aainstitutetext: Instituto de Física Corpuscular (IFIC, Universitat de
València and CSIC),
C/ Catedrático José Beltrán 2, E-46980 Paterna, Valencia,
Spainbbinstitutetext: PRISMA+ Cluster of Excellence, Johannes Gutenberg
University Mainz,
Staudingerweg 9, 55128 Mainz, Germany

# The top quark electro-weak couplings after LHC Run 2

Víctor Miralles<EMAIL_ADDRESS>a Marcos Miralles López
<EMAIL_ADDRESS>a María Moreno Llácer<EMAIL_ADDRESS>a,b
Ana Peñuelas<EMAIL_ADDRESS>a Martin Perelló
<EMAIL_ADDRESS>a and Marcel Vos<EMAIL_ADDRESS>
###### Abstract

Recent measurements at the Large Hadron Collider allow for a robust and
precise characterisation of the electro-weak interactions of the top quark. We
present the results of a global analysis at next-to-leading order precision
including LHC, LEP/SLD and Tevatron data in the framework of the Standard
Model Effective Field Theory. We include a careful analysis of the impact of
correlations among measurements, as well as of the uncertainties in the
Effective Field Theory setup itself. We find remarkably robust global fit
results, with central values in good agreement with the Standard Model
prediction, and 95% probability bounds on Wilson coefficients that range from
$\pm 0.35$ to $\pm 8$ TeV -2. This result represents a considerable
improvement over previous studies, thanks to the addition of differential
cross-section measurements in associated production processes of top quarks
and neutral gauge bosons.

###### Keywords:

top quark, electro-weak couplings, effective field theory

††preprint: IFIC/21-28

##  1 Introduction

The Standard Model (SM) of particle physics offers an accurate description of
the interactions of elementary particles. A large number of precise
measurements at high-energy colliders test the theory at the TeV scale. The
Standard Model Effective Field Theory (SMEFT) extends the theory with higher
dimension operators compatible with the gauge symmetry and provides a
systematic framework to analyze the wealth of data in terms of potential new
physics.

The top quark, the most massive particle of the SM, has tight connections to
the Higgs boson and plays a prominent role in many extensions of the SM. In
particular, the top and bottom quark electro-weak couplings can be modified in
composite Higgs/extra dimension models and scenarios with vector-like quarks
Richard:2014upa ; Durieux:2018ekg ; Dawson:2020oco . Its properties make it
particularly suited for experimental investigation and a rich experimental
programme has developed around the top quark since its discovery in 1995
PhysRevLett.74.2626 ; D0:1995jca . The Tevatron experiments could characterize
the top quark interactions with the gluon, through the QCD production
mechanism, and with the $W$ boson, through the electro-weak decay
$t\rightarrow Wb$ and single top quark production processes. The Large Hadron
Collider (LHC) experiments provide new measurements of these traditional
probes with increased precision and in an extended kinematic regime, and have
observed rare associated production processes that provide a direct probe for
the couplings of the top quark with the photon, the $Z$ boson and the Higgs
boson.

Compared to previous Effective Field Theory (EFT) analyses of the top quark
sector Durieux:2019rbz ; Brivio:2019ius ; Hartland:2019bjb ; Buckley:2015nca ;
Buckley:2015lku , we extend the set of measurements considerably. We
incorporate recent differential measurements in $pp\rightarrow t\bar{t}\gamma$
and $pp\rightarrow t\bar{t}Z$ production, and measurements of the
$pp\rightarrow t\bar{t}W$, $pp\rightarrow t\bar{t}H$, $pp\rightarrow tZq$ and
$pp\rightarrow t\gamma q$ inclusive cross-sections. The precise measurements
of the $Z\rightarrow b\bar{b}$ vertex at LEP and SLD ALEPH:2005ab  remain very
relevant and are included, as well as legacy Tevatron measurements.

In this paper we use this large set of measurements at high-energy colliders
to provide a complete characterisation of the electro-weak couplings of the
top quark. We derive bounds on eight Wilson coefficients of the relevant
dimension-six two-fermion operators. An important effort is made to ensure
that the results are robust. Importantly, we estimate the correlations between
measurements and evaluate their impact on the fit results. The main
limitations of the SMEFT analysis, due to the truncation of the expansion at
dimension-six and due to the suppression of certain degrees of freedom, are
also addressed in detail. As the main result of this paper, we present robust
limits on the effects of high-scale phenomena beyond the SM on the top quark
electro-weak couplings that include the most recent LHC Run 2 results.

Several groups have performed combined EFT fits of the top, Higgs and electro-
weak sectors Ellis:2020unq ; Ethier:2021bye , while others have explored the
interplay between top quark physics and the flavour factories
Bruggisser:2021duo ; Bissmann:2020mfi ; Bissmann:2019gfc ; Cirigliano:2016nyn
; Alioli:2017ces . In addition, three recent studies explore events involving
top quarks and $Z$ bosons to further constrain some electro-weak couplings
CMS:2021rvz ; Keaveney:2021dfa ; Ravina:2021kpr .

##  2 Effective Field Theory framework

Effects of new physics in the couplings of the top quarks can be described as
effective interactions of SM particles at energies below a new physics
matching scale $\Lambda$. These effective interactions can be parameterised in
terms of a set of Wilson coefficients $C_{i}$ of dimension-six operators
$O_{i}$ in the effective Lagrangian,

$\mathcal{L}_{\text{eff}}=\mathcal{L}_{\text{SM}}+\left(\frac{1}{\Lambda^{2}}\sum_{i}\mathcal{C}_{i}O_{i}+\text{h.c.}\right)+\mathcal{O}\left(\Lambda^{-4}\right),$
(1)

where the sum runs over a total of ten operators that involve top quarks, as
described below, and which can be interpreted in terms of new physics
mediators. This EFT preserves the local and gauge symmetries of the SM, and
operators with odd dimension are omitted since they will violate baryon or
lepton number.

The SMEFT predictions of a physical quantity, $X$, can be written in this
form:

$\begin{array}[]{cc}X=X_{SM}+\sum_{i}\frac{\mathcal{C}_{i}}{\Lambda^{2}}X_{i}^{(1)}+\sum_{ij}\frac{\mathcal{C}_{i}\mathcal{C}_{j}}{\Lambda^{4}}X_{ij}^{(2)}+\mathcal{O}\left(\Lambda^{-4}\right)=\\\
X_{SM}\times\left(1+\sum_{i}\frac{\mathcal{C}_{i}}{\Lambda^{2}}\frac{X_{i}^{(1)}}{X_{SM}}+\sum_{ij}\frac{\mathcal{C}_{i}\mathcal{C}_{j}}{\Lambda^{4}}\frac{X_{ij}^{(2)}}{X_{SM}}\right)+\mathcal{O}\left(\Lambda^{-4}\right).\end{array}$
(2)

This expression contains the SM part, the interferences of the effective
dimension-six operators with the SM (referred to as “linear terms”) which are
proportional to $\mathcal{C}_{i}/\Lambda^{2}$, and “quadratic terms”
proportional to $\mathcal{C}_{i}\mathcal{C}_{j}/\Lambda^{4}$. The effects of
dimension-eight operators and the so-called “double insertions” terms of
dimension-six operators that contribute at $\Lambda^{-4}$ order are not
included. To assess the validity of the EFT expansion, we compare fits with
and without $\Lambda^{-4}$ terms, as recommended by the LHC Top Working Group
AguilarSaavedra:2018nen , and discuss the robustness of the bounds in detail
in Sections 7 and 8.

The number of operators involved in the most general EFT description is
prohibitive for an analysis of this type. Therefore, here we select a subset
of operators that provides an adequate basis for a study of new physics
effects in the top and bottom quark electro-weak couplings. We focus on the
two-fermion operators that affect top and bottom quark interactions with the
vector, tensor or scalar Lorentz structures listed below, and we use the
Warsaw basis Grzadkowski:2010es  (see also Refs. AguilarSaavedra:2008zc ;
Zhang:2010dr ) following the conventions proposed by the LHC Top Working Group
AguilarSaavedra:2018nen :

$\begin{array}[]{@{}rlcc@{}}\underline{O_{\varphi
Q}^{1}}&\equiv\frac{1}{2}\left(\bar{\mathsfit{q}}\gamma^{\mu}\mathsfit{q}\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right),\\\
O_{\varphi
Q}^{3}&\equiv\frac{1}{2}\left(\bar{\mathsfit{q}}\tau^{I}\gamma^{\mu}\mathsfit{q}\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}^{I}\\!\\!\\!_{\mu}\>\varphi\right),\\\
O_{\varphi Q}^{-}&\equiv O_{\varphi Q}^{1}-O_{\varphi Q}^{3},\\\ O_{\varphi
u}&\equiv\frac{1}{2}\left(\bar{\mathsfit{u}}\gamma^{\mu}\mathsfit{u}\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right),\\\
O_{\varphi
d}&\equiv\frac{1}{2}\left(\bar{\mathsfit{d}}\gamma^{\mu}\mathsfit{d}\right)\left(\varphi^{\dagger}i\\!\overleftrightarrow{D}\\!_{\mu}\varphi\right),\\\
O_{\varphi
ud}&\equiv\frac{1}{2}\left(\bar{\mathsfit{u}}\gamma^{\mu}\mathsfit{d}\right)\left(\varphi^{T}\epsilon
iD_{\mu}\varphi\right),\end{array}\quad\begin{array}[]{@{}rlcc@{}}O_{uW}&\equiv\left(\bar{\mathsfit{q}}\tau^{I}\sigma^{\mu\nu}\mathsfit{u}\right)\left(\epsilon\varphi^{*}W_{\mu\nu}^{I}\right),\\\
O_{dW}&\equiv\left(\bar{\mathsfit{q}}\tau^{I}\sigma^{\mu\nu}\mathsfit{d}\right)\left(\varphi
W_{\mu\nu}^{I}\right),\\\
\underline{O_{uB}}&\equiv\left(\bar{\mathsfit{q}}\sigma^{\mu\nu}\mathsfit{u}\right)\left(\epsilon\varphi^{*}B_{\mu\nu}\right),\\\
\underline{O_{dB}}&\equiv\left(\bar{\mathsfit{q}}\sigma^{\mu\nu}\mathsfit{d}\right)\left(\varphi
B_{\mu\nu}\right),\\\
O_{uZ}&\equiv-\sin\theta_{W}O_{uB}+\cos\theta_{W}O_{uW},\\\
O_{dZ}&\equiv-\sin\theta_{W}O_{dB}+\cos\theta_{W}O_{dW},\\\
O_{u\varphi}&\equiv\left(\bar{\mathsfit{q}}\mathsfit{u}\right)\left(\epsilon\varphi^{*}\;\varphi^{\dagger}\varphi\right),\\\
O_{d\varphi}&\equiv\left(\bar{\mathsfit{q}}\mathsfit{d}\right)\left(\epsilon\varphi^{*}\;\varphi^{\dagger}\varphi\right).\end{array}$
(3)

