# Photospheric velocity evolution of SN 2020bvc: signature of $r$-process
nucleosynthesis from a collapsar

Long Li Deep Space Exploration Laboratory / Department of Astronomy,
University of Science and Technology of China, Hefei 230026, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Shu-Qing Zhong Deep Space Exploration Laboratory / Department of Astronomy,
University of Science and Technology of China, Hefei 230026, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Zi-Gao Dai Deep Space Exploration Laboratory / Department of Astronomy,
University of Science and Technology of China, Hefei 230026, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China

###### Abstract

Whether binary neutron star mergers are the only astrophysical site of rapid
neutron-capture process ($r$-process) nucleosynthesis remains unknown.
Collapsars associated with long gamma-ray bursts (GRBs) and hypernovae are
promising candidates. Simulations have shown that outflows from collapsar
accretion disks can produce enough $r$-process materials to explain the
abundances in the universe. However, there is no observational evidence to
confirm this result at present. SN 2020bvc is a broad-lined type Ic (Ic-BL)
supernova (SN) possibly associated with a low-luminosity GRB. Based on semi-
analytic SN emission models with and without $r$-process materials, we perform
a fitting to the multi-band light curves and photospheric velocities of SN
2020bvc. We find that in a $r$-process-enriched model the mixing of
$r$-process materials slows down the photospheric recession and therefore
matches the velocity evolution better. The fitting results show that
$r$-process materials with mass of $\approx 0.36~{}M_{\odot}$ and opacity of
$\approx 4~{}\rm cm^{2}~{}g^{-1}$ is needed to mix with about half of the SN
ejecta. Our fitting results are weakly dependent on the nebular emission.
Future statistical analysis of a sample of type Ic-BL SNe helps us understand
the contribution of collapsars to the $r$-process abundance.

Core-collapse supernovae (304); Type Ic supernovae (1730); R-process (1324);
Explosive nucleosynthesis (503); Nucleosynthesis (1131)

††software: emcee (Foreman-Mackey et al., 2013)

##  1 Introduction

The astrophysical origin of rapid neutron-capture process ($r$-process)
nucleosynthesis responsible for the production of about half of the elements
heavier than iron remains an open question (e.g., Cowan et al. 1991; Arnould
et al. 2007; Qian & Wasserburg 2007; Thielemann et al. 2011; Cowan et al.
2021; Siegel 2022). Many studies (see Cowan et al. 2021, for a review) suggest
that the extremely high neutron densities ($\gtrsim 10^{26}~{}\rm{cm^{-3}}$)
required for $r$-process nucleosynthesis can be realized in the core collapse
of massive stars or in compact binary mergers involving at least one neutron
star.

GW170817, the first gravitational-wave event from a binary neutron star
merger, was detected by the gravitational-wave observatories Advanced Laser
Interferometer Gravitational-Wave Observatory and Advanced Virgo
Interferometer on August 17, 2017 (Abbott et al., 2017a). Subsequently, a low-
luminosity short gamma-ray burst (GRB) GRB 170817A was detected about 1.7
seconds after the coalescence (Abbott et al., 2017b; Goldstein et al., 2017;
Savchenko et al., 2017). Roughly 11 hours later, an luminous optical
counterpart AT2017gfo was discovered in the galaxy NGC 4993 (Coulter et al.,
2017). Dozens of ground and space telescopes followed up the optical
counterpart in the ultraviolet, optical, and near-infrared bands for about 30
days after the merger (Arcavi et al., 2017; Chornock et al., 2017; Coulter et
al., 2017; Cowperthwaite et al., 2017; Drout et al., 2017; Evans et al., 2017;
Kasliwal et al., 2017; Nicholl et al., 2017; Pian et al., 2017; Smartt et al.,
2017; Soares-Santos et al., 2017; Tanvir et al., 2017). The luminosity and the
multiband light-curve evolution support the hypothesis that AT2017gfo is a
kilonova powered by the radioactive decay of $r$-process nuclei synthesized
within the neutron-rich merger ejecta (Li & Paczyński, 1998; Metzger et al.,
2010; Roberts et al., 2011; Barnes & Kasen, 2013; Rosswog et al., 2018).
Furthermore, Watson et al. (2019) directly identified the neutron-capture
element strontium in the spectra of AT2017gfo. Observations of GW170817, GRB
170817A, and AT2017gfo provide the first direct indication that $r$-process
elements can be produced in binary neutron star merger.

