Pu b l i s h e d f o r SISSA b y Sp r i n g e r R e c e i v e d: August 8, 2016
R e v i s e d: October 4, 2016
A c c e p t e d: November 10, 2016
P u b l i s h e d: November 21, 2016
Study of hard double-parton scattering in four-jet events in pp collisions at = 7 TeV with the ATLAS experiment
T h e A T LA S collaboration
E -m a il: a t l a s . p u b l i c a t i o n s @ c e r n . c h
A b s t r a c t : Inclusive four-jet events produced in proton -proton collisions at a centre-of- mass energy o f y fs = 7 T eV are analysed for the presence o f hard double-parton scatter
ing using data corresponding to an integrated lum inosity o f 37.3 p b - 1 , collected with the A T L A S detector at the L H C . T he contribution o f hard double-parton scattering to the produ ction o f four-jet events is extracted using an artificial neural network, assuming that hard double-parton scattering can be approxim ated by an uncorrelated overlaying o f dijet events. For events containing at least four jets with transverse m om entum p T > 20 G eV and pseudorapidity |n| < 4.4, and at least one having p T > 42.5 G eV , the contribution o f hard double-parton scattering is estim ated to be /d p s = 0.092 +0 o?1 (stat.) +°° o00 (syst.). After com bining this measurement with those o f the inclusive dijet and four-jet cross-sections in the appropriate phase space regions, the effective cross-section, aeff, was determ ined to be creff = 14.9 +1 q (stat.) + 51 (syst.) mb. This result is consistent within the quoted uncer
tainties with previous measurements o f aeff, perform ed at centre-of-m ass energies between 63 G eV and 8 TeV using various final states, and it corresponds to 21+0% o f the total in
elastic cross-section measured at yfs = 7 TeV . T he distributions o f the observables sensitive to the contribution o f hard double-parton scattering, corrected for detector effects, are also provided.
Ke y w o r d s: H adron-H adron scattering (experim ents)
ArXiy ePr i n t: 1608.01857
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Contents
1 Introduction 1
2 Analysis strategy 3
3 The A T L A S detector 5
4 M onte Carlo simulation 5
5 Cross-section measurements 6
5.1 D ata set and event selection 6
5.2 C orrection for detector effects 7
6 Determination of the fraction of D P S events 8
6.1 Tem plate samples 9
6.2 K inem atic characteristics o f event classes 10
6.3 E xtraction o f the fraction o f D P S events using an artificial neural network 13
6.4 M eth odology validation 14
7 Systematic uncertainties 16
8 Determination of o ff 17
9 Normalized differential cross-sections 21
10 Summary and conclusions 22
A Normalized differential cross-sections 25
The A T L A S collaboration 35
1 Introduction
Interactions involving more than one pair o f incident partons in the same collision have been discussed on theoretical grounds since the introduction o f the parton m odel to the description o f particle produ ction in hadron-hadron collisions [1- 3]. These first studies were followed by the generalization o f the A ltarelli-Parisi evolution equations to the case o f m ulti-parton states in refs. [4 , 5] and a discussion o f possible correlations in the colour and spin degrees o f freedom o f the incident partons [6]. In the first phenom enological studies o f such effects, the m ost prominent role was played by processes known as double-parton scattering (D P S ), which is the simplest case o f m ulti-parton interactions (M P I), leading to
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final states such as four leptons, four jets, three jets plus a photon, or a leptonically decaying gauge boson accom panied by tw o jets [7- 15] . These studies have been supplemented by experim ental measurements o f D P S effects in hadron collisions at different centre-of-m ass energies, which now range over tw o orders o f magnitude, from 63 GeV to 8 TeV [16- 30], and which have firmly established the existence o f this mechanism. T he abundance o f M P I phenom ena at the LH C and their im portance for the full picture o f hadronic collisions have reignited the phenom enological interest in D P S and have led to a deepening o f its theoretical understanding [31- 39]. D espite this progress, quantitative measurements o f the effect o f D PS on distributions o f observables sensitive to it are affected by large system atic uncertainties. This is a clear indication o f the experim ental challenges and o f the com plexity o f the analysis related to such measurements. Therefore, the cross-section o f D PS continues to be estimated by ignoring the likely existence o f com plicated correlation effects. For a process in which a final state A + B is produced at a hadronic centre-of-m ass energy yfs, the simplified formalism o f refs. [12, 13] yields
d ^ ’ w = d *A (s)d ? w . (1 .1)
A+ B W 1 + foB Off (s) V '
The quantity <Ab is the Kronecker delta used to construct a sym m etry factor such that for identical final states with identical phase space, the D P S cross-section is divided by two.
The O ff, usually referred to as the effective cross-section, is a purely phenom enological param eter describing the effective overlap o f the spatial distribution o f partons in the plane perpendicular to the direction o f m otion. In hadronic collisions it was typically found to range between 10 and 25 m b [16- 30]. In eq. ( 1.1) , the various O are the parton-level cross
sections, either for the D PS events, indicated by the subscript A + B, or for the production o f a final state A or B in a single parton scatter (SP S), given by
dOA w = 2 s T d x i d x2 f i ( x1, ^ f ) f j(X2, ) d$A \ M i j ^ A ( x i X
2
S, ^F, ^ r) \ 2 . (1 .2) ijHere the functions f i (x, ^ F) are the single parton distribution functions (P D F s) which at leading order param eterize the probability o f finding a parton i at a m om entum fraction x at a given factorization scale ^ f in the incident hadron; d $ A is the invariant differential phase-space element for the final state A; M is the perturbative m atrix element for the process i j ^ A; and ^ R is the renorm alization scale at which the couplings are evaluated.
To constrain the phase space to that allowed by the energy o f each incom ing proton, a simple tw o-parton P D F is defined as
f i j ( b , X i , X j, ^ f ) = r (b) fi( x i,^ F ) f j (x j , ^ f ) © ( 1 - Xi - X j ) , (1.3)
where 0 ( x ) is the Heaviside step function, r (b) the area overlap function, and the x and scale dependence o f the P D F are assumed to be independent o f the im pact param eter b. Eq. ( 1.3) reflects the om ission o f correlations between the partons in the proton. At high energy, eq. ( 1.1) can be derived using eq. ( 1.3) by neglecting the contribution o f the step function.
