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

P u b l i s h e d f o r SISSA b y S p r i n g e r R e c e i v e d: June 3, 2016

R e v i s e d: J u ly 29, 2016 A c c e p t e d: A u g u s t 20, 2016 P u b l i s h e d: A u g u s t 29, 2016

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

T h e A T L A S collaboration

E -m ail: atlas.publications@cern.ch

A b s t r a c t : The angular distributions of Drell-Yan charged lepton pairs in the vicinity of the Z -boson mass peak probe the underlying Q C D dynamics of Z -boson production.

This paper presents a measurement of the complete set of angular coefficients A0 - 7 de­

scribing these distributions in the Z-boson Collins-Soper frame. The data analysed cor­

respond to 20.3 fb- 1 of pp collisions at √ s = 8TeV, collected by the A T L A S detector at the C E R N L H C . The measurements are compared to the most precise fixed-order calcu­

lations currently available (O ( a i2)) and with theoretical predictions embedded in Monte Carlo generators. The measurements are precise enough to probe Q C D corrections beyond the formal accuracy of these calculations and to provide discrimination between different parton-shower models. A significant deviation from the O ( a 2) predictions is observed for Ao — A 2. Evidence is found for non-zero A5,6,7, consistent with expectations.

K e y w o r d s : Hadron-Hadron scattering (experiments)

A r X i y e P r i n t : 1606.00689

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

Contents

1 I n t r o d u c t i o n 1

2 T h e o r e t i c a l p r e d i c t i o n s 4

3 T h e A T L A S e x p e r i m e n t a n d i t s d a t a a n d M o n t e C a r l o s a m p l e s 8

3.1 A T L A S d e te c to r 8

3.2 D a ta a n d M o n te C arlo sam p les 8

4 D a t a a n a l y s i s 9

4.1 E v e n t selectio n 9

4.2 B a c k g ro u n d s 10

4.3 A n g u la r d is trib u tio n s 15

5 C o e f f ic i e n t m e a s u r e m e n t m e t h o d o l o g y 15

5.1 T e m p la te s 15

5.2 L ikelihood 17

5.3 C o m b in a tio n s a n d a lte r n a tiv e p a ra m e te ris a tio n s 20

6 M e a s u r e m e n t u n c e r t a i n t i e s 21

6.1 U n c e rta in tie s from d a t a sam p le size 21

6.2 U n c e rta in tie s from M o n te C arlo sam p le size 22

6.3 E x p e rim e n ta l sy s te m a tic u n c e rta in tie s 22

6.4 T h e o re tic a l s y s te m a tic u n c e rta in tie s 23

6.5 S y ste m a tic u n c e rta in tie s re la te d to th e m e th o d o lo g y 24

6.6 S u m m a ry of u n c e rta in tie s 25

7 R e s u l t s 31

7.1 C o m p a tib ility b etw e en ch a n n els 31

7.2 R e su lts in th e in d iv id u a l a n d com b in ed ch a n n els 31

7.3 C ross-checks 34

8 C o m p a r i s o n s w i t h t h e o r y 3 7

9 S u m m a r y 4 8

A T h e o r e t i c a l f o r m a l i s m 4 9

B A d d i t i o n a l t e m p l a t e s 51

C R e g u l a r i s a t i o n 52

D C a t e g o r i s a t i o n o f s t a t i s t i c a l u n c e r t a i n t i e s 5 7

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

E Q u a n t i f y i n g A 5,6,7 5 8

F A d d i t i o n a l r e s u l t s 61

T h e A T L A S c o l l a b o r a t i o n 8 3

1 In tr o d u c tio n

T h e a n g u la r d is trib u tio n s of ch a rg ed le p to n p a irs p ro d u c e d in h a d ro n -h a d ro n collisions v ia th e D rell-Y an n e u tra l c u rre n t p ro cess p ro v id e a p o rta l to p recise m e a s u re m e n ts o f th e p ro d u c tio n d y n a m ic s th ro u g h spin c o rre la tio n effects b etw e en th e in itia l- s ta te p a rto n s an d th e fin a l-s ta te lep to n s m e d ia te d by a spin-1 in te rm e d ia te s ta te , p re d o m in a n tly th e Z boson.

In th e Z -b o so n re st fram e, a p lan e sp a n n e d by th e d ire c tio n s o f th e inco m in g p ro to n s ca n b e defined, e.g. usin g th e C o llin s-S o p er (C S) referen ce fram e [1]. T h e le p to n p o la r an d a z im u th a l a n g u la r v ariab les, d e n o te d by cos 6 a n d 0 in th e follow ing fo rm alism , a re defined in th is reference fram e. T h e sp in c o rre la tio n s a re d e sc rib e d by a se t of n in e helicity d e n sity m a tr ix elem en ts, w hich c a n b e c a lc u la te d w ith in th e c o n te x t o f th e p a r to n m od el u sing p e r tu r b a tiv e q u a n tu m ch ro m o d y n a m ic s (Q C D ). T h e th e o re tic a l fo rm alism is e la b o ra te d in refs. [2- 5].

T h e full five-dim ensional d iffe ren tial cro ss-sectio n d esc rib in g th e k in e m a tic s of th e tw o B orn-lev el lep to n s from th e Z -b o so n d ecay c a n b e d ec o m p o sed as a su m o f n in e h a r ­ m onic p o ly n o m ia ls, w hich d e p e n d o n cos 6 a n d 0, m u ltip lie d by co rre sp o n d in g helicity cro ss-sectio n s t h a t d e p e n d o n th e Z -b o so n tra n s v e rs e m o m e n tu m ( p f ), ra p id ity (y Z ), an d in v a ria n t m ass (m Z ). I t is a s ta n d a r d co n v e n tio n to fa c to rise o u t th e u n p o la ris e d cro ss­

sectio n , d e n o te d in th e lite ra tu re by au + L , a n d to p re s e n t th e fiv e-d im en sion al differen tial cro ss-sectio n as a n ex p a n sio n in to n in e h a rm o n ic p o ly n o m ia ls P i (cos 6,0) a n d d im en sio n - less a n g u la r coefficients A 0 -7 ( p f , y Z , m Z ), w hich re p re s e n t ra tio s of helicity cross-sectio ns w ith re sp e c t to th e u n p o la rise d one, au + L , as ex p la in e d in d e ta il in a p p e n d ix A :

d a 3 ^{d a u + L}

d p f d y Z d m Z d cos 6 d0 16n d p f d y Z d m Z (1.1)

x I (1 + co s26) + 1 A0 (1 — 3 cos26) + A i s i n 26 c o s 0

+ 1 A2 sin26 cos 2 0 + ^{A 3} sin 6 cos 0 + A4 cos 6 + A 5 sin26 sin 2 0 + A 6 sin 26 sin 0 + A7 sin 6 sin 0 ^ .

T h e d e p e n d e n c e of th e d iffe ren tial cro ss-sectio n o n cos 6 a n d 0 is th u s c o m p letely m a n ­ ifest an a ly tica lly . In c o n tra s t, th e d e p e n d e n c e on p f , y Z , a n d m Z is e n tire ly c o n ta in e d in th e A i coefficients an d au + L . T h erefo re, all h a d ro n ic d y n a m ic s from th e p ro d u c tio n m ech a n ism are d e sc rib e d im p lic itly w ith in th e s tru c tu r e o f th e A i coefficients, a n d a re fac­

to ris e d fro m th e d ecay k in e m a tic s in th e Z -b o so n re st fram e. T h is allow s th e m e a su re m e n t

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

precision to be essentially insensitive to all uncertainties in Q C D , quantum electrodynam­

ics (QED), and electroweak ( E W ) effects related to Z-boson production and decay. In particular, E W corrections that couple the initial-state quarks to the final-state leptons have a negligible impact (below 0.05%) at the Z-boson pole. This has been shown for the L E P precision measurements [6, 7], w hen calculating the interference between initial-state and final-state Q E D radiation.

W h e n integrating over cos 0 or 0, the information about the A1 and A 6 coefficients is lost, so both angles must be explicitly used to extract the full set of eight coefficients.