On the other hand, Galactic chemical evolution models have been challenging
neutron star mergers as the only, if not the dominant, sites of $r$-process
nucleosynthesis (e.g., Wehmeyer et al. 2015; Côté et al. 2019; Haynes &
Kobayashi 2019; van de Voort et al. 2020; Molero et al. 2021). The rate of
neutron star mergers seems too low and the delay times relative to star
formation too long to explain the $r$-process elemental abundances of the
extremely metal-poor stars in the Milky Way. The astrophysical sites
associated with core-collapse supernovae (SNe) have long been promising
candidates as the source of main $r$-process contributions. For regular core-
collapse SNe, previous theoretical work has mainly focused on $r$-process
nucleosynthesis in neutrino-driven winds from proto-neutron stars (e.g.,
Takahashi et al. 1994; Woosley et al. 1994). However, both theoretical studies
(unlikely high entropy and/or moderate electron fraction environments; e.g.,
Qian & Woosley 1996; Thompson et al. 2001; Roberts et al. 2012; Martínez-
Pinedo et al. 2012; Curtis et al. 2019) and observations (incompatible with
production yields and occurrence frequencies; e.g., Sneden et al. 2008;
Roederer et al. 2014; Hansen et al. 2018) suggest that regular core-collapse
SNe are unlikely to undergo strong $r$-processes producing heavy elements up
to the actinides, and at most weak $r$-processes synthesising elements below
europium. Recent studies have linked the $r$-process to two rare types of SNe:
magnetorotational/jet-driven SNe (e.g., Nishimura et al. 2015, 2017), and SNe
from collapsars (e.g., Siegel et al. 2019; Barnes & Metzger 2022).

Collapsars, the collapse of rapidly rotating massive stars, are expected to
form black hole-accretion disk systems that produce long GRBs and hypernovae
(MacFadyen & Woosley, 1999). Siegel et al. (2019) found that at sufficiently
high accretion rates, the matter in the mid-plane of the collapsar accretion
disk is driven into a neutron-rich state due to charged-current weak
interactions, thus leading to a rapidly expanding neutron-rich disk outflows
sufficient to undergo strong $r$-process nucleosynthesis. However, when
adopting different configurations (e.g. the treatment of neutrino absorption)
for numerical simulations, the conditions of low electron fraction required
for strong $r$-process are not always be expected (Miller et al., 2020;
Fujibayashi et al., 2022; Just et al., 2022b, a). Due to the high opacity of
the $r$-process material, it is possible to find observational signatures of
the $r$-process in SN light curves and spectra. Depending on the degree of
mixing of the outflows with the ejecta, excess near-infrared emission is
expected to emerge at different stages of the SN light curves (Siegel et al.,
2019; Barnes & Metzger, 2022). Recently, Anand et al. (2023) performed a
systematic study of 25 broad-lined type Ic (Ic-BL) SNe (which are most likely
to be associated with collapasars) with optical/near-infrared bands. However,
fitting results show no convincing evidence of $r$-process enrichment in their
sample.

The SN photospheric velocity evolution derived from the spectra allows
independent constraints on the properties of ejecta and explosions. Previous
studies have shown that 56Ni mixing can affect the photospheric velocity
evolution in the stripped-envelope SNe (e.g., Dessart et al. 2012; Moriya et
al. 2020). In general, with increased 56Ni mixing, the photosphere recedes
slower, thus leading to the photosphere velocity staying at a higher level
over time. It can be expected that, due to the high opacity of the $r$-process
material, the mixing of $r$-process material may lead to a similar effect on
the photospheric velocity.

