#  Effects of underlying topology on quantum state discrimination

Aatif Kaisar Khan<EMAIL_ADDRESS>Perimeter Institute for Theoretical
Physics, Waterloo, Ontario N2L 2Y5, Canada  Yasir Hassan Dar
<EMAIL_ADDRESS>Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5, Canada  Elias C. Vagenas<EMAIL_ADDRESS>Department of Physics, College of Science, Kuwait University, Sabah Al Salem
University City, P.O. Box 2544, Safat 1320, Kuwait  Salman Sajad Wani
<EMAIL_ADDRESS>Qatar Center for Quantum Computing, College of Science
and Engineering,
Hamad Bin Khalifa University, Qatar  Saif Al-Kuwari<EMAIL_ADDRESS>Qatar Center for Quantum Computing, College of Science and Engineering,
Hamad Bin Khalifa University, Qatar  Mir Faizal  Canadian Quantum Research
Center 204-3002, 32 Ave Vernon, BC V1T 2L7 Canada  Irving K. Barber School of
Arts and Sciences, University of British Columbia Okanagan Campus, Kelowna, BC
V1V1V7, Canada<EMAIL_ADDRESS>
###### Abstract

In this work, we show that quantum state discrimination can be modified due to
a change in the underlying topology of a system. In particular, we explicitly
demonstrate that the quantum state discrimination of systems with underlying
discrete topology differs from that of systems with underlying continuous
topology. Such changes in the topology of a spacetime can occur in certain
quantum gravity approaches. In fact, all approaches to quantum gravity can be
classified into two types: those with underlying continuous topology (such as
string theory) and those with an underlying discrete topology (such as loop
quantum gravity). We demonstrate that the topology of these two types of
quantum gravity approaches has different effects on the quantum state
discrimination of low-energy quantum systems. We also show that any
modification of quantum mechanics, which does not change the underlying
topology, does not modify quantum state discrimination.

In quantum mechanics, any physical quantity is calculated using the quantum
state of the system. However, to perform such calculations (and henceforth,
measurements based on the results of such calculations), we need to identify
the quantum state of the system. Performing a small number of measurements on
the system to obtain its quantum state is called quantum state discrimination
5a ; 5b . Although quantum state discrimination has been used in important
physical settings, such as noise-robust quantum networks a0 , noisy quantum
neural networks a1 , quantitative wave-particle duality a2 , decomposable
entanglement witness a3 , entanglement swapping a4 , and no-signaling bound a5
, the effect of the underlying topology of the system on the quantum state
discrimination has never been studied. Therefore, in this paper, we analyze
the effect of the underlying topology on the quantum state discrimination and
deduce that the underlying topology of the system can non-trivially modify the
quantum state discrimination. This observation can have important implications
for quantum gravity phenomenology.

The foundations of modern physics are generally based on two important
theories: quantum mechanics and relativity. Although quantum mechanics and
special relativity can be reconciled in the framework of quantum field
theories 1 , this is not true for general relativity and quantum mechanics 2 .
Therefore, in the absence of a complete and self-consistent theory of quantum
gravity, various approaches have been developed, such as string theory 3 ,
loop quantum gravity 4 , Euclidean quantum gravity 5 , Wheeler-DeWitt approach
6 , spin foam 7 , and causal sets 8 . While the underlying topology of a
spacetime is taken as continuous, there are indications that this could only
be an approximation. Thus, all these approaches can be broadly classified into
two main classes. In the first class, the spacetime emerges from a discrete
structure, such as loop quantum gravity 4  and in the second class, the
spacetime is continuous, such as string theory 3 . We emphasize that the
spacetime structure could still have an intrinsic cutoff, in the form of a
minimal length, but it is not discrete spacetime. There are ways to
incorporate a cutoff in the form of a minimal length or even a minimal time
without making the spacetime topology discrete. This is done in the context of
string theory using T-duality 9 . Due to T-duality, there is a minimal scale
for gaining new information beyond which no new information is obtained.
Hence, string theory cannot be probed below the minimal scale. This scale is a
sophisticated way to incorporate the naturally expected result that string
theory cannot be probed below. This is also the case in perturbative string
theory, where a fundamental string is the only quantum object available, and
it is impossible to probe spacetime below the length of this fundamental
string 10 . However, due to T-duality, these results also hold when non-
perturbative point-like objects, such as D0-branes, are considered 11 . At
this point, it is worth of noting that it has been possible to study, the
T-duality in a covariant formulation of string theory ref1 ; ref2 ; ref3 . In
this formalism, the properties of string theory can be related to the
foundations of quantum theory in terms of modular variables. In addition, it
has been demonstrated that it can help understand aspects of quantum non-
locality and it is also consistent with causality ref4 .

In this context, the principle of relative locality has been proposed to
construct a spacetime with minimal length but without a discrete topology 1b .
The main difference between a cutoff in the form of discrete spacetime and a
minimal length is that, in discrete spacetime, the spacetime emerges from a
discrete lattice. The lattice points are fixed, and the emergent structure
appears as a continuum at a scale large enough to neglect this underlying
discrete topology. On the other hand, it is not possible to define any short-
distance structure from which spacetime emerges in other more sophisticated
approaches to quantum gravity that have an intrinsic minimal length. Thus,
spacetime can be measured or even meaningfully defined, only up to a certain
length scale. It is evident that this is a rather small difference, but it
does produce different types of corrections to low-energy quantum systems.
These corrections can then be used to discriminate between these two classes
of quantum gravity approaches. Although, at present, we do not know which
approach to quantum gravity is the correct one, here we propose that we can
observe completely different effects on the quantum state discrimination of
these two classes. Both these classes have low-energy implications; however,
the class of quantum gravity approaches with continuous spacetime topology
produces a quartic correction to the low-energy effective field theory
describing simple quantum systems 1c ; 1c1 ; 1c2 ; 1c4 , whereas the class of
quantum gravity approaches with discrete topology produces a cubic correction
to the low-energy effective field theory describing simple quantum systems 1d
; 1d1 ; 1d2 ; 1d4 ; Khan:2022yfm .

In fact, these cubic corrections occur in any quantum system with underlying
discrete topology as the leading order corrections of the effective theory are
considered. They have even been observed in effective field theory describing
condensed matter systems like graphene, due to the underlying discrete
structure of such condensed matter systems grp1 ; grp2 . Similarly, the
quartic corrections occur in any quantum system with underlying continuous
topology as the leading order correction of the effective field theory, such
as non-linear optics eff1 ; eff2 . This observation can be used to critically
modify the quantum state discrimination. Thus, quantum state discrimination
will be different for theories whose leading order correction term is a cubic
correction term compared to those whose leading order correction term is
quartic. Several works have been proposed to test the effects of both quartic
and cubic corrections on quantum systems, and various tabletop experiments
have been proposed to detect such corrections 2a ; 2b ; 2c ; 2d . Despite the
fact that these corrections have not been detected to date, such a
phenomenological approach to quantum gravity has been used to propose bounds
on the scale at which quantum gravitational effects become important. Here, we
do expect quantum gravitational effects to become important at the Planck
scale. However, no rigorous arguments prevent such effects from becoming
important at scales much larger than the Planck scale 4a ; 4b . Various
experiments have been proposed to test these effects, but this is the first
time that quantum state discrimination has been directly related to various
quantum gravitational approaches. In particular, we present a novel method
that can be used to at least show which class of quantum gravity approaches
actually “represents” quantum gravity. Since string theory and loop quantum
gravity are placed in different classes, we can at least deduce which of these
two theories can “represent” quantum gravity. However, we point out that
quantum state discrimination cannot be used to separate two quantum gravity
approaches that both have an underlying discrete structure or two quantum
gravity approaches with spacetime that is not discrete at any scale. Hence,
for example, the quantum state discrimination will not be modified between the
causal set approach and loop quantum gravity, or between string theory and
Euclidean quantum gravity. However, the subtle observation that quantum state
discrimination changes between a wide class of quantum gravity approaches is a
novel result and can help determine the future direction of research in
quantum gravity.

We start by analyzing the corrections to the standard quantum mechanics from
quantum gravity. These quantum gravity corrections depend on the underlying
topology of the theory and, thus, they will be different for different quantum
gravity approaches. It is known that quantum gravity approaches, such as
string theory, modify the Heisenberg algebra, and this modified Heisenberg
algebra produces a quartic correction term in the Schrödinger equation 1c ;
1c1 ; 1c2 ; 1c4 . However, quantum gravity approaches, such as loop quantum
gravity, modify the Heisenberg algebra in such a way that the Schrödinger
equation is modified by a cubic correction term 1d ; 1d1 ; 1d2 ; 1d4 . Here,
we will include both quartic 1c ; 1c1 ; 1c2 ; 1c4  and cubic 1d ; 1d1 ; 1d2 ;
1d4  correction terms to the Schrödinger equation, and, thus, we will write it
in the form:

$\biggr{(}-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-\frac{i\alpha\hbar^{3}}{m}\frac{d^{3}}{dx^{3}}+\frac{\beta\hbar^{4}}{m}\frac{d^{4}}{dx^{4}}+V(x)\biggl{)}\psi(x)=E\psi(x)$
(1)

where $\alpha$ and $\beta$ parameterize the scales at which such corrections
occur. These parameters are fixed by current experimental data 2a ; 2b ; 2c ;
2d . As stated, the cubic corrections occur due to the underlying discrete
topology and, thus, have been observed as higher order corrections to certain
condensed matter systems grp1 ; grp2 . Similarly, quartic corrections occur
for systems with underlying continuous topology, such as non-linear optics
eff1 ; eff2 . Hence, even though these corrections were initially motivated by
quantum gravity approaches 1c ; 1c1 ; 1c2 ; 1c4 ; 1d ; 1d1 ; 1d2 ; 1d4 , we
keep our discussion general, as they are not limited to quantum gravity. In
fact, we consider both of them for completeness, which allows us to analyze
their relative effects on the quantum system. At this point, we should point
out that in the framework of the statistical mechanics of D0-branes, a similar
to Eq. (1) expansion with higher derivative corrections in momenta was
obtained ref8 ; ref8a ; ref8b ; ref8c ; ref8d . As expected (due to the
spacetime being continuous in this model), the leading order correction is a
quartic correction term. In this model, explicit expansion has been obtained
for higher powers of momentum, and it has been observed that this model
contains only even powers of momentum.

