Search for anomalous couplings in the W tb vertex from the measurement of double differential angular decay rates of single top quarks produced in the t-channel 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: October 14, 2015

R e v i s e d: February 5, 2016 A c c e p t e d: M arch 2, 2016 P u b l i s h e d: A pril 5, 2016

Search for anomalous couplings in the W tb vertex from the measurement of double differential angular decay rates of single top quarks produced in the t -channel with the ATLAS detector

T h e A T LA S collaboration

E - m a i l : 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 : T he electroweak p ro d uction and subsequent decay of single to p quarks is de­

term ined by th e prop erties of th e ^{W tb} vertex. This vertex can be described by th e com plex p aram eters of an effective L agrangian. An analysis of angular d istrib u tio n s of th e decay p ro d u cts of single to p quarks produced in th e t-channel con strains these p aram eters si­

m ultaneously. T he analysis described in th is pap er uses 4.6 fb -1 of p ro to n -p ro to n collision d a ta a t y f s = 7T eV collected w ith th e ATLAS d etecto r at th e LHC. Tw o param eters are m easured sim ultaneously in this analysis. T he fraction f 1 of decays containing transversely polarised W bosons is m easured to be 0.37 ± 0.07 (sta t.® sy st.). T he phase ^{S -} betw een am plitu des for transversely and longitudinally polarised W bosons recoiling against left­

h anded b-quarks is m easured to be —0.014n ± 0.036n (sta t.® sy st.). T he correlation in th e m easurem ent of these p aram eters is 0.15. These values result in tw o-dim ensional lim its a t th e 95% confidence level on th e ratio of th e com plex coupling p aram eters g R and VL, yielding Re[gR/V L] € [—0.36, 0.10] and Im[gR/V L] € [—0.17, 0.23] w ith a correlation of 0.11.

T he results are in good agreem ent w ith th e predictions of th e S ta n d ard M odel.

Ke y w o r d s: H adron-H adron scatterin g ArXiy ePr in t: 1510.03764

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C o n te n ts

1 In tro d u ctio n 1

2 M ea su rem en t d efin itio n 3

3 T h e A T L A S d ete c to r 6

4 D a ta and sim u la ted sam p les 7

5 O b je c t d efin itio n s and even t selec tio n 8

6 B ack g ro u n d e stim a tio n and ev en t y ie ld s 10

7 A n a ly sis m eth o d 11

8 S ou rces of sy ste m a tic u n certa in ty 17

8.1 O bject m odelling 17

8.2 MC generators and P D F s 17

8.3 Signal and background norm alisation 18

8.4 D etecto r correction and background p aram eterisatio n 18

8.5 U n certain ty com bination 18

9 R e s u lts 18

10 C on clu sio n 21

A P a ra m e ter d ep en d en ce of th e fo ld in g and background m o d els 23

T h e A T L A S co lla b o ra tio n 29

1 I n tr o d u c t io n

T he to p q u ark is th e heaviest known fundam ental particle, m aking th e m easurem ent of its p ro d u ctio n and decay kinem atics a unique probe of physical processes beyond th e S tan d ard M odel (SM). T he to p q u ark was first observed in 1995 a t th e T evatron [1, 2] th ro u g h its d om in ant p ro d u ctio n channel, to p -q u a rk pair (ft) pro d uctio n via th e flavour-conserving strong interaction. An altern ativ e process produces single to p quarks th ro u g h th e weak interaction, first observed a t th e Tevatron in 2009 [3 , 4].

T he f-channel exchange of a v irtu al W boson is th e dom inant process con trib u tin g to single to p -q u ark p ro d u ctio n (see figure 1) . T he pro du ctio n cross-section is calculated for

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F ig u re 1. Representative Feynman diagrams for t-channel single top-quark production and decay.

Here q represents a u or d quark, and q' represents a d or U quark, respectively. The initial b-quark arises from (a) a sea b-quark in the 2 ^ 2 process, or (b) a gluon splitting into a bb pair in the 2 ^ 3 process.

p ro to n -p ro to n (pp) collisions a t y f s = 7 TeV using a next-to-leading-order (NLO) QCD pre­

diction w ith resum m ed next-to -n ex t-to-leading logarithm ic (NNLL) accuracy, referred to as appro xim ate next-to -nex t-to -lead in g order (NNLO). W ith a to p -q u ark m ass of 172.5 GeV and M S T W 2 0 0 8 N N L O [5] p a rto n d istrib u tio n function (P D F ) sets, th e cross-section for th e t-channel process is calculated to be 6 4 .6 -^7 pb [6]. T he un certain ties correspond to th e sum in q u a d ra tu re of th e error o b tained from th e M ST W P D F set at th e 90%

confidence level (C.L.) and th e factorisation and renorm alisation scale u ncertainties.

New physics resulting in corrections to th e W tb v ertex would affect t-channel single to p -q u a rk produ ction and decay, and can be probed th ro u g h processes which affect vari­

ables sensitive to th e angu lar d istrib u tio n s of th e final-state particles from th e to p -q u ark decay. W ith in th e effective field th eo ry fram ew ork, th e L agrangian can be expressed in full generality as [7, 8]:

L Wtb = (VLP L + VRP R) ^{t W -} g=5--- ^ (gLP L + gRP R) ^{t W -} + h X^ (1.1)

V 2 a/2 ^{m w}

w here g is th e weak coupling co n stan t, and m w and qv are th e m ass and th e four-m om entum of th e W boson, respectively. P L,R = (1 ^ 7 5) /2 are th e left- and rig ht-hand ed projection o p erato rs and = -[y % y v] / 2. VL,R and gL,R are th e com plex left- and right-handed vector and ten so r couplings, respectively. T he couplings could also be expressed using th e W ilson coefficients [9] of th e relevant dim ension-six o p e ra to rs,1 described in refs. [7, 10].

In th e SM a t tre e level, ^Vl = ^{V tb}, which is an elem ent in th e C abibbo-K obayashi-M askaw a (CKM ) m atrix, while th e anom alous couplings Vr = gL,R = 0. D eviations from these values would provide h ints of physics beyond th e SM, and com plex values could im ply th a t to p -q u a rk decay has a C P-violating com ponent [11].

Hn general the couplings can depend on the momentum transfer q2. Since this analysis is mainly sensitive to top-quark decays to an on-shell ^W boson, where ^q2 = mW, no ^q2 dependence is considered.

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Indirect co n strain ts on VR, gL, and gR were obtained [12, 13] from precision m easure­

m ents of B -m eson decays. These result in very tig h t co n strain ts on VR and b u t m uch looser co n strain ts on gR. C alculations of th e anom alous couplings in specific m odels pre­

dict a m uch larger co n trib u tio n to gR th a n to VR or [14]. However, w ithin th e m odels studied so far for ten so r couplings, th e m agnitude of gR is expected to be < 0.01 [14, 15].

