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arXiv:1310.0799v1 [hep-ph] 2 Oct 2013

Prepared for submission to JHEP

Determination of the top quark mass circa 2013:

methods, subtleties, perspectives

Aurelio Justea,b Sonny Mantryc,d Alexander MitoveAlexander Peninf,g Peter Skandse

Erich VarneshMarcel VosiStephen Wimpennyj

aInstituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), 08010 Barcelona, Spain

bInstitut de F´ısica d’Altes Energies, Universitat Auton`oma de Barcelona, 08193 Bellaterra, Spain

cHigh Energy Division, Argonne National Laboratory, Argonne, IL 60439, USA

dDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

eTheory Division, CERN, CH-1211 Geneva 23, Switzerland

fUniversity of Alberta, Edmonton AB T6G 2J1, Canada

gInstitut f¨ur Theoretische Teilchenphysik, KIT, 76128 Karlsruhe, Germany

hDepartment of Physics, University of Arizona, Tucson AZ 85721, USA

iIFIC (UVEG/CSIC), Ap. Correos 22085, E-46071 Valencia, Spain

jDepartment of Physics & Astronomy, University of California - Riverside, Riverside, USA

Abstract: We present an up-to-date overview of the problem of top quark mass determi-

nation. We assess the need for precision in the top mass extraction in the LHC era together

with the main theoretical and experimental issues arising in precision top mass determination.

We collect and document existing results on top mass determination at hadron colliders and

map the prospects for future precision top mass determination at e+e−colliders. We present

a collection of estimates for the ultimate precision of various methods for top quark mass

extraction at the LHC.

1Preprint numbers: CERN-PH-TH/2013-226

Contents

1 Introduction 1

2 Issues in precision top mass determination at hadron colliders 2

3 Top mass determination at hadron colliders 5

3.1 “Conventional” top mass determination techniques 7

3.2 CMS end-point method [76]8

3.3 ATLAS 3-dimensional template ﬁt method [77]9

3.4 Top mass determination from J/Ψ ﬁnal states [62]10

3.5 Top mass determination from kinematic distributions 11

4 Top mass determination at lepton colliders 12

4.1 Theory of t¯

tproduction near threshold at e+e−colliders 12

4.2 Resonance energy and top quark mass determination 14

4.3 Threshold cross-section 15

4.4 Threshold t¯

tproduction at e+e−colliders: experimental simulations 15

4.5 Top quark mass from a reconstruction of the top decay products 17

5 Conclusions 18

1 Introduction

The precision with which we determine the top quark mass impacts our understanding of

several phenomena. Examples are EW precision ﬁts [1], determination of the vacuum stability

in the Standard Model [2,3] as well as models with broad cosmological implications [4,5].

A number of measurements of mtfrom hadron colliders exist [6,7], utilizing all measured

decay modes of the top quark. The experimental extraction has accuracy of δmt.1 GeV.

The main task for this writeup is to map the steps that can clarify the relation between the

extracted value of the mass and a theoretically well-deﬁned top mass (like the pole mass).

The top quark mass mtis not a physical observable and, therefore, it cannot be measured

directly. Virtually all existing strategies for determining mt(see section 3) are based on its

extraction from observables that are directly sensitive to it, i.e. mtis deﬁned as the solution

to the following implicit relation:

σexp({Q}) = σth (mt,{Q}),(1.1)

where {Q}is a set of kinematical variables, σexp stands for the measured and σth for the

predicted value of some chosen observable σ. In a typical application, mtis adjusted in σth

– 1 –

to obtain the best ﬁt to the shape of σexp , as a function of the variables {Q}. This implicitly

assumes that σexp has been corrected for detector (and possibly acceptance) eﬀects, or that

the converse has been applied to σth, so that the observables on either side of Eq. (1.1) are

deﬁned at the same level, with the same cuts. Uncertainties in the theoretical prediction

due to missing higher-order eﬀects, ﬁnite-width eﬀects, and non-perturbative corrections are

generally present. We discuss them in more detail in section 2.

The top mass mtis scheme dependent and a large number of such schemes exist. Ex-

amples are the pole, the MS, and the 1S schemes [8]; see Ref. [9,10] for a discussion in the

context of hadron colliders. Diﬀerent mass schemes are perturbatively related to each other.

For example, the top mass mt(R, µ) in scheme “R” is related to the pole mass mpole

tthrough

a perturbative series

mpole

t=mt(R, µ) + δmt(R, µ), δmt(R, µ) = R

∞

X

n=1

n

X

k=0

ank [αs(µ)]nlnkµ2

R2,(1.2)

where Ris a scale associated with the scheme; for the MS scheme R∼mt. The relation

between the pole and MS masses is known to three loops in QCD [11,12]; a possible large

EW correction has recently been reported in Ref. [13]. Large logarithms can arise in converting

between schemes if the scale R≪mt, as seen in top resonance schemes [14] where R∼Γtop,

and can be resummed via an infrared renormalization group equation [15].

A reliable interpretation of top mass measurements requires understanding the connection

between the theory prediction, in a given top mass scheme, and the experimental observable as

shown schematically in Eq.(1.1). This connection between experimental observables and the

appropriate top mass schemes is well understood in e+e−colliders. Precision top quark mass

determinations at e+e−colliders have been studied for top pair production near threshold [16–

19] and in the boosted regime [14,20]. The expected uncertainty in the top mass from the

threshold scan method is δmt.100 MeV [21,22] and a few hundred MeV [23] for boosted

top quarks. See sections 4,4.1,4.4 for details. For hadron colliders, which are the main focus

of the current and near future research, and by extension of this document, the situation

is more complicated and a rigorous framework is still lacking. Below we review the current

status and issues in precision top mass extractions at hadron colliders.