To use quantum state discrimination 5a ; 5b , we consider a Schrödinger
equation with a double-delta well potential. This double-delta potential can
be used to realize several interesting physical systems, such as neutron-bound
states n1 , PT-symmetric Bose-Einstein condensate n2 , and the Ramsauer-
Townsend effect n4 . In these physical systems, the effects of the modified
quantum mechanical behavior on the quantum state discrimination can be tested.
The specific potential $V(x)$ that represents a double-delta well with the
wells to be at $\pm L$, is given by (see Fig. 1)

$V(x)=-W_{o}\,\delta(x+L)-W_{o}\,\delta(x-L)~{}.$ (2)

Figure 1: Double-Delta Potential Function

The quantum state of a system can be used to estimate any observable needs to
measure. However, to use a quantum state to estimate any observable, the
quantum state of the system itself has to be identified. In the absence of
prior information, it is not possible to identify such a state. This is
achieved by quantum state discrimination, which provides a way to distinguish
between the different quantum states of a system. In quantum state
discrimination, the discrimination error is minimized via a specific process
that returns the minimum error in determining the quantum states h1 ; h2 .
Other ways to approach quantum state discrimination exist, such as the maximum
confidence, which gives the most likelihood of event detection in a
measurement process. However, in this work, we will focus only on the minimum
error discrimination for a two-state system h4  and consider the minimum error
discrimination for a two-state double-delta well potential system. We start
with the prior information that the quantum system is in one of the two states
with associated probabilities of occurrence: $p_{0}$ and $p_{1}$,
respectively. The probability of making an error in determining a state,
$P_{Err}$, is given by 5a ; 5b

$P_{Err}=p_{0}-Tr\left((p_{0}|\Psi_{0}^{\alpha}\rangle\langle\Psi_{0}^{\alpha}|)-p_{1}|\Psi_{1}^{\alpha}\rangle\langle\Psi_{1}^{\alpha}|\pi_{0})\right)$
(3)

where, $p_{0}$ and $p_{1}$ denote the probability of occurrence of the states
$|\Psi_{0}^{\alpha}>$ and $|\Psi_{1}^{\alpha}>$, respectively. In addition,
$p_{0}+p_{1}=1$ since there are only these two states. Furthermore, we assume
that for the probability operator $\pi_{0}$, the outcome is $0$ with the
associated state being $\Psi_{0}^{\alpha}$ while the probablility outcome for
$\pi_{1}$ is $1$ with the associated state being $\Psi_{1}^{\alpha}$. The
minimum error for determining the state for this two-state system is given by
the Helstorm bound h1 ; h2

$P_{Min\,Err}=\frac{1}{2}(1-\sqrt{1-4p_{0}p_{1}|<\Psi^{\alpha}_{0}|\Psi^{\alpha}_{1}>|^{2}})~{}.$
(4)

In the case of the double-delta potential, these two wavefunctions represent
the two energy eigenfunctions, i.e, $|\psi^{\alpha}_{upper}>$ and
$|\psi^{\alpha}_{lower}>$. In general, for the standard Schrödinger equation,
$|\Psi^{\alpha=0}_{0}>$ and $|\Psi^{\alpha=0}_{1}>$ can be represented as
$|\Psi^{\alpha=0}_{0}>=\cos\theta_{0}|\psi^{\alpha=0}_{upper}>+\sin\theta_{0}|\psi^{\alpha=0}_{lower}>$
and
$|\Psi^{\alpha=0}_{1}>=\cos\theta_{1}|\psi^{\alpha=0}_{upper}>-\sin\theta_{1}|\psi^{\alpha=0}_{lower}>.$,
repspectively. Thus, we can write
$<\Psi^{\alpha=0}_{0}|\Psi^{\alpha=0}_{1}>=\cos(\theta_{0}+\theta_{1})$. Using
these expressions, we obtain
$P_{Min\,Err}^{\alpha=0}=(1-\sqrt{1-4p_{0}p_{1}\cos^{2}(\theta_{0}+\theta_{1})})/2$.

Despite the fact that quantum state discrimination has been used to address
various important physical problems, such as noise-robust quantum networks a0
, noisy quantum neural networks a1 , quantitative wave-particle duality a2 ,
decomposable entanglement witness a3 , entanglement swapping a4 , and no-
signaling bound a5 , the effect of the modification of quantum mechanics on
quantum state discrimination has never been studied. Quantum state
discrimination can be utilized to study effects due to quantum mechnical
modifications on such systems. In this work, we use the quantum discrimination
that has been formally developed as a field 5a ; 5b  to analyze the effect of
such correction terms. For the Schrödinger equation corrected by cubic and
quartic corrections terms, the wavefunction will be modified (see Appendix A
for cubic and Appendix B for quartic correction terms). We let
$|\Psi^{\alpha}_{0}>$ and $|\Psi^{\alpha}_{1}>$ to be general modified states
given by
$|\Psi^{\alpha}_{0}>=\cos\theta_{0}|\psi^{\alpha}_{upper}>+\sin\theta_{0}|\psi^{\alpha}_{lower}>$
and
$|\Psi^{\alpha}_{1}>=\cos\theta_{1}|\psi^{\alpha}_{upper}>-\sin\theta_{1}|\psi^{\alpha}_{lower}>$.
Thus, again we have
$<\Psi^{\alpha}_{0}|\Psi^{\alpha}_{1}>=\cos(\theta_{0}+\theta_{1})$. Using the
corrected Schrödinger equation (with both cubic and quartic corrections
obtained in Appendix A and B), we obtain
$P_{Min\,Err}^{\alpha}=(1-\sqrt{1-4p_{0}p_{1}\cos^{2}(\theta_{0}+\theta_{1})})/2$.
This expression is similar to the expressions obtained for the standard
Schrödinger equation. Thus, a modification of the Schrödinger equation does
not have any direct effect on quantum state discrimination.

It should be stressed that the cubic corrections are only consistent with the
discrete underlying topology (see Appendix A), and a change in the underlying
topology can produce a modification to the quantum state discrimination. On
the contrary, the quartic corrections do not produce such topological changes
and the topology of spacetime remains continuous. However, here we observe the
discrimination between the quantum states at shorter lengths between the wells
compared to the cubic corrections or the standard case (see Appendix B).
Therefore, for analyzing the effects of topology on quantum state
discrimination, we neglect the quartic correction terms. We note that cubic
correction terms follow directly from the discrete underlying topology of
spacetime. In fact, cubic correction terms in the low-energy effective field
theory of any system imply that the underlying topology is discrete 1d ; 1d1 ;
1d2 ; 1d4 ; Khan:2022yfm . This has been done by first observing that any
physical length can in principle be represented by an infinite well (as the
wavefunction of any particle used to probe such a length will vanish on its
boundaries). Then, it was observed that any such length is a multiple of a
fixed small quantized length if cubic corrections are considered. As this
result was demonstrated to hold for any arbitrary length, we conclude that
this could only be possible if any length (or even any physical quantity,
which is expressed as a function of length) is discrete, and hence such cubic
correction would require discrete underlying topology of spacetime.

Consequently, we set in Eq. (1) the scale parameters to $\alpha\neq 0$ and
$\beta=0$. First, we generalize the results to a double-delta potential n1 ;
n2 ; n4 , and, then, demonstrate that, like an infinite well, this potential
also leads to discreteness. With $k\equiv\sqrt{{-2mE}/{\hbar^{2}}}$, and
$V(x)=0$, we can write $\psi(x)=Ae^{i\gamma x}+Be^{k_{1}x}+Ce^{-k_{2}x}$
where, $A$, $B$, and $C$ are constants, $\gamma\equiv{1}/{2\alpha\hbar},\quad
k_{1}\equiv k(1-i\alpha\hbar k)$, and $k_{2}\equiv k(1+i\alpha\hbar k)$. After
imposing the bound state condition, namely $\psi(-\infty)=\psi(\infty)=0$, the
wave-function becomes $\psi(x)=s_{1}e^{i\gamma x}+c_{11}e^{k_{1}x}$ for
$x<-L$, $\psi(x)=s_{2}e^{i\gamma x}+c_{21}e^{k_{1}x}+c_{22}e^{-k_{2}x}$ for
$x\in(-L,L)$ and $\psi(x)=s_{3}e^{i\gamma x}+c_{32}e^{-k_{2}x}$ for $x>L$
where, $s_{1}$, $s_{2}$, $s_{3}$, $c_{11}$, $c_{21}$, $c_{22}$, and $c_{32}$
are all constants of integration. We restrict $c_{11}$ to real values (by
eliminating any imaginary phase associated with it) and, which implies further
calculations. For the standard (unmodified) system, the $s_{i}$’s are all
zero. This can also be seen as a necessary condition for the wavefunction to
be physical. Let us now take $s_{1}e^{i\gamma x}$ as an example to demonstrate
this. By Taylor expansion, we obtain
$s_{i}[ix/2\alpha\hbar)+-x^{2}/4\alpha^{2}\hbar^{2}+O(1/\alpha^{2})$]. Thus,
as $\alpha\to 0$, it is obvious that $s_{i}$ must also vanish in order to have
a well-defined wavefunction. This means that given $\alpha<<1$, we must also
have $s_{i}<<1$. Consequently, any term which has $s_{i}\alpha$ or any higher
power of $s_{i}$ and $\alpha$ will vanish (this will be used to simplify
expressions at various places, such as in Eq. (18)). Hence, we get
$\lim_{\alpha\to 0}s_{i}=0\text{, for i=1,2,3}$. Therefore, for sufficiently
small $\alpha$, $s_{i}$ will also be small
$s_{i}\alpha^{m}=0=s_{i}^{n}\quad\forall\,m\geq 1,\,n>1$. To simplify the
calculations, we define the following new quantities
$\bar{L}\equiv{mW_{0}L}/{\hbar^{2}},\,\,\bar{\gamma}\equiv\gamma
L={L}/{2\alpha\hbar},$ and $\bar{k}\equiv kL$. After fixing the arbitrary
constants, the new length $\bar{L}$ now reads (see Appendix A)

$\bar{L}=\frac{\bar{k}\mp\frac{\bar{k}s_{1}}{c_{11}}\bigg{[}\frac{2\bar{L}}{\bar{k}}e^{-(\bar{k}+i\bar{\gamma})}+\bigg{(}1-\frac{\bar{L}}{\bar{k}}\bigg{)}e^{\bar{k}}(e^{i\bar{\gamma}}\mp
e^{-i\bar{\gamma}})\bigg{]}}{(1\pm e^{-2\bar{k}})}~{}.$ (5)

As the above equation originates in the quadratic equation solution, it will
be represented by two different wavefunctions (which is also the case in the
non-deformed Heisenberg algebra). The modifications produced in them are
represented by the terms proportional to $s_{1}$. Here, $\bar{L}$ for the
upper sign and the lower sign in the equation will be represented by
$\bar{L}_{u}$ and $\bar{L}_{l}$, respectively. Equating the imaginary parts of
Eq. (5) and that $s_{1},c_{11}\in\mathbb{R}$, we find the condition, which
holds for both these wavefunctions: $L=2n\pi\alpha\hbar$ or
$\bar{L}=2n\pi\alpha mW_{0}/\hbar$, with $n\in\mathbb{N}$. Thus, any arbitrary
length $L$ can only be a multiple of a fixed quantized value. Similar to the
infinite well potential, we here, too, obtain the expected result, i.e., the
cubic terms are produced by a discrete underlying topology. This discrete
nature of space is constructed by a quantum of length, given by
$2\pi\alpha\hbar$ (similar results can be found in Das:2020ujn ). We also
observe that the quantum of length scales with $\alpha$, and, thus, it could
only come from the cubic correction terms. Furthermore, there is no
$\alpha^{2}$ contribution to the quantum of length, which means that there is
no contribution from quartic correction terms; this is expected as such terms
were motivated by theories with continuous underlying topology.