M easurem ents of th e W boson po larisation fractions [16- 19] are sensitive to th e m agni­

tu d e of com binations of anom alous couplings. In order to o b tain co n strain ts on R e [gR] in those m easurem ents, th e anom alous couplings are tak en to be purely real, VR = = 0 is assum ed and VL is fixed to 1. In this analysis, th e anom alous couplings VL and gR are allowed to be complex, and th e ratio s Re [gR/V L] and Im [gR/V L] are m easured.

T his p a p e r is organised as follows. Section 2 describes th e coord inate system and pa- ram eterisatio n used in th e m easurem ent. Section 3 gives a short description of th e ATLAS d etecto r, th e n section 4 describes th e sim ulated sam ples used to predict prop erties of th e t-channel signal and th e background processes, and th e d a ta sam ples used. Section 5 describes th e definitions of th e objects used to identify t-channel events, and th e set of c riteria applied to reduce th e num ber of background events. T he procedures for m odelling background processes are described in section 6 , along w ith com parisons betw een th e pre­

dictions and d a ta . Section 7 describes th e efficiency, resolution, and background m odels used to tra n s la te th e d istrib u tio n of tru e t-channel signal events to th e d istrib u tio n of recon­

stru c te d signal and background events, and how th e p aram eters of th e m odel are estim ated.

Section 8 quantifies th e sources of u n certain ty im p o rta n t in th is m easurem ent, section 9 presents th e resulting central value and covariance m atrix for th e m odel p aram eters and th e ratio s Re [gR/V L] and Im [gR/V L], and th e conclusions are given in section 10.

2 M e a s u r e m e n t d e fin itio n

An event-specific coo rd in ate system is defined for analysing th e decay of th e to p quark in its rest fram e using th e directions of th e jets, lepton, and W boson in th e final state, depicted in figure 2 . T he z-axis is chosen along th e direction of th e W boson boosted into th e to p -q u ark rest fram e, z = q = q /|q |. T he angle 9* betw een q and th e m om entum of th e lepton in th e W boson rest fram e, pig, is th e sam e angle used to m easure th e W boson helicity fractions in to p -q u ark decays [17- 19]. T he spin of single to p q uarks produced in th e t-channel, s t , is predicted to be polarised along th e d irection of th e light q u ark (q; in figure 1) in th e to p -q u ark rest fram e [20], called th e sp ectato r-q u ark direction, p s. If th is je t defines th e spin analysing direction, th e degree of p olarisation is shown in refs. [21, 22] to be P = pis ■ s t / |s t | 0.9 a t y f s = 7 TeV for SM couplings. A three-dim ensional coordinate system is defined from th e q -p s plane and th e p erpendicu lar direction, w ith y = p s ^x q and X = y x q. In th is coord in ate system , th e direction of th e lepton (t) in th e W boson rest fram e pg is specified by 9* and th e com plem entary azim uthal angle 0*.

These angles arise as a n a tu ra l choice for m easuring a norm alised double differential d istrib u tio n , in th e helicity basis (to p -quark rest fram e), of th e leptonic decay of th e W boson arising from th e to p quark. In th is basis, th e t ^{^} W b tra n sitio n is d eterm ined by four helicity am plitudes, w here th e W boson and b-quark are eith er b o th right-handed, b o th

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Figure 2. Definition of the coordinate system with x , y, and z defined as shown from the momen­

tum directions of the W boson, ^q = z, and the spectator jet, ps, in the top-quark rest frame. The angles 0* and 0* indicate the lepton direction py in this coordinate system.

left-handed, or th e W boson is longitudinal and th e b-quark is eith er left- or right-handed.

T he resulting angu lar d istrib u tio n of final-state leptons in 9* and 0* is expressed [23] as a series in spherical harm onics, Y lm ( 9 * , 0*), param eterised by ^a = f , f + , f + , S + , S - ) and P :

p (9* ,0 *; a , p ) = = £ £ a i,m ( a , P )YT (9*, 0 7 w ith

l=0 ^{m = — l}

a0,0 = 7 4 7 a1,0 = 7 4 n f i ( f + - 2 ) , a2,0 = 7 2 0 0 ( 2 f i - 1 , a i,i = - a i , - i = p 7 f i (i - f i ) j / f + f + ei5+ + /

A

a2,2 = 0-2,-2 = 0. (2.1)

T he restrictio n to ^{l <} 2 in E q u a tio n (2.1) is caused by th e lim ited spin sta te s of th e initial and final sta te ferm ions and th e vector boson a t th e weak vertex.

T he param eters a describing th is d istrib u tio n can be expressed a t tre e level in term s of four linear com binations of th e anom alous couplings. T he fractions, f i , f + , and f + , d epend on th e m agn itu de of these four com binations, while 5+ and 5- depend on th e phases betw een pairs of com binations. In ad d itio n th ere is a small correction due to th e finite mass of th e b-quark, m b. Defining x w = m w / m t and ^{x b} = m b/ m t , w here m t is th e to p -q u ark m ass, th e p aram eters are given by:

• f i , th e fraction of decays containing transversely polarised W bosons, f = 2 (|x w V l - gR|2 + |x w V r - gL12) + ^{O ( x b )}

i 2 (|xw Vl - gR |2 + |xw Vr - gL|2) + | VL - x w g R |2 + |Vr - x w gL |2 + O(xb) (2.2)

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• f + , th e fraction of transversely polarised W boson decays th a t are right-handed, f + = ________|x w v r - gL|2 + O (x b)________ (2 3)

1 |^{X W}Vl - gRp + |^{X W} V r - gL|2 + ^{O (X b})

• f + , in events w ith longitudinally polarised W bosons, th e fraction of b-quarks th a t are right-handed,

f + = ________|VR - x w gL |2 + O (x b)________ (2 4) 0 |V r - x w g L |2 + |Vl - x w g R p + ^{O ( x b})

• 5+, th e phase betw een am plitudes for longitudinally polarised and transversely po­

larised W bosons recoiling against right-handed b-quarks,

5+ = arg ((xw Vr - gL)(VR - xwgL)* + O(x&)) (2.5)

• 5_, th e phase betw een am plitudes for longitudinally polarised and transversely po­

larised W bosons recoiling against left-handed b-quarks,

5_ = arg ((x w V l - gR) ( Vl - xwgR)* + O(x&)) (2 .6)

• P , which is considered sep arately from ^a because it depends on th e pro d u ctio n of th e to p quark, ra th e r th a n th e decay. T here is no an alytical expression for P in term s of anom alous couplings, b u t a param eterisatio n is determ ined in ref. [24] by fitting sim ulated sam ples produced w ith th e leading-order (LO) P r o t o s [25] g en erato r 2 w ith different values for th e various couplings.

T h rou g h these expressions, m easurem ents of th e param eterisatio n variables, ( a , P ), can be used to set lim its on th e values of th e couplings VL,R and gL,R.

