aainstitutetext: Department of Physics, Oxford University, Keble Road, Oxford,
OX1 3RH, UKbbinstitutetext: Institut für Theoretische Teilchenphysik und
Kosmologie, RWTH Aachen University, D-52056 Aachen, Germanyccinstitutetext:
Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126,
Milan, Italyddinstitutetext: Cavendish Laboratory, University of Cambridge,
J.J. Thomson Avenue, CB3 0HE, Cambridge, UKeeinstitutetext: MRC Institute of
Genetics and Molecular Medicine, University of Edinburgh, Crewe Road,
Edinburgh EH4 2XU, UK

# Simultaneous extraction of $\alpha_{s}$ and $m_{t}$ from LHC $t\bar{t}$
differential distributions

Amanda M. Cooper-Sarkar b Michal Czakon c Matthew A. Lim d Alexander Mitov d,e
and Andrew S. Papanastasiou<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a joint extraction of the strong coupling $\alpha_{s}$ and the top-
quark pole mass $m_{t}$ from measurements of top-quark pair production
performed by the ATLAS and CMS experiments at the 8 TeV LHC. For the first
time, differential NNLO theory predictions for different values of the top-
quark mass are utilised for four kinematic distributions: the average
transverse momentum of the top-quark, its average rapidity and the pair
invariant mass and rapidity. The use of fastNLO tables for these distributions
allows rapid evaluation of the differential theory predictions for different
PDF sets. We consider the single differential distributions from the
experiments both separately and in combination in order to obtain the best fit
to theory. Our final values are $\alpha_{s}=0.1159^{+0.0013}_{-0.0014}$ and
$m_{t}=173.8^{+0.8}_{-0.8}~{}\mathrm{GeV}$ which are compatible with previous
extractions using top-quark measurements. In the case of $m_{t}$, our value is
also compatible with the world average value collated by the Particle Data
Group.

###### Keywords:

QCD, Hadronic Colliders, NNLO, top quarks, strong coupling, top-quark mass

††preprint:  CAVENDISH-HEP-20/12 TTK-20-31 P3H-20-051

##  1 Introduction

The strong coupling constant $\alpha_{s}$ and the top quark mass $m_{t}$ are
among the most important parameters of the Standard Model (SM). The former is
present in every perturbative calculation as the expansion parameter while the
latter plays an important rôle in governing the stability of the electroweak
vacuum Degrassi:2012ry ; Branchina:2014usa ; Bezrukov:2014ina ;
Markkanen:2018pdo  and enters into calculations of important SM backgrounds to
LHC processes such as single and pair top-quark production. Despite its
importance, the value of $\alpha_{s}$ is one of the least well-constrained
parameters in the SM and is a dominant uncertainty in many precision
calculations. The measured value of the top-quark mass is also accompanied by
a large uncertainty and is subject to assumptions made by event generators.

In order to minimise the size of the uncertainties on these parameters, it is
vital that extractions use theoretical predictions at the highest available
accuracy – for many important processes, this means including next-to-next-to-
leading order (NNLO) QCD corrections. Several extractions of $\alpha_{s}$ have
been performed using NNLO theory already. The CMS collaboration has performed
an extraction using measurements of the total $t\bar{t}$ cross section at 7
TeV Chatrchyan:2013haa  and 13 TeV Sirunyan:2018goh , while ATLAS has taken a
similar approach using 13 TeV measurements in ref. Aad:2019hzw . In ref.
Klijnsma:2017eqp , the authors considered a variety of measurements of
$\sigma_{t\bar{t}}$ at the LHC and the Tevatron compared to the NNLO
prediction in order to obtain their value. NNLO calculations of deep inelastic
scattering have also been compared to HERA data to extract $\alpha_{s}$ in
refs. Andreev:2017vxu ; Britzger:2019kkb , while CMS have used measurements of
the inclusive Drell-Yan process in refs. dEnterria:2019aat ; Sirunyan:2019crt
.

Measurements of $m_{t}$, on the other hand, may proceed either by direct
reconstruction of the top-quark decay products or by examination of the total
cross section or kinematic distributions for processes featuring top-quarks in
the final state. Differential/total cross section extractions require NNLO
calculations with $m_{t}$ dependence which are readily available for the total
inclusive cross section Czakon:2011xx  but are not yet accessible for
differential distributions. For this reason, in the past only the NNLO cross
section has been utilised. One of the goals of this paper is to produce
differential predictions with sufficiently flexible $m_{t}$ dependence.

Extractions using measurements of $\sigma_{t\bar{t}}$ have been performed in
refs. Sirunyan:2018goh ; Aad:2014kva ; Khachatryan:2016mqs . In addition, NLO
extractions using measurements of the invariant mass distribution
$m_{t\bar{t}}$ by CDF, D0, ATLAS and CMS appear in refs. Tevatron:2014cka ;
Aaboud:2018zbu ; Sirunyan:2018mlv . An approach which does use NNLO
differential information has been presented in ref. Yuan:2020nzd , where
single top-quark production has been studied.

In this work, we perform a simultaneous extraction of the parameters
$\alpha_{s}$ and $m_{t}$ using measurements of $t\bar{t}$ production from
ATLAS and CMS. Such an extraction cannot be performed via knowledge of the
total inclusive cross section alone, since an increase in the value of
$\alpha_{s}$ can be compensated by an increase in the value of $m_{t}$. It is
therefore impossible to meaningfully constrain both parameters simultaneously.
However, the non-trivial dependence on $\alpha_{s}$ and $m_{t}$ of the shapes
of kinematic distributions can provide the additional information required to
make this feasible. In ref. Sirunyan:2019zvx , the CMS collaboration exploit a
triple differential $t\bar{t}$ distribution measured at 13 TeV to
simultaneously fit $\alpha_{s}$ and $m_{t}$ by comparing to NLO predictions. A
simultaneous fit of $\alpha_{s}$, $m_{t}$ and PDFs is also performed. We take
a similar approach here, exploiting measurements of single differential
distributions by ATLAS and CMS at 8 TeV but for the first time performing the
extraction using differential NNLO QCD predictions. In this way we obtain full
information about possible correlations between the parameters.

The paper is arranged as follows. In section 2 we detail the experimental
measurements and theoretical calculations which we have used as input to this
analysis. In section 3, we describe how we extract the parameters from the
measured data, considering both single differential distributions and
combinations thereof. We present our results in section 4 and discuss them in
detail in section 5. Finally, we present some general conclusions and
opportunities for future improvement in section 6.

##  2 Differential $t\bar{t}$ production at 8 TeV LHC

###  2.1 Experimental measurements

In this work we consider ATLAS and CMS $t\bar{t}$ measurements at 8 TeV of the
total cross section Aad:2014kva ; Khachatryan:2016mqs  as well as of kinematic
distributions Aad:2015mbv ; Khachatryan:2015oqa . The differential
measurements we exploit are those of the invariant mass and rapidity of the
top-quark pair system, $m_{t\bar{t}}$ and $y_{t\bar{t}}$, and the average
transverse momentum and average rapidity of the top or antitop-quark,
$p^{t}_{T}$ and $y_{t}$. ATLAS and CMS used a set of common bin edges,
catalogued in table 1, for their measurements of these distributions. The
experimental results include both statistical and systematic uncertainties and
they enter our definition of the $\chi^{2}$ objective (see section 3). We make
use of the covariance matrices provided by ATLAS and CMS Aad:2015mbv ;
Khachatryan:2015oqa , as per the approach taken in ref. Czakon:2016olj .

Observable | Bin edges
---|---
$p^{t}_{T}$ | {0, 60, 100, 150, 200, 260, 320, 400, 500} GeV
$m_{t\bar{t}}$ | {345, 400, 470, 550, 650, 800, 1100, 1600} GeV
$y_{t}$ | {-2.5, -1.6, -1.2, -0.8, -0.4, 0.0, 0.4, 0.8, 1.2, 1.6, 2.5}
$y_{t\bar{t}}$ | {-2.5, -1.3, -0.9, -0.6, -0.3, 0.0, 0.3, 0.6, 0.9, 1.3, 2.5}
Table 1: ATLAS and CMS common bin edges for measurements of $p^{t}_{T}$,
$m_{t\bar{t}}$, $y_{t}$ and $y_{t\bar{t}}$ at 8 TeV.

###  2.2 Theoretical predictions

To quantitatively compare against experimental measurements and extract
parameters, theoretical predictions of the measured quantities are required at
as high precision as possible. For the total $t\bar{t}$ cross section we use
$\sigma^{\rm theory}(\alpha_{s},m_{t})=\sigma^{\rm
NNLO+NNLL}_{t\bar{t}}(\alpha_{s},m_{t})$, i.e. the NNLO-QCD fixed-order
prediction supplemented with soft-gluon resummation Cacciari:2011hy ;
Baernreuther:2012ws ; Czakon:2012zr ; Czakon:2012pz ; Czakon:2013goa . The
fast computation of this is provided through top++ Czakon:2011xx . For the
total cross section we set the renormalisation and factorisation scales to the
common scale $\mu=\mu_{R}=\mu_{F}=m_{t}$. The differential cross section for
this process is known up to complete NLO including NNLO QCD and resummation
effects Czakon:2017wor ; Czakon:2019txp  – in this work we use only the NNLO
QCD predictions $d\sigma^{\rm theory}(\alpha_{s},m_{t})=d\sigma^{\rm
NNLO}_{t\bar{t}}(\alpha_{s},m_{t})$ Czakon:2015owf ; Czakon:2016ckf ;
Czakon:2016dgf  computed using the Stripper framework Czakon:2010td ;
Czakon:2014oma ; Czakon:2019tmo .

Throughout this work we define $m_{t}$ in the pole mass scheme. NNLO
differential predictions for top-quark pair production in the
$\overline{\mathrm{MS}}$ mass scheme have recently been made available in ref.
Catani:2020tko , following the earlier implementation of the process at NNLO
in the $q_{T}$ subtraction framework Catani:2019hip . The renormalisation and
factorisation scales chosen are those found to be optimal in ref.
Czakon:2016dgf , namely,

$\displaystyle\mu_{R}$
$\displaystyle=\mu_{F}=H_{T}/4,\;\;\text{for}\;\;m_{t\bar{t}},\,y_{t},\,y_{t\bar{t}}\,,$
(1) $\displaystyle\mu_{R}$
$\displaystyle=\mu_{F}=M_{T}/2,\;\;\text{for}\;\;p^{t}_{T}\,.$ (2)

Normalised distributions were obtained from the corresponding absolute
distributions by dividing the weight of each bin by the sum of weights in all
bins.

Finding the best-fit values for $\alpha_{s}$ and $m_{t}$ requires theoretical
predictions made with many input values of these fundamental parameters.
Furthermore, since the value of $\alpha_{s}$ is typically taken from the
associated input PDF set used to evaluate cross sections, these predictions
must be made with multiple PDF sets. To allow for the fast computation of the
relevant differential quantities, we have made use of the fastNLO interface
Kluge:2006xs ; Britzger:2012bs ; Czakon:2017dip , producing tables that allow,
for a fixed set of observables and binnings and a fixed input top-quark mass,
$\mathcal{O}(1\;{\rm second})$ evaluations of the NNLO differential cross
section with any desired PDF set. We have produced fastNLO tables for the
distributions and binnings given in table 1 and for the following values of
the top-quark mass:

$\displaystyle m_{t}=\\{169.0,171.0,172.5,173.3,175.0\\}\;\text{GeV}.$ (3)

The fastNLO tables for these values of $m_{t}$ are available for download from
ref. website . Predictions for different input $\alpha_{s}$ values are
obtained by convolving the fastNLO tables with PDF sets of the same family
which have been fit using different values of $\alpha_{s}$.

####  2.2.1 Choice of input PDFs

In order to make an extraction of $\alpha_{s}$ we require a PDF family to have
PDF sets that have been fit with different values of $\alpha_{s}$. To
construct a robust fit of the theoretical predictions (see section 2.2.2) it
is preferable that the PDF family contains at least three values of
$\alpha_{s}$. The PDFs considered in this study (all NNLO fits) are the CT14
Dulat:2015mca , NNPDF3.0 Ball:2014uwa  and NNPDF3.1 Ball:2017nwa  sets, which
come with 13, 5 and 11 values of $\alpha_{s}$ respectively.

Of these sets, only CT14 was fit without top-quark data. NNPDF3.0 has used
measurements of the total top-pair cross section at the LHC, whilst NNPDF3.1
additionally includes differential measurements, namely the ATLAS $y_{t}$ and
CMS $y_{t\bar{t}}$ distributions at 8 TeV.

One might be concerned that the presence of top-quark data in the PDF may bias
the extraction of $m_{t}$. In order to estimate the impact this may have we
have considered various PDF sets containing varying degrees of top-quark data.
As we shall see, it appears that at least at present this is not a very large
effect in most cases. We shall, however, comment in some detail on the
inclusion of differential top-quark data in NNPDF3.1 in section 5. Even if no
top-quark data were included in the PDFs at all, one should recall that
certain bias is still present in the experimental data where modelling with
Monte Carlo codes introduces some $m_{t}$ dependence. A remedy to this problem
may be to produce PDF sets with various values of $m_{t}$, treating it in a
manner similar to $\alpha_{s}$, but such sets are not available at present.
Similarly, there are subtleties in the extraction of $\alpha_{s}$ from PDF
sets with fixed values of $\alpha_{s}$. This has been discussed in refs.
Ball:2018iqk ; Forte:2020pyp .

We have also considered PDF sets from the MMHT family Harland-Lang:2014zoa .
However, since the sets do not pass our internal consistency checks when
fitting theory predictions (discussed in the following section), we do not
present the corresponding extractions.

####  2.2.2 Two-dimensional interpolation in $\alpha_{s}$ and $m_{t}$

To aid the parameter extraction via minimisation of a $\chi^{2}$ objective
function, for each PDF family (CT14, NNPDF3.0, NNPDF3.1) it is convenient to
move from a discrete grid of predictions in $\alpha_{s}$ and $m_{t}$ to a
smooth two-dimensional surface. This is achieved by fitting functions to the
available sets of $(\alpha_{s},m_{t})$ points, both for the total cross
section as well as for the weight of each bin of every distribution. In the
one-dimensional case we fit functions in $\alpha_{s}$ or in $m_{t}$, while in
the two-dimensional case the fit takes a factorised form as the product of a
function of $\alpha_{s}$ with a function of $m_{t}$.

