QCD radiation effects on the signal at the LHC
The discovery of a Standard Model Higgs boson is possible when experimental cuts are applied which increase the ratio of signal and background cross-sections. In this paper we study the signal cross-section at the LHC which requires a selection of Higgs bosons with small transverse momentum. We compare predictions for the efficiency of the experimental cuts from a NNLO QCD calculation, a calculation of the resummation of logarithms in the transverse momentum of the Higgs boson at NNLL, and the event generator MC@NLO. We also investigate the impact of various jet-algorithms, the underlying event and hadronization on the signal cross-section.
Physical processes at collider experiments can be simulated using flexible event generators such as PYTHIA  and HERWIG . In this approach, the momenta of partons in hard scattering processes are distributed among hadrons using an approximate probabilistic algorithm for parton branching and hadronization. Traditionally, event generators compute the hard scattering partonic cross-section at leading order in fixed order perturbation theory which yields only a rough estimate.
Cross-sections for the hard interaction of three or four particles can be computed routinely through next-to-leading order (NLO) in the expansion. The results from NLO QCD calculations and parton shower event generators are often combined with an empiric method. First the efficiency of the experimental cuts and normalized differential distributions are computed with parton shower event generators. Then, they are multiplied with the result for the total cross-section from the NLO calculation 111Note that differential “-factors” to reproduce bin integrated differential distributions at higher orders in perturbation theory are also used [3, 4].. However, it is now understood how to combine parton shower generators and NLO results with theoretically sound methods so that (i) leading logarithms are resummed with the parton shower, and (ii) all differential cross-sections are exactly accurate through NLO upon an expansion of the Sudakov factors in [5, 6].
Fixed order perturbative computations have now advanced beyond the next to leading order, and there are two hadron collider processes which are known through next-to-next-to-leading order (NNLO) in . The NNLO total cross-section of the Drell-Yan process [7, 8] is the most precise theoretical prediction for a hadron collider observable with a scale variation uncertainty of about . The total cross-section for Higgs boson production is also known at NNLO [8, 9, 10, 11, 12]. For the LHC, the NLO [13, 14] and NNLO corrections are both important and increase the LO result by about and respectively. The perturbative series converges slowly and the remaining theoretical uncertainty is about [15, 8, 9, 10, 11, 12].
In Higgs boson [16, 17, 18] and electroweak gauge boson [19, 20, 21, 22] production there exist novel differential cross-section calculations at NNLO [16, 17, 19, 20, 21, 22, 18] 222In non-hadronic collisions the state of the art at NNLO is fully differential cross-sections for [23, 24, 25].. The cross-sections with arbitrary experimental cuts applied at the parton level can be computed exactly at this order in pertubration theory for the two processes. It is very instructive to compare the efficiencies of experimental cuts from the newly available fully differential NNLO calculations, merged NLO and parton-shower calculations with MC@NLO, and simple leading order event generators. This is valuable in order to estimate the inherent theoretical uncertainties of the above approaches.
Such a comparison can be made for the Drell-Yan process with the results from Refs. [21, 22, 26], as well as discussed in Ref. . MC@NLO and the NNLO Monte-Carlo FEWZ [21, 22] predict very similar experimental efficiencies for the entire kinematic range where the NNLO prediction retains its phenomenal scale variation of about . Significant differences, however, arise when the experimental cuts suppress contributions from the two-loop matrix elements.
A similar comparison  between MC@NLO and the NNLO partonic Monte-Carlo FEHiP has been made for the Higgs boson diphoton signal at the LHC. The signal cross-section is known with a scale uncertainty of about at NNLO . Besides a relatively large perturbative correction from NLO to NNLO of about , the efficiency of the experimental cuts turns out to be very similar in MC@NLO and NNLO .
A challenging channel for the Higgs boson discovery at the LHC is . This channel is contaminated by background processes, and , with much larger cross-sections. For a Higgs boson with a mass close to the W-pair threshold, the cross-sections for all other discovery signals are suppressed. In this case, the process becomes the only viable channel for the Higgs boson to be discovered at the LHC. It is thus indispensable to achieve a very good signal to background (S/B) ratio. An optimized selection of W-pair events  is then required. If appropriate cuts are applied, as in Refs. [29, 30, 31, 32], a discovery of a Standard Model Higgs boson with a mass close to the threshold should be achieved with an integrated luminosity of a few at the LHC.