We now consider the case where the scale parameters are set to $\alpha=0$ and
$\beta\neq 0$, and the potential is $V(x)=0$. Hence, we have $\psi(x)=Ce^{\eta
x}+D^{-\eta x}+Ae^{bx}+Be^{-bx}$, where
$\eta\equiv(\hbar\sqrt{2\beta})^{-1},\,b\equiv k+\hbar^{2}k^{3}\beta$ and
$A,\,B,\,C$, and $D$ are constants. After implementing the boundary conditions
$\psi(-\infty)=\psi(\infty)=0$, we get, $\psi(x)=s_{11}e^{\eta
x}+c_{11}e^{bx}\,\text{for}\,x<-L$, $\psi(x)=s_{21}e^{\eta x}+s_{22}e^{-\eta
x}+c_{21}e^{bx}+c_{22}e^{-bx}\,\text{for}\,x\in(-L,L)$, and
$\psi(x)=s_{32}e^{-\eta x}+c_{32}e^{-bx}\,\text{for}\,x>L$, where
$s_{11},\,s_{21},\,s_{22},\,s_{32},\,c_{11},\,c_{21},\,c_{22},\,\text{and}\,c_{32}$
are constants of integration. Also, we have $s_{11}/c_{11}>0$ (see Appendix
B). In addition, similar to the $s_{i}$’s in the cubic correction case, here
$s_{ij}\to 0$ as $\beta\to 0$. Therefore, when $\beta$ is small, $s_{ij}$’s
will also be small. Thus, we have $s_{ij}\beta^{m}=0=s_{ij}^{n},\forall\,m\geq
1$ and $n>1$. Here, again, we obtain a condition that the length has to
satisfy, which is given by (see Appendix B)

$\bar{L}=\frac{\bar{k}-\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}-\frac{s_{11}}{c_{11}}\bar{k}e^{-\frac{\bar{L}}{\sqrt{2\lambda}}+\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}}}{(1\pm
e^{-2(\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}})})}$ (6)

where $\lambda\equiv\beta m^{2}W_{0}^{2}/\hbar^{2}$. It is clear from this
equation that everything here is real and, thus, consistent with the
continuous underlying topology. Hence, we conclude that quartic correction
terms are consistent with the continuous underlying topology of a spacetime.
We now consider the minimum length between the wells, which is necessary for
the existence of a particular wavefunction, represented by $\bar{L}_{0}$. In
the standard (unmodified) case and the case involving cubic correction terms,
$\bar{L}_{0}$ is $0$ and $1/2$ for the wavefunction obtained by taking the
upper sign and the lower sign, respectively, in Eq. (5) (see Appendix A).
However, for the case where quartic correction terms are considered, we
observe a difference, namely $\bar{L}_{0}$ is $0$ and
$(1-(s_{11}/c_{11})\exp{({-\bar{L}_{0}/\sqrt{2\lambda}}}))/2$ for the
wavefunction obtained by taking the upper sign and the lower sign,
respectively, in Eq. (6) (see Appendix A). Hence, the effect of the quartic
correction terms on the wavefunction occurs before the effect of the quartic
cubic correction terms on the wavefunction.

We now show that the discrete underlying topology directly modifies the
quantum state discrimination. This can be observed by starting with the
probabilities $p_{0}=p_{1}={1}/{2}$, and analyzing the effect of the discrete
underlying topology on them. Due to the discrete underlying topology, we have
$P_{Min\,Err}=(1-|\sin(\theta_{0}+\theta_{1})|)/2$ . This expression is valid
only for discrete values of $\bar{L}$, which are $\bar{L}=n({2\pi\alpha
mW_{0}}/{\hbar})$. This, along with the observation that for the two states to
exist, we need $2\bar{L}\geq 1$, leads to the condition (see Appendix A)

$n\geq\bigg{\lceil}{\frac{\hbar}{4\pi\alpha
mW_{0}}}\bigg{\rceil}\,\,\,\,\,\,\,\,\text{ or }\,\,\,\,\,\,\,\,L\geq
2\pi\alpha\hbar\bigg{\lceil}{\frac{\hbar}{4\pi\alpha
mW_{0}}}\bigg{\rceil}~{}.$ (7)

This changes the behavior of the quantum state discrimination, which can be
analyzed by plotting it. Consider, for our case, the minimum value of $n=3$ as
depicted in Fig. 2. From these figures, it can easily be seen that the
presence of $\alpha$ breaks the continuity of the plot, rendering it to be
observable only at discrete values. We also note that the distance between the
discrete points is directly related to $\alpha$. Hence, the quantum state
discrimination is only affected by models of quantum gravity with discrete
underlying spacetime topology. It does not get modified for a spacetime with
continuous underlying topology, and, hence, a quantum gravity approach, such
as string theory, will not change the behavior of the low-energy quantum state
discrimination of quantum mechanics. On the contrary, a quantum gravity
approach, such as loop quantum gravity, can indeed affect it. Here, it is
important to note that the original modification of the Heisenberg algebra did
not directly effect the quantum state discrimination, but it was the
discreteness of spacetime that modified it. Consequently, it would also be
modified for any system with discrete underlying topology, leading to
applications even beyond quantum gravity.

(a) $\alpha=0$ and $\beta=0$

(b) $\alpha\neq 0$ and $\beta=0$

(c) $\alpha=0$ and $\beta\neq 0$

Figure 2: Plots between $P_{Min\,Err}$ and $\theta_{0}+\theta_{1}$ for
different values of $\alpha$ and $\beta$

From Fig. 2, it can also be observed that the standard (unmodified)
Schrödinger equation ($\alpha=0$ and $\beta=0$) generates a plot with a
continuous underlying topology and the second wavefunction comes into
existence at $2\bar{L}=1$. The underlying topology remains continuous even if
we include the quartic correction term, namely with $\beta\neq 0$. What
significantly changes the underlying topology is the cubic correction term,
namely $\alpha\neq 0$. This produces a discrete structure of space and gives a
shelved architecture to the graph. Knowing that $\bar{L}=2n\pi\alpha
mW_{0}/\hbar$ with $n\in\mathbb{N}$, we first fix the scale parameter $\alpha$
to $\alpha=0.15/(2\pi mW_{0}/\hbar)$) and, subsequently, we check the effect
at $\alpha_{1}=\alpha$, $\alpha_{2}=\alpha/2$, and $\alpha_{3}=\alpha/3$. It
is clear from the graph that as $\alpha$ decreases, i.e., first from $\alpha$
to $\alpha/2$ and then to $\alpha/3$, these shelves converge as the unit of
length is directly proportional to $\alpha$. Also, at $\alpha=0$, the shelves
merge and form a continuous underlying structure, as expected.

Observations similar to the unmodified case can be found when the quartic
correction term, namely with $\beta\neq 0$, is considered, but with a subtle
difference such that the graph is shifted towards the left, that is, the
second wave function appears before $2\bar{L}=1$. Since the minimum length for
the second wavefunction has an opposite relation with $s_{11}$ (see Appendix
B), the graph moves towards $2\bar{L}=1$ ( unmodified case as the value of
$\beta$ decreases). To simplify the calculation, and since $s_{11}$ and
$\lambda$ are both of the order of $\beta$, we equate them. We also consider
$c_{11}=1$. Therefore, as the value of $\beta$ decreases from
$\beta_{1}=\beta$ to $\beta_{2}=\beta/2$ and then to $\beta_{3}=\beta/3$, it
can be clearly seen that initially the plot is dislocated to the left compared
to the unmodified case, and as the value of $\beta$ decreases, the plot tends
to converge toward the unmodified case, which is expected (here we have taken
$\beta=0.3\hbar^{2}/(m^{2}W_{0}^{2})$).

In conclusion, in this work, we proposed a novel method to test the effects of
quantum gravity on information theory. This method is based on quantum state
discrimination, which has been developed in the context of quantum information
theory. This is the first application of this method to quantum gravity. Thus,
we used quantum state discrimination to demonstrate that quantum gravity
approaches with an underlying discrete topology fundamentally differ from
quantum gravity approaches for which spacetime has a continuous underlying
topology. Consequently, the low-energy implications of approaches such as
string theory (with continuous underlying topology) on quantum state
discrimination would be very different from the low-energy implications of
approaches such as loop quantum gravity (with discrete underlying topology).
However, any modification that does not change the topology will not affect
the quantum state discrimination. These classes of models produce different
types of corrections to the low-energy effective field theory describing
simple quantum systems, and these corrections can be used to affect the
quantum state discrimination in a different way. We note that even though
quantum gravitational effects are expected to become important at the Planck
scale, they can also become important at scales much larger than the Planck
scale. Hence, it is generally possible to experimentally test the results of
the present work using currently available technologies. As quantum state
discrimination has been used to study noise-robust quantum networks a0 , noisy
quantum neural networks a1 , quantitative wave-particle duality a2 ,
decomposable entanglement witness a3 , entanglement swapping a4  and no-
signaling bound a5 , a modification to quantum state discrimination would also
modify these studies. In other words, it would be interesting to precisely
analyze how discrete topology can change their behavior.

Finally, it should be pointed out that the low-energy consequences of quantum
gravity need not be limited to higher derivative effects, but can produce more
fundamental corrections to quantum mechanics, such as higher-order
interference ref5 . These higher-order interference effects could even, in
principle, be measured ref6 , and could help understand low-energy
implications of quantum gravity beyond higher derivative corrections.
Moreover, it has been argued that such effects could occur in non-linear
optics ref7 . Actually, the use of non-linear optics as an analogue for
quantum gravity is well known ref6a ; ref6b . Therefore, it would be
interesting to investigate if such higher-order interference can also act as
an analogue of quantum gravitational effects.

## Acknowledgments

We would like to thank Jan Kolodynski for constructive discussions on the
quantum state discrimination.

