aainstitutetext: Dept. of Physics, University of Oxford,
Oxford OX1 3RH, UKbbinstitutetext: Dept. of Physics, University of Liverpool,
Liverpool L69 7ZE, UKccinstitutetext: Tel Aviv University,
Tel Aviv 69978, Israelddinstitutetext: University of Warwick,
Coventry CV4 7AL, United Kingdomeeinstitutetext: University of Minnesota Twin
Cities,
Minneapolis, MN 55455, USAffinstitutetext: Dept. of Physics and Astronomy,
University of Pittsburgh,
Pittsburgh PA 15260, USAgginstitutetext: Dept. of Physics and Astronomy, Tufts
University,
Medford MA 02155, USAhhinstitutetext: Instituto de Física Corpuscular,
46980, Paterna, Valéncia, Spainiiinstitutetext: Fermi National Accelerator
Laboratory,
Batavia, Illinois 60510, USAjjinstitutetext: Massachusetts Institute of
Technology,
Dept. of Physics, Cambridge, MA 02139, USAkkinstitutetext: Joint Institute for
Nuclear Research (JINR),
Dubna, Moscow region, 141980, Russiallinstitutetext: Alikhanov Institute for
Theoretical and Experimental Physics (ITEP) of NRC “Kurchatov Institute”,
Moscow, 117218, Russiamminstitutetext: CENBG, Université de Bordeaux,
CNRS/IN2P3, 33175 Gradignan, Francenninstitutetext: Argonne National
Laboratory,
Argonne, IL 60439, USAooinstitutetext: University of the Chinese Academy of
Sciences,
Beijing 100864, China

#  First combined tuning on transverse kinematic imbalance data with and
without pion production constraints

Weijun Li b Marco Roda c Júlia Tena-Vidal b Costas Andreopoulos d Xianguo Lu c
Adi Ashkenazi e Joshua Barrow f Steven Dytman g Hugh Gallagher h Alfonso
Andres Garcia Soto i Steven Gardiner c Matan Goldenberg i Robert Hatcher j Or
Hen k Igor D. Kakorin k,l Konstantin S. Kuzmin m Anselmo Meregalia k Vadim A.
Naumov n Afroditi Papadopoulou i Gabriel Perdue b Komninos-John Plows c Alon
Sportes i Noah Steinberg g Vladyslav Syrotenko g Jeremy Wolcott o,d and Qiyu
Yan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the first combined tuning, using genie, of four transverse
kinematic imbalance measurements of neutrino-hydrocarbon scattering, both with
and without pion final states, from the T2K and MINERvA experiments. As a
proof of concept, we have simultaneously tuned the initial state and final-
state interaction models (SF-CFG and hA, respectively), producing a new
effective model that more accurately describes the data.

††arxiv: 2404.08510

##  1 Introduction

Neutrinos play a central role in advancing our understanding of physics and
addressing fundamental inquiries in contemporary science. The Hyper-Kamiokande
experiment Hyper-Kamiokande:2018ofw  aims to conduct precise measurements of
charge parity violation within the neutrino sector, a phenomenon believed to
be closely linked to the observed matter-antimatter asymmetry in the universe.
Similarly, the Deep Underground Neutrino Experiment (DUNE) DUNE:2016hlj ;
DUNE:2015lol ; DUNE:2016evb ; DUNE:2016rla ; DUNE:2021tad  promises the same,
in addition to elucidating the neutrino mass ordering. Whether seeking to
ascertain Standard Model parameters or probing for exotic phenomena,
significant enhancements in both theoretical frameworks and computational
simulations of neutrino-nucleus interactions are imperative. As neutrinos
interact mainly with nucleons inside the nucleus, the interaction is subject
to nuclear effects, namely the nucleon initial state and final-state
interactions (FSIs). These effects are difficult to calculate and can alter
the final event topology by changing the number of final state pions with
respect to the interactions on free nucleons. Notably, the neutrino flux at
DUNE yields a comparable proportion of events with and without pions in the
final state, highlighting the pressing need for a generator capable of
accurately describing both event topologies.

The large data samples and superb imaging capabilities of modern neutrino
experiments offer us a detailed new look at neutrino interaction physics.
Recently, the GENIE Andreopoulos:2009rq ; GENIE:2021npt  Collaboration has
made substantial progress towards a global tuning using neutrino, charged
lepton and hadron scattering data, in an attempt to integrate new experimental
constraints with state-of-the-art theories and construct robust and
comprehensive simulations of neutrino interactions with matter. Cross-
experiment and cross-topology analyses are challenging tasks as each
measurement features its unique selection criteria and various other aspects,
such as the neutrino flux. genie has built an advanced tuning framework that
enables the validation and tuning of comprehensive interaction models using an
extensive curated database of measurements of neutrino, charged lepton and
hadron scattering off nucleus and nuclei. So far, the non-resonant backgrounds
GENIE:2021zuu , hadronization GENIE:2021wox  and the quasielastic (QE) and
2-particle-2-hole (2p2h) components GENIE:2022qrc  of the neutrino-nucleus
interaction have been tuned with $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ charged-
current (CC) pionless (0$\pi$) data from MiniBooNE, T2K and MINERvA. A partial
tune was performed for each experiment, highlighting the neutrino energy
dependence on the QE and 2p2h tuned cross sections. Even though post-tune
predictions enhanced the data description for each experiment, the added
degrees of freedom were not sufficient to fully describe all CC0$\pi$ data and
exhibited tensions with some proton observables GENIE:2022qrc . More exclusive
measurements result in additional model constraints. In addition, observables
that are sensitive to targeted aspects of the complex dynamics of neutrino
interactions are invaluable for model tuning. The transverse kinematic
imbalance (TKI) Lu:2015hea ; Lu:2015tcr , a final-state correlation between
the CC lepton and the handronic system, is a good example since it is
sensitive to the initial-state nuclear environment and hadronic FSI. Our next
step is to incorporate TKI data from experiments where various exclusive
topologies at different energies are considered. This marks the first combined
tuning on TKI data with and without pions in the final states and serves as
the starting point of a more comprehensive tuning effort in the energy region
most relevant for future accelerator-based neutrino experiments.

This paper is structured as follows: In Sec. 2 and Sec. 3, we review the TKI
measurements and genie models respectively. In Sec. 4, we detail the tuning
considerations and procedures. Results are summarized in Sec. 5, highlighting
how the data-MC discrepancy in the MINERvA-$\pi^{0}$ TKI measurement
MINERvA:2020anu  is resolved while maintaining good data-MC agreement
elsewhere. We conclude in Sec. 6.

