Reconstructing a Z' Lagrangian using the LHC and low-energy data
RReconstructing a Z (cid:48) Lagrangian using the LHC andlow-energy data
Ye Li, Frank Petriello and Seth Quackenbush
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
Abstract
We study the potential of the LHC and future low-energy experiments to preciselymeasure the underlying model parameters of a new Z (cid:48) boson. We emphasize the com-plimentary information obtained from both on- and off-peak LHC dilepton data, fromthe future Q-weak measurement of the weak charge of the proton, and from a proposedmeasurement of parity violation in low-energy Møller scattering. We demonstrate theimportance of off-peak LHC data and Q-weak for removing sign degeneracies between Z (cid:48) couplings that occur if only on-peak LHC data is studied. A future precision mea-surement of low-energy Møller scattering can resolve a scaling degeneracy betweenquark and lepton couplings that remains after analyzing LHC dilepton data, permit-ting an extraction of the individual Z (cid:48) couplings rather than combinations of them.We study how precisely Z (cid:48) properties can be extracted for LHC integrated luminositiesranging from a few inverse femtobarns to super-LHC values of an inverse attobarn. Forthe several example cases studied with M Z (cid:48) = 1 . ±
30% with 30 fb − ofintegrated luminosity, while ±
50% is possible with 10 fb − . With SLHC luminositiesof 1 ab − , we find that products of quark and lepton couplings can be probed to ± a r X i v : . [ h e p - ph ] A ug Introduction Z (cid:48) gauge bosons arise in most constructions of physics beyond the Standard Model (SM).They appear in grand unified theories such as SO (10) [1] and E (6) [2], in Little Higgsmodels [3], and in theories with extra space-time dimensions [4]. They often appear asmessengers which connect the SM to hidden sectors, such as in Hidden Valley models [5].Due to their pervasiveness in models of new physics, significant effort has been devoted tosearching for them experimentally. Z (cid:48) states that decay to lepton pairs have a simple, cleanexperimental signature and can easily be searched for at high-energy colliders. Current directsearch limits from the Tevatron require the Z (cid:48) mass to be greater than about 1 TeV whenits couplings to SM fermions are identical to those of the Z boson [6].Since they are so prevalent in models of new physics, and it worthwhile to study whatcould be learned from a Z (cid:48) discovery at the LHC. This also serves as a useful LHC benchmarkfor how well new-physics Lagrangian parameters can be experimentally determined. Sincethe experimental signature is clean and the QCD uncertainties have been studied and foundto be fairly small [7–9], it is likely that the couplings of a discovered Z (cid:48) can be studiedwith reasonable accuracy to probe the high scale theory that gave rise to it. Many studiesof how to discover, identify, or measure Z (cid:48) properties and couplings to SM particles havebeen performed [9–16]. Recently, we performed a detailed analysis using a fully differentialnext-to-leading order QCD simulation to quantify how well all possible Z (cid:48) couplings couldbe extracted from on-peak data at the LHC assuming realistic QCD, parton distributionfunction (PDF), and statistical errors [8]. Our study found that four combinations thatprobe both the parity-symmetric and parity-violating couplings of the Z (cid:48) could be probed,with some combinations measurable with roughly 15% precision assuming 100 fb − of data.This previous study contains several limitations. More statistical power is possible ifdata from below the Z (cid:48) peak is utilized. Also, additional differentiation between various Z (cid:48) couplings is possible with off-peak data. On-peak, the only combinations of couplingsthat appears are ( q × e ) , where q and e denote arbitrary quark and lepton couplings. Twodegeneracies exist in this expression: the relative signs between various q × e combinationsare not determined, and the scaling q → yq, e → e/y , with y an arbitrary constant, leavesthe expression unchanged. The first can be removed by Drell-Yan production of the Z (cid:48) off peak at the LHC, where the cross section contains a linear dependence on q × e . Theremoval of the second scaling degeneracy requires observation of the Z (cid:48) in a channel otherthan Drell-Yan at the LHC, or in a different experiment.In this manuscript we extend the previous study to address several of these issues for theexample case of a Z (cid:48) with mass M Z (cid:48) = 1 . Z (cid:48) signal and SM Drell-Yan background to study the potential of the LHC for a precisionextraction of the model parameters. We include errors arising from statistics, uncertaintiesin PDFs, and missing higher-order QCD and electroweak corrections. We study how wellcouplings can be measured assuming different energy and luminosity scenarios at the LHC,including collisions at both √ s = 10 and 14 TeV, and integrated luminosities ranging from3 fb − to super-LHC amounts of 1 ab − . If a Z (cid:48) is found at the LHC, additional informationon its parameters can be obtained from various low-energy measurements. Much of this1nformation is complementary, as many low-energy observables depend on different couplingcombinations than LHC measurements. We study several planned future experiments withthe potential for observing and studying a deviation from their SM expectations, and thatmay help remove the degeneracies described above. We focus on the Q-weak experiment [17],which plans to measure with high precision the weak charge of the proton in electron-protonscattering, and a proposed Jefferson Laboratory measurement of the effective weak-mixingangle through parity violation in low-energy Møller scattering [18]. This second experimentis of special interest. Since it depends only upon the leptonic Z (cid:48) couplings, it can potentiallybreak the scaling degeneracy between quark and lepton couplings to the Z (cid:48) . Our primaryfindings are summarized below. • Discrimination between the studied test models at the 68% and 90% confidence levelwith as little as 3 fb − of LHC data at the center-of-mass energy √ s = 10 TeV. With10 fb − of integrated luminosity at this energy, measurements of the c q parametersdefined in Refs. [8, 15] with relative uncertainties of ±
