Top-Yukawa contributions to bbH production at the LHC
We study the production of a Higgs boson in association with bottom quarks () in hadronic collisions at the LHC, including the different contributions stemming from terms proportional to the top-quark Yukawa coupling (), to the bottom-quark one (), and to their interference (). Our results are accurate to next-to-leading order in QCD, employ the four-flavour scheme and the (Born-improved) heavy-top quark approximation. We find that next-to-leading order corrections to the component are sizable, making it the dominant production mechanism for associated production in the Standard Model and increasing its inclusive rate by almost a factor of two. By studying final-state distributions of the various contributions, we identify observables and selection cuts that can be used to select the various components and to improve the experimental sensitivity of production on the bottom-quark Yukawa coupling.
Keywords:QCD Phenomenology, NLO Computations
UniversitÃ© catholique de Louvain, B-1348 Louvain-la-Neuve, BelgiumTheoretical Physics Department, CERN, Geneva, SwitzerlandNikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands
After the discovery of a scalar resonance with a mass of about Aad:2012tfa (); Chatrchyan:2012ufa (), the accurate determination of its couplings to Standard-Model (SM) particles has become one of the major objectives of LHC Run II and beyond. Data collected at the LHC so far supports the hypothesis that this resonance is the scalar boson predicted by the Brout–Englert–Higgs mechanism of electroweak symmetry breaking Englert:1964et (); Higgs:1964pj () as implemented in the SM Weinberg:1967tq (): the Higgs couplings are universally set by the masses of the corresponding particles the Higgs boson interacts with. Global fits of various production and decay modes of the Higgs boson Aad:2015gba (); Aad:2015zhl (); Khachatryan:2016vau (); ATLAS:2018doi (); CMS:2018hhg () constrain its couplings to third-generation fermions and to vector bosons to be within % of the values predicted by the SM. In particular, the recent measurement of Higgs production in association with a top-quark pair Sirunyan:2018hoz (); Aaboud:2018urx () provides the first direct evidence of the coupling between the Higgs boson and the top quark, thereby proving that gluon–gluon Higgs production proceeds predominantly via top-quark loops. The coupling of the Higgs boson to leptons has also been established at the 5 level for some time Aad:2015vsa (); Sirunyan:2017khh (), while the Higgs coupling to bottom quarks has been observed only very recently ATLAS:2018nkp (). By contrast, to date, we have no experimental confirmation that the Higgs boson couples to first-/second-generation fermions, nor about the strength of the Higgs self-interaction.
The ability to probe elementary couplings and to improve the experimental sensitivity strongly relies on precise theoretical predictions for both production and decay. The bottom-quark Yukawa coupling () plays a rather special role in this context: despite having a relatively low coupling strength with respect to the couplings to vector bosons and top quarks, the decay dominates the total decay width in the SM for a Higgs-boson mass of about due to kinematical and phase space effects. The observation of this decay is, however, quite challenging because of large backgrounds generated by QCD, especially in the gluon-fusion production mode atlas2016 (), and has for now only been searched for in vector-boson fusion TheCMSCollaboration2015 (); Aaboud2016 () and Higgsstrahlung Chatrchyan2014 (); atlas2016 (). The latter is the most sensitive channel, yielding a signal strength for the decay branching ratio of ATLAS:2018nkp (). However, since the total Higgs width is dominated by , the corresponding branching ratio has a rather weak dependence on . As a result, the sensitivity of processes involving Higgs decays to bottom quarks on this parameter is in fact rather low.
Studying production modes featuring a coupling is a promising alternative: on the one hand, Higgs production in the SM (inclusive over any particles produced in association) proceeds predominantly via the gluon-fusion process, where the Higgs–gluon coupling is mediated by heavy-quark loops. In particular, bottom-quark loops have a contribution of about % to the inclusive cross section, which can become as large as % for Higgs bosons produced at small transverse momentum Mantler2013 (); Grazzini2013 (); Banfi2014 (); Bagnaschi2012 (); Frederix:2016cnl (); Bagnaschi:2015bop (); Mantler:2015vba (); Harlander:2014uea (). On the other hand, the associated production of a Higgs boson with bottom quarks ( production) provides direct access to the bottom-quark Yukawa coupling already at tree-level Raitio:1978pt (). It yields a cross section comparable to the one of the associated production with top quarks (roughly at ), which is about of the fully-inclusive Higgs-production rate in the SM. Furthermore, the inclusive rate decreases dramatically once conditions on the associated jets are imposed to make it distinguishable from inclusive Higgs-boson production.