T he p aram eters to which th is analysis is m ost sensitive are th e fraction f 1 € [0,1]

and th e phase 5_ € [ - n ,n ] . T he p a ra m ete r f 1 can be related to th e W boson helicity fractions via f 1 = F R + F L, w here F R = f 1f + and F L = f 1(1 - f + ) . Since f + and f + are small near th e SM point, th e term in E q u a tio n (2.1) th a t is p ro p o rtio nal to e lS+ is nearly zero. T hus 5+ can no t be constrain ed and f,+ cannot be sep arated from P . C o n straints on f + are cu rren tly b e tte r d eterm ined from lim its on F r [17, 18]. T he variations th a t th e param eters f 1 and 5_ induce in th e m odel are d e m o n strated in figure 3 for t-channel signal events generated w ith P r o t o s . For regions of th e p a ra m ete r space ( f 1,5 _ ) close to th e SM, only these param eters can be significantly constrained. T he dependence of 5_ on VR and gL is suppressed by x b, while b o th f 1 and 5_ are dep endent on VL and gR d irectly or th ro u g h x w . T hus, to sim plify th e analysis, only variations in VL and gR are considered, while VR and gL are assum ed to be zero. W ith th is assum ption, th e values of f 1+ and f 0+ are very small. T he value of P is also determ ined from th e values of VL and

2PRotos (PROgram for TOp Simulations) is a generator for studying new physics processes involving the top quark. It has generators for single top-quark and top-quark pair production with anomalous

Wtb couplings.

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(a)

(b) (c)

F ig u re 3. Projections into (a) 4>* in bins of cos 0*, (b) cos 0*, and (c) 4>* in Equation (2.1), for different values of f and 5_. The black points represent the P r o t o s t-channel signal generated with SM parameters, and the curves shown represent the signal model. For the three curves shown, the parameters f and 5_ are set to their values in the SM, f = 0.3, 5_ = 0 (solid red), and to two sets of beyond-the-SM values, f = 0.1, 5_ = 0 (dashed blue) and f = 0.3, 5_ = 0.1n (dotted green).

gR. T he highest-order dependence of f 1 and 5_ on th e couplings VL and gR ap p e ar as th e ratio p - , w here th e real and im aginary p a rts of th is ratio are m easured separately. This m otivates quoting th e results in b o th th e p a ra m ete r space ( f 1,5 _ ) and th e coupling space (R e[gR /V L ], Im [s r/Vl]).

3 T h e A T L A S d e t e c t o r

T he ATLAS d e te c to r [26] consists of a set of sub-detecto r system s, cylindrical in th e central region and p lan a r in th e two end-cap regions, th a t cover alm ost th e full solid angle around th e in teractio n p o in t.3 ATLAS is com posed of an inner trackin g d e te c to r (ID) close to th e in teractio n point, surrounded by a superconducting solenoid providing a 2 T axial m agnetic field, electrom agnetic and hadronic calorim eters, and a m uon sp ectrom eter (MS).

3ATLAS 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, d) are used in the transverse plane, d being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle 6 as n = — lntan(6/2).

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T he ID consists of a silicon pixel detecto r, a silicon m icrostrip d e te c to r (SC T ), providing trackin g inform ation w ithin p seudorapidity |n| < 2.5, and a straw -tu b e tra n sitio n rad iatio n track er (T R T ) th a t covers |n| < 2.0. T he central electrom agnetic calorim eter is a lead and liquid-argon (LAr) sam pling calorim eter w ith high granularity, and is divided into a barrel region th a t covers |n| < 1.475 and end-cap regions th a t cover 1.375 < |n| <

3.2. An iro n /scin tilla to r tile calorim eter provides hadronic energy m easurem ents in th e cen tral p seudorapidity range. T he forw ard regions are in stru m en ted w ith LA r calorim eters covering 3.1 < |n| < 4.9 for b o th th e electrom agnetic and hadronic energy m easurem ents.

T he MS covers |n| < 2.7 and consists of th ree large superconducting toroid m agnets w ith eight coils each, a system of trigger cham bers, and precision track ing cham bers. T he ATLAS d etecto r has a three-level trig ger system [27], used to select events to be recorded for offline analysis. T he first-level trigg er is hardw are-based and uses a subset of th e d etecto r inform ation to reduce th e physical event ra te from 40 MHz to a t m ost 75 kHz. T he second and th ird levels are softw are-based and to g eth e r reduce th e event ra te to ab o u t 300 Hz.

4 D a t a a n d s im u la t e d s a m p le s

T his analysis is perform ed using pp collision d a ta delivered by th e LHC [28] in 2011 at

a/s = 7 TeV and recorded by th e ATLAS experim ent. S tringent d e te c to r and d a ta quality requirem ents are applied, resulting in a d a ta sam ple corresponding to a to ta l integ rated lum inosity of 4.59 ± 0.08 fb- i [29]. T h e events are selected by un-prescaled single-lepton triggers [27, 30] th a t require a m inim um transverse energy, Et, of 20 GeV or 22 GeV for electrons and a m inim um tran sv erse m om entum , p t, of 18 GeV for m uons, depending on th e d a ta -ta k in g conditions.

Sam ples of events g enerated using M onte C arlo (MC) sim ulations are produced for t- channel signal and th e background processes, and are used to evaluate m odels of efficiency and resolution, and to estim ate sy stem atic uncertainties.

Sam ples of sim ulated t-channel single to p -q u ark events are produced w ith th e A c - e r M C m ulti-leg LO g en erato r [31] (version 3.8) using th e LO CTEQ 6L1 [32] P D F sets.

A c e r M C incorporates b o th th e 2 ^ 2 and 2 ^ 3 processes (see figure 1) and features an a u to m a te d procedure to remove th e overlap in phase space betw een th em [33]. T he factori­

satio n and renorm alisation scales are set to = p R = 172.5 GeV. A dditional t-channel sam ples w ith different anom alous couplings are produced w ith P r o t o s [25] (version 2.2) using th e CTEQ 6L1 P D F sets. E vents are generated using P r o t o s w ith th e four-flavour schem e,4 incorporating only th e 2 ^ 3 process, and anom alous couplings are enabled in b o th th e p ro d u ction and th e decay vertices, varying Re [VL] and Im [gR] sim ultaneously to keep th e to p -q u ark w id th invariant. T he facto risation scale is set to = - P w for th e light q u ark and = p f + m 2 for th e gluon. T hese sam ples are used to evaluate t-channel g en erator m odelling un certain ties as described in section 8.2. T hey are also used to de­

rive folding m odels w ith non-SM values of th e couplings, used for perform ing validation te sts in MC sim ulation and m easurem ents in real d a ta , as described in section 7. T he

4In the four-flavour scheme, the PDFs only contain the quarks lighter than the b-quark.

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p a rto n showering (PS), h ad ro n isatio n and underlying-event (UE) m odelling in these sam ­ ples are sim ulated w ith P y t h i a [34] (version 6.426) using th e P eru g ia 2011C set of tu n ed p aram eters (P2011C tu ne) [35] w ith th e CTEQ 6L1 [32] P D F sets.

Sam ples of ^{t t} events, s-channel single to p -q u ark events, and associated p ro d u ctio n of a W boson and to p q u ark (W t) are produced using th e P o w h e g - b o x NLO generato r (version 1.0) coupled w ith th e NLO CT10 [36] set of P D F s and interfaced w ith P y t h i a for showering and h ad ro nisatio n using th e P2011C tu n e w ith CTEQ 6L1 P D F sets. A dditional

t t sam ples are produced w ith P r o t o s (version 2.2) using th e CTEQ 6L1 P D F sets. These sam ples are used to evaluate th e background m odel w ith non-SM values of th e couplings.

