#  The Effective Theory of Quantum Black Holes

E. Binetti<EMAIL_ADDRESS>Dept. of Physics E. Pancini, Universita
di Napoli Federico II, via Cintia, 80126 Napoli, Italy M. Del Piano
<EMAIL_ADDRESS>Scuola Superiore Meridionale, Largo S.
Marcellino, 10, 80138 Napoli NA, Italy  INFN sezione di Napoli, via Cintia,
80126 Napoli, Italy  S. Hohenegger<EMAIL_ADDRESS>Institut de
Physique des 2 Infinis (IP2I), CNRS/IN2P3, UMR5822, 69622 Villeurbanne, France
Université de Lyon, Université Claude Bernard Lyon 1, 69001 Lyon, France  F.
Pezzella<EMAIL_ADDRESS>INFN sezione di Napoli, via Cintia, 80126
Napoli, Italy  F. Sannino<EMAIL_ADDRESS>Dept. of Physics E. Pancini,
Universita di Napoli Federico II, via Cintia, 80126 Napoli, Italy Scuola
Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli NA, Italy  INFN
sezione di Napoli, via Cintia, 80126 Napoli, Italy  $CP^{3}$-Origins & D-IAS,
Southern Denmark Univ., Campusvej 55, 5230 Odense M, Denmark  CERN,
Theoretical Physics Department, 1211 Geneva 23, Switzerland

###### Abstract

We explore the quantum nature of black holes by introducing an effective
framework that takes into account deviations from the classical results. The
approach is based on introducing quantum corrections to the classical
Schwarzschild geometry in a way that is consistent with the physical scales of
the black hole and its classical symmetries. This is achieved by organizing
the quantum corrections in inverse powers of a physical distance. By solving
the system in a self-consistent way we show that the derived physical
quantities, such as event horizons, temperature and entropy can be expressed
in a well defined expansion in the inverse powers of the black hole mass. The
approach captures the general form of the quantum corrections to black hole
physics without requiring to commit to a specific model of quantum gravity.

##  1 Introduction

Understanding the quantum nature of space-time is an open challenge both from
a theoretical and an experimental point of view. Quantum gravity effects are
thought to be relevant, for example, in gravitational collapse of
astrophysical objects as well as evaporation processes of Planck-size black
holes (BH). The goal of this work is to construct an effective framework that
allows to investigate quantum corrections for BH physics in order to extract
reliable predictions. Effective approaches have been applied extensively to
account for quantum corrections in gravity and particle physics, see [1] for
an overview.

Rather than considering a specific theory of quantum gravity, our philosophy
is to develop a general effective framework based on formulating BH metrics
via dimensionless quantities and their physical scalings (see [2, 3, 4] for
related ideas in different areas of physics). Although this approach can be
viewed as a renormalization improvement [5, 6, 7, 8, 9, 10, 11] of the BH
metrics, it differs from the Wilsonian interpretation [12, 13] of the running
of couplings of the effective action [14] further explored in [15], and
therefore it is compatible with the arguments of Ref. [16].

We elucidate our approach by focusing on the static spherically symmetric
classical Schwarzschild BH. After introducing the approach, we determine the
impact of the leading order quantum corrections on the physical quantities
such as event horizons, temperature and entropy in a consistent fashion. In
the way the framework is setup, quantum corrections to physical observables
appear as a well-defined expansion in the mass of the BH relative to the
classical results. We show that the approach can be consistently generalized
to higher order quantum corrections leading to higher mass suppressed
corrections. Although we do not discuss it in this work, our approach can be
further generalized to account for non-local corrections to effective gravity
actions [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Our
findings amount to establishing a self-consistent effective counting scheme,
based on the physical mass of the BH.

The work is organized as follows. In Section 2 we start by introducing the
effective framework and by setting up the notation. The Section is further
divided into several subsections. The rationale behind our way to upgrade the
classical metric to an effective quantum one is summarized in the first
subsection. We then move to determine the leading order quantum corrected
horizons and discuss their impact on the BH physics. We show that, depending
on the sign of the first leading order corrections, the geometry can develop a
second (internal) horizon. We then move to show the associated conformal
diagrams. The self-consistency of the approach, when considering the
backreaction stemming from the quantum corrected proper distance, is presented
in 2.4. In Section 3 we discuss how to take into account higher order quantum
gravity corrections to the metric. One of the main results is establishing an
effective consistent framework in computing quantum corrections to BH physics
organised in their mass expansion.

The quantum corrected thermodynamic properties, such as temperature and
entropy, are discussed in Section 4. Conclusions and outlook are offered in
Section 5 while in the Appendix A we provide further details on how to compute
the horizons in our framework.

##  2 Quantum Schwarzschild Black Hole

We focus on the simplest BH in four dimensions, featuring a spherical and
stationary geometry with Schwarzschild metric

$\differential s^{2}=-f_{0}(r)\differential t^{2}+\frac{\differential
r^{2}}{f_{0}(r)}+r^{2}\differential\theta^{2}+r^{2}\sin^{2}\theta\,\differential\phi^{2}\,,$
(1)

where we use spherical coordinates and the metric tensor depends only on the
radial one through the function

$f_{0}(r)=1-\frac{2G_{\mathrm{N}}M}{r}\,,$ (2)

with $M$ being the mass sourcing the gravitational field and $G_{\mathrm{N}}$
the Newton constant.

###  2.1 Quantum Framework

We now upgrade the metric (1) to a quantum one without committing to a
specific underlying quantum gravity theory***At the quantum level, we require
the existence of a spherically symmetric metric with a time-like Killing
vector. This ensures that the quantum metric still preserves the form (1)
$\differential s^{2}=-h(r)\differential t^{2}+\frac{\differential
r^{2}}{f(r)}+r^{2}\differential\theta^{2}+r^{2}\sin^{2}\theta\,\differential\phi^{2}\,.$
(3) Here $f$ and $h$ are two functions of $r$. In this work, since we compute
static properties we focus on the quantum corrections contained in $f(r)$.. At
the classical level the metric depends on two dimensionful quantities†††The
Planck scale hidden in the Newton constant at the classical level defines the
units, and thereby has no influence on the classical physics., the mass of the
BH and the coordinate $r$. In the following we describe the quantum framework
that we employ to determine quantum corrections to BH observables.

  1. 1.

The quantum corrections are controlled by the Planck length $\ell_{\rm
P}=1/M_{\mathrm{P}}$ (with $M_{\mathrm{P}}$ the Planck mass), which governs
the transition from the classical to the quantum regime. As such, now
$\ell_{\rm P}$ is upgraded to a physically relevant length beyond providing
just a unit of measure. To reflect this, we introduce the following
dimensionless quantities:

$z:=M_{\mathrm{P}}r=\frac{r}{\ell_{\mathrm{P}}}\,,\qquad\chi:=\frac{M}{M_{\mathrm{P}}}\,,$
(4)

and rewrite (2) as:

$f_{0}(z)=1-\frac{2\chi}{z}g\ ,\qquad{\rm with}\qquad
g:=G_{\mathrm{N}}M_{\mathrm{P}}^{2}=1\ .$ (5)

  2. 2.

Transitioning from the classical to the quantum regime requires to modify (5)
as follows:

$f(z,\frac{u}{\ell_{\mathrm{P}}})=1-\frac{2\chi}{z}g(z,\frac{u}{\ell_{\mathrm{P}}})\
,$ (6)

where $g$ is an a priori undetermined function.‡‡‡As already in the classical
case (2), we do not explicitly exhibit the dependence on $\chi$ Here $u$ is an
arbitrary renormalization scale required to compensate for the presence of a
fundamental length in the problem, i.e. $\ell_{\mathrm{P}}$. Since $u$ is
arbitrary no physical quantity can depend on it. This means that the
derivative of any such quantity with respect to $u$ must vanish, therefore
imposing non-trivial consistency conditions also on
$\displaystyle{g(z,\frac{u}{\ell_{\mathrm{P}}})}$ (see _e.g._ [2] for similar
arguments in other physical systems). In order for any allowed coordinate
transformation of $f_{0}(z)$ to be carried over at the quantum level one has
to conclude that $g$ is a protected quantity and therefore:

$g(z,\frac{u}{\ell_{\mathrm{P}}})\quad\longrightarrow\quad
g(d,\frac{u}{\ell_{\mathrm{P}}})\ ,$ (7)

for a suitable (dimensionless) physical quantity $d$ which is therefore
independent of $u$.

  3. 3.

The choice of the physical dimensionless quantity $d$ is not unique. A
candidate choice for it is the normalized proper distance from the center of
the BH [14]

$d(z):=\frac{1}{\ell_{\mathrm{P}}}\int_{0}^{z\ell_{\mathrm{P}}}\sqrt{|\differential
s^{2}|}=\int_{0}^{z}\frac{\differential z^{\prime}}{\sqrt{|f(z^{\prime})|}}\
.$ (8)

The first integral is understood for fixed values of the angular coordinates
($\theta,\phi$). Notice that, as remarked above, since $d(z)$ cannot depend on
$u$ this implies a constrain for
$\displaystyle{f(z,\frac{u}{\ell_{\mathrm{P}}})}$ which restricts the $u$
dependence of $\displaystyle{g(d,\frac{u}{\ell_{\mathrm{P}}})}$. We note that
quantum improvements of the metric based on unphysical quantities such as the
radial coordinate $z$ lead to quantum geometries depending on the specific
choice of coordinates. This issue was discussed in detail in [33].

  4. 4.

At large proper distances from the BH the function $g$ approaches
asymptotically unity. At the quantum level we have, therefore, for $f(z)$:

$f(z)=1-\frac{2\chi}{z}\sum_{n=0}^{\infty}\frac{\Omega_{n}}{d(z)^{2n}}\,.$ (9)

The specific values of the dimensionless coefficients
$\Omega_{n}(\frac{u}{\ell_{\mathrm{P}}})$ , with $\Omega_{0}=1$, are dictated
by a given theory of quantum gravity. The $u$ dependence of the $\Omega_{n}$
coefficients is constrained by requiring physical quantities to be independent
on this arbitrary scale. The expansion in (9) is built to incorporate the fact
that at large distances the metric must asymptotically approach the classical
one (2). The choice of even inverse powers of $d(z)$ comes from our
expectation that this quantum metric emerges from a(n effective) quantum
gravity action with only even powers of the derivatives. We have also
neglected subleading logarithmic terms. The approach can be readily extended
to include a different counting scheme if required by more general theories of
quantum gravity. A different definition of the physical distance $d(z)$ leads
to modified coefficients $\Omega_{n}$.

