# Shining a Light on the QGP -
Electroweak Probes Experimental Summary

CERN
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###### Abstract:

Objects which are only subject to the electroweak force are an ideal probe of
QCD in high density and temperature environments as they carry information
about the conditions during their production out of the QGP without
interacting with it. Thus they can be used to characterize the initial state
as well as several properties of the QGP and its evolution. Within this
article the recent results regarding these electroweak probes presented at the
9th Hard Probes Conference (Aix-les-Bains, France, 2018) will be summarized.
In particular the following questions will be addressed: Can we determine the
necessary energy and particle density for the QGP creation using
electromagnetic probes? Can we use light-by-light scattering in heavy-ion
collisions to do precision tests of QED and measure the magnetic field of the
QGP? How are quark jets modified by the presence of the QGP? What can we learn
about the initial state and scaling properties in pp, p(d)-A and A-A
collisions from the production of high $p_{\mathrm{T}}$ photons, Z0 and W±
bosons?
In the version provided on arXiv, additional plots are included in the
appendix with respect to the version submitted for publication in the journal.

##  1 Introduction

Ultrarelativistic heavy-ion collisions provide the perfect laboratory to study
QCD at high temperatures and high energy density. In these collisions a Quark-
Gluon-Plasma (QGP) is created and subsequently thermalizes. The plasma cools
down and expands, until it finally hadronizes and the hadrons stop interacting
among themselves and free-stream. Electroweak probes offer a unique
perspective on the properties of the QGP and the hot hadron gas (HG) as they
do not interact strongly with the medium and are produced at all stages of the
collision. Depending on their production mechanism the different probes allow
access to different aspects of the heavy-ion collision.
In analogy to black body radiation the thermal medium created in heavy-ion
collisions will emit electromagnetic radiation in form of low momentum photons
and di-leptons. The measurement of the low-$p_{\mathrm{T}}$ (virtual) direct
photon excess beyond the expectations from next-to-leading order (NLO) pQCD
calculations is one of the important signatures for establishing a QGP
creation. It has been found that the hadron production in high multiplicity
p-p and p(d)-A collisions exhibits similar collective behaviors as seen in A-A
collisions [1]. The exact nature of this effect is still unclear, however.
Measuring a significant direct photon signal and the corresponding effective
temperature of the source should provide more information about the nature of
the medium created in these collisions. Furthermore, the measurement of the
di-lepton spectrum in A-A collisions can give insights to the in-medium
electromagnetic spectral function of the $\rho^{0}$ and its possible
modification due to chiral-symmetry restoration.
If instead the electroweak probes are produced in the initial hard scattering
processes, like the weak bosons (W±, Z0) and most of the direct photons at
high-$p_{\mathrm{T}}$, they are ideal probes for the parton distribution
functions of the incoming nuclei and protons. Additionally, their yield at
mid-rapidity can be used to verify the Glauber model as their production
should scale with the number of binary collisions. Moreover, they can be used
as calibration probes for the recoiling jets and their in medium
modifications, as they serve as a measure of the initial energy of the jet and
allow to discriminate between quark and gluon induced jets.
Electromagnetic probes in heavy-ion collisions, however, do not only provide a
stringent test of QCD, but even test QED under extreme conditions. Due to the
large fluxes of quasireal photons emitted by the nuclei accelerated to TeV
energies at large distances from the center of the nuclei light-by-light
scattering can occur in heavy-ion collisions at RHIC and the LHC [2]. The
first evidence for the presence of these processes in ultraperipheral Pb-Pb
collisions at the LHC has been presented by the ATLAS collaboration [3].
Observing these processes and their modifications in more central collisions
could help determining the magnetic field strength of the QGP itself [4].
Within this article a summary of the results on electroweak probes presented
at the 9th Hard Probes Conference (Aix-les-Bains, France, 2018) will be given.
Particular focus lies on addressing the following questions and the results
which contributed to their answers. Each section serves as a summary on the
corresponding topic: Can we determine the necessary energy and particle
density for the QGP creation using electromagnetic probes (Section 2)? Can we
use light-by-light scattering in heavy-ion collisions to do precision tests of
QED and measure the magnetic field of the QGP (Section 3)? How are quark jets
modified by the presence of the QGP (Section 4)? What can we learn about the
initial state and scaling properties in pp, p(d)-A and A-A collisions from the
production of high $p_{\mathrm{T}}$ photons, Z0 and W± bosons (Section 5)?

##  2 Probing the QGP with Direct Photons and Di-Leptons

Figure 1: Left: Summary of $x_{\mathrm{T}}$-scaled (isolated) direct photon
invariant cross sections measured in pp and p$\bar{\mathrm{p}}$ collisions at
various collider energies [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26] scaled with a common $n=4.5$. Right: Summary
of $N_{\mathrm{coll}}$ and $x_{\mathrm{T}}$-scaled (isolated) direct photon
invariant yields measured in A-A collisions at various collider energies [27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] scaled with a common $n=4.5$
and the number of collisions ($N_{\mathrm{coll}}$) per collision system and
respective centrality.

While it has been established that a QGP is created in central heavy-ion
collisions at SPS, RHIC and LHC energies the question remains whether the
particle and energy density is sufficient for it to be also created in
peripheral A-A collisions or even smaller systems, like high multiplicity p-p
or p-A collisions. The measurements of the low momentum (virtual) direct
photons, which are presumed to be of thermal origin, served as one of the key
pillars for establishing the creation of a QGP. These low $p_{\mathrm{T}}$
photons can be measured as an excess ($R_{\gamma}$) above the expected decay
photon spectrum originating from $\pi^{0}$, $\eta$, $\omega$ and other mesons
and baryons decaying into photons. Furthermore, they can be extracted from the
di-lepton invariant mass spectrum using the Kroll-Wada-equation, after
subtraction of the resonance contributions. Within the excess, however, not
only thermal photons are contained, but also photons from hard scattering
processes as well as fragmentation photons. Above a transverse momentum of
about 4 GeV/$c$ the dominant source of photons are those from the later
processes, which are well described by next-to-leading order pQCD
calculations. Below those momenta, on the other hand, NLO pQCD calculations
are not constrained any longer and the photon yield originating from the hard
scatterings can only be extracted with large uncertainties. In order to reduce
these uncertainties the experimental collaborations are pursuing the same
measurement in pp collisions, where mainly the photons from the hard
scattering and the fragmentation of jets should be present, which presumably
scale with the number of collisions ($N_{\mathrm{coll}}$) in a heavy-ion
collision.
At this conference new results regarding the direct photon production in pp
collisions at $\sqrt{s}$$~{}=~{}0.2,$ $2.76,~{}7,~{}8$ and $13$ TeV have been
presented by the PHENIX and ALICE collaborations [40, 24, 41]. While the
PHENIX collaboration has been able to extract a direct photon spectrum down to
about 1.5 GeV/c at $\sqrt{s}$$=0.2$ TeV the same measurements have proven to
be challenging at the LHC due to the smaller fraction of direct photons with
respect to the decay photons at the LHC. Thus the extraction of the direct
photon spectra at LHC energies was only possible above $6$ GeV/c (Figure 8),
where they are well described by pQCD calculations, and upper limits were
obtained for lower transverse momenta. When taking into account all measured
direct and isolated photon spectra at current and previous colliders (Figure 6
(left)) it has been found that the high $p_{\mathrm{T}}$ spectra exhibit
$x_{\mathrm{T}}$-scaling [5]. Figure 1 (left) shows the updated compilation of
all direct and isolated photon spectra measured at mid-rapidity in pp or
p$\bar{\mathrm{p}}$ collisions from $\sqrt{s}$$~{}=0.019$ to $13$ TeV as a
function of $x_{\mathrm{T}}=2p_{\mathrm{T}}/\sqrt{s}$. The effective scaling
parameter $n$ was chosen to be $4.5$ as obtained in [5]. It can be seen that
$x_{\mathrm{T}}$-scaling seems to hold within $20-50\%$ if the spectra are
scaled with $\sqrt{s}^{n}$ and plotted as a function of $x_{\mathrm{T}}$.
However, a discontinuity is seen for the spectra obtained at the LHC for the
measurements from the ALICE collaboration using the $R_{\gamma}$-technique and
the isolated direct photon spectra from ALICE, CMS & ATLAS. This discontinuity
arises from the fact that the isolated photon spectra do not contain the
fragmentation photons, while prompt photons cannot be discriminated from the
fragmentation photons using the $R_{\gamma}$-technique. Even though a full
reanalysis of the obtained data regarding the effective scaling parameter
should be performed, it seems that the spectra obtained a higher $\sqrt{s}$
can be used to constrain the low momentum spectra at lower $\sqrt{s}$,
yielding stronger constrains than the upper limits provided by the
measurements for the NLO pQCD calculations.
In addition to the base line measurements in minimum bias pp collisions the
PHENIX and ALICE collaborations presented new measurements regarding the low
momentum direct photon production in p-A collisions as a function of
centrality. The compilation of the available measurements in p(d)-A collisions
ranging from $\sqrt{s_{\mathrm{NN}}}$$~{}=0.2-8.16$ TeV can be found in Figure
7. The minimum bias measurements in p-A collisions show no significant excess
of direct photons beyond the expected yield of prompt and fragmentation
photons calculated using NLO pQCD [42, 40]. In 0-5% central p-Au collisions on
the other hand a slight increase of direct photons below $3$ GeV/$c$ has been
found by the PHENIX collaboration, indicating that a thermal source might be
present in these collisions [40]. A similar analysis versus multiplicity in
p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, also showed a slight
increase in the direct photon excess (Figure 8) for the most central 0-20%
events. Within the systematic and statistical uncertainties, however, it is
not yet possible to claim a significant direct photon signal at this center of
mass energy for $p_{\mathrm{T}}~{}<~{}3$ GeV/c.
Increasing the system size even more the PHENIX collaboration also showed new
results on the direct photon production in Cu-Cu collisions at
$\sqrt{s_{\mathrm{NN}}}$$~{}=~{}0.2$ TeV and Au-Au collisions at
$\sqrt{s_{\mathrm{NN}}}$$~{}=~{}39$ and $62.4$ GeV.

Figure 2: Left: Direct photon integrated yield for $p_{\mathrm{T}}~{}>~{}1$
GeV/$c$ as a function of the number of charged particles ($N_{\mathrm{ch}}$)
at mid rapidity. The yield has been normalized to corresponding integrated
photon yield in pp collisions at $\sqrt{s}$$~{}=~{}0.2$ TeV scaled by the
number of charged particles to the power of $\alpha=1.25$ [40]. Middle:
Integrated photon yield for prompt and thermal photons above 1 GeV/$c$ as
function of $N_{\mathrm{ch}}$ for 0-5% central events at different collision
energies [43]. Right: Integrated photon yield for prompt and thermal photons
above 1 GeV/$c$ as function of $N_{\mathrm{ch}}$ for different centralities in
Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [43].