Integrating eq. (1.1) over cos 0 yields:

^ = -1 daU+L (1.2)

dpf d y Z d m Z d0 2n dpf d y Z d m Z

At leading order (LO) in Q C D , only the annihilation diagram qq ^ Z is present and only A4 is non-zero. At next-to-leading order (NLO) in Q C D (O(as)), A0 - 3 also become non-zero. The L a m - T u n g relation [8- 10], which predicts that A0 — A2 = 0 due to the spin-1 of the gluon in the qg ^ Zq and qq ^ Z g diagrams, is expected to hold up to O ( a s), but can be violated at higher orders. The coefficients A5,6 ,7 are expected to become non-zero, while remaining small, only at next-to-next-to-leading order ( N N L O ) in Q C D (O (a Ę )), because they arise from gluon loops that are included in the calculations [11, 12]. The coefficients A3 and A4 depend on the product of vector and axial couplings to quarks and leptons, and are sensitive to the Weinberg angle sin2 0W . The explicit formulae for these dependences can be found in appendix A .

The full set of coefficients has been calculated for the first time at O ( a ‘2) in refs. [2-5].

More recent discussions of these angular coefficients m a y be found in ref. [13], where the predictions in the N N L O P S scheme of the PowHEg [14- 17] event generator are shown for Z-boson production, and in ref. [18], where the coefficients are explored in the context of W-boson production, for which the same formalism holds.

The C D F Collaboration at the Tevatron published [19] a measurement of some of the angular coefficients of lepton pairs produced near the Z-boson mass pole, using 2 .1 fb- 1

of proton-anti-proton collision data at a centre-of-mass energy yfs = 1.96 TeV. Since the measurement was performed only in projections of cos 0 and 0, the coefficients A1 and A 6 were inaccessible. They further assumed A5 and A7 to be zero since the sensitivity to these coefficients was beyond the precision of the measurements; the coefficients A0 ,2,3 ,4

were measured as a function of p f . These measurements were later used by C D F [20] to infer an indirect measurement of sin20w, or equivalently, the W-boson mass in the on- shell scheme, from the average A4 coefficient. These first measurements of the angular

xi ₁₄

+

A

cos

+ A ₁₆

cos

+

A

sin

+ ? 3 n A7 sin

,

2

16 I

while integrating over 0 yields:

da 3 daU+ L

|(1 + c o s2 0) + 1A0 ( 1 — 3 cos2 0) + A4 cos 0 dpf dy Z d m Z d cos 0 8 dpf dy Z d m Z

(1.3)

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

coefficients d e m o n s tra te d th e p o te n tia l o f th is n o t-y e t-fu lly ex p lo red e x p e rim e n ta l avenue for in v e stig a tin g h a rd Q C D a n d E W physics.

M e a su re m e n ts of th e W -b o so n a n g u la r coefficients a t th e L H C w ere p u b lish e d by b o th A T L A S [21] a n d C M S [22]. M ore recently, a m e a su re m e n t o f th e Z -b o so n a n g u la r coefficients w ith Z ^ ^ d ecays w as p u b lish e d by C M S [23], w h ere th e first five coefficients w ere m e a su re d w ith 19.7 fb - 1of p ro to n -p ro to n (pp) collision d a t a a t a/s = 8 T eV . T h e m e a su re m e n t w as p erfo rm ed in tw o y Z bins, 0 < |y Z | < 1 a n d 1 < |y Z | < 2.1, each w ith eig h t bins in p f u p to 300 G eV . T h e v io la tio n of th e L a m -T u n g re la tio n w as o b served , as p re d ic te d by Q C D c a lc u la tio n s b ey o n d N L O .

T h is p a p e r p re se n ts an inclusive m e a su re m e n t o f th e full set of eig h t Ai coefficients usin g ch a rg ed le p to n p a irs (e le ctro n s o r m u o n s), d e n o te d h e re a fte r by E T h e m e a su re m e n t is p e rfo rm e d in th e Z -b o so n in v a ria n t m ass w indow of 8 0 -1 0 0 G eV , as a fu n c tio n of p f , a n d also in th re e bins of y Z . T h e se re su lts are b ase d o n 20.3 fb - 1 o f p p collision d a t a co llected a t ^{^ / s} = 8 T eV by th e A T L A S e x p e rim e n t [24] a t th e L H C . W ith th e m e a su re m e n t te c h n iq u e s develo p ed for th is an aly sis, th e co m p lete set of coefficients is e x tra c te d w ith fine g ra n u la rity over 23 bins of p f u p to 600 G eV . T h e m e a su re m e n ts, p erfo rm ed in th e CS reference fram e [1], are first p re se n te d as a fu n c tio n o f p f , in te g ra tin g over y Z . F u rth e r m e a s u re m e n ts d iv id ed in to th re e bins of y Z a re also p re sen ted : 0 < |y Z | < 1, 1 < |y Z | < 2, a n d 2 < |y Z | < 3.5. T h e Z /y * ^ e + e - a n d Z /y * ^ ß + ß - c h a n n els w h ere b o th lep to n s fall w ith in th e p s e u d o ra p id ity ra n g e |n| < 2.4 (h e re a fte r re ferred to as th e c e n tra l-c e n tra l o r e e c c a n d ^ c c ch an n els) a re used for th e y Z-in te g ra te d m e a su re m e n t a n d th e first tw o y Z bins. T h e Z / y * ^ e + e - c h a n n el w h ere one of th e e le c tro n s in s te a d falls in th e region

|n| > 2.5 (referred to h e re a fte r as th e c e n tra l-fo rw a rd o r eeCF ch a n n el) is u sed to e x te n d th e m e a su re m e n t to th e h ig h -y Z regio n en c o m p a ssed by th e th ir d y Z bin. In th is case, how ever, b ec au se of th e few er ev e n ts availab le for th e m e a su re m e n t itse lf an d to e v a lu a te th e b a c k g ro u n d s (see sec tio n 4) , th e m e a su re m e n t is o nly p erfo rm ed for p f u p to 100 G eV usin g p ro je c tio n s o f cos 6 a n d 0 , m a k in g A1 a n d A6 inaccessible in th e 2 < |y Z | < 3.5 bin.

T h e high g ra n u la rity a n d precisio n of th e specific m e a su re m e n ts p re se n te d in th is p a ­ p e r p ro v id e a s trin g e n t te s t of th e m o st precise p e r tu r b a tiv e Q C D p re d ic tio n s for Z -b o so n p ro d u c tio n in pp collisions a n d of M o n te C arlo (M C ) ev en t g e n e ra to rs used to sim u la te Z -b o so n p ro d u c tio n . T h is p a p e r is o rg a n ise d as follows. S ectio n 2 su m m arises th e th e o r e t­

ical fo rm alism used to e x tra c t th e a n g u la r coefficients a n d p re se n ts th e fix ed -o rd er Q C D p re d ic tio n s for th e ir v a ria tio n s as a fu n c tio n o f p f . S ectio n 3 d esc rib es briefly th e A TLA S d e te c to r a n d th e d a t a a n d M C sam p les u sed in th e analy sis, w hile sec tio n 4 p re se n ts th e d a t a an a ly sis a n d b a c k g ro u n d e s tim a te s for each of th e th re e ch a n n els co n sid ered . S ectio n 5 d esc rib es th e fit m e th o d o lo g y used to e x tra c t th e a n g u la r coefficients in th e full p h a se space as a fu n c tio n of p f a n d sec tio n 6 gives an overview o f th e s ta tis tic a l a n d s y s te m a tic u n c e r­

ta in tie s of th e m e a su re m e n ts. S ectio ns 7 a n d 8 p re se n t th e re su lts a n d co m p a re th e m to v ario u s p re d ic tio n s from th e o re tic a l c a lc u la tio n s a n d M C ev en t g e n e ra to rs, a n d sec tio n 9 su m m arises a n d co n cludes th e p a p e r.

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

F ig u r e 1. Sketch of the Collins-Soper reference frame, in which the angles 0cs and 0 cs are defined w ith respect to the negatively charged lepton ^I (see text). The notations ^{x , y} and z denote the unit vectors along the corresponding axes in this reference frame.

