# Neutrino-hydrogen interactions with a high-pressure time projection chamber

Philip Hamacher-Baumann<EMAIL_ADDRESS>III.
Physikalisches Institut, RWTH Aachen University, 52056 Aachen, Germany
Xianguo Lu<EMAIL_ADDRESS>Department of Physics, University of
Oxford, Oxford OX1 3PU, United Kingdom  Justo Martín-Albo  justo.martin-
<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC), Universitat de
València & CSIC, 46980 Valencia, Spain

###### Abstract

We investigate the idea of detecting pure neutrino-hydrogen interactions in a
multinuclear target using the transverse kinematic imbalance technique [Lu et
al., Phys. Rev. D92, 051302 (2015)] in a high-pressure time projection chamber
(HPTPC). With full solid-angle acceptance, MeV-level proton tracking
threshold, state-of-the-art tracking resolution, and an
$\mathcal{O}(100\,\mathrm{m^{3}})$ gas volume at 10 bar, an HPTPC could
provide an opportunity to realize this technique. We propose the use of
hydrogen-rich gases in the TPC to achieve high detection purity with large
hydrogen mass. With the projected neutrino beam exposure at the DUNE
experiment, neutrino-hydrogen events of the order of $10^{4}$ per year with
purity above 90 % could be achieved with such an HPTPC using methane gas. In
this paper, we present a systematic study of the event rate and purity for a
variety of argon-alkane mixtures, and examine these gas candidates for the TPC
tracking-related properties.

##  I Introduction

Neutrino-oscillation measurements rely on our understanding of neutrino
interactions in the GeV regime to infer the neutrino energy and flux. To
achieve the required interaction rates, neutrino detectors use materials that
can be practically scaled up, like water, plastics or liquid argon, at the
cost of dealing with complex neutrino-nucleus interactions that result in a
major source of systematic uncertainties Alvarez-Ruso  _et al._ (2018);
Betancourt  _et al._ (2018). Understanding neutrino interactions has become
crucial for T2K Abe  _et al._ (2020a) and NOvA Acero  _et al._ (2019), and so
will be for DUNE Abi  _et al._ (2020) and Hyper-Kamiokande Abe  _et al._
(2018a). As intranuclear effects can only be inferred from final-state
particles, detectors in future experiments are being designed for lower
detection thresholds and larger acceptance. One such detector concept is the
high-pressure time projection chamber (HPTPC), which is one of the components
of the future DUNE near detector Abi  _et al._ (2020). The DUNE HPTPC, as
currently envisioned, consists of a cylindrical detection volume of about 100
m3 (about 5.2 m in diameter and 5 m in length) that holds a pressurized gas at
10 bar and room temperature. To provide constraints on neutrino interactions
at the DUNE far detector, which uses liquid argon, the default gas mixture of
the HPTPC is P-10 (90% Ar + 10% $\mathrm{CH}_{4}$), providing argon mass of
about 1.5 t. Housed in a magnet with a field strength of 0.5 T, this HPTPC
provides tracking and charge separation for particles originating from the
neutrino-gas interactions. In addition to the full (4$\pi$ solid angle)
acceptance, its proton tracking threshold is 3 MeV of kinetic energy Abi  _et
al._ (2020), over an order of magnitude smaller than in solid or liquid
detectors (see Figure 1).

Figure 1: Calculated range (using the SRIM program Ziegler  _et al._ (2010))
of protons in gaseous methane, ethane and argon at 10 bar and 25 ∘C, as well
as in liquid argon and polystyrene, as a function of the kinetic energy. Note
the power-law dependence: $\textrm{range}\sim(\textrm{kinetic
energy})^{2}/\textrm{(material density)}$.

Due to the absence of nuclear effects in neutrino-hydrogen interactions,
hydrogen would be the ideal target for a neutrino oscillation experiment, if
it were possible to use it in large quantities without the presence of other
nuclides. A hydrogen target would significantly improve the reconstruction of
the neutrino energy spectra that is crucial for measuring neutrino oscillation
probabilities Lu  _et al._ (2015). While hydrogen bubble chambers were used to
detect neutrinos before the 1990s, in more recent experiments using plastic
scintillator, mineral oil, or water as target, neutrino-hydrogen interactions
are inseparable from carbon and oxygen background interactions (for a review,
see, for example, Ref. Tanabashi  _et al._ (2018)). Recently, it has been
proposed Lu  _et al._ (2015) that neutrino-hydrogen interactions from a
neutrino beam could be selected in an event-by-event basis from a compound
target that contains hydrogen if sufficient momentum resolution is achieved.
The idea was to use the transverse kinematic imbalance (TKI) of the final-
state particles with respect to the neutrino beam direction: with perfect
tracking, interactions on hydrogen would have balanced final-state transverse
momenta (that is, zero TKI) while the TKI on heavy nuclei is irreducibly wide
due to nuclear effects such as Fermi motion and final-state interactions
(FSIs). As the original method has been discussed in Ref. Lu  _et al._ (2015),
the focus of this work is the impact of the hydrogen amount in the target and
the tracking performance that all together help suppress the nuclear
background.

A large HPTPC, with hydrogen in its gas mixture, could be the ideal detector
to realize this technique and provide high quality data on neutrino-hydrogen
interactions. The default P-10 gas of the DUNE HPTPC contains only very
limited hydrogen mass, and the background from both carbon and argon is
overwhelming. However, a TPC has the unique advantage of being flexible in
switching the gas, the target material that neutrinos interact on. In this
work, we discuss the feasibility of hydrogen-rich gas mixtures in a HPTPC,
with a focus on the perspective of measuring neutrino-hydrogen interactions
given the state-of-the-art tracking performance, an
$\mathcal{O}(100\,\mathrm{m^{3}})$ gas volume at 10 bar, and a neutrino rate
as expected at DUNE.

This paper is organized as follows: In Section II we review the TKI technique
that allows the use of hydrogen-containing chemical compounds for pure
neutrino-hydrogen interactions. In Section III we introduce the HPTPC gas
mixture candidates, examining them in terms of their hydrogen mass and purity.
Because the TKI technique relies on the TPC tracking, we discuss the gas
mixture properties in terms of drift velocity, diffusion, and gas gain in
Section IV. In Section V we summarize this study and discuss the outlook
towards a full realization of measuring neutrino-hydrogen interactions with a
HPTPC.

##  II The Method of TKI

In neutrino interactions on nuclei other than hydrogen, the nuclear remnant
carries away energy and momentum. The kinematics between the incoming neutrino
and the outgoing particles are therefore imbalanced. If the neutrino energy is
unknown, only the imbalance among the momenta transverse to the neutrino
direction is experimentally accessible. The method of TKI uses the details of
this imbalance to precisely identify intranuclear dynamics Lu  _et al._
(2016); Furmanski and Sobczyk (2017); Abe  _et al._ (2018b); Dolan  _et al._
(2018); Lu  _et al._ (2018); Dolan (2018); Lu and Sobczyk (2019); Harewood and
Gran (2019); Cai  _et al._ (2019, 2020); Coplowe  _et al._ (2020) or the
absence thereof Lu  _et al._ (2015); Duyang  _et al._ (2018, 2019); Munteanu
_et al._ (2019). In order to observe the balanced transverse momenta on
hydrogen, all final-state particles need to be measured. While a gaseous TPC
has the optimal acceptance and detection threshold for interactions on its
gas, it is only sensitive to charged particles. Therefore, the particular
neutrino-hydrogen interaction channels to consider are the ones with only
charged final states, which dominantly are the following three-track events Lu
_et al._ (2015) (for similar ideas, cf. Refs. Duyang  _et al._ (2018, 2019);
for antineutrino-hydrogen quasielastic interactions with a neutron in the
final state, see Ref. Munteanu  _et al._ (2019)):

$\displaystyle\nu+\textrm{p}$
$\displaystyle\rightarrow\mu^{-}+\textrm{p}+\pi^{+},~{}\textrm{and}$ (1)
$\displaystyle\bar{\nu}+\textrm{p}$
$\displaystyle\rightarrow\mu^{+}+\textrm{p}+\pi^{-},$ (2)

where $\nu$ and $\bar{\nu}$ are, respectively, a neutrino and an antineutrino,
and $\mu$, p, and $\pi$ are a muon, proton and pion, respectively. The
$\textrm{p}\pi^{+}$ channel takes place primarily through the Delta resonance
$\Delta^{++}(1232)$ production, while for $\textrm{p}\pi^{-}$, in addition to
the Delta resonance $\Delta^{0}(1232)$, charge-neutral nucleon resonances with
higher mass also contribute significantly (see discussions below).