50% are possible, while ± − . • With high luminosities of 300 fb − at √ s = 14 TeV, the relative signs between thevarious q × e combinations are determined, and measurements of these combinationsapproaching 10% precision are possible. • An interesting feature we derive from our analysis is an independent extraction of thewidth Γ Z (cid:48) to a few GeV precision obtained without fitting the Z (cid:48) Breit-Wigner peak.Since the on-peak cross section scales as ∼ / Γ Z (cid:48) while the off-peak result contains nowidth dependence, comparison of data from different invariant mass bins allows a gooddetermination. • A precision measurement of the weak charge of the proton with Q-weak helps removesign degeneracies between the q × e that remain after LHC running. • With the inclusion of a future measurement of the effective weak-mixing angle in Møllerscattering, the individual q and e charges can be separately determined. We demon-strate for an example model how a combination of LHC Drell-Yan data and Møllerscattering make this possible. LHC measurements restrict the Z (cid:48) model parameters tolie in a wedge in the e L versus e R coupling plane, while Møller scattering restrict thecouplings to a hyperbolic region. A combination of the two experiments localizes thecouplings to a point in this plane, within errors.Our paper is organized as follows. In Section 2 we review the parameters which describe Z (cid:48) production at the LHC and introduce the various test models we use to illustrate ouranalysis procedure. We describe the LHC observables which we use to analyze Z (cid:48) propertiesin Section 3, and discuss the expected low-energy measurements of the proton weak chargeand the effective weak-mixing angle in Section 4. We present our analysis of Z (cid:48) couplingsat the LHC and in the Q-weak experiment, which are both sensitive to the q × e coupling2ombinations, in Section 5. We study both low and high integrated luminosities in thisSection, detailing the expected uncertainties in the parameter extractions for a variety ofdifferent running scenarios. A summary showing how well all q × e couplings can be measuredas a function of integrated luminosity is given in Section 5.6. In Section 6 we consider theconstraints imposed by a proposed Møller scattering experiment, and demonstrate how aprecision measurement at J-Lab allows the determination of the individual q and e charges.Finally, we conclude in Section 7. In this section we motivate a parametrization of Z (cid:48) models which can be probed by futureexperiments, while remaining as model-neutral as possible. We also describe examples ofcommon models which fit into our parametrization, and which we use to test our procedurefor extracting these parameters. Z (cid:48) parameters At the LHC, the smoking-gun signal for a Z (cid:48) is a new resonance in dileptons. Other particlesthat decay into such final states, such as Kaluza-Klein gravitons, can be distinguished froma Z (cid:48) by study of the lepton angular distributions [19, 20]. The production of this resonancedepends on the following parameters: • the location of the resonance, i.e. the mass of the state, M Z (cid:48) ; • the width of the resonance, Γ Z (cid:48) ; • the couplings of the quarks which produce the Z (cid:48) , u iL , u iR , d iL , d iR with the index i denoting the generation ; • the couplings of the charged leptons in the observed decay mode, e iL , e iR , and theneutrino coupling v iL .While the Z (cid:48) may couple to and decay into any number of other particles, from W bosonsto new exotic fermions, these parameters only enter into the Drell-Yan channel we considerthrough the width.One cannot hope to determine all of these parameters in a completely model-independentway. To proceed, we must make some assumptions on the parameter space. Fortunately, sev-eral restrictions are very well motivated by experimental data. We first make the assumptionthat the Z (cid:48) couplings are generation independent. If they were not, a Z (cid:48) light enough to bediscovered at the LHC would generate large flavor changing neutral currents in contradictionwith experiment. Z (cid:48) bosons with coupling strengths approximately the same as the SM Z and O (1) coupling differences between the first two generations must have masses greater than We have absorbed any overall coupling into these parameters to simplify the discussion and analysis. TeV due to constraints on the generated ∆ F = 2 four-fermion operators [21].Constraints on Z (cid:48) bosons with different couplings to third-generation fermions are weaker;deviations from our parameterization for top quarks, bottom quarks, or tau leptons can besearched for explicitly by observation of Z (cid:48) decay into these final states [22–24]. Second, weassign states in a given SU (2) L doublet the same coupling to the Z (cid:48) . If the generator ofthe new gauge group to which the Z (cid:48) belongs does not commute with SU (2) L , the Z (cid:48) wouldgenerically couple to the electroweak symmetry-breaking sector, inducing Z − Z (cid:48) mixing.Specific models which would protect against this mixing, such as Kaluza-Klein Z (cid:48) s, shouldbe distinguishable from the class of models considered here [14]. LEP Z -pole measurementsrestrict the mixing angle to be smaller than a few thousandths of a radian [25, 26]. We takethis magnitude of mixing to be negligible. We note that arranging the Z-Z’ mixing angle tobe this small requires, for the models considered, either fine-tuning the charges and vevs inthe Higgs sector which gives mass to the Z and Z’ bosons, or setting these parameters smallerthan their expected values by roughly an order of magnitude [15]. There are many modelsin the literature which avoid these issues, but the E models we use as examples remain auseful benchmark, particularly for comparing with other studies. These restrictions leavethe following seven Z (cid:48) parameters to be determined: • the mass M Z (cid:48) and the width Γ Z (cid:48) ; • e L and e R , the couplings to the lepton doublet and right-handed electron; • q L , the quark doublet coupling; • u R and d R , the right-handed quark couplings. To illustrate how well our analysis strategy determines Z (cid:48) properties, we test it on severalexample models. Several well-known grand unified theories (GUTs) fit into the frameworkdescribed above. We take three such examples to illustrate how well their parameters couldbe determined if they are found. Two of these arise from the exceptional group E , and onefrom SO (10). Below we briefly describe these models and list their couplings. For a morecomprehensive introduction to these specific examples, we refer the reader to several excellentreviews [1, 2]. Our procedure is by no means limited by these choices; we choose thesecommon examples simply to demonstrate the efficacy of our model-independent approach.This method can be used for any Z (cid:48) effective theory satisfying the coupling assumptionsoutlined above. • E Models: E models are described by the breaking chain E → SO (10) × U (1) ψ → SU (5) × U (1) χ × U (1) ψ → SM × U (1) β (1)where Z (cid:48) = Z (cid:48) χ cos β + Z (cid:48) ψ sin β (2)4s the lightest new boson arising from this breaking. In this paper we take the χ model( β = 0), and the ψ model ( β = π/ • Left-right models: We also consider a left-right model coming from the symmetry group SU (2) R × SU (2) L × U (1) B − L . Left-right models can arise from the following breakingof SO (10): SO (10) → SU (3) × SU (2) L × SU (2) R × U (1) B − L . (3)The Z (cid:48) in left-right models couples to the current J muLR = α LR J mu R − / α LR J muB − L , (4)with α LR = (cid:112) ( c W g R /s W g L ) −
1, and g L = e/ cos θ W . We examine the symmetric case g L = g R , where α LR (cid:39) .
59 if one takes the on-shell value of sin θ W .The fermionic couplings of these models are summarized in Table 1. For the standarddefinition of the E models, we take the overall coupling to have its GUT-scale relation to theEM coupling down to the Z (cid:48) scale. An overall e/ cos θ W has been factored out. We will lateralso consider a version of the χ model with a slightly larger overall coupling to demonstratewhat is gained from potential measurements in low-energy experiments. The overall factorof e/ cos θ W (cid:39) .
