aainstitutetext: Department of Physics Engineering, Istanbul Technical
University, Maslak Istanbul, Turkeybbinstitutetext: Istanbul Technical
University Artificial Intelligence, Data Science Research and Application
Center, Maslak Istanbul, Turkey

# Comparison of spin-correlation and polarization variables of spin density
matrix for top quark pairs at the LHC and New Physics implications

Altan Cakir111Corresponding author. a and Orcun Kolay<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Precise determination of top-quark pairs is an essential tool for
understanding the overall consistency of the standard model (SM) expectations,
understanding limited New Physics (NP) models, through spin-spin correlation
and polarization parameters, and has a critical impact on the analyses
strategies at upcoming LHC programs. In this work, we review and discuss
various state-of-the-art Monte Carlo (MC) methodologies as MadGraph5_aMC@NLO,
Sherpa, Powheg-Box and Pythia8, which are Matrix Element (ME)$/$Parton Shower
(PS) matching generators including a complete set of spin correlation and
polarization in top quark pair production with dileptonic final states. This
is the first such study that not only compares the effects of different MC
event generator approaches on spin density matrix elements and polarization
parameters, but also investigates the effects of leading order (LO) and next-
to-leading order (NLO) accuracy in QCD, and electroweak (EW) corrections via
Sherpa. Moreover, as a continuation of the work, the prospects for possible NP
scenarios through top-quark spin-spin correlation and polarization
measurements for Supersymmetry (R parity conserved and violated models) and
Dark Matter (top quarks associated mediator) models during upcoming LHC runs
are briefly outlined. We find that all SM MC predictions for the defined set
of variables are generally consistent with the experimental data and
theoretical predictions within the uncertainty variations. Besides, for the
distributions of the $\cos\varphi$, laboratory-frame observables
($\cos\varphi_{lab}$ and $|\Delta\phi_{ll}|$) and the observables generated by
parity (P) and charge-parity (CP) conserving interactions, we conclude some
clues that the considered signals and beyond may well be separated from
experimental data and located above the SM predictions.

###### Keywords:

Top Quark, Hadronic Colliders, MC Simulations

##  1 Introduction

Top quark pair production is still one of the most critical process in
collider physics today. As the heaviest known fundamental particle in the
standard model (SM), the top-quark provides a unique opportunity to understand
the critical behaviour of Yukawa coupling close to unity, making it a more
important subject for beyond standard model (BSM) scenarios and in discussions
of the stability of the SM universe. Top-quark properties is therefore an
increasingly important input to many analyses in the following LHC runs.
Understanding higher order effects together with electroweak corrections to
top-quark processes play a crucial role in precision measurements, such as
spin-correlation and polarization of top-quark pairs 0a ; 0b ; 0c ; 0d ; 0e ;
0f ; 0g , and many other unknown BSM signals, such as supersymmetry and
effective field theory (EFT) models 0h ; 0i . More generally top quarks are
unavoidable facts in BSM searches, both as signal objects and for backgrounds
(e.g. as a source of leptons and b quarks).

Monte Carlo (MC) event generators are critical tools for the analysis and
interpretation of LHC analyses in today research 0j ; 0j1 ; 0j2 . In
particular, detailed aspects of the final states produced in individual
processes, such as top-quark pair production, are to be required advanced
calculations 0k ; 0k1 ; 0k2 . Top-quark pair production comprise for example
the contribution of higher order matrix element calculations with multiple
partons, the evaluation of parton shower and fragmentation, parton shower
matching selection efficiencies, and the extrapolation of cross sections to
the full phase space. In this manner, considering recent developments in
higher-order perturbative corrections, in particular in QCD, but also in QED
and in the electroweak sector, comparison studies of MC approaches are a
critical subject that makes some precision measurements and data
interpretation more visible. Based on a high level of calculation - studied
with LHC data - these contributions allow for both the realistic simulation of
top-quark pairs precision measurements and the possible understanding of BSM
signals. Thus, complex contributions of various MC approaches to experimental
objects and variables are a vital subject of collider-based particle physics
studies.

In the following sections, we provide an overview of the up-to-date top-quark
pairs hadron level production with various MC methodologies, above all, we
will try to stress the main motivation of the extended discussion of a spin
density matrix for the top pair system existing consistency and controversies
concerning interpretation of the LHC measurements, as well as the sources of
theory uncertainty. All in all, we review the precision measurement of spin
density matrix of top quark-pairs; discuss the NP ambiguity; and the main
strategies to measure spin-correleation and polarization top-quark pairs will
be presented. The interpretation of 22 different measurements and the
theoretical uncertainties will be discussed while conclusion will contain some
final remarks for future LHC runs.

As it is known, even though the SM is a theory that has been proven many times
by various tests, it has problems that it cannot solve, such as hierarchy
problem and dark matter (DM). Supersymmetry (SUSY) as an extension of the SM
could help to elucidate these both problems by introducing additional
particles (e.g. bosonic superpartner of the top quark (stop, $\tilde{t}$))
that cancel large quantum corrections to the Higgs boson squared mass in the
SM and feasible DM candidates. Due to the central role of the top quark in
interaction with these hypothetical particles, collaborations like ATLAS and
CMS try to measure top quark properties accurately to find these signals in
the experiment. Searches for stop $\tilde{t}$ and DM candidate particles
concentrate generally on high missing transverse energy $E_{T}^{miss}$ region
stem from undetected lightest SUSY particles (LSP), as a decay product of
$\tilde{t}$, and DM candidate particles. With these studies, most of the high
$E_{T}^{miss}$ region, even low $E_{T}^{miss}$, was excluded by ATLAS and CMS
Collaborations 1a ; 1b ; 0e ; 1d . In the last step of the study, we will
focus on the interpretation of SUSY and DM signals in low $E_{T}^{miss}$
region, which are studied with generic searching methods, by measuring
precisely. The signals are based on simplified models, which are the similar
topology with the models in the ref. 1a , of stop pair production and of top
quark pair production in associated with fermionic DM candidate particles. In
the SUSY signal, in figure 1(a), a pair of stop $\tilde{t}$ decays an on-shell
top quark $t$ and an LSP, in this case neutralino $\tilde{\chi}_{1}^{0}$,
which is stable for this simplified model. In order to investigate the
deviations of the signal from SM $t\bar{t}$, one mass point which is $175$ GeV
for $\tilde{t}$ and $1$ GeV for $\tilde{\chi}_{1}^{0}$ will be used. An
alternative dark matter signal in figure 1(b) include a scalar $\phi$ or
pseudoscalar $a$ spin-0 mediator decayed into fermionic DM particles $\chi$
and an associated top quark pair. The coupling strengths of the DM-mediator
and the mediator-SM fermions are assumed to be equal to one in the simplified
model. Additionally, in this analysis, the scenario with $m_{\chi}=1$ GeV and
$m_{\phi/a}=10$ GeV will be focused.

