SoLid Collaboration

# Search for Very-Short-Baseline Oscillations of
Reactor Antineutrinos with the SoLid Detector

Y. Abreu  Universiteit Antwerpen, Antwerpen, Belgium  Y. Amhis  IJCLab, Univ
Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France  L. Arnold
University of Bristol, Bristol, United Kingdom  W. Beaumont  Universiteit
Antwerpen, Antwerpen, Belgium  I. Bolognino  Department of Physics, The
University of Adelaide, Adelaide, SA 5005, Australia  M. Bongrand  IJCLab,
Univ Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France  D.
Boursette  IJCLab, Univ Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay,
France  V. Buridon  Normandie Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen,
Caen, France  H. Chanal  Université Clermont Auvergne, CNRS/IN2P3, LPCA,
Clermont-Ferrand, France  B. Coupé  SCK CEN, Belgian Nuclear Research Centre,
Mol, Belgium  P. Crochet  Université Clermont Auvergne, CNRS/IN2P3, LPCA,
Clermont-Ferrand, France  D. Cussans  University of Bristol, Bristol, United
Kingdom  J. D’Hondt  Vrije Universiteit Brussel, Brussel, Belgium  D. Durand
Normandie Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, Caen, France  M.
Fallot  SUBATECH, Nantes Université, IMT Atlantique, CNRS/IN2P3, Nantes,
France  D. Galbinski<EMAIL_ADDRESS>Normandie Univ., ENSICAEN,
UNICAEN, CNRS/IN2P3, LPC Caen, Caen, France  S. Gallego  Johannes Gutenberg
University of Mainz, Institute of Physics, Mainz  L. Ghys  SCK CEN, Belgian
Nuclear Research Centre, Mol, Belgium  L. Giot  SUBATECH, Nantes Université,
IMT Atlantique, CNRS/IN2P3, Nantes, France  K. Graves  Imperial College
London, Department of Physics, London, United Kingdom  B. Guillon  Normandie
Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, Caen, France  S. Hayashida
Kings College London, London, United Kingdom  D. Henaff  SUBATECH, Nantes
Université, IMT Atlantique, CNRS/IN2P3, Nantes, France  B. Hosseini  Imperial
College London, Department of Physics, London, United Kingdom  S. Kalcheva
SCK CEN, Belgian Nuclear Research Centre, Mol, Belgium  L. N. Kalousis  Vrije
Universiteit Brussel, Brussel, Belgium  R. Keloth  Vrije Universiteit Brussel,
Brussel, Belgium  L. Koch  Johannes Gutenberg University of Mainz, Institute
of Physics, Mainz  M. Labare  Universiteit Gent, Gent, Belgium  G. Lehaut
<EMAIL_ADDRESS>Normandie Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC
Caen, Caen, France  S. Manley  University of Bristol, Bristol, United Kingdom
L. Manzanillas  IJCLab, Univ Paris-Sud, CNRS/IN2P3, Université Paris-Saclay,
Orsay, France  J. Mermans  SCK CEN, Belgian Nuclear Research Centre, Mol,
Belgium  I. Michiels  Universiteit Gent, Gent, Belgium  S. Monteil  Université
Clermont Auvergne, CNRS/IN2P3, LPCA, Clermont-Ferrand, France  C. Moortgat
Universiteit Gent, Gent, Belgium  D. Newbold  STFC Rutherford Appleton
Laboratory, Didcot, United Kingdom  V. Pestel  Normandie Univ., ENSICAEN,
UNICAEN, CNRS/IN2P3, LPC Caen, Caen, France  K. Petridis  University of
Bristol, Bristol, United Kingdom  I. Piñera  Université Clermont Auvergne,
CNRS/IN2P3, LPCA, Clermont-Ferrand, France  A. de Roeck  Universiteit
Antwerpen, Antwerpen, Belgium  N. Roy  IJCLab, Univ Paris-Sud, CNRS/IN2P3,
Université Paris-Saclay, Orsay, France  D. Ryckbosch  Universiteit Gent, Gent,
Belgium  N. Ryder  University of Oxford, Oxford, United Kingdom  D. Saunders
University of Bristol, Bristol, United Kingdom  M. H. Schune  IJCLab, Univ
Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France  M. Settimo
SUBATECH, Nantes Université, IMT Atlantique, CNRS/IN2P3, Nantes, France  H.
Rejeb Sfar  Universiteit Antwerpen, Antwerpen, Belgium  L. Simard  IJCLab,
Univ Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France  A.
Vacheret<EMAIL_ADDRESS>Normandie Univ., ENSICAEN, UNICAEN,
CNRS/IN2P3, LPC Caen, Caen, France  S. Van Dyck  SCK CEN, Belgian Nuclear
Research Centre, Mol, Belgium  P. Van Mulders  Vrije Universiteit Brussel,
Brussel, Belgium  N. Van Remortel  Universiteit Antwerpen, Antwerpen, Belgium
G. Vandierendonck  Universiteit Gent, Gent, Belgium  S. Vercaemer
Universiteit Antwerpen, Antwerpen, Belgium  M. Verstraeten  Ecole Royale
Militaire/Koninklijke Militaire School, Plasma Physics Laboratory, Brussel,
Belgium  B. Viaud  SUBATECH, Nantes Université, IMT Atlantique, CNRS/IN2P3,
Nantes, France  A. Weber  Fermi National Accelerator Laboratory, Batavia, USA
Johannes Gutenberg University of Mainz, Institute of Physics, Mainz  M.
Yeresko  Université Clermont Auvergne, CNRS/IN2P3, LPCA, Clermont-Ferrand,
France  F. Yermia  SUBATECH, Nantes Université, IMT Atlantique, CNRS/IN2P3,
Nantes, France

###### Abstract

In this letter we report the first scientific result based on antineutrinos
emitted from the BR2 reactor at SCK CEN. The SoLid experiment uses a novel
type of highly granular detector whose basic detection unit combines two
scintillators, PVT and 6LiF:ZnS(Ag), to measure antineutrinos via their
inverse-beta-decay products. An advantage of PVT is its highly linear response
as a function of deposited particle energy. The full-scale detector comprises
$12\,800$ voxels and operates over a very short 6.3–8.9 m baseline from the
reactor core. The detector segmentation and its 3D imaging capabilities
facilitate the extraction of the positron energy from the rest of the visible
energy, allowing the latter to be utilised for signal-background
discrimination. We present a result based on 280 reactor-on days (55
$\mathrm{MW}$ mean power) and 172 reactor-off days, respectively, of live
data-taking. A total of $$29\,479$\pm 603$ (stat.) antineutrino candidates
have been selected, corresponding to an average rate of 105 events per day and
a signal-to-background ratio of 0.27. A search for disappearance of
antineutrinos to a sterile state has been conducted using complementary model-
dependent frequentist and Bayesian fits, providing constraints on the allowed
region of the Reactor Antineutrino Anomaly.

††preprint: APS/123-QED

The observation of the Reactor Antineutrino Anomaly (RAA) in 2011 [1]
initiated a flurry of activity in very-short-baseline neutrino physics. It
followed from updated reactor-antineutrino flux predictions, known
collectively as the Huber-Mueller (HM) model [2, 3], which retroactively
revealed a deficit in the electron-antineutrino rate measured by numerous
short-baseline experiments. Since then, antineutrino flux predictions from
both the conversion [4, 5, 6] and “ab initio” summation [7, 8, 9] methods have
evolved substantially. The statistical significance of the RAA—initially
2.6—varies greatly depending on the reactor model [10]. The current state-of-
the-art calculations tentatively point to ^235U as the principle source of the
rate deficit, likely induced by biases in the ILL data [11, 12, 13] on which
the HM model is based, or poorly understood uncertainties [14] and unreliable
nuclear databases [15]. This is supported by results from Daya Bay [16, 17]
and RENO [18] on the correlation between antineutrino yield and fuel
composition, which suggest that adjusting either the predicted ^235U rate, or
all fissile isotopes equally, could resolve the RAA.

Alternatively, the RAA could be explained by disappearance of reactor
antineutrinos to an eV-scale sterile state. Sterile neutrinos would not
participate in any Standard Model interaction, making them accessible to
experiments only by mixing with ordinary active neutrinos. Assuming one new
neutrino mass eigenstate (3+1 model), the two-flavour antineutrino survival
probability as a function of energy ($E$) and distance ($L$) is given by:

$P=1-\sin^{2}(2\theta_{14})\sin^{2}\left(1.27\>\Delta
m^{2}_{41}[${\mathrm{eV}}^{2}$]\>\frac{L[$\mathrm{m}$]}{E[$\mathrm{MeV}$]}\right)\>,$
(1)

where $\theta_{14}$ and $\Delta m_{41}^{2}$ are the mixing angle and mass
splitting, respectively, between the active and sterile states. Given the
typical energy of reactor antineutrinos (1–10 $\mathrm{MeV}$), the mass
splitting implied by the RAA means that oscillations occurring over
$\mathcal{O}$(1) m distances can be searched for using detectors placed in
extreme proximity to reactor cores. The sterile-neutrino hypothesis of the RAA
has been thoroughly tested at both research [19, 20, 21] and commercial [22,
23] reactors. No evidence supporting the existence of sterile neutrinos has
been found, with the exception of the unconfirmed Neutrino-4 result [24, 25]
which is in strong tension with most limits. It is, however, consistent with
the best-fit region of the Gallium anomaly [26, 27], recently refreshed by
BEST [28]. SoLid (Search for Oscillations with a Lithium-6 Detector) is
another such very-short-baseline experiment [29, 30, 31, 32, 33, 34], whose
main goal is to search for disappearance of reactor antineutrinos.

SoLid is operated next to the BR2 reactor at SCK CEN in Mol, Belgium. BR2 is
an open-pool research reactor that burns highly enriched uranium (93.5% ^235U)
at 40–100 $\mathrm{MW}$ thermal power. The unique inclined-channel design
results in a compact core ($\diameter$ $50\text{\,}\mathrm{cm}$ $\times$
$80\text{\,}\mathrm{cm}$) for fission reactions. The reactor operates at power
in cycles of around 30 days each, alternating with shut-down periods of
comparable duration. The SoLid detector is based on a combination of two
scintillators: 5 cm polyvinyl toluene (PVT) cubes and 6LiF:ZnS(Ag) screens.
Antineutrinos interact in the fiducial volume of the detector through inverse
beta decay (IBD), ${\mkern 1.0mu\overline{\mkern-1.0mu\nu\mkern-1.0mu}\mkern
1.0mu_{e}+p\rightarrow n+e^{+}}$, producing a positron and a neutron. PVT acts
as a proton-rich target for the incoming antineutrino, scintillator for the
positron, and moderator of the neutron. A schematic of the detection principle
is displayed in Fig. 1. Accompanying the positron signal are energy deposits
from Compton scattering interactions of the $511\text{\,}\mathrm{keV}$ -rays
emitted when the positron annihilates (${e^{+}e^{-}\rightarrow\gamma\gamma}$).
The total prompt signal is referred to as the Electromagnetic Signal (ES). The
6LiF:ZnS(Ag) screens on two faces of each PVT cube are held in place by Tyvek
wrapping, which also ensures optical isolation of the voxels. The neutron
transport was optimised by tuning the number of 6LiF:ZnS(Ag) screens per cube,
the thickness of the screens, and their ZnS-6LiF mass ratio. After an average
delay ($\Delta t$) of $\sim 62$ $\mathrm{\SIUnitSymbolMicro s}$, thermalised
neutrons are captured on 6Li via the reaction
$\ce{{}^{6}Li+\text{n}\rightarrow^{3}_{2}\text{He}+\alpha}$, whose products
cause fluorescence in the ZnS(Ag). This is labelled the Nuclear Signal (NS).
An ES-NS time-coincidence is the primary signature of IBD-like events [35].

