The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# Measurement of nuclear effects in neutrino-argon interactions using
generalized kinematic imbalance variables with the MicroBooNE detector

P. Abratenko  Tufts University, Medford, MA, 02155, USA  O. Alterkait  Tufts
University, Medford, MA, 02155, USA  D. Andrade Aldana  Illinois Institute of
Technology (IIT), Chicago, IL 60616, USA  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  G. Barr  University of Oxford,
Oxford OX1 3RH, United Kingdom  D. Barrow  University of Oxford, Oxford OX1
3RH, United Kingdom  J. Barrow  Massachusetts Institute of Technology (MIT),
Cambridge, MA, 02139, USA  Tel Aviv University, Tel Aviv, Israel, 69978  V.
Basque  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
O. Benevides Rodrigues  Illinois Institute of Technology (IIT), Chicago, IL
60616, USA  S. Berkman  Fermi National Accelerator Laboratory (FNAL), Batavia,
IL 60510, USA  Michigan State University, East Lansing, MI 48824, USA  A.
Bhanderi  The University of Manchester, Manchester M13 9PL, United Kingdom  A.
Bhat  University of Chicago, Chicago, IL, 60637, USA  M. Bhattacharya  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Bishai
Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake
Lancaster University, Lancaster LA1 4YW, United Kingdom  B. Bogart  University
of Michigan, Ann Arbor, MI, 48109, USA  T. Bolton  Kansas State University
(KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University, Cambridge,
MA 02138, USA  M. B. Brunetti  University of Warwick, Coventry CV4 7AL, United
Kingdom  L. Camilleri  Columbia University, New York, NY, 10027, USA  Y. Cao
The University of Manchester, Manchester M13 9PL, United Kingdom  D. Caratelli
University of California, Santa Barbara, CA, 93106, USA  F. Cavanna  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Cerati
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  A.
Chappell  University of Warwick, Coventry CV4 7AL, United Kingdom  Y. Chen
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, 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  University of Pittsburgh, Pittsburgh, PA, 15260, USA  J. I.
Crespo-Anadón  Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Madrid E-28040, Spain  R. Cross  University of Warwick,
Coventry CV4 7AL, United Kingdom  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  Universität Bern, Bern CH-3012,
Switzerland  Z. Djurcic  Argonne National Laboratory (ANL), Lemont, IL, 60439,
USA  R. Dorrill  Illinois Institute of Technology (IIT), Chicago, IL 60616,
USA  K. Duffy  University of Oxford, Oxford OX1 3RH, United Kingdom  S. Dytman
University of Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly  University of
Southern Maine, Portland, ME, 04104, USA  P. Englezos  Rutgers University,
Piscataway, NJ, 08854, USA  A. Ereditato  University of Chicago, Chicago, IL,
60637, USA  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  J. J. Evans  The University of Manchester, Manchester M13 9PL, United
Kingdom  R. Fine  Los Alamos National Laboratory (LANL), Los Alamos, NM,
87545, USA  O. G. Finnerud  The University of Manchester, Manchester M13 9PL,
United Kingdom  W. Foreman  Illinois Institute of Technology (IIT), Chicago,
IL 60616, USA  B. T. Fleming  University of Chicago, Chicago, IL, 60637, USA
N. Foppiani  Harvard University, Cambridge, MA 02138, USA  D. Franco
University of Chicago, Chicago, IL, 60637, USA  A. P. Furmanski  University of
Minnesota, Minneapolis, MN, 55455, USA  F. Gao  University of California,
Santa Barbara, CA, 93106, 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  Los Alamos National Laboratory (LANL), Los Alamos,
NM, 87545, USA  E. Gramellini  The University of Manchester, Manchester M13
9PL, United Kingdom  P. Green  University of Oxford, Oxford OX1 3RH, United
Kingdom  H. Greenlee  Fermi National Accelerator Laboratory (FNAL), Batavia,
IL 60510, USA  L. Gu  Lancaster University, Lancaster LA1 4YW, United Kingdom
W. Gu  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  R.
Guenette  The University of Manchester, Manchester M13 9PL, United Kingdom  P.
Guzowski  The University of Manchester, Manchester M13 9PL, United Kingdom  L.
Hagaman  University of Chicago, Chicago, IL, 60637, USA  O. Hen  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  R. Hicks  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  C. Hilgenberg
University of Minnesota, Minneapolis, MN, 55455, USA  G. A. Horton-Smith
Kansas State University (KSU), Manhattan, KS, 66506, USA  Z. Imani  Tufts
University, Medford, MA, 02155, USA  B. Irwin  University of Minnesota,
Minneapolis, MN, 55455, USA  M. S. Ismail  University of Pittsburgh,
Pittsburgh, PA, 15260, 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  Nankai University, Nankai District,
Tianjin 300071, China  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  J. H. Jo  Brookhaven National Laboratory (BNL), Upton, NY, 11973, 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  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  M. B. Leibovitch  University of California,
Santa Barbara, CA, 93106, USA  I. Lepetic  Rutgers University, Piscataway, NJ,
08854, USA  J.-Y. Li  University of Edinburgh, Edinburgh EH9 3FD, United
Kingdom  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  Rutgers University, Piscataway, NJ, 08854, USA  B. R.
Littlejohn  Illinois Institute of Technology (IIT), Chicago, IL 60616, USA  H.
Liu  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  W. C. Louis
Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  X. Luo
University of California, Santa Barbara, CA, 93106, 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  N. Martinez  Kansas
State University (KSU), Manhattan, KS, 66506, USA  D. A. Martinez Caicedo
South Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701,
USA  S. Martynenko  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  A. Mastbaum  Rutgers University, Piscataway, NJ, 08854, USA  N. McConkey
University College London, London WC1E 6BT, United Kingdom  V. Meddage  Kansas
State University (KSU), Manhattan, KS, 66506, USA  J. Micallef  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  Tufts University,
Medford, MA, 02155, USA  K. Miller  University of Chicago, Chicago, IL, 60637,
USA  A. Mogan  Colorado State University, Fort Collins, CO, 80523, USA  T.
Mohayai  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
Indiana University, Bloomington, IN 47405, 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  M. M. Moudgalya  The
University of Manchester, Manchester M13 9PL, United Kingdom  S. Mulleriababu
Universität Bern, Bern CH-3012, Switzerland  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  N. Nayak  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  J. Nowak  Lancaster University,
Lancaster LA1 4YW, United Kingdom  N. Oza  Columbia University, New York, NY,
10027, USA  O. Palamara  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  N. Pallat  University of Minnesota, Minneapolis, MN,
55455, USA  V. Paolone  University of Pittsburgh, Pittsburgh, PA, 15260, USA
A. Papadopoulou  Argonne National Laboratory (ANL), Lemont, IL, 60439, USA  V.
Papavassiliou  New Mexico State University (NMSU), Las Cruces, NM, 88003, USA
H. B. Parkinson  University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
S. F. Pate  New Mexico State University (NMSU), Las Cruces, NM, 88003, USA  N.
Patel  Lancaster University, Lancaster LA1 4YW, United Kingdom  Z. Pavlovic
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E.
Piasetzky  Tel Aviv University, Tel Aviv, Israel, 69978  I. Pophale  Lancaster
University, Lancaster LA1 4YW, United Kingdom  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  Argonne National
Laboratory (ANL), Lemont, IL, 60439, 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. 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  Tufts University, Medford, MA, 02155, USA  M. Ross-Lonergan  Los
Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  C. Rudolf von
Rohr  Universität Bern, Bern CH-3012, Switzerland  I. Safa  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  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. Shi  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  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  A. M. Szelc
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  W. Tang
University of Tennessee, Knoxville, TN, 37996, USA  N. Taniuchi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  K. Terao  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  C. Thorpe  Lancaster
University, Lancaster LA1 4YW, United Kingdom  The University of Manchester,
Manchester M13 9PL, United Kingdom  D. Torbunov  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  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  J. Tyler  Kansas State University
(KSU), Manhattan, KS, 66506, USA  M. A. Uchida  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  T. Usher  SLAC National Accelerator
Laboratory, Menlo Park, CA, 94025, USA  B. Viren  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern, Bern
CH-3012, Switzerland  H. Wei  Louisiana State University, Baton Rouge, LA,
70803, USA  A. J. White  University of Chicago, Chicago, IL, 60637, 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  W. Wu  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  E. Yandel  University of California,
Santa Barbara, CA, 93106, USA  T. Yang  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  L. E. Yates  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, 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 present a set of new generalized kinematic imbalance variables that can be
measured in neutrino scattering. These variables extend previous measurements
of kinematic imbalance on the transverse plane, and are more sensitive to
modeling of nuclear effects. We demonstrate the enhanced power of these
variables using simulation, and then use the MicroBooNE detector to measure
them for the first time. We report flux-integrated single- and double-
differential measurements of charged-current muon neutrino scattering on argon
using a topolgy with one muon and one proton in the final state as a function
of these novel kinematic imbalance variables. These measurements allow us to
demonstrate that the treatment of charged current quasielastic interactions in
GENIE version 2 is inadequate to describe data. Further, they reveal tensions
with more modern generator predictions particularly in regions of phase space
where final state interactions are important.

