Non-standard top substructure
The top quark, being the heaviest particle of the Standard Model, is a prime candidate of where physics beyond the SM might currently hide before our eyes. There are many natural extensions of the SM that rely on top compositeness, and the top quark could follow the paradigm of revealing a substructure when it is probed at high enough momentum transfers. Observing high top final states naturally drives us towards boosted hadronic analyses that can be tackled efficiently with jet substructure techniques. In this paper we analyse the prospects of constraining exemplary non-standard QCD top interactions in this kinematical regime. We correctly include QCD modifications to additional gluon emission off the boosted top quark and keep track of the modified top tagging efficiencies. We conclude that non-standard top QCD interactions can be formidably constrained at the LHC 14 TeV. Experimental systematic uncertainties are a major obstacle of the described measurement. Unless significantly improved for the 14 TeV run, they will saturate the direct sensitivity to non-resonant BSM top physics at luminosities of around 100/fb.
After the discovery of a SM Higgs boson orig () at the LHC atlash (); cmsh () and preliminary measurements of its properties and couplings atlash2 (); cmsh2 () which indicate close resemblance to the SM hypothesis, hints for physics beyond the SM remain elusive. A puzzle that remains in the context of SM irrespective of a seemingly unnatural electroweak scale is the mass hierarchy in the fermion sector and the large mass of the top quark rather close to the Higgs vacuum expectation value. The restoration of chiral symmetry for vanishing Yukawa interactions guarantees that corrections to elementary fermion masses are proportional to the fermion mass themselves in the SM. Using the language of effective field theory, the Yukawa couplings are marginal operators, i.e. once their values are fixed by some UV dynamics Froggatt:1978nt (), they remain small at low energy scales. Hence, the large hierarchy among the Yukawa couplings largely determined by the top quark is typically considered a potential source of physics beyond the SM.
Indeed, the top typically plays a central role in most models that try to explain the electroweak scale at a more fundamental level. Supersymmetric constructions susy (), fixed-point gravity scenarios fpg (), and strong interactions Agashe:2004rs () are just three well-known and well-established examples. In the latter case, the large mass of the top can be understood as a (linear) mixing effect of light elementary states with composite fermions of a strongly interacting sector comph (); silh () that also provides a set of Nambu Goldstone bosons forming the Higgs doublet. The mixing effects together with fermion and gauge boson loops induce a Coleman-Weinberg Higgs potential that triggers breaking of electroweak symmetry at a scale much smaller than the strong interaction scale. In such pseudo-Nambu Goldstone Higgs scenarios, we can have a large resemblance of the Higgs phenomenology with the SM, whilst the composite effects are hidden in the fermionic sector. Phenomenological searches that target the potential substructure of the top quark are therefore also extremely important in the context of Higgs physics, since both phenomena, the (100 GeV) electroweak scale with the top quark in the same ball park, might point us towards a solution in terms of strong interactions.111It should be noted that such interactions typically also alter low energy observables (see, e.g., Ref. maggie ()), but we remind the reader that we focus on the prospects of direct measurements in this work.
Of course, the phenomenological implications of compositeness are not new to particle and, more broadly speaking, to nuclear physics (see Ref. holst () for a review). The deviation from the anticipated Rutherford scattering cross section at large angles observed by Geiger and Marsden gm () and the later resolution of atomic nuclei rutherford (); gm2 () is a well-known example of such a programme resolving point-like sources by probing the characteristic energy scale with high enough momentum transfers. The non-linear structure of QCD and the mismatch of the theory’s fundamental degrees of freedom with the experimental observables, however, introduces another layer of complexity when we deal with non-standard interactions of a colour-charged object. We usually parametrize the deviations from the SM via introducing higher dimensional operators in an effective field theory description that is guided by the low-energy gauge symmetry requirements. Since we can expect a separation between the new physics and the electroweak scale, it is customary to limit analyses to dimension six operator extensions to the SM hikasa (); wyler (); Grzadkowski:2010es (). However, since we cannot separate different partonic initial and final states and due to the gauge structure, all operators that introduce non-standard QCD properties will contribute simultaneously. Their different kinematical dependencies can be used to disentangle them degrande (); haberl (); others (); peter (), but modifications due to new interactions will also change the response of the measurement strategy.