2 Issues in precision top mass determination at hadron colliders

A unique property of the top quark is that it decays very quickly, before it can form strongly

interacting bound states. For this reason the top quark can be studied largely free of

non-perturbative eﬀects [16–18]. Still, a number of uncertainties of perturbative and non-

perturbative origin aﬀect the extraction of mt:

1. MC modeling. Most methods for extraction of mtrely on modeling the measured

ﬁnal state with typically LO+LL MC generators. The extracted mass then reﬂects

the mass parameter in the corresponding MC generator. Identifying the nature of this

– 2 –

mass parameter and relating it to common mass schemes, like the pole mass, is a non-

trivial and open problem, and may be associated with ambiguities of order 1 GeV [24,

Appendix C]; see also Ref. [9]. The eﬀect of the top and bottom masses on parton-shower

radiation patterns is generally included already in the LO+LL Monte Carlos [25–30] and

acts to screen the collinear singularities. NLO matching and non-perturbative eﬀects

are discussed separately below.

2. Reconstruction of the top pair. Typically, the existing methods for extraction of the

top quark mass implicitly or explicitly rely on the reconstruction of the top pair from

ﬁnal state leptons and jets. This introduces uncertainties of both perturbative origin

(through higher-order corrections) and non-perturbative origin (related to hadronization

and non-factorizable corrections). Methods that do not rely on such reconstruction are

therefore complementary and highly desirable; two examples are given in 3.4and 3.5.

3. Unstable top and ﬁnite top width eﬀects. These eﬀects have been studied extensively

in the context of top pair production at e+e−colliders [31–33]. In the context of

higher order corrections at hadron colliders, ﬁnite top (and W) width eﬀects have been

computed in [34,35] where comparisons versus the narrow width approximation can

be found. The conclusion is that these corrections are small, sub-1%, in inclusive

observables (like the total inclusive cross-section used in 3.3) but can be sizable in tails

of kinematical distributions. In particular, they signiﬁcantly aﬀect the tail of the B ℓ

invariant mass distribution used in the method 3.4(but not the central region of the

distribution which is most relevant for the mtdetermination described in 3.4).

4. Bound-state eﬀects in top pair production at hadron colliders. The eﬀect of bound state

formation on top pair production at hadron colliders has been studied in Refs. [36–39].

It can dramatically aﬀect the shape of diﬀerential distributions within a few GeV of ab-

solute threshold. Therefore, any mass measurement that is sensitive to this kinematical

region has to properly take these eﬀects into consideration. In the context of the total

cross-section, see Refs. [40,41], the eﬀect on the cross-section is sub-1% and is taken

into account in current higher order calculations of the total inclusive cross-section (and

thus in mass extractions based upon it).

5. Renormalon ambiguity in top mass deﬁnition. It is well known [15,42–44] that the pole

mass of the top quark suﬀers from the so-called renormalon ambiguity. This implies an

additional irreducible uncertainty of several hundred MeV’s on the top pole mass. The

short distance masses do not suﬀer from the renormalon ambiguity and the precision in

their determination is restricted only by experimental and theoretical uncertainties. At

hadron colliders, where currently δmt.1 GeV, the renormalon ambiguity is numerically

subdominant; see also Ref. [10].

6. Alternative top mass deﬁnitions. It is well understood from e+e−collider studies that by

using alternative top mass deﬁnitions one could improve the precision of the extracted

– 3 –

top quark mass. Similar studies for hadron colliders have been done in Refs. [9,10,45].

It has been argued in Ref. [9] that for top mass extractions in the peak region, the

appropriate short distance mass schemes correspond to the top resonance schemes where

R∼Rsc ∼1 GeV ∼Γtop , where Rsc is the shower cutoﬀ implemented in the MC.

An interpretation of this statement in the context of a factorization framework for

hadron colliders is still lacking. Ref. [45] advocates extracting directly the top MS

mass from the top pair production cross-section. The improvement at the current

level of precision δmt.1 GeV, however, is small [10] (see also the discussion about

renormalon ambiguity, above). The extracted top MS mass might be aﬀected by the

ﬁndings reported in Ref. [13].

7. Higher-order corrections. Missing higher-order corrections can be an important source

of uncertainty in the determination of the top mass. These are typically added through

NLO calculations [34,35,46,47] and for the case of the total cross-section through

approximate NNLO calculations [10,41,45,48] (for calculations in full NNLO, see the

discussion in 3.3below). A particularly sensitive issue is the matching of NLO top-quark

calculations to parton showers, see [49–51].

8. Non-perturbative corrections. Non-perturbative corrections mostly aﬀect the MC mod-

eling of the ﬁnal state. These include hadronization, in particular of the ﬁnal-state

partons that inherit the top quark color charges (which causes an unavoidable non-

perturbative exchange of energy with the rest of the event), hadron and τdecays (includ-

ing the Bhadron decays), underlying event, and possible additional non-perturbative

phenomena such as color reconnections or other collective phenomena. Depending on

how the corrections to the cross-sections in eq. (1.1) are performed, these uncertainties

enter either on the experimental or theoretical side of the equation. The underlying-

event, hadronization, and particle-decay corrections are typically dealt with at the jet-

calibration stage, and the resulting systematic uncertainties become part of the jet-

energy-scale (JES) systematics. A study of color-reconnection eﬀects in the special case

of e+e−collisions found very small eﬀects <100 MeV [52], but toy models show that

the eﬀect in hadron collisions may be as large as 0.5 GeV [53]. More physical models

and better constraints are required to reduce this uncertainty further, for instance by

allowing one to bound it, rather than merely switching it on and oﬀ. Non-perturbative

corrections can also be introduced through ﬁnal-state interactions in the presence of

strong jet vetoes [54]. Inclusive measurements like the methods described in sections

3.4and 3.5are likely to suﬀer least from such non-perturbative eﬀects.

9. Contributions from physics beyond the Standard Model. It is possible that some yet-

undiscovered physics beyond the Standard Model (BSM) might inﬂuence the various

measurements used to extract the top quark mass. Given that in the context of top

mass extraction experimental measurements have so far always been compared with

predictions based on the SM, the possibility arises that there might be a bias in the

– 4 –

determination of the top quark mass due to new physics. While it is unlikely that such

new physics can cause large corrections, 1O(1 GeV) modiﬁcations to mtcannot be

excluded at present. A ﬁrst dedicated study of BSM contributions to mtdetermination

is ongoing [55]. Application to top mass measurements of the work reported in Ref. [56]

may also be useful for disentangling BSM contributions (although this will likely require

the inclusion of NLO QCD corrections).