The total cross section is fit as a function of $\alpha_{s}$ using polynomials
of order 1, 2 and 3 for the NNPDF3.0, CT14 and NNPDF3.1 sets respectively. We
use the functional form of mass dependence recommended in ref. Czakon:2013goa

$\displaystyle\sigma^{\rm theory}_{i}(m)=\sigma^{\rm
theory}_{i}(m_{\mathrm{ref}})\left(\frac{m_{\mathrm{ref}}}{m}\right)^{4}\left(1+a_{1}\frac{(m-m_{\mathrm{ref}})}{m_{\mathrm{ref}}}+a_{2}\frac{(m-m_{\mathrm{ref}})^{2}}{m_{\mathrm{ref}}^{2}}\right)\,,$
(4)

where we take $m_{\rm ref}=172.5$ GeV as a reference value for the top-quark
mass.

The functional forms of the fits to the differential distributions are chosen
based on the number of available points in $m_{t}$ and $\alpha_{s}$ (the
latter depending on the PDF set used) and based on the quality and stability
of the interpolation and extrapolation. To parametrise the $\alpha_{s}$
dependence of each bin, we choose a polynomial of order 4 for the CT14 and
NNPDF3.1 sets and a linear fit for the NNPDF3.0 set. 111Since only five values
of $\alpha_{s}$ are available for NNPDF3.0, we impose a linear dependence to
avoid overfitting. In this way, we can fit the predictions to within $3\sigma$
for the rapidity distributions and to within $2\sigma$ for the $p^{t}_{T}$ and
$m_{t\bar{t}}$ distributions, where $\sigma$ refers to the MC error of the
NNLO calculation. Note that the NNLO MC error is negligible compared to the
NNLO scale uncertainty. For the mass dependences of the distributions, we use
the forms

$\displaystyle w_{m_{t\bar{t}}}^{i}(m)$
$\displaystyle=w_{m_{t\bar{t}}}^{i}(m_{\mathrm{ref}})\left(\frac{m_{\mathrm{ref}}}{m}\right)^{4}\left(1+a_{1}\frac{(m-m_{\mathrm{ref}})}{m_{\mathrm{ref}}}+a_{2}\frac{(m-m_{\mathrm{ref}})^{2}}{m_{\mathrm{ref}}^{2}}\right.$
$\displaystyle\left.\hskip
201.16118pt+a_{3}\Big{(}1-\frac{1}{\cosh^{6}(m-m_{\mathrm{ref}})}\Big{)}\right)\,,$
(5) $\displaystyle w_{p^{t}_{T}}^{i}(m)$
$\displaystyle=w_{p^{t}_{T}}^{i}(m_{\mathrm{ref}})\left(\frac{m_{\mathrm{ref}}}{m}\right)^{4}\left(1+a_{1}\frac{(m-m_{\mathrm{ref}})}{m_{\mathrm{ref}}}+\frac{a_{2}}{m}\right)\,,$
(6) $\displaystyle w_{y}^{i}(m)$
$\displaystyle=w_{y}^{i}(m_{\mathrm{ref}})\left(1+a_{1}e^{-a_{2}(m-m_{\mathrm{ref}})}\right)\,,$
(7)

where $w^{i}(m)$ is the weight of the $i^{\mathrm{th}}$ bin and we again take
a reference mass $m_{\mathrm{ref}}=172.5$ GeV. The same functional form is
used for both rapidity distributions $y_{t}$ and $y_{t\bar{t}}$. The
coefficients $a_{j}$ are fit independently for each distribution and PDF set.

Although for the extractions in section 4 we have consistently used the two-
dimensional fits (projecting out one dimension as required for the one-
dimensional extractions of $\alpha_{s}$ or $m_{t}$), performing also the one-
dimensional fits independently acts as a consistency check on our fitting
procedure. In this way we can verify that, by fixing either $m_{t}$ or
$\alpha_{s}$ to some value, the resulting projections of the two-dimensional
fits onto one dimension are equivalent to fitting in one dimension to within
the Monte Carlo error of the theory predictions. Some examples of the quality
of the fits can be seen in figure 1. The requirement of good quality fits and
extrapolations is one of the aspects which determined which PDF sets we have
chosen to use – our results for the CT14, NNPDF3.0 and NNPDF3.1 sets all show
very good agreement, as a function of $\alpha_{s}$ and $m_{t}$, between the
bin weight from the fit and the discrete set of the calculated points to
within Monte Carlo error.

Figure 1: Example plots showing the interpolations in $\alpha_{s}$ and
$m_{t}$. The first and second (third and fourth) plots show the interpolations
of theoretical predictions in $\alpha_{s}$ ($m_{t}$) for the second bin of
$p^{t}_{T}$ (third bin of $y_{t}$), using the CT14 or NNPDF3.0 PDF sets. The
blue points display the ratio of the computed NNLO predictions to the one-
dimensional interpolation. The vertical bars indicate the MC error of the NNLO
calculation. The red line is the projection of the corresponding two-
dimensional fit in $\alpha_{s}$ or $m_{t}$.

####  2.2.3 Dependence of differential observables on $\alpha_{s}$ and $m_{t}$

In figures 2 and 3 we show the sensitivity on $\alpha_{s}$ and $m_{t}$ of the
absolute and normalised distributions respectively, at NNLO QCD accuracy. It
is the non-trivial shape of these dependences that will allow us to perform
simultaneous extractions of $\alpha_{s}$ and $m_{t}$. The plots have been
produced using the CT14 PDF set – however, very similar patterns are observed
when using other PDF sets, in particular the size of the effects of varying
$\alpha_{s}$ and $m_{t}$. These sensitivities have previously been discussed
at NLO in refs. Czakon:2016vfr ; Czakon:2016olj .

Figure 2: The sensitivity on $\alpha_{s}$ of the $p^{t}_{T}$ (column 1),
$m_{t\bar{t}}$ (column 2), $y_{t}$ (column 3) and $y_{t\bar{t}}$ (column 4) of
the absolute (top) and normalised (bottom) distributions in NNLO QCD. All
curves are relative to the world average values of $\alpha_{s}=0.118$ and
$m_{t}=173.3$ GeV. The PDF set CT14 has been used to produce these curves. The
pattern for other PDF sets is similar.

Figure 3: As in fig. 2 but for the $m_{t}$ sensitivity.

For the absolute distributions, we observe that the sensitivity on
$\alpha_{s}$ closely follows the $\alpha_{s}$-dependence of the total cross
section $\sim\mathcal{O}(\alpha_{s}^{2})$. For all absolute distributions, the
sensitivity increases in the tails of distributions, where corrections beyond
the Born-level in the cross section will have larger effects. A similar
pattern can be observed in the $m_{t}$ sensitivity of the rapidity
distributions, $y_{t}$ and $y_{t\bar{t}}$, ie the behaviour of the weights of
each bin roughly follow the $\sim\mathcal{O}(m_{t}^{-4})$ dependence of the
total cross section, and once again the tails see slightly increased
sensitivity. On the other hand, the $p^{t}_{T}$ and $m_{t\bar{t}}$
distributions display very different sensitivities with respect to the
$\alpha_{s}$ case. The bins most sensitive to changes in $m_{t}$ are the
lowest bins in the ranges where the bulk of the cross section lies. In the
tails, however, the sensitivity is dramatically reduced – this can be
understood by the fact that in the tails the top-quark is effectively massless
(the finite-mass effects being suppressed by powers of $p^{t}_{T}$ or
$m_{t\bar{t}}$).

In the case of normalised distributions, the sensitivity plots shown in
figures 2 and 3 show a far more involved dependence on $\alpha_{s}$ and
$m_{t}$. These patterns arise because the bin-weights of the absolute
distributions are divided by the cross section over all bins. In general we
observe that, except in the tails of distributions, the effect of varying
$\alpha_{s}$ by $\pm 0.003$ around the baseline value of $\alpha_{s}=0.118$ is
at the $\sim 1$%-level or less. On the other hand, variations of $m_{t}$ by
$\pm(2-3)$ GeV around the value $m_{t}=173.3$ GeV can result in much larger
effects: for $p^{t}_{T}$ the variations in $m_{t}$ lead to changes of $\sim
2\%$ and for $m_{t\bar{t}}$ these are $\gtrsim 0-5$% in the bins with largest
weight. For $y_{t}$ and $y_{t\bar{t}}$ the effects of varying $m_{t}$ are much
smaller and fall below $0.5\%$ in the bins with largest weight. If
experimental uncertainties were negligible, then we would expect these
patterns to be reflected in the uncertainties on the extracted values of
$\alpha_{s}$ and $m_{t}$. That is, we would expect that since the magnitude of
the dependence on $\alpha_{s}$ is roughly the same across all distributions,
this should translate to similar uncertainties on values of $\alpha_{s}$
extracted from each distribution. In contrast, we would expect that the
uncertainties on the extracted values of $m_{t}$ would be largest when using
the normalised $y_{t}$ and $y_{t\bar{t}}$ data (and significantly larger than
the uncertainties obtained when using the normalised $p^{t}_{T}$ and
particularly $m_{t\bar{t}}$ data), due to the relative insensitivity on the
mass of these distributions. We indeed find evidence of such behaviour in
section 4.

####  2.2.4 Top mass systematics and the lower edge of $m_{t\bar{t}}$

Experimental measurements of the cross section and of distributions depend on
the value of the top-quark mass. This dependence arises because inclusive
measurements of stable top quarks are actually extrapolations from
measurements of top-quark decay products in fiducial regions, and these
extrapolations are derived from simulated Monte Carlo data which explicitly
carries a dependence on $m_{t}$.222The default choice of input $m_{t}$ used by
both ATLAS and CMS is $m_{t}=172.5$ GeV. Typically, this ‘experimental
response’ to the input value of $m_{t}$ is known to constitute a $\sim 1-2\%$
effect at the level of the inclusive cross section. For generic bins of a
kinematic distribution this is a similarly small effect. However, there are
kinematic regions where these effects could be significantly larger and are
perhaps currently an underestimated systematic effect. This is particularly
the case for the lowest bin in $m_{t\bar{t}}$ – for simulated stable top data,
the lowest possible value of $m_{t\bar{t}}$ is $2m_{t}$ and therefore when
using data simulated with $m_{t}=172.5$ GeV, one can never obtain values
$m_{t\bar{t}}<345$ GeV.

Considering the $m_{t\bar{t}}$ distribution, if the lowest value of
$m_{t\bar{t}}$ considered is $345$ GeV (as is done here, see table 1), then
for values of the top-quark mass $m_{t}<172.5$ GeV, the theoretical
predictions will contain events with kinematics such that $m_{t\bar{t}}<345$
GeV. In our extractions above, and consistently with what is done by the two
experiments, these ‘underflow’ events are not included. Since this lowest bin
of $m_{t\bar{t}}$ is the most sensitive to the value of $m_{t}$, this issue
deserves a careful examination. If in the extrapolation to stable-top quarks,
some underflow events do leak into the first bin of $m_{t\bar{t}}$, then this
could have serious effects on the extraction of $m_{t}$ from this
distribution. In order to study this further, we have considered how our
extractions react to the addition of underflow events to this lowest bin. For
the discrete set of values of $m_{t}$ considered, it is only the predictions
for $m_{t}=169.0$ GeV and $m_{t}=171.0$ GeV that contain underflow events, but
nevertheless, these do alter the fits of the theory predictions. In appendix C
we present extractions using the invariant mass distribution supplemented with
underflow events $m_{t\bar{t}}^{u}$ and compare these with the results
obtained using simply $m_{t\bar{t}}$.

##  3 Extraction Methodology

In order to quantitatively compare experimental measurements to theoretical
predictions we use a least-squares method. We consider two sets of
measurements: (a) measurements of normalised distributions supplemented with
measurements of the total cross section, or (b) measurements of absolute
distributions. In both cases we first include data only from a single
experiment in the $\chi^{2}$ objective (we comment later on using data from
both ATLAS and CMS). The definitions of the $\chi^{2}$ variables we use are

$\displaystyle\chi^{2}_{\rm{norm}}$
$\displaystyle=\frac{1}{N_{\mathrm{data}}}\left[\sum_{i,j=1}^{N_{\mathrm{data}}-1}\zeta_{i}C_{ij}^{-1}\zeta_{j}\quad+\frac{(\sigma_{\rm{theory}}-\sigma_{\mathrm{data}})^{2}}{\delta\sigma_{\mathrm{data}}^{2}}\right]\,,$
(8) $\displaystyle\chi^{2}_{\rm{abs}}$
$\displaystyle=\frac{1}{N_{\mathrm{data}}}\sum_{i,j=1}^{N_{\mathrm{data}}}\zeta_{i}C_{ij}^{-1}\zeta_{j}\,,$
(9)

where $\zeta_{i}=\zeta_{i}^{\mathrm{data}}-\zeta_{i}^{\mathrm{theory}}$ is the
difference between the measured and predicted weights in bin $i$ and $C_{ij}$
a covariance matrix encoding experimental sources of error and correlations
between bins $i$ and $j$ of the distributions. Our covariance matrices are
obtained from the correlation matrices made publicly available by the
experiments. In the case of the ATLAS experiment, they contain information
about all sources of systematic and statistical correlation. We were unable to
obtain the corresponding matrices from CMS—for extractions from these data
(and in cases where ATLAS and CMS data are combined) we use the so-called
experimental definition of the covariance matrix, mirroring the analysis
performed in section 3.3 of ref. Czakon:2016olj . This definition takes into
account correlated systematic uncertainties but is diagonal in the statistical
uncertainties.

In the cases where normalised shape distributions have been used the final bin
of each distribution has been removed in order to render the covariance matrix
invertible 333The choice of bin to be removed does not affect the value of the
$\chi^{2}$.. A possible advantage of using normalised distributions rather
than absolute distributions is the improved control of systematic errors on
the experimental measurements of the total cross section. There is also a
partial cancellation of uncertainties such as the luminosity which takes place
in normalising the distributions.