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Typically, the main challenge in measurements of D PS is to determine if the A + B final state was produced in an SPS via the 2 ^ 4 process or in DPS through two inde
pendent 2 ^ 2 interactions. In one of the first studies of DPS in four-jet production at hadron colliders [10] the kinematic configuration in which there is a pairwise balance of the transverse momenta (pT) o f the jets was identified as increasing the contribution of the DPS mechanism relative to the perturbative QCD production of four jets in SPS. The idea is that in typical 2 ^ 2 scattering processes the two outgoing particles — here the partons identified as jets — are oriented back-to-back in transverse plane such that their net trans
verse momentum is zero. Corrections to this simple picture include initial- and final-state radiation as well as fragmentation and hadronization. In addition, recoil against the under
lying event can modify the four-momentum of the overall final-state particle configuration.
In attempting to describe all o f these features, Monte Carlo (M C) event generators form an integral part, providing a link between the experimentally observed jets and the simple partonic picture of DPS as two almost independent 2 ^ 2 scatters.
An analysis of inclusive four-jet events produced in proton-proton collisions at a centre- of-mass energy of yfs = 7 TeV at the LHC and collected during 2010 with the ATLAS detector is presented here. The topology of the four jets is exploited to construct observ
ables sensitive to the DPS contribution. The DPS contribution to the four-jet final state is estimated and combined with the measured inclusive dijet and four-jet cross-sections in the appropriate phase space regions to determine neff. The normalized differential four-jet cross
sections as a function o f DPS-sensitive observables are measured and presented here as well.
2 Analysis strategy
To extract aeff in the four-jet final state, eq. ( 1.1) is rearranged as follows. T he differential cross-sections in eq. (1 .1 ) are rewritten for the four-jet and dijet final states and integrated over the phase space defined b y the selection requirements o f the dijet phase space regions A and B. This yields the following expression for the D P S cross-section in the four-jet final state:
1 ^ A^ B
„DPS 1 a 2ja 2j 1 n,
a 4j ---“ ---- > (2 .1)
1 + CAB Offf
where n^j and n^j are the cross-sections for dijet events in the phase space regions labelled A and B respectively. The assumed dependence of the cross-sections and neff on s is omitted for simplicity. The DPS cross-section may be expressed as
n4jPS = /dps ■ n4j, (2.2)
where n4j is the inclusive cross-section for four-jet events in the phase-space region A ® B, including all four-jet final states, namely both the SPS and DPS topologies, and where f DPS represents the fraction of DPS events in these four-jet final states. The expression for neff then becomes,
n = 1 1
ne" = 1 + &B /dps n4j ■ (2'3)
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To extract neff, it is therefore necessary to measure three cross-sections, n^j, n^j and n4j, and estimate / dps .
The four-jet and dijet final states are defined inclusively [40, 41] such that at least four jets or tw o jets respectively are required in the event, while no restrictions are applied to additional jets. W h en measuring the cross-section o f n-jet events, the leading (highest-pT) n jets in the event are considered. T he general expression for the measured four-jet and dijet cross-sections may be w ritten as
nnj = ^ , (2.4)
Cnj L nj
where the subscript nj denotes either dijet (2j) or four-jet (4j) topologies. For each nj channel, Nnj is the number o f observed events, Cnj is the correction for detector effects, particularly due to the jet energy scale and resolution, and L nj is the corresponding proton- proton integrated luminosity.
The D PS m odel contributes in tw o ways to the produ ction o f events with at least four jets, leading to tw o separate event classifications. In one contribution, the secondary scatter produces tw o o f the four leading jets in the event; such events are classified as com plete-D P S (cD P S ). In the second contribution o f D P S to four-jet production, three of the four leading jets are produced in the hardest scatter, and the fourth jet is produced in the secondary scatter; such events are classified as sem i-DPS (sD P S ). T he D P S fraction is therefore rewritten as /dps = f cDPS + f sDPS, and f cDPS and f sDPS are both determined from data. T he dijet cross-sections in eq. ( 2.3) do not require any m odification since they are all inclusive cross-sections, i.e., the three-jet cross-section accounting for the production o f an sDPS event is already included in the dijet cross-sections.
D enoting the observed cross-section at the detector level by
Snj = N nj , (2.5)
L nj
and the ratio o f the corrections for detector effects by
“2j = J j i , (2 .6 )
C2j C2j
yields the expression from which neff is determined,
1 a j
neff = T T ^ /cDPS + fsDPS S j - <27)
The main challenge o f the measurement is the extraction o f /dps = f cDPS + f sDPS from optim ally selected measured observables. A n artificial neural network (N N) is used for the classification o f events [42], using as input various observables sensitive to the contribution o f D PS. T he differential distributions o f these observables are also presented here.
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3 The AT L A S detector
T he A T L A S detector is described in detail in ref. [43]. In this analysis, the tracking detectors are used to define candidate collision events by constructing vertices from tracks, and the calorimeters are used to reconstruct jets.
T he inner detector used for tracking and particle identification has com plete azimuthal coverage and spans the pseudorapidity region \n\ < 2.5.1 It consists o f layers o f silicon pixel detectors, silicon m icrostrip detectors, and transition-radiation tracking detectors, surrounded by a solenoid magnet that provides a uniform axial field o f 2 T.
T he electrom agnetic calorim etry is provided by the liquid argon (L A r) calorimeters that are split into three regions: the barrel (\n\ < 1.475) and the endcap (1.375 < \n\ < 3.2) regions which consist L A r /P b calorim eter m odules, and the forward (FCal: 3.1 < \ n\ < 4.9) region which utilizes L A r /C u m odules. T he hadronic calorim eter is divided into four distinct regions: the barrel ( \n \ < 0.8), the extended barrel (0.8 < \ n \ < 1.7), b oth o f which are scintillator/steel sampling calorimeters, the hadronic endcap (1.5 < \ n \ < 3.2), which has L A r /C u calorim eter m odules, and the hadronic FCal (same n-range as for the E M -F C al) which uses L A r /W m odules. T he calorim eter covers the range \ n \ < 4.9.
T he trigger system for the A T L A S detector consists o f a hardware-based level-1 trigger (L1) and the software-based high-level trigger (H LT) [44]. Jets are first identified at L1 using a sliding-window algorithm from coarse granularity calorim eter towers. This is refined using jets reconstructed from calorim eter cells in the HLT. Three different triggers are used to select events for this measurement: the m inim um -bias trigger scintillators, the central jet trigger ( \ n \ < 3.2) and the forward jet trigger (3.1 < \ n \ < 4.9). T he jet triggers require
at least one jet in the event.
