# Nuclear de-excitations in low-energy charged-current $\nu_{e}$ scattering on
[40]Ar

Steven Gardiner<EMAIL_ADDRESS>Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 USA  Department of Physics, University of California,
Davis, California 95616 USA

###### Abstract

Background

Large argon-based neutrino detectors, such as those planned for the Deep
Underground Neutrino Experiment, have the potential to provide unique
sensitivity to low-energy (few to tens of MeV) electron neutrinos produced by
core-collapse supernovae. Despite their importance for neutrino energy
reconstruction, nuclear de-excitations following charged-current $\nu_{e}$
absorption on [40]Ar have never been studied in detail at supernova energies.

Purpose

I develop a model of nuclear de-excitations that occur following the
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ reaction. This model is
applied to the calculation of exclusive cross sections.

Methods

A simple expression for the inclusive differential cross section is derived
under the allowed approximation. Nuclear de-excitations are described using a
combination of measured $\gamma$-ray decay schemes and the Hauser-Feshbach
statistical model. All calculations are carried out using a novel Monte Carlo
event generator called MARLEY (Model of Argon Reaction Low Energy Yields).

Results

Various total and differential cross sections are presented. Two de-excitation
modes, one involving only $\gamma$-rays and the other including single neutron
emission, are found to be dominant at few tens-of-$\mathrm{MeV}$ energies.

Conclusions

Nuclear de-excitations have a strong impact on the achievable energy
resolution for supernova $\nu_{e}$ detection in liquid argon. Tagging events
involving neutron emission, though difficult, could substantially improve
energy reconstruction. Given a suitable calculation of the inclusive cross
section, the MARLEY nuclear de-excitation model may readily be applied to
other scattering processes.

###### pacs:

##  I Introduction

Core-collapse supernovae are exceptionally intense sources of tens-
of-$\mathrm{MeV}$ neutrinos and antineutrinos of all flavors. In a typical
supernova, about ${10}^{58}$ neutrinos are released in a burst lasting tens of
seconds. Although the first observation of supernova neutrinos by the
Kamiokande-II Hirata  _et al._ (1987), Baksan Alekseev  _et al._ (1987), and
Irvine-Michigan-Brookhaven Bionta  _et al._ (1987) detectors in 1987 yielded a
total of only two dozen events, the scientific impact of this dataset has been
tremendous, leading to numerous publications on a wide variety of subjects
Raffelt (1990); Schaeffer (1990); Jegerlehner  _et al._ (1996); Vissani
(2014); Mirizzi  _et al._ (2016); Branch and Wheeler (2017). In the years
since first detection, a worldwide network of large neutrino experiments, most
built primarily for other applications, stands ready to perform a second,
high-statistics measurement if a core-collapse supernova should occur within
the galaxy Scholberg (2012). Due to the slow rate of galactic core-collapse
supernovae (estimated to be about 1.6 per century Rozwadowska  _et al._
(2021)), the prospect of such a measurement represents a rare but valuable
opportunity to shed light on the details of core-collapse and nucleosynthesis,
study neutrinos under extreme conditions, search for evidence of physics
beyond the Standard Model, and explore many other topics Müller (2019);
Horiuchi and Kneller (2018); Raffelt (2011).

A full realization of the physics potential of the next galactic core-collapse
supernova will require a simultaneous measurement of neutrinos of all flavors.
While detectors based on water and hydrocarbon scintillator will primarily
detect $\bar{\nu}_{e}$ via inverse beta decay (IBD)

$\bar{\nu}_{e}+p\rightarrow e^{+}+n\,,$ (1)

liquid-argon-based detectors are anticipated to provide unique sensitivity Abi
_et al._ (2020a); Ankowski  _et al._ (2016) to $\nu_{e}$ via the charged-
current (CC) reaction

$\nu_{e}+\isotope[40]{Ar}\rightarrow e^{-}+\isotope[40]{K}^{*}$ (2)

which dominates the expected signal at supernova energies.

Within the decade, the Deep Underground Neutrino Experiment (DUNE) will begin
operating four ten-kiloton liquid argon time projection chambers (LArTPCs)
with the primary goals of studying long-baseline oscillations of accelerator
neutrinos, searching for proton decay, and measuring the $\nu_{e}$ flux from a
galactic core-collapse supernova if one should occur during the lifetime of
the experiment Abi  _et al._ (2020b). Initial studies of the sensitivity of
DUNE to supernova neutrinos, performed by the collaboration itself Abi  _et
al._ (2020a) and by smaller groups (e.g., ref. Li  _et al._ (2021)) show
promise, and the potential exists for measurements by DUNE of other low-energy
neutrinos, notably those produced by the Sun Capozzi  _et al._ (2019).

In addition to DUNE, three sub-kiloton LArTPCs, SBND Tufanli (2017);
Brailsford (2017), MicroBooNE Acciarri  _et al._ (2017), and ICARUS Amerio
_et al._ (2004), are currently operating or being installed in the Booster
Neutrino Beam at Fermilab. A joint effort between the three experimental
collaborations, known as the Short Baseline Neutrino (SBN) program Machado
_et al._ (2019); Acciarri  _et al._ (2015), will perform precision
measurements of neutrino oscillations. In addition to this primary mission,
the SBN detectors will pursue a variety of other physics measurements and are
expected to be sensitive to supernova neutrinos. To ensure that data from a
core-collapse supernova would be fully recorded over the ten-second burst, the
MicroBooNE collaboration operates a first-of-its-kind continuous readout
stream and has demonstrated its capabilities via reconstruction of Michel
electrons produced by decays of cosmic muons Abratenko  _et al._ (2021).

While much remains to be done to fully exploit the low-energy capabilities of
LArTPCs, a first demonstration by the ArgoNeuT Anderson  _et al._ (2012)
experiment of reconstruction of $\mathrm{MeV}$-scale activity due to
accelerator-neutrino-induced neutrons and de-excitation $\gamma$-rays achieved
a detection threshold of around $200300\text{\ }\mathrm{keV}$ Acciarri  _et
al._ (2019). These encouraging initial results have prompted further
experimental work by MicroBooNE MicroBooNE Collaboration  and multiple
simulation-based studies considering the implications for reconstruction of
both high- and low-energy physics events Friedland and Li (2019); Castiglioni
_et al._ (2020).

In future analyses of supernova neutrino data, the event-by-event
reconstructed neutrino energy obtained by each detector will be of primary
interest. For IBD events in water or scintillator, because only a single
hadronic final state (a free neutron) is accessible at tens-of-$\mathrm{MeV}$
energies, a measurement of the outgoing positron energy is sufficient to
reconstruct the antineutrino energy with high accuracy. Due to the use of a
complex nuclear target (argon) in LArTPCs, however, various nuclear
transitions may occur in response to CC $\nu_{e}$ absorption, and thus a one-
to-one mapping (up to nuclear recoil) between the neutrino and electron
energies no longer exists.

To fully reconstruct the neutrino energy in the argon case, the reaction
Q-value, i.e., the energy imparted to the nuclear transition, must be inferred
by detecting the nuclear de-excitation products. For transitions to bound
nuclear energy levels, the neutrino energy is in principle fully recoverable
by measuring the energies of all de-excitation $\gamma$-rays in addition to
the primary electron. For transitions to unbound nuclear states, a model is
needed to correct for missing energy associated with undetected nuclear
fragments. In practice, an experimental analysis that attempts to isolate the
simpler bound transitions will also need a detailed de-excitation model in
order to estimate the purity of the event selection.

With the exception of a recent first measurement Akimov  _et al._ (2021) of
coherent elastic neutrino-nucleus scattering, experimental data have not yet
been obtained for neutrino-argon cross sections in the supernova energy
regime. Furthermore, only a few measurements with limited precision are
available for low-energy inelastic neutrino scattering on any nuclear target
(Ajimura  _et al._ , 2017, table 3). Despite this, a substantial literature
exists for theoretical calculations of the
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ process. A review through
2018 is provided in ref. (Gardiner, 2018, sec. 7.1), with a notable recent
addition being two publications Van Dessel  _et al._ (2019, 2020) which employ
a Continuum Random Phase Approximation (CRPA) model to study this cross
section above the nucleon emission threshold.

While highly useful for providing competing estimates of event rates in DUNE
and other argon-based detectors, all published calculations for this cross
section to date share the limitation of being fully inclusive, i.e.,
predictions are made that consider only the kinematics of the outgoing
electron. At very low neutrino energies, where only transitions to bound
nuclear states are possible, this is not problematic: measured de-excitation
$\gamma$-ray branching ratios exist for many levels of the daughter [40]K
nucleus, and missing data may be addressed using straightforward theoretical
estimates. However, above about $1520\text{\ }\mathrm{MeV}$, kinematic access
to unbound nuclear states becomes appreciable, and a detailed treatment of the
competition between various de-excitation channels (including emission of both
$\gamma$-rays and nuclear fragments) is needed.

Although such a treatment has not previously been provided for [40]Ar,
detailed modeling of nuclear de-excitations induced by low-energy neutrino
interactions has been pursued for a number of other nuclei Kolbe  _et al._
(1992); Langanke  _et al._ (1996); Kolbe and Langanke (2001); Cheoun  _et al._
(2012a); Bandyopadhyay  _et al._ (2017); Vale  _et al._ (2016). A universal
assumption made by all of these calculations (as well as the present work) is
that of compound nucleus formation: following the primary interaction, the
nucleus is left in a thermally-equilibrated excited state that decays
independently of the details of its formation process. While further work is
needed to fully investigate the adequacy of this assumption for low-energy
neutrino-nucleus reactions, both theoretical calculations Kim and Cheoun
(2009); Cheoun and Kim (2011) and electron scattering data Flowers  _et al._
(1978) suggest that compound processes dominate over the direct nucleon knock-
out important at higher energies.

In this paper, I present the first calculations at tens-of-$\mathrm{MeV}$
energies for cross sections for exclusive final states of the reaction
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$, emphasizing the role of
nuclear de-excitation processes. In section II, I develop a simple model for
the inclusive differential cross section, relying on approximations that work
best at low momentum transfers. The derivation in section II fully determines
the cross section up to the values of two (Fermi and Gamow-Teller) nuclear
matrix elements, $B(\mathrm{F})$ and $B(\mathrm{GT})$, which are considered in
section III. While relevant neutrino scattering data are currently
unavailable, the needed values of these matrix elements at low excitation
energies may be extracted from measurements of related processes. I supplement
these measurements with the results of a theoretical calculation at high
excitation energies to obtain a full treatment of the inclusive cross section.
In section IV, I describe a detailed model of nuclear de-excitations that can
be used together with the inclusive cross section to obtain predictions for
exclusive final states. In section V, I present example results calculated
using the models developed in the previous sections.

To enable practical calculations that have already helped to inform studies of
DUNE’s sensitivity to supernova neutrinos Abi  _et al._ (2020a), all of the
physics models described herein have been implemented in a new Monte Carlo
event generator called MARLEY (Model of Argon Reaction Low Energy Yields). All
results shown in this work may be reproduced using version 1.2.0 of MARLEY
Gardiner (2021a), which is publicly available as an open-source software
project Gardiner (2021b). Documentation of the technical details of MARLEY and
usage instructions are available in ref. Gardiner (2021c).

Due to the compound nucleus assumption, the MARLEY de-excitation model may
easily be used in the future to provide exclusive predictions for a more
refined calculation of the inclusive CC $\nu_{e}$ absorption cross section for
[40]Ar. A similar approach to modeling de-excitations for other reaction modes
and target nuclei is likewise possible, and I welcome potential collaboration
on this topic. Prospects for improving MARLEY predictions beyond the proof-of-
concept reported here are briefly considered in section VI.

##  II Inclusive cross section model

For momentum transfers that are small compared to the $W$ boson mass, the
tree-level amplitude $\mathcal{M}$ for inclusive charged-current neutrino-
nucleus scattering may be represented diagramatically as

$i\mathcal{M}=\vbox{\hbox{ \includegraphics{images/feyn.pdf} }}.$ (3)

The corresponding differential cross section may be written in the form

$\frac{d\sigma}{dQ^{2}}=\frac{G_{F}^{2}\,|V_{\mathrm{ud}}|^{2}}{32\,\pi\,(s-m_{i}^{2})^{2}}\,F_{C}\,\mathsf{L}_{\mu\nu}\,\mathsf{W}^{\mu\nu}\,,$
(4)

where $q=k-k^{\prime}=p^{\prime}-p$ is the four-momentum transfer,
$Q^{2}\equiv-q^{2}$, $G_{F}$ is the Fermi constant, $V_{\mathrm{ud}}$ is the
Cabibbo–Kobayashi–Maskawa matrix element connecting the up and down quarks,
Mandelstam $s$ is the square of the total energy in the center-of-momentum
(CM) frame, and $m_{i}$ is the mass of the initial-state nucleus. Discussion
of the Coulomb correction factor $F_{C}$ is deferred to section II.2.

The leptonic ($\mathsf{L}_{\mu\nu}$) and hadronic ($\mathsf{W}^{\mu\nu}$)
tensors are defined by

$\displaystyle\mathsf{L}_{\mu\nu}$
$\displaystyle\equiv\operatorname{Tr}[\gamma_{\mu}\,(1-\gamma_{5})\,\cancel{k}\,\gamma_{\nu}\,(1-\gamma_{5})\,(\cancel{k}^{\prime}+m_{\ell})]$
(5)
$\displaystyle=8\left[k_{\mu}\,k^{\prime}_{\nu}+k_{\nu}\,k^{\prime}_{\mu}-g_{\mu\nu}\,(k\cdot
k^{\prime})-i\epsilon_{\mu\nu\rho\sigma}\,{k}^{\rho}{k^{\prime}}^{\sigma}\right]$
(6)

and

$\mathsf{W}^{\mu\nu}\equiv\frac{1}{2J_{i}+1}\sum_{M_{i}}\sum_{M_{f}}\mathcal{N}^{\mu}\,\mathcal{N}^{\nu*}.$
(7)

Here $m_{\ell}$ is the mass of the final-state lepton, $J_{i}$ ($J_{f}$) is
the initial (final) nuclear spin, and $M_{i}$ ($M_{f}$) is the third component
of the nuclear spin in the initial (final) state.

Under the impulse approximation, the nuclear matrix element may be written in
coordinate space as

$\mathcal{N}^{\mu}=\big{<}f\big{|}\,{\textstyle\sum_{n=1}^{A}}\,e^{i\mathbf{q}\cdot\mathbf{x}(n)}\,j^{\mu}(n)\,\big{|}i\big{>}$
(8)

where $\mathbf{q}$ is the three-momentum transfer and the sum runs over all
$A$ nucleons. The weak current operator $j^{\mu}(n)$ is understood to act only
on the $n$th nucleon, as is the position operator $\mathbf{x}(n)$. The state
vectors in eq. 8 are normalized relativistically, i.e.,

$\displaystyle\big{<}i\big{|}i\big{>}$ $\displaystyle=2E_{i}$
$\displaystyle\big{<}f\big{|}f\big{>}$ $\displaystyle=2E_{f}$ (9)

where $E_{i}$ ($E_{f}$) is the total energy of the nucleus in the initial
(final) state. Equation 4 contains an implied sum over the accessible final
nuclear states.

