aainstitutetext: Department of Physics and Astronomy, University of
Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

# Quantum entanglement and top spin correlations in SMEFT at higher orders

Claudio Severi a Eleni Vryonidou<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the first analysis of top spin polarizations, $t\bar{t}$ spin
correlations, and $t\bar{t}$ spin entanglement at the LHC in the context of
the Standard Model Effective Field Theory, that goes beyond Leading Order QCD
accuracy. The complete set of independent dimension-6 operators entering
$t\bar{t}$ production is identified, and their effects on all top spin
observables are extracted at linear and quadratic order in $c/\Lambda^{2}$. By
comparing results at LO and NLO, we note that the inclusion of higher orders,
while not dramatically changing the picture, often amounts to notable
numerical differences, that are not fully captured by LO scale variation. We
also find that the expected deviations from the SM have an intricate phase
space structure, and show up predominantly at large top $p_{T}$. For this
reason, we advocate for the measurement of spin observables differentially or
doubly-differentially in the $t\bar{t}$ phase-space. We show how the inclusion
of present and future top spin measurements will improve global fits to top
LHC data, by also addressing the issue of flat directions.

##  1 Introduction

The Standard Model Effective Field Theory (SMEFT) is an extension of the
Standard Model (SM), that includes a stack of higher–dimensional operators
$\mathcal{O}_{k}$,

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\sum_{d=5}^{\infty}\sum_{k}\frac{c_{k}}{\Lambda^{d-4}}\mathcal{O}_{k},$
(1.1)

parameterized by a set of Wilson coefficients $c_{k}$ and a global scale
$\Lambda$. This framework offers the significant advantage that only minimal
assumptions are put on the new physics (NP): $\mathcal{L}_{\text{SMEFT}}$ is
constructed to be the most general Lagrangian that reproduces the SM at low
energy, and is Lorentz and $SU(3)_{\text{C}}\times SU(2)_{\text{L}}\times
U(1)_{\text{Y}}$ gauge invariant. The SMEFT provides a promising framework to
identify and parameterize deviations from the SM predictions, and a huge
effort is underway to combine a wide range of experimental measurements with
theoretical predictions in global fits, aimed at constraining the Wilson
coefficients of SMEFT operators. The correlating nature of the SMEFT between
different observables, different scales, and sometimes different sectors,
mandates the inclusion of the largest possible dataset in the fits. Over the
last few years, global combinations of LHC data in the top, Higgs and EW
sectors have been achieved Buckley:2015lku ; Aguilar-Saavedra:2018ksv ;
Hartland:2019bjb ; Brivio:2019ius ; Ellis:2020unq ; Ethier:2021bye . While
there has been enormous progress in the SMEFT interpretation of LHC
measurements, several operators exist for which the current sensitivity is
still limited. Further, global fits are often plagued by the presence of flat
directions in their parameter space. One such example arises in the top
sector, where several 2-quark and 4-quark operators can modify top
interactions. The sheer number of these operators, and the consequent need to
break the degeneracies that arise, demands the consideration of the largest
possible set of observables.

We are particularly fortunate, in that respect, that almost 30 years after the
discovery of the top quark, the remarkable performance of the LHC has moved
the top sector of the SM in the realm of precision measurements. A broad
campaign is underway involving the study of processes observed at the LHC for
the first time: the production of tops in association with a vector boson,
with a Higgs, and the simultaneous production of four tops. At the same time,
the luminosity and energy reach of the LHC allows precise differential
measurements of the main top production channels, both $t\bar{t}$ and and
single tops, in several final state configurations.

A particularly interesting class of measurements is those focusing on the top
spin. Top spin measurements are experimentally very challenging. The first
measurement of top spin, establishing the presence of spin correlation in top
pairs, was completed in 2012 ATLAS:2012ao . Subsequent studies at
$\sqrt{s}=7\,\text{TeV}$ ATLAS:2014aus , $8\,\text{TeV}$ CMS:2015cal ;
CMS:2016piu ; ATLAS:2016bac , and $13\,\text{TeV}$ CMS:2019nrx ; ATLAS:2019hau
; ATLAS:2019zrq  considered more refined observables, and included several
different top decay topologies. However, to this date, only one experimental
measurement of the full $t\bar{t}$ spin density matrix, that completely
characterizes top spin polarization and $t\bar{t}$ spin correlations, has been
published CMS:2019nrx . Furthermore, all the existing experimental
measurements were performed inclusively in the $t\bar{t}$ phase space; as we
will argue in the following, this is a major limitation, as all top spin
degrees of freedom exhibit a complex structure in phase space, that is largely
washed out if no binning is applied.

Top spin correlation measurements can also be studied in the framework of pure
Quantum Mechanics (QM). As first noted in Afik:2020onf , sufficiently strong
top spin correlations can signal the presence of quantum entanglement between
top quarks. Entanglement, arguably the most essential and counter-intuitive
prediction of QM, has been detected experimentally in a variety of physical
systems, but never at the TeV scale. As argued in Afik:2020onf , a simple
selection based on the $t\bar{t}$ invariant mass will be enough to observe
entanglement at the LHC. The structure of spin entanglement in $t\bar{t}$
pairs has since been studied multiple times, both in the SM and in the SMEFT
Afik:2022kwm ; Aguilar-Saavedra:2022uye ; Aoude:2022imd . When entanglement
between two particles is sufficiently strong, measurements can be set up so as
to violate Bell Inequalities (BI), the ultimate signal of non-classical
behavior. A violation of BIs signals correlations so strong that no classical
theory, even with arbitrary and hidden degrees of freedom, can explain. Top
spin correlations grow with the transverse momentum of tops so that, at
extreme $p_{T}$, BIs are violated at parton level. A first analysis
Fabbrichesi:2021npl  suggested a violation of BIs may be observed at the LHC.
Further studies Severi:2021cnj  suggested that the experimental verification
of BIs will likely be very challenging, and if no new physics is present to
enhance spin correlations, the detection of a violation may just escape the
reach of the LHC. The investigation of possible BI measurements with top spin
in colliders is ongoing, with progress having been reported recently Aguilar-
Saavedra:2022uye . Other collider processes have also been investigated
Barr:2021zcp ; Barr:2022wyq ; Aguilar-Saavedra:2022wam ; Aguilar-
Saavedra:2022mpg ; Ashby-Pickering:2022umy ; Fabbrichesi:2022ovb .

Standard Model predictions for the observables with which experiments can
probe all the $t\bar{t}$ spin degrees of freedom are available at NLO QCD + EW
Bernreuther:2015yna  and NNLO QCD Czakon:2020qbd  accuracy. Before this work,
SMEFT corrections to these observables were available at LO in QCD and LO in
$c/\Lambda^{2}$ Bernreuther:2015yna . Quadratic SMEFT corrections for a
selection of dimension-6 operators were computed point-by-point in $t\bar{t}$
phase space in Aoude:2022imd , for the two LO QCD channels $gg\to t\bar{t}$
and $q\bar{q}\to t\bar{t}$.

In this work we will systematically consider the effect of dimension-six
operators on $t\bar{t}$ entanglement, top spin polarization and $t\bar{t}$
spin correlation observables, at the $c/\Lambda^{2}$ and $c^{2}/\Lambda^{4}$
level, while also including higher-order QCD corrections.

This paper is organized as follows. In Section 2 we describe the SMEFT
operators that enter $t\bar{t}$ production at the LHC. In Section 3 we
introduce the relevant top spin observables and show how these can be
extracted from collider data. Section 4 describes in detail the numerical
computation we carried. Results are presented in Section 5 at the inclusive
and differential level. In Section 6 we estimate the improvement to global
fits caused by the inclusion of spin observables, once more experimental
results will become available in the near future. We conclude in Section 7.
Finally Appendices A and B contain additional numerical results and plots.

##  2 SMEFT operators in top pair production

In this section we present the relevant operators entering top pair production
at dimension-6. Within this work, we will assume conservation of baryon number
and CP invariance. To keep under control the otherwise unmanageable number of
four-quark operators, a flavor assumption must be put in place. Several have
been proposed, and are studied in detail in Aguilar-Saavedra:2018ksv . We will
assume the flavor symmetry as in Degrande:2020evl :

$U(2)_{Q}\times U(2)_{u}\times U(3)_{d},$ (2.1)

that implies new physics couples predominantly to top quarks. This is rooted
on the unusually large value of the SM top Yukawa coupling $y_{t}\approx
0.993$, signaling a special relationship between the two heaviest SM
particles, and hinting at the presence of new physics close to the electroweak
scale.

We focus on operators containing at least one top quark field, assuming other
operators are well enough constrained from measurements not involving tops.
All such operators are considered, in a systematic approach, with the
exception of those that enter top production only in virtual corrections, and
those only entering in highly suppressed contributions (e.g. $b\bar{b}\to
t\bar{t}$).

###  2.1 Two-fermion operators

One can write down three dimension-six operators modifying the top-gluon
interaction and thus $t\bar{t}$ production. To highlight the chiral properties
of the induced $t\bar{t}g$ vertex these can be written as:

$\displaystyle\mathcal{O}_{gt}$
$\displaystyle=\overline{t}\,T_{A}\gamma^{\mu}D^{\nu}t\,G_{\mu\nu}^{A},$ (2.2)
$\displaystyle\mathcal{O}_{gQ}$
$\displaystyle=\overline{Q}T_{A}\gamma^{\mu}D^{\nu}QG_{\mu\nu}^{A},$ (2.3)
$\displaystyle\mathcal{O}_{tG}$
$\displaystyle=g_{S}\,\overline{Q}T_{A}\tilde{\varphi}\sigma^{\mu\nu}t\,G_{\mu\nu}^{A}=g_{S}\,\mathcal{Q}_{uG,33}.$
(2.4)

The second equality is in terms of the Warsaw basis Grzadkowski:2010es . Once
the additional symmetry (2.1) and the EOM are enforced, only
$\mathcal{O}_{tG}$ and $\mathcal{O}_{tG}^{\dagger}$ survive, as standard
manipulations yield:

$\displaystyle\mathcal{O}_{gt}+\mathcal{O}_{gt}^{\dagger}$
$\displaystyle=g_{s}\Big{(}\text{four-quarks}\Big{)}$ (2.5)
$\displaystyle\mathcal{O}_{gQ}+\mathcal{O}_{gQ}^{\dagger}$
$\displaystyle=g_{s}\Big{(}\text{four-quarks}\Big{)}$ (2.6)
$\displaystyle\mathcal{O}_{gt}-\mathcal{O}_{gt}^{\dagger}$
$\displaystyle=\mathcal{O}_{gQ}^{\dagger}-\mathcal{O}_{gQ}=\frac{y_{t}}{g_{S}}(\mathcal{O}_{tG}-\mathcal{O}_{tG}^{\dagger})$
(2.7)

This shows that $\mathcal{O}_{gt}$ and $\mathcal{O}_{gQ}$ reduce to four-
fermion operators; more specifically, they reduce to operators composed of two
light quarks and two tops, that will be considered in the following, and to
operators with four top quark fields, that we will neglect due to their large
suppression in the process at hand. The only surviving operator,
$\mathcal{O}_{tG}$, has the effect of deforming the top quark color charge
distribution, producing a chromo-magnetic and chromo-electric dipole moment
proportional to $\Re(c_{tG})$ and $\Im(c_{tG})$ respectively.

On top of the two-fermion operators listed above, there are five operators
involving two fermion fields and electroweak gauge bosons that are in
principle relevant for top physics,

$\displaystyle\mathcal{Q}_{\varphi u,\,33}$
$\displaystyle=i(\varphi^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{\mu}\varphi)(\overline{t}\gamma^{\mu}t),$
(2.8) $\displaystyle\mathcal{Q}_{\varphi q,\,33}^{(1)}$
$\displaystyle=i(\varphi^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{\mu}\varphi)(\overline{Q}\gamma^{\mu}Q),$
(2.9) $\displaystyle\mathcal{Q}_{\varphi q,\,33}^{(3)}$
$\displaystyle=i(\varphi^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{\mu}\sigma_{I}\varphi)(\overline{Q}\gamma^{\mu}\sigma^{I}Q),$
(2.10) $\displaystyle\mathcal{Q}_{uB,\,33}$
$\displaystyle=(\overline{Q}\sigma^{\mu\nu}t)\tilde{\varphi}B_{\mu\nu},$
(2.11) $\displaystyle\mathcal{Q}_{uW,\,33}$
$\displaystyle=(\overline{Q}\sigma^{\mu\nu}\sigma_{I}t)\tilde{\varphi}W_{\mu\nu}^{I}.$
(2.12)

Both at $\mathcal{O}(c/\Lambda^{2})$ and $\mathcal{O}(c^{2}/\Lambda^{4})$,
these operators only enter electroweak top production, and since this process
is heavily suppressed with respect to QCD production, their effect will be
neglected. An NLO computation including both QCD and EW corrections would be
required to fully take into account these contributions.

We note that $\mathcal{Q}_{\varphi q,\,33}^{(3)}$ and $\mathcal{Q}_{uW,\,33}$
also induce a modified $tWb$ vertex. Since spin measurements involve both the
production and the decay of the particle at hand, operators that modify the
decay of tops enter our study. In the following we will show that their actual
effect on the spin observables we consider is negligible.

