###### Abstract

In this analysis, we study the dynamics of quantum oscillator fields within
the context of a position-dependent mass (PDM) system situated in an Einstein-
Maxwell space-time, incorporating a non-zero cosmological constant. The
magnetic field is aligned along the symmetry axis direction. To analyze PDM
quantum oscillator fields, we introduce a modification to the Klein-Gordon
equation by substituting the four-momentum vector
$p_{\mu}\to\Big{(}p_{\mu}+i\,\eta\,X_{\mu}+i\,\mathcal{F}_{\mu}\Big{)}$ into
the Klein-Gordon equation, where the four-vector is defibed by
$X_{\mu}=(0,r,0,0)$, $\mathcal{F}_{\mu}=(0,\mathcal{F}_{r},0,0)$ with
$\mathcal{F}_{r}=\frac{f^{\prime}(r)}{4\,f(r)}$, and $\eta$ is the mass
oscillator frequency. The radial wave equation for the relativistic modified
Klein-Gordon equation is derived and subsequently solved for two distinct
cases: (i) $f(r)=e^{\frac{1}{2}\,\alpha\,r^{2}}$, and (ii) $f(r)=r^{\beta}$,
where $\alpha\geq 0,\beta\geq 0$. The resultant energy levels and wave
functions for quantum oscillator fields are demonstrated to be influenced by
both the cosmological constant and the geometrical topology parameter which
breaks the degeneracy of the energy spectrum. Furthermore, we observed
noteworthy modifications in the energy levels and wave functions when compared
to the results derived in the flat space background.

PDM relativistic quantum oscillator in Einstein-Maxwell-Lambda space-time

Faizuddin<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Physics, University of Science & Technology Meghalaya, Ri-Bhoi,
793101, India
Abdelmalek<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>
Laboratory of theoretical and applied Physics, Echahid Cheikh Larbi Tebessi
University, Algeria

keywords: Einstein-Maxwell Space-times; Cosmological Constant; Relativistic
Wave Equations; Position Dependent Mass System; Solutions of Wave Equation;
Special Functions.

PACS numbers: 04.40.Nr; 98.80.Es; 03.65.Pm ; 03.65.Ge; 02.30.Gp

##  1 Introduction

Quantum field theory (QFT) in curved space-times [1, 2, 3, 4, 5, 6] provides a
framework for analyzing quantum phenomena in the presence of gravitational
effects [7, 8, 9]. This theory describes quantum fields interacting with
classical gravitational fields, assuming that the quantum nature of gravity is
not significant in the considered regimes. In this context, gravity is modeled
as a classical, curved space-time, following the principles of general
relativity. The resulting theory involves the propagation of quantum fields on
a curved background manifold. This approach draws inspiration from the
successful application of similar methods in quantum electrodynamics, where
early calculations treated the electromagnetic field as a background
interacting with quantized matter. Results from this approximation were found
to be consistent with the full theory of quantum electrodynamics. Quantum
field theory in curved space-times is particularly valuable when analyzing
phenomena at scales where gravitational quantum effects are negligible, such
as in the standard model of particle physics, which is applicable at scales
around $10^{-19}$ meters. The theory’s range of validity extends to larger
scales, encompassing a wide array of interesting phenomena. One notable
example is Hawking radiation [10, 11, 12], discovered by Stephen Hawking,
which predicts particle creation near black holes [13, 14, 15, 16].

The initial success of QFT in curved space-time stemmed from discoveries like
particle creation in expanding universes [17], the prediction of black hole
radiation [18], and the related observation of a finite temperature
experienced by uniformly accelerating observers in a vacuum state [19]. Recent
applications include its use in inflationary cosmology, where quantum field
fluctuations contribute to understanding early universe density fluctuations
(see [20] for example). Beyond its applications, the study of QFT on manifolds
offers intrinsic value. It enhances our understanding of the structure of QFT,
shedding light on aspects tied to specific space-times and their symmetries
versus those that are more fundamental. Structures like the existence of a
vacuum state, a well-defined particle picture, and global equilibrium states
in Minkowski space-time do not straightforwardly generalize to curved space-
times [4, 5, 6].

Magnetic fields play a pivotal role in exploring various astrophysical
phenomena, ranging from neutron stars, white dwarfs, pulsars, and black holes
to galaxies. Numerous observations underscore the significance of magnetic
fields, especially in scenarios where their interaction with general
relativity becomes essential. One notable example is their presence in active
galactic nuclei [21, 22], where these nuclei exhibit elevated radiation levels
compared to the rest of the galaxy, directly influencing its structure and
evolutionary trajectory. Another compelling scenario involves the emergence of
relativistic collimated jets within the inner regions of accretion discs, a
phenomenon explained by magneto-centrifugal mechanisms [23, 24]. Magnetic
fields play a crucial role in various physical phenomena, such as thermal
processes within strong magnetic neutron stars [25, 26], primordial accretion
disks [27], molecular cloud formation and evolution [28], formation of direct
collapse of black holes [29], population III star formation [30], particle
motion and acceleration around a Schwarzschild black hole [31], investigation
of exact superposition of a central static black hole with a surrounding thin
disk [32], and planetary nebulae and post-AGB nebulae [33]. Additionally,
magnetic fields contribute to various aspects of stellar objects and
astronomical scenarios, as detailed in references [34, 35, 36, 37, 38, 39, 40,
41].

Analytical models describing astrophysical entities are often constructed
based on solutions derived from Einstein’s field equations. In the pursuit of
more realistic models for compact stellar systems, modifications to the
energy-momentum tensor-the source of Einstein’s equations-are introduced to
incorporate additional physical properties, such as electromagnetic fields.
Bonnor was the first to consider this approach for solution derivation,
addressing both radial and longitudinal fields [42, 43]. Subsequently, Melvin
revisited this solution, which is now recognized as Bonnor-Melvin universe
[44]. Numerous authors have attempted to construct exact solutions in the
presence of magnetic fields, including Manko solution [45, 46], axisymmetric
Einstein-Maxwell solutions with a cosmological constant [47, 48, 49],
analytical models of axially symmetric thin disks with finite extension based
on the solutions [50], and general magnetostatic axisymmetric exact solutions
[51], among others. In the context of quantum mechanical systems, the
Einstein-Maxwell solutions with a cosmological constant [47, 48, 49] has
recently attracted much attention. These investigations include quantum motion
of scalar particles [52], quantum oscillator fields [53], scalar particles
interacting with linear and Cornell-type potentials [54], and effects of
rainbow gravity on scalar bosons [55, 56]. In addition, the relativistic wave
equation for spin $1/2$ particles in the Melvin space-time was investigated in
Ref. [57]. This emerging body of research significantly contributes to our
understanding of the interplay between quantum systems and the geometries of
curved space-times.

In this study, our objective is to investigate dynamics of quantum oscillator
fields within a PDM system in the backdrop of axisymmetric Einstein-Maxwell
solution featuring a cosmological constant [47]. The PDM oscillator system is
examined by substituting the momentum operator
$p_{\mu}\to\Big{(}p_{\mu}+i\,\eta\,X_{\mu}+i\,\mathcal{F}_{\mu}\Big{)}$ into
the Klein-Gordon equation, where $\mathcal{F}_{\mu}=(0,\mathcal{F}_{r},0,0)$
with
$\mathcal{F}_{r}=\frac{m^{\prime}(r)}{4\,m(r)}=\frac{f^{\prime}(r)}{4\,f(r)}$,
and $\eta=m_{0}\,\omega$ with $\omega$ represents the oscillator frequency. We
consider two forms of position-dependent mass scalar multiplier
$m(r)=m_{0}\,f(r)$, namely (i) $f(r)=e^{\frac{1}{2}\,\alpha\,r^{2}}$ and (ii)
$f(r)\propto r^{\beta}$, with $\alpha\geq 0$ and $\beta\geq 0$. We solve the
Klein-Gordon equation within the chosen space-time background and obtain the
energy eigenvalues for each case. Importantly, we demonstrate that the
eigenvalue solutions undergo modifications due to the presence of geometrical
topology parameter and the cosmological constant, deviating from the results
obtained in Minkowski flat space. This emphasizes the influence of the
considered space-time background on the quantum properties of the oscillator
fields, revealing deviations from the outcomes predicted in a flat space-time
setting. We format this paper as follows: In section 2, we derive the radial
equation of the Klein-Gordon wave equation with a position-dependent mass. The
radial equation is solved using two different PDM scalar multiplier through
special functions, leading to the energy profile of the oscillator fields. In
section 3, we present and discuss the results obtained. Throughout the paper,
we adopt a system of units where $c=1=\hbar$.

