2022

[1]Erin Marshall Seroka

[1]Physics Department, The George Washington University, 725 21st Street NW,
Washington, DC 20052, USA

# Predicted Measurements of the Tensor-to-Scalar Transition in the CLAS12
Nuclear Targets Experiment

<EMAIL_ADDRESS>Axel Schmidt<EMAIL_ADDRESS>*

###### Abstract

Short-range correlated (SRC) nucleon pairs, which are strongly interacting
nucleons at short inter-particle distances, can reveal properties of the
effective nucleon-nucleon (NN) interaction at short distance scales. The
relative abundance of proton-proton (pp) pairs and proton-neutron (pn) pairs,
for example, is sensitive to the tensor contribution to the NN interaction.
Generalized Contact Formalism (GCF) theory, when used with realistic
phenomenological NN potential models, predicts a transition from a tensor-
dominated regime—at relative momenta of approximately 400 MeV/c where pp pairs
are suppressed relative to pn pairs—to a scalar-dominated regime at higher
momenta with no preferred isospin projection. While an increase in the
prevalence of pp pairs with increasing momentum has been observed in a few
experiments, difficulties associated with neutron detection have so far
hindered the observation of a corresponding reduction in the abundance of pn
pairs. High-precision measurements showing a simultaneous increase in the
abundance of pp pairs and change in the abundance of pn pairs with increasing
momentum would conclusively demonstrate the existence of the tensor-to-scalar
transition. In this work, we study the potential impact of the recently
conducted Nuclear Targets Experiment at the CLAS12 detector at Jefferson Lab,
using GCF simulations. We model the expected yields and relevant observables
for a carbon target with a beam energy of 6 GeV and show that sufficient
statistical precision can be obtained from the experimental data, both for pp
and pn pairs, to observe the tensor-to-scalar transition.

###### keywords:

Short-Range Correlations, NN Interaction, CLAS12, Generalized Contact
Formalism, Quasielastic Electron Scattering

##  1 Introduction

Experiments and ab initio calculations in recent years have shown that, in
nuclei heavier than carbon, about 20% of nucleons occupy high-momentum states
above the Fermi momentum $k_{F}$ Fomin:2011ng ; Schmookler2019 ;
Wiringa:2013ala ; Lonardoni:2017egu ; Lonardoni:2017hgs , which consist almost
entirely of pairs of strongly interacting nucleons in close proximity. These
short-range correlated (SRC) pairs move through the nucleus with a small
center-of-mass momentum, typically below $k_{F}$, that is comparable to the
momenta of slower, uncorrelated nucleons Cohen2018 . SRC pairs may play a
significant role in outstanding questions in nuclear and particle physics,
including the nuclear matrix elements in double beta decay Kortelainen2007 ;
Simkovic2009 ; Benhar2014 , nuclear charge radii Miller2019 , and the EMC
effect Schmookler2019 . Recent reviews include Refs. Arrington:2022sov ;
Hen:2016kwk .

One of the striking features of short range correlations is the strong
prevalence of neutron-proton ($np$) pairs, i.e., an isospin-0, spin-1,
configuration, over proton-proton ($pp$) or neutron-neutron ($nn$) pairs,
i.e., any isospin-1 configuration. This property, referred to as
$np$-dominance, has been observed through a wide-range of techniques,
including proton scattering reaction Piasetzky:2006ai , exclusive electron
scattering JeffersonLabHallA:2007lly ; Subedi:2008zz ; LabHallA:2014wqo ;
Hen:2014nza ; DuerPRL ; CLAS:2020mom ; Korover2021 , and most recently through
inclusive electron scattering on isospin mirror nuclei Li:2022fhh . Studies of
asymmetric nuclei have shown that $np$-dominance persists even when there is
an imbalance between proton and neutron numbers Hen:2014nza ; DuerPRL ;
Li:2022fhh , which has implications for understanding the structure of
neutron-rich matter like that found in the cores of neutron stars Li2018 ;
Frankfurt2008 .

In a nonrelativistic picture of nucleon-nucleon ($NN$) pairs, the origin of
$np$-dominance lies in the tensor part of the $NN$ interaction, which favors a
spin-1 configuration. At distance scales where the central part of the
$NN$-interaction transitions from long-range attraction to short-range
repulsion—on the order of the separation distance of nucleons in SRC pairs—the
tensor component plays a significantly larger role than the scalar component.
Theory calculations with realistic $NN$ potentials show that SRCs are
dominated by the spin-1 isospin singlet of $pn$ pairs and the spin-0 isospin
triplet of $pp$, $pn$, and $nn$ pairs. For two nucleons in a spin-0, isospin-1
configuration, including $pp$ pairs, the two-body momentum distribution has a
node in the momentum range of 300–600 MeV/$c$, while the spin-1, isospin-0
configuration does not CiofidegliAtti2015 ; Ryckebusch2019 ; Sargsian2005 ;
Schiavilla2007 ; Alvioli2008 ; Weiss2018 . Therefore, the abundance of $pp$
pairs is most suppressed relative to $pn$ pairs at approximately 400 MeV/$c$
due to the preference for spin-1 $pn$ pairs by the tensor part of the $NN$
interaction Weiss2018 . A dramatic increase can be expected in the abundance
of $pp$ pairs from momentum of about 400 MeV/$c$ to momentum of about 1000
MeV/$c$. Simultaneously, the abundance of $pn$ pairs decreases in the tensor-
to-scalar transition from approximately 400 MeV/$c$ to about 1000 MeV/$c$. In
the high-momentum limit, where the scalar part of the $NN$ interaction
dominates over the tensor part, the abundance of pairs is expected to be equal
for all Pauli-allowed spin-isospin projections.

The tensor-to-scalar transition can be investigated through scattering
experiments in which a probe interacts with a correlated nucleon in a target
nucleus, inducing a hard break-up of the pair. Though several different probes
have been used, in this work we focus on electron scattering. We specifically
consider quasielastic electron scattering reactions with large momentum
transfer, in which the struck nucleon is a proton from an SRC pair, with cross
section $(e,e^{\prime}p)$. We refer to the outgoing struck proton as the
“leading proton,” since at high momentum transfer it leaves the nucleus with
significantly higher momentum than the recoil nucleon from the same correlated
pair. Contributions from reactions with non-SRC leading protons, such as
multi-nucleon and meson-exchange currents and final state interactions (FSI),
can be suppressed by a set of kinematic criteria, including requiring that the
initial proton momentum be above the Fermi momentum $k_{F}$ and requiring high
momentum transfer Hen:2014nza ; DuerPRL ; CLAS:2020mom ; Korover2021 . As a
subset of the $(e,e^{\prime}p)$ reaction channel, we look for exclusive events
with a recoiling spectator nucleon that originated as the correlated partner
to the struck proton. These events have cross section $(e,e^{\prime}pN)$,
where the recoil nucleon $N$ may be either a proton ($p$) or a neutron ($n$).

As shorthand, we define $pp/p$ and $pn/p$ to be the cross section ratios
$(e,e^{\prime}pp)/(e,e^{\prime}p)$ and $(e,e^{\prime}pn)/(e,e^{\prime}p)$,
respectively. The tensor-to-scalar transition can be mapped by observing
changes in $pp/p$ and $pn/p$ as a function of missing momentum. The missing
momentum is defined to be $\vec{p}_{\rm{miss}}=\vec{p}_{1}-\vec{q}$, where
$\vec{p}_{1}$ is the measured outgoing momentum of the leading proton and
$\vec{q}$ is the momentum transferred by the electron. Under the plane wave
impulse approximation, $\vec{p}_{\rm{miss}}$ is a proxy for the initial
momentum, $\vec{p}_{i}$, of the struck proton. The tensor-to-scalar transition
has been successfully demonstrated through the $pp/p$ channel CLAS:2020mom .
However, measurements of $pp$ are susceptible to inflation due to single
charge exchange (SCX) reactions of $pn$ or $np$ pairs. Events categorized as
$pp$ may have originated as $pn$ pairs in which the neutron swapped isospin
projections with a proton through the exchange of a charged pion. This is
especially concerning since $pn$ pairs are an order of magnitude more abundant
than $pp$ pairs at momenta near 400 MeV/$c$. Therefore, observing an increase
in $pp/p$ alone is not sufficient to conclusively demonstrate the transition
from a tensor-dominated interaction to a scalar-dominated interaction.

In the cross section ratio $pn/p$, SCX contributions from proton-to-neutron
and neutron-to-proton interactions are expected to roughly cancel. $pn/p$ is a
complementary measurement to $pp/p$, which, although more ambitious due to the
experimental challenges of neutron detection, is crucial because it is
significantly less sensitive to contamination from SCX reactions.

