# Study of Differential Scattering Cross-section using Yukawa term of medium-
modified Cornell potential

Siddhartha Solankia<EMAIL_ADDRESS>Manohar Lala
<EMAIL_ADDRESS>Vineet Kumar Agotiyaa<EMAIL_ADDRESS>aDepartment of Physics, Central University of Jharkhand, Ranchi, India, 835205

###### Abstract

In the present work we have studied the Differential Scattering Cross-section
for ground states of charmonium and bottomonium in the frame work of the
medium modified form of quark-antiquark potential and Born-approximation using
the non-relativistic quantum chromo-dynamics approach. To reach this end,
quasi-particle (QP) Debye mass depending upon baryonic chemical potential
($\mu_{b}$) and temperature has been employed and hence the variation of
Differential Scattering Cross-section with baryonic chemical potential and
temperature at fixed value of the scattering angle ($\theta$=$90^{o}$) has
been studied. The variation of Differential Scattering Cross-section with
scattering angle $\theta$ (in degree) at fixed temperature and baryonic
chemical potential has also been studied. We have also studied the effect of
impact parameter and transverse momentum on differential scattering cross-
section at $\theta$=$90^{o}$.

Keywords : Differential Cross-section, Scattering theory, Strongly Coupled
Plasma, Heavy Quark Potential, baryonic chemical potential, Born-
approximation, Scattering amplitude, quasi-particle Debye mass.

##  I Introduction

Quantum Chromo-dynamics (QCD) is one among the most important theories which
well describes the strong interaction occurring at the subatomic level. Since
the cross-section is an important key that pave a way of communication between
the real world of the experiment and idealized theoretical models. In the high
energy physics the term cross-section is used to specify the interaction of
elementary particle quantitatively. Cross-section may also be thought of as
the area within which the reaction among the elementary particle takes place.
Theoretical predictions for the cross-section of the oppositely lepton pairs
in p-p collisions are accurate up to the next to leading order (NLO) and next
to next leading order (NNLO) in electroweak and perturbative quantum chromo-
dynamics (PQCD) 1 ; 2 ; 3 ; 4  respectively. Precise measurements of the
differential cross-section at LHC for the Drell Yan process (end up with
conformal test for the standard model in the perturbative regime) is an
important test for the standard model in the perturbative frame. ATLAS 5 ; 6 ;
7  and CMS 8 ; 9 ; 10  recently measured the single and double cross-section.
Inclusive quarkonia production in pp collisions, in pp collision at
$\sqrt{s}$=$5.02$ TeV. Both the perturbative and non-perturbative fact of QCD
can be studied easily in high-energy hadronic collision by considering the
quarkonia production 11 ; 12 . The main consequences of scattering process in
hadronic collision is to produce quarkonia. In such process the momentum
transfer should be two times the mass of heavy quark and hence it can
described under perturbative calculations. But on the other hand, the binding
energy of quarkonia comes under the non-perturbative process as it account for
large distance scales and soft momentum scales. Various properties of the
Quark-gluon plasma (QGP) in nucleus-nucleus collision at different energy
scales and that of cold matter nuclear effect appearing in A-A collision could
be investigated by the quarkonium production measurement 12 ; 13 . The
quarkonium production can be described by various approaches. One of the most
important countering problem in the QCD is to fully understand the production
mechanism of the heavy quarkonia since after the discovery of $J/\psi$ in
1973. According the colored singlet model the quarkonium states are color-less
and they possesses the same $J^{pc}$ quantum number 14 ; 15 ; 16 ; 17 ; 18 ;
19 ; 20 ; 21 ; 22 ; 23 .
Production of $J/\psi$ and $\psi(2s)$ cross-section at high $P_{T}$ was
underestimated by the leading order (LO) calculation in colored singlet model
by one order of magnitude 24  and this problem was overcome by considering the
next to leading order (NLO) correction but this result would not be still able
to reduce the gap between color singlet model (CSM) and experimental
measurements 25 ; 26 ; 27 . Thereafter the non-relativistic QCD model came
into existence, which includes both color singlet and color octet studies 28 ;
29 , describing production cross-section at all $P_{T}$ values but fails to
explain the polarization 30 ; 31 ; 32 ; 33 ; 34 ; 35 ; 36 ; 37 ; 38 ; 39 ; 40
; 41 ; 42 ; 43 ; 44 ; 45 . Finally, studies 46 ; 47 ; 48 ; 49 ; 50 ; 51
solved this countering problem via production of pairs of quarkonia, as the
cross-section could be easily interpreted. This quarkonium pair production
forbid the feed down of excited C-even states which are very crucial in the
single quarkonia production. This typically makes the interpretation of
polarization very difficult and hence to compare the data. The double parton
process has a significant role for understanding the new physics e.g multi-
jets and gave a pave for the transverse momenta of partons. Several new
physics phenomenon have been studied by keeping in view of double parton
process such as $4$-jets by AFS 52 , UA2 53 , CDF 54  and ATLAS 55
collaborations. $\gamma$+$3$ jets by the CDF 56  and $D_{o}$ 57 ; 58
collaborations $W+2jets$ 59  and $\Upsilon$+$\Upsilon$ 60  by CMS
collaborations $J/\psi$+W 61 , Z+open charm 62 , and $\Upsilon$+open charm 63
by the $LHC_{b}$ collaborations. Quarkonia pairs are independently produced by
different partonic interaction in the frame work of double parton process
could by estimated by the formula 64 ; 65 ; 66 .
In this present paper, we studied the Differential Scattering Cross-section
for the ground state of Quarkonia (i.e, $J/\psi$ and $\Upsilon$), in the
presence of temperature, baryonic chemical potential ($\mu_{b}$) and the
scattering angle ($\theta$). To carry out this, we use the Born approximation
for calculating the Scattering amplitude and Differential Scattering Cross-
section. For calculating the Differential Scattering Cross-section,the
potential we consider is the medium-modified form of Cornell potential (Here
we take only the Yukawa term and neglect the other terms of the potential).
Differential Scattering Cross-section mainly define the probability of finding
the particle in a certain area and hence scattering angle $\theta$ plays a
major role while studying the quarkonium production.
The manuscript organized in the following manner. In the section-II, we
provide a outline for the quark-antiquark potential. Section-III, deals with
the Debye mass depending upon temperature and baryonic chemical potential.
Section-IV, Formulation of Differential Scattering cross-section using Non-
relativistic limit of Quantum field theory. In section-V, we briefly discussed
about the result and Conclusions of this present work.

Figure 1: The variation of differential Scattering cross-section as a function
of theta after consider the values of charmonium and bottomonium masses at
constant temperature and $\mu_{b}$.

