# Entanglement and scattering in quantum electrodynamics: S-matrix information
from an entangled spectator particle

Juan D. Fonsecaa<EMAIL_ADDRESS>B. Hillerb<EMAIL_ADDRESS>J. B.
Araujo b jonas@colibri I. G. da Pazc<EMAIL_ADDRESS>M. Sampaio a
<EMAIL_ADDRESS>a CCNH, Universidade Federal do ABC, 09210-580 ,
Santo André - SP, Brazil  b CFisUC - Department of Physics, University of
Coimbra, 3004-516 Coimbra, Portugal  c Universidade Federal do Piauí,
Departamento de Física, 64049-550, Teresina, PI, Brazil

###### Abstract

We consider a general quantum field relativistic scattering involving two half
spin fermions, $A$ and $B$, which are initially entangled with another fermion
$C$ that does not participate in the scattering dynamics. We construct general
expressions for the reduced spin matrices for the out-state considering a
general tripartite spin-entangled state. In particular we study an inelastic
QED process at tree-level, namely $e^{-}e^{+}\rightarrow\mu^{-}\mu^{+}$ and a
half spin fermion $C$ as an spectator particle which can be entangled to the
$AB$ system in the following ways: W state, GHZ state,
$|\text{A}^{\alpha}\rangle\otimes|\Psi^{\pm}\rangle_{\text{BC}}$ and
$|\text{A}^{\alpha}\rangle\otimes|\Phi^{\pm}\rangle_{\text{BC}}$, where
$\\{|\Psi^{\pm}\rangle,|\Phi^{\pm}\rangle\\}$ are the Bell basis states and
$|\text{A}^{\alpha}\rangle$ is a spin superposition state of system $A$. We
calculate the von-Neumann entropy variation before and after the scattering
for the particle $C$ and show that spin measurements in $C$ contain numerical
information about the total cross section of the process. We compare the
initial states W and GHZ as well as study the role played by the parameter
$\alpha$ in the evaluation of the entropy variations and the cross section
encoded in the spectator particle.

##  I Introduction

The counterintuitive properties of entanglement were formalized in the famous
1935 paper of Einstein, Podolsky and Rosen EPR . This work paved to way to
study theoretically and experimentally the behaviour of global states of
composite systems. Rather than an incomplete description of physical reality
or the lack of extra variables to comply with local realism, the violation of
Bell’s inequalities BELL  and its variants crowns quantum mechanics as a non-
local and complete theory. For a recent discussion on how quantum non-locality
does not contradict relativistic relativistic causality, see GRANGIER . The
experimental verification of quantum non-locality became possible only in the
early eighties through an ingenious setup in Aspect’s laboratory in France
ASPECT . Later on, possible locality and detection loopholes were discarded by
state-of-the-art technological refinements such as in the experiment involving
entangled electron spins separated by 1.3 km LOOPHOLE1  as well as atoms
separated by 398 m via measurements of the atomic spin states LOOPHOLE2 .

A growing interest in relativistic aspects of quantum entanglement (see
RQIREVIEW  for a review) is reflected in many works that appeared at the turn
of the $21^{st}$ century. Relativistic quantum information sets out to answer,
amongst other questions, how quantum information processing is affected by
relativistic motion both in inertial and non-inertial frames FRIIS . Friis,
Bertlmann, Huber and Hiesmayr BERTLMANN  studied a composite momentum-spin
state with two half-spin particles, initially entangled in spin and momentum
spaces separately but globally factorized in an unboosted frame. They found
that, for their particular state, Lorentz boosts expressed through Wigner
rotations, change the entanglement content as different partitions of the four
qubits are considered by different inertial observers. Specifically, whilst
two particle entanglement corresponding to Alice and Bob subsystems remain
invariant, other partitions are affected by the Wigner rotation on the spin
variables leading to observer dependent entanglement between spin and momentum
degrees of freedom. As for the Lorentz frame dependence of Bell inequality
violations, it has been argued whilst the degree of violation can decrease as
the rapidity increases AHN , the maximal violation can be recovered through
the choice of specific directions by any inertial observer BERTLMANN . Besides
the obvious conceptual interest, studies of relativistic aspects of quantum
information is motivated by using relativistic effects to enhance current
quantum technologies through new relativistic quantum protocols RQIIF . For
instance, in HU  two moving point-like Unruh-DeWitt like detectors prepared in
a separable state and coupled to quantum fields have been considered. Under
specific arrangements, quantum entanglement between the detectors was created
by the quantum field. In contrast, in strong coupling or acceleration regime,
the vacuum fluctuations have a deleterious effect on entanglement between the
detectors. Non-inertial effects on entanglement generation are also relevant
as precision measurements in cosmology GW  and the technology involving
photonic interferometric devices PID  progress.

As for cosmological applications, spacetime expansion leads to quantum
correlations developed by particle creation which may encode information on
the underlying cosmic history. Entanglement encoded in the von Neumann entropy
considered as an entanglement measure for quantum fields in expanding
spacetimes was studied in BALL . Morever, for the case of bosonic modes
entangled in different frames with relative acceleration, it was shown that
the state’s distillable entanglement vanishes as a consequence of the Unruh
effect ALICE , whereas for fermionic modes there exists a residual
entanglement which has been proved beyond single mode approximation LOUKO .

Along with those developments, many contributions have appeared to study
quantum entanglement in spin and momentum degrees of freedom as a result of
particle collisions supposing non-interacting in and out states. In a non-
relativistic formulation, electron-electron spin and momentum entanglement was
studied in FEDER  using the von Neumann entropy of the one-electron reduced
density matrix as an entanglement quantifier. Interacting fermions in a delta
potential with a spin-spin coupling strength were studied in GHANBARI  using
both plane wave and Gaussian packets to describe asymptotic states. Symmetry
and dynamical invariant tensor product structures in a class of potentials
were seen to provide frame independent measures of entanglement in HARSHMAN4 .
Along the same lines, the study symmetry invariance with respect to the
rotation and the Galilean group in terms of unitary irreducible
representations showed that the S matrix is determined by scattering phase
shifts for spin $1$, $1/2$ and $3/2$ particles, and together with the boundary
conditions, constrain entanglement in the spin variables of the system
HARSHMAN1 ,HARSHMAN2 ,HARSHMAN3 ,HARSHMAN5 . Lamata and León in LAMATALEON
showed that in a spin-independent scattering between two fermions, bipartite
spin entanglement is generated as a function of the scattering angle.
Concomitantly, spatial degrees of freedom play the rôle of ancillas for the
creation of entanglement which may violate Bell’s inequality certain values of
the scattering angle. The entanglement produced by elastic scattering of non-
relativistic electrons quantified by pair concurrence was studied in KOUZAKOV
, where it was also studied a Bell’s inequality violation in the final channel
and the formation of an entangled Werner spin state under certain conditions.

From the perspective of a quantum field theoretical formulation, entanglement
and scattering in a $S$-matrix framework has been studied for many authors. A
bridge between the ab-initio quantum electrodynamics formulation and the non-
relativistic limit for entanglement generation and quantification was first
studied by Pachos and Solano in PACHOS . Lately, much attention has been
devoted to the role played by quantum entanglement in particle physics (see
LYKKEN for a review) in order to identify the physical signals of an
observable hidden noisy backgrounds. The description of in-particles as
momentum eigenstates in the $S$-matrix was used in MISHIMA  to study
entanglement in the final state and in SHI  in connection with spacetime
symmetries. The authors of reference LAMATASOLANO  described a QED scattering
process showing that continuous variable momentum entanglement grows in time
as virtual photons are exchanged. They have also verified the entanglement
transfer between momentum and spin variable as seen from different Lorentz
observers. The phenomenological description of entangled one-half spins in QED
carried out in WANG  was nonetheless studied numerically in order to analyse
the entanglement dynamics using linear entropy as an entanglement measure
BUSCEMI . As regards of the effects of initial state quantum correlations in
the cross-section, the elastic channel of interacting scalar particles was
studied under the hypothesis of unitarity at a given order in perturbation
theory in SEKI1 ,SEKI2 ,SEKI3  and it was calculated the change of
entanglement entropy from the initial to the final state. Moreover in FAN1
using a QED inelastic scattering, it was shown that such a variation of
entanglement entropy between the pair $e^{+}e^{-}$ in the initial state to the
final state composed of a pair $\mu^{+}\mu^{-}$ was proportional to the total
cross section. On the other hand, the same process was used in FAN2  to study
the change of entanglement entropy in the scattering process as seen from
different Lorentz observers. Because such a quantity is proportional to the
cross-section, it was explicitly verified through appropriate Wigner rotations
that it is Lorentz invariant whilst the correlations involving only spin
degrees of freedoms is not. Bell’s inequality violation in QED was verified in
the process $2\gamma\rightarrow e^{+}e^{-}$ YONGRAM  shedding light on its
experimental measurement.

