# Forward–backward asymmetry in the gauge-Higgs unification at the International Linear Collider

## Abstract

The signals of the \(SO(5)\times U(1)\) gauge-Higgs unification model at the International Linear Collider are studied. In this model, Kaluza–Klein modes of the neutral gauge bosons affect fermion pair productions. The deviations of the forward–backward asymmetries of the \(e^+e^-\rightarrow \bar{b}b\), \(\bar{t}t\) processes from the standard model predictions are clearly seen by using polarised beams. The deviations of these values are predicted for two cases, the bulk mass parameters of quarks are positive and negative case.

## 1 Introduction

After the discovery of the Higgs boson, search for new physics in the electroweak sector is one of the most important topics of the particle physics. For this purpose, high-precision measurements of the electroweak sector are necessary. The International Linear Collider (ILC) has the capabilities for the high-precision measurements and enables us to test the standard model (SM) [1, 2, 3, 4]. New physics at the ILC are predicted by many alternative models such as the Higgs portal dark matter models [5, 6, 7], two-Higgs doublet models [8, 9, 10], Georgi–Machacek model [11], supersymmetric models [12, 13, 14, 15], littlest Higgs model [16, 17], universal extra dimensional model [18], warped extra dimensional models [19, 20, 21], composite Higgs models [22, 23, 24] and other models [25].

The gauge-Higgs unification (GHU) models [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] are alternative models of the SM, in which the Higgs boson appears as an extra-dimensional component of the higher-dimensional gauge boson. Hence the Higgs sector is governed by the gauge principle and the Higgs boson mass is protected against the radiative corrections in the GHU models. The Higgs boson is massless at the tree-level and acquires the finite mass by the radiative corrections [26, 27, 28, 29, 30, 31]. Note that also the Yukawa interactions in the GHU models are the gauge interactions of the extra-dimensional component of the gauge boson and the fermions. The phenomenologically most well-studied GHU model is the \(SO(5)\times U(1)\) GHU models [34, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56], which are defined on the warped metric [64]. In the \(SO(5)\times U(1)\) GHU models, the Higgs boson mass is protected by the custodial symmetry and the Higgs boson potential is calculable. One of the features of the warped extra dimensional models is that the first Kaluza–Klein (KK) excited states of the gauge bosons have the large asymmetry between the couplings with the left-handed and the right-handed fermions. The \(SO(5)\times U(1)\) GHU models have the same feature.

There are several kinds of the \(SO(5)\times U(1)\) GHU models depending on symmetry breaking patterns and the embedding patterns of the quarks and leptons. In the \(SO(5)\times U(1)\) GHU model discussed here, the quarks and leptons are embedded in the **5** representation of *SO*(5). Then the Yukawa, *HWW* and *HZZ* couplings are suppressed by \(\cos \theta _H\) from the SM values, where \(\theta _H\) is the Wilson line phase. The KK excited states of the neutral gauge bosons appear as the so-called \(Z'\) bosons. Besides the KK excited states of the photon \(\gamma ^{(n)}\) and that of the *Z* boson \(Z^{(n)}\), that of the \(SU(2)_R\) gauge boson \(Z_R^{(n)}\), which does not have a zero mode, exist as the neutral gauge bosons. Therefore the \(\gamma ^{(n)}\), \(Z^{(n)}\) and \(Z_R^{(n)}\) appear as the \(Z'\) bosons. From the result at the Large Hadron Collider, the parameter region is constrained to be \(\theta _H\lesssim 0.1\), where \(m_\text {KK} > rsim 8\) TeV [51]. For this region, the contributions of the KK excited states to the decay widths, \(\Gamma (H\rightarrow \gamma \gamma )\) and \(\Gamma (H\rightarrow Z\gamma )\) are less than 0.2%, thus negligible [47, 50]. The decay widths \(\Gamma (H\rightarrow WW)\), \(\Gamma (H\rightarrow ZZ)\), \(\Gamma (H\rightarrow \bar{q}q)\), \(\Gamma (H\rightarrow l^+l^-)\), \(\Gamma (H\rightarrow \gamma \gamma )\) and \(\Gamma (H\rightarrow Z\gamma )\) are approximately suppressed by the common factor \(\cos ^2\theta _H\) at the leading order. Hence the branching ratio of the Higgs boson in this model is almost equivalent to that in the SM. At the ILC 500 GeV, the branching ratios of the processes \(H\rightarrow WW\), \(H\rightarrow ZZ\), \(H\rightarrow \bar{b}b\), \(H\rightarrow \bar{c}c\) and \(H\rightarrow \tau ^+\tau ^-\) are measured at the \(O(1)\%\) accuracy [2]. The branching ratios of the Higgs boson are consistent between the GHU model and the SM at the ILC.

At the ILC, it is possible to measure effects of new physics on the the cross sections and the forward–backward asymmetries forward-backward asymmetry of the \(e^+ e^- \rightarrow f\bar{f}\) processes [16, 17, 19, 20, 57, 58, 59, 60]. The forward-backward asymmetries are important observables for indirect search of physics beyond the SM as shown in the studies of that of top quark production at the Tevatron and Large Hadron Collider [61, 62, 63]. In this paper, the effects of \(Z'\) bosons in the GHU model on these values are shown. The same topic is studied previously [52] and clear signals are predicted although \(Z'\) masses are much larger than the centre-of-mass of the ILC. However in the previous study, the bulk mass parameters are assumed to be positive although negative values are also allowed. Thus the negative region of the bulk mass parameters is also checked in this study. The following calculations are done at the tree-level without the one-loop effective potential of the Higgs boson, so the corrections from the strong interaction are not included.

This paper is constructed as follows. In Sect. 2, the model is shortly introduced. In Sect. 3, the parameters and the couplings and decay widths of the \(Z'\) bosons are shown. The bulk mass parameter dependence of the fermion mode functions are also shortly reviewed. In Sect. 4, the cross sections and the forward-backward asymmetries at the ILC are shown. In Sect. 5, the results are summarised.