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##  I Appendix A

Here we shall analyze the effects of cubic correction terms 1d ; 1d1 ; 1d2 ;
1d4  on the wavefunction and the underlying topology of spacetime. Thus, we
start from the cubic correction term of the form

$\frac{d^{2}\psi}{dx^{2}}+2i\alpha\hbar\frac{d^{3}\psi}{dx^{3}}-k^{2}\psi(x)+\frac{2mW_{0}[\delta(x+L)+\delta(x-L)]}{\hbar^{2}}\psi(x)=0$
(8)

with $k\equiv\sqrt{{-2mE}/{\hbar^{2}}}$. Solving the above equation for
$V(x)=0$, we get $\psi(x)=Ae^{i\gamma x}+Be^{k_{1}x}+Ce^{-k_{2}x}$ where $A$,
$B$, $C$ are constants, and $\gamma\equiv{1}/{2\alpha\hbar},\quad k_{1}\equiv
k(1-i\alpha\hbar k),\quad k_{2}\equiv k(1+i\alpha\hbar k)$. After imposing the
bound state condition, i.e., $\psi(-\infty)=\psi(\infty)=0$, the wave-function
looks as $\psi(x)=s_{1}e^{i\gamma x}+c_{11}e^{k_{1}x}$ for $x<-L$,
$\psi(x)=s_{2}e^{i\gamma x}+c_{21}e^{k_{1}x}+c_{22}e^{-k_{2}x}$ for
$x\in(-L,L)$ and $\psi(x)=s_{3}e^{i\gamma x}+c_{32}e^{-k_{2}x}$ for $x>L$
where $s_{1}$, $s_{2}$, $s_{3}$, $c_{11}$, $c_{21}$, $c_{22}$, and $c_{32}$
are all constants of integration. We restrict $c_{11}$ to take real values (by
eliminating any imaginary phase associated with it), and simplify further
calculations. We can first use suitable boundary conditions to fix the
arbitrary constants in the expression for the wavefunctions. Thus, using the
continuity of wavefunctions at $x=-L$ we get

$(s_{1}-s_{2})e^{-i\gamma
L}+(c_{11}-c_{21})e^{-k_{1}L}-c_{22}e^{k_{2}L}=0~{}.$ (9)

Similarly, at $x=+L$, we get

$(s_{2}-s_{3})e^{i\gamma L}+c_{21}e^{k_{1}L}+(c_{22}-c_{32})e^{-k_{2}L}=0~{}.$
(10)

Now, we analyze the first jump condition. So, integrating Eq. (8) from $x=\pm
L^{-}$ to $x=\pm L^{+}$, we get

$\psi^{\prime}(x)\bigg{|}^{\pm L^{+}}_{\pm
L^{-}}+2i\alpha\hbar\psi^{\prime\prime}(x)\bigg{|}^{\pm L^{+}}_{\pm
L^{-}}+\frac{2mW_{0}}{\hbar^{2}}\psi(\pm L)=0~{}.$ (11)

At $x=-L$, after finding $\psi^{\prime}(x)$ and $\psi^{\prime\prime}(x)$, and
using the smallness of $\alpha$ in Eq. (11), we obtain

$\frac{2mW_{0}}{\hbar^{2}}s_{1}e^{-i\gamma
L}+\bigg{(}c_{21}k+\bigg{(}\frac{2mW_{0}}{\hbar^{2}}-k\bigg{)}c_{11}\bigg{)}e^{-k_{1}L}-c_{22}ke^{k_{2}L}=0~{}.$
(12)

Similarly, at $x=+L$, from Eq. (11), we get

$\frac{2mW_{0}}{\hbar^{2}}s_{3}e^{i\gamma
L}-c_{21}ke^{k_{1}L}+\bigg{(}c_{22}k+\bigg{(}\frac{2mW_{0}}{\hbar^{2}}-k\bigg{)}c_{32}\bigg{)}e^{-k_{2}L}=0~{}.$
(13)

Thus, multiplying Eq. (9) by $k$ and subtracting it from Eq. (12), we obtain

$\bigg{[}\bigg{(}1-\frac{2mW_{0}}{\hbar^{2}k}\bigg{)}s_{1}-s_{2}\bigg{]}e^{(k_{1}-i\gamma)L}+2\bigg{[}\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{11}-c_{21}\bigg{]}=0$

which leads to

$s_{2}=\bigg{(}1-\frac{2mW_{0}}{\hbar^{2}k}\bigg{)}s_{1},\quad
c_{21}=\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{11}~{}.$ (14)

These equations are a result of the assumption that $c_{11}$ and $c_{21}$ are
deformed due to $\alpha$ in the same way, i.e.,
$\frac{c_{11}(\alpha)}{c_{21}(\alpha)}=\frac{c_{11}(\alpha=0)}{c_{21}(\alpha=0)}$.
This is quite intuitive as both $c_{11}$ and $c_{21}$ are coefficients of
$e^{k_{1}x}$. Thus, multiplying Eq. (10) by $k$ and adding it to Eq. (13), we
get

$\bigg{[}\bigg{(}1-\frac{2mW_{0}}{\hbar^{2}k}\bigg{)}s_{3}-s_{2}\bigg{]}e^{(k_{2}+i\gamma)L}+2\bigg{[}\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{32}-c_{22}\bigg{]}=0$

which leads to

$s_{2}=\bigg{(}1-\frac{2mW_{0}}{\hbar^{2}k}\bigg{)}s_{3},\quad
c_{22}=\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{32}~{}.$ (15)

Akin to how Eq. (14) is obtained, here a similar assumption is made regarding
$c_{32}$ and $c_{22}$, i.e.,
$\frac{c_{32}(\alpha)}{c_{22}(\alpha)}=\frac{c_{32}(\alpha=0)}{c_{22}(\alpha=0)}$,
as both $c_{32}$ and $c_{22}$ are coefficients of $e^{-k_{2}x}$.
Hence, from Eqs. (14) and (15), we get $s_{3}=s_{1}$. Now, using Eqs. (14) and
(15) in Eq. (9), we get

$\frac{2mW_{0}}{\hbar^{2}k}s_{1}e^{-i\gamma
L}+\frac{mW_{0}}{\hbar^{2}k}c_{11}e^{-k_{1}L}=\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{32}e^{k_{2}L}~{}.$

Also, knowing $s_{3}=s_{1}$ and using Eqs. (14) and (15) in Eq. (10), we get

$\frac{2mW_{0}}{\hbar^{2}k}s_{1}e^{i\gamma
L}+\frac{mW_{0}}{\hbar^{2}k}c_{32}e^{-k_{2}L}=\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{11}e^{k_{1}L}~{}.$

Dividing the above two equations with each other and using the smallness of
$\alpha$, we have

$\frac{2s_{1}e^{(k_{1}-i\gamma)L}+c_{11}}{2s_{1}e^{(k_{2}+i\gamma)L}+c_{32}}=\frac{c_{32}}{c_{11}}~{}.$

Let $A_{1}\equiv e^{(k_{1}-i\gamma)L}$ and $A_{2}\equiv e^{(k_{2}+i\gamma)L}$,
and using them in the above equation, we get
${c_{32}}^{2}+(2A_{2}s_{1})c_{32}-(2A_{1}s_{1}c_{11}+c_{11}^{2})=0$. Thus,
this equation is quadratic in $c_{12}$, so after solving it, we get
$c_{12}=(-2A_{2}s_{1}\pm\sqrt{(2A_{2}s_{1})^{2}+4(2A_{1}s_{1}c_{11}+c_{11}^{2})})/2~{}.$
Now, since $s_{i}$ is small, it is legitimate to consider $s_{1}^{2}\approx 0$
and, hence, we have $c_{32}=-A_{2}s_{1}\pm
c_{11}(1+2A_{1}s_{1}/c_{11})^{1/2}$. Consequently, after using Taylor
expansion of the second term and with $s_{1}^{2}\approx 0$, we get
$c_{32}=\pm(c_{11}+(A_{1}\mp A_{2})s_{1})$.
Therefore, utilizing it in Eq. (15) to eliminate $c_{32}$, we get
$c_{22}=\pm(1-mW_{0}/\hbar^{2}k)(c_{11}+(A_{1}\mp A_{2})s_{1})$. Using now
this result and Eq. (14) in Eq. (12) to eliminate $c_{22}$ and $c_{21}$, we
obtain

$\frac{2mW_{0}}{\hbar^{2}}s_{1}e^{-i\gamma
L}+\bigg{(}\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}c_{11}k+\bigg{(}\frac{2mW_{0}}{\hbar^{2}}-k\bigg{)}c_{11}\bigg{)}e^{-k_{1}L}-(\pm(1-\frac{mW_{0}}{\hbar^{2}k})(c_{11}+(A_{1}\mp
A_{2})s_{1}))ke^{k_{2}L}=0~{}.$

Rearranging the terms of the above equation, the equation now reads

$\bigg{(}\frac{2mW_{0}}{\hbar^{2}}e^{-i\gamma L}+(A_{2}\mp
A_{1})(1-\frac{mW_{0}}{\hbar^{2}k})ke^{k_{2}L})\biggr{)}\frac{s_{1}}{c_{11}k}+\biggr{(}\frac{mW_{0}}{\hbar^{2}k}e^{-k_{1}L}\mp(1-\frac{mW_{0}}{\hbar^{2}k})e^{k_{2}L}\bigg{)}=0~{}.$

Thus, we get

$\pm\frac{s_{1}}{c_{11}}\frac{\eta}{k}e^{-k_{2}L}+\frac{mW_{0}}{\hbar^{2}k}(1\pm
e^{-2kL})=1,\quad\text{where}\quad\eta\equiv\frac{2mW_{0}}{\hbar^{2}}e^{-i\gamma
L}+(A_{2}\mp A_{1})\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}ke^{k_{2}L}~{}.$
(16)

At this point, we introduce the quantities

$\displaystyle\bar{L}\equiv\frac{mW_{0}L}{\hbar^{2}},\quad\quad\bar{\gamma}\equiv\gamma
L=\frac{L}{2\alpha\hbar},\quad\quad\bar{k}\equiv
kL,\quad\quad\bar{k_{1}}\equiv k_{1}L,\quad\quad\bar{k_{2}}\equiv k_{2}L~{}.$
(17)

Now, as $s_{1}\alpha\approx 0$, we can easily see that

$s_{1}e^{\pm c\alpha}=s_{1}(1\pm c\alpha...)\approx s_{1}~{}.$ (18)

Thus, using Eq. (18), we have

$\frac{1}{\bar{\gamma}^{2}}\approx 0\approx\frac{s_{1}}{\bar{\gamma}},\quad
s_{1}e^{\bar{k_{1}}}=s_{1}e^{\bar{k}}e^{-i\alpha\hbar k^{2}L}\approx
s_{1}e^{\bar{k}},\quad s_{1}e^{\bar{k_{2}}}\approx s_{1}e^{\bar{k}}~{}.$ (19)

Employing Eq. (17) in Eq. (16), we get

$\pm\frac{s_{1}}{c_{11}}\frac{\eta}{k}e^{-\bar{k_{2}}}+\frac{\bar{L}}{\bar{k}}(1\pm
e^{-2\bar{k}})=1~{}.$ (20)

Multiplying $\eta$ with $\frac{s_{1}}{c_{11}}\frac{e^{-\bar{k_{2}}}}{k}$ and
using Eq. (18), we get

$\frac{s_{1}}{c_{11}}\frac{\eta}{k}e^{-k_{2}L}=\frac{s_{1}}{c_{11}}\frac{2mW_{0}}{\hbar^{2}k}e^{-(k+i\gamma)L}\mp\frac{s_{1}}{c_{11}}(A_{1}\mp
A_{2})\bigg{(}1-\frac{mW_{0}}{\hbar^{2}k}\bigg{)}~{}.$ (21)