###  II.1 Allowed approximation

The full expression for the single-nucleon weak current operator $j^{\mu}$ is
well-known and is given in ref. Gardiner (2018) among other places. For this
study, however, I evaluate the current operator in the allowed approximation,
which combines the long-wavelength ($q\to 0$) limit and the slow-nucleon limit
(in which the momentum of the initial struck nucleon is neglected compared to
its mass).

Under this approximation, the weak current operator reduces to the simple form

$\displaystyle j^{0}$ $\displaystyle=g_{V}\,t_{-}$ $\displaystyle j^{a}$
$\displaystyle=-\sigma^{a}\,g_{A}\,t_{-}$ (10)

where $j^{0}$ is the time component and the three Cartesian spatial components
are labeled with $a\in\\{1,2,3\\}$. The time component of the nuclear matrix
element $\mathcal{N}^{\mu}$ is given by

$\mathcal{N}^{0}=\frac{g_{V}}{\sqrt{2J_{i}+1}}\,\delta_{J_{i}\,J_{f}}\,\delta_{M_{i}\,M_{f}}\,\big{<}f\big{\|}\mathcal{O}_{\mathrm{F}}\big{\|}i\big{>}$
(11)

while the spatial components may be written in spherical coordinates as

$\mathcal{N}^{w}=\frac{-g_{A}\,(-1)^{J_{i}-M_{i}}}{\sqrt{3}}\,\big{(}J_{f}\;M_{f}\;J_{i}\;{-M_{i}}\,\big{|}\,1\;w\big{)}\,\big{<}f\big{\|}\mathcal{O}_{\mathrm{GT}}\big{\|}i\big{>}$
(12)

where $w\in\\{-1,0,1\\}$ and $g_{V}$ ($g_{A}$) is the vector (axial-vector)
weak coupling constant of the nucleon. The Fermi (F) and Gamow-Teller (GT)
operators are defined by

$\displaystyle\mathcal{O}_{\mathrm{F}}$
$\displaystyle\equiv\sum_{n=1}^{A}t_{-}(n)$ (13)
$\displaystyle\mathcal{O}_{\mathrm{GT}}$
$\displaystyle\equiv\sum_{n=1}^{A}\boldsymbol{\sigma}(n)\,t_{-}(n)$ (14)

where $\boldsymbol{\sigma}$ is the Pauli vector, and $t_{-}$, the isospin-
lowering operator, converts a neutron into a proton. Double bars ($\|$) denote
matrix elements which have been reduced via the Wigner-Eckhart theorem.

Equations 11 and 12 may be used to evaluate the elements of the hadronic
tensor $\mathsf{W}^{\mu\nu}$. Under the allowed approximation, these become

$\displaystyle\mathsf{W}^{00}$ $\displaystyle=4\,E_{i}\,E_{f}\,B(\mathrm{F})$
(15) $\displaystyle\mathsf{W}^{ab}$
$\displaystyle=\frac{4}{3}\,\delta_{ab}\,E_{i}\,E_{f}\,B(\mathrm{GT})$ (16)
$\displaystyle\mathsf{W}^{0a}$ $\displaystyle=\mathsf{W}^{a0}=0$ (17)

where the reduced Fermi and Gamow-Teller matrix elements are given by

$\displaystyle B(\mathrm{F})$
$\displaystyle\equiv\frac{g_{V}^{2}}{2J_{i}+1}\Big{|}\big{<}J_{f}\,\big{\lVert}\,\mathcal{O}_{\mathrm{F}}\,\big{\rVert}\,J_{i}\big{>}\Big{|}^{2}$
(18) $\displaystyle B(\mathrm{GT})$
$\displaystyle\equiv\frac{g_{A}^{2}}{2J_{i}+1}\Big{|}\big{<}J_{f}\,\big{\lVert}\,\mathcal{O}_{\mathrm{GT}}\,\big{\rVert}\,J_{i}\big{>}\Big{|}^{2}\,.$
(19)

The state vectors labeled using the nuclear spin ($J_{i}$ or $J_{f}$) are
normalized to unity:

$\big{<}J_{i}\big{|}J_{i}\big{>}=\big{<}J_{f}\big{|}J_{f}\big{>}=1.$ (20)

The reduced matrix elements satisfy the spin-parity selection rules

$B(\mathrm{F})=0\text{ unless }J_{f}=J_{i}\text{ and }\Pi_{f}=\Pi_{i}$ (21)

and

$B(\mathrm{GT})=0\text{ unless }|J_{i}-1|\leq J_{f}\leq J_{i}+1\text{ and
}\Pi_{f}=\Pi_{i}\,.$ (22)

where $\Pi_{i}$ ($\Pi_{f}$) is the initial (final) nuclear parity.

Combining the results above leads to the following expression for the allowed
approximation differential cross section in the CM frame:

$\displaystyle\frac{d\sigma}{d\cos\theta_{\ell}}=$
$\displaystyle\;\frac{G_{F}^{2}\,|V_{\mathrm{ud}}|^{2}}{2\,\pi}\,F_{C}\,\bigg{[}\frac{E_{i}\,E_{f}}{s}\bigg{]}\,E_{\ell}\left|\mathbf{p}_{\ell}\right|\bigg{[}\Big{(}1+\beta_{\ell}\cos\theta_{\ell}\Big{)}\,B(\mathrm{F})+\,\Big{(}1-\frac{1}{3}\,\beta_{\ell}\cos\theta_{\ell}\Big{)}\,B(\mathrm{GT})\bigg{]}.$
(23)

Here $E_{\ell}$, $\mathbf{p}_{\ell}$, $\theta_{\ell}$, and
$\beta_{\ell}=E_{\ell}/|\mathbf{p}_{\ell}|$ are, respectively, the total
energy, three-momentum, scattering angle, and speed of the final-state lepton.
The factor $E_{i}\,E_{f}/s$ accounts for nuclear recoil and is commonly
neglected.

In the CM frame, the particle energies are independent of the scattering angle
$\theta_{\ell}$. As a result, integration of the total cross section is
trivial and leads to the expression

$\sigma=\frac{G_{F}^{2}\,|V_{\mathrm{ud}}|^{2}}{\pi}\,F_{C}\,\bigg{[}\frac{E_{i}\,E_{f}}{s}\bigg{]}\,E_{\ell}\left|\mathbf{p}_{\ell}\right|\bigg{[}B(\mathrm{F})+B(\mathrm{GT})\bigg{]}.$
(24)

As was the case for eq. 4, the cross-section formulas in eqs. 23 and 24
contain an implicit sum over nuclear final states.

###  II.2 Coulomb corrections

Final-state interactions (FSIs) of the outgoing charged lepton with the
Coulomb field of the nucleus have a significant effect on the cross section at
low energies. While a detailed treatment of Coulomb FSIs is achievable via the
distorted-wave Born approximation, a much more convenient approximation scheme
based on the work of Engel Engel (1998) is typically used, e.g., in refs.
Volpe  _et al._ (2002); Ydrefors and Suhonen (2012); Van Dessel  _et al._
(2019).

Under this approach, the Coulomb correction factor $F_{C}$ that appears in
eqs. 4, 23 and 24 is calculated using either the Fermi function Fermi (1934);
Wilson (1968) or the modified effective momentum approximation (MEMA) Engel
(1998). Since the former is known to overestimate Coulomb corrections at high
lepton energies while the latter does the same at low energies, the smaller of
the two alternatives is always chosen. This amounts to defining the Coulomb
correction factor as

$F_{C}\equiv\begin{cases}F_{\text{Fermi}}&|F_{\text{Fermi}}-1|<|F_{\text{MEMA}}-1|\\\
F_{\text{MEMA}}&\text{otherwise}\end{cases}$ (25)

where the Fermi function is given by

$\displaystyle
F_{\text{Fermi}}=\frac{2(1+S)}{\big{[}\Gamma(1+2S)\big{]}^{2}}\,(2\,$
$\displaystyle\gamma_{\text{rel}}\,\beta_{\text{rel}}\,m_{\ell}\,R)^{2S-2}\,$
$\displaystyle e^{-\pi\,\eta}\,\left|\Gamma\left(S-i\eta\right)\right|^{2}$
(26)

and

$F_{\text{MEMA}}\equiv\frac{\mathpzc{K}_{\text{\,\,eff}}\,\mathpzc{E}_{\text{\,eff}}}{\mathpzc{K}\,\mathpzc{E}}\,.$
(27)

In eq. 26 the quantity $S$ is defined in terms of the fine structure constant
$\alpha$ by

$S\equiv\sqrt{1-\alpha^{2}Z_{f}^{2}\,}$ (28)

where $Z_{f}$ is the proton number of the final nucleus. The nuclear radius
(in natural units) may be estimated as

$R\approx\frac{1.2\,A^{1/3}\,\text{fm}}{\hbar\,c}\,,$ (29)

and the Sommerfeld parameter $\eta$ is given by

$\eta=\frac{\alpha\,Z_{f}\,z_{\ell}}{\beta_{\text{rel}}}\,.$ (30)

where $z_{\ell}$ is the electric charge (in units of the elementary charge) of
the final-state lepton.

Typical presentations of the correction factors defined in eqs. 26 and 27
neglect the small recoil kinetic energy of the final nucleus in the laboratory
frame. This allows the use of expressions for $F_{\text{Fermi}}$ and
$F_{\text{MEMA}}$ which are derived in the rest frame of the final nucleus. I
opt instead for Lorentz-invariant forms of the correction factors written in
terms of the relative speed $\beta_{\text{rel}}$ of the two final-state
particles: Cannoni (2017)

$\displaystyle\beta_{\text{rel}}$ $\displaystyle=\frac{\sqrt{(k^{\prime}\cdot
p^{\prime})^{2}-m_{\ell}^{2}\,m_{f}^{2}}}{k^{\prime}\cdot p^{\prime}}$
$\displaystyle\gamma_{\text{rel}}$
$\displaystyle\equiv\left(1-\beta^{2}_{\text{rel}}\right)^{-1/2}.$ (31)

The symbols $\mathpzc{E}$ and $\mathpzc{K}$ from eq. 27 denote, respectively,
the total energy and momentum of the outgoing lepton in the rest frame of
recoiling nucleus:

$\displaystyle\mathpzc{E}$ $\displaystyle\equiv\gamma_{\text{rel}}\,m_{\ell}$
$\displaystyle\mathpzc{K}$
$\displaystyle\equiv\beta_{\text{rel}}\,\mathpzc{E}\,.$ (32)

The effective values of these variables are those that exist in the presence
of the nuclear Coulomb potential

$\displaystyle\mathpzc{K}_{\text{\,\,eff}}$
$\displaystyle\equiv\sqrt{\mathpzc{E}_{\text{\,eff}}^{2}-m_{\ell}^{2}}$
$\displaystyle\mathpzc{E}_{\text{\,eff}}$
$\displaystyle\equiv\mathpzc{E}-V_{C}(0),$ (33)

which is approximated by that at the center of a uniformly-charged sphere:

$V_{C}(0)\approx\frac{3\,Z_{f}\,z_{\ell}\,\alpha}{2\,R}\,.$ (34)

It should be noted that, as originally presented Engel (1998), the MEMA also
involves modifying the value of the momentum transfer used to evaluate the
amplitude $\mathcal{M}$. Since the cross section treatment presented here
involves use of the long-wavelength limit $q\to 0$, however, I neglect this
additional correction.

##  III Allowed nuclear matrix elements

Despite sustained community interest and a concrete proposal by the CAPTAIN
experiment Berns  _et al._ (2013) to perform direct measurements, no
experimental data for tens-of-$\mathrm{MeV}$ charged-current $\nu_{e}$
scattering on argon are currently available. However, in recent decades, three
separate experiments have obtained values of the allowed matrix elements
$B(\mathrm{F})$ and $B(\mathrm{GT})$ by considering related physics processes.

The first two experiments were performed in the late 1990s by separate teams
working at the Gesellschaft für Schwerionenforschung (Society for Heavy Ion
Research) in Darmstadt, Germany Liu  _et al._ (1998) and the Grand
Accélérateur National d’Ions Lourds (Large Heavy Ion National Accelerator) in
Caen, France Bhattacharya  _et al._ (1998). Both sought to study CC $\nu_{e}$
absorption on [40]Ar by measuring beta decays of its mirror nucleus [40]Ti:

$\isotope[40]{Ti}\rightarrow\isotope[40]{Sc}^{*}+e^{+}+\nu_{e}\,.$ (35)

In the limit of perfect isospin symmetry, the matrix element describing a
[40]Ti beta decay transition to a specific [40]Sc level is equal to the matrix
element accessing the level’s isobaric analog in [40]K via CC $\nu_{e}$
scattering on [40]Ar. The main difficulties in applying this technique to
neutrino cross sections are  (1) the beta decay Q-value limits the maximum
excitation energy that may be studied, and  (2) energy levels in the beta
decay daughter nucleus ([40]Sc) must be matched to their analogs in the final-
state nucleus for neutrino scattering ([40]K).

The third experiment Bhattacharya  _et al._ (2009), performed about a decade
later at the Indiana University Cyclotron Facility, extracted $B(\mathrm{GT})$
values from measurements of (p,n) scattering on [40]Ar. The extraction
technique relied on the observation, first put forward in 1980 Goodman  _et
al._ (1980) and subsequently refined Taddeucci  _et al._ (1982, 1987); Goodman
_et al._ (2001), that the (p,n) cross section at very forward angles
($\theta\approx 0^{\circ}$) for proton energies around $100\text{\
}\mathrm{MeV}$ is approximately proportional to the allowed matrix elements
$B(\mathrm{F})$ and $B(\mathrm{GT})$. While subject to some unique
difficulties of its own (see, e.g., ref. (Frekers  _et al._ , 2013, sec.
4.2)), this approach overcomes key limitations of [40]Ti beta decay:
transitions to excited levels of [40]K may be studied directly at energies
higher than the mirror beta decay Q-value.

###  III.1 Re-evaluation of existing measurements

Reasonable attempts were made in the original publications describing these
measurements to assign the extracted matrix elements to known [40]K levels
satisfying the spin-parity selection rules in eqs. 21 to 22. That is, the
[40]K isobaric analog state accessed via a Fermi transition must have
$J^{\pi}=0^{+}$, while GT transitions may only populate levels with
$J^{\pi}=1^{+}$. However, in light of new [40]K level data that became
available in 2017 Chen (2017), I revisited the level assignments for all three
measurements.

The results of this re-evaluation are shown in table 1. Level energies
($\mathrm{keV}$) and spin-parity assignments retrieved from the Evaluated
Nuclear Structure Data File (ENSDF) database ENS (2021) are listed in the
first and second columns, respectively. Excitation energies (for either [40]Sc
or [40]K as appropriate) and matrix element values are listed in the following
columns for each of the three experimental measurements. In the case of the
(p,n) scattering data, the matrix element values provided in the original
paper Bhattacharya  _et al._ (2009) have been scaled by a factor of
$g_{A}^{2}=1.26^{2}$. This scaling was done because the definition of
$B(\mathrm{GT})$ used by the experiment does not include the axial-vector weak
coupling constant $g_{A}$. The specific value $g_{A}=1.26$ was chosen for
consistency with the one assumed in the experimental analysis.

