Investigating the $Z^{\prime}$ gauge boson at the future lepton colliders

With the running of the large hadron collider (LHC), the exploration of the heavy resonance, such as additional Higgs boson, neutral gauge boson ($Z^{\prime}$) and charged ones ($W^{\prime\pm}$), has been reached Tera-eV era currently. The Standard Model (SM) Higgs particle has been discovered in 2012 at LHC Aad et al. (2012); Chatrchyan et al. (2012), which confirms the prediction of the standard model for fundamental particles and the role of Higgs in the electroweak symmetry breaking. Further precision measurements on the Higgs particles have been proposed at the future Higgs factories, including the Circular Electron Positron Collider (CEPC) Dong et al. (2018), the electron-positron stage of the Future Circular Collider (FCC-ee) Abada et al. (2019a, b), and the International Linear Collider (ILC) Bambade et al. (2019). It is attractive to study the new physics effects in Higgs production at the future Higgs factories.

A simple extension of the SM can be possible by adding an additional $U(1)$ group which may arise in models derived from grand unified theories (GUT). Additional $U(1)$ groups can also arise from higher dimensional constructions like string compactifications. In many models of GUT symmetry breaking, $U(1)$ groups survive at relatively low energies, leading to corresponding neutral gauge bosons, commonly referred to as $Z^{\prime}$ bosons Langacker (2009). Such $Z^{\prime}$ bosons typically couple to SM fermions via the electroweak interaction, and can be observed at hadron colliders as narrow resonances. This extra gauge boson $Z^{\prime}$ has been searched for many years at the Tevatron, LEP and LHC Aaltonen et al. (2009a); Abazov et al. (2011); Feldman et al. (2006a); Chiappetta et al. (1996); Barger et al. (1996); Sirunyan et al. (2020); Aaboud et al. (2017). Another interesting motivation is that the $Z^{\prime}$ boson as a mediator or candidate for the dark matter sectors, which is out of the reach of this paper.

There are many experimental searches for the extra gauge boson $Z^{\prime}$ and the strong constraints on the mass of $Z^{\prime}$. The primary studied mode for a $Z^{\prime}$ at a hadron collider is the Drell-Yan production of a dilepton resonance $pp$ $(p\bar{p})\to Z^{\prime}\to l^{+}l^{-}$, where $l=e$ or $\mu$. Other channels, such as $Z^{\prime}\to jj$ where $j$ is jet, $Z^{\prime}\to t\bar{t}$, $Z^{\prime}\to e\mu$, or $Z^{\prime}\to\tau^{+}\tau^{-}$, are also possible. The forward-backward asymmetry for $pp$ $(p\bar{p})\to l^{+}l^{-}$ due to $\gamma$, $Z$, $Z^{\prime}$ interference below the $Z^{\prime}$ peak is also important. Nowadays, the collider phenomenology of the $Z^{\prime}$ boson has been extensively studied (see, for example Rizzo (1986); Nandi (1986); Baer et al. (1987); Barger and Whisnant (1987); Gunion et al. (1988); Hewett and Rizzo (1989); Feldman et al. (2006b); Rizzo (2006); Lee (2009); Barger et al. (2009); Cao and Zhang (2016); Gulov et al. (2018)). Phenomenological implications of the $Z^{\prime}$ boson in an additional $U(1)$ extended model are enormous. They can be explored in fermion pair production processes in $e^{-}e^{+}$ collisions Funatsu et al. (2020); Gao et al. (2010); Das et al. (2021). With the accumulated events increasing on the Higgs particle, the interaction of $Z^{\prime}$ and Higgs boson draws ones attention. An important process to study the $Z^{\prime}$ boson is an associated production of the SM Higgs boson ($H$) with the SM $Z$ boson, such as $pp\to Z^{\prime}\to ZH$ at the LHC Li et al. (2013); Das and Okada (2020); Aaboud et al. (2018a); Sirunyan et al. (2021a) and $e^{+}e^{-}\to Z^{\prime}\to ZH$ at future $e^{+}e^{-}$ linear colliders Gutiérrez-Rodríguez and Hernández-Ruiz (2015).

