Measurements of t ¯ t spin correlations and top quark polarization using dilepton final states in pp collisions at

Measurements of the top quark-antiquark ( t ¯ t ) spin correlations and the top quark polarization are presented for t ¯ t pairs produced in pp collisions at ﬃﬃﬃ s p ¼ 8 TeV. The data correspond to an integrated luminosity of 19 . 5 fb − 1 collected with the CMS detector at the LHC. The measurements are performed using events with two oppositely charged leptons (electrons or muons) and two or more jets, where at least one of the jets is identified as originating from a bottom quark. The spin correlations and polarization are measured from the angular distributions of the two selected leptons, both inclusively and differentially, with respect to the invariant mass, rapidity, and transverse momentum of the t ¯ t system. The measurements are unfolded to the parton level and found to be in agreement with predictions of the standard model. A search for new physics in the form of anomalous top quark chromo moments is performed. No evidence of new physics is observed, and exclusion limits on the real part of the chromo-magnetic dipole moment and the imaginary part of the chromo-electric dipole moment are evaluated.


I. INTRODUCTION
The top quark is the heaviest known elementary particle, with mass m t ¼ 172. 44 AE 0.48 GeV [1]. The top quark lifetime has been measured as 3.29 þ0.90 −0.63 × 10 −25 s [2], shorter than the hadronization timescale 1=Λ QCD ≈10 −24 s, where Λ QCD is the quantum chromodynamics (QCD) scale parameter, and also shorter than the spin decorrelation time scale m t =Λ 2 QCD ≈ 10 −21 s [3]. Consequently, measurements of the angular distributions of top quark decay products give access to the spin of the top quark, allowing the precise testing of perturbative QCD in the top quark-antiquark pair (tt) production process.
At the CERN LHC, top quarks are produced abundantly, predominantly in pairs. In the standard model (SM), top quarks from pair production have only a small net polarization arising from electroweak corrections to the QCD-dominated production process, but the pairs have significant spin correlations [4]. For low tt invariant masses, the production is dominated by the fusion of pairs of gluons with the same helicities, resulting in the creation of top quark pairs with antiparallel spins in the tt center-ofmass frame. For larger tt invariant masses, the dominant production is via the fusion of gluons with opposite helicities, resulting in tt pairs with parallel spins [3]. For models beyond the SM, couplings of the top quark to new particles can alter both the top quark polarization and the strength of the spin correlations in the tt system [4][5][6][7].
The charged lepton (l) from the decay t → bW þ → bl þ ν l is the best spin analyzer among the top quark decay products [8], and is sensitive to the top quark spin through the helicity angle θ ⋆ l . This is the angle of the lepton in the rest frame of its parent top quark or antiquark, measured in the helicity frame (i.e., relative to the direction of the parent quark momentum in the tt center-of-mass frame) [4].
For the decay tt → bl þ ν lb l −ν l , the difference in azimuthal angle of the charged leptons in the laboratory frame, Δϕ l þ l − , is sensitive to tt spin correlations and can be measured precisely without reconstructing the full tt system [3]. With the tt system fully reconstructed, the opening angle φ between the two lepton momenta measured in the rest frames of their respective parent top quark or antiquark is directly sensitive to spin correlations, as is the product of the cosines of the helicity angles of the two leptons, cos θ ⋆ l þ cos θ ⋆ l − [4]. Recent spin correlation and polarization measurements from the CDF, D0, and ATLAS Collaborations used template fits to angular distributions, and their results were consistent with the SM expectations [9][10][11][12][13][14]. In this analysis, the measurements are made using asymmetries in angular distributions unfolded to the parton level, allowing direct comparisons between the data and theoretical predictions. The analysis strategy is similar to that presented in Ref. [15]; however, the larger data set used here and improvements in the tt system reconstruction techniques lead to a reduced statistical uncertainty in the measurements. Furthermore, an improved unfolding technique allows for differential measurements, which were not presented in Ref. [15].
