A New Paradigm for Precision Top Physics: Weighing the Top with Energy Correlators

Final states in collider experiments are characterized by correlation functions, $\langle E(\vec n_1) \cdots E(\vec n_k) \rangle$, of the energy flow operator $ E(\vec n_i)$. We show that the top quark imprints itself as a peak in the three-point correlator at an angle $\zeta \sim m_t^2/p_T^2$, with $m_t$ the top quark mass and $p_T$ its transverse momentum, providing access to one of the most important parameters of the Standard Model in one of the simplest field theoretical observables. Our analysis provides the first step towards a new paradigm for a precise top mass determination that is, for the first time, highly insensitive to soft physics and underlying event contamination whilst remaining directly calculable from the Standard Model Lagrangian.


I. INTRODUCTION
The Higgs and top quark masses play a central role both in determining the structure of the electroweak vacuum [1][2][3], and in the consistency of precision Standard Model fits [4,5]. Indeed, the near-criticality of the electroweak vacuum may be one of the most important clues from the Large Hadron Collider (LHC) for the nature of beyond the Standard Model physics [2,[6][7][8][9][10]. This provides strong motivation for improving the precision of Higgs and top mass measurements.
While the measurement of the Higgs mass is conceptually straightforward both theoretically and experimentally [11], this could not be further from the case for the top mass (m t ). Due to its strongly interacting nature, a field theoretic definition of m t , and its relation to experimental measurements, is subtle. In e + e − colliders, precision m t measurements can be made from the threshold lineshape [12][13][14][15][16][17][18][19]. However, this approach is not possible at hadron colliders, where, despite the fact that direct extractions have measured m t to a remarkable accuracy [20][21][22][23], there is a debate on the theoretical interpretation of the measured "Monte Carlo (MC) top mass parameter" [24]. This has been argued to induce an additional O(1 GeV) theory uncertainty on m t . For recent discussions, see [25,26]. It is therefore crucial to explore kinematic top-mass sensitive observables at the LHC where a direct comparison of the experimental data with first principles theory predictions can be carried out.
In this article, we present the first steps towards a new paradigm for precision m t measurements based on the simple idea of exploiting the mass dependence of the characteristic opening angle of the decay products of the boosted top, ζ ∼ m 2 t /p 2 T (see Figure 1). The motivation for rephrasing the question in this manner is twofold. First, this angle can be accessed via low point correlators, which are field theoretically drastically more simple than a groomed substructure observable sensitive to ζ. Second, while the jet mass is sensitive to soft contamination and UE, the angle ζ is not, since it is primarily determined by the hard dynamics of the top decay. In the following, we will present a numerical proof-of-principles analysis illustrating that the three-point correlator in the vicinity of ζ ∼ m 2 t /p 2 T provides a simple, but highly sensitive probe of m t , free of the typical challenges of jet-shape based approaches. Our goal is to provide the motivation for future precision studies and the motivation to find solutions to outstanding theoretical problems in the study of low point correlators.

II. THE THREE-POINT CORRELATOR
There has recently been significant progress in understanding the perturbative structure of correlation functions of energy flow operators. This includes the landmark calculation of the two-point correlator at next-toleading order (NLO) in QCD [114,115] and NNLO in N = 4 super Yang-Mills [92,116], as well as the first calculation of a three-point correlator [105] at LO (also further analyzed in [52,106,107]). The idea of using the three-point correlator to study the top quark is a natural one, and was considered early on in the jet substructure literature [117]. However, only due to this recent theoretical progress can we now make concrete steps towards a comprehensive program of using energy correlators as a precision tool for Standard Model measurements [73,118].
