Z' Searches: From Tevatron to LHC\address
Departamento de Física Teórica, Instituto de Física,
Universidad Nacional Autónoma de México, 04510 México D.F., México
School of Natural Sciences, Institute for Advanced Study,
Einstein Drive, Princeton, NJ 08540, USA
The CDF collaboration has set lower limits on the masses of the bosons occurring in a range of GUT based models. We revisit their analysis and extend it to certain other scenarios as well as to some general classes of models satisfying the anomaly cancellation conditions, which are not included in the CDF analysis. We also suggest a Bayesian statistical method for finding exclusion limits on the mass, which allows one to explore a wide range of the gauge coupling parameter. This method also takes into account the effects of interference between the and the SM gauge bosons.
A neutral gauge boson appears in numerous models containing the SM gauge symmetry group along with an additional symmetry (for a review, see ). Grand Unified Theories (GUTs) larger than the original model, such as  or [3, 4], break down to the SM as: , with in turn SM . The thus surviving at the electroweak (EW) scale can be written as the linear combination,
If kinetic mixing [6, 7, 8] with the hypercharge group is neglected by setting in eq. (1), one obtains some well–known bosons by adjusting . These include (), (), () , () , () [10, 11] and () . The inclusion of kinetic mixing results in certain other phenomenologically interesting cases, such as which does not couple to the –type quarks, ()  which couples to the right–handed fermions only and ) , where is the baryon number and the lepton number of an ordinary fermion. The which exists in models with left–right symmetry  is equivalent to the linear combination: , where , with being the weak mixing angle and being the coupling strengths, respectively. The studied here corresponds to the specific case of . In addition to these based models, a from a specific superstring model  and a sequential are also included in this analysis.
A number of other classes of one or more–parameter models have been discussed [17, 18, 1]. Without the assumption of unification at the GUT scale, but assuming nullification of anomalies using three families of exotics (which is not the case in some supersymmetric models), there are four ‘one-parameter’ models , with fermion charges , , and , with arbitrary. The last three of these correspond to the generalized charges in eq. (1). ( corresponds to arbitrary superpositions of and , while corresponds to the models without kinetic mixing.) We have, therefore, normalized the fermions charges, , in the –parameter models to their values, . These charges are given in Table 1.
2 production at CDF and limits on its mass
The total cross–section for the Drell–Yan (DY) process at a hadron collider, with a neutral gauge boson as the mediator and as the outgoing particles, is given as 
where is the invariant mass of the muon pair, is the center–of–mass (CM) energy, and
where is the quark color factor. and above are the momentum fractions of the ingoing partons having parton distribution functions (PDFs) for hadrons and . is the strong coupling constant and , with being the QCD –factor and being the multiplicative factor due to QED corrections. is equal to
with being the polar angle defined in the CM frame and the individual amplitudes given as
where are run over . and are the electric charges of the contributing quark and the muon, respectively. , with being the gauge coupling strength of the boson, the coupling and the EW and charges of the fermion . are the total decay widths of the and bosons having masses .
We first follow the CDF analysis  and use the LO expression given in eq. (2) to calculate the cross–section due to the alone, neglecting the and contributions to the amplitudes in eq. (5). We employ LO CTEQ6L PDFs  and mass–dependent NNLO  and NLO  values, and assume that the decays into SM fermions only, which are taken to be massless. We achieve up to 99.8% agreement on the 95% confidence level (C.L.) lower limits with the CDF analysis, for the models included therein, and obtain new limits for the rest of the models. The numerical values of the limits are given in Table 2 and the parametrization of the models with contours in is plotted in Fig. 1. In Table 3 we give limits obtained for the LHC with a similar approach for some expected integrated luminosity and CM energy values.
3 Bayesian statistical method
The CDF analysis uses signal templates generated with a fixed resonance pole width, . However, there is no fundamental reason to only look for such a narrow . A wide resonance, implying a strongly coupling boson, could well be scattered over a few bins and no significant enhancement above the background will be visible. Besides, the effects of interference between the various bosonic contributions to the propagator, i.e., between , and (see eq. (5)), are lost in their approach. These effects could in principle cause a considerable enhancement or dip in the number of events in several accompanying low bins, e.g., in the case of a strongly interacting boson with mass just beyond the kinematic reach of the CDF. Finally, the CDF limits assume a fixed GUT–based and it is not straightforward to extend the limits to other values, particularly in the strong coupling regime (see, however ).
