Top-pair production at the LHC through NNLO QCD and NLO EW

Journal of High Energy Physics, Oct 2017

Abstract In this work we present for the first time predictions for top-quark pair differential distributions at the LHC at NNLO QCD accuracy and including EW corrections. For the latter we include not only contributions of \( \mathcal{O}\left({\alpha}_s^2\alpha \right) \), but also those of order \( \mathcal{O}\left({\alpha}_s{\alpha}^2\right) \) and \( \mathcal{O}\left({\alpha}^3\right) \). Besides providing phenomenological predictions for all main differential distributions with stable top quarks, we also study the following issues. 1) The effect of the photon PDF on top-pair spectra: we find it to be strongly dependent on the PDF set used — especially for the top p T distribution. 2) The difference between the additive and multiplicative approaches for combining QCD and EW corrections: with our scale choice, we find relatively small differences between the central predictions, but reduced scale dependence within the multiplicative approach. 3) The potential effect from the radiation of heavy bosons on inclusive top-pair spectra: we find it to be, typically, negligible.

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Top-pair production at the LHC through NNLO QCD and NLO EW

Received: May Top-pair production at the LHC through NNLO QCD and NLO EW Michal Czakon 0 1 3 6 7 David Heymes 0 1 3 5 7 Alexander Mitov 0 1 3 5 7 Davide Pagani 0 1 2 3 7 Ioannis Tsinikos 0 1 3 7 Marco Zaro 0 1 3 4 7 Cambridge 0 1 3 7 HE U.K. 0 1 3 7 0 James-Franck-Str. 1, Garching, D-85748 Germany 1 Universite Catholique de Louvain , Chemin du Cyclotron 2, Louvain la Neuve, B-1348 Belgium 2 Technische Universitat Munchen 3 Aachen , D-52056 Germany 4 Sorbonne Universites, UPMC Universite Paris 06 , UMR 7589, LPTHE 5 Cavendish Laboratory, University of Cambridge 6 Institut fur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University 7 Paris , F-75005 France In this work we present for the rst time predictions for top-quark pair differential distributions at the LHC at NNLO QCD accuracy and including EW corrections. For the latter we include not only contributions of O( s2 ), but also those of order O( s 2 and O( 3). Besides providing phenomenological predictions for all main di erential distributions with stable top quarks, we also study the following issues. 1) The e ect of the NLO Computations; QCD Phenomenology - EW photon PDF on top-pair spectra: we nd it to be strongly dependent on the PDF set used | especially for the top pT distribution. 2) The di erence between the additive and multiplicative approaches for combining QCD and EW corrections: with our scale choice, we nd relatively small di erences between the central predictions, but reduced scale dependence within the multiplicative approach. 3) The potential e ect from the radiation of heavy bosons on inclusive top-pair spectra: we nd it to be, typically, negligible. Phenomenological predictions for the LHC at 13 TeV Comparison of two approaches for combining NNLO QCD predictions Multiplicative combination and comparison with the additive one Contributions from heavy boson radiation 1 Introduction 3.1 3.2 and EW corrections Additive combination 4 Impact of the photon PDF 2 3 5 6 Conclusions A Notation 1 Introduction The availability of NNLO QCD predictions for stable top-pair production at the LHC, both for the total cross-section [1{4] with NNLL soft-gluon resummation [5, 6] and for all the main di erential distributions [7, 8], has made it possible to compare Standard Model (SM) theory with LHC data at the few-percent level accuracy. Such a high precision has led, among others, to further scrutiny of the di erences between LHC measurements [9] and the ability of Monte-Carlo event generators to describe hadronic tt production. As a result of these ongoing studies, new MC developments are taking place, such as the incorporation of non-resonant and interference e ects [10, 11], which builds upon previous works that included NLO top decay corrections through- xed order [12{16] and/or showered [17{19] calculations. One of the remaining ways for further improving SM theory predictions is by consistently including the so called Electro-Weak (EW) corrections on top of the NNLO QCD ones. Weak [20{28], QED [29] and EW (weak+QED) [30{34] corrections to top-quark pair production have been known for quite some time, and also EW corrections to the fully o -shell dilepton signature are nowadays available [35]. As it has been documented in the literature, although EW e ects are rather small at the level of total cross-section, they can have a sizeable impact on di erential distributions and also on the top-quark charge asymmetry. The goal of this work is to consistently merge existing NNLO QCD predictions with EW corrections into a single coherent prediction and to study its phenomenological impact. { 1 { ing the photon content of the proton [ 36, 37 ]. As shown in ref. [34], depending on the PDF set, photon-initiated contributions can be numerically signi cant in some regions of phase space.1 If the photon density from the NNPDF3.0QED set [40, 41] is employed, the photon-initiated contribution is large in size and of opposite sign