The sources of uncertainty included in the covariance matrices, and hence in
our extractions, are at present purely experimental. These experimental
uncertainties are statistical and systematic, also including uncertainties due
to luminosity and beam energy and the bin-by-bin correlations are either fully
(in the case of the ATLAS experiment) or partially (in the CMS case) retained.

In a more complete treatment of uncertainties, sources of theoretical
uncertainty such as those due to scale variation and PDF uncertainties should
also be included. The fact that we have not included the theoretical
uncertainties in our extraction of $m_{t}$ and $\alpha_{s}$ deserves
clarification. We justify this omission by the fact that these uncertainties,
both due to scales and PDF, are not dominant for the observables and data sets
we study here. Indeed, since the scale uncertainty is greatly reduced at NNLO
when considering normalised (compared to absolute) distributions, we do not
expect that including these would increase the uncertainties on the extracted
parameters dramatically. On the other hand, in one-dimensional extractions of
$\alpha_{s}$, the scale uncertainty on the cross section is known to
contribute a significant fraction of the overall uncertainty on the best-fit
$\alpha_{s}$ Klijnsma:2017eqp , and therefore we would expect this to also
play a rôle for our study here.

Despite the fact that the scale error itself has no direct statistical
meaning, extensive experience with collider processes shows that scale
variation could reasonably be interpreted as a proxy for a hypothetical error
estimator which is statistical in nature. One could then imagine a modified
$\chi^{2}$ function where the experimental and theoretical errors are added in
quadrature. A subtlety arises when one tries to generalise the full covariance
matrices entering the $\chi^{2}$ functions we study. The accumulated
experience in estimating theory scale errors does not automatically translate
from scale variation to the correlation between scale variations. As we
observe here, the effect of correlations between different bins or different
distributions play a very significant role and for the sake of being
conservative we have decided not to include such theory correlations. The
subject has recently been explored in ref. AbdulKhalek:2019ihb  which may
represent a starting point for improved future studies. An alternative
interpretation of the scale uncertainty in a Bayesian fashion has also been
given in ref. Bonvini:2020xeo  which may also shed some light on this issue.

We consider individual extractions of the parameters in which $m_{t}$ is fixed
at the world average $m_{t}=173.3$ GeV and we extract $\alpha_{s}$ or,
conversely, fits where we fix $\alpha_{s}=0.118$ and extract $m_{t}$. In both
of these cases, the resulting $\chi^{2}$ is one-dimensional and the best-fit
value of $m_{t}$ or $\alpha_{s}$ is that which minimises the $\chi^{2}$
objective. The uncertainties on the extracted values of the parameters are
estimated through the standard condition

$\Delta\chi^{2}\equiv N_{\mathrm{data}}(\chi^{2}-\chi^{2}_{\rm min})=1\,,$
(10)

which naturally corresponds to a 1$\sigma$ deviation from the best-fit value.

While the above procedure is considered standard practice, this could
potentially lead to a bias in the extracted value of $m_{t}$ (or $\alpha_{s}$)
or yield extracted parameters that do not minimise the $\chi^{2}$ objective in
the full $(m_{t},\alpha_{s})$ plane. We therefore extend the one-dimensional
studies to the case where both parameters, $m_{t}$ _and_ $\alpha_{s}$, are
allowed to vary independently resulting in a two-dimensional $\chi^{2}$. Once
again the best-fit value is taken to be that which minimises this two-
dimensional $\chi^{2}$. Estimating the uncertainties on the extracted values
is slightly more involved in this case, since the condition eq. (10) traces
out a contour in the $(m_{t},\alpha_{s})$ plane (in the ideal case, an
ellipse). The estimate of the uncertainties on the extracted values of
$\alpha_{s}$ and $m_{t}$ are derived from these contours by setting $m_{t}$
and $\alpha_{s}$ respectively to their best-fit value and deducing the range
corresponding to $\Delta\chi^{2}=1$ once this choice has been made.

Hitherto we have only discussed extractions from a single distribution
measured by a single experiment. Given that the shape of the dependence on
$\alpha_{s}$ or $m_{t}$ is distribution-dependent, combining information from
different distributions in an extraction could provide complementary
constraints. Indeed, the combination of several observables has been advocated
in ref. Frixione:2014ala  as an effective way of suppressing possible
theoretical biases in the determination of $m_{t}$. Therefore, by combining
distributions, it may be possible to obtain more reliable best-fit parameters
as well as smaller overall uncertainties. To this end we perform extractions
based on combinations of distributions from the ATLAS experiment, making use
of the covariance matrices provided by the collaboration to include
information about the correlations between distributions. It was not possible
to perform such an extraction in the CMS case, since such information was not
made publicly available and its omission would undoubtedly bias the results
and underestimate the extraction uncertainties. Accordingly, we make a simple
extension of our $\chi^{2}$ definitions in eqs. (8) and (9),

$\displaystyle\chi^{2}_{\mathrm{norm}}$
$\displaystyle=\frac{1}{(N-1)}\Bigg{[}\sum_{i,j=1}^{N-2}\zeta_{i,\textsc{atlas}}C_{ij,\textsc{atlas}}^{-1}\zeta_{j,\textsc{atlas}}+\frac{(\sigma_{\mathrm{NNLO}}-\sigma_{\textsc{atlas}})^{2}}{\delta\sigma_{\textsc{atlas}}^{2}}\Bigg{]}\,,$
(11) $\displaystyle\chi^{2}_{\mathrm{abs}}$
$\displaystyle=\frac{1}{N}\Bigg{[}\sum_{i,j=1}^{N}\zeta_{i,\textsc{atlas}}C_{ij,\textsc{atlas}}^{-1}\zeta_{j,\textsc{atlas}}\Bigg{]}\,,$
(12)

where $N=N_{\textsc{atlas,1}}+N_{\textsc{atlas,2}}$. The vector
$\zeta_{i,\textsc{atlas}}$ now contains data from both distributions and the
covariance matrices $C_{ij}$ contain off-diagonal entries (see also section
4.2.2).

Finally, since it is realistic to assume that to a large extent measurements
performed by _different_ experiments are independent (and therefore that
correlations between the experiments are negligible), 444The assumption of
negligible correlated uncertainties does not hold for the luminosity
uncertainty. Although non-negligible, this uncertainty is not dominant in this
analysis. we also consider the combination of a single distribution from ATLAS
with one from CMS. For these extractions we define further straightforward
extensions of our $\chi^{2}$ definitions,

$\displaystyle\chi^{2}_{\mathrm{norm}}$
$\displaystyle=\frac{1}{(N_{\textsc{atlas}}+N_{\textsc{cms}})}\Bigg{[}\sum_{i,j=1}^{N_{\textsc{atlas}}-1}\zeta_{i,\textsc{atlas}}C_{ij,\textsc{atlas}}^{-1}\zeta_{j,\textsc{atlas}}+\sum_{i,j=1}^{N_{\textsc{cms}}-1}\zeta_{i,\textsc{cms}}C_{ij,\textsc{cms}}^{-1}\zeta_{j,\textsc{cms}}$
$\displaystyle\qquad\qquad+\frac{(\sigma_{\mathrm{NNLO}}-\sigma_{\textsc{atlas}})^{2}}{\delta\sigma_{\textsc{atlas}}^{2}}+\frac{(\sigma_{\mathrm{NNLO}}-\sigma_{\textsc{cms}})^{2}}{\delta\sigma_{\textsc{cms}}^{2}}\Bigg{]}\,,$
(13) $\displaystyle\chi^{2}_{\mathrm{abs}}$
$\displaystyle=\frac{1}{(N_{\textsc{atlas}}+N_{\textsc{cms}})}\Bigg{[}\sum_{i,j=1}^{N_{\textsc{atlas}}}\zeta_{i,\textsc{atlas}}C_{ij,\textsc{atlas}}^{-1}\zeta_{j,\textsc{atlas}}+\sum_{i,j=1}^{N_{\textsc{cms}}}\zeta_{i,\textsc{cms}}C_{ij,\textsc{cms}}^{-1}\zeta_{j,\textsc{cms}}\Bigg{]}\,.$
(14)

##  4 Results

In this section we present and discuss the results of our extractions of
$\alpha_{s}$ and $m_{t}$, both separately from one-dimensional $\chi^{2}$
functions, as well as simultaneously letting both parameters vary. We focus on
the extractions using measurements of normalised distributions supplemented
with information from the total cross sections. Extractions using experimental
results from absolute distributions can be found in Appendix B. We do not
consider the rapidity distributions using NNPDF3.1, for reasons that will be
discussed in section 5.3.

###  4.1 Independent (one-dimensional) extractions of $\alpha_{s}$ and $m_{t}$

####  4.1.1 Extraction from complete distributions

The first set of extractions we present are the separate extractions of
$\alpha_{s}$ and $m_{t}$ using the one-dimensional form of our $\chi^{2}$
objective in eq. (8). The results of the $\alpha_{s}$ extraction, where the
value of top mass has been fixed to $m_{t}=173.3$ GeV, are shown in table 2.
The results of the $m_{t}$ extraction, where the value of the strong-coupling
constant has been fixed to $\alpha_{s}=0.118$, are shown in table 3.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1157^{+0.0020}_{-0.0022}$ | 0.42 | $0.1162^{+0.0027}_{-0.0027}$ | 0.50 | $0.1171^{+0.0022}_{-0.0022}$ | 0.72
$m_{t\bar{t}}$ | $0.1158^{+0.0020}_{-0.0021}$ | 0.17 | $0.1157^{+0.0027}_{-0.0027}$ | 0.19 | $0.1173^{+0.0022}_{-0.0022}$ | 0.22
$y_{t}$ | $0.1149^{+0.0018}_{-0.0019}$ | 0.91 | $0.1155^{+0.0018}_{-0.0018}$ | 0.22 |  |
$y_{t\bar{t}}$ | $0.1162^{+0.0019}_{-0.0020}$ | 2.04 | $0.1154^{+0.0025}_{-0.0025}$ | 0.57 |  |
Average | $0.1158^{+0.0014}_{-0.0015}$ |  | $0.1157^{+0.0013}_{-0.0013}$ |  | $0.1172^{+0.0016}_{-0.0016}$ |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1164^{+0.0014}_{-0.0015}$ | 5.01 | $0.1121^{+0.0026}_{-0.0026}$ | 3.62 | $0.1178^{+0.0019}_{-0.0019}$ | 2.56
$m_{t\bar{t}}$ | $0.1147^{+0.0014}_{-0.0014}$ | 9.68 | $0.1100^{+0.0014}_{-0.0043}$ | 6.97 | $0.1161^{+0.0018}_{-0.0018}$ | 6.01
$y_{t}$ | $0.1165^{+0.0019}_{-0.0020}$ | 2.36 | $0.1199^{+0.0020}_{-0.0020}$ | 2.99 |  |
$y_{t\bar{t}}$ | $0.1154^{+0.0016}_{-0.0017}$ | 2.14 | $0.1149^{+0.0017}_{-0.0017}$ | 0.89 |  |

Table 2:  Tabulated values of best-fit $\alpha_{s}$ (with uncertainties) and
associated $\chi^{2}_{\rm min}$ from extractions of $\alpha_{s}$ using ATLAS
(upper table) and CMS (lower table) measurements of normalised distributions
and the total cross section. Results are shown for three different PDF sets
and $m_{t}$ has been set to the world average value of 173.3 GeV. The cells
highlighted in grey correspond to extractions that satisfy the quality
conditions described in sec 5.1. Where shown, the average values have been
calculated from the shaded cells in the manner described in sec. 5.1. As
explained in section 5.3, the rapidity distributions are not used for
extractions with the PDF set NNPDF3.1.

Looking at table 2, we notice that for a given experiment and distribution,
there can be non-trivial differences between extracted values of $\alpha_{s}$.
For example for $p^{t}_{T}$ from ATLAS, there is a 0.0014 difference in the
best-fit values when using CT14 and NNPDF3.1, although these values are
consistent within uncertainties. Comparing the results obtained when using
measurements from the two experiments, it is clear that there are significant
differences, particularly in the values of $\chi^{2}_{\rm min}$ for
$p^{t}_{T}$ and $m_{t\bar{t}}$ using CT14 or NNPDF3.0. As detailed in ref.
Czakon:2016olj , this indicates a certain tension between the two sets of
measurements.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $174.5^{+1.0}_{-1.0}$ | 0.39 | $174.6^{+1.0}_{-1.0}$ | 0.36 | $174.6^{+1.0}_{-1.0}$ | 0.54
$m_{t\bar{t}}$ | $173.4^{+0.6}_{-0.5}$ | 0.35 | $173.4^{+0.6}_{-0.5}$ | 0.28 | $173.5^{+0.5}_{-0.5}$ | 0.21
$y_{t}$ | $175.6^{+1.2}_{-1.2}$ | 0.87 | $174.8^{+1.3}_{-1.2}$ | 0.30 |  |
$y_{t\bar{t}}$ | $175.0^{+1.3}_{-1.3}$ | 1.85 | $174.4^{+1.3}_{-1.3}$ | 0.62 |  |
Average | $174.3^{+0.9}_{-0.8}$ |  | $174.3^{+0.7}_{-0.7}$ |  | $173.9^{+0.7}_{-0.7}$ |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $170.6^{+0.6}_{-0.6}$ | 2.29 | $170.9^{+0.6}_{-0.6}$ | 1.82 | $171.4^{+0.6}_{-0.6}$ | 1.09
$m_{t\bar{t}}$ | $170.4^{+0.6}_{-0.7}$ | 7.31 | $170.5^{+0.7}_{-0.7}$ | 5.98 | $170.8^{+0.7}_{-0.7}$ | 3.87
$y_{t}$ | $173.8^{+1.1}_{-1.1}$ | 2.40 | $173.0^{+1.1}_{-1.1}$ | 3.08 |  |
$y_{t\bar{t}}$ | $176.2^{+1.3}_{-1.3}$ | 1.87 | $174.8^{+1.3}_{-1.2}$ | 1.05 |  |

Table 3: As in table 2 but for $m_{t}$ rather than $\alpha_{s}$, which has
been set to the world average value of 0.118. Masses are quoted in GeV.

