# BSM Targets at a ”Target-less DUNE”

Vedran Brdar  Fermi National Accelerator Laboratory, Batavia, IL, 60510, USA
Northwestern University, Department of Physics & Astronomy, 2145 Sheridan
Road, Evanston, IL 60208, USA  Bhaskar Dutta  Mitchell Institute for
Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas
A&M University, College Station, TX 77843, USA  Wooyoung Jang  Department of
Physics, University of Texas, Arlington, TX 76019, USA  Doojin Kim  Mitchell
Institute for Fundamental Physics and Astronomy, Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA  Ian M.
Shoemaker  Center for Neutrino Physics, Department of Physics, Virginia Tech,
Blacksburg, VA 24061, USA  Zahra Tabrizi  Northwestern University, Department
of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA  Adrian
Thompson  Mitchell Institute for Fundamental Physics and Astronomy, Department
of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
Jaehoon Yu  Department of Physics, University of Texas, Arlington, TX 76019,
USA  Neutrino Department, CERN, Geneva, Switzerland

###### Abstract

In this work we demonstrate that a future accelerator-based neutrino
experiment such as DUNE can greatly increase its sensitivity to a variety of
new physics scenarios by operating in a mode where the proton beam impinges on
a beam dump. We consider two new physics scenarios, namely light dark matter
(LDM) and axion-like particles (ALPs) and show that by utilizing a dump mode
at a DUNE-like experiment, unexplored new regions of parameter space can be
probed with an exposure of only 3 months with half of its expected initial
beam power. Specifically, target-less configuration of future high intensity
neutrino experiments will probe the parameter space for thermal relic DM as
well as the QCD axion (DFSZ and KSVZ). The strength of such configuration in
the context of new physics searches stems from the fact that the neutrino flux
is significantly reduced compared to that of the target, resulting in much
smaller backgrounds from neutrino interactions. We have verified this in
detail by explicitly computing neutrino fluxes which we make publicly
available in order to facilitate further studies with a target-less
configuration.

††preprint: MI-HET-778, FERMILAB-PUB-22-453-T††preprint: NUHEP-TH/22-04

Introduction. The remaining unknowns in the standard three flavor oscillation
paradigm are expected to be measured by the next generation neutrino
experiments among which DUNE Abi  _et al._ (2020a, b, c, d), Hyper-Kamiokande
Abe  _et al._ (2014) and JUNO An  _et al._ (2016) stand out. While
determination of the mass ordering, the octant of the atmospheric mixing angle
and the level of CP violation in the lepton sector are indeed the main
priority, it should be stressed that such experiments can also be utilized in
the context of various new physics searches. Specifically, in addition to new
physics realizations (_e.g._ light eV-scale sterile neutrinos Dasgupta and
Kopp (2021), non-standard interactions Farzan and Tortola (2018) and ultra-
light dark matter (DM) Krnjaic  _et al._ (2018); Brdar  _et al._ (2018)) that
could make a direct impact to the measured oscillations, it was shown
particularly that such experiments will also be very competitive in
complementary beyond the Standard Model (BSM) tests (see for instance section
7 in Argüelles  _et al._ (2020); Abi  _et al._ (2021) and references therein).
Focusing on a liquid Argon near detector, let us stress searches for LDM De
Romeri  _et al._ (2019); Celentano  _et al._ (2020); Breitbach  _et al._
(2021), axion-like particles (ALPs) Brdar  _et al._ (2021); Kelly  _et al._
(2021), new vector bosons Ballett  _et al._ (2019); Berryman  _et al._ (2020);
Dev  _et al._ (2021); Capozzi  _et al._ (2021); Chauhan  _et al._ (2022),
heavy neutral leptons Ballett  _et al._ (2018, 2020); Berryman  _et al._
(2020); Coloma  _et al._ (2021); Atkinson  _et al._ (2021); Abdullahi  _et
al._ (2022) and model-independent BSM searches in the context of the Standard
Model Effective Field Theory Falkowski  _et al._ (2018).

One of the main challenges in performing such BSM searches at neutrino
experiments is that the corresponding signals often appear in the detector
with a very similar signature as neutrino interactions, and hence the
neutrinos act as the main source of background. One of the proposals to reduce
the effect of such background is to place the detector off-axis, the case in
which the ratio between the BSM signal and the neutrino background increases
Coloma  _et al._ (2016); De Romeri  _et al._ (2019). The challenge is that by
going off-axis, the signal statistics also decreases significantly, and
therefore, several years of data taking will be necessary to obtain
competitive constraints in the BSM parameter space.

In this work, we propose a complementary configuration for a future DUNE-like
experiment, inspired by the method performed at the MiniBooNE beam-dump
experiment Aguilar-Arevalo  _et al._ (2018). The idea is that if the proton
beam can be directed away from the target (and hence directly impinge on the
beam dump), the neutrino flux will be greatly reduced, as charged mesons get
absorbed in the dump before they decay. Further, this configuration provides
additional materials for photon flux to increase; hence the effective distance
between the production source at the dump and the detector decreases, which
increases the solid angle coverage and the number of BSM particles entering
the detector (for the case of DUNE the BSM flux will be roughly increased by a
factor of $4$). While the long term usage of the target-less configuration
would clearly not allow for precise measurement of neutrino parameters, in
this work we will demonstrate that for several new physics scenarios large and
hitherto unconstrained regions of parameter space can be probed with only
three months of data taking at half the expected initial beam intensity of
DUNE. The relatively short duration needed here will therefore not
dramatically influence neutrino program at DUNE-like experiments where the
remaining unknowns in the standard three-neutrino paradigm are expected to be
obtained.

Target-less Configuration. In a typical accelerator neutrino experiment,
neutrinos are produced from the decays of the secondary charged mesons
generated by the proton interactions in the target. The surviving hadrons are
captured in the hadron absorber or a beam dump. In this work, however, we
consider a target-less configuration for which the proton beam directly makes
collisions with the beam dump. To estimate the sensitivity of this
configuration for different physics cases, we first model the fluxes of
neutrinos, which is the main source of background to BSM physics signals, and
the progenitor particles ($\pi^{0}$, $\eta^{0}$, $e^{\pm}$, $p$), which is the
source of the LDM and/or ALP signals. We have simulated the fluxes using
GEANT4 Agostinelli  _et al._ (2003) with the QGSP_BIC_AllHP physics list for
the hadronic reactions and G4EmStandardPhysics for the electromagnetic
interactions. For the case of DUNE we have used a cube-shaped aluminum core of
$4\times 4\times 4~{}$m3 for the dump and we have assumed a 120-GeV proton
beam impinging on it.

The resulting flux of neutrinos in the target-less mode as a function of the
neutrino energy is shown in the top panel of Fig. 1 (solid curves), where the
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes are shown in red and blue,
respectively. To demonstrate the potential of the target-less mode in
suppressing the neutrino fluxes, we have also shown the expected neutrino
fluxes for the target case in dashed curves, taken from Ref. Abi  _et al._
(2020b). The bottom panel shows the ratios of the fluxes in the target-less
compared to the target mode. The peaks for both $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ in the first bin for the target-less configuration is due to
the charged meson decay-at-rest, and the dip around 3 GeV in the ratio can be
understood as the effect of the focusing horns which is not present in the
target-less configuration. One can see that an overall flux suppression of
$10^{-3}-10^{-4}$ is achievable for the dump case, which demonstrates the
potential of the DUNE target-less configurations to significantly suppress the
neutrino background. In order to verify these results, we have performed a
separate simulation based on the MiniBooNE beam dump assumptions and have
found general agreement with the neutrino fluxes in Ref. Aguilar-Arevalo  _et
al._ (2017).