##  2 Position-dependent mass quantum oscillator in Einstein-Maxwell-Lambda
solution

In this section, we consider the physical environment under which the
relativistic quantum oscillator field described by the Klein-Gordon oscillator
within the framework of PDM system in an axisymmetric Einstein-Maxwell
solution with a cosmological is investigated. This cosmological model in the
cylindrical system is described by the following line-element [47, 52, 53, 54,
55, 56]

$ds^{2}=-dt^{2}+dr^{2}+\sigma^{2}\,\sin^{2}\left(\sqrt{2\Lambda}\,r\right)\,d\varphi^{2}+dz^{2},$
(1)

where $\sigma$ is the topological parameter which produces an angular deficit
and $\Lambda$ is the cosmological constant. The magnetic field strength
associated with this metric is given by
$H(r)=\sigma\,\sqrt{\Lambda}\,\sin\left(\sqrt{2\Lambda}\,r\right)$ which acts
along the $z$-direction. The electromagnetic field tensor associated with this
metric is given by ${\bf F}=H(r)\,dr\wedge d\varphi$. This metric is analogous
but not exact form to the cosmic string space-time,
$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}\,r^{2}\,d\varphi^{2}+dz^{2}$ [58, 59].
Expressing the space-time (1) in the form
$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}$, where $\mu,\nu=0,123$, the covariant
metric tensor $g_{\mu\nu}$ and its contravariant form $g^{\mu\nu}$ are given
by

$g_{\mu\nu}=\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&0\\\
0&0&\sigma^{2}\,\sin^{2}\left(\sqrt{2\Lambda}\,r\right)&0\\\
0&0&0&1\end{array}\right),\quad
g^{\mu\nu}=\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&0\\\
0&0&\frac{1}{\sigma^{2}\,\sin^{2}\left(\sqrt{2\Lambda}\,r\right)}&0\\\
0&0&0&1\end{array}\right).$ (2)

Finally, the determinant of the metric tensor $g_{\mu\nu}$ for the above
space-time (1) is given by

$\det\,(g_{\mu\nu})=g=-\sigma^{2}\,\sin^{2}(\sqrt{2\Lambda}\,r)$ (3)

which vanishes on the symmetry axis $r=0$, thus, representing an example of
degenerate metrics.

The relativistic quantum motions of spin-0 scalar particles is described by
the following wave equation [52, 53, 54, 55, 56, 59, 60]

$\displaystyle\Big{[}\frac{1}{\sqrt{-g}}\,\partial_{\mu}\,(\sqrt{-g}\,g^{\mu\nu})\,\partial_{\nu})\Big{]}\,\Psi=m^{2}_{0}\,\Psi,$
(4)

where $m_{0}$ is the rest mass of the particles and $g$ is the determinant of
the metric tensor $g_{\mu\nu}$.

In quantum system, the Klein-Gordon oscillator [61] can be studied by
replacing the momentum four-vector: $p_{\mu}\to(p_{\mu}+i\,\eta\,X_{\mu})$
[62], where $\eta$ is the oscillator frequency, and the four-vector is defined
by $X_{\mu}=(0,r,0,0)$. This quantum oscillator field has been analyzed by
numerous authors in the backdrop of various curved space-time, such as the
Gödel-type space-time of Som-Raychaudhuri metric [63], cosmic strings space-
time [64, 65], point-like global monopoles [66, 67], topologically non-trivial
[68] and trivial space-times [69], and in the context of Kaluza-Klein theory
[70]. In this analysis, we introduce position-dependent mass (PDM) into the
quantum system described by the Klein-Gordon oscillator. The position-
dependent mass system has recently attracted attention in the theoretical
physics in the studies of motion of spin-0 quantum particles (see, Refs. [71,
72, 73]). Therefore, PDM system Klein-Gordon oscillator can studied by
replacing the momentum four-vector as follows:

$p_{\mu}\to\Big{(}p_{\mu}+i\,\eta\,X_{\mu}\Big{)}\to\Big{(}p_{\mu}+i\,\eta\,X_{\mu}+i\,\mathcal{F}_{\mu}\Big{)},$
(5)

where we have defined this new four-vector $\mathcal{F}_{\mu}$ as follows [71,
72, 73]:

$\mathcal{F}_{\mu}=(0,\mathcal{F}_{r},0,0),\quad\mathcal{F}_{r}=\frac{m^{\prime}(r)}{4\,m(r)}=\frac{f^{\prime}(r)}{4\,f(r)}.$
(6)

Here, our assumption is that $m(r)=m_{0}\,f(r)$, only axial or radial
coordinate dependent.

Thus, the relativistic wave equation (4) describing PDM Klein-Gordon
oscillator using (4) becomes

$\displaystyle\Bigg{[}\frac{1}{\sqrt{-g}}\,\Big{(}\partial_{\mu}+\eta\,X_{\mu}+\mathcal{F}_{\mu}\Big{)}\,\Big{\\{}\sqrt{-g}\,g^{\mu\nu}\,\Big{(}\partial_{\nu}-\eta\,X_{\nu}-\mathcal{F}_{\nu}\Big{)}\Big{\\}}\Bigg{]}\,\Psi=m^{2}_{0}\,\Psi.$
(7)

Expressing this wave equation (7) in the space-time background (1) and using
(2)–(3), we obtain the following second-order differential equation involving
both time and space coordinates as follows:

$\displaystyle\Bigg{[}-\frac{d^{2}}{dt^{2}}+\frac{d^{2}}{dr^{2}}+\frac{\kappa}{\tan(\kappa\,r)}\,\frac{d}{dr}+\mathcal{M}(r)+\frac{1}{\sigma^{2}\,\sin^{2}(\kappa\,r)}\,\frac{d^{2}}{d\phi^{2}}+\frac{d^{2}}{dz^{2}}-m^{2}_{0}\Bigg{]}\,\Psi=0.$
(8)

where $\kappa=\sqrt{2\,\Lambda}$ and $\mathcal{M}(r)$ is given by

$\mathcal{M}(r)=\frac{3}{16}\,\frac{f^{\prime
2}}{f^{2}}-\frac{f^{\prime\prime}}{4\,f}-\frac{\kappa}{\tan(\kappa\,r)}\,\frac{f^{\prime}}{4\,f}-\frac{\eta\,\kappa\,r}{\tan(\kappa\,r)}-\eta-\eta^{2}\,r^{2}-\eta\,r\,\,\frac{f^{\prime}}{2\,f}.$
(9)

One can see that for constant mass, $m(r)=m_{0}$, where $f(r)=1$, one will get
back the Klein-Gordon equation in the chosen cosmological space-time
background obtained in Ref. [53]. In this analysis, we are mainly interested
on the system where the mass depends only on the radial distance $r$, that is
$m=m(r)=m_{0}\,f(r)$ also called position-dependent mass system.