###  2.2 Four-fermion operators

There are seven independent dimension-6 operators mediating $t\overline{t}$
production by $q\overline{q}$ annihilation that result in a color-octet final
state. These operators are constructed with a combination of four quark
fields, two light ($u,d,c,s$) and two tops. Following the LHC Top WG notation
and convention Aguilar-Saavedra:2018ksv , we express SMEFT operators in a
basis that highlights their chiral structure:

$\displaystyle\mathcal{O}_{tu}^{8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{t}\gamma_{\mu}T^{A}t)(\overline{u}_{\text{f}}\gamma^{\mu}T_{A}u_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}-\frac{1}{6}\mathcal{Q}_{uu,\text{\text{ff33}}}+\frac{1}{2}\mathcal{Q}_{uu,\text{3ff3}},$
(2.13) $\displaystyle\mathcal{O}_{td}^{8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{3}(\overline{t}\gamma_{\mu}T_{A}t)(\overline{d}_{\text{f}}\gamma^{\mu}T^{A}d_{\text{f}})=\textstyle\sum_{\text{f}=1}^{3}\mathcal{Q}_{ud,\text{33ff}}^{(8)},$
(2.14) $\displaystyle\mathcal{O}_{tq}^{8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{q}_{\text{f}}\gamma_{\mu}T_{A}q_{\text{f}})(\overline{t}\gamma^{\mu}T^{A}t)=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qu,\text{ff33}}^{(8)},$
(2.15) $\displaystyle\mathcal{O}_{Qu}^{8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}T_{A}Q)(\overline{u}_{\text{f}}\gamma^{\mu}T^{A}u_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qu,\text{33ff}}^{(8)},$
(2.16) $\displaystyle\mathcal{O}_{Qd}^{8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{3}(\overline{Q}\gamma_{\mu}T_{A}Q)(\overline{d}_{\text{f}}\gamma^{\mu}T^{A}d_{\text{f}})=\textstyle\sum_{\text{f}=1}^{3}\mathcal{Q}_{qd,\text{33ff}}^{(8)},$
(2.17) $\displaystyle\mathcal{O}_{Qq}^{1,8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}T^{A}Q)(\overline{q}_{\text{f}}\gamma^{\mu}T_{A}q_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\frac{1}{4}\mathcal{Q}_{qq,\text{3ff3}}^{(1)}-\frac{1}{6}\mathcal{Q}_{qq,\text{33ff}}^{(1)}+\frac{1}{4}\mathcal{Q}_{qq,\text{3ff3}}^{(3)},$
(2.18) $\displaystyle\mathcal{O}_{Qq}^{3,8}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}T^{A}\sigma_{I}Q)(\overline{q}_{\text{f}}\gamma^{\mu}T_{A}\sigma^{I}q_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\frac{3}{4}\mathcal{Q}_{qq,\text{3ff3}}^{(1)}-\frac{1}{6}\mathcal{Q}_{qq,\text{33ff}}^{(3)}-\frac{1}{4}\mathcal{Q}_{qq,\text{3ff3}}^{(3)}.$
(2.19)

The second equality in (2.13) – (2.19) again refers to the expression in terms
of the Warsaw basis Grzadkowski:2010es . The operators (2.13) – (2.19) can
also be considered in their color-singlet version:

$\displaystyle\mathcal{O}_{tu}^{1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{t}\gamma_{\mu}t)(\overline{u}_{\text{f}}\gamma^{\mu}u_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{uu,\text{ff33}},$
(2.20) $\displaystyle\mathcal{O}_{td}^{1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{3}(\overline{t}\gamma_{\mu}t)(\overline{d}_{\text{f}}\gamma^{\mu}d_{\text{f}})=\textstyle\sum_{\text{f}=1}^{3}\mathcal{Q}_{ud,\text{33ff}}^{(1)},$
(2.21) $\displaystyle\mathcal{O}_{tq}^{1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{q}_{\text{f}}\gamma_{\mu}q_{\text{f}})(\overline{t}\gamma^{\mu}t)=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qu,\text{ff33}}^{(1)},$
(2.22) $\displaystyle\mathcal{O}_{Qu}^{1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}Q)(\overline{u}_{\text{f}}\gamma^{\mu}u_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qu,\text{33ff}}^{(1)},$
(2.23) $\displaystyle\mathcal{O}_{Qd}^{1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{3}(\overline{Q}\gamma_{\mu}Q)(\overline{d}_{\text{f}}\gamma^{\mu}d_{\text{f}})=\textstyle\sum_{\text{f}=1}^{3}\mathcal{Q}_{qd,\text{33ff}}^{(1)},$
(2.24) $\displaystyle\mathcal{O}_{Qq}^{1,1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}Q)(\overline{q}_{\text{f}}\gamma^{\mu}q_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qq,\text{33ff}}^{(1)},$
(2.25) $\displaystyle\mathcal{O}_{Qq}^{3,1}$
$\displaystyle=\textstyle\sum_{\text{f}=1}^{2}(\overline{Q}\gamma_{\mu}\sigma_{I}Q)(\overline{q}_{\text{f}}\gamma^{\mu}\sigma^{I}q_{\text{f}})=\textstyle\sum_{\text{f}=1}^{2}\mathcal{Q}_{qq,\text{ff33}}^{(3)}.$
(2.26)

Again the second equality in (2.20) – (2.26) refers to the expression in terms
of the Warsaw basis Grzadkowski:2010es . When considering only the leading-
order SM QCD amplitudes for the Standard Model, these operators effectively
enter top production at the $1/\Lambda^{4}$ level, since the
$\mathcal{O}(1/\Lambda^{2})$ interference arises only with the suppressed
electroweak SM production $q\bar{q}\,\to\,Z/\gamma\,\to\,t\bar{t}$, or with
QCD production at NLO.

##  3 Spin of top quarks

Top quark pairs are unique candidates for spin polarization and spin
correlation measurements. Top production at hadron colliders involves top
quarks with negligible individual spin polarization, but large pair spin
correlation with an elaborate phase-space structure. In fact, with appropriate
cuts, it has been argued that spin correlations become so strong that quantum
entanglement can be detected between top pairs Afik:2022kwm  and, in
principle, the violation of Bell inequalities can also be observed
Fabbrichesi:2021npl ; Severi:2021cnj ; Aguilar-Saavedra:2022uye . Furthermore,
the spin de-correlation time from non-perturbative strong effects, of order
$m_{t}\,\Lambda^{-2}_{\text{\tiny QCD}}$, is much larger than the top
lifetime, so top quarks decay before their initial spin state is lost in the
environment. In fact, since the hadronization timescale,
$\Lambda_{\text{\tiny{QCD}}}^{-1}$, is also about one order of magnitude
larger than the top lifetime, tops behave like free, spinning particles.
Thanks to the chiral nature of weak interactions, the momenta of daughters
emerging from the decay of tops are correlated with the spin of the initial
top, with the decay width given at LO by:

$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_{X}}=\frac{1+\alpha_{X}\cos\theta_{X}}{2}\,,$
(3.1)

where $\alpha_{X}$ is a parameter known as spin analyzing power of particle
$X$, and $\theta_{X}$ is the angle between the original top spin and the
direction of the emitted $X$ in the top rest frame. As a result of this
effect, individual decay products can be considered as proxies for the spin of
the corresponding top quarks, and correlations between different decay
products as proxies for those between the top quark spins.

The spin state of a $t\bar{t}$ pair is described by the density matrix:

$\rho=\frac{1}{4}\Big{(}\mathbbm{1}\otimes\mathbbm{1}+\sum_{i=1}^{3}B_{i}\,\sigma_{i}\otimes\mathbbm{1}+\sum_{i=j}^{3}\bar{B}_{j}\,\mathbbm{1}\otimes\sigma_{j}+\sum_{i=1}^{3}\sum_{j=1}^{3}C_{ij}\,\sigma_{i}\otimes\sigma_{j}\Big{)},$
(3.2)

where the first term in each tensor product refers to the top and the second
term to the anti-top. The parameters entering (3.2) have the physical
interpretation of being the expectation values of individual spins and spin
correlations,

$\langle
S_{i}\rangle=B_{i},\quad\langle\bar{S}_{i}\rangle=\bar{B}_{j},\quad\langle
S_{i}\bar{S}_{j}\rangle=C_{ij}.$ (3.3)

The parameters entering (3.2) have well defined C, P, and CP transformation
properties Bernreuther:2015yna , listed in Table 1. In particular, the linear
combinations $B_{i}-\bar{B}_{i}$ and $C_{ij}-C_{ji}$ are CP violating, and
will not be considered further in this work.

#### Spin analyzing power.

At leading order in the SM, the spin analyzing power of prompt $W$ bosons and
charged leptons emerging from the $W$ decay is given by:

$\displaystyle\alpha_{W}=\frac{m_{t}^{2}-2m_{W}^{2}-m_{b}^{2}}{(m_{b}^{2}-m_{t}^{2})^{2}+(m_{b}^{2}+m_{t}^{2})m_{W}^{2}-2m_{W}^{4}}\
\times$ $\displaystyle\hskip
14.22636pt\times\sqrt{(m_{b}-m_{t}-m_{W})(m_{b}+m_{t}-m_{W})(m_{b}-m_{t}+m_{W})(m_{b}+m_{t}+m_{W})}\
\approx\ 0.394,$ (3.4) $\displaystyle\alpha_{\ell}=1.$ (3.5)

It is curious that the charged lepton has a larger spin-analyzing power than
its mother, the $W$ boson. This is due to the constructive and destructive
interference between amplitudes with intermediate $W$ bosons of different
helicities; this information is lost when considering the direction of flight
of the prompt $Wb$ pair.

The presence of new physics in the top electroweak couplings, (2.10) and
(2.12), can affect the value of $\alpha$ Zhang:2010dr . The effect of
$\mathcal{Q}_{\varphi q,\,33}^{(3)}$ is to rescale the SM value of the $tWb$
vertex, with no effect on angular distributions, and thus on $\alpha$. On the
other hand, $\mathcal{Q}_{uW,\,33}$ can alter spin analyzing powers. The spin
analyzing power of the $Wb$ pair is modified at the
$c_{uW,\,33}v^{2}/\Lambda^{2}$ level, as expected from dimensional analysis.
However, interestingly, when considering the full decay $t\to
b\,\ell\,\nu_{\ell}$ the on-shell conditions for final-state leptons remove
the linear (and cubic) contribution, and we find:

$\displaystyle\alpha_{\ell}(c_{uW,\,33})$
$\displaystyle=1-\frac{c_{uW,\,33}^{2}v^{4}}{\Lambda^{4}}\frac{4(2m_{t}^{6}+3m_{t}^{4}m_{W}^{2}-6m_{t}^{2}m_{W}^{4}+m_{W}^{6}+12m_{t}^{4}m_{W}^{2}\log
m_{W}/m_{t})}{(m_{W}^{2}-m_{t}^{2})^{2}(m_{t}^{2}+2m_{W}^{2})},$ (3.6)

up to corrections of order $c_{uW,\,33}^{4}v^{8}/\Lambda^{8}$,
$m_{b}^{2}/m_{W}^{2}$, and $\Gamma_{W}^{2}/m_{W}^{2}$. This result applies to
both the positively and the negatively charged lepton. The quadratic change in
$\alpha_{\ell}$ due to $c_{uW\,33}$ was already calculated in Aguilar-
Saavedra:2006qvv , obtaining a result in agreement with ours, when considering
the different conventions. We note that the small pre-factor makes the
$\mathcal{Q}_{uW,\,33}$ contribution to the lepton spin analyzing power of
order $10^{-3}$ or less, assuming recent bounds for its Wilson coefficient.
This suggests the dilepton final state is an exceptionally clean channel to
investigate new physics in top production, with little contamination from
electroweak effects in the top decay.

Higher order QCD corrections to $\alpha_{W}$ and $\alpha_{\ell}$ are known in
the SM Czarnecki:1990pe ; Brandenburg:2002xr . NLO effects in the top decay
can be reduced to a shift in $\alpha$, that we consider, and to very limited
corrections to the functional relation (3.1), that are neglected.

#### Measurement of the spin state.

The differential cross section for the process $pp\to t\bar{t}\to ab$, where
$a$ is a decay product of the top quark and $b$ is a decay product of the
anti-top quark can be expressed as Bernreuther:2015yna :

$\displaystyle\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta_{i}}=\frac{1+\alpha_{a}B_{i}\cos\theta_{i}}{2},\quad\frac{1}{\sigma}\frac{d\sigma}{d\cos\bar{\theta}_{i}}=\frac{1+\alpha_{b}\bar{B}_{i}\cos\bar{\theta}_{i}}{2}$
(3.7)
$\displaystyle\frac{1}{\sigma}\frac{d\sigma}{d(\cos\theta_{i}\cos\bar{\theta}_{j})}=-\frac{1+C_{ij}\alpha_{a}\alpha_{b}\,\cos\theta_{i}\cos\bar{\theta}_{j}}{2}\log\big{|}\cos\theta_{i}\cos\bar{\theta}_{j}\,\big{|},$
(3.8)

where $\theta_{i}$ is the angle between $\vec{p}_{a}$ and the $i$-th axis in
the top rest frame, and $\bar{\theta}_{j}$ the angle between $\vec{p}_{b}$ and
the $j$-th axis in the anti-top rest frame.

A suitable reference frame must be defined in order to evaluate the angles
$\theta_{i}$, $\bar{\theta}_{j}$. An advantageous choice is the helicity basis
$\\{\hat{k},\hat{r},\hat{n}\\}$, defined in the $t\bar{t}$ pair reference
frame as:

$\hat{k}=\text{top
direction},\quad\hat{r}=\frac{\hat{p}-\hat{k}\,\cos\theta}{\sin\theta},\quad\hat{n}=\frac{\hat{p}\times\hat{k}}{\sin\theta}$
(3.9)

where $\hat{p}$ is the beam axis and $\theta$ is the top/anti-top scattering
angle in the $t\bar{t}$ center of mass frame. There is a sign ambiguity in
choosing the beam direction $\hat{p}$; for every event, $\hat{p}$ is chosen so
that:

$0\leq\theta\leq\frac{\pi}{2}.$ (3.10)

The integration of (3.7) and (3.8) gives explicit relations to measure the
spin state:

$B_{i}=\big{\langle}\frac{3}{\alpha_{a}}\cos\theta_{i}\big{\rangle},\quad\bar{B}_{i}=\big{\langle}\frac{3}{\alpha_{b}}\cos\bar{\theta}_{i}\big{\rangle},\quad
C_{ij}=\big{\langle}\frac{9}{\alpha_{a}\alpha_{b}}\cos\theta_{i}\cos\bar{\theta}_{j}\big{\rangle},$
(3.11)

where angle brackets denote the expectation value:

$S=\langle s\rangle=\frac{\int
s\,|\mathcal{M}|^{2}\,d\Pi}{\int|\mathcal{M}|^{2}\,d\Pi}\equiv\frac{\sigma_{S}}{\sigma}.$
(3.12)

The above relations show that an arbitrary spin observable $S$ can be
extracted as a ratio of two cross sections: the numerator $\sigma_{S}$ is the
phase-space integral of $s$, some dimensionless combination of angles and spin
analysing powers, weighted with the matrix element squared, and the
denominator is the total cross section. Further, as the various examples
proposed in Ref. Aguilar-Saavedra:2022uye  show, it is possible to construct
combination of angles $s$ so that $\langle s\rangle$ is directly sensitive to
several quantities of interest, e.g. $B_{i}\pm B_{j}$, $C_{ij}\pm C_{ji}$,
$C_{ii}+|C_{jj}+C_{kk}|$.

The observables used in this work, that are those commonly considered in spin
measurements, are listed in Tab. 1, together with their transformation
properties under P and CP. Standard Model predictions for the spin observables
of Tab. 1 for $t\bar{t}$ pairs produced at the LHC are already available at
NNLO QCD Czakon:2020qbd  and NLO QCD + EW Bernreuther:2015yna .

CP even | CP odd
---|---
$B_{k}+\overline{B}_{k}$ | P-odd | $B_{k}-\overline{B}_{k}$ | P-odd
$B_{r}+\overline{B}_{r}$ | P-odd | $B_{r}-\overline{B}_{r}$ | P-odd
$B_{n}+\overline{B}_{n}$ | P-even | $B_{n}-\overline{B}_{n}$ | P-even
$C_{kk}$ | P-even |  |
$C_{rr}$ | P-even |  |
$C_{nn}$ | P-even |  |
$C_{kr}+C_{rk}$ | P-even | $C_{kr}-C_{rk}$ | P-even
$C_{kn}+C_{rn}$ | P-odd | $C_{kn}-C_{rn}$ | P-odd
$C_{rn}+C_{rn}$ | P-odd | $C_{rn}-C_{rn}$ | P-odd
Table 1: Top spin polarization and $t\bar{t}$ spin correlation observables
with their C and CP transformation properties. The CP-even observables are
those considered in this work, the CP-even observables are needed in principle
for a full reconstruction of the density matrix $\rho$, but they are not
considered here.

