# Entanglement entropy in conformal quantum mechanics

Michele Arzano<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”,
Università di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy  Alessandra D’Alise
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università di
Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy  Domenico Frattulillo
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università
di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy

###### Abstract

We consider sets of states in conformal quantum mechanics associated to
generators of time evolution whose orbits cover different regions of the time
domain. States labelled by a continuous global time variable define the two-
point correlation functions of the theory seen as a one-dimensional conformal
field theory. Such states exhibit the structure of a thermofield double built
on bipartite eigenstates of generators of non-global time evolution. In terms
of the correspondence between radial conformal symmetries in Minkowski
spacetime and time evolution in conformal quantum mechanics proposed in Arzano
(2020, 2021) such generators coincide with conformal Killing vectors tangent
to worldlines of Milne and diamond observers at constant radius. The
temperature of the thermofield double states in conformal quantum mechanics
reproduces the temperatures perceived by such diamond and Milne observers. We
calculate the entanglement entropy associated to the thermofield double states
and obtain a UV divergent logarithmic behaviour analogous to known results in
two-dimensional conformal field theory in which the entangling boundary is
point-like.

##  I Introduction

Entanglement entropy plays a central role in our understanding of the
interplay between the quantum realm and the geometry of spacetime. In quantum
field theory, it can characterize the quantum correlations between degrees of
freedom inside and outside a given region of spacetime. Due to the presence of
short scale correlations such geometric entropy is generally UV divergent but
has the notable feature that the leading divergent term is proportional to the
area of the entangling surface. Such property suggests a link between
entanglement entropy and thermodynamic properties of black holes. Indeed,
according to the celebrated entropy-area relation Bekenstein (1973), black
holes are characterized by an entropy proportional to their horizon surface
area. While there is to date no general consensus on the origin of the degrees
of freedom responsible for such entropy Jacobson (1999); Carlip (2007), its
area scaling suggests that black hole entropy is intimately related to quantum
correlations across the horizon Solodukhin (2011).

Entanglement entropy is notoriously hard to compute in quantum field theory
(see Rangamani and Takayanagi (2017) for a review of the various techniques).
One notable exception where analytical results exist is two-dimensional
conformal field theory (CFT2) Holzhey et al. (1994); Calabrese and Cardy
(2004). In this letter we show that $0+1$-dimensional conformal field theory,
i.e. conformal quantum mechanics, provides yet another model in which
entanglement entropy associated to correlations of the two-point function can
be calculated analytically in a rather straightforward way.

Our analysis is motivated by the correspondence between radial conformal
Killing vectors in Minkowski space-time and the generators of conformal
transformations of the real line which describe alternative time evolutions in
conformal quantum mechanics. In Arzano (2020, 2021) it was shown that such
alternative time evolutions coincide with the tangent vectors to worldlines of
observers sitting at the origin within a causal diamond and in Milne space-
time. The definition of positive and negative frequency modes for a
conformally invariant field for such observers are different than the one of
inertial observers and lead to the construction of different Fock spaces with
different vacuum states Olson and Ralph (2011); Su and Ralph (2016); Higuchi
et al. (2017); Wald (2019). The inertial vacuum density matrix is an entangled
state between the modes associated to such observers and the ones defined in
the space-time region from which they cannot exchange signals with. Tracing
over the inaccessible degrees of freedom the vacuum density matrix becomes a
thermal state at a characteristic diamond or Milne temperature. In conformal
quantum mechanics one can identify states which have a similar structure
associated to time domains covered by orbits of different generators of time
evolution and find analogous temperatures Arzano (2020, 2021). In particular
states labelled by a continuous time variable are used to build a two-point
function which corresponds the restriction to wordlines of observers sitting
at the origin of the two-point function of a massless scalar field in
Minkwoski space-time defined on the inertial vacuum state. Such states exhibit
the structure of a thermofield double in terms of excitations of the
Hamiltonian which generates time translations in time domains with boundaries.
Here we show how it is possible to quantify the entanglement of such state in
terms of the Von Neumann entropy of the associated reduced density matrix
obtained by tracing over one set of degrees of freedom of the thermofield
double. The result diverges logarithmically when the UV regulator is sent to
zero as in two-dimensional conformal field theory, a result expected in models
where the entangling boundary is point-like.

In the next section we start by recalling the correspondence between radial
conformal Killing vectors in 3+1-dimensional Minkowski space time and
conformal transformations of the real line with particular focus on generators
whose orbits exhibit boundaries. In Section 3 we examine the role of such
generators in conformal quantum mechanics, we introduce the sets of states
naturally associated to them and spell out their significance in the
definition of the two-point function of the model seen as a one-dimensional
conformal field theory. In Section 4 we show how states labelled by a
continuous time variable, whose inner product gives the two-point function of
the theory, exhibit the structure of a thermofield double state in terms of
the excitations of Hamiltonians whose orbits exhibit boundaries on the time
domain. We point out how, after tracing over one set of degrees of freedom of
the thermofield double, one obtains thermal density matrices at the Milne and
diamond temperature. In Section 5 we proceed to evaluate the entanglement
entropy of the reduced vacuum density matrix. This requires an appropriate
regularization of the vacuum state in order to control its non-normalizability
which is directly related to the divergence of the two-point function at
coincident points. We conclude in Section 6 with a summary and comments on the
results obtained.

##  II Radial conformal Killing vectors in Minkowski spacetime and conformal
transformations of the real line

Let us consider the Minkowski metric in spherical coordinates

$ds^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,.$ (1)

A vector field $\xi$ is a conformal Killing vector if

$\mathcal{L}_{\xi}g_{\mu\nu}\propto g_{\mu\nu}$ (2)

where $g_{\mu\nu}$ is a generic metric and $\mathcal{L}_{\xi}$ denotes the Lie
derivative. For Minkowski spacetime all radial conformal Killing vectors were
classified in Herrero and Morales (1999) and they have the general form

$\xi=\left(a(t^{2}+r^{2})+bt+c\right)\,\partial_{t}+r(2at+b)\,\partial_{r}$
(3)

with $a,b,c$ real constants. Central to the present work is the fact that such
vectors can be written as

$\xi=aK_{0}+bD_{0}+cP_{0}\,,$ (4)

with $P_{0}$, $D_{0}$ and $K_{0}$ generating, respectively, time translations,
dilations and special conformal transformations

$P_{0}=\partial_{t}\,,\qquad D_{0}=r\,\partial_{r}+t\,\partial_{t}\,,\qquad
K_{0}=2tr\,\partial_{r}+(t^{2}+r^{2})\,\partial_{t}\,,$ (5)

whose commutators close the $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra

$[P_{0},D_{0}]=P_{0}\,,\qquad[K_{0},D_{0}]=-K_{0}\,,\qquad[P_{0},K_{0}]=2D_{0}\,.$
(6)

The causal character of different conformal Killing vectors changes according
to the values of the real constants $a,b$ and $c$ Herrero and Morales (1999).
For example, when $a=0$ and $b=0$ the conformal Killing vectors are everywhere
time-like. This is the case of the vector $P_{0}$ which generates evolution in
inertial time and whose integral lines are worldlines of static inertial
observers i.e. straight infinite lines with $r=\mathrm{const}$. In general,
however, such conformal Killing vectors are not everywhere time-like. For
example generators for which $a=0$ and $b\neq 0$ are null on the light-cone
emanating from the point $(t=-c/b,r=0)$, time-like inside such light-cone and
space-like outside. The generator of dilations $D_{0}$ is one of such vector
whose integral lines are straight lines emanating from the origin in the
$t$-$r$ plane and within the future oriented light-cone describe worldlines of
comoving observers in a expanding Milne universe Ling (2020) (contracting in
the past cone). The Milne universe is a flat FRLW space-time with scale factor
linear in time

$ds^{2}=-d\bar{t}^{\,2}+\bar{t}^{\,2}\left(d\chi^{2}+\sinh\chi^{2}d\Omega^{2}\right)\,.$
(7)

Such metric is simply the Minkowski space-time metric restricted to the future
cone and rewritten in hyperbolic slicing coordinates $t=\bar{t}\,\cosh\chi$
and $r=\bar{t}\,\sinh\chi$. Worldlines of comoving Milne observers are
straight lines with $\chi=\mathrm{const.}$. These observers move with radial
Minkowskian velocity $r/t=\tanh\chi$ which approaches the value of 1 as
$\chi\rightarrow\infty$ i.e. as the worldline of the comoving observer
approaches the light-cone. Notice how the time evolution of Milne comoving
observers is not eternal, albeit being still infinite, since integral lines of
$D_{0}$ have a beginning (and an end for the past cone) at the origin
$(t=0,r=0)$. Finally notice that the conformal Killing $D_{0}$ vector becomes
null on the light-cone which thus represents a conformal Killing horizon Dyer
and Honig (1979).

