Probing the top-Higgs boson FCNC couplings via the $h\to\gamma\gamma$ channel at the HE-LHC and FCC-hh

Although the anomalous FCNC couplings between the top quark and Higgs boson may arise from different sources, an effective field theory approach can describe the effects of NP beyond the SM in a model-independent way AguilarSaavedra:2004wm . The most general Lagrangian for the top-Higgs FCNC interactions is written as

where $\kappa_{tuH}$ and $\kappa_{tcH}$ represent the strength of top-Higgs FCNC interactions. In this study we take them as real and symmetric, i.e., $\kappa_{tqH}=\kappa^{\dagger}_{tqH}=\kappa_{qtH}=\kappa^{\dagger}_{qtH}$ ($q=u,c$), since we here do not intend to consider CP-violating effects.

The decay width of the dominant top quark decay mode $t\rightarrow Wb$ could be found in Ref. Li:1990qf . Neglecting the light quark masses and assuming the dominant top decay width $t\to Wb$, the Next-to-Leading Order (NLO) BR($t\to qh$) is given by Greljo:2014dka ; Liu:2018bxa

with the Fermi constant $G_{F}$ and $x_{i}=m_{i}/m_{t}~{}(i=W,\ h)$. Here the factor $\lambda_{\rm QCD}$ is the NLO QCD correction to BR$(t\to qh)$ and equals about 1.1 Zhang:2008yn ; Drobnak:2010wh ; Zhang:2013xya . In our work, we require $\kappa_{tqh}\leq 0.04$ to satisfy the direct constraint from the ATLAS result mentioned in the previous section.

At the LHC, the cross section for $pp\to thj$ involving top-Higgs FCNC couplings would be coming from two subprocesses: (i) top pair production followed by one FCNC top decay, $pp\to t\bar{t}\to thj$, shown in Fig. 1(a-b) (henceforth referred to as ‘top FCNC decay’); (ii) single top-Higgs associated production in presence of a jet, $pp\to thj$, as shown in Fig. 1(c-f), which includes a $gg$ (henceforth referred to as ‘tH associated production’) and a $qg$ (henceforth referred to as ‘qg fusion’) induced subchannels, respectively yielding a(n) (anti)quark or gluon in the final state. The contribution of other subprocesses, such as $q\bar{q}$ fusion channels, is smaller than the above ones due to the suppression from colour factors and Parton Distribution Functions (PDFs) and thus is not shown in the Feynman diagrams, but all the contributions are included in our calculations. Obviously, the conjugated processes can also occur at tree level and are accounted for.

For the simulations of the HE-LHC and FCC-hh dynamics, we first use the FeynRules package feynrules to extract the Feynman rules from the effective Lagrangian and to generate the Universal FeynRules Output (UFO) files and calculate the LO cross sections of $pp\to thj$ by using MadGraph5-aMC$@$NLO mg5 with NNPDF23L01 PDFs Ball:2014uwa , considering the renormalisation and factorisation scales to be $\mu_{R}=\mu_{F}=\mu_{0}/2=(m_{t}+m_{h})/2$. In our numerical calculations, the SM input parameters are taken as pdg :

In Fig. 2, we show the dependence of the cross sections for the three $thj$ subprocesses on the top-Higgs FCNC coupling parameter at the HE-LHC and FCC-hh for two scenarios, as follows: Case I is for $\kappa_{tqh}=\kappa_{tuh},\kappa_{tch}=0$ whereas Case II is for $\kappa_{tqh}=\kappa_{tch},\kappa_{tuh}=0$. The cuts on the transverse momentum ($p_{T}^{j}$) and pseudo-rapidity ($\eta_{j}$) of the extra jet are shown in the figures for both the HE-LHC and FCC-hh. From Fig. 2 one can see that, for a given coupling parameter $\kappa_{tqh}$, the production cross sections can be very significant at the higher c.m. energies of these two future machine. Besides, we also have the following observations.

1. 1.

   For both Case I and II, the dominant contribution to the $thj$ final state is from (resonant) pair production, $pp\to t\bar{t}\to thj$. However, the other two contributions from tH associated production and qg fusion cannot be neglected, especially for Case I. To be specific, in this scenario, when $\sqrt{s}=27~{}(100)$ TeV and $\kappa_{tuh}=0.04$, the cross section of the top FCNC decay process is about 3.5 (30.9) pb while the cross section of the tH associated production process is about 1.7 (10.7) pb with the one for qg fusion being 1.3 (8.4) pb.

2. 2.