In the SMEFT, allowing at most one NP insertion in each Feynman diagram, cross
sections are quadratic functions of the Wilson coefficients so that:

$S=\frac{\sigma_{S}^{(0)}+\hat{c}\,\sigma_{S}^{(1)}+\hat{c}^{2}\,\sigma_{S}^{(2)}}{\sigma^{(0)}+\hat{c}\,\sigma^{(1)}+\hat{c}^{2}\,\sigma^{(2)}},\qquad\hat{c}\equiv\frac{c}{\Lambda^{2}\,/\,(1\,\text{TeV})^{2}}.$
(3.13)

The expression in (3.13) can be series expanded around the SM value,

$S=\frac{\sigma_{S}^{(0)}}{\sigma^{0}}+\hat{c}\>\frac{\sigma_{S}^{(1)}\,\sigma^{(0)}-\sigma_{S}^{(0)}\,\sigma^{(1)}}{(\sigma^{0})^{2}}+\cdots,$
(3.14)

so that a linear, quadratic, and so on, shift can be extracted. However, to
maintain generality, we extract the six cross sections in (3.13) separately,
so that the full dependence of $S$ on $c$ can be evaluated.

Accurate SM predictions can be implemented in several ways, such as:

  * –

the SM cross sections $\sigma_{S}^{(0)}$ and $\sigma^{(0)}$ in (3.13) are
replaced by their most accurate values, EFT terms are not modified, and then
the ratio $S$ is taken;

  * –

numerator and denominator of (3.13) are globally rescaled by the SM K-factors
$K_{S}=\sigma_{S,\text{accurate}}^{(0)}/\sigma_{S}^{(0)}$ and
$K=\sigma_{\text{accurate}}^{(0)}/\sigma^{(0)}$, then and the ratio $S$ is
taken;

  * –

the ratio (3.13) is series expanded, the first term
$\sigma_{S}^{(0)}/\sigma^{(0)}$ is replaced with the accurate value, other
terms in the series are not modified – even if they contain $\sigma_{S}^{(0)}$
or $\sigma^{(0)}$.

It is not clear which one, if any, of the prescriptions outlined above entails
the best physical meaning, and the spread in results obtained by implementing
accurate SM predictions in different ways may be considered to be a
theoretical uncertainty. In this work we will report all six cross sections
that enter (3.13), leaving the possibility open for arbitrary implementations
of accurate SM calculations.

#### Quantum entanglement.

As discussed in the introduction, and in more detail in Refs. Afik:2020onf ;
Severi:2021cnj ; Aoude:2022imd ; Afik:2022kwm , top spin correlations can
signal the presence of quantum entanglement between tops. A calculation shows
that, in the case of $t\bar{t}$, entanglement is signaled by the condition:

$\Delta=|C_{kk}+C_{rr}|-C_{nn}-1>0.$ (3.15)

Inequality (3.15) is general, and independent of the possible presence of new
physics. We also note that the concurrence of the $t\bar{t}$ spin quantum
state (3.2) is given by:

$\mathcal{C}[\rho]=\begin{dcases}\Delta/2,&\Delta>0\\\ 0&\Delta\leq
0\end{dcases}$ (3.16)

The absolute value in the definition of $\Delta$ is non-trivial, as, already
in the SM, at $t\bar{t}$ threshold $C_{kk}+C_{rr}<0$, so
$\Delta=-\text{Tr}\,C-1$, while at large top $p_{T}$, $C_{kk}+C_{rr}>0$, so
$\Delta$ becomes $C_{kk}+C_{rr}-C_{nn}-1$. The use of $\text{Tr}\,C$ to detect
entanglement at $t\bar{t}$ threshold has been the topic of extended studies
Afik:2020onf ; Afik:2022kwm ; in fact, as shown in Ref. Aguilar-
Saavedra:2022uye , $\Delta$ can be measured directly using dedicated angular
observables, regardless of the sign of the absolute value. To appreciate
directly the effect of SMEFT operators on the amount of spin entanglement, in
the following we will consider:

$\Delta^{\pm}=\pm(C_{kk}+C_{rr})-C_{nn}-1=\frac{\pm(\sigma_{C_{kk}}+\sigma_{C_{rr}})-\sigma_{C_{nn}}-\sigma}{\sigma}$
(3.17)

as additional observables, as an extension to those appearing in (3.2). We
note that $\Delta^{-}$ is a well known top spin observable, usually called
$D\equiv(\Delta^{-}+1)/3$, while $\Delta^{+}$ has not been considered
experimentally before.

##  4 Computation Setup

Since the reconstruction of a spin quantum state requires the analysis of
decay products, the analysis of top spin polarization and correlations at NLO
would require the generation of events of the form:

$p\ p\to b\,\bar{b}\,\ell^{+}\,\ell^{-}\,\nu\,\bar{\nu}$ (4.1)

at order $\mathcal{O}(\alpha_{\text{s}}^{3}\,\alpha_{\text{EW}}^{4})$. This
would fully include virtual corrections, and all off-shell effects for both
tops and $W$ bosons. This computation is extremely challenging, and as of now
can only be handled by purpose-made tools, e.g. Campbell:2012uf ;
Campbell:2014kua , that lack the versatility needed to include arbitrary SMEFT
operators in the matrix element.

A useful proxy for the full NLO computation could be given by the process:

$p\ p\to W^{+}\,W^{-}\,b\,\bar{b}$ (4.2)

at order $\mathcal{O}(\alpha_{\text{s}}^{3}\,\alpha_{\text{EW}}^{2})$, since
the prompt $W\,b$ pair emitted by the electroweak decay of tops also has spin
analyzing power. For such a computation, suitable phase-space cuts help
isolate the doubly resonant contribution $p\ p\to t\,\bar{t}\to
W^{+}\,W^{-}\,b\,\bar{b}$. Generally speaking, one should cluster parton-level
QCD radiation in jets, impose a selection of standard jet cuts, and ask that
the invariant mass of the two $W$ \+ $b$-jet pairs is close to $m_{t}$. The
full SM amplitude for (4.2) has been evaluated at NLO in Denner:2010jp ;
Bevilacqua:2010qb , but the significant complexity of SMEFT amplitudes
prevented us from extracting precise enough results for this process.

To obtain our results we combine a NLO fixed-order calculation for the process
$p\ p\to t\,\bar{t}$ and the modeling of the electroweak decays $t\to
b\,\ell^{+}\,\nu$ and $\bar{t}\to\bar{b}\,\ell^{-}\,\bar{\nu}$ with MadSpin
Artoisenet:2012st . This yields predictions whose accuracy is summarized in
Table 2. Accuracy is NLO QCD, up to virtual corrections to the spin state,
that can not be handled by MadSpin. However, as the kinematic state is
constructed before the spin state, and the top quarks’ momentum is correlated
with their spin111See for instance the argument given in Severi:2021cnj  for
top pairs emitted at large $p_{T}$., the full-NLO accuracy of the $t\bar{t}$
kinematics improves the partial-NLO accuracy of the spin state. It is also
true that virtual corrections to the $t\bar{t}$ spin state are known to be
small in the SM Bernreuther:2015yna ; Frederix:2021zsh  and it is reasonable
to assume that the same holds true for the SMEFT.

|  LO QCD | NLO QCD
---|---|---
|  | Real emission | Virtual corrections
Kinematics | ✓ | ✓ | ✓
Spin state | ✓ | ✓ | Approximate

Table 2: Accuracy of the computation resulting from Madgraph5_aMC@NLO and
MadSpin.

Hard events featuring a top pair production are generated with
Madgraph5_aMC@NLO v3.4.0 at fixed-order NLO QCD accuracy,
$\mathcal{O}(\alpha_{s}^{3}\alpha_{\text{EW}}^{0})$. Electroweak corrections,
known to amount to a percent-level correction to the total cross section
Bernreuther:2006vg , and to very small shifts in the spin observables
Bernreuther:2015yna , are not considered. The hadronic center of mass energy
is set to $\sqrt{s}=13.0$ TeV, corresponding to the second run of the LHC. We
use the NNPDF3.1 parton distribution function NNPDF:2017mvq  at NLO QCD with
$\alpha_{s}(m_{Z})=0.118$. The dataset used to extract this PDF does not
include any top spin data. The SM renormalization and factorization scales
$\mu_{\text{R}}$ and $\mu_{\text{F}}$ are set on an event-by-event basis to
one fourth of the total transverse energy $H_{T}$ of the process, and the
SMEFT renormalization scale $\mu_{\text{EFT}}$ is set to the top mass. Events
are generated within the SMEFT@NLO model Degrande:2020evl , whose flavor
symmetry is (2.1). Note that the flavor symmetry implies that light SM
particles, including the $b$ quark, are massless, and the CKM matrix is the
identity. Other SM input parameters are set to their current experimental
values. The SMEFT@NLO model is restricted to only contain the operators of
interest, $\mathcal{O}_{tG}$ and the four-quark operators (2.13) – (2.19) and
(2.20) – (2.26), one at a time. Top quarks are kept on shell.

The leptonic decay of top quarks $t\to b\,\ell^{+}\,\nu$,
$\bar{t}\to\bar{b}\,\ell^{-}\,\bar{\nu}$ is modeled by MadSpin
Artoisenet:2012st , adapted to handle fixed-order calculations with minor
improvements to the technique described in Frederix:2021zsh . Spin observables
are extracted at parton level using charged leptons emerging from the
electroweak decay, integrating the expressions that enter in the numerator and
denominator of (3.12) separately. The individual contributions entering (3.13)
are then extracted by varying the Wilson coefficients and fitting a quadratic
dependence.

The sources of theoretical uncertainty we consider are MC statistical
uncertainty and scale variation. Statistical uncertainty is estimated with
standard procedures, scale variation is evaluated from the envelope of the
nine points $\mu_{R}=(H_{T}/8,\,H_{T}/4,\,H_{T}/2)$,
$\mu_{\text{f}}=(H_{T}/8,\,H_{T}/4,\,H_{T}/2)$, $\mu_{\text{EFT}}=(m_{t})$.

#### Validations.

Before moving to NLO accuracy, we validate our setup at LO. We compare the SM
$t\bar{t}$ spin density matrix we obtain with MadSpin with the one obtained
from the benchmark process (4.1), that at LO is feasible. In addition, we
compare results with and without the narrow width approximation for tops and W
bosons. All our checks show deviations at the $10^{-3}$ level, well below
current (and presumably future) experimental accuracy, and other sources of
theoretical uncertainty. The SMEFT effects on the $t\bar{t}$ spin density
matrix of the two-fermion operator (2.4) and of the four-fermion color-octet
operators (2.13) - (2.19) were computed at LO QCD and LO in $c/\Lambda^{2}$ in
Bernreuther:2015yna . After accounting for the different choice of operators,
we are able to replicate the results found in Bernreuther:2015yna , up to
relative deviations of order $10\%$. We note that computational setup in
Bernreuther:2015yna  features a different PDF, different choices of
factorization and renormalization scales, and different numerical input
parameters with respect to the one presented in this work, so perfect
numerical agreement is not expected. However, the pattern of deviations from
the SM, and their relative size, is found to be in good agreement. To further
check compatibility, we reproduced the exact setup of Bernreuther:2015yna  in
a LO run with the operator $\mathcal{O}_{tG}$, and obtained agreement at the
$10^{-4}$ level.

##  5 Results and discussion

Our numerical integration is carried:

  * •

Inclusive in $t\bar{t}$ phase space.

  * •

Binned in $t\bar{t}$ phase space, with a $3\times 3$ binning in the $t\bar{t}$
invariant mass $m_{t\bar{t}}$ and the center of mass scattering angle
$\theta$. This binning will be updated to match the actual resolution of
experimental measurements, once available.

In any case, we do not impose any cuts or detector acceptance effects on the
final state $b\,\ell^{+}\,\nu\,\bar{b}\,\ell^{-}\,\bar{\nu}$. The plots in
this Section collect results for the fifteen SMEFT operators inclusive in
phase space, and for selected operators binned in phase space, using the
sample $m_{t\bar{t}}\,-\,\cos\theta$ binning we considered. Appendix A
contains tables with all cross-sections needed to replicate our plots
inclusive in phase space. Plots with binned results for operators not included
in this Section are in Appendix B.

To highlight the improvement offered by our NLO calculation over the LO, in
Appendix A the cross-sections we obtained are provided separately at LO and
NLO accuracy. The large $K$-factor of the total cross-section $\sigma$ makes
the numerical comparison between LO and NLO not obvious. However, after taking
the appropriate ratios, we find that, frequently, NLO results are appreciably
different from LO ones. As expected, NLO results also tend to show a smaller
scale variation.

###  5.1 Results inclusive in $t\bar{t}$ phase space

The following Figures 1 \- 3 show the change with respect to the SM value,
defined as:

$\Delta
S=\frac{\sigma_{S}^{(0)}+\hat{c}\,\sigma_{S}^{(1)}+\hat{c}^{2}\,\sigma_{S}^{(2)}}{\sigma^{(0)}+\hat{c}\,\sigma^{(1)}+\hat{c}^{2}\,\sigma^{(2)}}-\frac{\sigma_{S}^{(0)}}{\sigma^{(0)}},$
(5.1)

for the top spin polarizations $B_{i}+\bar{B}_{i}$, the $t\bar{t}$ spin
correlations $C_{ii}$ and $C_{ij}+C_{ji}$, and the entanglement markers
$\Delta^{\pm}$, as a function of the Wilson coefficient of the operators we
consider, inclusive in $t\bar{t}$ phase space. Similarly to (3.13), in (5.1)
we defined $\hat{c}\equiv c\ (1\,\text{TeV})^{2}/\Lambda^{2}$, which is
equivalent numerically to setting $\Lambda=1\,\text{TeV}$. For the purposes of
plotting results, we employ our own NLO QCD SM predictions $\sigma_{S}^{(0)}$
and $\sigma^{(0)}$ in (5.1). Accurate SM predictions, as discussed above, may
also be used in (5.1). However, as the leading SM contribution is subtracted,
the actual numerical difference is negligible in the range we consider. While,
as we will see in the following, an inclusive measurement hides most of the
physical features present, we can still highlight several interesting points,
presented in the rest of this Section.

We begin by noting that quadratic terms $\mathcal{O}(c^{2}/\Lambda^{4})$, that
induce a non-linear behaviour of spin observables, are almost always
important, and often leading with respect to the SM-SMEFT interference
$\mathcal{O}(c/\Lambda^{2})$. This is particularly pronounced for the color-
singlet operators for which the deviations from the SM are nearly identical
for positive and negative values of the coefficients. This is expected as this
class of operators only interferes with the SM at NLO in QCD. For the color-
octet operators both linear and quadratic terms are important for the range of
coefficients that we consider. We also notice that the effect of color-singlet
operators is more prominent than the effect of color-octet operators, due to a
relative color factor of $9/2$ entering the respective squared amplitudes.

Our results are in qualitative agreement with the findings at LO QCD of
Bernreuther:2015yna ; Aoude:2022imd , signaling that NLO QCD contributions are
under control, and our curves appear stable under the inclusion of the NLO
with $\mathcal{O}(10\%)\sim\mathcal{O}(\alpha_{s})$ deviations. This is
particularly true for spin correlation observables, as they arise from ratios
of (weighted) cross-sections. It is frequently the case, however, that LO
scale variation is not a good estimate for missing higher orders. This is
especially true for color-octet operators, where the shift between LO QCD and
NLO QCD often exceeds $3\times$ the LO scale variation. This is despite the
fact that our scale variation is conservative, as it includes all nine points
obtained from the $1/2,1,2$ rescaling of $\mu_{R}$ and $\mu_{\text{f}}$, and
not the often used seven.

