# Production of twisted particles in heavy-ion collisions

Liping Zou1<EMAIL_ADDRESS>Pengming Zhang2<EMAIL_ADDRESS>Alexander J. Silenko3,4,5<EMAIL_ADDRESS>1Sino-French Institute of Nuclear
Engineering and Technology, Sun Yat-sen University, 519082 Zhuhai, China
2School of Physics and Astronomy, Sun Yat-sen University, 519082 Zhuhai, China
3Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna 141980, Russia  4Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  5Research Institute for Nuclear Problems,
Belarusian State University, Minsk 220030, Belarus

###### Abstract

A prevalence of production of twisted (vortex) particles in noncentral heavy-
ion collisions is shown. In such collisions, photons emitted due to the
rotation of charges are highly twisted. Charged particles are produced in
nonspreading multiwave states and have significant orbital angular momenta. It
can be expected that an emission of any twisted particles manifesting
themselves in specific effects is rather ubiquitous.

A twisted (vortex) particle possesses an intrinsic orbital angular momentum
(OAM) and can be presented by a wave beam or a packet. Wave beams and packets
are localized with respect to two and three dimensions, respectively and are
characterized by two discrete transverse quantum numbers. Free twisted
particle beams of photons, electrons, and neutrons can be described in the
cylindrical coordinates by the Laguerre-Gauss (LG) wave function Kogelnik ;
Siegman ; Alda ; Pampaloni  (see also Supplemental Material I
SupplementalMaterial ). Free twisted particles can possess huge intrinsic
OAMs. Photons with OAMs more than 10000$\displaystyle\hbar$ PNAS  and
electrons with OAMs up to 1000$\displaystyle\hbar$ VGRILLO  have been
obtained. Twisted photons have unusual properties like a subluminality and
nonzero effective masses SupplementalMaterial ; photonPRA . Charged twisted
particles can be recognized by their dynamics Manipulating ;
ResonanceTwistedElectrons , magnetic moments BliokhSOI , and specific effects
in external fields KarlovetsZhevlakov ; PhysRevLettLanzhou2019 ; snakelike ;
arXiv  and collisions OAMinteraction .

The importance of production of twisted particles in heavy-ion collisions
(HIC) is evident. The similar effect of the global polarization of produced
particles GlPolar  attracts a lot of attention. To study the above-mentioned
problem at noncentral HIC, one should take into account an appearance of a
strong magnetic field SkokovIllarionovToneev  and a fast rotation of nuclear
matter leading to its large vorticity Baznatetal .

The recent theoretical Katoh ; Epp ; theoretical  and experimental
experimental  papers unambiguously show that the photons radiated by electrons
in a helical or circular motion are twisted (i.e., have nonzero OAMs). Similar
effects takes place for rotating black holes BlackHoles ; BlackHoleE . Since
electrons move in the magnetic field $\displaystyle B$, parameters of
synchrotron radiation are expressed in terms of $\displaystyle B$. However,
the synchrotron radiation is caused by an electron acceleration and is defined
by an electron motion. Its dependence on $\displaystyle B$ is conditioned by
only the fact that the electron motion is defined by the magnetic field.
Certainly, the synchrotron radiation can de described in terms of the velocity
and acceleration of rotating charges. Therefore, a rotation of charges at HIC
should lead to a photon production equivalent to the synchrotron radiation.
This radiation is defined by the rotation of charges and is independent of the
magnetic field which also appears due to the above-mentioned rotation.

The synchrotron radiation spectrum consists of frequencies multiple to the
rotation frequency of the compound nuclei, $\displaystyle\Omega$:

$\omega_{j}=j\Omega,\qquad j=1,2,3,\dots$ (1)

Amazingly, the total angular momentum of a radiated photon is equal to
$\displaystyle\hbar j$ and has orbital and spin parts Katoh ; Epp . All
photons with $\displaystyle j>1$ are twisted. This effect confirmed by
experimental data experimental  clearly shows that twisted photons are
widespread in nature.

Massive particles can also be produced in twisted states. The vorticity of
nuclear matter is, to some extent, an effect which is similar to the twist and
leads to a production of particles in twisted states footnote . To describe
processes taking place in a rotating nuclear matter, one often uses uniformly
rotating frames (see, e.g., Ref. Baznatetal ). For a quantum-mechanical
description of particles in such frames, it is convenient to apply the Foldy-
Wouthuysen (FW) representation providing for the Schrödinger picture of
relativistic quantum mechanics (see Refs. PRA2015 ; PRAFW  and references
therein). The relativistic FW Hamiltonian describing a spin-1/2 particle in
the frame uniformly rotating with the angular velocity
$\displaystyle\bm{\omega}$ has the form PRD2

$\begin{array}[]{c}{\cal
H}_{FW}=\beta\sqrt{m^{2}+\bm{p}^{2}}-\bm{\omega}\cdot(\bm{L}+\bm{s}),\end{array}$
(2)

where $\displaystyle\beta$ is the Dirac matrix and $\displaystyle\bm{L}$ and
$\displaystyle\bm{s}$ are the OAM and spin operators. The eigenstates of this
operator have definite integer OAMs and the corresponding eigenfunctions
describe LG beams.

