# Breaking down the entire spectrum of spin correlations of a pair of
particles involving fermions and gauge bosons

Rafiqul Rahaman<EMAIL_ADDRESS>Regional Centre for Accelerator-
based Particle Physics, Harish-Chandra Research Institute, A CI of Homi Bhabha
National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India  Ritesh K.
Singh<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research Kolkata, Mohanpur, 741246, India

(HRI-RECAPP-2021-010)

###### Abstract

We discuss a formalism for the spin correlations and polarizations in two-
particle systems with spins half-half, half-one and one-one, and provide the
connections between the polarizations and correlations with the joint angular
distributions of decay products identifying the asymmetries for them. We
demonstrate the formalism in the processes $e^{-}e^{+}\to t\bar{t}$, $gb\to
tW^{-}$ and $e^{-}e^{+}\to ZZ$ in the standard model as examples. We
investigate the effect of some anomalous couplings on the polarizations and
spin correlations in the processes $e^{-}e^{+}\to t\bar{t}$, $gb\to tW^{-}$
and $u\bar{d}\to ZW^{+}$ at parton level and compare their strengths. The spin
correlations have the potential to provide a significant improvement over the
polarizations in probing the anomalous couplings.

##  1 Introduction

The scaler sector of the standard model (SM) particle spectrum will be
affected by possible new physics beyond the SM (BSM) needed to address
phenomena, such as dark matter, neutrino oscillation, baryogenesis etc. These
BSM physics will lead to a modified electroweak sector with modified
interactions among the Higgs bosons, top quark, and the gauge bosons. Precise
measurement of the interactions in terms of strength and tensorial structure,
thus, may uncover new physics at colliders.

Polarizations of top quark and massive gauge bosons are interesting tools for
such precision measurement, as they are sensitive to the modification of the
interactions involved. The top quark, $Z$ and $W$ bosons being massive, they
decay immediately after they are produced, letting their decay products carry
their polarization as well as spin correlation information. In general a
spin-$s$ particle contains $(2s+1)^{2}-1=4s(s+1)$ polarization parameters,
e.g., a spin-$1/2$ particle has $3$ vector polarizations, and a spin-$1$
particle carries $5$ tensor polarizations along with $3$ vector polarizations
[1, 2]. These polarization parameters can be calculated from the production
process as well as from the angular distributions of the decay products [2,
3]. Like the polarizations, a system of two particles $A$ and $B$ with spins
$s_{A}$ and $s_{B}$ contains $4s_{A}(s_{A}+1)\times 4s_{B}(s_{B}+1)$ spin
correlation parameters along with $4s_{A}(s_{A}+1)+4s_{B}(s_{B}+1)$
polarization parameters. These spin correlation parameters can also be
calculated from the production process of the two-particle system as well as
from the joint angular distributions of their decay products.

There have been a lot of interest in top quark polarizations [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32] along with spin correlations of $t\bar{t}$ system [33, 34, 35,
36, 37, 38, 39, 40, 41, 20, 42, 43, 44, 45, 46] in probing new physics. The
top quark polarizations and spin correlations have also been measured at the
Tevatron [47, 48, 49, 50, 51, 52, 53] as well as at the large hadron collider
( LHC) [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. Lately, polarization for
$Z$ and $W$ bosons also got some attention in probing new physics [65, 3, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. The predictions of $Z$ and $W$
polarizations are also being developed within the SM including next-to-leading
order effects [77, 78, 79, 80, 81, 82]. The polarizations of $Z$ and $W$ have
also been measured recently at the LHC in $ZW$ production process [83, 84].
The decay angular correlations, in case of vector boson pair production, are
discussed before in Refs. [85, 86, 87, 88]. For the spin correlation, although
the $t\bar{t}$, i.e., half-half spin system, correlation have been talked
about before, only a subset of the full nine correlations are discussed.
Furthermore, although spin-correlated spin density matrix (SDM) and decay
distribution with correlations have been discussed earlier for the $t\bar{t}$
pair production [89, 90, 91, 92, 93, 94], no clear connection between them has
been made before. For the $tV$ ($V=Z/W$) or $VV$ pair production, i.e., half-
one or one-one spin systems, there exist no formalism for the spin correlated
production matrix or SDM, and thus no existence of a connection between the
decay distributions and spin correlations. In this article, we present a
formalism for the spin correlated density matrix and the connection between
joint decay angular distributions and the full-rank spin correlations in three
spin systems, namely half-half, half-one, and one-one. The presented formalism
is validated in some simple SM processes comparing the analytically obtained
values for the spin correlations to those numerically extracted values from
the angular distributions of the decay products obtained in Monte-Carlo
simulation. The potential of the spin correlations is investigated in probing
anomalous couplings, parameterized by effective operators, in some simple
partonic processes without realistic effects to make an ansatz for real
scenarios with collider data.

We proceed with a general description of production and decay of two spin full
particles in section 2 followed by discussing the individual polarizations of
spin-$1/2$ and spin-$1$ particles in section 2.1. Next, we present the
formalism for spin correlation for the three spin systems, i.e., half-half,
half-one, and one-one successively in section 2.2. In section 2.4, we
demonstrate the formalism in the SM by showing the agreement between the
analytically calculated correlations from the production process and the
numerically calculated values from MadGraph5_aMC@NLO [95] simulations using
decay distributions in three processes $e^{-}e^{+}\to t\bar{t}$, $gb\to
tW^{-}$ and $e^{-}e^{+}\to ZZ$ as a proof of principle. In section 3, we
discuss how spin correlations can help in probing new physics in parallel to
polarization in three processes $e^{-}e^{+}\to t\bar{t}$, $gb\to tW^{-}$ and
$u\bar{d}\to ZW^{+}$ at parton level and without initial state of parton
distribution functions (PDFs) folding, for simplicity. We summarize in section
4.

##  2 Formalism

Figure 1:  Schematic diagram for the production of two particles $A$ and $B$
along with bunch of other particles $C_{1},~{}C_{2},\ldots,~{}C_{N}$ by the
collision of beam particles $B_{1}$ and $B_{2}$. The particles $A$ and $B$
decay subsequently as $A\to aa^{\prime}$ and $B\to bb^{\prime}$. The dashed
lines are shown to separate the production part from the decay parts.

To describe the spin correlations along with spin-polarizations of a system of
two spin-full particles, let us consider the production and decay of two
massive spin-full unstable particles $A$ and $B$ in a general process
$B_{1}B_{2}\to ABC_{1}C_{2}\ldots C_{N}$ with $A\to aa^{\prime}$ and $B\to
bb^{\prime}$, as shown in Fig. 1. The differential rate for such a process can
be expressed as [2],

$\displaystyle
d\sigma={{\sum}}_{\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}}$
$\displaystyle\left[\frac{1}{I_{B_{1}B_{2}}}\rho_{AB}^{\prime}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})(2\pi)^{4}\delta^{4}\left(p_{B_{1}}+p_{B_{2}}-p_{A}-p_{B}-\sum_{i=1}^{N}p_{C_{i}}\right)\right.$
$\displaystyle\times$
$\displaystyle\left.\left(\dfrac{d^{3}p_{A}}{(2\pi)^{3}2E_{A}}\right)\left(\dfrac{d^{3}p_{B}}{(2\pi)^{3}2E_{B}}\right)\prod_{i=1}^{N}\left(\dfrac{d^{3}p_{C_{i}}}{(2\pi)^{3}2E_{C_{i}}}\right)\right]$
$\displaystyle\times$
$\displaystyle\left[\frac{1}{\Gamma_{A}}\frac{(2\pi)^{4}}{2m_{A}}\Gamma_{A}^{\prime}(\lambda_{A},\lambda_{A}^{\prime})\delta^{4}\left(p_{A}-p_{a}-p_{a^{\prime}}\right)\left(\dfrac{d^{3}p_{a}}{(2\pi)^{3}2E_{a}}\right)\left(\dfrac{d^{3}p_{a^{\prime}}}{(2\pi)^{3}2E_{a^{\prime}}}\right)\right]$
$\displaystyle\times$
$\displaystyle\left[\frac{1}{\Gamma_{B}}\frac{(2\pi)^{4}}{2m_{B}}\Gamma_{B}^{\prime}(\lambda_{B},\lambda_{B}^{\prime})\delta^{4}\left(p_{B}-p_{b}-p_{b^{\prime}}\right)\left(\dfrac{d^{3}p_{b}}{(2\pi)^{3}2E_{b}}\right)\left(\dfrac{d^{3}p_{b^{\prime}}}{(2\pi)^{3}2E_{b^{\prime}}}\right)\right]$

in the narrow-width approximation of the unstable particles $A$ and $B$,
allowing the factorization of production part (in the first square bracket)
and the decay parts (in the second and third square brackets). Here, $p_{X}$,
$m_{X}$ and $E_{X}$ with $X\in\\{B_{i},A,B,C_{i},a,a^{\prime},b,b^{\prime}\\}$
are the four-momenta, mass and energy of the particles, respectively; the flux
factor $I_{B_{1}B_{2}}$ is given by
$I_{B_{1}B_{2}}=4\sqrt{\left(p_{B_{1}}\cdot
p_{B_{2}}\right)^{2}-m_{B_{1}}^{2}m_{B_{2}}^{2}}$ with $m_{B_{i}}$ the mass of
the particles $B_{i}$; $\Gamma_{A/B}$ are the total decay width of $A/B$. The
$\lambda_{A/B}$, $\lambda_{A/B}^{\prime}$ are the helicities of $A/B$ and they
can take values in the range of

$\lambda_{A/B},\lambda_{A/B}^{\prime}\in\left[-s_{A/B},-s_{A/B}+1,\ldots,s_{A/B}\right]$

with $s_{A}$ and $s_{B}$ as the spins of the particle $A$ and $B$,
respectively. The helicities of all particles other than $A$ and $B$ are
suppressed, i.e., helicities are summed over the $C_{i}$ and averaged over the
$B_{i}$***For polarized beams, one has to use the initial state polarization
density matrix in Eq. (2), e.g., see Ref. [66].. The phase space integration
of the decay products of $A$ and $B$ can be performed in the rest of $A$ and
$B$ separately without any loss of generality. The production part or the
production density matrix of $A$ and $B$ can be expressed as,

$\displaystyle\rho_{AB}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{I_{B_{1}B_{2}}}\int\rho_{AB}^{\prime}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})(2\pi)^{4}\delta^{4}\left(p_{B_{1}}+p_{B_{1}}-p_{A}-p_{B}-\sum_{i=1}^{N}p_{C_{i}}\right)$
(2) $\displaystyle\times$
$\displaystyle\left(\dfrac{d^{3}p_{A}}{(2\pi)^{3}2E_{A}}\right)\left(\dfrac{d^{3}p_{B}}{(2\pi)^{3}2E_{B}}\right)\prod_{i=1}^{N}\left(\dfrac{d^{3}p_{C_{i}}}{(2\pi)^{3}2E_{C_{i}}}\right)$

with

$\rho_{AB}^{\prime}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})={\cal
M}(\lambda_{A},\lambda_{B})\times{\cal
M}^{\dagger}(\lambda_{A}^{\prime},\lambda_{B}^{\prime}),$

${\cal M}(\lambda_{A},\lambda_{B})$ being the helicity matrix amplitude with
helicities $\lambda_{A}$ and $\lambda_{B}$. The cross section for the
production of $A$ and $B$ would be,

${}\sigma_{p}=\text{Tr}\left[\rho_{AB}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})\right]=\sum_{\lambda_{A},\lambda_{B}}\rho_{AB}(\lambda_{A},\lambda_{A},\lambda_{B},\lambda_{B}).$
(3)

Here, only the diagonal elements of the
$(2s_{A}+1)(2s_{B}+1)\times(2s_{A}+1)(2s_{B}+1)$ dimensional production
density matrix $\rho_{AB}$ enter, while all the elements in a certain
combination contain polarization and spin correlation information of $A$ and
$B$. Thus the normalized production density matrix can be compared to the spin
density matrix of the system $A$ and $B$. We rewrite,

$\rho_{AB}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})=\sigma_{p}\times
P_{AB(2s_{A},2s_{B})}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})$
(4)

with
$P_{AB(2s_{A},2s_{B})}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})$
as the normalized production density matrix, we call it polarization-
correlation density matrix. The decay part of $A$, after partial phase-space
integration, can be expressed as,

$\displaystyle{{\int}}\frac{1}{\Gamma_{A}}\frac{(2\pi)^{4}}{2m_{A}}\Gamma_{A}^{\prime}(\lambda_{A},\lambda_{A}^{\prime})\delta^{4}\left(p_{A}-p_{a}-p_{a^{\prime}}\right)\left(\dfrac{d^{3}p_{a}}{(2\pi)^{3}2E_{a}}\right)\left(\dfrac{d^{3}p_{a^{\prime}}}{(2\pi)^{3}2E_{a^{\prime}}}\right)$
(5) $\displaystyle=$ $\displaystyle\frac{Br\left(A\to
aa^{\prime}\right)(2s_{A}+1)}{4\pi}\Gamma_{A(2s_{A})}(\lambda_{A},\lambda_{A}^{\prime})d\Omega_{a/a^{\prime}},$

where $Br\left(A\to aa^{\prime}\right)$ is the branching fraction of $A$
decaying to $aa^{\prime}$. The quantity
$\Gamma_{A}(\lambda_{A},\lambda_{A}^{\prime})$ is the decay density matrix
normalized to unit trace, and $d\Omega_{a}=\sin\theta_{a}d\theta_{a}d\phi_{a}$
is the measure of solid angle of the daughter $a$. The decay part of $B$ can
also be expressed in the same way as done for $A$. Combining the production
density matrix of $A$ and $B$ in Eq. (2) and their decay density matrices in
Eq. (5) according to Eq. (2), one obtains the normalized joint angular
distribution of the decay products as,

$\displaystyle\dfrac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}=\frac{2s_{A}+1}{4\pi}\frac{2s_{B}+1}{4\pi}{{\sum}}_{\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}}$
$\displaystyle
P_{AB(2s_{A},2s_{B})}(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime})$
(6)
$\displaystyle\times\Gamma_{A(2s_{A})}(\lambda_{A},\lambda_{A}^{\prime})\times\Gamma_{B(2s_{B})}(\lambda_{B},\lambda_{B}^{\prime}),$

where $\sigma=\sigma_{p}\times Br\left(A\to aa^{\prime}\right)\times
Br\left(B\to bb^{\prime}\right)$ is the total cross section of the production
of $A$ and $B$ followed by their decays.