2 T h e o r e tic a l p r e d ic tio n s

T h e d iffe ren tial cro ss-sectio n in eq. ( 1.1) is w ritte n for p u re Z bo sons, a lth o u g h it also holds fo r th e c o n trib u tio n from 7* a n d its in terfe ren c e w ith th e Z boso n. T h e tig h t in v a ria n t m ass w indow of 80-1 0 0 G eV is chosen to m in im ise th e 7* c o n trib u tio n , a lth o u g h th e p re ­ d ic te d A i coefficients p re se n te d in th is p a p e r are effective coefficients, c o n ta in in g th is sm all c o n trib u tio n from 7*. T h is c o n trib u tio n is n o t a c c o u n te d for e x p lic itly in th e d e ta ile d fo rm alism d esc rib ed in a p p e n d ix A , w hich is p re se n te d for sim p licity for p u re Z -b o so n p ro d u c tio n . T h ro u g h o u t th is p a p e r, th e le p to n s from Z -b o so n d ecay s are d efined a t th e B o rn level, i.e. b efo re fin a l-s ta te Q E D ra d ia tio n , w h en d iscu ssin g th e o re tic a l c a lc u la tio n s o r p re d ic tio n s a t th e e v e n t-g e n e ra to r level.

T h e p f a n d y Z d e p e n d e n c e of th e coefficients varies stro n g ly w ith th e choice o f spin q u a n tis a tio n axis in th e Z -b o so n re st fra m e (z -a x is). In th e CS referen ce fra m e a d o p te d for th is p a p e r, th e z-ax is is d efined in th e Z -b o so n re st fram e as th e e x te rn a l b ise c to r o f th e an g le b etw e en th e m o m e n ta of th e tw o p ro to n s, as d e p ic te d in figure 1. T h e p o sitiv e d irec­

tio n o f th e z-ax is is defined by th e d ire c tio n of p o sitiv e lo n g itu d in a l Z -b o so n m o m e n tu m in th e la b o ra to ry fram e. To c o m p le te th e c o o rd in a te sy stem , th e y -axis is d efined as th e n o rm a l v e c to r to th e p lan e sp a n n e d by th e tw o in com ing p ro to n m o m e n ta a n d th e x -a x is is chosen to define a rig h t-h a n d e d C a rte s ia n c o o rd in a te sy ste m w ith th e o th e r tw o axes.

P o la r a n d a z im u th a l angles a re c a lc u la te d w ith re sp e c t to th e n eg a tiv e ly ch a rg ed le p to n a n d a re lab elled 0CS a n d 0 c s , resp ectively . In th e case w h ere p f = 0, th e d ire c tio n o f th e y -ax is a n d th e d efin itio n of 0 c s a re a rb itra ry . H istorically, th e re h as b ee n a n a m b ig u ity in th e d e fin itio n of th e sign of th e 0 CS ang le in th e CS fram e: th is p a p e r a d o p ts th e recen t co n v e n tio n followed by refs. [13, 23], w h e reb y th e coefficients A1 a n d ^{A 3} are p o sitiv e.

T h e coefficients are n o t e x p lic itly used as in p u t to th e th e o re tic a l c a lc u la tio n s n o r in th e M C ev en t g e n e ra to rs. T h e y can, how ever, be e x tra c te d from th e sh a p e s of th e a n g u la r d is trib u tio n s w ith th e m e th o d p ro p o se d in ref. [3], ow ing to th e o rth o g o n a lity of th e P i p o ly n o m ia ls. T h e w eig h ted av erag e o f th e a n g u la r d is trib u tio n s w ith re sp e c t to an y specific p o ly n o m ia l iso lates an average reference value o r m o m e n t o f its c o rre sp o n d in g

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

coefficient. T h e m o m en t o f a p o ly n o m ia l P (c o s 0 ,0 ) over a specific ra n g e o f p f , VZ , a n d m Z is defined to be:

( P (cos 0 ,0 ) ) = J ' P < T W 0 . (2.1)

J d a (c o s 0,0)d c o s 0d0

T h e m o m e n t of each h a rm o n ic p o ly n o m ia l c a n th u s be ex p ressed as (see eq. ( 1.1) ):

1 3 / 2 \ 1 1

( 2 ( 1 - 3 cos2 0)) = 2 0 M o - 3 ) ; (sin 20 cos 0 ) = 5 A i; (sin2 0 c o s 2 0 ) = — A2; ( s i n 0 cos 0) = 7A 3; (cos 0) = 1A 4; (sin2 0 s in 2 0 ) = - A 5; (2 .2)

4 4 5

(sin 20 sin 0 ) = - A 6; (sin 0 sin 0 ) = 1A 7.

5 4

O ne th u s o b ta in s a re p re s e n ta tio n of th e effective a n g u la r coefficients for Z / y * p ro ­ d u c tio n . T h ese effective a n g u la r coefficients d isp lay in c e rta in cases a d e p e n d e n c e o n y Z , w hich arises m o stly from th e fa ct t h a t th e in te ra c tin g q u a rk d ire c tio n is u n k n o w n o n an ev e n t-b y -ev e n t basis. As th e m e th o d of ref. [3] relies o n in te g ra tio n over th e full p h ase space of th e a n g u la r d is trib u tio n s , it c a n n o t b e ap p lied d ire c tly to d a ta , b u t is used to c o m p u te all th e th e o re tic a l p re d ic tio n s show n in th is p a p e r.

T h e inclusive fix ed -o rd er p e r tu r b a tiv e Q C D p re d ic tio n s for Z -b o so n p ro d u c tio n a t N L O a n d N N L O w ere o b ta in e d w ith D Y N N L O v1.3 [25]. T h ese inclusive c a lc u la tio n s a re fo rm a lly a c c u ra te to 0 (0^ ). T h e Z -b oson is p ro d u c e d , how ever, a t no n -zero tra n s v e rse m o m e n tu m only a t 0 ( a s), a n d th e re fo re th e c a lc u la tio n of th e coefficients as a fu n c tio n o f p f is o n ly N L O . E v en th o u g h th e fix ed -o rd er c a lc u la tio n s d o n o t p ro v id e re lia b le ab so ­ lu te p re d ic tio n s for th e p f s p e c tru m a t low values, th e y c a n b e used for p f > 2.5 G eV fo r th e a n g u la r coefficients. T h e re su lts w ere cross-checked w ith N N L O p re d ic tio n s from F E W Z v3.1.b2 [26- 28] a n d a g re e m e n t b etw e en th e tw o p ro g ra m s w as fo u n d w ith in u n ­ c e r ta in ties. T h e re n o rm a lisa tio n a n d fa c to ris a tio n scales in th e c a lc u la tio n s w ere set to

E ZZ = (m Z )2 + ( p f )2 [29] o n a n ev e n t-b y -ev e n t basis. T h e c a lc u la tio n s w ere d o n e u sing th e C T 1 0 N L O o r N N L O p a r to n d is trib u tio n fu n c tio n s (P D F s) [30], d e p e n d in g o n th e o rd e r of th e p re d ic tio n .

T h e N L O E W c o rre c tio n s affect m o stly th e le a d in g -o rd e r Q C D cro ss-sectio n n o rm a li­

s a tio n in th e Z -p o le region a n d h av e som e im p a c t on th e p f d is trib u tio n , b u t th e y d o n o t affect th e a n g u la r c o rre la tio n s a t th e Z -b o so n v erte x . T h e D Y N N L O c a lc u la tio n w as d o n e a t lead in g o rd e r in E W , usin g th e Gß schem e [31]. T h is choice d e te rm in e s th e value o f A4 a t low p f , a n d for th e p u rp o se of th e c o m p ariso n s p re se n te d in th is p a p e r, b o th A3 an d A4 o b ta in e d from D Y N N L O a re rescaled to th e values p re d ic te d w h en u sin g th e m easu red valu e of sin2 0 f f = 0.23113 [32].