Two previous studies extracted $pn/p$ from quasielastic electron scattering
data in helium-4 LabHallA:2014wqo  and carbon-12 Korover2021 , but
unfortunately, their large uncertainties precluded any definite conclusions
about the onset of the repulsive core from this channel. In fact, in Ref.
Korover2021 , the measurement is consistent with a constant value of $pn/p$.
To date, no analysis has reported simultaneous measurements of $pp/p$ and
$pn/p$ with a statistically significant change in both quantities across the
momentum range. A higher statistics measurement, made possible by the
increased rate capabilities of the CLAS12 Spectrometer and a new dedicated
neutron detection system Chatagnon:2020lwt , could provide confirmation of the
tensor-to-scalar transition, test the role played by single charge exchange
reactions, and test the effectiveness of different $NN$ interaction models
with the Generalized Contact Formalism.

##  2 The CLAS12 Nuclear Targets Experiment

Table 1:  CLAS12 Nuclear Targets Experiment Data Target | Energy (GeV) | Integrated Luminosity (fb-1) | Total Charge (mC)
---|---|---|---
H | 5.99 | 3.68 | 2.92
H | 2.07 | 0.39 | 0.31
D | 5.99 | 52.00 | 16.87
D | 2.07 | 1.81 | 0.59
He | 5.99 | 57.24 | 24.37
C | 5.99 | 80.31 | 48.56
C | 4.03 | 10.46 | 6.33
C | 2.07 | 1.00 | 0.60
Ar | 5.99 | 4.91 | 1.87
Ar | 4.03 | 6.67 | 2.54
Ar | 2.07 | 1.35 | 0.51
40Ca | 5.99 | 51.59 | 44.28
48Ca | 5.99 | 20.75 | 17.81
Sn | 5.99 | 9.29 | 12.06

Jefferson Lab Experiment E12-17-006A, titled, “Exclusive Studies of Short
Range Correlations in Nuclei using CLAS12,” and hereafter referred to as the
CLAS12 Nuclear Targets Experiment, collected data between November 2021 and
February 2022 at the Thomas Jefferson National Accelerator Facility (or
“Jefferson Lab”) in Newport News, Virginia.

Figure 1: A schematics cross section of the CLAS12 detector, approximately to
scale. The Central Detector wraps around the target (green) and includes the
Central Vertex Tracker (CVT, blue), Central Time-of-Flight Detector (CTOF,
orange), and Central Neutron Detector (CND, magenta). The Forward Detector,
arranged in six sectors around the beam line, includes the High Threshold
Cherenkov Counter (HTCC, yellow), Drift Chambers (DC, blue), Low Threshold
Cherenkov Counter (LTCC, yellow), Forward Time-of-Flight Detector (FTOF,
orange), and the Electromagnetic Calorimeter and Preshower Calorimeter (EC &
PCAL, red). Also shown is the Backward Angle Neutron Detector (BAND, magenta).
The gray regions indicate the magnetic fields of the solenoid (left) and torus
(right) magnets.

The experiment was conducted as part of the CLAS Collaboration’s Run Group M,
during which an electron beam at an energy of 2, 4, and 6 GeV, was scattered
from 1H, 2H, 4He, C, 40Ca, 48Ca, Ar, and Sn targets. For the 6 GeV beam
energy, a four foil carbon target with a total thickness of 2 mm was used.
Outgoing particles were detected by the CLAS12 Spectrometer nima-clas12 . This
experimental run was the first use of nuclear targets heavier than deuterium
with CLAS12. Information about the data accumulated on all targets is given in
Table 1. The primary goal of the CLAS12 Nuclear Targets Experiment was to
perform high-precision and exploratory SRC studies on 4He and C and to compare
SRC characteristics across a range of nuclei. More information can be found in
the experiment’s proposal rgm-proposal , although the plan and scope of the
experiment was updated somewhat between the proposal and data collection.

CLAS12 is a large-acceptance magnetic spectrometer situated in Jefferson Lab’s
Experimental Hall B. CLAS12 is the successor to the retired CLAS detector
CLAS:2003umf , which provided the data for several ground-breaking findings in
the study of short-range correlations DuerPRL ; Duer2018 ; CLAS:2020mom ;
Hen:2014nza ; Cohen2018 ; Schmookler2019 ; Korover2021 . CLAS12 is arranged
with a Forward Detector with a toroidal magnetic field and a Central Detector
with a barrel-shaped solenoid magnet surrounding the target, as shown in Fig.
1. The Forward Detector, which includes some sub-detectors from the previous
CLAS Spectrometer, covers scattering angles of approximately
$5^{\circ}<\theta<40^{\circ}$, and includes drifts chambers for charged-
particle tracking Mestayer:2020saf , Cherenkov detectors for particle
identification Sharabian:2020whm ; Contalbrigo:2020lnd , time-of-flight
scintillation counters Carman:2020fsv , and an electromagnetic calorimeter
Asryan:2020iqj . The Central Detector, an entirely new compact barrel-shaped
set of detectors, extends the polar angle acceptance to approximately
$40^{\circ}<\theta<140^{\circ}$. The Central Detector includes components for
charged particle tracking and timing Acker:2020qkv ; Carman:2020yma , as well
as a dedicated Central Neutron Detector (CND) Chatagnon:2020lwt .

The CND allows detection of neutrons with momenta from approximately 0.2 to
1.0 GeV/$c$ with a maximum efficiency of about 10% nima-clas12 . Precise
measurements of the momentum and polar angle dependence of the neutron
detection efficiency for the Nuclear Targets Experiment are underway. Neutron
detection in the former CLAS detector was accomplished using either the
electromagnetic calorimeter, with efficiency in the range of 35-60%
Amarian_EC_nim ; lachinet.gmn ; lachinet.thesis ; Duer2018 , or the time-of-
flight detector, with efficiency of 2-8% lachinet.thesis ; Korover2021 .
Simulations have indicated that under the conditions of the Nuclear Targets
Experiment, more recoil nucleons can be expected in the Central Detector than
in the Forward Detector, and very few recoil nucleons can be expected in the
Backward Angle Neutron Detector (BAND), making the CND the optimal choice for
neutron detection in this study.

The enhanced capabilities of the CLAS12 Spectrometer have the potential to
lead to significant progress on a number of open questions related to short-
range correlations. First, CLAS12 can accommodate a luminosity of $10^{35}$
cm-2s-1, a factor of 10 higher than the original CLAS, which is valuable for
improving the precision on relatively rare quasielastic knockout of correlated
nucleons, such as those at high $p_{\rm{miss}}$. Second, CLAS12 has a larger
angular acceptance than CLAS, which is valuable for the reconstruction of
exclusive multi-nucleon final states. Third, CLAS12’s Central Neutron Detector
provides neutron detection capabilites over a wide solid angle (approximately
polar angles of $40^{\circ}$ to 140∘), which is especially valuable for the
study described in this work. The combination of these advantages will
drastically improve searches for 3-nucleon short-range correlations, for
example rgm-proposal .

For our study, we specifically consider the portion of the experiment with a 6
GeV beam and a carbon target as this setting has the largest number of events
for the quasielastic SRC break-up reactions of interest, and for comparison to
Ref. Korover2021 . The same analysis could be extended to other beam/target
combinations.

##  3 Simulation

To investigate the expected measurements of the tensor-to-scalar transition in
the CLAS12 Nuclear Targets Experiment, we have performed simulation studies of
the experimental conditions using the theoretical predictions of Generalized
Contact Formalism (GCF), a factorized model of short-range correlations,
according to Ref. Pybus2020 . Our simulated events contain a scattered
electron from a 6 GeV beam, a leading proton (assumed to have interacted with
the electron through a virtual photon), and a recoil nucleon (assumed to be
the correlated spectator nucleon in an SRC pair) from a 12C nucleus. We have
studied the events both by integrating over the approximate experimental
acceptance (so-called “Fast Monte Carlo”), as well as by using the Geant4
simulation package and CLAS12 reconstruction programs to produce simulated
data to analyze in the same manner as the experimental data. The Geant4
simulation includes a detailed model of the experimental response, including
detector efficiency and resolution. The Fast Monte Carlo, on the other hand,
assumes perfect efficiency within the acceptance as well as perfect
resolution. By examining both, we can study the impact of detector resolution
and efficiency on the observables of interest.

