Forward-backward asymmetry of leptonic decays of tt̄ atthe Fermilab Tevatron

Ziqing Hong,1,* Ryan Edgar,2 Sarah Henry,1 David Toback,1 Jonathan S. Wilson,1,2 and Dante Amidei21Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University,

College Station, Texas 77843, USA2University of Michigan, Ann Arbor, Michigan 48109, USA

(Received 1 April 2014; published 22 July 2014)

We report on a study of the measurement techniques used to determine the leptonic forward-backwardasymmetry of top-antitop quark pairs in Tevatron experiments with a proton-antiproton initial state. Recentlyit was shown that a fit of the differential asymmetry as a function of qlηl (where ql is the charge of the leptonfrom the cascade decay of the top quarks and ηl is the final pseudorapidity of the lepton in the detector frame)to a hyperbolic tangent function can be used to extrapolate to the full leptonic asymmetry. We find thisempirical method towell reproduce the results from current experiments and present arguments as to why thisis the case. We also introduce two more models, based on Gaussian functions, that better model the qlηldistribution. With our better understanding, we find that the asymmetry is mainly determined by the shift ofthe mean of the qlηl distribution, the main contribution to the inclusive asymmetry comes from the regionaround jqlηlj ¼ 1, and the extrapolation from the detector-covered region to the inclusive asymmetry is stablevia a multiplicative scale factor, giving us confidence in the previously reported experimental results.

DOI: 10.1103/PhysRevD.90.014040 PACS numbers: 14.65.Ha, 11.30.Er, 13.85.Qk

I. INTRODUCTION

Recent measurements of the forward-backward asym-metry (AFB) of top-antitop quark pair (tt̄) production inproton-antiproton collisions with

ffiffiffis

p ¼ 1.96 TeV at theFermilab Tevatron [1–3] have shown anomalously largevalues compared to the predictions from the standard model(SM) of particle physics at next-to-leading order (NLO) [4].This is of great interest as new particles or interactionscould cause the AFB of tt̄ (Att̄

FB) to be different fromSM-only predictions [5]. An alternative observable thatcould be also affected is the forward-backward asymmetryof the leptons from the cascade decay of the top quarks, theso-called leptonic forward-backward asymmetry (Al

FB) [6].In addition, Al

FB can deviate further from its SM predictionin the scenarios that the top quarks are produced with acertain polarization. For example, resonant production of tt̄via a hypothesized axigluon could cause the Att̄

FB to varyfrom its SM value; while different chiral couplings betweenthe axigluons and the top quarks could produce the samevalue of Att̄

FB, but very different values of AlFB [7].

For a sample of tt̄ events that decay into one or morecharged leptons, the Al

FB is defined as

AlFB ¼ Nðqlηl > 0Þ − Nðqlηl < 0Þ

Nðqlηl > 0Þ þ Nðqlηl < 0Þ ð1Þ

where N is the number of charged leptons (electrons ormuons) in the sample, ql is the lepton charge, and ηl is thepseudorapidity of the charged lepton. The measurement of

AlFB has been done in both the leptonþ jets final state (where

only one W boson from the top quarks decays leptonically)and the dilepton final state (where both W bosons decayleptonically) at both the CDF [8,9] and the D0 [10,11]experiments using different methods. Of critical importancefor this measurement is the methodology to extrapolate fromthe finite coverage of the experiments (jηlj < 1.25 and jηlj <2.0 in the leptonþ jets and dilepton final states respectively)to the full pseudorapidity range (inclusive) parton-level result.A method, first proposed in Ref. [8], is to decompose themeasured qlηl distribution into a symmetric part (SðqlηlÞterm) and an asymmetric part (differential forward-backwardasymmetry,AðqlηlÞ term). Studies indicated that the SðqlηlÞterm was nearly model independent; using a distributionestimated with any sample of simulated events only intro-duces a small systematic uncertainty.Equally important is thattheAðqlηlÞ term was found to vary significantly from modelto model as a function of Al

FB, allowing for a measurement;this part is measured directly from data. Interestingly,empirical studies showed that a hyperbolic tangent functioncould be used to model theAðqlηlÞ termwith a measurementbias that was negligible compared to the other uncertainties.In this article, we first briefly describe the parametrization

introduced by Ref. [8], then introduce more detailed studiesof the parton level qlηl distribution to both understand whythe hyperbolic tangent function works so well, and to seewhat improvements could be made with a better under-standing. We find that the qlηl distribution is actually welldescribed by a double-Gaussian distribution, where theasymmetry arises from a shift in themean of the distribution.We conclude this manuscript with the implications of thismodeling, as well as our thoughts for future measurements.*zqhong@fnal.gov