One can see that the second-order differential equation (8) depends only on
the radial coordinate $r$ and independent of the coordinates $(t,\varphi,z)$.
In mathematical physics, this type of differential equation can easily be
solved using the method of separation of variables. In this work, we choose an
ansatz for the wave function ($\Psi$) in terms of different variables with the
radial function $\psi(r)$ as follows:

$\Psi(t,r,\varphi,z)=e^{i\,(-E\,t+\ell\,\phi+k_{z}\,z)}\,\psi(r),$ (10)

where $E$ is the particle’s energy, $\ell=0,\pm\,1,\pm\,2,....$ are the
eigenvalues of the angular quantum number, and $k_{z}$ is an arbitrary
constant. For axisymmetric space-time, we can set $k_{z}=0$ in the wave
function for simplicity.

Thereby, substituting the total wave function (10) into the differential
equation (8) results the following a linear, second-order homogeneous
differential equation form given by

$\psi^{\prime\prime}(r)+\frac{\kappa}{\tan(\kappa\,r)}\,\psi^{\prime}(r)+\Bigg{[}E^{2}-m^{2}_{0}+\mathcal{M}(r)-\frac{\ell^{2}}{\sigma^{2}\,\sin^{2}(\kappa\,r)}\Bigg{]}\,\psi(r)=0,$
(11)

where $\mathcal{M}(r)$ is given in Eq. (9).

Now, we solve this second-order radial wave equation (11) using different
cases of $f(r)$ other than unity. For that purpose, we choose two such
function or multiplier, namely, (i) $f(r)=\exp(\alpha\,r^{2}/2)$, and (ii) a
power law type dimensionless scalar multiplier $f(r)\propto r^{\beta}$, where
$\alpha\geq 0,\beta\geq 0$ [71, 72, 73]. Using these, we solve the radial
equation (11) and obtain the energy profiles and the radial wave function in
each cases.

###  2.1 PDM scalar multiplier: $f(r)=e^{\alpha\,r^{2}/2}$

To solve the radial equation (11), we choose the following scalar multiplier

$f(r)=e^{\alpha\,r^{2}/2},$ (12)

where $\alpha\geq 0$.

Thereby, substituting this $f(r)$ into the differential equation (11), we
obtain the following differential equation form:

$\displaystyle\psi^{\prime\prime}(r)+\frac{\kappa}{\tan(\kappa\,r)}\,\psi^{\prime}(r)+\Bigg{[}E^{2}-m^{2}_{0}-\Big{(}\frac{\alpha}{4}+\eta\Big{)}^{2}\,r^{2}-\Big{(}\frac{\alpha}{4}+\eta\Big{)}\,\Big{\\{}1+\frac{\kappa\,r}{\tan(\kappa\,r)}\Big{\\}}$
$\displaystyle-\frac{\ell^{2}}{\sigma^{2}\,\sin^{2}(\kappa\,r)}\Bigg{]}\,\psi(r)=0.$
(13)

In order to solve equation (13), one can follows different approximation
schemes known in the quantum systems. In this analysis, we choose an
approximation where the axial distance $r$ is small such that $\sin r\approx
r$ and $\tan r\approx r$. Therefore, using this approximation up to first
order, we obtain the following equation from the Eq. (13) given by

$\displaystyle\psi^{\prime\prime}(r)+\frac{1}{r}\,\psi^{\prime}(r)+\Big{(}\Theta-\omega^{2}\,r^{2}-\frac{\iota^{2}}{r^{2}}\Big{)}\,\psi(r)=0,$
(14)

where we set

$\displaystyle\Theta=E^{2}-m^{2}_{0}-2\,\omega,\quad\omega=\frac{\alpha}{4}+\eta,\quad\iota=\frac{|\ell|}{|\sigma|\,\kappa}.$
(15)

Noted that one may consider up to the second order approximation which we left
for the future investigations.

At this stage, we can consider a substitution via $\psi=\frac{R(r)}{\sqrt{r}}$
into the equation (14) results

$\Bigg{[}\frac{d^{2}}{dr^{2}}-\omega^{2}\,r^{2}-\frac{(\iota^{2}-1/4)}{r^{2}}+\Theta\Bigg{]}\,R(r)=0.$
(16)

This is the radial equation that describes the position-dependent mass
Klein–Gordon oscillator in the cosmological space-time (1). In order to obtain
solution to this differential equation (16), it is necessary to analyze the
asymptotic behavior of the radial wave function $f(r)$ at $r\to 0$ and
$r\to\infty$. In this way, a possible regular solution at the origin $r=0$ is
obtained if the solution of the differential equation (16) has the following
form

$R(r)=r^{\iota+1/2}\,e^{-\frac{1}{2}\,\omega\,r^{2}}\,G(r),$ (17)

where $G(r)$ is an unknown function.

Thereby, substituting the above possible solution (17) into the Eq. (16)
results the following differential equation form

$G^{\prime\prime}+\Big{(}1+2\,\iota-2\,r^{2}\,\omega\Big{)}\,G^{\prime}/r+\Big{(}\Theta-2\,(1+\iota)\,\omega\Big{)}\,G=0.$
(18)

Finally, performing a transformation to a new variable via $s=\omega\,r^{2}$
into this equation (18), we obtain a standard second-order differential
equation form given by

$s\,G^{\prime\prime}(s)+\Big{(}1+\iota-s\Big{)}\,G^{\prime}(s)-\Bigg{(}\frac{1}{2}+\frac{\iota}{2}-\frac{\Theta}{4\,\omega}\Bigg{)}\,G(s)=0.$
(19)

The above equation is the confluenct hypergeometric second-order differential
equation form [74, 75] and the solution of this Eq. (19) is given by the
confluent hypergeometric function as follows:

$G(s)={}_{1}F_{1}\Bigg{(}\frac{1}{2}+\frac{\iota}{2}-\frac{\Theta}{4\,\omega},1+\iota;s\Bigg{)}.$
(20)

(a) $n=1=\eta$

(b) $n=1=\eta$

(c) $\eta=1$, $\Lambda=0.1$

(d) $\eta=1$, $\sigma=0.1$

(e) $\Lambda=0.1$, $n=1$

(f) $\sigma=0.1$, $n=1$

Figure 1: The energy spectrum with parameter $\sigma$ and cosmological
constant $\Lambda$ for various values of other parameters. Here
$\ell=1=\alpha=m_{0}$.

From the asymptotic behavior of the hypergeometric function, it is necessary
that this function when expanding in a power series must be a finite degree
polynomial function of $r$ having degree $n$, and the parameter
$\Big{(}\frac{1}{2}+\frac{\iota}{2}-\frac{\Theta}{4\,\omega}\Big{)}$ must be a
non-positive integer. Therefore, in that scenario, we have the following
relation

$\frac{1}{2}+\frac{\iota}{2}-\frac{\Theta}{4\,\omega}=-n\quad(n=0,1,2,3,...).$
(21)

Simplification of the above relation results the following energy expression
given by

$E_{n,\ell}=\pm\,\sqrt{m^{2}_{0}+(\alpha+4\,\eta)\,\Bigg{(}n+1+\frac{|\ell|}{2\,\sqrt{2\,\Lambda}\,|\sigma|}\Bigg{)}}.$
(22)

(a) $\Lambda=0.5=\sigma,\eta=1$

(b) $\eta=1=n,\sigma=0.5$

(c) $\Lambda=0.5,n=1=\eta$

(d) $\Lambda=0.75=\sigma,n=1$

(e) $n=1,\eta=0.5$

(f) $n=1$

Figure 2: The radial wave function for various values of parameters. Here
$\ell=1,\alpha=0.1$.

The radial wave function of the quantum oscillator fields are given by

$\psi_{n,\ell}(r)=\mathcal{N}\,r^{\frac{|\ell|}{\sqrt{2\,\Lambda}\,|\sigma|}}\,e^{-\frac{1}{2}\,\Big{(}\frac{\alpha}{4}+\eta\Big{)}\,r^{2}}\,{}_{1}F_{1}\Bigg{(}-n,1+\frac{|\ell|}{\sqrt{2\,\Lambda}\,|\sigma|};\Big{(}\frac{\alpha}{4}+\eta\Big{)}\,r^{2}\Bigg{)},$
(23)

where $\mathcal{N}$ is the normalization constant.

