# Understanding the QCD medium by the diffusion of charm quarks using a Color
String Percolation Model

Kangkan Goswami  Dushmanta Sahu  Raghunath Sahoo111Corresponding Author Email:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Indore, Simrol, Indore 453552, India

###### Abstract

We study the drag and diffusion coefficients of the charm quark in the
deconfined matter produced in the ultra-relativistic collisions by taking the
Color String Percolation Model (CSPM) approach. CSPM, being a QCD-inspired
model, can give us essential information about the hot and dense system
produced in ultra-relativistic collisions. With the information on initial
percolation temperature and percolation density, we estimate the relaxation
time ($\tau_{c}$), drag coefficient ($\gamma$), transverse momentum diffusion
coefficient ($B_{0}$), and spatial diffusion coefficient ($D_{s}$) of charm
quark inside a deconfined medium. Finally, we compare the obtained results
with lattice QCD and with various other theoretical models. A good agreement
can be observed between the results obtained from CSPM and lattice QCD.

##  I Introduction

Heavy quarks, such as the charm and bottom quarks, can be used as essential
probes to study the initial phase of the system’s evolution in an ultra-
relativistic collision Xu:2017hgt ; Prino:2016cni ; Aarts:2016hap ;
Greco:2017rro . As the charm and bottom quarks have very high masses compared
to the lighter quarks, they are produced relatively early in the evolution of
the system. The masses of the heavy quarks are significantly larger than that
of the temperature of quark-gluon plasma (QGP), which means that the
probability of the charm and bottom quarks getting created or annihilated
during the deconfined phase is much less than that of the lighter quarks and
gluons. As a result, the heavy quarks witness the whole space-time evolution
of the system. These heavy quarks interact with the hot and dense medium
formed in the ultra-relativistic collisions, because of which their momentum
spectra get modified. However, these interactions do not completely thermalize
heavy quarks with intermediate and high transverse momentum ($p_{\rm T}$).
Thus, they can carry out important information about the initial stages of the
expanding fireball.

The energy loss mechanism of the heavy quarks in the medium is very distinct
from that of the lighter quarks. For light quark jets, the leading mechanism
for energy loss is due to the gluon radiation Baier:1996kr . On the contrary,
the gluon radiation mechanism will be suppressed for heavy quarks
Dokshitzer:2001zm , and their energy loss will be mainly due to the elastic
collisions with the lighter quarks in the medium Moore:2004tg . Moreover, the
energy of a heavy quark is not changed too much from collisions with a light
quark; thus, the thermalization time of the heavy quark will be substantially
larger than that of the lighter quarks. It is worth mentioning that the QGP
lifetime is estimated to be in the order of 4-5 fm/c Heinz:2002gs  at the
Relativistic Heavy Ion Collider (RHIC). Similarly, at the Large Hadron
Collider (LHC), it is around 10-12 fm/c Foka:2016vta . On the other hand,
thermalization time for charm quark is calculated to be in the order of 10-15
fm/c, and for bottom quarks, 25-30 fm/c Moore:2004tg ; vanHees:2005wb ;
Cao:2011et . The charm quark interaction with the thermalized lighter quarks
and gluons in the medium will lead to a Brownian motion, which can be
explained by the Fokker-Planck transport equation. Hence, information about
the interaction of a heavy quark in the deconfined medium is preserved within
its drag and diffusion coefficients, which can be estimated by solving the
Fokker-Planck equation. While traversing through the medium, the average
momentum of heavy quarks gets modified, which is incorporated in their drag
coefficient. Their momentum distribution is also broadened, which is embedded
in their momentum diffusion coefficient. These coefficients vary with the
temperature of the expanding matter, and in principle, they can give essential
information about the systems formed in high-energy collisions.

Apart from the drag and momentum diffusion coefficients, the spatial diffusion
coefficient ($D_{s}$) is also of enormous interest to the scientific
community. Theoretically, the heavy quark spatial diffusion coefficient has
been calculated from perturbative QCD (pQCD) vanHees:2004gq  and also from
Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence Gubser:2006qh .
AdS/CFT calculations give the value of $2\pi TD_{s}$ = 1 in the strong
coupling limit Kovtun:2003wp . As observed from several works in the
literature, there is a minimum for this parameter approaching the AdS/CFT
value near the critical temperature. In Ref. Scardina:2017ipo , the authors
have estimated the charm quark spatial diffusion coefficient from $D$ meson
spectra at energies available at RHIC and LHC. The heavy quark momentum and
spatial diffusion coefficient have also been obtained from lattice QCD by
investigating a color electric field correlation function using Monte-Carlo
techniques Banerjee:2011ra . Recently, Brambilla et al. have reported some
progress towards the computation of heavy quark momentum diffusion coefficient
from two chromoelectric fields correlator attached to a Polyakov loop in SU(3)
gauge theory Brambilla:2020siz . By studying the diffusion coefficient as a
function of temperature, they show that the results from lattice QCD agree
quite well with that of the next-to-leading order (NLO) perturbative results.
The effect of viscosity on the diffusion coefficient has also been studied by
taking a semi-QGP matrix model Singh:2019cwi , where the spatial diffusion
coefficient is observed to be decreasing with the increase in shear and bulk
viscosities. In addition, the diffusion of $D$ meson in the hot and dense
hadronic matter has been studied for both zero and finite baryochemical
potential cases Ozvenchuk:2014rpa . A strong baryochemical potential
dependency has been observed in the estimation of relaxation time for $D$
meson; for higher $\mu_{B}$, the relaxation time is observed to be reduced by
a factor of 2-3.

It has been observed that the Color String Percolation Model (CSPM) can
qualitatively explain the observed increase in the relative $J/\psi$ yield
with respect to relative charged-particle multiplicity in pp collisions at
$\sqrt{s}$ = 7 TeV Ferreiro:2012fb . This motivates us to look into heavy
flavor dynamics in the deconfined medium by using CSPM. In this work, we study
the relaxation time, drag, and momentum diffusion coefficient of the charm
quark, along with $D_{s}$ as functions of initial percolation temperature by
taking the CSPM approach. The paper is organized as follows. Sec. II
encompasses the CSPM framework along with the formalism of the drag and
diffusion coefficients of the charm quark. In Sec. III, the results obtained
using the formulation are discussed. Finally, Sec. IV concludes the findings
of this study.

##  II Formulation

###  II.1 Color String Percolation Model

Figure 1: (Color Online) Percolation of colored strings in the transverse
space.

The color string percolation model is a QCD-inspired, well-established model
to explain the multi-particle production in an ultra-relativistic collision
Braun:2015eoa . The model has been used to estimate various thermodynamic and
transport properties of the matter formed in ultra-relativistic collisions,
and the results have been found to be consistent with other existing models
such as lattice QCD Sahu:2020mzo ; Mishra:2021yer ; Sahu:2022bzg . In this
model, the production of particles can be understood as a consequence of color
strings stretching between the partons of the target and the projectile. In
transverse space, these strings occupy some finite area. With an increase in
collision energy and the number of colliding partons, the number of strings
grows, and they start to overlap, forming colored clusters in the transverse
space. A macroscopic cluster appears after a certain critical string density
($\xi_{\rm c}$), where 50$\%$ of the transverse space is occupied by the
clusters. This marks the percolation phase transition Braun:2015eoa , which is
shown schematically in Fig. 1. According to Scwhinger’s string breaking
mechanism, these strings stretch and break to produce color-neutral quark and
antiquark pairs. These particles finally hadronize to produce the final state
hadrons Braun:2015eoa .

In a 2D percolation theory, the total multiplicity due to $N_{\rm s}$ number
of overlapping strings is given as Braun:1999hv ,

$\mu_{N_{s}}/\mu_{1}=\sum_{n=1}^{N_{\rm s}}\sqrt{n}(S_{\rm n}/S_{\rm 1})$ (1)

where, $n$ is any interger from 1 to $N_{\rm s}$, $S_{1}$ is the transverse
area of a single string given by $S_{\rm 1}=\pi r_{\rm 0}^{2}$ with $r_{\rm
0}$ being the single string radius $\simeq$ 0.2 fm, and $S_{\rm n}$ is the
transverse area occupied by the overlapping strings.