4 Monte Carlo simulation
M ulti-jet events were generated using fixed-order Q C D m atrix elements (2 ^ n, with n = 2 ,3 ,4 , 5, 6) with A l p g e n 2.14 [45] utilizing the C TE Q 6L1 P D F set [46], interfaced to Jimmy [47] and H e r w ig 6.520 [48]. T he events were generated using the A U E T 2 [49]
set o f parameters (tune), optim ized to describe underlying-event distributions obtained from a subsample o f the 2010, 7 TeV A T L A S data as well as from the Tevatron and LEP experiments. T he M LM [50] m atching scale, which divides the parton emission phase space into regions m odelled either by the perturbative m atrix-element calculation or by the shower resummation, was set to 15 GeV. T he im plication o f this choice is that partons with p t > 15 GeV in the final state originate from m atrix elements, and not from the parton shower. Event-record inform ation was used to extract a sample o f SPS candidate
"ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r, 0) are used in the transverse plane, 0 being the azimuthal angle around the beam pipe, referred to the x-axis. The pseudorapidity is defined in terms of the polar angle 6 with respect to the beamline as n = — lntan(6/2). When dealing with massive jets and particles, the rapidity y = 2 ln ^ EE+P* ) is used, where E is the jet energy and pz is the z-component of the jet momentum.
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events from the sample generated with the A L p g e n + H e r w ig + Jimmy M C com bination (A H J ). A sample o f candidate D PS events was also extracted from A H J in order to study the top olog y o f such events and validate the measurement m ethodology.
A n additional A H J sample was available that differed only in its use o f the earlier A U E T 1 [51] tune. Because this sample contained three times as many events, it was used to derive the corrections for detector effects in all differential distributions in the data.
Tree-level m atrix elements with up to five outgoing partons were used to generate a sample o f m ulti-jet events w ithout m ulti-parton interactions using S h e r p a 1.4.2 [52, 53]
with the C T 10 P D F set [54] and the default S h e r p a tune. T he C K K W [55, 56] m atching scale, similarly to the M LM one, was set to 15 GeV. This SPS sample was com pared to the SPS sample extracted from the A H J sample for validation purposes.
In addition, a sample o f multi-jet events was generated with P y t h i a 6.425 [57] using a 2 ^ 2 matrix element at leading order with additional radiation m odelled in the leading- logarithm ic approxim ation by pT-ordered parton showers. T he sample was generated uti
lizing the m odified leading-order P D F set M R S T LO * [58] with the A M B T 1 [59] tune.
To account for the effects o f multiple proton -proton interactions in the LH C (pile- up), the multi-jet events were overlaid with inelastic soft Q C D events generated with P y t h i a 6.423 using the M R S T L O * P D F set with the A M B T 1 tune. All the events were processed through the A T L A S detector simulation framework [60], which is based on G e a n t 4 [61] . T h ey were then reconstructed and analysed by the same program chain used for the data.
5 Cross-section measurements
5.1 D ata set and event selection
The measurement presented here is based on the full A T L A S 2010 data sample from proton- proton collisions at a/s = 7 TeV. T he trigger conditions evolved during the year with changing thresholds and prescales. A full description o f the trigger strategy, developed and used for the measurement o f the dijet cross-section using 2010 data, is given in ref. [62] . For the events in the samples used in this study, the trigger was fully efficient. In total, the data used correspond to a lum inosity o f 37.3 p b - 1 , with a system atic uncertainty o f 3.5% [63].
This data set was chosen because it has a low number o f proton -proton interactions per bunch crossing, averaging to approxim ately 0.4. It was therefore possible to collect m ulti-jet events with low pT thresholds and to efficiently select events w ith exactly one reconstructed vertex (single-vertex events), thereby rem oving any contribution from pile-up collisions to the four-jet final-state topologies.
To reject events initiated by cosm ic-ray muons and other non-collision backgrounds, events were required to have at least one reconstructed prim ary vertex, defined as a vertex that is consistent with the beam spot and is associated with at least five tracks with transverse m om entum pTack > 150 MeV. T he efficiency for collision events to pass these requirements was over 99%, while the contribution from fake vertices was negligible [62, 64].
Jets were identified using the anti-kt jet algorithm [65] , im plem ented in the F a s t - J e t [66] package, with radius parameter R = 0.6. The inputs to jet reconstruction are
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the energies in three-dim ensional topological clusters [67, 68] built from calorim eter cells, calibrated at the electrom agnetic (E M ) scale.2 A je t energy calibration was subsequently applied at the jet level, relating the jet energy measured with the A T L A S calorim eter to the true energy o f the stable particles entering the detector. A full description o f the jet energy calibration is given in ref. [64]. For the M C samples, particle jets were built from particles with a lifetime longer than 30 ps in the M onte Carlo event record, excluding muons and neutrinos.
For the purpose o f measuring 0eff in the four-jet final state, three samples o f events were selected, tw o dijet samples and one four-jet sample. T he form er tw o samples have at least two, and the latter at least four, jets in the final state, where each jet was required to have pT > 20 GeV and \ n \ < 4.4. In each event, jets were sorted in decreasing order o f their transverse m om enta. T he transverse m om entum o f the i th jet is denoted by pT and the jet with the highest pT (pT) is referred to as the leading jet. T o ensure 100% trigger efficiency, the leading jet in four-jet events was required to have pT > 42.5 GeV.
The selection requirements for the dijet samples were dictated by those used to select four-jet events. In one class o f dijet events, the requirement on the transverse m om entum of the leading jet must be equivalent to the requirement on the leading jet in four-jet events, pT > 42.5 GeV. T he other type o f dijet event corresponds to the sub-leading pair o f jets in the four-jet event, with a requirement o f pT > 20 GeV. In the following, the cross-section for dijets selected with pT > 20 GeV is denoted by and the cross-section for dijets with pT > 42.5 GeV is denoted by 0^ .
T o summarize, the measurement was perform ed using the dijet A sample and its two subsamples (dijet B and four-jet), selected using the following requirements:
D ijet A D ijet B Four-jet
Njet > 2 , pT > 20 GeV , pT > 20 GeV , \ n1)2 \ < 4.4 ,
Njet > 2 , pT > 42.5 GeV , pT > 20 GeV , \ n1)2\ < 4.4 , (5.1) Njet > 4 , pT > 42.5 GeV , pT’3,4 > 20 GeV , \ ^ ,2,3,4 \ < 4.4 ,
where Njet denotes the number o f reconstructed jets. All o f the selected events were cor
rected for jet reconstruction and trigger inefficiencies, the corrections ranging from 2 % -4 % for low-pT jets to less than 1% for jets with pT > 60 GeV. T he observed distributions o f the p T and y o f the four leading jets in the events are shown in figures 1(a) and 1(b) respectively.
