The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# First Measurement of Energy-Dependent Inclusive Muon Neutrino Charged-
Current Cross Sections on Argon with the MicroBooNE Detector

P. Abratenko  Tufts University, Medford, MA, 02155, USA  R. An  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  J. Anthony  University
of Cambridge, Cambridge CB3 0HE, United Kingdom  L. Arellano  The University
of Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi  University of
Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv University, Tel Aviv,
Israel, 69978  S. Balasubramanian  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  B. Baller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  C. Barnes  University of Michigan,
Ann Arbor, MI, 48109, USA  G. Barr  University of Oxford, Oxford OX1 3RH,
United Kingdom  V. Basque  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Bathe-Peters  Harvard University, Cambridge, MA 02138, USA
O. Benevides Rodrigues  Syracuse University, Syracuse, NY, 13244, USA  S.
Berkman  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
A. Bhanderi  The University of Manchester, Manchester M13 9PL, United Kingdom
A. Bhat  Syracuse University, Syracuse, NY, 13244, USA  M. Bishai  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster
University, Lancaster LA1 4YW, United Kingdom  T. Bolton  Kansas State
University (KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University,
Cambridge, MA 02138, USA  L. Camilleri  Columbia University, New York, NY,
10027, USA  D. Caratelli  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  I. Caro Terrazas  Colorado State University, Fort
Collins, CO, 80523, USA  F. Cavanna  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  G. Cerati  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  Y. Chen  Universität Bern, Bern
CH-3012, Switzerland  D. Cianci  Columbia University, New York, NY, 10027, USA
J. M. Conrad  Massachusetts Institute of Technology (MIT), Cambridge, MA,
02139, USA  M. Convery  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  L. Cooper-Troendle  Wright Laboratory, Department of Physics, Yale
University, New Haven, CT, 06520, USA  J. I. Crespo-Anadón  Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid
E-28040, Spain  M. Del Tutto  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  S. R. Dennis  University of Cambridge, Cambridge CB3
0HE, United Kingdom  P. Detje  University of Cambridge, Cambridge CB3 0HE,
United Kingdom  A. Devitt  Lancaster University, Lancaster LA1 4YW, United
Kingdom  R. Diurba  University of Minnesota, Minneapolis, MN, 55455, USA  R.
Dorrill  Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  K.
Duffy  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
S. Dytman  University of Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly
University of Southern Maine, Portland, ME, 04104, USA  A. Ereditato
Universität Bern, Bern CH-3012, Switzerland  J. J. Evans  The University of
Manchester, Manchester M13 9PL, United Kingdom  R. Fine  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  G. A. Fiorentini Aguirre  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  R.
S. Fitzpatrick  University of Michigan, Ann Arbor, MI, 48109, USA  B. T.
Fleming  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  N. Foppiani  Harvard University, Cambridge, MA 02138, USA  D.
Franco  Wright Laboratory, Department of Physics, Yale University, New Haven,
CT, 06520, USA  A. P. Furmanski  University of Minnesota, Minneapolis, MN,
55455, USA  D. Garcia-Gamez  Universidad de Granada, Granada E-18071, Spain
S. Gardiner  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  G. Ge  Columbia University, New York, NY, 10027, USA  S. Gollapinni
University of Tennessee, Knoxville, TN, 37996, USA  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  O. Goodwin  The University of
Manchester, Manchester M13 9PL, United Kingdom  E. Gramellini  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  P. Green  The
University of Manchester, Manchester M13 9PL, United Kingdom  H. Greenlee
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W. Gu
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  R. Guenette
Harvard University, Cambridge, MA 02138, USA  P. Guzowski  The University of
Manchester, Manchester M13 9PL, United Kingdom  L. Hagaman  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  O. Hen
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  C.
Hilgenberg  University of Minnesota, Minneapolis, MN, 55455, USA  G. A.
Horton-Smith  Kansas State University (KSU), Manhattan, KS, 66506, USA  A.
Hourlier  Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139,
USA  R. Itay  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA
C. James  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
X. Ji  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  L. Jiang
Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA  J. H.
Jo  Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  R. A. Johnson  University of Cincinnati, Cincinnati, OH, 45221,
USA  Y.-J. Jwa  Columbia University, New York, NY, 10027, USA  D. Kalra
Columbia University, New York, NY, 10027, USA  N. Kamp  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  N. Kaneshige
University of California, Santa Barbara, CA, 93106, USA  G. Karagiorgi
Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Kobilarcik
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  I.
Kreslo  Universität Bern, Bern CH-3012, Switzerland  I. Lepetic  Rutgers
University, Piscataway, NJ, 08854, USA  K. Li  Wright Laboratory, Department
of Physics, Yale University, New Haven, CT, 06520, USA  Y. Li  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  K. Lin  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  B. R. Littlejohn  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  W. C. Louis  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  X. Luo  University of
California, Santa Barbara, CA, 93106, USA  K. Manivannan  Syracuse University,
Syracuse, NY, 13244, USA  C. Mariani  Center for Neutrino Physics, Virginia
Tech, Blacksburg, VA, 24061, USA  D. Marsden  The University of Manchester,
Manchester M13 9PL, United Kingdom  J. Marshall  University of Warwick,
Coventry CV4 7AL, United Kingdom  D. A. Martinez Caicedo  South Dakota School
of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  K. Mason  Tufts
University, Medford, MA, 02155, USA  A. Mastbaum  Rutgers University,
Piscataway, NJ, 08854, USA  N. McConkey  The University of Manchester,
Manchester M13 9PL, United Kingdom  V. Meddage  Kansas State University (KSU),
Manhattan, KS, 66506, USA  T. Mettler  Universität Bern, Bern CH-3012,
Switzerland  K. Miller  University of Chicago, Chicago, IL, 60637, USA  J.
Mills  Tufts University, Medford, MA, 02155, USA  K. Mistry  The University of
Manchester, Manchester M13 9PL, United Kingdom  A. Mogan  University of
Tennessee, Knoxville, TN, 37996, USA  T. Mohayai  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. Moon  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  M. Mooney  Colorado State
University, Fort Collins, CO, 80523, USA  A. F. Moor  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Mousseau  University of
Michigan, Ann Arbor, MI, 48109, USA  M. Murphy  Center for Neutrino Physics,
Virginia Tech, Blacksburg, VA, 24061, USA  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  R. K. Neely  Kansas State
University (KSU), Manhattan, KS, 66506, USA  D. A. Newmark  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  J. Nowak  Lancaster
University, Lancaster LA1 4YW, United Kingdom  M. Nunes  Syracuse University,
Syracuse, NY, 13244, USA  O. Palamara  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  V. Paolone  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  A. Papadopoulou  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  V. Papavassiliou  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  S. F. Pate  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  N. Patel  Lancaster
University, Lancaster LA1 4YW, United Kingdom  A. Paudel  Kansas State
University (KSU), Manhattan, KS, 66506, USA  Z. Pavlovic  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv
University, Tel Aviv, Israel, 69978  I. D. Ponce-Pinto  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  S. Prince
Harvard University, Cambridge, MA 02138, USA  X. Qian  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V. Radeka  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Rafique  Kansas State
University (KSU), Manhattan, KS, 66506, USA  M. Reggiani-Guzzo  The University
of Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New Mexico State
University (NMSU), Las Cruces, NM, 88003, USA  L. C. J. Rice  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  L. Rochester  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Rodriguez Rondon  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  M.
Rosenberg  University of Pittsburgh, Pittsburgh, PA, 15260, USA  M. Ross-
Lonergan  Columbia University, New York, NY, 10027, USA  G. Scanavini  Wright
Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA
D. W. Schmitz  University of Chicago, Chicago, IL, 60637, USA  A. Schukraft
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W.
Seligman  Columbia University, New York, NY, 10027, USA  M. H. Shaevitz
Columbia University, New York, NY, 10027, USA  R. Sharankova  Tufts
University, Medford, MA, 02155, USA  J. Shi  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  J. Sinclair  Universität Bern, Bern
CH-3012, Switzerland  A. Smith  University of Cambridge, Cambridge CB3 0HE,
United Kingdom  E. L. Snider  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  M. Soderberg  Syracuse University, Syracuse, NY,
13244, USA  S. Söldner-Rembold  The University of Manchester, Manchester M13
9PL, United Kingdom  P. Spentzouris  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  J. Spitz  University of Michigan, Ann Arbor,
MI, 48109, USA  M. Stancari  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  J. St. John  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  T. Strauss  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  K. Sutton  Columbia University, New
York, NY, 10027, USA  S. Sword-Fehlberg  New Mexico State University (NMSU),
Las Cruces, NM, 88003, USA  A. M. Szelc  University of Edinburgh, Edinburgh
EH9 3FD, United Kingdom  W. Tang  University of Tennessee, Knoxville, TN,
37996, USA  K. Terao  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  C. Thorpe  Lancaster University, Lancaster LA1 4YW, United Kingdom
D. Totani  University of California, Santa Barbara, CA, 93106, USA  M. Toups
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y.-T.
Tsai  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  M. A.
Uchida  University of Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  W. Van De
Pontseele  University of Oxford, Oxford OX1 3RH, United Kingdom  Harvard
University, Cambridge, MA 02138, USA  B. Viren  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern, Bern CH-3012,
Switzerland  H. Wei  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  Z. Williams  University of Texas, Arlington, TX, 76019, USA  S. Wolbers
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T.
Wongjirad  Tufts University, Medford, MA, 02155, USA  M. Wospakrik  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  K. Wresilo
University of Cambridge, Cambridge CB3 0HE, United Kingdom  N. Wright
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  W. Wu
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E.
Yandel  University of California, Santa Barbara, CA, 93106, USA  T. Yang
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G.
Yarbrough  University of Tennessee, Knoxville, TN, 37996, USA  L. E. Yates
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  H. W.
Yu  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  G. P. Zeller
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J.
Zennamo  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
C. Zhang  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We report a measurement of the energy-dependent total charged-current cross
section $\sigma\left(E_{\nu}\right)$ for inclusive muon neutrinos scattering
on argon, as well as measurements of flux-averaged differential cross sections
as a function of muon energy and hadronic energy transfer ($\nu$). Data
corresponding to 5.3$\times$1019 protons on target of exposure were collected
using the MicroBooNE liquid argon time projection chamber located in the
Fermilab Booster Neutrino Beam with a mean neutrino energy of approximately
0.8 GeV. The mapping between the true neutrino energy $E_{\nu}$ and
reconstructed neutrino energy $E^{rec}_{\nu}$ and between the energy transfer
$\nu$ and reconstructed hadronic energy $E^{rec}_{had}$ are validated by
comparing the data and Monte Carlo (MC) predictions. In particular, the
modeling of the missing hadronic energy and its associated uncertainties are
verified by a new method that compares the $E^{rec}_{had}$ distributions
between data and an MC prediction after constraining the reconstructed muon
kinematic distributions, energy and polar angle, to those of data. The success
of this validation gives confidence that the missing energy in the MicroBooNE
detector is well-modeled and underpins first-time measurements of both the
total cross section $\sigma\left(E_{\nu}\right)$ and the differential cross
section $d\sigma/d\nu$ on argon.