In HIC of Au-Au (RHIC) and Pb-Pb (SPS and LHC) nuclei, the beam energies of
colliding nucleons are up to 100 GeV, 158 GeV, and 5.02 TeV, respectively AuAu
; PbPb ; Noferini ; RHIC . In the collision picture, nucleus are nonspherical
and are mostly like a disc Schuetrumpf . The 79Au197 nucleus radius is equal
to $\displaystyle R=6.98$ fm and the Coulomb energy of two touching Au nuclei
is $\displaystyle W=Z^{2}e^{2}/R=1.29$ GeV. Evidently, the Coulomb repulsion
of colliding nuclei can be neglected.

Let us consider the collision with the impact factor $\displaystyle b=4$ fm.
We can determine the momentum components which are normal and tangent to the
plane tangent to the surfaces of the two touching nuclei. We suppose that the
normal and tangent momentum components of each colliding nucleus contribute to
internal and rotational energies of the compound nucleus, respectively. Since
these momentum components are defined by $\displaystyle
p_{n}=p\sqrt{1-b^{2}/(4R^{2})}\approx 0.71p,\,p_{\tau}=pb/(2R)\approx 0.29p$,
the average velocities of rotating charges (protons or quarks) should be
ultrarelativistic. It can be checked that a compound nucleus cannot rotate at
HIC as a solid ($\displaystyle\Omega=const$). Since $\displaystyle v=\Omega
r$, only thin external layers of solids ($\displaystyle R-r\ll R$) can be
ultrarelativistic ($\displaystyle v\approx c$). Velocities of other layers are
defined by $\displaystyle v\approx cr/R$. Such layers cannot be
ultrarelativistic. Energies of nucleons forming such layers cannot be much
bigger than $\displaystyle m_{p}c^{2}$. This result contradicts to the energy
and angular momentum conservation because rotational energies of the colliding
nuclei are ultrarelativistic. Therefore, all or almost all rotating charges
should be ultrarelativistic. We admit that a strongly interacting and rotating
nuclear fluid can contain some vortices, especially in the case of an
appearance of quark-gluon plasma. In this case, the radius of trajectory of
rotating charges, $\displaystyle\rho$, can be significantly less than the
nucleus radius. However, we suppose that $\displaystyle\rho$ cannot be less
than the proton charge radius ($\displaystyle\rho\geqslant r_{p}$,
$\displaystyle r_{p}=0.84$ fm ProtonRadius ) because confinement forces are
unsufficiently strong on a smaller distance. For the quark-gluon plasma, some
estimates can in principle be substantially different.

The synchrotron radiation spectrum is formed by lines which frequencies are
harmonics of the rotation frequency [see Eq. (1)] and is defined by well-known
formulas LL4 . When the charges are ultrarelativistic, the spectral maximum in
classical electrodynamics is defined by

$\omega\sim\Omega\gamma^{3}.$ (3)

This relation determines a large orbital polarization of radiated photons.
However, quantum electrodynamics (see Eq. (90.24) in Ref. LL4 ) leads to the
different connection between the spectral maximum $\displaystyle\omega$ and
the particle energy $\displaystyle\epsilon^{\prime}=\gamma m$:

$\begin{array}[]{c}\frac{\hbar\omega}{(\epsilon^{\prime}-\hbar\omega)\chi}\sim
1,\qquad\chi\approx\frac{\hbar\Omega{\epsilon^{\prime}}^{2}}{m^{3}}.\end{array}$
(4)

Our denotations slightly differ. The classical theory can be used, when the
photon energy $\displaystyle\hbar\omega$ is small as compared with the
particle energy. When $\displaystyle\hbar\omega\approx\epsilon^{\prime}$ and
$\displaystyle\epsilon^{\prime}-\hbar\omega\ll\epsilon^{\prime}$, we obtain
the evaluation of the photon OAM $\displaystyle
L\approx\hbar\omega/\Omega\approx\gamma m/\Omega$. To evaluate this quantity
in the general case, one can apply the classical equation (3) but should take
into account the quantum-mechanical cutoff:

$\begin{array}[]{c}L\sim\min{\left(\hbar\gamma^{3},\frac{\gamma
m}{\Omega}\right)}.\end{array}$ (5)

For ultrarelativistic protons and valence quarks, $\displaystyle\Omega\sim
c/\rho$. The ratio $\displaystyle m/(\hbar\Omega)$ is maximum in the case of
$\displaystyle m=m_{p},\,\rho=R$ and is approximately equal to 33.2. For
confined constituent quarks, $\displaystyle m=m_{q}=m_{p}/3,\,\rho=r_{p}$, and
the ratio $\displaystyle m/(\hbar\Omega)$ is approximately equal to 1.3. As a
result, $\displaystyle L$ is large enough ($\displaystyle\gamma\gg 1$) and
photons emitted due to electromagnetic interactions at noncentral HIC are
significantly twisted. Thus, our model-independent analysis rigorously proves
a significant orbital polarization of such photons.

