# The hyperon superfluidity and the hyperon couplings in neutron stars within
the relativistic mean field model

Zhong-Hao Tu<EMAIL_ADDRESS>Shan-Gui Zhou<EMAIL_ADDRESS>CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing, 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
Peng Huanwu Collaborative Center for Research and Education, Beihang
University, Beijing, 100191, China

###### Abstract

A systematic study of the effects of hyperon couplings on hyperon
superfluidity is conducted by using the relativistic mean field model.
Combining the slope of symmetry energy, the hyperon couplings are determined
in two ways—by the hypernuclear potentials or under the SU(3) symmetry. In
either way, the hyperon coupling constants cannot be fixed uniquely but vary
within a certain range due to the uncertainties in hypernuclear potentials or
the breaking of SU(6) to SU(3) symmetry. When the coupling constants are
constrained by the hypernuclear potentials, the pairings of $\Lambda$ and
$\Xi^{0,-}$ are strong and they each show little variations. The pairing of
$\Sigma^{-}$ is more sensitive to hyperon potentials and the slope of symmetry
energy. Under the SU(3) symmetry, the superfluidity of various hyperons differ
significantly. The dependence of the pairings of $\Lambda$ and $\Xi^{0,-}$ on
the additional parameters of SU(3) symmetry are the opposite to that of the
maximum mass of neutron stars on the additional parameters of SU(3) symmetry,
while the pairing of $\Sigma^{-}$ shows a similar trend in general. These
results suggest that the hyperon superfluidity associated with astrophysical
processes is an essential window to probe the physics of neutron star cores,
the hyperon-hyperon interactions and the SU(3) symmetry. Compared with other
hyperons, $\Sigma^{-}$ could serve as a cleaner glass for this purpose.

###### keywords:

Equation of state , neutron star , hyperon coupling , hyperon superfluidity

##  1 Introduction

Our knowledge of the interior of neutron stars (NSs), especially in the region
of the baryon number density $\rho_{B}$ higher than two times the saturation
density $\rho_{0}$, is uncertain yet. Hyperons could appear in NS cores once
the nucleon Fermi energy exceeds the (in-medium) rest masses of the hyperons
[1]. A number of works have been devoted to the study of NSs composed of
hyperons [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. However, when hyperons are
included, the direct Urca (dUrca) processes involving hyperons significantly
enhance the neutrino luminosity and rapidly cool NSs [13, 14, 15, 16, 17], and
finally result in the fact that the surface temperatures of NSs are much lower
than those observed [18, 19, 20]. For fixing this serious problem, hyperons
are postulated to form a superfluid to suppress neutrino emission and ensure
that the cooling curve of NSs and the astrophysical observations are
consistent.

Hyperons can be paired after they are populated inside NS cores [18, 21, 22,
23, 24, 25]. The paired hyperons exponentially suppress the neutrino
emissivity of hyperon dUrca processes far below the critical temperature
$T_{c}$ of hyperon pairing [26, 27, 28] and affect the dominant neutrino
emission mechanism, i.e., the Cooper pair breaking and formation (PBF)
mechanism, near $T_{c}$ [29, 26, 30]. Several studies have indicated that the
hyperon superfluidity plays an important role in NS cooling [31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].

The magnitude of the hyperon superfluidity in NSs is characterized by the
hyperon-hyperon ($YY$) pairing gap. The uncertainties in determining the
pairing gap arise from the ambiguous nuclear interactions above $\rho_{0}$
(including nucleon-nucleon ($NN$), nucleon-hyperon ($NY$), hyperon-hyperon,
etc.) and $YY$ pairing force that is limited to theoretical aspect currently.
We are mainly interested in the ${}^{1}S_{0}$ pairing gap inside NS cores in
this work. Although the $YY$ pairing forces are still uncertain, several
studies have given the values of the hyperon pairing gaps based on different
available $YY$ pairing forces [24, 46, 47, 38, 48, 49]. But the study of the
effects of the nuclear interactions on the hyperon superfluidity is still
rare, especially for $NY$ and $YY$ interactions.

The nuclear interaction also determines the equation of state (EoS), interior
composition and global properties of NSs. In this work, the relativistic mean
field (RMF) model is utilized to calculate the EoSs of NSs. The nuclear
interaction is characterized by the coupling constants between baryons and
mesons. For an NS composed of necleons and hyperons, its EoS is certainly
affected by the hyperon couplings, specifically the coupling constants between
hyperons and mesons in the RMF model. The main motivation of this work is to
build the relation between the hyperon couplings and hyperon superfluidity by
taking into account the uncertainties of nuclear interactions. Two approaches
are employed for determining the hyperon couplings. One is the empirical
hypernuclear potentials, e.g., $U_{\Sigma}^{(N)}$ and $U_{\Xi}^{(N)}$. These
hypernuclear potentials are uncertain so far and hence we can use them to
adjust the magnitude of the attractive interaction brought by the scalar
mesons within their uncertainties [50, 7]. The other one is the SU(3) model.
The SU(3) symmetry can provide a possibility of the existence of massive NS,
we can relax the SU(6) symmetry to a more general SU(3) symmetry and determine
the repulsive interactions originating from the couplings of the vector mesons
and hyperons [4, 51, 52, 53, 54, 55]. Combining with the indeterminate
hypernuclear potentials and the SU(3) symmetry, we systematically study the
effects of hyperon couplings on the hyperon superfluidity for the first time.

This paper is organized as follows. In Sec. 2, the methodology for calculating
the EoS of NSs and the pairing gaps of hyperons are given briefly. Sec. 3
shows how the empirical hypernuclear potentials affect the hyperon
superfluidity. In Sec. 4, we discuss the effects of hyperon couplings on the
hyperon superfluidity under the SU(3) symmetry. Finally, a brief summary and
perspective are given in Sec. 5.

##  2 Theoretical Framework

Using the well-known Bardeen-Cooper-Schrieffer (BCS) approximation [56, 21,
47] and the separable pairing force [57, 58], the ${}^{1}S_{0}{}$ pairing gap
$\Delta_{B}(k)$ of the baryon species $B$ at zero temperature is obtained by
solving the gap equation

$\displaystyle 1$ $\displaystyle=-\frac{1}{4\pi^{2}}\int
k^{2}\mathrm{d}k\frac{G_{B}p^{2}(k)}{\sqrt{[E_{B}(k)-\mu]^{2}+\Delta_{0}^{2}p^{2}(k)}},$
(1)

where $E_{B}(k)$ is the single particle energy of the baryon species $B$, the
chemical potential $\mu$ is determined by the value of $E_{B}(k)$ at the Fermi
surface, i.e., $\mu=E_{B}(k_{\mathrm{F}})$ with the Fermi momentum
$k_{\mathrm{F}}$. $G_{B}$ is the pairing strength of the baryon species $B$,
$p(k)=\exp(-\alpha^{2}k^{2})$ with the finite range $\alpha$ is the momentum
dependence of pairing force. The separable pairing force of $BB$ pairing is
$G_{B}(k,k^{\prime})=-G_{B}p(k)p(k^{\prime})=-G_{B}\exp(-\alpha^{2}k^{2})\exp(-\alpha^{2}k^{\prime
2})$. $\Delta(k)=\Delta_{0}p(k)$ is the trivial solution of the Eq. (1) and we
mainly refer the pairing gap at the Fermi surface
$\Delta^{\mathrm{F}}=\Delta_{0}p(k_{\mathrm{F}})$ in the present work. For
$NN$ pairing, the pairing strength $G_{N}=738~{}\mathrm{MeV\cdot fm}^{3}$ and
the finite range parameter $\alpha=0.636~{}\mathrm{fm}$. A new set of hyperon
pairing strengths have been proposed in the form of the separable pairing
force [49]. In this work, we choose $G_{\Lambda}=2G_{N}/3$ for $\Lambda$
hyperon, the upper limit of the $\Sigma\Sigma$ pairing strength
$G_{\Sigma}=2G_{N}/3$ for $\Sigma$ hyperon, and the lower limit of $\Xi\Xi$
pairing strength $G_{\Xi}=G_{N}$ for $\Xi$ hyperon [49]. The finite range of
$YY$ pairings are the same as that of $NN$ pairing.

