Distinct signals of the gauge-Higgs unification in $e^+e^-$ collider experiments

TL;DR

This paper predicts distinctive signals of gauge-Higgs unification models in future electron-positron collider experiments, focusing on deviations in cross sections and asymmetries caused by Kaluza-Klein $Z'$ bosons.

Contribution

It demonstrates how Kaluza-Klein excitations in gauge-Higgs unification lead to observable deviations in collider cross sections and asymmetries, providing potential experimental signatures.

Findings

Significant deviations in muon pair production cross sections.

Altered forward-backward asymmetry in leptons.

Changes in $R_b$ ratio and left-right asymmetry from SM predictions.

Abstract

Effects of Kaluza-Klein excited neutral vector bosons ($Z'$ bosons) in the gauge-Higgs unification on $e^+e^- \to \bar{q}q, \ell^+\ell^-$ cross sections are studied, particularly in future $e^+ e^-$ collider experiments with polarized beams. Significant deviations in the energy and polarization dependence in $\sigma(\mu^+\mu^-)$, the lepton forward-backward asymmetry, $R_b(\mu) \equiv \sigma(\bar{b}b)/\sigma(\mu^+\mu^-)$ and the left-right asymmetry from the standard model are predicted.

Figures

Tables

Table 1. Table 1: Masses and widths of Z ′ superscript 𝑍 ′ Z^{\prime} bosons, Z ( 1 ) superscript 𝑍 1 Z^{(1)} , γ ( 1 ) superscript 𝛾 1 \gamma^{(1)} , and Z R ( 1 ) superscript subscript 𝑍 𝑅 1 Z_{R}^{(1)} ( N F = 4 subscript 𝑁 𝐹 4 N_{F}=4 )

${\theta }_H$	$\frac{z_L}{{10}^4}$	$m_{KK}$	$m_{Z^{\left(1\right)}}$	${\Gamma }_{Z^{\left(1\right)}}$	$m_{{\gamma }^{\left(1\right)}}$	${\Gamma }_{{\gamma }^{\left(1\right)}}$	$m_{Z_R^{\left(1\right)}}$	${\Gamma }_{Z_R^{\left(1\right)}}$
[rad.]		[TeV]	[TeV]	[GeV]	[TeV]	[GeV]	[TeV]	[GeV]
$0.115$	$10$	$7.41$	$6.00$	$406$	$6.01$	$909$	$5.67$	$729$
$0.0917$	$3$	$8.81$	$7.19$	$467$	$7.20$	$992$	$6.74$	$853$
$0.0737$	$1$	$10.3$	$8.52$	$564$	$8.52$	$1068$	$7.92$	$1058$

Table 2. Table 2: Couplings of neutral vector bosons ( Z ′ superscript 𝑍 ′ Z^{\prime} bosons) to fermions in unit of g w = e / sin ⁡ θ W subscript 𝑔 𝑤 𝑒 subscript 𝜃 𝑊 g_{w}=e/\sin\theta_{W} for θ H = 0.115 subscript 𝜃 𝐻 0.115 \theta_{H}=0.115 . Corresponding Z 𝑍 Z -boson coupling in the SM are ( g Z ​ ν L , g Z ​ ν R ) = ( 0.57027 , 0 ) superscript subscript 𝑔 𝑍 𝜈 𝐿 superscript subscript 𝑔 𝑍 𝜈 𝑅 0.57027 0 (g_{Z\nu}^{L},g_{Z\nu}^{R})=(0.57027,0) , ( g Z ​ e L , g Z ​ e R ) = ( − 0.30651 , 0.26376 ) superscript subscript 𝑔 𝑍 𝑒 𝐿 superscript subscript 𝑔 𝑍 𝑒 𝑅 0.30651 0.26376 (g_{Ze}^{L},\,g_{Ze}^{R})=(-0.30651,0.26376) , ( g Z ​ u L , g Z ​ u R ) = ( 0.39443 , − 0.17584 ) superscript subscript 𝑔 𝑍 𝑢 𝐿 superscript subscript 𝑔 𝑍 𝑢 𝑅 0.39443 0.17584 (g_{Zu}^{L},\,g_{Zu}^{R})=(0.39443,-0.17584) and ( g Z ​ d L , g Z ​ d R ) = ( − 0.48235 , 0.08792 ) superscript subscript 𝑔 𝑍 𝑑 𝐿 superscript subscript 𝑔 𝑍 𝑑 𝑅 0.48235 0.08792 (g_{Zd}^{L},\,g_{Zd}^{R})=(-0.48235,0.08792) .

$f$	$g_{Zf}^L$	$g_{Zf}^R$	$g_{Z^{\left(1\right)}f}^L$	$g_{Z^{\left(1\right)}f}^R$	$g_{Z_R^{\left(1\right)}}^L$	$g_{Z_R^{\left(1\right)}f}^R$	$g_{{\gamma }^{\left(1\right)}f}^L$	$g_{{\gamma }^{\left(1\right)}f}^R$
${\nu }_e$	$0.57041$	$0$	$-0.1968$	$0$	$0$	$0$	$0$	$0$
${\nu }_{\mu }$	$0.57041$	$0$	$-0.1968$	$0$	$0$	$0$	$0$	$0$
${\nu }_{\tau }$	$0.57041$	$0$	$-0.1967$	$0$	$0$	$0$	$0$	$0$
$e$	$-0.30659$	$0.26392$	$0.1058$	$1.0924$	$0$	$-1.501$	$0.1667$	$-1.983$
$\mu$	$-0.30659$	$0.26391$	$0.1058$	$1.0261$	$0$	$-1.420$	$0.1667$	$-1.863$
$\tau$	$-0.30658$	$0.26391$	$0.1057$	$0.9732$	$0$	$-1.354$	$0.1666$	$-1.767$
$u$	$0.39453$	$-0.17594$	$-0.1361$	$-0.7152$	$0$	$0.9846$	$-0.1111$	$1.2983$
$c$	$0.39453$	$-0.17594$	$-0.1361$	$-0.6631$	$0$	$0.9205$	$-0.1111$	$1.2036$
$t$	$0.39339$	$-0.17712$	$0.5068$	$-0.4764$	$1.0314$	$0.6899$	$0.4158$	$0.8666$
$d$	$-0.48247$	$0.087972$	$0.1665$	$0.3576$	$0$	$-0.4923$	$0.05557$	$-0.6491$
$s$	$-0.48247$	$0.087970$	$0.1664$	$0.3315$	$0$	$-0.4602$	$0.05556$	$-0.6018$
$b$	$-0.48254$	$0.087964$	$-0.6303$	$0.2387$	$1.0292$	$-0.3446$	$-0.2082$	$-0.4331$

