EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-278 LHCb-PAPER-2019-042 Feb. 25, 2020

First observation

of excited ${{\mathchar 28938\relax}^{-}_{b}}$ states

LHCb collaboration†††Authors are listed at the end of this Letter.

We report four narrow peaks in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum obtained using $pp$ collisions
at center-of-mass energies of 7, 8 and 13 TeV, corresponding to a total
integrated luminosity of 9$\text{\,fb}^{-1}$ recorded by the LHCb experiment.
Referring to these states by their mass, the mass values are

$\displaystyle m({{\mathchar 28938\relax}_{b}(6316)^{-}})$
$\displaystyle=6315.64\pm 0.31\pm 0.07\pm 0.50\text{\,Me\kern-1.00006ptV},$
$\displaystyle m({{\mathchar 28938\relax}_{b}(6330)^{-}})$
$\displaystyle=6330.30\pm 0.28\pm 0.07\pm 0.50\text{\,Me\kern-1.00006ptV},$
$\displaystyle m({{\mathchar 28938\relax}_{b}(6340)^{-}})$
$\displaystyle=6339.71\pm 0.26\pm 0.05\pm 0.50\text{\,Me\kern-1.00006ptV},$
$\displaystyle m({{\mathchar 28938\relax}_{b}(6350)^{-}})$
$\displaystyle=6349.88\pm 0.35\pm 0.05\pm 0.50\text{\,Me\kern-1.00006ptV},$

where the uncertainties are statistical, systematic and the last is due to the
knowledge of the ${{\mathchar 28932\relax}^{0}_{b}}$ mass. The natural widths
of the three lower mass states are consistent with zero, and the 90%
confidence-level upper limits are determined to be ${\Gamma({{\mathchar
28938\relax}_{b}(6316)^{-}})<2.8\text{\,Me\kern-1.00006ptV}}$,
${\Gamma({{\mathchar
28938\relax}_{b}(6330)^{-}})<3.1\text{\,Me\kern-1.00006ptV}}$ and
${\Gamma({{\mathchar
28938\relax}_{b}(6340)^{-}})<1.5\text{\,Me\kern-1.00006ptV}}$. The natural
width of the ${{\mathchar 28938\relax}_{b}(6350)^{-}}$ peak is
$1.4^{+1.0}_{-0.8}\pm 0.1\text{\,Me\kern-1.00006ptV}$, which is 2.5$\sigma$
from zero and corresponds to an upper limit of
$2.8\text{\,Me\kern-1.00006ptV}$. The peaks have local significances ranging
from 3.6$\sigma$ to 7.2$\sigma$. After accounting for the look-elsewhere
effect, the significances of the ${\mathchar 28938\relax}_{b}(6316)^{-}$ and
${\mathchar 28938\relax}_{b}(6330)^{-}$ peaks are reduced to 2.1$\sigma$ and
2.6$\sigma$ respectively, while the two higher mass peaks exceed 5$\sigma$.
The observed peaks are consistent with expectations for excited ${{\mathchar
28938\relax}^{-}_{b}}$ resonances.

Published in Phys. Rev. Lett.

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

The study of hadrons containing heavy ($b$ or $c$) quarks has undergone a
renaissance over the last couple of decades. During this time a plethora of
new states have been observed, including candidates for four-quark
(tetraquark) states, and more recently five-quark (pentaquark) states [1, 2,
3] (see Refs. [4, 5, 6] for recent reviews). In addition, a number of
observations of peaking structures in the invariant-mass spectra of final
states containing ${{\mathchar 28932\relax}^{+}_{c}}{{K}^{-}}$ [7],
${{\mathchar 28932\relax}^{0}_{b}}{{\pi}^{-}}$ [8], ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{-}}$ [9], and ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ [10, 11] have provided valuable
experimental information to improve our understanding of quantum
chromodynamics (QCD), the theory of the strong interaction.

Fueled by these observations, there has been a renewed interest in gaining a
deeper theoretical understanding of hadronic structure. The constituent quark
model [12, 13, *Zweig:570209] has been very successful in describing the types
of hadrons that form in nature and how they fit into multiplets [15] based on
the quantum numbers of the states. While conventional baryons are understood
to be states that contain three valence quarks, a deep understanding of how
best to describe these and other multi-quark states in terms of their
fundamental constituents is still an open question. For example, in QCD, two
quarks can exhibit attraction when in a $J^{P}=0^{+}$ quantum state, giving
rise to the notion that conventional baryons can be described as the bound
state of a quark and a $qq^{\prime}$ diquark [16, 17]. These ideas are
naturally extensible to describe tetraquark and pentaquark candidates [4, 5,
6].

Recently, the LHCb experiment observed five narrow states, assumed to be
excited ${{\mathchar 28938\relax}^{0}_{c}}$ baryons , which decay into
${{\mathchar 28932\relax}^{+}_{c}}{{K}^{-}}$ [7]. These states have been
analyzed from the perspective of constituent quark models and lattice QCD [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 31], quark-
diquark models [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], as well as
molecular models [46, 47, 48, 49, 50, 51] and pentaquark states [52, 53, 54].
Several of the models that seek to describe these peaks also make predictions
for ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ resonances. Since the quark
contents of the ${{\mathchar 28938\relax}^{0}_{c}}$ and ${{\mathchar
28938\relax}^{-}_{b}}$ baryons are $css$ and $bss$, respectively, it is of
great interest to search for analogous states in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum.

This Letter reports on a search for narrow resonances in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum close to the kinematic
threshold. The search uses data collected in $pp$ collisions with the LHCb
detector at center-of-mass energies of 7, 8 and 13 TeV, corresponding to
integrated luminosities of 1, 2 and 6$\text{\,fb}^{-1}$, respectively. Charge-
conjugate processes are implicitly included, and natural units with
$\hbar=c=1$ are used throughout.

The LHCb detector [55, 56] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. Events are selected online by a trigger, which
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction [57, 58]. Simulated data samples are produced using the
software packages described in Refs. [59, 60, 61, 62, 63, 64,
*Agostinelli:2002hh, 66], and are used to optimize selection requirements and
to quantify the invariant-mass resolution of the LHCb detector.

Samples of ${{\mathchar 28932\relax}^{0}_{b}}$ candidates are formed by
pairing ${{\mathchar 28932\relax}^{+}_{c}}$ and ${{\pi}^{-}}$ candidates,
where the ${{\mathchar 28932\relax}^{+}_{c}}$ decays are reconstructed in the
$p{{K}^{-}}{{\pi}^{+}}$ final state. All final-state hadrons must have
particle-identification (PID) information consistent with the assigned
particle hypothesis. The final-state particles are also required to be
inconsistent with originating from a primary $pp$ collision vertex (PV) by
requiring that they have large $\chi^{2}_{\text{IP}}$ with respect to all PVs
in the event. The quantity $\chi^{2}_{\text{IP}}$ is the difference in
$\chi^{2}$ of the vertex fit of a given PV when the particle (here $p$,
${{K}^{-}}$ or ${{\pi}^{+}}$) is included and excluded from the fit.

The ${{\mathchar 28932\relax}^{+}_{c}}$ candidates must have a fitted vertex
that is significantly displaced from all PVs in the event and have an
invariant mass within 18 MeV of the known ${{\mathchar 28932\relax}^{+}_{c}}$
mass [15]. About 20% of the ${{\mathchar 28932\relax}^{+}_{c}}$ background
comprises misidentified
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}},~{}{{D}^{+}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$,
${{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{*+}}\rightarrow({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays, as well as misidentified $\phi$ mesons with
$\phi\rightarrow{{K}^{+}}{{K}^{-}}$ combined with an additional particle from
elsewhere in the event. These background contributions are removed by
employing tighter PID requirements on candidates that are consistent with any
of these decay hypotheses, resulting in about 1% loss of signal efficiency.
The $p{{K}^{-}}{{\pi}^{+}}$ invariant-mass distribution of ${{\mathchar
28932\relax}^{+}_{c}}$ candidates satisfying these selection requirements is
shown in Fig. 1 (left).

