Measurement of the angular coefficients in Z-boson events using electron and muon pairs from data taken at s = 8 TeV with the ATLAS detector

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1 EUROPEAN ORGANISAION FOR NUCLEAR RESEARCH (CERN) Submitted to: JHEP CERN-EP6-87 nd June 6 Measurement of the angular coefficients in -boson events using electron and muon airs from data taken at s = 8 ev with the ALAS detector he ALAS Collaboration Abstract he angular distributions of Drell Yan charged leton airs in the vicinity of the -boson mass eak robe the underlying QCD dynamics of -boson roduction. his aer resents a measurement of the comlete set of angular coefficients A 7 describing these distributions in the -boson Collins Soer frame. he data analysed corresond to. fb of collisions at s = 8 ev, collected by the ALAS detector at the CERN LHC. he measurements are comared to the most recise fixed-order calculations currently available (O(α s)) and with theoretical redictions embedded in Monte Carlo generators. he measurements are recise enough to robe QCD corrections beyond the formal accuracy of these calculations and to rovide discrimination between different arton-shower models. A significant deviation from the O(α s) redictions is observed for A A. Evidence is found for nonzero A,6,7, consistent with exectations. c 6 CERN for the benefit of the ALAS Collaboration. Reroduction of this article or arts of it is allowed as secified in the -BY-. license.

2 . Introduction he angular distributions of charged leton airs roduced in hadron hadron collisions via the Drell Yan neutral current rocess rovide a ortal to recise measurements of the roduction dynamics through sin correlation effects between the initial-state artons and the final-state letons mediated by a sin- intermediate state, redominantly the boson. In the -boson rest frame, a lane sanned by the directions of the incoming rotons can be defined, e.g. using the Collins Soer () reference frame []. he leton olar and azimuthal angular variables, denoted by cos θ and in the following formalism, are defined in this reference frame. he sin correlations are described by a set of nine helicity density matrix elements, which can be calculated within the context of the arton model using erturbative quantum chromodynamics (QCD). he theoretical formalism is elaborated in Refs. [ ]. he full five-dimensional differential cross-section describing the kinematics of the two Born-level letons from the -boson decay can be decomosed as a sum of nine harmonic olynomials, which deend on cos θ and, multilied by corresonding helicity cross-sections that deend on the -boson transverse momentum ( ), raidity (y ), and invariant mass (m ). It is a standard convention to factorise out the unolarised cross-section, denoted in the literature by σ U+L, and to resent the five-dimensional differential cross-section as an exansion into nine harmonic olynomials P i (cos θ, ) and dimensionless angular coefficients A 7 (, y, m ), which reresent ratios of helicity cross-sections with resect to the unolarised one, σ U+L, as exlained in detail in Aendix A: dσ d dy dm d cos θ d = 6π d dy dm { ( + cos θ) + A ( cos θ) + A sin θ cos () dσ U+L + A sin θ cos + A sin θ cos + A cos θ } +A sin θ sin + A 6 sin θ sin + A 7 sin θ sin. he deendence of the differential cross-section on cos θ and is thus comletely manifest analytically. In contrast, the deendence on, y, and m is entirely contained in the A i coefficients and σ U+L. herefore, all hadronic dynamics from the roduction mechanism are described imlicitly within the structure of the A i coefficients, and are factorised from the decay kinematics in the -boson rest frame. his allows the measurement recision to be essentially insensitive to all uncertainties in QCD, quantum electrodynamics (QED), and electroweak (EW) effects related to -boson roduction and decay. In articular, EW corrections that coule the initial-state quarks to the final-state letons have a negligible imact (below.%) at the -boson ole. his has been shown for the LEP recision measurements [6, 7], when calculating the interference between initial-state and final-state QED radiation. When integrating over cos θ or, the information about the A and A 6 coefficients is lost, so both angles must be exlicitly used to extract the full set of eight coefficients. Integrating Eq. () over cos θ yields: dσ d dy dm d = π dσ U+L d dy dm { + A cos + π 6 A cos + A sin + π } 6 A 7 sin, ()

