Predictions for the Top-Quark Forward-Backward Asymmetry at High Invariant Pair Mass Using the Principle of Maximum Conformality
Sheng-Quan Wang, Xing-Gang Wu, Zong-Guo Si, Stanley J. Brodsky
aa r X i v : . [ h e p - ph ] D ec Predictions for the Top-Quark Forward-Backward Asymmetry at High Invariant PairMass Using the Principle of Maximum Conformality
Sheng-Quan Wang , , ∗ Xing-Gang Wu , † Zong-Guo Si , ‡ and Stanley J. Brodsky § Department of Physics, Chongqing University, Chongqing 401331, P.R. China School of Science, Guizhou Minzu University, Guiyang 550025, P.R. China Department of Physics, Shandong University, Jinan, Shandong 250100, P.R. China and SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94039, USA (Dated: October 10, 2018)The D0 collaboration at FermiLab has recently measured the top-quark pair forward-backwardasymmetry in ¯ pp → t ¯ tX reactions as a function of the t ¯ t invariant mass M t ¯ t . The D0 resultfor A FB ( M t ¯ t >
650 GeV) is smaller than A FB ( M t ¯ t ) obtained for small values of M t ¯ t , which mayindicate an “increasing-decreasing” behavior for A FB ( M t ¯ t > M cut ). This behavior is not explainedusing conventional renormalization scale-setting, even by a next-to-next-to-leading order (N LO)QCD calculation – one predicts a monotonically increasing behavior.In the conventional scale-setting method, one simply guesses a single renormalization scale µ r forthe argument of the QCD running coupling and then varies it over an arbitrary range. However,the conventional method has inherent difficulties. For example, the resulting pQCD predictionsdepend on the choice of renormalization scheme, in contradiction to the principle of “renormal-ization scheme invariance” – predictions for physical observables cannot depend on a theoreticalconvention. The error estimate obtained by varying µ r is unreliable since it is only sensitive toperturbative contributions involving the pQCD β -function. Worse, guessing the renormalizationscale gives predictions for precision QED observables which are in contradiction to results obtainedusing the standard Gell-Mann-Low method. In contrast, if one fixes the scale using the Principle ofMaximum Conformality (PMC), the resulting pQCD predictions are renormalization-scheme inde-pendent since all of the scheme-dependent { β i } -terms in the QCD perturbative series are resummedinto the QCD running couplings at each order. The { β i } -terms at each order can be unambiguouslyidentified using renormalization group methods such as the R δ method. The PMC then determinesthe renormalization scales of the running coupling at each order and provides unambiguous scale-fixed and scheme-independent predictions. The PMC reduces in the N C → e + e − → µ + µ − cross-section.By using the rigorous PMC scale-setting procedure, one obtains a comprehensive, self-consistentpQCD explanation for the Tevatron measurements of the top-quark pair forward-backward asym-metry. In this paper we show that if one applies the PMC to determine the top versus anti-topquark forward-backward asymmetry by properly using the pQCD predictions up to N LO level, oneobtains the predictions without renormalization scheme or scale ambiguities. For example, the PMCpredicts A PMCFB ( M t ¯ t >