Similarly, for mean transverse momentum squared, $\langle p^{2}_{\rm
T}\rangle$, we can write,

$\langle p^{2}_{\rm T}\rangle/\langle p^{2}_{\rm T}\rangle_{1}=\frac{N_{\rm
s}}{\sum_{n=1}^{N_{\rm s}}\sqrt{n}(S_{\rm n}/S_{\rm 1})}.$ (2)

When the strings in the transverse plane are just touching each other such
that, $S_{\rm n}=nS_{1}$, the total multiplicity and the mean transverse
momentum squared changes as $\mu_{\rm n}=n\mu_{\rm 1}$ and $\langle p^{2}_{\rm
T}\rangle_{n}=\langle p^{2}_{\rm T}\rangle_{1}$ respectively. Another possible
scenario is that all the strings overlap perfectly over each other such that,
$S_{\rm n}=S_{1}$. In this case, we get $\mu_{\rm n}=\sqrt{n}\mu_{\rm 1}$ and
$\langle p^{2}_{\rm T}\rangle_{n}=\sqrt{n}\langle p^{2}_{\rm T}\rangle_{1}$.
This gives us the case where the multiplicity is maximally suppressed and the
mean transverse momentum squared is maximally enhanced. From this, we can
write a relation between multiplicity and mean transverse momentum squared,
$\mu_{\rm n}\langle p^{2}_{\rm T}\rangle_{\rm n}=n\mu_{\rm 1}\langle
p^{2}_{\rm T}\rangle_{1}$, which implies conservation of total transverse
momentum.

Now, assuming the transverse nuclear overlap area to be $S$ and density
$\rho$, we introduce a dimensionless percolation density parameter, $\xi$, is
given by Braun:2015eoa ,

$\xi=\rho S_{\rm 1}=\frac{N_{\rm s}S_{1}}{S},$ (3)

In the thermodynamic limit, the number of the strings $N_{s}\rightarrow\infty$
while $\xi$ is fixed, and the distribution of the overlaps of $n$ strings is
Poissonian with a mean of value $\xi$,

$p_{n}=\frac{\xi^{n}}{n!}e^{-\xi}.$ (4)

From the above expression, the fraction of the total area occupied by the
strings can be given by,

$\sum_{n=1}p_{n}=1-e^{-\xi}.$ (5)

Dividing the above equation by $\xi$, we get the compression factor.
Furthermore, according to the CSPM approach, the multiplicity becomes damped
as a result of overlapping by the square root of the compression factor, which
can be written as,

$\mu_{n}/\mu_{1}=N_{\rm s}\sqrt{\frac{1-e^{-\xi}}{\xi}}.$ (6)

Hence, finally the damping factor or the color suppression factor is given as
Braun:2015eoa ,

$F(\xi)=\sqrt{\frac{1-e^{-\xi}}{\xi}}.$ (7)

In ultra-relativistic heavy-ion collisions, thermalization can occur through
Hawking-Unruh effect Hawking:1975vcx ; Unruh:1976db , where fast
thermalization happens with the existence of an event horizon caused by the
rapid deceleration of the colliding nuclei Castorina:2007eb . In CSPM, the
intense color field inside the large cluster causes deceleration of the
$q\bar{q}$ pair, which can be perceived as a thermal temperature due to the
Hawking-Unruh effect. This suggests that the radiation temperature can be
determined by the transverse extension of the color flux tube in terms of the
string tension.

The initial temperature of the percolation cluster is expressed in terms of
$F(\xi)$. The Schwinger distribution for massless particles can be expressed
in terms of $p_{\rm T}^{2}$ as Hirsch:2018pqm ; Braun:2015eoa ; cywong:1994 ;
Schwinger:1962tp :

$dn/dp_{\rm T}^{2}\sim{\rm exp}(-\pi p_{\rm T}^{2}/x^{2})$ (8)

where $\langle x^{2}\rangle$ is the average value of string tension. As the
chromoelectric field is not constant, the tension of the macroscopic cluster
fluctuates around its mean value. Due to these fluctuations, we get a Gaussian
distribution of the string tension given as,

$\frac{dn}{dp_{\rm T}^{2}}\sim\sqrt{\frac{2}{\langle
x^{2}\rangle}}\int_{0}^{\infty}dx\exp\left(-\frac{x^{2}}{2\langle
x^{2}\rangle}\right)\exp\left(-\pi\frac{p_{\rm T}^{2}}{x^{2}}\right)$ (9)

which in turn gives rise to a thermal distribution,

$\frac{dn}{dp_{\rm T}^{2}}\sim\exp\left(-p_{\rm
T}^{2}\sqrt{\frac{2\pi}{\langle x^{2}\rangle}}\right).$ (10)

From Eq.8 and 10, the initial temperature in terms of $F(\xi)$ can be
expressed as Hirsch:2018pqm ; Sahoo:2017umy ; Mishra:2020epq ,

$T(\xi)=\sqrt{\frac{\langle p^{2}_{\rm T}\rangle_{1}}{2F(\xi)}},$ (11)

where, $\langle x^{2}\rangle=\pi\langle p_{\rm T}^{2}\rangle_{1}/F(\xi)$. By
using $T_{c}=167.7\pm 2.8$ MeV Braun-Munzinger:2003htr ; Becattini:2010sk  and
$\xi_{\rm c}\sim 1.2$ Braun:2015eoa , we can get the single string squared
average transverse momentum, $\sqrt{\langle p^{2}_{\rm T}\rangle_{1}}=207.2\pm
3.3$ MeV. Using this value in Eq. 7, we can get the initial temperature for
different $F({\xi})$ values.

With this information, let’s estimate the drag and diffusion coefficients
within CSPM formalism.

###  II.2 Drag and diffusion coefficients

To study the interaction of charm quark with the thermal quarks inside the
deconfined medium, we take advantage of the Fokker-Planck transport equation
which can be written as,

$\frac{\partial f(t,p)}{\partial t}=\frac{\partial}{\partial
p^{i}}\bigg{[}(A^{i}f(t,p))+\frac{\partial}{\partial
p^{j}}(B^{ij}f(t,p))\bigg{]}.$ (12)

Here, $f(t,p)$ is the time evolution phase-space distribution of charm quarks.
The kernels $A^{i}$ and $B^{ij}$ are given by Ozvenchuk:2014rpa ,

$A^{i}=\int dk~{}\omega(p,k)k^{i},$ (13)

$B^{ij}=\frac{1}{2}\int dk~{}\omega(p,k)k^{i}k^{j},$ (14)

where $\omega(p,k)$ is the collision rate of the charm quark, with initial
momenta $p$ and final momenta $(p-k)$, with the transferred momenta $k$. $i,j$
= 1,2,3 are the spatial indices. In the low transverse momentum limit $(p\to
0)$, the kernels reduce to Ozvenchuk:2014rpa ,

$A_{i}=\gamma p_{i},$ (15)

$B_{ij}=B_{0}P_{ij}^{\bot}+B_{1}P_{ij}^{\parallel},$ (16)

where $\gamma$ is the drag coefficient, $B_{0}$ is the transverse momentum
diffusion coefficient, and $B_{1}$ is the longitudinal momentum diffusion
coefficient. $P_{ij}^{\parallel}$ and $P_{ij}^{\bot}$ are the longitudinal and
transverse projection operators respectively. This drag and diffusion
coefficient definition is valid only when the transferred energy is small.
This is possible in the case of charm quark being non-relativistic. The NRQCD
model assumes the heavy particles to be non-relativistic, and it describes the
inclusive production and decay of quarkonia along with the $S$-wave charmonium
production at high transverse momentum Bodwin:1994jh . Going in line with
this, we can assume the charm quark inside the deconfined medium to be non-
relativistic. Thus the assumptions of Eq.15 and Eq.16 hold true in this case.

The average momentum of a particle at a given time is given by Torres-
Rincon:2013nfa ,

$\langle
p\rangle=\frac{\int_{-\infty}^{\infty}dp~{}p~{}f(t,p)}{\int_{-\infty}^{\infty}dp~{}f(t,p)}=p_{0}~{}e^{-\frac{t}{\tau}}$
(17)

where, $\tau$ is the relaxation time and $p_{\rm 0}$ is the initial momentum
of the particle. Relaxation time of a quark in the hot QCD medium is defined
as the time accounting for the exponential decay of the average momentum. In
other words, the average time interval between two successive collisions can
be termed as the relaxation time of the system. Assuming the velocity of the
thermal quarks to be nearly the speed of light, in natural units we can take
the relaxation time to be equal to the mean free path of the system. The mean
free path ($\lambda$) is inversely proportional to the number density ($n$) of
the system and the scattering cross-section ($\sigma$) between the particles
Sahoo:2017umy ; Braun:2015eoa ,

$\lambda=\frac{1}{n\sigma}.$ (18)

In CSPM formalism, the number density is expressed as the effective number of
color sources per unit volume:

$n=\frac{N_{\rm sources}}{SL},$ (19)

where $S$ is the nuclear overlap area and $L$ is the longitudinal extension of
a string $\sim$ 1 fm. This particular string length is chosen because no new
quark-antiquark pairs can form if the separation between the gluons is less
than 1 fm. In addition, the colliding nuclei are Lorentz contracted, making
their longitudinal dimensions almost negligible. Thus, we take the lower limit
of the string length, which is standard in CSPM studies.