##  2 TKI measurements

TKI is a methodology based on the conservation of momentum in neutrino
interactions. In essence, it involves quantifying the imbalance between the
observed transverse momentum of the final-state particles and the expected
transverse momentum from neutrino interactions with free nucleons Lu:2015hea ;
Lu:2015tcr . This “kinematic mismatch” together with its longitudinal and
three-dimensional variations Furmanski:2016wqo ; Lu:2019nmf , and the derived
asymmetry Cai:2019jzk , has been a crucial set of observables, establishing a
pathway to extract valuable information about the participating particles and
the underlying nuclear processes. Recent experimental results from neutrino
experiments such as T2K T2K:2018rnz ; T2K:2021naz , MINERvA MINERvA:2018hba ;
MINERvA:2019ope ; MINERvA:2020anu ; MINERvA:2021csy , and MicroBooNE
MicroBooNE:2022emb ; MicroBooNE:2023cmw ; MicroBooNE:2023tzj ;
MicroBooNE:2023wzy ; MicroBooNE:2024tmp , as well as electron scattering
experiments such as CLAS CLAS:2021neh , highlight the efficacy of TKI.

Figure 1:  Schematic illustration of the TKI variables. Diagram taken from
Ref. Lu:2015tcr .

In neutrino scattering off a free nucleon, the sum of the transverse
components of the final products is expected to be zero, visualized through a
back-to-back configuration between the final-state lepton and hadronic system
in the plane transverse to the neutrino direction. Hence, in a neutrino
interaction with a nucleus, the transverse momentum imbalance, $\delta
p_{\textrm{T}}$ Lu:2015tcr , results from intranuclear dynamics, including
Fermi motion and FSIs as shown in Fig. 1. The deviation from being back-to-
back is quantified by the coplanarity angle $\delta\phi_{\textrm{T}}$
Lu:2015tcr , while the transverse boosting angle, $\delta\alpha_{\textrm{T}}$
Lu:2015tcr , represents the direction of $\delta p_{\textrm{T}}$ within the
transverse plane. Furthermore, analyzing the energy and longitudinal momentum
budget Furmanski:2016wqo ; Lu:2019nmf  enables the conversion of $\delta
p_{\textrm{T}}$ to the emulated nucleon momentum, $p_{\textrm{N}}$, providing
further insight into the Fermi motion; this conversion amounts to a correction
on the order of $\mathcal{O}(20\%)$ Yang:2023dxk . With one-body currents in
the absence of FSIs, $\delta\alpha_{\textrm{T}}$ remains uniform (except for
second-order effects, such as variations in the center-of-mass energy), given
the isotropic nature of the initial nucleon motion. However, as the final
products propagate through the nuclear medium, they experience FSIs, thereby
disturbing the isotropy and the Fermi motion peak of the
$\delta\alpha_{\textrm{T}}$ and $p_{\textrm{N}}$ ($\delta p_{\textrm{T}}$)
distributions, respectively. Hence, $\delta p_{\textrm{T}}$ and
$p_{\textrm{N}}$ elucidate the Fermi motion details, while
$\delta\alpha_{\textrm{T}}$ characterizes the FSI strength—crucial for
understanding medium effects in neutrino interactions. A notable advantage of
these observables is their minimal dependence on neutrino energy Lu:2015tcr .
Moreover, the double TKI variable, $\delta p_{\textrm{TT}}$ Lu:2015hea , is
the projection of $\delta\vec{p}_{\textrm{T}}$ along the axis perpendicular to
the lepton scattering plane (hence “double”). In addition to its use for
extracting neutrino-hydrogen interactions Lu:2015hea ; Hamacher-
Baumann:2020ogq , it has also been applied to study nuclear effects in
neutrino pion productions MINERvA:2020anu ; T2K:2021naz . Its equivalent in
pionless production, $\delta p_{\textrm{Tx}}$, has been proposed and studied
together with its orthogonal companion, $\delta p_{\textrm{Ty}}$, in MINERvA
MINERvA:2019ope .

This work surveyed four TKI data sets: T2K $0\pi$ T2K:2018rnz , T2K $\pi^{+}$
T2K:2021naz , MINERvA $0\pi$ MINERvA:2018hba ; MINERvA:2019ope , and MINERvA
$\pi^{0}$ MINERvA:2020anu  measurements. All four measurements require the
presence of one CC muon and at least one proton in the final state. While the
T2K-$\pi^{+}$ measurement requires exactly one $\pi^{+}$, and
MINERvA-$\pi^{0}$ requires at least one $\pi^{0}$, the other two datasets
require the absence of any pions. The definitions of the kinematic cuts for
these samples are summarised in Table 1.

Variables | Cuts ($p$ in $\textrm{GeV}/c$)
---|---
T2K-$0\pi$ T2K:2018rnz
$\vec{p}_{\mu}$ |  $0.25<p_{\mu}$, $\cos\theta_{\mu}>-0.6$
$\vec{p}_{\textrm{p}}$ |  $0.45<p_{\text{p}}<1.0$ , $\cos\theta_{\text{p}}>0.4$
T2K-$\pi^{+}$ T2K:2021naz
$\vec{p}_{\mu}$ |  $0.25<p_{\mu}<7$ , $\theta_{\mu}<70^{\circ}$
$\vec{p}_{\textrm{p}}$ |  $0.45<p_{\text{p}}<1.2$ , $\theta_{\text{p}}<70^{\circ}$
$\vec{p}_{\pi}$ |  $0.15<p_{\pi}<1.2$, $\theta_{\pi}<70^{\circ}$
MINERvA-$0\pi$ MINERvA:2018hba ; MINERvA:2019ope
$\vec{p}_{\mu}$ |  $1.5<p_{\mu}<10$ , $\theta_{\mu}<20^{\circ}$
$\vec{p}_{\textrm{p}}$ |  $0.45<p_{\text{p}}<1.2$ , $\theta_{\text{p}}<70^{\circ}$
MINERvA-$\pi^{0}$ MINERvA:2020anu
$\vec{p}_{\mu}$ |  $1.5<p_{\mu}<20$ , $\theta_{\mu}<25^{\circ}$
$\vec{p}_{\textrm{p}}$ | $0.45<p_{\text{p}}$
Table 1:  Kinematic cuts for the samples of the TKI measurements.

Given the significant physics potential of TKI measurements, a wealth of
literature has emerged on the combined analysis of T2K and MINERvA CC0$\pi$
TKI data Dolan:2018zye ; Bourguille:2020bvw ; Franco-Patino:2021yhd ;
Ershova:2022jah ; Franco-Patino:2022tvv ; GENIE:2022qrc ; Chakrani:2023htw ;
Ershova:2023dbv . However, model studies incorporating TKI data with pion
production have been notably absent. Since the TKI measurements are sensitive
to the initial state and FSI, this work investigates a partial tune of these
two processes, in an effort to derive an effective model describing neutrino-
nuclear pion production, while maintaining the efficacy with the pionless
production.

##  3 genie model selection

genie uses formatted strings to uniquely define configurations, which are
usually called tunes. Regardless of the name, they can refer to configurations
either before or after an actual tuning procedure.

Free nucleon tuning effort in Ref. GENIE:2021zuu  has produced a tune,
G18_10a_02_11b, with better description of bubble chamber data. This is the
starting point of the model we want to tune. The first effect to apply to this
model is adding the proper initial state. Among the available genie initial
states, the better choice is the spectral-function-like Correlated Fermi gas
(SF-CFG) sfcfg-talk ; sfcfg-GitHubCommit ; GENIE:2021npt  because it has the
most advanced physics and also has the largest number of degrees of freedom
that can be adjusted by the tuning procedure. More details on the SF-CFG can
be found in Ref. GENIE:2021npt .