An interesting question is whether one can prepare correlated initial states
in such a way that the differential cross section can be affected. It is well
known for instance that $\gamma\gamma$ direct scattering, which is realized at
one loop order through the exchange of virtual electron-positron pairs in QED,
has a very tiny cross-section ($\approx 10^{-24}\,\text{m}^{2}$ for
$\lambda=30$pm INADA ). In RATZELQED , the authors related the differential
cross section of photon-photon scattering is a function of the degree of
polarization entanglement of the two-photon state. For a certain scattering
angle, the scattering becomes stronger (weaker) for the symmetric
(antisymmetric) Bell state than for a factorized initial state for photons
(see also RATZELQG ). Correlation functions can also be established in
scattering experiments such as the Møller scattering through the explicit
evaluation of the final state density matrix. In CABAN1 ,CABAN2  relativistic
spin correlations for initially polarized beam scattered off polarized and
unpolarized targets were calculated showing that the Clauser-Horne-Shimony-
Holt (CHSH) inequality can be violated for relativistic energies when both
scattering electrons are highly polarized in some specific directions. The
same system was used in TSURIKOV  to derive a spin correlation tensor for the
two electron state. Thereby the authors derives a geometric entanglement
measure based on the distance between this tensor calculated for separable and
entangled states.

The quantum field theoretical description of quantum information in $S$-matrix
calculations opens new venues for questions related to the amount of
information drawn off by soft radiation emitted by asymptotic states
representing charged particles. The subject becomes even more interesting as
finite observables can only be obtained after taking into account the emission
of soft photons as stated by the Bloch and Nordsieck (BN) and the Kinoshita,
Lee and Nauenberg (KLN) theorem KLN . At next-to-leading order (NLO) for
instance, a judicious cancellation of infrared divergences from loop
corrections and tree level processes that include soft emission renders the
cross section finite after the sum of real-real and real-virtual
contributions. The loss of coherence stemming from the integration of infrared
degrees of freedom has been studied for instance in GOMEZ . Moreover, in
CARNEY  it was claimed that unobserved soft photons decohere momentum
superpositions of charged particles and that the dressed state formalism can
lead to more sensible physical results, later applied in TOMARAS  in
perturbative QED to study the entanglement between the hard and soft particles
in the final state to find that entanglement entropy is free of any infrared
divergences only order by order in perturbation theory.

Finally, some of us has studied the QED inelastic process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ watched by an spectator particle,
entangled with one of the particles in the in-state PRD2019 . It was claimed
that partial information about the dynamics and the total cross-section of
particles $A$, $B$ could be obtained from spin measurements on the spectator
particle $C$. In the present contribution we provide a complete analysis by
considering a general tripartite spin $1/2$-state

$\displaystyle\ket{\mathscr{S}}$ $\displaystyle=$ $\displaystyle
c_{1}\ket{\uparrow\uparrow\uparrow}+c_{2}\ket{\uparrow\uparrow\downarrow}+c_{3}\ket{\uparrow\downarrow\uparrow}$
(1) $\displaystyle+$ $\displaystyle
c_{4}\ket{\uparrow\downarrow\downarrow}+c_{5}\ket{\downarrow\uparrow\uparrow}+c_{6}\ket{\downarrow\uparrow\downarrow}$
$\displaystyle+$ $\displaystyle
c_{7}\ket{\downarrow\downarrow\uparrow}+c_{8}\ket{\downarrow\downarrow\downarrow},$

in which Alice, Bob and Claire state
$|s_{\text{A}}s_{\text{B}}s_{\text{C}}\rangle\equiv|s_{\text{A}}\rangle\otimes|s_{\text{B}}\rangle\otimes|s_{\text{C}}\rangle$
is a three-particle spin-state constrained by $\sum_{j=1}^{8}c_{j}^{2}=1$. For
specific sets of values of $c_{j}$, a Greenberger-Horne-Zeilinger
($c_{1}=c_{8}=1/\sqrt{2}$ and $c_{i}=0$ otherwise)

$\ket{\textup{GHZ}}\equiv\frac{1}{\sqrt{2}}[\ket{\downarrow\downarrow\downarrow}+\ket{\uparrow\uparrow\uparrow}],$
(2)

or a Wolfgang Dür-Vidal-Cirac ($c_{4}=c_{6}=c_{7}=1/\sqrt{3}$ and $c_{i}=0$
otherwise)

$\ket{\textup{W}}\equiv\frac{1}{\sqrt{3}}[\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\uparrow\downarrow\downarrow}]$
(3)

state vidal  is recovered. GHZ and W three-qubit states are inequivalent
classes of states under local operations and classical communications. Whilst
entanglement in the W state is robust against one particle loss, the GHZ state
is maximally entangled as it is reduced to a product of two qubits. They have
been applied in various contexts, for instance in algorithms for the
generation of entangled GHZ and W states of up to 16 qubits useful in quantum
networks CRUZ , and in the study of quantum teleportation through noisy
channels JUNG . Of particular interest to our work is the product state formed
by a spin superposition of Alice’s spin state and a general entangled Bob and
Claire state, which we divide into two special cases:

  * •

$c_{1}=c_{4}=c_{5}=c_{8}=0,c_{2}=\cos\alpha\cos\eta,c_{3}=\cos\alpha\sin\eta,c_{6}=\sin\alpha\cos\eta,c_{7}=\sin\alpha\sin\eta$:

$\displaystyle\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{\eta}}$
$\displaystyle=$
$\displaystyle[\cos\alpha\ket{\uparrow}+\sin\alpha\ket{\downarrow}]$ (4)
$\displaystyle\otimes$
$\displaystyle[\cos\eta\ket{\uparrow\downarrow}+\sin\eta\ket{\downarrow\uparrow}].$

Particularly, if $\eta=\pi/4$, we get
$\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{+}}$ where
$\ket{\Psi^{+}}=(2)^{-1/2}[\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}]$;
if $\eta=3\pi/4$, we get $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{-}}$
where
$\ket{\Psi^{-}}=(2)^{-1/2}[\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow}]$.

  * •

$c_{2}=c_{3}=c_{6}=c_{7}=0,c_{1}=\cos\alpha\cos\eta,c_{4}=\cos\alpha\sin\eta,c_{5}=\sin\alpha\cos\eta,c_{8}=\sin\alpha\sin\eta$:

$\displaystyle\ket{\textup{A}^{\alpha}}\otimes\ket{\Phi^{\eta}}$
$\displaystyle=$
$\displaystyle[\cos\alpha\ket{\uparrow}+\sin\alpha\ket{\downarrow}]$ (5)
$\displaystyle\otimes$
$\displaystyle[\cos\eta\ket{\uparrow\uparrow}+\sin\eta\ket{\downarrow\downarrow}].$

Particularly, if $\eta=\pi/4$, we get
$\ket{\textup{A}^{\alpha}}\otimes\ket{\Phi^{+}}$ where
$\ket{\Phi^{+}}=(2)^{-1/2}[\ket{\uparrow\uparrow}+\ket{\downarrow\downarrow}]$;
if $\eta=3\pi/4$, we get $\ket{\textup{A}^{\alpha}}\otimes\ket{\Phi^{-}}$
where
$\ket{\Phi^{-}}=(2)^{-1/2}[\ket{\downarrow\downarrow}-\ket{\uparrow\uparrow}]$.