T he PS, h ad ron isatio n and U E in these sam ples are sim ulated w ith P y t h i a (version 6.426) using th e A U ET2B tu n e [37] w ith th e M R ST LO** [38] P D F sets. All processes involving to p quarks are produced assum ing m t = 172.5 GeV.

V ector-boson pro d uctio n in association w ith je ts (V + je ts) is sim ulated using th e m ulti­

leg LO A L p g en g en erato r [39] (version 2.13) using CTEQ 6L1 P D F sets and interfaced to H e r w ig [40] (version 6.5.20) to g eth e r w ith th e Jim m y U E m odel [41] (version 4.31).

T he contrib utio ns of W + lig h t-je ts and W + h eav y -jets (W +bb, W + cc, W + c) are sim ulated separately. To remove overlaps betw een th e n and n + 1 p a rto n sam ples th e MLM m atching scheme [39] is used. D ouble counting betw een th e inclusive W + n p a rto n sam ples and sam ples w ith associated heavy-quark pair productio n is removed using an overlap-rem oval algorithm based on p a rto n -je t A R m atching [42]. D iboson processes (W W , W Z and Z Z ) are produced using th e H e r w ig gen erato r w ith th e M R ST LO** P D F sets.

A fter th e event-generation step, th e sam ples are passed th ro u g h th e full sim ulation of th e ATLAS d e te c to r [43] based on G e a n t 4 [44]. A dditional P r o t o s sam ples are passed th ro u g h th e ATLFAST2 sim ulation [43, 45] of th e ATLAS detector, which uses a fast sim­

u latio n for th e calorim eters and th eir response. T hey are th e n recon stru cted using th e sam e procedure as for d a ta . T h e sim ulation includes th e effect of m ultiple pp collisions per bunch crossing (pile-up). T he events are weighted such th a t th e d istrib u tio n of th e average num ber of collisions per bunch crossing is th e sam e as in d a ta .

5 O b j e c t d e f in it io n s a n d e v e n t s e le c t io n

T he definitions of objects used in th e analysis, including electrons, muons, je ts and b-tagged jets, and m issing tran sv erse m om entum , as well as th e basic event selection are chosen to be identical to those used for th e t-channel cross-section m easurem ents in ref. [46].

T his analysis requires exactly one isolated charged light lepton (electron or m uon) w ith transverse m om entum p T > 25 GeV and pseudorapidity |n| < 2.5. E x actly one b-tagged je t w ith |n| < 2.5 and exactly one untagged je t w ith |n| < 4.5 are required, b o th w ith p T > 30 GeV. T he second b-quark com ing from gluon sp littin g as shown in figure 1(b) can result in an additio n al b-tagged je t. T his second b-tagged je t generally has a softer p T spectru m and a bro ad er n d istrib u tio n com pared to th e b-tagged je t produced in the to p -q u a rk decay. It is often not d etected in th e experim ent and is th u s not required in th e event selection. To reject je ts from pile-up collisions, th e jet-v erte x fraction ejvF, defined as th e ratio of th e Pt of all tracks in th e je t originating a t th e p rim ary v ertex to th e Pt

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of all tracks associated w ith th e je t, is required to satisfy eJVF > 0.75. T he m ag nitu de of th e m issing transverse m om entum m ust be E™ ss > 30 GeV. Two additional m ultijet background rejection c riteria are applied. T he transverse m ass of th e lep to n -E ™ ss system ,

is required to be larger th a n 30 GeV. Also, a m ore string ent cu t on th e lepton p T is applied to events in which th e lepton and leading je t, j i , are back-to-back,

m om enta of th e lepton and th e neu trin o by using four-m om entum conservation. T he miss­

ing transverse m om entum vector, w ith m agnitude ETpiss, is used to represent th e transverse com ponent of th e n eu trin o m om entum and th e longitudinal com ponent pV is chosen such th a t th e resulting W boson is on its m ass shell. A q u a d ra tic expression is found for pV.

If th ere are two possible real values, th e value closer to zero is taken. If th e solutions are complex, th e assum ption th a t th e neu trin o is th e only c o n trib u to r to th e E™ ss is not valid. Therefore, th e ETpiss is rescaled, preserving its direction, in order to have physical (real) solutions for pV. If two solutions for ETpiss are found, th e one resulting in th e sm aller value of |pZ | is taken. T h e to p -q u a rk m om entum is th e n reco nstructed from th is W boson m om entum and th e m om entum of th e b-tagged je t. Finally, th e m om enta of th e W boson and sp e cta to r je t are boosted into th e to p -q u ark rest fram e to o b tain ^q and p s, used to gen erate th e coo rd in ate system in figure 2 , and th e lepton is boosted into th e W boson rest fram e to o b tain q^.

In ad d itio n to th is basic event selection, fu rth e r discrim ination betw een t-channel single to p -q u a rk events and background events is achieved by applying ad ditio nal criteria:

• T he pseudorapidity of th e untagged je t m ust satisfy |n| > 2, since th e sp e cta to r je t tend s to be forw ard in th e t-channel signature.

• T he sum of th e p T of all final-state objects, H T , m ust be larger th a n 210 GeV, since th e Ht distrib u tio n s of th e backgrounds have th eir peaks ju s t below th is value.

• T he m ass of th e to p q u ark recon structed from its decay p ro ducts is required to be w ithin 150-190 GeV, to reject background events from processes not involving to p

• T he distance in n betw een th e untagged je t and th e b-tagged je t m ust be larger th a n 1, to fu rth e r reduce t t co ntributions.

Two control regions are defined, enhanced in each of th e two d om in ant backgrounds, tb and W + je ts, to validate th e m odelling of th e backgrounds by sim ulated events. In th e

mT (^,E jplss) = ^ /2p t (Ą ■ E™ ss [1 - cos ( A ^ , E™ ss) ) ] ,

w here A 0 ( j i, ^) is th e difference in azim uthal angle betw een th e lepton m om entum and th e leading jet.

T he W boson com ing from th e decay of th e to p q u ark can be recon structed from th e

quarks.

These criteria and th e basic event selection to g eth e r define th e signal region of th e analysis.

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

control region w here th e ^{t t} background is enriched, events w ith exactly four jets, exactly one of which is b-tagged, passing th e basic event selection to g eth er w ith th e to p -q u ark m ass and H t requirem ents are selected. In th e control region w here W + je ts backgrounds are enriched, events w ith exactly two jets passing th e basic event selection and an inversion of th e to p -q u a rk m ass criterion are required.