Turning our attention to the top mass extractions shown in table 3, we observe
similar behaviour to the $\alpha_{s}$ case: that is, we see large differences
$>1$ GeV between the extracted values of the mass, particularly when using the
data on $m_{t\bar{t}}$ for ATLAS or $y_{t}$/$y_{t\bar{t}}$ for CMS. We also
note that the uncertainties when extracting $m_{t}$ from the $p^{t}_{T}$
distributions are roughly a factor of two larger than those obtained when
extracting using $m_{t\bar{t}}$. As a consistency check we see that the
pattern of top-mass sensitivity of figure 3 is largely reflected in the
uncertainties on the extracted values, i.e. that extractions based on
measurements of $m_{t\bar{t}}$ have smaller uncertainties than extractions
based on $y_{t}$ and $y_{t\bar{t}}$.

####  4.1.2 Extraction from individual bins of distributions

In order to scrutinise the patterns observed in the one-dimensional
extractions we perform bin-by-bin extractions of $\alpha_{s}$ and $m_{t}$ from
bins of absolute distributions and compare to the best-fit values obtained
from an extraction from all bins. This exercise serves not only as a check on
our overall extraction methodology, but also allows us to assess the
importance of the effects of correlations across different bins. In this case,
the form of the $\chi^{2}$ objective that is minimised is the same as that of
eq. (9), with the sum over all bins replaced by just the $\chi^{2}$ function
for a single bin. The results for the $\alpha_{s}$ and $m_{t}$ extractions are
shown in figures 4 and 5 respectively, where the blue points indicate the
best-fit values using just the individual bin weight, whilst the red
horizontal lines indicate the overall best-fit values (see also tables 8 and 9
in appendix B). For comparison, the world average values of $\alpha_{s}$ and
$m_{t}$ are also shown in dashed green. The results have been obtained using
theoretical predictions computed with CT14 PDFs, but very similar results are
obtained using the other PDF sets we have considered.

Figure 4: Plots showing the extraction of $\alpha_{s}$ for $m_{t}=173.3$ GeV
using CT14 for each individual bin of every distribution. Extractions based on
ATLAS and CMS _absolute_ measurements are at the top and bottom respectively,
and extractions from measurements of $p^{t}_{T}$, $m_{t\bar{t}}$, $y_{t}$ and
$y_{t\bar{t}}$ are shown in columns 1–4. The blue points show the values
extracted from the single bins with their associated errors, while the dashed
green line shows the world average value. The equivalent result from a full 1D
extraction including correlations is shown as a red line with an associated
error band.

Figure 5: As in fig. 4 but for the extraction of $m_{t}$ for
$\alpha_{s}=0.118$.

There are a few important features to identify in these plots. Firstly, we
note that, as expected, the uncertainties on the extracted values of
$\alpha_{s}$ and $m_{t}$ on the whole tend to be significantly larger for the
individual-bin extractions compared to the overall extractions. Since the
sensitivity to $\alpha_{s}$ is largest in the tails of distributions (see
figure 2), if experimental uncertainties were negligible we might expect that
uncertainties on the extracted values of $\alpha_{s}$ would decrease if we use
the weights of bins in the tails. The fact that this is not more generally
observed indicates that experimental uncertainties are not negligible, and
compete directly with the $\alpha_{s}$-sensitivity to determine the overall
size of uncertainties on the best-fit values. In contrast, the extractions of
$m_{t}$ from $p^{t}_{T}$ and $m_{t\bar{t}}$ do roughly follow expectations and
the bin-by-bin extractions show increased uncertainties for bins with low
$m_{t}$-sensitivities (those in the tails of distributions). This is due to
the fact that the largest $m_{t}$ sensitivity for the absolute $p^{t}_{T}$ and
$m_{t\bar{t}}$ distributions lies in the bins with largest cross section and
therefore smallest relative experimental uncertainties.

Another interesting feature of these plots is the difference between the
values of $\alpha_{s}$ and $m_{t}$ extracted from individual bin weights, and
the best-fit values that result from an extraction using all bins. If there
were no correlations between different bins in a particular distribution, then
the overall extracted value would lie in the range of values from the
individual-bin extractions. However, we observe that for some distributions,
for example $m_{t}$ from the CMS measurements of $m_{t\bar{t}}$, the overall
extracted values lie outside the range of individual-bin extractions. This is
a clear indication that correlations between bins (via the off-diagonal terms
in the experimental covariance matrices) can have a large effect and in some
cases shift the overall extracted value significantly. This may also reflect a
d’Agostini bias in the construction of the covariance matrices
DAgostini:1993arp .

###  4.2 Simultaneous (two-dimensional) extractions of $\alpha_{s}$ and
$m_{t}$

####  4.2.1 Extraction from single distributions

We now turn our attention to the simultaneous extraction of $\alpha_{s}$ and
$m_{t}$ from differential measurements, which uses the two-dimensional form of
our $\chi^{2}$ objective in eq. (8). The results of the joint $\alpha_{s}$ and
$m_{t}$ extractions are shown in table 4.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1171^{+0.0020}_{-0.0021}$ | $174.2^{+1.0}_{-1.0}$ | $0.38$ | $0.1188^{+0.0028}_{-0.0028}$ | $174.7^{+1.0}_{-1.0}$ | $0.35$ | $0.1210^{+0.0024}_{-0.0023}$ | $175.6^{+1.0}_{-1.0}$ | $0.43$
$m_{t\bar{t}}$ | $0.1155^{+0.0020}_{-0.0022}$ | $173.1^{+0.6}_{-0.5}$ | $0.16$ | $0.1157^{+0.0027}_{-0.0027}$ | $173.2^{+0.6}_{-0.5}$ | $0.19$ | $0.1176^{+0.0022}_{-0.0022}$ | $173.5^{+0.5}_{-0.5}$ | $0.21$
$y_{t}$ | $0.1171^{+0.0016}_{-0.0018}$ | $175.0^{+1.3}_{-1.3}$ | $0.86$ | $0.1156^{+0.0018}_{-0.0018}$ | $173.4^{+1.3}_{-1.3}$ | $0.22$ |  |  |
$y_{t\bar{t}}$ | $0.1205^{+0.0017}_{-0.0019}$ | $176.6^{+1.3}_{-1.2}$ | $1.76$ | $0.1150^{+0.0025}_{-0.0024}$ | $173.0^{+1.3}_{-1.3}$ | $0.57$ |  |  |
Average | $0.1163^{+0.0015}_{-0.0016}$ | $173.8^{+0.8}_{-0.8}$ |  | $0.1164^{+0.0019}_{-0.0019}$ | $173.6^{+0.8}_{-0.8}$ |  | $0.1192^{+0.0023}_{-0.0023}$ | $174.5^{+1.2}_{-1.2}$ |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1096^{+0.0017}_{-0.0015}$ | $169.0^{+0.6}_{-0.6}$ | $0.68$ | $0.1109^{+0.0023}_{-0.0022}$ | $170.5^{+0.6}_{-0.6}$ | $0.67$ | $0.1136^{+0.0020}_{-0.0021}$ | $170.7^{+0.6}_{-0.6}$ | $0.64$
$m_{t\bar{t}}$ | $0.1108^{+0.0013}_{-0.0012}$ | $168.5^{+0.8}_{-0.8}$ | $4.43$ | $0.1055^{+0.0021}_{-0.0020}$ | $168.8^{+0.9}_{-0.9}$ | $2.02$ | $0.1100^{+0.0022}_{-0.0030}$ | $169.1^{+0.8}_{-0.9}$ | $2.41$
$y_{t}$ | $0.1100^{+0.0021}_{-0.0018}$ | $169.7^{+1.2}_{-1.1}$ | $2.20$ | $0.1233^{+0.0022}_{-0.0021}$ | $175.3^{+1.1}_{-1.0}$ | $2.89$ |  |  |
$y_{t\bar{t}}$ | $0.1191^{+0.0013}_{-0.0015}$ | $177.0^{+1.3}_{-1.3}$ | $1.85$ | $0.1132^{+0.0016}_{-0.0016}$ | $171.8^{+1.2}_{-1.2}$ | $0.85$ |  |  |

Table 4:  As table 2 but for the simultaneous extractions of $m_{t}$ and
$\alpha_{s}$ using ATLAS (upper table) and CMS (lower table) differential
distributions.

The first feature to emphasise is that, compared to the separate extractions
in tables 2 and 3, for virtually every combination of kinematic distribution
and PDF set, the $\chi^{2}_{\rm min}$ value is reduced. This indicates an
improvement in the agreement between theory and data. None of the combinations
show an increase in $\chi^{2}_{\rm min}$.

In figure 6 we show contour plots of $\Delta\chi^{2}$ corresponding to our
extractions using measurements of $p^{t}_{T}$, $m_{t\bar{t}}$ and $y_{t}$ by
ATLAS. The red vertical and horizontal lines indicate the world-average values
of $m_{t}$ and $\alpha_{s}$. We have indicated the regions in the
$(\alpha_{s},m_{t})$-plane inside which we have ‘exact’ NNLO theory
predictions (through interpolations of the discrete points computed) as the
white area. The outer blue-grey regions indicate the parameter space where
extrapolation of the theory predictions is required.

Figure 6: $\Delta\chi^{2}$ contour plots from a two-dimensional extraction of
$\alpha_{s}$ and $m_{t}$ using ATLAS $m_{t\bar{t}}$ (first column), ATLAS
$p^{t}_{T}$ (second column) and ATLAS $y_{t}$ distributions (third column).
The blue and black contours correspond to a $\Delta\chi^{2}$ of 0.1 and 1.0.
CT14 and NNPDF3.0 PDFs have been used in the upper and lower plots
respectively. The red lines denote the world average values for $\alpha_{s}$
and $m_{t}$. The white area shows the parameter space where interpolation is
possible while the blue-grey area denotes the parameter space where the fits
are extrapolated.

These plots highlight that, by including the distribution shape information
together with the measurements of the cross section, it is possible to obtain
joint constraints on $\alpha_{s}$ and $m_{t}$. The particular set of contour
plots shown in figure 6 correspond to extractions where the experimental data
is particularly well described by the theoretical predictions for both CT14
and NNPDF3.0 PDFs. The shapes of the $\Delta\chi^{2}$ contours for the
extraction using the ATLAS $y_{t}$ data (third column) indicate a large amount
of compensation between $\alpha_{s}$ and $m_{t}$, similar to the pattern
observed for the inclusive cross section. This is an indication that for these
distributions the $\chi^{2}$ objective is, to a large extent, driven by the
measurements of the cross section, and that the normalised distributions
provide relatively weak constraining power.

####  4.2.2 Extraction from a combination of two ATLAS distributions

In this section, we make use of the full information about correlations
between bins made available by the ATLAS experiment in order to extract values
of the parameters from combinations of distributions. We form covariance
matrices which are block diagonal in systematic uncertainties but which
contain off-diagonal entries encoding information about the correlation of
statistical uncertainties between bins of different distributions. Information
on systematic correlations between the distributions is available, but several
studies have established that some decorrelation of the largest systematic
uncertainties is needed to obtain acceptable PDF fits ATLAS:2018owm ;
Amat:2019upj ; Bailey:2019yze . This implies that no systematic correlation
between the distributions is a better approximation than full correlation. It
is beyond the scope of this paper to investigate ATLAS systematic correlations
further.

We consider pairwise combinations of distributions and present in table 5 the
values obtained from a simultaneous extraction. Corresponding contour plots
are shown in figure 7.

ATLAS | ATLAS | CT14 | NNPDF30 | NNPDF31
---|---|---|---|---
1 | 2 | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $m_{t\bar{t}}$ | $0.1155^{+0.0020}_{-0.0022}$ | $173.2^{+0.5}_{-0.5}$ | $0.34$ | $0.1165^{+0.0027}_{-0.0027}$ | $173.5^{+0.5}_{-0.5}$ | $0.39$ | $0.1180^{+0.0022}_{-0.0022}$ | $173.8^{+0.5}_{-0.5}$ | $0.49$
$p^{t}_{T}$ | $y_{t}$ | $0.1159^{+0.0016}_{-0.0017}$ | $174.6^{+0.9}_{-1.0}$ | $0.66$ | $0.1168^{+0.0015}_{-0.0015}$ | $174.5^{+0.9}_{-0.9}$ | $0.31$ |  |  |
$p^{t}_{T}$ | $y_{t\bar{t}}$ | $0.1187^{+0.0017}_{-0.0018}$ | $175.1^{+0.9}_{-1.0}$ | $1.28$ | $0.1170^{+0.0024}_{-0.0023}$ | $174.4^{+1.0}_{-1.0}$ | $0.53$ |  |  |
$y_{t}$ | $m_{t\bar{t}}$ | $0.1142^{+0.0017}_{-0.0017}$ | $173.2^{+0.6}_{-0.5}$ | $0.63$ | $0.1155^{+0.0015}_{-0.0014}$ | $173.3^{+0.6}_{-0.5}$ | $0.22$ |  |  |
$y_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1166^{+0.0017}_{-0.0018}$ | $173.3^{+0.6}_{-0.5}$ | $1.35$ | $0.1152^{+0.0023}_{-0.0022}$ | $173.2^{+0.6}_{-0.5}$ | $0.44$ |  |  |
$y_{t}$ | $y_{t\bar{t}}$ | $0.1182^{+0.0013}_{-0.0014}$ | $176.4^{+1.0}_{-1.0}$ | $1.32$ | $0.1156^{+0.0014}_{-0.0014}$ | $173.5^{+1.1}_{-1.2}$ | $0.40$ |  |  |
Average | $0.1159^{+0.0013}_{-0.0014}$ | $173.8^{+0.8}_{-0.8}$ |  | $0.1161^{+0.0011}_{-0.0010}$ | $173.7^{+0.6}_{-0.6}$ |  | $0.1180^{+0.0022}_{-0.0022}$ | $173.8^{+0.5}_{-0.5}$ |
Table 5:  As in table 2 but for the simultaneous extractions of $m_{t}$ and
$\alpha_{s}$ from two ATLAS distributions. Measured observables has been
combined with appropriate correlations included.

Figure 7: As in fig. 6 but for pairs of distributions from ATLAS:
{$p^{t}_{T}$, $m_{t\bar{t}}$} (left two) and {$m_{t\bar{t}}$, $y_{t\bar{t}}$}
(right two). CT14 and NNPDF3.0 PDF results are shown alternately.