3 while integrating over yields: dσ U+L dσ d dy dm d cos θ = 8 d dy dm { ( + cos θ) + } A ( cos θ) + A cos θ. () At leading order (LO) in QCD, only the annihilation diagram q q is resent and only A is non-zero. At next-to-leading order (NLO) in QCD (O(α s )), A also become non-zero. he Lam ung relation [8 ], which redicts that A A = due to the sin of the gluon in the qg q and q q g diagrams, is exected to hold u to O(α s ), but can be violated at higher orders. he coefficients A,6,7 are exected to become non-zero, while remaining small, only at next-to-next-to-leading order (NNLO) in QCD (O(α s )), because they arise from gluon loos that are included in the calculations [, ]. he coefficients A and A deend on the roduct of vector and axial coulings to quarks and letons, and are sensitive to the Weinberg angle sin θ W. he exlicit formulae for these deendences can be found in Aendix A. he full set of coefficients has been calculated for the first time at O(α s) in Refs. [ ]. More recent discussions of these angular coefficients may be found in Ref. [], where the redictions in the NNLOPS scheme of the Powheg [ 7] event generator are shown for -boson roduction, and in Ref. [8], where the coefficients are exlored in the context of W-boson roduction, for which the same formalism holds. he CDF Collaboration at the evatron ublished [9] a measurement of some of the angular coefficients of leton airs roduced near the -boson mass ole, using. fb of roton anti-roton collision data at a centre-of-mass energy s =.96 ev. Since the measurement was erformed only in rojections of cos θ and, the coefficients A and A 6 were inaccessible. hey further assumed A and A 7 to be zero since the sensitivity to these coefficients was beyond the recision of the measurements; the coefficients A,,, were measured as a function of. hese measurements were later used by CDF [] to infer an indirect measurement of sin θ W, or equivalently, the W-boson mass in the on-shell scheme, from the average A coefficient. hese first measurements of the angular coefficients demonstrated the otential of this not-yet-fully exlored exerimental avenue for investigating hard QCD and EW hysics. Measurements of the W-boson angular coefficients at the LHC were ublished by both ALAS [] and CMS []. More recently, a measurement of the -boson angular coefficients with µµ decays was ublished by CMS [], where the first five coefficients were measured with 9.7 fb of roton roton () collision data at s = 8 ev. he measurement was erformed in two y bins, < y < and < y <., each with eight bins in u to GeV. he violation of the Lam ung relation was observed, as redicted by QCD calculations beyond NLO. his aer resents an inclusive measurement of the full set of eight A i coefficients using charged leton airs (electrons or muons), denoted hereafter by l. he measurement is erformed in the -boson invariant mass window of 8 GeV, as a function of, and also in three bins of y. hese results are based on. fb of collision data collected at s = 8 ev by the ALAS exeriment [] at the LHC. With the measurement techniques develoed for this analysis, the comlete set of coefficients is extracted with fine granularity over bins of u to 6 GeV. he measurements, erformed in the reference frame [], are first resented as a function of, integrating over y. Further measurements divided into three bins of y are also resented: < y <, < y <, and < y <.. he /γ e + e and /γ µ + µ channels where both letons fall within the seudoraidity range η <. (hereafter referred to as the central central or ee and µµ channels) are used for the y -integrated measurement and the first two y bins. he /γ e + e channel where one of the electrons instead falls in the region

4 η >. (referred to hereafter as the central forward or ee CF channel) is used to extend the measurement to the high-y region encomassed by the third y bin. In this case, however, because of the fewer events available for the measurement itself and to evaluate the backgrounds (see Section ), the measurement is only erformed for u to GeV using rojections of cos θ and, making A and A 6 inaccessible in the < y <. bin. he high granularity and recision of the secific measurements resented in this aer rovide a stringent test of the most recise erturbative QCD redictions for -boson roduction in collisions and of Monte Carlo (MC) event generators used to simulate -boson roduction. his aer is organised as follows. Section summarises the theoretical formalism used to extract the angular coefficients and resents the fixed-order QCD redictions for their variations as a function of. Section describes briefly the ALAS detector and the data and MC samles used in the analysis, while Section resents the data analysis and background estimates for each of the three channels considered. Section describes the fit methodology used to extract the angular coefficients in the full hase sace as a function of and Section 6 gives an overview of the statistical and systematic uncertainties of the measurements. Sections 7 and 8 resent the results and comare them to various redictions from theoretical calculations and MC event generators, and Section 9 summarises and concludes the aer.. heoretical redictions he differential cross-section in Eq. () is written for ure bosons, although it also holds for the contribution from γ and its interference with the boson. he tight invariant mass window of 8 GeV is chosen to minimise the γ contribution, although the redicted A i coefficients resented in this aer are effective coefficients, containing this small contribution from γ. his contribution is not accounted for exlicitly in the detailed formalism described in Aendix A, which is resented for simlicity for ure -boson roduction. hroughout this aer, the letons from -boson decays are defined at the Born level, i.e. before final-state QED radiation, when discussing theoretical calculations or redictions at the event-generator level. he and y deendence of the coefficients varies strongly with the choice of sin quantisation axis in the -boson rest frame (z-axis). In the reference frame adoted for this aer, the z-axis is defined in the -boson rest frame as the external bisector of the angle between the momenta of the two rotons, as deicted in Fig.. he ositive direction of the z-axis is defined by the direction of ositive longitudinal -boson momentum in the laboratory frame. o comlete the coordinate system, the y-axis is defined as the normal vector to the lane sanned by the two incoming roton momenta and the x-axis is chosen to define a right-handed Cartesian coordinate system with the other two axes. Polar and azimuthal angles are calculated with resect to the negatively charged leton and are labelled θ and, resectively. In the case where =, the direction of the y-axis and the definition of are arbitrary. Historically, there has been an ambiguity in the definition of the sign of the angle in the frame: this aer adots the recent convention followed by Refs. [, ], whereby the coefficients A and A are ositive. he coefficients are not exlicitly used as inut to the theoretical calculations nor in the MC event generators. hey can, however, be extracted from the shaes of the angular distributions with the method roosed in Ref. [], owing to the orthogonality of the P i olynomials. he weighted average of the angular distributions with resect to any secific olynomial isolates an average reference value or moment

5 l θ y^ x ^ z^ Figure : Sketch of the Collins-Soer reference frame, in which the angles θ and are defined with resect to the negatively charged leton l (see text). he notations ˆx, ŷ and ẑ denote the unit vectors along the corresonding axes in this reference frame. of its corresonding coefficient. he moment of a olynomial P(cos θ, ) over a secific range of, y, and m is defined to be: P(cos θ, ) = P(cos θ, )dσ(cos θ, )d cos θd dσ(cos θ, )d cos θd. () he moment of each harmonic olynomial can thus be exressed as (see Eq. ()): ( cos θ) = (A ); sin θ cos = A ; sin θ cos = A ; sin θ cos = A ; cos θ = A ; sin θ sin = A ; () sin θ sin = A 6; sin θ sin = A 7. One thus obtains a reresentation of the effective angular coefficients for /γ roduction. hese effective angular coefficients dislay in certain cases a deendence on y, which arises mostly from the fact that the interacting quark direction is unknown on an event-by-event basis. As the method of Ref. [] relies on integration over the full hase sace of the angular distributions, it cannot be alied directly to data, but is used to comute all the theoretical redictions shown in this aer. he inclusive fixed-order erturbative QCD redictions for -boson roduction at NLO and NNLO were obtained with DYNNLO v. []. hese inclusive calculations are formally accurate to O(α s ). he -boson is roduced, however, at non-zero transverse momentum only at O(α s ), and therefore the calculation of the coefficients as a function of is only NLO. Even though the fixed-order calculations do not rovide reliable absolute redictions for the sectrum at low values, they can be used for >. GeV for the angular coefficients. he results were cross-checked with NNLO redictions from FEW v..b [6 8] and agreement between the two rograms was found within uncertainties. he