The compilation of the $N_{\mathrm{coll}}$-scalled direct photon spectra in
A-A collisions for various species and centralities can be found in Figure 6
(right) and Figure 1 (right) as function of $x_{\mathrm{T}}$. Once more it can
be seen that $x_{\mathrm{T}}$ and $N_{\mathrm{coll}}$-scaling holds at high
transverse momenta within $20-50\%$. Taking into account only the lower
transverse momentum points for the various collision energies they seem to
follow a second $x_{\mathrm{T}}$-scaling curve with a similar exponent. The
PHENIX collaboration investigated this scaling property in more detail as a
function of the number of charged particles at mid rapidity [30] by
integrating the extracted photon spectra above 1 GeV/$c$. Figure 2 (left)
shows the obtained scaling relation including the integrated yields measured
in pp and p-A collisions. The points are normalized to the expected scaling of
the photons based on the observed charged particle scaling. Considering only
the points from the PHENIX and ALICE collaborations the direct photon yields
seems to rise with increasing number of charged particles and then saturates
at a common value for Au-Au and Pb-Pb collisions at all center of mass
energies. The onset and later saturation is interpreted by the PHENIX
collaboration as an indication of the QGP being produced in a fraction of the
events and, when it reaches the plateau, in all of the collisions [30].
However, such an independence of the collision energy cannot be explained
easily by theoretical calculations [43] as seen in Figure 2 (middle), while it
can be understood that the direct photon yield scales as function of charged
particle multiplicity when only considering one collision system and energy as
seen in Figure 2 (right). Furthermore, the scaling relation is not seen as
clearly when also including the corresponding integrated direct photon yields
at $\sqrt{s_{\mathrm{NN}}}$$=0.2$ TeV measured by the STAR collaboration.
Their results are significantly lower than those obtained by the PHENIX
collaboration. Thus it cannot be determined conclusively whether these
measurements indicate that a QGP can be create in high multiplicity p-A
collisions without further measurements from the various experiments.
In addition to the direct photon spectra measurements the simultaneous
development of direct photon flow in A-A collisions was discussed at the
conference based on the new results obtained by the ALICE collaboration [44].
In contrast to the results obtained by the PHENIX collaboration [45] the
spectra and flow measurements can be described by the theoretical calculations
within $1-2\sigma$. However, the uncertainties at the LHC are significantly
larger than those achieved by the PHENIX collaboration due to the smaller
signal to background ratio in the thermal photon transverse momentum region.
Additionally, the first di-muon measurements from the STAR collaboration in pp
and p-Au at $\sqrt{s_{\mathrm{NN}}}~{}=~{}0.2$ TeV at mid-rapidity have been
presented using neural networks to cope with the low purity and statistics in
the $\mu^{+}\mu^{-}$ channel at low invariant masses [46]. While the di-muon
measurement does not seem to be feasible in Au-Au collisions, the same
techniques can be applied to the di-electron measurements to improve the
results for the large data sets obtained during the beam energy scan and using
isobaric ions. The ALICE collaboration was able to extract the di-electron
spectra in pp collisions at $\sqrt{s}=7$ and $13$ TeV as well as in Pb-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ and $5.02$ TeV [41, 47, 48, 49].
The obtained Pb-Pb data are in good agreement with cocktail calculations
without in medium modifications of the $\rho^{0}$ between
$150<M_{\mathrm{ee}}<700$ MeV/$c^{2}$. However, there is a strong indication
that the charm suppression needs to be taken into account for the cocktail
calculations at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV. From the pp
measurements constraints on the correlation between the total $c\bar{c}$ and
$b\bar{b}$ cross sections could be extracted. Furthermore, no enhanced di-
electron production with respect to the cocktail calculations is seen in high
multiplicity pp collisions at $\sqrt{s}=13~{}$TeV indicating that the possible
QGP contributions should be small if present.

##  3 Light-by-Light Scattering in Heavy Ion Collisions

Figure 3: Left: Distribution of the $\sqrt{\langle p_{\mathrm{T}}^{2}\rangle}$
of excess di-electron yields in 60-80% peripheral Au-Au and U-U collisions
measured by the STAR collaboration [50]. Right: Mean transverse momentum
broadening ($k_{\mathrm{T}}^{RMS}$) obtained from Gaussian fits to the
acoplanarity distribution of high momentum di-muon events in Pb-Pb collisions
at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02~{}$TeV by the ATLAS collaboration as a
function of the number of participants ($N_{\mathrm{part}}$) [51].

In heavy-ion collisions at RHIC and LHC energies strong magnetic fields are
created by the passing nuclei at large distances from the nuclei cores. These
fields generate large quasireal photon fluxes and can lead to light-by-light
scattering. The first evidence of this process has been presented by the ATLAS
collaboration with $4.4\sigma$ [3] and was now confirmed by the CMS
collaboration with $4.1\sigma$ [52] in the two photon final state in
ultraperipheral Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$$~{}=~{}5.02$ TeV.
The respective measured fiducial cross sections are $\sigma_{fid}~{}=~{}70\pm
24$ (stat.) $\pm~{}17$(syst) nb and $\sigma_{fid}~{}=~{}120~{}\pm~{}46$ (stat)
$\pm~{}4$ (theo) nb. Using these data the CMS collaboration was also able to
improve the limits on the axion-like particle production in the
$\gamma\gamma$-channel as no significant excess was observed in the di-photon
invariant mass distribution [52].
Light-by-light scattering can also occur in more central collisions, however,
it is more difficult to observe as the resulting final state particles are
observed in parallel to a large amount of particles produced in the actual ion
collisions. The STAR collaboration presented results on the di-electron
invariant mass spectra in peripheral events showing a significant excess in
the low and intermediate mass region above the hadronic cocktail [50]. The
excess could be explained by the superposition of the light-by-light
scattering into $e^{+}e^{-}$ in these events. In this case the electron pair
$p_{\mathrm{T}}$ distribution is significantly enhanced at low
$p_{\mathrm{T}}$ with respect to the expectations from the hadronic cocktail
calculations. Furthermore, the $p_{\mathrm{T}}$2 distribution of the excess
yield appears to be broadened beyond the expectations based on models
describing the contributions from UPC events. The additional broadening and
its invariant mass dependence can be described by introducing an additional
strong magnetic field perpendicular to the beam line (blue dashed line in
Figure 3 (left)).
The ATLAS collaboration, on the other hand, looked at di-muon events overlaid
with the normal heavy-ion collision requiring the events to have at least two
muons with single muon $p_{\mathrm{T}}>4$ GeV/$c$ and di-muon invariant masses
of $4<M_{\mathrm{\mu\mu}}<45$ GeV/$c^{2}$ [51]. The pair acoplanarity
($\alpha$) and the asymmetry distributions ($A$) have been corrected for the
background contributions from heavy flavor decays based on data driven
template fits outside the signal region and were afterwards compared to the
corresponding distributions obtained from ultraperipheral collisions using the
same criteria. It can be seen that the $\alpha$ distribution broadens
significantly when going to more central events. This can be interpreted as
the muons being slightly deflected by the presence of an additional
electromagnetic field obtaining a small amount of transverse momentum
$|k_{\mathrm{T}}|\ll p_{\mathrm{T}}^{\pm}$. The corresponding
$k_{\mathrm{T}}^{RMS}$ values obtained from Gaussian fits to the $\alpha$
distributions after subtraction of the expected UPC background in different
centrality intervals can be seen in Figure 3 (right). In 0-10% central
collisions an average transverse momentum transfer of $70~{}\pm~{}10$ MeV/$c$
can be observed.
While the results from both collaboration indicate a deflection of the leptons
by an electromagnetic source, it remains to be seen whether it will be
possible to measure the field strength of such a source taking into account
all other possible background contributions and its origin [53].

##  4 Isolated Photons as Calibration and Tagging Objects for Jet Modification
Studies in p-A and A-A Collisions

Through the study of the direct photons not only the QGP properties can be
explored, but also a reference for distinguishing initial and final state
effects seen in other probes can be established. The isolated prompt photons
mainly produced in the initial hard scattering can be used as calibration and
tagging probe for the recoiling jets. Those jets will mainly originate from
quarks and can thus help to distinguish the effects of the final state
interaction in the QGP for quarks and gluons. Furthermore, the energy of the
photon serves as a measure of the initial energy of the jet, as the photon
will leave the QGP unaffected by the medium, while the constituents of the
quark-jet will interact with it.
At this conference the first results on isolated photon-hadron and jet
correlations measured by the ALICE collaboration in pp and p-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$$~{}=~{}5.02$ TeV have been presented for intermediate
photon transverse momenta between $10$ and $40$ GeV/c [54]. It has been found
that the fragmentation functions do not seem to be modified in minimum bias
p-Pb collisions, while it is expected that the strongest modifications for the
recoiling jets in heavy-ion collisions at the LHC should be seen in this
transverse momentum region. Similar observations for pp and p(d)-Au collisions
at $\sqrt{s_{\mathrm{NN}}}~{}=~{}200$ GeV were presented by the STAR [55] and
PHENIX [56] collaborations at slightly lower $p_{\mathrm{T}}$ for the isolated
photons. Additionally, these collaborations presented the corresponding
results in Au-Au collisions at the same $\sqrt{s_{\mathrm{NN}}}$, which
strongly indicate a redistribution of the recoiling jet energy towards large
angles.

Figure 4: Left: Photon-jet $p_{\mathrm{T}}$-balance distribution
($x_{J\gamma}$) in pp (blue) and 0-10% central Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV for the inclusive jet (closed) and
photon-plus-leading-jet (open) selection for $p_{\mathrm{T}}^{\gamma}=100-158$
GeV [57]. Middle: Ratio of the $\xi_{\mathrm{T}}^{\gamma}$ distribution in
0-10% central Pb-Pb and pp collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$
TeV compared to various theoretical models [58]. Right: Ratio of the
differential jet shape $\rho(r)$ for jets associated with an isolated photon
with a $p_{\mathrm{T}}$ above $60$ GeV/c for 0-10% central Pb-Pb and pp
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV compared to different
theory calculations [59].