##  I Introduction

All current and upcoming accelerator neutrino oscillation experiments rely on
the precise modeling of neutrino-nucleus interactions to perform high-accuracy
measurements [1, 2, 3, 4, 5, 6]. The experimental sensitivity of these
measurements can be limited by interaction modeling uncertainties related to
nuclear effects [7]. Significant progress has been made in understanding these
effects and improving their modeling in neutrino event generators. Yet,
outstanding tensions between measurement and theory still remain unresolved
[8]. A major challenge in the study of neutrino-nucleus interactions
originates from the wide-band accelerator neutrino beams, since nuclear
effects cannot be easily disentangled when averaged over a broad energy
spectrum. Examples of these are Fermi motion, final state interactions (FSI),
and nucleon-nucleon correlations. However, it has been shown that kinematic
imbalance variables in the plane transverse to the neutrino direction of
travel are powerful tools that can be used to separate these nuclear effects
in a neutrino energy independent way [9].

In this work, we extend these measurements to generalized kinematic imbalance
(GKI) variables by considering the longitudinal components along the beam
direction. We illustrate with generator-level studies that these generalized
variables achieve improved sensitivity to nuclear effects. Furthermore, we
present the first flux-integrated single- and double-differential cross-
section measurements for muon-neutrino charged-current (CC) reactions on argon
($\nu_{\mu}$-Ar) as a function of these generalized variables. Here we focus
on reactions where a single muon-proton pair is reconstructed with no
additional detected particles, similar to previous measurements [10, 11] We
refer to these events as CC1p0$\pi$, and they are dominated by quasielastic
(QE) interactions as it is required that there are no pions above the
detection threshold. The results reported here use the MicroBooNE detector
[12] with an exposure of $6.79\times 10^{20}$ protons on target from the
Booster Neutrino Beam (BNB) [13] at Fermi National Accelerator Laboratory.

In Sec. II we define the GKI variables. Sec. III shows the enhanced
sensitivity of GKI variables to nuclear effects by presenting comparisons to
their transverse plane equivalents. In Sec. IV we present the first flux-
integrated single- and double-differential cross section measurements in these
variables using $\nu_{\mu}$-Ar CC1p0$\pi$ interactions recorded by the
MicroBooNE detector. Finally, conclusions are presented in Sec. V.

##  II Observables

Variables based on the transverse kinematic imbalance are powerful
discriminators between interaction models as reported by multiple experiments
using a number of final states [14, 15, 16, 17, 18, 9, 19, 10, 11]. The
simplest case is for charged current quasielastic (CCQE) interactions, where
the final state can be characterized by a muon with transverse momentum
$\vec{p}_{T}^{\,\mu}$ and a proton with transverse momentum
$\vec{p}_{T}^{\,p}$. To extend to other hadronic final states, the proton’s
momentum is replaced with the momentum of the combined hadronic system. In
this work, however, we assume a final state containing only a muon and a
proton. The kinematics of the two particle final state on the transverse plane
can be fully characterized by a magnitude $\delta p_{T}$ =
$|\delta\vec{p}_{T}|=|\vec{p}_{T}^{\,\mu}+\vec{p}_{T}^{\,p}|$, as well as two
angles,

$\displaystyle\delta\phi_{T}$
$\displaystyle=\cos^{-1}\left(\frac{-\vec{p}_{T}^{\,\mu}\cdot\vec{p}_{T}^{\,p}}{|\vec{p}_{T}^{\,\mu}||\vec{p}_{T}^{\,p}|}\right),$
(1) $\displaystyle\delta\alpha_{T}$
$\displaystyle=\cos^{-1}\left(\frac{-\vec{p}_{T}^{\,\mu}\cdot\delta\vec{p}_{T}}{|\vec{p}_{T}^{\,\mu}||\delta\vec{p}_{T}|}\right).$
(2)