The underlined operators $O_{\varphi Q}^{1},O_{uB}$ and $O_{dB}$ are not used
directly, but are present in the linear combinations $O_{\varphi Q}^{-}$,
$O_{uZ}$ and $O_{dZ}$. The operators $O_{\varphi Q}^{1}$ and $O_{\varphi
Q}^{3}$ modify the left-handed couplings of the $Z$ boson to down-type and up-
type quarks. Two further operators $O_{\varphi u}$ ($O_{\varphi d}$) modify
the right-handed couplings of the $Z$ boson to up-type (down-type) quarks. The
electro-weak dipole operators labeled $O_{uW}$ ($O_{dW}$) and $O_{uZ}$
($O_{dZ}$) give rise to tensor couplings of the photon and $Z$ boson to the
up-type (down-type) quarks and induce an anomalous dipole moment. The
$O^{3}_{\varphi Q}$ and $O_{uW}$ operators also modify the charged-current
interactions of the up-type quarks with a $W$ boson and left-handed down-type
quark, while $O_{\varphi ud}$ and $O_{dW}$ give rise to interactions between
the up-type quarks with a $W$ boson and right-handed down-type quark. Finally,
$O_{u\varphi}$ and $O_{d\varphi}$ modify the Yukawa couplings of up-type and
down-type quarks.

Operators that modify only the bottom quark electro-weak couplings,
$O_{\varphi d}$ and $O_{dZ}$, are taken into account in the fit since we
include measurements from electro-weak precision data, but limits on their
coefficients are not reported in this paper since the obtained values are not
competitive enough using only the observables considered in our fit. The
operator that shifts the bottom quark Yukawa, $O_{d\varphi}$, is not included
in the fit, as the measurements considered here are not sensitive to it.

For the fits without (with) $\Lambda^{-4}$ terms, we report results for six
(eight) operator coefficients that characterize the top quark couplings to the
$Z$ boson, the photon and the Higgs boson and the charged-current $tWb$
vertex. Substituting $u$ for $t$ and $d$ for $b$, the list reads:

$C_{\varphi Q}^{-},C_{\varphi Q}^{(3)},C_{\varphi
t},C_{tW},C_{tZ},C_{t\varphi},(C_{\varphi tb},C_{bW}).$

This choice of basis omits several degrees of freedom. The CP-violating
imaginary parts of those operators are left for future work. The four-fermion
operators with two light quarks and two heavy quarks ($qqQQ$), and the two-
fermion operator
$O_{tG}=\left(\bar{q}\sigma^{\mu\nu}T^{A}u\right)\left(\epsilon\varphi^{*}G_{\mu\nu}^{A}\right)$,
that modifies the $t\bar{t}g$ vertex, are not included in the baseline fit,
but their impact is studied in Section 8. Finally, four-fermion operators with
two leptons and two heavy quarks ($\ell\ell QQ$) are not included. These can
be constrained with carefully constructed measurements at the LHC
Sirunyan:2020tqm  and through loop effects in B-factories Bissmann:2020mfi ;
Bruggisser:2021duo . Very strong bounds can be derived at a future lepton
collider Durieux:2019rbz .

##  3 Measurements

In this work, we include for the first time the differential cross-section
measurements of $t\bar{t}Z$ and $t\bar{t}\gamma$ processes (sensitive to top
quark neutral-current interactions). Inclusive cross-section measurements are
considered for $t\bar{t}W$ and $t\bar{t}H$ production, single top production
in the t-channel ($tq$), associated production ($tW$) and s-channel
($t\bar{b}$), and single top quark production in association with $Z$ boson or
a photon ($tZq$ and $t\gamma q$). We also include $W$ boson helicity fractions
(sensitive to charged-current interactions). Finally, measurements of $R_{b}$
and $A_{FBLR}^{b\bar{b}}$ in bottom quark pair production at the $Z$-pole from
the LEP and SLD experiments (sensitive to operators that affect the left-
handed couplings of the top and bottom quarks) are included. The selected
measurements are summarized in Table 1. Counting the bins in the differential
cross-section measurements, the total number of observables ($n_{\text{obs}}$)
is 30.

Process | Observable | $\sqrt{s}$ | $\int\cal{L}$ | Experiment | SM | Ref.
---|---|---|---|---|---|---
$pp\rightarrow t\bar{t}H+tHq$ | $\sigma$ | 13 TeV | 140 fb-1 | ATLAS | deFlorian:2016spz  | ATLAS:2020qdt
$pp\rightarrow t\bar{t}Z$ |  $d\sigma/dp_{T}^{Z}$ (7 bins) | 13 TeV | 140 fb-1 | ATLAS | Broggio:2019ewu  | ATLAS:2020cxf
$pp\rightarrow t\bar{t}\gamma$ |  $d\sigma/dp_{T}^{\gamma}$ (11 bins) | 13 TeV | 140 fb-1 | ATLAS | Bevilacqua:2018woc ; Bevilacqua:2018dny  | Aad:2020axn
$pp\rightarrow tZq$ | $\sigma$ | 13 TeV | 77.4 fb-1 | CMS | Sirunyan:2017nbr  | Sirunyan:2018zgs
$pp\rightarrow t\gamma q$ | $\sigma$ | 13 TeV | 36 fb-1 | CMS | Sirunyan:2018bsr  | Sirunyan:2018bsr
$pp\rightarrow t\bar{t}W$ | $\sigma$ | 13 TeV | 36 fb-1 | CMS | deFlorian:2016spz ; Frederix:2017wme  | Sirunyan:2017uzs
$pp\rightarrow t\bar{b}$ (s-ch) | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Aliev:2010zk ; Kant:2014oha  | Aaboud:2019pkc
$pp\rightarrow tW$ | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Kidonakis:2010ux  | Aaboud:2019pkc
$pp\rightarrow tq$ (t-ch) | $\sigma$ | 8 TeV | 20 fb-1 | LHC | Aliev:2010zk ; Kant:2014oha  | Aaboud:2019pkc
$t\rightarrow Wb$ |  $F_{0}$, $F_{L}$ | 8 TeV | 20 fb-1 | LHC | PhysRevD.81.111503  | Aad:2020jvx
$p\bar{p}\rightarrow t\bar{b}$ (s-ch) | $\sigma$ | 1.96 TeV | 9.7 fb-1 | Tevatron | PhysRevD.81.054028  | CDF:2014uma
$e^{-}e^{+}\rightarrow b\bar{b}$ |  $R_{b}$ , $A_{FBLR}^{bb}$ |  $\sim$ 91 GeV | 202.1 pb-1 | LEP/SLD | - | ALEPH:2005ab

Table 1: Measurements included in the EFT fit of the top quark electro-weak
sector. For each measurement, the process, the observable, the center-of-mass
energy, the integrated luminosity and the experiment/collider are given. The
last two columns list the references for the predictions and measurements that
are included in the fit. LHC refers to the combination of ATLAS and CMS
measurements. In a similar way, Tevatron refers to the combination of CDF and
D0 results, and LEP/SLD to different experiments from those two accelerators.

For LHC observables at $\sqrt{s}=$ 13 TeV where no official combinations are
available, measurements either from ATLAS or CMS experiment (but not both) are
included. In case results are available for several final states depending on
the decay mode of the top quarks or associated bosons, measurements are chosen
to be (nearly) orthogonal. In addition, for observables that have been
measured at multiple center-of-mass energies, only one measurement is included
in order to avoid issues with correlations. The one exception is the
$s$-channel single top quark production, where both Tevatron and LHC results
are included since they have similar sensitivity and are independent data
sets. In most cases, the latest Run 2 results at 13
${\mathrm{Te\kern-1.00006ptV}}\,$ are chosen, but for the measurements of the
single top quark cross-sections and $W$ boson helicity fractions the combined
(ATLAS+CMS) LHC 8 ${\mathrm{Te\kern-1.00006ptV}}\,$ results are preferred. For
two single top quark production channels, $pp\rightarrow tq$ and
$pp\rightarrow tW$, the cross-sections have also been measured at 13 TeV
(although no LHC combinations are available). We have checked that including
such measurements at 13 TeV, in addition to those at 8 TeV, has a negligible
impact in our limits. This check reinforces our strategy of including only the
measurement at one center-of-mass energy.

