# Flavor vacuum entanglement in boson mixing

Massimo<EMAIL_ADDRESS>0.85358pt1,2}$, Fabrizio
<EMAIL_ADDRESS>0.85358pt2,3}$, Giuseppe Gaetano
<EMAIL_ADDRESS>0.85358pt1,2}$ and Luciano
<EMAIL_ADDRESS>0.85358pt2,3}$ 1Dipartimento
di Fisica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132
I-84084 Fisciano (SA), Italy.
2INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy.
3Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno,
Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy.

###### Abstract

Mixing transformations in quantum field theory are non-trivial, since they are
intimately related to the unitary inequivalence between Fock spaces for fields
with definite mass and fields with definite flavor. Considering the
superposition of two neutral scalar (spin-0) bosonic fields, we investigate
some features of the emerging condensate structure of the flavor vacuum. In
particular, we quantify the flavor vacuum entanglement in terms of the von
Neumann entanglement entropy of the reduced state. Furthermore, in a suitable
limit, we show that the flavor vacuum has a structure akin to the thermal
vacuum of Thermo Field Dynamics, with a temperature dependent on both the
mixing angle and the particle mass difference.

##  I Introduction

The analysis of quantum correlations in the context of particle physics (and
in particular for neutrino and meson systems) is currently gaining a rising
attention. The initial observation that flavor mixing and oscillations can be
associated with (single-particle) entanglement entmix ; entmix2  served as a
basis for several studies numecorr ; numecorr2 ; numecorr3 ; numecorr4 ;
numecorr5 ; numecorr6 ; numecorr7 ; numecorr8 ; numecorr9  in which violations
of Bell, Leggett-Garg and Mermin-Svetchlichny inequalities, non-locality,
gravity/acceleration degradation effects and other similar occurrences have
been investigated both theoretically and experimentally.

Most of the above studies have been carried out within the framework of
Quantum Mechanics (QM). The extension to Quantum Field Theory (QFT) has been
later considered in Ref. entQFT , thus leading to the discovery of non-trivial
properties of the mixing transformation annals . Indeed, whilst in QM such a
transformation acts as a simple rotation between flavor and mass states
pontecorvo , its QFT counterpart behaves as a rotation nested into a non-
commuting Bogoliubov transformation annals ; bogoliubov . As a result, the
vacua for fields with definite mass and fields with definite flavor become
orthogonal to each other, with the latter acquiring the structure of a $SU(2)$
coherent state perelomov  and turning into a condensate of massive
particle/antiparticle pairs annals ; smald . This gives rise to a deeeper
understanding of particle mixing, since the Fock spaces for flavor and mass
fields are found to describe unitarily inequivalent representations.
Corrections to the standard QM predictions also appear in the oscillation
probability formula, as explicitly shown in Ref. exactoscill .

Originally developed for neutrinos propagating in flat spacetime, the above
considerations have been later extended to bosonic fields bosonmix1 ;
bosonmix2  as well as to non-trivial spacetime backgrounds (see Refs. prd ;
Capelup  for more details). Recently, a further evidence for the complex
structure of the flavor vacuum has been provided in Ref. CUBA , where it has
been established that the Fock space for flavor fields cannot be obtained by
the direct product of the spaces for massive fields. Therefore, entanglement
connected with flavor mixing appears to be an exquisite concept, boiling down
to the non-factorizability of the flavor states in terms of the ones with
definite masses. In other words, it is possible to come across flavor
entanglement already at the level of the vacuum state. More generally, the
phenomenon of a non-vanishing entanglement even for free fields is ultimately
related to the quotient space structure of the tensor Hilbert spaces casini .

In fact, the investigation of the properties of the QFT vacuum is one of the
most significant (albeit difficult) tasks in a wide variety of physical
scenarios. For instance, the Bardeen-Cooper-Schrieffer ground state plays a
pivotal rôle in condensed matter, being a condensate of Cooper pairs which
underpins the phenomenon of superconductivity bcs . In the same fashion,
vacuum is crucially important to explain the spontaneous symmetry breaking
miransky  and the ensuing appearance of Nambu-Jona Lasinio nambu , Goldstone
goldstone ; higgs  and pseudo-Goldstone bosons pseudo , both in low and high
energy regimes. On the other hand, in QFT vacuum energy is notoriously
responsible for the existence of Casimir effect casimir ; Mitton , which has
been largely studied both in flat list1 ; list1b ; list1c ; list1d ; list1e
and curved list2b ; list2c ; list2d ; list2e ; list2f ; list2g  spacetime in
recent years. Moreover, the study of vacua in the presence of gravity leads to
the loss of the absolute concept of particle and the emergence of distinctive
phenomena such as the black hole radiation hawking  and the akin Unruh effect
unruh , which find application in many research areas App .

Starting from the above premises, in the present work we analyze some relevant
and yet unexplored features of the flavor vacuum condensate, with a particular
focus on its entanglement structure. For this purpose, we consider the case of
mixing neutral bosonic fields. Besides the intrinsic importance (i.e. as in
the case of meson mixing), this choice allows to reduce several technical
complications and render the physical insight as transparent as possible.
Given the mixing of two bosonic fields, in the following we will quantify the
entanglement content of the flavor vacuum by computing von Neumann
entanglement entropy of its reduced density matrix in the limit of small mass
difference and/or mixing angle. We will then compare this result with the
entanglement of the thermal vacuum of Thermo Field Dynamics (TFD) tfd ; BTFD ,
which exhibits the paradigmatic structure occurring in the case of black holes
Israel  and Unruh effect unruh , namely the doubling of Fock space and the
ensuing correlation between two different sets of modes (the particle states
inside and outside the horizon for Hawking-Unruh effect, the physical and
auxiliary modes in TFD). It must be stressed that entanglement in TFD has been
previously discussed in Ref. Hashizume  for both equilibrium and non-
equilibrium states. From the comparison between our results and the ones on
TFD entanglement, we find that the condensate in the flavor vacuum is in
general richer than the one in TFD, as it exhibits all types of contributions,
both thermal and non-thermal. Finally, we determine a suitable limit in which
non-thermal contributions are subdominant, thereby allowing us to recognize a
TFD-like structure in the flavor vacuum, with an effective temperature
proportional to the mixing angle and the mass difference between the two
fields.

The paper is organized as follows: in Sec. II we review the main aspects of
boson field mixing, focusing on the case of neutral scalar fields. In Sec. III
we quantify the entanglement content of the flavor vacuum by computing the
reduced von Neumann entropy, and we discuss the condensate structure of this
state in connection with the thermal vacuum of TFD. Conclusions and outlook
are provided in Sec. IV. Throughout the work, we use natural units $c=\hbar=1$
while keeping the Boltzmann constant $k_{B}$ explicit.

##  II Mixing of bosonic fields

In this Section, we review the crucial aspects associated with the mixing of
two scalar (spin-$0$) neutral fields. Clearly, the same considerations can be
extended to the case of three boson generations. To this aim, we closely
follow Refs. bosonmix1 ; bosonmix2  and write down the mixing relations
between fields with definite mass and flavor, that is555Strictly speaking, in
the case of bosons we should refer to the mixing of some other quantum number,
such as the strangeness or the isospin rather than the flavor. However, with
an abuse of notation, in the following we keep on denoting such intrinsic
properties as “flavor” and the corresponding mixed fields as “flavor fields”.
Furthermore, we work in a simplified two-flavor model.

$\displaystyle\phi_{A}(x)$ $\displaystyle=$
$\displaystyle\cos\theta\,\phi_{1}(x)+\sin\theta\,\phi_{2}(x)\,,$ (1)
$\displaystyle\phi_{B}(x)$ $\displaystyle=$
$\displaystyle-\sin\theta\,\phi_{1}(x)+\cos\theta\,\phi_{2}(x)\,,$ (2)

with an analogous set of equations for the conjugate momenta
$\pi(x)=\partial_{t}\phi(x)$. The subscripts $A$ and $B$ indicate the fields
in the flavor basis, whereas $1$ and $2$ the ones in the mass basis.
Consequently, the expansions for $\phi_{1}$ and $\phi_{2}$ take the form

$\phi_{j}(x)=\int\frac{d^{3}k}{\sqrt{2\left(2\pi\right)^{3}\omega_{k,j}}}\left(a_{k,j}e^{-i\omega_{k,j}t}+a^{\dagger}_{-k,j}e^{i\omega_{k,j}t}\right)e^{i\textbf{k}\cdot\textbf{x}}\,,\qquad
j=1,2,$ (3)

where $\omega_{k,j}=\sqrt{\textbf{k}^{2}+m_{j}^{2}}$ and $a_{k,j}$
$(a^{\dagger}_{k,j})$ are the bosonic annihilation (creation) operators of
field quanta with momentum $k$ and mass $m_{j}$.