Further improvements to the configuration to be tuned include: 1) using
$z$-expansion axial-vector form factor Hill:2010yb  rather than the dipole
form factor of the Valencia model in QE processes Nieves:2004wx ; and 2)
replacing the Valencia model Nieves:2011pp  in 2p2h processes with the SuSAv2
Model Gonzalez-Jimenez:2014eqa  since it covers larger region of the $q^{0}$,
$q^{3}$ phase space.

Given its simplicity and interpretability, the INTRANUKE hA FSI model (hA for
short) emerges as the FSI candidate for the model to be tuned. The complete
list of the model components are given in Table 2. In the genie code, this
configuration is identified as G24_20i_00_000 and it is currently the default
genie configuration for a number of LAr based experiments. As G24_20i_00_000
is widely used in the paper, we will call it G24-0 for simplicity.

Simulation component | Model
---|---
Nuclear state | SF-CFG sfcfg-talk ; sfcfg-GitHubCommit ; GENIE:2021npt
QE | Valencia Nieves:2004wx
2p2h | SuSAv2 Gonzalez-Jimenez:2014eqa
QE $\Delta S=1$ | Pais Pais:1971er
QE $\Delta C=1$ | Kovalenko Kovalenko:1990zi
Resonance (RES) | Berger-Sehgal Berger:2007rq
Shallow/Deep inelastic scattering (SIS/DIS) | Bodek-Yang Bodek:2002vp
DIS $\Delta C=1$ | Aivazis-Tung-Olness Aivazis:1991fy
Coherent $\pi$ production | Berger-Sehgal Berger:2008xs
Hadronization | AGKY Yang:2009zx
FSI | INTRANUKE hA Andreopoulos:2015wxa
Table 2:  Model components of G24-0. Processes with non-trivial $\Delta S$ and
$\Delta C$ are those with strangeness and charm production, respectively.

Comparing the genie predictions for G24-0 against TKI datasets, Figs. 3 and 3,
shows that the model fails to accurately describe the MINERvA $\pi^{0}$
MINERvA:2020anu  TKI measurement.

Figure 2: $\delta\alpha_{\textrm{T}}$ measurements decomposed in interaction
types, compared to G24-0 prediction.

Figure 3:  Similar to Fig. 3 but for the $p_{\textrm{N}}$ (T2K-$\pi^{+}$,
MINERvA-$0\pi$ and MINERvA-$\pi^{0}$) and $\delta p_{\textrm{T}}$ (T2K-$0\pi$)
measurements.

##  4 Tuning initial-state and FSI models on TKI data

This study adopts the tuning procedure outlined in Ref. GENIE:2022qrc ,
utilizing $N_{\textrm{par}}$ model parameters. The objective is to identify
the best fit—that is, the optimal set of parameter values—within the parameter
space that minimizes the $\chi^{2}$ between model predictions and data. Given
the high dimensionality of the parameter space, conducting a brute-force,
point-by-point scan on a grid is infeasible. Therefore, a sufficient number of
points is randomly sampled. Each point is used to run a full simulation. The
simulation output for each data observable bin is then parameterized using
Professor Buckley:2009bj  with a polynomial of the model parameters of a
chosen order, $N_{\textrm{ord}}$. The minimal number of points
($N_{\textrm{s}}$) required for tuning $N_{\textrm{par}}$ parameters with an
order $N_{\textrm{ord}}$ polynomial is determined by the combination formula,

$N_{\textrm{s}}=\binom{N_{\textrm{par}}+N_{\textrm{ord}}}{N_{\textrm{ord}}}.$
(1)

Given that $N_{\textrm{s}}$ increases factorially, caution is essential when
augmenting the number of parameters or the polynomial degree. Given that our
tuning incorporates all $2$ SF-CFG and $12$ hA parameters (see Sec. 4.2), a
degree-$4$ parameterization necessitates over 6,000 generations. Hence, only
up to order $4$ polynomials are explored, and it has shown to reproduce the MC
predictions well, as shown in Fig. 4.

Figure 4:  Fractional difference for the bin-by-bin cross sections between MC
truth and the parameterized approximation with order-4 polynomials, both in
the Norm-Shape (NS) DAgostini:1993arp ; Hanson:2005mrg  space with AllPar. See
Table 4 for the definition of AllPar. The residual has a mean of $0.003$ and a
standard deviation of $0.016$.

Furthermore, to circumvent Peelle’s Pertinent Puzzle PPP_FNL , the Norm-Shape
(NS) transformation prescription DAgostini:1993arp ; Hanson:2005mrg  is
applied. The extremal point is then found from a minimization of $\chi^{2}$
between this NS-transformed polynomial approximation and NS-transformed data.
In minimization, priors, usually based on systematic uncertainty, can be
imposed on each parameter to penalize it from getting too far from its default
value. The following subsections elaborate on the specific choice of
measurement observables and model parameters in this work.

###  4.1  Data points

The observables for the reported differential cross sections across the four
TKI measurements—T2K-$0\pi$, T2K-$\pi^{+}$, MINERvA-$0\pi$, and
MINERvA-$\pi^{0}$—are detailed in Table 3. To identify the most sensitive
variables, various combinations were evaluated for tuning purposes. A total of
26 combinations were assessed, with the superset comprising
$\delta\alpha_{\textrm{T}}$, $\delta p_{\textrm{T}}$,
$\delta\phi_{\textrm{T}}$, $p_{\textrm{N}}$, and $\delta p_{\textrm{TT}}$ (see
Table 6 in Appendix A for all combinations). Each combination was evaluated
such that non-selected variables—including proton momentum and angle
($p_{\text{p}}$ and $\theta_{\text{p}}$) MINERvA:2018hba , as well as $\delta
p_{\textrm{Tx}}$ and $\delta p_{\textrm{Ty}}$ MINERvA:2019ope  in
MINERvA-$0\pi$—were used solely for validation; the model is expected to
accurately predict these variables post-tuning. When constructing the
combinations, it was observed that $\delta\phi_{\textrm{T}}$ is strongly
dependent on neutrino beam energy Lu:2015tcr , which suppresses its
sensitivity to nuclear effects. Additionally, correlations exist between
various observables, notably between $\delta p_{\textrm{T}}$ and
$p_{\textrm{N}}$. Therefore, determining the optimal combination is not
straightforward; this study employs a criterion based on the $\chi^{2}$ with
the complete (tuned plus validation) observable set (Fig. 5).