Even among measurements of different processes, correlations might have a non-
negligible effect, as experimental and modelling systematic uncertainties have
common sources. The theory predictions can also be correlated, through the
parton density functions and the similarity of the matrix elements of the
several associated production processes. When available, we have included
published experimental correlations, for instance for the LEP and SLD
measurements ALEPH:2005ab  and the $W$ boson helicity fractions Aad:2020jvx .
In the case of $t\bar{t}\gamma$ and $t\bar{t}Z$ differential cross-section
measurements Aad:2020axn ; ATLAS:2020cxf  as function of the gauge boson
transverse momentum, the correlations among bins provided by the experiments
are also included. For the remaining observables, we have defined an ansatz
correlation matrix to test the robustness of the fit results (see Section 8).

Several processes deserve some further discussion. The $t\bar{t}\gamma$ cross-
section measurement receives contributions from the $tW\gamma$ process and
from $t\bar{t}$ events, where one of the top quark decay products (_i.e._ the
$b$ quark or $W$ boson or its decay products) emits a photon. Events without a
top-photon vertex dilute the sensitivity to the EFT operator coefficients
Bevilacqua:2019quz . The fiducial region definition and the differential
analysis in the ATLAS measurement Aad:2020axn  mitigate their impact, as the
photons emitted by decay products tend to be close to jets and are
predominantly soft. We account for $tW\gamma$ and photons from top decay
products fully in the SM prediction, but ignore these contributions in the EFT
parameterisation.

Similarly, the $t\bar{t}H$ cross-section measurement includes also $tH$ (due
to the difficulty to disentangle the two processes experimentally). The latest
ATLAS combination of several final states has been used ATLAS:2020qdt . The
$tH$ contribution is included in the SM prediction and in the EFT
parameterisation.

The $t\bar{t}W$ measurement from the CMS experiment Sirunyan:2017uzs  targets
the multi-lepton final state and is inclusive in the number of jets. Thus, the
parameterisation considers the $pp\rightarrow t\bar{t}W$ process at NLO in QCD
and additionally explicitly takes into account the contribution of the
electro-weak $t\bar{t}Wq$ production, as done in Ref. Banelli:2020iau . The
latter process includes $tW\rightarrow tW$ scattering diagrams which have a
strong sensitivity to $C_{\varphi t}$ operator Dror:2015nkp .

##  4 Fit setup

The dependence of the observables included in the fit on the Wilson
coefficients, that is the $X_{i}^{(1)}/X_{SM}$ and $X_{ij}^{(2)}/X_{SM}$ terms
in Eq. 2, is evaluated with the Monte Carlo generator MadGraph5_aMC@NLO
v.2.7.0 Alwall:2014hca  using a fixed order calculation except for $t\to Wb$
decay for which the analytical calculation at NLO is used Zhang:2014rja . Two
UFO models are employed. The SMEFT@NLO111Either
SMEFTatNLO_U2_2_U3_3_cG_4F_LO_UFO or v1.0.1 are used. model Degrande:2020evl
is used for most of the operators and the predictions are derived at NLO in
QCD, except for the observables from LEP, SLD and Tevatron colliders which are
parameterised at LO. The parameterisation for the bottom-quark operators
($\mathcal{O}_{bW}$, $\mathcal{O}_{\varphi tb}$, $\mathcal{O}_{bZ}$ and
$\mathcal{O}_{\varphi b}$) are obtained with the TEFT_EW model Bylund:2016phk
. Both linear and quadratic terms are considered, as well as the interference
between the different operators.

The SMEFT@NLO model uses the $m_{W},m_{Z},G_{F}$ input electro-weak scheme
while the TEFT_EW model uses the $\alpha_{EM}^{-1}(m_{Z}),m_{Z},G_{F}$ scheme:

$\alpha_{EM}^{-1}(m_{Z})=127.95,\qquad G_{F}=1.16637\times
10^{-5}~{}\text{GeV}^{-2},$ $m_{t}=173.3~{}\text{GeV},\quad
m_{H}=125~{}\text{GeV},\quad m_{Z}=91.1876~{}\text{GeV},\quad
m_{W}=79.8244~{}\text{GeV}.$

The other fermion masses are taken to be zero.

The value of $X_{\text{SM}}$ at the beginning of Eq. 2 is scaled to the best
theoretical prediction for each observable, as documented in the references
given in Table 1.

The fit to data is performed using the open source HEPfit package
deBlas:2019okz ; hepfitsite . HEPfit is a general tool designed to combine
direct and indirect constraints in the SM deBlas:2016ojx , in EFT
deBlas:2018tjm ; Durieux:2019rbz  or in particular BSM extensions
Eberhardt:2020dat ; Eberhardt:2021ebh . Its flexibility allows to easily
implement any BSM model or observable. The fit is performed as a Bayesian
statistical analysis of the model, in which both theoretical and experimental
uncertainties are included. HEPfit includes a Markov Chain Monte Carlo
implementation provided by the Bayesian Analysis Toolkit Caldwell:2008fw  to
explore the parameter space.

##  5 Compatibility with the SM predictions

Prior to the SMEFT fit, the quality of the fit of the selected observables
($n_{\text{obs}}$ = 30) to the SM predictions has been investigated. The
correlations among the different data points within a given process, such as
the different bins in the $t\bar{t}Z$ and $t\bar{t}\gamma$ differential cross-
section measurements, the two $W$ boson helicity fractions ($F_{0}$ and
$F_{L}$) and the LEP/SLD observables of the $Zb\bar{b}$ vertex, were published
by the experiments and thus are considered in the fit. The obtained chi-square
value is $\chi^{2}_{\text{SM}}/(n_{\text{obs}}-1)=$ 21.3/29, corresponding to
a $p$-value of 0.85. Overall, a good agreement between the SM predictions with
the experimental results is seen. However, some observables exhibit some
tension with the SM predictions, such as few of the $p_{T}$ bins of the
$t\bar{t}Z$ and $t\bar{t}\gamma$ cross-section measurements. Figure 1 shows
the $\chi^{2}_{\text{SM}}$ values for some of the input measurements as well
as their discrepancy.

Figure 1: Contributions to the $\chi^{2}_{\text{SM}}$ of some of the input
measurements, together with their discrepancy with respect to the SM
predictions. The full fit has $30$ data points and a cumulative
$\chi^{2}_{\text{SM}}\sim 21.3$. The negative values indicate that the
prediction is lower than the measurement. Both theoretical and experimental
uncertainties are considered.

##  6 Interplay between measurements and operator coefficients

Figure 2 shows the individual 95% probability bounds on the eight operator
coefficients considered in our fit with $\Lambda^{-4}$ terms. The bounds from
measurements in different processes, ordered from most to least constraining
(going from left to right), are presented for each coefficient.

Figure 2: Individual constraints on the eight Wilson coefficient resulting
from measurements in different processes. The y-axis corresponds to the full
width of the 95% probability interval for fits including $\Lambda^{-4}$ terms.
The observables are set to their SM predictions, maintaining the experimental
uncertainty. The dark shading indicates the bound from the $t\bar{t}\gamma$
and $t\bar{t}Z$ differential cross-sections. Besides QCD production, in the
case of $t\bar{t}W$ process, the contribution of the electro-weak $t\bar{t}Wq$
production is also included.

The associated production processes of a top quark pair with a $Z$ boson or a
photon play an important role in the fit, especially for $C_{\varphi t}$ and
$C_{tZ}$ where they provide the leading constraints. For these processes, the
bounds obtained from the inclusive cross-section, indicated by the full length
of the bar, are compared to the differential measurements, indicated with a
darker shading. The bound from the differential measurement is considerably
more stringent, especially for $t\bar{t}\gamma$, thanks to the enhanced
sensitivity in events where the boson is emitted with a large transverse
momentum.

The existing data provide multiple constraints on the different operator
coefficients. The one exception is $C_{t\varphi}$, the coefficient that
modifies the top quark Yukawa coupling, that is only bounded by the
measurement of the $t\bar{t}H$ production rate in our fit. In a fit including
more Higgs data, additional constraints arise from the loop diagrams in
$gg\rightarrow H$ production and $H\rightarrow\gamma\gamma$ decay222These two
processes give quite competitive individual bounds, but these are not very
robust when all other Higgs operators affected by those processes are included
in the analysis Vryonidou:2018eyv ; Durieux:2018ggn ; Jung:2020uzh . Recent
global fits of top, Higgs and electro-weak data Ellis:2020unq ; Ethier:2021bye
confirm this pattern..

The $Z$-pole data collected by the LEP/SLD experiments, as included in the
electro-weak precision observables ALEPH:2005ab , continue to yield very tight
bounds on three operator coefficients: $C_{\varphi Q}^{3}$, $C_{\varphi
Q}^{-}$ and $C_{bW}$. The bound on the former two is multiplied by a factor 10
to facilitate the comparison.

The electro-weak dipole operator $C_{tW}$ is mainly constrained by $W$ boson
helicity fractions ($F_{0}$ and $F_{L}$) and also by the $t\bar{t}\gamma$
differential cross-sections. The charged current interactions to right-handed
bottom quarks, described by $C_{\varphi tb}$ operator, are difficult to
constrain and recent $tZq$ cross-section measurements are as sensitive as the
$F_{0}$ and $F_{L}$ fractions.

Figure 3: Two-dimensional 95% probability bounds on pairs of Wilson
coefficients, $C_{\varphi Q}^{-}$ and $C_{\varphi t}$ in the upper leftmost
panel, $C_{tW}$ and $C_{tZ}$ in the upper rightmost panel, $C_{\varphi Q}^{3}$
and $C_{tW}$ in the middle leftmost panel, $C_{tW}$ and $C_{\varphi tb}$ in
the middle rightmost panel, $C_{\varphi Q}^{3}$ and $C_{\varphi Q}^{-}$ in the
lower leftmost panel, and $C_{\varphi tb}$ and $C_{\varphi Q}^{3}$ in the
lower rightmost panel. Bounds are presented for two-parameter fits to the most
constraining measurements. The global fit results, marginalising over all
other Wilson coefficients, are also shown (red area). All these fits include
$\Lambda^{-4}$ terms. Besides QCD production, in the case of $t\bar{t}W$
process, the contribution of the electro-weak $t\bar{t}Wq$ production is also
included.