We generated Monte Carlo events according to the theoretical cross sections of
Generalized Contact Formalism, a factorized, scale-separated model of short-
range correlations among nucleons. Theoretical studies have long demonstrated
that high-momentum nucleons (with momenta exceeding the Fermi momentum
$k_{F}$) are dominated by short-range correlated nucleon pairs, whose behavior
can be described using a factorized wave function based on the relative and
center-of-mass momentum distributions of the pairs Frankfurt1981 ;
CiofiDegliAtti1991 . Some of the hallmark features of SRC behavior, including
the scaling for ratios of inclusive cross sections of nuclei to the deuteron
at $x_{B}>1$ Frankfurt1993 , where $x_{B}$ is the Bjorken scaling parameter
defined in Section 4.1, were predicted in these early works and later
demonstrated experimentally at facilities like SLAC and Jefferson Lab. The
development of Generalized Contact Formalism drew upon the aforementioned work
and upon a contact formalism used to analyze the properties of ultracold Fermi
gases tan1 ; tan2 ; tan3 . Detailed information about the development of GCF
and its application to nucleon-nucleon correlations can be found in Refs.
Weiss2015 ; Weiss2018 ; Weiss2019 ; Pybus2020  and is briefly summarized here.

GCF predicts that, at high missing momentum, the $(e,e^{\prime}pN)$ scattering
cross section approximately factorizes according to three distinct momentum
scales: the high-momentum scale, defined by the momentum transferred to the
nucleus by the probe, typically greater than 1 GeV/$c$; the medium-momentum
scale, describing the relative momentum between the correlated nucleons,
approximately $300$–$1000$ MeV/$c$; and the low-momentum scale, describing the
center-of-mass momentum of the pair with respect to the rest of the nucleus,
on the order of 100–150 MeV/$c$. The differential cross section takes the
factorized form

$d\sigma=\sigma_{ep}\cdot
K\cdot\sum_{a}C_{a}\cdot\mathcal{P}_{\rm{rel}}^{a}\cdot\mathcal{P}_{\rm{CM}}^{a},$
(1)

where $\sigma_{ep}$ is the cross section for an electron to scatter from an
off-shell proton in the nucleus; $K$ is a kinematic factor that depends on the
final state momentum vectors; $a$ represents the possible quantum numbers for
an SRC pair; $C_{a}$ is the contact term, i.e., the abundance of pairs with
quantum numbers $a$ in the nucleus; $\mathcal{P}_{\rm{rel}}^{a}$ is a
distribution governing relative motion of the nucleons in a pair of type $a$;
and $\mathcal{P}_{\rm{CM}}^{a}$ is a distribution governing the center-of-mass
motion of a pair of type $a$. For the $(e,e^{\prime}pp)$ reaction, in which
the struck nucleon is a proton, and the correlated spectator is also a proton,
the sum over $a$ only has one term; $pp$ pairs with spin-0 (Pauli exclusion
suppresses spin-1 pairs). For the $(e,e^{\prime}pn)$ reaction, the sum over
$a$ includes $pn$ spin-1 and $pn$ spin-0 pairs. There are multiple approaches
to specifying $\mathcal{P}_{\rm{rel}}^{a}$ and $\mathcal{P}_{\rm{CM}}^{a}$,
depending on assumptions made. Both the center-of-mass and relative
distributions depend on the initial momentum (prior to the scattering
reaction) of the struck nucleon, $\vec{p}_{i}$, and the initial momentum of
the spectator nucleon, $\vec{p}_{2}$. We define

$\displaystyle\vec{p}_{\rm{CM}}$ $\displaystyle=\vec{p}_{i}+\vec{p}_{2}$
$\displaystyle\vec{p}_{\rm{rel}}$
$\displaystyle=\frac{\vec{p}_{i}-\vec{p}_{2}}{2}$

as the center-of-mass momentum vector of the pair and the relative momentum
vector of the pair, respectively.

In this work we adopt the “light-cone” GCF specification of Ref. Pybus2020 ,
since it more naturally handles relativistic effects, which are likely to be
important for nucleons moving with momenta approaching 1 GeV$/c$. In light
cone coordinates, four-momentum vectors $(p^{0},p^{1},p^{2},p^{3})$ are
represented in terms of transverse components,
$\vec{p}_{\perp}\equiv(p^{1},p^{2})$, as well as plus-momentum and minus-
momentum components $p^{\pm}\equiv p^{0}\pm p^{3}$. We use $\alpha\equiv
p^{-}/\tilde{m}$, where $\tilde{m}=m_{A}/A$ and $m_{A}$ is the mass of the
target nucleus, to represent the fractional minus-momentum; for example, we
use $\alpha_{\rm{CM}}$ to represent the fractional minus-momentum of the
center-of-mass of the pair. In the light-cone GCF specification, the 8-fold
differential cross section for the $(e,e^{\prime}pN)$ reaction can be written
as

$\frac{d^{8}\sigma}{d\Omega_{k^{\prime}}d^{3}\vec{p}_{\rm{CM}}d^{3}\vec{p}_{\rm{rel}}}=\sigma_{ep}\cdot\left[\frac{1}{{4\pi\alpha_{i}}\lvert
1-\frac{\vec{p}_{i}\cdot\vec{k}^{\prime}}{E_{i}E^{\prime}_{k}}\rvert}\right]\cdot\sum_{a}C_{a}\cdot\mathcal{P}_{\rm{rel}}^{\rm{LC},a}\cdot\mathcal{P}_{\rm{CM}}^{\rm{LC},a},$
(2)

where the outgoing electron momentum, solid angle, and energy are given by
$k^{\prime}$, $\Omega_{k^{\prime}}$, and $E_{k^{\prime}}$, respectively, and
$\alpha_{i}$ is the initial minus-momentum fraction of the struck nucleon
prior to the collision. Ref. Pybus2020  gives a light-cone expression for
$\mathcal{P}_{\rm{rel}}^{\rm{LC},a}$ as

$\mathcal{P}_{\rm{rel}}^{\rm{LC},a}=\frac{1}{E_{2}}\cdot\frac{\sqrt{m_{N}^{2}+k^{2}}}{2-\alpha_{\rm{rel}}}\cdot\frac{\lvert\tilde{\phi}^{a}(k)\rvert^{2}}{(2\pi)^{3}}$
(3)

where $\lvert\tilde{\phi}^{a}(k)\rvert^{2}$ is the relative momentum
distribution for nucleons in an SRC pair with quantum numbers $a$, which
should be universal across nuclei; $m_{N}$ is the nucleon mass;
$E_{2}=\sqrt{\vec{p}^{\ 2}_{2}+m^{2}_{N}}$ is the energy of the spectator
nucleon in the pair; and $k$ is given by

$k^{2}\equiv\frac{m_{N}^{2}+p_{\rm{rel},\perp}^{2}}{\alpha_{\rm{rel}}(2-\alpha_{\rm{rel}})}-m_{N}^{2}.$
(4)

Following Ref. Pybus2020 , we assume that the center-of-mass distribution is
well-modelled by a Gaussian distribution in light-cone coordinates,

$\mathcal{P}_{\rm{CM}}^{\rm{LC},a}=\frac{\tilde{m}\alpha_{A-2}}{E_{A-2}\left(2\pi\sigma_{\rm{CM}}^{a\
2}\right)^{3/2}}\exp\left\\{-\frac{\tilde{m}^{2}(2-\alpha_{\rm{CM}})^{2}+{\vec{p}_{\rm{CM},\perp}}^{\
2}}{2{\sigma_{\rm{CM}}^{a\ 2}}}\right\\},$ (5)

with a single parameter $\sigma_{\rm{CM}}^{a}$ governing the width of the
distribution. Here, $E_{A-2}$ and $\alpha_{A-2}$ represent the energy and
minus-momentum fraction, respectively, of the residual $A-2$ system.

For our simulations, we produced Monte Carlo events weighted according to the
cross section specified in Eq. 2. More details on GCF and its implementation
as a Monte Carlo event generator can be found in Ref. Pybus2020 .

The GCF requires several model inputs, and for consistency with respect to
previous calculations Korover2021 ; CLAS:2020mom , we have matched their
choices of values for those inputs. The inputs include the contact $C_{a}$ for
each set of nucleon pair quantum numbers, obtained theoretically and in
agreement with experiment Weiss2018 ; the width of the center-of-mass momentum
distribution, $\sigma_{\rm{CM}}^{a}$, which we take to be the same for various
SRC analyses with various targets CLAS:2020mom ; Korover2021 , with a value of
150$\pm$20 MeV/$c$ Cohen2018 ; the residual excitation energy $E^{*}$ for the
$A-2$ system, which we varied uniformly from 0-30 MeV following Korover2021 ;
CLAS:2020mom  (which based their estimate on Ref. Ulmer1987 ); and a model
$NN$ interaction used to determine the universal two-nucleon wave functions
$\tilde{\phi}^{\alpha}(k)$, for which we use the AV18 model Wiringa1995 . GCF
has so far proven quite effective in predicting a variety of nuclear
properties Weiss2018 ; Pybus2020 ; CLAS:2020mom ; Korover2021 ; Cruz-
Torres2021 ; Weiss2019 ; Cohen2018 .