The polarizations of $A$ and $B$ and their spin correlations, embedded into
their polarization-correlations density matrix $P_{AB(2s_{A},2s_{B})}$, are
transferred to the distributions of their decay products $a$ and $b$. One can
obtain the polarizations and spin correlations from the production part as
well as form the joint angular distributions of the decay products. In this
work, we consider a system with a pair of particles combining only with
spin-$1/2$ and spin-$1$ particles, i.e., half-half, half-one, and one-one spin
systems. In the next sections, we describe how the spin correlations along
with the polarizations of two particles in these systems can be computed from
the production part as well as from the decay distributions. First, we briefly
present the scenarios with a single spin-$1/2$ and spin-$1$ particles in the
next section for completeness.

###  2.1 Polarizations of a single particle

The polarization density matrix or the spin-density matrix (SDM) of a spin-$s$
particle can be expressed with the irreducible spin tensors up to rank $2s$,
i.e., identity matrix, linear ( for spin-$1/2$ and spin-$1$), bilinear (only
for spin-$1$) combinations of standard spin matrices. The SDM can be
represented in terms of multi-pole parameters or can be given in Cartesian
form [1]. The properties of the density matrix will then be specified by the
expansion coefficients. In Cartesian form, the polarization density matrix of
spin-$1/2$ and spin-$1$ particles can be expressed as [1, 2],

$\displaystyle P_{f(1)}(\lambda,\lambda^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\mathbb{I}_{2\times
2}+\vec{p}\cdot\vec{\tau}\Big{]},~{}~{}\lambda,\lambda^{\prime}\in[+1,-1],$
(9) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\begin{tabular}[]{cc}$1+p_{z}$&$p_{x}-ip_{y}$\\\
$p_{x}+ip_{y}$&$1-p_{z}$\end{tabular}\right]~{}~{}~{}~{}~{}~{}~{}\text{and}$
$\displaystyle P_{V(2)}(\lambda,\lambda^{\prime})$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\left[\mathbb{I}_{3\times
3}+\dfrac{3}{2}\vec{p}.\vec{S}+\sqrt{\dfrac{3}{2}}T_{ij}\big{(}S_{i}S_{j}+S_{j}S_{i}\big{)}\right],~{}~{}\lambda,\lambda^{\prime}\in[+1,0,-1],$
(13) $\displaystyle=$
$\displaystyle\left[\begin{tabular}[]{lll}$\frac{1}{3}+\frac{p_{z}}{2}+\frac{T_{zz}}{\sqrt{6}}$&$\frac{p_{x}-ip_{y}}{2\sqrt{2}}+\frac{T_{xz}-iT_{yz}}{\sqrt{3}}$&$\frac{T_{xx}-T_{yy}-2iT_{xy}}{\sqrt{6}}$\\\
$\frac{p_{x}+ip_{y}}{2\sqrt{2}}+\frac{T_{xz}+iT_{yz}}{\sqrt{3}}$&$\frac{1}{3}-\frac{2T_{zz}}{\sqrt{6}}$&$\frac{p_{x}-ip_{y}}{2\sqrt{2}}-\frac{T_{xz}-iT_{yz}}{\sqrt{3}}$\\\
$\frac{T_{xx}-T_{yy}+2iT_{xy}}{\sqrt{6}}$&$\frac{p_{x}+ip_{y}}{2\sqrt{2}}-\frac{T_{xz}+iT_{yz}}{\sqrt{3}}$&$\frac{1}{3}-\frac{p_{z}}{2}+\frac{T_{zz}}{\sqrt{6}}$\end{tabular}\right],$

respectively. Repeated indices are summed over. Here, $\tau_{i}$ and $S_{i}$
are the spin matrices in spin-$1/2$ and spin-$1$ basis (given in appendix A),
respectively. The $p_{i}$s, components of a three vector
$\vec{p}=\\{p_{x},p_{y},p_{z}\\}$, are vector polarizations of the particles;
$T_{ij}$, components of a second-rank symmetric trace-less tensor, are tensor
polarizations of the spin-$1$ particle. For a spin-$1/2$ particle, there are
three vector polarizations, while for a spin-$1$ particle, there are three
vector and five independent tensor polarizations. These polarization
parameters can be estimated by comparing the polarization density matrices to
the respective normalized production density matrices. Combining the
polarization density matrices with their respective normalized decay density
matrices obtained at their rest frame (given in appendix A in Eqs. (63) and
(64) ), one obtains the angular distributions of decay products for $f(1)$ and
$V(2)$ as [2],

$\frac{1}{\sigma_{f}}\frac{d\sigma_{f}}{d\Omega}=\frac{1}{4\pi}\Big{[}1+\alpha
p_{x}\sin\theta\cos\phi+\alpha p_{y}\sin\theta\sin\phi+\alpha
p_{z}\cos\theta\Big{]}~{}~{}~{}\text{and}$ (14)
$\displaystyle\frac{1}{\sigma_{V}}\frac{d\sigma_{V}}{d\Omega}$
$\displaystyle=$
$\displaystyle\frac{3}{8\pi}\left[\left(\frac{2}{3}-(1-3\delta)\
\frac{T_{zz}}{\sqrt{6}}\right)+\alpha\
p_{z}\cos\theta+\sqrt{\frac{3}{2}}(1-3\delta)\ T_{zz}\cos^{2}\theta\right.$
(15) $\displaystyle+$ $\displaystyle\left(\alpha\
p_{x}+2\sqrt{\frac{2}{3}}(1-3\delta)\ T_{xz}\cos\theta\right)\sin\theta\
\cos\phi$ $\displaystyle+$ $\displaystyle\left(\alpha\
p_{y}+2\sqrt{\frac{2}{3}}(1-3\delta)\ T_{yz}\cos\theta\right)\sin\theta\
\sin\phi$ $\displaystyle+$
$\displaystyle\left.(1-3\delta)\left(\frac{T_{xx}-T_{yy}}{\sqrt{6}}\right)\sin^{2}\theta\cos(2\phi)+\sqrt{\frac{2}{3}}(1-3\delta)\
T_{xy}\ \sin^{2}\theta\ \sin(2\phi)\right],$

respectively. Here, $\theta$ and $\phi$ are the polar and azimuthal angles of
the decay products in the rest frame of the mother particles with their would-
be momentum along $z$-direction. One can construct several asymmetries related
to the polarization parameters by partially integrating the angular
distributions with respect to $\theta$ and $\phi$. The complete list of
polarization asymmetries is given in Refs. [3, 96]. We will list them later
while discussing correlations of a pair of particles in the next section.

###  2.2 Spin correlations of a pair of particles

In the spin correlated density matrix of two spin-full particles, there are
correlations between two vector polarizations (vector-vector correlation),
correlation between a vector and a tensor polarization (vector-tensor
polarization) and correlation between two tensor polarizations (tensor-tensor
correlation) apart from the vector polarizations and tensor polarizations. We
proceed with, first, discussing the simple spin-$1/2$ – spin-$1/2$
correlations and then spin-$1/2$ – spin-$1$ and spin-$1$ – spin-$1$
subsequently.

####  2.2.1 Spin-$1/2$ – spin-$1/2$ correlations

The spin correlations for a pair of spin-$1/2$ particles can be accommodated
in the polarization-correlation density matrix with an outer product of two
Pauli spin-$1/2$ matrices ($\tau\otimes\tau$), one for each particle. Thus the
fully spin correlated polarization density matrix of a pair of spin-$1/2$
particles will be given by [89, 90]

$\displaystyle
P_{AB(1,1)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)=\frac{1}{\left(2\times\frac{1}{2}+1\right)^{2}}\Biggr{[}\underbrace{\mathbb{I}_{2\times
2}\otimes\mathbb{I}_{2\times 2}}_{\mathbb{I}_{4\times
4}}+\vec{p}^{A}\cdot\vec{\tau}\otimes\mathbb{I}_{2\times
2}+\mathbb{I}_{2\times 2}\otimes\vec{p}^{B}\cdot\vec{\tau}$
$\displaystyle+pp_{ij}^{AB}\tau_{i}\otimes\tau_{j}\Biggr{]},~{}\left(i,j\in[x\equiv
1,y\equiv 2,z\equiv 3]\right),$ (16)

where $\vec{p}^{A/B}$ are the vector polarizations of $A/B$; $pp_{ij}^{AB}$,
components of a second-rank tensor $pp^{AB}$, are the vector-vector
correlations of $A$ and $B$ †††Equation (2.2.1) can be extended for three
spin-half particles introducing a rank-three correlation parameter, let’s say
$ppp^{ABC}$, $C$ being the third spin-half particle, with a suitable outer
product of the spin matrices ($\tau$).. The elements of $pp^{AB}$ are
completely independent of each other. However, the $pp^{AB}$ tensor can be
symmetric if $A=B$, i.e., they are identical. Thus, there exists a total of
$3+3+9=15$ polarization and correlation parameters (see Table 1) in a
spin-$1/2$ – spin-$1/2$ pair of particles production.

Parameter | $p_{i}^{A}$ | $p_{i}^{B}$ | $pp_{ij}^{AB}$ | Total
---|---|---|---|---
Number of parameter | $3$ | $3$ | $9$ | $=15$
Table 1:  Number of polarization and spin correlation parameters in a
spin-$1/2$ – spin-$1/2$ pair of particles production.

The polarization ($p_{i}^{A}$, $p_{i}^{B}$) of $A$ and $B$ can be obtained
either from their individual production density matrices or from the combined
production density matrix of $A$ and $B$, while the vector-vector spin
correlations $pp_{ij}^{AB}$ can only be obtained from their combined
production density matrix. The expanded form of the spin correlated
polarization density matrix in Eq. (2.2.1) is given in appendix B. The
polarization and spin correlation parameters are also given in terms of the
production density matrix elements in appendix B.