Figure 2: Flux-averaged differential cross section as a function of the final-
state particle polar angle $\theta$ with respect to the incoming neutrino
(top) or antineutrino (bottom) interacting on a CH “molecule”. The cross
section is calculated using GiBUU Buss  _et al._ (2012) with the DUNE fluxes
Abi  _et al._ (2020). The respective signal channels are Eqs. (1) and (2)
where the kinetic energy (K.E.) of each final-state particle is greater than 3
MeV. Comparison is made among the final-state particles. The carbon
backgrounds are shown as shaded histograms.

Without loss of generality, consider neutrino interactions on a CH model
“molecule” which has the same hydrogen-carbon ratio as polystyrene
[$(\textrm{C}_{8}\textrm{H}_{8})_{n}$],

$\displaystyle\nu+\textrm{CH}$
$\displaystyle\rightarrow\mu^{-}+\textrm{p}+\pi^{+}+\textrm{X},~{}\textrm{and}$
(3) $\displaystyle\bar{\nu}+\textrm{CH}$
$\displaystyle\rightarrow\mu^{+}+\textrm{p}+\pi^{-}+\textrm{X},$ (4)

where $\mu$, p, and $\pi$ are required to have kinetic energy greater than 3
MeV, and X stands for the molecular remnant. Flux-averaged differential cross
sections in the polar angle $\theta$ with respect to the neutrino direction,
as well as in the particle momentum $p$, are calculated using the event
generator GiBUU (2019 version) Buss  _et al._ (2012) with the DUNE fluxes Abi
_et al._ (2020). As can be seen in Figures 2 and 3, the muons are mostly at
low angle and high momentum, the pions are at high angle and low momentum, and
the protons, between them. As neutrinos interact with the gas inside the TPC,
high-angle events could be detected. This is advantageous compared to the
forward angular acceptance imposed by an external target to the TPC; one such
example is the T2K near detector TPCs Abe  _et al._ (2020b) that measure the
final-state particles from neutrino interactions on polystyrene in upstream
detectors. With the full acceptance and the low threshold, a HPTPC could
detect the large majority of the final-state particles. Considering instead
thresholds of 100 MeV and 75 MeV for protons and pions, respectively, as in a
polystyrene tracker Lu  _et al._ (2018); Mislivec  _et al._ (2018), only 26%
of the neutrino and 18% of the antineutrino events would be below threshold.
The higher acceptance for the antineutrino channel is due to the additional
high-mass resonances [$N(1440)$, $N(1535)$ and $N(1650)$] that enhance the
high-momentum parts of the spectra and therefore reduce the impact by the
thresholds.

Figure 3: Flux-averaged differential cross section as a function of the final-
state particle momentum $p$ for (a) neutrino and (b) antineutrino interactions
on the CH “molecule”. The pion (and the correlated proton) spectral shape in
carbon results from the competition between the resonance structure and the
pion absorption Pinzon Guerra  _et al._ (2019) inside the nucleus.

Across the whole $\theta$-$p$ phase space, the hydrogen signal and carbon
background are indistinguishable. To identify the hydrogen, a three-track TKI
corresponding to Eqs. (1) and (2), the so-called double-transverse momentum
imbalance, was introduced Lu  _et al._ (2015):

$\displaystyle\delta\textit{p}_{\textrm{TT}}\equiv\left(\vec{p}_{\textrm{p}}+\vec{p}_{\pi}\right)\cdot\hat{z}_{\textrm{TT}},$
(5)

where $\hat{z}_{\textrm{TT}}$ is the unit vector along
$\vec{p}_{\nu}\times\vec{p}_{\mu}$, and $\vec{p}_{\kappa}$ denotes the
momentum vector of particle $\kappa$ (Figure 4).

Figure 4: Schematic diagram for the particle kinematics of Eqs. (1) and (2).
The neutrino and muon momentum vectors, $\vec{p}_{\nu}$ and $\vec{p}_{\mu}$,
define the double-transverse axis
$\hat{z}_{\textrm{TT}}\equiv\vec{p}_{\nu}\times\vec{p}_{\mu}/|\vec{p}_{\nu}\times\vec{p}_{\mu}|$,
onto which the proton and pion momentum vectors, $\vec{p}_{\textrm{p}}$ and
$\vec{p}_{\pi}$, are projected. The sum of these projections defines
$\delta\textit{p}_{\textrm{TT}}$ in Eq. (5).

While the intrinsic $\delta\textit{p}_{\textrm{TT}}$ on hydrogen is zero, on
heavy nuclei it is dominated by Fermi motion and has a typical width of $\sim
200\,\mathrm{\textrm{MeV}/\textit{c}}$. The measured
$\delta\textit{p}_{\textrm{TT}}$ distribution from hydrogen interactions is,
therefore, a function of detector resolution, in contrast to the one from
heavy nuclei, which has an irreducible width due to intranuclear dynamics.
This is the basis of the TKI technique for an event-by-event selection of
neutrino-hydrogen interactions with superb tracking detectors. The
reconstruction resolution of $\delta\textit{p}_{\textrm{TT}}$ by the T2K TPC
is estimated to be $\sim 20\,\mathrm{\textrm{MeV}/\textit{c}}$ Lu  _et al._
(2015). The T2K TPC transverse (to the magnetic field) momentum
($\textit{p}_{\textrm{T}}$) resolution is $\mathcal{O}(10\%)$ at
$\textit{p}_{\textrm{T}}=1\,\mathrm{\textrm{GeV}/\textit{c}}$ Abgrall  _et
al._ (2011). With state-of-the-art TPC tracking performance, like that
achieved with the ALICE TPC, whose $\textit{p}_{\textrm{T}}$-resolution is
$\mathcal{O}(1\%)$ at 1 $\textrm{GeV}/\textit{c}$ Dellacasa  _et al._ (2000);
Abelev  _et al._ (2014), one would expect that a
$\delta\textit{p}_{\textrm{TT}}$-resolution of
$\mathcal{O}(1\,\mathrm{\textrm{MeV}/\textit{c}})$ could be obtained.

To illustrate this idea, we will use the same GiBUU calculation shown above to
calculate a smeared $\delta\textit{p}_{\textrm{TT}}$:

$\displaystyle\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}\equiv\delta\textit{p}_{\textrm{TT}}+\epsilon,$
(6)

where $\epsilon$ is a random variable that follows a Cauchy-Lorentz
probability density function (p.d.f.) $\sim 1/(\epsilon^{2}+\Gamma^{2})$ to
mimic the effect of reconstruction resolution. The width parameter $\Gamma$
takes three values that represent different tracking performance: 20
$\textrm{MeV}/\textit{c}$ (as is for the T2K TPC), 10
$\textrm{MeV}/\textit{c}$ , and 5 $\textrm{MeV}/\textit{c}$ . While, in
practice, momentum resolution is commonly fit by two Gaussian functions, where
the second one is needed to describe the relatively small amount of events
that have large reconstruction bias, here we choose instead the Cauchy-Lorentz
p.d.f. to provide a unified description Lu  _et al._ (2015). The differential
cross sections in the smeared $\delta\textit{p}_{\textrm{TT}}$ (Figure 5) show
that, while the hydrogen $\delta\textit{p}_{\textrm{TT}}$ changes its
Lorentzian shape with the width, the background varies insignificantly.