(a)

(b)

(c)

(d)

(e)

Figure 1: Change from the SM value for spin observables for the operators
$\mathcal{O}_{tG}$ (1(a)), $\mathcal{O}_{Qq}^{(8,3)}$ (1(b)),
$\mathcal{O}_{Qq}^{(1,3)}$ (1(c)), $\mathcal{O}_{Qq}^{(8,1)}$ (1(d)),
$\mathcal{O}_{Qq}^{(1,1)}$ (1(e)), inclusive in $t\bar{t}$ phase space. Dashed
lines indicate results at LO, continuous lines indicate NLO. The shaded region
around each curve represents the combination of scale and MC uncertainty. The
MC uncertainty is always sub-leading compared to scale variation. Only curves
that deviate appreciably from zero are shown.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Results for the operators $\mathcal{O}_{Qu}^{8}$ (2(a)),
$\mathcal{O}_{Qu}^{1}$ (2(b)), $\mathcal{O}_{tq}^{8}$ (2(c)),
$\mathcal{O}_{tq}^{1}$ (2(d)), $\mathcal{O}_{Qd}^{8}$ (2(e)) and
$\mathcal{O}_{Qd}^{1}$ (2(f)), similar to Figure 1.

(a)

(b)

(c)

(d)

Figure 3: Results for the operators $\mathcal{O}_{tu}^{8}$ (3(a)),
$\mathcal{O}_{tu}^{1}$ (3(b)), $\mathcal{O}_{td}^{8}$ (3(c)),
$\mathcal{O}_{td}^{1}$ (3(d)), similar to Figure 1.

Looking at individual spin observables, we note that several are not affected
by dimension-6 operators, even in the regime where the NP-squared terms are
dominant. The $\hat{n}$ polarization $B_{n}+\overline{B}_{n}$ never deviates
appreciably from its SM value for any of the operators we considered. This is
because, as already noted in Aoude:2022imd , the only contributions to
$B_{n}+\overline{B}_{n}$ come from the absorptive parts of one-loop
amplitudes, and are thus heavily suppressed. Amongst those in Table 1,
$B_{n}+\overline{B}_{n}$ is the only spin observable for which this happens.

The quantum entanglement markers $\Delta^{\pm}$ exhibit a similar pattern of
deviations from the SM to the other spin observables. However, they are not
included in this work for pure academic interest: their experimental
measurement may be significantly simpler than the one of other observables.
The enhanced precision is due to the fact that $\Delta^{-}$ is accessible
directly from the opening angle between the leptons, with no need for (3.11).
The CMS Collaboration recently published a projection for top spin correlation
measurements at the HL-LHC CMS:2022cqm , where it is argued that $\Delta^{-}$
will be the spin observable more easily accessible, with a projected relative
accuracy at the end of HL-LHC of $3\%$. Given the prospects of precise
measurements, the entanglement markers are expected to provide good
sensitivity to the dimension-6 coefficients.

Proceeding operator by operator, $\mathcal{O}_{tG}$ immediately stands out as
exhibiting a special behaviour. As the only SMEFT operator that enters gluon-
fusion top production $g\,g\to t\,\bar{t}$, since $\sigma(g\,g\to
t\,\bar{t})/\sigma(q\,\bar{q}\to t\,\bar{t})\sim 10$ at the LHC, the effect of
$c_{tG}$ is approximately ten times larger than the effect of all other four-
fermion Wilson coefficients. (Note the $x$-axis scale of Fig. 1(a) is
different from the one in all other plots.) In fact, as noted below, in our
projections top spin correlation measurements yield a remarkable sensitivity
in constraining the value of $c_{tG}$. We also note that $\mathcal{O}_{tG}$ is
affecting $t\bar{t}$ spin correlations, that is entries of the $C_{ij}$
matrix, without modifying the SM values of each top’s polarization, $B_{i}$
and $\bar{B}_{j}$. This is understood analytically, as both the linear and
squared contribution of $c_{tG}$ to all polarization observables vanish at LO
QCD. The scale variation uncertainty on shifts induced by $\mathcal{O}_{tG}$
is significantly smaller than the one induced by four-fermion operators. As it
can be seen in Table 5, the scale variation of $c_{tG}$ and SM cross-sections
are highly correlated, and cancel in the ratio when computing spin
observables. This is due to the nature of $\mathcal{O}_{tG}$, which tends to
produce contributions whose structure is closely correlated to the ones
present in the SM.

#### A case study: $B_{k}+\overline{B}_{k}$.

In the case of four-fermion operators, $B_{k}+\overline{B}_{k}$ is the spin
observable that consistently deviates the most from its SM value. We therefore
take it as a case study, and analyse its properties in detail. While an NLO-
accurate analytical calculation of the SMEFT shifts of
$B_{k}+\overline{B}_{k}$ is laborious, and beyond the scope of this work,
interesting patterns emerge from our numerical results, shown in App. A for
all observables, and reported in Table 3 for the reader’s convenience.

As noted above, $\mathcal{O}_{tG}$ does not contribute significantly to
$B_{k}+\bar{B}_{k}$. The four-fermion operators, that all contribute, can be
grouped in two classes,
$\mathcal{O}_{Qq}^{(8,3)},\mathcal{O}_{Qq}^{(8,1)},\mathcal{O}_{Qu}^{8},\mathcal{O}_{Qd}^{8}$,
for which the SM shift is negative, and
$\mathcal{O}_{tq}^{8},\mathcal{O}_{tu}^{8},\mathcal{O}_{td}^{8}$, for which
the shift is positive. In the following we will focus on color-octet
operators, but our exact arguments apply to color-singlet operators.

At order $c/\Lambda^{2}$, the trend of positive/negative shifts with respect
to the SM is consistent with the phase space integration of the results
presented in Aoude:2022imd , where it is noted that at LO QCD the coupling of
new physics to $B_{k}+\overline{B}_{k}$, at linear order, is only through the
Axial-Vector (AV) combinations:

$\displaystyle-c^{(8,3)}_{Qq}-c_{Qq}^{(8,1)}-c_{Qu}^{8}+c_{tq}^{8}+c_{tu}^{8},\quad$
$\displaystyle\text{for }u\bar{u}\to t\bar{t}$ (5.2) $\displaystyle
c^{(8,3)}_{Qq}-c_{Qq}^{(8,1)}-c_{Qd}^{8}+c_{tq}^{8}+c_{td}^{8},\quad$
$\displaystyle\text{for }d\bar{d}\to t\bar{t}.$ (5.3)

The coefficient $c^{(8,3)}_{Qq}$ enters with opposite signs the $u\bar{u}$ and
$d\bar{d}$ processes; the fact that the $u\bar{u}$ contribution is observed to
dominate in our numerical calculation, as shown in Figure 1(b), is of course
understood in terms of PDF luminosity in $p\,p$ collisions.

|  | Color-octet | Color-singlet
---|---|---|---
Operators | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\mathcal{O}_{Qq}^{(8,3)}$ and $\mathcal{O}_{Qq}^{(1,3)}$ | LO | $-0.079\scriptstyle{(10)}\,\displaystyle{{}^{+0.008}_{-0.011}}$ | $-0.133\scriptstyle{(3)}\,\displaystyle{{}^{+0.011}_{-0.013}}$ | $0.036\scriptstyle{(23)}\,\displaystyle{{}^{+0.007}_{-0.008}}$ | $-0.584\scriptstyle{(5)}\,\displaystyle{{}^{+0.038}_{-0.043}}$
NLO | $-0.083\scriptstyle{(7)}\,\displaystyle{{}^{+0.004}_{-0.001}}$ | $-0.118\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.004\scriptstyle{(16)}\,\displaystyle{{}^{+0.007}_{-0.004}}$ | $-0.761\scriptstyle{(4)}\,\displaystyle{{}^{+0.031}_{-0.034}}$
$\mathcal{O}_{Qq}^{(8,1)}$ and $\mathcal{O}_{Qq}^{(1,1)}$ | LO | $-0.441\scriptstyle{(10)}\,\displaystyle{{}^{+0.054}_{-0.068}}$ | $-0.123\scriptstyle{(3)}\,\displaystyle{{}^{+0.009}_{-0.011}}$ | $-0.004\scriptstyle{(28)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.557\scriptstyle{(7)}\,\displaystyle{{}^{+0.035}_{-0.040}}$
NLO | $-0.469\scriptstyle{(7)}\,\displaystyle{{}^{+0.014}_{-0.003}}$ | $-0.129\scriptstyle{(2)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $-0.044\scriptstyle{(14)}\,\displaystyle{{}^{+0.011}_{-0.015}}$ | $-0.767\scriptstyle{(4)}\,\displaystyle{{}^{+0.032}_{-0.035}}$
$\mathcal{O}_{Qu}^{8}$ and $\mathcal{O}_{Qu}^{1}$ | LO | $-0.272\scriptstyle{(8)}\,\displaystyle{{}^{+0.034}_{-0.043}}$ | $-0.080\scriptstyle{(2)}\,\displaystyle{{}^{+0.006}_{-0.008}}$ | $0.019\scriptstyle{(16)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.357\scriptstyle{(4)}\,\displaystyle{{}^{+0.023}_{-0.026}}$
NLO | $-0.203\scriptstyle{(7)}\,\displaystyle{{}^{+0.021}_{-0.009}}$ | $-0.055\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.062\scriptstyle{(10)}\,\displaystyle{{}^{+0.017}_{-0.012}}$ | $-0.462\scriptstyle{(2)}\,\displaystyle{{}^{+0.018}_{-0.020}}$
$\mathcal{O}_{tq}^{8}$ and $\mathcal{O}_{tq}^{1}$ | LO | $0.446\scriptstyle{(10)}\,\displaystyle{{}^{+0.073}_{-0.057}}$ | $0.123\scriptstyle{(2)}\,\displaystyle{{}^{+0.012}_{-0.010}}$ | $0.012\scriptstyle{(28)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.562\scriptstyle{(6)}\,\displaystyle{{}^{+0.042}_{-0.036}}$
NLO | $0.332\scriptstyle{(7)}\,\displaystyle{{}^{+0.016}_{-0.043}}$ | $0.089\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $-0.062\scriptstyle{(15)}\,\displaystyle{{}^{+0.014}_{-0.019}}$ | $0.747\scriptstyle{(3)}\,\displaystyle{{}^{+0.031}_{-0.029}}$
$\mathcal{O}_{Qd}^{8}$ and $\mathcal{O}_{Qd}^{1}$ | LO | $-0.195\scriptstyle{(6)}\,\displaystyle{{}^{+0.026}_{-0.034}}$ | $-0.049\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $0.003\scriptstyle{(11)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.217\scriptstyle{(3)}\,\displaystyle{{}^{+0.015}_{-0.017}}$
NLO | $-0.150\scriptstyle{(7)}\,\displaystyle{{}^{+0.017}_{-0.008}}$ | $-0.033\scriptstyle{(2)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.036\scriptstyle{(8)}\,\displaystyle{{}^{+0.008}_{-0.007}}$ | $-0.282\scriptstyle{(2)}\,\displaystyle{{}^{+0.011}_{-0.012}}$
$\mathcal{O}_{tu}^{8}$ and $\mathcal{O}_{tu}^{1}$ | LO | $0.262\scriptstyle{(9)}\,\displaystyle{{}^{+0.040}_{-0.032}}$ | $0.081\scriptstyle{(2)}\,\displaystyle{{}^{+0.008}_{-0.007}}$ | $0.026\scriptstyle{(17)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.360\scriptstyle{(4)}\,\displaystyle{{}^{+0.026}_{-0.023}}$
NLO | $0.268\scriptstyle{(7)}\,\displaystyle{{}^{+0.002}_{-0.008}}$ | $0.078\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.024\scriptstyle{(11)}\,\displaystyle{{}^{+0.011}_{-0.007}}$ | $0.471\scriptstyle{(3)}\,\displaystyle{{}^{+0.023}_{-0.020}}$
$\mathcal{O}_{td}^{8}$ and $\mathcal{O}_{td}^{1}$ | LO | $0.188\scriptstyle{(9)}\,\displaystyle{{}^{+0.029}_{-0.023}}$ | $0.050\scriptstyle{(2)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $-0.005\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.215\scriptstyle{(3)}\,\displaystyle{{}^{+0.016}_{-0.014}}$
NLO | $0.191\scriptstyle{(7)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $0.044\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.020\scriptstyle{(9)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $0.288\scriptstyle{(2)}\,\displaystyle{{}^{+0.015}_{-0.013}}$

Table 3: Cross sections appearing in the numerator of (3.13) for
$B_{k}+\bar{B}_{k}$ for the four-fermion operators we consider in this work,
extracted from Appendix A and with the same notation,
$\text{central(stat)}^{+\text{scale}}_{-\text{scale}}$

At order $\mathcal{O}(c^{2}/\Lambda^{4})$ and LO QCD, a calculation based on
FeynCalc Shtabovenko:2020gxv  and the SMEFTatNLO model Degrande:2020evl  shows
that new physics enters in $B_{k}+\overline{B}_{k}$ through the combinations:

$\displaystyle-(c^{(8,3)}_{Qq}+c_{Qq}^{(8,1)})^{2}-(c_{Qu}^{8})^{2}+(c_{tq}^{8})^{2}+(c_{tu}^{8})^{2},\quad$
$\displaystyle\text{for }u\bar{u}\to t\bar{t},$ (5.4)
$\displaystyle-(-c^{(8,3)}_{Qq}+c_{Qq}^{(8,1)})^{2}-(c_{Qd}^{8})^{2}+(c_{tq}^{8})^{2}+(c_{td}^{8})^{2},\quad$
$\displaystyle\text{for }d\bar{d}\to t\bar{t},$ (5.5)

when no phase-space cuts are applied on the $t\bar{t}$ pair. This is confirmed
by our numerical results in Table 3. The relative signs between four-fermion
operators that enter the contributions to $B_{k}+\overline{B}_{k}$ at linear
and quadratic order in $c/\Lambda^{2}$ are not the same as those that enter in
the total $t\bar{t}$ cross-section. For instance, at LO QCD and linearly in
$c/\Lambda^{2}$, the inclusive cross section is only sensitive to the Vector-
Vector (VV) combination Bernreuther:2015yna ; Aoude:2022imd :

$\displaystyle
c^{(8,3)}_{Qq}+c_{Qq}^{(8,1)}+c_{Qu}^{8}+c_{tq}^{8}+c_{tu}^{8}.$ (5.6)

At quadratic SMEFT order $\mathcal{O}(c^{2}/\Lambda^{4})$ more combinations of
Wilson coefficients enter the inclusive cross section, but, again, not the
specific ones in (5.4) and (5.5). This simple observation already highlights
the importance of spin observables in breaking flat directions otherwise
present in the 4-fermion sector.