6 B a c k g r o u n d e s t im a t io n a n d e v e n t y ie ld s

T he largest background co ntrib u tio ns to single to p -q u a rk t-channel pro du ction in th e signal region arise from t i p ro d u ctio n and from W boson p ro d u ction in association w ith jets (W + je ts ). Events containing ^{t t} p ro d u ction are difficult to distinguish from single top- q u ark events, as th ey also have real to p quarks in th e final sta te . Also, th e p ro d uction of a W boson w ith two jets, w here one is identified as containing a b-hadron, can m im ic the

t-channel final sta te . M ultijet p ro d u ction via th e stron g in teractio n can en ter th e signal region as well, and data-d riv en techniques are required to accu rately m odel it, as explained a t th e end of th is section. O th er backgrounds originate from W t p ro d u ctio n and s-channel single top-qu ark , diboson (W W , W Z , and Z Z ) and Z + je ts production. M ost of these background m odels are derived directly from MC sim ulation, b u t specialised procedures are im plem ented for W + je ts and m u ltijet production.

T he t i cross-section is calculated a t NNLO in QCD including resum m ation of NNLL soft-gluon term s w ith T o p + + 2.0 [47- 53] and is found to be 177+10 pb. T he uncertainties due to th e P D F s a t 68% C.L. and a S are calculated following th e PD F4L H C [54] pre­

scription for th e M S T W 2 0 0 8 N N L O , C T 1 0 , and N N P D F 2 .3 [55] erro r P D F sets, and are added in q u a d ra tu re to th e scale u n certain ty to yield a to ta l uncertain ty of 6%. Single to p -q u a rk p ro d u ction cross-sections for ^{W t} and ^s-channel p ro d uctio n are calculated a t ap ­ proxim ate NNLO and are found to be 15.7±1.2 pb [56] and 4.63++00 pb [57], respectively.

T he un certain ties correspond to th e sum in q u a d ra tu re of th e u n certain ty derived from th e M S T W 2 0 0 8 N N L O erro r P D F sets a t 90% C.L. and th e scale uncertainties, yielding a final u n certain ty of ab o u t 8% for ^{W t} p ro d uction and ab o u t 4% for ^s-channel production.

T he Z + je ts inclusive pro du ctio n cross-section is estim ated w ith NNLO precision using th e F E W Z program [58]. D iboson events (W W , W Z , and Z Z ) are norm alised to th e NLO cross-section prediction calculated w ith M C FM [59]. T he u n certain ty on th e com bined Z + je ts and diboson background is estim ated to be 60% as in ref. [46].

For th e p ro d uctio n of a W boson in association w ith jets, th e shapes of th e relevant k inem atic distrib u tio n s are predicted from th e A L p g en sam ple. T hey are initially nor­

m alised to m ake th e inclusive W cross-section correspond to th e NNLO prediction using th e F E W Z program [58], w ith th e sam e scaling factor being applied to th e A l p g e n pre­

diction for th e W +bb, W + cc, and W + lig h t-je ts sam ples. T he A l p g e n prediction for the W + c process is scaled by a factor th a t is o b tain ed from a stu d y based on NLO calculations using M C FM [59]. D ata-d riv en techniques are th e n used to e stim ate th e flavour com po­

sition and th e overall n orm alisation as described in ref. [60], and an u n certain ty of 18%

(15%) for electron (m uon) selections is applied.

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

M u ltijet background events pass th e signal region selection if a je t is m is-identified as an isolated lepton or if th e event has a non-prom pt lepton from th e decay of a hadron th a t ap p ears to be isolated. Since it is n either possible to sim ulate a sufficient num ber of those events nor to calculate th e ra te precisely, dedicated techniques are used to m odel m u ltijet events and to estim ate th eir productio n rate, em ploying b o th collision d a ta and sim ulated events. In th e electron channel, m is-identified jets are th e m ain source of m ultijet background events. T his is m odelled by th e jet-le p to n m ethod [46], in which an electron-like je t is selected in Py t h ia di-jet MC w ith special requirem ents and redefined as a lepton.

In th e m uon channel, a d ata-d riv en m a trix m ethod [46, 61] based on th e m uon im pact p a ra m ete r significance is used. An u ncertain ty of 50% is assigned to th e estim ated yield based on com parisons of th e rates o b tained by using a ltern ativ e m ethods [46], i.e. th e m a trix m ethod in th e electron channel and th e jet-le p to n m eth o d in th e m uon channel, and using an altern ativ e variable, i.e. m T (% E™ ss) instead of E™ ss.

Table 1 provides th e event yields for th e electron and m uon channels after th e event selection. T he predictions are norm alised to 4 .6 fb - i and th e ir un certainties are derived tak in g into account th e sta tistic a l uncertainties of th e sim ulated sam ples. For th e m ultijet background, th e uncertain ties on th e prediction are th e sta tistic a l uncertain ties of th e d a ta and sim ulated sam ples of th e m uon and electron channel, respectively. F igure 4 shows th e angu lar distrib u tio n s cos 9* and ^4>* after th e event selection for th e electron and m uon chan­

nels in d a ta , com pared to predictions using SM couplings. T he uncertainties are derived tak in g into account th e sta tistic a l uncertain ties of th e d a ta sam ples for d a ta and sim ulated sam ples on th e prediction and th e 50% sy stem atic u n certain ty on th e norm alisation of th e m u ltijet background. Since th e num ber of events in W + lig h t-je ts and m ultijet samples are low, sta tistic a l fluctuations and events w ith large g en erator weights affect th e sam ple shapes. Therefore, shape tem p lates from events selected w ith out th e b-tagging requirem ent are used for these two backgrounds to sm ooth th e sim ulated models. In these events, th e h ard est je t in th e event is chosen to tak e th e place of th e b-tagged-like je t for th e purposes of th e event selection and reconstruction. Overall, good agreem ent betw een th e observed and expected d istrib u tio n s is found.

7 A n a ly s is m e t h o d

T he m odel describing th e t-channel signal in section 2 is connected to th e angles m easured in recon structed events via an an aly tic folding procedure. M odels of th e selection efficiency and d etecto r resolution are derived from sim ulated t-channel signal events, and a m odel of th e recon structed background is derived from th e sum of th e com bined background processes described in section 4 . Since th e values of th e p aram eters f i and ^{S -} depend on to p -q u a rk couplings, b o th th e t-channel signal and th e t t background are sensitive to th eir values. Only th e angu lar d istrib u tio n s for th e leptonic decay of th e W boson are m easured, b u t th e efficiency and resolution m ay also depend on o th er unm easured distrib u tio n s, such as those associated w ith th e n d istrib u tio n of th e to p or sp e c ta to r q u ark in th e lab o rato ry fram e. T he efficiency, resolution, and background m odels are co n stru cted such th a t th ey account for any dependence on th e values of f i and ^S- , which is required to remove biases

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

Figure 4. Angular distributions cos 0* (upper row) and 4>* (lower row) in the signal region for electrons (left) and muons (right), comparing data, shown as the black points with statistical uncertainties, to SM signal and background predictions. The uncertainties shown on the prediction take into account MC sample sizes and the 50% systematic uncertainty on the normalisation of the multijet background. The W + jets backgrounds are normalised to the observed number of events in the control region, and the muon multijet backgrounds are predicted from observed data. All other samples are scaled to their theoretical predictions. The lower plots show the ratio of data to prediction in each bin.