SN 2020bvc is an Ic-BL SN discovered on February 4, 2020 by the All Sky
Automated Survey of Supernovae (ASAS-SN) in the nearby ($z=0.025235$) galaxy
UGC 9379 (Perley et al., 2020; Stanek, 2020). In addition to optical
observations, radio and X-ray counterparts have been detected by Very Large
Array, Swift-XRT and Chandra, respectively (Izzo et al., 2020; Ho et al.,
2020a). The results of the search for historical GRB data show that there is
no typical GRB associated with 2020bvc, but the presence of a low luminosity
GRB is still possible (Ho et al., 2020b). Considering the very high expansion
velocity and the X-ray light curve, Izzo et al. (2020) suggests that 2020bvc
is associated with an off-axis GRB. However, the radio observations seem to be
inconsistent with the off-axis GRB scenario (Ho et al., 2020b). Ho et al.
(2020b) also present a double-peaked feature in the optical light curve
derived from the Zwicky Transient Facility data, where the first peak can be
attributed to the shock cooling emission. The most noticeable feature of SN
2020bvc is its spectral evolution. Although the early spectra of SN 2020bvc is
similar to that of other SNe Ic-BL (e.g., SN 1998bw), however, the overall
shape of the spectra unexpectedly remains almost constant from about day 20 to
day 73, which is quite different from other SNe Ic-BL (Ho et al., 2020b; Rho
et al., 2021). The measured Fe II $\lambda$5169 absorption velocities between
day 20 and day 73 are almost identical, which are about
$20,000~{}\rm{km~{}s^{-1}}$ (Ho et al., 2020b; Rho et al., 2021).

Recently, Barnes & Metzger (2022) studied the signatures of $r$-process
enrichment in the light curves of SNe from collapsars based on a semi-analytic
model. In this paper, we focus on the features of $r$-process enrichment in
the photospheric velocity evolution. Based on semi-analytic model, we fit the
multi-band light curves and photospheric velocities of SN 2020bvc. In Section
2, we introduce our semi-analytic model. In Section 3, we fit the light curves
and velocity evolution of SN 2020bvc using the semi-analytic model and the
Monte Carlo Markov chain approach, and present the fitting results. Our
discussion and conclusions can be found in Section 4.

##  2 The model

The basic framework of our $r$-process-enriched SN emission model is similar
to Barnes & Metzger (2022). After the SN explosion, a homologously expanding
spherical ejecta is assumed. The radioactive 56Ni synthesized in the explosion
and the $r$-process-rich outflows from the collapsar accretion disk are mixed
with the ejecta to a certain degree due to various hydrodynamic processes. The
observed emission consists of photospheric emission from the photosphere and
nebular emission from the optically thin region. Under the assumption of
homologous expansion, the photospheric velocity can be easily derived from the
photospheric radius. Next we show the details of our model.

###  2.1 Density Profile

A few seconds after the SN explosion, the ejecta begins to enter the
homologous expansion phase after expanding to a few times its radius. The
density profile of the SN ejecta can be approximated as a segmented power law,
with a steep power law at large radii and a flatter power law at small radii.
Given the total kinetic energy ($E_{\rm SN}$) and total mass ($M_{\rm ej}$) of
the ejecta, the density profile can be expressed as (Chevalier, 1982;
Chevalier & Soker, 1989; Matzner & McKee, 1999; Kasen & Bildsten, 2010)

$\rho(v,t)=\begin{dcases}\zeta_{\rho}\frac{M_{\mathrm{ej}}}{v_{\mathrm{t}}^{3}t^{3}}\left(\frac{r}{v_{\mathrm{t}}t}\right)^{-\delta}&v<v_{\mathrm{t}},\\\
\zeta_{\rho}\frac{M_{\mathrm{ej}}}{v_{\mathrm{t}}^{3}t^{3}}\left(\frac{r}{v_{\mathrm{t}}t}\right)^{-n}&v>v_{\mathrm{t}},\end{dcases}$
(1)

where $v_{t}$ is the transition velocity which can be derived from the density
continuity condition,

