# Testing neutrino electromagnetic properties at current and future dark
matter experiments

Carlo Giunti<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare
(INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy  Christoph
A. Ternes<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare
(INFN), Laboratori Nazionali del Gran Sasso, 67100 Assergi, L’Aquila (AQ),
Italy

###### Abstract

We analyze data from the dark matter direct detection experiments PandaX-4T,
LUX-ZEPLIN and XENONnT to place bounds on neutrino electromagnetic properties
(magnetic moments, millicharges, and charge radii). We also show how these
bounds will improve at the future facility DARWIN. In our analyses we
implement a more conservative treatment of background uncertainties than
usually done in the literature. From the combined analysis of all three
experiments we can place very strong bounds on the neutrino magnetic moments
and on the neutrino millicharges. We show that even though the bounds on the
neutrino charge radii are not very strong from the analysis of current data,
DARWIN could provide the first measurement of the electron neutrino charge
radius, in agreement with the Standard Model prediction.

##  I Introduction

The investigation of neutrino properties is one of the most active research
fields in particle physics. In the Standard Model neutrinos are massless
particles which interact only via weak interactions. Through the observation
of neutrino oscillations we know, however, that at least two neutrinos are
massive particles. Therefore, the Standard Model has to be extended to account
for neutrino masses.

In some of the extensions of the Standard Model neutrinos can acquire
electromagnetic properties through quantum loop effects. Therefore, some
models of physics beyond the Standard Model predict the interaction of
neutrinos with electromagnetic fields and electromagnetic interactions with
charged particles. Moreover, even the Standard Model predicts non-zero
neutrino charge radii due to radiative corrections. For detailed reviews on
the theories of neutrino electromagnetic properties we refer the reader to
Refs. [1, 2].

In this paper we test the neutrino electromagnetic properties (magnetic
moments, millicharges, and charge radii) by analyzing data from dark matter
direct detection experiments. These experiments aim to measure the nuclear or
electron recoils from dark matter interacting with the material in the
detector, which is Xenon for all experiments under consideration in this work.
In this paper we consider the data from the PandaX-4T experiment [3], from
LUX-ZEPLIN (LZ) [4], and from XENONnT [5]. We also show the sensitivity of the
future experiment DARWIN [6].

One of the background sources for the dark matter search in these experiments
is the elastic scattering of solar neutrinos on the electrons in the detector
material. Therefore, if some new physics model changes the standard electron-
neutrino elastic scattering (E$\nu$ES) cross section, it can be tested at dark
matter direct detection experiments. One possibility to alter this process is
the presence of neutrino electromagnetic properties.

Our paper is structured as follows: In Section II we discuss how neutrino
magnetic moments, neutrino millicharges and neutrino charge radii alter the
E$\nu$ES process. In Section III we detail the analysis procedure of the
experiments under consideration. We proceed to present and discuss our results
in Section IV, before concluding in Section V.

##  II Theoretical framework

In this paper we obtain bounds on neutrino electromagnetic properties from the
data of Xenon dark matter experiments through the elastic scattering of solar
neutrinos with electrons (E$\nu$ES). These experiments are sensitive to the
neutrino electromagnetic properties through their contributions to E$\nu$ES in
addition to the Standard Model E$\nu$ES cross section. Therefore, we first
present the Standard Model E$\nu$ES cross section in Subsection II.1, and then
we discuss the cross sections due to the neutrino electromagnetic properties
that we consider: magnetic moments in Subsection II.2, electric charges in
Subsection II.3, and charge radii in Subsection II.4. These three
electromagnetic properties are the observable effective electromagnetic
properties of ultrarelativistic neutrinos, for which the effective magnetic
moments include possible electric moments and the effective charge radii
include possible anapole moments [1, 2, 7].

Solar neutrinos oscillate and arrive at a detector on Earth as a mixture of
$\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$, whose fluxes are given by

$\Phi_{\nu_{e}}^{i}=\Phi_{\nu_{e}}^{i\,\odot}P_{ee},\quad\Phi_{\nu_{\mu}}^{i}=\Phi_{\nu_{e}}^{i\,\odot}\left(1-P_{ee}\right)\cos^{2}\vartheta_{23},\quad\Phi_{\nu_{\tau}}^{i}=\Phi_{\nu_{e}}^{i\,\odot}\left(1-P_{ee}\right)\sin^{2}\vartheta_{23},,$
(1)

where $\Phi_{\nu_{e}}^{i\,\odot}$ are the fluxes of $\nu_{e}$ produced by
thermonuclear reactions in the center of the Sun, with
$i=pp,\,^{7}\text{Be},\,\text{etc.}$, and $P_{ee}$ is the survival probability
of $\nu_{e}$ at the Earth. In our analysis we use the solar spectra taken from
Ref. [8, 9, 10, 11] using the normalizations for the high metallicity model
taken from the review in Ref. [12]. We consider $pp$ and 7Be neutrinos, which
give the main contribution to the event rates of the experiments under
consideration. For these low-energy solar neutrinos

$P_{ee}\simeq\left(1-\frac{1}{2}\,\sin^{2}2\vartheta_{12}\right)\cos^{4}\vartheta_{13}+\sin^{4}\vartheta_{13}.$
(2)

Using $\sin^{2}\vartheta_{12}=0.318\pm 0.016$ and
$\left.\sin^{2}\vartheta_{13}\right|_{\text{NO}}=0.02200^{+0.00069}_{-0.00062}$
or
$\left.\sin^{2}\vartheta_{13}\right|_{\text{IO}}=0.02225^{+0.00064}_{-0.00070}$
obtained in the global fit of Ref. [13] (see also the similar results obtained
in Refs. [14, 15]) in the case of normal ordering and inverted ordering of the
neutrino masses, we obtain $P_{ee}=0.542\pm 0.011$. The fluxes of $\nu_{\mu}$
and $\nu_{\tau}$ depend on the mixing angle $\vartheta_{23}$, which is close
to maximal mixing ($\vartheta_{23}=\pi/4$) [13, 14, 15]. For simplicity, we
consider $\sin^{2}\vartheta_{23}=0.5$, which implies
$\Phi_{\nu_{\mu}}^{i}=\Phi_{\nu_{\tau}}^{i}$. Therefore, we obtain equal
constraints on the electromagnetic properties of $\nu_{\mu}$ and $\nu_{\tau}$.

###  II.1 The Standard Model E$\nu$ES cross section

The Standard Model E$\nu$ES cross section per Xenon atom is given by

$\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM}}}{dT_{\text{e}}}(E_{\nu},T_{\text{e}})=Z_{\text{eff}}^{\text{Xe}}(T_{e})\,\dfrac{G_{\text{F}}^{2}m_{e}}{2\pi}\left[\left(g_{V}^{\nu_{\ell}}+g_{A}^{\nu_{\ell}}\right)^{2}+\left(g_{V}^{\nu_{\ell}}-g_{A}^{\nu_{\ell}}\right)^{2}\left(1-\dfrac{T_{e}}{E_{\nu}}\right)^{2}-\left((g_{V}^{\nu_{\ell}})^{2}-(g_{A}^{\nu_{\ell}})^{2}\right)\dfrac{m_{e}T_{e}}{E_{\nu}^{2}}\right],$
(3)

where $G_{\text{F}}$ is the Fermi constant, $m_{e}$ is the electron mass,
$E_{\nu}$ is the neutrino energy, and $T_{e}$ is the observable electron
recoil energy. The neutrino-electron couplings $g_{V,A}^{\nu_{\ell}}$ depend
on the neutrino flavor:

$\displaystyle\hbox{}g_{V}^{\nu_{e}}=2\sin^{2}\vartheta_{W}+1/2,\quad\hbox{}$
$\displaystyle\hbox{}g_{A}^{\nu_{e}}=1/2,$ (4)
$\displaystyle\hbox{}g_{V}^{\nu_{\mu,\tau}}=2\sin^{2}\vartheta_{W}-1/2,\quad\hbox{}$
$\displaystyle\hbox{}g_{A}^{\nu_{\mu,\tau}}=-1/2,$ (5)

with $\sin^{2}\vartheta_{W}=0.23863\pm 0.00005$ [16]. The coefficient
$Z_{\text{eff}}^{\text{Xe}}(T_{e})$ quantifies the effective number of
electrons which can be ionized at $T_{e}$ [17]. We calculate
$Z_{\text{eff}}^{\text{Xe}}(T_{e})$ using Table II of Ref. [18].

###  II.2 Neutrino magnetic moments

Neutrino magnetic moments are predicted by many theories with massive
neutrinos beyond the Standard Model (BSM) and have been constrained by many
observations (see, e.g., Refs. [1, 2, 7, 16]). The most stringent experimental
limit $|\mu_{\nu_{e}}|<2.9\times 10^{-11}\,\mu_{B}$ at 90% C.L., where
$\mu_{B}$ is the Bohr magneton, was obtained in the GEMMA experiment [19]
through the E$\nu$ES of reactor $\bar{\nu}_{e}$. This bound is more than eight
orders of magnitude larger than the prediction of neutrino magnetic moments in
the minimal extension of the SM with right handed neutrinos and Dirac neutrino
masses [20, 21, 22]. However, in more elaborate models (see, e.g., the review
in Ref. [1]), the neutrino magnetic moments can be larger and can be probed in
current experiments.

The contribution of the neutrino magnetic moments to E$\nu$ES is incoherent
with the Standard Model contribution because the neutrino magnetic moment
interaction flips the helicity of ultrarelativistic neutrinos, whereas the
Standard Model weak interaction is helicity-conserving. Therefore, for each
flavor neutrino $\nu_{\ell}$ the total cross section is given by the sum of
the Standard Model cross section (3) and the magnetic moment cross section

$\dfrac{d\sigma_{\nu_{\ell}\text{-}\text{Xe}}^{\text{MM}}}{dT_{\text{e}}}(E_{\nu},T_{\text{e}})=Z_{\text{eff}}^{\text{Xe}}(T_{\text{e}})\dfrac{\pi\alpha^{2}}{m_{e}^{2}}\left(\dfrac{1}{T_{\text{e}}}-\dfrac{1}{E_{\nu}}\right)\left|\dfrac{\mu_{\nu_{\ell}}}{\mu_{\text{B}}}\right|^{2},$
(6)

where $\alpha$ is the fine-structure constant. For the low values of
$T_{\text{e}}$ in the experiments under consideration ($T_{\text{e}}\lesssim
30\,\text{keV}$), the cross section (6) is approximately proportional to
$T_{\text{e}}^{-1}$. Therefore, the effects of the neutrino magnetic moments
are probed by the observation of an event excess near the $T_{\text{e}}$
threshold.

###  II.3 Neutrino electric charges

In the Standard Model neutrinos are neutral, but in BSM theories they can have
small electric charges, often called “millicharges” (see, e.g., the review in
Ref. [1]), which can be probed in neutrino scattering experiments. In general,
the three flavor neutrinos can have the millicharges $q_{\nu_{e}}$,
$q_{\nu_{\mu}}$, and $q_{\nu_{\tau}}$, and there can be also the three
transition electric charges $q_{\nu_{e\mu}}$, $q_{\nu_{e\tau}}$, and
$q_{\nu_{\mu\tau}}$.