##  II Medium-modified form of Cornell potential

Since from 67 , it has been seen that there is also non-perturbative
calculations at deconfinement temperature instead of the perturbative and
ideal gas behavior according to thermodynamical studies of the QCD. Following
67  one cannot drops the string tension arising between the quark-antiquark
pairs above the critical temperature $T_{c}$. Medium modification to the
quark-antiquark potential provide a reliable way to study the fate of the
quarkonia. Here the quark-antiquark potential has been corrected which
embodied the medium effect 68 . In 69 ; 70 , the authors assume that the
string is melting, keeping in view that there is phase transition from the
hadron matter to the quark gluon plasma. Accordingly, the potential has been
modified to study the deconfined state of matter. Also at vanishing baryon
density there is cross-over rather than a phase transition. These indications
comes from the lattice studies. Bound state solutions of both relativistic and
non-relativistic wave has attracted more intention from the last decades.
The energy of these bound states is negative due to the fact that the energy
of the quark gluon plasma is less than the potential energy 71 . Schrodinger
equation accounts for the non-relativistic wave whereas for the relativistic
wave Klein-Gordan and Dirac equation are of utmost important 72 ; 73 ; 74 ; 75
; 76 ; 77 ; 78 ; 79 . There are various potential has been used to study the
quarkonia bound states like Hulthen, Poschl Teller, Eckart and coulomb
potential etc. and these are studied using special techniques AEIM, SUSYQM and
NU method 80 ; 81 ; 82 ; 83 ; 84 ; 85 ; 86 ; 87 ; 88 ; 89 ; 90  etc. In the
present work, we preferred to work with the Cornell potential which has both
coulombic as well as string part 91 ; 92 , and here we take only the coulombic
part of the said potential. This is because of the fact that the mass of
quarkonia $m_{Q}\geq\lambda_{QCD}$, small velocity of the bound states push to
understand these phenomenon in terms of non-relativistic potential model.

Figure 2: The variation of $J/\psi$ Differential Scattering Cross-section as a
function of $\mu_{b}$ at different values of temperature (in left panel) and
as a function of $T/T_{c}$ at different values of baryonic chemical potential
(in right panel).

In case of finite temperature QCD, we employ the ansatz that the medium
modification enter in the Fourier transform of heavy quark potential V(k) as
93

$\tilde{V(k)}=\frac{V(k)}{\varepsilon(k)}$ (1)

.

where $\varepsilon(k)$ is dielectric permittivity which is obtain from the
static limit of the longitudinal part of the gluon self energy.

$\varepsilon(k)=\left(1+\frac{\pi_{L}(0,k,T)}{k^{2}}\right)\equiv\left(1+\frac{m^{2}_{D}(T,\mu_{b})}{k^{2}}\right)$
(2)

.

V(k) is the Fourier transform of the Cornell potential given as,

$\mbox{\boldmath$V$}(k)=-\sqrt{\frac{2}{\pi}}\frac{\alpha}{k^{2}}-\frac{4\sigma}{\sqrt{2\pi}k^{4}}$
(3)

.

substituting the value of Eq.(2) and Eq.(3) in equation Eq.(1), and solving
using inverse Fourier transform, we get the medium modified potential
depending upon ’r’ 94 ; 95 ; 96 ; 97 ; 98 .

$\mbox{\boldmath$V$}(r,T,\mu_{b})=\left(\frac{2\sigma}{m^{2}_{D}(T,\mu_{b})}-\alpha\right)\frac{exp(-m_{D}(T,\mu_{b})r)}{r}\\\
-\frac{2\sigma}{m^{2}_{D}(T,\mu_{b})r}+\frac{2\sigma}{m_{D}(T,\mu_{b})}-\alpha
m_{D}(T,\mu_{b})$ (4)

.

##  III The Debye mass with baryonic-chemical potential from a quasi-particle
picture of hot QCD

In studies of the quantum mechanical properties of the quarkonia, Debye mass
has played a significant role. Generalization of the Debye mass has been made
from the quantum electrodynamics (QED) to QCD because of the non-abelian
nature of the QCD. In QCD, the Debye mass obtained is gauge invariant and non-
perturbative whereas the temperature dependent Leading order Debye mass is
perturbative and is known from a long time ago 99 ; 100 . The Debye mass can
also be defined as the pole static quark propagator 101  instead of limit
$p{\longrightarrow}0$ in the gluon self energy. Authors in 101 ; 102  also
calculated Debye mass for the NLO in QCD using Polyakov loop correlator
matching with HTL result.
Several studies has been devoted to include all the interaction present in the
Hot QCD equation of states (EoS) in terms of the quasi-partons. Some of them
include effective mass models, effective mass models with Polyakov loop, PNJL,
NJL model and effective fugacity model 103 ; 104 ; 104 ; 106 ; 107 ; 109 .
To understand the non-ideal behavior of the Quark gluon plasma near the cross
over region, quasi particle model has played eminent role. The interacting
system of massless quarks and gluons considered as the massive system in
Quasi-Particle (QP) model 109 . In our present calculation we use the quasi
particle model to study the quarkonia properties. All the interaction effects
could be related to the $Z_{q,g}$ term in the distribution function of the
quasi partons.

Figure 3: The variation of $\Upsilon$ Differential Scattering Cross-section as
a function of $\mu_{b}$ at different values of temperature (in left panel) and
as a function of $T/T_{c}$ at different values of baryonic chemical potential
(in right panel).

In our calculation, we use the Debye mass $m_{D}$ for full QCD case is,

$\displaystyle m^{2}_{D}\left(T\right)$ $\displaystyle=$ $\displaystyle
g^{2}(T)T^{2}\bigg{[}\bigg{(}\frac{N_{c}}{3}\times\frac{6PolyLog[2,z_{g}]}{\pi^{2}}\bigg{)}$
(5)
$\displaystyle+{\bigg{(}\frac{\hat{N_{f}}}{6}\times\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}\bigg{)}\bigg{]}}$

and

$\displaystyle\hat{N_{f}}$ $\displaystyle=$
$\displaystyle\bigg{(}N_{f}+\frac{3}{\pi^{2}}\sum\frac{\mu_{q}^{2}}{T^{2}}\bigg{)}$
(6)

and we also know that, the quark-chemical potential is equal to,

$\displaystyle\mu_{q}$ $\displaystyle=$ $\displaystyle\frac{\mu_{b}}{3}$ (7)

Where, $(\mu_{q})$ defined the quark-chemical potential and $(\mu_{b})$ is
baryonic chemical potential. Introducing the value of $\hat{N_{f}}$ in the
Eq.(5), we get the full expression of quasi particle Debye mass in terms of
temperature and baryonic chemical potential.

