Using Final State Pseudorapidities to Improve s-channel Resonance Observables at the LHC
aa r X i v : . [ h e p - ph ] S e p Using Final State Pseudorapidities to Improve s -channel Resonance Observables atthe LHC Ross Diener , Stephen Godfrey , ∗ and Travis A. W. Martin † Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Canada K1S 5B6 TRIUMF, 4004 Wesbrook Mall, Vancouver BC Canada V6T 2A3 (Dated: November 3, 2018)We study the use of final state particle pseudorapidity for measurements of s -channel resonancesat the LHC. Distinguishing the spin of an s -channel resonance can, in principle, be accomplishedusing angular distributions in the centre-of-mass frame, possibly using a centre-edge asymmetrymeasurement, A CE . In addition, forward-backward asymmetry measurements, A F B , can be usedto distinguish between models of extra neutral gauge bosons. In this note we show how thesemeasurements can be improved by using simple methods based on the pseudorapidity of the finalstate particles and present the expected results for A F B and A CE for several representative models. PACS numbers: 14.70.Pw, 12.60.Cn, 12.15.-y, 12.15.J
I. INTRODUCTION
The startup of the CERN Large Hadron Collider(LHC) will allow the exploration of the TeV energyregime and the testing of the multitude of proposed the-ories of physics beyond the Standard Model. Many ofthese theories predict the existence of massive, neutral s -channel resonances [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. For somemodels of new neutral gauge bosons ( Z ′ ), the LHC is ex-pected to have a discovery reach upwards of 5 TeV with100 fb − of integrated luminosity [11]. This is a signif-icant improvement over the current experimental limitsfor most models, which constrain Z ′ masses to valuesgreater than ∼ s -channel resonance were discovered, theimmediate task would be to identify its origins. Manyobservables have been proposed to this end, primarily fo-cused on the dilepton channel ( e and µ ), which would pro-duce the cleanest and most easily measured signal for anon-leptophobic Z ′ [1, 2, 3, 4, 5] with the ATLAS [15, 16]and CMS [17] detectors. The proposed measurements forthe dilepton channel are the Z ′ width, total cross sec-tion, forward-backward asymmetry ( A F B ) [18], central-to-edge rapidity ratio [19], and a comprehensive analysisof all rapidity regions [20, 21].There are challenges associated with some of thesemeasurements that we argue can be alleviated by us-ing the pseudorapidity of the final state fermions. Inparticular, we focus on two measurements: determining(or at least constraining) the spin of an s -channel reso-nance, and determining the forward-backward asymme-try of a Z ′ . Distinguishing whether the new resonance isa scalar, such as an R-parity violating sneutrino [9, 10],a spin-2 boson, such as a KK graviton [6, 7, 8], or a spin-1 Z ′ [1, 2, 3, 4, 5] will be challenging and is typicallydetermined through the study of the angular distribu- ∗ Email: [email protected] † Email: [email protected] tion in the centre-of-mass frame of the initial state quarkand anti-quark (c.m.) [8, 22, 23, 24, 25]. The forward-backward asymmetry measurement at the LHC has todeal with the ambiguity in defining the forward direc-tion due to the inability to unambiguously determine thedirection of the initial state quark in a symmetric proton-proton collision.Presently, some solutions exist to deal with these chal-lenges. To distinguish the spin of the resonance, a centre-edge asymmetry, A CE , [23] can be defined that is sensi-tive to the the angular distribution of the events. Thecentre-edge asymmetry is a simple means of binning theevents in the central and edge regions of cos θ ∗ , the c.m.scattering angle, which will be weighted differently de-pending on the angular distribution. This has the benefitof eliminating some of the systematic uncertainties of afit to the angular distribution. However, the A CE observ-able still relies on boosting the particle four-momentumfrom the lab frame to the c.m. frame.The forward-backward ambiguity in a symmetric pp collision can be resolved by exploiting the fact thatthe valence quarks have, on average, larger momen-tum than the sea anti-quarks. The quark direction canthen be identified with the boost direction of the dilep-ton system. [26]. Restricting the measurement to thoseevents that have a large boost (i.e.: | Y Z ′ | > .