The SM picture outlined above might be significantly modified by beyond-SM effects: while a direct observation in the SM is challenging at the LHC, production plays a crucial role in models with modified Higgs sectors. In particular in a generic two Higgs-doublet-model (2HDM), or in a supersymmetric one such as the MSSM, the bottom-quark Yukawa coupling can be significantly increased, promoting to the dominant Higgs production mode Rainwater2002 (); Dittmaier2004 () in many benchmark scenarios, being any of the scalars or pseudo-scalars in such theories. Given that a scalar sector richer than that of the SM has not yet been ruled out experimentally, this is a fact that one must bear in mind, and that constitutes a strong motivation for theoretical studies of scalar-particle production in association with bottom quarks.
The production of final states receives additional contributions the loop-induced gluon-fusion process (proportional to ; being the top-quark Yukawa coupling), which in the SM is of similar size as contributions, but have rarely been studied in the literature. In this paper, we consider Higgs production in association with bottom quarks for all contributions proportional to and at NLO QCD, as well as their interference terms proportional to . The process is particularly interesting also from a theoretical viewpoint in many respects. First, as for all mechanisms that feature bottom quarks at the level of the hard process, there are two schemes applicable to performing the computation. These so-called four-flavour scheme (4FS) and five-flavour scheme (5FS) reflect the issue that arise from different kinematic regimes, where either the mass of the bottom-quark can be considered a hard scale or bottom quarks are treated at the same footing as the other light quarks. Hence, the bottom-quark is considered to be massive in the 4FS, while its mass can be set to zero in the 5FS. The advantages of either scheme in the context of production have been discussed in detail in ref. Wiesemann2015 (). We employ the 4FS throughout this paper, owing to its superior description of differential observables related to final-state bottom quarks and the definition of bottom-flavoured jets, which is particularly striking in fixed-order computations. Another theoretical motivation lies in the nature of the loop-induced gluon-fusion process that leads to the contributions proportional to . Being dominated by kinematical configurations where the Higgs boson recoils against a gluon which splits into a bottom-quark pair, this collider process features the cleanest and most direct access to splittings. Thus, as a bonus, our computation also allows us to study the effect of NLO corrections on such splittings.
Given that the NLO QCD corrections to production for contributions (and the LO terms) were studied in great detail in ref. Wiesemann2015 (), including the effect of parton showers, we focus here on the computation of NLO QCD corrections to the terms proportional to and analyse their behaviour with respect to the contribution. We note that our computation of NLO corrections to the and terms employs an effective field theory, where the top quark is integrated out from the theory and the Higgs directly couples to gluons, to which we refer as Higgs Effective Field Theory (HEFT). Besides a detailed description of the application of this approach to our problem, we will show that this approximation is quite accurate in the bulk of the phase space region which is relevant for this study.