Another conformal Killing vector which will be relevant for our discussion is
the following combination of translations and special conformal
transformations

$S_{0}=\frac{1}{2}\left(\alpha P_{0}-\frac{K_{0}}{\alpha}\right)\ .$ (8)

Such conformal Killing vector exhibits a more articulated causal structure
determined by the intersection of two light-cones emanating from the points
$(t=\pm\alpha,r=0)$: it is null on these light-cones, time-like inside or
outside both light-cones and space-like elsewhere. Thus conformal
transformations generated by the vector map the intersection of the two light-
cones into itself Jacobson (2016), such region is a causal diamond of radius
$\alpha$. The integral lines of $S_{0}$ within the diamond are worldlines of
accelerated observers Herrero and Morales (1999) with a finite life-time
(since they are restricted to the causal diamond) and take the form

$t^{2}-\left(r-\alpha\omega\right)^{2}=\alpha^{2}\left(1-\omega^{2}\right)$
(9)

parametrized by a dimensionless parameter $\omega\neq 0$. Notice how the
generators of boosts, which describe time evolution in Rindler space-time, are
not included in the above classification since they are not radial (written in
spherical coordinates they are not independent of angular variables).

The connection with conformal quantum mechanics stems from the observation
that along $r=\mathrm{const}$ worldlines (but also on light-cones
$u=t-r=\mathrm{const}$, $v=t+r=\mathrm{const}$) the conformal Killing vector
(3) takes the form

$\xi=\left(a\,t^{2}+b\,t+c\,\right)\partial_{t}\,.$ (10)

This is the general expression for a generator of conformal transformations of
the real (time) line. In particular $P_{0}=\partial_{t}$ generates
translations in “inertial time” $t$ covering the entire time line. This time
variable can be identified with the proper time of static inertial observers
in Minkowski space-time (observers with four-velocity parallel to the
(conformal) Killing vector $P_{0}$). The dilation Killing vector
$D_{0}=t\partial_{t}$ generates translation in “Milne time” $\tau$ defined by

$D_{0}=\alpha\partial_{\tau}\,.$ (11)

The only worldline at fixed radius in the Milne universe is the one at $r=0$
for which $\tau$ corresponds to the conformal time. One can easily show that

$t=\pm\alpha\,\exp(\frac{\tau}{\alpha})$ (12)

and thus the Milne time coordinate covers only half (the regions $t>0$ or
$t<0$) of the whole time domain. Finally the conformal Killing vector
$S_{0}=\frac{1}{2\alpha}\left(\alpha^{2}-t^{2}\right)\partial_{t}$ generates
translation in “diamond time” $\sigma$

$S_{0}=\alpha\partial_{\sigma}\,.$ (13)

In Minkowski space-time this vector corresponds to the restriction to the
worldline of a static diamond observer at $r=0$ of the conformal Killing
vector (8). On the time line this time variable is related to inertial time by
the transformation

$t=\alpha\,\tanh{\left(\frac{\sigma}{2\alpha}\right)}$ (14)

and thus it covers only the segment $|t|<\alpha$ of the time domain.

##  III Time evolution in conformal quantum mechanics and the CFT1 two-point
function

The quantum mechanical counterpart of the generator (10) is

$G=i\xi=aK+bD+cH\,,$ (15)

where $K=iK_{0}$, $D=iD_{0}$ and $H=iP_{0}$, corresponds to the most general
Hamiltonian of conformal quantum mechanics, a quantum mechanical model first
studied in de Alfaro et al. (1976) characterized by an inverse square
potential Lagrangian invariant under the one-dimensional group of conformal
transformations of the time axis $\mathrm{SL}(2,\mathbb{R})$. Such model can
be understood as a one-dimensional conformal field theory Chamon et al.
(2011); Jackiw and Pi (2012), as we briefly recall below.

In de Alfaro et al. (1976) two sets of states were constructed: the
eigenstates $|n\rangle$ of the elliptic operator $R$ which has discrete
spectrum and are normalizable, and states labelled by inertial time
$|t\rangle$ on which the operator generating inertial time translations $H$
acts as a derivative

$H|t\rangle=-i\derivative{t}|t\rangle\,.$ (16)

Such states are labelled by the continuous parameter $t$ and are non-
normalizable as we will show below. The action of the remaining
$\mathrm{SL}(2,\mathbb{R})$ generators is given by

$\displaystyle D\ket{t}$
$\displaystyle=-i\left(t\derivative{t}+r_{0}\right)\ket{t}$ (17)
$\displaystyle K\ket{t}$
$\displaystyle=-i\left(t^{2}\derivative{t}+2r_{0}t\right)\ket{t}\,.$ (18)

Introducing ladder operators

$L_{0}=R=\frac{1}{2}\left(\frac{K}{\alpha}+\alpha H\right)\qquad L_{\pm}=S\pm
iD=\frac{1}{2}\left(\frac{K}{\alpha}-\alpha H\right)\pm iD\,,$ (19)

whose commutators are

$\commutator{L_{0}}{L_{\pm}}=\pm
L_{\pm},\qquad\commutator{L_{-}}{L_{+}}=2L_{0}\,,$ (20)

we define sates $|n\rangle$ labelled by positive integers $n=0,1,...$ on which
the action of the operators (19) is given by

$\displaystyle L_{0}\ket{n}$ $\displaystyle=(n+r_{0})\ket{n}$ (21)
$\displaystyle L_{\pm}\ket{n}$ $\displaystyle=\sqrt{((n+r_{0})(n+r_{0}\pm
1))-r_{0}(r_{0}-1))}\ket{n\pm 1}\,,$ (22)

with the orthonormality relation

$\langle n|n^{\prime}\rangle=\delta_{nn^{\prime}}\,.$ (23)

The constant $r_{0}$, the eigenvalue of the ground state $|n=0\rangle$, is a
positive real number Jackiw and Pi (2012) and it is related to the eigenvalue
of the Casimir operator

$\mathcal{C}=R^{2}-S^{2}-D^{2}=\frac{1}{2}\left(HK+KH\right)-D^{2}\,,$ (24)

given by

$\mathcal{C}\ket{n}=r_{0}(r_{0}-1)\ket{n}\,.$ (25)

For $r_{0}$ integer and half integer with $r_{0}\geq 1$ the set of states
$\ket{n}$ provides an irreducible representation of the Lie algebra
$\mathfrak{sl}(2,\mathbb{R})$ belonging to the so-called discrete series Sun
(2021).

In what follows we focus on the case $r_{0}=1$ as in Arzano (2021, 2020) since
this is the case in which the two-point function conformal quantum mechanics,
seen as a one dimensional conformal field theory, is equivalent to the
restriction of the two-point function of a massless scalar field along the
worldline of observers sitting at $r=0$ in Minkowski space-time.

Wavefunctions representing the states $|n\rangle$ as functions of the inertial
time coordinate $t$ can be obtained by considering the action of the $R$
operator on the $\langle t|$ states obtaining the differential equation

$\matrixelement{t}{R}{n}=\frac{i}{2}\left[\left(\alpha+\frac{t^{2}}{\alpha}\derivative{t}\right)+2\frac{t}{\alpha}\right]\innerproduct{t}{n}=(n+1)\innerproduct{t}{n}$
(26)

whose solution is given by

$\innerproduct{t}{n}=-\frac{\alpha^{2}c_{n}e^{2i(n+1)\tan^{-1}\left(\frac{\alpha}{t}\right)}}{\alpha^{2}+t^{2}}\
.$ (27)

One can determine the normalization constants by iteration (see Appendix A)
obtaining

$c_{n}=\sqrt{\frac{\Gamma(2+n)}{n!}}\ .$ (28)

Plugging this expression in (27) we can obtain the following equation relating
$|t\rangle$ and $|n\rangle$ states111Where we used the relation
$2i\tan^{-1}{x}=\log{\frac{1+ix}{1-ix}}$ in (27) to write (29).