  5. 5.

By construction (9) is an involved equation for $f(z)$ which we attack in a
self consistent iterative manner: we shall add one order in $n$ of the series
at a time and include the backreaction stemming from the corrected $d(z)$ from
the previous order. In practice, this procedure mimics the potential expansion
of an effective quantum gravity action in local derivative operators. However,
the overall approach does not rely on this interpretation and can therefore be
further extended to include non-analytic terms, which we plan to explore in
the future.

In the following subsection we start by considering the leading quantum
correction.

###  2.2 Leading Order Quantum Metric

To determine the leading order quantum corrected function $f_{1}(z)$, we
introduce the classical proper distance $d_{0}(z)$ given by

$d_{0}(z)=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|f_{0}(z^{\prime})\right|}}=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|1-\frac{2\chi}{z^{\prime}}\right|}}\,.$ (10)

Performing the integration we have

$d_{0}(z)=\begin{cases}\displaystyle\pi\chi-2\chi\tan^{-1}\sqrt{\frac{2\chi}{z}-1}-\sqrt{z(2\chi-z)}\,,&0<z<2\chi\,,\\\
\\\
\displaystyle\pi\chi+2\chi\tanh^{-1}\sqrt{1-\frac{2\chi}{z}}+\sqrt{z(z-2\chi)}\,,&2\chi<z<\infty\,.\end{cases}$
(11)

The left panel of Fig. 1 is the graphical representation of the integrand in
(10) while the right panel represents (11). The integrand (10) has an
integrable singularity at $z_{S}=2\chi$ yielding a regular proper distance.

Figure 1: Left panel: We plot the integrand of (10), normalised to the
dimensionless Schwarzschild radius of the BH $z_{\mathrm{S}}=2\chi$. Right
panel: the regular proper distance $d_{0}(z)$ given in (11).

Near the center of the BH the distance function behaves like

$d_{0}(z)\simeq\frac{2}{3}\frac{z^{3/2}}{\sqrt{2\chi}}+\mathcal{O}\left(\frac{z^{5/2}}{(2\chi)^{3/2}}\right)\,,$
(12)

while a linear dependence is recovered at distances far from the horizon as
shown in Fig. 1. The leading order quantum corrected $f$ function reads

$f_{1}(z)=1-2\frac{\chi}{z}\left[1+\frac{\Omega_{1}}{d_{0}^{2}(z)}\right]\,.$
(13)

Figure 2: Plot of the quantum corrected function $f_{1}(z)$ given in (13) as a
function of the distance in Planck units $z$ and for different values of
$\Omega_{1}$ for a fixed mass ratio $\chi=10$.

###  2.3 Quantum Horizons

We are now ready to discuss the quantum corrections to the BH classical
horizon starting with the zeros of $f_{1}(z)$. Here, the location of the
zeroes depends on the sign of the parameter $\Omega_{1}$ that leads to
qualitatively different BH solutions. To remain general, we discuss both cases
separately. We plot in Fig. 2 the function $f_{1}(z)$ for different values of
$\Omega_{1}$ (while we fixed $\chi=10$). This plot shows two qualitatively
very important results

  * 1.

The position of the (external) horizon is a function of $\Omega_{1}$: positive
values of $\Omega_{1}$ shift the zero of $f_{1}$ to larger values of $z$,
while negative values of $\Omega_{1}$ move it to smaller values. However, in
both cases, the effect is small compared to the classical position of the
horizon $2\chi$, even for values of $|\Omega_{1}|\geq 1$.

  * 2.

For negative values of $\Omega_{1}$, the function $f_{1}$ allows for a second
zero in the physical region $z>0$, which can be interpreted as the formation
of a new _internal horizon_. The position of the latter depends much stronger
on the numerical value of $\Omega_{1}$ than the corrections to the external
horizon.

In the following we shall discuss both effects more quantitatively, by
treating separately the different signs of $\Omega_{1}$.

####  2.3.1 $\Omega_{1}>0$

As shown in Fig. 2, for positive values of $\Omega_{1}$ the function $f_{1}$
has a single zero for $z>0$ corresponding to a single horizon which can be
expanded around the classical solution as follows:

$z_{+}=2\chi\left[1+\left(\frac{\Omega_{1}}{\pi^{2}\chi^{2}}\right)+{\cal{O}}{\left(\frac{\Omega_{1}}{\pi^{2}\chi^{2}}\right)^{3/2}}\right]\equiv
2\chi\left[1+\alpha+\,{\mathcal{O}}\left(\alpha^{3/2}\right)\right]\,,\mbox{\quad
with\quad}\alpha:=\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\,,$ (14)

We can render $\alpha$ arbitrarily small by increasing $\chi$ (the BH mass)
for fixed $\absolutevalue{\Omega_{1}}$. While in practice the factor of
$\pi^{2}$ in the definition of $\alpha$ in (14) further suppresses the quantum
corrections, we remark that $\pi\chi=d_{0}(2\chi)$ is in fact the classical
distance of the classical BH horizon (see eq. (11)). Therefore, the expansion
in (14) is organised in terms of physical quantities of the classical BH
geometry. Naturally, we recover the classical horizon when we either switch
off the quantum corrections or increase the BH mass (such that $\alpha\to 0$).
We discuss the numerical range of validity of (14) as a function of the BH
mass in Appendix A.1 and higher order corrections in $\alpha$ in Appendix A.2.
As we shall demonstrate in the next subsection, the corrections to $z_{+}$
stemming from self consistently replacing $d_{0}(z)$ in $f_{1}(z)$ (see (13))
with

$d_{1}(z)=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|f_{1}(z^{\prime})\right|}}\ ,$ (15)

appear at ${\mathcal{O}}(\alpha^{3/2})$. Therefore, to this order in $\alpha$,
all the quantum corrections are taken into account for the external horizon.
The horizon location could depend on the unphysical scale $u$ through
$\Omega_{1}$ which, however, to the current quantum order is constrained to be
$u$ independent by requiring $d_{1}$ to be a physical quantity to the same
order. This can be seen from (63).

Overall, the horizon increases due to quantum corrections and these are
further suppressed at large masses.

####  2.3.2 $\Omega_{1}<0$

In this case the quantum corrected external horizon reads:

$\displaystyle
z_{+}=2\chi\left[1-\alpha+\,{\mathcal{O}}\left(\alpha^{3/2}\right)\right]\ ,$
(16)

where $\alpha$ is defined as in (15). The external horizon, now, decreases due
to quantum corrections, while the other remarks made for the $\Omega_{1}$
positive case still apply. For negative values of $\Omega_{1}$, as shown in
Fig. 2, an internal horizon forms at the position

$\displaystyle
z_{-}=\chi\left(\frac{9\pi}{2}\right)^{1/3}\,\left[\alpha^{1/3}+\frac{1}{5}\left(\frac{\pi^{2}}{6}\right)^{1/3}\,\alpha^{2/3}+\frac{61}{700}\,\left(\frac{3\pi^{4}}{4}\right)^{1/3}\,\alpha+\mathcal{O}(\alpha^{4/3})\right]\
.$ (17)

Clearly, the existence of the internal horizon has a quantum nature and
strongly depends on the underlying theory of quantum gravity.

Figure 3: Black hole event horizons indicated by $z_{H}$ corresponding to
either the classical (straight line), $z_{+}$ and $z_{-}$, as functions of the
mass ratio $\chi$ for the value of $\Omega_{1}=-1$ ($a$) and $\Omega_{1}=1$
($b$). The values are found solving numerically $f_{1}(z)=0$.

We display the zeroes of $f_{1}(z)$ in Fig. 3 as function of $\chi$ and
observe that the internal horizon is less dependent on this parameter when
compared to the external one. Furthermore, for masses close to the Planck
value the two horizons merge leading to an extremality condition that is
analytically approximated to be:

$\chi^{\rm ext}\simeq\frac{16\sqrt{-\Omega_{1}}}{\pi^{2}}\,.$ (18)

The ultimate fate for the existence of the internal horizon depends, as we
shall see, on the size and sign of the higher order corrections.

####  2.3.3 Conformal Diagrams

The global properties of the quantum space-time described by the metric with
the $f$ function in eq. (13) can be neatly summarised via Penrose’s diagrams
[34, 35] shown in Fig. 4. The positive $\Omega_{1}$ case can be summarised as
similar to the classical Schwarzschild one with a spacelike singularity while
the BH horizon is just slightly larger. The conformal diagram relative to the
maximal extension of this space-time is shown in the right panel of Fig. 4,
which is the Szekeres-Kruskal conformal diagram [34, 35].
Since for negative $\Omega_{1}$ the quantum BH has two event horizons, its
space-time structure qualitatively resembles the classical Reissner-Nordström
(RN) one [34, 35, 14]. To better appreciate the differences we note that, at
large distances, the RN dependence on the charge decreases as $z^{-2}$ while
for the quantum corrected one it goes as $z^{-3}$ in terms of the quantum
effects. The qualitative conformal diagram of the maximal extension of such a
space-time is therefore still expected to be of the form given in the left
panel of Fig. 4.

As it is shown, the singularity at the origin is a timelike one and the two
horizons $z_{+}$ and $z_{-}$ can be crossed by a timelike infalling observer
who can reach multiple space-times. The case of an extremal BH is not shown
and it can be found in the literature as its conformal diagram, as stated
before, is analogous to the classical RN one.

$z=0$$z_{+}$$z_{+}$$z_{-}$$z_{-}$$z_{-}$$z_{-}$$z_{+}$$z_{+}$$i^{0}$$i^{0}$$i^{0}$$i^{0}$$z=0$$\Omega_{1}<0$$\mathcal{J}^{+}$$\mathcal{J}^{+}$$\mathcal{J}^{+}$$\mathcal{J}^{+}$$\mathcal{J}^{-}$$\mathcal{J}^{-}$$\mathcal{J}^{-}$$\mathcal{J}^{-}$$\Omega_{1}>0$$\mathcal{J}^{-}$$\mathcal{J}^{+}$$i^{0}$$z_{+}$$z_{+}$$z_{+}$$z_{+}$$\mathcal{J}^{+}$$i^{0}$$\mathcal{J}^{-}$$z=0$$z=0$$i^{+}$$i^{+}$$i^{-}$$i^{-}$
Figure 4: Conformal diagrams of the quantum corrected space-time described by
a metric given by the function in eq. (13) for $\Omega_{1}<0$ (left) and
$\Omega_{1}>0$ (right). The case of negative $\Omega_{1}$ corresponds to a BH
with two distinct event horizons. The notation is the following:
$\mathcal{J^{+}}$ ($\mathcal{J^{-}}$) is the future (past) null infinity,
$i^{+}$ ($i^{-}$) is the future (past) timelike infinity and $i^{0}$ is the
spatial infinity, while $z=0,\ z_{+},\ z_{-}$ are respectively the singularity
at the origin, the external event horizon and the internal one.

