Probing $tqZ$ anomalous couplings in the trilepton signal at the HL-LHC, HE-LHC and FCC-hh

In the search for FCNC $tqZ$ anomalous interactions, the top quark FCNC coupling is explored in a model-independent way by considering the most general effective Lagrangian approach AguilarSaavedra:2008zc . The Lagrangian involving FCNC $tqZ$ interactions can be written as AguilarSaavedra:2008zc

	$\displaystyle\mathcal{L}_{\rm eff}$	$\displaystyle=$	$\displaystyle\sum_{q=u,c}[\frac{g}{4c_{W}m_{Z}}\kappa_{tqZ}\bar{q}\sigma^{\mu\nu}(\kappa_{L}P_{L}+\kappa_{R}P_{R})tZ_{\mu\nu}$		(1)
		$\displaystyle+$	$\displaystyle\frac{g}{2c_{W}}\lambda_{tqZ}\bar{q}\gamma^{\mu}(\lambda_{L}P_{L}+\lambda_{R}P_{R})tZ_{\mu}]+h.c.,$

where $c_{W}=\cos\theta_{W}$ and $\theta_{W}$ is the Weinberg angle, $P_{L,R}$ are the left- and right-handed chirality projector operators, $\kappa_{tqZ}$ and $\lambda_{tqZ}$ are effective couplings for the corresponding vertices. The Electro-Weak (EW) interaction is parameterised by the coupling constant $g$ and the mixing angle $\theta_{W}$. The complex chiral parameters $\kappa_{L,R}$ and $\lambda_{L,R}$ are normalised as $|\kappa_{L}|^{2}+|\kappa_{R}|^{2}=|\lambda_{L}|^{2}+|\lambda_{R}|^{2}=1$.

The partial widths for the FCNC decays, wherein we separate the contributions of the two tensor structures entering the above equation, are given by

	$\displaystyle\Gamma(t\to qZ)~{}(\sigma^{\mu\nu})$	$\displaystyle=$	$\displaystyle\frac{\alpha}{128s_{W}^{2}c_{W}^{2}}|\kappa_{tqZ}|^{2}\frac{m_{t}^{3}}{m_{Z}^{2}}\left[1-\frac{m_{Z}^{2}}{m_{t}^{2}}\right]^{2}\left[2+\frac{m_{Z}^{2}}{m_{t}^{2}}\right],$
	$\displaystyle\Gamma(t\to qZ)~{}(\gamma^{\mu})$	$\displaystyle=$	$\displaystyle\frac{\alpha}{32s_{W}^{2}c_{W}^{2}}|\lambda_{tqZ}|^{2}\frac{m_{t}^{3}}{m_{Z}^{2}}\left[1-\frac{m_{Z}^{2}}{m_{t}^{2}}\right]^{2}\left[1+2\frac{m_{Z}^{2}}{m_{t}^{2}}\right].$		(2)

After neglecting all the light quark masses and assuming the dominant top decay partial width to be that of $t\to bW$ Li:1990qf

$\displaystyle\Gamma(t\to bW^{+})=\frac{\alpha}{16s_{W}^{2}}|V_{tb}|^{2}\frac{m_{t}^{3}}{m_{W}^{2}}\left[1-3\frac{m_{W}^{4}}{m_{t}^{4}}+2\frac{m_{W}^{6}}{m_{t}^{6}}\right],$

(3)

then the BR($t\to qZ$) can be approximated by AguilarSaavedra:2004wm

	$\displaystyle{\rm BR}(t\to qZ)~{}(\sigma^{\mu\nu})$	$\displaystyle=$	$\displaystyle 0.172|\kappa_{tqZ}|^{2},$		(4)
	$\displaystyle{\rm BR}(t\to qZ)~{}(\gamma^{\mu})$	$\displaystyle=$	$\displaystyle 0.471|\lambda_{tqZ}|^{2}.$

Here, the Next-to-Leading Order (NLO) QCD corrections to the top quark decay via model-independent FCNC couplings are also included and the $k$-factor is taken as 1.02 Zhang:2008yn ; Drobnak:2010wh .