As a final remark here let us emphasize that the DUNE dump is designed to
survive two consecutive accidental deposition of 2.4 MW proton beams,
effectively. To estimate the sensitivity of the target-less mode to different
BSM cases, we consider two scenarios: $(i)$ a realistic case assuming the beam
power is 0.6 MW with a run time of 3 months, and $(ii)$ an optimistic case
with a beam power of 1.2 MW and a run time of 1 year. We show the results of
these two cases throughout the paper.

Figure 1:  The target-less configuration fluxes (solid) obtained in this work
and neutrino fluxes for the target (dashed), adopted from Ref. Abi  _et al._
(2020b). Here, the fluxes of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ in the target
mode are the results obtained in the forward horn current mode and the reverse
horn current mode, respectively. The red and cyan curves show the fluxes of
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$, respectively. Absolute fluxes for target
and dump case are shown in the upper panel while their ratio is shown in the
lower panel.

Benchmark Physics Cases. In what follows we demonstrate the potential of a
target-less configuration at DUNE-like experiments in BSM searches by focusing
on two new physics scenarios: Light Dark Matter (LDM) and Axion-Like Particles
(ALPs).

_Light Dark Matter._ In recent years, significant attention has been paid to
studying light MeV-GeV scale dark matter at neutrino experiments De Romeri
_et al._ (2019); Celentano  _et al._ (2020); Breitbach  _et al._ (2021); Naaz
_et al._ (2019). These studies are done by utilizing the intense proton beam
at these facilities which strike the target, producing charged and neutral
mesons; latter may then produce DM particles and this typically goes via meson
decay into a final state containing dark photon which couples to DM particles.
The DM can then travel to the detector and produce a signal by elastic
scattering on electrons. In this work we consider the following dark-sector
model by augmenting the SM with a complex scalar DM $\phi$, charged under
$U(1)_{D}$ symmetry, which communicates with the SM through a dark photon
$A^{\prime}$ (the $U(1)_{D}$ gauge boson), described by the Lagrangian Batell
_et al._ (2009); deNiverville  _et al._ (2011, 2012); Izaguirre  _et al._
(2013)

$\mathscr{L}_{\rm{DM}}\supset-\frac{1}{2}\epsilon
F_{\mu\nu}F^{\prime\mu\nu}+\frac{1}{2}m_{A^{\prime}}^{2}A^{\prime}_{\mu}A^{\prime\mu}+|D_{\mu}\phi|^{2}-m_{\phi}^{2}|\phi|^{2}\,,$
(1)

where $\epsilon$ is the kinetic mixing, $F_{\mu\nu}$ and $F^{\prime}_{\mu\nu}$
are the electromagnetic and dark photon field strength tensors, respectively,
$m_{A^{\prime}}$ and $m_{\phi}$ are the dark photon and DM masses, and
$D_{\mu}$ is the covariant derivative corresponding to $U(1)_{D}$ with the
coupling $g_{D}$ and dark-sector fine structure constant $\alpha_{D}\equiv
g_{D}^{2}/4\pi$. Given this Lagrangian, dark photons can be produced through
various mechanisms. Among them, dominant production channels include the
following ones: $(i)$ neutral mesons (e.g., $\pi^{0}$, $\eta$) can decay into
a SM photon and a dark photon $A^{\prime}$ through the kinetic mixing, i.e.,
$\pi^{0}/\eta\to\gamma\,A^{\prime}$, $(ii)$ the incoming proton beam can
scatter off a nucleus, emitting a dark photon, i.e., $p\,N\to
p\,N\,A^{\prime}$, and $(iii)$ positrons arising in the secondary
electromagnetic showering process of the charged particles created by the beam
collision can produce dark photons resonantly, together with electrons in
target atoms, i.e., $e^{+}\,e^{-}\to A^{\prime}$. We use GEANT4 with the
QGSP_BIC_ALLHP physics list to simulate sample fluxes of the progenitor
particles ($\pi^{0}$, $\eta^{0}$, $e^{\pm}$, $p$), which can then source the
DM flux, $\mathcal{F}_{DM}$.

Figure 2: LDM sensitivity for target-less run of 3 months-0.6 MW beam
(1year-1.2 MW beam) shown in solid (dashed) red curve, compared to the DUNE
target configurations discussed in the literature, namely the shape analysis
(blue dashed curve) of Ref. Breitbach  _et al._ (2021) and the PRISM analysis
of Refs. Breitbach  _et al._ (2021); De Romeri  _et al._ (2019) (cyan dot-
dashed curve). The “target” symbol in the legend denotes the target mode
measurements. The thermal relic density for scalar DM reproducing the observed
DM abundance is shown in orange. The existing constraints from various
experiments are shown in gray shaded regions. See text for more details.

Once produced, $A^{\prime}$ immediately decays to a DM pair, and the DM can
manifest itself as scattering on e.g., electrons,
$\phi+e^{-}\rightarrow\phi+e^{-}$. The differential cross section with respect
to the electron recoil energy, $E_{r}$, is given by deNiverville  _et al._
(2011)

$\displaystyle\frac{d\sigma_{\phi
e}}{dE_{r}}=4\pi\epsilon^{2}\frac{\alpha_{D}\alpha_{\rm{EM}}}{p_{\phi}^{2}}\frac{2E_{\phi}m_{e}(E_{\phi}-E_{r})-m_{\phi}^{2}E_{r}}{(2m_{e}E_{r}+m_{A}^{\prime
2})^{2}}\,,$ (2)

where $m_{e}$ is the electron mass, $\alpha_{\rm EM}\simeq 1/137$ is the QED
fine structure constant and $E_{\phi}$ ($p_{\phi}$) is the DM energy
(momentum). The expected number of DM signal $N_{\rm DM}$ at the detector is
given by

$N_{\rm DM}=N_{e}\int dE_{\phi}\frac{d\mathcal{F}_{\rm
DM}}{dE_{\phi}}\sigma_{\phi e}(E_{\phi})\,,$ (3)

where $N_{e}$ is the number of electron targets inside the detector fiducial
volume, for which we have assumed 50-t of liquid argon (LAr). We have set the
threshold energy to 30 MeV Abi  _et al._ (2020b).

In Fig. 2 we have shown the target-less configuration’s sensitivity to LDM
using the dimensionless parameter
$Y\equiv\epsilon^{2}\alpha_{D}(m_{\phi}/m_{A^{\prime}})^{4}$, for dark sector
coupling we chose $\alpha_{D}=0.1$, and assumed that the DM mass, $m_{\phi}$,
is three times smaller than the dark photon mass $m_{A^{\prime}}$. For the
realistic case of 0.6 MW beam power and a run time of 3 months (optimistic
case of 1.2-MW in 1 year), we require $N_{\rm DM}=3.1~{}(8.0)$ to obtain 90%
CL sensitivity projections. Note that neutrino fluxes in the target-less
configuration, shown in Fig. 1, yield 0.5 (4) neutrino-electron scattering
background events and that information has been included in setting
sensitivity projections. The red curves in Fig. 2 represent the sensitivity
projections for the two aforementioned study cases. One can see that for
$m_{\phi}\gtrsim 10~{}$MeV a run of only 3 months yields stronger sensitivity
projections than other DUNE configurations with a run time of 5 years, e.g. a
careful bin by bin analysis using the on-axis detector Breitbach  _et al._
(2021) (dashed blue curve), or using the PRISM concept to suppress the
background Breitbach  _et al._ (2021); De Romeri  _et al._ (2019) (dot-dashed
cyan curve).