Observables | No. of bins | Combi-Superset | Combi-Best-AllPar | Combi-Best-RedPar
---|---|---|---|---
T2K-$0\pi$
$\delta\alpha_{\textrm{T}}$ | $8$ | $\checkmark$ |  | $\checkmark$
$\delta p_{\textrm{T}}$ | $8$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\delta\phi_{\textrm{T}}$ | $8$ | $\checkmark$ |  |
T2K-$\pi^{+}$
$\delta\alpha_{\textrm{T}}$ | $3$ | $\checkmark$ |  | $\checkmark$
$p_{\textrm{N}}$ | $4$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\delta p_{\textrm{TT}}$ | $5$ | $\checkmark$ |  | $\checkmark$
MINERvA-$0\pi$
$\delta\alpha_{\textrm{T}}$ | $12$ | $\checkmark$ |  | $\checkmark$
$p_{\textrm{N}}$ | $24$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\delta p_{\textrm{T}}$ | $24$ | $\checkmark$ | $\checkmark$ |
$\delta\phi_{\textrm{T}}$ | $23$ | $\checkmark$ |  |
$p_{\textrm{p}}$ | $25$ |  |  |
$\theta_{\textrm{p}}$ | $26$ |  |  |
$\delta p_{\textrm{Tx}}$ | $32$ |  |  |
$\delta p_{\textrm{Ty}}$ | $33$ |  |  |
MINERvA-$\pi^{0}$
$\delta\alpha_{\textrm{T}}$ | $9$ | $\checkmark$ |  | $\checkmark$
$p_{\textrm{N}}$ | $12$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\delta p_{\textrm{TT}}$ | $13$ | $\checkmark$ |  | $\checkmark$
Table 3:  Observables of the TKI measurements and their binning. Those with
“$\checkmark$”’s are used for tuning, while those without are for the
respective validation. See Table 4 for definitions of Combi-Best-RedPar and
Combi-Best-AllPar.  Figure 5:  Change of $\chi^{2}$ calculated for the full
(i.e., tuned plus validation) observable set as a function of the tuned
combination (cf. Table 6 in Appendix A). The two model parameter sets (AllPar
and RedPar, see Table 4 for definitions) are compared and it can be seen that
the respective minima happen at Combi-15 and 26.

###  4.2  Model parameters

As outlined in Sec. 3, we focus exclusively on the currently tunable
parameters within the SF-CFG and hA models. Therefore, $14$ parameters
(collectively referred to as the AllPar set) are utilized for tuning, as
detailed in Table 4. Nominal values and associated uncertainties for the hA
model are taken from Table 17.3 in Ref. Andreopoulos:2015wxa .

Parameter | Nominal (G24-0) | Range In Tuning |  AllPar (G24-t) |  RedPar (G24-c)
---|---|---|---|---
SF-CFG
$R_{\textrm{SRC}}$ | 0.12 | (0.0, 0.5) | ✓ | ✓
$E_{\textrm{RM}}^{\textrm{C}}$ | 0.01 | (0.0, 0.2) | ✓ |
hA
$S_{\lambda}^{\pi^{\pm}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ |
$S_{\lambda}^{\pi^{0}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ | ✓
$S_{\lambda}^{\textrm{N}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ |
$S_{\textrm{CEX}}^{\pi}$ | $1.0\pm 0.5$ | (0.0, 3.0) | ✓ | ✓
$S_{\textrm{CEX}}^{\textrm{N}}$ | $1.0\pm 0.5$ | (0.0, 3.0) | ✓ | ✓
$S_{\textrm{INEL}}^{\pi}$ | $1.0\pm 0.4$ | (0.0, 3.0) | ✓ |
$S_{\textrm{INEL}}^{\textrm{N}}$ | $1.0\pm 0.4$ | (0.0, 3.0) | ✓ |
$S_{\textrm{ABS}}^{\pi^{\pm}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ |
$S_{\textrm{ABS}}^{\pi^{0}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ |
$S_{\textrm{ABS}}^{\textrm{N}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ | ✓
$S_{\textrm{PIPD}}^{\pi}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ |
$S_{\textrm{PIPD}}^{\textrm{N}}$ | $1.0\pm 0.2$ | (0.0, 3.0) | ✓ | ✓
Table 4:  Tuneable parameters and their ranges in the SF-CFG (uppermost group)
and hA (lower groups, uncertainties from Ref. Andreopoulos:2015wxa ) models.
Parameters to be tuned in the two sets are marked with “✓”’s. See later text
for definitions of G24-t and G24-c.

In our analysis of the SF-CFG model, the two most relevant parameters are
$R_{\textrm{SRC}}$, representing the fraction of the high-momentum tail from
short-range correlations (SRC), notably above the Fermi surface, and
$E_{\textrm{RM}}^{\textrm{C}}$, marking the commencement of the nuclear
removal energy distribution for carbon. A larger $R_{\textrm{SRC}}$ indicates
the presence of more energetic initial nucleons, while an increased
$E_{\textrm{RM}}^{\textrm{C}}$ implies that greater energy is necessary to
liberate a nucleon from the carbon nucleus, so the product particles will
possess lower energy. Given the novelty and limited constraints of the
spectral-function-like implementation, this study employs relatively relaxed
priors for $R_{\textrm{SRC}}$ and $E_{\textrm{RM}}^{\textrm{C}}$, at
0.12$\pm$0.12 and 0.01$\pm$0.005 GeV, respectively.

genie users employ the hA model Andreopoulos:2015wxa  for the majority of
neutrino analysis, owing to its straightforward reweighting capability. Unlike
cascade models such as the hN model Andreopoulos:2015wxa , this model
determines the FSI type for each hadron exactly once, without considering
further propagation of rescattered hadrons within the nucleus. Its model
parameters are explained as follows.

  1. 1.

The model evaluates a mean free path (MFP), $\lambda$, which varies with the
hadron’s energy, $E$, and its radial distance, $r$, from the nucleus center,
as follows:

$\lambda(E,r)=\frac{1}{\sigma_{\textrm{hN,tot}}(E)\rho(r)},$ (2)

where $\sigma_{\textrm{hN,tot}}$ is the total hadron-nucleon cross section and
$\rho$ is the number density of the nucleons inside the nucleus. Once a hadron
is generated inside the nucleus, it travels a distance $\lambda$ before a
probabilistic determination of rescattering occurs, inversely related to
$\lambda$. If no rescattering happens, the hadron will be propagated for
another $\lambda$ and the dice will be thrown again. The cycle repeats until a
rescattering happens or the hadron is propagated far enough to escape the
nucleus without any interactions. If a rescattering happens, the type of
rescattering is evaluated according to their cross section. The relative
probabilities of each type are extracted from stored hadron scattering data
LADS:1999dyv ; Navon:1983xj ; Carroll:1976hj ; Clough:1974qt ; BAUHOFF1986429
; Mashnik:2000up ; Ishibashi:1997gbe . After a specific rescattering type is
chosen, the rescattered product will be generated. These rescattered products
will not undergo further rescattering and will be considered out of the
nucleus as final state particles. The MFP values for pions and nucleons can be
modified using scaling factors ($S_{\lambda}^{\pi^{\pm}}$,
$S_{\lambda}^{\pi^{0}}$, and $S_{\lambda}^{\textrm{N}}$) detailed in Table 4.
Both changed pions, $\pi^{+}$ and $\pi^{-}$, are scaled by the same factor,
$S_{\lambda}^{\pi^{\pm}}$, due to isospin symmetry. The $\lambda$ for
$\pi^{0}$ is calculated from those of charged pions based on isospin symmetry
as well and can be adjusted separately by $S_{\lambda}^{\pi^{0}}$. If the
factor is larger than $1$, $\lambda$ will increase and hence the total number
of steps required for the hadron to escape the nucleus reduces, thereby
reducing the average probability of rescattering. Given the high precision
with which total hadron-nucleus cross sections are determined LADS:1999dyv ;
Navon:1983xj ; Carroll:1976hj ; Clough:1974qt ; BAUHOFF1986429 , strict
Gaussian priors (error—$\sigma$, not to be confused with the various cross
sections—of $0.2$, same as the systematic uncertainties shown in Table 4) are
applied to the MFP scaling factors. Varying $S_{\lambda}^{\pi^{0}}$ by $\pm
2.5~{}\sigma$ significantly affects the MINERvA-$\pi^{0}$
$\textrm{d}\sigma/\textrm{d}p_{\textrm{N}}$, as depicted in Fig. 6. Decreasing
MFP naturally leads to increased rescattering. Hence, fewer $\pi^{0}$s can
escape the nucleus and fewer events will have $\pi^{0}$ in the final states,
thereby reducing the cross section.