We note that the majority of photons is produced in strong interactions.
However, we suppose that these photons should also be twisted footnote .

The following analysis based on precedent studies Gori ; Brinkmann ; Edwards ;
Burov ; Kim ; Sun ; Crewe ; Wakamatsu ; FloettmannKarlovets ; Karlovets2021
allows us to predict that all charged particles produced in noncentral HIC
also have significant OAMs. It is very important to determine what quantum
states are suitable for a production of such particles. Hereinafter, basic and
nonbasic states denote quantum states suitable and unsuitable for such a
production ffootnote  and $\displaystyle\bm{\pi}=\bm{p}-e\bm{A}$ is the
kinetic momentum. Basic states in the quantum-mechanical description of
charged particles in a uniform magnetic field are the Landau states with a
definite longitudinal momentum. In the Landau states, the canonical
(generalized) and kinetic (mechanical) OAMs
($\displaystyle\bm{L}=\bm{r}\times\bm{p}$ and
$\displaystyle\bm{\mathcal{L}}=\bm{r}\times\bm{\pi}$, respectively) can be not
only parallel but also antiparallel. The canonical OAM is conserved. In a
uniform magnetic field with the symmetric gauge of the vector potential
$\displaystyle\bm{A}=\bm{B}\times\bm{r}/2$, its connection with the kinetic
OAM $\displaystyle\bm{\mathcal{L}}=\bm{r}\times\bm{\pi}$ is given by (see
Refs. Wakamatsu ; FloettmannKarlovets )

$\begin{array}[]{c}\bm{\mathcal{L}}=\bm{L}-\frac{e}{2}\bm{r}\times(\bm{B}\times\bm{r})=\bm{L}-\frac{e\bm{B}}{2}r^{2},\end{array}$
(6)

where $\displaystyle r=|\bm{r}|$ is the radial coordinate in the plane
orthogonal to $\displaystyle\bm{B}$. The kinetic OAM, unlike the canonical
one, is gauge-invariant (see Ref. Wakamatsu ). We denote
$\displaystyle\bm{B}=B\bm{e}_{z}$ and $\displaystyle L_{z}\equiv\ell\hbar$.
Relativistic wave functions in the FW representation defining the Landau
states are based on the nonrelativistic solution Landau ; LL3  and are
presented, e.g., in Refs. JINRLETTERS2008 ; paraxialLandau . For relativistic
particles with a negative charge, the Landau energy levels are defined by (see
Ref. paraxialLandau  and references therein)

$\begin{array}[]{c}E=\sqrt{m^{2}+p_{z}^{2}+(2n+1+|\ell|+\ell+2s_{z})|e|B},\end{array}$
(7)

where $\displaystyle n=0,1,2,\dots$ is the radial quantum number. For
relativistic particles with a positive charge, $\displaystyle\ell$ in this
equation should be replaced with $\displaystyle-\ell$. The _exact_
relativistic solution of the problem shows the infinite degeneracy of the
Landau states with $\displaystyle{\rm sgn}(e)\ell>0$ and an arbitrary
$\displaystyle n$.

It has been shown in Ref. JINRLett12  with the use of the Wentzel-Kramers-
Brillouin method that the classical limit of _relativistic_ quantum-mechanical
equations is reduced to the replacement of operators in the Hamiltonian and
quantum-mechanical equations of motion by the respective classical quantities.
In the classical picture, particles move in a helix and the radial motion does
not occur. Well-known relativistic classical formulas defining the particle
dynamics lead to the following relations:

$r=|r|=-\frac{\pi_{\phi}}{eB},\qquad\mathcal{L}_{z}=2L_{z}=-eBr^{2}=-\frac{\pi^{2}_{\phi}}{eB}.$
(8)

The Landau state radii with respect to the cychrotron motion center are
defined by integral ; Li

$\langle r^{2}\rangle=\frac{2(2n+|\ell|+1)}{|e|B},$ (9)

Therefore, the connection between the kinetic and canonical OAM operators in
the FW representation is given by (see Eq. (6) where $\displaystyle\bm{L}$ is
an invariant)

$\begin{array}[]{c}\langle\mathcal{L}_{z}\rangle=\ell\hbar-{\rm
sgn}(e)(2n+|\ell|+1)\hbar\\\ =L_{z}-{\rm sgn}(e)[(2n+1)\hbar+|L_{z}|],\\\
\langle\mathcal{L}_{z}\rangle-2L_{z}=-{\rm sgn}(e)[2n+|\ell|+{\rm
sgn}(e)\ell+1]\hbar.\end{array}$ (10)