Utilizing the values of $A_{1}$ and $A_{2}$, Eqs. (17) and (19), and the
smallness of $\alpha$, we have $s_{1}(A_{1}\mp
A_{2})=s_{1}(e^{-i\bar{\gamma}}\mp e^{i\bar{\gamma}})e^{\bar{k}}$. Thus,
combining this equation with Eqs. (17) and (19) in Eq. (21), we get

$\frac{s_{1}}{c_{11}}\frac{\eta}{k}e^{-k_{2}L}=\frac{s_{1}}{c_{11}}\bigg{[}\frac{2\bar{L}}{\bar{k}}e^{-(\bar{k}+i\bar{\gamma})}+\bigg{(}1-\frac{\bar{L}}{\bar{k}}\bigg{)}e^{\bar{k}}(e^{i\bar{\gamma}}\mp
e^{-i\bar{\gamma}})\bigg{]}~{}.$

Now, using the above equation in Eq. (20), we get

$\frac{\bar{L}}{\bar{k}}=\frac{1\mp\frac{s_{1}}{c_{11}}\bigg{[}\frac{2\bar{L}}{\bar{k}}e^{-(\bar{k}+i\bar{\gamma})}+\bigg{(}1-\frac{\bar{L}}{\bar{k}}\bigg{)}e^{\bar{k}}(e^{i\bar{\gamma}}\mp
e^{-i\bar{\gamma}})\bigg{]}}{(1\pm e^{-2\bar{k}})}~{}.$ (22)

For the wavefunction with the upper sign, we now get

$\frac{\bar{L}_{u}}{\bar{k}}=\frac{1-\frac{s_{1}}{c_{11}}\bigg{[}\frac{2\bar{L}_{u}}{\bar{k}}e^{-(\bar{k}+i\bar{\gamma})}+\bigg{(}1-\frac{\bar{L}_{u}}{\bar{k}}\bigg{)}e^{\bar{k}}(e^{i\bar{\gamma}}-e^{i\bar{\gamma}})\bigg{]}}{(1+e^{-2\bar{k}})}~{}.$

Using $e^{-i\gamma}=\cos(\gamma)-i\sin(\gamma)$ and
$e^{i\gamma}-e^{-i\gamma}=2i\sin(\gamma)$, we get

$\frac{\bar{L}_{u}}{\bar{k}}=\frac{1-\frac{S_{1}}{C_{11}}(\frac{2\bar{L}_{u}e^{-\bar{k}}}{\bar{k}}(\cos(\gamma)-i\sin(\gamma))+(1-\frac{\bar{L}_{u}}{\bar{k}})e^{\bar{k}}(2i\sin(\gamma)}{1+e^{2\bar{k}}}~{}.$

Finally, we obtain

$\frac{\bar{L}_{u}}{\bar{k}}=\frac{1-\frac{2s_{1}}{c_{11}}\bigg{[}\frac{\bar{L}_{u}}{\bar{k}}e^{-\bar{k}}\cos{\bar{\gamma}}+i\bigg{(}1-\frac{\bar{L}_{u}}{\bar{k}}(1+e^{-2\bar{k}})\bigg{)}e^{\bar{k}}\sin{\bar{\gamma}}\bigg{]}}{(1+e^{-2\bar{k}})}~{}.$
(23)

Similarly, for the wavefunction with the lower sign in Eq. (22), we obtain

$\frac{\bar{L}_{l}}{\bar{k}}=\frac{1+\frac{2s_{1}}{c_{11}}\bigg{[}\bigg{(}1-\frac{\bar{L}_{l}}{\bar{k}}(1-e^{-2\bar{k}})\bigg{)}e^{\bar{k}}\cos{\bar{\gamma}}-i\frac{\bar{L}_{l}}{\bar{k}}e^{-\bar{k}}\sin{\bar{\gamma}}\bigg{]}}{(1-e^{-2\bar{k}})}~{}.$
(24)

We keep $c_{11}$ real by taking out any imaginary phase and we assume
$s_{1}\in\mathbb{R}$. Hence, equating the imaginary parts of Eq. (23), we
obtain

$\bigg{(}1-\frac{\bar{L}_{u}}{\bar{k}}(1+e^{-2\bar{k}})\bigg{)}e^{\bar{k}}\sin{\bar{\gamma}}=0$

which tells us either
$\frac{\bar{L}_{u}}{\bar{k}}=\frac{1}{(1+e^{-2\bar{k}})},$ or
$\sin{\bar{\gamma}}=0$. As the former one is in contradiction to Eq. (23),
since $s_{1}\neq 0$ in general, we therefore accept the latter one. Similarly,
equating the imaginary parts of Eq. (24), we also obtain
$\sin{\bar{\gamma}}=0$. Therefore, for both Eqs. (23) and (24), and with
$s_{1}\in\mathbb{R}$, we have

$\sin{\bar{\gamma}}=0\implies\bar{\gamma}=n\pi\quad\forall\,n\in\mathbb{N}\implies
L=2n\pi\alpha\hbar\quad\text{or}\quad\bar{L}=n\alpha\frac{2\pi
mW_{0}}{\hbar}~{}.$ (25)

Therefore, Eq. (25) shows the discrete nature of space by demonstrating the
discreteness of the length $L$ between the two delta-wells.

Now, let us see how the two wavefunctions come into being near $\bar{k}\to 0$.
For $\bar{k}\to 0$, the condition for the wavefunction described by Eq. (23)
looks like
$\bar{L}_{u0}=-(s_{1}/c_{11})\bar{L}_{u0}(\cos{\bar{\gamma}}-2i\sin{\bar{\gamma}})$,
where, $\bar{L}_{u0}$ depicts the minimum value of $\bar{L}_{u}$ for the
wavefunction described by Eq. (23). Thus, from above equation, by comparing
the real parts, we see that $s_{1}$ and $c_{11}$ have opposite signs as we
have assumed that $s_{1}$ is real. Solving this equation gives
$\bar{L}_{u0}(1+(s_{1}/c_{11})(\cos{\bar{\gamma}}-2i\sin{\bar{\gamma}}))=0$.
Thus, either $\bar{L}_{u0}=0$, or
$c_{11}=-s_{1}(\cos{\bar{\gamma}}-2i\sin{\bar{\gamma}})$. The latter one
cannot be the case, as this implies that if $\alpha=0$, $c_{11}=0$ (as here
$s_{1}=0$). Therefore, we remain with the only choice that $\bar{L}_{u0}=0.$
Therefore, this wavefunction gets to exist from the beginning, i.e., from
$L=0$.

Now, for $\bar{k}\to 0$, the condition for the wavefunction described by Eq.
(24) takes the form

$\bar{L}_{l0}=\frac{1}{2}+\lim_{\bar{k}\to
0}\frac{s_{1}}{c_{11}}\bigg{[}(1-2\bar{L}_{l0})\cos{\bar{\gamma}}-i\frac{\bar{L}_{l0}}{\bar{k}}\sin{\bar{\gamma}}\bigg{]}$

where, $\bar{L}_{l0}$ depicts the minimum value of $\bar{L}_{l}$ for which the
wavefunction described by Eq. (24) exists. This equation can only be plausible
if $\sin{\bar{\gamma}}=0\implies\bar{\gamma}=n\pi$, which is what we obtained
in Eq. (25). Hence, the assumption of considering $s_{1}$ as real perfectly
works. Thus,
$(s_{1}/c_{11})(2\bar{L}_{l0}-1)\cos{\bar{\gamma}}\implies(2\bar{L}_{l0}-1)(1+2(s_{1}/c_{11})\cos{\bar{\gamma}})=0.$
Hence, as $s_{1}/c_{11}=-1/(2\cos{\bar{\gamma}})$ is not feasible, as
$\cos{\bar{\gamma}}=(-1)^{n}$ and $s_{1}$ is very near to zero unlike
$c_{11}$, we thus remain with only one option, i.e.,
$2\bar{L}_{l0}-1=0\implies 2\bar{L}_{l0}=1$.
Therefore, if we consider $L$ to be continuous, then it can be said that the
condition for the second bound state to exist is given by $2\bar{L}\geq
1\implies L\geq\hbar^{2}/(2mW_{0})$. This is quite intuitive, as the closer
the wells will get, they will seem to act like a single well, i.e., have a
single wavefunction. Thus, considering the above equation and employing Eq.
(25) in it gives the following condition for observing the second bound state
in the discrete space

$n\geq\bigg{\lceil}{\frac{\hbar}{4\pi\alpha mW_{0}}}\bigg{\rceil}\implies
L\geq 2\pi\alpha\hbar\bigg{\lceil}{\frac{\hbar}{4\pi\alpha
mW_{0}}}\bigg{\rceil}~{}.$ (26)

We took the least integer function here as $n$ is discrete. Thus, the above
equation gives the minimum value of $L$, namely the length between the two
delta wells, for the existence of two bound states.
Now, using Eq. (25) in Eqs. (23) and (24) to replace cosine and sine, we get

$\frac{\bar{L}_{u}}{\bar{k}}=\frac{1-\frac{2s_{1}}{c_{11}}\frac{\bar{L}_{u}}{\bar{k}}e^{-\bar{k}}(-1)^{n}}{(1+e^{-2\bar{k}})}\quad\text{and}\quad\frac{\bar{L}_{l}}{\bar{k}}=\frac{1+\frac{2s_{1}}{c_{11}}\bigg{(}1-\frac{\bar{L}_{l}}{\bar{k}}(1-e^{-2k})\bigg{)}e^{\bar{k}}(-1)^{n}}{(1-e^{-2\bar{k}})}~{}.$
(27)

The second terms in the numerators of both the above equations represent the
correction due to the presence of the scale parameter $\alpha$.

##  II Appendix B

Here we shall analyze the effects of quartic correction terms 1c ; 1c1 ; 1c2 ;
1c4  on the wavefunction and the underlying topology of spacetime. Thus, we
start with the quartic correction term of the form

$\biggr{(}-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{\beta\hbar^{4}}{m}\frac{d^{4}}{dx^{4}}-[W_{o}\delta(x+L)+W_{o}\delta(x-L)+E]\biggl{)}\psi(x)=0~{}.$
(28)

We can express $\eta$ as

$\eta\equiv\frac{1}{\hbar\sqrt{2\beta}}~{}.$ (29)

Now, as $V(x)=0$ for $x\neq\pm L$, we obtain
$\frac{d^{2}\psi(x)}{dx^{2}}-2\beta\hbar^{2}\frac{d^{4}\psi(x)}{dx^{4}}-k^{2}\psi(x)=0$
. Hence, solving this equation, we obtain $\psi(x)=Ce^{\eta x}+D^{-\eta
x}+Ae^{bx}+Be^{-bx}$ with $b\equiv k+\hbar^{2}k^{3}\beta$ .