Figure 5: Flux-averaged differential cross section as a function of the smeared $\delta\textit{p}_{\textrm{TT}}$ for (a) neutrino and (b) antineutrino interactions on CH. The smearing is done by adding to the true $\delta\textit{p}_{\textrm{TT}}$ a random variable $\epsilon$ following a Cauchy-Lorentz p.d.f. $\sim 1/(\epsilon^{2}+\Gamma^{2})$. Comparison is made among different widths, $\Gamma=5,10,\textrm{ and }20\,\mathrm{\textrm{MeV}/\textit{c}}$. $\left|\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}\right|<3\Gamma$ |  $\sigma$ ($10^{-39}$ cm2) |  |
---|---|---|---
|  $\Gamma$ ($\textrm{MeV}/\textit{c}$) | $S$ | $B$ | $S/B$ | purity (%)
$\nu$ CH | 20 | 5.4 | 5.9 | 0.92 | 48
10 | 5.4 | 3.2 | 1.7 | 63
5 | 5.4 | 1.7 | 3.2 | 76
$\bar{\nu}$ CH | 20 | 1.2 | 1.3 | 0.93 | 48
10 | 1.2 | 0.73 | 1.7 | 63
5 | 1.2 | 0.38 | 3.3 | 77
Table 1: Integrated cross section within $3\Gamma$ of
$\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}$ for neutrino and
antineutrino interactions on CH. For different $\Gamma$, the respective cross
section is calculated for both the hydrogen signal $S$ and the carbon
background $B$. The $S/B$-ratio and purity, $S/(S+B)$, are also calculated.

To select the neutrino-hydrogen interactions, one could cut on
$\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}$. To quantify the
performance of such a selection, the signal and background integrated cross
section, $S$ and $B$ respectively, within the region
$\left|\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}\right|<3\Gamma$ are
calculated in Table 1. In both neutrino and antineutrino channels, at
$\Gamma=20\,\mathrm{\textrm{MeV}/\textit{c}}$, the signal and background are
of a similar size, yielding a $S/B$-ratio about 1. At a four-fold reduction of
$\Gamma$, the calculated $S/B$-ratio reaches 3.2, the corresponding purity
[$S/(S+B)$] being 76%. On the one hand, it is important to point out that
these numbers depend on the modeled nuclear effects. More generally speaking,
the departure of the hydrogen-carbon cross-section ratio from $1/6$ is a
measure of the nuclear medium effects Lu  _et al._ (2015). As the $S/B$-ratio
is affected by FSI on top of the Fermi motion of the initial bound proton,
mismodeled FSI such as the elastic component of GENIE hA Lu  _et al._ (2016);
Abe  _et al._ (2018b); Lu  _et al._ (2018); Harewood and Gran (2019); Cai  _et
al._ (2020) could cause significant bias Pickering and Lu (2016). In the
current GiBUU calculation, because $\pi^{+}$ and $\pi^{-}$ experience very
similar FSI inside the carbon remnant, even though both the signal and
background size are different between the neutrino and antineutrino channels,
the $S/B$-ratio is shown to be very similar between the two. On the other
hand, regardless of the underlying nuclear effects, for the same size of
signal, the background size decreases with $\Gamma$. As the relative size of
the background is reduced—via an improvement of the tracking resolution and an
increase in the hydrogen content as discussed in the following sections—the
relative background uncertainty will decrease and become insignificant.

##  III Gas mixture candidates

###  III.1  Argon-Alkane Mixtures

Figure 6: Flux-averaged differential cross section as a function of the
smeared $\delta\textit{p}_{\textrm{TT}}$ for (a) neutrino and (b) antineutrino
interactions on $\mathrm{CH}_{4}$.

Gas mixtures for TPCs have long been studied in field regions suitable for
drift and gas amplification (cf. Refs. Peisert and Sauli (1984); Blum  _et
al._ (2008); Hilke (2010)). Their typical composition is a noble gas with one
or more admixtures of other gases to engineer drift properties, like drift
velocity and diffusion, for the intended detector geometry and event
characteristics. Organic molecules like alkane in the admixture stabilize the
gas amplification by suppressing UV photons in the avalanche gas amplification
process, hence the name quenchers. In the case of Ar-alkane mixtures, the
quencher reduces diffusion and can increase the drift velocity (see Section IV
for detail). Quenchers alone can also act as counter gases in TPCs. This turns
out to be advantageous for the measurement of neutrino-hydrogen interactions.
For example, with a pure $\mathrm{CH}_{4}$ target in comparison to CH, the
hydrogen mass is increased by a factor of four for the same amount of carbon
background. The calculated differential and integrated cross sections are
shown in Figure 6 and Table 2, respectively. An $S/B$-ratio of 13 and a
selection purity of 93% are achieved thanks to the four-fold increase in the
signal size.

$\Gamma=5\,\mathrm{\textrm{MeV}/\textit{c}}$ |  $\sigma$ ($10^{-39}$ cm2) |  |
---|---|---|---
$\left|\delta\textit{p}_{\textrm{TT}}^{\textrm{smeared}}\right|<3\Gamma$ | $S$ | $B$ | $S/B$ | purity (%)
$\nu$ | CH | 5.4 | 1.7 | 3.2 | 76
$\mathrm{CH}_{4}$ | 22 | 1.7 | 13 | 93
$\bar{\nu}$ | CH | 1.2 | 0.38 | 3.3 | 77
$\mathrm{CH}_{4}$ | 5.0 | 0.38 | 13 | 93
Table 2: Integrated cross section within $3\Gamma$
($\Gamma=5\,\mathrm{\textrm{MeV}/\textit{c}}$) of the smeared
$\delta\textit{p}_{\textrm{TT}}$ for neutrino and antineutrino interactions on
CH and $\mathrm{CH}_{4}$.

While P-10 has been a common choice as TPC gas (see, for example, Ref. Wieman
_et al._ (1997)), other gases have also been used. For example, the ALICE TPC
uses 90% Ne + 10% $\mathrm{CO}_{2}$ to cope with the high multiplicity
environment at very high event rates in heavy-ion collisions Dellacasa  _et
al._ (2000), while the T2K near detector TPCs use 95% Ar + 3%
$\mathrm{CF}_{4}$ \+ 2% $i\mathrm{C}_{4}\mathrm{H}_{10}$ Abgrall  _et al._
(2011). Both examples have been operating at atmospheric pressure.

In DUNE, in order to provide constraints on neutrino interactions on argon in
the far detectors, P-10 is the default gas mixture of the near detector HPTPC.
Therefore, we focus on argon-based, in particular Ar-alkane, mixtures as an
extrapolation of the default gas. Depending on the argon mass fraction (that
is, the argon purity in terms of mass), one could choose a certain argon
concentration for the desired argon mass. For example, as is shown in Figure
7, both P-50 (50% Ar + 50% $\mathrm{CH}_{4}$) and 50% Ar + 50%
$\mathrm{C}_{3}\mathrm{H}_{8}$ have the same argon mass, but the argon mass
fraction in P-50 is higher by a relative 50%.

Figure 7: Argon mass fraction vs. argon mass for different Ar-alkane mixtures
at 25 ∘C and 10 bar in a volume of 106.19 $\textrm{m}^{3}$
[$\pi\cdot(2.6\,\mathrm{m})^{2}\times 5\,\mathrm{m}$]. The different points
along respective solid lines are a scan of the alkane
($\mathrm{C}_{x}\mathrm{H}_{y}$) concentration: 0%, 10%, 25%, 50%, 75%, 90%,
and 100% (Ar-$\mathrm{C}_{3}\mathrm{H}_{8}$ only up to 90%, see text).
Different alkane points at the same fraction (50%) are connected by a dashed
line.

In the following, we discuss a range of Ar-alkane mixtures as HPTPC gas
candidates by examining their hydrogen content and tracking-related
properties. Unless otherwise specified, we fix the temperature at 25 ∘C
throughout the discussions.

###  III.2 Hydrogen content

In the previous calculation for a CH target, a
$5\,\mathrm{\textrm{MeV}/\textit{c}}$
$\delta\textit{p}_{\textrm{TT}}$-resolution leads to a selection $S/B$-ratio
of 3.2, corresponding to a purity of 76%. In a HPTPC, as one can choose a
variety of gas mixtures, the selection purity, which depends on the ratio
between the number of free protons and that of the bound ones, can be
optimized alongside with the hydrogen mass as follows.

