# Nuclear Modification Factor in Pb-Pb and p-Pb collisions at
$\sqrt{s_{NN}}$=5.02 TeV at LHC energies using Boltzmann Transport Equation
with Tsallis Blast Wave Description.

Aditya Kumar Singh  Aviral Akhil  Swatantra Kumar Tiwari
<EMAIL_ADDRESS>(Corresponding Author) Department of Physics,
University of Allahabad, Prayagraj- 211002, U.P.  Pooja Pareek  Variable
Energy Cyclotron Centre, 1/AF- Bidhannagar, Kolkata-700064, India

###### Abstract

In this article, we have studied the nuclear modification factor measured in
Pb-Pb collisions ($R_{PbPb}$) for $\pi^{\pm}$, $K^{\pm}$, $p+\bar{p}$,
$K^{*0}+\bar{K^{*0}}$, $\phi$ and in p-Pb collisions ($R_{pPb}$) for
$\pi^{\pm}$, $K^{\pm}$, $p+\bar{p}$ at Large hadron collider (LHC) energy of
$\sqrt{s_{NN}}$ = 5.02 TeV for the most central and peripheral collisions. We
have also analysed the experimental data of transverse momentum ($p_{T}$)
spectra for these identified hadrons at LHC for Pb-Pb as well as for p-Pb
collisions. We have used Boltzmann transport equation (BTE) in relaxation time
approximation (RTA) for this analysis. The Tsallis statistics is used as an
initial distribution function and The Tsallis blast wave (TBW) model is
employed as an equilibrium distribution in BTE. The present model fits the
measured transverse momentum spectra, $R_{PbPb}$, and $R_{pPb}$ successfully
upto $p_{T}$ = 8 GeV with a reasonable $\chi^{2}/ndf$ for all the considered
hadrons at various centralities. The experimental data for $R_{pPb}$ are
generated using the particle yields at pPb and pp collisions where number of
binary collisions are taken from Glauber model calculations. We find that the
average transverse flow velocity ($<\beta_{r}>$) follows the mass and
centrality ordering and decreases with the mass as well as when one move from
the central collisions to peripheral collisions. These findings are inline
with the results of the hydrodynamical calculations.

###### pacs:

25.75.-q,25.75.Nq,25.75.Gz, 25.75.Dw,12.38.Mh, 24.85.+p

##  I Introduction

The ultra-relativistic heavy-ion collision experiment at Relativistic Heavy-
Ion Collider (RHIC), BNL, Large hadron collider (LHC) at CERN, Facility for
Antiproton and Ion research (FAIR), Darmstadt, and Nuclotron-based Ion
Collider Facility (NICA), Dubna aim to create the deconfined state of quarks
and gluons at extreme conditions of temperature and/or energy densities called
Quark-gluon plasma (QGP) Heinz:2000bk ; BRAHMS:2004adc ; Steinberg:2011dj ;
Wyslouch:2011zz . The QGP formed in high-energy heavy-ion collisions exists
for a brief time. It rapidly undergoes a transformation into a state
resembling a gas of hadrons which are captured by detectors for the purpose of
studying these states. To gather information about the transient state of
matter, it is necessary to examine the remnants (hadrons) of this state of
matter. During the course of the evolution, the system engages in a series of
interactions involving multiple partons (quarks and gluons) causing energy
dissipation of the emitted particles which results in the modification of the
particle yields Harris:2023tti ; ALICE:2023dei  depending on centrality and
energy of the colliding nuclei. When both elastic and inelastic interactions
eventually cease, the measurement of the invariant particle yield in the final
state provides information about the characteristics of the quark-gluon
plasma. The invariant transverse momentum spectra are essential for testing
hydrodynamic models that describe the collective behaviour of particles and
thermalisation properties of QGP STAR:2005gfr ; PHENIX:2004vcz ;
Gyulassy:2004zy . The final particle yield is modified to some extent
representing a factor by which it is suppressed or enhanced known as nuclear
modification factor ($R_{AA}$). This is evaluated by dividing the actual yield
of particles observed in $A-A$ and $p-A$ collisions by the yield observed in a
scaled proton-proton collision, taking into account the number of binary
collisions ($N_{coll}$) in the heavy-ion collision. This ratio helps us to
understand the modifications that occur as particles pass through the
extremely hot and dense medium created in relativistic nuclear collisions. It
is a key observable in the study of heavy-ion collisions and given as
CMS:2016xef :

$R_{AA}=\frac{1}{N_{coll}}\frac{d^{2}N_{AA}/dydp_{T}}{d^{2}N_{pp}/dydp_{T}},$
(1)

where $N_{AA}$ and $N_{pp}$ is the particle yield in nucleon-nucleon and
proton-proton collision, respectively and $N_{coll}$ is the number of binary
nucleon-nucleon collisions.

In various papers STAR:2017sal ; STAR:2019vcp ; STAR:2008med , fitting of
$p_{T}$ spectra is done using Boltzmann Gibbs blast wave (BGBW) model. These
studies have demonstrated that the BGBW model effectively describes the
transverse momentum spectra within a specific $p_{T}$ range. This model draws
inspiration from hydrodynamics and posits that particles reach a state of
local thermal equilibrium at the kinetic freeze-out temperature. Additionally,
they have assumed that the particles have a common transverse flow velocity as
they propagate STAR:2019vcp . After reaching local equilibrium, the system
undergoes hydrodynamic evolution assuming all $p-p$ events to be same
Che:2020fbz . In studies Grassi:1994ng ; Sinyukov:2002if , particles emission
does not exclusively happen from a precisely defined freeze-out hypersurface.
Instead, it occurs continuously from various regions within the expanding
system, each at different temperatures and points in time. The conventional
equilibrium description becomes inadequate when dealing with the high
transverse momentum ($p_{T}$) phenomena. In these cases, the particle
production tends to be predominantly governed by non-equilibrium or hard
processes, and it often displays a distinct power-law tail pattern Wilk:1999dr
. Further, the significant fluctuations that vary from one event to another
are anticipated in these cases Socolowski:2004hw . To consider the influence
of these fluctuations many authors Shao:2009mu ; Tang:2008ud ; Che:2020fbz
have taken Tsallis Blast Wave model to fit the transverse momentum spectra. In
a recent study Che:2020fbz , a comprehensive examination of particle spectra
has been undertaken specifically employing the TBW model to accurately depict
spectra up to $p_{T}$= $3$ GeV/c in the context of Pb-Pb, p-Pb, and Xe-Xe
collisions. In references Tripathy:2017nmo ; Tripathy:2017kwb , the authors
have employed the Boltzmann transport equation within the framework of the
Relaxation Time Approximation. They have utilized the Tsallis distribution as
the initial distribution function and the BGBW distribution as the equilibrium
distribution function. The approach was effectively used to examine the
evolution of particle distributions and skillfully reproduce spectra upto
$p_{T}$=$5$ GeV/c. In the present research work, our approach involves
employing the Tsallis distribution as the initial distribution function within
the Boltzmann Transport Equation (BTE) framework and instead of utilizing the
BGBW model, we adopt the TBW model as the equilibrium distribution function.
Significantly, any departure from the value of the non-extensive parameter in
$A-A$ collisions ($q_{AA}$) from 1 indicates that the system does not achieves
a state of full equilibrium, contradicts the assumption made by the BGBW
model.

The manuscript is organized as follows: In Section II, we outline the process
of deriving the transverse momentum spectra and nuclear modification factor
using the Boltzmann Transport Equation within the relaxation time
approximation. Section III is dedicated to an in-depth exploration and
analysis of the obtained results. Finally, in Section IV, we offer a concise
recapitulation of the findings of this study, accompanied by potential avenues
for future research.

##  II The Nuclear Modification Factor in the BTE within the context of RTA

In the domain of scientific investigation, numerous challenges revolve around
understanding the temporal evolution of a statistical ensemble. Here, we
narrow our focus to a particle system described by the probability
distribution $f(r,p,t)$, where this distribution depends on position ($r$),
momentum ($p$), and time ($t$). In general, when dealing with a dynamically
evolving physical system operating irreversibly away from thermodynamic
equilibrium, it’s important to note that $f(r,p,t)$ typically differs from
that of a Boltzmannian ensemble. Researchers often investigate its temporal
evolution by employing the Boltzmann Transport Equation (BTE), whose general
form is utilized for this purpose and is given as huang2008statistical ,

$\displaystyle\frac{df(x,p,t)}{dt}=\frac{\partial f}{\partial
t}+\vec{v}.\nabla_{x}f+\vec{F}.\nabla_{p}f=C[f].$ (2)

Here, $\vec{v}$ represents velocity, while $\vec{F}$ stands as the external
force. The notations $\nabla_{x}$ and $\nabla_{p}$ denote partial derivatives
concerning position and momentum, respectively. The term $C[f]$ stands for the
collision term. The Boltzmann Transport Equation (BTE) within the framework of
Relaxation Time Approximation (RTA) has previously found applications in
investigating a range of phenomena. These applications encompass the study of
temperature fluctuations in non-equilibrium systems over time
Bhattacharyya:2015nwa , the analysis of anisotropic flow in identified hadrons
Akhil:2023xpb , and the evaluation of $R_{AA}$ for various light and heavy
flavors, particularly at energies relevant to the Large Hadron Collider (LHC)
Tripathy:2016hlg . In cases where the considered system is homogeneous
($\nabla_{x}f=0$) and devoid of external forces (i.e $\vec{F}=$0), Eq.(2)
simplifies to the following form.

$\frac{df(x,p,t)}{dt}=\frac{\partial f}{\partial t}=C[f].$ (3)

Due to the flexibility in choosing the functional form of $C[f]$, the equation
remains highly versatile, accommodating a wide range of circumstances.
Nevertheless, in many practical applications, it is customary to simplify it
further into what is known as the relaxation time approximation (RTA). This
involves adopting a more streamlined version of the collision term,
representing a simplified form often used in such situations given as
huang2008statistical ; Bhatnagar:1954zz ; PhysRevC.102.064910 ;
Florkowski:2016qig .