$\tilde{\chi}_{1}^{0}$$\tilde{\chi}_{1}^{0}$$\overline{\text{t}}$$\tilde{\text{t}}$$\overline{\tilde{\text{t}}}$
(a)

$\chi$$\overline{\chi}$$\overline{\text{t}}$$\phi/a$ (b)

$\overline{\text{t}}$$\tilde{\text{t}}$$\overline{\tilde{\text{t}}}$$\tilde{\chi}_{1}^{0}$$\tilde{\chi}_{1}^{0}$
(c)

Figure 1: Diagrams of the signal samples for stop quark pair production with
decay into top quarks and stable neutralinos (a), direct DM production
mediated by a scalar or pseudoscalar particle with associated top quark pair
(b), and stop quark pair production with decay into top quarks and neutralinos
decaying into light flavour quarks (c).

In the above models, we focused on a small mass point to ensure low
$E_{T}^{miss}$. However, the additional model of R-Parity 1e  violating (RPV)
SUSY provides more mass points to study for low $E_{T}^{miss}$ due to the
decaying $\tilde{\chi}_{1}^{0}$. The $\tilde{\chi}_{1}^{0}$ decays into three
quarks with the help of a trilinear Yukawa coupling between quarks and
superpartner of quarks (squark) 1f . In this study, the production chain
includes a $\tilde{t}$ pair decaying into a $t$ and a $\tilde{\chi}_{1}^{0}$
as shown in figure 1(c), and the $\tilde{\chi}_{1}^{0}$ goes into three light
flavor quarks $q$, $\tilde{\chi}_{1}^{0}\rightarrow uds$, as considered in
ref. 1g . For the RPV SUSY scenario, $\tilde{t}$ masses up to $670$ GeV have
been excluded, by CMS Collaboration 1g , at $95\%$ confidence level with a
maximum observed local significance of $2.8\sigma$ for $m_{\tilde{t}}=400$
GeV.

This paper structured as follows: in the next section, we describe the MC data
samples and the CMS data used in the analysis, with generator settings for all
$t\bar{t}$ and signal samples. In section 3, we introduce the set of
observables of spin correlation and polarization in top quark pair production
by using decomposition of spin density matrix. In section 4, we present our
results for the differential distributions of the observables. The results are
divided into two parts as the comparison of different MC event generator
configurations on the distributions of the observables of $t\bar{t}$ events
and the investigation of behaviours of signals having different topology on
the observables. Finally, our conclusions are given in section 5.

##  2 Event Samples and Data

A range of MC generators to provide predictions for top quark pair production
and related BSM signal processes are used to investigate the effects of
various approaches of MC generators on top pair spin correlation matrix and
polarization variables and to assess the differences between BSM signals and
top quark pairs.

A summary of the produced MC samples, describing various the SM top quark pair
production processes and the possible signal scenarios, is shown in table 1.
The table also lists details of the simulation samples used, including the
matrix element (ME) event generator with the order of the calculation and the
central parton distribution function (PDF) set, the parton shower (PS) and
hadronization model, and the merging scheme used to remove the overlap between
ME and PS.

Process |  ME PDF |  ME generator |  Order |  PS and |  Merging
---|---|---|---|---|---
|  |  |  (ME$/$PS) |  hadronization |  scheme
$t\bar{t}+jets$ |  NNPDF3.0 NLO |  MG5_aMC@NLO |  NLO$/$LL |  Pythia 8 |  FxFx
$t\bar{t}+jets$ |  NNPDF3.0 LO |  MG5_aMC@NLO |  LO$/$LL |  Pythia 8 |  MLM
$t\bar{t}+jets$ |  NNPDF3.0 NLO |  Powheg-Box |  NLO$/$LL |  Pythia 8 |  Powheg
$t\bar{t}+jets$ |  NNPDF3.0 NLO |  Sherpa 2 |  NLO($+$EW)$/$LL |  CSShower |  MEPS@NLO
$t\bar{t}+jets$ |  NNPDF3.0 LO |  Sherpa 2 |  LO$/$LL |  CSShower |  MEPS@LO
$\tilde{t}\tilde{t}\to t\tilde{\chi}_{1}^{0}\bar{t}\tilde{\chi}_{1}^{0}$ |  NNPDF3.1 LO |  MG5_aMC@NLO |  LO$/$LL |  Pythia 8 |  MLM
$t\bar{t}+DM$ |  NNPDF3.1 LO |  MG5_aMC@NLO |  LO$/$LL |  Pythia 8 |  MLM

Table 1:  Matrix Element (ME) event generator settings for $t\bar{t}$ and BSM
signal samples used in this study. The generator versions, the order of QCD
accuracy of associated predictions, the PDF sets used in ME calculations and
the merging scheme are shown.

In all background and signal samples $t\bar{t}$ events are produced with a top
quark mass of $m_{t}=172.5$ GeV. The nominal $t\bar{t}$ sample (with up to two
additional matched partons included in the ME calculations) is simulated using
MadGraph5_aMC@NLO (v2.6.7) b  at NLO QCD accuracy. To evaluate the effects of
different perturbative orders of the calculation, we have been used the
samples generated with MadGraph5_aMC@NLO at LO accuracy as well. The parton
showers are modelled using Pythia (v8.2.44) c  with the default showering
tunes at leading logarithmic (LL) accuracy. The matrix elements at LO and NLO
are merged with the parton shower using the MLM d  and the FxFx e  merging
scheme, respectively. In order to determine state-of-the-art theoretical
approaches between the different methods in MC generators, two additional
generator setups are incorporated in the analysis chain. Firstly, $t\bar{t}$
sample is generated using the Sherpa (v2.2.8) f  generator with up to three
extra partons at LO and NLO. The Sherpa, in production of $t\bar{t}$ events,
employs Amegic++ g1 , Comix g  and OpenLoops h ; i ; j  (for one-loop
amplitudes) in matrix elements calculation which are merged with CSShower k
for parton showering (with LL accuracy) and hadronization by using the
MEPS@NLO l ; m  prescription. To match the ME and PS results, Sherpa and
MadGraph5_aMC@NLO make use of the same normalization method, which is MC@NLO n
. The other alternative $t\bar{t}$ sample is generated with Powheg (v2) o ; p
; q ; r  including up to two extra matched partons at the ME level with NLO
accuracy. The parton shower is employed by Pythia (v8.2.40) with
$h_{damp}=272.72$ GeV parameter of Powheg.