One can infer that the reach in the target-less realization also “exceeds” the
orange line corresponding to the DM abundance in accord with present
observations. Hence, a run of only 3 months will have the ability to probe
thermal relic LDM up to $m_{\phi}\simeq 200$ MeV ($m_{A^{\prime}}\simeq 600$
MeV). Given the reach of the target-less projection relative to the DM
abundance line, scenarios in which thermal relic is a subdominant DM component
are also testable. Further, note that asymmetric DM models generically require
larger than thermal relic cross sections Graesser  _et al._ (2011), and hence
these models are more testable at a DUNE-like experiment compared to
(symmetric) thermal relic DM. We have also shown the existing constraints in
shaded grey regions from the MiniBooNE Aguilar-Arevalo  _et al._ (2018), BaBar
Lees  _et al._ (2017) and NA64 Andreev  _et al._ (2021a) experiments. Current
direct detection experiments, for example XENON1T, XENON10 and SENSEI (shown
in gray dashed curve) also cover the same parameter space, and are marginally
better than NA64 only for masses larger than $m_{\phi}\gtrsim 100~{}$MeV,
while for masses around a few MeV the sensitivity is only an order of
magnitude better than BaBar Barak  _et al._ (2020).

_Axion-Like Particles._ Axions and axion-like particles (ALPs) span a wide
range of well-motivated models, namely those that solve the strong CP problem
(QCD axions) and those that feature the ALP as a viable DM candidate (see
Refs. Chadha-Day  _et al._ (2022); Marsh (2016) for a review), but otherwise
arise more generally from the zero modes of string theory models Arvanitaki
_et al._ (2010). A vast effort is being made into searching for ALPs using
their couplings with the SM particles at the current and future experiments:
Zioutas  _et al._ (1999); Anastassopoulos  _et al._ (2017); Irastorza  _et
al._ (2013); Kahn  _et al._ (2016); Salemi (2019); Asztalos  _et al._ (2001);
Du  _et al._ (2018); Jackson Kimball  _et al._ (2017); Brubaker  _et al._
(2017); Droster and van Bibber (2019); Abdullahi  _et al._ (2022); Spector
(2019); Abdullahi  _et al._ (2022); Melissinos (2009); DeRocco and Hook
(2018); Liu  _et al._ (2019); Obata  _et al._ (2018); Feng  _et al._ (2018);
Berlin  _et al._ (2019); Åkesson  _et al._ (2018); Volpe (2019); Dusaev  _et
al._ (2020); Banerjee  _et al._ (2020); Berlin  _et al._ (2018); Alekhin  _et
al._ (2016); Bonivento  _et al._ (2020); Dent  _et al._ (2020a); Aristizabal
Sierra  _et al._ (2020); Aprile  _et al._ (2020); Dent  _et al._ (2020b);
Aralis  _et al._ (2020); Fu  _et al._ (2017); Dent  _et al._ (2021); Balkin
_et al._ (2021).

To investigate the ALP parameter space, we will focus on a generic model where
ALP couples to an electron-positron pair and photons as described by
interaction terms in the Lagrangian of the form

$\mathcal{L}_{\rm
int}~{}\supset~{}-i\,g_{ae}\,a\,\bar{\psi}_{e}\,\gamma_{5}\,\psi_{e}\,-\frac{g_{a\gamma}}{4}\,a\,F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$
(4)

where $a$ denotes the ALP field and $\psi_{e}$ is the electron field.
$F^{\mu\nu}$ ($\tilde{F}^{\mu\nu}$) is the electromagnetic (dual) field
strength. The couplings $g_{ae}$ and $g_{a\gamma}$ are assumed to be
independent for this analysis. These operators capture, for example Di Luzio
_et al._ (2020); Grilli di Cortona  _et al._ (2016), phenomenology of Dine-
Fischler-Srednicki-Zhitnitsky (DFSZ) type models Zhitnitsky (1980); Dine  _et
al._ (1981a, b); Dine and Fischler (1983) of QCD axions which feature
couplings to SM fermions generated by the dynamics of an extended Higgs sector
and PQ field after spontaneous symmetry breaking. Alternatively, Kim-Shifman-
Vainshtein-Zakharov (KSVZ) Kim (1979); Shifman  _et al._ (1980) variants can
lead to operators like $aF\tilde{F}$ through loop diagrams of heavy color-
charged fermions. We do not, however, restrict ourselves to the parameter
space of traditional QCD axion models, permitting any phenomenologically
accessible parts of the parameter space of Eq. 4.

Given the interaction Lagrangian in Eq. 4, we consider multiple ALP production
channels that arise from this operator. To compute production of ALPs via
$g_{ae}$ we employ the photons, electrons, and positrons produced from
secondary interactions of protons incident on the dump volume, again using
GEANT4 with the QGSP_BIC_ALLHP physics list. The explicit production channels
are Compton scattering ($\gamma\,e^{-}\to e^{-}\,a$) from $\gamma$ incident on
electrons at rest, associated production ($e^{+}\,e^{-}\to\gamma\,a$) and
resonant production ($e^{+}\,e^{-}\to a$) both arising from positron
annihilation on electrons at rest, and finally, ALP-Bremsstrahlung
($e^{\pm}\,N\to e^{\pm}\,N\,a$) producing ALPs from braking radiation of
$e^{\pm}$ through the atomic targets $N$ in the dump material; see refs.
Capozzi  _et al._ (2021); Aguilar-Arevalo  _et al._ (2021); Nardi  _et al._
(2018); Tsai (1986) for other uses of these channels and the analytical
techniques therein. ALPs coupling to photons ($g_{a\gamma}$ coupling) are
produced via the coherent Primakoff process ($\gamma\,N\to a\,N$) due to
$\gamma$ incident on atomic targets in the dump material.

From the detection side, we consider several signatures of ALPs in the DUNE-
like detector. ALPs coupling to electrons through $g_{ae}$ could decay to an
electron-positron pair with the well-known decay width, which in conjunction
with the ALP energy, fixes the decay length. Also, an $e^{+}\,e^{-}$ final
state can be produced from ALPs undergoing external pair production (similar
to SM photon pair production) $a\,N\to e^{+}\,e^{-}N$ via interaction with the
strong electric field of the nucleus. Lastly, ALPs could also yield the signal
by scattering in the detector via $a\,e^{-}\rightarrow\gamma\,e^{-}$, which
produces a soft, slightly off-forward photon as well as a hard and forward-
going electron. For ALPs coupling to photons through $g_{a\gamma}$, we
consider the ALP decays to two photons $a\to\gamma\gamma$ in the LAr volume as
well as inverse-Primakoff scattering, $a\,N\to\gamma\,N$.

Figure 3: The target-less configuration sensitivity to the ALP-electron and
ALP-photon coupling using 3 months-0.6 MW power (1 year-1.2 MW power) shown in
red solid (dashed) curves. _Top panel_ : For $g_{ae}$, Compton scattering,
associated production, resonant production, and bremsstrahlung production of
ALPs are considered via $\gamma$ and $e^{\pm}$ scattering in the dump
environment. Detection channels include ALP decay to $e^{+}e^{-}$ for
$m_{a}>2m_{e}$ as well as external pair conversion and inverse Compton
scattering in the detector (see text for details). Also displayed are the DFSZ
model preferred regions, where DFSZ-I and DFSZ-II are sub-models
differentiated by the relationship between the axion-lepton and axion-quark
couplings. _Bottom panel_ : For $g_{ae}$, the production occurs via Primakoff
scattering process. The detection channels involve the inverse-Primakoff
scattering process and two photon final states from the decay. KSVZ model
preferred regions are shown as well. The blue dashed curve is the 1 year
target result using the gas detector in Ref. Brdar  _et al._ (2021). The
“target” symbol in the legend denotes this target mode measurement.

