwards, while in the case of Herwig6 the skewing is rather leftwards. This shows that, when the MCs have no information about the intermediate resonance, large model-dependent e ects can be introduced in the resonance structure. In the region of M (W +; Jb) above the peak, Pythia8 appears to be signi cantly more sensitive than Herwig6 to the choice of xcut = 0. This is related to the di erent construction principle underlying the showering models. While Herwig6 rst generates initial-state radiation, and follows up by generating nal-state emissions, Pythia8 constructs both showers in a combined, interleaved sequence. Thus, initial- and nal-state radiation are in direct competition in Pythia8. This competition is drastically di erent for xcut = 0 and xcut = 35. In general, the PS can emit from the initial-state partons, the light nal-state partons, and from the top-quark decay products. In the xcut = 0 case, all of these possibilities compete with each other for phase space. In the xcut = 35 case, the evolution is split into showering the production process, and showering the decay products. Thus, while all partons associated with the production process compete with each other, the radiation o the decay products is not encumbered by any competition.15 These di erent mechanisms | due to the di erent evolution prescriptions in the xcut = 0 and nite xcut cases | lead to the xcut dependence seen in the Pythia8 results. It is beyond the scope of this work to decide which showering model (interleaved versus independent) is preferable. We have instead chosen to document the features 14This statement depends on the pseudo-top mass range considered. If plotted in a range wider than that of gure 4, the xcut = 1 and xcut = 5 results (the former to a much larger extent than the latter, as expected) would also exhibit increasingly large di erences w.r.t. the xcut = 35 one. 15Note that the absence of competition does not mean the absence of constraints, since overall phasespace boundaries and momentum conservation have to be respected. For example, if no radiation o other legs were present, then the constraints for radiation o the b quark would be independent of xcut, in the global recoil scheme. related to this choice for the process under consideration, and regard the xcut dependence as a way of parametrising this modelling uncertainty. The fact remains, that by never writing the top on the hard-event record, one becomes more sensitive to the di erent underlying shower mechanisms. Next, we examine the invariant mass of the primary b-jet, M (Jb), displayed in gure 5. Due to the analysis setup we have adopted (speci cally the requirement of both a b-jet and a light jet in the nal state), at fLO M (Jb) only receives contributions in the rst bin, M (Jb) = mb = 0. At fNLO, the region M (Jb) > 0 is lled by events where real radiation is clustered together with the b-quark to form the b-jet. Since only real radiation contributes non-trivially in this region, the fNLO prediction diverges as M (Jb) ! 0. Shower e ects are dramatic: the threshold is shifted from zero to the mass of the lowest-lying B hadron, and the low-mass divergence present at xed order is o set by the usual Sudakov damping. Consequently, discrepancies in this case are expected to be signi cant, both in the comparison between NLO+PS and fNLO results, and between di erent MCs. We point out that the characteristics of the present observable outlined above render it analogous to any quantity which has only kinematically-trivial contributions at fLO, meaning that it displays maximal sensitivity to real radiation and to the shower. In these cases, a byproduct of matching to showers is also that of featuring an NLO-type scale uncertainty in the region which receives solely non-hard real-emission contributions. This analogy only holds to a certain extent here since, for the b-jet mass, hard (as opposed to soft) real radiation cannot be associated with certainty to a speci c region of the phase space. For example, a hard emission from the initial state could be nearly collinear to the nal-state bquark, yielding a small b-jet mass. However, one expects the impact of these con gurations to be subdominant. Jet masses will, in general, aside from an increased sensitivity to perturbative effects (including NLO-matching systematics), also show a relatively strong dependence on non-perturbative and soft-physics modelling; this is particularly true at small masses. Therefore, any conclusion based on varying only \perturbative parameters" is potentially incomplete. a.) Including NLO corrections when matching to parton showers leads to a harder M (Jb) distribution in the case of Herwig6, while induces only a constant shift for Pythia8. In both cases the e ects are relatively mild (up to 20% for Herwig6, and 10% for Pythia8), which is remarkable if compared with the situation at xed order. b.) The e