Hence, from the above equations, after utilizing the bound state conditions
$\psi(-\infty)=\psi(\infty)=0$, we get the complete equation for the double-
delta potential well in the form

$\psi(x)=\begin{cases}s_{11}e^{\eta x}+c_{11}e^{bx},&x<-L\\\ s_{21}e^{\eta
x}+s_{22}e^{-\eta x}+c_{21}e^{bx}+c_{22}e^{-bx},&x\in(-L,L)\\\ s_{32}e^{-\eta
x}+c_{32}e^{-bx},&x>L\end{cases}~{}.$ (30)

It is important to note that the impact of terms with $\eta$ in them is small,
so $s_{11},\,s_{21},\,s_{22},\,s_{32}$ are all very small because if $\beta\to
0$, then these terms must get killed. Thus, we can write
$s_{ij}\beta^{m}=0=s_{ij}^{n}\quad\forall\,i,j$ with $m\geq 1$ and $n>1$. Now,
since the potential is symmetric here, so even-odd parity can be used, i.e.,
$\psi(x)=\pm\psi(-x)$. Thus, using this in the region $||x||>0$, we have
$\psi(-x:x<-L)=\pm\psi(x:x>L)$ which gives, $(s_{11}\mp s_{32})=-(c_{11}\mp
c_{32})e^{(\eta-b)x}$. Hence, knowing here that $x$ is a variable, it holds
only when

$s_{11}=\pm s_{32},\,c_{11}=\pm c_{32}~{}.$ (31)

Now, using the even-odd parity in the region $||x||<0$, we get
$\psi(-x:-L<x<0)=\pm\psi(x:0<x<L)$ which leads to

$(s_{22}\mp s_{21})=-(c_{22}\mp c_{21})\frac{(e^{bx}\mp e^{-bx})}{(e^{\eta
x}\mp e^{-\eta x})}~{}.$ (32)

Now, differentiating this expression, we obtain

$\psi^{(n)}(x)=\begin{cases}s_{11}\eta^{n}e^{\eta
x}+c_{11}b^{n}e^{bx},&x<-L\\\ s_{21}\eta^{n}e^{\eta
x}+(-1)^{n}s_{22}\eta^{n}e^{-\eta
x}+c_{21}b^{n}e^{bx}+(-1)^{n}c_{22}b^{n}e^{-bx},&x\in(-L,L)\\\
s_{32}\eta^{n}e^{-\eta x}+(-1)^{n}c_{32}b^{n}e^{-bx},&x>L\end{cases}~{}.$ (33)

Also, by rearranging Eq. (28), we get

$\psi^{\prime\prime}(x)-2\beta\hbar^{2}\psi^{\prime\prime\prime\prime}(x)+\bigg{[}\frac{2mW_{0}}{\hbar^{2}}[\delta(x+L)+\delta(x-L)]-k^{2}\bigg{]}\psi(x)=0~{}.$
(34)

After integrating from $x=\pm L^{-}$ to $x=\pm L^{+}$, we obtain the jump
condition

$\psi^{\prime}(x)\bigg{|}^{\pm L^{+}}_{\pm
L^{-}}-2\beta\hbar^{2}\psi^{\prime\prime\prime}(x)\bigg{|}^{\pm L^{+}}_{\pm
L^{-}}=-\frac{2mW_{0}}{\hbar^{2}}\psi(\pm L)~{}.$ (35)

Taking the lower sign in the above equation, we get

$[Q(c_{21}-c_{11})-c_{11}]e^{-bL}-Qc_{22}e^{bl}-s_{11}e^{-\eta L}=0$ (36)

while solving with the upper sign, we obtain

$[Q(c_{22}-c_{32})-c_{32}]e^{-bL}-Qc_{21}e^{bl}-s_{32}e^{-\eta L}=0$ (37)

where

$Q=-\frac{\hbar^{2}b(1-2\beta\hbar^{2}b^{2})}{2mW_{0}}~{}.$ (38)

Thus, we get $(c_{22}\mp c_{21})(e^{-bl}\mp e^{bl})=0\implies c_{22}=\pm
c_{21}$ . Using this result in Eq. (32), we can also obtain $s_{22}=\pm
s_{21}~{}.$ Therefore, summarizing the results, we have

$s_{21}=\pm s_{22},\,s_{32}=\pm s_{11},\,c_{21}=\pm c_{22},\,c_{32}=\pm
c_{11}~{}.$ (39)

Now, double integrating from $x=\pm L^{-}$ to $x=\pm L^{+}$ with respect to
$x$, we get the second jump condition

$\psi(x)\biggr{|}^{\pm L^{+}}_{x=\pm
L^{-}}=2\beta\hbar^{2}\psi^{\prime\prime}(x)\biggr{|}^{\pm L^{+}}_{x=\pm
L^{-}}~{}.$ (40)

Thus, we obtain the following result for both $+L$ and $-L$

$c_{22}=\frac{c_{11}}{e^{2bL}\pm 1}~{}.$ (41)

Also, we can write

$\biggr{[}Q(\frac{c_{11}}{e^{2bL}\pm 1}\mp c_{11})\mp
c_{11}\biggr{]}e^{-bL}\mp Q\frac{c_{11}}{e^{2bL}\pm 1}e^{bl}\mp s_{11}e^{-\eta
L}=0~{}.$

At this point, we introduce the parameters

$\bar{L}\equiv\frac{mW_{0}L}{\hbar^{2}},\quad\lambda\equiv\frac{m^{2}W_{0}^{2}\beta}{\hbar^{2}}=\frac{1}{2}\bigg{(}\frac{\bar{L}}{\eta
L}\bigg{)}^{2},\quad\bar{k}=kL,\quad\bar{b}\equiv
bL=\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}~{}.$ (42)

Thus, we obtain
$-2Q=(\frac{s_{11}}{c_{11}}e^{-\frac{\bar{L}}{\sqrt{2\lambda}}+\bar{b}}+1)(1\pm
e^{-2\bar{b}})$, and
$Q=-\frac{1}{2}\frac{\bar{b}}{\bar{L}}(1-2\lambda\frac{\bar{b}^{2}}{\bar{L}^{2}})$.
Therefore, from these two expressions, we obtain

$\frac{\bar{L}}{\bar{b}}=\frac{\biggr{(}1-2\lambda\frac{\bar{b}^{2}}{\bar{L}^{2}}\biggr{)}}{\bigg{(}\frac{s_{11}}{c_{11}}e^{-\frac{\bar{L}}{\sqrt{2\lambda}}+\bar{b}}+1\bigg{)}}\frac{1}{(1\pm
e^{-2\bar{b}})}~{}.$ (43)

The above equation gives the impression that this system has two
wavefunctions, as expected. Also, from this equation, we obtain
$s_{11}/c_{11}\in\mathbb{R}$, as the exponential multiplied with it cannot
make the term to vanish if it is imaginary. Furthermore, it is assumed
$\frac{s_{11}}{c_{11}}>0$, since if this quantity is negative and the
exponential takes the value that makes the term equal to $-1$, then this term
will blow up the expression, which should not be the case.

Now, let us find the minimum length between the wells necessary for the
existence of a particular wavefunction. Since
$\bar{b}=\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}$, it is obvious that
if $\bar{k}\to 0$, then $\bar{b}\to 0$. Therefore,

$\bar{L}_{\bar{b}=0}=\begin{cases}0\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\text{for
even wave-function}\\\
\frac{1}{2}\bigg{(}1-\frac{s_{11}}{c_{11}}e^{-\frac{\bar{L}_{\bar{b}=0}}{\sqrt{2\lambda}}}\bigg{)}\quad\text{for
odd wave-function}\end{cases}~{}.$ (44)

Figure 3: The plot here is $\lambda$ vs $\bar{L}_{b=0}$ which is obtained by
plotting Eq. (44) for odd wavefunction while assuming $C_{11}\sim 1$ and
$S_{11}\sim\lambda$ since both are of the order of $\beta$.

In Fig. 3, the plot clearly shows that as $\lambda$ or, equivalently, $\beta$
decreases, $\bar{L}$ increases and, finally, it saturates to the value $1/2$
when $\lambda=0$ or, equivalently, $\beta=0$) which is what is anticipated.

Now, simplifying Eq. (43) and taking into consideration the smallness of
$s_{11}$ and $\lambda$, we obtain

$\bar{L}=\bigg{(}1-\frac{s_{11}}{c_{11}}e^{-\frac{\bar{L}}{\sqrt{2\lambda}}+\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}}\bigg{)}\frac{\bar{k}-\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}}{(1\pm
e^{-2(\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}})})}=\frac{\bar{k}-\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}-\frac{s_{11}}{c_{11}}\bar{k}e^{-\frac{\bar{L}}{\sqrt{2\lambda}}+\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}}}}{(1\pm
e^{-2(\bar{k}+\lambda\frac{\bar{k}^{3}}{\bar{L}^{2}})})}~{}.$ (45)

Therefore, unlike the cubic correction terms, the underlying topology is
continuous here. However, the quartic correction term explicitly induces a
modification in the wavefunction.


# Hybrid star properties from an extended linear sigma model

János Takátsy, Péter Kovács and György Wolf  Institute for Particle and
Nuclear Physics, Wigner Research Center for Physics, Hungarian Academy of
Sciences, H-1525 Budapest, Hungary<EMAIL_ADDRESS>
###### Abstract

The equation of state provided by effective models of strongly interacting
matter should comply with the restrictions imposed by current astrophysical
observations on compact stars. Using the equation of state given by the
(axial-)vector meson extended linear sigma model, we determine the mass-radius
relation and study whether these restrictions are satisfied under the
assumption that most of the star is filled with quark matter. We study the
dependence of the mass-radius relation on the parameters of the model.

##  I Introduction

By the recent astrophysical observation of heavy neutron stars Demorest:2010
and neutron star collisons GW170817:2018  the interest for studying dense
strongly interacting matter increased substantially. In terrestial experiments
ALICE at CERN, and PHENIX and STAR at RHIC (Tlusty:2018rif, ) explored the
strongly interacting matter at low density and high temperature. In this case,
the situation is also satisfactory on the theoretical side since lattice
calculations are applicable. At high density the theoretical investigations
are not at such a satisfactory level, therefore, effective models are needed
in this region where the existing experimental data (NA61 at CERN, BES/STAR at
RHIC) are scarce and have rather bad statistics. Soon to be finished
experimental facilities (NICA at JINR and CBM at FAIR) are designed to explore
this region more precisely Kekelidze:2014dta ; Ablyazimov:2017guv .

Neutron stars are on the one side a challenge requiring to understand their
properties i.e mass, radius, tidal deformability, structure etc. On the other
hand, they can provide information about the dense, strongly interacting
matter in a region of the phase diagram that is inaccessible to terrestrial
experiments lattimer2007 ; Watts:2016uzu ; Tews:2018chv . Since the Tolman-
Oppenheimer-Volkoff (TOV) equations Tolman ; OppVolk  provide a direct
relation between the equation of state (EoS) of the compact star matter and
the mass-radius (M-R) relation of the compact star, these data can help to
select those effective models, used to describe the strongly interacting
matter, whose predictions are consistent with compact star observables. For
example, the EoS must support the existence of a two-solar-mass neutron star
with star radii in the permitted radius window of 11-12.5km, cf. Hell:2014xva
; lattimer2014 .