5.2 Correction for detector effects
T he correction for detector effects was estim ated separately for each class o f events using the Py t h i a6 M C sample. T he same restrictions on the phase space o f reconstructed jets, defined in eq. ( 5.1) , were applied to particle jets. T he correction is given by
NA,B reco
p A>B = nj_______ (5 2)
nj NA,B particle , ( . )
nj
2The electromagnetic scale is the basic calorimeter signal scale to which the ATLAS calorimeters are calibrated. It was established using test-beam measurements for electrons and muons to give the correct response for the energy deposited by electromagnetic showers, while it does not correct for the lower response to hadrons.
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Figure 1. Distributions of the (a) transverse momentum, px, and (b) rapidity, y, of the four highest-pT jets, denoted by pT 2,3,4 and yi , 2,3,4, in four-jet events in data selected in the phase space as defined in the legend.
where N,Aj’B reco ( yA’B partlcle) is the number of n-jet events passing the A-or-B selection requirements using reconstructed (particle) jets.
This correction is sensitive to the migration o f events into and out of the phase space o f the measurement. Due to the very steep jet-pT spectrum in dijet and four-jet events, it is crucial to have good agreement between the jet pT spectra in data and in MC simulation close to the selection threshold before calculating the correction. Therefore, the jet pT threshold was lowered to 10 GeV and the fiducial |n| range was increased to 4.5 for both the reconstructed and particle jets, and the MC events were reweighted such that the jet pT- y distributions reproduced those measured in data. The value of a^j (see eq. (2.6) ), as determined from the reweighted MC events, is
a4j = 0.93 ± 0.01 (stat.) , (5.3) where the uncertainty is statistical. The systematic uncertainties are discussed in section 7.
6 Determination of the fraction of DPS events
The main challenge in the measurement of neff is to estimate the DPS contribution to the four-jet data sample. It is impossible to extract cDPS and sDPS candidate events on an event-by-event basis. Therefore, the usual approach adopted is to fit the distributions of variables sensitive to cDPS and sDPS in the data to a combination of templates for the expected SPS, cDPS and sDPS contributions. The template for the SPS contribution is extracted from the AH J MC sample, while the cDPS and sDPS templates are obtained by overlaying two events from the data. In addition to assuming that the two interactions
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producing the four-jet final state in a D PS event are kinematically decoupled, the analysis relies on the assum ption that the SPS tem plate from A H J properly describes the expected top olog y o f four-jet produ ction in a single interaction. T he latter assum ption is supported by the observation o f good agreement between various distributions in the SPS samples in A H J and in Sh e r p a. T o exploit the full spectrum o f variables sensitive to the various contributions and their correlations, the classification was perform ed with an artificial neural network.
6.1 Template samples
Differences were observed when com paring the pT and y distributions in data with those in A H J . Therefore, before extracting tem plate samples, the events in the four-jet A H J sample selected with the requirements detailed in eq. ( 5.1) are reweighted such that they reproduce the distributions in data.
In events generated in A H J , the outgoing partons can be assigned to the primary interaction from the Al p g e n generator or to a secondary interaction, generated by Ji m m y, based on the M C generator’s event record. T he form er are referred to as prim ary-scatter partons and the latter as secondary-scatter partons. T he pT o f secondary-scatter partons was required to be p t > 15 GeV in order to match the minimum p t o f prim ary-scatter partons set by the M L M matching scale in A H J . Once the outgoing partons were classified, the jets in the event were m atched to outgoing partons and the event was classified as an SPS, cD P S or sDPS event.
The matching o f jets to partons is done in the 0 - y plane by calculating the angular distance, A R parton_ jet, between the jet and the outgoing parton as
A R parton_jet — y (yparton yjet)2 + (^parton ^jet)2 . (6 .1)
For 99% o f the prim ary-scatter partons, the parton can be m atched to a jet within A R parton_ jet < 1 .0 , which was therefore used as a requirement for the matching o f jets and partons. Jets were first m atched to prim ary-scatter partons and the remaining jets were m atched to secondary-scatter partons.
Events in which none o f the leading four jets match a secondary-scatter parton were assigned to the SPS sample. This m ethod o f obtaining an SPS sample is preferred over turning o ff the M P I m odule in the generator since it retains all o f the soft M P I and underlying activity in the selected SPS events. Events were classified as cD P S events if two o f the four leading jets match prim ary-scatter partons and the other tw o match secondary- scatter partons. Events in which three o f the leading jets match prim ary-scatter partons and the fourth jet matches a secondary-scatter parton were classified as sDPS events.
Four-jet D PS events were m odelled by overlaying tw o different events. T o reduce any dependence o f the measurement on the m odelling o f je t production, this construction used events from data rather than M C simulation. C om plete-D P S events were built using dijet events from the A and B samples selected from data (see eq. ( 5.1) ). T o build sDPS events,
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tw o other samples were selected with the following requirements:
One-jet: Njet > 1 , pT > 20 GeV , \n1\ < 4.4 ,
2 3 (6 .2)
Three-jet: Njet > 3 , pT > 42.5 GeV , pT’ > 20 GeV , \n1,2’3\ < 4.4 .
The overlay was perform ed at the reconstructed jet level. W hen constructing cD P S and sDPS events the following conditions were im posed for a given pair o f events to be overlaid:
• none o f the four jets contains the axis o f one o f the other jets, i.e., A R j et_ jet > 0.6;
• the vertices o f the tw o overlaid events are no more than 10 mm apart in the z direction;
• when building cD P S events, each o f the overlaid events contributes tw o jets to the four leading jets in the constructed event;
• when building sDPS events, one o f the overlaid events contributes three jets to the four leading jets in the constructed event and the other contributes one jet.
The first condition ensures that none o f the jets would be merged if the four-jet event had been reconstructed as a real event; the second condition avoids possible kinem atic bias due to events where tw o jet pairs originate from far-away vertices; the last tw o conditions enforce the appropriate com position o f the four leading jets in the constructed event.
As is discussed in section 6.4, the top olog y o f cD P S and sDPS events constructed by overlaying tw o events is com pared to the top olog y o f cD P S and sDPS events extracted from the A H J sample respectively.
6.2 Kinem atic characteristics o f event classes
In cD P S, double dijet production should result in pairwise pT-balanced jets with a distance 101 - 0 2 1 n between the jets in each pair. In addition, the azimuthal angle between the tw o planes o f interactions is expected to have a uniform random distribution. In SPS, the pairwise pT balancing o f jets is not as likely; therefore the top olog y o f the four jets is expected to be different for cD P S and SPS.
The top olog y o f three o f the jets in sDPS events would resemble the top olog y o f the jets in SPS interactions. T he fourth jet initiated by the prim ary interaction in an SPS is expected to be closer, in the 0 - y plane, to the other three jets originating from that interaction. In an sDPS event, the jet produced in the secondary interaction would be em itted in a random direction relative to the other three jets.