The three-point correlator (EEEC) with generic energy weights is defined, following the notation in [105], as with the measurement operator given by Hereζ ij = (1 − cos(θ ij ))/2, with θ ij the angle between particles i and j, the sum runs over all triplets of particles in the jet, and Q denotes the hard scale in the measurement. The EEEC is not an event-by-event observable, but rather is defined as an ensemble average. We are interested in the limit ζ 12 , ζ 23 , ζ 31 1, such that all directions of energy flow lie within a single jet. In the case of a CFT (or massless QCD up to the running coupling), the EEEC simplifies due to the rescaling symmetry along the light-like direction defining the jet. In this case, the EEEC can be written in terms of a scaling variable, ζ 31 and exhibits a featureless powerlaw scaling governed by the twist-2 spin-4 anomalous dimension, γ(4) [78,96,104,105,107,119]. This behavior has been measured [118] using publicly released CMS data [120,121].
In contrast, m t explicitly breaks the rescaling symmetry of the collinear limit. Thus m t appears as a characteristic scale imprinted in the three-point correlator. While the top quark has a three-body decay at leading order, higher-order corrections give rise to additional radiation, which is primarily collinear to the decay products leading to a growth in the distribution at anglesζ ij m 2 t /p 2 T . To extract m t , we therefore focus on the correlator in a specific energy flow configuration sensitive to the hard decay kinematics. Here we study the simplest configuration, that of an equilateral triangleζ ij = ζ allowing for a small asymmetry (δζ). Thus the key object of our analysis is the n th energy weighted cross section defined as where the measurement operator M (n) is For δζ ζ, where we have made the dependence on m t explicit. Three-body kinematics implies that the distribution is peaked at ζ peak ≈ 3m 2 t /Q 2 , exhibiting quadratic sensitivity to m t . At the LHC the peak is resilient to collinear radiation since lnζ peak < 1/α s , makings its properties computable in fixed order perturbation theory at the hard scale. In the region ζ < 2δζ the hard three-body kinematics is no longer identified, leading to a bulge in the distribution. In Figure 2 we show these features in the simplest case of e + e − → t + X simulated using Pythia 8.3 parton shower, with the details of the simulation described below. We explain in appendix A through a leading-order analysis how these features arise and motivate the definition of our observable stated above. Finally, we do not consider here the optimization of δζ and leave it to future work.

III. MASS SENSITIVITY
To illustrate the mass sensitivity of our observable, we consider the simplest case of e + e − collisions simulated in Pythia 8.3 at a center of mass energy of Q = 2000 GeV using the Pythia 8.3 parton shower [122]. We reconstruct anti-k T [123] jets with R = 1.2 using FastJet [124], and analyze them using the jet analysis software JETlib [125]. Although jet clustering is not required in e + e − , this analysis strategy is chosen to achieve maximal similarity with the case of hadron colliders. In Figure 3 we show the distribution of the three-point correlator in the peak region, both with and without the effects of hadronization. Agreement of the peak position with the leading-order expectation is found, showing that the observed behavior is dictated by the hard decay of the top. In Figure 3, linear (n = 1) and quadratic (n = 2) energy weightings are used, see eq. (2). The latter is not collinear safe, but the collinear IR-divergences can be absorbed into moments of the fragmentation functions or track functions [73,100].
Non-perturbative effects in energy correlators are governed by an additive underlying power law [76,90,91,126], which over the width of the peak has a minimal effect on the normalized distribution. This is confirmed by the small differences in peak position between parton and hadron level distributions. In Figure 3 we also show a zoomed-in version for n = 2. Taking m t = 170, 172 GeV with n = 2 as representative distributions, we find that the shift due to hadronization corresponds to a ∆m Had. t ∼ 250 MeV shift in m t . This is in contrast with the groomed jet mass case where hadronization causes peak shifts equivalent to ∆m Had.

IV. HADRON COLLIDERS
We now extend our discussion to the more challenging case of proton-proton collisions. This study illustrates the difference between energy correlators and standard jet shape observables, and also emphasizes the irreducible difficulties of jet substructure at hadron colliders.
