# Densities and momentum distributions in $A\leq 12$ nuclei from chiral
effective field theory interactions

M. Piarulli1,2<EMAIL_ADDRESS>S. Pastore1,2<EMAIL_ADDRESS>R.B.
Wiringa3<EMAIL_ADDRESS>S. Brusilow1 R. Lim1 1Department of Physics,
Washington University in Saint Louis, Saint Louis, MO 63130, USA
2McDonnell Center for the Space Sciences at Washington University in St.
Louis, MO 63130, USA
3Physics Division, Argonne National Laboratory, Argonne, IL 60439

###### Abstract

Current and future electron and neutrino scattering experiments will be
greatly aided by a better understanding of the role played by short-range
correlations in nuclei. Two-body physics, including nucleon-nucleon
correlations and two-body electroweak currents, is required to explain the
body of experimental data for both static and dynamical nuclear properties. In
this work, we focus on examining nucleon-nucleon correlations from a chiral
effective field theory perspective and provide a comprehensive set of new
variational Monte Carlo calculations of one- and two-body densities and
momentum distributions based on the Norfolk many-body nuclear Hamiltonians for
$A\leq 12$ systems. Online access to detailed tables and figures is available.

##  I Introduction

The coordinate and momentum distributions of nucleons in nuclei are one of the
key indicators of short-range correlations (SRCs) in multinucleon systems.
SRCs represent a fascinating aspect of nuclear dynamics; understanding their
formation mechanisms and specific characteristics is required to obtain a
comprehensive description of nuclei and nucleonic matter. SRCs tell us much
about i) nuclear forces at short distances and how they are generated from
quantum chromodynamics; ii) the limitations of mean-field models and how to
ameliorate them; iii) the properties of matter at high densities, such as
those found in compact stellar objects and in relativistic heavy ion
collisions; iv) the response functions in hadron and lepton scattering from
nuclei; v) the origin of the EMC effect, and vi) the sensitivity of
neutrinoless double beta decay matrix elements to short-range dynamics.

Since the 1950s, many efforts have been devoted to the study of SRCs and the
short-range properties of the nuclear force. It was only recently that
experimental and theoretical studies of these phenomena were placed on solid
ground, thanks to sophisticated high-energy and large momentum transfer
electron and proton scattering experiments Tang et al. (2003); Piasetzky et
al. (2006); Shneor et al. (2007); Subedi et al. (2008); Fomin et al. (2012);
Korover et al. (2014); Hen et al. (2014); Duer et al. (2019, 2018), allowing
for precision measurements of small cross sections, together with the enormous
progress made by many-body theories Schiavilla et al. (2007); Alvioli et al.
(2008); Feldmeier et al. (2011); Alvioli et al. (2013); Rios et al. (2014);
Ciofi degli Atti et al. (1991); Wiringa et al. (2014); Ciofi degli Atti et al.
(2017); Ciofi degli Atti and Morita (2017). For instance, experiments
involving high-energy, semi-inclusive triple coincidence measurements that
successfully probed the isospin composition of nucleon-nucleon ($N\\!N$) SRCs
in the relative momentum range of 300–600 MeV/c discovered a strong (by a
factor of 20) dominance of neutron-proton ($np$) pair SRCs in nuclei when
compared with proton–proton ($pp$) and neutron–neutron ($nn$) correlations in
both light and heavy nuclei Hen et al. (2014); Korover et al. (2014); Duer et
al. (2019). This was explained on the basis of the large tensor force in the
$N\\!N$ interaction at the above-mentioned momentum range. As a result of this
finding, it was predicted that the single momentum distributions of the proton
and neutron, weighted by their respective fractions, are nearly equal, and
that the probability of a proton or neutron being in high momentum $N\\!N$
correlation is inversely proportional to their relative fractions in the
nucleus. The validity of these predictions were confirmed by results of ab-
initio variational Monte Carlo (VMC) calculations of the momentum
distributions of light nuclei Wiringa et al. (2014) and of approximate schemes
like cluster expansions Alvioli et al. (2013, 2008, 2012) and correlated basis
function theory Arias de Saavedra et al. (2007); Bisconti et al. (2007);
Ryckebusch et al. (2015) for medium to heavy nuclei. Moreover, calculations of
the momentum distributions of different light nuclei showed high momentum
tails that resembled those of the deuteron, demonstrating a universal nature
of SRCs Wiringa et al. (2014); Alvioli et al. (2013, 2008, 2012); Arias de
Saavedra et al. (2007); Bisconti et al. (2007); Ryckebusch et al. (2015).

An extensive library of VMC one- and two-body densities and momentum
distributions for many different light nuclei using the phenomenological
Argonne $v_{18}$ (AV18) two-nucleon ($N\\!N$) Wiringa et al. (1995), and
Urbana X (UX) three-nucleon ($3N$) interactions was previously constructed and
posted online for the benefit of the nuclear physics community at large
Wiringa et al. (2014). Additionally, these calculations have contributed to a
novel study of many-body factorization and the position-momentum equivalence
of nuclear short-range correlations, using a Generalized Contact Formalism
(GCF), which was reported in Nature Physics Cruz-Torres et al. (2021).

In this paper, we provide a comprehensive set of new results of one- and two-
body densities and momentum distributions over a wide range of nuclei from 2H
up to 12C, using the Norfolk $N\\!N$ and $3N$ (NV2+3) forces Piarulli et al.
(2016, 2015); Baroni et al. (2018); Piarulli and Schiavilla (2021); Piarulli
and Tews (2020). These results feature new calculations of the pair density as
a function of both the pair separation and pair center-of-mass, and
calculations of the two-body momentum distribution coming from short- and
long-range pairs differentiated by a pair separation boundary. The full set of
calculations is accessible in graphical and tabular forms online at
www.phy.anl.gov/theory/research/QMCresults.html.

The paper is structured as follows: a brief review of Norfolk interactions is
given in Sec. II. In Sec. III we present results for the one- and two-body
densities calculated for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C. The pair
density as a function of both the pair separation and pair center-of-mass is
presented for 4He and 12C. In Sec. IV the results for the one- and two-body
momentum distribution are provided for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C.
Results for momentum distributions as functions of the relative momentum and
center-of-mass momentum without and with pair separation boundary are
displayed for 4He and 12C. Additional results are available online.

