Optimal observables for $Z^{\prime }$ models in annihilation leptonic processes

The optimal observables with the best ratio of signal to statistical uncertainty are proposed for a bunch of popular models of the $Z^{\prime }$ boson. They are the cross sections integrated over the phase space of the final particles with proper weight functions. It is shown that the proposed observables are completely equivalent to the ${\chi }^2$ fit of the differential cross section, so they could be used as an alternative of aggregating events into bins with further minimization of the ${\chi }^2$ function, especially in preliminary analysis of experimental data. Application of the observables to the maximum likelihood estimate of the $Z^{\prime }$ mass and the $Z$–$Z^{\prime }$ mixing angle as well as to the exclusion reach and statistical efficiency of the signal is investigated in details.

Figure 1

The SM differential cross section (unpolarized) in picobarns with 2% systematical error used in numeric calculations. The results of zfitter are plotted as lines.

Figure 2

Estimation of the NLO contribution in $\mu$ to the cross section. The $Z^{\prime }$ mass and the collision energy correspond to the maximal influence from $F_2$, $\mu =-6.7\times 10^{-4}$. The dots show the maximal values of $F_1$ used to set the level of systematic errors. In $\mathrm{U}(1{)}_{X,x=1}$ and $\mathrm{U}(1{)}_{X,x=-1.25}$ models (marked as ${\mathrm{X}}_1$ and ${\mathrm{X}}_{-1.25}$), the NLO term is below 4% of $F_1$. For other models, this term is below 1.5%. Lower energies and higher $M_{Z^{\prime }}$ suppress the relative contribution from $F_2$. The plotted factors for the $\mathrm{U}(1{)}_X$ model must be multiplied by 2.5. The factors are in picobarns.

Figure 3

The fine structure constant ${\alpha }_{\mathrm{em}}$ determined by the photon polarization operator in a wide interval of energies. The values at $m_Z$ as well as at the supposed ILC energies are shown separately.

Figure 4

The unpolarized factors $F_1(\sqrt{s},z)$ in picobarns for models at 250 GeV. The factors are calculated up to 5% systematic error. The plotted factors for the $\mathrm{U}(1{)}_X$ model must be multiplied by 2.5.

Figure 5

The weight functions $-w(\sqrt{s},z)$ to amplify the $Z^{\prime }$ signal and to measure $M_{Z^{\prime }}$ at the ILC energies. The uncertainty arises from the systematic errors on the SM cross section and $Z^{\prime }$ factors. The universal normalization ${\int }_{-1}^1{w}^{2}dz=1$ is used to compare different models. The negative sign at $w$ is chosen to obtain positive values at $z=1$. The actual systematic error might be significantly less, accounting for possible destructive interference between the weight function and the uncertainties of the SM and $Z^{\prime }$ contribution to the cross section. The lines correspond to the SM from zfitter.

Figure 6

The unpolarized factors $F_{\mathrm{mix}}(\sqrt{s},z)$ in picobarns at 250 GeV for the models with the mixing angle as a separate parameter. The factors are calculated up to 5% systematic error. The plotted factors for the $\mathrm{U}(1{)}_X$ model must be multiplied by 2.5.

Figure 7

The weight functions $\eta w(\sqrt{s},z)$ to measure $M_{Z^{\prime }}$ and ${\theta }_0$ at the ILC energies. The universal normalization ${\int }_{-1}^1{w}^{2}dz=1$ is used to compare different models. The sign $\eta =-1$ for the $\chi$ and $\mathrm{U}(1{)}_{X,x=1}$ models and $\eta =1$ otherwise. The lines correspond to the SM from zfitter. The complete phase space $z\in [-1,1]$ is used.

Figure 8

The weight functions $-w(\sqrt{s},z)$ to measure $\mu$ in the Abelian models with ${\theta }_0$ from Eq. (39). The universal normalization ${\int }_{-1}^1{w}^{2}dz=1$ is used to compare different models. The shorthand notations ${\mathrm{X}}_1$ and ${\mathrm{X}}_{-1.25}$ are used for $\mathrm{U}(1{)}_{X,x=1}$ and $\mathrm{U}(1{)}_{X,x=-1.25}$ models.