Figure 6: Effect of varying $S_{\lambda}^{\pi^{0}}$ by $\pm 2.5~{}\sigma$
compared to the MINERvA-$\pi^{0}$ measurement. Each band’s width indicates the
genie prediction’s statistical uncertainty from $10^{5}$ events.

  2. 2.

The relative probability of each rescattering type can be adjusted by a scale
factor. There are four rescattering types for both nucleons and pions
available for tuning. They are charge exchange (CEX), inelastic scattering
(INEL), absorption (ABS), and pion production (PIPD). The default energy-
dependent probability distributions come from hadron data tuning LADS:1999dyv
; Navon:1983xj ; Carroll:1976hj ; Clough:1974qt ; BAUHOFF1986429  and can be
adjusted by respective scaling factors, like $S_{\textrm{CEX}}^{\pi}$ for CEX
of all pions, listed in Table 4. The relative probabilities of all four
rescattering types will be normalized such that they sum up to one. Scaling
factors directly multiply the probability of each type, with new relative
probabilities subsequently renormalized. More specifically, suppose the
initial fraction of absorption, $f_{\textrm{ABS}}$, is $0.5$, which means that
the sum of the other fractions is
$f_{\textrm{other}}=f_{\textrm{INEL}}+f_{\textrm{CEX}}+f_{\textrm{PIPD}}=1-0.5=0.5$.
If $f_{\textrm{ABS}}$ is scaled up with $S_{\textrm{ABS}}=2$, the new
absorption fraction is

$f^{\prime}_{\textrm{ABS}}=\frac{f_{\textrm{ABS}}*S_{\textrm{ABS}}}{f_{\textrm{ABS}}*S_{\textrm{ABS}}+f_{\textrm{oth}}}=2/3,$
(3)

which is not equal to simply multiplying $f_{\textrm{ABS}}$ by
$S_{\textrm{ABS}}$ due to the presence of the other rescattering types. Hence,
it is unavoidable that a change of the scaling factor for one type would
affect others. Nonetheless, when the cross section is plotted with a breakdown
according to rescattering type, the interpretation will be clear regarding the
increase or decrease of a particular FSI type. This correlation implies that
the scaling parameters’ effective impact on FSI fate is often less pronounced
than what the raw numbers suggest. As illustrated in the above example,
$f_{\textrm{ABS}}$ is only scaled up by
$f^{\prime}_{\textrm{ABS}}/f_{\textrm{ABS}}=4/3=1.3$ instead of
$S_{\textrm{ABS}}=2$. Hence, a relatively more relaxed Gaussian prior, with a
$\sigma$ of $0.5$, is placed on the FSI fate scales, slightly larger than the
systematic uncertainties shown in Table 4.

Understanding each rescattering type is crucial for interpreting tuning
results. A detailed elaboration of the four tunable FSI types is given as
follows.

    1. (a)

CEX involves changing the charge of the participating particles; for example,

$\pi^{+}+\textrm{n}\rightarrow\pi^{0}+\textrm{p},$ (4)

or vice versa. This rescattering type is crucial for event topologies
requiring the presence of a pion; depending on the signal pion charge, CEX
could migrate events between signal and background.

    2. (b)

INEL is the case where the nucleus is left in an excited state after the
rescattering. This category only contains the situation where a single
additional nucleon is emitted/knocked-out after rescattering. Since it does
not affect the number of pions produced, it will not convert an event from a
pion-less topology to a pion-production topology. The effects on nucleons are
two-fold. Firstly, it can alter the number of signal events within each event
topology. If the inelastic rescattering leads to two low-momentum protons
below the detection threshold as opposed to a high-momentum proton, this
signal event will be discarded as no protons are observed. Secondly, INEL
invariably changes the kinematics of the rescattering particle. Be it the
leading proton or the leading pion, based on which the TKI observables are
calculated, the TKI distribution shape will be affected. Hence, while
$S_{\textrm{INEL}}^{\textrm{N}}$ would affect all four data sets,
$S_{\textrm{INEL}}^{\pi}$ would only affect T2K-$\pi^{+}$ and
MINERvA-$\pi^{0}$.

    3. (c)

ABS refers to the case where the particle undergoes an interaction so that it
does not emerge as a final particle. For example, $\pi^{+}$ can interact with
two or more nucleons, initially forming a baryon resonance that subsequently
interacts with other nucleons, emitting multiple nucleons rather than pions.
Hence, the $\pi^{+}$ would not emerge from the nucleus anymore.

    4. (d)

PIPD happens for energetic particles where an extra pion emerges as a result
of the rescattering, such as

$\textrm{p}+\textrm{p}\rightarrow\textrm{p}+\textrm{n}+\pi^{+}.$ (5)

Such an interaction significantly alters the event topology.