Since the electric charge interaction conserves the helicity of
ultrarelativistic neutrinos and the millicharges of the flavor neutrinos
conserve the neutrino flavor, they contribute coherently with the Standard
Model interaction, which is helicity and flavor conserving. On the other hand,
since the transition electric charges induce a change of flavor they
contribute incoherently with the Standard Model interaction. Therefore, the
total E$\nu$ES cross section is given by

$\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+EC}}}{dT_{\text{e}}}=\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+EC}}}{dT_{\text{e}}}\right)_{q_{\nu_{\ell}}}+\sum_{\ell^{\prime}\neq\ell}\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{EC}}}{dT_{\text{e}}}\right)_{q_{\nu_{\ell\ell^{\prime}}}},$
(7)

where
$\left(d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+EC}}/dT_{\text{e}}\right)_{q_{\nu_{\ell}}}$
is given by Eq. (3) with

$g_{V}^{\nu_{\ell}}\to
g_{V}^{\nu_{\ell}}-\dfrac{\sqrt{2}\pi\alpha}{G_{\text{F}}m_{e}T_{e}}\,q_{\nu_{\ell}},$
(8)

and

$\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{EC}}}{dT_{\text{e}}}\right)_{q_{\nu_{\ell\ell^{\prime}}}}=Z_{\text{eff}}^{\text{Xe}}(T_{e})\,\dfrac{\pi\alpha^{2}}{m_{e}T_{\text{e}}^{2}}\left[1+\left(1-\dfrac{T_{\text{e}}}{E_{\nu}}\right)^{2}-\dfrac{m_{e}T_{\text{e}}}{E_{\nu}^{2}}\right]|q_{\nu_{\ell\ell^{\prime}}}|^{2},$
(9)

for $\ell^{\prime}\neq\ell$. Hence, E$\nu$ES gives full information on the
charges $q_{\nu_{\ell}}$ of the flavor neutrinos, including their sign,
whereas only the absolute values of the transition electric charges
$q_{\nu_{\ell\ell^{\prime}}}$ can be probed. Note also that the effects of the
electric charges are enhanced at small values of $T_{e}$, leading to the
possibility to probe very small electric charges in low-threshold experiments.

###  II.4 Neutrino charge radii

Even if neutrinos are neutral, they can have charge radii. Indeed, even in the
Standard Model the massless and neutral flavor neutrinos have tiny charge
radii induced by radiative corrections, which are given by [23, 24, 25] (with
the definition of the charge radii in Refs. [26, 27])

$\langle{r}_{\nu_{\ell}}^{2}\rangle_{\text{SM}}=-\frac{G_{\text{F}}}{2\sqrt{2}\pi^{2}}\left[3-2\ln\left(\frac{m_{\ell}^{2}}{m^{2}_{W}}\right)\right],$
(10)

where $m_{W}$ and $m_{\ell}$ are, respectively, the $W$ boson and charged
lepton masses ($\ell=e,\mu,\tau$). Numerically, we have

$\displaystyle\hbox{}\langle{r}_{\nu_{e}}^{2}\rangle_{\text{SM}}=-0.83\times
10^{-32}\,\text{cm}^{2},$ (11)
$\displaystyle\hbox{}\langle{r}_{\nu_{\mu}}^{2}\rangle_{\text{SM}}=-0.48\times
10^{-32}\,\text{cm}^{2},$ (12)
$\displaystyle\hbox{}\langle{r}_{\nu_{\tau}}^{2}\rangle_{\text{SM}}=-0.30\times
10^{-32}\,\text{cm}^{2}.$ (13)

These diagonal charge radii of the flavor neutrinos are the only charge radii
that exist in the Standard Model, where neutrino flavor is conserved.

In BSM theories neutrinos can have also the transition charge radii
$\langle{r}_{\nu_{e\mu}}^{2}\rangle$, $\langle{r}_{\nu_{e\tau}}^{2}\rangle$,
and $\langle{r}_{\nu_{\mu\tau}}^{2}\rangle$ which induce flavor transitions in
scattering processes. We consider the general case with both diagonal and
transition charge radii. As for the electric charges discussed in Subsection
II.3, the diagonal charge radii contribute to the E$\nu$ES cross section
coherently with the Standard Model interaction, because both conserve the
helicity of ultrarelativistic neutrinos and neutrino flavors, whereas the
transition charge radii contribute incoherently. Therefore, the total E$\nu$ES
cross section is given by

$\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+CR}}}{dT_{\text{e}}}=\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+CR}}}{dT_{\text{e}}}\right)_{\langle{r}_{\nu_{\ell}}^{2}\rangle}+\sum_{\ell^{\prime}\neq\ell}\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{CR}}}{dT_{\text{e}}}\right)_{\langle{r}_{\nu_{\ell\ell^{\prime}}}^{2}\rangle},$
(14)

where
$\left(d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{SM+CR}}/dT_{\text{e}}\right)_{\langle{r}_{\nu_{\ell}}^{2}\rangle}$
is given by Eq. (3) with

$g_{V}^{\nu_{\ell}}\to
g_{V}^{\nu_{\ell}}+\dfrac{\sqrt{2}\pi\alpha}{3G_{\text{F}}}\,\langle{r}_{\nu_{\ell\ell^{\prime}}}^{2}\rangle,$
(15)

and

$\left(\dfrac{d\sigma_{\nu_{\ell}-\text{Xe}}^{\text{CR}}}{dT_{\text{e}}}\right)_{\langle{r}_{\nu_{\ell\ell^{\prime}}}^{2}\rangle}=Z_{\text{eff}}^{\mathcal{A}}(T_{e})\,\dfrac{\pi\alpha^{2}m_{e}}{9}\left[1+\left(1-\dfrac{T_{e}}{E_{\nu}}\right)^{2}-\dfrac{m_{e}T_{e}}{E_{\nu}^{2}}\right]|\langle{r}_{\nu_{\ell\ell^{\prime}}}^{2}\rangle|^{2},$
(16)

for $\ell^{\prime}\neq\ell$. As for the electric charges discussed in
Subsection II.3, E$\nu$ES gives full information on the diagonal charge radii
$\langle{r}^{2}_{\nu_{\ell}}\rangle$ of the flavor neutrinos, including their
sign, and only information on the absolute values of the transition electric
charges $\langle{r}_{\nu_{\ell\ell^{\prime}}}^{2}\rangle$.

##  III Data analysis

###  III.1 Current experiments

In the analyses presented in this paper we include data from LZ [4], XENONnT
[5] and PandaX-4T [3]. In this section we discuss the details of each
analysis. For an experiment $X$ the overall predicted number of events in a
given energy-bin $k$ is given by

$R^{X}_{k}=R^{E\nu ES}_{k}+\sum_{i}R^{i}_{k}\,,$ (17)

where $R^{E\nu ES}_{k}$ is the contribution from solar neutrinos which
elastically scatter on electrons, while $R^{i}_{k}$ are the remaining
background components. We have extracted the contributions $R^{i}_{k}$ for the
different experiments from Refs. [28, 5, 3] and $R^{E\nu ES}_{k}$ is obtained
from

$R^{E\nu
ES}_{k}=N~{}\int_{T_{e}^{k}}^{T_{e}^{k+1}}dT_{e}~{}\int_{0}^{\infty}dT_{e}^{\prime}~{}R(T_{e},T_{e}^{\prime})~{}A(T_{e}^{\prime})\sum_{i=pp,^{7}\text{Be}}\int_{E_{\nu}^{\text{min}}}^{E_{\nu,i}^{\text{max}}}dE_{\nu}~{}\sum_{\ell}~{}\Phi_{\nu_{\ell}}^{i}(E_{\nu})~{}\frac{d\sigma_{\nu_{\ell}}}{dT_{e}^{\prime}}\,.$
(18)

In this expression $T_{e}$ and $T_{e}^{\prime}$ are the reconstructed and true
electron recoil energies, while $E_{\nu}$ is the neutrino energy. The
electron-neutrino cross section for a given neutrino flavor $\nu_{\ell}$ is
given by $d\sigma_{\nu_{\ell}}/dT_{e}^{\prime}$ and
$\Phi_{\nu_{\ell}}^{i}(E_{\nu})$ are the solar neutrino fluxes from Eq. (1).
The minimal neutrino energy to produce an electron recoil of $T_{e}^{\prime}$
is given by
$E_{\nu}^{\text{min}}=(T_{e}^{\prime}+\sqrt{2m_{e}T_{e}^{\prime}+T_{e}^{\prime
2}})/2$, while the maximal neutrino energy $E_{\nu,i}^{\text{max}}$ depends on
the production process, indicated with the index $i$. Next,
$R(T_{e},T_{e}^{\prime})$ and $A(T_{e}^{\prime})$ are the detector resolution
and efficiency and are different for all experiments. Finally, $N$ is a
normalization constant which takes into account the exposure and detector
volume. Also this quantity is different for each experiment.

For the analysis of the experiments under consideration we use the detector
efficiencies from Refs. [29, 5, 3]. For the energy resolution at LZ we use the
same function that has been used in Refs. [18, 30]. In the case of XENONnT we
use the resolution function of Ref. [31] and for PandaX-4T we use the one from
Ref. [3].

The predicted number of events in Eq. (17) has to be compared with the data
$D^{X}$ accumulated in each experiment. We use the data presented in Fig. 6 of
Ref. [4] for LZ111We do not use the timing data from Ref. [29], which could
result only in a 5% improvement in the bounds. and the data in Fig. 3 of Ref.
[3] for PandaX-4T. For XENONnT we use the data from Fig. 4 (5) of Ref. [5] for
recoil energies above (below) 30 keV.

Due to the low statistics in some of the bins, for LZ and PandaX-4T we use the
Poissonian least-squares function

$\chi^{2}_{X}=\min_{\vec{\alpha},\vec{\beta}}\left\\{2\left(\sum_{k}R^{X}_{k}-D^{X}_{k}+D^{X}_{k}~{}\log
D^{X}_{k}/R^{X}_{k}\right)+\sum_{i}(\alpha_{i}/\sigma_{\alpha_{i}})^{2}+\sum_{i}(\beta_{i}/\sigma_{\beta_{i}})^{2}\right\\}\,,$
(19)

where $\alpha_{i}$ are normalization constants multiplied to each single
background component in Eq. (17). For LZ, the uncertainties
$\sigma_{\alpha_{i}}$ are extracted from Ref. [28]. For PandaX-4T, they are
taken from Ref. [3]. Note that some of them are left to vary freely in the
analysis. Also included are uncertainty coefficients $\beta_{i}$ of the solar
neutrino fluxes, with uncertainties $\sigma_{\beta_{i}}$ taken from Ref. [12].

In the case of XENONnT data, we use instead

$\chi^{2}_{\text{XENONnT}}=\min_{\vec{\alpha},\vec{\beta}}\left\\{\sum_{k}\left(\frac{R^{\text{XENONnT}}_{k}-D^{\text{XENONnT}}_{k}}{\sigma_{k}}\right)^{2}+\sum_{i}(\alpha_{i}/\sigma_{\alpha_{i}})^{2}+\sum_{i}(\beta_{i}/\sigma_{\beta_{i}})^{2}\right\\}\,,$
(20)

where the uncertainties in each bin $\sigma_{k}$ are extracted from Ref. [5].
The remaining components are equivalent to the corresponding ones for LZ and
PandaX-4T.