The EoS and the single particle energies of each baryons are obtained by using
the RMF model. In this model, the $J^{P}=\frac{1}{2}^{+}$ baryonic octet
interact with each other through the exchange of the scalar mesons ($\sigma$
and $\sigma^{*}$) and vector mesons ($\omega$, $\rho$ and $\phi$). The
Lagrangian density which baryons interact with mesons is given by

$\displaystyle\mathcal{L}_{\mathrm{int}}=$
$\displaystyle\sum_{B}\bar{\psi}_{B}\left[\gamma^{\mu}(-g_{\omega
B}\omega_{\mu}-g_{\rho B}\bm{\rho}_{\mu}\bm{\tau}_{B}-g_{\phi
B}\phi_{\mu})\right.$ (2) $\displaystyle-\left.(-g_{\sigma B}\sigma-
g_{\sigma^{*}B}\sigma^{*})\right]\psi_{B},$

where $\psi_{B}$ is the Dirac field of the baryon species $B$; $\bm{\tau}_{B}$
is the Pauli matrices for isospin of the baryon species $B$; $\omega_{\mu}$,
$\bm{\rho}_{\mu}$, $\sigma^{*}$ and $\phi_{\mu}$ denote the quantum fields of
mesons, respectively. In the density dependent (DD) parameterization, e.g.,
DD-ME2 [59], the coupling constant between a baryon $B$ and a meson $m$ is
density dependent and is parameterized by the relation
$g_{mB}(\rho_{B})=g_{mB}(\rho_{0})f_{m}(\rho_{B})$ with

$\displaystyle f_{m}(\rho_{B})$
$\displaystyle=a_{m}\frac{1+b_{m}(\rho_{B}/\rho_{0}+d_{m})^{2}}{1+c_{m}(\rho_{B}/\rho_{0}+e_{m})^{2}},\quad
m=\sigma~{}\mathrm{and}~{}\omega,$ (3) $\displaystyle f_{m}(\rho_{B})$
$\displaystyle=\mathrm{exp}[-a_{m}(\rho_{B}/\rho_{0}-1)],\quad m=\rho.$ (4)

The single particle energy in RMF model is written as

$\displaystyle E_{B}(k)=\sqrt{k^{2}+M_{B}^{*2}}+g_{\omega B}\omega+g_{\rho
B}\tau_{3}^{B}\rho+g_{\phi B}\phi+\Sigma_{R},$ (5)

where the meson fields have been replaced by the their expectation values
after the mean field approximation. $\Sigma_{R}$ is the rearrangement term
originating from the density dependence of the coupling constants. The
effective mass $M_{B}^{*}$ is given by

$\displaystyle M_{B}^{*}=M_{B}+U_{B}^{S}=M_{B}-g_{\sigma B}\sigma-
g_{\sigma^{*}B}\sigma^{*},$ (6)

where $M_{B}$ is the rest mass of the baryon species $B$.

The $NN$ interaction at density higher than $2\rho_{0}$ remains unknown. Due
to the onset and $\beta$ equilibrium conditions for hyperons [1, 12], the
properties of hyperonic matter are affected by the nucleonic matter and hence
are influenced by the $NN$ interaction. The characteristic coefficients of
nucleonic matter, e.g., $K_{\mathrm{sat}}$, $Q_{\mathrm{sat}}$,
$E_{\mathrm{sym}}$ and $L_{\mathrm{sym}}$, dominated by $NN$ interaction could
affect the behavior of EoS of NSs at different density ranges. The value of
$L_{\mathrm{sym}}$ determines the intermediate-density properties of nucleonic
matter and hence influences the radius of the canonical $1.4M_{\odot}$ NS and
the NS mass that allows the dUrca processes [60]. For considering the
uncertainty of $NN$ interaction in this work, based on DD-ME2, we choose four
$NN$ effective interactions with different $L_{\mathrm{sym}}$ by changing
$g_{\rho N}$ and $a_{\rho}$ but fixing $E_{\mathrm{sym}}=27.09$ MeV at
$\rho_{B}=0.11$ fm-3 [61, 55]: $L_{\mathrm{sym}}=30$ MeV, $g_{\rho N}=3.0314$,
$a_{\rho}=0.8889$; original DD-ME2 with $L_{\mathrm{sym}}=51.26$ MeV;
$L_{\mathrm{sym}}=80$ MeV, $g_{\rho N}=3.8892$, $a_{\rho}=0.2576$;
$L_{\mathrm{sym}}=100$ MeV, $g_{\rho N}=4.1544$, $a_{\rho}=0.0905$. It should
be noted that the changes in the properties of finite nuclei induced by this
adjustment are less than 2%, indicating that these effective interactions are
acceptable for this work.

The $NY$ and $YY$ interactions can be determined by using SU(3) symmetry and
several hypernuclear potentials. We adopt the SU(3) symmetric model [62, 63,
4, 54, 53, 55] that contains three lightest quarks and assume the mixing angle
$\theta_{V}$, which describes the mixing between singlet and octet vector
mesons, is ideal mixing. The mixing angle from the quadratic mass formula for
mesons, i.e., $\theta_{V}\approx 40^{\circ}$ [64], is very close to the ideal
mixing angle $\theta_{V}=35.3^{\circ}$, and thus it is reasonable that we keep
the ideal mixing for the vector mesons in the present work. The additional two
parameters, i.e., $z$ and $\alpha_{V}$, are used to adjust the coupling
constants of the vector mesons [4, 53]. The $z$, which lies in the range of
$0\leq z\leq 2/\sqrt{6}$ for keeping the interaction due to $\omega$ exchange
to be repulsive for all baryons, is the ratio of the singlet and octet
coupling constants in the SU(3) quark model. The $\alpha_{V}$ is the weight
factor for the contributions of symmetric and the antisymmetric couplings
relative to each other in the SU(3) symmetric model and lies in the range of
$0\leq\alpha_{V}\leq 1$. We take $z$ and $\alpha_{V}$ as two free parameters.
The ratios of $g_{mY}$ to $g_{mN}$ for vector mesons can be written out by $z$
and $\alpha_{V}$, see Refs. [4, 53]. The SU(6) quark model combines the flavor
SU(3) and spin SU(2) symmetries and gives $z=1/\sqrt{6}$ and $\alpha_{V}=1$
[65, 63].

The scalar meson couplings $g_{\sigma Y}$ and $g_{\sigma^{*}Y}$ are determined
by fitting empirical hypernuclear potentials $U_{Y}^{(N)}$ and $U_{Y}^{(Y)}$
in symmetric nuclear matter at the saturation density

$\displaystyle U_{Y}^{(N)}$ $\displaystyle=-g_{\sigma Y}\sigma+g_{\omega
Y}\omega+\Sigma_{R},$ (7) $\displaystyle U_{Y}^{(Y)}$
$\displaystyle=-g_{\sigma Y}\sigma+g_{\omega Y}\omega-
g_{\sigma^{*}Y}\sigma^{*}+g_{\phi Y}\phi+\Sigma_{R}.$ (8)

The $U_{Y}^{(Y)}$’s are much uncertain. For reducing the complexity of the
discussion, we assume $U_{\Xi}^{(\Xi)}\simeq U_{\Lambda}^{(\Xi)}\simeq
2U_{\Lambda}^{(\Lambda)}\simeq 2U_{\Xi}^{(\Lambda)}\simeq-10$ MeV [66, 53, 63]
and $g_{\sigma^{*}\Lambda}=g_{\sigma^{*}\Sigma}$. The $U_{\Lambda}^{(N)}$ is
about 30 MeV [47, 67]. The $U_{\Sigma}^{(N)}$ and $U_{\Xi}^{(N)}$ are still
uncertain and thus we leave them as free parameters. The ranges of
$U_{\Sigma}^{(N)}$ and $U_{\Xi}^{(N)}$ are taken as $-10~{}\mathrm{MeV}\leq
U_{\Sigma}^{(N)}\leq 30~{}\mathrm{MeV}$ [7] and $-23~{}\mathrm{MeV}\leq
U_{\Xi}^{(N)}\leq-11~{}\mathrm{MeV}$ [68] in this work. We hereinafter omit
the superscripts of $U_{\Sigma}^{(N)}$ and $U_{\Xi}^{(N)}$ for simplicity.

In the present work, for every $L_{\mathrm{sym}}$, we change one of two groups
parameters, i.e., ($z,~{}\alpha_{V}$) or ($U_{\Sigma},~{}U_{\Xi}$) while
keeping the other one fixed, i.e., ($U_{\Sigma},~{}U_{\Xi}$) = (+30 MeV, $-$17
MeV) or ($z,~{}\alpha_{V}$)=($1/\sqrt{6},~{}1$), to change the hyperon
coupling and investigate the effects of hyperon couplings on the hyperon
superfluidity with different $L_{\mathrm{sym}}$.