The power of the measurements to simultaneously constrain multiple Wilson
coefficients is illustrated in Figure 3. The 95% probability bounds obtained
in two-parameter fits to a single measurement are shown as areas of different
colours. The red area shows the result of the global fit (ten-operator
coefficients), where the result is marginalised over the remaining operator
coefficients.

The limits on the left-handed couplings of the $Z$ boson to quarks, rescaled
by $C_{\varphi Q}^{3}$ and $C_{\varphi Q}^{-}$, are dominated by the LEP and
SLD electro-weak precision measurements in $Z\rightarrow b\bar{b}$ decay
although they only constrain a linear combination. This linear combination is
disentangled once we add other processes like $tZq$, $tq$ or $t\bar{t}Z$. The
strongest limit on $C_{\varphi t}$, right-handed couplings of the quarks to
the $Z$ boson, is obtained from the measurements of the associated $t\bar{t}Z$
production cross-section. The $t\bar{t}Z$ cross-section also yields a powerful
constraint on $C_{tZ}$ coefficient. The helicity fractions of the $W$ boson in
top quark decays provide a tight bound on $C_{tW}$ and $C_{\varphi tb}$, while
the $t\bar{t}\gamma$ cross-section provides a complementary constraint on a
linear combination of the two dipole operator coefficients.

In both planes in the top, one can see that the differential measurements of
the associated $t\bar{t}Z$ and $t\bar{t}\gamma$ production processes improve
the bounds very considerably with respect to the inclusive cross-sections. The
differential $t\bar{t}Z$ cross-section disentangles $C_{\varphi Q}^{-}$ and
$C_{\varphi t}$, and improves the limit on $C_{tZ}$ by a factor two. The
differential $t\bar{t}\gamma$ cross-section restricts the allowed area in the
$C_{tW}-C_{tZ}$ plane.

The $tZq$ process and the $W$ boson helicity fractions constrain the
$C_{\varphi tb}-C_{tW}$ plane giving complementary limits. A similar behaviour
can be seen in the $C_{\varphi tb}-C_{\varphi Q}^{3}$ plane, where the limits
obtained from $tZq$ and $tq$ processes are complementary and once they are
combined a much stronger constraint is obtained.

##  7 Bounds of a global EFT fit (to all operator coefficients)

The main results of this work are the 68% and 95% probability intervals for
the Wilson coefficients that modify the top quark electro-weak couplings. Two
set of results are given for global fits with $O(\Lambda^{-2})$ and
$O(\Lambda^{-4})$ terms, as shown in Figure 4, in orange and blue
respectively. In this figure we also show the result for different stress
tests that we have performed to test the robustness of the results. The dotted
light brown lines show the result increasing the basis of the linear baseline
fit with $C_{tG}$ and seven additional four-fermion operators. The dotted dark
brown lines show the result of including correlations between the observables
considered in the linear baseline fit. Finally, the red lines show the
envelope of all the stress tests that we have performed. The discussion of
these stress tests is postponed to the next section. The coefficients
$C_{\varphi tb}$ and $C_{bW}$ coefficients can not be constrained in the
linear fit since their $X_{i}^{(1)}$ terms in Eq. 2 vanish in the limit
$m_{b}\rightarrow 0$.

Overall, the two global fits with $O(\Lambda^{-2})$ and $O(\Lambda^{-4})$
yield comparable results and all of the coefficients are centered at zero
within 95% probability intervals. The difference between the two sets of
results is sizeable only for the dipole operator coefficient $C_{tZ}$. This
bound is dominated by the $t\bar{t}\gamma$ and $t\bar{t}Z$ differential cross-
section measurements, where the term proportional to $\Lambda^{-2}$ is known
to be strongly suppressed Bylund:2016phk . A small shift (within 95%
probability intervals) is seen for $C_{\varphi t}$ driven by the sixth $p_{T}$
bin of the $t\bar{t}Z$ cross-section measurement in which data are below the
prediction. In addition, the $C_{\varphi Q}^{3}$ and $C_{\varphi Q}^{-}$
coefficients are slightly shifted in opposite directions. The dependence of
the fit result on terms proportional to $\Lambda^{-4}$ is significantly
reduced compared to previous analyses Durieux:2019rbz . As the LHC
measurements become more precise, the bounds remain more and more in the range
where the linear terms are dominant and the fit results are valid in full
generality.

The obtained limits (95% probability intervals) for the fitted operator
coefficients are presented in Table 2. For completeness, the limits obtained
when fitting only one parameter and fixing the others to zero (individual
fits) are also shown in this table. The obtained bounds for $C_{\varphi
Q}^{3}$ and $C_{\varphi Q}^{-}$ in the individual fits are much better than
for the global fits. For the rest of coefficients, the limits do not degrade
that much in the global fit.

Figure 4: Results of the global fits with $O(\Lambda^{-2})$ (in orange) and
$O(\Lambda^{-4})$ (in blue) terms. The two thin lines below each of them
correspond to additional fits performed to test the robustness of the results:
to account for the effects of the inclusion of further operators and for the
correlations between all the different measurements (as described in Section
8). The red markers correspond to the envelope of those additional fits plus
another one that accounts for the theoretical uncertainties on the
parameterisations.

$C/\Lambda^{2}$ | Linear (95% probability) | Lin.+Quad. (95% probability) | (95% probability)
---|---|---|---
(TeV-2) | Individual | Global-Baseline | Individual | Global-Baseline | Global-Robust
$C_{t\varphi}$ | [-3.17, 3.47] | [-3.13, 3.63] | [-3.05, 4.05] | [-2.82, 4.92] | [-121.82, 62.82]
$C_{\varphi Q}^{-}$ | [-0.038, 0.079] | [-2.84, 0.78] | [-0.038, 0.079] | [-2.42, 1.62] | [-2.84, 1.62]
$C_{\varphi Q}^{3}$ | [-0.019, 0.040] | [-0.41, 1.39] | [-0.019, 0.040] | [-0.94, 0.81] | [-0.94, 1.39]
$C_{\varphi t}$ | [-6.6, 1.8] | [-8.96, 0.96] | [-8.6, 1.5] | [-9.01, 1.11] | [-37.50, 21.50]
$C_{tW}$ | [-0.30, 0.38] | [-0.26, 0.44] | [-0.28, 0.32] | [-0.19, 0.50] | [-0.35, 0.50]
$C_{tZ}$ | [-0.82, 2.21] | [-0.75, 2.37] | [-0.39, 0.57] | [-0.35, 0.88] | [-2.43, 3.53]
$C_{\varphi tb}$ | – | – | [-6.61, 6.71] | [-7.55, 7.05] | –
$C_{bW}$ | – | – | [-0.47, 0.47] | [-0.91, 0.91] | –

Table 2:  Allowed ranges of the Wilson coefficients with a probability of 95%
expressed in TeV-2 including only linear terms or linear and quadratic terms.
We show, from left to right, the results of five fits: individual with linear
terms, global baseline with linear terms, individual with linear and quadratic
terms, global baseline with linear and quadratic terms and global robust
limits. The robust result accounts for the effects of the correlations between
the observables, the inclusion of further operators and the theoretical
uncertainties on the parameterisations.

The EFT fit with terms $O(\Lambda^{-4})$, with ten floating Wilson
coefficients, returns a log-likelihood value $\log{L_{\text{EFT}}}=-10.1$ (see
Table 3) and the additional degrees of freedom do not improve the already
excellent agreement of the fit. The observables that contribute most to the
log-likelihood are the differential cross-section measurements of $t\bar{t}Z$
($\Delta\log{L}=5.8$) and $t\bar{t}\gamma$ ($\Delta\log{L}=2.7$) processes as
a function of the boson transverse momentum, with 7 and 11 bins respectively.
Similar values are obtained for the fit with $O(\Lambda^{-2})$ terms.

Fit | Correlation | $\log L$ | $\chi^{2}/\text{d.o.f}$ |  $p$-value
---|---|---|---|---
SM | All published | -10.6 | 21.3/29 | 0.85
SM | \+ our ansatz | -11.6 | 23.2/29 | 0.77
EFT Quad. | All published | -10.1 | 20.1/19 | 0.39
EFT Quad. | \+ our ansatz | -10.2 | 20.4/19 | 0.37
EFT Lin. | All published | -10.7 | 21.5/22 | 0.49
EFT Lin. | \+ our ansatz | -11.0 | 22.1/22 | 0.45
Table 3: Values of $\log L,\chi^{2}/\text{d.o.f}$ and $p$-values for different
assumptions on the correlations among the measurements in the SM and EFT fits.
“All published” include the theoretical correlations and the published
experimental correlations of the $t\bar{t}Z$, $t\bar{t}\gamma$, $F_{0}-F_{L}$
and LEP/SLD measurements. The number of degrees of freedom (d.o.f.) is the
number of observables ($n_{\text{obs}}$) minus the number of fitted parameters
and minus one. We assume Gaussian errors to extract the $\chi^{2}$ from $\log
L$.