††preprint: APS/123-QED

Current and next-generation precision neutrino oscillation experiments aim to
answer several critical questions in particle physics [1] by: i) searching for
CP violation in the lepton sector [2, 3],ii) determining the neutrino mass
ordering [4], and iii) searching for light sterile neutrinos [5]. For this
purpose, the SBN [6] and DUNE [7, 8] experiments employ liquid argon time
projection chambers (LArTPCs) [9, 10, 11, 12], a tracking calorimeter that
enables excellent neutrino flavor identification and neutrino energy
($E_{\nu}$) reconstruction in the GeV energy range [13]. These experiments are
designed to measure the neutrino flavor oscillations as a function of
$E_{\nu}$, which requires a good understanding of the neutrino energy
spectrum, neutrino-argon interaction cross sections [14], and LArTPC detector
response. High-precision measurements of $\nu$-Ar cross sections, particularly
those related to energy reconstruction, are of paramount importance.

While historical accelerator-based neutrino experiments often reported
$E_{\nu}$-dependent cross sections [15, 16], recent experiments tend to limit
cross-section measurements to the directly observable lepton and/or hadron
kinematics [17]. This paradigm shift was triggered by concerns that quantities
not directly measurable in detectors (e.g. the missing hadronic energy of the
interaction from undetected neutral particles) may not be correctly modeled in
simulations, which is of particular concern in a broad-band neutrino beam. In
this letter, we demonstrate that the MicroBooNE tune model [18] (based on
GENIE-v3 [19]) of missing energy with its associated uncertainty can be
validated with inclusive muon neutrino charged-current ($\nu_{\mu}$CC)
interactions from the the MicroBooNE detector [20]. After constraining the
lepton kinematics distributions of Monte Carlo (MC) to those of data, the
comparison of reconstructed hadronic energy $E^{rec}_{had}$ distributions
between data and the updated MC prediction reveals whether the model is able
to describe the relationship between the lepton kinematics and the visible
hadronic energy. This procedure validates whether the missing hadronic energy
is sufficiently modelled given the prior knowledge of the neutrino flux and
detector effects. This new procedure enables a first measurement of the
differential cross section as a function of the energy transfer to the argon
$d\sigma/d\nu$. Together with the differential cross section as a function of
the muon energy ($d\sigma/dE_{\mu}$), the $E_{\nu}$-dependent cross sections
are extracted. These data could be used to isolate problems for low-$E_{\nu}$
cross sections and to reduce modelling uncertainties for the low-$\nu$ method
[21, 22, 23] to constrain the shape of neutrino energy spectrum in future
experiments.