##  5 The first TKI-driven genie tunes

Given the intricate correlations among model parameters, determining which
ones are most sensitive to the data proves challenging. Considerable
correlation and anti-correlation are observed between different FSI fates for
the same particle, e.g. $S_{\textrm{ABS}}^{\pi^{\pm}}$ and
$S_{\textrm{CEX}}^{\pi}$, as shown in Figs. 7(a) and 7(b). Our tuning began
with the AllPar parameter set, achieving the best result with Combi-15, which
is denoted as Combi-Best-AllPar in Table 3 (cf. Fig. 5 for the $\chi^{2}$
distribution of all combinations). Observations showed that for most
combinations, certain parameters remained close to their default values,
within $1$ $\sigma$ of the imposed priors, such as $S_{\textrm{PIPD}}^{\pi}$
since none of the data sets has a significant contribution from PIPD of
pions—Removing these parameters from the tuning process does not significantly
affect the outcome. Therefore, we then constructed a reduced set comprising
only $R_{\textrm{SRC}}$, $S_{\lambda}^{\pi^{0}}$, $S_{\textrm{CEX}}^{\pi}$,
$S_{\textrm{CEX}}^{\textrm{N}}$, $S_{\textrm{ABS}}^{\textrm{N}}$, and
$S_{\textrm{PIPD}}^{\textrm{N}}$ (denoted as RedPar in Table 4) and ran the
tuning on the 26 combinations with it. Note that the parameters most
correlated with $S_{\textrm{CEX}}^{\pi}$ and $S_{\lambda}^{\pi^{0}}$ are not
present in RedPar. The RedPar tuning proved more stable, with nearly all
combinations showing a more negative $\chi^{2}$ change than the AllPar tuning,
as shown in Fig. 5. Moreover, the single-observable (Combi-1 to 5), single-
measurement (Combi-6 to 9), single-experiment (Combi-10 and 11), and single-
topology (Combi-12 and 13) combinations all systematically yielded no or
limited overall (tuned plus validation) improvement post-tuning. The best
tuning then occurred with Combi-26 (Combi-Best-RedPar in Table 3), that is,
all TKI variables excluding $\delta\phi_{\textrm{T}}$ and $\delta
p_{\textrm{T}}$ (unless no $p_{\textrm{N}}$ is
available)—$\delta\alpha_{\textrm{T}}$, $p_{\textrm{N}}$ (or $\delta
p_{\textrm{T}}$ if $p_{\textrm{N}}$ is unavailable), and $\delta
p_{\textrm{TT}}$—from pionless and pion-production measurements in both T2K
and MINERvA.

(a) AllPar

(b) RedPar

Figure 7:  Post-fit Correlation coefficient on Combi-26.  Parameter | Nominal (G24-0) |  RedPar (G24-c) |  AllPar (G24-t)
---|---|---|---
SF-CFG
$R_{\textrm{SRC}}$ | 0.12 | 0.15 $\pm$ 0.08 | 0.30 $\pm$ 0.05
$E_{\textrm{RM}}^{\textrm{C}}$ | 0.01 | 0.01 | 0.011 $\pm$ 0.003
hA
$S_{\lambda}^{\pi^{\pm}}$ | $1.0\pm 0.2$ | 1.0 | 1.11 $\pm$ 0.16
$S_{\lambda}^{\pi^{0}}$ | $1.0\pm 0.2$ | 0.22 $\pm$ 0.07 | 0.17 $\pm$ 0.06
$S_{\lambda}^{\textrm{N}}$ | $1.0\pm 0.2$ | 1.0 | 1.20 $\pm$ 0.12
$S_{\textrm{CEX}}^{\pi}$ | $1.0\pm 0.5$ | 0.26 $\pm$ 0.12 | 1.53 $\pm$ 0.37
$S_{\textrm{CEX}}^{\textrm{N}}$ | $1.0\pm 0.4$ | 1.43 $\pm$ 0.34 | 1.41 $\pm$ 0.38
$S_{\textrm{INEL}}^{\pi}$ | $1.0\pm 0.4$ | 1.0 | 0.67 $\pm$ 0.30
$S_{\textrm{INEL}}^{\textrm{N}}$ | $1.0\pm 0.4$ | 1.0 | 1.26 $\pm$ 0.48
$S_{\textrm{ABS}}^{\pi^{\pm}}$ | $1.0\pm 0.2$ | 1.0 | 1.59 $\pm$ 0.31
$S_{\textrm{ABS}}^{\pi^{0}}$ | $1.0\pm 0.2$ | 1.0 | 0.90 $\pm$ 0.28
$S_{\textrm{ABS}}^{\textrm{N}}$ | $1.0\pm 0.2$ | 0.25 $\pm$ 0.28 | 0.28 $\pm$ 0.27
$S_{\textrm{PIPD}}^{\pi}$ | $1.0\pm 0.2$ | 1.0 | 1.12 $\pm$ 0.30
$S_{\textrm{PIPD}}^{\textrm{N}}$ | $1.0\pm 0.2$ | 2.05 $\pm$ 0.48 | 1.27 $\pm$ 0.48
$\chi^{2}$ for combi
untuned |  | 231.75 | 161.26
tuned |  | 174.84 | 122.53
diff |  | -56.91 | -38.73
$\chi^{2}$ for vald
untuned |  | 229.5 | 299.99
tuned |  | 214.7 | 263.41
diff |  | -14.8 | -36.58
$\chi^{2}$ for combi+vald
untuned |  | 461.25 | 461.25
tuned |  | 389.54 | 385.94
diff |  | -71.71 | -75.31
Table 5:  Parameters in G24-0, G24-c, and G24-t. Those without errors are not
tuned. Lower section: combi indicates that the following $\chi^{2}$ sums are
calculated for the tuned measurements (i.e. Combi-Best-RedPar (98 bins) and
Combi-Best-AllPar (72 bins) for G24-c and G24-t, respectively), while vald
indicates that the respective validation sets are used; untuned means that the
$\chi^{2}$ calculation uses the nominal values of the parameters, while tuned
denotes the tuned ones. diff displays the improvement achieved by the
respective tuning.

Table 5 summarizes the primary results of our tuning process. The tuned
RedPar, G24_20i_06_22c (G24-c), is derived from the observable set Combi-Best-
RedPar, and the tuned AllPar, G24_20i_fullpara_alt (G24-t), is from Combi-
Best-AllPar. The upper part of Table 5 contains the parameter values. For the
G24-c tune, changes to the SF-CFG model were moderately sized
($R_{\textrm{SRC}}$ increases from $0.12$ to $0.15$), while in the hA model,
$S_{\lambda}^{\pi^{0}}$ and $S_{\textrm{CEX}}^{\pi}$ are highly suppressed and
the nucleon FSI has larger CEX and PIPD and lower ABS. The tune G24-t differs
most significantly from G24-c in $R_{\textrm{SRC}}$ and
$S_{\textrm{CEX}}^{\pi}$. The table’s lower section illustrates $\chi^{2}$
improvements for selected observables and their corresponding validation sets.
Crucially, tuning should also enhance the accuracy of validation set
descriptions. Overall, both tunes have substantial reduction in the total
$\chi^{2}$. In the following, we will discuss the two tunes.

###  5.1 Reduced tune: G24-c

To assess the tune’s quality, we plot predictions using G24-c for
$\delta\alpha_{\textrm{T}}$ and $p_{\textrm{N}}$ in Figs. 9 and 9,
respectively. For T2K-$0\pi$, T2K-$\pi^{+}$ and MINERvA-$0\pi$, the new
$\chi^{2}$ values are comparable to those obtained with G24-0, changing less
than the number of bins. For MINERvA-$\pi^{0}$, G24-c is distinctly better
than G24-0, thereby demonstrating the possibility of simultaneous good
descriptions of both pion-less and pion production samples with constrained
parameters from cross-topology TKI tuning.

Figure 8:  Similar to Fig. 3 but with G24-c. The G24-0 prediction is also
plotted for comparison.

Figure 9:  Similar to Fig. 9 but for the $p_{\textrm{N}}$ (T2K-$\pi^{+}$,
MINERvA-$0\pi$ and MINERvA-$\pi^{0}$) and $\delta p_{\textrm{T}}$ (T2K-$0\pi$)
measurements.