6 renormalisation and factorisation scales in the calculations were set to E = (m ) + ( ) [9] on an event-by-event basis. he calculations were done using the C NLO or NNLO arton distribution functions (PDFs) [], deending on the order of the rediction. he NLO EW corrections affect mostly the leading-order QCD cross-section normalisation in the -ole region and have some imact on the distribution, but they do not affect the angular correlations at the -boson vertex. he DYNNLO calculation was done at leading order in EW, using the G µ scheme []. his choice determines the value of A at low, and for the urose of the comarisons resented in this aer, both A and A obtained from DYNNLO are rescaled to the values redicted when using the measured value of sin θw eff =. []. he theoretical redictions are shown in Fig. and tabulated in able for three illustrative bins. he binning in is chosen based on the exerimental resolution at low and on the number of events at high and has the following boundaries (in GeV) used consistently throughout the measurement:,boundary = {,.,., 8.,.,.9, 8.,.,., 9.,.6, 6.,.,.9,., 6., 6.9, 7., 8.,.,., 7.,., 6.}. (6) he redictions show the following general features. he A and A coefficients increase as a function of and the deviations from lowest-order exectations are quite large, even at modest values of = GeV. he A and A coefficients are relatively small even at large, with a maximum value of.8. In the limit where =, all coefficients excet A are exected to vanish at NLO. he NNLO corrections are tyically small for all coefficients excet A, for which the largest correction has a value of.8, in agreement with the original theoretical studies []. he theoretical redictions for A,6,7 are not shown because these coefficients are exected to be very small at all values of : they are zero at NLO and the NNLO contribution is large enough to be observable, namely of the order of. for values of in the range GeV. he statistical uncertainties of the calculations, as well as the factorisation and renormalisation scale and PDF uncertainties, were all considered as sources of theoretical uncertainties. he statistical uncertainties of the NLO and NNLO redictions in absolute units are tyically. and., resectively. he larger statistical uncertainties of the NNLO redictions are due to the longer comutational time required than for the NLO redictions. he scale uncertainties were estimated by varying the renormalisation and factorisation scales simultaneously u and down by a factor of two. As stated in Ref. [], the theoretical uncertainties due to the choice of these scales are very small for the angular coefficients because they are ratios of cross-sections. he resulting variations of the coefficients at NNLO were found in most cases to be comarable to the statistical uncertainty. he PDF uncertainties were estimated using the C NNLO eigenvector variations, as obtained from FEW and normalised to 68% confidence level. hey were found to be small comared to the NNLO statistical uncertainty, namely of the order of. for A and. for A. 6

7 A. ALAS Simulation Preliminary s = 8 ev A.. ALAS Simulation Preliminary s = 8 ev.8.6 DYNNLO (NNLO) DYNNLO (NLO).8.6 DYNNLO (NNLO) DYNNLO (NLO) A. ALAS Simulation Preliminary s = 8 ev A.. ALAS Simulation Preliminary s = 8 ev.8.6 DYNNLO (NNLO) DYNNLO (NLO).8.6 DYNNLO (NNLO) DYNNLO (NLO) A -A.. ALAS Simulation Preliminary s = 8 ev DYNNLO (NNLO) DYNNLO (NLO) A.. ALAS Simulation Preliminary s = 8 ev DYNNLO (NNLO) DYNNLO (NLO).... Figure : he angular coefficients A and the difference A A, shown as a function of, as redicted from DYNNLO calculations at NLO and NNLO in QCD. he NLO redictions for A A are comatible with zero, as exected from the Lam ung relation [8 ]. he error bars show the total uncertainty of the redictions, including contributions from statistical uncertainties, QCD scale variations and PDFs. he statistical uncertainties of the NNLO redictions are dominant and an order of magnitude larger than those of the NLO redictions. 7