PHYSICAL REVIEW D 90, 014040 (2014)

1550-7998=2014=90(1)=014040(10) 014040-1 © 2014 American Physical Society

II. LEPTONIC AFB MEASUREMENTS ATTHE TEVATRON

To study the qlηl distribution with different physicalscenarios, we used six benchmark Monte Carlo (MC)simulated samples. To model the SM we consider two

leading-order (LO) SM samples generated by PYTHIA [12]and ALPGEN [13], and for NLO effects we use a samplegenerated with POWHEG [14–17]; we note that the POWHEG

sample does have quantum chromodynamics (QCD)effects, but does not have electroweak (EWK) effects[18]. To test the measurement on a larger range of Al

FB,we consider three samples with physics beyond the SM,with a class of relatively light and wide axigluons(m ¼ 200 GeV=c2, Γ ¼ 50 GeV=c2) with left-handed,right-handed, and axial flavor-universal couplings to thequarks [7], generated with MADGRAPH [19]. These arechosen as they all predict an Att̄

FB value that is close to thevalue observed at CDF [1], but give very different pre-dictions of Al

FB [7,9]. The qlηl distributions at parton levelfor all six benchmark tt̄ samples are shown in Fig. 1. TheAlFB values predicted by the samples span the range of

−0.1 < AlFB < 0.2 and are listed in Table I along with a full

NLO SM calculation, together with the results of themeasurements from CDF and D0 in both the leptonþjets and dilepton final states. We note that the measurementfrom D0 in the leptonþ jets final state is limited to theregion where jqlηlj < 1.5. Later in this article we provide astable extrapolation of the Al

FB in this region to theinclusive Al

FB.As described in Ref. [8,9], the qlηl distribution of the

leptons can be decomposed into an SðqlηlÞ term and anAðqlηlÞ term using the following formulas in the rangeqlηl ≥ 0:

SðqlηlÞ ¼N ðqlηlÞ þN ð−qlηlÞ

2; and ð2aÞ

(a)

(b)

FIG. 1 (color online). The qlηl distribution of charged leptonsproduced from tt̄ cascade decay from simulations with variousphysics models at parton level, before any selection requirements.In (b) only the range between −2.5 and 2.5 is shown.

TABLE I. A collection of different predictions and measurements of AlFB from various sources. The uncertainties

for the simulated samples are statistical only. The uncertainty for the NLO SM calculation is due to the variation inthe scales in the calculation. The uncertainties for the CDF and D0 results are the overall uncertainties from themeasurements.

Source AlFB Description

MADGRAPH (AxiL) −0.063� 0.002Tree-level left-handed axigluon(m ¼ 200 GeV=c2, Γ ¼ 50 GeV)

MADGRAPH (AxiR) 0.151� 0.002Tree-level right-handed axigluon(m ¼ 200 GeV=c2, Γ ¼ 50 GeV)

MADGRAPH (Axi0) 0.050� 0.002Tree-level unpolarized axigluon(m ¼ 200 GeV=c2, Γ ¼ 50 GeV)

ALPGEN 0.003� 0.001 Tree-level Standard ModelPYTHIA 0.000� 0.001 LO Standard ModelPOWHEG 0.024� 0.001 NLO Standard Model with

QCD correctionsCalculation 0.038� 0.003 NLO SM with QCD and

EWK corrections [4]

CDF0.094þ0.032

−0.029 Leptonþ jets

0.072� 0.060 Dilepton0.090þ0.028

−0.026 Combination

D00.047þ0.025

−0.027 Leptonþ jets, jqlηlj < 1.50.044� 0.039 Dilepton

HONG et al. PHYSICAL REVIEW D 90, 014040 (2014)

014040-2

AðqlηlÞ ¼N ðqlηlÞ −N ð−qlηlÞN ðqlηlÞ þN ð−qlηlÞ

; ð2bÞ

where N ðqlηlÞ represents the number of events as afunction of qlηl. With this, the Al

FB defined in Eq. (1)can be rewritten in terms of SðqlηlÞ and AðqlηlÞ as

AlFB ¼

R∞0 dx½AðxÞ · SðxÞ�R∞

0 dx0Sðx0Þ : ð3Þ

The SðqlηlÞ term and the AðqlηlÞ term distributionsfrom the benchmark samples are shown in Fig. 2(a) andFig. 2(b), respectively. We can readily see that the variationof the SðqlηlÞ term among the benchmark tt̄ samples issmall, so choosing any one of them for the measurementintroduces an uncertainty that is tiny compared to thedominant uncertainties. We will come back to the smalldifferences for qlηl < 0.2 and show why they do not havemuch effect on the measurement. On the other hand, theAðqlηlÞ term varies significantly from model to model.The AðqlηlÞ term has been well described in the regionjqlηlj < 2.0 using the ansatz of