At the target case the main source of background for ALP decay is the
production of NC-$\pi^{0}$ which can decay into a pair of photons. Using a
gaseous argon detector (GAr) at a DUNE-like experiment one can veto most of
this background by the hadronic activity as well as employing kinematic
variables Brdar  _et al._ (2021). The situation at the LAr detector is vastly
different, and such backgrounds have never been studied before. Here we stress
again that for target-less configuration at a DUNE-like experiments all such
backgrounds are practically negligible, since the original neutrino flux is
suppressed, as shown in Fig. 1.

In Fig. 3, we show sensitivity of a target-less DUNE-like experiment run to
the $g_{ae}-m_{a}$ parameter space (top) and $g_{a\gamma}-m_{a}$ parameter
space (bottom) for our two benchmark scenarios (3 months-0.6 MW power case is
shown in solid and 1 year-1.2 MW realization is shown in dashed red). Existing
laboratory limits on $g_{ae}-m_{a}$ Bjorken  _et al._ (1988); Andreas  _et
al._ (2010); Bechis  _et al._ (1979); Andreev  _et al._ (2021b); Gninenko  _et
al._ (2018); Andreev  _et al._ (2021a); Riordan  _et al._ (1987); Bross  _et
al._ (1991) and $g_{a\gamma}-m_{a}$ Blumlein  _et al._ (1992, 1991); Jaeckel
and Spannowsky (2016) are shown in gray while the astrophysical probes from
supernovae Lucente and Carenza (2021); Payez  _et al._ (2015) and stellar
cooling Dolan  _et al._ (2021); Hardy and Lasenby (2017); Carenza  _et al._
(2020) are also shown (shades of orange). Regarding supernovae, we also point
out Lucente  _et al._ (2020) featuring more developments in the calculations
of the supernova limit as well as Refs. Caputo  _et al._ (2022a, b) where the
authors discuss ruling out the so-called “cosmological triangle” Brdar  _et
al._ (2021).

As one can see from the figure, a 3 month exposure in the target-less
realization can result in significantly improved sensitivity to ALPs, and test
viable QCD axion models of both the DFSZ and KSVZ benchmark scenarios. For the
$g_{a\gamma}$ coupling this can even improve over the 1 year study at the GAr
detector for most of the parameter space. We also notice the overlap of the
sensitivity with supernova limits, and given the aforementioned supernovae
uncertainties, the target-less realization of DUNE will clearly be able to
make statements about the validity of such astrophysical considerations.

Conclusions. In this work we have considered a scenario in which proton beam
from a DUNE-like experiment impinges directly on the beam dump. We have shown,
via explicit calculation, that neutrino fluxes in such configuration are
rather small, guaranteeing effectively background-free BSM searches. We have
investigated a simplified light dark matter (LDM) model as well as photophilic
and photophobic ALPs in this context and have shown that by running in the
dump mode for only 3 months with a beam power of 0.6 MW, presently
unconstrained portions in the parameter space can be probed. It should be
stressed that this target-less configuration can probe high-priority new
physics targets including DM interactions consistent with the thermal relic
hypothesis for the observed DM density, as well as DFSZ and KSVZ models of the
QCD axion.

Acknowledgements. BD and AT acknowledge support from the U.S. Department of
Energy (DOE) Grant DE-SC0010813. The work of DK is supported by DOE under
Grant No. DE-FG02-13ER41976/DE-SC0009913/DE-SC0010813. The work of IMS is
supported by DOE under the award number DE-SC0020250. The work of ZT is
supported by the Neutrino Theory Network Program Grant No. DE-AC02-07CHI11359
and the U.S. Department of Energy under the award number DE-SC0020250. The
work of AB, WJ and JY is supported by the U.S. Department of Energy under
Grant No. DE-SC0011686. JY thanks the support of the CERN neutrino department
during his stay in which majority of this work is performed. Fermilab is
operated by the Fermi Research Alliance, LLC under contract No. DE-
AC02-07CH11359 with the United States Department of Energy. Portions of this
research were conducted with the advanced computing resources provided by
Texas A&M High Performance Research Computing. This research was supported in
part by the National Science Foundation under Grant No. PHY-1748958.

Disclaimer: We speak for ourselves, not on behalf of the DUNE Collaboration.
This work is based on our ideas, our calculations, and publicly available
information.

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# The subterahertz solar cycle: Polar and equatorial radii derived from SST
and ALMA

Fabian Menezes Centro de Rádio Astronomia e Astrofísica Mackenzie (CRAAM),
Universidade Presbiteriana Mackenzie, São Paulo, Brazil  Caius L. Selhorst
Núcleo de Astrofísica, Universidade Cruzeiro do Sul / Universidade Cidade de
São Paulo, São Paulo, SP, Brazil  Carlos Guillermo Giménez de Castro Centro de
Rádio Astronomia e Astrofísica Mackenzie (CRAAM), Universidade Presbiteriana
Mackenzie, São Paulo, Brazil  Instituto de Astronomía y Física del Espacio,
UBA/CONICET, Buenos Aires, Argentina.  Adriana Valio Centro de Rádio
Astronomia e Astrofísica Mackenzie (CRAAM), Universidade Presbiteriana
Mackenzie, São Paulo, Brazil

###### Abstract

At subterahertz frequencies – i.e., millimeter and submillimeter wavelengths –
there is a gap of measurements of the solar radius as well as other parameters
of the solar atmosphere. As the observational wavelength changes, the radius
varies because the altitude of the dominant electromagnetic radiation is
produced at different heights in the solar atmosphere. Moreover, radius
variations throughout long time series are indicative of changes in the solar
atmosphere that may be related to the solar cycle. Therefore, the solar radius
is an important parameter for the calibration of solar atmospheric models
enabling a better understanding of the atmospheric structure. In this work we
use data from the Solar Submillimeter-wave Telescope (SST) and from the
Atacama Large Millimeter/submillimeter Array (ALMA), at the frequencies of
100, 212, 230, and 405 GHz, to measure the equatorial and polar radii of the
Sun. The radii measured with extensive data from the SST agree with the
radius-vs-frequency trend present in the literature. The radii derived from
ALMA maps at 230 GHz also agree with the radius-vs-frequency trend, whereas
the 100-GHz radii are slightly above the values reported by other authors. In
addition, we analyze the equatorial and polar radius behavior over the years,
by determining the correlation coefficient between solar activity and
subterahertz radii time series at 212 and 405 GHz (SST). The variation of the
SST-derived radii over 13 years are correlated to the solar activity when
considering equatorial regions of the solar atmosphere, and anticorrelated
when considering polar regions. The ALMA derived radii time series for 100 and
230 GHz show very similar behaviors with those of SST.

Solar radius — Solar atmosphere — Solar radio emission

††journal: ApJ

##  1 INTRODUCTION

Due to technological limitations until some decades ago, only optical
observations of the Sun were available. Observations at radio wavelengths
began to take place after 1950 (Coates, 1958), and many authors had been using
measurements of the solar disk size – i.e. the center-to-limb distance – as
ways to determine the solar disk radius (hereafter solar radius) at different
radio wavelengths (Coates, 1958; Wrixon, 1970; Swanson, 1973; Kislyakov et
al., 1975; Labrum et al., 1978; Fürst et al., 1979; Horne et al., 1981;
Bachurin, 1983; Pelyushenko & Chernyshev, 1983; Wannier et al., 1983; Costa et
al., 1986, 1999; Selhorst et al., 2004; Alissandrakis et al., 2017; Menezes &
Valio, 2017; Selhorst et al., 2019b, a). There are different techniques to
measure the solar radius at radio frequencies, such as the determination from
total solar eclipse observations (Kubo, 1993; Kilcik et al., 2009), and from
direct observations as the inflection point method (Alissandrakis et al.,
2017) and the the half-power method (Costa et al., 1999; Selhorst et al.,
2011; Menezes & Valio, 2017).