After combining the normalized spin-$1/2$ decay density matrices (Eq. (63))
with the polarization-correlation density matrix
$P_{AB\left(\frac{1}{2},\frac{1}{2}\right)}$ (Eq. (2.2.1)) of $A$ and $B$ in
accordance with the Eq. (6), we obtain the joint angular distribution of the
decay products as,

$\displaystyle\dfrac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}$
$\displaystyle=$
$\displaystyle\frac{2}{4\pi}\frac{2}{4\pi}{{\sum}}_{\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}}P_{AB(1,1)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)\times\Gamma_{A(1)}\left(\lambda_{A},\lambda_{A}^{\prime}\right)\times\Gamma_{B(1)}\left(\lambda_{B},\lambda_{B}^{\prime}\right),$
(17) $\displaystyle=$
$\displaystyle\dfrac{1}{16\pi^{2}}\Big{[}1+\alpha_{A}p_{i}^{A}c_{i}^{a}+\alpha_{B}p_{i}^{B}c_{i}^{b}+\alpha_{A}\alpha_{B}pp_{ij}^{AB}c_{i}^{a}c_{j}^{b}\Big{]},$

where $c_{x}$, $c_{y}$ and $c_{z}$ are angular functions of the daughters,
i.e.,

$\displaystyle
c_{x}^{a}=\sin\theta_{a}\cos\phi_{a},~{}c_{y}^{a}=\sin\theta_{a}\sin\phi_{a},~{}c_{z}^{a}=\cos\theta_{a}.$
(18)

Here, the superscript $a$ on $c_{i}$ denote the daughter particle of the
particle $A$. One recovers Eq. (14), the angular distribution of a spin-$1/2$
particle production and decay, by integrating the angles of $a(b)$ in Eq.
(17), i.e.,

$\int_{\text{full}}d\Omega_{a(b)}\left(\dfrac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)=\dfrac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{b(a)}}.$
(19)

We know that individual polarizations $p_{i}^{A}$ and $p_{i}^{B}$ can be
obtained from some asymmetries made by partially integrating the angular
distributions [3, 96]. We write these asymmetries for $A$ in the following
way,

$\displaystyle{\cal A}[p_{1}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=-\frac{\pi}{2}}^{\frac{\pi}{2}}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{\pi}{2}}^{\frac{3\pi}{2}}\right)d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{1}}d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)=\frac{1}{2}\alpha_{A}p_{1}^{A},$
$\displaystyle{\cal A}[p_{2}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=0}^{\pi}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\pi}^{2\pi}\right)d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{2}}d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)=\frac{1}{2}\alpha_{A}p_{2}^{A},$
$\displaystyle{\cal A}[p_{3}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\frac{\pi}{2}}\int_{\phi_{a}=0}^{2\pi}-\int_{\theta_{a}=\frac{\pi}{2}}^{\pi}\int_{\phi_{a}=0}^{2\pi}\right)d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)$
(20) $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{3}}d\Omega_{a}\left(\frac{1}{\sigma}\frac{d\sigma}{d\Omega_{a}}\right)=\frac{1}{2}\alpha_{A}p_{3}^{A}$

to be used later for the asymmetry of spin correlations. The $\int
d\Omega_{b}$ integration is assumed to be performed in the above equation. The
asymmetries can also be represented as,

${\cal
A}[p_{i}^{A}]=\dfrac{\sigma\left(c_{i}^{a}>0\right)-\sigma\left(c_{i}^{a}<0\right)}{\sigma\left(c_{i}^{a}>0\right)+\sigma\left(c_{i}^{a}<0\right)}=\frac{1}{2}\alpha_{A}p_{i}^{A}$
(21)

for numerical purpose. For the particle $B$, the polarization asymmetries are
obtained by $A\to B,~{}a\to b$ in Eq. (2.2.1) and Eq. (21). The correlations
$pp_{ij}^{AB}$ can be obtained from some asymmetries made of partial
integration constructed from the individual partial integration in Eq. (2.2.1)
as,

$\displaystyle{\cal A}[pp_{ij}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{i}}d\Omega_{a}\int_{b}^{p_{j}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)=\frac{1}{4}\alpha_{A}\alpha_{B}pp_{ij}^{AB}.$
(22)

For example, the asymmetry for the correlation $pp_{12}^{AB}$ will be given
by,

$\displaystyle{\cal A}[pp_{12}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{1}}d\Omega_{a}\int_{b}^{p_{2}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right),$
$\displaystyle=$
$\displaystyle\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=-\frac{\pi}{2}}^{\frac{\pi}{2}}d\Omega_{a}\left[\left(\int_{\theta_{b}=0}^{\pi}\int_{\phi_{b}=-\frac{\pi}{2}}^{\frac{\pi}{2}}-\int_{\theta_{b}=0}^{\pi}\int_{\phi_{b}=\frac{\pi}{2}}^{\frac{3\pi}{2}}\right)d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)\right]$
$\displaystyle-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{\pi}{2}}^{\frac{3\pi}{2}}d\Omega_{a}\left[\left(\int_{\theta_{b}=0}^{\pi}\int_{\phi_{b}=-\frac{\pi}{2}}^{\frac{\pi}{2}}-\int_{\theta_{b}=0}^{\pi}\int_{\phi_{b}=\frac{\pi}{2}}^{\frac{3\pi}{2}}\right)d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)\right].$

For numerical purpose, the asymmetries for the spin correlations can also be
represented as,

$\displaystyle{\cal A}[pp_{ij}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\dfrac{\sigma\left(c_{i}^{a}c_{j}^{b}>0\right)-\sigma\left(c_{i}^{a}c_{j}^{b}<0\right)}{\sigma\left(c_{i}^{a}c_{j}^{b}>0\right)+\sigma\left(c_{i}^{a}c_{j}^{b}<0\right)}=\frac{1}{4}\alpha_{A}\alpha_{B}pp_{ij}^{AB}.$
(24)

In the next section, we discuss the formalism for spin-$1/2$ – spin-$1$
correlations assuming $B$ to be a spin-$1$ particle.

####  2.2.2 Spin-$1/2$ – spin-$1$ correlations

In the spin-$1/2$ – spin-$1$ case, apart from the vector-vector correlations,
we also have vector-tensor correlations, which we accommodate in the
polarization-correlation density matrix with an outer product between a Pauli
spin-$1/2$ matrix ($\tau$) and bilinear combination of the spin-$1$ matrices
$S$. Thus the full spin correlated polarization density matrix of a system of
spin-$1/2$ and spin-$1$ particles will be given by,

$\displaystyle
P_{AB\left(1,2\right)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(2\times\frac{1}{2}+1\right)}\frac{1}{\left(2\times
1+1\right)}\Big{[}\underbrace{\mathbb{I}_{2\times 2}\otimes\mathbb{I}_{3\times
3}}_{\mathbb{I}_{6\times
6}}+\vec{p}^{A}\cdot\vec{\tau}\otimes\mathbb{I}_{3\times
3}+\dfrac{3}{2}\mathbb{I}_{2\times 2}\otimes\vec{p}^{B}\cdot\vec{S}$ (25)
$\displaystyle+$ $\displaystyle\sqrt{\dfrac{3}{2}}\mathbb{I}_{2\times
2}\otimes
T_{ij}^{B}\big{(}S_{i}S_{j}+S_{j}S_{i}\big{)}+pp_{ij}^{AB}\tau_{i}\otimes
S_{j}+pT_{ijk}^{AB}\tau_{i}\otimes(S_{j}S_{k}+S_{k}S_{j})\Big{]},$
$\displaystyle\hskip 195.12767pt\left(i,j,k=x,y,z\right).$

Here, $p_{i}^{A/B}$ are the vector polarizations of $A/B$, $T_{ij}^{B}$ are
the tensor polarizations of $B$ and $pp_{ij}^{AB}$ are the vector-vector
correlations as appeared in the previous section in Eq. (2.2.1). The
$pT_{ijk}^{AB}$, representing vector-tensor correlations of $A$ and $B$, are
the components of a third-rank tensor $pT^{AB}$. The tensor $pT^{AB}$ is
symmetric and traceless under the last two indices, similar to the case of
$T^{B}$, i.e.,

$\displaystyle T_{(ij)}^{B}\equiv T_{ij}^{B}=T_{ji}^{B},~{}T_{ii}^{B}=0;$
$\displaystyle pT_{i(jk)}^{AB}\equiv
pT_{ijk}^{AB}=pT_{ikj}^{AB},~{}pT_{ijj}^{AB}=0.$ (26)

The parentheses $\left(\right)$ enclosing the indices $i,j$ represent that the
tensor is symmetric under these two indices. Here, we have a total of $3\times
5=15$ independent vector-tensor correlations apart from $9$ vector-vector
correlations, see Table 2.

Parameter | $p_{i}^{A}$ | $p_{i}^{B}$ | $T_{(ij)}^{B}$ | $pp_{ij}^{AB}$ | $pT_{i(jk)}^{AB}$ | Total
---|---|---|---|---|---|---
Number of parameter | $3$ | $3$ | $5$ | $9$ | $15$ | $=35$
Table 2:  Number of independent polarization and spin correlation parameters
in a spin-$1/2$ – spin-$1$ pair of particles production.

After combining the polarization density matrix in Eq. (25) with the
normalized decay density matrices in Eqs. (63) for $A$ and (64) for $B$ in
accordance with Eq. (6), we obtain the joint angular distribution of the decay
products as,

$\displaystyle\dfrac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}$
$\displaystyle=$
$\displaystyle\frac{2}{4\pi}\frac{3}{4\pi}{{\sum}}_{\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}}P_{AB\left(1,2\right)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)\times\Gamma_{A\left(1\right)}\left(\lambda_{A},\lambda_{A}^{\prime}\right)\times\Gamma_{B\left(2\right)}\left(\lambda_{B},\lambda_{B}^{\prime}\right),$
(27) $\displaystyle=$
$\displaystyle\dfrac{1}{16\pi^{2}}\Biggr{[}1+\alpha_{A}p_{i}^{A}c_{i}^{a}+\frac{3}{2}\alpha_{B}p_{i}^{B}c_{i}^{b}+\sqrt{\frac{3}{2}}(1-3\delta_{B})T_{ij}^{B}c_{i}^{b}c_{j}^{b}~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{3}{2}}(1-3\delta_{B})\underbrace{\left(T_{11}^{B}-T_{22}^{B}\right)}_{T_{11-22}^{B}}\underbrace{\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)}_{\sin^{2}\theta_{b}\cos(2\phi_{b})}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{3}{2}}(1-3\delta_{B})T_{33}^{B}\left(3(c_{3}^{b})^{2}-1\right)$
$\displaystyle+$
$\displaystyle\alpha_{A}\alpha_{B}pp_{ij}^{AB}c_{i}^{a}c_{j}^{b}$
$\displaystyle+$
$\displaystyle\alpha_{A}(1-3\delta_{B})pT_{ijk}^{AB}c_{i}^{a}c_{j}^{b}c_{k}^{b}~{}(j\neq
k)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\alpha_{A}(1-3\delta_{B})\underbrace{\left(pT_{i11}^{AB}-pT_{i22}^{AB}\right)}_{pT_{i(11-22)}^{AB}}c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\alpha_{A}(1-3\delta_{B})pT_{i33}^{AB}c_{i}^{a}\left(3(c_{3}^{b})^{2}-1\right)\Biggr{]}.$

Here, systematization of indices on $T$ and $pT$ are not included. The
independent vector-tensor correlations ($pT^{AB}$) are nine ($3\times 3$)
$pT_{i(jk)}^{AB}~{}(j\neq k)$, three $pT_{i(11-22)}^{AB}$, and three
$pT_{i33}$. We recover angular distributions in Eq. (14) for spin-$1/2$ and
Eq. (15) for spin-$1$ particle production and decay after integrating the
angles of $b$ and $a$ in Eq. (27), respectively. In this case, the asymmetries
for vector polarizations and vector-vector correlations are given by,

$\displaystyle{\cal A}[p_{i}^{A}]=\frac{1}{2}\alpha_{A}p_{i}^{A},~{}~{}{\cal
A}[p_{i}^{B}]=\frac{3}{4}\alpha_{B}p_{i}^{B},~{}~{}{\cal
A}[pp_{ij}^{AB}]=\frac{1}{4}\alpha_{A}\alpha_{B}pp_{ij}^{AB}.$ (28)

The asymmetries for tensor polarizations ($T$) of $B$ are given in appendix C
in Eqs. (76)-(81). The independent vector-tensor correlations
$pT_{i(jk)}^{AB}$ can be obtained from the following asymmetries,

$\displaystyle{\cal A}[pT_{i(jk)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{i}}d\Omega_{a}\int_{b}^{T_{jk}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(j\neq
k)$ (29) $\displaystyle=$
$\displaystyle\dfrac{\sigma\left(c_{i}^{a}c_{j}^{b}c_{k}^{b}>0\right)-\sigma\left(c_{i}^{a}c_{j}^{b}c_{k}^{b}<0\right)}{\sigma\left(c_{i}^{a}c_{j}^{b}c_{k}^{b}>0\right)+\sigma\left(c_{i}^{a}c_{j}^{b}c_{k}^{b}<0\right)}$
$\displaystyle=$
$\displaystyle\frac{2}{3\pi}\alpha_{A}(1-3\delta_{B})pT_{i(jk)}^{AB},$
$\displaystyle{\cal A}[pT_{i(11-22)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{i}}d\Omega_{a}\int_{b}^{T_{11-22}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(30) $\displaystyle=$
$\displaystyle\dfrac{\sigma\left(c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)>0\right)-\sigma\left(c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)<0\right)}{\sigma\left(c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)>0\right)+\sigma\left(c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)<0\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{3\pi}\alpha_{A}(1-3\delta_{B})pT_{i(11-22)}^{AB},$
$\displaystyle{\cal A}[pT_{i33}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{p_{i}}d\Omega_{a}\int_{b}^{T_{33}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(31) $\displaystyle=$
$\displaystyle\dfrac{\sigma\left(c_{i}^{a}\sin(3\theta_{b})>0\right)-\sigma\left(c_{i}^{a}\sin(3\theta_{b})<0\right)}{\sigma\left(c_{i}^{a}\sin(3\theta_{b})>0\right)+\sigma\left(c_{i}^{a}\sin(3\theta_{b})<0\right)}$
$\displaystyle=$
$\displaystyle\frac{3}{16}\alpha_{A}(1-3\delta_{B})pT_{i33}^{AB}.$

Here, the partial integration rules $\int_{b}^{T}$s are for the asymmetries
related to tensor polarizations and they are given in appendix C in Eqs.
(76)-(80). In the next section, we discuss the formalism for spin-$1$ –
spin-$1$ correlations considering $A$ also to be spin-$1$.