36 in this model has been replaced by 1 /
2. We denote this model χ ∗ . χ ψ LRq L − √ √ − α LR u R √ −√ − α LR + α LR d R − √ −√ − α LR − α LR e L √ √ α LR e R √ −√ α LR − α LR Table 1: Fermion couplings to the Z (cid:48) for the considered models. An overall e/ cos θ W hasbeen factored out. The primary means by which we will probe Z (cid:48) properties will be through examining thedilepton mode at the LHC. We describe here the structure of the Z (cid:48) cross section, presentthe observables we utilize for various choices of integrated luminosity, and discuss the detailsof our simulation procedure. 5 .1 LHC measurements We would like to determine seven parameters: M Z (cid:48) , Γ Z (cid:48) , q L , u R , d R , e L , and e R . The locationof the resonance peak in the invariant mass of the dileptons should determine M (cid:48) Z extremelywell with a sufficient number of events; we take the error in this measurement to be negligible.For this analysis, we take as an example M Z (cid:48) = 1 . Z (cid:48) sare excluded by experiment [6, 25]. For significantly heavier Z (cid:48) s, testing shows that thereis not much improvement over our previous on-peak analysis, [8]. In principle, the widthcan be determined from the resonance shape if the experimental resolution in invariant massdoes not dominate the natural width of the Z (cid:48) . With enough statistics, widths smaller thanthe experimental resolution could possibly be probed by fitting the observed spectrum to aconvolution of Z (cid:48) resonance shape with the experimental Gaussian resolution. How well thiscan be done requires a detailed detector-level study and is beyond the scope of this analysis.We make no assumption on how well the width can be determined, beyond the requirementthat one be able to define an “on-peak” bin containing most of the resonance, to be explainedbelow. For an experimental study of both the Z (cid:48) mass and width measurement at the LHC,see Ref. [27].One can reconstruct the following kinematic variables from the dilepton measurementsat the LHC. Taking the four-momenta of the electron and positron to be k and k , we have,with k Z (cid:48) = k + k : • the invariant mass of the pair, M = k Z (cid:48) ; • the reconstructed rapidity of the Z (cid:48) , Y = log k Z (cid:48) + k Z (cid:48) k Z (cid:48) − k Z (cid:48) ; • the scattering angle of the electron, θ ∗ , measured in the Collins-Soper frame [28]. Thesign of cos θ ∗ is chosen to be positive if k and k Z (cid:48) have the same sign, i.e., the electronis scattered in the same direction as the Z (cid:48) is boosted down the beam pipe.The differential cross section of pp → e + e − can then be written in the following form: d σdY d cos θ ∗ dM = (cid:88) q = u,d [ a q ( M Z (cid:48) , Γ Z (cid:48) )( q R + q L )( e R + e L ) + a q ( M Z (cid:48) , Γ Z (cid:48) )( q R − q L )( e R − e L )+ b q ( M Z (cid:48) ) q R e L + b q ( M Z (cid:48) ) q R e R ]+ b ( M Z (cid:48) ) q L e L + b ( M Z (cid:48) ) q L e R + c. (5)The coefficients a, b, and c contain all kinematic and PDF dependence. The c term is thecontribution from the SM γ and Z . To predict a measurement, one only has to integratethem over the appropriate region, and then input the dependence on the Z (cid:48) couplings. Bychoosing the regions to integrate over (measurement bins) appropriately, one can solve thesystem of equations and gain coupling information. We now motivate the choice of bins. • Forwards/Backwards: At the parton level, terms symmetric in the quark-lepton scat-tering angle go as ( q iR q jR + q iL q jL )( e iR e jR + e iL e jL ), where i, j ∈ γ, Z, Z (cid:48) . Terms antisym-metric in the scattering angle go as ( q iR q jR − q iL q jL )( e iR e jR − e iL e jL ). We could separate6hese terms by binning events depending on whether they are forwards or backwardsscattered; F + B is symmetric and F − B is antisymmetric. We do not have directaccess to the partons, however, and the LHC is a pp collider, so there is no preferreddirection for the quarks. We define “forwards” to be F = (cid:82) d cos θ ∗ and “backwards”to be B = (cid:82) − d cos θ ∗ . The advantage of defining the direction in terms of the Z (cid:48) boostis that this direction correlates with the quark direction, and therefore one still getsuseful information from the asymmetry even though the LHC is a pp collider [29]. • Rapidity: The higher the rapidity Z (cid:48) , the more likely it is to have come from a valence u quark and sea ¯ u quark. This is due to the dominance of valence u at high x . Inprinciple, to separate u from d -type quarks, one needs only two rapidity bins, wherethe dividing line is chosen such that the bins contain approximately the same numberof events [11]. There is no penalty to choosing additional bins, as long as one does notbin so finely that detector resolution becomes an important issue. For the analysis with M Z (cid:48) = 1 . < | Y | < .
4, 0 . < | Y | < .
8, 0 . < | Y | < . . < | Y | < Y max , where Y max = 1 / s/M Z (cid:48) ) is the largest rapidity available.We later study the effect of the coarser two-bin analysis, and find that only slightimprovement is obtained by this finer binning. • Invariant Mass: Previous studies have examined the above observables and simulatedextraction of coupling information at the LHC [8, 11–13], or bounded couplings at theTevatron [15], by integrating the invariant mass around the Z (cid:48) resonance peak. Here,the b and c terms in Eq. (5) can be safely ignored, but any information they mightprovide is discarded also. Only squares of couplings are accessible on the resonancepeak. We aim to gain sign information by probing regions where the Z (cid:48) interferenceterms with the other gauge bosons have an effect, i.e., we want to study the b terms.We will do this by including invariant mass bins between the Z pole and the Z (cid:48) pole. Inthe 1 . < M < < M < < M < < M < on-peak henceforth. The appropriate size to use forthe on-peak bin can be determined with only a rough idea of the Z (cid:48) resonance shape.The other bins should be chosen far enough away from the pole that the a terms donot dominate. For our considered models and mass, observables in each invariant massbin receive contributions roughly equal in magnitude from the a, b , and c terms, with c being somewhat stronger in low mass bins and a being stronger in high mass bins.In total, we simulate 2 × × Y , M , and cos θ ∗ are shown in Table. 2. For illustrativepurposes a selection of numerical results for both the SM and the test models using thefactorization and renormalization scale choices µ F = µ R = M Z (cid:48) can be found in Table 3.7in 1 Bin 2 Bin 3 Bin 4 | Y | [0 , .
4] [0 . , .
8] [0 . , .
2] [1 . , Y max ] M (TeV) [0 . , .
0] [1 . , .
2] [1 . , . on-peak cos θ ∗ [ − ,
0] [0 , − − Table 2: Summary of bin endpoints used in our analysis for the studied variables: the rapidity Y , the invariant mass M , and the leptonic scattering angle cos θ ∗ . Each region is displayedin the format [ lower, upper ]. Further details are given in the text. SM χ LR ψ χ ∗ σ on − peak F < Y < . < M < B . < Y < . < M < F . < Y < . < M < F Y > . on-peak B Y > . on-peak F < Y < . on-peak B < Y < . on-peak To simulate observation of a Z (cid:48) at the LHC we perform a fully differential next-to-leading-order (NLO) QCD calculation of all observables described above. We use CTEQ 6.6 PDFs [30].The QCD corrections to the cross section are needed to obtain the proper normalization ofthe observables [8]. We impose the following basic acceptance cuts on the final state leptontransverse momenta and pseudorapidities: p lT >