In constructing possible differentiating variables, three guiding principles were followed:
1. use pairwise relations that have the potential to differentiate between SPS and cD P S topologies;
2. include angular relations between all jets in light o f the expected top olog y o f sDPS events;
3. attem pt to construct variables least sensitive to system atic uncertainties.
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The first two guidelines encapsulate the different characteristics of SPS and DPS events.
The third guideline led to the usage of ratios of pT in order to avoid large dependencies on the jet energy scale (JES) uncertainty. Various studies, including the use of a principal component analysis [69], led to the following list of candidate variables for distinguishing event topologies:
PT + PT
A P/T = Y~; A0 j = |0 i — 0 j1; A y j = \yi — yj 1 ; (6 3 )
P t + P t *
101+2 - 03+4 \; 101+3 - 02+4 \; 101+4 - 02+3 \;
where p ' , P i , y* and 0 i stand for the scalar and vectorial transverse m om entum , the rapidity and the azimuthal angle o f jet i respectively, with i = 1,2, 3, 4. T he variables with the subscript i j are calculated for all possible jet com binations. T he term 0*+^ denotes the azimuthal angle o f the four-vector obtained by the sum o f jets i and j .
In the following, the pairing notation { (i, j) (k , l) } is used to describe a cD P S event in which jets i and j originate from one interaction and jets k and l originate from the other.
In around 85% o f cD P S events, the tw o leading jets originate from one interaction and jets 3 and 4 originate from the other.
Norm alized distributions o f the A^T and ApjJ variables in the three samples (SPS, cD P S and sDPS) are shown in figures 2(a) and 2 (b ) . In the cD P S sample, the A^T and A3T distributions peak at low values, indicating that both the leading and the sub-leading jet pairs are balanced in pT . T he small peak around unity is due to events in which the appropriate pairing o f the jets is { (1 ,3) (2 ,4) } or { (1 ,4) (2 , 3) }. In the SPS and sDPS samples, the leading jet-pair exhibits a wider peak at higher values o f A^T com pared to that in the cD P S sample. This indicates that the tw o leading jets are not well balanced in P t since a significant fraction o f the hard-scatter m om entum is carried by additional jets.
The A 034 distributions in the three samples are shown in figure 2 (c ). The pT balance between the jets seen in the ApjJ distribution in the cD P S sample is reflected in the A 034 distribution. T he A 034 distribution is almost uniform for the SPS and sDPS samples.
The correlation between the distributions o f the A3jJ and A 034 variables can be readily understood through the following approxim ation: p ' ~ p ' ~ P t . The expression for ApjJ then becom es
APT = |pT + P t ^ 2 p t + 2 p t cos(A 034) = ^ /1 + c o s (A 0 3 4 ) (6 4)
34 = pT + pT ~ 2 p t = V 2 ‘ (
The peak around unity observed in the A3jJ distributions in the SPS and sDPS samples is thus a direct consequence o f the Jacobian o f the relation between A3jJ and A 034.
The set o f variables quantifying the distance between jets in rapidity, A y j , is partic
ularly im portant for the sDPS topology. T he colour flow is different in SPS leading to the four-jet final state and results in smaller angles between the sub-leading jets. Hence, on average, smaller distances between non-leading jets are expected in the SPS sample com pared to the sDPS sample. This is observed in the com parison o f the A y34 distributions
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(b) A 34 , (c ) A ^34 and (d) A y34, Figure 2. Normalized distributions of the variables, (a) A 14
defined in eq. (6.3) , for the SPS, cDPS and sDPS samples as indicated in the legend. The hatched areas, where visible, represent the statistical uncertainties for each sample.
shown in figure 2 (d ), where the distribution in the sDPS sample is slightly wider than in the other two samples.
The study of the various distributions in the three samples is summed up as follows:
• Strong correlations between all variables are observed. The APT and A ¢ ij variables are correlated in a non-linear way, while geometrical constraints correlate the A yij and A<pij variables. Transverse momentum conservation correlates the — fik+i variables with the A j and A 0 ij- variables.
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• None o f the variables displays a clear separation between all three samples. T he vari
ables in which a large difference is observed between the SPS and cD P S distributions, e.g., A 3 J , do not provide any differentiating power between SPS and sDPS.
• All variables are im portant — in cD P S events, where the pairing o f the jets is different from { ( 1 , 2 ) ( 3 ,4 )}, variables relating the other possible pairs, e.g., A 0 13, may indicate which is the correct pairing.
These conclusions led to the decision to use a multivariate technique in the form o f an NN to perform event classification.
6.3 Extraction of the fraction of D P S events using an artificial neural network
For the purpose o f training the NN, events from each sample were divided into two sta
tistically independent subsamples, the training sample and the test sample. T he former was used to train the NN and the latter to test the robustness o f the result. T o avoid bias during training, the events in the SPS, cD P S and sDPS training samples were reweighted such that each sample contributed a third o f the total sum o f weights. In all subsequent figures, only the test samples are shown.
The NN used is a feed-forward multilayer perceptron with tw o hidden layers, imple
mented in the R O O T analysis framework [70] . T he input layer has 21 neurons, corre
sponding to the variables defined in eq. ( 6.3) , and the first and second hidden layers have 42 and 12 neurons respectively. These choices represent the product o f a study conducted to optim ize the perform ance o f the NN and balance the com plexity o f the network w ith the com pu tation tim e o f the training. T he output o f the NN consists o f three variables, which are interpreted as the probability for an event to be more like SPS ( {sps), cD P S ( { cDPS) or sDPS ( { sDPS). During training, each event is marked as belonging to one o f the samples;
e.g., an event from the cD P S sample is marked as
{SPS = 0, {cDPS = 1, {sDPS = 0. (6.5)
For each event, the three outputs are plotted as a single point inside an equilateral triangle (ternary plot) using the constraint { SPS+ { cDPS+ { sDPS = 1 . A point in the triangle expresses the three probabilities as three distances from each o f the sides o f the triangle. T he vertices would therefore be populated by events with high probability to belong to a single sample.
Figure 3 shows an illustration o f the ternary plot, where the horizontal axis corresponds to { sDPS + { cDPS and the vertical axis to the value o f { sDPS. T he coloured areas illustrate where each o f the three classes o f events is expected to populate the ternary plot.