$\ket{t}=\left(\frac{\frac{\alpha+it}{\alpha-it}+1}{2}\right)^{2}\
\sum_{n}(-1)^{n}\left(\frac{\alpha+it}{\alpha-it}\right)^{n}\
\sqrt{\frac{\Gamma(2+n)}{n!}}\ket{n}\ .$ (29)

These states can be written in terms of the action of the creation operator
$L_{+}$ on the ground state $\ket{n=0}$

$\ket{t}=\left(\frac{\frac{\alpha+it}{\alpha-it}+1}{2}\right)^{2}\
\exp{-\left(\frac{\alpha+it}{\alpha-it}\right)\,L_{+}}\ket{n=0}\,,$ (30)

and in particular

$\ket{t=0}=\exp\left(-L_{+}\right)\ket{n=0}\,.$ (31)

As discussed in Chamon et al. (2011); Jackiw and Pi (2012) the inner product
between the $|t\rangle$ states can be interpreted as the two-point function of
a one-dimensional CFT. One can explicitly evaluate such two-point function
using (29)

$G(t_{1},t_{2})\equiv\innerproduct{t_{1}}{t_{2}}=-\frac{\alpha^{2}}{4\
(t_{1}-t_{2})^{2}}\,.$ (32)

Notice how such expression matches, modulo a constant factor, that of the two-
point function of a free massless scalar field in Minkowski space-time along
the trajectory of a static inertial observer sitting at $r=0$ Arzano (2020).
Moreover, since the Hamiltonian $H$ generates the time evolution, the
$t$-state can actually be obtained with a time translation from the $t=0$
vacuum

$\ket{t}=e^{iHt}\ket{t=0}=\frac{1}{4}e^{(\alpha+it)H}\ket{n=0}$ (33)

and thus the two-point function (32) can be written as

$G(t_{1},t_{2})=\innerproduct{t_{1}}{t_{2}}=\bra{t=0}e^{-iHt_{1}}\,e^{iHt_{2}}\ket{t=0}\,.$
(34)

It is instructive to look at the two-point function (34) in terms of
eigenstates of the generator $H$. These eigenstates $\ket{E}$, first
introduced in de Alfaro et al. (1976), are defined by

$H\ket{E}=E\ket{E}$ (35)

and satisfy the conditions

$\innerproduct{E}{E^{\prime}}=\delta(E-E^{\prime})\mbox{\quad
and\quad}\int_{0}^{+\infty}\ \differential E\ket{E}\bra{E}=\mathbb{1}\ .$ (36)

Following de Alfaro et al. (1976) we can write $\ket{t}$ as

$\ket{t}=e^{iHt}\ket{t=0}=\int_{0}^{\infty}\differential E\
\frac{\alpha\sqrt{E}}{2}e^{iEt}\ket{E}$ (37)

and obtain the overlap between $\ket{E}$ and $\ket{t}$

$\innerproduct{t}{E}=\frac{\alpha\sqrt{E}}{2}e^{-iEt}\ .$ (38)

Therefore, the states $\ket{E}$ are similar in spirit to the momentum
eigenstates $\ket{\mathbf{p}}$ that one introduces in QFT, in terms of which
the action of a field operator $\phi(\mathbf{x})$ on the vacuum state is

$\phi(\mathbf{x})\ket{0}=\int\frac{\differential^{3}p}{(2\pi)^{3}}\frac{1}{2E_{p}}e^{-i\mathbf{p}\cdot\mathbf{x}}\ket{\mathbf{p}}\
,$ (39)

where

$\innerproduct{\mathbf{p}}{\mathbf{p}^{\prime}}=2E_{\mathbf{p}}(2\pi)^{3}\
\delta^{(3)}(\mathbf{p}-\mathbf{p}^{\prime})\ .$ (40)

The analogy between (37) and (39) clearly suggests that the state $\ket{t=0}$
plays a role analogous to the inertial vacuum for quantum fields in Minkowski
space-time which is the averaging state on which one builds the two-point
function.

##  IV “Vacuum” states and horizon temperature

We have seen that conformal quantum mechanics can be interpreted as a $0+1$
dimensional field theory in which any generator of conformal transformations
of the time axis can be used to define time evolution. Such time evolution can
be mapped to motions along $r=\text{const.}$ orbits of time-like conformal
Killing vectors in Minkowski space-time. For massless fields in Minkowski
space-time one can construct a Fock space using any conformal Killing vector
in the domain where the latter is time-like. As in the more familiar case of
the Unruh effect Unruh (1976), where boost Killing vectors in the Rindler
wedge are used to define positive frequency field modes, Fock spaces
constructed using mode decompositions based on different conformal Killing
vectors will lead to different Fock spaces and thus to different notions of
particles and vacuum states. For the Milne cone and causal diamond in
Minkowski space-time these different quantizations were explored in Higuchi et
al. (2017); Wald (2019); Olson and Ralph (2011); Su and Ralph (2016). These
works suggest that, in analogy with the Unruh effect, for both Milne and
diamond observers the vacuum state for inertial observers appears as a thermal
state.

Let us go back to the correspondence between generators of time evolution in
conformal quantum mechanics and time-like conformal Killing vectors
determining the wordlines of Milne and diamond observers. In light of what we
recalled above, we do expect that in conformal quantum mechanics one should be
able to identify an inertial “vacuum” state which is thermally populated by
excitations of the Hamiltonian describing the conformal quantum mechanics
counterparts of the Milne and diamond time evolutions. This is indeed the case
as first shown in Arzano (2021). To see this, let us recall that the
$\mathfrak{sl}(2,\mathbb{R})$ Lie algebra (19) can be realized in terms of two
sets of creation and annihilation operators
$a^{\dagger}_{L},a^{\dagger}_{R},a_{L},a_{R}$

$L_{0}=\frac{1}{2}\left(a^{\dagger}_{L}a_{L}+a^{\dagger}_{R}a_{R}+1\right)\
,\quad L_{+}=a^{\dagger}_{L}a^{\dagger}_{R}\mbox{\quad
and\quad}L_{-}=a_{L}a_{R}\,.$ (41)

This shows that the ground state of the $R$-operator has a bipartite structure

$\ket{n=0}=\ket{0}_{L}\otimes\ket{0}_{R}\ ,$ (42)

and that the $\ket{t=0}$ state in (31) can be written as

$\ket{t=0}=e^{-a^{\dagger}_{L}a^{\dagger}_{R}}\ket{0}_{L}\ket{0}_{R}=\sum_{n}\
(-1)^{n}\ket{n}_{L}\ket{n}_{R}=-i\sum_{n}\ e^{i\pi
L_{0}}\ket{n}_{L}\ket{n}_{R}\,.$ (43)

From the last equality it is clear that the $\ket{t=0}$ state exhibits a
structure similar to that of a thermofield double state for a harmonic
oscillator (see e.g. Lykken (2021) for a pedagogical review). Such state can
be built by “doubling” the oscillator’s degrees of freedom and is defined by
the superposition

$|TFD\rangle=\frac{1}{Z(\beta)}\sum^{\infty}_{n=0}e^{-\beta
E_{n}/2}|n\rangle_{L}\otimes|n\rangle_{R}\,,$ (44)

where $Z(\beta)=\sum^{\infty}_{n=0}e^{-\beta E_{n}}$ is the partition function
at inverse temperature $\beta$. The state (44) is highly entangled and,
tracing over the degrees of freedom of one copy of the system, we obtain a
thermal density matrix

$Tr_{L}\\{|TFD\rangle\langle TFD|\\}=\frac{e^{-\beta H}}{Z(\beta)}$ (45)

at a temperature $T=1/\beta$. The Hamiltonian $H$ is known as modular
Hamiltonian. For a quantum field in Minkowski space-time the inertial vacuum
state can be seen as a thermofield double state built on two copies of the
Rindler Hilbert space Valdivia-Mera (2020). Tracing over the degrees of
freedom of one copy (i.e. looking at the state from the point of view of
Rindler observers whose worldlines are restricted to a space-like wedge) one
obtains a thermal state at the Unruh temperature. The modular Hamiltonian in
this case is the generator of boosts which can be identified, modulo a factor
with dimensions of inverse length related to the magnitude of the
acceleration, with the generator of time evolution for Rindler observers.

We see that, setting aside normalization issues which will be the focus of the
next section, in conformal quantum mechanics we are dealing with a similar
scenario. Indeed we see that tracing over one set of degrees of freedom the
reduced density matrix associated to the state (43) has the form of a thermal
density matrix for the modular Hamiltonian $-iL_{0}$ at a temperature
$T=1/2\pi$. Since $L_{0}$ can be identified with the elliptic generator $R$ of
the $SO(2)$ compact subgroup of $SL(2,\mathbb{R})$ its “Wick rotated”
counterpart $iL_{0}$ will generate non compact transformations. Indeed one
finds Arzano (2021) that the generators of the Lie algebra (20), besides the
identification (17), have two alternative realizations in terms of the
generators $H,D$ and $K$ given by

$L_{0}=iS\ ,\qquad L_{+}=i(D-R)\ ,\qquad L_{-}=-i(D+R)$ (46)

and

$L_{0}=iD\ ,\qquad L_{+}=-i\alpha H\ ,\qquad L_{-}=-i\frac{K}{\alpha}$ (47)

in which the the modular Hamiltonian $-iL_{0}$ coincides with the generators
$S$ and $D$. From our discussion in Section 2 we see that, when divided by the
constant $\alpha$ with dimensions of length, these two Hamiltonians generate,
respectively, translations on diamond and Milne times.