20 GeV and | η l | < .
5. Together, these con-straints result in acceptances of roughly 90% [8]. Previous studies have found that detectorresolution effects and other measurement errors are unlikely to have a significant effect onthe e + e − final state [13], and are neglected. A CMS simulation of Z (cid:48) production found recon-struction efficiencies near 90% in the electron channel and no significant detector systematicerrors [31]. In addition, electron energies can be measured to better than 1% accuracy, andinvariant masses can therefore be reconstructed very well. Reconstruction efficiencies above90% and invariant mass measurements to sub-1% precision have also been found for highinvariant-mass muon production at the LHC [32]. We include here only the statistics for the e + e − production, without reconstruction efficiencies imposed. In light of the possibility for afactor of two increase in statistics from precision measurements in the muon-pair channel, webelieve our study conservatively estimates the LHC potential. Equivalently, one can rescaleall required luminosities by a factor of 2 if including this channel, or use it to separately8xtract electron couplings and muon couplings.To construct these measurements, we evaluate Eq. (5) in the following manner. Theconsidered models have narrow widths. For computational efficiency, we calculate the a coefficients once using the χ width of 17.9 GeV. We choose this width because it is centrallylocated among those considered. On the resonance peak, observables scale as 1 / Γ in thenarrow-width approximation. Since we are not integrating over all invariant masses, wemust improve this approximation for a finite bin size denoted B . When we scan over thewidth, we scale the a coefficients integrated over the on-peak bins by replacing (cid:90) ( M + B/ ( M − B/ dM a (Γ) → Γ χ Γ (1 + 2 Γ χ − Γ πB ) (cid:90) ( M + B/ ( M − B/ dM a (Γ χ ) (6)to integrate a (Γ) for other widths. Here we have included an O (Γ Z (cid:48) /B ) correction, anddropped terms of O ( B /M Z (cid:48) ). For M = 1 . B = 200 GeV. Theremaining integral of a (Γ χ ) is computed using the actual propagator, not with the narrow-width approximation. This is accurate to within tenths of a percent over the range of widthsconsidered, and allows the a coefficients to be computed once rather than for all widthsscanned over. If the finite bin-size corrections are completely dropped, errors of the size5 −
10% are introduced. We can then calculate the integrated coefficients a, b, c once themass is known. We take c to be the SM Drell-Yan cross section. We will also consider subsets of the above measurements for use in discriminating modelswith early LHC data assuming √ s = 10 TeV. Our restriction in the number of bins is moti-vated by the desire to avoid low-count statistical issues and the need to account for detectormiscalibration, which would over-complicate our analysis. At low luminosities the reduc-tion in bins does not sharply reduce our sensitivity to differences in the underlying modelparameters. We merge the bins as follows for 10 TeV running. •
30, 100 fb − : The two highest off-peak invariant mass bins are merged (now 1000
10 fb − : All off-peak invariant mass bins are merged (800 < M < × × • For the pairwise model comparison at a few femtobarns of integrated luminosity tobe described in Section 5, we consider only measurements on the resonance peak.For simplicity we use four bins: 2 rapidity bins ( | Y | < . , | Y | > . M Z (cid:48) = 1 . √ s = 14 TeV. Model σ (fb) χ ψ χ ∗ √ s = 10 TeV. The upgrade of the Jefferson Lab polarized e − beam will advance the precision frontierfor low-energy parity violation. The Q-weak experiment [17] will probe the effective weakcharge of the proton in ep scattering, and the proposed Møller scattering experiment [18] candetermine sin θ effW through a precision measurement of the asymmetry in e − e − scattering.Deviations from predicted SM values due to a Z (cid:48) can provide further coupling informationonce the Z (cid:48) mass is known from the LHC. We describe here which Z (cid:48) parameters to whicheach experiment is sensitive, and review the expected errors on the measured quantitiesarising from both experimental issues and SM theoretical uncertainties. The Q-weak experiment will probe the effective weak charge of the proton, Q pW (cid:39) − θ W , by observing the parity-violating asymmetry in ep scattering. Due to the smallnessof 1 − θ W , higher-order corrections are important. The predicted low-energy value inthe SM (from running from Z -pole measurements) is [33] Q pW = 0 . ± . . (7)The anticipated experimental error after Q-weak running is 4.3% [17]. The experimentallyobserved value will differ due to the presence of a Z (cid:48) ; the shift is [34]∆ Q pW = 2 √ M Z (cid:48) G F (2 e A u V + e A d V ) , (8)where A and V denote the axial-vector and vector charges, respectively. We rewrite this interms of the parameters defined in Section 2.1 to derive∆ Q pW = √ M Z (cid:48) G F (3 q L + 2 u R + d R )( e R − e L ) . (9)10ike LHC observables in the Drell-Yan channel, the Q-weak measurement is sensitive onlyto the coupling combination q × e . The goal of the proposed J-Lab Møller experiment is to measure the effective weak mixingangle, sin θ effW , to an absolute precision of 0.00025, or in terms of the weak charge of theelectron, to within 2.3%. The running of sin θ W in the SM gives a prediction for thisquantity at low energies [33]. The result depends rather strongly on the scale at which γZ box diagrams are evaluated: Q eW (0) = − . Q eW ( M Z ) = − . Q eW = 0 . ± . Z − pole) ± . γ Z) ± . . ± . , (11)where we have added the various sources of error in quadrature. Shifts from the expectedasymmetry, similarly to Q-weak, go as∆ Q eW = 2 √ M Z (cid:48) G F e A e V = √ M Z (cid:48) G F ( e R − e L ) . (12)An interesting feature of this quantity is that it depends only on electron couplings, and notthose of the quarks. A measurement of this quantity would break the scaling degeneracythat plagues the other measurements discussed previously. To extract coupling information from the LHC from the measurements described in Section 3,we must evaluate the coefficients a, b , and c in Eq. (5). This requires knowing the Z (cid:48) massfrom experiment. For illustrative purposes we explore the possibility M Z (cid:48) = 1 . Z (cid:48) , q L e L , q L e R , u R e L , and d R e L . Two other combinations, u R e R and d R e R , appear in Eq. (5);however, these are not independent from those listed due to the following relations: q L e R q L e L = u R e R u R e L = d R e R d R e L = e R e L ≡ tan θ l . (13)11ere we have defined the parameter θ l , which is the angle in the plane of the leptoniccouplings. There is no way to separate q couplings from e couplings at this stage, as the Z (cid:48) is produced via quarks and decays into leptons in this channel. This limitation is addressedin the next section. We include a discussion of the Q-weak measurement in this section asit probes the same q × e combination as the LHC.We determine the five unknown parameters by performing a scan over this parameterspace. For each point in the five-dimensional parameter space, we can reconstruct the pre-dicted LHC measurement bins discussed in Sec. 3 using Eq. (5), and compare to the actualobserved values. In the absence of data, we construct measurements for the χ, ψ , and LRmodels described in Section 2.2 to test our procedure. As we present analyses for severalcoupling combinations with varying assumptions regarding integrated luminosity and energy,we organize our results into subsections as follows. • We first present a discussion of the relevant uncertainties affecting both the LHC andQ-weak measurements. • We discuss an initial three-parameter χ comparison between data and model hypothe-ses that can be performed with only a few fb − of data at the LHC. • We present results for integrated luminosities between 10 −