450 GeV) = 29 .
9% at the Tevatron, which is consistent with the CDF mea-surements. In addition, the PMC prediction for A FB ( M t ¯ t > M cut ) shows an “increasing-decreasing”behavior for increasing values of M cut which is not observed in the NLO and N LO predictions for A FB ( M t ¯ t > M cut ) with conventional scale-setting. This behavior could be tested by the future moreprecise measurements at the LHC. PACS numbers: 12.38.Aw, 11.10.Gh, 11.15.Bt, 14.65.Ha
I. INTRODUCTION
Measurements of the top versus anti-top quark asym-metry in ¯ pp → t ¯ tX reactions at the Tevatron haveprovided important tests of perturbative quantum chro-modynamics (pQCD) and the Standard Model. The ∗ email:[email protected] † email:[email protected] ‡ email:[email protected] § email:[email protected] forward-backward t ¯ t asymmetry is defined as A FB = N (∆ y > − N (∆ y < N (∆ y >
0) + N (∆ y < , (1)where ∆ y = y t − y ¯ t is the difference between the rapiditiesof top and anti-top quarks, and N stands for the numberof events. This asymmetry is due in QCD to the inter-ference of perturbative amplitudes with different chargeconjugation, such as the one-gluon and two-gluon anni-hilation contributions to the q ¯ q → t ¯ t subprocess.A review of the theoretical and experimental featuresof A FB can be found in Ref.[1]. The initial StandardModel (SM) predictions for A FB , which were based onpQCD at next-to-leading order (NLO) [2, 3], appearedto be in substantial disagreement with the TevatronCDF and D0 measurements [4–9]. The SM predictionsfor the t ¯ t asymmetry A FB were subsequently improvedby including electroweak contributions: (8 . +0 . − . )% [10],(8 . ± . . ± . LO) pQCD calcula-tion was performed; it predicts A FB = (9 . ± . LO QCD prediction to theasymmetry, i.e. A FB = (10 . ± . LO andhigher order predictions agree with the Tevatron mea-surement within errors, A D0FB = (10 . ± . A D0FB = (11 . ± . ± . A CDFFB = (16 . ± . A CDFFB = (12 ± A FB ( M t ¯ t >
450 GeV) are (12 . +1 . − . )% [10],(12 . ± . . +0 . − . )% [12], which deviatefrom the 2011 CDF measurement (47 . ± . . σ standard deviations and the 2013 CDF mea-surement (29 . ± . ± . . σ standarddeviation. While an update of A FB ( M t ¯ t >
450 GeV) us-ing the N LO and approximate N LO calculations withconventional scale-setting is not yet available, a mea-surement of the differential A FB ( M t ¯ t ) has been providedby both CDF and D0 Collaborations [6, 9] and com-pared with the N LO and N LO results with conven-tional scale-setting [13, 14]. Agreement was found withthe D0 data but not with the CDF data at large valuesof M t ¯ t . Moreover, the new D0 result for A FB ( M t ¯ t >
650 GeV) = − . ± . A FB ( M t ¯ t ) obtained for small values of M t ¯ t , may sug-gest an “increasing-decreasing” behavior for A FB ( M t ¯ t >M cut ). This “increasing-decreasing” behavior was not re-flected in the pQCD predictions with conventional scale-setting. It is interesting to show whether one can achievesuch an “increasing-decreasing” behavior by using thePMC scale-setting.All of the SM predictions discussed above have beenbased on conventional scale-setting; i.e., one simply takesthe renormalization scale as the top-quark mass m t and then varies the scale over an arbitrary range, i.e.[ m t / , m t ], in order to estimate the scale uncertainty.However, this procedure of guessing the renormaliza-tion scale, although conventional, gives renormalizationscheme-dependent predictions. It also leads to a non-convergent renormalon perturbative series. Moreover,one would obtain incorrect results if one applies thismethod to QED processes [17]. It is possible for con-ventional scale-setting to accidentally predict the correctvalue of a global observable such as the total cross-sectionat sufficiently high order; however, since one assumes thesame renormalization scale at each order in α s , it willoften give incorrect predictions for each perturbative or-der correction. In fact, the renormalization scale andeffective number of flavors are in general distinct at eachorder of pQCD, reflecting the different virtualities of the subprocesses as a function of phase-space. This providesthe underlying reason why a single ‘guessed’ scale can-not explain the “increasing-decreasing” behavior of A FB as the t ¯ t -pair mass is varied.In contrast, the Principle of Maximum Conformality(PMC) [18–22] provides a systematic way to eliminatethe renormalization scheme-and-scale uncertainties. ThePMC reduces in the ( N c →