To study the properties of $Z^{\prime}$, including one additional $U(1)_{X}$ extension of the SM, we investigate the $Z^{\prime}$ signal via the process of $e^{+}e^{-}\to Z^{\prime}/Z\to ZH$ at the electron-positron collision in this paper. The cross section has been calculated including the $Z^{\prime}$-$Z$ mixing effects. As the leptonic decay of $Z$ boson can be a good trigger for the signal process, we show the final lepton angular distribution with the subsequent decay of $Z\to l^{+}l^{-}$ and $H\to b\bar{b}$. The correlation of the final lepton angular distribution with the $Z^{\prime}ZH$ interactions has been investigated at the unpolarized/polarized electron-positron collision.

This paper is organized as follows. The theoretical framework is listed in details in Section II and the current experimental status on $Z^{\prime}$ has been summarized as well. In Section III, we investigate the $ZH$ production via $Z^{\prime}/Z$ mediators and give the final leptons’ angular distributions with various parameter sets. A forward-backward asymmetry has been proposed to be an observable for $Z^{\prime}$-$Z$ mixing. Finally, the summary and discussion are given.

The gauge group structure of the standard model (SM), $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$, can be extended with an additional $U(1)_{X}$ group. We follow the notations in reference Wells (2008) to display the interactions and mass relations. The new gauge sector $U(1)_{X}$ is only coupled with the standard model by kinetic mixing with hypercharge gauge boson $B_{\mu}$. The kinetic energy terms of the $U(1)_{X}$ gauge group are

$\displaystyle\begin{split}\mathcal{L}_{K}&=-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}+\frac{\chi}{2}\hat{X}_{\mu\nu}\hat{B}^{\mu\nu}\end{split},$

(1)

where $\chi\ll 1$ is helpful to keep precision electroweak predictions consistent with experimental measurements. And the Lagrangian for the gauge sector is given by

$\displaystyle\begin{split}\mathcal{L}_{G}&=-\frac{1}{4}\hat{B}_{\mu\nu}\hat{B}^{\mu\nu}-\frac{1}{4}\hat{W}_{\mu\nu}^{a}\hat{W}^{a\mu\nu}-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}+\frac{\chi}{2}\hat{X}_{\mu\nu}\hat{B}^{\mu\nu},\end{split}$

(2)

where $W_{\mu\nu}^{a}$, $B_{\mu\nu}$, $X_{\mu\nu}$ are the field strength tensors for $SU(2)_{L}$, $U(1)_{Y}$, $U(1)_{X}$, respectively. We can diagonalize the 3$\times$3 neutral gauge boson mass matrix, and write the mass eigenstates as

$\displaystyle\begin{split}\begin{pmatrix}B&\\ W^{3}&\\ X\end{pmatrix}&=\begin{pmatrix}\cos\theta_{W}&-\sin\theta_{W}\cos\alpha&\sin\theta_{W}\sin\alpha\\ \sin\theta_{W}&\cos\theta_{W}\cos\alpha&-\cos\theta_{W}\sin\alpha\\ 0&\sin\alpha&\cos\alpha\\ \end{pmatrix}\begin{pmatrix}A&\\ Z&\\ Z^{\prime}\end{pmatrix}\end{split},$

(3)

where the usual weak mixing angle and the new gauge boson mixing angle are

$\displaystyle\begin{split}\sin\theta_{W}&=\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime 2}}}\ ;\ \ \tan 2\alpha=\frac{-2\eta\sin\theta_{W}}{1-\eta^{2}\sin^{2}\theta_{W}-\Delta z}\ \end{split},$

(4)

with $\Delta z$ = $\frac{m_{X}^{2}}{m_{Z_{0}}^{2}}$, $m_{Z_{0}}^{2}$ = $\frac{(g^{2}+g^{\prime 2})v^{2}}{4}$, $\eta=\frac{\chi}{\sqrt{1-\chi^{2}}}$. $m_{Z_{0}}$ and $m_{X}$ are masses before mixing. The two heavier boson mass eigenvalues are

