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Let r ≡ M 2 Z ′ $_{X}$/M 2 $_{Z}$, then the one-loop radiative correction to Z → ℓ $^{+}$ℓ − from Z ′ X exchange in model (II) is given by [16] 

where 

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In the above, Li 2 ( x ) = − ∫ x 0 ( dt/t ) ln(1 − t ) is the Spence function. For r ≫ 1 (i.e. M 2 Z ′ X ≫ M 2 $_{Z}$) which will be required by Tevatron data in any case (as shown below), F 2 ≃ r − $^{1}$[11 / 9 + (2 / 3) ln r ] and the resulting numerical bound is 

to a good approximation. There is also an overall correc- tion to Γ( Z → hadrons) / Γ( Z → leptons), but that effect may be absorbed into the value of α$_{S}$ in QCD. 

The agreement of the SM and the LEP2 data provides the indirect bounds on the Z ′ X mass. The strongest one for our models (I) and (II), where µ and τ have the same charges and e has the charge of opposite sign, comes from the e $^{+}$e − → µ $^{+}$µ $^{−}$, τ $^{+}$τ − channel with Λ − V V = 16 TeV [17]. (See Ref. [18] for some useful discussions.) ′ 

The Z X mass is bounded by 

for sufficiently large M$_{Z}$ ′ compared to the LEP2 energy. It results 

for g$_{X}$ = 0 . 1. 

The direct production of Z ′ X is possible at hadron col- liders through its coupling to quarks. Its decay to leptons is then a clear signature. At present, there is a Tevatron limit on the cross section σ ( p ¯ p → Z ′ X → e $^{+}$e $^{−}$) at a center-of-mass energy E cm = 1 . 96 TeV, based on an inte- grated luminosity of L = 2 . 5 fb − 1 [19], and similarly for dimuons [20]. To compare against these results, we take 

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## LEP2 contact interaction 

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## Phenomenology of Z ′ X 

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g$_{X}$ = 0 . 1 for definiteness. We assume Γ$_{Z}$ ′ X = 0 . 01 M$_{Z}$ ′ X for model (I) and Γ$_{Z}$ ′ X = 0 . 04 M$_{Z}$ ′ X for model (II), which are approximately their true values if they only decay into particles of the Standard Model (SM), i.e. not their superpartners nor the additional singlets. 

For the numerical analysis, we use CompHEP/CalcHEP [21, 22] and the parton distribution functions of CTEQ6L [23]. Since the Tevatron bounds from dielectron and dimuon data are similar for the same coupling, we show only the dielectron result. We find M$_{Z}$ ′ X ∼ > 830 GeV for model (I) and M$_{Z}$ ′ X ∼ > 940 GeV for model (II) as shown in Fig. 1. (Since | x$_{µ}$ | < | x$_{e}$ | in model (II), the bound from the dimuon data is less stringent.) We note also that the e − µ − τ nonuniversality constraint of Eq. (14) is easily satisfied. 

For the LHC discovery reach (with the design energy E cm = 14 TeV) through the dilepton Z ′ X resonance, we use cuts p$_{T}$ > 20 GeV, | η | < 2 . 4 (for each lepton) and | m inv ( ℓ $^{+}$ℓ $^{−}$) | < 3Γ$_{Z}$ ′ $_{X}$. SM background at the LHC with these cuts is negligible, and we just require 10 signal events to claim its discovery at the LHC for a fixed flavor. 

Figs. 2(a) and 2(b) show the LHC discovery reach us- ing dileptons for models (I) and (II) respectively. From the dilepton resonance only, model (I) cannot be dis- tinguished from the flavor-independent case of B − L . In model (I), Z ′ X will be revealed by both e $^{+}$e − and µ $^{+}$µ − channels at the same luminosity ( L ≃ 1 fb − 1 for M$_{Z}$ ′ X = 1 . 5 TeV). In model (II), the µ $^{+}$µ − reso- nance will need a luminosity about an order of magni- tude larger than that for the e $^{+}$e − resonance because σ ( µ $^{+}$µ $^{−}$) /σ ( e $^{+}$e $^{−}$) ≃ ( − 3) $^{2}$/ (9) $^{2}$. 

Since Higgs doublets have zero charges under B − x$_{i}$L , the channels which require nonzero charges such as the 6-lepton resonance discussed in Ref. [24] will be absent. 

In the presence of Z ′ X at the TeV scale, a (predomi- nantly right-handed) sneutrino as well as the usual neu- tralino can be a good LSP dark-matter candidate [25]. 