Table 1: Level assignments and measured $B(\mathrm{F})$ and $B(\mathrm{GT})$
values for [40]Ar($\nu_{e}$,$e^{-}$)[40]K^*

Assigned [40]K $E_{x}$ ($\mathrm{keV}$) ENSDF Chen (2017) spin-parity
assignment111Parenthesized values are based upon weak arguments Tuli . Liu et
al. Liu  _et al._ (1998) [40]Ti $\beta^{+}$ decay Bhattacharya et al.
Bhattacharya  _et al._ (1998) [40]Ti $\beta^{+}$ decay Bhattacharya et al.
Bhattacharya  _et al._ (2009)
$\isotope[\text{40}]{\text{Ar}}(p,n)\isotope[\text{40}]{\text{K}}$ [40]Sc
$E_{x}$ ($\mathrm{keV}$) $B(\mathrm{F})\,+$ $B(\mathrm{GT})$ [40]Sc $E_{x}$
($\mathrm{keV}$) $B(\mathrm{F})\,+$ $B(\mathrm{GT})$ [40]K $E_{x}$
($\mathrm{keV}$) Weak222The data tabulated in ref. Bhattacharya  _et al._
(2009) were multiplied by $g_{A}^{2}=1.26^{2}$ to obtain the $B(\mathrm{GT})$
values shown here. $B(\mathrm{GT})$ $2289.868(11)$ 1+ $2287(10)$ $0.83(8)$
$2281(8)$ $0.90(4)$ $2333(30)$ $1.64(16)$ $2730.357(19)$ 1+ $2761(10)$
$1.40(10)$ $2752(8)$ $1.50(6)$ $2775(30)$ $1.49(14)$ $2950.9(6)$ $2966(40)$
$0.03(1)$ $2937(13)$ $0.11(2)$ $3109.56(4)$ 1+, 2+ $3121(46)$ $0.06(3)$
$3143(20)$ $0.06(1)$ $3146.50(5)$ 1(-) $3235(50)$ $0.16(4)$ $3204(32)$
$0.06(2)$ $3293(10)$ unnatural333A nuclear level with parity $\Pi$ and spin
$J$ has natural parity if $\Pi=(-1)^{J}$. Otherwise it has unnatural parity.
$3342(40)$ $0.11(8)$ $3334(19)$ $0.04(1)$ $3439.18(3)$ (2+) $3418(60)$
$0.05(2)$ $3517(15)$ $3521(40)$ $0.06(2)$ $3569(56)$ $0.01(1)$ $3503(30)$
$0.16(2)$ $3738.49(3)$444Another candidate [40]K level for this transition has
$E_{x}=$3663.88\text{\ }\mathrm{keV}$$ and $J^{\pi}=(1^{-},2,3,4^{+})$. 1+
$3662(40)$ $0.13(7)$ $3652(10)$ $0.16(2)$ $3797.48(3)$ 1+ $3782(40)$
$0.40(16)$ $3786(10)$ $0.26(3)$ $3840.27(3)$ (1, 2+) $3861(49)$ $0.01(1)$
$3870(30)$ $0.44(5)$ $3996(10)$ unnaturalfootnote 3 $4033(88)$ $0.07(4)$
$4067(24)$ $0.05(2)$ $4080(5)$ $4194(60)$ $0.10(6)$ $4111(30)$ $0.11(3)$
$4251.70(15)$ (1, 2-) $4264(46)$ $0.15(4)$ $4267(10)$ $0.29(3)$
$4383.7(7)$555This level is the isobaric analog of the [40]Ar ground state. 0+
$4365(10)$ $4.01(31)$ $4364(8)$ $3.84(17)$ $4508(15)$ $4540(86)$ $0.14(5)$
$4522(16)$ $0.31(5)$ $4421(30)$ $0.86(14)$ $4697(10)$ unnaturalfootnote 3
$4628(40)$ $0.33(9)$ $4655(12)$ $0.38(6)$ $4765(5)$ (1)+ $4782(60)$ $0.26(11)$
$4825(21)$ $0.47(8)$ $4763(30)$ $0.48(5)$ $4930(10)$ unnaturalfootnote 3
$4997(72)$ $0.24(10)$ $5017(27)$ $0.36(9)$ $5063.37(7)$ (2-, 3+) $5051(40)$
$0.25(11)$ $5080(35)$ $0.23(7)$ $5189.89(5)$ (2-) $5135(86)$ $0.20(6)$
$5223(32)$ $0.03(3)$ $5162(30)$ $0.59(6)$ $5247.1(6)$ $5362(60)$ $0.19(7)$
$5488.65(17)$ (2-, 3, 4-) $5574(40)$ $0.07(4)$ $5681(32)$ $5777(60)$
$0.21(15)$ $5696(23)$ $0.11(4)$ $5681(32)$ $0.21(3)$ $5870(20)$ $5886(80)$
$0.17(7)$ $6118(30)$ $6126(60)$ $0.13(7)$ $6006(21)$ $0.13(5)$ $6118(30)$
$0.48(5)$ $6790(30)$ $6426(60)$ $0.11(6)$ $6790(30)$ $0.71(8)$ $7468(37)$
$7468(37)$ $0.06(2)$ $7795(33)$ $7795(33)$ $0.14(2)$ $7952(32)$ $7952(32)$
$0.97(10)$ Total666Gamow-Teller transitions are assumed for all levels other
than the isobaric analog state. $B(\mathrm{GT})$ $5.84(39)$ $5.52(20)$
$8.29(31)$

Figure 1: Gamow-Teller strengths $B(\mathrm{GT})$ from two independent
measurements of [40]Ti $\beta^{+}$ decay by Liu et al. Liu  _et al._ (1998)
and Bhattacharya et al. Bhattacharya  _et al._ (1998). Figure 2: Comparison of
the Gamow-Teller strengths $B(\mathrm{GT})$ measured using [40]Ti beta decay
(see fig. 1) with those obtained using a measurement of $0^{\circ}$ (p,n)
scattering by Bhattacharya et al. Bhattacharya  _et al._ (2009).

Figure 1 shows the Gamow-Teller strengths obtained by the two [40]Ti beta
decay experiments mentioned previously. Excitation energies of analog states
in [40]K, represented on the horizontal axis, are chosen to match the
assignments made in table 1. The vertical axis is inverted for the second
dataset to facilitate comparisons. Rough consistency is seen between the two
measurements, although the results reported by Liu et al. involve several more
nuclear levels.

Figure 2 uses a similar format to individually compare each beta decay
measurement to the GT strengths extracted from (p,n) scattering. Substantially
more fragmentation of the strength is seen in the beta decay data, and there
are areas of significant tension. For instance, the two experimental methods
disagree on whether the GT strength to the [40]K level at $2.3\text{\
}\mathrm{MeV}$ is larger or smaller than the strength to the level at
$2.7\text{\ }\mathrm{MeV}$.

Differences between the beta decay and (p,n) data were examined in detail by
Karakoç et al. in 2014 Karakoç  _et al._ (2014). Based on a combination of
theoretical calculations and a currently unpublished
$\isotope[40]{Ar}(\mathrm{h},\mathrm{t})\isotope[40]{K}$ measurement, the
authors argued that the (p,n) data should be preferred over the [40]Ti beta
decay data for calculations of CC $\nu_{e}$ absorption cross sections on
argon. Rather than attempt to adjudicate between the conflicting datasets, I
have opted to allow each of the three measurements to be used as a source of
$B(\mathrm{GT})$ values in MARLEY cross-section calculations.

###  III.2 Extension to higher excitation energies

Beyond the maximum excitation energy of about $8\text{\ }\mathrm{MeV}$ probed
by the experiments mentioned in the previous section, the presence of
additional Gamow-Teller strength is predicted by the model-independent Ikeda
sum rule Ikeda  _et al._ (1963). This rule states that the summed GT strength
$S_{\text{GT}}^{-}$ ($S_{\text{GT}}^{+}$) over all possible nuclear final
states for CC $\nu_{e}$ ($\bar{\nu}_{e}$) absorption satisfies the relation

$\Delta S_{\text{GT}}\equiv
S_{\text{GT}}^{-}-S_{\text{GT}}^{+}=3\,g_{A}^{2}\,(N_{i}-Z_{i})$ (36)

where $N_{i}=22$ ($Z_{i}=18$) is the neutron (proton) number of the initial
[40]Ar nucleus. Equation 36 implies a minimum possible value of
$S_{\text{GT}}^{-}=12\,g_{A}^{2}\approx 19$ for the integrated GT strength
associated with the reaction
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$. Comparing this value to
the measured total GT strengths listed in the final row of table 1 reveals
that the majority of the expected GT strength for CC $\nu_{e}$ absorption on
[40]Ar is unmeasured and associated with transitions to high-lying, nucleon-
unbound states of [40]K.

To supplement the experimental measurements with an estimate of the remainder
of the GT strength, I rely on a Quasiparticle Random Phase Approximation
(QRPA) calculation by Cheoun, Ha, and Kajino Cheoun  _et al._ (2012b). To
avoid double-counting, theoretical GT matrix elements are included with the
experimental ones only for excitation energies at which the integrated QRPA GT
strength exceeds the experimental total.

Figure 3: Complete sets of
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ allowed approximation
matrix elements distributed as part of MARLEY 1.2.0. The name of the data file
in which each set of matrix elements may be found is given in the appropriate
legend heading. The QRPA $B(\mathrm{GT})$ strengths (violet) were calculated
by Cheoun et al. Cheoun  _et al._ (2012b). Citations for the experimental
$B(\mathrm{GT})$ strengths are given in the captions for figs. 1 and 2.

###  III.3 Adopted matrix elements

Figure 3 presents three complete sets of allowed
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ nuclear matrix elements
prepared as input for MARLEY based on the measurements and QRPA prediction
discussed above. The experimental $B(\mathrm{GT})$ values are shown as green,
red, or blue bars depending on the dataset. The theoretical QRPA
$B(\mathrm{GT})$ values are shown in violet.

In addition to extracting GT strengths, both beta decay experiments measured
values of the Fermi matrix element $B(\mathrm{F})$. Under the approximation
that isospin is a good quantum number, this matrix element is expected to have
the value

$B(\mathrm{F})=g_{V}^{2}(N_{i}-Z_{i})=4$ (37)

and to connect the ground state of [40]Ar to a single $0^{+}$ isobaric analog
state in [40]K, which has been identified as the level with excitation energy
$E_{x}=$4.3837\text{\ }\mathrm{MeV}$$. Since the experimental data are fully
consistent with these expectations, I adopt the value $B(\mathrm{F})=4$ for
this transition in all three sets of MARLEY matrix elements. The known Fermi
transition is represented in each panel of Figure 3 by a white bar with
horizontal hatch marks. Transitions to all other nuclear levels are assumed to
proceed via the Gamow-Teller operator.

Although the differences become important at neutrino energies near threshold,
the observables predicted in this paper are largely insensitive to the choice
between the three sets of MARLEY matrix elements for the
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ reaction. For
definiteness, all MARLEY calculations shown in this work (see section V) are
obtained using the ve40ArCC_Bhattacharya1998.react input file, which contains
the matrix elements shown in the middle panel of fig. 3.

##  IV Nuclear de-excitation model

To model nuclear de-excitations following CC $\nu_{e}$ absorption on [40]Ar, I
rely on the observation that, due to the selection rules in eqs. 21 and 22,
each nuclear final state accessed by the neutrino interaction has a well-
defined excitation energy, spin, and parity. Distinct treatments are used for
bound and unbound nuclear states, with the latter being those for which the
excitation energy exceeds the separation energy for at least one nuclear
fragment with mass number $A\leq 4$. Separation energies are computed in
MARLEY using atomic and particle mass data from refs. Mohr  _et al._ (2012);
Audi  _et al._ (2012). Untabulated atomic masses are estimated using the
liquid drop model of Myers and Swiatecki Myers and Swiatecki (1966); Koning
_et al._ (2008a). The full de-excitation cascade is treated as a sequence of
binary decays while neglecting the possibility of fission and emission of
heavy nucleon clusters ($A\geq 5$).

###  IV.1 Bound states: discrete level data

De-excitations of bound nuclear states are handled in MARLEY using a set of
nuclear structure data files originally prepared for use with version 1.6 of
the TALYS Koning  _et al._ (2008b); Koning and Rochman (2012); Koning  _et
al._ (a) nuclear reaction code.777The current release of TALYS is version
1.95. However, beginning with TALYS 1.8, some nuclear levels for nuclei with
an odd mass number (e.g., [39]K) appear in the structure data files with an
unphysical integer spin (as opposed to a half-integer spin). This bug, which
remains unfixed Koning  _et al._ (b), has prevented updates to the data files
distributed with MARLEY. These in turn are based on the level schemes included
in version 3 of the Reference Input Parameter Library (RIPL-3) Capote  _et
al._ (2009). For a large number of nuclei, the files provide tables of
discrete nuclear energy levels including their excitation energies, spin-
parities, and de-excitation $\gamma$-ray branching ratios. Missing
experimental measurements of these quantities are supplemented by theoretical
estimates. Although internal conversion coefficients are provided in the TALYS
data files, this process is neglected in MARLEY 1.2.0. In the unusual case
where discrete level data are not available for a particular nuclide,
$\gamma$-ray emission is simulated in the same manner as for unbound nuclear
states.

The TALYS structure data files are used with minor reformatting for all
nuclides except [40]K. To ensure consistency with the level assignments made
in section III.1, I prepared an original decay scheme for [40]K using the
experimental data in ref. Chen (2017), the TALYS 1.6 structure file, and
(where needed) estimated $\gamma$-ray branching ratios computed using the
strength function defined in section IV.2.3.

###  IV.2 Unbound states: statistical emission

The MARLEY approach to modeling de-excitations of unbound nuclear states rests
on the assumption of compound nucleus formation: the neutrino interaction
leaves the nucleus in a state of thermal equilibrium which de-excites
independently of the manner in which it was formed. The number of open decay
channels is taken to be large enough that competition between them can be
modeled statistically. That is, transitions to individual nuclear final states
can be neglected in favor of averaging over many states of approximately the
same energy Weisskopf (1937). This last assumption is not strictly true for
excitation energies slightly above the lowest fragment emission threshold. In
such cases it is adopted as an approximation.

Compound nucleus modeling is a key ingredient in nuclear reaction codes
designed to compute nucleon-nucleus and nucleus-nucleus cross sections, such
as TALYS, EMPIRE Herman  _et al._ (2007), CCONE Iwamoto  _et al._ (2016), and
CoH3 Kawano  _et al._ (2010). The treatment in MARLEY uses the Hauser-Feshbach
formalism Hauser and Feshbach (1952) common to these other codes.

