3 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 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 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]. 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. 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) $^{2}$. 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. 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. 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. FIG. 1: Tevatron bounds on Z ′ X mass in models (I) and (II). ΔΓ$_{e}$ Γ$_{µ,τ}$ = (81 − 9) g 2 X 8 π 2 F 2 ( r ) , (12) F 2 ( r ) = − 7 2 − 2 r − (2 r + 3) ln r + 2(1 + r ) 2 × [ π 2 6 − Li 2 ( r 1 + r ) − 1 2 ln 2 ( r 1 + r )] . (13) g 2 $_{X}$/M 2 Z ′ X ∼ < 0 . 05 TeV − $^{2}$, (14) M 2 Z ′ X ∼ > g 2 X 4 π | x$_{e}$x$_{µ}$ | (Λ − $_{V V}$) 2 (15) M$_{Z}$ ′ X ∼ > 1 . 4 TeV in model (I) , (16) M$_{Z}$ ′ X ∼ > 2 . 3 TeV in model (II) , (17) Phenomenology of Z ′ X LEP2 contact interaction