aainstitutetext: Centre for Cosmology, Particle Physics and Phenomenology
(CP3),
Université catholique de Louvain, 1348 Louvain-la-Neuve,
Belgiumbbinstitutetext:  Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israelccinstitutetext:
Dipartimento di Fisica e Astronomia, Università di Bologna and INFN,
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italyddinstitutetext:
DAMTP, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

# Probing new physics through entanglement in diboson production

Rafael Aoude b Eric Madge a,c Fabio Maltoni d and Luca Mantani
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Pair production of heavy vector bosons is a key process at colliders: it
allows to test our understanding of the Standard Model and to explore the
existence of new physics through precision measurements of production rates
and differential distributions. New physics effects can be subtle and often
require observables specifically designed for their detection. In this study,
we focus on quantum information observables that characterise the spin states
of the final diboson system. We analyse concurrence bounds, purity, and Bell
inequalities for a bipartite qutrit system representing two massive gauge
bosons. Our findings show that quantum spin observables can serve as
complementary probes for heavy new physics as parametrised by higher
dimensional operators in the Standard Model effective field theory. In
particular, we find that these observables offer increased sensitivity to
operators whose contributions do not interfere with the Standard Model
amplitudes at the level of differential cross sections.

††preprint: IRMP-CP3-23-37

##  1 Introduction

Our current understanding of fundamental interactions of elementary particles
is based on relativistic quantum field theories (QFTs) that are built upon
symmetry principles. Lorentz and Poincaré invariance, for example, not only
dictate the possible particle content of the theory (in terms of group
irreducible representations), but also strongly limit the possible form of
their interactions. Charge conservation and then gauge invariance, provide
further constraints and non-trivial connections between interactions among
different particles. Finally, the requirement of renormalisability further
reduces the number of allowed interactions to a handful. The possible
interactions in a renormalisable QFT is therefore very much constrained once
the field content of the theory and its (gauge) symmetries are imposed. The
most successful and famous example of a renormalisable QFT featuring a very
limited number of interactions encoded in a simple Lagrangian, invariant with
respect to $SU(3)\times SU(2)\times U(1)$ gauge symmetries, is the Standard
Model (SM) of particle physics.

In addition to symmetries, other fundamental properties of QFTs, such as
unitarity and positivity, have shown to provide very important constraints on
the form of scattering amplitudes, which can be obscure at the Lagrangian
level. More recently, it has been suggested that being at the core of quantum
mechanics, the entanglement properties of a system could be used to provide
constraints on the underlying dynamics. For example, it was observed that the
requirement of maximal entanglement puts constrains on the form of the
interactions in QED and the EW theory Cervera-Lierta:2017tdt , while in Refs.
Beane:2018oxh ; Low:2021ufv  a very interesting relation between minimisation
of entanglement and enhanced symmetries was observed for low energy QCD. These
works focus on the entanglement of evolution dictated by the $S$-matrix, which
acts a quantum logical gate, between the initial and the same final states. A
different approach consists in studying the pattern of entanglement of a given
final state in a generic scattering amplitude. The simplest example is to
consider the spin degrees of freedom, described by a correlation matrix
($R$-matrix) which is pertubatively computable in QFT, and see how predictions
change depending on the form of the interactions. Although different, the
above two approaches allow to explore the relation between symmetries and
entanglement.

Recently, the authors of Ref. Afik:2020onf  have pointed out that the quantum
information properties of the spin states of top-anti-top quark pairs at
proton colliders should be already accessible in current data. Using two
measures of entanglement, the concurrence and the Peres-Horodecki criterion,
they identified two phase space regions featuring maximal entanglement: at
threshold and at high-$p_{T}$. This has triggered a series of studies on
Standard Model $t\bar{t}$ production, that have further elaborated on the
experimental detection strategy Fabbrichesi:2021npl ; Severi:2021cnj ;
Afik:2022kwm ; Aguilar-Saavedra:2022uye ; Dong:2023xiw . Moreover, other
observables have been explored such as quantum steering and discord that allow
the top-anti-top spin correlations Afik:2022dgh  to be organised in graded
sets of quantum correlations characteristic of a two-qubit state:

$\displaystyle\text{Spin
correlations}\supseteq\text{Discord}\supseteq\text{Entanglement}\supseteq\text{Steering}\supseteq\text{Bell
Inequalities}\,.$

In Ref. Aoude:2022imd , we have proposed to use quantum observables in
$t\bar{t}$ to search for physics beyond the Standard Model (BSM), _i.e._ , to
study the structure and properties of fundamental interactions at very high
scales. Working in the SM Effective Field Theory (SMEFT) framework, which
allows to “deform” the SM in a consistent way (_i.e._ compatible with the
gauge symmetries and the particle content of the theory), we have calculated
the new physics contributions proportional to the Wilson coefficients (with a
linear and quadratic111Strictly speaking, by quadratic we here mean effects
from the square of linear EFT amplitudes. In this work, we do not consider
double dimension-six insertions or linear dimension eight, which are formally
at the same order in the EFT expansion, _i.e._ $\Lambda^{-4}$. dependence), to
the concurrence fully analytically at the tree level. We have found that, in
the SMEFT analysis of $t\bar{t}$ production, higher-dimensional operators
reduce the entanglement generated by the SM. The same conclusion was later
corroborated also at next-to-leading order (NLO) in $\alpha_{S}$, confirming
the expectation that loop effects do not drastically change the leading-order
(LO) picture Severi:2022qjy . BSM effects in $t\bar{t}$ have also been
explored in Ref. Fabbrichesi:2022ovb .

One of the reasons for interest in the top quark pair system is its
simplicity: top quarks being fermionic “bare” spin-1/2 states, form bipartite
qubit systems. Beyond $t\bar{t}$ production, other two-particle final states
that feature spin correlations described by two qubits have been proposed,
from $\tau\tau$ to diphoton Fabbrichesi:2022ovb ; Altakach:2022ywa .

Qubits can model most of the SM particles – fermions and massless bosons –
with the exception of the Higgs scalar and massive gauge bosons. Leaving out
the Higgs boson which has no spin, one notes that $W^{\pm}$ and $Z$ bosons
being massive are characterised by three polarisations and therefore can be
described by qutrits. Barr et al. Barr:2021zcp ; Barr:2022wyq ; Ashby-
Pickering:2022umy  initiated the quantum studies of final states involving
massive vector bosons, first by introducing the qutrit formalism for
entanglement at colliders and then exploring Higgs boson decays and diboson
production, the latter mainly studied using a numerical approach. The same
processes have further been studied in Refs. Fabbrichesi:2023cev ; Aguilar-
Saavedra:2022mpg ; Aguilar-Saavedra:2022wam ; Fabbrichesi:2023jep . Recently,
entanglement in diboson production was also studied in the context of vector-
boson fusion Morales:2023gow  and from decays of top pairs Aguilar-
Saavedra:2023hss . The latter Ref. studies also for the first time the
detection of entanglement between a $W$ boson and a top quark.

Quantifying entanglement through the concurrence $\mathcal{C}$ Hill:1997pfa ;
10.1063/1.2795840  is a challenging analytical task for qutrits, as it
involves an optimisation procedure, and closed analytic expressions can be
only obtained in special cases or configurations. It turns out, however, that
lower and upper bounds for the concurrence can be obtained in closed form,

$\displaystyle\mathcal{C}_{\mathrm{LB}}\leq\mathcal{C}(\rho)\leq\mathcal{C}_{\mathrm{UB}},$
(1)

where $\mathcal{C}_{\mathrm{LB}}$ is the lower bound PhysRevLett.98.140505
and $\mathcal{C}_{\mathrm{UB}}$ the upper bound Zhang_2008 , and that in some
cases the bounds are so effective that they coincide with the actual values.

In this work we study the spin density matrix of diboson production, in the SM
and in SMEFT, with the goal of understanding whether quantum observables may
provide a better probe of new interactions than usual “classical” observables.
This paper is organised as follows. In Section 2, we review the formalism used
to study quantum information observables for spin-1 particles, which slightly
differs from the one used for top quarks. This formalism is then applied to
electroweak diboson production at present and future colliders. Their
interactions in the SM and in the SMEFT are described in Section 3. We proceed
by studying perturbative unitarity and its relation with entanglement in
Section 4. We finally present the results for diboson production at lepton and
proton colliders in Section 5 and conclude in Section 6. Details on the
density matrix coefficients are given in the ancillary Mathematica notebook
accompanying the arXiv submission of this manuscript.

##  2 Qutrit formalism

The spin density matrix is a fundamental object in quantum mechanics as its
knowledge completely characterises the quantum system. In this context,
quantum tomography is the idea of conducting experiments with the objective of
determining the density matrix of the system at study. This objective was at
the heart of top quark pair spin studies in Refs. Aoude:2022imd ; Afik:2020onf
; Severi:2021cnj ; Afik:2022dgh ; Afik:2022kwm ; Aguilar-Saavedra:2022uye ;
Fabbrichesi:2021npl ; Severi:2022qjy ; Fabbrichesi:2022ovb . In the
aforementioned studies, several quantum information observables have been
analysed, from spin correlations to quantum entanglement and tests of Bell
inequalities, both in the SM and exploring effects from heavy NP through SMEFT
dimension six effects.

###  2.1 Spin density matrix and quantum observables

In the following we present the theoretical framework that is going to be used
throughout the paper to describe the spin density matrix of a bipartite system
consisting of two spin-1 massive particles, _i.e._ , qutrits. The formalism
employed closely follows that of Ref. Ashby-Pickering:2022umy . While for
spin-$1/2$ particles the measurement of the particle polarisation completely
defines the spin density matrix, this is not the case anymore for spin-$1$
particles, as the polarisation vector determination is not enough to fully
characterise the system. For a generic particle of spin $s$, the Hilbert space
has dimension $d=2s+1$. The density matrix $\rho$ is therefore a $d\times d$
matrix with $d^{2}-1$ free parameters (as $\operatorname{Tr}[\rho]=1$). This
means that we can always decompose the generic one-particle spin density
matrix with the generators of $SU(d)$, _i.e._ , the generalised Gell-Mann
matrices:

$\rho=\frac{1}{d}\mathbb{I}+\sum_{i=1}^{d^{2}-1}a_{i}\lambda_{i}\,.$ (2)

As made explicit from the above decomposition, in order to fully characterise
the quantum system one needs to determine the Bloch vector ${a_{i}}$. In the
case of spin-$1/2$ particles, the Bloch vector has dimension $3$, the same of
the spin operator $\vec{S}$. This means that the density matrix can be recast
in terms of the spin operator spatial components and therefore its
determination is in one-to-one correspondence with the quantum tomography of
the system.

The situation is more complicated for particles of higher spins. For instance,
in the case of massive spin-$1$ particles, the Bloch vector has dimension $8$
and the measurement of the spin components of the particle are not sufficient
anymore for the complete characterisation of the quantum state. It is however
possible to express the spin density matrix in terms of a spin matrix
representation. In particular, given the spin-$1$ matrices

$S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&1\\\
0&1&0\end{array}\right),\quad
S_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-\mathrm{i}&0\\\
\mathrm{i}&0&-\mathrm{i}\\\ 0&\mathrm{i}&0\end{array}\right),\quad
S_{z}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{array}\right)\,,$
(3)

one can build a set of six operators

$S_{\\{ij\\}}\equiv S_{i}S_{j}+S_{j}S_{i}$ (4)

that together with the spin matrices allow us to decompose the spin density
matrix in the form

$\rho=\frac{1}{3}\mathbb{I}+\sum_{i=1}^{3}\alpha_{j}S_{i}+\sum_{i,j=1}^{3}\beta_{ij}S_{\\{ij\\}}\,.$
(5)

Note that in this formalism not all of the coefficients are free parameters
since some of these operators are not linearly independent from the identity
matrix, _i.e._

$S_{\\{xx\\}}+S_{\\{yy\\}}+S_{\\{zz\\}}=2\left(S_{x}^{2}+S_{y}^{2}+S_{z}^{2}\right)=4\,\mathbb{I}\,.$
(6)

In order for $\rho$ to have unit trace we therefore have to impose the
constraint

$\sum_{i=1}^{3}\beta_{ii}=0\,.$ (7)

The generalisation to a bipartite system in this formalism is straightforward.
The spin density matrix for a pair of qutrits, in the Gell-Mann decomposition,
is given by

$\rho=\frac{1}{9}\,\mathbb{I}\otimes\mathbb{I}+\frac{1}{3}\sum_{i=1}^{8}a_{i}\,\lambda_{i}\otimes\mathbb{I}+\frac{1}{3}\sum_{j=1}^{8}b_{j}\,\mathbb{I}\otimes\lambda_{j}+\sum_{i=1}^{8}\sum_{j=1}^{8}c_{ij}\,\lambda_{i}\otimes\lambda_{j}\,.$
(8)

Similarly, one could express Eq. 8 in terms of the spin matrix representation,
but we find the Gell-Mann decomposition more straightforward and neat to use.

The parameters $a_{i}$, $b_{j}$ and $c_{ij}$ (called Fano coefficients)
determine the angular distributions of the decay products, and characterise
the interactions governing the decay. This feature allows us to perform the
quantum tomography of the system experimentally by measuring the angular
distributions of the decays and reconstructing the coefficients. The aim of
this work is to first explore the properties of the density matrix based on
observables222By observables, we mean final states angular distributions after
the gauge boson decays. and the conditions for entanglement, deferring the
tomography studies to a later stage.