The muon transverse momentum ($\vec{p}^{\,\mu}_{T}$) is equal and opposite to
the transverse component of the momentum transfer to the nucleus
($\vec{q}_{T}$). The vector $\delta\vec{p}_{T}$ is the struck nucleon
transverse momentum, and should be zero in the absence of initial state
nucleon transverse momentum and final state interactions. Any non-zero value
of $\delta\vec{p}_{T}$ can therefore be described as the missing transverse
momentum (or rather, the negative thereof). Fermi motion inside the nucleus
produces non-zero values of $\delta\vec{p}_{T}$ for QE interactions, but
values far above the Fermi momentum are usually due to final state
interactions. The variable $\delta\phi_{T}$ corresponds to the angle between
the transverse momentum transfer vector and the final state proton momentum
($\vec{p}_{T}^{\,p}$). These vectors would be aligned in the case of a free
stationary nucleon target and this angle would be zero. Small values of
$\delta\phi_{T}$ are produced by initial state motion, while larger values are
indicative to final state interactions. The angle $\delta\alpha_{T}$ is
defined between the transverse momentum transfer vector (–
$\vec{p}_{T}^{\,\mu}$) and the transverse missing momentum vector
($\delta\vec{p}_{T}$). The angle is sensitive to final state interactions but
much less sensitive to initial state nucleon motion. In the absence of final
state interactions, $\delta\alpha_{T}$ does not have a preferred orientation
and yields a fairly flat distribution. A representation of the kinematic
imbalance variables on the transverse plane is shown in Fig. 1(a).

A more recent investigation identifies an alternative representation of the
transverse missing momentum vector using the projection components
perpendicular and parallel to the momentum transfer [16] [see Fig. 1(b)],
given by

$\displaystyle\delta p_{T,x}$
$\displaystyle=(\hat{z}\times\hat{p}_{T}^{\mu})\cdot\delta\vec{p}_{T},$ (3)
$\displaystyle\delta p_{T,y}$
$\displaystyle=-\hat{p}_{T}^{\mu}\cdot\delta\vec{p}_{T}.$ (4)

These are the components of $\delta\vec{p}_{T}$ perpendicular and parallel,
respectively, to the transverse momentum transfer vector.

(a)$y$$z$$x$$p^{p}_{T}$$\delta\phi_{T}$$q_{T}$$\delta
p_{T}$$\delta\alpha_{T}$(b)$y$$z$$x$$\delta p_{Tx}$$\delta p_{Ty}$$\delta
p_{T}$ Figure 1:  (a) Representation of the kinematic imbalance variables on
the transverse plane and (b) alternative representation using the projections
parallel and perpendicular to the transverse momentum transfer vector. The
finely dashed lines correspond to the direction of the momentum transfer
(blue) and the proton (orange). The $z$ axis corresponds to the neutrino
direction of travel.

The interpretation of these variables on the transverse plane can be
generalized to their three-dimensional equivalents. To do this, the
longitudinal components of the missing momentum and momentum transfer vectors
are required, and therefore an assumption of the incoming neutrino energy has
to be made. An initial attempt to perform this generalization is reported in
Ref. [20], which assumes that a neutrino interacts via a CCQE interaction with
a bound stationary neutron at rest inside a nucleus. This is then used to
obtain an estimate for the neutrino energy and the three components of the
missing momentum vector, labeled $\vec{p}_{n}$.

We present a slightly different formalism which relies on conservation of
energy and momentum. For the notation presented below, the speed of light $c$
is assumed to be unity. For a massless neutrino, if the entire final state is
visible, then the total visible energy and the longitudinal momentum will be
the same. If any missing momentum is carried away by the residual nucleus, the
kinetic energy of the residual nucleus can be assumed negligible, given its
large mass. The “calorimetric visible energy” is constructed as the primary
neutrino energy estimator

$\displaystyle E_{\text{cal}}=E_{\mu}+K_{p}+B,$ (5)

where $E_{\mu}$ is the total muon energy, $K_{p}$ is the proton kinetic
energy, and $B$ the argon binding energy set to 30.9 MeV [21].

The second estimator of the neutrino energy is the total momentum along the
neutrino direction of travel. This should be equal to the calorimetric energy
if all final state particles have been observed. Therefore the difference
between the total longitudinal momentum and the calorimetric energy provides
the longitudinal component of the missing momentum,

$\displaystyle p_{L}=p_{L}^{\mu}+p_{L}^{p}-E_{\text{cal}},$ (6)

where $p_{L}^{\mu(p)}$ is the longitudinal component of the muon (proton)
momentum vector. This definition of $p_{L}$ is numerically very close to that
in Ref. [20], but it enables a generalization to other final states without
having to make an assumption about the underlying interaction.

The missing momentum vector $\vec{p}_{n}$ is obtained as the vector sum of the
transverse missing momentum $\delta\vec{p}_{T}$ and the longitudinal component
$p_{L}$. The definitions given here produce a vector that aligns with the
initial struck neutron momentum, consistent with the definitions used in
previous work [20]. Assuming a neutrino traveling in the $z$ direction of a
detector coordinate system, the momentum transfer $\vec{q}$ of the interaction
is derived as the difference between the inferred neutrino and the muon
momentum vectors

$\displaystyle\vec{q}$ $\displaystyle=E_{\text{cal}}\hat{z}-\vec{p}_{\mu}.$
(7)

With these definitions, we generalize the transverse missing momentum
variables to three dimensions [see Fig 2(a)] to be

$\displaystyle p_{n}$ $\displaystyle=|\vec{p}_{n}|=\sqrt{p_{L}^{2}+\delta
p_{T}^{2}},$ (8) $\displaystyle\phi_{3D}$
$\displaystyle=\cos^{-1}\left(\frac{\vec{q}\cdot\vec{p}_{\,p}}{|\vec{q}\,||\vec{p}_{\,p}|}\right),$
(9) $\displaystyle\alpha_{3D}$
$\displaystyle=\cos^{-1}\left(\frac{\vec{q}\cdot\vec{p}_{n}}{|\vec{q}\,||\vec{p}_{n}|}\right).$
(10)

The vector $\vec{p}_{n}$ accounts for the missing momentum and is an estimate
of the initial struck nucleon momentum; $\phi_{3D}$ is the opening angle
between the total momentum transfer vector $\vec{q}$ and the proton momentum
vector $\vec{p}^{\,p}$; and $\alpha_{3D}$ is the angle between the momentum
transfer vector $\vec{q}$ and the missing momentum vector $\vec{p}_{n}$. These
variables are the three-dimensional analogues to the ones on the transverse
plane $\vec{p}_{T}$, $\delta\alpha_{T}$, and $\delta\phi_{T}$.