Furthermore, the ATLAS and CMS collaborations presented their recent multi-
differential results on the isolated-photon jet tagged measurements [57, 60,
58, 59] at high $p_{\mathrm{T}}$. When measuring the $\gamma$-jet
$p_{\mathrm{T}}$-balance distributions in multiple centrality classes the
ATLAS collaboration observed a significant broadening of the $x_{j\gamma}$
distribution in central Pb-Pb events with an isolated photon and a recoiling
leading jet (Figure 4 (left)). The broadening increases further when
considering all jets in the $\gamma$-tagged events. Such a smearing can be
explained by a strong variation in the jet-by-jet energy loss for instance
depending on the path length in the medium of the recoiling quark.
Furthermore, a clear difference in the evolution of $\gamma$-tagged jet
fragmentation functions compared to those for inclusive jets has been found by
the ATLAS collaboration going from pp to central Pb-Pb collisions [61], which
might be partially explained by a flavor depended energy loss of the jets. The
CMS collaboration, on the other hand, presented their results regarding the
isolated photon tagged jet shapes in pp and Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [59]. The corresponding comparisons
for pp and central Pb-Pb collisions are shown in Figure 4 as function of
$\xi_{\mathrm{T}}^{\gamma}$ (middle) and $r$ (right) for photons with a
minimum $p_{\mathrm{T}}$ of 60 GeV/$c$. These indicate an enhancement of low
$p_{\mathrm{T}}$ particles and a depletion of high $p_{\mathrm{T}}$ particles
inside the jet, which is stronger for $\gamma$-tagged jets than for inclusive
jets. Furthermore, it can be seen that the jet momentum is distributed at
greater radial distance in Pb-Pb collisions, which can be interpreted as a
direct observation of jet broadening in the QGP. By using the $\gamma$-tagged
jets the energy scale of the jet is fixed which eases the comparisons to
theory calculations, which can qualitatively reproduce the distributions, but
still require some fine tuning in terms of the magnitude of the effects seen
in Pb-Pb collisions.

##  5 Photons and Bosons as Probes for the Initial State and Scaling
Properties

As the electroweak probes at high $p_{\mathrm{T}}$ and masses at mid-rapidity
are not modified by the QGP, they can also be used in order to calibrate the
Glauber model calculations and their estimated $N_{\mathrm{coll}}$ in
particular in p(d)-A collisions, where the correlation between
$N_{\mathrm{coll}}$ and $N_{\mathrm{ch}}$ is not that strong. At higher
rapidities the photons, W± and Z0 can help to constrain the nuclear parton
distribution function (nPDFs). At this conference recent results on the high
$p_{\mathrm{T}}$ isolated photon [62] and W± [63] production in p-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}8.16$ TeV have been discussed in
different rapidity intervals. While the isolated photons can only mildly
discriminate between different nPDFs within the current uncertainties.
Stronger constrains for the nPDFs can be derived from the W±, which favor the
nPDFs from [64] as can be seen in Figure 5 (right). Additionally, the pp
measurements of the weak bosons have been improved [65] and consequently an
updated nuclear suppression factor ($R_{\mathrm{AA}}$) could be obtained with
smaller uncertainties than those of the Glauber model. Thus it might now be
possible to replace the traditional $R_{\mathrm{AA}}$ measurements for hard
probes by
$Z_{\mathrm{AA}}~{}=~{}(N_{\mathrm{AA}}^{X}\cdot\sigma_{\mathrm{pp}}^{Z})/(N_{\mathrm{AA}}^{Z}\cdot\sigma_{\mathrm{pp}}^{X})$
measurements, which do not depend on model calculations any longer. The multi-
differential measurements of the W± production can also be used to test and
constrain the existing PDFs for the proton as well as NNLO pQCD calculations
for the weak boson production in pp collisions. The comparisons of the most
recent NNLO pQCD calculations to the measured W+ production cross section in
pp collisions at $\sqrt{s}~{}=~{}5.02$ TeV as a function of the lepton pseudo-
rapidity can be seen in Figure 5 (left). It can be seen that the best
agreement can be reached when using the HERAPDF2.0 PDF sets, while the
remaining calculations slightly underestimate the cross sections in nearly all
pseudo-rapidity bins.

Figure 5: Left: Differential W${}^{+}\rightarrow\mu^{+}+\nu_{\mu}$ production
cross section as a function of the lepton pseudo-rapidity in pp collisions at
$\sqrt{s}~{}=~{}5.02$ TeV compared to different NNLO pQCD calculations with
different PDFs as input [65]. Right: W${}^{+}\rightarrow\mu^{+}+\nu_{\mu}$
production cross section as a function of the muon pseudo rapidity in p-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}8.16$ TeV [63].