The top quark production cross section will receive modifications for energetic events if new physics in the top sector is present. This immediately motivates boosted top searches boostedtops () as a sensitive probe of modified QCD interactions on which we focus our analysis in the following. From previous analyses peter () it is expected that upon correlating inclusive and boosted measurements of we will be able to tightly constrain such non-standard interactions. However, there is a caveat: top quarks when produced at high are very likely to emit hard gluons before they decay Ferroglia:2013zwa (); topradiation (). In Ref. peter () it was shown that such an interaction has a decreased sensitivity to anomalous QCD top interactions. It is therefore crucial to include the anomalous top interactions to the proper modelling of the exclusive final state to correctly evaluate the prospects of the described measurement. By analyzing the fully hadronized final state in such a setup, we are also guaranteed to correctly reflect the different selection efficiencies for the boosted subject analysis that emerge from the BSM-induced modifications of the top spectrum. More precisely: we investigate the constraints that we can expect from adapted searches for anomalous top interactions in the busy QCD-dominated LHC environment using realistic simulation, analysis and limit setting techniques.
Especially experimental systematics are known to be large in the tails of top distributions where the deviations from the SM will be most pronounced. Unless these uncertainties are properly included in the formulation of the BSM limits we cannot trust the analysis. We discuss the present systematics and include them to our CLs CLS () projection for the 14 TeV LHC run in the most conservative way. To keep our analysis transparent we focus on two representative anomalous top-QCD operators that are characteristic for composite fermionic structures from a QCD point of view, namely colour charge radius and anomalous magnetic moment brod () (see Ref. schwingdeinding () for similar work on composite leptons). The generalisation to other non-standard top-related interactions is straightforward.
Ii A phenomenological approach to anomalous QCD top interactions
To get a quantitative estimate of the leading effects of non-standard top interactions at the LHC we focus on new physics contributions to production arising from modified QCD interactions. Non-standard electroweak properties do impact the top decay Bach:2012fb (), but can be studied separately in single top-production and interlaced with our findings.
Since the current LHC searches imply strong bounds on the masses of potential new degrees of freedom, it is expected to have a mass gap between the SM and the BSM fields (which, e.g., lift the top mass via mixing effects Grossman:1999ra ()).
In this case, the new physics effects can be parametrized via higher dimension operators involving only the SM particles and there is a number of new contact operators which impact +jets production wyler (); degrande (). Here we focus on some operators that allow an interpretation in terms of composite structures such as radii and anomalous magnetic dipole moments as a proof of principle. These non-standard properties can be introduced in a gauge-covariant way through the following effective dimension six interaction terms peter (); hikasa (); haberl (); others ()
where is the gluon field, its field strength and the covariant derivative. The convention of Eq. (1) follows Ref. schwingdeinding (); the top quark radius and the anomalous chromomagnetic and chromoelectric dipole moments are related to the new physics scale in the “traditional” dimension six extension approach by
where is a parameter.
To have a consistent treatment of the dimension six operator expansion the new physics contributions are manifest only through the interference of these new physics operators’ contribution with the SM amplitude, i.e. we do not include terms to the hadronic cross section other than the ones that formally scale as . Splitting the amplitude that results from Eqs. (1) into a SM and BSM piece
we have for the (partonic) cross section
The expansion of the cross section to removes the chromoelectric operator from the sample haberl () and the sensitivity to arises from the less dominant contribution. The squared BSM matrix elements has a dependence on haberl (). At , however, when becomes resolvable, we can also expect additional dimension eight operators to enter the stage via interference with the SM amplitude. In such a case it is not clear how to interpret a limit obtained on . Expanding of the cross section to will therefore only yield mild constraints on .
where is the squared partonic center of mass energy and . Notice that for initial states both new physics contributions and are present, whereas for -induced production (the main production mode for inclusive production at the LHC) there is only sensitivity to the anomalous chromomagnetic moment . This is due gauge invariance of the dimension six operator, i.e., there is a Ward identity that guarantees the cancellation of the dependence222An identical cancellation is required to ensure a massless gluon in the extended theory: by closing the top-loop we have a contribution to the gluon two-point function from the two diagrams on the right hand side of Fig. 1 which do not vanish in dimensional regularization. in the sum of Fig. 1. It can be shown that for the sample the same conclusion holds, i.e., the sub-channel still has no dependence on the parameter which originates from the and induced subprocesses peter ().