Due to the absence of the radial motion in the classical picture, Eqs. (8) and
(10) agree for $\displaystyle{\rm sgn}(e)\ell\leq 0$ and disagree for
$\displaystyle{\rm sgn}(e)\ell>0$. Thus, the Landau states with
$\displaystyle{\rm sgn}(e)\ell\leq 0$ and $\displaystyle{\rm sgn}(e)\ell>0$
are basic and nonbasic for the particle production (see Refs.
SupplementalMaterial ; preprint  for more details). This conclusion does not
mean that nonbasic states are unphysical. Particles can be created in the
basic Landau states in the magnetic field
$\displaystyle\widetilde{\bm{B}}\neq\bm{B}$ and then penetrate to the area of
the field $\displaystyle\bm{B}$. After the penetration, their quantum states
are described by the LG beams or, in the specific case of
$\displaystyle\widetilde{\bm{B}}=-\bm{B}$, by the nonbasic Landau solution
with $\displaystyle{\rm sgn}(e)\ell>0$. This result explains the division of
the Landau states into basic and nonbasic SupplementalMaterial ; preprint .
Nonuniform magnetic fields cover uniform ones. Therefore, charged particles
cannot be created in _nonuniform_ magnetic fields in quantum states with
opposite directions of canonical and kinetic OAMs SupplementalMaterial ;
preprint .

It has been demonstrated in Refs. FloettmannKarlovets ; Karlovets2021  on the
basis of the quantum Busch theorem Busch ; Reiser  that particle beams
generated inside a magnetic field take the form of LG beams outside of this
field. This important achievement predicts wide dissemination of twisted beams
in nature. It has been noted in Refs. FloettmannKarlovets ; Karlovets2021
that twisted particles are produced in states with
$\displaystyle\mathcal{L}_{z}=0$ and $\displaystyle L_{z}\neq 0$. This note is
absolutely correct for specific experimental conditions FloettmannKarlovets ;
Karlovets2021  and does not contradict to our analysis. For a real solenoid,
it is convenient to choose the symmetric gauge with the solenoid symmetry axis
corresponding to $\displaystyle r=0$ SupplementalMaterial ; preprint . When a
particle is produced near this axis, $\displaystyle\mathcal{L}_{z}=0$ at the
moment of creation and then increases or decreases up to the extreme value
$\displaystyle-eBr^{2}=-\pi^{2}_{\phi}/(eB)$. Unlike Refs. FloettmannKarlovets
; Karlovets2021 , we condider the case of coinciding the trajectory symmetry
axis with the solenoid one. It can be shown SupplementalMaterial ; preprint
that the average values $\displaystyle\langle\mathcal{L}_{z}\rangle$ do not
depend on a shift of the former axis from the latter one. For the geometry
used in Refs. FloettmannKarlovets ; Karlovets2021 , the vector potential gauge
should be changed SupplementalMaterial ; preprint .

Twisted electrons were produced by the photoemission in experiments at FNAL
Sun  and had rather large OAMs of the order of $\displaystyle 10^{8}\hbar$ in
magnetic fields $\displaystyle B\sim 10^{-2}-10^{-1}$ T.

It has been also shown in Refs. arXiv ; FloettmannKarlovets ; Karlovets2021
that twisted beams are not spreading in nonuniform electric and magnetic
fields and their propagation does not destroy the beam coherence.
Nevertheless, the beam coherence at HIC needs a separate analysis because
relativistic effects and spin interactions should be considered. A rigorous
relativistic FW transformation providing for the Schrödinger-like form of
quantum mechanics for particles with different spins and arbitrary energies in
arbitrarily strong external fields can be fulfilled by the method developed in
Refs. JMP ; PRA2015 ; PRA2008 . This method allows one to calculate _exactly_
leading terms in FW Hamiltonians proportional to the zero and first powers of
$\displaystyle\hbar$ and terms describing contact interactions which are
proportional to $\displaystyle\hbar^{2}$ PRA2015 . The corresponding
relativistic FW Hamiltonian has been obtained in Refs. PRAFW ; JMP ; RPJ . It
contains spin-independent and spin-dependent terms and is the same for
untwisted and twisted spin-1/2 particles.

In general, particle wave beams penetrating to free space do not satisfy the
paraxial approximation. In strong _stationary_ electric and magnetic fields,
particles are produced in some quantum states with the fixed energy,
$\displaystyle{\cal H}_{FW}\Psi_{FW}=E\Psi_{FW}$. These states generalize the
Landau ones and the corresponding wave functions describe nonparaxial and
paraxial twisted beams. The beams are formed by partial waves with the same
constant energy $\displaystyle E$ which become plane waves in free space.
Since the wave coherence in a given spatial point is provided for coinciding
the wave energies, the energy conservation in stationary fields leads to the
coherence of the partial waves in the whole space. Momenta of the partial
waves do not coincide, can differently vary from point to point, and depend on
the spin polarization. However, these properties are not important for the
beam coherence. In principle, the beam shape near colliding ions can be
deduced from its shape in the free space and slightly depends on the spin
polarization.

Colliding ions are nonstationary quantum systems and their fields are not
fully stationary. However, this property does not violate the beam coherence
in a given spatial point if the nonstationarity does not eliminate the
equality of energies of the partial waves in this point.