For all SM samples listed, the initial state partons are modeled with the
NNPDF3.0 PDF set s ; t  implemented by LHAPDF u . To determine the systematic
uncertainties in predictions, uncertainties coming from PDF and scale
variations are used as summed in quadrature. Parton distribution function
uncertainties are evaluated by reweighting the $t\bar{t}$ samples generated by
MadGraph5_aMC@NLO at LO and Sherpa for all setups using generator weights
related with each of the variations given by NNPDF3.0, CT10 v  and MMHT14 w
PDF sets. Renormalization and factorization scale uncertainties are calculated
using variations by a factor of two around the central scales which are equal
to $\frac{1}{2}\sum_{i=1}^{2}\sqrt{m_{t}^{2}+P_{T,i}^{2}}$ for
MadGraph5_aMC@NLO and $\sqrt{m_{t}^{2}+\frac{1}{2}\sum_{i=1}^{2}P_{T,i}^{2}}$
for Sherpa setup x , where $P_{T,i}$ is the transverse momentum of top quarks.

Signal samples of SUSY and RPV SUSY are simulated at LO with up to three and
two additional partons, respectively, with MadGraph5_aMC@NLO (v2.4.2)
interfaced to Pythia (v8.2.40) for parton showering and hadronization using
the MLM merging scheme. For the SUSY sample in figure 1(a), the masses of the
$\tilde{t}$ and the $\tilde{\chi}_{1}^{0}$ will be considered as
$m_{\tilde{t}}=175$ GeV and $m_{\tilde{\chi}_{1}^{0}}=1$ GeV. On the RPV SUSY
side, the masses of the $\tilde{t}$ and the $\tilde{\chi}_{1}^{0}$ are taken
as $400$ GeV and $100$ GeV, respectively.

The other signal samples are top quark pairs associated with DM particles
$\chi$ as shown in figure 1(b) which is included scalar $\phi$ or pseudoscalar
$a$ mediators. The DM signal samples are simulated at LO including up to one
additional parton with MadGraph5_aMC@NLO (v2.6.1) interfaced with the Pythia
(v8.2.40) PS model using the MLM merging scheme. The masses of the $\chi$ and
the $\phi/a$ will be considered as $m_{\chi}=1$ GeV and $m_{\phi/a}=10$ GeV.
In all signal samples, NNPDF3.1 PDF set is used in ME calculation.

To assess the truth of the theoretical predictions, experimental measurements
published by CMS Collaboration are directly used 0i . The CMS analysis
unfolded the data to compare with parton level predictions at full phase
space. In our analysis, the MC predictions have baseline selections compatible
with the CMS analysis.

##  3 Spin Correlations on Top Quark Pairs

The total cross section for the top pair production at accelerator energy $s$
can be written as

$\sigma(s,m_{t})=\sum_{i,j}\int_{0}^{1}\int_{0}^{1}\,dx_{1}\,dx_{2}\phi_{i,A}(x_{1},\mu_{F})\phi_{j,B}(x_{2},\mu_{F})\hat{\sigma}_{ij\to
t\bar{t}}(\frac{m_{t}^{2}}{\hat{s}},\mu_{R},\mu_{F},\alpha_{s}(\mu_{R}))$ (1)

by using factorization theorem. This theorem divides the total cross section
into a short distance (hard) part $\hat{\sigma}_{t\bar{t}}$ and a long
distance (soft) part, which is known as parton distribution function
$\phi_{i,A/j,B}$ (PDF). The PDF determine the probability of the existence of
a parton $i$ ($j$) in the hadron A (B) with a rate $x_{1}$ ($x_{2}$) of the
longitudinal momentum of the hadron at the factorization scale $\mu_{F}$. The
partonic cross section $\hat{\sigma}_{ij\to t\bar{t}}$ involves the spin
information of top quark and anti-quark. However the spin information of a top
quark is not detectable because of short life time of it. In spite of the fact
that the spin of an unstable particle produced in high energy reactions is no
straightforwardly noticeable quantity, it is extremely helpful to present the
idea of a spin density matrix z  for the top pair system (2). The decay width
of the top quark much smaller than its mass allows to use narrow width
approximation (NWA) aa ; ab . Using NWA, the matrix element of the top pair
production can be written as the trace of the production and the decay
factors:

$|M|^{2}\propto Tr[\rho R\bar{\rho}]$ (2)

where $R$ is the spin density matrix of the on-shell top pair production and
$\rho$ and $\bar{\rho}$ are the decay spin density matrices of the polarized
top quark and antiquark, respectively. To look at the spin properties of the
top quark and antiquark, spin density matrix can be decomposed in the spin
space of $t$ and $\bar{t}$.

$R\propto\tilde{A}\mathbb{I}\otimes\mathbb{I}+\tilde{B}_{i}^{+}\sigma^{i}\otimes\mathbb{I}+\tilde{B}_{i}^{-}\mathbb{I}\otimes\sigma^{i}+\tilde{C}_{ij}\sigma^{i}\otimes\sigma^{j}$
(3)

Here, the first terms which are a 2x2 unit matrix $\mathbb{I}$ and the Pauli
matrices $\sigma^{i}$ in the tensor products refer to top spin spaces, and
second terms refer to anti-top.

While the function $\tilde{A}$ is related to the total $t\bar{t}$ cross
section and the top quarks kinematic properties, $\tilde{B}_{i}^{\pm}$ and
$\tilde{C}_{ij}$ are 3-vectors that describe the polarizations of $t$ and
$\bar{t}$ in each direction, and 3x3 matrix that determine the spin
correlation between top quark and antiquark.