##  3 The hyperon superfluidity and the hypernuclear potentials

In this section, we keep the parameters ($z,~{}\alpha_{V}$) fixed at their
SU(6) values ($1/\sqrt{6},~{}1$). How the hyperon potentials affect the
properties of NSs has been discussed before [50, 7]. When $U_{\Sigma}$ and
$U_{\Xi}$ are considered at the same time, the maximum mass of NSs is
insensitive to $U_{\Sigma}$ where $U_{\Sigma}>U_{\Xi}$ [50]. In the parameter
space ($U_{\Sigma},~{}U_{\Xi}$) in this work, $U_{\Sigma}$ is always larger
than $U_{\Xi}$ and hence the maximum masses of NS for different
$L_{\mathrm{sym}}$ mainly depend on $U_{\Xi}$. We take 25 points at equal
intervals within the value range of $U_{\Sigma}$ and $U_{\Xi}$ respectively,
and finally get 625 combinations of ($U_{\Sigma},U_{\Xi}$). The mass-radius
($M$-$R$) relations of NSs for all ($U_{\Sigma},U_{\Xi}$) and different
$L_{\mathrm{sym}}$ are displayed in Fig. 1(a). We can see that
$L_{\mathrm{sym}}$ have a distinct impact on the radius of NSs but the maximum
masses of NSs are nearly independent of it ($\Delta
M_{\mathrm{max}}<0.1M_{\odot}$). Although these radii differ significantly but
all $M$-$R$ relations satisfy the constraints from NICER [69, 70]. The maximum
masses are slightly lower than the observed most massive NSs, e.g., MSP
J0740+6620 (2.08${}^{+0.07}_{-0.07}M_{\odot}$ with 68.3% credibility
interval)[71] and the estimation $M_{\mathrm{max}}>2.09M_{\odot}$ ($3\sigma$
confidence) of the minimum value for the maximum NS mass [72]. The
modification of $K_{\mathrm{sat}}$ and $Q_{\mathrm{sat}}$ can increase the
maximum mass, see Refs. [61, 55], we do not refer to these additional degrees
of freedom in the present work. The changes in the hyperon potentials affect
the hyperon couplings and therefor only influence the EoS above the onset
density of the first appearing hyperon. As discussed in Ref. [50], hyperon
potentials have slight impact on the mass and radius of NS. In our parameter
space ($U_{\Sigma},~{}U_{\Xi}$), the variation of maximum mass induced by
varying hyperon potentials is less than $0.03M_{\odot}$ and is below several
uncertainties of astrophysical observation and analysis, e.g., $2.01M_{\odot}$
of PSR J0348+0432 with an uncertainty $0.04M_{\odot}$ [73]. It means that it
is difficult to extract the information of hyperon potentials through
astrophysical observations (e.g., mass and radius of NSs) and further explore
the effects of these hypernuclear potentials on some microscopic physical
processes of NSs (e.g., neutrino emission).

Figure 1: The $M$-$R$ relations (a) obtained by varying
($U_{\Sigma},~{}U_{\Xi}$) and (b) under SU(3) symmetry. In the case of varying
hyperon potentials, the all $M$-$R$ relations calculated with 625 combinations
of ($U_{\Sigma},~{}U_{\Xi}$) for different $L_{\mathrm{sym}}$ are presented,
the different combinations for ($U_{\Sigma},~{}U_{\Xi}$) are obtained by
taking 25 points at equal intervals within the value range of $U_{\Sigma}$ and
$U_{\Xi}$ respectively. Under SU(3) symmetry, the $M$-$R$ relations calculated
by varying $\alpha_{V}$ but fixing $z=0.782476$ for different
$L_{\mathrm{sym}}$ are illustrated. In panel (b), the top lines and bottom
line stand for $\alpha_{V}=0.0$ and $\alpha_{V}=1.0$ respectively, the results
between $\alpha_{V}=0.0$ and $\alpha_{V}=1.0$ are shown by shadows. The error
bars are constraints from mass and radius of PSR J0030+0451 from NICER [69,
70].

It is a failure to use global properties to resolve the hyperon potentials and
interior compositions of neutron stars due to the approximate degeneracy of
$M$-$R$ relations in parameter space ($U_{\Sigma},~{}U_{\Xi}$). In Ref. [7],
the effects of $U_{\Sigma}$ on the interior composition (e.g., onset density
and fraction of hyperon) of NSs have been discussed. The onset density and
fraction of hyperon depend on $U_{\Sigma}$, especially for $\Sigma^{-}$ and
$\Xi^{0,-}$. Here we briefly review the main result about $U_{\Sigma}$ and
interior composition. With increasing of $U_{\Sigma}$, the onset density of
$\Sigma^{-}$ is shifted to a higher value and the fraction of $\Sigma^{-}$ is
suppressed. Those of $\Xi^{0,-}$ show the opposite trend compared to
$\Sigma^{-}$. The pairing gap of hyperon is sensitive to interior composition
and can potentially distinguish NSs with similar global properties but
different $U_{\Sigma}$ and $U_{\Xi}$.

Figure 2: The maximum pairing gaps of $\Lambda$, $\Sigma^{-}$, $\Xi^{0}$ and
$\Xi^{-}$ inside NSs for different $L_{\mathrm{sym}}$ in the
($U_{\Sigma},U_{\Xi}$) parameter space. For $\Sigma^{-}$, there is no pairing
at the zone with high $U_{\Sigma}$ and hence the heat map is used to display
its result. The same row represent the results of the same hyperon and the
same column represent those for the same $L_{\mathrm{sym}}$. The subgraph
representing the maximum pairing gaps of hyperon $Y$ for $L_{\mathrm{sym}}$ is
labeled by ($Y$, $L_{\mathrm{sym}}$).

In Fig. 2, the maximum pairing gaps $\Delta_{Y}^{\mathrm{F}}$ of $\Lambda$,
$\Sigma^{-}$, $\Xi^{0}$ and $\Xi^{-}$ inside NSs for different
$L_{\mathrm{sym}}$ within the ($U_{\Sigma},~{}U_{\Xi}$) parameter space are
shown. For $\Lambda$, the pairing gaps are several tenths of a MeV and their
variation
$\Delta_{\Lambda}=\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{max})-\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{min})$
in the parameter space are smaller than 0.08 MeV, where
$\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{max})$ is the maximum
$\Delta_{\Lambda}^{\mathrm{F}}$ and
$\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{min})$ is the minimum one. The
variations are small and the dependence of $\Delta_{\Lambda}^{\mathrm{F}}$ to
hyperon potentials are different from the maximum mass. Specifically,
$\Delta_{\Lambda}^{\mathrm{F}}$ is slightly sensitive to $U_{\Sigma}$ but is
insensitive to $U_{\Xi}$ at smaller $U_{\Sigma}$. As $U_{\Sigma}$ increases,
the $\Lambda\Lambda$ pairing becomes stronger and $\Delta_{\Lambda}$
decreases. The trend can be visualized in Fig. 3(a). Due to the small
variation of $\Delta_{\Lambda}^{\mathrm{F}}$, if only $L_{\mathrm{sym}}$ and
hyperon potentials are considered, they have no significant impact on the
$\Lambda\Lambda$ pairing gap and hence we can leave the $\Lambda\Lambda$
pairing as an invariant factor in NS cooling. Similar conclusions can be drawn
for $\Xi^{0,-}$ with two main differences: firstly, both pairing gaps of
$\Xi^{0,-}$ are an order of magnitude larger than that of $\Lambda$, see Fig.
3(a) and (b); secondly, the variation of $\Delta_{\Xi^{0}}^{\mathrm{F}}$ is
significantly larger than that of $\Delta_{\Xi^{-}}^{\mathrm{F}}$, as shown in
3(b). The reason for the latter is that the onset density of $\Xi^{0}$ is
considerably higher than those of $\Lambda$, $\Sigma^{-}$ and $\Xi^{-}$,
resulting in the population of $\Xi^{0}$ being almost exclusively affected by
$U_{\Xi}$. In fact, $\Delta_{\Xi^{0}}^{\mathrm{F}}$ is sensitive to
$U_{\Sigma}$ at lower $U_{\Sigma}$ but depends on $U_{\Xi}$ at higher
$U_{\Sigma}$. The variations for $\Lambda$ and $\Xi^{0,-}$ decreases as
$L_{\mathrm{sym}}$ increases, which indicates that higher $L_{\mathrm{sym}}$
weakens the effects of hyperon potentials on hyperon superfluidity.

The effects of hyperon potentials on the pairing of $\Sigma^{-}$ are more
interesting. From Fig. 2, there is no paired $\Sigma^{-}$ in the parameter
space that falls within the blank area. We draw the heat map to illustrate the
maximum pairing gaps of $\Sigma^{-}$ to avoid a smoothing problem. For
$L_{\mathrm{sym}}=30$ MeV, the $\Sigma^{-}\Sigma^{-}$ pairing almost appears
everywhere, the exception is the area where $U_{\Sigma}$ is close to its upper
boundary and $U_{\Xi}$ approaches its minimum value. We can see that
$\Delta_{\Sigma^{-}}^{\mathrm{F}}$ can reach about 1 MeV below a critical
$U_{\Sigma}^{\mathrm{c}}=20$ MeV and the pairing gaps are relatively constant.
As $U_{\Sigma}$ increases to around 51 MeV, the area where the paired
$\Sigma^{-}$ appears becomes smaller, leaving a larger blank area in the lower
right corner of the parameter space. The critical $U_{\Sigma}^{\mathrm{c}}$
also reduces to approximately 15 MeV. With the further increase of
$L_{\mathrm{sym}}$, the blank area is extended to a quarter and a half of the
parameter space for $L_{\mathrm{sym}}=80$ MeV and 100 MeV,
$U_{\Sigma}^{\mathrm{c}}$ is reduced to 10 MeV and 0 MeV, respectively. The
larger $L_{\mathrm{sym}}$ is unfavored to the $\Sigma^{-}\Sigma^{-}$ pairing.