We also note that several operators exhibit a relatively large $K$-factor
between the LO and NLO. Most notably, $\mathcal{O}_{Qu}^{8}$,
$\mathcal{O}_{tq}^{8}$, and $\mathcal{O}_{Qd}^{8}$, show a $\sim 50\%$
decrease in the $\sigma^{(2)}$ cross-section when going to NLO. This in turn
translates to a significant mismatch between the LO and NLO curves in Figures
2(a), 2(c), and 2(e), that is not predicted by the LO scale variation.

We conclude this Section with a remark: the difference from the SM values for
all spin observables we considered is of order $0.1$ over the range of Wilson
coefficients still allowed by the 2021 global fit Ethier:2021bye . The
measurement of spin polarizations and spin correlations in a hadron collider
is notoriously difficult, and a $\%$ level precision, needed to be competitive
on the global fit, may be out of reach. However, as we shall see in the next
section, the elaborate structure of spin observables in $t\bar{t}$ phase space
suggests a different route.

###  5.2 Results differential in $t\bar{t}$ phase space

The following Figures 5 \- 6, similarly to Figures 1 \- 3, show, for selected
SMEFT operators, the change at NLO with respect to the SM value for the top
spin polarizations $B_{i}+\bar{B}_{i}$, the $t\bar{t}$ spin correlations
$C_{ii}$ and $C_{ij}+C_{ji}$, and the entanglement markers $\Delta^{\pm}$, as
a function of Wilson coefficients, with $t\bar{t}$ phase space binned in the
pair invariant mass $m_{t\bar{t}}$ and in the c.m.f. scattering angle
$\theta$. Figure 4 shows our binning, and the fraction of the total $t\bar{t}$
cross-section included in each bin. Even after accounting for detector
effects, already after Run 2 of the LHC, all bins contain several million
$t\bar{t}$ events decaying leptonically.

$\cos\theta\leq 1/3$ | $1/3\leq\cos\theta\leq 2/3$ | $2/3\leq\cos\theta$
---|---|---
$m_{t\bar{t}}\leq 420\,\text{GeV}$ | $m_{t\bar{t}}\leq 420\,\text{GeV}$ | $m_{t\bar{t}}\leq 420\,\text{GeV}$
$\bm{\sigma_{\text{bin}}/\sigma=9.0\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=10.0\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=12.3\%}$
$\cos\theta\leq 1/3$ | $1/3\leq\cos\theta\leq 2/3$ | $2/3\leq\cos\theta$
$420\,\text{GeV}\leq m_{t\bar{t}}\leq 600\,\text{GeV}$ | $420\,\text{GeV}\leq m_{t\bar{t}}\leq 600\,\text{GeV}$ | $420\,\text{GeV}\leq m_{t\bar{t}}\leq 600\,\text{GeV}$
$\bm{\sigma_{\text{bin}}/\sigma=9.1\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=12.6\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=24.6\%}$
$\cos\theta\leq 1/3$ | $1/3\leq\cos\theta\leq 2/3$ | $2/3\leq\cos\theta$
$600\,\text{GeV}\leq m_{t\bar{t}}$ | $600\,\text{GeV}\leq m_{t\bar{t}}$ | $600\,\text{GeV}\leq m_{t\bar{t}}$
$\bm{\sigma_{\text{bin}}/\sigma=2.3\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=4.0\%}$ | $\bm{\sigma_{\text{bin}}/\sigma=16.1\%}$

Figure 4: Choice of binning of $t\bar{t}$ phase-space considered in this work,
and fraction of the total cross-section that falls in each bin. Layout
reproduces that of our differential plots.

This Section contains plots for selected SMEFT operators, that have been
chosen as their behaviour is representative of the others. Appendix B contains
plots for the remaining operators considered in this work. Contrary to the
inclusive plots, the doubly-differential plots split the sources of theory
uncertainty, with scale variation and Monte Carlo uncertainty shown
separately. This is to highlight that, should it become the limiting factor,
MC uncertainty is improvable arbitrarily.

We find that spin observables are heavily affected by the presence of new
physics, with a rich, and non–trivial structure in $t\bar{t}$ phase space.
Even a modest cut on the invariant mass, for example our lowest row
$m_{t\bar{t}}>600\,\text{GeV}$, can enhance the sensitivity to new physics by
a factor of $10$ over an inclusive measurement. On top of the expected growth
of deviations from the SM with energy, we find that the $t\bar{t}$ kinematical
regions that are most sensitive to new physics are those at high-$p_{T}$,
already investigated in the context of the observation of quantum entanglement
between tops. Aggressive phase space selections in the high-$p_{T}$ region
will yield increasing gains in sensitivity, but will realistically come with a
corresponding worsening of experimental uncertainty, both statistical and
systematic.

Figure 5: Change at NLO from the SM value for spin observables for the
operator $\mathcal{O}_{tG}$. Phase space is divided in the $t\bar{t}$
invariant mass along $y$, and in $\cos\theta$ along $x$, as indicated by the
insets. Continuous lines around each plotted curve indicate scale variation,
dashed lines indicate MC uncertainty. Only curves deviating appreciably from
zero are shown.

The variation of the spin correlation observables with different phase space
regions is most obvious for the four-fermion operator coefficients. In the
Wilson coefficient range we consider, the deviation from the SM predictions
varies from the percent level in the low invariant mass region to more than
10% for the high-mass and high-$p_{T}$ region for certain operators.

It is also interesting that in several phase-space regions the relation
between Wilson coefficients and spin observables is not only departing from
the linear regime, but it is also observed to be deviating from a quadratic
approximation. This is evident, for instance, for the entanglement marker
$\Delta^{-}$ in the high $m_{t\bar{t}}$ region of the $\mathcal{O}_{tG}$ plot,
where two saddle points can be seen. Such a rich and non-trivial behaviour
highlights the advantage of our full parametrization (3.13) over series
expansions.

To conclude this Section, we note that for certain operators, even after
enhancing the impact on the observables by focusing on the high-energy
regions, the deviation from the SM remains at the 2-3% level in the Wilson
coefficient range still allowed by other measurements. For these operators it
will be challenging to obtain stringent constraints on the coefficients using
spin observables, as this would require extremely precise measurements,
probably beyond the reach of present colliders.

Figure 6: Results for the operators $\mathcal{O}_{Qq}^{(8,1)}$ (top) and
$\mathcal{O}_{Qq}^{(1,1)}$ (bottom), similar to Figure 5.

##  6 Constraints on the Wilson coefficients

The early Run 2 top spin polarization and $t\bar{t}$ spin correlation
measurement from the CMS Collaboration CMS:2019nrx  is the only experimental
result, as of 2022, where the full spin density matrix is reconstructed. The
measurement covers $35.9\,\text{fb}^{-1}$ of integrated luminosity, and is
performed inclusively in phase space, which largely washes out the intricate
structure of spin observables.

As we do not expect that constraints obtained from CMS:2019nrx  will be
competitive with the global fit of top data from the LHC Ethier:2021bye , we
do not perform a global fit employing the state-of-the-art statistical
machinery developed in Ethier:2021bye , but only consider one SMEFT operator
at a time, and a $\chi^{2}$ parameter estimation. Our fit considers our most
accurate SMEFT results, at NLO QCD and quadratic in $c/\Lambda^{2}$. The most
accurate SM predictions available are at NLO QCD + EW Bernreuther:2015yna  and
NNLO QCD Czakon:2020qbd . Due to the large uncertainty of the experimental
measurement, the use of NNLO QCD or NLO QCD + EW SM makes a negligible
difference on the final fit results. The numbers we quote in the following use
Bernreuther:2015yna  as the SM central value. We consider as experimental
sources of uncertainty the full statistical and systematic covariance matrices
of CMS:2019nrx ; theory uncertainty includes SM scale variation, SMEFT scale
variation, and the full statistical covariance matrix of the SMEFT
contribution. We assume Gaussian probability density functions for the
uncertainties.

The $2\,\sigma$ allowed range for the Wilson coefficient of all SMEFT
operators considered in this work are in the first column of Table 4. The
range we obtain for $c_{tG}$ is similar to the one already obtained by the CMS
Collaboration in CMS:2019nrx  using a SMEFT theory prediction accurate at LO
QCD and linear in $c/\Lambda^{2}$. As noted above, this is due to the nature
of the top chromomagnetic moment operator $\mathcal{O}_{tG}$, which leads to a
significant interference with LO QCD SM amplitudes. A selection of color-octet
four-fermion operators has also been constrained in CMS:2019nrx . While it is
true that, after accounting for the different conventions and restricting our
results to LO QCD and linear in $c/\Lambda^{2}$, we are able to obtain similar
exclusion ranges, we find that quadratic SMEFT (and, to a lesser extent,
higher order QCD) contributions amount to sizable corrections. In our fit we
also constrain all seven directions in the four-fermion color-octet parameter
space, compared to the six considered in CMS:2019nrx , and constrain the four-
fermion color-singlet sector for the first time.

The range obtained from the global fit of LHC data Ethier:2021bye  is also
reported in Table 4. The bounds we obtain from the CMS inclusive measurement
CMS:2019nrx  are as expected significantly worse than the results obtained in
global analyses of the top sector as these typically involve over a hundred
data points constraining these operators. For most four-fermion operators the
constraints we obtain are less stringent than the global fit by up to one
order of magnitude, whilst for the chromomagnetic operator $\mathcal{O}_{tG}$
the bounds are different by a factor of a few.

To explore the potential of spin observables in constraining SMEFT operators
in the future, we obtain estimates for a binned measurement using LHC Run 3
data, proceeding as follows. We assume the binning used by experimental
collaborations will be the one presented above, $3\times 3$ in $\cos\theta-
m_{t\bar{t}}$. The experimental statistical covariance is obtained by
rescaling the one in CMS:2019nrx  to an integrated luminosity of
$300\,\text{fb}^{-1}$ at $\sqrt{s}=13.6\,\text{TeV}$. We note that in practice
this means setting the statistical uncertainty to zero, as the existing
measurement is already systematics-dominated. The systematic covariance matrix
is obtained by assuming the systematics of the inclusive measurement
CMS:2019nrx  are replicated in each bin, neglecting bin-to-bin correlations.
While it is more than likely that between the early Run 2 measurement
CMS:2019nrx  and the end of Run 3 significant improvements on the systematic
uncertainties will be obtained, moving from an inclusive to a differential
measurement will inevitably come with a worsening of the experimental
resolution. It is impossible, a priori, to assess precisely either of these
two effects. Assigning the early Run 2 systematic uncertainties in each bin of
our projected Run 3 measurement is a compromise we consider to be realistic
for the near future and conservative in the medium term. We also consider the
SM and SMEFT scale variation, as described above, as a source of theory
uncertainty. Monte Carlo uncertainty is not considered, as it can be lowered
as needed. The bounds we obtain in this way are in the second column of Table
4.

Operator | CMS CMS:2019nrx  | Run III Projection | Current Global Fit
---|---|---|---
$36\,\text{fb}^{-1}$ Inclusive |  $300\,\text{fb}^{-1}$ Differential
$\mathcal{O}_{tG}$ | $[-0.18,0.18]$ | $[-0.03,0.04]$ | $[0.00,0.11]$
$\mathcal{O}_{tu}^{8}$ | $[-5.8,3.6]$ | $[-1.0,0.7]$ | $[-0.9,0.3]$
$\mathcal{O}_{td}^{8}$ | $[-7.9,5.2]$ | $[-1.3,1.0]$ | $[-1.3,0.6]$
$\mathcal{O}_{tq}^{8}$ | $[-4.2,3.1]$ | $[-0.7,0.5]$ | $[-0.5,0.4]$
$\mathcal{O}_{Qu}^{8}$ | $[-9.4,4.6]$ | $[-0.7,0.6]$ | $[-1.0,0.5]$
$\mathcal{O}_{Qd}^{8}$ | $[-11.7,5.8]$ | $[-0.9,0.8]$ | $[-1.6,0.9]$
$\mathcal{O}_{Qq}^{(1,8)}$ | $[-5.8,-4.6]\cup[-1.7,2.5]$ | $[-0.4,0.3]$ | $[-0.4,0.3]$
$\mathcal{O}_{Qq}^{(3,8)}$ | $[-5.0,4.2]$ | $[-1.1,0.8]$ | $[-0.5,0.4]$
$\mathcal{O}_{tu}^{1}$ | $[-2.1,2.1]$ | $[-0.5,0.5]$ | $[-0.4,0.3]$
$\mathcal{O}_{td}^{1}$ | $[-2.7,2.6]$ | $[-0.6,0.6]$ | $[-0.4,0.4]$
$\mathcal{O}_{tq}^{1}$ | $[-1.7,1.8]$ | $[-0.4,0.4]$ | $[-0.2,0.3]$
$\mathcal{O}_{Qu}^{1}$ | $[-2.1,2.4]$ | $[-0.4,0.5]$ | $[-0.3,0.4]$
$\mathcal{O}_{Qd}^{1}$ | $[-2.8,3.0]$ | $[-0.6,0.6]$ | $[-0.3,0.4]$
$\mathcal{O}_{Qq}^{(1,1)}$ | $[-1.8,1.8]$ | $[-0.4,0.4]$ | $[-0.3,0.2]$
$\mathcal{O}_{Qq}^{(3,1)}$ | $[-1.8,1.8]$ | $[-0.4,0.4]$ | $[-0.1,0.2]$

Table 4: $2\sigma$ bounds on Wilson coefficients of SMEFT operators considered
in this work. The scale $\Lambda$ is set to 1 TeV. First column: bounds from
the inclusive early LHC Run 2 experimental measurement CMS:2019nrx . Second
column: projected bounds from one binned, LHC Run 3 measurement. Third column:
current individual bounds from the global fit of top LHC data Ethier:2021bye .

Whilst the bounds obtained from the CMS inclusive measurement CMS:2019nrx  are
typically approximately an order of magnitude broader than those obtained from
the global fit, a binned measurement with LHC Run 3 statistics and present-day
systematics would yield bounds competitive or better than those in the global
fit. Even though we do not perform a global fit, we advocate that spin
observables should be included in future global fits. In particular,
entanglement markers such as $\Delta^{\pm}$, being measurable to a
significantly greater accuracy than the individual $C_{ij}$’s, should also be
considered, and may in fact be the observables yielding the best sensitivity.
For instance, in CMS:2019nrx , $C_{rr}$ was measured to a relative accuracy of
$46\%$, $C_{rk}+C_{kr}$ to $31\%$, while $\Delta^{-}$ was measured to $5\%$.

The expected improvement in the global fit driven by the inclusion of
spin/entanglement measurements is a result of two concurring facts. First, the
effect of most operators on the cross-sections $\sigma_{S}$ are qualitatively
different from the effect of the same operators on the total $t\bar{t}$ cross-
section $\sigma$. In several cases, the effects on $\sigma_{S}$ and $\sigma$
have similar magnitude but opposite sign, so that the resulting shift on the
ratio $S=\sigma_{S}/\sigma$ is enhanced with respect to the shift of $\sigma$
alone. In other cases, as noted in previous Sections, $\sigma_{S}$ is
sensitive to particular combinations of Wilson coefficients, e.g. (5.4) and
(5.5) for $B_{k}+\overline{B_{k}}$, that do not appear in $\sigma$ at all.
Second, the sheer number of new observables, that in fact probe new, never
added before, $t\bar{t}$ degrees of freedom, significantly improves the fit
quality.