As mentioned earlier, decay of the top quarks before hadronization allows to
study the spins of them by looking at the angular distribution of the decay
products. Top quarks decay almost to a $W$ boson and a bottom quark. The final
states of the decay coming from $W$ boson can be a quark-antiquark pair or a
lepton and a neutrino. The angular distribution of the top decay products with
respect to the spin alignment of the top is given by

$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\chi_{i}}=\frac{1}{2}(1+\alpha_{i}\cos\chi_{i})$
(4)

with $\chi_{i}$ the angle between the 3-momentum of decay product $i$ in the
top rest frame and the direction of the top spin. $\alpha_{i}$ is the spin
analyzing power of $i$. The spin analyzing power of the charged lepton is
almost equal to $1$ which is maximum value. For a top antiquark, $\alpha_{i}$
is minus sign of the top quark one. It does not change much at NLO QCD ac . To
make use of high valued spin analyzing power, leptons have been used as a
proxy for spins of top quarks and anti-quarks.

Scattering
Plane$\hat{p}$$\hat{p}$$\hat{k}$$\bar{t}$$\hat{r}$$\hat{n}$$\Theta$$\hat{a}_{3}(\hat{b}_{3})$$\hat{a}_{1}(\hat{b}_{1})$$\hat{a}_{2}(\hat{b}_{2})$$\ell^{+(-)}$$\theta_{1(2)}^{n}$$\theta_{1(2)}^{k}$$\theta_{1(2)}^{r}$at
$t(\bar{t})$ rest frame Figure 2: Coordinate system illustrating $t$ and
$\bar{t}$ helicity spin basis. The $\hat{k}$ axis is described as the
direction of flight of top quark measured in the $t\bar{t}$ CM. The directions
$\hat{k}$, $\hat{r}$, $\hat{p}$ and $\bar{t}$ are in the scattering plane
which the normal is the direction $\hat{n}$. The signs of the $\hat{r}$ and
$\hat{n}$ axes can have inverse according to the angle $\Theta$ as shown in
eq. (5). The angles between the direction of flight of the positively
$\ell^{+}$ (negatively $\ell^{-}$) charged lepton in the top quark (anti-
quark) rest frame and the axes $\hat{a}$ ($\hat{b}$) determined in eq. (5) are
used to calculate the coefficients in eqs. (9)-(12).

The functions $\tilde{B}_{i}^{\pm}$ and $\tilde{C}_{ij}$ in equation (3) can
be written as a linear combination of coefficient functions associated with an
orthonormal basis. The basis in this analysis was chosen as the helicity basis
as it gives more spin correlation compared to other alternatives as discussed
in ref. 0c . This basis can be shown in figure 2. $\hat{k}$ is described as
helicity axis determined by the top quark direction in the $t\bar{t}$ center-
of-mass (CM) frame. The direction $\hat{p}$ indicates the direction of one of
the incoming partons in the same frame. Using the directions $\hat{k}$ and
$\hat{p}$, the normal $\hat{n}$ to the scattering plane and the direction
$\hat{r}$ perpendicular to $\hat{k}$ in the scattering plane are given by
$\hat{n}=(\hat{p}\times\hat{k})/\sin(\Theta)$ and
$\hat{r}=(\hat{p}-\hat{k}\cos(\Theta))/\sin(\Theta)$.

A redefinition of the $\hat{r}$ and $\hat{n}$ axes is necessitated owing to
the Bose-Einstein symmetry of the initial $gg$ state 0h . The reference axes
$\hat{a}$ and $\hat{b}$ are described with above regularization as

$\hat{a}\to\\{\hat{k},sign(\cos\Theta)\hat{r},sign(\cos\Theta)\hat{n}\\},\,\,\hat{b}\to\\{-\hat{k},-sign(\cos\Theta)\hat{r},-sign(\cos\Theta)\hat{n}\\}$
(5)

Using the orthonormal basis, $\tilde{B}_{i}^{\pm}$ and $\tilde{C}_{ij}$ are
decomposed as

$\tilde{B}_{i}^{\pm}=b_{k}^{\pm}\hat{k}_{i}+b_{r}^{\pm}\hat{r}_{i}+b_{n}^{\pm}\hat{n}_{i},$
(6)
$\begin{split}&\tilde{C}_{ij}=c_{kk}\hat{k}_{i}\hat{k}_{j}+c_{rr}\hat{r}_{i}\hat{r}_{j}+c_{nn}\hat{n}_{i}\hat{n}_{j}\\\
&+c_{rk}(\hat{r}_{i}\hat{k}_{j}+\hat{k}_{i}\hat{r}_{j})+c_{nr}(\hat{n}_{i}\hat{r}_{j}+\hat{r}_{i}\hat{n}_{j})+c_{kn}(\hat{k}_{i}\hat{n}_{j}+\hat{n}_{i}\hat{k}_{j})\\\
&+c_{n}(\hat{r}_{i}\hat{k}_{j}-\hat{k}_{i}\hat{r}_{j})+c_{k}(\hat{n}_{i}\hat{r}_{j}-\hat{r}_{i}\hat{n}_{j})+c_{r}(\hat{k}_{i}\hat{n}_{j}-\hat{n}_{i}\hat{k}_{j})\end{split}$
(7)

The coefficient functions $b_{i}^{\pm}$, $c_{ij}$ and $c_{i}$ which are
functions of the partonic center of mass-energy $\hat{s}$ and $\cos\Theta$ are
related to some discrete symmetries such as parity (P) and charge-parity (CP),
and they are summarized in table 2.