Based on the above considerations, we investigate mass-radius sequences given
by the EoS obtained for the quark phase in kovacs2016  from the $N_{f}=2+1$
flavor extended linear sigma model introduced in parganlija2013 . For the
hadronic phase we use for the outer crust the BPS EoS Baym:1971pw , for the
inner crust the NV EoS Negele:1973  and for the outer core until the phase
transition we use the TNI2u TNIu:2006  EoS. The model used here should be
regarded as a rather crude approximation since the hadron-quark phase
transition is introduced in a very ad hoc way. The present work has to be
considered only as our first attempt to study the problem.

The paper is organized as follows. In Section II we present the model and
discuss how its solution was obtained in kovacs2016 , which reproduced quite
well some thermodynamic quantities measured on the lattice. It can be used in
the presence of a vector meson introduced here to realize the short-range
repulsive interaction between quarks in the simplest possible way. In Section
III we compare our results for the EoS and the M-R relation (star sequences),
for various values of the unknown model parameters, such as the vector
coupling $g_{v}$, the quark-hadron crossover transition density $\bar{\rho}$.
We draw the conclusions in Section IV and discuss possible ways to improve the
treatment of the model.

##  II Methods

For the EoS of the hadronic phase we use models from nuclear physics. For very
low density, (outer crust, $\rho<0.01\rho_{0}$) we use the BPS EoS Baym:1971pw
that corresponds to a Coulomb lattice of different nuclei embedded in a gas of
electrons and that has a remarkable influence on the mass-radius relation of
the star.

For the region of $0.01\rho_{0}\leq\rho<\rho_{0}$ (inner crust) we use the NV
EoS Negele:1973  and above this until the phase transition we use the TNI2u or
TNI3u TNIu:2006 ; Masuda:2013  EoS, which differ in their stiffness (K=250 MeV
or K=300 MeV, respectively). These EoS’s even with their derivatives go over
each other continuously, so there is no need of any treatment at the
connections and the exact density of the transitions are irrelevant. The model
used in this paper for the quark phase is an $N_{f}=2+1$ flavor (axial)vector
meson extended linear sigma model (eLSM). The Lagrangian and the detailed
description of this model, in which in addition to the full nonets of
(pseudo)scalar mesons the nonets of (axial)vector mesons are also included,
can be found in kovacs2016 ; parganlija2013 . The model contains three flavors
of constituent quarks, with kinetic terms and Yukawa-type interactions with
the (pseudo)scalar mesons. An explicit symmetry breaking of the mesonic
potentials is realized by external fields, which results in two scalar
expectation values, $\phi_{\mathrm{N}}$ and $\phi_{\mathrm{S}}.$

Compared to kovacs2016 , the only modification to the model is that we include
in the Lagrangian a Yukawa term
$-g_{v}\sqrt{6}\bar{\Psi}\gamma_{\mu}V_{0}^{\mu}\Psi$, which couples the quark
field $\Psi^{\rm T}=(u,d,s)$ to the $U_{V}(1)$ symmetric vector field, that is
$V_{0}^{\mu}=\frac{1}{\sqrt{6}}\mathrm{diag}(v_{0}+\frac{v_{8}}{\sqrt{2}},v_{0}+\frac{v_{8}}{\sqrt{2}},v_{0}-\sqrt{2}v_{8})^{\mu}$.
The vector meson field is treated at the mean-field level as in the Walecka
model walecka , but as a simplification we assign a nonzero expectation value
only to $v_{0}^{0}$: $v_{0}^{\mu}\to v_{0}\delta^{0\mu}$ and $v_{8}^{\mu}\to
0$. In this way the chemical potentials of all three quarks are shifted by the
same amount, allowing us to use, as shown below, the result obtained in
kovacs2016 . With the parameters used in kovacs2016 , the mass of the vector
meson $v_{0}^{\mu}$ turns out to be $m_{v}=871.9$ MeV.

Since a compact star is relatively cold ($T\approx 0.1$ keV), we work at $T=0$
MeV. We have three background fields $\phi_{\mathrm{N}},\phi_{\mathrm{S}}$ and
$v_{0}$, and the calculation of the grand potential, $\Omega$, is performed
using a mean-field approximation, in which fermionic fluctuations are included
at one-loop order, while the mesons are treated at tree-level. Hence, the
grand potential can be written in the following form:

$\displaystyle\Omega(\mu_{q};$
$\displaystyle\\!\\!\\!\phi_{\mathrm{N}},\phi_{\mathrm{S}},v_{0})=U_{\mathrm{mes}}(\phi_{\mathrm{N}},\phi_{\mathrm{S}})-\frac{1}{2}m_{v}^{2}v_{0}^{2}$
(1)
$\displaystyle+\Omega^{(0)\mathrm{vac}}_{q\bar{q}}(\phi_{\mathrm{N}},\phi_{\mathrm{S}})+\Omega^{(0)\mathrm{matter}}_{q\bar{q}}(\tilde{\mu}_{q};\phi_{\mathrm{N}},\phi_{\mathrm{S}})\>,$

where $\tilde{\mu}_{q}=\mu_{q}-g_{v}v_{0}$ is the effective chemical potential
of the quarks, while $\mu_{q}=\mu_{\mathrm{B}}/3$ is the physical quark
chemical potential, with $\mu_{B}$ being the baryochemical potential. On the
right hand side of the grand potential (1), the terms are (from left to
right): the tree-level potential of the scalar mesons, the tree-level
contribution of the vector meson, the vacuum and the matter part of the
fermionic contribution at vanishing mesonic fluctuating fields. The fermionic
part is obtained by integrating out the quark fields in the partition
function. The vacuum part was renormalized at the scale $M_{0}=351$ MeV. More
details on the derivation can be found in kovacs2016 .

The background fields $\phi_{\mathrm{N}},\phi_{\mathrm{S}}$, and $v_{0}$ are
determined from the stationary conditions

$\frac{\partial\Omega}{\partial\phi_{\mathrm{N}}}\bigg{|}_{\phi_{\mathrm{N}}=\bar{\phi}_{\mathrm{N}}}=\frac{\partial\Omega}{\partial\phi_{\mathrm{S}}}\bigg{|}_{\phi_{\mathrm{S}}=\bar{\phi}_{\mathrm{S}}}=0\quad\textnormal{and}\quad\frac{\partial\Omega}{\partial
v_{0}}\bigg{|}_{v_{0}=\bar{v}_{0}}=0\>,$ (2)

where the solution is indicated with a bar. Since $\partial/\partial
v_{0}=-g_{v}\partial/\partial{\tilde{\mu}_{q}}$, the stationary condition with
respect to $v_{0}$ reads

$\bar{v}_{0}(\phi_{\mathrm{N}},\phi_{\mathrm{S}})=\frac{g_{v}}{m_{v}^{2}}\rho_{q}(\tilde{\mu}_{q}(\bar{v}_{0});\phi_{\mathrm{N}},\phi_{\mathrm{S}})\>,$
(3)

where
$\rho_{q}(x;\phi_{\mathrm{N}},\phi_{\mathrm{S}})=-\partial\Omega^{(0)\mathrm{matter}}_{q\bar{q}}(x;\phi_{\mathrm{N}},\phi_{\mathrm{S}})/\partial
x.$

When solving the model, the value of $g_{v}$ does not effect the properties of
the system at zero chemical potential, therefore, we cannot fit it to the
vacuum properties, so we leave it in the range of $[0,3)$ as a free parameter,
while for the remaining 14 parameters of the model Lagrangian we use the
values given in Table IV of kovacs2016 . These values were determined there by
calculating constituent quark masses, (pseudo)scalar curvature masses and
decay widths at $T=\mu_{q}=0$ and comparing them to their experimental PDG
values PDG . Parameter fitting was done using a multiparametric $\chi^{2}$
minimization procedure james1975 . In addition to the vacuum quantities, the
pseudocritical temperature $T_{\mathrm{pc}}$ at $\mu_{q}=0$ was also fitted to
the corresponding lattice result aoki2006 ; Bazavov:2011nk . We mention that
the model also contains the Polyakov-loop degrees of freedom (see kovacs2016
for details), but to keep the presentation simple we omitted them from (1), as
at $T=0$ they do not contribute to the EoS directly. Their influence is only
through the value of the model parameters taken from kovacs2016 : since they
influence the value of $T_{\mathrm{pc}}$ used for parameterization.

Figure 1: The equation of state (left) and the M-R curves (right) for
different values of $g_{v}$ using $\bar{\rho}=4\rho_{0}$ and $\Gamma=1$ with
TNI2u (K=250 MeV) nuclear force for different $g_{v}$’s. The $g_{v}=2.5$ case
is calculated with TNI3u (K=300 MeV), as well. In the right figure the
observational constraint set by observed pulsars for the permitted radius
window of $11.0$-$12.5$ km at $2$ $M_{\odot}$ (e.g. Hell:2014xva ;
lattimer2014 ) is represented by the horizontal line. The different shaded
regions are excluded by causality $R>2.9GM/c^{2}$ and the rotational
constraint based on the $716$ Hz pulsar J1748-2446ad, $M/M_{\odot}>4.6\cdot
10^{-4}\>(R/\mathrm{km})^{3}$ hessels2006

The solution of the model at $g_{v}=0$, obtained in kovacs2016 , can be used
to construct the solution at $g_{v}\neq 0$ (see e.g. Ch. 2.1 of almasi ): one
only has to interpret the solution at $g_{v}=0$ as a solution obtained at a
given $\tilde{\mu}_{q}$ and determine $\mu_{q}$ at some $g_{v}\neq 0$ using
(3). To see that the solutions $\bar{\phi}_{\mathrm{N,S}}$ for $g_{v}\neq 0$
can be related to the solution obtained at $g_{v}=0,$ where $\bar{v}_{0}=0$,
consider the grand potential at $g_{v}=0$. This potential, denoted as
$\Omega_{0}$, is subject to the stationary conditions (2) with solutions
$\bar{\phi}_{\mathrm{N,S}}^{0}(\mu_{q})$. It is then easy to see using (1),
that the solution $\bar{\phi}_{\mathrm{N,S}}(\mu_{q})$ of (2) satisfies
$\bar{\phi}_{\mathrm{N,S}}(\mu_{q}+g_{v}v_{0})=\bar{\phi}_{\mathrm{N,S}}^{0}(\mu_{q})$
or, changing the variable $\mu_{q}$ to $\tilde{\mu}_{q}$ the relation becomes

$\bar{\phi}_{\mathrm{N,S}}(\tilde{\mu}_{q}+g_{v}v_{0})=\bar{\phi}_{\mathrm{N,S}}^{0}(\tilde{\mu}_{q}).$
(4)

