# Observation of a new $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ state at ATLAS
and a new $\Xi_{b}$ baryon at CMS

Andrew S. Chisholm, on behalf of the ATLAS and CMS Collaborations  School of
Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom

###### Abstract

Two recent observations of new $b$ hadrons in $pp$ collisions at
$\sqrt{s}=7{\mathrm{\ Te\kern-1.00006ptV}}$ at the Large Hadron Collider (LHC)
are presented. The ATLAS collaboration has observed a new state in radiative
transitions to ${\mit\Upsilon}(1S)$ and ${\mit\Upsilon}(2S)$ and interprets
this as the first observation of the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$
states. The CMS collaboration has observed a new $b$ baryon decaying to
$\Xi_{b}^{-}\pi^{+}$ (plus charge conjugates). This is interpreted as a
neutral $J^{P}=3/2^{+}$ $\Xi_{b}^{*}$ baryon.

##  I INTRODUCTION

The high cross section for heavy flavour production at the Large Hadron
Collider (LHC) offers a fantastic opportunity to study the known heavy flavour
hadrons and to search for the many “missing” states predicted in the quark
model. In particular, hadrons containing a $b$ quark are expected to be
copiously produced at the LHC. Further to this, many heavy hadron states
(bottom baryons in particular) are only accessible at hadron colliders, and
with the end of data taking at the Tevatron, the LHC is fast becoming the one
of the best facilities to study heavy hadron spectroscopy. The LHC has already
delivered its first two discoveries of new $b$ hadrons within the first three
years of running. The ATLAS collaboration has observed a new state in
radiative transitions to ${\mit\Upsilon}(1S)$ and ${\mit\Upsilon}(2S)$ and
interprets this as the first observation of the $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)$ states ATLAS_chib . The CMS collaboration
observes a new $b$ baryon decaying to $\Xi_{b}^{-}\pi^{+}$ (plus charge
conjugates), interpreted as a neutral $\Xi_{b}^{*}$ baryon CMS_Xi .

##  II OBSERVATION OF A NEW $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ STATE AT
ATLAS

The $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ states represent the $S=1$ (parallel
quark spins) $P$-wave states of the bottomonium ($b\bar{b}$) system. The
$\raise 1.72218pt\hbox{$\chi$}_{{b}}$ comprise a triplet of states, $\raise
1.72218pt\hbox{$\chi$}_{{b0}}$, $\raise 1.72218pt\hbox{$\chi$}_{{b1}}$,
$\raise 1.72218pt\hbox{$\chi$}_{{b2}}$, with quantum numbers
$J^{PC}=0^{++},1^{++},2^{++}$. The three states are characterised by a small
hyperfine mass splitting of ${\cal{O}}(10{\mathrm{\ Me\kern-1.00006ptV}})$.
The branching fractions for the radiative decays $\raise
1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}\,\gamma$ are large,
${\cal{O}}(10\%)$. The $\raise 1.72218pt\hbox{$\chi$}_{{b}}(1P)$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}(2P)$ triplets (with spin-weighted masses of
around $9.90$ and $10.26$ GeV respectively) have been studied in detail at
$e^{+}e^{-}$ colliders, providing precise measurements of the hyperfine mass
structure and radiative branching fractions chib_ee1 ; chib_ee2 ; chib_ee3 .
Additionally, a third triplet of states, the $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)$, is expected below the open beauty threshold
at a mass around $10.525{\mathrm{\ Ge\kern-1.00006ptV}}$ predictions1 ;
predictions2 ; predictions3 .

Given the complex hadronic environment of the LHC, the radiative decays
$\raise 1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}\,\gamma$ with
${\mit\Upsilon}\to\mathrm{\mu^{+}\mu^{-}}$ represent a very clean experimental
channel to study the $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ states, where the
presence of two muons offers a clear signature to trigger upon. The aim of the
ATLAS analysis is to reconstruct the radiative decays $\raise
1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}(1S)\gamma$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}(2S)\gamma$ with two independent
analyses that reconstruct the photon with either a direct calorimetric
measurement or through the reconstruction of $e^{+}e^{-}$ conversions in the
ATLAS tracker. The two photon reconstruction methods have their own advantages
and disadvantages. In particular, converted photons offer better invariant
mass resolution compared to photons reconstructed by the calorimeter but at
the expense of a much lower reconstruction efficiency. The ATLAS collaboration
has recently reported the observation of a new structure decaying to
${\mit\Upsilon}(1S)\gamma$ and ${\mit\Upsilon}(2S)\gamma$ consistent with the
$\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ system ATLAS_chib . The following
sections summarise the ATLAS analysis and results, a detailed description of
the ATLAS detector can be found in ATLAS_Paper .

###  II.1 Data Sample and Selection of
${\mit\Upsilon}\to\mathrm{\mu^{+}\mu^{-}}$ decays

The ATLAS analysis uses a data sample, recorded by the ATLAS experiment during
the 2011 LHC proton-proton collision run at a centre of mass energy
$\sqrt{s}=7{\mathrm{\ Te\kern-1.00006ptV}}$, representing an integrated
luminosity of $4.4\,\mbox{fb${}^{-1}$}$. The data sample was collected by a
set of triggers designed to select events containing di-muon candidates or
single high transverse momentum muons.

Muon candidates are reconstructed from stand alone tracks reconstructed in the
ATLAS muon spectrometer combined with tracks reconstructed in the ATLAS Inner
Detector (ID). Two oppositely charged muon candidates, both with transverse
momentum $p_{T}>4{\mathrm{\ Ge\kern-1.00006ptV}}$ and pseudorapidity
$|\eta|<2.3$, are fitted to a common vertex to form di-muon candidates. The
di-muon vertex fit quality is required to satisfy $\chi^{2}/d.o.f<20$.
Finally, di-muon candidates are required to have transverse momentum
$p_{T}>12{\mathrm{\ Ge\kern-1.00006ptV}}$ and rapidity $|y|<2.0$. The
invariant mass distribution of di-muon candidates is shown in Figure 1.
${\mit\Upsilon}(1S)$ and ${\mit\Upsilon}(2S)$ candidates are selected from di-
muon candidates with an invariant mass within the range
$9.25<m({\mathrm{\mu^{+}\mu^{-}}})<9.65{\mathrm{\ Ge\kern-1.00006ptV}}$ and
$9.80<m({\mathrm{\mu^{+}\mu^{-}}})<10.10{\mathrm{\ Ge\kern-1.00006ptV}}$
respectively.

Figure 1: The invariant mass distribution of di-muon candidates selected as
described in Section II.1. The shaded areas A and B represent the invariant
mass selections for ${\mit\Upsilon}(1S)$ and ${\mit\Upsilon}(2S)$ candidates
respectively ATLAS_chib .

###  II.2 Converted Photon Selection

Candidate photon conversions are reconstructed from two oppositely charged
tracks reconstructed in the ATLAS inner detector (ID) that intersect at a
common vertex. The two tracks are fitted to a common conversion vertex where
the fit is required to converge with a $\chi^{2}$ probability greater than
0.01. Each electron track is required to have transverse momentum
$p_{T}>500{\mathrm{\ Me\kern-1.00006ptV}}$ and pseudorapidity $|\eta|<2.3$ and
to be reconstructed from at least 4 hits in the silicon layers of the ID.
Conversion candidates reconstructed from tracks associated to the the di-muon
candidate are rejected. To reduce background contamination from Dalitz decays
and fake conversions, conversion vertices are required to be reconstructed
with a radial distance from the beam axis of greater than $40\,\mathrm{mm}$.
Converted photons not associated with the di-muon vertex are rejected by
demanding that the impact parameter of the converted photon candidate with
respect to the di-muon vertex be less than $2\,\mathrm{mm}$.

###  II.3 Unconverted Photon Selection

Unconverted photons are reconstructed from energy deposits in the ATLAS
electromagnetic calorimeter that are not matched to any ID track. Unconverted
photon candidates are required to have a transverse energy greater than
$2.5{\mathrm{\ Ge\kern-1.00006ptV}}$ and pseudorapidity $|\eta|<2.37$. Photon
candidates are also required to satisfy the “loose” photon identification
selection described in ATLAS_photon  to reject backgrounds from
$\pi^{0}\to\gamma\gamma$ decays and narrow jets. Unconverted photons
reconstructed within the transition region ($1.37<|\eta|<1.52$) between the
barrel and end cap calorimeters are not selected. To improve the momentum
resolution of unconverted photons, the polar angle of the unconverted photon
is corrected to point back to the di-muon vertex, exploiting the longitudinal
segmentation of the ATLAS electromagnetic calorimeter. This procedure also
allows photons that are not compatible with having originated from the di-muon
vertex to be rejected through a loose cut on the fit quality of
$\chi^{2}/d.o.f<200$. This procedure is described in detail in ATLAS_pointing
.