Equation (22) represents the approximate relativistic energy eigenvalue, while
equation (23) characterizes the radial wave function of PDM quantum oscillator
fields in an axisymmetric Einstein-Maxwell space-time with a cosmological
constant. Here, the position-dependent mass (PDM) is considered by
$m(r)=m_{0}\,e^{\alpha\,r^{2}/2}$. Notably, the energy levels and the radial
wave functions of the oscillator fields influenced by the magnetic field
parameters ($\sigma,\Lambda$), the coefficient parameter $\alpha$ and
undergoes modification. Furthermore, the eigenvalue solution undergoes
alteration as influenced by the quantum numbers $\\{n,\ell\\}$. Importantly,
the modifications diverge from the outcomes derived in the flat space context
for the quantum oscillator field.

Figure 1 shows the energy spectrum, $E^{\pm}_{n,\ell}$ of the quantum
oscillator field as defined in Eq. (22), with respect to the geometric
topology parameter $\sigma$ and the cosmological constant $\Lambda$,
considering varying values of other additional parameters. Notably, an
observable trend is the gradual decrease in the energy levels as these
additional parameters increase. Subsequently, increasing values of these
parameters result in a downward shift in the energy levels as seen in the sub-
figures (a)-(b), whereas an upward shift is evident in the sub-figures
(c)–(f). In Figure 2, we visualized the behavior of the radial wave function
(23) of the quantum oscillator field, and highlighting its response to diverse
values of one or more parameters involving in it.

If one choose $\alpha=0$, then the PDM becomes constant, that is,
$m(r)=m_{0}$. In that case, considering $\eta=m_{0}\,\Omega$ one will get back
the quantum system known as the Klein-Gordon oscillator. The approximate
energy eigenvalue of the relativistic quantum oscillator fields is given by
the following expression

$E_{n,\ell}=\pm\,\sqrt{m^{2}_{0}+2\,m_{0}\,\Omega\,\Bigg{(}2\,n^{\prime}+\frac{|\ell|}{\sqrt{2\,\Lambda}\,|\sigma|}\Bigg{)}},\quad(n^{\prime}=n+1).$
(24)

And that the radial wave function is given by

$\psi_{n,\ell}(r)=\mathcal{N}\,r^{\frac{|\ell|}{\sqrt{2\,\Lambda}\,|\sigma|}}\,e^{-\frac{1}{2}\,m_{0}\,\Omega\,r^{2}}\,{}_{1}F_{1}\Bigg{(}-n,1+\frac{|\ell|}{\sqrt{2\,\Lambda}\,|\sigma|};m_{0}\,\Omega\,r^{2}\Bigg{)}.$
(25)

###  2.2 PDM scalar multiplier: $f(r)=c\,r^{\beta}$, where $\beta\geq 0$

In this part, we solve the radial equation (11) using the following scalar
multiplier

$f(r)=c\,r^{\beta},\quad\beta\geq 0.$ (26)

Thereby, substituting this into the equation (11), we obtain the following
differential equation

$\displaystyle\psi^{\prime\prime}(r)+\frac{\kappa}{\tan(\kappa\,r)}\,\psi^{\prime}(r)+\Bigg{[}E^{2}-m^{2}_{0}-\eta-\frac{1}{2}\,\eta\,\beta+\Big{(}-\frac{\beta^{2}}{16}+\frac{\beta}{4}\Big{)}\,\frac{1}{r^{2}}-\eta^{2}\,r^{2}$
$\displaystyle-\frac{\kappa\,\beta}{4\,\tan(\kappa\,r)}\,\frac{1}{r}-\frac{\ell^{2}}{\sigma^{2}\,\sin^{2}(\kappa\,r)}-\eta\,\frac{\kappa\,r}{\tan(\kappa\,r)}\Bigg{]}\,\psi(r)=0.$
(27)

Following the approximation done in the previous case, we obtain a standard
second-order differential equation from Eq. (27) given by

$\displaystyle\psi^{\prime\prime}(r)+\frac{1}{r}\,\psi^{\prime}(r)+\Big{(}\Pi-\eta^{2}\,r^{2}-\frac{\tau^{2}}{r^{2}}\Big{)}\,\psi(r)=0,$
(28)

where we set

$\Pi=E^{2}-m^{2}_{0}-2\,\eta\,-\frac{1}{2}\,\eta\,\beta,\quad\tau=\sqrt{\frac{\beta^{2}}{16}+\frac{\ell^{2}}{2\,\Lambda\,\sigma^{2}}}.$
(29)

Here, also performing a transformation to new function via
$\psi(r)=\frac{R(r)}{\sqrt{r}}$ into the equation (28) results

$\displaystyle\Bigg{[}\frac{d^{2}}{dr^{2}}-\eta^{2}\,r^{2}-\frac{(\tau^{2}-1/4)}{r^{2}}+\Pi\Bigg{]}\,\psi(r)=0,$
(30)

As stated earlier, regular solution is one of the requirement of the quantum
systems under investigation. Let a regular solution of the differential Eq.
(28) has the following form

$R(r)=r^{\tau+1/2}\,e^{-\frac{1}{2}\,\eta\,r^{2}}\,F(r),$ (31)

where $F(r)$ is an unknown function.

Substituting this solution (30) into the equation (30), we obtain the
following differential equation form:

$F^{\prime\prime}+\Big{(}1+2\,\tau-2\,r^{2}\,\eta\Big{)}\,F^{\prime}/r+\Big{(}\Pi-2\,(1+\tau)\,\eta\Big{)}\,F=0.$
(32)

Finally, performing a transformation to a new variable via $u=\eta\,r^{2}$
into this equation (32), we obtain a standard second-order differential
equation form given by

$u\,F^{\prime\prime}(u)+\Big{(}1+\tau-u\Big{)}\,F^{\prime}(u)-\Bigg{(}\frac{1}{2}+\frac{\tau}{2}-\frac{\Pi}{4\,\eta}\Bigg{)}\,F(u)=0.$
(33)

The above equation is again the confluent hypergeometric second-order
differential equation form [74, 75] and the solution of this Eq. (33) is given
by the confluent hypergeometric function:

$F(u)={}_{1}F_{1}\Bigg{(}\frac{1}{2}+\frac{\tau}{2}-\frac{\Pi}{4\,\eta},1+\tau;u\Bigg{)}.$
(34)

(a) $n=1,\eta=0.1$

(b) $n=1,\eta=0.1$

(c) $\eta=0.1=\Lambda$

(d) $\eta=0.1=\sigma$

(e) $\Lambda=0.1$, $n=1$

(f) $\sigma=0.1$, $n=1$

Figure 3: The energy spectrum with parameter $\sigma$ and cosmological
constant $\Lambda$ for various values of other parameters. Here
$\ell=1=m_{0},\beta=0.1$.

Applying similar argument on the confluent hypergeometric function as done in
the previous case, this function is a finite degree polynomial function of $u$
having degree $n$ provided we have the following relation

$\frac{1}{2}+\frac{\tau}{2}-\frac{\Pi}{4\,\eta}=-n\quad(n=0,1,2,3,...).$ (35)

Simplification of the above relation results the following energy expression

$E_{n,\ell}=\pm\,\sqrt{m^{2}_{0}+4\,\eta\,\Bigg{(}n+\frac{1}{2}\sqrt{\frac{\beta^{2}}{16}+\frac{\ell^{2}}{2\,\Lambda\,\sigma^{2}}}+1+\frac{\beta}{8}\Bigg{)}}\,.$
(36)

(a) $\Lambda=0.5=\sigma,\eta=0.5$

(b) $n=1,\sigma=0.5=\eta$

(c) $\Lambda=0.5,n=1=\eta$

(d) $\Lambda=0.5=\sigma,n=1$

Figure 4: The radial wave function for various values of parameters. Here
$\ell=1=\beta$.