A CH target has a proton free-to-bound ratio of $1/6$, the same as in P-50
(Figure 8). This ratio increases with the methane concentration and reaches
$1/2$ for P-90 (10% Ar + 90% $\mathrm{CH}_{4}$), and $2/3$ for pure
$\mathrm{CH}_{4}$. By replacing CH (or P-50) with pure $\mathrm{CH}_{4}$ as
the interaction target, the $S/B$ ratio is shown to be improved by a factor of
$(\nicefrac{{2}}{{3}})/(\nicefrac{{1}}{{6}})=4$ (Table 2).

Figure 8: Proton free-to-bound ratio vs. hydrogen mass for different Ar-alkane
mixtures. As a comparison, the ratio for polystyrene (CH) is indicated by a
horizontal line regardless of the hydrogen mass. The hydrogen mass is
converted to the hydrogen-equivalent polystyrene mass, as well as to the
neutrino-hydrogen event rate per year, assuming a cross section of
$10^{-43}\,\mathrm{\textrm{m}^{2}}$ which is typical for the signal channels
[Eqs. (1) and (2)], and an exposure of $10^{21}$ protons on target (POT) per
year with a flux of
$10^{-3}\,\mathrm{\textrm{neutrinos}/\textrm{m}^{2}/\textrm{POT}}$ that gives
the projected neutrino rate at DUNE Abi  _et al._ (2020).

One mole of P-50 has the same amount of hydrogen as one mole of
$\textrm{H}_{2}$. In addition, as is shown in Figure 8 at 10 bar in a volume
of 106.19 m3, the P-50 gas contains the same hydrogen mass as $\sim 1$ t of
polystyrene. This amounts to $\sim 10\%$ of the proposed DUNE 3DST detector
that is a polystyrene tracker with dimensions $2.4\times 2.4\times
2\,\mathrm{\textrm{m}^{3}}$ Abi  _et al._ (2020). With the projected DUNE beam
exposure, $10^{21}$ protons on target (POT) per year and
$10^{-3}\,\mathrm{\textrm{neutrinos}/\textrm{m}^{2}/\textrm{POT}}$ Abi  _et
al._ (2020), this hydrogen mass gives $\sim 5000$ three-track events [Eqs. (1)
and (2)] per year assuming a typical cross section of
$10^{-43}~{}\textrm{m}^{2}$ (Table 1, cf. also Ref. Tanabashi  _et al._
(2018)). It follows immediately that pure $\mathrm{CH}_{4}$ improves the
hydrogen event rate by a factor of 2, to $\sim 10^{4}$ per year.

As alkanes ($\mathrm{C}_{x}\mathrm{H}_{y}$, $y=2x+2$) are acyclic saturated
hydrocarbons, it follows that $\mathrm{CH}_{4}$ with $x=1$ provides the
highest proton free-to-bound ratio among all hydrocarbons. For a given
concentration, other members in the alkane series like ethane
($\mathrm{C}_{2}\mathrm{H}_{6}$) and propane ($\mathrm{C}_{3}\mathrm{H}_{8}$)
can provide larger hydrogen mass with a different proton free-to-bound ratio
(Figure 8). For example, 50% Ar + 50% $\mathrm{C}_{3}\mathrm{H}_{8}$ has twice
the hydrogen mass as P-50. However, this progress along the series is limited
by the phase boundaries of the gas candidates. At 25 ∘C and 10 bar, the
maximal concentration of $\mathrm{C}_{3}\mathrm{H}_{8}$ is 95 % and for
isobutane ($i\mathrm{C}_{4}\mathrm{H}_{10}$) it is 35 % pha —higher than these
they liquefy. Therefore, 5% Ar + 95% $\mathrm{C}_{3}\mathrm{H}_{8}$ provides
the maximal hydrogen mass among all Ar-alkane candidates.

##  IV Tracking-related gas properties

The measurement of neutrino-hydrogen interactions—Eqs. (1) and (2)—from an Ar-
alkane gas mixture via the TKI technique relies on the reconstruction of the
trajectories in the TPC of primary charged particles ($\mu^{\mp}$, p, and
$\pi^{\pm}$). When these particles traverse the detector volume, they ionize
the gas liberating electrons that are then driven towards the readout plane
under the influence of electric and magnetic fields. The arrival time,
position, and amplitude of the collected drift-electron signals are used to
reconstruct the trajectories and characteristic energy loss
($\nicefrac{{\textrm{d}\textit{E}}}{{\textrm{d}\textit{x}}}$) of the primary
particles. In the presence of a magnetic field, the particles’ charges and
momenta can also be measured.

During their propagation, the drift electrons collide with the gas molecules
at energy and time scales different from those of the primary ionization. The
rate of these collisions depends on the gas density, which is sensitive to
temperature ($T$) and pressure ($P$), and thus running detectors are regularly
calibrated towards a certain operational point via temperature and pressure
scaling (see, for example, Ref. Abgrall  _et al._ (2011)). Such density
corrections for the drift field ($E$) and the gas parameters that we will
discuss in this Section are given in Table 3 Peisert and Sauli (1984); Blum
_et al._ (2008); Gonzalez-Diaz  _et al._ (2018).

Drift field and gas parameters | Density correction
---|---
Electric field strength | $\nicefrac{{ET}}{{P}}$
Drift velocity | $v_{\mathrm{d}}$
Diffusion coefficients | $\sigma_{\mathrm{L,T}}\cdot\sqrt{\nicefrac{{P}}{{T}}}$
First Townsend coefficient | $\alpha\cdot\nicefrac{{T}}{{P}}$
Table 3:  Density corrections Peisert and Sauli (1984); Blum  _et al._ (2008);
Gonzalez-Diaz  _et al._ (2018) for the electric field and the gas parameters
that will be discussed in this Section. These scaling laws indicate that, at
the same $\nicefrac{{ET}}{{P}}$, the drift velocity is independent of $T$ and
$P$, while the diffusion decreases and the gas gain increases as
$\nicefrac{{P}}{{T}}$ increases.  Figure 9: Measured drift velocity of P-10 at
three pressures, from close to atmospheric up to $\sim 6\,\mathrm{bar}$ at
296–297 K Hamacher-Baumann . The data closely follow the pressure scaling law
(Table 3) over a span of 5 bar. For comparison, the calculated drift velocity
at 10 bar by MagBoltz Biagi (1999) (interfaced to Garfield++ Schindler, H. and
Veenhof, R. (2020)) is also shown. The error bars are statistical only and
smaller than the marker size.

The discussion of the Ar-alkane properties, including drift and gas gain, will
be focused on their impact on TPC performance. All calculations presented here
were performed using MagBoltz version 11.7 Biagi (1999) interfaced to
Garfield++ Schindler, H. and Veenhof, R. (2020). While the temperature was set
to 298 K, results with various electric field strengths at 1 bar or 10 bar are
compared. The $\mathrm{C}_{x}\mathrm{H}_{y}$ fraction in the Ar-alkane mixture
is scanned from 0 % to 100 %, with the exception of propane
($\mathrm{C}_{3}\mathrm{H}_{8}$) that liquefies above 95 % at 10 bar. The
effect of a magnetic field parallel to the electric field is explicitly
discussed only when relevant.

As a validation, the calculated drift velocity (more detail in Section IV.1)
for P-10 is compared to the measurements by the High Pressure Gas Monitoring
Chamber Hamacher-Baumann . Figure 9 shows the experimental data in three
pressure settings up to $\sim 6\,\mathrm{bar}$. After correcting for the
temperature and pressure, the data show the expected scaling behavior over the
full measurement range. The MagBoltz calculation reproduced the measurements
satisfactorily, except for fields below
$40\,\mathrm{\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}~{}\nicefrac{{\textrm{K}}}{{\mathrm{mbar}}}}$,
where the predicted drift velocity is higher by $\sim 5\,\mathrm{\%}$; a
similar deviation has been reported in Ref. Atoum  _et al._ (2019).

###  IV.1 Drift Velocity

Figure 10:  MagBoltz Biagi (1999) (interfaced to Garfield++ Schindler, H. and
Veenhof, R. (2020)) calculation of the drift velocity at 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ as a
function of the alkane concentration. Ar-alkane mixtures at 1 bar and 10 bar
are compared. The ALICE (90% Ne + 10% $\mathrm{CO}_{2}$) value Dellacasa  _et
al._ (2000) is indicated by a horizontal line. Note the
$\nicefrac{{E}}{{P}}$-scaling (see also Table 3).