with respect to the Sudakov EW corrections, leading to the almost complete cancellation of the two e ects. Nevertheless, large PDF uncertainties from the photon PDF are still present after this cancellation. On the other hand, theoretical consensus about the correctness of the novel approach introduced in ref. [ 37 ] appears to have emerged by now.2 The PDF set provided with ref. [ 37 ], named LUXQED, includes a photon PDF whose central value and relative uncertainty are both much smaller than in the case of NNPDF3.0QED. Thus, at variance with the NNPDF3.0QED set, neither large cancellation between Sudakov e ects and photon-induced contributions nor large photon PDF uncertainty is present in LUXQEDbased predictions. In order to document the ambiguity arising from the di erences between the photon densities in the available PDF sets, with the exception of section 2, in this work we always give predictions for top-pair di erential distributions at the LHC based on the LUXQED [ 37 ] and NNPDF3.0QED [40, 41] PDF sets. We believe that our ndings will provide a valuable input to future PDF determinations including EW e ects. This paper is organised as follows: section 2 is devoted to the phenomenological study of our combined QCD and EW predictions for the LHC at 13 TeV. The reasons behind some of the choices made in section 2 | like the choice of PDF set and combination approach | are revealed in section 3, where we compare in great detail two approaches for combining NNLO QCD and EW corrections in top-pair di erential distributions. The so-called additive approach is discussed in section 3.1, while the multiplicative one in section 3.2. Section 4 is dedicated to studying the impact of the photon PDF on toppair spectra. In section 5 we provide an estimate of the impact of inclusive Heavy Boson Radiation (HBR), namely the contribution from ttV nal states with V = H; W ; Z. While most of the notation is introduced in the main text some technical details are delegated to appendix A. 2 Phenomenological predictions for the LHC at 13 TeV In this section we present predictions for tt distributions for the LHC at 13 TeV at NNLO QCD accuracy including also EW corrections. We focus on the following distributions: the top-pair invariant mass m(tt), the top/antitop average transverse momentum (pT;avt) and 1This has been studied also in refs. [38, 39] for the case of neutral-current Drell-Yan production. 2The consensus has been also supported by a preliminary study in the determination of the photon PDF including new LHC data [42]. { 2 { rapidity (yavt) and the rapidity y(tt) of the tt system. The pT;avt (yavt) distributions are calculated not on an event-by-event basis but by averaging the results of the histograms for the transverse momentum (rapidity) of the top and the antitop. Our calculation is performed using the following input parameters mt = 173:3 GeV ; mH = 125:09 GeV ; mW = 80:385 GeV ; mZ = 91:1876 GeV ; while all other fermion masses are set to zero. All masses are renormalised on-shell and all decay widths are set to zero. The renormalisation of s is performed in the 5- avour scheme while EW input parameters and the associated renormalisation condition are in refs. [44, 45], and in ref. [46] for the calculation of the complete NLO corrections. We work with dynamical renormalisation ( r) and factorisation ( f ) scales. Their common central value is de ned as = = = mT;t mT;t 2 2 HT = 4 1 4 for the pT;t distribution; for the pT;t distribution; mT;t + mT;t for all other distributions; (2.1) (2.2) (2.3) (2.4) (2.5) where mT;t respectively. qmt2 + p2T;t and mT;t qmt2 + p2T;t are the transverse masses of the top and antitop quarks. As already mentioned, pT;avt and yavt distributions are obtained by averaging the top and antitop distributions for the transverse momentum and rapidity, These scale choices have been motivated and studied at length in ref. [8]. In all cases theoretical uncertainties due to missing higher orders are estimated via the 7-point variation of r and f in the interval f =2; 2 g with 1=2 r= f 2. We remark that the combination of QCD and EW corrections is independently performed for each value of f;r. For theoretical consistency, a set of PDFs including QED e ects in the DGLAP evolution should always be preferred whenever NLO EW corrections are computed. At the moment, the only two NNLO QCD accurate PDF sets that include them are NNPDF3.0QED and LUXQED.3 Both sets have a photon density, which induces additional contributions to tt production [29, 34]. As motivated and discussed at length in section 3, the phenomenological predictions in this section are based on the LUXQED PDF set and on the multiplicative approach for combining QCD and EW corrections, which we will denote as QCD EW. We invite the 3The PDF sets MRST2004QED [47] and CT14QED [48] also include QED e ects in the DGLAP evolution, but they are not NNLO QCD accurate. A PDF set including full SM LO evolution (not only QCD and QED but also weak e ects) has also recently become available [49]. { 3 { 1.2 total unc. QCD×EW PDF unc. scale unc. total unc. QCD×EW PDF unc. scale unc. impact is observed in the two rapidity distributions: the relative e ect for yavt is around 2 permil and is much smaller