The radial wave functions are given by

$\psi_{n,\ell}(r)=\mathcal{P}\,r^{\sqrt{\frac{\beta^{2}}{16}+\frac{\ell^{2}}{2\,\Lambda\,\sigma^{2}}}}\,e^{-\frac{1}{2}\,\eta\,r^{2}}\,{}_{1}F_{1}\Bigg{(}-n,1+\sqrt{\frac{\beta^{2}}{16}+\frac{\ell^{2}}{2\,\Lambda\,\sigma^{2}}};\eta\,r^{2}\Bigg{)},$
(37)

where $\mathcal{P}$ is the normalization constant.

In this analysis also, one can see that for $\beta=0$ and
$\eta=m_{0}\,\Omega$, the quantum system under investigation reduces to the
Klein-Gordon oscillator with constant mass. Therefore, the approximate energy
eigenvalue and the radial wave function of the relativistic quantum oscillator
field in the background of axisymmetric Einstein-Maxwell space-time with a
cosmological constant is given by Eqs. (24)–(25).

Equation (36) represents the approximate relativistic energy eigenvalue and
Eq. (37) corresponds to the radial wave function of quantum oscillator fields
in the backdrop of magnetic space-time within the framework of position-
dependent mass (PDM) characterized by $f(r)\propto r^{\beta}$. It is evident
that the energy levels are subject to the influence of geometrical parameters
($\sigma,\Lambda$), and the parameter $\beta$ and this solution undergo
modifications by the quantum numbers $\\{n,\ell\\}$. This modification stand
in contrast to the outcomes obtained in a flat space context, emphasizing the
impact of both the cosmological background and the position-dependent mass on
the eigenvalue solution of the quantum oscillator system.

Figure 3 shows the energy spectrum, $E^{\pm}_{n,\ell}$ of the quantum
oscillator field as defined in Eq. (36), with respect to the parameter
$\sigma$ and the cosmological constant $\Lambda$, considering varying values
of additional parameters. We see that energy level trend is gradually decrease
as additional parameters increase. Subsequently, increasing values to these
parameters result in a downward shift in the energy levels in sub-figures
(a)-(b), whereas an upward shift is evident in sub-figures (c)–(f). In Figure
4, we plot the radial wave function (37) of the oscillator field for diverse
values of the parameters involved in it.

##  3 Conclusions

In this study, we focused on a specific Einstein-Maxwell solution
characterized by a magnetic field and a non-zero positive cosmological
constant. Within this magnetic space-time background, we delved into the
relativistic dynamics of quantum oscillator fields within the framework of
position-dependent mass systems. This position-dependent mass (PDM) system was
examined by substituting the momentum four-vector into the Klein-Gordon
oscillator equation. We derived the radial equation for the PDM system of the
Klein-Gordon oscillator in the presence of this magnetic solution.

Two distinct scalar multipliers, denoted as $f(r)$, were considered, and we
determined the relativistic energy eigenvalues and the radial wave function of
the system. Significantly, our analysis revealed that various parameters
associated with the magnetic fields, such as the geometry’s topology
($\sigma$) and the positive cosmological constant ($\Lambda>0$), exerted
influence on these eigenvalue solutions, thereby modifying the results
compared to those in flat space. To illustrate these findings, we generated
several figures depicting the energy spectrum and the radial wave function for
various values of the parameters ($\sigma,\Lambda$) involved, as well as the
quantum numbers ${n,\ell}$. These figures provided insights into the behavior
of the energy spectrum and the wave function as the values of these parameters
increased. This comprehensive exploration enhances our understanding of how
the interplay between magnetic fields, space-time geometry, and cosmological
constants impacts the quantum properties of oscillator fields.

The investigation into position-dependent mass quantum oscillator fields
within the framework of Einstein-Maxwell-Lambda space-time has provided us
valuable insights into the intricate interplay between quantum mechanics and
the gravitational effects. The analysis of such systems not only deepens our
understanding of the fundamental principles governing particle dynamics but
also sheds light on the behavior of quantum fields in curved space-time
backgrounds. The results obtained in this study contribute to the broader
exploration of quantum phenomena in diverse gravitational contexts, offering a
stepping stone for further research into the rich interconnections between
quantum theory and general relativity (GR).

In exploring the interplay between curved space-time and quantum particles,
crucial questions arise, prompting further investigation. How does space-time
curvature influence particle behavior at the quantum level, and are there
distinct quantum signatures in regions of intense gravitational curvature?
Venturing into realms with pronounced gravitational effects raises inquiries
about its implications for particle dynamics and the nature of quantum fields.
Unraveling these mysteries requires bridging conceptual gaps between general
relativity and quantum mechanics. Future studies may decipher quantum
entanglement in curved space-time or investigate quantum coherence in
gravitational fields, guiding us toward a deeper understanding of the quantum-
gravitational interface and setting the stage for groundbreaking discoveries
reshaping our fundamental cosmos comprehension.

## Data Availability Statement

No new data were generated or analysed in this study.

## Conflict of Interest

Author declares there is no such conflict of interests.

## Funding Statement

No fund has received for this work.

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# Towards Sustainability of Systematic Literature Reviews

Vinicius dos Santos, Anderson Yoshiaki Iwazaki
University of São Paulo (ICMC-USP)
São Carlos - São Paulo - Brazil
{vinicius.dos.santos<EMAIL_ADDRESS>
Katia Romero Felizardo, Érica Ferreira de Souza
Federal University of Technology - Paraná (UTFPR)
Cornélio Procópio - Paraná - Brazil
{katiascannavino<EMAIL_ADDRESS>
Elisa Yumi Nakagawa
University of São Paulo (ICMC-USP)
São Carlos - São Paulo - Brazil
<EMAIL_ADDRESS>

###### Abstract

Background: The software engineering community has increasingly conducted
systematic literature reviews (SLR) as a means to summarize evidence from
different studies and bring to light the state of the art of a given research
topic. While SLR provide many benefits, they also present several problems
with punctual solutions for some of them. However, two main problems still
remain: the high time-/effort-consumption nature of SLR and the lack of an
effective impact of SLR results in the industry, as initially expected for
SLR. Aims: The main goal of this paper is to introduce a new view — which we
name Sustainability of SLR — on how to deal with SLR aiming at reducing those
problems. Method: We analyzed six reference studies published in the last
decade to identify, group, and analyze the SLR problems and their
interconnections. Based on such analysis, we proposed the view of
Sustainability of SLR that intends to address these problems. Results: The
proposed view encompasses three dimensions (social, economic, and technical)
that could become SLR more sustainable in the sense that the four major
problems and 31 barriers (i.e., possible causes for those problems) that we
identified could be mitigated. Conclusions: The view of Sustainability of SLR
intends to change the researchers’ mindset to mitigate the inherent SLR
problems and, as a consequence, achieve sustainable SLR, i.e., those that
consume less time/effort to be conducted and updated with useful results for
the industry.

_K_ eywords Systematic Literature Review, SLR, Sustainability, Sustainability
of SLR

##  1 Introduction

The community of software engineering (SE) has increasingly adopted Systematic
Literature Review (SLR) in recent years [1] as a reliable method to summarize
evidence from a number of studies and find out the state of the art in a given
research topic [2, 3]. SLR has presented many important benefits, including
the possibility of dealing with information from different studies in an
unbiased manner [3, 4], producing auditable and repeatable results [5, 6], and
identifying research gaps and also perspectives for future investigations [3].

At the same time, SLR has also suffered from diverse problems. Studies
reported some of them, e.g., poor documentation of SLR conducted [7, 8] and
lack of rigor in following well-experimented guidelines for SLR directly
impacting the SLR quality [9]. Important solutions have been proposed for
specific problems, such as techniques for better planning SLR [10, 11],
conducting SLR [12], reporting [13], and adopting supporting tools [14].
However, two main problems still remain with no suitable solutions: (i) the
SLR conduction is still a very time- and effort-consuming task [15]; and (ii)
the lack of effective impact of SLR results in the industry [16]. This current
scenario of SLR leads the SE community to question if it is worth continuing
to invest in the SLR conduction. Hence, the research question addressed in
this work is: Is it possible to reduce the high consumption of resources (time
and effort) while assuring to achieve more accessible and useful SLR for the
industry? Another question that intrigues us is: Should the SLR researchers
change the mindset on how to better deal with SLR conduction and
dissemination?