####  IV.2.1 Differential decay widths

For the present application to neutrino-induced de-excitations, the physics
content of the Hauser-Feshbach statistical model (HFSM) may be conveniently
summarized by the differential decay widths

$\displaystyle\frac{d\Gamma_{a}}{dE_{x}^{\prime}}$
$\displaystyle=\frac{1}{2\,\pi\,\rho_{i}(E_{x},J,\Pi)}\sum_{\ell=0}^{\infty}\;\sum_{j=|\ell-s|}^{\ell+s}\;\sum_{J^{\prime}=|J-j|}^{J+j}T_{\ell
j}(\varepsilon)\,\rho_{f}(E_{x}^{\prime},J^{\prime},\Pi^{\prime})$ (38) and
$\displaystyle\frac{d\Gamma_{\gamma}}{dE_{x}^{\prime}}$
$\displaystyle=\frac{1}{2\,\pi\,\rho_{i}(E_{x},J,\Pi)}\sum_{\lambda=1}^{\infty}\;\sum_{J^{\prime}=|J-\lambda|}^{J+\lambda}\;\sum_{\Pi^{\prime}\in\\{-1,1\\}}T_{X\lambda}(E_{\gamma})\,\rho_{f}(E_{x}^{\prime},J^{\prime},\Pi^{\prime})\,.$
(39)

which describe de-excitations of a compound nuclear state via emission of a
fragment $a$ or a $\gamma$-ray, respectively.

Here the initial (final) nucleus has excitation energy $E_{x}$
($E_{x}^{\prime}$), total spin $J$ ($J^{\prime}$), and parity $\Pi$
($\Pi^{\prime}$); $s$, $\ell$, and $j$ are the spin, orbital, and total
angular momentum quantum numbers of the emitted fragment; $\rho_{i}$
($\rho_{f}$) is the density of nuclear levels in the vicinity of the initial
(final) state; $\varepsilon$ is the total kinetic energy of the final
particles in the rest frame of the initial nucleus; and $E_{\gamma}$ is the
energy of the emitted $\gamma$-ray. For decays involving emission of a
fragment with parity $\pi_{a}$, the value of $\Pi^{\prime}$ is fixed by
conservation of parity:

$\Pi^{\prime}=(-1)^{\ell}\,\pi_{a}\,\Pi\,.$ (40)

The possible $\gamma$-ray transitions are labeled by their multipolarity
$\lambda\geq 1$ and by whether they are electric ($X=\mathrm{E}$) or magnetic
($X=\mathrm{M}$) in nature. These two alternatives are distinguished based on
the multipolarity and the nuclear parity:

$X=\begin{cases}\mathrm{E}&\Pi=(-1)^{\lambda}\,\Pi^{\prime}\\\
\mathrm{M}&\Pi=(-1)^{\lambda+1}\,\Pi^{\prime}\,.\end{cases}$ (41)

The transmission coefficients $T_{\ell j}$ and $T_{X\lambda}$ quantify how
likely each decay mode is to occur. The methods used to compute them are
described in sections IV.2.2 and IV.2.3. For practical calculations, the
infinite sums that appear in eqs. 38 and 39 must be truncated. Because the
value of $T_{\ell j}$ ($T_{X\lambda}$) falls off rapidly with increasing
$\ell$ ($\lambda$), terms beyond $\ell=\lambda=5$ are neglected.

The HFSM is often communicated in terms of nuclear scattering cross sections
instead of decay widths. To aid the reader in connecting the expressions given
here with more standard presentations (see, e.g., refs. Cole (2000); Thompson
and Nunes (2009)), a brief derivation of eq. 38 is provided in appendix A.

####  IV.2.2 Fragment transmission coefficients

The fragment transmission coefficients $T_{\ell j}$ used in eq. 38 are
computed by solving the radial Schrödinger equation (with relativistic
kinematics as recommended in ref. Koning and Delaroche (2003))

$\left[\frac{d^{2}}{dr^{2}}+\mathpzc{k}^{2}-\frac{\ell(\ell+1)}{r^{2}}-\frac{\mathpzc{k}^{2}}{\varepsilon}\,\mathcal{U}(r,\varepsilon_{\text{lab}},\ell,j)\right]u_{\ell
j}(r)=0$ (42)

where $u_{\ell j}$ is the fragment’s radial wave function,

$\mathpzc{k}=\sqrt{\frac{(2\,m_{a}+\varepsilon_{\text{lab}})\,{M^{\prime}}^{\,2}\,\varepsilon_{\text{lab}}}{(m_{a}+M^{\prime})^{2}+2\,M^{\prime}\,\varepsilon_{\text{lab}}}}$
(43)

is the magnitude of its three-momentum in the rest frame of the initial
nucleus, and $m_{a}$ ($M^{\prime}$) denotes the mass of the emitted fragment
(final nucleus). The global nuclear optical potential $\mathcal{U}$ developed
by Koning and Delaroche Koning and Delaroche (2003) is used in the present
calculations. A full description thereof is given in appendix C.

The quantity

$\varepsilon_{\text{lab}}=\frac{\varepsilon^{2}+2\,(m_{a}+M^{\prime})\,\varepsilon}{2M^{\prime}}$
(44)

is the fragment’s kinetic energy in the rest frame of the final nucleus. The
label lab is applied to this variable because it also represents the
laboratory-frame kinetic energy for the time-reversed process in which the
fragment is absorbed to form the compound nucleus (see appendix A).

The transmission coefficient $T_{\ell j}$ is related to the energy-averaged
S-matrix element $\left<S_{\ell j}\right>$ via

$T_{\ell j}=1-\left|\left<S_{\ell j}\right>\right|^{\,2}\,.$ (45)

The latter quantity may be determined by comparing the full solution $u_{\ell
j}$ of eq. 42 to the asymptotic form

$u_{\ell
j}(r)\to\frac{i}{2}\left[H_{\ell}^{-}(\eta,\mathpzc{k}r)-\left<S_{\ell
j}\right>H_{\ell}^{+}(\eta,\mathpzc{k}r)\right]$ (46)

valid for large radii $r$, where the nuclear optical potential approaches the
Coulomb potential. Here $H_{\ell}^{\pm}$ are the Coulomb wave functions (Olver
_et al._ , 2021, chap. 33). These depend on the Sommerfeld parameter

$\eta\equiv\frac{z\,Z^{\prime}\,\alpha}{\beta_{\text{rel}}}\,,$ (47)

which is evaluated in terms of the proton number $z$ ($Z^{\prime}$) of the
emitted fragment (final nucleus) and the relative speed

$\beta_{\text{rel}}=\frac{\sqrt{\varepsilon_{\text{lab}}^{2}+2\,m_{a}\,\varepsilon_{\text{lab}}}}{m_{a}+\varepsilon_{\text{lab}}}\,.$
(48)

The numerical techniques used to evaluate the fragment transmission
coefficients $T_{\ell j}$ are documented in ref. (Gardiner, 2021c, sec.
2.2.2).

####  IV.2.3 Gamma-ray transmission coefficients

The $\gamma$-ray transmission coefficients $T_{X\lambda}$ used in eq. 39 may
be written in terms of a strength function $f_{X\lambda}(E_{\gamma})$ such
that

$T_{X\lambda}(E_{\gamma})=2\,\pi\,E_{\gamma}^{\,2\,\lambda+1}\,f_{X\lambda}(E_{\gamma})\,.$
(49)

To compute $\gamma$-ray strength functions in this work, I adopt the Standard
Lorentzian model from RIPL-3 Capote  _et al._ (2009), which is based on early
studies by Brink Brink (1955) and Axel Axel (1962). This model assumes that
$\gamma$-ray emissions of type $X\lambda$ occur via de-excitation of the
corresponding giant multipole resonance, which is parameterized in terms of
its centroid excitation energy $E_{X\ell}$, width $\Gamma_{X\ell}$, and peak
cross section $\sigma_{X\ell}$. The strength function is evaluated according
to

$f_{X\lambda}(E_{\gamma})=\frac{\sigma_{X\lambda}}{(2\,\lambda+1)\,\pi^{2}}\bigg{[}\frac{\Gamma^{2}_{X\lambda}\,E_{\gamma}^{\,3-2\lambda}}{(E_{\gamma}^{\,2}-E_{X\lambda}^{2})^{2}+E_{\gamma}^{\,2}\,\Gamma^{2}_{X\lambda}}\bigg{]}$
(50)

with the values of the giant resonance parameters given in table 2. Note that
some peak cross sections are given in the table in units of $\mathrm{mb}$
while eq. 50 employs natural units.

Table 2: Giant resonance parameters used herein for $\gamma$-ray strength
function calculations. Centroid excitation energies $E_{X\lambda}$ and widths
$\Gamma_{X\lambda}$ are given in $\mathrm{MeV}$. Peak cross sections for
electric multipole resonances ($\sigma_{E\lambda}$) are given in
$\mathrm{mb}$, while those for magnetic resonances ($\sigma_{M\lambda}$) are
given in ${\mathrm{MeV}}^{-2}$.

Transition | Parameters
---|---
Electric dipole (E1) a | $E_{\text{E1}}=31.2\,A^{-1/3}+20.6\,A^{-1/6}$
| $\displaystyle\Gamma_{\text{E1}}=0.026\,E_{\text{E1}}^{1.91}$
| $\sigma_{\text{E1}}=1.2\,(120\,N\,Z)\,/\,(\pi\,A\,\Gamma_{\text{E1}})$
Electric quadrupole (E2) b | $E_{\text{E2}}=63\,A^{-1/3}$
| $\displaystyle\Gamma_{\text{E2}}=6.11-0.012\,A$
|
$\sigma_{\text{E2}}=0.00014\,Z^{2}\,E_{\text{E2}}\,/\,(A^{1/3}\,\Gamma_{\text{E2}})$
Magnetic dipole (M1) c d | $E_{\text{M1}}=41\,A^{-1/3}$
| $\displaystyle\Gamma_{\text{M1}}=4$
|
$\displaystyle\sigma_{\text{M}1}=3\,\pi^{2}\left[\frac{\left(B_{\text{n}}^{2}-E_{\text{M}1}^{2}\right)^{2}+B_{\text{n}}^{2}\,\Gamma_{\text{M}1}^{2}}{B_{\text{n}}\,\Gamma_{\text{M}1}^{2}}\right]\Bigg{[}\frac{f_{\text{E}1}(B_{\text{n}})}{0.0588A^{0.878}}\Bigg{]}$
Other electric transitions (E$\lambda$) e | $E_{\text{E}\lambda}=E_{\text{E2}}$
| $\Gamma_{\text{E}\lambda}=\Gamma_{\text{E2}}$
|
$\sigma_{\text{E}\lambda}=($8\text{\times}{10}^{-4}$)^{\lambda-2}\,\sigma_{\text{E2}}$
Other magnetic transitions (M$\lambda$) e | $E_{\text{M}\lambda}=E_{\text{M1}}$
| $\Gamma_{\text{M}\lambda}=\Gamma_{\text{M1}}$
|
$\sigma_{\text{M}\lambda}=($8\text{\times}{10}^{-4}$)^{\lambda-1}\,\sigma_{\text{M1}}$

  * a

See ref. (Belgya  _et al._ , 2006, p. 129)

  * b

See ref. (International Atomic Energy Agency, , p. 103)

  * c

See ref. (Belgya  _et al._ , 2006, p. 132)

  * d

$B_{\text{n}}$ = $7\text{\ }\mathrm{MeV}$ and $f_{\text{E}1}$ is calculated
using natural units and the E1 parameters above.

  * e

Default approximation used by version 1.6 of TALYS Koning  _et al._ (a)

####  IV.2.4 Transitions to discrete nuclear levels

The differential decay widths given in eqs. 38 and 39 are appropriate for use
at high excitation energies $E_{x}^{\prime}$ where the nuclear levels may be
modeled as a continuum. When a discrete level scheme is available for the
final-state nuclide, MARLEY uses the excitation energy of the last tabulated
level as the lower bound for the continuum. Otherwise, a continuum level
density $\rho_{f}$ is used all the way down to the ground state
($E_{x}^{\prime}=0$).

Decays to discrete levels of the final-state nucleus are considered by MARLEY
in terms of the HFSM partial decay widths

$\displaystyle\Gamma_{a}$
$\displaystyle=\frac{1}{2\,\pi\,\rho_{i}(E_{x},J,\Pi)}\sum_{j=|J-J^{\prime}|}^{J+J^{\prime}}\;\sum_{\ell=|j-s|}^{j+s}\delta_{\pi}^{\ell}\,T_{\ell
j}(\varepsilon)$ (51) and $\displaystyle\Gamma_{\gamma}$
$\displaystyle=\frac{1}{2\,\pi\,\rho_{i}(E_{x},J,\Pi)}\sum_{\lambda=\max(1,|J-J^{\prime}|)}^{J+J^{\prime}}T_{X\lambda}(E_{\gamma})\,.$
(52)

The symbol $\delta_{\pi}^{\ell}$, which enforces parity conservation, is equal
to one if eq. 40 is satisfied and zero if it is not. If $J+J^{\prime}<1$, then
the width $\Gamma_{\gamma}$ vanishes. The expressions in eqs. 51 and 52 may be
derived from eqs. 38 and 39 by treating $\rho_{f}$ as a delta function
centered on the nuclear level of interest.

####  IV.2.5 Nuclear level density

In the continuum, the final level density $\rho_{f}$ is computed according to
the Back-shifted Fermi gas model (BFM) described in appendix B. The initial
level density $\rho_{i}$ is evaluated according to the BFM at all excitation
energies. However, since the overall factor involving $\rho_{i}$ cancels out
in the evaluation of decay branching ratios, the specific model chosen for
$\rho_{i}$ does not have any impact on the simulation results.

##  V Results

In this section, the MARLEY 1.2.0 implementation of the theoretical models
described above is used to obtain predictions of total and differential cross
sections for the reaction
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$. Because MARLEY
calculates the four-momentum of every final-state particle for every event,
various additional distributions may be studied beyond those presented in this
work.

###  V.1 Inclusive cross section

Figure 4 shows MARLEY predictions of the total cross section for inclusive
charged-current $\nu_{e}$ absorption on [40]Ar. The important role played by
the Coulomb corrections discussed in section II.2 is illustrated by the
different curves in the plot. The default MARLEY approach to Coulomb effects,
defined in eq. 25, involves choosing the smaller of two correction factors
calculated using the Fermi function and using the modified effective momentum
approximation (MEMA). In fig. 4, the solid black line gives the cross section
calculated using the default approach, while the dotted cyan and dashed red
lines give, respectively, the corresponding cross sections obtained when the
Fermi function and MEMA are used unconditionally. Applying either correction
leads to an enhancement of the total cross section over the uncorrected
result, which is drawn as the blue dash-dotted line. The relationships between
the different approaches to Coulomb corrections in the present calculation are
qualitatively similar to those seen previously using a CRPA model Van Dessel
_et al._ (2019), but there are some details that are different, e.g., the
cross sections calculated using the Fermi function and the MEMA intersect at a
neutrino energy between $5060\text{\ }\mathrm{MeV}$, about $10\text{\
}\mathrm{MeV}$ lower than in the CRPA result.

Figure 4: Inclusive total cross sections for charged-current absorption of
$\nu_{e}$ on [40]Ar. Each curve shows the result obtained using a specific
approach to Coulomb corrections.

###  V.2 Exclusive cross sections

Figure 5 presents the first calculation at supernova energies of total cross
sections for exclusive final states in the reaction
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$. Each exclusive channel
is labeled in terms of its hadronic content, but zero or more de-excitation
$\gamma$-rays are allowed even when not explicitly listed. Below a neutrino
energy of about $10\text{\ }\mathrm{MeV}$, only transitions to bound nuclear
levels are energetically possible. These de-excite via $\gamma$-ray emission.
Single neutron emission becomes appreciable around $15\text{\ }\mathrm{MeV}$.
Although the proton ($7.58\text{\ }\mathrm{MeV}$) and alpha particle
($6.44\text{\ }\mathrm{MeV}$) separation energies for [40]K are comparable to
the neutron separation energy ($7.80\text{\ }\mathrm{MeV}$), the Coulomb
barrier experienced by these charged particles suppresses their emission
relative to neutrons.

