# Reply to ‘Comment on “Proper and improper chiral magnetic interactions” ’

Manuel dos Santos Dias<EMAIL_ADDRESS>Peter Grünberg
Institut and Institute for Advanced Simulation, Forschungszentrum Jülich &
JARA, 52425 Jülich, Germany  Faculty of Physics, University of Duisburg-Essen
and CENIDE, 47053 Duisburg, Germany  Sascha Brinker  Peter Grünberg Institut
and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, 52425
Jülich, Germany  András Lászlóffy  Wigner Research Centre for Physics, P.O.
Box 49, H-1525 Budapest, Hungary  Department of Theoretical Physics, Budapest
University of Technology and Economics, Budafoki út 8, H-1111 Budapest,
Hungary  Bendegúz Nyári  Department of Theoretical Physics, Budapest
University of Technology and Economics, Budafoki út 8, H-1111 Budapest,
Hungary  Stefan Blügel  Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, 52425 Jülich, Germany  László
Szunyogh  Department of Theoretical Physics, Budapest University of Technology
and Economics, Budafoki út 8, H-1111 Budapest, Hungary  MTA-BME Condensed
Matter Research Group, Budapest University of Technology and Economics,
Budafoki út 8, H-1111 Budapest, Hungary  Samir Lounis<EMAIL_ADDRESS>Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich & JARA, 52425 Jülich, Germany  Faculty of Physics,
University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany

###### Abstract

In our previous Letter [Phys. Rev. B 103, L140408 (2021)], we presented a
discussion of the fundamental physical properties of the interactions
parameterizing atomistic spin models in connection to first-principles
approaches that enable their calculation for a given material. This explained
how some of those approaches can apparently lead to magnetic interactions that
do not comply with the expected physical properties, such as Dzyaloshinskii-
Moriya interactions which are non-chiral and independent of the spin-orbit
interaction, and which we consequently termed ‘improper’. In the preceding
Comment [Phys. Rev. B 105, 026401], the authors present arguments based on the
distinction between global and local approaches to the mapping of the magnetic
energy using first-principles calculations to support their proposed non-
chiral Dzyaloshinskii-Moriya interactions and their dismissal of our
distinction between ‘proper’ and ‘improper’ magnetic interactions. In this
Reply, we identify the missing step in the local approach to the mapping and
explain how ignoring this step leads to the identification of magnetic
interactions which do not comply with established physical principles and that
we have previously termed ‘improper’.

_Context._ The main issue being addressed in our previous Letter dos Santos
Dias  _et al._ (2021) and the subsequent Comment Cardias  _et al._ (2022)
concerns the mapping of density functional theory (DFT) calculations to
atomistic spin models and the corresponding interpretation. This is an
intricate and somewhat subtle issue, so it is important to carefully define
the quantities involved and to give them informative names, and to relate the
magnetic interactions to fundamental microscopic models, as we stressed in our
Letter. In the Comment, the authors first present an overview of mapping
approaches, highlighting the prominent role played by the Liechtenstein-
Katsnelson-Antropov-Gubanov (LKAG) approach Liechtenstein  _et al._ (1984,
1987) and how its extension to noncollinear magnetic states leads to what is
therein termed the nonrelativistic or non-chiral Dzyaloshinskii-Moriya
interaction (DMI). In contrast to many other known magnetic interactions which
are achiral, the original DMI Dzyaloshinskii (1957); Moriya (1960) which is
familiar to the magnetism community is chiral and originates from spin-orbit
coupling (SOC). This is followed by a discussion of the content of the
corresponding LKAG-type formulae based on form of the Kohn-Sham hamiltonian
for noncollinear magnetic systems, which recaps arguments given in Refs.
Cardias  _et al._ (2020, 2020). The parametrization of atomistic spin models
using the noncollinear LKAG approach is then compared to our parametrization
given in Ref. dos Santos Dias  _et al._ (2021), which is based on the spin
cluster expansion, listing the corresponding advantages and disadvantages. The
essential differences between the noncollinear LKAG approach and the spin
cluster expansion are finally discussed in terms of local vs. global
approaches to the mapping of the magnetic energy, pointing to the recent Ref.
Streib  _et al._ (2021), with a final remark arguing in favor of the name
‘nonrelativistic or non-chiral DMI’. Overall, we find the Comment very
constructive and a valuable contribution to the scientific discussion of the
important problem of parameterizing the magnetic energy of real materials
using DFT approaches. However, it is our view that the authors have overlooked
important aspects of the mapping between first-principles calculations and
atomistic spin models using LKAG-type approaches which are central to the
interpretation of the calculations in terms of magnetic interactions. These
aspects were discussed in Ref. dos Santos Dias  _et al._ (2021) and here we
expand upon them in connection to the preceding Comment.

_Mapping from DFT to a spin model._ Within DFT, the central quantities
describing the energetics of a given material are the electronic charge and
spin densities, $n(\mathbf{r})$ and $\mathbf{m}(\mathbf{r})$, respectively,
while an atomistic spin model is specified in terms of classical or quantum
spins, $\mathbf{S}_{i}$. The mapping that we are discussing is then
represented by

$E[n(\mathbf{r}),\mathbf{m}(\mathbf{r});\mathbf{S}_{1},\ldots,\mathbf{S}_{N}]\;\longrightarrow\;\mathcal{E}(\mathbf{S}_{1},\ldots,\mathbf{S}_{N})\;.$
(1)

The mapping indicated by the arrow comprises three aspects: (i) how to define
the total energy functional for a target spin configuration,
$E[n(\mathbf{r}),\mathbf{m}(\mathbf{r});\mathbf{S}_{1},\ldots,\mathbf{S}_{N}]$,
(ii) how to find the parameters for the atomistic spin model $\mathcal{E}$
using the DFT total energy functional $E$, and (iii) under what conditions is
the mapping meaningful. We will focus on the relation between (ii) and (iii).

_LKAG-type approaches to the mapping._ The LKAG approach to point (ii)
consists in expanding the total energy of the magnetic material with respect
to small deviations of the orientations of the magnetic moments from a chosen
reference magnetic state. This was the ferromagnetic state in the original
publications Liechtenstein  _et al._ (1984, 1987) and was later generalized to
noncollinear reference magnetic states Antropov  _et al._ (1997, 1999); Lounis
and Dederichs (2010); Szilva  _et al._ (2013); Secchi  _et al._ (2015); Szilva
_et al._ (2017); Cardias  _et al._ (2020, 2020). Thus the LKAG approach is to
be understood as a Taylor expansion of the magnetic energy that gives a local
view of the magnetic energy landscape, as also advocated in the Comment and in
Ref. Streib  _et al._ (2021).

Considering the mapping expressed in Eq. (1) and deviations of a reference
state written as $\mathbf{S}_{i}=\mathbf{S}_{i}^{0}+\delta\mathbf{S}_{i}$, we
not only need to Taylor expand the DFT total energy,

$\displaystyle E[\\{\mathbf{S}_{i}\\}]$
$\displaystyle=E[\\{\mathbf{S}_{i}^{0}\\}]+\sum_{i}\sum_{\mu}\frac{\delta
E[\\{\mathbf{S}_{i}^{0}\\}]}{\delta S_{i}^{\mu}}\,\delta S_{i}^{\mu}$
$\displaystyle+\frac{1}{2}\sum_{i,j}\sum_{\mu,\nu}\frac{\delta^{2}E[\\{\mathbf{S}_{i}^{0}\\}]}{\delta
S_{i}^{\mu}\delta S_{j}^{\nu}}\,\delta S_{i}^{\mu}\delta S_{j}^{\nu}+\ldots$
(2)

where $i,j$ label the spin sites and $\mu,\nu=x,y,z$ the spin components, but
also the atomistic spin model $\mathcal{E}[\\{\mathbf{S}_{i}\\}]$. The mapping
in Eq. (1) then becomes a mapping between the matching coefficients of the
respective Taylor expansions,

$\frac{\delta E[\\{\mathbf{S}_{i}^{0}\\}]}{\delta
S_{i}^{\mu}}\longrightarrow\frac{\delta\mathcal{E}[\\{\mathbf{S}_{i}^{0}\\}]}{\delta
S_{i}^{\mu}}\;,\quad\frac{\delta^{2}E[\\{\mathbf{S}_{i}^{0}\\}]}{\delta
S_{i}^{\mu}\delta
S_{j}^{\nu}}\longrightarrow\frac{\delta^{2}\mathcal{E}[\\{\mathbf{S}_{i}^{0}\\}]}{\delta
S_{i}^{\mu}\delta S_{j}^{\nu}}\;\ldots$ (3)

To establish this mapping, the form of $\mathcal{E}[\\{\mathbf{S}_{i}\\}]$
must be specified a priori. In dos Santos Dias  _et al._ (2021) we showed that
applying these ideas to a model including isotropic four-spin interactions
results in a Taylor expansion that contains DMI-like terms and other
anisotropic terms if the reference magnetic state $\\{\mathbf{S}_{i}^{0}\\}$
is noncollinear (see e.g. Eq. 8 in dos Santos Dias  _et al._ (2021)). This is
a counterexample against the direct interpretation of the terms of the Taylor
expansion of the magnetic energy as magnetic interactions, as in this case the
interactions were given a priori and their contributions to the LKAG-type
expansion could be traced term by term.