(a)$y$$z$$x$$p_{\nu}$$p^{p}$$\phi_{3D}$$q$$p_{n}$$p^{\mu}$$\alpha_{3D}$(b)$y$$z$$x$$p_{\nu}$$p_{n\perp}$$q$$p_{n}$$p_{n\parallel}$
Figure 2:  (a) Representation of the generalized kinematic imbalance variables
and (b) alternative representation using the projections parallel and
perpendicular to the missing momentum vector. The $z$ axis corresponds to the
neutrino direction of travel.

Likewise, the alternative reprentation on the transverse plane can also be
extended to three dimensions [Fig. 2(b)], in the form

$\displaystyle p_{n\perp,x}$
$\displaystyle=(\hat{q}_{T}\times\hat{z})\cdot\vec{p}_{n},$ (11)
$\displaystyle p_{n\perp,y}$
$\displaystyle=(\hat{q}\times(\hat{q}_{T}\times\hat{z}))\cdot\vec{p}_{n},$
(12) $\displaystyle p_{n\perp}$
$\displaystyle=\sqrt{(p_{n\perp,x})^{2}+(p_{n\perp,y})^{2}}=|p_{n}|\sin(\alpha_{3D}),$
(13) $\displaystyle p_{n\parallel}$
$\displaystyle=\hat{q}\cdot\vec{p}_{n}=|p_{n}|\cos(\alpha_{3D}).$ (14)

Here, $p_{n\parallel}$ ($p_{n\perp}$) is the component of the missing momentum
vector parallel (perpendicular) to the momentum transfer vector,
$p_{n\perp,x}$ ($p_{n\perp,y}$) is the component of $p_{n\perp}$ in the
neutrino-muon scattering plane (perpendicular to the neutrino-muon scattering
plane), $\hat{q}$ is the unit vector aligned with $\vec{q}$, $\hat{q}_{T}$ is
the unit vector aligned with the transverse component of $\vec{q}$, and
$\hat{z}$ is the unit vector aligned with the neutrino direction of travel
(the z-axis).

These variables have been defined for a CCQE interaction with only a muon and
a proton in the final state, however this is easily extended to other final
states by summing the hadron momenta, and adjusting the assumed binding energy
depending on the nucleon multiplicity.

##  III Generator Comparisons

To demonstrate the power of these novel variables, we use the CC1p0$\pi$
signal definition included in Refs. [10, 11] with several commonly-used
neutrino interaction generators and model configurations, convoluted with the
MicroBooNE $\nu_{\mu}$ flux prediction [13]. A number of simulation
predictions used for comparison correspond to GENIE model configurations, an
event generator commonly used by Fermilab-based experiments. Our comparisons
include an older GENIE version which was extensively used in the past, as well
as more modern GENIE tunes currently used by MicroBooNE [11, 10, 22].
Additionally, comparisons to alternative event generators used by other
neutrino experiments, such as T2K, or in the theory community, are included.
Overflow (underflow) values are included in the last (first) bin. Table 1
lists a summary of the abbreviations of the generators and configurations used
in this analysis, which are presented in more detail below.

Table 1: Generators and configurations abbreviations used. Name |  Generator / Configuration
---|---
Gv2 |  GENIE v2.12.10 [23, 24]
G18 |  GENIE v3.0.6 G18_10a_02_11a [25]
G18T |  G18 with tune [22]
G21 |  GENIE v3.2.0 G21_11b_00_000 [25]
GiBUU |  GiBUU 2021 [26]
NuWro |  NuWro v19.02.1 [27]
NEUT |  NEUT v5.4.0 [28]

The GENIE configurations used are:

  * •

GENIE v2.12.10 (Gv2) [23, 24]: This version corresponds to a historical
reference extensively used in cross section analyses in the past and includes
the Bodek-Ritchie Fermi Gas model, the Llewellyn Smith CCQE scattering
prescription [29], the empirical meson exchange current (MEC) model [30], a
Rein-Sehgal resonance (RES) and coherent (COH) scattering model [31], the
Bodek-Yang deep inelastic scattering (DIS) model [32], and a data driven FSI
model denoted as “hA” [33].

  * •

GENIE v3.0.6 G18_10a_02_11a (G18) [25]: This more modern model configuration
uses the local Fermi gas (LFG) model [34], the Nieves CCQE scattering
prescription [35] which includes Coulomb corrections for the outgoing muon
[36], and random phase approximation (RPA) corrections [37]. Additionally, it
uses the Nieves MEC model [38], the KLN-BS RES [39, 40, 41, 42], Berger-Sehgal
COH [43], and Bodek-Yang DIS [32] scattering models, and the hA2018 FSI model
[44].

  * •

G18T [22]: Corresponds to the same G18 configuration with additional
MicroBooNE-specific tuning.

  * •

GENIE v3.2.0 G21_11b_00_000 (G21) [25]. This configuration includes the SuSAv2
prediction for the QE and MEC scattering modes [45] and the hN2018 FSI model
[46]. The modeling options for RES, DIS, and COH interactions are the same as
for G18.

The alternative event generator predictions are:

  * •

GiBUU 2021 (GiBUU) [26]: Uses similar models to GENIE, but they are
implemented in a coherent way by solving the Boltzmann-Uehling-Uhlenbeck
transport equation [26]. The modeling includes the LFG model [34], a standard
CCQE expression [47], an empirical MEC model, and a dedicated spin dependent
resonance amplitude calculation following the MAID analysis [26]. The DIS
model is from PYTHIA [48]. GiBUU’s FSI treatment propagates the hadrons
through the residual nucleus in a nuclear potential consistent with the
initial state.

  * •

NuWro v19.02.1 (NuWro) [27]: Includes the LFG model [34], the Llewellyn Smith
model for QE events [29], the Nieves model for MEC events [49], the Adler-
Rarita-Schwinger formalism to calculate the $\Delta$ resonance explicitly
[42], the BS COH [43] scattering model, and an intranuclear cascade model for
FSI [49].

  * •

NEUT v5.4.0 (NEUT) [28]: Corresponds to the combination of the LFG model [34],
the Nieves CCQE scattering prescription [35], the Nieves MEC model [38], the
BS RES [39, 40, 41, 42] and BS COH [43] scattering models, and FSI with medium
corrections for pions [23, 24].

If the interacting neutrino scatters off a stationary neutron and all final
state particles are observed, $p_{n}$ and $\delta p_{T}$ become zero. Within
the dense nuclear medium of a heavy nucleus like argon, these variables follow
a broad distribution due to the struck nucleon motion before the interaction,
with a high missing momentum tail from non-CCQE events. Figure 3 shows the
$p_{n}$ distribution for interaction types using G18 and shows the
corresponding breakdown for $\delta p_{T}$ for the selected CC1p0$\pi$ events.
Both results illustrate a CCQE dominance that is driven by the CC1p0$\pi$
signal definition. The non-CCQE events in the $p_{n}$ distribution are
concentrated at higher momentum values than in the $\delta p_{T}$
distribution. Therefore, the missing momentum $p_{n}$ illustrates enhanced
discrimination capabilities between CCQE and non-CCQE events when compared to
$\delta p_{T}$, as already demonstrated in Ref. [20].