|  Observable |  Measured |  Coefficient |  Symmetries
---|---|---|---|---
|  |  coefficient |  function |
polarization |  $\cos\theta_{1}^{k}$ |  $B_{1}^{k}$ |  $b_{k}^{+}$ |  P-odd, CP-even
$\cos\theta_{2}^{k}$ |  $B_{2}^{k}$ |  $b_{k}^{-}$ |  P-odd, CP-even
$\cos\theta_{1}^{r}$ |  $B_{1}^{r}$ |  $b_{r}^{+}$ |  P-odd, CP-even
$\cos\theta_{2}^{r}$ |  $B_{2}^{r}$ |  $b_{r}^{-}$ |  P-odd, CP-even
$\cos\theta_{1}^{n}$ |  $B_{1}^{n}$ |  $b_{n}^{+}$ |  P-even, CP-even
$\cos\theta_{2}^{n}$ |  $B_{2}^{n}$ |  $b_{n}^{-}$ |  P-even, CP-even
$\cos\theta_{1}^{k*}$ |  $B_{1}^{k*}$ |  $b_{k}^{+}$ |  P-odd, CP-even
$\cos\theta_{2}^{k*}$ |  $B_{2}^{k*}$ |  $b_{k}^{-}$ |  P-odd, CP-even
$\cos\theta_{1}^{r*}$ |  $B_{1}^{r*}$ |  $b_{r}^{+}$ |  P-odd, CP-even
$\cos\theta_{2}^{r*}$ |  $B_{2}^{r*}$ |  $b_{r}^{-}$ |  P-odd, CP-even
diagonal spin corr. |  $\cos\theta_{1}^{k}\cos\theta_{2}^{k}$ |  $C_{kk}$ |  $c_{kk}$ |  P-even, CP-even
$\cos\theta_{1}^{r}\cos\theta_{2}^{r}$ |  $C_{rr}$ |  $c_{rr}$ |  P-even, CP-even
$\cos\theta_{1}^{n}\cos\theta_{2}^{n}$ |  $C_{nn}$ |  $c_{nn}$ |  P-even, CP-even
cross spin corr. |  $\cos\theta_{1}^{r}\cos\theta_{2}^{k}+\cos\theta_{1}^{k}\cos\theta_{2}^{r}$ |  $C_{rk}+C_{kr}$ |  $c_{rk}$ |  P-even, CP-even
$\cos\theta_{1}^{r}\cos\theta_{2}^{k}-\cos\theta_{1}^{k}\cos\theta_{2}^{r}$ |  $C_{rk}-C_{kr}$ |  $c_{n}$ |  P-even, CP-odd
$\cos\theta_{1}^{n}\cos\theta_{2}^{r}+\cos\theta_{1}^{r}\cos\theta_{2}^{n}$ |  $C_{nr}+C_{rn}$ |  $c_{nr}$ |  P-odd, CP-even
$\cos\theta_{1}^{n}\cos\theta_{2}^{r}-\cos\theta_{1}^{r}\cos\theta_{2}^{n}$ |  $C_{nr}-C_{rn}$ |  $c_{k}$ |  P-odd, CP-odd
$\cos\theta_{1}^{n}\cos\theta_{2}^{k}+\cos\theta_{1}^{k}\cos\theta_{2}^{n}$ |  $C_{nk}+C_{kn}$ |  $c_{kn}$ |  P-odd, CP-even
$\cos\theta_{1}^{n}\cos\theta_{2}^{k}-\cos\theta_{1}^{k}\cos\theta_{2}^{n}$ |  $C_{nk}-C_{kn}$ |  $-c_{r}$ |  P-odd, CP-odd
|  $\cos\varphi$ |  $D$ |  $-(c_{kk}+c_{rr}+c_{nn})/3$ |  P-even, CP-even
lab. frame |  $\cos\varphi_{lab}$ |  $A_{\cos\varphi}^{lab}$ |  $-$ |  $-$
$|\Delta\phi_{ll}|$ |  $A_{|\Delta\phi_{ll}|}$ |  $-$ |  $-$

Table 2: Observable quantities of the spin density matrix and their
corresponding measured coefficients, coefficient functions and P and CP
symmetry properties.

According to table 2, the P- and CP-even coefficient functions ($c_{kk}$,
$c_{rr}$, $c_{nn}$ and $c_{rk}$) can have large values due to the parity
invariance of strong interaction. However, the presence of electroweak (EW)
corrections can result in non-zero P-odd and CP-even coefficients ($c_{nr}$
and $c_{kn}$). On the other hand, the coefficients produced by CP-violating
interactions are close to zero in the SM. The approximate CP invariance of the
SM limits the top and anti-top quarks to have the same polarization, so
$b_{i}^{+}-b_{i}^{-}$ has a value of zero222See ref. 0h  for further details..

After the equations (2) and (3), the normalized distribution for the two
leptons final state can be written in the form of spin coefficients and
direction of flight of the leptons. This can be simplified as the differential
distribution of the cross-section. The cross section distribution for the
double leptonic channel can be written as 0i

$\frac{1}{\sigma}\frac{d^{2}\sigma}{d\cos\theta_{1}^{i}d\cos\theta_{2}^{j}}=\frac{1}{4}(1+B_{1}^{i}\cos\theta_{1}^{i}+B_{2}^{j}\cos\theta_{2}^{j}-C_{ij}\cos\theta_{1}^{i}\cos\theta_{2}^{j})$
(8)

where $\theta_{1}^{i}(\theta_{2}^{j})$ is the angle between the direction of
flight of the positively (negatively) charged lepton in the top quark (anti-
quark) rest frame and a reference direction $\hat{a}$ ($\hat{b}$). The
coefficients $B_{1,2}^{i,j}$ and $C_{ij}$ in (8) reflect the properties of the
spin density components in (3). The elements of $B_{1,2}$ and $C$ matrices
have the same discrete symmetry properties with related coefficient functions,
see in table 2. To extract each coefficient separately one can reduce the
double differential cross section in equation (8) to single differential
distributions

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta_{1}^{i}}=\frac{1}{2}(1+B_{1}^{i}\cos\theta_{1}^{i}),$
(9)

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta_{2}^{i}}=\frac{1}{2}(1+B_{2}^{i}\cos\theta_{2}^{i}),$
(10)

$\frac{1}{\sigma}\frac{d\sigma}{d(\cos\theta_{1}^{i}\cos\theta_{2}^{i})}=\frac{1}{2}(1-C_{ii}\cos\theta_{1}^{i}\cos\theta_{2}^{i})\ln{\left(\frac{1}{|\cos\theta_{1}^{i}\cos\theta_{2}^{i}|}\right)},$
(11)

$\frac{1}{\sigma}\frac{d\sigma}{dx_{\pm}}=\frac{1}{2}\left(1-\frac{C_{ij}\pm
C_{ji}}{2}x_{\pm}\right)\cos^{-1}|x_{\pm}|\quad\quad(\mbox{for}\,i\neq j),$
(12)

where
$x_{\pm}=\cos\theta_{1}^{i}\cos\theta_{2}^{j}\pm\cos\theta_{1}^{j}\cos\theta_{2}^{i}$.
In addition to eqs. (9), (10), four further observables $\cos\theta_{1,2}^{i}$
with respect to modified axes $\hat{k}^{*}$ and $\hat{r}^{*}$ are calculated.
Axes $\hat{k}^{*}$ and $\hat{r}^{*}$ for top quark are equal to
$sign(\Delta|y|)\hat{k}$ and $sign(\Delta|y|)sign(\cos\Theta)\hat{r}$,
respectively, while they are inverse for anti-top. Here
$\Delta|y|=|y_{t}|-|y_{\bar{t}}|$ is the difference in absolute values of the
rapidities of top and anti-top quarks in the laboratory frame. The observables
in modified axes may sensitive to the NP contributions 0h .

