Resummed differential cross sections for top-quark pairs at the LHC
We present state of the art resummation predictions for differential cross sections in top-quark pair production at the LHC. They are derived from a formalism which allows the simultaneous resummation of both soft and small-mass logarithms, which endanger the convergence of fixed-order perturbative series in the boosted regime, where the partonic center-of-mass energy is much larger than the mass to the top quark. We combine such a double resummation at NNLL accuracy with standard soft-gluon resummation at NNLL accuracy and with NLO calculations, so that our results are applicable throughout the whole phase space. We find that the resummation effects on the differential distributions are significant, bringing theoretical predictions into better agreement with experimental data compared to fixed-order calculations. Moreover, such effects are not well described by the NNLO approximation of the resummation formula, especially in the high-energy tails of the distributions, highlighting the importance of all-orders resummation in dedicated studies of boosted top production.
The run of the LHC delivered about of integrated luminosity to both the ATLAS and CMS experiments. Among the many important results coming from these data, the properties of the top-quark have been measured with unprecedented precision. At the same time, theoretical calculations of top-quark related observables have seen significant advancements in the last few years. In particular, very recently the next-to-next-to-leading order (NNLO) QCD corrections to differential cross sections in top-quark pair () production have been calculated Czakon:2015owf (). In CMS-PAS-TOP-15-011 (), the CMS collaboration performed a comprehensive comparison between their measurements Khachatryan:2015oqa () of the differential cross sections and various theoretical predictions, including those from the NNLO calculation and those from Monte Carlo event generators with next-to-leading order (NLO) accuracy matched to parton showers. The overall agreement between theory and data is truly remarkable, which adds to the success of the Standard Model (SM) as an effective description of Nature.
However, a persistent issue in the 8 TeV results is that the transverse momentum () distribution of the top quark is softer in the data than in theoretical predictions, i.e., the experimentally measured differential cross section at high is lower than predictions from event generators or from NLO fixed-order calculations Khachatryan:2015oqa (); Aad:2015hna (). While the NNLO corrections bring the fixed-order predictions into better agreement with the CMS data, as noted in Czakon:2015owf () and CMS-PAS-TOP-15-011 (), there is still some discrepancy in the high- bins where . Given the importance of the production process as a standard candle for validating the SM and as an essential background for new physics searches, it would be disconcerting if this feature were to persist at higher values and with more data. It is therefore important to assess the effects of QCD corrections even beyond NNLO, in order to see whether the gap between theory and data at high can be bridged.
For boosted top-quark pairs with high there are two classes of potentially large contributions. The first is the Sudakov-type double logarithms arising from soft gluon emissions. The second comes from gluons emitted nearly parallel to the top quarks, resulting in large logarithms of the form , where is the top quark mass, and is the transverse mass of the top quark. In Ferroglia:2012ku (), some of the authors of the current work developed a formalism for the simultaneous resummation of both type of logarithms to all orders in the strong coupling constant . In this Letter, we report the first phenomenological applications of that formalism, giving predictions for the top-quark and the invariant mass distributions at the LHC, and comparing with experimental measurements as well as the NNLO calculations when possible. With an eye to the future, we also present predictions for the LHC, where NNLO results are not yet available.
Our main finding is that the higher-order effects contained in our resummation formalism significantly alter the high-energy tails of the and invariant mass distributions, softening that of the distribution but enhancing that of the invariant mass distribution. These effects bring our results into better agreement with the experimental data compared to pure NLO fixed-order calculations. Interestingly, for the case of the distribution, this softening of the spectrum is slightly stronger than the similar effect displayed in recent NNLO results, and leads to a better modeling of the GeV portion of the CMS data Khachatryan:2015oqa (). We comment further on this fact in the conclusions.
Our predictions are based on the factorization and resummation formula derived in Ferroglia:2012ku (). The technical details will be given in a forthcoming article, although the main elements have already been sketched out in Ferroglia:2015ivv (). In the kinematic situation where the top quarks are highly boosted and the events are dominated by soft gluon emissions, the resummed partonic differential cross section in Mellin space can be written as
where for simplicity, we have suppressed some variables in the functional arguments which are unnecessary for the explanations below. In the above formula, is the invariant mass of the pair (which can be related to the of the top quark in the soft limit through a change of variables), is the Mellin moment variable, and with the Euler constant. The soft limit corresponds to in Mellin space. The four coefficient functions , , and encode contributions from four widely-separated energy scales , , and , respectively. The presence of the four scales leads to the two types of large logarithms discussed in the introduction. In correspondence with these four physical scales, there are four unphysical renormalization scales , , and , one for each coefficient function. The philosophy of resummation is to choose the four unphysical scales to be around their corresponding physical scales, so that the four coefficient functions are free of large logarithms and are well-behaved in fixed-order perturbation theory. One can then use renormalization group (RG) equations to evolve these functions to the factorization scale in order to convolute with the parton distribution functions (PDFs) and obtain the hadronic cross sections. The effects of the RG running are encoded in the two evolution factors (for and ) and (for and ), which resum all the large logarithms to all orders in in an exponential form.