Figure 1: Invariant-mass spectrum for (left) ${{\mathchar
28932\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ and (right)
${{\mathchar 28932\relax}^{0}_{b}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{-}}$ candidates in data passing the selection
requirements described in the text. The arrows indicate the requirements on
the invariant masses that are applied in the subsequent stages of the
analysis.

The ${{\mathchar 28932\relax}^{0}_{b}}$ candidates are formed from
${{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$ combinations that have a
significantly displaced decay vertex from all PVs in the event and a
trajectory that is consistent with originating from one of them. The PV for
which the ${{\mathchar 28932\relax}^{0}_{b}}$ candidate has the smallest
$\chi^{2}_{\text{IP}}$ is assigned to be the associated PV, and it is used
subsequently to compute quantities such as the ${{\mathchar
28932\relax}^{0}_{b}}$ decay time. Candidates satisfying the requirement
$5.6<M({{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}})<6.0$ GeV are retained,
where $M$ designates the invariant mass of the system.

To further suppress background in the ${{\mathchar
28932\relax}^{0}_{b}}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$
sample, a boosted decision tree (BDT) discriminant [67] is used. The BDT
exploits 21 input variables: the decay times of the ${{\mathchar
28932\relax}^{+}_{c}}$ and ${{\mathchar 28932\relax}^{0}_{b}}$ candidates and
the $\chi^{2}$ values associated with their decay-vertex fits; the angle
between the ${{\mathchar 28932\relax}^{0}_{b}}$ momentum vector and the line
that joins the ${{\mathchar 28932\relax}^{0}_{b}}$ decay vertex and its
associated PV; and for each final state particle the momentum, transverse
momentum, $\chi^{2}_{\text{IP}}$ and a PID response variable. The PID response
for final-state hadrons in the signal decay is obtained from large
${{D}^{*+}}\rightarrow({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
and ${\mathchar 28931\relax}\rightarrow p{{\pi}^{-}}$ calibration samples in
data [68, 69]. Simulated signal decays and background from the ${{\mathchar
28932\relax}^{+}_{c}}$ mass sidebands
($30<|M(p{{K}^{-}}{{\pi}^{+}})-m_{{{\mathchar 28932\relax}^{+}_{c}}}|<50$ MeV)
in data are used to train the BDT, where $m$ refers to the mass of the
indicated particle [15]. The chosen requirement on the BDT response provides a
relative signal efficiency of 90%, and reduces the combinatorial background by
about a factor of 2.5. Overall, the offline selection requirements are about
75% efficient on simulated decays, while reducing the background by about a
factor of 40.

Figure 1 (right) shows the ${{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$ mass
spectrum for candidates passing the above selection criteria. The spectrum is
fit with the sum of two Crystal Ball [70] functions with a common mean and
opposite-side power-law tails to model the signal, and an exponential function
to describe the background distribution. The fitted ${{\mathchar
28932\relax}^{0}_{b}}$ signal yield is $19\,200\pm 200$.

To search for peaking structures in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum, a requirement that
$|M({{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}})-m_{{{\mathchar
28932\relax}^{0}_{b}}}|<40\text{\,Me\kern-1.00006ptV}$ is imposed, which
reduces the number of ${{\mathchar 28932\relax}^{0}_{b}}$ signal decays to
about 18 000. Each ${{\mathchar 28932\relax}^{0}_{b}}$ candidate is combined
with a ${{K}^{-}}$ candidate that is consistent with originating from a PV in
the event. The ${{\mathchar 28932\relax}^{0}_{b}}$ and ${{K}^{-}}$
trajectories are fitted to a common vertex, and that vertex is kinematically
constrained to coincide with the PV associated with the ${{\mathchar
28932\relax}^{0}_{b}}$ candidate [71]. The additional PV constraint improves
the resolution on the mass difference $\delta M\equiv M({{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}})-M({{\mathchar 28932\relax}^{0}_{b}})$ by about
a factor of two.

Random combinations of ${{\mathchar 28932\relax}^{0}_{b}}$ baryons with a
${{K}^{-}}$ candidate are the largest source of background in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum. To improve the expected signal-
to-background ratio, a figure of merit, $\epsilon/(\sqrt{B}+5/2)$ [72], is
used to optimize the requirements on the PID information of the ${{K}^{-}}$
candidates. Here, $\epsilon$ is the efficiency as determined from simulation,
and $B$ is the number of wrong-sign ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{+}}$ combinations in the region $520<\delta M<570$
MeV passing the PID requirement, scaled to a 10 MeV mass window. The 10 MeV
width is chosen based on the search for narrow peaks, since the low signal
yields expected would make wide peaks difficult to separate from the
combinatorial background. The optimal requirement on the ${{K}^{-}}$ PID
provides an efficiency of about 85% and suppresses the background by a factor
of about 2.5.

The decay of a resonance to ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ will
produce peaks in the $\delta M$ spectrum. The experimental $\delta M$
resolution is obtained from simulated samples generated at several masses,
$m_{\rm res}$. The resolution function is described by the sum of two Gaussian
functions with a common mean. In addition, the width of the narrower Gaussian
component, $\sigma_{\rm core}$, is fixed to be 45% of that of the wider
component, and its contribution is required to constitute 80% of the total
shape. A smooth, monotonically increasing function, denoted as $\sigma(m_{\rm
res})$, is then used to parameterize $\sigma_{\rm core}$ as a function of
$m_{\rm res}$. In the $\delta M$ interval of interest, $\sigma(m_{\rm res})$
is in the range of 0.7–0.8 MeV.

The $\delta M$ distributions for right-sign (RS) and wrong-sign (WS)
candidates are shown in Fig. 2, along with fits to the spectra as described
below. Four peaks are seen in the RS spectrum of ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ candidates (red curves), whereas no
significant peaks are seen in the corresponding WS ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{+}}$ distribution. To obtain the parameters of the
peaks, a simultaneous unbinned extended maximum-likelihood fit is performed to
the RS and WS spectra. Each signal peak is described by an $S$–wave
relativistic Breit–Wigner function [73] with a Blatt–Weisskopf barrier factor
[74], convoluted with the resolution function $\sigma(m_{\rm res})$ described
above. A common background shape is used to describe both the RS and WS
spectra, and is described by a smooth three-parameter monotonic function that
accounts for the ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ threshold.

The peak values of $\delta M$, natural widths, signal yields, and the local
and global significances are summarized in Table 1. The local significance is
obtained as ${{\mathcal{S}}_{\rm data}=\sqrt{2\log({{\mathcal{L}}_{\rm
max}/\mathcal{L}}_{0}})}$, where ${\mathcal{L}}_{\rm max}$ is the maximum
value of the fit likelihood and ${\mathcal{L}}_{0}$ is the value obtained when
a given peak’s yield is fixed to zero. All peaks have natural width consistent
with zero. The highest-mass peak has the largest width, which differs from
zero by 2.5 standard deviations, as determined from a likelihood scan of the
width parameter.

To account for the look-elsewhere effect [75], which considers that the peak
search extends over about a 200 MeV wide mass region, a large number of
pseudoexperiments (pe) are generated. The pseudoexperiments use the nominal
parameters from the fit to the data, with the signal yield of each peak, in
turn, set to zero. The full mass region is scanned in 0.5 MeV steps to
identify the most significant positive fluctuation outside of the region of
the three retained peaks, from which the significance ${\mathcal{S}}_{\rm pe}$
is computed. From the corresponding distribution of ${\mathcal{S}}_{\rm pe}$
and the value ${\mathcal{S}}_{\rm data}$, a $p$-value — expressed in Gaussian
standard deviations — is obtained for each peak, as shown in Table 1.