Before introducing our calculation in the next section, we briefly summarise the status of the results for production available in the literature. As far as 4FS computations are concerned, seminal NLO fixed-order parton-level predictions were obtained in refs. Dittmaier:2003ej (); Dawson:2003kb (), and later updated to the case of MSSM-type couplings Dawson:2005vi (), and to SUSY-QCD corrections in the MSSM Liu:2012qu (); Dittmaier:2014sva (). The presentation of differential results in these papers is very limited as the focus is on the total cross section. Given that computations in the 5FS are technically much simpler, far more results in this scheme exist in the literature: the total cross section are known at NLO Dicus:1998hs (); Balazs:1998sb () since a long time and even NNLO QCD Harlander:2003ai () predictions were among the first computations at this level of accuracy ever achieved relevant for LHC phenomenology. Parton-level distributions were obtained at NLO for + and +jet production Campbell:2002zm (); Harlander:2010cz (), and at NNLO for jet rates Harlander:2011fx () and fully differential distributions Buehler:2012cu (). The analytical transverse-momentum spectrum of the Higgs boson was studied up to in ref. Ozeren:2010qp (), while analytically resummed NLO+NLL and NNLO+NNLL results were presented in ref. Belyaev:2005bs () and ref. Harlander:2014hya (), respectively.111Even the ingredients for the full NLO prediction are already available Ahmed:2014pka (); Gehrmann:2014vha (); their combination is far from trivial though. NLO+PS predictions for both the 4FS and the 5FS were presented for the first time in ref. Wiesemann2015 (), including a comprehensive comparison of the two schemes and the discussion several differential distributions with NLO QCD accuracy. Other NLO+PS results were later obtained in Powheg Jager:2015hka () and Sherpa Krauss:2016orf (). At the level of the total cross section advancements have been made by first understanding the differences between results obtained in the two schemes refs. Maltoni:2012pa (); Lim:2016wjo () and then by consistently combining state-of-the-art 4FS and 5FS predictions in refs. Forte:2015hba (); Forte:2016sja (); Bonvini:2015pxa (); Bonvini:2016fgf ().
The paper is organised as follows. In section 2 our computation is described in detail. We first discuss the various contributions to the cross section (section 2.1), then introduce the HEFT approximation to determine the terms (section 2.2) and finally perform a comprehensive validation of the HEFT approximation for the cross section (section 2.3); phenomenological results are presented in section 3 — see in particular section 3.1 for the input parameters, section 3.2 for SM results, section 3.3 for how to obtain the best sensitivity to extract in the measurements, and section 3.4 for our analysis on NLO corrections to splitting. We conclude in section 4 and collect relevant technical information in the appendices.
2 Outline of the calculation
2.1 Coupling structure of the cross section
The leading contribution to the associated production of a Higgs boson with bottom quarks in the 4FS starts at in QCD perturbation theory, and is mediated by the bottom-quark Yukawa coupling. Hence, the coupling structure of the LO process is . A sample Feynman diagram is shown in figure (a)a. At the next order in the typical one-loop (figure (b)b) and real-emission (figure (c)c) diagrams are included, and yield a contribution of . At the same order in additional one-loop diagrams appear featuring a closed top-quark loop which the Higgs boson couples to (figure (c)c). These diagrams introduce for the first time a dependence on top-quark Yukawa coupling in the cross section and lead to contributions of through their interference with diagrams as shown in figure (a)a. At the next order in , the square of these amplitudes yields a contribution that starts at . Thus, it is suppressed by two powers of with respect to the first non-zero contribution to production of and could be formally considered a NNLO contribution. However, it is easy to understand that a naïve power counting just based on the single parameter is not suitable for describing production, since the strong hierarchy between the top-quark and the bottom-quark Yukawa couplings in the SM is such that terms turn out to be of a similar size as the contributions. In this respect, one also expects that corrections to the contributions might turn out to be important, which are of and formally part of the NLO corrections with respect to the leading terms. They enter via virtual and real diagrams of the type shown in figure (a)a and in figure (b)b, respectively.
Collecting all relevant terms at different orders in , one can express the cross section as
where the amplitudes are introduced with the respective sample diagrams in figures 5–8, and denotes the appropriate phase space of the extra real emission with all relevant factors in each case. An equivalent, yet more appropriate and transparent way of organising the computation above is to consider a double coupling expansion in terms of and and then to systematically include corrections to each of these terms. Up to NLO, the cross section can be written as
It is trivial to see that eq. (1) and eq. (2) feature exactly the same terms. In this formulation QCD corrections to , and terms, the contributions, are manifestly gauge invariant and can be calculated independently of each other at LO and NLO. All the coefficients up to (, , and ) were determined and studied already in ref. Wiesemann2015 (). Our focus here is therefore on the calculation of the contributions involving in the 4FS, i.e., , , and .222The contributions () are implicitly included in the computation of gluon–gluon fusion at NNLO (NLO) in the 5FS Anastasiou:2002yz (); Anastasiou:2015ema (). These calculations, however, cannot provide information on final states specifically containing quarks.