## 2 Model

*k*is the AdS curvature. The action has the \(SO(5)\times U(1)_X\) local symmetry. The bulk action is given by

*SO*(5), \(U(1)_X\) and \(SU(3)_C\) gauge fields, \(F_{MN}^{(A)} = \partial _M A_N - \partial _N A_M - i g_A [A_M, A_N]\), \(F_{MN}^{(B)} = \partial _M B_N - \partial _N B_M\) and \(F_{MN}^{(G)} = \partial _M G_N - \partial _N G_M - i g_C [G_M, G_N]\), \(g_A\), \(g_B\) and \(g_C\) are the five-dimensional gauge couplings of

*SO*(5), \(U(1)_X\) and \(SU(3)_C\). \(f_\text {gf}^{(A)}\), \(f_\text {gf}^{(B)}\) and \(f_\text {gf}^{(G)}\) are gauge-fixing functions and \(\xi _{(A)}\), \(\xi _{(B)}\) and \(\xi _{(G)}\) are gauge parameters. \(\mathcal {L}_{GH}^{(A)}\), \(\mathcal {L}_{GH}^{(B)}\) and \(\mathcal {L}_{GH}^{(G)}\) denote ghost Lagrangians, respectively. \(\Psi _a^{g}\) (\(a=1,2,3,4\) and \(g=1,2,3\)) are the four

*SO*(5)-vector (

**5**representation) fermions for each generation and \(\Psi _{F_i}\) (\(i=1,\ldots ,N_F\)) are the \(N_F\) number of

*SO*(5)-spinor (

**4**representation) fermions, which exist in the bulk space. The colour indices are not shown. The

*SO*(5) gauge fields \(A_M\) are decomposed as

*SO*(5) /

*SO*(4), respectively. For the fermion Lagrangian, \(\Gamma ^M\) denotes 5D gamma matrices which is defined by \(\{ \Gamma ^M,\Gamma ^N \} = 2\eta ^{MN}\) (\(\eta ^{55} = +1\)),

*SO*(5)-vector fermions are decomposed to the \(SU(2)_L\times SU(2)_R\) bidoublet and singlet. The multiplet of the third generation are denoted as

## 3 Parameters, couplings and decay widths

### 3.1 Bulk mass parameter dependence

*c*on the warped metric is expanded as [65]

From (13) and (14), it is straightforwardly derived that the product of the left- and right-handed fermion mode functions with the same KK number and the bulk mass, \(f_{L}^{(n)}(y)\times f_{R}^{(n)}(y)\) is invariant under changing the sign of the bulk mass parameter \(c\rightarrow -c\). Consequently, the Yukawa couplings and the fermion masses obtained from the Higgs boson vacuum expectation value are independent of the sign of the bulk mass parameters in the case that the left- and right-handed quarks have the same bulk mass. In this model, the above arguments are applicable.

Considering gauge boson, mode function of zero-mode is independent of *y*-coordinate and 1st KK gauge bosons have peaks near \(y=L\) [66]. Therefore the right-handed fermions with \(0<c\), the left-handed fermions with \(c<0\) and the left-handed fermions with \(-\frac{1}{2}<c<\frac{1}{2}\) couple to \(Z'\) bosons rather largely. As shown in the next subsection, the bulk mass parameter of the third generation quarks is \(|c_t|<\frac{1}{2}\). Therefore the left-handed third generation quarks couplings with \(Z'\) bosons are large.

### 3.2 Parameters

Input parameters. Masses of *Z* boson, the Higgs boson, leptons and quarks in the unit of GeV at the *Z* mass scale

\(\alpha _\text {EM}^{-1}\) | \(\sin ^2\theta _W\) | ||||
---|---|---|---|---|---|

127.96 | 0.23122 |

\(m_Z\) | \(m_H\) | \(m_\nu \) | \(m_e\) | \(m_\mu \) | \(m_\tau \) |
---|---|---|---|---|---|

91.188 | 125 | \(10^{-12}\) | \(0.48657 \times 10^{-3}\) | \(102.72 \times 10^{-3}\) | 1.7462 |

\(m_u\) | \(m_d\) | \(m_s\) | \(m_c\) | \(m_b\) | \(m_t\) |
---|---|---|---|---|---|

\(1.27 \times 10^{-3}\) | \(2.90 \times 10^{-3}\) | 0.055 | 0.619 | 2.89 | 171.7 |

The warp factor, the bulk mass parameters of the fermion and AdS curvature with each values of \(\theta _H\) are listed. The resultant *W*-boson mass, Higgs boson mass and KK scale given from the model parameters are also summarized

\(\theta _H\) | \(e^{kL}\) | \(\left| c_F\right| \) |
| \(m_W\) (GeV) | \(m_H\) (GeV) | \(m_\text {KK}\) (GeV) |
---|---|---|---|---|---|---|

0.10 | \(2.90 \times 10^4\) | 0.29617 | \(7.4431 \times 10^7\) | 79.957 | 125.1 | 8063 |

0.09 | \(1.70 \times 10^4\) | 0.27670 | \(4.7190 \times 10^7\) | 79.958 | 125.1 | 8721 |

0.08 | \(1.01 \times 10^4\) | 0.25356 | \(3.0679 \times 10^7\) | 79.951 | 125.4 | 9544 |

\(\theta _H\) | \(\left| c_e\right| \) | \(\left| c_\mu \right| \) | \(\left| c_\tau \right| \) | \(\left| c_u\right| \) | \(\left| c_c\right| \) | \(\left| c_t\right| \) |
---|---|---|---|---|---|---|