It is well-known that the paraxial LG beams are nonspreading. Nonparaxial
twisted (and untwisted) beams can be considered as a result of some Lorentz
boosts of LG beams Manipulating ; photonPRA  while other approaches are also
possible Karlovets2021 ; Karlovets2018 . Evidently, Lorentz boosts cannot lead
to the beam spread.

Beams of scalar and vector mesons also possess above-mentioned properties.

It has been obtained in Ref. SkokovIllarionovToneev  that the magnetic field
emerging in the SPS, RHIC, and LHC experiments is of the order of 0.1
$\displaystyle m_{\pi}^{2}/|e|$, $\displaystyle m_{\pi}^{2}/|e|$, and 15
$\displaystyle m_{\pi}^{2}/|e|$, respectively footnote2 . The maximum beam
radius is restricted by the area of the nonuniform magnetic field and is of
the order of $\displaystyle r_{max}\sim 10$ fm (see Ref.
SkokovIllarionovToneev ). As follows from Eq. (9) at $\displaystyle
n=n_{min}=0$,

$L_{z}^{max}=\frac{|e|Br^{2}_{max}}{2}-\hbar.$ (11)

For the above-mentioned experiments, the maximum values of
$\displaystyle|\pi_{\phi}|$ and $\displaystyle|\ell|$ defined by Eqs. (8),
(11) read $\displaystyle\pi_{max}\sim 0.1\,{\rm GeV}/c,\,|\ell|_{max}\sim 1$;
$\displaystyle\pi_{max}\sim 1\,{\rm GeV}/c,\,|\ell|_{max}\sim 10$; and
$\displaystyle\pi_{max}\sim 10\,{\rm GeV}/c,\,|\ell|_{max}\sim 10^{2}$,
respectively. The total longitudinal momentum can be much bigger than
$\displaystyle\pi_{max}$ if a produced particle is paraxial relative to the
magnetic field direction. Evidently, the orbital polarization of charged
particles produced in HIC is not negligible. Since it manifests in processes
studied in Ref. OAMinteraction , it can in principle be discovered. The
vorticity of nuclear matter is another reason of a production of not only
charged but also uncharged twisted particles footnote .

Our analysis relates to noncentral HIC in all experiments including planning
experiments at NICA (JINR, Dubna) and HIAF (IMP, Lanzhou).

In summary, we have rigorously proven that photons emitted at noncentral HIC
due to electromagnetic interactions are highly twisted. We have also
unambiguously shown that charged particles are produced in this case in
nonspreading multiwave states and have significant OAMs. We expect that an
emission of other particles in twisted states is rather ubiquitous. It should
be noted that twisted particles manifest themselves in specific effects.

###### Acknowledgements.

The authors are grateful to O.V. Teryaev for multiple discussions of the
problem and valuable comments. The work was supported by the National Natural
Science Foundation of China (grants No. 11975320 and No. 11805242), by the
Chinese Academy of Sciences President’s International Fellowship Initiative
(grant No. 2019VMA0019), and by the Fundamental Research Funds for the Central
Universities, Sun Yat-sen University. A. J. S. also acknowledges hospitality
and support by the Institute of Modern Physics of the Chinese Academy of
Sciences.

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# Deep-SWIM: A few-shot learning approach to classify Solar WInd Magnetic
field structures

Hala Lamdouar
University of Oxford
<EMAIL_ADDRESS>&Sairam Sundaresan
Intel Labs
<EMAIL_ADDRESS>
&Anna Jungbluth
University of Oxford
<EMAIL_ADDRESS>
&Sudeshna Boro Saikia
University of Vienna
<EMAIL_ADDRESS>
&Amanda Joy Camarata
Colorado School of Mines
<EMAIL_ADDRESS>
&Nathan Miles
University of California,Los Angeles
<EMAIL_ADDRESS>
&Marcella Scoczynski
Federal University of Technology – Paraná
<EMAIL_ADDRESS>
&Mavis Stone
Harvey Mudd College
<EMAIL_ADDRESS>
&Anthony Sarah
Intel Labs
<EMAIL_ADDRESS>
&Andrés Muñoz-Jaramillo
Southwest Research Institute
<EMAIL_ADDRESS>
&Ayris Narock
NASA Goddard Space Flight Center
ADNET Systems Inc
<EMAIL_ADDRESS>
&Adam Szabo
NASA Goddard Space Flight Center
<EMAIL_ADDRESS>

###### Abstract

The solar wind consists of charged particles ejected from the Sun into
interplanetary space and towards Earth. Understanding the magnetic field of
the solar wind is crucial for predicting future space weather and planetary
atmospheric loss. Compared to large-scale magnetic events, smaller-scale
structures like magnetic discontinuities are hard to detect but entail
important information on the evolution of the solar wind. A lack of labeled
data makes an automated detection of these discontinuities challenging. We
propose Deep-SWIM, an approach leveraging advances in contrastive learning,
pseudo-labeling and online hard example mining to robustly identify
discontinuities in solar wind magnetic field data. Through a systematic
ablation study, we show that we can accurately classify discontinuities
despite learning from only limited labeled data. Additionally, we show that
our approach generalizes well and produces results that agree with expert
hand-labeling.

