Pinning down top dipole moments with ultra-boosted tops

We investigate existing and future hadron-collider constraints on the top dipole chromomagnetic and chromoelectric moments, two quantities that are expected to be modified in the presence of new physics. We focus first on recent measurements of the inclusive top pair production cross section at the Tevatron and at the Large Hadron Collider. We then analyse the role of top-antitop events produced at very large invariant masses, in the context of the forthcoming 13-14 TeV runs of the LHC, and at a future 100 TeV proton-proton collider. In this latter case, the selection of semileptonic decays to hard muons allows to tag top quarks boosted to the multi-TeV regime, strongly reducing the QCD backgrounds and leading to a significant improvement in the sensitivity to anomalous top couplings.


I. INTRODUCTION
The first run of the Large Hadron Collider (LHC) at CERN has successfully confirmed the Standard Model (SM) of particle physics with the discovery of a Higgs boson with SM-like properties [1,2]. Although no sign of physics beyond the SM has yet been observed, new phenomena at energies not far from the electroweak scale are predicted in many theories of new physics. There is therefore a great expectation for the upcoming LHC runs at center-of-mass (CM) energies ffiffi ffi s p ¼ 13 and 14 TeV, and for experiments at future accelerator facilities running at larger CM energies and luminosities. Furthermore, even if new physics were not directly reachable at the LHC or future machines, it might still be indirectly probed through precision measurements of the properties of the SM particles. In this context, the top quark is believed to play an important role, due to the closeness of its mass to the electroweak scale. This has motivated an intense research program dedicated to the study of its properties at the LHC. Future colliders will moreover be able to exploit the increase in CM energy and luminosity to probe indirect effects of new physics at higher momentum transfers, increasing the sensitivity to heavy new physics. The conceptual design studies of new accelerator complexes for future circular colliders (FCC) have recently started, at CERN [3] and at IHEP [4]. Their ultimate goal is the operation of a proton-proton (pp) collider designed to operate at ffiffi ffi s p ¼ 100 TeV. This accelerator will allow the exploration of energy scales several times higher than at the LHC, and will also significantly increase the statistics of known particles. For example, one trillion top quarks should be available with an integrated luminosity of 10 ab −1 . This consequently opens the door to indirect searches of new physics in the top sector with an unprecedented sensitivity.
In this paper we explore some of the opportunities offered by such a huge statistics of top quarks, focusing on the sensitivity to anomalous couplings to the gluons. The leading indirect effects from new physics present at a heavy scale Λ can be parametrized by adding to the SM Lagrangian L SM a set of dimension-six operators O x invariant under the SM gauge symmetry [5][6][7], where the Wilson coefficients C x depend on the type of new physics and how it couples to the SM particles. After the spontaneous breaking of the electroweak symmetry, these operators generate corrections to the SM couplings included in L SM , as well as interactions not present at the tree level, such as electric dipole moments and explicit magnetic dipole moments. The effects on the top dipole moments can be parametrized by adding an effective term to the top-gluon gauge coupling, with G a μν (a ¼ 1; …; 8) being the gluon field strength tensor, g s the strong coupling constant, m t the top mass and λ a the Gell-Mann matrices. The second term above contains both gtt and ggtt interactions that arise, in the conventions of Refs. [7,8], from the dimension-six operator where q L3 denotes the weak doublet of left-handed quark fields of third generation, t R the right-handed top quark field and ϕ is a weak doublet of Higgs fields (we define hereφ ¼ iτ 2 ϕ Ã ). After electroweak symmetry breaking, this operator yields the top dipole moments where v ¼ 246 GeV is the vacuum expectation value of the neutral component of ϕ. For weakly-interacting new physics at the TeV scale, C 33 uGϕ ∼ Oð1Þ so that one expects d V;A ∼ 0.05. This exceeds both the chromomagnetic dipole moment generated in the SM at the one-loop level d V ¼ −0.007 [9] and the associated negligible chromoelectric moment [10].