## References

  * [1] C. Loizides, “Experimental overview on small collision systems at the LHC,” Nucl. Phys. A956 (2016) 200–207, arXiv:1602.09138 [nucl-ex].
  * [2] C. A. Bertulani, S. R. Klein, and J. Nystrand, “Physics of ultra-peripheral nuclear collisions,” Ann. Rev. Nucl. Part. Sci. 55 (2005) 271–310, arXiv:nucl-ex/0502005 [nucl-ex].
  * [3] ATLAS Collaboration, M. Aaboud et al., “Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC,” Nature Phys. 13 no.~9, (2017) 852–858, arXiv:1702.01625 [hep-ex].
  * [4] D. E. Kharzeev and H. J. Warringa, “Chiral Magnetic conductivity,” Phys. Rev. D80 (2009) 034028, arXiv:0907.5007 [hep-ph].
  * [5] PHENIX Collaboration, A. Adare et al., “Direct-Photon Production in $p+p$ Collisions at $\sqrt{s}=200$ GeV at Midrapidity,” Phys. Rev. D86 (2012) 072008, arXiv:1205.5533 [hep-ex].
  * [6] D0 Collaboration, S. Abachi et al., “Isolated photon cross-section in the central and forward rapidity regions in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV,” Phys. Rev. Lett. 77 (1996) 5011–5015, arXiv:hep-ex/9603006 [hep-ex].
  * [7] CDF Collaboration, F. Abe et al., “A Precision measurement of the prompt photon cross-section in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV,” Phys. Rev. Lett. 73 (1994) 2662–2666. [Erratum: Phys. Rev. Lett.74,1891(1995)].
  * [8] UA1 Collaboration, C. Albajar et al., “Direct Photon Production at the CERN Proton - anti-Proton Collider,” Phys. Lett. B209 (1988) 385–396.
  * [9] UA1 Collaboration, C. Albajar et al., “Comparison of W and direct photon crossections at large transverse momenta,” in Proceedings of the 8th Topical Workshop on Proton Antiproton Collider Physics: 1-5 Sep. 1989, Castiglione della Pescaia, Italy, pp. 254–267.  1989\.  [,254(1989)].
  * [10] UA2 Collaboration, J. Alitti et al., “A Measurement of the direct photon production cross-section at the CERN anti-p p collider,” Phys. Lett. B263 (1991) 544–550.
  * [11] E704 Collaboration, D. L. Adams et al., “Measurement of single spin asymmetry for direct photon production in p p collisions at 200-GeV/c,” Phys. Lett. B345 (1995) 569–575.
  * [12] Fermilab E706 Collaboration, L. Apanasevich et al., “Nuclear effects in high-p(T) production of direct photons and neutral mesons,” Phys. Rev. D72 (2005) 032003, arXiv:hep-ex/0506003 [hep-ex].
  * [13] Fermilab E706 Collaboration, L. Apanasevich et al., “Measurement of direct photon production at Tevatron fixed target energies,” Phys. Rev. D70 (2004) 092009, arXiv:hep-ex/0407011 [hep-ex].
  * [14] FERMILAB-E706 Collaboration, G. Alverson et al., “Production of direct photons and neutral mesons at large transverse momenta by $\pi^{-}$ and $p$ beams at 500-GeV/c,” Phys. Rev. D48 (1993) 5–28.
  * [15] NA24 Collaboration, C. De Marzo et al., “A Measurement of Direct Photon Production at Large Transverse Momentum in $\pi^{-}p$, $\pi^{+}p$ and $pp$ Collisions at 300-GeV/$c$,” Phys. Rev. D36 (1987) 8.
  * [16] WA70 Collaboration, M. Bonesini et al., “Production of High Transverse Momentum Prompt Photons and Neutral Pions in Proton Proton Interactions at 280-GeV/c,” Z. Phys. C38 (1988) 371.
  * [17] UA6 Collaboration, G. Sozzi et al., “Direct photon production in $\bar{p}p$ and $pp$ interactions at $\sqrt{s}$ = 24.3-GeV” Phys. Lett. B317 (1993) 243–249.
  * [18] UA6 Collaboration, G. Ballocchi et al., “Direct photon cross-sections in proton proton and anti-proton - proton interactions at S**(1/2) = 24.3-GeV,” Phys. Lett. B436 (1998) 222–230.
  * [19] ATLAS Collaboration, G. Aad et al., “Measurement of the inclusive isolated prompt photon cross-section in $pp$ collisions at $\sqrt{s}=$ 7 TeV using 35 pb-1 of ATLAS data,” Phys. Lett. B706 (2011) 150–167, arXiv:1108.0253 [hep-ex].
  * [20] ATLAS Collaboration, G. Aad et al., “Measurement of the inclusive isolated prompt photon cross section in $pp$ collisions at $\sqrt{s}=7$ TeV with the ATLAS detector,” Phys. Rev. D83 (2011) 052005, arXiv:1012.4389 [hep-ex].
  * [21] ATLAS Collaboration, G. Aad et al., “Measurement of the inclusive isolated prompt photon cross section in pp collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” JHEP 08 (2016) 005, arXiv:1605.03495 [hep-ex].
  * [22] ATLAS Collaboration, M. Aaboud et al., “Measurement of the cross section for inclusive isolated-photon production in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector,” Phys. Lett. B770 (2017) 473–493, arXiv:1701.06882 [hep-ex].
  * [23] ALICE Collaboration, M. Wilde, “Measurement of Direct Photons in pp and Pb-Pb Collisions with ALICE,” Nucl. Phys. A904-905 (2013) 573c–576c, arXiv:1210.5958 [hep-ex].
  * [24] ALICE Collaboration, S. Acharya et al., “Direct photon production at low transverse momentum in proton-proton collisions at $\mathbf{\sqrt{s}=2.76}$ and 8 TeV,” arXiv:1803.09857 [nucl-ex].
  * [25] CMS Collaboration, S. Chatrchyan et al., “Measurement of the Differential Cross Section for Isolated Prompt Photon Production in pp Collisions at 7 TeV,” Phys. Rev. D84 (2011) 052011, arXiv:1108.2044 [hep-ex].
  * [26] CMS Collaboration, V. Khachatryan et al., “Measurement of the Isolated Prompt Photon Production Cross Section in $pp$ Collisions at $\sqrt{s}=7$ TeV,” Phys. Rev. Lett. 106 (2011) 082001, arXiv:1012.0799 [hep-ex].
  * [27] C. Gale, R. Chatterjee, D. K. Srivastava, and S. Jeon, “A Reanalysis of Single Photon Data at CERN SPS,” Nucl. Phys. A830 (2009) 579C–582C, arXiv:0907.3546 [nucl-th].
  * [28] WA98 Collaboration, M. M. Aggarwal et al., “Observation of direct photons in central 158-A-GeV Pb-208 + Pb-208 collisions,” Phys. Rev. Lett. 85 (2000) 3595–3599, arXiv:nucl-ex/0006008 [nucl-ex].
  * [29] STAR Collaboration, L. Adamczyk et al., “Direct virtual photon production in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV,” Phys. Lett. B770 (2017) 451–458, arXiv:1607.01447 [nucl-ex].
  * [30] PHENIX Collaboration, A. Adare et al., “Beam-energy and centrality dependence of direct-photon emission from ultra-relativistic heavy-ion collisions,” Submitted to: Phys. Rev. Lett. (2018) , arXiv:1805.04084 [hep-ex].
  * [31] PHENIX Collaboration, A. Adare et al., “Low-momentum direct photon measurement in Cu$+$Cu collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV,” Phys. Rev. C98 no.~5, (2018) 054902, arXiv:1805.04066 [hep-ex].
  * [32] PHENIX Collaboration, A. Adare et al., “Centrality dependence of low-momentum direct-photon production in Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV,” Phys. Rev. C91 no.~6, (2015) 064904, arXiv:1405.3940 [nucl-ex].
  * [33] PHENIX Collaboration, S. Afanasiev et al., “Measurement of Direct Photons in Au+Au Collisions at $\sqrt{s_{NN}}=200$ GeV,” Phys. Rev. Lett. 109 (2012) 152302, arXiv:1205.5759 [nucl-ex].
  * [34] PHENIX Collaboration, A. Adare et al., “Detailed measurement of the $e^{+}e^{-}$ pair continuum in $p+p$ and Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV and implications for direct photon production,” Phys. Rev. C81 (2010) 034911, arXiv:0912.0244 [nucl-ex].
  * [35] PHENIX Collaboration, A. Adare et al., “Enhanced production of direct photons in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV and implications for the initial temperature,” Phys. Rev. Lett. 104 (2010) 132301, arXiv:0804.4168 [nucl-ex].
  * [36] PHENIX Collaboration, S. S. Adler et al., “Centrality dependence of direct photon production in s(NN)**(1/ 2) = 200-GeV Au + Au collisions,” Phys. Rev. Lett. 94 (2005) 232301, arXiv:nucl-ex/0503003 [nucl-ex].
  * [37] ATLAS Collaboration, G. Aad et al., “Centrality, rapidity and transverse momentum dependence of isolated prompt photon production in lead-lead collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV measured with the ATLAS detector,” Phys. Rev. C93 no.~3, (2016) 034914, arXiv:1506.08552 [hep-ex].
  * [38] CMS Collaboration, S. Chatrchyan et al., “Measurement of isolated photon production in $pp$ and PbPb collisions at $\sqrt{s_{NN}}=2.76$ TeV,” Phys. Lett. B710 (2012) 256–277, arXiv:1201.3093 [nucl-ex].
  * [39] ALICE Collaboration, J. Adam et al., “Direct photon production in Pb-Pb collisions at $\sqrt{s_{\tiny NN}}=$ 2.76 TeV,” Phys. Lett. B754 (2016) 235–248, arXiv:1509.07324 [nucl-ex].
  * [40] PHENIX Collaboration, V. Khachatryan, “PHENIX measurements of low momentum direct photon radiation,” in 27th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018) Venice, Italy, May 14-19, 2018.  2018\.  arXiv:1812.01841 [nucl-ex].
  * [41] ALICE Collaboration, S. Acharya et al., “Dielectron production in proton-proton collisions at $\sqrt{s}=7$ TeV,” JHEP 09 (2018) 064, arXiv:1805.04391 [hep-ex].
  * [42] ALICE Collaboration, N. Schmidt, “Direct photon production and flow at low transverse momenta in pp, p-Pb and Pb-Pb collisions,” in 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2018 (HP2018) Aix-Les-Bains, Savoie, France, October 1-5, 2018.  2018\.  arXiv:1812.08104 [nucl-ex].
  * [43] C. Gale, S. Jeon, S. McDonald, J.-F. Paquet, and C. Shen, “Centrality dependence of the direct photon multiplicity in heavy ion collisions,” in 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2018 (HP2018) Aix-Les-Bains, Savoie, France, October 1-5, 2018.  2019\.  arXiv:1901.07019 [nucl-th].
  * [44] ALICE Collaboration, S. Acharya et al., “Direct photon elliptic flow in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV,” Phys. Lett. B789 (2019) 308–322, arXiv:1805.04403 [nucl-ex].
  * [45] PHENIX Collaboration, A. Adare et al., “Azimuthally anisotropic emission of low-momentum direct photons in Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV,” Phys. Rev. C94 no.~6, (2016) 064901, arXiv:1509.07758 [nucl-ex].
  * [46] STAR Collaboration, J. Brandenburg, “Measurement of the $\mu^{+}\mu^{-}$ invariant mass spectra in p+p and p+au collisions at $\sqrt{s_{NN}}$ = 200 gev with the muon telescope detector at star,” in 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2018 (HP2018) Aix-Les-Bains, Savoie, France, October 1-5, 2018.  2019\.
  * [47] ALICE Collaboration, S. Acharya et al., “Dielectron and heavy-quark production in inelastic and high-multiplicity proton–proton collisions at $\sqrt{s_{NN}}=$ 13TeV,” Phys. Lett. B788 (2019) 505–518, arXiv:1805.04407 [hep-ex].
  * [48] ALICE Collaboration, S. Acharya et al., “Measurement of dielectron production in central Pb-Pb collisions at $\sqrt{{\textit{s}}_{\mathrm{NN}}}$ = 2.76 TeV,” arXiv:1807.00923 [nucl-ex].
  * [49] ALICE Collaboration, A. Caliva, “Low-mass dilepton measurements with alice at the lhc,” in 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2018 (HP2018) Aix-Les-Bains, Savoie, France, October 1-5, 2018.  2019\.
  * [50] STAR Collaboration, J. Adam et al., “Low-$p_{T}$ $e^{+}e^{-}$ pair production in Au$+$Au collisions at $\sqrt{s_{NN}}$ = 200 GeV and U$+$U collisions at $\sqrt{s_{NN}}$ = 193 GeV at STAR,” Phys. Rev. Lett. 121 no.~13, (2018) 132301, arXiv:1806.02295 [hep-ex].
  * [51] ATLAS Collaboration, M. Aaboud et al., “Observation of centrality-dependent acoplanarity for muon pairs produced via two-photon scattering in Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with the ATLAS detector,” Phys. Rev. Lett. 121 no.~21, (2018) 212301, arXiv:1806.08708 [nucl-ex].
  * [52] CMS Collaboration, A. M. Sirunyan et al., “Evidence for light-by-light scattering and searches for axion-like particles in ultraperipheral PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV,” arXiv:1810.04602 [hep-ex].
  * [53] S. Klein, A. H. Mueller, B.-W. Xiao, and F. Yuan, “Acoplanarity of Lepton Pair to Probe the Electromagnetic Property of Quark Matter,” arXiv:1811.05519 [hep-ph].
  * [54] ALICE Collaboration, M. Arratia, “Measurement of isolated photon-hadron and jet correlations in 5 TeV pp and p-Pb collisions with the ALICE detector at the LHC,” in 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2018 (HP2018) Aix-Les-Bains, Savoie, France, October 1-5, 2018.  2018\.  arXiv:1812.04158 [nucl-ex].
  * [55] STAR Collaboration, N. R. Sahoo, “Semi-inclusive jets recoiling from $\gamma_{dir}$ and $\pi^{0}$ triggers in central Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV in STAR,” arXiv:1812.08045 [nucl-ex].
  * [56] PHENIX Collaboration, J. D. Osborn, “PHENIX results on jet modification with $\pi^{0}$\- and photon-triggered two particle correlations in p+p, p(d)-Au and Au-Au collisions,” in 27th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018) Venice, Italy, May 14-19, 2018.  2018\.  arXiv:1807.05090 [hep-ex].
  * [57] ATLAS Collaboration, M. Aaboud et al., “Measurement of photon–jet transverse momentum correlations in 5.02 TeV Pb + Pb and $pp$ collisions with ATLAS,” Phys. Lett. B789 (2019) 167–190, arXiv:1809.07280 [nucl-ex].
  * [58] CMS Collaboration, A. M. Sirunyan et al., “Observation of medium induced modifications of jet fragmentation in PbPb collisions using isolated-photon-tagged jets,” Phys. Rev. Lett. 121 no.~24, (2018) 242301, arXiv:1801.04895 [hep-ex].
  * [59] CMS Collaboration, A. M. Sirunyan et al., “Jet shapes of isolated photon-tagged jets in PbPb and pp collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV,” Submitted to: Phys. Rev. Lett. (2018) , arXiv:1809.08602 [hep-ex].
  * [60] CMS Collaboration, A. M. Sirunyan et al., “Study of jet quenching with isolated-photon+jet correlations in PbPb and pp collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}=$ 5.02 TeV,” Phys. Lett. B785 (2018) 14–39, arXiv:1711.09738 [nucl-ex].
  * [61] ATLAS Collaboration, T. A. collaboration, “Measurement of the fragmentation function for photon-tagged jets in $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV Pb+Pb and $pp$ collisions with the ATLAS detector,”.
  * [62] ATLAS Collaboration, T. A. collaboration, “Prompt photon production in $\sqrt{s_{\mathrm{NN}}}=8.16$ TeV $p$+Pb collisions with ATLAS,”.
  * [63] CMS Collaboration, C. Collaboration, “Constraints on nuclear parton distributions from W boson production in pPb collisions at $\sqrt{s_{\mathrm{NN}}}=8.16~{}\mathrm{TeV}$,”.
  * [64] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, “EPPS16: Nuclear parton distributions with LHC data,” Eur. Phys. J. C77 no.~3, (2017) 163, arXiv:1612.05741 [hep-ph].
  * [65] ATLAS Collaboration, M. Aaboud et al., “Measurements of $W$ and $Z$ boson production in $pp$ collisions at $\sqrt{s}=5.02$ TeV with the ATLAS detector,” Submitted to: Eur. Phys. J. (2018) , arXiv:1810.08424 [hep-ex].
  * [66] PHENIX Collaboration, A. Adare et al., “Direct photon production in $d+$Au collisions at $\sqrt{s_{NN}}=200$ GeV,” Phys. Rev. C87 (2013) 054907, arXiv:1208.1234 [nucl-ex].
  * [67] C. Shen, J.-F. Paquet, G. S. Denicol, S. Jeon, and C. Gale, “Collectivity and electromagnetic radiation in small systems,” Phys. Rev. C95 no.~1, (2017) 014906, arXiv:1609.02590 [nucl-th].

## Appendix

Figure 6: Left: Summary of (isolated) direct photon invariant cross sections
measured in pp and p$\bar{\mathrm{p}}$ collisions at various collider energies
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26]. Right: Summary of $N_{\mathrm{coll}}$-scaled (isolated) direct photon
invariant yields measured in A-B collisions at various collider energies [27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. The yields from the different
experiments have been scaled by $N_{\mathrm{coll}}$ obtained by the
corresponding collaboration for the respective centrality interval.