####  2.2.3 Spin-$1$ – spin-$1$ correlations

For spin-$1$ – spin-$1$ correlations, the polarization-correlation density
matrix contains tensor-tensor correlations apart from the vector-vector and
vector-tensor correlations. We accommodate the tensor-tensor correlations with
an outer product between two bilinear combinations of spin-$1$ matrices ($S$).
Thus the full spin correlated polarization density matrix for a pair of
spin-$1$ particles will be given by,

$\displaystyle
P_{AB(2,2)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\left(2\times
1+1\right)^{2}}\Bigg{[}\underbrace{\mathbb{I}_{3\times
3}\otimes\mathbb{I}_{3\times 3}}_{\mathbb{I}_{9\times
9}}+\dfrac{3}{2}\vec{p}^{A}\cdot\vec{S}\otimes\mathbb{I}_{3\times
3}+\dfrac{3}{2}\mathbb{I}_{3\times 3}\otimes\vec{p}^{B}\cdot\vec{S}$ (32)
$\displaystyle+$
$\displaystyle\sqrt{\dfrac{3}{2}}T_{ij}^{A}\big{(}S_{i}S_{j}+S_{j}S_{i}\big{)}\otimes\mathbb{I}_{3\times
3}+\sqrt{\dfrac{3}{2}}\mathbb{I}_{3\times 3}\otimes
T_{ij}^{B}\big{(}S_{i}S_{j}+S_{j}S_{i}\big{)}$ $\displaystyle+$ $\displaystyle
pp_{ij}^{AB}S_{i}\otimes
S_{j}+pT_{ijk}^{AB}S_{i}\otimes(S_{j}S_{k}+S_{k}S_{j})$ $\displaystyle+$
$\displaystyle Tp_{ijk}^{AB}(S_{i}S_{j}+S_{j}S_{i})\otimes S_{k}$
$\displaystyle+$ $\displaystyle
TT_{ijkl}^{AB}\big{(}S_{i}S_{j}+S_{j}S_{i}\big{)}\otimes\big{(}S_{k}S_{l}+S_{l}S_{k}\big{)}\Bigg{]},~{}\left(i,j,k,l=x,y,z\right).$

Here $TT_{ijkl}^{AB}$, components of a fourth-rank tensor, are the tensor-
tensor correlations. The $Tp_{ijk}^{AB}$ are tensor-vector correlations, which
are identical to $pT_{kij}^{BA}$. The $TT_{ijkl}^{AB}$ are symmetric and
traceless under the first two and last two indices separately, i.e.,

$\displaystyle TT_{(ij)(kl)}^{AB}\equiv
TT_{ijkl}^{AB}=TT_{ijlk}^{AB}=TT_{jikl}^{AB}=TT_{jilk}^{AB},~{}~{}TT_{(ij)kk}^{AB}=0=TT_{ii(jk)}^{AB},$
(33)

which leaves us $5\times 5=25$ independent tensor-tensor ($TT$) correlations,
see Table 3.

Parameter | $p_{i}^{A}$ | $p_{i}^{B}$ | $T_{(ij)}^{A}$ | $T_{(ij)}^{B}$ | $pp_{ij}^{AB}$ | $pT_{i(jk)}^{AB}$ | $pT_{i(jk)}^{BA}$ | $TT_{(ij)(kl)}^{AB}$ | Total
---|---|---|---|---|---|---|---|---|---
Number of parameter | $3$ | $3$ | $5$ | $5$ | $9$ | $15$ | $15$ | $25$ | $=80$
Table 3:  Number of independent polarization and spin correlation parameters
in a spin-$1$ – spin-$1$ pair of particles production.

After combining the polarization density matrix in Eq. (32) with the spin-$1$
normalized decay density matrix in Eq. (64) in accordance with Eq. (6), we
obtain the joint angular distribution of the decay products as,

$\displaystyle\dfrac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}$
$\displaystyle=$
$\displaystyle\frac{3}{4\pi}\frac{3}{4\pi}{{\sum}}_{\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}}P_{AB(2,2)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)\times\Gamma_{A(2)}\left(\lambda_{A},\lambda_{A}^{\prime}\right)\times\Gamma_{B(2)}\left(\lambda_{B},\lambda_{B}^{\prime}\right)$
(34) $\displaystyle=$
$\displaystyle\dfrac{1}{16\pi^{2}}\Bigg{[}1+\frac{3}{2}\alpha_{A/B}p_{i}^{A/B}c_{i}^{a/b}+\sqrt{\frac{3}{2}}(1-3\delta_{A/B})T_{ij}^{A/B}c_{i}^{a/b}c_{j}^{a/b}~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{3}{2}}(1-3\delta_{A/B})\underbrace{\left(T_{11}^{A/B}-T_{22}^{A/B}\right)}_{T_{11-22}^{A/B}}\underbrace{\left((c_{1}^{a/b})^{2}-(c_{2}^{a/b})^{2}\right)}_{\sin^{2}\theta_{a/b}\cos(2\phi_{a/b})}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{\frac{3}{2}}(1-3\delta_{A/B})T_{33}^{A/B}\left(3(c_{3}^{a/b})^{2}-1\right)$
$\displaystyle+$
$\displaystyle\alpha_{A}\alpha_{B}pp_{ij}^{AB}c_{i}^{a}c_{j}^{b}$
$\displaystyle+$
$\displaystyle\alpha_{A}(1-3\delta_{B})pT_{ijk}^{AB}c_{i}^{a}c_{j}^{b}c_{k}^{b}+\alpha_{B}(1-3\delta_{A})pT_{ijk}^{BA}c_{i}^{b}c_{j}^{a}c_{k}^{a}~{}(j\neq
k)$ $\displaystyle+$
$\displaystyle\frac{1}{2}\alpha_{A}(1-3\delta_{B})\underbrace{\left(pT_{i11}^{AB}-pT_{i22}^{AB}\right)}_{pT_{i(11-22)}^{AB}}c_{i}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\alpha_{B}(1-3\delta_{A})\underbrace{\left(pT_{i11}^{BA}-pT_{i22}^{BA}\right)}_{pT_{i(11-22)}^{BA}}c_{i}^{b}\Big{(}(c_{1}^{a})^{2}-(c_{2}^{a})^{2}\Big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\alpha_{A}(1-3\delta_{B})pT_{i33}^{AB}c_{i}^{a}\left(3(c_{3}^{b})^{2}-1\right)+\frac{1}{2}\alpha_{B}(1-3\delta_{A})pT_{i33}^{BA}c_{i}^{b}\left(3(c_{3}^{a})^{2}-1\right)$
$\displaystyle+$
$\displaystyle(1-3\delta_{A})(1-3\delta_{B})TT_{ijkl}^{AB}c_{i}^{a}c_{j}^{a}c_{k}^{b}c_{l}^{b}~{}(i\neq
j,~{}k\neq l)$ $\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\delta_{A})(1-3\delta_{B})\underbrace{\left(TT_{ij11}^{AB}-TT_{ij22}^{AB}\right)}_{TT_{ij(11-22)}^{AB}}c_{i}^{a}c_{j}^{a}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\delta_{A})(1-3\delta_{B})\underbrace{\left(TT_{11ij}^{AB}-TT_{22ij}^{AB}\right)}_{TT_{(11-22)ij}^{AB}}c_{i}^{b}c_{j}^{b}\Big{(}(c_{1}^{a})^{2}-(c_{2}^{a})^{2}\Big{)}~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\delta_{A})(1-3\delta_{B})TT_{ij33}^{AB}c_{i}^{a}c_{j}^{a}\left(3(c_{3}^{b})^{2}-1\right)~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{2}(1-3\delta_{A})(1-3\delta_{B})TT_{33ij}^{AB}c_{i}^{b}c_{j}^{b}\left(3(c_{3}^{a})^{2}-1\right)~{}(i\neq
j)$ $\displaystyle+$
$\displaystyle\frac{1}{4}(1-3\delta_{A})(1-3\delta_{B})\underbrace{(TT_{1111}^{AB}-TT_{1122}^{AB}-TT_{2211}^{AB}+TT_{2222}^{AB})}_{TT_{(11-22)(11-22)}^{AB}}$
$\displaystyle\times$
$\displaystyle\Big{(}(c_{1}^{a})^{2}-(c_{2}^{a})^{2}\Big{)}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{4}(1-3\delta_{A})(1-3\delta_{B})\underbrace{(TT_{1133}^{AB}-TT_{2233}^{AB})}_{TT_{(11-22)33}^{AB}}\Big{(}(c_{1}^{a})^{2}-(c_{2}^{a})^{2}\Big{)}\left(3(c_{3}^{b})^{2}-1\right)$
$\displaystyle+$
$\displaystyle\frac{1}{4}(1-3\delta_{A})(1-3\delta_{B})\underbrace{(TT_{3311}^{AB}-TT_{3322}^{AB})}_{TT_{33(11-22)}^{AB}}\left((c_{1}^{b})^{2}-(c_{2}^{b})^{2}\right)\left(3(c_{3}^{a})^{2}-1\right)$
$\displaystyle+$
$\displaystyle\frac{1}{4}(1-3\delta_{A})(1-3\delta_{B})TT_{3333}^{AB}\left(3(c_{3}^{a})^{2}-1\right)\left(3(c_{3}^{b})^{2}-1\right)\Bigg{]}.$

In the above distribution, systematization of indices on $T$, $pT$, and $TT$
are not included. One recovers the angular distribution in Eq. (15) for
spin-$1$ particle production and decay by integrating the angles of $a$ or $b$
in Eq. (34). The independent tensor-tensor correlations ($TT^{AB}$) are nine
$TT_{(ij)(kl)}^{AB}~{}(i\neq j,k\neq l)$, three
$TT_{(ij)(11-22)}^{AB}~{}(i\neq j)$, three $TT_{(11-22)(ij)}^{AB}~{}(i\neq
j)$, three $TT_{(ij)33}^{AB}~{}(i\neq j)$, three $TT_{33(ij)}^{AB}~{}(i\neq
j)$, one $TT_{(11-22)(11-22)}^{AB}$, one $TT_{(11-22)33}^{AB}$, one
$TT_{33(11-22)}^{AB}$, and one $TT_{3333}^{AB}$. In this case, asymmetries for
the vector polarizations, vector-vector and vector-tensor correlations are
given by,

$\displaystyle{\cal A}[p_{i}^{A/B}]$ $\displaystyle=$
$\displaystyle\frac{3}{4}\alpha_{A/B}p_{i}^{A/B},$ $\displaystyle{\cal
A}[pp_{ij}^{AB}]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\alpha_{A}\alpha_{B}pp_{ij}^{AB},$
$\displaystyle{\cal A}[pT_{i(jk)}^{AB}]$ $\displaystyle=$
$\displaystyle\frac{2}{3\pi}\alpha_{A}(1-3\delta_{B})pT_{i(jk)}^{AB}~{}(j\neq
k),$ $\displaystyle{\cal A}[pT_{i(jk)}^{BA}]$ $\displaystyle=$
$\displaystyle\frac{2}{3\pi}\alpha_{B}(1-3\delta_{A})pT_{i(jk)}^{BA}~{}(j\neq
k),$ $\displaystyle{\cal A}[pT_{i(11-22)}^{AB}]$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi}\alpha_{A}(1-3\delta_{B})pT_{i(11-22)}^{AB},$
$\displaystyle{\cal A}[pT_{i(11-22)}^{BA}]$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi}\alpha_{B}(1-3\delta_{A})pT_{i(11-22)}^{BA},$
$\displaystyle{\cal A}[pT_{i33}^{AB}]$ $\displaystyle=$
$\displaystyle\frac{3}{16}\alpha_{A}(1-3\delta_{B})pT_{i33}^{AB},$
$\displaystyle{\cal A}[pT_{i33}^{BA}]$ $\displaystyle=$
$\displaystyle\frac{3}{16}\alpha_{B}(1-3\delta_{A})pT_{i33}^{BA}.$ (35)