The number of color sources can be defined as the transverse area occupied by
the strings divided by the area of an effective string:

$N_{\rm sources}=\frac{(1-e^{-\xi})S_{\rm N}}{S_{\rm 1}F(\xi)}.$ (20)

Thus, the number density becomes,

$n=\frac{(1-e^{-\xi})}{S_{\rm 1}F(\xi)L}.$ (21)

The cross-section ($\sigma$) can also be expressed as the transverse area of
the effective strings $S_{\rm 1}F(\xi)$. Finally, using Eq.18 and 21 we can
write,

$\lambda=\frac{L}{(1-e^{-\xi})}.$ (22)

The above equation can be used to define the relaxation time of the deconfined
medium, $\tau=\lambda$ Rapp:2008qc .

Now, for heavier quarks like charm quark, the particle dependent relaxation
time can be expressed by Petreczky:2005nh ,

$\tau_{\rm c}=\frac{m_{\rm c}}{T}\tau$ (23) $\Rightarrow\tau_{\rm
c}=\frac{m_{\rm c}}{T}\frac{L}{(1-e^{-\xi})},$ (24)

where $m_{\rm c}$ is the mass of the charm quark, $m_{c}$ $\simeq$ 1.275 GeV.
Due to the dependency of relaxation time on mass of the particle, at any
temperature, we expect the relaxation time of the charm quark to be
significantly higher than that of the lighter quarks. This, in turn, affects
the elliptic flow of charm quark to be smaller than that of the light hadrons
Scardina:2017ipo ; vanHees:2004gq .

The drag coefficient or drag force ($\gamma$), which incorporates average
momentum change, is inversely related to the thermal relaxation time of the
particle, $\gamma=\frac{1}{\tau_{\rm c}}$. From the Einstein’s relation, the
transverse momentum diffusion coefficient for charm quark, which accounts for
broadening of the final momentum distribution, is related to the drag
coefficient as Tolos:2016slr ,

$B_{\rm 0}=\gamma Tm_{\rm c}$ (25) $\Rightarrow B_{\rm
0}=\frac{(1-e^{-\xi})T^{2}}{L}.$ (26)

The diffusion coefficient $(D_{s})$ in position space can also be introduced.
It can be estimated by starting a particle at position and time $x=0$ and
$t=0$, and finding the mean squared position at a later time Torres-
Rincon:2013nfa ,

$\langle(x(t)-x(0))^{2}\rangle=2D_{s}t.$ (27)

Here, we can see that $D_{s}$ is the measure of speed of diffusion in space
and is called as“Einstein relation”. In static limit, the spatial diffusion
coefficient is given by the expression Torres-Rincon:2013nfa ; DsBook ,

$D_{\rm s}=\frac{T}{m_{\rm c}\gamma}$ (28) $\Rightarrow D_{\rm
s}=\frac{L}{(1-e^{-\xi})}.$ (29)

By taking inputs from the CSPM formalism and using the above formulation, we
have estimated the relaxation time, drag, and diffusion coefficients of the
charm quarks inside the deconfined medium, which are discussed in the
following section.

##  III Results and Discussion

To better understand the systems produced in the ultra-relativistic collisions
at RHIC and LHC, we have studied the relaxation time, drag, and diffusion
coefficients of such matter using the CSPM. The results are contrasted with
the results obtained from various other models. Firstly, the charm quark
relaxation time is plotted as a function of the initial temperature in fig. 2.
$\tau_{\rm c}$ signifies the amount of time it takes for the charm particle to
lose its total momentum by interaction with other quarks and gluons while
traversing through the QGP medium. We observe that the relaxation time
decreases with the increase in initial temperature. At temperature nearing the
critical value, the relaxation time of the charm quark is around 10 fm/c. With
the rise in initial temperature, the charm quark relaxation time decreases,
and at 300 MeV, it becomes about 4 fm/c. This is comparable with the result
obtained in ref. vanHees:2004gq , where a pQCD approach has been taken
considering resonant particles’ inclusion inside the medium. The charm quark
relaxation time is lower for a denser system such as the one formed in most
central heavy-ion collisions. In comparison, $\tau_{\rm c}$ is relatively
higher for a less dense medium, such as the one formed in high multiplicity pp
collisions or peripheral heavy-ion collisions. This is because the charm quark
will thermalize faster in a denser medium than in a less dense one due to
substantially more interactions in the medium.

By taking input from the charm quark relaxation time, we estimate the drag
force of the charm quark inside the medium, which is the inverse of relaxation
time. Fig. 3 shows the drag force or drag coefficient of charm quark in the
deconfined medium as a function of temperature. We observe an increasing trend
with the increase in temperature. This suggests that the drag on the charm
quark in denser systems, such as those produced in most central heavy-ion
collisions, will be more than that in the peripheral collisions. For
comparison, we have also plotted the drag force of charm quark estimated from
the quasi-particle model Scardina:2017ipo  and pQCD vanHees:2004gq . The drag
coefficient calculated from the quasi-particle model is a little higher than
the CSPM estimation, whereas $\gamma$ obtained from pQCD lies below the CSPM
results. But there is a considerable agreement between the CSPM and the pQCD
results with the inclusion of resonant particles inside the medium
vanHees:2004gq .

Figure 2: (Color Online) Relaxation time of charm quarks as a function of
initial percolation temperature. The dotted violet line is from pQCD
estimations with resonant particles vanHees:2004gq . Figure 3: (Color Online)
Drag coefficient of charm quark as a function of temperature. The dashed green
line is from the quasi-particle model Scardina:2017ipo , the dotted black and
violet lines are from pQCD estimations without and with the inclusion of
resonant particles, respectively vanHees:2004gq . Figure 4: (Color Online)
Transverse momentum diffusion coefficient of charm quark as a function of
temperature. The dotted black and violet lines are from pQCD estimations
without and with the inclusion of resonant particles, respectively
vanHees:2004gq .

From Einstein’s relation, the drag and momentum diffusion coefficients are
related by the Eq. 25. Fig. 4 shows the transverse momentum diffusion
coefficient as a function of temperature. We observe that $B_{0}$ increases
linearly with an increase in temperature, suggesting that for high
temperature, the momentum broadening of the charm quarks will be higher. We
have also plotted the transverse momentum diffusion coefficient from pQCD to
compare our results. The charm quark momentum diffusion from CSPM estimation
is substantially larger than that of pQCD calculation. This momentum diffusion
of charm quarks, in principle, can affect the elliptic flow of the final state
charmed hadrons. It is worthy to note that, in the hadronic medium, $D_{\rm
0}$ meson will diffuse considerably larger than $J/\psi$ Mitra:2014ipa . This
essentially results in a suppressed $v_{\rm 2}$ of $D_{\rm 0}$. But, because
$J/\psi$ remains largely undiffused in the hadronic phase, the elliptic flow
of $J/\psi$ will give unfiltered information about the deconfined phase,
making $J/\psi$ a cleaner probe to study QGP.

Figure 5: (Color Online) The spatial diffusion coefficient as a function of
temperature scaled with critical temperature. The red and green bands are the
results obtained from T-matrix calculations for U-pot and F-pot respectively
Riek:2010fk . The black dotted line is from pQCD calculations vanHees:2004gq ,
the triangle Banerjee:2011ra  and rectangle Ding:2012sp  markers are from
lattice QCD estimations. The violet dashed line Tolos:2013kva  and the dashed-
dotted green line Ozvenchuk:2014rpa  are the results from D meson diffusion in
a hadron gas.

Finally, in fig. 5 we have plotted the spatial diffusion coefficient as a
function of temperature scaled with critical temperature. Expressing the
transport coefficients in the units of thermal wavelength is very convenient;
thus, we have multiplied a factor of $(2\pi T)$, making the diffusion
coefficient dimensionless. This dimensionless quantity can characterize the
coupling strength of the charm quark to the thermal medium. We have compared
our results with various other results that already exist in the literature,
such as estimations from T-matrix Riek:2010fk , pQCD vanHees:2004gq  and
lattice QCD Banerjee:2011ra ; Ding:2012sp  calculations. Interesting results
have also been observed from the D meson diffusion in the hadron gas
Ozvenchuk:2014rpa ; Tolos:2013kva . As the temperature approaches $T_{\rm c}$,
the D meson diffusion decreases rapidly and is almost comparable with that of
charm quark diffusion values at the critical temperature, showing almost a
smooth transition from hadronic to partonic phase. From AdS/CFT calculations,
the shear viscosity to entropy density ratio gives the minimum value of
$1/4\pi$ Kovtun:2004de . When we study the change of $\eta/s$ with temperature
Sahu:2020mzo , we observe that $\eta/s$ decreases with an increase in
temperature and becomes minimum at $T=T_{c}$, and then again starts increasing
with temperature. In addition, from the study of $\zeta/s$ as a function of
temperature Sahu:2020mzo , we observe that at $T=T_{c}$ the bulk viscosity to
entropy density becomes minimum and almost close to zero. Similarly, AdS/CFT
calculation gives a minimum of $D_{s}(2\pi T)$ $\sim$ 1 Kovtun:2003wp  at the
critical temperature. From fig. 5 we can see that the value of $D_{s}(2\pi T)$
approaches this minima near $T_{\rm c}$. Furthermore, we observe a good
agreement between the CSPM and other model estimations.