### Concurrence

We state that a system is entangled if the concurrence $\mathcal{C}(\rho)$ is
non zero. For bipartite qubit systems, this measure is easily evaluated by
relating to the eigenvalues of the matrix
$\omega=\sqrt{\sqrt{\tilde{\rho}}\rho\sqrt{\tilde{\rho}}}$ where
$\tilde{\rho}=(\sigma_{2}\otimes\sigma_{2})\rho^{*}(\sigma_{2}\otimes\sigma_{2})$.
However, for more complicated bipartite systems, such as the two qutrits
explored in this work, conditions for entanglement cannot be calculated
analytically Uhlmann:1996mk ; Horodecki:2009zz ; Barr:2022wyq .

For higher-dimensionality mixed states, the concurrence is obtained by the
method of convex roof extension Uhlmann:1996mk . For a given decomposition of
the $\rho$ matrix in pure states, _i.e._

$\displaystyle\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|\,,\qquad\sum_{i}p_{i}=1\,,\qquad
p_{i}\geq 0\,,$ (9)

the concurrence is defined as

$\displaystyle\mathcal{C}(\rho)=\text{inf}\left[\sum_{i}p_{i}c(|\psi_{i}\rangle)\right]\,,$
(10)

where the infimum is obtained over all possible ensembles
$\\{p_{i},\psi_{i}\\}$ for the decomposition in Eq. 9. The particular ensemble
to which this infimum is reached is called optimal. The given concurrence is
then the average of the optimal ensemble states concurrence.333It is clear
form Eq. 10 that if we have a pure state, the concurrence will be calculable.
This will be relevant later for $WZ$ production. The concurrence, however,
cannot be calculated in a closed form for systems higher than $2\times 2$. In
these cases, one can rely on lower and upper bounds,
$\mathcal{C}_{\mathrm{LB}}$ and $\mathcal{C}_{\mathrm{UB}}$ respectively, to
quantify the entanglement.

A lower bound on the concurrence is given by PhysRevLett.98.140505

$(\mathcal{C}(\rho))^{2}\geq
2\max\left(0,\operatorname{Tr}\left[\rho^{2}\right]-\operatorname{Tr}\left[\rho_{A}^{2}\right],\operatorname{Tr}\left[\rho^{2}\right]-\operatorname{Tr}\left[\rho_{B}^{2}\right]\right)\equiv\mathcal{C}_{\mathrm{LB}}^{2}\,,$
(11)

where $\rho_{A}=\operatorname{Tr}_{B}(\rho)$ and
$\rho_{B}=\operatorname{Tr}_{A}(\rho)$ are the reduced density matrices,
obtained by tracing out one of the subsystems. The function
$\mathcal{C}_{\mathrm{LB}}(\rho)$ is a marker that is telling us that in case
of positive values, we have an entangled state. However, in the scenario of
negative values, the test is inconclusive. In particular, for the qutrit pair,
we have

$\mathcal{C}_{\mathrm{LB}}^{2}=-\frac{4}{9}+\max\left(-\frac{8}{3}\sum_{i=1}^{8}a_{i}^{2}+\frac{4}{3}\sum_{j=1}^{8}b_{j}^{2},\frac{4}{3}\sum_{i=1}^{8}a_{i}^{2}-\frac{8}{3}\sum_{j=1}^{8}b_{j}^{2}\right)+8\sum_{i,j=1}^{8}c_{ij}^{2}\,.$
(12)

One can also obtain an upper bound for the concurrence, given in Ref.
PhysRevA.78.042308  and recently explored also in Ref. Fabbrichesi:2023cev ,

$(\mathcal{C}(\rho))^{2}\leq
2\,\text{min}\,\left(1-\text{Tr}[\rho_{A}^{2}],1-\text{Tr}[\rho_{B}^{2}]\right)\equiv\mathcal{C}_{\mathrm{UB}}^{2}\,,$
(13)

which in terms of the Fano coefficients reads

$\mathcal{C}_{\mathrm{UB}}^{2}=\frac{4}{3}-4\,\text{min}\left(\sum_{i=1}^{8}a_{i}^{2},\sum_{j=1}^{8}b_{j}^{2}\right)\,.$
(14)

For a qutrit pair, the maximum value of the concurrence is obtained for a
totally symmetric and entangled pure state,

$\left|\Psi_{+}\right\rangle=\frac{1}{\sqrt{3}}\sum_{i=1}^{3}|i\rangle\otimes|i\rangle\,,$
(15)

with $\mathcal{C}(\rho)=2/\sqrt{3}$. This is different from a qubit pair, in
which all entangled pure states have concurrence $\mathcal{C}(\rho)=1$.

### Purity

When exploring the density matrices to assess entanglement, it is useful to
know if the state is pure or mixed. This is quantified by the purity $P$ given
by

$\displaystyle P(\rho)\equiv\text{tr}[\rho^{2}]\,,$ (16)

which is one in the case of pure states and bounded to the lower value of
$1/d$ for qudits. Given the Fano decomposition in Eq. 8, this means

$\displaystyle
P(\rho)=\frac{1}{9}+\frac{2}{3}\sum_{i=1}^{8}(a_{i}^{2}+b_{i}^{2})+4\sum_{i,j=1}^{8}c_{ij}^{2}\,.$
(17)

### Bell inequalities

Another interesting aspect of quantum system is the possibility of violating
Bell inequalities Bell:1964kc . This allows to distinguish classical local
realist theories from quantum mechanical ones. In particular, for a pair of
qubits, the Clauser-Horne-Shimony-Holt (CHSH) Clauser:1969ny  inequality holds

$\mathcal{I}_{2}=E(a,b)-E\left(a,b^{\prime}\right)+E\left(a^{\prime},b\right)+E\left(a^{\prime},b^{\prime}\right)\leq
2\,.$ (18)

Quantum mechanics allows $\mathcal{I}_{2}$ to have values higher than two.

Analogously, one can define an observable for pairs of qutrits, the Collins-
Gisin-Linden-Massar-Popescu (CGLMP) inequality PhysRevA.65.032118 ;
PhysRevLett.88.040404 . By defining the quantum operator PhysRevLett.68.3259 ;
Acin:2002zz

$\mathcal{B}=-\frac{2}{\sqrt{3}}\left(S_{x}\otimes S_{x}+S_{y}\otimes
S_{y}\right)+\lambda_{4}\otimes\lambda_{4}+\lambda_{5}\otimes\lambda_{5}\,,$
(19)

one finds the CGLMP inequality

$\mathcal{I}_{3}=\operatorname{Tr}[\rho\mathcal{B}]\leq 2\,.$ (20)

The above equation is valid in the $x-y$ plane, but it can be generalised to
any direction in the 3-dimensional space and arbitrary bases in spin space.
The generalised condition for violation of the Bell inequalities becomes

$\left\langle\mathcal{B}\right\rangle_{\mathrm{max}}=\max_{U,V}\left(\operatorname{Tr}\left(\rho\,(U^{\dagger}\otimes
V^{\dagger})\,\mathcal{B}\,(U\otimes V)\right)\right)\geq 2\,,$ (21)

where $U,V\in U(3)$ are unitary matrices.

###  2.2 EW boson production at colliders

We now turn our attention to electroweak production of spin-$1$ particles at
colliders, _i.e._ $W$ and $Z$ bosons. Following the approach of Ref.
Aoude:2022imd , we define the $R$-matrix from the matrix element amplitude:

$\displaystyle R_{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}^{I}$
$\displaystyle\equiv\frac{1}{N_{a}N_{b}}\sum_{\begin{subarray}{c}\text{ colors
}\\\ \text{ a,b spins
}\end{subarray}}\mathcal{M}_{\alpha_{2}\beta_{2}}^{*}\mathcal{M}_{\alpha_{1}\beta_{1}}$
(22) $\displaystyle\text{ with }\quad\mathcal{M}_{\alpha\beta}$
$\displaystyle\equiv\left\langle
V\left(k_{1},\alpha\right)\bar{V}\left(k_{2},\beta\right)|\mathcal{T}|a\left(p_{1}\right)b\left(p_{2}\right)\right\rangle\,,$

where $I=ab=\bar{q}q^{\prime},\bar{e}e$ are the possible initial states in
proton and lepton colliders at LO with $N_{a,b}$ degrees of freedom, and
$V=W^{\pm},\,Z$ are the respective vector bosons. For all these diboson
amplitudes, we can factor out the polarisation vectors, which carry the spin
dependence of the $R$-matrix as

$\displaystyle\mathcal{M}_{\alpha\beta}=\mathcal{M}_{\mu\nu}\,\,\varepsilon^{\dagger\mu}_{\alpha}(k_{1})\varepsilon^{\dagger\nu}_{\beta}(k_{2})$
(23)

where both polarisation tensors act as a map between the Lorentz tensor
structures in $\mathcal{M}_{\mu\nu}$ and the spin-space labelled by the index
$\\{\alpha,\beta\\}$ forming the $9\times 9$ qutrit matrix. The $R$-matrix is
in direct relation with the spin density matrix, _i.e._

$R=\tilde{A}\,\mathbb{I}\otimes\mathbb{I}+\sum_{i=1}^{8}\tilde{a}_{i}\,\lambda_{i}\otimes\mathbb{I}+\sum_{j=1}^{8}\tilde{b}_{j}\,\mathbb{I}\otimes\lambda_{j}+\sum_{i=1}^{8}\sum_{j=1}^{8}\tilde{c}_{ij}\,\lambda_{i}\otimes\lambda_{j}\,,$
(24)

with the coefficient $\tilde{A}$ encoding information on the differential
cross section

$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{9\beta}{64\pi^{2}\hat{s}}\tilde{A}(\hat{s},\boldsymbol{k})\,,$
(25)

where $\boldsymbol{k}$ is the direction of the $V$ boson, $\hat{s}$ the
invariant mass of the pair and $\beta=\sqrt{1-4\frac{m_{V}^{2}}{\hat{s}}}$ the
velocity of the $V$ boson in the centre of mass frame. Each coefficient can be
obtained by tracing with the element of decomposition, e.g.
$\tilde{a}_{i}=\text{tr}[R\,\lambda_{i}\otimes\mathbb{I}]$ and similarly for
$\tilde{b}_{j}$ and $\tilde{c}_{ij}$.

If we consider production at proton colliders, such as the LHC, the total
$R$-matrix is given by a weighted sum of the various different partonic
channels, _i.e._

$R(\hat{s},\boldsymbol{k})=\sum_{I}L^{I}(\hat{s})R^{I}(\hat{s},\boldsymbol{k})\,,$
(26)

with $L_{I}$ the luminosity functions Bernreuther:1997gs . The relevant
channels for diboson production at a proton collider are the quark
annihilation ones. Note that, given that the initial state particles are not
identical, both $q\bar{q}$ and $\bar{q}q$ channels are to be summed over. This
can be also taken into account by a symmetrisation over the polar angle, since
it can be shown that
$R^{\bar{q}q}(\hat{s},\theta)=R^{q\bar{q}}(\hat{s},\theta+\pi)$, _i.e._

$R(\hat{s},\theta)=\sum_{q}L^{q\bar{q}}(\hat{s})(R^{q\bar{q}}(\hat{s},\theta)+R^{q\bar{q}}(\hat{s},\theta+\pi))\,,$
(27)

where we made explicit that there is no dependence on the azimuthal angle
$\phi$ of the vector $\boldsymbol{k}$, given the cylindrical symmetry of the
problem. It is clear from the expression that the $R$-matrix of a proton
collider is by definition symmetric around $\theta=\pi/2$. The $R$-matrix is
then related to the spin density matrix simply by an overall normalisation
factor, _i.e._ $\rho=R/\operatorname{Tr}(R)$. In particular we obtain the
decomposition of the spin density matrix in terms of the $R$-matrix Fano
coefficients

$\displaystyle a_{i}=\frac{\tilde{a}_{i}}{3\tilde{A}}\,,\qquad
b_{i}=\frac{\tilde{b}_{i}}{3\tilde{A}}\,,\qquad
c_{ij}=\frac{\tilde{c}_{ij}}{9\tilde{A}}\,.$ (28)

In terms of these coefficients, the purity condition $P(\rho)=1$ reads

$\displaystyle
36\tilde{A}^{2}=3\sum_{i=1}^{8}(\tilde{a}_{i}^{2}+\tilde{b}_{i}^{2})+2\sum_{i,j=1}^{8}\tilde{c}_{i,j}^{2}\,.$
(29)

We conclude this section by commenting on the possible effects of higher-order
corrections which in general will change the $R$-matrix and the expected
entanglement. In specific cases, such as $ZZ$ and $W^{+}W^{-}$ final states,
IR/UV finite loop-induced processes could also contribute, e.g., $gg\to
ZZ/W^{+}W^{-}$. Even though suppressed, at the LHC gluon fusion production
provides interesting information on Higgs properties. The framework presented
here could be directly employed to perform a dedicated study. This is left for
future investigations. More in general, for QCD or QED corrections, real and
virtual contributions need to be considered together. In case of inclusive
predictions, the framework presented here can be can be straightforwardly
applied by simply tracing out (including integration over the phase space) the
unobserved degrees of freedom. Naively, one can expect some degree of
decoherence which should lower the entanglement compared to the leading-order
$R$-matrix. On the other hand, final states characterised by resolvable
emissions could be analysed on their own as three-body final states, where all
vector bosons are measured and the $R$-matrix has dimensionality higher than
$9\times 9$. In this case the an extension of the framework presented here
would be needed.

##  3 Diboson interactions

In this section, we discuss the relevant interactions for diboson production
at colliders. In particular, we consider the case of the SM as well as its
extension in the SMEFT, where higher dimensional operators will lead to the
introduction of SM parameter shifts and new Lorentz structures. The objective
is to present the structure of the EW couplings dictated by the SM symmetries
and how heavy new physics could potentially alter it.