3 Top mass determination at hadron colliders

A major collection of experimental methods is available in [57]. Here we highlight a few that

have already proven useful or appear to be promising:

1. Matrix element methods. The most precise measurements of mtfrom the Tevatron use

the matrix element method [58,59], in which the measured objects are compared with

expectations from the LO t¯

tproduction and decay diagrams convoluted with the detec-

tor response. The method derives much of its power from the fact that the likelihood

for each event to be consistent with both t¯

tand background production is calculated;

greater weight is assigned to events that are more likely to be from t¯

twhen measuring

mt. In addition, the hadronically-decaying Wboson in ℓ+ jets events provides an in

situ constraint on the jet response, substantially reducing the systematic uncertainty.

An NLO theory approach is currently being developed [60].

2. Ideogram and template methods. The current generation of CMS analyses, which are

among the most precise mtmeasurements, use Ideogram techniques. The ideogram

corresponding to the most probable solution for the mass is determined on an event-by-

event basis. These are then summed over the full dataset to determine an “integrated

ideogram”. The top mass is then determined by ﬁtting this to a Monte Carlo spectrum

for the same number of events. The MC spectrum is determined as a function of mt

(CMS - all jets) or mtand JES (CMS - lepton and jets). The dilepton channel is handled

in a similar way using the analytical matrix weighting technique (AMWT) to treat the

2-neutrino ambiguities. Regarding Monte Carlo generators, CMS uses Madrgraph (LO

ME generator) with Pythia for the parton showering.

The ATLAS collaboration uses similar “template” methods. The main diﬀerences

with respect to the CMS analyses are that the ATLAS Collaboration currently uses

3-parameters (mt, lightJES, bJES) for their lepton + jets analysis as well as MC@NLO

+ Herwig for event generation.

3. Extraction from the total cross-section σtot. The total inclusive t¯

tcross-section at a

given collider depends on mt, so the measured cross-section can be used to constrain

mt. Extractions of the top mass from σtot have been performed in [10,41,45,48] using

1For example, the CMS end-point top mass determination (see sec. 3.2) is based on kinematical considera-

tions, i.e. it has reduced sensitivity to the top quark production mechanism.

– 5 –

NLO+NNLL or approximate NNLO cross-section calculations. Very recently a ﬁrst

analysis performed in full NNLO+NNLL appeared [61]. The sensitivity of σtot to the

top mass is relatively low (few %), so this method is not competitive in precision with

other existing methods. On the other hand the method uses an observable based on a

well-deﬁned top mass, has small uncertainties due to perturbative and non-perturbative

eﬀects, and is not very sensitive to top width eﬀects.

4. The J/ψ method [62]. In about one in 105top quark decays, the fragmentation prod-

ucts of the bquark will include a J/ψ decaying to µ+µ−. If the Wboson from the

same top quark also decays leptonically, the three-lepton invariant mass is sensitive to

mt. The other top quark is only used to discriminate t¯

tproduction from background.

The strength of this method is that the main systematic uncertainties arise from dif-

ferent sources than in other methods (primarily bfragmentation), and may be smaller.

Moreover, no t¯

treconstruction takes place i.e. the method is inclusive at any order in

perturbation theory. These potential advantages must be weighed against the statisti-

cal limitations arising from requiring a J/ψ candidate. MC studies of this method are

reported in [63,64], and the uncertainty from bfragmentation was studied at NLO in

[65–67]. A NLO study, with factorized production and decay, was performed in Ref. [47].

The complete NLO result including production/decay interferences, oﬀ-shell eﬀects and

backgrounds, was computed in Ref. [34] (the Bmesons in this work are treated as b-

jets). Additional error estimates, performed within this study, can be found in sec. 3.4

below.

5. Dilepton-speciﬁc methods. In the same spirit as the J/ψ method, it may be advantageous

to measure mtusing kinematic properties (e.g. the invariant mass and pT) of the lepton

pair in dilepton t¯

tcandidates (selected as pair of leptons and possibly two b’s, without

requiring t¯

treconstruction) [68]. These observables should have a smaller sensitivity

to the modeling of hadronic observables (showering and jets). Such measurements can

be compared versus complete NLO calculations [34,35], as well as versus standard

MC generators. This approach may not be as sensitive to the value of mtas other

methods, but oﬀers very diﬀerent systematics, and therefore may help to reduce the

overall uncertainty on the world-average mt. First measurements of top pair diﬀerential

distributions in dilepton ﬁnal states have already appeared [69]. See also the related

discussion in sec. 3.5 below.

In the near to medium term (i.e. prior to the construction of a lepton collider capable of

performing a t¯

tthreshold scan), improvement in the precision with which we know mtwill

depend on:

•Extraction of the top mass with new methods that have alternative systematics (like

4and 5in section 3). Such extractions will either validate the current precision in the

available top mass measurements or highlight the need for additional scrutiny. Further

phenomenological and experiment studies of these new methods are needed.

– 6 –

•Decreasing the perturbative uncertainty in currently used Matrix Element methods by

applying future extension of the work in Ref. [60]. It remains an open question if top

width eﬀects and non-perturbative eﬀects can also be reduced this way.

•Improved understanding of the relation between MC mass and standard quark masses,

such as the pole mass. Work along these lines has been reported in [9]; see also Ref. [24,

Appendix C].

In the following we review, and present estimates, for the capabilities of various methods

for top mass determination. The methods can be split into “conventional” (sec. 3.1), “other

available” (sec. 3.2,3.3) or “under development” (sec. 3.4,3.5).

3.1 “Conventional” top mass determination techniques

As a model for the conventional collider mass measurements, we consider the CMS lepton-

plus-jets [70], dilepton [71] and all-hadronic analyses [72]. These are currently the most precise

measurements in each channel. The analyses use similar methods and result in measurements

with comparable systematic uncertainties. To estimate the potential precision for the various

14 TeV scenarios we have taken the CMS lepton-plus-jet result mt= 173.49 ±0.27(stat.)±

1.03(syst.) GeV as representative and have performed extrapolations based on this. The

results are presented in Table 1.