$m^{2}_{D}\left(T,\mu_{b}\right)=T^{2}\left\\{\bigg{[}\frac{N_{c}}{3}Q^{2}_{g}\bigg{]}+\bigg{[}\frac{N_{f}}{6}+\frac{1}{2\pi^{2}}\bigg{(}\frac{\mu_{b}^{2}}{9T^{2}}\bigg{)}\bigg{]}Q^{2}_{q}\right\\}$
(8)

Here, $g(T)$ is the QCD running coupling constant, $N_{c}=3$ is the number of
color and $N_{f}$ is the number of flavor, the function $PolyLog[2,z]$ having
form, $PolyLog[2,z]=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{2}}$ and $z_{g}$ is the
quasi-gluon effective fugacity and $z_{q}$ is quasi-quark effective fugacity.

$\displaystyle f_{g,q}=\frac{z_{g,q}exp(-\beta p)}{\left(1\pm
z_{g,q}exp(-\beta p)\right)}$ (9)

These distribution functions are isotropic in nature. These fugacities have
been introduced the all interaction effects present within the baryonic
chemical potential. Both $z_{g}$ and $z_{q}$ have a very complicated
temperature dependence and asymptotically reach to the ideal value unity 110 .
The temperature dependence $z_{g}$ and $z_{q}$ fits well to the form given
below,

$z_{g,q}=a_{q,g}\exp\bigg{(}-\frac{b_{g,q}}{x^{2}}-\frac{c_{g,q}}{x^{4}}-\frac{d_{g,q}}{x^{6}}\bigg{)}$
(10)

(Here $x=T/T_{c}$ and $a$, $b$ and $c$ and $d$ are fitting parameters), for
both EOS1 and EOS2. Here, EoS1 is the $O(g^{5})$ hot QCD and EoS2 is the
$O(g^{6}\ln(1/g)$ hot QCD EoS in the quasi-particle description 94 ; 107
respectively. Where, $Q_{g}$ and $Q_{q}$ are the effective charges given by
the equations:

$\displaystyle Q^{2}_{g}$ $\displaystyle=$ $\displaystyle
g^{2}(T)\frac{6PolyLog[2,z_{g}]}{\pi^{2}}$ $\displaystyle Q^{2}_{q}$
$\displaystyle=$ $\displaystyle g^{2}(T)\frac{-12PolyLog[2,-z_{q}]}{\pi^{2}}$
(11)

In our present analysis,the temperature and baryonic chemical potential
dependent quasi-particle Debye mass, $m_{D}^{QP}$ in full QCD with $N_{f}=3$
has been employed to study the Differential Scattering cross-section of the
ground states of quarkonia.

##  IV Formulation of Differential Scattering cross-section using Non-
relativistic limit of Quantum field theory (QFT)

In the non-relativistic limit, the QFT equation for the S-matrix reduce to the
Lippmann-Schwinger equation for the scattering amplitude. The Lippmann-
Schwinger equation is equivalent to the Schrodinger equation. In non-
relativistic quantum mechanics the first order Born approximation of the
elastic scattering amplitude is given by the Fourier transform of the
potential. Correspondingly, the potential is given as inverse Fourier
transform of the Scattering amplitude.
In classical scattering theory, the essential countering problem is: (a) to
measure the impact parameter, (b) to calculate the scattering angle. But in
the Quantum scattering theory, the solution to the Schrodinger equation pave a
good way to understand the scattering process for the proper wave function.
Here quantum description of scattering of non-relativistic particles of mass
$m_{1}$ and $m_{2}$ has been considered. For simplicity, we consider the case
of elastic scattering. The interacting potential between particles is supposed
to be time independent, and obviously time independent Schrodinger equation
will be used. The wave function has been obtained which has been used to
obtained the scattering amplitude and hence the differential scattering cross-
section.

Figure 4: This figure shows the interaction of scattered wave and plane wave
with the potential V(r).

$\left[-\frac{\hbar^{2}}{2m_{1}}\bigtriangledown_{1}^{2}-\frac{\hbar^{2}}{2m_{2}}\bigtriangledown_{2}^{2}+V(\vec{r_{1}},\vec{r_{2}})\right]\psi(\vec{r_{1}},\vec{r_{2}})=E_{T}\psi(\vec{r_{1}},\vec{r_{2}})$
(12)

Where, $E_{T}$ is the total energy of the system, and this body problem can be
reduced into a one-body problem then the Schrodinger equation will be,

$\left[-\frac{\hbar^{2}}{2\mu}\bigtriangledown^{2}+V(\vec{r})\right]\psi(\vec{r})=E_{T}\psi(\vec{r})$
(13)

Now, we have to find wave function by solving Eq.(13). And this is obtained by
complex calculations using green function as following,

$\psi(\vec{r})=\Phi_{inc}(\vec{r})-\frac{\mu}{2\pi\hbar^{2}}\int\frac{e^{ik|\vec{r}-\vec{{r}^{\prime}}|}}{|\vec{r}-\vec{{r}^{\prime}}|}V(\vec{{r}^{\prime}})\psi(\vec{{r}^{\prime}})d^{3}{r}^{\prime}$
(14)

Where, $\psi(\vec{r})$ is the wave function after scattering. To find
scattering amplitude we will apply asymptotic limit on Eq.(14) and will
compare this wave function to the wave function in asymptotic limit that is
discussed below for asymptotic limit is,

$\psi(r,\theta)\cong
A\left\\{e^{ikz}+f(\theta,\Phi)\frac{e^{ikr}}{r}\right\\}$ (15)

The wave function in asymptotic limit after scattering will contain a
unscattered plane wave plus a scattered spherical wave. We consider $A=1$
because it don’t contribute in $d\sigma/d\pi$.

Figure 5: The variation of Differential Scattering Cross-section as a function
of $\mu_{b}$ at fixed values of temperature and theta (in left panel) and as a
function of $T/T_{c}$ at fixed value of $\mu_{b}$ and theta (in right panel)
after consider the values of charmonium and bottomonium masses.