###  2.4 Quantum Proper Distance

Even including only the leading quantum corrections by restricting to an
effective second-derivative action (and thus truncating the series (9) at
$n=1$), eq. (13) is only an approximation, since it contains the classical
proper distance $d_{0}$. Self consistency requires to include the impact of
the quantum corrected geodesic distance, previously also indicated as proper
distance, on the quantum $f$ function given in (13) by substituting $d_{0}$
with

$\displaystyle d_{1}(z)=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|1-\frac{2\chi}{z^{\prime}}\left[1+\frac{\Omega_{1}}{d_{0}^{2}(z^{\prime})}\right]\right|}}\,.$
(19)

Figure 5: In the panels $a)$ and $b)$ we plot the dimensionless quantum
corrected proper distance $d_{1}$ for $\Omega_{1}=1$ and $\Omega_{1}=-1$, and
for two different values of the mass ratio $\chi=$ 5 ($a$), 10 ($b$). We also
plot $d_{0}$ corresponding to the classical proper distance ($\Omega_{1}=0$).
The ratio $d_{1}/d_{0}$ is displayed in the inserts $c)$ and $d)$. In the
bottom panel $e)$ the solid lines (blue for $\Omega_{1}>0$ and orange for
$\Omega_{1}<0$) correspond to the improved quantum function $\bar{f}_{1}(z)$
while the dotted lines correspond to the unimproved $f_{1}(z)$. The green
solid line represents the classical function $f_{0}(z)$.

We therefore obtain the quantum self improved $f$ function§§§The procedure is
straightforwardly generalized when considering higher order quantum
corrections, as we shall see below.

$\bar{f}_{1}(z)=1-2\frac{\chi}{z}\left[1+\frac{\Omega_{1}}{d_{1}^{2}(z)}\right]\,.$
(20)

Figure 6: In the subplot $a)$ we set $\Omega_{1}=-1$ and we represent the
external and internal event horizon dimensionless radii $z_{H}$ as a function
of the mass ratio $\chi$ in the case of a space-time with back-reaction on $f$
described by the function in eq. (20) (solid lines) and compare them with the
ones that come from eq. (13) (dashed lines). In subplot $b)$ we study the case
in which $\Omega_{1}=1$ and plot the event horizon radius as $\chi$ varies and
compare it with the one without back-reaction on the $f$ discussion. Eventual
oscillations in the plots are due to numerical errors.

For $\Omega_{1}<0$, one observes that $d_{1}$ remains smaller than the
classical $d_{0}$ for values of $z$ smaller than the internal horizon while it
is larger for any other value of $z$. Nevertheless the qualitative behaviour
of the quantum distance mimics the classical geodesic one. A similar analysis
for $\Omega_{1}>0$ is simplified by the fact that only the external horizon is
present. Here the quantum corrected proper distance $d_{1}$ again follows the
behaviour of the classical distance. In panels $a)$ and $b)$ of Fig. 5 we
display the classical and quantum proper distances. The two plots correspond
to two different values of the BH mass. We also observe that near the origin
the quantum corrected geodesic distance approaches zero faster than the
classical one, specifically it goes as $z^{3}$ rather than $z^{3/2}$. For
completeness we plot the ratio of $d_{1}/d_{0}$ as function of $z$ in the
panels $c)$ and $d)$ of Fig. 5 corresponding again to two different choices of
the BH mass.

To acquire a general understanding of the effects of the improved results for
the function $\bar{f}_{1}$ of (20) we plot it in Fig. 5. The solid green line
corresponds to the classical function $f_{0}$, the orange and blue to the
different signs of $\Omega_{1}$ taking into account the quantum corrected
proper distance. The dashed curves correspond to the unimproved quantum $f$
function obtained via the classical proper distance. As we had anticipated
earlier, the location of the horizons, shown in Fig. 6, are marginally
affected by the improvement due to the quantum proper distance back-reaction.
Specifically, the quantum proper distance improvement appear, for the external
horizon to the ${\cal O}(\alpha^{3/2})$, and for the internal one (for
$\Omega_{1}$ negative) a numerical investigation suggests that the improvement
appears beyond the order ${\cal O}(\alpha)$.

##  3 Higher Order Quantum Corrections

After having treated the leading quantum corrections in the previous section,
we shall now discuss the procedure to self consistently consider higher order
quantum corrections. That is, we consider higher corrections to the $f$
function, but still truncate the sum in eq. (9) at a finite $n$.¶¶¶Notice that
by keeping $n$ finite allows us to avoid questions about the radius of
convergence of the sum in eq. (9). The latter is equivalent to the question
whether the underlying theory of quantum gravity allows for a non-perturbative
definition beyond a (perturbative) effective approach. In this way, we assume
that the underlying quantum corrected gravitational theory can be approximated
via a local effective action featuring higher derivative operators up to order
$2n$. Thus, we expect the resulting $f$ function to assume the form

$\bar{f}_{n}(z)=1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{\bar{d}_{n}^{2}(z)}+\frac{\Omega_{2}}{\bar{d}_{n}^{4}(z)}+\cdots+\frac{\Omega_{n}}{\bar{d}_{n}^{2n}(z)}\right)\
,$ (21)

with

$\displaystyle\bar{d}_{n}(z)=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|\bar{f}_{n}(z^{\prime})\right|}}\,.$ (22)

To be able to compute this quantity in an iterative manner we approximate it
via

$\displaystyle{d}_{n}(z)=\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|{f}_{n}(z^{\prime})\right|}}\equiv\int_{0}^{z}\frac{\differential
z^{\prime}}{\sqrt{\left|1-\frac{2\chi}{z^{\prime}}\left(1+\frac{\Omega_{1}}{d_{n-1}^{2}(z\prime)}+\frac{\Omega_{2}}{d_{n-1}^{4}(z\prime)}+\cdots+\frac{\Omega_{n}}{d_{n-1}^{2n}(z\prime)}\right)\right|}}\,.$
(23)

where $d_{n-1}$ is the quantum corrected proper distance at order $n-1$.

Even for $n>1$, there are two limits where the full behavior of $d_{n}$ as
function of $z$ can be studied:

  * 1.

asymptotically large distance: Far away from the black hole $d_{n}$ approaches
$z$. This limiting behaviour is crucial for the self-consistency of our
approach: indeed, it is required to match the effective coefficients
$\Omega_{n}$ to specific predictions from a given underlying quantum gravity
theory. The universality of this limit

$\displaystyle\lim_{z\to\infty}d_{n}(z)=\lim_{z\to\infty}d_{0}(z)\,,$
$\displaystyle\forall n\geq 0\,,$ (24)

ensures that the coefficients $\Omega_{i\leq n}$ can be defined in a
consistent fashion, independent of the order $n$.

  * 2.

distances close to the center of the BH: Here the the dominant term in the
integrand of (23) is the last term in the denominator. This allows us to
deduce, up to a multiplicative number, the following relation

$\lim_{z\to 0}\frac{d_{n}}{d_{0}}\sim\lim_{z\to
0}\frac{d^{n}_{n-1}}{\sqrt{\absolutevalue{\Omega_{n}}}}\ ,$ (25)

with $d_{0}$ computed near the origin of the BH and therefore given by the
first term in (12). Iteratively, this relation suggests

$\displaystyle\lim_{z\to 0}d_{n}\sim\lim_{z\to
0}\frac{(d_{0})^{e\,\Gamma(n+1,1)}}{\sqrt{\prod_{i=1}^{n}|\Omega_{i}|^{n!/i!}}}\,,$
(26)

where $\Gamma(n+1,1)$ is the incomplete Gamma function (and we have implicitly
assumed that all $\Omega_{i=1,\ldots,n}\neq 0$). This implies that near the
origin, at each given order in $n$, the truncated physical quantum distance
approaches zero extremely fast.

Now, we consider the explicit case of $n=2$ to learn how it affects our
previous results ∥∥∥ More details on the anaylsis can be found in Appendix
A.3.

$\displaystyle f_{2}(z)$
$\displaystyle=1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{d_{1}^{2}(z)}+\frac{\Omega_{2}}{d_{1}^{4}(z)}\right)\
.$ (27)

Using (25) we have that

$\lim_{z\to 0}{d_{1}}\sim\lim_{z\to
0}\frac{d_{0}^{2}}{\sqrt{\absolutevalue{\Omega_{1}}}}\ ,\qquad{\rm
and}\qquad\lim_{z\to 0}{d_{2}}\sim\lim_{z\to
0}\frac{d_{0}^{5}}{{\absolutevalue{\Omega_{2}}^{\frac{1}{2}}}\absolutevalue{\Omega_{1}}}\
.$ (28)

Therefore, near the origin the corrections stemming from $d_{2}$ do not affect
$f_{2}$ since they enter at higher orders.****** These limits are also
recovered in a slightly different fashion in Appendix A.3. In Fig. 7 we
compare the quantum function $f_{2}$ with the improved $\bar{f}_{1}$, for
different values of $\Omega_{1}=\pm 1$ and $\Omega_{2}=-0.5,0.2,0.5$. Of
course, when comparing with $\bar{f}_{1}$ we use the same values of
$\Omega_{1}$ appearing in $f_{2}$.

Figure 7: In top panel $a)$ we present $f_{2}$ for $\Omega_{1}=1$ and three
values of $\Omega_{2}$ as shown in the legend on the top, with the improved
function $\bar{f}_{1}$ displayed as a solid orange line. In the bottom panel
$b)$ is as in $a)$ but with $\Omega_{1}=-1$. In both plots we have set the BH
mass relative to the Planck one to $\chi=10$.