Ref.[70] Projections

CM Energy 7 TeV 14 TeV

Cross Section 167 pb 951 pb

Luminosity 5fb−1100fb−1300fb−13000f b−1

Pileup 9.3 19 30 19 30 95

Syst. (GeV) 0.95 0.7 0.7 0.6 0.6 0.6

Stat. (GeV) 0.43 0.04 0.04 0.03 0.03 0.01

Total 1.04 0.7 0.7 0.6 0.6 0.6

Total (%) 0.6 0.4 0.4 0.3 0.3 0.3

Table 1. Extrapolations based on the published CMS lepton-plus-jets analysis

These are based on the 7 and 14 TeV cross-sections calculated using the full NNLO

framework [73] with an allowance for a decreased trigger eﬃciency due to higher event rates

and trigger thresholds. For the systematic errors, we assume that some of the soft QCD and

fragmentation uncertainties will be constrained using the data from future LHC runs. We

keep the initial and ﬁnal state radiation and pdf uncertainties unchanged. Without a full

simulation of the machine conditions, we are unable to model the eﬀects of the increased

merging of the top-decay products in moving to the higher energy. To allow for this and the

uncertainties in the extrapolations we add in an additional 300 MeV uncertainty to the mass

measurement.

– 7 –

Scenario Dominant Uncertainties

Ref.[70] Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR

100 fb−1/19 PU Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR

100 fb−1/30 PU Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR, Pileup

300 fb−1/19 PU Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR

300 fb−1/30 PU Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR, PIleup

3000 fb−1/95 PU Jet Energy Scale, Hadronization, Soft QCD, ISR/FSR, PIleup

Table 2. Dominant systemic uncertainties for each scenario

In Table 2we summarize the dominant uncertainties for each scenario. While these

are very similar, it should be noted that pileup and the associated uncertainties from the

missing transverse energy and contamination of the underlying event are expected to become

increasingly important as the collision energy and pileup are increased. We also note that the

ISR/FSR uncertainly, that is one of the sub-leading uncertainties for [70] becomes one of the

leading uncertainties for each of the 300 fb−1and 3000 f b−1scenarios.

Based on the comparison of the results from [70] and the CMS combined result from the

three channels shown at the TOP2012 Workshop [74], see also [75], we estimate that combina-

tions of diﬀerent channels for each of the 14 TeV scenarios may lead to a small improvement

in the projected precisions. We also note that the triggering on the all-hadronic events may

prove diﬃcult when running at very high luminosity and under high pileup conditions. This

may prevent the eﬀective use of this channel under these conditions.

3.2 CMS end-point method [76]

This method is kinematical in nature and utilizes the correlation between the end-points of

the Mb ℓ and the M221

T2perp distributions and mt. It gives a mass measurement mt= 173.90 ±

0.90(stat.)+1.70

−2.1(syst.) GeV. This was extrapolated using similar assumptions to that used

for the CMS lepton-plus-jet method. A summary of the results is given in table 3. As

this technique is insensitive to pileup eﬀects we only quote one extrapolation for each of the

luminosity scenarios.

In Table 4we summarize the dominant uncertainties for each scenario. As with the

conventional analysis, these are fairly similar as a function of increasing luminosity. We also

note that, unlike the conventional method, the ISR/FSR and pileup terms do not seem to

play a role in the precision of the measurements, even at high luminosity.

Although the terms listed in Tables 2and 4have a large overlap, we note that they are

not 100% correlated so that combining the results from the two methods may be beneﬁcial

to the overall precision. This follows from the fact that, unlike the conventional analyses, the

Endpoint method does not rely on Monte Carlo modeling to do an internal calibration. It is

largely analytical with a data-driven model for the background.

– 8 –

Ref.[76] Projections

CM Energy 7 TeV 14 TeV

Cross Section 167 pb 951 pb

Luminosity 5fb−1100fb−1300fb−13000f b−1

Syst. (GeV) 1.8 1.0 0.7 0.5

Stat. (GeV) 0.90 0.10 0.05 0.02

Total 2.0 1.0 0.7 0.5

Total (%) 1.2 0.6 0.4 0.3

Table 3. Extrapolations based on the published CMS Endpoint analysis

Scenario Dominant Uncertainties

Ref.[76] Jet Energy Scale, Hadronization, Soft QCD

100 fb−1Jet Energy Scale, Hadronization, Soft QCD

300 fb−1Jet Energy Scale, Hadronization, Soft QCD

3000 fb−1Jet Energy Scale, Hadronization

Table 4. Dominant systemic uncertainties for each scenario

We also note that the kinematical nature of this method makes it suitable to attempt top

mass determination which is less likely to be aﬀected by possible new physics contributions.

Nonetheless, this important aspect of mtdetermination needs further study. Finally, one

would like to study in more detail the eﬀect of higher order corrections, for example, by

comparing with the ﬁndings of Refs. [34,35].

3.3 ATLAS 3-dimensional template ﬁt method [77]

The ATLAS collaboration has recently published a new determination of the top quark mass

in the lepton+jets ﬁnal state [77]. This analysis uses a 3-dimensional template technique

which determines the top quark mass together with two important experimental systematic

uncertainties.

The result is mt= 172.31 ±0.23 (stat) ±0.27 (JSF) ±0.67 (bJSF) ±1.35 (syst)

GeV. The uncertainties labeled JSR and bJSF correspond to the statistical uncertainty of

the global jet energy scale factor (JSF) and the relative b-jet to light-jet energy scale factor

(bJSF). The in-situ determination of these uncertainties in the 3D ﬁt has allowed the two

dominant systematic uncertainties to be transformed into statistical uncertainties to a large

extent. The residual Jet Energy Scale uncertainty is combined together with a large number

of other sources of uncertainty into “syst”. The modeling of top quark production and decay

has a non-negligible contribution.

– 9 –

3.4 Top mass determination from J/Ψﬁnal states [62]

Our estimate of the theory error is based on the NLO QCD calculation of Ref. [47] performed

for LHC 14 TeV. The estimation of the statistical uncertainty is based on preliminary studies

by the CMS collaboration. Calculations for LHC 33 TeV in leading order QCD are also

available. 2From these results we conclude that hMBℓ i(mt) is not sensitive to the collider

energy, if the same cuts are used. More restrictive cuts for LHC 33 TeV lead to slight

modiﬁcation of the hMBℓ i(mt) dependence, but the theoretical error of the extracted mt

remains largely unchanged.