The opening angle $\cos\varphi$, which is obtained by the inner product of the
direction of flight of positively and negatively charged leptons in their own
top quark rest frames, is taken into account as a useful spin correlation
observable below. The analysis also involves two observables measured directly
in the laboratory-frame. First one is $\cos\varphi_{lab}$, which is defined as
$\cos\varphi$ but in the laboratory-frame. $|\Delta\phi_{ll}|$ is the other
observable measured in the laboratory-frame. It is the absolute azimuthal
difference between the two charged leptons measured in the laboratory-frame.
The observables measured in the laboratory-frame cannot be obtained from spin
correlation or polarization observables, but they could have high precision in
measurement and computation.

##  4 Results

In this section, the normalized differential distributions for the 22
observables mentioned in previous section are presented as a comparison of
various MC approaches and experimental measurements published by the CMS
Collaboration 0i . The physics objects in events predicted by MC generators
are considered at generator-level. In that sense, objects come from MC history
after decay but before hadronization. Last top quark in top quark decay chain
as a top quark and first lepton after decay of W boson as a lepton are used in
the calculations of observables. As leptons in dilepton events, prompt
electrons and muons (not taus or decay products of taus) are taken into
account.

Three different MC generators as MadGraph5_aMC@NLO, Sherpa and Powheg box at
LO and NLO QCD accuracy used in the analysis. As in the CMS analysis 0i , the
analysis with MC generators is in full phase space. As the nominal $t\bar{t}$
sample, we have been chosen the sample simulated by MadGraph5_aMC@NLO ME event
generator at NLO interfaced with Pythia8 for the parton shower and
fragmentation.

In figures 3-8, the differential distributions predicted by MadGraph5+Pythia8
at NLO is compared with MadGraph5+Pythia8 at LO accuracy, with Powheg+Pythia8
at NLO, with Sherpa at LO, and with Sherpa at NLO QCD accuracy and at NLO in
QCD with EW correction. To check the compatibility with experiment, whole that
theoretical predictions are compared with CMS measurements.

In figure 3, the diagonal spin correlation observables
$\cos\theta_{1}^{i}\cos\theta_{2}^{i}$ defined in eq. 11 are presented. There
are no significant deviations in different MC approaches and different QCD
corrections as well as EW correction. The experimental uncertainty comprise
almost all variations of predictions.

Figure 3: The normalized differential cross sections with respect to the
diagonal spin correlation observables $\cos\theta_{1}^{i}\cos\theta_{2}^{i}$,
$i=k,r,n$ compared to the CMS data and the predictions from MadGraph5_aMC@NLO,
Sherpa and Powheg box in full phase space. The ratio of the normalized
differential cross sections predicted by the generator+parton shower
configurations to the CMS data is shown in the lower panels. The statistical
uncertainty band is shown in grey. The yellow uncertainty bands represent the
overall uncertainty obtained by summing the systematic and statistical
uncertainties in quadrature. The systematic uncertainty includes PDF and scale
uncertainties.

The predictions for the differential distributions of the observables
$x_{\pm}$, eq. 12, are considered in figure 4. Non-zero coefficients
$(C_{nr}+C_{rn})$ and $(C_{nk}+C_{kn})$ are expected from mixed QCD-weak
corrections correspond to ref. 0h . Due to smallness of this effect, scale,
PDF and experimental uncertainties dominate it. Within the uncertainties, the
experimental data and predictions for cross spin correlation observables are
matched.

The normalized differential distributions in $\cos\theta_{1,2}^{i}$, eq. (9,
10), which is the indicator of the polarization of top quark and anti-quark
are shown in figure 5. The polarization coefficients for all predictions
behave similar in all distributions. CMS data and MC predictions are
compatible within uncertainties. In the same way, the normalized differential
distributions of $\cos\theta_{1,2}^{i*}$ predicted by MC, figure 6, are in
agreement with data.

Figure 4:  The normalized differential cross sections with respect to the
cross spin correlation observables
$\cos\theta_{1}^{i}\cos\theta_{2}^{j}\pm\cos\theta_{1}^{j}\cos\theta_{2}^{i}$,
$i\neq j$. The ratio panels compare the MC predictions to the CMS data.

Figure 5:  The normalized differential cross sections with respect to the
observables $\cos\theta_{1}^{i}$ and $\cos\theta_{2}^{i}$, $i=k,r,n$. The
ratio panels compare the MC predictions to the CMS data.

Figure 6:  The normalized differential cross sections with respect to the
observables $\cos\theta_{1}^{i*}$ and $\cos\theta_{2}^{i*}$, $i=k,r,n$. The
ratio panels compare the MC predictions to the CMS data.

The differential distributions in the observables, which are calculated with
kinematic variables of leptons in laboratory-frame, are in figure 7. When
$|\Delta\phi_{ll}|$ is considered with total uncertainty which is around
$6\%$, the shapes predicted by MadGraph5 and Sherpa ME generators are roughly
consistent. Sherpa and MadGraph5 at LO accuracy show better performance than
the samples simulated at NLO, this may be the result of real contributions at
NLO accuracy spoiling the back-to-back alignment of the top quarks. Powheg is
separated from the other MC predictions and the data about $6\%$ at low and
high $|\Delta\phi_{ll}|$ regions. For the distributions of
$\cos\varphi_{lab}$, all MC predictions and the data are in good agreement in
whole range, although the systematic uncertainties coming from PDF sets and
scales are more than $10\%$ in almost whole regions. The huge systematic
uncertainty as seen in the lower panel of the right plot in figure 7 is coming
from PDF variations in Sherpa samples.