By requiring that the fields and the conjugate momenta obey the canonical
commutation relation (CCR) at equal times,

$\Bigl{[}\phi_{i}(x),\pi_{j}(x^{\prime})\Bigr{]}_{t=t^{\prime}}=i\hskip
0.56905pt\delta_{ij}\hskip
0.56905pt\delta\left(\textbf{x}-\textbf{x}^{\prime}\right),$ (4)

it follows that the only non-trivial commutator between the ladder operators
is

$\left[a_{k,i},a^{\dagger}_{k^{\prime},j}\right]=\delta_{ij}\delta(\textbf{k}-\textbf{k}^{\prime}).$
(5)

Let us now observe that Eqs. (1) and (2) and the ones for momenta can also be
rewritten as

$\displaystyle\phi_{\sigma}(x)$ $\displaystyle=$ $\displaystyle
G_{\theta}^{-1}(t)\phi_{j}(x)G_{\theta}(t)\,,$ (6)
$\displaystyle\pi_{\sigma}(x)$ $\displaystyle=$ $\displaystyle
G_{\theta}^{-1}(t)\pi_{j}(x)G_{\theta}(t)\,,$ (7)

where $(\sigma,j)=\left\\{(A,1),(B,2)\right\\}$ and

$G_{\theta}(t)=\mathrm{exp}\left\\{\theta\left[S_{+}(t)-S_{-}(t)\right]\right\\}$
(8)

is the mixing generator and an element of $SU(2)$, whose algebra is built with
the following operators

$\displaystyle S_{+}(t)$ $\displaystyle=$ $\displaystyle-i\int
d^{3}x\,\pi_{1}(x)\phi_{2}(x)\,,$ (9) $\displaystyle S_{-}(t)$
$\displaystyle=$ $\displaystyle-i\int d^{3}x\,\pi_{2}(x)\phi_{1}(x)\,$ (10)
$\displaystyle S_{3}(t)$ $\displaystyle=$ $\displaystyle-\frac{i}{2}\int
d^{3}x\left[\pi_{1}(x)\phi_{1}(x)-\pi_{2}(x)\phi_{2}(x)\right]\,.$ (11)

In light of the above equations, the flavor fields can then be expressed as

$\phi_{\sigma}(x)=\int\frac{d^{3}k}{\sqrt{2\left(2\pi\right)^{3}\,\omega_{k,j}}}\left(a_{k,\sigma}(t)\hskip
0.56905pte^{-i\omega_{k,j}t}+a^{\dagger}_{-k,\sigma}(t)\hskip
0.56905pte^{i\omega_{k,j}t}\right)e^{i\textbf{k}\cdot\textbf{x}}\,,\qquad\sigma=A,B,$
(12)

and, with the aid of Eqs. (1) and (2), we recognize the Bogoliubov
transformation between the flavor and mass ladder operators

$\displaystyle a_{k,A}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,a_{k,1}+\sin\theta\left[U_{k}^{*}(t)\,a_{k,2}+V_{k}(t)\,a^{\dagger}_{-k,2}\right],$
(13) $\displaystyle a_{k,B}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,a_{k,2}-\sin\theta\left[U_{k}(t)\,a_{k,1}-V_{k}(t)\,a^{\dagger}_{-k,1}\right].$
(14)

The above relations exhibit the structure of rotations nested into Bogoliubov
transformations with coefficients $U_{k}(t)$ and $V_{k}(t)$ given by

$U_{k}(t)=\left|U_{k}\right|e^{i\left(\omega_{k,2}-\omega_{k,1}\right)t}\,,\quad
V_{k}(t)=\left|V_{k}\right|e^{i\left(\omega_{k,1}+\omega_{k,2}\right)t},$ (15)
$\left|U_{k}\right|=\frac{1}{2}\left(\sqrt{\frac{\omega_{k,1}}{\omega_{k,2}}}+\sqrt{\frac{\omega_{k,2}}{\omega_{k,1}}}\right)\,,\quad\left|V_{k}\right|=\frac{1}{2}\left(\sqrt{\frac{\omega_{k,1}}{\omega_{k,2}}}-\sqrt{\frac{\omega_{k,2}}{\omega_{k,1}}}\right)\,.$
(16)

Accordingly, the flavor vacuum is provided with with a $SU(2)$ coherent state
structure perelomov

$|00(t)\rangle_{A,B}=G_{\theta}^{-1}(t)|00\rangle_{1,2}\,,$ (17)

with a condensation density given by

${}_{A,B}\langle
00(t)|a_{k,j}^{\dagger}\,a_{k,j}|00(t)\rangle_{A,B}\,=\,\sin^{2}\theta\left|V_{k}\right|^{2},\qquad
j=1,2\,.$ (18)

In the infinite volume limit bosonmix1 ; bosonmix2 , the two sets of vacua
become orthogonal, namely ${}_{1,2}\langle
00|00(t)\rangle_{A,B}\overset{V\to\infty}{\longrightarrow}0$, $\forall t$,
giving rise to physically inequivalent Fock spaces (i.e. unitarily
inequivalent representations of the canonical commutation relations for
fields).

Let us now cast the mixing generator in terms of mass definite annihilators
and creators. Straightforward calculations lead to

$|00(t)\rangle_{A,B}=\exp\Bigl{\\{}-\theta\int
d^{3}k\left[U^{*}_{k}(t)\,a^{\dagger}_{k,1}a_{k,2}-U_{k}(t)\,a_{k,1}a^{\dagger}_{k,2}+V_{k}(t)\,a^{\dagger}_{k,1}a^{\dagger}_{-k,2}-V_{k}^{*}(t)\,a_{k,1}a_{-k,2}\right]\Bigr{\\}}|00\rangle_{1,2}\,.$
(19)

Without harming the generality of our results, henceforth we perform
calculations for $t=0$. Thus, we have
$|00(t=0)\rangle_{A,B}\equiv|00\rangle_{A,B}$ with666Notice that, for $t=0$,
$U_{k}(t)=\left|U_{k}\right|$ and $V_{k}(t)=\left|V_{k}\right|$ (see Eq.
(15)).

$|00\rangle_{A,B}=\exp\Bigl{\\{}-\theta\int
d^{3}k\left[U_{k}\left(a^{\dagger}_{k,1}a_{k,2}-a_{k,1}a^{\dagger}_{k,2}\right)+V_{k}\left(a^{\dagger}_{k,1}a^{\dagger}_{-k,2}-a_{k,1}a_{-k,2}\right)\right]\Bigr{\\}}|00\rangle_{1,2}\,.$
(20)

Equation (20) allows us to immediately identify the generator of the rotation
(the operator in the brackets which multiplies $U_{k}$) and the one
responsible for the Bogoliubov transformation (the operator in the brackets
which multiplies $V_{k}$), in complete agreement with the fermion case
bogoliubov .

For later convenience, we can now further manipulate Eq. (20) by considering
the case of a discrete set of modes. Apart from an irrelevant numerical
factor, Eq. (20) becomes

$|00\rangle_{A,B}=\exp\Bigl{\\{}-\theta\sum_{k}\left[U_{k}\left(a^{\dagger}_{k}b_{k}-a_{k}b^{\dagger}_{k}\right)+V_{k}\left(a^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}b_{-k}\right)\right]\Bigr{\\}}|00\rangle_{1,2}\,.$
(21)

where we have introduced the shorthand notation $a_{k,1}\equiv a_{k}$ and
$a_{k,2}\equiv b_{k}$. In addition, since the transformations Eqs. (1) and (2)
are valid for any value of the rotation angle, we can focus on the case of
small values of $\theta$ to the $\mathcal{O}(\theta^{2})$-order, at which the
first non-trivial contribution is expected. In this framework, we can use
Zassenhaus formula

$e^{\lambda\left(X+Y\right)}=e^{\lambda X}e^{\lambda
Y}e^{-\frac{\lambda^{2}}{2}\left[X,Y\right]}\,,$ (22)

up to $\mathcal{O}(\lambda^{2})$ to approximate the flavor vacuum in Eq. (21).
In fact, if we identify $\lambda\to-\theta$,
$X\to\sum_{k}U_{k}\left(a^{\dagger}_{k}b_{k}-a_{k}b^{\dagger}_{k}\right)$ and
$Y\to\sum_{k}V_{k}\left(a^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}b_{-k}\right)$,
we are led to

$\displaystyle|00\rangle_{A,B}$ $\displaystyle=$ $\displaystyle
e^{-\theta\sum_{k}U_{k}\left(a^{\dagger}_{k}b_{k}-a_{k}b^{\dagger}_{k}\right)}e^{-\theta\sum_{k}V_{k}\left(a^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}b_{-k}\right)}e^{-\frac{\theta^{2}}{2}\sum_{k}U_{k}V_{k}\left(a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}a_{-k}+b_{k}b_{-k}\right)}|00\rangle_{1,2}$
$\displaystyle=$ $\displaystyle
e^{-\frac{\theta^{2}}{2}\sum_{k}U_{k}V_{k}\left(a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}\right)}e^{-\theta\sum_{k}U_{k}\left(a^{\dagger}_{k}b_{k}-a_{k}b^{\dagger}_{k}\right)}e^{-\theta\sum_{k}V_{k}\left(a^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}b_{-k}\right)}|00\rangle_{1,2}\,,$

where in the second step we have made use of a simplification allowed only in
the given approximation for $\theta$. Furthermore, by applying the identity
(22) to the two operators in Eq. (II) that depend on $U_{k}$ and $V_{k}$, we
get