The negative influence of the larger $L_{\mathrm{sym}}$ on the strength of
$\Sigma^{-}\Sigma^{-}$ pairing can be discussed from global and local
perspectives. From Fig. 2 and Fig. 3(a), although the dependence of
$\Delta_{\Sigma^{-}}^{\mathrm{F}}$ on ($U_{\Sigma},~{}U_{\Xi}$) is different
quickly, the maximum of $\Delta_{\Sigma^{-}}^{\mathrm{F}}$ within the
parameter space decreases slightly as $L_{\mathrm{sym}}$ increases. It means
that the strong $\Sigma^{-}\Sigma^{-}$ pairing is always found below a
critical $\Sigma$ hyperon potential $U_{\Sigma}^{\mathrm{c}}$. If
$L_{\mathrm{sym}}$ is constrained well, the upper limit of $U_{\Sigma}$ (or
$U_{\Sigma}^{\mathrm{c}}$) can be estimated by the verified of the existence
of strong $\Sigma^{-}\Sigma^{-}$ inside NSs. From a local perspective, for a
fixed ($U_{\Sigma},~{}U_{\Xi}$), the variation of
$\Delta_{\Sigma^{-}}^{\mathrm{F}}(\mathrm{fixed})$ may be obvious. In Fig.
3(a), we can see that $\Delta_{\Sigma^{-}}^{\mathrm{F}}(\mathrm{fixed})$ at
$U_{\Sigma}=20$ MeV and $U_{\Xi}=-15$ MeV drops to zero as $L_{\mathrm{sym}}$
increases. When $U_{\Sigma}$ and $U_{\Xi}$ are constrained, due to the
sensitivity of $\Delta_{\Sigma^{-}}^{\mathrm{F}}$ to $L_{\mathrm{sym}}$, the
value $L_{\mathrm{sym}}$ could be inferred by the determination of
$\Delta_{\Sigma^{-}}^{\mathrm{F}}$.

Figure 3: The several pairing gaps of (a) $\Lambda$, $\Sigma^{-}$ and (b)
$\Xi^{0,-}$ with the parameter space ($U_{\Sigma},~{}U_{\Xi}$) as a function
of $L_{\mathrm{sym}}$ . The pink area is the constraint of $L_{\mathrm{sym}}$,
i.e., $L_{\mathrm{sym}}=0$–51 MeV, from CREX [74]. The constraint of
$L_{\mathrm{sym}}$ with $83.1\pm 24.7$ MeV from Ref. [75] is shown by green
area. $\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{max})$ is the maximum
$\Delta_{\Lambda}^{\mathrm{F}}$, $\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{min})$
is minimum one and $\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{fixed})$ is the
value of $\Delta_{\Lambda}^{\mathrm{F}}$ at $U_{\Sigma}=20$ MeV and
$U_{\Xi}=-15$ MeV. There is no overlap between $L_{\mathrm{sym}}$ from CREX
and Ref. [75], the $L_{\mathrm{sym}}$ of original DD-ME2 lies this gap and
those of other effective interactions cover the two constraints of
$L_{\mathrm{sym}}$.

##  4 The hyperon superfluidity under the SU(3) symmetry

The massive $M\gtrsim 2.0M_{\odot}$ NSs featuring hyperons can be obtained if
the value of $Q_{\mathrm{sym}}$ is large enough [61]. The modification of
$Q_{\mathrm{sym}}$ is related to $NN$ interaction. In the $NY$ and $YY$
sectors, the repulsive interaction which increases the maximum mass of NS can
be enhanced by relaxing SU(6) symmetry to the more general SU(3) symmetry.
Indeed, the global properties and interior compositions of NSs within the
SU(3) symmetry have been studied [4, 53, 54, 55]. We review the main results
again. The effects of $z$ or $\alpha_{V}$ on EoS and particle fraction of NSs
are qualitatively similar [54]. As $z$ decreases from its upper bound, the
onset densities of hyperons are pushed to the higher densities and hyperons
may appear in a small core of NSs. It means that there have more nucleonic
matter in NSs for smaller $z$, leading to stiffer EoS and larger maximum
masses. The massive NSs, which mass could up to the mass $M\backsimeq
2.5M_{\odot}$ of the secondary object of GW190814 [76], calculated with SU(3)
symmetry are essentially nucleonic stars. It implies that hyperons are less
important for the massive NSs. The problem is how the hyperon pairings inside
NSs under the SU(3) symmetry.

We take ($U_{\Sigma},~{}U_{\Xi}$) as ($+30$ MeV, $-17$ MeV) in this section.
In Fig. 1(b), the $M$-$R$ relations calculated by varying $\alpha_{V}$ and
fixing $z=0.782476$ for different $L_{\mathrm{sym}}$ are illustrated. The top
and bottom lines represent the results for $\alpha_{V}=0.0$ and
$\alpha_{V}=1.0$, respectively. Contrary to the results displayed in Fig.
1(a), the $M$-$R$ relations of NSs are distinctly different ($>0.85M_{\odot}$)
within SU(3) symmetry. It provides a possibility to probe the interior
composition through measurements of the global properties of NSs. The
feasibility could be further improved by incorporating observations of NS
cooling connected to hyperon pairings.

In Fig. 4, we demonstrate the maximum pairing gaps $\Delta_{Y}^{\mathrm{F}}$
of $\Lambda$, $\Sigma^{-}$, $\Xi^{0}$ and $\Xi^{-}$ inside NSs for different
$L_{\mathrm{sym}}$ under SU(3) symmetry. For $\Lambda$ with a low baryon mass
and the most depth potential and $\Xi^{-}$ with the negative charge and the
moderate depth potential, they are the domain hyperons inside NSs and their
onset densities are usually close to each other. From Fig. 4, we can see that
both the larger pairing gaps for $\Lambda$ and $\Xi^{-}$ locate in the upper
right corner of SU(3) parameter space. Similar to EoS or NS mass, $z$ and
$\alpha_{V}$ yield similar effects on the pairing gaps of $\Lambda$ and
$\Xi^{-}$: the reducing of $z$ or $\alpha_{V}$ lead to the smaller pairing
gaps. The massive NSs are unfavorable to $\Lambda$ and $\Xi^{-}$ superfluidity
because their pairing gap and the maximum mass of NSs increase in opposite
directions. The low fractions of $\Lambda$ and $\Xi^{-}$ reduce their state
densities at the Fermi surface and further decrease the corresponding pairing
gaps. The variations of pairing gaps are huge and their magnitudes are the
same as those of the max pairing gaps, as shown in Fig. 5. From Fig. 5, we
also find that $L_{\mathrm{sym}}$ has little impact on the pairing gap at
every fixed ($z,~{}\alpha_{V}$) for $\Lambda$ and the maximum of pairing gaps
of $\Xi^{-}$. Note that the larger $L_{\mathrm{sym}}$ ($>80$ MeV) leads to
larger pairing gaps of $\Xi^{-}$ in the area where close to its minimum, like
near extreme point.

Figure 4: The maximum pairing gaps of $\Lambda$, $\Sigma^{-}$, $\Xi^{0}$ and
$\Xi^{-}$ inside NSs for different $L_{\mathrm{sym}}$ under SU(3) symmetry.
For $\Sigma^{-}$ and $\Xi^{0}$, there is no pairing at the blank area and the
heat map is used to display their results. The arrangement and label of the
subgraphs are the same as those of Fig. 2. The black border in the results for
$\Sigma^{-}$ and $\Xi^{0}$ means that the density corresponding to the maximum
pairing gap exceed the central density of the most massive NSs and hence we
use the pairing gap at the central density to replace the maximum pairing gap.

For $\Xi^{0}$, the results are more complicated. In this work, the central
densities of NSs we discussed are bounded below those of the theoretical most
massive NSs ($\rho_{\mathrm{cM}}^{\mathrm{max}}$). For $\Xi^{0}$, the
dependence of $\Delta_{\Xi^{0}}^{\mathrm{F}}$ on $z$ or $\alpha_{V}$ is
similar, but $\Xi^{0}$ is not paired in the lower $z$ and lower $\alpha_{V}$
zone because the onset density is shifted to the density range above
$\rho_{\mathrm{cM}}^{\mathrm{max}}$. Note that the density corresponding to
the maximum pairing gap ($\rho_{\mathrm{cp}}^{\mathrm{max}}$) might surpass
$\rho_{\mathrm{cM}}^{\mathrm{max}}$ and the pairing gap at
$\rho_{\mathrm{cM}}^{\mathrm{max}}$ is plotted in Fig. 4 by using black
border. Depending on $L_{\mathrm{sym}}$, the black border is also plotted in
the strong $\Xi^{0}\Xi^{0}$ pairing zone, $\rho_{\mathrm{cM}}^{\mathrm{max}}$
is smaller and even below $\rho_{\mathrm{cp}}^{\mathrm{max}}$ because the EoS
is softer as shown in Ref. [4]. As $L_{\mathrm{sym}}$ increases, the area
where $\Xi^{0}\Xi^{0}$ pairing exists gets larger but changes little. From
Fig. 5(b), the maximum pairing gap of $\Xi^{0}$ under SU(3) symmetry varies
little with $L_{\mathrm{sym}}$. However, the pairing gap near the boundary
with and without $\Xi^{0}\Xi^{0}$ pairing may alter significantly. For
example, the pairing gap at $z=0.170103$ and $\alpha_{V}=1.0$ increases with
increasing $L_{\mathrm{sym}}$. Due to both the existence of massive NSs and
the vanishing $\Xi^{0}$ superfluidity in the region with lower $z$ and lower
$\alpha_{V}$, it is unnecessary to consider $\Xi^{0}$ superfluidity in the
cooling of massive NSs, the lower bound of NS mass depends on
$L_{\mathrm{sym}}$.