The polarization P AE of the top quark (antiquark) in the helicity basis is given by P AE ¼ 2A PAE [4], where the asymmetry variable A PAE is defined as where the numbers of events Nðcos θ ⋆ l AE > 0Þ and Nðcos θ ⋆ l AE < 0Þ are counted using the helicity angle of the positively (negatively) charged lepton in each event. Assuming CP invariance, these two measurements can be combined to give the SM polarization P ¼ 2A P ¼ ðA Pþ þ A P− Þ. Alternatively, the variable P CPV ¼ 2A CPV P ¼ ðA Pþ − A P− Þ measures possible polarization introduced by a maximally CP-violating process [4].
For tt spin correlations, the variable discriminates between correlated and uncorrelated t andt spins, while the variable A c 1 c 2 ¼ Nðc 1 c 2 > 0Þ − Nðc 1 c 2 < 0Þ Nðc 1 c 2 > 0Þ þ Nðc 1 c 2 < 0Þ ; where c 1 ¼ cos θ ⋆ l þ and c 2 ¼ cos θ ⋆ l − , provides a direct measure of the spin correlation coefficient C hel through the relationship C hel ¼ −4A c 1 c 2 [4]. The variable provides a direct measure of the spin correlation coefficient D by the relation D ¼ −2A cos φ [4]. In addition to the inclusive measurements, we determine the asymmetries differentially as a function of three variables describing the tt system in the laboratory frame: its invariant mass M tt , rapidity y tt , and transverse momentum p T tt . The results presented in this paper are based on data collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 19.5 fb −1 from pp collisions at ffiffi ffi s p ¼ 8 TeV.

II. THE CMS DETECTOR
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity coverage provided by the barrel and endcap detectors. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. The first level of the CMS trigger system, composed of custom hardware processors, uses information from the calorimeters and muon detectors to select the most interesting events in a fixed time interval of less than 4 μs. The high-level trigger processor farm further decreases the event rate from around 100 kHz to less than 1 kHz, before data storage. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [16].

A. Object definition and event selection
Events are selected using triggers that require the presence of at least two leptons (electrons or muons) with transverse momentum (p T ) greater than 17 GeV for the highest-p T lepton and 8 GeV for the second-highest p T lepton. The trigger efficiency per lepton, measured relative to the full offline lepton selection detailed in this section using a data sample of Drell-Yan (Z=γ ⋆ → ll) events, is about 98% (96%) for electrons (muons), with variations at the level of several percent depending on the pseudorapidity η and p T of the lepton.
The particle-flow (PF) algorithm [17,18] is used to reconstruct and identify each individual particle with an optimized combination of information from the various elements of the CMS detector. The offline selection requires events to have exactly two leptons of opposite charge with p T > 20 GeV and jηj < 2.4. Electron candidates are reconstructed starting from a cluster of energy deposits in the electromagnetic calorimeter. The cluster is then matched to a reconstructed track. The electron selection is based on the shower shape, track-cluster matching, and consistency between the cluster energy and the track momentum [19]. Muon candidates are reconstructed by performing a global fit that requires consistent hit patterns in the silicon tracker and the muon system [20].
The events with an e þ e − or μ þ μ − pair having an invariant mass, M ll , within 15 GeV of the Z boson mass are removed to suppress the Drell-Yan background. For all events, we require M ll > 20 GeV. Leptons are required to be isolated from other activity in the event. The lepton isolation is measured using the scalar p T sum (p sum T ) of all PF particles not associated with the lepton within a cone of radius ΔR ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðΔηÞ 2 þ ðΔϕÞ 2 p ¼ 0.3, where Δη (Δϕ) is the distance in η (ϕ) between the directions of the lepton and the PF particle at the primary interaction vertex [21]. The average contribution of particles from additional pp interactions in the same or nearby bunch crossings (pileup) is estimated and subtracted from the p sum T quantity. The isolation requirement is p sum T < minð5 GeV; 0.15 p l T Þ, where p l T is the lepton p T . Typical lepton identification and isolation efficiencies, measured in samples of Drell-Yan events [22], are 76% for electrons and 91% for muons, with variations at the level of several percent within the p T and η ranges of the selected leptons. The PF particles are clustered to form jets using the antik T clustering algorithm [23] with a distance parameter of 0.5, as implemented in the FASTJET package [24]. The contribution to the jet energy from pileup is estimated on an event-by-event basis using the jet-area method described in Ref. [25], and is subtracted from the overall jet p T . Jets from pileup interactions are suppressed using a multivariate discriminant based on the multiplicity of objects clustered in the jet, the jet shape, and the impact parameters of the charged tracks in the jet with respect to the primary interaction vertex. The jets must be separated from the selected leptons by ΔR > 0.4.