###  3.1 SM couplings

Figure 1: $W^{+}W^{-}$ production in the SM.

Figure 2: $ZZ$ production in the SM.

Figure 3: $WZ$ production in the SM.

As we are interested in both lepton and hadron colliders, we now discuss the
couplings that EW bosons have with quarks and with electrons. In particular,
we consider the processes $e^{+}e^{-}\to W^{+}W^{-}$ and $e^{+}e^{-}\to ZZ$,
for a lepton collider as well as $pp\to W^{+}W^{-}$, $pp\to ZZ$, and $pp\to
W^{+}Z$ for an hadron collider, respectively. In the latter case, the relevant
partonic channels at LO are given by $u\bar{u}$ and $d\bar{d}$ for the neutral
final states, while for $W^{+}Z$ it is $u\bar{d}$. We work in the 5-flavour
scheme, so $u$ comprises both up and charm quark, while $d$ includes down,
strange and bottom quarks. We choose to discuss $W^{+}Z$, in analogy with Ref.
Fabbrichesi:2023cev , as it is the dominant channel in a proton collider, but
kinematic features are similar for $W^{-}Z$.

In Figs. 3, 3 and 3 we show the topology of interest in terms of Feynman
diagrams for all the processes considered. The properties of each process
depend on the values and form of couplings, which, in the SM, are completely
determined by gauge symmetry and the EWSB pattern. Specifically, the relevant
couplings of the fermions are the coupling to the $Z$ boson, the coupling to
the $W$ boson and the coupling to the $h$ boson. The latter is proportional to
the mass of the fermions, so it will be highly suppressed and substantially
irrelevant for the phenomenology at hadron or $e^{+}e^{-}$ colliders. We will
however discuss their role in the high energy limit in Section 4. Moreover,
the processes are sensitive to the triple gauge coupling (TGC) between the $W$
and the $Z$ boson, as well as the coupling of the $W$ with a photon. The
coupling of EW bosons to the $h$ boson is always accompanied by the Yukawa
coupling of the fermions, therefore our sensitivity to that is
negligible444However it could be interesting at a future Higgs factory at
$\sqrt{s}=m_{h}$. The couplings of the fermions to the EW bosons read

$\displaystyle\mathcal{L}_{\bar{f}fZ}\propto\left(g_{V}^{Z}\right)-\left(g_{A}^{Z}\right)\gamma_{5}\,,\qquad\text{with}\qquad
g_{V}^{Z}=\frac{T_{3}}{2}-Q\sin^{2}{\theta_{W}}\,,\quad
g_{A}^{Z}=\frac{T_{3}}{2},$ (30)

for the $Z$ boson and

$\displaystyle\mathcal{L}_{\bar{f}fW}$ $\displaystyle\propto
g_{W}\left(1-\gamma_{5}\right)\,,$ (31)

for the $W$ boson, with $T_{3}=-1/2$ for down-type quarks and leptons,
$T_{3}=1/2$ for up-type quarks and the electric charge $Q=-1,2/3,-1/3$ for
$e$, $u$ and $d$ respectively. In the SM, $g_{V}^{Z}\approx-0.027,0.1,-0.17$
and $g_{A}^{Z}\approx-0.25,0.25,-0.25$ for $e$, $u$ and $d$ respectively and
$g_{W}=1/2$. Finally, the TGCs are specified by the gauge symmetries of the SM
through the gauge Lagrangian

$\mathcal{L}_{\text{gauge
}}=-\frac{1}{4}W_{\mu\nu}^{a}W^{a\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\,.$
(32)

In the SM, the couplings are given by

$\displaystyle g_{WW\gamma}=e\,,\qquad g_{WWZ}=e\cot{\theta_{W}}\,,$ (33)

where the electric charge $e$ and the trigonometric functions of the Weinberg
angle can be determined in terms of the chosen EW input parameters.

###  3.2 Modified interactions: SMEFT framework

Operator | Coefficient | Definition | 95 % CL bounds
---|---|---|---
two-fermion operators
$\mathcal{O}_{\varphi u}$ | $c_{\varphi u}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\varphi\big{)}\big{(}\bar{u}\gamma^{\mu}u\big{)}$ | $\left[-0.17,0.14\right]$
$\mathcal{O}_{\varphi d}$ | $c_{\varphi d}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\varphi\big{)}\big{(}\bar{d}\gamma^{\mu}d\big{)}$ | $\left[-0.07,0.09\right]$
$\mathcal{O}_{\varphi q}^{(1)}$ | $c_{\varphi q}^{(1)}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\,\varphi\big{)}\big{(}\bar{q}\,\gamma^{\mu}\,q\big{)}$ | $\left[-0.06,0.22\right]$
$\mathcal{O}_{\varphi q}^{(3)}$ | $c_{\varphi q}^{(3)}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\,\tau_{I}\varphi\big{)}\big{(}\bar{q}\,\gamma^{\mu}\,\tau^{I}q\big{)}$ | $\left[-0.21,0.05\right]$
$\mathcal{O}_{\varphi e}$ | $c_{\varphi e}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\varphi\big{)}\big{(}\bar{e}\gamma^{\mu}e\big{)}$ | $\left[-0.21,0.26\right]$
$\mathcal{O}_{\varphi l}^{(1)}$ | $c_{\varphi l}^{(1)}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\varphi\big{)}\big{(}\bar{l}\gamma^{\mu}l\big{)}$ | $\left[-0.11,0.13\right]$
$\mathcal{O}_{\varphi l}^{(3)}$ | $c_{\varphi l}^{(3)}$ | $i\big{(}\varphi^{\dagger}\overset{\leftrightarrow}{D}_{\mu}\tau_{I}\varphi\big{)}\big{(}\bar{l}\gamma^{\mu}\tau^{I}l\big{)}$ | $\left[-0.21,0.05\right]$
bosonic operators
$\mathcal{O}_{W}$ | $c_{W}$ |  $\varepsilon_{IJK}W^{I}_{\mu\nu}W^{J,\nu\rho}W^{K,\mu}_{\rho}$, | $\left[-0.18,0.22\right]$
$\mathcal{O}_{\varphi W}$ | $c_{\varphi W}$ | $\bigg{(}\varphi^{\dagger}\varphi-\frac{v^{2}}{2}\bigg{)}W_{I}^{\mu\nu}W_{\mu\nu}^{I}$ | $\left[-0.15,0.30\right]$
$\mathcal{O}_{\varphi B}$ | $c_{\varphi B}$ | $\bigg{(}\varphi^{\dagger}\varphi-\frac{v^{2}}{2}\bigg{)}B_{\mu\nu}B^{\mu\nu}$ | $\left[-0.11,0.11\right]$
$\mathcal{O}_{\varphi WB}$ | $c_{\varphi WB}$ | $(\varphi^{\dagger}\tau_{I}\varphi)B^{\mu\nu}W^{I}_{\mu\nu}$ | $\left[-0.17,0.27\right]$
$\mathcal{O}_{\varphi D}$ | $c_{\varphi D}$ | $(\varphi^{\dagger}D^{\mu}\varphi)^{\dagger}(\varphi^{\dagger}D_{\mu}\varphi)$ | $\left[-0.52,0.43\right]$
four-fermion operator
$\mathcal{O}_{ll}$ | $c_{ll}$ | $\big{(}\bar{l}\gamma_{\mu}l\big{)}\big{(}\bar{l}\gamma^{\mu}l\big{)}$ | $\left[-0.16,0.02\right]$
Table 1: Definition of the dimension-six SMEFT operators relevant for this
analysis. The bounds assume a scale of $\Lambda=$1\text{\,}\mathrm{TeV}$$ and
are taken from the global fit of Ref. Ethier:2021bye  at order
$\mathcal{O}(\Lambda^{-4})$. The bound on $c_{ll}$ comes from EWPO fits and
here we quote the result from Ref. Brivio_2017 .

We now discuss the effects of heavy new physics to the production of diboson
at colliders. We do so within the framework of the SMEFT where the SM gauge
symmetries are all preserved and the Higgs mechanism is realised linearly. The
SMEFT is an extension of the SM in which higher order operators modify the SM
interactions, characterised by a Lagrangian of the kind

$\mathcal{L}_{\mathrm{SMEFT}}=\mathcal{L}_{\mathrm{SM}}+\sum_{n=1}^{N}\frac{c_{n}}{\Lambda^{2}}\mathcal{O}_{n}+\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)\,,$
(34)

where $\mathcal{O}_{n}$ indicates a higher dimensional operator, $c_{n}$ is
the associated Wilson coefficient, a free parameter that in a bottom-up
approach needs to be determined experimentally, and $N$ is the number of
operators at this order in the expansion.

The operators are suppressed by the scale of new physics $\Lambda$, assumed to
be at least at the order of a TeV. This scale is what allows us to truncate
the EFT series by means of its power counting. For this analysis, we focus on
the SMEFT dim-6 operators Grzadkowski:2010es ; Aguilar-Saavedra:2018ksv  at
order $1/\Lambda^{2}$ and neglect higher order corrections. Note that we can
only be sensitive to the ratio $c_{n}/\Lambda^{2}$ and therefore we will
absorb $\Lambda^{2}$ into the definition of the Wilson coefficient for the
rest of the paper, _i.e._

$\frac{c_{n}}{\Lambda^{2}}\to c_{n}\,.$ (35)

For the sake of simplicity, we assume flavour universality for the few
operators that could, in principle, have a more involved structure. Results
will be presented in the $m_{W}$ input parameter scheme, where the
experimentally determined EW parameters of choice are
$\\{m_{W},m_{Z},m_{h},G_{F}\\}$. In Table 1 the relevant CP-even operators for
leading-order diboson production, both in lepton and proton colliders are
shown. We note that when considering a massless initial state,
$\mathcal{O}_{\varphi W}$ and $\mathcal{O}_{\varphi B}$ which modify the
coupling of the EW bosons to the Higgs, do not enter the processes because of
the aforementioned pairing with the Yukawa couplings. We will therefore not
discuss further these two operators, but we listed them for completeness.

The operators act in a multitude of ways, leading to different
phenomenological consequences. In particular, some operators act by shifting
the SM value of the coupling of the fermions to the EW bosons. Specifically we
have

$\displaystyle\delta g_{V}^{Z}(e)=$
$\displaystyle\left(\sin^{2}{\theta_{W}}-\frac{1}{4}\right)\delta g_{Z}-\delta
s^{2}_{\theta}-\frac{c_{\varphi l}^{(3)}+c_{\varphi e}+c_{\varphi
l}^{(1)}}{4\sqrt{2}G_{F}}\,,$ $\displaystyle\delta g_{A}^{Z}(e)=$
$\displaystyle\frac{\delta g_{Z}}{4}-\frac{c_{\varphi l}^{(3)}-c_{\varphi
e}+c_{\varphi l}^{(1)}}{4\sqrt{2}G_{F}}\,,$ $\displaystyle\delta
g_{V}^{Z}(u)=$
$\displaystyle\left(\frac{1}{4}-\frac{2\sin^{2}{\theta_{W}}}{3}\right)\delta
g_{Z}+\frac{2}{3}\delta s^{2}_{\theta}-\frac{c_{\varphi q}^{(-)}+c_{\varphi
u}}{4\sqrt{2}G_{F}}\,,$ (36) $\displaystyle\delta g_{A}^{Z}(u)=$
$\displaystyle\frac{\delta g_{Z}}{4}-\frac{c_{\varphi q}^{(-)}-c_{\varphi
u}}{4\sqrt{2}G_{F}}\,,$ $\displaystyle\delta g_{V}^{Z}(d)=$
$\displaystyle\left(\frac{\sin^{2}{\theta_{W}}}{3}-\frac{1}{4}\right)\delta
g_{Z}-\frac{1}{3}\delta s^{2}_{\theta}-\frac{2c_{\varphi q}^{(3)}+c_{\varphi
d}+c_{\varphi q}^{(-)}}{4\sqrt{2}G_{F}}\,,$ $\displaystyle\delta
g_{A}^{Z}(d)=$ $\displaystyle\frac{\delta g_{Z}}{4}-\frac{2c_{\varphi
q}^{(3)}-c_{\varphi d}+c_{\varphi q}^{(-)}}{4\sqrt{2}G_{F}}\,,$

where we defined $c_{\varphi q}^{(-)}=c_{\varphi q}^{(1)}-c_{\varphi
q}^{(3)}$, which is the combination of Wilson coefficients that modifies the
coupling of up-type quarks to the $Z$ boson. The variations of the couplings
are written as function of the quantities

$\displaystyle\begin{aligned} \delta g_{Z}&=-\frac{4c_{\varphi
l}^{(3)}-2c_{ll}+c_{\varphi D}}{4\sqrt{2}G_{F}}\,,\\\ \delta
s^{2}_{\theta}&=\frac{c_{\varphi
D}\,m_{W}^{2}}{2\sqrt{2}G_{F}\,m_{Z}^{2}}+\frac{c_{\varphi
WB}\,m_{W}\sqrt{1-\frac{m_{W}^{2}}{m_{Z}^{2}}}}{\sqrt{2}G_{F}\,m_{Z}}\,,\end{aligned}$
(37)

which are SMEFT induced universal shifts specific to the EW input parameter
scheme, see Ref. Brivio_2017  for more details. For the coupling to the $W$
boson, we have