than the scale uncertainty. The y(tt) distribution is slightly more sensitive, with a relative impact of slightly above 1% for large values of y(tt). This correction is also well within the scale-variation uncertainty band. The impact of EW corrections on the m(tt) distribution is larger. Relative to NNLO QCD it varies between +2% at the absolute threshold and 6% at high energies. Still, this correction is well within the scale variation uncertainty. The small sensitivity of the yavt and y(tt) distributions to EW corrections supports the ndings of ref. [9] where these two distributions were used for constraining the PDF of the gluon. The largest correction due to EW e ects is observed in the pT;avt distribution. Relative to NNLO QCD, the correction ranges from +2% at low pT;avt to 25% at pT;avt 3 TeV. The correction is signi cant and is comparable to the scale variation band already for pT;avt 500 GeV. Overall, the EW contribution to the pT;avt distribution is as large as the total theory uncertainty band in the full kinematic range pT;avt 3 TeV considered in this work. The fraction of the theory uncertainty induced by PDFs is strongly dependent on kinematics. For the yavt and y(tt) distributions, the PDF error is slightly smaller than the scale uncertainty for central rapidities, but is larger in the peripheral region, especially for the y(tt) distribution. The PDF uncertainty becomes the dominant source of theory error in the pT;avt distribution for pT;avt as large as 500 GeV, while for the m(tt) distribution it begins to dominate over the scale uncertainty for m(tt) 2:5 TeV. There are many applications for the results derived in this work. Examples are: inclusion of EW e ects in PDF determinations from LHC tt distributions, high-mass LHC searches, precision SM LHC measurements and benchmarking of LHC event generators. A practical and su ciently accurate procedure for the utilisation of our results could be as follows. One starts by deriving an analytic t for the QCD EW=QCD K-factor (all K-factors in gure 1 are available in electronic form4); as evident from gure 1 it is a very smooth function for all four di erential distributions. Under the assumption that this K-factor is PDF independent, such an analytic t could then be used to rescale the NNLO-QCD-accurate di erential distributions derived with any PDF set from existing NNLO QCD [ 50 ] fastNLO [ 51, 52 ] tables. Regarding the PDF error of NNLO QCD differential distributions, it can be calculated very fast with any PDF set with the help of the fastNLO tables of ref. [ 50 ]. As we show in the following the PDF error of the QCD and combined QCD and EW predictions is almost the same, especially for the LUXQED PDF set used for our phenomenological predictions. 3 Comparison of two approaches for combining NNLO QCD predictions and EW corrections In this work we compare two approaches for combining QCD and EW corrections. For brevity, we will refer to them as additive and multiplicative approaches. As already men4Repository with results and additional plots of NNLO QCD + EW tt di erential distributions: { 5 { tioned, the results presented in section 2 have been calculated using the multiplicative approach. In the additive approach the NNLO QCD predictions (de ned as the complete set of O( sn) terms up to n = 4) are combined with all possible remaining LO and NLO terms arising from QCD and electroweak interactions in the Standard Model. In other words, at LO we include not only the purely QCD O( s2) contribution, but also all O( s ) and O( 2) terms. Similarly, at NLO we take into account not only the NLO QCD O( s3) contribution but also the O( s2 ) one, the so-called NLO EW, as well as the subleading contributions of O( s 2) and O( 3). For brevity, we will denote as \EW corrections" the sum of all LO and NLO terms of the form O( sm n) with n > 0. Moreover, when we will refer to \QCD" results, we will understand predictions at NNLO QCD accuracy. For a generic observable in the additive approach we denote the prediction at this level of accuracy as In the multiplicative approach one assumes complete factorisation of NLO QCD and NLO EW e ects. This approach is presented in section 3.2 and is denoted as The precise de nition of the various quantities mentioned in the text is given in appendix A where an appropriate notation for the classi cation of the di erent contributions is introduced. Here, we just state the most relevant de nitions for the following discussion, QCD+EW. QCD EW. QCD EW QCD+EW QCD EW LO QCD + denotes a generic observable in tt production and KQNCLOD is the standard NLO/LO K-factor in QCD. A variation of the multiplicative approach denoted as QCD2 EW will also be considered; it is de ned similarly to QCD EW in eq. (3.1) but with NNLO/LO QCD K-factor. As it has been discussed in ref. [34], the usage of di erent PDF sets leads to a very di erent impact of the photon-induced contribution on tt distributions. While in the case of NNPDF3.0QED the impact of photon-induced contributions is relatively large and with very large uncertainties, in the case of LUXQED it is expected to be negligible. For this reason in the rest of this work we always show predictions with both PDF sets. 3.1 Additive combination