Z' physics with early LHC data
We discuss the prospects for setting limits on or discovering spin-1 bosons using early LHC data at 7 TeV. Our results are based on the narrow width approximation in which the leptonic Drell-Yan boson production cross-section only depends on the boson mass together with two parameters and . We carefully discuss the experimental cuts that should be applied and tabulate the theoretical next-to-next-to-leading order corrections which must be included. Using these results the approach then provides a safe, convenient and unbiased way of comparing experiment to theoretical models which avoids any built-in model dependent assumptions. We apply the method to three classes of perturbative boson benchmark models: models, left-right symmetric models and sequential standard models. We generalise each class of model in terms of mixing angles which continuously parametrize linear combinations of pairs of generators and lead to distinctive orbits in the plane. We also apply this method to the strongly coupled four-site benchmark model in which two bosons are predicted. By comparing the experimental limits or discovery bands to the theoretical predictions on the - plane, we show that the LHC at 7 TeV with integrated luminosity of 500 pb will greatly improve on current Tevatron mass limits for the benchmark models. If a is discovered our results show that measurement of the mass and cross-section will provide a powerful discriminator between the benchmark models using this approach.
The end of the first decade of the millenium is an exciting time in particle physics, with the CERN LHC enjoying an extended run at 7 TeV, and the Fermilab Tevatron collecting unprecedented levels of integrated luminosity, eventually up to perhaps 10 fb, in the race to discover the first signs of new physics Beyond the Standard Model (BSM). Since spin-1 bosons are predicted by dozens of such models, and are very easy to discover in the leptonic Drell-Yan mode, this makes them good candidates for an early discovery at the LHC. For a review see Refs. Langacker (2008); Erler et al. (2009); Nath et al. (2010) and references therein. Furthermore high mass bosons are more likely to be discovered at the LHC than the Tevatron Carena et al. (2004), since energy is more important than luminosity for the discovery of high mass states. This makes the study of bosons both timely and promising and has led to widespread recent interest in this subject (see for example Chen and Dobrescu (2008); Coriano et al. (2008a); King et al. (2006a, b); Howl and King (2008); Li et al. (2009); Diener et al. (2010); Erler et al. (2010); Petriello and Quackenbush (2008); Accomando et al. (2010); Basso et al. (2010); Athron et al. (2009a, b); Appelquist et al. (2003); Hays et al. (2009); Salvioni et al. (2009, 2010); Coriano et al. (2008a)).
Since one of the purposes of this paper is to facilitate the connection between experiment and theory, it is worth being clear at the outset precisely what we shall mean by a boson. To an experimentalist a is a resonance “bump” more massive than the of the Standard Model (SM) which can be observed in Drell-Yan production followed by its decay into lepton-antilepton pairs. To a phenomenologist a boson is a new massive electrically neutral, colourless boson (equal to its own antiparticle) which couples to SM matter. To a theorist it is useful to classify the according to its spin, even though actually measuring its spin will require high statistics. For example a spin-0 particle could correspond to a sneutrino in R-parity violating supersymmetric (SUSY) models. A spin-2 resonance could be identified as a Kaluza-Klein (KK) excited graviton in Randall-Sundrum models. However a spin-1 is by far the most common possibility usually considered, and this is what we shall mean by a boson in this paper.