This behavior of spatial diffusion coefficient can be understood in terms of
the interaction strength in the system. In the hadronic phase, at a lower
temperature, the interaction will be less; thus, the spatial diffusion
coefficient is higher. As the temperature increases and we go towards the
critical temperature, the interaction also increases, resulting in lower
values of $D_{s}(2\pi T)$. At $T=T_{\rm c}$, the interaction will be maximum
due to the onset of the QGP medium, which corresponds to the minimum of
$D_{s}(2\pi T)$. After this, the spatial diffusion coefficient increases again
with an increase in temperature. This is because, at a higher temperature, the
patrons will be asymptotically free, resulting in a lower interaction
strength.

##  IV Conclusion

In this work, for the first time, we study the relaxation time, drag, and
diffusion coefficients of charm quark in the deconfined medium by taking the
color string percolation approach. Our result is comparable with the pQCD
approach, where the resonant heavy-light quark interactions are introduced.
The observations from the spatial diffusion coefficient state that there is a
minimum at the phase transition. We observe that our findings from the CSPM
agree with the results obtained from lattice QCD for the spatial diffusion
coefficient.

In the view of ALICE RUN-3 going towards high luminosity, the heavy flavor
sector will be of particular interest. With higher statistics, the study of
hadron production containing charm and bottom quarks can be done with high
precision for a broad multiplicity range. Our work, along with other
theoretical works will hopefully help to understand heavy flavor dynamics in
ultra-relativistic collisions as case studies.

## Acknowledgment

KG acknowledges the doctoral fellowship from UGC, Government of India. The
authors gratefully acknowledge the DAE-DST, Govt. of India funding under the
mega-science project – “Indian participation in the ALICE experiment at CERN”
bearing Project No. SR/MF/PS-02/2021-IITI (E-37123).

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conditionHiExposure_Undisclosed:interest_grouptechnology |  $-$1.522 (1.875) |  $-$1.870 (1.465) |  $-$1.926 (1.495)
|  t = $-$0.811 |  t = $-$1.277 |  t = $-$1.288
Constant |  23.789∗∗∗ (2.704) |  17.655∗∗∗ (2.185) |  17.609∗∗∗ (2.235)
| t = 8.796 | t = 8.080 | t = 7.880
Observations | 2,618 | 2,618 | 2,618
Log Likelihood |  $-$10,155.720 |  $-$9,508.078 |  $-$9,561.834
Akaike Inf. Crit. | 20,371.430 | 19,076.160 | 19,183.670
Bayesian Inf. Crit. | 20,547.540 | 19,252.260 | 19,359.770
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01

##  Appendix L Implementation Complications

Running a massive online networked experiment open to any user on the Internet
will often invite implementation challenges. In the interest of disclosure—and
for the benefit of future researchers running similar experiments—we share the
challenges we faced, solutions we implemented, and rationales for our
decisions.

###  L.1. Server Capacity

Overall, we received far more responses than we expected. At several points
throughout the experiment, we experienced more concurrent traffic than the
application was designed to handle. Hence, we had to temporarily turn off the
experiment to wait out high demand, add more resources, or implement and test
changes described in Content Moderation. We note that these capacity issues
did not affect the responses we collected.

###  L.2. Content Moderation

Initially, we did not implement any content moderation. But on July 8th, we
received an influx of responses from Reddit. Several participants were trolls,
providing repetitive profane responses. We then implemented a form of content
moderation, flagging any idea that contained a word in a list of words banned
by Google as of July 8, 2023131313https://github.com/coffee-and-fun/google-
profanity-words and subsequently added two more words to the list. If an idea
was flagged, it was written to our database but not shown to future
participants. Some profane ideas were already shown to participants in between
the time when the responses were submitted and we saw and implemented our
solution. Also, the content moderation strategy was imperfect: 10 of 46
flagged ideas were false positives. As discussed earlier, condition was
unrelated to the number of flagged ideas ($\chi^{2}$(4) = 6.06, p = 0.19) or
total number of excluded ideas ($\chi^{2}$(4) = 3.87, p = 0.42).

We acknowledge there are more advanced and nuanced content moderation
strategies, but this one was the best option given our specific circumstances
and constraints. First, this bag-of-words method is very transparent. Second,
we deployed this experiment through Heroku, which imposes a CPU limit on the
project, precluding the use of pre-trained classifiers such as BERT. Third, we
did not want to use APIs like Jigsaw or OpenAI moderation endpoints because
these APIs have rate limits, which can slow down the experiment.

###  L.3. Small Deviations From 20 Trials Per Chain

We intended for each response chain to contain 20 responses. The average
number of trials per response chain was 19.73 (SD = 1.45) and the median
number per chain was 20. We concluded the experiment before the last round of
response chains was completely finished for all condition and item
combinations, so the minimum number of trials in a response chain (occurring
for an item and condition combination in the last round) was 14. The maximum
number of trials was 24. These minor deviations occurred due to server
overload, very high traffic leading to race conditions and excluding several
responses based on the criteria described in D. Based on a two-way ANOVA, we
concluded that response chain lengths did not differ by item,
$(F(4)=0.38,p=0.82)$, condition $(F(4)=0.01,p=1.00)$, or the interaction
between items and conditions $(F(16)=0.01,p=1.00)$,

##  Appendix M Experiment Screenshots

Figure 12. Screenshot of a trial participants are told to complete. This is
the [Low Exposure, Disclosed] condition since there are two AI ideas and the
AI ideas are labeled.

Figure 13. After each trial, participants were given feedback as motivation to
continue.

Figure 14. Screenshots of the experiment and feedback we provided after each
response


# Quantum scalar field propagator in a stochastic gravitational plane wave

Z. Haba
Institute of Theoretical Physics, University of Wroclaw,
50-204 Wroclaw, Plac Maxa Borna 9, Poland
<EMAIL_ADDRESS>
###### Abstract

We show that if the linearized stochastic Gaussian gravitational plane wave
has the frequency spectrum $\omega^{4\gamma-1}$ ( $0\leq\gamma<1$ ) then the
equal-time propagator of the scalar field behaves as $p^{-\frac{1}{1-\gamma}}$
for large momenta.

##  1 Introduction

The Standard Model on a stochastic background of gravitational waves or the
one supplemented with quantum gravity can be viewed as a theory of matter
fields in a random geometry which can be singular at Planck scale
distances.There are some results [1][2][3][4][5][6] indicating that the
gravity can modify the short distance (equivalently large momentum) behavior
of matter fields. There are some simplified Lorentz non-invariant models where
such a modified short distance behavior in various directions in space-time is
realized [7][8][9]. The question appears whether the large momentum behavior
of quantum fields in the background of quantized gravity or in the stochastic
background could be experimentally verified, e.g., in high energy scattering.
The detection of gravitational waves [10] raises the questions whether these
waves can be considered as a stream of gravitons and whether their interaction
with particles can be treated as a particle-graviton scattering. There is a
suggestion [11] that the effect of gravitons can be observed as a noise in the
interferometers applied for gravitational waves detection. In such a model the
quantum equation for geodesic deviation is studied. It is shown that gravitons
transform this equation into a stochastic differential equation changing the
particle evolution.

In this paper we are interested in the problem how the dynamics of the quantum
scalar field can be changed in a linearly polarized quantum plane wave moving
in the $x$-direction. In such a model the gravitons have the spectral function
$\rho({\bf k})$ restricted to $k_{x}\geq 0$ with $k_{y}=k_{z}=0$.We show that
if the graviton correlation function is singular at short distances then the
scalar field propagator decays faster in the momentum space (in comparison to
the free field propagator) in the direction orthogonal to the wave
propagation. Such an improved behavior of propagators of matter fields in the
background of quantum gravitational field is important for renormalization and
ultraviolet stability of these theories.

##  2 Scalar field in a stochastic metric

We consider a perurbation $h_{\mu\nu}(\xi)$ ( $\xi=(t,{\bf x})$) of the
Minkowski metric in the traceless transverse (TT) gauge requiring in addition
that the metric tensor is diagonal and assuming that $h_{\mu\nu}$ is
propagating only along the $x$ axis. Then, the metric is

$\begin{array}[]{l}ds^{2}=g_{\mu\nu}d\xi^{\mu}d\xi^{\nu}\equiv
dt^{2}-(\delta_{jk}+h_{jk})dx^{j}dx^{k}\cr\equiv
dt^{2}-dx^{2}-(1-h(t,x))dy^{2}-(1+h(t,x))dz^{2},\end{array}$ (1)

Such a metric can describe a gravitational wave moving along the $x$-axis. It
can be considered as a solution of the $TT$-conditions $\partial_{j}h_{jk}=0$
and $h^{j}_{j}=0$. For the plane wave moving in the positive direction of the
$x$-axis [12] $h(t,x)=h(u)$ where $u=t-x$ (the velocity of light $c=1$) .