$C[f]=-\frac{f-f_{eq}}{\tau},$ (4)

where $f_{eq}$ is the Boltzmann local equilibrium distribution characterized
by a temperature $T$. The parameter $\tau$ signifies the relaxation time,
representing the duration required for a non-equilibrium system to transition
into equilibrium. By incorporating Eq.(4) into the context, Eq.(3) transforms
as follows,

$\frac{\partial f}{\partial t}=-\frac{f-f_{eq}}{\tau}.$ (5)

Solving the above equation with the initial conditions i.e. at $t=0,f=f_{in}$
and at $t=t_{f},f=f_{fin}$, we get,

$f_{fin}=f_{eq}+(f_{in}-f_{eq})e^{-\frac{t_{f}}{\tau}},$ (6)

where $t_{f}$ is the freeze-out time and $\tau$ is the relaxation time. Using
Eq.(6) the nuclear modification factor can be given as,

$R_{AA}=\frac{f_{fin}}{f_{in}}=\frac{f_{eq}}{f_{in}}+\big{(}1-\frac{f_{eq}}{f_{in}}\big{)}e^{\frac{-t_{f}}{\tau}}$
(7)

Equation 7 represents the nuclear modification factor obtained by
incorporating the relaxation time approximation, serving as the foundation for
our analysis in this current paper. In this context, we have selected the
Tsallis blast wave as the equilibrium function denoted by $f_{eq}$ which can
be written as Che:2020fbz ,

$\displaystyle
f_{eq}=\frac{d^{3}N}{p_{T}dp_{T}dyd\phi_{p}}=D\int_{\sum_{f}}dy_{s}rdrd\phi_{b}\times\cosh(y_{s}-y)\Big{[}1+\frac{q_{AA}-1}{T}[m_{T}\cosh\rho\cosh(y_{s}-y)$
$\displaystyle-
p_{T}\sinh\rho\cos(\phi_{p}-\phi_{b})]\Big{]}^{\frac{\displaystyle-1}{\displaystyle
q_{AA}-1}}.$ (8)

Where, $D=\frac{g\tau_{0}m_{T}}{(2\pi)^{3}}$, $\sum_{f}$ is the decoupling
hypersurface. Here the parameter $g$ corresponds to the degeneracy factor
while temperature denoted as $T$ represents the kinetic freeze-out
temperature. The variables $y$ and $m_{T}$ are employed to denote the rapidity
and transverse mass of the identified particles, respectively. Additionally,
$y_{s}$ is utilized to represent the rapidity of the emitting source, while
$\phi_{p}$ and $\phi_{b}$ describe the azimuthal angles of the emitted
particle velocity and the flow velocity relative to the x-axis within the
reaction plane. The parameter $q_{AA}$ serves to encapsulate non-extensivity,
providing a quantitative measure of the extent of deviation from equilibrium.
A departure from unity in this parameter signifies the non-equilibrium nature
of the system. Additionally, the parameter $\rho$, known as the transverse
expansion rapidity Che:2020fbz , is mathematically expressed as
$\rho=\tanh^{-1}\beta_{r}$ Huovinen:2001cy , where $\beta_{r}$ is defined as
$\beta_{s}(\xi)^{n}$. Here, $\beta_{s}$ denotes the maximum surface velocity,
and $\xi$ is defined as $\big{(}r/R\big{)}$, with $r$ representing the radial
distance and $R$ is the maximum radius of the expanding source at freeze-out
($0<\xi<1$). The parameter $n$ characterizes the flow profile. It is
noteworthy that within the Tsallis blast-wave (TBW) model, particles located
closer to the center of the fireball exhibit slower motion compared to those
at the periphery. The calculation of the average transverse velocity is
conducted according to PHENIX:2003wtu .

$<\beta_{r}>=\frac{\int\beta_{s}\xi^{n}\xi\;d\xi}{\int\xi\;d\xi}=\Big{(}\frac{2}{2+n}\Big{)}\beta_{s}.$
(9)

In our calculations, we have varied the parameter $n$ to explore a range of
flow profiles within the Tsallis Blast-Wave model.

We adopt Bjorken’s longitudinal expansion assumption, implying that the
measured particle yield remains constant across rapidity, as we integrate over
the entire rapidity range of the source Schnedermann:1993ws . This
approximation is reasonably valid for mid-rapidity measurements at both the
RHIC and LHC energies BRAHMS:2004adc . While we simplify our analysis by
assuming isotropic azimuthal emission within each local source, it’s essential
to acknowledge that, in reality, the source’s distribution may exhibit
variations or dependencies in the azimuthal direction STAR:2000ekf . We also
assume that the emission source maintains uniformity in both density and
degree of non-equilibrium at the moment of kinetic freeze-out. However, it’s
important to note that this assumption doesn’t hold for high-$p_{T}$ particles
(jets) as they often exhibit emission patterns concentrated near the surface,
deviating from the assumed uniformity Loizides:2006cs ; Zhang:2007ja . We have
omitted the contributions from resonance decay when examining the $p_{T}$
spectra of stable particles, as their impact becomes significant primarily at
extremely low $p_{T}$. A thorough investigation of resonance decay kinematics
and its influence on the spectra can be found in the references
Schnedermann:1993ws  and STAR:2008med . With the mentioned assumptions and by
defining $\phi_{p}-\phi_{b}$ as $\phi$, equation 8 is expressed as Tang:2008ud

$\displaystyle\frac{d^{3}N}{2\pi
p_{T}dp_{T}}=D\int_{-Y}^{+Y}\cosh(y)dy\int_{-\pi}^{+\pi}d\phi\times\int_{0}^{R}rdr\Big{[}1+\frac{(q_{AA}-1)}{T}[m_{T}\cosh(\rho)\cosh(y)$
$\displaystyle-
p_{T}\sinh(\rho)\cos(\phi)]\Big{]}^{\frac{\displaystyle-1}{\displaystyle
q_{AA}-1}}.$ (10)

Here, we have used Jacobian for the transformation of the coordinates and
integrated it over $d\phi_{p}$. $Y$ is the rapidity of the emitting beam. At
mid-rapidity i.e y $\simeq$ 0 above equation becomes,

$\displaystyle f_{eq}=\frac{d^{3}N}{2\pi
p_{T}dp_{T}dy}=D\int_{-\pi}^{+\pi}d\phi\times\int_{0}^{R}rdr\Big{[}1+\frac{(q_{AA}-1)}{T}[m_{T}\cosh(\rho)-p_{T}\sinh(\rho)\cos(\phi)]\Big{]}^{\frac{\displaystyle-1}{\displaystyle
q_{AA}-1}}.$ (11)

Following the details in reference Rybczynski:2012vj , the system formed right
after high-energy collisions tends to exist far from thermal equilibrium. The
temperature of this system exhibits variations from event to event. This
unique circumstance is characterized by the application of non-extensive
statistics, specifically the Tsallis statistics Tsallis:1987eu . Therefore, as
done in ref. Tripathy:2017kwb , we set the initial distribution as the Tsallis
distribution given as,

$f_{in}=D^{\prime}[1+(q_{pp}-1)\,\frac{m_{T}}{T_{ts}}]^{\frac{-q_{pp}}{q_{pp}-1}}.$
(12)

Here, $\displaystyle D^{\prime}=\frac{gVm_{T}}{(2\pi)^{2}}$. V is the volume
of the fireball formed in the heavy- ion collisions. Consequently, we have
employed the Tsallis distribution to derive both the final particle
distribution and the nuclear modification factor ($R_{AA}$). Using equations
11 and 12 in equation 6, the final distribution can be expressed as,

$f_{fin}=\Bigg{[}D\int_{-\pi}^{+\pi}d\phi\times\int_{0}^{R}rdr\Big{[}1+\frac{(q_{AA}-1)}{T}[m_{T}\cosh(\rho)-p_{T}\sinh(\rho)\cos(\phi)]\Big{]}^{\frac{\displaystyle-1}{\displaystyle
q_{AA}-1}}\\\
+\Bigg{(}D^{\prime}\left[1+(q_{pp}-1)\,\frac{m_{T}}{T_{ts}}\right]^{\frac{\displaystyle-
q_{pp}}{\displaystyle
q_{pp}-1}}-\Bigg{(}D\int_{-\pi}^{+\pi}d\phi\times\int_{0}^{R}rdr\Big{[}1+\frac{(q_{AA}-1)}{T}[m_{T}\cosh(\rho)\\\
-p_{T}\sinh(\rho)\cos(\phi)]\Big{]}^{\frac{\displaystyle-1}{\displaystyle
q_{AA}-1}}\Bigg{)}\Bigg{)}\exp^{-t_{f}/\tau}\Bigg{]}.$ (13)

Using Eq.(12) and Eq.(13) in Eq.(7), the nuclear modification factor can be
expressed as,

$R_{AA}=\frac{f_{fin}}{f_{in}}=\frac{D\int_{-\pi}^{+\pi}\,d\phi\times\int_{0}^{R}r\,dr\bigg{[}1+\frac{(q_{AA}-1)}{T}\big{[}m_{T}\cosh(\rho)-p_{T}\sinh(\rho)\cos(\phi)\big{]}\bigg{]}^{\frac{-1}{q_{AA}-1}}}{D^{\prime}\bigg{[}1+(q_{pp}-1)\frac{m_{T}}{T_{ts}}\bigg{]}^{\frac{-q_{pp}}{q_{pp}-1}}}+\\\
\left(1-\frac{D\int_{-\pi}^{+\pi}\,d\phi\times\int_{0}^{R}r\,dr\bigg{[}1+\frac{(q_{AA}-1)}{T}\big{[}m_{T}\cosh(\rho)-p_{T}\sinh(\rho)\cos(\phi)\big{]}\bigg{]}^{\frac{-1}{q_{AA}-1}}}{D^{\prime}\bigg{[}1+(q_{pp}-1)\frac{m_{T}}{T_{ts}}\bigg{]}^{\frac{-q_{pp}}{q_{pp}-1}}}\right)e^{\frac{-t_{f}}{\tau}}.$
(14)

The equations 13 and 14 are used to fit the experimental results as discussed
in the following section.