At the moment, the four coefficient functions are known to NNLO Ferroglia:2012ku (); Broggio:2014hoa (); Ferroglia:2012uy (), while the two evolution factors are known to next-to-next-to-leading logarithmic (NNLL) accuracy Ferroglia:2012ku (). Such a level of accuracy is usually referred to as NNLL in the literature, and we adopt that nomenclature here. While the formula (II) is only applicable in the boosted soft limit, we can extend its domain of validity by combining it with information from NNLL soft gluon resummation derived in Ahrens:2010zv () (recast into Mellin space) as well as the NLO fixed-order result calculated in NLO () and implemented in MCFM Campbell:2010ff (). The precise matching formula can be found in Ferroglia:2015ivv (). After such a matching procedure, we denote the final accuracy of our predictions, which are valid throughout phase space, as NLO+NNLL.
It would be desirable to match with the recent NNLO results in Czakon:2015owf () to achieve NNLO+NNLL accuracy. However, at the moment NNLO results are only available for fixed (i.e., kinematics-independent) factorization and renormalization scales , whereas for the study of differential distributions over large ranges of phase space we consider it important to follow common practice and use dynamical (i.e., kinematics-dependent) scale choices. Therefore, such an improvement over our result is not currently possible, and we leave it for the future.
In the following we present NLO+NNLL predictions for the and distributions at the LHC. In all our numerics we choose and use MSTW2008NNLO PDFs Martin:2009iq (). For distributions, the default values for the factorization scale and the four renormalization scales are chosen as , , , and . For distributions, the only difference is . We estimate scale uncertainties by varying the five scales around their default values by factors of two and combining the resulting variations of differential cross sections in quadrature; we do not consider uncertainties from PDFs and in this Letter. The hadronic differential cross sections are first evaluated in Mellin space at a given point in phase space, and we then perform the inverse Mellin transform numerically using the Minimal Prescription Catani:1996yz (). This procedure relies on an efficient construction of Mellin-transformed parton luminosities, for which we use methods outlined in Bonvini:2014joa (); Bonvini:2012sh ().
The differential cross sections considered below span several orders of magnitude when going from low to high values of or . In order to better display the relative sizes of various results, we show in the lower panel of each plot the differential cross sections normalized to our default prediction, i.e., the ratio defined by
Fig. 1 compares our NLO+NNLL resummed prediction for the normalized top-quark distribution to the CMS measurement Khachatryan:2015oqa () in the lepton+jet channel at the LHC with a center-of-mass energy . Also shown is the NNLO result from Czakon:2015owf (), which adopted by default the renormalization and factorization scales , and also used a slightly different top-quark mass, . At low , it is clear that both the NLO+NNLL and the NNLO results describe the data fairly well. With the increase of , it appears that the NNLO prediction systematically overestimates the data, although there is still agreement within errors. On the other hand, with the simultaneous resummation of the soft gluon logarithms and the mass logarithms and also with the dynamical scale choices, our NLO+NNLL resummed formula produces a softer spectrum which agrees well with the data.
In Aad:2015hna (), the ATLAS collaboration carried out a measurement of the top-quark spectrum in the highly-boosted region using fat-jet techniques. Although the experimental uncertainty is rather large due to limited statistics, it is interesting to compare it with the theoretical predictions here, since it is expected that the soft and small-mass logarithms become more relevant at higher energies. In Fig. 2 we show such a comparison. The NNLO result for such high values is not yet available, so we compare instead with the NLO result computed using MCFM with MSTW2008NLO PDFs and dynamical renormalization and factorization scales, whose default values are . Scale uncertainties of the NLO results are estimated through variations of by a factor of two around the default value. From the plot one can see that the NLO result calculated in this way does a good job in estimating the residual uncertainty from higher order corrections, as the resummed band lies almost inside the NLO one up to . On the other hand, the inclusion of the higher-order logarithms in the NLO+NNLL result significantly reduces the theoretical uncertainty, which is crucial for future high precision experiments at the LHC.