Figure 1: SoLid antineutrino detection principle. The positron and neutron
from inverse beta decay create two scintillation signals, separated by a
characteristic time delay ($\Delta t\sim 62$ $\mathrm{\SIUnitSymbolMicro s}$)
during which the neutron thermalises over $\mathcal{O}$(10) cm distances.

The 1.6 $\mathrm{t}$ detector comprises $12\,800$ cubes arranged into five
mechanically independent modules of ten planes each, where a plane is a grid
of $16\times 16$ cubes. Each cube is crossed by four wavelength-shifting
fibres in the directions perpendicular to the reactor-detector axis. One end
of each fibre is capped by a Hamamatsu S12572-050P multi-pixel photon counter
(MPPC) and the other by reflective foil. The detector is housed at BR2 in a
light-proof, environmentally controlled shipping container to ensure
consistent data-taking conditions. This is topped by passive
$50\text{\,}\mathrm{cm}$ HDPE shielding to add to the minimal 8 m.w.e.
overburden of the BR2 building and a $50\text{\,}\mathrm{cm}$ water wall on
all sides of the container helps to thermalise cosmic neutrons and block
reactor-induced gamma background. During normal data-taking, an FPGA-based
trigger system is deployed [33]. For recording IBD-like events, the approach
taken is to trigger on NS rather than ES waveforms which are dominated by
gamma-background events. Upon firing of this trigger, a space-time window
large enough to encapsulate the entire ES-NS event is read out and saved to
disk.

The digitised signals from the readout are organised into discrete clusters
based on the natural temporal and spatial correlations between waveforms from
the same physics event, whilst neglecting dark counts via an amplitude
threshold. Through-going muons deposit energy in the cubes along their path,
leaving highly distinguishable tracks. Other clusters are tagged as either ES
or NS by a simple integral-over-amplitude cut, which enhances the purity of
the NS sample from the 20% trigger-only output to over 99%. It is then
necessary to allocate the MPPC signals to the cubes from where they originated
and assign them corresponding energies. Since physics events typically
generate energy deposits that span multiple cubes and channels, this problem
is non-trivial. Defining $a_{ij}$ as the projector of cube $j$ to MPPC $i$,
the equation ${AE=p}$ must be solved for every plane in which energy was
deposited, where $p$ is the vector of MPPC signals ($p_{1},p_{2},...,p_{64}$),
$E$ is the list of unknown cube energies ($E_{1},E_{2},...,E_{256}$), and $A$
is a $64\times 256$ matrix dubbed the system matrix. Solutions are obtained
using a custom algorithm called CCube, which combines Maximum-Likelihood
Expectation-Maximisation [36] with a simplified version of the Orthogonal
Matching Pursuit algorithm [37]. An extensive description of CCube is provided
in Ref. [38].

The system matrix parameterises vital information about the detector
response—such as scintillation efficiency and photon attenuation—and provides
a relative calibration. The responses of individual cubes can be equalised
detector-wide by fitting the $\text{d}E/\text{d}x$ energy-loss distributions
of horizontal muons (i.e. those that cross only one cube per plane). Moreover,
the fractions of light shared between cubes ($\mathcal{O}$(1)% due to the
fibre-slot clearance) are evaluated and factored into the matrix. Muons can
also be used to estimate the detector energy scale, but the associated
uncertainties are relatively large. The absolute light yield is determined
more precisely using in-situ radioactive sources, ^22Na and AmBe, by fitting
the Compton edge profile of their emitted gamma radiation [34]. The
corresponding energy resolution is approximately 16%. Alternatively, the
energy calibration requirements can be fulfilled by the monoenergetic 3.43
$\mathrm{MeV}$ electron-positron pair produced by the 4.44 $\mathrm{MeV}$ AmBe
-ray. This relatively low-rate signal ($\sim 30$ times less likely than
Compton scattering) can be identified via the same techniques as employed for
antineutrino events [39] and allows the light yield to be computed with a
percent-level uncertainty. Validations of the energy scale and MC detector
simulation are provided by the -spectra of cosmogenic ^12B and _83^214Bi (of
BiPo background), as shown in Fig. 2 along with the measured PVT response. The
high degree of linearity attained in the 1–11 $\mathrm{MeV}$ region
demonstrates how the SoLid technology complements similar liquid-scintillator-
based experiments.

Figure 2: Top: Measured response as a function of energy of a subset of the
PVT detector cubes, from September 2018 calibration data. The linear fit is
derived from the blue points and constrained to intercept the vertical axis at
zero; it is validated by the red points which align well with the fit. The
grey band represents the error on the fitted gradient. All but the first ^12B
point are incorporated in the $\chi^{2}$. Bottom: Data-MC comparisons of BiPo
and boron-12 spectra.

The dataset on which this analysis is based spans data taken between 11 April
2018 to 2 July 2020. It comprises 13 reactor-on (ROn) cycles with a total
live-time of 280.3 days, and a total reactor-off (ROff) live-time of 171.8
days. Detector stability was continuously monitored through low-level DAQ
parameters including MPPC pedestal and gain, which were regularly re-equalised
across all channels. The dataset has been filtered through stringent data-
quality criteria based on various reconstructed quantities and restrictions on
environmental conditions such as temperature and humidity. Periodic
calibration campaigns with AmBe and ^22Na were used to track the time-
evolution of the neutron reconstruction efficiency and energy scale,
respectively. The former remained stable across the detector planes to within
1%, while both exhibited drifts of a few percent per year caused by
degradation of the scintillating materials. The muon-derived system matrix for
the ES calibration was updated roughly every ten data-taking days to
accommodate shifts in detector response, particularly the light yield. For the
analysis, the detector system and surrounding environment are accurately
implemented in Geant4 [40], including all materials that could influence
particle interactions within the detector volume. A second software framework
then applies readout effects to the simulated energy deposits according to a
system matrix. Fission yields are simulated using an MCNPX [41] representation
of the BR2 reactor core coupled with MCNPX/CINDER90 burn-up code; off-
equilibrium effects are taken into account in MURE [42]. Conversion
antineutrino spectra are then constructed by linking the fission vertices to
the ILL data and summation spectra by aggregating the -decay branches of all
relevant isotopes.

SoLid faces three IBD-like backgrounds: fast neutrons, from cosmic-ray-induced
spallation of atmospheric nuclei and materials surrounding the detector;
radioactive decays, both from airborne ^222Rn and ^238U contamination in the
6LiF:ZnS(Ag) screens; and a low rate of “accidental” events from time-
coincidences of uncorrelated signals. Fast neutrons cause proton recoils (ES)
prior to their capture on 6LiF (NS) after thermalisation. The radioactive
background, BiPo, produces ES-NS coincidences from radiative -decay of
_83^214Bi, followed by emission of an -particle from _84^214Po—which has a
decay time of $235.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$—generating
scintillation light in the ZnS(Ag). The exceedingly high initial background
rate is suppressed by application of the selection cuts outlined in Tab. 1.
Cuts 1–6 are based on low-level event-reconstruction variables. In particular,
the NS-ES displacement cuts reduce the number of accidental events to an
almost negligible level. The muon veto decreases the number of fast-neutron
events created after muon spallation in materials in and around the detector;
all NS clusters that follow within $200\text{\,}\mathrm{\SIUnitSymbolMicro s}$
of a muon track are rejected.

| Description | Applied Limits
---|---|---
1) | NS-ES Time Diff. |  $1$ $\mathrm{\SIUnitSymbolMicro s}$ $\leq\Delta t\leq 141$ $\mathrm{\SIUnitSymbolMicro s}$
2) | NS-ES Distance | $1\leq\Delta R\leq 4$
3) | NS-ES Displacement | $-3\leq\Delta X\leq 3$
4) | " | $-3\leq\Delta Y\leq 3$
5) | " | $-2\leq\Delta Z\leq 3$
6) | Muon Veto |  $t_{\text{nearest muon}}\geq 200$ $\mathrm{\SIUnitSymbolMicro s}$
7) | Prompt Energy |  $1$ $\mathrm{MeV}$ $\leq E_{\text{MEC}}\leq 6$ $\mathrm{MeV}$
8) | Gamma Multiplicity | $N_{\text{gammas}}\geq 1$
9) | CNN for NS signal |  $0.70\leq$ Score $\leq 1.00$
10) | BDT for ES signal |  $0.73\leq$ Score $\leq 1.00$
Table 1: Selection cuts of the IBD analysis.

For cuts 7 and 8, the detector granularity is exploited to isolate the
positron energy, which carries the antineutrino spectrum information, and
reconstruct the annihilation photons ($\gamma_{\text{a}}$). Simulations of
antineutrino events (IBD MC) indicate that the most energetic cube (MEC) in
the ES cluster contains on average 91.5% and 12.5% of the deposited positron
and $\gamma_{\text{a}}$ energy, respectively. The positron is confined to a
single voxel for around 78% of IBD MC events. In other cases, the majority of
the remaining positron energy is located in cubes directly adjacent to the
MEC, which can also contain significant quantities of $\gamma_{\text{a}}$
energy. As such, the positron energy is defined as the energy of the MEC plus
any bordering cubes that contain at least 20% of the total ES energy
(${E_{\text{cube}}/E_{\text{tot}}\geq 0.2}$). This estimator contains 96.7%
and 13.8% of positron and $\gamma_{\text{a}}$ energy on average. Furthermore,
low-energy (${E_{\text{cube}}/E_{\text{tot}}<0.2}$) deposits in cubes not
bordering the MEC are dominated by contributions from the annihilation gammas.
Capitalising on the anti-parallel emission directions of the two photons,
detached clusters of gamma deposits are searched for in opposing half-spheres
around the MEC. The “gamma multiplicity” variable can thus take values 0, 1,
or 2; background-rich 0- events are discarded.

The final two cuts are applied on the outputs of machine-learning models. A
one-dimensional Convolutional Neural Net (CNN) has been trained on pure data
samples to distinguish between neutron- and alpha-induced NS waveforms using
only pulse shape information. The CNN architecture is identical to the one
described in Ref. [43] and it achieves 80% neutron efficiency for rejection of
95% of alpha waveforms. The second model is a Boosted Decision Tree (BDT) made
with XGBoost [44], primarily designed to discriminate between the ES clusters
of IBD events and fast neutrons. It is trained on a dataset containing
reactor-off events (background, labelled 0) and IBD MC events (signal,
labelled 1) in equal proportions, after having first applied selection cuts
1–9. The twenty BDT features mainly comprise topological variables (e.g. from
the gamma reconstruction) that have no significant correlation with the
positron energy or Z-position. Discrimination is also provided by the spatial-
displacement parameters in Tab. 1 and the event coordinates in the X–Y plane.
The signal-background separation granted by these models is illustrated in
Fig. 3. Overall, cuts 1–9 remove 61% of signal events and 98% of background
events; the BDT cut then removes an additional 39% and 90%, respectively. A
comprehensive summary of the antineutrino selection may be found in Ref. [45].