###  II.4 Selection of $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ Candidates

Reconstructed ${\mit\Upsilon}\to\mathrm{\mu^{+}\mu^{-}}$ candidates are
associated with reconstructed photons to form $\raise
1.72218pt\hbox{$\chi$}_{{b}}$ candidates. To minimise the effects of the
experimental di-muon mass resolution, the invariant mass difference $\Delta
m=m(\mathrm{\mu^{+}\mu^{-}}\gamma)-m(\mathrm{\mu^{+}\mu^{-}})$ is calculated.
The $\Delta m$ distributions for ${\mit\Upsilon}(1S)\gamma$ and
${\mit\Upsilon}(2S)\gamma$ candidates can be shown on the same mass scale
through the definition of the $\Delta m+m_{{\mit\Upsilon}(kS)}$ distribution
(the $m_{{\mit\Upsilon}(k=1,2S)}$ represent the current world average masses
of the ${\mit\Upsilon}(1S)$ and ${\mit\Upsilon}(2S)$ states) PDG . The $\Delta
m+m_{{\mit\Upsilon}(1,2S)}$ distributions for $\raise
1.72218pt\hbox{$\chi$}_{{b}}$ candidates reconstructed from unconverted and
converted photons are shown in Figures 2 and 3 respectively. A final selection
requirement on the transverse momentum of the di-muon system of
$p_{T}>20{\mathrm{\ Ge\kern-1.00006ptV}}$ is imposed for unconverted photon
candidates to maximise the signal significance of the $\raise
1.72218pt\hbox{$\chi$}_{{b}}(1P)$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}(2P)$ peaks.

Figure 2: The $\Delta m+m_{{\mit\Upsilon}(1S)}$ distribution for $\raise
1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}(1S)\gamma$ candidates
reconstructed from unconverted photons ATLAS_chib . Figure 3: The $\Delta
m+m_{{\mit\Upsilon}(1S)}$ (filled points) and $\Delta
m+m_{{\mit\Upsilon}(2S)}$ (open triangles) distributions for $\raise
1.72218pt\hbox{$\chi$}_{{b}}\to{\mit\Upsilon}(1S,2S)\gamma$ candidates
reconstructed from converted photons. The data points have not been corrected
for energy losses due to bremsstrahlung ATLAS_chib .

The mass distributions for ${\mit\Upsilon}(1S)\gamma$ candidates reconstructed
from both unconverted and converted photons shown in Figures 2 and 3 exhibit
clear peaks at approximately $9.9{\mathrm{\ Ge\kern-1.00006ptV}}$ and
$10.2{\mathrm{\ Ge\kern-1.00006ptV}}$ consistent with $\raise
1.72218pt\hbox{$\chi$}_{{b}}(1P)\to{\mit\Upsilon}(1S)\gamma$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}(2P)\to{\mit\Upsilon}(1S)\gamma$ decays. In
addition to these peaks, a third structure is also observed at a mass of
approximately $10.5{\mathrm{\ Ge\kern-1.00006ptV}}$. This additional structure
is also observed in the mass distribution for ${\mit\Upsilon}(2S)\gamma$
candidates. The mass and decay modes of these additional structures is
consistent with the expectations for the $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)$ states decaying in the modes $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)\to{\mit\Upsilon}(1S)\gamma$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)\to{\mit\Upsilon}(2S)\gamma$. The higher
transverse momentum threshold for the reconstruction of unconverted photons
prohibits the reconstruction of soft photons from $\raise
1.72218pt\hbox{$\chi$}_{{b}}(2P,3P)\to{\mit\Upsilon}(2S)\gamma$ decays.

###  II.5 Fit Description and Results

Unbinned maximum likelihood fits are performed to the $\Delta
m+m_{{\mit\Upsilon}(1,2S)}$ distributions for both the converted and
unconverted $\raise 1.72218pt\hbox{$\chi$}_{{b}}$ candidates to measure the
mass of the new structure under its interpretation as the $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)$ states, both fits are described in detail in
ATLAS_chib .

The $\Delta m+m_{{\mit\Upsilon}(1S)}$ distribution for unconverted photon
candidates is described by three Gaussian probability density functions
(PDFs), each with independent mean value, width and normalisation parameters.
The background distribution is described by the smooth function,
$\exp{\left(A\Delta m+B{\Delta m}^{-2}\right)}$ with two free parameters, $A$
and $B$. The mass barycenter of the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$
signal is measured to be $\overline{m}_{3}=10.541\pm
0.011\mbox{$\;$(stat.)}\pm 0.030\mbox{$\;$(syst.)}{\mathrm{\
Ge\kern-1.00006ptV}}$ from the fit to unconverted photon candidates alone
ATLAS_chib . The systematic uncertainty on the unconverted photon mass
measurement is dominated by the modelling of the background distribution and
the uncertainty associated with the unconverted photon energy scale.

Both the $\Delta m+m_{{\mit\Upsilon}(1S)}$ and $\Delta
m+m_{{\mit\Upsilon}(2S)}$ distributions for converted photon candidates are
fitted together in a simultaneous fit. Each $\raise
1.72218pt\hbox{$\chi$}_{{b}}(nP)$ peak is described by a pair of Crystal Ball
(CB) functions. This is motivated by the fact that the converted photon mass
resolution is comparable to the hyperfine splitting between the $J=1,2$
states. The background distribution for both the $\Delta
m+m_{{\mit\Upsilon}(1S)}$ and $\Delta m+m_{{\mit\Upsilon}(2S)}$ distributions
is described by the function $(\Delta
m-q_{0})^{\alpha}\cdot\exp\left\\{(\Delta m-q_{0})\cdot\beta\right\\}$ where
$q_{0}$, $\alpha$ and $\beta$ are all free parameters (an independent set of
parameters for both the $\Delta m+m_{{\mit\Upsilon}(1,2S)}$ distributions).
The mass barycenter of the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ signal,
determined from converted photon candidates alone, is measured to be
$\overline{m}_{3}=10.530\pm 0.005\mbox{$\;$(stat.)}\pm
0.009\mbox{$\;$(syst.)}{\mathrm{\ Ge\kern-1.00006ptV}}$ ATLAS_chib . The
systematic uncertainty associated with the mass measurement is dominated by
the various assumptions made in the simultaneous fit.

The mass measurements using both converted and unconverted photons are
compatible with each other and with the theoretical expectations for the
$\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ system. The mass measurement from
the converted photon analysis with the lower statistical and systematic
uncertainty is chosen to represent the final measurement of the mass
barycenter of the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ system.

The significance of the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ signal is
assessed from the logarithmic likelihood ratio
$\log{\left(L_{max}/L_{0}\right)}$, where $L_{max}$ and $L_{0}$ represent
likelihood values calculated from fits with and without a $\raise
1.72218pt\hbox{$\chi$}_{{b}}(3P)$ signal included respectively. The
significance is re-assessed for each set of systematic variations and is
consistently found to be in excess of $6$ standard deviations for both the
unconverted and converted photon analyses independently.

The observation of a new structure in the ${\mit\Upsilon}(1S)\gamma$ spectrum
was recently confirmed by the $D\O$ collaboration D0 . $D\O$ measure the mass
of the structure to be $\overline{m}_{3}=10.551\pm 0.014\mbox{$\;$(stat.)}\pm
0.017\mbox{$\;$(syst.)}{\mathrm{\ Ge\kern-1.00006ptV}}$, consistent with the
ATLAS measurement.

###  II.6 Conclusion

The ATLAS collaboration has observed the known $\raise
1.72218pt\hbox{$\chi$}_{{b}}(1P)$ and $\raise
1.72218pt\hbox{$\chi$}_{{b}}(2P)$ states in radiative transitions to
${\mit\Upsilon}(1S)\gamma$ in proton-proton collisions at the LHC. In addition
to this, ATLAS observe a new structure in radiative decays to
${\mit\Upsilon}(1S)\gamma$ and ${\mit\Upsilon}(2S)\gamma$ consistent with
theoretical expectations for the $\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$
system. The mass barycenter of this structure, under the interpretation as the
$\raise 1.72218pt\hbox{$\chi$}_{{b}}(3P)$ system, is measured to be
$\overline{m}_{3}=10.530\pm 0.005\mbox{$\;$(stat.)}\pm
0.009\mbox{$\;$(syst.)}{\mathrm{\ Ge\kern-1.00006ptV}}$ ATLAS_chib . Further
measurements by ATLAS and the other LHC experiments will hopefully shed more
light on this new state in the near future.