The states $\ket{\Psi^{\pm}}$ and $\ket{\Phi^{\pm}}$ are the Bell basis states
bellstates  for Bob and Claire spins. We shall verify in this work that the
parameter $\alpha$ plays a crucial role in the quantum coherences developed by
the spectator Claire which contain information on Alice and Bob dynamics. In
the description of relativistic quantum information, the identification of
observer independent quantities is crucial. As discussed in BERTLMANN , the
entanglement between the spins of two particles is not Lorentz invariant in
general. Bringing in the momenta of the relativistic particles, in a bipartite
Alice and Bob system, Lorentz invariance of entanglement can be claimed for
the Hilbert space partition into Alice and Bob particles. The entanglement in
other spin and momentum partitions turns out to be observer dependent, or in
another words it can be transferred between them. Obviously it is interesting
and equally more involved to study the effect of entanglement transformation
in a relativistic multi-particle setting where different partitions of
multipartite entanglement are studied including Lorentz frame dependence. In
this sense, the system we present here turns to be an ideal playground for
studying how the dynamics of Alice and Bob scattering particles is sensed by
Claire’s which is only entangled in spin degrees of freedom throughout the
scattering process, providing indirect information of the cross section
through spin measurements in the out-state. We explore how the entanglement is
distributed as measured by the von Neumann entropy.

This contribution is organized in the following manner: In Sec. II,
definitions are made, including the in-state proposed with all possible spin
configurations and, with the help of the scattering matrix, the out-state is
constructed. In Sec. III, final reduced density matrix of spectator particle
$C$ is calculated. In Sec. IV, the theory is applied for the particular QED
process $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$, emphasizing the results found
on $C$ restricted to the tripartite and Bob and Claire spin entanglements.
Final discussion and perspectives are commented in Sec. V.

##  II In-state and 2-particle scattering

In this section we set up our notation. As usual, one describes asymptotic
particle states in the distant past ($t\rightarrow-\infty$) and future
($t\rightarrow+\infty$) as wave packets which are well separated in position
space and narrowly peaked in momentum space. Here, without loss of generality,
we take them plane waves with definite momenta. The unitary scattering matrix
$\bm{\widehat{\mathcal{S}}}$ is defined as the time evolution operator in the
interaction picture in terms of asymptotic states. Therefore for an initially
factorized state, we have

$\displaystyle\ket{\psi_{\text{in}}^{\text{fact}}}$ $\displaystyle=$
$\displaystyle\bigotimes_{j=1}^{n}\ket{\textbf{{p}}_{\textup{i}}^{j},s_{\textup{i}}^{j}},$
$\displaystyle\ket{\psi_{\textup{out}}}$ $\displaystyle=$
$\displaystyle\bigg{[}\bigotimes_{k=1}^{m}\mathcal{\widehat{I}}^{(k)}_{\textup{f}}\bigg{]}\bm{\widehat{\mathcal{S}}}\ket{\psi_{\textup{in}}^{\text{fact}}},$
(6)

for $n(m)$ initial (final) particles, with

$\mathcal{\widehat{I}}_{\textup{f}}=\sum_{s_{\textup{f}}}\int_{\textbf{{p}}_{\textup{f}}}\ket{\textbf{{p}}_{\textup{f}},s_{\textup{f}}}\bra{\textbf{{p}}_{\textup{f}},s_{\textup{f}}},$
(7)

in which for short
$\int_{\textbf{{p}}_{\textup{f}}}\equiv[(2\pi)^{3}2E_{\textbf{{p}}_{\textup{f}}}]^{-1}d^{3}\textbf{{p}}_{\textup{f}}$
is the Lorentz invariant measure over final momentum and
$\sum_{s_{\textup{f}}}$ is the sum over final spin possibilities. It is useful
to define the transition matrix $\bm{\widehat{\mathcal{T}}}$ such that
$\bm{\widehat{\mathcal{S}}}=\bm{\widehat{\mathcal{I}}}+i\bm{\widehat{\mathcal{T}}}$
which accounts for the non-trivial of scattering amplitudes and is related to
the transition amplitude $\mathcal{M}$ as given by Feynman diagrams as

$\displaystyle\braket{\textbf{{p}}_{\textup{f}}^{(1)},s_{\textup{f}}^{(1)};\textbf{{p}}_{\textup{f}}^{(2)},s_{\textup{f}}^{(2)}}{i\bm{\widehat{\mathcal{T}}}}{\textbf{{p}}_{\textup{i}}^{(1)},s_{\textup{i}}^{(1)};\textbf{{p}}_{\textup{i}}^{(2)},s_{\textup{i}}^{(2)}}$
$\displaystyle=$ $\displaystyle
i(2\pi)^{4}\delta^{4}(p_{\text{i}}^{(1)}+p_{\text{i}}^{(2)}-p_{\text{f}}^{(1)}-p_{\text{f}}^{(2)})\mathcal{M}_{\text{i}\rightarrow\text{f}},$
(8)

for $n=m=2$, and

$\mathcal{M}_{\text{i}\rightarrow\text{f}}=\mathcal{M}^{\textbf{\text{p}}_{\text{i}}^{(1)},\textbf{\text{p}}_{\text{i}}^{(2)}}_{\textbf{\text{p}}_{\text{f}}^{(1)},\textbf{\text{p}}_{\text{f}}^{(2)}}(s_{\text{i}}^{(1)}s_{\text{i}}^{(2)}\rightarrow
s_{\text{f}}^{(1)}s_{\text{f}}^{(2)}).$ (9)

We study the 2-particle scattering of the tripartite in- spin-$1/2$ state
which we call Alice, Bob and Claire states, factorized in the momentum
variables and whose spin is described by the general state in equation (70),
namely

$\ket{\psi_{\text{in}}}=\bigg{[}\bigotimes_{j=1}^{3}\ket{\textup{{p}}_{j}}\bigg{]}\otimes\ket{\mathscr{S}},$
(10)

where, for shorthand notation, we set
$\ket{\textup{{p}}_{1}}\equiv\ket{\textup{{p}}_{\textup{i}}^{(1)}}$,
$\ket{\textup{{p}}_{2}}\equiv\ket{\textup{{p}}_{\textup{i}}^{(2)}}$ and
$\ket{\textup{{q}}}\equiv\ket{\textup{{p}}_{\textup{i}}^{(3)}}$. The out-state
then is

$\displaystyle\ket{\psi_{\text{out}}}$ $\displaystyle=$
$\displaystyle\ket{\psi_{\text{in}}}+i\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{\textup{{p}}_{3},\textup{{p}}_{4},r,s}(2\pi)\delta(E_{\textbf{\text{p}}_{1}}+E_{\textbf{\text{p}}_{2}}-E_{\textbf{\text{p}}_{3}}-E_{\textbf{\text{p}}_{4}})$
(20)
$\displaystyle(2\pi)^{3}\delta^{3}(\textbf{\text{p}}_{1}+\textbf{\text{p}}_{2}-\textbf{\text{p}}_{3}-\textbf{\text{p}}_{4})\Big{[}c_{1}\mathcal{M}(\uparrow\uparrow\rightarrow
rs)\ket{\textup{{q}},\uparrow}$
$\displaystyle+c_{2}\mathcal{M}(\uparrow\uparrow\rightarrow
rs)\ket{\textup{{q}},\downarrow}+c_{3}\mathcal{M}(\uparrow\downarrow\rightarrow
rs)\ket{\textup{{q}},\uparrow}$
$\displaystyle+c_{4}\mathcal{M}(\uparrow\downarrow\rightarrow
rs)\ket{\textup{{q}},\downarrow}+c_{5}\mathcal{M}(\downarrow\uparrow\rightarrow
rs)\ket{\textup{{q}},\uparrow}$
$\displaystyle+c_{6}\mathcal{M}(\downarrow\uparrow\rightarrow
rs)\ket{\textup{{q}},\downarrow}+c_{7}\mathcal{M}(\downarrow\downarrow\rightarrow
rs)\ket{\textup{{q}},\uparrow}$
$\displaystyle+c_{8}\mathcal{M}(\downarrow\downarrow\rightarrow
rs)\ket{\textup{{q}},\downarrow}\Big{]}\otimes\ket{\textbf{\text{p}}_{3},r}\otimes\ket{\textbf{\text{p}}_{4},s}$
$\displaystyle\equiv\ket{\psi_{\text{in}}}+\ket{\psi_{\text{trans}}},$

where the subscript “trans” stands for transition, the non-trivial part of the
$out$-state and we omitted the momentum indices
$\textbf{\text{p}}_{1},\textbf{\text{p}}_{2},\textbf{\text{p}}_{3},\textbf{\text{p}}_{4}$
in the $\mathcal{M}$’s. It is noteworthy that in the case of an inelastic QED
scattering such as $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$, the amplitude
$\mathcal{M}$ has the s-channel only. Thus we call the particle states
$\ket{\textup{{p}}_{1},\uparrow(\downarrow)}$ and $\ket{\textup{{p}}_{3},r}$
as system $A$, whereas the antiparticle states
$\ket{\textup{{p}}_{2},\uparrow(\downarrow)}$ and $\ket{\textup{{p}}_{4},s}$
are called system $B$. The spectator particle
$\ket{\textup{{q}},\uparrow(\downarrow)}$, is called system $C$. The total
density matrix is given by