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Electrons Muons

t-channel 274

±

±

s-channel 4

±

±

Wt-channel 13

±

±

tt 119

±

±

W +heavy-jets 92

±

±

W+light-jets 13

±

±

Z+light-jets 4

±

±

Diboson 1

±

±

Multijet 20

±

±

Total expected 538 ± 11 660 ± 10

Data 576 691

Table 1. Event yields for the electron and muon channels in the signal region. Individual predic­

tions are rounded to integers while “Total expected” corresponds to the rounding of the sum of full precision individual predictions. The uncertainties shown are statistical only. Uncertainties of less than 0.5 events appear as zero.

dependent on the values of these parameters (see appendix A) . A probability density is derived for all events in the signal region, as a function of the reconstructed angles cos 0*

and **, and conditional on the parameters. This density is then used to construct a likelihood, from which / i and S- are measured in data.

The signal model described by Equation (2.1) is a series in spherical harmonics, only containing terms up to a maximum value of l, lmfx = 2. Describing the efficiency and reso­

lution models similarly allows the use of the orthogonality properties of spherical harmonics to construct analytically-folded distributions [23]. An efficiency function, given by

7eff lmax

e(0T , 0 T ; a , P) = &i>,m'( a , P)Yim ($ t (^-1) l',m'

describes the probability that a t-channel signal event with the given true angles $T and

*T will be reconstructed in the signal region. The series contains all allowed values of l and m/ up to a maximum spherical harmonic degree, ieffax. The selected signal density, ps, is then defined as the product of the efficiency function and the signal model, p, normalised to the overall rate,

Ps(0T, *1; a, P ) = ;0 fT: ^ ; , a pp,); <(fT- f i ; , a PP ] o . = £ cl,m(a . p )Ylm(0T, ¢1), / e ( 0 T,* T; a ,P )p (0 T,* T; a, P)dOT ^M

E ei',m' (A ,P K m ( A ,P )G r^m^ ⁿL^M

t-

n,

.

for ^{c l,m} ( a , p ) =

w

’ E ( - 1 ) ei,-m (a,P )ai,m (a,P ) l,m

The Gaunt coefficients G are defined in terms of Clebsch-Gordan coefficients Cm ,

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

given by,

= I (2l + 1)(2l' + 1) C o,o,o c m,m! ,M

l'V'L Y 4n(2L + 1) ■

T he efficiency function is d eterm ined from a likelihood fit to sim ulated t-channel events.

A value is chosen for l ^ x by com paring th e associated values of th e corrected Akaike Info rm atio n C riteria, AICc [62, 63], a likelihood-ratio te s t w ith an add ition al p en alty term which increases w ith th e num ber of p aram eters. From th is te st, lm |x = 6 is chosen. T he deviation in th e results from th e selection of efficiency m odels w ith lm |x varied by ± 2 are included as system atic u ncertainties. O th e r criteria, such as th e Schwarz C riteria [64], select values of ieffax w ithin th e range chosen for this uncertainty.

W hen reco nstru ctin g events in th e signal region, th e finite resolution of th e d etecto r results in a m igration from th e tru e angles to m easured angles 9* and 4>*. This m igration is m odelled by th e resolution function,

l reco itrue lmax lmax

R (9*,0*|9T , 4 ; a , P ) = £ £ r x , , , L 'M ' ( 0 , P ) Y £ ( 9 * , $ * ) ¥ # ' (9*T , ^ ) - (7.3)

A,^ L ',M '

T his series contains all term s w ith allowed values of A, p, L ' , and M ' up to th e m axim um degree p aram eters l^T^x, associated w ith th e dependence on 9T and ^T, and lmax, associated w ith th e dependence on 9* and ^4>*. T he reconstructed signal density, pr , is defined as th e convolution of th is function w ith th e selected signal density,

f

p r (9*,4>*; a ,p ) = r ( 9 * ,^ * |9 T ,^ T ; a , p ) p s (9T ,^T ; a ,p ) d Q T

= £ d x ^ ( a , P )Y /(9 * ,0 * ), a,m

for d x , ^ ( a , P ) = £ ( - 1 ) Mc l,m( a , P ) r x , ^ , L , - M ( a , P ) . (7.4)

L ,M

T he resolution function is d eterm ined using a spherical Fourier technique [65]. In principle, describing a very narrow resolution function could require large values of l ^ x and lmi£.

In practice, th e m ath em atics of angular-m om entum ad d itio n g u aran tee th a t th ere are no term s in E q u a tio n (7.2) w ith L > lm |x + lmix. Thus, l ^ x = lmlx + l ^ | x. T he p aram eter lmax is d eterm ined using th e m ean in teg rated squared error, M ISE = / ( p ( x ) — p (x ))2 dx, w here p(x) is th e tru e probab ility density and p(x) is a d istrib u tio n estim atin g th a t density.

C alculating th e M ISE w ith pr yields a broad m inim um for lmix > 4, and for this analysis lmax = 6 is chosen. T he deviation in th e results from th e selection of resolution m odels w ith lmix varied by ± 2 are included as system atic un certainties.

T his density of recon stru cted t-channel signal events is a series in spherical harm onics, w ith coefficients which are functions of th e physics p aram eters describing th e p roduction (P ) and decay (ao ) of th e to p quark. A background model,

lbkgimax

P(9*,4>*; a ,p ) = £ b x A & , P) Y / ( 9 * £ ) , (7.5) A,m

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is derived which is also a series in spherical harm onics, and is added to th e recon structed signal density via th e signal fraction, f s. T his background function is determ ined from a likelihood fit to all sim ulated background events in th e signal region, and contains all term s w ith allowed values of A and ^p up to th e m axim um degree p a ra m ete r T his p aram eter is determ ined like th e equivalent p a ra m ete r in th e efficiency function, using th e AICc. T he v ariations in th e results from th e selection of background m odels w ith l^lax ± 2 are included as system atic un certainties.

T he probab ility density of all events in th e signal region, pt , is th e sum of th e recon­

stru c te d signal and background densities w ith signal fraction f s,

max(irecomaxi 1 max ,1 max ?bkg \ j

pt(9*,0*; a , P , fs) = ^ A \ ^ ( a , P ) y £ ( 0 * , 4 > * ) for A,^

Aa^(<3, P, fs) = f s d \ ^ ( a , P ) + (1 - f s ) b \ , ^ (7.6)

£ ( - 1 ) Me v m a i , m ( a , P , Mt a^ l -m

f 1,1',L ,m ,m ',M | ^ f ^

= fs Z ( - 1 ) m e i , - m a i m ( d , P) + ( - fs) A^

l,m

F igure 5 shows th e d a ta sum m ed over th e electron and m uon channels for th e signal region. T hree representative m odels are also shown: one for th e SM expectatio n s, one th a t is near th e expected 95% C.L. sensitivity in f i , and one near th e expected sensitivity in

S- . These values for f 1 and ^{5 _} are th e sam e as those used in figure 3, b u t th e curves now include th e effects of efficiency, resolution and background. T he m ain differences betw een th e two figures are due to th e isolation requirem ents placed on th e leptons. For cos 9* = - 1 , th e lepton overlaps th e b-tagged je t, while for 0* = n , th e lepton m ay overlap th e sp e c ta to r je t. T he region w here th e efficiency is less th a n 0.05% (cos 9* < - 0 .9 5 , and -0 .9 5 < cos 9* < - 0 .8 5 for n — 0.1 < 0 * < n + 0.1), is excluded from fu rth e r analysis. For o th er regions, th e efficiency varies, as a function of 0* and c o s 9*, from 0.1% up to 1.7%

for larger cos 9* and away from th e plane containing th e sp e cta to r jet. As th e features of th e angular distrib u tio n s are broad, th e effects of resolution are small. T he m ain effect is to sm ooth out some of th e effects of th e acceptance and to increase th e num ber of events a t large cos 9* w here th e SM co n trib u tio n is small. As seen in figure 4 , th e background is also som ew hat larger for large cos 9*.