For the simulations of the ensuing collider phenomenology, we first use the FeynRules package feynrules to generate the Universal FeynRules Output (UFO) files Degrande:2011ua . The LO cross sections are obtained by using MadGraph5-aMC$@$NLO Alwall:2014hca with NNPDF23L01 Parton Distribution Functions (PDFs) Ball:2014uwa taking the renormalisation and factorisation scales to be $\mu_{R}=\mu_{F}=\mu_{0}/2=(m_{t}+m_{Z})/2$. The numerical values of the input parameters are taken as follows pdg :

	$\displaystyle m_{t}$	$\displaystyle=173.1{\rm~{}GeV},\quad m_{Z}=91.1876{\rm~{}GeV},\quad m_{W}=80.379{\rm~{}GeV},$		(5)
	$\displaystyle\alpha_{s}(m_{Z})$	$\displaystyle=0.1181,\quad G_{F}=1.16637\times 10^{-5}\ {\rm GeV^{-2}}.$

In Fig. 2, we show the total cross sections $\sigma$ in pb versus the two kinds of coupling parameters, $\kappa_{tqZ}$ and $\lambda_{tqZ}$, at LO. One can see that the dipole $\sigma^{\mu\nu}$ terms lead to larger cross sections with the same coupling values. For the two kinds of couplings, the cross sections of $\bar{u}g\to\bar{t}Z$ are overwhelmed by $ug\to tZ$ due to the difference between the $u$-quark and $\bar{u}$-quark PDF of the proton. Thus, if we consider the leptonic top decay modes, more leptons will be observed than anti-leptons for a given c.m. energy and integrated luminosity. Due to the similarly small PDFs of the $c$-quark and $\bar{c}$-quark, the cross section of $\bar{c}g\to\bar{t}Z$ is essentially the same as that of $cg\to tZ$ and they are much smaller than the cross section of $ug\to tZ$ for the same values of the coupling parameter. This implies that the sensitivity to the FCNC coupling parameter $\kappa_{tuZ}~{}(\lambda_{tuZ})$ will be better than $\kappa_{tcZ}~{}(\lambda_{tcZ})$.

The signal is produced through the processes (herein, all charge conjugated channels are included)

	$\displaystyle pp$	$\displaystyle\to$	$\displaystyle t(\to bW^{+}\to b\ell^{+}\nu)Z(\to\ell^{+}\ell^{-}),$		(6)
	$\displaystyle pp$	$\displaystyle\to$	$\displaystyle t(\to bW^{+}\to b\ell^{+}\nu)\bar{t}(\to\bar{q}Z(\to\ell^{+}\ell^{-})),$		(7)

where $\ell=e,\mu$ and $q=u,c$, the latter eventually generating a jet $j$.

As intimated, the final state for the signal is characterised by three leptons (electrons and/or muons), one $b$-tagged jet plus missing transverse energy from the escaping undetected neutrino in the $tZ$-FCNC case. In the final state from the $t\bar{t}$-FCNC process, there is an additional jet arising from the hadronisation of the quark $q$. Furthermore, notice that the interference between the $tZ$-FCNC (with an additional $q$ emission) and $t\bar{t}$-FCNC processes can be neglected Barros:2019wxe .

The main backgrounds which yield identical final states to the signal ones are $W^{\pm}Z$ production in association with jets, $t\bar{t}V$ ($V=W^{\pm},Z$) and the irreducible $tZj$ process, where $j$ denotes a non-$b$-quark jet. Besides, in the top pair production case (where the top quark pairs decay semi-leptonically), a third lepton can come from a semi-leptonic $B$-hadron decay inside the $b$-jet. Here, we do not consider multijet backgrounds where jets can fake electrons, since they are generally negligible in multilepton analyses Khachatryan:2014ewa . Other processes, such as the $t\bar{t}h$, tri-boson events or $W^{\pm}$ + jets are not included in the analysis due to the very small cross sections after applying the cuts.

The signal and background samples are generated at LO by interfacing MadGraph5-aMC$@$NLO to the the Monte Carlo (MC) event generator Pythia 8.20 Sjostrand:2014zea for the parton showering. All produced jets are forced to be clustered using FASTJET 3.2 Cacciari:2011ma assuming the anti-$k_{t}$ algorithm with a cone radius of $R=0.4$ Cacciari:2008gp . All event samples are fed into the Delphes 3.4.2 package deFavereau:2013fsa with the default HL-LHC, HE-LHC and FCC-hh detector cards. Finally, the event analysis is performed by using MadAnalysis5 Conte:2012fm . To include inclusive QCD contributions, we generate the hard scatterings of signal and backgrounds with up to one additional jet in the final state, followed by matrix element and parton shower merging with the MLM matching scheme Frederix:2012ps . Furthermore, we renormalise the LO cross sections for the signals to the corresponding higher order QCD results of Refs. Li:2011ek ; Zhang:2010dr ; Degrande:2014tta . For the SM backgrounds, we generated LO samples renormalised to the NLO or next-NLO (NNLO) order cross sections, where available, taken from Refs. Lazopoulos:2008de ; Kardos:2011na ; Czakon:2013goa ; Mangano:2016jyj ; Campanario:2010hp ; Campbell:2012dh ; Frederix:2017wme ; Frixione:2015zaa ; Pagani:2020mov ; Azzi:2019yne . For instance, the LO cross section for the $W^{\pm}Z$ + jets background (one of the most relevant ones overall) is renormalised to the NLO one through a $k$-factor of 1.3 Campanario:2010hp at 14 TeV LHC and, as an estimate, we assume the same correction factor at the HE-LHC and FCC-hh. The LO cross section for the $t\bar{t}$ process is renormalised to the NNLO one by a $k$-factor of 1.6 Azzi:2019yne at the HL-LHC as well as HE-LHC and 1.43 Mangano:2016jyj at the FCC-hh.