0) Abelian limit [23] to thestandard Gell-Mann-Low method [24], where all vacuumpolarization contributions are associated with the dressedphoton propagator and are thus resummed to determinethe optimal scale of the running coupling. The PMC pro-vides the underlying principle for the Brodsky-Lepage-Mackenzie approach [25], extending it unambiguously toall orders using renormalization group methods.The authors of Ref.[13] have noted that an alter-native scale-setting procedure, called the “large β -approximation” [26, 27], leads to incorrect pQCD n f -terms at NNLO. It should be emphasized that this ana-lytic error is a defect of the “large β -approximation”; itdoes not occur if one uses PMC scale-setting [17], i.e. thecorrect n f -series at each perturbative order can be deter-mined via its one-to-one correspondence to the β -seriesat the same order.The PMC has a solid theoretical foundation, satis-fying renormalization group invariance [28] and all theother self-consistency conditions derived from the renor-malization group [29]. The PMC scales at each orderare formed by shifting the arguments of the running cou-pling to eliminate all non-conformal { β i } -terms. Thiselimination is done by using the β -pattern determinedby renormalization group equations; one then obtainscorrect behavior of the running coupling at each orderand at each phase-space point. A systematic method foridentifying the { β i } -terms is given in Refs. [21, 22]. Thisprocedure also predicts some of the previously unknownhigher-order { β i } -terms via the degeneracy patterns ap-pearing at different orders [30]. After applying the PMC,the divergent renormalon series such as P n n ! β n α ns doesnot appear and thus the pQCD convergence is automat-ically improved. The PMC has been successfully appliedto many high-energy processes, including processes whichinvolve multiple physical scales, such as hadronic Z de-cays [31] and Υ(1 S ) leptonic decays [32].In this paper we will apply the PMC procedure tothe calculation of the A FB asymmetry by applying thepQCD results up to N LO level with the goal of achiev-ing precise pQCD predictions without renormalizationscale ambiguities. We shall show that the PMC providesa self-consistent explanation of the recently reported D0and CDF measurements. In our numerical calculations,we will assume the top-quark mass is m t = 173 . .Thus, the LO and NLO PMC scales can be unambigu-ity determined by the higher information in HATHORprogram. We shall show that the pQCD convergence forthe t ¯ t production cross sections are greatly improved; our PMC predictions, even at low orders, is in good agree-ment with the complete NNLO prediction [13], and pro-vide a self-consistent pQCD explanation for the Tevatronmeasurements. Conventional scale-setting PMC scale-settingLO NLO N LO Total
LO NLO N LO Total ( q ¯ q )-channel 4.901 0.960 0.481 6.346 4.760 1.728 -0.0621 6.383( gg )-channel 0.542 0.434 0.156 1.132 0.540 0.516 0.149 1.230( gq )-channel 0.000 -0.0361 0.0051 -0.0309 0.000 -0.0361 0.0051 -0.0309( g ¯ q )-channel 0.000 -0.0361 0.0051 -0.0310 0.000 -0.0361 0.0051 -0.0310sum 5.444 1.322 0.647 7.416 5.300 2.172 0.0971 7.552TABLE I: The top-quark pair production cross-sections (in unit: pb) before and after PMC scale-setting at the Tevatron withthe collision energy √ S = 1 .
96 TeV. The initial renormalization scale and the factorization scale are taken as µ r = µ f = m t . II. COMPARISONS OF PMC ANDCONVENTIONAL SCALE-SETTINGPREDICTIONS FOR THE t ¯ t PRODUCTIONCROSS-SECTION
We will first compare the pQCD predictions up tothe N LO level for the total top-quark pair productioncross-sections for the Tevatron collision energy √ s = 1 . q ¯ q )-, ( gq )-, ( g ¯ q )-and ( gg )- channels are also presented. The initial scale µ r is set as m t and then varied over the range µ r ∈ [ m t / , m t ]. The total cross-sections from all the produc-tion channels, before and after PMC scale-setting, are σ Total | Conv . = 7 . +0 . − . pb , (2) σ Total | PMC ≃ .
55 pb . (3)Both the PMC and conventional scale-setting proceduresagree with the CDF and D0 measurements within er-rors [36–39]; the recent combined cross-section given bythe CDF and D0 collaborations is 7 . ± .
41 pb [39]. ThePMC total cross-section is almost unchanged when onevaries the starting scale µ r . The dependence of the totalcross-section on the choice of renormalization scale is alsosmall using conventional scale-setting if one incorporatesN LO QCD corrections.However, using a single guessed scale does not pre-dict the cross-sections for individual channels correctly ateach perturbative order. In fact, by analyzing the pQCDseries in detail, we find that the errors for the separatecross-sections at each perturbative order from conven-tional scale-setting are large in all of the contributingchannels.