(a) Transverse momentum spectra of pions, kaons and protons for 0-5$\%$
centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2019hno

(b) Transverse momentum spectra of $(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ meson
for 0-10$\%$ centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
ALICE:2019xyr

(c) Transverse momentum spectra of pions, kaons and protons for 70-80$\%$
centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2019hno

(d) Transverse momentum spectra of $(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ meson
for 70-80$\%$ centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
ALICE:2019xyr

Figure 1: Transverse momentum spectra of pions, kaons, protons,
$(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ meson for Pb-Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV.

##  III Results and Discussions

In this section, we present the results of the transverse momentum ($p_{T}$)
spectra, $R_{PbPb}$ for $\pi^{\pm}$, $K^{\pm}$, $p+\bar{p}$,
$(K^{*0}+\bar{K^{*0}})/2$, $\phi$ and $R_{pPb}$ for $\pi^{\pm}$, $K^{\pm}$,
$p+\bar{p}$ at $\sqrt{s_{NN}}$ = 5.02 TeV for the most central and peripheral
collisions. We have fitted the $p_{T}$ spectra, $R_{PbPb}$ and $R_{pPb}$ and
find a remarkably favourable $\chi^{2}/ndf$ values. We have performed the
fittings for identified particles in both the most central and peripheral Pb-
Pb as well as p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV. It is noted that,
we have kept the parameters such as $\beta_{s}$, $n$, $t_{f}/\tau$, $q_{pp}$,
$q_{AA}$ and $T_{ts}$ as free parameters throughout this analysis. To fit the
experimental data, we have employed the TF2 cernroot  class provided by the
ROOT library cernroot  to obtain a convergent solution. This fitting process
involved minimization of the chi-squared ($\chi^{2}$) value, a technique
commonly used for obtaining a consistent and well-fitting solution. In this
analysis, the kinetic freeze-out temperature ($T$) is regarded as a fixed
parameter. It has been noted that $T$ decreases as one transitions from highly
central to more peripheral collisions Tang:2008ud ; Ristea:2013ara . This
temperature is observed to be 110 MeV for the most central Pb-Pb collisions
and decreases to 96 MeV for the peripheral ones. Similarly, for p-Pb
collisions, $T$ is observed to be 120 MeV for the most central collisions and
100 MeV for the peripheral ones Che:2020fbz .

In Figures 1 and 2, we have shown the $p_{T}$ spectra of light flavors and
resonances for Pb-Pb collision and light flavors for p-Pb collision,
respectively for the most central and peripheral ones at center-of-mass energy
of $\sqrt{s_{NN}}$ = 5.02 TeV at mid-rapidity. Transverse momentum spectra are
indispensable for unraveling the intricate dynamics associated with the Quark-
Gluon Plasma (QGP) and the transitions between quarks and hadrons
Gupta:2021efj . Several theoretical models Tang:2008ud ; Tripathy:2017nmo ;
Schnedermann:1993ws  have been developed specifically for the characterization
of these $p_{T}$ spectra explaining well upto a limited $p_{T}$ range.
Nonetheless, the formulation we have put forth, as indicated by equation 13,
provides an effective and comprehensive description of the transverse momentum
spectra for all identified hadrons, offering a well-matched fit up to a
transverse momentum ($p_{T}$) = $8$ GeV/c. The parameters derived from the
analysis of these spectra consistently align with the hydrodynamic behaviour
manifested within the system. This concordance lends significant weight to the
credibility of our theoretical formulation.

Pions, being one of the lightest hadrons, manifest distinct resonance effects
due to their relatively small mass Andronic:2008gu ; Schnedermann:1993ws .
Importantly, we opted not to include the contribution of pion yields
originating from resonance decays in our analysis. These resonance decays have
a significant impact on the spectral shape at very low momentum values. To
address the influence of pions resulting from resonance decays in the low
$p_{T}$ range, we have introduced a lower boundary for the pion spectrum,
setting it at 0.5 GeV/c Che:2020fbz .

(a) Transverse momentum spectra of pions, kaons and proton for 0-5$\%$
centrality in p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2016dei .

(b) Transverse momentum spectra of pions, kaons and protons for 60-80$\%$
centrality in p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2016dei .

Figure 2: Transverse momentum spectra of pions, kaons, and protons for p-Pb
collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.

Figure 3 shows the nuclear modification factor ($R_{AA}$) spectra as a
function of $p_{T}$ of $(\pi^{+}+\pi^{-})$, $(K^{+}+K^{-})$, $(p+\bar{p})$,
$(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ mesons in Pb-Pb for the most central and
peripheral collisions at $\sqrt{s_{NN}}$ = 5.02 TeV. Proton-nucleus ($pA$)
collisions occupy an intermediate position between proton-proton ($pp$) and
nucleus-nucleus ($AA$) collisions, both in terms of system size and the
quantity of produced particles. A common approach in research involves
comparing particle production in $pp$, $pA$, and $AA$ reactions to distinguish
between initial state effects, associated with the utilization of the nuclear
beams or targets, and final state effects, associated with the existence of
high-temperature and high-density matter ALICE:2013wgn . Officially ALICE
collaboration have not released data of $R_{pPb}$ of identified hadrons yet.
However in reference ALICE:2016dei , the ALICE collaboration has released the
transverse momentum spectra for pions, kaons, and protons for p-Pb collisions
at $\sqrt{s_{NN}}$ = 5.02 TeV. Additionally, they have presented the $p_{T}$
spectra of these particles in pp collisions at the same energy. Consequently,
we have generated the nuclear modification factor $R_{pPb}$ from the yields at
p-Pb and p-p collisions using equation 1 at $\sqrt{s_{NN}}$ = 5.02 TeV
considering the number of binary collisions ($N_{coll}$). The values for
$N_{coll}$ are taken from the Glauber model calculations ALICE:2014xsp .
Figure 4 shows the generated $R_{pPb}$ spectra of $(\pi^{+}+\pi^{-})$,
$(K^{+}+K^{-})$ and $(p+\bar{p})$ for both the most central and the peripheral
p-Pb collisions. The parameters extracted from fitting of $R_{PbPb}$ and
$R_{pPb}$ are tabulated in the table 1.

The proposed model explains $R_{AA}$ spectra upto $p_{T}$ = $8$ GeV/c. It is
observed that the average transverse flow velocity, $<\beta_{r}>$ decreases
with the particle mass along with the centrality progressing from the most
central to the peripheral collisions Tang:2008ud ; Chen:2020zuw  as shown in
the figure 5. This trend is consistent in both the Pb-Pb and p-Pb collisions.
In central collisions, a substantially higher energy density is generated
within the region where the two colliding nuclei overlap. This elevated energy
density results in intensified particle interactions and a more pronounced
transverse expansion when compared to the peripheral collisions. It is also
noticed that Pb-Pb collisions involve two heavy lead nuclei colliding at high
energies, creating a much hotter and denser medium compared to p-Pb collisions
resulting in the high multiplicity, thus a higher transverse flow Che:2020fbz
. In p-Pb collisions, the transverse flow is suggested to be primarily driven
by the cold nuclear matter effect Albacete:2017qng .

(a) $R_{AA}$ spectra of pions, kaons and protons for 0-5$\%$ centrality in Pb-
Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2019hno .

(b) $R_{AA}$ spectra of $(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ mesons for
0-10$\%$ centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
ALICE:2021ptz .

(c) $R_{AA}$ spectra of pions, kaons and protons for 60-80$\%$ centrality in
Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ALICE:2019hno .

(d) $R_{AA}$ spectra of $(K^{*0}+\bar{K^{*0}})/2$ and $\phi$ mesons for
60-80$\%$ centrality in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
ALICE:2021ptz .

Figure 3: $R_{AA}$ spectra of pions, kaons, protons, $(K^{*0}+\bar{K^{*0}})/2$
and $\phi$ mesons for Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.

(a) $R_{pPb}$ spectra of pions, kaons and protons for 0-5$\%$ centrality in
p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.

(b) $R_{pPb}$ spectra of pions, kaons and protons for 60-80$\%$ centrality in
p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.