Of course, reaction mechanisms not accounted for in the formalism described
above also occur. These effects may include elastic and inelastic final state
interactions and recoil events in which a knocked-out neutron interacts with
other nucleons before leaving the nucleus CLAS:2020mom . The Glauber
approximation is used to estimate the probability of FSI under different
kinematic conditions and to establish event selection criteria to choose
events with a low probability of FSI. We therefore chose to conduct our study
in antiparallel kinematics, in which the angle between the missing momentum
$\vec{p}_{\rm{miss}}$ and electron momentum change $\vec{q}$ is close to 180∘.
In Refs. Hen:2014nza ; Cohen2018 ; Duer2018 ; Duer2019 ; LabHallA:2014wqo ;
Korover2021 , suppression of non-plane wave reaction mechanisms in
antiparallel kinematics was accomplished by a common set of event selection
criteria or “cuts.” The contribution from some reaction effects, including
SCX, are somewhat more difficult to estimate, underscoring the importance of
the $pn/p$ measurement.

To account for the changes in the experimental cross sections due to nuclear
transparency and SCX, we used a simplified version of the procedure described
in Ref. Pybus2020  and in the Supplementary Information of Ref. CLAS:2020mom .
Because the transparencies and SCX probabilities given in Ref. Colle2016  are
fairly rough estimates, we neglect for the time being other kinematic-
dependent effects, such as the decrease in the effect of SCX with increasing
momentum of the struck proton Gibbs1994 .

The precise effect of SCX on the $(e,e^{\prime}p)$ and $(e,e^{\prime}pN)$
reactions is difficult to calculate. We estimate the effect of SCX on the
measured cross sections, which we denote by $(e,e^{\prime}p)_{\rm{SCX}}$ and
$(e,e^{\prime}pN)_{\rm{SCX}}$, by forming a mixture of the $(e,e^{\prime}pp)$
and $(e,e^{\prime}pn)$ channels predicted by our GCF simulations. Glauber
calculations Colle2016  performed to support measurements in similar
kinematics DuerPRL ; CLAS:2020mom  found single-charge exchange probabilities
for leading and recoiling nucleons being ejected from carbon to be on the
order of 4–5%. Assuming, for simplicity, that the probability of SCX,
$P_{\text{SCX}}$, is the same for leading and recoiling nucleons, and for both
protons and neutrons, with a value of 5%, we find

$\displaystyle(e,e^{\prime}pp)_{\rm{SCX}}$
$\displaystyle=(1-2P_{\text{SCX}})\cdot(e,e^{\prime}pp)+\left(1+\frac{\sigma_{en}}{\sigma_{ep}}\right)\cdot(e,e^{\prime}pn)\cdot
P_{\rm{SCX}}$ (6) $\displaystyle(e,e^{\prime}pn)_{\rm{SCX}}$
$\displaystyle=(1-2P_{\text{SCX}})\cdot(e,e^{\prime}pn)+\left(1+\frac{\sigma_{en}}{\sigma_{ep}}\right)\cdot(e,e^{\prime}pp)\cdot
P_{\rm{SCX}}$ (7) $\displaystyle(e,e^{\prime}p)_{\rm{SCX}}$
$\displaystyle\approx(e,e^{\prime}p)$ (8)

where $\sigma_{ep}/\sigma_{en}$ is the ratio of electron-proton to electron-
neutron cross sections. In the kinematics of this experiment,
$\sigma_{ep}/\sigma_{en}\approx 2.5$, according to the nucleon form factor
parameterization of Ref. Kelly:2004hm . We neglect the effect of SCX on the
$(e,e^{\prime}p)$ cross section, since it is relatively small compared to that
on $(e,e^{\prime}pn)$ and $(e,e^{\prime}pp)$. A derivation of the above
equations can be found in the Appendix. Given the aforementioned numerical
values, $(1-2P_{\text{SCX}})$ evaluates to 0.9, and
$(1+\frac{\sigma_{en}}{\sigma_{ep}})P_{\text{SCX}}$ is 0.07.

We modeled the attenuation of outgoing nucleons by the nuclear medium as a
constant transparency factor, with values taken from Ref. Duer_thesis ;
CLAS:2020mom  based on the Glauber calculations described in Ref. Colle2016 .
In this context, nuclear transparency is the probability that an outgoing
proton or neutron emerges from the nucleus rather than failing to emerge due
to scattering from the surrounding nuclear material. We used a transparency
value of $T^{N}$=0.53 for the $(e,e^{\prime}p)$ yields to indicate the
probability that the leading proton escapes the nucleus, and we used a value
of $T^{NN}$=0.44 for the $(e,e^{\prime}pN)$ yields to indicate that both the
leading proton and recoil nucleon escape. These transparency factors were
applied to the eight cross sections given in Eqs. 6-8.

##  4 Results

###  4.1 Fast Monte Carlo Event Selection

Figure 2: Distribution of the missing momentum for $(e,e^{\prime}p)$,
$(e,e^{\prime}pn)$, and $(e,e^{\prime}pp)$ for an arbitrary number of physics
events.

To understand the distribution of events without any influence from
experimental or detector effects, we first performed a “Fast Monte Carlo”
analysis, integrating carbon events from the Monte Carlo event generator at a
beam energy of 6 GeV over bounds defined by the approximate acceptance of the
detector with an approximate set of event selection criteria. We stratified
these events over the different sets of selection criteria, labeling them as
$(e,e^{\prime}p)$, $(e,e^{\prime}pp)$, and $(e,e^{\prime}pn)$, depending on
which criteria are passed. The definitions of the criteria are explained
below. The missing momentum distribution for each channel, based on the Fast
Monte Carlo analysis, is shown in Fig. 2.

For this study, we require that leading protons pass through the Forward
Detector rather than the Central Detector by requiring their polar angle to be
$\theta_{p}<$ 40∘. This requirement is partly due to the limited momentum
resolution of the Central Detector, which limits the experimental resolution
of $p_{\rm{miss}}$ for Central Detector protons. Additionally, our simulation
indicates that at least as many leading protons will pass through the Forward
Detector as through the Central Detector, making forward protons advantageous
for high event counts.

Figure 3:  Estimates of the $(e,e^{\prime}pn)/(e,e^{\prime}p)$ (solid red) and
$(e,e^{\prime}pp)/(e,e^{\prime}p)$ ratios (solid blue) for simulated SRC
events. The dark error bands indicate simulation statistical error, the light
error bands indicate systematic error due to uncertainty in nuclear
properties, and the black error bars indicate the expected statistical error
from the Nuclear Targets Experiment. The dotted red and blue lines indicate
$(e,e^{\prime}pn)_{\rm{SCX}}/(e,e^{\prime}p)$ and
$(e,e^{\prime}pp)_{\rm{SCX}}/(e,e^{\prime}p)$, respectively, and the
associated gray bands indicate their systematic error. Figure 4:  Estimates of
the $(e,e^{\prime}pn)/(e,e^{\prime}pN)$ (solid red) and
$(e,e^{\prime}pp)/(e,e^{\prime}pN)$ ratios (solid blue) for simulated SRC
events. The dark error bands indicate statistical error, the light error bands
indicate systematic error due to uncertainty in nuclear properties, and the
black error bars indicate the expected statistical error from the Nuclear
Targets Experiment. The dotted red and blue lines indicate
$(e,e^{\prime}pn)_{\rm{SCX}}/(e,e^{\prime}pN)_{\rm{SCX}}$ and
$(e,e^{\prime}pp)_{\rm{SCX}}/(e,e^{\prime}pN)_{\rm{SCX}}$, respectively, and
the associated gray bands indicate their systematic error.

Our $(e,e^{\prime}p)$ sample consists of leading protons that also meet
several additional criteria, constraining the interaction to so-called
antiparallel kinematics, in which $\vec{p}_{\rm{miss}}$ is roughly
antiparallel to the momentum transfer $\vec{q}$. This is enforced by selecting
events with $x_{B}>1.1$, where $x_{B}$ is the Bjorken scaling parameter
defined by $x_{B}\equiv Q^{2}/2m_{N}\omega$, $\omega$ is the energy
transferred by the electron, and $Q^{2}$ is the squared four-momentum transfer
given by $Q^{2}\equiv q^{2}-\omega^{2}$. Under the Plane Wave Impulse
Approximation (PWIA), we assume that the electron scatters from a single
nucleon that leaves the nucleus with momentum $\vec{p}_{1}$ without
interaction with other nucleons, so we can approximate the initial proton
momentum as $\vec{p}_{i}\approx\vec{p}_{\rm{miss}}$. We accept only leading
protons with $p_{\rm{miss}}>$ 0.3 GeV/$c$ to restrict the yields to events
with high-momentum nucleons. We also require the angle between $\vec{p}_{1}$
and $\vec{q}$ to be $\theta_{pq}<25^{\circ}$, and we restrict the ratio of
their magnitudes to be $0.62<p_{1}/q<1.1$, both so that the struck proton
carries a large fraction of the momentum transfer. Finally, we select events
with missing mass
$M_{\rm{miss}}=\sqrt{(q^{\mu}-p^{\mu}_{1}+p_{d}^{\mu})^{2}}<$ 1.1 GeV/$c^{2}$,
where $p_{d}^{\mu}=(2m_{N},\vec{0})$ in order to suppress contributions from
non-quasielastic events, such as those involving pion production and nucleon
resonances. These event selection criteria, which are presented succinctly in
Table 2, ensure that we are selecting events with a sufficiently high $Q^{2}$,
that is, at least 1.5 GeV/$c^{2}$.