Figure 2: Distribution of the mass difference for (top) right-sign ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ candidates, and (bottom) wrong-sign ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{+}}$ candidates, as described in the text. Table 1: Peak positions, widths, signal yields, and local and global significances of the four mass peaks observed in the ${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectra, as described in the text. The uncertainties are statistical only. Peak of $\delta M$ | Width | Signal | Significances [$\sigma$]
---|---|---|---
$[\\!\text{\,Me\kern-1.00006ptV}]$ | $[\\!\text{\,Me\kern-1.00006ptV}]$ | yield | Local | Global
$523.74\pm 0.31$ | $0.00\,^{+\,0.7}_{-\,0.0}$ | $15\,_{-\,5}^{\,+6\phantom{0}}$ | 3.6 | 2.1
$538.40\pm 0.28$ | $0.00\,^{+\,0.4}_{-\,0.0}$ | $18\,_{-\,5}^{+\,6\phantom{0}}$ | 3.7 | 2.6
$547.81\pm 0.26$ | $0.47\,^{+\,0.6}_{-\,0.5}$ | $47\,_{-\,10}^{+\,11}$ | 7.2 | 6.7
$557.98\pm 0.35$ | $1.4\,^{+\,1.0}_{-\,0.8}$ | $57\,_{-\,13}^{+\,14}$ | 7.0 | 6.2

The sources of systematic uncertainty that affect the measured masses are
summarized in Table 2. The momentum scale uncertainty is assessed by shifting
the momentum scale of all charged tracks by $\pm$0.03% [76] in simulated
decays, and evaluating the change in $\delta M$. The imperfect modeling of the
energy loss in the detector material results in a systematic uncertainty of
0.04 MeV [77]. The uncertainty due to the choice of signal model is assigned
by fitting the data with an alternative signal model composed of two Gaussian
functions with a common mean. The largest change, 0.02 MeV, is assigned as a
systematic uncertainty to all of the peak positions. The background shape
uncertainty is assessed by removing the influence of the WS data on the
background shape, and fitting only the RS data; the difference in the peak
positions with respect to the nominal fit is assigned as a systematic
uncertainty. The relativistic Breit–Wigner signal shape in the nominal fit
assumes that the decay proceeds through an $S$-wave, with an interaction
radius in the Blatt–Weisskopf barrier factor of
$R=3\text{\,Ge\kern-1.00006ptV}^{-1}$. Changing the angular momentum in the
decay to $L=2$ ($D$-wave), and separately varying $R$ between 1 and 5
$\text{\,Ge\kern-1.00006ptV}^{-1}$, leads to a negligible change in the peak
positions. For the absolute mass determination, the world-average ${{\mathchar
28932\relax}^{0}_{b}}$ mass of $5791.9\pm 0.5$ MeV [15] is used. The
uncertainty of 0.5 MeV on this mass dominates the systematic uncertainty and
is quoted separately in the final results.

Table 2: Systematic uncertainties on the measured peak positions in the ${\delta M=M({{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}})-M({{\mathchar 28932\relax}^{0}_{b}})}$ spectrum. The peaks are numbered in order of increasing mass.  Source | Peak 1 | Peak 2 | Peak 3 | Peak 4
---|---|---|---|---
| [MeV] | [MeV] | [MeV] | [MeV]
Momentum scale | 0.01 | 0.02 | 0.02 | 0.03
Energy loss | 0.04 | 0.04 | 0.04 | 0.04
Signal shape | 0.02 | 0.02 | 0.02 | 0.02
Background | 0.05 | 0.05 | 0.01 | 0.01
Total | 0.07 | 0.07 | 0.05 | 0.05

The primary source of systematic uncertainty on the natural widths of the
observed peaks is from an imperfect knowledge of the $\delta M$ resolution,
which is obtained from simulation. Based on previous studies of
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays [78], the $\delta M$
resolution in simulation agrees with that of data within 10%. The impact of a
$\pm$10% variation in the resolution is evaluated using pseudoexperiments,
where each experiment is generated using the nominal signal resolution
function, and fitted with a 10% smaller or larger $\delta M$ resolution.
Deviations of $\pm$0.10 MeV relative to the true value of the width are found
for a range of input widths corresponding to that which is observed in data.
The upper limits on the natural width of the observed peaks are evaluated by
convoluting the likelihoods with this 0.10 MeV uncertainty, and finding the
values of the widths that contain 90% and 95% of the integrated probability.
For both the mass differences and widths, the total uncertainty is dominated
by the statistical component.

Table 3: Summary of the peak parameters of the four peaks, showing the peak positions of ${\delta M=M({{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}})-M({{\mathchar 28932\relax}^{0}_{b}})}$, the masses, and 90% (95%) confidence level upper limits on the natural widths. The indicated uncertainties are statistical, systematic, and due to the world-average value of the ${{\mathchar 28932\relax}^{0}_{b}}$ mass (for the masses). For the ${{\mathchar 28938\relax}_{b}(6350)^{-}}$ peak, the central value of the width is also indicated. |  $\delta M_{\rm peak}$ [MeV] | Mass [MeV] | Width [MeV]
---|---|---|---
${{\mathchar 28938\relax}_{b}(6316)^{-}}$ | $523.74\pm 0.31\pm 0.07$ | $6315.64\pm 0.31\pm 0.07\pm 0.50$ | $<2.8~{}(4.2)$
${{\mathchar 28938\relax}_{b}(6330)^{-}}$ | $538.40\pm 0.28\pm 0.07$ | $6330.30\pm 0.28\pm 0.07\pm 0.50$ | $<3.1~{}(4.7)$
${{\mathchar 28938\relax}_{b}(6340)^{-}}$ | $547.81\pm 0.26\pm 0.05$ | $6339.71\pm 0.26\pm 0.05\pm 0.50$ | $<1.5~{}(1.8)$
${{\mathchar 28938\relax}_{b}(6350)^{-}}$ | $557.98\pm 0.35\pm 0.05$ | $6349.88\pm 0.35\pm 0.05\pm 0.50$ | $<2.8~{}(3.2)$
|  |  | $~{}~{}\,\,1.4\,^{+\,1.0}_{-\,0.8}\pm 0.1$

The measured masses and widths of the four peaks in the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum are summarized in Table 3. They
are qualitatively similar to those observed in the ${{\mathchar
28932\relax}^{+}_{c}}{{K}^{-}}$ mass spectrum [7]. Arguably, the simplest
interpretation of these peaks is that they correspond to excited ${{\mathchar
28938\relax}^{-}_{b}}$ states, in particular the $L=1$ angular momentum
excitations of the ground state, or possibly $n=2$ radial excitations. Many of
the quark model calculations predict $L=1$ states in this mass region [18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 29, 34], and at least some of the states
should be narrow [22, 24, 34]. In particular, using the ${}^{3}P_{0}$ model,
five states in this mass region are predicted, with approximately 8 MeV mass
splittings; the four lightest have partial width, $\Gamma({{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}})$, below 1 MeV, while that with the largest
mass has $\Gamma({{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}})=1.49\text{\,Me\kern-1.00006ptV}$ [24]. On the
other hand, predictions using the chiral quark-model indicate that the
$J^{P}=\frac{3}{2}^{-}$ and $\frac{5}{2}^{-}$ states are narrow, but the
$\frac{1}{2}^{-}$ states are wide, in the 50–100 MeV range [34].

Quark-diquark models have also predicted several excited ${{\mathchar
28938\relax}^{-}_{b}}$ states in the region around 6.3 GeV [35, 36, 43, 79],
with mass splittings similar to those observed here. In an implementation of
the ${}^{3}P_{0}$ model, the $J^{P}=\frac{3}{2}^{-}$ and $\frac{3}{2}^{-}$ are
predicted to be narrow [79]. Molecular models have also been employed, where
two narrow $J^{P}=\frac{1}{2}^{-}$ states are predicted at 6405 and 6465 MeV
[80]; no statistically significant peaks are seen at those masses with the
current dataset.