T h e th e o re tic a l p re d ic tio n s a re show n in figure 2 a n d ta b u la te d in ta b le 1 for th re e illu s tra tiv e p f bins. T h e b in n in g in p f is chosen b ase d o n th e e x p e rim e n ta l re so lu tio n a t low p f a n d on th e n u m b e r of ev e n ts a t h ig h p f a n d h as th e follow ing b o u n d a rie s (in G eV )

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

Pt = 5 —

NLO

8 GeV NNLO

pT — 22 — NLO

25.5 GeV NNLO

pT — 132 - NLO

173 GeV NNLO A 0

A2

A0 ^{— A} 2

00 1 1 5 + 0' 0006

° . 0 1 1 5 — 0.0003 00 1 1 3+0 .° 004 0 1 1 3— 0.0004 00 0 0 2 + 0.0007

° . ° 0 0 2 — 0.0005

0.0150—^

0.00 60—0.0017 0.00 90—0.0013

0 1 5 8 3 + 0.0008 0 .1 5 80—0.0 0 0g 0 .1 5 8 8 —0.0009

— 0 .0 0 0 5 _ ° .° ° 1 2

01 5 7 7 + 0.0041 0 . 1 5 M —0 .0018 0 1 1 6 1 + 0.00g2 0 .1161 — 0 .0028

+ 0 .0 0 3 6

0 4 —0 .0067

0 8 6 5 5 + 0.0008 0 . 8 6 5 5 — 0.0006 0 8632+ 0.0013 0 .8 6 3 2 — 0.0009 0 0 0 2 3 + 0.0015 0 .0 0 2 3 —0.0 0 1 1

0.8697—0'.0°26 0.8012—0■.021|

0.0685—0'.0°82 Ai

A3 A4

0 0 0 5 2 + 0 . 0004

° . ° ° 52 — 0.0 0 0 3 0 0 0 0 4 + 0 . 0002

° . °0 0 4—0.0001 0 0 7 2 Q + 0 . 0023 0 .0 ' 29 —0.0006

00 0 7 4 + 0.0020 0 . 0 0 7 4 _ 0 .0008 00 0 1 2 + 0.0003 0 .0012 —0.0006 00 7 5 7 + 0.0021 0 . 0 7 5 7 _ 0.0025

0 0301_{+ 0.0013} 0 .0 3 0 1 — 0.0013 0.0 0 6 6—0.0005 0.0 6 5 9—0.0003

+0.0014

0.0405 — 0.0038 0 007 0 + 0 .0017 0.0 0'0—0.0020 0 0 6 7 2 + 0 .0018 0 .0 6 7 2 — 0.0050

+ 0 .0 0 1 3 0 . 0 6 0 0 _ 0.0 0 1 5 00 5 4 5 + 0.0003 0 .0545 —0.0 0 1 6 O .O 2 5 3 _ ° . ° 0 0 2

+0.0018 0.0 6 1 1_0.0023 0 0 5 8 4 + 0.0018 0 . 0 58 4 — 0.0047 0 0247+0.0024 0.0247 —0.0018

A5 A6 A7

0.0001—0:0002 -0.0002—0;000!

< 0.0001

00001+ 0.0007 0 . 0 0 0 1 — 0 .0007 0 0 0 1 3 + 0.0006 0 . 0 0 1 0 —0.0005

0 0 0 1 4 + 0 .0007 0 . 0 0 1 4 —0 .0004

< 0.0001 0 0004+0.0006 0.0004_ 0.0004 0 0002+0.0003 0.0002 — 0.0007

00011^{+ 0.0013} 0.0 0 11 ^{— 0.0030} 0 0 0 1 7 + 0.0043

0 . 0 0 1 ' —0.0015 0 00 2 4 + 0.0013

0 . 0 0 2 4 — 0.0013

—0.0004—°:0°°5 0 0003+0.0003 0.0003—0.0006 0 0003+0.0004 0.0003—0.0007

0.0044—0■.0°2i 0.0028—0'.0°18 0.0048—0'.0°12

T a b le 1. Summ ary of predictions from DYNNLO at NLO and NNLO for A0, A2, A 0 — A 2, A 1, A3, A4, A5, A6, and A7 at low (5-8 GeV), mid (22-25.5 GeV), and high (132-173 GeV) pT for the y Z-integrated configuration. The uncertainty represents th e sum of statistical and system atic uncertainties.

used consistently throughout the measurement:

{0, 2.5, 5.0, 8.0, 11.4, 14.9, 18.5, 2 2 .0 , 25.5, 29.0, 32.6, 36.4, 40.4, 44.9, 50.2, 56.4, 63.9, 73.4, 85.4, 105.0, 132.0, 173.0, 253.0, 600.0}

(2.3)

The predictions show the following general features. T h e Ao and A2 coefficients in­

crease as a function of pf and the deviations from lowest-order expectations are quite large, even at modest values of pf = 20-50 GeV. The Ai and A3 coefficients are relatively small even at large p f , with a m a x i m u m value of 0.08. In the limit where pf = 0, all coefficients except A4 are expected to vanish at N L O . The N N L O corrections are typically small for all coefficients except A 2, for which the largest correction has a value of — 0.08, in agreement with the original theoretical studies [2]. The theoretical predictions for A5,6,7 are not shown because these coefficients are expected to be very small at all values of pf : they are zero at N L O and the N N L O contribution is large enough to be observable, namely of the order of 0.005 for values of pf in the range 20-200 GeV.

The statistical uncertainties of the calculations, as well as the factorisation and renor­

malisation scale and P D F uncertainties, were all considered as sources of theoretical un­

certainties. The statistical uncertainties of the N L O and N N L O predictions in absolute units are typically 0.0003 and 0.003, respectively. The larger statistical uncertainties of the N N L O predictions are due to the longer computational time required than for the N L O pre­

dictions. The scale uncertainties were estimated by varying the renormalisation and factori­

sation scales simultaneously up and d o w n by a factor of two. As stated in ref. [2], the the­

oretical uncertainties due to the choice of these scales are very small for the angular coeffi­

cients because they are ratios of cross-sections. The resulting variations of the coefficients at N N L O were found in most cases to be comparable to the statistical uncertainty. The P D F

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

p T >boundary[G eV] =

F ig u r e 2 . The angular coefficients A0-4 and the difference A0 — A 2, shown as a function of p T , as predicted from D Y N N L O calculations at NLO and NNLO in QCD. The NLO predictions for A0 — A2 are com patible w ith zero, as expected from the Lam-Tung relation [8- 10]. The error bars show the total uncertainty of the predictions, including contributions from statistical uncertainties, QCD scale variations and PDFs. The statistical uncertainties of the NNLO predictions are dom inant and an order of m agnitude larger th an those of the NLO predictions.

uncertainties were estimated using the C T 1 0 N N L O eigenvector variations, as obtained from F E W Z and normalised to 6 8% confidence level. They were found to be small compared to the N N L O statistical uncertainty, namely of the order of 0.001 for A0 - 3 and 0 . 0 0 2 for A4.

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

3 T h e A T L A S e x p e r im e n t an d its d a ta an d M o n te C arlo sa m p les

3 .1 A T L A S d e t e c t o r

T h e A T L A S e x p e rim e n t [24] a t th e L H C is a m u lti-p u rp o se p a rtic le d e te c to r w ith a fo rw a rd ­ b ac k w ard sy m m e tric c y lin d rica l g e o m e try a n d a n e a r 4 n coverage in solid a n g le. 1 It co n sists o f a n in n e r tra c k in g d e te c to r, e le c tro m a g n e tic (E M ) a n d h a d ro n ic c a lo rim e te rs, a n d a m u o n s p e c tro m e te r. T h e in n e r tra c k e r p ro vides precisio n tra c k in g o f ch a rg ed p artic le s in th e p se u d o ra p id ity ra n g e |n| < 2.5. T h is region is m a tc h e d to a h ig h -g ra n u la rity E M sa m p lin g c a lo rim e te r covering th e p s e u d o ra p id ity ra n g e |n| < 3.2 a n d a c o a rse r g ra n u la rity c a lo rim e te r u p to |n| = 4 . 9 . A h a d ro n ic c a lo rim e te r sy ste m covers th e e n tire p s e u d o ra p id ity ra n g e u p to |n| = 4.9. T h e m u o n sp e c tro m e te r p rov id es trig g e rin g a n d tra c k in g c a p a b ilitie s in th e ra n g e |n| < 2.4 a n d |n| < 2.7, respectiv ely . A first-level trig g e r is im p le m e n te d in h a rd w a re , followed by tw o so ftw a re-b a sed trig g e r levels t h a t to g e th e r re d u ce th e ac cep ted ev en t r a te to 400 Hz o n average. F o r th is p a p e r, a c e n tra l le p to n is one fo u n d in th e regio n |n| < 2.4 (exclu ding, for elec tro n s, th e e le c tro m a g n e tic c a lo rim e te r b a r re l/e n d -c a p tra n s itio n region 1.37 < |n| < 1.52), w hile a fo rw ard e le c tro n is one fo u n d in th e reg ion 2.5 <

|n| < 4.9 (e x clu d in g th e tra n s itio n regio n 3.16 < |n| < 3.35 b etw e en th e e le c tro m a g n e tic e n d -c a p a n d fo rw ard c a lo rim e te rs).