Table 3. Table 3: Z ′ superscript 𝑍 ′ Z^{\prime} couplings of fermions for θ H = 0.0917 subscript 𝜃 𝐻 0.0917 \theta_{H}=0.0917 . Unit is the same as in Table  2 .

$f$	$g_{Zf}^L$	$g_{Zf}^R$	$g_{Z^{\left(1\right)}f}^L$	$g_{Z^{\left(1\right)}f}^R$	$g_{Z_R^{\left(1\right)}}^L$	$g_{Z_R^{\left(1\right)}f}^R$	$g_{{\gamma }^{\left(1\right)}f}^L$	$g_{{\gamma }^{\left(1\right)}f}^R$
${\nu }_e$	$0.57037$	0	$-0.2092$	0	0	0	0	0
${\nu }_{\mu }$	$0.57037$	0	$-0.2092$	0	0	0	0	0
${\nu }_{\tau }$	$0.57037$	0	$-0.2092$	0	0	0	0	0
$e$	$-0.30656$	$0.26387$	$0.1124$	$1.0443$	0	$-1.438$	$0.1769$	$-1.899$
$\mu$	$-0.30656$	$0.26387$	$0.1124$	$0.9804$	0	$-1.361$	$0.1769$	$-1.783$
$\tau$	$-0.30656$	$0.26387$	$0.1124$	$0.9278$	0	$-1.296$	$0.1768$	$-1.687$
$u$	$0.39450$	$-0.17591$	$-0.1447$	$-0.6838$	$0$	$0.9438$	$-0.1179$	$1.2433$
$c$	$0.39450$	$-0.17591$	$-0.1477$	$-0.6328$	$0$	$0.8818$	$-0.1179$	$1.1505$
$t$	$0.39367$	$-0.17678$	$0.5635$	$-0.4245$	$1.1239$	$0.6258$	$0.4606$	$0.7734$
$d$	$-0.48243$	$0.087957$	$0.1769$	$0.3419$	$0$	$-0.4719$	$0.05897$	$-0.6216$
$s$	$-0.48243$	$0.087955$	$0.1769$	$0.3164$	$0$	$-0.4409$	$0.05896$	$-0.5753$
$b$	$-0.48249$	$0.087951$	$-0.6959$	$0.2127$	$1.1220$	$-0.3127$	$-0.2304$	$-0.3866$

Table 4. Table 4: Z ′ superscript 𝑍 ′ Z^{\prime} couplings of fermions for θ H = 0.0737 subscript 𝜃 𝐻 0.0737 \theta_{H}=0.0737 . Unit is the same as in Table  2 .

$f$	$g_{Zf}^L$	$g_{Zf}^R$	$g_{Z^{\left(1\right)}f}^L$	$g_{Z^{\left(1\right)}f}^R$	$g_{Z_R^{\left(1\right)}}^L$	$g_{Z_R^{\left(1\right)}f}^R$	$g_{{\gamma }^{\left(1\right)}f}^L$	$g_{{\gamma }^{\left(1\right)}f}^R$
${\nu }_e$	$0.57034$	0	$-0.2225$	0	0	0	0	0
${\nu }_{\mu }$	$0.57034$	0	$-0.2225$	0	0	0	0	0
${\nu }_{\tau }$	$0.57034$	0	$-0.2225$	0	0	0	0	0
$e$	$-0.30655$	$0.26384$	$0.1196$	$0.9981$	0	$-1.376$	$0.1880$	$-1.817$
$\mu$	$-0.30655$	$0.26384$	$0.1196$	$0.9369$	0	$-1.303$	$0.1880$	$-1.705$
$\tau$	$-0.30655$	$0.26384$	$0.1195$	$0.8847$	0	$-1.240$	$0.1879$	$-1.610$
$u$	$0.39448$	$-0.17589$	$-0.1539$	$-0.6536$	$0$	$0.9034$	$-0.1253$	$1.1896$
$c$	$0.39448$	$-0.17589$	$-0.1539$	$-0.6041$	$0$	$0.8439$	$-0.1253$	$1.0994$
$t$	$0.39379$	$-0.17661$	$0.6888$	$-0.3431$	$1.3208$	$0.5253$	$0.5616$	$0.6258$
$d$	$-0.48241$	$0.087947$	$0.1882$	$0.3268$	$0$	$-0.4517$	$0.06267$	$-0.5948$
$s$	$-0.48241$	$0.087946$	$0.1882$	$0.3021$	$0$	$-0.4320$	$0.06266$	$-0.5497$
$b$	$-0.48246$	$0.087941$	$-0.8470$	$0.1720$	$1.3189$	$-0.2625$	$-0.2808$	$-0.3129$