The decay mode was recently implemented in the NNLO Monte-Carlo FEHiP  and a calculation of the cross-section with the experimental cuts of Ref. [29, 30] was performed in Ref. . The very good agreement of the MC@NLO and NNLO calculations for the efficiency of the experimental cuts in the diphoton signal [27, 17, 5] does not guarantee that a similarly good agreement will be found in the channel. In this paper, we will compare the NNLO predictions from Ref.  with resummation calculations and event generators.
In the channel, the experimental cuts select events “democratically”, irrespectively of the transverse momentum of the jets associated with the Higgs boson production. The event selection in the channel [30, 31] imposes an explicit jet-veto and other cuts which reject events with large transverse momentum for the Higgs boson. This may turn out to be problematic for an agreement in experimental efficiencies between MC@NLO and NNLO for two reasons. First, the NNLO/NLO -factor is sensitive to the cutoffs imposed on the [34, 16, 17]; this effect is treated only in the parton shower approximation in MC@NLO. Second, by selecting events with low , multiple gluon radiation effects which are not included in the fixed order NNLO calculation may be important.
The resummation through next-to-next-to-leading logarithms (NNLL) of is now achieved [35, 36, 37]. The resummed spectrum, after matching to fixed order, integrates to the NNLO total cross-section. This calculation takes into account consistently both multiple gluon radiation effects at low transverse momentum and fixed order high transverse momentum contributions.
A theoretical prediction for the signal cross-section cannot be made directly from the spectrum, since the experimental cuts restrict many phase-space variables. The cross-section must therefore be computed using fully differential Monte-Carlo programs. However, we will use the resummation calculation of the spectrum [38, 36, 37] to validate fixed order Monte-Carlo’s and parton shower event generators in the low kinematic region, which is favored by the experimental selection cuts.
We first compare in Section 2 the theoretical calculations from the NNLL resummation , MC@NLO  and NNLO  for the spectrum. Then we compare in Section 3 the MC@NLO , HERWIG  and NNLO  Monte-Carlo predictions for kinematic distributions of variables which are restricted by the experimental selection.333In the case of MC@NLO and HERWIG, we use a modified version of HERWIG with the correct decay angular correlations. We perform a similar comparison for the signal cross-sections when all cuts are applied. We find a good agreement for the efficiencies of experimental cuts and normalized kinematic distributions. This gives us confidence that the selection of events for the signal cross-section does not invalidate the approximations in the used Monte-Carlo programs.
In Section 4 we study the sensitivity of the signal cross-section with all experimental cuts applied to the choice of jet algorithm. We find a mild change on the cross-section () by using the SISCone or the -algorithm. We also study the effect of hadronization (using the model in HERWIG ) and of the underlying event (using the JIMMY model ). We find that proposed experimental cuts render the cross-section sensitive to these effects at the level of up to . The changes to the cross-section are with opposite signs and the combined effect is rather mild.
2 Integrating over the transverse momentum of the Higgs boson
In this Section we compare fixed order  and parton shower calculations [2, 5] against the theoretical predictions for the transverse momentum spectrum of the Higgs boson from resummation . Fixed order perturbation theory is invalid for small values of . Nevertheless, observables can be reliably computed if their definition allows for an integration over a sufficiently large range in values. Here, we study the cumulative cross-section
This observable mimics the effect of selection cuts with a cutoff on the maximum Higgs transverse momentum. Since in the transverse plane the Higgs boson balances the associated jet radiation, the cross-section with a jet-veto is similar to the cross-section with a veto on high values of .
In Fig. 1 we plot the cross-section in the fixed NLO and NNLO expansion. As in all plots of this paper, we show the scale variation in the interval . The fixed order cross-sections at NLO and NNLO become negative and tend to infinity when the cutoff is small (below ). For such small values of perturbation theory breaks down. We observe that for larger cutoffs (above and up to ) the NLO and NNLO results are in very good agreement. The NNLO cross-section increases however faster than NLO with higher cutoffs leading to the known by larger NNLO result with respect to NLO for the total cross-section [8, 9, 10]. The analyses in Ref. [29, 30] show that a better S/B ratio is achieved if stricter than 40 cutoffs are used for the jet transverse momenta; for cutoff values in between and we observe large perturbative corrections.
In Fig. 2 we compare the fixed NLO result with the resummed NLL spectrum from Ref.  and with MC@NLO  without hadronization and underlying event. We find that for the parton shower or the NLL resummation change significantly the integrated NLO distribution. All results converge for higher and higher cutoffs and agree with each other for the fully inclusive cross-section. Notably, the MC@NLO and the NLL resummation are in a rather good agreement with each other within the uncertainties from scale variation.