The asymmetries for the independent tensor-tensor correlations are given by,

$\displaystyle{\cal A}[TT_{(ij)(kl)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{ij}}d\Omega_{a}\int_{b}^{T_{kl}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(i\neq
j,~{}k\neq l)$ (36) $\displaystyle=$
$\displaystyle\left(\frac{4}{3\pi}\right)^{2}(1-3\delta_{A})(1-3\delta_{B})TT_{(ij)(kl)}^{AB},$
$\displaystyle{\cal A}[TT_{(ij)(11-22)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{ij}}d\Omega_{a}\int_{b}^{T_{11-22}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(i\neq
j)$ (37) $\displaystyle=$
$\displaystyle\left(\frac{8}{9\pi^{2}}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{(ij)(11-22)}^{AB},$
$\displaystyle{\cal A}[TT_{(11-22)(ij)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{11-22}}d\Omega_{a}\int_{b}^{T_{ij}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(i\neq
j)$ (38) $\displaystyle=$
$\displaystyle\left(\frac{8}{9\pi^{2}}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{(11-22)(ij)}^{AB},$
$\displaystyle{\cal A}[TT_{(ij)33}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{ij}}d\Omega_{a}\int_{b}^{T_{33}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(i\neq
j)$ (39) $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi}\right)^{2}(1-3\delta_{A})(1-3\delta_{B})TT_{(ij)33}^{AB},$
$\displaystyle{\cal A}[TT_{33(ij)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{33}}d\Omega_{a}\int_{b}^{T_{ij}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)~{}(i\neq
j)$ (40) $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi}\right)^{2}(1-3\delta_{A})(1-3\delta_{B})TT_{33(ij)}^{AB}~{}(i\neq
j),$ $\displaystyle{\cal A}[TT_{(11-22)(11-22)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{11-22}}d\Omega_{a}\int_{b}^{T_{11-22}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(41) $\displaystyle=$
$\displaystyle\left(\frac{4}{9\pi^{2}}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{(11-22)(11-22)}^{AB},$
$\displaystyle{\cal A}[TT_{(11-22)33}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{11-22}}d\Omega_{a}\int_{b}^{T_{33}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(42) $\displaystyle=$
$\displaystyle\left(\frac{1}{4\pi}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{(11-22)33}^{AB},$
$\displaystyle{\cal A}[TT_{33(11-22)}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{33}}d\Omega_{a}\int_{b}^{T_{11-22}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(43) $\displaystyle=$
$\displaystyle\left(\frac{1}{4\pi}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{33(11-22)}^{AB},$
$\displaystyle{\cal A}[TT_{3333}^{AB}]$ $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{33}}d\Omega_{a}\int_{b}^{T_{33}}d\Omega_{b}\left(\frac{1}{\sigma}\dfrac{d^{2}\sigma}{d\Omega_{a}d\Omega_{b}}\right)$
(44) $\displaystyle=$
$\displaystyle\left(\frac{9}{64}\right)(1-3\delta_{A})(1-3\delta_{B})TT_{3333}^{AB}.$

For the numerical purpose, all the independent ($25$) tensor-tensor
correlation asymmetries can be obtained as,

${\cal A}_{mn}[TT^{AB}]=\dfrac{\sigma\left({\cal C}_{m}^{a}{\cal
C}_{n}^{b}>0\right)-\sigma\left({\cal C}_{m}^{a}{\cal
C}_{n}^{b}<0\right)}{\sigma\left({\cal C}_{m}^{a}{\cal
C}_{n}^{b}>0\right)+\sigma\left({\cal C}_{m}^{a}{\cal
C}_{n}^{b}<0\right)},~{}~{}m,n\in[1,2,3,4,5],$ (45)

where ${\cal C}_{m}$ are the combinations of $c_{x},c_{y},c_{z}$ given in Eq.
(82) in appendix C.

The formalism we presented here for the spin correlations in all three cases,
namely half-half, half-one and one-one spin cases, gives the method to
estimate all the spin correlations along with the polarizations from the
normalized production density matrix as well as from the joint angular
distribution by constructing several asymmetries. We test the formalism in
some SM scattering processes in section 2.4 for all three scenarios for the
purpose of sanity checking. We will use some compact notations of the
polarization and correlation parameters and their asymmetries for aesthetic
visibility. These notations are $T_{x^{2}-y^{2}}=T_{11-22}$,
$TT_{(mn)^{2}}=TT_{(mn)(mn)}$. For example, we will use
$TT_{(xy)^{2}}=TT_{(12)(12)}$, $TT_{(x^{2}-y^{2})^{2}}=TT_{(11-22)(11-22)}$,
$TT_{(zz)^{2}}=TT_{3333}$ etc.

###  2.3 Spin correlations in laboratory frame

The values of the polarization parameters $p_{i}$ and $T_{ij}$ depend on the
choice of the reference frame, and thus, the values of spin correlations do
so. The above formalism of spin polarization and correlations is based on the
helicity frame, equivalent to the center-of-mass (CM) frame. For an
$e^{-}$-$e^{+}$ collider, CM frame and laboratory (Lab) frame are the same,
while for a hadron collider such as the LHC, they are different due to the
involvement PDFs. In a hadron collider, the polarization density matrix of a
single particle receives an effective total rotation comprising boost and
angular rotations, leaving the trace invariant going from CM to Lab frame. As
a result, the polarization parameters $p_{i}$ and $T_{ij}$ get transformed as
[1, 97, 98],

$\displaystyle p_{i}^{\text{Lab}}$ $\displaystyle=$ $\displaystyle
R_{ij}^{Y}(\omega)p_{j}^{\text{CM}},$ $\displaystyle T_{ij}^{\text{Lab}}$
$\displaystyle=$ $\displaystyle
R_{ik}^{Y}(\omega)R_{jl}^{Y}(\omega)T_{kl}^{\text{CM}},$ (46)

where

$\displaystyle\cos\omega$ $\displaystyle=$
$\displaystyle\cos\theta_{\text{CM}}\cos\theta_{\text{Lab}}+\gamma_{\text{CM}}\sin\theta_{\text{CM}}\sin\theta_{\text{Lab}},$
$\displaystyle\sin\omega$ $\displaystyle=$
$\displaystyle\frac{m}{E}\left(\sin\theta_{\text{CM}}\cos\theta_{\text{Lab}}-\gamma_{\text{CM}}\cos\theta_{\text{\text{CM}}}\sin\theta_{\text{Lab}}\right).$
(47)

The spin correlation parameters $pp$, $pT$ and $TT$ of a system of two
particles $A$ and $B$, thus, get transformed as,

$\displaystyle\left[pp_{ij}^{AB}\right]^{\text{Lab}}$ $\displaystyle=$
$\displaystyle
R_{ik}^{Y}(\omega_{A})R_{jl}^{Y}(\omega_{B})\left[pp_{kl}^{AB}\right]^{\text{CM}},$
$\displaystyle\left[pT_{ijk}^{AB}\right]^{\text{Lab}}$ $\displaystyle=$
$\displaystyle
R_{il}^{Y}(\omega_{A})R_{jm}^{Y}(\omega_{B})R_{kn}^{Y}(\omega_{B})\left[pT_{lmn}^{AB}\right]^{\text{CM}},$
$\displaystyle\left[TT_{ijkl}^{AB}\right]^{\text{Lab}}$ $\displaystyle=$
$\displaystyle
R_{im}^{Y}(\omega_{A})R_{jn}^{Y}(\omega_{A})R_{kp}^{Y}(\omega_{B})R_{lq}^{Y}(\omega_{B})\left[TT_{mnpq}^{AB}\right]^{\text{CM}}.$
(48)

Here, $R_{ij}^{Y}$ is the usual rotational matrix w.r.t. $y$-direction and
$\gamma_{\text{CM}}=1/\sqrt{1-\beta_{\text{CM}}^{2}}$ with $\beta_{\text{CM}}$
being boost of the CM frame. The quantities $m$ and $E$ are the mass and
energy of the particle in consideration, respectively.

###  2.4 Examples of spin correlations in the SM

Figure 2: . The SM values of the asymmetries for spin correlations and
polarizations are shown as a function of center-of-mass energy (c.m.e) for
$e^{-}e^{+}\to t\bar{t}$ in the left-top-panel, $e^{-}e^{+}\to ZZ$ in the
right-top-panel and $gb\to tW^{-}$ in the bottom-panel. The data points with
errorbars correspond to $10^{7}$ events generated in MadGraph5_aMC@NLO.

As a demonstration of the formalism discussed above for the spin correlations
along with polarizations, we choose three processes namely $e^{-}e^{+}\to
t\bar{t}$, $gb\to tW^{-}$ and $e^{-}e^{+}\to ZZ$ as examples of all three
scenarios, i.e., half-half ($t\bar{t}$), half-one ($tW^{-}$) and one-one
($ZZ$) spin systems. The correlation and polarization parameters are
constructed at the production level of the particles using the polarization-
correlation matrices in Eqs. (2.2.1), (25), (32) as well as in their decay
level using the normalized angular distributions of the decay products (Eqs.
(17), (27), (34)) using the partial integration assuming on-shell production
and decay of the mother particles. We calculate all the asymmetries related to
spin correlations and polarizations of the particles at the production level
using the helicity amplitude technique as a function of center-of-mass energy
(c.m.e) in all three processes. At first, the polarization and correlations
parameters are obtained with the method explained in appendix B. The
asymmetries are then obtained with the appropriate coefficients given above.

We find all the $CP$-odd polarization and correlation parameters, i.e., all
the parameters with only one $y$\- suffix, e.g., $p_{y}$, $T_{xy}$, $pp_{yz}$,
$pT_{x(yz)}$, $TT_{(xy)zz}$ to be zero as they should be for the SM being $CP$
conserving. However, correlations appearing with two $y$\- suffix, e.g.,
$pp_{yy}$, can be non zero in general.

In the $t\bar{t}$ production process, all the polarization and correlations
except the $CP$-odd ones are non vanishing. Furthermore, both $t$ and
$\bar{t}$ have same values of $p_{x}$ and opposite values of $p_{z}$, i.e.,
$p_{x}^{t}=p_{x}^{\bar{t}}$ and $p_{z}^{t}=-p_{z}^{\bar{t}}$. The correlations
$pp_{xz}$ and $pp_{zx}$ are equal and opposite because of the nature of
individual $p_{x}$ and $p_{z}$.

In the $tW^{-}$ production process, all the $CP$-even polarization and
correlations are non-vanishing. There are no relations among the polarizations
and correlations here, as the final state particles are entirely different.

In the $ZZ$ production process, final state being symmetric, there are only
eight independent polarizations and thirty six independent correlations‡‡‡
Independent parameters are $3$ piece $p$, $5$ piece $T$, $6$ independent $pp$,
$15$ independent $pT$ and $15$ independent $TT$.. Owing to the initial state
symmetry, there are only three non-vanishing polarizations ($p_{x}$,
$T_{x^{2}-y^{2}}$, $T_{zz}$), three non-vanishing vector-vector correlations
(diagonals only, i.e., $pp_{ii},~{}i=x,y,z$), three non-vanishing vector-
tensor correlations ($pT_{y(xy)}$, $pT_{x(x^{2}-y^{2})}$, $pT_{xzz}$), and six
non-vanishing tensor-tensor correlations ($TT_{(xy)^{2}}$, $TT_{(xz)^{2}}$,
$TT_{(yz)^{2}}$, $TT_{\left(x^{2}-y^{2}\right)^{2}}$, $TT_{(x^{2}-y^{2})zz}$,
$TT_{(zz)^{2}}$) totaling $3+12=15$ non-vanishing polarization and
correlations. Note that the correlation $TT_{(xz)^{2}}$ is non-vanishing even
the $T_{xz}$ vanishes. Besides these, correlation of two $CP$ odd
polarizations ($pp_{yy}$, $pT_{y(xy)}$, $TT_{(xy)^{2}}$, $TT_{(yz)^{2}}$) came
out to be non-vanishing.