We also perform a combined analysis of all three experiments. In this case we
correlate the uncertainties regarding the neutrino flux among the experiments.
In addition, several of the background components are common to at least two
of the three experiments. In these cases we also correlate the normalizations
of the background components. While such a combined analysis has not been
performed yet, we have noticed that this correlated analysis produces only
slightly tighter bounds than simply summing up the individual $\chi^{2}$.

###  III.2 DARWIN sensitivity

We also compute the sensitivity to electromagnetic neutrino properties for the
future experiment DARWIN. The calculation of the event rate at DARWIN is
basically the same as for the current experiments, given in Eqs. (17) and
(18), but we also include the contributions from solar N, O and $pep$
neutrinos. Note, however, that their contribution is mostly negligible in
comparison with some of the background contributions, as shown in Fig. 1 of
Ref. [6]. We include them, nevertheless, since we use the full spectrum as
shown in Ref. [6]. The energy dependence of the neutrino oscillation
probability is taken into account for these higher-energy neutrinos.

For the individual background components we use the spectra given in Ref. [6],
which need to be normalized to the considered exposure. Due to lack of more
detailed information, we use the same resolution function and detector
efficiency as for XENONnT. We assume the efficiency to remain constant for
$T_{e}>T_{e,\text{max}}^{\text{XENONnT}}$. With these assumptions, we are able
to reproduce the E$\nu$ES spectra for all five neutrino species in Fig. 1 of
Ref. [6], which validates our choice of efficiency and resolution. As was done
in Ref. [6], we consider different scenarios for the DARWIN sensitivity. First
we assume an exposure of 30 ty. Next, we assume an exposure of 300 ty.
Finally, we assume an exposure of 300 ty again, but also that the 136Xe
background component is depleted to 1%.

When generating the mock data, always compatible with the Standard Model, we
use 51 logarithmically spaced bins between 1 and 1500 keV recoil energy. Note
that the spectrum at higher energies is not sensitive to any BSM effect
considered in this paper, because the E$\nu$ES rate is much smaller than some
of the background rates. We still use the full spectrum, since the inclusion
of events at high energies can help to control the effect of background
uncertainties.

Figure 1: $\Delta\chi^{2}$ profiles for the different neutrino magnetic
moments obtained from the analyses of the data of PandaX (magenta), LZ (blue),
XENONnT (orange) and from the combination of all experiments (green). Also
shown is the sensitivity range for DARWIN (red shaded), where the worst (best)
case scenario corresponds to the analysis with 30 ty (300 ty) exposure without
(with) depleted 136Xe background. The dashed line corresponds to 300 ty
exposure without depleted 136Xe background.

##  IV Results

In this section we present the bounds that can be obtained from current data
and the sensitivity at DARWIN to neutrino electromagnetic properties.

###  IV.1 Neutrino magnetic moments

We first discuss neutrino magnetic moments. Our results are presented in Fig.
1 and Table 1. In the first column of Table 1 we present the bounds on the
magnetic moment $\mu_{\nu_{e}}$. In the second we present the results for
$\mu_{\nu_{\mu}}$ and $\mu_{\nu_{\tau}}$, which are the same in our analysis.
Slightly different bounds for the two moments have been found, e.g., in Refs.
[18, 30], where non maximal atmospheric mixing ($\sin^{2}\theta_{23}\neq 0.5$)
was assumed to calculate the neutrino oscillation probability. Since we chose
maximal atmospheric mixing, the expected number of events is the same for a
non-zero $\mu_{\nu_{\mu}}$ or $\mu_{\nu_{\tau}}$, hence producing the same
bound. Finally, we consider the case of a single effective magnetic moment,
$\mu_{\nu}^{eff}=\mu_{\nu_{e}}=\mu_{\nu_{\mu}}=\mu_{\nu_{\tau}}$.

Our analysis of PandaX-4T updates the analysis of Ref. [32], where the Panda
collaboration analyzed the data from a smaller version of the current
detector. In any case, due to the larger background rates observed in this
experiment the bounds are always weaker than those obtained from LZ or
XENONnT. It should be noted that the LZ bound obtained in our analysis is a
bit weaker than those obtained in Refs. [18, 30]. This is due to the more
conservative use of systematic uncertainties in our analysis, since we treat
each background component individually with an individual nuisance parameter.
Also our XENONnT bound is slightly weaker than that from the collaboration [5]
and the one from Ref. [33]. This is not worrying since we used a different
statistical method and since our bound lies in any case within the XENONnT
sensitivity band, see Ref. [5]. Also in the case of XENONnT, we are more
conservative in the treatment of systematic uncertainties than other
phenomenological analyses [30, 33].

The bounds obtained from the combined analysis, which at 90% confidence level
read

$\displaystyle|\mu_{\nu_{e}}|$ $\displaystyle<$ $\displaystyle 10.3\times
10^{-12}~{}\mu_{B}\,,$ (21) $\displaystyle|\mu_{\nu_{\mu/\tau}}|$
$\displaystyle<$ $\displaystyle 15.6\times 10^{-12}~{}\mu_{B}\,,$ (22)
$\displaystyle|\mu_{\nu}^{eff}|$ $\displaystyle<$ $\displaystyle 7.5\times
10^{-12}~{}\mu_{B}\,,$ (23)

are among the strongest laboratory bounds on neutrino magnetic moments [1, 2,
7, 16]. They are about one or two orders of magnitude stronger than bounds
from COHERENT or Dresden-II data [34, 35, 36, 37, 38], or Borexino data [39].
They are also stronger than older bounds reported in the literature [40, 41,
19, 42, 43, 44, 45, 46], as the best bound on $\mu_{\nu_{e}}$ obtained in the
GEMMA experiment ($|\mu_{\nu_{e}}|<2.9\times 10^{-11}\,\mu_{B}$ at 90% C.L.)
[19], the best bound on $\mu_{\nu_{\mu}}$ obtained in the LSND experiment
($|\mu_{\nu_{\mu}}|<6.8\times 10^{-10}\,\mu_{B}$ at 90% C.L.) [44], and the
best bound on $\mu_{\nu_{\tau}}$ obtained in the DONUT experiment
($|\mu_{\nu_{\tau}}|<3.9\times 10^{-7}\,\mu_{B}$ at 90% C.L.) [45].

As also shown in Fig. 1 and Table 1, these bounds can be further improved by
about a factor of 2–5 by the DARWIN experiment which would make dark matter
direct detection experiments competitive with astrophysical observations,
which constrain the neutrino magnetic moments below a few $10^{-12}\,\mu_{B}$
[47, 48, 49, 50, 51].

Experiment | $|\mu_{\nu_{e}}|~{}[10^{-12}\mu_{B}]$ | $|\mu_{\nu_{\mu/\tau}}|~{}[10^{-12}\mu_{B}]$ | $|\mu_{\nu}^{eff}|~{}[10^{-12}\mu_{B}]$
---|---|---|---
PandaX-4T | $<38.7$ | $<58.6$ | $<28.3$
LZ | $<17.1$ | $<25.9$ | $<12.5$
XENONnT | $<11.5$ | $<17.5$ | $<8.4$
combined | $<10.3$ | $<15.6$ | $<7.5$
DARWIN 30 ty | $<4.0$ | $<6.0$ | $<2.9$
DARWIN 300 ty | $<2.3$ | $<3.5$ | $<1.7$
DARWIN 300 ty depl. | $<2.1$ | $<3.2$ | $<1.5$
Table 1: The 90% bounds ($\Delta\chi^{2}=2.71$) that can be obtained on the
different neutrino magnetic moments.

###  IV.2 Neutrino millicharge

Dark matter direct detection experiments are sensitive to all of the neutrino
millicharges. Note that due to the choice $\sin^{2}\theta_{23}=0.5$ the bounds
on the muon and tau neutrino millicharges are going to be the same for the
same reason as explained in the previous section. The results from our
analyses are presented in Figs. 2 and 3 and in Table 2. An interesting feature
can be seen in Fig. 2. While the preferred regions of the current experiments
have circular shape (left panel), the region of DARWIN does not (right panel).
The reason for these shapes is the following: If one substitutes Eq. (8) into
Eq. (3), one sees that the cross section depends on terms which are
proportional to $q_{\nu_{\alpha}}$ and to $q_{\nu_{\alpha}}^{2}$. In the case
of $q_{\nu_{\mu/\tau}}$, the dominating new physics contribution to the cross
section comes always from the terms which are proportional to
$q_{\nu_{\mu/\tau}}^{2}$. Also in the case of electron neutrinos the
dominating contribution comes from the terms which are proportional to
$q_{\nu_{e}}^{2}$ when considering current experiments. Therefore, no
cancellations can occur among the new physics parameters and the preferred
regions obtain the circular shape. However, in the case of DARWIN, which is
sensitive to smaller millicharges, the contributions of both terms are of
similar strength for electron neutrinos. Hence, correlations can appear among
the parameters, which is reflected in the shape of the allowed regions. In our
analyses we have taken the possible correlations among the parameters into
account.

Figure 2: Left: The 90% C.L. (2 d.o.f.) allowed regions in the
$q_{\nu_{e}}-q_{\nu_{\mu}}$-plane from PandaX (magenta), LZ (blue), XENONnT
(orange) and from the combination of all experiments (green) in comparison
with the DARWIN sensitivity assuming 30 ty of exposure. Right: The 90% C.L. (2
d.o.f.) expected sensitivity of DARWIN for different exposures and background
assumptions. The star denotes the SM value.

As can be seen in Figs. 2 and 3, again the PandaX-4T bound is weaker than the
one from LZ, while this one is slightly weaker than that from XENONnT. Due to
the conservative approach of background treatment, our bounds are again
slightly weaker than those obtained in Refs. [18, 30].

At 90% confidence level we obtain from the combined analysis:

$\displaystyle q_{\nu_{e}}$ $\displaystyle\in$ $\displaystyle(-2.0,7.0)\times
10^{-13}~{}e\,,$ (24) $\displaystyle q_{\nu_{\mu/\tau}}$ $\displaystyle\in$
$\displaystyle(-7.5,7.3)\times 10^{-13}~{}e\,,$ (25)
$\displaystyle|q_{\nu_{e\mu/e\tau}}|$ $\displaystyle<$ $\displaystyle
4.1\times 10^{-13}~{}e\,,$ (26) $\displaystyle|q_{\nu_{\mu\tau}}|$
$\displaystyle<$ $\displaystyle 5.2\times 10^{-13}~{}e\,.$ (27)

Our bound on $q_{\nu_{e}}$ is stronger than the bounds from the analyses of
the data from TEXONO ($|q_{\nu_{e}}|<1.0\times 10^{-12}\,e$ at 90% C.L.) [52,
41, 53, 54] or GEMMA ($|q_{\nu_{e}}|<1.5\times 10^{-12}\,e$ at 90% C.L.) [55,
19]. Regarding $q_{\nu_{\mu/\tau}}$, the bound is more than one order of
magnitude more stringent than that obtained from solar neutrinos by the XMASS
collaboration ($|q_{\nu_{\mu/\tau}}|<1.1\times 10^{-11}\,e$ at 90% C.L.) [56].
In the case of the non-diagonal millicharges, we improve the DRESDEN-II
CE$\nu$NS bound [34] on $|q_{\nu_{e\mu/e\tau}}|$ by more than one order of
magnitude and we improve the CE$\nu$NS COHERENT bound [34] on
$|q_{\nu_{\mu\tau}}|$ by about three orders of magnitude.