Figures 4 (a ), 4 (b ) and 4 (c) show the NN output distribution for the test samples in the ternary plot, presenting the separation power o f the NN. T he SPS-type events are m ostly found in the b ottom left corner in figure 4 (a ). However, a ridge o f SPS events extending towards the sDPS corner is observed as well. A contribution from SPS events is also visible in the b ottom right corner. T he clearest peak is seen for events from the cD P S sample in the b ottom right corner in figure 4 (b ) . A visible cluster o f sDPS events is seen in figure 4(c) concentrated around { sDPS ~ 0.75 and there is a tail o f events along the side connecting
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F i g u r e 3 . Illu stration o f th e tern ary p lot c on stru cted from three N N o u tp u ts, £s p s; scDPS, a nd £sd p s, w ith th e con strain t, £sp s + £cd p s + ssDPS = 1- T h e vertical and h orizon tal axes are defined in the figure. T h e coloured areas illustrate th e classes o f events e x p e c te d to p o p u la te the correspon din g vertices.
the SPS and sDPS corners. The NN output distribution in the data, shown in figure 4(d), is visually consistent with a superposition of the three components, SPS, cDPS and sDPS.
Based on these observations, it is clear that event classification on an event.-by-event.
basis is not possible. However, the differences between the SPS, cDPS and sDPS distri
butions suggest that an estimation of the different contributions can be performed. To estimate the cDPS and sDPS fractions in four-jet. events, the ternary distribution in data.
(V) is fitted to a. weighted sum of the ternary distributions in the SPS ( A 4s p s) , cDPS (AtcDPs) and sDPS ( A 4sd p s) samples, each normalized to the measured four-jet. cross- section in data, with the fractions as free parameters. The optimal fractions were obtained using a. fit. o f the form,
V = (I - /cDPS - /sDPs)A4sPS + /cDPS-AdcDPS + /sDPS-A4sDPS , (6.6) where a. \ 2 minimization was performed, as implemented in the M i n u i t [71] package in R O O T, taking into account, the statistical uncertainties of all the samples in each bin. The results of the fit. are presented in section 8, after the methodology validation and discussion o f systematic uncertainties.
6.4 M ethodology validation
A sizeable discrepancy was found in the AgJ and A(/>34 distributions between the data, and A H J (See section 9 for details), suggesting that there are more sub-leading jets (jets 3 and 4) that are baek-t.o-ba.ck in AH J than in the data. In order to test, that the discrepancies are not. from mis-modelling o f SPS in A H J, the AgJ and A(/>34 distribu
tions in the SPS sample extracted from A H J were compared to the distributions in the SPS sample generated in Sh e r p a. G ood agreement, between the shapes of the distributions was observed for both variables. This and further studies indicate that the excess of events
J H E P l l ( 2 0 1 6 ) 1 1 0
Figure 4. Normalized distributions of the NN outputs, mapped to a ternary plot as described in the text, in the (a) SPS, (b) cDPS, (c) sDPS test samples and (d) in the data.
with jets 3 and 4 in the back-to-back topology is due to an excess of DPS events in the A H J sample compared to the data.
In order to verify that the topologies of cDPS and sDPS events can be reproduced by overlaying two events, the overlay samples are compared to the cDPS and sDPS samples extracted from AH J. An extensive comparison between the distributions of the variables used as input to the NN in the overlay samples and in AH J was performed and good agreement was observed. This can be summarized by comparing the NN output distribu
tions. The NN is applied to the cDPS and sDPS samples extracted from AH J and the
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Figure 5. Comparison between the normalized distributions of the NN outputs — CsDPS + — CcDPS, integrated over all £sDPS values 0.0 < £sDPS < 1.0, in DPS events extracted from AH J and in the DPS samples constructed by overlaying events from data, for (a) cDPS events and (b) sDPS events.
In the AH J distributions, statistical uncertainties are shown as the hatched area and the shaded area represents the sum in quadrature of the statistical and systematic uncertainties.
output distributions are compared to the output distributions in the corresponding sam
ples constructed by overlaying events selected from data. Normalized distributions of the projection o f the full ternary plot on the horizontal axis are shown in figures 5(a) and 5(b) for the cDPS and sDPS samples respectively. G ood agreement is observed between the distributions. Based on these results, it is concluded that the topology of the four jets in the overlaid events is comparable to that of the four leading jets in DPS events extracted from A H J. The added advantage of using overlaid events from data to construct the DPS samples is that the jets are at the same JES as the jets in four-jet events in data, leading to a smaller systematic uncertainty in the final result.
As an additional validation step, the NN is applied to the inclusive AH J sample and the resulting distribution is fitted with the NN output distributions of the SPS, cDPS and sDPS samples. The fraction obtained from the fit, /Dp^ , is compared to the fraction at parton level, /D(P)ps , extracted from the event record,
/DMS) = 0.129 ± 0.007 (s ta t.), /DPs = 0.142 ± 0.001 (s ta t.). (6.7) Fair agreement is observed between the value obtained from the fit and that at parton level. The larger statistical uncertainty in /D P ^ compared to /Dp)S reflects the loss of statistical power due to the use of a template fit to estimate the former.
7 Systematic uncertainties
For jets with 20 < pT < 30 GeV, the fractional JES uncertainty is about 4.5% in the central region of the detector, rising to about 10% in the forward region [64]. The overall
JHEP11(2 01 6)1 10
Source o f system atic uncertainty A /dps A a2j A Oeff
Lum inosity ± 3 .5 %
M odel dependence for detector corrections ±2 % ±2 %
Reweighting o f A H J ±6 % ±6 %
Jet reconstruction efficiency ± 0 .1 %
Single-vertex events selection ± 0 .1 %
Jet energy and angular resolution ± 1 5 % ± 3 % ± 1 5 %
JES uncertainty -37 % ± 1 2 % -19 %
T otal system atic uncertainty -40 % ± 1 3 % -25 %
T able 1. Summary of the relative systematic uncertainties in /d p s , «2j and °eff •
im pact o f the JES on the distributions, /dps and a^j was estim ated by shifting the jet energy upwards and downwards in the M C samples by the JES uncertainty and repeating the analysis. Similarly, the overall im pact o f the jet energy and angular resolution was determ ined by varying the jet energy and angular resolution in the M C samples by the corresponding resolution uncertainty [72].
T h e system atic uncertainties in the measured cross-sections due to the integrated lu
m inosity measurement uncertainty (± 3 .5 % ), the jet reconstruction efficiency uncertainty ( ± 2 % ) and the uncertainty as a result o f selecting single-vertex events (± 0 .5 % ) were prop
agated to the uncertainty in creff.
T he statistical uncertainty in the A H J sample was translated to a system atic uncer
tainty in /dps by varying the reweighting function used to reweight A H J and repeating the analysis.
T he statistical uncertainty in a^j (~ 1 % ) was propagated as a system atic uncertainty in creff. T he system atic uncertainty in a^j arising from m odel-dependence (± 2 % ) was de
term ined from deriving a^j using Sh e r p a.