AðqlηlÞ ¼ a · tanh

�1

2qlηl

�; ð4Þ

where a is a free parameter that is directly related to thefinal asymmetry. Best fits of the data to the a · tanh model

from Eq. (4) are also shown in Fig. 2(b). While theAðqlηlÞterm is well modeled in the region where qlηl < 2.5, it isnot as good above 2.5. The comparison between thepredicted Al

FB and the AlFB obtained with a measured value

of a in Eq. (4) from the AðqlηlÞ term (restricting the fitwithin the region qlηl < 2.0 to simulate a detector) isshown in Fig. 3. The differences are on the order of afraction of a percent, which is tiny compared to thedominant uncertainties listed in Table I [8,9].While the methodology works well, the parametrization

of Eq. (4) is purely empirical. In the following sections,we provide a partial explanation of where the hyperbolictangent functional form comes from as well as thebetter parametrization to which the new understandingleads.We will further see that the choice of the observable Al

FBis advantageous because it uses the precise measurement ofthe value of qlηl for each lepton. Thus, the bin-to-binmigration of events due to detector smearing is small, andhas no measurable effect on the final value of Al

FB.However, we will also see that judicious choices of thebinning, especially at large jqlηlj, are important when usingfitting and extrapolation techniques.

III. SINGLE AND DOUBLEGAUSSIAN MODELING

The qlηl distributions in Fig. 1 appear to be roughlyGaussian distributed with a nonzero mean. However, theGaussian model is only good in the small-jqlηlj region.Figure 4 shows the qlηl distribution at parton level fromthe POWHEG tt̄ sample with a fit to a Gaussian function, butwith the fit restricted to jqlηlj < 1.4. Note that the fit isnot good for jqlηlj > 1.4. This simple model is clearlyinsufficient.Before moving on to a better model, we use this simple

model to illustrate the methodology. We note that thenumber of events in the interval (qlηl, qlηl þ δðqlηlÞ) canbe readily calculated using

(a)

(b)

FIG. 2 (color online). The SðqlηlÞ term (a) and theAðqlηlÞ term(b) of the qlηl distribution from various physics models. The linesin (b) correspond to the best fits from the a · tanh model.

FIG. 3 (color online). A comparison between the predicted AlFB

from simulations and the AlFB as measured using the a · tanh

parametrization with parton level information from jqlηlj < 2.0.The dashed line indicates the location of the equal values, whilethe points are superimposed at their measured locations. All thepoints lie along the line within uncertainties.

FORWARD-BACKWARD ASYMMETRY OF LEPTONIC DECAYS … PHYSICAL REVIEW D 90, 014040 (2014)

014040-3

(a)

(b)

FIG. 5 (color online). The qlηl distribution from the POWHEG tt̄sample at parton level, overlaid with the double-Gaussian fit.Note that both the tails and the central part of the distribution arewell described.

(a)

(b)

FIG. 4 (color online). The qlηl distribution from the POWHEG tt̄sample at parton level, with a fit to a single Gaussian function inthe region jqlηlj < 1.4 (indicated by the dashed lines). Note thatthe agreement is not good for jqlηlj > 1.4.

(a) (b)

(c) (d)

FIG. 6 (color online). Fit parameters from our benchmark samples as a function of AlFB.

HONG et al. PHYSICAL REVIEW D 90, 014040 (2014)

014040-4

N ðqlηl; qlηl þ δðqlηlÞÞ ¼Z

qlηlþδðqlηlÞ

qlηl

dx C · Exp

�−ðx − μÞ22σ2

�

¼ C · Exp

�−ðqlηl − μÞ2

2σ2

�δðqlηlÞ; when δðqlηlÞ → 0; ð5Þ

where C is a normalization constant, μ is the mean of the distribution and σ is the width of the distribution. We can thencalculate AðqlηlÞ with this function:

AðqlηlÞ ¼ limδðqlηlÞ→0

N ðqlηl; qlηl þ δðqlηlÞÞ −N ð−qlηl − δðqlηlÞ;−qlηlÞN ðqlηl; qlηl þ δðqlηlÞÞ þN ð−qlηl − δðqlηlÞ;−qlηlÞ