Table 2:  Event Selection Criteria for SRC Events $(e,e^{\prime}p)$ Selection
Criteria
---
$\theta_{p}<$ 40∘
$x_{B}>$ 1.1
$p_{\rm{miss}}>$ 0.3 GeV/$c$
$\theta_{pq}<$ 25∘
0.62 $<p_{1}/q<$ 1.1
$M_{\rm{miss}}<$ 1.1 GeV/$c^{2}$

Finally, the $(e,e^{\prime}pN)$ sample contains the subset of
$(e,e^{\prime}p)$ events with a recoil nucleon, either a proton or a neutron,
with a momentum larger than 0.3 GeV/$c$ and a polar angle
$\theta_{\text{rec}}$ in the range
$40^{\circ}<\theta_{\text{rec}}<140^{\circ}$, which is the approximate
coverage of the Central Detector, through which our simulations have indicated
that most recoil nucleons will pass.

Our final observables, the resulting $pn/p$ and $pp/p$ ratios, are shown in
Fig. 3 with their statistical and systematic errors. The statistical error
bands are not meant to reflect expectations for the Nuclear Targets Experiment
but to show convergence of the GCF prediction. We simulated a sufficiently
high number of events in order to make the simulation statistical error
negligible compared to the systematic error and in order to obtain a more
precise value of the high-$p_{\rm{miss}}$ limit of the ratios. The systematic
error was obtained by taking the standard deviation of the $pp/p$ and $pn/p$
ratios for several hundred simulations with randomized nuclear properties,
namely $C_{a}$, $E^{*}$, and $\sigma^{a}_{CM}$. We also show the
$(e,e^{\prime}pn)_{\rm{SCX}}/(e,e^{\prime}p)$ and
$(e,e^{\prime}pp)_{\rm{SCX}}/(e,e^{\prime}p)$ ratios as an estimate of how the
$pn/p$ and $pp/p$ may deviate from the predictions of GCF given the effects of
single charge exchange. $(e,e^{\prime}pn)_{\rm{SCX}}/(e,e^{\prime}p)$ and
$(e,e^{\prime}pp)_{\rm{SCX}}/(e,e^{\prime}p)$ are shown with their respective
systematic error, calculated in the same way as for $pn/p$ and $pp/p$.

Additionally, as an alternative observable, we show in Fig. 4 the cross
section ratios $pp/pN$ and $pn/pN$, that is,
$(e,e^{\prime}pp)/(e,e^{\prime}pN)$ and $(e,e^{\prime}pn)/(e,e^{\prime}pN)$,
respectively. We also present the expected ratio $pp/2pn$ =
$(e,e^{\prime}pp)/2(e,e^{\prime}pn)$ in Fig. 5. Both figures also include an
estimate of how the observables may change due to SCX.

The black statistical error bars shown in Figs. 3, 4, and 5 are those we
expect for the 12C data under a 6 GeV beam from the CLAS12 Nuclear Targets
Experiment. To obtain the statistical error estimates, the analysis described
above, with additional event selection as described in Section 4.2, was
performed on a sample of preliminary 4He data from the Nuclear Targets
Experiment. The resulting 4He yields were scaled to obtain expected 12C yields
using the total experiment integrated luminosity for 4He, the relative
integrated luminosity between 4He and 12C, the nuclear transparency factors
for the two nuclei, and their values of $a_{2}$, a scaling factor which
quantifies the abundance of SRC pairs in a given nucleus Hen2012 .

Figure 5:  Estimates of the $(e,e^{\prime}pp)/2(e,e^{\prime}pn)$ ratio (solid
purple) for simulated SRC events. The dark error bands indicate statistical
error, the light error bands indicate systematic error due to uncertainty in
nuclear properties, and the black error bars indicate the expected statistical
error from the Nuclear Targets Experiment. The dotted purple line indicates
$(e,e^{\prime}pn)_{\rm{SCX}}/2(e,e^{\prime}pp)_{\rm{SCX}}$, and the associated
gray band indicate its systematic error.

###  4.2 Geant4 Event Selection

We have also estimated the carbon $(e,e^{\prime}p)$, $(e,e^{\prime}pp)$, and
$(e,e^{\prime}pn)$ missing momentum distributions at a beam energy of 6 GeV
for Monte Carlo simulations run through CLAS12’s Geant4 program and
reconstructed with CLAS12 software. We define these yields based on
reconstructed quantities. They include the effects of detector efficiency and
resolution and are intended to approximate the results of this analysis on
experimental data in the absence of corrections due to the momentum, momentum
resolution, and detection inefficiency. The results are shown in Fig. 6.

Figure 6:  Distribution of the missing momentum for $(e,e^{\prime}p)$,
$(e,e^{\prime}pn)$, and $(e,e^{\prime}pp)$ for an arbitrary number of physics
events after CLAS12 event reconstruction, with error bands for statistical
error.

The event selection for the Fast Monte Carlo yields all apply to analysis of
the simulated events reconstructed after Geant4, but the events passing
through the simulated detector require that some additional criteria be
applied for particle identification.

For electron identification, we require between 1 and 100 photoelectrons to be
detected in the High Threshold Cherenkov Counter, and we require electron
detection to occur in the fiducial region of the electromagnetic calorimeter.
Additionally, we require that the energy deposition match the momentum of the
associated electron track by restricting the sampling fraction to be between
0.18 and 0.28. We also require the electron to have momentum in the range
$0.5-6$ GeV/$c$. An electron detection efficiency exceeding 99% is expected
nima-clas12 .

To identify the leading proton with momentum $p_{1}$, we require a hit in the
Forward Time-of-Flight detector, with leading proton polar angle in the range
$\theta_{p}<40^{\circ}$. Based on recent upgrades using Machine Learning in
particle tracking, we expect at least 98% proton detection efficiency in the
Forward Detector, depending on luminosity gavalian1 ; gavalian2 ; gavalian3 .
We also restrict the difference in the $z$ component of the vertex position
between the leading proton and electron to be between -3 and 3 cm. We allow
only protons with well-reconstructed momentum by requiring
$-0.05<\Delta\beta<0.05$, where $\Delta\beta$ is the difference between the
reconstructed velocity and $p_{\rm{1}}/\sqrt{p^{2}_{\rm{1}}+m^{2}_{p}}$, where
$m_{p}$ is the proton mass.

For recoil nucleons, we select protons with tracks in the CVT and neutrons
with hits in the CND. For both, the momentum must be at least 300 MeV/$c$, and
the polar angle must fall between 40∘ and 140∘. For recoil protons, we expect
the detection efficiency to drop from a maximum of 75-80% for protons with a
momentum of 1 GeV/$c$ to a negligible efficiency at about 0.3 GeV/$c$ nima-
clas12 . As previously mentioned, we expect a neutron detection efficiency of
up to 10% nima-clas12 .

These criteria are meant to be minimal and they lead to a negligible reduction
in yield. In the analysis of data, in which various backgrounds may be of
concern, more stringent criteria may become necessary.

##  5 Discussion

The theoretical predictions presented here differ slightly from previous
predictions of the tensor-to-scalar transition, primarily due to detector
acceptance. Our results from Section 4.2 are not acceptance- or efficiency-
corrected, i.e., they reflect predictions for the raw yield measured in the
experiment under CLAS12 reconstruction. Ref. CLAS:2020mom  similarly presented
non-corrected GCF predictions for an analysis of data from CLAS, which has a
different acceptance, particularly for recoil nucleons. Ref. Korover2021
performed an acceptance correction, though with large systematic
uncertainties. Older works, such as Ref. LabHallA:2014wqo  predate the
development of GCF and instead used a simple model of the abundance and
center-of-mass distribution of SRC pairs in the nucleus.

We have neglected several factors that will be important in making the
measurement. First, the proton tracking efficiency in the CLAS12 Central
Vertex Tracker and the neutron detection efficiency in the Central Neutron
Detector are the largest sources of systematic uncertainty, especially for the
$pn/p$ measurement. Second, the Forward Detector proton momentum resolution is
expected to be on the order of 1%, while the Central Detector proton momentum
resolution is expected to be on the order of 5%. The Central Detector neutron
momentum resolution has not yet been precisely measured. Finally, the geometry
and imperfect proton efficiency of the Central Detector allow improperly
tracked protons to be mis-reconstructed as neutrons, leading to confusion
between the $(e,e^{\prime}pp)$ and $(e,e^{\prime}pn)$ samples.