Implicit in the definition of energy correlators, ψ|E( n 1 ) · · · E( n k )|ψ , is a characterization of the QCD final state |ψ . In the correlator literature, |ψ is usually defined by a local operator of definite momentum acting on the QCD vacuum, |ψ = O|0 , giving rise to a perfectly specified hard scale, Q. This is the case of e + e − collisions. In hadronic final states at proton-proton collisions, the states on which we compute the energy correla- tors are necessarily defined through a measurement, e.g. by selecting anti-k T jets with a specific p T,jet . Due to the insensitivity of the energy correlators to soft radiation, we will show that it is in fact the non-perturbative effects on the jet p T selection that are the only source of complications in a hadron collider environment. This represents a significant advantage of our approach, since it shifts the standard problem of characterizing non-perturbative corrections to infrared jet shape observables, to characterizing non-perturbative effects on a hard scale. This enables us to propose a methodology for the precise extraction of m t in hadron collisions by independently measuring the universal non-perturbative effects on the p T,jet spectrum. We now illustrate the key features of this approach. The three-point correlator in hadron collisions is defined as whereζ (pp) ij = ∆R 2 ij = ∆η 2 ij + ∆φ 2 ij , with η, φ the standard rapidity, azimuth coordinates. The peak of the EEEC distribution is determined by the hard kinematics and is found at ζ (pp) To clearly illustrate the distinction between the infrared measurement of the EEEC and the hard measurement of the p T,jet spectrum, we present a two-step analysis using data generated in  We then performed a proof-of-principles analysis to illustrate that a characterization of non-perturbative corrections to the p T,jet spectrum allows us to extract m t , with small uncertainties from non-perturbative physics. While we will later give a factorization formula for the observable dΣ(δζ)/dp T,jet dζ, for the present discussion it is useful to write it as dΣ(δζ) dp T,jet dζ = dΣ(δζ) dp T,t dζ dp T,t dp T,jet .
This formula, combined with Figure 4, illustrates that the source of complications in the hadron-collider environment lies in the observable-independent function of hard scales dp T,t /dp T,jet , which receives both perturbative and non-perturbative contributions. To extract a value of m t , we write the peak position as Here F pert incorporates the effects of perturbative radiation. At leading order, F pert = m 2 t . Corrections from hadronization and MPI are encoded through the shifts ∆ NP (R) and ∆ MPI (R). Crucially, in the factorization limit that we consider, these are not a property of the EEEC observable, but can instead be extracted directly from the non-perturbative corrections to the jet p T spectrum [128]. This is a unique feature of our approach.
To illustrate the feasibility of this procedure, we used Pythia 8.3 (including hadronization and MPI) to extract ζ (pp) peak as a function of p T,jet , over an energy range within the expected reach of the high luminosity LHC. As a proxy for a perturbative calculation, we used parton level data to extract F pert . To the accuracy we are working, F pert is independent of the jet p T , and can just be viewed as an effective top mass F pert (m t ). We also extract ∆ NP (R) + ∆ MPI (R) independently from the p T,jet spectrum. Note that an error of ±δ on ∆ NP/MPI in a given p T,jet bin leads to an error on F pert (m t ) of ±δ F pert /p T,jet . Using eq. (8) we fit ζ (pp) peak as a function of p T,jet for an effective value of F pert (m t ). An example of the distribution in the peak region is shown in Figure 5, which also highlights the insensitivity of the peak position to the use of charged particles only (tracks). A fit to ζ (pp) peak for several p T,jet bins is shown in Figure 6. With a perfect characterization of the non-perturbative corrections to the EEEC observable, the value of F pert (m t ) extracted when hadronization and MPI are included should exactly match its extraction at parton level. This would lead to complete control over m t . In Table I we show the extracted value of F pert (m t ) from our parton level fit, and from our hadron+MPI level fit for two values of the Pythia 8.3 m t . The errors quoted are the statistical errors on the parton shower analysis. The Hadron+MPI fit is quoted with two errors: the first originates from the statistical error on the EEEC measurement, the second originates from the statistical error on the determination of ∆ NP (R) + ∆ MPI (R) from the p T,jet spectrum. A more detailed discussion of this procedure is provided in appendix B. Thus we find promising evidence that theoretical control of m t , with conservative errors 1GeV, is possible with an EEEC-based measurement. Our analysis also emphasizes the importance of understanding nonperturbative corrections to the jet p T spectrum. Theory errors are contingent upon currently unavailable NLO computations, discussed in the following section, and so are not provided. However, we expect observable dependent NLO theory errors on m t to be better than those in other inclusive measurements wherein in the dominant theory errors are from PDFs+α s [129,130] and which mostly affect the normalisation of the observable. By contrast the EEEC is also inclusive but the extracted m t is only sensitive to the observable's shape.