We notice again that the inclusion of $m_{t\bar{t}}$ data seems to lower the
extracted value of $m_{t}$ by about 1 GeV compared to combinations where it is
not included and does so consistently across PDF sets.

####  4.2.3 Extraction from a combination of one ATLAS and one CMS
distribution

The final extraction we discuss in this section is one that is based on using
measurements of two normalised distributions, one from ATLAS and one from CMS.
As mentioned in section 3, we have performed these extractions assuming that
the correlations between measurements of the two experiments are negligible.
While this exercise may not be fully complete in terms of including
uncertainties such as luminosity systematics, it illustrates the potential of
exploiting measurements from _both_ experiments. Unlike the other ATLAS
extractions presented thus far, the covariance matrices used in this section
are diagonal in statistical uncertainties in order to match those used for the
CMS extractions. For completeness we present results for all 16 possible
combinations of pairs of kinematic distributions from the two experiments –
however, we mainly focus our attention and draw insights from the combinations
where _different_ distributions are combined (avoiding potential shared
systematics that may enter in measuring a given distribution). The extractions
we present are two-dimensional extractions based on the definition of
$\chi^{2}$ given in eq. (13). The results of these extractions are shown in
table 6.

ATLAS | CMS | CT14 | NNPDF30 | NNPDF31
---|---|---|---|---
|  | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $p^{t}_{T}$ | $0.1142^{+0.0014}_{-0.0014}$ | $172.1^{+0.4}_{-0.4}$ | $2.64$ | $0.1148^{+0.0017}_{-0.0017}$ | $172.7^{+0.4}_{-0.4}$ | $2.60$ | $0.1176^{+0.0015}_{-0.0015}$ | $173.3^{+0.4}_{-0.4}$ | $2.57$
$p^{t}_{T}$ | $m_{t\bar{t}}$ | $0.1135^{+0.0013}_{-0.0013}$ | $172.3^{+0.3}_{-0.4}$ | $4.61$ | $0.1129^{+0.0017}_{-0.0017}$ | $172.6^{+0.3}_{-0.3}$ | $4.28$ | $0.1158^{+0.0015}_{-0.0015}$ | $172.9^{+0.4}_{-0.3}$ | $4.14$
$p^{t}_{T}$ | $y_{t}$ | $0.1178^{+0.0013}_{-0.0014}$ | $174.4^{+0.5}_{-0.5}$ | $1.62$ | $0.1209^{+0.0016}_{-0.0016}$ | $174.8^{+0.5}_{-0.5}$ | $1.93$ |  |  |
$p^{t}_{T}$ | $y_{t\bar{t}}$ | $0.1172^{+0.0012}_{-0.0013}$ | $174.7^{+0.5}_{-0.5}$ | $1.33$ | $0.1173^{+0.0015}_{-0.0015}$ | $174.7^{+0.5}_{-0.5}$ | $0.78$ |  |  |
$m_{t\bar{t}}$ | $p^{t}_{T}$ | $0.1162^{+0.0012}_{-0.0013}$ | $172.9^{+0.2}_{-0.2}$ | $3.06$ | $0.1158^{+0.0017}_{-0.0017}$ | $173.0^{+0.2}_{-0.2}$ | $2.87$ | $0.1186^{+0.0015}_{-0.0015}$ | $173.2^{+0.2}_{-0.2}$ | $2.93$
$m_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1148^{+0.0013}_{-0.0013}$ | $172.9^{+0.2}_{-0.2}$ | $5.25$ | $0.1140^{+0.0017}_{-0.0017}$ | $172.9^{+0.2}_{-0.2}$ | $4.79$ | $0.1172^{+0.0015}_{-0.0015}$ | $173.2^{+0.2}_{-0.2}$ | $4.75$
$m_{t\bar{t}}$ | $y_{t}$ | $0.1168^{+0.0014}_{-0.0015}$ | $173.2^{+0.2}_{-0.2}$ | $1.94$ | $0.1198^{+0.0015}_{-0.0015}$ | $173.3^{+0.2}_{-0.2}$ | $2.50$ |  |  |
$m_{t\bar{t}}$ | $y_{t\bar{t}}$ | $0.1161^{+0.0013}_{-0.0014}$ | $173.3^{+0.2}_{-0.2}$ | $1.80$ | $0.1166^{+0.0014}_{-0.0014}$ | $173.3^{+0.2}_{-0.2}$ | $1.36$ |  |  |
$y_{t}$ | $p^{t}_{T}$ | $0.1128^{+0.0009}_{-0.0008}$ | $170.8^{+0.5}_{-0.5}$ | $5.83$ | $0.1109^{+0.0010}_{-0.0010}$ | $170.5^{+0.5}_{-0.5}$ | $1.11$ |  |  |
$y_{t}$ | $m_{t\bar{t}}$ | $0.1127^{+0.0008}_{-0.0008}$ | $170.9^{+0.6}_{-0.6}$ | $8.71$ | $0.1091^{+0.0009}_{-0.0009}$ | $169.3^{+0.7}_{-0.7}$ | $2.34$ |  |  |
$y_{t}$ | $y_{t}$ | $0.1175^{+0.0010}_{-0.0011}$ | $176.4^{+0.8}_{-0.8}$ | $4.79$ | $0.1142^{+0.0010}_{-0.0010}$ | $172.4^{+0.8}_{-0.8}$ | $3.24$ |  |  |
$y_{t}$ | $y_{t\bar{t}}$ | $0.1186^{+0.0008}_{-0.0009}$ | $178.6^{+0.8}_{-0.8}$ | $3.71$ | $0.1120^{+0.0009}_{-0.0009}$ | $171.4^{+0.9}_{-0.9}$ | $1.19$ |  |  |
$y_{t\bar{t}}$ | $p^{t}_{T}$ | $0.1140^{+0.0008}_{-0.0008}$ | $171.7^{+0.5}_{-0.5}$ | $11.00$ | $0.1115^{+0.0008}_{-0.0008}$ | $170.7^{+0.5}_{-0.5}$ | $1.76$ |  |  |
$y_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1139^{+0.0008}_{-0.0008}$ | $172.1^{+0.5}_{-0.7}$ | $14.27$ | $0.1105^{+0.0008}_{-0.0008}$ | $169.9^{+0.6}_{-0.6}$ | $3.29$ |  |  |
$y_{t\bar{t}}$ | $y_{t}$ | $0.1176^{+0.0009}_{-0.0009}$ | $176.7^{+0.7}_{-0.7}$ | $8.11$ | $0.1134^{+0.0008}_{-0.0008}$ | $172.0^{+0.8}_{-0.8}$ | $3.74$ |  |  |
$y_{t\bar{t}}$ | $y_{t\bar{t}}$ | $0.1179^{+0.0008}_{-0.0009}$ | $178.0^{+0.7}_{-0.7}$ | $7.03$ | $0.1125^{+0.0008}_{-0.0008}$ | $172.0^{+0.8}_{-0.8}$ | $1.66$ |  |  |
Average | $0.1172^{+0.0009}_{-0.0010}$ | $174.5^{+0.5}_{-0.5}$ |  | $0.1172^{+0.0013}_{-0.0013}$ | $174.5^{+0.5}_{-0.5}$ |  | $0.1178^{+0.0013}_{-0.0013}$ | $173.3^{+0.3}_{-0.3}$ |
Table 6:  As in table 2 but for the simultaneous extractions of $m_{t}$ and
$\alpha_{s}$ from one ATLAS and one CMS differential distribution. Data from
the two experiments has been combined assuming no correlations between
distributions.

Figure 8: As in fig. 6 but for pairs of distributions from {ATLAS, CMS}:
{$m_{t\bar{t}}$, $y_{t\bar{t}}$} (left two) and {$p^{t}_{T}$, $y_{t\bar{t}}$}
(right two). CT14 and NNPDF3.0 PDF results are shown alternately.

Examining table 6, we note that compared to the other extractions presented so
far, the $\chi^{2}_{\rm min}$ values returned by the combinations of ATLAS and
CMS data are rather large. Indeed, taking for example the ATLAS $y_{t\bar{t}}$
and CMS $m_{t\bar{t}}$ distributions together, with the CT14 PDF set one
obtains the highest value of $\chi^{2}_{\rm min}$ encountered thus far, at
14.27. These large values clearly indicate a poor description of the data by
the theory, both across PDF sets and combinations of distributions.

Bearing this in mind, we note that the combination of distributions yielding
the best results for the CT14 and NNPDF3.0 PDF sets are $p^{t}_{T}$ or
$m_{t\bar{t}}$ from ATLAS combined with $y_{t\bar{t}}$ from CMS. This
observation is consistent with the results of the simultaneous extractions of
$\alpha_{s}$ and $m_{t}$ from each individual experiment (discussed in the
previous section and tabulated in table 4). These two combinations give best-
fit values of $\alpha_{s}$ and $m_{t}$ that differ by about 1% – in the case
of the $\alpha_{s}$ values, these remain fully consistent within their
uncertainties but for $m_{t}$ the values lie slightly outside the $1\sigma$
interval.

In figure 8 we show the $\Delta\chi^{2}$ contours for the extractions based on
the combinations of distributions {$p^{t}_{T}$, $y_{t\bar{t}}$} and
{$m_{t\bar{t}}$, $y_{t\bar{t}}$}. Compared to the contour plots arising from
other extractions, we do see reduced uncertainties despite the slightly higher
$\chi^{2}_{\rm min}$ values, particularly in the $m_{t}$ direction.

We comment further on the combined CMS and ATLAS data extractions in section
5.2.

##  5 Discussion

###  5.1 Averaging of $\alpha_{s}$ and $m_{t}$ extractions

A large number of $\alpha_{s}$ and/or $m_{t}$ extractions were presented in
section 4 and appendices A,B,C. Upon reviewing them it quickly becomes
apparent that some extractions can be of low quality (indicated by a high
value of $\chi^{2}_{\rm min}$) and some can be non-sensible (the extracted
parameters differ greatly from current world average values). This inability
of theory to simultaneously describe all data sets used in this work has
already been noted in ref. Czakon:2016olj . To proceed with our extraction of
$\alpha_{s}$ and $m_{t}$ we follow the following strategy. We produce a
weighted average for the values of the extracted parameters over as many data
sets as possible in order to reduce the role of biases from theory and
modelling (as already discussed in section 3). In practical terms, for each
type of extraction discussed in section 4 we produce one value of $\alpha_{s}$
and $m_{t}$ for each PDF set used. We do not combine extractions based on
different PDF sets because they contain different systematic uncertainties,
but we check if the extractions based on different PDF sets are compatible
with each other. We exclude from this approach the extractions based on the
NNPDF3.1 PDF sets since they already contain fits to the top-quark data used
in this work. We return to make further comments on the use of this PDF set in
section 5.3 below.

In implementing our averaging procedure, described in detail in Appendix D, we
have borne the following considerations in mind. We have decided to exclude
from the averaging procedure extractions which return values of the parameters
significantly different from previous results, restricting ourselves to those
which yield best-fit values of $\alpha_{s}$ and $m_{t}$ that lie roughly
within $\pm 3\sigma$ of the current world average values. Specifically, we
require that:

$\displaystyle 0.115\leq\alpha_{s}\leq
0.121\;\;\;\text{and}\;\;\;170.0\;\text{GeV}\leq m_{t}\leq
176.0\;\text{GeV}\,.$ (15)

To aid the reader, in all tables in this work we have highlighted in grey the
extractions that satisfy the criteria in eq. (15). Our motivation for
excluding such extractions is that they indicate the presence of some serious
difference between theory and data as opposed to a legitimate statistical
fluctuation which one may hope to remove with the help of a weighted average.

In principle we do not neglect extractions which return a high value of
$\chi^{2}_{\rm min}$. Our averaging procedure is designed in such a way that
it weights values based on their inverse $\chi^{2}_{\rm min}$, i.e.
extractions with large $\chi^{2}_{\rm min}$ which are less likely to be
reliable receive smaller weight in the determination of both the average value
as well as in the uncertainty estimate on the average. This procedure fails,
of course, in cases where all $\chi^{2}_{\rm min}$ are large. In such cases we
have not performed averaging over the values of a given data set.
Specifically, we do not average the CMS entries in tables 2 and 3 or the CMS
entries in table 4. In addition, we exclude the entries in all of the tables
appearing in Appendix B from the averaging procedure due to the larger
uncertainties we obtain relative to the normalised cases.

To account for the likely systematic spread between the various extractions we
have introduced a systematic component to the uncertainty estimate. It is also
weighted by the inverse $\chi^{2}_{\rm min}$ in order to reduce the effect of
outliers (see Appendix D for details). This error is designed such that for a
large number $n$ of measurements to be averaged, all with similar values of
$\chi^{2}_{\rm min}$, the error tends to a constant value and, unlike the
estimate of the statistical uncertainty, does not decrease as $1/\sqrt{n}$.

Finally, we would like to mention that when averaging over $\alpha_{s}$ or
$m_{t}$ values we do not introduce any correlations between them. The reasons
for this are: first, all correlations available to us have already been
accounted for in the individual extractions and, second, improper modelling of
correlations can have a significant detrimental effect on the final
uncertainty estimate.

###  5.2 Summary of findings

In this section, we summarise the $\alpha_{s}$ and $m_{t}$ values obtained
with the help of the averaging procedure outlined in section 5.1 above. We
only use extractions derived from normalised differential distributions
supplemented with measurements of the total cross section, since they have a
higher precision than those based on absolute distributions.