Figure 5: The several pairing gaps of (a) $\Lambda$, $\Sigma^{-}$ and (b)
$\Xi^{0,-}$ under SU(3) symmetry as a function of $L_{\mathrm{sym}}$.
$\Delta_{\Lambda}^{\mathrm{F}}(\mathrm{fixed})$ is the value of
$\Delta_{\Lambda}^{\mathrm{F}}$ at $z=0.170103$ and $\alpha_{V}=1.0$. The
other definitions in figure are the same as Fig.3.

The result of $\Sigma^{-}\Sigma^{-}$ pairing is also interesting as we
discussed in the case of varying hyperon potential. Contrary to
$\Lambda\Lambda$ or $\Xi^{0,-}\Xi^{0,-}$ pairing, with the decrease of $z$ or
$\alpha_{V}$, the maximum pairing gap increases in general. Unlike $\Xi^{0}$,
the area where $\Sigma^{-}\Sigma^{-}$ pairing exists becomes smaller as
$L_{\mathrm{sym}}$ increases. A peak of $\Delta_{\Sigma^{-}}^{\mathrm{F}}$ in
the parameter space can be found near the extreme point ($2/\sqrt{6},~{}1.0$),
the other peak can be found near the SU(6) point if $L_{\mathrm{sym}}$ is
small ($<80$ MeV). The pairing gaps around peaks are several tenths of a MeV
which can be compared with the pairing gaps of $\Lambda$. From Fig. 5(a),
$\Delta_{\Sigma^{-}}^{\mathrm{F}}(\mathrm{max})$ is insensitive to
$L_{\mathrm{sym}}$ but a $\Delta_{\Sigma^{-}}^{\mathrm{F}}(\mathrm{fixed})$
which is fixed at $z=0.170103$ and $\alpha_{V}=1.0$ is sensitive to
$L_{\mathrm{sym}}$. It means that, if the breaking of SU(6) is constrained
well, the $L_{\mathrm{sym}}$ can be estimated by determining of
$\Delta_{\Sigma^{-}}^{\mathrm{F}}(\mathrm{fixed})$ in NSs like we discussed in
Sec. 3.

##  5 Summary and perspective

We explored the effects of hyperon couplings on the hyperon superfluidity
inside NSs. Combining $NN$ effective interactions with different
$L_{\mathrm{sym}}$, the coupling constants between the vector mesons and
hyperons are obtained under the SU(3) symmetry; the coupling constants of the
scalar mesons are adjusted by the hypernuclear potentials to be confirmed. A
set of separable pairing forces of finite range is used in this work.

By fixing the ($z,~{}\alpha_{V}$) to the SU(6) values, we find that the
variations of the pairing gaps of $\Lambda$ and $\Xi^{0,-}$ are relatively
small but $\Delta_{\Sigma^{-}}^{\mathrm{F}}$ can range from 0.0 to around 1.0
MeV in the ($U_{\Sigma},~{}U_{\Xi}$) parameter space. $L_{\mathrm{sym}}$ has
little effect on the maximum pairing gaps of all hyperons but could change
significantly the pairing gap of $\Sigma^{-}$ at some fixed hyperon
potentials. Next we fixed ($U_{\Sigma},~{}U_{\Xi}$) as (+30 MeV, $-$17 MeV),
the pairing gaps of $\Lambda$, $\Sigma^{-}$ and $\Xi^{0,-}$ differ
significantly under SU(3) symmetry. The massive NS ($>2M_{\odot}$) is
unfavorable to the superfluidity of $\Lambda$ and $\Xi^{0,-}$ but is favorable
to that of $\Sigma^{-}$. $L_{\mathrm{sym}}$ also has less effects on the
pairing gaps of $\Lambda$ and $\Xi^{-}$ but obviously influences those of
$\Sigma^{-}$ and $\Xi^{0}$ at some fixed additional SU(3) parameters.

The $\Sigma^{-}$ superfluidity could be a better pointer to detect the
properties of NSs: both the cases of hyperon potentials and SU(3) symmetry,
the pairing gap of $\Sigma^{-}$ changes significantly and hence is sensitive
to ($U_{\Sigma},~{}U_{\Xi}$) and ($z,~{}\alpha_{V}$); when
($U_{\Sigma},~{}U_{\Xi}$) or ($z,~{}\alpha_{V}$) are constrained well, the
pairing gap of $\Sigma^{-}$ could depend on $L_{\mathrm{sym}}$, the smaller
$L_{\mathrm{sym}}$ is favored to $\Sigma^{-}$ superfluidity; the tendency of
the pairing gap of $\Sigma^{-}$ to change with $z$ or $\alpha_{V}$ is the
opposite of those of $\Lambda$ and $\Xi^{0,-}$. The effects of $\Sigma^{-}$ on
the thermodynamic and dynamic evolutions of NSs could not be ignored.

Although plausible quantitative results are presented in this work, our main
conclusions are more qualitative because many uncertainties are not included.
In the $NN$ sector, except for $L_{\mathrm{sym}}$, $K_{\mathrm{sat}}$ and
$Q_{\mathrm{sat}}$ even other higher order characteristic coefficients also
have significant effects on the EoS beyond twice the saturation density. In
the $NY$ and $YY$ sectors, we keep $U_{Y}^{(Y)}$ fixed but the properties of
NSs are also affected by them, see Ref. [54]. Uncertainties in pairing forces
for various hyperons are also disregarded. Fortunately, the strength of the
pairing force can be decomposed from the integral in Eq. 1 so that the
relation between the pairing gap and the strength of the pairing gap is
monotonic, the main conclusions obtained in this work still hold for different
pairing strengths. We apply the standard BCS approximation in this work and
neglect the many-body correlations, e.g., short- and long-range correlations.
In Ref. [77], the authors have confirmed that the short-range correlations
suppress the pairing gap of nucleon. The theoretical framework that accounts
for many-body correlations is more realistic. However, our understanding of
the correlations referring to hyperon-hyperon or nucleon-hyperon is lacking,
we discard the treatment of many-body correlations to avoid introducing
additional uncertainties. More quantitative results are expected to be
obtained in our future works after some additional factors we mentioned above
are well-constrained.

## Acknowledgements

Valuable effective interaction parameters from Jia-Jie Li are gratefully
acknowledged. This work has been supported by the National Natural Science
Foundation of China (Grants No. 12070131001, No. 12047503, No. 12175151, No.
12375118, and No. 11961141004), the Strategic Priority Research Program of
Chinese Academy of Sciences (Grant No. XDB34010000), and the IAEA Coordinated
Research Project “F41033”. The results described in this paper are obtained on
the High-performance Computing Cluster of ITP-CAS and the ScGrid of the
Supercomputing Center, Computer Network Information Center of Chinese Academy
of Sciences.

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# Observations of Mini Coronal Dimmings Caused by Small-scale Eruptions in the
Quiet Sun

Rui Wang (王瑞) State Key Laboratory of Space Weather, National Space Science
Center, Chinese Academy of Sciences, Beijing 100190, People’s Republic of
China  Ying D. Liu State Key Laboratory of Space Weather, National Space
Science Center, Chinese Academy of Sciences, Beijing 100190, People’s Republic
of China  University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China  Xiaowei Zhao Key Laboratory of Space Weather, National
Satellite Meteorological Center (National Center for Space Weather), China
Meteorological Administration, Beijing 100081, People’s Republic of China
School of Earth and Space Sciences, Peking University, Beijing 100871,
People’s Republic of China  Huidong Hu State Key Laboratory of Space Weather,
National Space Science Center, Chinese Academy of Sciences, Beijing 100190,
People’s Republic of China

###### Abstract

Small-scale eruptions could play an important role in coronal heating,
generation of solar energetic particles (SEPs), and mass source of the solar
wind. However, they are poorly observed, and their characteristics,
distributions, and origins remain unclear. Here a mini coronal dimming was
captured by the recently launched Solar Orbiter spacecraft. The observations
indicate that a minifilament eruption results in the dimming and takes away
approximately $(1.65\pm 0.54)\times 10^{13}$ g of mass, which also exhibits
similar features as the sources of SEP events. The released magnetic free
energy is of the order of $\sim 10^{27}$ erg. Our results suggest that weak
constraining force makes the flux rope associated with the minifilament easily
enter a torus-unstable domain. We discuss that weak magnetic constraints from
low-altitude background fields may be a general condition for the quiet-Sun
eruptions, which provide a possible mechanism for the transport of coronal
material and energy from the lower to the middle or even higher corona.