We observe that, for both signs of $\Omega_{1}$, the external horizon
expansion works extremely well yielding only minor corrections stemming from
$f_{2}$ when keeping $\Omega_{2}$ of order unity. This result is confirmed by
the analytic expression of the external horizon in the expansion in the
inverse BH mass which reads, for either positive or negative values of
$\Omega_{1,2}$:

$z_{+}=2\chi\left[1+\frac{\Omega_{1}}{\pi^{2}\chi^{2}}+{\bar{a}_{3}}{\left(\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\right)^{3/2}}+{\bar{a}_{4}}{\left(\frac{\Omega_{1}}{\pi^{2}\chi^{2}}\right)^{2}}+\frac{\Omega_{2}}{\pi^{4}\chi^{4}}+{\cal{O}}{\left(\frac{|\Omega_{1}|^{5/2},\,|\Omega_{1}|^{1/2}\Omega_{2}}{\pi^{5}\chi^{5}}\right)}\right]\
.$ (29)

The leading $f_{2}$ corrections appear at the order $\Omega_{2}/(\pi\chi)^{4}$
while the corrections stemming from the quantum corrected geodesic $d_{1}$
appear at the order $(\Omega_{1}/(\pi^{2}\chi^{2}))^{3/2}$ which is one order
less in $1/\pi\chi$, consistently with this expansion. The coefficient
$\bar{a}_{3}/4$ of $(\Omega_{1}/(\pi^{2}\chi^{2}))^{3/2}$ is numerically
evaluated in Appendix A.2 (see eq. (46) for $\Omega_{1}>0$ and eq. (50) for
$\Omega_{1}<0$).

Figure 8: Phase diagram defining the regions in the $(\Omega_{1},\Omega_{2})$
plane with different numbers of internal horizons emerging from the quantum
function $f_{2}$ for $\chi=5$ (panel $a$) and $\chi=10$ (panel $b$).

The situation changes for the interior of the BH: as shown in Fig. 7, the
structure of zeroes of the function $f_{2}$ depends crucially on the sign and
magnitude of $\Omega_{2}$ (and $\Omega_{1}$). The BH geometry can therefore
have zero, one or two internal horizons. Concretely, the phase diagram,
representing the regions in the $(\Omega_{1},\Omega_{2})$ plane featuring
different numbers of internal horizons stemming from the $f_{2}$ function, is
shown in Fig. 8 for $\chi=5$ in the left panel $a)$ and $\chi=10$ in the right
panel $b)$. One observes that the majority of the phase diagram features
either none (upper pink region) or at most one (lower orange region) internal
horizon. The light blue region supports two internal horizons.

##  4 Thermodynamics

So far we investigated the static properties of the BH in the effective
quantum regime. We now move to determine its leading quantum thermodynamic
properties.

We start by computing the Hawking’s [36] equilibrium temperature around the
external horizon via its surface gravity parameter $\kappa$. For the present
BH metric the temperature is given in terms of the $f$ function, by

$T_{\mathrm{H}}=\frac{\kappa}{2\pi}=\frac{1}{4\pi}\left.\derivative{f(r)}{r}\right|_{r=r_{+}}\,,$
(30)

where $r_{+}$ is the radius of the external event horizon. Specializing this
expression to the case in (13) with $r_{+}=\ell_{\rm P}z_{+}$, we have

$T_{\mathrm{H}}=\frac{1}{4\pi\ell_{\mathrm{P}}}\left[\frac{2\chi}{z_{+}^{2}}\left(1+\frac{\Omega_{1}}{d_{0}(z_{+})^{2}}\right)+\frac{4\chi}{z_{+}}\frac{\Omega_{1}}{d_{0}(z_{+})^{3}}\left.\derivative{d_{0}}{z}\right|_{z=z_{+}}\right]\,.$
(31)

The prefactor

$T_{\mathrm{P}}^{(0)}=\frac{1}{4\pi\ell_{\mathrm{P}}}\,,$ (32)

can be naturally interpreted as the Hawking temperature for a classical
Schwarzschild BH of Planck mass. From now on, we shall therefore work with the
normalized ratio $T_{\mathrm{H}}/T_{\mathrm{P}}^{(0)}$.

As expected, for large enough values of $\chi$, the temperature in eq. (31)
tends to the _semiclassical_ one, which is defined as

$\frac{T_{\mathrm{H}}^{(0)}}{T_{\mathrm{P}}^{(0)}}=\frac{1}{2\chi}\,.$ (33)

This is shown in Fig. 9. The consistent quantum expansion of eq. (31) reads:

$\frac{T_{\mathrm{H}}}{T_{\mathrm{P}}^{(0)}}=\begin{cases}\displaystyle\frac{1}{2\chi}\left[1-\frac{4}{\pi^{2}}\frac{\sqrt{|\Omega_{1}|}}{\chi}+\left(1-\frac{48}{\pi^{2}}\right)\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}+\mathcal{O}\left(\left(\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\right)^{3/2}\right)\right]&\Omega_{1}<0\
,\\\ \\\
\displaystyle\frac{1}{2\chi}\left[1+\frac{4}{\pi^{2}}\frac{\sqrt{|\Omega_{1}|}}{\chi}-\left(1+\frac{48}{\pi^{2}}\right)\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}+\mathcal{O}\left(\left(\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\right)^{3/2}\right)\right]&\Omega_{1}>0\
.\end{cases}$ (34)

At the quantum level the corrected BH temperature decreases or increases,
depending on whether the first order corrections to the metric are negative or
positive. Additionally, in contrast to the classical case the decrease
(increase) to the external horizon due to the quantum corrections leads to a
decrease (increase) in the associated quantum temperature.

Although our counting scheme for quantum corrections limits the validity of
our analysis to the results above it is interesting to discuss the small
$\chi$ limit for negative $\Omega_{1}$. We consider still the leading order
correction to the metric (_i.e._ we consider $n=1$ in eq. (9)), which we
approximate as (12). In this case, the quantum BH temperature achieves a
maximum around $\chi\sim 2$ and rapidly decreases to zero for smaller $\chi$.
At this point the internal and external horizons merge and the BH becomes
extremal (18). If this picture holds for the full quantum result this would
suggest that the evaporation process leaves behind a stable remnant rather
than observing a complete evaporation of the BH. We hasten to add, however,
that higher order corrections cannot consistently be neglected in this regime
and may qualitatively change the picture.

Figure 9: We plot the value of the Hawking temperature ratio
$T_{\mathrm{H}}/T_{\mathrm{P}}^{(0)}$ as a function of $\chi$ in the case for
$\Omega_{1}=-1$ (left) and $\Omega_{1}=1$ (right). It is noticeable that the
temperature drops to zero at the extremal value of the mass while the _(semi)_
classical result continues evaporating.

We finally turn our attention to the BH entropy $S$ which is obtained by
integrating the first law of BH thermodynamics

$\differential M=T_{\mathrm{H}}\differential S\implies\differential
S=\frac{\differential
M}{T_{\mathrm{H}}}=M_{\mathrm{P}}\frac{\differential\chi}{T_{\mathrm{H}}}\,.$
(35)

The temperature depends on the mass ratio $\chi$ once $\Omega_{1}$ is fixed
and therefore we have

$S=M_{\mathrm{P}}\int\frac{\differential\chi}{T_{\mathrm{H}}(\chi)}\,.$ (36)

Inserting the mass expansion of the temperature in eq. (34), we have that the
entropy assumes, up to a reference value, the following form

$S=\begin{cases}\displaystyle
4\pi\chi^{2}\left[1+\frac{8}{\pi^{2}}\frac{\sqrt{|\Omega_{1}|}}{\chi}-4\left(1-\frac{64}{\pi^{2}}\right)\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\log\chi+\mathcal{O}\left(\left(\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\right)^{3/2}\right)\right]&\Omega_{1}<0\
,\\\ \\\ \displaystyle
4\pi\chi^{2}\left[1-\frac{8}{\pi^{2}}\frac{\sqrt{|\Omega_{1}|}}{\chi}+4\left(1+\frac{64}{\pi^{2}}\right)\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\log\chi+\mathcal{O}\left(\left(\frac{|\Omega_{1}|}{\pi^{2}\chi^{2}}\right)^{3/2}\right)\right]&\Omega_{1}>0\
.\end{cases}$ (37)

Thus, the entropy increases (decreases) for negative (positive) values of
$\Omega_{1}$. The leading quantum correction is linear in $\chi$ while only
the subleading quantum correction receives corrections in the logarithm of the
mass. In fact, the latter vanish at the Planck mass. Logarithmic corrections
to the entropy of quantum BHs have also been found by using various other
methods, see for example [37, 38, 39].

##  5 Conclusions

We investigated the quantum nature of black holes by employing an effective
approach able to describe quantum deviations from the classical results. We
have studied in detail the Schwarzschild black hole, which is the simplest
example in four dimensions, however, our approach can readily be extended to
other types of geometries (which we plan to discuss in the future). Here we
notably assumed that no other physical quantum gravity scale emerges besides
the Planck one.

Upon setting up the framework we determined the quantum corrections to the
event horizon structure. To leading order in the quantum corrections, and
depending on their sign, we showed that the black hole can either have a
single horizon or develop a second internal one. We tested the robustness of
our results by further considering both the backreaction on the quantum proper
distance as well as the effects of higher order corrections. In this way we
have demonstrated that the quantum corrections can be consistently organised
into expansions dictated by inverse powers of the mass. We also note that
these results do not hinge on a particular model of quantum gravity: indeed,
different models simply provide specific values of the coefficients
$\Omega_{n}$ for the quantum corrections. For example, according to the
radiative computations in [40], one would have
$\Omega_{1}=-167/(30\pi)\sim-1.77$ and therefore the physics is the one
stemming from a negative value of $\Omega_{1}$. Similarly one could match the
coefficients to the prediction for the quantum metric stemming from different
quantum gravity actions featuring, for example, higher curvature terms such as
$f(R)$ theories [41, 42, 43]. For a recent discussion about potential
constraints on some of the $\Omega_{n}$ coefficients see [44]. Alternatively,
in the future, some of these coefficients could be experimentally determined.
To further test the robustness of our quantum framework we have shown how to
take into account higher order corrections to the metric. We have even
provided the explicit form of the next to next leading order quantum
corrections to the external horizon. The results demonstrate the effectiveness
and reliability of the expansion. We have also observed that the fate of the
internal horizon, for $\Omega_{1}$ negative, is sensitive to higher order
corrections. We have consequently provided, to the next-to-next leading
quantum order, the parameter space diagram illustrating the various scenarios
for the internal horizon. Because of the nature of the expansion in (9) one
can only address the ultimate fate of the singularity at the origin of the
black hole within a specific model of quantum gravity which would allow to
resum the entire series nonperturbatively. We also provided the conformal
diagrams for the quantum corrected black holes and determined the impact of
the quantum corrections on the thermodynamic properties such as temperature
and entropy.