Figure 5: Exclusive total cross sections for charged-current absorption of
$\nu_{e}$ on [40]Ar.

Throughout the remainder of this paper, calculations of flux-averaged cross
sections will be reported for two distinct sources of low-energy electron
neutrinos. The first of these is the approximate supernova neutrino energy
spectrum

$\phi(E_{\nu})\propto\left(\frac{E_{\nu}}{\left<E_{\nu}\right>}\right)^{\alpha}\exp\left[-(\alpha+1)\,\frac{E_{\nu}}{\left<E_{\nu}\right>}\right]$
(53)

described in ref. Nikrant  _et al._ (2018). Here the dependence on the
neutrino energy $E_{\nu}$ is expressed in terms of the mean energy
$\left<E_{\nu}\right>$ and a shape parameter $\alpha$. Based on an analysis of
a simulated supernova, the authors of ref. Nikrant  _et al._ (2018) report
values of $\left<E_{\nu}\right>=$14.1\text{\ }\mathrm{MeV}$$ and $\alpha=2.67$
for the time-integrated $\nu_{e}$ spectrum, which I denote by SNT. I also
consider four instantaneous spectra estimated using fig. 1 of ref. Nikrant
_et al._ (2018). These are labeled SN1 through SN4 in chronological order.
Table 3 gives the values of the spectral parameters and the approximate
elapsed time since the start of the supernova for each of these
configurations.

Table 3: Parameters used to compute the supernova neutrino spectrum described in the text. Values of the mean $\nu_{e}$ energy $\left<E_{\nu}\right>$ and the elapsed time $t$ are given in $\mathrm{MeV}$ and seconds, respectively. The shape parameter $\alpha$ is dimensionless. Configuration | $\left<E_{\nu}\right>$ | $\alpha$ | $t$
---|---|---|---
SNT | 14.1 | 2.67 |
SN1 | 9 | 5 | 0
SN2 | 12 | 4 | 0.5
SN3 | 15 | 3.25 | 3
SN4 | 17 | 2.9 | 6.25

The second source of low-energy electron neutrinos considered in this work is
the decay

$\mu^{+}\rightarrow e^{+}+\nu_{e}+\bar{\nu}_{\mu}\,.$ (54)

For an antimuon decaying at rest ($\mu$DAR), the $\nu_{e}$ energy spectrum is
given by Kosmas  _et al._ (2017)

$\phi(E_{\nu})\propto E_{\nu}^{2}\,m_{\mu}^{-4}\,(m_{\mu}-2E_{\nu})\,,$ (55)

where $m_{\mu}$ is the muon mass and the neutrino energy $E_{\nu}$ satisfies

$0<E_{\nu}<m_{\mu}/2\,.$ (56)

Experimental facilities which generate large numbers of stopped muons, such as
the Spallation Neutron Source at Oak Ridge National Laboratory, provide a
valuable opportunity to study tens-of-$\mathrm{MeV}$ neutrino interactions
using a terrestrial source Bolozdynya  _et al._ (2012).

Table 4 reports a wide variety of flux-averaged total cross sections for each
of the electron neutrino spectra $\phi(E_{\nu})$ described above. For each
entry in the table, the flux-averaged total cross section
$\left<\sigma\right>_{f}$ for a specific final state $f$ was obtained via the
expression

$\left<\sigma\right>_{f}=\frac{1}{\Phi}\,\int\phi(E_{\nu})\left[\sum_{L}\sigma_{L}(E_{\nu})\,R_{L}(f)\right]dE_{\nu}$
(57)

where

$\Phi\equiv\int\phi(E_{\nu})\,dE_{\nu}$ (58)

and the integrals in eqs. 57 and 58 are over the entire neutrino spectrum.
Here $\sigma_{L}$ is the inclusive total cross section for transitions to a
particular [40]K nuclear level $L$ and $R_{L}(f)$ is the branching ratio for
the final state $f$ when the de-excitation cascade begins at the level $L$.
The sum in eq. 57 runs over all energetically-accessible nuclear levels. All
quantities in eq. 57 are calculated analytically except for $R_{L}(f)$, which
is estimated using Monte Carlo simulations of de-excitations from every
nuclear level listed in the MARLEY input file ve40ArCC_Bhattacharya1998.react.
The statistical uncertainty associated with each entry in table 4 never
exceeds 10% and is typically much smaller.

Table 4: Flux-averaged total cross sections computed for several different $\nu_{e}$ spectra described in the text. All numerical values are given in $10^{-42}$ ${\mathrm{cm}}^{2}$ $/$ [40]Ar. For example, a table entry of $4.5\times 10^{1}$ should be interpreted as $4.5\times 10^{-41}\,${\mathrm{cm}}^{2}$\,/\,\isotope[40]{Ar}$. Channel | SNT |  $\mu$DAR | SN1 | SN2 | SN3 | SN4
---|---|---|---|---|---|---
$\isotope[40]{Ar}(\nu_{e},e^{-})X$ | $4.485\,471\text{\times}{10}^{01}$ | $3.561\,866\text{\times}{10}^{02}$ | $7.722\,795\text{\times}{10}^{00}$ | $2.248\,090\text{\times}{10}^{01}$ | $5.015\,778\text{\times}{10}^{01}$ | $7.655\,412\text{\times}{10}^{01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\gamma)\isotope[40]{K}$ | $3.292\,455\text{\times}{10}^{01}$ | $1.941\,676\text{\times}{10}^{02}$ | $7.335\,443\text{\times}{10}^{00}$ | $1.896\,931\text{\times}{10}^{01}$ | $3.678\,237\text{\times}{10}^{01}$ | $5.176\,244\text{\times}{10}^{01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{n})\isotope[39]{K}$ | $8.221\,183\text{\times}{10}^{00}$ | $1.044\,471\text{\times}{10}^{02}$ | $3.128\,591\text{\times}{10}^{-01}$ | $2.570\,796\text{\times}{10}^{00}$ | $9.232\,876\text{\times}{10}^{00}$ | $1.669\,965\text{\times}{10}^{01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{p})\isotope[39]{Ar}$ | $2.161\,494\text{\times}{10}^{00}$ | $3.140\,200\text{\times}{10}^{01}$ | $4.918\,244\text{\times}{10}^{-02}$ | $5.786\,560\text{\times}{10}^{-01}$ | $2.420\,270\text{\times}{10}^{00}$ | $4.613\,263\text{\times}{10}^{00}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{d})\isotope[38]{Ar}$ | $7.752\,669\text{\times}{10}^{-02}$ | $1.372\,191\text{\times}{10}^{00}$ | $1.018\,324\text{\times}{10}^{-03}$ | $1.699\,856\text{\times}{10}^{-02}$ | $8.613\,345\text{\times}{10}^{-02}$ | $1.773\,532\text{\times}{10}^{-01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{t})\isotope[37]{Ar}$ | $9.703\,949\text{\times}{10}^{-04}$ | $2.316\,395\text{\times}{10}^{-02}$ | $3.368\,466\text{\times}{10}^{-06}$ | $1.417\,373\text{\times}{10}^{-04}$ | $1.060\,080\text{\times}{10}^{-03}$ | $2.491\,569\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{h})\isotope[37]{Cl}$ | $7.203\,451\text{\times}{10}^{-05}$ | $1.869\,776\text{\times}{10}^{-03}$ | $2.031\,919\text{\times}{10}^{-07}$ | $9.583\,260\text{\times}{10}^{-06}$ | $7.818\,488\text{\times}{10}^{-05}$ | $1.912\,196\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\alpha)\isotope[36]{Cl}$ | $1.310\,415\text{\times}{10}^{00}$ | $2.062\,336\text{\times}{10}^{01}$ | $2.383\,907\text{\times}{10}^{-02}$ | $3.240\,405\text{\times}{10}^{-01}$ | $1.463\,157\text{\times}{10}^{00}$ | $2.875\,723\text{\times}{10}^{00}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{n}\,\mathrm{p})\isotope[38]{Ar}$ | $5.801\,653\text{\times}{10}^{-02}$ | $1.617\,583\text{\times}{10}^{00}$ | $1.371\,793\text{\times}{10}^{-04}$ | $7.082\,974\text{\times}{10}^{-03}$ | $6.257\,303\text{\times}{10}^{-02}$ | $1.588\,299\text{\times}{10}^{-01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{n}\,\mathrm{d})\isotope[37]{Cl}$ | $4.098\,699\text{\times}{10}^{-05}$ | $1.710\,917\text{\times}{10}^{-03}$ | $6.910\,478\text{\times}{10}^{-09}$ | $1.922\,290\text{\times}{10}^{-06}$ | $4.142\,292\text{\times}{10}^{-05}$ | $1.454\,654\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{n}\,\alpha)\isotope[35]{Cl}$ | $8.954\,764\text{\times}{10}^{-02}$ | $2.187\,162\text{\times}{10}^{00}$ | $2.938\,477\text{\times}{10}^{-04}$ | $1.275\,645\text{\times}{10}^{-02}$ | $9.765\,275\text{\times}{10}^{-02}$ | $2.320\,456\text{\times}{10}^{-01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{p}\,\alpha)\isotope[35]{S}$ | $7.864\,443\text{\times}{10}^{-03}$ | $2.232\,536\text{\times}{10}^{-01}$ | $1.783\,766\text{\times}{10}^{-05}$ | $9.391\,102\text{\times}{10}^{-04}$ | $8.469\,125\text{\times}{10}^{-03}$ | $2.168\,641\text{\times}{10}^{-02}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,2\mathrm{n}\,\mathrm{p})\isotope[37]{Ar}$ | $1.547\,900\text{\times}{10}^{-04}$ | $6.712\,654\text{\times}{10}^{-03}$ | $2.659\,833\text{\times}{10}^{-09}$ | $3.782\,569\text{\times}{10}^{-06}$ | $1.500\,282\text{\times}{10}^{-04}$ | $6.260\,851\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,2\alpha)\isotope[32]{Si}$ | $6.369\,740\text{\times}{10}^{-04}$ | $2.349\,068\text{\times}{10}^{-02}$ | $3.360\,093\text{\times}{10}^{-07}$ | $4.680\,675\text{\times}{10}^{-05}$ | $6.666\,756\text{\times}{10}^{-04}$ | $1.988\,936\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,2\mathrm{n})\isotope[38]{K}$ | $9.181\,615\text{\times}{10}^{-04}$ | $3.614\,841\text{\times}{10}^{-02}$ | $2.800\,241\text{\times}{10}^{-07}$ | $5.770\,515\text{\times}{10}^{-05}$ | $9.516\,680\text{\times}{10}^{-04}$ | $2.976\,387\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,\mathrm{n}\,2\mathrm{p})\isotope[37]{Cl}$ | $2.095\,507\text{\times}{10}^{-04}$ | $9.116\,601\text{\times}{10}^{-03}$ | $4.588\,246\text{\times}{10}^{-09}$ | $5.372\,580\text{\times}{10}^{-06}$ | $2.036\,794\text{\times}{10}^{-04}$ | $8.406\,027\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,2\mathrm{p})\isotope[38]{Cl}$ | $6.404\,862\text{\times}{10}^{-04}$ | $2.376\,367\text{\times}{10}^{-02}$ | $3.315\,336\text{\times}{10}^{-07}$ | $4.643\,078\text{\times}{10}^{-05}$ | $6.695\,140\text{\times}{10}^{-04}$ | $2.009\,746\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,3\mathrm{n})\isotope[37]{K}$ | $4.154\,733\text{\times}{10}^{-06}$ | $1.772\,791\text{\times}{10}^{-04}$ | $6.181\,527\text{\times}{10}^{-11}$ | $9.776\,563\text{\times}{10}^{-08}$ | $4.015\,836\text{\times}{10}^{-06}$ | $1.694\,344\text{\times}{10}^{-05}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,3\mathrm{p})\isotope[37]{S}$ | $3.591\,039\text{\times}{10}^{-07}$ | $1.460\,174\text{\times}{10}^{-05}$ | $4.681\,105\text{\times}{10}^{-12}$ | $8.012\,472\text{\times}{10}^{-09}$ | $3.454\,761\text{\times}{10}^{-07}$ | $1.485\,352\text{\times}{10}^{-06}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[28]{Al}$ | $2.235\,610\text{\times}{10}^{-08}$ | $6.552\,005\text{\times}{10}^{-07}$ | $1.844\,574\text{\times}{10}^{-13}$ | $4.013\,386\text{\times}{10}^{-10}$ | $2.107\,268\text{\times}{10}^{-08}$ | $9.823\,915\text{\times}{10}^{-08}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[30]{P}$ | $2.569\,873\text{\times}{10}^{-09}$ | $2.930\,011\text{\times}{10}^{-09}$ | $1.097\,592\text{\times}{10}^{-15}$ | $1.966\,075\text{\times}{10}^{-11}$ | $2.267\,468\text{\times}{10}^{-09}$ | $1.346\,372\text{\times}{10}^{-08}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[30]{Si}$ | $1.539\,868\text{\times}{10}^{-07}$ | $3.154\,833\text{\times}{10}^{-06}$ | $8.630\,280\text{\times}{10}^{-13}$ | $2.300\,615\text{\times}{10}^{-09}$ | $1.427\,498\text{\times}{10}^{-07}$ | $7.093\,741\text{\times}{10}^{-07}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[31]{P}$ | $4.332\,934\text{\times}{10}^{-05}$ | $1.881\,430\text{\times}{10}^{-03}$ | $8.678\,348\text{\times}{10}^{-10}$ | $1.089\,998\text{\times}{10}^{-06}$ | $4.206\,660\text{\times}{10}^{-05}$ | $1.744\,072\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[31]{Si}$ | $3.066\,393\text{\times}{10}^{-06}$ | $1.301\,006\text{\times}{10}^{-04}$ | $4.608\,830\text{\times}{10}^{-11}$ | $7.207\,959\text{\times}{10}^{-08}$ | $2.963\,083\text{\times}{10}^{-06}$ | $1.251\,621\text{\times}{10}^{-05}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[32]{P}$ | $6.369\,792\text{\times}{10}^{-04}$ | $2.349\,069\text{\times}{10}^{-02}$ | $3.360\,093\text{\times}{10}^{-07}$ | $4.680\,679\text{\times}{10}^{-05}$ | $6.666\,801\text{\times}{10}^{-04}$ | $1.988\,962\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[33]{P}$ | $2.602\,884\text{\times}{10}^{-07}$ | $8.538\,772\text{\times}{10}^{-06}$ | $3.053\,539\text{\times}{10}^{-12}$ | $5.213\,132\text{\times}{10}^{-09}$ | $2.471\,098\text{\times}{10}^{-07}$ | $1.121\,934\text{\times}{10}^{-06}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[33]{S}$ | $2.822\,196\text{\times}{10}^{-06}$ | $1.015\,368\text{\times}{10}^{-04}$ | $5.948\,095\text{\times}{10}^{-11}$ | $6.504\,966\text{\times}{10}^{-08}$ | $2.707\,282\text{\times}{10}^{-06}$ | $1.180\,065\text{\times}{10}^{-05}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[34]{Cl}$ | $2.756\,524\text{\times}{10}^{-05}$ | $1.182\,512\text{\times}{10}^{-03}$ | $4.249\,723\text{\times}{10}^{-10}$ | $6.552\,929\text{\times}{10}^{-07}$ | $2.666\,446\text{\times}{10}^{-05}$ | $1.121\,527\text{\times}{10}^{-04}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[34]{P}$ | $1.359\,927\text{\times}{10}^{-06}$ | $5.610\,901\text{\times}{10}^{-05}$ | $1.885\,264\text{\times}{10}^{-11}$ | $3.100\,535\text{\times}{10}^{-08}$ | $1.310\,547\text{\times}{10}^{-06}$ | $5.596\,867\text{\times}{10}^{-06}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[34]{S}$ | $3.416\,978\text{\times}{10}^{-04}$ | $1.489\,999\text{\times}{10}^{-02}$ | $1.214\,284\text{\times}{10}^{-08}$ | $9.619\,152\text{\times}{10}^{-06}$ | $3.339\,035\text{\times}{10}^{-04}$ | $1.349\,221\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[35]{Cl}$ | $8.954\,772\text{\times}{10}^{-02}$ | $2.187\,163\text{\times}{10}^{00}$ | $2.938\,477\text{\times}{10}^{-04}$ | $1.275\,645\text{\times}{10}^{-02}$ | $9.765\,283\text{\times}{10}^{-02}$ | $2.320\,460\text{\times}{10}^{-01}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[35]{S}$ | $7.864\,456\text{\times}{10}^{-03}$ | $2.232\,537\text{\times}{10}^{-01}$ | $1.783\,766\text{\times}{10}^{-05}$ | $9.391\,103\text{\times}{10}^{-04}$ | $8.469\,137\text{\times}{10}^{-03}$ | $2.168\,648\text{\times}{10}^{-02}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[36]{Ar}$ | $1.306\,030\text{\times}{10}^{-05}$ | $5.298\,376\text{\times}{10}^{-04}$ | $6.763\,852\text{\times}{10}^{-10}$ | $3.891\,657\text{\times}{10}^{-07}$ | $1.276\,353\text{\times}{10}^{-05}$ | $5.163\,422\text{\times}{10}^{-05}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[36]{Cl}$ | $1.310\,425\text{\times}{10}^{00}$ | $2.062\,375\text{\times}{10}^{01}$ | $2.383\,908\text{\times}{10}^{-02}$ | $3.240\,407\text{\times}{10}^{-01}$ | $1.463\,167\text{\times}{10}^{00}$ | $2.875\,764\text{\times}{10}^{00}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[36]{S}$ | $5.319\,084\text{\times}{10}^{-07}$ | $2.025\,986\text{\times}{10}^{-05}$ | $9.538\,410\text{\times}{10}^{-12}$ | $1.222\,793\text{\times}{10}^{-08}$ | $5.112\,601\text{\times}{10}^{-07}$ | $2.209\,211\text{\times}{10}^{-06}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[37]{Ar}$ | $1.166\,172\text{\times}{10}^{-03}$ | $3.158\,752\text{\times}{10}^{-02}$ | $3.378\,036\text{\times}{10}^{-06}$ | $1.474\,422\text{\times}{10}^{-04}$ | $1.251\,531\text{\times}{10}^{-03}$ | $3.263\,119\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[37]{Cl}$ | $3.034\,012\text{\times}{10}^{-04}$ | $1.189\,716\text{\times}{10}^{-02}$ | $2.113\,727\text{\times}{10}^{-07}$ | $1.594\,868\text{\times}{10}^{-05}$ | $3.038\,462\text{\times}{10}^{-04}$ | $1.110\,044\text{\times}{10}^{-03}$
$\isotope[40]{Ar}(\nu_{e},e^{-}\,X)\isotope[38]{Ar}$ | $1.355\,432\text{\times}{10}^{-01}$ | $2.989\,774\text{\times}{10}^{00}$ | $1.155\,503\text{\times}{10}^{-03}$ | $2.408\,154\text{\times}{10}^{-02}$ | $1.487\,065\text{\times}{10}^{-01}$ | $3.361\,831\text{\times}{10}^{-01}$