100 fb − for the c q and e q coupling combinations defined in [8], which are most relevant when only on-peak datagives significant information. • We then discuss extraction of the q × e coupling combinations assuming large LHCintegrated luminosities of 300 − − , and demonstrate how the Z (cid:48) width can bedetermined by comparing on-peak and off-peak bins. We add the expected Q-weakmeasurement to the analysis at this stage. • Finally, we summarize our analysis and present results showing how well all Z (cid:48) param-eters relevant at the LHC can be determined as a function of integrated luminosity. Our ultimate goal is to extract the underlying Z (cid:48) couplings. We will need to perform a χ test between hypothesis points in the coupling parameter space and the “observed” valuesfor each of our example models. For early LHC running, we will want to know whetherone can start eliminating canonical models, and we do a pairwise comparison between ourtest models to see how well this can be done. For later LHC running, we will demonstratethe efficacy of our coupling extraction procedure. To do so we must study the uncertaintieswhich affect our analysis procedure. We classify errors into four categories. • PDF. We determine the PDF errors using the CTEQ 6.6 PDF error sets [30] accordingto the prescription of [37], and account for correlations among our measurement bins.To be conservative, we take the larger magnitude shifts from each of the 22 eigenvectordirections in PDF parameter space, and when forming the χ , interpret these errors12s 1 σ . If the Z cross section is used as a standard candle, one should instead findthe PDF error in the ratios of our bins to the expected number of Z events. If therewere significant correlations between our bins and the Z cross section, this wouldreduce PDF errors. Testing indicates this is not the case, and for simplicity we do notnormalize to Z production to determine PDF errors. • Statistical. For each considered value of integrated luminosity we determine the count-ing error in each bin. The bin choices for each luminosity are described in Section 3.1. • Theory. We have calculated all cross sections at NLO in the strong coupling. Residualscale errors are negligible compared to our other errors [8], and it is known that theNNLO corrections have a very small impact on the central value of the NLO result [9,38]. QED corrections in the final state may be important in predicting the invariantmass distribution [39, 40], though with our coarse binning this issue is less important.Further electroweak corrections come in two forms: vertex corrections, which can beabsorbed into the effective couplings extracted, and box corrections (where two sides ofthe box are
W, Z , or Z (cid:48) ), which cannot. The latter have large logarithmic corrections inthe energy, log(ˆ s/M W,Z ) [41], and should be determined when examining off-peak bins.We assume that a full analysis will have computed these corrections; the actual shiftsin the coefficients a, b , and c should not be large enough to affect the efficiency of ourmethod. However, we do include, conservatively, a 5% error in the overall normalizationof the off-peak bins to account for remaining errors after EW corrections have beenincluded. We have found that including this error does not substantially affect thequality of the coupling extraction. • Detector. Our binning is coarse enough that we take detector issues such as energymis-measurement, cracks, etc., to be sub-dominant. We neglect them in our study.
The Q-weak experiment will also probe combinations of quark × lepton couplings; fromEq. (9), it will measure the combination(3 q L e L + 2 u R e L + d R e L )( q L e R q L e L −
1) = √ M Z (cid:48) G F ∆ Q pW . (14)The Z (cid:48) mass will be known from the LHC. The anticipated uncertainty after combining bothexperimental and theoretical errors is large compared to the expected deviations of our testmodels: δ ∆ Q pW Q pW = 4 . , (15) δ ∆ Q pW = 0 . Z (cid:48) would have to be fairly strongly coupled to be observable by Q-weak. The Q-weak experimentmay add somewhat to the discriminating power of the LHC, however. We assume that Q-weak finds a value consistent with our test model, and for each test point, we generate ameasurement and add it to the χ appropriately. For simplicitly, the Z (cid:48) parameters are notrun from the Z (cid:48) scale down, but compared directly. If the running is O (10%), as expected,this can be safely neglected compared to the experimental error.∆ Q pW × χ LR -0.45 ψ χ ∗ Z (cid:48) , in unit of 10 − . This shouldbe compared to the expected measurement precision of δ ∆ Q pW = 0 . χ comparisons We first assume a center-of-mass energy of only √ s = 10 TeV, consistent with early-runningscenarios. At low luminosities, the statistical errors are too large for a meaningful couplingextraction. Therefore we ask the question, is it possible to begin to discriminate models? Howcan one rule out a “model”? High-scale models typically determine the Z (cid:48) fermion charges,but the overall coupling running or the leptonic branching fraction may depend on otherparameters of the theory not considered here, or even predicted by the model. Therefore,in this first pass to distinguish models, we wish to consider observables independent ofthese parameters. We first note the following. For very early running, the error in off-peak observables is too large for them to be useful. Secondly, the overall on-peak crosssection, σ peak , is proportional to the overall coupling squared and the branching fraction: σ peak ∝ g Z (cid:48) Br ( Z (cid:48) → e + e − ). Therefore, if one takes on-peak observables normalized by theoverall on-peak cross section, ˆ σ i = σ i /σ peak , they are independent of the overall coupling andleptonic branching fraction. Considering these observables only will allow us to differeniatemodels defined only up to these ambiguous parameters. In forming our χ comparison usingthe four bins described in Sec. 3.3, we normalize the bins to the on-peak cross section,and we restrict ourselves to on-peak bins. As these observables are no longer independent( (cid:80) i σ i = σ peak ), one bin is dropped, for three observables, and three degrees of freedom. Wepresent the χ comparison of experimentally “found” modes versus other model hypothesesin Table 6. For reference, 68% confidence corresponds to χ = 3 .
5, 90% to 6.3, and 95% to7.8.In performing this test, we have assumed statistics consistent with our prior assumptionsabout the overall coupling and width for the experimentally “found” model; we remind the14 fb −
10 fb −
30 fb − χ ψ LR χ ψ LR χ ψ LR χ - 1.7 6.7 - 5.1 23.1 - 15.7 54.7 ψ χ comparison (with three degrees of freedom) between models for early LHCrunning. Hypothesis models are in columns, and are tested against the experimentally foundrow models. Errors are determined from the hypothesis model.reader that our procedure is independent of these assumptions. We note that the ψ model isdifficult to differentiate under these assumptions due to its lower production rate. We notethat “nice” models with clean signatures such as the χ model and left-right model can bedistinguished from each other at over 90% confidence with as little as 3 fb − . For slightly higher luminosities, we can use the off-peak bins to begin to reduce the allowedregion in coupling space. As an example, we plot the 95% confidence region for two of thecoupling combinations ( q L × e R and q L × e L ) for measurements corresponding to the χ modelin Fig. 1. If all couplings and the width are fit simultaneously, this corresponds to a 68%confidence region. We make the following observations. • At 10 fb − , one only gets a vague notion of the size of the couplings. Since we do notassume the width is known from the Breit-Wigner scan at this stage, this is not dueto knowledge of the on-peak total cross section. It is instead coming from the limitedsize of the deviations of the off-peak bins from SM predictions. • At 