8 able : Summary of redictions from DYNNLO at NLO and NNLO for A, A, A A, A, A, A, A, A 6, and A 7 at low ( 8 GeV), mid (. GeV), and high ( 7 GeV) for the y -integrated configuration. he uncertainty reresents the sum of statistical and systematic uncertainties. = 8 GeV =. GeV = 7 GeV NLO NNLO NLO NNLO NLO NNLO A A A A A A A A < A A 7 < he ALAS exeriment and its data and Monte Carlo samles.. ALAS detector he ALAS exeriment [] at the LHC is a multi-urose article detector with a forward-backward symmetric cylindrical geometry and a near π coverage in solid angle. It consists of an inner tracking detector, electromagnetic (EM) and hadronic calorimeters, and a muon sectrometer. he inner tracker rovides recision tracking of charged articles in the seudoraidity range η <.. his region is matched to a high-granularity EM samling calorimeter covering the seudoraidity range η <. and a coarser granularity calorimeter u to η =.9. A hadronic calorimeter system covers the entire seudoraidity range u to η =.9. he muon sectrometer rovides triggering and tracking caabilities in the range η <. and η <.7, resectively. A first-level trigger is imlemented in hardware, followed by two software-based trigger levels that together reduce the acceted event rate to Hz on average. For this aer, a central leton is one found in the region η <. (excluding, for electrons, the electromagnetic calorimeter barrel/end-ca transition region.7 < η <.), while a forward electron is one found in the region. < η <.9 (excluding the transition region.6 < η <. between the electromagnetic end-ca and forward calorimeters). ALAS uses a right-handed coordinate system with its origin at the nominal interaction oint (IP) in the centre of the detector and the z-axis along the beam ie. he x-axis oints from the IP to the centre of the LHC ring, and the y-axis oints uwards. Cylindrical coordinates (r, ) are used in the transverse lane, being the azimuthal angle around the z-axis. he seudoraidity is defined in terms of the olar angle θ as η = ln tan(θ/). Angular distance is measured in units of R ( η) + ( ). 8

9 .. and Monte Carlo samles he data were collected by the ALAS detector in at a centre-of-mass energy of s = 8 ev, and corresond to an integrated luminosity of. fb. he mean number of additional interactions er bunch crossing (ile-u events) in the data set is aroximately. he simulation samles used in the analysis are shown in able. he four event generators used to roduce the /γ ll signal events are listed in able. he baseline PowhegBox (v/r9) samle [ 7], which uses the C NLO set of PDFs [], is interfaced to Pythia 8 (v.8.7) [] with the AU set of tuned arameters [] to simulate the arton shower, hadronisation and underlying event, and to Photos (v.) [6] to simulate QED final-state radiation (FSR) in the -boson decay. he alternative signal samles are from PowhegBox interfaced to Herwig (v.6..) [7] for the arton shower and hadronisation, Jimmy (v.) [8] for the underlying event, and Photos for FSR. he Shera (v...) [9 ] generator is also used, and has its own imlementation of the arton shower, hadronisation, underlying event and FSR, and uses the C NLO PDF set. hese alternative samles are used to test the deendence of the analysis on different matrix-element calculations and arton-shower models, as discussed in Section 6. he Powheg (v.) + MiNLO event generator [] was used for the +jet rocess at NLO to normalise certain reference coefficients for the ee CF analysis, as described in Section. he number of events available in the baseline PowhegBox + Pythia 8 signal samle corresonds to aroximately () times that in the data below (above) = GeV. Backgrounds from EW (diboson and γγ ll roduction) and to-quark (roduction of to-quark airs and of single to quarks) rocesses are evaluated from the MC samles listed in able. he W + jets contribution to the background is instead included in the data-driven multijet background estimate, as described in Section ; W-boson samles listed in able are thus only used for studies of the background comosition. All of the samles are rocessed with the Geant-based simulation [] of the ALAS detector []. he effects of additional collisions in the same or nearby bunch crossings are simulated by the addition of so-called minimum-bias events generated with Pythia 8.. analysis.. Event selection As mentioned in Sections and, the data are slit into three orthogonal channels, namely the ee channel with two central electrons, the µµ channel with two central muons, and the ee CF channel with one central electron and one forward electron. Selected events are required to be in a data-taking eriod in which the beams were stable and the detector was functioning well, and to contain a reconstructed rimary vertex with at least three tracks with >. GeV. Candidate ee events are obtained using a dielectron trigger requiring two electron candidates with > GeV, combined with high- single-electron triggers. Electron candidates are required to have > GeV and are reconstructed from clusters of energy in the electromagnetic calorimeter matched to inner detector tracks. he electron candidates must satisfy a set of medium selection criteria [, ], which have been otimised for the level of ile-u resent in the data. Events are required to contain exactly two electron candidates of oosite charge satisfying the above criteria. 9

10 able : MC samles used to estimate the signal and backgrounds in the analysis. Signature Generator PDF Refs. /γ ll PowhegBox + Pythia 8 C NLO [ 7,, ] /γ ll PowhegBox + Jimmy/Herwig C NLO [7] /γ ll Shera C NLO [9 ] /γ ll + jet Powheg + MiNLO C NLO [] W lν PowhegBox + Pythia 8 C NLO W lν Shera C NLO t t air MC@NLO + Jimmy/Herwig C NLO [8, 6] Single to quark: t channel AcerMC + Pythia 6 CEQ6L [7, 8] s and Wt channels MC@NLO + Jimmy/Herwig C NLO Dibosons Shera C NLO Dibosons Herwig CEQ6L γγ ll Pythia 8 MRSQED NLO [9] Candidate µµ events are retained for analysis using a dimuon trigger requiring two muon candidates with > 8 GeV and 8 GeV, resectively, combined with single high- muon triggers. Muon candidates are required to have > GeV and are identified as tracks in the inner detector which are matched and combined with track segments in the muon sectrometer []. rack-quality and longitudinal and transverse imact-arameter requirements are imosed for muon identification to suress backgrounds, and to ensure that the muon candidates originate from a common rimary interaction vertex. Events are required to contain exactly two muon candidates of oosite charge satisfying the above criteria. Candidate ee CF events are obtained using a single-electron trigger, requiring an isolated central electron candidate with > GeV, combined with a looser high- single-electron trigger. he central electron candidate is required to have > GeV. Because the exected background from multijet events is larger in this channel than in the ee channel, the central electron candidate is required to satisfy a set of tight selection criteria [], which are otimised for the level of ile-u observed in the data. he forward electron candidate is required to have > GeV and to satisfy a set of medium selection criteria, based only on the shower shaes in the electromagnetic calorimeter [] since this region is outside the accetance of the inner tracker. Events are required to contain exactly two electron candidates satisfying the above criteria. Since this analysis is focused on the -boson ole region, the leton air is required to have an invariant mass (m ll ) within a narrow window around the -boson mass, 8 < m ll < GeV. Events are selected for y -integrated measurements without any requirements on the raidity of the leton air (y ll ). For the y -binned measurements, events are selected in three bins of raidity: y ll <.,. < y ll <., and. < y ll <.. Events are also required to have a dileton transverse momentum ( ll ) less than the value of 6 () GeV used for the highest bin in the ee and µµ (ee CF ) channels. he variables m ll, y ll, and ll, which are defined using reconstructed leton airs, are to be distinguished from the variables m, y, and, which are defined using leton airs at the Born level, as described in Section. he simulated events are required to satisfy the same selection criteria, after alying small corrections to account for the differences between data and simulation in terms of reconstruction, identification and trigger efficiencies and of energy scale and resolution for electrons and muons [ ]. All simulated