Distributions for pT;avt and m(tt) are shown in gure 2, while the yavt and y(tt) distributions are shown in gure 3. The format of the plots for all distributions is as follows: for each observable, we show two plots side-by-side, with the same layout. The plot on the lefthand side shows predictions obtained using the LUXQED set, while for the one on the right the NNPDF3.0QED set is employed. Results at NNLO QCD accuracy are labelled as \QCD" while the combination of NNLO QCD predictions and EW corrections in the additive approach are labelled as \QCD+EW". { 6 { 1 (QCD+EW)/QCD; scale unc. QCD EW 1 (QCD+EW)/QCD; scale unc. QCD EW 1 (QCD+EW)/QCD; scale unc. QCD 1 (QCD+EW)/QCD; scale unc. QCD In each plot the three insets display ratios of di erent quantities5 over the central-scale QCD result (i.e., in the case of LUXQED, the black line in the main panel of gure 1). In the rst inset we show the scale uncertainty due to EW corrections alone (red band), without QCD contributions ( EW using the notation of appendix A). This quantity can be compared to the scale uncertainty of the QCD prediction at NNLO accuracy (grey band). In the second inset we present the scale-uncertainty band (red) for the combined QCD+EW prediction. The grey band corresponds to the NNLO QCD scale-uncertainty band already shown in the rst inset. The third inset is equivalent to the second one, but it shows the PDF uncertainties. We combine, for each one of the PDF members, the QCD prediction and the EW corrections into the QCD+EW result. The PDF uncertainty band of the QCD+EW prediction is shown in red while the grey band corresponds to the PDF uncertainty of the QCD prediction. For all insets, when the grey band is covered by the red one, its borders are displayed as black dashed lines. As can be seen in gures 2 and 3, the e ect of EW corrections is, in general, within the NNLO QCD scale uncertainty. A notable exception is the case of the pT;avt distribution with LUXQED. In the tail of this distribution the e ect of Sudakov logarithms is large and 5It is actually in all cases the QCD+EW= QCD ratio, but the bands refer to three di erent quantities, as explained in the text. { 7 { 1.05 (QCD+EW)/QCD; scale unc. QCD EW negative, of the order of (10{20%), and is not compensated by the photon-induced contribution. On the contrary, in the case of NNPDF3.0QED, photon-induced contributions mostly compensate the negative corrections due to Sudakov logarithms. As it has already been noted in ref. [34], with this PDF set, the e ect of photon-induced contributions is not negligible also for large values of m(tt), yavt and y(tt). As it can be seen in the rst inset, in the large pT;avt regime the scale dependence of the EW corrections alone is of the same size as, or even larger than, the scale variation at NNLO QCD accuracy. For this reason, as evident from the second inset, the scale uncertainty of the combined QCD+EW prediction is much larger than in the purely QCD case, both with the LUXQED and NNPDF3.0QED PDF sets. This feature is present only in the tail of the pT;avt distribution. The PDF uncertainties (third inset) for all distributions do not exhibit large di erences between QCD and QCD+EW predictions, despite the fact that the photon-induced contribution in NNPDF3.0QED is large and has very large PDF uncertainty (relative to LUXQED). 3.2 Multiplicative combination and comparison with the additive one The additive approach QCD+EW for combining QCD and EW corrections discussed in section 3.1 is exact to the order at which the perturbative expansion of the production cross{ 8 { section is truncated. An alternative possibility for combining QCD and EW corrections is what we already called the multiplicative approach, QCD EW. This approach is designed to approximate the leading EW corrections at higher orders. In the case of tt production these are NNLO EW contributions of order O( s3 ). The multiplicative approach is motivated by the fact that soft QCD and EW Sudakov logarithms factorise, with the latter typically leading to large negative corrections for boosted kinematics. Thus, when dominant NLO EW and NLO QCD corrections are at the same time induced by these two e ects, the desired xed order can be very well approximated via rescaling NLO EW corrections with NLO QCD K-factors.6 Otherwise, if one is in a kinematical regime for which the dominant NLO EW or NLO QCD corrections are of di erent origin (i.e. not Sudakov or soft), the di erence between the multiplicative and additive approaches given by the term mixed in eq. (A.8) can be considered as an indication of theory uncertainty in that kinematics. It must be stressed that the perturbative orders involved in the additive approach are included exactly also in the multiplicative approach; the only addition the multiplicative approach introduces on top of the additive one is the approximated O( s3 ) contribution. One of the advantages of the multiplicative approach is the stabilisation of scale dependence. As we saw in section 3.1, when QCD and EW corrections are combined in the additive approach, the scale dependence at large pT;avt can exceed that of the NNLO QCD prediction. On the other hand, the large pT;avt limit is precisely the kinematic regime