Quiet Sun — Solar coronal transients — Solar filaments — Solar magnetic fields

##  1 Introduction

Solar eruptive activities, including flares, eruptive filaments, coronal mass
ejections (CMEs), and coronal jets, are widely distributed in energy scales.
The observed solar eruptions span a range of at least a factor of $10^{8}$ in
energy (Schrijver et al., 2012). Large, highly energetic “X-class” eruptions
can release energy substantially exceeding $\sim 10^{33}$ erg. With the advent
of high-resolution observations, the scaled-down versions of the eruptive
activities are getting more attention, e.g., nanoflares (Bahauddin et al.,
2021), minifilaments (Wang et al., 2000; Innes et al., 2009, 2010; Sterling et
al., 2015; Panesar et al., 2016, 2018; McGlasson et al., 2019), and the newly
discovered “campfires” (Berghmans et al., 2021), which are at lower energies
probably below the current detection limit. Even-smaller-scale eruptive
structures have been speculated to exist (Hermans & Martin, 1986; Sterling &
Moore, 2016). The latest observations of Parker Solar Probe near the Sun
(McComas et al., 2019) indicate that small solar energetic particle (SEP)
events, which cannot be captured by 1 au spacecraft but are only observable
close to the Sun, may be much more common than previously thought. Where do
these small SEP events come from? Is it possible that they are generated by
these small-scale solar eruptions?

Jackson & Howard (1993) and Webb & Howard (1994) suggest that the ratio of the
annualized CME to solar wind mass flux is no more than $16\%$, and Lamy et al.
(2017) indicate that this ratio is only up to $6\%$. However, it should be
noted that the mass of these counted CMEs is usually in the range of $\sim
10^{14}-10^{15}$ g. Then what about small solar eruptions with less mass?
Aschwanden et al. (2000) and Schrijver et al. (2012) indicate that the events
at the lowest energies of $\sim 10^{24}$ erg can occur millions of times each
day. Namely, smaller eruptions generally have higher frequency of occurrence.
How do the small-scale eruptions, with the advantages of large numbers and
high occurrence rate, contribute to the mass of the solar wind? Most of our
knowledge of the energy-mass relation of eruptions is based on the
observations of large-scale eruptions. It remains unclear what the actual
relationship is between the mass and energy of small-scale eruptions in the
source region.

Coronal dimmings are considered to be an important characteristic of solar
eruptions, which can provide mass information of eruptions (Rust & Hildner,
1976; Thompson et al., 1998). It is generally accepted that they are caused by
plasma evacuation during the eruption of a CME (Harrison & Lyons, 2000). On
the Sun, the dimming regions correspond to the footprint of the CME (Jin et
al., 2022). Upflowing expanding plasma has been observed in the dimming region
(Harra & Sterling, 2001; Tian et al., 2012). Harrison & Lyons (2000) find that
the mass loss in the coronal dimmings is on the same order as the estimated
mass of the associated CMEs. Therefore, the mass of small-scale eruptions can
also be estimated by their associated coronal dimmings. However, small-scale
coronal dimmings have been poorly studied. A small dimming associated with an
eruption is not easy to identify, since the eruption itself and the associated
characteristic structures, such as post-flare arcades, flare ribbons, and
coronal waves, are difficult to recognize under the previous observational
limits. Therefore, it is difficult to determine whether the observed dimming
is related to an eruption or not. In short, we do not know much about how
small-scale coronal dimmings are produced, what their characteristics are, and
how they are distributed.

In this Letter, we identify the characteristic structures associated with a
small-scale eruption with greater certainty, owing to the 17.4 nm EUV High
Resolution Imager (HRIEUV) of the Extreme Ultraviolet Imager (EUI; Rochus et
al., 2020) 111We used level 2 (L2) EUI data, which can be accessed via
http://sidc.be/EUI/data/releases/202112_release_4.0. Information about the
data processing can be found in the release notes. DOI:
https://doi.org/10.24414/s5da-7e78. on board Solar Orbiter (Müller et al.,
2020). We estimate the evacuated mass and the released energy of the eruption.
A survey of mini coronal dimmings from Solar Orbiter observations is also
carried out for the first time to investigate the distributions and occurrence
frequency. These results provide crucial information for understanding small-
scale eruptions and their contributions to SEPs, the solar wind, and space
weather.

##  2 Results

###  2.1 High-resolution Observations by Solar Orbiter

On 2020 May 20 21:20 universal time (UT), a small-scale eruption with coronal
dimming occurred near the central meridian, which was captured by the recently
launched Solar Orbiter spacecraft. At this moment, Solar Orbiter was located
at a distance of 0.612 au from the Sun during its perihelion pass with a
separation angle of $\sim 17^{\circ}$ from the Sun-Earth line (Figure 1a) and
was almost in quadrature with STEREO-A. From this vantage point, the HRIEUV
angular pixel size of $0^{\prime\prime}.492$ corresponds to 217 km on the
solar surface. Even under such a high spatial resolution, the eruption itself
is still too small (length scale $\sim$10 Mm) to be noticed. By contrast, the
associated coronal dimming is relatively easier to be observed (length scale
$\sim$34 Mm; see Figure 1b, c, d, and the animation of Figure 1; images for
making the video aligned by means of a cross-correlation method to remove the
effect of jitter in the data). Solar Dynamics Observatory (SDO; Pesnell et
al., 2012) allows us to carry out a joint observation of this small event. The
HRIEUV of EUI provides a more distinct EUV image than the Atmospheric Imaging
Assembly (AIA; Lemen et al., 2012) on board SDO. Figure 1e shows clearly
discernible strip structures in an HRIEUV bandpass centered at 17.4 nm. The
distinct bright strips traced by the red dotted curves are considered as the
post-eruption arcades, which are generally associated with newly reconnected
field lines caused by the rise of filaments during the eruption process
(Shibata et al., 1995; Reeves & Golub, 2011). We remap the SDO’s AIA image at
1600 Å and the Helioseismic and Magnetic Imager (HMI; Scherrer et al., 2012;
Schou et al., 2012) line-of-sight (LOS) magnetic fields from the SDO view to
the Solar Orbiter view. It shows that the footpoints of the bright strips are
also brightened in AIA 1600 Å (Figure 1f), which are considered as the flare
ribbons following the eruption. The ribbons are aligned with the relatively
strong photospheric magnetic fields where the post-eruption arcades are rooted
(Figure 1g).

Figure 1: Joint observations of Solar Orbiter and SDO on 2020 May 20. (a)
Heliospheric positions of planets and spacecraft. (b) Location of the mini
coronal dimming (red arrow) in HRIEUV 174 Å. (c) and (d) Enlarged views of the
dimming at different scales. (e) Post-eruption arcades (traced by red dotted
curves) in HRIEUV 174 Å, with location shown by the green arrow in (c). (f)
Flare ribbons in SDO/AIA 1600 Å. (g) SDO/HMI LOS magnetic fields. An animation
of this figure is available. It shows the EUI animation of the dimming from
21:20 to 22:16 UT without annotations. Its real-time duration is 5 s. It has
an FOV slightly larger than (d) but smaller than (c).

###  2.2 Temperature and Mass in the Mini Dimming Region

We use a differential emission measure (DEM) analysis to examine the
temperature and density properties of the coronal dimming. In this work, we
adopt the “simple_reg_dem.pro” routine (Plowman & Caspi, 2020) in SSW packages
to compute the DEM with the AIA EUV images at six passbands (94, 131, 171,
193, 211, and 335 Å) in the temperature range of 5.5 $\leq$ log${}_{10}(T)$
$\leq$ 7.5. This algorithm is fast and relatively free from idiosyncrasies,
which is suitable for long time series analysis of the DEM of numerous pixels.

Figure 2 shows the evolution of EM for 0.7 MK plasma. We select a heart-shaped
area (contoured in red), which almost contains all the dimming areas during
the eruption and meanwhile excludes the bright plasma (i.e., the core region
of the eruption under the post-eruption arcades within the black contours).
The cool-color region within the heart-shaped area (lower density) gradually
expanded (Figures 2a-c). At 21:07 UT, the beginning of the eruption, there are
some preexisting dim locations in the corona (within the heart-shaped area)
that are not associated with the eruption. At 21:40 UT, we can observe a small
dimming area appearing near the core region. Then, the dimming area rapidly
expanded around 22:00 UT and filled the whole heart-shaped area by 22:26 UT.

Figure 2e shows that the changes of the normalized intensities of the flare
region exhibit a trend of increase before 21:00 UT. Figure 2f shows a
significant decrease in the intensity curve of the heart-shaped area after
21:40 UT which is attributed to the occurrence of coronal dimming in that
area. Prior to the flares associated with the dimming, the flare region
exhibited continuous EUV brightenings, which may have been caused by magnetic
reconnection resulting from the motion of the photospheric magnetic field.

Figure 2g shows the averaged EM within the dimming region as a function of
time, which is calculated using

$EM=\int DEM(T)dT.$ (1)

The EM has an obvious decrease after 21:40 UT and almost keeps constant after
22:00 UT, which indicates that coronal plasma probably escaped from the
dimming region during the eruption. The EM-weighted median temperature (red
curve) in Figure 2e is defined as

$\overline{T}=\frac{\int DEM(T)\times TdT}{\int DEM(T)dT}.$ (2)

It almost keeps constant below $10^{6.5}$ K, which is a typical temperature in
a dimming region (Cheng et al., 2012). This implies that the EUV dimming
mainly resulted from plasma depletion during the eruption. By the DEM
analysis, the mass loss within the heart-shaped area is estimated to be
$(1.65\pm 0.54)\times 10^{13}$ g (see Appendix). This mass is 1 to 2 orders
lower than the normal CME mass of $\sim 10^{14}-10^{15}$ g (Vourlidas et al.,
2010).