An alternate interpretation for one or more of the observed peaks is that they
arise from the decay of a higher-mass excited ${{\mathchar
28938\relax}^{-}_{b}}$ state, ${{\mathchar
28938\relax}^{**-}_{b}}\rightarrow{{\mathchar 28932\relax}^{\prime
0}_{b}}(\rightarrow{{\mathchar 28932\relax}^{0}_{b}}{{\pi}^{0}}){{K}^{-}}$,
where the ${{\pi}^{0}}$ meson is undetected. While the ${{\mathchar
28932\relax}^{\prime-}_{b}}$, ${{\mathchar 28932\relax}^{*-}_{b}}$ [78] and
${{\mathchar 28932\relax}^{*0}_{b}}$ [81, 82] baryons have been observed, the
${{\mathchar 28932\relax}^{\prime 0}_{b}}$ resonance is yet to be seen. If the
${{\mathchar 28932\relax}^{\prime 0}_{b}}$ mass is in the interval
$m_{{{\mathchar 28932\relax}^{0}_{b}}}+m_{{{\pi}^{0}}}<m_{{{\mathchar
28932\relax}^{\prime 0}_{b}}}<m_{{{\mathchar 28932\relax}^{\prime-}_{b}}}$,
each of the observed narrow peaks can be interpreted as having originated from
the above decay, provided that the corresponding ${{\mathchar
28938\relax}^{**-}_{b}}$ state is narrow. In this case, their masses can be
evaluated as $m_{{{\mathchar 28938\relax}^{**-}_{b}}}=m_{{{\mathchar
28932\relax}^{\prime 0}_{b}}}+\delta M_{\rm peak}$, where the values of
$\delta M_{\rm peak}$ are taken from Table 3. If the ${{\mathchar
28932\relax}^{\prime 0}_{b}}$ baryon can only decay electromagnetically to
${{\mathchar 28932\relax}^{0}_{b}}\gamma$, then the ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ peaks would be significantly broader and
inconsistent with our data.