##  II Norfolk Many-body Interactions

The Norfolk interactions are obtained from a chiral effective field theory
($\chi$EFT) that uses pions, nucleons and $\Delta$’s as fundamental degrees of
freedom, and consists of long-range parts mediated by one- and two-pion
exchange, and contact terms specified by unknown low-energy constants (LECs).
The LECs entering the $N\\!N$ contact interactions are constrained to
reproduce $N\\!N$ scattering data from the most recent and up-to-date database
collected by the Granada group Navarro Pérez et al. (2013, 2014); Navarro
Perez et al. (2014). The contact terms are regularized via a Gaussian cutoff
function with $R_{S}$ as the Gaussian parameter Piarulli et al. (2016, 2015);
Baroni et al. (2018). The divergences at high-value of momentum transfer in
the pion-range operators are removed via a special radial function
characterized by the cutoff $R_{L}$ Piarulli et al. (2016, 2015); Baroni et
al. (2018). There are two classes of NV2 potentials. Class I (II) has been
fitted to data up to $125$ MeV ($200$ MeV). For each class, two combinations
of short- and long-range regulators have been used, namely ($R_{S}$,
$R_{L}$)=(0.8, 1.2) fm (models NV2-Ia and NV2-IIa) and ($R_{S}$,
$R_{L}$)=(0.7, 1.0) fm (models NV2-Ib and NV2-IIb). Class I (II) fits about
2700 (3700) data points with a $\chi^{2}$/datum $\lesssim 1.1$ ($\lesssim
1.4$) Piarulli et al. (2016, 2015). The short-range component of the $3N$
interactions is parametrized in terms of two LECs, $c_{D}$ and $c_{E}$. In the
first generation of Norfolk potentials (NV2+3-Ia/b and NV2+3-IIa/b), these
LECs have been determined by simultaneously reproducing the experimental
trinucleon ground-state energies and $nd$ doublet scattering length Piarulli
et al. (2018). Within the $\chi$EFT framework, $c_{D}$ is related to the LEC
entering the axial two-body contact current Gazit et al. (2009); Marcucci et
al. (2012); Schiavilla (2017). This allows one to adopt a different strategy
to constrain $c_{D}$ and $c_{E}$. In particular, in Ref. Baroni et al. (2018)
they have been constrained to reproduce the trinucleon binding energies and
the empirical value of the Gamow-Teller matrix element in tritium $\beta$
decay. Norfolk models that use this fitting procedure are designated with a
‘*’ namely, NV2+3-Ia*/b* and NV2+3-IIa*/b*.

These interactions have been recently employed in the VMC and Green’s function
Monte Carlo (GFMC) approaches Carlson et al. (2015); Gandolfi et al. (2020) to
calculate energies Piarulli et al. (2018), charge radii and electromagnetic
form factors Gandolfi et al. (2020), beta-decay transitions Baroni et al.
(2018); King et al. (2020a, b), neutrinoless double beta-decay Cirigliano et
al. (2019, 2018) of light nuclei, beta decay spectra King et al. (2022a),
muon-capture rates King et al. (2022b) and with the auxiliary field diffusion
Monte Carlo (AFDMC) Gandolfi et al. (2020) to study the equation of state of
pure neutron matter Piarulli et al. (2020); Lovato et al. (2022).

##  III density distributions

The one- and two-body densities are evaluated as simple $\delta$-function
expectation values given by

$\displaystyle\rho_{N}(r)$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r^{2}}\big{\langle}\Psi\big{|}\sum_{i}\mathcal{P}_{N_{i}}\delta(r-|\bf{r}_{i}-\bf{R}_{\mbox{cm}}|)\big{|}\Psi\big{\rangle}\,,$
(1) $\displaystyle\rho_{N\\!N}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
r^{2}}\big{\langle}\Psi\big{|}\sum_{i<j}\mathcal{P}_{N_{i}}P_{N_{j}}\delta(r-|\bf{r}_{i}-\bf{r}_{j}|)\big{|}\Psi\big{\rangle}\,,$
(2)

where $\mathcal{P}_{N_{i}}$ represents the projector operator onto protons
($+$) or neutrons ($-$) defined as $\mathcal{P}_{N_{i}}=(1\pm\tau_{z_{i}})/2$,
$\bf{r}_{i}$ is the position of nucleon $i$ and $\bf{R}_{\mbox{cm}}$ is the
coordinate of the center of mass.

A detailed survey of one- and two-body densities have been calculated for a
variety of nuclei in the range $A=2-12$ using variational Monte Carlo wave
functions developed for the AV18+UX and the Norfolk local chiral interactions.
The corresponding tables and figures are available online at
www.phy.anl.gov/theory/research/density/, for the one-nucleon densities, and
at www.phy.anl.gov/theory/research/density2/, for the two-nucleon densities.

###  III.1 One-body density results

In Fig. 1 we present the neutron and proton densities calculated for 3H,
3,4,8He, 6,7Li, 9Be, 10B, and 12C using the AV18+UX and the NV2+3-Ia,
NV2+3-Ia*, and NV2+3-IIb* local chiral interactions. Additional densities for
2H, 6He, 8,9Li, 8,10,12Be, 11B and 10,11C may be found in the online tables,
as well as results for the NV2+3-Ib* and NV2+3-IIa* interactions. We also give
neutron and proton rms radii there.

Figure 1: One-body neutron (left panel) and proton (right panel) densities are
shown for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C using the phenomenological
AV18+UX and the local chiral NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* interactions.

The VMC wave functions are treated as states of unique isospin $T$. Thus for
$N=Z$ nuclei, proton and neutron densities are the same and only proton
densities are given in the online tables. However, the wave functions for
nuclei with $T>0$ can be different for different isospin projections $T_{z}$,
so mirror nuclei are not isospin symmetric. This allows the proton-rich nuclei
to be slightly more diffuse than neutron-rich nuclei due to their greater
repulsive Coulomb interaction.

Spin-up and spin-down densities are also provided in the online tables. In
$J=0$ nuclei, spin-up and spin-down densities are identical, but not for $J>0$
nuclei. If spin-up and spin-down projections are the same, as in $0^{+}$
states, we give only totals. The total number of spin-up/down protons and
neutrons in $J>0$ nuclei with $M_{J}=J$ are reported in Table 1. Unless
otherwise indicated by an error in parentheses, variation among the different
interaction models is less than 0.01. We note that for these nuclei, the
subset with an odd number of neutrons has
$(n\\!\uparrow\\!-n\\!\downarrow)\approx$ 0.7-0.9, while those with an even
number of neutrons have $(n\\!\uparrow\\!-n\\!\downarrow)\approx$ -0.02.
Similar results hold for nuclei with odd and even proton numbers. The sole
exception is 9Li which has an exceptionally large error bar.