For the one-dimensional extractions from single distributions we obtain:

  * •

Averaging the ATLAS results for $\alpha_{s}$ (see table 2) for $m_{t}=173.3$
GeV:

$\displaystyle\alpha_{s}$
$\displaystyle=0.1158^{+0.0014}_{-0.0015}\qquad\mathrm{CT14}$ (16)
$\displaystyle\alpha_{s}$
$\displaystyle=0.1157^{+0.0013}_{-0.0013}\qquad\mathrm{NNPDF3.0}$ (17)

  * •

Averaging the ATLAS results for $m_{t}$ (see table 3) for $\alpha_{s}=0.118$:

$\displaystyle m_{t}=$ $\displaystyle
174.3^{+0.9}_{-0.8}~{}\mathrm{GeV}\qquad\mathrm{CT14}$ (18) $\displaystyle
m_{t}=$ $\displaystyle
174.3^{+0.7}_{-0.7}~{}\mathrm{GeV}\qquad\mathrm{NNPDF3.0}$ (19)

For the simultaneous (two-dimensional) extraction of $\alpha_{s}$ and $m_{t}$
we obtain:

  * •

Extraction from a single (ATLAS) distribution (see table 4)

$\displaystyle\alpha_{s}$ $\displaystyle=0.1163^{+0.0015}_{-0.0016},\quad
m_{t}=173.8^{+0.8}_{-0.8}~{}\mathrm{GeV}\qquad\mathrm{CT14}$ (20)
$\displaystyle\alpha_{s}$ $\displaystyle=0.1164^{+0.0019}_{-0.0019},\quad
m_{t}=173.6^{+0.8}_{-0.8}~{}\mathrm{GeV}\qquad\mathrm{NNPDF3.0}$ (21)

  * •

Extraction from multiple ATLAS distributions (see table 5)

$\displaystyle\alpha_{s}$ $\displaystyle=0.1159^{+0.0013}_{-0.0014},\quad
m_{t}=173.8^{+0.8}_{-0.8}~{}\mathrm{GeV}\qquad\mathrm{CT14}$ (22)
$\displaystyle\alpha_{s}$ $\displaystyle=0.1161^{+0.0011}_{-0.0010},\quad
m_{t}=173.7^{+0.6}_{-0.6}~{}\mathrm{GeV}\qquad\mathrm{NNPDF3.0}$ (23)

The above results have a number of interesting features and show an impressive
mutual consistency. In particular, they appear to be almost identical for the
two PDF sets we consider, CT14 and NNPDF3.0, both in terms of central values
and uncertainties. This is not neccessarily the case for the individual
extractions, as can be noted by inspection of the tables appearing in section
4.

The independent extractions of $\alpha_{s}$ and $m_{t}$ agree well with the
simultaneous extractions, both from one differential distribution and from a
pair of differential distributions. The size of the uncertainties in all cases
is comparable. Perhaps the only (mild) exception we observe is the uncertainty
on the $\alpha_{s}$ extraction with NNPDF3.0, which increases by about 50%
from the independent extraction to the simultaneous extraction from a single
distribution but then decreases for the simultaneous extraction from a pair of
distributions. In general, as expected, the uncertainty on the extractions
from a pair of distributions is smallest, both for $\alpha_{s}$ and $m_{t}$
and for each PDF set. It is interesting to compare the above numbers to the
extraction from the total cross section only (see appendix A). Regarding
$\alpha_{s}$, while all central values are fairly close and compatible within
uncertainties, the extractions from the differential distributions have
notably smaller uncertainties, in some cases by as much as a factor of two.
This is consistent with the expectation that the inclusion of differential
data and correlations should improve the extraction of $\alpha_{s}$. In the
case of $m_{t}$, we find very good consistency between the extraction from the
total cross section and that from the independent (one-dimensional) $m_{t}$
fit. The difference in central values is larger when compared to two-
dimensional extractions and can reach up to 1 GeV (although this difference is
within the uncertainties). In all cases, as expected, the uncertainties for
the differential extractions are smaller, sometimes by a factor of two.

Finally, it is interesting to consider the extraction from the simultaneous
fit to one ATLAS and one CMS distribution given in table 6 and to compare it
to the above extractions. As emphasised in section 4, the fits produced are of
low quality as indicated by the large $\chi^{2}_{\rm min}$ values which they
return. Since one might a priori consider this particular extraction to be the
most reliable due to the overlapping data sets from different experiments, we
have listed in table 6 the average extracted values. We observe that the
extractions with both PDF sets, CT14 and NNPDF3.0, lead to almost identical
central values and uncertainty estimates. The central values for $\alpha_{s}$
are closer to the world average than those discussed above, while the central
value for the average $m_{t}$ extraction is further away. Notably, the
estimated uncertainties are rather small and are, in fact, significantly lower
than in the cases discussed above. On the one hand these results show the
power of averaging (reasonably independent) results. On the other hand, one
might infer the low quality of the fits from the fact that the central value
for $m_{t}$ returned is 2.4$\sigma$ away from the world average.

###  5.3 Extraction from PDF sets which have already fitted top-quark data

In this section we briefly consider the question of consistency of parameter
extraction with a PDF set that has itself been fit to the same data. Of the
three PDF sets used in this work, only one, NNPDF3.1, has fit top-quark pair
differential distributions. The general expectation regarding $m_{t}$ is that
one should obtain from the fits using this set a value closer to $m_{t}=173.3$
GeV, which was used to fit the PDF initially. It is therefore interesting to
examine whether this expectation is borne out by the fits. Regarding
$\alpha_{s}$, the expectation is less clear than in the case of $m_{t}$ since
the strong coupling is not a parameter relevant only to the process of top-
quark pair production and in fact features in the theoretical predictions for
the majority of processes which enter into the PDF fit.

A second point one might hope to assess is whether any impact on the parameter
extraction appears in fits from all differential distributions, or only in
fits from distributions directly exploited by the PDF collaboration. Of the
distributions which we consider in this work, the NNPDF3.1 sets contain
information only on the $y_{t}$ data from ATLAS and the $y_{t\bar{t}}$ data
from CMS. In the following we shed some light on this issue, however, we
caution that we cannot offer a comprehensive answer to this question here,
since that would require detailed information about the specifics of the
NNPDF3.1 PDF fit that are not available to us. To that end one would need to
examine, for example, the relative impacts of different data sets not related
to top-quark data, any modification of fitting methodology etc. which
distinguish NNPDF3.1 with respect to say NNPDF3.0 but are not directly related
to the inclusion of top-quark data.

In figure 9 we show the results of several two-dimensional $\alpha_{s}$ and
$m_{t}$ extractions from various differential distributions computed with the
NNPDF3.1 PDF set. To emphasise the important features of these plots, we have
increased the displayed area in parameter space and have shown several
contours corresponding to the values $\Delta\chi^{2}=$ 1.0 (black), 0.5
(green), 0.1 (blue) and 0.02 (orange). In the upper left and upper centre
plots we see the extractions from the distributions directly fitted by the PDF
set. In line with our general expectations above, we observe that the
behaviour of the fits is strikingly different compared to the fits performed
with CT14 or NNPDF3.1. In particular we observe strong, nearly perfect,
correlation between the two fitted parameters as indicated by the $\chi^{2}$
contours that do not show a localised minimum in the $\alpha_{s}-m_{t}$ plane
but rather a degenerate flat ‘valley’ of minima. Clearly, the distributions
used in the PDF fits are not suitable for extraction of top-quark-related
parameters. In the lower left and lower centre plots we show the fits to the
same two rapidity distributions but now fitted from the opposite experiment:
$y_{t\bar{t}}$ from ATLAS and $y_{t}$ from CMS. We again observe strong
correlation between $\alpha_{s}$ and $m_{t}$, although the degeneracy is not
quite as marked. Inferring that the two rapidity distributions are rather
correlated, we conclude that we cannot use either for $\alpha_{s}$ and $m_{t}$
fits. For this reason we have not shown any results in the tables in section 4
for NNPDF3.1 with $y_{t}$ or $y_{t\bar{t}}$.

It is next interesting to consider the extraction of $\alpha_{s}$ and $m_{t}$
from distributions that have not been directly used in the PDF fits. For the
case at hand these would be the $m_{t\bar{t}}$ and $p^{t}_{T}$ distributions.
The resulting $\chi^{2}$ fit can be seen in figure 9 (upper right). From that
figure we conclude that the fit based on these two differential distributions
does not appear to have any obvious degeneracy between the two parameters as
was observed in the cases of the rapidity distributions. This may be an
indication that distributions which have not been directly included in PDF
fits can be considered to be unaffected. This is especially relevant for the
$m_{t\bar{t}}$ distribution, which is very important to $m_{t}$ determination
and searches for BSM physics. We caution that this finding should not be
considered as definitive proof of a lack of correlation but rather as an
evidence of absence of obvious correlation. Lastly, it is interesting to
consider an intermediate case where $\alpha_{s}$ and $m_{t}$ are fit to two
distributions, one of which has been included in the PDF fit and one of which
has not. In figure 9 (lower right) we show the $\Delta\chi^{2}$ contours for
the two-dimensional extraction from the combined ATLAS data on $y_{t}$ and
$m_{t\bar{t}}$. We see only a small change relative to the $m_{t\bar{t}}$ and
$p^{t}_{T}$ case and observe $\Delta\chi^{2}$ contours which do not show
obvious signs of a degeneracy. Nevertheless, in order to be conservative, we
have consistently omitted extractions using a combination of either $y_{t}$ or
$y_{t\bar{t}}$ and any other distribution from the NNPDF3.1 column of the
tables in section 4.

Figure 9:  Examples of simultaneous $\alpha_{s}$ and $m_{t}$ extractions from
differential distributions using the PDF set NNPDF3.1. Some of the plots show
extractions from distributions that have been used in the PDF fit while others
are from distributions that have not. See section 5.3 for details.

##  6 Conclusions

In this work we have presented the first joint extraction using NNLO theory
predictions of $\alpha_{s}$ and the top-quark pole mass $m_{t}$ from
measurements of both the top-quark pair production cross section as well as
differential distributions. We have taken experimental measurements of the
kinematic variables $p^{t}_{T},m_{t\bar{t}},y_{t}$ and $y_{t\bar{t}}$ in
$t\bar{t}$ events provided by the ATLAS and CMS experiments at the 8 TeV LHC.
These have been compared with theoretical predictions including NNLO QCD
corrections for different values of the top-quark mass and using different PDF
sets. The fastNLO tables used to produce the results in this paper, for values
of the top-quark pole mass $m_{t}=\\{169.0,171.0,172.5,173.3,175.0\\}$ GeV,
are available for download from ref. website . Functional forms of the bin
weights of each predicted distribution have been determined by smooth
interpolation. A least-squares method has been used to perform individual and
simultaneous extractions of the SM parameters $m_{t}$ and $\alpha_{s}$—we have
also considered combining distributions from different experiments in the
absence of correlations.

Our best result is thus a two-dimensional extraction, taking into account
$\alpha_{s}$ and $m_{t}$ correlations and using two distributions to maximise
the differential information included. We select the case of two different
ATLAS distributions since the experimental correlations between the
distributions are properly included and the overall quality of the fits is
good. Within this, we choose the more conservative estimate using CT14 (which
contains no top-quark data). Our final values are therefore

$\alpha_{s}=0.1159^{+0.0013}_{-0.0014},\quad
m_{t}=173.8^{+0.8}_{-0.8}~{}\mathrm{GeV}\,.$ (24)

These values are compatible with those obtained from extractions using the
total cross section for top-quark pair production. The $\alpha_{s}$ extraction
is about 1.5$\sigma$ below the world average of $0.118$; it is of high
precision which is competitive with measurements from other processes (for
example, the CMS Drell-Yan measurement in ref. dEnterria:2019aat ). Subtleties
related to extracting $\alpha_{s}$ from PDFs with fixed $\alpha_{s}$ value
aside Forte:2020pyp , this indicates that the determination of the strong
coupling constant from $t\bar{t}$ production has significant potential. The
extracted $m_{t}$ value is in perfect agreement with the world average of
$173.3$ GeV and has impressively small uncertainty.

In performing our analysis, we have noticed several features of the data which
hinder a clean extraction. We observe certain tensions between ATLAS and CMS
measurements of the same distribution (in keeping with previous observations
Czakon:2016olj ), which may point to different treatments by the two
experiments. Furthermore, in some cases we also observe considerable
differences between PDF sets, which may indicate that a simultaneous fit of
$\alpha_{s}$, $m_{t}$ and the PDFs must be considered. In any case, despite
these difficulties we believe that the approach presented in this work is
promising and is likely to bear greater fruit when provided with the
significantly larger data sets offered by newer LHC runs.

There are a few ways in which our treatment could be further refined in a
future study. Firstly, as we have highlighted in section 3, the uncertainties
on the extracted values of $\alpha_{s}$ and $m_{t}$ which we have presented
only include experimental effects, both systematic and statistical. In a more
complete treatment, theory uncertainties due to the choice of renormalisation
and factorisation scales as well as PDF uncertainties should be included in
the definition of the $\chi^{2}$ objective function. Since the scale
uncertainty is greatly reduced at NNLO when considering normalised (compared
to absolute) distributions, we expect that including these will not increase
the uncertainties on the extracted parameters dramatically. On the other hand,
in one-dimensional extractions of $\alpha_{s}$, the scale uncertainty on the
cross section is known to contribute a significant fraction of the overall
uncertainty on the best fit $\alpha_{s}$ Klijnsma:2017eqp , and we would
therefore expect this to also play a rôle for our study here. Additionally, in
the case of our extractions from combined measurements of ATLAS and CMS, a
complete treatment would also include systematic uncertainties correlated
between the two experiments, such as that of luminosity.

Finally, given the constraining power that differential measurements of top-
pair production have on PDFs Czakon:2016olj ; Sirunyan:2017azo ;
Bailey:2019yze ; Czakon:2019yrx ; Kadir:2020yml , a joint fit at NNLO in QCD
of $\alpha_{s}$ and $m_{t}$ together with PDFs using differential data would
simultaneously provide useful constraints on all these three fundamental
ingredients of QCD.

###### Acknowledgements.

M.A.L. is grateful to Emma Slade for useful discussions. A.M. thanks the
Department of Physics at Princeton University for hospitality during the
completion of this work. A.M.C.S. acknowledges the support of the Leverhulme
Trust. The work of M.C. was supported in part by a grant of the BMBF and by
the Deutsche Forschungsgemeinschaft under grant 396021762 – TRR 257. The work
of M.A.L., A.M. and A.S.P. has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 683211); it was also supported by the UK STFC
grants ST/L002760/1 and ST/K004883/1. M.A.L. acknowledges funding from
Fondazione Cariplo and Regione Lombardia, grant 2017-2070 as well as from the
Cambridge Philosophical Society. A.P. is a cross-disciplinary postdoctoral
fellow supported by funding from the University of Edinburgh and Medical
Research Council (core grant to the MRC Institute of Genetics and Molecular
Medicine).