In summary, $pp$ collision data collected with the LHCb experiment at center-
of-mass energies of $7$, $8$ and $13\text{\,Te\kern-1.00006ptV}$,
corresponding to integrated luminosities of 1, 2 and 6$\text{\,fb}^{-1}$,
respectively, have been used to search for near-threshold ${{\mathchar
28932\relax}^{0}_{b}}{{K}^{-}}$ resonances. Four new peaks are seen. Two of
the peaks, the ${{\mathchar 28938\relax}_{b}(6340)^{-}}$ and ${{\mathchar
28938\relax}_{b}(6350)^{-}}$, are observed with global (local) significance of
6.7 (7.2) and 6.2 (7.0), respectively, while the two lower-mass peaks have
global (local) significance of 2.1 (3.6) and 2.6 (3.7). The peaks are
consistent with expectations for excited ${{\mathchar 28938\relax}^{-}_{b}}$
resonances.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala80, S. Aiola25, Z. Ajaltouni9, S. Akar66, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, J.
Arnau Romeu10, A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M.
Atzeni49, B. Audurier26, S. Bachmann16, J.J. Back55, S. Baker60, V.
Balagura11,b, W. Baldini20,47, A. Baranov41, R.J. Barlow61, S. Barsuk11, W.
Barter60, M. Bartolini23,47,h, F. Baryshnikov77, G. Bassi28, V. Batozskaya35,
B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1,
A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43,
I. Belyaev38, G. Bencivenni22, E. Ben-Haim12, S. Benson31, S. Beranek13, A.
Berezhnoy39, R. Bernet49, D. Berninghoff16, H.C. Bernstein67, C. Bertella47,
E. Bertholet12, A. Bertolin27, C. Betancourt49, F. Betti19,e, M.O. Bettler54,
Ia. Bezshyiko49, S. Bhasin53, J. Bhom33, M.S. Bieker14, S. Bifani52, P.
Billoir12, A. Bizzeti21,u, M. Bjørn62, M.P. Blago47, T. Blake55, F. Blanc48,
S. Blusk67, D. Bobulska58, V. Bocci30, O. Boente Garcia45, T. Boettcher63, A.
Boldyrev78, A. Bondar42,x, N. Bondar37, S. Borghi61,47, M. Borisyak41, M.
Borsato16, J.T. Borsuk33, T.J.V. Bowcock59, C. Bozzi20, M.J. Bradley60, S.
Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A. Brossa
Gonzalo55, D. Brundu26, E. Buchanan53, A. Buonaura49, C. Burr47, A. Bursche26,
J.S. Butter31, J. Buytaert47, W. Byczynski47, S. Cadeddu26, H. Cai72, R.
Calabrese20,g, L. Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,i, M.
Calvo Gomez44,m, P. Camargo Magalhaes53, A. Camboni44,m, P. Campana22, D.H.
Campora Perez31, L. Capriotti19,e, A. Carbone19,e, G. Carboni29, R.
Cardinale23,h, A. Cardini26, P. Carniti24,i, K. Carvalho Akiba31, A. Casais
Vidal45, G. Casse59, M. Cattaneo47, G. Cavallero47, S. Celani48, R. Cenci28,p,
J. Cerasoli10, M.G. Chapman53, M. Charles12,47, Ph. Charpentier47, G.
Chatzikonstantinidis52, M. Chefdeville8, V. Chekalina41, C. Chen3, S. Chen26,
A. Chernov33, S.-G. Chitic47, V. Chobanova45, M. Chrzaszcz33, A. Chubykin37,
P. Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, F. Cindolo19,
P.E.L. Clarke57, M. Clemencic47, H.V. Cliff54, J. Closier47, J.L.
Cobbledick61, V. Coco47, J.A.B. Coelho11, J. Cogan10, E. Cogneras9, L.
Cojocariu36, P. Collins47, T. Colombo47, A. Comerma-Montells16, A. Contu26, N.
Cooke52, G. Coombs58, S. Coquereau44, G. Corti47, C.M. Costa Sobral55, B.
Couturier47, D.C. Craik63, J. Crkovska66, A. Crocombe55, M. Cruz Torres1,ab,
R. Currie57, C.L. Da Silva66, E. Dall’Occo14, J. Dalseno45,53, C.
D’Ambrosio47, A. Danilina38, P. d’Argent16, A. Davis61, O. De Aguiar
Francisco47, K. De Bruyn47, S. De Capua61, M. De Cian48, J.M. De Miranda1, L.
De Paula2, M. De Serio18,d, P. De Simone22, J.A. de Vries31, C.T. Dean66, W.
Dean80, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P. Dembinski15, M.
Demmer14, A. Dendek34, V. Denysenko49, D. Derkach78, O. Deschamps9, F.
Desse11, F. Dettori26,f, B. Dey7, A. Di Canto47, P. Di Nezza22, S. Didenko77,
H. Dijkstra47, V. Dobishuk51, F. Dordei26, M. Dorigo28,y, A.C. dos Reis1, L.
Douglas58, A. Dovbnya50, K. Dreimanis59, M.W. Dudek33, L. Dufour47, G.
Dujany12, P. Durante47, J.M. Durham66, D. Dutta61, M. Dziewiecki16, A.
Dziurda33, A. Dzyuba37, S. Easo56, U. Egede69, V. Egorychev38, S.
Eidelman42,x, S. Eisenhardt57, R. Ekelhof14, S. Ek-In48, L. Eklund58, S.
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Falabella19, J. Fan3, N. Farley52, S. Farry59, D. Fazzini11, P. Fedin38, M.
Féo47, P. Fernandez Declara47, A. Fernandez Prieto45, F. Ferrari19,e, L.
Ferreira Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49,
M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g, M. Firlej34,
K.M. Fischer62, C. Fitzpatrick47, T. Fiutowski34, F. Fleuret11,b, M.
Fontana47, F. Fontanelli23,h, R. Forty47, V. Franco Lima59, M. Franco
Sevilla65, M. Frank47, C. Frei47, D.A. Friday58, J. Fu25,q, M. Fuehring14, W.
Funk47, E. Gabriel57, A. Gallas Torreira45, D. Galli19,e, S. Gallorini27, S.
Gambetta57, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4, L.M. Garcia
Martin46, J. García Pardiñas49, B. Garcia Plana45, F.A. Garcia Rosales11, J.
Garra Tico54, L. Garrido44, D. Gascon44, C. Gaspar47, D. Gerick16, E.
Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph. Ghez8, V.
Gibson54, A. Gioventù45, O.G. Girard48, P. Gironella Gironell44, L. Giubega36,
C. Giugliano20, K. Gizdov57, V.V. Gligorov12, C. Göbel70, D. Golubkov38, A.
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Harrison59, R. Hart31, C. Hasse47, M. Hatch47, J. He5, M. Hecker60, K.
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Hu16, W. Hu7, W. Huang5, W. Hulsbergen31, T. Humair60, R.J. Hunter55, M.
Hushchyn78, D. Hutchcroft59, D. Hynds31, P. Ibis14, M. Idzik34, P. Ilten52, A.
Inglessi37, A. Inyakin43, K. Ivshin37, R. Jacobsson47, S. Jakobsen47, E.
Jans31, B.K. Jashal46, A. Jawahery65, V. Jevtic14, F. Jiang3, M. John62, D.
Johnson47, C.R. Jones54, B. Jost47, N. Jurik62, S. Kandybei50, M. Karacson47,
J.M. Kariuki53, N. Kazeev78, M. Kecke16, F. Keizer54,47, M. Kelsey67, M.
Kenzie55, T. Ketel32, B. Khanji47, A. Kharisova79, K.E. Kim67, T. Kirn13, V.S.
Kirsebom48, S. Klaver22, K. Klimaszewski35, S. Koliiev51, A. Kondybayeva77, A.
Konoplyannikov38, P. Kopciewicz34, R. Kopecna16, P. Koppenburg31, I.
Kostiuk31,51, O. Kot51, S. Kotriakhova37, L. Kravchuk40, R.D. Krawczyk47, M.
Kreps55, F. Kress60, S. Kretzschmar13, P. Krokovny42,x, W. Krupa34, W.
Krzemien35, W. Kucewicz33,l, M. Kucharczyk33, V. Kudryavtsev42,x, H.S.
Kuindersma31, G.J. Kunde66, T. Kvaratskheliya38, D. Lacarrere47, G.
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H. Li71, X. Li66, Y. Li6, Z. Li67, X. Liang67, R. Lindner47, V. Lisovskyi14,
G. Liu71, X. Liu3, D. Loh55, A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H.
Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,o, M. Lucio
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X. Lyu5, S. Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48, P.
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S. Perazzini19, D. Pereima38, P. Perret9, L. Pescatore48, K. Petridis53, A.
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Placinta36, S. Playfer57, J. Plews52, M. Plo Casasus45, F. Polci12, M. Poli
Lener22, M. Poliakova67, A. Poluektov10, N. Polukhina77,c, I. Polyakov67, E.
Polycarpo2, G.J. Pomery53, S. Ponce47, A. Popov43, D. Popov52, S.
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Puig Navarro49, H. Pullen62, G. Punzi28,p, W. Qian5, J. Qin5, R. Quagliani12,
B. Quintana9, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34, J.H.
Rademacker53, M. Rama28, M. Ramos Pernas45, M.S. Rangel2, F. Ratnikov41,78, G.
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Renaudin62, S. Ricciardi56, S. Richards53, K. Rinnert59, P. Robbe11, A.
Robert12, A.B. Rodrigues48, E. Rodrigues64, J.A. Rodriguez Lopez73, M.
Roehrken47, S. Roiser47, A. Rollings62, V. Romanovskiy43, M. Romero Lamas45,
A. Romero Vidal45, J.D. Roth80, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J.
Ruiz Vidal46, J. Ryzka34, J.J. Saborido Silva45, N. Sagidova37, B. Saitta26,f,
C. Sanchez Gras31, C. Sanchez Mayordomo46, R. Santacesaria30, C. Santamarina
Rios45, M. Santimaria22, E. Santovetti29,j, G. Sarpis61, A. Sarti30, C.
Satriano30,s, A. Satta29, M. Saur5, D. Savrina38,39, L.G. Scantlebury Smead62,
S. Schael13, M. Schellenberg14, M. Schiller58, H. Schindler47, M.
Schmelling15, T. Schmelzer14, B. Schmidt47, O. Schneider48, A. Schopper47,
H.F. Schreiner64, M. Schubiger31, S. Schulte48, M.H. Schune11, R. Schwemmer47,
B. Sciascia22, A. Sciubba30,k, S. Sellam68, A. Semennikov38, A. Sergi52,47, N.
Serra49, J. Serrano10, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M.
Shangase80, M. Shapkin43, L. Shchutska48, T. Shears59, L. Shekhtman42,x, V.
Shevchenko76,77, E. Shmanin77, J.D. Shupperd67, B.G. Siddi20, R. Silva
Coutinho49, L. Silva de Oliveira2, G. Simi27,o, S. Simone18,d, I. Skiba20, N.
Skidmore16, T. Skwarnicki67, M.W. Slater52, J.G. Smeaton54, A. Smetkina38, E.
Smith13, I.T. Smith57, M. Smith60, A. Snoch31, M. Soares19, L. Soares Lavra1,
M.D. Sokoloff64, F.J.P. Soler58, B. Souza De Paula2, B. Spaan14, E. Spadaro
Norella25,q, P. Spradlin58, F. Stagni47, M. Stahl64, S. Stahl47, P. Stefko48,
O. Steinkamp49, S. Stemmle16, O. Stenyakin43, M. Stepanova37, H. Stevens14, S.
Stone67, S. Stracka28, M.E. Stramaglia48, M. Straticiuc36, S. Strokov79, J.
Sun3, L. Sun72, Y. Sun65, P. Svihra61, K. Swientek34, A. Szabelski35, T.
Szumlak34, M. Szymanski5, S. Taneja61, Z. Tang3, T. Tekampe14, G. Tellarini20,
F. Teubert47, E. Thomas47, K.A. Thomson59, M.J. Tilley60, V. Tisserand9, S.
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Machado1, D.Y. Tou12, E. Tournefier8, M. Traill58, M.T. Tran48, C. Trippl48,
A. Trisovic54, A. Tsaregorodtsev10, G. Tuci28,47,p, A. Tully48, N. Tuning31,
A. Ukleja35, A. Usachov11, A. Ustyuzhanin41,78, U. Uwer16, A. Vagner79, V.
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Cardona44,m, A. Vitkovskiy31, V. Volkov39, A. Vollhardt49, D. Vom Bruch12, A.
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J. Wang72, J. Wang6, M. Wang3, Y. Wang7, Z. Wang49, D.R. Ward54, H.M. Wark59,
N.K. Watson52, D. Websdale60, A. Weiden49, C. Weisser63, B.D.C. Westhenry53,
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Williams54, M. Williams63, M.R.J. Williams61, T. Williams52, F.F. Wilson56, W.
Wislicki35, M. Witek33, L. Witola16, G. Wormser11, S.A. Wotton54, H. Wu67, K.
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Xu5, Z. Xu8, Z. Xu4, Z. Yang3, Z. Yang65, Y. Yao67, L.E. Yeomans59, H. Yin7,
J. Yu7,aa, X. Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c, M.
Zdybal33, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, W.C. Zhang3,z, Y.
Zhang47, A. Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V.
Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,e.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
77National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
78National Research University Higher School of Economics, Moscow, Russia,
associated to 41
79National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
80University of Michigan, Ann Arbor, United States, associated to 67