Therefore, we propose a Bayesian statistical method which allows one to vary in order to obtain the corresponding limit on . It is based on the likelihood function , written as
where is the Poisson probability of finding events given expected events in the th bin with total bins. in eq. (6) is then evaluated using two hypotheses: the null hypothesis, , assumes that the DY process occurs only via and , and the signal hypothesis, , with , wherein the boson also contributes to the cross–section along with the SM gauge bosons. The SM events, , expected in an invariant mass bin are calculated as
where fb is the integrated luminosity at the CDF, is the detector efficiency and is the CDF acceptance, which is a mass–dependent multiplicative factor. in the above equation is the muon–pair mass measured by the detector, refers to the non–DY events and the probability density is given as
with , , where TeV is the variance. The purpose of the above probability function is to smear over the DY background before distributing it into bins, hence accounting for the mis–identification of an event in a bin where it does not actually belong. For , a function is constructed as
and is minimized to obtain the best–fit values in the given range of and , which correspond to a boson with cross–section best favored by the data. The contours in and , for a certain C.L. value specified by the allowed number of standard deviations, , from the minimum, can then also be drawn, giving the exclusion limits on these parameters. Our preliminary contours are given in Fig. 2 for as a representative model. The numerical value of the 95% C.L. limit is 913 GeV for , which is about 21 GeV higher than the CDF value. We next plan to undertake a global analysis including constraints from electroweak precision data . Eventually, a similar statistical analysis of the LHC data, as soon as it is released, will also be performed.
The work at IF-UNAM is supported by CONACyT project 82291–F. The work of P.L. is supported by an IBM Einstein Fellowship and by NSF grant PHY–0969448.
- P. Langacker, Rev. Mod. Phys. 81, 1199 (2009).
- R.W. Robinett and J.L. Rosner, Phys. Rev. D 26, 2396 (1982).
- P. Langacker, R.W. Robinett and J.L. Rosner, Phys. Rev. D 30, 1470 (1984).
- J.L. Hewett and T.G. Rizzo, Phys. Rept. 183, 193 (1989).
- H. Georgi and S.L. Glashow, Phys. Lett. B 32, 438 (1974).
- B. Holdom, Phys. Lett. B 166, 196 (1986).
- F. del Aguila, M. Masip and M. Perez-Victoria, Nucl. Phys. B 456, 531 (1995).
- K.S. Babu, C.F. Kolda and J. March-Russell, Phys. Rev. D 54, 4635 (1996) and Phys. Rev. D 57, 6788 (1998).
- P. Candelas, G.T. Horowitz, A. Strominger, and E. Witten, Nucl. Phys. B 258, 46 (1985).
- J. Erler, P. Langacker and T. Li, Phys. Rev. D 66, 015002 (2002).
- J. Kang, P. Langacker, T. Li and T. Liu, Phys. Rev. Lett. 94, 061801 (2005).
- E. Ma, Phys. Lett. B 380, 286 (1996).
- J. Erler, P. Langacker, S. Munir and E. Rojas, JHEP 06, 017 (2009).
- T. Appelquist, B.A. Dobrescu and A.R. Hopper, Phys. Rev. D 68, 035012 (2003).
- R.N. Mohapatra in Unification And Supersymmetry (Springer–Verlag, Berlin, 1986).
- S. Chaudhuri, S.W. Chung, G. Hockney and J.D. Lykken, Nucl. Phys. B 456, 89 (1995).
- J. Erler, Nucl. Phys. B 586, 73 (2000).
- M.S. Carena, A. Daleo, B.A. Dobrescu and T.M.P. Tait, Phys. Rev. D 70, 093009 (2004).
- J. Erler, P. Langacker, E. Rojas and S. Munir, work in progress.
- R.K. Ellis, W.J. Stirling and B.R. Webber, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8, 1 (1996).
- CDF collaboration, T. Aaltonen et al., Phys. Rev. Lett. 102, 091805 (2009).
- J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P.M. Nadolsky and W.K. Tung, JHEP 07, 012 (2002).
- U. Baur, S. Keller and W.K. Sakumoto, Phys. Rev. D 57, 199 (1998).
- CDF Collaboration: D. Whiteson et al., conference note CDF/PHYS/EXO/PUBLIC/ 10165.