_Interpretation of the Taylor expansion._ What is discussed in the Comment and
in Refs. Cardias  _et al._ (2020, 2020); Mankovsky  _et al._ (2021) is the
microscopic origin of the different coefficients in the Taylor expansion of
the magnetic energy for a noncollinear magnetic state, by relating the
electronic Green functions to charge and spin currents. We find this very
interesting, noting existing interpretations of the DMI in terms of spin
currents Kikuchi  _et al._ (2016); Freimuth  _et al._ (2017), and we highlight
the strong parallel with the linear response theory of electronic transport in
magnetic materials, which has a similar interpretation. Quantities such as
anomalous or topological Hall conductivities are theoretically obtained by
perturbing a reference magnetic state with an electric field, and a strong
emphasis is placed in their variation with a change of the magnetic state
(e.g. rotation of the magnetic moments with respect to the lattice or change
in their relative alignment), as this offers clues concerning the different
underlying physical mechanisms (SOC, emergent magnetic fields, etc.) and is
experimentally feasible. Both the conductivities and the coefficients in the
LKAG-type expansion of Eq. (Reply to ‘Comment on “Proper and improper chiral
magnetic interactions” ’) are thus expected to depend on and change with the
choice of reference magnetic state $\\{\mathbf{S}_{i}^{0}\\}$, and the precise
nature of these variations is what contains information about the underlying
interactions.

_Explicit and implicit constructions of the spin model._ The connection in Eq.
(3) is needed to interpret the magnetic interactions obtained from the Taylor
expansion of the DFT total energy, which in turn requires the form of the
atomistic spin model $\mathcal{E}[\\{\mathbf{S}_{i}\\}]$, i.e. which magnetic
interactions it contains. This can be established in several ways, using
phenomenological arguments based on symmetry or by explicit construction
starting from a microscopic electronic model, as mentioned both in our Letter
and in the Comment. These constructions produce magnetic interactions that
comply with general requirements such as invariance under time-reversal and
compliance with the spatial symmetries of the material. The microscopic
approach has the advantage of also identifying the mechanism behind each type
of magnetic interaction. For instance, Refs. Takahashi (1977); MacDonald  _et
al._ (1988); Hoffmann and Blügel (2020) derive systematically the magnetic
interactions from a Hubbard model without considering SOC, and find isotropic
four-spin interactions but no non-chiral DMI. The Kramers-Anderson
superexchange theory employed by Moriya also required SOC in order to generate
the DMI Moriya (1960).

As an alternative, we can extract the implicit form of the atomistic spin
model directly from Eq. (Reply to ‘Comment on “Proper and improper chiral
magnetic interactions” ’). Adopting the electronic hamiltonian given in Eq. 1
of the Comment and splitting the spin-independent ($\mathcal{H}^{0}$) from the
spin-dependent parts while omitting the spatial arguments:

$\mathcal{H}^{\mathrm{el}}=\mathcal{H}^{0}+\sum_{i}\,B_{i}^{\mathrm{xc}}\,\mathbf{S}_{i}\cdot\boldsymbol{\upsigma}\;.$
(4)

Here we neglect SOC and make explicit the dependence of the exchange-
correlation magnetic fields on the directions of the spin moments, with
$B_{i}^{\mathrm{xc}}$ the local magnitude of the fields and
$\boldsymbol{\upsigma}$ the vector of Pauli matrices. The second-order
coefficients of the Taylor expansion in Eq. (Reply to ‘Comment on “Proper and
improper chiral magnetic interactions” ’) follow from the general LKAG-type
expression Liechtenstein  _et al._ (1984, 1987); Antropov  _et al._ (1997,
1999); Udvardi  _et al._ (2003); Ebert and Mankovsky (2009); Lounis and
Dederichs (2010); Szilva  _et al._ (2013); Secchi  _et al._ (2015); Szilva
_et al._ (2017); Cardias  _et al._ (2020, 2020); Mankovsky  _et al._ (2020);
Lounis (2020); Streib  _et al._ (2021), here given for a pair of magnetic
moments:

$\delta^{2}E_{12}=-\frac{1}{\pi}\,\mathrm{Im}\,\mathrm{Tr}\int^{E_{\text{F}}}\mathrm{d}E\;B_{1}^{\mathrm{xc}}\,\delta\mathbf{S}_{1}\cdot\boldsymbol{\upsigma}\,G_{12}\,B_{2}^{\mathrm{xc}}\,\delta\mathbf{S}_{2}\cdot\boldsymbol{\upsigma}\,G_{21}\;.$
(5)

The trace is over spin and orbital degrees of freedom and
$G(E)=\left(E-\mathcal{H}^{\mathrm{el}}\right)^{-1}$. We follow the strategy
of Ref. Brinker  _et al._ (2019) in order to extract the dependence of the
coefficients on the reference magnetic state: Introduce
$G^{0}(E)=\left(E-\mathcal{H}^{0}\right)^{-1}$, expand the Dyson equation
relating $G$ to $G^{0}$ in the local fields
$B_{i}^{\mathrm{xc}}\mathbf{S}_{i}^{0}$ present in the reference state, and
evaluate the spin trace. The lowest order contribution is found to be

$\delta^{2}E_{12}^{(0)}=-\frac{1}{\pi}\,\mathrm{Im}\,\mathrm{Tr}\int^{E_{\text{F}}}\mathrm{d}E\;B_{1}^{\mathrm{xc}}\,G_{12}^{0}\,B_{2}^{\mathrm{xc}}\,G_{21}^{0}\,\big{(}\delta\mathbf{S}_{1}\cdot\delta\mathbf{S}_{2}\big{)}\;.$
(6)

It has the expected form of the isotropic Heisenberg interaction and is
independent of the reference magnetic state. The next non-vanishing
contributions involve the spin directions of the reference state twice, with
one example being (note that $\mathbf{S}_{i}^{0}\cdot\delta\mathbf{S}_{i}=0$)

$\displaystyle\delta^{2}E_{12}^{(2)}$
$\displaystyle=-\frac{1}{\pi}\,\mathrm{Im}\,\mathrm{Tr}\int^{E_{\text{F}}}\mathrm{d}E\;\left(B_{1}^{\mathrm{xc}}\,G_{12}^{0}\,B_{2}^{\mathrm{xc}}\,G_{21}^{0}\right)^{2}$
$\displaystyle\left(\big{(}\mathbf{S}_{1}^{0}\cdot\mathbf{S}_{2}^{0}\big{)}\big{(}\delta\mathbf{S}_{1}\cdot\delta\mathbf{S}_{2}\big{)}-\big{(}\mathbf{S}_{1}^{0}\times\mathbf{S}_{2}^{0}\big{)}\cdot\big{(}\delta\mathbf{S}_{1}\times\delta\mathbf{S}_{2}\big{)}\right)\;.$
(7)

The first term has the form of the isotropic Heisenberg exchange if only the
dependence on $\delta\mathbf{S}_{1}\cdot\delta\mathbf{S}_{2}$ is considered,
but together with the dependence on
$\mathbf{S}_{1}^{0}\cdot\mathbf{S}_{2}^{0}$ it resembles the isotropic
biquadratic interaction. Likewise, the second term has the form of the non-
chiral DMI advocated by the authors of the Comment if only the dependence on
$\delta\mathbf{S}_{1}\times\delta\mathbf{S}_{2}$ is considered, but together
with $\mathbf{S}_{1}^{0}\times\mathbf{S}_{2}^{0}$ it resembles in fact a
rewriting of the isotropic biquadratic interaction, as we discussed in the
Letter. The perturbative expansion can be easily generalized to multiple
magnetic moments and to include SOC Brinker  _et al._ (2019), and as we just
showed it provides the form of the atomistic spin model directly from the
LKAG-type expressions. It also explains why the coefficients in Eq. (Reply to
‘Comment on “Proper and improper chiral magnetic interactions” ’) can vary
strongly with a change in the reference magnetic state.