The goal of this article has been to introduce our novel approach to top mass measurements, illustrating its theoretical feasibility and advantages. Our promising results motivate developing a deeper theoretical understanding of the three-point correlator of boosted tops in the hadron collider environment. Nevertheless, there remain many areas in which our methodology could be improved to achieve greater statistical power and bring it closer to experimental reality. These include the optimization of δζ, the binning of p T,jet and ζ (pp) , and including other shapes on the EEEC correlator. Regardless, our analysis does demonstrate the observable's potential for a precision m t extraction when measured on a sufficiently large sample of boosted tops. We are optimistic that such a sample will be accessible at the HL-LHC where it is forecast that ∼ 10 7 boosted top events with p T > 500 GeV will be measured [131].

V. FACTORIZATION THEOREM
Combining factorization for massless energy correlators [104] with the bHQET treatment of the top quark near its mass shell [27,28,43,67] allows us to separate the dynamics at the scale of the hard production, the jet radius R, the angle ζ, and the top width Γ t . While factorization is generically violated for hadronic jet shapes (see [132]), our framework is based on the rigorous factorization for single particle massive fragmentation [133][134][135][136][137][138][139][140]. Assuming ζ R, we perform a matching at the perturbative scale of the jet radius, using the fragmenting jet formalism [141][142][143], which captures the jet algorithm dependence. The final jet function describing the collinear dynamics at the scale of ζ is therefore free of any jet algorithm dependence. Correspondingly, we expect to obtain the following factorized expression dΣ dp T,jet dη dζ for the energy-weighted cross section differential in p T,jet , rapidity η, and ζ. This can be used to compute F pert (m t ) in a systematically improvable fashion. Obvious dependencies, such as on factorization scales, have been suppressed for compactness. Here f i are parton distribution functions, and H i,j→t is the hard function for inclusive massive fragmentation [144,145], which is known for LHC processes at NNLO [146]. J t→t is the fragmenting jet function, which is known at NLO for antik T jets [142,143], but can be extended to NNLO using the approach of [147]. The convolutions over f i,j H i,j→t and J t→t alone determine the p T,jet spectrum, independent of the EEEC measurement. Finally, J EEEC , is the energy correlator jet function, which can be computed in a well defined short-distance top mass scheme (such as the MSR mass [39,41,148]), and can include information from track or fragmentation functions. Around the top peak, J EEEC is almost entirely determined by perturbative physics and is currently known at LO. The NLO determination of J EEEC is an outstanding theoretical problem and is very involved, thus beyond the scope of this article, though a road map towards its completion has recently become available [105,114,115]. In the region of on-shell top, J EEEC can be matched onto a jet function defined in bHQET [27,28,39,40,67]. The functions in the factorization formula above exhibit standard DGLAP [149][150][151] evolution in the momentum fractions z J and z h = p T,hadron /p T,jet , and the ⊗ denote standard fragmentation convolutions. A more detailed study of the structure of the factorization will be provided in a future publication.

VI. CONCLUSIONS
We have proposed a new paradigm for jet-substructure based measurements of the top mass at the LHC in a rigorous field theoretic setup. Instead of using standard jet shape observables, we have analyzed the three-point correlator of energy flow operators, and have illustrated a number of its remarkable features. Our results support the possibility of achieving complete theoretical control over an observable with top mass sensitivity competitive with direct measurements whilst avoiding the ambiguities associated with the usage of MC event generators.