11 Exected events ALAS 8 ev,. fb ee µµ ee CF Full hase sace..... y Accetance * efficiency ALAS s = 8 ev ee µµ ee CF..... y Exected events ALAS 8 ev,. fb ee µµ ee CF Full hase sace Accetance * efficiency ALAS s = 8 ev ee µµ ee CF Figure : Comarison of the exected yields (left) and accetance times efficiency of selected events (right) as a function of y (to) and (bottom), for the ee, µµ, and ee CF events. Also shown are the exected yields at the event generator level over the full hase sace considered for the measurement, which corresonds to all events with a dileton mass in the chosen window, 8 < m < GeV. events are reweighted to match the distributions observed in data for the level of ile-u and for the rimary vertex longitudinal osition. Figure illustrates the different ranges in and y exected to be covered by the three channels along with their accetance times selection efficiencies, which is defined as the ratio of the number of selected events to the number in the full hase sace. he difference in shae between the ee and µµ channels arises from the lower reconstruction and identification efficiency for central electrons at high values of η and from the lower trigger and reconstruction efficiency for muons at low values of η. he central central and central forward channels overla in the region. < y <.... Backgrounds In the -boson ole region, the backgrounds from other rocesses are small, below the half-ercent level for the ee and µµ channels and at the level of % for the ee CF channel. he backgrounds from romt isolated leton airs are estimated using simulated samles, as described in Section, and consist redominantly of leton airs from to-quark rocesses and from diboson roduction with a smaller contribution from ττ decays. he other background source arises from events in which at least one of the leton candidates is not a romt isolated leton but rather a leton from heavy-flavour hadron

12 able : For each of the three channels, yield of events observed in data and exected background yields (multijets, to+electroweak, and total) corresonding to the data set and an integrated luminosity of. fb. he uncertainties quoted include both the statistical and systematic comonents (see text). Channel Observed Exected background Multijets (from data) o+electroweak (from MC) otal ee. 6 6 ± ± 9 ± µµ ± 9 ± 8 ± 6 ee CF. 6 8 ± ± 9 ± decay (beauty or charm) or a fake leton in the case of electron candidates (these may arise from charged hadrons or from hoton conversions within a hadronic jet). his background consists of events containing two such letons (multijets) or one such leton (W + jets or to-quark airs) and is estimated from data using the leton isolation as a discriminating variable, a rocedure described for examle in Ref. [] for electrons. For the central central channels, the background determination is carried out in the full twodimensional sace of (cos θ, ) and in each bin of ll. In the case of the central forward channel, the multijet background, which is by far the dominant one, is estimated searately for each rojection in cos θ and because of the limited amount of data. his is the main reason why the angular coefficients in the central forward channel are extracted only in rojections, as described in Section. Figure shows the angular distributions, cos θ and, for the three channels for the data, the - boson signal MC samle, and the main sources of background discussed above. he total background in the central central events is below.% and its uncertainty is dominated by the large uncertainty in the multijet background of aroximately %. he uncertainty in the to+electroweak background is taken conservatively to be %. In the case of the central forward electron airs, the to+electroweak background is so small comared to the much larger multijet background that it is neglected for simlicity in the fit rocedure described in Section. able summarises the observed yields of events in data for each channel, integrated over all values of ll, together with the exected background yields with their total uncertainties from multijet events and from to+electroweak sources. More details of the treatment of the background uncertainties are discussed in Section 6. here are also signal events that are considered as background to the measurement because they are resent in the data only due to the finite resolution of the measurements, which leads to migrations in mass and raidity. hese are denoted Non-fiducial events and can be divided into four categories: the dominant fraction consists of events that have m at the generator level outside the chosen m ll mass window but ass event selection, while another contribution arises from events that do not belong to the y bin considered for the measurement at generator level. he latter contribution is sizeable only in the ee CF channel. Other negligible sources of this tye of background arise from events for which the central electron has the wrong assigned charge in the ee CF channel or both central electrons have the wrong assigned charge in the ee channel, or for which at the generator level is larger than 6 GeV. hese backgrounds are all included as a small comonent of the signal MC samle in Fig.. heir contributions amount to one ercent or less for the ee and µµ channels, increasing to almost 8% for the ee CF channel because of the much larger migrations in energy measurements in the case of forward electrons. For the < y <. bin in the ee CF channel, the y migration contributes % to the non-fiducial background. he fractional contribution of all backgrounds to the total samle is shown exlicitly for each channel as a function of ll in Fig. together with the resective contributions of the multijet and