The selected events are required to contain at least two jets with p T > 30 GeV and jηj < 2.4. At least one of these jets must be consistent with containing the decay of a bottom (b) flavored hadron, as identified using the medium operating point of the combined secondary vertex (CSV) b quark tagging algorithm [26]. We refer to such jets as b-tagged jets. The efficiency of this algorithm for b quark jets in the p T range 30-400 GeV is 60%-75% for jηj < 2.4. The misidentification rate for light-quark or gluon jets is approximately 1% for the chosen working point [26].
The missing transverse momentum vectorp miss T is defined as the projection on the plane perpendicular to the beam direction of the negative vector sum of the momenta of all reconstructed particles in the event. Its magnitude is referred to as E miss T . The calibrations that are applied to the energy measurements of jets are propagated to a correction ofp miss T . The E miss T value is required to exceed 40 GeV in events with same-flavor leptons in order to further suppress the Drell-Yan background. There is no E miss T requirement for e AE μ ∓ events.
For both signal and background events, pileup interactions are simulated with PYTHIA and superimposed on the hard collisions using a pileup multiplicity distribution that reflects the luminosity profile of the analyzed data. The CMS detector response is simulated using a GEANT4-based model [48]. The simulated events are reconstructed and analyzed with the same software used to process the collision data.
The measured trigger efficiencies are used to weight the simulated events to account for the trigger requirement. Small differences between the b tagging efficiencies measured in data and simulation [26] are accounted for by using data-to-simulation correction factors to adjust the b tagging probability in simulated events, while the lepton selection efficiencies (reconstruction, identification, and isolation) are found to be consistent between data and simulation [22].

IV. BACKGROUND ESTIMATION
Control regions (CR) are used to validate the background estimates from simulation and derive scale factors (SF) and systematic uncertainties for some background processes. Each SF multiplies the simulated background yield for the given process in the signal region (SR) to obtain the final background prediction. The CRs are designed to have similar kinematics to the SR, but with one or two selection requirements reversed, thus enhancing different SM contributions. The main CRs used in this analysis and the values of the derived SFs are summarized in Table I. For Drell-Yan events, the SF accounts for mismodeling of the E miss T distribution (coming largely from mismeasured jets) and mismodeling of the heavy-flavor content. Only the latter is relevant for Z=γ ⋆ ð→ ττÞ þ jets, where the E miss T is dominated by the well-modeled undetected neutrinos, so we omit the E miss T mismodeling in the derivation of the SF for this process. The systematic uncertainties in the SFs are taken from the envelope of the variation observed between the three dilepton flavor combinations and in various alternative CRs. The CR for single top quark production in association with a W boson (tW) is still dominated by signal events (75%), with only a 16% contribution from tW production, which is an enhancement by a factor of 4 compared to the SR. Given the good agreement between data and simulation in this CR, we assume a SF of unity for tW production, with an uncertainty of 25% based on the recent CMS tW cross section measurement of 23.4 AE 5.4 pb [49].
Contributions to the background from diboson and triboson production, as well as tt production in association with a boson, are estimated from simulation. Recent measurements from the CMS Collaboration [50][51][52] indicate agreement between the predicted and measured cross sections for these processes, and we assign a systematic uncertainty of 50%.

V. EVENT YIELDS AND MEASUREMENTS AT THE RECONSTRUCTION LEVEL
The expected background and observed event yields for different dilepton flavor combinations are listed in Table II. The total predicted yield in the eμ channel is significantly larger than for the same-flavor channels because of the additional requirements on E miss T and M ll described in Sec. III that are applied to suppress the Drell-Yan background. After subtraction of the predicted background yields, the remaining yield in the data is assumed to be a signal from dileptonic tt decays, including τ leptons that decay leptonically. All other tt decay modes are treated as background and are included in the tt → l þ jets category. The largest background comes from tW production with dileptonic decays.