   For the same values of $\kappa_{tuh}$ and $\kappa_{tch}$, the cross sections coming from the tH associated production and qg fusion processes in Case I are much larger than those in Case II: this is because the $u$-quark has a larger PDF than that of the $c$-quark. To be specific, in Case II, when $\sqrt{s}=27~{}(100)$ TeV and $\kappa_{tch}=0.04$, the cross section of the tH associated production process is only about 0.3 (3.1) pb while for qg fusion the rates are 0.19 (1.96).

In this section, we present the numerical calculations at the HE-LHC and FCC-hh of the processes

where $\ell=e,\mu$ and $j$ represents (a(n) (anti)quark or gluon) jet, interfaced to the subsequent parton shower by using the MLM matching scheme Alwall:2007fs ; Alwall:2008qv . The final state topology is thus characterised by two photons appearing as a narrow resonance centered around the SM-like Higgs boson mass, at least two jets with exactly one being tagged as $b$-jet, one charged lepton and missing transverse momentum from the undetected neutrino. The main sources of background events that include both a Higgs boson decaying into di-photons in association with other particles and non-resonant production of $\gamma\gamma$ pairs are accounted for here:

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  $pp\to t\bar{t}h$,

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  $pp\to thj$,

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  $pp\to W^{\pm}jjh$,

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  $pp\to t\bar{t}\gamma\gamma$,

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  $pp\to tj\gamma\gamma$,

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  $pp\to\gamma\gamma W^{\pm}jj$.

The parton level events for the signal and the SM backgrounds are interfaced to parton shower, fragmentation and hadronisation by using PYTHIA8.20 pythia8 . Then, we have passed all generated events through Delphes3.4.2 delphes3 for detector simulation. Finally, event analysis is performed by using MadAnalysis5 ma5 . As far as jet reconstruction is concerned, the anti-$k_{t}$ algorithm Cacciari:2011ma with a jet radius of 0.4 is used. For the HE-LHC and FCC-hh analysis, we have used the default HL-LHC and FCC-hh detector card configuration implemented into the aforementioned detector emulator.

The cross sections of the signal and dominant backgrounds at LO are adjusted to NLO QCD through $K$-factors, i.e., $K=1.4$ for the $pp\to t\bar{t}\to thj$ process Mangano:2016jyj , $K=1.5$ for the tH associated production process Wang:2012gp ; zhangcen and $K=1.3$ for the $pp\to t\bar{t}h$ process Mangano:2016jyj ; Cepeda:2019klc ; Yu:2014cka . For the sake of simplicity, we have rescaled the other SM background processes by a $K$-factor of 1.5. This approximation does not have a significant impact on our derived sensitivities and can be fully addressed in a future analysis.

In order to identify objects, we impose the following basic cuts to select the events Mandrik:2018yhe :

where $\Delta R$ is the angular distance between any two objects.

In order to choose appropriate kinematic cuts, in Fig. 3²²2Hereafter, in figures and tables, by using ‘thj_tuh ($pp\to thj$ via $tuh$)’ and ‘thj_tch($pp\to thj$ via $tch$)’, we will intend the contribution to the signal due to tH associated production plus $qg$ fusion when only including the $tuh$ or $tch$ coupling on its own, respectively., we plot some differential distributions for signals and SM backgrounds at the HE-LHC at 27 TeV, such as the (ordered) transverse momentum distributions of the two photons, $p_{T}^{\gamma_{1,2}}$, the separation, $\Delta R_{\gamma_{1},\gamma_{2}}$ and $\Delta R_{\gamma_{1}\gamma_{2},j}$, and invariant mass, $M_{\gamma\gamma}$, distributions of the two photons, the transverse mass distribution for the $\ell\not{E}_{T}$, $M_{T}(l)$, and $b\ell\not{E}_{T}$, $M_{T}(bl)$ systems, and the invariant mass, $M_{\gamma\gamma j}$. Based on these distributions, we impose a further set of cuts.

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  Cut 1: Exactly one isolated lepton (electron or muon), at least two jets and one of which must be $b$-tagged.

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  Cut 2: At least two photons with $p_{T}^{\gamma_{1}}>60\rm~{}GeV$, $p^{\gamma_{2}}_{T}>30\rm~{}GeV$, since the two photons in the signal and resonant SM backgrounds come from the Higgs boson they have a harder $p_{T}$ spectrum than those in the non-resonant SM backgrounds.

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  Cut 3: The distance between two photons lies in $1.8<\Delta R_{\gamma_{1},\gamma_{2}}<3.5$, the distance between the di-photon system and the extra light jet lies in $\Delta R_{\gamma\gamma,j}<1.8$.

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  Cut 4: The invariant mass of the di-photon system, $M_{\gamma\gamma}$, is peaked in both the signals and resonant backgrounds, thus we require $M_{\gamma\gamma}$ to be in the range $|M_{\gamma\gamma}-m_{h}|<2$ GeV.