0.10 | 1.8734 | 1.3139 | 1.0060 | 1.6796 | 1.1200 | 0.16116 |

0.09 | 1.9504 | 1.3599 | 1.0348 | 1.7459 | 1.1552 | 0.11646 |

0.08 | 2.0342 | 1.4100 | 1.0663 | 1.8180 | 1.1936 | 0.0089140 |

The free parameters of this model are the warp factor \(e^{kL}\) and \(N_F\) which is \(\Psi _F\)’s degrees of the freedom, so once the \(e^{kL}\) and \(N_F\) are set, \(\theta _H\) is determined. The physics of the quarks and leptons are almost independent of \(N_F\) and determined by \(\theta _H\) [47, 48]. The input parameters and the model parameters to realise the input parameters at the tree level are listed in Tables 1 and 2 respectively, where \(c_e\), \(c_\mu \), \(c_\tau \), \(c_u\), \(c_c\), \(c_t\) and \(c_F\) are the bulk mass parameters of leptons and quarks for the each generations and \(\Psi _F\). As explained in previous subsection, the sign of the bulk mass parameter *c* does not affect the fermion mass in this model. Therefore only the absolute values of *c* is determined from the input values. In the following, the bulk mass parameters of leptons and quarks are abbreviated as \(c_l\equiv (c_e, c_\mu , c_\tau )\) \(c_q\equiv (c_u, c_c, c_t)\). The resultant *W*-boson mass at the tree-level calculated by the boundary condition is \(m_W^\text {tree}=80.0\) GeV. To realise the input parameters, the parameter region of \(\theta _H\) is found to be \(0.08\le \theta _H \le 0.10\). The lower limit of \(\theta _H\) becomes slightly smaller for \(N_F>4\). In \(N_F=8\) case, lower limit of \(\theta _H=0.078\).

### 3.3 Couplings and decay widths

Couplings of neutral vector bosons (\(Z'\) bosons) to fermions in unit of \(g_w = e/\sin \theta _W\) for \(\theta _H = 0.10\), \(c_l\), \(c_q>0\)

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57037\) | 0 | \(-\,0.20943\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _\mu \) | \(+\,0.57037\) | 0 | \(-\,0.20943\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _\tau \) | \(+\,0.57037\) | 0 | \(-\,0.20928\) | 0 | 0 | 0 | 0 | 0 |

| \(-\,0.30661\) | \(+\,0.26384\) | \(+\,0.11258\) | \(+\,1.04332\) | 0 | \(-\,1.4357\) | \(+\,0.17720\) | \(-\,1.8962\) |

\(\mu \) | \(-\,0.30661\) | \(+\,0.26384\) | \(+\,0.11258\) | \(+\,0.97948\) | 0 | \(-\,1.3582\) | \(+\,0.17720\) | \(-\,1.7801\) |

\(\tau \) | \(-\,0.30661\) | \(+\,0.26383\) | \(+\,0.11250\) | \(+\,0.92684\) | 0 | \(-\,1.2940\) | \(+\,0.17708\) | \(-\,1.6844\) |

| \(+\,0.39453\) | \(-\,0.17589\) | \(-\,0.14486\) | \(-\,0.68311\) | 0 | \(+\,0.94208\) | \(-\,0.11813\) | \(+\,1.2415\) |

| \(+\,0.39453\) | \(-\,0.17589\) | \(-\,0.14485\) | \(-\,0.63219\) | 0 | \(+\,0.88013\) | \(-\,0.11812\) | \(+\,1.1489\) |

| \(+\,0.39353\) | \(-\,0.17694\) | \(+\,0.57109\) | \(-\,0.42117\) | \(+\,1.1369\) | \(+\,0.62142\) | \(+\,0.46722\) | \(+\,0.76730\) |

| \(-\,0.48245\) | \(+\,0.087946\) | \(+\,0.17715\) | \(+\,0.34156\) | 0 | \(-\,0.47104\) | \(+\,0.059066\) | \(-\,0.62077\) |

| \(-\,0.48245\) | \(+\,0.087945\) | \(+\,0.17713\) | \(+\,0.31609\) | 0 | \(-\,0.44006\) | \(+\,0.059060\) | \(-\,0.57445\) |

| \(-\,0.48252\) | \(+\,0.087939\) | \(-\,0.70659\) | \(+\,0.21112\) | \(+\,1.1347\) | \(-\,0.31045\) | \(-\,0.23377\) | \(-\,0.38353\) |

\(Z'\) couplings of fermions for \(\theta _H = 0.09\), \(c_l\), \(c_q>0\). The same unit as in Table 3

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57035\) | 0 | \(-\,0.21569\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _{\mu }\) | \(+\,0.57035\) | 0 | \(-\,0.21569\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _{\tau }\) | \(+\,0.57035\) | 0 | \(-\,0.21553\) | 0 | 0 | 0 | 0 | 0 |

| \(-\,0.30660\) | \(+\,0.26382\) | \(+\,0.11595\) | \(+\,1.02101\) | 0 | \(-\,1.4062\) | \(+\,0.18238\) | \(-\,1.8568\) |

\(\mu \) | \(-\,0.30660\) | \(+\,0.26382\) | \(+\,0.11595\) | \(+\,0.95843\) | 0 | \(-\,1.3307\) | \(+\,0.18238\) | \(-\,1.7430\) |