$\rho_{\text{out}}=\mathcal{N}^{-1}\ket{\psi_{\text{out}}}\bra{\psi_{\text{out}}},$
(21)

and the reduced density matrices of a system are obtained by tracing out the
other systems, with the trace defined by
$\text{Tr}[\rho]=\sum_{\sigma}\int_{\textbf{\text{k}}}\bra{\textbf{\text{k}},\sigma}\rho\ket{\textbf{\text{k}},\sigma}$,
to yield

$\displaystyle\rho_{\text{A}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{B}}\text{Tr}_{\text{C}}[\rho_{\text{out}}],$
$\displaystyle\rho_{\text{B}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{A}}\text{Tr}_{\text{C}}[\rho_{\text{out}}],$
$\displaystyle\rho_{\text{C}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{A}}\text{Tr}_{\text{B}}[\rho_{\text{out}}].$
(22)

The density matrix normalization $\mathcal{N}$, complying with
$\text{Tr}[\rho_{\text{out}}]=1$, is
$\mathcal{N}=\text{Tr}_{\text{C}}\text{Tr}_{\text{A}}\text{Tr}_{\text{B}}[\rho_{\text{out}}]$.
The ordering of the traces is immaterial but we start off tracing out systems
$A$ and $B$ as given in equation (22) since we are interested in evaluating
expectation values on the system $C$. Also, it is useful to write

$\mathcal{N}=\mathcal{N}_{\text{in}}+\mathcal{N}_{\text{cross}}+\mathcal{N}_{\text{trans}},$
(23)

with

$\displaystyle\mathcal{N}_{\text{in}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{C}}\text{Tr}_{\text{A}}\text{Tr}_{\text{B}}[\ket{\psi_{\text{in}}}\bra{\psi_{\text{in}}}],$
$\displaystyle\mathcal{N}_{\text{cross}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{C}}\text{Tr}_{\text{A}}\text{Tr}_{\text{B}}[\ket{\psi_{\text{in}}}\bra{\psi_{\text{trans}}}+\ket{\psi_{\text{trans}}}\bra{\psi_{\text{in}}}],$
$\displaystyle\mathcal{N}_{\text{trans}}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\text{C}}\text{Tr}_{\text{A}}\text{Tr}_{\text{B}}[\ket{\psi_{\text{trans}}}\bra{\psi_{\text{trans}}}].$
(24)

In the trace operations we use a Lorentz invariant particle spin-momentum
state scalar product
$\braket{\textbf{\text{k}},\sigma}{\textbf{\text{p}},s}=(2\pi)^{3}2E_{\textbf{\text{p}}}\delta^{3}(\textbf{\text{k}}-\textbf{\text{p}})\delta_{\sigma,s}$,
such that

$\displaystyle\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{\bf{k},\sigma}\frac{1}{2E_{\textbf{\text{k}}}}\braket{\textbf{\text{k}},\sigma}{\textbf{\text{p}},s}\braket{\textbf{\text{p}}^{\prime},s^{\prime}}{\textbf{\text{k}},\sigma}$
$\displaystyle=$ (33) $\displaystyle
2E_{\textbf{\text{p}}}(2\pi)^{3}\delta^{3}(\textbf{\text{p}}^{\prime}-\textbf{\text{p}})\delta_{s^{\prime},s}.$
(34)

Since $\sum_{j=1}^{8}c_{j}^{2}=1$ in the initial state (10), and using
equation (34), we get the dimensionless quantity

$\mathcal{N}_{\text{in}}=8E_{\textbf{\text{q}}}E^{2}V^{3},$ (35)

where $V=(2\pi)^{3}\delta^{3}({\bm{\epsilon}})$ with
${\bm{\epsilon}}\rightarrow\bf{0}$ is the space volume, $E$ the energy of
colliding particles ($A,B$) in the centre-of-mass frame, and
$E_{\textbf{\text{q}}}$ the energy of the spectator particle C, whereas
$\mathcal{N}_{\text{cross}}$ can be easily shown to give zero. As for
$\mathcal{N}_{\text{trans}}$, using that the spin-$1/2$ state in equation (70)
is diagonal in the $z$-direction, which is also the direction of the initial
momenta before collision, as well as energy momentum conservation and
spherical symmetry of the final state momenta to simplify the integrand and
obtain

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\text{in}}+\mathcal{N}_{\text{trans}}$ (44)
$\displaystyle=$ $\displaystyle
8E_{\textbf{\text{q}}}E^{2}V^{3}+\frac{E_{\textbf{\text{q}}}TV^{2}|\textbf{\text{p}}_{3}|}{16\pi^{2}E}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{r,s}d\Omega\Lambda(r,s),$

as the total normalization. In the equation above, $\Lambda(r,s)$ stands for

$\displaystyle\Lambda(r,s)$ $\displaystyle=$
$\displaystyle(c_{1}^{2}+c_{2}^{2}+c_{7}^{2}+c_{8}^{2})|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow
rs)|^{2}$ $\displaystyle+$
$\displaystyle(c_{3}^{2}+c_{4}^{2}+c_{5}^{2}+c_{6}^{2})|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow
rs)|^{2}$ $\displaystyle+$ $\displaystyle
2(c_{3}c_{5}+c_{4}c_{6})\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow
rs)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow rs),$

where, in the centre-of-mass frame, each $\mathcal{M}_{cm}$ represents
$\mathcal{M}^{\bf{p}_{1},-\bf{p}_{1}}_{\bf{p}_{3},-\bf{p}_{3}}$ evaluated at
$|\textbf{\text{p}}_{3}|=\sqrt{E^{2}-m_{3}^{2}}$ and the infinitesimal solid
angle is $d\Omega=\sin\theta d\theta d\phi$, with $\theta$ being the
scattering angle. A barred quantity means Hermitian conjugation and $T$ is the
“time duration” such that
$2\pi\delta(E_{\text{f}}-E_{\text{i}})=\int_{-T/2}^{+T/2}\,dt\,\exp[i(E_{\text{f}}-E_{\text{i}})t]$.

##  III Total spin density matrix of system $\bm{C}$

In order to extract information about the scattering from the system $C$, we
need to evaluate its reduced density matrix $\rho_{\text{C}}$ from equation
(20). By tracing over system $B$ first, we get only two contributions