In the case of the diamond Hamiltonian we notice that the identification (46)
can be obtained by “Wick rotating” the length parameter $\alpha\rightarrow
i\alpha$. Under this map the generator $R$ turns into $iS$ and the
wavefunctions (27) into eigenfunctions of the operator $L_{0}=iS$. Following
steps analogous to the ones leading to (43) we find that the state
$|t=0\rangle$ has the structure of a thermofield double state for the modular
Hamiltonian $S/\alpha$ at a temperature $T=1/(2\pi\alpha)$.

For the Milne Hamiltonian the picture is less straightforward since the
generator $D$ is ill defined at $t=0$. One can solve the eigenvalue equation

$(n+1)\,{}_{D}\langle
t|n\rangle=\matrixelement{t}{iD}{n}=-\left[t\derivative{t}+1\right]\,{}_{D}\langle
t|n\rangle$ (48)

to obtain the eigenstates (see Appendix B)

${}_{D}\langle t|n\rangle=\frac{(-1)^{n}}{2}\
\alpha^{n+2}\sqrt{\frac{\Gamma(2+n)}{n!}}\,t^{-n-2}\ .$ (49)

Since the conformal transformation

$t^{\prime}=\frac{\alpha(t-\alpha)}{\alpha+t}$ (50)

maps the generator $S$ written as a differential operator in terms of the time
variable $t^{\prime}$ into the generator $D$ as a differential operator in the
variable $t$, one can obtain the eigenfunctions (49) starting from the
eigenfunctions of the $R$ operator (27), performing the Wick rotation
$\alpha\rightarrow i\alpha$ and then the conformal transformation (50) on the
time variable. The states $|t\rangle_{D}$ can be now written in terms of
eigenstates of the $L_{0}=iD$ operator as

$\begin{split}\ket{t}_{D}&=\frac{1}{2}\left(\frac{\alpha}{t}\right)^{2}\sum_{n}(-1)^{n}\left(\frac{\alpha}{t}\right)^{n}\sqrt{\frac{\Gamma(2+n)}{n!}}\ket{n}=\frac{1}{2}\left(\frac{\alpha}{t}\right)^{2}e^{-\frac{\alpha}{t}L_{+}}\ket{n=0}\
.\end{split}$ (51)

We see that now the state $t=\alpha$ exhibits the structure of a thermofield
double state for the modular Hamiltonian $D/\alpha$ at the temperature
$T=1/(2\pi\alpha)$. The point $t=\alpha$ is the image of the origin under the
conformal mapping (50) and it corresponds to the origin of the conformal time
$\tau$ variable defined by $t=\alpha\ e^{\frac{\tau}{\alpha}}$.

The state $\ket{t=0}$, as evidenced by eq. (43), is an entangled state with
respect to the bi-partition of the Hilbert space in terms of $L$ and $R$
degrees of freedom. In analogy with the case of the inertial vacuum written as
a thermofield double state over the excitations of the left and right Rindler
modes (the two complementary domains of the evolution of the boost modular
Hamiltonian), we can think of the two sets of degrees of freedom in (43) as
belonging to the domain of diamond and Milne time evolution and their
complements (the restriction to $r=0$ wordlines of the entanglement considered
in Higuchi et al. (2017) and Olson and Ralph (2011)).

We can quantify this entanglement by calculating the Von Neumann entropy of
the reduced density matrix obtained by tracing over one set of degrees of
freedom in the density matrix associated to the inertial vacuum $\ket{t=0}$.
Such entanglement entropy can be seen as the $0+1$-dimensional analogue of the
entanglement entropy a quantum field across space-time regions. Unlike its
higher dimensional counterparts, the simple structure of the state $\ket{t=0}$
makes it rather straightforward to calculate the entanglement entropy
associated to the diamond and Milne time domains.

##  V Entanglement entropy

In order to derive the entanglement entropy associated to the partition of the
$\ket{t=0}$ state we first notice that such state is non-normalizable. This is
to be expected given the correspondence between the inner product
$\innerproduct{t_{1}}{t_{2}}$ and the restriction of the two-point function of
a massless field to the $r=0$ worldline since it reflects the UV divergence of
the latter for coincident points.

We can regularize the state $\ket{t=0}$ via an infinitesimal translation in
imaginary time. We consider a state at time $t=i\epsilon$

$\ket{t=i\epsilon}=\left(\frac{\frac{\alpha-\epsilon}{\alpha+\epsilon}+1}{2}\right)^{2}\
e^{-\frac{\alpha-\epsilon}{\alpha+\epsilon}\ L_{+}}\ket{n=0}$ (52)

where $\epsilon$ can be interpreted as a short-distance cut-off scale. We have

$\ket{t=i\epsilon}=\left(\frac{\alpha}{\alpha+\epsilon}\right)^{2}\sum_{n=0}^{\infty}(-1)^{n}\left(\frac{\alpha-\epsilon}{\alpha+\epsilon}\right)^{n}\ket{n}_{L}\ket{n}_{R}$
(53)

so that

$\bra{t=i\epsilon}\ket{t=i\epsilon}=\frac{\alpha}{4\epsilon}\left(\frac{\alpha}{\alpha+\epsilon}\right)^{2}\equiv\frac{1}{\mathcal{N}^{2}}\
.$ (54)

In order to derive the reduced density matrix we normalize the
$\ket{t=i\epsilon}$ and introduce a state $\ket{\delta}$ with unitary norm

$\ket{\delta}\equiv\mathcal{N}\ket{i\epsilon}\ .$ (55)

Let us now consider the density matrix

$\rho_{RL}=\ket{\delta}\bra{\delta}$ (56)

and compute the reduce density matrix $\rho_{L}$ explicitly by tracing over
the R degrees of freedom

$\rho_{L}=\Tr_{R}{\rho_{LR}}=\mathcal{N}^{2}\left(\frac{\alpha}{\alpha+\epsilon}\right)^{4}\sum_{n=0}^{\infty}\left(\frac{\alpha-\epsilon}{\alpha+\epsilon}\right)^{2n}\ket{n}_{L}\bra{n}_{L}\
.$ (57)

The Von Neumann entropy of the reduced density matrix is then given by

$S=-\Tr{\rho_{L}\log\rho_{L}}=-\frac{(\alpha-\epsilon)^{2}\log\left(\frac{(\alpha-\epsilon)^{2}}{(\alpha+\epsilon)^{2}}\right)}{4\alpha\epsilon}-\log\left(\frac{4\alpha\epsilon}{(\alpha+\epsilon)^{2}}\right)\,,$
(58)

which considering the limit $\epsilon\rightarrow 0$ leads to

$S=\log\left(\frac{\alpha}{\epsilon}\right)+\text{const}+\order{\epsilon^{2}}\,.$
(59)

We see that the result obtained exhibits a logarithmic divergence when the UV
cut-off scale $\epsilon$ is sent to zero. Let us recall that in a
d-dimensional free quantum field theory the entanglement entropy associated to
a spatial region $\mathcal{A}$ is UV divergent with a leading divergent term
proportional to

$\frac{\text{Area}(\partial\mathcal{A})}{\epsilon^{d-2}}\ ,$ (60)

where $\text{Area}(\partial\mathcal{A})$ is the area of the boundary of the
region (entangling surface) and $\epsilon$ is a UV cut-off Rangamani and
Takayanagi (2017). Two-dimensional conformal theories $\mathrm{CFT_{2}}$ are a
special case since they predict a logarithmic divergence and therefore the
entanglement entropy fails to follow an area law. For example the entanglement
entropy between a shorter line-segment with length $\alpha$ and a longer one
with length $L$ containing it in the limit $\frac{\alpha}{L}\ll 1$ reads
Calabrese and Cardy (2004); Saravani et al. (2014); Rangamani and Takayanagi
(2017)

$S=\frac{c}{3}\log{\frac{\alpha}{\epsilon}}+\text{const}\ ,$ (61)

where $c$ is the central charge of the CFT which is equal to $1$ in a quantum
field theory of a massless bosonic field. This peculiar behaviour can be
understood heuristically by arguing that the logarithm arises as a limiting
case of a power law divergence and it is consistent with the entangling
surface comprising of a set of disconnected points.