The propagator of the scalar field (with mass $m$) in a stochastic metric is
defined by a functional integral

$\begin{array}[]{l}G(\xi,\xi^{\prime})=Z^{-1}\Big{<}\int{\cal
D}\phi\exp\Big{(}\frac{i}{2\hbar}\int
d^{4}x\sqrt{|g|}(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-m^{2}\phi^{2}\Big{)}\phi(x)\phi(y)\Big{>}\cr=\Big{<}{\cal
A}^{-1}(\xi;\xi^{\prime})\Big{>}\end{array}$ (2)

where $Z$ is a normalization factor, $g=\det(g_{\mu\nu})$ is the determinant
of the metric $g_{\mu\nu}$ and ${\cal A}^{-1}(x,y)$ denotes the kernel (Green
function) of the inverse of the operator defined by the bilinear form in the
Lagrangian of the scalar field in eq.(2)

$2{\cal
A}=-|g|^{-\frac{1}{2}}\partial_{\mu}|g|^{\frac{1}{2}}g^{\mu\nu}\partial_{\nu}+m^{2}$
(3)

In eq.(2) we take an average $<...>$ over the metric. The average can be
understood as an average over a stochastic background of gravitational waves
[17] which may have a primordial origin or come from the ongoing phenomena in
the universe. We assume that $h$ can be approximated by a Gaussian variable.
Such an assumption, in the case of a stochastic background, is justified by
the central limit theorem of probability theory. If we have small stochastic
sources of gravitational radiation then the probability distribution of an
infinite sum of stochastic variables (no matter what are their individual
probability distributions) can be approximated by the Gaussian distribution.
The Gaussian (complex) measure could also result from the Feynman integral
quantization of $F(R)$ gravity (with a certain function $F$ of $R$, where $R$
denotes the Riemannian scalar curvature)

$\int{\cal D}g_{\mu\nu}\exp\Big{(}i\int d^{4}\xi\sqrt{|g}|F(R)\Big{)},$ (4)

when a quadratic (in $h$ ) approximation to $F(R)$ is justified.

We assume that we have a small perturbation of the metric, so that
$h^{2}\simeq 0$. In such a case the term
$\partial_{t}\ln|g|\partial_{t}-\partial_{x}\ln|g|\partial_{x}$ which is of
the first order in derivatives for the scalar wave operator in eq.(3) is
absent because $g=\det(g_{jk})=1-h^{2}\simeq 1$. For the same reason we make
the approximation $(1\pm h)^{-1}\simeq 1\mp h$. If $h$ depends only on $u=t-x$
then we can change the variable $t\rightarrow t-x$ in ${\cal A}$. Then, the
coefficients of the differential operator ${\cal A}$ in eq.(3) depend on one
variable $u$

${\cal
A}=-\frac{1}{2}(\partial_{u}^{2}-\triangle+m^{2})+\frac{1}{2}h(u)\partial_{y}^{2}-\frac{1}{2}h(u)\partial_{z}^{2}.$
(5)

We can take the Fourier transform in spatial variables of the operator (5) and
its kernel

$\tilde{\cal A}^{-1}(u,u^{\prime},{\bf p})=\int d{\bf
x}\exp(ip_{x}(x-x^{\prime})+ip_{y}(y-y^{\prime})+ip_{z}(z-z^{\prime})){\cal
A}^{-1}(\xi;\xi^{\prime})$ (6)

It follows that $\tilde{\cal A}^{-1}(u,u^{\prime},{\bf p})$ is the kernel of
the operator

$\tilde{{\cal A}}=-\frac{1}{2}(\partial_{u}^{2}+{\bf
p}^{2}+m^{2})+\frac{1}{2}h(u)p_{+}p_{-}\equiv\tilde{{\cal A}}_{0}+V$ (7)

acting in the space of functions $\psi(u,{\bf p})$ (where the action of the
momentum ${\bf p}$ in $\tilde{{\cal A}}$ is just the multiplication). Here,
${\bf p}^{2}=p_{x}^{2}+p_{y}^{2}+p_{z}^{2}$ ,
$p_{+}=p_{z}+p_{y}$,$p_{-}=p_{z}-p_{y}$,

$\tilde{{\cal A}}_{0}=-\frac{1}{2}\partial_{u}^{2}-\frac{1}{2}{\bf
p}^{2}-\frac{1}{2}m^{2}$ (8)

and

$V=\frac{1}{2}h(u)p_{-}p_{+}.$ (9)

Our estimates are based on explicit calculations. The dependence of the
operator $\tilde{{\cal A}}$ solely on one variable $u$ enables such
calculations. We represent the propagator ${\cal
A}^{-1}=i\int_{0}^{\infty}d\tau\exp(-i\tau{\cal A})$ in terms of the proper
time Hamiltonian evolution. The operator $\tilde{{\cal A}}$ has the form of a
Hamiltonian in quantum mechanics. We assume that it has a self-adjoint
extension. The unitary evolution $\exp(-i\tilde{{\cal A}}\tau)$ can be
expressed in the interaction picture. We write

$U_{\tau}=\exp(-i\tilde{{\cal A}}\tau)=\exp(-i\tilde{{\cal
A}}_{0}\tau)U^{I}_{\tau}.$ (10)

Then,

$\partial_{\tau}U_{\tau}^{I}=-iV_{\tau}U_{\tau}^{I},$ (11)

where

$V_{\tau}=\exp(i\tilde{{\cal A}}_{0}\tau)V\exp(-i\tilde{{\cal A}}_{0}\tau).$
(12)

We are interested in an average over the random field $h$. We can calculate
such averages expanding the solution of eq.(11) in the Dyson series

$U_{\tau}=\exp(-i\tilde{{\cal A}}_{0}\tau)U_{\tau}^{I}=\exp(-i\tilde{{\cal
A}}_{0}\tau)\Big{(}1-i\int_{0}^{\tau}dsV_{s}-\int_{0}^{\tau}ds_{2}\int_{0}^{s_{2}}ds_{1}V_{s_{2}}V_{s_{1}}+...\Big{)}$
(13)

The expectation value over the metric of eq.(1) in the lowest order is

$\begin{array}[]{l}<(U_{\tau}\psi)(u)>=(\exp(-i\tau
H_{0})\psi)(u)\cr-\int_{0}^{\tau}ds_{2}\int_{-\infty}^{\infty}du_{1}\int_{-\infty}^{\infty}du_{2}(\exp(-i(\tau-
s_{2})H_{0}))(u,u_{1})<V(u_{2})V(u_{1})>\cr\times(\exp(-i(s_{2}-s_{1})H_{0}))(u_{1},u_{2})\int_{0}^{s_{2}}ds_{1}(\exp(-is_{1}H_{0}))(u_{2},u_{3})\psi(u_{3})du_{3},\end{array}$
(14)

where

$\begin{array}[]{l}(\exp(-is\tilde{{\cal A}}_{0}))(u,u^{\prime})=(2i\pi
s)^{-\frac{1}{2}}\exp\Big{(}\frac{i}{2s}|u-u^{\prime}|^{2}\Big{)}\exp(i\frac{s}{2}(m^{2}+{\bf
p}^{2}))\cr\equiv\exp(i\frac{s}{2}(m^{2}+{\bf
p}^{2}))p(s;u-u^{\prime})\end{array}$ (15)

Going from eq.(13) to eq.(14) we have changed the proper time integration
variables $s_{1}\rightarrow\tau-s_{2}$ and $s_{2}\rightarrow\tau-s_{1}$.