While the jΔϕ l þ l − j measurement relies only on the leptonic information, the measurements based on cos φ and cos θ ⋆ l AE require the reconstruction of the entire tt system. Each signal event has two neutrinos in the final state, and there is also a twofold ambiguity in combining the b quark jets with the leptons. In the case of events with only one b-tagged jet (62% of the selected events), the untagged jet with the highest b quark likelihood from the CSV algorithm is assumed to be the second b quark jet. Analytical solutions for the two neutrino momenta are obtained from the measuredp miss T with constraints on the invariant masses of the top quark and W boson decay systems of m t ¼ 172.5 GeV and m W ¼ 80.385 GeV. Each event can have up to 8 possible solutions. The one most likely to represent the correct tt configuration is chosen based on the probabilities to observe the extracted Bjorken x values of the initial-state partons and the measured lepton energies in their parent top quark rest frames [53]. For events with no physical solutions, a method is used to find a solution with the vector sum of the p T of the two neutrinos as close as possible to the measuredp miss T [54,55]. Nevertheless, in 16% of the events, no solutions can be found, both in the data and the simulation. These events are not used, except in the inclusive measurement of jΔϕ l þ l − j.
A comparison of the distributions for the reconstructed tt system variables M tt , y tt , and p T tt between data and simulation is shown in Fig. 1, where the signal yield from the simulation has been normalized to the number of signal events in the data after background subtraction. In general, the shapes of the distributions from data and simulation show reasonable agreement, with the small discrepancies covered by the systematic variations in the top quark p T modeling, PDFs, and μ R and μ F values, which will be discussed in Sec. VII. A similar comparison of the angular distributions is shown in Fig. 2. The corresponding inclusive asymmetry values, uncorrected for background, from the data and simulation are given in Table III.   TABLE I. Descriptions of the various control regions, their intended background process, and the scale factors derived from them, including either the statistical and systematic uncertainties or the total uncertainty. The last row gives the scale factor used for all the remaining backgrounds, whose contributions are estimated from simulation alone.

VI. UNFOLDING THE DISTRIBUTIONS
The observed angular distributions are distorted compared to the underlying distributions at the parton level (for which theoretical predictions exist) by the detector acceptance and resolution and the trigger and event selection efficiencies. To correct the data for these effects, we apply an unfolding procedure that yields the corrected jΔϕ l þ l − j, cos φ, c 1 c 2 , and cos θ ⋆ l AE distributions at the parton level. In the context of theoretical calculations and parton-shower event generators, the parton-level top quark is defined before it decays, and its kinematics include the effects of recoil from initial-and final-state radiation in the rest of the event and from final-state radiation from the top quark itself. The parton-level charged lepton, produced from the decay of the intermediate W boson, is defined before the lepton radiates any photons or the muon or tau lepton decays.
In order to unfold the observed distributions it is necessary to choose a binning scheme. Aiming to have bins with widths well matched to the reconstruction resolution and with approximately uniform event contents, we select six bins for each parton-level angular distribution except that of Δϕ l þ l − . This variable depends only on the lepton momentum measurements, not on the reconstruction of the tt system, and the superior resolution allows us to use 12 bins. For the reconstruction-level distributions we use twice as many bins as for the parton-level distributions.
The background-subtracted distribution for each variable, considered as a vectorỹ, is related to the underlying parton-level distributionx through the equationỹ ¼ SAx, where A is a diagonal matrix describing the fraction (acceptance times efficiency) of all produced signal events that are expected to be selected in each of the measured bins, and S is a nondiagonal "smearing" matrix describing the migration of events between bins caused by imperfect detector resolution and reconstruction techniques. The A and S matrices are constructed using simulated MC@NLO tt events. The smearing in cos φ, c 1 c 2 , and cos θ ⋆ l AE can be large in some events because of the uncertainties in the reconstruction of the tt kinematic quantities, but the smearing matrices are still predominantly diagonal. The smearing matrix for jΔϕ l þ l − j is nearly diagonal because of the excellent angular resolution of the lepton momentum measurements.
To determine the parton-level angular distribution in data, we employ a regularized unfolding algorithm implemented in the TUNFOLD package [56]. The effects of large statistical fluctuations in the algorithm are greatly reduced by introducing a term in the unfolding procedure that regularizes the output distribution based on the curvature of the simulated signal distribution. In general, unfolding introduces negative correlations between adjacent bins, while regularization introduces positive correlations, and the regularization strength is optimized by minimizing the average global correlation coefficient in the unfolded distribution. The regularization strength obtained here is relatively weak, contributing at the 10% level to the total χ 2 minimized by the algorithm.