Figure 4: $R_{pPb}$ spectra of pions, kaons and protons for p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV Table 1: The parameters obtained by fitting the nuclear modification spectra for Pb-Pb and p-Pb collisions at LHC energies for the most central and peripheral collisions. | $\beta_{r}$ | n | $q_{pp}$ | $T_{ts}$ | $q_{AA}$ | $t_{f}/\tau$ | $\chi^{2}/ndf$
---|---|---|---|---|---|---|---
Pb-Pb | (0-5)% | ${\pi^{+}+\pi^{-}}$ | 0.639 | 2.446 | 1.100 | 0.087 | 1.030 | 2.155 | 0.253
${K^{+}+K^{-}}$ | 0.584 | 0.601 | 1.139 | 0.129 | 1.029 | 2.178 | 0.533
${p+\bar{p}}$ | 0.536 | 0.400 | 1.108 | 0.130 | 1.037 | 3.45 | 0.714
(0-10)% | $(K^{*0}+\bar{K^{*0}})/2$ | 0.546 | 0.587 | 1.138 | 0.120 | 1.053 | 3.00 | 1.365
$\phi$ | 0.423 | 0.399 | 1.135 | 0.119 | 1.062 | 4.00 | 1.055
(70-80)% | ${\pi^{+}+\pi^{-}}$ | 0.481 | 1.077 | 1.150 | 0.033 | 1.047 | 1.70 | 0.940
$K^{+}+K^{-}$ | 0.469 | 1.000 | 1.146 | 0.120 | 1.100 | 1.80 | 1.22
$p+\bar{p}$ | 0.351 | 0.400 | 1.156 | 0.097 | 1.099 | 7.41 | 0.831
$(K^{*0}+\bar{K^{*0}})/2$ | 0.372 | 0.727 | 1.140 | 0.105 | 1.099 | 2.10 | 0.509
$\phi$ | 0.268 | 0.300 | 1.141 | 0.110 | 1.100 | 8.00 | 0.823
p-Pb | (0-5)% | ${\pi^{+}+\pi^{-}}$ | 0.536 | 3.00 | 1.110 | 0.107 | 1.080 | 1.50 | 0.170
$K^{+}+K^{-}$ | 0.501 | 1.57 | 1.109 | 0.126 | 1.075 | 2.92 | 0.458
$p+\bar{p}$ | 0.447 | 0.84 | 1.100 | 0.127 | 1.065 | 6.21 | 0.134
(60-80)% | ${\pi^{+}+\pi^{-}}$ | 0.415 | 1.49 | 1.21 | 0.098 | 1.097 | 1.98 | 0.579
$K^{+}+K^{-}$ | 0.325 | 1.38 | 1.113 | 0.119 | 1.098 | 2.27 | 0.524
$p+\bar{p}$ | 0.313 | 1.12 | 1.110 | 0.119 | 1.092 | 3.49 | 5.114

As shown in the table 1, it is evident that the non-extensive parameter,
$q_{AA}$ exhibits a minor departure from the value of 1. This observation
unequivocally indicates that the system is not in a state of complete
equilibrium, a critical consideration in the Boltzmann transport equation
under the relaxation time approximation. This enhances our model, rendering it
more adept at describing data across a wide range of transverse momentum.
Furthermore, it is observed that the Tsallis parameters $q_{pp}$ and $q_{AA}$
demonstrates a significant rise as the transition is made from central to
peripheral collisions. This observation implies that central collisions create
conditions characterized by a greater degree of particle interactions and
higher energy density, thereby fostering a system behavior that more closely
resembles the equilibrium in contrast to the behavior observed in the
peripheral collisions. Notably, this result aligns with the patterns
previously reported in references Tang:2008ud ; Ristea:2013ara  for different
energies. In contrast, the Tsallis temperature, denoted as $T_{ts}$, exhibits
an opposite trend to that of $q_{pp}$ as it decreases progressively from
central to peripheral collisions Che:2020fbz .

(a) $\beta_{r}$ as a function of particle mass in Pb-Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV at 0-5$\%$, 0-10$\%$ and 70-80$\%$ centrality
classes.

(b) $\beta_{r}$ as a function of particle mass in p-Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV at 0-5$\%$ and 60-80$\%$ centrality classes.

Figure 5: Variation of average transverse flow velocty $<\beta_{r}>$ as a
function of particle mass in Pb-Pb and p-Pb at $\sqrt{s_{NN}}$ = 5.02 TeV for
most central and peripheral collisions.

We have noticed that the ratio $t_{f}/\tau$ increases with the mass of the
particles in both the Pb-Pb and p-Pb collisions. This may be due to that
lighter particles tend to have higher transverse flow velocities which leads
to a higher transverse energy (or transverse mass) compared to heavier
particles. This higher transverse energy is associated with a more rapid
expansion of the fireball created in heavy-ion nuclear collisions. Due to
their higher transverse momenta, lighter particles gain kinetic energy more
quickly and move outward more rapidly, which can lead to an earlier freeze-out
or hadronization of these particles in the collision process. It is also seen
that it does not exhibit any centrality-dependent behaviour. This observation
warrants further investigation, and we plan to delve into this aspect in our
future research endeavours. We have conducted an extensive study by examining
a broad range of flow profiles, each characterized by a different value of the
parameter $n$. The velocity profile $n$ is seen to be decreasing with the
particle mass and shows no trend with the centrality.

In reference Qiao:2020yry , the authors employed a fitting approach to analyze
the nuclear modification factors, $R_{PbPb}$ and $R_{pPb}$ at $\sqrt{s_{NN}}$
= 2.76 TeV and 5.02 TeV, using a combination of the Tsallis distribution as
the initial distribution and the Boltzmann-Gibbs Blast-Wave (BGBW)
distribution as the equilibrium distribution within the Boltzmann transport
equation with RTA. Their results showed that this formulation yielded a good
fit to the data, as indicated by favorable $\chi^{2}/ndf$ values, and was
consistent with hydrodynamics upto transverse momenta ($p_{T}$) = $3$ GeV/c.
In our earlier work Tripathy:2017kwb , the same formulation was employed to
successfully fit the nuclear modification factor, $R_{PbPb}$ at
$\sqrt{s_{NN}}$ = 2.76 TeV upto $p_{T}$= 8 GeV/c. However, the parameter,
$t_{f}/\tau$ does not show any mass ordering and the value of $<\beta_{r}>$
for proton and $K^{*0}$ does not follow the hydrodynamic behaviour. In
contrast, our model distinguishes itself by incorporating a departure from
local equilibrium distribution. Specifically, we employ the TBW model within
the Boltzmann transport equation, which differentiates our approach from the
BGBW model. As a result, when we fit the experimental data for the nuclear
modification factor across a broader $p_{T}$ range, from 0 to 8 GeV/c, our
model achieves an impressive agreement with hydrodynamic behaviour. Our
approach demonstrates that the transverse velocity follows a clear ordering
based on mass and centrality, while the Tsallis parameters $q_{pp}$, $q_{AA}$
and $T_{ts}$ exhibit a well-defined centrality-dependent pattern Che:2020fbz ;
Tang:2008ud . This comprehensive consideration of the system’s dynamics,
including the departure from equilibrium, contributes significantly to the
improved agreement with hydrodynamics in our approach.

##  IV Summary and Outlook

In summary, we have presented the variation of transverse momentum spectra,
nuclear modification factor with respect to transverse momentum in Pb-Pb and
p-Pb collisions for the most central and peripheral collisions at LHC energy
of $\sqrt{s_{NN}}$ = 5.02 TeV. We fitted the experimental data using the
present formulation where Boltzmann transport equation with relaxation time
approximation is used. Here, we have considered Tsallis statistics as an
initial while Tsallis blast wave model is used as equilibrium distribution
function and find a reasonable $\chi^{2}/ndf$ for all the particles species at
both the Pb-Pb and p-Pb collisions in the most central and peripheral
collisions. The outcome of this exercise are succinctly summarized as:

  1. 1.

$<\beta_{r}>$ is consistently observed to decrease with the increasing
particle mass and the collision centrality transitioning from central to
peripheral in both the Pb-Pb and p-Pb collisions at a center-of-mass energy of
$\sqrt{s_{NN}}$ = 5.02 TeV. This may be due to elevated energy density leads
to heightened particle interactions and a more conspicuous transverse
expansion when contrasted with collisions occurring in the peripheral regime.

  2. 2.

The parameter $t_{f}/\tau$ displays a notable trend as it increases with
particle mass in both Pb-Pb and p-Pb collisions, while it does not exhibit any
trend with respect to collision centrality. This may be due to lighter
particles possess higher transverse momenta, they acquire kinetic energy at a
faster rate and undergo a more rapid outward motion. Consequently, this
accelerated motion can lead to an earlier freeze-out or hadronization process
for these particles within the collision dynamics.

  3. 3.

The Tsallis parameters $q_{pp}$ and $q_{AA}$ exhibit an upward trend when
transitioning from the most central to peripheral collisions, whereas $T_{ts}$
displays a contrasting pattern, decreasing under the same conditions. This
phenomenon could be attributed to the notion that central collisions tend to
approach a state of closer equilibrium compared to peripheral collisions.

  4. 4.

We have taken a wide spectrum of flow profiles, with a distinct parameter
value $n$. Notably, the velocity profile demonstrates a decreasing trend as a
function of particle mass, while it exhibits no correlation with the
centrality of the collisions under scrutiny.

## ACKNOWLEDGEMENTS

SKT acknowledges the financial support of the seed money grant provided by the
University of Allahabad, Prayagraj.

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# Random shots phase retrieval by angular streaking of XUV ionization

A. S. Kheifets1 R. Wielian1 V. V. Serov2 I. A. Ivanov3 A. Li Wang4 A.
Marinelli4 J. P. Cryan4 1Research School of Physics, The Australian National
University, Canberra ACT 2601, Australia<EMAIL_ADDRESS>2General,
Theoretical and Computer Physics, Saratov State University, Saratov 410012,
Russia  3Center for Relativistic Laser Science, Institute for Basic Science
(IBS), Gwangju 61005, South Korea  4Stanford PULSE Institute, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025 USA

# Ionization phase retrieval by angular streaking from random shots of XUV
radiation

A. S. Kheifets1 R. Wielian1 V. V. Serov2 I. A. Ivanov3 A. Li Wang4 A.
Marinelli4 J. P. Cryan4 1Research School of Physics, The Australian National
University, Canberra ACT 2601, Australia<EMAIL_ADDRESS>2General,
Theoretical and Computer Physics, Saratov State University, Saratov 410012,
Russia  3Center for Relativistic Laser Science, Institute for Basic Science
(IBS), Gwangju 61005, South Korea  4Stanford PULSE Institute, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025 USA

###### Abstract

We demonstrate an accurate phase retrieval of XUV atomic ionization by
streaking the photoelectron in a circularly polarized IR laser field. Our
demonstration is based on a numerical solution of the time-dependent
Schrödinger equation. We test this technique using the hydrogen atom ionized
by isolated attosecond XUV pulses across a wide range of the photon energies.
Importantly, the proposed method works in a random shot mode when the time
delay between the ionizing and streaking XUV and IR pulses may vary from shot
to shot. This is a significant advantage over the existing interferometric
techniques which require a systematic and controllable scan of the XUV/IR
delay in one set of measurements. Such a scan may not be possible to take
because of the arrival time jitter inherent to the stochastic nature of self-
amplified FEL radiation.