The main goal of this work is to introduce a new view — which we name
Sustainability of SLR — to better deal with SLR and reduce those two main
problems (high resources consumption and lack of impact in industry) and, as a
consequence, achieve sustainable SLR (i.e., SLR that consume fewer resources
to be conducted and updated with useful results for the industry). To achieve
our goal and answer the research question, we carefully examined six reference
studies published in the last decade [8][6][17][18][19][7] to identify, group,
and systematically analyze the main problems in both SLR conduction and
update. We found four groups of problems and 31 barriers (i.e., possible
causes of the problems). Following this, we proposed the view of
Sustainability of SLR, which connects these problems among them as well as
their causes and encompasses three perspectives (social, economic, and
technical) that could somehow deal with the SLR problems. The social
perspective addresses human aspects, such as reviewers’ communication and
stakeholders’ participation during the SLR conduction. The economic one is
related to the resources (effort and time) to conduct and update an SLR. The
technical perspective is related to the supporting tools and technologies used
to conduct and update SLR.

It is worth highlighting that this prospective view does not intend to solve
all current problems of SLR; instead, the main contribution of this work is to
raise the awareness of researchers to change the mindset when proposing new
solutions for SLR problems through a more holistic view.

The remainder of this paper is organized as follows: Section 2 presents the
research method. Section 3 presents the SLR problems identified in the
literature and their possible causes; Section 4 presents the view of
Sustainability of SLR; Section 5 discuss some challenges for achieving the
Sustainability of SLR as well as the threats to validity; and Section 6
presents the final remarks.

##  2 Research Method

To propose the view of Sustainability of SLR, it was necessary first to survey
the literature and understand the current problems of both SLR conduction and
update. For this, we selected six reference studies published over the last
decade (2010 - 2020) [8, 6, 7, 19, 18, 17] based on our group experience in
conducting SLR. We also used experts’ opinions to validate our choice and
mainly avoid bias.

The six studies selected are tertiary studies (SLR of secondary studies) and
experience reports from researchers’ perspectives to identify the main
problems of conducting and updating SLR. Riaz et al. (2010) [19] presented an
experience report involving three Ph.D. students that identified the main
difficulties of novices while they conduct SLR and compared the results with
SLR expert experience. Imtiaz et al. (2013) [17] conducted a tertiary study
that evaluated 116 secondary studies published from 2005 to 2011, summarized
the experience of researchers, and highlighted the challenges in conducting
SLR. Kitchenham and Brereton (2013) [18] conducted a tertiary study that
assessed 68 secondary studies published from 2005 to 2012 to summarize the
researchers’ opinions about their experiences of conducting SLR and discuss
techniques that could be used to improve the SLR process. Bugden et al. (2018)
[6] conducted another tertiary study assessing 178 studies published from 2010
to 2015 summarizing lessons learned, researchers’ main problems, and focusing
on how good SLR could be reported.

We can also interpret threats to validity as problems in SLR that could not be
solved providing clues about the researchers’ difficulties during the SLR
conduction. Hence, two tertiary studies [8, 7] identified and discussed the
most common threats to validity and strategies to mitigate them. Zhou et al.
(2016) [7] assessed 316 studies published from 2004 to 2015 and Ampatzoglou et
al. (2019) [8] analyzed 165 studies published from 2007 to 2016.

Figure 1: Research method

Based on the six studies aforementioned, we performed the four main steps, as
shown in Figure 1, which summarized the research method adopted. In Step 1, we
systematically identified the main problems that occur during the SLR
conduction and reported in the six studies. We also deeply checked the studies
cited by these six studies. In Step 2, to understand the source of those
problems, we carefully identified the current barriers, i.e., the possible
causes for these problems, that could be hampering authors to conduct good and
effective SLR. In Step 3, we deeply analyzed and summarized those barriers
removing duplication and classifying them111Replication package:
https://github.com/CSM-Research/06-LR-TSInSLR according to three dimensions
(or perspectives): social, economic, and technical. This classification was
based on well-known sustainability dimensions already explored in other areas
[20, 21, 22, 23, 24, 25], as described below:

  * •

Social barriers are mainly associated with human aspects of SLR (e.g.,
communication, culture) and how SLR results can impact the society (academia
and industry);

  * •

Economic barriers are directly associated with the high consumption of time
and effort while planning, conducting, or updating SLR; and

  * •

Technical barriers address problems with tools and technologies used to
support SLR conduction.

It is also worth highlighting that Steps 1, 2, and 3 were performed
considering our long-term experience researching, conducting, and updating
SLR. Besides, we based on evidence from the literature to make decisions.
Finally, in Step 4, we analyzed the SLR problems and proposed a new view that
connects these problems and their interactions as well as a possible means to
deal with these problems. For this, we inspired ourselves on the views of
sustainability from other areas, in particular, the ecology area [26] and SE
area [20], to derive the view of Sustainability of SLR.

##  3 SLR problems and barriers

After conducting Steps 1 to 3 of our research method, Table 1 summarizes the
four main problems were identified: (i) lack of industry’s interest in SLR
results, (ii) SLR conduction is still very, time and effort consuming (iii)
poor documentation of SLR, (iv) lack of SLR’s quality. For each large problem,
we identified the barriers (i.e., possible causes for these problems and
listed in column 4 of Table 1) and classified them in social, economic, or
technical dimensions (shown in column 3).

Table 1: Problems of SLR and possible causes Problems | ID | Dimension |  Barriers (Possible causes) |  Reference
---|---|---|---|---
Lack of industry’s interest in SLR results | 1 | Economic/Social |  Data synthesis, interpretation, and presentation of results are damaged when researchers have different interpretations of data extracted or when are conducted by a single researcher. |  [8] [19] [7]
2 | Social |  Cultural differences may impact the validity results |  [19] [7]
3 | Social |  Inappropriate research questions are defined |  [8] [17] [18] [19] [7]
4 | Social |  SLR does not present generalizable results which can be used in industry or other areas |  [8][6]
SLR conduction/update is still very time- and effort-consuming | 5 | Economic |  Lack of standardization and incorrect/incomplete keywords hinder the search string construction |  [8][17] [18][19][7]
6 | Economic |  Inadequate number of studies are evaluated hindering the validity of results |  [19] [7]
7 | Economic/Technical |  Use of not credible or very specific/broad digital databases may return many irrelevant studies or miss relevant studies |  [8][17] [18] [19] [7]
8 | Economic/Technical |  Primary studies may be duplicated in different databases |  [7]
9 | Economic/Technical |  Inaccessibility of resources (papers/databases) |  [7]
10 | Economic/Technical |  Database limitations and inefficiencies (e.g., interface, search string syntax) |  [17] [18][19]
11 | Economic/Technical |  Need tool to support SLR |  [17][18]
Poor documentation of SLR | 12 | Economic |  Lack of information about the SLR conduction process (e.g, initial date, duplicated studies). |  [6] [18] [19]
13 | Economic |  Lack information about the use of inclusion/exclusion criteria. |  [6] [18]
14 | Economic |  Insufficient details/report about the quality evaluation |  [6]
15 | Economic |  Additional search is not being reported correctly |  [6]
16 | Economic |  Lack of details about the primary studies selected (e.g. context, participants, source material) |  [6]
17 | Economic |  Lack of information about the exclusion of studies impacting SLR data extraction and synthesis |  [6]
18 | Economic |  Studies do not specify important SLR details, causing problems to repeatability and replicability |  [8][19][7]
19 | Economic/social |  The synthesis process is not clear |  [6] [19]
20 | Economic/social |  Lack information about the reviewers’ participation during the SLR conduction process |  [6][17][19]
Lack of quality of SLR | 21 | Economic |  Many studies do not perform the quality evaluation or do not use it correctly in selection process. |  [6] [19]
22 | Economic |  Few studies had a good search coverage. |  [6]
23 | Economic |  Inappropriate search methods leading to problems in SLR coverage |  [17][7]
24 | Economic |  Selection process can be difficult when inclusion/exclusion criteria are generic or inappropriate. |  [8][7]
25 | Economic |  The wrong classification of the primary studies may cause the secondary study to lack robustness. |  [8][19][7]
26 | Economic |  Delimiting a time span affect the coverage of SLR |  [7]
27 | Economic |  Lack of unpartiality of researchers resulting in bias in study selection |  [7]
28 | Economic |  Extraction process is difficult when quality assessment is biased by reviewer subjectivity or they do not completely understand the data extraction items |  [8][7]
29 | Social |  Lack of expert evaluation of the results |  [18] [19] [7]
30 | Economic |  Data model and data extraction forms may change during extraction |  [18]
31 | Economic |  Difficulties in deciding when to stop the piloting process |  [19]