$\displaystyle\begin{split}m_{Z,Z^{\prime}}^{2}&=\frac{m_{Z_{0}}^{2}}{2}\left[\left(1+\eta^{2}\sin^{2}\theta_{W}+\Delta Z\right)\pm\sqrt{\left(1-\eta^{2}\sin^{2}\theta_{W}-\Delta Z\right)^{2}+4\eta^{2}\sin^{2}\theta_{W}}\right]\end{split}.$

(5)

With the assumption of $\eta\ll 1$, we can take the mass eigenvalues as $m_{Z}\approx m_{Z_{0}}=91.19$GeV and $m_{Z^{\prime}}\approx m_{X}$. Therefore, the $Z$ and $Z^{\prime}$ coupling to SM fermions are

$\displaystyle\begin{split}\bar{\psi}\psi Z&=\frac{ig}{\cos\theta_{W}}\left[\cos\alpha\left(1-\eta\sin\theta_{W}\tan\alpha\right)\right]\left[T_{L}^{3}-\frac{\left(1-\eta\tan\alpha/\sin\theta_{W}\right)}{\left(1-\eta\sin\theta_{W}\tan\alpha\right)}\sin^{2}\theta_{W}Q\right]\end{split},$

(6)

$\displaystyle\begin{split}\bar{\psi}\psi Z^{\prime}&=\frac{-ig}{\cos\theta_{W}}\left[\cos\alpha\left(\tan\alpha+\eta\sin\theta_{W}\right)\right]\left[T_{L}^{3}-\frac{\left(\tan\alpha+\eta/\sin\theta_{W}\right)}{\left(\tan\alpha+\eta\sin\theta_{W}\right)}\sin^{2}\theta_{W}Q\right]\end{split},$

(7)

and $Z^{\prime}ZH$ interaction can be written as

$\displaystyle\begin{split}Z^{\prime}ZH&=2i\frac{m_{Z_{0}}^{2}}{v}\left(-\cos\alpha+\eta\sin\theta_{W}\sin\alpha\right)\left(\sin\alpha+\eta\sin\theta_{W}\cos\alpha\right).\end{split}$

(8)

The above is a very common model including $Z^{\prime}$ boson. There are also such kinds of models which differ from the coupling strength, e.g., $Z^{\prime}_{SSM}$, $Z^{\prime}_{\psi}$, etc., which have been studied extensively at high energy colliders.

The processes of $pp\to Z^{\prime}\to e^{+}e^{-}$ and $pp\to Z^{\prime}\to\mu^{+}\mu^{-}$ are studied by the D0 Abazov et al. (2011) and CDF collaborations Aaltonen et al. (2009b), respectively, and lower limit on the mass of the Sequential Standard Model (SSM) $Z^{\prime}$ boson of 1023 GeV is given by D0. The CDF excludes the mass region of 100 GeV $<m_{Z^{\prime}}<$ 982 GeV for a $Z^{\prime}_{\eta}$ boson in the E6 model.

For the SSM $Z^{\prime}$ bosons decaying to a pair of $\tau$-leptons with $m_{Z^{\prime}}$ $\textless$ 2.42 TeV is excluded at 95$\%$ confidence level by the ATLAS experiment, while $m_{Z^{\prime}}$ $\textless$ 2.25 TeV is excluded for the non-universal $G(221)$ model that exhibits enhanced couplings to third-generation fermions Aaboud et al. (2018b). And searches are performed for high-mass resonances in the dijet invariant mass spectrum. $Z^{\prime}$ gauge bosons are excluded in the SSM with $Z^{\prime}\to b\bar{b}$ for masses up to 2.0 TeV, and excluded in the leptophobic model with SM-value couplings to quarks for masses up to 2.1 TeV Aaboud et al. (2018c).