A convenient metric to quantify the strength of the constraints in the
$n$-dimensional parameter space of effective-operator coefficients is the so-
called global determinant parameter (GDP) defined as the $2n$ root of the
Gaussian covariance matrix determinant Durieux:2017rsg : GDP = $\sqrt[2n]{\det
V}$. This figure of merit measures the hypervolume of the allowed parameter
space, taking into account the correlations among coefficients. As it assumes
that the errors are Gaussian, we cannot use this determinant in the fit
including $O(\Lambda^{-4})$ terms but it is still interesting to show the
results for the fit including only $O(\Lambda^{-2})$ terms. For the partial
Run 2 result that we reported in 2019 Durieux:2019rbz  we obtain a GDP score
of 1.37. The new Run 2 data included in the current fit improves the GDP to
0.58 with inclusive measurements only, and to 0.49 when the differential
measurements are used. The bounds on the Wilson coefficients are improved, on
average, to 70% thanks to the addition of the new LHC Run 2 measurements.

##  8 Robust bounds

In this section, a number of effects have been investigated to test the
validity of the obtained limits. We assess the quantitative impact of the
inclusion of further degrees of freedom of the SMEFT, of correlations among
the different observables, and of missing higher-orders (uncertainty on the
scale choice) for the parameterisation between the observables and the Wilson
coefficients.

###  8.1 Theory uncertainties on the parameterisation

Theoretical uncertainties due to missing higher orders in $\alpha_{s}$ in the
predictions for the terms $X_{i}^{(1)}/X_{SM}$ and $X_{ij}^{(2)}/X_{SM}$ in
Eq. 2, that parameterise the dependence of the observables on the Wilson
coefficients, have also been considered. However, this effect does not have an
important impact on the fit (maximum of 5$\%$) given that most of the
predictions are extracted at NLO.

###  8.2 Limitations of the basis

The set of operators included in the fit ignores the impact of the two-fermion
operator $C_{tG}$ and of the four-fermion operators with two light quarks and
two heavy quarks. In global fits, these are constrained by the precise
$t\bar{t}$ cross-section measurements at the Tevatron and LHC, but may still
have an impact on the $t\bar{t}X$ production cross-sections. The sensitivity
to these operator coefficients is assessed by extending the operator basis
with $C_{tG}$ and the seven four-fermion operators that affect the $t\bar{t}X$
processes included in the analysis via $O(\Lambda^{-2})$ terms at LO QCD
Brivio:2019ius . The Wilson coefficients of these four-fermion operators are

$C_{tu}^{8},\,C_{td}^{8},\,C_{Qq}^{1,\,8},\,C_{Qq}^{3,\,8},\,C_{Qu}^{8},\,C_{Qd}^{8},\,C_{tq}^{8},$

which are defined with the same notation of Ref. Brivio:2019ius .

To provide a constraint on these additional degrees of freedom, the inclusive
$p\bar{p}\rightarrow t\bar{t}$ cross-section measurement from the Tevatron
experiments CDF:2013hmv  and the $pp\rightarrow t\bar{t}$ cross-section at 13
TeV ATLAS:2019hau  are included in this extended fit. In order to perform a
competitive global fit on these additional degrees of freedom we should
include more observables, like differential measurements of $pp\rightarrow
t\bar{t}$, the $t\bar{t}$ charge asymmetry Rosello:2015sck  or the top energy
asymmetry Basan:2020btr . The purpose of this analysis is just to see the
degradation of our limits due to extending the basis, not to find a
competitive constraint on $C_{tG}$ and/or the four-fermion operators.

Looking at the dotted light brown lines of Figure 4, we clearly see how
extending the basis has a slight effect on $C_{tW}$, $C_{\varphi Q}^{3}$ and
$C_{\varphi Q}^{-}$. The limits on $C_{tZ}$ are degraded by a factor of two
but they are still competitive. For $C_{\varphi t}$ and $C_{t\varphi}$ the
limits are totally ruined. Indeed, these three operators ($C_{tZ}$,
$C_{\varphi t}$ and $C_{t\varphi}$) that are mostly affected by extending our
basis are mainly constrained by $t\bar{t}X$ cross-section measurements in the
baseline fits, as can be seen in Figure 2. Precisely, these cross-sections are
the ones affected by the presence of $C_{tG}$ and the four-fermion operators.

Operators with two heavy quarks and two charged leptons (type
$t\bar{t}l^{+}l^{-}$) are also ignored in our fit. Their impact on the
operators that are constrained by the measurements in the $t\bar{t}Z$ process
must be assessed carefully in future work Sirunyan:2020tqm .

###  8.3 Correlations between measurements

The selection of measurements included in our fit aims to minimise the
statistical correlation among them. However, the correlation of experimental
and theoretical systematic uncertainties among the different processes could
be sizeable. Since such correlations have not been provided by the
experiments, we vary them from zero to the ansatz discussed here.

For the experimental correlations, we estimate an ansatz as follows. For each
pair of measurements we break down the uncertainties reported in the
experimental publications into: i) statistical, ii) mildly correlated, and
iii) highly correlated systematic uncertainties. The combined correlation
between two observables $i$ and $j$ is then given by:

$\bar{\rho}_{ij}^{\alpha}=\frac{\sum_{\alpha}\rho_{ij}u_{i}^{\alpha}u_{j}^{\alpha}}{\sqrt{\left(\sum_{\alpha}u_{i}^{\alpha}u_{i}^{\alpha}\right)\left(\sum_{\alpha}u_{j}^{\alpha}u_{j}^{\alpha}\right)}}\,,$
(4)

where $u_{i}^{\alpha}$ is the uncertainty of the observable $i$, $\alpha$ runs
over the three categories of uncertainties and $\rho_{ij}$ is an ansatz which
varies from 0 to 0.5 (depending on the source of the uncertainty).

In addition, theoretical uncertainties on the predictions can be correlated,
due to the PDFs and the similarity of certain production processes. For the
observables measured at the LHC, the ansatz for this correlation is: 100%
between the different bins of $t\bar{t}Z$ and $t\bar{t}\gamma$ differential
cross-sections, $-100$% between the two $W$ helicity fractions $F_{0,L}$, and
either 20% or 50% between the other 13 TeV measurements and some 13 TeV - 8
TeV and 8 TeV - 8 TeV pairs. The correlations of a given observable with
$F_{L}$ have opposite sign compared to the correlations of such observable
with $F_{0}$.

The ansatz for the correlations is a rough estimate, but allows us to
investigate the robustness of the result. The most significant effects are the
shifts observed for $C_{\varphi t}$, $C_{\varphi Q}^{-}$ and $C_{\varphi
Q}^{3}$ coefficients. The shifts are small compared to the width of the 95%
probability interval, which are $\cal{O}$(1 TeV -2) for the $C_{\varphi
Q}^{-}$ and $C_{\varphi Q}^{3}$ coefficients and $\cal{O}$(10 TeV -2) for
$C_{\varphi t}$. The shifts on $C_{\varphi t}$ increase the tension with the
SM. The log-likelihood of the fit in Table 3 goes to larger negative values as
the degree of correlation among the measurement increases, but this
corresponds to a very mild decrease in the $p$-values.

###  8.4 Discussion

We have attempted to perform a robust analysis and have presented several
stress tests to assess the impact of the limitations of EFT analyses. The
“robust limits”, obtained as the envelope of all these fits, are indicated
with red vertical lines in Figure 4 and in the right column of Table 2.

In the baseline fits we have included all published correlations between
measurements, taking advantage in particular of LHC combinations whenever
available. To avoid strong - and unknown - correlations between measurements,
we have included only the most precise measurement of each observable,
discarding less precise measurements by other experiments, and at other
centre-of-mass energies. We have also checked that, whenever available,
including measurements at two center-of-mass energies, has a negligible impact
in our limits. This reinforces our strategy of including them only at one
center-of-mass energy. To estimate the residual impact of correlations, we
include a complete covariance matrix with plausible ansatz values. The fit is
found to be relatively robust to the assumed correlations, with minor changes
in the results for three operator coefficients. Compared to Ref.
Bissmann:2019gfc , correlations seem relatively harmless. We consider that
this is due to our choice to select a single result among redundant
measurements. Eventually, with the legacy combinations by the ATLAS and CMS
collaborations, this unsatisfactory reduction of the data set will no longer
be necessary.

##  9 Conclusions

The most precise measurements of top quark associated production with bosons
from ATLAS and CMS experiments at LHC, the latest LHC combined results of $W$
boson helicity fractions and single top quark production, as well as a few
legacy LEP/SLD and Tevatron measurements have been considered to perform a
global analysis of the top quark electro-weak sector in the SMEFT.

We find that the Standard Model offers an adequate description of the 30
observables included in the fit. The total $\chi^{2}/\text{d.o.f.}$ is
21.3/29, corresponding to a $p$-value greater than 80%. The 95% probability
intervals of the SMEFT fit include the SM prediction ($C=0$) for the eight
Wilson coefficients that modify the top quark electro-weak couplings. The
global 95% intervals vary from $\pm 0.35$ to $\pm 8$ TeV -2.

The results of our fit improve significantly with respect to a previous
analysis of partial Run 2 data Durieux:2019rbz . This demonstrates the value
of recent LHC results based on analyses of the complete Run 2 data set. In
particular, the differential cross-section measurements for the $t\bar{t}Z$
and $t\bar{t}\gamma$ processes represent a significant step forward.

Following LHC Top Working Group recommendations AguilarSaavedra:2018nen , we
report the results of two EFT fits, with parameterisations including only
terms proportional to $\Lambda^{-2}$ or with both $\cal{O}$$(\Lambda^{-2})$
and $\cal{O}$$(\Lambda^{-4})$ terms. The largest difference between the two
fits is observed for the coefficient $C_{tZ}$, where the bound still relies
strongly on quadratic $\cal{O}$$(\Lambda^{-4})$ terms (as signalled already by
Ref. Bylund:2016phk , the $\Lambda^{-2}$ term is suppressed in $t\bar{t}X$
production).