3 .2 D a t a a n d M o n t e C a r l o s a m p l e s

T h e d a t a w ere co llected by th e A T L A S d e te c to r in 2012 a t a ce n tre-o f-m ass en e rg y of

y / s = 8 TeV , a n d co rre sp o n d to an in te g ra te d lu m in o sity of 20.3 f b - 1 . T h e m e a n n u m ­ b e r of a d d itio n a l p p in te ra c tio n s p e r b u n c h cro ssin g (p ile-u p ev en ts) in th e d a t a set is a p p ro x im a te ly 2 0.

T h e sim u la tio n sam p les used in th e an a ly sis a re show n in ta b le 2 . T h e fo u r even t g e n e ra to rs used to p ro d u c e th e Z / y * ^ ^H signal ev e n ts a re liste d in ta b le 2 . T h e b aselin e P o w h e g B o x (v 1 /r2 1 2 9 ) sam p le [14- 17], w hich uses th e C T 1 0 N L O set o f P D F s [33], is in terfa ced to P y t h i a 8 (v.8.170) [34] w ith th e AU2 set of tu n e d p a ra m e te rs [35] to sim u la te th e p a r to n show er, h a d ro n is a tio n a n d u n d e rly in g ev en t, a n d to P h o t o s (v2.154) [36] to sim u la te Q E D fin a l-s ta te ra d ia tio n (F S R ) in th e Z -b o so n decay. T h e a lte r n a tiv e signal sam p les a re from P o w h e g B o x in terfa ced to H E rw iG (v.6.520.2) [37] for th e p a r to n show er a n d h a d ro n is a tio n , Jim m y (v4.31) [38] for th e u n d e rly in g ev en t, a n d P h o t o s for F S R . T h e SH ErpA (v.1.4.1) [39- 42] g e n e ra to r is also used, a n d h as its ow n im p le m e n ta tio n of th e p a r to n show er, h a d ro n isa tio n , u n d e rly in g ev en t a n d F S R , a n d uses th e C T 1 0 N L O P D F set. T h ese a lte r n a tiv e sam p les are u sed to te s t th e d e p e n d e n c e of th e an aly sis on d ifferen t m a trix -e le m e n t c a lc u la tio n s a n d p a rto n -sh o w e r m o dels, as discu ssed in sec tio n 6. T h e P o w h e g (v2.1) + M iN L O ev en t g e n e ra to r [43] w as u sed for th e Z + j e t p ro cess a t N LO to n o rm alise c e rta in reference coefficients for th e e e CF an aly sis, as d e sc rib e d in sec tio n 5.

T h e n u m b e r of ev e n ts av ailab le in th e b aselin e P o w h e g B o x + P y t h i a 8 signal sam p le c o rre sp o n d s to a p p ro x im a te ly 4 (25) tim e s t h a t in th e d a t a below (above) p f = 105 G eV .

1ATLAS 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 upwards. Cylindrical coordinates (r, Y) are used in the transverse

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

Signature G enerator PD F Refs.

Z /7* ^ t t P o w h e g B o x + P y t h i a 8 C T 1 0 NLO [14- 17, 33, 34]

Z /7* ^ t t P o w h e g B o x + Jim m y/H e r w ig C T 1 0 NLO [37]

Z /7* ^ t t S h e rp a C T 1 0 NLO [39- 42]

Z /7* ^ t t + jet P o w h e g + M iN LO C T 1 0 NLO [43]

W ^ tv P o w h e g B o x + P y t h i a 8 C T 1 0 NLO

W ^ tv S h e rp a C T 1 0 NLO

t t pair M C@ NLO + J im m y /H e rw ig C T 1 0 NLO [38, 46]

Single top quark:

t channel A c e rM C + P y t h i a 6 C T E Q 6 L 1 [47, 48]

s and W t channels M C@ NLO + J im m y /H e rw ig C T 1 0 NLO

Dibosons S h e rp a C T 1 0 NLO

Dibosons H e rw ig C T E Q 6 L 1

7 7 ^ t t P y t h i a 8 M R S T 2 0 0 4 Q E D NLO [49]

T a b le 2. MC samples used to estim ate the signal and backgrounds in the analysis.

B a c k g ro u n d s from E W (d ib o so n an d 7 7 ^ t t p ro d u c tio n ) a n d to p -q u a r k (p ro d u c tio n o f to p -q u a r k p a irs a n d o f single to p q u a rk s) processes are e v a lu a te d from th e M C sam p les liste d in ta b le 2 . T h e W + je ts c o n trib u tio n to th e b a c k g ro u n d is in ste a d in clu d ed in th e d a ta -d riv e n m u ltije t b a c k g ro u n d e s tim a te , as d esc rib ed in sec tio n 4 ; W -b o so n sam p les liste d in ta b le 2 a re th u s o n ly used fo r stu d ie s of th e b a c k g ro u n d co m p o sitio n .

A ll of th e sam p les a re processed w ith th e G e a n t 4 - b a s e d sim u la tio n [44] o f th e A T L A S d e te c to r [45]. T h e effects o f a d d itio n a l pp collisions in th e sam e o r n e a rb y b u n c h crossings a re sim u la te d by th e a d d itio n of so-called m in im u m -b ia s ev e n ts g e n e ra te d w ith P y t h i a 8.

4 D a ta a n a ly sis

4 .1 E v e n t s e l e c t i o n

As m e n tio n e d in sectio n s 1 a n d 3 , th e d a t a a re sp lit in to th re e o rth o g o n a l ch an n els, n am ely th e e e c c c h a n n e l w ith tw o c e n tra l elec tro n s, th e ^ c c ch a n n el w ith tw o c e n tra l m uons, a n d th e eecF ch a n n el w ith one c e n tra l e le c tro n a n d one fo rw ard elec tro n . S elected ev en ts a re re q u ire d to b e in a d a ta - ta k in g p e rio d in w h ich th e b ea m s w ere s ta b le a n d th e d e te c to r w as fu n c tio n in g well, a n d to c o n ta in a re c o n s tru c te d p rim a ry v e rte x w ith a t le a st th re e tra c k s w ith p T > 0.4 G eV .

C a n d id a te eec c ev en ts a re o b ta in e d u sin g a logical O R of a d ie le c tro n trig g e r re q u irin g tw o e le c tro n c a n d id a te s w ith p T > 12 G eV a n d of tw o h ig h -p T sin g le-electro n trig g e rs (th e m a in one co rre sp o n d in g to a p T th re s h o ld of 24 G eV ). E le c tro n c a n d id a te s a re re q u ire d to h ave p t > 25 G eV a n d a re re c o n s tru c te d from clu ste rs of en e rg y in th e ele c tro m a g n e tic c a lo rim e te r m a tc h e d to in n e r d e te c to r tra c k s. T h e e le c tro n c a n d id a te s m u st sa tisfy a set o f “m e d iu m ” selectio n c rite ria [5 0 , 51], w hich have b ee n o p tim ise d for th e level o f p ile-u p p re se n t in th e 2012 d a ta . E v e n ts a re re q u ired to c o n ta in e x a c tly tw o e le c tro n c a n d id a te s o f o p p o site ch a rg e sa tisfy in g th e abo ve c rite ria .

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

C a n d id a te ^ c c ev en ts are re ta in e d fo r an a ly sis u sin g a logical O R o f a d im u o n tr ig ­ g er re q u irin g tw o m u o n c a n d id a te s w ith p f > 18 G eV a n d 8 G eV , respectively, a n d of tw o h ig h -p f single-m uo n trig g e rs (th e m a in one co rre sp o n d in g to a p f th re s h o ld of 24 G eV ).

M u o n c a n d id a te s are re q u ired to have p f > 25 G eV a n d are iden tified as tra c k s in th e in n er d e te c to r w hich are m a tc h e d a n d com b in ed w ith tra c k seg m en ts in th e m u o n s p e c tro m e ­ te r [52]. T ra c k -q u a lity a n d lo n g itu d in a l a n d tra n s v e rs e im p a c t-p a ra m e te r re q u ire m e n ts are im p o sed for m u o n id e n tific a tio n to su p p re ss b ac k g ro u n d s, a n d to en su re t h a t th e m u on c a n d id a te s o rig in a te from a co m m o n p rim a ry p p in te ra c tio n v e rte x . E v e n ts are re q u ire d to c o n ta in e x a c tly tw o m u o n c a n d id a te s of o p p o site ch a rg e sa tisfy in g th e abov e c rite ria .