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras
†Deceased


# Observation of spontaneous x-ray magnetic circular dichroism in a chiral
antiferromagnet

Shoya Sakamoto  These two authors contributed equally to this work The
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Tomoya Higo  These two authors contributed equally to this
work Department of Physics, University of Tokyo, Hongo, Bunkyo, Tokyo
113-0033, Japan  CREST, Japan Science and Technology Agency (JST), 4-1-8
Honcho Kawaguchi, Saitama 332-0012, Japan  Masanobu Shiga  The Institute for
Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Kenta Amemiya  Institute of Materials Structure Science, KEK, Tsukuba
305-0801, Japan  Satoru Nakatsuji  The Institute for Solid State Physics, The
University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Department of Physics,
University of Tokyo, Hongo, Bunkyo, Tokyo 113-0033, Japan  CREST, Japan
Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012,
Japan  Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo,
Tokyo 113-0033, Japan  Institute for Quantum Matter and Department of Physics
and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
Shinji Miwa  The Institute for Solid State Physics, The University of Tokyo,
Kashiwa, Chiba 277-8581, Japan  CREST, Japan Science and Technology Agency
(JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan  Trans-scale Quantum
Science Institute, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan

###### Abstract

X-ray magnetic circular dichroism (XMCD) signals are usually absent in
antiferromagnets. In this letter, we report the observation of spontaneous
XMCD spectra originating from the inverse triangular spin structure, or the
polarization of the cluster magnetic octupole, in the chiral antiferromagnet
Mn3Sn thin film. The result is consistent with the recent theoretical
predictions that the inverse triangular spin structure can give rise to finite
XMCD signals in the absence of net magnetization [J. Phys. Soc. Jpn. 89,
083703 (2020) and Phys. Rev. Lett. 126, 157402 (2021)].

##  I Introduction

Antiferromagnets do not possess a net magnetic moment and are hence invisible
to common magnetic probes. X-ray magnetic circular dichroism (XMCD) Stöhr
(1995); Funk  _et al._ (2005); van der Laan and Figueroa (2014), which
typically probes spin and orbital magnetic moments, is usually inapplicable to
antiferromagnets in the absence of external magnetic fields. Recently,
Yamasaki et al. Yamasaki  _et al._ (2020) proposed that the inverse triangular
spin (ITS) structure (also referred to as the triangular spin structure with
negative chirality) of a noncollinear antiferromagnet can give rise to finite
XMCD signals through the non-vanishing magnetic dipole term $T$ even without
net magnetization, as schematically shown in Fig. 1(a). The magnetic dipole
term is defined as $T=S-3(S\cdot\hat{r})\hat{r}$—where $S$ and $\hat{r}$
denote the spin angular momentum and unit electron position operators,
respectively—and becomes locally finite when the spin density distribution is
anisotropic Stöhr and König (1995). The spin magnetic moments in the
$d_{z^{2}}$ orbitals are shown as an example in Fig. 1(a). Sasabe et al.
Sasabe  _et al._ (2021) extended this idea and showed that the spontaneous
XMCD signals can arise even without the local magnetic dipole term as a result
of spin- and orbital-direction-dependent x-ray absorption.

A chiral antiferromagnet Mn3Sn with the $D0_{19}$ structure Nakatsuji  _et
al._ (2015) is an ideal material for observing the aforementioned XMCD signals
originating from the ITS structure because the direction of the ITS structure
can be controlled by external magnetic fields, as explained below. Very
recently the observation of finite XMCD signals has been reported using single
crystalline Mn3Sn Kimata $et$ $al.$ (2020). Mn3Sn consists of the AB stacking
of two-dimensional Mn Kagome lattices, depicted by the colored planes in Fig.
1(b); the inherent geometrical frustration of the Kagome lattice stabilizes
the ITS structure, as shown in Fig. 1(c). This spin structure can also be
viewed as the ferroic order of the cluster magnetic octupoles Suzuki  _et al._
(2017), which is represented by hexagons with blue and red colors in Fig.
1(c). Each Mn spin has a local easy axis toward its in-plane nearest-neighbor
Sn sites, and only one of the three Mn spins in each Mn triangle is parallel
to the local easy axis Tomiyoshi and Yamaguchi (1982). As a result, the other
two Mn spins show spin canting toward their local easy axis and induce weak
ferromagnetism with a net magnetic moment ($\sim 0.002$ $\mu_{\rm B}$/Mn atom)
parallel to the octupole polarization Tomiyoshi and Yamaguchi (1982);
Nakatsuji  _et al._ (2015). This weak ferromagnetic moment allows one to
control the direction of the octupole polarization by external magnetic
fields.

It is worth mentioning that $D0_{19}$ Mn3Sn has attracted considerable
attention because the octupole polarization bestows Mn3Sn with pairs of Weyl
nodes in momentum space Kuroda  _et al._ (2017); Yang  _et al._ (2017); Chen
_et al._ (2021) and induces strong ferromagnetic-like responses such as the
anomalous Hall effect Nakatsuji  _et al._ (2015); Zhang  _et al._ (2017),
anomalous Nernst effect Ikhlas  _et al._ (2017); Li  _et al._ (2017), and
magneto-optical Kerr effect Higo  _et al._ (2018a); Miwa  _et al._ (2021).
Octupole polarization switching or manipulation by electric means has also
recently been demonstrated Tsai  _et al._ (2020, 2021); Takeuchi  _et al._
(2021).

In this article, we report the observation of XMCD signals originating from
the ITS structure, or the polarization of the cluster magnetic octupole, in a
Mn3Sn thin film with the Kagome planes standing normal to the sample surface.
The standing Kagome planes allow us to probe the octupole polarization
directly.

Figure 1: (a) Inverse triangular spin ($S$) structure with net magnetic dipole
term ($T$). (b) Crystal structure of $D0_{19}$ Mn3Sn. (c) Top view of Mn3Sn
crystal structure. Mn atoms form two-dimensional Kagome lattices with an
inverse triangular spin structure. This spin structure can be viewed as a
ferroic order of cluster magnetic octupoles indicated by arrows and hexagons
with blue and red gradation. (d) Thin film structure with dominant Mn3Sn
orientation is indicated. (e) Out-of-plane x-ray diffraction profiles of the
Mn3Sn thin film and an MgO(110) substrate. (f) Out-of-plane magnetic
hysteresis loop measured using the magneto optical Kerr effect (MOKE).

##  II Methods

The Mn3Sn thin film with the Kagome planes standing normal to the sample
surface You  _et al._ (2019); Yoon  _et al._ (2019); Takeuchi  _et al._ (2021)
was grown on a single-crystalline MgO(110) substrate with a W seed layer. The
sample structure, MgO(110) substrate/W (15 nm)/Mn3Sn (40 nm)/MgO (3 nm), is
schematically shown in Fig. 1(d). Mn and Sn atoms were co-deposited using
effusion cells. The sample was annealed at 600 ∘C for 30 min after MgO
deposition. The actual atomic composition was estimated to be Mn3.07Sn0.93.
The detailed sample characterization will be reported elsewhere.

X-ray absorption spectroscopy (XAS) and XMCD measurements were performed on
the BL-16A2 beamline at the Photon Factory Amemiya  _et al._ (2013). The
measurement temperature was room temperature, and magnetic fields ($H$) of up
to 5 T were applied. The measurements with 0.1 T and 1 T were performed after
applying 2 T along the $z$ direction to saturate the octupole polarization.
The XAS and XMCD spectra were recorded in total electron yield mode. XMCD
signals were measured by reversing the helicity of x rays with 10-Hz frequency
at each photon energy under a fixed magnetic field Amemiya  _et al._ (2013).
The measurements were repeated in the opposite magnetic field direction to
eliminate artifacts. The XAS is defined as
$(\sigma_{R,+H}+\sigma_{L,+H})+(\sigma_{R,-H}+\sigma_{L,-H})$, and XMCD is
defined as $(\sigma_{R,+H}-\sigma_{L,+H})-(\sigma_{R,-H}-\sigma_{L,-H})$,
where $\sigma_{R/L,\pm H}$ denotes the absorption coefficient measured with
right- or left-circularly polarized x-rays under positive or negative magnetic
fields.