Differences between cross-section predictions with fixed order perturbation theory and resummation are expected to become smaller when the fixed order calculations are extended to higher orders.
In  it was found that the average of the Higgs boson, when discovery selection cuts are applied, can be as low as . This leads to the question whether the NNLO result, unlike the NLO result, is a reliable prediction for such small values of the average . In Fig. 3 we compare the integrated distribution at NNLO against the resummed NNLL spectrum. We find a very good agreement between the two approaches for surprisingly low values of . We conclude that higher than NNLO perturbative contributions, which are accounted for with the resummation, remain small for the integrated spectrum, when the maximum Higgs transverse momentum is restricted even down to .
We have now validated the NNLO perturbative calculation  in a challenging case for fixed order perturbation theory which similarly emerges, due to the jet-veto and other cuts favoring small values, in the search for a Higgs boson in the decay channel. However, the simulation of processes at NNLO is only performed at the parton level. We would like to investigate whether parton shower Monte-Carlo programs, which can also model non-perturbative effects and are computationally more flexible than NNLO Monte-Carlo’s, provide realistic estimates of the signal cross-section.
We first discuss the problem of the normalization of the event generators. Parton shower Monte-Carlo programs predict the same total cross-section as the cross-section for their encoded partonic hard scattering at fixed order in perturbation theory. Therefore, HERWIG predicts the Higgs boson total cross-section with LO accuracy (underestimating it by a factor of ) and MC@NLO provides NLO precision (underestimating the total cross-section by a factor of ). A matching of parton showers to NNLO fixed order calculations is not yet developed. Following a practical approach, we will validate whether the efficiency of experimental cuts and normalized differential distributions are in agreement with the NNLO calculations of Ref. . We will then rescale the predictions of the MC@NLO and HERWIG event generators with a global -factor in order to reproduce the fixed order result for the total cross-section. We will denote that the results of the Monte-Carlo have been multiplied with a -factor using the notation .
Now we will test how well event generators agree with resummation results for the spectrum. In Fig. 4 we compare the integrated spectrum of MC@NLO and HERWIG against the resummed NNLL prediction. We observe that both generators are in very good agreement with the NNLL spectrum. This is especially surprising for HERWIG which aims to describe the salient physics features of the process. Note, however, that MC@NLO gives slightly larger and HERWIG slightly smaller values than the NNLL resummation .
Before we conclude our analysis of the integrated distribution we wish to comment further on the scale variation of the fixed order results. In Fig. 1 we find a with no scale variation. A similar behavior is also observed for the accepted cross-section with all experimental cuts [29, 30] in Ref. . It is therefore necessary to investigate in better detail the variation of the NNLO result. We show the value of the cross-section at NNLO for , varying independently the renormalization and factorization scales (the errors correspond to the numerical integration):
The cross-section is more sensitive to independent variations of the renormalization and factorization scales; however, this variation is significantly smaller than in the total cross-section. A detailed study of the cross-section when all experimental discovery cuts are applied, which shows a similar scale-variation pattern, has been made in Ref. .
Summarizing, in this Section we found that the integrated spectrum is predicted reliably at NNLO for the kinematic range of which is relevant in the search . On the contrary, the NLO fixed order calculation is unreliable. We have also established that for the same observable, MC@NLO and HERWIG are in a very good agreement with the resummed NNLL spectrum when they are normalized to a common NNLO total cross-section.
3 Kinematic distributions and signal cross-section
The main backgrounds for the process are and . These backgrounds are sufficiently suppressed to allow for the discovery of a Higgs boson with a combination of experimental cuts [29, 30], exploiting the spin-correlations in the decay of the Higgs boson and the high average transverse momentum of jets in top-pair events. In Ref. , the signal cross-section with these cuts was computed at NNLO. In this section we will compare the NNLO results of Ref.  with MC@NLO. The public version of MC@NLO includes only partial spin-correlations for the decay. Here, the full spin-correlations for the decay of the Higgs boson have been implemented in MC@NLO. All the results of this paper correspond to a Standard Model Higgs boson mass of .
, the invariant mass of the charged lepton pair,
, the angle between the two charged leptons in the plane transverse to the beam axis,
, the transverse momentum of the harder lepton, and
, the missing transverse energy.
The four distributions show an excellent agreement for the efficiencies among NNLO and MC@NLO. This is a remarkable result and could not have been easily foreseen; there is a significant change in the shape of the , , and distributions from NLO to NNLO, as seen in Ref. .