Figure 2: DEM analysis of the coronal dimming. (a-d) Time evolution of AIA 0.7
MK DEMs at four typical times. (e-f) EUV light curves of the normalized
integrated intensities of the flare region (black contour) and the dimming
region within the heart-shaped area (red contour). (g) Averaged EM (black) and
EM-weighted median temperature (red) within the dimming region as a function
of time. The green vertical dashed lines mark the corresponding times in
(a-d).

###  2.3 Minifilament Eruption and the Associated Magnetic Fields

We do not observe a filament structure in EUI/HRIEUV 174 Å. Fortunately, the
304 Å data from SDO/AIA can provide supplementary information for observing
filament structures that may not be visible in EUI/HRIEUV 174 Å images.
Although AIA has a relatively lower spatial resolution, we were able to track
the erupting filament structure at 304 Å as it rose above the bright
structures at the source region of the eruption (see the animation of Figure
3a). The S-shaped filament (contoured by the dotted line) is probably related
to a rising, twisted flux-rope structure (Figure 3a). Figure 3b shows the
time-distance profile of the rising filament along the slit (3 pixel width) in
Figure 3a. The pattern in Figure 3b implies that the filament rises in a
rolling manner expanding to higher altitudes. The maximum of the local
projected speed is around 12 km $s^{-1}$; the average projected speed is only
around 3.6 km $s^{-1}$. Although it is rather slow compared with regular
filament eruptions (Wang et al., 2016; Cheng et al., 2020), this speed is
considered to be relatively typical for minifilament eruptions (Wang et al.,
2000; Panesar et al., 2020; Sterling et al., 2022). Panesar et al. (2020)
indicate that the speed of the quiet-region jet-making minifilament eruptions
is at least $\sim$ 1 km s-1 during the slow-rise phase. The average projected
speed in our case is comparable to that. Sterling et al. (2022) suggest that
the slow speed reported by Panesar et al. (2020) may be attributed to a
projection effect caused by differences in viewing angles. On the other hand,
time-distance plot helps us to roughly determine the start time of the
eruption, which is around 21:15 UT.

Figure 3: Small-scale filament eruption and the associated magnetic
cancellation. (a) Rising minifilament in AIA 304 Å (contoured by the cyan
dotted line). The black square corresponds to the boundary for extrapolation
in Figure 4. (b) Time-distance profile of the rising filament along the slit
(cyan) in (a). The blue vertical line marks the onset of the filament rise.
(c) Positive (blue) and negative (red) magnetic fields scaled to $\pm 150$ G
corresponding to the black contours in (a). (d) HMI time-distance map along
the slit in (c). The integrated negative magnetic flux as a function of time
computed inside the dotted box in (c) is overplotted in black. A least-squares
polynomial fit curve (red) indicates the trend of the flux. The vertical
dotted-dashed and dashed lines correspond to the times of the flare and
dimming, respectively. An animation of this figure is available. It shows the
AIA 304 Å animation from 21:00 to 21:57 UT, and has an FOV larger than (a).
Its real-time duration is 7 s. The green arrow in the first frame of the
animation indicates the approximate position of the rising filament.

Figure 3c shows the distribution of the SDO/HMI LOS magnetic fields. The
length scale of the major magnetic polarity is around 10 Mm. Filaments with
lengths ranging from 10 to 20 Mm are generally classified as minifilaments
(Wang et al., 2000; Sterling et al., 2015). Therefore, the filament in our
case should fall into this category. The fast moving negative magnetic
polarities cancel the continually emerging positive polarities in the dotted
box region of Figure 3c. We make a narrow slit (3 pixel width) along the
converging direction of the moving negative polarities. The time-distance plot
of Figure 3d displays that the negative magnetic polarities converge with the
emerging positive ones around 19:30 UT. This is similar to the magnetic
cancellation in a normal active region. Flux emergence and magnetic
cancellation are a coupled process and have always been observed to appear
together and interact with each other (Wang et al., 2018, 2022; Chintzoglou et
al., 2019). Due to the emergence of the positive flux, we can only calculate
the change of the negative flux in the dotted box region (black curve in
Figure 3d). The red curve in Figure 3d presents a significant drop in the flux
in the hour before the eruption (the vertical dotted-dashed line). The time-
distance plot patterns indicate a closer convergence of the positive and
negative magnetic poles at the onset of the eruption. Moreover, the
calculation region corresponds to the EUV brightenings at 304 Å observed after
the onset of the filament eruption (see Figure 3a and 3c). These results
suggest that magnetic cancellation occurred before the eruption and likely
resulted in the eruption. Magnetic cancellation is generally related to the
buildup of a magnetic flux rope and is also related to magnetic reconnection
resulting in the final eruption (van Ballegooijen & Martens, 1989; Moore et
al., 2001; Wang et al., 2018, 2022). In fact, eruptions caused by cancellation
in the quiet Sun often manifest as minifilament eruptions with coronal jets
(Sterling et al., 2015; Panesar et al., 2016, 2018; McGlasson et al., 2019;
Muglach, 2021).

Figure 4a shows that the extrapolated magnetic fields exhibiting twisted
morphology along the polarity inversion line (PIL), which is regarded as a
flux rope. A magnetic dip of the flux rope is just above the cancellation
region (see Figure 4a). The cool, dense filament material is probably located
here via a magnetic tension force (Gilbert et al., 2000). Meanwhile, it is
often related to the appearance of bald patches in the PIL. The field lines
threading the bald patch form a separatrix surface where reconnection
preferentially occurs (Titov et al., 1993). Figure 4a shows a hook structure
of the flux rope in the southeast corresponding to the EUV filament at 304 Å
in Figure 4b, which should be part of the entire filament and is located at a
higher altitude above the bright magnetic reconnection region.

We adopt two different methods to reconstruct the flux-rope structure, i.e., a
magnetohydrostatic (MHS) method (Zhu & Wiegelmann, 2018) and a widely used
optimization approach (Wiegelmann, 2004). Both methods give similar twisted
S-shaped flux-rope structures. Figure 4a only displays the reconstructed
results of the MHS for its better performance in the lower corona.

SDO/HMI does not provide the direct boundary data for extrapolation in our
case. We adopt the “bvec2cea.pro” routine in SSW packages to convert the
disambiguated full-disk vector magnetic field “hmi.B_720s” series from the
native CCD coordinates to the cylindrical equal area heliographic coordinates
(Hoeksema et al., 2014), which is appropriate for extrapolation. We centered
the remapped data on Carrington coordinates $(22^{\circ}W,12^{\circ}N)$ where
the major magnetic polarities are located. The “bvec2cea.pro” routine uses a
radial-acute method (Hoeksema et al., 2014) to resolve the 180∘ azimuthal
uncertainty in the transverse field direction. Meanwhile, we cut out a
relatively small region from the remapped data for extrapolation in order to
keep flux balance of the boundary condition (see Figure 3a).

The magnetic free energy is also calculated by the two methods, which give
relatively similar results, i.e., $\Delta E_{free}^{MHS}=(4.87\pm 0.80)\times
10^{27}$ erg, and $\Delta E_{free}^{NLF}=(3.10\pm 0.68)\times 10^{27}$ erg. An
eruption of $\Delta E_{free}\sim 10^{27}$ erg is classified as a microflare by
the flare-energy power law summarized from early flare statistics (Aschwanden
et al., 2000; Schrijver et al., 2012).

Figure 4: Magnetic topological structure of the small-scale filament
(prominence) and its ascending process in the STEREO-A/EUVI view. (a)
Reconstructed flux-rope structure with the EUV background at 304 Å. The
positive (white) in [20, 50] G and negative (black) fluxes in [-20, -50] G are
overplotted. The rising part of the minifilament is contoured in cyan dashed
lines in (b). (c) Side view from STEREO-A/EUVI of the ascending filament (red
arrows) along a red radial line, which shows the slit for the time-distance
map in (d). The red cross marks the approximate position of the eruption
source. An animation of this figure is available. The EUVI animation starts at
2020 May 20 18:06 UT and ends at 2020 May 22 01:58 UT. Its real-time duration
is 7 s. With the exception of the arrows, it has the same annotations as (c)
but a smaller FOV. The red cross disappears as the source rotates to the far
side of the Sun after 2020 May 21 08:00 UT.

##  3 Summary and Discussion

Our results indicate that the small-scale coronal dimming results from an
eruption of a minifilament, which is likely to have similar eruption features
(e.g., post-eruption arcades and flare ribbons), trigger mechanisms
(associated with magnetic cancellation), and magnetic topology (e.g., flux-
rope structures and magnetic dips) to large eruptions. We compare the mass and
energy loss with previous statistics on the power-law relationship between the
flare X-ray fluence and CME mass (Drake et al., 2013). Our results fit the
energy-mass relation, i.e., an eruption of $\Delta E_{free}\sim 10^{27}$ erg
carries the mass in the order of $\sim 10^{13}$ g.