Figure 7: Left: Summary of $N_{\mathrm{coll}}$-scaled (isolated) direct photon
invariant yields measured in p(d)-B collisions at various collider energies
[66, 40, 42]. The yields from the different experiments have been scaled by
$N_{\mathrm{coll}}$ obtained by the corresponding collaboration for the
respective multiplicity interval. Right: Summary of $N_{\mathrm{coll}}$ and
$x_{\mathrm{T}}$-scaled (isolated) direct photon invariant yields measured in
p(d)-B collisions at various collider energies using a common $n=4.5$.

Figure 8: Left: Compilation of the direct photon excess ratio ($R_{\gamma}$)
measurements provided by the ALICE collaboration in pp collisions at
$\sqrt{s}~{}=~{}2.76,7$ and $8$ TeV [23, 24, 41]. The vertical errors bars
represent the total uncertainties. Right: Direct photon excess ratio in the
0-20% most central p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV
measured by the ALICE collaboration [42] together with the most recent
predictions from NLO pQCD and hydrodynamic calculations [67].


#  Quasi-normal modes of naked singularities in presence of non-linear scalar
fields

O. S. Stashko  Princeton University, Princeton, NJ 08544  Goethe Universität,
Max-von-Laue Str. 1, Frankfurt am Main, 60438, Germany  O. V. Savchuk
Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI
48824 USA  Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine
V. I. Zhdanov  Taras Shevchenko National University of Kyiv, Ukraine  Igor
Sikorsky Kyiv Polytechnic Institute, Ukraine

###### Abstract

We study linear perturbations against static spherically symmetric background
configurations of General Relativity with a real scalar field (SF), which is
minimally coupled with gravity; it is non-linear due to the presence of the
self-action potential. The background solutions have a naked singularity at
the center of the configuration. The focus is on the stability of the
background and fundamental frequencies of the quasi-normal modes (QNM) of the
axial perturbations in the Regge-Wheeler gauge. The problem is reduced to one
hyperbolic master equation with an effective potential $W_{\rm eff}$, which
turns out to be positive for a general non-negative SF potential; this ensures
the linear stability with respect to this kind of perturbations.

For numerical simulations, the SF potential was chosen in the power-law form
$V(\phi)\sim\phi^{2n}$ with $2<n\leq 40$. We extracted the fundamental
frequencies of QNM for different $n$ and various sets of the background
configuration parameters. The results show that even for a small background
SF, there is a significant difference between the fundamental frequencies and
ones in case of the Schwarzschild background. The results are also compared
with the case of the Fisher-Janis-Newman-Winicour background dealing with a
massless linear scalar field.

Naked singularities, quasi-normal modes, tests of General Relativity

###### pacs:

11

## Introduction

Successful detections of gravitational waves LIGO Scientific Collaboration and
Virgo Collaboration (2016, 2021) and the emerging prospects for multi-
messenger astronomy Ronchini  _et al._ (2022) have opened up new possibilities
to test the gravitational theories in the strong field regime. This stimulated
a surge of attention to damped oscillations of perturbed astrophysical objects
as a source of gravitational radiation. These oscillations, known as quasi-
normal modes Nollert (1999); Kokkotas and Schmidt (1999); Berti  _et al._
(2009); Konoplya and Zhidenko (2011) carry important information about
properties of relativistic objects, which, in particular, can be used, for
example, to distinguish the black holes from the naked singularities arising
in General Relativity (GR) and/or competing theories. In this paper, we
analyze theories with minimally coupled self-interacting scalar field (SF),
which can be considered as simplest modifications of GR. On the other hand,
self-interacting SF is a workhorse of huge number of models dealing with the
inflation of the Universe, dark energy etc.

In GR, if static spherically symmetric configuration contains only
gravitational and scalar fields satisfying the conditions of asymptotic
flatness, then the appearance of a naked singularity (NS) is expected in this
system. An example is the well-known Fisher-Janis-Newman-Winicour (FJNW)
solution Fisher (1948); Janis  _et al._ (1968); Virbhadra (1997). The
existence of NS has been demonstrated in case of a spherically symmetric
static configuration with a fairly general SF potential Zhdanov and Stashko
(2020); Stashko and Zhdanov (2021). Here we are faced with the main question:
whether there are static configurations with SF and NS in our Universe at all.
The no-hair theorems Bekenstein (1972a, b) say that the black hole with a
regular non-zero SF cannot exist; so one can suppose that the scalar hair must
simply dissipate during the collapse into the black hole (BH). This would be
in accordance with the Cosmic Censorship hypothesis Penrose (1965). On the
other hand, there exist solutions in GR and the other theories that describe
NS (see, e.g., Christodoulou (1984); Ori and Piran (1987); Joshi and Dwivedi
(1993); Joshi  _et al._ (2013); Ong (2020)), though, how they can be formed in
our Universe is still a moot point. In this regard, it is appropriate to look
for observational effects that can distinguish between normal black holes and
their mimickers that contain NS.

The quasinormal modes (QNM) of the space-time vibrations emerging around
relativistic objects can be good probes of different gravitational theories.
Much attention has been paid to perturbations of the black holes and regular
compact astrophysical objects (see, e.g., reviews Kokkotas and Schmidt (1999);
Berti  _et al._ (2009); Konoplya and Zhidenko (2011)). QNM near NS have been
studied in Chirenti  _et al._ (2012); Khamesra and Suneeta (2014); Santos  _et
al._ (2019); Chowdhury and Banerjee (2020); Chowdhury  _et al._ (2022); here,
most works essentially deal with spherically symmetric background space-times,
where an analytic representation of a background metric is available, such as
in the Reissner-Nordstrom Chirenti  _et al._ (2012); Santos  _et al._ (2019)
and FJNW Chowdhury and Banerjee (2020) cases. Obviously, considerations of the
linear perturbations are deeply connected with the stability analysis (see,
e.g., Khamesra and Suneeta (2014); Chowdhury  _et al._ (2022); Dotti (2022)
and references therein).

In the present paper, we study axial perturbations against the background
formed by a isolated static spherically symmetric configuration of GR in
presence of a minimally-coupled non-linear real SF. Our findings extend the
results of Chowdhury et al. Chowdhury and Banerjee (2020), which deal with
pure FJNW (massless linear case). In our case, the nonlinearity is introduced
using a general SF-potential $V(\phi)$ satisfying the condition $\phi
V^{\prime}(\phi)\geq 0$, which grows more slowly that exponent for
$|\phi|\to\infty$. In the numerical simulations we are dealing with the power-
law SF potential.

In Section I the basic relations for static spherically symmetric background,
are presented. In Section II we turn to numerical methods for the case of the
power-law SF potential. We use the method of backwards integration Zhdanov and
Stashko (2020); Stashko  _et al._ (2021) to obtain the solutions of Einstein +
SF equations for the background. In Section III we use the time-domain
integration method Chirenti and Rezzolla (2007) to solve the wave equations
for the perturbations. Given these solutions, we use the Prony method Berti
_et al._ (2007) to extract the frequencies of the fundamental QNM. The final
section summarizes the results.

##  I Analytic relations for general SF potential

The action of standard GR in the presence of a minimally-coupled real SF
$\phi$ is given by111Units: $c=1$; the metric signature is (+ - - -),
$R^{\alpha}_{\,\,\beta\gamma\delta}=-R^{\alpha}_{\,\,\beta\delta\gamma}=\partial_{\gamma}\Gamma^{\alpha}_{\beta\delta}-...\,;\,\,R_{\mu\nu}=R^{\alpha}_{\,\,\mu\alpha\nu}$.
Below we rescale units of mass, length and SF so as to put $G=1$ and to remove
a coupling constant in the SF potential.

$S=\int d^{4}\,x\sqrt{-g}\left(-\frac{R}{16\pi
G}+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right)\,.$ (1)

The background metric describes the static spherically symmetric space-time in
Schwarzschild-like (curvature) coordinates:

$ds^{2}=e^{\alpha(r)}dt^{2}-e^{\beta(r)}dr^{2}-r^{2}d\Omega^{2},$ (2)

where $d\Omega^{2}=d\theta^{2}+(\sin\theta)^{2}d\varphi^{2}$ and the radial
variable $r>0$.

In static spherically symmetric case we deal with two independent first-order
Einstein equations

$\frac{d}{dr}\left[r\left(e^{-\beta}-1\right)\right]=-8\pi
r^{2}[e^{-\beta}\phi^{\prime 2}/2+V(\phi)],$ (3)
$re^{-\beta}\frac{d\alpha}{dr}+e^{-\beta}-1=8\pi r^{2}[e^{-\beta}\phi^{\prime
2}/2-V(\phi)],$ (4)

and one second order equation for SF $\phi=\phi(r)$

$\frac{d}{dr}\left[r^{2}e^{\frac{\alpha-\beta}{2}}\frac{d\phi}{dr}\right]=r^{2}e^{\frac{\alpha+\beta}{2}}V^{\prime}(\phi)\,.$
(5)

The equations (3, 4, 5) define the background solutions under appropriate
asymptotic conditions.

A small perturbation of the background solution can be considered in the
linear approximation. The space-time metric and SF then:

$\displaystyle g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu},\leavevmode\nobreak\
\phi(x^{\mu})=\phi_{0}(r)+\delta\phi(x^{\mu}),$ (6)

where $g_{\mu\nu}^{(0)}$ and $\phi_{0}(r)$ and represent the static
spherically symmetric solution of (3-5), while
$h_{\mu\nu}=h_{\mu\nu}(x^{\mu})$ and $\delta\phi(x^{\mu})$ represent the
perturbations. There is extensive literature on the gravitational
perturbations against spherically symmetric background (see, e.g.,
Chandrasekhar (1998); Berti  _et al._ (2009)).