$\displaystyle|00\rangle_{A,B}$ $\displaystyle=$ $\displaystyle
e^{-\frac{\theta^{2}}{2}\sum_{k}U_{k}V_{k}\left(a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}\right)}e^{-\theta\sum_{k}U_{k}a^{\dagger}_{k}b_{k}}e^{\theta\sum_{k}U_{k}a_{k}b^{\dagger}_{k}}e^{\frac{\theta^{2}}{2}\sum_{k}U_{k}^{2}\left(a_{k}a_{k}^{\dagger}-b_{k}b_{k}^{\dagger}\right)}$
$\displaystyle\times$ $\displaystyle
e^{-\theta\sum_{k}V_{k}a^{\dagger}_{k}b^{\dagger}_{-k}}e^{\theta\sum_{k}V_{k}a_{k}b_{-k}}e^{-\frac{\theta^{2}}{2}\sum_{k}V_{k}^{2}\left(a_{k}a_{k}^{\dagger}+b_{k}^{\dagger}b_{k}\right)}|00\rangle_{1,2}$
$\displaystyle\simeq$ $\displaystyle
e^{-\frac{\theta^{2}}{2}\sum_{k}U_{k}V_{k}\left(a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}\right)}e^{-\frac{\theta^{2}}{2}\sum_{k}\left(V_{k}^{2}b^{\dagger}_{k}b_{k}+U_{k}^{2}b_{k}b^{\dagger}_{k}\right)}e^{-\theta\sum_{k}V_{k}a^{\dagger}_{k}b^{\dagger}_{-k}}|00\rangle_{1,2}\,.$

In performing the second passage, we have omitted an unimportant constant
factor and we have made use of the current approximation to streamline the
shape of the total operator. We will take advantage of the form (II) of the
flavor vacuum in the next Section, when we compare the condensate structure of
this state to the one of the thermal vacuum of Thermo Field Dynamics.

##  III Entanglement of the flavor vacuum

Let us now quantify the entanglement between the massive particle states in
the flavor vacuum. As said earlier, we consider the case $t=0$. For notational
convenience, it comes in handy to rewrite Eq. (20) as

$|00\rangle_{A,B}\equiv|\psi\rangle=\exp\Bigl{\\{}-\theta\int
d^{3}k\left[U_{k}\left(a^{\dagger}_{k}b_{k}-a_{k}b^{\dagger}_{k}\right)+V_{k}\left(a^{\dagger}_{k}b^{\dagger}_{-k}-a_{k}b_{-k}\right)\right]\Bigr{\\}}|00\rangle\,,$
(25)

where $|00\rangle\equiv|00\rangle_{1,2}$ and we have used the same notation as
in Eq. (21) for ladder operators. Interestingly, we observe that the above
transformation is the result of a simultaneous coexistence of a beam splitter
and a two-mode squeezing transformation.

As a preliminary analysis, we can perform a first-order approximation in
$\theta$ and in the case of small mass difference, namely when $m_{1}=m$,
$m_{2}=m+\delta m$ and hence $\varepsilon\equiv(m_{2}-m_{1})/m_{1}=\delta
m/m\ll 1$. This is in line with many works regarding flavor mixing, and in
particular with Ref. bogoliubov , in which the neutrino flavor vacuum is
written up to the second order in $\theta$ and $\varepsilon$. Here, we make
the same considerations with the purpose of seeking an analogous result.

Starting from Eq. (25), it is immediate to derive that

$|\psi\rangle=\exp\\{-\theta\int
d^{3}k\left[\left(a_{k}^{\dagger}b_{k}-a_{k}b_{k}^{\dagger}\right)-\frac{\varepsilon}{2}\frac{m^{2}}{\omega_{k}^{2}}\left(a_{k}^{\dagger}b_{-k}^{\dagger}-a_{k}b_{-k}\right)\right]\\}|00\rangle\,,$
(26)

and thus expand the exponential operator as

$|\psi\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}\left\\{-\theta\int
d^{3}k\left[\left(a_{k}^{\dagger}b_{k}-a_{k}b_{k}^{\dagger}\right)-\frac{\varepsilon}{2}\frac{m^{2}}{\omega_{k}^{2}}\left(a_{k}^{\dagger}b_{-k}^{\dagger}-a_{k}b_{-k}\right)\right]\right\\}^{n}|00\rangle\,.$
(27)

Since terms of the order $\mathcal{O}\left(\varepsilon^{2}\right)$ are
neglected, we observe that it is possible to identify two recursive formulas
in the expression (27), one in front of the operator $a^{\dagger}b^{\dagger}$
and another one next to
$\left(a^{\dagger}a^{\dagger}-b^{\dagger}b^{\dagger}\right)$. The exact
computation yields

$|\psi\rangle=\left\\{\textbf{1}+\frac{\varepsilon}{2}\left[\sum_{n=0}^{\infty}\frac{(-1)^{n+1}\theta^{2n+2}\,4^{n}}{(2n+2)!}\right]\int
d^{3}k\,\frac{m^{2}}{\omega_{k}^{2}}\left[a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}\right]+\frac{\varepsilon}{2}\left[\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{2n+1}\,4^{n}}{(2n+1)!}\right]\int
d^{3}k\,\frac{m^{2}}{\omega_{k}^{2}}\,a^{\dagger}_{k}b^{\dagger}_{-k}\right\\}|00\rangle\,.$
(28)

It is straightforward to realize that the series in Eq. (28) converge to two
simple analytic functions:

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n+1}\theta^{2n+2}\,4^{n}}{(2n+2)!}$
$\displaystyle=$ $\displaystyle-\frac{\sin^{2}\theta}{2}\,,$ (29)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{2n+1}\,4^{n}}{(2n+1)!}$
$\displaystyle=$ $\displaystyle\frac{\sin 2\theta}{2}\,.$ (30)

As a result, in the limit of small mass difference the flavor vacuum takes the
form

$|\psi\rangle=\left\\{\textbf{1}-\frac{\varepsilon}{4}\sin^{2}{\theta}\int
d^{3}k\,\frac{m^{2}}{\omega_{k}^{2}}\left[a^{\dagger}_{k}a^{\dagger}_{-k}-b^{\dagger}_{k}b^{\dagger}_{-k}\right]+\frac{\varepsilon}{4}{\sin{2}\theta}\int
d^{3}k\,\frac{m^{2}}{\omega_{k}^{2}}\,a^{\dagger}_{k}b^{\dagger}_{-k}\right\\}|00\rangle\,.$
(31)

Before proceeding with the computation of von Neumann entropy, we pause to
compare the above form of the flavor vacuum with the thermal vacuum of Thermo
Field Dynamics defined in the Appendix (see in particular Eq. (52)). The
similarity between these two states sinks its roots in the fact that in TFD
there is the need to double the physical Hilbert space by introducing a dual
space (and hence an auxiliary field) whose excitations are holes from the
point of view of the physical field (a detailed mathematical explanation of
these notions can be found in the Appendix). As a consequence, the thermal
vacuum appears as a condensate of excitations of the physical and auxiliary
fields, which thus resembles the structure of the flavor vacuum being a
condensate of particle/antiparticle pairs with different masses.

In this vein, we emphasize that a first attempt to perform a comparison
between the flavor and thermal vacua was carried out in Ref. Dimauro  and
later in bogoliubov , arguing that the two states cannot be exactly matched,
since the would-be entropy operator defined for mixed fields does not possess
the same properties as the one introduced in TFD (see Eq. (54)). However, here
we tackle this problem from a different perspective; indeed, we directly look
at the inherent structure of the two vacua. Even though the result of Ref.
bogoliubov  remains in general valid since the thermal state (52) does not
contain terms of the form $a_{k}^{\dagger}a_{-k}^{\dagger}$,
$b_{k}^{\dagger}b_{-k}^{\dagger}$ which instead appear in the flavor vacuum
(31), for small values of the mixing angle we can approximate Eq. (31) to the
leading order as

$|\psi\rangle=\left\\{\textbf{1}+\frac{\varepsilon}{2}{\theta}\int
d^{3}k\,\frac{m^{2}}{\omega_{k}^{2}}\,a^{\dagger}_{k}b^{\dagger}_{-k}\right\\}|00\rangle.$
(32)

Then, from comparison with the TFD vacuum in Eq. (56),

$|0(\vartheta)\rangle\simeq\textbf{1}+\sum_{k}a^{\dagger}_{k}\tilde{a}^{\dagger}_{k}\vartheta_{k}|0\rangle\,,$

one can recognize in the state (32) a thermal-like vacuum with an inverse
temperature $\beta=(k_{B}T)^{-1}$ given by (up to a scaling factor due to the
conversion of the $k$-integral into a discrete sum, as viewed in Eq. (21) and
following)

$\beta\simeq\frac{1}{\omega_{k}}\ln\left(\frac{2\hskip
0.56905pt\omega_{k}^{2}}{\varepsilon\hskip 0.56905pt\theta\hskip
0.56905ptm^{2}}\right).$ (33)

Clearly, for $\theta$ and/or $\varepsilon=0$, we have
$\beta\rightarrow\infty$, or equivalently $T\rightarrow 0$. This is somehow
expected since, for vanishing mixing, the flavor and mass vacua coincide with
each other, which corresponds in the TFD language to the case where the
doubling of degrees of freedom (and thus the temperature) disappears.