The preceding discussion leads us to conclude that the non-chiral DMI is not a
fundamental magnetic interaction, as it cannot be derived from any of those
well-established approaches. It arises from considering the coefficients of
the Taylor expansion in Eq. (Reply to ‘Comment on “Proper and improper chiral
magnetic interactions” ’) without constructing the associated mapping given by
Eq. (3), which obscures the dependence of the coefficients on the choice of
the reference magnetic state $\\{\mathbf{S}_{i}^{0}\\}$.

_Global vs. local mapping approaches to the magnetic energy._ We agree with
the distinction between global and local approaches to the mapping of the
magnetic energy discussed at the end of the Comment and in Ref. Streib  _et
al._ (2021). While local approaches characterize the energy landscape near a
reference magnetic state (in the spirit of LKAG), global approaches
parameterize models that reproduce the energies of different magnetic states.
It follows that local and global approaches have different ranges of validity
and applicability (our point (iii) in connection to Eq. (1)). For instance,
local approaches are suitable to compute the magnon spectrum of the magnetic
ground state, while global approaches are required to establish the complete
magnetic phase diagram in an unbiased way. As explained in our Letter and is
motivated by the preceding discussion, it is our view that the atomistic spin
model parametrized through a global approach is more fundamental, as it
complies with all the physical requirements that we used to define ‘proper’
magnetic interactions. The authors of the Comment are correct to point out
that the coefficients of the LKAG expansion worked out and computed in Refs.
Cardias  _et al._ (2020, 2020); Mankovsky  _et al._ (2020) do not have to
comply with those physical requirements, as we argued that these coefficients
should not be directly interpreted as specific magnetic interactions.

_Comparing parameterizations._ On the section ‘Comparing Parametrizations’ of
the Comment, the authors write that the global approach to the magnetic energy
used in our Letter does not have a unique mapping onto a multi-spin model.
This is true concerning the calculations for Mn3Sn, where the fitted
interaction parameters correspond to effective sublattice interactions that
sum over many different types of multi-spin and multi-site interactions. We
remark that we also included interactions among three sublattices and that
this was important for a good fit to the angular dependence of the DFT total
energies. However, our global approach based on the spin cluster expansion is
unique, systematic and complete for the magnetic trimers, due to the finite
magnetic configuration space. Thus it was quite puzzling that we could not
reproduce the very large non-chiral DMI reported in Ref. Cardias  _et al._
(2020) for the Cr trimers on Au(111) using our global approach, c.f. also
previous work in Refs. Antal  _et al._ (2008); Brinker  _et al._ (2020). This
disagreement might be settled by comparing different parameterizations to the
angular dependence of the total energy for the case of the trimers.

_Conclusions._ In this Reply we addressed the various points raised by the
preceding Comment by focusing on the nature of the mapping between DFT total
energy calculations and atomistic spin models, as indicated by Eq. (1). The
local approach to this mapping selects a given magnetic reference state and
proceeds by Taylor expanding the DFT total energy for small spin deviations
around that reference state, which is the spirit of the LKAG approach
Liechtenstein  _et al._ (1984, 1987). Eq. (3) identifies the missing step in
this approach, which demands that the atomistic spin model be subjected to the
same Taylor expansion and the expansion coefficients matched to the ones from
the DFT total energy. In our Letter we already provided a simple example of
how the Taylor expansion of the DFT total energy can contain coefficients that
have the appearance of improper magnetic interactions, such as the non-chiral
DMI, a puzzle which is resolved in general by the procedure encoded in Eq. (3)
and with a concrete example based on the LKAG-type expressions in Eq. (Reply
to ‘Comment on “Proper and improper chiral magnetic interactions” ’).

It is our understanding that the authors of the Comment interpret the
coefficients of the DFT-based LKAG expansion in Eq. (Reply to ‘Comment on
“Proper and improper chiral magnetic interactions” ’) directly as magnetic
interactions, according to the form of the combination of spin components
involved, and do not separately and independently specific the form of the
atomistic spin model according to a set of physical principles, thus skipping
the step given in Eq. (3). The resulting ‘local’ magnetic interactions have
the symmetry properties of the magnetic space group of the reference state
chosen for the Taylor expansion, in analogy with the well-known properties of
electronic transport coefficients in magnetic materials, such as the anomalous
or topological Hall conductivities, and similarly can be given microscopic
interpretations in terms of charge and spin currents, as also done recently in
Ref. Mankovsky  _et al._ (2021) for three-spin interactions.

Our conclusion is that taking the coefficients of the LKAG expansion to
represent directly the magnetic interactions of a given material needs to be
reconsidered, and that the terminology of ‘improper’ vs. ‘proper’ magnetic
interactions introduced in our Letter should be retained given the raised
issues with this interpretation. In particular, we showed that the proposed
‘non-chiral DMI’ cannot be justified either from fundamental microscopic
models or even through the analysis of LKAG-type expressions, and so should
not be considered as a fundamental magnetic interaction, in contrast to the
original chiral one introduced by Dzyaloshinskii and Moriya.

_Acknowledgements._ We gratefully acknowledge financial support from the DARPA
TEE program through grant MIPR (# HR0011831554) from DOI, from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program (Grant No. 856538, project “3D MAGiC” and ERC-consolidator
grant 681405 — DYNASORE), from Deutsche Forschungsgemeinschaft (DFG) through
SPP 2137 “Skyrmionics” (Project BL 444/16 and LO 1659/8-1), Priority Programme
SPP 2244 “2D Materials - Physics of van der Waals Heterostructures” (project
LO 1659/7-1), and the Collaborative Research Centers SFB 1238 (Project C01).
B.N., A.L. and L.S. acknowledge the support provided by the Ministry for
Innovation and Technology and by the National Research, Development, and
Innovation Office under Project No. K131938 and within the Quantum Information
National Laboratory of Hungary. The authors gratefully acknowledge the
computing time granted through JARA-HPC on the supercomputer JURECA at the
Forschungszentrum Jülich Jülich Supercomputing Centre (2018).

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# A Search for Coincident Neutrino Emission from Fast Radio Bursts with Seven
Years of IceCube Cascade Events