The study of the radio solar radius provides important information about the
solar atmosphere and activity cycle (Swanson, 1973; Costa et al., 1999;
Menezes & Valio, 2017; Selhorst et al., 2019b). Using radio measurements of
the solar radius derived from eclipse and direct observations one can probe
the solar atmosphere, since these measurements show the height above the
photosphere at which most of the emission at determined observation frequency
is generated (Swanson, 1973; Menezes & Valio, 2017; Selhorst et al., 2019b).
However, as the observation frequency changes, the height changes as well
(Selhorst et al., 2004, 2019b; Menezes & Valio, 2017). To determine the height
above the photosphere of the radio emission, we consider the optical solar
radius. At optical wavelengths the canonical value of the mean apparent solar
radius is $R_{\theta}=$${959.63}^{\prime\prime}$ corresponding to
$R_{\odot}=6.9599\times 10^{8}$ m. This value has been widely used in the
literature, hence we adopt it as the reference value. In this work we focus on
the solar disk radius at subterahertz radio frequencies – i.e., millimeter and
submillimeter wavelengths.

Therefore, with observations at several frequencies, different layers of the
solar atmosphere can be observed and studied. Furthermore, these parameters
can be used to improve and calibrate solar atmosphere models as an input
parameter and boundary condition. In other words, solar radius at radio
frequencies reflect the changes in the local distribution of temperature and
density of the solar atmosphere. In Table 1 and Figure 3 we compiled data of
the solar radius at several radio frequencies from different authors. We note,
however, that the different works use different definitions and methods for
determining the solar radius.

Table 1: Solar radius and altitude values at different frequencies and wavelengths. Authors | Wavelength | Frequency | Radius | Altitude
---|---|---|---|---
|  |  | (arcsec) | ($10^{6}$ m)
Fürst et al. (1979) | 1 dm | 3 GHz | $1070\pm 17$ | $80\pm 12$
Fürst et al. (1979) | 6 cm | 5 GHz | $1020\pm 9$ | $44\pm 7$
Bachurin (1983) | 3.3 cm | 9 GHz | $989\pm 2$ | $21\pm 1$
Fürst et al. (1979) | 2.7 cm | 11 GHz | $991\pm 5$ | $23\pm 4$
Bachurin (1983) | 2.3 cm | 13 GHz | $989\pm 2$ | $21\pm 1$
Wrixon (1970) | 1.9 cm | 16 GHz | $990\pm 4$ | $22\pm 3$
Selhorst et al. (2004) | 1.8 cm | 17 GHz | $976.6\pm 1.5$ | $12.3\pm 1.1$
Costa et al. (1986) | 1.4 cm | 22 GHz | $981.7\pm 0.8$ | $16.0\pm 0.6$
Fürst et al. (1979) | 1.2 cm | 25 GHz | $979\pm 4$ | $14\pm 3$
Wrixon (1970) | 1 cm | 30 GHz | $979\pm 4$ | $14\pm 3$
Pelyushenko & Chernyshev (1983) | 8.6 mm | 35 GHz | $979\pm 3$ | $14\pm 2$
Selhorst et al. (2019a) | 8.1 mm | 37 GHz | $979\pm 5$ | $14\pm 4$
Costa et al. (1986) | 6.8 mm | 44 GHz | $978.1\pm 1.3$ | $13.4\pm 0.9$
Costa et al. (1999) | 6.2 mm | 48 GHz | $983.6\pm 1.9$ | $17.4\pm 1.4$
Pelyushenko & Chernyshev (1983) | 6.2 mm | 48 GHz | $973.1\pm 2.9$ | $9.8\pm 2.1$
Coates (1958) | 4.3 mm | 70 GHz | $969\pm 5$ | $7\pm 4$
Kislyakov et al. (1975) | 4 mm | 74 GHz | $967\pm 4$ | $5\pm 3$
Swanson (1973) | 3.2 mm | 94 GHz | $972\pm 5$ | $9\pm 4$
Alissandrakis et al. (2017) | 3 mm | 100 GHz | $964.1\pm 4.5$ | $3.2\pm 3.3$
Labrum et al. (1978) | 3 mm | 100 GHz | $966\pm 1$ | $5.0\pm 0.7$
Selhorst et al. (2019b) | 3 mm | 100 GHz | $965.9\pm 3.2$ | $4.5\pm 2.3$
Wannier et al. (1983) | 2.6 mm | 115 GHz | $969.3\pm 1.6$ | $7.0\pm 1.2$
Menezes & Valio (2017) | 1.4 mm | 212 GHz | $966.5\pm 2.8$ | $5.0\pm 2.0$
Alissandrakis et al. (2017) | 1.3 mm | 230 GHz | $961.1\pm 2.5$ | $1.1\pm 1.8$
Selhorst et al. (2019b) | 1.3 mm | 230 GHz | $961.6\pm 2.1$ | $1.4\pm 1.5$
Horne et al. (1981) | 1.3 mm | 231 GHz | $968.2\pm 1.0$ | $6.2\pm 7.3$
Menezes & Valio (2017) | 0.7 mm | 405 GHz | $966.5\pm 2.7$ | $5.0\pm 2.0$

Another aspect to be considered is that the Sun’s radius measured at the same
radio frequency over time shows slight variations. Temporal series of
observations obtained over many years show that the radius can be modulated
with the 11-year activity cycle (mid-term variations) as well as longer
periods (long term variations), as suggested by Rozelot et al. (2018) and
references therein. Costa et al. (1999) measured the radius from solar maps at
48 GHz taken with a 13.7-meter dish at the Pierre Kaufmann Radio Observatory
(ROPK, former Itapetinga Radio Observatory, ROI), and reported an average
radius of ${983.6}^{\prime\prime}$ $\pm$ ${1.9}^{\prime\prime}$. In a period
of 3 years (from 1990 to 1993), temporal variations were observed following
the linear relation

$R_{48\mathrm{GHz}}=[1.029-0.0015(\mathrm{year}-1990)]\;R_{\odot}\;,$ (1)

which yields a total decrease of ${8}^{\prime\prime}$ if extrapolated for a
period of 5.5 yr (half a cycle). Considering this short period, the data
suggest that the radius decreases in phase with the monthly mean sunspot
number and the soft X-ray flux from GOES (Geostationary Operational
Environmental Satellite). At 37 GHz using data from Metsähovi Radio
Observatory from 1989 to 2015, Selhorst et al. (2019a) measured a radius of
${979}^{\prime\prime}$$\pm$${5}^{\prime\prime}$ and obtained a positive
correlation coefficient of 0.44 between the monthly averages of the solar
radius and the solar flux at 10.7 cm. In Selhorst et al. (2004), the average
solar radius from NoRH (Nobeyama Radioheliograph) daily solar maps at 17 GHz
is found to be ${976.6}^{\prime\prime}$ $\pm$ ${1.5}^{\prime\prime}$. Over 11
years (one solar cycle), from 1992 to 2003, the variation in the solar radius
is correlated with the sunspot cycle with a coefficient $\rho=0.88$. However,
the polar radius – measurements above ${60}^{\circ}$ N and below
${60}^{\circ}$ S of the solar disk – is anticorrelated with the sunspot cycle.
The anticorrelation between polar radius and sunspot number yields a
coefficient $\rho=-0.64$.