We observe that the analytical behaviour of our result (59) for the
entanglement entropy in $\mathrm{CFT_{1}}$ is the same as in
$\mathrm{CFT_{2}}$ and in particular it shows the same logarithmic divergence.
This is in line with the point-like nature of the entangling surface.

##  VI Conclusions

Conformal quantum mechanics is a simple one-dimensional model which, as we
have seen, possesses enough structure to mimic the non-trivial vacuum
structure of higher dimensional free quantum field theories. Such non-trivial
structure is due to the existence of excitations in the theory which are
associated to Hamiltonians whose orbits posses a boundary, the one dimensional
counterparts of horizons in higher dimensions. As in quantum field theories in
higher dimensions we have seen that one can associate temperatures to these
horizons. Moreover, we evaluated the entanglement entropy associated to the
bipartite decomposition of the states on which one builds the two-point
function of the theory into modes of Hamiltonians whose orbits do not cover
the entire time domain. The possibility of entanglement between time domains
in Minkowski space-time has been considered before Olson and Ralph (2011);
Higuchi et al. (2017). Here we provide an explicit calculation of entanglement
entropy in conformal quantum mechanics for partitions of states in terms of
modular Hamiltonians which define evolution in time domains with boundaries.
These domains can be embedded in Minkowski space-time as wordlines of diamond
and Milne observers at the origin and, thus, one could interpret our result as
quantifying a worldline entanglement entropy (see Anninos et al. (2012);
Nakayama (2012) for a similar interpretation of conformal quantum mechanics as
worldline quantum mechanics for static patch observers in de Sitter space-
time) for observers who cannot have access to the past beyond a certain point
(the initial “singularity” for Milne observers) or to the future and the past
beyond two points (the future and past tips of the diamond for diamond
observers). The worldline boundaries are point-like and the entanglement
entropy exhibits a logarithmic divergence similar to that found in two-
dimensional CFTs, where the boundaries of the spatial region considered are
also point-like. It is tempting to speculate that such worldline entropy could
play a role as a tool for facilitating the calculation of entanglement entropy
for conformally invariant quantum field theories in higher space-time
dimensions. We leave this task for future investigations.

## Acknowledgements

We acknowledge support from the INFN Iniziativa Specifica QUAGRAP. This
research was carried out in the frame of Programme STAR Plus, financially
supported by the University of Napoli Federico II and Compagnia di San Paolo.
This work also falls within the scopes of the European Union COST Action
CA18108 Quantum gravity phenomenology in the multi-messenger approach.

##  Appendix A Determining the coefficients $c_{n}$

In order to find the coefficients $c_{n}$ we act with the ladder operators. We
start with the action of $L_{+}$

$\displaystyle\sqrt{(n+1)(n+2)}\innerproduct{t}{n+1}$
$\displaystyle=\matrixelement{t}{L_{+}}{n}=\left[\left(i\frac{t}{\alpha}-1\right)+\left(i\frac{t^{2}}{2\alpha}-i\frac{\alpha}{2}-t\right)\derivative{t}\right]\innerproduct{t}{n}$
(A.62)

which gives

$\frac{c_{n}}{c_{n+1}}=\frac{(n+1)}{\sqrt{(n+1)(n+2)}}$ (A.63)

while acting with $L_{-}$

$\displaystyle\sqrt{n(n+1)}\innerproduct{t}{n-1}$
$\displaystyle=\matrixelement{t}{L_{-}}{n}=\left[\left(i\frac{t}{\alpha}+1\right)+\left(i\frac{t^{2}}{2\alpha}-i\frac{\alpha}{2}+t\right)\derivative{t}\right]\innerproduct{t}{n}$
(A.64)

we arrive at

$\frac{c_{n-1}}{c_{n}}=\frac{n}{\sqrt{n(n+1)}}\ ,$ (A.65)

hence we conclude that

$c_{n}=\sqrt{\frac{\Gamma(2+n)}{\Gamma(n+1)}}=\sqrt{\frac{\Gamma(2+n)}{n!}}\
.$ (A.66)

##  Appendix B Eigenstates of the $iD$ generator

The differential equation

$(n+1)\innerproduct{t}{n}=\matrixelement{t}{iD}{n}=-\left[t\derivative{t}+1\right]\innerproduct{t}{n}$
(B.67)

admits solutions

$\innerproduct{t}{n}=c_{n}\ t^{-n-2}\ .$ (B.68)

To determine the coefficients $c_{n}$ we act with $L_{+}$ on these functions

$\displaystyle\sqrt{(n+1)(n+2)}\innerproduct{t}{n+1}$
$\displaystyle=\matrixelement{t}{L_{+}}{n}=\alpha\derivative{t}\innerproduct{t}{n}$
(B.69)

and obtain

$\sqrt{(n+1)(n+2)}\ c_{n+1}=-\alpha\ c_{n}(2+n)\ .$ (B.70)

The action with $L_{-}$ gives

$\displaystyle\sqrt{n(n+1)}\innerproduct{t}{n-1}$
$\displaystyle=\matrixelement{t}{L_{-}}{n}=\frac{1}{\alpha}\left[t^{2}\derivative{t}+2t\right]\innerproduct{t}{n}$
(B.71)

whose solution is

$-\alpha\sqrt{n(n+1)}c_{n-1}=n\ c_{n}\ .$ (B.72)

By combining the two results obtained we arrive at the following expression
for the coefficients $c_{n}$

$c_{n}=\frac{(-1)^{n}}{2}\ \alpha^{n+2}\sqrt{\frac{\Gamma(2+n)}{n!}}\ .$
(B.73)

## References

  * Arzano (2020) M. Arzano, JHEP 05, 072 (2020), eprint 2002.01836.
  * Arzano (2021) M. Arzano, JHEP 07, 003 (2021), eprint 2103.07228.
  * Bekenstein (1973) J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * Jacobson (1999) T. Jacobson, AIP Conf. Proc. 493, 85 (1999), eprint gr-qc/9908031.
  * Carlip (2007) S. Carlip, J. Phys. Conf. Ser. 67, 012022 (2007), eprint gr-qc/0702094.
  * Solodukhin (2011) S. N. Solodukhin, Living Rev. Rel. 14, 8 (2011), eprint 1104.3712.
  * Rangamani and Takayanagi (2017) M. Rangamani and T. Takayanagi, _Holographic Entanglement Entropy_ , vol. 931 (Springer, 2017), eprint 1609.01287.
  * Holzhey et al. (1994) C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424, 443 (1994), eprint hep-th/9403108.
  * Calabrese and Cardy (2004) P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406, P06002 (2004), eprint hep-th/0405152.
  * Olson and Ralph (2011) S. J. Olson and T. C. Ralph, Phys. Rev. Lett. 106, 110404 (2011), eprint 1003.0720.
  * Su and Ralph (2016) D. Su and T. C. Ralph, Phys. Rev. D 93, 044023 (2016), eprint 1507.00423.
  * Higuchi et al. (2017) A. Higuchi, S. Iso, K. Ueda, and K. Yamamoto, Phys. Rev. D 96, 083531 (2017), eprint 1709.05757.
  * Wald (2019) R. M. Wald, Phys. Rev. D 100, 065019 (2019), eprint 1908.06363.
  * Herrero and Morales (1999) A. Herrero and J. A. Morales, J. Math. Phys. 40, 3499 (1999).
  * Ling (2020) E. Ling, Found. Phys. 50, 385 (2020), eprint 1810.06789.
  * Dyer and Honig (1979) C. Dyer and E. Honig, J.Math.Phys. 20, 409 (1979).
  * Jacobson (2016) T. Jacobson, Phys. Rev. Lett. 116, 201101 (2016), eprint 1505.04753.
  * de Alfaro et al. (1976) V. de Alfaro, S. Fubini, and G. Furlan, Nuovo Cim. A 34, 569 (1976).
  * Chamon et al. (2011) C. Chamon, R. Jackiw, S.-Y. Pi, and L. Santos, Phys. Lett. B 701, 503 (2011), eprint 1106.0726.
  * Jackiw and Pi (2012) R. Jackiw and S. Y. Pi, Phys. Rev. D 86, 045017 (2012), [Erratum: Phys.Rev.D 86, 089905 (2012)], eprint 1205.0443.
  * Sun (2021) Z. Sun (2021), eprint 2111.04591.
  * Unruh (1976) W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  * Lykken (2021) J. Lykken, PoS TASI2020, 010 (2021), eprint 2010.02931.
  * Valdivia-Mera (2020) G. Valdivia-Mera (2020), eprint 2001.09869.
  * Saravani et al. (2014) M. Saravani, R. D. Sorkin, and Y. K. Yazdi, Class. Quant. Grav. 31, 214006 (2014), eprint 1311.7146.
  * Anninos et al. (2012) D. Anninos, S. A. Hartnoll, and D. M. Hofman, Class. Quant. Grav. 29, 075002 (2012), eprint 1109.4942.
  * Nakayama (2012) R. Nakayama, Prog. Theor. Phys. 127, 393 (2012), eprint 1112.1267.