All of these bounds will be further improved significantly by the DARWIN
experiment, up to a factor of more than 20 in the most optimistic scenario in
the case of $q_{\nu_{e}}$, making dark matter direct detection experiments
again competitive with astrophysical observations which constrain the neutrino
millicharges below a few $10^{-14}\,e$ [57].

Figure 3: $\Delta\chi^{2}$ profiles for the different neutrino millicharges obtained from the analysis of data of PandaX (magenta), LZ (blue), XENONnT (orange) and from the combination of all experiments (green). Also shown is the sensitivity range for DARWIN (red shaded), where the worst (best) case scenario corresponds to the analysis with 30 ty (300 ty) exposure without (with) depleted 136Xe background. The dashed line corresponds to 300 ty exposure without depleted 136Xe background. Experiment | $q_{\nu_{e}}~{}[10^{-13}~{}e]$ | $q_{\nu_{\mu}}~{}[10^{-13}~{}e]$ | $|q_{\nu_{e\mu/e\tau}}|~{}[10^{-13}~{}e]$ | $|q_{\nu_{\mu\tau}}|~{}[10^{-13}~{}e]$
---|---|---|---|---
PandaX-4T | $(-12.6,16.4)$ | $(-22.3,22.2)$ | $<12.2$ | $<15.7$
LZ | $(-4.6,9.9)$ | $(-11.5,11.3)$ | $<6.3$ | $<8.1$
XENONnT | $(-2.5,7.4)$ | $(-8.1,8.0)$ | $<4.4$ | $<5.7$
combined | $(-2.0,7.0)$ | $(-7.5,7.3)$ | $<4.1$ | $<5.2$
DARWIN 30 ty | $(-0.4,1.0)$ | $(-4.1,4.1)$ | $<2.3$ | $<2.9$
DARWIN 300 ty | $(-0.2,0.4)$ | $(-2.4,2.5)$ | $<1.3$ | $<1.7$
DARWIN 300 ty depl. | $(-0.1,0.3)$ | $(-2.2,2.3)$ | $<1.2$ | $<1.6$
Table 2: The 90% bounds ($\Delta\chi^{2}=2.71$) that can be obtained on the
different neutrino millicharges.

###  IV.3 Neutrino charge radius

Finally, we can use the data to place bounds on the neutrino charge radii.
Regarding the charge radii, similar correlations as discussed in the context
of millicharges have to be taken into account, as can be seen in Fig. 4.

Figure 4: Left: The 90% C.L. (2 d.o.f.) allowed regions in the $\langle
r_{\nu_{e}}^{2}\rangle-\langle r_{\nu_{\mu}}^{2}\rangle$-plane from PandaX
(magenta), LZ (blue), XENONnT (orange) and from the combination of all
experiments (green) in comparison with the DARWIN sensitivity assuming 30 ty
of exposure. Right: The 90% C.L. (2 d.o.f.) expected sensitivity of DARWIN in
the $\langle r_{\nu_{e}}^{2}\rangle-\langle r_{\nu_{\mu}}^{2}\rangle$-plane
for different exposures and background assumptions. The star denotes the SM
value, while the black circle denotes the position of a secondary minimum
found in the analysis.

The results of the analyses of PandaX-4T, LZ and XENONnT and from the combined
analysis are shown in Table 3, in the left panels of Figs. 4 and 5 for the
diagonal charge radii, and Fig. 6 for the non-diagonal ones. As can be seen in
Table 3, the bound on $\langle r_{\nu_{e}}^{2}\rangle$ is stronger than that
on $\langle r_{\nu_{\mu/\tau}}^{2}\rangle$. Unfortunately, current dark matter
direct detection experiments are not competitive with the bounds from other
experiments [34, 58, 42] for neither of the two charge radii and they are far
from the Standard Model values in Eqs. (11)–(13). Also the bounds on the non-
diagonal parameters are weaker than those obtained in CE$\nu$NS experiments
[34].

However, we have found that DARWIN will improve significantly the allowed
region of parameter space. In the left panel of Fig. 4 we show the expected
allowed region from DARWIN for a 30 ty exposure. As can be seen, DARWIN could
significantly reduce the volume of the allowed parameter space. With this
relatively small exposure it would be possible to set the strongest upper
limit on $\langle r_{\nu_{e}}^{2}\rangle$, although the lower limit would
remain weaker than the most stringent current bound obtained in the TEXONO
experiment ($\langle r_{\nu_{e}}^{2}\rangle\in(-4.2,6.6)\times
10^{-32}~{}\text{cm}^{2}$ at 90% C.L.) [58]. The size of the sensitivity
region shrinks when considering a larger exposure or the better background
model, as shown in the right panel of Fig. 4. In the analysis with a 300 ty
exposure, values around $\langle r_{\nu_{e}}^{2}\rangle\approx-10\times
10^{-32}~{}\text{cm}^{2}$ and $\langle
r_{\nu_{\mu/\tau}}^{2}\rangle\approx-35\times 10^{-32}~{}\text{cm}^{2}$ become
more disfavored, but there is still a secondary minimum present, as can be
seen in the right panels of Figs. 4 and 5. The secondary solution remains even
when considering the better background model. The bounds that could be
obtained at 90% confidence level are

$\displaystyle\langle r_{\nu_{e}}^{2}\rangle$ $\displaystyle\in$
$\displaystyle(-45.3,0.6)\times 10^{-32}~{}\text{cm}^{2},\text{ DARWIN
30~{}ty}\,,$ (28) $\displaystyle\langle r_{\nu_{e}}^{2}\rangle$
$\displaystyle\in$
$\displaystyle\left\\{(-32.9,-14.8)~{}\&~{}(-3.6,-0.2)\right\\}\times
10^{-32}~{}\text{cm}^{2},\text{ DARWIN 300~{}ty}\,,$ (29)
$\displaystyle\langle r_{\nu_{e}}^{2}\rangle$ $\displaystyle\in$
$\displaystyle\left\\{(-29.1,-20.7)~{}\&~{}(-1.6,-0.3)\right\\}\times
10^{-32}~{}\text{cm}^{2},\text{ DARWIN 300~{}ty, depleted}\,.$ (30)

Thus, DARWIN 300 ty could improve the current best limit on $\langle
r_{\nu_{e}}^{2}\rangle$ obtained in the TEXONO experiment ($\langle
r^{2}\rangle_{\nu_{e}}\in(-4.2,6.6)\times 10^{-32}$ at 90% C.L.) [58] (taking
into account that the secondary solution is excluded by TEXONO and other
bounds; see, e.g., Refs. [1, 2, 7, 16]) and could indicate that $\langle
r_{\nu_{e}}^{2}\rangle$ is negative. This would be the first indication of a
non-zero value of a neutrino charge radius, in agreement with the Standard
Model prediction (11).

In the lower panels of Fig. 5 we show the sensitivity of DARWIN to the charge
radius $\langle r_{\nu_{\mu}}^{2}\rangle$. Although not as strong as for
$\langle r_{\nu_{e}}^{2}\rangle$, DARWIN could provide strong bounds on this
quantity, which read at 90% confidence level

$\displaystyle\langle r_{\nu_{\mu}}^{2}\rangle$ $\displaystyle\in$
$\displaystyle(-62.9,30.4)\times 10^{-32}~{}\text{cm}^{2},\text{ DARWIN
30~{}ty}\,,$ (31) $\displaystyle\langle r_{\nu_{\mu}}^{2}\rangle$
$\displaystyle\in$
$\displaystyle\left\\{(-59.5,-44.6)~{}\&~{}(-19.9,11.7)\right\\}\times
10^{-32}~{}\text{cm}^{2},\text{ DARWIN 300~{}ty}\,,$ (32)
$\displaystyle\langle r_{\nu_{\mu}}^{2}\rangle$ $\displaystyle\in$
$\displaystyle\left\\{(-57.8,-51.4)~{}\&~{}(-8.6,5.7)\right\\}\times
10^{-32}~{}\text{cm}^{2},\text{ DARWIN 300~{}ty, depleted}\,.$ (33)

Such bounds would be complementary to the bounds obtained in other experiments
[34, 42, 59, 60]. Note that the secondary minimum obtained in our DARWIN
analyses requires both charge radii to be different from the Standard Model
values. We could eliminate the secondary minimum by using external constraints
on $\langle r_{\nu_{e}}^{2}\rangle$ (e.g the TEXONO bound [58]) or $\langle
r_{\nu_{\mu}}^{2}\rangle$ (e.g. the one from Ref. [42]).

Figure 5: $\Delta\chi^{2}$ profiles for the diagonal charge radii obtained
from the analyses of current data (left panels) and the DARWIN sensitivity
(right panels). For comparison, we also show in the left panels the DARWIN-
sensitivity corresponding to an exposure of 30 ty.

Regarding the transition charge radii, the DARWIN sensitivity is shown in
comparison with the current bounds in Fig. 6 and in the last two columns of
Table 3. In this case the bounds that DARWIN could set assuming an exposure of
30 ty are similar in strength to those obtained with CE$\nu$NS in Ref. [34].
Only with a larger exposure and a better background model the current bounds
could be improved by a factor of about 2–3.

Figure 6: $\Delta\chi^{2}$ profiles for the non-diagonal charge radii obtained from the analyses of current data (left panels) and the DARWIN sensitivity (right panels). For comparison, we also show in the left panels the DARWIN-sensitivity corresponding to an exposure of 30 ty. Experiment |  $\langle r_{\nu_{e}}^{2}\rangle~{}[10^{-32}]$ cm2 |  $\langle r_{\nu_{\mu/\tau}}^{2}\rangle~{}[10^{-32}]$ cm2 |  $|\langle r_{\nu_{e\mu/e\tau}}^{2}\rangle|~{}[10^{-32}]$ cm2 |  $|\langle r_{\nu_{\mu\tau}}^{2}\rangle|~{}[10^{-32}]$ cm2
---|---|---|---|---
PandaX-4T | $(-134.5,48.2)$ | $(-135.3,141.3)$ | $<76.2$ | $<97.8$
LZ | $(-110.4,26.4)$ | $(-101.8,105.5)$ | $<57.1$ | $<73.3$
XENONnT | $(-113.7,34.1)$ | $(-112.9,112.3)$ | $<62.2$ | $<79.9$
combined | $(-99.5,12.8)$ | $(-82.2,88.7)$ | $<47.3$ | $<60.7$
DARWIN 30 ty | $(-45.3,0.6)$ | $(-62.9,30.4)$ | $<28.6$ | $<37.9$
DARWIN 300 ty |  $(-32.9,-14.8)$ & $(-3.6,-0.2)$ |  $(-59.5,-44.6)$ & $(-19.9,11.7)$ | $<28.6$ | $<37.9$
DARWIN 300 ty depl. |  $(-29.1,-20.7)$ &$(-1.6,-0.3)$ |  $(-57.8,-51.4)$&$(-8.6,5.7)$ | $<12.0$ | $<15.7$
Table 3: The 90% bounds ($\Delta\chi^{2}=2.71$) that can be obtained on the
different neutrino charge radii.