In $\cos\varphi$ distribution, figure 8, the predictions of MC generators are
more similar compared to laboratory-frame distributions. While the
distributions of different MC variations have considerably the same shape,
Sherpa with NLO QCD accuracy and EW correction deviate from the others,
especially at low $\cos\varphi$, but are still in total uncertainty.

In second part of the analysis, we have investigated various new physics
models with the help of the elements of spin density matrix. For top quark
pair channel, most searches for SUSY and DM particles comprise generally high
missing transverse energy $E_{T}^{miss}$ region stem from undetected LSP and
DM 1a ; 1b ; 0e ; 1d . The main point for the BSM analysis in this study is
the models with low $E_{T}^{miss}$ but also the same models with high
$E_{T}^{miss}$, which are not excluded by CMS and ATLAS Collaborations yet,
will be mentioned for the sake of broad phase space.

Figure 7:  The normalized differential cross section as a function of
laboratory frame observable which are $|\Delta\phi_{ll}|$ (left plot) and
$\cos\varphi_{lab}$ (right plot). The ratio panels compare the MC predictions
to the CMS data. Figure 8:  The normalized differential cross section with
respect to $\cos\varphi$ predicted by various MC event generators at LO, NLO
and NLO-EW accuracy. The ratio of the MC predictions to the CMS data is shown
in the lower panel.

Figure 9:  The normalized differential cross sections with respect to the
diagonal spin correlation observables $\cos\theta_{1}^{i}\cos\theta_{2}^{i}$,
$i=k,r,n$. The ratio panels compare the BSM predictions to the CMS data.

Figure 10:  The normalized differential cross sections with respect to the
cross spin correlation observables
$\cos\theta_{1}^{i}\cos\theta_{2}^{j}\pm\cos\theta_{1}^{j}\cos\theta_{2}^{i}$,
$i\neq j$. The ratio panels compare the BSM signals to the CMS data.

Figure 11:  The normalized differential cross sections with respect to the
observables $\cos\theta_{1}^{i}$ and $\cos\theta_{2}^{i}$, $i=k,r,n$. The
ratio panels compare the BSM signals to the CMS data.

As mentioned earlier, we take into account three different BSM signal
topologies to compare with SM predicted results and the data. The details of
the event samples of the signals can be seen in section 1. The SUSY samples
were simulated as unpolarized. To check importance of it, the R-Parity
conserved SUSY sample is determined as two separate configurations having
polarized stop quarks. To polarize the $\tilde{t}$ as fully left- (SUSY LH)
and right-handed (SUSY RH), the distributions in the SUSY sample is multiplied
by a weight obtained from the reweighting algorithm detailed in af . The
second signal is that DM model having scalar or pseudoscalar mediator
associated with top quark pair. Within this model, the collision process has
missing transverse energy $E_{T}^{miss}$ as in the model above. For both
sample, the study is focused on a narrow phase space in which mass difference
between $\tilde{t}$ and $\tilde{\chi}_{1}^{0}$ is slightly bigger than top or
the mediator mass is relatively small, in this case 10 GeV. To ensure the
larger phase space in the $\tilde{t}$ mass, the R-Parity violating (RPV) SUSY
is included in the analysis framework. The mechanism of the R-Parity violation
gives us three quarks decayed from $\tilde{\chi}_{1}^{0}$. Thus $E_{T}^{miss}$
in the $\tilde{t}$ pair production consists of only neutrinos from top quark
decay chain. The following results for RPV SUSY are performed with a signal
point of $m_{\tilde{t}}=400$ GeV. Furthermore, $\tilde{t}$ masses from $300$
GeV to $1000$ GeV for RPV SUSY were analyzed and all mass points appear
similarly.

Figure 12:  The normalized differential cross sections with respect to the
observables $\cos\theta_{1}^{i*}$ and $\cos\theta_{2}^{i*}$, $i=k,r,n$. The
ratio panels compare the BSM signals to the CMS data.

The distributions of spin correlation and polarization observables are
presented in figures 9-14. The figures involve related BSM signals, CMS data
and the nominal $t\bar{t}$ sample with systematic and statistical
uncertainties calculated in the previous section. In the lower panels, the
statistical uncertainties of the signals are specified on each signal sample.

The differential distributions of the diagonal spin correlation observables in
BSM signals are shown in figure 9. Almost all signals for
$\cos\theta_{1}^{k}\cos\theta_{2}^{k}$ and
$\cos\theta_{1}^{n}\cos\theta_{2}^{n}$ observables deviate up to $40\%$ from
SM nominal $t\bar{t}$ predictions and the CMS data. For
$\cos\theta_{1}^{r}\cos\theta_{2}^{r}$, the difference between signals and the
nominal sample can be seen despite the domination of the uncertainties. It is
seen that the scalar-mediated DM signal apart from the others has a non-zero
diagonal spin correlation coefficient.

Figure 13:  The normalized differential cross section as a function of
laboratory frame observable which are $|\Delta\phi_{ll}|$ (left plot) and
$\cos\varphi_{lab}$ (right plot).The ratio panels compare the BSM signals to
the CMS data. Figure 14:  The normalized differential cross section with
respect to $\cos\varphi$. The ratio panel compares the BSM signals to the CMS
data.

In figure 10, the distributions of the observables $x_{\pm}$ defined in
equation 12 are discussed for signals with the nominal $t\bar{t}$ sample and
the CMS data. The distribution in the upper left corner has a difference
between signals and nominal sample because the coefficient of this observable
is generated by P and CP conserving interactions as in the diagonal spin
coefficients. For the other cross spin correlation observables, the
distributions of signals and the nominal $t\bar{t}$ sample are compatible with
the experimental data when considered total uncertainty.

The observables in equations (9, 10) are not sensitive to small amount of top
quark polarization predicted in both BSM and SM as seen in figures 11-12. The
distributions are very flat for all predictions and the data. Furthermore,
polarized stops do not have any effect on the distributions as shown in fully
left and right-handed separated SUSY samples.

Figure 15:  Deviations of the MC $t\bar{t}$ predictions from the CMS data for
all observables as a matrix. Values correspond to differences in per cent
between each bin of the rebinned distributions of the predictions and the
data, shown in figures 3-8. Figure 16:  Deviations of the BSM signals from the
CMS data for all observables as a matrix. Values correspond to differences in
per cent between each bin of the rebinned distributions of the signals and the
data, shown in figures 9-14.