R. Abbasi Department of Physics, Loyola University Chicago, Chicago, IL 60660,
USA  M. Ackermann Deutsches Elektronen-Synchrotron DESY, Platanenallee 6,
15738 Zeuthen, Germany  J. Adams  Dept. of Physics and Astronomy, University
of Canterbury, Private Bag 4800, Christchurch, New Zealand  N. Aggarwal  Dept.
of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1  J. A.
Aguilar Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels,
Belgium  M. Ahlers Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  J.M. Alameddine Dept. of Physics, TU Dortmund University,
D-44221 Dortmund, Germany  A. A. Alves Jr  Karlsruhe Institute of Technology,
Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany  N. M. Amin
Bartol Research Institute and Dept. of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA  K. Andeen  Department of Physics, Marquette
University, Milwaukee, WI, 53201, USA  T. Anderson  Dept. of Astronomy and
Astrophysics, Pennsylvania State University, University Park, PA 16802, USA
Dept. of Physics, Pennsylvania State University, University Park, PA 16802,
USA  G. Anton Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-
Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany  C. Argüelles
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  Y. Ashida  Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  S. Athanasiadou  Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  S. N.
Axani Bartol Research Institute and Dept. of Physics and Astronomy, University
of Delaware, Newark, DE 19716, USA  X. Bai Physics Department, South Dakota
School of Mines and Technology, Rapid City, SD 57701, USA  A. Balagopal V.
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  M. Baricevic  Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  S. W. Barwick Dept. of Physics and
Astronomy, University of California, Irvine, CA 92697, USA  V. Basu Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  R. Bay  Dept. of Physics,
University of California, Berkeley, CA 94720, USA  J. J. Beatty Dept. of
Astronomy, Ohio State University, Columbus, OH 43210, USA  Dept. of Physics
and Center for Cosmology and Astro-Particle Physics, Ohio State University,
Columbus, OH 43210, USA  K.-H. Becker  Dept. of Physics, University of
Wuppertal, D-42119 Wuppertal, Germany  J. Becker Tjus Fakultät für Physik &
Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany  J. Beise Dept.
of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden
C. Bellenghi  Physik-department, Technische Universität München, D-85748
Garching, Germany  S. BenZvi Dept. of Physics and Astronomy, University of
Rochester, Rochester, NY 14627, USA  D. Berley  Dept. of Physics, University
of Maryland, College Park, MD 20742, USA  E. Bernardini Dipartimento di Fisica
e Astronomia Galileo Galilei, Università Degli Studi di Padova, 35122 Padova
PD, Italy  D. Z. Besson  Dept. of Physics and Astronomy, University of Kansas,
Lawrence, KS 66045, USA  G. Binder  Dept. of Physics, University of
California, Berkeley, CA 94720, USA  Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  D. Bindig  Dept. of Physics, University of Wuppertal,
D-42119 Wuppertal, Germany  E. Blaufuss Dept. of Physics, University of
Maryland, College Park, MD 20742, USA  S. Blot Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  F. Bontempo
Karlsruhe Institute of Technology, Institute for Astroparticle Physics,
D-76021 Karlsruhe, Germany  J. Y. Book Department of Physics and Laboratory
for Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138,
USA  J. Borowka  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  C. Boscolo Meneguolo Dipartimento di Fisica e Astronomia
Galileo Galilei, Università Degli Studi di Padova, 35122 Padova PD, Italy  S.
Böser Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099
Mainz, Germany  O. Botner Dept. of Physics and Astronomy, Uppsala University,
Box 516, S-75120 Uppsala, Sweden  J. Böttcher  III. Physikalisches Institut,
RWTH Aachen University, D-52056 Aachen, Germany  E. Bourbeau  Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  J. Braun
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  B. Brinson  School of
Physics and Center for Relativistic Astrophysics, Georgia Institute of
Technology, Atlanta, GA 30332, USA  J. Brostean-Kaiser  Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  R. T. Burley
Department of Physics, University of Adelaide, Adelaide, 5005, Australia  R.
S. Busse  Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
D-48149 Münster, Germany  M. A. Campana Dept. of Physics, Drexel University,
3141 Chestnut Street, Philadelphia, PA 19104, USA  E. G. Carnie-Bronca
Department of Physics, University of Adelaide, Adelaide, 5005, Australia  C.
Chen School of Physics and Center for Relativistic Astrophysics, Georgia
Institute of Technology, Atlanta, GA 30332, USA  Z. Chen  Dept. of Physics and
Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA  D. Chirkin
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  S. Choi  Dept. of
Physics, Sungkyunkwan University, Suwon 16419, Korea  B. A. Clark Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
L. Classen  Institut für Kernphysik, Westfälische Wilhelms-Universität
Münster, D-48149 Münster, Germany  A. Coleman Dept. of Physics and Astronomy,
Uppsala University, Box 516, S-75120 Uppsala, Sweden  G. H. Collin  Dept. of
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  A.
Connolly  Dept. of Astronomy, Ohio State University, Columbus, OH 43210, USA
Dept. of Physics and Center for Cosmology and Astro-Particle Physics, Ohio
State University, Columbus, OH 43210, USA  J. M. Conrad Dept. of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  P. Coppin
Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium  P.
Correa Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium
S. Countryman  Columbia Astrophysics and Nevis Laboratories, Columbia
University, New York, NY 10027, USA  D. F. Cowen  Dept. of Astronomy and
Astrophysics, Pennsylvania State University, University Park, PA 16802, USA
Dept. of Physics, Pennsylvania State University, University Park, PA 16802,
USA  C. Dappen  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  P. Dave School of Physics and Center for Relativistic
Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA  C. De
Clercq Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium
J. J. DeLaunay Dept. of Physics and Astronomy, University of Alabama,
Tuscaloosa, AL 35487, USA  D. Delgado López Department of Physics and
Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge,
MA 02138, USA  H. Dembinski Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  K. Deoskar  Oskar
Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm,
Sweden  A. Desai Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin–Madison, Madison, WI 53706, USA  P. Desiati
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  K. D. de Vries Vrije
Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium  G. de
Wasseige Centre for Cosmology, Particle Physics and Phenomenology - CP3,
Université catholique de Louvain, Louvain-la-Neuve, Belgium  T. DeYoung Dept.
of Physics and Astronomy, Michigan State University, East Lansing, MI 48824,
USA  A. Diaz Dept. of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  J. C. Díaz-Vélez Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin–Madison,
Madison, WI 53706, USA  M. Dittmer  Institut für Kernphysik, Westfälische
Wilhelms-Universität Münster, D-48149 Münster, Germany  H. Dujmovic Karlsruhe
Institute of Technology, Institute for Astroparticle Physics, D-76021
Karlsruhe, Germany  M. A. DuVernois Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI
53706, USA  T. Ehrhardt  Institute of Physics, University of Mainz, Staudinger
Weg 7, D-55099 Mainz, Germany  P. Eller Physik-department, Technische
Universität München, D-85748 Garching, Germany  R. Engel  Karlsruhe Institute
of Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
Karlsruhe Institute of Technology, Institute of Experimental Particle Physics,
D-76021 Karlsruhe, Germany  H. Erpenbeck  III. Physikalisches Institut, RWTH
Aachen University, D-52056 Aachen, Germany  J. Evans  Dept. of Physics,
University of Maryland, College Park, MD 20742, USA  P. A. Evenson  Bartol
Research Institute and Dept. of Physics and Astronomy, University of Delaware,
Newark, DE 19716, USA  K. L. Fan  Dept. of Physics, University of Maryland,
College Park, MD 20742, USA  A. R. Fazely Dept. of Physics, Southern
University, Baton Rouge, LA 70813, USA  A. Fedynitch Institute of Physics,
Academia Sinica, Taipei, 11529, Taiwan  N. Feigl  Institut für Physik,
Humboldt-Universität zu Berlin, D-12489 Berlin, Germany  S. Fiedlschuster
Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, D-91058 Erlangen, Germany  A. T. Fienberg  Dept. of
Physics, Pennsylvania State University, University Park, PA 16802, USA  C.
Finley Oskar Klein Centre and Dept. of Physics, Stockholm University, SE-10691
Stockholm, Sweden  L. Fischer  Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany  D. Fox Dept. of Astronomy and
Astrophysics, Pennsylvania State University, University Park, PA 16802, USA
A. Franckowiak Fakultät für Physik & Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany  E. Friedman  Dept. of Physics, University of
Maryland, College Park, MD 20742, USA  A. Fritz  Institute of Physics,
University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  P. Fürst  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  T.
K. Gaisser Bartol Research Institute and Dept. of Physics and Astronomy,
University of Delaware, Newark, DE 19716, USA  J. Gallagher  Dept. of
Astronomy, University of Wisconsin–Madison, Madison, WI 53706, USA  E. Ganster
III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany
A. Garcia Department of Physics and Laboratory for Particle Physics and
Cosmology, Harvard University, Cambridge, MA 02138, USA  S. Garrappa Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  L.
Gerhardt  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  A.
Ghadimi Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL
35487, USA  C. Glaser  Dept. of Physics and Astronomy, Uppsala University, Box
516, S-75120 Uppsala, Sweden  T. Glauch Physik-department, Technische
Universität München, D-85748 Garching, Germany  T. Glüsenkamp Erlangen Centre
for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany  N. Goehlke  Karlsruhe Institute of Technology,
Institute of Experimental Particle Physics, D-76021 Karlsruhe, Germany  J. G.
Gonzalez  Bartol Research Institute and Dept. of Physics and Astronomy,
University of Delaware, Newark, DE 19716, USA  S. Goswami  Dept. of Physics
and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  D. Grant
Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI
48824, USA  S. J. Gray Dept. of Physics, University of Maryland, College Park,
MD 20742, USA  T. Grégoire  Dept. of Physics, Pennsylvania State University,
University Park, PA 16802, USA  S. Griffin  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin–Madison,
Madison, WI 53706, USA  S. Griswold Dept. of Physics and Astronomy, University
of Rochester, Rochester, NY 14627, USA  C. Günther  III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  P. Gutjahr Dept. of
Physics, TU Dortmund University, D-44221 Dortmund, Germany  C. Haack  Physik-
department, Technische Universität München, D-85748 Garching, Germany  A.
Hallgren Dept. of Physics and Astronomy, Uppsala University, Box 516, S-75120
Uppsala, Sweden  R. Halliday  Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  L. Halve III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  F. Halzen Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  H. Hamdaoui  Dept. of Physics and
Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA  M. Ha Minh
Physik-department, Technische Universität München, D-85748 Garching, Germany
K. Hanson  Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin–Madison, Madison, WI 53706, USA  J. Hardin
Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  A. A. Harnisch  Dept.
of Physics and Astronomy, Michigan State University, East Lansing, MI 48824,
USA  P. Hatch  Dept. of Physics, Engineering Physics, and Astronomy, Queen’s
University, Kingston, ON K7L 3N6, Canada  A. Haungs Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
K. Helbing Dept. of Physics, University of Wuppertal, D-42119 Wuppertal,
Germany  J. Hellrung  III. Physikalisches Institut, RWTH Aachen University,
D-52056 Aachen, Germany  F. Henningsen Physik-department, Technische
Universität München, D-85748 Garching, Germany  L. Heuermann  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  S.
Hickford  Dept. of Physics, University of Wuppertal, D-42119 Wuppertal,
Germany  A. Hidvegi  Oskar Klein Centre and Dept. of Physics, Stockholm
University, SE-10691 Stockholm, Sweden  C. Hill Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  G. C. Hill  Department of Physics, University of Adelaide,
Adelaide, 5005, Australia  K. D. Hoffman  Dept. of Physics, University of
Maryland, College Park, MD 20742, USA  K. Hoshina  also at Earthquake Research
Institute, University of Tokyo, Bunkyo, Tokyo 113-0032, Japan  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  W. Hou Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
T. Huber Karlsruhe Institute of Technology, Institute for Astroparticle
Physics, D-76021 Karlsruhe, Germany  K. Hultqvist Oskar Klein Centre and Dept.
of Physics, Stockholm University, SE-10691 Stockholm, Sweden  M. Hünnefeld
Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany  R.
Hussain  Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  K. Hymon  Dept. of
Physics, TU Dortmund University, D-44221 Dortmund, Germany  S. In  Dept. of
Physics, Sungkyunkwan University, Suwon 16419, Korea  N. Iovine Université
Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  A.
Ishihara  Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  M. Jansson  Oskar Klein
Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden
G. S. Japaridze CTSPS, Clark-Atlanta University, Atlanta, GA 30314, USA  M.
Jeong  Dept. of Physics, Sungkyunkwan University, Suwon 16419, Korea  M. Jin
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  B. J. P. Jones Dept. of Physics,
University of Texas at Arlington, 502 Yates St., Science Hall Rm 108, Box
19059, Arlington, TX 76019, USA  D. Kang Karlsruhe Institute of Technology,
Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany  W. Kang Dept.
of Physics, Sungkyunkwan University, Suwon 16419, Korea  X. Kang  Dept. of
Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
A. Kappes Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
D-48149 Münster, Germany  D. Kappesser  Institute of Physics, University of
Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  L. Kardum  Dept. of Physics,
TU Dortmund University, D-44221 Dortmund, Germany  T. Karg Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  M. Karl
Physik-department, Technische Universität München, D-85748 Garching, Germany
A. Karle Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  U. Katz Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, D-91058 Erlangen, Germany  M. Kauer Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin–Madison,
Madison, WI 53706, USA  J. L. Kelley Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI
53706, USA  A. Kheirandish Department of Physics & Astronomy, University of
Nevada, Las Vegas, NV, 89154, USA  Nevada Center for Astrophysics, University
of Nevada, Las Vegas, NV 89154, USA  K. Kin  Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  J. Kiryluk Dept. of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794-3800, USA  S. R. Klein Dept. of Physics,
University of California, Berkeley, CA 94720, USA  Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  A. Kochocki Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  R. Koirala
Bartol Research Institute and Dept. of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA  H. Kolanoski Institut für Physik, Humboldt-
Universität zu Berlin, D-12489 Berlin, Germany  T. Kontrimas  Physik-
department, Technische Universität München, D-85748 Garching, Germany  L.
Köpke  Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099
Mainz, Germany  C. Kopper Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  D. J. Koskinen Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark  P. Koundal Karlsruhe
Institute of Technology, Institute for Astroparticle Physics, D-76021
Karlsruhe, Germany  M. Kovacevich Dept. of Physics, Drexel University, 3141
Chestnut Street, Philadelphia, PA 19104, USA  M. Kowalski Institut für Physik,
Humboldt-Universität zu Berlin, D-12489 Berlin, Germany  Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  T. Kozynets  Niels
Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  K.
Kruiswijk  Centre for Cosmology, Particle Physics and Phenomenology - CP3,
Université catholique de Louvain, Louvain-la-Neuve, Belgium  E. Krupczak
Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI
48824, USA  A. Kumar Deutsches Elektronen-Synchrotron DESY, Platanenallee 6,
15738 Zeuthen, Germany  E. Kun  Fakultät für Physik & Astronomie, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  N. Kurahashi Dept. of Physics,
Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA  N. Lad
Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
C. Lagunas Gualda Deutsches Elektronen-Synchrotron DESY, Platanenallee 6,
15738 Zeuthen, Germany  M. Lamoureux  Centre for Cosmology, Particle Physics
and Phenomenology - CP3, Université catholique de Louvain, Louvain-la-Neuve,
Belgium  M. J. Larson Dept. of Physics, University of Maryland, College Park,
MD 20742, USA  F. Lauber Dept. of Physics, University of Wuppertal, D-42119
Wuppertal, Germany  J. P. Lazar Department of Physics and Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  J. W. Lee Dept. of
Physics, Sungkyunkwan University, Suwon 16419, Korea  K. Leonard DeHolton
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  A. Leszczyńska Bartol
Research Institute and Dept. of Physics and Astronomy, University of Delaware,
Newark, DE 19716, USA  M. Lincetto  Fakultät für Physik & Astronomie, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  Q. R. Liu Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  M. Liubarska  Dept. of Physics,
University of Alberta, Edmonton, Alberta, Canada T6G 2E1  E. Lohfink
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  C. Love  Dept. of Physics, Drexel University, 3141 Chestnut Street,
Philadelphia, PA 19104, USA  C. J. Lozano Mariscal  Institut für Kernphysik,
Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany  L. Lu
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  F. Lucarelli
Département de physique nucléaire et corpusculaire, Université de Genève,
CH-1211 Genève, Switzerland  A. Ludwig Department of Physics and Astronomy,
UCLA, Los Angeles, CA 90095, USA  W. Luszczak Dept. of Astronomy, Ohio State
University, Columbus, OH 43210, USA  Dept. of Physics and Center for Cosmology
and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  Y. Lyu Dept. of
Physics, University of California, Berkeley, CA 94720, USA  Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  W. Y. Ma Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  J. Madsen Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  K. B. M. Mahn  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  Y. Makino
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  S. Mancina  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  W. Marie Sainte  Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  I. C. Mariş Université Libre de
Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  S. Marka  Columbia
Astrophysics and Nevis Laboratories, Columbia University, New York, NY 10027,
USA  Z. Marka  Columbia Astrophysics and Nevis Laboratories, Columbia
University, New York, NY 10027, USA  M. Marsee  Dept. of Physics and
Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  I. Martinez-Soler
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  R. Maruyama Dept. of Physics,
Yale University, New Haven, CT 06520, USA  F. Mayhew  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  T. McElroy
Dept. of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1  F.
McNally Department of Physics, Mercer University, Macon, GA 31207-0001, USA
J. V. Mead  Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  K. Meagher Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI
53706, USA  S. Mechbal  Deutsches Elektronen-Synchrotron DESY, Platanenallee
6, 15738 Zeuthen, Germany  A. Medina  Dept. of Physics and Center for
Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH
43210, USA  M. Meier Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  S. Meighen-Berger
Physik-department, Technische Universität München, D-85748 Garching, Germany
Y. Merckx  Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels,
Belgium  L. Merten  Fakultät für Physik & Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany  J. Micallef  Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  D. Mockler  Université Libre de
Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  T. Montaruli
Département de physique nucléaire et corpusculaire, Université de Genève,
CH-1211 Genève, Switzerland  R. W. Moore Dept. of Physics, University of
Alberta, Edmonton, Alberta, Canada T6G 2E1  Y. Morii  Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  R. Morse  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
M. Moulai Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  T. Mukherjee
Karlsruhe Institute of Technology, Institute for Astroparticle Physics,
D-76021 Karlsruhe, Germany  R. Naab Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany  R. Nagai Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  U. Naumann  Dept. of Physics, University of Wuppertal,
D-42119 Wuppertal, Germany  A. Nayerhoda Dipartimento di Fisica e Astronomia
Galileo Galilei, Università Degli Studi di Padova, 35122 Padova PD, Italy  J.
Necker Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen,
Germany  M. Neumann  Institut für Kernphysik, Westfälische Wilhelms-
Universität Münster, D-48149 Münster, Germany  H. Niederhausen Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
M. U. Nisa Dept. of Physics and Astronomy, Michigan State University, East
Lansing, MI 48824, USA  A. Noell  III. Physikalisches Institut, RWTH Aachen
University, D-52056 Aachen, Germany  S. C. Nowicki  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  A.
Obertacke Pollmann Dept. of Physics, University of Wuppertal, D-42119
Wuppertal, Germany  M. Oehler  Karlsruhe Institute of Technology, Institute
for Astroparticle Physics, D-76021 Karlsruhe, Germany  B. Oeyen Dept. of
Physics and Astronomy, University of Gent, B-9000 Gent, Belgium  A. Olivas
Dept. of Physics, University of Maryland, College Park, MD 20742, USA  R.
Orsoe  Physik-department, Technische Universität München, D-85748 Garching,
Germany  J. Osborn  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
E. O’Sullivan Dept. of Physics and Astronomy, Uppsala University, Box 516,
S-75120 Uppsala, Sweden  H. Pandya Bartol Research Institute and Dept. of
Physics and Astronomy, University of Delaware, Newark, DE 19716, USA  D. V.
Pankova  Dept. of Physics, Pennsylvania State University, University Park, PA