ects of parton showering for M (Jb) lead to completely di erent distributions with respect to xed-order results; the latter are indeed not particularly sensible for an observable of this type. As discussed before, this stems from two main reasons. Firstly, at xed order the bins for M (Jb) > 0 only receive contributions from real corrections. Secondly, b-jets are reconstructed at the hadron level, and hence their mass threshold is close to the physical mass of B hadrons. We remark, however, that even if b-jets were reconstructed at the parton level in (N)LO+PS simulations, one would obtain a very similar threshold, owing to the fact that the MCs need to turn b quarks into massive objects with mMC description of b-physics phenomena. b 5 GeV, in order to give a realistic c.) The comparison between Pythia8 and Herwig6 reveals sizable di erences, compatible with the fact that this observable receives large contributions both from higher perturbative orders and from the underlying showers. Nevertheless, given the behaviour of the xed-order results, the (N)LO+PS predictions appear to be reasonably close to each other and, in addition, by including the information on the NLO matrix elements the di erences seen at LO+PS are reduced. In particular, it is reassuring that the agreement between Herwig6 and Pythia8 improves to about 15% in the medium- and high-mass regions. There is also an improvement at low jet masses (not visible in gure 5), which however should not be over-interpreted, since non-perturbative e ects are expected to be substantial in this region. d.) The solid-red and dashed-magenta lines in the second inset indicate that the MadSpin results roughly follow similar shape patterns as those of LO+PS W +bj, namely that they are softer than NLO+PS W +bj for Herwig6, and at for Pythia8. However, in absolute value they are in much better agreement than the LO+PS results with the NLO+PS W +bj predictions. The MadSpin results show no dependence on the BWcut parameter, indicating that o -shell e ects do not have an impact on the shape of M (Jb). e.) The third inset shows that Herwig6 and Pythia8 have a vastly di erent dependence on xcut. The results showered with Herwig6 are mostly independent of xcut (except in the low-mass region), while those showered with Pythia8 are highly sensitive to this parameter over the whole range considered. As for the case of M (W +; Jb), this behaviour can be attributed to the di erent showering models. The Pythia8 sensitivity of the b-jet kinematics on xcut does indeed largely drive the xcut variation of the reconstructed-top mass that we have observed previously.16 Radiation from the b-quark is the primary source of mass increase in the region of moderate b-jet masses, i.e. M (Jb) & 10 GeV, with other phenomena playing only a subdominant role there (contributions from splash-in radiation o other legs become more important at larger masses, which we do not show in gure 5). For a vanishing xcut value, as discussed in item 5 of section 3.2.3, this radiation is always in direct competition with radiation o all other initial and nal state partons, forming a single evolution chain. Conversely, for non-null values of xcut the radiation o the b starts without competition. Thus, it turns out that in this case non-competing radiation o the b quark lls the b-jet more substantially, and leads to a heavier jet. Such an xcut dependence diminishes at larger M (Jb), owing to o -b radiation no longer being dominant. The behaviour at low M (Jb) . 10 GeV is also due to the choice of cuts on the b-jet. Lowering the pT (Jb)-cut in particular leads to a less marked shape di erence between the low- and high-xcut results. Crucially, we have checked (with LO simulations) that the same xcut-variation pattern in Pythia8 is found when using the local recoil. Hence, the striking feature seen in the comparison of the Pythia8 and Herwig6 results is not due to the recoil scheme we have adopted. We conclude this section by re-iterating the message that gure 5 must be seen in its entirety: di erences between generators are as important as perturbative uncertainties, and can suggest strategies to improve the description of M (Jb). Furthermore, we point out that this observable is also quite sensitive to e ects controlled by tuning and underlyingevent modelling, which we have not studied here. 3.2.5 Relative transverse momentum of primary b-jet, pT;rel(Jb) We now turn our attention to the transverse momentum of the primary b-jet in the reconstructed top quark rest frame, relative to the direction of ight of the reconstructed top quark. We denote this quantity by pT;rel(Jb), and display it in gure 6. This observable 16We remark that the xcut dependence of M (Jb) is subdominant as far as the behaviour of the reconstructed top mass is concerned. In the case of M (W +; Jb), the xcut dependence is chie y driven by changes in the direction