$\displaystyle\begin{aligned} \delta g_{W}(e)&=\frac{c_{\varphi
l}^{(3)}}{2\sqrt{2}G_{F}}-\frac{\delta G_{F}}{2\sqrt{2}}\,,\\\ \delta
g_{W}(q)&=\frac{c_{\varphi q}^{(3)}}{2\sqrt{2}G_{F}}-\frac{\delta
G_{F}}{2\sqrt{2}}\,,\end{aligned}$ (38)

with the fractional shift to the Fermi constant originating from the muon
decay measurement given by

$\delta G_{F}=\frac{2c_{\varphi l}^{(3)}-c_{ll}}{2G_{F}}\,.$ (39)

Finally, the operators can also alter the TGCs. The modified Lagrangian reads

$\frac{\mathcal{L}_{WWV}}{-ig_{WWV}}=g_{1}^{V}\left(W_{\mu\nu}^{+}W^{-\mu}V^{\nu}-W_{\mu}^{+}V_{\nu}W^{-\mu\nu}\right)+\kappa_{V}W_{\mu}^{+}W_{\nu}^{-}V^{\mu\nu}+\frac{i\delta\lambda_{V}}{m_{W}^{2}}V^{\mu\nu}W_{\nu}^{+\rho}W_{\rho\mu}^{-}\,,$
(40)

where $V=Z,\gamma$ and we have defined
$V_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ and
$W_{\mu\nu}^{\pm}=\partial_{\mu}W_{\nu}^{\pm}-\partial_{\nu}W_{\mu}^{\pm}$.
The dimension-6 SMEFT operators introduce a dependence on the couplings
$g_{1}^{V}=1+\delta g_{1}^{V}$ and $\kappa_{V}=1+\delta\kappa_{V}$, _i.e._

$\displaystyle\begin{aligned} \delta
g_{1}^{\gamma}&=\frac{1}{4\sqrt{2}G_{F}}\left(c_{\varphi
D}\frac{m_{W}^{2}}{m_{W}^{2}-m_{Z}^{2}}-4c_{\varphi
l}^{(3)}+2c_{ll}-c_{\varphi
WB}\frac{4m_{W}}{\sqrt{m_{Z}^{2}-m_{W}^{2}}}\right)\,,\\\ \delta
g_{1}^{Z}&=\frac{1}{4\sqrt{2}G_{F}}\left(c_{\varphi D}-4c_{\varphi
l}^{(3)}+2c_{ll}+4\frac{m_{Z}}{m_{W}}\sqrt{1-\frac{m_{W}^{2}}{m_{Z}^{2}}}c_{\varphi
WB}\right)\,,\\\
\delta\kappa_{\gamma}&=\frac{1}{4\sqrt{2}G_{F}}\left(c_{\varphi
D}\frac{m_{W}^{2}}{m_{W}^{2}-m_{Z}^{2}}-4c_{\varphi
l}^{(3)}+2c_{ll}\right)\,,\\\
\delta\kappa_{Z}&=\frac{1}{4\sqrt{2}G_{F}}\left(c_{\varphi D}-4c_{\varphi
l}^{(3)}+2c_{ll}\right)\,.\end{aligned}$ (41)

Note that while $c_{\varphi l}^{(3)}$, $c_{ll}$ and $c_{\varphi D}$
universally shift the TGC of the SM, $c_{\varphi WB}$ does not contribute to
$\kappa_{V}$ but only to $g_{1}^{V}$, changing the symmetrical structure of
the SM interactions. Only one operator in Table 1 leads to a new Lorentz
structure by generating a term proportional to $\delta\lambda_{V}$, with a
dependence given by

$\displaystyle\begin{aligned}
\delta\lambda_{\gamma}&=-6\sin{\theta_{W}}\frac{m_{W}^{2}}{g_{WW\gamma}}c_{W}\,,\\\
\delta\lambda_{Z}&=-6\cos{\theta_{W}}\frac{m_{W}^{2}}{g_{WWZ}}c_{W}\,.\end{aligned}$
(42)

$\mathcal{O}_{W}$ is therefore of particular interest, since it modifies the
interactions among EW bosons in a way that could potentially induce different
helicity structures. This is of relevance for this study, given that the
density matrix of the diboson system could markedly change if the EW bosons
are produced in configurations not present in the SM.

##  4 Perturbative unitarity and entanglement

Perturbative unitarity and the role of the Higgs boson in the SM can be
effectively studied by considering multi-boson longitudinal scattering
amplitudes and their cross-section Romao:2016ien . Conservation of probability
in QFT requires the unitarity of the $S$-matrix, which imposes bounds on the
energy dependence of the corresponding terms in the perturbative expansion.
The scattering amplitudes and cross-sections involving weak bosons violate the
perturbative bound if gauge invariance is not respected or Higgs-mediated
interactions are not included. How these violations appear and then cancel
among different contributions in usual observables has been studied
extensively. However, to the best of our knowledge, the perturbative unitarity
constraints on entanglement and quantum information observables has not been
explored. Before proceeding, we note that for this study and at variance with
what done in the previous sections, here we keep the fermion masses non-zero,
to allow non-vanishing Higgs couplings and investigate their role too.

It is possible to express 2$\to$2 scattering amplitudes in powers of the
normalised energy $\sqrt{s}/(2m_{V})$:

$\displaystyle\mathcal{M}=\mathcal{M}^{(2)}\left(\frac{\sqrt{s}}{2m_{V}}\right)^{2}+\mathcal{M}^{(1)}\left(\frac{\sqrt{s}}{2m_{V}}\right)+\mathcal{M}^{(0)}+\mathcal{O}\left(\frac{2m_{V}}{\sqrt{s}}\right)\,,$
(43)

noting that this expansion is relevant also for the cross-section and the
$R$-matrix Romao:2016ien . Perturbative unitarity requires the bad high-energy
behaviour to cancel and the $2\to 2$ amplitude to go at most as a constant in
the high-energy limit ($s\sim-t\sim-u\gg M_{V})$

$\displaystyle\lim_{\frac{\sqrt{s}}{2m_{V}}\rightarrow\infty}\mathcal{M}=\mathcal{M}^{(0)},$
(44)

_i.e._ , any amplitude respecting perturbative unitarity should have vanishing
$\mathcal{M}^{(2)}$ and $\mathcal{M}^{(1)}$ at high energies.

###  4.1 $WW$ final state

Let us consider $e^{+}e^{-}\rightarrow W^{+}W^{-}$. We have four amplitudes
contributing to the process, depicted in Fig. 3: $\mathcal{M}_{\gamma}$ and
$\mathcal{M}_{Z}$ with an $s$-channel photon or $Z$-boson (left),
$\mathcal{M}_{\nu}$ with a $t$-channel neutrino (centre), and
$\mathcal{M}_{h}$ with a Higgs boson in the $s$-channel (right). The sum of
the $\mathcal{M}_{\gamma}$, $\mathcal{M}_{Z}$, and $\mathcal{M}_{\nu}$
amplitudes grows with energy, displaying a unitarity violating behaviour. At
high energies we find

$\displaystyle\mathcal{M}_{\gamma+Z+\nu}(\pm\pm\rightarrow
00)\sim-\mathcal{M}_{h}(\pm\pm\rightarrow
00)\sim-\frac{e^{2}}{2\sin^{2}\theta_{W}}\frac{m_{e}}{m_{W}}\,\frac{\sqrt{s}}{2m_{W}}\,.$
(45)

However, the Higgs exchange $\mathcal{M}_{h}$ has the same behaviour in the
high-energy limit and exactly cancels the growth of the amplitude, recovering
perturbative unitarity. As one might expect, this cancellation occurs only for
the longitudinal final states, since the other cases do not grow with energy.
At the cross-section level, a similar cancellation occurs (see Ref.
Romao:2016ien  for an easy-to-access review).

Let us now turn our attention to the Fano coefficients. As one might expect,
the cancellation for the $\tilde{A}$ is the same as that happening at the
cross-section level. At high energy and keeping the electron mass finite, the
squared contribution of the sum of the $\gamma$, $Z$ and $\nu$ mediated
diagrams to the Fano coefficient is

$\displaystyle\tilde{A}_{|\gamma+Z+\nu|^{2}}=\frac{e^{4}}{288\sin^{4}\theta_{W}}\frac{m_{e}^{2}}{m_{W}^{2}}\frac{s}{m_{W}^{2}}+\mathcal{O}(s^{0})\,.$
(46)

Adding the contribution of the Higgs diagram, we obtain three more terms,

$\displaystyle\tilde{A}_{|h|^{2}}\sim-\tilde{A}_{(\gamma+Z+\nu)^{*}h}\sim-\tilde{A}_{(\gamma+Z+\nu)h^{*}}\sim\tilde{A}_{|\gamma+Z+\nu|^{2}}\quad\Longrightarrow\quad\tilde{A}_{|\gamma+Z+\nu+h|^{2}}\sim\mathcal{O}(s^{0})\,,$
(47)

where now, due to the sign of the amplitudes, the interference terms of
$(\gamma+Z+\nu)$ with the Higgs cancel with the matrix-element-squared
contributions, resulting in a $\mathcal{O}(s^{0})$ dependence. This
cancellation happens similarly for the $\tilde{a}_{i}$, $\tilde{b}_{i}$ and
$\tilde{c}_{ij}$ coefficients. Specifically, we obtain the following non-zero
results at order $s/m_{W}^{2}$ for the $(\gamma+Z+\nu)$ diagrams

$\displaystyle\tilde{a}_{3,|\gamma+Z+\nu|^{2}}\sim-\frac{1}{\sqrt{3}}\tilde{a}_{8,|\gamma+Z+\nu|^{2}}\sim\frac{e^{4}}{288\sin^{4}\theta_{W}}\frac{m_{e}^{2}}{m_{W}^{2}}\frac{s}{m_{W}^{2}}+\mathcal{O}(s^{0})\,,$
(48)

which in the case of the third component cancels when summed with the Higgs-
mediated-squared and interference contributions,
$\tilde{a}_{3,(\gamma+Z+\nu)^{*}h}+\tilde{a}_{3,(\gamma+Z+\nu)h^{*}}+\tilde{a}_{3,|h|^{2}}$,
and likewise for the eighth component. The same expressions hold when
considering $\tilde{b}_{i}$. With respect to the correlation matrix
$\tilde{c}_{ij}$, we have that for the $\gamma+Z+\nu$ diagrams

$\displaystyle\begin{aligned}
\tilde{c}_{33,|\gamma+Z+\nu|^{2}}&=-\frac{1}{\sqrt{3}}\tilde{c}_{38,|\gamma+Z+\nu|^{2}}=-\frac{1}{\sqrt{3}}\tilde{c}_{83,|\gamma+Z+\nu|^{2}}=\frac{1}{3}\tilde{c}_{88,|\gamma+Z+\nu|^{2}}\\\
&=\frac{e^{4}}{288\sin^{4}\theta_{W}}\frac{m_{e}^{2}}{m_{W}^{2}}\frac{s}{m_{W}^{2}}+\mathcal{O}(s^{0})\,.\end{aligned}$
(49)

Adding the Higgs contribution leads to the same cancellation of all the
coefficients at $s/m_{W}^{2}$ order, and perturbative unitarity is restored in
the $R$-matrix. It is interesting to note that this cancellation occurs only
for the coefficients of the third and eighth Gell-Mann matrix, while the
others do not exhibit an energy growing behaviour. This occurs because in the
high energy limit the $R$-matrix is dominated by the longitudinal
polarisations and these are the Fano coefficients sensitive to those.

###  4.2 $ZZ$ final state

We now turn to $ZZ$ production. As in the previous case, let us focus on the
lepton-initiated channel. The story for $ZZ$ is similar but now we only have
two types of diagrams, $t$ and $u$-channel mediated by electrons and the Higgs
$s$-channel. The cancellation in the amplitude occurs for the same helicities
as before

$\displaystyle\mathcal{M}_{e}(\pm\pm\rightarrow
00)\sim\mathcal{M}_{h}(\pm\pm\rightarrow
00)\sim-e^{2}\csc^{2}{2\theta}\,\frac{m_{e}}{2m_{Z}}\,\frac{\sqrt{s}}{2m_{Z}}+\mathcal{O}(s^{0})\,,$
(50)

where $\mathcal{M}_{e}$ represents both $t$ and $u$ channels. At the
$R$-matrix level, the cancellation happens in a similar fashion to the $WW$
case. The $\tilde{A}$ coefficient for the electron diagrams,

$\displaystyle\tilde{A}_{e}=\frac{e^{4}}{288\cos^{4}\theta_{W}\sin^{4}\theta_{W}}\frac{m_{e}^{2}}{m_{Z}^{2}}\frac{s}{m_{Z}^{2}}+\mathcal{O}(s^{0})\,,$
(51)

cancel when adding the Higgs diagram. For the coefficients $\tilde{b}_{i}$ and
$\tilde{c}_{ij}$, we again only have contributions to the third and eighth
components and the cancellations repeat the previous pattern.

###  4.3 Non-interference EFT effects

The study of perturbative unitarity fits well in renormalisable theories where
one does not expect growing amplitudes in the high-energy limit. This
discussion is different for EFTs, in which higher-dimensional operators are
included and they are allowed to grow with energy. This is actually a feature
used in SMEFT analyses to probe deviations from the SM in tails of
distributions. This growth can happen due to the particular new Lorentz
structure of the operators or because of the spoiling of the SM cancellations.
To understand the high-energy limit of the $R$-matrix and spin-related
observables, let us first look at the amplitudes.