\(\tau \) | \(-\,0.30660\) | \(+\,0.26382\) | \(+\,0.11586\) | \(+\,0.90600\) | 0 | \(-\,1.2671\) | \(+\,0.18225\) | \(-\,1.6476\) |

| \(+\,0.39452\) | \(-\,0.17588\) | \(-\,0.14919\) | \(-\,0.66857\) | 0 | \(+\,0.92287\) | \(-\,0.12159\) | \(+\,1.2159\) |

| \(+\,0.39452\) | \(-\,0.17588\) | \(-\,0.14918\) | \(-\,0.61829\) | 0 | \(+\,0.86208\) | \(-\,0.12157\) | \(+\,1.1244\) |

| \(+\,0.39363\) | \(-\,0.17681\) | \(+\,0.61316\) | \(-\,0.39018\) | \(+\,1.2038\) | \(+\,0.58325\) | \(+\,0.50090\) | \(+\,0.71132\) |

| \(-\,0.48244\) | \(+\,0.087940\) | \(+\,0.18244\) | \(+\,0.33428\) | 0 | \(-\,0.46144\) | \(+\,0.060793\) | \(-\,0.60793\) |

| \(-\,0.48244\) | \(+\,0.087939\) | \(+\,0.18242\) | \(+\,0.30915\) | 0 | \(-\,0.43104\) | \(+\,0.060787\) | \(-\,0.56219\) |

| \(-\,0.48250\) | \(+\,0.087933\) | \(-\,0.75660\) | \(+\,0.19561\) | \(+\,1.2016\) | \(-\,0.29141\) | \(-\,0.25054\) | \(-\,0.35559\) |

\(Z'\) couplings of fermions for \(\theta _H = 0.08\), \(c_l\), \(c_q>0\). The same unit as in Table 3

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57034\) | 0 | \(-\,0.22233\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _{\mu }\) | \(+\,0.57034\) | 0 | \(-\,0.22233\) | 0 | 0 | 0 | 0 | 0 |

\(\nu _{\tau }\) | \(+\,0.57034\) | 0 | \(-\,0.22216\) | 0 | 0 | 0 | 0 | 0 |

| \(-\,0.30659\) | \(+\,0.26380\) | \(+\,0.11952\) | \(+\,0.99861\) | 0 | \(-1.3762\) | \(+\,0.18789\) | \(-1.8171\) |

\(\mu \) | \(-\,0.30659\) | \(+\,0.26380\) | \(+\,0.11952\) | \(+\,0.93739\) | 0 | \(-1.3029\) | \(+\,0.18789\) | \(-1.7057\) |

\(\tau \) | \(-\,0.30659\) | \(+\,0.26380\) | \(+\,0.11943\) | \(+\,0.88524\) | 0 | \(-1.2401\) | \(+\,0.18775\) | \(-1.6107\) |

| \(+\,0.39451\) | \(-\,0.17587\) | \(-\,0.15379\) | \(-\,0.65398\) | 0 | \(+\,0.90342\) | \(-\,0.12526\) | \(+1.1900\) |

| \(+\,0.39451\) | \(-\,0.17587\) | \(-\,0.15377\) | \(-\,0.60444\) | 0 | \(+\,0.84393\) | \(-\,0.12524\) | \(+1.0998\) |

| \(+\,0.39365\) | \(-\,0.17675\) | \(+\,0.71984\) | \(-\,0.33029\) | \(+1.3690\) | \(+\,0.50941\) | \(+\,0.50941\) | \(+\,0.58717\) |

| \(-\,0.48242\) | \(+\,0.087934\) | \(+\,0.18806\) | \(+\,0.32699\) | 0 | \(-\,0.45171\) | \(+\,0.062629\) | \(-\,0.59500\) |

| \(-\,0.48242\) | \(+\,0.087934\) | \(+\,0.18804\) | \(+\,0.30222\) | 0 | \(-\,0.43197\) | \(+\,0.062622\) | \(-\,0.54990\) |

| \(-\,0.48248\) | \(+\,0.087927\) | \(-\,0.88581\) | \(+\,0.16571\) | \(+1.3666\) | \(-\,0.25452\) | \(-\,0.29359\) | \(-\,0.30139\) |

\(Z'\) couplings of fermions for \(\theta _H = 0.10\), \(c_l\), \(c_q<0\). The same unit as in Table 3

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57054\) | 0 | \(+2.2561\) | 0 | \(-3.1047\) | 0 | 0 | 0 |

\(\nu _\mu \) | \(+\,0.57053\) | 0 | \(+2.1181\) | 0 | \(-2.9371\) | 0 | 0 | 0 |

\(\nu _\tau \) | \(+\,0.57052\) | 0 | \(+2.0042\) | 0 | \(-2.7982\) | 0 | 0 | 0 |

| \(-\,0.30384\) | \(+\,0.26662\) | \(-1.2015\) | \(-\,0.09790\) | \(-3.1130\) | 0 | \(-1.8962\) | \(+\,0.17720\) |

\(\mu \) | \(-\,0.30384\) | \(+\,0.26662\) | \(-1.1280\) | \(-\,0.09790\) | \(-2.9450\) | 0 | \(-1.7801\) | \(+\,0.17720\) |

\(\tau \) | \(-\,0.30383\) | \(+\,0.26662\) | \(-1.0674\) | \(-\,0.09783\) | \(-2.8057\) | 0 | \(-1.6844\) | \(+\,0.17708\) |

| \(+\,0.39419\) | \(-\,0.17630\) | \(+1.5309\) | \(+\,0.06473\) | \(-1.3595\) | 0 | \(+1.2415\) | \(-\,0.11813\) |

| \(+\,0.39180\) | \(-\,0.17868\) | \(+1.4082\) | \(+\,0.06560\) | \(+2.3878\) | 0 | \(+1.1489\) | \(-\,0.11812\) |

| \(+\,0.39281\) | \(-\,0.17766\) | \(+\,0.9400\) | \(-\,0.25520\) | \(+1.7082\) | \(+\,0.41431\) | \(+\,0.7673\) | \(+\,0.46722\) |

| \(-\,0.48019\) | \(+\,0.09032\) | \(-1.8649\) | \(-\,0.03316\) | \(-1.3614\) | 0 | \(-\,0.62077\) | \(+\,0.05907\) |

| \(-\,0.48256\) | \(+\,0.08794\) | \(-1.7344\) | \(-\,0.03229\) | \(+2.3805\) | 0 | \(-\,0.57445\) | \(+\,0.05906\) |

| \(-\,0.48255\) | \(+\,0.08793\) | \(-1.1585\) | \(+\,0.12877\) | \(+1.7027\) | \(-\,0.20689\) | \(-\,0.38353\) | \(-\,0.23377\) |