of ight of the b-jet, which renders the M (Jb) function of xcut and of the mass ranges considered. M (W +; Jb) correlation a rather non-trivial direction of ight of the reconstructed top quark. is challenging to simulate accurately because its shape is | already at xed order | the result of a balance between di erent kinematical e ects. The sharp edge present in this distribution corresponds to the value pT;rel(Jb) = (mt2 m2W )=2mt. In the NWA at fLO, transverse momenta larger than this threshold are kinematically forbidden; the tail beyond the edge starts appearing at fNLO, due to real corrections to the production subprocess. NLO corrections to the decay become important near the peak of the distribution, whilst at the peak and above it becomes crucial to treat the top quark as o -shell [37]. The shoulder of the distribution and the region above the peak are shaped by an interplay of contributions from real emissions that originate from the production subprocess, as well as resonant/non-resonant interference e ects and pure non-resonant e ects | the latter increasing in importance the further one goes into this region of phase space. The sensitivity of the shoulder and the tail of this distribution to mt make it potentially a good observable for mt-extraction, provided that the theoretical systematics are under control. Additionally, observables such as this one are well-suited to disentangling top signals from QCD backgrounds. Therefore, it is imperative that MC predictions are fully understood, and faithfully describe any signi cant e ects over the full range of pT . a.) Overall, it is apparent that the xed-order K-factor, as in the case of M (W +; Jb), is smoothed out by the showers. In the region of the shoulder and below the peak, LO+PS and NLO+PS predictions are similar in shape for both showers. However, at the kinematic threshold and beyond, the NLO corrections result in a \step up" in Herwig6 whilst in Pythia8 they do not induce such a change in shape. This is an indication of how copiously the Pythia8 shower populates this region with radiation already at LO. b.) The e ects of parton showering are mild in the low-pT;rel(Jb) region (especially for Pythia8), while they are very large at the sharp edge and also visible in the highpT;rel(Jb) tail. The sharp edge at xed order is made less steep through the combination of two e ects. Firstly, near the edge (i.e. close to the xed-order kinematical threshold) multiple FSR emissions o the b-quark leaking out of the b-jet lead to the lowering of the peak. Secondly, emissions from the production process captured inside Jb enhance the region beyond the sharp edge. We note that the high-pT;rel(Jb) region is predominantly LO-accurate in the context of the simulations performed in this paper,17 hence the results are expected to have a larger sensitivity to the various approximations here than elsewhere. This is re ected in the shape of the scale-uncertainty band in the top-left inset. c.) Considering the balance of di erent e ects resulting in the shape at xed order, the relative agreement between Pythia8 and Herwig6 for this observable is encouraging. There is a clear improvement in the agreement when passing from LO+PS to NLO+PS, even in the high-tail region, where the cross section is reduced by 2{3 orders of magnitude with respect to the peak. The pattern of the Herwig6 over Pythia8 ratio at the LO+PS level can be understood as due to the larger amount of radiation (from production) in Pythia8, which enhances the region above threshold, and thus, by unitarity, decreases the cross section below threshold. At NLO+PS this feature is very much reduced, with discrepancies smaller than 15% in most bins. d.) For values of pT;rel(Jb) below the sharp edge, all MadSpin results do a very good job in approximating the shape of the full result in both showers. However, beyond threshold, the di erences between the two MadSpin BWcut values, as well as the di erences of these w.r.t. the full result, begin to grow signi cantly, reaching 50% or more at the edge of the range considered. By moving deeper into this region of phase space, as also pointed out in ref. [37], the result becomes increasingly sensitive to o -shell and non-resonant e ects. Non-resonant contributions are missing in the MadSpin results and therefore the di erences between these and the complete NLO+PS W +bj result can be expected. This comparison indicates that for this observable and in these regions of phase space the full NLO+PS W +bj computation is a prerequisite for a reliable description. 