Let us now come back to the quantification of von Neumann entropy of the
flavor vacuum. To this aim, we focus on the $k=0$ mode, for which the
condensation density reaches its maximal value in the case of boson mixing
(see Eq. (18)) bosonmix2 . Therefore, we expect that the effects of the mixing
transformation are maximally non-trivial in this case. However, for this
purpose we need to go beyond the linear order approximation in the mass
difference, as it can be easily shown that the linear term in $\varepsilon$
gives a vanishing contribution. Then, by resorting to Eq. (27) and retaining
only the factors that do not exceed $\mathcal{O}(\varepsilon^{2})$, we find a
recurrence series of the form

$\displaystyle|\psi\rangle=\Bigl{\\{}\textbf{1}+\left(\frac{\varepsilon}{2}-\frac{\varepsilon^{2}}{4}\right)\left[\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{2n+1}\,4^{n}}{(2n+1)!}\right]a^{\dagger}b^{\dagger}+\left(\frac{\varepsilon}{2}-\frac{\varepsilon^{2}}{4}\right)\left[\sum_{n=0}^{\infty}\frac{(-1)^{n+1}\theta^{2n+2}\,4^{n}}{(2n+2)!}\right]\left[(a^{\dagger})^{2}-(b^{\dagger})^{2}\right]$
(34)
$\displaystyle+\,\frac{\varepsilon^{2}}{4}\left[\sum_{n=0}^{\infty}\frac{(-1)^{n}\theta^{2n+2}\,4^{n}}{(2n+2)!}\right]\left[(a^{\dagger})^{2}(b^{\dagger})^{2}-\textbf{1}\right]+\frac{3\,\varepsilon^{2}}{4}\left\\{\sum_{n=0}^{\infty}\left[\frac{(-1)^{n+1}\theta^{2n+3}4^{n}}{(2n+3)!}\sum_{m=0}^{n}\left(3\sqrt{2}+\sqrt{6}\right)^{m}\right]\right\\}\left[(a^{\dagger})^{3}b^{\dagger}-a^{\dagger}(b^{\dagger})^{3}\right]$
$\displaystyle+\,\frac{3\,\varepsilon^{2}}{4}\left\\{\sum_{n=0}^{\infty}\left[\frac{(-1)^{n}\theta^{2n+4}4^{n}}{(2n+4)!}\sum_{m=0}^{n}\left(3\sqrt{2}+\sqrt{6}\right)^{m}\right]\right\\}\left[(a^{\dagger})^{4}-6\sqrt{2}(a^{\dagger})^{2}(b^{\dagger})^{2}+(b^{\dagger})^{4}\right]\Bigr{\\}}|00\rangle\,.$

The series appearing in Eq. (34) converge to trigonometric functions in
$\theta$. More precisely, we have

$\displaystyle|\psi\rangle$
$\displaystyle=\left[1-\frac{\varepsilon^{2}}{8}\sin^{2}{\theta}\right]|00\rangle-\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin^{2}{\theta}\Bigl{(}|20\rangle-|02\rangle\Bigr{)}+\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin{2}\theta\,|11\rangle$
$\displaystyle+$
$\displaystyle\frac{\varepsilon^{2}}{4}\left[\frac{\sin^{2}\theta}{2}-18\sqrt{2}\,\chi(\theta)\right]|22\rangle+\frac{3\,\varepsilon^{2}}{4}\chi(\theta)\Bigl{(}|40\rangle+|04\rangle\Bigr{)}+\frac{3\,\varepsilon^{2}}{4}\xi(\theta)\Bigl{(}|31\rangle-|13\rangle\Bigr{)}\,,$

where

$\displaystyle\chi(\theta)$ $\displaystyle=$
$\displaystyle\frac{\left(6+2\sqrt{3}\right)\left(1-\cos
2\theta\right)-\sqrt{2}\left[1-\cos\left(2^{\frac{5}{4}}\sqrt{3+\sqrt{3}}\,\theta\right)\right]}{32\left(3+\sqrt{3}\right)\left(3\sqrt{2}+\sqrt{6}-1\right)}\,,$
(36) $\displaystyle\xi(\theta)$ $\displaystyle=$
$\displaystyle\frac{2^{\frac{3}{4}}\sin\left(2^{\frac{5}{4}}\sqrt{3+\sqrt{3}}\,\theta\right)-2\sqrt{3+\sqrt{3}}\sin
2\theta}{16\sqrt{3+\sqrt{3}}\left(3\sqrt{2}+\sqrt{6}-1\right)}\,.$ (37)

The state in Eq. (III) is not normalized, since $\langle\psi|\psi\rangle\neq
1$. For this reason, before computing the projector
$\rho=|\psi\rangle\langle\psi|$, we must divide the above state by a suitable
normalization factor, which is explicitly given by

$N=\sqrt{\langle\psi|\psi\rangle}=\sqrt{1-\frac{\varepsilon^{2}}{4}\left[\sin^{2}{\theta}-\frac{\sin^{4}{\theta}}{2}-\frac{\sin^{2}{2}\theta}{4}\right]}\,.$
(38)

Accordingly, the normalized density matrix reads

$\displaystyle\rho$
$\displaystyle=\frac{|\psi\rangle\langle\psi|}{N^{2}}=\left[1-\frac{\varepsilon^{2}}{8}\sin^{4}{\theta}-\frac{\varepsilon^{2}}{16}\sin^{2}{2}\theta\right]|00\rangle\langle
00|+\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin{2}\theta\Bigl{(}|00\rangle\langle
11|+|11\rangle\langle 00|\Bigr{)}$ $\displaystyle-$
$\displaystyle\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin^{2}{\theta}\Bigl{(}|00\rangle\langle
20|-|00\rangle\langle 02|+|20\rangle\langle 00|-|02\rangle\langle
00|\Bigr{)}+\frac{\varepsilon^{2}}{16}\sin^{2}{2}\theta\,|11\rangle\langle
11|$ $\displaystyle+$
$\displaystyle\frac{\varepsilon^{2}}{4}\left[\frac{\sin^{2}\theta}{2}-18\sqrt{2}\,\chi(\theta)\right]\Bigl{(}|00\rangle\langle
22|+|22\rangle\langle
00|\Bigr{)}+\frac{3\,\varepsilon^{2}}{4}\chi(\theta)\Bigl{(}|00\rangle\langle
40|+|00\rangle\langle 04|+|40\rangle\langle 00|+|04\rangle\langle 00|\Bigr{)}$
$\displaystyle+$
$\displaystyle\frac{\varepsilon^{2}}{16}\sin^{4}{\theta}\Bigl{(}|20\rangle\langle
20|-|20\rangle\langle 02|-|02\rangle\langle 20|+|02\rangle\langle
02|\Bigr{)}+\frac{3\,\varepsilon^{2}}{4}\xi(\theta)\Bigl{(}|00\rangle\langle
31|-|00\rangle\langle 13|+|31\rangle\langle 00|-|13\rangle\langle 00|\Bigr{)}$
$\displaystyle-$
$\displaystyle\frac{\varepsilon^{2}}{16}\sin^{2}{\theta}\sin{2}\theta\Bigl{(}|20\rangle\langle
11|-|02\rangle\langle 11|+|11\rangle\langle 20|-|11\rangle\langle
02|\Bigr{)}\,.$

By partial tracing $\rho$ with respect to either one of the two subsystems,
say $\mathcal{S}_{1}$, we obtain the reduced density matrix

$\displaystyle\rho^{(2)}_{r}=\mathrm{Tr}_{\mathcal{S}_{1}}\rho$
$\displaystyle=$
$\displaystyle\left[1-\frac{\varepsilon^{2}}{16}\sin^{2}2\theta-\frac{\varepsilon^{2}}{16}\sin^{4}\theta\right]|0\rangle\langle
0|+\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin^{2}\theta\Bigl{(}|0\rangle\langle
2|+|2\rangle\langle
0|\Bigr{)}+\frac{\varepsilon^{2}}{16}\sin^{2}2\theta\,|1\rangle\langle 1|$
$\displaystyle+$
$\displaystyle\frac{3\,\varepsilon^{2}}{4}\chi(\theta)\Bigl{(}|0\rangle\langle
4|+|4\rangle\langle
0|\Bigr{)}+\frac{\varepsilon^{2}}{16}\sin^{4}\theta\,|2\rangle\langle 2|\,,$

with the same result for $\rho^{(1)}_{r}=\mathrm{Tr}_{\mathcal{S}_{2}}\rho$.
From Eq. (III), we can compute von Neumann entropy up to
$\mathcal{O}(\varepsilon^{2})$. Notice that the matrix
$\rho^{(1)}_{r}=\rho^{(2)}_{r}\equiv\rho_{r}$, given by