After unfolding, each distribution is normalized to unit area to give the normalized differential cross section for each variable. We use an analogous unfolding procedure to measure the normalized double-differential cross section, using three bins of M tt , jy tt j, and p T tt for each variable. The full covariance matrix is used in the evaluation of the statistical uncertainty in the asymmetry measured from each distribution.

VII. SYSTEMATIC UNCERTAINTIES
The systematic uncertainties coming from the detector performance and the modeling of the signal and background processes are evaluated from the difference between the nominal measurement and that obtained by repeating the unfolding procedure using simulated events with the appropriate systematic variation.
The uncertainty from the jet energy scale (JES) corrections affects the tt final-state reconstruction, as well as the event selection. It is estimated by varying the energies of jets within their uncertainties [57], and propagating this to the E miss T value. Similarly, the jet energy resolution is varied by 2%-5%, depending on the η of the jet [57], and the electron energy scale is varied by AE0.6% (AE1.5%) for barrel (endcap) electrons (the uncertainty in muon energies is negligible), as estimated from comparisons between measured and simulated Drell-Yan events [58].
The uncertainty in the background contribution is obtained by varying the normalization of each background component by the uncertainties described in Sec. IV.
Many of the signal modeling and simulation uncertainties are evaluated by using weights to vary the MC@NLO tt sample: the simulated pileup multiplicity distribution is changed within its uncertainty; the correction factors between data and simulation for the b tagging [26], trigger, and lepton selection efficiencies are shifted up and down by their uncertainties; and the PDFs are varied using the PDF4LHC procedure [59,60]. Previous CMS studies [61,62] have shown that the p T distribution of the top quark measured from data is softer than that in the NLO simulation of tt production. Since the origin of the discrepancy is not fully understood, the change in the measurement when reweighting the MC@NLO tt sample to match the top quark p T spectrum in data is taken as a systematic uncertainty associated with signal modeling. The remaining signal modeling uncertainties are separately evaluated with dedicated tt samples: μ R and μ F are varied together up and down by a factor of 2; the top quark mass is varied by AE1 GeV, to be consistent with the uncertainty used in other CMS measurements with the ffiffi ffi s p ¼ 8 TeV data set (the effect on the total systematic uncertainty of using the reduced uncertainty from the recent CMS combined m t measurement [1] would be negligible); and the S matrix is rederived from a tt sample generated with POWHEG and PYTHIA, while the A matrix is unchanged, in order to estimate the difference in hadronization modeling between HERWIG and PYTHIA. To avoid underestimation of systematic uncertainties caused by statistical fluctuations, which can be significant in the estimates evaluated using dedicated tt samples, for each source of uncertainty the maximum of the estimated systematic uncertainty and the statistical uncertainty in that estimate is taken as the final systematic uncertainty. The uncertainty in the unfolding procedure is dominated by the statistical uncertainty arising from the finite number of events in the MC@NLO tt sample. The uncertainty owing to the unfolding regularization is evaluated by using the reconstucted distribution of a variable in data to reweight the corresponding simulated signal distribution used to regularize the curvature of the unfolded distribution. Using this method, the regularization uncertainty is found to be negligible for all measurements.
The systematic uncertainties in the inclusive asymmetry variables obtained from the unfolded distributions are summarized in Table IV. The systematic uncertainties are evaluated for each bin of the unfolded distributions, from which the covariance matrix is constructed, assuming 100% correlation or anticorrelation between bins for each individual source of uncertainty. The total systematic uncertainty is calculated by adding in quadrature the listed uncertainties.
For A Δϕ , the top quark p T modeling uncertainty dominates; this arises from the dependence of the jΔϕ l þ l − j distribution shape on the top quark p T (through the spin correlations and event kinematics); that, in turn, introduces a significant dependence of the acceptance correction on the top quark p T . For A P , the JES and hadronization systematic uncertainties are dominant. Both affect the reconstructed b quark jet energy, and can therefore bias the boost from the laboratory frame to the top quark centerof-mass frame, and thus the measurement of cos θ ⋆ l AE . For similar reasons, the same two uncertainties are large for A c 1 c 2 and A cos φ , which are also significantly affected by the top quark p T modeling uncertainty through its effect on the spin correlations. For A CPV P , the similar systematic uncertainties in A Pþ and A P− largely cancel when A P− is subtracted from A Pþ ; the remaining contributions to the systematic uncertainty are dominated by the statistical uncertainty in the simulation.