The MicroBooNE detector is a 10.4$\times$2.6$\times$2.3 m3 LArTPC. It consists
of approximately 85 ton of liquid Ar in the active TPC volume for ionization
charge detection, along with 32 photomultiplier tubes (PMTs) [24] for
scintillation light detection. This work makes use of a data set corresponding
to an exposure of 5.3$\times 10^{19}$ protons on target (POT) from the Booster
Neutrino Beamline (BNB), which produces a neutrino flux with an estimated
93.6% $\nu_{\mu}$ purity [25] and a mean $E_{\nu}$ of 0.8 GeV. At these
energies, $\nu$-Ar interactions are dominated by quasielastic and meson-
exchange current interactions as well as resonant pion productions, and the
final-state hadrons consist mostly of protons and neutrons with some charged
and neutral pions. The $\mathcal{O}(1)$ MeV energy threshold [26] of LArTPC
allows detecting these particles down to low kinetic energies.

Compared to earlier work [27], this measurement incorporates an improved TPC
detector simulation and signal processing procedure [28, 29], the Wire-Cell
tomographic event reconstruction [30, 31], and a many-to-many TPC-charge to
PMT-light matching algorithm for cosmic-ray rejection [31]. In particular, the
“generic neutrino detection” [32, 33], which limits the cosmic-ray muon
backgrounds to below 15% at over 80% $\nu_{\mu}$CC selection efficiency, is
used as a pre-selection. The $\nu_{\mu}$CC event selection is further improved
using a set of pattern recognition techniques, including i) neutrino vertex
identification, ii) track/shower topology separation, iii) particle
identification, and iv) particle flow reconstruction in the Wire-Cell
reconstruction package [34]. Since many of the analysis details in this work
including the event reconstruction, event selection, the overall model
prediction along with its systematic uncertainties, and the model validation
are in common with those in searching for an anomalous low-energy excess in
inclusive charged-current $\nu_{e}$ channel that was documented in Ref. [35],
they are only briefly reviewed in this letter.

First, the reconstructed neutrino vertex is required to be inside a fiducial
volume, defined to be 3 cm inside the effective detector boundary [33].
Second, a set of dedicated background taggers are constructed to further
reject residual muon backgrounds that entered the detector from outside based
on directional information. Finally, neutral-current (NC) events are
substantially reduced by requiring a reconstructed primary muon candidate to
be longer than 5 cm. Some limited charged pion rejection is achieved by
detecting large-angle scattering in reconstructed track trajectories. Using
input variables from the background taggers, a multivariate classifier is
constructed using the modern boosted decision tree (BDT) library XGBoost [36]
that yields a $\nu_{\mu}$CC selection with an estimated 92% purity and 68%
efficiency [35]. In total, 11528 $\nu_{\mu}$CC candidates are selected and
used for cross-section extraction. About 1/3 of the events are fully contained
(FC) and 2/3 are partially contained (PC). Here, the FC events are defined to
be events with their main TPC cluster [31] fully contained within the fiducial
volume [33] and PC events are mostly because of exiting muons.

\begin{overpic}[width=216.81pt]{xspaper_fig1_v2.pdf}
\put(11.0,97.0){{MicroBooNE $\mathsf{5.3\times 10^{19}}$POT}} \end{overpic}
Figure 1: (Top) Distribution of the selected FC $\nu_{\mu}$CC events as a
function of reconstructed neutrino energy. The inset figure is the smearing
matrix from true neutrino energy to reconstructed neutrino energy. (Middle)
Data/prediction ratio. The pink band represents the total uncertainty
(statistical and systematic) of the MC prediction. (Bottom) Selection
efficiency of the $\nu_{\mu}$CC events in the fiducial volume as a function of
true neutrino energy. At high neutrino energy, muons are more likely to exit
the TPC, which leads an increase (decrease) in the efficiency of PC (FC)
samples.

Three methods are used to reconstruct the energy of tracks and electromagnetic
(EM) showers [34, 35]: i) The energy of a stopped charged particle track can
be estimated by its travel range using the NIST PSTAR database [37]. ii) The
kinetic energy of a charged particle track can be estimated by integrating
over the reconstructed energy loss per unit length $dE/dx$, which is
calculated from the measured $dQ/dx$ (ionization charge per unit length) using
a recombination model [38]. iii) The energy of an EM shower can be estimated
calorimetrically by scaling the total reconstructed charge of the EM shower
with a factor of 2.50, which is derived from simulation and includes the bias
in the reconstructed charge [31] and the average recombination factor. This
factor is validated with the reconstructed invariant mass of the neutral pion
[39]. For stopping tracks with trajectories longer than 4 cm, the range method
is used to estimate the energy. For short tracks ($<$4 cm), tracks exiting the
detector, tracks with “wiggled” topology [34] (e.g. low-energy electrons), and
muon tracks with identified $\delta$ rays, the recombination method is used to
estimate its kinetic energy.

The reconstructed neutrino energy $E^{rec}_{\nu}$ per event is estimated by
summing the kinetic energies of each reconstructed (visible) final-state
particle. For each reconstructed muon, charged pion, or electron candidate,
its mass is added to the energy reconstruction. An average binding energy of
8.6 MeV [40] was added for each proton identified. Figure 1 shows the FC
$\nu_{\mu}$CC distribution as a function of reconstructed neutrino energy, the
selection efficiency as a function of true neutrino energy, and the smearing
matrix between $E^{rec}_{\nu}$ and $E_{\nu}$ according to the Monte Carlo
simulation. The predicted energy resolution using the MicroBooNE MC for FC
$\nu_{\mu}$CC events is $\sim$10% for muon energy, $\sim$20% for neutrino
energy, and $\sim$30-50% for hadronic energy. The hadronic energy resolution
is dominated by the missing hadronic energy and imperfect event
reconstruction. For events well reconstructed, the resolution of the
reconstructed visible hadronic energy approaches $\sim$10%. Among all events,
the average bias (towards low energy) of $E^{rec}_{\nu}$ for FC $\nu_{\mu}$CC
events is less than 10% for $E_{\nu}<$ 800 MeV and increases to $\sim$25% at
$E_{\nu}=$ 2.5 GeV.