16802, USA  N. Park Dept. of Physics, Engineering Physics, and Astronomy,
Queen’s University, Kingston, ON K7L 3N6, Canada  G. K. Parker  Dept. of
Physics, University of Texas at Arlington, 502 Yates St., Science Hall Rm 108,
Box 19059, Arlington, TX 76019, USA  E. N. Paudel Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  L. Paul  Department of Physics, Marquette University, Milwaukee, WI,
53201, USA  C. Pérez de los Heros Dept. of Physics and Astronomy, Uppsala
University, Box 516, S-75120 Uppsala, Sweden  J. Peterson  Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  S. Philippen  III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  S. Pieper  Dept. of
Physics, University of Wuppertal, D-42119 Wuppertal, Germany  A. Pizzuto Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  M. Plum Physics Department, South
Dakota School of Mines and Technology, Rapid City, SD 57701, USA  Y. Popovych
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  A. Porcelli Dept. of Physics and Astronomy, University of Gent,
B-9000 Gent, Belgium  M. Prado Rodriguez  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin–Madison,
Madison, WI 53706, USA  B. Pries Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  R. Procter-Murphy  Dept. of
Physics, University of Maryland, College Park, MD 20742, USA  G. T. Przybylski
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  C. Raab
Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium
J. Rack-Helleis  Institute of Physics, University of Mainz, Staudinger Weg 7,
D-55099 Mainz, Germany  M. Rameez Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  K. Rawlins  Dept. of Physics and
Astronomy, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK
99508, USA  Z. Rechav  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
A. Rehman Bartol Research Institute and Dept. of Physics and Astronomy,
University of Delaware, Newark, DE 19716, USA  P. Reichherzer  Fakultät für
Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany  G.
Renzi  Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels,
Belgium  E. Resconi Physik-department, Technische Universität München, D-85748
Garching, Germany  S. Reusch  Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany  W. Rhode Dept. of Physics, TU
Dortmund University, D-44221 Dortmund, Germany  M. Richman  Dept. of Physics,
Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA  B.
Riedel Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  E. J. Roberts
Department of Physics, University of Adelaide, Adelaide, 5005, Australia  S.
Robertson  Dept. of Physics, University of California, Berkeley, CA 94720, USA
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  S. Rodan
Dept. of Physics, Sungkyunkwan University, Suwon 16419, Korea  G. Roellinghoff
Dept. of Physics, Sungkyunkwan University, Suwon 16419, Korea  M. Rongen
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  C. Rott Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112, USA  Dept. of Physics, Sungkyunkwan University, Suwon
16419, Korea  T. Ruhe  Dept. of Physics, TU Dortmund University, D-44221
Dortmund, Germany  L. Ruohan  Physik-department, Technische Universität
München, D-85748 Garching, Germany  D. Ryckbosch  Dept. of Physics and
Astronomy, University of Gent, B-9000 Gent, Belgium  D. Rysewyk Cantu Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
S.Athanasiadou  Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738
Zeuthen, Germany  I. Safa Department of Physics and Laboratory for Particle
Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  J. Saffer  Karlsruhe Institute of
Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe,
Germany  D. Salazar-Gallegos Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  P. Sampathkumar  Karlsruhe Institute
of Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
S. E. Sanchez Herrera  Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  A. Sandrock Dept. of Physics, TU
Dortmund University, D-44221 Dortmund, Germany  M. Santander Dept. of Physics
and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  S. Sarkar
Dept. of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1  S.
Sarkar Dept. of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
J. Savelberg  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  P. Savina  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
M. Schaufel  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  H. Schieler  Karlsruhe Institute of Technology, Institute for
Astroparticle Physics, D-76021 Karlsruhe, Germany  S. Schindler Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, D-91058 Erlangen, Germany  B. Schlueter  Institut für Kernphysik,
Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany  T.
Schmidt  Dept. of Physics, University of Maryland, College Park, MD 20742, USA
J. Schneider Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-
Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany  F. G. Schröder
Karlsruhe Institute of Technology, Institute for Astroparticle Physics,
D-76021 Karlsruhe, Germany  Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  L. Schumacher
Physik-department, Technische Universität München, D-85748 Garching, Germany
G. Schwefer  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  S. Sclafani Dept. of Physics, Drexel University, 3141
Chestnut Street, Philadelphia, PA 19104, USA  D. Seckel  Bartol Research
Institute and Dept. of Physics and Astronomy, University of Delaware, Newark,
DE 19716, USA  S. Seunarine  Dept. of Physics, University of Wisconsin, River
Falls, WI 54022, USA  A. Sharma  Dept. of Physics and Astronomy, Uppsala
University, Box 516, S-75120 Uppsala, Sweden  S. Shefali  Karlsruhe Institute
of Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe,
Germany  N. Shimizu  Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  M. Silva Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  B. Skrzypek  Department of Physics
and Laboratory for Particle Physics and Cosmology, Harvard University,
Cambridge, MA 02138, USA  B. Smithers Dept. of Physics, University of Texas at
Arlington, 502 Yates St., Science Hall Rm 108, Box 19059, Arlington, TX 76019,
USA  R. Snihur  Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin–Madison, Madison, WI 53706, USA  J.
Soedingrekso  Dept. of Physics, TU Dortmund University, D-44221 Dortmund,
Germany  A. Søgaard  Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  D. Soldin Karlsruhe Institute of Technology, Institute of
Experimental Particle Physics, D-76021 Karlsruhe, Germany  C. Spannfellner
Physik-department, Technische Universität München, D-85748 Garching, Germany
G. M. Spiczak Dept. of Physics, University of Wisconsin, River Falls, WI
54022, USA  C. Spiering Deutsches Elektronen-Synchrotron DESY, Platanenallee
6, 15738 Zeuthen, Germany  M. Stamatikos  Dept. of Physics and Center for
Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH
43210, USA  T. Stanev  Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  R. Stein Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  T.
Stezelberger Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
T. Stürwald  Dept. of Physics, University of Wuppertal, D-42119 Wuppertal,
Germany  T. Stuttard Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  G. W. Sullivan Dept. of Physics, University of Maryland,
College Park, MD 20742, USA  I. Taboada School of Physics and Center for
Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332,
USA  S. Ter-Antonyan Dept. of Physics, Southern University, Baton Rouge, LA
70813, USA  W. G. Thompson Department of Physics and Laboratory for Particle
Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA  J.
Thwaites  Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  S. Tilav  Bartol
Research Institute and Dept. of Physics and Astronomy, University of Delaware,
Newark, DE 19716, USA  K. Tollefson Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  C. Tönnis  Dept. of Physics,
Sungkyunkwan University, Suwon 16419, Korea  S. Toscano Université Libre de
Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  D. Tosi  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  A. Trettin  Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  C. F. Tung School
of Physics and Center for Relativistic Astrophysics, Georgia Institute of
Technology, Atlanta, GA 30332, USA  R. Turcotte  Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
J. P. Twagirayezu  Dept. of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824, USA  B. Ty  Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI
53706, USA  M. A. Unland Elorrieta Institut für Kernphysik, Westfälische
Wilhelms-Universität Münster, D-48149 Münster, Germany  K. Upshaw  Dept. of
Physics, Southern University, Baton Rouge, LA 70813, USA  N. Valtonen-Mattila
Dept. of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala,
Sweden  J. Vandenbroucke Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
N. van Eijndhoven Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050
Brussels, Belgium  D. Vannerom  Dept. of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  J. van Santen Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany  J. Vara  Institut
für Kernphysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster,
Germany  J. Veitch-Michaelis  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA
S. Verpoest Dept. of Physics and Astronomy, University of Gent, B-9000 Gent,
Belgium  D. Veske  Columbia Astrophysics and Nevis Laboratories, Columbia
University, New York, NY 10027, USA  C. Walck  Oskar Klein Centre and Dept. of
Physics, Stockholm University, SE-10691 Stockholm, Sweden  T. B. Watson Dept.
of Physics, University of Texas at Arlington, 502 Yates St., Science Hall Rm
108, Box 19059, Arlington, TX 76019, USA  C. Weaver Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  P. Weigel
Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  A. Weindl  Karlsruhe Institute of Technology, Institute for Astroparticle
Physics, D-76021 Karlsruhe, Germany  J. Weldert  Institute of Physics,
University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  C. Wendt Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  J. Werthebach  Dept. of Physics, TU
Dortmund University, D-44221 Dortmund, Germany  M. Weyrauch  Karlsruhe
Institute of Technology, Institute for Astroparticle Physics, D-76021
Karlsruhe, Germany  N. Whitehorn Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA  C. H. Wiebusch III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  N.
Willey  Dept. of Physics and Astronomy, Michigan State University, East
Lansing, MI 48824, USA  D. R. Williams  Dept. of Physics and Astronomy,
University of Alabama, Tuscaloosa, AL 35487, USA  M. Wolf Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  G. Wrede  Erlangen Centre for
Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany  J. Wulff  Fakultät für Physik & Astronomie, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  X. W. Xu  Dept. of Physics,
Southern University, Baton Rouge, LA 70813, USA  J. P. Yanez  Dept. of
Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1  E. Yildizci
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin–Madison, Madison, WI 53706, USA  S. Yoshida Dept. of
Physics and The International Center for Hadron Astrophysics, Chiba
University, Chiba 263-8522, Japan  S. Yu  Dept. of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  T. Yuan Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin–Madison, Madison, WI 53706, USA  Z. Zhang  Dept. of Physics and
Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA  P. Zhelnin
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA

###### Abstract

This paper presents the results of a search for neutrinos that are spatially
and temporally coincident with 22 unique, non-repeating Fast Radio Bursts
(FRBs) and one repeating FRB (FRB121102). FRBs are a rapidly growing class of
Galactic and extragalactic astrophysical objects that are considered a
potential source of high-energy neutrinos. The IceCube Neutrino Observatory’s
previous FRB analyses have solely used track events. This search utilizes
seven years of IceCube’s cascade events which are statistically independent of
the track events. This event selection allows probing of a longer range of
extended timescales due to the low background rate. No statistically
significant clustering of neutrinos was observed. Upper limits are set on the
time-integrated neutrino flux emitted by FRBs for a range of extended time-
windows.

##  1 Introduction

The IceCube Neutrino Observatory, located at the geographic South Pole, is the
largest neutrino detector in the world. Encompassing a cubic kilometer of
instrumented ice, IceCube is comprised of 5,160 digital optical modules (DOMs)
situated on 86 read-out and support cables or “strings” to detect the
Cherenkov radiation from charged particles created by neutrino interactions in
the Antarctic ice (Abbasi et al., 2009; Aartsen et al., 2017). IceCube’s
instrumentation density is optimized for the detection of neutrinos with
energies from 100 GeV – 10 PeV. It contains a higher density sub-volume,
enabling the detection of neutrinos down to 10 GeV (Abbasi et al., 2012).
IceCube has observed a diffuse flux of high-energy astrophysical neutrinos
(Aartsen et al., 2013, 2014a, 2015). A study of data collected between April
6, 2008 and July 10, 2018 revealed a 3.3$\sigma$ inconsistency with background
expectations for four sources that include NGC 1068 and TXS 0506+056 (Aartsen
et al., 2020b). In 2022, IceCube found further evidence of neutrino emission
from NGC 1068 at a significance of 4.2$\sigma$ (Abbasi et al., 2022). Despite
this growing evidence, the origin of the majority of the diffuse astrophysical
flux remains unexplained. In this paper, we present a search for time-
dependent neutrino emission from various Fast Radio Bursts (FRBs) using seven
years of IceCube’s cascade events and set upper limits on the associated
neutrino flux for flares of varying duration.

Transient astrophysical objects are among the primary candidates for producing
the astrophysical neutrino flux (Murase & Bartos, 2019). The first evidence of
neutrino emission from a flaring object came from the blazar TXS 0506+056
(Aartsen et al., 2018a, b). FRBs are a class of transient astrophysical
objects that could contribute to the diffuse neutrino flux (Metzger et al.,
2020). FRBs are periodic or non-periodic transient radio bursts that have
Galactic or extragalactic origins. To date, hundreds of FRBs across the entire
sky have been detected. Recently, FRB200448 was localized to the Galactic
magnetar SGR 1935+2154, suggesting magnetars may be a source of FRBs (Anderson
et al., 2020). This has been further supported by the polarization of some
FRBs (Wang et al., 2021). While the underlying mechanism that creates FRBs is
unknown, it is predicted that this coherent radio emission is the result of an
ultra-relativistic shock that propagates into a baryon filled medium. A by-
product of this scenario are TeV – PeV neutrinos that are produced by photo-
hadronic interactions on timescales of varying duration after the FRB (Metzger
et al., 2020; Qu & Zhang, 2022). Here, we perform a time-dependent stacking
search to test if two catalogs of FRBs, one repeating and the other non-
repeating, are producing a statistically significant number of neutrinos.

##  2 Search for Correlated Cascade Events

The majority of events detected by IceCube can be divided into two topological
classes: cascades and tracks. Cascade signatures are produced by charged-
current electron neutrino and tau neutrino interactions as well as all-flavor
neutral-current interactions. These interactions typically produce
electromagnetic and hadronic showers that have a range of 20 meters or less;
this range scales with energy (Aartsen et al., 2014b). Due to the average DOM
spacing and light scattering within the ice, these particle showers lead to
cascade events having angular resolutions $\sim$10∘ – 15∘ (Aartsen et al.,
2019). In contrast, track events are the result of muons that are produced by
charged-current muon neutrino interactions and have an angular resolution of
less than 1∘ at TeV – PeV energies (Abbasi et al., 2021a). Track events are
subject to larger background rates that stem from the atmospheric muons. To
reduce this, track events below a certain energy threshold are filtered out
(Abbasi et al., 2021a). The distinctive topology of cascade events allows our
selections to include events that have energies down to hundreds of GeV
(Aartsen et al., 2019).

To search for correlations between FRBs and cascade events, the energy,
spatial, and temporal information of each cascade event is used with the
spatial and temporal information of each FRB. This method is similar to that
of IceCube’s previous FRB analyses, referred to as Six-year Southern Tracks
(Aartsen et al., 2018c) and All-Sky Tracks (Aartsen et al., 2020c), which
analyzed track data for correlations.

###  2.1 Dataset and FRB Catalogs

This analysis uses IceCube’s Medium Energy Starting Cascade (MESC) data-set. A
starting event is classified as an event that contains the neutrino
interaction vertex in the detector. This data-set was originally developed to
search for all-sky signatures of potential neutrino sources such as the
Galactic plane (Aartsen et al., 2019). It contains 1,980 cascade events
detected from May 2010 to May 2017. This corresponds to 2428 days of IceCube
livetime. The first year of data was taken with a 79-string detector
configuration and the remaining six years use the complete 86-string
configuration. The events in this data-set have energies that range between
270 GeV and 1.6 PeV.

Figure 1: The spatial distribution of FRBs in equatorial coordinates. The 22
unique, non-repeating FRBs are shown in red and FRB121102 is shown in blue.
FRB121102 is an extragalactic source located at a declination of 33.15∘
(Spitler et al., 2014).

This analysis focuses on 48 unique bursts that were detected between May 2010
- May 2017; the bursts are separated into two catalogs (Petroff et al., 2016).
As shown in Fig. 1, the first catalog contains 22 non-repeating FRBs that are
located at a declination of six degrees or below. The second catalog contains
one repeating FRB (FRB121102) that produced 26 bursts within the livetime of
the MESC data-set. Since FRBs are an observational class of astrophysical
phenomena, non-repeating and repeating FRBs could be associated with different
classes of astrophysical objects. As the underlying physical processes may
differ, we treat them in this analysis as independent phenomena. These
catalogs are a subset of the FRBs that were analyzed in the Six Year Southern
Tracks and All-Sky Tracks analyses. This is because the MESC dataset has a
livetime period that does not fully overlap with the track datasets. The
catalogs of non-repeating and repeating FRBs are independently analyzed
through two time-dependent stacking analyses.

Figure 2: The 90% confidence level upper limits as a function of time window
duration. We assume power law spectra of E-2 (left) and E-3 (right) for the
time-integrated neutrino fluxes. The Six-Year Southern Tracks and All-Sky
Tracks analyses used statistically independent events; they are included in
addition to our limits. Note that previous IceCube analyses used symmetric
time windows that are centered on the FRB to search for neutrino emission;
this analysis only searches for neutrino emission after the FRB has occurred.
Hence $\Delta T$is offset by $\frac{\Delta T}{2}$ between this analysis and
IceCube’s previous analyses and the time windows do not completely overlap.