Our formalism is flexible and can be applied to other differential distributions as well. To demonstrate this fact, in Fig. 3 we show the NLO+NNLL resummed prediction for the top-quark pair invariant mass distribution along with a measurement from the ATLAS collaboration Aad:2015mbv () at the LHC. Since the NNLO result in Czakon:2015owf () for this distribution has an incompatible binning, it is currently not possible to include it in the plot, so we show instead the NLO result computed with the same input as in Fig. 2, but this time with the default scale choice . One can see from the plot that the NLO result with this scale choice is consistently lower than the experimental data. The resummation effects significantly enhance the differential cross sections, especially at high . As a result, the NLO+NNLL prediction agrees with data quite well. We have found that choosing the default renormalization and factorization scales to be half the invariant mass increases the fixed-order cross section and therefore mimics to some extent the resummation effects. In fact, this procedure has been extensively employed in the literature for processes such as Higgs production Anastasiou:2015ema (), where higher-order corrections are also large. Consequently, it may be advisable to employ a renormalization and factorization scale of the order of in fixed-order calculations (and Monte Carlo event generators), and we shall use this choice when studying the distribution at the 13 TeV LHC below.
The LHC has started the run in 2015. So far there are only two CMS measurements CMS:2015toa (); CMS:2015bta () of differential cross sections for production, based on just of data. The resulting experimental uncertainties are therefore quite large and it is not yet possible to probe higher or values. Nevertheless, in the near future there will be a large amount of high-energy data, which will enable high-precision measurements of kinematic distributions, also in the boosted regime. In Fig. 4 we show our predictions for the and spectrum up to and , contrasted with the NLO results. Note that for the distribution, we have changed the default to a lower value for the reasons explained above. The plots exhibit similar patterns as observed at , namely that the higher-order resummation effects serve to soften the tail of the distribution but enhance that of the distribution compared to a pure NLO calculation.
As mentioned before, we would like to match our calculations with the NNLO results when they become available in the future. We end this section by discussing the expected effects of such a matching, by estimating the size of resummation corrections beyond NNLO. We do this in Fig. 5, where the relative sizes of the beyond-NNLO corrections generated through the resummation formula are displayed as a function of or with the default scale choices. The exact NNLO results for these scale choices are not yet available, so we show in comparison the relative sizes of the approximate NNLO (aNNLO) corrections obtained by expanding and truncating our resummation formula to that order. More precisely, the blue and black curves in Fig. 5 correspond to
where refers to the approximate NNLO result. The figure clearly shows that corrections beyond NNLO are significant in the tails of the distributions, especially in the case of the distribution.
Iv Conclusions and outlook
In this Letter we have presented new resummation predictions for differential cross sections in production at the LHC. The predictions include the simultaneous resummation to NNLL accuracy of both soft and small-mass logarithms, which endanger the convergence of the fixed-order perturbative series in the boosted regime where the partonic center-of-mass energy is much larger than the mass of the top quark. This resummation is matched with both standard soft-gluon resummation at NNLL accuracy and fixed-order NLO calculations, so that our results are applicable in the whole phase space. Such predictions for differential distributions at the LHC are not only the first to be calculated in Mellin space, but also represent the highest resummation accuracy achieved to date, namely NLO+NNLL. Our results are thus a major step forward in the modeling of high-energy tails of distributions, which is of great importance for new physics searches.
The agreement of NLO+NNLL predictions with data indicates the value of including resummation effects and using dynamical scale settings correlated with or when studying differential distributions. Interestingly, in the case of normalized distribution measured by the CMS collaboration Khachatryan:2015oqa (), the NLO+NNLL calculation produces a slightly softer spectrum than recent NNLO predictions (which use a fixed scale setting where by default), thus achieving a better agreement with the data. However, we emphasize that the optimal use of resummation is to supplement NNLO calculations, not to replace them. With this in mind, we have studied the size of corrections beyond NNLO encoded in our resummation formula, and found that their effects are significant in the high-energy tails of distributions, especially for the invariant mass distribution where they enhance the differential cross section. It will therefore be an essential and informative exercise to produce NNLO+NNLL predictions once NNLO calculations are available with dynamical scale settings.
Acknowledgments: We would like to thank Alexander Mitov for providing us the results of the NNLO calculations in Czakon:2015owf (). We are grateful to Andrea Ferroglia for collaboration on many related works. X. Wang and L. L. Yang are supported in part by the National Natural Science Foundation of China under Grant No. 11575004. D. J. Scott is supported by an STFC Postgraduate Studentship.
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