8) re-duces the misidentification of the initial state quarksand antiquarks, resulting in greater than 70% of dileptonevents being correctly identified as being boosted by thequark [26]. Both of these methods have been explored ingreat detail and remain the standard approach used inthe literature [20, 21, 24, 25].Both the A CE and the A F B measurements requireanalysis of the centre-of-mass (c.m.) angular distribu-tion of the dilepton events - directly for A CE , and whentagging forward or backward events in A F B . In thisnote we propose a simpler method of measuring theseasymmetries without reconstructing the angular distribu-tions. Specifically, we exploit the direct measurements ofthe lepton pseudorapidities to calculate the observables,rather than using derived quantities that may propagateuncertainties into the result. The proposed methods alsotake advantage of the fact that differences in pseudora-pidities are Lorentz invariant quantities, so that all cal-culations can be performed using quantities measured inthe lab frame.The point of these methods is not to provide newphenomenological insight into the models, but rather todemonstrate how the use of final state pseudorapiditiesprovides a simpler and cleaner means of obtaining the A CE and A F B values. The dimuon signal is very cleanand error propagation should not be a big issue. Thereal power of this approach will be seen when applied toheavy quark final states [27]. In the following sections,we give some calculational details which are followed by adescription of our approaches to the Centre-Edge Asym-metry and the Forward-Backward Asymmetry. We con-clude with some final comments.
II. CALCULATIONAL DETAILS
The basic ingredients in our calculations are the crosssections, σ ( pp → R → µ + µ − ), where R = Z ′ , ˜ ν or G . The cross section for R = Z ′ is described by theDrell-Yan process with the addition of a Z ′ [11, 18, 28].Analagous expressions for the spin-0 ˜ ν and spin-2 gravi-ton are given in Ref. [9, 10] and [7], respectively. We com-puted the cross sections using Monte-Carlo phase spaceintegration with weighted events and imposed kinematiccuts to take into account detector acceptances, as de-scribed in the following sections.In our numerical results we take α = 1 / .
9, sin θ w =0 . M Z = 91 .
188 GeV, and Γ Z = 2 .
495 GeV [29]. Weused the CTEQ6M parton distribution functions [30] andincluded a K-factor to account for NLO QCD corrections[31]. We neglected NNLO corrections, which are notnumerically important to our results [32, 33], and finalstate QED radiation effects, which are potentially impor-tant [34] but require a detailed detector level simulationthat is beyond the scope of the present analysis. The Z ′ widths only include decays to standard model fermionsand include NLO QCD and electroweak radiative correc-tions [35]. For the ˜ ν width, we take Γ ˜ ν = 1 GeV followingRef. [10]. Expressions for the G width can be found inRef. [22, 24, 36, 37]. III. SPIN DISCRIMINATION USINGCENTRE-EDGE ASYMMETRY, A CE The parton level angular distributions, d ˆ σ/d cos θ ∗ , ofthe spin-0, -1, -2 bosons, shown in Fig. 1, are distinctenough that, in principle, such a measurement woulduniquely identify the spin [8, 22]. However, these dis-tributions are not directly accessible due to the convo-lution with the parton distributions of the protons, theboosting of measured lab frame quantities to the centre-of-mass frame, detector limitations and finite statistics, -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.300.350.400.450.500.550.600.650.700.750.80 * θ cos * θ d c o s σ d σ FIG. 1: Normalized parton-level angular distribution ofspin-0 (black), spin-1 (dark grey/orange) and spin-2 (lightgrey/yellow) bosons decaying to fermions. all of which will make the measurement challenging [22].The centre-edge asymmetry is almost entirely model-independent for spin-0 and spin-1 bosons. For example,assuming the narrow width approximation for a Z ′ , wefind that A CE ≈ / z (1+1 / z ) − /