T he stability o f the value o f aeff relative to the various param eter values used in the measurement was studied. Parameters such as pTart° n and A R j e t-jet were varied and the requirement A R part°n -jet < 0.6 was applied, leading to a relative change in creff o f the order o f a few percent. Since the observed relative changes are small com pared to the statistical uncertainty in aeff, no system atic uncertainty was assigned due to these parameters.
T he relative system atic uncertainties in /d p s , a^j and aeff are summarized in table 1.
T he dom inant system atic uncertainty on /d p s originates from the JES variation. A varia
tion in the JES results in a m odification o f the NN output distribution for the SPS tem plate used in the fit, which directly im pacts the value o f /d p s .
8 Determination of
aeff
T o determine / dps and aeff and their statistical uncertainties taking into account all o f the correlations, many replica fits were perform ed by random sampling from the NN output
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distributions. T he system atic uncertainties were obtained by propagating the expected variations into this analysis, and the resulting shifts were added in quadrature. T he result for /dps is
where the contribution o f f sDPS to /dps was found to be about 40%. T he fraction o f D PS
(see eq. (6.7) ). Taking into account the system atic uncertainties in the calculation o f the goodness-of-fit x 2, a value for x2/ N DF o f 112/84 = 1.3 is obtained, where N DF is the number o f degrees o f freedom in the fit.
In order to visualize the results o f the fit, the ternary distribution is divided into five slices,
• 0.0 < /sDPS < 0.1,
• 0.1 < /sDPS < 0.3,
• 0.3 < /sDPS < 0.5,
• 0.5 < /sDPS < 0.7,
• 0.7 < /sdps < 1.0.
A com parison o f the fit distributions with the distributions in data in the five slices of the ternary plot is shown in figure 6 . Considering the system atic uncertainties, the most significant difference between the data and the fit is seen for the tw o left-m ost bins in the range 0.0 < / sDPS < 0.1 (figure 6 (a )) o f the ternary plot. These bins are dom inated by the SPS contribution. Thus, a discrepancy between the data and the fit result in these bins is expected to have a negligible effect on the measurement o f the D P S rate. A discrepancy between the data and the fit result is also observed in the three rightmost bins in figure 6 (a ). These bins have about a 30% contribution from cD PS. To test the effect o f this discrepancy on the description o f observables in data, the distributions o f the various variables in data were com pared to a com bination o f the distributions in the SPS, cD P S and sDPS samples, norm alizing the latter three distributions to their respective fractions in the data as obtained in the fit. This com parison for the A|J and A /3 4 variables is shown in figure 7, where a g o o d description o f the data is observed. T he same level o f agreement is seen for all the variables.
Before calculating creff, the sym m etry factor in eq. ( 2.3) has to be adjusted because there is an overlap in the cross-sections 0 +- and 0^ when the leading jet in sample A has p T > 42.5 GeV (see eq. ( 5.1) ). T he adjusted sym m etry factor is
as determ ined from the measured dijet cross-sections. This factor was also determined using P y t h i a6 and good agreement was observed between the two values. The relative difference in the value o f oeff obtained by using the sym m etry factors extracted from the
f DPs = ° . ° 92 (sta t.) I 0;03? (sy st.) , (8 .1)
estim ated in data is 65+2?% o f the fraction in A H J as extracted from the event record
1 (8 .2)
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F igu re 6 . Distributions of the NN outputs, — CsDPS + — CcDPS, in the £sDPS ranges indicated in the panels, for four-jet events in data, selected in the phase space defined in the legend, compared to the result of fitting a combination of the SPS, cDPS and sDPS contributions, the sum of which is shown as the solid line. In the fit distribution, statistical uncertainties are shown as the dark shaded area and the light shaded area represents the sum in quadrature of the statistical and systematic uncertainties. The ratio of the fit distribution to the data is shown in the bottom panels.
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Figure 7. Comparison between the distributions of the variables (a) AgJ and (b) A ^ 34, defined in eq. (6.3) , in four-jet events in data and the sum of the SPS, cDPS and sDPS contributions, as indicated in the legend. The sum of the contributions is normalized to the cross-section measured in data and the various contributions are normalized to their respective fractions obtained from the fit. In the sum of contributions, statistical uncertainties are shown as the dark shaded area and the light shaded area represents the sum in quadrature of the statistical and systematic uncertainties.
The ratio of the sum of contributions to the data is shown in the bottom panels.
JHEP11(2 01 6)1 10
data and from P y t h i a6 was o f the order o f 0.2%, a negligible difference com pared to the statistical uncertainty o f aeff.
A n additional correction o f + 4 % is applied to the measured D P S cross-section due to the probability o f jets from the secondary interaction overlapping with jets from the primary interaction. In this configuration, the anti-kt algorithm merges the tw o overlapping jets into one, and hence the event cannot pass the four-jet requirement. T he value o f this correction was determ ined from the fraction o f phase space occupied by a jet. It was also determ ined directly in A H J and g ood agreement between the two values was observed.
Finally, the measurements o f the dijet and four-jet cross-sections can be used to cal
culate the effective cross-section, yielding
oeff = 14.9 (stat.) +0.5 (syst.) mb . (8.3) This value is consistent within the quoted uncertainties with previous measurements, per
form ed by the A T L A S collaboration and by other experim ents [16- 30] , all o f which are summarized in figure 8 . Figure 9 shows aeff as a function o f a/s, where the A F S result and some o f the L H C b results are om itted for clarity. W ithin the large uncertainties, the measurements are consistent with no y fs dependence o f aeff. T he aeff value obtained is 21+6% o f the inelastic cross-section, ajnel, measured by A T L A S at y fs = 7 TeV [73] .
9 Normalized differential cross-sections
T o allow the results o f this study to be used in future com parisons with M P I models, the distributions o f the variables used as input to the NN were corrected for detector effects. T he corrections were derived using an iterative unfolding, producing an unfolding m atrix for each observable, relating the particle-level and reconstructed-level quantities.
These matrices were derived using samples o f four-jet events selected from the A H J and P y t h i a6 samples by im posing the cuts detailed in eq. ( 5.1) on particle jets. The A H J sample generated with the A U E T 1 tune was used to derive the unfolding m atrix. The distributions were unfolded with the Bayesian unfolding algorithm, im plem ented in the R ooU n fold package [74], using tw o iterations.
T he unfolding matrices derived from A H J were taken as the nominal matrices and the differences observed when using the matrices derived from P y t h i a6 were used as an additional system atic uncertainty, typically o f the order o f a few percent in each bin.