13 Entries /. ALAS Preliminary 8 ev,. fb ee ee Multijet Diboson o ττ Entries /. ALAS Preliminary 8 ev,. fb ee ee Multijet Diboson o ττ cos θ 6 Entries /. ALAS Preliminary 8 ev,. fb µµ µµ Multijet Diboson o ττ Entries /. ALAS Preliminary 8 ev,. fb µµ µµ Multijet Diboson o ττ cos θ 6 Entries /. ALAS Preliminary 8 ev,. fb ee CF ee Multijet Diboson o Entries /. ALAS Preliminary 8 ev,. fb ee CF ee Multijet Diboson o ττ ττ cos θ 6 Figure : he cos θ (left) and (right) angular distributions, averaged over all -boson, for the ee (to), µµ (middle) and ee CF (bottom) channels. he distributions are shown searately for the different background sources contributing to each channel. he multijet background is determined from data, as exlained in the text.

14 Background fraction..... ALAS 8 ev,. fb ee : y -integrated otal Multijet o+ew Non-fiducial.. ll Background fraction... ALAS 8 ev,. fb µµ : y -integrated otal Multijet o+ew Non-fiducial.. ll Background fraction ALAS 8 ev,. fb ee CF : y -integrated otal Multijet Non-fiducial ll Figure : Fractional background contributions as a function of ll, for the ee (to), µµ (middle) and ee CF (bottom) channels. he distributions are shown searately for the relevant background contributions to each channel together with the summed total background fraction. he label Non-fiducial refers to signal events which are generated outside the hase sace used to extract the angular coefficients (see text).

15 to+electroweak backgrounds. he sum of all these backgrounds is also shown and temlates of their angular distributions are used in the fit to extract the angular coefficients, as described in Section... Angular distributions he measurement of the angular coefficients is erformed in fine bins of and for a fixed dileton mass window on the same samle as that used to extract from data the small corrections alied to the leton efficiencies and calibration. he analysis is thus largely insensitive to the shae of the distribution of, and also to any residual differences in the modelling of the shae of the dileton mass distribution. It is, however, imortant to verify qualitatively the level of agreement between data and MC simulation for the cos θ and angular distributions before extracting the results of the measurement. his is shown for the three channels searately in Fig. 6, together with the ratio of the observed data to the sum of redicted events. he data and MC distributions are not normalised to each other, resulting in normalisation differences at the level of a few ercent. he measurement of the angular coefficients is, however, indeendent of the normalisation between data and simulation in each bin of. he differences in shae in the angular distributions reflect the mismodelling of the angular coefficients in the simulation (see Section 7).. Coefficient measurement methodology he coefficients are extracted from the data by fitting temlates of the P i olynomial terms, defined in Eq. (), to the reconstructed angular distributions. Each temlate is normalised by free arameters for its corresonding coefficient A i, as well as an additional common arameter reresenting the unolarised cross-section. All of these arameters are defined indeendently in each bin of. he olynomial P 8 = + cos θ in Eq. () is only normalised by the arameter for the unolarised cross-section. In the absence of selections for the final-state letons, the angular distributions in the gauge-boson rest frame are determined by the gauge-boson olarisation. In the resence of selection criteria for the letons, the distributions are sculted by kinematic effects, and can no longer be described by the sum of the nine P i olynomials as in Eq. (). emlates of the P i terms are constructed in a way to account for this, which requires fully simulated signal MC to model the accetance, efficiency, and migration of events. his rocess is described in Section.. Section. then describes the likelihood that is built out of the temlates and maximised to obtain the measured coefficients. he methodology for obtaining uncertainties in the measured arameters is also covered there. he rocedure for combining multile channels is covered in Section., along with alternative coefficient arameterisations used in various tests of measurement results from different channels... emlates o build the temlates of the P i olynomials, the reference coefficients A ref i for the signal MC samle are first calculated with the moments method, as described in Section and Eq. (). hese are obtained in each of the bins in Eq. (6), and also in each of the three y bins for the y -binned measurements. he information about the angular coefficients in the simulation is then available through the corresonding functional form of Eq. (). Next, the MC event weights are divided by the value of this function

16 Entries / ALAS Preliminary 8 ev,. fb ee Prediction Entries / ALAS Preliminary 8 ev,. fb ee Prediction /Pred. Entries / cos θ ALAS Preliminary 8 ev,. fb µµ Prediction /Pred. Entries / ALAS Preliminary 8 ev,. fb µµ Prediction /Pred. Entries /.. cos θ cos θ 8 6 ALAS Preliminary 8 ev,. fb ee CF Prediction /Pred. Entries / ALAS Preliminary 8 ev,. fb ee CF Prediction /Pred.. cos θ cos θ /Pred Figure 6: he cos θ (left) and (right) angular distributions, averaged over all ll, for the ee (to), µµ (middle) and ee CF (bottom) channels. In the anels showing the ratios of the data to the summed signal+background redictions, the uncertainty bars on the oints are only statistical. 6