In this paper, then, we shall discuss electrically neutral colourless spin-1 bosons, which are produced by the Drell-Yan mechanism and decay into lepton-antilepton pairs, yielding a resonance bump more massive than the . We shall be particularly interested in the prospects for discovering or setting limits on such bosons using early LHC data. By early LHC data we mean the present 2010/11 run at the LHC at 7 TeV, which is anticipated to yield an integrated luminosity approximately of 1 fb. Since the present LHC schedule involves a shut-down during 2012, followed by a restart in 2013, the early LHC data will provide the best information possible about bosons over the next three years, so in this paper we shall focus exclusively on what can be achieved using these data, comparing the results with current Tevatron limits. In order to enable contact to be made between early LHC experimental data and theoretical models, we advocate the narrow width approximation, in which the leptonic Drell-Yan boson production cross-section only depends on the boson mass together with two parameters and Carena et al. (2004). Properly defined experimental information on the boson cross-section may then be recast as limit or discovery contours in the plane, with a unique contour for each value of boson mass. In order to illustrate how this formalism enables contact to be made with theoretical models we study three classes of boson benchmark models: models, left-right (LR) symmetric models and sequential standard models (SSM). We also apply this method to the strongly coupled four-site benchmark model in which two bosons are predicted Accomando et al. (2010). Each benchmark model may be expressed in the plane which enables contact to be made with the experimental limit or discovery contours. Working to next-to-next-leading order (NNLO) we show that the LHC at 7 TeV with as little data as 500 pb can either greatly improve on current Tevatron mass limits, or discover a , with a measurement of the mass and cross-section providing powerful resolving power between the benchmark models using this approach. We also briefly discuss the impact of the boson width on search strategies. Although the width into standard model particles may in principle be predicted as a function of , in practice there may be considerable uncertainty concerning due for example to the possible decay into other non-standard particles, including supersymmetric (SUSY) partners and exotic states, for example.
At the outset we would like to highlight some of the strengths and
limitations of our approach to the benchmark models. One of the strengths of
our approach is that the considered benchmark models encompass two quite
different types of models: perturbative gauge models and strongly
coupled models, where the perturbative models generally involve relatively
narrow widths (which however can get larger if SUSY and exotics are included
in the decays in addition to SM particles), while the strongly coupled models
involve multiple bosons with rather broad widths. The perturbative
benchmark gauge models are defined in terms
of continuous mixing angles,
in analogy to the class of models
expressed through the linear combinations of and generators. This
approach is generalised to the case of LR models involving linear
combinations of the generators and
The layout of the rest of the paper is as follows. In section II we describe our model independent approach based on the narrow width approximation and the variables , and . Higher order corrections to the cross-section are tabulated in the narrow width approximation and new K factors are defined which enable the and approach to be reliably extended to NNLO. In this section we consider finite width effects and discuss the choice of invariant mass window around which matches the narrow width approximation, showing that in the case of the LHC this choice is only weakly constrained. We also comment on the effects of interference and show that they may become important for invariant mass cuts close to 100 GeV but are negligible for the a suitable invariant mass cut around , which we therefore advocate. In section III we define our benchmark models based on , LR and SSM, generalised using variables which continuously parametrize mixing of the respective generators.
For these perturbative models we specify the gauge coupling and calculate the vector and axial couplings in terms of the mixing angle variable. We tabulate the results for some special choices of the mixing angle variable which reproduce the models commonly quoted and analysed in the literature. For the four-site model we discuss the parameter space describing the two bosons allowed by electroweak precision measurements and unitarity. In section IV we apply the model independent approach to the benchmark models discussing first the current Tevatron limits using the latest results with collected luminosity , then the expected LHC potential based on the projected CMS limits on the boson cross section normalized to the SM -boson one for an integrated luminosity . We use the results rather than since they are more closely related to the narrow width approximation that we advocate, and we use CMS rather than ATLAS since the projected limits are publicly available. In both cases we express the experimental cross-section limits in the - plane and compare these limits to the benchmark models also displayed in the - plane. In the case of CMS we also show the discovery limits. We tabulate some of the results for some special choices of the mixing angle variable, including the width, the leptonic branching ratio, and values, our derived current direct limit on the mass based on results, as well as other indirect and mixing limits where available. Finally in section V we discuss the impact of on search strategies. Section VI summarises and concludes the paper.
Ii Model Independent Approach
At collider energies, the gauge group of a typical model predicting a single extra boson is:
where the Standard Model is augmented by an additional gauge group. The gauge group is broken near the TeV scale giving rise to a massive gauge boson with couplings to a SM fermion given by:
The values of depend on the particular choice of and on the particular fermion . We assume universality amongst the three families, , and , as well as , and similarly for the couplings, which means that there are eight model dependent couplings to SM fermions , with . These eight couplings are not all independent since they are related to the left (L) and right (R) couplings as follows:
where , and . The couplings are not all independent since the left-handed fermions are in doublets with the same charges and . Excluding the right-handed neutrinos (which we assume to be heavier than the ) there are really five independent couplings . However we prefer to work with the eight vector and axial couplings . In addition, the strength of the gauge coupling is model dependent. Throughout, we follow the conventions of Langacker (2008).