17This is strictly true in the NWA, while o -shell e ects partially ll the region beyond threshold already at fLO in the full W +bj computation. e.) There is a striking di erence in the dependence of the Herwig6 and Pythia8 results on the xcut parameter. This is particularly pronounced beyond the edge in the spectrum, i.e. in the region where both real radiation and non-resonant e ects are very important. The Herwig6 distribution displays at most a 20% dependence on xcut. On the other hand, beyond the sharp edge the results showered with Pythia8 become very sensitive to xcut, a feature that stems from the same reasons as those already described in detail in item 5 of sections 3.2.3 and 3.2.4. Invariant mass of lepton+b-jet system, M (l+; Jb) The relative transverse momentum discussed in section 3.2.5 can be seen as a member of a class of observables characterised by the presence of sharp edges due to thresholds in the NWA. Other examples are the invariant mass of the b-jet-lepton pair, M (l+; Jb), and the transverse mass of the system composed of the b-jet, the charged lepton, and the neutrino.18 These observables may have fairly di erent properties from the experimental viewpoint (in particular as far as the reconstruction of the candidate top pseudo-particle is concerned), the discussion of which is beyond the scope of the present work. On the other hand, because of the kinematic features they have in common, they display similar patterns in terms of the various theoretical approximations that can be used to predict them. To exemplify this fact, in gure 7 we show the results for M (l+; Jb) (we remind the reader that spin correlations have been neglected in the leptonic decay of the W boson). This observable is often used as a discriminating variable to help disentangle top-quark signal events from backgrounds, and interestingly it has been employed to extract the top mass in t-channel-enhanced events [14]. As can be seen from the plots of gure 7, the conclusions of section 3.2.5 also apply, largely unchanged, to the present case. Given that the di erent predictions have been studied in detail and are now available, it would be interesting to exploit these to quantify the systematic error on the extraction of mt via M (l+; Jb) that is introduced through the use of generators that do not include full o -shell e ects and NLO top-decay corrections. 4 Conclusions and outlook In this paper we have studied t-channel single-top hadroproduction, where full o -shell and non-resonant e ects are computed at the matrix-element level with NLO accuracy in QCD, and matching to parton showers is included. We have done so in the context of the automated MadGraph5 aMC@NLO framework, where we have implemented our solutions in a process-independent way. This constitutes the rst example, in the MC@NLO approach, of the matching of an NLO calculation to a parton shower that features an improved treatment of intermediate coloured resonances. We have considered two Monte Carlos, Pythia8 and Herwig6, as representatives of di erent behaviours with regard to several characteristics, most notably shower evolution 18Strictly speaking, the pseudo-top mass itself also belongs to this category, but being extremely peculiar it constitutes a case on its own. and handling of resonances. This has allowed us to deal in detail with a few resonancespeci c aspects of the matching. Among these were the de nition of the MC counterterms necessary in the MC@NLO formalism, the treatment of resolved MC emissions o intermediate top quarks, and the writing of information on these tops in Les-Houches event les. The latter item, to a large extent, pertains to MC modelling, and it is thus important to keep in mind that the writing of an intermediate top in the hard-event le can signi cantly a ect how the events look after parton showering. Although we have argued that including such information is certainly physically motivated, and is consistent with results obtained in the t ! 0 limit, we have studied its consequences by parametrising it by means of an arbitrary dimensionless quantity xcut. It will be interesting to compare theoretical predictions, and their dependence on xcut, with actual data. We have obtained results based on a generic nal-state analysis, chosen to be as similar as possible to that of the xed-order calculation presented in ref. [37], in order to allow for a direct comparison to the latter paper. Overall, it is observed that for this typical analysis the di erential K-factors at the hadron level can be large and non-constant in shape. However, as expected, the scale dependence of the results generally decreases when going to NLO+PS accuracy. Our comparison to the fNLO predictions illustrates that e ects of parton showering and hadronisation can be large (>10-15%). We also nd that the agreement between the predictions of the two showers improves at NLO, though the di erences themselves are found to be sizeable for some observables (in particular for the invariant mass of the primary b-jet). An important consequence of the availability of predictions at NLO+PS accuracy, with full o -shell e ects, is that existing approximations can be scrutinised and validated. We have made a