For the first problem “Lack of industry’s interest in SLR results”, four
social barriers were identified and one can be also interpreted as economic.
Studies reported that SLR are conducted using inappropriate research questions
[8, 17, 18, 19, 7] and highlighted that the cultural differences among
researchers threats the validity of results (e.g., preferences for the studies
of some researchers’ nationality) [7, 19]. In addition, the lack of a clear
synthesis of SRL results and lack of practical recommendations for SE
practitioners impact somehow the generalizability of the results and hamper
the use of these results in the industry and academia [6, 8].

For the second problem “SLR conduction/update is still very time- and effort-
consuming”, seven barriers were found and all of them are economic barriers,
while five of them can also be interpreted as technical barriers. A possible
cause for this problem is the lack of standardization of terms to be used in
the search string (as the case of SE area [7, 8, 18, 19, 17]) and as
consequence, the search might return a large number of primary studies
(including many irrelevant ones) or miss some relevant studies [8]. Another
cause is the inefficiency of electronic databases [17, 19, 18] and, despite
the improvements in many aspects over the years (e.g., usability, search
engine, coverage, interface, etc), several studies still mention problems
indicating that the choice of databases is still a challenge [8]. Five studies
mention that non-credible or very specific or broad databases can return many
irrelevant studies or can even miss relevant studies, increasing the effort
for conducting SLR [7, 8, 18, 17, 19]. In this context, many supporting tools
were developed to mitigate such problem [27], especially aiming at reducing
the additional effort caused by databases inefficiency or integration problems
(e.g., studies indexed by multiple databases). However, many of these tools
solve only punctual problems. Hence, the lack of reliable and integrated tools
is still considered an important factor for the excessive time and effort
consumption.

For the third problem “Poor documentation of SLR”, nine economic barriers were
found and two of them can also be interpreted as social barriers. Results
indicate a lack of commitment of researchers in following the well-
experimented guidelines proposed for reporting SLR [6]. Three studies reported
that SLR do not report important details about the conduction process that
allows SLR to be updated or replicated [6, 18, 19]. The poor documentation
directly impacts academia and industry by creating a lack of credibility and
problems in auditability and reproducibility of SLR [7, 8, 19]. In addition,
this problem extends to the social dimension since a clear data synthesis is
considered very important to disseminate useful results of SLR to industry and
also academia [6].

For the fourth problem “Lack of quality of SLR”, 10 economic barriers and one
social barrier were found. The economic barriers indicate that authors have
difficulty in following the guidelines proposed for SLR and ensuring the
quality of selection [7, 8], data extraction [19, 8], synthesis and
documentation [6]. The wrong classification of the primary studies or
application of inappropriate search methods, for example, cause problems in
the SLR coverage and robustness. In addition, the lack of quality can also be
considered a social problem because it is recommended that SLR include experts
to evaluate the SLR results [18, 19, 7], because when they are not involved in
the SLR process, results cannot be adequately synthesized, even generalized,
and useful for SE practitioners.

Observing from a broader perspective, all problems and their possible causes
mentioned previously are somehow interconnected among them. For example, using
the economic perspective, the poor documentation hampers the auditability of
SLR, resulting in a lack of confidence in the SLR results [7, 8, 19]. It also
hampers the repeatability of SLR; hence, SLR cannot be easily updated [5], and
researchers need sometimes to reconduct the SLR from “scratch” and, as a
consequence, consuming extra resources (time and effort). This problem is also
connected with a social problem because the low quality of SLR also directly
impacts the confidence in the results and generates doubts in readers about
biases that could be occurred during the SLR conduction. Jointly to the lack
of practical recommendations for SE practitioners in such documentation [16,
6], in many cases, SLR are not used source of evidence for decision-making
process [13], which could improve the productivity of industry and academia.
Aiming to reduce the time and effort spent on the SLR conduction and deliver
fast results, researchers have ignored important well-experimented guidelines
[6], which could ensure such quality, and have created poorly documented SLR
with shallow results, which are not useful in the end.

Based on the set of problems, their dimensions, and barriers, we proposed a
new view that could suggest to the researchers how to better deal with these
problems in an integrated way, as presented in the next section.

##  4 Sustainability of Systematic Literature Review

Before presenting the view of Sustainability of SLR, we provide a brief
overview of the topic of sustainability that has been already researched in
other areas and has served as a basis for our work.

###  4.1 Brief Overview of Sustainability

Sustainability started in the ecology context [23] and became popular in 1987
when the UN World Commission on Environment and Development published the
report “Our Common Future” (Brundtland Report) defining “sustainable
development” as a development that meets the needs of the present without
compromising the ability of future generations to meet their own needs [22].

Next, computer science researchers realized that sustainability could be
adapted to other contexts, such as SE [28]. In 2009, the International
Conference on Software Engineering (ICSE) started a special track called
“software engineering for the
planet”222https://www.cs.uoregon.edu/events/icse2009/specialSessions/#planet
discussing how software can interact with sustainability, including energy
consumption and the impact of software production in society [28]. In the next
years, the term “Sustainable Software” was interpreted in two ways [29]: (i)
the software code being sustainable; and (ii) the purpose of the software
being to support sustainability goals, i.e., to improve the sustainability of
humanity on our planet. Both interpretations must coincide in a software
system that contributes to a more sustainable life, i.e., sustainable software
is energy efficient, minimizes the environmental impact of the processes it
supports and has a positive impact on social sustainability and/or economics.
In 2015, the Karlskrona manifesto333http://www.sustainabilitydesign.org/ was
signed [20], creating awareness of the SE community on the need for
sustainability and becoming a non-functional requirement for software [30] and
aiming to support software engineers to reduce unnecessary resource
consumption.

To adequately address sustainability, it is usually broken down in different
dimensions [23]. The Brundtland Report [22] described sustainability in
ecology with three dimensions: social, economical, and environmental. Although
these dimensions have become mainstream throughout the literature, they are
not universal [23]. Some authors consider necessary to have additional
dimensions (e.g., institutional [24], cultural [25], or technical [21]). In
the SE community, the Karlskrona Manifesto included the individual and
technical dimensions to maintain human capital and ensure the longevity of
information, systems, and infrastructure and their adequate evolution with
changing surrounding conditions.

In summary, software construction has a well-defined process, consumes time
and effort, uses human resources, and its success depends on efficient tools,
human skills, and social aspects (e.g., participation of stakeholders). In
this perspective, researchers in the SE area already understood that this
problem must be solved using a systemic solution [20]. Coming from this
context, we have used all knowledge previously accumulated and well-
experimented in the environmental and SE area to propose a view of
sustainability that could be valid to SLR.