##  V Conclusions

In this paper we have analyzed the electron recoil data from the dark matter
direct detection experiments PandaX-4T, LZ and XENONnT to place constraints on
neutrino electromagnetic properties. We also explored the sensitivity of the
future experiment DARWIN. We implemented a more realistic treatment of
systematic uncertainties than in other phenomenological analyses previously
performed in the literature, following closer what is done by the
collaborations.

We have set strong limits on the neutrino magnetic moments and on the neutrino
millicharges. These are still weaker than those from astrophysical probes.
However, we have shown that the next generation experiment DARWIN will put the
sensitivity of dark matter direct detection experiments into the same
ballpark.

We have shown that in the case of the neutrino charge radii current dark
matter direct detection experiments are not competitive with other types of
experiments. However, this situation will change dramatically for DARWIN.
Under ideal circumstances a measurement of $\langle r_{\nu_{e}}^{2}\rangle$
could be possible for the first time.

An interesting feature of dark matter direct detection experiments which can
detect solar neutrinos is that part of the neutrinos arrives as $\nu_{\tau}$,
hence allowing to set bounds also on the $\nu_{\tau}$ related quantities,
which is not possible with other experiments.

Summarizing, we have shown that current and particularly future dark matter
direct detection experiments provide a powerful tool to test neutrino
electromagnetic properties.

## Acknowledgments

We are very thankful to Dimitris Papoulias for helful discussions. C.G. and
C.A.T. acknowledge support from Departments of Excellence grant awarded by
MIUR and the research grant TAsP (Theoretical Astroparticle Physics) funded by
Istituto Nazionale di Fisica Nucleare (INFN).

## References

  * [1] C. Giunti and A. Studenikin, “Neutrino electromagnetic interactions: a window to new physics,” Rev. Mod. Phys. 87 (2015) 531, arXiv:1403.6344 [hep-ph].
  * [2] C. Giunti, K. A. Kouzakov, Y.-F. Li, A. V. Lokhov, A. I. Studenikin, and S. Zhou, “Electromagnetic neutrinos in laboratory experiments and astrophysics,” Annalen Phys. 528 (2016) 198–215, arXiv:1506.05387 [hep-ph].
  * [3] PandaX Collaboration, D. Zhang et al., “Search for Light Fermionic Dark Matter Absorption on Electrons in PandaX-4T,” Phys. Rev. Lett. 129 no. 16, (2022) 161804, arXiv:2206.02339 [hep-ex].
  * [4] LZ Collaboration, J. Aalbers et al., “First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment,” Phys. Rev. Lett. 131 no. 4, (2023) 041002, arXiv:2207.03764 [hep-ex].
  * [5] XENON Collaboration, E. Aprile et al., “Search for New Physics in Electronic Recoil Data from XENONnT,” Phys. Rev. Lett. 129 no. 16, (2022) 161805, arXiv:2207.11330 [hep-ex].
  * [6] DARWIN Collaboration, J. Aalbers et al., “Solar neutrino detection sensitivity in DARWIN via electron scattering,” Eur. Phys. J. C 80 no. 12, (2020) 1133, arXiv:2006.03114 [physics.ins-det].
  * [7] C. Giunti, J. Gruszko, B. Jones, L. Kaufman, D. Parno, and A. Pocar, “Report of the Topical Group on Neutrino Properties for Snowmass 2021,” arXiv:2209.03340 [hep-ph].
  * [8] J. Bahcall. http://www.sns.ias.edu/~jnb/SNdata/sndata.html.
  * [9] J. N. Bahcall and R. K. Ulrich, “Solar Models, Neutrino Experiments and Helioseismology,” Rev. Mod. Phys. 60 (1988) 297–372.
  * [10] J. N. Bahcall, “The Be-7 solar neutrino line: A Reflection of the central temperature distribution of the sun,” Phys. Rev. D 49 (1994) 3923–3945, arXiv:astro-ph/9401024.
  * [11] J. N. Bahcall, E. Lisi, D. E. Alburger, L. De Braeckeleer, S. J. Freedman, and J. Napolitano, “Standard neutrino spectrum from B-8 decay,” Phys. Rev. C 54 (1996) 411–422, arXiv:nucl-th/9601044.
  * [12] F. L. Villante and A. Serenelli, “The relevance of nuclear reactions for Standard Solar Models construction,” Front. Astron. Space Sci. 7 (2021) 112, arXiv:2101.03077 [astro-ph.SR].
  * [13] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martinez-Mirave, O. Mena, C. A. Ternes, M. Tortola, and J. W. F. Valle, “2020 Global reassessment of the neutrino oscillation picture,” JHEP 2021 (2020) 071, arXiv:2006.11237 [hep-ph].
  * [14] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” JHEP 09 (2020) 178, arXiv:2007.14792 [hep-ph].
  * [15] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri, and A. Palazzo, “The unfinished fabric of the three neutrino paradigm,” Phys.Rev.D 104 (7, 2021) 083031, arXiv:2107.00532 [hep-ph].
  * [16] Particle Data Group Collaboration, R. L. Workman, “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * [17] S. A. Fayans, L. A. Mikaelyan, and V. V. Sinev, “Weak and magnetic inelastic scattering of anti-neutrinos on atomic electrons,” Phys. Atom. Nucl. 64 (2001) 1475–1480, hep-ph/0004158.
  * [18] M. Atzori Corona, W. M. Bonivento, M. Cadeddu, N. Cargioli, and F. Dordei, “New constraint on neutrino magnetic moment and neutrino millicharge from LUX-ZEPLIN dark matter search results,” Phys. Rev. D 107 no. 5, (2023) 053001, arXiv:2207.05036 [hep-ph].
  * [19] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. Medvedev, V. S. Pogosov, M. V. Shirchenko, and A. S. Starostin, “The results of search for the neutrino magnetic moment in GEMMA experiment,” Adv. High Energy Phys. 2012 (2012) 350150.
  * [20] K. Fujikawa and R. Shrock, “The Magnetic Moment of a Massive Neutrino and Neutrino Spin Rotation,” Phys. Rev. Lett. 45 (1980) 963.
  * [21] P. B. Pal and L. Wolfenstein, “Radiative Decays of Massive Neutrinos,” Phys. Rev. D25 (1982) 766.
  * [22] R. E. Shrock, “Electromagnetic properties and decays of Dirac and Majorana neutrinos in a general class of gauge theories,” Nucl. Phys. B206 (1982) 359.
  * [23] J. Bernabeu, L. G. Cabral-Rosetti, J. Papavassiliou, and J. Vidal, “On the charge radius of the neutrino,” Phys. Rev. D62 (2000) 113012, hep-ph/0008114.
  * [24] J. Bernabeu, J. Papavassiliou, and J. Vidal, “On the observability of the neutrino charge radius,” Phys. Rev. Lett. 89 (2002) 101802, hep-ph/0206015.
  * [25] J. Bernabeu, J. Papavassiliou, and J. Vidal, “The neutrino charge radius is a physical observable,” Nucl. Phys. B680 (2004) 450, hep-ph/0210055.
  * [26] M. Cadeddu, C. Giunti, K. A. Kouzakov, Y. F. Li, A. I. Studenikin, and Y. Y. Zhang, “Neutrino Charge Radii from COHERENT Elastic Neutrino-Nucleus Scattering,” Phys.Rev. D98 (2018) 113010, arXiv:1810.05606 [hep-ph].
  * [27] M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, and Y. Y. Zhang, “Neutrino, Electroweak and Nuclear Physics from COHERENT Elastic Neutrino-Nucleus Scattering with Refined Quenching Factor,” Phys.Rev. D101 (2020) 033004, arXiv:1908.06045 [hep-ph].
  * [28] LZ Collaboration, J. Aalbers et al., “Background determination for the LUX-ZEPLIN dark matter experiment,” Phys. Rev. D 108 no. 1, (2023) 012010, arXiv:2211.17120 [hep-ex].
  * [29] LZ Collaboration, J. Aalbers et al., “A search for new physics in low-energy electron recoils from the first LZ exposure,” arXiv:2307.15753 [hep-ex].
  * [30] S. K. A., A. Majumdar, D. K. Papoulias, H. Prajapati, and R. Srivastava, “Implications of first LZ and XENONnT results: A comparative study of neutrino properties and light mediators,” Phys. Lett. B 839 (2023) 137742, arXiv:2208.06415 [hep-ph].
  * [31] XENON Collaboration, E. Aprile et al., “Excess electronic recoil events in XENON1T,” Phys. Rev. D 102 no. 7, (2020) 072004, arXiv:2006.09721 [hep-ex].
  * [32] PandaX-II Collaboration, X. Zhou et al., “A Search for Solar Axions and Anomalous Neutrino Magnetic Moment with the Complete PandaX-II Data,” Chin. Phys. Lett. 38 no. 1, (2021) 011301, arXiv:2008.06485 [hep-ex]. [Erratum: Chin.Phys.Lett. 38, 109902 (2021)].
  * [33] A. N. Khan, “Light new physics and neutrino electromagnetic interactions in XENONnT,” Phys. Lett. B 837 (2023) 137650, arXiv:2208.02144 [hep-ph].
  * [34] M. Atzori Corona, M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, C. A. Ternes, and Y. Y. Zhang, “Impact of the Dresden-II and COHERENT neutrino scattering data on neutrino electromagnetic properties and electroweak physics,” JHEP 09 (2022) 164, arXiv:2205.09484 [hep-ph].
  * [35] P. Coloma, I. Esteban, M. C. Gonzalez-Garcia, L. Larizgoitia, F. Monrabal, and S. Palomares-Ruiz, “Bounds on new physics with data of the Dresden-II reactor experiment and COHERENT,” JHEP 05 (2022) 037, arXiv:2202.10829 [hep-ph].
  * [36] J. Liao, H. Liu, and D. Marfatia, “Implications of the first evidence for coherent elastic scattering of reactor neutrinos,” Phys. Rev. D 106 no. 3, (2022) L031702, arXiv:2202.10622 [hep-ph].
  * [37] A. N. Khan, “Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data,” Nucl. Phys. B 986 (2023) 116064, arXiv:2201.10578 [hep-ph].
  * [38] V. De Romeri, O. G. Miranda, D. K. Papoulias, G. Sanchez Garcia, M. Tórtola, and J. W. F. Valle, “Physics implications of a combined analysis of COHERENT CsI and LAr data,” JHEP 04 (2023) 035, arXiv:2211.11905 [hep-ph].
  * [39] P. Coloma, P. Coloma, M. C. Gonzalez-Garcia, M. C. Gonzalez-Garcia, M. Maltoni, M. Maltoni, J. a. P. Pinheiro, J. a. P. Pinheiro, S. Urrea, and S. Urrea, “Constraining new physics with Borexino Phase-II spectral data,” JHEP 07 (2022) 138, arXiv:2204.03011 [hep-ph]. [Erratum: JHEP 11, 138 (2022)].
  * [40] MUNU Collaboration, Z. Daraktchieva et al., “Final results on the neutrino magnetic moment from the MUNU experiment,” Phys. Lett. B 615 (2005) 153–159, arXiv:hep-ex/0502037.
  * [41] TEXONO Collaboration, H. T. Wong et al., “A Search of Neutrino Magnetic Moments with a High-Purity Germanium Detector at the Kuo-Sheng Nuclear Power Station,” Phys. Rev. D 75 (2007) 012001, arXiv:hep-ex/0605006.
  * [42] L. A. Ahrens et al., “Determination of electroweak parameters from the elastic scattering of muon-neutrinos and anti-neutrinos on electrons,” Phys. Rev. D 41 (1990) 3297–3316.
  * [43] R. C. Allen et al., “Study of electron-neutrino electron elastic scattering at LAMPF,” Phys. Rev. D 47 (1993) 11–28.
  * [44] LSND Collaboration, L. B. Auerbach et al., “Measurement of electron - neutrino - electron elastic scattering,” Phys. Rev. D 63 (2001) 112001, arXiv:hep-ex/0101039.
  * [45] DONUT Collaboration, R. Schwienhorst et al., “A New upper limit for the tau - neutrino magnetic moment,” Phys. Lett. B 513 (2001) 23–29, arXiv:hep-ex/0102026.
  * [46] B. C. Canas, O. G. Miranda, A. Parada, M. Tortola, and J. W. F. Valle, “Updating neutrino magnetic moment constraints,” Phys. Lett. B 753 (2016) 191–198, arXiv:1510.01684 [hep-ph]. [Addendum: Phys.Lett.B 757, 568–568 (2016)].
  * [47] A. Ayala, J. C. D’Olivo, and M. Torres, “Bound on the neutrino magnetic moment from chirality flip in supernovae,” Phys. Rev. D 59 (1999) 111901, arXiv:hep-ph/9804230.
  * [48] N. Viaux, M. Catelan, P. B. Stetson, G. Raffelt, J. Redondo, A. A. R. Valcarce, and A. Weiss, “Neutrino and axion bounds from the globular cluster M5 (NGC 5904),” Phys. Rev. Lett. 111 (2013) 231301, arXiv:1311.1669 [astro-ph.SR].
  * [49] A. H. Córsico, L. G. Althaus, M. M. Miller Bertolami, S. O. Kepler, and E. García-Berro, “Constraining the neutrino magnetic dipole moment from white dwarf pulsations,” JCAP 08 (2014) 054, arXiv:1406.6034 [astro-ph.SR].
  * [50] F. Capozzi and G. Raffelt, “Axion and neutrino bounds improved with new calibrations of the tip of the red-giant branch using geometric distance determinations,” Phys. Rev. D 102 no. 8, (2020) 083007, arXiv:2007.03694 [astro-ph.SR].
  * [51] K. Mori, M. Kusakabe, A. B. Balantekin, T. Kajino, and M. A. Famiano, “Enhancement of Lithium in Red Clump Stars by the Additional Energy Loss Induced by New Physics,” Mon. Not. Roy. Astron. Soc. 503 no. 2, (2021) 2746–2753, arXiv:2009.00293 [astro-ph.SR].
  * [52] TEXONO Collaboration, H. B. Li et al., “Limit on the electron neutrino magnetic moment from the Kuo-Sheng reactor neutrino experiment,” Phys. Rev. Lett. 90 (2003) 131802, arXiv:hep-ex/0212003.
  * [53] S. N. Gninenko, N. V. Krasnikov, and A. Rubbia, “Search for millicharged particles in reactor neutrino experiments: A Probe of the PVLAS anomaly,” Phys. Rev. D 75 (2007) 075014, arXiv:hep-ph/0612203.
  * [54] J.-W. Chen, H.-C. Chi, H.-B. Li, C. P. Liu, L. Singh, H. T. Wong, C.-L. Wu, and C.-P. Wu, “Constraints on millicharged neutrinos via analysis of data from atomic ionizations with germanium detectors at sub-keV sensitivities,” Phys. Rev. D 90 no. 1, (2014) 011301, arXiv:1405.7168 [hep-ph].
  * [55] A. Studenikin, “New bounds on neutrino electric millicharge from limits on neutrino magnetic moment,” EPL 107 no. 2, (2014) 21001, arXiv:1302.1168 [hep-ph]. [Erratum: EPL 107, 39901 (2014), Erratum: Europhys.Lett. 107, 39901 (2014)].
  * [56] XMASS Collaboration, K. Abe et al., “Search for exotic neutrino-electron interactions using solar neutrinos in XMASS-I,” Phys. Lett. B 809 (2020) 135741, arXiv:2005.11891 [hep-ex].
  * [57] G. G. Raffelt, “Limits on neutrino electromagnetic properties: An update,” Phys. Rept. 320 (1999) 319–327.
  * [58] TEXONO Collaboration, M. Deniz et al., “Measurement of Nu(e)-bar -Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor,” Phys. Rev. D 81 (2010) 072001, arXiv:0911.1597 [hep-ex].
  * [59] CHARM-II Collaboration, P. Vilain et al., “Experimental study of electromagnetic properties of the muon-neutrino in neutrino - electron scattering,” Phys. Lett. B 345 (1995) 115–118.
  * [60] A. N. Khan, “$\sin^{2}\theta_{W}$ Estimate and Neutrino Electromagnetic Properties from Low-Energy Solar Data,” J. Phys. G 46 no. 3, (2019) 035005, arXiv:1709.02930 [hep-ph].