Conventional PMC µ r m t / m t m t m t / m t m t σ LO q ¯ q σ NLO q ¯ q σ N LO q ¯ q q ¯ q )-channel cross-sections (in unit: pb) ateach perturbative order under the conventional and PMCscale-settings, where three typical renormalization scales µ r = m t / m t and 2 m t are adopted. The factorization scale istaken as µ f = m t . As an example, the contributions of the dominant ( q ¯ q )-channel with and without PMC scale-setting are pre-sented in Table II. This subprocess is the dominant sourceof the t ¯ t asymmetry. In agreement with expectations,the scale dependence of the total ( q ¯ q )- cross-section upto N LO level is small; i.e., ∆ σ Total q ¯ q = N LO P i =LO ∆ σ iq ¯ q ≃ ± µ r ∈ [ m t / , m t ]. For clarity, we define a ratio κ i to illuminate the scale dependence of individual cross-sections σ iq ¯ q at each order: κ i = σ iq ¯ q (cid:12)(cid:12) µ r = m t / − σ iq ¯ q (cid:12)(cid:12) µ r =2 m t σ iq ¯ q (cid:12)(cid:12) µ r = m t , where i =LO, NLO and N LO, respectively. Using con-ventional scale-setting, we obtain κ LO = 39% , κ NLO = − , κ N LO = − . These results show that if one uses conventional scale-setting, then the dependence on the choice of initial scaleat each order is very large. For example, the scale de-pendence of σ NLO q ¯ q , which gives the dominant compo-nent of the asymmetry A FB , reaches up to − σ iq ¯ q using conventional scale-setting. Un-der conventional scale-setting, σ NLO q ¯ q increases and σ Total q ¯ q decreases with the increment of µ r . Thus in order toagree with the measured total cross-section, it prefers asmaller scale ≃ m t /
2, which however leads to the predic-tion of a small t ¯ t asymmetry, well below the data. Onthe other hand, Table II shows that all of the κ i -valuesbecome less than 0 .
1% if one uses the PMC. Thus thescale errors for both the total cross-section and the in-dividual cross-sections at each order are simultaneouslyeliminated using the PMC scale-setting, and the resid-ual scale dependence due to unknown higher-order { β i } -terms become negligible [40]. III. PREDICTIONS FOR THE t ¯ t FORWARD-BACKWARD ASYMMETRY
The top-quark pair forward-backward asymmetry in¯ pp → t ¯ tX collisions is also sensitive to the renormaliza-tion scale-setting procedure. As shown in Table I the( q ¯ q )-channel provides the dominant contribution to A FB at the Tevatron. It is important to note the NLO PMCscale µ PMC , NLO r of the ( q ¯ q )-channel is much smaller than m t (as shown by Table III). This reflects the fact thatthe two virtual gluons in the s -channel in the annihila-tion amplitude q ¯ q → g ∗ g ∗ → t ¯ t share the virtuality of thesubprocess. The resulting NLO ( q ¯ q ) cross-section is infact about twice as large as the cross-section predicted byconventional scale-setting; the precision of the predictedasymmetry A FB is also greatly improved. Moreover, af-ter applying the PMC, the predicted ratio of the cross-section at the N LO level to the NLO cross-section for the q ¯ q -channel, i.e. | σ N LO q ¯ q /σ NLO q ¯ q | , is reduced from ∼
50% tobe less than ∼ t ¯ t asymmetry. If we further include the O ( α s α ) and the O ( α ) electroweak contributions, we achieve a precise SM“NLO-asymmetry” predictions [41], A PMCFB = α s N + α s α ˜ N + α ˜ N α s D + α s D , (4)where the D i -terms stand for the total cross-sectionsat each α s -order and the N i -terms stand for the corre-sponding asymmetric contributions. The term labeled˜ N corresponds to the QCD-QED interference contribu-tion at the order O ( α s α ), and ˜ N stands for the pure elec-troweak antisymmetric O ( α ) contribution arising from |M q ¯ q → γ → t ¯ t + M q ¯ q → Z → t ¯ t | [10].Under the conventional scale-setting, the N LO N -term provides large contribution for A FB [13]. Given that after applying the PMC scale-setting, the pQCD conver-gence is greatly improved and the D -term is lowered byone order of magnitude in comparison to the D -term,we assume the same behavior for N and consider theasymmetric NLO N -term to provide the dominant con-tribution. We therefore neglected the asymmetric N LO N -term when using