Figure 3: Separation between signal (IBD MC) and background (reactor-off)
events achieved by the BDT and CNN models, expressed as the product of their
two output scores. Figure 4: Overview of the SoLid dataset in terms of daily
variations of different event types and atmospheric-pressure corrections.

The oscillation fits are performed on a background-subtracted IBD spectrum.
Subtraction of accidentals and BiPo is realised by estimating their respective
rates and spectrum shapes in dedicated sideband regions of parameter space.
For example, BiPo is characterised in a high-purity sideband that is nearly
identical to the “signal region” (as defined by the Tab. 1 selections) but
which occupies a different region of $\Delta t$-CNN space. An equivalent
region exists for isolating accidental background. The relative rates of
different event types in a given sample are deduced from a triple-exponential
fit on the $\Delta t$ distribution, adopting the known ES-NS time constants of
each. To subtract fast-neutron events, it is essential to account for their
dependence on atmospheric pressure. This is done by applying a daily
correction computed from a linear fit of the fast-neutron rate
$N^{\text{ROff}}_{\text{Signal}-\text{Acc}-\text{BiPo}}$ (accidental- and
BiPo-subtracted reactor-off data) as a function of pressure $P$, both relative
to their respective means over a set period:

$\displaystyle f(P,\mkern 1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern
1.0mu)=$
$\displaystyle\>\>N^{\text{ROff}}_{\text{Signal}-\text{Acc}-\text{BiPo}}-\mkern
1.0mu\overline{\mkern-1.0muN\mkern-1.0mu}\mkern
1.0mu^{\text{ROff}}_{\text{Signal}-\text{Acc}-\text{BiPo}}$ (2)
$\displaystyle=$ $\displaystyle\>\>p_{0}+p_{1}(P-\mkern
1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern 1.0mu)\>,$

where $p_{0}$ and $p_{1}$ are the parameters of the fit. In practice, the
robustness of the subtraction is improved by splitting the dataset into five
independent “epochs”, each with its own pressure fit $f_{e}$. The correction
for a given day (time $t$) is thus determined by the composite model
$F(t,P,\mkern 1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern
1.0mu)=\sum_{e}\delta(t,e)\>f_{e}(P,\mkern
1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern 1.0mu)$, where the Kronecker-
like function $\delta$ ensures the correct epoch. Denoting the number of
events in a particular parameter-space region as $N$ and using a tilde to
signify a pressure-corrected quantity, the number of IBD events can be
expressed as:

$\displaystyle N_{\text{IBD}}=$
$\displaystyle\>\>\tilde{N}^{\text{ROn}}_{\text{Signal}-\text{Acc}-\text{BiPo}}-\frac{t_{\text{ROn}}}{t_{\text{ROff}}}\cdot\tilde{N}^{\text{ROff}}_{\text{Signal}-\text{Acc}-\text{BiPo}}$
(3) $\displaystyle=$
$\displaystyle\>\>(N^{\text{ROn}}_{\text{Signal}}-N^{\text{ROn}}_{\text{Signal}\>\text{Acc}})-\beta^{\text{ROn}}\cdot(N^{\text{ROn}}_{\text{BiPo}}-N^{\text{ROn}}_{\text{BiPo}\>\text{Acc}})-\sum\limits_{i}^{I}F(t_{i},P_{i},\mkern
1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern 1.0mu_{\text{ROn}})$
$\displaystyle-$
$\displaystyle\>\frac{t_{\text{ROn}}}{t_{\text{ROff}}}\cdot[(N^{\text{ROff}}_{\text{Signal}}-N^{\text{ROff}}_{\text{Signal}\>\text{Acc}})-\beta^{\text{ROff}}\cdot(N^{\text{ROff}}_{\text{BiPo}}-N^{\text{ROff}}_{\text{BiPo}\>\text{Acc}})-\sum\limits_{j}^{J}F(t_{j},P_{j},\mkern
1.0mu\overline{\mkern-1.0muP\mkern-1.0mu}\mkern 1.0mu_{\text{ROff}})]\>,$

where $N_{\text{Signal}}$ and $N_{\text{BiPo}}$ refer to the number of events
in the signal region and BiPo sideband, respectively, with $\beta$ correcting
for the BiPo-rate difference between these two regions, and
$t_{\text{ROn}}=\sum\limits t_{i}$ ($t_{\text{ROff}}=\sum\limits t_{j}$) is
the total reactor-on (reactor-off) live-time corresponding to $I$ ($J$) data-
taking days. Finally, the dataset is bisected after epoch 3 to create two
halves (of roughly equal size in ROn data) for which the subtractions are
performed using separate ROff data. The resulting spectra are then combined.
This procedure was found to mitigate the need for an extra systematic
uncertainty related to changes in detector efficiency over time.

Presented in Fig. 4 are the time-evolution of the backgrounds rates, the
stability of the IBD-like excess (i.e. events remaining after subtraction),
and the daily pressure corrections applied. Good control over the background
is maintained across the data-taking period and it is clear that fast neutrons
constitute the largest background after selection. The significance of the
IBD-like excess typically lies in the 3–5 range for reactor-on days and, with
the exception of a few outliers, the reactor-off excess lies within 2 of zero.
The variations in atmospheric pressure over this period result in adjustments
of up to 75 events per day to the number of fast neutrons subtracted. The
total number of antineutrino candidates is $$29\,479$\pm 603$ (stat.) with a
signal-to-background ratio of 0.27.

Figure 5: Reconstructed positron spectrum per detector module compared to the
Huber-Mueller model without (red) and with (blue) the best-fit oscillations of
the data applied. The two predictions have been scaled by their individually
optimised normalisation parameter. Error bars represent combined statistical
and systematic uncertainties.

A model-dependent search for oscillation of reactor antineutrinos has been
conducted using frequentist and Bayesian methods. In both cases, the
prediction for a particular oscillation scenario is built by re-weighting the
MC events in the Huber-Mueller truth-space histogram, according to equation 1,
and propagating the result to reconstructed space using a probability matrix
[46]. The reconstructed events are sorted into ten 500 keV energy bins and
five position bins.

The Bayesian analysis is based on a Markov chain Monte Carlo (MCMC) procedure
that assumes a uniform prior for the oscillation amplitude
$\sin^{2}(2\theta_{14})$ (between 0.005 and 1) and a log-uniform prior for
$\Delta m_{41}^{2}$ (between 0.05 and 40 ${\mathrm{eV}}^{2}$). Credible
posterior regions are estimated with a Metropolis-Hasting sampling algorithm
[47, 48] based on a Poisson likelihood. The frequentist analysis follows the
standard Feldman-Cousins technique [49], with test statistic
${\Delta\chi^{2}=\chi^{2}_{\text{H}_{\text{x}}}-\chi^{2}_{\text{H}_{\text{BF}}}}$
for oscillation hypothesis H${}_{\text{x}}$. The best-fit value
$\chi^{2}_{\text{H}_{\text{BF}}}$ is obtained by minimising globally over the
sensitive region of the $\Delta m_{41}^{2}$-$\sin^{2}(2\theta_{14})$ plane. We
choose a $\chi^{2}$ metric of the form:

$\chi^{2}=r^{T}\>V_{\text{cov}}^{-1}\>r\>,$ (4)

where $V_{\text{cov}}$ is the matrix of statistical and systematic covariances
and the residual $r$ is defined for position-energy bin $ij$ as:

$r_{ij}=D_{ij}-\eta\>P_{ij}\>,$ (5)

where $D_{ij}$ and $P_{ij}$ refer to the data and prediction bin contents,
respectively, and $\eta$ is a free-floating normalisation parameter which
renders the fit insensitive to the overall antineutrino rate. Frequentist
exclusion regions are thus defined by points in the discretised parameter-
space grid whose $\Delta\chi^{2}$ exceeds some critical threshold at some
confidence level (CL). The thresholds are calculated by generating
$\mathcal{O}$(1000) pseudo-experiments at each grid point—via Cholesky
decomposition of the covariance matrix—to build empirical distributions of the
test statistic.

Figure 6: Frequentist exclusion and sensitivity contours (Feldman-Cousins) and
Bayesian MCMC credible region. RAA contour and best-fit taken from Ref. [1].

The background-subtracted IBD spectra per module are shown in Fig. 5. The
final spectrum follows an unblinding process in which the selection cuts were
frozen prior to inspection of individual reactor cycles, which had to pass a
data-MC goodness-of-fit check to be included. The prevailing systematic
uncertainty originates from the precision of the PVT light yield—approximately
2% on the first position-energy bin compared to 10% statistical—but overall
the bin errors are dominated by the statistical uncertainty. The reduced
$\chi^{2}$ between the data and the no-oscillation HM prediction (H0) is
49.8/49, indicating excellent agreement. Minimisation of the $\chi^{2}$
results in a $\Delta\chi^{2}$ of 0.915 at $\sin^{2}(2\theta_{14})$ = 0.123 and
$\Delta m_{41}^{2}$ = $3.76\text{\,}{\mathrm{eV}}^{2}$, which cannot be
rejected at 90% CL. Therefore, the data exhibits a very mild preference for
oscillations over the null hypothesis. Likewise, the Bayesian fit finds
insufficient evidence to reject H0 and identifies a most probable value at
$\sin^{2}(2\theta_{14})$ = 0.02 and $\Delta m_{41}^{2}$ =
$0.69\text{\,}{\mathrm{eV}}^{2}$. The frequentist exclusion contours and
Bayesian credible regions (CR) are drawn in Fig. 6. No significant evidence of
an energy- and length-dependent oscillation pattern is observed and a portion
of the 3+1 parameter space is rejected. The border of the credible region
closely follows the shape of the frequentist contours but excludes a slightly
greater area of parameter space. Our results are not sensitive to the
Neutrino-4 best-fit point.

In conclusion, this first analysis by the SoLid collaboration, conducted on a
dataset comprising 280.3 live reactor-on days, has selected $$29\,479$\pm 603$
(stat.) antineutrino candidates with a signal-to-background ratio of 0.27.
Uniquely amongst similarly motivated experiments, SoLid has direct access to
the positron energy which permits the definition of high-level topological
variables for event characterisation. Consequently, machine-learning tools for
background suppression are both well-suited and have been necessary to achieve
a reasonable background level given the initial rates. The stability of the
detector response and background rates throughout the data-taking period were
ensured by custom calibration tools. A model-dependent search for antineutrino
disappearance has been performed, which finds no evidence of active-to-sterile
oscillations in the 3+1 extension to the standard 3-neutrino paradigm and
disfavours at 90% CL a small section of the RAA allowed region.

This work was supported by the following funding sources: Science & Technology
Facilities Council (STFC), United Kingdom; FWO-Vlaanderen and the Vlaamse
Herculesstichting, Belgium; Agence Nationale de la Recherche grant
ANR-16CE31001803, Institut Carnot Mines, CNRS/IN2P3 and Region Pays de Loire,
France. We are also deeply indebted to the SCK CEN staff for their generous
assistance and accommodation. Individuals have received support from the FWO-
Vlaanderen and the Belgian Federal Science Policy Office (BelSpo) under the
IUAP network programme, the STFC Rutherford Fellowship program, and the
European Research Council under the European Union’s Horizon 2020 Programme
(H2020-CoG)/ERC Grant Agreement no. 682474.