(a)

(b)

Figure 3: The flux-integrated single-differential cross section interaction
breakdown as a function of (a) $p_{n}$ and (b) $\delta p_{T}$ for the selected
CC1p0$\pi$ events. Colored lines show the results of theoretical cross section
calculations using the G18 prediction for QE (blue), MEC (red), RES (orange),
and DIS (green) interactions.

To quantify this improved separation ability, we form signal acceptance-
background rejection curves for both kinematic imbalance variables using G18,
as shown in Fig. 4(a). Assuming events with a value of $p_{n}$ less than a
given “cut value” are retained, the fraction of true CCQE events accepted and
the fraction of true non-CCQE events rejected can be calculated as a function
of this cut value. An improved ability to isolate true CCQE events is observed
for $p_{n}$ compared to $\delta p_{T}$, while rejecting a larger fraction of
non-CCQE background events. The evolution of the product between the signal
acceptance and the background rejection denoted by “rejection $\times$
acceptance” as a function of the cut value yields an optimal requirement at
$p_{n}$ $\approx$ 0.3 GeV/c and at $\delta p_{T}$ $\approx$ 0.2 GeV/c [Fig.
4(b)]. The application of the corresponding selection criteria results in a
CCQE loss of 25.3% for $p_{n}$ and 29.7% for $\delta p_{T}$. The looser
requirement on $p_{n}$ results in a slightly more pure CCQE sample (95% CCQE
purity) with higher statistics compared to the equivalent case with $\delta
p_{T}$ (92% CCQE purity). The same behavior is observed across all event
generators and is shown in the Supplemental Material.

(a)

(b)

Figure 4:  (a) Signal acceptance fraction vs background rejection fraction as
a function of the minimum requirement on either the missing momentum $p_{n}$
(orange) or the transverse missing momentum $\delta p_{T}$ (blue) for
CC1p0$\pi$ events using the G18 prediction. (b) Evolution of the product
between the signal acceptance and the background rejection denoted as
“rejection $\times$ acceptance” as a function of the minimum requirement.

If a neutrino scatters off a moving, but unbound, neutron, $\alpha_{3D}$
becomes the angle of the struck nucleon direction before the interaction
relative to the momentum transfer vector. There is no directional preference
since the nucleus is at rest, so $\alpha_{3D}$ follows an approximately
sinusoidal curve due to the phase space for a randomly distributed three-
dimensional direction. However, final state interactions [46] in the nucleus
introduce missing momentum, which is transferred from the hadronic system to
the residual nucleus, reducing the magnitude of $\vec{p}^{\,p}$ and enhancing
the magnitude of the components transverse to $\vec{q}$. This effect enhances
the contribution of events with higher values of $\alpha_{3D}$. Like
$\alpha_{3D}$, the angular orientation $\delta\alpha_{T}$ has been shown to be
sensitive to FSI effects [9]. The distribution of $\delta\alpha_{T}$
illustrates a transition from a uniform angular orientation in the absence of
FSI to one that peaks close to 180∘ due to the reduction of $\vec{p}^{\,p}$ in
the presence of FSI. Figure 5 shows the distribution of $\alpha_{3D}$ and
$\delta\alpha_{T}$ for the G18 predictions with and without FSI, illustrating
the impact of FSI on the shapes. Figure 6 shows the simulation predictions
using several event generators for $\alpha_{3D}$ and $\delta\alpha_{T}$. The
Gv2 model predicts a significantly different shape that peaks at the edges of
the distribution. The GiBUU distribution is more sharply peaked in
$\alpha_{3D}$ than the other generators and shows a larger discrepancy. All
other event generators yield consistent predictions.

(a)

(b)

Figure 5:  Comparison of the flux-integrated single-differential cross section
as a function of (a) $\alpha_{3D}$ and (b) $\delta\alpha_{T}$ with (solid) and
without (dashed) final state interaction effects using the G18 prediction for
the selected CC1p0$\pi$ events.

(a)

(b)

Figure 6:  The flux-integrated single-differential cross section as a function
of (a) $\alpha_{3D}$ and (b) $\delta\alpha_{T}$ for the selected CC1p0$\pi$
events. Colored lines show the results of cross section calculations using the
G18 (solid black), Gv2 (blue), NEUT (dashed pink), NuWro (red) and GiBUU
(green) predictions.

The third transverse kinematic imbalance variable, $\delta\phi_{T}$, has the
benefit of not depending on the magnitude of the particles’ momenta, but only
their direction. Therefore, it is often more precisely measured. The other two
variables in the transverse plane rely on an accurate momentum reconstruction
for the muon and the hadronic system. Unlike $\delta\phi_{T}$, the definition
of $\phi_{3D}$ requires an estimation of the momentum transfer. Figure 7 shows
the interactions using the G18 prediction for $\phi_{3D}$ with a turnover at
low values. The corresponding $\delta\phi_{T}$ distribution monotonically
decreases at higher values.

(a)

(b)

Figure 7:  The flux-integrated single-differential cross section interactions
as a function of (a) $\phi_{3D}$ and (b) $\phi_{T}$ for the selected
CC1p0$\pi$ events. Colored lines show the results of cross section
calculations using the G18 prediction for QE (blue), MEC (red), RES (orange),
and DIS (green) predictions.

The ratios with and without FSI are shown in Fig. 8 for the stuck nucleon-
missing momentum opening angles ($\delta\alpha_{T}$ or $\alpha_{3D}$) and the
proton-missing momentum opening angles ($\delta\phi_{T}$ or $\phi_{3D}$) using
several event generator predictions. Shape differences become more pronounced
and yield a larger range of ratios when the GKI variables are used when
compared to the equivalent TKI results, indicating greater sensitivity to the
details of FSI modeling.

(a)

(b)

Figure 8:  Ratios with and without FSI for G18, NEUT, and NuWro predictions as
a function of (a) the struck nucleon-missing momentum angle
($\delta\alpha_{T}$ or $\alpha_{3D}$) and (b) the proton-missing momentum
angle ($\delta\phi_{T}$ or $\phi_{3D}$) for the selected CC1p0$\pi$ events.