##  III Results

Figure 1(e) shows the out-of-plane $\theta/2\theta$ x-ray diffraction (XRD)
patterns of the Mn3Sn thin film and MgO(110) substrate. The XRD pattern shows
a relatively strong Mn3Sn (2$\overline{2}$00) peak and weak (0002) and
(2$\overline{2}$01) peaks, indicating that most Kagome planes stand normal to
the sample surface. That is, the [$1\overline{1}00$] direction is parallel to
the MgO [110] direction, as shown in Fig. 1(d). The XRD $\phi$ scan (not
shown) revealed that the Mn3Sn [0001] direction was parallel to the MgO [001]
direction, and the Mn3Sn [$11\overline{2}0$] direction was parallel to the MgO
[1$\overline{1}$0] direction, which is consistent with findings of previous
studies Takeuchi  _et al._ (2021); Yoon  _et al._ (2021). Note that the XRD
pattern did not show peaks of ferromagnetic WMn2Sn impurities, which were
reported to appear in W/M$3$Sn heterostructures Takeuchi  _et al._ (2021);
Yoon  _et al._ (2021).

The magnetic properties were characterized by magneto-optical Kerr effects
(MOKE), and the measured magnetic hysteresis loop at room temperature is shown
in Fig. 1(f). The magnetic fields were applied along the surface normal
direction. The MOKE showed hysteresis with a coercive field of 0.35 T, which
is larger than the bulk value of $\sim$0.05 T Nakatsuji  _et al._ (2015) but
smaller than the value for randomly-oriented polycrystalline thin films, 0.6 T
Higo  _et al._ (2018b). The sizable Kerr rotation most likely originates from
the octupole polarization in Mn3Sn Higo  _et al._ (2018a); Miwa  _et al._
(2021) because the sign of the Kerr rotation angle was opposite to that of
typical ferromagnets such as Fe and because the ferromagnetic moment induced
by spin canting ($<10$ m$\mu_{\rm B}$/Mn atom, as described below) is almost
negligible.

Figure 2: (a) Typical XAS spectrum of the Mn3Sn thin film. The inset shows the
measurement geometry. (b) XMCD spectra taken with $H$ = 0.1 T and $H$ = 5 T
under the normal-incidence (NI) geometry with $\theta=0^{\circ}$. The spin
canting component (blue curve) is extracted by subtracting the 0.1 T spectrum
from the 5 T spectrum. The second derivative spectra are shown in the inset.
(c) XMCD spectra taken under the NI geometry and the grazing-incidence
geometry ($\theta=70^{\circ}$). The inset shows the magnified view around the
Mn $L_{3}$ edge.

Figure 2(a) shows a typical XAS spectrum of the Mn3Sn thin film. The
background, consisting of a double step function and a linear function, is
subtracted. The XAS spectral line shape did not depend on the magnetic field
strength or direction. The inset of Fig. 2(a) shows the measurement geometry,
where the incident x ray is in the $yz$ plane, and the angle between the
incident x ray and the sample normal direction is denoted by $\theta$.

The XMCD spectra obtained with $H$ = 0.1 T and 5 T in the normal incidence
(NI) geometry ($\theta=0^{\circ}$) are shown in Fig. 2(b). The spectra before
performing magnetic-field average are shown in the Supplementary material Sup
. The spectral line shapes significantly differ from each other: the spectrum
obtained with $H$ = 0.1 T shows a strong positive pre-edge peak, denoted by
$\alpha$ in Fig. 2(b), while the spectrum obtained with $H$ = 5 T shows an
intense negative peak at the XAS peak position, denoted by $\beta$ in Fig.
2(b). The XMCD spectra are expected to consist of the spin-canting component,
which develops with increasing magnetic fields, and the field-independent
octupole component. The peak $\beta$ most likely originates from the spin-
canting component, as the peak develops with increasing magnetic fields. If
the spectrum taken with $H$ = 0.1 T is subtracted from the spectrum taken with
$H$ = 5 T, a broad featureless XMCD spectrum without peak $\alpha$ is
obtained, as shown by the blue curve in Fig. 2(b). This broad spectrum
resembles the XMCD spectra of metallic systems, suggesting that it reflects
the spin component. Hereafter, we refer to this difference spectrum as the
spin-canting XMCD spectrum. It follows that the spectrum taken with $H$ = 0.1
T, which shows multiple sharp peaks, predominantly reflects the octupole
component. Note that a previous study on a Mn3Sn ($40\overline{4}3$) thin film
observed field-induced XMCD signals, the magnitude of which is comparable to
that obtained in the present study, but the pre-edge peak $\alpha$, a
signature of the octupole polarization, was not observed Taylor  _et al._
(2020).

To examine the spectral line shapes in more detail, the second derivatives of
the XMCD spectra are shown in the inset of Fig. 2(b). The second derivative of
the spin-canting XMCD spectrum does not exhibit a peak at position $\alpha$,
indicating that the XMCD signals originating from the octupole polarization do
not depend on the magnetic field strength. In contrast, the XMCD spectrum
obtained with $H$ = 0.1 T is expected to contain a finite spin-canting
component in addition to the octupole component. To estimate how much the
spin-canting component contributes to the spectrum taken with $H$ = 0.1 T, we
subtracted a fraction of the spin-canting XMCD spectrum from the $H$ = 0.1 T
spectrum and minimized the kink feature $\gamma$ around 642 eV and the main
feature $\beta$. From this analysis, it was determined that approximately
$13\pm 13$ % of the spin canting component contributes to the XMCD spectrum
taken with $H$ = 0.1 T. The XMCD spectrum that only reflects the octupole
polarization can thus be obtained as ${\rm XMCD(0.1\ T)}-0.13\times{(\rm
spin}$-canting XMCD), which we refer to as the octupole XMCD.

The XMCD spectra captured in the NI geometry ($\theta=0^{\circ}$) and the
grazing-incidence (GI) geometry ($\theta=70^{\circ}$) with a magnetic field of
0.1 T are shown in Fig. 2(c). Here, the probing depth in the GI geometry is
almost the same as that in the NI geometry because a typical x-ray penetration
depth of $\sim$ 100 nm is much longer than a typical electron escaping depth
of $\sim$ 5 nm. The XMCD signals captured in the GI geometry are smaller than
those captured in the NI geometry, and they seem to scale with $\cos\theta$,
as can be seen in the inset of Fig. 2(c). This scaling behavior suggests that
the XMCD signals at 0.1 T originate from the octupole polarization, which
always lies in the Kagome plane, projected onto the x-ray incidence axis. The
small difference in the spectral line shape may have originated from the spin-
canting component. Note that the XMCD spectrum (not shown) taken with the
magnetic field along the [$\overline{1}2\overline{1}0$] direction (30-degree
away from the sample normal in the $xz$ plane) looked almost the same as that
taken in the NI geometry as expected Yamasaki  _et al._ (2020); Sasabe  _et
al._ (2021).