##  Appendix A Extractions using measurements of the total cross section

In this appendix, see table 7, we present the results for the extraction of
$m_{t}$ and $\alpha_{s}$ using the total cross section alone. Results for the
central value of the $\alpha_{s}$ extraction can be compared with Table 6 of
ref. Klijnsma:2017eqp , where full agreement is found.

| CT14 | NNPDF30 | NNPDF31
---|---|---|---
| $\alpha_{s}$ | $m_{t}$ | $\alpha_{s}$ | $m_{t}$ | $\alpha_{s}$ | $m_{t}$
ATLAS | $0.1158^{+0.0021}_{-0.0020}$ | $174.8^{+1.4}_{-1.3}$ | $0.1156^{+0.0027}_{-0.0027}$ | $174.4^{+1.4}_{-1.3}$ | $0.1172^{+0.0021}_{-0.0022}$ | $173.8^{+1.4}_{-1.3}$
CMS | $0.1163^{+0.0020}_{-0.0019}$ | $174.4^{+1.3}_{-1.2}$ | $0.1164^{+0.0026}_{-0.0026}$ | $174.1^{+1.3}_{-1.2}$ | $0.1178^{+0.0021}_{-0.0021}$ | $173.4^{+1.3}_{-1.2}$
Table 7: Best-fit $\alpha_{s}$ and $m_{t}$ from the total cross section:
extracted $\alpha_{s}$ or $m_{t}$ (with uncertainties) from ATLAS and CMS
measurements at 8 TeV. The extractions have been performed after fixing
$m_{t}=173.3$ GeV and $\alpha_{s}=0.118$ respectively. We show results
obtained from three different PDF sets. Masses are quoted in GeV.

##  Appendix B Extractions using measurements of absolute distributions

In this appendix, we present results for the extraction of $m_{t}$ and
$\alpha_{s}$ using measurements of absolute distributions by ATLAS and CMS. We
show the results of one-dimensional extractions for $\alpha_{s}$ in table 8
and for $m_{t}$ in table 9. Results from two-dimensional extractions from a
single distribution appear in table 10, while those from combinations of
distributions appear in table 11 (two ATLAS) and table 12 (ATLAS and CMS).

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1189^{+0.0035}_{-0.0041}$ | 0.32 | $\pagecolor{Gray}0.1203^{+0.0055}_{-0.0055}$ | 0.32 | $0.1194^{+0.0043}_{-0.0046}$ | 0.49
$m_{t\bar{t}}$ | $0.1211^{+0.0020}_{-0.0026}$ | 0.20 | $0.1247^{+0.0047}_{-0.0047}$ | 0.21 | $0.1251^{+0.0040}_{-0.0038}$ | 0.25
$m_{t\bar{t}}^{u}$ | $0.1211^{+0.0020}_{-0.0026}$ | 0.17 | $0.1247^{+0.0047}_{-0.0047}$ | 0.18 | $0.1251^{+0.0041}_{-0.0038}$ | 0.22
$y_{t}$ | $0.1154^{+0.0029}_{-0.0033}$ | 2.55 | $0.1135^{+0.0033}_{-0.0033}$ | 0.72 |  |
$y_{t\bar{t}}$ | $0.1184^{+0.0029}_{-0.0042}$ | 2.48 | $0.1127^{+0.0054}_{-0.0054}$ | 0.66 |  |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1153^{+0.0016}_{-0.0017}$ | 1.63 | $0.1138^{+0.0023}_{-0.0023}$ | 1.30 | $0.1164^{+0.0018}_{-0.0019}$ | 1.01
$m_{t\bar{t}}$ | $0.1142^{+0.0016}_{-0.0017}$ | 6.20 | $0.1100^{+0.0020}_{-0.0034}$ | 4.38 | $0.1157^{+0.0021}_{-0.0022}$ | 3.91
$m_{t\bar{t}}^{u}$ | $0.1142^{+0.0016}_{-0.0017}$ | 6.18 | $0.1100^{+0.0020}_{-0.0034}$ | 4.36 | $0.1157^{+0.0021}_{-0.0022}$ | 3.89
$y_{t}$ | $0.1152^{+0.0017}_{-0.0019}$ | 1.82 | $0.1177^{+0.0021}_{-0.0021}$ | 2.80 |  |
$y_{t\bar{t}}$ | $0.1161^{+0.0015}_{-0.0016}$ | 1.57 | $0.1154^{+0.0018}_{-0.0018}$ | 0.74 |  |

Table 8:  Tabulated values of best-fit $\alpha_{s}$ (with uncertainties) and
associated $\chi^{2}_{\rm min}$ from extractions of $\alpha_{s}$ using ATLAS
(upper table) and CMS (lower table) measurements of absolute distributions.
Results are shown for three different PDF sets and $m_{t}$ has been set to the
world average value of 173.3 GeV. The cells highlighted in grey correspond to
extractions that satisfy the quality conditions described in sec 5.1. As
explained in section 5.3, the rapidity distributions are not used for
extractions with the PDF set NNPDF3.1. The quantity $m_{t\bar{t}}^{u}$ is
defined in sec. 2.2.4.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $173.3^{+1.3}_{-1.2}$ | 0.32 | $173.5^{+1.3}_{-1.3}$ | 0.34 | $173.7^{+1.3}_{-1.3}$ | 0.49
$m_{t\bar{t}}$ | $172.9^{+0.5}_{-0.5}$ | 0.30 | $173.0^{+0.5}_{-0.5}$ | 0.45 | $173.1^{+0.6}_{-0.5}$ | 0.71
$m_{t\bar{t}}^{u}$ | $172.9^{+0.5}_{-0.5}$ | 0.28 | $173.0^{+0.5}_{-0.5}$ | 0.41 | $173.1^{+0.5}_{-0.5}$ | 0.67
$y_{t}$ | $178.2^{+2.3}_{-2.6}$ | 2.07 | $176.2^{+2.8}_{-3.7}$ | 0.96 |  |
$y_{t\bar{t}}$ | $178.3^{+3.5}_{-4.0}$ | 2.18 | $173.7^{+4.5}_{-4.8}$ | 0.85 |  |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $173.1^{+0.9}_{-0.8}$ | 2.04 | $173.0^{+0.9}_{-0.8}$ | 1.69 | $173.0^{+0.9}_{-0.8}$ | 1.10
$m_{t\bar{t}}$ | $168.8^{+1.1}_{-1.2}$ | 4.36 | $169.1^{+1.1}_{-1.2}$ | 3.59 | $169.9^{+1.0}_{-1.1}$ | 2.47
$m_{t\bar{t}}^{u}$ | $170.1^{+0.7}_{-0.7}$ | 4.57 | $170.3^{+0.7}_{-0.7}$ | 3.77 | $170.7^{+0.8}_{-0.7}$ | 2.58
$y_{t}$ | $174.8^{+1.1}_{-1.1}$ | 1.91 | $174.7^{+1.1}_{-1.1}$ | 2.62 |  |
$y_{t\bar{t}}$ | $175.3^{+1.2}_{-1.1}$ | 1.43 | $174.5^{+1.1}_{-1.1}$ | 0.81 |  |

Table 9: As in table 8 but for $m_{t}$ rather than $\alpha_{s}$, which has
been set to the world average value of 0.118. Masses are quoted in GeV.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1199^{+0.0031}_{-0.0039}$ | $173.8^{+1.2}_{-1.2}$ | $0.31$ | $0.1229^{+0.0057}_{-0.0057}$ | $174.2^{+1.2}_{-1.2}$ | $0.28$ | $0.1240^{+0.0052}_{-0.0041}$ | $175.0^{+1.2}_{-1.2}$ | $0.37$
$m_{t\bar{t}}$ | $0.1211^{+0.0021}_{-0.0028}$ | $173.0^{+0.5}_{-0.5}$ | $0.14$ | $0.1244^{+0.0046}_{-0.0046}$ | $173.1^{+0.5}_{-0.5}$ | $0.19$ | $0.1256^{+0.0042}_{-0.0039}$ | $173.5^{+0.5}_{-0.5}$ | $0.24$
$m_{t\bar{t}}^{u}$ | $0.1211^{+0.0021}_{-0.0028}$ | $173.0^{+0.5}_{-0.5}$ | $0.12$ | $0.1243^{+0.0046}_{-0.0046}$ | $173.1^{+0.5}_{-0.4}$ | $0.16$ | $0.1254^{+0.0042}_{-0.0038}$ | $173.4^{+0.5}_{-0.4}$ | $0.22$
$y_{t}$ | $0.1183^{+0.0024}_{-0.0036}$ | $178.3^{+2.3}_{-2.5}$ | $2.07$ | $0.1116^{+0.0027}_{-0.0027}$ | $168.0^{+3.4}_{-5.7}$ | $0.47$ |  |  |
$y_{t\bar{t}}$ | $0.1195^{+0.0028}_{-0.0042}$ | $178.4^{+3.3}_{-3.7}$ | $2.15$ | $0.1121^{+0.0048}_{-0.0048}$ | $170.8^{+5.8}_{-4.3}$ | $0.62$ |  |  |

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $0.1083^{+0.0018}_{-0.0017}$ | $169.1^{+0.8}_{-0.8}$ | $0.72$ | $0.1088^{+0.0022}_{-0.0022}$ | $170.5^{+0.9}_{-0.8}$ | $0.66$ | $0.1115^{+0.0018}_{-0.0018}$ | $170.6^{+0.8}_{-0.8}$ | $0.62$
$m_{t\bar{t}}$ | $0.1114^{+0.0018}_{-0.0018}$ | $166.6^{+1.5}_{-1.6}$ | $2.81$ | $0.1089^{+0.0026}_{-0.0026}$ | $168.3^{+1.3}_{-1.4}$ | $1.88$ | $0.1134^{+0.0022}_{-0.0024}$ | $169.0^{+1.2}_{-1.3}$ | $1.94$
$m_{t\bar{t}}^{u}$ | $0.1134^{+0.0018}_{-0.0018}$ | $169.7^{+0.8}_{-0.7}$ | $3.56$ | $0.1098^{+0.0026}_{-0.0026}$ | $170.1^{+0.8}_{-0.7}$ | $2.38$ | $0.1147^{+0.0022}_{-0.0023}$ | $170.5^{+0.8}_{-0.7}$ | $2.26$
$y_{t}$ | $0.1086^{+0.0017}_{-0.0014}$ | $169.6^{+1.0}_{-1.0}$ | $1.60$ | $0.1241^{+0.0024}_{-0.0024}$ | $177.0^{+1.2}_{-1.0}$ | $2.36$ |  |  |
$y_{t\bar{t}}$ | $0.1195^{+0.0012}_{-0.0013}$ | $176.4^{+1.2}_{-1.1}$ | $1.39$ | $0.1152^{+0.0018}_{-0.0018}$ | $173.2^{+1.1}_{-1.1}$ | $0.74$ |  |  |