\(Z'\) couplings of fermions for \(\theta _H = 0.09\), \(c_l\), \(c_q<0\) The same unit as in Table 3

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57050\) | 0 | \(+2.2079\) | 0 | \(-3.0408\) | 0 | 0 | 0 |

\(\nu _{\mu }\) | \(+\,0.57050\) | 0 | \(+2.0725\) | 0 | \(-2.8775\) | 0 | 0 | 0 |

\(\nu _{\tau }\) | \(+\,0.57048\) | 0 | \(+1.9592\) | 0 | \(-2.7400\) | 0 | 0 | 0 |

| \(-\,0.30436\) | \(+\,0.26607\) | \(-1.1779\) | \(-\,0.10062\) | \(-3.0474\) | 0 | \(-1.8568\) | \(+\,0.18238\) |

\(\mu \) | \(-\,0.30436\) | \(+\,0.26607\) | \(-1.1057\) | \(-\,0.10062\) | \(-2.8838\) | 0 | \(-1.7430\) | \(+\,0.18238\) |

\(\tau \) | \(-\,0.30436\) | \(+\,0.26607\) | \(-1.0452\) | \(-\,0.10055\) | \(-2.7460\) | 0 | \(-1.6476\) | \(+\,0.18225\) |

| \(+\,0.39424\) | \(-\,0.17621\) | \(+1.4986\) | \(+\,0.06664\) | \(-1.3314\) | 0 | \(+1.2159\) | \(-\,0.12159\) |

| \(+\,0.39231\) | \(-\,0.17813\) | \(+1.3792\) | \(+\,0.06736\) | \(+2.3375\) | 0 | \(+1.1244\) | \(-\,0.12157\) |

| \(+\,0.39322\) | \(-\,0.17723\) | \(+\,0.8717\) | \(-\,0.27396\) | \(+1.6023\) | \(+\,0.43863\) | \(+\,0.7113\) | \(+\,0.50090\) |

| \(-\,0.48061\) | \(+\,0.08986\) | \(-1.8270\) | \(-\,0.03398\) | \(-1.3329\) | 0 | \(-\,0.60793\) | \(+\,0.06079\) |

| \(-\,0.48253\) | \(+\,0.08794\) | \(-1.6963\) | \(-\,0.03325\) | \(+2.3317\) | 0 | \(-\,0.56219\) | \(+\,0.06079\) |

| \(-\,0.48252\) | \(+\,0.08793\) | \(-1.0734\) | \(+\,0.13788\) | \(+1.5982\) | \(-\,0.21909\) | \(-\,0.35559\) | \(-\,0.25054\) |

\(Z'\) couplings of fermions for \(\theta _H = 0.08\), \(c_l\), \(c_q<0\) The same unit as in Table 3

| \(g^L_{Zf}\) | \(g^R_{Zf}\) | \(g^L_{Z^{(1)}f}\) | \(g^R_{Z^{(1)}f}\) | \(g^L_{Z_R^{(1)}}\) | \(g^R_{Z_R^{(1)}f}\) | \(g^L_{\gamma ^{(1)}f}\) | \(g^R_{\gamma ^{(1)}f}\) |
---|---|---|---|---|---|---|---|---|

\(\nu _e\) | \(+\,0.57046\) | 0 | \(+2.1594\) | 0 | \(-2.9760\) | 0 | 0 | 0 |

\(\nu _{\mu }\) | \(+\,0.57045\) | 0 | \(+2.0271\) | 0 | \(-2.8174\) | 0 | 0 | 0 |

\(\nu _{\tau }\) | \(+\,0.57045\) | 0 | \(+1.9143\) | 0 | \(-2.6816\) | 0 | 0 | 0 |

| \(-\,0.30483\) | \(+\,0.26557\) | \(-1.1539\) | \(-\,0.10353\) | \(-2.9811\) | 0 | \(-1.8171\) | \(+\,0.18789\) |

\(\mu \) | \(-\,0.30482\) | \(+\,0.26557\) | \(-1.0832\) | \(-\,0.10353\) | \(-2.8223\) | 0 | \(-1.7057\) | \(+\,0.18789\) |

\(\tau \) | \(-\,0.30482\) | \(+\,0.26557\) | \(-1.0229\) | \(-\,0.10345\) | \(-2.6862\) | 0 | \(-1.6107\) | \(+\,0.18775\) |

| \(+\,0.39429\) | \(-\,0.17612\) | \(+1.4626\) | \(+\,0.06866\) | \(-1.3031\) | 0 | \(+1.1900\) | \(-\,0.12526\) |

| \(+\,0.39277\) | \(-\,0.17764\) | \(+1.3499\) | \(+\,0.06924\) | \(+2.2871\) | 0 | \(+1.0998\) | \(-\,0.12524\) |

| \(+\,0.39363\) | \(-\,0.17678\) | \(+\,0.7390\) | \(-\,0.32167\) | \(+1.3984\) | \(+\,0.49872\) | \(+\,0.6028\) | \(+\,0.5872\) |

| \(-\,0.48099\) | \(+\,0.08945\) | \(-1.7886\) | \(-\,0.03487\) | \(-1.3042\) | 0 | \(-\,0.59500\) | \(+\,0.06269\) |

| \(-\,0.48250\) | \(+\,0.08793\) | \(-1.6583\) | \(-\,0.03427\) | \(+2.2826\) | 0 | \(-\,0.54990\) | \(+\,0.06262\) |

| \(-\,0.48248\) | \(+\,0.08793\) | \(-\,0.9093\) | \(+\,0.16143\) | \(+1.3959\) | \(-\,0.24918\) | \(-\,0.30139\) | \(-\,0.29359\) |