The projection variables $p_{n\parallel}$ and $p_{n\perp}$ vary when studied
using several generators. Like its transverse equivalent, $\delta p_{T_{y}}$
[11, 16], $p_{n\parallel}$ shows an asymmetric behavior due to the tendency
for FSI to decelerate reinteracting hadrons [Fig. 9(a)]. In addition, GiBUU
exhibits an offset by 0.15 GeV/c to smaller values, unlike any of the other
event generators where the peak is centered around 0 GeV/c. The GiBUU
interaction breakdown in Fig. 10(a) shows that the shift is driven both by the
QE and MEC contributions. In the absence of FSI, the QE offset is no longer
present and the asymmetric behavior is driven by the MEC contribution.

GiBUU shows a shift to higher values compared to the other generators for the
$p_{n\perp}$ (Fig. 9). The Gv2 distributions are significantly different
compared to the other more modern generators, especially at lower values. In
this older version, the hadronic and leptonic kinematics are generated
independently which violates required correlations. This is modified for QE
interactions in more recent versions (such as G18), resulting in smooth
distributions that are similar to the other generators. The corresponding
$p_{n\perp,x}$ and $p_{n\perp,y}$ distributions predict a much sharper peak
around 0 GeV/c for Gv2 compared to the other generators (Fig. 11). The results
without FSI are presented in the Supplemental Material.

The two perpendicular projection variables, $p_{n\perp,x}$ and $p_{n\perp,y}$,
show some differences which naively might not be expected. Due to the way
these variables are defined, $p_{n\perp,y}$ is always perpendicular to the
neutrino-muon scattering plane, which contains the vector $\vec{q}$. As there
is no directional preference for nuclear motion, this is symmetric. Similarly,
$p_{n\perp,x}$ is also perpendicular to $\vec{q}$, so naively it should also
be symmetric. However, the variable is defined in the scattering plane. In
this case missing energy can lead to small differences between the estimated
direction of $\vec{q}$ and the true momentum transfer vector. This leads to a
slightly wider distribution with a tail at negative $p_{n\perp,x}$ values.

(a)

(b)

Figure 9:  The flux-integrated single-differential cross section as a function
of (a) $p_{n\,\parallel}$ and (b) $p_{n\,\perp}$ for the selected CC1p0$\pi$
events. Colored lines show the results of cross section calculations using the
G18 (solid black), Gv2 (blue), NEUT (dashed pink), NuWro (red) and GiBUU
(green) generators.

(a)

(b)

Figure 10:  The flux-integrated single-differential cross section interaction
breakdown as a function of $p_{n\parallel}$ (a) with FSI and (b) without FSI
for the selected CC1p0$\pi$ events. Colored lines show the results of cross
section calculations using the GiBUU prediction for QE (blue), MEC (red), RES
(orange), and DIS (green) generators.

(a)

(b)

Figure 11:  The flux-integrated single-differential cross section as a
function of (a) $p_{n\,\perp,x}$ and (b) $p_{n\,\perp,y}$ for the selected
CC1p0$\pi$ events. Colored lines show the results of cross section
calculations using the G18 (solid black), Gv2 (blue), NEUT (dashed pink),
NuWro (red) and GiBUU (green) generators.

The sensitivity to nuclear effects of these variables becomes even more
pronounced when performing double-differential measurements. As mentioned
earlier, the angle $\alpha_{3D}$ is sensitive to FSI, with higher values of
$\alpha_{3D}$ corresponding to events that primarily undergo FSI. The double-
differential $p_{n}$ cross section illustrates significant differences
depending on whether FSI have been added or not (Fig. 12). For events with
$\alpha_{3D}<$ 45∘, FSI leads to a significant reduction in the normalization
of the peak, while the high missing momentum tail is largely unaffected.
Conversely, events with 135∘ $<\alpha_{3D}<$ 180∘ yield a significantly
enhanced high-$p_{n}$ tail when FSI are included. That feature makes the
high-$\alpha_{3D}$ region ideal to study the impact of FSI effects. We can
study the FSI impact on $p_{n}$ by isolating two groups of events, those with
$\alpha_{3D}$ values $>$ 135∘ and with $\alpha_{3D}$ $<$ 45∘. Figure 13(a)
shows the QE-dominated low-$\alpha_{3D}$ region with a tail that vanishes at
$\approx 0.5$ GeV/c. The primary cause of the high-$p_{n}$ tail in this region
is interactions that produce additional undetected particles, and therefore
consist mainly of non-QE interactions. Conversely, the high-$\alpha_{3D}$
region shown in Fig. 13(b) illustrates a much wider tail that extends up to
$\approx 1$ GeV/c. This tail has a significant contribution from MEC and RES
events, as well as a large population of events from QE interactions that
undergo FSI and therefore yield higher $p_{n}$ values.

(a)

(b)

Figure 12:  Comparison of the flux-integrated double-differential cross
section as a function of $p_{n}$ for (a) $\alpha_{3D}<$ 45∘ and (b) 135∘
$<\alpha_{3D}<$ 180∘ with (solid) and without (dashed) final state interaction
effects using the G18 prediction for the selected CC1p0$\pi$ events.

(a)

(b)

Figure 13:  The flux-integrated double-differential cross section interaction
breakdown as a function of $p_{n\,\parallel}$ for (a) $\alpha_{3D}<$ 45∘ and
(b) 135∘ $<\alpha_{3D}<$ 180∘ for the selected CC1p0$\pi$ events. Colored
lines show the results of theoretical cross section calculations using the G18
prediction for QE (blue), MEC (red), RES (orange), and DIS (green)
interactions.

A complementary way to group the selected events is by missing momentum
($p_{n}<$ 0.2 GeV/c and $p_{n}>$ 0.4 GeV/c). The double-differential cross
section in $\alpha_{3D}$ yields a distribution that approximately follows the
expected sine-curve behavior when events with low missing momentum are used
[Fig. 14(a)]. The same shape is observed when FSI are turned off [Fig. 15(a)].
A different shape is observed in Fig. 14(b) for events with high missing
momentum with a pronounced peak at high $\alpha_{3D}$ values, which is driven
by QE events. On the other hand, MEC and RES events result in a wider range of
$\alpha_{3D}$ angles and less peaked distributions, illustrating that the
missing momentum has less directional preference. Furthermore, the region
where $\alpha_{3D}<$ 90∘ is dominated by MEC events across all event
generators due to the missing momentum introduced by undetected particles. The
prediction without FSI yields a smoother distribution [Fig. 15(b)].

(a)

(b)

Figure 14:  The flux-integrated double-differential cross sections as a
function of $\alpha_{3D}$ for (a) $p_{n}<$ 0.2 GeV/c and (b) $p_{n}>$ 0.4
GeV/c for the selected CC1p0$\pi$ events. Colored lines show the results of
cross section calculations using the G18 prediction for QE (blue), MEC (red),
RES (orange), and DIS (green) predictions.