# KG-EmpiRE: A Community-Maintainable Knowledge Graph for a Sustainable
Literature Review on the State and Evolution of Empirical Research in
Requirements Engineering

Oliver Karras1  1TIB - Leibniz Information Centre for Science and Technology,
Germany, Email<EMAIL_ADDRESS>
###### Abstract

In the last two decades, several researchers provided snapshots of the
“current” state and evolution of empirical research in requirements
engineering (RE) through literature reviews. However, these literature reviews
were not sustainable, as none built on or updated previous works due to the
unavailability of the extracted and analyzed data. KG-EmpiRE is a Knowledge
Graph (KG) of empirical research in RE based on scientific data extracted from
currently 680 papers published in the IEEE International Requirements
Engineering Conference (1994-2022). KG-EmpiRE is maintained in the Open
Research Knowledge Graph (ORKG), making all data openly and long-term
available according to the FAIR data principles. Our long-term goal is to
constantly maintain KG-EmpiRE with the research community to synthesize a
comprehensive, up-to-date, and long-term available overview of the state and
evolution of empirical research in RE. Besides KG-EmpiRE, we provide its
analysis with all supplementary materials in a repository. This repository
contains all files with instructions for replicating and (re-)using the
analysis locally or via executable environments and for repeating the research
approach. Since its first release based on 199 papers (2014-2022), KG-EmpiRE
and its analysis have been updated twice, currently covering over 650 papers.
KG-EmpiRE and its analysis demonstrate how innovative infrastructures, such as
the ORKG, can be leveraged to make data from literature reviews FAIR, openly
available, and maintainable for the research community in the long term. In
this way, we can enable replicable, (re-)usable, and thus sustainable
literature reviews to ensure the quality, reliability, and timeliness of their
research results.

###### Index Terms:

Knowledge graph, empirical research, requirements engineering, FAIR,
sustainability, literature review

##  I Introduction

For 20 years, various researchers conducted literature reviews to examine the
state and evolution of empirical research in requirements engineering (RE)
with the shared goal of providing a comprehensive, up-to-date, and long-term
available overview [1, 2]. However, these literature reviews were not
sustainable, as none built on or updated previous ones, which are known
challenges of literature reviews [3]. While recent research addresses these
challenges by providing social and economic decision support and guidance [3],
the underlying problem is the unavailability of the extracted and analyzed
data. Researchers need technical support, i.e., infrastructures, to conduct
sustainable literature reviews so that all data is openly and long-term
available according to the FAIR data principles [3] and corresponding to open
science in SE [4].

In their joint work, Wernlein [5] and Karras et al. [1, 6] examined the use of
the Open Research Knowledge Graph (ORKG) [7], as such technical support by
building, publishing, and analyzing a Knowledge Graph (KG) of empirical
research in RE (KG-EmpiRE) based on currently 680 research track papers of the
IEEE International Requirements Engineering Conference (1994-2022).

In this paper, we present the KG-EmpiRE, available in the
ORKG111https://orkg.org/observatory/Empirical_Software_Engineering, and its
analysis, available on GitHub [8], Zenodo [9], and on
Binder222https://tinyurl.com/empire-analysis for interactive replication and
(re-)use.

KG-EmpiRE contains scientific data on the six themes research paradigm,
research design, research method, data collection, data analysis, and
bibliographic metadata. We plan to expand these themes in the long term. For
more details on the themes, refer to the supplementary materials [9, 8]. Since
its first release based on 199 papers (2014-2022) [5], KG-EmpiRE and its
analysis have been updated twice. Karras et al. [1] published the first update
with 570 papers (2000-2022) at the 17th ACM/IEEE International Symposium on
Empirical Software Engineering and Measurement 2023, where they received the
best paper award. The second update is ongoing and covers 680 papers
(1994-2022) so far. The goal for the second update is to cover all 748
research track papers from the IEEE International Requirements Engineering
Conference (1993-2023).

The analysis provides answers to 16 out of 77 competency questions (cf.
supplementary materials [9, 8]) regarding empirical research in RE that we
derived from the vision of Sjøberg et al. [10] on the role of empirical
methods in SE, including RE, for 2020-2025. While the number of competency
questions answered reflects the coverage of the curated topic in KG-EmpiRE,
the answers to competency questions provide insights into the state and
evolution of empirical research in RE. For each competency question answered,
we provide all details of the analysis with its data, visualizations,
explanations, and answers in a repository [9, 8] that is also hosted on Binder
for interactive replication and (re-)use.

Overall, this repository contains all files with detailed explanations and
instructions for replication and (re-)use of KG-EmpiRE and its analysis
locally or via executable environments (Binder and GitHub Codespaces), as well
as for repeating the research approach for sustainable literature reviews with
the ORKG. The repository also contains all generated visualizations with their
data, exported as PNG and CSV files, as well as supplementary materials on the
themes, their structuring in the ORKG, and all 77 competency questions.

##  II Structure of KG-EmpiRE and the Repository

###  II-A KG-EmpiRE

We developed an ORKG template333https://orkg.org/template/R186491 to organize
the scientific data extracted from the papers in the ORKG. ORKG templates
implement a subset of the Shapes Constraint Language (SHACL) and allow
specifying the underlying (graph) structure to organize the data in a
structured manner [11]. In this way, we determined which data to extract and
standardized their description to ensure they are FAIR, consistent, and
comparable across all papers. The developed ORKG template covers the six
themes investigated. For more details on the ORKG template, refer to the
supplementary materials [9, 8].

By applying the ORKG template to the papers, KG-EmpiRE currently consists of
almost 35,000 triples, which are made up of over 51,000 resources and almost
19,000 literals (see Table I). While these statistics reflect the efforts to
provide a solid structured description of the extracted data, they also show
that KG-EmpiRE is relatively small compared to the entire ORKG and other well-
known knowledge graphs, e.g., Wikidata or DBpedia, which include millions of
entities.