At high energy, the SM and the SMEFT induce a specific helicity pattern for
diboson production. In Table 2, as representative, we report the helicity
amplitudes for $e^{+}e^{-}\to W^{+}W^{-}$. The helicity states are specified
by the notation $\mathcal{M}(\lambda_{1}\lambda_{2}|\alpha\beta)$, where
$\lambda_{1},\lambda_{2}$ are the helicities of the initial state electrons
and $\alpha,\beta$ are the helicities of the final state EW bosons. We retain
contributions up to order $\mathcal{O}(x^{0})$ (see also
Refs.Falkowski:2016cxu  and Helset:2017mlf ).

$(\lambda_{1}\lambda_{2}|\alpha\,\beta)$ | SM |  EFT $\Lambda^{-2}:c_{WWW}$
---|---|---
$+-00$ | $-2\sqrt{2}G_{F}m_{Z}^{2}\sin\theta$ | -
$+--+$ | $2\sqrt{2}G_{F}m_{W}^{2}\sin\theta$ | -
$+-+-$ | $-\frac{1}{\sqrt{2}}G_{F}m_{W}^{2}\sin^{3}\theta\csc^{4}(\theta/2)$ | -
$+-\pm\pm$ | - |  $3\cdot 2^{1/4}\sqrt{G_{F}}m_{W}\sin\theta\,(4m_{W}^{2}x^{2}-m_{Z}^{2})$
$+-0\pm$ | - | $-3\cdot 2^{3/4}\sqrt{G_{F}}m_{W}^{3}(\pm 1+\cos\theta)\,x$
$+-\pm 0$ | - | $-3\cdot 2^{3/4}\sqrt{G_{F}}m_{W}^{3}(\mp 1+\cos\theta)\,x$
$-+00$ | $2\sqrt{2}G_{F}(m_{Z}^{2}-m_{W}^{2})\sin\theta$ | -
$-+\pm\pm$ | - | $6\cdot 2^{1/4}\sqrt{G_{F}}m_{W}(m_{Z}^{2}-m_{W}^{2})\sin\theta$
Table 2: Helicity pattern in the high-energy limit of electron-initiated
diboson production amplitudes for the SM and the SMEFT amplitudes induced by
the $\mathcal{O}_{W}$ operator, which has a distinguished Lorentz structure.
The $\pm\pm\alpha\beta$ case has all zero entries, regardless of
$\alpha\beta$. The short-hand notation $x=\sqrt{s}/(2m_{W})$ is used.
Contributions to the helicity amplitude that are sub-leading and energy
suppressed are indicated with -.

It is clear that the SM and the $\mathcal{O}_{W}$ operator induce different
helicity amplitudes. When computing a cross-section, where the on-shell final
states are the gauge bosons, this leads to a vanishing EFT linear correction
due to non-interference of the amplitudes. The cancellation for massless
particles can be proven by applying helicity selection rules, see Ref.
Azatov:2016sqh . However, one can show that the interference can be recovered
exploiting the angular distributions of the decay products, e.g. considering
the full process $e^{+}e^{-}\rightarrow VV\rightarrow 4f$, with $f$ either a
lepton or a quark. For in-depth phenomenological studies of this aspect we
refer the reader to the literature Panico:2017frx ; Franceschini:2017xkh ;
Azatov:2017kzw ; Helset:2017mlf ; Azatov:2019xxn ; Aoude:2019cmc ;
Degrande:2021zpv .

In the $R$-matrix formulation, this translates into the fact that the diagonal
terms at the linear EFT level vanish, while the off-diagonal terms allow for a
resurrection of the interference between the SM and the operator
$\mathcal{O}_{W}$. In the high energy and massless limit, the Fano coefficient
$\tilde{A}$ has a vanishing linear EFT contribution

$\displaystyle\tilde{A}(\mathcal{O}_{W})\sim 0\,\,,$ (52)

but the other Fano coefficients can be different from zero and potentially
allow for increased sensitivity. Defining $x=\sqrt{s}/(2m_{V})$ and in the
limit $m_{W}\sim m_{Z}\sim m_{V}$ we have

$\displaystyle\tilde{a}_{1}(\mathcal{O}_{W})$
$\displaystyle\simeq\tilde{b}_{1}(\mathcal{O}_{W})\simeq\bar{c}_{W}\,2^{5/4}\,x\;\cos^{4}(\theta/2)(\cos\theta+3)\csc\theta\,,$
(53a) $\displaystyle\tilde{a}_{4}(\mathcal{O}_{W})$
$\displaystyle\simeq\tilde{b}_{4}(\mathcal{O}_{W})\simeq-\bar{c}_{W}\,2^{3/4}(\cos\theta((4x^{2}-3)\cos\theta+4x^{2}+1)+2)\,,$
(53b) $\displaystyle\tilde{a}_{6}(\mathcal{O}_{W})$
$\displaystyle\simeq\tilde{b}_{6}(\mathcal{O}_{W})\simeq\bar{c}_{W}\,2^{1/4}\,x\,\sin^{2}(\theta/2)\sin\theta\,,$
(53c)

with $\bar{c}_{W}=c_{W}\,G_{F}^{3/2}\,m_{V}^{5}$. The spin-spin Fano
coefficients $\tilde{c}_{ij}(=\tilde{c}_{ji})$ for the operator
$\mathcal{O}_{W}$ are given by

$\displaystyle\tilde{c}_{13}$ $\displaystyle\simeq 3\,\bar{c}_{W}\cdot
2^{3/4}\cos^{2}(\theta/2)(3\cos\theta+1)\cot(\theta/2)\,x$ (54a)
$\displaystyle\tilde{c}_{14}$
$\displaystyle\simeq-\tilde{c}_{25}\simeq\tilde{c}_{46}\simeq-\tilde{c}_{57}\simeq-3\,\bar{c}_{W}\cdot
2^{3/4}\sin\theta(1+\cos\theta)\,x\,,$ (54b) $\displaystyle\tilde{c}_{16}$
$\displaystyle\simeq\tilde{c}_{27}\simeq 3\,\bar{c}_{W}\cdot
2^{-5/4}\sin^{2}(\theta/2)((3-4x^{2})\cos\theta-4x^{2}+1)\,,$ (54c)
$\displaystyle\tilde{c}_{18}$ $\displaystyle\simeq\bar{c}_{W}\,\sqrt{3}\,\cdot
2^{-3/4}\cos^{2}(\theta/2)(\cos\theta+2)\cot(\theta/2)\,x\,,$ (54d)
$\displaystyle\tilde{c}_{35}$ $\displaystyle\simeq\bar{c}_{W}\,\sqrt{3}\,\cdot
2^{-3/4}\sin^{2}(\theta/2)\sin\theta\,x\,,$ (54e)
$\displaystyle\tilde{c}_{48}$ $\displaystyle\simeq\sqrt{3}\,\bar{c}_{W}\cdot
2^{-9/4}(8(1-x^{2})\cos\theta+(4x^{2}-3)\cos(2\theta)-5(4x^{2}-1))\,,$ (54f)
$\displaystyle\tilde{c}_{68}$ $\displaystyle\simeq-5\sqrt{3}\,\bar{c}_{W}\cdot
2^{-3/4}\sin^{2}(\theta/2)\sin\theta\,.$ (54g)

The omitted ones are vanishing, showing that this matrix is rather sparse and
several of the energy growing terms are closely related between each other.
Here, we note also that the relevant coefficients for the perturbative
unitarity cancellation vanish in the non-interference analysis, as they are
pertinent to the longitudinal polarisations, while the $\mathcal{O}_{W}$
induces energy growing behaviour in transverse helicity configurations.

##  5 Entanglement at particle colliders

In the following we will study the spin density matrix and the presence of
entanglement in diboson production for both lepton and proton colliders. In
the case of the lepton collider, we consider centre of mass energies up to
$1\text{\,}\mathrm{TeV}$, while in the case of the proton collider we focus on
the LHC setup with a centre of mass energy of $13\text{\,}\mathrm{TeV}$. In
the latter we work in the $5$ flavour scheme, _i.e._ , all quarks are massless
aside for the top quark, and we use the latest NNPDF4.0 NNLO PDF set Ball_2022
. The input parameters used for the calculations are the following

$\displaystyle m_{W}$ $\displaystyle=$80.377\text{\,}\mathrm{GeV}$\,,$
$\displaystyle m_{Z}$ $\displaystyle=$91.1876\text{\,}\mathrm{GeV}$\,,$
$\displaystyle m_{h}$ $\displaystyle=$125.35\text{\,}\mathrm{GeV}$\,,$
$\displaystyle G_{F}$
$\displaystyle=$1.166\,378\,8\text{\times}{10}^{-5}\text{\,}\mathrm{G}\mathrm{eV}^{-2}$\,.$

Calculations are performed analytically, taking advantage of the Feynrules
Darme:2023jdn ; Alloul_2014 , FeynArts Hahn_2001  and FeynCalc
Shtabovenko:2020gxv  tool chain. Amplitude computations are also validated
numerically with MadGraph5_aMC@NLO Alwall:2014hca  and SMEFT@NLO
Degrande:2020evl . The results are presented in the centre of mass frame of
the diboson pair, as function of the kinematical variables $m_{VV}$, the
invariant mass of the system, and $\cos{\theta}$, the angle between the
initial state anti-particle and the $W^{+}$ or $Z$ boson. In the case of the
SM calculations, we have validated our results against the ones obtained in
Ref. Fabbrichesi:2023cev , finding excellent agreement.

For each process considered, we computed the density matrix including EFT
corrections (both linear and quadratic) and consequently the various Fano
coefficients, from which we determine analytically the entanglement related
markers $\mathcal{C}_{\mathrm{LB}}$ and $\mathcal{C}_{\mathrm{UB}}$, the
purity $P$ and the indicator $\langle\mathcal{B}\rangle_{\mathrm{max}}$ for
Bell inequality violation.555The minimisation in the calculation of the latter
is, however, done numerically.

###  5.1 Lepton collider

In this section we discuss the entanglement pattern in diboson production at a
lepton collider, focusing in particular on its dependence on the couplings in
the context of the SMEFT. In a lepton collider, the collision energy is fixed.
This means that at LO, without considering initial state QED (or more in
general EW) radiation, the diboson pair is produced with an invariant mass
$m_{VV}$ which is identical to the initial state energy. In this setup, we can
therefore study the behaviour of the entanglement by fixing the collider
energy and looking at its angular dependence. This remains true if we consider
lepton colliders with energies up to a few $\mathrm{TeV}$. In the scenario of
a multi-$\mathrm{TeV}$ lepton collider, such as a circular muon collider, the
machine would effectively behave as an EW boson collider and the main mode of
production for diboson pair will be vector boson scattering Costantini:2020stv
, see also Refs. Al_Ali_2022 ; Accettura:2023ked ; Aime:2022flm ;
MuonCollider:2022xlm ; Chiesa:2020awd  for in depth studies of muon collider
physics. The study of this kind of processes, which would be
phenomenologically quite different and would add considerable sensitivity on
dimension-8 operators, is left for future work.

####  $WW$ production

Figure 4: Entanglement in the $e^{+}e^{-}\to W^{+}W^{-}$ channel in the SM. We
show the lower bound $\mathcal{C}_{\mathrm{LB}}$ (bottom left) and upper bound
$\mathcal{C}_{\mathrm{UB}}$ (bottom right) on the concurrence $\mathcal{C}$,
the purity $P$ (top left) as well as the indicator
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ for Bell inequality violation (top
right) as a function of the invariant mass $m_{WW}$ (or equivalently the
collider energy) and the cosine of the angle between the positron and the
$W^{+}$ in the centre of mass frame.

Figure 5: For benchmark fixed collider energies, we show the expected value
for the concurrence as a function of $\cos{\theta}$ in the SM. The bands are
determined by the lower and upper bounds of the concurrence, _i.e._ ,
$\mathcal{C}_{\mathrm{LB}}$ and $\mathcal{C}_{\mathrm{UB}}$. Left:
$e^{+}e^{-}\to W^{+}W^{-}$. Right: $e^{+}e^{-}\to ZZ$.

The production of a pair of $W$ bosons is of most interest in a lepton
collider, as it is characterised by a considerably high cross section even at
high energies. For instance, at $1\text{\,}\mathrm{TeV}$, the total cross
section is still of the order of a few $\mathrm{pb}$. This means that the
process is going to be particularly advantageous in terms of statistics and
could prospectively allow to detect entanglement with strong significance
Fabbrichesi:2023cev .

The relevant couplings for the process are the coupling of the lepton to the
$W$ boson $g_{W}$, the coupling to the $Z$ boson $g_{V}^{Z}$ and $g_{A}^{Z}$,
and the TGCs. The process is therefore sensitive to several of the EW
interactions. Furthermore, in the SMEFT framework strong correlations are
present among the parameters.

In Fig. 4 we show the entanglement pattern we can expect from the production
of a pair of $W$ bosons in the SM. Results are shown as a function of the
kinematical variables $m_{WW}$ and $\cos{\theta}$. Note that there is no
symmetry around $\theta=\pi/2$, as the EW interactions are not parity
invariant. This is not the case when considering for instance $t\bar{t}$
production, where the process is dominated by QCD Aoude:2022imd . In the upper
left figure we show the values for the purity indicator $P$, which depicts a
scenario where the majority of the phase space is characterised by a density
matrix close to maximal purity. In the bottom left and bottom right figure, we
show the value of the lower and upper bound for the concurrence. The plots
demonstrate that the diboson pair has in general a very high value of the
concurrence across the phase space, indicating that entanglement is present
almost everywhere, with the exception of the collinear region with $\theta=0$,
where low values of entanglement are expected.