$k|\vec{r}-\vec{{r}^{\prime}}|=kr-\vec{k}\vec{{r}^{\prime}}$ (16)

$\frac{1}{|\vec{r}-\vec{{r}^{\prime}}|}=\frac{1}{r}$ (17)

$\psi(\vec{r})=\Phi_{inc}(\vec{r})-\frac{\mu}{2\pi\hbar^{2}}\int\frac{e^{i(kr-\vec{k}\vec{{r}^{\prime}})}}{r}V(\vec{{r}^{\prime}})\psi(\vec{{r}^{\prime}})d^{3}{{r}^{\prime}}$
(18)

$\psi(\vec{r})=\Phi_{inc}(\vec{r})-\frac{\mu}{2\pi\hbar^{2}}\frac{e^{ikr}}{r}\int
e^{-ikr^{\prime}}V(\vec{{r}^{\prime}})\psi(\vec{{r}^{\prime}})d^{3}{{r}^{\prime}}$
(19)

Now, compare it with,

Figure 6: The variation of $J/\psi$ Differential Scattering Cross-section as a
function of impact parameter at different values of temperature (in left
panel) and at different values of baryonic chemical potential (in right
panel).

$\psi(\vec{r})=\Phi_{inc}(\vec{r})+f(\theta,\Phi)$ (20)

So, Scattering amplitude is,

$f(\theta,\Phi)=-\frac{\mu}{2\pi\hbar^{2}}\int
e^{-i\vec{k}\vec{{r}^{\prime}}}V(\vec{{r}^{\prime}})\psi(\vec{{r}^{\prime}})d^{3}{r}^{\prime}$
(21)

Now, using first order Born Approximation then the above expression is,

$f(\theta,\Phi)=-\frac{\mu}{2\pi\hbar^{2}}\int
e^{i\vec{q}\vec{{r}^{\prime}}}V(\vec{{r}^{\prime}})d^{3}{r}^{\prime}$ (22)

Now, we consider the spherically symmetric potential,

Figure 7: The variation of differential Scattering cross-section as a function
of impact parameter after consider the values of charmonium and bottomonium
masses.

$f(\theta,\Phi)=-\frac{\mu}{2\pi\hbar^{2}}\int
e^{i\vec{q}\vec{{r}^{\prime}}}V(r)r^{2}sin\theta drd\theta d\Phi$ (23)

This is the modified form of Born approximation mainly not necessarily at low
energy. But the simplest form of spherical symmetry amplitude after complete
solution we get,

$f(\theta,\Phi)\cong-\frac{2m_{{Q\bar{Q}}}}{q\hbar^{2}}\int_{0}^{\infty}rV(r)sinqrdr$
(24)

The angular dependence of $f$ is carried by $q$,

Figure 8: Variation of the differential Scattering cross-section with the
transverse momentum ($P_{T}$) at $\mu_{b}$=500 MeV, $\theta$=$90^{o}$,and
impact parameter b=5 fm and temperature T=150 MeV.

$q=2ksin\frac{\theta}{2}$ (25)

We consider the r-dependence of the medium modified Cornell potential Eq.(4),
for the calculation of differential scattering cross-section, we consider only
Yukawa term and neglect the constant term of the equation Eq.(4) then we get,

$V(r)=\left[\frac{2\sigma}{m_{D}^{2}}-\alpha\right]\frac{exp(-m_{D}r)}{r}-\frac{2\sigma}{m_{D}^{2}r}$
(26)

Now, we used Eq.(24) (scattering amplitude formula) for the above potential
Eq.(26) to calculate the differential scattering cross-section, and hence the
result of differential scattering cross-section is shown below,

$\frac{d\sigma}{d\Omega}=\left|f(\theta,\mu_{b})\right|^{2}=\left[\frac{2\sigma}{m_{D}^{2}}-\alpha\right]^{2}\left[\frac{m_{q}}{m_{D}^{2}+(10sin\frac{\theta}{2})^{2}}\right]^{2}\\\
+\left[\frac{2m_{q}\sigma}{m_{D}^{2}(10sin\frac{\theta}{2})^{2}}\right]^{2}$
(27)

We also calculate the Differential Scattering Cross-section in terms of impact
parameter and momentum transfer which is described below,

$b=\frac{a}{2}cot\frac{\theta}{2}$ (28)

After consider the semi-classical consideration, for H-atom like problem
($z_{1}e$=$z_{2}e$ $\approx$ $1$) the value of a is,

$a=\frac{z_{1}z_{2}e^{2}}{mv^{2}}$ (29)

So, after consider the Eq.(29) the Eq.(28) is changes into the Eq.(30) i.e,

$sin\frac{\theta}{2}=\frac{m_{q}cos\frac{\theta}{2}}{2P_{T}^{2}sin\frac{\theta}{2}}$
(30)

Now, put the value of Eq.(30) into the Eq.(27), then the Differential
Scattering Cross-section is converted into the Eq.(31) in terms of impact
parameter and momentum transfer.

$\frac{d\sigma}{d\Omega}=|f(\theta,\mu_{b})|^{2}=\left[\frac{2\sigma}{m_{D}^{2}}-\alpha\right]^{2}\left\\{\frac{m_{q}}{m_{D}^{2}+\left[\frac{10m_{q}cos\frac{\theta}{2}}{2bP_{T}^{2}}\right]^{2}}\right\\}^{2}\\\
+\left\\{\frac{2m_{q}\sigma}{m_{D}^{2}\left[\frac{10m_{q}cos\frac{\theta}{2}}{2bP_{T}^{2}}\right]^{2}}\right\\}^{2}$
(31)