17 on an event-by-event basis. When the MC events are weighted in this way, the angular distributions in the full hase sace at the event generator level are flat. Effectively, all information about the -boson olarisation is removed from the MC samle, so that further weighting the events by any of the P i terms yields the shae of the olynomial itself, and if selection requirements are alied, this yields the shae of the selection efficiency. he selection requirements, corrections, and event weights mentioned in Section are then alied. Nine searate temlate histograms for each and y bin j at generator level are finally obtained after weighting by each of the P i terms. he temlates t i j are thus three-dimensional distributions in the measured cos θ,, and ll variables, and are constructed for each and y bin. Eight bins in cos θ and are used, while the binning for the reconstructed ll is the same as for the bins defined in Eq. (6). By construction, the sum of all signal temlates normalised by their reference coefficients and unolarised cross-sections agrees exactly with the three-dimensional reconstructed distribution exected for signal MC events. Examles of temlates rojected onto each of the dimensions cos θ and for the y -integrated ee channel in three illustrative ranges, along with their corresonding olynomial shaes, are shown in Fig. 7. he olynomials P and P 6 are not shown as they integrate to zero in the full hase sace in either rojection (see Section.). he effect of the accetance on the olynomial shae deends on because of the event selection, as can be seen from the difference between the temlate olynomial shaes in each corresonding bin. his is articularly visible in the P 8 olynomial, which is uniform in, and therefore reflects exactly the accetance shae in the temlated olynomials. In Aendix B, two-dimensional versions of Fig. 7 are given for all nine olynomials in Figs.. hese two-dimensional views are required for P and P 6, as discussed above. emlates B are also built for each of the multijet, to+electroweak, and non-fiducial -boson backgrounds discussed in Section.. hese are normalised by their resective cross-sections times luminosity, or data-driven estimates in the case of the multijet background. he temlates for the rojection measurements in the ee CF channel are integrated over either the cos θ or axis at the end of the rocess. emlates corresonding to variations of the systematic uncertainties in the detector resonse as well as in the theoretical modelling are built in the same way, after varying the relevant source of systematic uncertainty by ± standard deviation (σ). If such a variation changes the A ref i coefficients in the MC rediction, for examle in the case of PDF or arton shower uncertainties, the varied A ref i coefficients are used as such in the weighting rocedure. In this way, the theoretical uncertainties on the redictions are not directly roagated to the uncertainties on the measured A i coefficients. However, they may affect indirectly the measurements through their imact on the accetance, selection efficiency, and migration modelling... Likelihood A likelihood is built from the nominal temlates and the varied temlates reflecting the systematic uncertainties. A set of nuisance arameters (NPs) θ = {β, γ} is used to interolate between them. hese are constrained by auxiliary robability density functions and come in two categories: β and γ. he first category β are the NPs reresenting exerimental and theoretical uncertainties. Each β m in the set β = { β,..., β M} are constrained by unit Gaussian robability density functions G( β m, ) and linearly interolate between the nominal and varied temlates. hese are defined to have a nominal value of zero, with β m = ± corresonding to ±σ for the systematic uncertainty under consideration. he total number of β m is M = 7 for the ee + µµ channel and M = for the ee CF channel. he second category γ are NPs that handle systematic uncertainties from the limited size of the MC samles. For 7

18 Polynomial value ALAS P 8 P P Polynomial value 8 ALAS P P 7 P P P cos θ 6 emlate value /. ALAS Simulation emlated P. 8 s = 8 ev emlated P emlated P ee : y -integrated = -8 GeV.8.6. emlate value / ALAS Simulation s = 8 ev ee : y -integrated = -8 GeV emlated P emlated P emlated P emlated P emlated P cos θ emlate value / ALAS Simulation s = 8 ev ee : y -integrated = -. GeV emlated P emlated P emlated P cos θ emlate value / ALAS Simulation s = 8 ev ee : y -integrated = -. GeV emlated P emlated P emlated P emlated P emlated P emlate value /. ALAS Simulation emlated P. 8 s = 8 ev emlated P emlated P ee : y -integrated = 7 GeV cos θ emlate value / ALAS Simulation emlated P 8 s = 8 ev emlated P 7 ee emlated P : y -integrated emlated P = 7 GeV emlated P 6 Figure 7: Shaes of olynomials P,,8 as a function of cos θ (to left) and P,,,7,8 as a function of (to right). Shown below are the temlated olynomials for the y -integrated ee events at low ( 8 GeV), medium (. GeV), and high ( 7 GeV) values of rojected onto each of the dimensions cos θ and. he ll dimension that normally enters through migrations is also integrated over. he differences between the olynomials and the temlates reflect the accetance shae after event selection. 8

19 each bin n in the reconstructed cos θ,, and ll distribution, γn in the set γ = { γ,..., γ bins} N, where N bins = 8 8 is the total number of bins in the reconstructed distribution, has a nominal value of one and normalises the exected events in bin n of the temlates. hey are constrained by Poisson robability density functions P(Neff n γn Neff n ), where Nn eff is the effective number of MC events in bin n. he meaning of effective here refers to corrections alied for non-uniform event weights. When all signal and background temlates are summed over with their resective normalisations, the exected events Nex n in each bin n can be written as: where: Nex(A, n σ, θ) = j= A i j : Coefficient arameter for bin j A: Set of all A i j σ j : Signal cross-section arameter σ: Set of all σ j θ: Set of all NPs β: Set of all Gaussian-constrained NPs γ n : Poisson-constrained NP t i j : P i temlate B : Background temlates L: Integrated luminosity constant. 7 bkgs σ j L tn 8 j (β) + A i j ti n j (β) + B n (β) γn, (7) he summation over the index j takes into account the contribution of all bins at generator level in each reconstructed ll bin. his is necessary to account for migrations in ll. he likelihood is the roduct of Poisson robabilities across all N bins bins and of auxiliary constraints for each nuisance arameter β m : i= B N bins { L(A, σ, θ N obs ) = P(N n obs Nex(A, n σ, θ))p(neff n γn Neff n )} n M G( β m, ). (8) m Unlike in the ee and µµ channels that use both angular variables simultaneously, the ee CF measurements are erformed in rojections (see Eq. () and Eq. ()), and therefore the A and A 6 coefficients are not measured in this channel. he P i olynomials that normally integrate to zero when rojecting onto one angular variable in full hase sace may, however, not integrate to zero if their shae is distorted by the event selection. he residual shae is not sufficient to roerly constrain their corresonding A i, and therefore an external constraint is alied to them. For the A i that are largely indeendent of y (A and A ), the constraints are taken from the indeendent y -integrated measurements in the combined ee + µµ channel. For the y -deendent coefficients A, A, and A, which are inaccessible to the ee + µµ channels in the y bin in which ee CF is used, redictions from Powheg + MiNLO [] are used. 9