Table 5. Table 5: R f , RL ​ ( P ¯ ) subscript 𝑅 𝑓 RL ¯ 𝑃 R_{f,\text{RL}}(\overline{P}) in the SM, and deviations of R f , RL ​ ( P ¯ ) GHU / R f , RL ​ ( P ¯ ) SM subscript 𝑅 𝑓 RL superscript ¯ 𝑃 GHU subscript 𝑅 𝑓 RL superscript ¯ 𝑃 SM R_{f,\text{RL}}(\overline{P})^{\rm GHU}/R_{f,\text{RL}}(\overline{P})^{\rm SM} from unity are tabulated for P ¯ = 0.8 ¯ 𝑃 0.8 \overline{P}=0.8 . Statistical uncertainties of R f , RL SM superscript subscript 𝑅 𝑓 RL SM R_{f,\rm RL}^{\rm SM} is estimated with L i ​ n ​ t subscript 𝐿 𝑖 𝑛 𝑡 L_{int} data for both σ ​ ( f ¯ ​ f ; P e − = + P ¯ ) 𝜎 ¯ 𝑓 𝑓 subscript 𝑃 superscript 𝑒 ¯ 𝑃 \sigma(\bar{f}f;P_{e^{-}}=+\overline{P}) and σ ​ ( f ¯ ​ f ; P e − = − P ¯ ) 𝜎 ¯ 𝑓 𝑓 subscript 𝑃 superscript 𝑒 ¯ 𝑃 \sigma(\bar{f}f;P_{e^{-}}=-\overline{P}) , namely with 2 ​ L i ​ n ​ t 2 subscript 𝐿 𝑖 𝑛 𝑡 2L_{int} data in all.

$f$	$\sqrt{s}$ ,  $L_{int}$	SM	GHU
		$R_{f,RL}^{SM}$ (uncertainty)	${\theta }_H=0.0917$	${\theta }_H=0.0737$
$\mu$	$250\text{ GeV}$ $250{\text{ fb}}^{-1}$	$0.890$ ($0.3\%$)	$-3.4\%$	$-2.2\%$
	$500\text{ GeV}$ $500{\text{ fb}}^{-1}$	$0.900$ ($0.4\%$)	$-13.2\%$	$-8.6\%$
$b$	$250\text{ GeV}$ $250{\text{ fb}}^{-1}$	$0.349$ ($0.3\%$)	$-3.1\%$	$-2.1\%$
	$500\text{ GeV}$ $500{\text{ fb}}^{-1}$	$0.340$ ($0.5\%$)	$-12.3\%$	$-8.3\%$
$t$	$500\text{ GeV}$ $500{\text{ fb}}^{-1}$	$0.544$ ($0.4\%$)	$-13.0\%$	$-8.2\%$

Table 6. Table 6: ( A FB q , GHU − A FB q , SM ) / A FB q , SM superscript subscript 𝐴 FB 𝑞 GHU superscript subscript 𝐴 FB 𝑞 SM superscript subscript 𝐴 FB 𝑞 SM (A_{\rm FB}^{q,\rm GHU}-A_{\rm FB}^{q,\rm SM})/A_{\rm FB}^{q,\rm SM} ( q = b , t 𝑞 𝑏 𝑡 q=b,t )

$q\overline{q}$	${\theta }_H$	$\sqrt{s}$	$(P_{e^{-}},P_{e^{+}})$
			$(0,\mathrm{\hspace{0.17em}0})$	$(+0.8,-0.3)$	$(-0.8,+0.3)$
$b\overline{b}$	$0.0917$	$250\text{ GeV}$	$+0.8\%$	$+3.3\%$	$+0.1\%$
		$500\text{ GeV}$	$+2.9\%$	$+12.2\%$	$+0.2\%$
	$0.0737$	$250\text{ GeV}$	$+0.7\%$	$+3.2\%$	$+0.1\%$
		$500\text{ GeV}$	$+2.5\%$	$+11.2\%$	$+0.2\%$
$t\overline{t}$	$0.0917$	$500\text{ GeV}$	$+0.9\%$	$+4.5\%$	$+0.1\%$
	$0.0737$	$500\text{ GeV}$	$+1.2\%$	$+4.2\%$	$+0.2\%$

Table 7. Table 7: Deviations of the ratio R t ​ ( μ ) GHU / R t ​ ( μ ) SM subscript 𝑅 𝑡 superscript 𝜇 GHU subscript 𝑅 𝑡 superscript 𝜇 SM R_{t}(\mu)^{\rm GHU}/R_{t}(\mu)^{\rm SM} from the unity.

${\theta }_H$	$\sqrt{s}$	$(P_{e^{-}},P_{e^{+}})$
		$(0,\mathrm{\hspace{0.17em}0})$	$(+0.3,\mathrm{\hspace{0.17em}0.0})$	$(-0.3,0.0)$
$0.0917$	$500\text{ GeV}$	$+2.7\%$	$+2.7\%$	$+2.2\%$
$0.0737$	$500\text{ GeV}$	$+1.8\%$	$+1.9\%$	$+1.5\%$