where the multiplicative approach is a good approximation and can be trusted: at large pT;avt the NLO EW and NLO QCD corrections are mainly induced by Sudakov logarithms and soft emissions, respectively, and as we just pointed out these two contributions factorise. The presence of large Sudakov logarithms in the NLO EW result at large pT;avt is easy to see since for Born kinematics large pT;t implies large pT;t which, in turn, implies large s^; t^ and u^ Mandelstam variables. That NLO QCD corrections at large pT;avt are mainly of soft origin can be shown with an explicit NLO calculation; by applying appropriate cuts on the jet, the top and/or the antitop one can easily see that the di erential cross-section is dominated by kinematic con gurations containing almost back-to-back hard top and antitop and a jet with small pT . Plots demonstrating this can be found in footnote 4. In the following, for all observables considered in this work, we present predictions in the multiplicative approach denoted as QCD EW. As a further check of the stability of the multiplicative approach we display also the quantity QCD2 EW, whose precise de nition can be found in appendix A. QCD2 EW is de ned analogously to QCD EW, but by rescaling NLO EW corrections via NNLO QCD K-factors. By comparing QCD EW and QCD2 EW one can further estimate the uncertainty due to mixed QCD-EW higher orders. Figure 4 shows the pT;avt and m(tt) distributions, while gure 5 refers to yavt and y(tt). As in section 3.1, the plots on the left are produced using the LUXQED PDF set, while those on the right using the NNPDF3.0QED PDF set. We next describe the format of the plots. Each plot consists of ve insets, which all show ratios of di erent quantities over the central value of QCD. In the rst inset we compare the central-scale results for the 6The precise de nitions of QCD EW is given in eq. (A.13). { 9 { (QCD×EW)/QCD (QCD2×EW)/QCD 1.2 (QCD+EW)/QCD (QCD×EW)/QCD (QCD2×EW)/QCD di erential distributions at 13 TeV. The format of the plots is described in the text. three alternative predictions: QCD+EW= QCD (red line), QCD EW= QCD (green line) and QCD2 EW= QCD (violet line). These quantities are further displayed in the second, third and fourth inset, respectively, where not only the central value but also the scale dependence of the numerator is shown. In all cases we calculate the scale-uncertainty band as a scale-by-scale combination and subsequent variation in the 7-point approach. Scale variation bands have the same colour as the corresponding central-value line. For comparison we also display (grey band) the relative scale uncertainty of QCD. Thus, the second inset is exactly the same as the second inset in the corresponding plots in section 3.1. The last inset shows a comparison of the ratio QCD+EW= QCD including (red line) or not (orange line) the contribution res, where \res" stands for residual and denotes the fact that res are contributions to EW that are expected to be small, regardless of the PDF set used (see eq. (A.6)). 1.5 (QCD×EW)/QCD; scale unc. QCD (QCD+EW)/QCD (QCD×EW)/QCD (QCD2×EW)/QCD (QCD+EW)/QCD (QCD×EW)/QCD (QCD2×EW)/QCD As expected, the multiplicative approach shows much smaller dependence on the scale variation. This is particularly relevant for the tail of the pT;avt distribution, where the scale uncertainty of EW alone is comparable in size with the one of QCD; with this reduction of the scale uncertainty the QCD EW and QCD uncertainty bands do not overlap when LUXQED is used. In the case of m(tt) and yavt distributions, the QCD EW central-value predictions are typically larger in absolute value than those of QCD+EW, while they are all almost of the same size for the y(tt) distribution. In the case of yavt the di erence between the additive and multiplicative approaches is completely negligible compared to their scale uncertainty. Therefore, besides the kinematic region where Sudakov e ects are the dominant contribution, the multiplicative and additive approaches are equivalent. Moreover, the di erence between QCD EW and QCD2 EW is in general small; a sizeable di erence between their scale dependences can be noted only in the tail of the pT;avt distribution. For all the reasons mentioned above we believe that the multiplicative approach should be preferred over the additive one and, indeed, it has been used for the calculation of our best predictions in section 2. As can be seen from gures 4 and 5 and their thresholdzoomed-in versions in footnote 4 the di erence between QCD+EW and QCD EW for nonboosted kinematics is much smaller than the total theory uncertainty (scale+PDF) shown in gure 1. Thus, the di erence between the two approaches can be safely ignored in the estimation of the theory uncertainty. One should bear in mind that this conclusion depends on the choice of scale, which in our case, as explained in ref. [8], is based on the principle of fastest convergence. A di erent scale choice with larger K factors would likely arti cially enhance the di erence between QCD+EW and QCD EW. In the last inset in gures 4 and 5 we compare the quantities EW and EW res, where the res contribution is exactly included in both the additive and multiplicative approaches. As expected, one can