Table I: Statistics of KG-EmpiRE and the ORKG on 30.04.24. Type | Paper | Triples | Resources | Literals
---|---|---|---|---
KG-EmpiRE | 680 | 34,912 | 51,035 | 18,789
ORKG | 29,379 | 1,407,164 | 432,323 | 768,212

###  II-B Repository

In the repository, there are three folders and six files, with the Jupyter
Notebook empire-analysis.ipynb as the main file. The Jupyter Notebook
encapsulates the entire analysis of KG-EmpiRE and provides visualizations,
explanations, and answers for each of the 16 competency questions. The
visualizations are exported as PNG files per competency question to the
Figures folder. The data retrieved by KG-EmpiRE for analysis is stored as CSV
files for each competency question in the SPARQL-Data folder by date. In this
folder, we also provide CSV files of the latest release to replicate the
results of the related publication [1]. The last folder Supplementary
materials provides additional materials for detailed overviews of the content
for data extraction regarding the themes, the developed ORKG template, all 77
competency questions derived, and the research approach. The second most
important file is README.md, which contains detailed explanations and
instructions about the project, the repository, its installation (locally and
via executable environments), the replication of the analysis, and the (re)use
of KG-EmpiRE with its most recent data. The remaining four files support the
installation (requirements.txt, runtime.txt), clarify the copyright (LICENSE),
and ensure the citability of the repository (CITATION.cff)444https://citation-
file-format.github.io/.

##  III Conclusion

Overall, KG-EmpiRE and its analysis lay the foundation for a sustainable
literature review on the state and evolution of empirical research in
requirements engineering. They can be used to replicate the results from the
related publication [1], (re-)use the data for further studies, and repeat the
research approach for sustainable literature reviews on other topics. KG-
EmpiRE and its analysis demonstrate how innovative infrastructures, such as
the ORKG, can be leveraged to make data from literature reviews FAIR and
openly available in the long term. In this way, researchers can build on and
update the data ideally collaboratively, enabling sustainable literature
reviews for comprehensive, up-to-date, and long-term available overviews, true
to the principle: Divide et Impera.

In summary, the special feature of KG-EmpiRE lies in the proof that data from
literature reviews can already be prepared during data extraction in such a
way that they are understandable and processable by humans and machines to
update, replicate, and (re-)use them sustainably. KG-EmpiRE and the underlying
research approach using technical infrastructures, such as the ORKG, have the
potential to be used on a large scale to establish sustainable literature
reviews and thus ensure the quality, reliability, and timeliness of their
research results.

## Acknowledgment

The authors thank the Federal Government, the Heads of Government of the
Länder, as well as the Joint Science Conference (GWK), for their funding and
support within the NFDI4Ing and NFDI4DataScience consortia. This work was
funded by the German Research Foundation (DFG) project numbers 44214671 and
460234259 and by the European Research Council for the project ScienceGRAPH
(Grant agreement ID: 819536).

## References

  * [1] O. Karras, F. Wernlein, J. Klünder, and S. Auer, “Divide and Conquer the EmpiRE: A Community-Maintainable Knowledge Graph of Empirical Research in Requirements Engineering,” in _International Symposium on Empirical Software Engineering and Measurement_ , 2023.
  * [2] O. Karras, F. Wernlein, J. Klünder, and S. Auer, “A Comparison of Scientific Publications on the State of Empirical Research in Requirements Engineering and Software Engineering,” 2023. [Online]. Available: https://orkg.org/comparison/R650023/
  * [3] V. dos Santos _et al._ , “Towards Sustainability of Systematic Literature Reviews,” in _15th ACM/IEEE International Symposium on Empirical Software Engineering and Measurement_ , 2021.
  * [4] D. Mendez _et al._ , “Open Science in Software Engineering,” in _Contemporary Empirical Methods in Software Engineering_. Springer, 2020.
  * [5] F. Wernlein, “Acquisition and Analysis of Research Practices Using Semantic Structures,” Bachelor Thesis, Gottfried Wilhelm Leibniz Universität, 2022\. [Online]. Available: https://doi.org/10.15488/12796
  * [6] O. Karras, F. Wernlein, J. Klünder, and S. Auer, “KG-EmpiRE: A Community-Maintainable Knowledge Graph of Empirical Research in Requirements Engineering,” in _Software Engineering 2024_ , 2024.
  * [7] M. Stocker _et al._ , “FAIR Scientific Information with the Open Research Knowledge Graph,” _FAIR Connect_ , vol. 1, no. 1, 2023.
  * [8] O. Karras, “Divide and Conquer the EmpiRE: A Community-Maintainable Knowledge Graph of Empirical Research in Requirements Engineering - A Sustainable Literature Review for Analyzing the State and Evolution of Empirical Research in Requirements Engineering,” 2023. [Online]. Available: https://github.com/okarras/EmpiRE-Analysis
  * [9] ——, “Divide and Conquer the EmpiRE: A Community-Maintainable Knowledge Graph of Empirical Research in Requirements Engineering - A Sustainable Literature Review for Analyzing the State and Evolution of Empirical Research in Requirements Engineering,” 2024. [Online]. Available: https://doi.org/10.5281/zenodo.11092471
  * [10] D. I. Sjoberg _et al._ , “The Future of Empirical Methods in Software Engineering Research,” in _Future of Software Engineering_ , 2007.
  * [11] H. Hussein _et al._ , “Increasing Reproducibility in Science by Interlinking Semantic Artifact Descriptions in a Knowledge Graph,” in _Leveraging Generative Intelligence in Digital Libraries: Towards Human-Machine Collaboration_. Springer, 2023.


11institutetext: Department of Computer Science, University of Manchester,
Manchester, UK
11email<EMAIL_ADDRESS>
{caroline.jay<EMAIL_ADDRESS>22institutetext: Idiap Research
Institute
Rue Marconi 19, Martigny, 1920, Switzerland

# Structuralist analysis for neural network system diagrams

Guy Clarke Marshall 11 Caroline Jay 11 and André Freitas 1122

###### Abstract

This short paper examines diagrams describing neural network systems in
academic conference proceedings. Many aspects of scholarly communication are
controlled, particularly with relation to text and formatting, but often
diagrams are not centrally curated beyond a peer review. Using a corpus-based
approach, we argue that the heterogeneous diagrammatic notations used for
neural network systems has implications for signification in this domain. We
divide this into (i) what content is being represented and (ii) how relations
are encoded. Using a novel structuralist framework, we use a corpus analysis
to quantitatively cluster diagrams according to the author’s representational
choices. This quantitative diagram classification in a heterogeneous domain
may provide a foundation for further analysis.

###### Keywords:

Neural network Scholarly diagrams Empirical Semiotics

##  1 Introduction

Currently, there is no consistent model for visually or formally representing
the architecture of neural networks (NNs). This lack of representation brings
interpretability, correctness and completeness challenges in the description
of existing models and systems engineering. In the context of scientific
communication, most approaches and systems today are described by a
combination of arbitrary diagrammatic elements, algorithms, formulae and
natural language descriptions. In this paradigm, there is little consistency
on abstraction levels or notation. From the perspective of scientific
practice, these limitations challenge dialogue, transparency and
reproducibility.

We will show that authors of applied NN systems papers could be getting more
communicative value from their diagrams. Focusing on the diagrammatic
representations used, it seems the author’s notion of relevance is skewed,
resulting in missing context and content. Diagrams often feature partial
information on, for example, the key novelty or contribution the paper is
articulating. This is unlike other disciplines such as mechanical, electronic,
or structural engineering, where standardised and complete diagrams are
utilised.

This short paper is an empirical analysis based on semiotic theory,
introducing a semiotic viewpoint to the analysis of NN diagrams used at
leading Natural Language Processing (NLP) conferences, applying a novel
structuralist approach, inspired by structural linguistics [8] to survey a
corpus of NN diagrammatic representations. Our contributions are:

  * •

To introduce diagrammatic structuralism as a method for analysing diagrammatic
semiotics.

  * •

To show the heterogeneity of neural network diagrams through an empirical
corpus-based analysis.

  * •

To identify clusters of diagrams through quantitative analysis.

##  2 Related work

###  2.1 Scientific diagrams

In an interview study involving academics about NN system diagrams, Marshall
et al. [10] found differing views on whether precision was meaningful: “If you
have four, it means you have four” contrasts with “it cannot be four
dimensions, it must be much more than that.”

Specific semiotic principles for diagrams can be found in other technical
applications. In the domain of Business Process Mapping, it was found that the
shape of the graphic symbols improved ease of understanding more effectively
than colour or number of symbols [3]. In a small study of mechanical aviation
diagrams, Kim et al. [9] found that highlighting/shading to identify key
components was more usable and useful, compared to other visual techniques
such as “zooming in”. Additionally, Heiser and Tversky [6] showed in
mechanical diagrams that arrows added to static structural (parts and
relations) diagrams increase the capability of the representation to convey
function (temporal, dynamic, or causal process).

###  2.2 Diagrammatic representations

According to Peirce [13], the concept of a diagram as an icon is malleable to
Kant’s concept of Schema (the procedural rule by which a Kantian category is
associated with a sense impression). Applying Saussurian semiotic thinking
beyond the linguistic domain, in a diagram we might wish to minimise the
distance between the signifier and the signified. There is agreement that, at
the holistic and iconic level, a diagram should attempt to represent the thing
it is representing. Tylén et al. [17] concisely summarise that diagrams:

  * •

are external representational support to cognitive processes [2].

  * •

make abstract properties and relations accessible [7].

  * •

can be in a public space, therefore enabling collective and temporally
distributed forms of thinking [13].

  * •

are manipulated in order to profile known information in an optimal fashion.

Diagrams and text, even if together in the same media, are different
modalities. Due to the prevalence of their combined usage, when referencing
diagrams we assume dual coding of text and graphics as part of our working
definition of a diagram. We build on Morris [12] by partitioning semiotics,
the field examining the process by which something functions as a sign, into
the following categories:

  * •

Semantics: relation between signs and the things to which they refer, or their
meaning

  * •

Syntactics: relations among or between signs

  * •

Pragmatics: relation between signs and sign-users [12]

  * •

Empirics: the statistical analysis of repeated use of signs, adapted from the
Information Systems domain [16]

##  3 Neural network diagram empirics

###  3.1 Context

Neural approaches are popular in AI research, and many of the diagrammatic
design decisions are centred on the machine-learning mechanisms. We performed
a corpus-based analysis on the use of diagrams to represent NNs, with NLP as
an application domain.

This study applies an empirical approach to diagrammatic representations,
inspired by structuralist linguistics, in order to uncover the potential
implicit semiotic processes and laws. This approach has a potential
shortcoming in revealing only partial truths [14], but allows us to access a
complex conceptual space. By analogy to Saussure’s 2011 linguistic concept of
parole (speech as concrete instances of the uses of language), the proposed
diagrammatic structuralism purely considers the sign itself and does not
include the act of using it. To assess the similarity of diagrams from a
visual perception standpoint, we should consider the overall structure rather
than substructures [4]. Following Saussure, the American Structuralist school
(e.g. Bloomfield [1]) advocate a mechanistic approach to the analysis of
linguistics, which we apply to NN diagrams. We believe this approach to be
pragmatic and not excessively reductionist, as the establishment of common
graphical elements is beneficial for understanding what is being articulated.