## References

  * Mention  _et al._ [2011] G. Mention, M. Fechner, T. Lasserre, T. A. Mueller, D. Lhuillier, M. Cribier, and A. Letourneau, Physical Review D 83, 073006 (2011).
  * Mueller  _et al._ [2011] T. A. Mueller, D. Lhuillier, M. Fallot, A. Letourneau, S. Cormon, M. Fechner, L. Giot, T. Lasserre, J. Martino, G. Mention, A. Porta, and F. Yermia, Physical Review C 83, 054615 (2011).
  * Huber [2011] P. Huber, Physical Review C 84, 024617 (2011).
  * Hayen  _et al._ [2019] L. Hayen, J. Kostensalo, N. Severijns, and J. Suhonen, Physical Review C 99, 031301 (2019).
  * Li and Zhang [2019] Y.-F. Li and D. Zhang, Physical Review D 100, 053005 (2019).
  * Kopeikin  _et al._ [2021] V. Kopeikin, M. Skorokhvatov, and O. Titov, Physical Review D 104, L071301 (2021).
  * Estienne  _et al._ [2019] M. Estienne _et al._ , Physical Review Letters 123, 022502 (2019).
  * Perissé  _et al._ [2023] L. Perissé, A. Onillon, X. Mougeot, M. Vivier, T. Lasserre, A. Letourneau, D. Lhuillier, and G. Mention, Phys. Rev. C 108, 055501 (2023).
  * Letourneau  _et al._ [2023] A. Letourneau, V. Savu, D. Lhuillier, T. Lasserre, T. Materna, G. Mention, X. Mougeot, A. Onillon, L. Perisse, and M. Vivier, Physical Review Letters 130, 021801 (2023).
  * Giunti  _et al._ [2022] C. Giunti, Y. Li, C. Ternes, and Z. Xin, Physics Letters B 829, 137054 (2022).
  * Schreckenbach  _et al._ [1981] K. Schreckenbach, H. Faust, F. von Feilitzsch, A. Hahn, K. Hawerkamp, and J. Vuilleumier, Physics Letters B 99, 251 (1981).
  * von Feilitzsch  _et al._ [1982] F. von Feilitzsch, A. Hahn, and K. Schreckenbach, Physics Letters B 118, 162 (1982).
  * Schreckenbach  _et al._ [1985] K. Schreckenbach, G. Colvin, W. Gelletly, and F. V. Feilitzsch, Physics Letters B 160, 325 (1985).
  * Hayes and Vogel [2016] A. C. Hayes and P. Vogel, Annual Review of Nuclear and Particle Science 66, 219 (2016).
  * Sonzogni  _et al._ [2016] A. A. Sonzogni, E. A. McCutchan, T. D. Johnson, and P. Dimitriou, Physical Review Letters 116, 132502 (2016).
  * An  _et al._ [2017] F. P. An _et al._ , Physical Review Letters 118, 251801 (2017).
  * An  _et al._ [2023] F. P. An _et al._ , Physical Review Letters 130, 211801 (2023).
  * Bak  _et al._ [2019] G. Bak _et al._ , Physical Review Letters 122, 232501 (2019).
  * Andriamirado  _et al._ [2021] M. Andriamirado _et al._ , Physical Review D 103, 032001 (2021).
  * Almazán  _et al._ [2020a] H. Almazán _et al._ , Physical Review D 102, 052002 (2020a).
  * Serebrov  _et al._ [2017] A. P. Serebrov _et al._ , Journal of Physics: Conference Series 888, 012089 (2017).
  * Alekseev  _et al._ [2018] I. Alekseev _et al._ , Physics Letters B 787, 56 (2018).
  * Ko  _et al._ [2017] Y. J. Ko _et al._ , Physical Review Letters 118, 121802 (2017).
  * Serebrov  _et al._ [2021] A. P. Serebrov _et al._ , Physical Review D 104, 032003 (2021).
  * Almazán  _et al._ [2020b] H. Almazán _et al._ , arXiv e-prints 10.48550/arXiv.2006.13147 (2020b).
  * Abazov  _et al._ [1991] A. I. Abazov _et al._ , Physical Review Letters 67, 3332 (1991).
  * Hampel  _et al._ [1998] W. Hampel, G. Heusser, J. Kiko, T. Kirsten, M. Laubenstein, E. Pernicka, W. Rau, U. Rönn, C. Schlosser, M. Wójcik, R. v. Ammon, K. Ebert, T. Fritsch, D. Heidt, E. Henrich, _et al._ , Physics Letters B 420, 114 (1998).
  * Barinov  _et al._ [2022] V. V. Barinov _et al._ , Physical Review C 105, 065502 (2022).
  * Abreu  _et al._ [2017] Y. Abreu _et al._ , Journal of Instrumentation 12 (04), P04024.
  * Abreu  _et al._ [2018a] Y. Abreu _et al._ , Journal of Instrumentation 13 (05), P05005.
  * Abreu  _et al._ [2018b] Y. Abreu _et al._ , Journal of Instrumentation 13 (09), P09005.
  * Abreu  _et al._ [2019a] Y. Abreu _et al._ , Journal of Instrumentation 14 (02), P02014.
  * Abreu  _et al._ [2019b] Y. Abreu _et al._ , Journal of Instrumentation 14 (11), P11003.
  * Abreu  _et al._ [2021] Y. Abreu _et al._ , Journal of Instrumentation 16 (02), P02025.
  * Reines and Cowan [1953] F. Reines and C. L. Cowan, Physical Review 92, 10.1103/PhysRev.92.830 (1953).
  * Simón  _et al._ [2016] A. Simón, P. Ferrario, and A. Izmaylov, Nuclear and Particle Physics Proceedings 273-275, 2624 (2016).
  * Mallat and Zhang [1993] S. Mallat and Z. Zhang, IEEE Transactions on Signal Processing 41, 3397 (1993).
  * Abreu  _et al._ [2024] Y. Abreu _et al._ , arXiv e-prints https://doi.org/10.48550/arXiv.2404.03580 (2024).
  * Yeresko [2022] M. Yeresko, Ph.D. thesis, Université Clermont Auvergne (2022).
  * Agostinelli  _et al._ [2003] S. Agostinelli _et al._ , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 506, 250 (2003).
  * Waters  _et al._ [2007] L. S. Waters, G. W. McKinney, J. W. Durkee, M. L. Fensin, J. S. Hendricks, M. R. James, R. C. Johns, and D. B. Pelowitz (AIP, 2007) pp. 81–90.
  * Méplan  _et al._ [2009] O. Méplan _et al._ , Tech. Rep., LPSC 0912 and IPNO-09-01 (2009).
  * Griffiths  _et al._ [2020] J. Griffiths, S. Kleinegesse, D. Saunders, R. Taylor, and A. Vacheret, Machine Learning: Science and Technology 1, 045022 (2020).
  * Chen and Guestrin [2016] T. Chen and C. Guestrin, in  _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_, KDD ’16 (2016) pp. 785–794.
  * Galbinski [2024] D. Galbinski, Ph.D. thesis, Imperial College London (2024).
  * Koch [2019] L. Koch, Journal of Instrumentation 14 (09), P09013.
  * Metropolis  _et al._ [1953] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, The Journal of Chemical Physics 21, 1087 (1953).
  * Hastings [1970] W. K. Hastings, Biometrika 57, 97 (1970).
  * Feldman and Cousins [1998] G. J. Feldman and R. D. Cousins, Physical Review D 57, 3873 (1998).


* (138) Shankaranarayanan S. Temperature and entropy of Schwarzschild-de Sitter space-time. Physical Review D 2003; 67: 084026.
  * (139) Boonserm P, Visser M. Analytic bounds on transmission probabilities. Annals of Physics 2010, 325: 1328.
  * (140) Miao YG, Xu ZM. Hawking Radiation of Five-dimensional Charged Black Holes with Scalar Fields. Physics Letter B 2017; 772: 542-546.
  * (141) Barman S. The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes. European Physical Journal C 2020; 80: 50.
  * (142) Casana R, Cavalcante A, Poulis FP, Santos EB. Exact Schwarzschild-like solution in a bumblebee gravity model. Physical Review D 2018; 97: 104001.
  * (143) Boonserm P, Visser M. Transmission probabilities and the Miller-Good transformation. Journal of Physics A 2009; 42: 045301.
  * (144) Rincón Á, Santos V, Greybody factor and quasinormal modes of Regular Black Holes. European Physical Journal C 2020; 80: 910.
  * (145) Ngampitipan T, Boonserm P. Bounding the Greybody Factors for Non-rotating Black Holes. International Journal of Modern Physics D 2013; 22: 1350058.
  * (146) Barman S. The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes. European Physical Journal C 2020; 80: 50.
  * (147) Boonserm P, Ngampitipan T, Visser M. Bounding the greybody factors for scalar field excitations on the Kerr-Newman spacetime. Journal of High Energy Physics 2014; 03: 113.
  * (148) Boonserm P, Ngampitipan T, Wongjun P. Greybody factor for black holes in dRGT massive gravity. European Physical Journal C 2018; 78: 492.
  * (149) Boonserm P, Chen CH, Ngampitipan T, Wongjun P. Greybody factor for massive fermion emitted by a black hole in de Rham-Gabadadze-Tolley massive gravity theory.
  * (150) Boonserm P, Ngampitipan T, Wongjun P. Greybody factor for black string in dRGT massive gravity. European Physical Journal C 2019; 330: 330.
  * (151) Heydar-Fard M, Razmi H, Sepangi HR. Brane-World Black Hole Solutions via a Confining Potential Physical Review D 2007; 76: 066002.
  * (152) Al-Badawi A, Kanzi S, Sakallı İ. Greybody Radiation of scalar and Dirac perturbations of NUT Black Holes. European Physical Journal Plus 2022; 137: 94.
  * (153) Al-Badawi A, Kanzi S, Sakallı İ. Fermionic and Bosonic Greybody Factors as well as Quasinormal Modes for Charged Taub NUT Black Holes. [arXiv:2203.04140 [hep-th]].
  * (154) Al-Badawi A, Sakalli İ. Solution of the Dirac equation in the rotating Bertotti-Robinson spacetime. Journal of Mathematical Physics 2008; 49: 052501.
  * (155) Dariescu C, Dariescu M. A, Stelea C. Dirac Equation on the Kerr–Newman Spacetime and Heun Functions. Advances in High Energy Physics 2021; 2021; 5512735.
  * (156) Nakamura T, Nakano H, Tanaka T. Detecting quasinormal modes of binary black hole mergers with second-generation gravitational-wave detectors. Physical Review D 2016; 93: 044048.
  * (157) Ghosh A, Brito R, Buonanno A. Constraints on quasinormal-mode frequencies with LIGO-Virgo binary–black-hole observations. Physical Review D 2021; 103: 124041.
  * (158) Chatrchyan S. et al. [CMS], Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Physics Letters B 2012; 716: 30.
  * (159) Aad G. et al. [ATLAS], Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Physics Letters B 2012; 716: 1.
  * (160) Akiyama K. et al. [Event Horizon Telescope]. First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole. The Astrophysical Journal Letters 2019; 875: L1.
  * (161) Snider E, Dasenbrock-Gammon N, McBride, R. et al, Room-temperature superconductivity in a carbonaceous sulfur hydride. Nature 2020; 586: 373.
  * (162) Kolobov VI, Golubkov K, Muñoz de Nova JR, Steinhauer J. Observation of stationary spontaneous Hawking radiation and the time evolution of an analogue black hole. Nature Physics 2021; 17: 362.