It shows that the results imply a lower limit of 4.5 (5.1) TeV on $m_{Z^{\prime}}$ for the E6-motivated $Z^{\prime}_{\psi}$ ($Z^{\prime}_{SSM}$) boson in the dilepton channel Aad et al. (2019). Upper limits are set on the production cross section times branching fraction for the $Z^{\prime}$ boson in the top color-assisted-technicolor model, resulting in the exclusion of $Z^{\prime}$ masses up to 3.9 TeV and 4.7 TeV for decay widths of 1$\%$ and 3$\%$ , respectively Aad et al. (2020). The heavy $Z^{\prime}$ gauge bosons decay into $e\mu$ final states in proton-proton collisions are excluded for masses up to 4.4 TeV by the CMS experiment Sirunyan et al. (2018). By analyzing data of proton-proton collision collected by the CMS experiment from 2016 to 2018, the limit of $Z^{\prime}$ mass is obtained for the combination of the dielectron and dimuon channels. The limits are 5.15 TeV for the $Z^{\prime}_{SSM}$ and 4.56 TeV for the $Z^{\prime}_{\psi}~{}$Sirunyan et al. (2021b).

The studies on the $Z^{\prime}$ boson also conducted at the lepton colliders especially in the small mass regions. The experiments of ALEPH and OPAL at the LEP limited the mass of the $Z^{\prime}$ boson with the process of $e^{+}e^{-}\to ff$ where $f$ is a $\tau$ lepton or $b$ quark. The lower limits of $Z^{\prime}$ from $\tau$-pair production are 365 GeV at ALEPH and 355 GeV at OPAL, and those from $b\bar{b}$ production are 523 GeV at ALEPH and 375 GeV at OPAL Lynch et al. (2001). With the proposal of the future lepton colliders, the mass region and the coupling strength of the $Z^{\prime}$ boson can be extended extensively, especially in the process of Higgs boson production.

The extensions of the standard model have been studied extensively at the frontier of energy and luminosity of high energy collision experiments. $Z^{\prime}$ boson as a new gauge boson has been proposed in many new physics models. As a proton-proton collider, the LHC has reported the searching results of $Z^{\prime}$ with the mass of a few TeV. The interactions for $Z^{\prime}$ coupling to fermions have been investigated in detail. The next generation of high energy colliders possibly focuses on the Higgs physics, i.e., Higgs-factory, which provides an opportunity to study the interaction of $Z^{\prime}$ coupling to Higgs boson.

In this paper we investigated the process of $e^{+}e^{-}\to Z^{\prime}/Z\to ZH\to l^{+}l^{-}b\bar{b}$. The interactions and couplings are followed up the models proposed by J.D. Wells et al.. The $Z^{\prime}$ couplings to the standard models are related to the $Z^{\prime}$-$Z$ mixing. We investigate the $e^{+}e^{-}\to Z^{\prime}\to ZH$ production cross section. The cross section will be at the same order as the standard model process $e^{+}e^{-}\to Z\to ZH$ with the mixing angle $\sin\alpha\sim 10^{-2}$. The angular distributions of the leptons decaying from the $Z$-boson rely on the mixing angle and $Z^{\prime}$ mass, we have investigated the distributions with the parameters variation of $0.001<\sin\alpha<0.1$, $200~{}\text{GeV}<m_{Z^{\prime}}<1000$ GeV and $\sqrt{s}=500,1000$ GeV. The forward-backward asymmetry can reach 0.0729 for $\sin\alpha=0.01$ and $m_{Z^{\prime}}=\sqrt{s}=500$ GeV. The beam polarization effects have been investigated for the signal process $e^{+}e^{-}\to Z^{\prime}/Z\to ZH\to l^{+}l^{-}b\bar{b}$. The final particles distributions change obviously with some special polarization comparing to the unpolarized condition. As the raising of the precision measurement on the Higgs boson, these studies will promote the understanding of the interactions between Higgs particle and the new physics ones.

This work was supported by the National Natural Science Foundation of China (NNSFC) under grant Nos. 11635009 and 11805081, Natural Science Foundation of Shandong Province under grant No. ZR2019QA021, and the Open Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology, No. NLK2021-07.