A thorough understanding of the proton and neutron efficiencies and momentum
resolutions will be crucial to the analysis, as will a method to reject
charged particle contamination in the neutron sample. These studies are
already underway, with promising results. With corrections made for these
effects, the estimates on our statistical uncertainty indicate that we are
likely to be able to measure the change in the $pn/p$ ratio. The results
reported in this study offer a promising outlook on the possibility of
measuring the transition from a tensor-dominated $NN$ interaction at 300–600
MeV/$c$ to a scalar-dominated interaction at high missing momentum using the
CLAS12 detector.

Acknowledgments

This work was supported by the US Department of Energy Office of Science,
Office of Nuclear Physics, under contract no. DE-SC0016583; a Columbian
Fellowship from The George Washington University; and a JSA/JLab Graduate
Fellowship. The authors thank Jackson Pybus for the use of his GCF event
generator code.

Data availability statement

This manuscript has no associated data or the data will not be deposited. The
physics events used in this analysis are simulated using Generalized Contact
Formalism cross sections as described in the text. Further information about
the generator code is available upon reasonable request.

## Declarations

Conflict of interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  Appendix A Simple Estimate of Single Charge Exchange

Single charge exchange can result in an SRC pair being detected in an isospin
configuration other than the one in which it occurred when the leading nucleon
is struck by the electron. For example, a $pn$ pair can be detected as a $pp$
pair, an $nn$ pair as a $pn$ pair, etc. Estimates based on the Glauber
approximation have found approximately a 4-5% probability of single charge
exchange in CLAS kinematics, depending on the pair’s isospin configuration
Colle2016 ; DuerPRL ; CLAS:2020mom . For simplicity, we use 5% for all pairs,
and we neglect the possibility of both nucleons in the pair undergoing SCX.
Here we derive our simple estimate found in Eqs. 6, 7, and 8.

The number of $pp$-pairs observed in an experiment, $N_{pp}^{\text{exp.}}$,
can be written

$N_{pp}^{\rm{exp.}}=N_{\rm{SRC}}\cdot\left[R_{pp}\cdot
2\sigma_{ep}\cdot(1-2P_{\text{SCX}})+(1-2R_{pp})\cdot(\sigma_{ep}+\sigma_{en})P_{\rm{SCX}}\right],$
(9)

where $R_{pp}$ is the fraction of SRC pairs that are $pp$ pairs,
$P_{\rm{SCX}}$ is the probability for a nucleon (either leading or recoil) to
undergo single charge exchange, $\sigma_{ep}$ and $\sigma_{en}$ are
respectively the off-shell electron-proton and electron-neutron cross
sections, and $N_{\rm{SRC}}$ is an overall normalization factor. This
expression assumes that the number of $pp$ pairs and the number of $nn$ pairs
are equal (in other words, $2R_{pp}+R_{pn}=1$) and neglects the simultaneous
charge exchange of both leading and recoil nucleons. The first term of Eq. 9
represents scattering from a true $pp$ pair, while the second term represents
scattering from either the $p$ or the $n$ of an $np$ pair, which then
undergoes single charge exchange so as to appear like a $pp$ pair in the
experiment. If there were no single charge exchange, i.e. $P_{\rm{SCX}}=0$,
then the experiment would have instead observed

$N_{pp}^{\text{no SCX}}=N_{\rm{SRC}}\cdot R_{pp}\cdot 2\sigma_{ep}.$ (10)

We can write a similar expression for the number of $np$-pairs observed
through the $(e,e^{\prime}pn)$ reaction, i.e., with a leading proton, in an
experiment,

$N_{pn}^{\rm{exp.}}=N_{\rm{SRC}}\cdot\left[(1-2R_{pp})\cdot\sigma_{ep}\cdot(1-2P_{\text{SCX}})+2R_{pp}\cdot(\sigma_{ep}+\sigma_{en})\cdot
P_{\text{SCX}}\right].$ (11)

In the limit of no single charge exchange, this reduces to

$N_{pn}^{\text{no SCX}}=N_{\rm{SRC}}\cdot(1-2R_{pp})\cdot\sigma_{ep}.$ (12)

By substituting Eqs. 10 and 12 into Eqs. 9 and 11, we find

$\displaystyle N_{pp}^{\rm{exp.}}$ $\displaystyle=N_{pp}^{\text{no
SCX}}(1-2P_{\text{SCX}})+N_{pn}^{\text{no
SCX}}\left(1+\frac{\sigma_{en}}{\sigma_{ep}}\right)P_{\text{SCX}}$ (13)
$\displaystyle N_{pn}^{\rm{exp.}}$ $\displaystyle=N_{pn}^{\text{no
SCX}}(1-2P_{\text{SCX}})+N_{pp}^{\text{no
SCX}}\left(1+\frac{\sigma_{en}}{\sigma_{ep}}\right)P_{\text{SCX}}.$ (14)

These expressions are used for the numerical estimates given in Eqs. 6 and 7.

Adding together Eqs. 9 and 11, we obtain an expression for the expected number
of observed $(e,e^{\prime}p)$ SRC pairs,

$N_{p}^{\rm{exp.}}=N_{\rm{SRC}}\cdot[\sigma_{ep}(1-2P_{\rm{SCX}})+(\sigma_{en}+\sigma_{ep})P_{\rm{SCX}}].$
(15)

We compare $N_{p}^{\rm{exp.}}$ to the number of $(e,e^{\prime}p)$ pairs
expected in the absence of SCX, namely, $N_{p}^{\text{no
SCX}}=N_{\rm{SRC}}\sigma_{ep}$, by examining the fractional difference between
the two,

$\frac{\Delta_{p}}{N_{p}^{\text{no
SCX}}}=\frac{N_{p}^{\text{exp.}}-N_{p}^{\text{no SCX}}}{N_{p}^{\text{no
SCX}}}=P_{\rm{SCX}}\left(\frac{\sigma_{en}}{\sigma_{ep}}-1\right).$ (16)

Numerically, Eq. 16 evaluates to -0.03. We therefore make the approximation
that the change in the $(e,e^{\prime}p)$ cross section due to SCX is
negligible.

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# Learning velocity model for complex media with deep convolutional neural
networks

A.Stankevich  Moscow Institute of Physics and Technology, Russia
I.Nechepurenko  Moscow Institute of Physics and Technology, Russia
A.Shevchenko  Moscow Institute of Physics and Technology, Russia
L.Gremyachikh  HSE University, Russia  A.Ustyuzhanin  HSE University, Russia
Moscow Institute of Physics and Technology, Russia  A.Vasyukov  Moscow
Institute of Physics and Technology, Russia

###### Abstract

The paper considers the problem of velocity model acquisition for a complex
media based on boundary measurements. The acoustic model is used to describe
the media. We used an open-source dataset of velocity distributions to compare
the presented results with the previous works directly. Forward modeling is
performed using the grid-characteristic numerical method. The inverse problem
is solved using deep convolutional neural networks. Modifications for a
baseline UNet architecture are proposed to improve both structural similarity
index measure quantitative correspondence of the velocity profiles with the
ground truth. We evaluate our enhancements and demonstrate the statistical
significance of the results.

Keywords— acoustics, inverse problems, velocity model, convolutional neural
networks

##  1 Introduction

The problem of identifying elastic media properties based on their measured
response is a well-known one. This problem has many applications and
variations in industrial non-destructive testing, seismic exploration,
biomedical engineering, and other areas. This paper considers methods based on
acoustic or elastic wave excitation in a media under consideration, recording
the media’s response and identifying the media’s properties from this
response. This problem statement is typical for ultrasonic techniques and
seismic imaging.

There are many different approaches for solving an inverse problem to
determine the spatial distribution of mechanical properties from the recorded
response. New methods have emerged recently based on the success in deep
convolutional neural networks research and development. The media’s response
is used as an input for the neural network that predicts the required
properties. The feasibility of this approach was demonstrated for different
problems.

The works [1, 2] successfully applied deep learning for non-destructive
testing data interpretation and defect classification. The approaches
presented can work in real-time and provide new options compared with
traditional imaging and interpretation methods.

Several cases demonstrate the convolutional network’s feasibility for
different biomedical imaging problems. The paper [3] studied an application of
deep learning to artifacts correction on single-shot ultrasound images
obtained with sparse linear arrays. The initial results demonstrated an image
quality comparable or better to that obtained from conventional beamforming.
The work [4] used neural networks to identify the shape of the aberration
prism that distorts the ultrasonic signal. The work [5] applied convolutional
networks to elasticity imaging to distinguish benign tumors from their
malignant counterparts based on measured displacement fields on the boundary
of the domain.