2.2 HEFT approximation in production
NLO corrections to the contributions proportional to the top-quark Yukawa coupling require the computation of two-loop amplitudes with internal massive fermion lines, see figure (a)a. The evaluation of such diagrams is beyond current technology. Hence, in this section, we introduce the heavy top-mass approximation that can be employed for the computation of these amplitudes, and we rearrange the SM cross section in section 2 in the HEFT. In this effective theory, the top quark is integrated out and yields an effective point-like interaction between the Higgs boson and gluons. This approximation has been used successfully to compute a number of observables in the Higgs sector, with the gluon fusion cross section through as the most notable example Mistlberger:2018etf (); Anastasiou:2016cez (); Anastasiou:2015ema (). By substituting top loops with a point-like coupling, the HEFT allows for significant simplifications of Higgs-related observables at the price of a limited range of applicability: the approximation is expected to break down when one of the scales appearing in the process, and in particular in the massive loop integrals, becomes comparable with the top-quark mass. The case at hand corresponds to +jet (+) production with splitting either in the initial or in the final state. It has been shown that the HEFT provides an excellent approximation in that case as long as the scales of the process remain moderate Harlander:2012hf (); Neumann:2014nha (), for example as long as the Higgs transverse momentum () is below . In section 2.3, we provide a detailed assessment of the goodness of the heavy top-mass approximation. As we will show, the heavy-top mass approximation works extremely well (with differences from the full computation below 10%) as long as the probed momentum scales (Higgs or leading b-jet transverse momentum, or invariant mass of the b-jet pair) do not exceed .
Working in the HEFT allows us to avoid the computation of the highly complicated amplitudes in figure (d)d and figure 8, and evaluate instead the diagrams shown in figure 12, which have a much lower complexity, being at most at the one-loop level. In addition, the HEFT has been implemented in an Universal Feynrules Output (UFO) model Artoisenet:2013puc (), and this calculation can be performed using existing automated Monte Carlo tools. Nonetheless, present implementations neglect power-suppressed corrections to SM parameters generated by the heavy-top mass approximation, which play a crucial role in the case at hand. In particular the bottom-quark Yukawa coupling must be corrected in the following way:
which generates additional terms of and entering the cross section at the perturbative order we are interested in. As a result, we insert into eq. (2) to yield the cross section in the HEFT and rearrange it as follows:
where top-quark loops have been replaced by the HEFT contact interaction in quantities with a hat. In this cross section, the only contribution that could not be directly calculated using automated tools is the power-suppressed bottom-quark Yukawa correction, which we derive in the Appendix B. We find
where and are understood to be renormalised in the scheme at a scale .
We have implemented by hand this modification in the HEFT model at NLO. This enables a complete calculation of the QCD corrections and in the heavy-top mass approximation in a fully automated way. We therefore can employ MadGraph5_aMC@NLO Alwall:2014hca () to perform the calculation of the cross section in the 4FS at parton level. We use the recently-released version capable of computing a mixed-coupling expansion Frederix:2018nkq () of the cross section in order to compute all six contributions (, and both at LO and NLO) with the appropriate renormalisation of simultaneously.333A similar computation was performed in the context of charged-Higgs production in the intermediate-mass range Degrande:2016hyf ().
Besides computing eq. (4) in the HEFT, we also calculate the LO contributions in the full theory in order to rescale the contributions and to provide the best approximation of the cross section in eq. (2). We refer to this approach as the Born-improved HEFT (BI-HEFT) in the following:
For differential distributions, eq. (6) is applied bin-by-bin.
2.3 Assessment of the HEFT approximation
In this section we assess the accuracy of the heavy-top quark approximation. To this end, we compare the LO cross section against its approximation in the HEFT . We use the same input parameters as for our phenomenological results in section 3, and refer to section 3.1 for details. We perform a validation for both the inclusive cross section and differential distributions. Since the topology of the process at LO is very similar to that of the +jet process, we expect the HEFT to provide a good description in the relevant phase-space regions, in particular concerning the shapes of distributions. We stress again that in our best prediction, the BI-HEFT, we use the the HEFT only to determine the radiative corrections in terms of the NLO -factor. Total cross section and kinematic distributions, obtained in the HEFT, are reweighted (bin-by-bin) by a factor equal to the ratio between the full theory and the HEFT, both evaluated at LO. This has been shown to be an excellent approximation for +jet production as long as the relevant scales do not become too large Harlander:2012hf (); Neumann:2014nha ().