The main sources of theoretical error in the J/Ψ method are scale variation and B-

fragmentation. Modeling of hMBℓ iin NNLO QCD could become possible during the LHC 13

TeV run, which would reduce the scale variation by a factor of 2.5. We estimate this possible

improvement by comparing in Table 5the scale and pdf uncertainty of the total inclusive

cross-section for LHC 13 and 33 TeV at NLO and NNLO [73]. We use m= 173.3 GeV with

LHC 13 TeV LHC 33 TeV

δscale[%] δpdf [%] δscale [%] δpdf [%]

MSTW NNPDF MSTW NNPDF MSTW NNPDF MSTW NNPDF

NLO +12.1

−12.1

+11.8

−11.9

+1.9

−2.3

+1.8

−1.8

+11.5

−10.3

+11.2

−10.0

+1.2

−1.5

+1.0

−1.0

NNLO +3.4

−5.6

+3.5

−5.7

+1.8

−2.0

+1.8

−1.8

+3.1

−4.7

+3.1

−4.7

+1.0

−1.4

+1.0

−1.0

Table 5. Scale and pdf uncertainty for the total inclusive t¯

tcross-section at 13 and 33 TeV.

MSTW2008 [78] (with 68cl) and NNPDF2.3 [79] (with αs(MZ) = 0.118 and nf= 5) NLO

and NNLO pdf sets.

The long-term limiting factor would be the uncertainty in B-fragmentation. As a bench-

mark, we take the DELPHI measurement [80] of the ﬁrst moment of the fragmentation func-

tion hxi= 0.7153 ±0.0052, which has an uncertainty of about 0.7% (completely dominated

by systematics). Such error in hMBℓ iimplies δmt≈0.9 GeV. A future dedicated ILC run at

the Z-pole should be able to improve this measurement signiﬁcantly. Such a measurement

is likely to occur only after the end of the currently foreseen LHC operations and before the

dedicated top threshold scan during the later phases of the ILC where, for the ﬁrst time,

measurement of mtwith very high precision O(100 MeV) will be performed (see sections 4,

4.1,4.4).

The estimates for the total error are given in Table 6. The theoretical error is estimated as

follows: for LHC 8 and 14 TeV and luminosity up to 300fb−1we take the error as estimated in

Ref. [47]. For 3000fb−1at 14 TeV we assume that NNLO calculation will be available, which

will decrease the scale uncertainty by a factor of 2.5. At this point the dominant uncertainty

is the one from B-fragmentation. For LHC at 100TeV we assume that the B-fragmentation

2We thank the authors of Ref. [47] for providing us with these additional estimates.

– 10 –

uncertainty is reduced by a factor of 2 with the help of a dedicated future lepton collider

measurement.

Ref. analysis Projections

CM Energy 8 TeV 14 TeV 33 TeV 100 TeV

Cross Section 240 pb 951 pb 5522 pb 25562 pb

Luminosity 20fb−1100fb−1300fb−13000fb−13000f b−13000f b−1

Theory (GeV) - 1.5 1.5 1.0 1.0 0.6

Stat. (GeV) 7.00 1.8 1.0 0.3 0.1 0.1

Total - 2.3 1.8 1.1 1.0 0.6

Total (%) - 1.3 1.0 0.6 0.6 0.4

Table 6. Extrapolations based on the J/Ψ method.

3.5 Top mass determination from kinematic distributions

The top quark mass can be extracted from σtot. The advantage of this method is that a

mass is obtained in a rigorously deﬁned mass scheme. The D0 experiment has attempted

this approach [81]. Preliminary results have been presented by both the ATLAS and CMS

Collaborations. The uncertainty on the extracted top quark mass amounts to approximately

3%. Although the recently derived NNLO result [73] has not yet been fully utilized in this

regard (however see Ref. [61]), signiﬁcant future improvements within this approach are un-

likely given that the uncertainty in σtot at present arises from a number of competing sources

[82]. Ultimately the potential of this method is expected to be limited by the relatively small

sensitivity of the cross section with respect to the top quark mass.

Kinematic diﬀerential distributions oﬀer improved sensitivity to mt. Ref. [83] suggested

mtextraction from the invariant mass distribution of t¯

tpairs produced in events in association

with a hard jet. The sensitivity is improved well beyond what can be achieved with the

total cross-section. The authors claim that uncertainties related to uncalculated higher order

corrections or uncertainties in the parton distribution functions are expected to aﬀect the mass

measurement by less than 1 GeV. The impact of top decays and experimental uncertainties

- evaluated in a generic detector simulation - is also expected to be sub-GeV.

The extraction of mtfrom leptonic kinematic distributions in dilepton events [68] is

less aﬀected by MC modeling and non-perturbative corrections, thus reducing an important

source of uncertainty in the current top mass extractions. The only currently available study

of mtextraction from dilepton events has been performed for LHC 14 TeV in Ref. [47] where

the authors ﬁnd the possibility for extracting mtwith precision of about 1.5 GeV. Such a

precision is similar to the one from the J/Ψ method. Further exploration of the systematics

in this method is needed and studies are currently underway [68].

– 11 –

4 Top mass determination at lepton colliders

Current theoretical understanding of top quark threshold production at lepton colliders sug-

gests (see sec. 4.1 below) that it is feasible to determine the top quark mass with a precision

of about 100 MeV, the top quark width with a precision of about 40 MeV and the top quark

Yukawa coupling with a precision of about 50%. Such a precision is substantially higher than

the ultimate precision expected at hadron colliders.

Several proposals for lepton colliders – mainly linear e+e−colliders – have been put

forward so far. The International Linear Collider (ILC [84]) is a e+e−machine based on

superconducting radio-frequency cavities. The Compact Linear Collider (CLIC [85]) has

drive beam scheme capable of operating at multi-TeV energies. Both ILC and CLIC are

expect to collect 100 fb−1after only few months of operation. A circular e+e−collider with

a circumference of approximately 80-100 km could also reach the t¯

tproduction threshold

(TLEP [86]). Research and Development towards a muon collider is also ongoing [87].