##  3 Feynman path integral representation

For a non-perturbative averaging over $h$ it is useful to express the
evolution $U_{\tau}$ by means of the Feynman path integral

$\begin{array}[]{l}(U_{\tau}\psi)(u)=\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int_{q(0)=u}{\cal
D}q(.)\cr\times\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}ds(\frac{dq}{ds})^{2}\Big{)}\exp(-i\int_{0}^{\tau}dsV(q(s)))\psi(q(\tau)).\end{array}$
(16)

The mean value of the evolution operator in the Gaussian random metric can be
evaluated as

$\begin{array}[]{l}<(U_{\tau}\psi)(u)>=\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int_{q(0)=u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}(\frac{dq}{ds})^{2}\Big{)}\cr\exp(-\frac{1}{2}\int_{0}^{\tau}ds\int_{0}^{\tau}ds^{\prime}<V(q(s))V(q(s^{\prime}))>)\psi(q(\tau)).\end{array}$
(17)

The Feynman path integral (16) can be considered as a resummation of the Dyson
series (13). The lowest order term in the expansion of the Feynman formula for
$U_{\tau}\psi$ in eq.(17) is

$\begin{array}[]{l}<(U_{\tau}\psi)(u)>=\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int_{q(0)=u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}(\frac{dq}{ds})^{2}\Big{)}\cr\times\Big{(}1-\int_{0}^{\tau}ds_{2}\int_{0}^{s_{2}}ds_{1}<V_{s_{2}}V_{s_{1}}>+...\Big{)}\psi(q(\tau)).\end{array}$
(18)

It agrees with the Dyson expansion (14). The equality of the expansion in $V$
of the Feynman integral (17)-(18) and the Dyson expansion (13)-(14) can be
shown using the expression for the calculation of the Feynman integral of
”cylinder functions” [13][14]. If $0\leq s_{1}\leq s_{2}\leq.....\leq
s_{n-1}\leq\tau$ then the Feynman integral of functions of paths starting from
$q(0)=u$ and depending on a finite set of points (cylinder functions)
$(q(s_{1}),q(s_{2}),....,q(s_{n-1}),q(\tau))$ is

$\begin{array}[]{l}\int_{q(0)=u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}(\frac{dq}{ds})^{2}\Big{)}F(q(s_{1}),q(s_{2}),....,q(s_{n-1}),q(\tau))\cr=\int
p(s_{1},u_{1}-u)p(s_{2}-s_{1};u_{2}-u_{1})....p(\tau-
s_{n-1};u_{n}-u_{n-1})\cr\times F(u_{1},...,u_{n})du_{1}...du_{n}\end{array}$
(19)

for any function $F(u_{1},...,u_{n})$ of $n$-variables. Eqs.(16)-(17) coincide
with eqs.(13)-(14) if in the $(n-1)$th order of Dyson perturbation expansion
we change the proper time integration variables $s_{j}\rightarrow\tau-s_{n-j}$
(as we did in eq.(14) for $n=3$).

We may express the time evolution in terms of the evolution kernel $K_{\tau}$

$(U_{\tau}\psi)(u)=\int du^{\prime}K_{\tau}(u,u^{\prime};{\bf
p})\psi(u^{\prime}).$ (20)

The two-point function for the scalar field is obtained as the resolvent of
${\cal A}$

$G({\cal A})=i\int_{0}^{\infty}d\tau\exp(-i{\cal A}\tau)={\cal A}^{-1}.$

Its kernel is (in time coordinates $u$ and in spatial momenta)

$\begin{array}[]{l}G(\tilde{{\cal A}},u,u^{\prime};{\bf
p})=i\int_{0}^{\infty}d\tau\exp(-i\tilde{{\cal
A}}\tau)(u,u^{\prime})=i\int_{0}^{\infty}d\tau K_{\tau}(u,u^{\prime};{\bf
p}).\end{array}$ (21)

As discussed in sec.2 we assume that $h(u)$ is a Gaussian random field defined
by the measure $d\mu(h)$ characterized by its generating functional

$\int d\mu(h)\exp\Big{(}i\int duJ(u)h(u)\Big{)}=\exp\Big{(}-\frac{1}{2}\int
dudu^{\prime}J(u)g(u-u^{\prime})J(u^{\prime})\Big{)}$ (22)

where $g(u-u^{\prime})$ is the correlation function of $h(u)$ which we take in
a scale invariant form

$g(u-u^{\prime})=<h(u)h(u^{\prime})>=\kappa^{2}|u-u^{\prime}|^{-4\gamma}$ (23)

where $0<\gamma<\frac{1}{4}$ (the restriction $\gamma<\frac{1}{4}$ is
discussed at the end of this section). The exact scale invariance is not
necessary for our results. It is sufficient to assume that $g(u)$ has the form
(23) for $u\rightarrow 0$.We make the assumption of an exact scale invariance
for simplicity of the argument. The singularity of $g(u)$ at small $u$ is
common in quantum field theory as the two-point function is the kernel of the
inverse of the second order differential operator on $R^{4}$. For a stochastic
background of gravitational waves the scale invariant correlation functions
(23) have been derived in some models of gravitational radiation [15][16]. The
singularity of the correlations at small time means a large contribution of
high-frequency waves to the spectrum (as the frequency spectrum of the
covariance (23) is $\omega^{4\gamma-1}$) . At present only low frequency
gravitational waves have been detected [17].

We have for a Gaussian field $h(u)$

$\begin{array}[]{l}<G(\tilde{{\cal A}},u,u^{\prime};{\bf
p})>=i\int_{0}^{\infty}d\tau\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int_{q(0)=u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}(\frac{dq}{ds})^{2}\Big{)}\cr\times\exp(-\frac{1}{2}\int_{0}^{\tau}ds\int_{0}^{\tau}ds^{\prime}<V(q(s))V(q(s^{\prime}))>)\delta(u^{\prime}-q(\tau)),\end{array}$
(24)

so that the path $q(s)$ starts at $u$ and ends in $u^{\prime}$.

Let us note the identity for $q(s)$ (in the sense of the equality of
expectation values; a consequence of the transition function (19))

$q(s)=\sqrt{\tau}q(\frac{s}{\tau}).$ (25)

Then, in eq.(24) with the correlations (23) we have

$\begin{array}[]{l}<G(\tilde{{\cal A}},u,u^{\prime};{\bf
p})>=i\int_{0}^{\infty}d\tau\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int_{q(0)=\sqrt{\tau}u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{1}(\frac{dq}{ds})^{2}\Big{)}\cr\exp(-\frac{1}{2}\tau^{2(1-\gamma)}(p_{+}p_{-})^{2}\kappa^{2}\int_{0}^{1}ds\int_{0}^{1}ds^{\prime}<g({\bf
q}(s))-{\bf q}(s^{\prime}))>\delta(u^{\prime}-\sqrt{\tau}{\bf
q}(1)).\end{array}$ (26)

The kernel $K_{\tau}(u,u^{\prime};{\bf p})$ (20) can be expressed in terms of
the Brownian bridge $Q$[14]( its Feynman version is derived in [18]) defined
as the Gaussian process on the interval $[0,1]$ ($Q(0)=0$ and $Q(1)=0$) with
the covariance ($s,s^{\prime}\geq 0$)

$<Q(s)Q(s^{\prime})>=is(1-s^{\prime})\theta(s^{\prime}-s)+is^{\prime}(1-s)\theta(s-s^{\prime}),$
(27)

where $\theta$ is the Heaviside step function. Let us denote by $d\nu(Q)$ the
Gaussian measure with the covariance (27). Then,

$\begin{array}[]{l}K_{\tau}(u,u^{\prime};{\bf
p})=(2i\pi\tau)^{-\frac{1}{2}}\exp\Big{(}\frac{i}{2\tau}|u-u^{\prime}|^{2}\Big{)}\exp(\frac{i\tau}{2}({\bf
p}^{2}+m^{2}))\cr\int d\nu(Q)\exp\Big{(}i\tau
p_{+}p_{-}\int_{0}^{1}dsh(u^{\prime}s+(1-s)u+\sqrt{\tau}Q(s))\Big{)}\end{array}$
(28)

The expectation value of the propagator (26) is

$\begin{array}[]{l}<G(\tilde{{\cal A}};u,u^{\prime};{\bf
p})>=i\int_{0}^{\infty}d\tau(2i\pi\tau)^{-\frac{1}{2}}\exp\Big{(}\frac{i}{2\tau}|u-u^{\prime}|^{2}\Big{)}\cr\exp(i\tau\frac{1}{2}({\bf
p}^{2}+m^{2}))\int
d\nu(Q)\exp\Big{(}-\frac{1}{2}\tau^{2(1-\gamma)}(p_{-}p_{+})^{2}\kappa^{2}\int_{0}^{1}ds\int_{0}^{1}ds^{\prime}\cr\times
g(\frac{1}{\sqrt{\tau}}(u^{\prime}-u)(s-s^{\prime})+Q(s)-Q(s^{\prime}))\Big{)}.\end{array}$
(29)

In eqs.(26)and (29) we applied the scaling property of $g(u)$ (23).