A. Unfolded distributions
The background-subtracted, unfolded, and normalizedto-unit-area angular distributions for the selected data events are shown in Fig. 3, along with the parton-level predictions obtained with the MC@NLO event generator and from calculations at NLO in the strong and weak gauge couplings for tt production, with and without spin correlations [4,63].
The measured asymmetries, obtained from the angular distributions unfolded to the parton level, are presented with their statistical and systematic uncertainties in Table V, where they are compared to predictions from MC@NLO and Normalized differential cross section as a function of jΔϕ l þ l − j, cos φ, cos θ ⋆ l þ cos θ ⋆ l − , and cos θ ⋆ l from data (points); partonlevel predictions from MC@NLO (dashed histograms); and theoretical predictions at NLO [4,63] with (SM) and without (no spin corr.) spin correlations (solid and dotted histograms, respectively). For the cos θ ⋆ l distribution, CP conservation is assumed in the combination of the cos θ ⋆ l AE measurements from positively and negatively charged leptons. The ratio of the data to the MC@NLO prediction is shown in the lower panels. The inner and outer vertical bars on the data points represent the statistical and total uncertainties, respectively. The hatched bands represent variations of μ R and μ F simultaneously up and down by a factor of 2. the NLO calculations. Correlations between the contents of different bins, introduced by the unfolding process and from the systematic uncertainties, are accounted for in the calculation of the experimental uncertainties. The uncertainties in the NLO predictions come from varying μ R and μ F simultaneously up and down by a factor of 2. For A cos φ and A c 1 c 2 , these scale uncertainties are summed in quadrature with the difference between the NLO predictions from Ref.
[4] when the ratio in the calculation is expanded in powers of the strong coupling constant and when the numerator and denominator are evaluated separately.
Using the relationships between the asymmetry variables and spin correlation coefficients given in Sec. I, we find C hel ¼ 0.278 AE 0.084 and D ¼ 0.205 AE 0.031, where the uncertainties include the statistical and systematic components added in quadrature. Similarly, the CP-conserving and CP-violating components of the top quark polarization are found to be P ¼ −0.022 AE 0.058 and P CPV ¼ 0.000 AE 0.016, respectively. All measurements are consistent with the expectations of the SM.
The NLO predictions for jΔϕ l þ l − j, cos φ, and c 1 c 2 with and without spin correlations in Table Vare used to translate the measurements into determinations of f SM , the strength of the spin correlations relative to the SM prediction, with f SM ¼ 1 corresponding to the SM and f SM ¼ 0 corresponding to uncorrelated events. The measurements of f SM are shown in Table VI and are derived under the assumption that the A matrix used for the unfolding is independent of spin correlations. This is found to give conservative estimates for the experimental uncertainties.
The dependence of each asymmetry on M tt , jy tt j, and p T tt is extracted from the measured normalized doubledifferential cross section, and the results are shown in Fig. 4. The measurements are all consistent with the MC@NLO predictions, and with the SM NLO prediction for the M tt and jy tt j dependencies. No comparison is made with the NLO prediction for the p T tt dependence because the substantial effect of the parton shower on the p T tt distribution means fixed-order NLO calculations are not a sufficiently good approximation of the data.
Compared to the measurement of A Δϕ in Table V, the differential measurement in bins of M tt (Fig. 4, top row, left plot) has a significantly reduced (factor of 2.3) systematic uncertainty associated with the top quark p T modeling. When the acceptance correction is binned in a variable that is correlated with the top quark p T (e.g., M tt ), the top quark p T reweighting affects the numerator and denominator in the acceptance ratio similarly, leading to a reduction in the associated systematic uncertainty. The inclusive asymmetry measured from the projection in jΔϕ l þ l − j of the normalized double-differential cross section is A Δϕ ¼ 0.095 AE 0.006ðstatÞ AE 0.007ðsystÞ, which is converted into the value of f SM ¼ 1.12 þ0.12 −0.15 given in Table VI.