$v_{\mathrm{t}}=\zeta_{v}\left(\frac{E_{\mathrm{SN}}}{M_{\mathrm{ej}}}\right)^{1/2}.$
(2)

The coefficients $\zeta_{\rho}$ and $\zeta_{v}$ are given by

$\zeta_{\rho}=\frac{(n-3)(3-\delta)}{4\pi(n-\delta)},$ (3)
$\zeta_{v}=\left[\frac{2(5-\delta)(n-5)}{(n-3)(3-\delta)}\right]^{1/2}.$ (4)

A dimensionless radius of the break in the SN ejecta density profile from the
inner flatter component to the outer steeper component $x_{0}$ is introduced
to link $v_{t}$ and the characteristic velocity ($v_{\rm SN}$) of the SN
ejecta (Chatzopoulos et al., 2012),

$v_{\mathrm{SN}}=\frac{v_{\mathrm{t}}}{x_{0}}.$ (5)

Equation (1) is valid only if $n>5$ and $\delta<3$. The parameter $n$ depends
on the properties of the progenitor and the convection in the outer layers of
the star. For the progenitors of type Ib/c and Ia SNe, one has $n\simeq 10$
(Matzner & McKee, 1999; Kasen & Bildsten, 2010; Moriya et al., 2013). For the
progenitors of red supergiants, one has $n\simeq 12$ (Matzner & McKee, 1999;
Moriya et al., 2013). Typical values of the slope of the inner density profile
of the ejecta is generally taken as $\delta\simeq 0-2$. In our model, we adopt
$\delta=1,n=10$ as default values and obtain
$\zeta_{\rho}=0.124,\zeta_{v}=1.69$.

###  2.2 Material Mixing

Due to various hydrodynamic instabilities (e.g., Rayleigh-Taylor instability),
large-scale mixing may occur between the core and the envelope of the ejecta,
the collapsar disk outflows and the ejecta. The degree of mixing is highly
uncertain. In particular, mixing of the radioactive 56Ni and $r$-process
material alters the internal energy and opacity profiles, thus changing the
observational behaviour of SNe. In our model, we put 56Ni and $r$-process
material in the ejecta with different degrees of mixing. We introduce a
maximum dimensionless mixing radius of 56Ni ($x_{\rm Ni}$) and $r$-process
material ($x_{\rm rp}$). Under $x_{\rm Ni}$ and $x_{\rm rp}$, 56Ni and
$r$-process material are evenly distributed within the ejecta. Considering a
more realistic scenario, we impose the requirement that $x_{\rm rp}<x_{\rm
Ni}$.

###  2.3 Radioactive Heating

The decay of 56Ni and $r$-process material heats up the ejecta and provides
the energy source for SNe. The specific radioactive heating rate due to the
decay chain 56Ni$\rightarrow^{56}$Co$\rightarrow^{56}$Fe can be written as
(Colgate & McKee, 1969; Colgate et al., 1980; Arnett, 1980, 1982)

$\dot{q}_{\mathrm{Ni}}=\left(\epsilon_{\mathrm{Ni}}-\epsilon_{\mathrm{Co}}\right)e^{-t/\tau_{\mathrm{Ni}}}+\epsilon_{\mathrm{Co}}e^{-t/\tau_{\mathrm{Co}}},$
(6)

where $\epsilon_{\mathrm{Ni}}=3.9\times
10^{10}~{}\rm{erg}~{}\rm{g}^{-1}~{}\rm{s}^{-1}$,
$\epsilon_{\mathrm{Co}}=6.78\times
10^{9}~{}\rm{erg}~{}\rm{g}^{-1}~{}\rm{s}^{-1}$. $\tau_{\mathrm{Ni}}=8.8~{}\rm
days$ and $\tau_{\mathrm{Co}}=111.3~{}\rm days$ are the decay time of 56Ni
decays to 56Co and 56Co decays to 56Fe, respectively.