This paper contributed to the same area. We present the results primarily for
seismic use cases, but this approach, together with models, methods, and a
numerical pipeline, can be easily generalized and transferred to other problem
domains.

The success of deep convolutional neural networks for image processing caused
a significant interest in applying them to the problem of seismic segmentation
and interpretation. The works [6, 7] presented an approach to discriminate
between salt and other geological structures based on the seismic response.
Related results were presented in [8, 9] and many other works.

This initial success of deep convolutional neural networks in seismic
interpretation caused complementary works that contributed to open datasets.
The paper [10] covered the facies classification problem and published an
open-source, fully annotated 3D geologic model of a genuine Dutch F3 block.
The work [11] contributed Penobscot – another open dataset for seismic
interpretation problems.

These datasets became de-facto benchmarks for comparing different machine-
learning approaches for facies classification. The availability of these
benchmarks empowered further research in this area. It caused an open-source
domain-specific framework [12] that applies state-of-the-art segmentation
algorithms for seismic interpretation.

Similar ideas are also applied to seismic impedance inversion and acquiring
velocity model of a media. Novel methods complement classical approaches for
seismic inversion problems. The techniques based on neural networks don’t
require initial guesses and are extremely fast. These features make them
attractive for a quick assessment of large datasets.

The works [13, 14, 15] presented a promising approach based on two
complementary neural networks that allowed the fast adaptation of the baseline
model for the site under investigation using transfer learning techniques. The
authors demonstrated the high quality of the predicted results. The approach
relies on a cleaned dataset and is fine-tuned exclusively for seismic
exploration cases when drilling is possible.

Another approach was taken in [16, 17, 18, 19, 20]. These works consider
boundary observations only. This technique is more demanding than the previous
one from an inverse problem point of view. However, it is preferable from the
engineering point of view for a wide range of use cases when drilling is
complicated. Moreover, this approach can be generalized to other areas besides
seismic exploration, where invasive data acquisition is impossible.

Our work contributes to the area of velocity model acquisition for the complex
heterogeneous media-based exclusively on boundary measurements. We present
several enhancements for the baseline neural networks and demonstrate them
using an open-source dataset of 1600 velocity models shown in [16]. We believe
that this will also contribute to establishing domain-specific benchmark
datasets and empower further research in this area.

##  2 Model and method

###  2.1 Numerical pipeline

The dataset from [16] is used in this paper. Each sample has a length along
the OX axis of 3000 meters and a depth of 2000 meters along the OY axis. Sound
speed in this dataset varies typically from 2500 m/s in softer media to 4500
m/s in harder media. Figure 2 presents velocity models for 2 of 1600 samples.
The Black area is an acoustically hard zone located in relatively soft
background media.

(a) Harder inclusion on the bottom

(b) Harder inclusion in the middle

Figure 2: Sample velocity models

The seismic response was calculated numerically for each velocity distribution
as described in the section 2.2. Thus we created 1600 pairs of ground truth
velocity models and corresponding seismic responses recorded on the domain’s
boundary. The data was split into training, validational, and testing subsets
using random sampling with 70%/15%/15% split ratio. The training data sample
was fed to a deep convolutional neural network described in the section 2.3.
The network trains to predict velocity distribution from the seismic response.
The quality of the predictions was evaluated on the testing dataset.

###  2.2 Forward modeling

Direct problem uses an acoustic model following the velocity models in the
dataset. The media is described with the following equations:

$\displaystyle\rho\dot{\vec{v}}$ $\displaystyle=-\nabla p$
$\displaystyle\dot{p}$ $\displaystyle=-K(\nabla\cdot\vec{v}).$ (1)

In these equations $p$ \- pressure, $\vec{v}$ \- velocity vector, $\rho$ \-
medium density, $K$ \- bulk modulus.

Forward modeling uses grid-characteristic numerical method, that demonstrated
good quality of the numerical solution for various wave dynamics problems,
including wave propagation in complex composite structures [21, 22], classical
seismic migration [23], wave propagation in a fractured [24] and
multicomponent [25] geological media.

The condition of the free boundary was specified at the upper boundary of the
computational domain. An absorbing boundary condition was used on all the
other borders. The emitter of the scanning pulse was moved along the upper
boundary of the domain, and 9 equidistant emitter positions were used for each
velocity model sample. There were 300 equidistant receivers on the upper
boundary of the domain to record the response.

Figure 4 presents an example of forward modeling results. The media has the
velocity distribution as presented in Figure 2(a), the modeling covers 2
seconds of physical time. One can see that during an initial phase of scanning
pulse propagation ($t=0.3s$), there are only low amplitude responses from the
low contrast boundaries between the layers of the background media. The
scanning pulse reaches the left peak of the hard object at the moment
$t=0.35s$, and the first high amplitude response forms. A medium amplitude
response from the medium contrast boundary in the background media happens
later ($t=0.45s$). The high amplitude response from the right peak of the hard
object starts to form around $t=0.5s$. Figure 6 shows the wave patterns in the
media at the same moments for a different position of the emitter.

These single-shot responses for the different positions of the emitter for the
same velocity distribution are stacked together and form a complete multi-shot
seismic response data for the media under investigation.

The dataset produced by forward modeling was published as [26]. We believe the
availability of these data will enable community contribution to future
research.

(a) $t=0.3s$

(b) $t=0.35s$

(c) $t=0.4s$

(d) $t=0.45s$

(e) $t=0.5s$

(f) $t=0.55s$

Figure 4: Sample wave propagation, emitter located in the center of the domain

(a) $t=0.3s$

(b) $t=0.35s$

(c) $t=0.4s$

(d) $t=0.45s$

(e) $t=0.5s$

(f) $t=0.55s$

Figure 6: Sample wave propagation, emitter located in the left part of the
domain

###  2.3 Inverse problem

Following previous research, we utilize UNet [27], a fully convolutional
encoder-decoder neural network architecture, to predict seismic velocity
distribution. Originally UNet was designed to solve problems of semantic image
segmentation, taking RGB image as input and producing segmentation mask as an
output. Our adaptation assumes stacking rescaled seismograms, obtained from
different source positions, to form an N-channeled image to feed into UNet.
The number of channels equals the number of emitter positions. The network
output, respectively, is interpreted as the velocity distribution in the
media.

Our work additionally contributes three modifications to the default pipeline
of solving such problems, increasing its overall precision.

First, we suggest adding the real- and complex-valued part of the 2D Fourier
images of seismograms as additional channels to the input of the neural
network. Different seismogram sources contribute as separate channels to the
input tensor.

Second, we experimented with analogs for classical regularization techniques.
In classical methods for solving inverse problems, the uniqueness of the
solution is often achieved by adding a regularizing term to the loss function,
which endows the solution with additional properties, such as smoothness. To
achieve a similar effect in application to neural network training, we suggest
the regularization, obtained by calculating the $L_{2}$ norm of the result of
the application of Sobel filter [28] to the output of the network. The impact
of the Sobel filter is similar to that of the operator of numerical
differentiation. Thus, we expect the obtained solutions to have a smoother
structure than those without regularization.

Third, we combined several instances of the neural network into an ensemble by
averaging their outputs. Ensembling is a well-known technique for various
machine learning problems. In this context, ensembling serves as another
analog for classical regularization to obtain a smoother structure of the
predicted velocity model.

##  3 Numerical Experiments

###  3.1 Data representation

The dataset $\mathcal{D}$ consists of pairs $(d_{i},s_{i})$, where $s_{i}$ is
the seismic response from the velocity distribution $d_{i}$, calculated as
presented in Section 2.2. Velocity distributions are represented with
$300\times 200$ real-valued arrays, corresponding to values of $c_{p}$ in
nodes of a regular mesh, covering the computational domain. In turn,
seismograms are described with $300\times 200\times n$ real-valued arrays,
containing vertical velocity component of the upper boundary of the domain
recorded at 100 Hz frequency. Here $n$ is the total number of shots (emitter
positions) that make up a complete seismic response for the sample. We
additionally rescale seismograms to match 0. - 1. range.

###  3.2 Neural network training and evaluation

We train all the models for a maximum of 50 epochs using Adam optimization
algorithm [29] with constant learning rate 1e-4, using mini-batches of size
10. The training was performed using the MSE loss function.

Due to the general deficiency of literature on machine learning applications
to geological inversion, there is no precedence for using the exact metrics
for the quality assessment of neural network predictions. We utilize
structural similarity index measure (SSIM) [30] to evaluate the performance of
model instances since the same metric was mentioned in [16] for the
experiments with the same dataset. Hence, the choice of the resulting neural
network weights was made by the epoch when it reaches a maximum value of SSIM
metric [30] on the validational data fold. The whole training pipeline is
implemented using PyTorch deep learning framework [31].