Our approach differs from the renormalization group improvement of a black
hole space time in which the Newton constant is upgraded to an effective
running coupling [14, 15]. Within this latter framework it has been recently
shown [33] that the renormalization group improvement at the level of the
metric is coordinate-dependent while the approach is applicable at the level
of curvature invariants. Although in our framework we still work at the level
of the black hole metric our quantum modified metric depending only on
physical quantities leads to coordinate independent observables.

If the quantum scale for gravity turns out to be lower than the Planck scale,
our framework can take this into account by a simple rescaling of the
dimensionless proper distance used in the definition of the quantum $f$
function of (6). A smaller quantum gravity scale can lead to sizable
phenomenological effects.

Overall, we have showed that quantum corrections to black hole physics can be
organised in a powerful expansion in their mass that, already at the leading
order, allows us to explore the quantum nature of black holes at distances
that can be as close to the origin of the black hole, as few times the Planck
length. The framework can be employed to investigate quantum corrections for
other extended gravitational objects.

### Acknowledgments

We thank Matthias Blau, Alfio Bonanno, Cliff Burgess, Salvatore Capozziello,
John Donoghue, Domenico Orlando, Roberto Percacci, Alessia Platania and Manuel
Reichert for comments and/or enlightening discussions.

##  Appendix A Quantum Corrected Horizons of the Schwarzschild Metric

###  A.1 Range of Validity of the Quantum Corrected Horizon Position

We provide a numerical estimate of the range of validity of the approximation
(14) for the position of the (external) horizon calculated as the zeroes of
the $f$ function (12). For simplicity (and since these computations only act
as an order of magnitude estimate), we limit ourselves to the case
$\Omega_{1}$. Specifically, we consider the zero (14) as an entire series of
the form

$\displaystyle z_{+}=2\chi\sum_{n=0}^{\infty}a_{n}\,\alpha^{n/2}\,,$ with
$\displaystyle\begin{array}[]{l}a_{0}=1\,,\\\ a_{1}=0\,,\\\
a_{2}=1\,.\end{array}$ (41)

An estimate for the range of validity of the result (14) can be obtained by
the radius of convergence of this series. Indeed, in Fig. 10 we have plotted
(norm) of the ratio $|a_{n}/a_{n+1}|$ which asymptotically approaches to the
radius of convergence $r\sim 0.1968\pm 0.0004$. This suggests, that the result
(14) can be trusted for BH masses

$\displaystyle\chi\geq\frac{\sqrt{\Omega_{1}}}{\pi\sqrt{r}}\sim
0.718\,\sqrt{\Omega_{1}}\,.$ (42)

Figure 10: Absolute value of the quotient $|a_{n}/a_{n+1}|$ of the
coefficients in eq. (41). The black line interpolates the coefficients by a
function of the form $\frac{a}{\alpha^{b}}+c$ for $a=0.475\pm 0.003$,
$b=1.279\pm 0.008$ and $c=0.1968\pm 0.0004$.

###  A.2 Numerical Calculation of the Position of the Horizon(s)

Here we perform a numerical analysis of the zeroes of the quantum self
improved metric function (20) with $d_{1}$ defined in (19). We are in
particular interested in the order in the parameter $\alpha$ (defined in eq.
(14)) at which this modification changes the position of the horizon(s) of the
black hole compared to $f_{1}$ defined in (13). To this end, we distinguish
the cases $\Omega_{1}>0$ and $\Omega_{1}<0$

  * •

$\Omega_{1}>0$: In this case, the zeroes of $\bar{f}_{1}$ as a function of
$\alpha$ (for $\chi=2$) are plotted in Figure 11 for the choice $\chi=2$. The
solid black line is approximated by

Figure 11: Numerical position of the position of the horizon $z_{+}$ as a
function of $\alpha$ for $\chi=2$ and $\Omega_{1}>0$.

$\displaystyle
z_{+}=4(\bar{a}_{1}+\bar{a}_{2}\,\alpha+\bar{a}_{3}\,\alpha^{3/2})\,.$ with
$\displaystyle\begin{array}[]{l}4\bar{a}_{1}=4\pm 2.2*10^{-9}\,,\\\
4\bar{a}_{2}=4.0036\pm 0.007\,,\\\ 4\bar{a}_{3}=7.853\pm 2.523\,.\end{array}$
(46)

The numerical results for $\bar{a}_{1}$ and $\bar{a}_{2}$ are compatible with
the coefficients obtained for the zeroes of $f_{1}$, while the coefficient
$\bar{a}_{3}$ is not (namely $a_{1}=1$, $a_{2}=1$ and
$a_{3}=-\frac{2}{\pi}\sim-0.637$). This suggests, that the corrections to the
position of the horizon $z_{+}$ that stem from using the improved metric
function $\bar{f}_{1}$ are of order $\mathcal{O}(\alpha^{3/2})$.

  * •

$\Omega_{1}<0$: In this case, the two zeroes of $\bar{f}_{1}$ as a function of
$\alpha$ (for $\chi=2$) are plotted in Figure 12 for $\chi=2$. The solid black
line is approximated by

Figure 12: Numerical position of the position of the horizon $z_{+}$ (left
panel) and $z_{-}$ (right panel) as a function of $\alpha$ for $\chi=2$ and
$\Omega_{1}<0$.

$\displaystyle
z_{+}=4(\bar{a}_{1}+\bar{a}_{2}\,\alpha+\bar{a}_{3}\,\alpha^{3/2})\,,$ with
$\displaystyle\begin{array}[]{l}4\bar{a}_{1}=4\pm 4.8*10^{-7}\,,\\\
4\bar{a}_{2}=-3.979\pm 0.045\,,\\\ 4\bar{a}_{3}=9.947\pm 2.685\,,\end{array}$
(50) $\displaystyle
z_{-}=\bar{b}_{1}\,\alpha^{\frac{1}{3}}+\bar{b}_{2}\,\alpha^{\frac{2}{3}}+\bar{b}_{3}\,\alpha+\bar{b}_{4}\,\alpha^{\frac{4}{3}}\,,$
with $\displaystyle\begin{array}[]{l}\bar{b}_{1}=7.0827\pm 0.00005\,,\\\
\bar{b}_{2}=1.6724\pm 0.0114\,.\\\ \bar{b}_{3}=2.26\pm 0.60\,,\\\
\bar{b}_{4}=4.45\pm 9.08\,.\end{array}$ (55)

For the position of the outer horizon $z_{+}$, the situation is similar to the
case $\Omega_{1}>0$: the coefficients $\bar{a}_{1,2}$ are numerically
compatible with the values $a_{1,2}$ coming from $f_{1}$, while $\bar{a}_{3}$
is not. This suggests, that the corrections to the position of the horizon
$z_{+}$ that stem from using the improved metric function $\bar{f}_{1}$ are of
order $\mathcal{O}(\alpha^{3/2})$. For $z_{-}$ the coefficients
$\bar{b}_{1,2,3}$ are numerically compatible with the values obtained from
$f_{1}$. In view of the large numerical uncertainty, this suggests that the
corrections to the position of the inner horizon appear at an order larger
than $\mathcal{O}(\alpha)$.

###  A.3 Second Order Quantum Corrections

Before treating the second order quantum corrections, we first provide a
relation between the proper distances $d_{1}$ and $d_{0}$: we start from the
definition of $d_{1}$ in (19), however, instead of a function of $z$ we
consider it as a function of $d_{0}$ defined in (11).††††††This is possible
since $d_{0}$ is a monotonic function in $z$. We then find for the first
derivative

$\displaystyle\frac{\differential d_{1}}{\differential z}=\frac{\differential
d_{1}}{\differential d_{0}}\,\frac{\differential d_{0}}{\differential
z}=\frac{1}{\sqrt{\left|1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{d_{0}^{2}}\right)\right|}}\,.$
(56)

Using (10) we therefore find

$\displaystyle\frac{\differential d_{1}}{\differential
d_{0}}=\sqrt{\frac{\left|1-\frac{2\chi}{z}\right|}{\left|1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{d_{0}^{2}}\right)\right|}}$
(57)

where implicitly $z$ is understood as a function of $d_{0}$. We can consider
two limits of this equation

  * •

$\chi\ll z$ (or equivalently $\chi\ll d_{0}$): in this case, the equation (57)
becomes

$\displaystyle\frac{\differential d_{1}}{\differential d_{0}}=1\,,$ (58)

which has as solution $d_{1}\sim d_{0}$, _i.e._ for distances far away from
the BH the two distances become equivalent.

  * •

$\chi\gg z$ (or equivalently $\chi\gg d_{0}$): in this case, the equation (57)
becomes

$\displaystyle\frac{\differential d_{1}}{\differential
d_{0}}=\sqrt{\frac{1}{\left|1+\frac{\Omega_{1}}{d_{0}^{2}}\right|}}\,,$ (59)

which is a differential equation for $d_{1}$ and can be integrated up in a
direct fashion‡‡‡‡‡‡Requiring $d_{1}$ to be $u$ independent implies also
independence of $\Omega_{1}$ on $u$.

$\displaystyle
d_{1}=\left\\{\begin{array}[]{lcl}d_{0}\sqrt{1+\frac{\Omega_{1}}{d_{0}^{2}}}-\sqrt{\Omega_{1}}&\text{if}&\Omega_{1}>0\,,\\\
\sqrt{-\Omega_{1}}-d_{0}\sqrt{-\frac{\Omega_{1}}{d_{0}^{2}}-1}&\text{if}&\Omega_{1}<0\text{
and }d_{0}<\sqrt{-\Omega_{1}})\,,\\\
\sqrt{-\Omega_{1}}+d_{0}\sqrt{1+\frac{\Omega_{1}}{d_{0}^{2}}}&\text{if}&\Omega_{1}<0\text{
and }d_{0}>\sqrt{-\Omega_{1}})\,.\end{array}\right.$ (63)

Here the integration constants have been chosen in such a way that $d_{1}$ is
a continuous function of $d_{0}$ and $\lim_{d_{0}\to 0}d_{1}=0$. Graphically,
the solutions are shown in Figure 13. For small values of $d_{0}$ we find

Figure 13: Distance $d_{1}$ in eq. (63) as a function of $d_{0}$ (left panel)
and of $z$ assuming that $\chi=20$ (right panel). The dashed black line
corresponds to $d_{0}$ for comparison.