* [45] F. Marques, _Isolated singularities of solutions to the Yamabe equation._ Calc. Var. Partial Differential Equations 32 (2008), no. 3, 349–371.
  * [46] R. Mazzeo, _Regularity for the singular Yamabe problem._ Indiana Univ. Math. J. 40 (1991), 1277–1299.
  * [47] R. Mazzeo and F. Pacard, _Constant scalar curvature metrics with isolated singularities._ Duke Math. J. 99 (1999), 353–418.
  * [48] R. Mazzeo, D. Pollack and K. Uhlenbeck, _Moduli spaces of singular Yamabe metrics._ J. Am. Math. Soc. 9 (1996), 303–344.
  * [49] R. Mazzeo and N. Smale, _Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere._ J. Diff. Geom. 34 (1991), 581–621.
  * [50] M. Niu, Z. Peng and J. Xiong, _Compactness of solutions to nonlocal elliptic equations._ J. Funct. Anal. 275 (2018), 2333–2372.
  * [51] M. Obata, _The conjectures on conformal transformations of Riemannian manifolds._ J. Differential Geometry 6 (1971/72), 247–258.
  * [52] R. Schoen, _Conformal deformation of a Riemannian metric to constant scalar curvature._ J. Differential Geom. 20 (1984), 479–495.
  * [53] R. Schoen, _The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation._ Commun. Pure Appl. Math. 41 (1988), 317–392.
  * [54] R. Schoen, Course notes on ‘Topics in differential geometry’ at Stanford University, (1988), available at https://www.math.washington.edu/ pollack/research/Schoen-1988-notes.html.
  * [55] R. Schoen and S.-T. Yau, _Conformally flat manifolds, Kleinian groups and scalar curvature._ Invent. Math. 92 (1988), no. 1, 47–71.
  * [56] A. Silva Santos, _A construction of constant scalar curvature manifolds with Delaunay-type ends._ Ann. Henri Poincaré 10 (2010),1487–1535.
  * [57] G. Stampacchia, _Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus._ Annales de l’institut Fourier, tome 15, no. 1 (1965), p. 189–257.
  * [58] S. D. Taliaferro and L. Zhang, _Asymptotic symmetries for conformal scalar curvature equations with singularity._ Calc. Var. Partial Differential Equations 26 (2006), no. 4, 401–428.
  * [59] N. Trudinger, _Remarks concerning the conformal deformation of Riemannian structures on compact manifolds._ Ann. Scuola Norm. Sup. Cl. Sci. (3) 22 (1968), 265–274.
  * [60] J. Xiong and L. Zhang, _Isolated singularities of solutions to the Yamabe equation in dimension $6$._ Int. Math. Res. Not., 22 (2022), 9571–9597.
  * [61] H. Yamabe, _On a deformation of Riemannian structures on compact manifolds._ Osaka Math. J. 12 (1960), 21–37.
  * [62] L. Zhang, _Refined asymptotic estimates for conformal scalar curvature equation via moving sphere method._ J. Funct. Anal. 192 (2002), 491–516.

Z.-C. Han

Department of Mathematics, Rutgers University
Hill Center-Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854
Email<EMAIL_ADDRESS>
J. Xiong

School of Mathematical Sciences, Beijing Normal University
Beijing 100875, China
Email<EMAIL_ADDRESS>
L. Zhang

Department of Mathematics, University of Florida
358 Little Hall P.O. Box 118105
Gainesville FL 32611-8105, USA
Email<EMAIL_ADDRESS>


# Scattering off a junction

Eric Tan<EMAIL_ADDRESS>University of Western Ontario, London,
Ontario N6A 3K7, Canada  Department of Physics, Brock University, St.
Catharines, Ontario L2S 3A1, Canada  R. Ganesh<EMAIL_ADDRESS>Department
of Physics, Brock University, St. Catharines, Ontario L2S 3A1, Canada

###### Abstract

Scattering off a potential is a fundamental problem in quantum physics. It has
been studied extensively with amplitudes derived for various potentials. In
this article, we explore a setting with no potentials, where scattering occurs
off a junction where many wires meet. We study this problem using a tight-
binding discretization of a star graph geometry – one incoming wire and $M$
outgoing wires intersecting at a point. An incoming wave arrives at the
junction and scatters. One part is reflected along the same wire while the
rest is transmitted along the others. Remarkably, the reflectance increases
monotonically with $M$, i.e., the greater the number of outgoing channels, the
more the particle bounces back. In the $M\rightarrow\infty$ limit, the wave is
entirely reflected back along the incoming wire. We rationalize this
observation by establishing a quantitative mapping between a junction and an
on-site potential. To each junction, we assign an equivalent potential that
produces the same reflectance. As the number of wires ($M$) increases, the
equivalent potential also increases. A recent article by one of us has drawn
an equivalence between junctions and potentials from the point of view of
bound state formation. Our results here show that the same equivalence also
holds for scattering amplitudes. We verify our analytic results by simulating
wavepacket motion through a junction. We extend the wavepacket approach to two
dimensions where analytic solutions cannot be found. An incoming wave travels
on a sheet and scatters off a point where many sheets intersect. As in the
one-dimensional problem, we assign an equivalent potential to a junction.
However, unlike in 1D, the equivalent potential is momentum-dependent.
Nevertheless, for any given momentum, the equivalent potential grows
monotonically with the number of intersecting sheets. Our findings can be
tested in ultracold atom setups and semiconductor structures.

##  I Introduction

The problem of scattering has a long, illustrious history in classical and
quantum physics. In its simplest form, a wave travels through free space and
reaches a local potential. A part of the wave is then transmitted while the
rest is reflected or scatteredSchiff (1968); Sakurai and Napolitano (2017);
Goldberger and Watson (2004); Newton (1982); Joachain (1975). Reflection and
transmission coefficients have been calculated for various potentials (e.g.,
see Ref. Ahmed and Bhattacharya, 2021). In this article, we discuss scattering
off a junction and compare it to that off a potential. A junction is similar
to a potential in that it breaks translational symmetry. As the strength of a
potential can be tuned, so can the ‘degree of connectivity’ at a junction.
This quantity represents the number of wires that meet at the junction. Our
key result is that in one dimension, a junction and a local potential are
quantitatively equivalent. In particular, as the connectivity of a junction
increases, the equivalent potential increases. This leads to a counter-
intuitive result: the larger the number of outgoing channels at a junction,
the more is the reflection back along the incoming channel.

Junctions have been experimentally studied in various settings. Ultracold
atoms have been loaded onto Y-junction and X-junction geometriesCassettari
_et al._ (2000); Dumke  _et al._ (2002). Semiconductor structures with T
geometries (three wires meeting at a point)Goñi  _et al._ (1992); Hasen  _et
al._ (1997) and X geometries (four wires meeting at a point)Berggren and Ji
(1991) have been explored. Networks built from arrangements of junctions have
also been studied using confined electron gasesNaud  _et al._ (2001) and
microwavesHul  _et al._ (2004). To understand transport in junctions, it is
crucial to understand the scattering process where a wavepacket impinges on a
junction. This acquires significance in fields such as coherent matter wave
physicsBongs and Sengstock (2004); Fortágh and Zimmermann (2007) or
atomtronicsAmico  _et al._ (2022). Many studies have focussed on complex
effects caused by interactions, transverse potentials, etc. In this article,
we demonstrate that the simplest version of the problem – a free particle
scattering off a simple junction – already shows interesting and surprising
properties.

From a theoretical point of view, transmission and scattering at junctions
have been extensively studied in the field of quantum graphsAndrade  _et al._
(2016). Typically, the Schrödinger equation is solved on each smooth wire. At
a junction where several wires meet, wavefunctions are related to one another
by a scattering matrix. Unitarity imposes strong constraints on this matrix.
For example, in a star graph where $M$ wires meet at a point, the net
transition amplitude must scale as $\sim 1/M$Gratus  _et al._ (1994). In other
words, for large $M$, an incoming wave is entirely reflected back along the
same wire. However, in order to explicitly find the scattering matrix, a
boundary condition must be imposed at the junction – for example, see Ref.
Andrade  _et al._ , 2016 and references therein. The boundary condition is not
fully determined by physical considerations (such as probability
conservation). For example, the Robin-type boundary condition involves a free
parameter which cannot be fixed or tuned by any known experimental means. As
such, the choice of boundary condition is arbitrary and can bias the outcome.
In this study, we circumvent this issue by taking a tight binding approach
where the space of intersecting wires is discretized. As we show below, this
allows for finding scattering coefficients in an unbiased manner.

The tight binding setup can also be directly realized with ultracold atomic
gasesZakrzewski (2007), superconducting circuitsWeiss  _et al._ (2021), etc.
It can handle potentials and junctions on the same footing. Recently, one of
the present authors has studied the formation of bound states at junctionsHe
and Ganesh (2023). Using a tight binding description, junctions of thin wires
were argued to generically host a bound state. The bound state arises from
additional kinetic-energy terms involving shuttling among wires in the
vicinity of the junction. This phenomenon is also seen in higher dimensions,
e.g., when multiple 2D sheets or 3D spaces intersect one another. Bound states
are intimately related to the scattering problem. As is well known, bound
states can be viewed as poles in the scattering coefficientsSakurai and
Napolitano (2017); Ahmed and Bhattacharya (2021). In this article, we extend
junction-potential equivalence from bound states to scattering amplitudes.

Figure 1: Scattering off a potential. A wave travels along a wire, shown as a
collection of sites connected by bonds. It encounters an attractive potential
at one particular site and scatters.

##  II Scattering off a potential: tight-binding approach

To set the stage, we first discuss scattering off a delta-function potential.
While textbooks describe this problem in the continuum using appropriate
boundary conditions, we take a tight-binding approach. This will allow us to
handle the junction problem on the same footing in later sections.

We consider an infinite wire with sites labelled by integers $j$ as shown in
Fig. 1. A particle moves on this wire by ‘hopping’ between nearest neighbours.
When the particle is at site $j=0$, it experiences an attractive potential of
amplitude $g$. In second quantized notation, this problem is described by the
Hamiltonian

$\displaystyle
H=-t\sum_{j}\big{[}c_{j+1}^{\dagger}c_{j}+c_{j-1}^{\dagger}c_{j}\big{]}-g~{}c_{0}^{\dagger}c_{0}.$
(1)

###  II.1 The scattering state

We seek to find stationary states that encode scattering. We propose the
following ansatz for a stationary state,

$\displaystyle\psi_{j}=\left\\{\begin{array}[]{c}Ae^{ikj}+Be^{-ikj},~{}~{}j<0,\\\
C,~{}~{}j=0,\\\ De^{ikj},~{}~{}j>0.\end{array}\right.$ (5)

Here, $A$ represents the amplitude of the incoming wave, coming in from
$-\infty$. The momentum of the wave is denoted as $k$. $B$ and $D$ represent
the amplitudes of the reflected and transmitted waves. We assume that this
ansatz represents an eigenstate with energy $E$.