We also note that the s-shell nuclei ($A\leq 4$) exhibit large peaks at small
separation, while the p-shell nuclei ($A\geq 6$) are much reduced at small $r$
and more spread out. This can be attributed to the cluster structure of these
light p-shell nuclei, e.g., $\alpha d$ in 6Li, $\alpha t$ in 7Li,
$\alpha\alpha n$ in 9Be, and $3\alpha$ in 12C. This puts the center of mass of
these nuclei in between clusters and thus reduces the central density.

Table 1: Total number of spin-up/down protons and neutrons in $J>0$ nuclei with $M_{J}=J$ for the local chiral Norfolk NV2+3 interactions. Variation among the different interactions NV2+3-Ia, -Ia*, -Ib*, -IIa*, and -IIb* is less than 0.01 unless otherwise indicated by an error in parentheses. Nucleus | $N_{\uparrow p}$ | $N_{\downarrow p}$ | $N_{\uparrow n}$ | $N_{\downarrow n}$
---|---|---|---|---
2H($1^{+}$) | $0.96$ | $0.04$ | $0.96$ | $0.04$
3He($\frac{1}{2}^{+}$) | $0.98$ | $1.02$ | $0.94$ | $0.06$
6Li($1^{+}$) | $1.93$ | $1.07$ | $1.93$ | $1.07$
7Li($\frac{3}{2}^{-}$) | $1.94$ | $1.06$ | $1.99$ | $2.01$
8Li($2^{+}$) | $1.91$ | $1.09$ | $2.85(1)$ | $2.15(1)$
9Li($\frac{3}{2}^{-}$) | $1.91$ | $1.09$ | $3.12(7)$ | $2.88(7)$
9Be($\frac{3}{2}^{-}$) | $2.00$ | $2.00$ | $2.85(2)$ | $2.15(2)$
10B($3^{+}$) | $2.90(1)$ | $2.10(1)$ | $2.90(1)$ | $2.10(1)$
11B($\frac{3}{2}^{-}$) | $2.87(2)$ | $2.13(2)$ | $2.99(1)$ | $3.01(1)$

###  III.2 Two-body density results

In Fig. 2, we present the relative-distance pair densities, with neutron-
proton ($np$) in the left panel and proton-proton ($pp$) in the right panel,
for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C using the phenomenological AV18+UX
and the local chiral NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* interactions. The
online tables contain additional results for the NV2+3-Ib* and NV2+3-IIa*
interactions.

Figure 2: Relative-distance $np$ (left panel) and $pp$ (right panel) pair
densities for several nuclei. Distributions are shown for 3H, 3,4,8He, 6,7Li,
9Be, 10B, and 12C using the phenomenological AV18+UX and the local chiral
NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* interactions. Figure 3: Relative-distance
$np$ (left panel) and $pp$ (right panel) pairs densities for several nuclei.
Distributions are shown for 3H, 3,4,8He, 6,7Li, 9Be, 10B, and 12C using the
phenomenological AV18+UX and the local chiral NV2+3-Ia, NV2+3-Ia*, and
NV2+3-IIb* interactions. For each potential, all calculations are scaled to
have the same value at $\sim 1$ fm.

We can see that within a fixed interaction model, the two-nucleon densities at
$r\lesssim 1.5$ fm for various nuclei exhibit a similar behavior, generated by
the cooperation of the short-range repulsion and the intermediate-range tensor
attraction of the $N\\!N$ interaction, with the tensor force governing the
large overshoot at $r\sim 1.0$ fm between $np$ pairs.

As shown in Fig. 3, where all calculations are scaled to have the same value
at $\sim 1$ fm, the two-nucleon densities at short separations appears to be
the same for all values of $A$, which leads to the nontrivial conclusion that
at short ranges the two-nucleon motion is not affected by the presence of the
other particles. This is what has been called universality of SRCs Feldmeier
et al. (2011). Moreover, at large separation the asymptotic behavior of the
two-nucleon densities for different nuclei differs due to the different
surface effects.

While the short-distance behaviour is the same for all nuclei, it differs for
each interaction. Indeed, the probability of finding two nucleons at short
distances is finite for the ”soft” NV2+3-Ia and NV2+3-Ia* chiral models, but
approaches zero as we progress to the ”hard” local chiral interaction
NV2+3-IIb* and the ”hardest” phenomenological AV18+UX.

Nucleon pair distributions in different combinations of $ST$ for different
nuclei can also be found online. In Table 2, we report the number of pairs
$N_{ST}$ for 3,4,6,8He, 6,7,8,9Li, 8,9,10Be, 10,11B, and 12C using the NV2+3
potentials. We show both independent pair (IP) numbers for the highest spatial
symmetry states and for the fully correlated (cor) wave functions. Correlated
pair counts for the AV18+UX interaction fall within these ranges for all but
two cases.