detailed comparison to the results of stable single-top production at NLO+PS accuracy, where the spin-correlated decay of the top quark and the leading o -shell e ects are included by using MadSpin. On the whole, it is observed that the MadSpin results describe the full results remarkably well, which is certainly encouraging in view of the fact that the predictions of the former type presently constitute the benchmark for simulations that involve coloured resonances at LHC experiments. However, we have observed some notable di erences between stable-top and full results in some of the distributions we have studied, with an observable-dependent pattern. We have attributed these to NLO e ects in the top-quark decay and to non-resonant e ects, neither of which can be described by the MadSpin procedure, and to o -shell e ects, which MadSpin can simulate only in an approximate manner. We thus conclude that, when targeting a less-than-10% accuracy, a better description of intermediate resonances is a necessity. There are several aspects of this paper that will be interesting to consider in future work. The most obvious is the application of our ndings to the matching of tt production with full o -shell e ects (W +W bb) to parton showers. Furthermore, in view of the sometimes large di erences between the Pythia8 and Herwig6 predictions, it will also be worthwhile performing a careful study of the impact of shower initial conditions. This is because single-top production is an example of a multi-scale process for which a more sophisticated choice of shower scales might be required for an improved phenomenology treatment (as recently observed e.g. in ref. [71], which also deals with the presence of nalstate b quarks). Finally, we point out that our results matched to the Pythia8 shower show qualitative agreement with those presented in ref. [45], which employ the POWHEG matching scheme. A thorough comparison of the two approaches using the same inputs and analysis setup would obviously be of great interest, and one which we intend to pursue. Acknowledgments We would like to thank Bryan Webber for many useful and illuminating discussions on this topic as well as for providing us with a modi ed version of Herwig6 that allowed the vetoing of PS emissions o intermediate top quarks. The work of RF is supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovaleskaja Award Project \Event Simulation for the Large Hadron Collider at High Precision," endowed by the German Federal Ministry of Education and Research. SF is grateful to CERN TH division for hospitality during the course of this work. The work of AP is supported by the U.K. Science and Technology Facilities Council [grant ST/L002760/1]. SP is supported by the US Department of Energy under contract DEAC02-76SF00515. The work of PT has received funding from the European Union Seventh Framework programme for research and innovation under the Marie Curie grant agreement N. 609402-2020 researchers: Train to Move (T2M). This work has been supported in part by the ERC grant 291377 \LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier". A Technicalities on the treatment of resonances In this appendix we sketch the implementation in MadGraph5 aMC@NLO of the typeIIb solution alluded to in section 2.1.1, for the case of nal-state singularities relevant to the FKS sector where the FKS pair belongs to the tree whose root is the resonance . In order to simplify the discussion as much as possible, we shall proceed as in section 2.1, i.e. pretending that only soft singularities are present. However, as mentioned before, in MadGraph5 aMC@NLO all counterevents associated with a given event have the same reduced kinematics. Therefore, the actual formulae implemented in the program di er from those given below only by marginal technical aspects. We also recall that in the case of the singularities we are interested in, the phase-space parametrisation employed by MadGraph5 aMC@NLO is that of section 5.2 of ref. [28], and its generalisation to the case of a massive FKS sister. This implies that, using the labelling of gure 1: solution. k2 (b; ) = (k + k )2 ; k2 (b; 0) = k + k 2 ; where the barred four-momenta k conventionally denote those of the counterevent conguration associated with the event con guration whose four-momenta are denoted by un-barred symbols k. Furthermore k = B k ; k 6= B k ; with B a boost. Therefore, eq. (2.3) is not ful lled, whence the necessity of a type-II In order to proceed, let us start from the basic expression of the subtracted cross section, eq. (2.2), that we can replace with: Z db Z 0 M(b) d 1 k2 (b; ) j b; k2 (b; 0) j b; 0 ; &(b; ) = (b; ) ( M(b) ) : Equation (A.3) di ers from eq. (2.2) by a contribution due to the integral of the counterterm in the range > M(b) (owing to the fact that for the soft counterterm the function of eq. (A.4) is identically equal to one). As