We start by reporting the result for the inclusive cross section:
The results lie within of each other. Considering that the perturbative uncertainties are one order of magnitude larger, we conclude the inclusive cross section is well described by the heavy-top quark approximation. Furthermore, the accuracy of the BI-HEFT result can be assumed to be considerably better than this value, since top-mass effects are included at LO by the rescaling in eq. (6). As in the case of +jet production, the dominant configurations are with the Higgs at low transverse momentum, which explains the quality of the approximation.
Let us now turn to differential cross sections in figure 19. The main frame shows the SM (blue dash-dotted) and HEFT (green dotted) predictions. The lower inset shows their bin-by-bin ratios. The first three plots, figures (a)a-(c)c feature the transverse-momentum spectra of the Higgs boson, the leading and the subleading jet respectively. As expected, we find that the HEFT provides a good description of the SM result, especially in terms of shapes. Only at large transverse the two curves start deviating with the HEFT result becoming harder. This happens after transverse momenta of for the Higgs and the leading jet, and a bit earlier for the second-hardest jet.
The Higgs rapidity distribution in figure (d)d is hardly affected by the HEFT approximation, with the HEFT/SM ratio being essentially flat. Also for the invariant mass of the two jets in figure (e)e, the heavy-top quark result provides a good description as long as . Finally, for the separation in the – plane between the two jets, shown in figure (f)f , the agreement between HEFT and SM is very good up to . Above this value, the distribution is dominated by large invariant-mass pairs, and the HEFT/SM ratio follows what happens for the invariant-mass distribution.
Overall, the heavy-top quark approximation used in the HEFT results works extremely well for this process over a large fraction of the phase space and in particular where the majority of events are produced. For the goals of our study, it is especially important to verify that the comparison of the angular separation of the jets and of their invariant mass is well reproduced, as it indicates that we can safely explore the regime in which the two bottom jets merge into a single one. This regime is particularly interesting to study for the terms as we will see in section 3. Furthermore, there is a reasonable range of -jet transverse momentum where the process is correctly described, so that we can trust the prediction to study the impact of -jet requirements on the relative importance of the and contributions. It should be noted, however, that in the two -jet configuration, the HEFT prediction is rather poor over a larger range of transverse momenta for the subleading jet. Nevertheless, this is not expected to have an impact on our phenomenological study in the upcoming section.
3 Phenomenological results
In this section we present differential results for production at the 13 TeV LHC including all contributions proportional to , , and at NLO QCD, see eq. (2). We analyze the importance of radiative corrections and the relative size of the three contributions. Although we work in the SM, thanks to the separation of the cross section by the Yukawa coupling structure, our predictions are directly applicable to 2HDM-type extensions of the SM (for with a neutral Higgs boson ) by an appropriate rescaling of the top and bottom-quark Yukawa. Even for the MSSM such rescaling has been shown to be an excellent approximation of the complete result Dittmaier:2006cz (); Dawson:2011pe ().
3.1 Input parameters
Our predictions are obtained in the four-flavour scheme throughout. We use the corresponding NNPDF 3.1 Ball:2017nwa () sets of parton densities at NLO with the corresponding running and values.444More precisely, NNPDF31_nlo_as_0118_nf_4 (lhaid=320500 in LHAPDF6 Buckley:2014ana ()) corresponding to . The central values of the renormalisation () and factorisation () scales are set on an event-to-event basis to
where the index runs all the final-state particles, possibly including the extra parton from the real emission. Scale uncertainties are computed without extra runs using a reweighting technique Frederix:2011ss (), and correspond to independent (nine-point) variations in the range . Internal masses are set to their on-shell values , and . The top-quark Yukawa is renormalised on-shell; for the bottom-quark Yukawa, instead, we compute by adopting the scheme, with a four-loop evolution Marquard:2015qpa () from up to the central value of the renormalisation scale, and two-loop running for the scale variations, as recommended by the LHC Higgs cross section working group deFlorian:2016spz ().