30 fb − , we see multiple “islands” emerge; these are due to the nonlinearity ofthe cross section in the q × e parameters. The allowed coupling space has shrunkconsiderably, and we see the beginnings of a measurement for the correct island. Note,however, that all sign degeneracies remain in the couplings; it is still the on-peakmeasurements leading to reduction of the allowed region. To determine these signs,one must wait for more integrated luminosity.At this stage, the couplings themselves are not measured particularly well, though we areseeing quite a reduction in allowed parameter space. In Ref. [8], the anticipated errors for thecombinations of the couplings measurable on peak were derived. Given the success of thatextraction, we wish to revisit these results using the chosen binning and lower luminosity,again choosing the χ model as an example. We plot in Fig. 2 the 68% confidence regions forthe four parameters c q and e q ( q = u, d ), defined in Ref. [8] as15igure 1: 95% confidence-level region for the q L × e R and q L × e L couplings extracted frommeasurements corresponding to the χ model, at 10, 30, and 100 fb − . Other model valuesfor these couplings are shown for orientation.16 q = ( q R + q L )( e R + e L ) M Z (cid:48) π Γ Z (cid:48) ,e q = ( q R − q L )( e R − e L ) M Z (cid:48) π Γ Z (cid:48) . (17)These have the advantage that they are directly related to the cross section near the resonancepeak, d σdY d cos θ ∗ dM (cid:39) (cid:88) q = u,d [ a (cid:48) q ( M Z (cid:48) , Γ Z (cid:48) ) c q + a (cid:48) q ( M Z (cid:48) , Γ Z (cid:48) ) e q ] (18)where we have rescaled the coefficients a (cid:48) qi = π Γ Z (cid:48) M Z (cid:48) a qi of Eq. (5) to move most of the widthdependence into the coupling combinations c q and e q . For this analysis, since we use a fixedbin size, we also fold a factor of (1 + 2 Γ χ − Γ πB ) from Eq. (6) into the definitions of c q and e q tokeep the results practically width-independent. This is relatively unimportant if the widthis small relative to the on-peak bin size, or known a priori.The on-peak bins are doing almost all the work discriminating c q and e q , as these quan-tities are directly related to the on-peak bins. The off-peak bins do not significantly help inreducing the allowed coupling space at these luminosities. As such, we see nearly ellipticalerrors as one would expect from a linear propagation of errors from Eq. (18). However, theerrors differ slightly from our previous analysis [8], and we wish to mention the followingdifferences. • These are 68% confidence regions for 2 parameters, and not 1 σ in each parameter. • √ s = 10 TeV, not 14 TeV, which affects both statistical and PDF errors. • The physical constraint u L = d L is explicit in the parameter space scan, as are c q > | e q | < c q , etc. • CTEQ 6.6 PDFs are used instead of CTEQ 6.5, which allows for both s quark differ-ences and improves high x errors [30]. • Less conservative assumptions are made on the correlations of differing PDF eigenvec-tor directions.We note that under these assumptions u and d coupling measurements are very anti-corellated on peak; the sum is known much better. This is to be expected; their separationdepends on subtracting low and high rapidity Z (cid:48) bins, thereby increasing errors, while thesum depends on combining bins. 17igure 2: 68% confidence regions for the coupling combinations c q and e q extracted frommeasurements corresponding to the χ model, at 10, 30, and 100 fb − . Other model valuesare shown for orientation. We note that c q and e q also have correlations with each other thatare not displayed. 18 .5 Late LHC and Q-weak results We finally consider the improvements possible if integrated luminosities of 300 fb − or 1ab − are available. We now add information from a Q-weak measurement of Q pW , assumingit finds a value consistent with that predicted by a test model. We demonstrate in Fig. 3the successive improvement upon adding different data in our extraction for 1 ab − at theLHC, by first applying the off-peak bins only to determine the couplings for the χ model,then adding on-peak bins, and finally the Q-weak experiment. We choose to show u R e L versus d R e L to illustrate this improvement. Due to the nonlinear nature of the cross sectionin the Z (cid:48) parameters, multiple islands are possible, where one region of parameter space canmimic the signal of another within the allowed errors. The Q-weak experiment is helpful inbreaking degeneracies such as these that may remain, where the false islands have a differentenough value of ∆ Q pW to rule them out in conjunction with the LHC measurements. Thedifference is not in general substantial enough to rule out the islands if the LHC did notalready disfavor them. χ LR ψ Act. LL HL Act. LL HL Act. LL HLΓ Z (cid:48) (GeV) 17.9 ± . . ± . . ± . . ± . . ± . . ± . . × q L e L -1.63 ± . . ± . . -0.43 ± . . ± . . ± . . ± . . × q L e R -0.55 ± . . ± . . ± . . ± . . -0.91 ± . . ± . . × u R e L ± . . ± . . ± . . ± . . -0.91 ± . . ± . . × d R e L ± . . ± . . -3.70 ± . . ± . . -0.91 ± . . ± . . θ l . ◦ ± . . ± . . − . ◦ ± . . ± . . − ◦ ± ± Table 7: Results of the LHC/Q-weak coupling extraction. We use the following abbreviationsfor the integrated luminosity: LL = 300 fb − , HL = 1 ab − .We use the Q-weak experiment in conjunction with the LHC at high luminosities to fitthe couplings and width for all models. In Table 7 we list the results of the extraction of theseparameters for integrated luminosities of 300 fb − and 1 ab − . The errors listed are 1 σ foreach individual parameter, while allowing the other values to float. For brevity we omit the χ ∗ model. Interestingly, the width is measured extremely well by the extraction. This essentiallycomes from comparing the size of on-peak bins to what would be predicted by extractingthe couplings off-peak only; the size of the on-peak cross section is directly controlled bythe width. We note that the precision of this measurement is not directly controlled bythe calorimeter resolution, and is therefore controlled by different systematic errors thana fit to the resonance peak. The errors here are competitive with a previous detector-level analysis [27] of the Breit-Wigner fit with a fixed number of events corresponding toluminosities of 100-300 fb − . We have tested that adding knowledge of the width from theBreit-Wigner fit following the study in Ref. [27] does not significantly improve the qualityof our results.The couplings, especially the larger ones, are measured to good precision. The ψ model is19igure 3: Reduction in allowed coupling space with successive application of off-peak mea-surements only, on-peak measurements, and the Q-weak expected result for the χ model,at 95% CL. We note the removal of the small island in the upper-right quadrant upon theinclusion of Q-weak. These results include 1000 fb − of integrated luminosity at the LHC.20n exception; several parameters are measured poorly. This is due to the lack of asymmetry,where the differences F − B are zero, giving a large relative statistical error. For non-leptophobic, narrow-width Z (cid:48) models, we consider this a worst-case scenario. However,the table above does not tell the whole story. The 5D 68% CL region includes strongcorrelations between the parameters, and portrays a dramatic reduction of parameter spacecompared to a naive estimate from the table alone. We plot several 2D projections of the5D confidence region for our test models in Figs. 4 and 5 to demonstrate. These regions canalso be interpreted as 95% confidence regions for the two parameters displayed in each plotseparately, allowing the others to float.Figure 4: Width versus q L e L fits for the considered models. Widths tend to correlate withlarger couplings.We make a few remarks on these results. First, the χ model measurement improves sub-stantially with increased (super-LHC) luminosity. Remaining degeneracies in the parameterspace are removed. However, for the other models, there is only minor improvement. Theallowed volume of coupling space is small, and our test models are easily distinguished. Also,21igure 5: Coupling fits for the considered models. The actual value for each model is shownby the labeled black symbols. 