###  2.2 Analysis Methods

This analysis uses an unbinned extended maximum likelihood method, similar to
the previous analyses (Braun et al., 2010; Aartsen et al., 2018c). The
likelihood is comprised of spatial, temporal, and energy probability density
functions (PDFs). These PDFs characterize how signal-like or background-like a
given cascade is. To test if a cascade and FRB are temporally correlated, a
search time window, $\Delta T,$ is constructed in the interval
$[t_{FRB},\>t_{FRB}+\Delta T]$ where $t_{FRB}$ denotes when the FRB was
detected. Motivated by Metzger et al. (2020), we test eight different $\Delta
T$that are logarithmically spaced from [10-2, 107] seconds after the FRB has
occurred.111The Six Year Southern Tracks and All-Sky Track analyses searched
for neutrino emission in the following intervals $[t_{FRB}$ \- $\frac{\Delta
T}{2}$, $t_{FRB}$ \+ $\frac{\Delta T}{2}$]. The spatial PDF accounts for the
angular distance between a given FRB and the reconstructed cascade direction.
A 2-D Gaussian spatial PDF is used for cascades with an angular resolution
less than 7∘, while for a resolution greater than 7∘ a Kent Distribution is
used. At larger angular resolutions, a Kent Distribution characterizes a
probability distribution in curved space and cannot be approximated by a
Gaussian distribution in flat space. The energy PDF characterizes the neutrino
flux in terms of a powerlaw, $E^{-\gamma}$, where $\gamma$ is the spectral
index. We assume a single power-law for the energy flux. From the likelihood,
we construct a test statistic (TS) that encompasses our null and alternative
hypotheses,

$\textrm{TS}=-2\log\frac{\mathcal{L}(n_{s}=0)}{\mathcal{L}(n_{s},\gamma)}.$
(1)

The null hypothesis represents background expectations $(n_{s}=0)$whereas the
alternative hypothesis fits for two parameters: the number of signal events,
$n_{s},$and $\gamma$. The number of events in a given time window can vary
according to Poisson statistics. In each $\Delta T,$a small number of events
are required to observe a statistically significant correlation. To perform a
stacking search, the TS in Eq. 1 is additionally summed over each FRB in the
catalog.

##  3 Results

We observed no significant emission in this analysis resulting in TS = 0 for
every time-window. Upper limits are calculated for both catalogs at the 90%
confidence level for the time-integrated flux per FRB for every $\Delta T$
(Figure 2). The upper limits assume a flavor ratio
($\nu_{e}$:$\nu_{\mu}$:$\nu_{\tau}$) of 1:1:1 with equal parts from $\nu$ and
$\overline{\nu}$. From 10-2 to approximately 106 seconds, the upper limits are
relatively constant due to a low background rate. The background saturation
occurs 106 seconds after the FRB.

For $E^{-2}$(Figure 2 \- left) and $E^{-3}$(Figure 2 \- right) spectra, we
find that the stacked upper limits with cascade events are comparable to Six
Year Southern Tracks and All-Sky Tracks results (Figure 2). For $E^{-3},$this
analysis offers improved upper limits with respect to the Six-year Tracks. For
$E^{-2}$this analysis is not quite as sensitive as the Six-Year Tracks when
searching for neutrino emission in $\Delta T$less than 104 seconds. The flux
allowed by the All-sky tracks analysis increases by an order of magnitude
between $\Delta T$that have durations of sub-second to days. This is due to
the rapidly increasing background rate which requires more signal events to
see a statistically significant correlation. As noted above, this is in
contrast with cascade events. Finally, we find that for harder spectra, the
catalog of non-repeating FRBs offers slightly more constraining upper limits
when compared with FRB121102. This is in contrast with our upper limits for
soft spectra where FRB121102 has slightly stronger upper limits.

Figure 3: The 90% confidence level upper limits as a function of time windows
assuming the best-fit diffuse spectrum $(E^{-2.53})$measured in cascades
(Aartsen et al., 2020a). The Diffuse Astrophysical Flux Constraint assumes
that FRBs are solely responsible for the diffuse neutrino flux. The constraint
is calculated by dividing the entire diffuse astrophysical flux equally
amongst 820 homogeneous FRBs. Note that we assume 820 FRBs per day from the
CHIME experiment’s estimations of the FRB all-sky rate (Amiri et al., 2021).

These results are extended to $E^{-2.53}$to draw comparisons to the measured
diffuse flux that has been observed in IceCube’s cascade analyses (Aartsen et
al., 2020a). The diffuse flux is an all-sky high-energy neutrino flux. Since
FRBs are an all-sky phenomena, these results are relevant. Figure 3 compares
the per-burst upper limits for each catalog to the diffuse cascade
astrophysical flux by integrating the diffuse flux over both the duration of
each $\Delta T$and the entire sky then dividing by the Canadian Hydrogen
Intensity Mapping Experiment’s (CHIME) estimated FRB all-sky rate of 820 FRBs
per day (Amiri et al., 2021). This assumes that each burst contributes equally
to the diffuse all-sky cascades. This flux per FRB is then compared to the
$E^{-2.53}$ upper limits from this analysis. Using this method of comparison,
we see that the measured diffuse cascade flux establishes the most stringent
limit on neutrino emission from FRBs.

##  4 Conclusion and Future Outlook

Figure 4: Projected sensitivities for various time-windows durations as the
number of FRBs in a catalog increases. We assume a powerlaw spectrum of
$E^{-3}$for the time-integrated neutrino fluxes. Future catalogs are expected
to have orders of magnitude more FRBs. The projected sensitivities are
calculated at the 90% confidence level and use simulated FRBs that are
uniformly distributed in the southern sky.

In two independent searches for neutrino emission from 22 unique, non-
repeating FRBs and 26 unique bursts from FRB121102, no significant correlation
was found. We provide upper limits on the time integrated neutrino flux at the
90% confidence level for various spectral indices, as shown in Figures 2 and
3. We also provide estimates on the neutrino flux for FRB catalogs of various
sizes (Figure 4). These estimates show that for non-periodic FRBs that are
isotropically distributed throughout the southern sky, the neutrino flux per
FRB decreases as the catalog size increases. In recent years more FRB
observatories, such as CHIME, have come online. This has increased the
detection rate of both repeating and non-repeating FRBs. However, we were only
able to analyze a subset of FRBs that overlapped with our data-set. New data-
sets are in preparation that would overlap with 100 – 1,000 FRBs and
potentially offer more stringent upper limits (Petroff et al., 2016; Amiri et
al., 2021).

Overall, this analysis shows that cascades offer sensitive upper limits when
performing transient stacking searches. Given that cascades have a low
background rate and are considered independent of track events, IceCube’s
future analyses would benefit from combining track and cascade events when
searching for time-dependent neutrino emission from FRBs. This approach can be
extended to IceCube’s potential searches that aim to observe transient
neutrino emission in realtime as well as gamma ray bursts (Abbasi et al.,
2021b; Aartsen et al., 2016). In addition, the next-generation of IceCube,
IceCube-Gen2, will provide the opportunity to conduct more sensitive searches
for neutrino emission from transient sources (Aartsen et al., 2021). In turn,
this will allow us to uncover the origin of high-energy astrophysical
neutrinos.

## Acknowledgements

The IceCube collaboration acknowledges the significant contributions to this
manuscript from Michael Kovacevich. USA – U.S. National Science Foundation-
Office of Polar Programs, U.S. National Science Foundation-Physics Division,
U.S. National Science Foundation-EPSCoR, Wisconsin Alumni Research Foundation,
Center for High Throughput Computing (CHTC) at the University of
Wisconsin–Madison, Open Science Grid (OSG), Advanced Cyberinfrastructure
Coordination Ecosystem: Services & Support (ACCESS), Frontera computing
project at the Texas Advanced Computing Center, U.S. Department of Energy-
National Energy Research Scientific Computing Center, Particle astrophysics
research computing center at the University of Maryland, Institute for Cyber-
Enabled Research at Michigan State University, and Astroparticle physics
computational facility at Marquette University; Belgium – Funds for Scientific
Research (FRS-FNRS and FWO), FWO Odysseus and Big Science programmes, and
Belgian Federal Science Policy Office (Belspo); Germany – Bundesministerium
für Bildung und Forschung (BMBF), Deutsche Forschungsgemeinschaft (DFG),
Helmholtz Alliance for Astroparticle Physics (HAP), Initiative and Networking
Fund of the Helmholtz Association, Deutsches Elektronen Synchrotron (DESY),
and High Performance Computing cluster of the RWTH Aachen; Sweden – Swedish
Research Council, Swedish Polar Research Secretariat, Swedish National
Infrastructure for Computing (SNIC), and Knut and Alice Wallenberg Foundation;
European Union – EGI Advanced Computing for research; Australia – Australian
Research Council; Canada – Natural Sciences and Engineering Research Council
of Canada, Calcul Québec, Compute Ontario, Canada Foundation for Innovation,
WestGrid, and Compute Canada; Denmark – Villum Fonden, Carlsberg Foundation,
and European Commission; New Zealand – Marsden Fund; Japan – Japan Society for
Promotion of Science (JSPS) and Institute for Global Prominent Research (IGPR)
of Chiba University; Korea – National Research Foundation of Korea (NRF);
Switzerland – Swiss National Science Foundation (SNSF); United Kingdom –
Department of Physics, University of Oxford.

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