The total and differential cross sections are extracted using the Wiener-SVD
unfolding method [41] as follows:

$M_{i}-B_{i}=\sum_{j}R_{ij}\cdot
S_{j}=\sum_{j}\widetilde{\Delta}_{ij}\cdot\widetilde{F}_{j}\cdot S_{j}.$ (1)

$M_{i}$ is the measured number of events in bin $i$ of the reconstructed
energy space, and $B_{i}$ is the expected number of backgrounds.
$R_{ij}=\widetilde{\Delta}_{ij}\cdot\widetilde{F}_{j}$ is the overall response
matrix. $S_{j}$, to be extracted, is the average (differential) cross section
in bin $j$ of the true energy, weighted by the nominal $\nu_{\mu}$ neutrino
flux, which is tabulated in Ref. [27]. This definition of $S_{j}$ with the
nominal neutrino flux coincides with a recommendation from Ref. [42] in
addressing a concern on the treatment of neutrino flux uncertainty.
$\widetilde{\Delta}_{ij}$, the ratio between the selected number of events in
reconstructed energy bin $i$ that originate from the true energy bin $j$ and
the generated number of events in bin $j$, is calculated using central-value
MC. This encapsulates both the smearing between reconstructed and true space
and the efficiency. $\widetilde{F}_{j}$ is a constant that is calculated with
the POT, number of Ar nuclei, the integrated nominal $\nu_{\mu}$ flux in bin
$j$, and the bin width (for differential cross sections only).

The Wiener-SVD unfolding is performed based on a

$\chi^{2}=\left(\bm{M}-\bm{B}-\bm{R}\cdot\bm{S}\right)^{T}\cdot\bm{V}^{-1}\cdot\left(\bm{M}-\bm{B}-\bm{R}\cdot\bm{S}\right)$
(2)

test statistics and an additional regularization constructed from a Wiener
filter [41]. $\bm{V}$ is the covariance matrix on the measured number of
events in the reconstructed energy bins, encoding the statistical and
systematic uncertainties for both signal and background events. Statistical
uncertainties on the data are calculated following the combined Neyman-Pearson
procedure [43].

The covariance matrix also includes several systematic uncertainties. The
neutrino flux model uncertainty (5-15%) follows the work in Ref. [27]. It
includes effects from hadron production of $\pi^{+}$, $\pi^{-}$, $K^{+}$,
$K^{-}$, and $K^{0}_{L}$, together with total, inelastic, and quasielastic
cross sections of pion and nucleon re-scattering on beryllium and aluminum. In
addition, modeling of the horn current distribution and calibration is
included. The neutrino-argon interaction cross section model uncertainties
($\sim$20%) are described in Ref. [18]. Particularly, the uncertainties
associated with the hadronic interactions, which are important in modeling
missing energy, are conservatively estimated: the proton to neutron charge
exchange and the proton knockout have 50% and 20% uncertainties, respectively
[44, 45]. The uncertainties on the GEANT4 models [46] used to simulate
secondary interactions of protons and charged pions outside the target nucleus
($\sim$1.5%) follows Ref. [47]. These uncertainties on the flux, cross
section, and GEANT4 models are estimated using a _multisim_ technique [48] in
which parameters that govern interaction models are simultaneously varied in
generating hundreds of universes to construct covariance matrices.

The detector response uncertainty follows the work in Ref. [49], considering
the effects of variations in the TPC waveform, light yield and propagation,
space charge effect [50, 51], and ionization recombination model. For each
source, the same set of MC interactions are re-simulated through the detector
response simulation with a $1\sigma$ change to the corresponding detector
model parameter. The differences in the selected number of events between the
modified and original simulations are used to construct a covariance matrix
with a bootstrapping [52] procedure. The uncertainty of modeling the “dirt”
events that originate outside the cryostat follows the work in Ref. [35]. The
statistical uncertainty of the Monte-Carlo sample is treated using the methods
described in Ref. [53]. The uncertainties on the POT (2% based on _in-situ_
proton flux measurements [25]) and the number of target nuclei ($\sim$1%) are
also included.

Given Eq. (1), the uncertainties on the neutrino flux, GEANT4 model, detector
model, and POT enter through $B_{i}$ and the numerator of
$\widetilde{\Delta}_{ij}$. “Dirt” uncertainties enter through $B_{i}$. In
comparison, the cross section uncertainty enters through $B_{i}$ and both
numerator and denominator of $\widetilde{\Delta}_{ij}$. Although the
uncertainty on the predicted inclusive cross section is $\sim$20%, it is
reduced to $\sim$5% because of the cancellation between numerator and
denominator of $\widetilde{\Delta}_{ij}$.

\begin{overpic}[width=208.13574pt]{canv_spectra_GoF_total_40001_Emuon_PC_by_FC.pdf}
\put(43.0,4.0){\footnotesize{a)}} \put(12.0,68.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}}
\end{overpic}\begin{overpic}[width=208.13574pt]{canv_spectra_GoF_total_40002_theta_by_Emuon.pdf}
\put(44.0,3.5){\footnotesize{b)}} \put(50.0,36.0){\footnotesize{{FC}}}
\put(92.0,36.0){\footnotesize{{PC}}} \put(12.0,68.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}}
\end{overpic}\begin{overpic}[width=208.13574pt]{canv_spectra_GoF_total_40003_Ehad_FCbyEmuonTheta.pdf}
\put(43.0,4.0){\footnotesize{c)}} \put(12.0,68.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}} \end{overpic}

Figure 2: Data are compared with MC predictions as a function of: a)
reconstructed muon energy $E^{rec}_{\mu}$ for the partially contained (PC)
sample. The MC prediction after applying constraints from the fully-contained
(FC) sample in $E^{rec}_{\mu}$ is shown. The last bin represents all events
with $E^{rec}_{\mu}>1.5$ GeV. The blue (red) points represent the ratio
between data and the MC prediction with (without) constraint, and the bands
with same colors depict the total (include statistical) uncertainty of the MC
prediction. b) reconstructed $\cos\theta^{rec}_{\mu}$ for the FC (first half)
and PC (second half) sample. The MC prediction after applying constraints on
both FC and PC samples in $E^{rec}_{\mu}$ is shown. c) reconstructed hadronic
energy $E^{rec}_{had}$ for the FC sample. The MC prediction after applying
constraints on muon kinematics ($E^{rec}_{\mu}$ and $\cos\theta^{rec}_{\mu}$)
is shown. The last bin represents all events with $E^{rec}_{had}>1.5$ GeV.