$\rho_{r}=\begin{pmatrix}1-\frac{\varepsilon^{2}}{16}\sin^{2}2\theta-\frac{\varepsilon^{2}}{16}\sin^{4}\theta&0&\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin^{2}\theta&\frac{3\,\varepsilon^{2}}{4}\chi(\theta)\\\
0&\frac{\varepsilon^{2}}{16}\sin^{2}2\theta&0&0\\\
\left(\frac{\varepsilon}{4}-\frac{\varepsilon^{2}}{8}\right)\sin^{2}\theta&0&\frac{\varepsilon^{2}}{16}\sin^{4}\theta&0\\\
\frac{3\,\varepsilon^{2}}{4}\chi(\theta)&0&0&0\end{pmatrix},$ (41)

has eigenvalues

$\lambda_{i}=\left\\{\mathcal{O}(\varepsilon^{3}),\mathcal{O}(\varepsilon^{3}),1-\frac{\varepsilon^{2}}{32}\left(1-\cos
4\theta\right),\frac{\varepsilon^{2}}{16}\sin^{2}2\theta\right\\}\,.$ (42)

Therefore, to order $\varepsilon^{2}$, von Neumann entanglement entropy reads

$S_{V}=-\sum_{i}\lambda_{i}\log_{2}{\lambda_{i}}=-\left[1-\frac{\varepsilon^{2}}{32}\left(1-\cos
4\theta\right)\right]\log_{2}\left[1-\frac{\varepsilon^{2}}{32}\left(1-\cos
4\theta\right)\right]-\frac{\varepsilon^{2}}{16}\sin^{2}2\theta\log_{2}\left(\frac{\varepsilon^{2}}{16}\sin^{2}2\theta\right)\,.$
(43)

Figure 1: Behavior of von Neumann entanglement entropy $S_{V}$ as a function
of $\varepsilon\in\left[0,10^{-2}\right]$ and
$\theta\in\left[0,\frac{\pi}{2}\right]$. As explicitly shown by the level
curves, $S_{V}$ monotonically decreases with decreasing $\varepsilon$.

As expected, in the limit of either $\varepsilon\to 0$ or $\theta\to 0$, we
recover $S_{V}=0$, as it should correctly be, because in these cases the
flavor vacuum reduces to the tensor product
$|00\rangle_{1,2}=|0\rangle_{1}\otimes|0\rangle_{2}$. In Fig. 1 we report the
behavior of von Neumann entropy as a function of both $\varepsilon$ and
$\theta$. In order to better convey the dependence of $S_{V}$ on the mixing
angle, it is also appropriate to exhibit how it varies by keeping
$\varepsilon$ fixed. This is done in Fig. 2. In particular, we can notice that
for arbitrary $\varepsilon$ the maximum of $S_{V}$ always occurs at the
perfectly balanced mixing angle $\theta_{max}=\pi/4$. Increasing $\varepsilon$
only results in an upper shift in the magnitude of $S_{V}$, with an
essentially invariant shape. The value of $\theta_{max}$ is exactly the one
that could have been initially predicted, since $\pi/4$ corresponds to maximal
mixing.

##  IV Conclusions

In this paper, we have explored some non-trivial features of the flavor vacuum
in the case of mixing of two neutral scalar fields. A first preliminary
analysis has shown that we cannot quantify entanglement by resorting to a
direct comparison between $|00\rangle_{A,B}$ and the free thermal vacuum
introduced in Thermo Field Dynamics, since the nature of the two states is not
exactly the same. Note that this is consistent with the result found in Ref.
bogoliubov  in the context of neutrino mixing. A similar outcome has also been
exhibited in Refs. prd , where it has been shown that the flavor vacuum for an
accelerated observer is not strictly a thermal state, in contrast with the
case of the standard Unruh effect. However, in spite of these differences, we
have found that, in a proper limit, the flavor vacuum can be identified with
the vacuum of TFD, with a temperature dependent on the mixing angle and the
mass difference of the two mixed fields.

To further investigate the condensate structure of the flavor vacuum, we have
restricted our attention to the mode $k=0$, for which the condensation density
in $|00\rangle_{A,B}$ is maximal. In this setting, we have quantified von
Neumann entropy for small values of the mass difference. The shape exhibited
in Fig. 1 is the one derived from our analysis, and the picture of Fig. 2
shows that there is an angle in correspondence of which von Neumann entropy is
maximal for a given value of the small mass difference. It is worth observing
that such angle is precisely the one responsible for maximal mixing, namely
$\theta_{max}=\pi/4$. Clearly, it would be interesting to go beyond the
single-mode analysis and derive the full expression for the von Neumann
entropy. However, we expect that the result will exhibit some ultraviolet
divergence of the same kind as in Ref. Capozz , where the contribution of the
flavor vacuum energy to the cosmological constant was shown to diverge only
logarithmically, in contrast with the standard asymptotic behavior of free-
field vacuum energy.

It is worth remarking that entanglement entropy and particle mixing have
already been analyzed together in several papers in recent years entmix ;
entmix2 , but only in the context of flavor transitions and for one-particle
flavor states. In this respect, it is interesting to compare the field-
theoretical approach to flavor entanglement developed in the present work with
the exact methods for the quantification of entanglement introduced in the
framework of quantum information and quantum optics of non-relativistic
continuous variable systems (see for instance Refs. gaussian ; gaussian2 ;
gaussian3 ; review  and references therein). Such methods revolve around the
transformation of the quadrature operators (namely, position and momentum)
under a given operation, which in our case is the mixing transformation. Since
at the level of these operators mixing acts merely as a rotation, we would not
achieve the desired result, thus preventing us from reaching an accurate
evaluation of the entanglement entropy of the flavor vacuum via the procedures
adopted for quantum optical systems gaussian ; gaussian2 ; gaussian3 .

Figure 2: Behavior of von Neumann entanglement entropy $S_{V}$ as a function
of $\theta\in\left[0,\frac{\pi}{2}\right]$ for different values of
$\varepsilon$.

Finally, we emphasize that the present analysis may have several implications
in a wide range of diversified contexts. For instance, once fully extended to
non-inertial frames accCa ; accAl ; acc1  and, more generally, to gravity
scenarios gravity ; gravity2 ; gravity3 ; gravity4 ; gravity5 ; gravity6 ;
gravity7 ; gravity8 ; gravity9 ; gravity10 ; gravity11 , one could investigate
how the information content of the flavor vacuum is degraded for increasing
values of the acceleration/gravity. Note that a similar analysis has been
carried out in Ref. fuentes1  for the case of the entanglement between two
free modes of a scalar field as seen by an inertial observer detecting one of
the modes and a uniformly accelerated observer detecting the other one. These
and other aspects will be thoroughly investigated in future works.

##  Appendix A Thermo Field Dynamics

This Appendix is devoted to review the basics of Thermo-Field-Dynamics (TFD)
tfd , which represents one of the approaches to QFT at finite temperature and
density. The reason for this study lies in the fact the TFD and the formalism
of mixing analyzed in Sec. II exhibit some non-trivial conceptual
similarities, the most prominent one being the doubling of the degrees of
freedom of the original system.

In order to introduce the TFD from a more quantitative point of view, let us
consider the Hamiltonian of a free physical system written in the usual form

$H=\sum_{k}\epsilon_{k}a^{\dagger}_{k}a_{k}\,.$ (44)

We now define a completely identical fictitious system, which we name “tilde”
system, sharing the same properties of the physical one. The former can then
be written as

$\tilde{H}=\sum_{k}\epsilon_{k}\tilde{a}^{\dagger}_{k}\tilde{a}_{k}\,.$ (45)

The above relation allows us to establish a new set of “thermal” operators via
the Bogoliubov transformations

$\displaystyle a_{k}(\vartheta)$ $\displaystyle=$ $\displaystyle
a_{k}\cosh\vartheta_{k}-\tilde{a}^{\dagger}_{k}\sinh\vartheta_{k}\,,$ (46)
$\displaystyle\tilde{a}_{k}(\vartheta)$ $\displaystyle=$
$\displaystyle\tilde{a}_{k}\cosh\vartheta_{k}-a^{\dagger}_{k}\sinh\vartheta_{k}\,,$
(47)

with $\theta_{k}$ encoding the information about the original and thermal
ladder operators. Notice that the transformations (46) and (47) leave the
total Hamiltonian of the system $H_{tot}=H-\tilde{H}$ invariant, provided that
the Hamiltonians (44) and (45) have the same spectrum.