The perturbation can be separated into axial and polar parts222In this case
“axial” and “polar” Berti  _et al._ (2009); Chandrasekhar (1998) corresponds
to “odd” and “even” according to the initial paper Regge and Wheeler (1957).,
which can be treated independently. In this paper, we focus solely on the
axial perturbations. The metrics of the axial gravitational perturbations
$h_{\mu\nu}$ in the Regge-Wheeler gauge Regge and Wheeler (1957) take the
following form

$\displaystyle
h^{axial}_{\mu\nu}=\left[\begin{array}[]{cccc}0&0&0&h_{0}(t,r)\\\
0&0&0&h_{1}(t,r)\\\ 0&0&0&0\\\
h_{0}(t,r)&h_{1}(t,r)&0&0\end{array}\right]\left(\sin\theta\frac{\partial}{\partial\theta}\right)P_{l}(\cos\theta)\,,$
(11)

where $h_{0}(t,r)$ and $h_{1}(t,r)$ are two unknown functions, and $P_{l}(x)$
is the Legendre polynomial333Due to the spherical symmetry of the background
configuration, more general case that involves $Y_{lm}$, leads to the same
master equation and the same QNM frequencies. The perturbations with $l=1$ can
be removed by gauge transformation Vishveshwara (1970).  with $l\geq 2$. After
substitution of (11) to the Einstein-SF equations and retaining only linear
terms, we obtain:

$\displaystyle-{h_{0}^{\prime}}\left(\alpha^{\prime}+\beta^{\prime}\right)+2h_{0}^{\prime\prime}-{h_{0}}\left(\frac{2{l(l+1)}}{r^{2}}e^{\beta}+2\alpha^{\prime\prime}-\alpha^{\prime}\beta^{\prime}+\alpha^{\prime
2}\right)+\dot{h_{1}}\left(\alpha^{\prime}+\beta^{\prime}-\frac{4}{r}\right)-2\dot{h_{1}^{\prime}}=0,$
(12)
$\displaystyle\dot{h_{0}^{\prime}}-\frac{2}{r}\dot{h_{0}}-\ddot{h_{1}}-\frac{l(l+1)-2}{r^{2}}e^{\alpha}{h_{1}}=0,$
(13) $\displaystyle
2{h_{1}^{\prime}}+{h_{1}}\left(\alpha^{\prime}-\beta^{\prime}\right)-2e^{\beta-\alpha}\dot{h_{0}}=0.$
(14)

In agreement with Kobayashi  _et al._ (2012), the perturbed part of SF equals
to zero

$\displaystyle\delta\phi=0.$ (15)

Equation (12) is redundant, because for perturbations $\sim e^{-i\omega t}$ it
can be obtained from (13, 14).

After substitution

$\displaystyle h_{1}(t,r)=re^{(\beta-\alpha)/2}\Psi(t,r),$ (16)

and introducing the “tortoise” coordinate $r^{*}$

$\displaystyle\frac{dr^{*}}{dr}=e^{(\beta-\alpha)/2}\,,\quad r^{*}(0)=0\,,$
(17)

we can combine (13, 14) into a single master equation

$\left(\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial
r^{*}{}^{2}}\right)\Psi(t,r^{*})+W_{\rm eff}(r,l)\Psi(t,r^{*})=0,$ (18)

where

$W_{\rm
eff}(r,l)=e^{\alpha-\beta}\left(\frac{\beta^{\prime}-\alpha^{\prime}}{2r}+e^{\beta}\frac{(l-1)(l+2)}{r^{2}}+\frac{2}{r^{2}}\right),$
(19)

will be referred to as the “wave potential”.

Using equations (3, 4), we can rewrite (19) as

$W_{\rm eff}=e^{\alpha}\left[8\pi
V(\phi)+\frac{(l-1)(l+2)-1}{r^{2}}\right]+\frac{3}{r^{2}}e^{\alpha-\beta}\,.$
(20)

Evidently, $\forall r$: $W_{\rm eff}(r)>0$, for $l\geq 2$ in case of an
arbitrary non-negative SF potential. This leads to the absence of bound states
and stability of solutions of equation (18).

Under the condition of convergence of the integrals involved and for

$\lim_{r^{*}\to 0+0}\left[\frac{\partial\Psi(t,r^{*})}{\partial
t}\frac{\partial\Psi(t,r^{*})}{\partial r^{*}}\right]=0\quad{\rm
and}\quad\lim_{r^{*}\to\infty}\left[\frac{\partial\Psi(t,r^{*})}{\partial
t}\frac{\partial\Psi(t,r^{*})}{\partial r^{*}}\right]=0\,,$ (21)

we have a conserved quantity

$E(t)=\frac{1}{2}\int\limits_{0}^{\infty}dr^{*}\left\\{\left(\frac{\partial\Psi}{\partial
t}\right)^{2}+\left(\frac{\partial\Psi}{\partial r^{*}}\right)^{2}+2W_{\rm
eff}(\phi)\Psi^{2}\right\\}.$ (22)

So far, we have formally dealt with arbitrary $V(\phi)$. Now, we need more
detailed information concerning the properties of the system near the singular
center. To do so, we impose the conditions $V(0)=0$, $\phi
V^{\prime}(\phi)\geq 0$ as in Zhdanov and Stashko (2020), and
$|V^{\prime}(\phi)|\leq C\exp(\kappa|\phi|^{p})$, where $C,\kappa,p$ are
positive constants, and $p<1$. Under the latter condition, the terms with
$V(\phi)$ in (3,4,5) do not directly contribute into the first two orders of
the asymptotic expansion of $\phi,\alpha,\beta$ for $r\to 0$ yielding

$\phi(r)=-\xi\ln\left(\frac{r}{r_{g}}\right)+\phi_{0}+O(r^{\eta+1})\,,$ (23)
$\alpha(r)=(\eta-1)\ln\left(\frac{r}{r_{g}}\right)+\alpha_{0}+O(r^{\eta+1}),\quad\beta(r)=(\eta+1)\ln\left(\frac{r}{r_{g}}\right)+\beta_{0}+O(r^{\eta+1}),$
(24)

where $\eta=4\pi\xi^{2}>0$, and the constants
$\xi\,,\eta,\alpha_{0}\,,\beta_{0}$ and $\phi_{0}$, which depend on $Q,M,n$,
have been used to substantiate the asymptotic of $W_{\rm eff}(r^{*})$ below.
The numerical values can be determined for the explicit form of $V(\phi)$.
Note that the the conditions imposed on $V$ are satisfied in case of the
power-law potential considered below.

Taking into account (23, 24), we get asymptotic relations near the singularity
($r\to{0}$)

$r_{*}=\frac{r^{2}}{2r_{g}}e^{(\beta_{0}-\alpha_{0})/2}\left[1+O\left(r^{\eta+1}\right)\right],\quad
W_{\rm
eff}(r)=\frac{3e^{\alpha_{0}-\beta_{0}}r_{g}^{2}}{r^{4}}\left[1+O\left(r^{\eta}\right)\right],\quad
W_{\rm
eff}(r^{*})=\frac{3}{4r^{*}{}^{2}}\left[1+O\left({r^{*}{}^{\eta/2}}\right)\right].$
(25)

In view of (25) for $r^{*}\to 0$, we see that the asymptotic properties of the
problem near the naked singularity are essentially the same as in the FJNW
case. Therefore, one can use considerations of Chowdhury and Banerjee (2020)
to justify the choice of the null Dirichlet boundary condition for $r^{*}\to
0$, which we use in the computed simulation below.

On account of (25), there are two kinds of wave solutions of (18) with
asymptotics $\sim r^{*}{}^{-1/2}$ and $\sim r^{*}{}^{3/2}$ as $r^{*}\to 0$. We
choose the latter yielding regular $h_{0}$ and $h_{1}$. Having this in mind,
consider $C^{(2)}$-solutions $\Psi(t,r^{*})$ of (18) defined on
$[0,T)\otimes(0,\infty)$, ($T>0$), and assume that for each $t$, solution
$\Psi(t,\cdot)\in S$: $S\subset C^{(2)}(0,\infty)$ is a space of functions
$f(x)$, such that (i) $f(0)=0$, (ii) $f(x)$ and $df/dx$ decay fast enough as
$x\to\infty$, so that the integrals in (22) converge and $E(t)$ is well
defined. Because $E(t)$ is conserved, this ensures the uniqueness of solutions
of (18) with initial data from $S$ and a linear stability with respect to
axial perturbations. This justifies the null boundary condition at the center,
used below in the numerical calculations.

##  II Power-law potential

Scalar field self-interation in the form of the power-law potential is
considered:

$V(\phi)=V_{0}|\phi|^{2n},$ (26)

where $V_{0}>0$ and $n>2$. The latter condition means that we restrict
ourselves to the case of the massless SF with the Coulomb behavior at spatial
infinity, which requires $n>2$ Zhdanov and Stashko (2020); Stashko  _et al._
(2021); otherwise the asymptotic condition for $\phi$ will be different.
Because the solution $\phi(r)$ of (5) has no zeros Zhdanov and Stashko (2020),
we can choose the sign by setting $\phi(r)>0$. From now on the units are set
to $G=1$ and $V_{0}=1$.

We are interested in isolated configurations with spatially-flat asymptotics,
which satisfy the conditions

$\lim_{r\to\infty}[r(e^{\alpha}-1)]=\lim_{r\to\infty}[r(e^{-\beta}-1)]=-r_{g},\quad\lim_{r\to\infty}\left[r\phi(r)\right]=Q\,,$
(27)

where $r_{g}=2M$ and $M>0$ is the configuration mass, $Q$ is the “scalar
charge”. The global and asymptotic properties of the solutions satisfying (27)
have been studied in Zhdanov and Stashko (2020); Stashko  _et al._ (2021). For
fixed $n>2$ the solution of system (3–5) is uniquely defined by parameters $M$
and $Q$ Stashko  _et al._ (2021). There exists the naked singularity at the
center ($r=0$); the asymptotic behaviour of the functions $\alpha(r),\beta(r)$
and $\phi(r)$ is (23, 24). We use the backwards integration starting from a
sufficiently large initial radius $r_{\rm in}$ to smaller $r$ up to the
origin. The “initial” conditions at $r_{\rm in}$ were specified using the
asymptotic relations Zhdanov and Stashko (2020); Stashko  _et al._ (2021) for
$r\to\infty$, $n>2$ that can be inferred from (27):

$e^{\alpha}=\left(1-\frac{r_{g}}{r}\right)\left[1+O\left(\frac{\mu(r)}{r^{2}}\right)\right],\leavevmode\nobreak\
e^{-\beta}=\left(1-\frac{r_{g}}{r}\right)\left[1+\frac{4\pi
Q^{2}}{r^{2}}+O\left(\frac{\mu(r)}{r^{2}}\right)\right]\,,$ (28)
$\phi(r)=\frac{Q}{r}\left[1+\frac{r_{g}}{2r}+\frac{n|Q|^{2n-2}}{(n-2)(2n-3)r^{2n-4}}+O\left(\frac{\mu(r)}{r^{2}}\right)\right]\,.$
(29)

where $\mu(r)=1/r$ for $n\geq 3$ and $\mu(r)=1/r^{2n-4}$ for $2<n<3$. This
yields for $r\to{\infty}$

$\displaystyle r^{*}=r+r_{g}\ln(r/r_{g})+O\left(\frac{1}{r}\right),\quad
W_{\rm eff}(r)=\frac{l(l+1)}{r^{2}}+O\left(\frac{1}{r^{3}}\right),\quad W_{\rm
eff}(r^{*})=\frac{l(l+1)}{r^{*}{}^{2}}\left(1+O\left(\frac{\ln
r^{*}}{r^{*}}\right)\right).$ (30)

In our numerical calculations we have considered $r_{\rm in}=10^{5}$. Examples
of the qualitative behavior of the solutions are shown in Fig. 1 for some
values of $Q$. The scalar field $\phi(r)$ decreases monotonically for all
possible values of $M,Q,n$. As for the metric coefficients, $e^{\alpha(r)}$
increases monotonically from zero, whereas $e^{\beta(r)}$ increases form zero
to some maximum and then decreases to $1$. For large $Q$ the maximum of
$e^{\beta(r)}$ is almost invisible, but for small $Q\to 0$ the maximum of
$e^{\beta}$ increases indefinitely. Also, for $Q\to 0$ the metric in the
region $r>r_{g}$ tends to the Schwarzschild values, so all physical effects
occurring in the this region will be indistinguishable from those of the
Schwarzschild black hole. On the other hand, in the inner region ($r<r_{g}$),
there is a cardinal difference from the Schwarzschild black hole even for
small $Q$. This gives us a motivation to look for observational effects
associated with the inner region, that may arise in the analysis of
perturbations.