At subterahertz frequencies there is a gap of measurements of the radius and
other parameters of the solar atmosphere (Table 1 and Figure 3). In this work
we determine the mean equatorial and mean polar radii of the Sun at 100, 212,
230, and 405 GHz using single-dish observations from the Solar Submillimeter-
wave Telescope (SST) and the Atacama Large Millimeter/submillimeter Array
(ALMA).

Moreover, we present an investigation about the relation between subterahertz
radii time series and solar activity cycle proxies – solar flux at 10.7 cm,
$F_{10.7}$, and mean magnetic field, $|B|$, of the Sun. We analyze the
equatorial and polar radius behavior over time, with qualitative analysis for
100 and 230 GHz (ALMA) from 2015 to 2018. Also, we determine the correlation
coefficient between the solar activity proxies and equatorial and polar radii
time series at 212 and 405 GHz (SST), from 2007 to 2019. The methodology used
for that is an improved version of the methodology presented in Menezes &
Valio (2017) with a new approach and different analysis. In this work we
focused on the behavior of polar and equatorial solar radius over time.

##  2 OBSERVATIONS AND DATA

The data used for the determination of the solar radii were provided by daily
solar observations of the Solar Submillimeter-wave Telescope (Kaufmann et al.,
2008) at 212, and 405 GHz, and the Atacama Large Millimeter/submillimeter
Array (Wootten & Thompson, 2009) at 100 and 230 GHz. The SST radio telescope
was conceived to monitor continuously the submillimeter spectrum of the solar
emission in quiescent and explosive conditions. In operation since 1999, the
instrument is located at CASLEO observatory (lat.:
${31}^{\circ}$${47}^{\prime}$${54.7}^{\prime\prime}$ S; lon.:
${69}^{\circ}$${17}^{\prime}$${44.1}^{\prime\prime}$ W), at 2552 m elevation,
in the Argentine Andes. From its six receivers, we used data from one
radiometer at 405 GHz and three at 212 GHz, which have nominal half-power beam
widths, HPBW, of ${2}^{\prime}$ and ${4}^{\prime}$ (arcminutes), respectively
(Kaufmann et al., 2008).

The SST radiometers were upgraded in 2006 to improve bandwidth, noise and
performance, and in 2007 the SST reflector was repaired to provide better
antenna efficiency, according to Kaufmann et al. (2008). Therefore, we used
data from 2007 onward. From 2007 to 2019 there were 3093 days of solar
observation, with an average of approximately 17 solar maps per day
considering all receivers, resulting in an extensive data set of 36 034 maps
(27 109 at 212 GHz and 8925 at 405 GHz). Most of the maps are obtained from
azimuth and elevation scans from a ${60}^{\prime}$$\times$ ${60}^{\prime}$
area, with a ${2}^{\prime}$ separation between scans and tracking speed in the
range of ${0.1}^{\circ}$ and ${0.2}^{\circ}$ per second (Giménez de Castro et
al., 2020). For an integration of 0.04 second, for example, it results in
rectangular pixels of ${0.48}^{\prime}$$\times$ ${2}^{\prime}$ which are then
interpolated to obtain a square matrix (600$\times$ 600) as shown in the top
panels of Figure 1.

ALMA is an international radio interferometre located in the Atacama Desert of
northern Chile (lat.: ${23}^{\circ}$${1}^{\prime}$${44.40}^{\prime\prime}$ S;
lon.: ${67}^{\circ}$${45}^{\prime}$${18.00}^{\prime\prime}$ W), at 5000 m
elevation. From four solar single-dish observation campaigns between 2015 and
2018, we use 196 fast-scan maps (125 at 100 and 71 at 230 GHz). These maps are
derived from full-disk solar observations, which consist of a circular field
of view of diameter ${2400}^{\prime\prime}$ using a “double-circle” scanning
pattern (White et al., 2017). The nominal spatial resolutions, HPBW, are
${58}^{\prime\prime}$ and ${25}^{\prime\prime}$ at 100 (Band 3) and 230 GHz
(Band 6), respectively. Figure 1, bottom panels, shows examples of the ALMA
maps obtained at 100 GHz (left panel) and 230 GHz (right panel), where the
color represents the brightness temperature.

Figure 1: SST maps (upper panels) obtained on 2008-02-09 and ALMA maps (lower
panels) obtained on 2015-12-17. (a) 212 GHz; (b) 405 GHz; (c) 100 GHz (green
lines and crosses are the constraints of the polar region and in blue, the
equatorial ones); (d) 230 GHz.

###  2.1 Radius Determination

Two widely used methods for measuring the radio solar radius are the
inflection point method (Alissandrakis et al., 2017) and the the half-power
method (Costa et al., 1999; Selhorst et al., 2011; Menezes & Valio, 2017). In
Menezes et al. (2021), both methods are compared and it is shown how the
combination of limb brightening of the solar disk with the radio-telescope
beam width and shape can affect the radius determination depending on the
method. Menezes et al. (2021) showed that the inflection point method is less
susceptible to the irregularities of the telescope beams and to the variations
of the brightness temperature profiles of the Sun (e. g. limb brightening
level and active regions). Thus, the inflection point method brings a low bias
to the calculation of the solar radius and, therefore, we use this method to
determine the solar radius.

The first step is to extract the solar limb coordinates from each map which
are defined as the maximum and minimum points of the numerical differentiation
red curve in Figure 2-a of each scan that the telescope makes on the solar
disk. All solar maps are rotated so that the position of the solar North
points upwards, and the coordinates are corrected according to the
eccentricity of the Earth’s orbit – the apparent radius of the Sun varies
between ${975.3}^{\prime\prime}$ (perihelion) and ${943.2}^{\prime\prime}$
(aphelion) during the year. During the limb points extraction, some criteria
are adopted to avoid extracting limb points associated with active regions,
instrumental errors or high atmospheric opacity, which may increase the
calculated local radius in that region and hence the average radius. With this
radius determination filter, only points with a center–to–limb distance
between ${815}^{\prime\prime}$ ($0.85R_{\theta}$) and ${1100}^{\prime\prime}$
($1.15R_{\theta}$) are considered.

Figure 2: Steps of solar radius measurement; (a) minimum and maximum points of
the numerical differentiation of a scan (red curve) corresponding to the limb
coordinates (black crosses); (b) limb coordinates (black crosses) extracted
from a solar map with a circle fit (red dashed line).

The second step is to calculate the average radius of each map. The solar limb
coordinates are fit by a circle and the radius is calculated as the average of
the center-to-limb distances. Successive circle fits are made until certain
conditions are met. For each fit, the points with center-to-limb distance
(obtained from the fitting) outside the interval
($\bar{R}-$${10}^{\prime\prime}$, $\bar{R}+$${10}^{\prime\prime}$) are
discarded, and then a new circle fit is performed with the remaining points.
This process is repeated until there are at least 35% of the points, usually 6
out of 16 depending on the map and the latitude region – polar or equatorial.
If there are fewer points remaining, the entire map is discarded; otherwise,
the radius is calculated. If the radius value is between
${800}^{\prime\prime}$ and ${1300}^{\prime\prime}$ and the standard deviation
is below ${20}^{\prime\prime}$, then the calculated radius is stored and the
next map is submitted to this process. The same method is applied to both
telescopes, so that a comparison between the 212 GHz radius of SST and the 230
GHz radius of ALMA could be made.