$\rho_{\text{AC}}=\frac{1}{\mathcal{N}}\Big{(}\rho_{\text{AC(in)}}+\rho_{\text{AC(trans)}}\Big{)},$
(46)

since the crossed term vanishes, $\mathcal{N}$ is given by (44) and

$\rho_{\text{AC(in)}}=8E_{\textbf{\text{q}}}E^{2}V^{3}\times$ (47)
$\times\begin{pmatrix}c_{1}^{2}+c_{3}^{2}&c_{1}c_{2}+c_{3}c_{4}&c_{1}c_{5}+c_{3}c_{7}&c_{1}c_{6}+c_{3}c_{8}\\\
\\\
c_{1}c_{2}+c_{3}c_{4}&c_{2}^{2}+c_{4}^{2}&c_{2}c_{5}+c_{4}c_{7}&c_{2}c_{6}+c_{4}c_{8}\\\
\\\
c_{1}c_{5}+c_{3}c_{7}&c_{2}c_{5}+c_{4}c_{7}&c_{5}^{2}+c_{7}^{2}&c_{5}c_{6}+c_{7}c_{8}\\\
\\\
c_{1}c_{6}+c_{3}c_{8}&c_{2}c_{6}+c_{4}c_{8}&c_{5}c_{6}+c_{7}c_{8}&c_{6}^{2}+c_{8}^{2}\end{pmatrix},$

and

$\rho_{\text{AC(trans)}}=\frac{E_{\textbf{\text{q}}}TV^{2}|\textbf{\text{p}}_{3}|}{16\pi^{2}E}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{s}d\Omega\times$
(48)
$\times\begin{pmatrix}\Lambda^{1,1}_{\text{AC}}(s)&\Lambda^{1,2}_{\text{AC}}(s)&\Lambda^{1,3}_{\text{AC}}(s)&\Lambda^{1,4}_{\text{AC}}(s)\\\
\\\
\Lambda^{1,2}_{\text{AC}}(s)&\Lambda^{2,2}_{\text{AC}}(s)&\Lambda^{2,3}_{\text{AC}}(s)&\Lambda^{2,4}_{\text{AC}}(s)\\\
\\\
\Lambda^{1,3}_{\text{AC}}(s)&\Lambda^{2,3}_{\text{AC}}(s)&\Lambda^{3,3}_{\text{AC}}(s)&\Lambda^{3,4}_{\text{AC}}(s)\\\
\\\
\Lambda^{1,4}_{\text{AC}}(s)&\Lambda^{2,4}_{\text{AC}}(s)&\Lambda^{3,4}_{\text{AC}}(s)&\Lambda^{4,4}_{\text{AC}}(s)\end{pmatrix};$

where the matrix elements $\Lambda^{i,j}_{\text{AC}}(s)$ are given by:

$\displaystyle\Lambda^{1,1}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{1}^{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{3}^{2}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{5}^{2}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{7}^{2}|\mathcal{M}(\downarrow\downarrow\rightarrow\uparrow s)|^{2}$
$\displaystyle+$ $\displaystyle
2c_{3}c_{5}\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\uparrow s),$
$\displaystyle\Lambda^{2,2}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{2}^{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{4}^{2}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{6}^{2}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{8}^{2}|\mathcal{M}_{cm}(\downarrow\downarrow\rightarrow\uparrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
2c_{4}c_{6}\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\uparrow s),$
$\displaystyle\Lambda^{3,3}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{1}^{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{3}^{2}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{5}^{2}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{7}^{2}|\mathcal{M}_{cm}(\downarrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
2c_{3}c_{5}\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow s),$
$\displaystyle\Lambda^{4,4}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{2}^{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{4}^{2}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{6}^{2}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{8}^{2}|\mathcal{M}_{cm}(\downarrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
2c_{4}c_{6}\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow s),$
$\displaystyle\Lambda^{1,2}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{1}c_{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{3}c_{4}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{5}c_{6}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\uparrow
s)|^{2}+c_{7}c_{8}|\mathcal{M}_{cm}(\downarrow\downarrow\rightarrow\uparrow
s)|^{2}$ $\displaystyle+$
$\displaystyle(c_{3}c_{6}+c_{4}c_{5})\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\uparrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\uparrow s),$
$\displaystyle\Lambda^{3,4}_{\text{AC}}(s)$ $\displaystyle=$ $\displaystyle
c_{1}c_{2}|\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{3}c_{4}|\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$ $\displaystyle
c_{5}c_{6}|\mathcal{M}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)|^{2}+c_{7}c_{8}|\mathcal{M}_{cm}(\downarrow\downarrow\rightarrow\downarrow
s)|^{2}$ $\displaystyle+$
$\displaystyle(c_{3}c_{6}+c_{4}c_{5})\mathcal{M}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow s),$
$\displaystyle\Lambda^{1,3}_{\text{AC}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)[(c_{1}c_{3}+c_{5}c_{7})\mathcal{\bar{M}}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)$ $\displaystyle+$
$\displaystyle(c_{1}c_{5}+c_{3}c_{7})\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)],$ $\displaystyle\Lambda^{1,4}_{\text{AC}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)[(c_{1}c_{4}+c_{5}c_{8})\mathcal{\bar{M}}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)$ $\displaystyle+$
$\displaystyle(c_{1}c_{6}+c_{3}c_{8})\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)],$ $\displaystyle\Lambda^{2,3}_{\text{AC}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)[(c_{2}c_{3}+c_{6}c_{7})\mathcal{\bar{M}}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)$ $\displaystyle+$
$\displaystyle(c_{2}c_{5}+c_{4}c_{7})\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)],$ $\displaystyle\Lambda^{2,4}_{\text{AC}}(s)$ $\displaystyle=$
$\displaystyle\mathcal{M}_{cm}(\uparrow\uparrow\rightarrow\uparrow
s)[(c_{2}c_{4}+c_{6}c_{8})\mathcal{\bar{M}}_{cm}(\uparrow\downarrow\rightarrow\downarrow
s)$ (49) $\displaystyle+$
$\displaystyle(c_{2}c_{6}+c_{4}c_{8})\mathcal{\bar{M}}_{cm}(\downarrow\uparrow\rightarrow\downarrow
s)].$

Then, tracing over $A$ using the orthogonality relation (34) yields the
$2\times 2$ reduced matrix for the system $C$,

$\rho_{\text{C}}=\frac{1}{\mathcal{N}}\Big{(}\rho_{\text{C(in)}}+\rho_{\text{C(trans)}}\Big{)}\equiv\begin{pmatrix}\tilde{\Lambda}_{C}^{1,1}&\tilde{\Lambda}_{C}^{1,2}\\\
\\\ \tilde{\Lambda}_{C}^{2,1}&\tilde{\Lambda}_{C}^{2,2}\\\ \end{pmatrix},$
(50)

with

$\rho_{\text{C(in)}}=8E_{\textbf{\text{q}}}E^{2}V^{3}\times$ (51)
$\times\begin{pmatrix}c_{1}^{2}+c_{3}^{2}+c_{5}^{2}+c_{7}^{2}&c_{1}c_{2}+c_{3}c_{4}+c_{5}c_{6}+c_{7}c_{8}\\\
\\\
c_{1}c_{2}+c_{3}c_{4}+c_{5}c_{6}+c_{7}c_{8}&c_{2}^{2}+c_{4}^{2}+c_{6}^{2}+c_{8}^{2}\end{pmatrix}$

and

$\rho_{\text{C(trans)}}=\frac{E_{\textbf{\text{q}}}TV^{2}|\textbf{\text{p}}_{3}|}{16\pi^{2}E}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{s}d\Omega\times$
(52)
$\times\begin{pmatrix}\Lambda^{1,1}_{\text{AC}}(s)+\Lambda^{3,3}_{\text{AC}}(s)&\Lambda^{1,2}_{\text{AC}}(s)+\Lambda^{3,4}_{\text{AC}}(s)\\\
\\\
\Lambda^{1,2}_{\text{AC}}(s)+\Lambda^{3,4}_{\text{AC}}(s)&\Lambda^{2,2}_{\text{AC}}(s)+\Lambda^{4,4}_{\text{AC}}(s)\end{pmatrix}.$

In principle, if there are modifications in the matrix elements of the reduced
density matrix $\rho_{\text{C}}$ after the scattering, as given by equations
(50), (51) and (52), the variation of the spin measurement on spectator
particle $C$, reads:

$\displaystyle\Delta\langle S_{x,y,z}\rangle=\langle
S_{x,y,z}\rangle_{\mathrm{out}}-\langle S_{x,y,z}\rangle_{\mathrm{in}}$
$\displaystyle\equiv\frac{1}{2}\text{Tr}[\sigma_{x,y,z}\,\,\rho_{\text{C}}]-\frac{1}{2}\text{Tr}[\sigma_{x,y,z}\,\,\rho_{\text{C(in)}}],$
(53)

from which we may infer dynamical information about the scattering process, as
we shall see in next sections.