We further calculate all the asymmetries related to the polarizations and
correlations using the angular distributions of the daughter particles in each
production process by generating large number of events ($10^{7}$) in
MadGraph5_aMC@NLO (mg5_aMC) Monte-Carlo event generator. The full processes,
including decay, are chosen as follows,

$\displaystyle e^{-}e^{+}\to t\bar{t}$ $\displaystyle:$
$\displaystyle~{}~{}t\to bW^{+},~{}W^{+}\to
l^{+}\nu_{l};~{}\bar{t}\to\bar{b}W^{-},~{}W^{-}\to l^{-}\bar{\nu_{l}},$
$\displaystyle gb\to tW^{-}$ $\displaystyle:$ $\displaystyle~{}~{}t\to
bW^{+},~{}W^{+}\to l^{+}\nu_{l};~{}W^{-}\to l^{-}\bar{\nu_{l}},$
$\displaystyle e^{-}e^{+}\to ZZ$ $\displaystyle:$ $\displaystyle~{}~{}Z\to
e^{-}e^{+},~{}Z\to\mu^{+}\mu^{-}.$ (49)

We use the $b$ (for $t$) and $\bar{b}$ (for $\bar{t}$) angular distributions
in $e^{-}e^{+}\to t\bar{t}$ process, $b$ (for $t$) and $l^{-}$ (for $W^{-}$)
angular distributions in $gb\to tW^{-}$ process, and $e^{-}$ (for $Z_{1}$) and
$\mu^{-}$ (for $Z_{2}$) angular distributions in $e^{-}e^{+}\to ZZ$ process.
For the top ($t$) polarization, one can also use the angular distributions of
secondary lepton ($l$) with different analyzing power [7]. The SM values for
some chosen asymmetries (in $\%$) (away from zero) related to spin
correlations and polarizations are shown in Fig. 2 as a function of c.m.e
($\sqrt{s}$) in all three processes. The values for the same asymmetries from
the Monte-Carlo simulation are shown with data points with an errorbar
corresponding to $10^{7}$ events. One easily finds an excellent agreement
between the analytical values shown by lines and the Monte-Carlo simulated
values shown by the points for a range of c.m.e. The asymmetries tend to
saturate at some value as the energy increases in all three cases. The reasons
are the following: The polarization and spin-correlation parameters are ratios
of two quantities (cross sections), both depending on the velocity, which
approaches unity as the energy increases. As a result, the polarization and
spin-correlation parameters saturate to some values. We note that different
cuts such as transverse momentum ($p_{T}$), pseudorapidity ($\eta$), etc.,
which are necessary for realistic scenarios, on the daughter particles will
reduce the full angular phase-space. In that case, one can not recover the
actual polarization and spin-correlation parameters from the asymmetries
discussed above. Nevertheless, the asymmetries will be related to the
polarizations and spin-correlations in the process, and they can be used to
study BSM physics. We note that, although the spin density matrices assume on-
shell production and decay, the finite width effect included in the
MadGraph5_aMC@NLO (see Eq. (2.4)) simulation gives the identical results. In
the next section, we investigate how these polarizations and correlations are
affected by BSM physics and how the correlations perform over the
polarizations in probing BSM physics.

##  3 New physics effect on spin correlations and polarizations

We investigate the effect of possible new physics on the spin correlations and
polarizations in $e^{-}e^{+}\to t\bar{t}$, $gb\to tW^{-}$ and $u\bar{d}\to
ZW^{+}$ processes as example of all three scenarios, namely half-half
($t\bar{t}$), half-one ($tW^{-}$) and one-one ($ZW^{+}$) spin systems. We
consider $\gamma t\bar{t}$, $gt\bar{t}$ and $W^{+}W^{-}Z$ anomalous couplings
as examples of new physics for $t\bar{t}$, $tW^{-}$ and $ZW^{+}$ production
processes, respectively. The BSM Lagrangians for the $\gamma t\bar{t}$ [99],
$gt\bar{t}$ [45] and $W^{+}W^{-}Z$ [85] anomalous couplings are given by,

$\displaystyle{\cal L}_{\gamma t\bar{t}}$ $\displaystyle=$ $\displaystyle-
ie\bar{t}\frac{\sigma^{\mu\nu}q_{\nu}}{m_{t}}\left(d_{V}^{\gamma}+id_{A}^{\gamma}\gamma_{5}\right)tA_{\mu},$
$\displaystyle{\cal L}_{gt\bar{t}}$ $\displaystyle=$
$\displaystyle-\frac{g_{s}}{m_{t}}\bar{t}\sigma^{\mu\nu}\left(d_{V}^{g}+id_{A}^{g}\gamma_{5}\right)\frac{\lambda^{a}}{2}tG_{\mu\nu}^{a},$
$\displaystyle{\cal L}_{WWZ}$ $\displaystyle=$
$\displaystyle-\frac{ie\cos\theta_{W}}{\sin\theta_{W}}\left[\frac{\lambda^{Z}}{m_{W}^{2}}W_{\mu}^{+\nu}W_{\nu}^{-\rho}Z_{\rho}^{\mu}+\frac{\widetilde{\lambda^{Z}}}{m_{W}^{2}}W_{\mu}^{+\nu}W_{\nu}^{-\rho}\widetilde{Z}_{\rho}^{\mu}\right].$
(50)

Here, $q$ is the four-momentum transfer of the photon in $\mathscr{L}_{\gamma
t\bar{t}}$; $G_{\mu\nu}^{a}$ are the gluon field strength tensor;
$\lambda^{a}$ are the Gell-Mann matrices; $g_{s}$ is the strong coupling
constant; $m_{t}$ is the top quark mass; $\theta_{W}$ is the weak mixing
angle:
$W_{\mu\nu}^{\pm}=\partial_{\mu}W_{\nu}^{\pm}-\partial_{\nu}W_{\mu}^{\pm}$,
$Z_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}$,
$\widetilde{Z}^{\mu\nu}=1/2\epsilon^{\mu\nu\rho\sigma}Z_{\rho\sigma}$. The
$\gamma t\bar{t}$, $gt\bar{t}$ and $W^{+}W^{-}Z$ vertices can have more
anomalous couplings, but we restrict to only those in Eq. (3) for simplicity.
All the couplings $d_{V,A}^{\gamma/g}$, $\lambda^{Z}$ and
$\widetilde{\lambda^{Z}}$ are zero in the SM; $d_{V}^{\gamma/g}$ and
$\lambda^{Z}$ are $CP$-even; $d_{A}^{\gamma/g}$ and $\widetilde{\lambda^{Z}}$
are $CP$-odd. The coupling $d_{V}^{\gamma}$ ($d_{V}^{g}$) corresponds to top
quark magnetic (chromomagnetic) dipole moment, where as $d_{A}^{\gamma}$
($d_{A}^{g}$) correspond to electric (chromoelectric) dipole moment.

##### Effect of anomalous $\gamma t\bar{t}$ couplings in $e^{-}e^{+}\to
t\bar{t}$ :

Figure 3:  The absolute deviation of the asymmetry from the SM, i.e.,
$|\Delta{\cal A}|=|{\cal A}_{SM+BSM}-{\cal A}_{SM}|$ and their square sum,
i.e., $\sum|\Delta{\cal A}|^{2}$ are shown as a function of anomalous
couplings in the process $e^{-}e^{+}\to t\bar{t}$ at $\sqrt{s}=500$ GeV.

We calculate all the polarizations of $t$ and $\bar{t}$ and their spin
correlations in $e^{-}e^{+}\to t\bar{t}$, analytically from the production
process, as a function of the anomalous couplings $d_{V}^{\gamma}$ and
$d_{A}^{\gamma}$ for a fixed c.m.e of $\sqrt{s}=500$ GeV in the same decay
channel as considered in the SM example given in Eq. (2.4). The $CP$-odd
coupling $d_{A}^{\gamma}$ appears linearly on the numerator of $CP$-odd
parameters ($p_{y}^{t/\bar{t}}$, $pp_{ij},~{}(i/j=y)$) and only quadratically
on the diagonal correlations $pp_{ii}$. The $CP$-even coupling
$d_{V}^{\gamma}$ appears linearly on the numerators of $CP$-even polarizations
($p_{x}^{t/\bar{t}}$, $p_{z}^{t/\bar{t}}$) and off diagonal correlations
$pp_{ij},~{}(i\neq j)$, while the numerators of diagonal correlations
($pp_{ii}$) have linear as well as quadratic contribution of $d_{V}^{\gamma}$.
The polarization asymmetries for $t$ and $\bar{t}$ are identical except ${\cal
A}[p_{x}]$ being opposite. We calculate the deviations of all the asymmetries
from the SM, i.e., $\Delta{\cal A}={\cal A}_{SM+BSM}-{\cal A}_{SM}$ as
functions of the couplings and show their absolute values ($|\Delta{\cal A}|$)
in percentage ($\%$) in Fig. 3 in the top-row to see their relative strengths.
Here, $|\Delta{\cal A}[p_{i}^{t}]|=|\Delta{\cal A}[p_{i}^{\bar{t}}]|$ and
$|\Delta{\cal A}[pp_{ij}^{t\bar{t}}]|=|\Delta{\cal A}[pp_{ji}^{t\bar{t}}]|$.
In Fig. 3 top-row, only one asymmetry is shown in the case of degenerate
asymmetries. The $|\Delta{\cal A}|$ can be considered as sensitivity if we
assume $1\%$ errors for the asymmetries, which corresponds to $10^{4}$ events
in the SM. We see that, polarization asymmetries ${\cal
A}[p_{z}^{t/\bar{t}}]$, ${\cal A}[p_{x}^{t/\bar{t}}]$ show large deviation for
small values of $d_{V}^{\gamma}$ and ${\cal A}[p_{y}^{t/\bar{t}}]$ for small
values of $d_{A}^{\gamma}$. The correlation asymmetries also show similar
behavior, especially for $d_{A}^{\gamma}$. We compare the correlation
asymmetries and the polarization asymmetries by taking quadratic sum of the
$|\Delta{\cal A}|$, i.e., $\sum_{i}|\Delta{\cal A}[p_{i}]|^{2}$ and
$\sum_{ij}|\Delta{\cal A}[pp_{ij}]|^{2}$. These quadratic sums are equivalent
to $\chi^{2}$ for $10^{4}$ events in the SM. In Fig. 3 bottom-row, we show the
$\sum|\Delta{\cal A}|^{2}$ for polarization asymmetries, correlation
asymmetries and their sum. The correlation asymmetries offer significant
improvement over the polarization asymmetries in probing the anomalous
couplings, especially for the $CP$-odd coupling $d_{A}^{\gamma}$.

##### Effect of anomalous $gt\bar{t}$ couplings in $gb\to tW^{-}$ :

Figure 4:  The absolute deviation of the asymmetry from the SM ($|\Delta{\cal
A}|$) and their square sum ($\sum|\Delta{\cal A}|^{2}$) are shown as a
function of anomalous couplings in the process $gb\to tW^{-}$ at
$\sqrt{\hat{s}}=1$ TeV.

In $gb\to tW^{-}$ partonic process, we choose the same decay channel chosen
for the SM example in sec. 2.4 given in Eq. (2.4). In this case, we use decay
distribution of the daughters of $t$ and $W^{-}$ to obtain their polarization
and correlation asymmetries by generating large number of events ($10^{6}$) in
mg5_aMC for a range of couplings $d_{V/A}^{g}$ at partonic $\sqrt{\hat{s}}=1$
TeV .

In this process, all the polarization and correlation asymmetries are entirely
different from each other. In Fig. 4 top-row, we show the absolute deviation
of the asymmetries from the SM ($|\Delta{\cal A}|$) as a function of couplings
for all the asymmetries for completeness. The legends are shown only for the
asymmetries having $|\Delta{\cal A}|>1$ within the range of couplings shown.
There are many correlation asymmetries along with polarization asymmetries
that show large $|\Delta{\cal A}|$ for small values of couplings. For example,
along with the polarization asymmetries ${\cal A}[p_{x}^{W}]$ and ${\cal
A}[p_{x}^{t}]$, the correlation asymmetries ${\cal A}[pp_{yy}^{tW}]$ and
${\cal A}[pp_{xx}^{tW}]$ also show large $\Delta{\cal A}$ for small values of
$d_{V}^{g}$, see Fig. 4 left-top-panel. Similarly for the $CP$-odd coupling
$d_{A}^{g}$, shown in Fig. 4 right-top-panel, the correlation asymmetries
${\cal A}[pp_{xy}^{tW}]$ and ${\cal A}[pp_{yx}^{tW}]$ along with the
polarization asymmetries ${\cal A}[p_{y}^{W}]$ and ${\cal A}[p_{y}^{t}]$ have
large $\Delta{\cal A}$ for small values of coupling. Thus it is expected that
the correlation asymmetries will improve over the polarization asymmetries in
$\sum|\Delta{\cal A}|^{2}$, which are shown in Fig. 4 in the bottom-row for
both couplings $d_{V}^{g}$ and $d_{A}^{g}$. The $\sum|\Delta{\cal A}|^{2}$ for
correlation asymmetries (${\cal A}^{tW}$) are better than that of top
polarization asymmetries (${\cal A}^{t}$) for both couplings although not
better than $W$ polarization asymmetries (${\cal A}^{W}$). However, the
combined $|\Delta{\cal A}|^{2}$ of polarization and correlation asymmetries
improve significantly over the polarization asymmetries.