The Bogoliubov transformation (46) and (47) can also be recast as

$\displaystyle a_{k}(\vartheta)$ $\displaystyle=$ $\displaystyle
e^{-iG}\,a_{k}\,e^{iG}\,,$ (48) $\displaystyle\tilde{a}_{k}(\vartheta)$
$\displaystyle=$ $\displaystyle e^{-iG}\,\tilde{a}_{k}\,e^{iG}\,,$ (49)

with

$G=i\sum_{k}\vartheta_{k}\left(a_{k}^{\dagger}\tilde{a}^{\dagger}_{k}-\tilde{a}_{k}a_{k}\right)\,,$
(50)

being the generator of the transformation, which is also a conserved quantity
(namely $\left[G,H_{tot}\right]=0$) since the total Hamiltonian is conserved
under its action. By means of $G$, we can now build a new vacuum state for the
operators $a_{k}(\vartheta)$ and $\tilde{a}_{k}(\vartheta)$ defined as tfd

$|0(\vartheta)\rangle=e^{-iG}|0\rangle\,,$ (51)

where $|0\rangle$ is the vacuum associated with $a_{k}$ and $\tilde{a}_{k}$.
If we explicitly act with the above operator on $|0\rangle$, we are left with

$|0(\vartheta)\rangle=\prod_{k}\frac{e^{a^{\dagger}_{k}\tilde{a}^{\dagger}_{k}\tanh\vartheta_{k}}}{\cosh\vartheta_{k}}\,|0\rangle\,.$
(52)

The above expression conveys the idea that the “theta” vacuum is actually a
condensate of $a$\- and $\tilde{a}$-particles, which somehow resembles the
situation we have already encountered with mixing. However, in order to
construct an effective comparison with the latter framework, we need to
reformulate Eq. (52) as

$|0(\vartheta)\rangle=e^{-\frac{\mathcal{K}}{2}}e^{\sum_{k}a_{k}^{\dagger}\tilde{a}^{\dagger}_{k}}|0\rangle=e^{-\frac{\tilde{\mathcal{K}}}{2}}e^{\sum_{k}a_{k}^{\dagger}\tilde{a}^{\dagger}_{k}}|0\rangle\,,$
(53)

where

$\mathcal{K}=-\sum_{k}\Bigl{(}a^{\dagger}_{k}a_{k}\ln\sinh^{2}\vartheta_{k}-a_{k}a^{\dagger}_{k}\ln\cosh^{2}\vartheta_{k}\Bigr{)}\,,$
(54)

is called the entropy operator tfd , because its vacuum expectation value
multiplied by $k_{B}$ yields precisely the entropy of the physical system. As
a matter of fact, in order to make contact with a feasible physical picture
and describe QFT for free fields at finite temperature, it is possible to
prove that the arbitrary factors $\vartheta_{k}$ should satisfy the relation

$\beta\,\omega_{k}=-\ln\tanh^{2}\vartheta_{k}\,,$ (55)

with $\beta=(k_{B}T)^{-1}$ being the inverse of the emerging temperature $T$
of the thermal vacuum $|0(\vartheta)\rangle$ and
$\omega_{k}=\epsilon_{k}-\mu$, where $\mu$ is the chemical potential.
Remarkably, in the limit $\theta_{k}\rightarrow 0$ (i.e. for $T$ small
enough), Eq. (52) reads

$|0(\vartheta)\rangle\simeq\textbf{1}+\sum_{k}a^{\dagger}_{k}\tilde{a}^{\dagger}_{k}\vartheta_{k}|0\rangle\,.$
(56)

As a final remark, we notice that the thermal vacuum (52), obtained by
augmenting the physical Fock space by a fictitious “tilde” space, has the same
condensate structure of the vacuum perceived by stationary (uniformly
accelerating) observers outside black holes (in Minkowski spacetime). Of
course, in that case the dual space can be interpreted in terms of the
particle states on the hidden side of the horizon Israel ; unruh  and the
entanglement arises among modes across the horizon.

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# Interacting Tsallis and Rényi Holographic Dark Energy with Hybrid Expansion
Law

Snehasish Bhattacharjee 111Email<EMAIL_ADDRESS>
ORCID: https://orcid.org/0000-0002-7350-7043
Department of Astronomy, Osmania University, Hyderabad-500007, India

###### Abstract

The manuscript presents the dynamics of Tsallis holographic dark energy (THDE)
and Rényi holographic dark energy (RHDE) prescribed by a non-linear
interaction in the FRW spacetime and for a scale factor evolving with a
composite power law-exponential (hybrid) form. To construct the energy
densities of these holographic dark energy models, I assume the Hubble cutoff
to be the IR limit. I find that the deceleration parameter undergoes a
signature flipping at a redshift $z$ consistent with observations. The EoS
parameter $\omega_{de}$ for both the HDE models exhibit quite contrasting
dynamical behavior despite assuming values close to $-1$ at $z=0$ and
therefore consistent with current observations. Next, I find the squared sound
speed $c_{s}^{2}$ to be positive for the THDE model ensuring stability against
perturbations, whereas for the RHDE model, $c_{s}^{2}<0$ implying instability
against perturbations. Furthermore, I analyzed the evolutionary behavior of
the EoS parameter of the HDE models by constructing the
$\omega_{de}-\omega_{de}^{{}^{\prime}}$ plane and find that the plane lies in
the freezing region for the THDE model and in the thawing region for the RHDE
model.

holographic dark energy; hybrid expansion law

##  I Introduction

The fact that the universe at the present epoch is experiencing a period of
accelerated expansion with an EoS parameter $\omega\simeq-1$ is well
established and agreed upon note2 . However, there is no general consensus
regarding the source which is fueling the acceleration. A plethora of well
motivated theories have been reported which aim at sufficing this
acceleration. Amongst them, Dark Energy seems to be the most plausible
explanation. Many models of dark energy have been proposed in literature such
as quintessence ratra , K-essence picon . Some alternate theories have also
been reported such as the backreaction mechanism buchert , voids pandey ,
configurational entropy of the universe pandey2 , extra dimension milton ,
entropic force easson  and entropy maximization pavon  and modified gravity
note17 .
Holographic dark energy (HDE) is an intriguing candidate which can elegantly
describe the current acceleration constructed from the holographic hypothesis
note27  and is also compatible with multiple cosmological observations note30
. In these models, the horizon entropy is the most important aspect which upon
altering, alters the HDE model drastically note .
Gravitational systems can be studied through generalized statistical mechanics
since gravity is a long range force note61 ; note64 ; note65 ; note66 ; note78
. Since, black hole entropy can be obtained through the application of Tsallis
statistics note61 , HDE models such as Tsallis HDE and Rényi HDE have recently
been reported note1 ; note64 ; note65 . Readers are encouraged to see newHDE
for a comprehensive review of HDE models (also see reviewHDE  for some recent
work in HDE). Amognst the two HDE models, without the interaction between the
dark sectors, the RHDE constructed from the Rényi entropy is reported to be
stable note65 . However, when an interaction between the dark sectors is
presumed, both of these HDE models attains stability.
The fact that our universe may be filled with an interacting fluid exchanging
energy between dark matter and dark energy cannot be ruled out completely
note80to87 ; note88to90 . If such a fluid exists, it can lead to a possible
solution to the coincidence problem note88to90 ; note91to96 . In this work I
shall study the dynamics of interacting THDE and RHDE models in a FRW universe
where the scale factor evolves with a composite power law-exponential (hybrid)
form. To construct the energy density of these HDE models, I use the Hubble
cutoff as the IR limit.
The paper is organized as follows: In Section II I describe the hybrid
expansion law and constrain the free parameters of the ansatz from the
redshift of transition. In Section III I outline the non-linear interaction
between HDE models and cold dark matter. In Section IV I explain in detail the
holographic dark energy models used in the work and study the behavior of dark
energy EoS parameter for both the models. In Section V I present the
expressions of the sound speed for the HDE models to study their stability
against perturbations. In Section VI I study the evolutionary behavior of the
models by constructing the $\omega_{de}-\omega_{de}^{{}^{\prime}}$ plane and
in Section VII I present our results and conclusions.

##  II Hybrid Expansion Law

I shall now consider an ansatz of scale factor often refereed to as the hybrid
expansion law (HEL). The motivation to employ such an ansatz is driven by the
fact that it is both theoretically consistent and observationally viable non .
Additionally, in bhatta  the authors reported viable estimates of baryon to
entropy ratio employing HEL in $f(R,T)$ gravity. The ansatz reduces to the
usual power law and de Sitter solutions for special cases and also depicts the
transition from a decelerated to an accelerated phase elegantly.
HEL has been tested against multiple cosmological observations such as BBN,
CMB and BAO non35 . Remarkably, the cosmological parameters obtained from HEL
lie well within $1-\sigma$ confidence level to that obtained from the standard
$\Lambda$CDM model non35 .
Bearing that in mind, I consider the HEL to be of the form

$a(t)=e^{rt}t^{s}$ (1)

where $r$ and $s$ are constants. The Hubble parameter and deceleration
parameter reads

$H(t)=r+\frac{s}{t},\hskip 18.06749ptq(t)=-1+\frac{s}{(rt+s)^{2}}$ (2)

$r$ and $s$ needs to be carefully constrained so as to support the transition
from a decelerated to an accelerated phase at a redshift of about
$z_{r}\approx 0.55$.
From Eq. 2, it is obvious that the transition occurs at time
$t=-\frac{s}{r}\pm\frac{\sqrt{s}}{r}$ with $0<s<1$. Since for negative
$\frac{\sqrt{s}}{r}$, time becomes negative, I therefore infer that the
transition must have taken place when $t=\frac{\sqrt{s}-s}{r}$.
using the relation $a(t)=\frac{1}{1+z}$, I arrive the following time-redshift
relation

$t=\frac{s\mathcal{W}\left[\frac{r(\frac{1}{1+z})^{\frac{1}{s}}}{s}\right]}{r}$
(3)

where $\mathcal{W}$ denotes the Lambert function. Employing Eq. 3, I plot the
deceleration parameter as a function of redshift for different values of $r$
and $s$ in Fig. 1. It is observed that when $r=s=0.5$, the transition occurs
at a much smaller redshift which is unsuited with observations. Alternatively,
when $r=s=0.7$, the signature flipping occurs at $z>1$ which again is ill-
matched with observations. Fortuantely, I find that when $r=0.7$ and $s=0.5$
the transition occurs at $z_{r}\approx 0.50575$ in consistent with
observations non36  and therefore throughout the analysis, I use this
particular combination of $r$ and $s$.