##  IV Inelastic scattering $\bm{e^{+}e^{-}}\rightarrow\mu^{+}\mu^{-}$

Consider the inelastic scattering $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ in
quantum electrodynamics taken in the centre-of-mass frame, where the
electron/positron collision along the $z$-axis gives rise to a muon/anti-muon
pair with centre-of-mass momenta $p$ and $P$, respectively as depicted in
figure 1. The scattering angle is $\theta$, between $\bf{p_{1}}(\bf{p_{2}})$
and $\bf{p_{3}}(\bf{p_{4}})$ and $\phi$ is the azimuthal angle. In the initial
state expressed by equation (10), we will admit an spectator particle $C$
(Claire) which may be entangled in spin degrees of freedom with the positron
(Bob) as represented by the blue curved line, or with both the positron and
the electron (Alice) as represented by the red line, determined by specific
values of $c_{i}$’s in (70).

FIG. 1: Collision diagram for the inelastic scattering
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$.

For specific spin quantum numbers $s_{i}$, the tree level contribution to this
scattering contains only the $s$-channel represented by the Feynman diagram
depicted in figure 2, whose analytic expression reads

$\mathcal{M}_{e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}}=\frac{e^{2}}{(p_{1}^{\alpha}+p_{2}^{\alpha})^{2}}[\bar{u}_{3}^{s_{3}}\gamma^{\mu}v_{4}^{s_{4}}][\bar{v}_{2}^{s_{2}}\gamma_{\mu}u_{1}^{s_{1}}],$
(54)

FIG. 2: $s$ channel for the inelastic scattering
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$.

where

$\displaystyle u_{1}^{\uparrow}=N_{1}\begin{pmatrix}1\\\ 0\\\
\frac{p}{E+m_{e}}\\\ 0\end{pmatrix},\quad
u_{1}^{\downarrow}=N_{1}\begin{pmatrix}0\\\ 1\\\ 0\\\
\frac{-p}{E+m_{e}}\end{pmatrix},$
$\displaystyle\bar{v}_{2}^{\uparrow}=N_{1}\begin{pmatrix}0,\frac{p}{E+m_{e}},0,1\end{pmatrix}\gamma^{0},\quad\bar{v}_{2}^{\downarrow}=-N_{1}\begin{pmatrix}\frac{-p}{E+m_{e}},0,1,0\end{pmatrix}\gamma^{0},$
$\displaystyle\bar{u}_{3}^{\uparrow}=N_{2}\begin{pmatrix}1,0,\frac{P}{E+m_{\mu}}\cos\theta,\frac{P}{E+m_{\mu}}\sin\theta
e^{-i\phi}\end{pmatrix}\gamma^{0},$
$\displaystyle\bar{u}_{3}^{\downarrow}=N_{2}\begin{pmatrix}0,1,\frac{P}{E+m_{\mu}}\sin\theta
e^{i\phi},\frac{-P}{E+m_{\mu}}\cos\theta\end{pmatrix}\gamma^{0},$
$\displaystyle
v_{4}^{\uparrow}=N_{2}\begin{pmatrix}\frac{-P}{E+m_{\mu}}\sin\theta
e^{-i\phi}\\\ \frac{P}{E+m_{\mu}}\cos\theta\\\ 0\\\
1\end{pmatrix},v_{4}^{\downarrow}=-N_{2}\begin{pmatrix}\frac{-P}{E+m_{\mu}}\cos\theta\\\
\frac{-P}{E+m_{\mu}}\sin\theta e^{i\phi}\\\ 1\\\ 0\end{pmatrix},$ (55)

with normalizations $N_{1}=\sqrt{E+m_{e}}$, and $N_{2}=\sqrt{E+m_{\mu}}$ in
terms of the masses of the electron $m_{e}$ and the muon $m_{\mu}$. In order
to the produce a $\mu^{+}\mu^{-}$ pair, the centre of mass energy $\sqrt{s}$
must be such that $s=(p_{1}+p_{2})^{2}=4E^{2}\geq 4m_{\mu}^{2}$.

###  IV.1 Entangled spin tripartite initial state

Consider firstly GHZ and W initial spin in-states. In the scattering of $AB$
system, the composite states obey
$\rho_{\text{AC}}^{\text{GHZ}}=\rho_{\text{BC}}^{\text{GHZ}}$ and
$\rho_{\text{AC}}^{\text{W}}=\rho_{\text{BC}}^{\text{W}}$ before and after
collision. In other words, the collision affects equally the bi-partitions
$AC$ and $BC$, resulting in equivalent spin density matrices. This result is
expected because both $A$ and $B$ are equally entangled with $C$. On the other
hand, the von Neumann entropy $\text{S}_{\text{Neumann}}$,

$\text{S}^{\text{XC}}_{\text{Neumann}}=-\text{Tr}[\rho_{\text{XC}}\ln\rho_{\text{XC}}],$
(56)

$\text{X}=\\{A,B\\}$, will always increase (since the total system $ABC$
starts as a pure state with a null entropy horodick ), and it does so more
significantly after the collision where there are more contributing states for
all spin configurations defined by the constants $c_{(j)}$. This in general
leads to a change in the entropy of individual systems. Let us specifically
study the variation

$\Delta\text{S}^{\text{C}}_{\text{Neumann}}=(\text{S}^{\text{C}}_{\text{Neumann}})_{\text{out}}-(\text{S}^{\text{C}}_{\text{Neumann}})_{\text{in}}$
(57)

with
$\text{S}^{\text{C}}_{\text{Neumann}}=-\text{Tr}[\rho_{\text{C}}\ln\rho_{\text{C}}],$
for a spin configuration such that the in-state is characterised by GHZ and W
tripartite spin states. For the state GHZ, although the bipartite $AC$ or $BC$
state becomes more mixed, the entropy of the reduced system will not change as
it started at its maximal mixedness degree. Hence the von Neumann entropy of
the spectator particle $C$ will not be affected by the collision for the GHZ
configuration. On the other hand for the state W, the systems do not start
with a maximum mixedness, and thus the collision at centre-of-mass energy $E$
can increase the entropy of particle $C$. We can schematically represent these
features in terms of the reduced density matrices. For the GHZ state,

$\displaystyle\begin{pmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{pmatrix}_{\text{GHZ}}^{\text{A,B,C}}\xrightarrow{\text{Scattering}}\begin{pmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{pmatrix}_{\text{GHZ}}^{\text{A,B,C}}$ (58)

whereas for the W state, for the systems $A$ and $B$,

$\displaystyle\begin{pmatrix}\frac{1}{3}&0\\\
0&\frac{2}{3}\end{pmatrix}_{\text{W}}^{\text{A,B}}\xrightarrow{\text{Scattering}}\begin{pmatrix}f_{1}(E)&0\\\
0&f_{2}(E)\end{pmatrix}_{\text{W}}^{\text{A,B}}$ (59)

in which $f_{1}(E)$ and $f_{2}(E)$ are functions of the energy $E$, and for
the spectator particle C,

$\displaystyle\begin{pmatrix}\frac{1}{3}&0\\\
0&\frac{2}{3}\end{pmatrix}_{\text{W}}^{\text{C}}\xrightarrow{\text{Scattering}}\begin{pmatrix}f_{3}(E)&0\\\
0&f_{4}(E)\end{pmatrix}_{\text{W}}^{\text{C}}$ (60)

where here we write out

$f_{3}(E)\equiv\frac{24\pi VE^{7}+E^{2}\mathscr{E}^{2}}{72\pi
VE^{7}+(2m^{2}_{e}+E^{2})\mathscr{E}^{2}}$ (61)

and

$f_{4}(E)\equiv\frac{48\pi VE^{7}+2m_{e}^{2}\mathscr{E}^{2}}{72\pi
VE^{7}+(2m^{2}_{e}+E^{2})\mathscr{E}^{2}},$ (62)

with $\mathscr{E}^{2}\equiv
e^{4}T\sqrt{E^{2}-m_{\mu}^{2}}(m_{\mu}^{2}+2E^{2})$. The graph in figure 3
shows the variations of the von Neumann entropy and the measured spin of the
spectator particle $C$ in the $z$-direction as well as the cross section for
the $AB$ collision, which in the center-of-mass frame reads:

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{2}s}\frac{P}{p}\int
d\Omega\,|\mathcal{M}_{\text{i}\rightarrow\text{f}}|^{2}.$ (63)
$\displaystyle\equiv$
$\displaystyle\frac{e^{4}(m_{e}^{2}+E^{2})(m_{\mu}^{2}+2E^{2})}{48\pi
E^{6}}\sqrt{\frac{E^{2}-m_{\mu}^{2}}{E^{2}-m_{e}^{2}}},$

all as a function of the energy $E$. We display in the left vertical axis the
values associated with the dimensionless quantities
$\Delta\text{S}_{\text{Neumann}}$ and $\Delta\langle S_{z}\rangle$
($\hbar=1$), whilst the right vertical axis corresponds to the total cross
section values in $\text{MeV}^{-2}$. Notice that as no off-diagonal elements
are developed for the reduced matrix $\rho_{\text{C}}$, $\Delta\langle
S_{x,y}\rangle=0$. Furthermore, starting from the threshold energy for this
process where $E=m_{\mu}$, figure 3 shows a common peak around $1.18m_{\mu}$
for the three curves.