Jets are reconstructed with the anti- algorithm Cacciari:2008gp (), as implemented in FastJet Cacciari:2011ma (), with a jet radius of , and subject to the condition and . Results with a larger jet radius, , are available in appendix A. Bottom-quark flavoured jets ( jets) are defined to include at least one bottom quark among the jet constituents. A jet containing a pair of bottom quarks is denoted as a jet. Within a fixed-order computation, we will use the word hadrons to identify bottom quarks (the notation will refer to bottom quarks, while the notation to bottom-tagged jets). Bottom-quark observables are infrared safe owing to the finite bottom-quark mass in the 4FS.
|LO (acceptance)||NLO (acceptance)|
3.2 Predictions for production in the SM
We start by discussing integrated cross sections in table 1, both fully inclusive and within cuts. As far as the latter are concerned, we have considered various possibilities: the requirement that there be at least one or two jet(s); that there be at least one jet containing a pair of bottom quarks ( jets); and that the transverse momentum of the Higgs boson be larger than , , and (boosted scenarios). The residual scale uncertainties are computed by varying the scales as indicated in section 3.1. We present separately the results for terms proportional to , , and . The contributions are provided in two approximations: using the HEFT, on the one hand, and our BI-HEFT prediction, computed by rescaling the HEFT result at NLO by the LO evaluated in the full theory, on the other hand. For completeness, we also quote the BI-HEFT prediction for the sum of all individual contributions. Besides LO and NLO of the cross sections we also provide the NLO/LO -factor to assess the importance of QCD corrections. Inside the bracket after the LO and NLO cross sections we quote the acceptance of the respective scenario, defined as the ratio of the cross section within cuts divided by the inclusive one. We refrain from quoting the acceptance for the interference terms since this quantity is meaningless on its own. The conclusions that can be drawn from the table are the following:
Already at LO the terms yield a significant contribution to the SM cross section. Due to sizable QCD corrections to the terms (), the inclusive NLO cross section is a factor of three larger after including the loop-induced gluon-fusion component than when considering only contributions. Hence, the cross section in the SM is substantially larger than generally assumed from computations, which could make its observation much easier. At the same time, however, the sensitivity to the extraction of the bottom-quark Yukawa coupling is diminished. Below, we discuss how suitable phase-space cuts can be used to enhance the over the contributions and to retain sensitivity to the extraction of .
The relative size of contributions further increases when considering the various scenarios with additional phase-space cuts. The reason is that the loop-induced gluon-fusion component generates harder (-)jet activity and the cuts favour hard configurations. For example, tagging one jet has the effect of decreasing the NLO cross section by %, while for it is only %, and the NLO cross section is four times as large as in the -jet scenario, to be compared to the factor of two in the inclusive case. Tighter -jet requirements or the requirements only have the effect of further increasing the relative size of the contributions.
It is interesting to notice that the jet category, which requires one jet containing two bottom quarks, receives contributions essentially only from terms. This can be understood easily: a major part of the events for the loop-induced gluon fusion component features the Higgs recoiling against a hard gluon, which splits into a -pair. The two bottom quarks in these configurations are boosted and generally close together, which makes it more likely for them to end up inside the same jet.
Two opposed effects render the measurement of the bottom-quark Yukawa coupling in production complicated: as pointed before the relative size of terms proportional to decreases as soon as jets are tagged. Nonetheless, tagging at least one of the jets is essential to distinguish production from inclusive Higgs production, which predominantly proceeds via gluon fusion.555Inclusive Higgs production may be used to extract only through the measurement of the inclusive Higgs transverse-momentum spectrum at very small , see ref. Bishara:2016jga () for example. Therefore, it is necessary to select suitable phase-space requirements which increase the relative contribution even in presence of at least one jet without loosing too much statistics. Given our findings for the -jet category, one can require at least one jet and veto all jets. This decreases the -jet rate for by roughly %, while having a negligible effect on the rate. Below, we study differential distributions in order to find further requirements to enhance the relative size of the contributions.