The most promising method for high-precision extraction of the top quark mass is through

a scan of the t¯

tproduction threshold [88]. The authors of Ref. [89] ﬁnd that a 4-parameter ﬁt

including the top quark mass and width, the strong coupling constant and the top Yukawa

coupling can yield a statistical precision of several tens of MeV on the top quark mass. Calcu-

lations of the production cross-section in the threshold region [31,90,91] have since reached

a precision of few percent. The potentials of ILC and CLIC have been revisited [92] with

realistic luminosity spectra for both machines, a detailed simulation of the detector response

and an evaluation of the dominant systematic uncertainties. Assuming total integrated lu-

minosity of 100 fb−1, statistical uncertainty of 34 MeV on the (1S) top quark mass when

extracted from a 10-step threshold scan was found there.

Top quark mass measurements can also be performed at center-of-mass energies away

from threshold. Above threshold (i.e. for √s > 2mt) the top mass extracted from the invariant

mass distribution of the reconstructed top quark decay products has excellent statistical

precision; Ref. [92] quotes statistical uncertainty of 80 MeV combining the events collected

in the semi-leptonic and fully hadronic decay channels for 100 fb−1at √s= 500 GeV.

The rate for single top production (e+e−→t¯

bW −and the charge conjugate process)

depends strongly on the top quark mass for √s < 2mt. The cross-section for this process

is very small (less than a femtobarn for √sbelow 300 GeV). Given the likely prospect that

a future ILC will be operating for several years at energy around 250 GeV before any top

threshold measurement can be done, an exhaustive study of the possibilities for top mass

determination below threshold is highly desirable.

4.1 Theory of t¯

tproduction near threshold at e+e−colliders

The dynamics of top pair production at threshold is controlled mainly by two opposing eﬀects.

Firstly, due to the strong interactions, the non-relativistic quark–antiquark pair tends to form

a series of Coulomb-like bound states below threshold (“toponium”). Secondly, due to the

weak interactions, the large decay width of the top quark (which is comparable to its Coulomb

– 12 –

E (GeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-6 -4 -2 0 2 4 6

(a) (b)

Figure 1. (a) Typical threshold behavior of the (normalized) total e+e−→t¯

tcross-section with

Coulomb and ﬁnite width eﬀects taken into account, E=ps−4m2

t. (b) Realistic simulations of

various observables for the t¯

tthreshold production in e+e−collisions with the beam eﬀects taken

account [89]. Sensitivity to the top quark mass is indicated with the diﬀerent symbols denoting

200 MeV steps in top mass.

binding energy) smears out the sharp would-be resonances in the cross-section. The interplay

of these two eﬀects leaves a single well-pronounced peak at √sr es ≈2mtwhich roughly

corresponds to the would-be toponium ground state (see ﬁg. 1a).

The expression for the resonance cross-section, σres ∼α3

s/(mtΓt), reveals strong depen-

dence on the top quark mass and width as well as on the strong coupling constant. Since

Γt≫ΛQCD , the top quark decays well before it hadronizes, i.e. the top quark width serves

as an infrared cutoﬀ which makes the process perturbative in the whole threshold region

[16–18]. With non-perturbative eﬀects fully under control, perturbative QCD gives a reliable

theoretical description of the t¯

tthreshold production.

The accuracy of the approximation for σres is limited mainly by its convergence, i.e.

by the number of known terms in its perturbative expansion. Systematic calculation of the

higher-order corrections in heavy quarkonium systems is based on the non-relativistic eﬀective

theory of (potential) NRQCD [93–95] which involves simultaneous expansions in the strong

coupling constant and in the heavy quark velocity. The perturbative analysis has been pushed

up to the NNLO by several groups [22]. The NNLO corrections to the cross-section turned

out to be huge despite the renormalization group suppression of the strong coupling at the

characteristic mass scales.

A few conjectures have been made relating the slow convergence of the perturbation the-

ory to the infrared renormalon contribution to the top quark pole mass, and to the corrections

enhanced by powers of the logarithms of the heavy quark velocity in the case of the cross-

section. Estimates of the missing higher order corrections have been done based on these

– 13 –

NLO

NNLO

N3LO

µ (GeV)

Ep.t.

t (GeV)

S=1

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8

10 15 20 25 30 35 40 45 50

LO

NLO

NNLO

NNNLO

20

40

60

80

100

120

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Μ@GeVD

(a) (b)

Figure 2. Structure of perturbative expansion for (a) the resonance energy counted from the threshold

[96] and (b) the (normalized) resonance cross-section [97] in e+e−→t¯

t. Subsequent approximations

are plotted as functions of the strong coupling normalization scale. The shaded area represent the

uncertainty due to yet unknown three-loop Wilson coeﬃcient.

assumptions. In particular, the use of various “threshold” or “short-distance” mass param-

eters free of infrared renormalon have been suggested in order to improve the convergence

of the series for the resonance energy [22]. As it turns out, however, complete control over

the N3LO corrections is ultimately necessary for a rigorous quantitative analysis of threshold

production. Signiﬁcant progress has been achieved in this ﬁeld [96,98–109] and the main

results are reviewed below.

4.2 Resonance energy and top quark mass determination

The total O(α3

s) correction to the leading order toponium ground state energy has been

obtained in [96]. The renormalon, logarithmic, and “generic” third order contributions turn

out to be comparable in magnitude with no particular contribution saturating the total result.

As shown in ﬁg. 2a, the third order correction stabilizes the series in the pole mass scheme

and considerably reduces the scale dependence.

The numerical analysis of Ref. [96] produces a simple relation between the resonance

energy and the top quark pole mass

√sres =1.9833 + 0.007 mt−174.3 GeV

174.3 GeV ±0.0009×mt,(4.1)

including the eﬀect from the ﬁnite top quark width and the uncertainties in αs(MZ) =

0.118 ±0.003 and from unknown high-order terms. This corresponds to a theoretical uncer-

tainty of about 80 MeV in the extracted pole mass. The use of a threshold mass parameter

may apparently further reduce the error; for example, an uncertainty of 40 MeV in the deter-

mination of the “conventional” short-distance MS mass mt(mt) is quoted in [110]. However,

– 14 –

the N3LO analysis requires the O(α4

s) perturbative relation between the pole and the MS

mass (currently known to O(α3

s) [11,12]) and one has to rely on an assumption about the

structure of the corresponding perturbative series [111], which may introduce an additional

uncertainty. The calculation of the four-loop mass relation is, therefore, crucial for the deter-

mination of the short-distance mass with such an accuracy. At the same time the pole mass

is a natural parameter for the description of the invariant mass distribution of the top quark

decay product. The comparison of the values extracted from the invariant mass distribution

and from the threshold energy scan may give a realistic estimate of the experimental and

theoretical uncertainties.