Figure 1: Deceleration parameter as a function of redshift $z$.

##  III Non-Linear Interaction between dark matter and dark energy

If a mutual exchange of matter-energy take place between dark matter and dark
energy, ther energy densities would assume a constant. Such scenarios could
lead to a possible solution to the coincidence problem frame64 . For this
work, I shall consider a non-gravitational, non-linear interaction between the
dark sectors first studied in nonlinear . The authors in nonlinear  argued
that such type of interactions could lead to a future evolution which differs
drastically from the de Sitter universe.
The general expression of the interaction term takes the form nonlinear

$Q=3bH\rho_{eff}^{(i+j+k)}\rho_{dm}^{k}\rho_{de}^{-(j+k)}$ (4)

where $i,j,k$ and $b$ are constants, $\rho_{eff}=\rho_{dm}+\rho_{de}$ is the
total energy density, $\rho_{dm}$ the energy density of cold dark matter and
$\rho_{de}$ the same for dark energy.
Now the conservation equations for dark matter and dark energy assumes the
form

$\dot{\rho}_{dm}+3H\rho_{dm}=-3bH\rho_{eff}^{(i+j+k)}\rho_{dm}^{k}\rho_{de}^{-(j+k)}$
(5)
$\dot{\rho}_{de}+3H(\rho_{de}+p_{de})=3bH\rho_{eff}^{(i+j+k)}\rho_{dm}^{k}\rho_{de}^{-(j+k)}$
(6)

where $p_{de}$ denote pressure of dark energy. Note that the strength of
interaction depends on $i,j,$ and $k$. For example, for $(i,j,k)=(1,-1,1)$, I
obtain $Q=3bH\rho_{dm}$ while for $(i,j,k)=(1,-1,1)$ I get $Q=3bH\rho_{de}$.
For this work, I shall set $(i,j,k)=(1,-2,0)$ and therefore the interaction
$Q$ assumes the form

$Q=3bH\left(\frac{\rho_{de}^{2}}{\rho_{dm}+\rho_{de}}\right).$ (7)

Thence, Eqs. 5 and 6 become

$\dot{\rho}_{dm}+3H\rho_{dm}=-3bH\left(\frac{\rho_{de}^{2}}{\rho_{dm}+\rho_{de}}\right)$
(8)
$\dot{\rho}_{de}+3H(\rho_{de}+p_{de})=3bH\left(\frac{\rho_{de}^{2}}{\rho_{dm}+\rho_{de}}\right).$
(9)

The constant $b$ determines the direction in which energy flow take place. For
positive $b$, energy flows from dark energy to dark matter and otherwise.
Nonetheless, I shall assume positive $b$ in this work as it is favorable
observationally and thermodynamically.

##  IV Holographic Dark Energy

The energy density of holographic dark energy can be written as frame64

$\rho_{de}=\frac{3B^{2}m_{p}^{2}}{L^{2}},$ (10)

where the constraint on the energy density of the holographic dark energy
reads $Lm_{p}^{2}\geq L^{3}\rho_{de}$. Here $m_{p}$ denote Planck mass, $B$ a
dimensionless constant and $L$ the IR cutoff. Since the energy density depends
on the choice of the IR cutoff, many such cutoffs have been proposed such as
the Hubble cutoff, future event horizon cutoff, Ricci cutoff and the Granda-
Oliveros cutoff. However, in this work I shall set the Hubble cutoff as the IR
limit.
For a flat FRW background coupled with holographic dark energy and
pressureless cold dark matter, the first Friedmann equation can be expressed
as

$H^{2}=\frac{\rho_{de}+\rho_{dm}}{3m_{p}^{2}}.$ (11)

I shall now study the dynamics two holographic dark energy models namely the
Tsallis holographic dark energy and the Renýi holographic dark energy in the
following sections.

###  IV.1 Tsallis Holographic Dark Energy

The horizon entropy of a black hole has been modified in note78  and assumes
the form $S_{\delta}=\gamma A_{\delta}$, where $\delta$ denote the non-
additivity parameter, $\gamma$ is a constant and $A=4\pi L^{2}$ represent the
surface area of the event horizon. Note that Bekenstein entropy can be
obtained for $\delta=1$. In note27 , the authors reported a mutual
relationship between the IR cutoff $L$, UV cutoff $\Gamma$ and the entropy $S$
as

$S^{3/4}\geq(L\Gamma)^{3}.$ (12)

Upon substituting the value of $S$, yields

$\gamma 4\pi^{\delta}L^{(2\delta-4)}\geq\Gamma^{4}.$ (13)

Now since the holographic dark energy $\rho_{de}\sim\Gamma^{4}$, the Tsallis
holographic dark energy takes the final form note1

$\rho_{de}=CL^{(2\delta-4)},$ (14)

where $C$ is a constant. Now to obtain the expression of the energy density of
the THDE model prescribed by a non linear interaction and for a scale factor
evolving with a composite power law-exponential (hybrid) form, I Substitute
Eq. 14 and Eq. 2 in Eq. 11 to obtain

$\rho_{de}=C\left(r+\frac{s}{t}\right)^{4-2\delta}.$ (15)

The expression of pressure associated with this energy density can be obtained
by substituting Eq. 15 and Eq. 2 in Eq. 9 and reads

$p_{de}=-C(r+\frac{s}{t})^{(4-2\delta)}+\frac{1}{3}(r+\frac{s}{t})\left[C^{2}b(r+\frac{s}{t})^{(7-4\delta)}+\left(\frac{Cs(r+\frac{s}{t})^{(3-2\delta)}(4-2\delta))}{t^{2}}\right)\right].$
(16)

The EoS parameter $\omega_{de}=p_{de}/\rho_{de}$ reads

$\omega_{de}=\frac{Cb\left(r+\frac{s}{t}\right)^{-2\delta}(s+rt)^{4}+t^{2}(-3t^{2}-2s(\delta-2))}{3t^{4}}.$
(17)

Figure 2: EoS parameter as a function of redshift $z$ with $r=0.7$, $s=0.5$,
$C=1$ and $b=0.01$.

In Fig. 2 I show the evolution of THDE dark energy EoS parameter $\omega_{de}$
for some rational values of Tsallis parameter $\delta$. For all $\delta$,
$\omega_{de}$ remains in the Quintessence region and approaches the Phantom
divide line in future (i.e, for $z<0$). It can also be noted that as $\delta$
decreases the profiles shift towards higher values at redshift $z=0$ and
beyond. The solid black line corresponds to the dark energy EoS parameter of
the cosmological constant $\Lambda$ which assumes a value of $-1$ at all
times.

###  IV.2 Rényi Holographic Dark Energy

Rényi entropy can be written as note65

$S=\frac{1}{\delta}\log\left[1+\pi\delta L^{2}\right].$ (18)

Now considering $\rho_{de}dV\propto TdS$, where $T$ and $V$ denote the volume
and temperature of the system, the expression of Rényi holographic dark energy
assumes the form note65

$\rho_{de}=\frac{3D^{2}}{8\pi L^{2}}\left(1+\pi\delta L^{2}\right)^{-1}$ (19)

where $D$ is a constant. Similar to the previous section, I shall substitute
Eq. 19 and Eq. 2 in Eq. 9, to obtain the expression of the energy density for
the RHDE model prescribed by a non linear interaction and for a scale factor
evolving with a composite power law-exponential (hybrid) form and reads

$\rho_{de}=\frac{3D^{2}(s+rt)^{4}}{8\pi t^{2}\left((s+rt)^{2}+\pi
t^{2}\delta\right)}.$ (20)

The expression of pressure and EoS parameter for the RHDE model can now be
obtained in a similar way and reads respectively as

$p_{de}=\frac{D^{2}(s+rt)^{4}\left(3bD^{2}(s+rt)^{4}+8\pi(s+rt)^{2}(2s-3t^{2})+8\pi^{2}t^{2}(4s-3t^{2})\delta\right)}{64\pi^{2}t^{4}\left((s+rt)^{2}+\pi
t^{2}\delta\right)^{2}}$ (21)
$\omega_{de}=\frac{\left(3bD^{2}(s+rt)^{4}+8\pi(s+rt)^{2}(2s-3t^{2})+8\pi^{2}t^{2}(4s-3t^{2})\delta\right)}{24\pi
t^{2}\left((s+rt)^{2}+\pi t^{2}\delta\right)}.$ (22)

Figure 3: EoS parameter as a function of redshift $z$ with $r=0.7$, $s=0.5$,
$D=-1$ and $b=3.65$.