# Study of water Cherenkov detector design for ground-based gamma-ray
experiments

F. Bisconti11footnotetext: Corresponding author. and A. Chiavassa

###### Abstract

In the framework of the development of the SWGO experiment we have performed a
detailed study of the single unit of an extensive air shower observatory based
on an array of water Cherenkov detectors. Indeed, one of the possible water
Cherenkov detector unit configurations for SWGO consists of tanks, and to
reach a high detection efficiency and discrimination capability between gamma-
ray and hadronic air showers, different tank designs are under investigation.
In this study, we considered double-layer tanks with several sizes, shapes and
number of photo-multiplier tubes (PMTs). Muons, electrons, and gamma-rays with
energies typical of secondary particles in extensive air showers have been
simulated entering the tanks with zenith angles from 0 to 60 degrees. The tank
response was evaluated considering the number of photoelectrons produced by
the PMTs, the detection efficiency, and the time resolution of the measurement
of the first photon. This analysis allowed to compare the performance of tanks
with different size, configuration of PMTs, and with circular, hexagonal and
square geometry. The method used and the results will be discussed in this
paper.

##  1 Introduction

Wide field of view gamma-ray observatories can be realized by an array of
water Cherenkov detectors, covering areas ranging from $10^{4}$ to $10^{6}$
square meter, usually located in desertic areas. Secondary particles produced
in extensive air showers induced by astrophysical gamma-rays or hadrons (that
represent a background source), can be detected measuring the Cherenkov light
produced when they cross the detectors filled with clean water. A next-
generation gamma-ray observatory is the SWGO experiment [1, 2], which will be
realized at high altitude in the Southern Hemisphere, to be complementary to
other gamma-ray experiments in the Northern Hemisphere, like HAWC [3] and
LHAASO [4], for the observation of the entire sky. It will operate with close
to 100% duty cycle and order steradian field of view. The site has to be at
high altitude (above $4\,400$ m a.s.l.), in order to be closer to the maximum
of the extensive air showers induced by astrophysical gamma-rays with primary
energy in the range of interest (between 100 GeV and a few PeV). The SWGO
design will be primarily based on water Cherenkov detectors, and the final
array and detector unit configurations are still to be defined [1]. One
configuration under study consists of and array of (surface) water Cherenkov
tanks arranged in a high fill-factor core (with area considerably larger than
HAWC) and a low density outer array.

To study the single tank behaviour, we performed simulations of particles
crossing tanks with different size and configuration of PMTs. We simulated
double-layer tanks [5], in which the lower layer helps in the gamma/hadron
discrimination, as muons are more abundant in hadronic showers and they can
cross the upper layer reaching the lower layer where they are measured. We
considered tanks of different shape, with circular (Circular-DLT), hexagonal
(Hexagonal-DLT) and square (Square-DLT) base.

To simulate the particles crossing the tanks and their response we used the
HAWCSim framework [6], which makes use of GEANT4 [7] to simulate the
interaction of the particle with the tank itself and the water volume,
including the production of the Cherenkov photons that can be detected by the
PMTs inside the tank.

##  2 Simulations

###  2.1 Particles

In this analysis we considered the most abundant particles contained in an
extensive air shower generated by 400 GeV protons and 200 GeV photons at an
observation level of $4\,500-5\,000$ m a.s.l.. Therefore, we performed
simulations of electrons, gamma-rays and muons with fixed energies: 10 MeV,
100 MeV and 1 GeV electrons and photons, and 1 GeV and 10 GeV muons. To define
the directions, we used azimuth angles $\phi$ uniformly distributed in the
range $0-360\deg$ and zenith angles $\theta$ in the range $0-60\deg$ sampled
on a $\cos^{2}\theta$ distribution.

The particles were generated on a large circular surface 10 cm above the tank
and centered with it. The size of the generation area is such that even the
most inclined particles could enter the tank from the lateral walls of the
upper layer, to avoid the detection of particles entering in the tank directly
from the lower layer that would affect the overall performance of a single
tank study. This has to be considered in the context of a sparse array of
tanks, while in a dense array the nearby tanks contribute to the detection
capability of the large scale experiment. Therefore, for each tank design,
particle type and energy, 10000 particles entering the upper layer of the
tanks have been analyzed.

###  2.2 Specifications of the tanks

####  2.2.1 Shapes and dimensions of the tanks

In this analysis, Circular-DLTs, Hexagonal-DLTs and Square-DLTs were
considered. In Fig. 1 examples of the Geant4 visualization of the three tank
designs crossed by a muon are shown.

Figure 1: Geant4 visualization of a Circular-DLT, an Hexagonal-DLT and a
Square-DLT, crossed by a 1 GeV vertical muon. All tanks have widths of 3 m
(diameter for Circular-DLT, two times the side for Hexagonal-DLT, and side for
Square-DLT) and lower layers 1 m high. The upper layers were simulated with
non-reflective walls, while the lower layer with reflective walls. The green
line represents the simulated muon and the red lines a sample of Cherenkov
photons.

The height of the upper layer was chosen allowing the Cherenkov photons to
reach any PMT at the base of the upper layer. Assuming a vertical particle
entering the tank from the center of the roof, the Cherenkov photons should be
able to reach the lateral walls of a Circular-DLT or the corners of an
Hexagonal-DLT and a Square-DLT at the base of the upper layer. For Circular-
DLTs the height $h$ and radius $r$ follow the relation $h=r/\tan{\theta_{C}}$,
where $\theta_{C}=41.2\deg$ is the emission angle of the Cherenkov photons
with respect to the trajectory of the particle crossing the water. Similarly,
for Hexagonal-DLTs with side $L$, the height is $h=L/\tan{\theta_{C}}$, and
for Square-DLTs with half side $l$, $h=\sqrt{2}l/\tan{\theta_{C}}$. To the
height calculated with previous formulas, 1 m of water is added to have 90%
probability that gamma-rays interact by pair production. The lower layer, with
height independent of the radius, is dedicated to muon measurements, allowing
for the gamma/hadron discrimination and the separation of mass groups of
charged primaries (from 2 to 4). For the lower layer, we chose heights of 0.5
m, 0.75 m and 1 m. The dimensions of the tanks are collected in Tab. 1.

Tank | Width (m) | Cir.&Hex. Height u.l. (m) | Sqr. Height u.l. (m) | Height l.l. (m)
---|---|---|---|---
T1 | 3 | 2.7 | 3.4 | 0.5, 0.75, 1
T2 | 3.5 | 3.0 | 3.8 | 0.5, 0.75, 1
T3 | 4 | 3.3 | 4.2 | 0.5, 0.75, 1
T4 | 4.5 | 3.6 | 4.6 | 0.5, 0.75, 1
T5 | 5 | 3.9 | 5.0 | 0.5, 0.75, 1
T6 | 5.5 | 4.2 | 5.4 | 0.5, 0.75, 1
Table 1: Size of the tanks. “Width” is the diameter of Circular-DLT, the side
of Square-DLT and two times the side of Hexagonal-DLT; “Cyl.&Hex. Height u.l.”
is the height of the upper layer of Circular-DLT and Hexagonal-DLT; “Sqr.
Height u.l.” is the height of the upper layer of Square-DLT; “Height l.l.” is
the height of the lower layer.

####  2.2.2 Properties of the inner walls

For the inner walls of the upper layers, we used both reflective (Tyvek) and
non-reflective (Polypropylene) materials. The reflectivity of the materials
depends on the wavelength of the incident photons. Tyvek has a reflectivity of
$0.63-0.92$ in the wavelength range $250-650$ nm; polypropylene has a
reflectivity of 0.10 over the same wavelength range. Reflective walls allow
for a better detection capability, but might extend the detection time, due to
possible consecutive reflections of photons on the walls before they reach the
PMTs. This results in a higher detection efficiency as photons that would not
be detected with non-reflective walls are instead detected with reflective
walls, but also widen the time resolution for the detection of the first
photon. For the lower layer we used reflective walls, as the priority was
given to the detection efficiency of particles entering the lower layer rather
than the timing.