We have made an important effort to study the resilience of the bounds against
uncertainties inherent in the EFT framework. We present robust bounds that
take into account the uncertainties due to missing higher orders in
$\alpha_{s}$, potential correlations among the different observables and
missing degrees of freedom. The analysis is found to be quite robust against
the first two effects but only robust for some operators against the inclusion
of other degrees of freedom. The most sensible observables to the extension of
the basis are $C_{t\varphi}$, $C_{\varphi t}$ and $C_{tZ}$, although
competitive limits can still be found for the latter coefficient. This result
is in agreement with what is observed in global SMEFT fits Ellis:2020unq ;
Ethier:2021bye .

We take into account the effect of published correlations between
measurements. Large and unknown correlations are avoided by selecting a single
measurement for each observable. For pairs of measurements where no public
information is available, we estimate a plausible ansatz for the correlation.
The resulting shifts of the central value are small, which shows that the EFT
analysis is reasonably robust. Of course, a rigorous treatment of carefully
estimated correlation is a necessary ingredient for a SMEFT analysis of LHC
legacy measurements that aims to reach the ultimate precision.

The bounds reported in this paper agree well with the results of more
comprehensive analyses that include data on top quark and Higgs boson
production and electro-weak precision measurements Ellis:2020unq ;
Ethier:2021bye , once the differences in the operator basis and treatment of
$\Lambda^{-4}$ terms are accounted for. The main exception is the bound on
$C_{tZ}$ that is significantly stronger in our analysis. This difference can
be traced back to the omission of the $t\bar{t}\gamma$ measurement in Ref.
Ethier:2021bye  and in the initial result of Ref. Ellis:2020unq . On the other
hand, Refs. Ellis:2020unq ; Ethier:2021bye  report tighter individual bounds
on the coefficients $C_{\varphi t}$ and $C_{t\varphi}$ from the Higgs and
electro-weak data. This effect remains limited to individual bounds: the
global bounds on the top electro-weak couplings remain clearly dominated by
top physics data. The interplay between Higgs, electro-weak and top physics
(and flavour Bruggisser:2021duo ) definitely merits further exploration.

## Acknowledgements

This project builds heavily on previous work in collaboration with Gauthier
Durieux (CERN) and Cen Zhang (IHEP). We acknowledge their help and support in
setting up the EFT fit. Cen Zhang passed away unexpectedly in June 2021 and
will be missed dearly. We would also like to thank Antonio Pich for his useful
comments on the manuscript. This work is supported by the projects:
FPA2017-84445-P, PGC2018-094856-B-100 and RTI2018-100863-B-I00 (Spanish
Ministry MICINN and FEDER); PROMETEO-2018/060 and SEJI-2020/037 (Generalitat
Valenciana); LCF/BQ/PI19/11690014 (La Caixa Banking Foundation); and the iLINK
grant LINKB20065 (CSIC). The work of V.M. is supported by the FPU doctoral
contract FPU16/0191, funded by the Spanish Ministry MICIU.

## Appendix: Correlation Matrices

Figure 5: Experimental correlation matrix used. The boxes in white correspond
to the correlations published by the experiments for $t\bar{t}Z$,
$t\bar{t}\gamma$ and $W$ boson helicity fractions ATLAS:2020cxf ; Aad:2020axn
; Aad:2020jvx . The rest of the entries correspond to our ansatz, as described
in Section 8.3. Cells are filled if the correlation is higher than $10\%$ in
absolute value. Figure 6: Theoretical correlation matrix used. The boxes in
white correspond to the correlations between the differential bins in
$t\bar{t}Z$ and $t\bar{t}\gamma$ ($W$ boson helicity fractions) and a 100%
($-100$%) correlation is assumed among them. The rest of the entries
correspond to our ansatz, as described in Section 8.3. Cells are filled if the
correlation is higher than $10\%$ in absolute value. Figure 7: Correlation
matrix between the different EFT operators obtained in the baseline linear
$(\Lambda^{-2})$ fit. Cells are filled if the correlation is higher than
$10\%$ in absolute value. The operator $O_{\varphi b}$, that modifies only the
bottom quark electro-weak couplings, is taken into account in the fit but
limits on its coefficients are not reported since the obtained values are not
competitive using only the observables considered in our fit. Figure 8:
Correlation matrix between the different EFT operators obtained in the
baseline quadratic $(\Lambda^{-4})$ fit. Cells are filled if the correlation
is higher than $10\%$ in absolute value. The operators that modify only the
bottom quark electro-weak couplings, $O_{\varphi d}$ and $O_{dZ}$, are taken
into account in the fit but limits on their coefficients are not reported
since the obtained values are not competitive using only the observables
considered in our fit. Figure 9: Correlation matrix between the different EFT
operators obtained in the linear $(\Lambda^{-2})$ “All correlations”
robustness test fit described in Section 8.3. Cells are filled if the
correlation is higher than $10\%$ in absolute value. The operator $O_{\varphi
b}$, that modifies only the bottom quark electro-weak couplings, is taken into
account in the fit but limits on its coefficients are not reported since the
obtained values are not competitive using only the observables considered in
our fit. Figure 10: Correlation matrix between the different EFT operators
obtained in the linear $(\Lambda^{-2})$ “4F” robustness test fit described in
Section 8.2. Cells are filled if the correlation is higher than $10\%$ in
absolute value. The operator $O_{\varphi b}$, that modifies only the bottom
quark electro-weak couplings, is taken into account in the fit but limits on
its coefficients are not reported since the obtained values are not
competitive using only the observables considered in our fit.

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††thanks: Contributed equally to this work and the names are written
alphabetically††thanks: Contributed equally to this work and the names are
written alphabetically††thanks: Contributed equally to this work and the names
are written alphabetically

# Simple generic picture of toughness in solid polymer blends

Debashish Mukherji<EMAIL_ADDRESS>Quantum Matter Institute,
University of British Columbia, Vancouver BC V6T 1Z4, Canada  Shubham Agarwal
Department of Mechanical Engineering, University of British Columbia,
Vancouver BC V6T 1Z4, Canada  Tiago Espinosa de Oliveira  Departamento de
Farmacociências, Universidade Federal de Ciências da Saúde de Porto Alegre,
Porto Alegre 90050-170, Brazil  Céline Ruscher  Department of Mechanical
Engineering, University of British Columbia, Vancouver BC V6T 1Z4, Canada
Jörg Rottler  Quantum Matter Institute, University of British Columbia,
Vancouver BC V6T 1Z4, Canada  Department of Physics and Astronomy, University
of British Columbia, Vancouver BC V6T 1Z1, Canada

###### Abstract

Toughness $\mathcal{T}$ of a brittle polymeric solid can be enhanced by
blending another compatible and ductile polymer. While this common wisdom is
generally valid, a generic picture is lacking that connects the atomistic
details to the macroscopic non-linear mechanics. Using all-atom and
complementary generic simulations we show how a delicate balance between the
side group contact density of the brittle polymers $\rho_{\rm c}$ and its
dilution upon adding a second component controls $\mathcal{T}$. A broad range
of systems follows a universal trend in $\mathcal{T}$ with ${\rm d}\rho_{\rm
c}/{\rm d}\varepsilon$, where $\varepsilon$ is the tensile strain. The
simulation data is consistent with a simple model based on the parallel spring
analogy.

Polymers are widely used in designing light weight, high performance organic
materials [1, 2, 3, 4, 5] that find use in common household utensils to
complex nano-materials [1, 2, 6, 7, 8]. Because of their wide applicability,
these materials are often exposed to a range of environmental conditions, such
as temperature [9, 10, 11, 12], pressure [10, 13], and/or mechanical
deformation [9, 14, 7, 15, 16]. Here, one important materials property is the
ability to sustain large deformation [5, 17, 16, 18, 19].

Most known commodity polymers, such as poly(methyl methacrylate) (PMMA) [15,
16], poly(lactic acid) (PLA) [16, 5], and polystyrene (PS) [20], are brittle
in their glassy states, i.e., small strain-to-fracture $\varepsilon_{\rm f}$
and low yield stress $\sigma_{\rm y}\simeq 0.1$ GPa due to the weak van der
Waals (vdW) monomer-monomer interactions, the strength of which is about
$k_{\rm B}T$ at room temperature, where $k_{\rm B}$ is the Boltzmann constant
[21]. On the other hand, $\sigma_{\rm y}$ of a polymer can be enhanced by
about an order of magnitude when the monomer-monomer hydrogen bonds
(H$-$bond), with an interaction strength of about $4k_{\rm B}T$ [4], dominate
the materials properties, such as in poly(acrylamide) (PAM), poly(acrylic
acid) (PAA), and poly(vinyl alcohol) (PVA). These H$-$bonded systems, however,
can either be brittle or ductile dependent on their respective
(macro-)molecular architectures. Given the above discussion, mechanical
response remains restricted in the single component polymeric systems and thus
often limits their broad applications.

A simple route to tune the tensile toughness
$\mathcal{T}=\int_{0}^{\infty}\sigma(\varepsilon){\rm d}\varepsilon$ of a
brittle organic solid is by blending in another polymer with a relatively
larger ductility and component-wise interactions. Here, $\sigma$ and
$\varepsilon$ are the tensile stress and tensile strain, respectively.
Experimentally relevant examples include, but are not limited to, PMMA-PLA
[16, 22], PAA-PVA blends [18], and/or PMMA-PVA [23]. These systems are assumed
to be fairly miscible, while abstaining to discuss the effects of
immiscibility.