¼ Expð− ðqlηl−μÞ22σ2

Þ − Expð− ð−qlηl−μÞ22σ2

ÞExpð− ðqlηl−μÞ2

2σ2Þ þ Expð− ð−qlηl−μÞ2

2σ2Þ

¼ tanh

�μ · qlηlσ2

�: ð6Þ

We note that it has the form of a hyperbolic tangentfunction, but with the parameter inside the function argu-ment, not an overall scaling factor as in Eq. (4).Since the single Gaussian function works only in the

small jqlηlj region, we tried a more sophisticated model,and found that the sum of two Gaussian functions with acommon mean works very well at describing the data, evenat large values of qlηl. We have not uncovered an a prioriexplanation why this should be so, but it appears to be truefor all the models we considered [20]. We use the func-tional form,

dN ðqlηlÞdðqlηlÞ

¼ C ·

�Exp

�−ðqlηl − μÞ2

2σ21

�

þ r · Exp

�−ðqlηl − μÞ2

2σ22

��; ð7Þ

where C is a normalization constant, r is a multiplicativefactor that covers the relative normalization of the twocomponents and σ1 and σ2 are the widths of the two differentdistributions. Figure 5 shows a comparison between the bestfit and the parton level data. This functional form works wellfor all our benchmark signal samples; the two σ terms andthe r term are very consistent as shown in Fig. 6. We findσ1 ¼ 0.91, σ2 ¼ 1.61 and r ¼ 0.11. More importantly, themean (μ) varies significantly from one sample to another,and appears to be linear with Al

FB. From here on, we assumethe two σ terms and the r term have the best-fit values fromthe benchmark samples for further studies.The double-Gaussian modeling allows for closed form

calculations of the SðqlηlÞ and theAðqlηlÞ terms as well asthe inclusive Al

FB using just the μ, σ1, σ2 and r parameters.We find the SðqlηlÞ term and the AðqlηlÞ term have thefunctional forms of

SðqlηlÞ ¼C2·

�e−ðqlηl−μÞ2

2σ21 þ e

−ðqlηlþμÞ22σ2

1 þ r · e−ðqlηl−μÞ2

2σ22 þ r · e

−ðqlηlþμÞ22σ2

2

�; and ð8aÞ

0.1 0.1 0.2

0.1

0.1

0.2

AFBl

(a)

3 2 1 1 2 3

1

1

AFBl

(b)

FIG. 7 (color online). With the double-Gaussian modeling, and constraining the two σ values and the r to the best estimated valuesfrom the benchmark simulations, Al

FB appears to be linear as a function of the mean of the double-Gaussian function in the small AlFB

region. In a larger region, AlFB asymptotes to �1.

FORWARD-BACKWARD ASYMMETRY OF LEPTONIC DECAYS … PHYSICAL REVIEW D 90, 014040 (2014)

014040-5

AðqlηlÞ ¼e−ðqlηl−μÞ2

2σ21 − e

−ðqlηlþμÞ22σ2

1 þ r · e−ðqlηl−μÞ2

2σ22 − r · e

−ðqlηlþμÞ22σ2

2

e−ðqlηl−μÞ2

2σ21 þ e

−ðqlηlþμÞ22σ2

1 þ r · e−ðqlηl−μÞ2

2σ22 þ r · e

−ðqlηlþμÞ22σ2

2

: ð8bÞ

It is not clear how to simplify these. However, theinclusive Al

FB from Eq. (3) can be simplified to

AlFB ¼

σ1 · erfð μffiffi2

pσ1Þ þ r · σ2 · erfð μffiffi

2p

σ2Þ

σ1 þ r · σ2: ð9Þ

This functional form is shown in Fig. 7, and, in the limit ofμ ≪ σ1, which corresponds to jAl

FBj≲ 0.2, in Fig. 7(a), wefind that Al

FB ¼ 0.82 · μ which approximates the data well.The SM prediction and most models of new physics (and

the current data) all have values of AlFB < 0.2, so this can

have a significant impact in simplifying the measurements.We can show the distribution of the SðqlηlÞ and AðqlηlÞfrom Eq. (8) with μ ¼ −0.1, 0.02 and 0.2 in Fig. 8. TheSðqlηlÞ term is largely unchanged except for small valuesof qlηl as previously noted, and the AðqlηlÞ term variessignificantly. We also note that the distribution looks like ahyperbolic tangent function for qlηl < 2, but has differentstructures for larger values of qlηl.A second set of important results comes from a descrip-

tion of how much contribution there is to the totalasymmetry as a function of qlηl (the differential contribu-tion). It can be calculated as