##  V Results and conclusions

In the present paper, we have studied the Differential Scattering Cross-
section for quarkonium ground states i.e, charmonium (J/$\psi$ state) and
bottomonium ($\Upsilon$ state) using Non-relativistic limit of Quantum field
theory (QFT). Figure 1, shows the variation of Differential Scattering Cross-
section as a function of theta for different masses of the quarkonia (for
J/$\psi$=1.5 GeV and for $\Upsilon$=4.5 GeV). It has been seen from figure 1
that the variation of Differential Scattering Cross-section for $\Upsilon$ is
greater than that of J/$\psi$ at T=250 MeV and $\mu_{b}$=500 MeV. It has been
also observed that if the value of $\theta$ increases, the separation between
the Differential scattering cross-section of $\Upsilon$ and J/$\psi$ is
decreases. This is because of the fact, with increase in the values of
$\theta$, the probability of finding the particle (J/$\psi$ and $\Upsilon$)
decreases.
Figure 2 and Figure 3, shows that the variation of Differential scattering
Cross-section as a function of $\mu_{b}$ (in left panel) at different values
of temperature and as a function of temperature (in right panel) at different
values of $\mu_{b}$ at fixed value of $\theta$=$90^{o}$ for the J/$\psi$ and
for $\Upsilon$ respectively. It has been seen that the Differential Scattering
Cross-section as a function of $\mu_{b}$ and temperature is decreases
exponentially. If we increase the values of temperature (in left panel), then
variation of Differential Scattering cross-section is also decreases. With
increases the values of baryonic chemical potential $\mu_{b}$ (in right
panel), the variation of Differential Scattering cross-section is decreases
but the effect of $\mu_{b}$ is smaller as compared to the temperature which
can be seen from the right panel of figure 2 and 3 respectively.
Figure 4, shows the interaction of plane wave with the potential V(r), after
interaction, the plane wave scattered spherically. The interacting potential
between particles is considered as time independent potential given in Eq.(26)
and here we use time independent Schrodinger equation to calculate the
Differential scattering cross-section expression which is given in Eq.(27).
Whereas, figure 5, shows the variation of Differential scattering cross-
section as a function of baryonic chemical potential at fixed value of
$\theta=90^{o}$ and temperature (T=250 MeV) (in left panel) and in right panel
as a function of $T/T_{c}$ for the fixed value of $\mu_{b}$ and $\theta$
(i.e., $\mu_{b}$=500 MeV and $\theta=90^{o}$) and for mass $m_{J/\psi}$=1.5
GeV and $m_{\Upsilon}$=4.5 GeV respectively. It has been clearly seen from
figure 5 that the variation of Differential scattering cross-section of
$\Upsilon$ is greater than in comparison to $J/\psi$ because mass of
$\Upsilon$ is greater than as compared to J/$\psi$.
Figure 6 shows the variation of Differential Scattering Cross-section for the
fixed value of transverse momentum $P_{T}$=5 GeV and $\theta$=$90^{o}$ as a
function of impact parameter at different values of temperature and fixed
baryonic chemical potential $\mu_{b}$=500 MeV (in left panel) and at different
values of the baryonic chemical potential and fixed temperature T=300 MeV (in
right panel). Also, differential cross section of the quarkonium production is
decreases with impact parameter. In the case of impact parameter, the
variation of differential scattering cross-section is also decreases with the
increases of temperature and baryonic chemical potential.
Figure 7 shows the variation of differential Scattering cross-section for the
fixed values of the temperature (T=300 MeV), baryonic chemical potential
($\mu_{b}$=500 MeV), transverse momentum ($P_{T}$=5 GeV) and $\theta$=$90^{o}$
with impact parameter for charmonium ($m_{J/\psi}$=1.5 GeV) and bottomonium
($m_{\Upsilon}$=4.5 GeV) masses. It has been clearly seen from figure 7 that
the variation of differential scattering cross-section of J/$\psi$ is greater
than in comparison to $\Upsilon$ with respect to impact parameter.
Finally, figure 8 shows that how the differential Scattering cross-section
varies with the transverse momentum at $\mu_{b}$=500 MeV, $\theta$=$90^{o}$,
$b$=5 fm and $T$=150 MeV. It has been also observed that there is strong
decrease in the differential cross section with the transverse momentum for
higher masses, and same behavior of deferential scattering cross-section is
observed like impact parameter.
Usually the scales one encounters are $P_{T}$, $m_{Q}$, $m_{Q}\lambda$ and
$m_{Q}\lambda^{2}$, where, $P_{T}$, $m_{Q}$ and $\lambda$ are the transverse
momentum, heavy quark mass and heavy quark-anti-quark pair relative velocity
in the quarkonium rest frame ( $\lambda^{2}$ is 0.1 for bottomonium and 0.3
for charmonium). For moderate and high transverse momentum $P_{T}\gtrsim
2m_{Q}$ the established and most successful theory that describes quarkonium
production and decays in non-relativistic QCD 34 ; 38 , and this theory is
very useful for showing the accurate description of this kind of purpose.
In recent years, different phenomenological approaches have been proposed to
describe the modification of the production cross-section of moderate and high
transverse momentum quarkonia. Theoretical guidance on the relative
significance of the various nuclear effects in the currently accessible
transverse momentum range can be very useful.
Finally, we have concluded that the probability of finding the particle
(charmonium $m_{J/\psi}$=1.5 GeV and bottomonium $m_{\Upsilon}$=4.5 GeV)
depends upon the scattering angle ($\theta^{o}$), temperature ($T/T_{c}$) and
baryonic chemical potential $\mu_{b}$. Although the baryonic chemical
potential show small effect in comparison to the temperature. Moreover,
Differential Scattering Cross-section shows the strong decrease with increase
in the impact parameter as well as in the transverse momentum. This work might
be helpful in understanding the process of quarkonia production under
different parameters such as temperature, baryonic chemical potential etc. It
is also useful to investigate the scattering rates of quarkonia. It would also
provide a large amount of information regarding the internal structure of the
colliding particles.

## Data Availability

The data used to support the findings of this study are available from the
corresponding author upon request.

## Conflicts of Interest

The author declare that there is no conflict of interest regarding the
publication of this paper.

## Acknowledgement

One of the author, VKA acknowledge the Science and Engineering research Board
(SERB) Project No. EEQ/2018/000181 New Delhi for providing the financial
support. We record our sincere gratitude to the people of India for their
generous support for the research in basic sciences.

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11institutetext: Process and Data Science (PADS), RWTH Aachen University,
Germany 11email<EMAIL_ADDRESS>
22institutetext: Knowledge-Based Systems Group, RWTH Aachen University,
Germany 22email<EMAIL_ADDRESS>
33institutetext: Learning Technologies Research Group, RWTH Aachen University,
Germany
33email<EMAIL_ADDRESS>

# A Combined Approach of Process Mining and Rule-based AI for Study Planning
and Monitoring in Higher Education

Miriam Wagner 11 Hayyan Helal 22 Rene Roepke 33 Sven Judel 33 Jens Doveren 33
Sergej Goerzen 33 Pouya Soudmand 11 Gerhard Lakemeyer 22 Ulrik Schroeder 33
Wil van der Aalst 11

###### Abstract

This paper presents an approach of using methods of process mining and rule-
based artificial intelligence to analyze and understand study paths of
students based on campus management system data and study program models.
Process mining techniques are used to characterize successful study paths, as
well as to detect and visualize deviations from expected plans. These insights
are combined with recommendations and requirements of the corresponding study
programs extracted from examination regulations. Here, event calculus and
answer set programming are used to provide models of the study programs which
support planning and conformance checking while providing feedback on possible
study plan violations. In its combination, process mining and rule-based
artificial intelligence are used to support study planning and monitoring by
deriving rules and recommendations for guiding students to more suitable study
paths with higher success rates. Two applications will be implemented, one for
students and one for study program designers.

###### Keywords:

Educational Process Mining Conformance Checking Rule-based AI Study Planning
Study Monitoring.