Equations

$$\frac{g_{HWW}^{GHU}}{g_{HWW}^{\mathrm{SM}}},\frac{g_{HZZ}^{\mathrm{GHU}}}{g_{HZZ}^{\mathrm{SM}}},\frac{y_{\bar{f}f}^{\mathrm{GHU}}}{y_{\bar{f}f}^{\mathrm{SM}}}\simeq \cos {\theta }_H.$$

$$A_{\mathrm{FB}}$$

$${\sigma }^{\mathrm{meas}}(\bar{q}q)/{\sigma }^{\mathrm{fit}}(\bar{q}q)$$

$$A_{\mathrm{FB}}^{\ell }{}^{\mathrm{meas}}/A_{\mathrm{FB}}^{\ell }{}^{\mathrm{fit}}$$

$$R_{\ell }^{0,\mathrm{meas}}/R_{\ell }^{0,\mathrm{fit}}$$

$$A_{\mathrm{FB}}^b{}^{\mathrm{meas}}/A_{\mathrm{FB}}^b{}^{\mathrm{fit}}$$

$$R_b^{\mathrm{meas}}/R_b^{\mathrm{fit}}$$

$${\sigma }^{\mathrm{GHU}}(\bar{q}q)/{\sigma }^{\mathrm{SM}}(\bar{q}q)$$

$$A_{\mathrm{FB}}^{\ell }{}^{\mathrm{GHU}}/A_{\mathrm{FB}}^{\ell }{}^{\mathrm{SM}}$$

$$R_{\ell }^0{}^{\mathrm{GHU}}/R_{\ell }^0{}^{\mathrm{SM}}$$

$$A_{\mathrm{FB}}^b{}^{\mathrm{GHU}}/A_{\mathrm{FB}}^b{}^{\mathrm{SM}}$$

$$R_b^{\mathrm{GHU}}/R_b^{\mathrm{SM}}$$

$$\frac{d\sigma }{d\cos \theta }$$

$$R_{f,\text{RL}}(\overline{P})=\frac{\sigma (\bar{f}f;P_{e^{-}}=+\overline{P},P_{e^{+}}=0)}{\sigma (\bar{f}f;P_{e^{-}}=-\overline{P},P_{e^{+}}=0)}$$

Full text

With the establishment of the standard model (SM) by the discovery of the Higgs boson, searching for physics beyond the SM and understanding the electroweak phase transition have become a few of the main topics in particle physics. Not only large hadron colliders, but also $e^{+}e^{-}$ colliders play an important role for this purpose. In this letter we study distinct signals of the gauge-Higgs unification (GHU) [1]-[10] in the future $e^{+}e^{-}$ collider experiments.

In GHU the Higgs boson is a part of the extra-dimensional component of the gauge potentials, appearing as a fluctuation mode of an Aharonov-Bohm (AB) phase $\theta_{H}$ in the fifth dimension. As a consequence the Higgs couplings $HWW$, $HZZ$ and Yukawa couplings deviate from those in the SM in a universal manner.[11] They are suppressed by a common factor $\cos\theta_{H}$;

[TABLE]

For $\theta_{H}={\cal O}(0.1)$, probable values in the model, the deviation of the couplings amounts to $1-\cos\theta_{H}={\cal O}(0.005)$, and is small. At the ILC at $\sqrt{s}=250\text{ GeV}$, the $ZZH$ coupling can be measured in the $0.6\%$ accuracy with $2\text{ ab}^{-1}$ data [12]. Another prominent feature of the model is that the first Kaluza-Klein (KK) excited states of the neutral gauge bosons, $Z^{\prime}$, have large couplings to right-handed components of quarks and leptons, viable signals of which can be seen in hadron collider experiments [8, 10].

The main purpose of this letter is to check the effect of such $Z^{\prime}$ bosons using lepton collider experiments in the past and future. We first examine the GHU model with precision measurements in LEP1 experiment at $\sqrt{s}=M_{Z}$, and LEP2 experiments for $130\text{ GeV}\leq\sqrt{s}\leq 207\text{ GeV}$. Then we predict several signals of $Z^{\prime}$ bosons in GHU in $e^{+}e^{-}$ collider experiments designed for future with collision energy $\sqrt{s}\geq 250\text{ GeV}$ with polarized electron and positron beams.

The GHU model we consider is the $SO(5)\times U(1)_{X}$ gauge theory in the Randall-Sundrum warped space with metric $ds^{2}=e^{-2k|y|}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}$ ($0\leq|y|\leq+L$) where $k$ is the $AdS_{5}$ curvature. The warp factor $z_{L}\equiv e^{kL}$ is large ($\gg 1$). $SO(5)$ symmetry is broken to $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$ by the orbifold boundary conditions at $y=0$ and $L$. The $SO(5)/SO(4)$ part of the gauge fields, $A_{y}^{\hat{a}}$ ($a=1\sim 4$), plays the role of the Higgs field in the SM. $SU(2)_{R}\times U(1)_{X}$ symmetry is spontaneously broken to $U(1)_{Y}$ by a brane-localized scalar field at $y=0$. Finally the $SU(2)_{L}\times U(1)_{Y}$ symmetry is dynamically broken to $U(1)_{\rm em}$ by the Hosotani mechanism.

5D fields are expanded in KK series. In particular, there are four KK towers of the neutral vector bosons, $\gamma^{(m)}$, $Z^{(m)}$, $Z_{R}^{(n)}$ and $A^{\hat{4}(n)}$ ($m=0,1,2,\cdots$, $n=1,2,3\cdots$) where $\gamma^{(0)}$ and $Z^{(0)}$ correspond to the photon and $Z$ boson, respectively. These fields except for $A^{\hat{4}}$ couple to the SM fields and can be observed as neutral $Z^{\prime}$ vector bosons.

In addition to the quark-lepton multiplets in the vector representation of $SO(5)$, $N_{F}$ dark fermions in the spinor representation are introduced. As a consequence the electroweak symmetry breaking is achieved at the one loop level. The Higgs boson, which is massless at the tree level, acquires a finite mass $m_{H}$, independent of the cutoff scale. The gauge hierarchy problem is thus solved.[2]

There remain two free parameters, $N_{F}$ and $z_{L}$. Given $N_{F}$ and $z_{L}$, the effective potential $V_{\rm eff}(\theta_{H})$ is fixed. From the location of the minimum of $V_{\rm eff}(\theta_{H})$, the value $\theta_{H}$ is determined. There is the property called the universality such that many of the physical quantities are determined by $\theta_{H}$, but do not depend on $N_{F}$ and $z_{L}$ independently. In the following we take $N_{F}=4$ and parameterize the model by $\theta_{H}$.