SðqlηlÞ ·AðqlηlÞR∞0 SðxÞdx ; ð10Þ

where the denominator normalizes the area under the curveto be the total asymetry. The results are shown in Fig. 9(a)for the same three μ values. In some ways the threecurves look very different, but they do share some commonfeatures. While the area under the curve is stronglydependent on μ, the shape of the distribution looksremarkably similar for all three curves. To see the sim-ilarity, we plot the normalized shape by rewriting Eq. (10)such that the integral under the curve is equal to unity.Specifically:

(a)

(b)

FIG. 8 (color online). The SðqlηlÞ term and the AðqlηlÞ termfrom the double-Gaussian model, with the μ parameter varied.

(a)

(b)

FIG. 9 (color online). Figures showing the differential contri-bution to the total asymmetry as a function of qlηl using thedouble-Gaussian model. This is estimated using the SðqlηlÞ termtimes the AðqlηlÞ term, with the μ parameter varied, withdifferent overall normalizations. (a) The curves are normalizedso that

RSðqlηlÞdqlηl ¼ 1 as in Eq. (10). In this case, the areas

under the curves give the inclusive asymmetry. (b) The curves arenormalized to

RSðqlηlÞ ·AðqlηlÞdqlηl ¼ 1 as in Eq. (11). In this

case, we can see that the differential contribution to theasymmetry as a function of qlηl is largely independent of thevalue of μ for small values of μ.

HONG et al. PHYSICAL REVIEW D 90, 014040 (2014)

014040-6

SðqlηlÞ ·AðqlηlÞR∞0 SðxÞ ·AðxÞdx : ð11Þ

The results are shown in Fig. 9(b) and we note that theshape of the differential contribution stays remarkablystable.We are now able to make a number of further observa-

tions. First, the dominant contribution to the overallasymmetry comes from the region around jqlηlj ¼ 1, whichis the place where the detectors have excellent coverage andresolution. We can also see why the slight mismodeling inthe vicinity of qlηl ¼ 0 in the SðqlηlÞ term, as shown inFig. 2(a), and the mismodeling from the a · tanh descriptionin the region where qlηl > 2.5 in the AðqlηlÞ term wouldonly introduce small biases in the overall measurementcompared to the dominant uncertainties. Specifically, eventhough most of the events have jqlηlj < 0.1, the contribu-tion to Al

FB from this region is ∼2%. Similarly, the qlηlregion where there is no detector coverage at CDF or D0,jqlηlj > 2.0, contributes ∼11% to the inclusive Al

FB; con-versely, the region where the a · tanh fit performs poorly,jqlηlj > 2.5, contributes only 4%. In addition, the con-stancy of the shape of the differential contribution providesan explanation for why the extrapolation technique fromthe measured Al

FB to the inclusive AlFB is robust. The

fraction of the AlFB within certain jqlηlj ranges are shown in

Fig. 10, and some interesting numbers corresponding to

typical lepton coverages at CDF and D0 are listed inTable II.

IV. COMPARING THE SENSITIVITY OFTHE a · tanh, SINGLE-GAUSSIAN AND

DOUBLE-GAUSSIAN MODELS

We compare the sensitivity of the possible measurementtechniques in a number of ways. First we compare themvisually, then we consider how well the different meas-urement techniques would work. Fig 11 shows the AðqlηlÞterm and the differential contribution to the inclusive Al

FB asa function of qlηl from the POWHEG sample, overlaid withthe best fit from the a · tanh model, the single-Gaussianmodel and the double-Gaussian model described in thisarticle, when we only consider events with jqlηlj < 2.0. Allthree models fit this qlηl region well. Since the regionjqlηlj < 2.0 is where most of the contribution to Al

FB comesfrom, all three models (including the single-Gaussianmodel) get back to the inclusive Al

FB of the samplereasonably well. The double-Gaussian model fits theasymmetric part better in the qlηl region above 2.0 thanthe tanh model, thus the differential contribution predictedby the double-Gaussian model lines up with the POWHEG

predicted points marginally better. However, as statedearlier, the improvement is in the region where thecontribution to the inclusive Al

FB is small, thus the

TABLE II. Fraction of AlFB within typical qlηl coverage at CDF

and D0.

qlηl Coverage AlFB Fraction

1.25 0.731.5 0.822.0 0.93

(a)

(b)

FIG. 11 (color online). Comparison among the a · tanh model,the single-Gaussian model, the double-Gaussian model and thePOWHEG simulation. (a) shows the best fits of the AðqlηlÞdistribution (done only using events with jqlηlj < 2.0), while(b) shows the differential contribution to the Al

FB as a function ofqlηl from different models.