ACKNOWLEDGMENTS
We are particularly grateful to G. Salam for insightful questions that led us to (hopefully) significantly improve our presentation. We also thank B. Nachman, M. Schwartz, I. Stewart and W. Waalewijn for feedback on the manuscript. We thank H. Chen, P. Komiske, K. Lee, Y. Li, F. Ringer, J. Thaler, A. Venkata, X.Y. Zhang and H.X. Zhu for many useful discussions and collaborations on related topics that significantly influenced the philosophy of this work. A.P. is grateful to M. Vos, M. LeBlanc, J. Roloff and J. Aparisi-Pozo for many helpful discussions about subtleties of the top mass extraction at the LHC. This work is supported in part by the GLUODYNAMICS project funded by the "P2IO LabEx (ANR-10-LABX-0038)" in the framework "Investissements d'Avenir" (ANR-11-IDEX-0003-01) managed by the Agence Nationale de la Recherche (ANR), France. I.M. is supported by startup funds from Yale University. A.P. is a member of the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics, which is supported FIG. 7. The effect of applying different δζ cuts to ensure an equilateral configuration for e + e − → t + X and pp → t + X processes. The δζ cuts isolate the peak, which is governed by the hard decay of the top, from the "bulge" contribution.
by the UK Science and Technology Facilities Council (STFC) under grant number ST/T001038/1.

Appendix A: Leading-Order Analysis
Here we perform a leading-order analysis of the observable which suffices to explain the general features of the spectrum in Figure 2. For concreteness, we will define the kinematics assuming a e + e − → t(→ bqq )+X process where we take the b, q,q partons to be massless. No further complications, beyond the need for more ink, are introduced by using the longitudinally invariant kinematics needed for measurements at the LHC. At leading order, we can factorize the Born cross-section dσ (0) /σ (0) into the dimensionless three-body phase space for the top's decay products, dΦ 3 , and the dimensionless weighted squared matrix element, σ t |M (t → bW → bqq )| 2 /σ (0) where σ t is the cross section to produce a top quark. As |M (t → bW → bqq )| 2 ∼ O(1), we can approximate the differential EEEC distribution in eq. (3) as reducing the problem of understanding the observable of interest to studying three-body kinematics. Before directly working with eq. (A1), let us develop some intuition for the three-body kinematics. Consider the decay of a top quark in its rest frame, withp t =p b + p q +pq . Here we are usingp i as a rest-frame momentum and p i as a lab-frame momentum. In the top rest frame, the angular parameters on which the EEEC depends are given bỹ Momentum conservation requires thatζ 12 +ζ 23 +ζ 31 ∈ [2, 2.25]. Let the lab frame top momentum be p t = (E t , p t ). In the boost between the lab and rest frame, To first order in m t /E t 1, we also have sinh β ≈ cosh β. Hence the sum of lab frame EEEC parameters is withθ ti denoting the angle between parton i and the boost axis in the top's rest frame. The function g ≡x tbxtqζ12 +x tbxtq ζ 31 +x tqxtq ζ 23 is also kinematically bounded so that g ∈ [0, 3]. Upon averaging over the possible boost axes one finds that g ∈ [1, 2.25]. Thus, returning to eq. (A1), we expect the partially integrated EEEC distribution . However, this peak will have a large width (of the order of 3m 2 t /(4E 2 t )), whose origin can be understood by interpreting the parametersx ti ∈ [0, 2] as three sources of (correlated) random noise in the shape of the flow of energy which 'smears' the EEEC distribution. We can largely remove the noise by constraining the shape of the energy flow on the celestial sphere. This is most simply done by requiring that ζ ij approximately form the sides of an equilateral triangle ( ζ ij ≈ √ ζ ik ). Consequently, removing two of the noisy degrees of freedom from the distribution. Upon including this constraint, we find that g ≈ 2.1 with a small variance. This motivates us to introduce an EEEC distribution on equally spaced triplets of partons and allow for small asymmetries around this configuration governed by the parameter δζ: where the operator M (n) in the collinear limit is l,m,n∈{1,2,3} Θ(δζ − |ζ lm − ζ mn |).
As previously explained, three-body kinematics determines that this distribution is peaked at ζ peak ≈ 3m 2 t /(4E 2 t ) ∼ m 2 t /Q 2 . Furthermore, at the LHC the peak should be resilient to collinear radiation since lnζ peak < 1/α s .