We note that some of the composite Higgs models (CHM) have similar features to those in the GHU. In particular, CHM based on $SO(5)$ gauge group has almost the same gauge structure as the $SO(5)\times U(1)_{X}$ GHU [4, 13]. The $SO(5)$ gauge invariance is reduced to $SO(4)$ by orbifold boundary conditions in both cases. However, there are many differences between the two. The 4D Higgs boson in GHU is a fluctuation mode of the AB phase $\theta_{H}$ in the fifth dimension, but is not a pseudo-Nambu-Goldstone boson supposed in CHM. Secondly, in most of the CHM, $SO(4)$-breaking boundary conditions are imposed on fermion fields by hand to obtain the quark-lepton spectrum. In the GHU theory based on the action principle the $SO(5)\times U(1)_{X}$ gauge invariance in the bulk and the $SO(4)\times U(1)_{X}$ gauge invariance on the UV and IR branes are strictly preserved. GHU is more restrictive than CHM, and is powerful to make predictions.

In GHU the relevant parameter for physics of SM particles is $\theta_{H}$. With $\theta_{H}$ given, the KK spectra of various fields, the couplings of quarks and leptons to KK gauge bosons, and the Higgs couplings are all determined. The Higgs boson mass $m_{H}\sim 125\,$GeV and $m_{KK}=7\sim 10\,$TeV are naturally realized for $\theta_{H}\sim 0.1$ without fine-tuning of the parameters. It has been shown that corrections to $H\rightarrow\gamma\gamma,Z\gamma$ due to an infinite number of KK states of $W$, $t$ et al. running in the loop are finite and tiny for $\theta_{H}\sim 0.1$. It has been recognized that the $SO(5)\times U(1)_{X}$ GHU in the RS space gives nearly the same phenomenology at low energies as the SM for $\theta_{H}\lesssim 0.1$.

The phase $\theta_{H}$ in GHU corresponds to the vacuum misalignment angle in CHM. A bound $\theta_{H}<0.3$ has been derived in CHM from the $S$ parameter constraint.[13, 14] In GHU much stronger constraint $\theta_{H}\lesssim 0.1$ is obtained from the current non-observation of $Z^{\prime}$ signals at LHC. It should be stressed in this connection that in GHU in the RS space right-handed quarks and leptons and KK gauge bosons are localized near the IR brane whereas left-handed quarks and leptons are localized near the UV brane so that right-handed quarks and leptons have larger couplings to KK gauge bosons than left-handed quarks and leptons.

Masses and widths of $Z^{\prime}$ bosons are tabulated in Table 1. Fermion couplings to $Z^{\prime}$ for $\theta_{H}=0.115$, $0.0917$ and $0.0737$ are given in Tables 2, 3 and 4, respectively. In the evaluation $\sin^{2}\theta_{W}=0.23126$ and $M_{Z}=91.1876\text{ GeV}$ are adopted. The $Z$ couplings of quarks and leptons except for top quark are almost the same as in the SM within the accuracy of one part in $10^{4}$. The deviation of the $Zt\bar{t}$ couplings are less than 1%, whereas the deviation of the $Zb\bar{b}$ couplings are very tiny in GHU.

We evaluate $e^{+}e^{-}\to\bar{f}f$ cross sections $\sigma(\bar{f}f)$ where $f$ is a lepton or quark. In addition to leptonic and hadronic cross sections, forward-backward asymmetry defined by

[TABLE]

the ratio of hadronic and leptonic cross sections $R_{\mu}\equiv\sigma(\bar{q}q)/\sigma(\mu^{+}\mu^{-})$, and the asymmetry of $\sigma(\bar{f}f)$ with right- and left-handed polarized electron beams are investigated.111In the numerical evaluation in this letter we have used the values of the various couplings obtained for $m_{H}=126$GeV. With $m_{H}=125$GeV the value of $M_{Z^{\prime}}$, for instance, decreases by 1.3 %.

Cross sections are evaluated to the leading order, which may receive quantum corrections. Such corrections are parametrised as $\sigma\to\delta_{\rm QCD}\cdot\delta_{\rm QED}\cdot\sigma+r_{\rm nf}$ where $\delta_{\rm QCD}=1+{\cal O}(\alpha_{s}/\pi)$ and $\delta_{\rm QED}=1+{\cal O}(\alpha_{EM}/\pi)$ are factorizable QCD and QED corrections, whereas $r_{\rm nf}$ denotes non-factorizable corrections. In this paper we assume that $\delta_{\rm QCD,QED}^{\rm GHU}\simeq\delta_{\rm QCD,QED}^{\rm SM}$ and $r_{\rm nf}$ for both GHU and SM are small. We have taken only the first KK states into account. The second KK states are approximately twice as heavy as the first KK states. The magnitudes of couplings of the second KK states are at most a half of the couplings of the first KK states. Thus the contributions of the second KK states are expected to be small.

In the LEP1 experiment [15] at the $Z$-pole ($\sqrt{s}=M_{Z}$) the measured and fitted values of cross sections, forward-backward asymmetries of charged leptons $A_{\rm FB}^{\ell}$, $R_{\ell}^{0}\equiv\Gamma_{\rm hadrons}/\Gamma_{\rm\ell}$ ($\ell=e$, $\mu$) and $R_{b}\equiv\Gamma_{b}/\Gamma_{\rm hadrons}$ are given by

[TABLE]

where $\sigma^{\rm fit}(\bar{q}q)=41.478\text{ nb}$, $A_{\rm FB}^{\ell}{}^{\rm fit}=0.01645$, $R_{\ell}^{0}{}^{\rm,fit}=20.742$, $A_{\rm FB}^{b}{}^{\rm fit}=0.1038$ and $R_{b}^{\rm fit}=0.21579$. In GHU, we obtain

[TABLE]

for $\theta_{H}=0.115$, $0.0917$ and $0.0737$, respectively. For $\sqrt{s}=M_{Z}$, cross section is dominated by the $Z$ boson resonance and effects of $Z^{\prime}$ are very small. $Z$-boson couplings are very close to the SM value so that the deviation of the cross sections from the SM is very tiny. No significant deviations are seen. In LEP1, the measured $A_{\rm FB}^{b}$ value deviates from the fit value at nearly $3\sigma$ level. In GHU, $A_{\rm FB}^{b}$ is close to the SM value.