Table 10: As in table 8 but for the simultaneous extractions of $\alpha_{s}$ and $m_{t}$. ATLAS | ATLAS | CT14 | NNPDF30 | NNPDF31
---|---|---|---|---
obs. 1 | obs. 2 | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p^{t}_{T}$ | $m_{t\bar{t}}$ | $0.1202^{+0.0020}_{-0.0024}$ | $173.1^{+0.5}_{-0.4}$ | $0.28$ | $0.1227^{+0.0036}_{-0.0036}$ | $173.2^{+0.5}_{-0.5}$ | $0.31$ | $0.1233^{+0.0028}_{-0.0029}$ | $173.6^{+0.5}_{-0.5}$ | $0.43$
$p^{t}_{T}$ | $m_{t\bar{t}}^{u}$ | $0.1202^{+0.0020}_{-0.0024}$ | $173.1^{+0.4}_{-0.4}$ | $0.26$ | $0.1226^{+0.0036}_{-0.0036}$ | $173.2^{+0.4}_{-0.4}$ | $0.29$ | $0.1230^{+0.0028}_{-0.0029}$ | $173.5^{+0.5}_{-0.4}$ | $0.43$
$p^{t}_{T}$ | $y_{t}$ | $0.1174^{+0.0016}_{-0.0019}$ | $174.6^{+1.2}_{-1.2}$ | $1.55$ | $0.1140^{+0.0021}_{-0.0021}$ | $172.7^{+1.3}_{-1.3}$ | $0.61$ |  |  |
$p^{t}_{T}$ | $y_{t\bar{t}}$ | $0.1194^{+0.0017}_{-0.0021}$ | $174.4^{+1.2}_{-1.2}$ | $1.48$ | $0.1146^{+0.0031}_{-0.0031}$ | $172.9^{+1.3}_{-1.2}$ | $0.58$ |  |  |
$y_{t}$ | $m_{t\bar{t}}$ | $0.1171^{+0.0016}_{-0.0019}$ | $173.1^{+0.6}_{-0.5}$ | $1.73$ | $0.1154^{+0.0021}_{-0.0021}$ | $172.9^{+0.6}_{-0.5}$ | $0.75$ |  |  |
$y_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1194^{+0.0016}_{-0.0019}$ | $173.1^{+0.6}_{-0.5}$ | $1.57$ | $0.1170^{+0.0029}_{-0.0029}$ | $172.9^{+0.5}_{-0.5}$ | $0.68$ |  |  |
$y_{t}$ | $m_{t\bar{t}}^{u}$ | $0.1171^{+0.0015}_{-0.0019}$ | $173.1^{+0.5}_{-0.5}$ | $1.72$ | $0.1154^{+0.0021}_{-0.0021}$ | $172.9^{+0.5}_{-0.5}$ | $0.74$ |  |  |
$y_{t\bar{t}}$ | $m_{t\bar{t}}^{u}$ | $0.1194^{+0.0015}_{-0.0019}$ | $173.1^{+0.5}_{-0.5}$ | $1.55$ | $0.1170^{+0.0029}_{-0.0029}$ | $172.9^{+0.5}_{-0.5}$ | $0.66$ |  |  |
$y_{t}$ | $y_{t\bar{t}}$ | $0.1188^{+0.0014}_{-0.0018}$ | $177.9^{+1.4}_{-1.5}$ | $1.85$ | $0.1120^{+0.0018}_{-0.0018}$ | $169.0^{+2.4}_{-2.8}$ | $0.53$ |  |  |
Table 11: As in table 8 but for the simultaneous extractions of $\alpha_{s}$ and $m_{t}$ from two ATLAS distributions. ATLAS | CMS | CT14 | NNPDF30 | NNPDF31
---|---|---|---|---
|  | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$p_{T}^{t}$ | $p_{T}^{t}$ | $0.1084^{+0.0019}_{-0.0017}$ | $169.6^{+0.7}_{-0.7}$ | $0.67$ | $0.1115^{+0.0020}_{-0.0020}$ | $171.6^{+0.7}_{-0.7}$ | $0.71$ | $0.1140^{+0.0017}_{-0.0017}$ | $171.8^{+0.7}_{-0.7}$ | $0.69$
$p_{T}^{t}$ | $m_{t\bar{t}}$ | $0.1124^{+0.0016}_{-0.0016}$ | $169.7^{+0.8}_{-0.8}$ | $1.99$ | $0.1098^{+0.0023}_{-0.0023}$ | $170.2^{+0.8}_{-0.8}$ | $1.55$ | $0.1138^{+0.0020}_{-0.0021}$ | $170.7^{+0.8}_{-0.8}$ | $1.52$
$p_{T}^{t}$ | $y_{t}$ | $0.1086^{+0.0018}_{-0.0014}$ | $170.0^{+0.8}_{-0.8}$ | $1.13$ | $0.1216^{+0.0021}_{-0.0021}$ | $175.2^{+0.8}_{-0.8}$ | $1.56$ |  |  |
$p_{T}^{t}$ | $y_{t\bar{t}}$ | $0.1177^{+0.0012}_{-0.0014}$ | $174.3^{+0.9}_{-0.8}$ | $1.01$ | $0.1160^{+0.0018}_{-0.0018}$ | $173.4^{+0.9}_{-0.8}$ | $0.59$ |  |  |
$m_{t\bar{t}}$ | $p_{T}^{t}$ | $0.1150^{+0.0015}_{-0.0016}$ | $172.5^{+0.5}_{-0.6}$ | $1.02$ | $0.1148^{+0.0020}_{-0.0020}$ | $172.7^{+0.4}_{-0.5}$ | $0.98$ | $0.1171^{+0.0016}_{-0.0017}$ | $173.0^{+0.5}_{-0.4}$ | $0.91$
$m_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1144^{+0.0016}_{-0.0017}$ | $171.3^{+0.8}_{-0.7}$ | $2.74$ | $0.1123^{+0.0023}_{-0.0023}$ | $171.6^{+0.8}_{-0.7}$ | $2.28$ | $0.1167^{+0.0019}_{-0.0019}$ | $172.3^{+0.5}_{-0.7}$ | $2.05$
$m_{t\bar{t}}$ | $y_{t}$ | $0.1157^{+0.0015}_{-0.0017}$ | $172.9^{+0.5}_{-0.5}$ | $1.30$ | $0.1192^{+0.0019}_{-0.0019}$ | $173.5^{+0.5}_{-0.5}$ | $1.84$ |  |  |
$m_{t\bar{t}}$ | $y_{t\bar{t}}$ | $0.1167^{+0.0013}_{-0.0014}$ | $173.1^{+0.6}_{-0.5}$ | $1.15$ | $0.1163^{+0.0017}_{-0.0017}$ | $173.0^{+0.5}_{-0.5}$ | $0.71$ |  |  |
$y_{t}$ | $p_{T}^{t}$ | $0.1124^{+0.0014}_{-0.0014}$ | $170.9^{+0.9}_{-0.8}$ | $1.71$ | $0.1104^{+0.0018}_{-0.0018}$ | $170.9^{+0.9}_{-0.8}$ | $0.70$ |  |  |
$y_{t}$ | $m_{t\bar{t}}$ | $0.1127^{+0.0014}_{-0.0014}$ | $167.9^{+1.3}_{-1.4}$ | $3.08$ | $0.1101^{+0.0019}_{-0.0019}$ | $168.1^{+1.3}_{-1.4}$ | $1.34$ |  |  |
$y_{t}$ | $y_{t}$ | $0.1167^{+0.0014}_{-0.0016}$ | $174.4^{+1.1}_{-1.0}$ | $2.05$ | $0.1199^{+0.0019}_{-0.0019}$ | $175.7^{+1.1}_{-1.0}$ | $2.06$ |  |  |
$y_{t}$ | $y_{t\bar{t}}$ | $0.1193^{+0.0010}_{-0.0012}$ | $176.7^{+1.1}_{-1.1}$ | $1.68$ | $0.1142^{+0.0016}_{-0.0016}$ | $172.6^{+1.1}_{-1.1}$ | $0.74$ |  |  |
$y_{t\bar{t}}$ | $p_{T}^{t}$ | $0.1127^{+0.0015}_{-0.0015}$ | $171.0^{+0.9}_{-0.8}$ | $1.75$ | $0.1097^{+0.0020}_{-0.0020}$ | $170.8^{+0.8}_{-0.8}$ | $0.67$ |  |  |
$y_{t\bar{t}}$ | $m_{t\bar{t}}$ | $0.1132^{+0.0016}_{-0.0015}$ | $168.1^{+1.2}_{-1.3}$ | $3.10$ | $0.1096^{+0.0022}_{-0.0022}$ | $168.5^{+1.2}_{-1.3}$ | $1.39$ |  |  |
$y_{t\bar{t}}$ | $y_{t}$ | $0.1180^{+0.0015}_{-0.0016}$ | $175.0^{+1.1}_{-1.0}$ | $2.05$ | $0.1207^{+0.0021}_{-0.0021}$ | $175.7^{+1.1}_{-1.0}$ | $1.99$ |  |  |
$y_{t\bar{t}}$ | $y_{t\bar{t}}$ | $0.1197^{+0.0011}_{-0.0012}$ | $176.8^{+1.1}_{-1.1}$ | $1.66$ | $0.1145^{+0.0017}_{-0.0017}$ | $172.8^{+1.1}_{-1.1}$ | $0.72$ |  |  |
Table 12: As in table 8 but for the simultaneous extractions of $\alpha_{s}$
and $m_{t}$ from one ATLAS and one CMS distribution.

##  Appendix C Extractions using $m_{t\bar{t}}$ supplemented with underflow
events

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $0.1158^{+0.0020}_{-0.0021}$ | 0.17 | $0.1157^{+0.0027}_{-0.0027}$ | 0.19 | $0.1173^{+0.0022}_{-0.0022}$ | 0.22
$m_{t\bar{t}}^{u}$ | $0.1158^{+0.0020}_{-0.0021}$ | 0.15 | $0.1157^{+0.0027}_{-0.0027}$ | 0.17 | $0.1173^{+0.0022}_{-0.0022}$ | 0.21

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $0.1147^{+0.0014}_{-0.0014}$ | 9.68 | $0.1100^{+0.0014}_{-0.0043}$ | 6.97 | $0.1161^{+0.0018}_{-0.0018}$ | 6.01
$m_{t\bar{t}}^{u}$ | $0.1147^{+0.0014}_{-0.0014}$ | 9.66 | $0.1100^{+0.0014}_{-0.0043}$ | 6.94 | $0.1161^{+0.0018}_{-0.0018}$ | 5.99

Table 13:  As in table 2 but the case with and without underflow events is
considered.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $173.4^{+0.6}_{-0.5}$ | 0.35 | $173.4^{+0.6}_{-0.5}$ | 0.28 | $173.5^{+0.5}_{-0.5}$ | 0.21
$m_{t\bar{t}}^{u}$ | $173.3^{+0.5}_{-0.5}$ | 0.33 | $173.4^{+0.5}_{-0.5}$ | 0.27 | $173.5^{+0.5}_{-0.5}$ | 0.20

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $170.4^{+0.6}_{-0.7}$ | 7.31 | $170.5^{+0.7}_{-0.7}$ | 5.98 | $170.8^{+0.7}_{-0.7}$ | 3.87
$m_{t\bar{t}}^{u}$ | $170.8^{+0.6}_{-0.5}$ | 7.31 | $170.9^{+0.6}_{-0.5}$ | 5.98 | $171.2^{+0.6}_{-0.6}$ | 3.91

Table 14:  As in table 13 but for $m_{t}$ rather than $\alpha_{s}$, which has
been set to the world average value of 0.118.

In table 13 we show the extraction of $\alpha_{s}$ for a fixed value of the
top mass, $m_{t}=173.3$ GeV, using either the invariant mass distribution,
$m_{t\bar{t}}$, or the invariant mass distribution supplemented with underflow
events, $m_{t\bar{t}}^{u}$. In table 14 we show the corresponding table for
the extraction of $m_{t}$ for $\alpha_{s}=0.118$. In the extraction of
$\alpha_{s}$, for all PDF sets and for both experiments it is clear that the
addition of the underflow events only slightly increases the value of
$\chi^{2}_{\rm min}$ and barely affects the best-fit value and the associated
uncertainties. In the extraction of $m_{t}$, when using data from either
experiment, the addition of underflow events again has a very small effect on
$\chi^{2}_{\rm min}$. Interestingly however, in the case of CMS, the best-fit
value of $m_{t}$ is shifted upwards by 0.3 GeV in all cases and the associated
uncertainties decrease.

| ATLAS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $0.1155^{+0.0020}_{-0.0022}$ | $173.1^{+0.6}_{-0.5}$ | 0.16 | $0.1157^{+0.0027}_{-0.0027}$ | $173.2^{+0.6}_{-0.5}$ | 0.19 | $0.1176^{+0.0022}_{-0.0022}$ | $173.5^{+0.5}_{-0.5}$ | 0.21
$m_{t\bar{t}}^{u}$ | $0.1156^{+0.0020}_{-0.0022}$ | $173.1^{+0.5}_{-0.5}$ | 0.13 | $0.1157^{+0.0027}_{-0.0027}$ | $173.2^{+0.5}_{-0.5}$ | 0.17 | $0.1176^{+0.0022}_{-0.0022}$ | $173.4^{+0.5}_{-0.5}$ | 0.20

| CMS
---|---
| CT14 | NNPDF30 | NNPDF31
| $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$ | $\alpha_{s}$ | $m_{t}$ | $\chi^{2}_{\rm min}$
$m_{t\bar{t}}$ | $0.1108^{+0.0013}_{-0.0012}$ | $168.5^{+0.8}_{-0.8}$ | 4.43 | $0.1055^{+0.0021}_{-0.0020}$ | $168.8^{+0.9}_{-0.9}$ | 2.02 | $0.1100^{+0.0023}_{-0.0030}$ | $169.1^{+0.8}_{-0.9}$ | 2.41
$m_{t\bar{t}}^{u}$ | $0.1119^{+0.0014}_{-0.0013}$ | $169.9^{+0.6}_{-0.6}$ | 5.03 | $0.1069^{+0.0022}_{-0.0022}$ | $170.2^{+0.6}_{-0.6}$ | 2.71 | $0.1125^{+0.0020}_{-0.0023}$ | $170.5^{+0.6}_{-0.6}$ | 2.91

Table 15:  As in table 2 but comparing the case with and without underflow
events for the simultaneous extractions of $m_{t}$ and $\alpha_{s}$ using
ATLAS (upper table) and CMS (lower table).

We have also investigated the effects of the addition of underflow events in
simultaneous extractions of $\alpha_{s}$ and $m_{t}$, the results of which are
tabulated for each experiment in table 15. Reflecting the results of the
individual extractions, we see that adding the underflow events has only a
very small effect on the extractions when using ATLAS data. This is not the
case when using the CMS data, where we now observe significant increases in
both the best-fit values of $\alpha_{s}$ and $m_{t}$, with $m_{t}$ in
particular being shifted by almost 1$\sigma$ for all PDFs. These shifts are
consistent within uncertainties, however it is interesting that they push the
extracted values upwards towards the respective world averages.

Perhaps the most surprising feature of the numbers presented in this section
is that the change in the extractions is not the same for both experiments.
This may suggest a different treatment by the two experiments, of underflow
events either in the direct measurements themselves or in the process of the
extrapolation of direct measurements to stable tops. Given that we have found
that different treatments of underflow events can have a significant effect on
extractions of $m_{t}$ and $\alpha_{s}$, this aspect of the measurement of
$m_{t\bar{t}}$ deserves careful consideration in future measurements. This
issue has been considered with respect to resummed calculations in ref.
Ju:2020otc , where similar conclusions were drawn.

##  Appendix D $\alpha_{s}$ and $m_{t}$ averaging procedure

The averaging procedure introduced in section 5.1 works as follows. For the
parameter $\beta=\\{\alpha_{s},m_{t}\\}$ we obtain an average value as

$\bar{\beta}=\frac{\sum_{i}u_{i}\beta_{i}}{\sum_{i}u_{i}}\,,$ (25)

where $\beta_{i}$ is the $i$th measurement of $\beta$. The weights $u_{i}$ are
given by the $p$-values associated with the $\chi^{2}_{\mathrm{min},i}$,

$u_{i}=\int^{\infty}_{\chi^{2}_{\mathrm{min},i}}f(z;n_{\mathrm{d}})\mathrm{d}z\,,$
(26)

where $f(z;n_{\mathrm{d}})$ is the $\chi^{2}$ probability distribution
function and $n_{\mathrm{d}}$ is the appropriate number of degrees of freedom.
The estimated uncertainty of the average $\bar{\beta}$ reads

$\delta\bar{\beta}^{\mathrm{up/down}}=\sqrt{(\delta\beta_{\mathrm{stat}}^{\mathrm{up/down}})^{2}+(\delta\beta_{\mathrm{sys}})^{2}}\,.$
(27)

It includes both statistical and systematic contributions:

$\displaystyle\delta\beta_{\mathrm{stat}}^{\mathrm{up/down}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\sum_{i}u_{i}^{2}(\delta\beta_{i}^{\mathrm{up/down}})^{2}}}{\sum_{i}u_{i}}\,,$
(28) $\displaystyle\delta\beta_{\mathrm{sys}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}u_{i}|\beta_{i}-\bar{\beta}|}{\sum_{i}u_{i}}\,,$
(29)

where $\delta\beta_{i}^{\mathrm{up/down}}$ is the uncertainty on $\beta_{i}$
derived from the condition eq. (10). The pairs
$\left(\bar{\beta}_{i},\delta\beta_{i}^{\mathrm{up/down}}\right)$ are given in
the tables in section 4 and appendices A,B,C.

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