The eigenvalue equation reduces to the following condition on each site,
$-t\sum_{m(n)}\psi_{m}=E\psi_{n}$. Here, $m(n)$ represents the neighbours of
site $n$. For a stationary state, this condition must be satisfied at every
site $n$. This leads to three independent relations: one arising from sites to
the left of the potential ($n<0$), one at the potential itself and the last
arising from sites to the right ($n>0$). These relations allow us to solve for
B, C and D in terms of A, as shown in the appendix. We write $B=R(k,g)A$ and
$D=T(k,g)A$, where $R$ and $T$ are the reflection and transmission
coefficients respectively. We find

$\displaystyle R(k,g)=\left[\frac{-g}{g+2it\sin
k}\right];~{}T(k,g)=\left[\frac{2it\sin k}{g+2it\sin k}\right].$ (6)

The ‘reflectance’ or reflection probability is given by

$\displaystyle\left|R(k,g)\right|^{2}=\frac{\tilde{g}^{2}}{\tilde{g}^{2}+\sin^{2}k},$
(7)

where $\tilde{g}=g/2t$ is a dimensionless measure of the potential strength.

###  II.2 Bound state from scattering coefficients

It is well known that poles of scattering coefficients correspond to bound
statesNewton  _et al._ (2012). A bound state occurs when there is no incoming
wave, but reflected and transmitted pieces survive – with exponential decay
rather than propagating waves. This can be interpreted as an extension of the
wavevector $k$ to complex values, $k\rightarrow i\kappa$. The reflected
component would take the form $Be^{\kappa j}$ $(j<0)$ while the transmitted
component would be given by $De^{-\kappa j}$ $(j>0)$. The two forms lead to an
exponentially localized solution, provided $\kappa$ is real and positive. To
have a non-zero value of B when A vanishes, $R(i\kappa,g)$ must diverge. This
corresponds to a pole in the reflection coefficient. In the same manner, a
bound state must also correspond to a pole in $T(i\kappa,g)$.

As seen in Eq. 6 above, $R(i\kappa,g)$ and $T(i\kappa,g)$ share the same
denominator. We have a pole when this denominator vanishes. We find a single
pole with

$\displaystyle{\kappa}=\ln\\{\tilde{g}+\sqrt{\tilde{g}^{2}+1}\\}.$ (8)

Here, $\kappa$ is a decay constant. When the potential strength $\tilde{g}$
increases, $\kappa$ increases monotonically; the bound state becomes
progressively more bound. Note that a pole only occurs if the potential is
attractive, i.e., when $g<0$.

##  III Scattering off a junction

Figure 2: Scattering off a junction. A wave travels along the ‘incoming’ wire.
It encounters a junction with $M$ outgoing wires and scatters. The incoming
wire is labelled as ‘$0$’. A part is reflected along the same wire, while the
rest is transmitted through the remaining $M$ wires.

We take the junction geometry shown in Fig. 2. We discretize this space with
points represented as $(j,\ell)$, where $\ell$ identifies a wire and $j$
refers to points within the wire. We designate the incoming wire as $\ell=0$
and outgoing wires as $\ell=1,2,\ldots,M$. Sites of the form $(0,\ell)$ are
identified, i.e., they represent the same site irrespective of $\ell$. This
common site serves as the junction. On the outgoing wires, sites are labelled
as $j=1,2,\ldots$ as we move away from the junction. On the incoming wire, we
label sites as $j=-1,-2,\ldots$ as we move away from the junction.

A particle moving in this geometry is described by a tight binding
Hamiltonian,

$\displaystyle H=H_{\ell=0}+\sum_{\ell=1}^{M}H_{\ell}+H_{junction}.$ (9)

The first piece represents hopping on the incoming wire,

$\displaystyle
H_{\ell=0}=-t\sum_{j=-1,-2,-3,\ldots}\big{[}c_{j,0}^{\dagger}c_{j-1,0}+c_{j-1,0}^{\dagger}c_{j,0}\big{]}.$
(10)

The second represents hopping on each outgoing wire,

$\displaystyle H_{\ell\neq
0}=-t\sum_{j=1,2,3,\ldots}\big{[}c_{j,\ell}^{\dagger}c_{j+1,\ell}+c_{j+1,\ell}^{\dagger}c_{j,\ell}\big{]}.$
(11)

The third encodes hopping from the junction to each wire,

$\displaystyle
H_{junction}=-t\big{\\{}c_{-1,0}^{\dagger}c_{0}+c_{0}^{\dagger}c_{-1,0}\big{\\}}$
$\displaystyle-t\sum_{\ell=1,2,\ldots,M}\big{\\{}c_{1,\ell}^{\dagger}c_{0}+c_{0}^{\dagger}c_{1,\ell}\big{\\}}.$
(12)

Note that all terms in the Hamiltonian represent kinetic energies of hopping.
There are no potential terms.

###  III.1 The scattering state

To describe a steady state, we propose a scattering ansatz of the form

$\displaystyle\psi(j,\ell)=\left\\{\begin{array}[]{c}Ae^{ikj}+Be^{-ikj},~{}~{}j<0,~{}\ell=0\\\
C,~{}~{}j=0,\\\ De^{ikj},~{}~{}j>0,~{}\ell=1,2,\ldots,M.\end{array}\right.$
(16)

As with the potential problem, $A$ is the amplitude of the incoming wave. $B$
and $D$ are amplitudes of reflected and transmitted waves respectively. We
have taken the transmitted wave to be symmetrically distributed among the $M$
outgoing wires. This assumption follows from the symmetry of the junction
geometry. It is also in agreement with numerical results presented in Sec. V
below.

We express $B=R(k,M)A$ and $D=T(k,M)A$, where $R$ and $T$ are the reflection
and transmission coefficients. We demand that this ansatz be an eigenstate of
the Hamiltonian given in Eq. 9. Relegating details to the appendix, we find

$\displaystyle R(k,M)$ $\displaystyle=$
$\displaystyle\Big{[}\frac{(1-M)e^{ik}}{Me^{ik}-e^{-ik}}\Big{]},$
$\displaystyle T(k,M)$ $\displaystyle=$ $\displaystyle\Big{[}\frac{2i\sin
k}{Me^{ik}-e^{-ik}}\Big{]}.~{}~{}~{}$ (17)

This leads to a reflection probability

$\displaystyle\left|R(k,M)\right|^{2}=\frac{f_{M}^{2}}{f_{M}^{2}+\sin^{2}k},$
(18)

where $f_{M}=\frac{1}{2}\Big{(}\sqrt{M}-\frac{1}{\sqrt{M}}\Big{)}$. This
quantity is a dimensionless measure of the ‘degree of connectivity’ of the
junction. When $M=1$, we have one incoming wire and one outgoing wire; the
system forms one smooth wire with no junction. In this limit, $f_{M}$
vanishes. For $M>1$, $f_{M}$ increases monotonically with $M$.

###  III.2 Bound state from scattering coefficients

As in Sec. II.2, bound states can be found from poles of $R(k,M)$ and
$T(k,M)$. To allow for exponential decay, we extend the wavevector $k$ to
complex values, $k\rightarrow i\kappa$. We then set the denominators in Eq. 17
to zero. We find a single bound state with

$\displaystyle\kappa=\frac{\ln M}{2}.$ (19)

Note that $\kappa$ is positive for any $M>1$, indicating a bound state. This
pole represents a localized state that is peaked at the junction, decaying
exponentially along each wire. The decay constant, denoted by $\kappa$, is the
same on all wires. As the junction’s degree of connectivity ($M$) increases,
so does $\kappa$. This indicates that the bound state becomes progressively
more bound. This expression for $\kappa$ agrees with the result in Ref. Khatua
_et al._ , 2021 (after replacing $M\rightarrow 2M-1$, to align with the
notation therein).

##  IV Equivalence between a junction and a potential

Figure 3: Equivalent potential strength vs. degree of connectivity of a
junction, $\tilde{g}_{M}$ vs. $M$.

We have found expressions for reflectance in potential-scattering and
junction-scattering. Comparing Eqs. 7 and 18, we find the same functional
form. This reveals a quantitative equivalence. A junction with $M$ outgoing
wires has the same reflectance as a potential of strength $g$, where $g$ and
$M$ satisfy $\tilde{g}=f_{M}$. Note that $\tilde{g}$ and $f_{M}$ are monotonic
functions of $g$ and $M$ respectively. This relation is the same as the
equivalence condition derived in Ref. He and Ganesh, 2023 on the basis of
bound state properties. Here, we have demonstrated that the same equivalence
holds for scattering (reflectance) as well. In Fig. 3, we plot the equivalent
potential strength ($\tilde{g}_{M}$) vs. the degree of connectivity of a
junction, $M$.

The equivalence here is based on reflectance. A potential and a junction are
considered equivalent if they produce the same reflection probability.
However, the phase of the reflected wave may be different in the two cases.
For example, when $k=\pi/2$, $R(k=\pi/2,M)$ is real for any $M$. However,
$R(k=\pi/2,g)$ carries a non-zero phase. This difference can have observable
consequences in the case of a multichromatic incoming wave (with contributions
from many frequencies). The junction and its equivalent potential will reflect
each frequency component with a different phase shift. As a result, the
reflected wavefront may look very different in the two cases.

In the case of a potential, it is easy to understand that reflectance
increases with potential strength. In the limit of a very large potential
($g/t\rightarrow\infty$), the reflectance reaches unity – an incoming particle
is entirely reflected, with no transmission across the potential. The
equivalence discussed above leads to a surprising observation for junctions.
As the degree of connectivity ($M$) increases, so does the reflectance. For a
junction with two outgoing channels, the minimum reflection probability is
$\sim$ 11% (from Eq. 7 with $k\rightarrow\pi/2$). When the number of outgoing
channels increases to ten, minimum reflectance reaches $\sim$ 67%. With a
hundred outgoing channels, this quantity touches $\sim$ 96%. For any $M>1$,
the maximum reflection probability is unity which occurs in the long
wavelength limit ($k\rightarrow 0$), as seen from Eq. 7.

##  V Wavepacket scattering

Secs. II and III discuss scattering from a steady-state perspective.
Reflectance and transmittance were obtained from a scattering-eigenstate of
the Hamiltonian. We now discuss scattering from a dynamical perspective with
wavepackets that impinge on a potential or a junction. We take a numerical
tight binding approach following Ref. Staelens and Marsiglio, 2021. We begin
with the geometry of Fig. 2, with wires consisting of $L$ sites. Each wire, at
one end, is connected to a potential or a junction. At the other end, we have
an open boundary. The open end does not affect our results as the wavepacket
does not reach the boundary during the course of the simulation.