The LO contribution of the mixed terms is negative in all scenarios, as has been observed already in ref. Wiesemann2015 (). At NLO it yields a positive contribution only to the -jet scenario. The NLO/LO -factor of the term strongly depends on the scenario under consideration, which is expected given its interference-type contribution. By and large the impact of the terms is minor though, reaching at most a few percent at NLO.
Overall, QCD corrections have an even larger impact on the terms than on the terms, but they are quite sizable in either case. This can be understood as follows: as pointed out in ref. Wiesemann2015 () potentially large logarithmic terms of enter the perturbative expansion of contributions in the 4FS and cause large perturbative corrections. Contributions proportional to , on the other hand, feature a logarithmic enhancement of splittings. These logarithms are taken into account for the first time up to NLO QCD in this paper, and yield an important correction to the cross section.
Given the large QCD corrections, it is not surprising that perturbative uncertainties estimated from scale variations are relatively large as well. As expected they are largest for terms. The inclusion of NLO corrections reduces the uncertainties significantly, but they are still at the level % to %. Their main source is again the logarithmic enhancement of the individual contributions pointed out above.
We now turn to discussing differential distributions. We first consider the NLO/LO -factor of the different contributions to the cross section in figures 25 and 25. These figures are organised according to the following pattern: there is a main frame, which shows histograms of the LO (green dashed), NLO (blue solid), LO (purple dash-dotted), and NLO (red dotted) predictions as cross section per bin (namely, the sum of the values of the bins is equal to the total cross section, possibly within cuts). In an inset we display the -factor for each contribution by taking the bin-by-bin ratio of the NLO histogram which appears in the main frame over the LO one. The bands correspond to the residual uncertainties estimated from scale variations according to section 3.1.
Figure (a)a shows the transverse momentum distribution of the Higgs boson. As expected from the general hardness of the two different production processes leading to and (the Higgs being radiated off a top-quark loop, and the Higgs boson being coupled to a bottom-quark line), the latter features the significantly softer spectrum. The -factors for both contributions grow with the value of similarly, and become quite flat at large transverse momenta. However, as observed before, the size of QCD corrections is larger for the terms, ranging from at small to for . For contributions they are and in the same regions.
Also the transverse-momentum distribution of the leading jet in figure (b)b displays a harder spectrum for the contributions. Note that for the distribution, as we select the leading jet, the integral of the distribution corresponds to the -jet rate. The behaviour of the -factor is quite different in this case. While it is essentially flat and about for the terms, it is about for at low , decreases to for , and turns flat afterwards.
In figure 25 we consider two observables which require the presence of at least two jets: the left panel, figure (a)a, shows the invariant-mass distribution of the -jet pair, , and the right panel, figure (b)b, shows their distance in the – plane, . It is interesting to notice the very different behaviour of and terms in the main frame of these two distributions. While clearly prefers small invariant-masses and small separations between the jets, peaks around and . The reason is clear: the dominant contribution for originates from the splittings, which generate bottom-quark pairs that are hardly separated and, hence, also have a rather small invariant mass. Looking at the -factors in the lower inset, the one of the terms turns out to be rather close to one for a large part of the phase space. It slightly increases with both and . For , on the other hand, the -factor is around , and shows an even milder increase with and .
The scale-uncertainty bands in all four plots show the same features as the scale uncertainties discussed in table 1: Their size decreases upon inclusion of higher-order corrections, but overall they are rather large even at NLO. The contributions feature a stronger scale dependence due to the logarithmic enhancement of splittings. By and large, LO and NLO at least have some overlap in most cases. Nevertheless, contributions show the better converging perturbative series in that respect, which of course is directly related to their smaller QCD corrections.