Table 2: Total number of spin-isospin $ST$ pairs in different nuclei for the NV2+3 potentials, showing both independent pair (IP) for the highest spatial symmetry states and the fully correlated (cor) wave functions. Correlated pair counts for AV18+UX are consistent within error bars for all but a few cases. Nucleus | $\Psi$ | $N_{01}$ | $N_{11}$ | $N_{10}$ | $N_{00}$
---|---|---|---|---|---
3He($\frac{1}{2}^{+}$) | IP | $1.5$ | $0.0$ | $1.5$ | $0.0$
| cor | $1.37$ | $0.13$ | $1.49$ | $0.01$
4He($0^{+}$) | IP | $3.0$ | $0.0$ | $3.0$ | $0.0$
| cor | $2.57(1)$ | $0.43(1)$ | $2.99$ | $0.01$
6He($0^{+}$) | IP | $5.5$ | $4.5$ | $4.5$ | $0.5$
| cor | $4.95(1)$ | $5.05(1)$ | $4.49$ | $0.51$
6Li($1^{+}$) | IP | $4.5$ | $4.5$ | $5.5$ | $0.5$
| cor | $4.07(2)$ | $4.93(2)$ | $5.47$ | $0.53$
7Li($\frac{3}{2}^{-}$) | IP | $6.75$ | $6.75$ | $6.75$ | $0.75$
| cor | $6.13(2)$ | $7.37(2)$ | $6.73$ | $0.77$
8He($0^{+}$) | IP | $9.0$ | $12.0$ | $6.0$ | $1.0$
| cor | $8.16(2)$ | $12.84(2)$ | $6.00$ | $1.00$
8Li($2^{+}$) | IP | $8.0$ | $11.0$ | $8.0$ | $1.0$
| cor | $7.42(3)$ | $11.58(3)$ | $7.94$ | $1.06$
8Be($0^{+}$) | IP | $9.0$ | $9.0$ | $9.0$ | $1.0$
| cor | $8.09(3)$ | $9.91(3)$ | $8.97$ | $1.03$
9Li($\frac{3}{2}^{-}$) | IP | $10.5$ | $15.0$ | $9.0$ | $1.5$
| cor | $9.46(12)$ | $16.04(11)$ | $8.99(1)$ | $1.51(1)$
9Be($\frac{3}{2}^{-}$) | IP | $10.5$ | $13.5$ | $10.5$ | $1.5$
| cor | $9.57(4)$ | $14.43(4)$ | $10.46$ | $1.54$
10Be($0^{+}$) | IP | $13.0$ | $18.0$ | $12.0$ | $2.0$
| cor | $11.73(4)$ | $19.27(4)$ | $11.98$ | $2.02$
10B($3^{+}$) | IP | $12.0$ | $18.0$ | $13.0$ | $2.0$
| cor | $11.01(7)$ | $18.99(7)$ | $12.94(1)$ | $2.06(1)$
11B($\frac{3}{2}^{-}$) | IP | $15.0$ | $22.5$ | $15.0$ | $2.5$
| cor | $13.84(2)$ | $23.66(3)$ | $14.91(3)$ | $2.59(3)$
12C($0^{+}$) | IP | $18.0$ | $27.0$ | $18.0$ | $3.0$
| cor | $16.54(6)$ | $28.46(6)$ | $17.91(1)$ | $3.09(1)$

A common feature in the $ST$ pair counts is that there is a moderate 10-15%
depletion of the $ST=01$ pairs going from IP to correlated wave functions,
with a corresponding increase in the number of $ST=11$ pairs. This is
attributable to the many-body tensor correlations, which can flip spins (in
exchange for orbital angular momentum) but not change isospin. Because the
$ST=01$ interactions are more attractive than $ST=11$, this depletion
mechanism is a source of saturation of the nuclear binding. The $ST=10$ pairs
also show a depletion going from IP to correlated wave functions, with an
increase of $ST=00$ pairs, but the effect is much smaller, probably because
$ST=00$ interaction is generally much more repulsive than $ST=11$.

The probability of finding two nucleons with relative separation $r$ and
center-of-mass distance $R$ is described by the calculation of the full
probability density $\rho_{N\\!N}(r,R)$. These densities are computationally
demanding and are not available for all nuclei and interactions, but they can
be generated upon request.

In Fig. 4, we present the the $np$ and $pp$ densities, multiplied by
$r^{2}R^{2}$, as a function of $r$ and $R$ for 4He and 12C using the
phenomenological AV18+UX interaction. The curves are normalized to obtain the
corresponding $np$ and $pp$ pairs, 4 $np$ and 1 $pp$ pairs in 4He, 36 $np$ and
15 $pp$ pairs in 12C.

Figure 4: The surface plots show the $np$ and $pp$ densities as function of
the relative distance $r$ and center-of-mass $R$ for 4He (left panel) and 12C
(right panel) using the phenomenological AV18+UX. The curves are normalized to
obtain the corresponding $np$ and $pp$ pairs.  Figure 5: Total one-body
neutron (left panel) and proton (right panel) momentum distributions for 3H,
3He, 4He, 6Li, 7Li, 8He, 9Be, 10B, and 12C using the AV18+UIX phenomenological
potentials, and the NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* local chiral
interactions.

##  IV Momentum distributions

###  IV.1 One-body momentum distributions

The probability of finding a nucleon with momentum $k$ and spin-isospin
projection $\sigma$,$\tau$ in a given nuclear state is obtained by the Fourier
transform of the one-nucleon nondiagonal density matrix

$\displaystyle\rho_{\sigma\tau}({\bf k})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int d{\bf r}^{\prime}_{1}\,d{\bf r}_{1}\,d{\bf
r}_{2}\cdots d{\bf r}_{A}\,\psi^{\dagger}_{JM_{J}}({\bf r}_{1}^{\prime},{\bf
r}_{2},\dots,{\bf r}_{A})\,$ (3) $\displaystyle\times$
$\displaystyle\,e^{-i{\bf k}\cdot({\bf r}_{1}-{\bf
r}^{\prime}_{1})}\,P_{\sigma\tau}(1)\,\psi_{JM_{J}}({\bf r}_{1},{\bf
r}_{2},\dots,{\bf r}_{A})\,.$

where $P_{\sigma\tau}(i)$ is the spin-isospin projection operator for nucleon
$i$, and $\psi_{JM_{J}}$ is the nuclear wave function with total spin $J$ and
spin projection $M_{J}$. The normalization is

$N_{\sigma\tau}=\int\frac{d{\bf k}}{(2\pi)^{3}}\,\,\rho_{\sigma\tau}({\bf k})\
,$ (4)

where $N_{\sigma\tau}$ is the number of spin-up or spin-down protons or
neutrons.

Monte Carlo (MC) integration is used to construct the Fourier transform in Eq.
(3). A conventional Metropolis walk, guided by $|\psi_{JM_{J}}({\bf
r}_{1},\dots,{\bf r}_{i},\dots,{\bf r}_{A})|^{2}$, is used to sample
configurations Pudliner et al. (1997). We average across all particles $i$ in
each configuration, and for each particle, the Fourier transform is computed
using a grid of Gauss-Legendre points $\bf x_{i}$. Instead of just moving the
position ${\bf r}_{i}^{\prime}$ in the left-hand wave function away from a
fixed position ${\bf r}_{i}$ in the right-hand wave function, both positions
are moved symmetrically away from ${\bf r}_{i}$, so Eq. (3) becomes

