Abstract

We assess the scope of a Future Circular Collider operating in \(e^{+}e^{-}\) mode (FCC-ee) in accessing the parameter space of the 4-Dimensional Composite Higgs Model (4DCHM), which represents a realistic implementation of Electro-Weak Symmetry Breaking (EWSB) dynamics triggered by a pseudo-Nambu-Goldstone Boson (pNGB) emerging from the breaking of a global symmetry group \(SO(5)\to SO(4)\) describing new strong interactions eventually resposible for Higgs compositeness. In fact, in such a framework, other composite states exist, like heavy spin-1 bosons (\(W^{\prime}\)s and \(Z^{\prime}\)s) and spin-1/2 fermions (\(b^{\prime}\)s and \(t^{\prime}\)s). Herein, we concentrate initially on the neutral gauge boson sector of this scenario, by attempting to extract the (modified) couplings of the Standard Model (SM) \(Z\) state as well as the masses and quantum numbers of the additional \(Z^{\prime}\) objects present in the 4DCHM. We establish the sensitivity of an FCC-ee to such parameters via the processes \(e^{+}e^{-}\to\mu^{+}\mu^{-}\) and \(e^{+}e^{-}\to t\bar{t}\) for a varieties of foreseen energies and luminosities, by exploiting both cross section and asymmetry observables. We finally combine the results obtained in the \(Z^{\prime}\) sector of the 4DCHM with those emerging from foreseen Higgs measurements, as previously assessed, so as to delineate an analysis programme aimed at confirming or disproving once and for all the validity of the compositeness paradigm.

^{1}^{1}We can insert here the determination of the Higgs total width in the 4DCHM, and also the dependence of the Higgs coupling extraction from the \(Z^{\prime}\) exchanges .This is true a fortiori for the top EW coupling measurements. In fact, an \(e^{+}e^{-}\) collider can separately extract the left- and right-handed components of such couplings. This is particularly relevant for models with a partially composite top quark, like the 4DCHM, where the \(Zt\bar{t}\) coupling modifications come not only from the mixing of the \(Z\) with the \(Z^{\prime}\)s but also from the mixing of the top (antitop) with the extra fermions (antifermions). For the rather “natural” scan described in the caption of Fig. \ref{tLR}, the typical deviations lie within the region uncovered by the HL-LHC but are well inside the reach of a polarized ILC running at 500 GeV (ILC-500) and the FCC-ee where the lack of initial polarisation is compensated by the presence of a substantial final state polarisation combined with a larger integrated luminosity (Barducci 2015, Janot 2015, Andreazza 2015).

But this is not the end of the story, in fact, in CHMs the modifications to the \(e^{+}e^{-}\to HZ\) and \(e^{+}e^{-}\to t\bar{t}\) processes arise not only via the alterations to the \(HZZ\) coupling for the former and to the \(Zt\bar{t}\) vertex for the latter, but also from the presence of new particles, namely the \(s\)-channel exchange of \(Z^{\prime}\)s, which can be sizeable also for large \(Z^{\prime}\) masses (compared to the \(e^{+}e^{-}\) energy) due to their interference with the SM states (\(\gamma\) and \(Z\)). This effect can thus be crucial at high Center-of-Mass (CM) energies of the collider but also important at moderate \(\sqrt{s}\) values (Barducci 2015). In particular, it is impressive how the FCC-ee with \(\sqrt{s}=365\) GeV and 2.6 ab\({}^{-1}\) (corresponding to 3 years of operation) could discover the presence of extra \(Z^{\prime}\) particles through their (effective) contribution to the EW top coupling deviations. This result emerges from the optimal-observable analysis of the lepton angular and energy distributions from top-quark pair production with semi-leptonic decays (Grzadkowski 2000, Grzadkowski 2015, Janot 2015, Janot 2015a). Things go as follow. The \(Vt\bar{t}\) (\(V=\gamma,Z\)) vertices can be expressed in terms of 8 form factors:

\begin{aligned} \label{asym} \label{asym}\Gamma_{Vt\bar{t}}^{\mu}=\frac{g}{2}\bar{u}(p_{t})[\gamma^{\mu}\{A_{V}+\delta A_{V}-(B_{V}+\delta B_{V})\gamma_{5}\}+\frac{(p_{t}-p_{\bar{t}})^{\mu}}{2m_{t}}(\delta C_{V}-\delta D_{V}\gamma_{5})]v(p_{\bar{t}})\\ \end{aligned}and the differential cross section for the process \(e^{+}e^{-}\to t\bar{t}\to(bW^{+})(\bar{b}W^{-})\to(bqq^{\prime})(\bar{b}l\nu)\) can be “expanded” around the SM prediction

\begin{aligned} \label{xsec} \label{xsec}\frac{d^{2}\sigma}{dxd\cos\theta}\sim S^{0}(x,\theta)+\sum_{i=1}^{8}\delta_{i}f^{i}(x,\cos\theta),~{}\delta_{i}=\delta(A,B,C,D)_{V},~{}f^{i}=f^{A,B,C,D}_{V}(x,\cos\theta).\\ \end{aligned}Here, \(S^{0}\) gives the SM contribution, \(x\) and \(\theta\) are the lepton reduced energy and the polar angle. By considering only the 6 CP-conserving form factors (\(A_{V},B_{V},C_{V}\)), the elements of the covariance matrix (the statistical uncertainties) are derived from a likelihood fit to the lepton angular/energy distributions and the total event rate (Janot 2015). The result for the top-quark left- and right-handed couplings to the \(Z\) is represented by the continous green ellipses in Fig. \ref{fig:tLR}.

As a check, we have considered the general expression of the polarised squared matrix element for the process \(f\bar{f}\to Z^{\prime}_{i}/Z^{\prime}_{j}\to F\bar{F}\) where \(f\) and \(F\) are massless and massive fermions respectively and \(Z^{\prime}_{i}\) denotes a vector boson with mass \(m_{i}\) and width \(\Gamma_{i}\). Assuming chiral interaction terms \(Z^{\prime}_{i}\bar{f}f\) and \(Z^{\prime}_{i}\bar{F}F\) of the form

\begin{aligned} \label{coup} \label{coup}\frac{1}{2}Z^{\prime}_{i\mu}\{\bar{f}\gamma^{\mu}[(1-\gamma^{5})f^{i}_{L}+(1+\gamma^{5})f^{i}_{R}]f+\bar{F}\gamma^{\mu}[(1-\gamma^{5})F^{i}_{L}+(1+\gamma^{5})F^{i}_{R}]F\}\\ \end{aligned}and using the helicity projection operators, we get the squared matrix element \({\cal{M}}(\lambda_{F},\lambda_{\bar{F}})\) in terms of the helicities of the final state fermions,

\begin{aligned} \label{ME} \label{ME}{\cal{M}}(\lambda_{F},\lambda_{\bar{F}})=C_{F}\frac{s^{2}}{8}\sum_{i,j}\Delta_{ij}{\cal{\overline{M}}}(\lambda_{F},\lambda_{\bar{F}}),\\ \end{aligned}where

\begin{aligned} \label{ME2} {\cal{\overline{M}}}(\pm,\pm) & = & (1-\beta^{2})s^{2}_{\theta_{p}}(f^{i}_{L}f^{j}_{L}+f^{i}_{R}f^{j}_{R})(F^{i}_{L}+F^{j}_{R})(F^{i}_{L}+F^{j}_{R}), \\ {\cal{\overline{M}}}(\pm,\mp) & = & [(1+c_{\theta_{p}})^{2}f^{i}_{L}f^{j}_{L}+(1-c_{\theta_{p}})^{2}f^{i}_{R}f^{j}_{R}][(1\mp\beta)F^{i}_{L}+(1\pm\beta)F^{i}_{R}] \\ & \times & \label{ME2}[(1\mp\beta)F^{j}_{L}+(1\pm\beta)F^{j}_{R})],\nonumber \\ \end{aligned}with \(C_{F}\) a color factor for the final state fermions (equal to 1 for leptons and 3 for quarks), \(\theta_{p}\) is the polar angle of \(F\) with respect to \(f\), \(s\) is the CM energy squared, \(\beta^{2}=1-4m_{F}^{2}/s\) and \(\Delta_{ij}\) the propagator squared