In Fig. 3 I show the evolution of RHDE dark energy EoS parameter $\omega_{de}$
for some values of Rényi parameter $\delta$. From the figure it is clear that
for $\delta=0.5$ and $0.6$, $\omega_{de}$ lies in the Phantom region for the
illustrated redshift range. Interestingly, for $\delta=0.7$, $\omega_{de}$
assume the value of $-1$ at $z=0$ in harmony with observations. The profile
then enters the Quintessence region for some time and then reverts back to
being Phantom in nature. The solid black line corresponds to the dark energy
EoS parameter of the cosmological constant $\Lambda$ which assumes a value of
$-1$ at all times.

##  V Evolution of Sound Speed

The square of sound speed ($c_{s}^{2}$) is a useful parameter to understand
the stability of a dark energy model against perturbations. If $c_{s}^{2}>0$,
the model is stable and otherwise unstable peebles . The expression of
$c_{s}^{2}$ reads

$c_{s}^{2}=\frac{\partial p}{\partial\rho}.$ (23)

###  V.1 For Tsallis Holographic Dark Energy

For the THDE model, the expression of $c_{s}^{2}$ reads

$c_{s}^{2}=\frac{2Cb(r+\frac{s}{t})^{-2\delta}(s+rt)^{4}+t^{2}\left((2r-3t)t-2s(\delta-3)\right)}{3t^{4}}.$
(24)

Figure 4: Squared sound speed as a function of redshift $z$ with $r=0.7$,
$s=0.5$, $C=1$ and $b=0.01$.

In Fig. 4 I show the evolution of squared sound speed ($c_{s}^{2}$) for the
THDE model for some values of Tsallis parameter $\delta$. It is evident that
for all $\delta$, $c_{s}^{2}>0$ and therefore ensures stability against
perturbations. However in future (particularly for $z\lesssim-0.5$), the model
becomes unstable. It can be noted that $c_{s}^{2}$ is not very sensitive to
$\delta$.

###  V.2 For Rényi Holographic Dark Energy

For the RHDE model, the expression of $c_{s}^{2}$ reads

$c_{s}^{2}=\frac{3bD^{2}(s+rt)^{6}+12\pi^{2}t^{2}(s+rt)^{2}\left[4s+(2r-3t)t\right]\delta+8\pi^{3}t^{4}\left[6s+(2r-3t)t\right]\delta^{2}+2\pi(s+rt)^{4}(8s+t)(4r-6t+3bD^{2}t\delta)}{12\pi
t^{2}\left[(s+rt)^{4}+3\pi
t^{2}(s+rt)^{2}\delta+2\pi^{2}t^{4}\delta^{2}\right]}.$ (25)

Figure 5: Squared sound speed as a function of $z$ with $r=0.7$, $s=0.5$,
$D=-1$, and $b=3.65$.

In Fig. 5 I show the evolution of squared sound speed ($c_{s}^{2}$) for the
RHDE model for some values of Rényi parameter $\delta$. Unlike the THDE model,
the RHDE model is unstable against perturbations since $c_{s}^{2}<0$ for all
$\delta$. Moreover, $c_{s}^{2}$ is highly sensitive to $\delta$ unlike the
THDE model. It can also be noted that negativity of $c_{s}^{2}$ increases as
$\delta$ is decreased.

##  VI $\omega_{de}-\omega_{de}^{{}^{\prime}}$ Plane

To explain the dynamical properties of dark energy models, frame70  proposed
the $\omega-\omega^{{}^{\prime}}$ plane where $\omega_{de}^{{}^{\prime}}$
represent the derivative of $\omega_{de}$ with respect to lna. The plane is
sectioned into two parts: The thawing region where
($\omega_{de}<0,\omega_{de}^{{}^{\prime}}>0$) and the freezing region where
($\omega_{de}<0,\omega_{de}^{{}^{\prime}}<0$).

###  VI.1 For Tsallis Holographic Dark Energy

For the THDE model, the expression of $\omega_{de}^{{}^{\prime}}$ reads

$\omega_{de}^{{}^{\prime}}=\frac{d\omega_{de}}{\text{dlna}}=\frac{2r^{2}(\delta-2)\left[2(\frac{tr}{s})^{2}+Cdr^{2}s(r+\frac{s}{t})^{-2\delta}(1+\frac{tr}{s})^{3}\right]}{3s^{2}(\frac{tr}{s})^{4}(1+\frac{tr}{s})}.$
(26)

Figure 6: $\omega_{de}-\omega_{de}^{{}^{\prime}}$ plane with $r=0.7$, $s=0.5$,
$C=1,b=0.01,$ and $\delta=1.95$.

###  VI.2 For Rényi Holographic Dark Energy

For the RHDE model, the expression of $\omega_{de}^{{}^{\prime}}$ reads

$\omega_{de}^{{}^{\prime}}=\frac{d\omega_{de}}{\text{dlna}}=\frac{\left[\Sigma\left(\Pi\left(\Theta(16\pi+3bD^{2}s)\delta-2r^{2}(r^{2}(32\pi+15bD^{2}s))+\pi(40\pi+9bD^{2}s)\frac{tr}{s}-\Theta(8\pi+3bD^{2}s)\delta+\Upsilon(\frac{tr}{s})^{3}\right)\right)\right]}{\left[12\pi
s^{2}(\frac{tr}{s})^{2}(1+\frac{tr}{s})(r^{2}+2r^{2}(\frac{tr}{s})+(r^{2}+\pi\delta)\frac{tr}{s})^{2}\right]}$
(27)

where

$\Sigma=-r^{6}(16\pi+3bD^{2}s)+r^{2}\frac{tr}{s}$ (28)
$\Pi=-r^{4}(64\pi+15bD^{2}s)+\frac{tr}{s}$ (29)
$\Theta=-6r^{4}(16\pi+5bD^{2}s)-2\pi r^{2}$ (30)
$\Upsilon=32\pi^{3}\delta^{2}(\frac{tr}{s})^{2}-3bD^{2}r^{2}s(r^{2}+2\pi\delta).$
(31)

Figure 7: $\omega_{de}-\omega_{de}^{{}^{\prime}}$ plane with $r=0.7$, $s=0.5$,
$D=-1,b=3.65,\delta=0.6$.

##  VII Results & Conclusions

Holographic dark energy (HDE) models are a class of dark energy models
constructed from the holographic hypothesis and is becoming widely popular
owing to its elegant explanation of the current acceleration of the universe
in harmony with multiple cosmological observations. In HDE models, the horizon
entropy plays the most crucial part in constructing the dark energy density.
Different IR cutoffs result in different dark energy densities which are then
confronted with observations.
In this work I investigated the dynamics of two recently proposed holographic
dark energy models namely the Tsallis holographic dark energy (THDE) and the
Rényi holographic dark energy (RHDE) in the FRW spacetime with the scale
factor evolving as a composite power law-exponential (hybrid) form. I also
assumed a non-linear interaction between dark matter and dark energy. To
construct the expressions of the energy densities of the HDE models, I assumed
the Hubble cutoff to be the IR limit. The results can be summarized as
follows:

  * •

The deceleration parameter undergoes a signature flipping for suitable
combinations of $r$ and $s$. For $r=0.7$ and $s=0.5$ the transition occurs at
$z_{r}\approx 0.50575$ in consistent with observations non36 .

  * •

For all values of Tsallis parameter $\delta$, $\omega_{de}$ of the THDE model
remains in the Quintessence region and approaches the Phantom divide line in
future (i.e, for $z<0$). I also find that as $\delta$ decreases the profiles
shift towards higher values at redshift $z=0$ and beyond.
In the case of RHDE model, I find that when the Rényi parameter $\delta=0.5$
and $0.6$, $\omega_{de}$ lies in the Phantom region. Nontheless, when
$\delta=0.7$, $\omega_{de}$ assume the value of $-1$ at $z=0$ in harmony with
observations. The profile then enters the Quintessence region for some time
and then reverts back to being Phantom in nature.

  * •

Next, for the THDE model, the squared sound speed $c_{s}^{2}>0$ for all
$\delta$ and therefore ensures stability against perturbations. Interestingly,
for $z\lesssim-0.5$, the model becomes unstable. I also find that $c_{s}^{2}$
is not very sensitive to $\delta$.
The squared sound speed for the RHDE model is negative implying instability
against perturbations. Moreover, $c_{s}^{2}$ is highly sensitive to $\delta$.
It can also be noted that as $\delta$ decreases the negativity of $c_{s}^{2}$
increases.

  * •

Finally I analyze the evolutionary behavior of the EoS parameter of the HDE
models by constructing the $\omega_{de}-\omega_{de}^{{}^{\prime}}$ plane and
find that $\omega_{de}^{{}^{\prime}}<0$ for $\omega_{de}<0$ for the THDE model
implying the plane to lie in the freezing region while for the RHDE model,
$\omega_{de}^{{}^{\prime}}>0$ for $\omega_{de}<0$ entailing the plane to lie
in the thawing region.

## Acknowledgments

I thank DST, New-Delhi, Government of India for the provisional INSPIRE
fellowship selection [Code: DST/INSPIRE/03/2019/003141]. I also thank the
anonymous referee for his/her constructive comments and useful criticisms
which greatly improved the quality and presentation of the work.

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