$\displaystyle\rho_{\sigma\tau}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{A}\sum_{i}\int d{\bf r}_{1}\cdots d{\bf r}_{i}\cdots
d{\bf r}_{A}\int d{\bf\Omega}_{x}\int_{0}^{x_{\rm max}}x^{2}dx$
$\displaystyle\psi^{\dagger}_{JM_{J}}({\bf r}_{1},\dots,{\bf r}_{i}+{\bf
x}/2,\dots,{\bf r}_{A})\,e^{-i{\bf k}\cdot{\bf x}}$ $\displaystyle\times$
$\displaystyle\,P_{\sigma\tau}(i)\,\psi_{JM_{J}}({\bf r}_{1},\dots,{\bf
r}_{i}-{\bf x}/2,\dots,{\bf r}_{A})\,.$

Here the polar angle $d{\bf\Omega}_{x}$ is also sampled by MC integration,
with a randomly chosen direction for each particle in each MC configuration.
This approach is analogous to that used in studies of the nucleon-pair
momentum distribution, see Refs. Schiavilla et al. (2007); Wiringa et al.
(2014), and has the benefit of significantly decreasing statistical errors
caused by the rapidly oscillating nature of the integrand for large values of
$k$. To reach momenta $k\sim 10$ fm-1 in 4He with good statistics requires
integrating to $x_{\rm max}$=20 fm using 200 Gauss-Legendre points.

The results for a variety of nuclei in the range $A=2-12$ are available on the
web page at www.phy.anl.gov/theory/research/momenta/. They are generated as
distributions for neutron spin-down $\rho_{n\downarrow}(k)$, neutron spin-up
$\rho_{n\uparrow}(k)$, proton spin-down $\rho_{p\downarrow}(k)$, and proton
spin-up $\rho_{p\uparrow}(k)$, for the $M_{J}=J$ state. Where proton and
neutron momentum distributions are the same, as in $T=0$ nuclei, only one set
is given, and similarly, if spin-up and spin-down projections are the same, as
in $0^{+}$ states, we give totals only.

In Fig. 5 we show the total one-body neutron (left panel),
$\rho_{n\downarrow}(k)+\rho_{n\uparrow}(k)$, and proton (right panel),
$\rho_{p\downarrow}(k)+\rho_{p\uparrow}(k)$, momentum distributions for 3H,
3,4He, 6,7Li, 8He, 9Be, 10B, and 12C using the phenomenological AV18+UX, and
the local chiral NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* interactions. Additional
results for 2H, 6He, 8,9Li, 8,10Be, and 11B are shown in the online tables.

All models show the progressive high-momentum behavior in $k$ as the number of
nucleons increases. Adding nucleons to the p-shell widens the distribution at
low momenta and creates a peak at a finite $k$. All of these nuclei have a
dramatic shift in slope at $k=2$ fm-1 to a broad shoulder, which is attributed
to the large tensor correlation caused by the pion-exchange component of the
nuclear force. As expected, the differences between the models are most
noticeable in the high-momentum tails, which decay more rapidly with
increasing $k$ for the ”soft” NV2+3-Ia and NV2+3-Ia* interactions than the
”hard” NV2+3-IIb* and AV18+UX potentials. The difference in momentum
distributions observed for the NV2+3-Ia and NV2+3-Ia* models, which have the
same two-body interaction but different the three-nucleon force
parametrization, is small. For NV2+3-Ia, there is a minor overall shift in
$\rho_{n}(k)$ and $\rho_{p}(k)$ toward bigger $k$.

###  IV.2 Two-body momentum distributions

The probability of finding two nucleons in a nucleus with relative momentum
${\bf q}=({\bf k}_{1}-{\bf k}_{2})/2$ and total center-of-mass momentum ${\bf
Q}={\bf k}_{1}+{\bf k}_{2}$ in a given spin-isospin state is given by:

$\displaystyle\rho_{ST}({\bf q},{\bf Q})$ $\displaystyle=$ $\displaystyle\int
d{\bf r}^{\prime}_{1}d{\bf r}_{1}d{\bf r}^{\prime}_{2}d{\bf r}_{2}d{\bf
r}_{3}\cdots d{\bf r}_{A}$ $\displaystyle\psi^{\dagger}_{JM_{J}}({\bf
r}_{1}^{\prime},{\bf r}^{\prime}_{2},{\bf r}_{3},\dots,{\bf r}_{A})$
$\displaystyle e^{-i{\bf q}\cdot({\bf r}-{\bf r}^{\prime})}e^{-i{\bf
Q}\cdot({\bf R}-{\bf R}^{\prime})}$ $\displaystyle
P_{ST}(12)\psi_{JM_{J}}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3},\dots,{\bf
r}_{A})\,,$

where ${\bf r}={\bf r}_{1}-{\bf r}_{2}$, ${\bf R}=({\bf r}_{1}+{\bf
r}_{2})/2$, and $P_{ST}(12)$ is a projector onto pair spin $S=0$ or 1, and
isospin $T=0$ or 1. The total normalization is:

$N_{ST}=\int\frac{d{\bf q}}{(2\pi)^{3}}\frac{d{\bf
Q}}{(2\pi)^{3}}\rho_{ST}({\bf q},{\bf Q})\,,$ (7)

where $N_{ST}$ is the total number of nucleon pairs with given spin-isospin.
Alternate projectors can also be used, e.g., for $N\\!N$ pairs $pp$, $np$, and
$nn$ (and each of these with spin S) with corresponding normalizations.

The nucleon-pair momentum distributions can be examined in a number of
different ways. One way is to integrate over all values of ${\bf Q}$ and
reduce the total pair density to a function $\rho_{N\\!N}(q)$ of the relative
momentum $q$ only. In this case, Eq.(IV.2) reduces to a form similar to
Eq.(IV.1), with a sum over all configurations in the Monte Carlo walk
controlled by $|\Psi_{JM_{J}}|^{2}$, and a Gauss-Legendre integration over the
relative separation ${\bf x}={\bf r}-{\bf r}^{\prime}$. Again, the polar angle
${\bf\Omega}_{x}$ is sampled by randomly choosing the direction of ${\bf x}$
in space, and an average over all pairs in every MC configuration is made.

Many results for $\rho_{ST}(q)$ and $\rho_{N\\!N}(q)$ obtained for various
light nuclei in the range $A=3-12$ are recorded in the online tables at
www.phy.anl.gov/theory/research/momenta2/. These are from VMC calculations
using different Norfolk NV2+3 potentials, including -Ia, -Ia*, -Ib*, -IIa*,
and -IIb*, as well as results obtained with the AV18+UX. The nuclei covered
include 3H, 3,4,6,8He, 6,7,8,9Li, 8,9Be, 10B, and 12C.