FIG. 3: Variations in the state W of the von Neumann entropy and the measured
spin in the $z$-direction for $C$, and cross section for the scattering $AB$.

It is also possible to relate the von Neumann entropy and the cross section as
well as the variation of the expectation value of the spectator particle spin
in the $z$-direction as both observables, since they can be expressed in terms
of the energy $E$ via equations (61), (62) and (63). This is displayed in
figure 4.

FIG. 4: Variations in the state W of the von Neumann entropy and the measured
spin in the $z-$direction for $C$ in terms of the cross section.

Moreover, in FAN1 , the inelastic scattering
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ was studied and the variation of
entanglement entropy between the initial state and final state was seen to be
proportional to the total cross section. In our case, we can also see in
figure 4 that the variation of von Neumann entropy of the spectator particle
is proportional to the cross section.

###  IV.2 Bob and Claire initial entangled spin state

Contrarily to the GHZ and W initial entangled states, in this case the
bipartitions $AC$ and $BC$ are not symmetrical. That is because $AC$ begins as
a mixed state namely $\rho_{\text{AC}}=\rho_{\text{A}}\otimes\rho_{\text{C}}$,
where $\rho_{\text{A}}$ is a pure and $\rho_{\text{C}}$ (just as
$\rho_{\text{B}}$) is a mixed state derived of entanglement, whereas BC begins
as a pure entangled state. The degree of the mixedness associated to the spin
density matrix of any individual system ($A$, $B$ and $C$) must comply with
the fact that the mixedness of a bipartitions ($AC$, $BC$ and $AB$) increase
due to the scattering. In the case of the system $C$, the mixedness as
measured by the von Neumann entropy clearly decreases as seen in figure 5.

FIG. 5: Variation of the von Neumann entropy for $C$ in the state
$\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{\pm}(\Phi^{\pm})}$.

For $\alpha=n\pi/4$, $n$ being and integer number, this variation decreases
with the energy $E$ when $n$ is even and vanishes for $n$ odd. In our general
state $\ket{\text{A}^{\alpha}}\otimes\ket{\Psi^{\pm}(\Phi^{\pm})}$ state,
system $A$ starts off maximally pure, that is its initial entropy
$\text{S}^{\text{A}}_{\text{Neumann}}$ is zero. On the other hand given that
the mixedness of the bipartitions $AB$ and $AC$ must increase after the
scattering, so does the entropy of system $A$ (as it cannot decrease). On the
other hand, for the individual systems $B$ and $C$ the opposite happens: they
cannot increase their mixedness beyond the limit
$\text{S}_{\text{Neumann}}^{\text{B,C}}=\ln 2$, and thus a decrease of their
mixedness is the only way to satisfy the constraints imposed by the increase
of mixedness of the respective composite bipartite systems due to scattering.

The effect of the parameter $\alpha$ on the variation of spin expectation
value for system $C$ is deduced from its reduced density matrices:

$\displaystyle\begin{pmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{pmatrix}_{\Psi^{\pm}}^{\text{C}}\xrightarrow{\text{Scattering}}\begin{pmatrix}g_{1}(E,\alpha)&\frac{1}{2}\sin(2\alpha)g_{3}^{\pm}(E)\\\
\\\
\frac{1}{2}\sin(2\alpha)g_{3}^{\pm}(E)&g_{2}(E,\alpha)\end{pmatrix}_{\Psi^{\pm}}^{\text{C}}$
(64) $\displaystyle\begin{pmatrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{pmatrix}_{\Phi^{\pm}}^{\text{C}}\xrightarrow{\text{Scattering}}\begin{pmatrix}g_{2}(E,\alpha)&\frac{1}{2}\sin(2\alpha)g_{3}^{\pm}(E)\\\
\\\
\frac{1}{2}\sin(2\alpha)g_{3}^{\pm}(E)&g_{1}(E,\alpha)\end{pmatrix}_{\Phi^{\pm}}^{\text{C}}$
(65)

where

$g_{1}(E,\alpha)=\frac{48\pi
VE^{7}+\mathscr{E}^{2}[m^{2}_{e}\cos^{2}\alpha+2E^{2}\sin^{2}\alpha]}{96\pi
VE^{7}+(m^{2}_{e}+2E^{2})\mathscr{E}^{2}},$ (66) $g_{2}(E,\alpha)=\frac{48\pi
VE^{7}+\mathscr{E}^{2}[m^{2}_{e}\sin^{2}\alpha+2E^{2}\cos^{2}\alpha]}{96\pi
VE^{7}+(m^{2}_{e}+2E^{2})\mathscr{E}^{2}},$ (67)
$g_{3}^{\pm}(E)=\pm\frac{m^{2}_{e}\mathscr{E}^{2}}{96\pi
VE^{7}+(m^{2}_{e}+2E^{2})\mathscr{E}^{2}};$ (68)

the superscript $+$ ($-$) refers to $\Psi^{+},\Phi^{+}$ ($\Psi^{-},\Phi^{-}$).
Interestingly, the off-diagonal values of the reduced density matrix for the
system $C$ in equations (64) and (65) are identical and proportional to the
off diagonal elements of the density matrix for system $A$ before scattering:

$\displaystyle\begin{pmatrix}\cos^{2}\alpha&\frac{1}{2}\sin(2\alpha)\\\ \\\
\frac{1}{2}\sin(2\alpha)&\sin^{2}\alpha\end{pmatrix}_{\Psi^{\pm},\Phi^{\pm}}^{\text{A}}$

In other words, the $\alpha$ parameter which specifies the spin state of the
system $A$ has an important role in the computation of spin expectation values
for the system $C$ in the $x$-direction after the scattering. Therefore we
expect the same behavior of $\Delta\langle S_{x}\rangle$ for
$\ket{\text{A}^{\alpha}}\otimes\ket{\Psi^{+}},\ket{\text{A}^{\alpha}}\otimes\ket{\Phi^{+}}$
as well as for
$\ket{\text{A}^{\alpha}}\otimes\ket{\Psi^{-}},\ket{\text{A}^{\alpha}}\otimes\ket{\Phi^{-}}$
as can be seen in figures 6(a) and 6(b). When the systems $B$ and $C$ are
coupled in the Bell bases $\ket{\Psi^{+}(\Phi^{+})}$ the dependence on
$\alpha$ is such that for $\alpha=n\pi/4$, $n$ being and integer number,
$\Delta\langle S_{x}\rangle$ decreases with the energy $E$ when $n$ is odd
(with maxima and minima close to $E=m_{\mu}$) and vanishes for $n$ even. For
the Bell bases $\ket{\Psi^{-}(\Phi^{-})}$ we have a similar pattern, except
for the interchange between maxima and minima with respect to the other Bell
basis.

(a) $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{+}(\Phi^{+})}$ state.

(b) $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{-}(\Phi^{-})}$ state.

FIG. 6: Variation of the measured spin in the $x-$direction for $C$ in the
states $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{\pm}(\Phi^{\pm})}$.