$\displaystyle
d_{1}\sim\frac{d_{0}^{2}}{2\sqrt{|\Omega_{1}|}}+\mathcal{O}(d_{0}^{4})\,.$
(64)

We can now turn to the second order quantum corrections. To this end, we
generalise the differential equation (57) for $d_{2}$ (as a function of
$d_{1}$)

$\displaystyle\frac{\differential d_{2}}{\differential
d_{1}}=\sqrt{\frac{\left|1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{d_{0}^{2}}\right)\right|}{\left|1-\frac{2\chi}{z}\left(1+\frac{\Omega_{1}}{d_{1}^{2}}+\frac{\Omega_{2}}{d_{1}^{4}}\right)\right|}}\,.$
(65)

As before, we can consider two limits of this equation

  * •

$\chi\ll z$ (or equivalently $\chi\ll d_{1}$): in this case, the equation (65)
becomes

$\displaystyle\frac{\differential d_{2}}{\differential d_{1}}=1\,,$ (66)

which has as solution $d_{2}\sim d_{1}$, _i.e._ for distances far away from
the BH the two distances become equivalent.

  * •

$\chi\gg z$ (or equivalently $\chi\gg d_{1}$): in this case, the equation (65)
becomes

$\displaystyle\frac{\differential d_{2}}{\differential
d_{1}}=\sqrt{\frac{\left|1+\frac{\Omega_{1}}{d_{0}^{2}}\right|}{\left|1+\frac{\Omega_{1}}{d_{1}^{2}}+\frac{\Omega_{2}}{d_{1}^{4}}\right|}}\,.$
(67)

which together with (63) is a differential equation for $d_{2}$ and can in
principle be integrated as a function for $d_{1}$. Since this is, however,
technically difficult, we focus on expanding $d_{2}$ around $d_{1}=0$ and for
simplicity focus on the case $\Omega_{1,2}>0$******Other cases can be analysed
in a similar fashion:

$\displaystyle\frac{\differential d_{2}}{\differential
d_{1}}=\sqrt{\frac{1+\frac{\Omega_{1}}{d_{1}(d_{1}+2\sqrt{\Omega_{1}})}}{1+\frac{\Omega_{1}}{d_{1}^{2}}+\frac{\Omega_{2}}{d_{1}^{4}}}}\,.$
(68)

Expanding for small $d_{1}$ (or equivalently small $d_{0}$), we find

$\displaystyle
d_{2}\sim\frac{\sqrt{2}}{5}\,\frac{\Omega_{1}^{1/4}}{\sqrt{\Omega_{2}}}\,d_{1}^{5/2}+\mathcal{O}(d_{1}^{7/2})\sim\frac{d_{0}^{5}}{20\,\Omega_{1}\,\sqrt{\Omega_{2}}}+\mathcal{O}(d_{0}^{7})\,,$
(69)

which is indeed compatible with (28).

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# Process Comparison Using Object-Centric Process Cubes

Anahita Farhang Ghahfarokhi  Alessandro Berti  Wil M.P. van der Aalst Anahita
Farhang Ghahfarokhi, Alessandro Berti, Wil M.P. van der Aalst are with the
Process and Data Science institute, RWTH Aachen University, Germany (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>aachen.de).

###### Abstract

Process mining provides ways to analyze business processes. Common process
mining techniques consider the process as a whole. However, in real-life
business processes, different behaviors exist that make the overall process
too complex to interpret. Process comparison is a branch of process mining
that isolates different behaviors of the process from each other by using
process cubes. Process cubes organize event data using different dimensions.
Each cell contains a set of events that can be used as an input to apply
process mining techniques. Existing work on process cubes assume single case
notions. However, in real processes, several case notions (e.g., order, item,
package, etc.) are intertwined. Object-centric process mining is a new branch
of process mining addressing multiple case notions in a process. To make a
bridge between object-centric process mining and process comparison, we
propose a process cube framework, which supports process cube operations such
as slice and dice on object-centric event logs. To facilitate the comparison,
the framework is integrated with several object-centric process discovery
approaches.

Process mining, multidimensional process mining, multi-perspective business
processes, OLAP, process cubes, process discovery.

##  1 Introduction

Every organization has to manage business processes such as the Purchase-to-
Pay (P2P) and Order-to-Cash (O2C) processes. To manage real-life processes we
need to consider two challenges. First, the nature of the business processes
is not static due to the environmental changes (e.g., seasonal demands,
changing in customer preferences). Thus, different behaviors may exist in
business processes [1]. We can compare different process behaviors using
process cubes. Process cubes are inspired by the notion of OLAP [2]. Similar
operations to OLAP are defined for process cubes, i.e., slice, dice, drill-
down, and roll-up [3].

Second, multiple objects interact with each other in the business processes
[4]. Considering simple P2P and O2C processes, multiple objects, e.g., order,
item, and package are involved. Moreover, different relations exist between
these objects. For example, an order contains multiple items and multiple
items are packed in one package for the delivery. Each of these objects can be
considered as a case notion (i.e., a process instance). Therefore, in real
business processes, extracted from ERP systems, we have multiple case notions.
However, like most existing process mining techniques, current process cube
approaches can only handle one case notion at a time [1, 5, 6]. Thus, they
cannot cover multiple case notions involved in business processes such as
orders and items. The interaction between multiple case notions can be
analyzed using object-centric process mining [4]. This emerging subfield of
process mining provides a more general vision of business processes by
considering multiple case notions in processes.

In this paper, we address the problem of handling multiple interacting case
notions in process cubes. In order to fit object-centric event logs into
process cubes, some challenges arise due to the nature of object-centric event
logs, where we need to consider multiple case notions. In traditional event
logs, each event refers to a single case notion, an activity, a timestamp, and
some possible additional attributes (e.g., location, cost, etc.). However, in
object-centric event logs, each event may refer to multiple case notions. For
example, suppose we have an event with a confirm order, related to one order
but multiple items. In fact, there is a one-to-many relationship between order
and item. Then, each event is related to one order and many items, i.e.,
convergence [4]. Considering slicing based on item dimension, the values of
item dimension are not atomic. However, in the existing process cubes [1, 5,
7, 8], we need atomic values to apply each operation.

In this paper, we extend the notion of process cubes to be applicable to non-
atomic values and provide an open-source implementation of our object-centric
process cube approach. Moreover, to facilitate the comparison, we use object-
centric process discovery approaches. We can extract process models of the
cells of the process cubes and by comparing them against each other, we are
able to do performance analysis from different angles.

Figure 1: Schematic overview of the proposed approach. In the proposed
framework, we feed the process cube with object-centric event logs

A schema of the proposed approach is shown in Fig. 1. In the first step, we
have operational and business processes running in the real world. The
information of these business processes is recorded in information systems
such as SAP or ERP systems. We can extract object-centric event logs, i.e.,
event logs with multiple case notions from data stored in information systems.
Then, we use the object-centric event log as an input for the process cube. In
the process cube, different operations such as slicing and dicing are
applicable. Through the application of different operations, we obtain object-
centric sublogs. We can apply the object-centric process discovery methods on
the object-centric sublogs to have the model of the process. Using such
object-centric sublogs and the discovered models, we can analyze the process
and find the pitfalls of the process.

The structure of the paper is as follows. In Section 2, we discuss related
work. In Section 3, we describe the running example that is used throughout
the paper. In Section 4, we extend the definitions of the process cube to
support object-centric event logs. Section 5 provides our implementation of
the proposed framework. In Section 6, we provide performance related results
of our framework. Finally, Section 7 concludes the paper and discusses the
future work.

##  2 Related Work

In this section, first, we present the work related to object-centric process
mining. Then, we discuss the developed approaches on process comparison.

One of the approaches developed to model the processes with multiple case
notions is artifact-centric modeling. Artifacts combine process and data as
the building blocks [9]. In [10], the authors formulate artifact-based
business models and develop an initial framework that enables the automated
construction of the processes. In [11], the authors introduce an artifact-
centric process model, showing business objects and their life-cycles. The
proposed techniques do not show the whole process in one diagram, which leads
to losing a general vision over the process. Object-Centric Behavioral
Constraint (OCBC) models show the whole process in one diagram and incorporate
data perspective in the process model [12]. The main challenge for OCBC models
is complexity, which leads to the development of MVP (Multiple Viewpoint)
Models [13]. MVP models are graphs, annotated by frequency and performance
information, where relationships between activities are shown by colored arcs.
Object-centric Petri nets are another type of object-centric process models
that can be extracted from object-centric event logs which provide the
execution semantics [14]. In this paper, comparison between the MVP models and
Object-centric Petri nets extracted from cube operations against the whole
cube is possible, which is helpful in process analysis.

Process cubes are inspired by OLAP, where event data are organized using
dimensions. Each cell contains events, which can be used as an input to apply
process mining techniques such as process discovery. In [2], the event cube is
introduced with the application of OLAP operations on event data. The first
notion for process cubes was proposed in [6] and then enhanced in [1]. In [1]
an approach for interactive analysis of the event data is also proposed.
Process cubes were used for analysis in several case studies for different
purposes [7, 5, 15]. Although, none of them addresses handling object-centric
event logs.

##  3 Running Example

The example process is from the publicly available SAP IDES system which
belongs to a real purchasing process (available in https://gitlab.com/Anahita-
Farhang/process-cube). It contains 17500 events, 4 object types (i.e., case
notions), 4 attributes, and 10 number of activities recorded from 2000 until
2020. A fragment of a the object-centric event log is shown in Table 1. There
is one-to-many relationship between order and item and one-to-one relationship
between order and invoice. A part of the MVP model of the process is shown in
Fig. 2. MVPs are graphs in which nodes represent activities and there is an
edge between two nodes if there is at least one event in the event log where
the source activity is followed by the target activity. In the MVP models,
arcs with different colors represent different case notions [4]. Following the
blue and purple arcs, the order and item go through create purchase order, and
enter incoming invoice and a few of them leave the process at the _cancel
invoice document_.

Table 1:
A Fragment of an Object-Centric Event Log

event | attributes | object types
---|---|---
identifier | activity | timestamp | resource | order | item | invoice
0001 | _create purchase order_ | 01-01-2000:00.00 | USER01 | $\left\\{o_{1}\right\\}$ | $\left\\{i_{1},i_{2},i_{3}\right\\}$ | $\left\\{\right\\}$
0002 | _post document_ | 01-01-2000:08.05 | USER01 | $\left\\{\right\\}$ | $\left\\{\right\\}$ | $\left\\{inv_{1}\right\\}$
… | … | … | … | … | … | …
17499 | _enter incoming invoice_ | 20-05-2020:15.11 | USER50 | $\left\\{o_{12000}\right\\}$ | $\left\\{i_{27005}\right\\}$ | $\left\\{inv_{800}\right\\}$
17500 | _park invoice_ | 20-05-2020:15.14 | USER42 | $\left\\{o_{12000}\right\\}$ | $\left\\{\right\\}$ | $\left\\{\right\\}$

We also use Object-centric Petri nets in this paper. These discovery
techniques focus on solitary processes. There may exist different behaviors in
the processes. The variability in this purchasing process motivates us to
extend the process cubes notion to support object-centric event logs to
compare different processes.