In the LEP2 experiment, cross sections of $\bar{q}q$, $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ for twelve different collision energies $\sqrt{s}$ ($130\text{ GeV}\leq\sqrt{s}\leq 207\text{ GeV}$) were measured. For the energy $120\text{ GeV}\lesssim\sqrt{s}\lesssim 207\text{ GeV}$, $\sigma(\bar{q}q)$ is much larger than the cross sections for lepton final states. The average of $\sigma^{\rm exp}/\sigma^{\rm SM}(\bar{q}q)$ is $1.0092\pm 0.0076$ [16]. In GHU, we obtain $\sigma^{\rm GHU}/\sigma^{\rm SM}(\bar{q}q)=0.9975$, $0.9985$ and $0.9993$ at $\sqrt{s}=130\text{ GeV}$, and $0.9882$, $0.9923$ and $0.9953$ at $\sqrt{s}=207\text{ GeV}$ for $\theta_{H}=0.115$, $0.0917$ and $0.0737$, respectively. Using the ratios $\sigma^{\rm exp}/\sigma^{\rm SM}$ we perform the $\chi^{2}$-test for the $\sigma^{\rm model}/\sigma^{\rm SM}$. When $\sigma^{\rm model}=\sigma^{\rm SM}$, the $\chi^{2}$-value is $\chi^{2}/\text{d.o.f.}=7.3/12$. In GHU ($\sigma^{\rm model}=\sigma^{\rm GHU}$), $\chi^{2}/\text{d.o.f.}=14.4/12$ ($p$-value is $28\%$) [ $11.2/7$ ($p$-value is $13\%$)], $12.0/12$ [$8.9/7$ (26%)] and $10.6/12$ [$7.6/7$ (37%)] for with 12-bin [7-bin] fit for $\theta_{H}=0.115$, $0.0917$ and $0.0737$, respectively. Here in the 7-bin fit we have chosen seven largest energies ($189\text{ GeV}\leq\sqrt{s}\leq 207\text{ GeV}$). In the fit, correlations among the data are taken into account. GHU is within the allowed range of the experimental uncertainty.

Recently at LHC with $36.4$ fb*-1* $pp$-collision data $pp\to\mu^{+}\mu^{-}$ invariant mass distribution $d\sigma/dM_{\mu\mu}$ has been obtained[17]. The number of observed events and the expected number of the Drell-Yang process in the SM are $N^{\rm exp}=4$ and $N^{SM}_{\rm DY}=5.4\pm 0.8$ for $1800\text{ GeV}\leq M_{\mu\mu}\leq 3000\text{ GeV}$, respectively. In GHU with $\theta_{H}=0.115$, $0.0917$ and $0.0737$ the expected numbers of events are $N^{\rm GHU}_{\rm DY}/N^{\rm SM}_{\rm DY}=1.8$, $1.1$ and $0.8$ for $1800\text{ GeV}\leq M_{\mu\mu}\leq 3000\text{ GeV}$ [10], respectively.222In the ratio the K-factors in numerators and denominators are cancelled. For $\theta_{H}=0.115$ the GHU prediction deviates from the SM value (the observed number of events) at 1.8-sigma (2.4-sigma) level. In [10], we have evaluated the expected number of events of Drell-Yang process with $\sqrt{s_{pp}}=14\text{ TeV}$. For $300\text{ fb}^{-1}$ LHC data, we predict for $\theta_{H}=0.0737$ the excess of Drell-Yang events as $N_{\rm DY}^{\rm GHU}/N_{\rm DY}^{\rm SM}$ = $42/47$, $6.9/3.6$, $2.6/0.4$ and $1.1/0.04$ for bins [2000, 3000], [3000, 4000], [4000,5000] and [5000, 6000], respectively.

At LEP2, experimental values of $\sigma(\mu^{+}\mu^{-})$, $\sigma(\tau^{+}\tau^{-})$ and $A_{\rm FB}^{\ell}$ have rather large statistical errors and no significant deviations of the ratios $\sigma^{\rm GHU}/\sigma^{\rm SM}$ and $A_{\rm FB}^{\rm GHU}/A_{\rm FB}^{\rm SM}$ from the experimental data for these modes are seen.

The LHC results put the limit $\theta_{H}\lesssim 0.1$ on GHU. To explore GHU one has to go to future $e^{+}e^{-}$ colliders at higher energies, with $250\text{ GeV}\leq\sqrt{s}\lesssim\text{a few }\text{TeV}$ [14, 18, 19, 20, 21]. Although with such energy $Z^{\prime}$s cannot be directory produced, the effects of interference among $\gamma$, $Z$ and $Z^{\prime}$s can be seen. Furthermore, polarized electron and/or positron beams can be produced at future $e^{+}e^{-}$ colliders. Since right-handed fermions have larger couplings to $Z^{\prime}$s in GHU, right-handed polarized electron beam will be sensitive to the $Z^{\prime}$s effects.