see that the res contribution is typically at and very small. The only exception is the m(tt) distribution where a visible di erence between the two curves ( EW and EW res) is present, especially in the tail. The res contribution includes the squared EW tree-level diagrams, the O( 2) contribution denoted as in (A.2), and the two subleading NLO corrections of respectively O( s 2) and O( 3), LO;3 denoted as negligible, the O( 2) and O( s 2) ones both lead to positive non-negligible contributions of similar size to the m(tt) distribution. Indeed, the O( 2) contribution involves bb ! tt squared diagrams with a W boson in the t-channel, which at large m(tt) are not as much suppressed as the contributions from the other initial states. Similarly, the O( s 2 ) contributions contain QCD corrections to them, featuring the same beahaviour. Relevant plots displaying individually all the aforementioned contributions can be found at 4. 4 Impact of the photon PDF In this section we quantify the impact on tt di erential distributions of the di erence between the photon densities provided by the LUXQED and NNPDF3.0QED PDF sets. In other words, we repeat the study performed in ref. [34] for these two PDF sets since they were not considered in that work. We compare the size of the electroweak corrections with and without the photon PDF for both PDF sets. In each plot of gure 6 we show the relative impact induced by the electroweak corrections (the ratio EW= QCD; see de nitions in appendix A) for four cases: NNPDF3.0QED setting the photon PDF equal to zero (red) or not (green), and LUXQED setting the photon PDF equal to zero (violet) or not (blue). For the cases including the photon PDF, we also show the PDF-uncertainty band of The impact of the photon-induced contribution can be evaluated via the di erence between the green and red lines in the case of NNPDF3.0QED and the di erence between the blue and violet lines in the case of LUXQED. As can be seen in gure 6, the impact of the photon PDF on the pT;avt, m(tt), yavt and y(tt) distributions is negligible in the case of LUXQED, while it is large and with very large uncertainties for the case of NNPDF3.0QED, as already pointed out in ref. [34] for NNPDF2.3QED. At very large pT;avt and m(tt) also HJEP10(27)86 yavt at 13 TeV. The format of the plots is described in the text. LUXQED show a non-negligible relative PDF uncertainty, which is not induced by the photon but from the PDFs of the coloured partons at large x. We checked that a similar behaviour is exhibited also by NNPDF3.0QED when its photon PDF is set to zero. 5 Contributions from heavy boson radiation In the calculation of EW corrections to QCD processes the inclusion of real emissions of massive gauge bosons (heavy boson radiation or HBR) is not mandatory since, due to the nite mass of the gauge bosons, real and virtual weak corrections are separately nite (albeit the virtual corrections are enhanced by large Sudakov logarithms). Furthermore, such emissions are typically resolved in experimental analyses and are generally considered as a di erent process ttV (+X) with V = H; W ; Z. For these reasons, the results in section 2 do not include HBR contributions. It is, nonetheless, interesting to estimate the contribution of HBR to inclusive tt production. Our motivation is threefold: rst, resolved or not, HBR is a legitimate contribution to the tt(+X) nal state considered in this work. Secondly, it is clear that one cannot guarantee that HBR is resolved with 100% e ciency. Therefore, it is mandatory to have a prior estimate for the size of the e ect. Finally, we are unaware of prior works where the HBR contribution has been estimated in inclusive tt production. Recently, refs. [44, 53] have provided estimates for HBR in the processes ttV (+X), with V = H; Z; W . We have investigated the impact of HBR on all four distributions considered in this work: pT;avt, m(tt), yavt and y(tt). Our results are shown in gure 7, where we plot the e ect of HBR on the central scale normalised to the QCD prediction. We show separately the LO HBR e ect of order O( s2 ) as well as the NLO QCD HBR prediction which includes terms of order O( s3 ). As a reference we also show the EW corrections for tt. In our calculations we include HBR due to H; W and Z. We are fully inclusive in HBR, i.e., no cuts on the emitted heavy bosons are applied. Clearly, any realistic experi0.05 0 yavt 0 y(t­t) 1 1 0.1 EW/QCD HBR(αs2α)/QCD HBR(αs2α+αs3α)/QCD 0.1 EW/QCD HBR(αs2α)/QCD HBR(αs2α+αs3α)/QCD 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 yavt 0 y(t­t) 1 1 2 2 0 −0.1 −0.2 distributions at 13 TeV. The format of the plots is described in the text. mental analysis will require an estimate of HBR subject to experimental cuts, but such an investigation would be well outside the scope of the present work. From gure 7 we conclude that the e ect of HBR is generally much smaller than the EW corrections. In particular, higher-order QCD corrections to HBR are completely negligible, i.e. HBR is well described in LO for all the tt inclusive distributions and for the full kinematic ranges considered here. The absolute e ect of HBR on the pT;avt distribution is positive and small; it never exceeds 2{3% (relative to the