Figure 2: MVP model for a fragment of the process

##  4 Object-Centric Process Cubes

In this section, we formalize the notion of object-centric process cubes.
Using the running example, described in Section 3, we provide examples for the
object-centric process cube notion. A process cube is formed by its structure
and an object-centric event log. The structure describes the distribution of
the cells and the object-centric event log is used to materialize the cells of
cube by events.

###  4.1 Object-Centric Event Log

The object-centric event log, shown in Table 1, represents a collection of
events and that is totally ordered based on the timestamp. Each event consists
of an event identifier, attributes, and object types (i.e., case notions). To
formalize the object-centric event log, we define the universes to be used
throughout the paper.

######  Definition 1 (Universes).

We define the following universes:

  * •

$\mathbb{U}_{ei}$ is the universe of event identifiers,

  * •

$\mathbb{U}_{att}$ is the universe of all possible attribute names,

  * •

$\mathbb{U}_{v}$ is the universe of all possible attribute values,

  * •

$\mathbb{U}_{vmap}=\mathbb{U}_{att}\nrightarrow\mathbb{U}_{v}$ is the universe
of functions mapping attributes on attribute values,

  * •

$\mathbb{U}_{ot}$ is the universe of all object types,

  * •

$\mathbb{U}_{oi}$ is the universe of object identifiers,

  * •

$\mathbb{U}_{s}=(\mathcal{P}(\mathbb{U}_{v})\cup\mathcal{P}(\mathbb{U}_{oi}))\setminus\left\\{\emptyset\right\\}$
is the universe of value sets excluding the set containing the empty set,

  * •

$\mathbb{U}_{h}=(\mathcal{P}(\mathcal{P}(\mathbb{U}_{v}))\cup\mathcal{P}(\mathcal{P}(\mathbb{U}_{oi})))\setminus\left\\{\emptyset,\left\\{\emptyset\right\\}\right\\}$
is the universe of value set collections excluding the set and the set
collection containing the empty set,

  * •

$type\in\mathbb{U}_{oi}\rightarrow\mathbb{U}_{ot}$ assigns precisely one type
to each object identifier,

  * •

$\mathbb{U}_{omap}$ is the universe of all the object mappings indicating
which object identifiers are included per type. $\mathbb{U}_{omap}$ is defined
like:

$\displaystyle\mathbb{U}_{omap}=\\{omap\in\mathbb{U}_{ot}\rightarrow\mathcal{P}(\mathbb{U}_{oi})~{}\mid~{}\forall_{ot\in\mathbb{U}_{ot}}~{}$
$\displaystyle\forall_{oi\in omap(ot)}~{}type(oi)=ot\\}$

  * •

$\mathbb{U}_{event}=\mathbb{U}_{ei}\times\mathbb{U}_{vmap}\times\mathbb{U}_{omap}$
is the universe of events.

$e=(ei,vmap,omap)\in\mathbb{U}_{event}$ is an event with event identifier
$ei$, referring to the objects specified in $omap$, and having attribute
values specified by $vmap$. Each row in the Table 1 refers to an event, which
contains an event-identifier, attribute values, and object identifiers. Note
that $\mathbb{U}_{s}$ does not include $\left\\{\emptyset\right\\}$ and
$\mathbb{U}_{h}$ does not include
$\left\\{\emptyset,\left\\{\emptyset\right\\}\right\\}$. These values are
created after the power set generation. However, it is not meaningful for
these sets to contain such values.

######  Definition 2 (Object-Centric Event Log).

An object-centric event log is a tuple
$E=(EI,ATT,OT,\pi_{vmap},\pi_{omap})\in\mathcal{P}(\mathbb{U}_{ei})\times\mathcal{P}(\mathbb{U}_{att})\times\mathcal{P}(\mathbb{U}_{ot})\times\mathcal{P}(\mathbb{U}_{vmap})\times\mathcal{P}(\mathbb{U}_{omap})$
where $\pi_{vmap}\in EI\rightarrow U_{vmap}$ is a function mapping each event
to its attribute mapping, and $\pi_{omap}\in EI\rightarrow U_{omap}$ is a
function mapping each event to its object mapping. Event identifiers are
unique and time is non-descending.

The first column in the object-centric event log of the running example shows
unique event identifiers. Consider $e_{1}$, the first event in Table 1.
$\pi_{vmap}(e_{1})(activity){=}\text{\emph{create purchase order}}$,
$\pi_{vmap}(e_{1})(resource){=}\text{\emph{USER01}}$,
$\pi_{omap}(e_{1})(order){=}\left\\{o_{1}\right\\}$, and
$\pi_{omap}(e_{1})(item){=}\left\\{i_{1},i_{2},i_{3}\right\\}$.

###  4.2 Process Cube Structure

We define the structure of the process cube independent from the object-
centric event log. A process cube structure is fully specified by the set of
dimensions.

######  Definition 3 (Process Cube Structure).

A process cube structure is a triplet $PCS=(D,val,gran)$ where:

  * •

$D$ is a set of dimensions,

  * •

$val\in D\rightarrow\mathbb{U}_{s}$ is a function associating a dimension to a
set of values.

  * •

$gran\in D\rightarrow\mathbb{U}_{h}$ defines a level for each dimension such
that for any $d\in D:val(d)=\cup gran(d)$.

A dimension $d$ has a value $val(d)$ and a granularity $gran(d)$. The possible
set of values for each dimension is specified by $val(d)$ and a subset of
these values exist in a sample of the process cube. For example,
$val(item){=}\left\\{i_{1},i_{2},i_{3},...,i_{27005}\right\\}$ and
$val(activity){=}\left\\{\text{\emph{create purchase
order}},...,\text{\emph{park invoice}}\right\\}$. A dimension also has a
granularity $gran(d)$ which is a set of sets. For example, $gran(timestamp)$
contains sets such as $T_{2017}$ and $T_{2018}$ each showing all the
timestamps in a particular year. These sets form levels based on set inclusion
(e.g., $T_{2019}$ dominates $T_{Apr-2019}$).

The content of the cube is the object-centric event log. Therefore, we make
the _process cube structure_ and the _object-centric event log_ compatible.

######  Definition 4 (Compatible).

Let $E=(E,ATT,OT,\pi_{vmap},\pi_{omap})$ be an object-centric event log and
$PCS=(D,val,gran)$ be a process cube structure. They are compatible if:

  * •

$D\subseteq OT\cup ATT$, dimensions should correspond to attributes or object
types,

  * •

for any $d\in D\cap ATT$ and $e\in E:\pi_{vmap}(e)(d)\in val(d)$,

  * •

for any $d\in D\cap OT$ and $e\in E:\pi_{omap}(e)(d)\subseteq val(d)$.

In making the process cube structure and the object-centric event log
compatible, there is a difference between dimensions that correspond to object
types and dimensions that correspond to attributes. This difference arises due
to the non-atomic values for object types. Consider the activity as the
dimension, which is an attribute, $activity{\in}D{\cap}ATT$, then
$\pi_{vmap}(e_{1})(activity){\in}\left\\{\text{\emph{create purchase
order}},...,\text{\emph{park invoice}}\right\\}$. However, if we consider item
as the dimension, which is an object type, $item{\in}D{\cap}OT$, then
$\pi_{omap}(e_{1})(item){\subseteq}\left\\{i_{1},...,i_{27005}\right\\}$.

Different operations are possible in process cubes. By applying process cube
operations such as slicing, the content of the process cube structure and
object-centric event log do not change. We only change the fragment of the
cube that we visualize. A process cube view defines the events that are
visible for us.

######  Definition 5 (Process Cube View).

Let $PCS=(D,val,gran)$ be a process cube structure. A process cube view is a
pair $PCV=(D_{sel},sel)$ such that:

  * •

$D_{sel}\subseteq D$ are the selected dimensions,

  * •

$sel\in D\rightarrow\mathbb{U}_{h}$ is a function selecting the part of the
level considered per dimension. The function $sel$ is such that for any $d\in
D$:

    * –

$sel(d)\subseteq gran(d)$

    * –

for any $S_{1},S_{2}\in sel(d):S_{1}\subseteq S_{2}$ implies $S_{1}=S_{2}$.

A process cube view is a cube with $|D_{sel}|{\leq}|D|$ dimensions. Fig. 3(a)
shows an example of a process cube view with three dimensions. The selected
dimensions are activity, item and timestamp
($D_{sel}{=}\left\\{item,timestamp,activity\right\\}$). Function $sel$ selects
values for all dimensions regardless of whether they are in $D_{sel}$ or not.
For the $D{\setminus}D_{sel}$ we cannot see the values of $sel$ in the
_process cube view_. However, the values of $sel$ exist for these dimensions
and these dimensions may have been used in filtering. For example, in slicing
the dimension is no longer visible but it is used in filtering. In the process
cube view shown in Fig. 3(a),
$sel(timestamp){=}\left\\{T_{2000},...,T_{2020}\right\\}$. Figure 3(b) shows
another view where
$sel(timestamp){=}\left\\{T_{Jan-2000},...,T_{June-2020}\right\\}$. Through
the requirement $S_{1}{\subseteq}S_{2}$ implies $S_{1}{=}S_{2}$, we ensure
that elements of $sel(d)$ do not intersect, e.g.,
$sel(timestamp){=}\left\\{T_{Jan-2017},T_{2017},T_{2018},T_{2019}\right\\}$ is
not possible. By having the process cube view and object-centric event log, we
materialize the cells by events. We can extract event logs from cells of the
process cube to apply process mining techniques such as process discovery.