C a n d id a te eecF ev e n ts are o b ta in e d u sin g th e logical O R of th e tw o h ig h -p f single­

e le c tro n trig g e rs u sed for th e eec c ev en ts, as d esc rib ed above. T h e c e n tra l e le c tro n c a n ­ d id a te is re q u ire d to have p f > 25 G eV . B ecau se th e e x p e c te d b a c k g ro u n d fro m m u ltije t ev e n ts is la rg e r in th is c h a n n el th a n in th e e e c c ch a n n el, th e c e n tra l e le c tro n c a n d id a te is re q u ire d to sa tisfy a set of “t ig h t” selectio n c r ite ria [50], w hich a re o p tim ise d for th e level o f p ile-u p o b serv ed in th e 2012 d a ta . T h e fo rw ard e le c tro n c a n d id a te is re q u ired to have p f > 20 G eV a n d to sa tisfy a set of “m e d iu m ” selectio n c rite ria , b ase d o nly o n th e show er sh a p e s in th e e le c tro m a g n e tic c a lo rim e te r [50] since th is regio n is o u ts id e th e a c c e p ta n c e of th e in n e r tra c k e r. E v e n ts a re re q u ire d to c o n ta in e x a c tly tw o e le c tro n c a n d id a te s satisfy in g th e abo v e c rite ria .

Since th is an a ly sis is focused o n th e Z -b o so n pole region, th e le p to n p a ir is re q u ired to have a n in v a ria n t m ass (m??) w ith in a n a rro w w in do w a ro u n d th e Z -b o so n m ass, 80 < m?? < 100 G eV . E v e n ts a re selected for yZ-in te g ra te d m e a su re m e n ts w ith o u t an y re­

q u ire m e n ts on th e ra p id ity o f th e le p to n p a ir (y??). F o r th e y Z-b in n e d m e a su re m e n ts, ev en ts a re selected in th re e bins of ra p id ity : |y??| < 1.0, 1.0 < |y??| < 2.0, a n d 2.0 < |y??| < 3.5.

E v e n ts are also re q u ire d to have a d ile p to n tra n s v e rs e m o m e n tu m (pf?) less th a n th e value o f 600 (100) G eV u sed for th e h ig h est bin in th e e e c c a n d ^ c c (e ec F ) ch an n els. T h e v aria b les m??, y??, a n d pf?, w hich are defined u sin g re c o n s tru c te d le p to n p airs, a re to be d istin g u ish e d from th e v aria b les m Z , y Z , a n d p f , w hich are defined u sin g le p to n p a irs a t th e B o rn level, as d e sc rib e d in sec tio n 2 .

T h e sim u la te d ev e n ts are re q u ired to sa tisfy th e sam e selectio n c rite ria , a fte r a p p ly in g sm all c o rre c tio n s to a c c o u n t for th e differences b etw e en d a t a a n d sim u la tio n in te rm s of re c o n stru c tio n , id e n tific a tio n a n d trig g e r efficiencies a n d o f en e rg y scale a n d re so lu tio n for e le c tro n s a n d m uons [50- 53]. All sim u la te d ev e n ts are rew eig h ted to m a tc h th e d is trib u tio n s o b serv ed in d a t a for th e level o f p ile-u p a n d for th e p rim a ry v e rte x lo n g itu d in a l p o sitio n .

F ig u re 3 illu s tra te s th e d ifferent ra n g es in p f a n d y Z e x p e c te d to b e covered by th e th re e ch a n n els alo n g w ith th e ir a c c e p ta n c e tim e s selectio n efficiencies, w hich is d efined as th e ra tio of th e n u m b e r o f selected ev e n ts to th e n u m b e r in th e full p h a se space. T h e difference in sh a p e b etw e en th e eec c a n d ^ c c ch a n n els arises from th e low er re c o n s tru c tio n an d id en tific a tio n efficiency for c e n tra l e lec tro n s a t high values o f |n| a n d from th e low er trig g e r a n d re c o n stru c tio n efficiency for m uon s a t low values o f |n |. T h e c e n tra l-c e n tra l a n d c e n tra l­

fo rw ard ch a n n els o v erla p in th e reg ion 1.5 < |y Z | < 2.5.

4 .2 B a c k g r o u n d s

In th e Z -b o so n pole region, th e b ac k g ro u n d s fro m o th e r pro cesses a re sm all, below th e h alf­

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

F ig u r e 3. Comparison of the expected yields (left) and acceptance times efficiency of selected events (right) as a function of yZ (top) and pT (bottom ), for the eec c , MMcc, and eecF events.

Also shown are the expected yields at the event generator level over the full phase space considered for th e measurem ent, which corresponds to all events with a dilepton mass in the chosen window, 80 < m Z < 100 GeV.

The backgrounds from prompt isolated lepton pairs are estimated using simulated sa m ­ ples, as described in section 3, and consist predominantly of lepton pairs from top-quark processes and from diboson production with a smaller contribution from Z ^ t t decays.

The other background source arises from events in which at least one of the lepton candi­

dates is not a prompt isolated lepton but rather a lepton from heavy-flavour hadron decay (beauty or charm) or a fake lepton in the case of electron candidates (these m a y arise from charged hadrons or from photon conversions within a hadronic jet). This background consists of events containing two such leptons (multijets) or one such lepton ( W + jets or top-quark pairs) and is estimated from data using the lepton isolation as a discriminating variable, a procedure described for example in ref. [50] for electrons. For the central-central channels, the background determination is carried out in the full two-dimensional space of (cos0cs,^cs) and in each bin of p f In the case of the central-forward channel, the multijet background, which is by far the dominant one, is estimated separately for each projection in cos 0cs and 0 cs because of the limited amount of data. This is the main reason w h y the angular coefficients in the central-forward channel are extracted only in projections, as described in section 1.

Figure 4 shows the angular distributions, cos 0cs and 0 cs, for the three channels for the data, the Z-boson signal M C sample, and the main sources of background discussed

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

F ig u r e 4 . The cos 0cs (left) and ^ cs (right) angular distributions, averaged over all Z-boson p T, for the eec c (top), p,p,c c (middle) and eeCF (bottom ) channels. The distributions are shown separately for the different background sources contributing to each channel. The m ultijet background is determ ined from data, as explained in the text.

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

Channel Observed Expected background

M ultijets (from data) Top+electroweak (from MC) Total eecc

MMcc eecF

5.5 x 106 7.0 x 106 1.5 x 106

6000 ± 3000 9000 ± 4000 28000 ± 14000

13000 ± 3000 19000 ± 4000 1000 ± 200

19000 ± 4000 28000 ± 6000 29000 ± 14000 T a b le 3. For each of the three channels, yield of events observed in d a ta and expected background yields (m ultijets, top+electroweak, and total) corresponding to the 2012 d a ta set and an integrated luminosity of 20.3 fb- 1 . The uncertainties quoted include b oth the statistical and system atic com­

ponents (see text).

above. T h e to ta l b a c k g ro u n d in th e c e n tra l-c e n tra l ev en ts is below 0.5% a n d its u n c e rta in ty is d o m in a te d by th e larg e u n c e rta in ty in th e m u ltije t b a c k g ro u n d o f a p p ro x im a te ly 50%.

T h e u n c e rta in ty in th e to p + e le c tro w e a k b a c k g ro u n d is ta k e n co n se rv ativ ely to be 20%. In th e case of th e c e n tra l-fo rw a rd e le c tro n p a irs, th e to p + e le c tro w e a k b a c k g ro u n d is so sm all c o m p a re d to th e m uch la rg e r m u ltije t b ac k g ro u n d t h a t it is n eg lected for sim p licity in th e fit p ro c e d u re d esc rib ed in sec tio n 5 . T ab le 3 su m m arises th e o b serv ed yields o f ev en ts in d a ta for each ch a n n el, in te g ra te d over all values of p f to g e th e r w ith th e e x p e c te d b a c k g ro u n d yields w ith th e ir to ta l u n c e rta in tie s from m u ltije t ev en ts an d from to p + e le c tro w e a k sources.