We can enhance the fraction of the initial state and still probe at the LHC by requiring boosted top events.333A similar strategy has been discussed in the context of the central-forward top asymmetry afb (). This is because energetic events probe the incoming partons at high momentum fractions where the proton’s valence quarks’ parton densities peak. We illustrate this in Fig. 2, where we present the fractional contribution of each partonic subprocess to the hadronic SM cross section as a function of the reconstructed mass and the top transverse momentum . We can invoke cuts on either observable to suppress the initial state although is more effective and the more crucial observable in the context of top tagging afb (); heptoptagger ().
Iii Details, Analysis and Results
In our analysis we focus production with one top decaying semi-leptonically and the other hadronically. As this process involves the production of heavy coloured particles and we are selecting the boosted kinematical regime, we can expect an important contribution from initial and final-state jet radiation alwall (); boosted (). To take this sufficiently into account we include the BSM-mediated hard radiation effects via jet merging, keeping the full BSM dependence on the non-standard parameters of the respective samples to .
As already mentioned, the dependencies on the top radius arise entirely from the and initial states. Therefore, to constrain this operator it is necessary to suppress the dominant sub-channel at the LHC, namely the initial state. The boosted high selection serves two purposes in this sense: it removes the less sensitive initial states and focuses on regions where deviations from the SM are large, Eq. (5).444It is worth noticing that for boosted final states we do not need to worry about trigger issues Aad:2012dpa (); TheATLAScollaboration:2013kha ()..
Our implementation starts by including the new interactions presented in Eqs. (1) through FeynRules feynrules (), which outputs a Ufo model file ufo () that is further used into MadGraph5 mg5 (). MadGraph performs the event generation that is subsequently showered with Pythia6 pythia () where we take into account the initial and final state radiation, hadronization and underlying event. The hard matrix elements have been adapted to only include the interference of the new physics amplitude with the SM counterpart; this way we guarantee a consistent expansion of the cross section up to as discussed earlier when QCD emission is hard and sensitive to the BSM effects. We have validated our parton level matrix element implementation against existing analytic calculations as well as an independent Monte Carlo implementation peter (); degrande (); haberl ().
The jet merging is subsequently performed by employing the MLM scheme mlm () as implemented in the MadGraph package. Throughout the analysis we consider the LHC running at and the SM cross section normalization is re-scaled to the NNLO value, pb moch (). We find that for our boosted selection that the background is completely dominated by SM production. All other background contributions are negligible and well below the SM uncertainty.
We include the expected dominant NLO shape modifications via aMC@NLO amcatnlo (): we construct a re-weighting function with respect to the sample (the SM) to account for differential QCD corrections in the BSM histograms. This is a necessary procedure to have a well-defined limit . Throughout, we choose the renormalization and factorization scales as the transverse mass since this choices yield a rather flat scale dependence of the NLO matched cross section, Fig. 3.
Instead of proceeding as in a “traditional” semi-leptonic analysis we take advantage of the efficient top tagging for high fat jets. This is facilitated by defining a fat jet with a large cone size using the Cambridge/Aachen algorithm as implemented in Fastjet fastjet (). We require at least one of these objects to have a transverse momentum larger than . We choose this exemplary value due to a large top tagging efficiency and small fake rate . For this threshold the cross section is also still large enough to perform measurements with small statistical uncertainties; the eventual value of by the experiments will optimise the systematic uncertainty. This fat jet is then further processed by the HEPTopTagger heptoptagger (). Initially the HEPTopTagger was designed to reconstruct only mildly boosted top quarks () using a very large fat jet cone size. However, in searches for heavy resonances heptopexp () it was shown that due to its flexible reconstruction algorithm and jet grooming procedures the HEPTopTagger is an effective tool to reconstruct highly boosted top quarks while maintaining a small background fake rate. Other top taggers, designed to tag highly-boosted top quarks, can be similarly effective othertoptaggers (); topradiation (). Top tagging is sensitive to the top’s BSM spectrum modification and modified hard shower profile that results from including at precision. Hence, the top tag efficiency itself is a function of the anomalous parameters.