the PMC.Furthermore, to compare the PMC prediction with theasymmetry assuming conventional scale-setting, we fur-ther rewrite the PMC asymmetry as [41] A PMCFB = ( σ Conv ., LOtot σ PMC , NLOtot ) ( α s (cid:0) µ PMC , NLO r (cid:1) α s ( µ Conv .r ) A (Conv . )FB (cid:12)(cid:12)(cid:12) α s + α s (cid:0) µ PMC , NLO r (cid:1) α s ( µ Conv .r ) A (Conv . )FB (cid:12)(cid:12)(cid:12) α s α + A (Conv . )FB (cid:12)(cid:12)(cid:12) α ) , (5)where the symbol “Conv.” stands for the predictioncalculated by using the conventional scale-setting, and“PMC” stands for the corresponding value after apply-ing the PMC. By using Eq.(5), we obtain precise predic-tion for A FB without renormalization scale uncertainty: A PMCFB = 9 . IV. PREDICTIONS FOR THE t ¯ t FORWARD-BACKWARD ASYMMETRY AS AFUNCTION OF THE PAIR MASS A F B ( M t ¯ t > G e V ) PMCCDF(2013)CDF(2011)weighted average
PRD 86 034026 JHEP 1201 063 PRD 84 093003
FIG. 1: Comparison of the PMC prediction for the top-pairasymmetry A FB ( M t ¯ t >
450 GeV) with the CDF measure-ment [5, 6]. The NLO results predicted by Refs. [10–12] un-der conventional scale-setting are presented as a comparison,which are shown by shaded bands.
We will next discuss predictions for the top-quark pairasymmetry A FB ( M t ¯ t > M cut ) as a function of the top-pair invariant mass lower limit M cut . In the case of M cut = 450 GeV, the predicted asymmetry using con-ventional scale-setting is A FB ( M t ¯ t >
450 GeV) | Conv . = Top-quark pair asymmetries A FB ( M t ¯ t > M cut ) M cut (GeV) 350 400 450 500 550 600 650 700 750 800 A FB ( M t ¯ t > M cut ) | Conv . A FB ( M t ¯ t > M cut ) | PMC 9.6% 17.1% 29.9% 43.5% 45.1% 37.8% 33.5% 31.4% 30.5% 30.1% α s ( µ PMC r ) 0.123 0.131 0.146 0.157 0.153 0.138 0.129 0.123 0.120 0.117 µ PMC r (GeV) 71 48 26 18 20 35 53 69 83 94TABLE III: Top-quark pair asymmetries A FB ( M t ¯ t > M cut ) using conventional (Conv.) and PMC scale-setting procedures,respectively. The Conv. predictions are for the NLO pQCD predictions with O ( α s α ) and the O ( α ) electroweak contributionsand the PMC predictions are calculated by Eq.(5). The predictions are shown for typical values of M cut . The last two linesgive the values of the effective couplings α s ( µ PMC r ) and the underlying effective scale µ PMC r , respectively. The initial scale istaken as µ r = m t . . A FB ( M t ¯ t >
450 GeV) | PMC =29 . A FB ( M t ¯ t ) is not availablefor us, we thus cannot directly compare with the resultsfor A FB ( M t ¯ t ) given in the literature, and instead, we givethe asymmetry A FB ( M t ¯ t > M cut ) for several choices ofminimum pair invariant mass. The results are given inTable III, where the conventional results are for the NLOpQCD predictions with the electroweak corrections cal-culated by using the Bernreuther-Si program [12]. Thisbehavior reflects the running behavior of the QCD cou-pling α s ( Q ) for different kinematic regions. If one as-sumes conventional scale-setting and the fixed scale m t ,then A FB ( M t ¯ t > M cut ) monotonically increases with in-creasing M cut . This trend is consistent with the behaviorof the N LO differential A FB ( M t ¯ t ) with the conventionalscale-setting [13]. In contrast, if one employs the PMCscale-setting, then A FB ( M t ¯ t > M cut ) first increases andthen decreases as the lower limit of the pair mass M cut is increased. These trends are more clearly shown byFig.(2), in which the Standard Model predictions usingconventional and PMC scale-settings are compared withthe CDF [6] and D0 [9] measurements.This “increasing-decreasing” behavior can be under-stood in terms of the effective pQCD coupling ¯ α s ( µ PMC r )introduced in Ref.[41]; it is the weighted average of therunning couplings entering the ( q ¯ q )-channel, the subpro-cess underlying the asymmetry in pQCD. The effectivecoupling can be unambiguously determined because theNLO-level asymmetric contribution from ( q ¯ q )-channelonly involves a single PMC scale. The effective coupling¯ α s ( µ PMC r ) depends in detail on the kinematics. More ex-plicitly, as shown by Table III, the non-monotonic behav-ior of the effective coupling accounts for the “increasing-decreasing” behavior of A FB ( M t ¯ t > M cut ). We have adopted the partonic center-of-mass frame to estimatethe PMC total cross-section for various M t ¯ t cuts; the re-sults, however, agree with the results obtained that of t ¯ t -rest frame within high accuracy since events near thepartonic threshold give the dominant contribution to thecross-section at the Tevatron [42, 43]. V. CONCLUSIONS