##  1 Introduction

In higher education, study programs usually come with an idealized,
recommended study plan. However, given how students have different capacities
to study due to circumstances like part-time jobs or child care, and how one
deviation from the intended study plan can have ripple effects spanning
several semesters, in reality, a large number of different study paths can be
observed. Further, capacity limits like room sizes or the amount of
supervision that lecturers can provide make the planning of study paths more
complex. Even though individualized study paths are possible due to the
flexibility in study programs and their curriculum, students may need
assistance and guidance in planning their studies. Software systems that
assist students and study program designers in planning might do so by
analyzing the large amounts of data in higher education institutions [12]. Of
particular interest in this context are event data extracted from Campus
Management Systems (CMS) including course enrollments, exam registrations and
grade entries. To this purpose the AIStudyBuddy project - a cooperation
between RWTH Aachen University, Ruhr University Bochum and University of
Wuppertal - is set up. For preliminary analyses, we received access to the CMS
data of two Bachelor programs, Computer Science and Mechanical Engineering, at
RWTH Aachen University. Within the project, it will be investigated how to
preprocess the data of all partners to apply the preliminary as well as the
further intended analyses.

Figure 1: Overview of the project, showing the two parts: StudyBuddy and
BuddyAnalytics and their relationships to the different systems and
techniques.

The aim of the project is to develop two applications: an intelligent planning
tool for students and an analytics dashboard for study program designers (see
Figure 1). Both will be powered by a combination of rule-based Artificial
Intelligence (AI) and Process Mining (PM) approaches. The implementation and
evaluation of this combination’s ability to efficiently generate rich feedback
when checking the conformance to formal study plans is a key aspect of this
project. This feedback will include PM results in the form of recommendations,
which do not result from explicit regulations but rather historic study path
data.

The planning tool for students, StudyBuddy, will use rule-based AI to check
preliminary plans against an encoding of study program regulations. It will be
able to provide immediate high-quality feedback regarding any potential
conflicts in students’ study plans. In addition to the rules explicitly
codified in institutional regulations, the tool will have a notion of
recommendations, which result from analyzing historical CMS data using PM
approaches and finding characterizations of successful paths, e.g., finished
in standard period of study.

The analytics dashboard, BuddyAnalytics, will enable study program designers
to explore the PM results for the process of Curriculum Analytics. Process
models of recommended study plans can be compared to study paths in the data
to detect possible deviations or favorable routes. Various study path analyses
could support monitoring and help study program designers as well as student
counseling services to support successful study paths and intervene in
misguided study planning by providing individualized plans.

The paper is structured as follows: Section 2 presents relevant related work
in the fields of PM, rule-based AI and curriculum analytics. Section 3
introduces the aim of addressing individualized study planning for students
and data-driven study monitoring for study program designers in a combined
approach. The current state as well as challenges of the project are described
in Section 4, while Section 5 presents objectives of future work. Section 6
concludes the paper.

##  2 Related Work

###  2.1 Process Mining in Education

Educational Process Mining (EPM) [4, 27] is a sub-field of PM [28], using
various, commonly known PM techniques in the educational context, e.g. higher
education. While we focus on CMS data, most work in EPM has been done using
Learning Management Systems (LMS) data with similar aims. In [20], two online
exams have been analyzed using dotted chart analysis and process discovery
with various miners. The applicability of standard methods provided in ProM in
the context of LMS data is shown. In [5], course-related student data has been
extracted to visualize the learning processes using an inductive miner to help
preventing failing the course. “Uncover relationships between usage behavior
and students’ grades” is the aim of [13] by using Directly-Follow Graph (DFG).
In [11], a case study is described in which the LMS data of one course is
analyzed using among other things DFG. Also, in [18], data from an LMS is used
and the creation of the event log is described in detail. Those event logs are
used for the creation of DFG.

Analyses of LMS data show that the PM techniques can be used in the
educational context but while concentrating on the behavior of students in one
course, Curriculum Mining analyzes the different study paths a student can
take [19] which is a substantial aspect in our work. Here, different
approaches exist: [29, 25] describe ways to use curriculum data to uncover the
de-facto paths students take to, in the next step, recommend suitable follow-
up courses. To our knowledge, this next step has not been done. [8] focuses on
the study counselor perspective and uses, e.g., Fuzzy Miner and Inductive
Visual Miner, to visualize the de-facto study paths and use those insights to
improve the curriculum. In [23], the influence of failing a course on the
study success is analyzed using mainly DFGs, while in [24], the analysis is
done by modeling how students retake courses and the influence on study
program dropouts.

Further, we will explore the application of conformance checking [10].
Therefore, similar approaches to ours are reviewed. An extended approach to
conformance checking is multi-perspective conformance checking as in [17]. For
our purpose, one reason to not extend this technique is that the Petri nets
representing the study plan are hard to read when including all allowed
behavior. For example, allowing a course to be in different semesters might
lead to repositioning other courses as well. Furthermore, some rules that need
to be represented are not connected to the model itself, e.g., credit point
thresholds belonging to a semester and not to a course. Those could be modeled
using invisible transitions, which makes the model more complicated and less
intuitive.

###  2.2 Related Work on Rule-based AI

The goal of rule-based AI is to model the examination regulations and the
module catalog in a machine-readable language that allows for dealing with and
planning events. For such scenarios, the combination of Answer Set Programming
(ASP) and Event Calculus (EC) is applied. Both are based on a wider concept
called non-monotonic reasoning, which differentiates from monotonic reasoning
by the ability to retract already made implications based on further evidence
[6].

Non-monotonic reasoning can model defaults as described in [22]. Defaults are
assumed to hold, but do not necessarily have to. For instance, Students
typically take course X after they do course Y will be modeled as a default,
as it is a recommendation, not a requirement. As long as the student does not
plan anything against it, it will be considered in their planning. Else, it
will be ignored. A requirement on the other hand must be valid for all plans.

Looking for similar approaches, in [2], the problem of curriculum-based course
timetabling was solved using ASP, however using a mechanism other than EC.
While we consider recommendations to be defaults that must be typically
followed, they should only ever result in a warning to the student, still
giving the freedom to be deviated from. In [2], recommendations come in handy
for planning, where the number of violations on them should be minimized.
Furthermore, the timetabling problem focuses much more on the availability
requirement for courses rather than also considering the results (e.g. success
or failure, Credit Points (CPs) collected, …) of these courses, which is the
main focal point for us.

More generally, Declarative Conformance Checking [10] is a common application
of rule-based AI to process models. In [16, 9], declarative rules are used
instead of classical conformance checking based on Petri nets. While [16] just
covers the activities for constraints, [9] extended it with a data- and time-
perspective. Furthermore, [1] has a wider model for requirements. It specifies
three kinds of requirements, which refer to the relation in time between
events, e.g. an event has a succession requirement if there is an event that
must be done in the future after doing it. All three approaches use Linear
Temporal Logic instead of ASP and EC, as it it suitable for modeling the three
mentioned requirements. For our purposes though, it makes the modeling of the
contribution of an event to a specific result (e.g., CPs) harder, because our
approach does not focus on the relation in time between events as much as the
contributions of these events.