The laboratory frame observables are indicator of presence of spin
correlations as in P- and CP-even observables. Especially the
$|\Delta\phi_{ll}|$ distribution in figure 13 reveals this with the difference
between BSM and SM. Moreover, it can be seen that the dark matter model with
scalar mediator has spin correlation though not as much as SM. In the
$\cos\varphi_{lab}$ with high systematic uncertainties, some BSM models act as
in the SM while others deviate much more.

The last spin correlation sensitive observable is that the inner product of
flight direction of two leptons in its own top quark center of mass frame
($\cos\varphi$). In the figure 14, the sensitivity to spin correlation is seen
in the DM with scalar mediator while not in the other BSM samples.

To get a general picture, all distributions of various MC event generator
configurations, BSM signals and the CMS data for all observables presented in
figures 3-14 are summarized as heat maps in figures 15-16. In these figures,
each normalized distribution of each configuration is used as reduced bins
from $6$ to $2$ to observe the degree of deviation of the distributions.
Values on each cell in figures 15-16 indicate the percentage difference
between rebinned normalized distributions of $t\bar{t}$ predictions or BSM
signals and of the data. In figure 15, it can be explicitly seen that all MC
event generator configurations explain the data, within total uncertainty, for
distributions of the observables extracted from the spin density matrix.
However, $\cos\varphi$ and laboratory-frame observables ($|\Delta\phi_{ll}|$
and $\cos\varphi_{lab}$) show deviations of up to $5.6\%$ in NLO QCD accuracy
in Sherpa, as well as the inclusion of EW correction. The positive or negative
deviation (in percentage) of rebinned normalized distributions of the BSM
signals from the data is shown in figure 16. The first two columns indicate,
for each bin, absolute total uncertainty (statistical and systematic sum)
calculated in each of the above distributions. This table is a beneficial tool
to see the degree of deviation of the BSM signals from the experimental data
in a particular region of each observable. With this sense, it is clearly seen
that the $\cos\theta_{1}^{k}\cos\theta_{2}^{k}$,
$\cos\theta_{1}^{n}\cos\theta_{2}^{n}$ and
$\cos\theta_{1}^{r}\cos\theta_{2}^{k}+\cos\theta_{1}^{k}\cos\theta_{2}^{r}$
observables as elements of the spin density matrix are sensitive to the
considered BSM signals. In addition, all BSM signals for $\cos\varphi$ and
$|\Delta\phi_{ll}|$ seem to differ significantly from the experiment. Finally,
it appears that the left or right-handed polarization of the stop quarks does
not cause any notable change for these observables, whereas the DM signal with
the scalar mediator has more spin correlation between the top quarks compared
to the pseudoscalar mediator.

##  5 Conclusions and Outlook

In this work, we have made for the first time a comprehensive analysis of the
spin correlation and polarization observable set and the direct angle-related
observables between leptons in the $t\bar{t}$ production with dileptonic final
states at $13$ TeV. This has provided us to investigate top quark pairs
advanced properties for proton-proton collisions by using various MC event
generator methods at LO, NLO QCD accuracy and NLO QCD with EW corrections in
detail. For the distributions of the set of observables at the full phase
space, we have compared our results with the unfolded data performed by CMS
collaboration. Our study is mainly composed of two parts: the first is to
compare different MC configurations with the experimental data and determine
their effects on each variable, and the second is to find the degree of
deviation of $\tilde{t}\tilde{t}$ and DM signals from the data for each spin
correlation observable. For a clear view of deviations from the data, we have
created two heat map tables of SM $t\bar{t}$ and BSM samples with the set of
observables, in figures 15-16.

Considering the different theoretical approaches, the polarization observables
(including terms with modified axes) and the cross terms of the spin
correlation observables are in agreement within the systematic and statistical
uncertainty (around the $1.5\%$). Especially samples of Sherpa at NLO accuracy
(and with electroweak corrections) for diagonal terms in the spin correlation
observables tend to have more deviation (from $1.4\%$ to $2.5\%$) compared to
the others (almost all under $1\%$) but they are still in total uncertainty.
The biggest contribution to the uncertainty comes from the PDF variations of
predictions at LO accuracy. Separation might be seen better in NLO
calculations due to the less dependency of higher order calculations to the
PDF variations. Furthermore, to be more accurate in predictions, it is needed
to increase the statistics in experimental data, and upcoming LHC runs could
provide this. The behavior of the different configurations can be clearly seen
in the $\cos\varphi$ and $|\Delta\phi_{ll}|$ distributions, the observables
most sensitive to spin correlation in the top quark pair. The deviations up to
$5.6\%$ in Sherpa samples (NLO and NLO-EW) for these observables, in figure
15, could be arisen from its approximation in hard and large angle emissions.

Figure 16 provides a complete map of the BSM signals according to the
experiment, in the phase space of the related observables. Polarization and
cross spin correlation observables (except
$\cos\theta_{1}^{r}\cos\theta_{2}^{k}+\cos\theta_{1}^{k}\cos\theta_{2}^{r}$)
are compatible for the BSM signals and the data, while the signals have good
separation for P- and CP-even observables (up to $10\%$). Only the DM sample
with a scalar mediator has spin correlation, unlike other BSM signals, but the
spin correlation power in its top-quark pair is not as much as in the SM.
Furthermore, polarized stop quarks of different composition have no impact
observed on top quark spin correlation.

The different MC event generator configurations are compatible with each
other, within total uncertainty, for the observables of spin correlation and
polarization in the top quark pair. However, for some observables as mentioned
above, there are small separations dominated by systematic uncertainties.
Next-to-next-to leading order (NNLO) calculations can make these differences
more visible by reducing systematic uncertainties, or they can reduce the
difference between configurations by making more accurate calculations with
higher-order ME calculations. Thereby, this gets NNLO calculations for
theoretical predictions more important. Additionally, the observables may be
used, especially in low missing transverse energy region, for searching BSM
signals due to their sensitivity of contributions from parity and/or CP
conserving or violating interactions.

###### Acknowledgements.

We thank Alexander Josef Grohsjean for discussions on spin correlation
analysis, and Gurpreet Singh Chahal for contributions about Sherpa
productions. This work is supported by Istanbul Technical University Research
Fund under grant number TGA-2020-42345 and TUBITAK grant 121F065.

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