To study the variance of model predictions depending on data, we utilize
bootstrapping procedure to resample training subsets. The training procedure
for the neural network was repeated 10 times for each suggested modification
of the network. So, each new model instance was trained with a different
training subset and demonstrated different results. For the rest of this
section, we measure network performance on a testing sample after it was
trained on a training and validation samples.

###  3.3 Ablation study

The following ablation study demonstrates the causations in the proposed
architecture. We compare the quality of the solution by varying or disabling
the pipeline steps one by one. This test identifies the parts of the setup
that are essential for the quality of the solution and unnecessary features
that are not statistically significant.

Table 1 summarizes experiments conducted with Fourier images and proposed
regularization. The table provides average SSIM values for models trained on
multishot seismograms. Distributions of SSIM values for model trained using
different bootstrapped datasets on single-shot / triple-shot / nine-shot
seismograms are depicted by Figure 8. Aggregated data allows us to hypothesize
that complementing input tensors with Fourier images of seismograms positively
affects the quality of predictions. At the same time, the impact of adding a
regularization term is negligible.

We conduct a series of statistical tests on populations of SSIM test values to
confirm this statement. First, we verify that SSIM values for models instances
may be considered normally distributed, employing Wilk-Shapiro test [32] with
0.05 alpha-level (see Table 2). Further, we examine the equality of
populations variances with Levene’s test, grouping populations by the number
of seismogram shots (Table 3). As neither of the tests rejects the respective
null hypotheses, we assume that model populations satisfy all the required
criteria for ANOVA [33] testing.

Number of shots | Fourier | Regularization | Average SSIM
---|---|---|---
1 | False | False | $0.912\pm 0.003$
1 | True | False | $\mathbf{0.915}\pm\mathbf{0.002}$
1 | False | True | $0.912\pm 0.003$
1 | True | True | $0.915\pm 0.001$
3 | False | False | $0.910\pm 0.003$
3 | True | False | $0.916\pm 0.002$
3 | False | True | $0.911\pm 0.003$
3 | True | True | $\mathbf{0.917}\pm\mathbf{0.002}$
9 | False | False | $0.909\pm 0.005$
9 | True | False | $\mathbf{0.916}\pm\mathbf{0.003}$
9 | False | True | $0.911\pm 0.004$
9 | True | True | $0.915\pm 0.002$
Table 1: Average SSIM for the test dataset for models trained with MSE
criterion

(a) Single-shot data

(b) Triple-shot data

(c) Nine-shot data

Figure 8: Distributions of SSIM values for bootstrapped model training Shots | Fourier | Regularization | p-value
---|---|---|---
1 | False | False | 0.990
1 | True | False | 0.254
1 | False | True | 0.784
1 | True | True | 0.712
3 | False | False | 0.365
3 | True | False | 0.439
3 | False | True | 0.376
3 | True | True | 0.545
9 | False | False | 0.363
9 | True | False | 0.771
9 | False | True | 0.186
9 | True | True | 0.647
Table 2: p-values for Shapiro-Wilk statistics for sets of model instances trained on bootstrapped data Shots | p-value for Levene’s test
---|---
1 | 0.060
3 | 0.371
9 | 0.070
Table 3: p-values for Levene’s test for sets of model instances grouped by
number of shots in seismograms

Table 4 demonstrates that the contribution of regularization from the point of
view of increasing the value of SSIM metric is statistically negligible. On
the contrary, ANOVA-testing rejects the hypothesis about the equality of
population means for models trained with and without adding Fourier images of
seismograms to input tensors (see 5). Thus, we conclude that additional
information provided by Fourier images enhances the framework’s performance on
a statistically significant level.

Shots | Fourier | p-value for ANOVA statistic
---|---|---
1 | False | 0.770
1 | True | 0.541
3 | False | 0.291
3 | True | 0.630
9 | False | 0.321
9 | True | 0.375
Table 4: p-values for ANOVA statistic, calculated for model populations trained with and without regularization Shots | Regularization | p-value for ANOVA statistic
---|---|---
1 | False | 1.42e-2
1 | True | 1.01e-2
3 | False | 4.93e-5
3 | True | 6.02e-4
9 | False | 7.62e-4
9 | True | 4.80e-2
Table 5: p-values for ANOVA statistic, calculated for model populations
trained with and without adding fourier images

The research has shown that suggested neural network architecture has enough
statistical capacity to achieve low bias on the training dataset. However, it
turned out that regularization methods, presented earlier, cannot resolve the
issue of relatively poor generalization of the model. We suggest combining
model instances, trained before, into an ensemble, by averaging their outputs
to overcome the issue.

Table 6 showcases the value of MSE on test subset for ensembles, composed from
model instances trained before. Compared to constituent models, such
modification yields an additional $1.\%$ gain in the value of target metrics.

Number of shots | Fourier | Regularization | SSIM
---|---|---|---
1 | False | False | 0.929
1 | True | False | 0.928
1 | False | True | 0.929
1 | True | True | 0.928
3 | False | False | 0.926
3 | False | True | 0.927
3 | True | False | 0.930
3 | True | True | 0.929
9 | False | False | 0.927
9 | True | False | 0.931
9 | False | True | 0.928
9 | True | True | 0.930
Table 6: Average SSIM for the test dataset for ensembles of models trained
with MSE criterion

The ablation study results showed that Fourier images and ensembles contribute
to the quality of the solution, and the proposed regularization approach is
not significant.

###  3.4 Velocity distribution prediction results

Figure 10 presents the prediction results for the ground truth velocity
distribution shown on Figure 2(b). One can see that adding Fourier images
reduces artifacts for both single model instances and ensembles.

(a) Baseline single model

(b) Single model with Fourier images

(c) Ensembled model

(d) Ensembled model with Fourier images

Figure 10: Prediction results for different network architectures

The difference between single model instances and ensembles can be noted from
1D vertical velocity distribution profiles. Figure 12 presents such profiles
along different cross-sections of the same sample depicted on Figure 10. These
profiles reveal how the prediction of an ensemble of models differs from a
single model participating in it. Ensemble predictions are closer to the
ground truth values, and this effect becomes more prominent with the depth (on
the right parts of the images). They also tend to be more smooth than a single
model’s predictions, which is expected due to the aggregating nature of
ensembles.

(a) $x=750m$

(b) $x=1500m$

(c) $x=2250m$

Figure 12: Vertical profiles of predicted velocity distributions

###  3.5 Discussion

The presented method of velocity model acquisition from the boundary
measurements demonstrated a good solution quality and can work in real-time.
The resulting architecture defined based on this study includes two
significant modifications compared with the baseline architecture [16]: adding
Fourier images of the seismograms and using ensembling. These changes allowed
us to reach an SSIM value of 0.93 compared with 0.4 mentioned in the previous
work.

It was demonstrated that adding Fourier images of raw measured seismic
response improves the predicted acoustic impedance distribution quality. This
statement passes statistical tests regarding its significance for SSIM metric
value. Also, one can see that this modification qualitatively leads to cleaner
velocity profiles and removes shady artifacts from the main object.

The regularization approach based on the Sobel filter and implemented as
described in the present work was statistically insignificant. Another
regularization approach was based on the ensembling technique and was proven
to increase the quality of the neural network predictions. Ensembling gives a
slight increase in traditional SSIM metric value but provides qualitatively
smoother acoustic impedance profiles and better quantitative correspondence
with the ground truth properties of the media.

##  4 Conclusion

This work targeted the problem of velocity model acquisition for complex media
using convolutional neural networks and based exclusively on boundary
measurements. We considered the variation of this problem that is typical for
seismic exploration. An open-source dataset of velocity distributions was used
to directly compare our results with the results from the previous works.
Forward modeling was performed using the grid-characteristic numerical method,
the dataset with the numerically obtained seismic records was made publicly
available to empower further research in this area.

The modifications of the neural network model proposed in this work allowed us
to achieve a structural similarity index measure of $0.93\pm 0.01$, which is
significantly better than the baseline result. We demonstrated that the
changes presented contribute not only to the structural similarity but also to
the quantitative correspondence with the ground truth.

The approach, pipeline, and network architecture presented in this work are
not specific for seismic exploration and can be transferred to similar areas,
such as non-destructive testing and biomedical engineering. Experimenting with
these areas is subject to future works.

## Acknowledgements

AV and AS are supported by RFBR project 18-29-02127 for their work on
establishing a numerical pipeline and creating the dataset. LG is supported by
the Basic Research Program at the National Research University Higher School
of Economics for his work on forward-modeling, and inverse problem approaches.
AU is supported by the Russian Science Foundation under grant agreement
19-71-30020 for his work on ablation study.

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