###  V.3 Electron angle and energy distributions

Figure 6 shows flux-averaged differential cross sections predicted by MARLEY
for the laboratory-frame scattering cosine of the outgoing electron. The upper
panel shows the total result (solid black) for the SNT $\nu_{e}$ spectrum
together with the separate contributions arising from Fermi (dashed blue) and
Gamow-Teller (dotted red) transitions. The lower panel presents the same
quantities for the $\mu$DAR $\nu_{e}$ spectrum. Competition between the two
linear components of the cross section gives rise to a total angular
distribution that is nearly flat in both cases, with SNT being slightly
forward and $\mu$DAR slightly backward.

Figure 6: Flux-averaged differential cross sections with respect to the
laboratory-frame scattering cosine $\cos\theta_{\mathrm{e}}$ of the final-
state electron. Top: Calculation for the time-integrated supernova $\nu_{e}$
spectrum described in the text. Bottom: Calculation for $\nu_{e}$ produced by
$\mu^{+}$ decay at rest.

A recent theoretical study Van Dessel  _et al._ (2020) has pointed out that
forbidden nuclear transitions, which are neglected in the present calculation,
have an increasingly strong effect on the electron angular distribution as the
neutrino energy grows beyond a few tens of $\mathrm{MeV}$. Deviations from the
linear behavior shown in fig. 6 signal the breakdown of the allowed
approximation used by MARLEY. A future measurement of the
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ angular differential
cross section will thus provide a powerful constraint on the nuclear modeling
needed to predict the relative contributions of the allowed and forbidden
transitions.

Figure 7 shows the flux-averaged differential cross section with respect to
the kinetic energy of the outgoing electron. The inclusive prediction for the
SNT ($\mu$DAR) spectrum is shown by the solid black line in the upper (lower)
panel, with the other line styles used to represent individual contributions
from four exclusive final states. While the cross sections for both spectra
are dominated by de-excitation modes involving only $\gamma$-rays (loosely
dotted blue) or single neutron emission (densely dotted red), the contribution
of the latter is much more pronounced for the $\mu$DAR case. The small cross
sections for single proton and single $\alpha$ emission are also noticeably
enhanced as one moves from the SNT spectrum to the $\mu$DAR spectrum.

Figure 7: Flux-averaged differential cross sections with respect to the final-
state electron kinetic energy $\mathrm{T_{e}}$. Top: Calculation for the time-
integrated supernova $\nu_{e}$ spectrum described in the text. Bottom:
Calculation for $\nu_{e}$ produced by $\mu^{+}$ decay at rest.

###  V.4 Neutrino energy reconstruction

The energy of the incident neutrino is distributed among the final products of
the $\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ reaction according to
the relation

$E_{\nu}=E_{\text{bind}}+E_{\mathrm{e}}+\mathcal{T}_{\gamma}+\mathcal{T}_{\text{ch}}+\mathcal{T}_{\mathrm{n}}\,,$
(59)

where $E_{\mathrm{e}}$ is the total energy of the outgoing electron and
$\mathcal{T}_{\gamma}$, $\mathcal{T}_{\text{ch}}$, and
$\mathcal{T}_{\mathrm{n}}$ are, respectively, the total kinetic energies of
all de-excitation $\gamma$-rays, charged hadrons, and neutrons in the final
state. The small recoil kinetic energy of the remnant nucleus is included in
$\mathcal{T}_{\text{ch}}$. The change in binding energy $E_{\text{bind}}$ is
given by the expression

$E_{\text{bind}}=\Delta_{\mathrm{RT}}-m_{\mathrm{e}}+\sum_{\mathrm{k}}m_{\mathrm{k}}+\sum_{\mathrm{k}}Q_{\mathrm{k}}\,m_{\mathrm{e}}$
(60)

if electron binding energies are neglected. Here

$\Delta_{\mathrm{RT}}\equiv
m_{\text{atom}}(\mathrm{R})-m_{\text{atom}}(\mathrm{T})$ (61)

is the difference of ground-state atomic masses between the remnant nucleus
($\mathrm{R}$) and the nuclear target ($\mathrm{T}=\isotope[40]{Ar}$),
$m_{\mathrm{e}}$ is the electron mass, and $m_{k}$ ($Q_{k}$) is the mass
(electric charge) of the $k$th nuclear de-excitation product. The sums in eq.
60 run over all particles emitted during nuclear de-excitations.

The minimum possible change in binding energy,

$\displaystyle E_{\text{bind}}^{\text{min}}$ $\displaystyle\equiv
m_{\text{atom}}(\mathrm{F})-m_{\text{atom}}(\mathrm{T})-m_{\mathrm{e}}$ (62)
$\displaystyle=m_{\text{atom}}(\isotope[40]{K})-m_{\text{atom}}(\isotope[40]{Ar})-m_{\mathrm{e}}=$0.99\text{\
}\mathrm{MeV}$$ (63)

occurs for final states in which only $\gamma$-rays are emitted during nuclear
de-excitations. In this case, the nuclide $\mathrm{F}=\isotope[40]{K}$
produced immediately after the primary neutrino interaction and the nuclide
$\mathrm{R}$ remaining after de-excitations are identical.

Since an a priori correction for $E_{\text{bind}}^{\text{min}}$ may be applied
when reconstructing the neutrino energy for any
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ event, eq. 59 may be
usefully rewritten in the form

$E_{\nu}=E_{\text{bind}}^{\text{min}}+\epsilon_{\text{bind}}+E_{\mathrm{e}}+\mathcal{T}_{\gamma}+\mathcal{T}_{\text{ch}}+\mathcal{T}_{\mathrm{n}}\,.$
(64)

Here I have defined the excess binding energy

$\epsilon_{\text{bind}}\equiv E_{\text{bind}}-E_{\text{bind}}^{\text{min}}\,.$
(65)

For [40]K∗ de-excitation modes involving only $\gamma$-rays ($\gamma$), single
neutron emission ($1\mathrm{n}$), single proton emission ($1\mathrm{p}$), or
the emission of both a single neutron and a single proton
($1\mathrm{n}1\mathrm{p}$), the excess binding energy takes the values

$\displaystyle\epsilon_{\text{bind}}^{\gamma}$ $\displaystyle=0$ (66)
$\displaystyle\epsilon_{\text{bind}}^{\mathrm{1n}}$
$\displaystyle=$7.80\text{\ }\mathrm{MeV}$$ (67)
$\displaystyle\epsilon_{\text{bind}}^{\mathrm{1p}}$
$\displaystyle=$7.58\text{\ }\mathrm{MeV}$$ (68)
$\displaystyle\epsilon_{\text{bind}}^{\mathrm{1n1p}}$
$\displaystyle=$14.18\text{\ }\mathrm{MeV}$\,.$ (69)

A useful property of the excess binding energy is that only a few discrete
values of this variable are likely to occur at supernova energies. A future
analysis of supernova neutrino data may therefore attempt to correct for
nonzero values of $\epsilon_{\text{bind}}$ by tagging events in which a
nucleon or a heavy fragment was emitted from the struck nucleus.

Beyond the binding energy contributions, the other terms in eq. 64 vary in the
degree to which they may be reconstructed by a detector. In a liquid argon
time projection chamber (LArTPC), the primary electron will produce a
$\mathrm{cm}$-scale ionization track which may be used to determine its energy
and direction. De-excitation $\gamma$-rays will produce isolated small energy
depositions within several tens of $\mathrm{cm}$ of the interaction vertex,
primarily via Compton scattering on atomic electrons. Reconstruction of both
of these features for supernova neutrino interactions is considered in ref.
Castiglioni  _et al._ (2020), with the conclusion that the energy associated
with each can largely be recovered under realistic detector performance
assumptions. Neutron tagging and calorimetry, on the other hand, were found to
be far more challenging.

Low-energy charged nuclear fragments, such as protons and alpha particles, may
also produce observable ionizations in a LArTPC. A key challenge for
identifying the activity induced by these particles is that, at the energies
relevant for supernova neutrinos, charged hadrons will produce
$\mathrm{mm}$-scale or smaller ionization tracks. These will likely be
difficult to distinguish from the longer track produced by the primary
electron. However, if events involving charged nuclear fragment emission can
be successfully tagged, perhaps by identifying unusually large charge deposits
near the start of the primary electron track, then at least some of the
charged hadron kinetic energy may be recoverable.

To assess the relative importance of the various terms on the right-hand side
of eq. 64, I define several observables, all of which may be interpreted as a
reconstructed neutrino energy $E_{\nu}^{\text{reco}}$ under different, often
quite optimistic, assumptions. The simplest reconstruction method involves
adding the outgoing electron’s total energy to the minimum possible change in
binding energy for CC $\nu_{e}$ absorption on [40]Ar:

$E_{\mathrm{e}}^{\text{reco}}\equiv
E_{\text{bind}}^{\text{min}}+E_{\mathrm{e}}\,.$ (70)

This estimate of the neutrino energy may be refined by adding the summed
energies of all de-excitation $\gamma$-rays

$E_{\mathrm{e}+\gamma}^{\text{reco}}\equiv
E_{\mathrm{e}}^{\text{reco}}+\mathcal{T}_{\gamma}$ (71)

and further refined by adding the summed kinetic energies of all final-state
charged hadrons

$E_{\text{vis}}^{\text{reco}}\equiv
E_{\mathrm{e}+\gamma}^{\text{reco}}+\mathcal{T}_{\mathrm{ch}}\,.$ (72)

I call the last of these variables the visible energy while recognizing that
low-energy neutrons may nevertheless produce some observable signals in a
LArTPC.

Finally, I consider three possible strategies for implementing a binding
energy correction via tagging of final-state nuclear fragments. All three
involve conditionally adding one or more terms to the expression for the
visible energy above. Under the assumption that de-excitation neutrons may be
successfully tagged, I define the reconstructed neutrino energy

$E_{\text{tag}}^{\text{n}}\equiv
E_{\text{vis}}^{\text{reco}}+\delta_{\mathrm{n}}\,\epsilon_{\text{bind}}^{1\mathrm{n}}$
(73)

in which the symbol $\delta_{\mathrm{n}}$ is defined to be unity when a MARLEY
event contains at least one final-state neutron and zero otherwise. Similarly,
under the assumption that charged nuclear fragment emission may be
successfully identified, I define

$E_{\text{tag}}^{\text{p}}\equiv
E_{\text{vis}}^{\text{reco}}+\delta_{\mathrm{ch}}\,\epsilon_{\text{bind}}^{1\mathrm{p}}$
(74)

in which $\delta_{\mathrm{ch}}$ is unity when a MARLEY event contains a
charged nuclear fragment in the final state and zero when it does not. In an
ideal scenario in which both of these tagging techniques are reliable, a still
more refined estimate of the neutrino energy may be obtained via

$\displaystyle E_{\text{tag}}^{\text{n}+\text{p}}\equiv
E_{\text{vis}}^{\text{reco}}$
$\displaystyle+\delta_{\mathrm{n}}\,(1-\delta_{\mathrm{ch}})\,\epsilon_{\text{bind}}^{1\mathrm{n}}$
$\displaystyle+\delta_{\mathrm{ch}}\,(1-\delta_{\mathrm{n}})\,\epsilon_{\text{bind}}^{1\mathrm{p}}+\delta_{\mathrm{n}}\,\delta_{\mathrm{ch}}\,\epsilon_{\text{bind}}^{1\mathrm{n}1\mathrm{p}}\,.$
(75)

Figure 8 shows the MARLEY prediction for flux-averaged differential sections
with respect to each of these observables. The top (bottom) panel of the
figure presents results for the SNT ($\mu$DAR) energy spectrum defined
earlier. A solid black line is used to draw the differential cross section
with respect to the true neutrino energy, while the other line styles
represent the various methods for reconstructing it. The
$E_{\text{vis}}^{\text{reco}}$ result is not shown in the top panel since it
is difficult to distinguish from the $E_{\mathrm{e}+\gamma}^{\text{reco}}$ one
on the scale of the plot.