###### pacs:

32.80.Rm 32.80.Fb 42.50.Hz

##  I Introduction

Angular streaking of extreme ultraviolet (XUV) atomic ionization with a
circularly polarized IR laser radiation has become a useful tool for
characterizing isolated attosecond pulses (IAP) from free-electron laser (FEL)
sources. First suggested theoretically Zhao et al. (2005); Kazansky et al.
(2016); Li et al. (2018); Kazansky et al. (2019), this method has been
implemented in practice for a shot-to-shot characterization of IAP at FEL
Hartmann et al. (2018); Duris et al. (2020). Angular streaking of XUV
ionization (ASXUVI or ASX for brevity) combines elements of the two previously
developed techniques: attosecond angular streaking known as the attoclock
Eckle et al. (2008a, b); Pfeiffer et al. (2012) and the attosecond streak
camera (ASC) Constant et al. (1997); Itatani et al. (2002); Goulielmakis et
al. (2004); Kienberger et al. (2004); Yakovlev et al. (2005); Frühling et al.
(2009); Zhang and Thumm (2011); Ivanov and Smirnova (2011). It is analogous to
ASC in that XUV pulses are the primary source of ionization while it is common
to the attoclock in that the ionized electrons interact with a circularly
polarized laser field which rotates the photoelectron momentum distribution
(PMD). This rotation is most graphic in the plane perpendicular to the laser
propagation direction. The difference with the attoclock is that the latter is
a self-referencing technique where both tunneling ionization and steering are
driven by the same elliptical infrared (IR) laser pulse. In its original form
Kazansky et al. (2016); Li et al. (2018); Kazansky et al. (2019); Hartmann et
al. (2018); Duris et al. (2020), ASX was considered in an intense IR laser
field and analyzed within the strong field approximation (SFA) Zhao et al.
(2022). An alternative view within the lowest order perturbation theory (LOPT)
Dahlström et al. (2012, 2013); Maquet et al. (2014) considers IR streaking as
an interference phenomenon which opens a natural access to the streaking phase
$\Phi_{S}$. The latter is typically decomposed into the XUV ionization (or
Wigner) phase and the laser induced (or continuum-continuum - CC) phase. These
two phases can be converted to the corresponding time delay components, which
add up to the atomic time delay. This delay is often interpreted as the time
it takes for the electron to be photoionized plus the time it takes for the
measurement process to occur. This timing analysis has been used to determine
the attosecond time delay in photoemission from atoms Schultze et al. (2010)
and solid surfaces Cavalieri et al. (2007). The latter seminal works opened up
the rapidly growing field of attosecond chronoscopy of photoemission Pazourek
et al. (2013a); Pazourek et al. (2015); Thumm et al. (2015).

A straightforward extension of time-resolved ionization studies to novel FEL
sources of XUV radiation is not possible at present. The existing
interferometric techniques rely on a systematic and controllable scan of the
XUV/IR pulse delay in one set of measurements. The stochastic nature of self-
amplified FEL radiation and the inherent timing jitter do not permit such a
scan. In the meantime, ASX allows for the retrieval of the XUV ionization
phase and the associated timing information from a series of XUV shots
arriving at the target atom randomly. In this Letter, we demonstrate this
useful phase retrieval capability of ASX over a wide range of photoelectron
energies right from the ionization threshold and exceeding it many times. We
demonstrate the accuracy of the proposed technique by considering the hydrogen
atom driven by a femtosecond XUV pulse across a wide range of carrier
frequencies. Our demonstration is based on a numerical solution of the time-
dependent Schrödinger equation (TDSE). It is verified through comparison with
the well established RABBITT (reconstruction of attosecond beating by
interference of two-photon transitions) technique. Both methods are shown to
provide comparable phase and timing information.

Figure 1: Left: Schematic representation of IR streaking of XUV ionization.
The first order direct process (d) and the second order processes aided by an
IR photon absorption (a) and emission (e) are labeled accordingly (adopted
from Dahlström et al. (2013)). Right: photoelectron spectra of H driven by XUV
and XIR are overlapped with the Fourier transform of the electric field.

##  II Physical interpretation

An interference character of IR streaking of XUV ionization is illustrated in
the left panel of Fig. 1. There are three ionization channels marked (d), (a)
and (e) leading to the same final state with the photoelectron energy $E$.
While the direct channel (d) contains an unassisted XUV ionization, two other
channels are aided by an IR absorption (a) or emission (e). For these three
channels to interfere, the spectral width of the XUV pulse should be large
enough to accommodate the $\pm\omega$ laser assisted processes. In the time
domain, this means that the XUV pulse is shorter than the laser period. The
spectrum of such a pulse is illustrated in the right panel of Fig. 1 and
corresponds to ionization of the hydrogen atom at the central XUV and IR
photon energies $\Omega_{0}=0.7$ au and $\omega=0.038$ au, respectively, while
FWHM of the XUV pulse is set to 2 fs. The photoelectron spectrum of an
unassisted XUV photoionization (process d) is an exact replica of the Fourier
transform of the electric $E$-field shifted by the ionization potential.
Meanwhile, the photoelectron spectrum of XUV+IR ionization (XIR for brevity)
is significantly broadened by the second-order (a) and (e) processes.

The proposed phase retrieval by ASX is based on the following consideration.
We apply the SFA and write the photoionization amplitude as Kitzler et al.
(2002)

$b({\bm{k}},\tau)=i\int_{t_{0}}^{\infty}\\!\\!dt\
E_{x}(t-\tau)D_{x}\left[{\bm{k}}-{\bm{A}}(t)\right]e^{-i\Phi(t)}\ .$ (1)

Here the electric field of the XUV pulse $E_{x}$ is advancing the streaking
pulse by the time $\tau$. The streaking field is described by its vector
potential

${\bm{A}}(t)=A_{0}\cos(\omega t)\hat{{\bm{x}}}+A_{0}\sin(\omega
t)\hat{{\bm{y}}}\ ,$

while the photoelectron momentum is confined to the polarization plane
${\bm{k}}=k\cos\phi~{}\hat{{\bm{x}}}+k\sin\phi~{}\hat{{\bm{y}}}$, where $\phi$
is the emission angle.

The exponential term contains the so-called Volkov phase

$\Phi(t)=\frac{1}{2}\left[\int_{t}^{\infty}dt^{\prime}\left[{\bm{k}}-{\bm{A}}(t^{\prime})\right]^{2}-k_{0}^{2}\,t\right]\
,$ (2)

in which the photoelectron energy in the absence of streaking
$E_{0}=k_{0}^{2}/2=\Omega-I_{p}$. This phase can be estimated by the saddle
point method (SPM) which selects the most probable photoelectron trajectories
leaving the atom at the time $t_{\rm st}$ and keeping the phase stationary:

$\Phi^{\prime}(t_{\rm st})=\frac{1}{2}|{\bm{k}}-{\bm{A}}(t_{\rm
st})|^{2}-E_{0}=0$ (3)

We assume that the XUV pulse is short relative to the IR pulse and shifted
relative to its peak position by the time $\tau$. Under these conditions, Eq.
(3) is transformed to the following  isochrone equation Kazansky et al.
(2016):

$k^{2}/2-E_{0}=kA_{0}\cos(\phi-\omega\tau);\
\left\\{\begin{array}[]{rl}k>k_{0}&\phi-\omega\tau\sim 0\\\
k<k_{0}&\phi-\omega\tau\sim 180^{\circ}\end{array}\right.$ (4)

Here we neglected the ponderomotive energy $U_{p}=A_{0}^{2}/2$ in a weak
streaking field. The bottom panel of Fig. 2 illustrates the isochrone equation
(4) most straightforwardly. The PMD extrema, which correspond to $\phi=0$ and
$180^{\circ}$ at $\tau=0$, are displaced relative to the photoelectron
momentum $k_{0}$ (dashed line) by the amount $\pm A_{0}$.

In the above derivation it was implicitly assumed that the phase of the dipole
matrix element $D\left[{\bm{k}}-{\bm{A}}(t)\right]$ does not depend on the
photoelectron energy. In a more general case, this phase contains an energy
dependent term

$\arg\left\\{D\left[{\bm{k}}-{\bm{A}}(t)\right]\right\\}\propto\alpha|{\bm{k}}-{\bm{A}}(t)|^{2}/2\
,$ (5)

where

$\alpha=\partial\arg D(\sqrt{2E})/\partial E$ (6)

is a generalized definition of the photoelectron group delay in photoemission
(see e.g. Eq. (S10) of Schultze et al. (2010)). The introduction of this delay
modifies the stationary phase equation:

$\frac{1}{2}\left|{\bm{k}}-{\bm{A}}(t_{st})\right|^{2}-E_{0}+\frac{\alpha}{2}\frac{d}{dt}\left[\left({\bm{k}}-{\bm{A}}(t_{\rm
st})\right)^{2}\right]=0$ (7)

This leads to a modified isochrone equation

$\displaystyle k^{2}/2-E_{0}$ $\displaystyle=$ $\displaystyle
kA_{0}\left[\cos(\phi-\omega\tau)-\alpha\omega\sin(\phi-\omega\tau)\right]$
(8) $\displaystyle\approx$ $\displaystyle
kA_{0}\cos[\phi-\omega\tau+\omega\alpha]$

In the second line, we used the identity $b\cos x-a\sin
x=\sqrt{1+(ab)^{2}}\cos(x+y),\ y=\tan^{-1}(a/b)$ and the long wavelength
approximation $\omega\alpha\ll 1$ for the streaking field. Thus the isochrone
acquires an additional phase shift $\Phi_{S}=\omega\alpha$ which can be
extracted from the photoelectron momentum distribution (PMD). In the
following, we shall demonstrate that $\alpha=\Phi_{S}/\omega=\tau_{a}$ under
certain XUV and IR pulse parameters. For simplicity of derivation, $\alpha$
was introduced in Eq. (6) solely as the dipole phase derivative. In practice,
we will show that it also contains the measurement induced CC components.