##  III OBSERVATION OF A NEW $\Xi_{b}$ BARYON AT CMS

Until recently, the $\Xi_{b}$ states represented the only experimentally
observed baryons to contain one strange and one bottom valence quark within
the quark model of baryons. The first direct observation of the $\Xi_{b}$, of
which there exists both neutral $usb$ and negatively charged $dsb$ varieties,
came from the Tevatron experiments Tev1 ; Tev2 ; Tev3 , although indirect
evidence for the $\Xi_{b}^{-}$ was seen at LEP LEP1 ; LEP2 . In addition to
the the $J^{P}=1/2^{+}$ $\Xi_{b}$ ground states, the quark model predicts the
$J^{P}=1/2^{+}$ $\Xi_{b}^{\prime}$, $J^{P}=3/2^{+}$ $\Xi_{b}^{*}$ and two
further states with $L=1$ orbital angular momentum between the $b$ quark and
the light di-quark system XiTheory1 ; XiTheory2 ; XiTheory3 ; XiTheory4 ;
XiTheory5 . The strong decay $\Xi_{b}^{\prime}\to\Xi_{b}\pi$ is expected to be
kinematically forbidden due to an expected $\Xi_{b}^{\prime}-\Xi_{b}$ mass
difference below the pion mass. However, the $\Xi_{b}^{*}\to\Xi_{b}\pi$ decay
is expected to be kinematically allowed, in analogy with the better studied
$\Xi_{c}$ baryon system.

The CMS analysis represents a search for $\Xi_{b}^{*0}$ baryons in
$\Xi_{b}^{-}\pi^{+}$ (plus charge conjugate) decays with $\Xi_{b}^{-}\to
J/\psi\,\Xi^{-}$, $J/\psi\rightarrow\mathrm{\mu^{+}\mu^{-}}$,
$\Xi^{-}\to\Lambda^{0}\,\pi^{-}$ and $\Lambda^{0}\to p\pi^{-}$ (plus charge
conjugates). For brevity, the reconstruction of the charge conjugate decay
modes will be implied throughout this summary. The following sections
summarise the CMS analysis and results presented in CMS_Xi . A detailed
description of the CMS detector can be found in CMS_Paper .

###  III.1 Data Sample and Event Selection

The CMS analysis is based on a sample of $pp$ collision data representing an
integrated luminosity of $5.3\,\mbox{fb${}^{-1}$}$ collected during the 2011
LHC run at a centre of mass energy $\sqrt{s}=7{\mathrm{\
Te\kern-1.00006ptV}}$. The data used by the CMS analysis are collected by
specialised triggers designed to record events containing two oppositely
charged muons that are compatible with having been produced in the decay of a
$J/\psi$. Separate triggers are used to select events containing
$J/\psi\rightarrow\mathrm{\mu^{+}\mu^{-}}$ candidates that are promptly
produced and those that are displaced from the primary vertex.

###  III.2 Selection of $\Xi_{b}^{-}$ candidates

The foundation of the analysis is the reconstruction of a pure sample of
$\Xi_{b}^{-}\to J/\psi\,\Xi^{-}$ decays (with $\Xi^{-}\to\Lambda^{0}\,\pi^{-}$
and $\Lambda^{0}\to p\pi^{-}$). Candidate $J/\psi$ are formed from pairs of
oppositely charged muons reconstructed from tracks in the silicon tracker,
matched to independent tracks in the muon detectors. The two muons are
required to pass the trigger selection and the di-muon candidate must have an
invariant mass within $150{\mathrm{\ Me\kern-1.00006ptV}}$ of the world
average $J/\psi$ mass PDG .

The reconstruction of $\Xi^{-}$ decays begins with the identification of
candidate $\Lambda^{0}\to p\pi$ decays from pairs of oppositely charged
tracks. The track with the higher momentum is taken to be the proton. The two
tracks are required to have been reconstructed from at least 6 six hits in the
silicon tracker and to have a track fit $\chi^{2}/d.o.f<5$. The two tracks are
fitted to a common decay vertex and the fit result is required to satisfy
$\chi^{2}/d.o.f<7$. The decay vertex is required to be displaced from the beam
line by more than ten times the uncertainty on the displacement. To remove
possible contamination from $K^{0}_{s}$ decays, the invariant mass of the
candidate (with both tracks assigned the pion mass) is rejected if it lies
within $20{\mathrm{\ Me\kern-1.00006ptV}}$ of the $K^{0}_{s}$ mass. $\Xi^{-}$
candidates are then reconstructed through the combination of a candidate
$\Lambda^{0}$ with a track (denoted $\pi_{\Xi}$) with the same charge as the
pion from the $\Lambda^{0}\to p\pi$ candidate (denoted $\pi_{\Lambda}$). The
three tracks are then subjected to a kinematic vertex fit where the invariant
mass of the two tracks from the $\Lambda^{0}$ candidate are constrained to the
world average $\Lambda^{0}$ mass. To reject backgrounds from mis-reconstructed
$\Omega^{-}\to\Lambda^{0}\,K^{-}$ decays, the $\Xi^{-}$ candidate is rejected
if its invariant mass, with the $\pi_{\Xi}$ track given the charged kaon mass,
lies within $20{\mathrm{\ Me\kern-1.00006ptV}}$ of the $\Omega^{-}$ mass.

Finally, $\Xi_{b}^{-}$ candidates are formed by combining candidate $\Xi^{-}$
and $J/\psi$ decays in a kinematic vertex fit where the masses of the
$\Xi^{-}$and $J/\psi$ candidates are constrained to the world average values.

To select a high yield sample of $\Xi_{b}^{-}$ candidates with high purity,
the $\Xi_{b}^{-}$ selection (characterised by the values of thirty selection
variables) is optimised by an iterative algorithm that maximises both the
signal yield and significance. The thirty optimised variables include the
transverse momentum threshold of the $J/\psi$, $p$, $\pi_{\Lambda}$,
$\pi_{\Xi}$, $\Xi^{-}$, $\Xi_{b}^{-}$ and muon candidates. The values of the
latter two thresholds ($\Xi_{b}^{-}$ and both muons) can take different values
depending on whether the candidates were reconstructed in the barrel or endcap
regions on the detector. The requirements on the difference between the
reconstructed invariant masses and the world average values for the
$\Lambda^{0}$, $\Xi^{-}$ and $J/\psi$ candidates are also optimised. Other
important optimised variables include the impact parameter significance of the
$p$, $\pi_{\Lambda}$ and $\pi_{\Xi}$ tracks and transverse decay length (and
its significance) of the $\Lambda^{0}$, $\Xi^{-}$ and $\Xi_{b}^{-}$ decay
vertices. The details of the optimisation procedure are described in CMS_Xi .

Figure 4 shows the invariant mass of selected $\Xi_{b}^{-}$ candidates. The
$m(J/\psi\,\Xi^{-})$ distribution exhibits a clear peak at approximately
$5.8{\mathrm{\ Ge\kern-1.00006ptV}}$, consistent with a $\Xi_{b}^{-}$ signal.
The $m(J/\psi\,\Xi^{-})$ distribution is fit with a Gaussian PDF to describe
the signal and a second order polynomial to describe the background. The fit
results in a yield of $108\pm 14$ $\Xi_{b}^{-}$ candidates and a fitted
$\Xi_{b}^{-}$ mass of $5795.0\pm 3.1\mbox{$\;$(stat.)}{\mathrm{\
Me\kern-1.00006ptV}}$, a value which agrees well with the current world
average PDG .

Figure 4: The $J/\psi\,\Xi^{-}$ invariant mass distribution for $\Xi_{b}^{-}$
candidates (filled points). The invariant mass distribution of
$J/\psi\,\Xi^{-}$ pairs where the proton and $\pi_{\Xi}$ have the same charge
is shown also shown (open squares) CMS_Xi .

###  III.3 Search for $\Xi_{b}^{*0}$ baryons

$\Xi_{b}^{-}$ candidates with a mass within $2.5\sigma$ of the fitted
$\Xi_{b}^{-}$ mass are associated with a track, given the pion mass, with a
charge opposite to that of the $\pi_{\Xi}$. The tracks are required to be
compatible with having been produced at the primary vertex and to have a
transverse momentum greater than $0.25{\mathrm{\ Ge\kern-1.00006ptV}}$. The
events in the CMS data sample contain on average eight primary vertices. The
primary vertex reconstructed closest to the $\Xi_{b}^{-}$ line of flight is
assumed to be associated with the production of the $\Xi_{b}^{-}$.