In order to identify objects, we impose the following basic or generation (parton level) cuts for the signals and SM backgrounds:

$\displaystyle p_{T}^{\ell}>25~{}\text{GeV},\quad p_{T}^{j/b}>30~{}\text{GeV},\quad|\eta_{i}|<2.5,\quad\Delta R_{ij}>0.4~{}~{}(i,j=\ell,b,j),$

(8)

where $j$ and $b$ denote light-flavour jets and a $b$-tagged jet, respectively. Here, $\Delta R=\sqrt{\Delta\Phi^{2}+\Delta\eta^{2}}$ is the separation in the rapidity-azimuth plane. Next, we discuss the events selection by focusing on two cases: the $pp\to t\bar{t}\to tZj$ (henceforth referred to as ‘Case A’) process and the $pp\to tZ$ (henceforth referred to as ‘Case B’) process, respectively. As mentioned, the main difference is whether there is a light jet in the final state. We first discuss the selection cuts for Case A and then for Case B.

For case A, the trilepton analysis aims to select $t\bar{t}$ events where one of the top quarks decays via the FCNC process ($t\to qZ\to q\ell_{1}\ell_{2}$) while the other top quark decays leptonically ($t\to Wb\to\ell_{3}\nu b$). Here, the leptons $\ell_{1}$ and $\ell_{2}$ are the two Opposite-Sign and Same-Flavour (OSSF) leptons that are assumed to be the product of the $Z$-boson decay, whereas the third lepton, $\ell_{3}$, is assumed to originate from the leptonically decaying top quark, with the $b$-tagged jet emerging from the $t\to bW^{+}$ decay and the light jet $j$ is the non-$b$-tagged one. Therefore, the following preselection is used for Case A (Cut 1):

• •

  exactly three isolated leptons with $p_{T}>30\rm~{}GeV$, in which at least one OSSF lepton pair;

• •

  at least two jets with $p_{T}>40\rm~{}GeV$, with exactly one of them $b$-tagged;

• •

  the missing transverse energy $E_{T}^{\rm miss}>30~{}\rm GeV$.

In Fig. 3, we plot some differential distributions for signals and SM backgrounds at the HL-LHC, such as the invariant mass distributions of the two leptons, $M_{\ell_{1}\ell_{2}}$, the transverse mass distribution for $M_{T}(\ell_{3})$ and $M_{T}(b\ell_{3})$ as well as the triple invariant mass, $M_{\ell_{1}\ell_{2}j}$. Furthermore, the top quark transverse cluster mass can be defined as Conte:2014zja

$\displaystyle M_{T}^{2}\equiv(\sqrt{(p_{\ell_{3}}+p_{b})^{2}+|\vec{p}_{T,\ell_{3}}+\vec{p}_{T,b}|^{2}}+|\vec{\not{p}}_{T}|)^{2}-|\vec{p}_{T,\ell_{3}}+\vec{p}_{T,b}+\vec{\not{p}}_{T}|^{2},$

(9)

where $\vec{p}_{T,\ell_{3}}$ and $\vec{p}_{T,b}$ are the transverse momenta of the third charged lepton and $b$-quark, respectively, and $\vec{\not{p}}_{T}$ is the missing transverse momentum determined by the negative sum of the visible momenta in the transverse direction.

According to the above analysis, we can impose the following set of cuts.

• •

  (Cut 2) Two of the same-flavour leptons in each event are required to have opposite electric charge and have an invariant mass, $M_{\ell_{1}\ell_{2}}$, compatible with the $Z$ boson mass, i.e., $|M(\ell_{1}\ell_{2})-m_{Z}|<15\rm~{}GeV$.

