Bounds on the Number of Light Neutrinos Species , ′ 1 g Coupling and Z − Z ′ Mixing Angle in a U ( 1 ) B − L Model

The constraints on the number of neutrinos generations, ′ g1 coupling and ′ Z Z − mixing angle through the invisible width method, and in the framework of a ( )B L U 1 − model are obtained. Based on the experimental value reported by the LEP for the rate LEP exp inv ll R 5.942 0.016 = Γ Γ = ± , we obtained a bound on the ′ g1 coupling, ′ g1 0.65 ≤ . In addition, we derive 90% C.L. bounds on the ′ Z Z − mixing angle B L 4 4 2.2 10 1.0 10 − − − − × ≤ ≤ × θ , improving the existing bounds by one order of magnitude.


Introduction
The number of fermion generations, which is associated to the number of light neutrinos, is one of the most important predictions of the Standard Model of the electroweak interactions (SM) [1]- [3].In the SM, the decay width of the Z boson into each neutrino family is calculated to be 166.3 1.5 MeV νν Γ = ± [4].Additional generations, or other new weakly interacting particles with masses below 2 Z M , would lead to a decay width of the Z into invisible channels larger than the SM prediction for three families while a smaller value could be produced, for example, by the presence of one or more right-handed neutrinos mixed with the left-handed ones [5].Thus the number of light neutrinos generations N ν , defined as the ratio between the measured invisible decay width of the Z, inv Γ , and the SM expectation νν Γ for each neutrino family, needs not be an integer number and has to be measured with the highest possible accuracy.
The most precise measurement of the number of light ( ) active neutrino types, and therefore the number of associated fermion families, comes from the invisible Z width inv Γ , obtained by subtracting the observed width into quarks and charged leptons from the total width obtained from the lineshape.The number of effective neutrinos N ν is given by [4]   , where , the SM expression for the ratio of widths into a charged lepton and a single active neutrino, is introduced to reduce the model dependence and the experimental value for the ratio [4] 5.942 0.016.
In the SM the experimental value from the four LEP experiments for the number of light neutrinos species is 2.9841 0.0083 , excluding the possibility of a fourth family unless the neutrino is very heavy.This result is in agreement with cosmological constraints on the number of relativistic species around the time of Big Bang nucleosynthesis, which seems to indicate the existence of three very light neutrinos species [9]- [12].
The existence of a heavy neutral ( ) Z ′ vector boson is a feature of many extensions of the standard model.In particular, one (or more) additional ( ) 1 U ′ gauge factor provides one of the simplest extensions of the SM.Additional Z ′ gauge bosons appear in Grand Unified Theories (GUTs) [13], Superstring Theories [14], Left- Right Symmetric Models (LRSM) [15]- [17] and in other models such as models of composite gauge bosons [18].In particular, it is possible to study some phenomenological features associates with this extra neutral gauge boson through of the minimal B L − (baryon minus lepton number) extension of the SM [19] [20], where the gauge group is given by ( ) ( ) ( ) . This model is considered as one of the candidates to study physics beyond the standard model, contain a significant set of particles and interactions whose existence could be proven both at the LHC [21] and future Linear Colliders ILC/CLIC [22].Detailed discussions on the minimal

B L
− model can be found in the literature [23].
Our aim in this paper is to estimate constraints on the number of neutrinos generations, 1 g′ coupling and Z Z′ − mixing angle through the invisible width method, and in the framework of a ( ) This paper is organized as follows.In Section 2, we present the theoretical framework.In Section 3 we present the numerical computation.Finally, we summarize our conclusions in Section 4.

Theoretical Framework
We consider a ( ) ( ) ( ) model consisting of one doublet Φ and one singlet χ and briefly describe the lagrangian including the scalar, fermion and gauge sector.The Lagrange for the gauge sector is given by [24] [25] where a W µν , B µν and Z µν ′ are the field strength tensors for ( ) ( ) − , respectively.The Lagrangian for the scalar sector of the ( ) ( ) ( ) where the potential term is [23], with Φ and χ as the complex scalar Higgs doublet and singlet fields, respectively.The covariant derivatives for the doublet and singlet are given by [23] 1 1 where the doublet and singlet scalars are with G ± , Z G and z′ the Goldstone bosons of W ± , Z and Z ′ , respectively.From the Lagrangian of the ( ) ( ) ( )

the terms for the interactions between neutral gauge bosons ,
Z Z′ to a pair of fermions of the SM can be written in the form ( ) ( ) where the couplings of the , Z Z′ bosons with the SM fermions are given by where sin W g e θ = and B L θ − is the Z Z′ − mixing angle [23].The current bound on this parameter is

Results
In order to compare the respective expressions with the experimental result for the number of light neutrinos species N ν , we will use the definition for N ν in a SM analysis [26] given in Equation ( 1) and the LEP result for the Z invisible width [4] [6]- [8] given in Equation ( 2).
Since Equation ( 2) reduces the influence of the top quark mass, the expression ( ) g′ coupling of the model ( ) is presented in Figure 3. From the plot we see that there is a strong correlation between B L θ − and 1 g′ .By reversing the process we determined a limit on the 1 g′ coupling from the expression for the number of light neutrinos species ( ) B L N ν − given in Equation (11).The bound on the 1 g′ coupling have obtained by using the upper bound on the mixing angle B L θ − reported in the literature [4].In Figure 4, we show the dependence of the number of light neutrinos species with respect to the 1 g′ coupling.Using which is consistent with that obtained through from a Renormalisation Group Equation (RGE) analysis [30]- [32].Finally, is clear that the effects induced by the tree level Ze e + − and Zνν couplings in the ( ) model increase the decay widths ll Γ and νν Γ , and the predictions on the number of light neutrinos species are better estimated.Therefore of the e V g coupling constant given in Equation ( 9) we estimated bounds on the Z Z′ − mixing angle to different values of 1 g′ .The plot of this quantity as a function of B L θ − is show in

Conclusion
From the invisible width method and and in the framework of a ( )  1 compare favorably and improve the existing bounds reported in the literature [4] by one order of magnitude.Our work complements other studies on the number of light neutrinos species, 1 g′ coupling and Z Z′ − mixing angle.In the decoupling limit, that is to say, when 0 B L θ − = and 1 0 g′ = , we recover the bounds on N v for the SM previously reported in the literature [4] [6]- [8], as well as the f V g and f A g couplings of the SM.
through the invisible width method, and in the framework of a ( ) B L U 1 − model are obtained.Based on the experimental value reported by the LEP for the rateLEP . In the decoupling limit, that is to say, when 1 0 g′ = and 0 B L θ − = the couplings of the SM are recovered.
information on the meaning of N ν in the reduces the influence of the top quarks mass.To get information on the meaning of N ν in the

Figure 2 .
Figure 2. Allowed region for ( ) B L N ν − as a function of the mixing angle B L θ − with the experimental value LEP exp R .The dashed line shows the SM allowed region for N ν at 90% C.L.

Figure 5 .
Figure 5.The horizontal lines give the experimental region at 90% C.L. From this figure the bounds obtained for B L θ − are show inTable1.The bounds obtained in Table 1 for the mixing angle B L θ − are one order of

Figure 4 .
Figure 4.The curve shows the shape for ( ) B L N ν − as a function of the 1 g′

Figure 5 .Table 1 .
Figure 5.The curves shows the shape for e V g as a function of the mixing angle B L θ − .The curves are for 1 0.2,0.4,0.6,0.8,1g′ = .

.
[4] bounds obtained in Table1for the mixing angle B L θ − are one order of magnitude stronger than the one obtained in the literature[4].