\begin{aligned} \label{Deltaij} \label{Deltaij}\Delta_{ij}=\frac{(s-m_{i}^{2})(s-m_{j}^{2})+m_{i}m_{j}\Gamma_{i}\Gamma_{j}}{((s-m_{i}^{2})^{2}+m_{i}^{2}\Gamma_{i}^{2})((s-m_{j}^{2})^{2}+m_{j}^{2}\Gamma_{j}^{2})}.\\ \end{aligned}In our case we must include the \(s\)-channel exchange of four spin-1 particles: \(\gamma,~{}Z,~{}Z^{\prime}_{2},~{}Z^{\prime}_{3}\) (let us label these as \(i=0,1,2,3\), respectively). We can then expand \(\Delta_{ij}\) in the limit \(s>>M_{Z^{\prime}}^{2}\). At the first order in \(s/M_{Z^{\prime}}^{2}\) we get:

\begin{aligned} \label{DD} \label{DD}\Delta_{00}=\frac{1}{s^{2}},~{}~{}~{}~{}\Delta_{11}=\frac{1}{(s-m_{1}^{2})^{2}+\Gamma^{2}_{1}m^{2}_{1}},~{}~{}~{}~{}\Delta_{01}=\frac{s-m_{1}^{2}}{s[(s-m_{1}^{2})^{2}+\Gamma^{2}_{1}m^{2}_{1}]} \\ \Delta_{02}\sim-\frac{1}{sm_{2}^{2}}\sim-\frac{s}{m_{2}^{2}}\Delta_{00},~{}~{}~{}~{}\Delta_{03}\sim-\frac{1}{sm_{3}^{2}}\sim-\frac{s}{m_{3}^{2}}\Delta_{00},~{}~{}~{}\Delta_{22}\sim\Delta_{33}\sim\Delta_{23}\sim 0,\nonumber \\ \Delta_{12}\sim-\frac{s-m_{1}^{2}}{m_{2}^{2}[(s-m_{1}^{2})^{2}+\Gamma^{2}_{1}m^{2}_{1}]}\sim-\frac{s}{m_{2}^{2}}\Delta_{01},\nonumber \\ \Delta_{13}\sim-\frac{s-m_{1}^{2}}{m_{3}^{2}[(s-m_{1}^{2})^{2}+\Gamma^{2}_{1}m^{2}_{1}]}\sim-\frac{s}{m_{3}^{2}}\Delta_{01},\nonumber \\ \end{aligned}with \(\Gamma_{1}=\Gamma_{Z}\), \(m_{1}=M_{Z}\), \(m_{2}=M_{Z^{\prime}_{2}}\) and \(m_{3}=M_{Z^{\prime}_{3}}\).

As intimated, the left- and right-handed EW couplings of the \(Z\) boson to the top quark get modified in tne 4DCHM and this is mainly due to the partial compositeness of the top itself. On the contrary, the modifications of the couplings to leptons are quite small and we can neglect them here^{1}^{1}This will no longer be possible though in the forthcoming LEP and SLC
analysis.. The coupling of the photon to all the fermions is given by the Electro-Magnetic (EM) interaction and determined by the electric charge. Referring to eq. \ref{coup} we will parameterise

where \(g^{\gamma,Z}\) are the SM couplings to the electron, \(G^{\gamma,Z}\) are the SM couplings to the top-quark and \(\Delta G\) is the modification of the \(Z\) coupling. We will show that the \(Z^{\prime}_{2,3}\) \(s\)-channel exchange can be expressed as an effective modification of the couplings. Let us insert eqs. (\ref{DD})-(\ref{DG}) in eq. \ref{ME} and expand to first order in \(\Delta G\). For simplicity, let us consider the influence of the \(Z^{\prime}_{2,3}\) exchanges in the \({\cal{M}}(+,+)\) squared amplitude:

\begin{aligned} \label{m++} {\cal{M}}(+,+) & \simeq & {\cal{M}}(+,+)^{\rm SM}+\frac{s^{2}}{4}(1-\beta^{2})s^{2}_{\theta_{p}}\Big{[}c(\Delta G_{L}+\Delta G_{R})\Delta_{01} \\ & & +d(\Delta G_{L}+\Delta G_{R})\Delta_{11}+\sum_{i=2,3}\alpha_{i}(F^{i}_{L}+F^{i}_{R})\Delta_{0i}+\beta_{i}(F^{i}_{L}+F^{i}_{R})\Delta_{1i}\Big{]}\nonumber \\ & \simeq & {\cal{M}}(+,+)^{\rm SM}+\frac{s^{2}}{4}(1-\beta^{2})s^{2}_{\theta_{p}}\Big{[}\sum_{i=2,3}\alpha_{i}(-\frac{s}{m_{i}^{2}})(F^{i}_{L}+F^{i}_{R})\Delta_{00}\nonumber \\ & & \label{m++}+[c(\Delta G_{L}+\Delta G_{R})+\sum_{i=2,3}\beta_{i}(-\frac{s}{m_{i}^{2}})(F^{i}_{L}+F^{i}_{R})\Delta_{01}+d(\Delta G_{L}+\Delta G_{R})\Delta_{11}\Big{]}\nonumber \\ \end{aligned}with

\begin{aligned} c=(g^{\gamma}_{L}g^{Z}_{L}+g^{\gamma}_{R}g^{Z}_{R})(G^{\gamma}_{L}+G^{\gamma}_{R}), \\ d=((g^{Z}_{L})^{2}+(g^{Z}_{R})^{2})(G^{Z}_{L}+G^{Z}_{R}), \\ \alpha_{i}=(g^{\gamma}_{L}f^{i}_{L}+g^{\gamma}_{R}f^{i}_{R})(G^{\gamma}_{L}+G^{\gamma}_{R}), \\ \beta_{i}=(g^{Z}_{L}f^{i}_{L}+g^{Z}_{R}f^{i}_{R})(G^{Z}_{L}+G^{Z}_{R}).\\ \end{aligned}The deviation with respect the SM value can be recast in a form as due to an effective coupling correction, namely,

\begin{aligned} \label{eff} {\cal{M}}(+,+) & \simeq & {\cal{M}}(+,+)^{\rm SM}+\frac{s^{2}}{4}(1-\beta^{2})s^{2}_{\theta_{p}}\Big{[}a(\Delta G^{\gamma{\rm eff}}_{L}+\Delta G^{\gamma{\rm eff}}_{R})\Delta_{00} \\ & & \label{eff}+[b(\Delta G^{\gamma{\rm eff}}_{L}+\Delta G^{\gamma{\rm eff}}_{R})+c(\Delta G^{Z{\rm eff}}_{L}+\Delta G^{Z{\rm eff}}_{R})]\Delta_{01}+d(\Delta G^{Z{\rm eff}}_{L}+\Delta G^{Z{\rm eff}}_{R})\Delta_{11}\Big{]}\nonumber \\ \end{aligned}with

\begin{aligned} a=((g^{\gamma}_{L})^{2}+(g^{\gamma}_{R})^{2})(G^{\gamma}_{L}+G^{\gamma}_{R}), \\ b=(g^{\gamma}_{L}g^{Z}_{L}+g^{\gamma}_{R}g^{Z}_{R})(G^{Z}_{L}+G^{Z}_{R}).\\ \end{aligned}By comparing eq. (2) with eq. (\ref{eff}) we get the expression for the effective coupling corrections:

\begin{aligned} \label{eff1} \Delta G^{\gamma{\rm eff}}_{L,R}=\sum_{i=2,3}\frac{\alpha_{i}G^{i}_{L,R}(-s/m^{2}_{i})}{a+b\Delta_{01}/\Delta_{00}},\nonumber \\ \label{eff1}\Delta G^{Z{\rm eff}}_{L,R}=\Delta G^{Z}_{L,R}+\sum_{i=2,3}\frac{\beta_{i}G^{i}_{L,R}(-s/m^{2}_{i})}{c+d\Delta_{11}/\Delta_{01}}.\\ \end{aligned}Notice that the expressions for the effective modifications depend on the CM energy of the process and are obtained for \(s<<m^{2}_{i}\) and at the first order in \(\Delta G^{Z}\). Also notice that, while the same effective corrections are obtained by considering \({\cal M}(-,-)\), corrections proportional to the \(\beta\) factor are present in the expressions derived from \({\cal M}(\pm,\mp)\). Nevertheless, we checked numerically that these further corrections are negligible. What is clear from eq. (\ref{eff1}) is that the EW couplings of the photon to the top quark are effectively modified due to the \(Z^{\prime}\) exchange. To verify numerically our derivation we compare in Tab. 1 the effective coupling modifications derived from eq. (\ref{eff1}) with the 4-parameter fit results in different configurations: without \(Z^{\prime}\) exchanges, with only \(Z^{\prime}_{2}\), with only \(Z^{\prime}_{3}\) and in the complete configuration of the benchmark point-A.

no \(Z^{\prime}\) | \(Z^{\prime}_{2}\)-only | \(Z^{\prime}_{3}\)-only | all \(Z^{\prime}\)s | |
---|---|---|---|---|

\(\Delta G^{\gamma{\rm eff}}_{L}\) | 0 | 0.00136 | 0.00428 | 0.00563 |

Fit | -0.00056\(\pm\)0.00224 | 0.00095\(\pm\)0.00229 | 0.00359\(\pm\)0.00226 | 0.00512\(\pm\)0.00231 |

\(\Delta G^{\gamma{\rm eff}}_{R}\) | 0 | 0.00566 | 0.00010 | 0.00576 |

Fit | 0.00035\(\pm\)0.00201 | 0.00640\(\pm\)0.00205 | 0.00043\(\pm\)0.00203 | 0.00647\(\pm\)0.00206 |

\(\Delta G^{Z{\rm eff}}_{L}\) | -0.00713 | -0.00759 | -0.00185 | -0.00231 |

Fit | -0.00716\(\pm\)0.00474 | -0.00802\(\pm\)0.00483 | -0.00043\(\pm\)0.00479 | -0.00126\(\pm\)0.00488 |

\(\Delta G^{Z{\rm eff}}_{R}\) | -0.00703 | -0.00878 | -0.00691 | -0.00866 |

Fit | -0.00701\(\pm\)0.00358 | -0.01052\(\pm\)0.00365 | -0.00687\(\pm\)0.00362 | -0.01086\(\pm\)0.00368 |

It is now clear that the optimal-observable statistical analysis at the FCC-ee, based on the lepton polar angle and reduced energy distributions offers a unique possibility to disentangle the effects of top coupling modifications (always taken into account in NP searches) from \(Z^{\prime}\) interference effects (often neglected). Thus, it is mandatory to extract the EW effective couplings of the photon and \(Z\) to the top quark. They depend on 12 parameters of the 4DCHM, namely: the left- and righ-handed EW couplings of the \(Z,Z^{\prime}_{2},Z^{\prime}_{3}\) to the top-quark, the left- and right-handed couplings of the \(Z^{\prime}_{2},Z^{\prime}_{3}\) to the electron, the masses of the \(Z^{\prime}_{2},Z^{\prime}_{3}\). In order to do so, additional observables are required.

To this end, we now turn to the \(e^{+}e^{-}\to\mu^{+}\mu^{-}\) process.

TO BE HAPPILY CONCLUDED

SM is supported in part through the NExT Institute.

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