Direct limits on the top chromomagnetic and chromoelectric dipole moments can be derived from measurements of the tt inclusive cross section at the Tevatron and the LHC [11][12][13], and of several tt differential distributions [14][15][16]. Moreover, weaker bounds have been recently calculated [17] from a CMS measurement of the tt spin correlation in LHC data at 8 TeV [18], as this observable is also modified by the anomalous interactions of Eq. (2). The most stringent limits on d V and d A however arise nowadays from low-Q 2 probes, such as measurements of the neutron electric dipole moment, which constrain jd A j ≤ 9.5 × 10 −4 at the 90% confidence level (C.L.) [16], and rare B-meson decays, which imply −3.8 × 10 −3 ≤ d V ≤ 1.2 × 10 −3 at the 95% C.L. [19].
At the LHC running at 14 TeV, and even more at 100 TeV, a significant amount of top-antitop pairs with a multi-TeV invariant mass will be produced, with contributions dominated by the gluon-gluon fusion channel. These kinematical configurations with very large momentum transfer allow one to explore the structure of the ttg couplings at the shortest distances, and should then be particularly sensitive probes of the top dipole moments [20,21]. After reviewing the constraints that can be obtained from the measurements of total production cross sections, in this paper we therefore focus on the study of very high mass top-antitop final states, where the top quarks are necessarily highly boosted. We consider a simple-minded approach to extract the top-antitop signal from the large QCD background, and verify that, at 100 TeV, this is sufficient to significantly push the sensitivity to both chromoelectric and chromomagnetic dipole moments.

II. TEVATRON AND LHC LIMITS
The combination of inclusive tt cross section measurements at the Tevatron and the LHC provides much stronger limits on the top dipole moments than the individual measurements. The complementarity of these two colliders is due to a very different functional dependence of the total cross section on d V and d A at the Tevatron (pp collisions at 1.96 TeV) and the LHC (pp collisions at 7, 8 TeV), owing to the dominance of qq → tt at the former collider and gg → tt at the latter. Making use of the FEYNRULES package [22,23] to import the Lagrangian of Eq. (2) into MADGRAPH5_AMC@NLO [24], we evaluate the tt total production cross section at the Tevatron and the LHC with 8 TeV, σ ð2Þ tt and σ ð8Þ tt respectively, including the leadingorder contributions of the top dipole moments. Since the amplitudes contain at most two insertions of the anomalous vertices in Eq. (2), the dependence on d A and d V can be parametrized by a fourth-order polynomial in these two variables. 1 We find when employing the NNPDF 2.3 set of parton densities [25]. We extract the limits on d V and d A in Fig. 1 by using the most recent total rate measurements at the Tevatron, σ ð2Þ exp ¼ 7.60 AE 0.41 pb [26], and the LHC with 8 TeV, σ ð8Þ exp ¼ 241.5 AE 8.5 pb [27], together with the most precise SM predictions at the next-to-next-to-leading order accuracy in QCD, σ The shaded area corresponds to the overlap of the corresponding 95% C.L. allowed regions from the Tevatron (dashed) and the LHC (solid) measurements, and implies that −0.012 ≤ d V ≤ 0.023 and jd A j ≤ 0.087. Our results are found compatible with the ones previously obtained in Ref. [13] and given at the 68.3% C.L., and are stronger than 1 We have made the choice of keeping all terms in the expansion of Eq. (5). Third-order and fourth-order terms do not play any role in the regions relevant for the combined limits. Quadratic terms correspond to contributions of order Λ −4 . Concerning the d 2 V term, dimension-eight operators could generate additional contributions at the same Λ −4 order, and including them would change the interpretation of the limits obtained on d V . On the other hand, dropping quadratic and higher-order terms, or equivalently truncating the series at order Λ −2 , would imply to neglect effects that formally appear at order Λ −4 but that are relevant in the extraction of the limits. We have therefore adopted the expansion of Eq. (5). the bounds derived from the spin correlation measurements [17,18]. 