tt prediction at NNLO QCD accuracy) and is always much smaller than the EW correction. The only distribution where the HBR contribution is not negligible compared to the EW one is m(tt) computed with LUXQED. For this distribution the HBR correction is positive and only about half the absolute size of the (negative) EW correction. Still, the absolute size of the HBR, relative to the prediction at NNLO QCD accuracy, is within 1% and so its phenomenological relevance is unclear. The impact of HBR on the two rapidity distributions is tiny, typically within 3 permil of the NNLO QCD prediction. { 14 { In this work we derive for the rst time predictions for all main top-quark pair di erential distributions7 with stable top quarks at the LHC at NNLO QCD accuracy and including the following EW corrections: the NLO EW e ects of O( s2 ), all subleading NLO terms of order O( s 2) and O( 3) as well as the LO contributions of order O( s ) and O( 2). We present a detailed analysis of top-pair production at the LHC at 13 TeV and we nd that the e ect of EW corrections on di erential distributions with stable top quarks is in general within the current total (scale+PDF) theory uncertainty. A notable exception is the pT;avt distribution in the boosted regime where the e ect of EW corrections is signi cant with respect to the current total theory error. We have checked that similar conclusions apply also for LHC at 8 TeV. All results derived in this work in the multiplicative approach, for both 8 and 13 TeV, are available in electronic form 4 as well as with the ArXiv submission of this paper. Providing phenomenological predictions for the LHC is only one of the motivations for the present study. In this work we also quantify the impact of the photon PDF on top-pair di erential distributions and study the di erence between the additive and multiplicative approaches for combining QCD and EW corrections. Moreover, we analyse the contribution from inclusive Heavy Boson Radiation on inclusive top-pair di erential distributions. In order to quantify the impact of the photon PDF, we use two recent PDF sets whose photon components are constructed within very di erent approaches. The rst set, LUXQED, is based on the PDF4LHC15 set [54] and adds to it a photon contribution that is derived from the structure function approach of ref. [ 37 ]. The second set, NNPDF3.0QED, is based on the NNPDF3.0 family of PDFs and adds a photon component that is extracted from a t to collider data. NNPDF3.0QED photon density has both a much larger central value and PDF uncertainty than those of LUXQED. On the other hand, the two sets are compatible within PDF errors and they both include QED e ects in the DGLAP evolution on top of the usual NNLO QCD evolution. We con rm the observations already made in ref. [34], namely, the way the photon PDF is included impacts all di erential distributions. The size of this impact is di erent for the various distributions; the most signi cant impact can be observed in the pT;avt distribution at moderate and large pT where the net e ect from EW corrections based on NNPDF3.0QED is rather small and with large PDF uncertainties, while using LUXQED it is negative, with small PDF uncertainties and comparable to the size of the NNLO QCD scale error. The m(tt) distribution displays even larger e ects, but only at extremely high m(tt). The y(tt) distribution is also a ected at large y(tt) values.8 It seems to us that a consensus is emerging around the structure-function approach of ref. [ 37 ]. Given its appealing predictiveness, this approach will likely be utilised in the 7One distribution we do not consider is pT;tt which is not known in NNLO QCD, and for which resummation is mandatory in order to have reliable predictions. are even more pronounced at 8 TeV. 8As it has been lengthly motivated and discussed in ref. [34], e ects due to the photon PDF a la NNPDF future in other PDF sets. Therefore, at present, it seems to us that as far as the photon PDF is concerned predictions based on the LUXQED set should be preferred. Our best predictions in this work are based on the so-called multiplicative approach for combining QCD and EW corrections. We have also presented predictions based on the standard additive approach. In general, we nd that the di erence between the two approaches is small and well within the scale uncertainty band. The di erence between the two approaches is more pronounced for the m(tt) and pT;avt distributions. Nevertheless, both approaches agree within the scale variation. The scale uncertainty is smaller within the multiplicative approach and, especially in the case of the pT;avt distribution, does not overlap with the NNLO QCD uncertainty band. We stress that these features may be sensitive to the choice of factorisation and renormalisation scales. Since we are unaware of a past study of Heavy Boson Radiation (i.e. H; W and Z) in inclusive tt production, for completeness, we have also presented the impact of inclusive HBR on the inclusive top-pair di erential spectrum. While it is often assumed that additional HBR emissions can be removed in the measurements, it is nevertheless instructive to consider the contribution of such nal states. We nd that, typically, the