A prior condition of using the Wiener-SVD unfolding method to extract cross
sections is that the data must be well-described by the overall model
prediction within its uncertainties. In Fig. 1 and Fig. 2, data and simulation
are shown for key reconstructed kinematic variables including i) neutrino
energy $E^{rec}_{\nu}$, ii) muon energy $E^{rec}_{\mu}$, iii) cosine of muon
polar angle $\cos\theta^{rec}_{\mu}$, and iv) hadronic energy $E^{rec}_{had}$.
The compatibility between the data and prediction is demonstrated
quantitatively by decent $\chi^{2}/\mathrm{ndf}$ values (ndf is the number of
degrees of freedom) with corresponding p-values larger than 0.05 considering
full uncertainties using the Pearson $\chi^{2}$ [54]. To examine different
components of systematic uncertainties, we further utilize the conditional
covariance matrix formalism [55] to adjust the model prediction and reduce its
uncertainties by applying constraints from data. Figure 2a) shows the
comparison of the $E^{rec}_{\mu}$ distribution for PC $\nu_{\mu}$CC in data to
that of the model prediction after applying constraints from the FC
$E^{rec}_{\mu}$ events. While the uncertainties are largely reduced, there is
only a small change to $\chi^{2}/\mathrm{ndf}$. The data and constrained model
agree within uncertainties, verifying the modeling of the invisible energy of
muons outside the active detector volume for PC events. Figure 2b) shows the
comparison of the $\cos\theta^{rec}_{\mu}$ distribution for both FC and PC
$\nu_{\mu}$CC candidates in data with the model prediction after applying a
constraint from the $E^{rec}_{\mu}$ distributions of the same set of
$\nu_{\mu}$CC candidate events. Compared to the previous case, the correlated
statistical uncertainties between the $\cos\theta^{rec}_{\mu}$ distributions
and the $E^{rec}_{\mu}$ distributions are estimated with a bootstrapping
procedure. While the uncertainties are significantly reduced after applying
the constraint, the change to $\chi^{2}/\mathrm{ndf}$ is small, showing well-
modeled muon kinematics.

\begin{overpic}[width=142.65869pt]{ccinc_svd_enu.pdf}
\put(50.0,-3.0){\footnotesize{a)}} \put(15.0,88.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}}
\end{overpic}\begin{overpic}[width=142.65869pt]{ccinc_svd_emu.pdf}
\put(50.0,-3.0){\footnotesize{b)}} \put(16.0,88.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}}
\end{overpic}\begin{overpic}[width=142.65869pt]{ccinc_svd_nu.pdf}
\put(50.0,-3.0){\footnotesize{c)}} \put(15.0,88.0){{MicroBooNE
$\mathsf{5.3\times 10^{19}}$POT}} \end{overpic}

Figure 3: a) The extracted $\nu_{\mu}$CC inclusive scattering cross section
per nucleon divided by the bin-center neutrino energy, as a function of
neutrino energy. b) The measured $\nu_{\mu}$CC differential cross section per
nucleon as a function of muon energy $d\sigma/dE_{\mu}$. c) The measured
$\nu_{\mu}$CC differential cross section per nucleon as a function of energy
transfer $d\sigma/d\nu$. Various model predictions are compared to all three
measurements (see text for details).

We will examine the modeling of the mapping between the reconstructed energy
of the hadronic system $E^{rec}_{had}$ and the energy transfer to the argon
nucleus $\nu=E_{\nu}-E_{\mu}$ after taking into account the muon results. The
mapping of $E^{rec}_{had}$ to $\nu$ (or $E^{rec}_{\nu}$ to true $E_{\nu}$)
relies on the overall cross section model to correct for the missing energy
going into undetected neutrons, low-energy photons and other particles below
the detection threshold. To validate the model, we examine the $E^{rec}_{had}$
distribution for the FC $\nu_{\mu}$CC candidates in data with that of the
model prediction after applying constraints from two one-dimensional
distributions in muon kinematics: $E^{rec}_{\mu}$ and $\cos\theta^{rec}_{\mu}$
in Fig. 2c). After applying constraints, the uncertainties on the model
prediction for $E^{rec}_{had}$ are significantly reduced because of the
cancellation of common systematic uncertainties, such as neutrino flux. At the
lowest energies, it reduces from 20% to 5%. Nevertheless, the
$\chi^{2}/\mathrm{ndf}$ values still yields p-value above 0.5, indicating that
the model describes the relationship between $E^{rec}_{had}$ and
$E^{rec}_{\mu}$ well within its uncertainty. In particular, the difference
between the data and the model prediction in the first three bins of
$E^{rec}_{had}$ is significantly reduced after applying the constraints. This
test further validates that the modeling of the missing hadronic energy can
describe data within its associated uncertainty. We note the conditional
covariance matrix formalism, which is used to update the MC predictions and
their uncertainties given the data constraints (more details can be found in
Ref. [35]), is only used in validating the overall model, and is not used in
extracting cross sections through the unfolding procedure. With fake data, we
show that the $\chi^{2}/\mathrm{ndf}$ has a significant increase with a shift
of $\sim$15% in the hadronic energy fraction allocated to protons (mimicking a
variation of the proton-inelastic cross section), and this procedure is also
able to distinguish between two GENIE models (see Supplemental Material [56]).
In addition, the model validation procedure is shown to be much more sensitive
to detect an insufficient input model compared to the extracted cross
sections.

With the overall model validated, the total and differential cross sections
per nucleon are extracted. The binning of the unfolded results is chosen by
considering the energy resolution and the number of samples in the true space.
Considering both FC and PC samples, the total cross section divided by the
bin-center neutrino energy is shown as a function of neutrino energy in Fig.
3a), where the bin-center is calculated as the flux-weighted average neutrino
energy. Excluding the PC sample does not change the overall behaviour of the
cross sections, but increases their uncertainties for neutrino energy above
1.2 GeV modestly. Besides the nominal cross-section model used in the
“MicroBooNE MC” [18], predictions from GENIE v3.0.6 [44, 19], NuWro 19.02.01
[57], NEUT 5.4.0.1 [58], and GiBUU 2019.08 [59] after applying the Wiener
filter are quantitatively compared with the measurement through calculating
$\chi^{2}/{\rm ndf}$ with the uncertainty covariance matrix obtained from the
unfolding procedure. Note that these comparisons only incorporate the central
predictions from various generators without their theoretical uncertainties,
which are particularly important in constructing predictions in analysis. The
central predictions of GENIE v3 and NuWro are disfavored compared to the other
three. Particularly, the “MicroBooNE MC” (tuned GENIE-v3 model [18]) has
better agreement than GENIE v3.0.6, given the tuned GENIE-v3 model is
constructed by fitting T2K data [60] in a similar energy range.