A slightly more complicated setup is needed to describe the four-site model which, in this paper, has been chosen to represent Higgsless multiple -boson theories. The corresponding framework will be given in Sec.III.2.1.
Throughout the paper we shall ignore the couplings of the to beyond SM particles such as right-handed neutrinos, SUSY partners and any exotic fermions in the theory, which all together may increase the width of the by up to about a factor of five Kang and Langacker (2005) and hence lower the branching ratio into leptons by the same factor.
ii.2 production and decay in the narrow width approximation
The contribution to the Drell-Yan production cross-section of fermion-antifermion pairs in a symmetric mass window around the mass () may be written as:
In the narrow width approximation (NWA), it becomes
where the parton luminosities are written as and is the peak cross-section given by:
The branching ratio of the boson into fermion-antifermion pairs is
where is the total width and the partial widths into a particular fermion-antifermion pair of colours is given by
Assuming only SM fermions in the final state and neglecting in first approximation their mass, one finds the total width:
where the coefficients and are given by:
and and are related to the parton luminosities and and therefore only depend on the collider energy and the mass. All the model dependence of the cross-section is therefore contained in the two coefficients, and . These parameters can be calculated from and , assuming only SM decays of the boson. The corresponding values for all models which predict a single boson purely decaying into SM fermions are given in Table 2.
A slight complication arises in Higgsless theories, which in the present paper are represented by the four-site model. Here in fact the two neutral extra gauge bosons, , decay preferebly into di-boson intermediate states. Their total width gets therefore two contributions:
where the two terms on the right-hand side represent the fermionic and bosonic decay, respectively. More in detail,
with i=1,2, where in this case we have included the coupling in the definition of and . In the above formulas and are the masses of the two extra gauge bosons, , while is their ratio, i.e. . The direct consequence of this peculiarity is that the leptonic branching ratio acquires a not trivial dependence on the boson mass which reflects in an intrinsic mass dependence of the and coefficients. In addition, there is an external source of variation with mass. As all vector and axial couplings in the four-site model can be expressed in terms of the three independent free parameters (, , ), and are completely specified by these quantities as well: . This means that at fixed masses, and , these coefficients get constrained by the EWPT bounds acting on . As these limits vary with mass (see Fig. 6), and acquire this extra dependence. The net result opens up a parameter space in the plane which will be dispalyed at due time.
As emphasized in Carena et al. (2004), the plane parametrization is a model-independent way to create a direct correspondance between the experimental bounds on cross sections and the parameters of the Lagrangian. An experimental limit on for a given mass gives in fact a linear relation between and ,
where can be regarded as known numbers given by:
where represents the 95 C.L. upper bound on the experimental Drell-Yan cross section which can be derived from observed data.
In practice, it is more convenient to use a log-log scale resulting in the limits appearing as contours for a fixed mass in the plane. We use this representation in the next subsections.
ii.3 Higher-order corrections
At higher-orders, the expression for production given by Eq. (II.8) strictly speaking is no longer valid. However, as it was shown in Ref. Carena et al. (2004), the additional terms which are not proportional to and in Eq. (II.8) can be neglected at NNLO. Therefore, Eq. (II.8) gives a quite accurate description of the approach we are discussing here even at NNLO.
In the following, we take into account QCD NNLO effects as implemented in
the WZPROD program Hamberg et al. (1991); van Neerven and Zijlstra (1992); Hamberg et al. ()
as a correction to the total production cross section
in the NWA.
The QCD NNLO production cross sections obtained using CTEQ6.6 and MSTW08 PDFs are in a good agreement. Their difference is in fact at the 2-3% level over a wide mass spectrum as shown in Fig. 1, where we plot the total cross section at the Tevatron and LHC@7TeV versus the mass. Here, we have taken as factorization scale the value . The further detailed analysis of the cross section variation with the scale is outside of the scope of the current paper.