(a)

(b)

Figure 15:  The flux-integrated double-differential cross sections as a
function of $\alpha_{3D}$ for (a) $p_{n}<$ 0.2 GeV/c and (b) $p_{n}>$ 0.4
GeV/c with (solid) and without (dashed) final state interaction effects using
the G18 prediction for the selected CC1p0$\pi$ events.

##  IV MicroBooNE Cross Section Measurement

To perform the first measurement of the cross section as a function of these
variables, we use three years of data collected by the MicroBooNE detector.
MicroBooNE is an 85-tonne active mass liquid argon time projection chamber in
the Booster Neutrino Beam (BNB). The MicroBooNE detector is described in
detail in Ref. [12]. We use the same event selection and measurement strategy
as used in Refs. [10, 11], and the same CC1p0$\pi$ definition for the
variables of interest as described in Sec. II.

Data are processed by filtering noise and deconvolving the wire response to
produce unipolar signals, which are then fitted with Gaussian functions to
produce hits. The Pandora multi-algorithm pattern recognition package [50] is
used to cluster hits, match them across wire planes, and construct tracks and
showers. Topological and optical information are used to identify and remove
cosmic tracks. The remaining tracks and showers are grouped into neutrino
candidates. Events are selected by requiring a neutrino candidate with exactly
two reconstructed tracks by applying particle identification requirements
based on $dE/dx$ measurements to ensure the tracks are muon-like or proton-
like, respectively. More details on the selection can be found in Ref. [11].

Uncertainties related to the incident neutrino flux [13], interaction model
[22], particle propagation [51], and detector response [52] are assessed
separately to produce a covariance matrix describing the uncertainty of the
predicted event rate. The binning is chosen to balance resolution and
statistics. The Wiener-Singular Value Decomposition unfolding technique [53]
is used to transform both the data measurement and covariance matrix into a
regularized phase space. Each measurement is accompanied by an additional
smearing matrix $A_{C}$ which performs the conversion from the true to the
regularized phase space. The $A_{C}$ matrix is included in the Supplemental
Material and needs to be applied to all theory predictions in order to compare
to the data measurements.

The unfolded event rate has the predicted background subtracted. It is further
divided by the integrated neutrino flux and number of argon nuclei in the
fiducial volume to report a differential cross section. In the results
presented below, the inner error bars on the cross sections correspond to the
statistical uncertainties. The systematic uncertainties are decomposed into
shape- and normalization-related sources following the procedure outlined in
Ref. [54]. The cross-term uncertainties are incorporated in the normalization.
The outer error bars on the reported cross sections correspond to statistical
and shape uncertainties added in quadrature. The normalization uncertainties
are presented as a band at the bottom of each plot. Overflow (underflow)
values are included in the last (first) bin. The degrees of freedom correspond
to the number of bins. The $\chi^{2}$/ndf data comparison for each generator
shown on all the figures takes into account the total covariance matrix. More
details on the systematic uncertainties and the cross-section extraction
technique can be found in Ref. [11].

Figures 16-18 show the single-differential cross sections as a function of the
$p_{n}$, $\alpha_{3D}$, and $\phi_{3D}$ compared to several predictions. We
conclude that Gv2 is a poor description of the data and results in large
$\chi^{2}$ values. Among the other generator predictions, GiBUU provides the
best description of the data in $\alpha_{3D}$. There is a spread in $p_{n}$
and $\phi_{3D}$ with G21 describing the data best. Unlike these variables, G21
shows a poor agreement with the data in $\alpha_{3D}$. G18T illustrates a
similar pattern as G21 with a better agreement in $p_{n}$ and $\phi_{3D}$ and
a worse performance in $\alpha_{3D}$.

(a)

(b)

Figure 16:  The flux-integrated single-differential cross sections as a
function of $p_{n}$ with (a) generator and (b) GENIE configuration predictions
compared to data. Inner and outer error bars show the statistical and total
(statistical and shape systematic) uncertainty at the 1$\sigma$, or 68%,
confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

(a)

(b)

Figure 17:  The flux-integrated single-differential cross sections as a
function of $\alpha_{3D}$ with (a) generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

(a)

(b)

Figure 18:  The flux-integrated single-differential cross sections as a
function of $\phi_{3D}$ with (a) generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

The different projections of $p_{n}$ parallel and perpendicular to the
momentum transfer are shown in Figs. 19 \- 22. Again Gv2 provides a poor
description of the data, particularly in the case of $p_{n\perp}$. The data
show a large tail at negative values of $p_{n\parallel}$ as expected from the
effects of FSI. The predictions in this region show large variation due to the
different ways of modeling FSI. Based on the $\chi^{2}$/ndf, GiBUU provides
the best description of this data in most variables. G18T provides the best
performance in the case of $p_{\perp}$ and $p_{\perp,x}$. Conversely, there is
a spread in performance among the generators in the description of
$p_{\perp,y}$ with G21 yielding the lowest $\chi^{2}$/ndf ratio.

(a)

(b)

Figure 19:  The flux-integrated single-differential cross sections as a
function of $p_{n\perp}$ with (a) generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

(a)

(b)

Figure 20:  The flux-integrated single-differential cross sections as a
function of $p_{n\perp,x}$ with (a) generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

(a)

(b)

Figure 21:  The flux-integrated single-differential cross sections as a
function of $p_{n\perp,y}$ with (a) Generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

(a)

(b)

Figure 22:  The flux-integrated single-differential cross sections as a
function of $p_{n\parallel}$ with (a) Generator and (b) GENIE configuration
predictions compared to data. Inner and outer error bars show the statistical
and total (statistical and shape systematic) uncertainty at the 1$\sigma$, or
68%, confidence level. The gray band shows the normalization systematic
uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf calculation
for each one of the predictions.

As discussed in Sec. III, the sensitivity of the generalized kinematic
imbalance variables can be further enhanced when performing multi-differential
measurements. Following the approach outlined in Refs. [10, 11], we present
double-differential measurements as a function of $p_{n}$ and $\alpha_{3D}$.

FSI effects are minimal for the double-differential cross section of $p_{n}$
with $\alpha_{3D}<45^{\circ}$ and, as expected, the tail of the $p_{n}$
distribution is significantly suppressed (Fig. 23). The $\chi^{2}$/ndf is
reasonably consistent across all event generators apart from Gv2. This
observation could be driven by the fact that the more modern CCQE models used
by GENIE v3 and alternative event generators are very similar. This improved
picture originates from the fairly well-understood QE interaction channel that
has been extensively investigated. Conversely, the region
$\alpha_{3D}>135^{\circ}$ (Fig. 24) contains a large fraction of events that
undergo FSI, leading to an enhanced high-$p_{n}$ tail, which is under-
predicted by most generators. Furthermore, GiBUU shows an offset to the right
compared to other event generators and demonstrates the best agreement with
the data.