##  1 Introduction

Every star has an extended atmosphere of charged particles reaching far into
interplanetary space. For the Sun, this extended atmosphere is called the
solar wind and it significantly impacts Earth’s space weather. Understanding
the magnetic field that drives the solar wind and the different magnetic
structures it contains is essential to predicting the effect of the solar wind
on Earth and future extraterrestrial missions. Furthermore, knowledge on solar
wind properties helps constrain our host star’s impact on planetary
atmospheres to inform our understanding of exoplanet habitability. Over the
past decades, instruments like the WIND spacecraft have collected measurements
of the magnetic field of the solar wind. While some magnetic events, e.g.
interplanetary coronal mass ejections, are high-strength and therefore easy to
detect [1, 2], smaller-scale events, like discontinuities, are more difficult
to spot. Discontinuities describe "discontinuous" spatial changes in the solar
magnetic field. Although they only last for a few seconds or a few minutes,
and are usually weak in amplitude, the occurence rate of discontinuties is
quite high, ranging up to 10,000s over a few months [3]. Experts who can
identify these magnetic structures are limited, and hand-labeling is too time-
consuming to catalog all available data.

Applying machine learning (ML) to solar wind magnetic field data allows us to
quickly and accurately classify these small-scale structures to create a
comprehensive catalog of all measured discontinuities. To do this, we employ
both supervised and semi-supervised machine learning approaches using 1D and
2D convolutional neural networks (CNNs). As the availability of labeled data
is limited, and the classes are heavily imbalanced, we apply various data
augmentation techniques and leverage recent advances in contrastive learning
[4], pseudo-labeling [5], and online hard example mining [6]. We show that
contrastive learning and pseudo-labeling significantly improve the performance
of the model and allow it to generalize better to unseen data.

##  2 Data sets

We use data from the Magnetic Field Investigation (MFI) fluxgate magnetometer
on board the WIND spacecraft [7, 8] 111The data is publicly available on
NASA’s Coordinated Data Analysis Web at https://cdaweb.gsfc.nasa.gov/. WIND is
positioned in a halo orbit around Lagrange point 1, measuring small-scale
magnetic structures embedded in the solar wind [9] at a constant distance from
the Sun and at sub-second resolution. To remove artifacts of the 3s rotational
period of the instrument, we smooth and re-sample the original 11 Hz data to a
3s cadence. The time series are split into 5-minute intervals, and the x, y,
and z components of the magnetic field are stacked as inputs to the CNN. For
our supervised approach we assign binary labels to each segment specifying
whether it contains a discontinuity or not. The labels are provided by expert
hand-labeling for one day (11-18-2018) or using a non-machine learning based
heuristic approach which compares the rotation angle of neighboring points to
identify discontinuities (2006-2021) 222The hand-labeled and heuristic catalog
were provided by Dr. Adam Szabo and Dr. Ayris Narock who are both authors of
this work.. Importantly, not all labels in the heuristic catalog are expected
to be correct. For the one day of joint labeling, both the heuristic approach
and hand-labeling identified 39 discontinuities out of which 24
discontinuities were identified by both approaches. In addition to noise in
the labeling, our data set is highly imbalanced. Only around 15% of our
5-minute intervals contain discontinuities. To perform our ablation study, we
select a representative sample of 3 months of data (May - July 2018). We split
the data using stratified sampling to ensure the correct proportional
representation of positive and negative examples and randomly allocate
segments to the training and validation sets. 20% of the data (5428 5s
segments) was used for training the supervised component of our approach, 70%
(18492 5s segments) were used for training the semi-supervised component, and
10% (2576 5s segments) was used for validation. The one day of expert hand-
labeled data (11-18-2018, 286 5s segments) was used as a final test set.

##  3 Methodology

Figure 1: Schematic of our machine learning pipeline. We compare and evaluate
two modalities (1) a supervised approach using a standard data-label pairing
as input, and (2) a semi-supervised approach using contrastive learning where
the input data consists of an anchor (data of interest), a positive example,
and a negative example. Additional model improvements are achieved by
employing pseudo-labeling and online hard example mining during training.

A schematic of our ML pipeline is shown in Figure 1. We compare the
performance of our solar wind magnetic field classification based on two
modalities. Firstly, we use a standard data-label pairing as input to a four
layer 1D-CNN. We also test the performance of 2D-CNN with a ResNet-18 backbone
after converting the original time series into images using a Gramian Angular
Field (GAF) conversion [10]. The successful use of GAF conversions for ml-
based astrophysics research has been demonstrated e.g. in [11]. While we
evaluated other backbone architectures including AlexNet, Inception and larger
ResNets, we found that they either quickly overfit the data or do not
converge. Secondly, we employ contrastive learning [12] using three examples
as simultaneous inputs to the model. In this case, the classification head of
each model is replaced with a fully connected layer which has a size equal to
the size of the embedding dimension desired. A downstream classifier head is
appended to this for fine tuning to produce classification results once the
contrastive model has been properly trained. The training pipeline is made
increasingly more sophisticated through the additions of online hard example
mining and pseudo-labeling.