It is also convenient to define customary NLO and NNLO -factors which can be useful for experimentalists in establishing exclusion limits:
where the index corresponds to NLO and NNLO -factors, respectively. As an example, in Fig. 2 we present the values of these -factors for Standard Model-like production at the Tevatron (left panel) and the LHC@7TeV (right panel) for CTEQ6.6 and MSTW08 PDF’s.
Oppositely to what happens for the aforementioned exact NNLO production cross section, where the agreement between CTEQ6.6 and MSTW08 PDF predictions is optimal, in this case there is a noticable difference in the behavior of and factors as a function of the mass when convoluting the production cross section with CTEQ6.6 or MSTW08 PDF’s. This difference is related to the way of fitting the LO PDF’s of CTEQ and MSTW collaborations (see e.g. Lai et al. (2010); Martin et al. (2009)). Furthemore, both and factors display a strong dependence on the mass. As an example, varies between 10-40% at the Tevatron and 10-30% at the LHC@7TeV for potentially accessible masses.
Applying a universal -factor can be highly misleading. As shown above, the -factor has indeed a two-fold source of dependence: PDF set and energy scale (i.e. ). A uniform setup must be fixed when comparing experimental limits on different models.
Since Eq. (II.8) gives an accurate description even at NNLO Carena et al. (2004), and noting that QCD NNLO corrections are universal for up- and down-quarks, one can effectively apply the same -factor derived for SM-like to generic models without loosing of generality. Owing to the remarkable mass dependence of the -factor, we first convolute the LO production cross section with the respective LO PDF’s and then we multiply it by .
For convenience and clarity, we provide in Tables 3 and 4 shown in Appendix A the values of -factors and cross sections for the SM-like -boson production process at the Tevatron and the LHC@7TeV: . The first table contains the results obtained with MSTW08 PDF, the latter with CTEQ6.6 PDF. The quoted numbers correspond to the curves visualised in Figs.1,2.
In narrow width approximation, the two-fermion cross section is the product of the production cross section and the respective branching ratio. When considering the complete -boson production and decay in the Drell-Yan channel, one has to keep in mind that QCD NNLO corrections also affect the branching ratio even for purely leptonic decays, , since the total decay width will be corrected at NNLO. This reflects into an higher order correction to the and coefficients, through which explicitly enters the expression for and given in Eq.(II.9). The NNLO Drell-Yan cross section can be thus written as:
The leading NLO QCD correction to the total width is known to be Gorishnii et al. (1988); Kataev (1992). This gives an enhancement of the order of 2-3% to the width for in the range 500-2000 GeV. The will thus decrease accordingly by . The net result corresponds to a deplection of the leptonic branching ratio within the SM-like model. An effect of the same order is expected for the other classes of models under consideration. In the current study, we neglect this effect and use the following formula for establishing limits on models:
ii.4 Finite width effects
So far we have have discussed the boson production using narrow width approximation. However, the experimental search for an extra boson and the descrimination of the SM backgrounds could strongly depend on the realistic width. Moreover the theoretical prediction of the production cross section also depends on its width as we discuss below.
We start this discussion with Fig. 3 where we present the di-lepton invariant mass distribution for the boson production within various models at the Tevatron (left panel) and the LHC@7TeV (right panel). We consider three representative models: the SM-like model (black line), the N-type model defined in Table 1 (red line), and the weakly coupled SM-like model where the boson gauge coupling to SM fermions is reduced by a factor 10 (blue line). From top to bottom, the last two distributions are normalized to the integral under the first one. We first consider the SM-like model distribution at the Tevatron. It is important to stress that the total cross section of process integrated over the entire range is actually is almost as twice as large as the SM-like in the narrow width approximation. The main reason for this effect is the specific shape of the distribution in the region of small far away from . This region is exhibited by a non-negligible tail due to the steeply rising PDF in the region of low even though the boson isextremely far off mass-shell in this region. The integral over this region can even double the cross section evaluated in the NWA in the case of production at the Tevatron.
This effect, which is related to the off-shellness of the extra gauge boson, varies according to the total width. In the model, it brings an additional 20% contribution to the narrow width approximation cross section at the Tevatron. In the weakly coupled SM-like model, the far off-shellness effects are effectively negligible (below 1%).