##  III Numerical results

Figure 2: The PMD of the hydrogen atom in the polarization plane displayed in
the Cartesian) and polar (bottom) coordinates. The dashed circle (top) and the
horizontal straight line (bottom) display the momentum value
$k_{0}=\sqrt{2(\Omega_{0}-I_{p})}$ from the energy conservation. The solid
vertical lines (bottom) define the angular integration limits when calculating
the radial momentum profiles $P_{\pm}(k,\tau)$. The XUV/IR time delay $\tau=0$
in this figure.  Figure 3: Top: radial momentum displacements
$k_{\pm}^{2}/2-k_{0}^{2}/2$ are shown at various XUV/IR delays $\tau$. The
dashed line represents the fit with Eq. (9). The arrow indicates the streaking
phase $\Phi_{S}$. Bottom: the fit with Eq. (9) is applied to individual $\tau$
values to determine the instantaneous $\Phi_{S}(\tau)$. The average $\Phi_{s}$
is shown as a solid line with error bars visualized by dotted lines.

In Fig. 2 we display the PMD of the hydrogen atom ionized with a 2 fs XUV
pulse at $\rm{6}\times 10^{{13}}~{}W/cm^{2}$ and $\Omega_{0}=0.7$ au, in the
presence of a long IR pulse with $\omega=0.038$ au ($\lambda=1200$ nm), FWHM =
25 fs at $\rm{1.5}\times 10^{{11}}~{}W/cm^{2}$. The XUV pulse is linearly
polarized along the $\hat{\bm{x}}$ axis whereas the IR pulse is circularly
polarized in the $(xy)$ plane. The PMD is projected on this plane. The top
panel illustrates the dipole angular pattern $\propto\cos^{2}\phi$ of the
primary XUV ionization of the hydrogen $1s$ initial state which is largely
retained in the XIR ionization. The bottom panel shows the polar coordinate
representation highlighting the momentum offset induced by the vector
potential of the laser pulse. In the exhibited case, the XUV/IR delay
$\tau=0$. To simulate XIR ionization numerically, we run a series of TDSE
calculations using the computer code Serov (2011). At each XUV photon energy,
we scan the delay between the XUV pulse and the IR laser field ($\tau$) in the
range of 0 to 70 au in 8 increments.

The modified isochrone equation (8) allows, in principle, for the ASX phase
extraction at each pair of the variables $k,\phi$. For better clarity, it is
more advantageous to conduct such an extraction at the extrema of the PMD.
When a limited statistics is available experimentally, the whole lobe of the
PMD centered at these extrema can be used. We demonstrate this by integrating
the PMD over the $\pm 90^{\circ}$ angular intervals centered around $\phi=0$
and $\pm 180^{\circ}$ (shown by vertical lines in the bottom panel of Fig. 2)
. We carry out this procedure at several fixed increments of the XUV/IR delay
$\tau$. The corresponding radial momentum profiles $P_{+}(k,\tau)$ are shown
in the top panel of Fig. 3 for the angular integration range
$|\phi|<90^{\circ}$ The analogous profiles $P_{-}(k,\tau)$ corresponding to
the $|180^{\circ}-\phi|<90^{\circ}$ integration range mirror closely the set
shown in the figure. The two sets of the momentum profiles allow to determine
the mean photoelectron momenta at each $\tau$

$k_{\pm}(\tau)=\int kP_{\pm}(k,\tau)dk\Big{/}\int P_{\pm}(k,\tau)dk\ ,$

which are then used to obtain the isochrone phase offset:

$k_{\pm}^{2}(\tau)/2-E_{0}=\pm A_{0}\,k_{\pm}(\tau)\cos(\omega\tau+\Phi_{S})\
.$ (9)

In our simulation, the ansatz (9) can be applied to the whole set of the time
delays $\tau$ treating $\Phi_{s}$ as a common fitting parameter. This way an
average $\Phi_{S}$ value is obtained as illustrated in the top panel of Fig3.
Alternatively, we can solve Eq. (9) for each $\tau$ value and to obtain an
instantaneous $\Phi_{S}(\tau)$ value. This value would mimic a streaking phase
retrieved in a single shot measurement at a given $\tau$. Both sets of
streaking phases are shown in the bottom panel of Fig. 3 where the average
$\Phi_{S}$ is displayed with the straight solid line while its error bars are
exhibited with the dotted lines. While the deviation of $\Phi_{S}(\tau)$ from
$\Phi_{S}$ is exceeding the error bars of the latter, the method works quite
satisfactorily even if a single $\tau$ determination of the streaking phase is
made. The attained accuracy of about 20% is on par with that achieved in other
streaking measurements (see e.g. Schmid et al. (2019)).

We note that Eq. (9) contains the vector potential magnitude which may not be
known experimentally because of the laser focus averaging effect. After a
sufficient number of XUV shots at random $\tau$ is collected, an effective
vector potential magnitude can be determined from the maximum vertical
displacement between the upwards and downwards shifted lobes
$A_{0}=0.5|k_{+}-k_{-}|_{\max}$ across the whole series. Statistical nature of
such a determination will not be detrimental to the accuracy of the phase
retrieval which is rather insensitive to the IR pulse intensity. For instance,
at $\Omega=0.7$ au and the IR pulse intensity of $\rm{1.5}\times
10^{{11}}~{}W/cm^{2}$, the streaking phase $\Phi_{s}=-0.196$ rad. This value
changes to $-0.177$ rad at $\rm{1.2}\times 10^{{11}}~{}W/cm^{2}$ and to
$-0.211$ rad at $\rm{1.8}\times 10^{{11}}~{}W/cm^{2}$. While these two
intensities differ by 50%, the streaking phase $\Phi_{s}$ varies by only 10%.
The streaking phase is similarly insensitive to ellipticity. The cited value
of $\Phi_{s}=-0.196$ rad changes only in the fourth significant figure when
the circular polarization is reduced to an elliptical one with $\epsilon=0.9$.
At a sufficiently large XUV photon energy, the streaking phase tends to zero
as a fast photoelectron is represented by a plane wave. Hence the maximum
vertical displacement $2A_{0}$ is attained at $\tau=0$. This provides a
convenient calibration of the XUV pulse arrival time.

The avarage streaking phases $\Phi_{S}$ at various XUV photon energies are
displayed in Fig. 4. These phases are benchmarked against our present RABBITT
calculations on hydrogen at the corresponding XUV energies and the same IR
intensity and wavelength. Numerical details of these calculations can be found
in preceding works Bray et al. (2018); Kheifets (2021). From these
simulations, we obtain the RABBITT phase $\Phi_{R}$ by fitting the angular
integrated photoelectron spectra at various $\tau$ with the following
expression:

$\displaystyle S_{2q}(\tau)$ $\displaystyle=$ $\displaystyle
A+B\cos[2\omega\tau-\Phi_{R}(E)]$ (10)

Here the sideband index $2q$ defines the photoelectron energy $E=2q\omega-
I_{p}$. We note that the amplitudes of the RABBITT peaks (10) oscillate with
twice the laser frequency $\propto 2\omega\tau$ and the atomic time delay is
defined as $\tau_{R}=\Phi_{R}/(2\omega)$. In the meantime the streaking signal
(9) contains the $\propto\omega\tau$ oscillation and we expect the atomic time
delay to be expressed as $\tau_{S}=\Phi_{S}/\omega$. Therefore, we make a
comparison of the RABBITT phase $\Phi_{R}$ with twice the streaking phase
$2\Phi_{S}$. This comparison is shown in Fig. 4. The error bars in the figure
indicate the accuracy of the fitting procedure with the isochrone and RABBITT
equations (4) and (10), respectively, by employing the Marquardt-Levenberg
nonlinear algorithm. We observe a rather close agreement between $\Phi_{R}$
and $2\Phi_{S}$ in Fig. 4. However, various numerical factors stop us short
from claiming these two phases leading to the identical atomic time delays.
Such an identity is established in linear streaking and RABBITT Dahlström et
al. (2013) . We note that the RABBITT phase $\Phi_{R}$ cannot be determined
from Eq. (10) using a single value of $\tau$ as the magnitude coefficients
$A,B$ are not known ab initio and cannot be determined experimentally. Only
the whole set of RABBITT measurements or simulations can be fit by Eq. (10)
with a common $\Phi_{R}$ value. Thus RABBITT is not suitable for a random shot
phase determination. In addition, to cover a wide photoelectron energy range
of Fig. 4, the large harmonic orders in their 40s should be used in the
RABBITT simulations for the hydrogen and Yukawa atoms. This large order leads
to progressively increasing error bars in the RABBITT data. We note that the
typical harmonic orders in their 20s are employed in most RABBITT experiments.
This observation demonstrates another advantage of the proposed scheme which
can probe large photoelectron energies without compromising accuracy since the
XUV and IR photon energies are entirely decoupled.

Figure 4: Left axis: RABBITT phase $\Phi_{R}$ as extracted from Eq. (10) is
compared with the twice the streaking phase $2\Phi_{s}$ obtained from the
isochrone ansatz Eq. (4). Right axis: RABBITT time delay
$\tau_{R}=\Phi_{R}/(2\omega)$ and the streaking time delay
$\tau_{S}=\Phi_{S}/\omega$ are compared with the Coulombic Wigner time delay
augmented by the CC correction from Pazourek et al. (2013b); Serov et al.
(2015). The RABBITT phase and time delay in the Yukawa atom (Yu) are also
shown.