A crucial experimental cut for suppressing the top-pair contribution to the background is a jet-veto. We veto events which have a transverse momentum of the leading jet in the central rapidity region () that is larger than . For the jet definition we use here a algorithm with a jet-radius parameter ; later we will also use a cone algorithm (SISCone ). The two algorithms are identical for the LO and NLO calculation where only up to one parton can be present in the final state, if the same jet-radius is used. They differ, however, in the parton shower calculations (MC@NLO and HERWIG) and at NNLO as more partons are generated. In the envisaged experimental analysis a jet-veto with a rather small value of is considered. We will investigate whether the NNLO and MC@NLO predictions are consistent with each other for such small values of the jet-veto.
The good agreement of the integrated distribution between NNLO, MC@NLO and NNLL resummation suggests that a good agreement between MC@NLO and the NNLO cross-sections with a jet-veto may also hold. The jet-veto cross-section should be qualitatively similar to the cross-section with a cutoff on the since at NLO the Higgs transverse momentum corresponds exactly to the transverse momentum of the additional jet. However, the two cuts are not exactly the same and they compare only qualitatively. The jet-veto applies only at central rapidities; in addition, beyond NLO the is not the same variable as the maximum transverse momentum of the jets.
In Fig. 9 we present the cross-section with a jet-veto applied. Indeed, we find a very good agreement between the NNLO result and MC@NLO (rescaled with the appropriate NNLO/NLO -factor for the total cross-section).
In Table 1 we list the cross-section after all signal cuts as described in Ref.  are applied. We have used both the and SISCone algorithm of Ref.  and their implementation from Ref. . The jet radius in the azimuth-rapidity plane was set to and the merging parameter for the SISCone algorithm to 444The merging parameter defines, how much two separate proto-jets need to overlap in order to be merged into one jet.. The two algorithms yield formally identical results for the fixed order calculation through NLO and indistinguishable results at NNLO within our Monte-Carlo integration precision 555We thank Gavin Salam for pointing out to us that the SISCone and algorithms are formally different already at NNLO.. In the first section of the Table we present the results obtained using a fixed-order LO computation and the LO+parton-shower event generator HERWIG. We find a much larger fixed order result than when a parton shower is added. At fixed leading order all events have a Higgs boson with zero transverse momentum, and the jet-veto rejects none of the events. On the contrary, HERWIG generates a large fraction of events with .
In the second section of Table 1 we present the results obtained using a fixed-order NLO computation, the event generator MC@NLO and HERWIG after we have rescaled it with an inclusive NLO/LO factor. While in MC@NLO and fixed order (LO, NLO and NNLO) we can set the renormalization and factorization scales for the hard scattering at will, we use HERWIG at the default scale since this affects the triggering of the hadronization procedure. We then rescale the HERWIG result using a -factor, taking the NLO fixed order result to be at the scale or . The NLO result is quite different from the one obtained with MC@NLO. We can attribute this failure of the NLO computation to the poor modeling of the low region, as is shown in Fig 2.
In the last part of Table 1 we present the results obtained using a fixed-order NNLO computation and the results from MC@NLO and HERWIG, rescaled to the NNLO total cross-section. The NNLO result and the rescaled MC@NLO give consistent results, albeit with different behavior when varying the renormalization and factorization scales. A detailed analysis of the NNLO scale dependence when all cuts are applied can be found in Ref. .
We note that for all the results of this section we use MC@NLO and HERWIG at the parton level, switching off the hadronization and without using a model for the underlying event. We observe that the -algorithm gives larger cross-sections than the SISCone algorithm for MC@NLO and HERWIG; as mentioned before, the results for the two algorithms are very similar at fixed order through NNLO (within our integration precision). Additionally MC@NLO gives slightly smaller values for the cross-sections than HERWIG.
We have established in this section that the efficiency of experimental cuts computed with MC@NLO and HERWIG is similar to the efficiency obtained at NNLO. There are rather dramatic changes in differential distributions when going from NLO to NNLO . It is only at NNLO that the fixed order calculation is consistent with the parton shower efficiency of the experimental cuts. In the following Section we will study the dependence of the cross-section on effects that are not captured by the fixed order NNLO calculation. We will study the effect of hadronization and of the underlying event. We will also investigate further the differences in the cross-section due to the two different jet algorithms.
4 Jet algorithms, hadronization and the underlying event
In this Section we perform a study of the signal cross-section with all cuts applied using MC@NLO. We will analyze the impact of different jet clustering methods, hadronization and the underlying event.