Masses of \(Z^{(1)}\), \(Z_R^{(1)}\) and \(\gamma ^{(1)}\) and total decay width of \(\gamma ^{(1)}\) in the unit of GeV. \(\Gamma _{\gamma ^{(1)}}\) is independent of the sign of the bulk fermion parameters. \(\Gamma _{Z^{(1)}/Z_R^{(1)}}(\pm ,\pm )\) represent that left and right sign is sign of \(c_l\) and \(c_q\)

\(\theta _H\) | \(m_{Z^{(1)}}\) | \(m_{Z_R^{(1)}}\) | \(m_{\gamma ^{(1)}}\) | \(\Gamma _{\gamma ^{(1)}}\) |
---|---|---|---|---|

0.10 | 6585 | 6172 | 6588 | 905 |

0.09 | 7149 | 6676 | 7152 | 940 |

0.08 | 7855 | 7305 | 7858 | 986 |

\(\theta _H\) | \(\Gamma _{Z^{(1)}}(+,+)\) | \(\Gamma _{Z^{(1)}}(+,-)\) | \(\Gamma _{Z^{(1)}}(-,+)\) | \(\Gamma _{Z^{(1)}}(-,-)\) |
---|---|---|---|---|

0.10 | 429 | 1632 | 959 | 2162 |

0.09 | 463 | 1674 | 1014 | 2225 |

0.08 | 534 | 1705 | 1112 | 2283 |

\(\theta _H\) | \(\Gamma _{Z_R^{(1)}}(+,+)\) | \(\Gamma _{Z_R^{(1)}}(+,-)\) | \(\Gamma _{Z_R^{(1)}}(-,+)\) | \(\Gamma _{Z_R^{(1)}}(-,-)\) |
---|---|---|---|---|

0.10 | 784 | 2437 | 2398 | 4051 |

0.09 | 856 | 2480 | 2529 | 4153 |

0.08 | 1005 | 2485 | 2758 | 4238 |

## 4 Cross section and forward–backward asymmetry

The parameters are constrained by the experimental results of the forward-backward asymmetries at the *Z*-pole. For \(c_l>0\), the deviations of the *Z*-boson couplings are \(O(0.01) \%\). In contrast, for \(c_l<0\), their deviations are \(O(0.1) \%\). Thus the forward-backward asymmetry of \(e^+e^-\rightarrow \mu ^+\mu ^-\) process at the *Z*-pole in the GHU model deviate nearly 10 % from the observed value for \(c_l<0\). Consequently, the value \(\sin ^2\theta _W=0.23122\) is not valid and the value of \(\theta _W\) which consistently explain the experimental results at the *Z*-pole must be searched. In this paper, \(c_l<0\) case is not considered further.

Considering the \(e^+e^-\rightarrow \mu ^+\mu ^-\) process, the difference of the cross sections between \(c_q>0\) case (\(\sigma ^{c_q>0}\)) and \(c_q<0\) case (\(\sigma ^{c_q<0}\)) arises from only the \(Z'\) decay widths. Consequently the deviation of \(\sigma ^{c_q<0}(\mu ^+\mu ^-)\) from \(\sigma ^{c_q>0}(\mu ^+\mu ^-)\) is small. As shown in [52], at \(\sqrt{s} = 250\) GeV with 250 \(\text {fb}^{-1}\) unpolarised beam, \(4.66 \times 10^5\) events are expected in the SM. Therefore the statistical uncertainty is 0.15 %. The difference of the cross sections of the two cases over the SM value, \((\sigma ^{c_q>0}-\sigma ^{c_q<0})/\sigma ^{\text {SM}}(\mu ^+\mu ^-)\) is less than 0.11% at \(\sqrt{s}=250\) GeV with unpolarised beam. For the forward-backward asymmetry \((A_\text {FB}^{c_q>0}-A_\text {FB}^{c_q<0})/A_\text {FB}^{\text {SM}}(\mu ^+\mu ^-)\) is less than 0.04 % at \(\sqrt{s}=250\) GeV with unpolarised beam. Thus the two cases are difficult to distinguish at the \(e^+e^-\rightarrow \mu ^+\mu ^-\) process. The detailed analysis of the \(e^+e^-\rightarrow \mu ^+\mu ^-\) process in the GHU model is shown in [52].

Deviations of the cross sections at \(\sqrt{s}=250\) GeV with \(P_{e^-}=+0.8\) and \(P_{e^+}=-0.3\) and 250 fb\(^{-1}\) luminosity for \(e^+e^-\rightarrow \bar{q}q ~(q\ne b, t)\) and \(e^+e^-\rightarrow \bar{b}b\) processes. The statistical uncertainties calculated by the SM prediction are 0.074 % for \(\bar{q}q\) and 0.20 % for \(\bar{b}b\)

\(\theta _H\) | \(\sigma ^{c_q>0}/\sigma ^\text {SM}(\bar{q}q)-1\) | \(\sigma ^{c_q<0}/\sigma ^\text {SM}(\bar{q}q)-1\) |
---|---|---|

0.10 | \(-4.56 \%\) (\(-61.5\sigma \)) | \(+\,0.16 \%\) (\(+2.21\sigma \)) |

0.09 | \(-3.70 \%\) (\(-49.9\sigma \)) | \(+\,0.14 \%\) (\(+1.90\sigma \)) |

0.08 | \(-2.98 \%\) (\(-40.0\sigma \)) | \(+\,0.11 \%\) (\(+1.53\sigma \)) |

\(\theta _H\) | \(\sigma ^{c_q>0}/\sigma ^\text {SM}(\bar{b}b)-1\) | \(\sigma ^{c_q<0}/\sigma ^\text {SM}(\bar{b}b)-1\) |
---|---|---|