Figure 1: Typical behaviour of the background solutions of Einstein–SF
equations. The examples are given for $M=1$, $n=3$ and (a) $Q=0.5$; (b)
$Q=0.3$, (c) $Q=0.2$. For smaller $Q$, the metric gets closer to the abscissa
near the origin and approach the Schwarzschild values for $r>r_{g}$. Right
panel shows how $|r\,d\phi/dr|$ tends to asymptotic value $\xi$ of (23).
Figure 2: Typical behaviour of the effective potential $W_{\rm eff}(r,l=2)$:
(a) $M=1$, $n=3$ for different Q; (b) $Q=0.25$, $M=1$, different $n$; (c)
$Q=1$, $n=3$, different $M$.

##  III Numerical solutions to the master equation and frequencies of
fundamental QNM

Our aim is to extract the fundamental QNM frequencies444Here the fundamental
$\omega$ is that of QNM with the least damping, that is with the minimal value
$\omega_{I}=\text{Im}(\omega)$ of the axial space-time perturbations governed
by (18). To do this, we solve this equation keeping in mind that the QNM
frequencies reflect the internal properties of the background configuration;
they do not depend on the form of the initial perturbation, which can be
chosen with a large degree of arbitrariness.

The background configuration manifests itself by means of the wave potential
in (18), which depends on the configuration parameters. Typical behaviour of
$W_{\rm eff}$ is shown in Fig. 2 for different $Q$, $M$ and $n$, in panels
(a), (b), (c), respectively. For a certain set of parameters, the effective
potential can have a local maximum, which can disappears with increasing $Q$
or decreasing $M$.

In order to integrate the master equation (18) and analyze the QNMs spectrum
we use the time-domain integration method Konoplya and Zhidenko (2011).

It is convenient to rewrite the wave equation (18) in terms of the null
coordinates $u=t-r^{*}$, $v=t+r^{*}$:

$\displaystyle 4\frac{\partial^{2}}{\partial u\partial v}\Psi(u,v)+W_{\rm
eff}(u,v,l)\Psi(u,v)=0.$ (31)

Taking into account the remarks at the end of Section I in connection with
asymptotic relations (25), following Wald (1980), we impose the null Dirichlet
condition at $r^{*}=0$:

$\Psi(u=v,v)=0\,.$ (32)

Assuming that the ringtone frequencies are not sensitive to the form of the
initial perturbation we chose a condition on characteristic $u=0$
corresponding to the Gaussian wave packet:

$\Psi(u=0,v)=\exp\left[-\frac{(v-v_{c})^{2}}{2\sigma^{2}}\right]\,,$ (33)

centered at $v=v_{c}=50$, the width $\sigma=1$.

Figure 3: Typical behaviour of $|\Psi(t,r^{*}=200)|$ for $l=2$, $n=3$, $M=1$.
The left panel shows QNM signal with echo noise due to the existence of the
potential peak. For $Q=0.15$ (blue graph) the echo dominates for $t\lesssim
500$; then the ringdown dominates. The blue and yellow curves on panel (b)
show, how after the ringdown time we have typical power-law tails. The panel
(c) also illustrates that for larger values of $Q$, the values of $\omega_{R}$
and $\omega_{I}$ becomes smaller and then, after finishing oscillations we
have power-law tails.

Figure 4: Fundamental frequencies $\omega$ as functions of $Q$ for three $n$
values (indicated in the upper right corner of the right panel); $M=1$, $l=2$.
Solid blue curves correspond to the FJNW case. Left panel: trajectories
$\omega(Q)$ in the complex plane; black arrow shows the direction of
increasing $Q$ and the numbers near corresponding points of the dashed and
solid blue curves indicate the values of $Q$; the blue star shows the
Schwarzschild case. Right panel shows separately $\omega_{R}(Q)$ and
$\omega_{I}(Q)$; the horizontal dashed lines correspond to the Schwarzschild
case.

Figure 5: The same as on Fig. 4 for $M=1$, $l=3$.

Figure 6:  Fundamental frequencies $\omega$ as functions of $M$ for different
$n$ values (indicated in the right lower corner of the right panel); $l=2$,
$Q=1$. Left panel: trajectories $\omega(M)$ in the complex plane; black arrow
shows the direction of increasing $M$ and the numbers near the corresponding
points of the dashed and solid blue curves indicate the values of $M$; the
blue star shows the Schwarzschild case. Right panel shows separately
dependencies $\omega_{R}(M)$ and $\omega_{I}(M)$; the horizontal dashed lines
correspond to the Schwarzschild case.

Figure 7: The same as on Fig. 6 for $Q=1$, $l=3$.

Figure 8: The fundamental frequencies $\omega$ in the complex plane
$\omega_{R}\times\omega_{I}$ as functions of $n$ for $l=2$. Left panel:
trajectories $\omega(n)$ for fixed $M=1$ and several different $Q$; the
numbers near the corresponding points of some curves indicate the values of
$n$. Right panel: the same for $Q=1$ and several different $M$. Black arrows
show the direction of increasing $n$. As $n$ increases, $\omega(n)$ approach
FJNW values (orange stars); however, for sufficiently large $n$ we found
typical small features shown in the insets.

To obtain numerical solutions, we use a second-order discretization scheme
proposed by Chirenti and Rezzolla Chirenti and Rezzolla (2007).

$\displaystyle\Psi_{N}=(\Psi_{W}+\Psi_{E})\frac{16-\Delta^{2}W_{\rm
eff}(S)}{16+\Delta^{2}W_{\rm eff}(S)}-\Psi_{S}+O(\Delta^{4}),$ (34)

where the indices $(N,W,E,S)$ correspond to the points of the space-time
triangular grid and as follows: $N=(u+\Delta,v+\Delta)$, $W=(u+\Delta,v)$,
$E=(u,v+\Delta)$, $S=(u,v)$; $\Delta$ is the spacing between the grid points.

To extract fundamental frequencies $\omega=\omega_{R}+i\omega_{I}$, we fitted
the time-domain profiles for a sufficiently large times and fixed $r^{*}$ by a
sum of complex exponentials
$\Psi(t)\simeq\sum\limits_{j=1}^{p}A_{j}e^{-i\omega_{j}t}$ using the Prony
method Berti  _et al._ (2007).

Some typical examples of numerical solutions of (31) are presented on Fig. 3.
The left panel (a) of this figure shows situation with small $Q$, when there
is a series of echoes. The ringdown profile of damped QNM can be retrieved at
a later time when the echo contribution is negligible. The series of the
ringdown oscillations is followed by a power-law damping.

The derived fundamental QNM frequencies are presented in Tables 1–2. We
plotted the trajectories of these frequencies in the complex $\omega$-plane as
functions of one of the parameters $M,Q,n$ with other parameters fixed (Figs.
5– 8). As can be seen from Figs. 4, 5, for smaller $Q$ the fundamental
frequencies get closer the FJNW values for all fixed $n$ and $M$. Analogously,
Figs. 6, 7 show how $\omega$ tend to the FJNW value as $M$ grows.

As $n$ grows, it seems that the fundamental frequencies approach FJNW ones
(see Fig. 8). However, for large $n$ we found a small feature shown in the
inserts in Fig.8. To be sure that this is not an artefact of calculations, we
have checked the occurrence of analogous features for different choices of the
parameters. We explain these features by an influence of the region near the
singularity. In fact, the QNM are formed mainly far from the center; in this
region, for large $n$ the wave potential (18) is almost zero, as in the FJNW
case. On the other hand, the effect of the region where $|\phi(r)|>1$, becomes
more tangible as $n$ grows, but it is suppressed due to the null boundary
condition (32); that is why these features are almost imperceptible.

##  IV Discussion

We have studied axial linear perturbations against static spherically
symmetric asymptotically flat background described by the Einstein – SF
equations with the power-law SF potential. The background solutions
necessarily have NS at the origin. The characteristic behavior of the
perturbations induced by an initial pulse, has three main stages: (i) possible
domination of echo-signals, (ii) the ringdown stage followed by (iii)
monotonous decay. The role and duration of these stages differ depending on
the configuration parameters affecting the wave potential (20); for example,
sometimes the stage (i) may be substantially absent. We were mostly interested
in stage (ii) of the QNM domination yielding the fundamental frequencies of
these modes. The results of the numerical simulations are presented in Tables
1, 2; they are used in Figs. 4-8 for different sets of the background
configuration parameters.

We also draw attention to the analytic result described by equations (18, 20)
that have been used for the general SF potential $V(\phi)\geq 0$ to study
stability issues. In particular, we used this in case of the rather general
$V(\phi)$, when the background solutions have an appropriate asymptotic near
the naked singularity. Of course, this does not yet solve the entire problem
of the stability of the NS, since we have limited ourselves to axial
perturbations. The polar perturbations deserve a separate study and we plan to
consider this issue elsewhere.

The presence of the non-zero SF potential manifests itself in the values
$\omega$ of the fundamental QNM frequencies that are different for different
background configuration parameters. There is also a considerable difference
from the FJNW values. However, the qualitative behavior of trajectories in the
$\omega$-plane is rather similar to the FJNW curves. For fixed $M$, the
$\omega$-trajectories get closer to the FJNW curves in the intervals of
smaller $Q$ (Figs. 4 \- 5). Therefore, for smaller SF the influence of the
power-law potential decreases. A similar trend is also observed for a fixed
$Q$, when $M$ gets larger (Figs. 6 \- 7). For large $n$, the wave potential
and $\omega$-trajectories can be well approximated by the FJNW curves;
nevertheless the nonlinear effects manifest themselves though the small
features that are almost imperceptible. On the whole, the FJNW case
satisfactory describes the qualitative behavior of $\omega$-trajectories. Of
course, this cannot be said about the numerical values of the fundamental QNM
frequencies.