FIG. 10 (color online). Fraction of AlFB within a certain qlηl

coverage. The vertical lines show qlηl ¼ 1.25, 1.5, and 2.0corresponding to the typical detector coverages at CDF andD0. The numbers are given in Table II. The horizontal lineindicates that the fraction asymptotes to one as the qlηl coveragegoes to infinity.

FORWARD-BACKWARD ASYMMETRY OF LEPTONIC DECAYS … PHYSICAL REVIEW D 90, 014040 (2014)

014040-7

improvement in the resultant AlFB using the double-

Gaussian model is very small. Figure 12 shows thedouble-Gaussian model fit to the AðqlηlÞ distribution forall the six benchmark samples at parton level. A compari-son with Fig. 2(b) shows that the double-Gaussian modelmatches all the simulated samples better than the a · tanhmodel, although the differences are mostly in the high-qlηlregion where the contribution to the inclusive Al

FB is small,and there is no data from the experiments in this region.We next compare how well the various methods will

work for real data by considering just the set of POWHEG

simulated events within jqlηlj < 2.0 and employing differ-ent methodologies to see how well each reproduces theinclusive Al

FB of 0.0236. We performed 10 000 pseudoex-periments by varying the dN ðqlηlÞ=dðqlηlÞ distributionwith statistical fluctuations for about 1 million simulated

events. We then measured AlFB for each pseudoexperiment

using each of the four methods:(1) A pure counting of the number of events with

positive and negative qlηl values, with a correctionfor the limited detector coverage using the correctionfactor of 0.93 (see Table II) to extrapolate to theinclusive value

(2) Fitting the a · tanh model to the AðqlηlÞ term ofthe distribution for the parameter a and calculatingthe inclusive Al

FB using the SðqlηlÞ distributionand Eq. (3)

(3) Fitting the asymmetric part of the double-Gaussianmodel to the AðqlηlÞ term of the distribution for theparameter μ and calculating the inclusive Al

FBwith Eq. (9)

(4) Fitting the double-Gaussian model to the qlηl dis-tribution itself for the parameter μ and calculatingthe inclusive Al

FB again with Eq. (9)The results of the pseudoexperiments are shown inTable III.The average of the pseudoexperiments for each method

is always within one standard deviation of the input AlFB

value, indicating none have noticeable bias. As expected,the pure counting method has the largest uncertainty, as thefits incorporate the additional shape information to reducethe uncertainties. While there does not seem to be muchdifference in the sensitivity of the fitting methods, we notethat the fit on theAðqlηlÞ term has the systematic advantageover the pure fit to the mean, μ, of the full dN ðqlηlÞ=dðqlηlÞ distribution as most of the systematic uncertaintiesdue to the acceptance of the detector are expected to cancel

FIG. 12 (color online). Best fit of double-Gaussian model to theAðqlηlÞ distribution for various tt̄ samples at generator level. Thisfigure can be compared directly to Fig. 2(b) where we fit the samedata, but using the a · tanh function.

TABLE III. Results of pseudoexperiments using the different methods to reproduce AlFB of the POWHEG simulation

(0.0236), but only using events with jqlηlj < 2.0. Note that the uncertainties listed are statistical only and are due tothe size of the simulated data sample.

Method Mean Mean-expected Uncertainty

Counting 0.0241 0.0004 0.0008a · tanh AðqlηlÞ fit 0.0243 0.0006 0.0006Double Gaussian AðqlηlÞ fit 0.0236 −0.0001 0.0006Double Gaussian direct fit 0.0238 0.0002 0.0006

TABLE IV. Comparison of the predicted AlFB values and the corresponding measured Al

FB values with the a · tanhmodel and the double-Gaussian model. The uncertainties are statistical only and are always small compared to theexpected statistical uncertainty in data collected by the CDF and D0 experiments.