In Fig. 6, we display the $np$ and $pp$ momentum distributions for selected
nuclei using the AV18+UX phenomenological potentials, and the NV2+3-Ia,
NV2+3-Ia*, and NV2+3-IIb* local chiral interactions. They have been calculated
for relative momentum $q$ from 0 to 10 fm-1 and integrated over all values of
${\bf Q}$. All the four Hamiltonians show the high-momentum tail in $q$, but
it decays more rapidly for the soft NV2+3-Ia and NV2+3-Ia*.

Figure 6: Total $np$ (left panel) and $pp$ (right panel) momentum
distributions as function of the relative momentum $q$ for 3H, 3He, 4He, 6Li,
7Li, 8He, 9Be, 10B, and 12C using the AV18+UIX phenomenological potentials,
and the NV2+3-Ia, NV2+3-Ia*, and NV2+3-IIb* local chiral interactions.

In Fig. 7, we show the ratio of the $np$ and $pp$ momentum distributions for
several nuclei relative to the total $np/pp$ ratio. For $q\leq 2$ fm-1 the
ratios are virtually identical for the different interactions and close to the
total number of $np/pp$ pairs. Beyond that point, the $np/pp$ ratio gets
larger, with the soft interactions showing a larger peak at smaller $q$, while
the hard interactions interactions show a lower but broader peak at larger
$q$. This behavior is probably due to the strong tensor correlations in the
$np$ channel. Note these ratios are integrated over all $Q$, while the much
larger $np/pp$ ratios, mentioned above from Refs. Hen et al. (2014); Korover
et al. (2014); Duer et al. (2019) are for small $Q$.

Figure 7: Ratio of $np$ to $pp$ pairs as function of relative momentum $q$ for
various nuclei; the constant line in each panel is the total pair ratio.
Figure 8: The surface plots show the the $np$ (left panels) and $pp$ (right
panels) momentum distributions as functions of the relative momentum $q$ and
center-of-mass momentum $Q$ for the alpha particle obtained with the NV2+3-Ia*
and AV18+UX interactions. Figure 9: Same as Fig. 8 but for 12C using the
AV18+UX.

We can also integrate Eq.(IV.2) over all ${\bf q}$, leaving a function
$\rho_{N\\!N}(Q)$ of the total pair momentum $Q$ only. In general, the
$\rho_{N\\!N}(Q)$ for a given nucleus has a smaller falloff at large momenta
than the $\rho_{N\\!N}(q)$ and the ratios of different $N\\!N$ components vary
less over the range of $Q$. These distributions are generally not as
interesting, but they can be generated on request.

Calculations of the full $\rho_{N\\!N}({\bf q},{\bf Q})$ are more challenging
as they require a double Gauss-Legendre integral over two randomly chosen
directions for each pair in each MC sample: ${\bf x}$ for ${\bf r}$ and ${\bf
X}$ for ${\bf R}$. In Fig. 8, we display the surface plots of $\rho_{np}({\bf
q},{\bf Q})$ and $\rho_{pp}({\bf q},{\bf Q})$ as functions of the relative
momentum $q$ and center-of-mass momentum $Q$ for 4He using the NV2+3-Ia* and
AV18+UX interactions. In Fig. 9 we show similar plots for 12C but only for the
AV18+UX interaction. In the online tables we present $\rho_{N\\!N}({\bf
q},{\bf Q})$ results for 3,4He, 6Li, 12Be, and 12C for the AV18+UX and one or
both of the NV2+3-Ia* and NV2+3-IIb* interactions.

In addition, we can differentiate between short-range (SR) and long-range (LR)
pair contributions by simply sorting our MC samples into two sets, where $r<a$
for SR pairs and $r>a$ for LR pairs. For example, in 4He, a boundary of $a=2$
fm divides the six $N\\!N$ pairs into approximately two equal groups. Fig. 10
shows the $np$ (left panels) and $pp$ (right panels) SR and LR pair
distributions for the alpha particle using the NV2+3-Ia* and AV18+UX
interactions. Taking the integral of these distributions we find that in the
case of the NV2+3-Ia* the number of $np$ SR and LR pairs are 2.04 and 1.96,
respectively, while the number of $pp$ SR and LR pairs are 0.53 and 0.47. For
the case of the AV18+UX the of $np$ SR and LR pairs are 2.06 and 1.94 while
the number of $pp$ SR and LR pairs are 0.48 and 0.52.

Figure 10:  $np$ (left panel) and $pp$ (right panel) momentum distributions in
4He from pairs with separation $r\leq 2.0$ fm of each other and from pairs
with $r\geq 2.0$ fm.

Similarly, The $np$ and $pp$ SR and LR pair distributions for 12C employing
the AV18+UX interactions are shown in Fig. 11 for $a=2.5$ fm. By integrating
these distributions, we find that the number of $np$ SR and LR pairings is
12.9 and 23.1 respectively, while the number of $pp$ SR and LR pairs is 4.3
and 10.7, respectively. In the online figures and tables we also provide the
breakdown for 3,4He and 6Li for the AV18+UX, NV2+3-Ia*, and NV2+3-IIb* with
the break point $a=2.5$ fm. These figures clearly show that the LR pairs
dominate at low pair momenta but fall off rapidly beyond $q\approx 1.5$ fm-1,
while the SR pairs provide the high-momentum tail. For the pair center of mass
momentum, the total number of pairs declines significantly beyond $Q\approx 2$
fm-1 but there continues to be a high-momentum tail in $q$.

Figure 11:  Same as Fig. 10 but for 12C using the AV18+UX with separation
boundary $a=2.5$ fm.

##  V Conclusions

We have performed VMC calculations of one- and two-body density distributions
and one- and two-body momentum distributions for a wide variety of nuclei from
2H up to 12C, using the $\Delta$-full Norfolk interactions obtained from
$\chi$EFT. Results are compared to those obtained with the conventional
AV18+UX interaction, some of which were previously reported in Ref. Wiringa et
al. (2014). New features in the present work include i) calculations of the
pair density $\rho_{N\\!N}(r,R)$ as a function of both the pair separation $r$
and pair center-of-mass $R$; and ii) calculations of the two-body momentum
distribution $\rho_{N\\!N}(q,Q,a)$ coming from short- and long-range pairs
differentiated by a pair separation boundary $a$.