The specific radioactive heating rate from the $r$-process material can be
written as (Metzger et al., 2010; Korobkin et al., 2012)

$\dot{q}_{\rm rp}=4\times 10^{18}\left[\frac{1}{2}-\frac{1}{\pi}{\rm
arctan}\left(\frac{t-t_{0}}{\sigma}\right)\right]^{1.3}~{}\rm
erg~{}s^{-1}~{}g^{-1}$ (7)

with $t_{0}=1.3$ s and $\sigma=0.11$ s.

We simply assume that the energy from the decay of 56Ni and $r$-process
material heats the ejecta with 100% efficiency. This assumption is roughly
satisfied in the optically thick region. This may result in a slight
underestimation of the mass of 56Ni in our model, but will not affect the mass
of $r$-process material because the heating from $r$-process decay is
secondary with respect to the heating due to the 56Ni decay chain in almost
all stages of SN evolution (Siegel et al., 2019). For the optically thin
region, this assumption would lead to an overestimation of the nebular
luminosity. However, as discussed in Sections 3 and 4, the uncertainty of the
nebular emission has a minor influence on the results of this work.

###  2.4 Opacity

A wavelength-independent gray opacity is assumed in our model. The opacity of
any region in the ejecta depends on its composition and can be written as

$\kappa=\kappa_{\rm sn}(1-X_{\rm rp})+\kappa_{\rm rp}X_{\rm rp},$ (8)

where $\kappa_{\rm sn}$ is the opacity of the $r$-process-free ejecta, $X_{\rm
rp}$ and $\kappa_{\rm rp}$ are the the mass fraction and the opacity of the
$r$-process-rich outflows, respectively. $\kappa_{\rm rp}$ depends mainly on
the electron fraction $Y_{e}$ of the outflows, which is quite uncertain
considering the complexity of neutrino transport and angular momentum
transport in numerical simulations. Siegel et al. (2019) found that for
neutrino-cooled disks with sufficiently high accretion rates ($\sim
0.003-0.1~{}M_{\odot}~{}\rm s^{-1}$), the electron fraction in the disk
midplane (and then the outflows) can reach $Y_{e}\lesssim 0.2$. Simulations
employing different neutrino transport solutions show that the electron
fractions of the outflows can be raised to $Y_{e}\gtrsim 0.25$ (Just et al.,
2022b), $Y_{e}\gtrsim 0.3$ (Miller et al., 2020), and $Y_{e}\gtrsim 0.4$
(Fujibayashi et al., 2022), respectively. The electron fraction determines the
abundance of lanthanides and actinides that dominate the opacity of the
mixture. Tanaka et al. (2020) found that the opacities for the mixture of
$r$-process elements to be $\kappa_{\rm rp}\sim 20-30~{}\rm cm^{2}~{}g^{-1}$
for $Y_{e}\lesssim 0.2$, $\kappa_{\rm rp}\sim 3-5~{}\rm cm^{2}~{}g^{-1}$ for
$Y_{e}\approx 0.25-0.35$, and $\kappa_{\rm rp}\sim 1~{}\rm cm^{2}~{}g^{-1}$
for $Y_{e}\approx 0.4$ at temperature $T=5-10\times 10^{3}~{}\rm K$.
Therefore, as a result of the numerical simulations, $\kappa_{\rm rp}$ ranges
roughly from $1$ to $30~{}\rm cm^{2}~{}g^{-1}$. Barnes & Metzger (2022) fixes
the opacity of the ejecta without 56Ni and $r$-process material to $0.05~{}\rm
cm^{2}~{}g^{-1}$, and expresses the opacity of pure 56Ni as a temperature-
dependent segmentation function varying between $0.01~{}\rm cm^{2}~{}g^{-1}$
and $0.1~{}\rm cm^{2}~{}g^{-1}$. In this work, we set $\kappa_{\rm
sn}=0.05~{}\rm cm^{2}~{}g^{-1}$ by assuming that the two cases above have the
same opacity. This difference is negligible considering the large opacity of
the $r$-process material.