Examining individual datasets closely helps to understand the origin of the
improvement. The decomposition of the cross section for
$\delta\alpha_{\textrm{T}}$ according to the FSI fates of the $\pi^{0}$ is
presented in Fig. 10. The hA model rescatters primary interaction products
only once, and records this event with a rescattering code for these
particles. The rescattered products, which are the final outputs, are stored
as daughters of the primary interaction products. Hence, the FSI fate that
produces the final products can be checked by the rescattering code of their
first parent. More specifically, we loop through all final-state particles to
look for the leading $\pi^{0}$ and check the rescattering code of its first
parent.

Figure 10:  MINERvA-$\pi^{0}$ $\delta\alpha_{\textrm{T}}$ measurement compared
to genie predictions decomposed in $\pi^{0}$ FSI fates with (a) G24-0 and (b)
G24-c.

In MINERvA-$\pi^{0}$, the number of $\pi^{0}$s undergoing no FSI (“No
Interaction”, as shown in the figures) is adjustable via the
$S_{\lambda}^{\pi^{0}}$ parameter. A decrease in $S_{\lambda}^{\pi^{0}}$
reduces the size of the “No Interaction” events, as is shown in Fig. 10b when
compared to Fig. 10a. Meanwhile, an increase in $\pi^{0}$ rescattering only
manifests in pion-less measurements through ABS. There is indeed an increase
of ABS for $\pi^{0}$ in T2K-$0\pi$ and MINERvA-$0\pi$, but the fraction is
small to begin with, so the impact is small. The increase of $\pi^{0}$
rescattering can increase T2K-$\pi^{+}$ via CEX as discussed below. However,
due to the significant suppression of CEX, its impact on T2K-$\pi^{+}$ is also
minimal (for detailed breakdown, see Figs. 15-17 in Appendix B).

As shown in Fig. 10a, the MINERvA-$\pi^{0}$ prediction from the nominal tune
has considerable contributions from CEX controlled by
$S_{\textrm{CEX}}^{\pi}$. It does not impact T2K-$0\pi$ and MINERvA-$0\pi$ as
CEX only changes the pion type without removing them. Hence, events with pions
in the final state would be rejected in these two data sets regardless of the
pion charge. In principle, CEX will also migrate events between the signal and
background definitions for T2K-$\pi^{+}$ when a $\pi^{0}$ is converted to a
$\pi^{+}$ and vice verse. However, T2K-$\pi^{+}$ does not have a considerable
CEX fraction to begin with, so changing CEX will have a minimal impact on its
prediction. Hence, this is one of the most effective paths to decrease the
MINERvA-$\pi^{0}$ cross section prediction without affecting other
measurements, and indeed the $S_{\textrm{CEX}}^{\pi}$ is heavily suppressed as
shown in Table 5 and illustrated in Fig. 10b compared to Fig. 10a. However,
due to the intricate correlation between the FSI fates in the hA
implementation discussed in Sec. 4.2, although $S_{\textrm{INEL}}^{\pi}$ and
$S_{\textrm{PIPD}}^{\pi}$ are not modified explicitly, a considerable increase
is observed in both INEL and PIPD for MINERvA-$\pi^{0}$.

Figure 11:  Similar to Fig. 10 but for $p_{\textrm{N}}$ with proton FSI fate
decomposition. Note that the “Absorption” of the proton is referring to the
absorption of the $\pi^{+}$ from the decay of $\Delta^{++}$, which could lead
to emission of nucleons as discussed in Sec. 4.2.

Figure 12:  MINERvA-$\pi^{0}$ $p_{\textrm{N}}$ measurement compared to genie
predictions decomposed in (a) $\nu$-N interaction, (b) $\pi^{0}$ FSI, and (c)
proton FSI for the alternative tune, G24-t.

Suppressing both “No Interaction” and CEX for $\pi^{0}$ adjusts the
MINERvA-$\pi^{0}$ prediction to the appropriate magnitude with small impacts
on other data sets. The effects of the other parameter changes are more
transparent in the comparison of the $p_{\textrm{N}}$ distribution of
MINERvA-$\pi^{0}$ between Figs. 11a and 11b. A relatively large increase in
$S_{\textrm{CEX}}^{\textrm{N}}$ and $S_{\textrm{PIPD}}^{\textrm{N}}$ moves
events away from the Fermi motion peak at $p_{\textrm{N}}\leq
0.25~{}\textrm{GeV}/c$; the same effect can be achieved by an increased
$R_{\textrm{SRC}}$.

###  5.2 Alternative tune: G24-t

A similar improvement is achieved by the intermediate tune, G24-t, illustrated
in Fig. 12. Instead of significantly increasing proton PIPD to elevate the
$p_{\textrm{N}}$ tail, this tune notably enhances $R_{\textrm{SRC}}$—less-
energetic final states are present and therefore an increased RES/DIS (Fig.
12a) ratio. Rather than suppressing $S_{\textrm{CEX}}^{\pi}$ (cf. Fig. 17d in
Appendix B), this tune significantly increases it (Fig. 12b) to offset the
further reduction of “No Interaction” $\pi^{0}$ events (smaller
$S_{\lambda}^{\pi^{0}}$ in AllPar, cf. Table 5). Overall, this leads to a
reduction of $75.31$ in $\chi^{2}$ (Table 5), similar to G24-c, as well as a
good data-MC agreement across all data sets.

Figure 13:  Change in MC prediction for $\pi^{0}$ cross section between G24-0
and G24-c.

###  5.3 Discussion

For both G24-c and G24-t the fit results suggest extreme values, but the
effect of the parameter change is less effective than the change in the
parameter suggests. For the individual FSI fate cross section, due to the re-
normalization step after scaling, the effective change for one particular fate
is smaller than the scaling parameter magnitude as explained in Sec. 4.2. As
for the total FSI cross sections, only that of $\pi^{0}$ is changed. The total
$\pi^{0}$-C scattering cross section for pion kinetic energy is plotted in
Fig. 13 to assess the overall impact. The shape remains similar and consistent
with Fig. 2.23 ($\pi^{+}$-C reactions) in Ref. Andreopoulos:2015wxa  while the
peak magnitude increases from $380$ to $538$ mb, a $48\%$ increase,
considerably less than the scaling factor, $S_{\lambda}^{\pi^{0}}=0.22$,
seemingly suggests. Moreover, the default $\pi^{0}$ parameter values are
calculated from charged pion data assuming isospin symmetry rather than
extracted directly from experimental data. Hence, this modification does not
violate any existing agreement with hadron scattering data.

In addition, agreement of this tune with most of the non-TKI neutrino datasets
available is unchanged before and after the tune, thereby demonstrating the
physicality of the tune. In conclusion, this tune is a valid and effective
model that can be used as a starting point for an analysis. Further refinement
will be available once more data will be included in the fit, including non-
TKI observables.