As mentioned in Section 1, both the equatorial and polar radii are calculated.
First, the radius of each map is determined using only the equatorial
latitudes of the solar disk – points between ${30}^{\circ}$ N and
${30}^{\circ}$ S. Then, the polar radius is determined considering only points
above ${60}^{\circ}$ N and below ${60}^{\circ}$ S. These latitude boundaries
are depicted in Figure 1-c, in dashed blue (equatorial) and green (polar)
lines. Finally, the visible solar radius is subtracted from the the mean
subterahertz radii to determine the altitudes in the atmosphere at which the
100, 212, 230, and 405 GHz emissions are predominantly produced.

Even with the strict criteria adopted in the mean radius determination, there
is still a large scattering in the distribution of SST radius values. Thus, we
applied a sigma clipping on the distribution, subtracting from the
distribution a running mean of 300 points, and then discarding values that are
outside the $\pm 2.5\sigma$ range. From the remaining values, we obtain the
average radii.

Giménez de Castro et al. (2020) have shown the influence of the SST irregular
beams in the study of quiescent solar structures. To assess the quality of the
radius determinations, here we carry out a series of simulations convolving
2-D antenna beam matrix representations with a solar disk with uniform
temperature. The results show that azimuth-elevation maps increase the
uncertainty of radius determination in the direction perpendicular to scans.
Since SST has an altazimuthal mount and maps are obtained at different times
of the day, the uncertainty is uniformly distributed in the data set.

###  2.2 Observational Time Series

To analyze the radius variation over the solar activity cycle, we use the
average taken every 6 months from the solar radii and solar proxies. We use
radius daily averages, smoothed over a 100-day period to build time series at
radio frequencies. As solar activity proxies, we used the 10.7-centimeter
solar flux, $F_{10.7}$, and the intensity of the mean photospheric magnetic
field, $|B|$, both smoothed by a 396-day running mean (13 months) to avoid the
influence of annual modulations. Daily flux values of the 10.7-centimeter
solar radio emission (Dominion Radio Astrophysical Observatory, DRAO), given
in solar flux units ($\mathrm{1\;SFU=10^{-22}Wm^{-2}Hz^{-1}}$), is a very good
proxy for solar activity cycle, as it is always measured by the same
instruments, and has a smaller intrinsic scatter than the sunspot number.
Another proxy used was the mean solar magnetic field, given in $\mu{T}$,
provided by The Wilcox Solar Observatory (WSO; Scherrer et al., 1977).

##  3 RESULTS

We used 29 088 SST maps (from a total of 36 034 maps) to calculate the radii
at 212 GHz (24 186 maps) and 405 GHz (4902 maps), and 196 ALMA maps to
calculate the radii at 100 GHz (125 maps) and 230 GHz (71) maps, with the
method described in Section 2.

###  3.1 Subterahertz Radii

The average equatorial and polar radii are calculated at 100, 212, 230, and
405 GHz. Moreover, the corresponding height with respect to the photosphere is
deduced from the angular radii. In Table 2 the radii are listed by frequency
and latitude.

Table 2: Measured average radii and altitudes at subterahertz frequencies and radii derived from SSC model Frequency | Latitude | Radius | Radius | Altitude | SSC radius
---|---|---|---|---|---
(GHz) |  | (arcsec) | ($10^{3}$ km) | ($10^{3}$ km) | (arcsec)
100 | Equatorial | 968$\pm$3 | 702.1$\pm$2.2 | 6.1$\pm$2.2 | 964.2
Polar | 968.4$\pm$2.3 | 702.4$\pm$1.7 | 6.4$\pm$1.7
212 | Equatorial | 963$\pm$4 | 699$\pm$3 | 3$\pm$3 | 963.8
Polar | 963$\pm$4 | 698$\pm$3 | 2$\pm$3
230 | Equatorial | 963.7$\pm$1.8 | 698.9$\pm$1.3 | 3.0$\pm$1.3 | 963.2
Polar | 963.7$\pm$1.6 | 698.9$\pm$1.2 | 3.0$\pm$1.2
405 | Equatorial | 963$\pm$5 | 699$\pm$4 | 3$\pm$4 | 962.8
Polar | 963$\pm$6 | 698$\pm$4 | 2$\pm$4

Our results are plotted with those from other authors (listed in Table 1) in
Figure 3. To guide the eye, an exponential curve (dashed line) is over plotted
to show the trend of the radius as a function of the observing frequency or
wavelength, indicating that the radius decreases exponentially at radio
frequencies. Note that the trend curve is just a least-square exponential fit,
not a physical model. Our results are shown with green (100 GHz), blue (212
GHz), yellow (230 GHz) and red (405 GHz) crosses. SST and ALMA radii seem to
agree within uncertainties with the trend of Figure 3 and the solar
atmospheric model predictions.

Figure 3: Solar radius as a function of frequency or wavelength. The dashed
line represents the exponential trend of the radius. The black crosses are
previous measurements from other authors (listed in Table 1), the blue (212
GHz) and red crosses (405 GHz) are the present radius values derived from SST
maps, the green (100 GHz) and yellow (230 GHz) crosses are the radii derived
from ALMA.

Next, we used a 2-D solar atmospheric model developed by Selhorst et al.
(2005) (hereafter referred to as the SSC model) to generate profiles of
temperature brightness, $T_{B}$, at 100, 212, 230, and 405 GHz, which yield
radii of ${964.2}^{\prime\prime}$, ${963.8}^{\prime\prime}$,
${963.2}^{\prime\prime}$, and ${962.8}^{\prime\prime}$ respectively (also
listed in the last column of Table 2). Our results at 212, 230 and 405 GHz are
very close to the radii derived from the model, whereas the 100-GHz radii are
about ${4}^{\prime\prime}$ bigger, not agreeing with the model.

###  3.2 Correlation with Solar Activity

We investigated the temporal variation of the solar radius and its
relationship with the 11-year solar cycle. The subterahertz radius time series
was analyzed from 2007 to 2016 using radii derived from the whole solar disk
only (Menezes & Valio, 2017). Here, we analyze a 13-year period – from January
2007 to December 2019 – using radii derived from equatorial and polar latitude
regions. As solar activity proxies, we used the 10.7-cm solar flux,
$F_{10.7}$, and the mean solar magnetic field intensity, $|B|$. The results
are plotted in Figure 4.

Figure 4:  Solar radius time series, $R_{\theta}$, at 212 and 405 GHz from
January 2007 to December 2019. The top panel (a) show time series for
equatorial (dark blue) and polar (light blue) radii at 212 GHz. The bottom
panel (b) show time series for equatorial (red) and polar (pink) radii at 405
GHz. The orange and gray lines represent $F_{10.7}$ and $|B|$, respectively.
Every box represents data for a 6-month period: the horizontal line inside the
box is the period median and the crosses represent the outliers of the
distribution. Figure 5:  Boxplot time series for equatorial $R_{\theta}$ at
100 (green), 212 (dark blue) and 230 GHz (orange), and for polar $R_{\theta}$
at 100 (light green), 212 (light blue) and 230 GHz (yellow), from January 2014
to December 2019. Every box represents data for a 6-month period: the
horizontal line inside the box is the period median. The dots represent the
outliers of the distribution. The red line represents $F_{10.7}$ and the gray
line represents $|B|$.