Following Ref. [22] we define the longitudinal polarization $P_{e^{\pm}}$ ($-1\leq P_{e^{\pm}}\leq 1$) so that the electron [positron] is purely right-handed when $P_{e^{-}}=1$ [$P_{e^{+}}=1$]. When the vector bosons dominate in the mediators, the cross section at the center-of-mass frame is given by

[TABLE]

where $\sigma_{LR}$ ($\sigma_{RL}$) is $e_{L}^{-}e_{R}^{+}(e_{R}^{-}e_{L}^{+})\to f\bar{f}$ scattering cross section. Hereafter we consider $\sigma(\bar{q}q)$, $A_{\rm FB}(\mu^{+}\mu^{-})$ and $R_{\mu}\equiv\sigma(\bar{q}q)/\sigma(\mu^{+}\mu^{-})$. Although these quantities depend on both $P_{e^{-}}$ and $P_{e^{+}}$, the dependence is parameterized by one effective polarization $P_{\rm eff}=(P_{e^{-}}-P_{e^{+}})/(1-P_{e^{-}}P_{e^{+}})$. When $\sigma$ is given by (13), $\sigma(P_{\rm eff},0)=\sigma(P_{e^{-}},P_{e^{+}})/(1-P_{e^{-}}P_{e^{+}})$ is satisfied so that one finds that $O(P_{\rm eff},0)=O(P_{e^{-}},P_{e^{+}})$ where $O=\sigma^{\rm GHU}(\bar{q}q)/\sigma^{\rm SM}(\bar{q}q)$, $A_{FB}$, $R_{\mu}$. As typical values one finds $P_{\rm eff}=\pm 0.887$ for $(P_{e^{-}},P_{e^{+}})=(\pm 0.8,\mp 0.3)$. In the following study we parameterize the polarization in terms of $P_{\rm eff}$ instead of $(P_{e^{-}},P_{e^{+}})$.

At $\sqrt{s}=250\text{ GeV}$ with unpolarized beam (with polarized beam with $P_{\rm eff}=0.877$), $\sigma^{\rm SM}(\mu^{+}\mu^{-})=1.87\text{ pb}$ ($2.16\text{ pb}$). In Figure 1, the relative cross section $\sigma(\mu^{+}\mu^{-})^{\rm GHU}/\sigma(\mu^{+}\mu^{-})^{\rm SM}$ is plotted as a function of $P_{\rm eff}$ at $\sqrt{s}=250\text{ GeV}$ and $500\text{ GeV}$. At $\sqrt{s}=250$ GeV, $\sigma(\mu^{+}\mu^{-})$ in GHU is smaller than the SM value by $4.0\%$ [$2.5\%$] for $\theta_{H}=0.0917$ [$0.0737$] when $P_{\rm eff}=0.877$. At $\sqrt{s}=500$ GeV with polarization $P_{\rm eff}=0.877$, $15\%$ [$9\%$] decrease of $\sigma^{\rm GHU}(\bar{q}q)/\sigma(\bar{q}q)^{\rm SM}$ due to the interference will be observed. At $\sqrt{s}=250\text{ GeV}$ with $250\text{ fb}^{-1}$ unpolarized $e^{+}e^{-}$ beam, we expect $4.66\times 10^{5}$ $\mu^{+}\mu^{-}$ events in the SM. In GHU the expected number of events and statistical significance are estimated to be $4.57\times 10^{5}$ [$4.60\times 10^{5}$] and $13.3$ [$8.5$] for $\theta_{H}=0.0917$ [$0.0737$].

Systematic errors in the normalization of the cross sections can be reduced by measuring

[TABLE]

where the electron beams are polarized with $P_{e^{-}}=+\overline{P}$ and $-\overline{P}$. We note that the left-right asymmetry $A_{\rm LR}^{f}\equiv[\sigma_{LR}-\sigma_{RL}]/[\sigma_{LR}+\sigma_{RL}]$ is related to $R_{f,RL}$ by $A_{LR}^{f}=(\overline{P})^{-1}[1-R_{f,RL}]/[1+R_{f,RL}]$. In Table 5, the effects of GHU on the $R_{f,\rm RL}$ are tabulated. GHU predicts a significant deficit in $R_{f,\text{RL}}(\overline{P})$ in the early stage of the ILC experiment.

In Figure 2, $\sigma(\mu^{+}\mu^{-})^{\rm GHU}/\sigma(\mu^{+}\mu^{-})^{\rm SM}$ up to $\sqrt{s}=3\text{ TeV}$ is displayed. For $1\text{ TeV}\lesssim\sqrt{s}\lesssim 3\text{ TeV}$ large deficit is expected for right-handed electron (and/or left-handed positron) beams. We have also plotted the case (“e” in the figure) with $P_{\rm eff}=-0.877$, namely the case with left-handed electron (and/or right-handed positron) beams. In this case, the interference effect of $Z^{\prime}$s is hardly seen for $\sqrt{s}<2\text{ TeV}$. In all cases the ratios grow for $\sqrt{s}\gtrsim 3\text{ TeV}$ up to the large $Z^{\prime}$ resonances.

Forward-backward asymmetry (FBA) in the SM at $\sqrt{s}=250\text{ GeV}$ is $A_{\rm FB}^{\rm SM}(\mu^{+}\mu^{-})=0.522$ ($0.506$) for $P_{\rm eff}=0$ ($+0.877$). In Figure 3, deviations of $A_{\rm FB}(\mu^{+}\mu^{-})$ from the SM values are plotted as functions of $P_{\rm eff}$. At $\sqrt{s}=250\text{ GeV}$ and for $P_{\rm eff}=0.877$, $A_{\rm FB}$ deviates by $-2.1\%$ ($-1.3\%$) from the SM for $\theta_{H}=0.0917$ ($0.0737$). At $\sqrt{s}=500\text{ GeV}$ and $P_{\rm eff}=0.877$, $A_{FB}$ deviates by $-12.0\%$ ($-7.4\%$) for $\theta_{H}=0.0917$ ($0.0737$). Signals of GHU will be seen at $2\,\sigma$ [$4\sigma$] level at $\sqrt{s}=250\text{ GeV}$ with $250\text{ fb}^{-1}$ unpolarized [polarized] beams.