B. Limits on new physics
Anomalous ttg couplings can lead to a significant modification of the polarization and spin correlations in tt events. A model-independent search can be performed using an effective model of chromo-magnetic and chromoelectric dipole moments (denoted CMDM and CEDM, respectively). This study follows the proposal in Ref. [4]. For an anomalous ttg interaction arising from heavyparticle exchange characterized by a mass scale M ≳ m t , one can write an effective Lagrangian as TABLE V. Inclusive asymmetry measurements obtained from the angular distributions unfolded to the parton level, and the partonlevel predictions from the MC@NLO simulation and from NLO calculations with (SM) and without (no spin corr.) spin correlations [4,63]. For the data, the first uncertainty is statistical and the second is systematic. For the MC@NLO Table V. The last row shows an additional measurement of f SM made from the projection in jΔϕ l þ l − j of the normalized double-differential cross section as a function of jΔϕ l þ l − j and M tt . The uncertainties shown are statistical, systematic, and theoretical, respectively. The total uncertainty in each result, found by adding the individual uncertainties in quadrature, is shown in the last column.  , jy tt j (middle), and p T tt (right), obtained from the unfolded double-differential distributions; parton-level predictions from MC@NLO (dashed histograms); and theoretical predictions at NLO [4,63] with (SM) and without (no spin corr.) spin correlations (solid and dotted histograms, respectively). The inner and outer vertical bars on the data points represent the statistical and total uncertainties, respectively. The hatched bands represent variations of μ R and μ F simultaneously up and down by a factor of 2. The last bin of each plot includes overflow events.
whereμ t andd t are the CMDM (CP-conserving) and CEDM (CP-violating) dipole moments, G a μν is the gluon field strength, and T a are the QCD fundamental generators. It is usually preferred to define dimensionless parameterŝ where g s is the QCD coupling constant [4]. The parameterŝ μ t andd t correspond to the form factors in the timelike kinematic domain and are therefore complex quantities, here assumed to be constant. In general, both the real and imaginary parts ofμ t andd t can be determined, but the spin correlations and polarization measured in this paper are only sensitive to Reðμ t Þ and Imðd t Þ, respectively [4]. We begin with the determination of Reðμ t Þ using the measured normalized differential cross section ð1=σÞðdσ= djΔϕ l þ l − jÞ. In the presence of a small new physics (NP) contribution such that Reðμ t Þ ≪ 1, one can linearly expand the normalized differential cross section as [4] The predicted shapes of the SM and NP terms in Eq.
(3) are shown in Fig. 5. The NP term arises from interference with SM tt production, and therefore gives both positive and negative contributions to the differential cross section.
To measure Reðμ t Þ, the SM and NP contributions to Eq. (3) are parametrized by polynomial functions (shown in Fig. 5), which are then used in a template fit to the measured normalized differential cross section. We use the projection in jΔϕ l þ l − j of the measured normalized double-differential cross section in bins of M tt to minimize the systematic uncertainty from top quark p T modeling, as for the extraction of f SM . The measurement is made under the assumption that the A matrix is unchanged by the presence of NP. Studies of the effects of our selection criteria at the parton level show that this leads to conservative estimates of the experimental uncertainties. The fit is performed using a χ 2 minimization, accounting for both statistical and systematic uncertainties and their correlations, with Reðμ t Þ as the only free parameter. The systematic uncertainty arising from the choice of μ R and μ F in the theoretical calculations from Ref.
[4] is estimated by repeating the fit after varying both scales together up and down by a factor of 2. This constitutes the dominant source of uncertainty. The proper behavior of the fit is verified using pseudoexperiments. The result of the fit is Reðμ t Þ ¼ −0.006 AE 0.024 and is shown graphically in Fig. 5. The corresponding 95% confidence level (C.L.) interval is −0.053 < Reðμ t Þ < 0.042.
The spin correlation coefficient D is also sensitive to Reðμ t Þ, and the CP-violating component of the top quark polarization P CPV is sensitive to Imðd t Þ. Studies of the effects of our selection criteria at the parton level show that the presence of anomalous top quark chromo moments has no significant effect on the A matrix for either of these variables, and we use this assumption in the derivation of limits on Reðμ t Þ and Imðd t Þ.