After a successful tag, the corresponding jet is removed from the event and we proceed by re-clustering the remaining hadronic activity as usual, i.e. by applying the Cambridge/Aachen algorithm with . Jets are selected with properties and . We also require an isolated lepton in the final state with and where the lepton is defined isolated if the transverse energy deposit inside a cone around the lepton of size is less than 20% of its transverse energy .
On the one hand, the small theoretical uncertainties on the invariant mass motivates this observable as a suitable choice to examine our BSM hypotheses Ferroglia:2013zwa (). From Eq. (5) it becomes clear that dominant BSM corrections are directly reflected in the distributions (it is also the variable which typically enters as the only kinematical parameter in total cross section and re-summation calculations, see Ferroglia:2013zwa (); moch ()). On the other hand, the transverse fat jet momentum and lepton pseudorapidity determine the kinematics to a large extent for boosted final states. From a boosted top reconstruction point of view, is the crucial observable as the threshold largely determines the working point. Since we choose a specific value for in our analysis, we turn to and in the following.
Missing energy of the final state from the leptonic top decay is not a drawback: the final state neutrino momentum can be reconstructed by requiring transverse momentum conservation and by imposing that the invariant mass –neutrino is equal to . These conditions define respectively the neutrino transverse and longitudinal momentum components. To suppress the combinatorics in the mass reconstruction we need to identify which jet is the most likely to be the -jet, despite of not using -tagging in this analysis. This can be efficiently done by identifying the -jet as the closest jet to the lepton with an invariant bottom-lepton mass that satisfies the top decay kinematics Plehn:2011tf ()
After these steps we end up with distributions as depicted in Fig. 4; the BSM-induced shape modification includes a lot of information that we would like to exploit in a binned hypothesis test based on sampling the log-likelihood
There is a caveat. The uncertainties, especially in the tails of the distributions can be large, and are currently driven by experimental systematics Aad:2012dpa () rather than theoretical limitations (for a recent high precision calculation see Ferroglia:2013zwa ()). To get a feeling of the size of the systematics we include the relative systematic uncertainty from Aad:2012dpa () for TeV to Fig. 4; the theoretical uncertainty of Ferroglia:2013zwa () is negligible compared to the systematics of Aad:2012dpa (). We map the integrated uncertainty to a flat uncertainty; for central tops at transverse momenta of the order of 600 GeV this is a reasonable approximation. It becomes immediately clear that the shape uncertainty will be the limiting factor of this analysis, especially if we want push limits .
The standard way of including such an uncertainty is via nuisance parameters of the null hypothesis (SM production in our case) CLS (); junk (); cranmer (). When computing the confidence level, these nuisance parameters are marginalized or profiled. However, it can happen that the process of marginalization can stealth the systematic uncertainty entirely. By, e.g., including a shape uncertainty to only the null hypothesis and not to the alternative hypothesis, marginalization will shift the median of the toy-sampled log-likelihood distribution for the null hypothesis away from the alternative hypothesis’ median. The exclusion in this case appears to be larger than it should be, especially when the uncertainty bands overlap with the difference of null- and alternative hypothesis. To avoid issues of this type we include only bins which exceed the SM uncertainty to the log-likelihood; i.e. our null hypothesis is the one sigma upwards fluctuated SM hypothesis. This way we reflect the systematic uncertainty in an extremely conservative way; profiling or marginalization will correctly reduce the uncertainty when correlations with other signal regions (e.g. total cross sections and subsidiary top measurements using the ABCD method) are taken into account. This is information which requires access to the LHC data samples is not available to us and also somewhat beyond the scope of this work. We remind the reader to keep in mind that the outlined analysis when performed by the experiments is likely to yield improved constraints eventually.
From Eq. (7) it is clear that the binned log-likelihood approach will pick up sensitivity from regions in the single-valued discriminant where is large but still resolvable according to our definition. Hence, the sensitivity is dominated by the threshold behavior of the sample and jet radiation. There the uncertainty is comparably low and the absolute cross section modification large (keep in mind that the tails of the parton-level distribution grow according to Eq. (5), which does not include the pdf suppression, which quickly limits the considered analysis statistically).