If SF is small enough, its influence in the outer region ($r>r_{g}$) is
becomes negligible. In particular, this concerns the effects in motion of
particles and photons, structure of accretion disks etc. On the other hand, it
is important to note that in the region $r<r_{g}$, the significant differences
from the Schwarzschild case persist even for small $Q$. This can be explained
by different boundary conditions: the Dirichlet condition at the naked
singularity in presence of SF versus the in-going wave condition at the black
hole horizon in the Schwarzschild case. The boundary condition at the origin
feels the difference of NS from the black hole even when SF is arbitrarily
small. Thus, the fundamental frequencies of the configuration considered,
differ qualitatively and numerically from those in case of the Schwarzschild
solution even for a small scalar field. One can suggest that this is a fairly
general property of static systems with the scalar field leading to the
occurrence of naked singularity.

###### Acknowledgements.

We are indebted to Luciano Rezzolla for helpful discussions. The work of
V.I.Z. was supported by the National Research Foundation of Ukraine under
Project No. 2020.02/0073.

##  V Appendix: tables

Table 1: The fundamental quasinormal mode $M\omega$ of the NS with nonlinear SF ($M=1$, $n=(2.1,3)$) and FJNW for various values of $Q$. The first line corresponds to the Schwarzschild BH. | $n=2.1$ | $n=3$ | FJNW
---|---|---|---
$Q$ | $l=2$ | $l=3$ | $l=2$ | $l=3$ | $l=2$ | $l=3$
$0$ | $0.3730-0.0891i$ | $0.5993-0.0927i$ | $0.3730-0.0891i$ | $0.5993-0.0927i$ | $0.3730-0.0891i$ | $0.5993-0.0927i$
$0.2$ | $0.4303-0.2425i$ | $0.6917-0.2536i$ | $0.3815-0.0077i$ | $0.489-0.0002i$ | $0.3683-0.0052i$ | $0.4643-8.881\cdot 10^{-5}i$
$0.25$ | $0.2172-0.2558i$ | $0.4005-0.322i$ | $0.4639-0.04798i$ | $0.6413-0.01445i$ | $0.45481-0.0381i$ | $0.6196-0.0084i$
$0.3$ | $0.0874-0.1371i$ | $0.1731-0.1748i$ | $0.5051-0.112i$ | $0.7293-0.0701i$ | $0.5042-0.09739i$ | $0.7179-0.0534i$
$0.35$ | $0.0346-0.0573i$ | $0.0692-0.0712i$ | $0.5131-0.1799i$ | $0.7703-0.1484i$ | $0.5213(5)-0.166i$ | $0.7723-0.1268i$
$0.45$ | $0.0066-0.0112i$ | $0.01323-0.01374i$ | $0.4682-0.2874i$ | $0.7567-0.2992i$ | $0.4888-0.2839i$ | $0.7824-0.286i$
$0.55$ | $0.00194-0.00335i$ | $0.0039-0.0041i$ | $0.3933-0.3449i$ | $0.6803-0.3998i$ | $0.4161-0.3558i$ | $0.7172-0.4067i$
$0.65$ | $0.00081-0.0014i$ | $0.0016-0.0017i$ | $0.3203-0.3653i$ | $0.5898-0.4512i$ | $0.3388-0.388i$ | $0.6269-0.4791i$
Table 2: The fundamental quasinormal mode $M\omega$ of the NS with nonlinear SF ($M=1$, $Q=(0.15,\leavevmode\nobreak\ 0.3,\leavevmode\nobreak\ 0.45)$) and FJNW for various values of $n$. The first and second lines correspond to the Schwarzschild BH and FJNW solutions, respectively.  | $Q=0.15$ | $Q=0.3$ | $Q=0.45$
---|---|---|---
$n$ | $l=2$ | $l=3$ | $l=2$ | $l=3$ | $l=2$ | $l=3$
$Schw$ | $0.3730-0.0891i$ | $0.5993-0.0927i$ | $0.3730-0.0891i$ | $0.5993-0.0927i$ | $0.3730-0.0891i$ | $0.5993-0.0927i$
FJNW | $0.2436-0.00006i$ | $-$ | $0.5042-0.0974i$ | $0.7183-0.0535i$ | $0.4888-0.2839i$ | $0.7824-0.286i$
2.05 | $0.467-0.1918i$ | $0.721-0.1773i$ | $0.0073-0.0123i$ | $0.0146-0.0151i$ | $0.0006-0.001i$ | $0.0012-0.0013i$
2.1 | $0.4517-0.0484i$ | $0.6305-0.0158i$ | $0.0876-0.1369i$ | $0.173-0.1748i$ | $0.0066-0.0112i$ | $0.01324-0.0137i$
2.3 | $0.3178-0.0012i$ | $-$ | $0.4698-0.2222i$ | $0.7352-0.2153i$ | $0.2654-0.2943i$ | $0.4839-0.3661i$
2.5 | $0.2802-0.00029i$ | $-$ | $0.5016-0.1558i$ | $0.7475-0.1226i$ | $0.3952-0.306i$ | $0.6675-0.3476i$
3 | $0.2532-0.00009i$ | $-$ | $0.5052-0.112i$ | $0.7293-0.0701i$ | $0.4682-0.2874i$ | $0.7567-0.2992i$
4 | $0.245-0.00007i$ | $-$ | $0.5043-0.0995i$ | $0.7201-0.056i$ | $0.4867-0.2837i$ | $0.7792-0.2868i$
5 | $0.2439-6.318\cdot 10^{-5}i$ | $-$ | $0.5042-0.0979i$ | $0.7188-0.0541i$ | $0.4884-0.2838i$ | $0.7817-0.2861i$
7 | $0.2436-6.242\cdot 10^{-5}i$ | $-$ | $0.5042-0.0976i$ | $0.7184-0.0536i$ | $0.4888-0.2839i$ | $0.7823-0.286i$
10 | $0.2436-6.234\cdot 10^{-5}i$ | $-$ | $0.5042-0.0975i$ | $0.7184-0.0535i$ | $0.4888-0.2839i$ | $0.78235-0.286i$

## References

  * LIGO Scientific Collaboration and Virgo Collaboration (2016) LIGO Scientific Collaboration and Virgo Collaboration, Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * LIGO Scientific Collaboration and Virgo Collaboration (2021) LIGO Scientific Collaboration and Virgo Collaboration, arXiv e-prints , arXiv:2108.01045 (2021), arXiv:2108.01045 [gr-qc] .
  * Ronchini  _et al._ (2022) S. Ronchini, M. Branchesi, G. Oganesyan, B. Banerjee, U. Dupletsa, G. Ghirlanda, J. Harms, M. Mapelli, and F. Santoliquido, A&A 665, A97 (2022), arXiv:2204.01746 [astro-ph.HE] .
  * Nollert (1999) H.-P. Nollert, Classical and Quantum Gravity 16, R159 (1999).
  * Kokkotas and Schmidt (1999) K. D. Kokkotas and B. G. Schmidt, Living Reviews in Relativity 2, 2 (1999).
  * Berti  _et al._ (2009) E. Berti, V. Cardoso, and A. O. Starinets, Classical and Quantum Gravity 26, 163001 (2009), arXiv:0905.2975 [gr-qc] .
  * Konoplya and Zhidenko (2011) R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011).
  * Fisher (1948) I. Z. Fisher, Zh. Exp. Theor. Phys., 18, 636 (1948), arXiv:gr-qc/9911008 [gr-qc] .
  * Janis  _et al._ (1968) A. I. Janis, E. T. Newman, and J. Winicour, Phys. Rev. Lett. 20, 878 (1968).
  * Virbhadra (1997) K. S. Virbhadra, International Journal of Modern Physics A 12, 4831–4835 (1997).
  * Zhdanov and Stashko (2020) V. I. Zhdanov and O. S. Stashko, Phys. Rev. D 101, 064064 (2020).
  * Stashko and Zhdanov (2021) O. Stashko and V. I. Zhdanov, Galaxies 9, 72 (2021), arXiv:2109.01931 [gr-qc] .
  * Bekenstein (1972a) J. D. Bekenstein, Physical Review Letters 28, 452 (1972a).
  * Bekenstein (1972b) J. D. Bekenstein, Physical Review D 5, 1239 (1972b).
  * Penrose (1965) R. Penrose, Physical Review Letters 14, 57 (1965).
  * Christodoulou (1984) D. Christodoulou, Communications in Mathematical Physics 93, 171 (1984).
  * Ori and Piran (1987) A. Ori and T. Piran, Physical Review Letters 59, 2137 (1987).
  * Joshi and Dwivedi (1993) P. S. Joshi and I. H. Dwivedi, Physical Review D 47, 5357 (1993).
  * Joshi  _et al._ (2013) P. S. Joshi, D. Malafarina, and R. Narayan, Classical and Quantum Gravity 31, 015002 (2013).
  * Ong (2020) Y. C. Ong, International Journal of Modern Physics A 35, 2030007 (2020).
  * Chirenti  _et al._ (2012) C. Chirenti, A. Saa, and J. Skákala, Phys. Rev. D 86, 124008 (2012), arXiv:1206.0037 [gr-qc] .
  * Khamesra and Suneeta (2014) B. Khamesra and V. Suneeta, Phys. Rev. D 90, 024044 (2014), arXiv:1405.6821 [gr-qc] .
  * Santos  _et al._ (2019) E. C. Santos, J. C. Fabris, and J. A. de Freitas Pacheco, arXiv e-prints , arXiv:1903.04874 (2019), arXiv:1903.04874 [gr-qc] .
  * Chowdhury and Banerjee (2020) A. Chowdhury and N. Banerjee, Phys. Rev. D 102, 124051 (2020).
  * Chowdhury  _et al._ (2022) A. Chowdhury, S. Devi, and S. Chakrabarti, Phys. Rev. D 106, 024023 (2022).
  * Dotti (2022) G. Dotti, Universe 8, 38 (2022).
  * Stashko  _et al._ (2021) O. S. Stashko, V. I. Zhdanov, and A. N. Alexandrov, Phys. Rev. D 104, 104055 (2021), arXiv:2107.05111 [gr-qc] .
  * Chirenti and Rezzolla (2007) C. B. M. H. Chirenti and L. Rezzolla, Classical and Quantum Gravity 24, 4191 (2007).
  * Berti  _et al._ (2007) E. Berti, V. Cardoso, J. A. González, and U. Sperhake, Phys. Rev. D 75, 124017 (2007).
  * Chandrasekhar (1998) S. Chandrasekhar,  _The Mathematical Theory of Black Holes_ (1998).
  * Regge and Wheeler (1957) T. Regge and J. A. Wheeler, Physical Review 108, 1063 (1957).
  * Vishveshwara (1970) C. V. Vishveshwara, Phys. Rev. D 1, 2870 (1970).
  * Kobayashi  _et al._ (2012) T. Kobayashi, H. Motohashi, and T. Suyama, Phys. Rev. D 85, 084025 (2012), arXiv:1202.4893 [gr-qc] .
  * Wald (1980) R. M. Wald, Journal of Mathematical Physics 21, 2802 (1980).