####  2.2.3 PMTs

In the upper layer we used two configurations of PMTs looking upwards: one
central 10" PMT or four peripheral 5" PMTs placed at half radius in Circular-
DLTs, half the apothem in Hexagonal-DLTs, and half diagonal in the Square-DLT.
Signals of the peripheral PMTs are summed in one unique output. In the lower
layer we used one central 10" PMT or 5" PMT looking downwards. In each layer,
the two PMT configurations have to be considered independently. In the
simulations, we used two models of PMTs from Hamamatsu: the 10" R7081HQE PMT,
and the 8" R5912 PMT, then re-scaled to a 5" PMT during the analysis phase.

##  3 Analysis

For the evaluation of the tank response, the parameters taken into account
are:

((a))

((b))

((c))

((d))

((e))

((f))

((g))

((h))

((i))

Figure 2: Distributions used in the analysis of the tank performance: number
of PEs (a-c), arrival time of the first photon (d-f), and arrival time of any
photon (g-i). These plots refer to simulations of 1 GeV electrons, gamma-rays
and muons crossing a Circular-DLT with non-reflective walls in the upper layer
and reflective walls in the lower layer. The statistics boxes shown on the
right-side of the plots can be used to analyze the results for the different
configurations of PMTs. For example, the “Mean” value in (a-c) can be used to
estimate the average number of PEs, while the width of the distribution “Std
Dev” in (d-f) can be used to evaluate the time resolution of the measurement
of the first photon.

  * •

The number of photoelectrons (PEs) produced in both layers. In the upper layer
we considered separately the configuration with one central 10" PMT or four
peripheral 5" PMTs. For the lower layer we considered individually a central
10" PMT or 5" PMT. This allowed to understand which of the two configurations
in the upper layer gives the higher detection efficiency and the better time
resolution of the first detected photon and, for the lower layer, how a
different size of the PMT changes the performance.

  * •

The time resolution of the measurement of the first photon in the upper layer,
evaluated as the standard deviation of the distribution of the first photon
arrival time.

  * •

The detection efficiency of both layers. The efficiency is calculated as the
number of detected particles (events) divided by the number of particles
entering the upper layer of the tank (10000). The latter is based on simple
geometrical considerations, based on the height of the entrance point of the
particles. Due to the random direction of particles, a fraction of the
particles that enter the upper layer does not enter the lower layer. We tried
to evaluate the number of particles actually entering the lower layer based on
the initial direction of the particles, but this was not possible due to non-
tracked deflections of the particle trajectories occurring while they cross
the tank. We performed a set of simulations where only 10 GeV vertical muons
were thrown through a Circular-DLT, and in this case the detection efficiency
was $\sim$1 for both the upper and lower layer. This demonstrated that the
inefficiency of the lower layer is only due to geometrical constraints, and
would be effectively reduced considering a joint detection of inclined
particles by neighbouring tanks in a dense array. Therefore, also in the
calculation of the detection efficiency of the lower level we used as
reference the number of particles entering the upper layer of the tank. This
effect underestimates the detection efficiency of the lower layer for any type
of tank and particle, but the comparison between the different configurations
remains valid. For the upper layer, we considered as threshold both 1 PE and
the coincidence of 2 PEs produced within 30 ns by the central 10" PMT or by
the four peripheral 5" PMTs, while for the lower layer the threshold was only
of 1 PE.

In Fig. 2 sample distributions of the number of PEs (a-c), the arrival time of
the first photon (d-f), and the arrival time of photons (g-i) of are shown.
They refer to simulations of 1 GeV electrons, gamma-rays and muons crossing a
Circular-DLT with non-reflective walls in the upper layer and reflective walls
in the lower layer. The distribution of the number of PEs in the lower layer
has a higher average for muons than for electrons and gamma-rays, as the
latter are absorbed by the water of the upper layer. The PE distributions also
show that particles generate in average more PEs on the central 10" PMT than
on the four peripheral 5" PMTs. The distributions of the arrival time of the
first photon are similar. In the sample of distributions reported, the timing
resolution for the four peripheral PMTs is slightly narrower than for the
central PMT.

The time distributions of photons are also presented to show the effect of the
reflective walls in the lower layer. Especially for muons that reach easily
the lower layer, a long tail and a bump after the main peak are due to the
consecutive reflections on the walls before photons reach the PMT. This effect
is more visible when reflective covers are used also for the upper layer, due
to the higher statistics of PEs generated in the upper layer. Depending on the
distributions of the impact point on the tank and the direction of the
particles, a sequence of bumps might appear instead of the tail.

##  4 Results

###  4.1 Comparison of tanks with different size, reflective properties of the
inner walls and PMT configuration

In this section, plots are referred to Circular-DLTs, but the results are
similar for all tank geometries. In Fig. 3–5 the performance of the upper
layer is shown, considering a central 10" PMT or four peripheral 5" PMTs.
Panels (a) are relative to non-reflecting walls, while panels (b) are for
reflecting walls. Similarly, in Fig. 6–7 the performance of the lower layer,
which has always reflective walls, is shown considering a 10" PMT or a 5" PMT.

The number of detected PEs in the upper layers (see Fig. 3) and consequently
the detection efficiency (see Fig. 4) decrease while increasing the size of
the tank. This is due to the decrease of the ratio between the area of the PMT
and that of the base of the tank. To verify this, we made some test
simulations using different tank widths and rescaling the PMT size in order to
have a constant ratio between the area of the PMT and the base of the tank,
and the detection efficiency remained almost constant. With 1 PE threshold,
the detection efficiency of the upper layers considering one central 10" PMT
or four peripheral 5" PMTs are comparable, although more PEs are produced in
the central PMT. The sensitive area based on the size of the PMTs is similar
for the two configurations. Therefore, the difference is related to the
position of the PMTs. The efficiency for 10 MeV and 100 MeV particles is a few
ten percent, while it is above 70% for higher energies. With 2 PEs threshold
(plots not shown in this work), the efficiency is reduced by a few ten percent
for 10 MeV and 100 MeV particles, while it is similar for particles with
higher energy. With reflective walls in the upper layers, the number of PEs
and the detection efficiency are higher than those for non-reflective walls,
even for low energy particles (compare Fig. 3 (a) with (b) and Fig. 4 (a) with
(b)). Using 2 PEs threshold, the effect is the same as that for non-reflecting
walls.

((a))

((b))

Figure 3: Number of PEs detected in the upper layer with non-reflecting walls
(a) and reflective walls (b) in Circular-DLT, note the different scales on the
y-axis.

((a))

((b))

Figure 4: Detection efficiency of the upper layer with non-reflecting walls
(a) and reflective walls (b) in Circular-DLT.

The time resolution of the measurement of the first photon worsen, i.e. the
standard deviation of the distribution gets larger, with the increase of the
size of the tank (see Fig. 5). In average, using non-reflecting walls it
ranges from $\sim$2.5 ns in small tanks to $\sim$3.5 ns in large tanks.
Considering 2 PEs threshold, it has smaller values, between $\sim$1 ns and
$\sim$2 ns. Using reflective walls, it slightly increases for 100 MeV and 1
GeV particles, and rises up to $\sim$18 ns for particles of 10 MeV, because
the time distribution of the first photon shows a long tail for these
particles. It has similar values for the central 10" PMT and the four
peripheral 5" PMTs. Considering 2 PEs threshold, it has slightly lower values.

((a))

((b))

Figure 5: Time resolution of the measurement of the first photon in the upper
layer with non-reflective walls (a) and reflecting walls (b) in Circular-DLT.

Like in the upper layer, the number of detected PEs (see Fig. 6) and the
detection efficiency (see Fig. 7) in the lower layers decrease with the size
of the tank. Electrons and gamma-rays of 10 MeV and 100 MeV are rarely
detected in lower layers. Considering a 5" PMT instead of a 10" PMT, the
efficiency slightly decreases although the number of produced PEs is the 25%,
proportional to the area of the photocathode. Both kinds of PMTs are placed at
the center of the ceiling of the lower layer, so there is no effect due to
different positioning like it happens in the upper layer. The height of the
lower layer influences the number of PEs, which is lower for 0.5 m and
comparable for 0.75 m and 1 m, but does not affect the detection efficiency
(plots not shown in this work). In all the configurations, the detection
efficiency of the lower layer is underestimated in the same way due to
geometrical constraints (see details about the calculation of the detection
efficiency in Section 3).