Model True AlFB

Measured AlFB

(a · tanh model)Measured Al

FB (Double-Gaussian model)

AxiL −0.063ð2Þ −0.064ð2Þ −0.064ð2ÞAxiR 0.151(2) 0.148(2) 0.150(2)Axi0 0.050(2) 0.048(2) 0.048(2)ALPGEN 0.003(1) −0.004ð1Þ 0.002(1)PYTHIA 0.001(1) −0.005ð1Þ 0.001(1)POWHEG 0.023(1) 0.024(1) 0.023(1)

HONG et al. PHYSICAL REVIEW D 90, 014040 (2014)

014040-8

out [8]. Thus, we favor the use of the fit on the AðqlηlÞdistribution over the simple counting for resolution reasons,and over the fit on the full distribution for robustnessreasons. Between the two fits on the AðqlηlÞ term, we seethat the a · tanh formulation is easier to work with, butfurther checks to see if there are other effects due to detectorresponse should be considered.We next test how well the a · tanh and the double-

Gaussian methods reproduce the inclusive AlFB values

for all six simulated samples with only events withinjqlηlj < 2.0. A comparison of results is given inTable IV. Though the double-Gaussian model works betterin the high qlηl region, the impact on the Al

FB measurementis negligible compared to the dominant uncertainties in themeasurement (∼0.02 in the leptonþ jets final state [8] and∼0.05 in the dilepton final state [9]).Finally, on a related measurement note we point out that,

because of the predicted structure in the high-qlηl region,when computing AðqlηlÞ the choice of the qlηl bincentroids and widths should be made with care. Eachbin should contain a reasonable number of events to avoidstatistical fluctuations; on the other hand, as shown inFigure 12, the curve is changing drastically aboveqlηl ∼ 1.5, thus a simple fit through either the bin centeror the bin centroid could introduce a sizeable systematicuncertainty if the bin is overly wide.

V. CONCLUSION

We have described the qlηl distribution from the leptoniccascade decays of tt̄ events produced at the Tevatron andthe corresponding Al

FB that can be determined from it.Many data measurements have been produced in this finalstate, all of which only have coverage of jηj < 2.0, andsome have used a empirical functional form of a · tanh toextrapolate from the limited detector η coverage to aninclusive parton-level estimate. We now understand thatthis excellent approximation is fortuitous but robust. Thea · tanh parametrization is an approximation that is onlygood for values of jqlηlj < 2.5, but it is more than goodenough for the Tevatron experiments. It may well be usefulfor all Tevatron experiments to report their Al

FB in the

restricted qlηl regimes as well as measurements of the aterm in the a · tanh formulation if possible.Our studies show that a more sophisticated empirical

function, which takes the form of the sum of two Gaussianfunctions with a common mean, and with empiricallydetermined values of the two σ and r parameters, describesthe qlηl distribution better at all qlηl values. This functionalform has not yielded a simple closed form for the AðqlηlÞterm. While the double-Gaussian parametrization is betterin principle, in practice using it does not provide additionaluseful measurement sensitivity and it is more cumbersometo use. On the other hand, this better understanding of theexpected shapes lead to some interesting and usefulconclusions in addition to the confidence we now havein the methods previously being employed. First, it isadvantageous to think of the asymmetry as coming from theshift of the mean of the qlηl distribution. To a good degreeof approximation, measuring the Al

FB is equivalent tomeasuring the mean, μ, in the limit of small Al

FB; measuringthe AðqlηlÞ term of the distribution is one of a number ofways to do so, which also takes advantage of the cancellingof the systematic uncertainties caused by the detectorresponse in the measurement. Ultimately, we now under-stand that the dominant contribution to the inclusive Al

FBcomes from the region within the CDF and D0 detectorswhich are best covered, and that the extrapolation proce-dures allow for a robust measurement.

ACKNOWLEDGMENTS

The authors would like to thank Dr. MichelangeloMangano for the discussion about the origin of the double-Gaussian model. We also would like to thank HamiltonCarter and Dr. Ilarion Melnikov for the useful discussionsabout turning the integrations of Gaussian distributions intohyperbolic tangent functions. We would like to thankFNAL and the CDF Collaboration for their support whilethis work was done. S. H., Z. H., and D. T. would also liketo thank the Mitchell Institute for Fundamental Physics andAstronomy and the Department of Physics and Astronomyat Texas A&M University for its support.

[1] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D 87,092002 (2013).

[2] V. Abazov et al. (D0 Collaboration), Phys. Rev. D 84,112005 (2011).

[3] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett.111, 182002 (2013).

[4] W. Bernreuther and Z.-G. Si, Phys. Rev. D 86, 034026 (2012);J. H. Kühn and G. Rodrigo, Phys. Rev. Lett. 81, 49 (1998).