###  2.5 Internal Energy Evolution and Emerging Radiation

We divide the homologously expanded ejecta into a series of concentric shell
layers. The mass of each shell layer can be derived from the density profile
in Section 2.1. The evolution of the internal energy of the $i$th layer is
given by the following energy conservation equation

$\frac{dE_{{\rm int},i}}{dt}=m_{{\rm Ni},i}\dot{q}_{{\rm Ni}}+m_{{\rm
rp},i}\dot{q}_{{\rm rp}}-L_{i}-\frac{E_{{\rm
int},i}}{r_{i}}\frac{dr_{i}}{dt},$ (9)

where $E_{{\rm int},i}$ is the internal energy, $L_{i}$ is the luminosity of
the net flow of energy from the $i$th layer, $m_{{\rm Ni},i}$ and $m_{{\rm
rp},i}$ are the mass of 56Ni and $r$-process material, $r_{i}$ is the radius
of the layer.

The output luminosity contributed by the $i$th layer can be estimated by

$L_{i}=\frac{E_{{\rm
int},i}}{\max\left[t_{\mathrm{d},i},t_{\mathrm{lc},i}\right]}$ (10)

where $t_{\mathrm{d},i}$ is the radiation diffusion timescale,
$t_{\mathrm{lc},i}=r_{i}/c$ limits the energy loss time to the light crossing
time. The radiation diffusion timescale of the $i$th layer is given by

$t_{\mathrm{d},i}=\frac{3\kappa}{4\pi
r_{i}c}\sum_{i^{\prime}=i}^{n}m_{i^{\prime}},$ (11)

where the effect of the shell outside the $i$th layer is considered by summing
the masses of layers $i$ to $n$. Finally, by summarising the contributions of
all layers, the emerging bolometric luminosity of the entire ejecta can be
obtained as follows

$L_{\rm bol}(t)=\sum\limits_{i}L_{i}.$ (12)

In optically thick regions, where matter is in local thermal equilibrium,
radiation tends towards the blackbody spectrum. Assuming that all radiation in
the optically thick regions is emitted from the photosphere in the form of a
standard blackbody spectrum, one can define an effective temperature

$T_{\rm eff}=\left(\frac{L_{\rm ph}}{4\pi\sigma_{\rm SB}R_{\rm
ph}^{2}}\right)^{1/4},$ (13)

where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant, $L_{\rm ph}$ is the
photospheric luminosity by summing the luminosity below the photospheric
radius $R_{\rm ph}$. The photosphere is defined as the position where the
optical depth is 2/3 from the outermost layer of the ejecta inwards, thus
$R_{\rm ph}$ can be given by

$\int_{R_{\text{ph}}}^{R_{\text{SN}}}\rho\kappa dr=\frac{2}{3},$ (14)

where $R_{\text{SN}}$ is the radius of the outermost layer of the ejecta. For
homologously expanding ejecta, the photospheric velocity can be easily derived
from

$v_{\text{ph}}=\frac{v_{\text{SN}}R_{\text{ph}}}{R_{\text{SN}}}.$ (15)

The flux density of the photospheric emission at photon frequency $\nu$ can be
given by

$F_{\nu}(t)=\frac{2\pi h\nu^{3}}{c^{2}}\frac{1}{\exp(h\nu/kT_{\rm
eff})-1}\frac{R_{\rm ph}^{2}}{D_{\rm L}^{2}},$ (16)

where $h$ is the Planck constant, $k$ is the Boltzmann constant, $D_{\rm
L}=114$ Mpc is the luminosity distance of SN 2020bvc.

The nebular emission of ejecta mixed with $r$-process material from optically
thin regions is highly uncertain. We follow Barnes & Metzger (2022) in
treating nebular emission as a combination of of two different spectral energy
distributions (SEDs) associated with $r$-process and $r$-process-free
material. The blackbody spectrum at a temperature of 2500 K is treated as an
$r$-process SED. However, for the $r$-process-free material, we loose the
limitation on the $r$-process-free SED and express it as an empirical
quadratic function

$\mathcal{F}(\nu)=-f_{a}(\nu-f_{h})^{2}+f_{k},$ (17)

where $f_{a},f_{h},f_{k}$ are positive coefficients. As shown in Figure 1, the
empirical function can roughly reproduce the late photometry of type Ib/c SNe.