We can see that in general experimental limits would and should strongly depend on the particular model predicting a specific width. On the other hand, if one requires a di-lepton mass window cut around the mass, one can establish a quasi model-independent experimental upper limit on versus and apply this limit to constraint different classes of models.
In Fig. 4 we present the effect of a symmetric mass window cut around for the SM-like model and two other representative models (see Table 1) at the Tevatron and the LHC@7TeV. We fix the mass to be TeV. We plot the relative difference between the full cross section for the process evaluated taking into account the finite width () and the cross section computed in narrow width approximation (). The relative difference is presented as a function if the symmetric mass window cut () applied to the full cross section . One can see that for the SM-like model at the Tevatron, a cut in the 9-25% range brings the agreement between and down to 5% level while exactly matches and . At the LHC, the corresponding range of the cut is 15-80%, and and are matched for . The model has a narrower width, making the choice of the cut more insensitive, while the model width is broader leading to a more sensitive choice of the mass window cut to reproduce the narrow width approximation. Note that all lines cross the abscissa at about the same value of , meaning that there will be an optimal mass window cut consistent with all models. The choice of the mass window cut to gain agreement with the narrow width approximation also depends on . This dependence is defined by proton parton densities and is therefore model-independent. The net effect is again to make all the lines cross the abscissa at about the same value of , where this point depends on . Therefore for every given mass one can work out a quasi model-independent mass window cut where the full cross section matches the narrow width approximation. The additional advantage of this choice is that in the selected mass window around the mass the model-dependent interference effect between signal and SM background is highly suppressed.
The experimental limits would be quasi model-independent if one would apply this cut on the around the : it brings in agreement the cross section calculated in the narrow width approximation and in the finite width approximation as well as removes model-dependent shape of the distributions in the region of low especially for the case of large width effects as, for example, take place for SM-like . Moreover, the cut on around the plays an important role in reducing an effect of interference with down to a few% level, which again, allows to establish and use experimental limits in model-independent way.
For example, in case of SM-like production at the Tevatron, the relative interference, which is defined as is as large as about (meaning of interference(!)) for GeV cut but it drops down to for cut which matches NWA and finite width cross sections. The effect of the mass window cuts is also quite large for the case of SM-like production at the LHC, where interference is about for GeV cut and only about for cut.
We can see, that there is a strong motivation to use an invariant mass window cut for conducting a model-independent analysis. The size of this cut, if one aims to match the NWA and finite width cross sections, is collider dependent: it is about of for the Tevatron and about of for the LHC7TeV.
In this paper we are using results of experimental analysis which are based on mass window cut similar to what we are advocating. This would allow us to use precise NNLO model predictions and perform a respective model-independent interpretation of the experimental limits.
Iii Benchmark Models
In this section, we extend and classify the benchmark models present in the literature. We divide such classes into two main types: perturbative and strongly coupled gauge theories.
iii.1 Perturbative gauge theories
In these models one envisages that at the GUT scale the gauge group is . The gauge group is broken at the GUT scale to and a gauge group,
The is further broken at the GUT scale to and a gauge group,
Finally the is broken at the GUT scale to the Standard Model (SM) gauge group,
All these breakings may occur at roughly the GUT scale. The question which concerns us here is what happens to the two Abelian gauge groups and with corresponding generators and . Do they both get broken also at the GUT scale, or may one or other of them survive down to the TeV scale? In general it is possible for some linear combination of the two to survive down to the TeV scale,
where . More correctly the surviving generator should be written as,
Some popular examples of such are shown in Table 1.
The resulting heavy couples as . Note that in models it is reasonable to assume that the gauge coupling is equal to the GUT normalized gauge coupling of the SM, where and where the value is . Thus we take . GUT normalization also implies that the charges of the fermions in the representation for the case are all equal to , while for the case the charges of the representations are . Recalling that , and and , and that have the opposite charges to , this results in the values of the charges for the and cases as shown in Table 1. The general charges as a function of are then simply given as,
where the numerical charges for the popular models quoted in the literature are listed in Table 1.