2, for some value ¯ z that separates the centre and edge regions of z ∗ , indepen-dent of the couplings to fermions. Spin-2 KK gravitonshave contributions from gg and q ¯ q processes that haveslightly different angular distributions, and the A CE de-pends on the weighted contribution of each. The spe-cific model will have an effect on the expected statisti-cal uncertainties, but this should not be significant tothe measurement due to the low backgrounds associatedwith leptonic final states. Thus, with limited statistics,an A CE measurement could have an advantage over a fitto the angular distribution.For a hadron collider, the centre-of-mass angle of theoutgoing fermion is not directly measurable on an eventby event basis due to the unknown values of the par-ton momentum fractions. However, there exists a directmapping between the c.m. angular distribution and thedifference in pseudorapidity of the final state lepton andanti-lepton, ∆ η . Furthermore, it is straightforward toshow that ∆ η is a Lorentz invariant quantity, so thatmeasuring this quantity in the lab frame is equivalent tomeasuring it in the centre-of-mass frame:∆ η lab = ∆ η ∗ = ln (cid:18) θ ∗ − cos θ ∗ (cid:19) (1)The normalized ∆ η lab distributions for spin-0, -1, -2 res-onances are shown in Fig. 2, where it is clear that theyare distinct from one another. One can therefore con-struct a new centre-edge asymmetry using the lab frame∆ η lab distribution in place of the c.m. frame angulardistribution.Using the mapping given by Eq. (1), we define the Δη -5 -4 -3 -2 -1 0 2 Δ η d σ d σ FIG. 2: Normalized ∆ η distribution including detector ac-ceptance cuts ( | η l | < . p T l >
20 GeV) and only includingevents within | M R − M l + l − | < . M . These cuts reducethe number of measurable events with large values of | ∆ η | . R = ˜ ν (black), Z ′ (dark grey/orange), G (light grey/yellow),where only one spin-1 distribution is shown due to the modelindependent nature of the spin-1 measurement. centre-edge asymmetry:˜ A CE = Z Π − Π − Z − Π −∞ − Z ∞ Π ! dσd ∆ η d ∆ η Z ∞−∞ dσd ∆ η d ∆ η . (2)Following Osland, et al. [24], we take ¯ z = 0 .
5, whichthey find to be the “optimal” value, and translates toΠ = ∆ η = 1 . | η | < .
5, which limits | ∆ η | < ν with λλ ′ = (0 . as an example of a spin-0 resonance [9, 10], and an RS graviton with c = 0 . Z ′ case we explored the E models ( ψ , χ and η ) [1], the Left-Right Symmetric model (LRSM, g R = g L ) [39], both the Littlest Higgs (LHM, tan θ H =1 .
0) [40] and Simplest Little Higgs (SLHM) [41] models,and the Sequential Standard Model (SSM).The spin-0 model, spin-2 and some Z ′ models we studypredict narrow resonances, such that including eventswithin several widths of the peak will be impossible inpractice due to detector resolution effects smearing theBreit-Wigner distribution. Instead, we examine eventswithin one dilepton invariant mass bin as defined in theATLAS TDR [16], using ∆ M = 42 . TABLE I: ˜ A CE values with corresponding statistical uncer-tainties for 100 fb − integrated luminosity, p T l >
20 GeV, | η l | < .
5, within one bin ∆ M l + l − = 42 . M R =1 . − integrated luminosity. Model ˜ A CE ± δ ˜ A CE N Events E χ − . ± E ψ − . ± E η − . ± − . ± − . ± − . ± − . ± . ± ν +0 . ± In Table I we show the expected centre-edge asymme-try for a spin-0, spin-1 and spin-2 resonance, analogousto the study performed by Dvergsnes, et al. [23], assum-ing muon final states with 96% detection efficiency [15].From Table I one sees that if a Z ′ were observed, a G or ˜ ν could be ruled out. Likewise, an A CE measurementwould strongly discriminate against the Z ′ or ˜ ν hypoth-esis if a G were observed. However, the Z ′ and G hy-pothesis could only be ruled out at approximately 2 . σ ifa ˜ ν signal was observed. The primary limitation in dis-tinguishing between the different possibilities is the lowstatistics for ˜ ν production, as shown in the table, whichis due to the tight constraints on the allowed values ofits couplings. Other hypothetical spin-0 resonances maynot be as tightly constrained and could therefore be dis-tinguished from a Z ′ or G with higher statistical signifi-cance. IV. FORWARD-BACKWARD ASYMMETRY
The forward-backward asymmetry is a well-establishedmeasurement for distinguishing between models of Z ′ ’s[18]. For p ¯ p collisions at the Tevatron, the proton direc-tion provides an obvious choice to define the “forward”direction. The choice of forward direction at the LHC ismore subtle, and is conventionally defined as the direc-tion of the Z ′ rapidity, Y Z ′ . The Z ′ rapidity is chosenbecause the parton distribution functions for the valencequarks peak at a higher momentum fraction than those ofthe anti-quarks, so the system has a higher probabilityof being boosted in the quark direction. This observa-tion is statistical in nature and is more likely to holdtrue for larger values of | Y Z ′ | . For smaller values of | Y Z ′ | ,the momentum fractions of the quark and anti-quark aregenerally closer in magnitude, so that using | Y Z ′ | in the Z' Y -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 ( F ( Y )- B ( Y )) N -0.004-0.0020.0000.0020.0040.0060.0080.010 Z ' Z ' LHME6 χ SSMSLHMLRSME6 η E6 ψ FIG. 3: A F B as a function of the Z ′ rapidity following Eq. (3)except that these results are not integrated over rapidity.From top to bottom, the models are LHM, LRSM, SLHM,SSM, E ψ , E η , E χ . low rapidity region is less likely to correctly identify thequark direction.A simpler method of defining a “forward” or “back-ward” event uses pseudorapidity. As before, we definethe quark direction to be that of the higher momentumparton, or equivalently the direction of the Z ′ rapidity.One can then show that a “forward” event is one in which | η f | > | η ¯ f | in the lab frame, and vice-versa for a “back-ward” event. Using these definitions for forward andbackward, one can define the forward-backward asym-metry: A F B = Z [ F ( y ) − B ( y )] dy Z [ F ( y ) + B ( y )] dy (3)where F ( y ) is the number of forward events and B ( y ) isthe number of backward events for a given y , the Z ′ ra-pidity (i.e.: Y Z ′ ). The F ( y ) − B ( y ) distribution under thisdefinition is clearly shown in Fig. 3 to be symmetric in Z ′ rapidity. This method of finding A F B has the advan-tage of being very straightforward and clean. It simplyrelies on counting events with | η f | > | η ¯ f | and those with | η f | < | η ¯ f | . We note that a related technique is employedby the CDF collaboration [42] for the Z A F B in p ¯ p col-lisions at the Tevatron. However, the natural choice ofthe quark direction in p ¯ p collisions at the Tevatron incontrast to pp collisions at the LHC results in importantdifferences between the methods.As in the conventional method for finding A F B , forsmall values of | Y Z ′ | , there is a higher probability towrongly assume that the quark is the parton with thehigher momentum fraction. This results in incorrectlyassigning the forward or backward direction and givesa small “wrong” contribution to the A F B measurement.For this reason, it has been suggested that the central re-gion, | Y Z ′ | < Y min , be excluded in the measurement [26]. (on-peak) FB A -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 ( o ff - pea k ) F B A -0.2-0.10.00.10.20.30.4 χ E6 Model - ψ E6 Model - η E6 Model - LR SymmetricSequential SM Littlest Higgs ModelSimplest Little HiggsStandard Model
FIG. 4: A F B off-peak versus on-peak for a variety of models,including detector acceptance limits and kinematic cuts aspreviously listed. Standard Model measurement determinedfrom the standard model Drell-Yan cross section, with on-peak events within | M l + l − − M Z ′ | <
100 GeV and off-peakevents within 2 / M Z ′ < M l + l − < M Z ′ −
300 GeV to includelarge enough statistics.
However, the coupling dependency can still be deter-mined without this constraint on | Y Z ′ | [19].Another consideration for excluding the central regionis that the number of events that remain after subtract-ing F − B is small, as shown in Fig. 3, while the totalnumber of events in this region is large. Excluding theevents in the central region would increase the magnitudeof A F B , potentially making models more distinguishable.However, we found that increasing Y min resulted in an in-crease in the relative uncertainty. We therefore concludethat little is gained by excluding events with small Y Z ′ ,and suggest that the whole rapidity region be included todecrease uncertainty and further simplify the A F B mea-surement.Using this method, we calculate A F B for the E mod-els ( ψ , χ , η ) [1], the Left Right Symmetric model [39],the Littlest Higgs model [40], the Simplest Little Higgsmodel [41], and the Sequential Standard model. The on-peak versus off-peak A F B are shown in Fig. 4 , where on-peak includes events which satisfy | M l + l − − M Z ′ | < Z ′ and off-peak includes events which satisfy 2 / M Z ′ In this paper we described an approach for discriminat-ing between various spin hypotheses for a newly discov-ered s -channel resonance at the LHC using a centre-edgesymmetry, ˜ A CE , that is based on the difference of therapidities of final state fermions. We also described asimple way to measure the forward-backward asymme- try, A F B , using the properties of pseudorapidity. Bothof these measurements have an advantage over previousapproaches as they rely solely on the measurement ofpseudorapidity, a fairly basic quantity. The new mea-surements require simple counting and should propagatefewer errors than previous approaches that rely on boost-ing the four momentum into the centre-of-mass frame inorder to perform the analysis. 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