At the end of this section let us return to the expansion (18) and calculate
it with the correlation function (23)

$\begin{array}[]{l}\int_{q(0)=u}{\cal
D}q(.)\exp\Big{(}\frac{i}{2}\int_{0}^{\tau}(\frac{dq}{ds})^{2}\Big{)}\int_{0}^{\tau}ds_{2}\int_{0}^{s_{2}}ds_{1}<V_{s_{2}}V_{s_{1}}>+...\Big{)}\psi(q(\tau))\cr=\kappa^{2}(p_{+}p_{-})^{2}\int_{0}^{\tau}ds_{2}\int_{0}^{s_{2}}ds_{1}p(s_{1};u-u_{1})p(s_{2}-s_{1};u_{2}-u_{1})\cr\times
p(\tau-
s_{2};u_{3}-u_{2})|u_{2}-u_{1}|^{-4\gamma}\psi(u_{3})du_{3}+....\end{array}$
(30)

The integral over $u_{2}-u_{1}$ exists as the Lebesgue integral only if
$\gamma<\frac{1}{4}$. We encounter the same problem when calculating
(perturbatively) the expectation value of the exponential in eq.(24). Such
integrals could possibly be interpreted in the sense of generalized functions
[19] allowing an extension to $\gamma>\frac{1}{4}$.

An explicit calculation of the propagator (29) can be done in a perturbation
expansion in $\kappa$. The result is equivalent to the calculation by means of
the Dyson series (18). When we set $u=u^{\prime}$ then we can obtain from
eq.(29) some non-perturbative estimates. Let us change the proper time
variable as $\tau=\tau^{\prime}(\kappa p_{+}p_{-})^{-\frac{1}{(1-\gamma)}}$.
Then, at $u=u^{\prime}$ in the exponential in eq.(29) if $0<\gamma<1$ the term
${\bf p}^{2}\tau={\bf p}^{2}\tau^{\prime}(\kappa
p_{+}p_{-})^{-\frac{1}{(1-\gamma)}}$ becomes small in comparison to the term
$\tau^{\prime-2(1-\gamma)}$ in the limit of large $p_{z}$ (or $p_{y}$).
Performing the $\tau^{\prime}$ integral (with the negligence of the ${\bf
p}^{2}\tau$ term) we obtain

$\begin{array}[]{l}<G(\tilde{{\cal A}};u,u;{\bf
p})>=C(p_{+}p_{-})^{-\frac{1}{2(1-\gamma)}}\end{array}$ (31)

with a certain constant $C$ (this constant can be an infinite renormalization
constant if $\gamma\geq\frac{1}{4}$ because of the divergence of the integral
(30)). If there is no stochastic metric ($\kappa=\gamma=0$ in eq.(29) ) then
by means of the integration over $\tau$ we obtain the propagator of the scalar
field at equal time $u=u^{\prime}$ as

$({\bf p}^{2})^{-\frac{1}{2}}.$ (32)

We can see that if $1>\gamma>0$ then the propagator (31) is decaying faster
for large $p_{3}$ than the free propagator (32).

Although the propagator in a stochastic metric (1) is not Lorentz invariant it
is instructive to calculate the Lorentz invariant propagators with a non-
canonical scaling as discussed in [20][21]

$\begin{array}[]{l}G(\xi)=i\int dp_{0}d{\bf
p}\exp(i\xi^{\mu}p_{\mu})\int_{0}^{\infty}d\tau(i\tau)^{\alpha}\exp(i\frac{\tau}{2}p^{2})\cr=(2\pi)^{-2}\int_{0}^{\infty}d\tau(i\tau)^{-2+\alpha}\exp(\frac{i}{2\tau}\xi^{2})\end{array}$
(33)

where $p^{2}=p_{0}^{2}-{\bf p}^{2}$ and $\xi^{2}=t^{2}-{\bf x}^{2}$. From
eq.(33) the propagator in the momentum space is $(p^{2})^{-1-\alpha}$ and the
propagator at $t=0$ is $({\bf p}^{2})^{-\frac{1}{2}-\alpha}$. Hence, in order
to obtain the behavior (31) of the equal time propagator (33) for a large
$p_{z}$ we need $(1-\gamma)^{-1}=1+2\alpha$.

##  4 Summary

Our result (31), which follows from direct calculations, is in contradiction
with the Källen-Lehmann representation of the propagator. As discussed in [21]
under the usual assumptions of quantum field theory the propagator cannot
decay at high momentum faster than $(p^{2})^{-1}$ (then the equal-time
propagator behaves as in eq.(32)). This raises the question which assumptions
of Källen and Lehmann are violated in [3][5][20]. In our case the metric
explicitly violates the Lorentz invariance. It may be that the result showing
that field theory with quantized gravity has more regular short distance
behavior implicitly violates either the Lorentz invariance or causality. From
the functional integration point of view it is difficult to restrict the
functional integral over the metric in a way preserving the causal structure
of the space-time (in particular the signature of the metric). In our case,
assuming that $h$ is Gaussian, we implicitly allowed $h$ to have arbitrarily
large fluctuations with an arbitrary sign. Nevertheless, there remains the
prospect of a construction of quantum field theory including gravity which is
more regular at short distances (below the Planck scale) at the cost of
changing the standard scheme of treating the matter fields as the building
blocks which satisfy the principles leading to the Källen-Lehmann
representation.

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# Prototype-oriented Unsupervised Change Detection
for Disaster Management

Youngtack Oh, Minseok Seo, Doyi Kim, Junghoon Seo
SI Analytics
70, Yuseong-daero 1689beon-gil, Yuseong-gu, Daejeon, Republic of Korea
<EMAIL_ADDRESS>

###### Abstract

Climate change has led to an increased frequency of natural disasters such as
floods and cyclones. This emphasizes the importance of effective disaster
monitoring. In response, the remote sensing community has explored change
detection methods. These methods are primarily categorized into supervised
techniques, which yield precise results but come with high labeling costs, and
unsupervised techniques, which eliminate the need for labeling but involve
intricate hyperparameter tuning. To address these challenges, we propose a
novel unsupervised change detection method named Prototype-oriented
Unsupervised Change Detection for Disaster Management (PUCD). PUCD captures
changes by comparing features from pre-event, post-event, and prototype-
oriented change synthesis images via a foundational model, and refines results
using the Segment Anything Model (SAM). Although PUCD is an unsupervised
change detection, it does not require complex hyperparameter tuning. We
evaluate PUCD framework on the LEVIR-Extension dataset and the disaster
dataset and it achieves state-of-the-art performance compared to other methods
on the LEVIR-Extension dataset.

##  1 Introduction

Climate change is leading to more extreme weather events [23]. With the
acceleration of global warming, damage from climate-related disasters such as
floods and tropical cyclones is escalating [2, 19]. As a result, there is
heightened emphasis on the critical role of disaster management in both
prevention and response [28]. Employing remote sensing data for disaster
management proves efficient as it enables the monitoring of large areas [11,
34]. However, even though remote sensing is efficient, the large area affected
by natural disasters makes it difficult for humans to analyze [18]. Addressing
these challenges, unsupervised change detection (UCD) methods have been
introduced, utilizing change vector analysis (CVA) of pre- and post-disaster
images [20, 5, 22, 25, 32, 6]. While UCD allows for rapid responses, it
necessitates the adjustment of numerous hyperparameters. This is because UCD
uses pixel statistical differences such as brightness, color, and clarity of
the image, so it is difficult to generalize differences across various
disaster types and regions. On the other hand, supervised change detection
[12, 1, 35, 26] can be used without the need to adjust hyperparameters.
However, it is difficult to build large-scale datasets because disaster
situations are rare [13]. Specifying and labeling targets of interest also
requires a significant cost [33]. In addition, these models cannot respond to
disasters that are not trained.

Figure 1: Overview of our PUCD framework. The framework receives a bi-temporal
pair of images: the pre-disaster image $x_{1}$ and the post-disaster image
$x_{2}$. (a) A prototype is then synthesized as $\hat{x}_{1}$ and
$\hat{x}_{2}$ . (b) Subsequent steps involve feature extraction and
subtraction them output $S_{x_{2}x_{1}}$, $S_{\hat{x}_{1}x_{1}}$ and
$S_{\hat{x}_{2}x_{2}}$. (c) change vector creation through CVA. (d) A change
map emerges from change clustering. The final output is a fusion of this map
and the results from the SAM.

In this paper, we introduce a Prototype-oriented Unsupervised Change Detection
for Disaster Management (PUCD), illustrated as Figure 1. This approach
designates a target-of-interest as a prototype and detects changes relative to
it. PUCD uses the DINOv2 foundational model to extract features representing
the context and structure of pre-, post-event imageries and prototypes.
Following this, CVA is applied to the features of the pre-, post-event images
and the prototype to identify changed regions. The preliminary coarse output
is then enhanced using the segment anything technique [16]. To evaluate the
PUCD framework, we conducted qualitative assessments on events like the Libyan
flood, the United States hurricane, and the Japan tsunami. Subsequent
quantitative evaluations were executed using the LEVIR-Extension dataset. Our
experimental results showed that PUCD is qualitatively and quantitatively
efficient.

##  2 Method

In this section, we provide a detailed explanation of the PUCD framework.
Initially, we summarize the components constituting PUCD, specifically the
DINOv2 and the segment anything model. Subsequently, we delve into the
contents corresponding to Figure 1-(a), (b), (c), and (d) in sequence.