A potential $\Xi_{b}^{*0}$ signal is expected to appear as a peak in the
$Q=m(J/\psi\,\Xi^{-}\pi^{+})-m(J/\psi\,\Xi^{-})-m(\pi)$ distribution. In order
to search for such peaks, the background contribution to the $Q$ distribution
must first be reliably estimated. This is done through the preparation of a
background sample of $\Xi_{b}^{-}$ candidates associated with prompt pion
tracks of the same charge as the $\Xi_{b}^{-}$. The $Q$ distribution for this
background sample is shown in Figure 5. The momentum distributions of the
$\Xi_{b}^{-}$ and pions ($p(\Xi_{b})$ and $p(\pi)$) and the distribution of
angle between them ($\alpha$) from the (same sign) background sample are used
to randomly generate an uncorrelated set of values for $p(\Xi_{b})$, $p(\pi)$
and $\alpha$. The uncorrelated set of values is used to calculate a value for
$Q$, this is repeated $10^{8}$ times to give a $Q$ distribution which is
expected to predict the shape of the combinatorial background. This
distribution is then fitted with the function
$Q^{c_{1}}\left(e^{-c_{2}Q}+e^{-c_{3}Q}+e^{-c_{4}Q}\right)$ where the $c_{i}$
are all free parameters. The fit result is shown by the red dashed line in
Figure 5.

Figure 5: The $Q$ distribution for $\Xi_{b}^{-}$ candidates associated with
prompt pions of the same charge CMS_Xi . Figure 6: The $Q$ distribution for
$\Xi_{b}^{-}\pi^{+}$ candidates where the $\Xi_{b}^{-}$ and prompt pion
candidates have opposite charges CMS_Xi .

The $Q$ value distribution for opposite sign $\Xi_{b}^{-}\pi^{+}$ combinations
is shown in Figure 6. The $Q$ value distribution in Figure 6 exhibits a clear
excess above the background expectation in the region $12<Q<18{\mathrm{\
Me\kern-1.00006ptV}}$. An unbinned maximum likelihood fit is performed to the
$Q$ value distribution in Figure 6 where the excess is modelled by a Breit-
Wigner distribution convolved with a Gaussian resolution function (with a
width fixed to $1.91\pm 0.11{\mathrm{\ Me\kern-1.00006ptV}}$, a value derived
from Monte Carlo (MC) simulation). The background model discussed earlier is
used to describe the combinatorial background where the $c_{i}$ parameters are
free to vary within their total uncertainties. The fitted mean value of the
signal is $14.84\pm 0.74\mbox{$\;$(stat.)}{\mathrm{\ Me\kern-1.00006ptV}}$
with a Breit-Wigner width of $2.1\pm 1.7\mbox{$\;$(stat.)}{\mathrm{\
Me\kern-1.00006ptV}}$ CMS_Xi .

The significance of the signal is assessed though the calculation of
$\sqrt{\ln{\left(L_{s+b}/L_{b}\right)}}$ where $L_{s+b}$ is the likelihood
value of the nominal (signal and background) fit and $L_{b}$ is the likelihood
value for a fit performed with a background only model. The statistical
significance calculated in the way is $6.9$ standard deviations. The
significance is also evaluated though a number of other approaches (including
a treatment of the “look elsewhere effect”), all of which result in a
significance in excess of $5$ standard deviations.

Various cross checks are performed using MC simulation samples of $B^{+}$,
$B^{0}$, $B_{s}$ and $\Lambda_{b}$ decays. The analysis procedure performed on
these samples does not exhibit any excess in the $Q$ distribution that might
arise from the mis-reconstruction of known $b$ hadrons.

The systematic uncertainty on the measured $Q$ value of the signal receives
contributions from the observation of a small upwards shift in the $Q$
distribution in MC simulation and uncertainties associated with the background
model. Together these effects constitute a systematic uncertainty of
$0.28{\mathrm{\ Me\kern-1.00006ptV}}$.

With the charged pion and $\Xi_{b}^{-}$ masses taken from the PDG the mass of
the new baryon is measured to be $5945.0\pm 0.7\mbox{$\;$(stat.)}\pm
0.3\mbox{$\;$(syst.)}\pm 2.7~{}(\mathrm{PDG}){\mathrm{\ Me\kern-1.00006ptV}}$
CMS_Xi .

###  III.4 Conclusion

The CMS collaboration has observed a new $\Xi_{b}$ baryon, decaying to
$\Xi_{b}^{-}\pi^{+}$ (plus charge conjugate) with a statistical significance
in excess of $5$ standard deviations. The mass of the new baryon is measured
to be $5945.0\pm 0.7\mbox{$\;$(stat.)}\pm 0.3\mbox{$\;$(syst.)}\pm
2.7~{}(\mathrm{PDG}){\mathrm{\ Me\kern-1.00006ptV}}$ CMS_Xi . The measured
mass and decay mode are consistent with theoretical expectations for the
$\Xi_{b}^{*0}$ state, predicted to have quantum numbers $J^{P}=3/2^{+}$.

###### Acknowledgements.

The author is grateful for the financial support provided by the Science and
Technology Facilities Council (STFC) in the UK.

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# Étale triviality of finite equivariant vector bundles

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>and  Peter O’Sullivan  Mathematical Sciences Institute, The Australian
National University, Canberra ACT 2601, Australia<EMAIL_ADDRESS>
  *     * scAbstract. Let $H$ be a complex Lie group acting holomorphically on a complex analytic space $X$ such that the restriction to $X_{s}c\mathrm{red}$$ofevery$H$-invariantregularfunctionon$X$isconstant.Weprovethatan$H$-equivariantholomorphicvectorbundle$E$over$X$is$H$-finite,meaning$f_1(E) = f_2(E)$as$H$-equivariantbundlesfortwodistinctpolynomials$f_1$and$f_2$whosecoefficientsarenonnegativeintegers,ifandonlyifthepullbackof$E$alongsome$H$-equivariantfinite\'{e}talecoveringof$X$istrivialasan$H$-equivariantbundle.\vskip 3.0pt plus 1.0pt minus 1.0pt\par\vskip 12.0pt plus 4.0pt minus 4.0pt\par\noindent sc\bf Keywords.$ Rigid tensor category; tensor functor; finite bundle; étale triviality; equivariant bundle

sc2020 Mathematics Subject Classification. 32L10, 53C55, 14D21, 16B50

sc[Français]

scÉtale trivialité des fibrés vectoriels équivariants finis

scRésumé. Soit $H$ un groupe de Lie complexe agissant holomorphiquement sur un
espace analytique complexe $X$ de telle sorte que la restriction à
$X_{s}c\mathrm{red}$$detoutefonctionr\'{e}guli\\`{e}re$H$-invariantesoitconstante.Nousmontronsqu^{s}c\prime$un
fibré vectoriel holomorphe $H$-invariant $E$ sur $X$ est $H$-fini, c’est-à-
dire vérifie $f_{1}(E)\,=\,f_{2}(E)$ comme $H$-fibrés équivariants pour deux
polynômes distincts $f_{1}$ et $f_{2}$ à coefficients entiers positifs, si et
seulement si l’image réciproque de $E$ par un revêtement étale fini
$H$-équivariant est triviale comme $H$-fibré équivariant.

  *
cSeptember 14, 2021Received by the Editors on March 17, 2021\.
Accepted on October 27, 2021.

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha
Road, Mumbai 400005, India

sce-mail<EMAIL_ADDRESS>
Mathematical Sciences Institute, The Australian National University, Canberra
ACT 2601, Australia

sce-mail<EMAIL_ADDRESS>

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Finite bundles and reduction of structure group

## 1\. Introduction

Given a vector bundle $E$, and a polynomial
$f(t)\,=\,\sumop\displaylimits_{i=0}^{n}a_{i}t^{i}$, where $a_{i}$ are
nonnegative integers, define the vector bundle
$f(E)\,:=\,\bigoplusop\displaylimits_{i=0}^{n}(E^{\otimes i})^{\oplus a_{i}}$.
A vector bundle $E$ is called finite if there are two such distinct
polynomials $f_{1}$ and $f_{2}$ satisfying the condition that the vector
bundles $f_{1}(E)$ and $f_{2}(E)$ are isomorphic. This notion was introduced
by Weil in [Wei38]. He proved that a vector bundle $E\,\longrightarrow\,X$
over a complex projective variety is finite if there is a finite étale
covering $\varpi\,:\,\widehat{X}\,\longrightarrow\,X$ such that the pulled
back vector bundle $\varpi^{*}E$ is trivial. Nori in his thesis proved a
converse of it. He proved that a vector bundle $E$ over a projective variety
$X$, defined over an algebraically closed field of characteristic zero, is
finite if and only if there is a finite étale covering
$\varpi\,:\,\widehat{X}\,\longrightarrow\,X$ such that $\varpi^{*}E$ is
trivial [Nor76], [Nor82]. Nori’s proof involves reducing the problem to the
case of curves by taking hyperplanes in $X$. Note that there is a finite étale
covering $\varpi\,:\,\widehat{X}\,\longrightarrow\,X$ such that $\varpi^{*}E$
is trivial if and only if $E$ admits a reduction of structure group to a
finite group.