This can be seen more explicitly in the left plot of Fig. 5, where four
collider energies, 170, 250, 500 and 1000 GeV, are chosen as benchmarks. The
plot displays bands for the concurrence, where the lower and the upper bound
are given by $\mathcal{C}_{\mathrm{LB}}$ and $\mathcal{C}_{\mathrm{UB}}$
respectively, as a function of $\cos{\theta}$. Entanglement is high and stable
for most of the angles, but decreases sharply as we approach the forward
collinear limit at high energies. Also, the closer we are to threshold the
tighter the bands get, indicating that the quantum state of the process goes
towards maximal purity, as confirmed from the purity plot in Fig. 4. In fact,
as previously discussed, in this limit $\mathcal{C}_{\mathrm{LB}}$ and
$\mathcal{C}_{\mathrm{UB}}$ coincide.

Finally, in the upper right plot of Fig. 4 we display the Bell inequality
violation indicator $\langle\mathcal{B}\rangle_{\mathrm{max}}$. The pattern in
the figure closely resembles that shown for the concurrence marker
$\mathcal{C}_{\mathrm{LB}}$ and as expected, the violation is ubiquitous. In
particular, Bell inequalities are severely violated when high values of
entanglement are present, _i.e._ , at high energy in the central
region.666Note that we validated our results against Ref. Fabbrichesi:2023cev
finding excellent agreement for most of phase space but some discrepancy in
the forward region $\cos{\theta}=1$, where the authors find a slight violation
of Bell inequalities, _i.e._ $\langle\mathcal{B}\rangle_{\mathrm{max}}>2$.
However, as explicitly discussed above, we find that in the phase space region
in question the density matrix of the system is described by a pure separable
state. Therefore, no Bell inequality violation is expected, in agreement with
Fig. 4. This property has been also verified with a numerical simulation in
MadGraph5_aMC@NLO, finding good agreement with the analytical calculation.

In order to gain a better insight, it is useful to see how the density matrix
decomposes in terms of quantum states at particular phase space points. For
instance, we find that the diboson pair is produced at threshold in a pure and
entangled quantum state, _i.e._ ,

$\ket{\Psi(m_{WW}=2\,m_{W})}=\frac{1}{\sqrt{2}}\left(\ket{+0}_{\boldsymbol{p}}+\ket{0+}_{\boldsymbol{p}}\right)=\ket{\Psi_{0+}}_{\boldsymbol{p}}\,,$
(55)

where
$\ket{s_{1}s_{2}}_{\boldsymbol{p}}=\ket{s_{1}}_{\boldsymbol{p}}\otimes\ket{s_{2}}_{\boldsymbol{p}}$
and $\ket{s}_{\boldsymbol{p}}$ is the eigenstate of the spin operator in the
direction of the beam line $\boldsymbol{p}$ with eigenvalue $s$. This means
that at threshold, the diboson pair is produced in an entangled state,
characterised by total spin $2$ and spin component along $\boldsymbol{p}$
equal to $1$. However, despite the fact that this is an entangled state, the
value of the concurrence at threshold is $1$, not reaching the maximal value
of $2/\sqrt{3}$. We find that this quantum state is unaffected by the presence
of new physics. Even when EFT effects are taken into account, the quantum
state of the system is still expressed by $\ket{\Psi_{0+}}_{\boldsymbol{p}}$.
The reason for that is that most of the contributions are identically zero at
threshold and the ones that are not, are simply shifting the absolute value of
$g_{W}$, resulting in an increased total cross section but not affecting the
spin correlation patterns. It turns out indeed that at threshold, the only
relevant coupling for the process is $g_{W}$. Equally interesting is the
situation at high energy. In particular, in the central region, we find that
the density matrix is characterised by a mixed quantum state but dominated by
the presence of a pure entangled quantum state, which explains the high
concurrence. Specifically, the density matrix can be defined with respect to
the $\boldsymbol{k}$ direction, the momentum of the $W^{+}$ boson, in the
following way

$\rho(m_{WW}\to\infty,\cos{\theta}=0)=p_{1}\,\ket{1}\bra{1}+p_{2}\,\ket{2}\bra{2}\,,$
(56)

with

$\displaystyle\begin{aligned}
\ket{1}&=0.64\ket{++}_{\boldsymbol{k}}-0.64\ket{--}_{\boldsymbol{k}}+0.43\ket{00}_{\boldsymbol{k}}\,,\\\
\ket{2}&=0.3\ket{++}_{\boldsymbol{k}}-0.3\ket{--}_{\boldsymbol{k}}-0.9\ket{00}_{\boldsymbol{k}}\,,\end{aligned}$
(57)

and $p_{1}\approx 0.97$ and $p_{2}\approx 0.03$.

On the other hand, in the collinear region, $\theta=0$, the concurrence goes
to zero (see Fig. 4) and we find that the diboson pair is produced in a
separable state, _i.e._

$\ket{\Psi(m_{WW}\to\infty,\cos{\theta}=1)}=\ket{++}_{\boldsymbol{k}}\,.$ (58)

Figure 6: The changes in the marker $\mathcal{C}_{\mathrm{LB}}$ is shown for a
selection of operators and benchmark Wilson coefficient values for the
production of $W^{+}W^{-}$ at a lepton collider. Only one operator at the time
is switched on. Top left: $c_{\varphi
e}=$0.1\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, top right: $c_{\varphi
l}^{(1)}=$0.1\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, bottom left: $c_{\varphi
WB}=$0.25\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, bottom right:
$c_{W}=$0.25\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$.

We now move to discuss the effects of dimension-6 operators on the spin
density matrix. In Fig. 6, we show the effects of a selected number of
operators on the marker $\mathcal{C}_{\mathrm{LB}}$. For each operator we
choose values that are currently allowed or at the boundary of the most up to
date global fit studies (see Table 1 and for example Refs. Ethier:2021bye ;
Ellis:2020unq ). In the context of the EFT, here and in the following
sections, we decide to limit ourselves mostly to show the effects only for the
$\mathcal{C}_{\mathrm{LB}}$ indicator, as a good representative metric for the
entanglement pattern across phase space. In some cases, we also show the
purity for comparison. Note however that the calculation performed and the
expressions provided in the ancillary files allow for a complete determination
of the density matrix in the SMEFT and consequently for the calculation of
every quantum observable derived from it. We show the effects of the operators
by turning on only one of them at the time, in order to display how different
modifications of the couplings alter the entanglement pattern. It is
particularly interesting to see that not only the effects of the operators are
substantial for the value of choice of the Wilson coefficients, but that the
pattern of modification severely changes from operator to operator. The
ultimate reason for that has to be tracked down to the way they induce shifts
to the EW couplings defined in Section 3. For instance, the operator
$\mathcal{O}_{ll}$ (not displayed) is affecting the SM couplings in a
universal way, inducing an overall rescaling factor, and therefore does not
affect the density matrix in any way, leaving the $\mathcal{C}_{\mathrm{LB}}$
marker unchanged. On the other hand, the four Wilson coefficients shown in
Fig. 6, $c_{\varphi e}$, $c_{\varphi l}^{(1)}$, $c_{\varphi WB}$ and $c_{W}$,
alter the EW interactions in such a way that the density matrix of the final
state spins is sensibly affected. We can see for instance that a positive
value of $c_{\varphi e}$, which shifts the value of the right-handed coupling
to the $Z$ boson, induces an augmented level of entanglement at high energy
both in the central region and in the backward one. Very different behaviour
is instead produced by a positive $c_{\varphi l}^{(1)}$, which by shifting the
left-handed coupling to the $Z$ boson decreases the value of the
$\mathcal{C}_{\mathrm{LB}}$ marker in the same region. The effects of the
$c_{\varphi WB}$ and $c_{W}$ operators is instead milder. In particular, one
could have potentially expected a big impact from the $\mathcal{O}_{W}$
operator given that it induces the presence of the non-SM Lorentz structure
$\delta\lambda_{V}$, but that does not seem to be the case. Note that in
general, to an opposite sign value of the Wilson coefficient corresponds an
opposite effect, _i.e._ , if for
$c_{W}=$0.25\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$ the high entanglement region
increases, for $c_{W}=$-0.25\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$ we would see
a decrease. Finally, it is worth noticing that all of the operators leave the
entanglement pattern in the forward region unchanged and have mostly sensible
effects in the high energy region, as one would have naively expected.

Figure 7: The relative changes in the marker $\mathcal{C}_{\mathrm{LB}}$
(lower triangle) and purity $P$ (upper triangle) compared to the SM values
$\mathcal{C}_{\mathrm{LB}}^{\mathrm{SM}}=1.0$ and $P^{\mathrm{SM}}=0.94$ as a
function of the Wilson coefficients $c_{\varphi e}$, $c_{\varphi l}^{(1)}$,
$c_{\varphi WB}$ and $c_{W}$ for $W^{+}W^{-}$ production at a lepton collider,
at $m_{WW}=$500\text{\,}\mathrm{GeV}$$ and $\cos\theta=0$. The lines indicate
the relative change of the cross section.

Finally, in Fig. 7, we depict the relative change of
$\mathcal{C}_{\mathrm{LB}}$ (lower triangle) and the purity (upper triangle)
as a function of the Wilson coefficients, varying two coefficients at a time
and considering the fixed phase space point
$m_{WW}=$500\text{\,}\mathrm{GeV}$$ and $\theta=\pi/2$. Here,
$\Delta\mathcal{C}_{\mathrm{LB}}$ and $\Delta P$ denote the difference between
the marker $\mathcal{C}_{\mathrm{LB}}$ and the purity, respectively,
calculated within the SMEFT, and the SM values,
$\mathcal{C}_{\mathrm{LB}}^{\mathrm{SM}}=1.0$ and $P^{\mathrm{SM}}=0.94$. The
SMEFT values are calculated including dimension-6 and dimension-6 squared
contributions. In addition, the contours depict the relative change in
$\tilde{A}$, _i.e._ , the relative change of the differential cross-section
with respect to the SM. Notably, we see that that the spin-related observables
generally probe different parameter directions than the cross-section,
potentially offering complementary probes of NP. This could be of fundamental
importance both for discovery, enhancing the sensitivity to EFT corrections,
and for characterisation in the event of a clear deviation from the SM.
Additionally, one can clearly see from the $c_{W}$ plots that the spin-related
observables display a resurrection of the interference, as expected, while the
differential cross-section contour lines are mostly dominated by quadratic
corrections.

####  $ZZ$ production

Figure 8: Entanglement in the $e^{+}e^{-}\to ZZ$ channel in the SM. We show
the lower bound $\mathcal{C}_{\mathrm{LB}}$ (bottom left) and upper bound
$\mathcal{C}_{\mathrm{UB}}$ (bottom right) on the concurrence $\mathcal{C}$,
the purity $P$ (top left) as well as the indicator
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ for Bell inequality violation (top
right) as a function of the invariant mass $m_{ZZ}$ (or equivalently the
collider energy) and the cosine of the angle between the positron and the $Z$,
in the centre of mass frame.

One key difference between $ZZ$ and $W^{+}W^{-}$ production is the fact that
they probe complementary couplings of the fermions to the diboson system. In
particular, in the case of $ZZ$, only the coupling of the fermions to the $Z$
boson are relevant, while in the case of $W^{+}W^{-}$ a more intricate
coupling dependence is present, including the triple gauge coupling. For $Z$
pair production, the scattering amplitudes are completely determined by the
values of the vectorial and axial couplings to the $Z$ boson in Eq. 30.

In Fig. 8 we show the entanglement pattern in the SM. Contrary to the
$W^{+}W^{-}$ case, we see a symmetry with respect to $\cos{\theta}=0$, given
that this time the final state has identical particles and consequently the
system exhibits a symmetry under parity transformations. The plot in the upper
left corner further indicates that, in contrast to $W^{+}W^{-}$ production,
the majority of the phase space for $ZZ$ production is characterised by a
mixed quantum state. High purity $P$ is reached only in the high energy
central region. The plot on the lower left depicts the entanglement pattern in
terms of the lower bound marker $\mathcal{C}_{\mathrm{LB}}$. According to
that, the entanglement is expected to be high at high energy in the central
region, but quite low in the forward region. However, the picture gets more
complicated if we look at the right panel of Fig. 5, which displays bands for
the concurrence, making use of both the lower and the upper bound, for
benchmark collider energies as a function of $\cos{\theta}$. In the figure, we
see that the lower bound goes towards zero in the collinear regime, but the
upper bound does not, giving us enormous uncertainty on the determination of
the entanglement with this approach. This is also confirmed by the plot in the
lower right panel of Fig. 8 displaying the upper bound marker
$\mathcal{C}_{\mathrm{UB}}$ across phase space. Almost all of the phase space
is characterised by $\mathcal{C}_{\mathrm{UB}}$ close to maximal. Finally, in
the upper right plot of Fig. 8 we report on the expected value for the Bell
inequality violation marker $\langle\mathcal{B}\rangle_{\mathrm{max}}$. In
contrast to the $W^{+}W^{-}$ final state, the region of phase space with
$\langle\mathcal{B}\rangle_{\mathrm{max}}>2$ is rather limited and slight
violations are only present in the high energy central region.

More information on the entanglement can be gathered by directly inspecting
the density matrix and its decomposition in terms of quantum states. In
particular, we find that at threshold the $Z$ boson pair is produced in a
mixed state

$\rho(m_{ZZ}=2\,m_{Z})=p_{1}\,\ket{\Psi_{0+}}_{\boldsymbol{p}}\bra{\Psi_{0+}}_{\boldsymbol{p}}+p_{2}\,\ket{\Psi_{0-}}_{\boldsymbol{p}}\bra{\Psi_{0-}}_{\boldsymbol{p}}\,,$
(59)

with $p_{1}=0.7$ and $p_{2}=0.3$ and

$\displaystyle\begin{aligned}
\ket{\Psi_{0+}}_{\boldsymbol{p}}&=\frac{1}{\sqrt{2}}\left(\ket{+0}_{\boldsymbol{p}}+\ket{0+}_{\boldsymbol{p}}\right)\,,\\\
\ket{\Psi_{0-}}_{\boldsymbol{p}}&=\frac{1}{\sqrt{2}}\left(\ket{-0}_{\boldsymbol{p}}+\ket{0-}_{\boldsymbol{p}}\right)\,.\end{aligned}$
(60)

The two states are both fully entangled, but, given the fact that the density
matrix is in a mixed state, the value of the concurrence is not maximal.