The phase of the angular integrated RABBITT signal can be converted to the
atomic time delay as $\tau_{R}=\Phi_{R}/(2\omega)$. Similarly, the streaking
phase can be converted to the time delay as $\tau_{S}=\Phi_{S}/\omega$. Both
conversions are shown on the right axis of Fig. 4. As the hydrogen atom is
free from many-electron correlation and resonances, its atomic time delay is
known analytically Pazourek et al. (2013b); Serov et al. (2015). Comparison of
both the ASX and RABBITT time delays with this analytic test is rather
accurate across a broad range of XUV energies. As an additional test, we run a
set of analogous streaking and RABBITT calculations on the Yukawa atom in
which the Coulomb potential is screened as $(Z/r)\exp(-r/a)$, where $Z=2.785$
and $a=0.5$, thus maintaining the hydrogen atom ionization potential. Both the
streaking and RABBITT phases vanish in the Yukawa atom thus indicating the
Coulomb origin of the atomic time delay in hydrogen. Besides solving the TDSE,
we obtain the PMD from the SPM simulations as described in Kheifets et al.
(2021). The streaking phase vanishes once the same phase extraction method is
applied to the SPM data. This provides additional credibility to the proposed
technique.

Figure 5: Top: the PMD of the hydrogen atom in the polarization plane at
$\Omega=0.7$ au, $\lambda=10.6~{}\mu$m and $\tau=2.5$ fs is displayed in the
Cartesian coordinates. Bottom: radial momentum displacements
$k_{\pm}^{2}/2-k_{0}^{2}/2$ are shown at various XUV/IR delays $\tau$. The
dashed line represents the fit with Eq. (9). The arrow indicates the streaking
phase $\Phi_{S}$.

Furthermore, we extended our ASX simulations to a different regime utilizing a
mid-IR laser pulse at $\lambda=10.6~{}\mu$m as in the experiment Hartmann et
al. (2018). The IR intensity was varied between $\rm{1.5}\times
10^{{9}}~{}W/cm^{2}$ and $\rm{2.4}\times 10^{{10}}~{}W/cm^{2}$ to demonstrate
the stability of the streaking phase extraction. Results of these simulations
at a fixed XUV photon energy $\Omega=0.7$ au are shown in Fig. 5. The top
panel displays the PMD in the polarization plane at the IR intensity of
$\rm{2.4}\times 10^{{10}}~{}W/cm^{2}$ and the XUV/IR time delay $\tau=2.5$ fs.
The PMD displays both the radial and angular displacement of the maxima away
from XUV polarization direction. With the radial displacements being
determined in these maxima directions, the isochrone analysis is shown in the
bottom panel of Fig. 5. While the streaking phase is decreased to
$\Phi_{S}=-0.046$ rad, the corresponding time delay grows to $\tau_{S}=260$
as. This compares with the time delay value $\tau_{S}=125$ as at 1200 nm. Such
an increase is expected as the CC correction contains the term

$\tau_{\rm cc}\propto-{Z\over k^{3}}\left[\ln 1.16{k^{2}\over\omega}-1\right]\
,$ (11)

where $k\simeq 0.63$ is the photoelectron momentum and $Z\simeq 1$ the Coulomb
charge of the ion remainder Serov et al. (2015).

##  IV Conclusions

In conclusion, we demonstrate an accurate retrieval of the streaking phase
$\Phi_{S}$ from XUV atomic ionization in the presence of a circularly
polarized IR laser field. The rational of this method is based on introduction
of the streaking phase $\Phi_{S}$ into the isochrone equation (4) derived
earlier in Kazansky et al. (2016) within the SFA. The modified isochrone (8)
has two phase shifts. One is due to the XUV/IR delay $\omega\tau$ which can be
controlled by an incremented displacement of the short XUV pulse relative to
the long IR pulse. An additional term $\Phi_{S}$ is due to the energy
dependent phase of the dipole ionization amplitude (6).  Our numerical
simulations on hydrogen indicate that such an extracted phase can be converted
to the atomic time delay $\tau_{S}=\Phi_{S}/\omega$ similarly to the analogous
conversion of the phase of the RABBITT oscillation
$\tau_{R}=\Phi_{R}/(2\omega)\ .$ Even though the numerical values of
$\tau_{S}$ and $\tau_{R}$ may differ insignificantly in our simulations, their
similarity across a wide range of photon energies, right from the ionization
threshold, is quite noticeable .

Even though the rational of the proposed phase extraction is drawn from the
SFA, its validity is actually much wider and applies to photoelectrons with
the energy as low as only 10% of the ionization potential. Simplicity of the
hydrogen atom allows us to test the ASX and RABBITT time delay $\tau_{a}$
against the analytical results Pazourek et al. (2013b); Serov et al. (2015).
These results allow to interpret the atomic time delay as the time it takes
for the electron to be photoionized and the time it takes for the measurement
process to occur. This interpretation was challenged recently for the
Coulombic systems Saalmann and Rost (2020) but seems to be confirmed by the
present study. In addition, the proposed technique was extended to the H2
molecule Serov and Kheifets (2022). Not only did we show an accurate phase
extraction. We also reproduced orientation and two-center interference effects
which are specific to homo-nuclear diatomic molecules.

The proposed ASX phase retrieval can be utilized in FEL by collecting a
sequence of XUV radiation shots with random arrival times $\tau$. This
sequence allows to determine the effective vector potential magnitude $A_{0}$
from the maximum vertical displacement of the two PMD lobes. This
determination, conducted at a sufficiently large photoelectron energy, will
point to the zero net displacement of the XUV and IR pulses $\tau=0$. This
arrival time calibration along with the known effective $A_{0}$ will allow to
conduct the phase determination at the whole set of collected data points.
Importantly, the magnitude and phase of the XUV pulse, which may vary from
shot to shot, do not affect the proposed phase determination as these
parameters do not enter the isochrone equations (4) and (8).

In parallel with the absolute streaking phase determination, one can use a
reference of a second ionization state in the same target or ionization from
an additional, reference target mixed with the original target, to determine
their relative ionization phase. The reference and target photolines are
produced by the same XUV pulse, so they share a common global streaking angle.
Their difference in streaking angle will give $\tau_{a}-\tau_{r}$, where
$\tau_{r}$ is the reference EWP group delay. If we choose a reference
photoline that is much higher in energy and has a clear continuum, then
$\tau_{r}$ will be negligible. This technique has been implemented and
analyzed in the SFA framework Driver et al (2019), where the radial ansatz Eq.
(9) can be used to directly read on a single-shot basis.

VVS was supported by the Australian Research Council Discovery Project
DP190101145. IAI was supported by the Institute for Basic Science under the
grant IBS-R012-D1. JPC, AM and ALW were supported by the U.S. Department of
Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES),
Chemical Sciences, Geosciences, and Biosciences Division (CSGB).

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# Quantum singularities in conformally static spacetimes

Deborah A. Konkowski1 and Thomas M. Helliwell2 1 Mathematics Department, U.S.
Naval Academy, Annapolis, Maryland 21402 USA  2 Physics Department, Harvey
Mudd College, Claremont, California 91711 USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

After a brief review of the standard definition and analysis of classical
singularities in general relativistic spacetimes, and of quantum singularities
in static spacetimes with timelike classical singularities, an extension of
quantum singularities to conformally static spacetimes is summarized and
applied to two test cases. The timelike classical singularities in a
Friedmann-Robertson-Walker (FRW) universe with a cosmic string, and in Roberts
spacetime, are shown to be quantum mechanically singular when tested by either
minimally coupled or conformally coupled scalar waves. In the Roberts case,
however, non-minimally coupled scalar waves with a coupling constant $\xi\geq
2$ do not detect the classical singularity.

##  1 Introduction

We study quantum wave packet propagation in conformally static spacetimes with
timelike classical singularities. If the wave propagation turns out to be well
defined, the spacetimes are said to be quantum mechanically non-singular.

The order of the paper is as follows: First, classical and quantum
singularities are defined with the latter restricted (as usual) to static
spacetimes with timelike singularities. Next, the definition of quantum
singularity is extended to conformally static spacetimes with a timelike
singularity (spacelike singularities, if present, are not tested). In
particular, two spacetimes are tested with generally coupled scalar waves: a
Friedmann-Robertson-Walker (FRW) spacetime with a cosmic string and the
Roberts spacetime. Finally, conclusions are given, together with ideas for
further research.

##  2 Classical singularities

A spacetime $(M,g)$ is taken to be a paracompact, $C^{\infty}$, connected,
Hausdorff manifold $M$ with a Lorentzian metric $g$ [3]. So what is a
classical _singularity_? A spacetime is by definition smooth, so ‘singular’
points are not part of the spacetime; they must be cut out of the spacetime
manifold. This leaves a ‘hole’, with incomplete curves, a seeming boundary to
spacetime. How do we complete spacetime, and how do we define a boundary
$\partial M$ to spacetime? There have been a number of attempts, none of them
entirely satisfactory. Note that Cauchy completeness works only in Riemannian
metrics, not Lorentzian. Boundary definitions have included the
a(abstract)-boundary of Scott and Szekeres [17], the b(bundle)-boundary of
Schmidt [16], the c(causal)- boundary of Geroch, Kronheimer, and Penrose [5]
and the g(geodesic)-boundary of Geroch [6]. In this discussion we will use
Geroch’s 1968 description of a classical singularity. He states that “a
singularity is indicated by incomplete geodesics or incomplete curves of
bounded acceleration in a maximal spacetime.” This is closest to the
definition of classical singularity used in the famous singularity theorems of
Hawking and Penrose, which predict that singularities are ubiquitous in exact
solutions of Einstein’s equations (see, e.g., [7])).

Ellis and Schmidt have classified singular points into three types according
to their strength [3]: quasi-regular (mild, topological singularities), non-
scalar curvature (diverging tidal forces on curves ending at the singularity;
finite tidal forces on some nearby curves) and scalar curvature (diverging
scalars – usually one considers only $C^{0}$ scalar polynomial invariants).
Conical singularities, as in idealized cosmic strings, are a good example of
quasiregular singularities. The other two types of singularities are stronger,
curvature singularities. Nonscalar curvature singularities include those in
whimper cosmologies and certain plane-wave spacetimes, whereas scalar
curvature singularities are the best-known, occurring at the centers of black
holes or the beginning of big bang cosmologies.