((a)) Figure 6: Number of PEs detected in the lower layer with reflective
walls. ((a)) Figure 7: Detection efficiency of the lower layer with reflective
walls. In all the configurations, it is underestimated due to geometrical
constraints.

###  4.2 Comparison of tank shapes

The analysis described in the previous section was performed for the three
tank shapes of interest: Circular-DLTs, Square-DLTs and Hexagonal-DLTs. In
addition to tanks with circular base, which is the most commonly used shape
for the construction of ground based astrophysical experiments due to their
stronger strain resistance to the water pressure, ease of construction, and
lower cost, tanks with square and the hexagonal bases have been considered
because they would offer a higher fill factor, particularly important when it
is necessary to cover areas with high density of tanks, like the inner array
of the SWGO experiment.

The plots shown in this section represent the aforementioned parameters in
function of the size of the tanks with different geometries and were made for
1 GeV particles, in order to compare the tank performance in the same
conditions. The size of the tanks is represented by their width, i.e. diameter
for Circular-DLT, two times the side for Hexagonal-DLT, and side for Square-
DLT. In Fig. 8–13 the performance of the upper layers of tanks with different
shapes are shown, considering a central 10" PMT or four peripheral 5" PMTs.
Panels (a) are relative to non-reflecting walls, while panels (b) are for
reflecting walls. Similarly, in Fig. 14–17 the performance of the lower layer
is shown considering a 10" PMT or a 5" PMT.

In general the Circular-DLT and Hexagonal-DLT produce more PEs than the
Square-DLT.

((a))

((b))

Figure 8: Comparison of the number of PEs detected in the upper layer with
non-reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the central 10"PMTs.

((a))

((b))

Figure 9: Comparison of the number of PEs detected in the upper layer with
non-reflective walls (a), and reflective walls (b) for 1 GeV particles, for
different geometries and considering only the peripheral 5"PMTs.

With reflective walls, the number of PEs is about three times that obtained
with non-reflective walls, in any kind of tank (compare panels (a) with panels
(b) in Fig. 8 and Fig. 9). Comparing the response of the central 10" PMT and
the four peripheral 5" PMTs in the upper layer, the number of PEs for the
first configuration is higher than the number of PEs for the latter
configuration for all tank geometries (compare panel (a) and panel (b) in Fig.
8 with the same panels in Fig. 9). The detection efficiency of the upper layer
is similar for Circular-DLTs and Hexagonal-DLTs, while it is slightly lower
for Square-DLTs. As already shown, it is higher using reflective walls
(compare panels (a) with panels (b) in Fig. 10 and Fig. 11). Although the
number of PEs is higher for the central 10" PMT, the detection efficiency for
particles of 1 GeV is similar for both configurations of PMTs (compare panels
(a) and (b) in Fig. 10 with those in Fig. 11).

((a))

((b))

Figure 10: Comparison of detection efficiency of the upper layer with non-
reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the central 10"PMTs.

((a))

((b))

Figure 11: Comparison of the detection efficiency of the upper layer with non-
reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the peripheral 5"PMTs.

The temporal resolution of the measurement of the first photon considering the
central 10" PMT is $\sim$0.5 ns larger than that of the four peripheral 5"
PMTs (compare panels (a) and panels (b) in Fig. 12 with those in Fig. 13).
However, such values are slightly higher for Square-DLT than the others. In
general, with reflective walls the temporal resolution of the measurement of
the first photon in upper layers is about $1-2$ ns larger than with non-
reflective walls, in all kinds of tanks (compare panels (a) with panels (b) in
Fig. 12 and Fig. 13).

((a))

((b))

Figure 12: Comparison of the time resolution of the measurement of the first
photon in case of non-reflecting walls (a) and reflective walls (b) in the
upper layer, for 1 GeV particles, for different geometries and considering
only the peripheral 10" PMTs in the upper layer.

((a))

((b))

Figure 13: Comparison of the time resolution of the measurement of the first
photon in case of non-reflecting walls (a) and reflective walls (b) in the
upper layer, for 1 GeV particles, for different geometries and considering
only the peripheral 5" PMTs in the upper layer.

In the lower layer of the three kinds of tanks, the number of PEs produced by
the 10" PMT is four times that produced by the 5" PMT, simply because of the
ratio between the active areas of the sensors (see Fig. 14 and Fig. 15). The
height of the lower layer influences the number of PEs, which is lower for 0.5
m but comparable for 0.75 m and 1 m.

((a)) Figure 14: Comparison of the number of PEs in the lower layer with
reflective walls, for 1 GeV particles, for different geometries and
considering the 10" PMT. ((a)) Figure 15: Comparison of the number of PEs in
the lower layer with reflective walls, for 1 GeV particles, for different
geometries and considering the 5" PMT.

Despite the difference in the number of PEs detected with the two PMT
configurations, the detection efficiency of the lower layer does not vary (see
Fig. 16 and Fig. 17). It is similar for Circular-DLT and Hexagonal-DLT and
higher than for Square-DLT, and it is similar also for the three different
heights considered. In all the configurations, the detection efficiency of the
lower layer is underestimated in the same way due to geometrical constraints
(see details about the calculation of the detection efficiency in Section 3).

((a)) Figure 16: Comparison of the detection efficiency of the lower layer
with reflective walls, for 1 GeV particles, for different geometries and
considering the 10" PMT. In all the configurations, it is underestimated due
to geometrical constraints. ((a)) Figure 17: Comparison of the detection
efficiency of the lower layer with reflective walls, for 1 GeV particles, for
different geometries and considering the 5" PMT. In all the configurations, it
is underestimated due to geometrical constraints.

##  5 Conclusion

This study allowed to compare the response of double-layer tanks with
different shapes, i.e. with circular, hexagonal and square base of different
size, to the passage of single particles of different type and energy.
Moreover, it offered the possibility to compare tanks with reflective and non-
reflective walls in the upper layer.

We found that regardless of the tank design and the reflective properties of
the walls in the upper layer, the performance of the tanks worsen while
increasing the width of the tank, because the “sensitive area”, i.e. the area
covered by the PMTs, decreases with respect to that of the base of the tank.

By using reflective walls instead of non-reflective walls in the upper layers,
the detection efficiency increases, but the time resolution of the measurement
of the first photon widen, in particular for particle with low energy.

In lower layers, electrons and gamma-rays of 10 MeV and 100 MeV are rarely
detected. The height of the lower layer influences the number of PEs, which is
lower for 0.5 m but comparable for 0.75 m and 1 m. Howewer, the detection
efficiency of the lower layer does not vary much with its height.

Comparing the performance of the central 10" PMT and the four peripheral 5"
PMTs in the upper layer, the first configuration produces more PEs than the
second one, although both sensitive areas are similar, but the detection
efficiencies are comparable. In a similar comparison for the lower layer, the
10" PMT produces more PEs than the 5" PMT due to the larger area of the
photocathode, but the detection efficiencies are similar.

The comparison of the performance of tanks with different geometries revealed
that the Circular-DLTs and Hexagonal-DLTs have similar performance, which is
better than that of the Square-DLTs. Nevertheless, for the final design of
ground based astrophysical observatories like the SWGO array, it should be
taken into account also that with Hexagonal-DLT and Square-DLT it is possible
to achieve a higher fill factor, although they are potentially more expensive
solutions.

## Acknowledgments

We thank our colleagues within the SWGO Collaboration for the discussions and
the software framework used in this work. We thank the HAWC Collaboration for
providing the AERIE software.

## References

  * [1] U. Barres de Almeida (for the SWGO Collaboration), The Southern Wide-Field Gamma-ray Observatory (SWGO), arXiv e-prints: arXiv:2012.13740 (2020)
  * [2] J. Hinton et al. (for the SWGO Collaboration), The Southern Wide-field Gamma-ray Observatory: Status and Prospects, in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)023 (2021)
  * [3] T. DeYoung (for the HAWC Collaboration), The HAWC observatory, Nuclear Instruments and Methods in Physics Research A, 692, 72 (2012)
  * [4] X. Bai et al., The Large High Altitude Air Shower Observatory (LHAASO) Science White Paper, arXiv e-prints: arXiv:1905.02773 (2019)
  * [5] S. Kunwar et al. (for the SWGO Collaboration), Double-layered Water Cherenkov Detector for the Southern Wide-field-of-view Gamma-ray Observatory (SWGO), in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)902 (2021)
  * [6] H. Schoorlemmer et al. (for the SWGO Collaboration), Simulating the performance of the Southern Wide-view Gamma-ray Observatory, in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)903 (2021)
  * [7] J. Allison et al., Geant4 developments and applications, IEEE Transactions on Nuclear Science 53 No. 1 (2006) 270-278