In a TPC, the electron drift velocity ($v_{\mathrm{d}}$) is used to convert
the signal arrival time to a position along the drift direction, enabling the
three-dimensional reconstruction of the primary ionization spatial
coordinates. The ALICE TPC has a drift velocity of 2.83
$\nicefrac{{\mathrm{cm}}}{{\mu\mathrm{s}}}$ for an electric field strength
$E=400\,\mathrm{V/cm}$ across a drift length of 2.5 m and for a gas mixture of
90% Ne + 10% $\mathrm{CO}_{2}$ at one atmospheric pressure Dellacasa  _et al._
(2000). Due to the $\nicefrac{{E}}{{P}}$-scaling (Table 3), to maintain the
same drift velocity in the same gas at 10 bar, a field strength of
$4\,\mathrm{kV/cm}$ is required, implying a cathode voltage of 1 MV across a
2.5-m drift length. However, as commercial power supplies are not readily
available above 500 kV, we consider $\nicefrac{{E}}{{P}}\sim
40\textrm{--}200\,\mathrm{\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}}$
a practical operational region in an ALICE-sized TPC at 10 bar. In comparison,
the T2K TPC has a drift velocity of 7.8
$\nicefrac{{\mathrm{cm}}}{{\mu\mathrm{s}}}$ at 275 V/cm for a mixture of 95%
Ar + 3% $\mathrm{CF}_{4}$ \+ 2% $i\mathrm{C}_{4}\mathrm{H}_{10}$ at
atmospheric pressure Abgrall  _et al._ (2011).

Figure 11: Fractional change (in %) of drift velocity for every percentage
increase of the alkane concentration.

Figure 12: Drift velocity as a function of the scaled electric field strength
for different gas mixtures with alkane concentrations (a) 10%, (b) 50%, and
(c) 100% (90% for $\mathrm{C}_{3}\mathrm{H}_{8}$). The reference values from
ALICE and T2K (95% Ar + 3% $\mathrm{CF}_{4}$ \+ 2%
$i\mathrm{C}_{4}\mathrm{H}_{10}$) Abgrall  _et al._ (2011) are shown.

The calculated drift velocity at 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ is shown
in Figure 10. At a few percentage of alkane concentration, the drift velocity
dramatically increases from the pure-argon value 0.2
$\nicefrac{{\mathrm{cm}}}{{\mu\mathrm{s}}}$ by an order of magnitude. This is
due to the so-called Ramsauer minimum of argon Ramsauer (1921): the low
excitation energy of alkanes (due to the presence of the vibrational and
rotational degrees of freedom) effectively reduces the energy of the drifting
electrons such that the collisional cross section of electrons on argon
reaches a minimum, making the gas as a whole more transparent to the drifting
electrons. The drift velocity then falls back as the concentration increases,
approaching 0.5 $\nicefrac{{\mathrm{cm}}}{{\mu\mathrm{s}}}$ for pure methane
and ethane, and even lower for propane-rich mixtures (still higher than for
pure argon). This level of drift velocity corresponds to sub-millisecond drift
time across a 2.5 m drift length, which would allow for a pile-up-free event
rate of $\mathcal{O}(1\,\mathrm{kHz})$, much higher than the ones foreseen in
future accelerator neutrino experiments Abi  _et al._ (2020).

While the magnitude of the drift velocity is not critical here, an optimal
tracking performance relies on a uniform and stable drift velocity in the
large gas volume, which in turn poses a constraint on the gas system Kotchenda
_et al._ (2003). Figure 11 shows the fractional change of the drift velocity
for every percentage increase of the alkane concentration. At 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ for any
$\mathrm{C}_{x}\mathrm{H}_{y}$ concentration above 5 %, the change of the
drift velocity is greater than 1 %, which indicates that a per-mil-level
stability of the drift velocity requires a control on the quencher
concentration at the per-mil level.

In the practical operational region 40–200
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ , the
drift velocity generally increases with $E$. Figure 12 shows the drift
velocity as a function of the pressure-scaled $E$ for the alkane
concentrations 10%, 50%, and 100% (90% for $\mathrm{C}_{3}\mathrm{H}_{8}$). In
particular, at
$\nicefrac{{E}}{{P}}<50\,\mathrm{\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}}$,
$v_{\mathrm{d}}\propto E^{a}$, where $a\sim 1.0\textrm{--}1.2$. In this
quasilinear region, the electron mobility Blum  _et al._ (2008),

$\displaystyle\mu\equiv\frac{v_{\mathrm{d}}}{E},$ (7)

is largely field-independent. For pure alkane at 10 bar [Figure 12 (c)], the
typical mobility is $\sim
0.001\,\mathrm{\nicefrac{{\mathrm{cm}^{2}}}{{\mathrm{V}}}\nicefrac{{}}{{\mu\mathrm{s}}}}$.
Furthermore, the drift velocity variation is $\mathcal{O}(\%)$ for every 1
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ change,
as can be seen in Figure 12. Compared to TPCs operating at atmospheric
pressure, the pressure variation in a high-pressure TPC is relatively better
under control as the pressurized vessel is not connected to the atmosphere.

###  IV.2 Diffusion

Once liberated, the primary ionization electrons start to diffuse in all
directions through scattering on gas molecules. The size of the spread grows
with time $t$ as $\sim\sqrt{t}$. Under the influence of an electric field, the
diffusing electron clouds drift and the spread in the transverse and
longitudinal direction to the field are characterized by
$\sigma_{\mathrm{L,T}}\sqrt{l_{\mathrm{d}}}$, where $\sigma_{\mathrm{T}}$ and
$\sigma_{\mathrm{L}}$ are the transverse and longitudinal diffusion
coefficients, respectively, and $l_{\mathrm{d}}=v_{\mathrm{d}}t$ is the drift
length. Diffusion limits the TPC point resolution and track separation
threshold. For its momentum reconstruction in a high-multiplicity environment,
ALICE chose
$\sigma_{\mathrm{T}}=\sigma_{\mathrm{L}}=220\,\mathrm{\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}}$
Dellacasa  _et al._ (2000). In T2K, the near detector TPC has
$\sigma_{\mathrm{T}}=265\,\mathrm{\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}}$
Abgrall  _et al._ (2011); Abe  _et al._ (2019).

The calculated $\sigma_{\mathrm{T}}$ for various Ar-alkane mixtures at 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ is shown
in Figure 13. Because of the $1/\sqrt{P}$-suppression at the same
$\nicefrac{{E}}{{P}}$ (Table 3), the diffusion in 10 bar for most of the
mixtures is smaller than in ALICE—nearly by half for concentrations above 20
%. It slowly decreases with the alkane concentration and approaches the
thermal limit at 113 $\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}$ for
$\nicefrac{{E}}{{P}}$ =40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ Blum
_et al._ (2008). In Figure 14, the transverse diffusion is shown to decrease
with $E$ in the practical region 40–200
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ (except
for P-10, where it becomes stable). In addition, at higher concentration, as
it approaches the thermal limit, the dependence on $E$ of different alkane
becomes similar.

Figure 13: Transverse diffusion coefficient calculated at 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ as a
function of the alkane concentration. The reference values from ALICE and T2K
and the thermal limit for 10 bar are shown.

Figure 14: Scaled transverse diffusion coefficient as a function of the scaled
electric field strength. Note the
$\sqrt{P}\sigma_{\mathrm{T}}(\nicefrac{{E}}{{P}})$-scaling (see also Table 3).
The reference values from ALICE and T2K are shown.