It is not inconceivable that, assuming differential and doubly-differential
spin measurements turn out to be experimentally feasible, in the future the
bounds for several Wilson coefficients that enter $t\bar{t}$ physics will
receive significant contributions from spin polarization, spin correlation,
and spin entanglement observables.

##  7 Conclusions

We have presented a numerical simulation of the effect of new physics on top
spin polarization, $t\bar{t}$ spin correlation, and $t\bar{t}$ spin
entanglement at the LHC, in the framework of the Standard Model Effective
Field Theory. Our simulation is accurate at NLO QCD in the $t\bar{t}$
production, and considers both $\mathcal{O}(c/\Lambda^{2})$ and
$\mathcal{O}(c^{2}/\Lambda^{4})$ effects, which is state of the art. All
dimension-6 operators containing top quarks have been considered, with the
exception of those highly suppressed by the proton PDF and those only entering
in NLO corrections. This amounts to 15 operators in total. We have presented
results for the expected deviations from the SM as a function of all Wilson
coefficients for all non-CP-violating terms in the $t\bar{t}$ density matrix.

We find that the inclusion of NP-squared $\mathcal{O}(c^{2}/\Lambda^{4})$
contributions to cross-sections is almost always important for describing the
effects of SMEFT operators on all spin observables. This is true not only for
color-singlet operators, whose interference with the LO SM QCD amplitude
vanishes, but also for the color-octets, in the range of Wilson coefficients
still allowed by other measurements. The notable exception is the top
chromomagnetic moment operator $\mathcal{O}_{tG}$, for which a LO analysis is
found to be adequate. Next-to-leading order QCD corrections, evaluated here
for the first time, do not radically change the patterns seen at LO, but the
appearance of relatively large $K$-factors on several cross-sections often
makes LO scale variation a poor estimate of missing higher orders. As
expected, NLO corrections also significantly reduce the uncertainties due to
missing higher orders as quantified by scale variations.

We presented results both for an inclusive measurement, and for a doubly-
differential measurement in the top pair phase space. Already at the inclusive
level, we find that spin observables depend on Wilson coefficients in a
largely different way than the total $t\bar{t}$ cross-section, thus providing
a novel sensitivity to new physics in the top sector. Our second scenario is
for a measurement binned in the top pair invariant mass and in the top
scattering angle. All spin observables show an intricate structure in phase
space. The kinematical region that seems most promising for the detection of
NP is the one at large top $p_{T}$, already considered in the context of
detection of quantum effects, such as entanglement between the tops’ spin and
possible violations of Bell Inequalities.

Only one experimental result is available as of this writing, an early Run 2
measurement from the CMS Collaboration CMS:2019nrx , where the $t\bar{t}$ spin
density matrix has been fully measured, however without any phase space cuts.
No significant deviation from the SM was observed. We argue that this should
not be taken as discouraging, as the effects induced by new physics are
localized in phase-space, and are washed out by an inclusive measurement.

Using our results and precision SM calculations, we extract bounds for Wilson
coefficients from the CMS measurement CMS:2019nrx , and produce a projection
for an LHC Run 3 differential measurement. We find that just one doubly-
differential measurement of the $t\bar{t}$ spin density matrix would yield
constraints competitive with the current full global fit to LHC data.

Apart from the prospects of detecting quantum entanglement at the TeV scale
Afik:2020onf ; Afik:2022kwm ; Aguilar-Saavedra:2022uye , and possibly
observing Bell Inequalities violations Fabbrichesi:2021npl ; Severi:2021cnj ,
we have shown how the inclusion of spin measurements in the physics program of
experimental collaborations would yield a novel and promising road for
detecting beyond the SM effects, and provided a systematic and complete
framework for these studies.

##  8 Acknowledgements

We are grateful to Fabio Maltoni and Luca Mantani for comments on the
manuscript, and to Rafael Aoude for useful discussions. We also thank Ioannis
Tsinikos for collaboration at the early stages of this work, and Rikkert
Frederix and Timea Vitos for the valuable help. The Authors are supported by
the European Union’s Horizon 2020 research and innovation program under the
EFT4NP project (grant agreement no. 949451) and by a Royal Society University
Research Fellowship through grant URF/R1/201553.

##  Appendix A Numerical results inclusive in phase space

The following Tables 5 \- 12 collect numerical results needed to reproduce our
plots in Figures 1 \- 3. We present all cross sections at LO and NLO appearing
in the numerator and denominator of (3.13) for the spin polarization
observables $B_{i}+\bar{B}_{i}$, and the spin correlation observables $C_{ii}$
and $C_{ij}+C_{ji}$, for all operators we considered. The first quoted
uncertainty, in brackets, is Monte Carlo, in units of the least significant
digit of the central value. The second one, ${}^{+x}_{-y}$, is scale
variation222The scale variation we quote on (3.13) is not obtained by
propagating the scale variation from each term, but it is extracted by
evaluating the full ratio at each scale choice..

|  | SM | $\mathcal{O}_{tG}$
---|---|---|---
Observable | Order |  $\sigma^{(0)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $29.12\scriptstyle{(2)}\,\displaystyle{{}^{+9.10}_{-6.49}}$ | $9.32\scriptstyle{(7)}\,\displaystyle{{}^{+2.90}_{-2.08}}$ | $1.63\scriptstyle{(1)}\,\displaystyle{{}^{+0.52}_{-0.37}}$
NLO | $36.23\scriptstyle{(2)}\,\displaystyle{{}^{+1.70}_{-3.14}}$ | $11.50\scriptstyle{(6)}\,\displaystyle{{}^{+0.51}_{-0.97}}$ | $2.01\scriptstyle{(1)}\,\displaystyle{{}^{+0.09}_{-0.17}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.09\scriptstyle{(5)}\,\displaystyle{{}^{+0.02}_{-0.03}}$ | $-0.01\scriptstyle{(19)}\,\displaystyle{{}^{+0.00}_{-0.00}}$ | $-0.00\scriptstyle{(4)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
NLO | $-0.04\scriptstyle{(6)}\,\displaystyle{{}^{+0.00}_{-0.01}}$ | $0.00\scriptstyle{(24)}\,\displaystyle{{}^{+0.01}_{-0.00}}$ | $0.01\scriptstyle{(5)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $0.04\scriptstyle{(6)}\,\displaystyle{{}^{+0.01}_{-0.01}}$ | $-0.01\scriptstyle{(23)}\,\displaystyle{{}^{+0.00}_{-0.00}}$ | $-0.00\scriptstyle{(4)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
NLO | $0.03\scriptstyle{(7)}\,\displaystyle{{}^{+0.03}_{-0.02}}$ | $-0.01\scriptstyle{(27)}\,\displaystyle{{}^{+0.01}_{-0.01}}$ | $-0.00\scriptstyle{(5)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $0.11\scriptstyle{(6)}\,\displaystyle{{}^{+0.03}_{-0.02}}$ | $-0.02\scriptstyle{(23)}\,\displaystyle{{}^{+0.01}_{-0.01}}$ | $-0.01\scriptstyle{(4)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
NLO | $0.04\scriptstyle{(4)}\,\displaystyle{{}^{+0.03}_{-0.02}}$ | $0.03\scriptstyle{(16)}\,\displaystyle{{}^{+0.01}_{-0.01}}$ | $0.00\scriptstyle{(3)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $9.63\scriptstyle{(8)}\,\displaystyle{{}^{+2.96}_{-2.13}}$ | $6.68\scriptstyle{(31)}\,\displaystyle{{}^{+2.13}_{-1.52}}$ | $0.40\scriptstyle{(5)}\,\displaystyle{{}^{+0.13}_{-0.09}}$
NLO | $11.27\scriptstyle{(6)}\,\displaystyle{{}^{+0.43}_{-0.80}}$ | $8.30\scriptstyle{(27)}\,\displaystyle{{}^{+0.42}_{-0.72}}$ | $0.56\scriptstyle{(5)}\,\displaystyle{{}^{+0.05}_{-0.06}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $0.19\scriptstyle{(8)}\,\displaystyle{{}^{+0.19}_{-0.13}}$ | $4.37\scriptstyle{(31)}\,\displaystyle{{}^{+1.44}_{-1.02}}$ | $0.65\scriptstyle{(5)}\,\displaystyle{{}^{+0.21}_{-0.15}}$
NLO | $1.66\scriptstyle{(6)}\,\displaystyle{{}^{+0.65}_{-0.45}}$ | $5.75\scriptstyle{(26)}\,\displaystyle{{}^{+0.37}_{-0.57}}$ | $0.81\scriptstyle{(5)}\,\displaystyle{{}^{+0.04}_{-0.07}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $9.42\scriptstyle{(7)}\,\displaystyle{{}^{+3.12}_{-2.22}}$ | $4.60\scriptstyle{(30)}\,\displaystyle{{}^{+1.47}_{-1.05}}$ | $0.97\scriptstyle{(5)}\,\displaystyle{{}^{+0.31}_{-0.22}}$
NLO | $11.83\scriptstyle{(6)}\,\displaystyle{{}^{+0.68}_{-1.03}}$ | $5.66\scriptstyle{(24)}\,\displaystyle{{}^{+0.29}_{-0.47}}$ | $1.17\scriptstyle{(5)}\,\displaystyle{{}^{+0.05}_{-0.10}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-6.81\scriptstyle{(9)}\,\displaystyle{{}^{+1.45}_{-2.01}}$ | $-0.65\scriptstyle{(36)}\,\displaystyle{{}^{+0.11}_{-0.15}}$ | $-0.03\scriptstyle{(6)}\,\displaystyle{{}^{+0.01}_{-0.01}}$
NLO | $-7.36\scriptstyle{(11)}\,\displaystyle{{}^{+0.44}_{-0.13}}$ | $-1.07\scriptstyle{(46)}\,\displaystyle{{}^{+0.14}_{-0.18}}$ | $-0.04\scriptstyle{(9)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $-0.06\scriptstyle{(8)}\,\displaystyle{{}^{+0.01}_{-0.02}}$ | $-0.02\scriptstyle{(33)}\,\displaystyle{{}^{+0.01}_{-0.01}}$ | $-0.00\scriptstyle{(6)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
NLO | $-0.03\scriptstyle{(10)}\,\displaystyle{{}^{+0.04}_{-0.03}}$ | $-0.03\scriptstyle{(41)}\,\displaystyle{{}^{+0.01}_{-0.02}}$ | $-0.01\scriptstyle{(8)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $0.11\scriptstyle{(10)}\,\displaystyle{{}^{+0.04}_{-0.03}}$ | $-0.04\scriptstyle{(40)}\,\displaystyle{{}^{+0.01}_{-0.02}}$ | $-0.01\scriptstyle{(7)}\,\displaystyle{{}^{+0.00}_{-0.00}}$
NLO | $0.16\scriptstyle{(10)}\,\displaystyle{{}^{+0.05}_{-0.04}}$ | $0.01\scriptstyle{(44)}\,\displaystyle{{}^{+0.03}_{-0.01}}$ | $0.00\scriptstyle{(9)}\,\displaystyle{{}^{+0.00}_{-0.00}}$

Table 5: Numerical results for the Standard Model and for the operator
$\mathcal{O}_{tG}$.

|  | $\mathcal{O}_{Qq}^{(8,3)}$ | $\mathcal{O}_{Qq}^{(1,3)}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.073\scriptstyle{(2)}\,\displaystyle{{}^{+0.008}_{-0.007}}$ | $0.075\scriptstyle{(0)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $0.006\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.334\scriptstyle{(1)}\,\displaystyle{{}^{+0.019}_{-0.017}}$
NLO | $0.059\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.005}}$ | $0.072\scriptstyle{(1)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.476\scriptstyle{(1)}\,\displaystyle{{}^{+0.025}_{-0.022}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $0.002\scriptstyle{(7)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.003\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.002\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.009\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.007}}$ | $0.000\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.022\scriptstyle{(13)}\,\displaystyle{{}^{+0.010}_{-0.006}}$ | $0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $-0.008\scriptstyle{(7)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.011\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $-0.002\scriptstyle{(15)}\,\displaystyle{{}^{+0.003}_{-0.005}}$ | $-0.050\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.003}}$
NLO | $-0.011\scriptstyle{(11)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $-0.005\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.005\scriptstyle{(14)}\,\displaystyle{{}^{+0.006}_{-0.004}}$ | $-0.059\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $-0.079\scriptstyle{(10)}\,\displaystyle{{}^{+0.008}_{-0.011}}$ | $-0.133\scriptstyle{(3)}\,\displaystyle{{}^{+0.011}_{-0.013}}$ | $0.036\scriptstyle{(23)}\,\displaystyle{{}^{+0.007}_{-0.008}}$ | $-0.584\scriptstyle{(5)}\,\displaystyle{{}^{+0.038}_{-0.043}}$
NLO | $-0.083\scriptstyle{(7)}\,\displaystyle{{}^{+0.004}_{-0.001}}$ | $-0.118\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.004\scriptstyle{(16)}\,\displaystyle{{}^{+0.007}_{-0.004}}$ | $-0.761\scriptstyle{(4)}\,\displaystyle{{}^{+0.031}_{-0.034}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.019\scriptstyle{(10)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.011\scriptstyle{(14)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $-0.002\scriptstyle{(10)}\,\displaystyle{{}^{+0.008}_{-0.015}}$ | $-0.004\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.006\scriptstyle{(13)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.031\scriptstyle{(11)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $-0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.001\scriptstyle{(14)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.029\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $-0.027\scriptstyle{(10)}\,\displaystyle{{}^{+0.008}_{-0.005}}$ | $-0.004\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.010\scriptstyle{(13)}\,\displaystyle{{}^{+0.003}_{-0.006}}$ | $-0.040\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.036\scriptstyle{(12)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.055\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $0.085\scriptstyle{(34)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $-0.246\scriptstyle{(8)}\,\displaystyle{{}^{+0.013}_{-0.015}}$
NLO | $-0.080\scriptstyle{(10)}\,\displaystyle{{}^{+0.012}_{-0.017}}$ | $-0.041\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.002\scriptstyle{(22)}\,\displaystyle{{}^{+0.015}_{-0.007}}$ | $-0.294\scriptstyle{(5)}\,\displaystyle{{}^{+0.008}_{-0.008}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $0.001\scriptstyle{(15)}\,\displaystyle{{}^{+0.007}_{-0.004}}$ | $-0.006\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.052\scriptstyle{(23)}\,\displaystyle{{}^{+0.011}_{-0.008}}$ | $-0.059\scriptstyle{(5)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
NLO | $-0.005\scriptstyle{(18)}\,\displaystyle{{}^{+0.020}_{-0.010}}$ | $0.001\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.015\scriptstyle{(25)}\,\displaystyle{{}^{+0.007}_{-0.013}}$ | $-0.076\scriptstyle{(6)}\,\displaystyle{{}^{+0.005}_{-0.008}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.006\scriptstyle{(13)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.005\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.009\scriptstyle{(18)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.003\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.013\scriptstyle{(16)}\,\displaystyle{{}^{+0.004}_{-0.007}}$ | $0.001\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.013\scriptstyle{(22)}\,\displaystyle{{}^{+0.009}_{-0.004}}$ | $0.004\scriptstyle{(5)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $-0.008\scriptstyle{(14)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.011\scriptstyle{(20)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $-0.003\scriptstyle{(5)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $-0.009\scriptstyle{(18)}\,\displaystyle{{}^{+0.007}_{-0.013}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.026\scriptstyle{(23)}\,\displaystyle{{}^{+0.014}_{-0.009}}$ | $-0.004\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.002}}$