Sterling et al. (2015) and Sterling & Moore (2016) indicate that solar coronal
jets probably result from eruptions of small-scale filaments. Panesar et al.
(2016, 2018) suggest that the continuous cancellation can destabilize
minifilaments and result in eruptions, where internal magnetic reconnection
occurs. The minifilament as shown here has similar size and takes comparable
energy of $\sim 10^{27}$ erg with the small filaments for driving jets.
Moreover, the EUI/HRIEUV observations present jet-like structures at the
beginning of the eruption and post-flare arcades (see animation of Figure 1).
Therefore, this eruption is likely to have similar formation and trigger
mechanisms as coronal jets. McComas et al. (2019) indicate that small,
longitudinally distributed SEP events are associated with jet-like coronal
emissions, some of which are not observed at 1 au but only detected close to
the Sun. Thus, the mini eruptions as shown here become the potential
candidates for these small SEP events. There have been quite a few reports of
small-scale eruptions with coronal dimming (e.g., Innes et al., 2009, 2010;
Yang et al., 2012; Innes & Teriaca, 2013; Zhang & Ji, 2014; Sterling et al.,
2022), but due to limitations in observation conditions, it is difficult to
discern the intricate features of the eruption sources associated with these
events, such as the post-flare arcades or the temperature and density of the
corona. These difficulties result in challenges in determining, for instance,
whether the coronal dimming is a long-lasting effect caused by mass evacuation
or just a wave-like dimming and whether they are associated with the
aforementioned on-disk jets. The combined observations of Solar Orbiter and
SDO in this work have helped to address some of these challenges.

The small-scale eruptions are omnipresent features in the quiet Sun, estimated
with as many as 1400 events per day over the whole Sun (Innes et al., 2009;
Madjarska et al., 2022). Meanwhile, the total number of mini dimmings
(associated with eruption or not) for a full disk in a day (e.g., Alipour et
al., 2012) is much more than twice of Innes et al. (2009). We make a survey of
mini coronal dimmings from EUI/HRIEUV high-resolution observations as shown in
Table 1, which gives the approximate time and location of the dimmings. These
dimmings are all associated with small-scale eruptions. Although the valid
EUI/HRIEUV observations are only available on certain days in certain months,
we still identify at least one dimming event from the valid 1 or 2 hr
observation periods for a day. We only select the relatively obvious events in
the field of view (FOV) of a local area. There should be more events on a
global scale.

On the other hand, we check the torus instability of the flux rope in the
lower coronal (below $\sim$70 Mm) by the decay index $n=-\partial
ln\|\mathbf{B}_{p}\|/\partial lnR$ (Kliem & Török, 2006), where
$\|\mathbf{B}_{p}\|$ is an external poroidal field and $R$ is the height of
the flux-rope axis’s apex above the photosphere, which shows a low critical
height $\sim$8 Mm above the PIL. This height is favorable for the escape of a
CME from its source region. Understandably, weaker quiet-Sun magnetic fluxes
result in weaker constraining force above the flux rope. However, it does not
mean that CMEs can successfully escape into interplanetary space. Figure 4c
and 4d show that the minifilament propagated slowly along a radial direction
in the lower corona with an average speed only $\sim$5.4 km s-1. In fact, the
slow speed is not unexpected, which is also attributed to the weaker quiet-Sun
magnetic fluxes. Namely, they generate a non-strong toroidal current inside
the flux rope, which results in a relatively weak upward hoop force. On the
other hand, quiet-Sun eruptions in the current solar cycle are mainly under
streamer belts where closed background fields are generally dominated (see
Figure 4c). Nonetheless, it remains challenging to determine whether quiet-Sun
eruptions are confined or eruptive. In our case, we find many narrow blobs
moving outwards in the STEREO-A/COR2 view. However, due to the lack of usable
COR1 data, we cannot determine the exact origin of these blobs. Regardless,
quiet-Sun eruptions are more likely to get into torus-unstable regions due to
the relatively weak quiet-Sun magnetic fluxes. The mass and magnetic free
energy can be transferred from the lower corona to the middle or higher
corona, which could become a possible source of the solar wind. Chitta et al.
(2023) gave a direct observational evidence for a coronal separatrix web
driving highly structured slow solar wind through complex middle-coronal
dynamic processes. Perhaps the minifilament eruption we observed could
transport mass and energy to the middle or higher corona, which then becomes
the solar wind through the mechanism proposed by their research.

Table 1: List of the small-scale eruptions with coronal dimmings from EUI/HRIEUV observations at 174 Å Event | Date/TimeaaApproximate onset time of the eruptions. | LocationbbHelioprojective coordinates from the Solar Orbiter perspective. (arcsec) | Event | Date/Time | Location (arcsec)
---|---|---|---|---|---
1 | 2020-05-20/21:20 | -228, 288 | 10 | 2021-03-22/11:40 | 98, 556
2 | 2020-05-21/16:50 | -307, -203 | 11 | 2021-08-21/21:33 | -400, -5
3 | 2020-10-19/20:14 | 192, 147 | 12 | 2021-08-21/22:35 | 152, -68
4 | 2020-10-19/20:14 | 562, 71 | 13 | 2021-08-31/08:54 | 322, 172
5 | 2020-10-21/11:39 | 22, 209 | 14 | 2021-08-31/10:47 | 46, 204
6 | 2020-11-19/12:39 | -526, 375 | 15 | 2022-02-08/13:48 | -32, -115
7 | 2021-02-22/03:51 | -63, 117 | 16 | 2022-02-08/14:00 | -105, -85
8 | 2021-03-22/10:16 | -300, 313 | 17 | 2022-03-27/21:30 | -318, -123
9 | 2021-03-22/11:13 | -154, -119 |  |  |

We thank Xudong Sun for providing suggestions on HMI data processing and D.
Berghmans for valuable suggestions on EUI data. The research was supported by
National Key R&D Program of China No. 2021YFA0718600 and No. 2022YFF0503800,
NSFC under grants 12073032, 42274201, 42004145 and 42150105, the Specialized
Research Fund for State Key Laboratories of China. We acknowledge the use of
data from Solar Orbiter and SDO. Solar Orbiter is a space mission of
international collaboration between ESA and NASA, operated by ESA. The EUI
instrument was built by CSL, IAS, MPS, MSSL/UCL, PMOD/WRC, ROB, LCF/IO with
funding from the Belgian Federal Science Policy Office (BELSPO/PRODEX PEA
4000134088); the Centre National d’Etudes Spatiales (CNES); the UK Space
Agency (UKSA); the Bundesministerium für Wirtschaft und Energie (BMWi) through
the Deutsches Zentrum für Luft- und Raumfahrt (DLR); and the Swiss Space
Office (SSO).

##  Appendix A Mass-loss estimate of the eruption

We use the DEM techniques to determine the CME masses from coronal dimmings.
Considering the limitations posed by the signal-to-noise ratio of the AIA
passbands, we only utilize the passbands centered at 171, 193, and 211 Å,
which are sufficient to encompass the temperature range of the quiet Sun
within the heart-shaped area.

To determine the total evacuated mass $M$ of the eruption, we use

$\begin{split}M&=\sum_{i=0}^{n}\mu m_{p}A_{s}\int_{0}^{L}N(z)dz\\\
&=\sum_{i=0}^{n}\mu
m_{p}A_{s}\lambda_{p}N_{0}I\left(\frac{\lambda_{p}}{R_{\odot}}\right),\end{split}$
(A1)

where $n$ is the total number of the pixels within the heart-shaped area,
$\mu$ the mean molecular weight of the hydrogen ion ($\mu$ = 1.27), mp the
proton mass, As the area of each pixel, $\int_{0}^{L}N(z)dz$ the total number
of coronal plasma at each pixel in the dimming region along the LOS, and
$\lambda_{p}$ the pressure scale height defined as

$\lambda_{p}=\frac{2k_{B}T_{e}}{\mu m_{p}g_{\odot}},$ (A2)

where $k_{B}$ is the Boltzmann constant, $T_{e}$ the electron temperature at
each pixel calculated from the DEM analysis by Equation 2, and $g_{\odot}$ the
gravitational acceleration at the solar surface.

We obtain the value of the basal electron density $N_{0}$ with the help of the
EM measurement (for detailed derivation, please refer to the previous
implementation about CME mass determination of López et al., 2017), i.e.,

$N_{0}=\sqrt{\frac{EM}{\frac{\lambda_{p}}{2}I\left(\frac{\lambda_{p}}{2R_{\odot}}\right)}},$
(A3)

where $R_{\odot}$ is the solar radius, and $I$ the integral quantity defined
as

$I(\alpha)=\int_{0}^{2L/\lambda_{p}}exp\left(-\frac{x}{\alpha x+1}\right)dx.$
(A4)

We have set $L=5\lambda_{p}$ in order to include as much mass as possible
along the LOS.

The total evacuated mass $\Delta M$ is determined by the difference between
the mass $M_{before}$ before the eruption and the mass $M_{after}$ after the
eruption, i.e.,

$\Delta M=M_{before}-M_{after}.$ (A5)

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