2 With the important amount of tt data to be collected at the upcoming LHC run with pp collisions at 14 TeV, top dipole moments could be probed by going beyond the use of inclusive total cross sections, and could benefit from differential cross section measurements. For illustration, we consider three representative cases and focus on inclusive cross section measurements as well as on the production of tt pairs with an invariant mass m tt required to be larger than either 1 or 2 TeV. These last two observables are expected to exhibit an enhanced sensitivity to the top dipole moments because of the larger momentum transfers that are now (phase-space) favored and the specific Lorentz structure of the top dipole operators in Eq. (2). For the inclusive measurement, our predictions rely on standard tt reconstruction techniques so that the tt signal is considered well separable from the background, in a way similar to what has been achieved with collision data at 7 and 8 TeV. We subsequently assume an overall uncertainty of 5% on the would-be measurement at 14 TeV. (For comparison, the 8 TeV measurement has a 3.5% uncertainty.) When the tt system has an invariant mass m tt > 1 TeV or 2 TeV, the produced top quarks are usually boosted. This renders any associated measurement more difficult because of both the smaller statistics and the large QCD multijet contribution where a pair of boosted top and antitop quarks is faked. In order to realistically estimate the uncertainty that would be associated with measurements in such regimes, we restrict our analysis to tt pairs produced in the central region of the detector (with a pseudorapidity satisfying jηj < 2). This ensures a better performance of the top tagging algorithms due to a finer detector granularity, so that one could aim for a better rejection of the QCD background. We employ, in the m tt > 1 TeV (2 TeV) case, the third working point of CMS for top quarks with a transverse momentum p T > 400 GeV (800 GeV) [30], so that a boosted top quark will be correctly tagged with an efficiency of about 12.5%, for a mistagging rate of a QCD jet as a top quark of about 0.03% (the studies of top-tagging performance by ATLAS are documented in Ref. [31]). We present, in Table I, the corresponding fiducial cross sections for both the top-antitop signal and the multijet background, together with the sensitivity defined as ffiffiffiffiffiffiffiffiffiffiffiffi S þ B p =S where S and B are the numbers of signal and background events normalized to a luminosity of 100 fb −1 , respectively.
We use the above results to deduce the statistical uncertainties that would be related to a fiducial cross section measurement in the large m tt region for pp collisions at 14 TeV. Adding in quadrature systematic uncertainties, assumed to be 5%, 3 we show, in Fig. 2, the limits expected to be extracted by using inclusive cross sections (solid black) and after imposing that the top-antitop systems under consideration have an invariant-mass larger than 1 TeV (solid blue annulus) and 2 TeV (solid red ellipse). For comparison, we superimpose the limits derived from inclusive cross section measurements at the Tevatron (dashed) while the bounds derived from the first LHC run, already presented in Fig. 1, are omitted as they are superseded by the potential LHC result at 14 TeV. Aside from the fact that limits derived when m tt is required to be large are more constraining, the shape of the allowed region in the ðd V ; d A Þ plane changes for m tt > 2 TeV, turning out to be an ellipse instead of an annulus. This can be traced back to the smaller importance of the cubic terms Furthermore, despite the large coefficients, the quartic terms d 4 V , d 4 A and d 2 V d 2 A are always subleading for d V;A ≲ 0.03.
We estimate, with the shaded area in the figure, the 95% C.L. bounds derived from combining the inclusive and high-m tt measurements at the LHC Run 2. The top dipole moments are constrained to fulfill −0.0086 ≤ d V ≤ 0.012 and jd A j ≤ 0.019, so that the future run of the LHC is expected to improve the limits on d V by a factor of 2. Moreover, the sensitivity to a CP-violating chromoelectric moment using a CP-even observable-such as the tt fiducial cross section at high m tt -is found remarkably similar to the expected one when measuring CP-odd triple product asymmetries (jd A j ≤ 0.02 [17]). Finally, assuming the Wilson coefficient C 33 uGϕ to be of at most 4π, one can translate the bounds on d V and d A into a lower limit on the new physics scale Λ that is found to be Λ ≳ 5 TeV. [This ensures the validity of the effective field theory approach used in Eq. (2).] For smaller C 33 uGϕ the limits on Λ are correspondingly looser.