In summary, we have shown that the PMC providesa systematic and unambiguous way to set the optimalrenormalization pQCD scales for top-quark pair produc-tion at each order of α s . The resulting pQCD predic-tions are renormalization-scheme independent, since allof the scheme-dependent { β i } -terms in the QCD pertur-bative series are resummed into the QCD running cou-plings at each order. By applying the PMC, one obtainsnot only scheme-independent predictions but also a moreconvergent pQCD series without factorial renormalon di-vergences.After applying the PMC, the uncertainties from renor-malization scale-setting for both the total cross-sectionand the individual cross-sections at each order are simul-taneously eliminated, and the residual scale dependencedue to unknown higher-order { β i } -terms is found to benegligible. After applying the PMC, we improved theprediction respect to renormalization scale dependenceand obtain the top-quark pair forward-backward asym-metry A PMCFB = 9 .
2% and A FB ( M t ¯ t >
450 GeV) = 29 .
350 400 450 500 550 600 650 700 750 800 850 900−60−40−20020406080 M t ¯ t (GeV) A F B ( % ) Conv.Conv.CDFD0
350 400 450 500 550 600 650 700 750 800 850 900−60−40−20020406080 M t ¯ t (GeV) A F B ( % ) Conv.PMCCDFD0
FIG. 2: A comparison of SM predictions of A FB using con-ventional (Conv.) and PMC scale-settings with the CDF [6]and D0 [9] measurements. The Conv. predictions are forthe NLO pQCD predictions with O ( α s α ) and the O ( α ) elec-troweak contributions and the PMC predictions are calcu-lated by Eq.(5). The upper diagram is for conventional scale-setting, and the lower one is for PMC scale-setting. The initialscale is taken as µ r = m t . backward asymmetry A FB ( M t ¯ t > M cut ) displays an“increasing-decreasing” behavior as M cut is increased.This behavior however cannot be explained even by aN LO QCD calculation using conventional scale-setting,since in contrast, it predicts a monotonically increasingbehavior. We have also shown in a recent paper [44] thatthe PMC predictions are in agreement with the availableATLAS and CMS data.Thus, the proper setting of the renormalization scaleprovides a consistent Standard Model explanation of thetop-quark pair asymmetry measurements at both theTevatron and LHC. It is noted that the large discrepancybetween the conventional lower-order prediction andthe data can be cured to a certain degree by includinghigh-order terms, such as the state-of-art results ofRefs.[13, 14], however in those works the renormalizationscale uncertainty is “suppressed” but not “solved”. ThePMC results demonstrate that the application of thePMC eliminates a major theoretical uncertainty forpQCD predictions, thus increasing the sensitivity of theLHC and other colliders to possible new physics beyondthe Standard Model.
Acknowledgments : This work was supported inpart by the Natural Science Foundation of Chinaunder Grant No.11275280, No.11325525, No.11547010and No.11547305, the Department of Energy Con-tract No.DE-AC02-76SF00515, and by Fundamental Re-search Funds for the Central Universities under GrantNo.CDJZR305513. SLAC-PUB-16368. [1] J. A. Aguilar-Saavedra, D. Amidei, A. Juste andM. Perez-Victoria, “Asymmetries in top quark pair pro-duction at hadron colliders,” Rev. Mod. Phys. , 421(2015).[2] J. H. Kuhn and G. Rodrigo, “Charge asymmetry inhadroproduction of heavy quarks,” Phys. Rev. Lett. ,49 (1998).[3] J. H. Kuhn and G. Rodrigo, “Charge asymmetry of heavyquarks at hadron colliders,” Phys. Rev. D , 054017(1999).[4] T. Aaltonen et al. [CDF Collaboration], “Forward-Backward Asymmetry in Top Quark Production in p ¯ p Collisions at sqrts = 1 .