Figure 3b) and c) show the flux-averaged differential cross sections as a
function of muon energy ($d\sigma/dE_{\mu}$) and energy transfer to the argon
nucleus ($d\sigma/d\nu$). The same set of model predictions are compared to
these measurements. The model comparison of $d\sigma/dE_{\mu}$ shows a shape
agreement with most models, although the normalization predictions differ. The
central predictions of GENIE v3 and NuWro are more disfavored. The model
predictions in $d\sigma/d\nu$ show large variations, particularly in the low
energy transfer ($\nu$) region, where the shape difference contributes
considerably to the $\chi^{2}/{\rm ndf}$ given the correlations in the
uncertainty covariance matrix. The central prediction from GiBUU has the best
agreement with data in the low $\nu$ region, but is systematically lower than
data at high $\nu$ region, which could be originated from an underestimation
of the cross sections in the nucleon resonance region beyond $\Delta$.
Considering all three cross-section results, the GiBUU prediction has the best
agreement with acceptable $\chi^{2}/{\rm ndf}$ values, while the performance
of the NEUT prediction is comparable. The central predictions of the other
three models show larger disagreement.

In summary, we present a measurement of cross section as a function of the
neutrino energy based on data from a broad-band neutrino beam. We report the
nominal-flux weighted total inclusive $\nu_{\mu}$CC cross sections
$\sigma\left(E_{\nu}\right)$, and the nominal flux-averaged differential cross
sections as a function of muon energy $d\sigma/dE_{\mu}$ and energy transfer
$d\sigma/d\nu$ using the Wiener-SVD unfolding method [41]. A new procedure
based on the conditional covariance matrix formalism [55] and the
bootstrapping method [52] is used to validate the model of missing energies,
which enables the first measurement of $d\sigma/d\nu$ on argon and
significantly adds value to the measurement of the total cross section as
function of neutrino energy $\sigma\left(E_{\nu}\right)$. These results
provide a detailed way to compare data and calculations beyond what is
possible with existing flux-averaged total cross section results. With
additional accumulated data statistics (up to 1.2$\times 10^{21}$ POT from
BNB) in the MicroBooNE detector, additional neutrino cross-section
measurements are expected that will lead to further model development and
generator improvements for neutrino scattering in argon.

###### Acknowledgements.

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); and The European
Union’s Horizon 2020 Marie Sklodowska-Curie Actions. Additional support for
the laser calibration system and cosmic ray tagger was provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses,
without whom this work would not have been possible.

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Fixity of elusive groups and the polycirculant conjecture

Majid Arezoomand

University of Larestan, Larestan, 74317-16137, Iran

<EMAIL_ADDRESS>
> Abstract. Let $G\leq{\rm Sym}(\Omega)$ be transitive. Then $G$ is called
> elusive on $\Omega$ if it has no fixed point free element of prime order.
> The $2$-closure of $G$, denoted by $G^{(2),\Omega}$, is the largest subgroup
> of ${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$ are the same
> orbits of $G$. $G$ is called $2$-closed on $\Omega$ if $G=G^{(2),\Omega}$.
> The polycirculant conjecture states that there is no $2$-closed elusive
> group. In this paper, we study the fixity of elusive groups, where the
> fixity of $G$ is the maximal number of fixed points of a non-trivial element
> of $G$. In particular, we prove that there is no $2$-closed elusive solvable
> group of fixity at most $5$, a partial answer to the polycirculant
> conjecture.

Keywords: Fixity, elusive group, polycirculant conjecture.

2000 Mathematics subject classification:  20B25, 05E18.

##  1\. Introduction

The concept of fixity was introduced by Rosen in 1980 to study of permutation
groups of prime power order. Let $G$ be a permutation group on a set $\Omega$.
We say that $G$ has fixity $f=f(G)$ if no non-trivial element of $G$ fixes
more than $f$ letters, and there is a non-trivial element of $G$ fixing
exactly $f$ letter [11].

Let $G$ be a transitive permutation group. Then $f(G)=0$ if and only if $G$ is
regular. Furthermore, $f(G)=1$ if and only if $G$ is a Frobenius group. For
some applications of the fixity of permutation groups, we refer the reader to
[8, 9, 11].

A permutation $g$ of a set $\Omega$ is said to be a derangement if it has no
fixed-point on $\Omega$, equivalently for each $\alpha\in\Omega$,
$\alpha^{g}\neq\alpha$. A transitive permutation group $G\leq{\rm
Sym}(\Omega)$ is called elusive on $\Omega$ if it has no fixed point free
element of prime order. Let $G$ be a permutation group on a set $\Omega$. The
$2$-closure of $G$, denoted by $G^{(2),\Omega}$, is the largest subgroup of
${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$ are the same orbits
of $G$. Then $G$ is called $2$-closed on $\Omega$ if $G=G^{(2),\Omega}$.

In 1981, Marušič asked whether there exists a vertex-transitive digraph
without a non-identity automorphism having all of its orbits of the same
length [10, Problem 2.4]. In 1988, independently, the above problem was again
proposed by Jordan [6]. In the 15th British combinatorial conference, in 1995,
Klin proposed a more general question in the context of $2$-closed groups [7]:
“Is there a $2$-closed transitive permutation group containing no fixed-point-
free element of prime order?” After decades, not only has a positive answer to
Klin’s question not been found, based on the evidence, mathematicians
conjecture that there is no positive answer to this question. This conjecture
is known as polycirculant conjecture. Equivalently, the polycirculant
conjecture states that no $2$-closed transitive permutation group is elusive.
The conjecture is still open. We refer the reader to [1] to a survey on the
recent results and future directions of the polycirculant conjecture.

In this paper, we study the fixity of elusive groups. Then we prove that every
transitive $2$-closed solvable permutation group of fixity at most $5$
confirms the polycirculant conjecture.