###  2.1 Preliminaries

#### DINOv2

DINOv2 [24] is a foundational model rooted in representation learning, widely
used in various computer vision tasks. It utilizes the Vision Transformer
(ViT) architecture [10, 14] and is trained on a comprehensive compilation of
diverse datasets [24]. DINOv2 is trained using contrastive learning, assessing
the similarity between objects in two images based on their context and shape
[4, 24]. Consequently, DINOv2 is not only capable of capturing changes in
context and shape but also exhibits robustness to stylistic change. This
capability underpins our rationale for employing DINOv2, particularly for UCD
tasks that necessitate detecting changes in context and shape.

#### Segment Anything

The Segment Anything Model (SAM) [16], built upon the MaskFormer architecture
[8], is a universal segmentation solution. It is recognized as a foundational
model in the segmentation field, having been trained on a massive instance
segmentation dataset comprised of 1 billion masks and 11 million images. SAM
consistently produces fine-grained segmentation results across diverse
applications, ranging from satellite imagery and medical imaging to autonomous
driving [21, 27, 9, 31]. Given its capabilities, SAM holds significant promise
for enhancing the precision of change detection in the remote sensing domain.

###  2.2 PUCD framework

#### Prototype-oriented Change Synthesis

Satellite imagery exhibits significant variation in resolution, and even
within the same resolution category, the visual representation of target
objects can substantially differ [29]. Given that disasters can occur
globally, satellite images with varying resolutions play a crucial role in
disaster management. However, the context and structural attributes of objects
differ depending on the region. Addressing this complexity necessitates a
dynamic change threshold, which considers both satellite resolution and
specific object attributes. To meet this need, we introduce a prototype-
oriented change detection.

A prototype ($p$) refers to a predefined single-sample image representing a
target of interest, such as a house or sports field. By storing these images
in advance, the system can define changes with respect to the prototype,
offering a structured method to handle different imaging contexts. The PUCD
framework, while allowing for the specification of a prototype, remains
adaptable enough to operate without manually defined one. Notably, if $p$ is
not provided from user’s choices, an object from either $x_{1}$ or $x_{2}$ is
randomly selected as $p$ from results of SAM, streamlining change detection
across various regions or targets and obviating exhaustive hyperparameter
tuning. Note that the prototype aids in the assignment of change clusters,
discussed in the subsequent section.

#### Change Feature Extraction

Given the bi-temporal images $(x_{1},x_{2})$ and the images generated through
prototype-oriented change synthesis $(\hat{x}_{1},\hat{x}_{2})$, PUCD extracts
the corresponding features. Specifically, PUCD divides these images into
multiple patches, each of size $14\times 14$. Each patch is then processed by
DINOv2 to extract its features. Essentially, PUCD derives features of context
and structure from a confined $14\times 14$ area for image comparison. This
approach is especially advantageous in situations where the same area of
interest is captured in bi-temporal images from satellites, but spatial
discrepancies, referred to as co-registration errors, exist between these
images [17, 30].

Figure 2: Results of qualitative analysis of PUCD. Note that all qualitative
experiments were performed by selecting a random object from images $x_{1}$
and $x_{2}$ using the SAM model, rather than manually selecting a specific
prototype $p$.

#### Change Vector Analysis & Change Cluster proposal

The CVA of PUCD identifies change features using the $\hat{x}_{1}$ and
$\hat{x}_{2}$, in conjunction with the $x_{1}$ and $x_{2}$. Specifically,
$S_{x_{2}x_{1}}$, $S_{\hat{x}_{1}x_{1}}$, and $S_{\hat{x}_{2}x_{2}}$ represent
the feature differences between $x_{2}$ and $x_{1}$, the $\hat{x}_{1}$ and
$x_{1}$, and the $\hat{x}_{2}$ and $x_{2}$, respectively. After concatenating
the features $S_{x_{2}x_{1}}$, $S_{\hat{x}_{1}x_{1}}$, and
$S_{\hat{x}_{2}x_{2}}$, they undergo dimensionality reduction using Principal
Component Analysis (PCA) [15]. Beyond its primary role in reducing dimensions,
it accentuates pertinent features while diminishing data noise, ensuring the
feature space concentrates on the most crucial variances. Once significant
features have been extracted via PCA, K-means clustering categorizes these
features, distinguishing changes from unchanges. Within this scheme, the
cluster that contains the prototype is identified as the change cluster.

#### Refinement with SAM Model

A single pixel within the feature map from DINOv2 corresponds to a $14\times
14$ area in the original image. Consequently, detected changes often manifest
over an excessively larger region than a size of $14\times 14$. To address
this issue, we utilize the SAM model. Upon obtaining a coarse change map
$S_{\text{map}}$, PUCD refines the $x_{1}$ image. Specifically, only those
objects with an overlap exceeding 70% with the semantic SAM segmentation map
are retained.

##  3 Experiment Results

In our experiments, we employed the DINOv2 ViT-L/14 distilled model [24] for
DINOv2 and utilized the SAM-H [16] model for SAM. Within the PUCD framework,
the number of PCA components was set to 1, and for k-means clustering, the
value of $k$ was set to 2(change and unchange).

To assess the efficacy of the PUCD framework, we conducted a quantitative
evaluation using the LEVIR-Extension dataset. Additionally, qualitative
evaluations were carried out on events such as the Libyan flood, the hurricane
in the United States, and the tsunami in Japan. The LEVIR-CD dataset [7]
comprises 637 bi-temporal pairs of high spatial resolution (HSR) optical
images with labels indicating building changes. Each image has a spatial size
of 1,024$\times$1,024 pixels and a ground sample distance (GSD) of 0.5 $m$. To
capture all major changes in the LEVIR-CD dataset, we added labels for roads,
parking lots, lakes, and other features. This augmented dataset, named LEVIR-
Extension, will be available online shortly. The pre/post-disaster dataset for
qualitative results was sourced from MAXAR’s WorldView-2 and GeoEye-1 imagery,
with GSDs of $0.46m$ and $0.25m$, respectively.

Table 1: Performance comparison of unsupervised change detection methods (CVA,
IRMAD, PCAKmeans, SFA, DCVA) that require co-registered bi-temporal images on
the LEVIR-Extension dataset. Please note that the best scores are indicated in
bold, and the second-best scores are underlined.

Method | Param(M) | LEVIR-Extension
---|---|---
Pre. (0/1) | Rec. (0/1) | F1 (0/1) | IoU (0/1) | ACC
CVA [3] | - | 45.4/60.9 | 92.7/9.3 | 60.9/16.2 | 43.8/8.8 | 46.7
IRMAD [22] | - | 73.6/46.3 | 93.7/13.9 | 82.5/21.4 | 70.1/12.0 | 71.3
PCAKmeans [5] | - | 63.7/43.6 | 92.5/9.9 | 75.4/16.2 | 60.6/8.8 | 62.0
SFA [32] | - | 76.9/48.0 |  94.1/16.1 | 84.7/24.1 | 73.4/13.7 | 74.5
DCVA [25] | 16.15 | 94.1/14.3 | 92.3/18.2 |  93.2/16.0 |  87.2/8.7 | 87.41
SAM-CD | 632 | 92.7/10.9 | 66.9/43.7 | 77.7/17.4 | 63.6/9.5 | 65.0
PUCD w/o SAM | 932 |  97.4/23.1 | 75.8/78.6 | 85.2/35.6 | 74.3/21.7 | 76.0
PUCD w/ SAM | 932 |  95.0/47.7 |  95.3/46.2 |  95.1/46.8 |  90.8/30.7 | 91.1

#### Qualitative Results

Figure 2 showcases the change detection results for various disasters,
including floods, hurricanes, and typhoons. The figure clearly demonstrates
PUCD’s ability to detect signs of damage, such as destroyed buildings, flooded
areas, and damaged trees. Remarkably, these results were achieved without any
training procedure on the target dataset, highlighting its capability as an
UCD model and its adaptability across different regions and disaster types.

#### Quantitative Results

Table 1 presents the performance comparison of PUCD with other UCD
methodologies using the LEVIR-Extension dataset. The table results reveal that
PUCD consistently outperforms existing methods, achieving state-of-the-art
results with a significant margin. These findings underscore the PUCD
framework’s potential in the domain of UCD.

##  4 Conclusion

In this paper, we introduced the Prototype-oriented Unsupervised Change
Detection (PUCD), which delivers robust performance regardless of satellite
image resolution or the specific subject of interest. By leveraging features
from both pre-event and post-event imagery and incorporating prototypes,
followed by refining results through the Segment Anything technique, PUCD
achieves superior accuracy over the other unsupervised change detectors.
Qualitative assessments across various disaster types further attest to its
robust performance across different regions. We foresee PUCD becoming a
cornerstone tool in future disaster management efforts.

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