The longitudinal diffusion was also calculated and shows similar size and
trends as the transverse diffusion. A comparison between the two at 10 bar and
40 $\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ is
shown in Table 4.

| Concentration (%) |  $\sigma_{\mathrm{L}}$ ($\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}$) |  $\sigma_{\mathrm{T}}$ ($\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}$)
---|---|---|---
$\mathrm{CH}_{4}$ | 10 | 224 | 190
50 | 174 | 133
| 100 | 151 | 124
$\mathrm{C}_{2}\mathrm{H}_{6}$ | 10 | 189 | 183
50 | 150 | 134
| 100 | 133 | 123
$\mathrm{C}_{3}\mathrm{H}_{8}$ | 10 | 174 | 160
50 | 130 | 122
| 90 | 122 | 118
Table 4:  Longitudinal and transverse diffusion coefficients for Ar-alkane
mixtures calculated at 10 bar and 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ .

With an additional magnetic field parallel to the electric field,
$\vec{B}\,||\,\vec{E}$, the transverse diffusion is suppressed Blum  _et al._
(2008):

$\frac{\sigma_{\mathrm{T}}(B)}{\sigma_{\mathrm{T}}(0)}=\frac{1}{\sqrt{1+(B\mu)^{2}}},$
(8)

where $\mu$ is the electron mobility [Eq. (7)]. For pure alkane at 10 bar
where $\mu\sim
0.001\,\mathrm{\nicefrac{{\mathrm{cm}^{2}}}{{\mathrm{V}}}\nicefrac{{}}{{\mu\mathrm{s}}}}$
(see Section IV.1), the suppression by a 0.5 T magnetic field is less than 1%.
Furthermore, with the mobility at 1 bar $\sim
0.01\,\mathrm{\nicefrac{{\mathrm{cm}^{2}}}{{\mathrm{V}}}\nicefrac{{}}{{\mu\mathrm{s}}}}$,
the magnetic field required to produce the same suppression achieved at 10 bar
is $B=3/\mu=3\,\mathrm{T}$. It is interesting to note that, for the
longitudinal diffusion, while it can also be reduced by pressure scaling, it
is not affected by the parallel magnetic field.

###  IV.3 Gas Gain

After propagation through the drift region, electrons are multiplied in strong
electric fields that start ionization avalanches. Electrodes pick up the
amplified signal which can then be more easily digitized by a number of
electronics. In an amplification region with a spatial coordinate $s$
($s_{0}<s<s_{1}$), the gas gain $G$ depends on the path of the electrons Blum
_et al._ (2008):

$G=\exp\left[\int_{s_{0}}^{s_{1}}{\left(\alpha-\eta\right)}\mathrm{d}s\right],$
(9)

where $\alpha$ is the first Townsend coefficient and $\eta$ the attachment
coefficient, both being functions of the electric field strength $E(s)$. In
our calculation, there are no impurities in the Ar-alkane mixtures, so
attachment can not proceed via three-body processes Blum  _et al._ (2008).
Generally, the attachment is a small correction to the amplification; however,
at amplification onset, the attachment cannot be neglected and the effective
Townsend coefficient $\alpha-\eta$ is considered. There is no attachment for
the fields relevant in the drift region.

Following the density correction in Table 3, the effective Townsend
coefficient is enhanced by the pressure but delayed in onset field due to a
shortened electron mean free path—a larger field strength is needed to provide
enough energy to initiate the avalanche.

The calculations in this work consider Penning transfer contributions to
$\alpha$. Penning transfers are ionizing energy transfers between gas
molecules or atoms and can be summarized by a single transfer coefficient
$\mathcal{R}$ that enhances $\alpha$ Şahin  _et al._ (2010), in which case
$\alpha$ in Eq. (9) is replaced by $(1+\mathcal{R})\alpha$. Values for
$\mathcal{R}$ have been calculated from gas gain measurements for some common
argon-based drift gases, but not for quencher fractions above 10 % Şahin  _et
al._ (2010).

Figure 15:  Effective Townsend coefficient calculated at 10 bar as a function
of the electric field strength. The dashed line is to guide the eye for
$\alpha-\eta=0$. In panel (a), the simulated values for 90% Ne + 10%
$\mathrm{CO}_{2}$ at 1 bar ($\mathcal{R}$ determined from Şahin  _et al._
(2016)) are shown for comparison. The error bands for the Ar-alkane mixtures
show the effect of varying $\mathcal{R}$ from 0 % (lower bound) to 100 %
(upper bound), while the lines correspond to $\mathcal{R}=50\,\mathrm{\%}$. In
panel (c), the $\mathrm{C}_{2}\mathrm{H}_{6}$ curve has no error band due to
the absence of Penning transfer, whereas the Ar-$\mathrm{C}_{3}\mathrm{H}_{8}$
curve has an error band that is too small to be visible.

The calculated effective Townsend coefficients for different Ar-alkane
mixtures at 10 bar are shown in Figure 15. For Ar-alkane mixtures, the
difference between no and maximal $\mathcal{R}$ can be very large, especially
for $\mathrm{CH}_{4}$, but is expected to decrease with quencher fraction as
transfers from excited Ar to $\mathrm{C}_{x}\mathrm{H}_{y}$ become less
frequent. It is interesting to note that, pure methane has excited states that
exceed the ionization threshold of the molecule, opening the possibility for
ionizing energy transfers within the quenching gas itself Şahin  _et al._
(2010) [Figure 15 (c)].

Due to the $\nicefrac{{E}}{{P}}$-scaling, the onset field strength at 10 bar
is a factor of 10 larger than at 1 bar, as exemplified by the comparison with
the ALICE TPC gas (90% Ne + 10% $\mathrm{CO}_{2}$, 1 bar) in Figure 15 (a).
The onset is shown to increase along with the alkane concentration, from 50
$\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}$ at 10 % to 150–250
$\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}$ for pure alkanes. High
concentrations of propane require significantly larger amplification fields to
reach $\alpha-\eta$ values comparable to methane and ethane. In comparison,
the gain onset for the ALICE TPC gas is at around 4
$\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}$ Dellacasa  _et al._ (2000); Lippmann
(2014).

The need for such high voltages poses a challenge to currently existing gas
amplification structures. As examples, wire chambers Dellacasa  _et al._
(2000) and MicroMegas Giomataris  _et al._ (1996) are typically operated at
about
$10-100\,\mathrm{\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}}$.
The biasing voltage needed to achieve these fields varies between
$\mathcal{O}(100-1000\,\mathrm{V})$. At 10 bar, the bias voltage would be up
to $\mathcal{O}(10\,\mathrm{kV})$; a challenge for high voltage safety from
spark protection to electrostatic distortion of wires. The significantly
higher voltages needed for high fractions of $\mathrm{C}_{3}\mathrm{H}_{8}$
might prove prohibitive in order to reach sufficient gas gain. A new
technology, the resistive MicroMegas, has proven to be operational under such
high fields close to $80\,\mathrm{\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}}$
and 1 bar Procureur  _et al._ (2012); meanwhile, conventional MicroMegas have
been successfully used in pressurized TPCs (see, for example, Ref. Cebrian
_et al._ (2013)).

##  V Summary and Discussions

In this paper, we study the feasibility of measuring neutrino-hydrogen
interactions in a HPTPC using argon-alkane gas mixtures. The charged-particle
sensitivity of the TPC and its full acceptance and low threshold make it ideal
for a measurement of the neutrino exclusive $\mu\textrm{p}\pi$ production
[Eqs. (1) and (2)] that could be used to identify interactions on the hydrogen
component out of other nuclear target backgrounds Lu  _et al._ (2015). With
event-generator calculations, we confirm the efficient phase-space coverage of
the detector. By modeling the detector response to the
$\delta\textit{p}_{\textrm{TT}}$ observable [Eqs. (5) and (6) ], we
demonstrate that the signal-background ratio could be efficiently enhanced by
improving the tracking resolution. A hydrogen-enriched TPC gas in an
$\mathcal{O}(100\,\mathrm{m^{3}})$ volume at 10 bar could not only further
increase the signal-background ratio but also deliver a significant event
rate. Methane, for example, would provide $\sim 10^{4}$ event per year (at the
projected DUNE exposure) with purity above 90 % (assuming a
$\delta\textit{p}_{\textrm{TT}}$ resolution of
$\Gamma=5\,\mathrm{\textrm{MeV}/\textit{c}}$). Such an event rate is twice
what pure hydrogen would yield. The highest hydrogen mass would be provided by
$5\%~{}\textrm{Ar}+95\%~{}\mathrm{C}_{3}\mathrm{H}_{8}$, nearly doubling the
signal yield of pure $\mathrm{CH}_{4}$ and with a signal-background ratio
twice as good as by polystyrene.