##  6  Summary and outlook

This work represents the first global tuning effort on TKI data. Our partial
tune of the SF-CFG and hA models, G24_20i_06_22c (G24-c), provides an
effective theory to better describe both the neutrino-hydrocarbon pionless and
pion production. The largest change in the model is demanded by the
MINERvA-$\pi^{0}$ TKI measurement MINERvA:2020anu  that was significantly
overestimated in genie. The improvement is crucial for future precision GeV
neutrino experiments. This tuning configuration has been integrated into the
master branch of genie and is slated for inclusion in the upcoming release.

To develop an effective model, we focused on the most sensitive parameters,
$R_{\textrm{SRC}}$, $S_{\lambda}^{\pi^{0}}$, $S_{\textrm{CEX}}^{\pi}$,
$S_{\textrm{CEX}}^{\textrm{N}}$, $S_{\textrm{ABS}}^{\textrm{N}}$, and
$S_{\textrm{PIPD}}^{\textrm{N}}$, of the SF-CFG and hA models, reducing the
total number of parameters from $14$ to $6$. The corresponding best
combination of observables is $\delta\alpha_{\textrm{T}}$, $p_{\textrm{N}}$
(or $\delta p_{\textrm{T}}$ if no $p_{\textrm{N}}$ is available), and $\delta
p_{\textrm{TT}}$, from pionless and pion-production measurements in both T2K
and MINERvA. In doing so, the pion production model is held fixed such that
the tuned values of $S_{\lambda}^{\pi^{0}}$, $S_{\textrm{CEX}}^{\pi}$,
$S_{\textrm{ABS}}^{\textrm{N}}$, and $S_{\textrm{PIPD}}^{\textrm{N}}$ depart
from the established constraints from hadron measurements. A different pion
production model might compensate part of these model tuning effects
Yan:2024kkg . We also chose the hA FSI model for practical reasons. Its
sequential treatment of MFP and rescattering should be considered as a
numerical advantage (as was also pointed out in Ref. GENIE:2022qrc ). Other
more sophisticated models could be considered in future tunes. With the
existing four TKI measurements from T2K and MINERvA, we have also derived
other effective tunes like G24_20i_fullpara_alt (G24-t). The degeneracy can be
resolved with additional data, particularly from argon-based measurements
MicroBooNE:2022emb ; MicroBooNE:2023cmw ; MicroBooNE:2023tzj ;
MicroBooNE:2023wzy ; MicroBooNE:2024tmp ; MicroBooNE:2015bmn .

## Acknowledgment

We thank the CC-IN2P3 Computing Center providing computing resources and for
their support. X.L. is supported by the STFC (UK) Grant No. ST/S003533/2.
J.T.-V. acknowledges the support of the Raymond and Beverly Sackler
scholarship.

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##  Appendix A Combinations of observables

Combi- |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |  10 |  11 |  12 |  13 |  14 |  15 |  16 |  17 |  18 |  19 |  20 |  21 |  22 |  23 |  24 |  25 |  26
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
T2K-$0\pi$
$\delta\alpha_{\textrm{T}}$ |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$\delta p_{\textrm{T}}$ |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$\delta\phi_{\textrm{T}}$ |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |
T2K-$\pi^{+}$
$\delta\alpha_{\textrm{T}}$ |  |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$p_{\textrm{N}}$ |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$\delta p_{\textrm{TT}}$ |  |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$
MINERvA-$0\pi$
$\delta\alpha_{\textrm{T}}$ |  |  |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$p_{\textrm{N}}$ |  $\checkmark$ |  |  |  |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$\delta p_{\textrm{T}}$ |  |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |
$\delta\phi_{\textrm{T}}$ |  |  |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |
MINERvA-$\pi^{0}$
$\delta\alpha_{\textrm{T}}$ |  |  |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$p_{\textrm{N}}$ |  $\checkmark$ |  |  |  |  |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
$\delta p_{\textrm{TT}}$ |  |  |  |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$
Table 6:  Specifications of observable combinations within the tuning superset
in Table 3. Combi-15 is Best-AllPar, Combi-24 is Superset, and Combi-26 is
Best-RedPar.

Table 6 displays the tested combinations of observables:

  * •

Combi-1 through 5 are cross-experiment selections of a single observable. For
example, Combi-$3$ uses only $\delta\alpha_{\textrm{T}}$ from all four data
sets. If a chosen observable is absent from a dataset, that dataset is
excluded from the specific combination. For example, T2K-$0\pi$ is not used in
Combi-1 due to the absence of $p_{\textrm{N}}$.

  * •

Combi-6 through 9 incorporate all variables from a single measurement. For
example, Combi-$9$ uses MINERvA-$\pi^{0}$ only.

  * •

Combi-10 to 13 uses two out of four measurements according to the experiment
or topology. For example, Combi-$10$ uses only the two data sets from T2K, and
Combi-$13$ only with pion production.

  * •

Combi-14 to 17 combine two separate combinations:

Combi-14 $\displaystyle=\texttt{Combi-3}\cup\texttt{Combi-5},$ Combi-15
$\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-2},$ Combi-16
$\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-3},$ Combi-17
$\displaystyle=\texttt{Combi-2}\cup\texttt{Combi-3}.$

  * •

Combi-18 through 22 result from merging three combinations:

Combi-18
$\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-3}\cup\texttt{Combi-5},$
Combi-19
$\displaystyle=\texttt{Combi-2}\cup\texttt{Combi-3}\cup\texttt{Combi-5},$
Combi-20
$\displaystyle=\texttt{Combi-3}\cup\texttt{Combi-4}\cup\texttt{Combi-5},$
Combi-21
$\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-2}\cup\texttt{Combi-3},$
Combi-22
$\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-2}\cup\texttt{Combi-4}.$

  * •

Combi-23 merges four different combinations:

Combi-23 $\displaystyle=\texttt{Combi-1}\cup\texttt{Combi-2}\cup$
$\displaystyle\texttt{Combi-3}\cup\texttt{Combi-5}.$

  * •

Combi-24 encompasses all variables, acting as the superset.

  * •

Combi-25 and 26 are the same as Combi-$21$ and $23$, respectively, except that
$\delta p_{\textrm{T}}$ in MINERvA-$0\pi$ is removed to avoid correlation with
$p_{\textrm{N}}$ in the same data set.

Figure 5 shows the change in $\chi^{2}$ for the complete observable set (tuned
plus validation) as a function of the tuning combination, where the two model
parameter sets, AllPar and RedPar, are compared and it can be seen that the
respective minima happen at Combi-15 and 26.

##  Appendix B $\pi^{0}$ FSI fate decomposition for G24-0 and G24-c

Figures 15-17 display comparisons of G24-0 and G24-c predictions to data,
detailing the composition by $\pi^{0}$ FSI fate.

Figure 14:  Extension of figure 10a to all four $\delta\alpha_{\textrm{T}}$
measurements.

Figure 15:  Extension of figure 10a but to all four $p_{\textrm{N}}$
measurements.

Figure 16:  Extension of figure 10b to all four $\delta\alpha_{\textrm{T}}$
measurements.

Figure 17:  Extension of figure 10b but to all four $p_{\textrm{N}}$
measurements.