In general, $\mathcal{T}$ of a blend varies monotonically between the two pure
phases as a function of their relative mixing ratios [18]. However, a set of
recent experiments on PMMA-PLA blends have reported a non-monotonic variation
in $\mathcal{T}$ with concentration [16]. This behavior was attributed to the
formation of “so called” co-continuous phase, with an estimated length scale
on the order of several $\mu$m [16, 22]. However, a set of relatively much
smaller scale all-atom simulations have found a similar trend in $\mathcal{T}$
[24], suggesting that the formation of a co-continuous phase may not be a
necessary criterion for the toughness enhancement in these materials.

Traditionally, extensive research has been conducted to understand the
mechanics of polymeric materials, both from the experimental [16, 20, 18, 5]
and the simulation [14, 15, 24, 17] communities. Here, however, most
simulation studies usually deal with single component systems and also
predominantly at the (mesoscopic) generic level, where all-atom details are
drastically coarse-grained [21]. While such models are extremely useful in
dealing with generic polymer properties, they often (in their pristine form)
lack the atomic-level details that play a key role in materials properties,
unless of course the generic model is specifically tuned to reproduce certain
properties of interest. In particular, when dealing with polymer blends having
very specific macromolecular structures and interactions [18, 16, 24], special
attention should be paid. Therefore, a better understanding of the monomer-
level (atomistic) interaction is needed to achieve a predictive mechanical
response of the polymer blends, which to the best of our knowledge is lacking.

Figure 1: Stress $\sigma$ as a function of strain $\varepsilon$ for three
different atomistic blends, namely poly(methyl methacrylate) (PMMA) and
poly(venyl alcohol) (PVA) (panel a), poly(acitic acid) (PAA) and PVA (panel
b), PMMA and poly(lactic acid) (PLA) (panel c), and the generic model (panel
d). The data is shown for five different PVA and PLA and four (generic) linear
chain monomer mole fractions $x_{\rm i}$. Here, $x_{\rm i}=0.0$ corresponds to
the pure brittle phase. The corresponding monomer structures are shown in the
insets. $\sigma$ for the generic model is converted using a factor
$p_{\circ}=40$ MPa [21]. Note that the data for PMMA-PLA blends are taken from
Ref. [24].

In this work, we investigate the mechanics of commodity organic solids with
the goals to: (1) understand the effect of blending on the mechanics of
brittle systems, (2) show how the monomer (atomistic) structures play a key
role in dictating the materials properties, (3) connect the atomistic
interaction details to the macroscopic non-linear mechanics, and (4) propose
how the behavior of many different chemically specific polymeric materials can
be understood within one universal framework. To achieve the above goals, we
combine all-atom and complementary generic molecular dynamics simulations with
a simple parallel spring model.

For the all-atom simulations, we have chosen both vdW and H$-$bonded
(experimentally relevant) commodity polymers (i.e., PMMA, PAM, PAA, PLA, and
PVA) and polymer blends (i.e., PMMA-PVA, PAA-PVA, and PMMA-PLA). Note that we
have performed the simulations for all systems in this work, aside from the
PMMA-PLA trajectories which were obtained earlier [24] and reanalyzed here.
The monomer mole fractions are varied between $0.0\leq x_{{}_{\rm i}}\leq 1.0$
in steps of 0.25. In this study, $x_{{}_{\rm i}}=0.0$ always corresponds to
the pure phase of brittle polymer (BP) and the pure second polymer (SP) at
$x_{{}_{\rm i}}=1.0$. The OPLS-AA parameters are used for PAA, PLA, and PVA
[25], while the modified force-field parameters are used for PAM [26] and PMMA
[27]. The chain length $N_{\ell}=30$ monomers is chosen for all systems except
PAM, where $N_{\ell}=32$. The simulations are performed using the GROMACS
molecular dynamics package [28]. More system specific details are shown in the
Supplementary Section S1A [29].

The well known bead-spring polymer model is used for the generic simulations
[21]. Note that the default bead-spring model is highly ductile [14].
Therefore, inspired by the all-atom BP architectures, we have parameterized a
brush-like polymer with short (stiff) side groups of the dimers, blended with
the (ductile) linear chains. The monomer-monomer interactions are tuned to
weakly mimic the atomistic PMMA-PVA blends. The backbone chain length is taken
as $N_{\ell}=30$. The generic simulations are performed using the LAMMPS
package [30]. A detailed discussion of the generic model is described in the
Supplementary Section S1B [29]. The numbers that are representative of
hydrocarbons are: the units of length $d=0.5$ nm, energy $\epsilon=30$ meV,
time $t_{\circ}=3$ ps, and pressure $p_{\circ}=40$ MPa [21]. Note that for
simplicity of presentation, we have converted all generic units into the real
units.

In Figs. 1(a-c) we show stress-strain behavior of three different all-atom
blends under uniaxial tensile deformation. As expected, blending in SPs drive
the BP-based systems to a greater ductility, especially because the pure SPs
are relatively more ductile than BPs. Ideally one can argue, if SPs have their
glass transition temperatures $T_{\rm g}$ much smaller than BP (thus remain in
their rubbery phases), the ductility or ${\mathcal{T}}$ may increase via
rubber plasticity. In our case, however, all polymers have $T_{\rm g}\simeq
355-390$K (as measured in the experiments) [16, 31], while the simulations
have reported $T_{\rm g}\simeq 400$K [24, 32]. In this context, our
simulations are performed at $T=300$K and thus $T_{\rm g}-T$ are such that the
individual components of a blend remain deep in their respective glass phases
[31, 32]. This readily excludes any explanation purely based on $T_{\rm g}$,
and it becomes apparent that a nanoscopic picture is needed to better
understand the nonlinear mechanics of the multi-component polymeric materials.

A comparative analysis of the monomer structures of BP and SP reveals that one
major difference between these two sets is the presence of bulky side groups
in BPs (see PAA and PMMA structures in Fig. 1), while SPs are smooth (see PVA
and PLA structures in Fig. 1). In this context, it is known that the side
groups of BPs are rather stiff and thus form extremely rugged (instantaneous)
structural corrugation along the chain contour. Corrugation (or more generally
speaking, the breaking of translational Galilean invariance) is a necessary
“ingredient” to exert the shear forces upon deformation. Note also that the
side groups can flip-flop normal to the chain contour in the polymer melt,
while these structural fluctuations are frozen in the glassy state. In this
context, it has been shown previously that the presence of stiff side chains
may serve as a necessary criterion to control the brittleness in a solid PMMA
[15].

Figure 2: Schematic representations of the direct contacts between the brittle
polymers (BP) before (panel a) and after including the second polymer (SP)
(panel b). For the simplicity of presentation, the one-dimensional projection
of the effective roughness (or the effective free energy landscape) of the
molecules along the chain contours is shown. The grey and red lines represent
(rugged) BP and (smooth) SP molecules, respectively.

Within the picture discussed above, the side groups of two or more neighboring
chains can interlock, shown by the schematic in Fig. 2(a). Here, the
interlocking can be dictated by either vdW or H$-$bond interactions. When a
pure BP glass is deformed, the rigid side chain contacts break via relative
motion of the chains and thus the sliding molecules overcome the effective
free energy barriers. Such broken contacts initiate cavitation at around the
yield strain, i.e., $\varepsilon_{\rm y}\simeq 0.1-0.2$, see Fig. 1. Beyond
$\varepsilon_{\rm y}$, the systems go into the flow regime with growing voids,
leading to their coalescence and ultimate fracture. The stronger the side
chain contacts, the larger the $\sigma_{\rm y}$ values. In this context, we
find $\sigma_{\rm y}^{\rm PMMA}<\sigma_{\rm y}^{\rm PAA}<\sigma_{\rm y}^{\rm
PAM}$, see the supplementary Fig. S8 [29]. This is expected because two PMMA
side chains interact via weak vdW interaction, while two PAA and two PAM can
form 0.84 and 1.52 H$-$bonds per monomer, respectively.

When the SP molecules are added, they lubricate and also dilute the density of
side chain contacts $\rho_{\rm c}$ between the neighboring BPs, see the
schematic in Fig. 2(b). This microscopic molecular arrangement increases
$\varepsilon_{\rm f}$ and also hinders the growing voids via rearrangements of
SPs. Here, the extent of molecular ruggedness that a molecule feels, while
moving in the homogeneous bulk, is directly related to $x_{\rm i}$ and thus
has a direct implication on the polymer motion. The mean-square-displacement
$C(t)$ data, shown in the Supplementary Figs. S2 & S3, provide evidence that
the molecular diffusion $D$ increases with increasing $x_{\rm i}$. Within the
linear response, the different damping contributions are linearly additive and
thus the effective damping $\gamma\propto 1/D$. Furthermore, following $C(t)$
in the Supplementary Figs. S2 & S3, it becomes apparent that the atomic level
ruggedness (or corrugation) reduces with increasing $x_{\rm i}$. In the glassy
state, we also expect the relative ruggedness across different samples to
remain the same. The statement above does not aim to make a direct
quantitative comparison between the melt and the glassy states, rather only a
qualitative discussion without attempting to discuss the time-temperature
superposition.

We note in passing that the systems investigated in this study have larger
ductility than the experimentally reported values [16, 20, 18, 5]. This is
particularly because the quenching rates (i.e., from melt-to-glass) in the
simulations are usually several orders of magnitude faster than the
corresponding experiments and thus lead to poorer annealed samples [33]. Such
samples usually have residual $\sigma$ distributions that are rather broad,
while well-annealed samples have narrow distribution in $\sigma$ and thus all
sites fail almost at once upon loading, leading to a smaller $\varepsilon_{\rm
f}$ in experiments in comparison to the standard simulations. Here, however,
it is important to note that our simulation samples are all prepared using an
identical protocol and considering that this study aims at the relative
trends, we believe that the data sets presented in Fig. 1 give a reasonable
relative picture.