Figure 9: The changes in the marker $\mathcal{C}_{\mathrm{LB}}$ is shown for a
selection of operators and benchmark Wilson coefficient values for the
production of $ZZ$ at a lepton collider. Only one operator at the time is
switched on. Left: $c_{\varphi
D}=$0.5\text{\,}\,\mathrm{T}\mathrm{e}\mathrm{V}^{-2}$$, right: $c_{\varphi
l}^{(3)}=$-0.25\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}^{-2}$$. Figure 10: The
relative changes in the marker $\mathcal{C}_{\mathrm{LB}}$ (lower triangle)
and purity $P$ (upper triangle) compared to the SM values
$\mathcal{C}_{\mathrm{LB}}^{\mathrm{SM}}=0.86$ and $P^{\mathrm{SM}}=0.83$ as a
function of the Wilson coefficients $c_{\varphi l}^{(3)}$, $c_{\varphi D}$,
$c_{\varphi e}$ and $c_{\varphi WB}$ for $ZZ$ production at a lepton collider,
at $m_{ZZ}=500\,\text{GeV}$ and $\cos\theta=0$. The lines indicate the
relative change of the cross section.

On the other hand, at high energy the picture is different. In the central
region, $\theta=\pi/2$, the diboson pair is produced in a pure spin-2
maximally entangled state

$\ket{\Psi_{+-}}_{\boldsymbol{k}}=\frac{1}{\sqrt{2}}\left(\ket{++}_{\boldsymbol{k}}-\ket{--}_{\boldsymbol{k}}\right)\,.$
(61)

This is fully consistent with what we observe in Fig. 8. The concurrence study
based on the lower and upper bounds is inconclusive in the forward region. On
the other hand, by inspecting the density matrix directly at $\theta=0$ in the
high energy limit, we find that the $Z$ pair is produced in a mixed ensemble
of separable quantum states, _i.e._

$\rho(m_{ZZ}\to\infty,\cos{\theta}=1)=p_{1}\,\ket{++}_{\boldsymbol{p}}\bra{++}_{\boldsymbol{p}}+p_{2}\,\ket{--}_{\boldsymbol{p}}\bra{--}_{\boldsymbol{p}}\,,$
(62)

with $p_{1}=0.7$ and $p_{2}=0.3$. We are therefore able to conclude that in
the forward region, the diboson pair is indeed not entangled as suggested by
the $\mathcal{C}_{\mathrm{LB}}$ marker behaviour.

As for the case of $W^{+}W^{-}$ production, in Fig. 9 we show the effects of a
selected number of operators on the marker $\mathcal{C}_{\mathrm{LB}}$. Since
for $ZZ$ production the dependence on the couplings is considerably simpler,
the only defining parameter of the entanglement pattern is the balance between
the vector and the axial coupling, or, to be more precise, the ratio between
the two. We observe that, aside for the operators that universally rescale
both couplings leaving the density matrix unchanged, the behaviour of all the
operators is phenomenologically identical. In the figure, we show the two
possible deviating patterns from the SM, choosing as benchmark Wilson
coefficients $c_{\varphi D}=$0.5\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$ and
$c_{\varphi l}^{(3)}=$-0.25\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}^{-2}$$. In
the case of the former, the entanglement marker is augmented almost
everywhere, indicating that the Wilson coefficient shifts the couplings in
such a way that the dominant coupling prevails a bit more and the density
matrix is slightly “less mixed”. On the other hand, for $c_{\varphi
l}^{(3)}=$-0.25\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$ we observe a decrease of
the entanglement across the board. Note that at variance with $W^{+}W^{-}$
production, the effects close to threshold are a bit more pronounced. We
stress again that none of these behaviours is pertinent to a specific
operator, but both the increase and decrease of entanglement can be produced
by any of the operators by switching on the corresponding Wilson coefficient
(with negative and positive values for opposite effects). Finally, in Fig. 10
we show the relative change of $\mathcal{C}_{\mathrm{LB}}$ and the purity as a
function of a selection of Wilson coefficients, considering the fixed phase
space point $m_{ZZ}=$500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ and
$\theta=\pi/2$. As for the case of $W$ pair production, the complementarity
between differential cross section and spin-related observables is self-
evident. Additionally, it is interesting to see that purity and
$\mathcal{C}_{\mathrm{LB}}$ probe the same direction in the Wilson coefficient
parameter space, indicating that there is a strong correlation between the two
observables at the spin density matrix level.

###  5.2 Hadron collider

We now present results for a hadron collider, specifically for a proton
collider corresponding to the LHC in Run 2. Studying the processes at LO, the
relevant channels are the quark annihilation ones. At the level of the hard
scattering, the kinematic dependence and therefore the density matrices are
fairly similar to the corresponding ones at a lepton collider. The main
differences are given by the different charges, which induce different
couplings to the EW bosons. We find these effects to be relevant but not
particularly disruptive of the entanglement pattern. In particular, one can
see that the explicit expressions for the density matrix in the benchmark
regions analysed in the previous section are fairly similar.

What really changes the picture in a proton collider are the contributions
from different partonic channels, weighted by the corresponding parton
luminosity. Also, since we are now colliding identical particles, the system
will be invariant under the transformation $\theta\to\pi+\theta$, showing
therefore a symmetry with respect to $\cos{\theta}=0$. This can also be
understood at the level of the $R$-matrix from the symmetrisation in Eq. 27.
As a general remark, the most evident effect of having to sum over the
different partonic channels is that the presence of entanglement is
considerably diluted.

####  $WW$ production

Figure 11: The lower bound for the concurrence $\mathcal{C}_{\mathrm{LB}}$ in
the partonic channels as a function of the invariant mass $m_{VV}$ of the
diboson pair and the cosine of the angle between the $\bar{q}$ and the $W^{+}$
(or $Z$ for $ZZ$), in the centre of mass frame, in the SM. Bottom:
$W^{+}W^{-}$ production in the $u\bar{u}$ (left) and $d\bar{d}$ (right)
channel. Middle: $ZZ$ production in the $u\bar{u}$ (left) and $d\bar{d}$
(right) channel. Top: $u\bar{d}\to ZW^{+}$ production.  Figure 12:
Entanglement in the $pp\to W^{+}W^{-}$ channel in the SM. We show the lower
bound $\mathcal{C}_{\mathrm{LB}}$ (bottom left) and upper bound
$\mathcal{C}_{\mathrm{UB}}$ (bottom right) on the concurrence $\mathcal{C}$,
the purity $P$ (top left) as well as the indicator
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ for Bell inequality violation (top
right) as a function of the invariant mass $m_{WW}$ and the cosine of the
angle between the proton and the $W^{+}$, in the centre of mass frame.

As in the case of a lepton collider, the production of a pair of $W$ bosons is
the dominant diboson production channel. The high cross section makes it an
ideal candidate to probe the EW interactions and potentially uncover signs of
NP in the tails of the distributions. With this aim, the study of the spin
density matrix can offer a complementary approach which could be helpful to
disentangle degeneracies and characterise potential signals.

In the bottom panel of Fig. 11, we show the value of the marker
$\mathcal{C}_{\mathrm{LB}}$ across phase space for the two independent
channels in proton collisions, $u\,\bar{u}\to W^{+}\,W^{-}$ and $d\,\bar{d}\to
W^{+}\,W^{-}$. As opposed to the lepton collider, we decide to show results up
to $2\text{\,}\mathrm{TeV}$ in invariant mass. The physics regulating the
production of $W$ bosons is fairly similar for all initial states and the only
differences are given by the different couplings the particles have with the
$Z$ boson, _i.e._ the $s$-channel diagram. As a consequence, the balance
between the right-handed and left-handed couplings of the various initial
states is slightly different and this translates into a slight difference for
the density matrix. Note that the mirroring of the $u\bar{u}$ channel is
fictitious and simply given by the conventional choice of the angle between
the anti-up quark and the $W^{+}$. In fact, in the case of the positron and
the $d$ quark, that angle is the one between the initial and final state
particles that share the same-sign charge, while in the $u\bar{u}$ case it is
the angle between the opposite charge particles.

In the lower panel of Fig. 12, we plot the lower limit of the concurrence
$\mathcal{C}_{\mathrm{LB}}$ for the parton luminosity weighted combination of
the channels, as described in Eq. 27. As expected, the result is symmetric
with respect to $\cos{\theta}=0$, given the symmetrisation over the polar
angle. It is interesting to observe the strong dilution of the entanglement
pattern, which is caused precisely by summing over the initial state and
considering both $q\bar{q}$ and $\bar{q}q$ channels. This can be intuitively
understood by looking at the plots of the individual channels, where we can
observe that the two collinear regions, $\cos{\theta}=1$ and $\cos{\theta}=-1$
are characterised by opposite behaviours and therefore the high entanglement
of one region is washed out by the other when summing over the two different
polar angles in Eq. 27. Because of this, the diboson pair produced will mostly
be in a mixed state, and a high level of entanglement will only be found in
the central region and at high energy, around $\theta=\pi/2$.

As for the lepton collider case, in Fig. 12 we also report on the value of the
quantum observable indicators for the purity $P$, the Bell inequality
violation marker $\langle\mathcal{B}\rangle_{\mathrm{max}}$ and the upper
bound on the concurrence $\mathcal{C}_{\mathrm{UB}}$. The latter is not
bringing much information to the table, since it shows values of order 1
across phase space. On the other hand, from the purity plot we learn that the
density matrix is characterised by a highly mixed state and it goes towards a
purer state in the high-energy central region. We find that this is a common
feature across all proton collider processes, mostly an effect of the state
mixing dictated by Eq. 27. Finally, in the upper right plot of Fig. 12 we
display the marker for Bell inequality violation, which follows closely the
pattern indicated by the marker $\mathcal{C}_{\mathrm{LB}}$, sign once again
that states violating Bell inequalities are a subset of entangled states.

Figure 13: The changes in the marker $\mathcal{C}_{\mathrm{LB}}$ is shown for
a selection of operators and benchmark Wilson coefficient values for
$W^{+}W^{-}$ production at a proton collider. Only one operator at the time is
switched on. Top left: $c_{\varphi
u}=$0.05\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, top right: $c_{\varphi
d}=$0.05\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, bottom left: $c_{\varphi
q}^{(3)}=$0.05\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, bottom right:
$c_{W}=$0.03\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$. Figure 14: The relative
changes in the marker $\mathcal{C}_{\mathrm{LB}}$ (lower triangle) and purity
$P$ (upper triangle) compared to the SM values
$\mathcal{C}_{\mathrm{LB}}^{\mathrm{SM}}=0.73$ and $P^{\mathrm{SM}}=0.64$ as a
function of the Wilson coefficients $c_{\varphi d}$, $c_{\varphi q}^{(3)}$,
$c_{\varphi u}$ and $c_{W}$ for $W^{+}W^{-}$ production at a proton collider,
at $m_{WW}=$500\text{\,}\mathrm{GeV}$$ and $\cos\theta=0$. The lines indicate
the relative change of the cross section.

We now move on to the study of the EFT effects to the density matrix of the
$W$ pair produced in a proton collider. In Fig. 13 we display the changes in
the $\mathcal{C}_{\mathrm{LB}}$ marker for some benchmark Wilson coefficients.
As can be seen in the two top figures, the effects of $c_{\varphi u}$ and
$c_{\varphi d}$ are very similar, enhancing the right-handed coupling of the
$Z$ boson with the quarks and consequently decreasing the level of
entanglement at high energy. The different intensity of the two operators has
to be traced back to the different weight at the level of the PDFs, _i.e._ ,
the $u\bar{u}$ luminosity is higher than the $d\bar{d}$ one. As expected,
switching on the $c_{\varphi q}^{(3)}$, which instead modifies the left-handed
coupling to the $Z$ of the $d$-quarks and the coupling to the $W$ boson of
both $u$ and $d$, has the opposite effect, increasing the value of the
concurrence in the central region where we see the emergence of maximal level
of entanglement. A similar effect is found for the $c_{\varphi q}^{-}$ Wilson
coefficient (not displayed) which modifies the left-handed coupling of the $u$
quarks with the $Z$ boson. Finally, in the lower-right plot in Fig. 13 we show
the effects of the $\mathcal{O}_{W}$ operator. Interestingly, we find that the
presence of the pure BSM coupling $\delta\lambda_{V}$ can be quite disruptive,
especially at high energy, inducing a decrease of the level of entanglement.
The effects of the $c_{\varphi D}$ and $c_{\varphi WB}$, which modify the SM
TGCs, are found to be sensibly smaller in this case. We observe that the EFT
effects are mostly in the central region while the collinear regions keep
being characterised by a substantial absence of entanglement.