already discussed in section 2.1, Born-like terms do not pose signi cant problems in the presence of resonances. We shall thus ignore this contribution in what follows, and deal solely with eq. (A.3). In MadGraph5 aMC@NLO, (A.1) (A.2) (A.3) (A.4) the parametrisation of Born-level integration variables in the context of an NLO computation is identical to that adopted for a tree-level computation of the same multiplicity (see ref. [52]). This implies that, by construction, one of the variables b will coincide with the virtuality of the resonance , computed with the counterevent kinematics. Let us denote this variable by b , and all of the other integration variables collectively by bA: By construction, one has: whereas, owing to eqs. (A.1) and (A.2): b = fb ; bAg : b = k2 (b; 0) k2 (b ; bA; 0) ; b 6= k2 (b ; bA; ) : Equation (A.7) suggests that k2 (b ; bA; ), seen as a function of b at xed (bA; ), can be identi ed with of eq. (2.5) for a one-dimensional change of integration variables for a type-IIa solution; its inverse can be used for a type-IIb solution. We have considered both types of approaches, and found that the type-IIa one did not perform in a satisfactory manner from the numerical viewpoint. The reason is the following: with a type-IIa solution, both event and (re-mapped) counterevents have a Breit-Wigner peak at k2 (b ; bA; ) ' m2 . This implies that the integration variable b peaked at m2 , but at a somewhat di erent (typically lower) value. Moreover, and more will not be importantly, the position of such a peak will be correlated with the value of (and, in the actual QCD case where collinear singularities are present, with that of the angle between the FKS parton and its sister). The rst issue renders it di cult to guess analytically an e cient change of variables from the relevant \Vegas x" to b , which implies a longer-thandesired grid optimisation, while the second issue e ectively hampers such an optimisation (since correlations are notoriously di cult to handle in adaptive integrations). For this reason, our solution of choice in MadGraph5 aMC@NLO is a type-IIb one, which we now proceed to describe in greater detail. As discussed in section 2.1, type-IIb solutions entail the manipulation of the event contribution. We single out such contribution in eq. (A.3), which we implicitly and temporarily regularise (e.g. with a cuto ) in order to avoid divergences, and rewrite it as follows: (A.5) (A.6) (A.7) db0 Z Z db0 db0 Z 0 M(b0) d Z 1 d A 0 k2 (b0; ) j b0; A k2 (b0 ; b0 ; ) j b0 ; b0 ; A M(b0 ; b0 ) A ; (A.8) having trivially renamed b as b0. We then perform the following change of integration fb0 ; b0Ag ! fb ; bAg ; A b0 = b ; A b0 = k 2 1 (b ; bA; ) ; (A.9) whence eq. (A.8) becomes: E = Z db dbA 0 M(k2 1 (b ; bA; ); bA) b j k 2 1 (b ; bA; ); bA; : (A.10) The rst argument of in eq. (A.10) shows that the event now has the desired property (thanks to eq. (A.6)), namely that the reconstructed invariant mass of the resonance is equal to that of the counterevent (generated with the same fb ; bAg and not re-mapped), in keeping with the general derivation of type-IIb solutions. A drawback of eq. (A.10) is the possible di culty of computing the jacobian analytically; in MadGraph5 aMC@NLO we bypassed this problem by resorting to entirely numerical methods (employing fast and readily available numerical-derivative routines, such as those of CERNLIB), with excellent performances in terms of stability and accuracy. The re-mapped subtracted cross section can nally be obtained by replacing the event contribution to eq. (A.3) with the r.h.s. of eq. (A.10). Such a form is essentially what is implemented in MadGraph5 aMC@NLO, barring numerically-small contributions due to the following features. The support of k2 (b ; bA; ) is in general di erent w.r.t. that of its inverse. In this case, where either the event or the counterevents are equal to zero, no re-mapping is performed. It may happen that, for certain values of (typically far from zero, and with a massive FKS sister) and close to the borders of the support of k2 (b ; bA; ), such a function is not monotonic. Although one could carry out the procedure outlined above in a piece-wise manner, we have opted for not performing the re-mapping in such a case. Finally, we point out that in our code the integration variable relevant to the FKS soft subtraction is not , but actually its rescaled version ^, de ned so that = ^ M(b). This guarantees a better numerical performance, chie y owing to the complete absence of correlations between ^ and other integration variables. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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R. Frederix, S. Frixione, A.S. Papanastasiou, S. Prestel, P. Torrielli. Off-shell single-top production at NLO matched to parton showers, Journal of High Energy Physics, 2016, 27, DOI: 10.1007/JHEP06(2016)027