Here, we estimate the spin and orbital magnetic moments using the XMCD sum
rules Thole  _et al._ (1992); Carra  _et al._ (1993); Chen  _et al._ (1995):

$\displaystyle\displaystyle m_{\rm orb}=-\frac{4}{3}\frac{S^{\rm
XMCD}_{L_{3}}+S^{\rm XMCD}_{L_{2}}}{S^{\rm XAS}}n_{h},$ (1)
$\displaystyle\displaystyle m_{\rm spin}^{\rm eff}=m_{\rm spin}+7m_{\rm
T}=-2\frac{S^{\rm XMCD}_{L_{3}}-2S^{\rm XMCD}_{L_{2}}}{S^{\rm XAS}}n_{h},$ (2)

where $m_{\rm orb}$ and $m_{\rm spin}$ are the orbital and spin magnetic
moments in units of $\mu_{\rm B}$/atom, respectively, and $\mu_{\rm B}$ is the
Bohr magneton; $m_{\rm T}$ is the expectation value of the magnetic dipole
term $m_{\rm T}=-\langle T\rangle\mu_{\rm B}/\hbar$, with $\hbar$ being the
reduced Planck constant Stöhr (1999); $S^{\rm XMCD}_{L_{2,3}}$ represents the
XMCD integral over the $L_{2}$ or $L_{3}$ edges, and $S^{\rm XAS}$ represents
the XAS integral over the $L_{2,3}$ edges; $n_{h}$ is the number of 3$d$
holes, which was assumed to be five in the present study.

Figure 3: Magnetic field dependence of (a) the effective spin magnetic moment
and (b) the orbital magnetic moment deduced using the XMCD sum rules in the NI
geometry (black circles). The black error bars represent uncertainty in the
spectral background subtraction procedure. The data are decomposed into the
spin canting component (blue squares) and octupole component (red diamonds).
The shaded areas represent the uncertainty in the spectral decomposition
procedure.

The deduced effective spin magnetic moments $m_{\rm spin}^{\rm eff}$ and
orbital magnetic moments $m_{\rm orb}$ are plotted with black circles in Figs.
3(a) and 3(b), respectively. The effective spin magnetic moment increases as
the magnetic field increases, and changes the sign at around 1 T.

The spectral decomposition procedure mentioned above can separate the deduced
magnetic moments into the spin canting component and the octupole component,
as shown by the blue squares and red diamonds, respectively, in Figs. 3(a) and
3(b). Here, we assume that the octupole component does not depend on the
magnetic field, which is evidenced by the fact that the spin-canting XMCD
spectrum does not show any octupole XMCD features. The effective spin magnetic
moment, or the magnetic dipole term, of the octupole component is found to be
antiparallel to that of the spin-canting component. These negative field-
independent effective spin magnetic moments indicate that the octupole XMCD
indeed originates from the inverse triangular spin structure of Mn3Sn because
if the octupole XMCD originated from ferromagnetic or superparamagnetic
impurities, the effective spin magnetic moments would have been positive and
magnetic field dependent. The orbital magnetic moment is almost absent for the
octupole component but is finite for the spin-canting component. The absence
of the orbital magnetic moment is consistent with the fact that the magnetic
octupole polarization does not possess a net magnetic moment.

The decomposition analysis showed that the spin canting component was no more
than 0.04 $\mu_{\rm B}$/Mn—even at 5 T—and $0.004\pm 0.005$ $\mu_{\rm B}$/Mn
in the remanence state. This almost negligible remanent spin magnetic moment
is consistent with the bulk value of 0.002 $\mu_{\rm B}$/Mn Nakatsuji  _et
al._ (2015). Despite this consistency, there still is a possibility that the
tiny field-dependent spin magnetic moments were induced by minor paramagnetic
or ferromagnetic second phases or the spin canting of Mn3Sn grains with
different orientations. Note that the magnetic moments are expected to be
slightly underestimated because of the existence of a small number of Mn3Sn
grains with different orientations and the spectral overlap between the Mn
$L_{2,3}$ edges Teramura  _et al._ (1996).

Figure 4: Comparison between the octupole XMCD spectrum (red) and the
theoretical spectrum (orange) Sasabe  _et al._ (2021). The integral XMCD
spectra are also shown at the bottom.

##  IV Discussion

Figure 4 compares the octupole XMCD spectrum obtained in this study with a
previously reported theoretical spectrum Sasabe  _et al._ (2021). The overall
line shapes roughly agree with each other, especially for the Mn $L_{3}$ edge.
However, the XMCD signals were more than an order of magnitude weaker than
those of the calculations. This discrepancy may be due to the fact that the
theoretical calculations assumed an isolated ionic Mn2+ high-spin state with
$\langle S\rangle/\hbar=5/2$, $\langle L\rangle/\hbar\sim 0$, $\langle
T\rangle/\hbar\sim 0$, which is quite different from the local electronic
structure of the Mn atoms in Mn3Sn with a metallic nature and a local magnetic
moment of $\sim 3$ $\mu_{\rm B}$ Brown  _et al._ (1990).

In theoretical calculations, it was argued that the sizable XMCD signals arise
even without the net spin magnetic moment or net magnetic dipole term. This
can be seen in the XMCD integral becoming zero above both the Mn $L_{3}$ and
$L_{2}$ edges ($S^{\rm XMCD}_{L_{3}}=S^{\rm XMCD}_{L_{2}}=0$). The finite
oscillating XMCD spectrum can be considered to originate from the magnetic
dipole term of each Mn 3$d$ orbital, which is split in energy by a crystal
field and has a different x-ray absorption process. For example, when the
occupied $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals with the opposite magnetic
dipole terms are non-degenerate, they would exhibit XMCD spectra that are
different in line shape and hence a finite superimposed XMCD spectrum
regardless of whether or not net magnetic dipole term exists.

Experimentally, the XMCD integral over the Mn $L_{3}$ edge remains finite and
approaches zero above the Mn $L_{2}$ edge ($S^{\rm XMCD}_{L_{3}}\neq 0$ and
$S^{\rm XMCD}_{L_{3}}+S^{\rm XMCD}_{L_{2}}=0$). As shown in Figs. 3(a) and
3(b), this integral results in a finite effective spin magnetic moment but a
negligible orbital magnetic moment. We attribute the origin of the finite
effective spin magnetic moment to the non-vanishing magnetic dipole term $T$
in the cluster magnetic octupole. Note that the finite net magnetic dipole
term does not seem necessary to observe XMCD signals as described above.

The present findings demonstrate that non-collinear antiferromagnetic spin
structures can be probed by XMCD. This enables the resolution of
antiferromagnetic domains and antiferromagnetic domain walls using the
scanning XMCD technique Fischer and Ohldag (2015); Kotani  _et al._ (2018).
The local anisotropic spin distribution may also be elucidated by analyzing
the spectral line shape in greater detail Shibata  _et al._ (2018); Sakamoto
_et al._ (2021).

##  V Summary

In this letter, we report the observation of XMCD signals arising from the
non-collinear antiferromagnetic spin structure of a Mn3Sn thin film with the
Kagome planes standing normal to the sample surface. We reveal that the finite
net magnetic dipole term in the inverse triangular spin structure, or the
cluster magnetic octupole, induces the XMCD signals.

## Supplementary Information

###  V.1 XMCD spectra recorded with positive and negative magnetic fields

Figure 5 shows the x-ray magnetic circular dichroism (XMCD) spectra captured
with positive and negative magnetic fields. The XMCD spectra reversed their
signs with the magnetic-field directions, which rules out the possibility that
the XMCD signals originated from non-magnetic artifacts. For example, when
photon energies of right- and left-circularly polarized x-rays are slightly
different, the photon-energy derivative of absorption spectra can be obtained.
Such artifact can be removed by averaging XMCD spectra taken with positive and
negative magnetic fields, as shown in the main text.

Figure 5: XMCD spectra of the Mn3Sn thin film recorded with $\pm 0.1$ T and
$\pm 5$ T.

###### Acknowledgements.

We thank K. Kondou of RIKEN for discussion. This work was performed under the
approval of the Photon Factory Program Advisory Committee (Proposal No.
2019S2-003). This work was supported by JSPS KAKENHI (Nos. JP18H03880,
JP19H00650, JP20K15158), JST CREST (JPMJCR18T3), JST-Mirai Program
(JPMJMI20A1), and Spintronics Research Network of Japan (Spin-RNJ). Institute
for Quantum Matter, an Energy Frontier Research Center was funded by DOE,
Office of Science, Basic Energy Sciences under Award No. DE-SC0019331.

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