Figure 8: Flux-averaged differential cross sections with respect to various
definitions of the reconstructed neutrino energy $E_{\nu}^{\text{reco}}$. Top:
Calculation for the time-integrated supernova $\nu_{e}$ spectrum described in
the text. Bottom: Calculation for $\nu_{e}$ produced by $\mu^{+}$ decay at
rest.

For both spectra studied, the agreement between the reconstructed and true
neutrino energy distributions improves most dramatically as one moves from
using only the primary electron ($E_{\mathrm{e}}^{\text{reco}}$, thin dashed
green) to using both the electron and the de-excitation $\gamma$-rays
($E_{\mathrm{e}+\gamma}^{\text{reco}}$, densely dotted brown) in the
reconstruction. Although inclusion of information about charged hadrons is
also seen to be helpful, the next most important improvement comes from the
inclusion of binding energy corrections related to neutron tagging
($E_{\text{tag}}^{\mathrm{n}}$, thick dashed violet). Due to the higher mean
energy of the $\mu$DAR spectrum, nuclear fragment emission becomes more
important relative to SNT, and the impact of the tagging-based binding energy
corrections on neutrino energy reconstruction becomes more pronounced.

To further quantify the performance of each of these energy reconstruction
methods, fig. 9 reports the fractional root mean square (RMS) resolution

$f_{\text{RMS}}(E_{\nu})\equiv\sqrt{\left<\left(\frac{E_{\nu}^{\text{reco}}-E_{\nu}}{E_{\nu}}\right)^{\\!2}\right>}$
(76)

for each definition of the reconstructed neutrino energy
$E_{\nu}^{\text{reco}}$ above. Here, the angle brackets denote the arithmetic
mean of the enclosed quantity. The choice of $f_{\text{RMS}}$ as a metric is
intended to facilitate comparisons with fig. 4 of ref. Castiglioni  _et al._
(2020) and fig. 7 of ref. Abi  _et al._ (2020a), both of which use the same
quantity to study energy reconstruction in a full LArTPC detector simulation.
The results are shown in small bins of the true neutrino energy $E_{\nu}$. To
obtain the curves shown in fig. 9, a large sample of MARLEY events was
generated, and a Monte Carlo estimator for $f_{\text{RMS}}$ was evaluated via

$f_{\text{RMS}}(E_{\nu}\in
b)\approx\sqrt{\frac{1}{N_{b}}\,\sum_{j=1}^{N_{b}}\left[\frac{E_{\nu}^{\text{reco}}(j)-E_{\nu}(j)}{E_{\nu}(j)}\right]^{2}}$
(77)

where the sum runs over over the $N_{b}$ simulated events which fell into the
neutrino energy bin $b$ of interest.

Figure 9: Fractional RMS resolution for the neutrino energy reconstruction
methods described in the text.

The vertical line seen around $4\text{\ }\mathrm{MeV}$ for the
$E_{\mathrm{e}}^{\text{reco}}$ curve (thin dashed green) corresponds to the
MARLEY energy threshold for CC $\nu_{e}$ absorption. Because the third-
forbidden transition between the ground states of [40]Ar ($J^{\pi}=0^{+}$) and
[40]K ($J^{\pi}=4^{-}$) is neglected under the allowed approximation, MARLEY
predicts a finite resolution for $E_{\mathrm{e}}^{\text{reco}}$ even at
threshold.

The considerable improvements in energy resolution seen between
$E_{\mathrm{e}}^{\text{reco}}$ and $E_{\mathrm{e}+\gamma}^{\text{reco}}$
(densely dotted brown) and between $E_{\mathrm{e}+\gamma}^{\text{reco}}$ and
$E_{\text{tag}}^{\mathrm{n}}$ (thick dashed violet) further highlight the
conclusions mentioned above with respect to fig. 8: while $\nu_{e}$ energy
reconstruction in CC absorption on argon will benefit from increased
information about any final-state particle species, determining the de-
excitation $\gamma$-ray energies and tagging neutrons are both particularly
impactful.

##  VI Summary and conclusions

Due to the potential for DUNE to obtain a once-in-a-lifetime large-statistics
measurement of supernova electron neutrinos, achieving a detailed
understanding of tens-of-$\mathrm{MeV}$ neutrino-argon interactions is an
investment that may yield a high scientific return. This paper expands the
ability to model these interactions by providing a first calculation of
exclusive cross sections for the
$\isotope[40]{Ar}(\nu_{e},e^{-})\isotope[40]{K}^{*}$ reaction at supernova
energies. The implementation of the models underlying this calculation in the
MARLEY event generator enables studies of neutrino energy reconstruction to be
carried out easily. The simple approach pursued in section V.4 reveals the
substantial role that measuring the energies of de-excitation $\gamma$-rays
and (though difficult) neutron tagging may play in optimizing supernova
$\nu_{e}$ energy resolution in a future analysis by DUNE. Further insights are
available by using MARLEY in conjunction with a realistic detector simulation
Abi  _et al._ (2020a); Castiglioni  _et al._ (2020).

Two major approximations adopted in MARLEY 1.2.0 constitute limitations on the
present study that should be revisited in future research. The first of these
is the allowed approximation invoked during derivation of the inclusive
differential cross section in section II.1. In a more detailed calculation of
this cross section, the factor $e^{i\mathbf{q}\cdot\mathbf{x}(n)}$ that
appears in the nuclear matrix element from eq. 8 is expanded in a series of
multipoles Walecka (1975) that depend on the spherical Bessel function
$j_{J}(|\mathbf{q}|\,r_{n})$, where $r_{n}$ is the magnitude of
$\mathbf{x}(n)$ and $J$ is the multipole order. Terms representing forbidden
nuclear transitions ($J>0$) vanish in the $|\mathbf{q}|\to 0$ limit imposed by
the allowed approximation, but their contribution to the cross section becomes
increasingly important as the momentum transfer grows. Since the centroid
energy of the multipole giant resonances grows with $J$ roughly like
$41\,J\,A^{-1/3}$ $\mathrm{MeV}$ Kolbe  _et al._ (2003), the inclusion of
forbidden transitions should enhance neutrino scattering to high-lying unbound
nuclear states which de-excite primarily via fragment emission. The degree to
which this observation affects the present results may be studied in the
future by combining a more detailed calculation of the inclusive differential
cross section with the MARLEY de-excitation model.

The second major approximation used in this work, which is shared by nearly
all calculations of exclusive cross sections for tens-of-$\mathrm{MeV}$
neutrino-nucleus scattering, is the compound nucleus assumption discussed in
section IV.2. Further investigation, both theoretical and experimental, will
be needed to clarify the degree to which direct knock-out and pre-equilibrium
processes may safely be neglected in models of low-energy neutrino-nucleus
reactions. A key question is how the transition between the compound nucleus
picture, which is standard for low-energy neutrinos, and the intranuclear
cascade picture, which is commonly used in models of accelerator neutrino
interactions, should be handled as a function of neutrino energy.

Although the current discussion has focused specifically on the description of
nuclear de-excitations following CC $\nu_{e}$ absorption on [40]Ar, the model
presented in section IV is sufficiently general that it may be applied
unaltered in a variety of other contexts. A natural next step is the use of
MARLEY together with an inclusive description of inelastic neutral-current
scattering on argon, a process for which de-excitations provide the only
experimental observables apart from nuclear recoil. While measurements of
tens-of-$\mathrm{MeV}$ neutrino-argon inelastic cross sections must be pursued
to meet the needs of the DUNE supernova neutrino program, more immediate
opportunities for confronting MARLEY with data may become available if the
code is used to obtain predictions for other nuclei. Near-future measurements
that could provide a detailed test of MARLEY include studies of CC $\nu_{e}$
absorption on carbon by JSNS2 Ajimura  _et al._ (2017) and neutrino-induced
neutron production on lead, iron, and copper by COHERENT Akimov  _et al._
(2018). Measurements of exclusive cross sections and decay rates for processes
that are closely related to neutrino interactions, such as electron-nucleus
scattering Flowers  _et al._ (1978) and muon capture Klinskih  _et al._
(2007); Pârvu  _et al._ (2018), may also provide helpful model constraints.
Finally, the capabilities of MARLEY may prove useful in simulating nuclear de-
excitations induced by processes beyond the Standard Model, including nucleon
decay Heeck and Takhistov (2020); Djurcic  _et al._ and the absorption of
fermionic dark matter Dror  _et al._ (2020a, b).

##  VII Acknowledgements

I thank Myung-Ki Cheoun for providing the QRPA matrix elements used in this
work, and I am grateful to the TALYS authors for the decision to release their
nuclear structure data files under the terms of the GNU General Public
License.

Robert Svoboda, Ramona Vogt, and Michael Mulhearn provided helpful feedback on
the doctoral thesis Gardiner (2018) that was a precursor to this publication.

I gratefully acknowledge financial support while at the University of
California, Davis from the John Jungerman-Charles Soderquist Graduate
Fellowship and from the DOE National Nuclear Security Administration through
the Nuclear Science and Security Consortium under award number DE-NA0003180.

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

The high-statistics MARLEY calculations reported herein were performed using
resources provided by the Open Science Grid, which is supported by the
National Science Foundation and the U.S. Department of Energy’s Office of
Science.

##  Appendix A Decay width for fragment emission

In this appendix, I give a brief derivation of the expression in eq. 38 for
the nuclear fragment emission differential decay width of a compound nucleus.
A similar approach can be used to obtain the result in eq. 39 for $\gamma$-ray
emission. The argument presented here is a modern version of one originally
given by Weisskopf in ref. Weisskopf (1937).

Consider the decay process $i\to a+f$ in which a compound nucleus $i$ emits a
fragment $a$ to become a final-state nucleus $f$. Adopt the same notation as
in section IV.2.1: the initial (final) nucleus has spin $J$ ($J^{\prime}$),
parity $\Pi$ ($\Pi^{\prime}$), and mass $M$ ($M^{\prime}$). The emitted
fragment has spin $s$, orbital (total) angular momentum $\ell$ ($j$), three-
momentum magnitude $\mathpzc{k}$, mass $m_{a}$, and parity $\pi_{a}$. Denote
the initial nuclear excitation energy by $E_{x}$, and let the final nuclear
excitation energy lie on the small interval
$[E_{x}^{\prime},E_{x}^{\prime}+dE_{x}^{\prime}]$. The symbol $\rho_{i}$
($\rho_{f}$) denotes the spin-parity dependent level density (see section B.2)
in the vicinity of $E_{x}$ ($E_{x}^{\prime}$) for the initial (final) nucleus.

Within an arbitrary volume $V$ and in the rest frame of the initial nucleus,
the number of states $n_{a+f}$ that may be populated by the decay is given by

$n_{a+f}=n_{a}\,n_{f}$ (78)

where

$n_{a}=(2s+1)\,\frac{V}{2\pi^{2}}\,\mathpzc{k}^{2}\left|\frac{d\mathpzc{k}}{dE_{x}^{\prime}}\right|dE_{x}^{\prime}$
(79)

and

$n_{f}=(2J^{\prime}+1)\,\rho_{f}(E_{x}^{\prime},J^{\prime},\Pi^{\prime})\,dE_{x}^{\prime}\,.$
(80)

By detailed balance, the decay width $\Gamma_{a+f}$ is related to the width
$\Gamma_{i}$ of the time-reversed absorption process $a+f\to i$ via

$\Gamma_{a+f}=\frac{n_{a+f}}{n_{i}}\,\Gamma_{i}$ (81)

where

$n_{i}=(2J+1)\,\rho_{i}(E_{x},J,\Pi)\,\frac{dE_{x}}{dE_{x}^{\prime}}\,dE_{x}^{\prime}$
(82)

is the number of states in which the compound nucleus $i$ may be formed. The
absorption width may be written as

$\Gamma_{i}=\phi\,\sigma$ (83)

where

$\phi=\frac{M\,\mathpzc{k}}{V\,E_{a}\,E_{f}}=\frac{1}{V}\frac{dE_{x}}{d\mathpzc{k}}$
(84)

is the particle flux and

$\sigma=\frac{\pi\,(2J+1)}{\mathpzc{k}^{2}\,(2s+1)(2J^{\prime}+1)}\sum_{\ell=0}^{\infty}\;\sum_{j=|\ell-s|}^{\ell+s}T_{\ell
j}(\epsilon)$ (85)

is the compound nucleus formation cross section. Here $E_{a}$ ($E_{f}$) is the
total energy of the emitted fragment (final nucleus) and

$\epsilon=M-m_{a}-M^{\prime}$ (86)

is the total kinetic energy of the $a+f$ system. A derivation of the
expression in eq. 85 is given888Note that there is a misprint in equations
(A.5) and (A.6) from ref. Gardiner (2018). The quantities $\ell\pm s_{A}$,
$\ell^{\prime}\pm s_{B}$, and $\ell\pm s_{b}$ should be replaced wherever they
occur with, respectively, $j\pm s_{A}$, $j^{\prime}\pm s_{B}$, and
$\ell^{\prime}\pm s_{b}$. in ref. Gardiner (2018). Similar derivations can
also be found in, e.g., refs. Cole (2000); Thompson and Nunes (2009).

Combining the results above and summing over the allowed values of
$J^{\prime}$, which satisfies the triangle relation

$|J-j|\leq J^{\prime}\leq J+j\,,$ (87)

leads immediately to eq. 38.

##  Appendix B Level density model

The nuclear level density model used in the present calculations is the RIPL-3
parameterization Capote  _et al._ (2009) of the Back-shifted Fermi gas Model
(BFM), which is based on the work of Koning, Hilaire, and Goriely Koning  _et
al._ (2008a). The “back shift” used by this model, which accounts for nucleon
pairing effects, involves replacing the nuclear excitation energy $E_{x}$ by
an effective value $U$ defined by

$U\equiv E_{x}-\Delta$ (88)

where the energy shift

$\Delta=\chi_{\text{pair}}\,\frac{$12\text{\ }\mathrm{MeV}$}{\sqrt{A}}+\delta$
(89)

is adjusted to fit experimental data using the empirical parameter $\delta$.
The pairing factor $\chi_{\text{pair}}$ is defined by

$\chi_{\text{pair}}\equiv\begin{cases}1&\text{even-even}\\\
0&\text{odd-$A$}\\\ -1&\text{odd-odd}.\end{cases}$ (90)

###  B.1 Total level density

Under the Back-shifted Fermi gas model (BFM), the total density of nuclear
levels near excitation energy $E_{x}$ is given by the expression Koning  _et
al._ (2008a)

$\rho(E_{x})=\left[\frac{1}{\rho_{F}(E_{x})}+\frac{1}{\rho_{0}(E_{x})}\right]^{-1}$
(91)

where

$\rho_{F}(E_{x})\equiv\frac{1}{\sqrt{2\pi}\sigma}\,\frac{\sqrt{\pi}}{12}\,\frac{\exp(2\sqrt{a_{\rm
LD}\,U})}{a_{\rm LD}^{1/4}\,U^{5/4}}$ (92)

is the Fermi gas level density and

$\rho_{0}(E_{x})=\frac{a_{\rm LD}}{12\,\sigma}\,\exp(a_{\rm LD}\,U+1)$ (93)

is a correction intended to suppress the unphysical divergence of
$\rho_{F}(E_{x})$ at low excitation energies.