(a)

(b)

Figure 23:  The flux-integrated double-differential cross sections as a
function of $p_{n}$ for $\alpha_{3D}<$ 45∘ with (a) generator and (b) GENIE
configuration predictions compared to data. Inner and outer error bars show
the statistical and total (statistical and shape systematic) uncertainty at
the 1$\sigma$, or 68%, confidence level. The gray band shows the normalization
systematic uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf
calculation for each one of the predictions.

(a)

(b)

Figure 24:  The flux-integrated double-differential cross sections as a
function of 135∘ $<\alpha_{3D}<$ 180∘ with (a) Generator and (b) GENIE
configuration predictions compared to data. Inner and outer error bars show
the statistical and total (statistical and shape systematic) uncertainty at
the 1$\sigma$, or 68%, confidence level. The gray band shows the normalization
systematic uncertainty. The numbers in parentheses give the $\chi^{2}$/ndf
calculation for each one of the predictions.

Figures 25 and 26 show the double-differential measurement in $\alpha_{3D}$
for events with low and high values of $p_{n}$, respectively. For events with
low missing momentum, the distribution is very symmetric and approximately
follows the $\sin(\alpha_{3D})$ shape as expected. Final state interactions
predominantly remove events from this region of phase space, leading to
normalization differences between generators, with the exception of Gv2 which
illustrates a significantly different behavior. Conversely, events with high
missing momentum (Fig. 26) show a large asymmetry with strong enhancement in
the FSI-driven region at high values of $\alpha_{3D}$. Apart from Gv2, all
generator predictions show similar shapes with some normalization differences.
Gv2 predicts an enhanced tail at low values of $\alpha_{3D}$ which does not
appear in the data. This low-$\alpha_{3D}$, high-$p_{n}$ region contains a
large contribution ($\geq$ 50%) from MEC according to generator predictions.
Most models provide a reasonable description of the data in this region.

(a)

(b)

Figure 25:  The flux-integrated double-differential cross sections as a
function of $\alpha_{3D}$ for $p_{n}<$ 0.2 GeV/c with (a) generator and (b)
GENIE configuration predictions compared to data. Inner and outer error bars
show the statistical and total (statistical and shape systematic) uncertainty
at the 1$\sigma$, or 68%, confidence level. The gray band shows the
normalization systematic uncertainty. The numbers in parentheses give the
$\chi^{2}$/ndf calculation for each one of the predictions.

(a)

(b)

Figure 26:  The flux-integrated double-differential cross sections as a
function of $\alpha_{3D}$ for $p_{n}>$ 0.4 GeV/c with (a) Generator and (b)
GENIE configuration predictions compared to data. Inner and outer error bars
show the statistical and total (statistical and shape systematic) uncertainty
at the 1$\sigma$, or 68%, confidence level. The gray band shows the
normalization systematic uncertainty. The numbers in parentheses give the
$\chi^{2}$/ndf calculation for each one of the predictions.

A simultaneous extraction of the double-differential results across all parts
of the available phase-space is presented in the Supplemental Material.

##  V Conclusions

We report measurements of flux-integrated differential cross sections for
event topologies with a single muon and a single proton in the final state
using the Booster Neutrino Beam at Fermi National Accelerator Laboratory. The
data are recorded with the MicroBooNE detector and studied for the first time
in the form of single- and double-differential cross sections in novel
generalized kinematic imbalance variables that consider not only the
transverse, but also the longitudinal component of the missing momentum. These
generalized kinematic imbalance variables show sensitivity to nuclear effects,
leading to some significant differences between generator predictions and
data.

The GENIE v2.12.10 (Gv2) cross section predictions are systematically a poor
fit to data with significant shape and normalization differences across almost
all variables of interest. This is in contrast to several recent measurements
(e.g., [55, 56, 57, 58]) where Gv2 and the more modern GENIE v3 predictions
show similar shapes across many variables that only depend on either the muon
or the proton kinematics, whereas these kinematic imbalance variables also
depend on correlations between the leptonic and hadronic system. Gv2 is
therefore no longer able to provide a good description of the data.

GiBUU 2021 (GiBUU) is able to describe most distributions well, with the
exception of $\phi_{3D}$, $p_{n}$, and $p_{n\perp}$. GiBUU agrees particularly
well in areas of phase space where final state interactions have a very large
effect, for example $p_{n}$ for events with $\alpha_{3D}>135^{\circ}$, whereas
all of the other generators show large disagreements in those regions of phase
space.

The GENIE v3.0.6 G18_10a_02_11a cross section predictions with the MicroBooNE-
specific tuning (G18T), on the other hand, fit the $\phi_{3D}$, $p_{n}$, and
$p_{n\perp}$ data well. This contrasts with the GENIE v3.0.6 G18_10a_02_11a
configuration without additional tuning (G18) which shows a systematic deficit
of $\approx 20$%. The MicroBooNE-specific tuning modifies the normalization of
QE and MEC events, as well as the shape of MEC and RPA suppression, and none
of the tuning relied on data that included hadron kinematics. This difference
indicates that the underlying physics models in G18 can describe the
correlation between lepton and hadron kinematics, although the normalization
of some components need to be adjusted to better agree with data. However, in
regions of phase space sensitive to FSI, this model does not perform as well
as GiBUU, and the tuning does not improve the agreement.

The GENIE v3.2.0 G21_11b_00_000 configuration (G21) serves as an example of a
GENIE configuration that shows good agreement with data in most variables
without the need for additional tuning. NEUT produces similar agreement as G21
across many distributions. However, NEUT provides a better description of the
$\alpha_{3D}$ data, though a worse description of the $p_{n}$ data. In regions
of phase space sensitive to FSI, both of these models provide poor agreement
with data. In most variables and parts of the phase space, NuWro results in
$\chi^{2}$/ndf ratios fairly higher than unity.

The reported results provide precision data in new GKI variables that are more
sensitive to nuclear effects for the first time. This data could be used to
benchmark and tune neutrino-nucleus interaction models, particularly the
modeling of final state interactions.

##  VI 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.
This material is based upon work supported by Laboratory Directed Research and
Development (LDRD) funding from Argonne National Laboratory, provided by the
Director, Office of Science, of the U.S. Department of Energy under Contract
No. DE-AC02-06CH11357. 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); the UK Research and Innovation (UKRI) Future Leaders Fellowship; 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. For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC BY)
license to any Author Accepted Manuscript version arising from this
submission.

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