We also examine the gas-mixture properties related to TPC tracking. At high
pressure, the effective drift field is reduced. Due to the limitation on
megavolt power supplies, the electric field strength across several meters of
drift length will not be strong enough to saturate the drift velocity to reach
the stable maximum. In the practical operational region
($\nicefrac{{E}}{{P}}\sim
40\textrm{--}200\,\mathrm{\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}}$)
we consider, the drift velocity is (quasi)linear to the field strength with
electron mobility $\sim
0.001\,\mathrm{\nicefrac{{\mathrm{cm}^{2}}}{{\mathrm{V}}}\nicefrac{{}}{{\mu\mathrm{s}}}}$
at 10 bar. The resulting drift time could comfortably cope with the highest
event rates foreseen in future accelerator-neutrino experiments. The sensitive
drift velocity poses a constraint on the gas system: a per-mil-level stability
requires a per-mil-level control on the gas composition, the drift field
strength, as well as the temperature and the pressure.

The high pressure also reduces both transverse and longitudinal diffusion to
significantly below the ALICE values. At
$\nicefrac{{E}}{{P}}=40\,\mathrm{\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}}$
and high alkane concentration, the diffusion coefficients approach the thermal
limit $\sim 100\,\mathrm{\nicefrac{{\mu\mathrm{m}}}{{\sqrt{\mathrm{cm}}}}}$
and become almost independent of the alkane type. The impact on the diffusion
by a parallel magnetic field is shown to be negligible due to the small
electron mobility.

One further impact on the gas properties by the high pressure is the much
stronger amplification field required for gas gain to set in. For pure alkane
the onset field strength is 150-250 $\nicefrac{{\mathrm{kV}}}{{\mathrm{cm}}}$
, about 50 times of the ALICE value. And yet, the increased primary ionization
density at high pressure (see Appendix B) reduces the overall gain needed to
produce usable signals.

In this work, we used the GiBUU neutrino-event generator for the calculation
of the signal and background rates. The underlying nuclear effects belong to a
currently very active research area and the resulting uncertainties on our
estimation need to be addressed both theoretically and most importantly by
dedicated experiments. We modeled the detector response to the TKI observable
by a one-parameter smearing function. For the next order accuracy, a detailed
tracking model (and eventually a full detector simulation taking into account
the detector geometry) could be applied to the particle-by-particle momentum
vectors given by an event generator. Having these potential future
improvements in mind, we emphasize in this paper the scaling behavior of the
signal and background with the tracking resolution and the hydrogen content of
the gas. We argue that with the state-of-the-art tracking performance foreseen
in a future HPTPC (see Appendix A for further discussions) and the existing
hydrogen-rich gas mixtures, neutrino-nucleus interaction background could be
reduced to less than 10 %. It would be crucial at this early stage to estimate
a more realistic $\Gamma$ value that a near-future HPTPC could achieve, with
the help of detailed detector simulations, so that further discussions could
proceed on the physics opportunities provided by a high-purity neutrino-
hydrogen sample.

In the search of hydrogen-rich gas, we start with argon-hydrocarbon mixtures.
The main purpose of the argon component is to provide early-stage synergy with
the DUNE argon program. For example, the first-stage hydrogen program could
proceed with P-50 to establish the baseline performance while still providing
high-statistics neutrino-argon events (the carbon background might need to be
constrained or statistically subtracted with the help of auxiliary
measurements). Except for this practical concern, the argon component could be
replaced by helium (see Appendix C), for example, to study neutrino
interactions on light nuclei. The carbon base, on the other hand, is motivated
by its small number of (bound) protons. In addition, hydrocarbon, in
particular alkane, is a well-studied TPC gas. As is shown in this work, the
drift and gas gain properties with high-concentration alkane do not raise
serious concerns in the TPC design. Yet, as it has been mentioned in this
paper, the existing calculation of the gas properties could be further
improved. In addition to developing better models for higher order accuracy,
dedicated measurements of the gas properties are valuable. It is important to
point out that, while we have demonstrated the gas-searching strategy, the gas
mixture candidates that we discussed can be further improved, for example, by
adding a third component: the ternary mixture
Ar-$i\mathrm{C}_{4}\mathrm{H}_{10}$-$\mathrm{C}_{3}\mathrm{H}_{8}$ with the
isobutane fraction up to 35 % can provide even more hydrogen events than by
the Ar-$\mathrm{C}_{3}\mathrm{H}_{8}$ mixture. In fact, if we only consider
alkane that is gaseous at 1 bar, the theoretical limit for maximal hydrogen
mass at 10 bar is reached by a mixture that is equivalent to C3.93H9.86: 17 %
$\mathrm{C}(\mathrm{C}\mathrm{H}_{3})_{4}$ (neopentane), 35 %
$i\mathrm{C}_{4}\mathrm{H}_{10}$, 24 % $\mathrm{C}_{4}\mathrm{H}_{10}$
(butane), and 24 % $\mathrm{C}_{3}\mathrm{H}_{8}$ (the first three fractions
are determined by the respective vapor pressure at 10 bar). Another prospect
would be to include gas components that have additional merits like UV
transparency Azevedo  _et al._ (2018). In addition, the use of flammable gas
such as alkane in underground laboratories requires extra precautions; a
successful search for alternative hydrogen-rich nonflammable gas mixtures
would ease this practical concern.

Finally, we would like to emphasize the impact on neutrino oscillation
programs by such a hydrogen-rich HPTPC Lu  _et al._ (2015). Around the
neutrino energy of the first oscillation maximum at DUNE, pion production is
the dominant dynamics. The exclusive processes
$\smash{\overset{\scalebox{0.3}{(}\raisebox{-1.7pt}{--}\scalebox{0.3}{)}}{\nu}}\textrm{p}\rightarrow\mu^{\mp}\textrm{p}\pi^{\pm}$
[Eqs. (1) and (2)], which the hydrogen-extraction technique relies on, could
provide constraints to both the neutrino and antineutrino fluxes. The
symmetric final states between the neutrino and antineutrino interactions
might provide further experimental advantages over the highly asymmetric
quasi-elastic dynamics. Measurements of pion production on nucleons also
provide critical input for the study of nuclear effects in neutrino-nucleus
interactions. In addition, there are several possible physics opportunities
beyond the oscillation program, as a hydrogen-rich HPTPC revives the
possibility of neutrino-hydrogen interaction measurements after 30 years. The
processes, Eqs. (1) and (2), are the ideal channels to study neutrino deeply
virtual meson production ($\nu$DVMP) where Generalized Parton Distributions
(GPDs) could be extracted Kopeliovich  _et al._ (2012); Siddikov and Schmidt
(2019). Because (anti)neutrinos probe different quark flavors and spins,
$\nu$DVMP unfolds the nucleon structure in a complementary way to the GPD
program in the proposed Electron-Ion Collider Accardi  _et al._ (2016). In
addition, because the TKI technique can also be applied to electron and muon
beams—the corresponding leading exclusive channel being
$\ell\textrm{p}\rightarrow\ell\textrm{p}$, where $\ell$ is the electron or
muon—electron/muon-hydrogen interactions Bernauer  _et al._ (2010); Adams  _et
al._ (2019) could be studied by a HPTPC. Furthermore, because of the common
detector technology, the extraction technique for $\nu/\ell$-hydrogen
interactions could be tested with a small-scale prototype detector at
electron/muon beam lines at, for example, Mainz Microtron (MAMI) Kaiser  _et
al._ (2008) or CERN.

###### Acknowledgements.

We would like to thank Diego González-Díaz for helpful comments on the
manuscript. PH-B would like to thank Stefan Roth for helpful discussions. PH-B
is supported by DFG (Germany) Grant No. RO 3625/2-1. XL is supported by STFC
(United Kingdom) Grant No. ST/S003533/1. JM-A is supported by a fellowship
from Fundación Bancaria “la Caixa” (ID 100010434), code LCF/BQ/PI19/11690012.