An unbinned likelihood function is co nstructed from th e probab ility density in E q u a ­ tio n (7.6) and from a set of events D = {9*, 0*, W i }, i = 1, 2 , . . . N ,

N

L ( a) = ^ exp (wipt(9*, 0 - 1 a , P , f s ) ) , (7.7) i= 1

w here 9ⁱ* and 0ⁱ* can be tak en eith er from d a ta or MC sim ulation, and th e event weights Wⁱ are 1 for m easured d a ta . T he likelihood is evaluated on a grid w ith spacing 0.01 in f 1 and 0.01n in £_. T he efficiency and resolution m odels a t each point are derived from P r o t o s t-channel sim ulated events, and th e background m odel uses P r o t o s ^{t t} sim ulated events.

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

(a)

(b) (c)

F ig u re 5. Projections into (a) 4>* in bins of cos 0*, (b) cos 0*, and (c) 4>* of the function described in Equation (7.6), for different values of ^{f i} and ^{5- .} The black points shown are for the selected data events with statistical uncertainties. The curves shown represent the model at the SM point f i = 0.3, 5- = 0 (solid red), and two sets of beyond-the-SM values, f l = 0.1, 5- = 0 (dashed blue) and f i = 0.3, 5- = 0.1n (dotted green).

In te rp re tin g these p aram eters as varying th e coupling ratio g R / V L , th e p olarisation P is varied sim ultaneously according to th e param eterisatio n in ref. [24].

A cen tral value is m easured a t th e m axim um value of L on this grid, d eterm ined from a G aussian fit to th e points at which L is evaluated. T he 68% and 95% confidence lim its are defined by th e region w here th e likelihood ratio, L / L max, is larger th a n th e value th a t would yield th e sam e likelihood for a tw o-dim ensional G aussian d istrib u tio n . T hey are refined to increase th e accessible precision by in terp o latin g betw een points on eith er side of th e contours determ ined from these evaluated points. T he sta tistic a l u ncertain ty is estim ated from th e sym m etrised 68% C.L. interval of each p aram eter.

T his m odel is checked for closure b o th w ith MC sam ples, and by defining toy-M C sam ples based on E q u a tio n (7.6) w ith th e sam e num ber of events as in d a ta . These samples were generated a t various points in ( f , 5- ) space. In all cases th ey are found to reproduce th e expected values of f and 5- . T hese toy-M C sam ples were also used to derive pull d istrib u tio n s for f and 5- to check th e sta tistic a l uncertain ties retu rn ed by th e fit.

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

8 S o u r c e s o f s y s t e m a t ic u n c e r t a in t y

System atic uncertainties are evaluated in th e full ( f i , 5 - ) p a ra m ete r space. Efficiency, resolution, and background m odels are determ ined from MC sam ples w ith a p aram eter varied by its uncertainty, or a subset w here ap p ro p riate. A likelihood is con stru cted from th e resulting m odel, using events generated w ith nom inal values of th e varied param eters.

T he difference betw een th e central values estim ated a t th e nom inal value of a p a ra m ete r and a t th e value varied by its uncertainty, or half th e difference betw een central values estim ated w ith th e p a ra m ete r varied up and down by its uncertainty, is used to co n struct a covariance m atrix for each source of system atic uncertainty. T he to ta l covariance m atrix for th e system atic uncertain ties and its correlation m a trix are found from th e sum of th e covariance m atrices determ ined for individual uncertainties.

8.1 O b ject m o d ellin g

T he uncertainties in th e reco nstructio n of jets, leptons, and Em iss are p ro p ag ated to th e analysis, following th e same procedures as described in ref. [46]. T he m ain source of u n certain ty from these physics objects is th e je t energy scale (JE S) [66], which is th e largest system atic u n certain ty on th e m easurem ent of f i . To estim ate th e im pact of th e JE S u n certain ty on th e result, th e je t energy is scaled up and down by its u n certain ty [66], which ranges from 2.5% in th e cen tral region w ith high-px jets to 14% in th e far forw ard region w ith low-px jets. U ncertainties are also estim ated for je t energy resolution and reco nstruction efficiency; th e im pact of varying th e jet-v erte x fraction requirem ent; Em iss reco nstruction and th e effect of pile-up collisions on th e Em iss; 6-tagging efficiency and m istagging rate; lepton trigger, identification, and recon stru ctio n efficiencies; and lepton m om entum , energy scale, and resolution.

8.2 M C g en era to rs and P D F s

M ultiple MC event g enerators are used to m odel th e t-channel and t t processes in this analysis, and th e differences betw een these generators are included as system atic u n c e rtain ­ ties. C om paring th e A o e r M C gen erato r used for t-channel events and th e P o w h e g - b o x g en erator used for ^{t t} events to th e P r o t o s g enerator used for b o th th e t-channel single to p -q u a rk events and ^{t t} events yields th e largest u n certain ty in 5- . A dditional t-channel com parisons are perform ed betw een th e A c e r M C and P o w h e g - b o x generators and be­

tw een A c e r M C events w ith showering by P y t h i a and H e r w ig . T he renorm alisation and facto risatio n scales in th e P o w h e g - b o x t-channel sam ple are also varied indepen­

d ently by factors of 0.5 and 2.0. A dditional ttt com parisons are perform ed betw een th e P o w h e g - b o x g en erator and th e M C @ N L O [67] g enerato r w ith showering by H e r w ig , and betw een P o w h e g - b o x events w ith showering by P y t h i a and H e r w ig . T he variations betw een P D F sets and w ithin individual sets are used to estim ate a system atic un certain ty following th e PD F 4L H C prescription.

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

8.3 Signal and backgrou n d n orm alisation

Cross-sections for background processes given in section 6 are varied up and down by th eir u ncertainties. T h e m u ltijet n orm alisation is varied by 50% as discussed in section 6 , and th e rem aining backgrounds are varied sim ultaneously to produce a conservative estim ate.

T he im pact of ob ject m odelling and o th er uncertain ties on th e W + je ts background nor­

m alisation are considered in parallel w ith th e variations m ade in t-channel and ^{t t} samples.