We compared the equatorial and polar radius time series at 212 GHz (SST) with
the time series of the equatorial and polar radii at 230 GHz (ALMA), with
6-month averages. The radii derived from SST’s maps are found to have both
average values and behavior over time very close to ALMA’s, which can be seen
in Figure 5. Moreover, besides the lower frequency and higher radius values,
equatorial and polar radii at 100 GHz seem to have similar behavior with those
of 212 and 230 GHz time series.

The comparison of the solar radii in time with the solar proxies is summarized
in Table 3, where the calculated correlation coefficients, $\rho$, are
organized by frequency, solar latitude regions and solar proxies. The
correlation coefficients between equatorial radius, $R_{212}^{equat.}$, and
solar proxies are very low (0.05 and 0.16, respectively). However, for
$R_{212}^{polar}$ the coefficients indicate weak anticorrelation (-0.36 for
$F_{10.7}$ and -0.23 for $|B|$). $R_{405}^{equat.}$ is moderately correlated
with $F_{10.7}$ (0.64) and $|B|$ (0.50), while $R_{405}^{polar}$ is weakly
anticorrelated with $F_{10.7}$ (-0.39) and $|B|$ (-0.23). In summary, the
radii – $R_{212}^{equat.}$, $R_{212}^{polar}$, $R_{405}^{equat.}$, and
$R_{405}^{polar}$ – have stronger correlation with $|B|$ than with $F_{10.7}$.

Table 3: Linear correlation coefficients, $\rho$, between solar radii and solar proxies. Frequency | Latitude | $\rho_{F_{10.7}}$ | $\rho_{|B|}$
---|---|---|---
212 GHz | Equatorial | 0.05 | 0.16
212 GHz | Polar | -0.36 | -0.23
405 GHz | Equatorial | 0.64 | 0.50
405 GHz | Polar | -0.39 | -0.23

##  4 DISCUSSION AND CONCLUSIONS

From the extensive SST and the ALMA data set we determined the polar and
equatorial radii of the Sun at 100, 212, 230, and 405 GHz. The average radii
are in agreement with the radius-vs-frequency trend (Fig. 3), however the
values obtained for the ALMA maps are higher than those obtained by Selhorst
et al. (2019b) and Alissandrakis et al. (2017). The 100-GHz average radius is
about ${4}^{\prime\prime}$ larger then that measured by Alissandrakis et al.
(2017), and about ${2}^{\prime\prime}$ larger then those measured by Labrum et
al. (1978) and Selhorst et al. (2019b). Nevertheless, the 100-GHz radius was
measured by Alissandrakis et al. (2017) and Selhorst et al. (2019b) using maps
observed only in December 2015, and by Labrum et al. (1978) using observations
of the total eclipse of 1976 October 23. Also there is a difference of roughly
${4}^{\prime\prime}$ between our 100-GHz measurements and the radius derived
from the SSC model. Menezes et al. (2021) showed that the radius increases as
a function of the limb brightening intensity. By convolving the ALMA beam with
the SSC model (limb brightening level 33.6% above the quiet Sun level), they
estimated an increase of ${2.3}^{\prime\prime}$ on the radius at 100 GHz. The
limb brightening levels could be increasing over time as the solar activity
decreases, and therefore affecting the solar radius.

Moreover, we have analyzed the subterahertz solar radius time series for more
than a solar cycle, over 13 years (2007 – 2019), at 212 and 405 GHz. The radii
time series are not strongly correlated with the solar proxies, however one
can observe particular behaviors of these time series, with the polar radius
being anticorrelated and the equatorial radius being correlated with $|B|$.
This is a similar behavior of the radio radius presented in the literature for
lower frequencies Costa et al. (1999); Selhorst et al. (2004, 2019a), which
are expected to be correlated to the solar cycle – positively for the average
radius and negatively for the polar radius. The equatorial radii time series
are expected to be positively correlated to the solar cycle, since the
equatorial regions are more affected by the increase of active regions during
solar maxima, making the solar atmosphere warmer in these regions. On the
other hand, the anticorrelation between polar radius time series and the solar
activity proxies could be explained by a possible increase of polar limb
brightening during solar minima. In a previous study of polar limb brightening
seen in 17 GHz solar maps, Selhorst et al. (2003) concluded that the intensity
of this brightening was anticorrelated with the solar cycle.

The 212 and 405-GHz radius time series do not seem to be as well defined as at
lower frequencies such as 17, 37, and 48 GHz (Costa et al., 1999; Selhorst et
al., 2004, 2019a). The subterahertz radii measured in this work correspond to
emission altitudes much closer to the photosphere. Therefore, our results
correspond to a frequency range (212 to 405 GHz) that probably reflects the
behavior from both the photosphere and the lower chromosphere. In part, this
would explain the weak (212 GHz) and the moderate (405 GHz) correlation of the
radii with the solar proxies.

Moreover, we could also observe analogous behavior between SST and ALMA time
series, even with the shorter period of ALMA solar observations. Besides lower
frequency and higher radius values, equatorial and polar radii at 100 GHz seem
to have similar variations with those of 212 and 230 GHz time series. At both
ALMA frequencies, the equatorial radii decrease from 2015 to 2017 and increase
from 2017 to 2018 – roughly close to $|B|$. The polar radii time series just
increases from 2015 to 2018, which is the opposite of $|B|$. As the
subterahertz radiation is influenced by Bremsstrahlung emission, the observed
variations are expected since the solar atmosphere’s density, temperature, and
magnetic field change with time. Longer periods of future observations at
these frequencies will reveal the polar and equatorial trends of the solar
atmosphere.

Measuring the solar radius at subterahertz frequencies is not an easy task as
well as to compare the values from different studies, since different
instruments observe at different band widths. Our results are important to
test atmospheric models, and better understand the solar cycle, since they
probe directly different layers of the solar atmosphere over time. More
studies of such kind at other frequencies and for longer periods of time are
needed to achieve this goal.

The authors thank J. Valle, P. J. A. Simões, and D. Cornejo for fruitful
discussions. F. M. thanks MackPesquisa and CAPES for the scholarship. We
acknowledge the financial support for operation of the Solar Submillimeter
Telescope (SST) from São Paulo Research Foundation (FAPESP) Proc.
#2013/24155-3 and AFOSR Grant #FA9550-16-1-0072. A. V. acknowledges partial
financial support from the FAPESP, grant number 2013/10559-5. C.L.S.
acknowledges financial support from the FAPESP, grant number 2019/ 03301-8.
C.G.G.C. is grateful to CNPq for providing support to this research (grant
307722/2019-8). This work is based on data acquired at Complejo Astronómico El
Leoncito, operated under agreement between the Consejo Nacional de
Investigaciones Científicas y Técnicas de la República Argentina and the
National Universities of La Plata, Córdoba and San Juán. This paper makes use
of the following ALMA data: ADS/JAO.ALMA #2011.0.00020.SV #2016.1.00050.S,
#2016.1.00070.S, #2016.1.00156.S, #2016.1.00166.S, #2016.1.00182.S,
#2016.1.00201.S, #2016.1.00202.S, #2016.1.00423.S, #2016.1.00572.S,
#2016.1.00788.S, #2016.1.01129.S, #2016.1.01408.S, #2016.1.01532.S,
#2017.1.00009.S, #2017.1.00870.S, #2017.1.01138.S. ALMA is a partnership of
ESO (representing its member states), NSF (USA) and NINS (Japan), together
with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ. This work makes use of data provided by The Wilcox
Solar Observatory, which was obtained via the web site
http://wso.stanford.edu. Also, we acknowledge Dr. Ken Tapping, Research
Council Officer, DRAO, and the Solar Radio Monitoring Program, operated
jointly by the National Research Council Canada and Natural Resources Canada,
for the radio flux data.

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