96 TeV,” Phys. Rev. Lett. ,202001 (2008).[5] T. Aaltonen et al. [CDF Collaboration], “Evidence fora Mass Dependent Forward-Backward Asymmetry inTop Quark Pair Production,” Phys. Rev. D , 112003(2011). [6] T. Aaltonen et al. [CDF Collaboration], “Measurement ofthe top quark forward-backward production asymmetryand its dependence on event kinematic properties,” Phys.Rev. D , 092002 (2013).[7] V. M. Abazov et al. [D0 Collaboration], “First measure-ment of the forward-backward charge asymmetry in topquark pair production,” Phys. Rev. Lett. , 142002(2008).[8] V. M. Abazov et al. [D0 Collaboration], “Forward-backward asymmetry in top quark-antiquark produc-tion,” Phys. Rev. D , 112005 (2011).[9] V. M. Abazov et al. [D0 Collaboration], “Measurement ofthe forward-backward asymmetry in top quark-antiquarkproduction in ppbar collisions using the lepton+jetschannel,” Phys. Rev. D , 072011 (2014).[10] W. Hollik and D. Pagani, “The electroweak contributionto the top quark forward-backward asymmetry at theTevatron,” Phys. Rev. D , 093003 (2011). [11] J. H. Kuhn and G. Rodrigo, “Charge asymmetries of topquarks at hadron colliders revisited,” JHEP , 063(2012).[12] W. Bernreuther and Z. G. Si, “Top quark and leptoniccharge asymmetries for the Tevatron and LHC,” Phys.Rev. D , 034026 (2012).[13] M. Czakon, P. Fiedler and A. Mitov, “Resolving theTevatron top quark forward-backward asymmetry puz-zle,” Phys. Rev. Lett. , 052001 (2015).[14] N. Kidonakis, “The top quark forward-backward asym-metry at approximate N LO,” Phys. Rev. D , 071502(R) (2015).[15] V. M. Abazov et al. [D0 Collaboration], “Simultaneousmeasurement of forward-backward asymmetry and toppolarization in dilepton final states from t ¯ t production atthe Tevatron,” arXiv:1507.05666 [hep-ex].[16] T. Aaltonen et al. [CDF Collaboration], “Measurement ofthe Forward-Backward Asymmetry of Top-Quark Pairsin the Dilepton Final State and Combination at CDF,”Conf. Note 11161.[17] X. G. Wu, S. J. Brodsky and M. Mojaza, “The Renor-malization Scale-Setting Problem in QCD,” Prog. Part.Nucl. Phys. , 44 (2013).[18] S. J. Brodsky and X. G. Wu, “Scale Setting Using theExtended Renormalization Group and the Principle ofMaximum Conformality: the QCD Coupling Constantat Four Loops,” Phys. Rev. D , 034038 (2012) [Phys.Rev. D , 079903 (2012)].[19] S. J. Brodsky and X. G. Wu, “Eliminating the Renor-malization Scale Ambiguity for Top-Pair Production Us-ing the Principle of Maximum Conformality,” Phys. Rev.Lett. , 042002 (2012).[20] S. J. Brodsky and L. Di Giustino, “Setting the Renormal-ization Scale in QCD: The Principle of Maximum Con-formality,” Phys. Rev. D , 085026 (2012).[21] M. Mojaza, S. J. Brodsky and X. G. Wu, “SystematicAll-Orders Method to Eliminate Renormalization-Scaleand Scheme Ambiguities in Perturbative QCD,” Phys.Rev. Lett. , 192001 (2013).[22] S. J. Brodsky, M. Mojaza and X. G. Wu, “SystematicScale-Setting to All Orders: The Principle of MaximumConformality and Commensurate Scale Relations,” Phys.Rev. D , 014027 (2014).[23] S. J. Brodsky and P. Huet, “Aspects of SU(N(c)) gaugetheories in the limit of small number of colors,” Phys.Lett. B , 145 (1998).[24] M. Gell-Mann and F. E. Low, “Quantum electrodynam-ics at small distances,” Phys. Rev. , 1300 (1954).[25] S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, “On theElimination of Scale Ambiguities in Perturbative Quan-tum Chromodynamics,” Phys. Rev. D , 228 (1983).