We can now complete our leading-order discussion by computing the Born contribution to eq. (A7). Expanding for δζ ζ, we obtain where z 1 = E b /E t and z 2 = Eq /E t . The delta function causes the distribution to be sharply peaked at ζ = 3m 2 t /(4E 2 t ). This matches the intuition we have developed from considering pure kinematics. Looking at eq. (3) to all orders in α s , up to power corrections in δζ, where the latter to leading order in δζ ζ can be written as, whilst in the region where 2δζ > ζ dΣ(δζ) dQdζ ≈ dζ 12 dζ 23 dζ 31 G (n) (ζ 12 , ζ 23 , ζ 31 ) . (A12) Figure 7 demonstrates that this dependence on δζ is born out in simulation for e + e − and pp collisions. To conclude our discussion of δζ, the limiting cases discussed above motivate that an optimal choice of this parameter will be a function of Q that strikes a balance between statistics and constraining the three-body kinematics (δζ optimal ≈ κ ζ peak /2 for κ 1). A more sophisticated analysis may also sum over several shapes of energy flow on the celestial sphere to increase statistics -perhaps allowing for smaller values of δζ optimal .
Finally, in Figure 8 we show the top peak in the 3point correlator for n = 2 in e + e − → t + X simulations in Vincia 2.3. We find the peak position almost in line with that of Pythia 8.3, justifying our earlier assumption that the features of the observable are largely determined by the fixed-order expansion in α s .

Appendix B: Details of the EEEC Analysis at Hadron Colliders
Here we describe the details of the proof-of-principles peak position analysis outlined in section IV. The longitudinally boost invariant measurement operator for the

EEEC observable is
As before, the peak of the M (n) EEEC distribution is determined by the top quark hard kinematics and is found at ζ (pp) peak ≈ 3m 2 t /p 2 T,t , where p T,t is the hard top p T , not p T,jet . Consequently, the basic properties of the dΣ(δζ)/dp T,t dζ distribution are completely insensitive to non-perturbative physics. In sections III and IV we demonstrated this insensitivity by parton shower simulation wherein we showed evidence that the top decay peak is nearly entirely independent of hadronization and UE. Consequently, in the limit that p T,t /(∆ NP + ∆ MPI ) → ∞, the top decay peak position is exactly independent of non-perturbative effects. However, since p T,t is not directly accessible, the observable we consider is dΣ(δζ) dp T,jet dζ = dΣ(δζ) dp T,t dζ dp T,t dp T,jet , where p T,jet is the p T of an identified anti-k T top-jet. The top peak position in the distribution dΣ(δζ)/dp T,jet dζ will be shifted by hadronization and UE due to shifts in the jet p T distribution. This shift can be measured independently from our observable and will be universal to all measurements of energy correlators on top quarks at the LHC.