M o re d e ta ils of th e tr e a tm e n t of th e b a c k g ro u n d u n c e rta in tie s a re d iscu ssed in sec tio n 6 . T h e re are also signal ev e n ts t h a t a re co n sid ered as b a c k g ro u n d to th e m e a su re m e n t b ec au se th e y are p re se n t in th e d a ta o nly d u e to th e fin ite re so lu tio n o f th e m e a su re m e n ts, w hich leads to m ig ra tio n s in m ass a n d ra p id ity . T h ese a re d e n o te d “N o n-fid ucial Z ” ev en ts a n d c a n b e d iv id ed in to fo u r categ o ries: th e d o m in a n t fra c tio n co n sists o f ev e n ts t h a t have m Z a t th e g e n e ra to r level o u ts id e th e chosen m f l m ass w indow b u t p ass ev en t selection, w hile a n o th e r c o n trib u tio n arises from ev en ts t h a t d o n o t b elo n g to th e y^Z b in co n sid ered for th e m e a su re m e n t a t g e n e ra to r level. T h e l a t t e r c o n trib u tio n is sizeable o n ly in th e e e cF ch a n n el. O th e r negligible sources of th is ty p e of b a c k g ro u n d arise from ev e n ts for w hich th e c e n tra l e le c tro n h as th e w ro n g assign ed ch arg e in th e eecF c h a n n el o r b o th c e n tra l e lec tro n s have th e w ro n g assig n ed ch a rg e in th e e e c c ch a n n el, o r for w hich p f a t th e g e n e ra to r level is la rg e r th a n 600 G eV . T h ese b ac k g ro u n d s are all in clu d ed as a sm all c o m p o n e n t o f th e signal M C sam p le in figure 4 . T h e ir c o n trib u tio n s a m o u n t to o ne p e rc e n t o r less for th e eec c a n d ^ c c ch an n els, in cre asin g to a lm o st 8% for th e e e cF ch a n n el b ec au se o f th e m uch la rg e r m ig ra tio n s in en e rg y m e a su re m e n ts in th e case of fo rw ard e lec tro n s. F or th e 2 < |y Z | < 3.5 b in in th e eecF ch a n n el, th e y Z m ig ra tio n c o n trib u te s 2%

to th e non-fiducial Z b ac k g ro u n d . T h e fra c tio n a l c o n trib u tio n o f all b ac k g ro u n d s to th e t o ta l sam p le is show n e x p lic itly for each ch a n n el as a fu n c tio n of pf? in figure 5 to g e th e r w ith th e re sp ectiv e c o n trib u tio n s of th e m u ltije t a n d to p + e le c tro w e a k b ac k g ro u n d s. T h e sum of all th e se b a c k g ro u n d s is also show n a n d te m p la te s of th e ir a n g u la r d is trib u tio n s are used in th e fit to e x tra c t th e a n g u la r coefficients, as d e sc rib e d in sec tio n 5 .

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

PT [GeV]

F ig u r e 5. Fractional background contributions as a function of p r , for the eec c (top), c c (middle) and eecF (bottom ) channels. The distributions are shown separately for the relevant background contributions to each channel together w ith the summed to tal background fraction.

The label “Non-fiducial Z ” refers to signal events which are generated outside the phase space used to extract the angular coefficients (see text).

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

4 .3 A n g u l a r d i s t r i b u t i o n s

T h e m e a su re m e n t of th e a n g u la r coefficients is p erfo rm ed in fine bins o f p f a n d for a fixed d ile p to n m ass w indow o n th e sam e sam p le as t h a t u sed to e x tra c t from d a t a th e sm all c o rre c tio n s a p p lie d to th e le p to n efficiencies a n d c a lib ra tio n . T h e an a ly sis is th u s largely in sen sitiv e to th e sh a p e of th e d is trib u tio n of p f , a n d also to an y re sid u a l differences in th e m o d ellin g o f th e sh a p e of th e d ile p to n m ass d is trib u tio n . I t is, how ever, im p o rta n t to verify q u a lita tiv e ly th e level of a g reem en t b etw e en d a t a a n d M C sim u la tio n for th e cos 0c s a n d 0 c s a n g u la r d is trib u tio n s befo re e x tra c tin g th e re su lts of th e m e a su re m e n t. T h is is show n for th e th re e ch a n n els s e p a ra te ly in figure 6, to g e th e r w ith th e ra tio of th e o b serv ed d a t a to th e sum o f p re d ic te d ev en ts. T h e d a t a a n d M C d is trib u tio n s are n o t n o rm alised to each o th e r, re su ltin g in n o rm a lis a tio n differences a t th e level of a few p e rc e n t. T h e m e a su re m e n t o f th e a n g u la r coefficients is, how ever, in d e p e n d e n t of th e n o rm a lisa tio n b etw e en d a t a an d sim u la tio n in each bin of p f . T h e differences in sh a p e in th e a n g u la r d is trib u tio n s reflect th e m ism o d e llin g o f th e a n g u la r coefficients in th e sim u la tio n (see sec tio n 7) .

5 C o efficien t m e a su r em e n t m e th o d o lo g y

T h e coefficients are e x tra c te d from th e d a t a by fittin g te m p la te s of th e P i p o ly n o m ia l te rm s, defined in eq. ( 1.1) , to th e re c o n s tru c te d a n g u la r d is trib u tio n s . E a c h te m p la te is n o rm alised by free p a ra m e te rs for its co rre sp o n d in g coefficient Ai , as well as a n a d d itio n a l co m m o n p a r a m e te r re p re se n tin g th e u n p o la rise d cro ss-sectio n . All of th e se p a ra m e te rs are defined in d e p e n d e n tly in each b in of p f . T h e p o ly n o m ia l P8 = 1 + co s2 0c s in eq. ( 1.1) is o n ly n o rm alised by th e p a r a m e te r for th e u n p o la ris e d cross-section .

In th e ab se n ce of selectio ns for th e fin a l-s ta te lep to n s, th e a n g u la r d is trib u tio n s in th e g au g e -b o so n re st fram e are d e te rm in e d by th e g au g e -b o so n p o la ris a tio n . In th e p resen ce o f selectio n c rite ria for th e lep to n s, th e d is trib u tio n s a re s c u lp te d by k in e m a tic effects, a n d c a n no lo n g er be d esc rib ed by th e su m of th e n in e P i p o ly n o m ia ls as in eq. ( 1.1) . T e m p la te s of th e P i te rm s are c o n s tru c te d in a w ay to ac c o u n t for th is, w h ich re q u ires fully sim u la te d signal M C to m odel th e ac c e p ta n c e , efficiency, a n d m ig ra tio n of ev en ts. T h is pro cess is d e sc rib e d in sec tio n 5 .1 . S ectio n 5.2 th e n d escrib es th e likelihood t h a t is b u ilt o u t o f th e te m p la te s a n d m ax im ise d to o b ta in th e m e a su re d coefficients. T h e m e th o d o lo g y for o b ta in in g u n c e rta in tie s in th e m e a su re d p a ra m e te rs is also covered th e re . T h e p ro c e d u re fo r co m b in in g m u ltip le ch a n n els is covered in sec tio n 5 .3 , alo ng w ith a lte r n a tiv e coefficient p a ra m e te ris a tio n s u sed in v ario u s te s ts o f m e a su re m e n t re su lts fro m d ifferen t ch an n els.

5 .1 T e m p l a t e s

To b u ild th e te m p la te s of th e P i p o ly n o m ials, th e referen ce coefficients AJJef for th e sig­

n al M C sam p le a re first c a lc u la te d w ith th e m o m en ts m e th o d , as d e sc rib e d in sec tio n 2 a n d eq. (2.2) . T h e se are o b ta in e d in each o f th e 23 p f bins in eq. (2.3) , a n d also in each o f th e th re e y Z b in s for th e y Z-b in n e d m e a su re m e n ts. T h e in fo rm a tio n a b o u t th e a n g u la r coefficients in th e sim u la tio n is th e n availab le th ro u g h th e co rre sp o n d in g fu n c tio n a l form o f eq. ( 1.1) . N e x t, th e M C ev en t w eights a re d iv id ed by th e v alu e of th is fu n c tio n o n an

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

F ig u r e 6 . The cos 0cs (left) and ^ cs (right) angular distributions, averaged over all p ^ , for the eec c (top), p p c c (middle) and eecF (bottom ) channels. In th e panels showing the ratios of the d a ta to the summed signal+background predictions, the uncertainty bars on the points are only statistical.