##  2\. Main Results

First we collect some notations we need later. Let $G$ be a finite group and
$\Omega$ be a non-empty set. We denote the center of $G$ and the set of all
prime divisors the order of $G$ by $Z(G)$ and $\pi(G)$, respectively. Also
${\rm Sym}(\Omega)$ denotes the group of all permutations on $\Omega$. Let $G$
acts on $\Omega$ and $\alpha\in\Omega$. We denote the stabilizer of $\alpha$
in $G$ and the orbit of $G$ containing $\alpha$ by $G_{\alpha}$ and
$\alpha^{G}$, respectively. Also ${\rm Fix}_{\Omega}(G)$ denotes the fixed
points of $G$ on $\Omega$, the set of all elements of $\Omega$ which fixes by
all elements of $G$. Furthermore, $G^{\Omega}$ denotes the homomorphic image
of the action of $G$ on $\Omega$ which is a subgroup of ${\rm Sym}(\Omega)$.
Recall that $G$ is called a permutation group on $\Omega$ if $G$ is isomorphic
to $G^{\Omega}$. For the notations and terminology not defined here, we refer
the reader to [2].

Let $G$ be a permutation group on a set $\Omega$. Recall that fixity of $G$ is
the maximal number of fixed points of a non-trivial element of $G$ on
$\Omega$. We start with the following lemma:

######  Lemma 2.1.

Let $G$ be a finite permutation group on $\Omega$ of fixity $f\geq 2$, $p>f$ a
prime and $\alpha\in\Omega$. If $p\in\pi(G_{\alpha})$ then $G_{\alpha}$
contains a Sylow $p$-subgroup of $G$. In particular, if $G$ is transitive and
contains a non-trivial normal $p$-subgroup then $p\notin\pi(G_{\alpha})$.

###### Proof.

Let $p$ be a prime divisor of order of $G_{\alpha}$. Then there exists a Sylow
$p$-subgroup $P$ of $G$ such that $P_{\alpha}\neq 1$. We claim that $Z(P)\leq
G_{\alpha}$. Suppose towards a contradiction, that our claim is not true. Then
there exists $g\in Z(P)$ such that $g\notin G_{\alpha}$. Hence
$\alpha,\alpha^{g},\ldots,\alpha^{g^{p-1}}$ are distinct elements of ${\rm
Fix}_{\Omega}(P_{\alpha})$, because if $\alpha^{g^{i}}=\alpha^{g^{j}}$, for
some $0\leq i<j\leq p-1$, then $g^{j-i}\in G_{\alpha}$ and so $(j-i,p)=1$
implies that $g\in G_{\alpha}$, a contradiction. So $P_{\alpha}\neq 1$ implies
that $p\leq f$, a contradiction. This proves that $Z(P)\leq G_{\alpha}$.

Now we prove that $P\leq G_{\alpha}$. Suppose, by a contrary, that $P\nleq
G_{\alpha}$. Then $x\notin G_{\alpha}$ for some $x\in P$. Hence, by a similar
argument to the above paragraph, $\alpha,\alpha^{x},\ldots,\alpha^{x^{p-1}}$
are distinct. Now $Z(P)\leq G_{\alpha}$ implies that these elements are all in
${\rm Fix}_{\Omega}(Z(P))$. Let $y$ be a non-trivial element of $Z(P)$. Then

$p\leq|{\rm Fix}_{\Omega}(Z(P))|=|\bigcap_{x\in Z(P)}{\rm
Fix}_{\Omega}(x)|\leq|{\rm Fix}_{\Omega}(y)|\leq f,$

which is a contradiction.

Finally, suppose that $G$ is transitive and $N$ is a non-trivial normal
$p$-subgroup of $G$. Then is contained in every Sylow $p$-subgroup of $G$. Now
if $p$ is a divisor of $|G_{\alpha}|$, then by the above paragraph, $N\leq
G_{\alpha}$. Since $G$ is transitive, this implies that $N$ is contained in
any point-stabilizer of $G$ which implies that $N=1$, because $G$ is a
permutation group. This completes the proof. ∎

######  Corollary 2.2.

Let $G$ be an elusive group on $\Omega$ with fixity $f$. If $G$ contains a
non-trivial normal $p$-subgroup then $p\leq f$.

###### Proof.

Suppose, towards a contradiction, that $p>f$. Then by Lemma 2.1,
$p\notin\pi(G_{\alpha})$, where $\alpha\in\Omega$. On the other hand, by [4,
Lemma 2.1], $\pi(G)=\pi(G_{\alpha})$ which is a contradiction. ∎

######  Corollary 2.3.

Let $G$ be an elusive group on $\Omega$ of fixity $f$. Then $f\geq 3$.

###### Proof.

Suppose, towards a contradiction, that $f\leq 2$. Since $f=0$ if and only if
$G$ is regular on $\Omega$, and since regular groups are not elusive, we have
$f=1,2$. On the other hand, $f=1$ if and only if $G$ is a Frobenius group with
Frobenius complement $G_{\alpha}$ on $\Omega$, where $\alpha\in\Omega$. Since
the Frobenius kernel of any finite Frobenius group is a regular subgroup, we
conclude that $f=2$.

Let $\alpha\in\Omega$. Then, by [4, Lemma 2.1], $\pi(G)=\pi(G_{\alpha})$.
Hence, by Lemma 2.1, $|\Omega|=2^{k}$, for some $k\geq 1$, which contradicts
[1, Lemma 2.6]. ∎

######  Lemma 2.4.

Let $G$ be an elusive group of fixity $f\geq 3$ on a finite set $\Omega$. Let
$|\Omega|=p_{1}^{n_{1}}\ldots p_{k}^{n_{k}}$, where $n_{i}\geq 1$ and $k\geq
2$. Then

  * $(i)$

for each $1\leq i\leq k$, $p_{i}\leq f$,

  * $(ii)$

if $p\in\\{p_{1},\ldots,p_{k}\\}$ then every $p$-element in $G$ is either
fixed-point free or fixes $mp$ points, where $1\leq m\leq f/p$.

###### Proof.

$(i)$ Let $\alpha\in\Omega$ and $p$ be a prime divisor of
$|\Omega|=|G:G_{\alpha}|$. By Lemma 2.1, if $p>f$ then $G_{\alpha}$ contains a
Sylow $p$-subgroup of $G$. Hence $p$ can not divide $|\Omega|$, a
contradiction.

$(ii)$ Let $p\in\\{p_{1},\ldots,p_{k}\\}$ and $x\in G$ be a $p$-element. Then
$x$ fixes at most $f$ points. Since the length of any orbit of $\langle
x\rangle$ on $\Omega$ is a power of $p$, by $(i)$, $|\Omega|=|{\rm
Fix}_{\Omega}(\langle x\rangle)|+lp$, for some positive integer $l$. Hence $p$
divides $|{\rm Fix}_{\Omega}(\langle x\rangle)|$. On the other hand, ${\rm
Fix}_{\Omega}(\langle x\rangle)={\rm Fix}_{\Omega}(x)$ which implies that $x$
is either fixed-point free or fixes $mp$ points, where $1\leq m\leq f/p$. ∎

######  Corollary 2.5.

Let $G$ be an elusive group on $\Omega$ with fixity $f$. If $|\Omega|$ is odd,
then $f\geq 5$. In particular, if $|G|$ is odd then $f\geq 5$.