We can parameterize the all-orders peak position in dΣ(δζ)/dp T,jet dζ as Mainly, ∆ receives three additive contributions from perturbative radiation, hadronization, and from UE/MPI: Some simple manipulations can be made so as to minimize the sensitivity to ∆ in an extracted value of m t . We define the following function of measurable and perturbatively calculable quantities, is the peak position in a fixed reference p T bin, p ref T,jet , and ζ (pp)v peak is the peak position for a variable p T,jet value, p v T,jet , larger than the reference value (we require p v T,jet > p ref T,jet to avoid divergences). ρ 2 is defined so that, in the limit p v T,jet , p ref T,jet → ∞, we have ρ 2 → m 2 t . In the analysis below we set 3(1 + O(α s )) → 3 so that, in the limit p v T,jet , p ref T,jet → ∞, we find ρ 2 → F pert as defined in eq. (8). Now let us make a further definition, We can substitute eq. (B3) into eq. (B5) to find which is plotted in Figure 10, left. ρ has an asymptote as p v T,jet → ∞ around which we perform a series expansion: Thus a fit of the asymptote of ρ, and its first non-zero correction, can be used to extract F pert and ∆ ref . All dependence on ∆ v enters in the higher order terms. However, in the limit that p ref T,jet → ∞, ρ 2 → F pert and so while the fit for F pert will become exact, the error on a fit for ∆ ref will diverge. In practice it will be necessary to perform the EEEC measurement with boosted tops in order to get a well-defined peak. Consequently, fits for ∆ ref will suffer from parametrically large errors (as can be seen in the large deviation between the exact and expanded curves at low p ref T,jet in Figure 9). However, as previously stated, ∆ ref can be extracted from an inde- pendent measurement of the top-jet p T distribution, dσ pp→t(→bqq )+X dp ref One can parameterize the non-pertubative effects in D in the same way as we did in ζ where D pert. (m t , p T,jet , α s ) is the all-orders perturbative top-jet p T distribution, and g(p T,jet , . . .) captures all the non-perturbative modifications to p T,jet . As before, we parameterize the modifications via introducing a shift function ∆ defined as It is required for consistency with the factorization in eq. (9)  Thus, we fit for F pert using the following procedure: 1. Following eq. (B8) we fit for the asymptote of ρ (which we label ρ asy ) using a polynomial in (1/p v T,jet ) n . In this paper we found that a third degree polynomial, ρ(p T,jet ) = ρ asy + c 2 (p v T,jet ) −2 + c 3 (p v T,jet ) −3 , (B13) optimized the reduced χ 2 . The value of ρ asy was found to be stable, within our statistical accuracy, against the inclusion of further higher order terms, c n (p v T,jet ) −n . Figure 10 shows one such fit. No error bars are shown in this Figure 10 as it was produced from a single Monte Carlo sample. Fits of five samples are averaged over to produce the results and their errors in Table II. 2. We extract ∆ ref from the top-jet p T spectrum as shown in Figure 11.
3. Finally, we compute F pert using the asymptote of ρ, ρ asy , defined above in eq. (B13) as The outcome of this procedure is given in Table II which shows the extracted F pert from Pythia 8.3 with m t = 172 GeV and 173 GeV. The important outcome of this analysis is that the differences between the measured masses with parton, hadron and hadron+MPI data are 1GeV and are smaller than the statistical errors. This analysis was not optimized to give a good statistical error and certainly can be improved. Thus we find promising evidence that complete theoretical control of the top mass, up to errors < 1GeV, is possible with an EEEC based measurement.
To cross-check our result, purely to demonstrate selfconsistency, in Figure 12 we illustrate a theory fit of ρ using parton shower data from Pythia 8.3 with m t = 172 GeV at parton level and hadron level. The curves in Figure 12 are not the third degree polynomial used to extract ρ asy in eq. (B13). Rather, the curves are, truncated at second order, using the values of F pert given in Table II and the values of ∆ ref given in Figure 11. Error bars correspond to the errors on F pert and ∆ ref . To illustrate the partonic curve, a value of ∆ ref pert = (11 ± 3) GeV has been used which was extracted from the fit for ρ (2) (i.e. c 2 in eq. (B13)). This ∆ ref pert is not used in any of the preceding analysis (or anywhere else in this article) where all dependence on ∆ ref pert is absorbed in to the definition of F pert . Each error band shows the combined statistical error from the determination of the asymptote and of ∆ ref (including the dominant 3 GeV error on ∆ ref pert ). We find agreement between the MC data and our theory fit. Figure 13 along with Figure 6 further demonstrates the excellent agreement between theory and parton shower data wherein we fit ζ peak (p t,jet ) with the ansatz in eq. (8), also using the values for F pert in Table II and the values of ∆ ref given in Figure 11. One such data set and its fit is shown in Figure 10. In each column the first error is from the fit of the ρ asymptote and is statistical. The second error (when given) is also statistical and is the error from using the parton shower to determine ∆ ref ≈ ∆ ref as extracted from the top jet pt distribution. Errors have been combined in quadrature in the final row. No theory errors are given.  Table II and ∆ ref extracted in Figure 11. Excellent agreement between the theoretical fit and Pythia 8.3 is observed in all cases.