Recall that for the initial state discussed in the previous section, namely
the state W, the reduced density matrix of system $C$ has not developed off-
diagonal elements after the scattering and hence $\Delta\langle
S_{x}\rangle=0$. Notice that diagonal elements of the reduced density matrix
$C$ in equations (64) and (64) are defined by the functions
$g_{1,2}(E,\alpha)$ in (66) and (67). Such diagonal elements, which play a key
role in the determination of $\langle S_{z}^{C}\rangle$, have terms
proportional to diagonal elements of the initial density matrix of the system
$A$, namely $\cos^{2}\alpha$ and $\sin^{2}\alpha$ besides a term proportional
to $E^{7}$. These diagonal elements are interchanged in the states
$\ket{\Psi^{\pm}}$ and $\ket{\Phi^{\pm}}$ as can be seen from the density
matrices given by the equations (64) and (65), respectively. This feature can
be observed in the graphs depicted in figure 7(a) and 7(b), where the
variation of the spin measured for system $C$ in the $z$-direction is plotted
as a function of the energy $E$ and the angle $\alpha$ that characterises the
initial spin state of system $A$. For the Bell bases $\ket{\Psi^{\pm}}$ taking
for $\alpha=n\pi/4$, $\Delta\langle S_{z}\rangle$ decreases with the energy
$E$ when $n$ is even (with maxima and minima ranging from $E=m_{\mu}$ to
$E=1.5m_{\mu}$) and vanishes for $n$ odd. For the Bell bases
$\ket{\Phi^{\pm}}$ in figure 7(b) we have a similar pattern, except for the
interchange between maxima and minima with respect to figure 7(a). It is
noteworthy that the $\Delta\langle S_{z}\rangle$ is 5 orders of magnitude
greater than $\Delta\langle S_{x}\rangle$ in the Bell state configurations and
1 order of magnitude than $\Delta\langle S_{z}\rangle$ calculated for particle
$C$ taking a state W as the initial spin state.

(a) $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{\pm}}$ state.

(b) $\ket{\textup{A}^{\alpha}}\otimes\ket{\Phi^{\pm}}$ state.

FIG. 7: Variation of the measured spin in the $z-$direction for $C$ in the
states $\ket{\textup{A}^{\alpha}}\otimes\ket{\Psi^{\pm}(\Phi^{\pm})}$.

We may summarize the case in which Bob and Claire are initially entangled
whereas Alice is in a superposition spin state in the following table.

(r,1cm,0.5cm)[2pt]—c—c—c—c—c—c——c—c— Initial Spin of A &
$\bm{\braket{S_{x}^{\text{A}}}}$ $\bm{\Delta\braket{S_{x}^{\text{C}}}}$ in
$\bm{\ket{\Psi^{\pm}(\Phi^{\pm})}}$ $\bm{\braket{S_{z}^{\text{A}}}}$
$\bm{\Delta\braket{S_{z}^{\text{C}}}}$ in $\bm{\ket{\Psi^{\pm}}}$
$\bm{\Delta\braket{S_{z}^{\text{C}}}}$ in $\bm{\ket{\Phi^{\pm}}}$
$\cos\alpha\ket{\uparrow}+\sin\alpha\ket{\downarrow}$
$\frac{1}{2}\sin(2\alpha)$ $\frac{1}{2}\sin(2\alpha)\cdot g_{3}^{\pm}(E)$
$\frac{1}{2}\cos(2\alpha)$ $\frac{1}{2}\cos(2\alpha)\cdot h_{3}(E)$
$-\frac{1}{2}\cos(2\alpha)\cdot h_{3}(E)$

Table 1: $\alpha$-dependence in the spin expectation values of the spectator
particle $C$ given the initial state of the system $A$ governed by $\alpha$.

where

$h_{3}(E)=\frac{(m^{2}-2E^{2})\mathscr{E}^{3}}{96\pi
VE^{7}+(m^{2}+2E^{2})\mathscr{E}^{3}}.$ (69)

Just as W state, it is possible to parametrize the cross section and
expectation value through the energy $E$ and $\alpha$. However, $\alpha$ is
chosen such that $A$ is initially polarized in order to compare the
expectation value in the $z$-direction:

FIG. 8: $\Delta S_{z}$ for particle C as a function of the cross section for
$\ket{\Psi^{\pm}(\Phi^{\pm})}$ states.

There is a direct (inverse) linear proportionality between the two quantities
as the W state.

##  V Final discussion and perspectives

As seen in all the entangled states, the mixedness of the spectator particle
$C$ is premeditated by the restrictions imposed by the composite systems $AC$
and $BC$, causing its entropy (and consequently its expected value) to
increase, decrease or remain invariant depending on the entangled state in
question. Thus, for the state GHZ, its mixedness remains invariant, supported
by the invariance of its density matrix before and after the collision; in
that case, $C$ has no information, through the expectation value, in neither
direction. For the state W, its mixedness increases favored by the emergence
of values in the diagonal elements that depend on the energy $E$. For the
states $\ket{\Psi^{\pm}(\Phi^{\pm})}$, its mixture decreases since it started
as a maximally mixed state; secondly, the creation of elements inside and
outside the diagonal dependent on the energy $E$ makes to $C$ have information
about the collision in either direction. Particularly, the expectation values
are proportional to those shown by the system $A$ before the collision as was
shown in the table 1. On the other hand, we could consider system $A$
initially in a non polarized state, say a statistical mixture
$\rho_{\text{A}}=(2)^{-1}[\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}]$
and $BC$ entangled in spin as

$\ket{\mathscr{S}}_{\text{BC}}=d_{1}\ket{\uparrow\uparrow}+d_{2}\ket{\uparrow\downarrow}+d_{3}\ket{\downarrow\uparrow}+d_{4}\ket{\downarrow\downarrow}.$
(70)

In PRD2019 , it was studied the case $d_{2}=d_{3}=0$ and $d_{1}=\cos\eta$,
$d_{2}=e^{i\beta}\sin\eta$ which yields, for the initial state,

$\rho_{\text{in}}^{\text{ABC}}=\begin{pmatrix}\frac{1}{2}&0\\\ \\\
0&\frac{1}{2}\end{pmatrix}\otimes\begin{pmatrix}\cos^{2}\eta&0&0&e^{-i\beta}\sin\eta\cos\eta\\\
0&0&0&0\\\ 0&0&0&0\\\
e^{i\beta}\sin\eta\cos\eta&0&0&\sin^{2}\eta\end{pmatrix}.$

Interestingly, with such an initial state as input, after the scattering the
resulting reduced density matrices of $A$, $B$ and $C$ systems turn out to be
identical to the GHZ density matrices seen in equation (58). Consequently, for
this particular state we would have $\Delta\langle
S_{x,z}^{\text{C}}\rangle=0$, contrarily to the conclusion in PRD2019 . This
result is guaranteed for any other combination of $d_{j}$, since any other
entanglement of two particles implies the same behavior in the individual
systems $B$ and $C$ as seen in $\ket{\Psi^{\pm}(\Phi^{\pm})}$; a combination
in which all the $d_{j}$ are different from $0$ would lead to a product state
in which $C$ could clearly be isolated from the scattering process between $A$
and $B$.

The collision analyzed above did not present any mathematical difficulties, as
the density matrices were fully analytical in their domain; i.e.,
$0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$. This is supported by the fact
that an inelastic collision of this type is given by the s-channel, where
$\mathcal{M}\propto[2E^{2}]^{-1}$, and the energy required for this collision
begins with the mass of the muon: 105.7. For an elastic collision, the
scatterings Møller ($e^{-}e^{-}\rightarrow e^{-}e^{-}$) and Bhabha
($e^{-}e^{+}\rightarrow e^{-}e^{+}$) for example, where channels t and u given
by $\mathcal{M}\propto[-2p^{2}(1\pm\cos\theta)]^{-1}$ represent the
scattering, it could have a different behavior when including the non-analytic
variable $\theta$ in its entire domain ($\theta=0,\pi$); in this case, the
density matrices could not be integrated into the variable $\theta$. However,
this does not mean the impossibility of carrying out a new analysis: an
equivalence could be planned with the differential cross section per unit of
polar angle $d\sigma/d\theta$, as well as one among the total cross section
$\sigma$ was raised for the inelastic case. In these circumstances, a
contribution from the variable $\theta$ could be expected, such that the
expectation values in the witness particle are much more significant than
those of the inelastic collision.

###### Acknowledgements.

MS thanks CNPq for a research grant 302790/2020-9. IGP thanks CNPq for the
Grant No.307942/2019-8. We acknowledge support from CFisUC and FCT through the
project UID/FIS/04564/ 2020, grant CERN/FIS-COM/0035/2019. JDF was financed in
part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES).

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