20 he migration of events between ll bins leads to anti-correlations between A i in neighbouring bins which enhance the effects of statistical fluctuations. o mitigate this effect and aid in resolving underlying structure in the A i sectra, the A i sectra are regularised by multilying the unregularised likelihood by a Gaussian enalty term, which is a function of the significance of higher-order derivatives of the A i with resect to. he covariance terms between the A i j coefficients are taken into account and comuted first with the unregularised likelihood. his has arallels with, for examle, regularised Bayesian unfolding, where additional information is added through the rior robability of unfolded arameter values [, ]. As is the case there, the choice of enalty term (or in the Bayesian case, the choice of added information) must be one that leads to a sound result with minimal bias. See Aendix C for more details. he uncertainties in the arameters are obtained through a likelihood scan. For each arameter of interest A i j, a likelihood ratio is constructed as Λ(A i j ) = L(A i j, Â(A i j ), ˆθ(A i j )). (9) L(Â, ˆθ) In the denominator, the likelihood is maximised unconditionally across all arameters of interest and NPs. In the numerator, the likelihood is maximised for a secific value of a single A i j. he maximum likelihood estimators for the other arameters of interest  and NPs ˆθ are in general a function of A i j, hence the exlicit deendence is shown in the numerator. he Minuit ackage is used to erform numerical minimisation [6] of log Λ(A i j ), and a two-sided test statistic is built from the likelihood ratio: q Ai j = log Λ(A i j ). () his is asymtotically distributed as a χ with one degree of freedom [7]. In this case, the ±σ confidence interval of A i j is defined by the condition q A ± i j =, where A ± i j  i j ± σ ±... Combinations and alternative arameterisations When alicable, multile channels are combined through a simle likelihood multilication. Each likelihood can be decomosed into three tyes of terms: those that contain the observed data in each channel, denoted L i (A, σ, θ), the auxiliary terms that constrain the nuisance arameters θ, denoted A i (θ i ), and the auxiliary term that imoses the regularisation, A reg (A). here are a total of M cb NPs, corresonding to the total number of unique NPs, including the total number of bins, across all combined channels. With this notation the combined likelihood can be written as: L cb (A, σ, θ) = channels i M cb L i (A, σ, θ) A i (θ i ) A reg(a). () here are several instances in which a combination of two channels is erformed. Within these combinations, the comatibility of the channels is assessed. he measurements in the first two y bins and the y -integrated configuration are obtained from a combination of the ee and µµ channels. he y - integrated µµ and ee CF channels are also combined in order to assess the comatibility of the high y region robed by the ee CF channel and the lower raidity region robed by the central central channels. i

21 he comatibility of channels is assessed through a rearameterisation of the likelihood into arameters that reresent the difference between the coefficients in two different channels. For coefficients A a i j and Ab i j in resective channels a and b, difference arameters A i j A a i j Ab i j are defined that effectively reresent the difference between the measured coefficients in the two channels. Substitutions are made in the form of A a i j A i j + A b i j. hese new arameters are measured with the same methodology as described in Section.. Similar rearameterisations are also done to measure the difference between the A and A coefficients. hese rearameterisations have the advantage that the correlations between the new arameters are automatically taken into account. 6. Measurement uncertainties Several sources of statistical and systematic uncertainty lay a role in the recision of the measurements resented in this aer. In articular, some of the systematic uncertainties imact the temlate-building rocedure described in Section.. For this reason, temlates are rebuilt after each variation accounting for a systematic uncertainty, and the difference in shae between the varied and nominal temlates is used to evaluate the resulting uncertainty. A descrition of the exected statistical uncertainties (both in data in Section 6. and in simulation in Section 6.) and systematic uncertainties (exerimental in Section 6., theoretical in Section 6., and those related to the methodology in Section 6.) associated with the measurement of the A i coefficients is given in this section. hese uncertainties are summarised in Section 6.6 in three illustrative bins for the ee, µµ (and their combination), and ee CF channels. he evolution of the uncertainty breakdown as a function of is illustrated there as well. 6.. Uncertainties from data samle size Although the harmonic olynomials are comletely orthogonal in the full hase sace, resolution and accetance effects lead to some non-zero correlation between them. Furthermore, the angular distributions in a bin of reconstructed ll have contributions sanning several generator-level bins. his leads to correlations between the measured coefficients which increase their statistical uncertainties. he amount of available data is the largest source of uncertainty, although the resolution and binning in the angular variables also lay a role. A discussion of the categorisation of this uncertainty may be found in Aendix D. 6.. Uncertainties from Monte Carlo samle size Statistical uncertainties from the simulated MC samles are treated as uncorrelated between each bin of the three-dimensional ( ll, cos θ, ) distribution. Although the events used to build each temlate are the same, they receive a different weight from the different olynomials, and are therefore only artially correlated. It was verified that assuming that the temlates are fully correlated yields slightly more conservative uncertainties, but central values identical to those obtained using the fully correct treatment. For simlicity, this assumtion is used for this uncertainty.