In Figure 4, $A_{\rm FB}(\mu^{+}\mu^{-})$ is displayed up to $\sqrt{s}=3\text{ TeV}$. At $\sqrt{s}=1\sim 2\text{ TeV}$, the effect of the interference among $\gamma$, $Z$ and $Z^{\prime}$ becomes maximum. In particular for right-handed polarized electron beams very large deviation from the SM is expected.

One can also measure $A_{\rm FB}(\bar{b}b)$, $A_{\rm FB}(\bar{t}t)$. They are tabulated in Table. 6. We note that $A_{\rm FB}(\bar{b}b)$ and $A_{\rm FB}(\bar{t}t)$ become larger than those in the SM, in quite contrast with the $A_{\rm FB}(\mu^{+}\mu^{-})$ case.

The effect of the differences in the couplings of $Z^{\prime}$ to leptons and quarks can be seen in the ratio of the cross sections $R_{b}(\mu)\equiv\sigma(\bar{b}b)/\sigma(\mu^{+}\mu^{-})$.333The $e^{+}e^{-}\to\bar{b}b$ scattering process contains not only the process mediated by neutral vector bosons, but also the $W$-fusion process $e^{+}e^{-}\to H\nu\bar{\nu}$ followed by $H\to\bar{b}b$. We have assumed that these processes are efficiently separated and we consider only the vector-boson mediated process. In the SM with unpolarized $e^{+}e^{-}$ beams, $R_{b}(\mu)^{\rm SM}=0.95$, $0.84$ and $0.82$ for $\sqrt{s}=250\text{ GeV}$, $500\text{ GeV}$ and $\infty$, respectively. In Figure 5, deviations of $R_{b}(\mu)$ from the SM value $R_{b}(\mu)^{\rm GHU}/R_{b}(\mu)^{\rm SM}$ are plotted as functions of $P_{\rm eff}$. The excess in $R_{b}(\mu)$ becomes maximum for $P_{\rm eff}\sim 0.3$. At $\sqrt{s}=250\text{ GeV}$, $1.1\%$ [$0.8\%$] excess for unpolarized beams and $1.3\%$ [$0.9\%$] excess for $P_{\rm eff}=0.4$ polarized beams for $\theta_{H}=0.0917$ [$0.0737$] are expected. In GHU with $\theta_{H}\simeq 0.09$, $3\sigma$ deviation is expected with $250\text{ fb}^{-1}$ data. At $\sqrt{s}=500\text{ GeV}$, $5.3\%$ [3.3%] excess is expected for $P_{\rm eff}=0.4$ polarized beams. In Table. 7, deviation of $R_{t}(\mu)^{\rm GHU}$ from $R_{t}(\mu)^{\rm SM}$ is tabulated. The deviation becomes largest around $P_{\rm eff}\simeq+0.3$.

In this letter we have studied the effects of the $Z^{\prime}$ bosons in GHU in the $e^{+}e^{-}$ collider experiments. At the $Z$ pole ($\sqrt{s}=M_{Z}$), the effects of the $Z^{\prime}$ bosons are small and both cross sections and lepton forward-backward asymmetries are consistent with the experiments. At the energies $130\text{ GeV}\leq\sqrt{s}\leq 207\text{ GeV}$, $e^{+}e^{-}\to\bar{f}f$ cross sections and forward-backward asymmetry in GHU are found to be consistent with the LEP2 results. Recent LHC results put the limit $\theta_{H}\lesssim 0.1$ in GHU. Large deviations from the SM in $\sigma(\mu^{+}\mu^{-})$, $A_{\rm FB}$, $R_{b}(\mu)$ and $R_{f,RL}$ are predicted at higher energies. In the future $e^{+}e^{-}$ collider experiments, measurements of $\sigma(\mu^{+}\mu^{-})$, $\sigma(\bar{q}q)$, $A_{\rm FB}(\mu^{+}\mu^{-})$ and $R_{f,RL}$ with polarized beams will well discriminate GHU from the SM. In particular, $\sigma(\mu^{+}\mu^{-})$ measurement, even with unpolarized beams, can discriminate the GHU with $\theta_{H}\simeq 0.09\,(0.07)$ at 11 (8) times of the statistical uncertainty level at $\sqrt{s}=250\text{ GeV}$ with $250\text{ fb}^{-1}$ data. In the left-right asymmetry $R_{f,RL}$, for which systematic uncertainty is reduced, signals of GHU can be observed at 8 (5) times of the statistical uncertainty level . The characteristic dependence of $A_{\rm FB}^{\mu}$ and $R_{b}(\mu)$ on the electron-positron polarization can also be used to study the couplings of the $Z^{\prime}$ bosons to quarks and leptons as well.

The gauge-Higgs unification is promising. It predicts many signals in $e^{+}e^{-}$ collider experiments. The left-right asymmetry $R_{f,RL}=\sigma(\bar{f}f;P_{e^{-}}=\overline{P})/\sigma(\bar{f}f;P_{e^{-}}=-\overline{P})$ will exhibit a distinct deviation from the SM in the early stage of 250 GeV ILC with polarized $e^{-}$ beams. At 1 TeV ILC or CLIC, clear signals of GHU will be seen in the forward-backward asymmetry $A_{\rm FB}(\mu^{+}\mu^{-})$.

Acknowledgments

The authors would like to thank M. Tanaka and S. Kanemura for valuable comments. This work was supported in part by the Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research No 15K05052 (HH and YH).

References