3.3 Accessing the bottom-quark Yukawa coupling
We now return to the question of how to improve the sensitivity to the bottom-quark Yukawa coupling in production. The goal is to increase the relative contribution of the terms by suitable selections, while keeping the absolute value of the cross section as large as possible. First, in order to be able to distinguish from inclusive Higgs production, we require to observe at least one jet. Second, we have already noticed that by removing all jets containing a pair of bottom quarks, we can decrease rate to some extent, with a negligible impact on the rate. The combination with additional phase-space requirements provides the most promising approach to further improve the sensitivity to the bottom-quark Yukawa coupling and to recover production as the best process to measure directly. To this end, we consider the relative contribution of and terms to the cross section for various differential observables in figures 29–37. All the plots in these figures have a similar layout: the main frame shows NLO predictions for the contribution (blue dash-dotted), the contribution (red dotted), and the sum of all contributions, including the interference (black solid). The ratio inset shows the relative contributions of the and terms to cross section. The bands reflect the residual uncertainties estimated from scale variations according to section 3.1.
We start in figure 29 with the transverse-momentum distribution of the Higgs boson. The three panels show this distribution with different requirements: figure (a)a displays the inclusive spectrum, figure (b)b is in the -jet category, and figure (c)c is in the same category, but vetoing jets. This observable constitutes one of the strongest discriminators between and terms. The reason is the significantly softer spectrum of the terms proportional to , which we already observed before. By looking at the three plots in figure 29 one can infer that the relative contribution is maximal in the inclusive case and at low Higgs transverse momentum, even exceeding 50% in the lowest part of the spectrum. If we require at least one jet the situation becomes worse, with the term reaching at most 40%, again in the lowest transverse-momentum bins. If we require at least one jet and veto jets, the relative contribution of at low transverse momentum is mildly increased. In the three cases (inclusive, jet and jet without jets) the relative contribution quickly decreases with , being less than 20% already at . At the level of the cross section, in the jet category, the relative and contributions are respectively 81% and 19%. In the jet and no -jet category their relative contributions become and , respectively. All in all, the gain coming from vetoing jets is moderate. Another strategy, which can be combined with the -jet veto, consists in discarding events with the Higgs transverse momentum larger than a given value. For example, with an upper cut on at , in the category with at least one jet and no jet, we can increase the relative contribution of terms to about , while keeping about of its rate. Hence, restricting the phase space to small values allows us to increase the relative size of terms, while the impact on the rate is moderate due to the quite strong suppression at large .
We continue in figure 32 with the transverse-momentum distribution of hardest jet. The general features of the spectrum are similar to the ones of . However, as this observable clearly does not help very much in distinguishing between and contributions, we do not suggest any additional cut on . It becomes clear from these plots, though, that a lower threshold used in the definition of jets would increase the relative size of the terms. In the present study jets are defined with a threshold of . A value of or even could be feasible at the LHC, and would further increase the sensitivity to the bottom-quark Yukawa coupling in production. We note that additional modifications of the -jet definition, for example the usage of a different jet radius (as shown in appendix A), or of jet-substructure techniques, can provide further handles to improve the discrimination of the contribution.
Finally, we consider figure 37, where we show the invariant-mass distributions of the two jets, figure (a)a, and their distance , figure (b)b, and the corresponding distributions for hadrons, figures (c)c and (d)d. Note that the and distributions by construction require the presence of two jets, while and are shown with at least one jet and no jet. Clearly, all of these observables, especially those related to hadrons, could in principle provide information to discriminate between and contributions. However, in practice, their usefulness is limited, due two main reasons, both related to statistics. First, the two -jet distributions require the presence of at least two jets and the corresponding rate is significantly reduced (by roughly one order of magnitude) with respect to the -jet one. Second, the bulk of the -hadron distributions feature hadrons which are quite soft, and therefore possibly not accessible in the measurements. We therefore conclude that, despite showing useful features, neither of these distributions can significantly help in obtaining additional sensitivity to the bottom-quark Yukawa coupling.
3.4 QCD corrections to splitting
Besides its phenomenological relevance for the extraction of the bottom-quark Yukawa or as a background to other Higgs production processes at the LHC, production induced by the top-quark Yukawa coupling offers a clean and simple theoretical setting to study the dynamics of splitting in presence of a hard scale. Cases of interest at the LHC where such splitting plays an important role, and is in fact one of the main sources uncertainties, are