22e point out that it is often possible to mimic the signals of our test models within errorswhile keeping q L zero, as can be seen in the upper panel of Fig. 5. If the average asymmetrycoming from the quarks is small enough, one can account for it in the leptonic couplings,and get the relative amount of u and d correct by adjusting the values of u R and d R withoutaffecting other measurements. For example, the ψ model is extremely difficult to pin downdue to the lack of asymmetries. One cannot even favor a sign on d R e L in this model, as canbe seen from the lower panel of Fig. 5. As our analysis contained many parameters, we review here the combinations that can beprobed by the various data sets. At low luminosity, we have seen that the majority of theanalyzing power comes from the on-peak bin. The natural coupling combinations to utilizewith this data set are the c q and e q combinations defined in Eq. (17). These quantities areformed from ( q × e ) . At higher luminosities, the off-peak bins contain enough events toallow a direct measurement of the linear combinations q × e . We summarize this informationin Table 8. Only six of seven possible parameters appear in this table; this reflects the q → yq, e → e/y degeneracy discussed earlier.on-peak LHC: M Z (cid:48) , c u , c d , e u , e d on-peak+off-peak LHC: M Z (cid:48) , Γ Z (cid:48) , q L e L , q L e R , u R e L , d R e L Q-weak: (3 q L + 2 u R + d R )( e R − e L ) /M Z (cid:48) Table 8: Summary of which Z (cid:48) coupling combinations can be probed with the various datasets considered.We summarize the analyzing power of LHC and Q-weak by displaying in Figs. 6, 7, 8, and 9the improvement in the coupling extraction for the χ model, one parameter at a time, asthe luminosity is increased. For integrated luminosities of 100 fb − or less, we conservativelyuse a center-of-mass energy of 10 TeV; for higher luminosities we assume the full 14 TeV hasbeen reached. For the c q , e q couplings, relative uncertainties of roughly ±
50% for non-zerocouplings are possible with 10 fb − , while ±
30% errors become possible with 30 fb − . At300 fb − and above, the q × e combinations are directly measurable with precisions of 10%or better. We note that a sign degeneracy in all q × e parameters remains at integratedluminosities of less than 100 fb − ; this is resolved at 1 σ for all parameters at 300 fb − . Our goal is to probe all Z (cid:48) parameters to fully determine the underlying theory; thus far, wehave only determined the couplings q × e . A linear collider can probe the leptonic couplingsdirectly, even below the resonance peak, if the mass is known [42]. In this paper, we do notassume access to a linear collider, and attempt to determine how well the last parameter can23igure 6: 1 σ errors for the on-peak coupling combinations c u and c d as a function of integratedluminosity. To the left of the dashed line, √ s = 10 TeV, while √ s = 14 TeV to its right.24igure 7: 1 σ errors for the on-peak coupling combinations e u and e d as a function of inte-grated luminosity. To the left of the dashed line, √ s = 10 TeV, while √ s = 14 TeV to itsright. 25igure 8: 1 σ errors for the coupling combinations q L e L , q L e R as a function of integratedluminosity. Note that all signs are selected at 300 fb − , but not before. To the left of thedashed line, √ s = 10 TeV, while √ s = 14 TeV to its right.26igure 9: 1 σ errors for the coupling combinations u R e L , d R e L as a function of integratedluminosity. Note that all signs are selected at 300 fb − , but not before. To the left of thedashed line, √ s = 10 TeV, while √ s = 14 TeV to its right.27e measured by other means. It is possible that observation of the Z (cid:48) decaying to bottom ortop quark final states can help break this degeneracy. We pursue here instead the alternatepossibilities of determining the invisible width of the Z (cid:48) , or by using a proposed low energyMøller scattering experiment. The Møller scattering analysis in particular demonstrates anice complimentarity between low and high-energy experiments. A study of the sensitivityof future low energy Møller scattering experiments to Z (cid:48) effects has also been considered inRef. [43], although the analysis performed there did not focus on the interplay with LHCmeasurements. If we assume that the leptonic couplings are flavor-universal, we can write the Z (cid:48) width inthe limit of vanishing SM fermion masses in terms of parameters already determined, andthe leptonic coupling:Γ Z (cid:48) = M Z (cid:48) π (6 q L + 3 u R + 3 d R + 2 e L + e R ) + Γ new = M Z (cid:48) πe L (6( q L e L ) + 3( u R e L ) + 3( d R e L ) + (2 + tan θ l ) e L ) + Γ new . (19)We have denoted the decay width of the Z (cid:48) into particles other than SM fermions as Γ new ,even if those decays are into SM modes such as W + W − . If this extra width is determine bymeasuring additional visible and invisible decays, we can extract e L up to a 2-fold degeneracy,with no further input : e L = 4 π Γ Z (cid:48) − Γ new M Z (cid:48) ± (cid:113) (4 π Γ Z (cid:48) − Γ new M Z (cid:48) ) − (2 + tan θ l )(6( q L e L ) + 3( u R e L ) + 3( d R e L ) )2 + tan θ l . (20)The invisible width can be probed in associated production with a Z [44] or photon [45]by comparing the branching fraction of invisible Z (cid:48) decays to leptonic Z (cid:48) decays, where theinvisible decays due to a Z (cid:48) have been identified with a missing p T cut. This would allow usto determine Γ new . The quality of this probe is highly model-dependent, so we do not includeit in the analysis, but point out that it could be used to determine the leptonic couplings.If the invisible width cannot be determined to any degree of accuracy, then in the worstcase we have bounds for e L . For a given set of coupling combinations q × e , raising (orlowering) e will raise the partial width into e (or q ). Since we have measured the width,there is a natural ceiling in either direction, corresponding to the bounds e L ≥ π Γ Z (cid:48) M Z (cid:48) − (cid:113) (4 π Γ Z (cid:48) M Z (cid:48) ) − (2 + tan θ l )(6( q L e L ) + 3( u R e L ) + 3( d R e L ) )2 + tan θ l , (21) e L ≤ π Γ Z (cid:48) M Z (cid:48) + (cid:113) (4 π Γ Z (cid:48) M Z (cid:48) ) − (2 + tan θ l )(6( q L e L ) + 3( u R e L ) + 3( d R e L ) )2 + tan θ l . (22) If e L →
0, one should instead solve for e R . Z (cid:48) can decay [46]. As such, we refrain from putting these bounds in our plots, thoughthey are always present. We also note that if the Z (cid:48) width is determined, upper limits onall couplings can be derived by requiring that a given partial width no more than saturatesthe measured value. The future J-Lab Møller scattering experiment may be able to observe a shift in the asym-metry as discussed previously. Measuring the combination e V e A amounts to measuring ahyperbola in the e L − e R plane: e V e A = ( e R − e L ) /