###  2.3 Curriculum Analytics and Planning

Having emerged as a sub-field of Learning Analytics, curriculum analytics aims
to use educational data to drive evidence-based curriculum design and study
program improvements [15]. Leveraging the data gathered in educational
institutions, it can help identify student’s needs and reduce dropout rates
[12]. As such, different approaches and tools (e.g., [3, 7, 14, 21]) have been
developed to support the analysis of CMS or LMS data with the aim of helping
instructors and program coordinators reflect on the curriculum and teaching
practices. While various data and PM approaches have been used to analyze
study paths provided through CMS event data [3, 21], curriculum sequencing and
study planning was explored using semantic web concepts applied on examination
regulations, with the overall aim of supporting curriculum authoring, i.e.,
the design of personalized curricula fulfilling a set of constraints [1].
Other approaches include recommender systems [30] or genetic algorithms [26]
to support students in course selection processes and fulfilling requirements
of a study program. To the best of our knowledge, however, no joint approach
of PM and rule-based AI has yet been explored in order to support study
planning and monitoring for students and study program designers.

##  3 Approach

The aim of AIStudyBuddy is to support individualized study planning (for
students) and monitoring (for study program designers). Study planning
describes the students’ activities of planning and scheduling modules, courses
and exams throughout the complete course of a study program. The examination
regulations provide recommendations and requirements to describe a study
program and the conditions for students to successfully earn a degree. These
may include a sample study plan recommending when to take which module or
course and attempting to distribute CPs evenly over the standard period of
study. Students choose from the module catalog, a list of mandatory and
elective modules.

While most students may start with the same recommended plan in their first
semesters, deviations due to various reasons can occur at any time, e.g.,
working part-time may result in a reduced course load and delaying courses to
the next year, thus, changing the complete plan and its duration. Therefore,
support for individualized planning as well as recommendations of suitable
study paths are needed. Further, the diversity of study paths and deviations
from recommended study plans raises questions of how different students move
through a study program, if certain modules or courses cause delays in the
study plan, or whether a study program may need revisions. Here, study
monitoring can be provided by analyzing students’ traces in various systems
used in the university. In our project, we will initially focus on CMS data
and might include LMS data later.

In order to support students and study program designers in their respective
tasks, a modular infrastructure (see Figure 1) with two primary applications
for the target groups will be implemented. The application StudyBuddy presents
a web interface to guide and engage students in study planning activities. As
in many programs students do not necessarily have to follow a recommended plan
and in later phases not even have recommendations. To help finding suitable
courses historic data can be used to give hints which combinations have been
successful. Furthermore, course-content is not always independent from other
courses and a specific order might help to pass with higher chance. It offers
an overview of a student’s study program and allows for creation and
validation of individual study plans. ASP and EC are used to model these
regulations. Given a study plan, they can be used to generate feedback
regarding violations and give recommendations. These recommendations are the
result of mining historic data of previous study paths for those with high
success rates.

For study program designers, the application BuddyAnalytics presents an
interactive, web-based dashboard visualizing PM data analysis results.
Different methods, i.e., process discovery and conformance checking, can help
to understand how different student cohorts behave throughout the course of
the study program and identify deviations from recommended study plans. Based
on different indicators and questions by study program designers, student
cohorts can be analyzed and insights into their paths can be gained. Study
program designers can evaluate and compare different study paths and further
develop new redesigned study plans in an evidence-based way.

##  4 Current State & Challenges

The main data source for this project is the CMS of a university, which
contains information about the students, courses, exams and their combination.
Later, the possibility to integrate LMS data will be explored. As the project
aims to be independent from the systems and study programs at different
universities, a general data model has been created (see Figure 2). This model
is the starting point for our project work and shows the general relation
between courses and students as well as study programs. The diagram does not
include all possible data fields as they differ depending on the available
data of a university.

Figure 2: A basic and generic data model for CMS data

Students can have multiple study programs, e.g., first do a Bachelor in
Computer Science followed by a Master. Each semester a student has a study
status, e.g., enrolled or semester on leave. The same offered course is
scheduled in different semesters, e.g., Programming is offered every winter
semester, and in different study programs, e.g., Introduction to Data Science
is mandatory for a Master in Data Science but elective for a Master in
Computer Science. Students also have a status for scheduled courses during
their study program, e.g., course passed.

Until now, we explored data on exam information (ie., registrations and
results). The analyzed data includes Bachelor and Master Computer Science
students as well as Mechanical Engineering Bachelor of RWTH Aachen University.
Some standard KPIs used in various departments of universities that give
meaningful insights about students, study programs or cohorts are:

  * •

Success rate of a course [in specific semesters] [for a cohort]

  * •

Number of attempts a course is taken [on average] [for a cohort]

  * •

Exams taken/passed in a [specific] semester [on average] [of a cohort]

  * •

Average study duration [of a cohort]

  * •

Percentage of dropouts [of a cohort] [in a predefined period]

A cohort can be defined based on the semester a group of students started,
e.g., cohort WS21 refers to all students that started in winter semester
2021/2022. It can also be defined by the amount of semesters students already
studied or the examination regulations they belong to. Different cohort
definitions are needed to answer various questions about the behavior of
students.

For more insights exceeding simple SQL queries used for descriptive
statistics, the data is transferred into specific event logs, in which
activities can be based just on courses and exams, or may even include
additional information. First, we concentrated on events describing the final
status of exams for students. A student can have multiple occurrences of a
course, e.g. when they do not pass the exam in the first try or when they
registered first, but in the end, they did not take it. As a timestamp, the
semester or the exact exam date can be used. Also, some activities may have
specific status dates, e.g., the date of the (de-)registration. Those event
logs can be used to create de-facto models showing the actual behavior of a
group of students. As model we use DFG, BPMN models, process trees and Petri
nets, as shown in Figure 3, because they are easy to read also for non-
specialists in PM.