##### Effect of $WWZ$ anomalous couplings in $u\bar{d}\to ZW^{+}$ :

Figure 5:  The absolute deviation of the asymmetry from the SM ($|\Delta{\cal
A}|$) and their square sum ($\sum|\Delta{\cal A}|^{2}$) are shown as a
function of anomalous couplings in the process $u\bar{d}\to ZW^{+}$ at
$\sqrt{\hat{s}}=1$ TeV.

We choose the $u\bar{d}\to ZW^{+}$ partonic process as an example to see the
effect of anomalous $WWZ$ couplings, one $CP$-even coupling $\lambda^{Z}$ and
one $CP$-odd coupling $\widetilde{\lambda^{Z}}$ (see Eq. (3)), on the
polarizations and spin correlations of $Z$ and $W^{+}$. The $ZW^{+}$
production process including decay is chosen to be,

$\displaystyle u\bar{d}\to ZW^{+}$ $\displaystyle:$ $\displaystyle~{}~{}Z\to
e^{+}e^{-};~{}W^{+}\to\mu^{+}\nu_{\mu}.$ (51)

In this process too, we use decay distribution of the daughters of $Z$ and
$W^{+}$ to obtain their polarization and correlation asymmetries by generating
large number of events ($10^{6}$) in mg5_aMC at $\sqrt{\hat{s}}=1$ TeV with
varying the anomalous couplings $\lambda^{Z}$ and $\widetilde{\lambda^{Z}}$.
Here also, all the asymmetries are entirely independent of each other, as the
final states are entirely different. In Fig. 5, we show the absolute deviation
of all the polarization and correlation asymmetries from the SM ($|\Delta{\cal
A}|$) as a function of the couplings in top-row. We labeled the plot lines
only for the asymmetries having $|\Delta{\cal A}|>1$ within the range of
couplings shown. For the range of couplings shown in Fig. 5, $|\Delta{\cal
A}|$ for some parameters for $Z$ and $W^{+}$ happen to be same or very close,
e.g., $|\Delta{\cal A}[T_{x^{2}-y^{2}}^{W}]|\simeq|\Delta{\cal
A}[T_{x^{2}-y^{2}}^{Z}]|$ appearing for $\lambda^{Z}$ in the left-top-panel
and $|\Delta{\cal A}[T_{xy}^{W}]|\simeq|\Delta{\cal A}[T_{xy}^{Z}]|$ appearing
for $\widetilde{\lambda^{Z}}$ in the right-top-panel. The correlation
asymmetries also show large and comparable deviations as the polarization
asymmetries do. For example, $|\Delta{\cal A}|$ for the correlation
asymmetries ${\cal A}[pp_{zz}^{ZW}]$, ${\cal A}[TT_{(xy)^{2}}^{ZW}]$, and
${\cal A}[TT_{\left(x^{2}-y^{2}\right)^{2}}^{ZW}]$ are large and comparable to
the polarization asymmetries ${\cal A}[p_{x}^{W}]$, ${\cal A}[p_{z}^{W}]$, and
${\cal A}[T_{x^{2}-y^{2}}^{W/Z}]$ for small values of $\lambda^{Z}$, see left-
top-panel in Fig. 5. Similar argument is true for $\widetilde{\lambda^{Z}}$
shown in the right-top-panel of Fig. 5. For example, apart from the
polarization asymmetries ${\cal A}[p_{y}^{W}]$, ${\cal A}[p_{z}^{W}]$, and
${\cal A}[T_{xy}^{W/Z}]$, the correlation asymmetries ${\cal
A}[pp_{zz}^{ZW}]$, ${\cal A}[TT_{(xy)^{2}}^{ZW}]$, ${\cal
A}[TT_{\left(x^{2}-y^{2}\right)^{2}}^{ZW}]$, and ${\cal A}[pp_{yx}^{ZW}]$ also
show comparable deviation for small value of couplings, see right-top-panel in
Fig. 5. The quadratic sum of $|\Delta{\cal A}|$ for the correlations
asymmetries (${\cal A}^{ZW}$) is comparable to the polarization asymmetries of
(${\cal A}^{Z}$) and the polarization asymmetries of $W$ (${\cal A}^{W}$) for
both the couplings $\lambda^{Z}$ and $\widetilde{\lambda^{Z}}$, see bottom-
panel in Fig. 5. The combinations of the correlations and the polarizations
show significant improvement over the combined polarizations (${\cal
A}^{Z}+{\cal A}^{W}$) for both the couplings $\lambda^{Z}$ and
$\widetilde{\lambda^{Z}}$.

From the three cases we discussed above, it is evident that the spin
correlations play a significant role along with the polarizations in probing
new physics. The overall effect is more prominent for $CP$-odd couplings
($d_{A}^{\gamma}$, $d_{A}^{g}$ ,$\widetilde{\lambda^{Z}}$) than the $CP$-even
couplings ($d_{V}^{\gamma}$, $\lambda^{Z}$, $d_{V}^{g}$). The partonic
$ZW^{+}$ and $tW^{-}$ production processes are possible in a hadronic collider
such as the LHC, where initial states are folded with parton distribution
functions. In this case, the Lab frame will be different from the CM frame,
and we have to consider the boost and rotations of the polarization and spin
correlations through the relations given in section 2.3. Realistic effects,
such as initial state radiation (ISR), final state radiation (FSR),
hadronization and detector effects, will affect the polarization and
correlations. All three processes contain missing neutrinos, which need to be
reconstructed to obtain the rest frame of the particles whose polarization and
correlations are to be obtained. Reconstruction of missing neutrinos along
with realistic effects would change the way $|\Delta A|$ depends on the
anomalous couplings. Usually, these effects lead to lower sensitivity to
anomalous couplings, but in some cases, the sensitivity can go up, e.g., see
Ref. [76] for $W$ polarizations.

###  3.1 Note on higher order effect

Besides the realistic effects at colliders, higher order perturbative effects
are important and should be taken into consideration while estimating the
polarization and spin correlations in scattering processes [70, 76, 77].
Though the polarization and spin correlations receive corrections from higher
order effect and changes from the tree level value, the formalism we discuss
here holds in SM and BSM [7]. Higher order radiative corrections to a full
$2\to n$ process involve correction to the production part alone, the decay
part alone, and the non-factorizable correction connecting the production and
decay part.

Corrections only to the production process will change the polarization and
spin correlation and hence change the decay product distributions. But the
asymmetries from the decay product distribution do measure the polarization
and spin correlation, including higher order effects.

On the other hand, real emission from the decay products changes the energy
and angle factorization of decay density matrices, hence changing the decay
product distributions. However, whether QCD or QED, these corrections are very
small and can be neglected [100, 101] if the decay products are color-neutral.

Non-factorizable higher order corrections are also negligible for off-shell
weak boson [102, 103] as well as top quark [104, 105, 106, 107] production,
while zero for on-shell production case and will not affect the decay product
distributions.

##  4 Summary

To summarize, we presented a formalism for the spin correlations of two-
particle systems with spins half-half, half-one, and one-one and showed the
connection of these correlations with the joint angular distributions of the
decay products by identifying the asymmetries for them. We validated the
formalism in the SM in three processes, such as $e^{-}e^{+}\to t\bar{t}$,
$gb\to tW^{-}$ and $e^{-}e^{+}\to ZZ$ by calculating the correlations from the
production process as well as from the decay angular distributions by
generating events in mg5_aMC with complete decay chains with finite width
effects of the mother particles. We find that although some individual
polarizations vanish in the SM, their correlations do not, e.g., $CP$-odd
polarizations $p_{y}$ vanish for all particles, but $pp_{yy}$ do not vanish in
any of the three scenarios. We then investigated the effect of some possible
new physics on the correlations and polarizations in three processes, such as
$\gamma t\bar{t}$ anomalous couplings in $e^{-}e^{+}\to t\bar{t}$ process,
$gt\bar{t}$ anomalous couplings in $gb\to tW^{-}$ process, and $W^{+}W^{-}Z$
anomalous couplings in $u\bar{d}\to ZW^{+}$ process. We compare the
polarization and correlation asymmetries individually in terms of absolute
deviation from the SM ($|\Delta{\cal A}|$) as well as combined way by taking
the quadratic sum of $|\Delta{\cal A}|$ in all three processes. The spin
correlations have the potential to provide a significant improvement over the
polarizations in probing the anomalous couplings. With a large set of spin
correlations asymmetries, it will help to study a large number of anomalous
couplings simultaneously in a given process. It is straightforward to extend
this method of describing spin correlations of two particles to describe spin
correlations of more than two particles [108], which is beyond the scope of
this paper.

Acknowledgment: The work of RR is supported by funding available from the
Department of Atomic Energy, Government of India, for the Regional Centre for
Accelerator-based Particle Physics (RECAPP), Harish-Chandra Research
Institute. The work of RKS is partially supported by SERB, DST, Government of
India through the project EMR/2017/002778.

##  Appendix A Spin matrices and normalized decay density matrices

The Pauli spin-$1/2$ matrices are

$\tau_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},~{}~{}\tau_{y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},~{}~{}\tau_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (52)

The spin-$1$ matrices are given by,

$\displaystyle S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\\\
\end{array}\right),~{}S_{y}=\frac{i}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-1&0\\\
1&0&-1\\\ 0&1&0\\\
\end{array}\right),~{}S_{z}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\ \end{array}\right).$ (62)

The normalized decay density matrices at the helicity rest frame for
spin-$1/2$ particles and spin-$1$ particles are given by [2],

$\Gamma_{(1)}(\lambda,\lambda^{\prime})=\left[\begin{tabular}[]{ll}$\frac{1+\alpha\cos\theta}{2}$&$\frac{\alpha\sin\theta}{2}\
e^{i\phi}$\\\ $\frac{\alpha\sin\theta}{2}\
e^{-i\phi}$&$\frac{1-\alpha\cos\theta}{2}$\\\
\end{tabular}\right]~{}\text{and}$ (63)
$\Gamma_{(2)}(\lambda,\lambda^{\prime})=\left[\begin{tabular}[]{lll}$\frac{1+\delta+(1-3\delta)\cos^{2}\theta+2\alpha\cos\theta}{4}$&$\frac{\sin\theta(\alpha+(1-3\delta)\cos\theta)}{2\sqrt{2}}\
e^{i\phi}$&$(1-3\delta)\frac{(1-\cos^{2}\theta)}{4}\ e^{i2\phi}$\\\
$\frac{\sin\theta(\alpha+(1-3\delta)\cos\theta)}{2\sqrt{2}}\
e^{-i\phi}$&$\delta+(1-3\delta)\frac{\sin^{2}\theta}{2}$&$\frac{\sin\theta(\alpha-(1-3\delta)\cos\theta)}{2\sqrt{2}}\
e^{i\phi}$\\\ $(1-3\delta)\frac{(1-\cos^{2}\theta)}{4}\
e^{-i2\phi}$&$\frac{\sin\theta(\alpha-(1-3\delta)\cos\theta)}{2\sqrt{2}}\
e^{-i\phi}$&$\frac{1+\delta+(1-3\delta)\cos^{2}\theta-2\alpha\cos\theta}{4}$\end{tabular}\right],$
(64)

respectively with $\theta$ and $\phi$ being the polar and azimuthal angle of
the decay products, in the rest frame or helicity rest frame of the mother
particles. The $\alpha$s are called spin analyzing power of the decay
products. For the spin-$1/2$ particle $f$ decaying to another spin-$1/2$
fermion $f^{\prime}$ and a spin-$1$ vector boson $V$ through the vertex
structure $\bar{f}\gamma^{\mu}\
(C_{L}P_{L}+C_{R}P_{R})f^{\prime}V_{\mu},~{}~{}P_{L/R}=\frac{1}{2}\left(1\mp\gamma_{5}\right)$,
the analyzing power $\alpha$ is given by [2],

$\alpha_{\left(1\right)}=\frac{(C_{R}^{2}-C_{L}^{2})(1-x_{1}^{2}-2x_{2}^{2})\sqrt{1+(x_{1}^{2}-x_{2}^{2})^{2}-2(x_{1}^{2}+x_{2}^{2})}}{(C_{R}^{2}+C_{L}^{2})(1-2x_{1}^{2}+x_{2}^{2}+x_{1}^{2}x_{2}^{2}+x_{1}^{4}-2x_{2}^{4})-12C_{L}C_{R}x_{1}x_{2}^{2}},$
(65)

where $x_{i}=m_{i}/m$ with $m_{i}$ as the mass of daughters and $m$ as the
mass of mother particle. In the case of $t\to bW^{+}$ decay, within the SM we
have $\alpha\sim-0.396$. For the case of $V\to ff^{\prime}$ decay through the
same decay vertex as above, the analyzing power $\alpha$ and $\delta$ are
given by [2],

$\alpha_{(2)}=\frac{2(C_{R}^{2}-C_{L}^{2})\sqrt{1+(x_{1}^{2}-x_{2}^{2})^{2}-2(x_{1}^{2}+x_{2}^{2})}}{12C_{L}C_{R}x_{1}x_{2}+(C_{R}^{2}+C_{L}^{2})[2-(x_{1}^{2}-x_{2}^{2})^{2}+(x_{1}^{2}+x_{2}^{2})]},$
(66)
$\delta=\frac{4C_{L}C_{R}x_{1}x_{2}+(C_{R}^{2}+C_{L}^{2})[(x_{1}^{2}+x_{2}^{2})-(x_{1}^{2}-x_{2}^{2})^{2}]}{12C_{L}C_{R}x_{1}x_{2}+(C_{R}^{2}+C_{L}^{2})[2-(x_{1}^{2}-x_{2}^{2})^{2}+(x_{1}^{2}+x_{2}^{2})]}.$
(67)

For massless final state fermions, $x_{1}\to 0,\ x_{2}\to 0$; one obtains
$\delta\to 0$ and $\alpha\to(C_{R}^{2}-C_{L}^{2})/(C_{R}^{2}+C_{L}^{2})$.
Furthermore, for the $W$ decay, within the SM we have $C_{R}=0$ and thus
$\alpha=-1$. For the $Z$ decay to $l^{+}l^{-}$, $\alpha\sim-0.2193$.