• •

  (Cut 3) The transverse mass of the $W^{\pm}$ candidate is required to be $50~{}{\rm GeV}<M_{T}^{\ell_{3}}<100~{}{\rm GeV}$ whereas the transverse mass of the top quark is required to be $100~{}{\rm GeV}<M_{T}^{b\ell_{3}}<200~{}{\rm GeV}$.

• •

  (Cut 4) The triple invariant mass $M_{\ell_{1}\ell_{2}j}$ cut is such that $140~{}{\rm GeV}<M_{\ell_{1}\ell_{2}j}<200~{}{\rm GeV}$.

We use the same selection cuts for the HE-LHC and FCC-hh analysis because the distributions are very similar to the case of the HL-LHC. The effects of the described cuts on the signal and SM background processes are illustrated in Tabs. 2–4. Due to the different $b$-tagging rates for $u$- and $c$-quarks, we give the events separately for $q=u,c$ for the signals. One can see that, at the end of the cut flow, the largest SM background is the $pp\to tZj$ process, which is about 0.048 fb, 0.144 fb and 1.45 fb at the HL-LHC, HE-LHC and FCC-hh, respectively. Besides, the $W^{\pm}Z$ + jets and $t\bar{t}Z$ processes can also generate significant contributions to the SM background. Obviously, the dominant signal contribution comes from the $t\bar{t}$-FCNC process, but the contribution from the $tZ$-FCNC production process cannot be ignored, especially for the $tuZ$ couplings.

For this case, we will mainly concentrate on the signal from the $ug\to tZ$ process due to the relative large cross section and we will veto extra jets in the following analysis. However, the final signals for Case A could also be considered as a source for Case B if the light quark is missed by the detector. Hence, we combine these processes into the complete signal events.

The process $ug\to tZ$ should include two leptons with positive charge, one coming from the decay $Z\to\ell^{+}\ell^{-}$ and the other from the top quark decay $t\to W^{+}b\to\ell^{+}\nu b$. Because the distributions for the signal and backgrounds are similar for the invariant mass $M_{\ell_{1}\ell_{2}}$ as well as the transverse masses $M_{T}(\ell_{3})$ and $M_{T}(b\ell_{3})$, we only plot the distributions for the distance of the OSSF lepton pair, $\Delta R(\ell_{1},\ell_{2})$, and the rapidity of the OSSF lepton pair, $y_{\ell_{1}\ell_{2}}$, in Fig. 4. (Here, the distributions are obtained at the HL-LHC, but the pattern is very similar at the HE-LHC and FCC-hh.) One can see that, for Case B, the $Z$ boson from the $ug\to tZ$ process concentrates in the forward and backward regions since the partonic c.m. frame is highly boosted along the direction of the $u$-quark.

Thus, we can impose the following set of cuts for Case B.

• •

  (Cut 1) There are three leptons in which at least two with positive charge and $p_{T}>30\rm~{}GeV$, exactly one $b$-tagged jet with $p_{T}>40\rm~{}GeV$ and the event is rejected if the $p_{T}$ of the subleading jet is greater than 25 GeV.

• •

  (Cut 2) The distance between the OSSF lepton pair should lie within $\Delta R(\ell_{1},\ell_{2})\in[0.4,1.2]$ while the corresponding invariant mass is required to be $|M(\ell_{1}\ell_{2})-m_{Z}|<15\rm~{}GeV$.

• •

  (Cut 3) The transverse masses of the reconstructed $W^{\pm}$ boson and top quark masses are required to satisfy $50~{}{\rm GeV}<M_{T}^{\ell_{3}}<100~{}{\rm GeV}$ and $100~{}{\rm GeV}<M_{T}^{b\ell_{3}}<200~{}{\rm GeV}$, respectively.

• •

  (Cut 4) The rapidity of the OSSF lepton pair is required to be $|y_{\ell_{1}\ell_{2}}|>1.0$.

The effects of these cuts on the signal and background processes for Case B are illustrated in Tabs. 5–7. One can see that all backgrounds can be suppressed efficiently after imposing such a selection. At the end of the cut flow, the $W^{\pm}Z$ + jets and $t\bar{t}$ production processes are the dominant SM backgrounds mainly due to the initially large cross sections.