In Fig. 10 we plot the cross-section using MC@NLO as a function of the value for the and the SISCone algorithm with a jet-radius and . The clustering is applied to all final state particles before hadronization.
We find that for small values of the jet-veto parameter the choice of the jet clustering method is more significant. For a jet-veto at the choice of jet-algorithm changes the cross-section by with MC@NLO. A similarly large variation of is observed when we vary the jet-radius from to . For a jet veto value larger than about the sensitivity of the cross-section to the choice of the jet-algorithm or the jet-radius falls below .
We now study the effect of hadronization as it is modeled in HERWIG and of the underlying event as implemented in JIMMY . In Fig. 11 we present the relative difference of the cross-section with respect to the partonic cross-section when the hadronization or/and the underlying event models are switched on. We have used here the SISCone algorithm with a merging parameter and two values for the jet-radius R=0.4 (left) and R=0.7 (right). We apply the signal cuts set to the values which are used in Ref. . We vary, however, the allowed maximum value of . Of interest are values of the jet-veto between 25 and 40 , which are envisaged in the Higgs boson search.
Qualitatively, we anticipate that the hadronization and the underlying event change the partonic cross-section with opposite signs (we refer the reader to the recent analysis in Ref.  for a detailed study). Hadronization reduces the average of (gluonic) jets by roughly . The underlying event increases the jet by roughly at the LHC. The slope of the partonic cross-section with the jet-veto cutoffs (Fig. 9) is large for small values of the jet-veto. The shifts from hadronization and the underlying event can therefore induce significant changes to the cross-section. A jet-veto after hadronization corresponds to a looser effective jet-veto at the parton level. We therefore anticipate the cross-section to increase by switching on the hadronization model. Similarly, we anticipate a decrease of the cross-section due to the underlying event.
The trends can be verified in Fig. 11. A smaller jet-radius increases the impact of hadronization and decreases the impact of the underlying event. The two effects are not linearly additive. However, we find that a cancelation between the two effects, which varies according to the jet-radius, takes place. For a jet veto and a radius , the hadronization shift is about and the underlying event shift is . For a larger radius , the two shifts are and correspondingly.
In Fig. 12 we show the effects of hadronization and the underlying event for the algorithm. We find the same features qualitatively for the two effects as in the SISCone algorithm. We note however, that the -algorithm shows an overall reduced sensitivity.
In this paper we have studied QCD effects for the process . This process is of particular interest at the LHC since it is possible that for a range of mass values of the Higgs boson, this channel is the only viable one for a discovery.
The cross-section with all envisaged experimental cuts applied was computed in an earlier publication  at NNLO in QCD. In this paper we compared these NNLO results with the leading order event generator HERWIG  and the event generator MC@NLO  which performs a matching of HERWIG with NLO fixed order perturbation theory. We found very good agreement in efficiencies of all experimental cuts that are relevant in the search for the Higgs boson. This is rather spectacular given that there are significant corrections in the total cross-section and the shape of kinematic distributions from NLO to NNLO.
The experimental cuts select events with small transverse momentum of the Higgs boson. This is important in order to reduce the selection of events from top-quark production, which is a major background. We have compared a NNLO computation and the result of NNLL resummation  for the cumulative distribution. We found that NNLL resummation does not induce significant corrections with respect to the NNLO calculation for the kinematic range which is favoured by the selection cuts. We have also found that, within the uncertainty from scale variations, MC@NLO and HERWIG are in very good agreement with the NNLL result. On the contrary, fixed order NLO perturbation theory provides a rather poor approximation for the required distributions and efficiencies.
Finally, we investigated the magnitude of effects due to hadronization, the underlying event, and the choice of jet algorithms. For typical choices of parameters and cutoffs in the experimental cuts we find a mild dependence of the cross-section on these effects. In this paper we did not examine a variety of remaining uncertainties, such as uncertainties in the parton densities, which may be relevant at a precision level of 10%. Studies of electroweak corrections can be found in Refs. [43, 44, 45] and of the background-signal interference in Ref. .
The results and comparisons made in this paper provide a firm validation of the fixed order NNLO results and the event generator tools which are available for simulating the process at the LHC.
We thank Michael Dittmar, Stefano Frixione, Massimiliano Grazzini and
Giulia Zanderighi for useful discussions and suggestions.
This research was supported by the Swiss
National Science Foundation under contracts 200021-117873
We thank Gavin Salam for pointing out to us that the SISCone and algorithms are formally different at NNLO.
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