We construct a Gaussian wavepacket that serves as the initial wavefunction,

$\displaystyle\psi_{j,\ell=0}=\frac{1}{(2\pi\alpha^{2})^{1/4}}\exp\Big{\\{}-\frac{1}{4}\frac{(j-j_{0})^{2}}{\alpha^{2}}+ik_{0}(j-j_{0})\Big{\\}}.~{}~{}$
(20)

The entire weight lies on the incoming wire, with zero amplitude on outgoing
wire(s). The amplitude is peaked at site $j_{0}$ and spread on either side
over a width set by $\alpha$. The parameter $k_{0}$ encodes momentum of the
wavepacket. When $k_{0}=0$, we expect the wavefunction to remain centred at
$j_{0}$, while spreading on either side with time. With a positive value of
$k_{0}$, the wavepacket will move towards the potential/junction. As shown in
Ref. Staelens and Marsiglio, 2021, $k_{0}=\pi/2$ is an ideal choice for
simulations. It leads to minimal spread of the wavefunction, with coherence
retained over a relatively long time scale.

Starting with this initial wavefunction, we time-evolve using the tight
binding Hamiltonian. We take the Hamiltonian of Eq. 1 for potential scattering
and that of Eq. 9 for junction scattering. We have performed simulations with
various values of $\alpha$ (initial width), $k_{0}$ (initial velocity), $g$
(potential strength) and $M$ (degree of connectivity at a junction). We obtain
consistent results that are in line with the steady-state arguments in
previous sections.

Figure 4: Scattering off a potential. We have a 60-site wire, with sites
labelled by $x\in[1,60]$. An incoming wavepacket moves towards increasing $x$
(left) and scatters off a potential at $x=30$ (centre). Subsequently, a
reflected wavepacket moves left while a transmitted component moves to the
right (right). In each plot, the wavefunction amplitude is shown as height,
while the marker colour represents phase. As the phase is $k_{0}=\pi/2$, we
see that the phase varies with a period of four sites.

Fig. 4 shows the evolution of the wavepacket as it impinges on a potential.
After the scattering process, we are left with two propagating wavepackets –
one moving back along the incoming wire and one moving forward on the outgoing
wire. We identify these as reflected and transmitted components. Reflectance
or reflection probability is simply the weight of the reflected component. As
we show below, the reflectance grows monotonically with potential strength.

Figure 5: Scattering off a junction. The top row shows a junction with two
outgoing wires. The wavefunction on each wire during scattering is shown in
(a). The wavefunction after scattering is shown in (b). The bottom row shows a
junction with four outgoing wires. The wavefunction during scattering is shown
in (c). The post-scattering wavefunction is shown in (d).

Fig. 5 shows the wavepacket impinging on a junction. After the scattering
process, we are left with $(M+1)$ propagating pieces – one moving back along
the incoming wire and $M$ moving forward on outgoing wires. They represent
reflected and transmitted components respectively. On each outgoing wire, we
find the same transmitted component. This is consistent with our ansatz in
Sec. III.1. Reflectance is the weight of the reflected component. Fig. 5 shows
two cases: $M=2$ and $M=4$. As seen from the figure, reflection is stronger
for $M=4$. Our simulations show that this is a general trend: reflectance
increases with increasing $M$.

Figure 6: Comparing potential and junction scattering using wavepacket
dynamics. Junction scattering is represented by circles, as reflectance vs.
$f_{M}$. Potential scattering is shown by squares, as reflectance vs.
$\tilde{g}$. We show data for two choices of $k_{0}$, the momentum of the
incoming wavepacket. Open circles/squares correspond to $k_{0}=\pi/2$ while
solid markers correspond to $k_{0}=\pi/4$. The curves plotted are $R(x)$ vs.
$x$ (see text) for the two values of $k_{0}$.

We compare the reflectance from potentials and junctions in Fig. 6. In line
with the equivalence discussed in Sec. IV above, we consider $\tilde{g}$ and
$f_{M}$ to be independent variables. They serve as dimensionless measures of
potential strength/junction connectivity. They are plotted along the X axis in
the figure. The reflectance (weight of reflected component) is the dependent
variable, plotted along the Y axis. For a given value of the incoming momentum
($k_{0}$), the potential-reflectance and junction-reflectance data collapse
onto the same curve. This curve is $R(x)\sim x^{2}/(x^{2}+\sin^{2}k_{0})$,
where $x$ is $\tilde{g}$ or $f_{M}$. This is precisely the reflectance
function found in Eqs. 7 and 18. The figure shows reflectance data for
$k_{0}=\pi/4$ and $\pi/2$ as well as $R(x)$ curves for the two momenta. Data
points for each $k_{0}$ value collapse onto the corresponding $R(x)$ curve.
This provides numerical verification for the arguments of Sec. IV, with each
junction producing the same reflectance as its equivalent potential. Fig. 6
also shows that reflectance increases monotonically with $\tilde{g}$ or
$f_{M}$.

##  VI Scattering in two-dimensions

We have established an equivalence between junctions and potentials in one-
dimensional wires. It is natural to ask if an equivalence also exists in
higher dimensions. We consider the analogous problem in two-dimensions. On the
one hand, we consider a single two-dimensional sheet with a localized (delta-
function-like) potential. On the other, we consider multiple sheets that meet
at a common point. This point of intersection serves as a scattering centre.
We restrict our attention to point-like rather than line-like intersections.
The latter case effectively reduces to a one-dimensional problem as momentum
parallel to the intersection is conserved.

In this two-dimensional setting, the ‘reflected’ component takes the form of a
scattered wave that propagates outward from the scattering centre. Simple
ansätze such as Eqs. 5 and 16 cannot be used. For example, at long
wavelengths, we may expect scattered waves to have circular symmetry with
amplitudes decaying as $1/\sqrt{r}$Adhikari (1986). At the scale of the
lattice, an analytic form is not known for the scattered wavefront. We
therefore restrict ourselves to a numerical wavepacket-based approach.

In the case of potential scattering, we have a single sheet. We describe this
sheet as an $L$x$L$ grid, indexed by $x$ and $y$ coordinates that each range
from $-L/2$ to $+L/2$. An attractive local potential of strength $g$ is placed
at the origin. A two-dimensional Gaussian state is initialized in one of the
quadrants,

$\displaystyle\psi_{j,initial}^{2D}=\frac{1}{\sqrt{2\pi\alpha^{2}}}\exp\Big{\\{}-\frac{1}{4}\frac{(x-x_{0})^{2}+(y-y_{0})^{2}}{\alpha^{2}}\Big{\\}}$
$\displaystyle\times\exp\Big{\\{}ik_{0,x}(x-x_{0})+ik_{0,y}(y-y_{0})\Big{\\}}.~{}~{}$
(21)

This wavefunction is peaked at $(x_{0},y_{0})$. We choose $x_{0}=y_{0}<0$,
with the initial peak lying along the $x=y$ line. The momenta, $k_{0,x}$ and
$k_{0,y}$, are chosen so that the initial velocity points towards the
potential. The spread of the wavefunction ($\alpha$) is chosen to be less than
the initial distance to the origin.

Figure 7: Scattering off a potential in two dimensions. An incoming wavepacket
scatters off a potential located at the origin. Wavefunctions before (left),
during (centre) and after (right) scattering are shown. Post-scattering, we
find a scattered component that moves outward from the origin with square-like
wavefront.

The scattering process is shown in Fig. 7. After the wavepacket scatters off
the potential at the origin, the wavefunction has two parts. One part
resembles a Gaussian moving along the same direction – this is identified as
the ‘unscattered’ component. The second moves away from the origin in the form
of a square-circular ripple – this is the ‘scattered’ component. By comparing
the wavefunction in different quadrants, we may separate the scattered and
unscattered components. For example, near the $x=-y$ line, we only have
contributions from the scattered component. In analogy with reflectance in 1D,
we define the ‘scattered weight’ in 2D. This represents the fraction that has
been deflected away from the initial direction of motion. We extract this
quantity by measuring the weight remaining in the original quadrant after
scattering has taken place. Multiplying this quantity by four yields the
scattered weight.

Fig. 9 (left) plots the scattered weight as a function of potential strength.
As $g$ increases, scattered weight increases monotonically. Unlike reflectance
in 1D, the scattered weight does not approach unity for large $g$. This is
consistent with having a point-like potential in two dimensions: a part of the
wavepacket always passes through without hitting the potential.

We next consider scattering off a junction, with one incoming sheet and $M$
outgoing sheets. All sheets share a common point, which is taken to be the
origin. We initialize a Gaussian wavepacket on the incoming sheet, as given by
Eq. 21. We time-evolve this state using a tight binding Hamiltonian that is
analogous to Eq. 9. After scattering, the resulting wavefunction has
identifiable parts. One part resembles a Gaussian that continues to move in
the same direction. We identify this as the ‘unscattered’ component –
unaffected by the junction. Other parts are square-circular ripples that
propagate outward from the junction. They occur on every sheet with equal
amplitude. In particular, the ripple on the incoming sheet has the same weight
as that on each of the outgoing sheets. We identify these ripples as the
‘scattered’ component. We calculate the scattered weight as the total weight
in all ripples. This represents the probability of the incoming particle being
deflected by the junction.

Figure 8: Scattering off a junction in two dimensions. We have two sheets –
one represented in the top row and the other shown in the bottom row. The two
sheets share a common point at the origin. At top left, we see an incoming
wave on sheet 1 as it moves towards the origin. At this time, there is no
weight on sheet 2 as shown at bottom left. Wavefunctions at the time of
scattering are shown at centre-top and centre-bottom. Post scattering, the
wavefunctions on each sheet are shown in the panels on the right. An
unscattered component proceeds in the same direction on sheet 1. Scattered
wavefronts are identical between the two sheets, proceeding outwards from the
origin.

Fig. 9 (centre) shows the scattered weight as a function of $M$, the degree of
connectivity of the junction. It increases monotonically with $M$ but does not
approach unity.

By comparing Figs. 9 (left) and (centre), we may deem a junction to be
equivalent to a potential if it produces the same scattered weight. For a
given $M$, an equivalent potential $g_{M}$ can be easily found as the
scattered weight increases monotonically with $M$ as well as with $g$. The
obtained equivalence is shown in Fig. 9 (right) . On the X axis, we have the
degree of connectivity (number of outgoing sheets) of the 2D junction. On the
Y axis, we have the equivalent potential ($g$) that produces the same
scattered weight. We show three data sets corresponding to three amplitudes of
the incoming momentum, $k_{0}$. Unlike 1D, the equivalent potential varies
with the incoming momentum. We compare these plots with the equivalence found
in Ref. He and Ganesh, 2023 which compared 2D junctions and potentials in
terms of bound state properties. As seen from the plot, the equivalent
potential à la Ref. He and Ganesh, 2023 is different from that found by
comparing scattered weights.

The 2D equivalence shown in Fig. 9 (right) plot can be compared to the 1D
result in Fig. 6. In both cases, as the junction’s connectivity ($M$)
increases, the equivalent potential increases monotonically. In 1D, the
equivalence is robust and non-dispersive as it does not depend on the incoming
momentum. However, in 2D, the equivalent potential shifts with momentum. This
suggests that a limited notion of equivalence holds in 2D. For a coherent
incoming wave with a dominant frequency, junction-scattering is equivalent to
potential scattering. However, with a generic incoming wavepacket, there is no
equivalence.

Figure 9: Scattering in two dimensions. At left, we represent potential
scattering with scattered weight plotted against potential strength for
$k_{0}=\pi/2$. At the centre, we depict junction-scattering with scattered
weight plotted against $M$, also for $k_{0}=\pi/2$. Both functions are
increase monotonically. At right, we plot the ‘equivalent potential’ $g_{M}$
vs. $M$. The equivalent potential changes with incoming momentum, with data
for three values of $k_{0}$ shown. The plot shows a fourth data set, labelled
‘Bound state $g_{M}$’. This is the equivalent potential found in Ref. He and
Ganesh, 2023 – which defines an equivalent potential as that which produces
the same bound state wavefunction.