Data Augmentations.  To mitigate limitations of our imbalanced data set, we
employ data augmentations to increase the number of examples containing
discontinuities. We perform horizontal mirroring, random scaling between $0.5$
and $2$, random inverting, and random channel shuffling of the time series
data. These data augmentations all preserve the physics of the solar magnetic
field and present scenarios that could realistically be measured by the
instrument; for instance, an inverted magnetic field could be observed during
a different solar cycle with reversed magnetic poles.

Contrastive Learning.  By employing contrastive learning [12, 4], rather than
learning from a single input-output pair, the model aims to learn the
similarities and differences between examples. For this, three examples are
paired; the anchor ($x$), i.e. the example of interest, a positive example of
the same class as the anchor ($x^{+}$), and a negative example from a
different class ($x^{-}$). The model learns an embedding space to group
similar examples close together and dissimilar examples far apart. To do this,
we employ a triple margin loss [13, 14].

$L_{triple}(x,x^{+},x^{-})=\sum\max(0,\|f(x)-f(x^{+})\|^{2}-\|f(x)-f(x^{-})\|^{2}+\alpha),$
(1)

where $\alpha$ is the margin between the positive and negative pairs (here,
$\alpha=1$), and $f$ is the learned embedding function. Further for the
triplet margin loss, the positive and negative examples are chosen at random.

Pseudo-Labeling.  Pseudo-labeling [5] is a semi-supervised learning approach
to maximize the potential of small labeled data sets. The model is first
trained on limited labeled data and subsequently used to predict labels of
unlabeled data. The pseudo labels are refreshed every epoch. The total loss is
calculated as the sum of the loss of the labeled and unlabeled data
($\mathrm{L_{Total}=L_{labeled}+\alpha\cdot L_{unlabeled}}$) with a weighting
factor $\alpha$ that changes with epoch $t$ according to equation 2. Here,
$\alpha_{f}=3$, $T_{1}=5$, and $T_{2}=100$.

$\alpha(t)=\begin{cases}0,&t<T_{1}\\\
\frac{t-T_{1}}{T2-T1}\alpha_{f},&T_{1}<t<T_{2}\\\
\alpha_{f},&t>T_{2}\end{cases}$ (2)

Online Hard example mining.  Most machine learning tasks have combinations of
easy and hard examples to learn from. Online Hard example mining (OHEM) [6]
focuses on the model’s performance on examples that are either misclassified
with high confidence or correctly classified with reduced confidence. The
total loss is calculated as $\mathrm{L_{total}=\omega\cdot
L_{OHEM}+(1-\omega)\cdot L_{raw}}$, where $\mathrm{L_{raw}}$ is the standard
loss across all samples, $\mathrm{L_{OHEM}}$ is the loss computed on the top
$70\%$ of hardest examples, i.e. examples with the highest loss values, and
$\omega$ is a weighting factor. In our experiments, $\omega$ was set to $0.8$.

##  4 Experiments

Figure 2: Example performance of our best model. The model was tested on one
day of expert hand-labeled data (11-18-2018). Here we show a) a true negative,
b) true positive, and c) false positive example of the model’s labeling. The
colors in each graph present the x, y, and z component of the solar wind
magnetic field. The gray shading in panel b) shows the time stamp of the
discontinuity.

### Training.

In this section, we compare our experiments to a non deep learning baseline
using Support-Vector Machines (SVMs) [15, 16]. The SVM baseline is obtained
using a linear kernel with $\gamma=1.0$, $c=0.1$. These parameters are chosen
through a grid search over the parameter space and selecting the combination
which yields the best Area Under the receiver operating characteristic Curve
(AUC) score on the validation set. We have also tested SVM with a Radial Basis
Function (RBF) kernel as well as polynomial and linear and found the linear
kernel to provide the best performance. Note that we apply the SVM directly to
time series data, but extract Oriented Rotated BRIEF (ORB) features [17] from
images prior to using the SVM. Both our time-series and image based CNNs are
trained on a single NVIDIA A100 GPU and use the Adam optimizer, a learning
rate of $10^{-3}$, and a batch size of 16. These parameters are obtained
through hyper-parameter search runs. Additionally, the 2D-CNN is initialized
with ImageNet weights to facilitate transfer learning.