If the observation discussed above is generic, the trends in Figs. 1(a-c)
should also be visible in a chemically independent model by only incorporating
the monomer structures. Therefore, we have also conducted one set of generic
simulations. Here, a BP consists of a bottle brush-like polymer with short,
stiff side groups, while a SP is represented by a standard (fully flexible)
linear bead-spring chain [21]. In Fig. 1(d) the generic data is presented. It
can be appreciated that the trend observed in Fig. 1(a-c) is also reproduced
by the generic model.

Figure 3: Density of side chain contacts ${\Delta{\rho}}_{{}_{\rm
c}}={{\rho}}_{{}_{\rm c}}(\varepsilon)-{{\rho}}_{{}_{\rm c}}(0)$ as a function
of strain $\varepsilon$ between the neighboring monomers of the brittle
polymers in the blends of poly(methyl methacrylate) (PMMA) and poly(vinyl
alcohol) (PVA) (panel a), poly(acrylic acid) (PAA) and PVA (panel b), PMMA and
poly(lactic acid) (PLA) (panel c) and the generic model (panel d). For better
visibility of the relative variation in the data, we have shifted all curves
with respect to ${\rho}_{{}_{\rm c}}(0)$ of the unstrained system, i.e., at
$\varepsilon=0$. In the generic model, ${\rm d}\rho_{{}_{\rm c}}/{\rm
d}\varepsilon$ is scaled using the unit conversion $d=0.5$ nm [21]. Data for
PLA-PMMA blends is derived from our earlier simulation trajectories in Ref.
[24]. The vertical dashed lines show the $\varepsilon$ values at the yield
points, see Fig. 1.

So far we have only presented a qualitative picture of the results. It will,
however, be useful to draw more quantitative comparisons between the side
chain contact density $\rho_{\rm c}(\varepsilon)$ and $\mathcal{T}$. Here,
$\rho_{\rm c}(\varepsilon)=N_{\rm c}(\varepsilon)/v(\varepsilon)$, with
$N_{\rm c}(\varepsilon)$ and $v(\varepsilon)$ being the number of side chain
contacts and the instantaneous system volume, respectively. In the all-atom
simulations, $N_{\rm c}(\varepsilon)$ is calculated when the center-of-masses
of two neighboring (inter-molecular) side chains are within a minimum distance
$r_{\rm min}\leq 0.65$ nm, i.e., the first peak of the radial distribution
function [34, 32]. In the generic model, $r_{\rm min}\leq 1.5~{}\sigma$ [35].

In Fig. 3 we show $\rho_{\rm c}(\varepsilon)$ for the all-atom and the generic
models. It can be appreciated that: (a) $\rho_{\rm c}$ decreases with
increasing $\varepsilon$, which is expected because the tensile deformation
breaks the side chain contacts between BPs that lead to the ultimate fracture
of the samples. (b) Increasing $x_{\rm i}$ generally dilutes $\rho_{\rm
c}(\varepsilon)$ and thus leads to the weaker decay rates for
$\varepsilon>\varepsilon_{\rm y}$ (shown by the vertical dashed lines in Fig.
3).

The individual data sets in Fig. 3 show two different ${\rm d}\rho_{\rm
c}/{\rm d}\varepsilon$ regions. The first is a relatively rapid decay for
$\varepsilon_{\rm y}<0.2$ (in all-atom) and $\varepsilon_{\rm y}<0.1$ (in
generic), where the ${\rm d}\rho_{\rm c}(\varepsilon)/{\rm d}\varepsilon$
behavior is dominated by the initial disruption between the side groups, see
Fig. 1. The second linear decay beyond $\varepsilon_{\rm y}$ occurs when the
individual systems reach the flow (stationary) regime, where the voids grow
and also SPs rearrange to control $\mathcal{T}$. We calculate the slopes
$S={\rm d}\rho_{\rm c}(\varepsilon)/{\rm d}\varepsilon$ in the stationary flow
regimes, where the behavior is reasonably well described by the simple linear
form $\rho_{\rm c}(\varepsilon)=\rho_{\circ}+S\varepsilon$.

Figure 4: A master curve relating the normalized toughness
$\mathcal{T}/\sigma_{\rm y}$ with the change in side chain contact density
$\rho_{\rm c}$ upon deformation, as quantified by the slope $S={\rm
d}\rho_{\rm c}(\varepsilon)/{\rm d}\varepsilon$. The data is normalized with
respect to the yield stress $\sigma_{\rm y}$, calculated from Fig. 1. The line
is a fit to the data using Eq. 2. While most data sets are shown for
uniaxially deformed samples, we also present one case study in which a set of
PMMA-PVA blends is biaxially deformed.

Fig. 4 shows the variation of toughness-to-strength ratio
$\mathcal{T}/\sigma_{\rm y}$ with $S$. It can be appreciated that a broad
range of chemical specific systems and the generic model data fall onto one
empirical master curve. Here, most data sets are obtained for uniaxial
loading, where the mechanics is dominated by the shear force exerted by the
sliding molecules. However, the data for the biaxial loading, where the
cavitation is enhanced (see the Supplementary Figs. S6 & S7), also support the
same empirical behavior.

While the data sets in Fig. 4 show a correlation across several samples, it is
important to investigate if this behavior can be understood within a model by
incorporating some microscopic descriptions, which is achieved using a simple
(mechanical) model. This is inspired by the parallel spring model analogy
[36]. The model specific details are described in the Supplementary section S2
[29], here we only sketch the important ingredients. Within this model, an
organic solid is modeled as a three-dimensional network of springs in
parallel, where one side chain interlocking represents one spring contact with
an effective spring constant $k_{i}$. Force $F$ and local displacement
$\delta$ then follows $F(\delta)=\sum_{i=1}^{N_{\rm c}}k_{i}\delta$. If we
consider that the all contacts contribute equally to the loading, i.e.,
$k_{i}\simeq k$, we get $F(\delta)=kN_{\rm c}(\delta)\delta$. Under the affine
assumption, the local (microscopic) displacement $\delta$ relates to the
macroscopic displacement via $\delta=c\varepsilon L_{\circ}$. Here $L_{\circ}$
is the unstrained box dimension and $c$ is a (small) dimensionless constant.
This assumes that each contact carry the same load and $\delta\ll\varepsilon
L_{\circ}$, i.e., $c\ll 1$. Following $\sigma$ from $F$ (see the Supplementary
section S2 [29]), we obtain

$\frac{\mathcal{T}}{\sigma_{\rm y}}=\frac{1}{\sigma_{\rm
y}}\int_{0}^{\varepsilon_{\rm f}}\sigma(\varepsilon){\rm
d}\varepsilon=\frac{ckL_{\circ}^{2}}{\sigma_{\rm y}}\int_{0}^{\varepsilon_{\rm
f}}\rho_{\rm c}(\varepsilon)\varepsilon\left(1+\varepsilon\right){\rm
d}\varepsilon.$ (1)

Eq. 1 leads to an empirical solution,

$S\simeq\frac{\left[\frac{\mathcal{T}}{\sigma_{\rm
y}}-\frac{ckL_{\circ}^{2}\rho_{\circ}}{\sigma_{\rm
y}}\left\\{\frac{1}{2}\left(\frac{\mathcal{T}}{\sigma_{\rm
y}}\right)^{2}+\frac{1}{3}\left(\frac{\mathcal{T}}{\sigma_{\rm
y}}\right)^{3}\right\\}\right]}{\frac{ckL_{\circ}^{2}}{\sigma_{\rm
y}}\left[\frac{1}{3}\left(\frac{\mathcal{T}}{\sigma_{\rm
y}}\right)^{3}+\frac{1}{4}\left(\frac{\mathcal{T}}{\sigma_{\rm
y}}\right)^{4}\right]}.$ (2)

Eq. 2 is plotted as the solid line in Fig. 4, with
$ckL_{\circ}^{2}\rho_{\circ}/\sigma_{\rm y}\simeq 0.01$. Note that the factor
$k/\sigma_{\rm y}$ remains constant for a given sample. It can be appreciated
that Eq. 2 follows a reasonable agreement with the simulation data in Fig. 4.
This further reinforces a picture directing at a possible route towards the
tunability in $\mathcal{T}/\sigma_{\rm y}$ and its links to the atomic level
details.

It will certainly require more detailed experiments to validate our scenario.
One plausible path might be to follow the protocol presented in an earlier
study of one of us [37], where a proton nuclear magnetic resonance (NMR) setup
was used to obtain the local side chain organizations between the neighboring
non-bonded monomers. Here, we expect that the on-the-fly NMR measurements (at
different $\varepsilon$) might give direct information about $\rho_{\rm c}$
and thus can lead to a better understanding of the underlying atomistic
picture in these commodity organic solids.

In conclusion, we have performed large scale molecular dynamics simulations to
study the mechanics of solid commodity polymer blends. We show how a delicate
balance between the (macromolecular) side chain organizations, resultant
corrugation along the chain contour, and their interlocking controls the
toughness $\mathcal{T}$ of a polymeric material. Our study establishes a
relationship between the microscopic interactions and the macroscopic
nonlinear mechanics, which follows a universal empirical relationship across a
wide range of polymeric systems. To better understand this behavior, we have
also formulated a simple model based on the springs in parallel analogy. The
simplified picture presented here may serve as a guiding tool for the
development of advanced functional materials with tunable and predictive
mechanical properties.

D.M. thanks Carlos Marques and Kurt Kremer for stimulating discussions. This
research was undertaken thanks, in part, to the Canada First Research
Excellence Fund (CFREF), Quantum Materials and Future Technologies Program.
D.M. further thanks ARC Sockeye facility of the University of British Columbia
where the simulations are performed.

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