A sep arate shape u n certain ty is assigned to th e W + je ts sam ples by varying th e m atching and factorisation scales in th e A l p g e n generator. T he in tegrated lum inosity is varied up and down by its uncertainty, ±1.8% , derived as detailed in ref. [29].

8.4 D e te c to r co rrectio n and b ackground p a ra m eterisa tio n

A n u n certain ty due to th e lim ited size of th e MC sam ples used to e stim ate th e efficiency and resolution m odels is estim ated from th e sta tistic a l uncertain ties derived from a m ea­

surem ent of / i and 5 _ in th e t-channel signal MC sam ple alone. T he background statistical u n certain ty is estim ated by varying th e background m odel according to th e eigenvectors of its covariance m atrix. T he background sta tistic a l u ncertain ty dom inates th e to ta l “MC s ta tistic s” u n certain ty listed in tab le 2 . Its significance reflects th e sm all size of some background sam ples in th e signal region, and th e resulting d isp a ra te values of th e sam ple weights. T he effect of varying th e cutoff degree lmax in th e d eterm in atio n of these models is also estim ated, and is found to be sm all com pared to MC sta tistic a l uncertainty.

8.5 U n c erta in ty co m b in a tio n

Table 2 shows th e co n trib u tio n of each source of u n certain ty to th e m easurem ent of the p aram eters / i and 5 _ and th eir correlation, p ( / i , 5—). T he to ta l system atic u n certain ty and correlation is obtained from th e sum of th e covariance m atrices d eterm ined for each source.

It is com bined w ith th e covariance m a trix of th e sta tistic a l u n certain ty via th e following m ethod. A t each point (¾ in th e ( / i ,^ - ) space, a m u ltivariate norm al d istrib u tio n N i is co n stru cted w ith th e covariance m atrix representing th e system atic uncertainties, S syst, and th e m ean, a^ T h e resulting d istrib u tio n is evaluated at a point a j and m ultiplied by th e likelihood at th is point, L jta t. T he m axim um m odified likelihood value, over all possible p oints ^{( j}, is kept, and th e resulting broadened likelihood d istrib u tio n ^ stat+syst is used to represent th e m easurem ent w ith b o th sta tistic a l and system atic variation incorporated:

T he resulting confidence lim its are tak en as th e full u n certain ty in th e m easurem ent.

9 R e s u lt s

T he result for ( / 1, £_) and th e coupling ratio s (Re [gR/V L] , Im [gR/V L]) is shown in figure 6.

T he 68% co ntou r represents th e to ta l u n certain ty on th e m easurem ent.

T he p aram eters / 1 and ^{5 _} and th e ir uncertain ties are m easured to be

L stat+syst = m axj { N i(a j j; ^ , £ syst) ■ L jta t} . (8.1)

/1 = 0.37 ± 0.05 (sta t.) ± 0.05 (syst.),

S _ = — 0.014n ± 0.023n (sta t.) ± 0.028n (syst.). (9.1)

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

Source a(f1) a ( £ _ ) / n p ( f 1 , M

D a ta statistics 0.05 0.023 0.01

Je ts 0.03 0.015 0.39

b-tagging < 0.01 < 0.001 -0 .7 0

Leptons 0.02 0.007 0.39

zT'miss

E x 0.01 0.004 - 0 .2 7

G en erator 0.02 0.017 0.40

P a rto n shower 0.02 0.001 0.98

P D F variations 0.01 0.009 0.23

Cross-sections < 0.01 < 0.001 1.00

W + je ts shape < 0.01 0.001 -0 .5 9 M u ltijet norm alisation < 0.01 0.002 -1 .0 0 Lum inosity < 0.01 < 0.001 -1 .0 0 M odel 1max variation 0.01 0.001 -0 .7 0

MC statistics 0.02 0.011 0.14

Com bined system atic 0.05 0.028 0.27

Total 0.07 0.036 0.15

T ab le 2. Sources of systematic uncertainty on the measurement of f 1 and S_ using A c e rM C t- channel single top-quark simulated events and backgrounds estimated from both MC simulation and data, including P o w h eg -b o x ^{t t} simulation. Individual sources are evaluated separately for shifts up and down, and symmetrised uncertainties a (f i), a (J_ ) and correlation coefficients p (fi,J _ ) are given.

T he correlation in th e m easurem ent of these p aram eters is p ( f 1 , 5 - ) = 0.15. T he results are com patible w ith th e SM expectations a t LO, derived from expressions in refs. [11, 68]

w ith m t = 172.5 GeV, m W = 80.399 GeV, and m b = 4.95 GeV: f 1 = 0.304 and ^{5 _} = 0.

T he dependence of th e p aram eters f 1 and 5 _ on th e to p -q u ark m ass is evaluated using t-channel and ttt sim ulation sam ples w ith a range of different to p -q u a rk masses. A linear dependence is found, resulting from changes in acceptance a t different masses, w ith a slope of -0 .0 1 9 GeV-1 for f 1 and a negligible slope for 5 - . T he u n certain ty due to th e top- q u ark m ass dependence is not included in th e to ta l system atic u n certain ty since it has no significant im pact on th e results.

T he p ro p ag ation of th e un certainties to th e (Re [gR/V L] , Im [gR/V L]) space gives

Re Im

9 R

Vl 9r

Vl

= - 0 .1 3 ± 0.07 (sta t.) ± 0.10 (syst.),

= 0.03 ± 0.06 (sta t.) ± 0.07 (syst.).

(9.2)

T he correlation in th e m easurem ent of these coupling ratios is p (Re [gR/V L] , Im [gR/VL]) = 0.11. T he effect on th e p ropagatio n due to th e cu rren t

J H E P 0 4 ( 2 0 1 6 ) 0 2 3

Figure 6. Projections of the likelihood function constructed from the signal region probability density Equation (7.6) and data events into (a) f i, (b) □ , (c) f i vs. □ , and (d) Re [#r /V l]

vs. Im[gR/VL], with systematic uncertainties incorporated. The black points indicate the largest evaluated likelihood in each bin of the projected variable. Gaussian fits to the one-dimensional projections were performed, displayed as the red curve. Regions shown in green and yellow represent the 68% and 95% confidence level regions, respectively. A black line or cross indicates the observed value, and the grey line or point indicates the SM expectation.

uncertainty in the top-quark, W boson and b-quark masses [69] is < 0.01 in Re[gR/VL], and < 0.0001 in Im [gR/VL].

Limits are placed simultaneously on the possible complex values of the ratio of the anomalous couplings gR and VL at 95% C.L.,

Re ^5R

Vl € [-0.36,0.10] and Im

5R

Vl

€ [-0.17, 0.23]. (9.3) The best constraints on Re [$r] come from W boson helicity fractions in top-quark decays, with Re [gR] of [-0.08, 0.04] and [-0.08,0.07], both at 95% C.L., from ATLAS [17]

and from CMS [18], respectively. However, these limits use the measured single top-quark production cross-section [46, 70] along with the assumption th at VL = 1 and Im [gR] = 0.

W ithout these assumptions no value within the range 0.0 < Re [gR/VL] < 0.8 can be

J H E P 0 4 ( 2 0 1 6 ) 0 2 3