[26] M. Neubert, “Scale setting in QCD and the momen-tum flow in Feynman diagrams,” Phys. Rev. D , 5924(1995).[27] M. Beneke and V. M. Braun, “Naive nonAbelianizationand resummation of fermion bubble chains,” Phys. Lett.B , 513 (1995).[28] X. G. Wu, Y. Ma, S. Q. Wang, H. B. Fu, H. H. Ma,S. J. Brodsky and M. Mojaza, “Renormalization GroupInvariance and Optimal QCD Renormalization Scale-Setting,” arXiv:1405.3196 [hep-ph].[29] S. J. Brodsky and X. G. Wu, “Self-Consistency Re- quirements of the Renormalization Group for Setting theRenormalization Scale,” Phys. Rev. D , 054018 (2012).[30] H. Y. Bi, X. G. Wu, Y. Ma, H. H. Ma, S. J. Brodskyand M. Mojaza, “Degeneracy Relations in QCD and theEquivalence of Two Systematic All-Orders Methods forSetting the Renormalization Scale,” Phys. Lett. B ,13 (2015).[31] S. Q. Wang, X. G. Wu and S. J. Brodsky, “Reanaly-sis of the Higher Order Perturbative QCD correctionsto Hadronic Z Decays using the Principle of MaximumConformality,” Phys. Rev. D , 037503 (2014).[32] J. M. Shen, X. G. Wu, H. H. Ma, H. Y. Bi andS. Q. Wang, “Renormalization group improved pQCDprediction for (1S) leptonic decay,” JHEP , 169(2015).[33] P. M. Nadolsky, H. L. Lai, Q. H. Cao, J. Huston,J. Pumplin, D. Stump, W. K. Tung and C.-P. Yuan,“Implications of CTEQ global analysis for collider ob-servables,” Phys. Rev. D , 013004 (2008).[34] M. Aliev, H. Lacker, U. Langenfeld, S. Moch, P. Uwerand M. Wiedermann, “HATHOR: HAdronic Top andHeavy quarks crOss section calculatoR,” Comput. Phys.Commun. , 1034 (2011).[35] M. Czakon and A. Mitov, “Top++: A Program for theCalculation of the Top-Pair Cross-Section at Hadron Col-liders,” Comput. Phys. Commun. , 2930 (2014).[36] T. Aaltonen et al. [CDF Collaboration], “First Measure-ment of the Ratio σ ( t ¯ t ) /σ ( Z/γ ∗ ∗ → ℓℓ ) and Precise Ex-traction of the t ¯ t Cross Section,” Phys. Rev. Lett. ,012001 (2010).[37] V. M. Abazov et al. [D0 Collaboration], “Measurementof the top quark pair production cross section in thelepton+jets channel in proton-antiproton collisions at √ s =1.96 TeV,” Phys. Rev. D , 012008 (2011).[38] V. M. Abazov et al. [D0 Collaboration], “Measurementof the t ¯ t production cross section using dilepton eventsin p ¯ p collisions,” Phys. Lett. B , 403 (2011).[39] T. A. Aaltonen et al. [CDF and D0 Collaborations],“Combination of measurements of the top-quark pairproduction cross section from the Tevatron Collider,”Phys. Rev. D , 072001 (2014).[40] S. J. Brodsky and X. G. Wu, “Application of the Princi-ple of Maximum Conformality to Top-Pair Production,”Phys. Rev. D , 014021 (2012) [Phys. Rev. D , 099902(2013)].[41] S. J. Brodsky and X. G. Wu, “Application of thePrinciple of Maximum Conformality to the Top-QuarkForward-Backward Asymmetry at the Tevatron,” Phys.Rev. D , 114040 (2012).[42] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak andL. L. Yang, “RG-improved single-particle inclusive crosssections and forward-backward asymmetry in t ¯ t produc-tion at hadron colliders,” JHEP , 070 (2011).[43] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak andL. L. Yang, “The top-pair forward-backward asymmetrybeyond NLO,” Phys. Rev. D , 074004 (2011).[44] S. Q. Wang, X. G. Wu, Z. G. Si and S. J. Brodsky, “Ap-plication of the Principle of Maximum Conformality tothe Top-Quark Charge Asymmetry at the LHC,” Phys.Rev. D90