###  2.1 Quantum Singularities

What happens if instead of classical particle paths (e.g., null and timelike
geodesics) one uses quantum mechanical particles (quantum wave packets) to
identify singularities? Following pioneering work by Wald [19], Horowitz and
Marolf answered this question for static spacetimes with timelike classical
singularities. In their 1995 paper they posit that a spacetime is quantum
mechanically (QM) _non_ singular if the evolution of a test scalar wave
packet, representing a quantum particle, is uniquely determined by the initial
wave packet, the manifold and the metric, without having to place boundary
conditions at the classical singularity. Technically, a static spacetime is
QM-singular if the spatial portion of the Klein-Gordon operator is not
essentially self-adjoint on $C_{o}^{\infty}(\Sigma)$ in the space of square
integrable functions $L^{2}(\Sigma)$, where $\Sigma$ is a spatial
hypersurface.

The term “essentially self adjoint” arises in functional analysis [14]. An
operator $A$ is called self-adjoint if (i) $A=A^{*}$ and (ii) $Dom(A)$ =
$Dom(A^{*})$, where $A^{*}$ is the adjoint of $A$ and $Dom$ is short for
domain. An operator is _essentially_ self-adjoint if (i) is met and (ii) can
be met by expanding the domain of the operator $A$ or its adjoint $A^{*}$ so
that it is true.

There are two basic tests for essential self-adjointness [14]. The first uses
the von Neumann criterion of deficiency indices [18]; one studies solutions of
$A\Psi=\pm i\Psi$, where $A$ is the spatial portion of the Klein-Gordon
operator, and finds the number of solutions for each sign that are self-
adjoint. The second technique uses the so-called Weyl limit point - limit
circle criterion [20], which relates essential self-adjointness of the
Hamiltonian operator to the behavior of the “potential” in an effective one-
dimensional Schrödinger equation, which in turn determines the behavior of the
scalar wave packet. Relevant theorems that simplify the analysis can be found
in Reed and Simon [14].

Many authors have used the definition of quantum singularity to study the
singularity structure of spacetimes. For a summary, see, for example, the
review article by Pitelli and Letelier [13] or the conference proceeding by
the authors [12] and the references therein. Also, there is the alternative
concept of ‘wave regularity’ introduced by Ishibashi and Hosoya [10], which is
relevant to the discussion. It uses a non-standard Hilbert space, $H^{1}$, the
first Sobolev space.

##  3 Conformally Static Spacetimes

A spacetime $g_{\mu\nu}(x^{\alpha})$ that is conformally static is related to
a static spacetime $\bar{g}_{\mu\nu}(x^{a})$ by a conformal transformation
$C(\eta)$ of the metric. Here $C(\eta)$ is the conformal factor, where $\eta$
is the conformal time, related to the time $t$ by $dt=Cd\eta$. Simply put,
$g_{\mu\nu}(x^{\alpha})=C^{2}(\eta)\bar{g}_{\mu\nu}(x^{a})$. Here Greek
letters $\alpha,\beta,...$ label spacetime indices and have the range over 0,
1, 2, 3, and Latin letters $a,b,c,...$ label spatial indicies that range over
1, 2, 3.

The Lagrangian density for a generally coupled scalar field is [1],

$\mathcal{L}=1/2(-g)^{1/2}[g^{\mu\nu}\Phi,_{\mu}\Phi,_{\nu}-(M^{2}+\xi
R)\Phi^{2}],$ (1)

where $M$ is the mass if the scalar particle, $R$ is the scalar curvature, and
$\xi$ is the coupling (in particular, $\xi=0$ for minimal coupling and
$\xi=1/6$ for conformal coupling). Varying the action $S=\int\mathcal{L}\
d^{4}x$ gives the Klein-Gordon field equation,

$|g|^{-1/2}\left(|g|^{1/2}g^{\mu\nu}\Phi,_{\nu}\right),_{\mu}-\xi
R\Phi=M^{2}\Phi.$ (2)

.

In the massless case with conformal coupling, the field equation above is
conformally invariant under a conformal transformation of the metric and
field; in this case the inner product respecting the stress tensor for the
field is also conformally invariant. This led Ishibashi and Hosoya to state
[10], in the case of wave regularity, that “the calculation is as simple as
that in the static case when singularities in conformally static space-times
are probed with conformally coupled scalar fields.”

Here we study the quantum particle propagation in spacetimes with massive
scalar particles described by the Klein-Gordon equation and the limit point -
limit circle criterion of Weyl [20] [14]. In particular, after separating
variables we study the radial equation in a one-dimensional Schrödinger form
with a ‘potential’ and determine the number of solutions that are square
integrable. If we obtain a unique solution, without placing boundary
conditions at the location of the classical singularity, we can say that the
solution to the full Klein-Gordon equation is quantum mechanically (QM)
nonsingular. The results depend on the spacetime metric parameters and wave
equation modes.

After separating variables we take the spatial portion to be an operator
equation on a Hilbert space $L^{2}(\Sigma)$ with inner product (see, e.g.,
[11]),

$(\chi,\zeta)=\int d^{3}x|\bar{g}_{3}/g_{00}|^{1/2}\chi(x^{a})\zeta(x^{b}),$
(3)

where $\bar{g}_{3}$ is the determinant of the spatial portion of the static
metric, $\chi$ and $\zeta$ are spatial mode solutions and $a,b$ range over 1,
2, 3. Then we consider the radial portion alone, change variables and write
the radial equation in one-dimensional Schrodinger form, $Hu(x)=Eu(x)$, where
the operator $H=-d^{2}/dx^{2}+V(x)$ and $E$ is a constant, with the
singularity at $x=0$. The inner product here is simply $\int dx|u(x)|^{2}$ and
the Hilbert space is $L^{2}(0,\infty)$. At this point one can simply apply the
limit point - limit circle criterion as easily as in the static case in order
to determine the quantum singularity structure.

###  3.1 FRW with a Cosmic String

A simple metric modeling a Friedmann-Robertson-Walker cosmology with a cosmic
string [2] is given by

$ds^{2}=a^{2}(t)(-dt^{2}+dr^{2}+\beta^{2}r^{2}d\phi^{2}+dz^{2})$ (4)

where $\beta=1-4\mu$ and $\mu$ is the mass per unit length of the cosmic
string. This metric is conformally static (actually conformally flat).
Classically it has a scalar curvature singularity when $a(t)$ is zero and a
quasiregular singularity when $\beta^{2}\neq 1$. Here we will consider the
timelike quasiregular singularity alone. The Klein-Gordon equation with
general coupling can be separated into mode solutions

$\Phi=T(t)H(r)e^{im\phi}e^{ikz}$ (5)

where

$\ddot{T}+2\left(\frac{\dot{a}}{a}\right)\dot{T}+(M^{2}a^{2}+\xi Ra^{2}-q)T=0$
(6)

and

$H^{\prime\prime}+\frac{1}{r}H^{\prime}+(-k^{2}-q-\frac{m^{2}}{\beta^{2}r^{2}})H=0.$
(7)

The $T$-equation alone contains $M$ and $R$. Rewriting the dependent and
independent variables as $r=x$ and $H=xu(x)$, we get the correct inner product
form and a one-dimensional Schrödinger equation,

$u^{\prime\prime}+(E-V(x))u=0$ (8)

where $E=-k^{2}-q$ and

$V(x)=\frac{m^{2}-\beta^{2}/4}{\beta^{2}x^{2}}.$ (9)

.

Near zero one can show that the potential $V(x)$ is limit point if
$m^{2}/\beta^{2}\geq 1$. Therefore any modes with sufficiently large $m$ are
limit point, but $m=0$ is limit circle; thus generically this conformally
static space-time is quantum mechanically singular.

###  3.2 Roberts Spacetime

The Roberts metric [15] is

$ds^{2}=e^{2t}(-dt^{2}+dr^{2}+G^{2}(r)d\Omega^{2})$ (10)

where $G^{2}(r)=(1/4)[1+p-(1-p)e^{-2r}](e^{2r}-1)$. The spacetime is
conformally static, spherically symmetric, and self-similar (see, e.g., [10]).
It has a timelike classical scalar curvature singularity at $r=0$ for $0<p<1$.
The Klein-Gordon equation can be solved by separation of variables with mode
solutions given by $\Phi=T(t)H(r)Y_{lm}(\theta,\phi)$. The radial operator can
be put in one-dimensional Schrödinger form and the limit point - limit circle
criterion applied. Details are given in [8]. One finds that the spacetime is
quantum mechanically singular if $\xi<2$ and quantum mechanically non-singular
if $\xi\geq 2$. Therefore, the classical timelike singularity remains singular
when probed by minimally coupled ($\xi=0$) waves or by conformally coupled
($\xi=1/6$) waves.

##  4 Conclusions

After a brief review of the standard definition and analysis of classical
singularities in general relativistic spacetimes, and of quantum singularities
in static spacetimes with timelike classical singularities, an extension of
quantum singularites to conformally static spacetimes was summarized and
applied to two test cases. The timelike classical singularities in a FRW
universe with a cosmic string and in Roberts spacetime were shown to be
quantum mechanically singular when tested by either minimally coupled or
conformally coupled scalar waves. In the Roberts case, however, non-minimally
coupled scalar waves with a coupling constant $\xi\geq 2$ did not detect the
classical singularity.

Further analysis of the singularity structure of conformally static spacetimes
is underway [8]. A class of spherically symmetric conformally static
spacetimes is being analyzed; this class includes the spacetimes of HMN[9] and
Fonarev[4], as well as the Roberts spacetime.

## Acknowledgements

One of us (DAK) thanks B. Yaptinchay for useful discussions.

## References

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