##  Appendix A Multiple Scattering

The ultimate TPC tracking performance is limited by multiple scattering,
diffusion, the geometry of the readout unit, and the field distortion in the
drift volume. The first two depend on the gas. As is shown in Section IV.2,
the diffusion is suppressed at high pressure and approaches the thermal limit
at high alkane concentration. For completeness, in this section we estimate
the size of multiple scattering in the Ar-alkane gas mixtures.

Multiple scattering is commonly quantified by the (r.m.s.) angular deflection,
$\theta_{\textrm{MS}}$. It depends on the radiation length $X_{0}$ [measured
in $\textrm{g}/\textrm{cm}^{2}$ (length$\times$density)] Tanabashi  _et al._
(2018):

$\displaystyle\theta_{\textrm{MS}}=\frac{13.6\,\mathrm{\textrm{MeV}/\textit{c}}}{p}\sqrt{F}\left(1+0.038\ln
F\right),$ (10)

with the particle momentum $p$ and

$\displaystyle F=\frac{x}{X_{0}\beta^{2}},$ (11)

where $x/X_{0}$ is the thickness of material measured in $X_{0}$ and $\beta$
is the particle velocity in unit of $c$. The gas-dependent part is

$\displaystyle F\propto\frac{\rho}{X_{0}}\propto\frac{PA}{X_{0}},$ (12)

where $\rho$ is the gas density, $P$ is the pressure, and $A$ is the atomic
mass number. The weighted $A/X_{0}$ for different mixtures is shown in Figure
16: it decreases with an increasing alkane concentration. For 10-bar
$\mathrm{CH}_{4}$, the $F$ factor is about 4.6 times as large as the one for
the 1-bar ALICE gas; the corresponding $\theta_{\textrm{MS}}$ is therefore
about a factor of 2 larger, assuming the log-term in Eq. (10) is negligible.

Figure 16: Weighted inverse radiation length, $A/X_{0}$, as a function of the
alkane concentration. The ALICE (90% Ne + 10% $\mathrm{CO}_{2}$) and T2K (95%
Ar + 3% $\mathrm{CF}_{4}$ \+ 2% $i\mathrm{C}_{4}\mathrm{H}_{10}$) values
Dellacasa  _et al._ (2000) are indicated by the horizontal lines.

##  Appendix B Primary Ionization Density

The primary ionization density ($\mathrm{N}_{\mathrm{I,e}}$) along a track
depends on the particle energy lost
($\nicefrac{{\textrm{d}\textit{E}}}{{\textrm{d}\textit{x}}}$) and the average
energy spent for the creation of an electron-ion pair
($\mathrm{W}_{\mathrm{I}}$). The value of $\mathrm{W}_{\mathrm{I}}$ is a
property of the drift gas and independent of the pressure Gonzalez-Diaz  _et
al._ (2018), whereas the gas density, and therefore
$\mathrm{N}_{\mathrm{I,e}}$, increases with pressure. This implies an
improvement in the $\nicefrac{{\textrm{d}\textit{E}}}{{\textrm{d}\textit{x}}}$
measurement with high pressure due to reduced statistical fluctuations.

For the primary ionization density in Ar-alkane mixtures
($\mathrm{N}_{\mathrm{I,e}}^{q}$, $q$ being the quencher fraction), one
expects a simple sum rule from the gas components assuming that individual
$\mathrm{W}_{\mathrm{I}}$ remains unchanged in the mixture:

$\displaystyle\mathrm{N}_{\mathrm{I,e}}^{q}$
$\displaystyle=\left[\left(1-q\right)\cdot\mathrm{N}_{\mathrm{I,e}}^{\text{Ar}}+q\cdot\mathrm{N}_{\mathrm{I,e}}^{\mathrm{C}_{x}\mathrm{H}_{y}}\right]\cdot
P\left[\mathrm{bar}\right].$ (13)

Simulations of minimum-ionizing particles with Heed Smirnov (2005) are
consistent with this expectation, with a deviation smaller than 1 %. Fitting
Eq. (13) to the simulations with various $q$, we obtain

$\displaystyle\mathrm{N}_{\mathrm{I,e}}^{\text{Ar}}$
$\displaystyle=26\,\mathrm{\nicefrac{{1}}{{\mathrm{cm}}}},$ (14)
$\displaystyle\mathrm{N}_{\mathrm{I,e}}^{\mathrm{CH}_{4}}$
$\displaystyle=29\,\mathrm{\nicefrac{{1}}{{\mathrm{cm}}}},$ (15)
$\displaystyle\mathrm{N}_{\mathrm{I,e}}^{\mathrm{C}_{2}\mathrm{H}_{6}}$
$\displaystyle=48\,\mathrm{\nicefrac{{1}}{{\mathrm{cm}}}},\textrm{~{}and}$
(16) $\displaystyle\mathrm{N}_{\mathrm{I,e}}^{\mathrm{C}_{3}\mathrm{H}_{8}}$
$\displaystyle=59\,\mathrm{\nicefrac{{1}}{{\mathrm{cm}}}}.$ (17)

However, this simple assumption of unchanged $\mathrm{W}_{\mathrm{I}}$ in
mixtures is inadequate. Measurements show that $\mathrm{W}_{\mathrm{I}}$ in
Ar-alkane mixtures is minimal when $q$ lies round 2–3.5 % Blum  _et al._
(2008).

##  Appendix C Helium-Alkane Mixtures

Figure 17: Proton free-to-bound ratio vs. hydrogen mass for different He-
alkane mixtures. See Figure 8 for comparison.

The hydrogen content of helium-alkane mixtures is summarized in Figure 17. At
a given alkane concentration, the hydrogen event rate remains the same
regardless of the noble gas (see Figure 8 for comparison). On the other hand,
at low concentration, the hydrogen purity is more sensitive to the added
alkane because of the small number of bound protons in helium; a high
concentration mainly contributes to the increase of the event rate.

The calculated drift velocity and transverse diffusion coefficient as a
function of the alkane concentration are shown in Figure 18. The axis ranges
of the figures are chosen to be the same as in Figures 10 and 13 respectively
for a direct comparison. With a helium base, the drift velocity increases much
more slowly with the added alkane due to the absence of the Ramsauer minimum
in helium Ramsauer (1921); Blum  _et al._ (2008). In pure argon and helium,
the drift velocities are similar (both $\sim
0.2\,\mathrm{\nicefrac{{\mathrm{cm}}}{{\mu\mathrm{s}}}}$), but the transverse
diffusion coefficients are very different—for 10 bar helium it is about a
factor of 5 smaller and already close to the thermal limit.

Figure 18: (a) Drift velocity and (b) transverse diffusion coefficient
calculated at 40
$\nicefrac{{\mathrm{V}}}{{\mathrm{cm}}}\nicefrac{{}}{{\mathrm{bar}}}$ as a
function of the alkane concentration of He-alkane mixtures. See Figures 10 and
13 for comparison.

As the difference between helium and argon is more relevant at low alkane
concentration, the gas parameters ($v_{\mathrm{d}}$, $\sigma_{\mathrm{T}}$,
and $\alpha-\eta$) in He-alkane mixtures are shown for the 10 % concentration
in Figure 19. As discussed previously, because of the absence of the Ramsauer
minimum, the drift velocity is significantly smaller across the whole range of
$\nicefrac{{E}}{{P}}$. For the transverse diffusion coefficient, the
difference between helium and argon is most visible at high
$\nicefrac{{E}}{{P}}$. This strong $\nicefrac{{E}}{{P}}$-dependence is also
seen in the gas gain. The gain onset is earlier and much steeper in helium-
alkane mixtures. In Figure 19 (c), Penning transfers always happen for helium
mixtures in MagBoltz; in the case of $\mathrm{CH}_{4}$, auto-transfers cause
the broadening of the curve [see Figure 15 (c) and related discussions].

Figure 19: (a) Drift velocity, (b) scaled transverse diffusion coefficient,
and (c) effective Townsend coefficient as a function of the (scaled) electric
field strength for He-alkane mixtures at 10% alkane concentration. See Panels
(a) of Figures 12, 14, and 15 for comparison.

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