Figure 3: Model created by ProM plugin "Mine Petri net with Inductive Miner"
for data of students studying in examination regulation 2018 just using their
mandatory courses

For useful insights, the multiple occurrence and the partial order of courses
must be treated. The partial order is caused by using, e.g., the scheduled
semester, instead of the arbitrarily set exam dates, based on among others
room capacities. We tried out different solutions with the setting depending
on the underlying questions that should be answered by the resulting model,
e.g., when using a combination of exam attempt and course ID as the activity,
the resulting de-facto model shows how courses are retried and visualizes
better the real workload per semester. In Figure 3, just the first occurrence
of the course is used and all exams of a semester have the same date.
Semester-blocks are visible, especially when the offered semester of a course
is in mind, e.g., Programming and Calculus are offered in the winter semester.
The examination regulation of 2018 states that it should be taken in the first
semester. Compared to the (simplified) recommended plan (see Figure 4)
Mentoring occurs two semesters before Calculus, while they should be
concurrent. Data Communication and Security is taken two semesters earlier
than planned and before courses that should precede it, e.g., Computability
and Complexity. Those models give a first impression of the actual study path
but need interpretation.

As a simpler approach to the later proposed combination of ASP and classical
PM conformance checking, we explored the possibility of creating de-jure
models based on the recommended study plan. We used Petri nets since they can
cover course concurrency and are still understandable by non-experts. The de-
jure model in Figure 4 shows the main recommended path. Note, the data was
just available including the third semester and later courses are invisible.
Using Petri nets and conformance checking this recommendation becomes a
requirement.

Figure 4: Conformance checking result using ProM plugin "Replay a Log on Petri
Net for Conformance Analysis" on data of students studying in examination
regulation 2018 and a simplified Petri net model of the regulation

The results of classical conformance checking are still useful to find typical
deviation points, e.g., Linear Algebra tends to be taken in a different
semester than proposed. Also, when filtering on the first exam attempts, the
resulting insights are different from filtering on the successful passing of
exams. Filtered on the first attempt, we can see how many students actually
tried to follow the plan, while filtered on the passed exams indicates the
success route. When we have a high percentage of students that try to follow
the recommended study plan, but just a low percentage that succeeds, this may
be a warning for study program designers that the rules may need to be adapted
to improve the recommendation and thereby increase the study success of
students.

Our findings show that in later semesters, students deviate more from the
recommended study plan, which can be explained by delays occurring earlier in
their study. What is not modeled by the Petri net here is that for Seminar
(semester 5), Proseminar (semester 2) is a prerequisite. Therefore, Proseminar
has to be taken before Seminar and forms a requirement. Including those
additional requirements and all already planned exceptions from the original
plan, those models are fast becoming so called spaghetti models and lose a lot
of their readability. Lastly, additional constraints, e.g., credit point
constraints such as at the end of the third semester, at least 60 CPs must
have been earned, are not taken into account using just the described
approach.

For that matter, we used the combination of ASP and EC such that e.g. defaults
can model recommendations. The first main issues concerning modeling study
requirements in general and using EC was translating examination regulations
given in natural languages into formal languages. We encountered the following
problems and challenges:

  * •

There are rules that are understood by any human and thus not written.

  * •

There is a lot of human interference that allows for exceptions. Exceptions in
study plans are not rare.

  * •

There are older versions of the examination regulations, which certain
students still follow.

The second problem we encountered with EC is that almost all events contribute
to a single result (e.g. CPs), instead of a majority of events, each
initiating new kinds of results. EC is designed for the latter, but in study
plans the former holds. We thus adjusted the EC. One modification was to
differentiate between events that happened and events that are planned. For
planning in the future, one needs to follow the rules. For events in the past,
a record is sufficient and there is no need for further requirement checking.
This allows to add exceptions that are actually inconsistent with the
examination regulations. It was also important to keep track of certain
relevant numbers a student has at any point in time, in order to be able to do
requirement checking. This was achieved through results, which events can
contribute to. Mathematics 1, for example, adds 9 units to the result credit
points, after the event of success at it. A requirement on CPs should consider
the general number of CPs collected or just within a field or a time frame.
For that matter we created the notion of a result requirement, which makes
sure that the sum of results caused by a subset of events is less than,
greater than, or equal to some value. With all of this in mind, we separated
the required rules into three categories:

  * •

Invariant: Rules about the requirements and the EC modified axiom system.

  * •

Variant by Admins: Rules about modules and their availability.

  * •

Variant by Student: Rules about the plan of the student.

After that, we were able to translate the examination regulations, module
catalogs, and student event logs into rules. This enables us to perform model
as well as conformance checking.

##  5 Future steps

Until now, the data are limited to information about exams and is exclusively
derived from the CMS. In a next step, course enrollments will be added to
further analyze study behavior of students. This additional information will
give more concrete insights about the students’ intended study plan, since at
many universities, course enrollments are not automatically coupled to exam
registrations. While students might start to take a course in the intended
semester, thus enroll in it, they might realize that the workload is too high
or exam qualification requirements are not fulfilled and refrain from
registering for the exam in the end. This may also be valuable information
considering the instructors’ workload as more course enrollments indicate more
work during the course and may require larger lecture halls or additional
support staff. As such, this workload needs to be balanced out when planning
courses for upcoming semesters

The information stored in the LMS contains valuable information to understand
students’ learning behavior, as shown in related work. When combined with
activities in the CMS, a more complete view on students’ behavior and more
direct feedback about the success of the intended plan can be provided. This
feedback can then be used in BuddyAnalytics to help study program designers in
improving curricula and recommended study plans, as well as give more informed
suggestions for individual study plans. Possibly, in StudyBuddy, students
might be informed about their behavior deviating from a recommended plan and
are presented with suggestions suitable to their individual circumstances.

On the theoretical side, the possibilities of a combination of AI and PM are
further explored and implemented. The main focus will be to improve the
conformance checking results. Also, PM conformance checking possibilities will
be further explored. One planned aspect is the extraction of constraints from
event logs directly. We expect to learn rules that are not intended but are
beneficial, e.g., Statistics is a good preparation for Introduction to Data
Science and when taken in order, the grade and success rate of the latter
improves. Those rules could be added to the examination regulations rules as
defaults.

##  6 Conclusion

The AIStudyBuddy project will combine different existing AI and PM frameworks
and extend them with new features, making use of the already existing data at
universities, to help students and study program designers make more informed
decisions about study paths and curricula. The first results get positive
feedback from students and study program designers. Currently, only a small
fraction of available CMS data was used to produce these results, leaving a
lot of potential for future steps. PM techniques already give valuable new
insights to the study program designers and the combination of AI and PM for
conformance checking in particular helps overcome restrictions due to the data
and rule properties. Having requirements and recommendations, credit point
boundaries, and long-term relations between courses should be included in the
system to model examination regulations in a more accurate manner.

####  6.0.1 Acknowledgements:

The authors gratefully acknowledge the funding by the Federal Ministry of
Education and Research (BMBF) for the joint project AIStudyBuddy (grant no.
16DHBKI016).

####  6.0.2 Printversion

This paper is a postprint version. The published version is ©Springer (DOI
pending).

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