###  4.2 Sustainability of SLR and their Dimensions

Considering that the problems and barriers found in this work are associated
with economic, social, and technical perspectives, we broke down the
Sustainability of SLR into these three dimensions, as described below:

  * •

Social dimension: it addresses the human aspects associated with SLR, such as
communication and collaboration, and ensures effective participation of
stakeholders in the SLR process. This dimension focuses on ensuring that
current and future stakeholders (including both, academic community and SE
practitioners) have open access to the SLR results and ensuring that these
results could be easily used.

  * •

Economic dimension: it preserves the resources (time and effort) during the
conduction, audition, and update or replication of SLR by creating high-
quality SLR, i.e., this dimension ensures high-quality documentation, credible
databases, efficient search engines, and rigor in following the guidelines to
preserve reproducibility and auditability, minimizing unnecessary efforts.

  * •

Technical dimension: it preserves the reliability of supporting tools used to
conduct SLR, facilitating their use for the update or reconduction of those
SLR. This dimension must assure technical ways to minimize the efforts by
automating tasks and supporting their execution.

Based on these three dimensions, the view of Sustainability of SLR can be
understood as “the process and a set of actions to make it possible to
preserve SLR that endure over time (i.e., longevity) with less possible time
and effort consumed and an effective impact to the industry.”

Regarding other sustainability dimensions, we believe they are not directly
related to SLR. For instance, the environmental dimension handles the impacts
of human activities in natural resources (e.g., water, land, air) or, in SE,
it manages the excessive amount of energy consumption caused by software. We
did not include such a dimension because there is no clear connection between
SLR and the environment. Therefore, for the while, we believe that the three
dimensions seem to be sufficient to connect SLR problems and understand their
connections and possible ways to achieve sustainable SLR.

Following this, we develop a discussion about how to better represent
graphically the Sustainability of SLR. For this, we revisit studies in the
literature from other areas [31, 22, 23] and observed that the sustainability
dimensions are described interconnected among them [23]. Therefore, an
important discussion in the sustainability context is how to make a fair
graphical representation including all dimensions defined and how they
interact with each other. The Brundtland Report [22] proposed a representation
(i.e., a Venn diagram with interlaced circles and the intersection between
them) that became the most popular one for sustainability in the ecological
area. The key idea is that human society is only sustainable if it can be
sustained in all dimensions.

Figure 2 shows a graphical representation of Sustainability of SLR using the
Venn diagram. Figure 2.A represents the Ballanced View with an intersection of
all dimensions. We can observe that attributing the same weight for all
dimensions and balancing them to achieve sustainability may lead to questions
about its feasibility, due to the large number of factors that impact each
dimension. An alternative is the Unballanced View that considers the technical
dimension as a common concern that cannot be dissociated from social and
economic perspective, as shown in Figure 2.C.We consider that both
representations (shown in Figures 2.A and 2.C) present a “Weak Sustainability”
(which was previously defined in [32]) because solving all problems associated
only with, for instance, social dimension does not assure to achieve a
sustainable SLR.

In an opposite perspective, Giddings et al [31] proposed a different
representation that considers that dimensions must be nested and with
different weights. This representation assumes that it is not possible for the
society and economy to develop outside the biosphere [33] and this perspective
warns that a finite planet cannot sustain human life with an economy that
intends unlimited growth. It is necessary to acknowledge that there are
fundamental biophysical limits that constrain the natural resources on the
planet [34]. This perspective arranges the sustainability dimensions providing
a hierarchic view, i.e., problems of each dimension must be treated by
different disciplines and each decision must consider the constraints of each
dimension [20] and this representation was later called “Strong
Sustainability” [33] .

Figure 2.B presents the Sociocentric View that nests the dimensions
considering the social dimension is the broadest and technical one is the most
specific, so attributing different weights for them. This representation
assumes technical problems are essentially connected with economic problems,
and the social constraints must guide the decision-making of researchers while
conducting SLR. Despite this representation seems to be fair for SLR, it may
be a problem to put social priorities higher than economic needs. For example,
the software industry (which is tightly associated with the social dimension)
needs to speed up the SLR conduction; however, it may cause a lack of rigor in
documentation and reliability that could directly impact the feasibility and
value the of SLR and, as a consequence, it may result in unsustainable SLR.
Figure 2.D presents the Econocentric View that considers that social needs are
constrained by economic aspects. Due to the resources constraints sometimes
imposed by the economic dimension, researchers are prone to conduct SLR
ignoring the social concerns. An example is not to include a given search
question that could indeed answer the doubt from a wider community only to
speed up the SLR process.

In the end, the graphical representations of Sustainability of SLR intend to
provide us the opportunity for better understanding the inherent difficulties
to create hierarchies among the dimensions and to better balance them.
Considering the current scenario of SLR, we believe that it is necessary to
move backward and recognize that the problems cannot be solved without
integrated thinking.

Figure 2: An View of Sustainability of SLR

##  5 Discussion

We believe that this paper provides a good starting point and raises awareness
of the community about the current problems of SLR from a holistic panorama.
Many of the barriers have been mentioned since 2010 but not solved yet, in
particular, the need for adequate synthesis of SLR results to be suitable for
the industry. Our work indicates there is also a lack of: (i) knowledge about
the SLR conduction process; (ii) application of the good practices for SLR
conduction; (iii) details and important information to reproduce and update
SLR; and (iv) tools’ integration. We also observed many barriers found are
related to the conduction process, while few social barriers are reported
indicating a lack of knowledge about social problems.

We believe this paper is provocative in the sense that it could bring a
disruptive view and, as a consequence, new and coordinated actions, and
solutions to make SLR reduce the high consumption of resources while producing
useful results for the industry and academia. In this context, create
effective methods to include the stakeholders (i.e., members of the industry)
could be one of the possible solutions for social dimension problems. Economic
dimension must create feasible methods to support the production of high-
quality SLR consuming fewer resources (time and effort) to the conduction and
update. For this, it is necessary to overcome many challenges including the
researcher’s mindset about the importance of correctly documenting SLR using
the best practices available for reporting and include SLR sustainability as a
quality criterion. The technical dimension must support tools and technologies
used to plan/conduct/report aiming to ensure its quality, accessibility, and
integration. Furthermore, this dimension must handle the slow and fragmented
development of tools to support SLR and propose solutions towards integration.
For this, it is necessary an international collaboration to provide long-term
support and integration of SLR tools. Finally, the main challenge of
Sustainability of SLR is to change the researchers’ mindset about the
importance of considering this integrated and holistic view while conducting
SLR and proposing new solutions to overcome the current problems.

Regarding the threats to the validity, we did not perform an exhaustive search
for studies that could have reported other problems and barriers in SLR. It is
possible some problems and barriers have been missed. Besides, our work
considered a 10-year period (2010-2020); hence, it is possible that problems
and barriers reported before this period have not been considered. To mitigate
these two threats, we used the experts’ opinions to validate the set of
studies considered in our analysis. Moreover, the results presented in this
work could have been influenced by our knowledge and experience in researching
and conducting SLR. To mitigate this threat, we strictly based on evidence
from the literature to make any decisions.

##  6 Final Remarks

The academic community has increasingly conducted SLR, but they still suffer
from critical problems, and the several punctual solutions being proposed for
those problems have not been widely effective. This paper is placed in this
context and intends to somehow be provocative to call the attention of this
community to change its mindset regarding the conduction and use of SLR
results. Derived from the sustainability concept from other areas, the
Sustainability of SLR primarily intends to achieve sustainable SLR, i.e., low
consumption of resources (time and effort) and with useful results for the
industry. As future work, the community needs to work together to promote
sustainable SLR and, as a consequence, to achieve the initial goal of SLR that
is the support of the industry.

##  7 Acknowledgements

This study was supported by FAPESP (grants: 2015/24144-7, 2019/23663-1), Capes
(grants: PROEX-11308091/D, PROEX-11357580/D), and CNPq (grant: 312634/2018-8).

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