To estimate the exclusion significance, we use the following expression Cowan:2010js :

$\displaystyle Z_{\text{excl}}=\sqrt{2\left[S-B\ln\left(\frac{B+S+x}{2B}\right)-\frac{1}{\delta^{2}}\ln\left(\frac{B-S+x}{2B}\right)\right]-\left(B+S-x\right)\left(1+\frac{1}{\delta^{2}B}\right)},$

(10)

with

$\displaystyle x=\sqrt{(S+B)^{2}-4\delta^{2}SB^{2}/(1+\delta^{2}B)}.$

(11)

Here, $S$ and $B$ represent the total signal and SM background events, respectively. Furthermore, $\delta$ is the percentage systematic error on the SM background estimate. Following Refs. Cowan:2010js ; Kling:2018xud , we define the regions with $Z_{\text{excl}}\leq 1.645$ as those that can be excluded at 95% CL. In the case of $\delta\to 0$, the above expressions are simplified as

$\displaystyle Z_{\text{excl}}=\sqrt{2[S-B\ln(1+S/B)]}.$

(12)

Using the results from Case A and Case B, we combine the significance for $Br(t\to uZ)$ with two kinds of couplings,

$\displaystyle Z_{\text{comb}}=\sqrt{Z_{\text{A}}+Z_{\text{B}}}$

(13)

while, for $Br(t\to cZ)$, we only use the results from Case A.

In Figs. 5–6, we plot the $95\%$ CL lines as a function of the integrated luminosity and BR$(t\to qZ)$ for the two kinds of couplings with three typical values of systematic uncertainties: $\delta=0,5\%$ and $10\%$. One can see from Fig. 5 that, for the tensor (vector) terms, the combined $95\%$ CL limits without systematic error on ${\rm BR}(t\to uZ)$ are $2.3~{}(5.3)\times 10^{-6}$ and $0.76~{}(1.2)\times 10^{-6}$ at the HE-LHC and FCC-hh with an integrated luminosity of 10 ab^-1, respectively. For this value of integrated luminosity, by taking into account a $5\%$ systematic error, the obtained limits are about $0.34~{}(1.47)\times 10^{-5}$ and $0.27~{}(1.21)\times 10^{-5}$, respectively, while, for the case $\delta=10\%$, the $95\%$ CL limits on ${\rm BR}(t\to uZ)$ change to $0.51~{}(2.53)\times 10^{-5}$ and $0.48~{}(2.2)\times 10^{-5}$, respectively. From Fig. 6, one can see that for the Case A, the $95\%$ CL limits without systematic error on ${\rm BR}(t\to cZ)$ are $0.45~{}(0.64)\times 10^{-5}$ and $1.13~{}(1.54)\times 10^{-6}$ at the HE-LHC and FCC-hh with an integrated luminosity of 10 ab^-1, respectively. By taking into account a $5\%$ systematic error, the obtained limits are about $1.43~{}(2.06)\times 10^{-5}$ and $1.35~{}(1.82)\times 10^{-5}$, respectively.

In Tab. 8, we list the exclusion limits at 95% CL at the future HL-LHC with 3 ab^-1, at the HE-LHC with 15 ab^-1 and at the FCC-hh with 30 ab^-1, respectively, with two systematic error: $\delta=0\%$ and $\delta=10\%$. From Tab. 8, we have the following observations to make,

• •

  More stringent limits are obtained on the $tuZ$ coupling compared to the $tcZ$ coupling due to the larger cross section in the corresponding signal.

• •

  For the $tuZ$ coupling, the sensitivities for the tensor couplings are smaller than those for the vector terms, being of the order of $10^{-6}$ at the 95% CL by considering a $10\%$ systematic uncertainty.

• •

  For both channels, the sensitivities are weaker than those without any systematic error. This means that those searches will be dominated by systematic uncertainties and will not benefit further from the energy and luminosity upgrades.

Very recently, many phenomenological studies available in literature have extensively investigated the top FCNC anomalous couplings at various future high energy colliders, including $e^{+}e^{-}$ and $e^{-}p$ machines: see, e.g. Refs. Basso:2014apa ; Aguilar-Saavedra:2017vka ; Khanpour:2019qnw ; Behera:2018ryv ; Cakir:2018ruj ; Khanpour:2014xla ; AguilarSaavedra:2001ab ; deBlas:2018mhx for the most recent reviews. Besides, the expected limits of the four-fermion coefficients at the LHeC and CEPC are obtained in Refs. Shi:2019epw ; Liu:2019wmi . It is then worth comparing the limits on ${\rm BR}(t\to qZ)$ obtained in this study with those obtained by other groups, which are summarised in Tab. 9. One can see that the limits on the BRs are expected to be of $\mathcal{O}(10^{-4}-10^{-6})$. Therefore, we expect our advocated signatures to provide competitive complementary information to that from the above studies in detecting $tqZ$ ($q=u,c$) anomalous couplings at future hadronic colliders.