We show the expected 95% exclusion as a function of the integrated luminosity in Fig. 5 for three different samples that can be excluded with a data sample of 100/fb at a 14 TeV LHC. The width of the 1 and 2 sigma bands being rather large indicates that we are very close to the border of the discriminable parameter region (in terms of our definition laid out in the previous section). Indeed, for smaller individual values we cannot formulate constraints as the BSM distribution is entirely covered by the SM uncertainty band. We therefore conclude that an improvement beyond the shown parameter choices depends crucially on the reduction of the experimental systematics (which should be well-possible when larger data samples are available). As expected the expected constraints from using as a single discriminant are superior to integrated sensitivity observables such as , Fig. 6.
Comparing to the preliminary investigations of Ref. peter (), we find that applying statistical algorithms as applied by the experiments and realistic simulation and analysis approaches, we find constraints in roughly the same parameter region: and at 95% CL. And extrapolation into the plane is shown in Fig. 7. Since we include a differential shape information of the top spectrum and a lower threshold that guarantees a quick saturation of the statistical uncertainty at comparably small luminosities we obtain more stringent expected constraints than simple correlations of inclusive and exclusive measurements, even when the systematic uncertainty is larger. Working in a consistent expansion to , we can only obtain unrealistically large values on that feed into our results through higher jet multiplicities exclusively.555Going beyond the approximation will be unavoidable if an excess in the tail will be observed with the described limit-setting analysis that implements a practitioners’ approach.
Iv Conclusions and Outlook
After the discovery of a Higgs boson that seems to follow the SM-paradigm and the lack of any hints towards natural physics completions at the TeV scale prompts us study the heavy degrees of freedom of the SM more carefully. Top quark physics, typically considered an impediment for new physics searches by providing a major background contribution, is a well-motivated candidate for such analyses. On the one hand, the properties of the top quark are still largely unknown, even after it was discovered nearly twenty years ago. On the other hand, the abundant production of top pairs at the LHC allows us to tightly constrain smallest resolvable deviations from the SM-predicted coupling pattern that is expected to be observed if the top quark arises (partially) as a bound state of a strongly interacting sector. This option is widely discussed in the literature and investigating anomalous QCD interactions in the top sector provides a path to either observe our strongly constrain such a scenario.
Resolving a potential composite structure with large momentum transfers in the top sector naturally motivates boosted top analyses as highly sensitive channels. Reconstruction techniques are under good theoretical control and have successfully been applied in resonance searches Aad:2012dpa (). Such resonances are expected in strongly interacting theories, too, but typical composite interactions can be expected to predominantly manifest themselves in a large deviation of the spectrum’s tail and experimental and theoretical uncertainties become major limitations of such searches.
In this paper we have computed the expected 95% confidence level constraints on a set of non-SM effective top QCD interactions resulting from an exemplary boosted top analyses and a representative set of operators. We have included the dominant first hard gluon radiation effects in a matched approach. Systematic differential uncertainties are taken into account in the most conservative way, and are based on current 7 TeV measurements. We therefore expect our constraints to be on the conservative end and believe that the actual analysis when performed by the experiments can indeed improve on our results.
Our hadron-level analysis correctly captures the top tagging’s varying efficiency as function of the anomalous parameters. This together with a state-of-the-art binned log-likelihood formulation of the expected confidence level constraints shows that differential shape information supersedes the naive extrapolation of earlier theoretical work, even when errors are considerably larger. We find that we should be able to probe an anomalous chromomagnetic moment at the per cent level and QCD-induced top radii at .
In summary, the search for a potential top substructure strongly benefits from recent developments in jet substructure analysis techniques. Adapting existing boosted top searches to BSM scenarios of this type is a straightforward exercise in the light of the results of Ref. Aad:2012dpa (). Given that this is an alternative route to study theoretically well-motivated scenarios beyond the SM we hope that this is incentive enough for the experiments to eventually perform measurements as outlined here.
We thank James Ferrando and Olivier Mattelaer for helpful conversations. We also thank Ben Pecjak for providing the results of Ref. Ferroglia:2013zwa (). CE thanks David Miller, Liam Moore, Michael Russell, and Chris White for discussions on the topic. CE is supported in parts by the IPPP Associateship programme.
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