Comparing results among the different NV2+3 and AV18+UX interactions, we find
the one-body densities $\rho_{N}(r)$ for a given nucleus are very similar for
all cases. Also, the total number of spin-up/down protons and neutrons is
remarkably constant. In contrast, the two-body densities $\rho_{N\\!N}(r)$
vary significantly at short distances, depending on whether the interaction is
‘soft’ like NV2+3-Ia or ‘hard’ like NV2+3-IIb*. However, the total number of
spin-isospin $ST$ pairs for a given nucleus shows little variation among the
different interactions.

One-body momentum distributions $\rho(k)$ all share the same characteristics,
with a maximum at $k=0$ fm-1, a rapid fall off to $k\approx 2$ fm-1, followed
by a high-momentum tail that is more prominent for ‘hard’ interactions, and
less so for ‘soft’ interactions. For a given interaction, the low-$k$ behavior
varies with the nucleus, but the high-$k$ tails are essentially parallel for
all $A=2$-12 nuclei. Two-body momentum distributions $\rho_{N\\!N}(q)$ are
similar, but tend to change slope near $q\approx 1.5$ fm-1. Again, the
high-$q$ tail is larger for ‘hard’ interactions. We also note that the ratio
$\rho_{np}(q)/\rho_{pp}(q)$ is relatively flat and proportional to the number
of pairs of each type at lower $q$, but much larger at higher $q$, reflecting
the importance of the stronger tensor correlations in $np$ versus $pp$ pairs.
Finally, our studies of $\rho_{N\\!N}(q,Q,a)$, which separates the
contributions of short- from long-range pairs, clearly indicates the high-$q$
tails are due to SRCs.

Concurrent to these studies, are QMC studies of nuclear electroweak response
densities and response functions Pastore et al. (2020); Barrow et al. (2021);
Andreoli et al. (2022) where the interaction of the external probes (both
electron and neutrinos) is accounted for at one- and two-body level along with
SRCs. In particular, within the Short-Time-Approximation Pastore et al. (2020)
it is possible to analyze electroweak nuclear responses in terms of the
kinematic variables, that is relative and center of mass momenta, associated
with a pair of correlated nucleons struck by the probe. Many-body effects in
the coupling of electroweak probes with correlated nucleons are being
vigorously investigated due to their relevance to both electron and neutrino
scattering processes.

While this paper provides examples of the densities and momentum
distributions, the full set of results is accessible in graphical and tabular
forms online at www.phy.anl.gov/theory/research/QMCresults.html. We expect to
continue expanding and updating these results in future.

## Acknowledgments

This work is supported by the U.S. Department of Energy under contract DE-
SC0021027 (S. P.), DE-AC02-06CH11357 (R. B. W.), a 2021 Early Career Award
number DE-SC0022002 (M. P.), the FRIB Theory Alliance award DE-SC0013617 (S.
P. and M. P.), and the NUCLEI SciDAC program (R. B. W.).

The many-body calculations were performed on the parallel computers of the
Laboratory Computing Resource Center, Argonne National Laboratory, and the
computers of the Argonne Leadership Computing Facility via the INCITE grant
“Ab-initio nuclear structure and nuclear reactions”, the 2019/2020 ALCC grant
“Low Energy Neutrino-Nucleus interactions” for the project NNInteractions, the
2020/2021 ALCC grant “Chiral Nuclear Interactions from Nuclei to Nucleonic
Matter” for the project ChiralNuc, and by the 2021/2022 ALCC grant “Quantum
Monte Carlo Calculations of Nuclei up to ${}^{16}{\rm O}$ and Neutron Matter”
for the project QMCNuc.

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# ORES-Inspect: A technology probe for machine learning audits on enwiki

Zachary Levonian
Digital Harbor Foundation
&Lauren Hagen
University of Minnesota
&Lu Li
University of Pennsylvania
Jada Lilleboe
University of Minnesota
&Solvejg Wastvedt
University of Minnesota
&Aaron Halfaker
Microsoft Research
&Loren Terveen
University of Minnesota

###### Abstract

Auditing the machine learning (ML) models used on Wikipedia is important for
ensuring that vandalism-detection processes remain fair and effective.
However, conducting audits is challenging because stakeholders have diverse
priorities and assembling evidence for a model’s [in]efficacy is technically
complex. We designed an interface to enable editors to learn about and audit
the performance of the ORES edit quality model. ORES-Inspect111https://ores-
inspect.toolforge.org is an open-source web tool and a provocative technology
probe for researching how editors think about auditing the many ML models used
on Wikipedia. We describe the design of ORES-Inspect and our plans for further
research with this system.

Keywords: machine learning, auditing, tools, ORES, edit quality

## Introduction

Figure 1: The ORES-Inspect interface and login page, as accessible via
Toolforge.

ORES is a widely-used service for building and hosting machine learning models
requested by the Wikipedia community [Halfaker and Geiger, 2020]. Of
particular relevance is the edit quality model, which makes predictions about
the quality of individual Wikipedia edits and is used in other systems for
vandalism detection and removal. ORES’ edit quality predictions directly
influence the likelihood of an edit being reverted [TeBlunthuis et al., 2020].
This impact is a notable success for community-centered and participatory
machine learning processes: ORES is hosting an increasing number of
models.222ORES is being replaced with LiftWing, but this work is applicable to
any revscoring model.

A key challenge for ORES and other machine learning services is that it is
hard to determine if a model is consistently producing reasonable outputs. In
other words, it is hard to audit these models. There are many barriers to
auditing complex machine learning systems like ORES: (a) identifying a
relevant sample of incorrect predictions, (b) determining if those incorrect
predictions represent a pattern of undesired behavior (a “bug”), and (c)
convincing system designers to fix the undesired behavior. To address those
barriers, we are building ORES-Inspect, an open-
source333https://github.com/levon003/wiki-ores-feedback tool to audit the
behavior of the ORES edit quality model for English Wikipedia.

In the consensus-driven Wikipedia context, the developers of ML-driven systems
like ORES are enthusiastic about receiving community input on problems or
potential areas for improvement. Thus, the key design objective for ORES-
Inspect is to address problems (a) and (b) by making it easy to identify high-
quality quantitative evidence of the ORES edit quality model’s behaviors. We
are developing ORES-Inspect as a “technology probe” to reflect on the process
of conducting ML audits in the Wikipedia context by highlighting the benefits
and challenges of collecting quantitative evidence of system bugs [Hutchinson
et al., 2003].