T he total system atic uncertainty o f the differential distributions in data was obtained by summing in quadrature the uncertainty due to M C m odelling in a given bin w ith the system atic uncertainties in this bin due to the JES and jet energy and angular resolution uncertainties, while preserving correlations between bins. Figure 10 shows the normalized differential cross-section distribution in data for the A3J and A ^ 34 variables com pared to the particle-level distributions in the A H J samples generated with the A U E T1 and A U E T 2 tunes. T he particle-level distributions in the A U E T 2 A H J sample overestimate the normalized differential cross-section distributions in data in the regions A3J < 0.15 and A0 34 > 2.8, dem onstrating the excess o f the D PS contribution in this sample com pared to the data. On the other hand, the D P S contribution in the data is underestim ated by
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F igu re 8 . The effective cross-section, <reff, determined in various final states and in different exper
iments [16- 30]. The inner error bars (where visible) correspond to the statistical uncertainties and the outer error bars represent the sum in quadrature of the statistical and systematic uncertainties.
Dashed arrows indicate lower limits and the vertical line represents the AFS measurement published without uncertainties.
the prediction obtained with the AUET1 tune. These comparisons demonstrate the power o f these distributions to constrain MPI models and tunes. In section A , the normalized differential cross-sections in data for the remaining variables are compared to the particle- level distributions in the AH J samples generated using the AUET1 and AUET2 tunes.
10 Summary and conclusions
A measurement of the rate of hard double-parton scattering in four-jet events was per
formed using a sample of data collected with the ATLAS experiment at the LHC in 2010, with an average of approximately 0.4 proton-proton interactions per bunch crossing, cor
responding to an integrated luminosity o f 37.3 ± 1.3 pb- 1 . Three different samples were selected, all consisting of single-vertex events from proton-proton collisions at a centre- of-mass energy of yfs = 7 TeV. Four-jet events were defined as those containing at least four reconstructed jets with pT > 20 GeV and |n| < 4.4, and at least one jet having pT > 42.5 GeV. Two additional dijet samples were selected with the requirement of having
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F igu re 9. The effective cross-section, <reff, as a function of the centre-of-mass energy, a/s, for a representative set of measurements [17- 30]. The inner error bars (where visible) correspond to the statistical uncertainties and the outer error bars represent the sum in quadrature of the statistical and systematic uncertainties. Dashed arrows indicate lower limits. For clarity, measurements at identical centre-of-mass energies are slightly offset in a/s.
at least two jets with pT > 20 GeV and |n| < 4.4. One of the dijet samples was further constrained such that it contained at least one jet with pT > 42.5 GeV.
The contribution of hard double-parton scattering to the production of four-jet events was extracted using an artificial neural network. The four-jet topology originating from hard double-parton scattering was represented by a random combination o f events selected in data. The fraction of events corresponding to the contribution made by hard double- parton scattering in four-jet events was determined to be,
/ d p s = 0.092 (stat.) -0.007 (s y s t.). (10.1)
After combining this result with measurements of the dijet and four-jet cross-sections in the appropriate phase space regions, the effective cross-section was determined to be
O f f = 14.9 - - 0 (stat.) -0- (syst.) m b.
This value is 21+ 0% of the measured value of o ; nei at / s = 7 TeV and is consistent with previous measurements performed at various centre-of-mass energies and in various final states. It is compatible with a model in which oeff is a universal parameter that does not depend on the process or phase space. To facilitate future studies o f the dynamics o f multi-parton interactions, distributions of observables sensitive to the presence of hard double-parton scattering are also presented.
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Figure 10. Distributions of the variables (a) A3J and (b) A ^34, defined in eq. (6.3), in data after unfolding to particle level, compared to the M C prediction from A H J at the particle level, generated using the AUET1 and AUET2 tunes, as indicated in the legend. The hatched area represents the sum in quadrature of the statistical and systematic uncertainties in the normalized differential cross-sections and all histograms are normalized to unity. The ratio of the particle-level distribution to the normalized differential cross-section is shown in the bottom panels, where the shaded areas represent statistical uncertainties.
Acknowledgments
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.
We acknowledge the support of A N PCyT, Argentina; YerPhI, Armenia; ARC, Aus
tralia; B M W F W and FW F, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CO N ICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS, CEA -D SM /IRFU , France;
GNSF, Georgia; BM BF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF, I-CO RE and Benoziyo Center, Israel; INFN, Italy; M E XT and JSPS, Japan; CNRST, M orocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; M N E/IFA, Romania; MES of Russia and NRC KI, Russian Fed
eration; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZS, Slovenia; D ST /N R F , South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, the Canada Council, CANARIE, CRC, Compute Canada, FQRNT, and the Ontario Innovation Trust, Canada; EPLANET, ERC, FP7, Horizon 2020 and Marie Sklodowska-Curie Actions, European Union; Investissements
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d ’Avenir L abex and Idex, A N R , R egion Auvergne and Fondation Partager le Savoir, France;
D F G and A vH Foundation, Germany; Herakleitos, Thales and Aristeia program m es co financed by E U -E SF and the Greek N SRF; BSF, G IF and Minerva, Israel; B R F , Norway;
Generalitat de Catalunya, Generalitat Valenciana, Spain; the R oyal Society and Lever- hulme Trust, United K ingdom .
The crucial com puting support from all W L C G partners is acknowledged gratefully, in particular from C E R N , the A T L A S Tier-1 facilities at T R IU M F (C anada), N D G F (Denmark, Norway, Sweden), C C -IN 2P3 (France), K I T /G r id K A (G erm any), IN F N -C N A F (Italy), N L -T 1 (Netherlands), P IC (Spain), A S G C (Taiw an), R A L (U .K .) and BNL (U .S .A .), the Tier-2 facilities w orldwide and large n on -W L C G resource providers. M a
jo r contributors o f com puting resources are listed in ref. [ 75] .
A Normalized differential cross-sections
Figures 11- 15 show the normalized differential cross-sections in data for all the observables used as input to the NN, com pared to the particle-level distributions in the A H J samples generated using the A U E T 1 and A U E T 2 tunes. T he hatched areas in the distributions represent the total uncertainty o f the normalized differential cross-section, obtained by adding in quadrature the statistical and system atic uncertainties.
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F igu re 11. Distributions of the variables (a) ApJ, (b) A^T, (c) ApT and (d) ApJ, defined in eq. (6.3) , in data after unfolding to particle level, compared to the M C prediction from A H J at the particle level, generated using the AUET1 and AUET2 tunes, as indicated in the legend. The hatched areas represent the sum in quadrature of the statistical and systematic uncertainties in the normalized differential cross-sections and all histograms are normalized to unity. The ratio of the particle-level distribution to the normalized differential cross-section is shown in the bottom panels, where the shaded areas represent statistical uncertainties.