###  3.2 Method

Table 1: Surveyed Papers
Source | Papers | Sampled | Diagrams
---|---|---|---
COLING 2016 | 338 | 16 | 6
COLING 2018 | 330 | 16 | 8
EMNLP 2016 | 275 | 9 | 4
EMNLP 2017 | 323 | 14 | 6
NAACL 2016 | 181 | 8 | 4
NAACL 2018 | 332 | 14 | 7
NIPS 2016 | 569 | 14 | 2
NIPS 2017 | 679 | 23 | 3

In order to map the current state of diagrammatic representations of NNs, we
conducted systematic literature analysis, categorising NN diagrams based on
Diagrammatic and Content features (see Marshall et al. [11] for a complete
list, including observed frequencies). This partition is inspired by
Jakobson’s [8] “axis of selection” and “axis of combination” applied in
structural linguistics. Prior to the sampling, we identified repeated
concepts, divided into “Content” and “Diagrammatic” primitives, a diagrammatic
equivalent of Jakobson’s “selection” and “combination”. The categorisation is
in this way because the sets of symbols and their relationships together can
also form a symbol themselves, in the same way that the set of low-level
symbols come together to for the diagram symbol. This intertwining means it is
not possible to distill semantic and syntactic information independently, as
argued by Jakobson for linguistics.

We then sampled and applied our coding based on content and diagrammatic
features (see Table 1 for content, diagrammatic features table omitted for
brevity), analysing the results.

With a corpus-based approach, we randomly sampled papers and manually
extracted 40 diagrams (see Table 1). Where multiple diagrams were identified,
we have taken the diagram including the most features. We excluded Computer
Vision venues to avoid complication with the visual nature of their input and
high specificity of their architectures, and as such this is not
representative of all scholarly NN diagrams.

The diagrams were assessed by one NN practitioner according to pre-defined
categories covering Content (what features have been selected for diagrammatic
representation) and Diagrammatic concepts (their visual and relational
encoding).

###  3.3 Analysis

The low occurrence of data-centric content features such as operations
indicates that these diagrams are not describing transformations the data
undergoes, but instead prioritise the NN system, particularly the layers,
dimensionality, input and output.

In order to extract 40 diagrams, we sampled 114 papers. Table 1 details the
paper counts from each conference. From papers at these recent top neural
network-centric venues, we expect that at least 25% will contain NN diagrams
(with 99% confidence, due to our sampling of 40 diagrams from 114 sampled
papers from a total population of 3027). Note that the sampling was not
filtered for NN content, and the prevalence of NN diagrams underscores the
community requirement for effective NN diagrams.

Most diagrams are representing the same content. In 82.5% of papers, the role
of NN diagrams in the sample was to communicate the function of the NN, rather
than context, the whole system, or a component. In the sample, no diagram
included identical Diagrammatic features. Two of the sampled diagrams shared
the same Content features (lower semantic content, NN-only diagrams, with an
output and an explicit example). Fig. 2 shows the quantity of semiotic
elements according to our schema, which were found to be normally distributed
(13.07) and have a large standard deviation (3.96). Details are available
[11]. Some highlights:

  * •

92.5% used directional arrows, and 42.5% dotted arrows. Given the utility of
arrows to provide a sense of functional composition, this uniformity is as
expected.

  * •

60% had a caption over one sentence long, perhaps suggesting the authors
struggled to get the relevant content into the diagram.

  * •

60% used an ellipsis in order to indicate “incomplete sets of objects.”

  * •

47.5% included labels on neural layers.

  * •

42.5% made use of an explicit example input.

  * •

52.5% did not indicate dimensionality.

  * •

27.5% indicated the key features by boundary lines, shading or labeling.

  * •

10% indicated data resources of any kind, and a different 10% utilised
symbolic mathematical tensor operation.

  * •

None used UML, SysML, OML or other existing formal diagrammatic language.

Figure 1: Surveyed Semiotic Content of Diagrams

Fig. 1 shows the relationship between NN Content and Diagrammatic features.
Diagrams with additional feature content are more likely to have additional
diagrammatic features. It may be that there is variable author diagramming
efficiency, or that these systems are intrinsically more complex in both
dimensions.

Figure 2: Distribution of the total number of features in each diagram

The study found little evidence of standardisation. In the language of Peirce,
the level of heterogeneity in representing such similar _objects_ implies that
it is more appropriate to consider the components of NN diagrams as _symbols_
rather than _icons_ , since conventions appear to be utilised over physical
resemblance of system or data structures. Despite this heterogeneity, common
themes have been identified in order to classify and group diagram types.
Based on k-means clustering on the 69-dimensional space created by applying
diagrammatic structuralism (24 content and 45 diagrammatic), together with
manual adjustment, we identified four potential NN diagram classes:

  1. 1.

Visual using coloured circles to represent tensors, mostly left-to-right
shape, mostly including an input and output example, a short caption, and
featuring an embedding.

  2. 2.

Mathematical with more mathematical notation, often with an upward linear
shape.

  3. 3.

Lightweight with less content and fewer visual encoding mechanisms, relying on
text, often with labelled layers, and sometimes in a Block diagram format.

  4. 4.

Unorthodox not fitting into the above groups.

Applying this classification manually, we identified 8 Visual, 4 Mathematical,
12 Lightweight and 16 Unorthodox diagrams. This classification may help
structure future research.

###  3.4 Limitations

  * •

The dimensions were chosen to have a good coverage of important features in
the space. Whilst this was intended to be comprehensive, because the
dimensions were identified before the sampling (a methodological choice),
several features were inadvertently omitted. This was partly due to the
heterogeneity of the representations and included colour relationship, the
presence of keys, the usage of standard notation in non-standard ways,
presence of ambiguous syntactics.

  * •

It would be an improvement to have multiple annotators. However, due to the
reasonably unambiguous nature of the assessment, this was not deemed critical
in this early stage of analysis.

  * •

We do not include bibliometrics such as citation count due to the small sample
size and the sensitivity in adjustment for different publication venues.

###  3.5 Discussion

The consequence of variable signification methods within the same domain is
that readers are required to switch modalities even between very similar
papers. Readers may also be unable to access the information they require, as
the diagrammatic signifier may not contain some important Content-related
aspects of the signified system.

We found that additional visual objects were correlated with additional
diagrammatic grammar features. This suggests that either the expressions were
more complex and required both types of encoding, or that some author were
more likely to use a larger variety of encoding objects generally (not
discriminating between objects and relations). We have not shown that this
heterogeneity is a problem per se, but it does differ from many other Computer
Science domains where diagrammatic representation of systems is more
standardised.

Grice [5] distinguishes between “what is said” and “what is implicated”. In
this domain, the figure caption text usually refers to a system rather than a
scholarly contribution. It may be that in these diagrams we are observing an
ambiguity between “what is drawn” and “what is implicated”, even for the
author’s own mind. The system diagram may be functioning as a Peician index
for a scholarly contribution, rather than a system, the functioning of that
system, or the manipulation of data. This ambiguity introduces complexity in
identifying the object of the diagram. Despite this ambiguity, a semiotic lens
has allowed quantitative exploration of diagram components and poses further
questions about the intent of the authors.

##  4 Conclusion

Using a structuralist framework we gathered, sampled and analysed neural
network diagrams from a variety of recent neural network-centric conferences,
using this to demonstrate aspects of the heterogeneity of diagrammatic
representations employed. Using a k-means clustering algorithm on the high-
dimensional visual semiotic space, we derived clusters of NN diagram types.

Beyond diagram classification and hetergeneity quantitification, this work
aims to encourage increase attention on scholarly diagramming practices, which
may help authors to better fulfil their communicative intent.

## References

  * Bloomfield [1984] Bloomfield, L.: Language. University of Chicago Press (1984), ISBN 0226060675
  * Clark and Chalmers [1998] Clark, A., Chalmers, D.: The Extended Mind. Analysis 58(1), 7–19 (1 1998), ISSN 0003-2638, doi:10.1093/analys/58.1.7
  * Gabryelczyk and Jurczuk [2017] Gabryelczyk, R., Jurczuk, A.: Does Experience Matter? Factors Affecting the Understandability of the Business Process Modelling Notation. Procedia Engineering 182, 198–205 (2017), ISSN 18777058, doi:10.1016/j.proeng.2017.03.164
  * Goldmeier [1972] Goldmeier, E.: Similarity in visually perceived forms. Psychological issues 8(1), 1–136 (1972), ISSN 0048-5748, URL http://www.ncbi.nlm.nih.gov/pubmed/4661205
  * Grice [1989] Grice, P.: Studies in the Way of Words. Harvard University Press (1989)
  * Heiser and Tversky [2006] Heiser, J., Tversky, B.: Arrows in Comprehending and Producing Mechanical Diagrams. Tech. rep., Department of Psychology, Stanford University (2006), doi:10.1207/s15516709cog0000–“˙˝70, URL https://www.ncbi.nlm.nih.gov/pubmed/21702825
  * Hutchins [1995] Hutchins, E.: How a Cockpit Remembers Its Speeds. Cognitive Science 19(3), 265–288 (7 1995), ISSN 03640213, doi:10.1207/s15516709cog1903–“˙˝1
  * Jakobson and Halle [2010] Jakobson, R., Halle, M.: Fundamentals of language, vol. 1. Walter de Gruyter (2010)
  * Kim et al. [2010] Kim, S., Woo, I., Maciejewski, R., Ebert, D.S., Ropp, T.D., Thomas, K.: Evaluating the Effectiveness of Visualization Techniques for Schematic Diagrams in Maintenance Tasks. In: Proceedings of the 7th Symposium on Applied Perception in Graphics and Visualization - APGV ’10, p. 33, New York, New York, USA (2010), ISBN 9781450302487, doi:10.00
  * Marshall et al. [2020] Marshall, G., Freitas, A., Jay, C.: How researchers use diagrams in communicating neural network systems. arXiv preprint arXiv:2008.12566 (2020)
  * Marshall et al. [2021] Marshall, G., Freitas, A., Jay, C.: Tables supporting “semiotic classification ofneural network system diagrams” (2021), URL https://aidiagramscom.files.wordpress.com/2021/03/philosophy_paper_tables.pdf
  * Morris [1938] Morris, C.W.: Foundations of the theory of signs. The Journal of Symbolic Logic 3(04), 158 (12 1938), doi:10.2307/2267781
  * Peirce [1966] Peirce, C.S.: The collected papers of Charles S. Peirce. Harvard University Press (1931–1966)
  * Rutherford [2016] Rutherford, D.: How structuralism matters. Journal of Ethnographic Theory 6(3), 61–77 (2016), doi:10.14318/hau6.3.008
  * Saussure et al. [2011] Saussure, F.d., Baskin, W., Meisel, P., Saussy, H.: Course in general linguistics. Columbia University Press (2011), ISBN 0231157274
  * Stamper [1987] Stamper, R.: Semantics. In: Critical issues in information systems research, pp. 43 – 78, John Wiley & Sons, Inc., New York, NY (1987), ISBN 0-471-91281-6
  * Tylén et al. [2014] Tylén, K., Fusaroli, R., Bjørndahl, J.S., Ra̧czaszek-Leonardi, J., Østergaard, S., Stjernfelt, F.: Diagrammatic reasoning Abstraction, interaction, and insight. Pragmatics & Cognition 22(2), 264–283 (2014), ISSN 1569-9943, doi:10.1075/pc.22.2.06tyl