##  Appendix B Spin correlations from production process in spin-$1/2$ –
spin-$1/2$ case

The harmitian spin-polarization density matrix for spin-$1/2$ – spin-$1/2$
case in Eq. (2.2.1) will be expanded into

$\displaystyle
P_{AB(1,1)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)=$
$\displaystyle\dfrac{1}{4}\left[\begin{array}[]{llll}1+p_{3}^{A}+p_{3}^{B}+pp_{33}^{AB}&~{}~{}p_{1}^{B}+pp_{31}^{AB}-i(pp_{32}^{AB}+p_{2}^{B})&~{}~{}p_{1}^{A}+pp_{13}^{AB}-i(pp_{23}^{AB}+p_{2}^{A})&pp_{11}^{AB}-pp_{22}^{AB}-i(pp_{12}^{AB}+pp_{21}^{AB})\\\
&1+p_{3}^{A}-p_{3}^{B}-pp_{33}^{AB}&pp_{11}^{AB}+pp_{22}^{AB}+i(pp_{12}^{AB}-pp_{21}^{AB})&~{}~{}p_{1}^{A}-pp_{13}^{AB}-i(p_{2}^{A}-pp_{23}^{AB})\\\
&&1-p_{3}^{A}+p_{3}^{B}-pp_{33}^{AB}&p_{1}^{B}-pp_{31}^{AB}-i(p_{2}^{B}-pp_{32}^{AB})\\\
&&&1-p_{3}^{A}-p_{3}^{B}+pp_{33}^{AB}\\\ \end{array}\right],$ (72)

which can be compared to the normalized production density matrix ($\rho$) of
$A$ and $B$ given below,

$P_{AB(1,1)}\left(\lambda_{A},\lambda_{A}^{\prime},\lambda_{B},\lambda_{B}^{\prime}\right)=\dfrac{1}{\sigma_{p}}\left[\begin{array}[]{llll}\rho_{(++++)}&\rho_{(+++-)}&\rho_{(+-++)}&\rho_{(+-+-)}\\\
\rho_{(++-+)}&\rho_{(++--)}&\rho_{(+--+)}&\rho_{(+---)}\\\
\rho_{(-+++)}&\rho_{(-++-)}&\rho_{(--++)}&\rho_{(--+-)}\\\
\rho_{(-+-+)}&\rho_{(-+--)}&\rho_{(---+)}&\rho_{(----)}\\\
\end{array}\right].$ (74)

Here, we have used the notations $(+)\equiv(+1/2)$ and $(-)\equiv(-1/2)$ for
the helicities of the spin-$1/2$ particles $A$ and $B$. One obtains the
following relations for the polarizations and spin-correlations in terms of
$\rho$:

$\displaystyle p_{1}^{A}$ $\displaystyle=$
$\displaystyle\left[\rho_{(-+--)}+\rho_{(-+++)}+\rho_{(+---)}+\rho_{(+-++)}\right]/\sigma_{p},$
$\displaystyle p_{2}^{A}$ $\displaystyle=$
$\displaystyle-i\left[\rho_{(-+--)}+\rho_{(-+++)}-\rho_{(+---)}-\rho_{(+-++)}\right]/\sigma_{p},$
$\displaystyle p_{3}^{A}$ $\displaystyle=$
$\displaystyle\left[-\rho_{(----)}-\rho_{(--++)}+\rho_{(++--)}+\rho_{(++++)}\right]/\sigma_{p},$
$\displaystyle p_{1}^{B}$ $\displaystyle=$
$\displaystyle\left[\rho_{(---+)}+\rho_{(--+-)}+\rho_{(++-+)}+\rho_{(+++-)}\right]/\sigma_{p},$
$\displaystyle p_{2}^{B}$ $\displaystyle=$
$\displaystyle-i\left[\rho_{(---+)}-\rho_{(--+-)}+\rho_{(++-+)}-\rho_{(+++-)}\right]/\sigma_{p},$
$\displaystyle p_{3}^{B}$ $\displaystyle=$
$\displaystyle\left[-\rho_{(----)}+\rho_{(--++)}-\rho_{(++--)}+\rho_{(++++)}\right]/\sigma_{p},$
$\displaystyle pp_{11}^{AB}$ $\displaystyle=$
$\displaystyle\left[\rho_{(-+-+)}+\rho_{(-++-)}+\rho_{(+--+)}+\rho_{(+-+-)}\right]/\sigma_{p},$
$\displaystyle pp_{12}^{AB}$ $\displaystyle=$
$\displaystyle-i\left[\rho_{(-+-+)}-\rho_{(-++-)}+\rho_{(+--+)}-\rho_{(+-+-)}\right]/\sigma_{p},$
$\displaystyle pp_{13}^{AB}$ $\displaystyle=$
$\displaystyle\left[-\rho_{(-+--)}+\rho_{(-+++)}-\rho_{(+---)}+\rho_{(+-++)}\right]/\sigma_{p},$
$\displaystyle pp_{21}^{AB}$ $\displaystyle=$
$\displaystyle-i\left[\rho_{(-+-+)}+\rho_{(-++-)}-\rho_{(+--+)}-\rho_{(+-+-)}\right]/\sigma_{p},$
$\displaystyle pp_{22}^{AB}$ $\displaystyle=$
$\displaystyle\left[-\rho_{(-+-+)}+\rho_{(-++-)}+\rho_{(+--+)}-\rho_{(+-+-)}\right]/\sigma_{p},$
$\displaystyle pp_{23}^{AB}$ $\displaystyle=$ $\displaystyle
i\left[\rho_{(-+--)}-\rho_{(-+++)}-\rho_{(+---)}+\rho_{(+-++)}\right]/\sigma_{p},$
$\displaystyle pp_{31}^{AB}$ $\displaystyle=$
$\displaystyle\left[-\rho_{(---+)}-\rho_{(--+-)}+\rho_{(++-+)}+\rho_{(+++-)}\right]/\sigma_{p},$
$\displaystyle pp_{32}^{AB}$ $\displaystyle=$ $\displaystyle
i\left[\rho_{(---+)}-\rho_{(--+-)}-\rho_{(++-+)}+\rho_{(+++-)}\right]/\sigma_{p},$
$\displaystyle pp_{33}^{AB}$ $\displaystyle=$
$\displaystyle\left[\rho_{(----)}-\rho_{(--++)}-\rho_{(++--)}+\rho_{(++++)}\right]/\sigma_{p}.$
(75)

In case of spin-$1/2$ – spin-$1$ and spin-$1$ – spin$1$ correlations, one
needs to use the trace equations and symmetry conditions (Eq. (2.2.2), Eq.
(33)) along with Eq. (4) to solve for the polarization and correlation
parameters.

##  Appendix C Asymmetries for tensor polarizations of spin-$1$ particles

The asymmetries for the five independent tensor ($T$) polarizations of a
spin-$1$ particle $A$ are given by the following equations [3, 96],

$\displaystyle{\cal A}[T_{12}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=0}^{\frac{\pi}{2}}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{\pi}{2}}^{\pi}\right.$
(76)
$\displaystyle\left.+\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\pi}^{\frac{3\pi}{2}}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{3\pi}{2}}^{2\pi}\right)d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{12}}d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta_{A})T_{12}^{A},$
$\displaystyle{\cal A}[T_{13}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\frac{\pi}{2}}\int_{\phi_{a}=-\frac{\pi}{2}}^{\frac{\pi}{2}}-\int_{\theta_{a}=0}^{\frac{\pi}{2}}\int_{\phi_{a}=\frac{\pi}{2}}^{\frac{3\pi}{2}}\right.$
(77)
$\displaystyle\left.+\int_{\theta_{a}=\frac{\pi}{2}}^{\pi}\int_{\phi_{a}=\frac{\pi}{2}}^{\frac{3\pi}{2}}-\int_{\theta_{a}=\frac{\pi}{2}}^{\pi}\int_{\phi_{a}=-\frac{\pi}{2}}^{\frac{\pi}{2}}\right)d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{13}}d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta_{A})T_{13}^{A},$
$\displaystyle{\cal A}[T_{23}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\frac{\pi}{2}}\int_{\phi_{a}=0}^{\pi}-\int_{\theta_{a}=0}^{\frac{\pi}{2}}\int_{\phi_{a}=\pi}^{2\pi}\right.$
(78)
$\displaystyle\left.+\int_{\theta_{a}=\frac{\pi}{2}}^{\pi}\int_{\phi_{a}=\pi}^{2\pi}-\int_{\theta_{a}=\frac{\pi}{2}}^{\pi}\int_{\phi_{a}=0}^{\pi}\right)d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{23}}d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)=\frac{2}{\pi}\sqrt{\frac{2}{3}}(1-3\delta_{A})T_{23}^{A},$
$\displaystyle{\cal A}[T_{11-22}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=-\frac{\pi}{4}}^{\frac{\pi}{4}}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{\pi}{4}}^{\frac{3\pi}{4}}\right.$
(79)
$\displaystyle\left.+\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{3\pi}{4}}^{\frac{5\pi}{4}}-\int_{\theta_{a}=0}^{\pi}\int_{\phi_{a}=\frac{5\pi}{4}}^{\frac{7\pi}{4}}\right)d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)$
$\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{11-22}}d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)=\frac{1}{\pi}\sqrt{\frac{2}{3}}(1-3\delta_{A})\left(T_{11-22}^{A}\right),$
$\displaystyle{\cal A}[T_{33}^{A}]$ $\displaystyle\equiv$
$\displaystyle\left(\int_{\theta_{a}=0}^{\frac{\pi}{3}}\int_{\phi_{a}=0}^{2\pi}-\int_{\theta_{a}=\frac{\pi}{3}}^{\frac{2\pi}{3}}\int_{\phi_{a}=0}^{2\pi}+\int_{\theta_{a}=\frac{2\pi}{3}}^{\pi}\int_{\phi_{a}=0}^{2\pi}\right)d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)$
(80) $\displaystyle\equiv$
$\displaystyle\int_{a}^{T_{33}}d\Omega_{a}\left(\frac{1}{\sigma}\dfrac{d\sigma}{d\Omega_{a}}\right)=\frac{3}{8}\sqrt{\frac{3}{2}}(1-3\delta_{A})T_{33}^{A},$

with $a$ being a daughter of the particle $A$. For the numerical purpose,
these five asymmetries can also be obtained as,

${\cal A}_{m}[T^{A}]=\dfrac{\sigma\left({\cal
C}_{m}^{a}>0\right)-\sigma\left({\cal C}_{m}^{a}<0\right)}{\sigma\left({\cal
C}_{m}^{a}>0\right)+\sigma\left({\cal
C}_{m}^{a}<0\right)},~{}~{}m\in[1,2,3,4,5],$ (81)

with

${\cal
C}_{m}\in\left[c_{x}c_{y},~{}c_{x}c_{z},~{}c_{y}c_{z},~{}c_{x}^{2}-c_{y}^{2},~{}|\sqrt{c_{x}^{2}+c_{y}^{2}}|(4c_{z}^{2}-1)=\sin(3\theta)\right].$
(82)

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