III. SENSITIVITY AT 100 TEV
A significantly large number of tt pairs with a multi-TeV invariant mass are expected to be produced in several ab −1 of pp collisions at 100 TeV. With the opening of new kinematical regimes that have never been probed so far, the performance of the standard boosted top reconstruction techniques, developed in the LHC context and relying of the top-jet substructure [21,[30][31][32][33][34][35][36][37][38][39][40][41], needs to be reassessed, also in view of the potential improvements in the features of future detectors. Considering that a top quark with p T ¼ 5 TeV would have its three primary decay products contained within a cone of R ≲ 0.05, it is clear that the effectiveness of a tagging based on the jet substructure would be strongly tied to the details of the detectors, such as tracking performance and calorimetric granularity. 4 We therefore focus our study on a rather safe feature, in principle usable with any conceivable detector design, which should enable the differentiation of highly boosted top quarks from generic QCD jets. That is the spectrum of muons coming from the top decays, already discussed in the context of top tagging in Refs. [43][44][45]. We show here that the study of this observable provides a proof of principle that a large-invariant-mass tt signal can be isolated from the otherwise overwhelming QCD background, in spite of the limited efficiency due to the branching ratio for a muonic decay. It is likely that more efficient top taggers, along the lines of what developed for the TeV regime of relevance to the LHC will significantly improve the usable statistics and our final sensitivity to anomalous top dipole moments.
We generate hard top-pair and dijet events with MADGRAPH5_AMC@NLO and further match them to the parton showering and hadronization algorithms included in the PYTHIA 8 program [46]. We then consider all central (jηj < 2) jets with p T > 1 TeV reconstructed by making use of an anti-k T algorithm with a radius parameter R ¼ 0.2 [47], as implemented in the FASTJET package [48]. We finally analyze the reconstructed events with the MADANALYSIS 5 framework [49]. We preselect events featuring at least two reconstructed jets, and the invariant mass of the system made of the two leading jets (generically denoted by m tt ) is demanded to be greater than some threshold M min . In Fig. 3 (upper plot) we show the cross section as a function of the minimum dijet invariant mass M min , both for the top signal and for the inclusive multijet background, which is over 2 orders of magnitude larger. As indicated above, in order to extract a top-antitop signal from the multijet QCD background, we further require at least one muon lying within a cone of R ¼ 0.2 around any of the selected jets. This final step of the selection relies on the different properties of the muons arising from multijet and tt events. In the former case, they are found to only carry a small fraction of the jet transverse momentum, as inferred by their production from B-meson and D-meson decays, whereas in the latter case they are induced by  2 (color online). Expected 95% C.L. limits on d V and d A at the future LHC run, with ffiffi ffi s p ¼ 14 TeV. We show results using inclusive tt cross sections (solid black) and after considering only top-antitop pairs with an invariant mass larger than 1 TeV (solid blue annulus) and 2 TeV (solid red ellipse). For comparison, the Tevatron limit is also displayed (dashed). 4 After completion of this paper, a study appeared [42], focused on the issue of tagging multi-TeV top jets. That paper confirms the potential for good tagging efficiency and background rejection, provided the detector performance can match the challenge of dealing with these high-density track environments. prompt top decays and can get a significant fraction of the top transverse momentum. Events are consequently selected by requesting a minimum value z min for the variable z μ , defined by where we maximize, over the n muons possibly present in a given event, the ratio of the muon transverse momentum p T ðμ i Þ to the corresponding jet transverse momentum p T ðj i Þ. The efficiency of signal and background, as a function of the requirement on z μ , are shown in the central plot of Fig. 3. In the case of the top signal, we removed from the definition of this efficiency the trivial branching ratio factor for the decay tt → μ þ X (the contribution from muonic decays of the b hadrons is negligible in the relevant regions of z μ ). Since the z μ distribution has a slight dependence on the transverse momentum of the jets, we show, as an example, the result averaged over the set of events with dijet invariant mass larger than 6 TeV. Convoluting the efficiencies with the appropriate rates results in the cross sections shown in the bottom plot of Fig. 3. As we can see, imposing z μ ≳ 0.5 reduces the background by orders of magnitude, down to a level comparable to the signal. Of course the experimental implementation of a selection like this will require a good muon identification efficiency in the dense jet environment, and a good momentum resolution in the multi-TeV momentum range. The study of the dijet rate, in the region of z min where the background dominates, will provide nevertheless a good control sample for a robust data-driven determination of the absolute background normalization.