4.3 Threshold cross-section

The evaluation of the threshold cross-section through N3LO is one of the most challenging

problems of perturbative QCD. Currently the bulk of the third order corrections to the

threshold cross-section is available [98,100–107] with only a few Wilson coeﬃcients still

missing. The analysis is likely to be completed in the nearest future.

The structure of the perturbative series for the cross-section is shown in Fig. 2b. As in the

case of the resonance energy, the third order correction stabilizes the series and the accuracy

of the N3LO approximation is likely to be about 3%, or even better. Further reduction of

the renormalization scale dependence may be achieved by resummation of the higher order

logarithmically enhanced corrections through eﬀective theory renormalization group methods

[112–114]. At this level of accuracy the electroweak eﬀects become important. A consistent

treatment of the top quark ﬁnite lifetime beyond the resonance approximation has been

obtained through N2LO [32,33]. The one-loop electroweak corrections to the cross-section

have been considered in [115,116]. Besides the total cross-section, diﬀerential observables

including forward-backward asymmetry and the top quark momentum distribution are known

through NNLO up to non-factorizable eﬀects in the top quark ﬁnite lifetime [8,117].

4.4 Threshold t¯

tproduction at e+e−colliders: experimental simulations

Realistic simulations of the t¯

tthreshold production have been performed in [89]. This study

assumes a 9-point energy scan around the t¯

tthreshold where the nominal center-of-mass en-

ergy is varied between 346 GeV and 354 GeV, in 1 GeV steps, with an additional energy point

taken well below threshold to measure the background. The assumed integrated luminosity

per energy point is 30 fb−1, for a total of 300 fb−1used in the full scan. This simulation

takes into account the experimental uncertainties related to the detector eﬀects, event selec-

tion eﬃciency, and the statistics, as well as an estimated theoretical uncertainty of 3% in the

normalization of the cross-section.

At each energy point, three observables are considered: the total cross-section, the peak

of the top quark momentum distribution, and the forward-backward asymmetry. The simula-

tions show the total cross-section to have an estimated experimental error of about 3%, much

below the one of the diﬀerential observables. No theoretical uncertainties on the diﬀerential

– 15 –

observables have been taken into account yet. The results of the simulated scan for these

three observables are shown in ﬁg. 1b.

As it can be appreciated, the beam energy spread, bremsstrahlung and beamstrahlung

signiﬁcantly smear the measured cross-section and the precise determination of the (machine-

dependent) luminosity spectrum is crucial for the reconstruction of the actual energy depen-

dence of the cross-section from the threshold scan. A multi-parameter ﬁt including the top

quark mass, top quark width and top quark Yukawa coupling is performed considering simul-

taneously the three observables mentioned above. The strong coupling constant αs(MZ) is

used as an input value with an assumed uncertainty of ±0.001. The resulting uncertainties

on the top quark mass and width are 31 MeV and 34 MeV, respectively. Note that these

estimates do not account for any uncertainties on the nominal beam energy or the luminosity

spectrum, which must be accurately known [118].

More recent studies have evaluated the potential precision on the top quark mass consid-

ering realistic luminosity spectra generated with the GuineaPig [119] program. In particular,

Ref. [120] reports a detailed evaluation of the sensitivity of the top quark mass measurement

to the ILC accelerator parameters. The nominal ILC parameters (Nominal) are compared

to two alternative machine parameter known as LowQ and LowP, that have reduced and

increased beamstrahlung, respectively. Reference [92] has compared the top quark mass ex-

traction form the threshold scan using luminosity spectra of the (nominal) ILC and CLIC,

where beamstrahlung plays a more important role.

[GeV]s

340 345 350 355 360

[pb]

t

t

σ

0

0.2

0.4

0.6

0.8

1Bare TOPPIK

Nominal

LowQ

LowP

Figure 3. Top quark pair production cross-section in e+e−scattering near the t¯

tthreshold. The

NNLO prediction based on the TOPPIK program [8], not including beam eﬀects, is shown as the

dashed line. Also shown are the predicted cross sections after convolution of the beam eﬀects (beam

energy spread, bremsstrahlung and beamstrahlung) corresponding to three diﬀerent sets of ILC accel-

erator parameters (see text for details).

As an example, Figure 3shows the bare t¯

tthreshold as a function of centre of mass energy

near threshold, as well as the eﬀective cross-sections after convolution with the total lumi-

– 16 –

nosity spectrum, for the Nominal,LowQ and LowP ILC machine parameters. The eﬀective

luminosity of the machine is clearly reduced due to the combined eﬀects of bremsstrahlung,

beamstrahlung and energy spread. The impact on the sensitivity is rather small: the statis-

tical uncertainty on the top quark mass extracted at CLIC, with a very substantial increase

in the level of beamstrahlung level, is degraded by a few MeV with respect to the ILC [92].

An accurate knowledge of the eﬀect on the shape of the cross section in the threshold region

is however required to avoid a large systematic contribution to the extracted mass. While

bremsstrahlung can be accurately predicted, the impact of beamstrahlung and beam energy

spread (a much smaller contribution to the luminosity spectrum) must be determined exper-

imentally. A detailed study [121] has been performed on how to reconstruct the luminosity

spectrum from Bhabha events measured with the tracking detectors and calorimeters, taking

into all relevant theoretical and experimental eﬀects. This study shows that, in the context of

the CLIC accelerator at √s= 3 TeV, the luminosity spectrum can be reconstructed to better

than 5% between the nominal and about half the nominal centre-of-mass energy. Pending a

precise estimate of the resulting systematic uncertainty on the top quark mass measurement,

a conservative 50 MeV uncertainty based on early studies is assumed here.

The uncertainty on the nominal beam energy contributes a further systematic uncertainty.