HBR contribution is negligible, except for the m(tt) distribution, where it tends to partially o set the EW correction (when computed with LUXQED). We have also checked that NLO QCD corrections to the LO HBR result are negligible for all inclusive tt distributions considered by us. Acknowledgments We thank Stefano Frixione for his suggestions and interest at the early stage of this project. D.P., I.T. and M.Z. acknowledge also Fabio Maltoni for his strong encouragement and support in pursuing this study. The work of M.C. is supported in part by grants of the DFG and BMBF. The work of D.H. and A.M. is supported by the U.K. STFC grants ST/L002760/1 and ST/K004883/1. A.M. is also supported by the European Research Council Consolidator Grant \NNLOforLHC2". The work of D.P is partially supported by the ERC grant 291377 \LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier" and by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja Award Project \Event Simulation for the Large Hadron Collider at High Precision". The work of I.T. is supported by the F.R.S.-FNRS \Fonds de la Recherche Scienti que" (Belgium) and in part by the Belgian Federal Science Policy O ce through the Interuniversity Attraction Pole P7/37. The work of M.Z. is supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodovska-Curie grant agreement No. 660171 and in part by the ILP LABEX (ANR-10-LABX-63), in turn supported by French state funds managed by the ANR within the \Investissements d'Avenir" programme under reference ANR-11-IDEX-0004-02. A Notation In this appendix we specify how EW corrections and NNLO QCD results are combined in the additive and multiplicative approaches. The notation matches the one introduced in [53]. The phenomenology of tt production within the additive approach is presented in section 3.1. The multiplicative approach is studied in section 3.2 where it is also compared to the additive one. A generic observable tt in the process pp ! tt(+X) can be expanded simultaneously in the QCD and EW coupling constants as: X In order to simplify the notation, we further de ne the following purely QCD quantities and those involving EW corrections res EW LO;3 + NLO;3 + NLO;4 ; Throughout this work with the term \EW corrections" we refer to the quantity while the term \NLO EW corrections" will only refer to NLO EW. In the additive approach, which is presented in section 3.1, QCD and electroweak corrections are combined through the linear combination (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) EW, (A.7) (A.8) The so called \multiplicative approach", which has been discussed in section 3.1, is precisely de ned in the following. The purpose of the multiplicative approach is to estimate the size of NNLO;2, which for convenience we rename mixed and assuming complete factorisation of NLO QCD and NLO EW e ects we estimate as mixed NNLO;2 NLO QCD NLO EW : LO QCD In the regime where NLO QCD corrections are dominated by soft interactions and NLO EW by Sudakov logarithms, eq. (A.8) is a very good approximation, since the two e ects factorise and are dominant. In other regimes mixed can be used as an estimate of the leading missing mixed QCD{EW higher orders. The advantage of the inclusion of mixed is the stabilisation of the scale dependence of the term NLO EW, which in tt production has almost9 the same functional form of LO QCD. To this end we de ne the multiplicative approach as res res res 1) NLO EW where we used the standard K-factors KQNCLOD In order to test the stability of the multiplicative approach under even higher mixed QCD-EW orders, we combine NNLO QCD corrections and NLO EW corrections in order to estimate, besides the mixed term, also NNNLO contributions of order s4 . For this purpose we de ne the quantity QCD2 EW KENWLO QCD + res = KQNCNDLO ( LO QCD + = = QCD + KQNCNDLO QCD+EW + (KQNCNDLO NLO EW) + res res where we introduced the K-factor KQNCNDLO QCD LO QCD : Finally, we brie y describe how the dependence on the photon PDF enters the di erent perturbative orders. At LO and NLO accuracy, all contributions, with the exception of to LO QCD and NLO QCD, depend on the photon PDF. The dominant photon-induced process is g ! tt, which contributes to LO EW and, via QCD corrections to this order, NLO EW. In addition, NLO EW, but also NLO;4, receive contributions from the q also the ! ttq and q ! ttq processes. Moreover, in the case of LO;3 and NLO;4, initial state contributes. As already discussed in ref. [34], almost all of the photon-induced contribution arises form LO EW. In this work, at variance with ref. [34], we also include the term res in our calculations. However, since the size of res is in general small, the previous argument still applies. The numerical impact of res is discussed in section 3.2. 9We say \almost" because this order receives also QCD corrections to the LO EW contributions from the g and bb initial states. Besides these e ects NLO EW( 2) = NLO EW( 1) LLOO QQCCDD(( 12)) . 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Michał Czakon, David Heymes, Alexander Mitov, Davide Pagani, Ioannis Tsinikos, Marco Zaro. Top-pair production at the LHC through NNLO QCD and NLO EW, Journal of High Energy Physics, 2017, 186, DOI: 10.1007/JHEP10(2017)186