### Results and Discussion.

We conduct an ablation study on the different components of our training
pipeline and present the results in Table 1. We find that the SVM approach,
c.f. experiments (A1) and (B1), provides considerably lower performance
compared to CNN-based approaches, especially for images combined with ORB
features that fail to extract distinguishing attributes from the Gramian
Angular Field conversions. The raw time series and image data types yield
comparable results when the CNNs without additions are considered, with
slightly better performance for the raw time series. However, once we
introduce pseudo-labeling, raw times series (B4) clearly outperform image data
types (A4) by over $27\%$ AUC. Moreover, we found that using contrastive
learning did not alleviate the convergence issues in the image based models
while pseudolabeling resulted in divergence. Overall, we achieve the best
performance, i.e. experiment (B5) with $90\%$ AUC, using a combination of
contrastive learning, pseudo-labeling, and OHEM. While the AUC scores for B3
to B7 are close, we found qualitatively, B3 appeared to be robust on certain
hard examples. In our future work, we plan to substantiate this with a
statistical significance test. All experiments presented in Table 1 are run
with a random seed of $42$. Varying the random seed of our best model across 2
orders of magnitude resulted in the following median absolute deviations
(MAD): $\mathrm{{MAD}_{Precision}}=0.014$, $\mathrm{{MAD}_{Recall}}=0.000$ and
$\mathrm{{MAD}_{AUC}}=0.003$.

To further evaluate our best model, we compare the model’s labels to the
expert hand-labeled data set (11-18-2018). An example of a true negative, true
positive, and false positive labeling is provided in Figure 2. The false
positive labeling is caused by the sharp dip in the $\mathrm{B_{Z}}$ component
around $10:54$ pm, which is incorrectly understood to be a discontinuity. Our
best model achieves a precision of $0.52$, a recall of $0.73$, and an AUC of
$0.82$ on this final test set.

| Architecture | data type | Contrastive Learning | Pseudo-labeling | OHEM | Precision | Recall | AUC
---|---|---|---|---|---|---|---|---
A1 | SVM | images | ✗ | ✗ | ✗ | 0.00 | 0.00 | 0.50
A2 | ResNet-18 | images | ✗ | ✗ | ✗ | 0.51 | 0.65 | 0.79
A3 | ResNet-18 | images | ✗ | ✓ | ✗ | 1.00 | 0.03 | 0.51
A4 | ResNet-18 | images | ✗ | ✓ | ✓ | 0.8 | 0.26 | 0.62
B1 | SVM | time series | ✗ | ✗ | ✗ | 1.00 | 0.23 | 0.62
B2 | 1D-CNN | time series | ✗ | ✗ | ✗ | 0.66 | 0.61 | 0.79
B3 | 1D-CNN | time series | ✗ | ✓ | ✗ | 0.63 | 0.84 | 0.89
B4 | 1D-CNN | time series | ✗ | ✓ | ✓ | 0.67 | 0.84 | 0.89
B5 | 1D-CNN | time series | ✓ | ✓ | ✓ | 0.59 | 0.87 | 0.9
B6 | 1D-CNN | time series | ✓ | ✓ | ✗ | 0.47 | 0.90 | 0.89
B7 | 1D-CNN | time series | ✓ | ✗ | ✗ | 0.77 | 0.77 | 0.87
Table 1: Ablation study of our proposed approach to classify discontinuities.
We select the best model based on the higher AUC (Area Under the receiver
operating characteristic Curve).

##  5 Conclusions

In this work, we present a few-shot learning approach for classifying the
magnetic structures of solar wind magnetic field data. Specifically, we focus
on identifying discontinuities, which present "discontinuous" spatial changes
in the magnetic field. Our proposed approach combines contrastive learning and
pseudo-labeling to overcome the large imbalance of our data set. We show that
incorporating online hard example mining further improves the overall
performance. Finally, our method, trained on heuristic labels, generalizes
well to expert hand-labeling, circumventing the labor-intensive and time-
consuming process of manual annotation of such complex data.

## Broader Impact

At the present moment, state-of-the-art applications of deep learning to solar
wind measurements are primarily confined to supervised classification of
large-scale structures like interplanetary coronal mass ejections (ICME) [2,
1]. The primary driver behind this is the occurrence rate of these structure.
Large-scale structures like ICMEs occur at a rate of about 10 to 50 times per
year depending on solar activity and are very pronounced and easily detectable
in in-situ measurements. While this facilitates the creation of hand-labeled
data sets, there are still issues associated with misclassification of these
structures due to human bias [18]. For small-scale structures like
discontinuities, creating a hand-labeled catalog is infeasible since
discontinuities occur at rates of more than 10,000 times per year. Here we
show that we can apply pseudo-labeling to leverage the potential of small data
sets and accurately identify these target structures. For future research,
this enables to hand-label small amounts of data, or apply costly heuristic
approaches to only a subset of the data to create a small training set for the
supervised component of our proposed approach.

## Acknowledgments and Disclosure of Funding

This work was conducted at the Frontier Development Laboratory (FDL) USA 2021.
The FDL USA is a public / private research partnership between NASA, the SETI
Institute and private sector partners including Google Cloud, Intel, IBM,
Lockheed Martin, and NVIDIA. These partners provide the data, expertise,
training, and compute resources necessary for rapid experimentation and
iteration in data-intensive areas.

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