Functionally, ORES-Inspect is a labeling interface for individual Wikipedia
edits. The key intuition is that any Wikipedia user may be interested in
auditing a system like ORES, but different auditors will have different
priorities (e.g. are new editors unfairly targeted, is vandalism on stubs
missed more often than on larger articles, etc.). For that reason, the process
of auditing is the process of quantifying one’s intuitions and identifying
evidence that a single misclassification represents a pattern that should be
changed. Therefore, we designed ORES-Inspect as a provocation: it is designed
to educate editors about how ML models can be audited and how to translate
intuitions into high-quality evidence.

To fulfill this educational objective and to make auditing tractable for
users, we designed the interface (Figure 1) around four phases of activity. We
will describe our design decisions, the data, and our future analysis plans in
the remainder of this extended abstract, but we conclude this introduction
with the verbatim contents of the info panel shown to ORES-Inspect users on
first login:

ORES finds vandalism. ORES is a machine learning model that gives every edit
on Wikipedia a score from 0 (least likely to be damaging) to 1 (most likely to
be damaging). Score predictions are used to highlight the Recent Changes feed
and in other places to find and revert vandalism. ORES-Inspect helps you audit
ORES by looking at score predictions and determining if they are correct.
Audit ORES in four steps:

  1. 1.

Filter: Choose which edits to look at. ORES-Inspect shows you all human edits
on mainspace articles by default, but you can filter down to look only at
edits on particular pages (such as pages related to LGBT history) or from
particular editors (such as newcomers).

Or, use the filter controls to choose something else entirely, like bot edits
on Talk pages!

  2. 2.

Focus: When an edit is damaging, it is usually reverted by the editor
community. ORES-Inspect helps you focus on cases where the community behavior
disagrees with the ORES prediction.

If you choose to look at Unexpected Reverts, you’re looking at edits that ORES
thinks are non-damaging… but that the community reverted anyway.

If you choose to look at Unexpected Consensus, you’re looking at edits that
ORES thinks are damaging… but that the community didn’t revert.

  3. 3.

Inspect: Look at individual edits and label them as damaging (“I would revert
this.”) or not damaging. See if you can find a pattern of errors in ORES’
predictions.

  4. 4.

Discuss: View a summary of your edit labels by clicking “View Annotation
History”. How often did ORES misclassify the edits you looked at?

You can discuss your results with the ORES developers. If you change the
filters, you can compare two groups of edits to identify bias (“Are newcomers’
edits misclassified more often than experienced editors’?”)

## Implementation & Data

ORES-Inspect is a React and Python app hosted on Toolforge. Auditors use
filters to focus their attention on specific properties of articles
(namespace, category, size), of edits (size, marked as minor), or of users
(registration status, bot status). ORES-Inspect is based on the 35.6 million
non-bot enwiki edits in 2019 and the corresponding prediction made by ORES at
the time of the edit,444Historical ORES predictions were only available until
the end of 2019. but auditors focus on only those revisions that have already
received attention by the community: Unexpected Consensus edits are predicted
to be damaging by ORES but were not reverted within 1 year, while Unexpected
Reverts were predicted to be non-damaging by ORES but were reverted. By
focusing on these two categories, we focus on identifying false positives and
false negatives respectively with a much higher precision than random sampling
of revisions. By then inspecting and labeling specific revisions as damaging
or not damaging, auditors create quantitative estimates of the prevalance of
false positives and/or false negatives for a specific subset of pages, edits,
or editors.

## Discussion & Future Work

Other research-driven interfaces for working with ORES include Wikibench for
curating and discussing training data [Kuo et al., 2024] and ORES Explorer for
exploring fairness trade-offs induced by model thresholding decisions [Ye et
al., 2021]. We focus on auditing models that are already in use with an
emphasis on building quantitative evidence of ML system bugs. In our
experience as Wikipedia editors, we observe that most feedback on ML systems
happens on the basis of a single bad prediction noticed while focused on other
editing work. ORES-Inspect aims to be a tool for turning those singletons into
rigorous and useful audits, and we have already found the tool helpful for
reflecting on how one’s opinions on edit quality might diverge from consensus.
As we continue to develop ORES-Inspect, we intend to conduct interviews with
editors and share the results of audits conducted with the tool, aiming to
generate discussion on how and when ML model efficacy should be evaluated by
editors.

## Acknowledgements

We would like to thank Haiyi Zhu, Phyllis Gan, and Isaac Johnson for their
contributions and the Wiki Workshop reviewers for useful feedback. This work
was partially supported by the National Science Foundation under Grant No.
1908688.

## References

  * [Halfaker and Geiger, 2020]  Aaron Halfaker and R. Stuart Geiger.  2020\.  ORES: Lowering Barriers with Participatory Machine Learning in Wikipedia.  ACM Hum.-Comput. Interact., 4(CSCW2):148:1–148:37, October.
  * [Hutchinson et al., 2003]  Hilary Hutchinson, Wendy Mackay, Bo Westerlund, Benjamin B. Bederson, Allison Druin, Catherine Plaisant, Michel Beaudouin-Lafon, Stéphane Conversy, Helen Evans, Heiko Hansen, Nicolas Roussel, and Björn Eiderbäck.  2003\.  Technology probes: inspiring design for and with families.  In CHI’03, pages 17–24. ACM, April.
  * [Kuo et al., 2024]  Tzu-Sheng Kuo, Aaron Halfaker, Zirui Cheng, Jiwoo Kim, Meng-Hsin Wu, Tongshuang Wu, Kenneth Holstein, and Haiyi Zhu.  2024\.  Wikibench: Community-Driven Data Curation for AI Evaluation on Wikipedia, February.  arXiv:2402.14147 [cs].
  * [TeBlunthuis et al., 2020]  Nathan TeBlunthuis, Benjamin Mako Hill, and Aaron Halfaker.  2020\.  The effects of algorithmic flagging on fairness: quasi-experimental evidence from Wikipedia.  arXiv:2006.03121 [cs], June.
  * [Ye et al., 2021]  Zining Ye, Xinran Yuan, Shaurya Gaur, Aaron Halfaker, Jodi Forlizzi, and Haiyi Zhu.  2021\.  Wikipedia ORES Explorer: Visualizing Trade-offs For Designing Applications With Machine Learning API.  In DIS ’21, DIS ’21, pages 1554–1565. ACM, June.