[5] D.-W. Jung, P. Ko, and J. S. Lee, Phys. Lett. B 701, 248(2011); D.-W. Jung, P. Ko, J. S. Lee, and S. hyeon Nam,ibid. 691, 238 (2010); P. H. Frampton, J. Shu, and K. Wang,ibid. 683, 294 (2010); E. Álvarez, L. Rold, and A.Szynkman, J. High Energy Phys. 05 (2011) 070; C.-H.Chen, G. Cvetic, and C. Kim, Phys. Lett. B 694, 393 (2011);Y.-k. Wang, B. Xiao, and S.-h. Zhu, Phys. Rev. D 82,094011 (2010); A. Djouadi, G. Moreau, F. Richard, and

FORWARD-BACKWARD ASYMMETRY OF LEPTONIC DECAYS … PHYSICAL REVIEW D 90, 014040 (2014)

014040-9

R. K. Singh, ibid. 82, 071702 (2010); R. S. Chivukula, E. H.Simmons, and C.-P. Yuan, ibid. 82, 094009 (2010); B. Xiao,Y.-k. Wang, and S.-h. Zhu, ibid. 82, 034026 (2010); Q.-H.Cao, D. McKeen, J. L. Rosner, G. Shaughnessy, and C. E.M. Wagner, ibid. 81, 114004 (2010); I. Doršner, S. Fajfer,J. F. Kamenik, and N. Košnik, ibid. 81, 055009 (2010); S.Jung, H. Murayama, A. Pierce, and J. D. Wells, ibid. 81,015004 (2010); J. Shu, T. M. P. Tait, and K. Wang, ibid. 81,034012 (2010); A. Arhrib, R. Benbrik, and C.-H. Chen,ibid. 82, 034034 (2010); J. Cao, Z. Heng, L. Wu, and J. M.Yang, ibid. 81, 014016 (2010); V. Barger, W.-Y. Keung, andC.-T. Yu, ibid. 81, 113009 (2010); P. Ferrario and G.Rodrigo, ibid. 78, 094018 (2008); P. Ferrario and G.Rodrigo, Phys. Rev. D 80, 051701 (2009); M. Bauer, F.Goertz, U. Haisch, T. Pfoh, and S. Westhoff, J. High EnergyPhys. 11 (2010) 039; K. Cheung, W.-Y. Keung, and T.-C.Yuan, Phys. Lett. B 682, 287 (2009).

[6] W. Bernreuther and Z.-G. Si, Nucl. Phys. B837, 90 (2010).[7] A. Falkowski, M. L. Mangano, A. Martin, G. Perez, and

J. Winter, Phys. Rev. D 87, 034039 (2013).[8] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D 88,

072003 (2013).[9] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett.

113, 042001 (2014).[10] V. Abazov et al. (D0 Collaboration), Phys. Rev. D 88,

112002 (2013).[11] V. Abazov et al. (D0 Collaboration), D0 Note 6394-CONF,

2013.

[12] T. Sjöstrand, S. Mrenna, and P. Z. Skands, J. High EnergyPhys. 05 (2006) 026.

[13] M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau, andA. D. Polosa, J. High Energy Phys. 07 (2003) 001.

[14] S. Frixione, P. Nason, and G. Ridolfi, J. High Energy Phys.09 (2007) 126.

[15] P. Nason, J. High Energy Phys. 11 (2004) 040.[16] S. Frixione, P. Nason, and C. Oleari, J. High Energy Phys.

11 (2007) 070.[17] S. Alioli, P. Nason, C. Oleari, and E. Re, J. High Energy

Phys. 06 (2010) 043.[18] O. Antunano, J. H. Kühn, and G. Rodrigo, Phys. Rev. D 77,

014003 (2008); W. Hollik and D. Pagani, Phys. Rev. D 84,093003 (2011); A. V. Manohar and M. Trott, Phys. Lett. B711, 313 (2012); J. Kühn and G. Rodrigo, J. High EnergyPhys. 01 (2012) 063.

[19] J. Alwall, P. Demin, S. de Visscher, R. Frederix, M. Herquet,F. Maltoni, T. Plehn, D. L. Rainwater, and T. Stelzer, J. HighEnergy Phys. 09 (2007) 028.

[20] After communication with Dr. Michelangelo Mangano, werealized that this shape may not have a first-principleanalytical explanation, but rather be a combined effect fromthe behavior of the PDFs, the matrix element, and the topdecay kinematics. There is some evidence that the charge-weighted rapidity distribution of the top quark is actuallyGaussian distributed, so the second Gaussian may be just theboosts as part of the decay processes of the top and the Wboson.

HONG et al. PHYSICAL REVIEW D 90, 014040 (2014)

014040-10