(a) A process cube view with the granularity of year for the time dimension

(b) A process cube view with the granularity of month for the time dimension

Figure 3: Example of different process cube views of the same process cube
structure

######  Definition 6 (Materialized Process Cube View).

Let process cube structure $PCS=(D,val,gran)$ and object-centric event log
$E=(E,ATT,OT,\pi_{vmap},\pi_{omap})$ be compatible. The materialized process
cube for some view $PCV{=}(D_{sel},sel)$ of $PCS$ is
$M_{E,PCV}=\left\\{(c,events(c))~{}|~{}c\in cells\right\\}$ with
$cells=\left\\{c\in D_{sel}{\rightarrow}\mathbb{U}_{s}~{}|~{}\forall_{d\in
D_{sel}}c(d)\in sel(d)\right\\}$ being the cells of the cube and

$\displaystyle events(c)=$ $\displaystyle\\{e{\in}E~{}|~{}\forall_{d\in
D_{sel}\cap ATT}~{}\pi_{vmap}(e)(d){\in}c(d)$
$\displaystyle\wedge\forall_{d{\in}D\cap ATT}~{}\pi_{vmap}(e)(d){\in}\cup
sel(d)$ $\displaystyle\wedge\forall_{d{\in}D_{sel}\cap
OT}~{}c(d){\subseteq}\pi_{omap}(e)(d)$
$\displaystyle\wedge\forall_{d{\in}D\cap OT}~{}\pi_{omap}(e)(d){\subseteq}\cup
sel(d)\\}$

In materializing, we add content to the cells. In other words, we create an
event log for the cells in the process cube view. In a $c\in cells$, each
visible dimension is assigned to a value of that dimension, e.g.,
$c(activity){=}\text{\emph{create purchase order}}$,
$c(timestamp){=}T_{2020}$, and
$c(item){=}\left\\{i_{1},i_{2},...,i_{13502}\right\\}$. We materialize the
cells of the process cube with events. $events(c)$ are all the events
corresponding to the first requirement (i.e., $\forall_{d{\in}D_{sel}\cap
ATT}~{}\pi_{vmap}(e)(d){\in}c(d)$, $\forall_{d{\in}D_{sel}\cap
OT}~{}c(d){\subseteq}\pi_{omap}(e)(d)$) which is different for object types
and attributes. The second requirement (i.e., $\forall_{d{\in}D\cap
ATT}~{}\pi_{vmap}(e)(d){\in}\cup sel(d)$, $\forall_{d{\in}D\cap
OT}~{}\pi_{omap}(e)(d){\subseteq}\cup sel(d)$), which is also different for
object types and attributes, makes sure that events are not filtered out.
Events in the cell can be converted to an object-centric event log and used to
apply process mining techniques.

Using earlier formalizations, we define process cube operations such as
_slice_.

Figure 4: The process cube view after slicing based on dimension item Figure
5: The process cube view after dicing based on dimensions item and time

######  Definition 7 (Slice).

Let $PCS=(D,val,gran)$ be a process cube structure and $PCV=(D_{sel},sel)$ a
view of $PCS$. For any $d\in D_{sel}$ and $V\in
sel(d):slice_{d,V}(PCV)=(D_{sel}^{\prime},sel^{\prime})$ with
$D_{sel}^{\prime}=D_{sel}\setminus\left\\{d\right\\},sel^{\prime}(d)=\left\\{V\right\\}$
and $sel^{\prime}(d^{\prime})=sel(d^{\prime})$ for $d^{\prime}\in
D\setminus\left\\{d\right\\}$.

Through slicing a new cube view is produced and a dimension _d_ is removed
from the cube. As shown in Fig. 4, in slicing for the item dimension and value
set $\left\\{i_{13503},i_{13504},...,i_{27005}\right\\}$, the item dimension
is removed and only events in which the item is a subset of
$\left\\{i_{13503},i_{13504},...,i_{27005}\right\\}$ remain in the cube view.

######  Definition 8 (Dice).

Let $PCS=(D,val,gran)$ be a process cube structure and $PCV=(D_{sel},sel)$ a
view of $PCS$. Let $fil\in D_{sel}\nrightarrow U_{h}$ be a filter such that
for any $d\in dom(fil):fil(d)\subseteq sel(d)$.
$dice_{fil}(PCV)=(D_{sel},sel^{\prime})$ with $sel^{\prime}(d)=fil(d)$ for
$d\in dom(fil)$ and $sel^{\prime}(d)=sel(d)$ for $d\in D\setminus dom(fil)$.

The difference between dice and slice is that through dicing, no dimension is
removed and it limits the values for one or more dimensions. For example
dicing is applicable based on two dimensions time and item, as shown in Fig.
5. In this example, we have new process cube view based on two filters:
$fil(time)=\left\\{T_{2018},T_{2019}\right\\}$ and
$fil(item)=\left\\{i_{1},i_{2},...,i_{13502}\right\\}$.

######  Definition 9 (Change Granularity).

Let $PCS=(D,val,gran)$ be a process cube structure and $PCV=(D_{sel},sel)$ a
view of $PCS$. Let $d\in D_{sel}$ and $G\in\mathbb{U}_{h}$ such that:
$G\subseteq gran(d)$ and $\cup G=\cup sel(d)$.
$chgr_{d,G}(PCV)=(D_{sel},sel^{\prime})$ with $sel^{\prime}(d)=G$, and
$sel^{\prime}(d^{\prime})=sel(d^{\prime})$ for $d^{\prime}\in
D\setminus\left\\{d\right\\}$.

The dimensions in process cube view do not change during changing granularity,
but the dimensions are shown in a more fine-grained or coarse-grained vision.
In Fig. 3(a) the granularity for time dimension is year. However, in Fig.
3(b), the granularity for time dimension is month. Having different levels of
the granularity, we can compare processes in different levels of granularity.

##  5 Implementation

The approach has been implemented as a standalone Python application
111available in https://gitlab.com/Anahita-Farhang/process-cube by using
PM4PY-MDL. The user can easily install the framework in Python 3.6. The
procedure to create a cube in this framework is shown in Fig. 6. The
functionalities of the framework are:

  * $\bullet$

It is possible to import object-centric event logs (The framework also accepts
CSV and XES formats and automatically converts them to an object centric event
log with one case notion).

  * $\bullet$

The created cube can be exported in a dump file and stored in the memory for
future exploration.

  * $\bullet$

At any point in time, it is possible to export the object-centric event log as
an object-centric event log or a traditional event log by selecting a case
notion. It is also possible to show the process models of the selected
cell(s).

Figure 6: An overview showing how the framework can be used. The input is an
object-centric event log. By choosing the dimensions, the user can build the
process cube and through process cube operations explore the cube. It is
possible to have the output as an event log, a process model, and a sub cube
Figure 7: Process Mining Cube: process comparison approach

It is possible to explore the cube interactively through process cube
operations such as slicing. The user can discover an MVP model, enhanced with
performance/frequency information, and Object-centric Petri net for each cell.
Fig. 7 shows a comparison between MVP models of a dice and the whole cube:

  * $\bullet$

In Box 1, it is possible to specify for each object type the activities that
are considered for that object type. For example, for the activity _create
purchase order_ , _orders_ are involved, but _items_ are not involved in the
_post document_.

  * $\bullet$

The MVP model of the whole cube is shown in Box 2 by considering the filtering
options in Box 1. MVP models are DFGs with colored arcs. In this figure, the
color of the arcs related to the _order_ is red.

  * $\bullet$

The MVP model of the specific _slice_ /_dice_ of the cube is represented in
Box 3. Putting this model near the whole cube’s model makes the comparison
easier.

  * $\bullet$

In Box 4, the user can change the frequency of the nodes and edges appearing
in the MVP model. There is a performance annotated version of MVP models that
is reachable only by clicking on performance. It is possible to export the
object-centric process model as an image in the desired address.

##  6 Evaluation

In this section, we analyze the scalability of the object-centric process cube
tool. To assess the performance of our approach, we measure the scalability of
the approach from two perspectives: creation time (e.g., the time required to
create and materialize cells of the cube), and the loading time (e.g., the
time required to import the cube). Results, shown in Fig. 8, have been done in
three different settings: the time for creating/loading the cube in terms of
the number of events in the event log (while keeping the number of object
types and attributes constant), the number of object types in the event log
(while keeping the number of events and attributes constant), and the number
of attributes in the event log (while keeping the number of events and object
types constant). Performance analysis of the cube with different settings
shows the time required for creating/loading the cube increases linearly, non-
linearly, and linearly when increasing number of events, object types, and
attributes respectively. The proposed framework creates a process cube for an
event log with 17500 events in almost 4 minutes.

$0$$0.5$$1$$1.5$$\cdot 10^{4}$$0$$100$$200$Number of eventsCreate the cube
(sec)

a.1) The difference in time required for creating the cube for different
numbers of events: n_object types = 4, and n_attributes = 4

$0$$0.5$$1$$1.5$$\cdot 10^{4}$$0$$500$$1{,}000$$1{,}500$$2{,}000$Number of
eventsLoad the cube (msec)

a.2) The difference in time required for loading the cube for different
numbers of events: n_object types = 4, and n_attributes = 4

$1$$1.5$$2$$2.5$$3$$3.5$$4$$0$$100$$200$Number of object typesCreate the cube
(sec)

b.1) The difference in time required for creating the cube for different
numbers of object types: n_events = 17500, and n_attributes = 4

$1$$1.5$$2$$2.5$$3$$3.5$$4$$500$$1{,}000$$1{,}500$$2{,}000$Number of object
typesLoad the cube (msec)

b.2) The difference in time required for loading the cube for different
numbers of object types: n_events = 17500, and n_attributes = 4

$1$$1.5$$2$$2.5$$3$$3.5$$4$$100$$150$$200$$250$Number of attributesCreate the
cube (sec)

c.1) The difference in time required for creating the cube for different
numbers of attributes: n_events= 17500 , and n_object types = 4.

$1$$1.5$$2$$2.5$$3$$3.5$$4$$500$$1{,}000$$1{,}500$$2{,}000$Number of
attributesLoad the cube (msec)

c.2) The difference in time required for loading the cube for different
numbers of attributes: n_events = 17500, and n_object types = 4.

Figure 8: Performance analysis of creating (diagrams with red dots)/loading
(diagrams with blue dots) the cube for the proposed approach based on a) the
number of events b) the number of object types, and c) the number of
attributes

##  7 Conclusion

This paper bridges the gap between object-centric process mining and process
comparison. Therefore, we proposed an object-centric process cube that
organizes data through dimensions referring to case notions (i.e., process
instances) and attributes in the event log. The proposed framework allows the
users to explore the object-centric event logs interactively through the
process cube operations such as slice. Furthermore, the proposed framework,
which is publicly available, is able to discover object-centric process models
from object-centric event logs extracted from the cells of the cube, which
speeds up process comparison. For the future work, we aim to add more features
related to performance to the object-centric process cube framework to
facilitate the process analysis

## Acknowledgments

We thank the Alexander von Humboldt (AvH) Stiftung for supporting our
research. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy–EXC-2023 Internet of
Production – 390621612.

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