Attempts were made to prove the above theorem of Nori in the context of
complex manifolds. A vector bundle $E$ over a compact Kähler manifold $X$ is
finite if and only if there is a finite étale covering
$\varpi\,:\,\widehat{X}\,\longrightarrow\,X$ such that $\varpi^{*}E$ is
trivial [BHS00]. When $X$ is a compact complex manifold admitting a Gauduchon
astheno-Kähler metric, the same result was proved in [BP18]. We recall that
astheno-Kähler metrics, introduced in [JY93], are weaker than Kähler metrics.
More recently, in [Bis20] this result was proved for holomorphic vector
bundles over any compact complex manifold.

Our aim here is to prove a similar result in a more general setting:

  * •

We allow $X$ to be a complex analytic space, so it can have singularities and
may not be reduced,

  * •

$X$ is allowed to be noncompact, and

  * •

a complex Lie group $H$ acts on $X$, and the vector bundles considered are
$H$-equivariant.

Let $H$ be a complex Lie group acting holomorphically on a complex analytic
space $X$. An $H$-equivariant holomorphic vector bundle ${\mathcal{V}}$ over
$X$ is called $H$-finite if there are two distinct polynomials $f_{1}$ and
$f_{2}$, whose coefficients are nonnegative integers, such that $f_{1}(E)$ and
$f_{2}(E)$ are isomorphic as $H$-equivariant bundles, and $H$-trivial if it is
$H$-equivariantly isomorphic to ${\mathcal{O}}^{\oplus r}_{X}$ for some $r$.

We prove the following theorem (see Theorem 2.3):

######  Theorem 1.1.

Let $H$ be a complex Lie group acting holomorphically on a complex analytic
space $X$ such that the restriction to $X_{\mathrm{red}}$ of every
$H$-invariant regular function on $X$ is constant. Then an $H$-equivariant
holomorphic vector bundle over $X$ is $H$-finite if and only if its pullback
along some $H$-equivariant finite étale covering of $X$ is $H$-trivial.

As mentioned above, when $X$ is a compact complex manifold, and
$H\,=\,\\{e\\}$, Theorem 1.1 was proved in [Bis20]. The proof of Theorem 1.1
given here is an application of the splitting theorem of [AK02], [O’S05] and
[O’S19]. It is simpler and much shorter than the proof given in [Bis20]. It
resembles [And09] in the sense that it constitutes a simple category theoretic
proof of a result proved using elaborate analytical methods.

## 2\. Finite bundles and reduction of structure group

Throughout, all spaces are complex analytic, and all maps are holomorphic.

By a tensor category we mean a $\mathbb C$-linear category with $\mathbb
C$-bilinear tensor product, together with a unit and associativity and
commutativity constraints satisfying the usual compatibilities [DM82, p. 105,
Definition 1.1]. A tensor functor between tensor categories is a $\mathbb
C$-linear functor together with constraints ensuring that unit and tensor
product are preserved up to isomorphism [DM82, pp. 113–114, Definition 1.8].

Let $X$ be a complex analytic space. We identify holomorphic vector bundles on
$X$ with locally free $\mathcal{O}_{X}$-modules of finite type. The tensor
category of holomorphic vector bundles over $X$ will be denoted by
$\operatorname{Mod}(X)$. If $X$ is a point, then $\operatorname{Mod}(X)$ is
the category $\operatorname{Mod}(\mathbb C)$ of finite-dimensional $\mathbb
C$-vector spaces.

Let $G$ be a finite group. We write $\operatorname{Rep}_{\mathbb C}(G)$ for
the tensor category of finite dimensional representations of $G$. If we regard
the algebra $\mathbb C[G]$ of functions on $G$ as the left regular
representation of $G$, where any $g\,\in\,G$ sends any $w$ in $\mathbb C[G]$
to $w(g^{-1}\cdot)$, then $\mathbb C[G]$ is a commutative $G$-algebra under
pointwise multiplication. The action of $G$ on a representation $V$ of $G$ is
an isomorphism

(2.1) $V_{0}\otimes_{\mathbb C}\mathbb
C[G]\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,V\otimes_{\mathbb
C}\mathbb C[G]$

of modules over $\mathbb C[G]$ in $\operatorname{Rep}_{\mathbb C}(G)$, where
$V_{0}$ is the underlying $\mathbb C$-vector space for $V$ equipped with the
trivial action of $G$.

Let $H$ be a complex Lie group, and let $X$ be a complex analytic space with a
holomorphic action of $H$. An $H$-equivariant vector bundle over $X$ is a
vector bundle over $X$ equipped with an action of $H$ above that on $X$. If
${\mathcal{V}}$ is such a vector bundle we write

$H^{0}_{H}(X,\,{\mathcal{V}})$

for the $\mathbb C$-vector space of $H$-invariant cross-sections of
${\mathcal{V}}$, _i.e._ those cross-sections whose pullback onto $H\times X$
along the projection to $X$ coincides with the pullback by the map $H\times
X\,\longrightarrow\,X$ giving the action of $H$ on $X$.

We denote by $\operatorname{Mod}_{H}(X)$ the tensor category of
$H$-equivariant vector bundles over $X$, where a morphism is a morphism of
vector bundles compatible with the actions of $H$, and the tensor product is
the tensor product of vector bundles equipped with an appropriate action of
$H$. Given also $H^{\prime}$ acting on $X^{\prime}$, together with a
homomorphism

$H^{\prime}\,\longrightarrow\,H$

and a morphism $X^{\prime}\,\longrightarrow\,X$ compatible with the actions,
pullback along $X^{\prime}\,\longrightarrow\,X$ defines a tensor functor from
$\operatorname{Mod}_{H}(X)$ to $\operatorname{Mod}_{H^{\prime}}(X^{\prime})$.
When $H$ is a finite group and $X$ is a point, the tensor category
$\operatorname{Mod}_{H}(X)$ is $\operatorname{Rep}_{\mathbb C}(\Gamma)$.

Let $G$ be a finite group. If $P$ is a holomorphic principal $G$-bundle over
$X$, there is an action of $G$ on $P$ with $gp\,=\,pg^{-1}$ for $g$ in $G$ and
$p$ in $P$. Thus we have pullback tensor functors from
$\operatorname{Rep}_{\mathbb C}(G)$ and from $\operatorname{Mod}(X)$ to
$\operatorname{Mod}_{G}(P)$. Given $V$ in $\operatorname{Rep}_{\mathbb C}(G)$,
a pair consisting of ${\mathcal{V}}$ in $\operatorname{Mod}(X)$ and a
$G$-isomorphism from the pullback of $V$ onto $P$ to the pullback of
${\mathcal{V}}$ onto $P$ exists, and is unique up to unique isomorphism: for
the existence reduce by arguing locally over $X$ to the case where $P$ is
trivial. We write

(2.2) $P\times^{G}V$

for ${\mathcal{V}}$. Formation of $P\times^{G}V$ commutes with pullback of
principal $G$-bundles.

By an $H$-equivariant principal $G$-bundle over $X$ we mean a principal
$G$-bundle $P$ over $X$ together with an action of $H$ on $P$ above that on
$X$, such that $(hp)g\,=\,h(pg)$ for points $h$, $p$ and $g$ of respectively
$H$, $P$ and $G$. There is an action of $H\times G$ on such a $P$ above the
action of $H$ on $X$, with $(h,\,g)p\,=\,hpg^{-1}$. Again we have pullback
tensor functors from $\operatorname{Rep}_{\mathbb C}(G)$ and from
$\operatorname{Mod}_{H}(X)$ to $\operatorname{Mod}_{H\times G}(P)$. Given $V$
in $\operatorname{Rep}_{\mathbb C}(G)$, a pair consisting of ${\mathcal{V}}$
in $\operatorname{Mod}_{H}(X)$ and an $(H\times G)$-isomorphism from the
pullback of $V$ onto $P$ to the pullback of ${\mathcal{V}}$ onto $P$ exists,
and is unique up to unique isomorphism. Indeed ${\mathcal{V}}$ is
$P\times^{G}V$ in (2.2) with an appropriate $H$-equivariant structure. We then
have a tensor functor $P\times^{G}-$ from $\operatorname{Rep}_{\mathbb C}(G)$
to $\operatorname{Mod}_{H}(X)$.