Table 6: Numerical results for the operators $\mathcal{O}_{Qq}^{(8,3)}$ (left)
and $\mathcal{O}_{Qq}^{(1,3)}$ (right).

|  | $\mathcal{O}_{Qq}^{(8,1)}$ | $\mathcal{O}_{Qq}^{(1,1)}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.385\scriptstyle{(2)}\,\displaystyle{{}^{+0.046}_{-0.039}}$ | $0.075\scriptstyle{(0)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.335\scriptstyle{(1)}\,\displaystyle{{}^{+0.019}_{-0.017}}$
NLO | $0.345\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.013}}$ | $0.078\scriptstyle{(1)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.019\scriptstyle{(3)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $0.478\scriptstyle{(1)}\,\displaystyle{{}^{+0.026}_{-0.022}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.013\scriptstyle{(7)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $-0.006\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.005\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.001}}$ | $0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.005\scriptstyle{(12)}\,\displaystyle{{}^{+0.003}_{-0.005}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $-0.062\scriptstyle{(9)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.012\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.009\scriptstyle{(11)}\,\displaystyle{{}^{+0.004}_{-0.002}}$ | $-0.054\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $-0.074\scriptstyle{(11)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.031\scriptstyle{(13)}\,\displaystyle{{}^{+0.008}_{-0.006}}$ | $-0.055\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $-0.441\scriptstyle{(10)}\,\displaystyle{{}^{+0.054}_{-0.068}}$ | $-0.123\scriptstyle{(3)}\,\displaystyle{{}^{+0.009}_{-0.011}}$ | $-0.004\scriptstyle{(28)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.557\scriptstyle{(7)}\,\displaystyle{{}^{+0.035}_{-0.040}}$
NLO | $-0.469\scriptstyle{(7)}\,\displaystyle{{}^{+0.014}_{-0.003}}$ | $-0.129\scriptstyle{(2)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $-0.044\scriptstyle{(14)}\,\displaystyle{{}^{+0.011}_{-0.015}}$ | $-0.767\scriptstyle{(4)}\,\displaystyle{{}^{+0.032}_{-0.035}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.089\scriptstyle{(10)}\,\displaystyle{{}^{+0.010}_{-0.009}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.004\scriptstyle{(13)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
NLO | $0.073\scriptstyle{(9)}\,\displaystyle{{}^{+0.007}_{-0.013}}$ | $-0.003\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $-0.030\scriptstyle{(13)}\,\displaystyle{{}^{+0.007}_{-0.010}}$ | $0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.201\scriptstyle{(10)}\,\displaystyle{{}^{+0.019}_{-0.022}}$ | $-0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.019\scriptstyle{(14)}\,\displaystyle{{}^{+0.006}_{-0.004}}$ | $-0.026\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
NLO | $-0.206\scriptstyle{(11)}\,\displaystyle{{}^{+0.003}_{-0.001}}$ | $-0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.017\scriptstyle{(13)}\,\displaystyle{{}^{+0.010}_{-0.006}}$ | $-0.032\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.202\scriptstyle{(10)}\,\displaystyle{{}^{+0.020}_{-0.022}}$ | $-0.047\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $0.004\scriptstyle{(40)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.219\scriptstyle{(10)}\,\displaystyle{{}^{+0.012}_{-0.013}}$
NLO | $-0.286\scriptstyle{(11)}\,\displaystyle{{}^{+0.022}_{-0.030}}$ | $-0.047\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.026\scriptstyle{(18)}\,\displaystyle{{}^{+0.010}_{-0.015}}$ | $-0.305\scriptstyle{(5)}\,\displaystyle{{}^{+0.010}_{-0.012}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.194\scriptstyle{(14)}\,\displaystyle{{}^{+0.022}_{-0.028}}$ | $-0.011\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.007\scriptstyle{(20)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.046\scriptstyle{(5)}\,\displaystyle{{}^{+0.005}_{-0.005}}$
NLO | $-0.167\scriptstyle{(19)}\,\displaystyle{{}^{+0.009}_{-0.005}}$ | $-0.002\scriptstyle{(5)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.040\scriptstyle{(23)}\,\displaystyle{{}^{+0.020}_{-0.012}}$ | $-0.065\scriptstyle{(6)}\,\displaystyle{{}^{+0.002}_{-0.003}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.024\scriptstyle{(14)}\,\displaystyle{{}^{+0.007}_{-0.005}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.025\scriptstyle{(19)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $0.003\scriptstyle{(5)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $-0.012\scriptstyle{(15)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $-0.001\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.012\scriptstyle{(20)}\,\displaystyle{{}^{+0.009}_{-0.006}}$ | $-0.005\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $0.002\scriptstyle{(13)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.010\scriptstyle{(16)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.009\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $0.005\scriptstyle{(20)}\,\displaystyle{{}^{+0.010}_{-0.005}}$ | $-0.001\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.001\scriptstyle{(21)}\,\displaystyle{{}^{+0.008}_{-0.013}}$ | $0.003\scriptstyle{(6)}\,\displaystyle{{}^{+0.001}_{-0.000}}$

Table 7: Numerical results for the operators $\mathcal{O}_{Qq}^{(8,1)}$ (left)
and $\mathcal{O}_{Qq}^{(1,1)}$ (right).

|  | $\mathcal{O}_{Qu}^{8}$ | $\mathcal{O}_{Qu}^{1}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.230\scriptstyle{(2)}\,\displaystyle{{}^{+0.027}_{-0.023}}$ | $0.046\scriptstyle{(0)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.207\scriptstyle{(1)}\,\displaystyle{{}^{+0.012}_{-0.011}}$
NLO | $0.193\scriptstyle{(2)}\,\displaystyle{{}^{+0.005}_{-0.012}}$ | $0.034\scriptstyle{(1)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.020\scriptstyle{(3)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $0.296\scriptstyle{(1)}\,\displaystyle{{}^{+0.016}_{-0.013}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.003\scriptstyle{(8)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.003\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.006\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.003\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.004\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $-0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $-0.033\scriptstyle{(10)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $-0.008\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.000\scriptstyle{(11)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.030\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.002\scriptstyle{(11)}\,\displaystyle{{}^{+0.015}_{-0.009}}$ | $-0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.011\scriptstyle{(11)}\,\displaystyle{{}^{+0.009}_{-0.006}}$ | $-0.027\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $-0.272\scriptstyle{(8)}\,\displaystyle{{}^{+0.034}_{-0.043}}$ | $-0.080\scriptstyle{(2)}\,\displaystyle{{}^{+0.006}_{-0.008}}$ | $0.019\scriptstyle{(16)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.357\scriptstyle{(4)}\,\displaystyle{{}^{+0.023}_{-0.026}}$
NLO | $-0.203\scriptstyle{(7)}\,\displaystyle{{}^{+0.021}_{-0.009}}$ | $-0.055\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.062\scriptstyle{(10)}\,\displaystyle{{}^{+0.017}_{-0.012}}$ | $-0.462\scriptstyle{(2)}\,\displaystyle{{}^{+0.018}_{-0.020}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.056\scriptstyle{(13)}\,\displaystyle{{}^{+0.007}_{-0.006}}$ | $0.000\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.006\scriptstyle{(15)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.055\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.014\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.123\scriptstyle{(10)}\,\displaystyle{{}^{+0.011}_{-0.013}}$ | $-0.006\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.004\scriptstyle{(14)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.016\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.111\scriptstyle{(10)}\,\displaystyle{{}^{+0.011}_{-0.004}}$ | $-0.000\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.001\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.021\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.004}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.140\scriptstyle{(15)}\,\displaystyle{{}^{+0.016}_{-0.019}}$ | $-0.033\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.048\scriptstyle{(20)}\,\displaystyle{{}^{+0.006}_{-0.004}}$ | $-0.147\scriptstyle{(5)}\,\displaystyle{{}^{+0.008}_{-0.009}}$
NLO | $-0.138\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.015\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.044\scriptstyle{(14)}\,\displaystyle{{}^{+0.011}_{-0.009}}$ | $-0.179\scriptstyle{(4)}\,\displaystyle{{}^{+0.005}_{-0.005}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.116\scriptstyle{(17)}\,\displaystyle{{}^{+0.012}_{-0.015}}$ | $-0.007\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.002\scriptstyle{(23)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.034\scriptstyle{(6)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $-0.090\scriptstyle{(20)}\,\displaystyle{{}^{+0.009}_{-0.005}}$ | $0.002\scriptstyle{(5)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.044\scriptstyle{(20)}\,\displaystyle{{}^{+0.015}_{-0.009}}$ | $-0.037\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.003}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.014\scriptstyle{(14)}\,\displaystyle{{}^{+0.007}_{-0.004}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.029\scriptstyle{(23)}\,\displaystyle{{}^{+0.008}_{-0.006}}$ | $0.001\scriptstyle{(6)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.012\scriptstyle{(16)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.001\scriptstyle{(16)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $-0.003\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $0.006\scriptstyle{(16)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.017\scriptstyle{(22)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $0.004\scriptstyle{(6)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $-0.010\scriptstyle{(20)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.001\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.003\scriptstyle{(19)}\,\displaystyle{{}^{+0.009}_{-0.015}}$ | $-0.006\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.002}}$

Table 8: Numerical results for the operators $\mathcal{O}_{Qu}^{8}$ (left) and
$\mathcal{O}_{Qu}^{1}$ (right).

|  | $\mathcal{O}_{tq}^{8}$ | $\mathcal{O}_{tq}^{1}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.380\scriptstyle{(1)}\,\displaystyle{{}^{+0.045}_{-0.038}}$ | $0.074\scriptstyle{(0)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.335\scriptstyle{(1)}\,\displaystyle{{}^{+0.019}_{-0.017}}$
NLO | $0.340\scriptstyle{(3)}\,\displaystyle{{}^{+0.005}_{-0.015}}$ | $0.056\scriptstyle{(1)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $-0.030\scriptstyle{(3)}\,\displaystyle{{}^{+0.005}_{-0.008}}$ | $0.475\scriptstyle{(1)}\,\displaystyle{{}^{+0.025}_{-0.022}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.002\scriptstyle{(9)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.007\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.013\scriptstyle{(11)}\,\displaystyle{{}^{+0.003}_{-0.005}}$ | $0.000\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.011\scriptstyle{(11)}\,\displaystyle{{}^{+0.007}_{-0.004}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.009\scriptstyle{(13)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $0.049\scriptstyle{(9)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.008\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.006\scriptstyle{(12)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.050\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $0.014\scriptstyle{(12)}\,\displaystyle{{}^{+0.010}_{-0.015}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.032\scriptstyle{(13)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $0.043\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.004}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $0.446\scriptstyle{(10)}\,\displaystyle{{}^{+0.073}_{-0.057}}$ | $0.123\scriptstyle{(2)}\,\displaystyle{{}^{+0.012}_{-0.010}}$ | $0.012\scriptstyle{(28)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.562\scriptstyle{(6)}\,\displaystyle{{}^{+0.042}_{-0.036}}$
NLO | $0.332\scriptstyle{(7)}\,\displaystyle{{}^{+0.016}_{-0.043}}$ | $0.089\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $-0.062\scriptstyle{(15)}\,\displaystyle{{}^{+0.014}_{-0.019}}$ | $0.747\scriptstyle{(3)}\,\displaystyle{{}^{+0.031}_{-0.029}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.099\scriptstyle{(12)}\,\displaystyle{{}^{+0.012}_{-0.010}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.007\scriptstyle{(14)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
NLO | $0.065\scriptstyle{(11)}\,\displaystyle{{}^{+0.009}_{-0.020}}$ | $-0.005\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.009\scriptstyle{(14)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.191\scriptstyle{(10)}\,\displaystyle{{}^{+0.016}_{-0.019}}$ | $-0.000\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.003\scriptstyle{(14)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.022\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
NLO | $-0.206\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.005\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.008\scriptstyle{(13)}\,\displaystyle{{}^{+0.006}_{-0.007}}$ | $-0.042\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.005}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.235\scriptstyle{(13)}\,\displaystyle{{}^{+0.027}_{-0.034}}$ | $-0.045\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $-0.030\scriptstyle{(43)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.217\scriptstyle{(10)}\,\displaystyle{{}^{+0.012}_{-0.013}}$
NLO | $-0.232\scriptstyle{(9)}\,\displaystyle{{}^{+0.012}_{-0.017}}$ | $-0.027\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.026\scriptstyle{(21)}\,\displaystyle{{}^{+0.016}_{-0.014}}$ | $-0.295\scriptstyle{(5)}\,\displaystyle{{}^{+0.010}_{-0.011}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.167\scriptstyle{(15)}\,\displaystyle{{}^{+0.015}_{-0.018}}$ | $-0.013\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.011\scriptstyle{(18)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.051\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.003}}$
NLO | $-0.166\scriptstyle{(20)}\,\displaystyle{{}^{+0.007}_{-0.010}}$ | $-0.009\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.003}}$ | $-0.013\scriptstyle{(23)}\,\displaystyle{{}^{+0.006}_{-0.005}}$ | $-0.070\scriptstyle{(5)}\,\displaystyle{{}^{+0.004}_{-0.006}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.016\scriptstyle{(17)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.010\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.011\scriptstyle{(16)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.002\scriptstyle{(17)}\,\displaystyle{{}^{+0.004}_{-0.009}}$ | $0.000\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.011\scriptstyle{(19)}\,\displaystyle{{}^{+0.006}_{-0.007}}$ | $0.001\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $-0.016\scriptstyle{(18)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $0.000\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.028\scriptstyle{(18)}\,\displaystyle{{}^{+0.011}_{-0.007}}$ | $-0.004\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.003\scriptstyle{(21)}\,\displaystyle{{}^{+0.005}_{-0.009}}$ | $0.001\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.002\scriptstyle{(24)}\,\displaystyle{{}^{+0.013}_{-0.006}}$ | $-0.003\scriptstyle{(5)}\,\displaystyle{{}^{+0.004}_{-0.005}}$

Table 9: Numerical results for the operators $\mathcal{O}_{tq}^{8}$ (left) and
$\mathcal{O}_{tq}^{1}$ (right).