0.10 | \(-4.18 \%\) (\(-21.1\sigma \)) | \(-3.96 \%\) (\(-19.9\sigma \)) |

0.09 | \(-3.41 \%\) (\(-17.2\sigma \)) | \(-3.29 \%\) (\(-16.6\sigma \)) |

0.08 | \(-2.84 \%\) (\(-14.3\sigma \)) | \(-2.65 \%\) (\(-13.3\sigma \)) |

Deviation of \(A_\text {FB}(\bar{c}c)\) and \(A_\text {FB}(\bar{b}b)\) at \(\sqrt{s}=250\) GeV with \(P_\text {eff}=+0.887\) and 250 fb\(^{-1}\) luminosity. The statistical uncertainties \(\sigma \) calculated by the event number of the SM prediction are 0.16 % and 0.70 %, respectively

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) |
---|---|---|

0.10 | \(-\,0.38 \%~(-2.32\sigma )\) | \(+\,1.68 \%~(+10.1\sigma )\) |

0.09 | \(-\,0.31 \%~(-1.88\sigma )\) | \(+\,1.25 \%~(+7.58\sigma )\) |

0.08 | \(-\,0.25 \%~(-1.49\sigma )\) | \(+\,1.03 \%~(+6.21\sigma )\) |

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) |
---|---|---|

0.10 | \(+4.24 \%~(+6.04\sigma )\) | \(+7.33 \%~(+10.4\sigma )\) |

0.09 | \(+4.07 \%~(+5.81\sigma )\) | \(+5.69 \%~(+8.10\sigma )\) |

0.08 | \(+4.15 \%~(+5.92\sigma )\) | \(+3.83 \%~(+5.45\sigma )\) |

Deviations of \(A_\text {FB}(\bar{c}c)\), \(A_\text {FB}(\bar{b}b)\) and \(A_\text {FB}(\bar{t}t)\) at \(\sqrt{s}=500\) GeV with \(P_\text {eff}=+0.887\) and 500 fb\(^{-1}\) luminosity. The statistical uncertainties calculated by the event number of the SM prediction are 0.26 %, 0.79 % and 0.53 %, respectively

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) |
---|---|---|

0.10 | \(-3.25 \%~(-12.3\sigma )\) | \(+7.49 \%~(+28.3\sigma )\) |

0.09 | \(-2.58 \%~(-9.77\sigma )\) | \(+6.50 \%~(+24.6\sigma )\) |

0.08 | \(-2.01 \%~(-7.59\sigma )\) | \(+5.53 \%~(+20.9\sigma )\) |

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) |
---|---|---|

0.10 | \(+15.4 \%~(+19.4\sigma )\) | \(+23.1 \%~(+29.1\sigma )\) |

0.09 | \(+14.3 \%~(+18.1\sigma )\) | \(+18.4 \%~(+23.3\sigma )\) |

0.08 | \(+14.1 \%~(+17.8\sigma )\) | \(+12.9 \%~(+16.3\sigma )\) |

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{t}t)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{t}t)-1\) |
---|---|---|

0.10 | \(+5.36 \%~(+9.96\sigma )\) | \(+9.25 \%~(+17.2\sigma )\) |

0.09 | \(+5.03 \%~(+9.35\sigma )\) | \(+7.23 \%~(+13.4\sigma )\) |

0.08 | \(+5.14 \%~(+9.55\sigma )\) | \(+4.91 \%~(+9.13\sigma )\) |

Deviation of \(A_\text {FB}(\bar{c}c)\), \(A_\text {FB}(\bar{b}b)\) and \(A_\text {FB}(\bar{t}t)\) at \(\sqrt{s}=1\) TeV with \(P_\text {eff}=+0.887\) and 1000 fb\(^{-1}\) luminosity. The statistical uncertainties calculated by the event number of the SM prediction are 0.39 %, 1.07 % and 0.45 %, respectively

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{c}c)-1\) |
---|---|---|

0.10 | \(-25.9 \%~(-67.3\sigma )\) | \(-2.46 \%~(-6.37\sigma )\) |

0.09 | \(-18.3 \%~(-47.6\sigma )\) | \(+4.61 \%~(+12.0\sigma )\) |

0.08 | \(-12.8 \%~(-33.3\sigma )\) | \(+9.02 \%~(+23.4\sigma )\) |

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{b}b)-1\) |
---|---|---|

0.10 | \(+46.1 \%~(+43.1\sigma )\) | \(+21.1 \%~(+19.7\sigma )\) |

0.09 | \(+45.7 \%~(+42.8\sigma )\) | \(+36.8 \%~(+34.4\sigma )\) |

0.08 | \(+44.9 \%~(+42.0\sigma )\) | \(+38.8 \%~(+36.3\sigma )\) |

\(\theta _H\) | \(A_\text {FB}^{c_q>0}/A_\text {FB}^\text {SM}(\bar{t}t)-1\) | \(A_\text {FB}^{c_q<0}/A_\text {FB}^\text {SM}(\bar{t}t)-1\) |
---|---|---|