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

ev e n t-b y -ev e n t basis. W h e n th e M C ev en ts are w eig h ted in th is way, th e a n g u la r d is trib u ­ tio n s in th e full p h a se space a t th e ev e n t g e n e ra to r level a re flat. Effectively, all in fo rm a tio n a b o u t th e Z -b o so n p o la ris a tio n is rem oved from th e M C sam p le, so t h a t fu r th e r w eig h tin g th e ev en ts by an y of th e P j te rm s yield s th e sh a p e o f th e p o ly n o m ia l itself, a n d if selectio n re q u ire m e n ts are ap p lied , th is yields th e sh a p e of th e selectio n efficiency. T h e selection re q u ire m e n ts, co rrec tio n s, a n d ev en t w eights m e n tio n e d in sec tio n 4 are th e n ap p lied . N ine s e p a ra te te m p la te h isto g ra m s for each pT a n d b in j a t g e n e ra to r level are finally o b ta in e d a fte r w eig h tin g by each of th e P i te rm s. T h e te m p la te s are th u s th re e -d im e n sio n a l d is­

tr ib u tio n s in th e m e a su re d cos 0CS, ^ c s , a n d p T v ariab les, a n d a re c o n s tru c te d for each pT a n d bin. E ig h t b ins in cos 0CS a n d 0 CS are used, w hile th e b in n in g for th e re c o n s tru c te d p T is th e sam e as for th e 23 bins d efined in eq. (2.3) . B y c o n s tru c tio n , th e sum o f all signal te m p la te s n o rm alised by th e ir referen ce coefficients a n d u n p o la rise d cro ss-sectio ns agrees e x a c tly w ith th e th re e -d im e n sio n a l re c o n s tru c te d d is trib u tio n e x p e c te d for sign al M C ev en ts. E x a m p le s o f te m p la te s p ro je c te d o n to each o f th e d im en sio n s cos 0CS a n d 0 CS for th e -in te g ra te d eeCC ch a n n e l in th re e illu s tra tiv e pT ran g es, alon g w ith th e ir c o rre­

sp o n d in g p o ly n o m ia l sh ap e s, are show n in figure 7. T h e p o ly n o m ia ls P i a n d P6 are n o t show n as th e y in te g ra te to zero in th e full p h a se sp ace in e ith e r p ro je c tio n (see sec tio n 5.2) . T h e effect o f th e a c c e p ta n c e on th e p o ly n o m ia l sh a p e d e p e n d s on pT b ec au se o f th e ev en t se­

lection , as ca n be seen from th e d ifference b etw e en th e te m p la te p o ly n o m ia l sh ap e s in each co rre sp o n d in g pT bin. T h is is p a rtic u la rly visible in th e P8 p o ly n o m ial, w hich is u n ifo rm in 0CS, a n d th e re fo re reflects e x a c tly th e a c c e p ta n c e s h a p e in th e te m p la te d p o ly n o m ials.

In a p p e n d ix B , tw o -d im en sio n al version s o f figure 7 a re given for all nin e p o ly n o m ia ls in fig­

ures 21- 2 3 . T h ese tw o -d im en sio n al view s are re q u ired for P1 a n d P 6, as discu ssed above.

T e m p la te s T B are also b u ilt for each o f th e m u ltije t, to p + e le c tro w e a k , a n d n o n-fidu cial Z -b o so n b a c k g ro u n d s d iscu ssed in sec tio n 4 .2 . T h e se a re n o rm alised by th e ir re sp ectiv e cro ss-sectio n s tim e s lum in osity, or d a ta -d riv e n e s tim a te s in th e case o f th e m u ltije t b ac k ­ g ro u n d . T h e te m p la te s for th e p ro je c tio n m e a su re m e n ts in th e eeCF ch a n n el are in te g ra te d over e ith e r th e cos 0CS o r 0CS axis a t th e en d of th e process.

T e m p la te s co rre sp o n d in g to v a ria tio n s of th e sy s te m a tic u n c e rta in tie s in th e d e te c to r re sp o n se as well as in th e th e o re tic a l m o d ellin g are b u ilt in th e sam e way, a fte r v ary in g th e re le v an t so urce of sy ste m a tic u n c e rta in ty by ± 1 s ta n d a r d d e v ia tio n (a ) . If such a v a ria tio n chan g es th e coefficients in th e M C p re d ic tio n , for ex a m p le in th e case of P D F o r p a r to n show er u n c e rta in tie s , th e v aried AJjef coefficients are used as such in th e w eig h tin g p ro c e d u re . In th is way, th e th e o re tic a l u n c e rta in tie s on th e p re d ic tio n s a re n o t d ire c tly p ro p a g a te d to th e u n c e rta in tie s o n th e m e a su re d A coefficients. H ow ever, th e y m ay affect in d ire c tly th e m e a su re m e n ts th ro u g h th e ir im p a c t o n th e a c c e p ta n c e , selectio n efficiency, a n d m ig ra tio n m odelling.

5 .2 L i k e l i h o o d

A likelihood is b u ilt from th e n o m in al te m p la te s a n d th e v aried te m p la te s reflectin g th e s y s te m a tic u n c e rta in tie s. A set of n u isan ce p a ra m e te rs (N P s) d = { ß ,Y } is used to in te r­

p o la te b etw e en th e m . T h ese a re c o n s tra in e d by a u x ilia ry p ro b a b ility d e n s ity fu n c tio n s an d com e in tw o catego ries: ß a n d y . T h e first c a te g o ry ß are th e N P s re p re se n tin g ex p e rim en -

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

F ig u r e 7. Shapes of polynomials Po,4,8 as a function of cos 0cs (top left) and P2,3,5,7,8 as a function of ^ cs (top right). Shown below are the tem plated polynomials for the yZ-integrated eec c events at low (5-8 GeV), m edium (22-25.5 GeV), and high (132-173 GeV) values of pT projected onto each of the dimensions cos 0cs and ^ c s . The pT dimension th a t norm ally enters through m igrations is also integrated over. The differences between the polynomials and the tem plates reflect the acceptance shape after event selection.

J H E P 0 8 ( 2 0 1 6 ) 1 5 9

tal and theoretical uncertainties. Each ßm in the set ß = { ß1,...,ßM } are constrained by unit Gaussian probability density functions G(0|ßm , 1) and linearly interpolate between the nominal and varied templates. These are defined to have a nominal value of zero, with ß m = ± 1 corresponding to ± 1 o for the systematic uncertainty under consideration. The total number of ß m is M = 171 for the eecc + W c c channel and M = 105 for the eeCF channel. The second category 7 are N P s that handle systematic uncertainties from the limited size of the M C samples. For each bin n in the reconstructed cos 0CS, 0 Cs, and p T distribution, yn in the set 7 = {71,... ,7M m s j , where N bins = 8 x 8 x 23 is the total number of bins in the reconstructed distribution, has a nominal value of one and normalises the expected events in bin n of the templates. They are constrained by Poisson probability density functions P (Nff |7n Neff), where Nff is the effective number of M C events in bin n. The meaning of “effective” here refers to corrections applied for non-uniform event weights. W h e n all signal and background templates are s u m m e d over with their respective normalisations, the expected events N x p in each bin n can be written as:

• A ij : coefficient parameter for pT bin j

• A: set of all A j

• Oj : signal cross-section parameter

• o: set of all Oj

• 0: set of all N P s

• ß: set of all Gaussian-constrained N P s

• Yn : Poisson-constrained N P

• tij : Pi template

• T B : background templates

• L: integrated luminosity constant.

The summation over the index j takes into account the contribution of all pT bins at generator level in each reconstructed p^ bin. This is necessary to account for migrations in pTp. The likelihood is the product of Poisson probabilities across all N bins bins and of auxiliary constraints for each nuisance parameter ß m :

{ ^{£ 23}

L x tn (ß) + £ Aij x tn (ß ) + £ TB (ß ) x ^{r 7 1 bkgs ï}

n , (5.1)

j=1 L

0 J

)

where:

N

C ( A , a , e \ N 0 b s ) = n {P(NO bs|NeXp(A, a , d ) ) P (Nff |7nNen ff)} x ^ G(0|ßm , 1). (5.2)

n m

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