To generate the following results, we considered the three reference M min thresholds of 6, 10 and 15 TeV. The corresponding requirements on the z μ -variable, which we used to optimize the signal significance, are given in Table II. We also include here the luminosities that are necessary for a signal extraction at the 5σ level. We have verified that the results are similar when using HERWIG++ [50] instead of PYTHIA, or Alpgen þ Herwig6 [51,52].
Using the above results to deduce the statistical uncertainties associated with a would-be tt cross section measurement at 100 TeV, we additionally account for systematic uncertainties of 5% before deriving the expectation for constraining the top dipole moments. (As previously discussed, we use a 5% systematic uncertainty as a reasonable reference value.) Assuming an integrated luminosity of 10 ab −1 , we display our results on Fig. 4, together with the limits expected from the LHC run at 14 TeV when using tt pairs with m tt > 2 TeV. As for the LHC case, the selection on the top-antitop invariant mass enforces the contributions to the tt cross section that are quadratic in d V and d A to dominate, so that the allowed regions of the ðd V ; d A Þ plane are ellipses. As also suggested by Table II, the statistical uncertainties start to be important for large m tt thresholds, so that the bounds derived when m tt > 15 TeV are similar to those obtained when m tt > 10 TeV. An optimal m tt selection would however strongly depend on FIG. 3. Top: Cross sections for inclusive tt and dijet final states, with the mass of the two leading jets above M min (jets as defined in the text). Central: efficiency of the cut z μ > z min , for tt → μ þ X and for dijet events, with M min > 6 TeV. Bottom: z min dependence of the total rates for tt → μ þ X and dijets, with M min > 6 TeV.

IV. SUMMARY
Direct limits on the top dipole couplings improve greatly by probing higher momentum transfers, as a consequence of their Lorentz structure. A 100 TeV pp collider is therefore a suitable machine to explore these anomalous interactions, in order to expose the indirect effects of new heavy states. In this paper, we have investigated the sensitivity of the future run of the LHC at 14 TeV and of a 100 TeV collider to anomalous top chromoelectric and chromomagnetic dipole moments d V and d A . We have considered both the study of tt inclusive cross sections and of top-antitop pairs with a multi-TeV invariant mass. The summary of our results is shown in Fig. 5 where we compare the current direct bounds to the projected limits for 100 fb −1 at 14 TeV and 10 ab −1 at 100 TeV.
The sensitivity at 100 TeV is expected to allow for a very important improvement of the bounds derived from measurements at the Tevatron and at the previous and future LHC runs, so that the d V and d A allowed ranges could be reduced by more than 1 order of magnitude with respect to the current direct limits. Furthermore, one may also wonder whether the expected LHC limits on d V and d A , obtained from tt spin correlation measurements and CP-odd triple product asymmetries, respectively, could be improved at 100 TeV (at the LHC, they are of the same order as the expected ones from cross section measurements in Fig. 2). Clearly, considering large energy scales-by selecting, for instance, high-m tt top-antitop pairs as done in this paperenhances the effect on the anomalous contributions, not only in the fiducial cross section but also in the angular distributions. However, the angular distributions that may be measured in a typical LHC boosted top kinematical regime of m tt ≲ 2 TeV (see, e.g., Ref. [53]) may not be useful for ultraboosted tops, and deserve further investigations. Further studies would also be desirable to evaluate the complementarity of the measurements discussed in this paper, with those possible with e þ e − collisions at the top-antitop threshold, where a large statistics is foreseen by the e þ e − option of the FCC complex [54], and at higher energies (e.g., at CLIC [55] or ILC [56]).
As a by-product of our study, we developed and described a new robust and effective approach to the problem of tagging top-antitop final states of large invariant mass, exploiting the hard muon spectrum in top decays. We are confident that more detailed studies of top-tagging algorithms, made possible also by the more aggressive detector technologies envisaged for the future collider experiments, can further improve our results. What we showed is just an example of possible opportunities offered by the huge samples of top quarks available at a 100 TeV pp collider. Other areas that would certainly benefit include the study of rare or forbidden decays (e.g. t → q þ g=Z=γ=H (q ¼ u; c), tests of electroweak couplings (e.g. in s-channel single top production pp → W Ã → tb at very large invariant mass), and high-precision measurements of production asymmetries, spin correlations, etc. We look forward to future studies addressing these observables.