For the D coefficient, Eq. (3) simplifies to D ¼ D SM þ Reðμ t ÞD NP [4]. Using the values from Table V, the relationship D ¼ −2A cos φ , and taking D NP ¼ −1.712 AE 0.019 from Ref.
[4], we find Reðμ t Þ ¼ −0.014 AE 0.020, with the corresponding 95% C.L. interval −0.053 < Reðμ t Þ < 0.026. The constraints on Reðμ t Þ from D are stronger than those [4] (points) and polynomial parametrization (line) for the contribution from new physics with a nonzero CMDM to the normalized differential cross section ð1=σÞðdσ=djΔϕ l þ l − jÞ, for Reðμ t Þ ≪ 1. Bottom : normalized differential cross section from data (points). The solid line corresponds to the result of the fit to the form given in Eq. (3), and the dashed lines show the parametrized SM NLO predictions for μ R and μ F equal to m t , 2m t , and m t =2. The vertical bars on the data points represent the total uncertainties. from the jΔϕ l þ l − j fit because the smaller theoretical uncertainty in the SM NLO calculation of D compared to that in the jΔϕ l þ l − j distribution outweighs the slightly larger experimental uncertainty.
The jΔϕ l þ l − j distribution is potentially sensitive to pairproduced scalar top quark partners (top squarks) that decay to produce a top quark and antiquark with no additional visible particles. The spin-zero particles transmit no spin information from the initial state to the final-state top quarks, meaning such events look much like uncorrelated tt events. We assess the sensitivity of the measured jΔϕ l þ l − j distribution to pair-produced top squarks with mass equal to m t . As seen from the measurement of f SM in the last row of Table VI, the dominant source of uncertainty is the theoretical scale uncertainty in the jΔϕ l þ l − j distribution. The result is that no exclusion limits on top squarks can be set using the jΔϕ l þ l − j normalized differential cross section alone, and the additional sensitivity it would bring in combination with the inclusive measurement of the cross section would be marginal.

IX. SUMMARY
Measurements of the tt spin correlations and the top quark polarization have been presented in the tt dilepton final states (e þ e − , e AE μ ∓ , and μ þ μ − ), using angular distributions unfolded to the parton level and as a function of the tt-system variables M tt , jy tt j, and p T tt . The data sample corresponds to an integrated luminosity of 19.5 fb −1 from pp collisions at ffiffi ffi s p ¼ 8 TeV, collected by the CMS experiment at the LHC. For the spin correlation coefficients, we measure C hel ¼ 0.278 AE 0.084 and D ¼ 0.205 AE 0.031. The measurements sensitive to spin correlations are translated into determinations of f SM , the strength of the spin correlations relative to the SM prediction. The most precise result comes from the measurement of A Δϕ ¼ 0.095 AE 0.006 ðstatÞ AE 0.007 ðsystÞ, yielding f SM ¼ 1.12 þ0.12 −0.15 . The SM (CPconserving) top quark polarization is measured to be P ¼ −0.022 AE 0.058, while the CP-violating component is found to be P CPV ¼ 0.000 AE 0.016. All measurements are in agreement with the SM expectations, and help constrain theories of physics beyond the SM.
The measured top quark spin observables are compared to theoretical predictions in order to search for hypothetical top quark anomalous couplings. No evidence of new physics is observed, and exclusion limits on the real part of the chromo-magnetic dipole moment Reðμ t Þ and the imaginary part of the chromo-electric dipole moment Imðd t Þ are evaluated. Values outside the intervals −0.053 < Reðμ t Þ < 0.026 and −0.068 < Imðd t Þ < 0.067 are excluded at the 95% confidence level, the first such measurements to date.

ACKNOWLEDGMENTS
We would like to thank W. Bernreuther and Z.-G. Si for calculating the theoretical predictions for this paper, and for studies of the effect of anomalous top quark chromo moments on the acceptance of our selection criteria at the parton level. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies:   [50] CMS Collaboration, Measurement of the sum of WW and WZ production with W þ dijet events in pp collisions at ffiffi ffi s p ¼ 7 TeV, Eur. Phys. J. C 73, 2283 (2013).
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