##  VII Discussion

Our central result is that a junction is quantitatively equivalent to a
potential in 1D, in the sense that it produces the same reflectance. In 2D, an
equivalence exists but is momentum-dependent. We have used the tight binding
approach to describe scattering. Previous studies on junction-scattering have
used a continuum approach. The junction is implemented via a boundary
conditionKostrykin and Schrader (1999) or an ersatz inter-wire couplingTokuno
_et al._ (2008). The continuum and tight binding approaches have been compared
in previous studiesKowal  _et al._ (1990); Aharony and Entin-Wohlman (2009);
Soori (2023). One of our results (reflectance of a junction scales as $\sim
1/M$) agrees with unitarity-based constraints derived using a continuum
description of a star graphGratus  _et al._ (1994). This suggests an
interesting question for future studies: does the junction-potential
equivalence hold in the continuum?

Transmission through quantum networks is an issue of wide interestKottos and
Smilansky (1999); Hennig and Tsironis (1999); Vidal  _et al._ (2000); Naud
_et al._ (2001); Yang  _et al._ (2007); Almeida and Macêdo (2013); Caudrelier
_et al._ (2013); Wang  _et al._ (2022). Abstract networks with fractal
geometries have also evoked interest for their transmission propertiesBell and
MacKinnon (1994); Lin  _et al._ (2007); Monthus and Garel (2011); Xu  _et al._
(2021). As networks are constructed with junctions as building blocks, our
results may hold relevance. Our results suggest that the degree of
connectivity at each junction plays a strong role. For example, an $n$-Cayley
tree is built from junctions where $n$ wires meetWeisstein . Our results
suggest that transmission through such as network will fall as $n$ increases.

Apart from physical networks, junction transport can play a role in abstract
spaces. One example is the phenomenon of Fermi surface oscillations in metals
(e.g., de Haas van Alphen effect)Ashcroft and Mermin (2011). In the
semiclassical limit, it is understood as the motion of electronic wavepackets
on the Fermi surface. In materials such as highly-doped grapheneNikolaev
(2021), multiple Fermi surfaces intersect at junctions. Fermi surface
oscillations must then be understood in terms of junction-scattering. Our
results for scattering amplitudes may be of use here.

###### Acknowledgements.

We thank Diptiman Sen, Kirill Samokhin, Abhiram Soori and Eric He for
insightful discussions. This work was supported by a Discovery Grant
2022-05240 from the Natural Sciences and Engineering Research Council of
Canada.

##  Appendix A Scattering coefficients from tight binding

We first consider scattering off a potential in one dimension as described in
Sec. II. We demand that the ansatz of Eq. 5 must be an eigenstate of the
Hamiltonian with eigenvalue $E$. On sites that are not connected to the
potential ($j<-1$ and $j>1$), the eigenstate condition yields

$\displaystyle E=-2t\cos k,$ (22)

which fixes the energy eigenvalue. The eigenstate condition on sites connected
to the potential ($j=\pm 1$) yields a continuity relation,

$\displaystyle A+B=C=D.$ (23)

Precisely at the potential site, we obtain

$\displaystyle-t\big{[}Ae^{-ik}+Be^{+ik}+De^{ik}\big{]}-gC=EC.$ (24)

From these relations, the reflection coefficient $C/A$ and the transmission
coefficient $D/A$ can be found. This leads to the expressions in Eq. 6.

We next consider scattering off a junction, described in Sec. III. The ansatz
in Eq. 16 is to be an eigenstate of the Hamiltonian with energy eigenvalue
$E$. Away from the junction ($|j|>1$), the eigenstate condition reduces to
$E=-2t\cos k$. In the immediate neighbourhood of the junction ($j=\pm 1$), the
eigenstate condition yields a continuity equation, $A+B=C=D$. At the junction
itself ($j=0$), the eigenstate condition yields

$\displaystyle-t\big{[}Ae^{-ik}+Be^{ik}+MDe^{ik}\big{]}=EC.$ (25)

From these relations, we obtain the expressions for reflectance and
transmission in Eq. 17.

## References

  * Schiff (1968) L. Schiff,  _Quantum Mechanics_ (Tokyo, 1968).
  * Sakurai and Napolitano (2017) J. J. Sakurai and J. Napolitano,  _Modern Quantum Mechanics_, 2nd ed. (Cambridge University Press, 2017).
  * Goldberger and Watson (2004) M. Goldberger and K. Watson,  _Collision Theory_, Dover books on physics (Dover Publications, 2004).
  * Newton (1982) R. G. Newton,  _Scattering Theory of Waves and Particles_, 2nd ed. (Springer Berlin, Heidelberg, 1982).
  * Joachain (1975) C. Joachain,  _Quantum Collision Theory_ (North-Holland Publishing Company, 1975).
  * Ahmed and Bhattacharya (2021) Z. Ahmed and K. Bhattacharya, European Journal of Physics 42, 055405 (2021).
  * Cassettari  _et al._ (2000) D. Cassettari, B. Hessmo, R. Folman, T. Maier, and J. Schmiedmayer, Phys. Rev. Lett. 85, 5483 (2000).
  * Dumke  _et al._ (2002) R. Dumke, T. Müther, M. Volk, W. Ertmer, and G. Birkl, Phys. Rev. Lett. 89, 220402 (2002).
  * Goñi  _et al._ (1992) A. R. Goñi, L. N. Pfeiffer, K. W. West, A. Pinczuk, H. U. Baranger, and H. L. Stormer, Applied Physics Letters 61, 1956 (1992).
  * Hasen  _et al._ (1997) J. Hasen, L. Pfeiffer, A. Pinczuk, H. Baranger, K. West, and B. Dennis, Superlattices and Microstructures 22, 359 (1997).
  * Berggren and Ji (1991) K.-F. Berggren and Z.-l. Ji, Phys. Rev. B 43, 4760 (1991).
  * Naud  _et al._ (2001) C. Naud, G. Faini, and D. Mailly, Phys. Rev. Lett. 86, 5104 (2001).
  * Hul  _et al._ (2004) O. Hul, S. Bauch, P. Pakoński, N. Savytskyy, K. Życzkowski, and L. Sirko, Phys. Rev. E 69, 056205 (2004).
  * Bongs and Sengstock (2004) K. Bongs and K. Sengstock, Reports on Progress in Physics 67, 907 (2004).
  * Fortágh and Zimmermann (2007) J. Fortágh and C. Zimmermann, Rev. Mod. Phys. 79, 235 (2007).
  * Amico  _et al._ (2022) L. Amico, D. Anderson, M. Boshier, J.-P. Brantut, L.-C. Kwek, A. Minguzzi, and W. von Klitzing, Rev. Mod. Phys. 94, 041001 (2022).
  * Andrade  _et al._ (2016) F. M. Andrade, A. Schmidt, E. Vicentini, B. Cheng, and M. da Luz, Physics Reports 647, 1 (2016).
  * Gratus  _et al._ (1994) J. Gratus, C. J. Lambert, S. J. Robinson, and R. W. Tucker, Journal of Physics A: Mathematical and General 27, 6881 (1994).
  * Zakrzewski (2007) J. Zakrzewski, Acta Physica Polonica B 38, 1673 (2007).
  * Weiss  _et al._ (2021) D. K. Weiss, W. DeGottardi, J. Koch, and D. G. Ferguson, Phys. Rev. Res. 3, 033244 (2021).
  * He and Ganesh (2023) E. He and R. Ganesh, “Bound states without potentials: localization at singularities,” (2023), arXiv:2302.03065 [quant-ph] .
  * Newton  _et al._ (2012) R. Newton, K. Chadan, and P. Sabatier,  _Inverse Problems in Quantum Scattering Theory_, Theoretical and Mathematical Physics (Springer Berlin Heidelberg, 2012).
  * Khatua  _et al._ (2021) S. Khatua, S. Srinivasan, and R. Ganesh, Phys. Rev. B 103, 174412 (2021).
  * Staelens and Marsiglio (2021) M. Staelens and F. Marsiglio, American Journal of Physics 89, 693 (2021), https://doi.org/10.1119/10.0003689 .
  * Adhikari (1986) S. K. Adhikari, American Journal of Physics 54, 362 (1986), https://pubs.aip.org/aapt/ajp/article-pdf/54/4/362/11969961/362_1_online.pdf .
  * Kostrykin and Schrader (1999) V. Kostrykin and R. Schrader, Journal of Physics A: Mathematical and General 32, 595 (1999).
  * Tokuno  _et al._ (2008) A. Tokuno, M. Oshikawa, and E. Demler, Phys. Rev. Lett. 100, 140402 (2008).
  * Kowal  _et al._ (1990) D. Kowal, U. Sivan, O. Entin-Wohlman, and Y. Imry, Phys. Rev. B 42, 9009 (1990).
  * Aharony and Entin-Wohlman (2009) A. Aharony and O. Entin-Wohlman,  _The Journal of Physical Chemistry B_ , The Journal of Physical Chemistry B 113, 3676 (2009).
  * Soori (2023) A. Soori, Solid State Communications 360, 115034 (2023).
  * Kottos and Smilansky (1999) T. Kottos and U. Smilansky, Annals of Physics 274, 76 (1999).
  * Hennig and Tsironis (1999) D. Hennig and G. Tsironis, Physics Reports 307, 333 (1999).
  * Vidal  _et al._ (2000) J. Vidal, G. Montambaux, and B. Douçot, Phys. Rev. B 62, R16294 (2000).
  * Yang  _et al._ (2007) S. Yang, Z. Song, and C.-p. Sun, Frontiers of Physics in China 2, 1 (2007).
  * Almeida and Macêdo (2013) F. Almeida and A. Macêdo, Journal of Computational Physics 243, 1 (2013).
  * Caudrelier  _et al._ (2013) V. Caudrelier, M. Mintchev, and E. Ragoucy, Physics Letters A 377, 1788 (2013).
  * Wang  _et al._ (2022) X. Wang, M. Laav, I. Volotsenko, A. Frydman, and B. Kalisky, Phys. Rev. Appl. 17, 024073 (2022).
  * Bell and MacKinnon (1994) P. M. Bell and A. MacKinnon, Journal of Physics: Condensed Matter 6, 5423 (1994).
  * Lin  _et al._ (2007) Z. Lin, Z. Hou, and Y. Liu, Physics Letters A 365, 240 (2007).
  * Monthus and Garel (2011) C. Monthus and T. Garel, Journal of Physics A: Mathematical and Theoretical 44, 145001 (2011).
  * Xu  _et al._ (2021) X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. M. Smith, and X.-M. Jin, Nature Photonics 15, 703 (2021).
  * (42) E. W. Weisstein, “Cayley tree. From MathWorld—A Wolfram Web Resource,” .
  * Ashcroft and Mermin (2011) N. Ashcroft and N. Mermin,  _Solid State Physics_ (Cengage Learning, 2011).
  * Nikolaev (2021) A. V. Nikolaev, Phys. Rev. B 104, 035419 (2021).