A commutative $\mathcal{O}_{X}$-algebra ${\mathcal{R}}$ will be called finite
étale if locally on $X$ it is a finite product of copies of $\mathcal{O}_{X}$.
It is equivalent to require that the underlying $\mathcal{O}_{X}$-module of
${\mathcal{R}}$ lie in $\operatorname{Mod}(X)$ and that each fibre of
${\mathcal{R}}$ be a finite product of copies of $\mathbb C$. Indeed since
$\mathcal{O}_{X,x}$ is Henselian for each $x$, a decomposition of the fibre of
${\mathcal{R}}$ at $x$ into a product of copies of $\mathbb C$ will lift to a
decomposition of the stalk of ${\mathcal{R}}$ at $x$, and the idempotents
defining it will then extend to some neighbourhood of $x$.

A morphism $X^{\prime}\,\longrightarrow\,X$ of complex analytic spaces will be
called finite étale if locally on $X$ it is isomorphic to the projection onto
$X$ of the product of $X$ with a finite set. A surjective finite étale
morphism will also be called a finite étale covering of $X$. The assignment to
$X^{\prime}$ with structural morphism $a$ of the commutative
$\mathcal{O}_{X}$-algebra $a_{*}\mathcal{O}_{X^{\prime}}$ defines an anti-
equivalence from finite étale complex analytic spaces over $X$ to finite étale
$\mathcal{O}_{X}$-algebras, and from $H$-equivariant finite étale complex
analytic spaces over $X$ to $H$-equivariant finite étale
$\mathcal{O}_{X}$-algebras.

We say that $X$ is $H$-connected if it is not the disjoint union of two non-
empty $H$-equivariant open subspaces. It is equivalent to require that
$H^{0}_{H}(X,\,\mathcal{O}_{X})$ have no idempotent other than $0$ or $1$. If
$X$ is $H$-connected, any $H$-equivariant vector bundle on $X$ has constant
rank.

######  Lemma 2.1.

Let $H$ be a complex Lie group and $X$ be a non-empty complex analytic space
with an action of $H$. Then the following conditions are equivalent:

  1. (1)

the restriction to $X_{\mathrm{red}}$ of any $H$-invariant regular function on
$X$ is constant;

  2. (2)

$H^{0}_{H}(X,\,\mathcal{O}_{X})$ is a local $\mathbb C$-algebra;

  3. (3)

$H^{0}_{H}(X,\,\mathcal{O}_{X})$ is a Henselian local $\mathbb C$-algebra with
residue field $\mathbb C$;

  4. (4)

$H^{0}_{H}(X,\,\mathcal{O}_{X})$ is the limit of a sequence $(R_{n})_{n\geq
1}$ of commutative $\mathbb C$-algebras with surjective transition
homomorphisms and with $R_{1}\,=\,\mathbb C$, such that for each $n$ the
kernel of $R_{n}\,\longrightarrow\,R_{1}$ is nilpotent of degree at most $n$.

If these conditions hold, then they hold with $X$ replaced by any
$H$-connected $H$-equivariant finite étale cover.

###### Proof.

Write $R\,=\,H^{0}_{H}(X,\,\mathcal{O}_{X})$. If ${\mathcal{N}}$ is the
nilradical of $\mathcal{O}_{X}$, then for each $n$ the action of $H$ on $X$
induces one on the complex analytic subspace $X_{n}$ of $X$ with the same
underlying topological space and structure sheaf
$\mathcal{O}_{X}/{\mathcal{N}}^{n}$, with $X_{1}\,=\,X_{\mathrm{red}}$. Since
${\mathcal{N}}$ is locally nilpotent on $X$,

$R\,=\,\lim_{n}H^{0}_{H}(X_{n},\,\mathcal{O}_{X_{n}}),$

where the kernel of
$H^{0}_{H}(X_{n},\,\mathcal{O}_{X_{n}})\,\longrightarrow\,H^{0}_{H}(X_{1},\,\mathcal{O}_{X_{1}})$
is nilpotent of degree at most $n$. Thus (1) implies (4) with $R_{n}$ the
image of

$R\,\longrightarrow\,H^{0}_{H}(X_{n},\,\mathcal{O}_{X_{n}}).$

If (4) holds, then any finite commutative $R$-algebra $R^{\prime}$ which is a
free $R$-module is the limit of the
$R^{\prime}\\!{}_{n}\,=\,R^{\prime}\otimes_{R}R_{n}$. Since the transition
homomorphisms of $(R^{\prime}\\!{}_{n})$ are surjective with nilpotent kernel,
it follows that any idempotent of $R^{\prime}\\!{}_{1}$ lifts uniquely to an
idempotent of each $R^{\prime}\\!{}_{n}$, and hence to $R^{\prime}$, so that
(3) holds. That (3) implies (2) is clear. If (2) holds, then the evaluation
homomorphism $R\,\longrightarrow\,\mathbb C$ at each point of $X$ has kernel
the unique maximal ideal of $R$, so that (1) holds.

Let $X^{\prime}$ be an $H$-connected $H$-equivariant finite étale cover of
$X$. Then the push forward $a_{*}\mathcal{O}_{X^{\prime}}$ along
$a:X^{\prime}\,\longrightarrow\,X$ is locally free of constant rank over $X$.
For any element $\alpha$ of
$H^{0}_{H}(X^{\prime},\,\mathcal{O}_{X^{\prime}})$, the Cayley–Hamilton
theorem applied to the endomorphism of $a_{*}\mathcal{O}_{X^{\prime}}$ induced
by $\alpha$ thus shows that $\alpha$ is annihilated by a monic polynomial with
coefficients in $R$. Suppose that (1) holds. Then the restriction
$\overline{\alpha}$ of any $\alpha$ to $X^{\prime}\\!{}_{\mathrm{red}}$ is
annihilated by a monic polynomial with coefficients in $\mathbb C$. Since
$X^{\prime}\\!{}_{\mathrm{red}}$ is reduced and $H$-connected, it follows that
$\overline{\alpha}$ lies in $\mathbb C$. Thus (1) holds with $X^{\prime}$ for
$X$. ∎

Let $X$ be complex analytic space with an action of $H$. If $Y$ and $Z$ are
$H$-equivariant finite étale coverings of $X$, and $Y$ is $H$-connected, then
any two $H$-equivariant morphisms from $Y$ to $Z$ over $X$ which send a given
point $y$ of $Y$ to the same point of $Z$ actually coincide over the whole of
$Y$, because their equaliser is an open and closed $H$-equivariant subset of
$Y$ containing $y$.

Suppose that $X$ is $H$-connected, and let $X^{\prime}$ be an $H$-equivariant
finite étale covering of $X$. If $x_{1},\,x_{2},\,\ldots,\,x_{n}$ are the
points of $X^{\prime}$ above the point $x$ of $X$, write $\mathcal{P}$ for the
$H$-connected component that contains the point

$p\,=\,(x_{1},\,x_{2},\,\ldots,\,x_{n}).$

of the fibre product over $X$ of $n$ copies of $X^{\prime}$. For each point
$q$ of $\mathcal{P}$ above $x$, there exists an $H$-equivariant endomorphism
of $\mathcal{P}$ over $X$ sending $p$ to $q$, because for each $i$ there
exists an $H$-equivariant morphism from $\mathcal{P}$ to $X^{\prime}$ over $X$
sending $p$ to $x_{i}$. Since $\mathcal{P}$ is $H$-connected, such a
endomorphism is unique, and is an automorphism. The group $G$ of
$H$-equivariant automorphisms of $\mathcal{P}$ over $X$ thus acts simply
transitively on the fibre of $\mathcal{P}$ above $x$ and hence on every fibre.
Using the inverse involution of the finite group $G$, we obtain an
$H$-connected $H$-equivariant principal $G$-bundle $\mathcal{P}$ from which
there exists an $H$-equivariant morphism to $X^{\prime}$ over $X$.

An $H$-equivariant vector bundle over $X$ will be called $H$-trivial if it is
$H$-isomorphic to the pullback onto $X$ of a finite-dimensional $\mathbb
C$-vector space $V$ on which $H$ acts trivially.