Finally, in Fig. 14, we depict the relative change of
$\mathcal{C}_{\mathrm{LB}}$ (lower triangle) and the purity (upper triangle)
as a function of the Wilson coefficients, varying two coefficients at a time
and considering the fixed phase space point
$m_{WW}=$500\text{\,}\mathrm{GeV}$$ and $\theta=\pi/2$. Contrary to the case
of a lepton collider, we do not find that we gain much from the chosen spin-
related observables, _i.e._ , the probed directions in parameter space are
very similar to the ones probed by the differential cross section. This has to
be traced back to the fact that in a proton collider we sum over the different
partonic channels and in doing so we lose sensitivity to the spin related Fano
coefficients. The quantum spin observables are indeed highly dependent on the
Fano coefficient $\tilde{A}$, which controls the abundance of a channel over
the other, and therefore affects the spin density matrix of the total system.
We verified indeed that if one were to single out a specific channel (for
example only $u\bar{u}$ and its $\bar{u}u$ counterpart), one would find
results much more similar to those observed in Fig. 7 for the case of a lepton
collider.

####  $ZZ$ production

Figure 15: Entanglement in the $pp\to ZZ$ channel in the SM. We show the lower
bound $\mathcal{C}_{\mathrm{LB}}$ (bottom left) and upper bound
$\mathcal{C}_{\mathrm{UB}}$ (bottom right) on the concurrence $\mathcal{C}$,
the purity $P$ (top left) as well as the indicator
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ for Bell inequality violation (top
right) as a function of the invariant mass $m_{ZZ}$ and the cosine of the
angle between the proton and the $Z$, in the centre of mass frame.

We now discuss $ZZ$ production at a proton collider. In the middle panel of
Fig. 11, we show the entanglement pattern for the case of quark annihilation
and in Fig. 15 their combination in a proton collider (lower left plot). It is
interesting to notice that the quark channels present a different pattern of
entanglement with respect to the lepton collider, which is caused by the
simple fact that the value of the $\bar{g}_{V}$ coupling depends on the charge
of the particle and is sensibly different in the three cases, _i.e._
$\bar{g}_{V}\approx-0.027,0.1,-0.17$ for $e$, $u$ and $d$ respectively. This
is non-trivial as one could have naively expected the patterns to be the same
given that the EW couplings involved are the same. Also in the case of a
proton collider, the general feature remains that high entanglement in $ZZ$
production can be found at high invariant mass and in the central region. Once
again the plot of the upper bound marker $\mathcal{C}_{\mathrm{UB}}$ does not
deliver any information, as the indicator reaches close to maximal values
everywhere in phase space. The plot of the Bell violating marker
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ in the upper right corner of Fig.
15 confirms that the density matrix in the central high-energy region is
characterised by highly entangled quantum states.

Moving on to the EFT effects, as we already discussed in the corresponding
section on $ZZ$ production at a lepton collider, the only possible
modifications are given by the shift of the vectorial and axial couplings to
the $Z$ boson, in particular from operators that spoil the balance between the
two. We already saw that the effects are more subtle compared to the case of
$W^{+}W^{-}$ production. However, rather surprisingly we observe that in the
case of the proton collisions there is even more sturdiness towards EFT
effects. For values of the Wilson coefficients within the current bounds
coming from global fits, no visible effect on the pattern of entanglement is
present. For this reason we believe that $Z$ pair production is the least
promising process to probe dimension-6 effects at the level of the spin
density matrix.

####  $WZ$ production

Figure 16: Entanglement in the $pp\to W^{+}Z$ channel in the SM. We show the
lower bound $\mathcal{C}_{\mathrm{LB}}$ (bottom left) and upper bound
$\mathcal{C}_{\mathrm{UB}}$ (bottom right) on the concurrence $\mathcal{C}$,
the purity $P$ (top left) as well as the indicator
$\langle\mathcal{B}\rangle_{\mathrm{max}}$ for Bell inequality violation (top
right) as a function of the invariant mass $m_{WZ}$ and the cosine of the
angle between the proton and the $W^{+}$, in the centre of mass frame.

Contrary to the previous diboson production modes, $WZ$ cannot be produced at
a $e^{+}e^{-}$ collider without the emission of additional charged particles.
At a proton collider, only one relevant partonic channel exists, _i.e._ ,
$u\bar{d}\to W^{+}Z$ and the charge conjugated one. In the following we will
focus on $W^{+}Z$ production as representative of the two processes, which can
distinguished in experiments. In this process, the relevant couplings are
those of the $W$ and $Z$ to the fermions, as well as the triple gauge
coupling. Note that in the case of $WZ$, at the partonic level depicted in the
top panel of Fig. 11, the expressions for $\mathcal{C}_{\mathrm{LB}}$ and
$\mathcal{C}_{\mathrm{UB}}$ are identical (in the SM), indicating that
$\mathcal{C}_{\mathrm{LB}}$ is precisely the value of the concurrence and not
just a lower bound. This is ultimately due to the fact that the trace of
$\rho^{2}$ is equal to 1 (the system is a pure state everywhere in phase
space) and therefore the expressions in Eqs. 12 and 14 coincide. The statement
does not necessarily hold true anymore in the presence of modified
interactions, as we verified for the dimension-6 operators considered in this
work. However, this property is not maintained once the symmetrisation of the
$\theta$ angle in Eq. 27 is performed and consequently the density matrix for
proton collisions is ultimately highly mixed, as can be seen in the upper left
plot in Fig. 16. To improve the purity, one could consider events boosted in
the forward/backward regions and try to infer on a statistical basis the
directions of the quark and anti-quark in the initial states.

In Fig. 11 (top plot) we show the $\mathcal{C}_{\mathrm{LB}}$ pattern for the
individual channel, while the proton collider one is displayed in the bottom
left plot in Fig. 16, where the main difference is due to the fact that we
have to take into account both $q\bar{q}$ and $\bar{q}q$ initial states, cf.
Eq. 27. We notice once again, that the entanglement is much lower in the
proton collider with respect to the individual channel. Surprisingly, we find
high entanglement at threshold as well as for $\cos{\theta}\approx\pm 0.5$,
while in the central region, for $\cos{\theta}=0$, the value of the
concurrence is low. The overall pattern differs substantially with respect to
the previously analysed diboson processes. As a matter of fact, as can be seen
from the marker $\langle\mathcal{B}\rangle_{\mathrm{max}}$ the violation of
Bell inequalities is really weak even at high energy, mostly a consequence of
the fact that the density matrix is in a high mixture of states (see upper
left plot in Fig. 16).

With respect to the quantum state produced by the $u\bar{d}$ channel, we find
that at threshold, the density matrix is described by a pure state

$\ket{\Psi(m_{WZ}=m_{W}+m_{Z})}\approx
0.75\ket{0+}_{\boldsymbol{p}}+0.66\ket{+0}_{\boldsymbol{p}}\,.$ (63)

Notably, the quantum state is not symmetric under label exchange, as a
consequence of the fact that we are not dealing any more with pairs of
particles sharing the same mass and interactions. At high energy, in the
collinear limit $\theta=0$, the density matrix is described by a pure
separable state, _i.e._ , $\ket{++}_{\boldsymbol{p}}$. On the other hand, in
the central region, we still have a pure but partially entangled state

$\ket{\Psi(m_{WZ}\to\infty,\cos{\theta}=0)}\approx
0.164\ket{++}_{\boldsymbol{k}}-0.973\ket{00}_{\boldsymbol{k}}-0.164\ket{--}_{\boldsymbol{k}}\,.$
(64)

Figure 17: The changes in the marker $\mathcal{C}_{\mathrm{LB}}$ is shown for
a selection of operators and benchmark Wilson coefficient values for $W^{+}Z$
production at a proton collider. Only one operator at the time is switched on.
Left: $c_{\varphi q}^{(3)}=$0.03\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$, Right:
$c_{W}=$0.01\text{\,}\mathrm{T}\mathrm{eV}^{-2}$$. Figure 18: The relative
changes in the marker $\mathcal{C}_{\mathrm{LB}}$ (lower triangle) and purity
$P$ (upper triangle) compared to the SM values
$\mathcal{C}_{\mathrm{LB}}^{\mathrm{SM}}=0.38$ and $P^{\mathrm{SM}}=0.55$ as a
function of the Wilson coefficients $c_{W}$ and $c_{\varphi q}^{(3)}$ for $WZ$
production at a proton collider, at $M_{WZ}=$500\text{\,}\mathrm{GeV}$$ and
$\cos\theta=1/2$. The lines indicate the relative change of the cross section.

Finally, we discuss the effects of the dimension-6 EFT operators. We point out
that, for this specific process, we set $m_{W}=m_{Z}$ when computing the EFT
corrections. This approximation considerably speeds up the computation, and
given that higher dimensional operator corrections are mostly relevant at high
energy, we expect the difference $m_{Z}-m_{W}$ to be negligible. We have
verified the former statement for a subset of the operators, finding that the
naive expectation holds true.

In Fig. 17 we display the changes in the $\mathcal{C}_{\mathrm{LB}}$ marker
for some benchmark Wilson coefficients. We find that the density matrix is
particularly sensitive to higher dimensional operators for this process, even
within the current bounds set by global EFT fits. In particular in the plot on
the right, we observe a strong effect coming from the presence of the Wilson
coefficient $c_{W}$, which is unique in its kind inducing the BSM coupling
$\delta\lambda_{V}$. The effect of this operator is to increase the
entanglement in the central region at high energy, indicating that the
dominant configuration in that phase space region is enhanced by the presence
of the operator. As previously discussed, the $\mathcal{O}_{W}$ operator is of
particular interest from an EFT perspective and we find that the $W^{+}Z$
final state seems to be the best probe among the diboson final states with
respect to the spin density matrix. In the plot on the left in Fig. 17 we
instead show the changes coming from a modification of the left-handed
coupling to the EW bosons by the $\mathcal{O}_{\varphi q}^{(3)}$ operator. The
effect in this case is opposite and the value of the
$\mathcal{C}_{\mathrm{LB}}$ marker is generally decreased in the high energy
phase space region, indicating that the two operators enhance different spin
configurations. We find that other operators affecting the process
($\mathcal{O}_{\varphi q}^{(1)}$, $\mathcal{O}_{\varphi D}$ and
$\mathcal{O}_{\varphi WB}$) have close to negligible effects on the
entanglement pattern when considering non-excluded values of the Wilson
coefficients. We therefore do not show the corresponding plots.

To conclude the section, in Fig. 18 we display the relative change of
$\mathcal{C}_{\mathrm{LB}}$ (left) and the purity (right) as a function of the
Wilson coefficients in two-dimensional planes. Note that contrary to
previously discussed processes, here the phase space point chosen is different
as in the central phase space region $\theta=\pi/2$, low values of
entanglement are found and specifically for
$m_{WZ}=$500\text{\,}\mathrm{GeV}$$, the value of $\mathcal{C}_{\mathrm{LB}}$
is equal to $0$. We therefore opt for showing plots for $\cos{\theta}=1/2$.
Once again, the added value of the spin observables in searches for new heavy
physics is evident and the plot displays a significant sensitivity both at the
cross section level and at the level of the spin density matrix, with
deviations from the SM predictions as big as $30\text{\,}\mathrm{\char
37\relax}$ for values of the Wilson coefficients well within the current
limits from global fits.

##  6 Conclusions

In this work, we have explored the sensitivity of quantum observables to the
strength and structure of couplings entering diboson production, both in the
context of the SM and of the SMEFT. Our main objective was to gauge the power
of the quantum spin density matrix and related observables to probe the
existence of NP.

After setting up the formalism, we studied the behaviour of scattering
amplitudes in the high energy regime, where longitudinal polarisations
dominate the production mode, assessing the consequences for the spin density
matrix. In particular, we exploited the known fact that spin observables allow
to study effects that are subdominant at the level of single particle
distributions, and depend on the interference between higher-dimensional
operators and the SM. These observables display a sensitivity to deviations
from the SM predictions in the high-energy tails of the distributions, which
will be further explored with Run-3 and the HL-LHC in the coming decade.

The main results of our study are reported in Section 5, where different
processes have been analysed. We have considered both lepton and proton
colliders, finding that the former offer a much cleaner setup for spin density
matrix probes. This is mostly due to the fact that in a proton collider the
quantum state of the system is the incoherent sum of different partonic
channels and therefore tends to be mixed. Nonetheless, considerable
sensitivity to NP is also found at proton colliders, which, featuring higher
centre of mass energies, can take full advantage of the energy growth of the
dimension-6 amplitudes.

In general, we find that the $ZZ$ production is the least interesting process
when it comes to NP sensitivity, as the phenomenology is completely determined
by only two possibly anomalous couplings (the right-handed and the left-handed
coupling to the $Z$ boson) and the dimension-6 operators do not introduce new
Lorentz structures. We note, however, the potential interest in studying the
effects of the neutral TGC which arise at dimension-8. In this case, spin-
observables could help in gaining sensitivity, especially because of the
possibility to fully reconstruct the final state, something which is
experimentally more challenging for final states involving $W$ bosons. On the
other hand, we find that $WW$ and $WZ$ production show a rather large
sensitivity to heavy NP effects in the spin density matrix already at
dimension-6 with significant changes expected in the entanglement pattern
across phase space. For example, interference effects due to the triple gauge
operator $\mathcal{O}_{W}$ are clearly identified by quantum observables.

Our results motivate an experimental feasibility study for performing the
detailed quantum tomography of the four-fermion final states arising from $VV$
production in the SMEFT framework at the LHC and at future lepton colliders.

## Acknowledgements

RA’s research was supported by the F.R.S.-FNRS project no. 40005600 and the
FSR Program of UCLouvain. FM is partially supported by the F.R.S.- FNRS under
the “Excellence of Science” EOS be.h project no. 30820817. LM is supported by
the European Research Council under the European Union’s Horizon 2020 research
and innovation Programme (grant agreement n.950246).

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