Finally, we determine the monochromatic AB magnitude by
$M_{\nu}=-2.5\log_{10}(F_{\nu}/3631\rm Jy)$.

Figure 1: The late-time photometry of SN 2007gr (Ic, 140 days and 380 days;
Hunter et al. 2009), SN 2007C (Ib, 90 days; Bianco et al. 2014), SN 2007I (Ic-
BL, 80 days; Bianco et al. 2014), SN 2007uy (Ib, 100 days; Bianco et al.
2014). We fit the photometry for each epoch with a parabola and the results
are represented by solid lines in the corresponding colors. For comparison, we
plot the scaled empirical $r$-process-free nebular SEDs derived from fitting
the observations of SN 2020bvc based on $r$-process-enriched and $r$-process-
free models, represented by the red and blue solid lines, respectively.

###  2.6 Shock Cooling Emission

In order to explain the first peak of SN 2020bvc, we introduced the shock
cooling emission in our model. We assume that the progenitor was surrounded by
extended material with mass $M_{e}$ and radius $R_{e}$. As the SN shock
powered by the explosion energy $E_{\rm SN}$ moves from the ejecta to the
extended material, it will transition to a new velocity $v_{e}$ which can be
expressed in the following (Nakar & Piro, 2014)

$v_{e}\approx 1.5\times 10^{9}E_{\rm
SN,51}^{0.5}\left(\frac{M_{\mathrm{ej}}}{3M_{\odot}}\right)^{-0.35}\left(\frac{M_{\mathrm{e}}}{0.01M_{\odot}}\right)^{-0.15}\mathrm{~{}cm}\mathrm{~{}s}^{-1},$
(18)

where $E_{\rm SN,51}=E_{\rm SN}/10^{51}$ erg, then the energy carried by the
extended material is

$E_{e}\approx 2\times 10^{49}E_{\rm
SN,51}\left(\frac{M_{\mathrm{ej}}}{3M_{\odot}}\right)^{-0.7}\left(\frac{M_{\mathrm{e}}}{0.01M_{\odot}}\right)^{0.7}\mathrm{erg}.$
(19)

Assuming that the extended material has a uniform density profile
$\rho_{e}(t)=3M_{\mathrm{e}}/4\pi R(t)^{3}$, where
$R(t)=R_{\mathrm{e}}+v_{\mathrm{e}}t$, the luminosiy of shock cooling emission
can be expressed analytically as (Piro, 2015)

$L_{e}(t)=\frac{t_{e}E_{e}}{t_{p}^{2}}\exp\left[-\frac{t\left(t+2t_{e}\right)}{2t_{p}^{2}}\right],$
(20)

where $t_{p}$ is the diffusion timescale and can be roughly estimated as
(Arnett, 1982)

$t_{p}\approx\left(\frac{M_{e}\kappa_{e}}{v_{e}c}\right)^{1/2},$ (21)

where $\kappa_{e}$ is the opacity of the extented material, which we set equal
to the opacity of the ejecta without $r$-process material, i.e.,
$\kappa_{e}=\kappa_{\rm sn}=0.05~{}\rm cm^{2}~{}g^{-1}$.

##  3 Fitting and results

The multi-band light curves of SN 2020bvc are taken from Ho et al. (2020b) and
Rho et al. (2021), where the early double-peaked light curves are mainly
contributed by Ho et al. (2020b). The $UBVgriz$-band data used for the fitting
are all presented in the AB magnitude system (the $UBV$-band data from Rho et
al. (2021) have been calibrated from Vega magnitudes to AB magnitudes), as
shown in Figure 2. In addition, the Fe II $\lambda$5169 absorption velocities
of SN 2020bvc from Rho et al. (2021) are regarded as part of the data used for
the fitting111Here we consider the Fe II $\lambda$5169 line velocities of SN
2020bvc as photospheric velocities (e.g., Branch et al. 2002; for a caveat see
Dessart et al. 2015., as shown in Figure 3.