Generalised Left-Right Symmetric Models (GLRs)
These models are motivated by the left-right (LR) extensions of the SM gauge group with the symmetry breaking,
which, from the point of view of models essentially involves the symmetry breaking,
where involves the generator corresponding to the third component of , while involves the generator . The hypercharge generator is then just given by . Assuming a left-right symmetry, the gauge couplings of are then equal, and the resulting heavy then couples as where
with and as before.
The left-right symmetric models therefore motivate a which is a particular linear combination of and with a specific gauge coupling. From this perspective the special cases where the corresponds to a pure or a pure are not well motivated. Nevertheless these types of have been well studied in the literature and so it is useful to propose a generalization of the LR models which includes these special cases. To this end we propose a generalized left-right (GLR) symmetric model in which the corresponds to a general linear combination of the generators of and ,
where . The gauge coupling is fixed so that for a particular value of the of the GLR may be identified with the of the LR symmetric model above. To be precise, we identify, for a particular value of :
which implies which corresponds to for and we find . Keeping fixed, we are then free to vary over its range where gives the LR model, but other values of define new models.
Clearly gives a model while gives a model. In the GLR model the value of also defines a which couples to hypercharge (not to be confused with the sequential SM which couples like the ). The couplings of the for the special cases of the GLR models are give in Table 1. The general charges as a function of are then simply given as,
where the numerical charges for particular models are shown in Table 1.
Generalised Sequential Models (GSMs)
No study is complete without including the sequential standard model (SSM) which is defined to have identical couplings as for the usual , namely given by and where and imply that . Similar to the GLR models, it is useful to define a generalised version of the SSM called GSM where the heavy gauge boson then couples as where corresponds to a general linear combination of the generators of and ,
and where . The gauge coupling is fixed so that for a particular value of the of the GSM may be identified with the of the SSM above. To be precise, we identify, for a particular value of :
This implies that the GSM reduces to the SSM case for and which corresponds to . Keeping fixed, we are then free to vary over its range where gives the usual SSM, but other values of define new models. Clearly gives a model while gives a model.
The couplings of the for the special cases of the GLR models are give in Table 1. The general charges as a function of are then simply given as,
where the numerical charges for particular models are shown Table 1.
iii.2 Strongly coupled gauge theories
Strongly interacting gauge theories provide an alternative mechanism for the electroweak symmetry breaking (EWSB). The EWSB is not driven by a light Higgs boson anymore, but it happens in a dynamical way. Such theories date back to decades. However, even if they predict the existance of new gauge bosons in order to delay at high energy the perturbative unitarity violation in vector boson scattering amplitudes, they are not considered when performing searches of bosons in the dilepton Drell-Yan channel. The reason is that historically the predicted new resonances must be fermiophobic in order to evade EWPT constraints. However, in recent years, new models have been proposed that are able to satisfy the EWPT bounds without imposing such a strong condition. Both the Minimal Walking Technicolour Foadi et al. (2007); Belyaev et al. (2009) and the four site Higgsless model Accomando et al. (2009, 2010, 2008) predict extra bosons with sizeable couplings to SM matter. Hence, they could be tested in the favoured Drell-Yan channel at the Tevatron and during the early stage of the LHC.
The Four Site Higgsless Model
Higgsless models emerge naturally from local gauge theories in five dimensions. Their major outcome is delaying the unitarity violation of vector-boson scattering (VBS) amplitudes to higher energies, compared to the SM without a light Higgs, by the exchange of Kaluza-Klein excitations Sekhar Chivukula et al. (2002). Their common drawback is to reconcile unitarity with the ElectroWeak Precision Test (EWPT) bounds. Within this framework, and in the attempt to solve this dichotomy, many models have been proposed Csaki et al. (2004a); Agashe et al. (2003); Csaki et al. (2004b); Barbieri et al. (2004a); Nomura (2003); Cacciapaglia et al. (2004a); Cacciapaglia et al. (2005); Cacciapaglia et al. (2004b); Contino et al. (2007).
In this paper, we consider the four site Higgsless model Casalbuoni et al. (2005) as represen