######  Lemma 2.2.

Let $H$ be a complex Lie group and $X$ be a non-empty complex analytic space
with an action of $H$ such that the equivalent conditions of Lemma 2.1 hold.
Then the following conditions on an $H$-equivariant vector bundle
${\mathcal{V}}$ over $X$ are equivalent:

  1. (1)

There exists an $H$-equivariant finite étale covering $X^{\prime}$ of $X$ such
that the pullback of ${\mathcal{V}}$ onto $X^{\prime}$ is $H$-trivial.

  2. (2)

The $H$-equivariant vector bundle ${\mathcal{V}}$ is $H$-isomorphic to
$P\times^{G}V$ for some finite group $G$, finite-dimensional representation
$V$ of $G$, and $H$-equivariant principal $G$-bundle $P$ over $X$.

###### Proof.

If (2) holds then (1) holds with $X^{\prime}=P$.

Conversely suppose that (1) holds. Since $X$ is $H$-connected, there is as
above a finite group $G$ and an $H$-connected $H$-equivariant principal
$G$-bundle $P$ (denoted above by $\mathcal{P}$) over $X$ with an
$H$-equivariant morphism to $X^{\prime}$ over $X$. The pullback
$a^{*}{\mathcal{V}}$ of ${\mathcal{V}}$ along $a\,:\,P\,\longrightarrow\,X$ is
thus $H$-trivial. To see that (2) holds, it is to be shown that for some
finite-dimensional representation $V$ of $G$ there exists an $(H\times
G)$-equivariant isomorphism from the pullback $V\otimes_{\mathbb
C}\mathcal{O}_{P}$ of $V$ onto $P$ to $a^{*}{\mathcal{V}}$.

The equivalent conditions of Lemma 2.1 hold with $X$ replaced by $P$. It
follows that

$R\,=\,H^{0}_{H}(P,\,\mathcal{O}_{P})$

is the limit of a sequence $(R_{n})_{n\geq 1}$ of $\mathbb C$-algebras as in
(4) of Lemma 2.1. Let

$\rho\,:\,G\,\longrightarrow\,{\rm GL}_{n}(R)$

be a group homomorphism, and write $\rho_{r}$ for $\rho$ followed by the
projection onto ${\rm GL}_{n}(R_{r})$. The group $G$ acts on the kernel
$J_{r+1}$ of the projection from ${\rm GL}_{n}(R_{r+1})$ to ${\rm
GL}_{n}(R_{r})$, with $g$ in $G$ acting as conjugation by the element
$\rho_{1}(g)$ of the subgroup ${\rm GL}_{n}(\mathbb C)$ of ${\rm
GL}_{n}(R_{r+1})$. Since $G$ is finite and $J_{r+1}$ is the additive group of
a $\mathbb C$-vector space, we have

$H^{1}(G,\,J_{r+1})\,=\,0.$

Thus there is an element $\theta\,=\,(\theta_{r})$ of ${\rm
GL}_{n}(R)\,=\,\lim_{r}{\rm GL}_{n}(R_{r})$ such that the conjugate of $\rho$
by $\theta$ factors through ${\rm GL}_{n}(\mathbb C)$, because the obstruction
to lifting a given $\theta_{r}$ to a $\theta_{r+1}$ lies in
$H^{1}(G,\,J_{r+1})$.

The $G$-equivariant structure of $a^{*}{\mathcal{V}}$ defines an action of $G$
on the finitely generated free $R$-module $H^{0}_{H}(P,\,a^{*}{\mathcal{V}})$.
Choosing a basis $e_{1},\,e_{2},\,\cdots,\,e_{n}$ of
$H^{0}_{H}(P,\,a^{*}{\mathcal{V}})$ thus defines a homomorphism $\rho$ as
above. If we modify the basis by a $\theta$ in ${\rm GL}_{n}(R)$ such that the
conjugate of $\rho$ by $\theta$ factors through ${\rm GL}_{n}(\mathbb C)$,
then the action of $G$ on $H^{0}_{H}(P,\,a^{*}{\mathcal{V}})$ restricts to an
action on the $\mathbb C$-vector subspace $V$ generated by the $e_{i}$. The
embedding of $V$ then induces the required $(H\times G)$-equivariant
isomorphism. ∎

Let ${\mathcal{C}}$ be a tensor category which is rigid (_i.e._ every object
has a dual). Then the trace

${\rm tr}(e)\,\in\,\operatorname{End}_{\mathcal{C}}(\mathbf{1})$

of an endomorphism $e$ in ${\mathcal{C}}$ is defined. If the $\mathbb
C$-algebra $\operatorname{End}_{\mathcal{C}}(\mathbf{1})$ is local, then the
morphisms $r\,:\,A\,\longrightarrow\,B$ in ${\mathcal{C}}$ for which ${\rm
tr}(s\circ r)$ lies in the maximal ideal of
$\operatorname{End}_{\mathcal{C}}(\mathbf{1})$ for every
$s\,:\,B\,\longrightarrow\,A$ form the unique maximal tensor ideal
${\mathcal{J}}$ of ${\mathcal{C}}$ [O’S19, p. 73]. We then write

(2.3) $\overline{\mathcal{C}}$

for the quotient ${\mathcal{C}}/{\mathcal{J}}$. It has the same objects as
${\mathcal{C}}$, and there is a projection
${\mathcal{C}}\,\longrightarrow\,\overline{\mathcal{C}}$ which is the identity
on objects and is a full tensor functor with kernel ${\mathcal{J}}$.

The group algebra of the symmetric group of degree $d$ acts on the $d$-th
tensor power of every object of ${\mathcal{C}}$, and if ${\mathcal{C}}$ is
pseudo-abelian we may define for example the $d$-th exterior power as the
image of the antisymmetrising idempotent.

Suppose that ${\mathcal{C}}$ is essentially small (_i.e._ has a small
skeleton) and pseudo-abelian, that the $\mathbb C$-algebra
$\operatorname{End}_{\mathcal{C}}(\mathbf{1})$ is Henselian local with residue
field $\mathbb C$, and that for every object of ${\mathcal{C}}$ some exterior
power is $0$. Then $\overline{\mathcal{C}}$ is a semisimple Tannakian category
over $\mathbb C$ ([O’S19, Proposition 11.2(i)] and [Del90, p. 165, Théorème
7.1]), and the projection
${\mathcal{C}}\,\longrightarrow\,\overline{\mathcal{C}}$ reflects isomorphisms
[O’S19, Proposition 11.1(ii)]. It follows that $\mathcal{C}$ has the
Krull–Schmidt property (_i.e._ the commutative monoid under direct sum of
isomorphism classes of objects of $\mathcal{C}$ is freely generated by the
classes of the indecomposable objects) because $\overline{\mathcal{C}}$ does.
The splitting theorem for ${\mathcal{C}}$ ([AK02, p. 225, Théorème 16.1.1(a)],
[O’S05, Theorem 1.1], [O’S19, Theorem 11.7]) states that the projection has a
right quasi-inverse, _i.e._ a tensor functor
$\overline{\mathcal{C}}\,\longrightarrow\,{\mathcal{C}}$ with the composite

$\overline{\mathcal{C}}\,\longrightarrow\,{\mathcal{C}}\,\longrightarrow\,\overline{\mathcal{C}}$

tensor isomorphic to the identity. In [AK02] and [O’S05] this result is proved
under the stricter hypothesis that
$\operatorname{End}_{\mathcal{C}}(\mathbf{1})\,=\,\mathbb C$. A reader who
wishes to confine attention to this case may replace the hypothesis on $H$ and
$X$ in Theorem 2.3 by the stricter one that
$H^{0}_{H}(X,\,\mathcal{O}_{X})\,=\,\mathbb C$. However it is still convenient
to prove Lemmas 2.1 and 2.2 in the generality given, because unless $X$ is
reduced the condition $H^{0}_{H}(X,\,\mathcal{O}_{X})\,=\,\mathbb C$ is not
stable under passage to $H$-connected $H$-equivariant finite étale covers.

Let $H$ and $X$ be as in Lemma 2.2. Then $X$ is $H$-connected so that every
$\mathcal{V}$ in $\operatorname{Mod}_{H}(X)$ has constant rank. Since
$\operatorname{End}(\mathbf{1})$ in $\operatorname{Mod}_{H}(X)$ is
$H^{0}_{H}(X,\,\mathcal{O}_{X})$, it thus follows from (3) of Lemma 2.1 that
the above conditions on $\mathcal{C}$ are satisfied when $\mathcal{C}$ is
$\operatorname{Mod}_{H}(X)$.