0.10 | \(+11.3 \%~(+24.3\sigma )\) | \(+9.36 \%~(+20.6\sigma )\) |

0.09 | \(+11.6 \%~(+25.7\sigma )\) | \(+12.3 \%~(+27.1\sigma )\) |

0.08 | \(+12.2 \%~(+26.9\sigma )\) | \(+11.3 \%~(+25.0\sigma )\) |

For the \(e^+e^-\rightarrow \bar{b}b\) process, the cross section in the SM is \(\sigma ^\text {SM}(\bar{b}b)\)= 1.77 pb and 1.02 pb at \(\sqrt{s}=250\) GeV with unpolarised and polarised (\(P_{e^-}=+0.8\) and \(P_{e^-}=-0.3\)) beams, respectively. The statistical uncertainty at \(\sqrt{s}=250\) GeV and 250 fb\(^{-1}\) luminosity with \(P_{e^-}=+0.8\) and \(P_{e^+}=-0.3\) beam are 0.20 %. For this process, the cross sections in the GHU model with \(c_q>0\) and \(c_q<0\) cases both decrease from that in the SM. The deviations at \(\sqrt{s} = 250\) GeV are summarised in Table 10. The deference between the \(c_q>0\) and \(c_q<0\) cases more obviously appear at the forward-backward asymmetry. In the SM, \(A_\text {FB}^\text {SM}(\bar{b}b)=0.618\) and 0.366 at \(\sqrt{s}=250\) GeV with unpolarised and \(P_\text {eff}=+0.887\) beams, respectively. In the GHU model the \(A_\text {FB}(\bar{b}b)\) increase from the SM value, \(A_\text {FB}^{c_q>0}(\bar{b}b)\) increase \(4.24 \%\), \(4.07 \%\), \(4.15 \%\) and \(A_\text {FB}^{c_q<0}(\bar{b}b)\) increase \(7.33 \%\), \(5.69 \%\), \(3.83 \%\) at \(\sqrt{s}=250\) GeV with \(P_\text {eff}=+0.887\) for \(\theta _H=0.10\), 0.09 and 0.08. In Fig. 4, the ratio of the \(A_\text {FB}(\bar{b}b)\) in the GHU model to that in the SM with polarised beams are plotted. The \(c_q<0\) case predicts larger deviation than the \(c_q>0\) case for \(\theta _H=0.10\) and 0.09. At \(\sqrt{s}=250\) GeV with \(P_\text {eff}=+0.887\) and 250 fb\(^{-1}\) luminosity, the statistical uncertainty of the \(A_\text {FB}(\bar{b}b)\) in the SM is 0.70%. \(A_\text {FB}(\bar{b}b)\) in the GHU model deviates from the that in the SM larger than \(5.4\sigma \). The deviations at \(\sqrt{s} = 250\) GeV, 500 GeV and 1 TeV are summarised in Tables 11, 12 and 13, respectively.

The forward-backward asymmetry of the \(e^+e^-\rightarrow \bar{t}t\) process is measured at \(\sqrt{s}=500\) GeV. The polarisation dependence of \(A_\text {FB}^\text {GHU}/A_\text {FB}^\text {SM}(\bar{t}t)\) is qualitatively similar to that of \(A_\text {FB}^\text {GHU}/A_\text {FB}^\text {SM}(\bar{b}b)\), which also increase from that in the SM. \(A_\text {FB}^{c_q>0}(\bar{t}t)/A_\text {FB}^\text {SM}(\bar{t}t)-1=5.36 \%\), \(5.03 \%\), \(5.14 \%\) and \(A_\text {FB}^{c_q<0}(\bar{t}t)/A_\text {FB}^\text {SM}(\bar{t}t)-1=9.25 \%\), \(7.23 \%\), \(4.91 \%\) at \(\sqrt{s}=500\) GeV with \(P_\text {eff}=+0.887\) for \(\theta _H=0.10\), 0.09 and 0.08. At \(\sqrt{s}=500\) GeV with \(P_\text {eff}=+0.887\) and 500 fb\(^{-1}\) luminosity, \(\sigma ^\text {SM}(\bar{t}t)=479\) \(\text {fb}^{-1}\), \(A_\text {FB}^\text {SM}(\bar{t}t)=0.463\) and the uncertainty of the \(A_\text {FB}^\text {SM}(\bar{t}t)\) is 0.538 %. The deviations of the forward-backward asymmetries of the \(e^+e^-\rightarrow \bar{c}c,\ \bar{b}b,\ \bar{t}t\) processes at \(\sqrt{s}=500\) GeV and 1 TeV are summarised in Tables 12 and 13.

## 5 Summary

In the above calculations, the quark bulk mass parameters \((c_u, c_c, c_t)\) are assumed to be all positive or all negative. It is also allowed to be only one of them is positive or negative. In the case, the \(Z'\) decay widths change from the values shown in Table 9, therefore the cross sections and the forward-backward asymmetries slightly change from the results in this paper. Neglecting the difference arising from the \(Z'\) decay widths, the sign of the \(c_c\) is determined by measuring the \(A_\text {FB}(\bar{c}c)\) and the sign of the \(c_u\) is determined by the \(A_\text {FB}(\bar{c}c)\) and \(\sigma (\bar{q}q)\) at \(\sqrt{s}=250\) GeV. It is difficult to determine the sign of the \(c_t\) by measuring \(A_\text {FB}(\bar{b}b)\) and \(A_\text {FB}(\bar{t}t)\). At the ILC 250 GeV, the \(c_q<0\) case predicts \(4 \sigma \) larger deviation of the \(A_\text {FB}(\bar{b}b)\) than the \(c_q<0\) case for \(\theta _H=0.10\), and at the ILC 500 GeV \(5 \sigma \) larger deviation for \(\theta _H=0.09\). For \(\theta _H=0.08\), to clearly determine the sign of \(c_t\) by observing the \(A_\text {FB}(\bar{t}t)\), higher energy and luminosity, such as the ILC 1 TeV are necessary.

In this paper, the forward-backward asymmetries of the \(e^+e^-\rightarrow \bar{c}c,\ \bar{b}b,\ \bar{t}t\) processes in the GHU model are studied for two cases where all of the quark bulk mass parameters are positive or negative. The GHU model predicts large deviations at the \(\sqrt{s}=250\) GeV with polarised beams. Therefore the GHU model is testable at the ILC 250 GeV. The signs of the bulk mass parameters are distinguished at the ILC 500 GeV or ILC 1 TeV. For the case where the lepton bulk mass parameters are negative, detail is going to be analysed in near future.

## Notes

### Acknowledgements

I thank Yutaka Hosotani, Hisaki Hatanaka, Yuta Orikasa and Naoki Yamatsu for many advices and important discussions. This work is supported by the National Natural Science Foundation of China (Grant nos. 11775092, 11675061, 11521064 and 11435003), and the International Postdoctoral Exchange Fellowship Program (IPEFP).

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