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  Cut 5: The transverse mass $M_{T}(\ell)$ and $M_{T}(b\ell)$ cuts are $M_{T}(\ell)>30\rm~{}GeV$ and $100~{}{\rm GeV}<M_{T}(b\ell)<180\rm~{}GeV$.

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  Cut 6: The invariant mass $M_{\gamma\gamma j}$ cut is $160~{}{\rm GeV}<M_{\gamma\gamma j}<190\rm~{}GeV$.

For the FCC-hh analysis at 100 TeV, we use the same selection cuts for the signal and SM backgrounds because the distributions are very similar to the case of HE-LHC presented in Fig. 3. In fact, the difference between the HE-LHC and FCC-hh mainly comes from the different detector configurations. The effects of the suitable cuts on the signal and SM background processes are illustrated in Tab. 2 and Tab. 3 at the HE-LHC and FCC-hh, respectively. Due to the different $b$-tagging rates for $u$- and $c$-quarks, the signal efficiencies of the two top (anti)quark decays differ after applying requirements on the $b$-tagged jet multiplicity. Thus we give the events separately for $q=u,c$. One can see that, at the end of the cut flow, the largest SM background is the $pp\to t\bar{t}h$ process, which is about 0.001 fb and 0.02 fb at the HE-LHC and FCC-hh, respectively. Besides, the $tj\gamma\gamma$ process can also generate significant contributions to the SM background due to the large production cross section. Finally, notice that, after the final cuts, the dominant signal contribution comes from the FCNC top (anti)quark decay process, so that one can safely ignore the contribution from the top-Higgs associated production channels.

In Figs. 4-5, we plot the exclusion limits at 95% CL in the plane of the integrated luminosity and the BR$(t\to qh)$’s at the HE-LHC and FCC-hh with the aforementioned two systematic error cases of $\delta=0$ and $\delta=10\%$. One can see that, our signals are rather robust against the systematic uncertainties on the background determination, though they differ between the HE-LHC (where limits change within a factor of $\approx 1.1$) and FCC-hh (where limits change within a factor of $\approx 3.1$) due to the relatively different number of SM background events. The values for 95% CL upper limits are summarised in Tab. 4. With a realistic 10% systematic error, the sensitivities are slightly weaker than those without any systematic error, being of the order of $10^{-5}$ at the 95% CL both at the HE-LHC and FCC-hh. For comparison, the recent 95% upper limits on BR$(t\to qh)$ obtained at the HL-LHC with an integrated luminosity of 3 ab^-1 by the ATLAS Collaboration atlas-14-3000 are also presented, which are obtained via the decay mode $t\to qh(\to b\bar{b})$. Besides, the upgraded ATLAS experiment has also estimated top-Higgs FCNC couplings via the decays $t\to ch(\to\gamma\gamma)$ at the HL-LHC and obtained an expected upper limit of BR$(t\to ch)<1.5\times 10^{-4}$ at 95% CL atlas1-14-3000 . Altogether, the sensitivity to the BR of the $t\to qh$ are two order of magnitude better than the most recent direct limits reported by the ATLAS Collaboration at the $13\,{\rm TeV}$ LHC.

Before closing, let us also review competing limits from other authors. Very recently, the author of Ref. Khanpour:2019qnw has studied the top-Higgs FCNC couplings in the triple-top signal at the HE-LHC and FCC-hh. The 95% CL upper limits on BR$(t\to uh)$ (and BR$(t\to ch)$) were found, respectively, as $7.01\times 10^{-4}~{}(3.66\times 10^{-4})$ at the HE-LHC with 15 ab^-1 and as $2.49\times 10^{-5}~{}(5.85\times 10^{-5})$ at the FCC-hh with 10 ab^-1. Furthermore, in the context of the 2-Higgs Doublet Model (2HDM), the authors of Ref. Jain:2019ebq have recently investigated the prospect for $t\to ch$ decay in top quark pair production via the $h\to WW^{\ast}\to\ell^{+}\ell^{-}+\not{E}_{T}^{miss}$ channel. For the HE-LHC and FCC-hh, the 95% CL upper limits on BR$(t\to ch)$ was found to be at the order of $10^{-4}$ with an integrated luminosity of 3 ab^-1 and such limits would be increased by an higher integrated luminosity. Finally, at the FCC-hh with an integrated luminosity of 10 ab^-1, Ref. Papaefstathiou:2017xuv has investigated the $t\to ch$ decay and the its BR can be constrained to $\mathcal{O}(10^{-5})$ either with or without considering $c$-jet tagging.