|  | $\mathcal{O}_{Qd}^{8}$ | $\mathcal{O}_{Qd}^{1}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.158\scriptstyle{(1)}\,\displaystyle{{}^{+0.018}_{-0.016}}$ | $0.028\scriptstyle{(0)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.128\scriptstyle{(1)}\,\displaystyle{{}^{+0.007}_{-0.007}}$
NLO | $0.148\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.004}}$ | $0.023\scriptstyle{(1)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.182\scriptstyle{(1)}\,\displaystyle{{}^{+0.009}_{-0.008}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.007\scriptstyle{(9)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.003\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.000\scriptstyle{(9)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.005\scriptstyle{(10)}\,\displaystyle{{}^{+0.006}_{-0.003}}$ | $-0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.003\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.007}}$ | $0.003\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $-0.020\scriptstyle{(9)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.005\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.011\scriptstyle{(9)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.020\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.003\scriptstyle{(11)}\,\displaystyle{{}^{+0.006}_{-0.004}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.008\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $-0.022\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $-0.195\scriptstyle{(6)}\,\displaystyle{{}^{+0.026}_{-0.034}}$ | $-0.049\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $0.003\scriptstyle{(11)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.217\scriptstyle{(3)}\,\displaystyle{{}^{+0.015}_{-0.017}}$
NLO | $-0.150\scriptstyle{(7)}\,\displaystyle{{}^{+0.017}_{-0.008}}$ | $-0.033\scriptstyle{(2)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.036\scriptstyle{(8)}\,\displaystyle{{}^{+0.008}_{-0.007}}$ | $-0.282\scriptstyle{(2)}\,\displaystyle{{}^{+0.011}_{-0.012}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.050\scriptstyle{(11)}\,\displaystyle{{}^{+0.009}_{-0.007}}$ | $0.008\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.011\scriptstyle{(11)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $-0.000\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.022\scriptstyle{(10)}\,\displaystyle{{}^{+0.005}_{-0.008}}$ | $0.000\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.006\scriptstyle{(10)}\,\displaystyle{{}^{+0.006}_{-0.007}}$ | $0.005\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.083\scriptstyle{(10)}\,\displaystyle{{}^{+0.007}_{-0.008}}$ | $-0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.002\scriptstyle{(11)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.011\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.103\scriptstyle{(11)}\,\displaystyle{{}^{+0.008}_{-0.013}}$ | $-0.000\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.006\scriptstyle{(10)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.016\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.106\scriptstyle{(10)}\,\displaystyle{{}^{+0.013}_{-0.017}}$ | $-0.022\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.007\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.078\scriptstyle{(3)}\,\displaystyle{{}^{+0.005}_{-0.005}}$
NLO | $-0.095\scriptstyle{(8)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.008\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.029\scriptstyle{(9)}\,\displaystyle{{}^{+0.007}_{-0.006}}$ | $-0.106\scriptstyle{(2)}\,\displaystyle{{}^{+0.004}_{-0.005}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.060\scriptstyle{(15)}\,\displaystyle{{}^{+0.005}_{-0.005}}$ | $-0.004\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.016\scriptstyle{(15)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.015\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $-0.078\scriptstyle{(21)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $-0.004\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.009\scriptstyle{(17)}\,\displaystyle{{}^{+0.008}_{-0.006}}$ | $-0.022\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.005\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.006\scriptstyle{(16)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.008\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.001\scriptstyle{(15)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.000\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.010\scriptstyle{(16)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $0.012\scriptstyle{(20)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $0.003\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.002\scriptstyle{(16)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.007\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $0.002\scriptstyle{(19)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.005\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.002\scriptstyle{(17)}\,\displaystyle{{}^{+0.003}_{-0.003}}$ | $0.005\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.001}}$

Table 10: Numerical results for the operators $\mathcal{O}_{Qd}^{8}$ (left)
and $\mathcal{O}_{Qd}^{1}$ (right).

|  | $\mathcal{O}_{tu}^{8}$ | $\mathcal{O}_{tu}^{1}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.220\scriptstyle{(2)}\,\displaystyle{{}^{+0.025}_{-0.021}}$ | $0.045\scriptstyle{(0)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.206\scriptstyle{(0)}\,\displaystyle{{}^{+0.012}_{-0.011}}$
NLO | $0.205\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.008}}$ | $0.047\scriptstyle{(1)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.000\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.296\scriptstyle{(1)}\,\displaystyle{{}^{+0.015}_{-0.013}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $0.007\scriptstyle{(7)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.005\scriptstyle{(9)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.000\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.002}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.012\scriptstyle{(10)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $-0.004\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $0.031\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.009\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.006\scriptstyle{(9)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.032\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.029\scriptstyle{(11)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.004\scriptstyle{(12)}\,\displaystyle{{}^{+0.001}_{-0.004}}$ | $0.032\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.001}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $0.262\scriptstyle{(9)}\,\displaystyle{{}^{+0.040}_{-0.032}}$ | $0.081\scriptstyle{(2)}\,\displaystyle{{}^{+0.008}_{-0.007}}$ | $0.026\scriptstyle{(17)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.360\scriptstyle{(4)}\,\displaystyle{{}^{+0.026}_{-0.023}}$
NLO | $0.268\scriptstyle{(7)}\,\displaystyle{{}^{+0.002}_{-0.008}}$ | $0.078\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.024\scriptstyle{(11)}\,\displaystyle{{}^{+0.011}_{-0.007}}$ | $0.471\scriptstyle{(3)}\,\displaystyle{{}^{+0.023}_{-0.020}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.036\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.000\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.012\scriptstyle{(11)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
NLO | $0.042\scriptstyle{(11)}\,\displaystyle{{}^{+0.005}_{-0.010}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.020\scriptstyle{(10)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.132\scriptstyle{(11)}\,\displaystyle{{}^{+0.013}_{-0.016}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.007\scriptstyle{(13)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $-0.019\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.125\scriptstyle{(12)}\,\displaystyle{{}^{+0.005}_{-0.006}}$ | $-0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.017\scriptstyle{(11)}\,\displaystyle{{}^{+0.006}_{-0.003}}$ | $-0.021\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.127\scriptstyle{(12)}\,\displaystyle{{}^{+0.012}_{-0.013}}$ | $-0.032\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.032\scriptstyle{(24)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $-0.151\scriptstyle{(6)}\,\displaystyle{{}^{+0.008}_{-0.010}}$
NLO | $-0.155\scriptstyle{(10)}\,\displaystyle{{}^{+0.010}_{-0.014}}$ | $-0.027\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.001\scriptstyle{(16)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $-0.177\scriptstyle{(4)}\,\displaystyle{{}^{+0.006}_{-0.006}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.105\scriptstyle{(14)}\,\displaystyle{{}^{+0.010}_{-0.012}}$ | $-0.008\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.018\scriptstyle{(17)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $-0.034\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $-0.112\scriptstyle{(18)}\,\displaystyle{{}^{+0.008}_{-0.012}}$ | $-0.009\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $-0.011\scriptstyle{(20)}\,\displaystyle{{}^{+0.013}_{-0.023}}$ | $-0.043\scriptstyle{(5)}\,\displaystyle{{}^{+0.003}_{-0.004}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $-0.017\scriptstyle{(15)}\,\displaystyle{{}^{+0.004}_{-0.006}}$ | $-0.006\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.008\scriptstyle{(14)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.011\scriptstyle{(16)}\,\displaystyle{{}^{+0.015}_{-0.009}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.009\scriptstyle{(17)}\,\displaystyle{{}^{+0.007}_{-0.003}}$ | $0.005\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $-0.006\scriptstyle{(14)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.020\scriptstyle{(18)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $0.012\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.002}}$
NLO | $-0.001\scriptstyle{(20)}\,\displaystyle{{}^{+0.012}_{-0.007}}$ | $0.008\scriptstyle{(5)}\,\displaystyle{{}^{+0.006}_{-0.003}}$ | $0.013\scriptstyle{(21)}\,\displaystyle{{}^{+0.014}_{-0.008}}$ | $0.004\scriptstyle{(5)}\,\displaystyle{{}^{+0.000}_{-0.000}}$

Table 11: Numerical results for the operators $\mathcal{O}_{tu}^{8}$ (left)
and $\mathcal{O}_{tu}^{1}$ (right).

|  | $\mathcal{O}_{td}^{8}$ | $\mathcal{O}_{td}^{1}$
---|---|---|---
Observable | Order |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb] |  $\sigma^{(1)}$ [pb] |  $\sigma^{(2)}$ [pb]
$\sigma$ | LO | $0.161\scriptstyle{(2)}\,\displaystyle{{}^{+0.020}_{-0.017}}$ | $0.030\scriptstyle{(0)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.129\scriptstyle{(1)}\,\displaystyle{{}^{+0.007}_{-0.007}}$
NLO | $0.139\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.009}}$ | $0.030\scriptstyle{(1)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $-0.018\scriptstyle{(3)}\,\displaystyle{{}^{+0.004}_{-0.005}}$ | $0.183\scriptstyle{(1)}\,\displaystyle{{}^{+0.010}_{-0.008}}$
$\sigma_{\displaystyle B_{n}+\bar{B}_{n}}$ | LO | $-0.005\scriptstyle{(8)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.005\scriptstyle{(8)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.006\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.011\scriptstyle{(10)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $-0.002\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
$\sigma_{\displaystyle B_{r}+\bar{B}_{r}}$ | LO | $0.026\scriptstyle{(8)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.005\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.006\scriptstyle{(9)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.021\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $0.011\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.005}}$ | $0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.020\scriptstyle{(11)}\,\displaystyle{{}^{+0.007}_{-0.005}}$ | $0.020\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
$\sigma_{\displaystyle B_{k}+\bar{B}_{k}}$ | LO | $0.188\scriptstyle{(9)}\,\displaystyle{{}^{+0.029}_{-0.023}}$ | $0.050\scriptstyle{(2)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $-0.005\scriptstyle{(10)}\,\displaystyle{{}^{+0.001}_{-0.002}}$ | $0.215\scriptstyle{(3)}\,\displaystyle{{}^{+0.016}_{-0.014}}$
NLO | $0.191\scriptstyle{(7)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $0.044\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.000}}$ | $0.020\scriptstyle{(9)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $0.288\scriptstyle{(2)}\,\displaystyle{{}^{+0.015}_{-0.013}}$
$\sigma_{\displaystyle C_{nn}}$ | LO | $0.023\scriptstyle{(12)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.009\scriptstyle{(12)}\,\displaystyle{{}^{+0.002}_{-0.002}}$ | $0.001\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$
NLO | $0.029\scriptstyle{(10)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.016\scriptstyle{(10)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{rr}}$ | LO | $-0.073\scriptstyle{(9)}\,\displaystyle{{}^{+0.006}_{-0.007}}$ | $-0.011\scriptstyle{(2)}\,\displaystyle{{}^{+0.002}_{-0.003}}$ | $0.008\scriptstyle{(14)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $-0.007\scriptstyle{(4)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
NLO | $-0.082\scriptstyle{(10)}\,\displaystyle{{}^{+0.003}_{-0.001}}$ | $-0.001\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.000\scriptstyle{(11)}\,\displaystyle{{}^{+0.003}_{-0.004}}$ | $-0.013\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$
$\sigma_{\displaystyle C_{kk}}$ | LO | $-0.095\scriptstyle{(11)}\,\displaystyle{{}^{+0.009}_{-0.010}}$ | $-0.022\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $-0.028\scriptstyle{(14)}\,\displaystyle{{}^{+0.005}_{-0.007}}$ | $-0.080\scriptstyle{(4)}\,\displaystyle{{}^{+0.005}_{-0.005}}$
NLO | $-0.122\scriptstyle{(10)}\,\displaystyle{{}^{+0.013}_{-0.019}}$ | $-0.017\scriptstyle{(2)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.040\scriptstyle{(10)}\,\displaystyle{{}^{+0.010}_{-0.014}}$ | $-0.106\scriptstyle{(3)}\,\displaystyle{{}^{+0.003}_{-0.003}}$
$\sigma_{\displaystyle C_{rk}+C_{kr}}$ | LO | $-0.099\scriptstyle{(15)}\,\displaystyle{{}^{+0.012}_{-0.014}}$ | $-0.002\scriptstyle{(3)}\,\displaystyle{{}^{+0.001}_{-0.001}}$ | $0.014\scriptstyle{(17)}\,\displaystyle{{}^{+0.005}_{-0.003}}$ | $-0.025\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.001}}$
NLO | $-0.063\scriptstyle{(19)}\,\displaystyle{{}^{+0.011}_{-0.006}}$ | $0.013\scriptstyle{(4)}\,\displaystyle{{}^{+0.004}_{-0.004}}$ | $0.003\scriptstyle{(20)}\,\displaystyle{{}^{+0.000}_{-0.002}}$ | $-0.026\scriptstyle{(5)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{nr}+C_{rn}}$ | LO | $0.006\scriptstyle{(14)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.006\scriptstyle{(3)}\,\displaystyle{{}^{+0.002}_{-0.001}}$ | $0.002\scriptstyle{(17)}\,\displaystyle{{}^{+0.000}_{-0.001}}$ | $-0.006\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.002}}$
NLO | $-0.019\scriptstyle{(15)}\,\displaystyle{{}^{+0.009}_{-0.015}}$ | $-0.004\scriptstyle{(3)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $-0.027\scriptstyle{(17)}\,\displaystyle{{}^{+0.006}_{-0.007}}$ | $-0.001\scriptstyle{(4)}\,\displaystyle{{}^{+0.001}_{-0.002}}$
$\sigma_{\displaystyle C_{nk}+C_{kn}}$ | LO | $0.011\scriptstyle{(15)}\,\displaystyle{{}^{+0.004}_{-0.003}}$ | $0.000\scriptstyle{(4)}\,\displaystyle{{}^{+0.000}_{-0.000}}$ | $0.020\scriptstyle{(16)}\,\displaystyle{{}^{+0.005}_{-0.004}}$ | $-0.009\scriptstyle{(5)}\,\displaystyle{{}^{+0.002}_{-0.003}}$
NLO | $0.015\scriptstyle{(19)}\,\displaystyle{{}^{+0.016}_{-0.007}}$ | $-0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.006}}$ | $-0.017\scriptstyle{(18)}\,\displaystyle{{}^{+0.008}_{-0.012}}$ | $0.002\scriptstyle{(4)}\,\displaystyle{{}^{+0.003}_{-0.002}}$

Table 12: Numerical results for the operators $\mathcal{O}_{td}^{8}$ (left)
and $\mathcal{O}_{td}^{1}$ (right).

##  Appendix B Additional differential plots

This Appendix completes Section 5.2, showing results differential in
$t\bar{t}$ phase space for all other dimension-6 operators considered in this
work.

Figure 7: Results for the operators $\mathcal{O}_{Qq}^{(8,3)}$ (top) and
$\mathcal{O}_{Qq}^{(1,3)}$ (bottom), similar to Figure 5.

Figure 8: Results for the operators $\mathcal{O}_{Qu}^{8}$ (top) and
$\mathcal{O}_{Qu}^{1}$ (bottom), similar to Figure 5.

Figure 9: Results for the operators $\mathcal{O}_{tq}^{8}$ (top) and
$\mathcal{O}_{tq}^{1}$ (bottom), similar to Figure 5.

Figure 10: Results for the operators $\mathcal{O}_{Qd}^{8}$ (top) and
$\mathcal{O}_{Qd}^{1}$ (bottom), similar to Figure 5.

Figure 11: Results for the operators $\mathcal{O}_{tu}^{8}$ (top) and
$\mathcal{O}_{tu}^{1}$ (bottom), similar to Figure 5.

Figure 12: Results for the operators $\mathcal{O}_{td}^{8}$ (top) and
$\mathcal{O}_{td}^{1}$ (bottom), similar to Figure 5.

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