The value of the grand potential $\Omega$ at the extremum can be given in
terms of the value of the grand potential with $g_{v}=0$, that is
$\Omega_{0}$, at its extremum. Using that the extrema of
$\Omega_{0}(\tilde{\mu}_{q},\phi_{\mathrm{N}},\phi_{\mathrm{S}},v_{0}=0)$ are
$\bar{\phi}_{\mathrm{N}}^{0}$ and $\bar{\phi}_{\mathrm{S}}^{0}$, one has

$\displaystyle\Omega(\mu_{q};$
$\displaystyle\bar{\phi}_{\mathrm{N}}(\mu_{q}),\bar{\phi}_{\mathrm{S}}(\mu_{q}),\bar{v}_{0})$
(5)
$\displaystyle=\Omega_{0}(\tilde{\mu}_{q},\phi_{\mathrm{N}}^{0}(\tilde{\mu}_{q}),\phi_{\mathrm{S}}^{0}(\tilde{\mu}_{q}),v_{0}=0)-\frac{1}{2}m^{2}_{v}\bar{v}_{0}^{2}\>,$

where

$\displaystyle\bar{v}_{0}$ $\displaystyle\equiv$
$\displaystyle\bar{v}_{0}(\bar{\phi}_{\mathrm{N}}(\mu_{q}),\bar{\phi}_{\mathrm{S}}(\mu_{q}))=\frac{g_{v}}{m^{2}_{v}}\rho_{q}(\tilde{\mu}_{q};\bar{\phi}_{\mathrm{N}}(\mu_{q}),\bar{\phi}_{\mathrm{S}}(\mu_{q}))$
(6) $\displaystyle=$
$\displaystyle\frac{g_{v}}{m^{2}_{v}}\rho_{q}(\tilde{\mu}_{q};\bar{\phi}_{\mathrm{N}}^{0}(\tilde{\mu}_{q}),\bar{\phi}_{\mathrm{S}}^{0}(\tilde{\mu}_{q}))$

and $\mu_{q}=\tilde{\mu}_{q}+g_{v}\bar{v}_{0}$. The pressure $p$ and the
energy density $\varepsilon$ are calculated from the grand potential. At
$v_{0}\neq 0$ they can be expressed in terms of the pressure obtained at
$g_{v}=0$:

$\displaystyle p(\mu_{q})$ $\displaystyle=$
$\displaystyle\Omega(\mu_{q}=0;\bar{\phi}_{\mathrm{N}}(0),\bar{\phi}_{\mathrm{S}}(0),\bar{v}_{0}(0))-\Omega(\mu_{q};\bar{\phi}_{\mathrm{N}},\bar{\phi}_{\mathrm{S}},\bar{v}_{0})$
(7) $\displaystyle=$
$\displaystyle\Omega_{0}(\tilde{\mu}_{q}=0;\bar{\phi}_{\mathrm{N}}^{0}(0),\bar{\phi}_{\mathrm{S}}^{0}(0),v_{0}=0)$
$\displaystyle-\Omega_{0}(\tilde{\mu}_{q};\bar{\phi}_{\mathrm{N}}^{0},\bar{\phi}_{\mathrm{S}}^{0},v_{0}=0)+\frac{1}{2}m^{2}_{v}\bar{v}_{0}^{2}$
$\displaystyle=$ $\displaystyle
p(\tilde{\mu}_{q})|_{g_{v}=0}+\frac{1}{2}m^{2}_{v}\bar{v}_{0}^{2}\>,$

where
$\bar{v}_{0}=\frac{g_{v}}{m^{2}_{v}}\rho_{q}(\tilde{\mu}_{q};\bar{\phi}_{\mathrm{N}}^{0}(\tilde{\mu}_{q}),\bar{\phi}_{\mathrm{S}}^{0}(\tilde{\mu}_{q}))$,
and then $\varepsilon=-p+\mu_{q}\rho_{q}$, where
$\mu_{q}=\tilde{\mu}_{q}+g_{v}\bar{v}_{0}$.

Figure 2: The equation of state (left) and the M-R curves (right) using
$\bar{\rho}=4\rho_{0}$ and $g_{v}=2.5$ with TNI2u (K=250 MeV) nuclear force
for different $\Gamma$’s. The $\Gamma=1.0$ case is calculated with TNI3u
(K=300 MeV), as well. The shaded regions in the right figure are explained in
Fig.1.

To connect the nuclear phase (TNI2u) and the quark phase EoS we assume a
hadron-quark crossover, and use the $P$-interpolation Asakawa:1997 :

$\displaystyle P(\rho)$ $\displaystyle=$ $\displaystyle
P_{H}(\rho)f_{-}(\rho)+P_{Q}(\rho)f_{+}(\rho),$ (8) $\displaystyle
f_{\pm}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1\pm\mathrm{tanh}\left(\frac{\rho-\bar{\rho}}{\Gamma}\right)\right)$
(9) $\displaystyle\varepsilon(\rho)$ $\displaystyle=$
$\displaystyle\varepsilon_{H}(\rho)f_{-}(\rho)+\varepsilon_{Q}(\rho)f_{+}(\rho)+\Delta\varepsilon$
(10) $\displaystyle\Delta\varepsilon$ $\displaystyle=$
$\displaystyle\rho\int^{\rho}_{\bar{\rho}}(\varepsilon_{H}(\rho^{\prime})-\varepsilon_{Q}(\rho^{\prime}))\frac{g(\rho^{\prime})}{\rho^{\prime}}d\rho^{\prime}$
$\displaystyle g(\rho^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2\Gamma}\mathrm{cosh}^{-2}\left(\frac{\rho^{\prime}-\bar{\rho}}{\Gamma}\right).$
(11)

where $\bar{\rho}$ is the transition density and $\Gamma$ is the width of the
crossover transition. Note that neither the transition density - $\bar{\rho}$
\- nor the order of the phase transition between the hadron and the quark
phases are known, therefore, we will investigate the dependence of the M-R
curves on these parameters, as well.

By solving the TOV equation Tolman ; OppVolk  using a fourth-order Runge-Kutta
differential equation integrator with adaptive stepsize-control for a specific
EoS, one can obtain the radial dependence of the energy density (and thus of
the pressure) for a certain central energy density, $\varepsilon_{0}$. One can
then determine the mass and radius of the compact star for that central energy
density. By changing $\varepsilon_{0}$, one gets a sequence of compact star
masses and radii parameterized by the central energy density, as shown in Fig.
1 for various models. The sequence of stable compact stars ends when the
maximum compact star mass is reached with increasing central energy density.

##  III Results

As described above we use an equation of state (EoS) sewn together from 3
nuclear (BPS, NV and TNIu) and from the EoS for the quark phase. We consider
that the low density (below $\rho<\rho_{0}$) nuclear EoS’s are safe, their
ingredients are based on well known and tested nuclear physics data, and their
slight modifications change only the propeties of the crust, and do not show
up significantly in those neutron star properties which we study here. Here we
investigate the effect of the not very well determinable parameters
$\bar{\rho},\Gamma,g_{v}$ on the M-r curves and on the tidal deformability. We
study moreover, what the effect of the stiffnes of the hadronic EoS is, so we
compare results using TNI3u (K=300 MeV) and TNI2u (K=250 MeV) interactions.

Inclusion of the repulsive interaction between quarks in the eLSM renders the
EoS stiffer compared to the $g_{v}=0$ case, as expected, and at lower
densities one can observe the effect of the stiffnes of the hadronic phase. It
is worth noting that relatively small differences in the $p(\epsilon)$ lead to
significant differences in the M-R curves, as we shall see in Figs. 1, 2 and
3. It can also be seen that the maximum possible compact star mass is lower
for models with less stiff EoSs. The highest mass compact star has a mass of
$\sim 1.5$ $M_{\odot}$ with $g_{v}=0$. The effect of hadronic compressibility
is rather large, and the higher stiffnes brings the M-R curve closer to the
observed radius window.

The effect of the crossover width $\Gamma$ is sizeable for either on the EoS
or in the M-R sequence 2, however, it does not effect the maximal neutron star
mass and the corresponding radius. The EOS for $\Gamma=0.2$ is acausal
($c_{s}>1$) around the transition point, so that solution is not physical.

Figure 3: The equation of state (left) and the M-R curves (right) for
different values of $\bar{\rho}$ using $g_{v}=2.5$ and $\Gamma=1$ with TNI2u
(K=250 MeV) nuclear force for different $\bar{\rho}$’s. The
$\bar{\rho}=4\rho_{0}$ case is calculated with TNI3u (K=300 MeV), as well. The
shaded regions are explained in Fig.1.

As it could be expected the variation of transition density $\bar{\rho}$ has a
large effect on the EoS and on the M-R curve (Fig. 3), too. Since the hadronic
part of the EoS is softer than the quark one, the maximal neutron star mass is
higher for smaller $\bar{\rho}$. With decreasing $\bar{\rho}$ also the radius
corresponding to the maximal mass neutron star is increasing. So in this model
the parameter set: $g_{v}=2.5,\Gamma=0-1,\bar{\rho}=2\rho_{0}-3\rho_{0}$ and
the nuclear TNI2u interaction (with K=250 MeV compressibility) fulfills the
M-R constraints of neutron star observations.

Figure 4: Tidal deformability parameter, $\Lambda$ as a function of the
neutron star mass for various crossover densities with $g_{v}=2.5,\Gamma=1$
(left), and various coupling strengths between the quarks and vector mesons,
$g_{v}$s with $\Gamma=1,\bar{\rho}=4\rho_{0}$ (right). The experimental point
is from GW170817:2018 .

The tidal deformability parameter $\Lambda$ as a function of the neutron star
mass is shown for various parameter sets in Fig. 4. Since the experimental
constraint, $70<\Lambda_{1.4M_{\odot}}<580$ is not very conclusive it does not
constrain our parameterizing. However, more precise estimate of $\Lambda$
could rule out models and parameter sets.

##  IV Conclusions

We employed the zero-temperature EoS to determine the mass-radius relation of
compact stars. Including the repulsive interaction in the eLSM model makes the
EoS stiff enough to support in some narrow range of the Yukawa coupling
compact stars with masses larger than $2$ $M_{\odot}$ and in the permitted
radius window of $11.0$-$12.5$ km at $M=2$ $M_{\odot}$.

In the future, we would like to go beyond the mean-field approximation, used
for the mesons in the eLSM, in a way that takes into account the effect of
fermions in the mesonic fluctuations. At lowest order, this can be done by
expanding to quadratic order the fermionic determinant obtained after
integrating out the quark fields in the partition function and performing the
Gaussian integral over the mesonic fields. In order to have a physically more
reliable description, we also plan to improve the treatment of the interaction
between vector mesons and quarks employed here. Furthermore, to reduce the
uncertainty emerging from the artificially performed phase transition, we plan
to derive the phase transition parameters from the model itself.

##  V Acknowledgment

P. Kovács acknowledges support by the János Bólyai Research Scholarship of the
Hungarian Academy of Sciences. This work was supperted by an NKFIH-OTKA grant
K131982.

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