Recent studies in the context of the ILC [122] suggest that beam energy resolutions of 10−4

should be readily achievable. Therefore, the uncertainty in √sres /2 induced from the beam

energy measurement is assumed to be 35 MeV and independent of luminosity and machine

parameter sets.

In summary, for a 300 fb−1threshold scan, the total expected uncertainty on the top quark

mass is ∼100 MeV, resulting from the sum in quadrature of the following contributions: a

statistical uncertainty of order 30 MeV (from Ref. [89], conﬁrmed to be possible also with

100 fb−1from a 2-parameter ﬁt in a recent study in Ref. [92]), 35 MeV (beam energy),

50 MeV (luminosity spectrum) and 80 MeV (from the conversion of sres into mtaccording

Eq. 4.1). Given the dominance of systematic uncertainties, it should be possible to reduce

the integrated luminosity used in the threshold scan without signiﬁcantly degrading the total

uncertainty.

4.5 Top quark mass from a reconstruction of the top decay products

At an e+e−collider the top quark mass can also be measured via reconstruction in the

continuum, following approaches similar to those being pursued at the Tevatron and the LHC.

One could a priori hope that the cleaner environment at an e+e−collider would allow smaller

systematic uncertainties and thus improve upon the measurements from hadron colliders.

Full simulation studies on the top quark mass via direct reconstruction at an e+e−collider

have been carried out in both the fully hadronic (e+e−→t¯

t→q¯qbq ¯qb) and semi-leptonic

(e+e−→t¯

t→ℓνbq¯qb) decay channels [123–125]. These studies have shown that statistical

uncertainties on the top quark mass below 100 MeV per decay channel are possible assuming

an integrated luminosity of 100 fb−1at √s= 500 GeV. A similar statistical uncertainty is

obtained for the measurement of the top width.

– 17 –

Similarly to the case of hadron colliders, systematic uncertainties are again expected to

be the limiting factor. At present only limited information on the anticipated experimental

and theoretical systematic uncertainties at an e+e−collider exists. Nevertheless, it is possible

to obtain a rough lower limit on the total systematic uncertainty. The expected uncertainty

due to fragmentation/hadronization modeling is ∼250 (400) MeV in case of the semi-leptonic

(fully hadronic) decay channel [126]. Reconnection eﬀects in the ﬁnal state could contribute

uncertainties at the level of few hundred MeV. Preliminary studies suggest that Bose-Einstein

correlations could contribute an uncertainty of ∼100−250 MeV [126], while color reconnection

eﬀects could also lead to an uncertainty of O(100) MeV [127]. Finally, there is a theoretical

uncertainty in the relation between the maximum of the invariant mass distribution and the

mass parameter in the QCD Lagrangian.

It would be desirable to update these estimates taking advantage of the most recent

developments in both event generators and experimental techniques for in situ constraining

systematic uncertainties at hadron colliders. Taking into account all these contributions, and

the fact that we have not considered experimental systematic uncertainties (e.g. jet energy

calibration), it is diﬃcult to imagine that the total systematic uncertainty would be less than

(∆mt)syst ∼500 MeV, completely dominating this measurement. Thus the threshold scan

clearly beats the direct reconstruction of the top quark mass in precision. The latter, however,

can be used for additional control of systematic uncertainty in the threshold measurements.

5 Conclusions

In the course of the 2013 Snowmass process, and during the preparation of this document,

we have analyzed the theoretical and experimental aspects of the problem of top quark mass

determination. We have reached the following conclusions that reﬂect the past developments

and future prospects in this ﬁeld:

•Need for precision in mtdetermination. The current precision with which mtis

known, δmt.1 GeV [6,7], is already impressive; indeed the EW precision tests [1] are

currently limited by the uncertainty in mWrather than in mt. Nonetheless, motivation

for increased precision may come from cosmology [4,5], more fundamental issues in

particle physics [2,3], or a discovery of beyond the Standard Model physics at the

LHC.

We estimate that some methods for top mass determination at the LHC might lead to

top mass extraction with uncertainty as low as 500-600 MeV. Delivering such precision

at the LHC will, however, be challenging and it remains to be seen if it can be achieved

in practice. In the meantime, the most pressing issue is the relationship between the top

quark mass measured at hadron colliders and a well-deﬁned quark mass. Meaningful

improvement in the precision will therefore likely require the application of several

current and novel experimental methods that are sensitive to diﬀerent eﬀects, and also

– 18 –

advances in the theoretical understanding of the relationship between measured and

fundamental quantities.

A signiﬁcant increase in precision, reaching δmt.100 MeV, can b e achieved at a future

lepton collider.

•A comprehensive collection of mtdetermination techniques. This paper con-

tains a comprehensive collection of top mass extraction methods for hadron colliders.

These are methods that have been used in the past, are in current use or are under

development. We discuss the salient features of each method and present estimates for

the precision reach for some of them.

•Recommendations for further studies. Going beyond the methods discussed in

this paper, we point to two problems that have not been studied so far and that we

think will be playing an increasingly important role in the future.

1. The possibility of BSM “contamination” in the various top mass measurements

[55]. Both model–dependent and model–independent studies would be very useful.

2. The most precise known method for extracting mtis from a threshold scan at a

future lepton collider. At present, however, it appears that the most likely lepton

collider to be built is an ILC with a ﬁrst stage operating at c.m. energy signiﬁcantly

below the t¯

tthreshold. The current expectation is that such ﬁrst stage will be

operational for a number of years; moreover, its energy upgrade might be aﬀected

by future considerations (like funding, for example). For this reason it is important

to fully explore the possibility for top mass extraction at below–threshold energies

through, for example, single top production. Such studies are lacking at present.

Acknowledgments

Paper written within the Snowmass Energy Frontier working group HE3: Fully Understanding

the Top Quark. We would like to thank Andre Hoang, Pedro Ruiz Femenia, Andre Sailer and

Frank Simon for discussions. The work of S. Mantry is supported by the U.S. National Science

Foundation under grant NSF-PHY- 0705682. The work of A. Mitov is supported by ERC

grant 291377 LHCtheory: Theoretical predictions and analyses of LHC physics: advancing

the precision frontier.

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