Although a constant value for the level density parameter $a_{\rm LD}$ is
sometimes used, I adopt the energy-dependent functional form Ignatyuk  _et
al._ (1975) recommended by RIPL-3 to correct for the damping of shell effects
at high excitation energies:

$\displaystyle a_{\rm LD}$ $\displaystyle\equiv a_{\rm LD}(E_{x},Z,A)$
$\displaystyle=\tilde{a}(A)\left\\{1+\frac{\delta
W(Z,A)}{U}\left[1-\exp(-\gamma\,U)\right]\right\\}.$ (94)

Here $\delta W(Z,A)$ is the shell correction energy, $\tilde{a}(A)$ is the
asymptotic value of $a_{\rm LD}$ at high excitation energies, and $\gamma$ is
a damping parameter that represents how quickly $a_{\rm LD}(E_{x},Z,A)$
approaches $\tilde{a}(A)$. The values of these three parameters are given by
the relations

$\displaystyle\delta W(Z,A)$ $\displaystyle=\delta M_{\text{exp}}(Z,A)-\delta
M_{\text{LDM}}(Z,A)$ (95) $\displaystyle\tilde{a}$
$\displaystyle=\alpha\,A+\beta\,A^{2/3}$ (96) $\displaystyle\gamma$
$\displaystyle=\gamma_{0}\,A^{-1/3}$ (97)

where $\delta M_{\text{exp}}(Z,A)$ is the measured nuclear mass excess Audi
_et al._ (2012); Wang  _et al._ (2012) for the nuclide with proton number $Z$
and mass number $A$, and $\delta M_{\text{LDM}}(Z,A)$ is the corresponding
prediction for the nuclear mass excess using the liquid drop model (Capote
_et al._ , 2009, p. 3164) Mengoni and Nakajima (1994); Myers and Swiatecki
(1966).

This work uses the global “BFM effective” values of the empirical parameters
$\alpha$, $\beta$, $\delta$, and $\gamma_{0}$ obtained in ref. Koning  _et
al._ (2008a) using fits to nuclear level data:

$\displaystyle\alpha$ $\displaystyle=$0.072\,239\,6\text{\
}{\mathrm{MeV}}^{-1}$$ $\displaystyle\beta$ $\displaystyle=$0.195\,267\text{\
}{\mathrm{MeV}}^{-1}$$ (98) $\displaystyle\gamma_{0}$
$\displaystyle=$0.410\,289\text{\ }{\mathrm{MeV}}^{-1}$$ $\displaystyle\delta$
$\displaystyle=$0.173\,015\text{\ }\mathrm{MeV}$.$ (99)

###  B.2 Spin dependence

The density of nuclear levels $\rho(E_{x},J,\Pi)$ with total spin $J$ and
parity $\Pi$ near excitation energy $E_{x}$ may be written in the form

$\rho(E_{x},J,\Pi)=\pi(E_{x},J,\Pi)\,R(E_{x},J)\,\rho(E_{x})$ (100)

where $R(E_{x},J)$ is the nuclear spin distribution and $\pi(E_{x},J,\Pi)$ is
the parity distribution.

Under the assumption that the individual nucleon spins are pointing in random
directions, it can be shown Bethe (1937) that the spin distribution
$R(E_{x},J)$ is given by Koning  _et al._ (2008a)

$R(E_{x},J)=\frac{2J+1}{2\,\sigma^{2}}\exp\left[-\frac{(J+\frac{1}{2})^{2}}{2\,\sigma^{2}}\right].$
(101)

The spin cutoff parameter $\sigma$ determines the width of $R(E_{x},J)$. To
calculate this parameter, I adopt the expression recommended by RIPL-3 Capote
_et al._ (2009) in the absence of discrete level data:

$\displaystyle\sigma^{2}$ $\displaystyle=\sigma^{2}(E_{x})$
$\displaystyle=\begin{cases}\sigma_{d,\text{global}}^{2}+\frac{E_{x}}{S_{n}}\left[\sigma^{2}_{F}(S_{n})-\sigma_{d,\text{global}}^{2}\right]&\text{for
}E_{x}<S_{n}\\\ \sigma^{2}_{F}(E_{x})&\text{for }E_{x}\geq S_{n}.\end{cases}$
(102)

Here $S_{n}$ is the neutron separation energy,

$\sigma^{2}_{F}(E_{x})\equiv\left($0.013\,89\text{\
}{\mathrm{MeV}}^{-1}$\right)\frac{A^{5/3}}{\tilde{a}}\sqrt{a_{\rm LD}\,U}\,,$
(103)

and

$\sigma_{d,\text{global}}\equiv 0.83\,A^{0.26}\,.$ (104)

###  B.3 Parity dependence

Most level density calculations assume equipartition of parity, i.e.,

$\pi(E_{x},J,\Pi)=\frac{1}{2}\,.$ (105)

I adopt this assumption in agreement with RIPL-3. However, I note that more
sophisticated treatments of $\pi(E_{x},J,\Pi)$ have been proposed (see, e.g.,
ref. Al-Quraishi  _et al._ (2003)).

##  Appendix C Optical potential

For the statistical model calculations reported here and implemented in
MARLEY, the global nuclear optical potential developed by Koning and Delaroche
Koning and Delaroche (2003) has been adopted. This phenomenological potential
is based on fits to nucleon-nucleus scattering data and may be written in the
form

$\displaystyle\mathcal{U}=$
$\displaystyle-\mathcal{V}_{V}-i\,\mathcal{W}_{V}-i\,\mathcal{W}_{D}+d_{\ell
s}\,\big{(}\mathcal{V}_{SO}+i\,\mathcal{W}_{SO}\big{)}+\mathcal{V}_{C}\,.$
(106)

Here

$d_{\ell s}\equiv j(j+1)-\ell(\ell+1)-s(s+1).$ (107)

is the eigenvalue of the spin-orbit operator
$2\,\boldsymbol{\ell}\cdot\boldsymbol{s}$ for a projectile with definite total
angular momentum $j$, orbital angular momentum $\ell$, and spin $s$.

The Coulomb potential $\mathcal{V}_{C}$ is taken to be that of a uniformly-
charged sphere:

$\mathcal{V}_{C}(r)=\begin{cases}\frac{Z\,z\,e^{2}}{2\,R_{C}}\left(3-\frac{r^{2}}{R_{C}^{2}}\right)&r<R_{C}\\\\[8.5359pt]
\frac{Z\,z\,e^{2}}{r}&r\geq R_{C}\end{cases}$ (108)

In the expression above, $r$ is the radial coordinate of the projectile, $Z$
($z$) is the proton number of the target nucleus (projectile), $e$ is the
elementary charge, and $R_{C}$ is the Coulomb radius of the nucleus.

###  C.1 Nucleon projectiles

The volume ($V$), surface ($D$), and spin-orbit ($SO$) terms of the optical
potential are functions that may be expressed as the product of an energy-
dependent well depth and an energy-independent radial part:

$\displaystyle\mathcal{V}_{V}$
$\displaystyle=V_{V}(\varepsilon_{\text{lab}})\,f(r,R_{V},a_{V})$ (109)
$\displaystyle\mathcal{W}_{V}$
$\displaystyle=W_{V}(\varepsilon_{\text{lab}})\,f(r,R_{V},a_{V})$ (110)
$\displaystyle\mathcal{W}_{D}$
$\displaystyle=-4\,a_{D}{N}\,W_{D}(\varepsilon_{\text{lab}})\,\frac{d}{dr}f(r,R_{D},a_{D}{N})$
(111) $\displaystyle\mathcal{V}_{SO}$
$\displaystyle=V_{SO}(\varepsilon_{\text{lab}})\,\frac{1}{m_{\pi}^{2}\,r}\,\frac{d}{dr}f(r,R_{SO},a_{SO})$
(112) $\displaystyle\mathcal{W}_{SO}$
$\displaystyle=W_{SO}(\varepsilon_{\text{lab}})\,\frac{1}{m_{\pi}^{2}\,r}\,\frac{d}{dr}f(r,R_{SO},a_{SO})$
(113)

Here $m_{\pi}$ is the mass of a charged pion, and the well depths $V_{V}$,
$W_{V}$, etc., are real-valued functions of the laboratory kinetic energy
$\varepsilon_{\text{lab}}$ of the projectile.

The radial dependence in eqs. 109 to 113 is given by the Woods-Saxon Woods and
Saxon (1954) shape

$f(r,R,a)=\left(1+\exp\left[(r-R)/a\right]\right)^{-1}$ (114)

with effective radius $R$ and diffuseness parameter $a$. Table 5 lists the
values of the parameters needed to compute the radially-dependent parts of the
nuclear optical potential. Each effective radius $R_{j}$ is related to its
tabulated parameter $r_{j}$ via

$\displaystyle R_{j}$ $\displaystyle=r_{j}\,A^{1/3}$ $\displaystyle
j\in\\{V,D,SO,C\\}\,.$ (115)

Note that all parameter values listed in table 5 are given in $\mathrm{fm}$,
while the expressions given in the text assume natural units ($\hbar=c=1$).

The expressions for the well depths are most conveniently written in terms of

$\mathcal{E}\equiv\varepsilon_{\text{lab}}-E_{F}{N}\,,$ (116)

the difference between the laboratory-frame kinetic energy of the projectile
$\varepsilon_{\text{lab}}$ and the nuclear Fermi energy $E_{F}{N}$ for the
projectile species $\mathpzc{N}\in\mathrm{\\{p,n\\}}$ of interest:

$\displaystyle V_{V}(\varepsilon_{\text{lab}})$
$\displaystyle=v_{1}{N}\big{(}1-v_{2}{N}\,\mathcal{E}+v_{3}{N}\,\mathcal{E}^{2}-v_{4}{N}\,\mathcal{E}^{3}\big{)}+V_{\text{Coul}}$
(117) $\displaystyle W_{V}(\varepsilon_{\text{lab}})$
$\displaystyle=\frac{w_{1}{N}\,\mathcal{E}^{2}}{\mathcal{E}^{2}+(w_{2}{N})^{2}}$
(118) $\displaystyle W_{D}(\varepsilon_{\text{lab}})$
$\displaystyle=\frac{d_{1}{N}\,\mathcal{E}^{2}}{\mathcal{E}^{2}+(d_{3}{N})^{2}}\,\exp\\!\big{(}-d_{2}{N}\,\mathcal{E}\big{)}$
(119) $\displaystyle V_{SO}(\varepsilon_{\text{lab}})$
$\displaystyle=v_{so1}{N}\,\exp\\!\big{(}-v_{so2}{N}\,\mathcal{E}\big{)}$
(120) $\displaystyle W_{SO}(\varepsilon_{\text{lab}})$
$\displaystyle=\frac{w_{so1}{N}\,\mathcal{E}^{2}}{\mathcal{E}^{2}+(w_{so2}{N})^{2}}$
(121)

Here the Coulomb contribution to $V_{V}$ is given by

$V_{\text{Coul}}\equiv\delta_{\mathrm{p}\mathpzc{N}}\,\overline{V}_{\\!C}\,v_{1}^{\mathrm{p}}\big{(}v_{2}^{\mathrm{p}}-2\,v_{3}^{\mathrm{p}}\,\mathcal{E}+3\,v_{4}^{\mathrm{p}}\,\mathcal{E}^{2}\big{)}$
(122)

where the symbol $\delta_{\mathrm{p}\mathpzc{N}}$ is defined by

$\delta_{\mathrm{p}\mathpzc{N}}\equiv\begin{cases}0&\mathpzc{N}=\mathrm{n}\\\
1&\mathpzc{N}=\mathrm{p}\,.\end{cases}$ (123)

Tables 10 and 11 from ref. Koning and Delaroche (2003) list the parameters
needed to calculate the well depths for a nucleon projectile.

Table 5: Radial parameters for the global nuclear optical potential defined in ref. Koning and Delaroche (2003). Parameter | Value ($\mathrm{fm}$)
---|---
$r_{V}$ | $1.3039-0.4054\,A^{-1/3}$
$a_{V}$ | $0.6778-$1.487\text{\times}{10}^{-4}$\,A$
$r_{D}$ | $1.3424-0.01585\,A^{1/3}$
$a_{D}^{\mathrm{n}}$ | $0.5446-$1.656\text{\times}{10}^{-4}$\,A$
$a_{D}^{\mathrm{p}}$ | $0.5187+$5.205\text{\times}{10}^{-4}$\,A$
$r_{SO}$ | $1.1854-0.647\,A^{-1/3}$
$a_{SO}$ | 0.59
$r_{C}$ | $1.198+0.697\,A^{-2/3}+12.994\,A^{-5/3}$

###  C.2 Complex projectiles

To compute the nuclear optical potential for complex projectiles ($A>1$),
MARLEY implements a superposition model based on a recommendation by Madland
Madland . It is equivalent to the default treatment used by TALYS. Under this
approach, the radial optical model parameters for a projectile with mass
number $A$ and proton (neutron) number $Z$ ($N$) are computed by weighting the
corresponding parameters for individual nucleons:

$\displaystyle r_{V}$ $\displaystyle=\frac{N\,r_{V}^{n}+Z\,r_{V}^{p}}{A}$
$\displaystyle r_{D},\text{}r_{SO}$ likewise (124) $\displaystyle a_{V}$
$\displaystyle=\frac{N\,a_{V}^{n}+Z\,a_{V}^{p}}{A}$ $\displaystyle
a_{D},\text{}a_{SO}$ likewise. (125)

The Coulomb radius parameter $r_{C}$ remains unchanged from the nucleon case.
The well depths are evaluated according to the relations

$\displaystyle V_{V}(\varepsilon_{\text{lab}})=$ $\displaystyle
N\,V_{V}^{n}(\varepsilon_{\text{lab}}/A)+Z\,V_{V}^{p}(\varepsilon_{\text{lab}}/A)$
(126) $\displaystyle W_{V},$ $\displaystyle W_{D}\text{ likewise}$
$\displaystyle V_{SO}(\varepsilon_{\text{lab}})=$
$\displaystyle\frac{V_{SO}^{n}(\varepsilon_{\text{lab}})+V_{SO}^{p}(\varepsilon_{\text{lab}})}{2A}$
(127) $\displaystyle W_{SO}$ likewise

In the expressions above, the superscript n (p) denotes the value of the
corresponding quantity for an individual neutron (proton), e.g.,
$V_{SO}^{n}(\varepsilon_{\text{lab}})$ is the spin-orbit well depth for a
neutron projectile.

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