4. Using the anticipated experimentaland theoretical errors, δ e R − e L M Z (cid:48) = 0 . − , (23)or, for a 1.5 TeV Z (cid:48) , δ ( e R − e L ) = 0 . . (24)The shifts due to our considered models are listed below. As with Q-weak, we ignore runningdown from the Z (cid:48) scale. e R − e L χ -0.0436 LR ψ χ ∗ -0.0833Table 9: Coupling combinations contributing to Møller scattering for our various test cases.None of the test models with the standard coupling induces a 1 σ deviation in Møller. InFig. 10, we plot these models with the 1 σ , 1 ab − LHC errors in θ l , as well as hyperbolicupper limits on the size of the leptonic couplings assuming no deviation from the StandardModel is found. The allowed regions do not actually extend to the origin; there is somemininum size of the leptonic couplings due to the argument of Sec. 6.1. An overall sign inthe couplings is unphysical, so we have chosen the sign of e L for each model for conveniencein displaying the plot.Our original test models should be consistent with the Standard Model in the Møllerexperiment for a mass of 1.5 TeV at 1 σ , though the χ model is very close to the limit. Wetherefore ask the question, if the leptonic couplings were somewhat larger, what could we see?As an example, we consider a test model like the previously considered χ model, except witha larger coupling; we call this the χ ∗ model. Specifically, we replace the e/ cos θ W appearing29igure 10: Measurements of the leptonic couplings from the LHC and J-Lab Møller experi-ment. Møller bounds assume no measured deviation from the Standard Model. The overall,unphysical, sign of the coupling is chosen for each model for convenient display of the results.30n E models with a somewhat larger value, 1/2. Such shifts in the overall normalizationfrom that predicted by the canonical breaking pattern has been considered elsewhere in theliterature [5]. In this case, the leptonic asymmetry is large enough to produce a significantdeviation from the standard model in the Møller experiment. This produces upper andlower hyperbolic bounds in the e L - e R plane. We plot the expected results in Fig. 11.Møller asymmetry gives us a hyperbola in the e L - e R plane. The LHC gives us the angle.Together, they pinpoint the Z (cid:48) leptonic couplings to a small region in the e L versus e R plane.Figure 11: Measurements of the leptonic couplings from the LHC and J-Lab Møller exper-iment, for a model with larger couplings. There are upper and lower hyperbolic bounds onthe size of the couplings. Once the leptonic couplings are determined, they can be input into the q × e measurementcoming from the LHC and Q-weak to extract all Z (cid:48) couplings. Explicitly, the quark charges31re given by q L = q L e L e L ,u R = u R e L e L ,d R = d R e L e L . (25)We list in Table 10 the final expected results after solving these equations for the quarkcharges and propagating through the anticipated Møller errors. We include all LHC data at1 ab − , and the Q-weak and Møller experiments. Each parameter range given is 68% CL.Note that for typical models, we only get upper bounds on the sizes of the leptonic couplings,and lower bounds for the sizes of the quark couplings. The signs are determined from theLHC. We note that the bounds derivable from width constraints discussed above have notbeen included in this table. χ LR ψ χ ∗ e L .
323 0 to 0 .
228 + 0 . ± . . e R . − .
343 to 0 - 0 . ± . . q L < − . < − .
009 + − . ± . . u R > . > .
126 - 0 . ± . . d R < − . < .
162 ? − . ± . . Table 10: Final determinations of the couplings using data from the LHC, Q-weak, and lowenergy Møller scattering. The results assume 1 ab − of integrated luminosity at the LHC.The sign of e L is chosen to be positive to fix the overall sign degeneracy. d R remains totallyunknown for the ψ model extraction, and for other parameters only the sign is determined. We have presented a thorough look at the ability of the LHC to analyze the underlying modelparameters of a new Z (cid:48) boson in the dilepton channel. In addition, we have studied thecomplimentary information that can be obtained from upcoming high-precision, low-energyexperiments. Our analysis is as model-independent as possible, and includes the majorsources of error which hinder the parameter extraction. We have discussed the importantrole played by off-peak data and the future Q-weak measurement of the proton weak chargein removing sign degeneracies that remain if only LHC data on the Z (cid:48) peak is analyzed.We also have shown how a precision measurement of low-energy Møller scattering can breakthe q → yq, e → e/y scaling degeneracy inherent to the dilepton channel at the LHC, andpermit an extraction of the individual Z (cid:48) charges. Our study demonstrates that a global32nalysis of data from many different observations is a necessary component of any attemptto determine Z (cid:48) couplings from the bottom up.Our study has elucidated the how precisely Z (cid:48) couplings can be measured as a function ofintegrated luminosity. Our LHC analysis relies upon appropriate binning of dilepton eventsin rapidity and scattering angle: u and d couplings are correlated with higher and lower Z (cid:48) rapidities respectively, while left- and right-handed couplings yield different lepton angulardistributions. As the parameters extracted and the bins used vary as a function of integratedLHC lumunosity, we summarize below how we believe the analysis of a dilepton signal atthe LHC should proceed. This template assumes that the resonance has a mass, width andtotal cross section roughly in the ranges considered here. For significantly different values,the timeline presented below must be adjusted. • With less than 10 fb − , the first property to establish is the spin of the observedresonance [19, 20]. At this point, a χ comparison between data and model hypothesesbased on the three on-peak observables discussed in Section 5.3 allows initial guessesas to the underlying Z (cid:48) model to be tested. This analysis is less sensitive to additionaldecay modes and overall coupling than comparisons of the total cross sections. • With 10 −
30 fb − , enough on-peak data is available to attempt an extraction ofthe couplings. Sufficient data exists only on-peak at this point, indicating that thenatural parameters to study are the c q , e q defined in Eq. (17). This analysis can beperformed in a simple, model-independent fashion by following the procedure describedin Section 5.4. We have found that the c q coupling combinations can be measured withrelative uncertainties of ±
30% for the M Z (cid:48) = 1 . • On-peak observables are sensitive to only the combinations ( q × e ) . The sign ambigu-ities can be removed by studying off-peak data, which probes the q × e combinationsdirectly. This becomes possible with 100 fb − of integrated luminosity. We have foundthat q × e combinations can be determined with relative uncertainties of approximately10 to 100%, depending on the particular model and coupling combination, for luminosi-ties of 300 − − . The couplings display very strong correlations, which indicate amuch stronger reduction in parameter space than the individual errors would suggest.A precision measurement from Q-weak helps remove sign degeneracies that remainafter LHC running. Another interesting aspect we have seen in our study is that thewidth can be determined precisely at this stage without a direct measurement of theBreit-Wigner shape. • Some direct measurement of either lepton or quark couplings is crucial to break the q → yq, e → e/y degeneracy that exists in the LHC dilepton mode. We have foundthat a proposed Møller experiment can determine the combination e R − e L directlyin some situations, and bound it in others. Measuring this last parameter can tell usall the Z (cid:48) couplings to Standard Model fermions, by separating out the well-measuredcombinations q × e from the LHC. 33ur study can be extended in several interesting ways. It would be interesting to applyour analysis procedure to other Z (cid:48) models to more fully determine its robustness. Anotherpossible extension would be to add other LHC final states such as t ¯ t , which could alsopotentially break the scaling degeneracy from which the dilepton channel suffers. We lookforward to studying these possibilities and hope that Nature provides on opportunity toapply our procedure to LHC data. Acknowledgements
We thank J. Erler, M. Herndon, K. Kumar, M. Ramsey-Musolf and T. Rizzo for manyhelpful discussions and suggestions. This work is supported by the DOE grant DE-FG02-95ER40896, by the University of Wisconsin Research Committee with funds provided by theWisconsin Alumni Research Foundation, and by the Alfred P. Sloan Foundation.
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