###### Proof.

Since $|\Omega|$ is a divisor of $|G|$, the second part is a direct
consequence of the first part. Suppose, by contrary, that $f\leq 4$. Then by
Corollary 2.3, $f\in\\{3,4\\}$. Now by Lemma 2.4, $|\Omega|=2^{m}3^{n}$, where
$m,n\geq 0$ are integers. On the other hand, by [1, Lemma 2.6], $m,n\neq 0$.
Let $x$ be an element of order $3$ of $G$. Then, by Lemma 2.4, $x$ is fixed-
point free, which is a contradiction. ∎

For a finite group $1\neq G$, we denote the smallest prime divisor of $|G|$ by
$p_{G}$. Then we have the following lemma.

######  Lemma 2.6.

Let $G$ be an elusive group on a finite set $\Omega$ with fixity $f$. Then

$|\Omega|\leq f\min\\{\frac{|H|-1}{p_{H}-1}\mid 1\neq H\unlhd G\\}.$

###### Proof.

Let $H$ be a non-trivial normal subgroup of $G$. Let
$\Omega_{1},\ldots,\Omega_{m}$ be all $H$-orbits on $\Omega$. By [2, Theorem
1.6. A], for $i\neq j$, $H^{\Omega_{i}}$ is permutation isomorphic to
$H^{\Omega_{j}}$. If $H$ acts regularly on one orbit, then it acts regularly
on all of its orbits and so every element of prime order in $H$ must be fixed-
point free which is a contradiction. Hence $H$ does not act regularly on its
orbits. Thus by [11, Lemma 2.6], $|\Omega|\leq f(|H|-1)/(p_{H}-1)$, which
completes the proof. ∎

######  Lemma 2.7.

Let $G$ be an elusive group on a finite set $\Omega$ with fixity $f$. If $G$
has a non-trivial normal abelian subgroup $N$ and $p$ is the smallest prime
divisor of $|N|$, then

  * $(1)$

$|N|\leq pf$,

  * $(2)$

$|\Omega|\leq\frac{f(pf-1)}{p-1}\leq f(2f-1)$.

###### Proof.

It is obvious that the fixity is at most $f$. On the other hand, by the proof
of Lemma 2.6, $N$ has no regular orbit on $\Omega$. Hence, by [11, Lemma 2.7],
$|N|\leq pf$. The second part follows from Lemma 2.6. ∎

######  Corollary 2.8.

Let $G$ be a $2$-closed elusive solvable group of fixity $f$. Then $f\geq 6$.

###### Proof.

Let $G$ be a $2$-closed elusive group on $\Omega$. Suppose that $N$ is a
minimal normal subgroup of $G$. Then $N\cong\mathbb{Z}_{p}^{k}$ for some
$k\geq 1$, where $p$ is a prime.

Suppose, towards a contradiction, that $f\leq 5$. Then, by Lemma 2.7,
$|\Omega|\leq 45$ which contradicts [5, Proposition 6.1]. ∎

######  Corollary 2.9.

Let $G$ be an elusive group of fixity $f$ on $\Omega$ and $N\neq 1$ be an
abelian normal subgroup of $G$ of order $p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots
p_{k}^{n_{k}}$, where $p_{1}<p_{2}<\ldots<p_{k}$ are primes, $n_{i}\geq 1$ and
$k\geq 1$. Then

  * $(1)$

If $k=1$ then $n_{1}\neq 1$. Also $(n_{1},\ldots,n_{k})\neq(1,\ldots,1)$.

  * $(2)$

$p_{1}^{n_{1}+\ldots+n_{k}-1}\leq f$. In particular, $p_{1}^{k-1}\leq f$.

  * $(3)$

If $f=3$, then $p_{1}=2$ or $3$ and $N\cong\mathbb{Z}_{2}^{2}$ or
$\mathbb{Z}_{3}^{2}$, respectively.

  * $(4)$

If $f=4$ then $p_{1}=2$ or $3$. In the first case $N$ is isomorphic to one of
the groups $\mathbb{Z}_{2}^{2}$, $\mathbb{Z}_{2}\times\mathbb{Z}_{p}^{2}$ or
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{p}$, where $p$ is an odd prime. In the
later case, $N\cong\mathbb{Z}_{3}^{2}$.

###### Proof.

$(1)$ If $|N|=p_{1}$ or $p_{1}p_{2}\ldots p_{k}$ then there exists a non-
trivial normal cyclic subgroup of $G$ which contradicts [1, Lemma 2.20].

$(2)$ It is an immediate consequence of Lemma 2.7.

$(3)$ Since $G$ is elusive, $N$ is not cyclic by [1, Lemma 2.20]. Now $(3)$ is
a consequence of $(1)$ and $(2)$.

$(4)$ Let $f=4$. Then $(2)$ implies that $p_{1}=2$ or $p_{1}=3$. If $k=1$ then
by Lemma 2.7, $N\cong\mathbb{Z}_{2}^{2}$. If $k\geq 2$ the the result follow
from $(1)$ and $(2)$. ∎

######  Corollary 2.10.

Let $G$ be a transitive $2$-closed permutation group on a set $\Omega$ of
fixity $4$ and $N\neq 1$ be a normal $p$-subgroup of $G$. Then $G$ admits
fixed-point free element.

###### Proof.

Suppose, towards a contradiction, that $G$ is elusive. Then, by Corollary 2.9,
we have $N\cong\mathbb{Z}_{2}^{2}$ or $N\cong\mathbb{Z}_{3}^{2}$. Let
$\alpha^{N}$ be an orbit of $N$ on $\Omega$. Then
$|\alpha^{N}|\in\\{1,p,p^{2}\\}$, where $p\in\\{2,3\\}$. If $|\alpha^{N}|=1$,
then $N=N_{\alpha}\leq G_{\alpha}$ which implies that $N=1$, a contradiction.
If $|\alpha^{N}|=p^{2}$ then $N_{\alpha}=1$ which contradicts [1, Lemma 2.7].
Hence $|\alpha^{N}|=p$ which contradicts [1, Theorem 2.11]. ∎

## Acknowledgments

The author gratefully appreciate an anonymous referee for constructive
comments and recommendations which definitely helped to improve the
readability and quality of the article.

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