Reducing using (1) of Lemma 2.2 to the $H$-trivial case shows that the full
subcategory of $\operatorname{Mod}_{H}(X)$ consisting of the $\mathcal{V}$
satisfying the conditions of Lemma 2.2 is stable under the formation of tensor
products, duals, direct sums and (since $\operatorname{End}(\mathbf{1})$ is
local) direct summands. When $X$ is reduced, the full tensor subcategory of
such $\mathcal{V}$ is Tannakian over $\mathbb C$, with each point of $X$
defining a fibre functor, because by Lemma 2.1 we then have
$\operatorname{End}(\mathbf{1})\,=\,\mathbb C$ in
$\operatorname{Mod}_{H}(X^{\prime})$ for any $H$-connected $H$-equivariant
étale cover $X^{\prime}$ of $X$, so that any morphism between $H$-trivial
objects in $\operatorname{Mod}_{H}(X^{\prime})$ factors as the projection onto
a direct summand followed by the embedding of a direct summand.

If ${\mathcal{V}}$ is an $H$-equivariant vector bundle over $X$ and
$f(t)\,=\,\sumop\displaylimits_{i=0}^{n}a_{i}t^{i}$ is a polynomial whose
coefficients $a_{i}$ are nonnegative integers, define the $H$-equivariant
vector bundle $f({\mathcal{V}})$ as
$\bigoplusop\displaylimits_{i=0}^{n}({\mathcal{V}}^{\otimes i})^{\oplus
a_{i}}$; note that ${\mathcal{V}}^{\otimes 0}\,=\,{\mathcal{O}}_{X}$ and
$({\mathcal{V}}^{\otimes i})^{\oplus 0}\,=\,0$. Then ${\mathcal{V}}$ will be
called $H$-finite if there exist distinct polynomials $f_{1}$ and $f_{2}$ with
$f_{1}({\mathcal{V}})$ $H$-isomorphic to $f_{2}({\mathcal{V}})$. Since
$\operatorname{Mod}_{H}(X)$ has the Krull–Schmidt property, it is equivalent
to require that the ${\mathcal{V}}^{\otimes n}$ should contain only finitely
many non-isomorphic indecomposable direct summands in
$\operatorname{Mod}_{H}(X)$. Thus the full subcategory of $H$-finite
$\mathcal{V}$ in $\operatorname{Mod}_{H}(X)$ is stable under tensor products,
duals, direct sums and direct summands.

######  Theorem 2.3.

Let $H$ and $X$ be as in Lemma 2.2. Then an $H$-equivariant vector bundle
${\mathcal{V}}$ over $X$ is $H$-finite if and only if it satisfies the
equivalent conditions of Lemma 2.2.

###### Proof.

Suppose that ${\mathcal{V}}$ satisfies (2) of Lemma 2.2. Then class of $V$ in
the representation ring of $G$ is annihilated by a non-zero polynomial with
coefficients in ${\mathbb Z}$, because the representation ring is finite over
${\mathbb Z}$. Thus $P\times^{G}V$ and hence ${\mathcal{V}}$ is $H$-finite.

Conversely if ${\mathcal{V}}$ is $H$-finite, we show that ${\mathcal{V}}$
satisfies (1) of Lemma 2.2. Denote by ${\mathcal{C}}$ the full pseudo-abelian
rigid tensor subcategory of $\operatorname{Mod}_{H}(X)$ generated by
${\mathcal{V}}$. Then $\overline{\mathcal{C}}$ (defined in (2.3)) is
semisimple Tannakian over $\mathbb C$. The image $\overline{\mathcal{V}}$ of
$\mathcal{V}$ in $\overline{\mathcal{C}}$ generates it as a semisimple abelian
tensor category. Thus for some (not necessarily connected) reductive algebraic
group $G$ over $\mathbb C$ there is by [DM82, p. 130–131, Theorem 2.11] a
tensor equivalence

(2.4) $\operatorname{Rep}_{\mathbb
C}(G)\,\longrightarrow\,\overline{\mathcal{C}}.$

Since $\mathcal{C}$ contains only finitely many non-isomorphic indecomposable
objects by $H$-finiteness of $\mathcal{V}$, the same holds for
$\overline{\mathcal{C}}$ and $\operatorname{Rep}_{\mathbb C}(G)$. Thus $G$ is
finite.

By the splitting theorem, the projection
${\mathcal{C}}\,\longrightarrow\,\overline{\mathcal{C}}$ has a right quasi-
inverse

$\overline{\mathcal{C}}\,\longrightarrow\,{\mathcal{C}}.$

The image of $\overline{\mathcal{V}}$ in ${\mathcal{C}}$ is isomorphic to
${\mathcal{V}}$, because the projection is full and reflects isomorphisms.
Composing with (2.4) and the embedding into $\operatorname{Mod}_{H}(X)$ gives
a tensor functor

$T\,:\,\operatorname{Rep}_{\mathbb
C}(G)\,\longrightarrow\,\operatorname{Mod}_{H}(X)$

such that $\mathcal{V}$ is $H$-isomorphic to $T(V)$ for some $V$ in
$\operatorname{Rep}_{\mathbb C}(G)$.

Applying $T$ to (2.1) gives an $H$-equivariant isomorphism of $T(\mathbb
C[G])$-modules

(2.5) $T(V_{0})\otimes_{\mathcal{O}_{X}}T(\mathbb
C[G])\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,T(V)\otimes_{\mathcal{O}_{X}}T(\mathbb
C[G]),$

where $T(V_{0})$ is $H$-trivial. Since $T$ followed by passage to the fibre at
a point of $X$ is tensor isomorphic to the forgetful functor, the fibres of
$T(\mathbb C[G])$ are isomorphic to the underlying $\mathbb C$-algebra of
$\mathbb C[G]$. Hence $T(\mathbb C[G])$ is a non-zero $H$-equivariant finite
étale $\mathcal{O}_{X}$-algebra. There thus exists an $H$-equivariant finite
étale covering $a:X^{\prime}\,\longrightarrow\,X$ of $X$ with
$a_{*}\mathcal{O}_{X^{\prime}}$ $H$-isomorphic to $T(\mathbb C[G])$. Replacing
$T(\mathbb C[G])$ by $a_{*}\mathcal{O}_{X^{\prime}}$ in $\eqref{e:TGact}$ and
applying $a^{*}$ and then
$a^{*}a_{*}\mathcal{O}_{X^{\prime}}\,\longrightarrow\,\mathcal{O}_{X^{\prime}}$
shows that $a^{*}T(V)$ and hence $a^{*}{\mathcal{V}}$ is $H$-trivial. ∎

It follows from Theorem 2.3 and the remarks preceding it that if $H$ and $X$
are as in Lemma 2.2 and $X$ is reduced, then the full tensor subcategory of
$\operatorname{Mod}_{H}(X)$ consisting of the $H$-finite objects is a neutral
Tannakian category over $\mathbb C$, and hence by the $H$-finiteness is tensor
equivalent to $\operatorname{Rep}_{\mathbb C}(G)$ for a profinite group $G$.
It can be shown more generally using the structure theorem of [O’S05, § 7]
that a tensor category $\mathcal{C}$ as above is tensor equivalent to
$\operatorname{Rep}_{\mathbb C}(G)$ for a profinite group $G$ if and only if
$\operatorname{End}_{\mathcal{C}}(\mathbf{1})\,=\,\mathbb C$, every object of
$\mathcal{C}$ satisfies a condition analogous to $H$-finiteness, and
$\mathcal{C}$ is reduced, in the sense that $h^{\otimes n}\,=\,0$ implies
$h\,=\,0$ for every morphism $h$ of $\mathcal{C}$.

Much more general results than Theorem 2.3 can be obtained by combining the
splitting theorem of [AK02], [O’S05] and [O’S19] with the complex analytic
analogue of the well-known dictionary between principal bundles and tensor
functors from categories of group representations to categories of vector
bundles. In particular we obtain in this way complex analytic analogues of the
results on uniqueness up to conjugacy of minimal reductions of principal
bundles with reductive structure group proved by Bogomolov [Bo94, p. 401,
Theorem 2.1] and in a more general form in [O’S19, Corollary 12.13].

Similar techniques are potentially applicable in many other geometric
contexts. As well as rigid analytic analogues, we may for example consider
vector bundles or principal bundles equipped with a connection along the
leaves of a foliated differentiable manifold on which every leafwise constant
function is constant. For the analogue of Theorem 2.3 in the latter case, the
splitting theorem again gives an algebra $R$ in an $\mathbb R$-linear tensor
category of vector bundles with connection along the leaves of the foliated
manifold. The fibres of $R$ are isomorphic to ${\mathbb R}[G]$ for a finite,
but not necessarily discrete, algebraic group $G$ over $\mathbb R$, and the
points of the required étale cover are the $\mathbb R$-points of their
spectra.

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