We develop a Fortran-based numerical model based on the description in Section
2 and implement the Markov Chain Monte Carlo (MCMC) techniques by using the
emcee Python package (Foreman-Mackey et al., 2013). F2PY is employed to
provide a connection between Python and Fortran languages. We perform the MCMC
with 26 walkers for running at least 100,000 steps, until the step is longer
than 50 times the integrated autocorrelation time $\tau_{\rm ac}$, i.e.
$N_{\rm step}=\max(10^{5},50\tau_{\rm ac})$ to make sure that the fitting is
sufficiently converged (Foreman-Mackey et al., 2013). Once the MCMC is done,
the best-fitting values and the $1\sigma$ uncertainties are computed as the
50th, 16th, and 84th percentiles of the posterior samples.

The free parameters required for the fitting include: the total kinetic energy
of SN ejecta ($E_{\rm SN}$), the total mass of SN ejecta ($M_{\rm ej}$), the
mass of radioactive 56Ni ($M_{\rm Ni}$), the mass of $r$-process material
($M_{\rm rp}$), the opacity of $r$-process material ($\kappa_{\rm rp}$), the
dimensionless radius of the break in the SN ejecta density profile from the
inner component to the outer component ($x_{0}$), the maximum dimensionless
mixing radius of 56Ni ($x_{\rm Ni}$), the maximum dimensionless mixing radius
of $r$-process material ($x_{\rm rp}$), the SN explosion time ($t_{\rm
shift}$), the total mass of extended material ($M_{e}$), the maximum radius of
extended material ($R_{e}$), the coefficients of the empirical function of the
$r$-process-free nebular SED ($f_{a}$, $f_{h}$, and $f_{k}$). Uniform priors
on $E_{\rm SN}/10^{51}~{}\rm erg$, $M_{\rm ej}/M_{\odot}$, $M_{\rm
Ni}/M_{\odot}$, $M_{\rm rp}/M_{\odot}$, $\kappa_{\rm rp}/{\rm
cm^{2}~{}g^{-1}}$, $x_{0}$, $x_{\rm Ni}$, $x_{\rm rp}$, $t_{\rm shift}/\rm
day$, $M_{e}/M_{\odot}$, $R_{e}/R_{\odot}$, $\log f_{a}$, $\log f_{h}$, and
$\log f_{k}$ are adopted in this paper. In order to explore a parameter space
as large as possible, we set a wide enough range for the priors. As a
comparison, we also perform a fitting based on a $r$-process-free model that
simply makes $M_{\rm rp}=0$. The free parameters and priors in our model can
be seen in Table 1.

Figure 2 shows the best-fitting multi-band light curves based on the
$r$-process-enriched model and the $r$-process-free model. We find that for
both models the overall quality of the fitting is good, except for a few
photometric data that deviate slightly from the best-fitting light curves.
This deviation is mainly caused by the inconsistency of the photometric data.
After showing the photometric data from different telescopes with different
markers, one can find a significant offset between the late-time $BVgri$-band
light curves from LCO (filled circles) and that from Konkoly (open circles).
The late-time $z$-band light curve from Konkoly also shows two different
trends. Due to the difference in filters, there is also an inconsistency of
$U$-band photometry between LCO from Rho et al. (2021) and Swift/UVOT from Ho
et al. (2020b). Figure 3 shows the best-fitting photospheric velocity
evolution based on both models. The best-fitting photospheric velocities from
the $r$-process-enriched model matches well with the Fe II $\lambda$5169
absorption velocities of SN 2020bvc, while those from $r$-process-free model
drop off so rapidly, in contrast to the almost identical high Fe II velocities
at late times.

Figure 2: Multi-band light curves of SN 2020bvc and the best-fitting results
for SN models with (left) and without (right) $r$-process material. Data for
LCO (filled circles), Konkoly (open circles), and ZTF (filled Triangles) are
from Rho et al. 2021. Early photometric data from Ho et al. 2020 are
represented by open squares. The time zero point is set to the most recent
nondetection before the first detection (MJD = 58,882.67; Ho et al. 2020). The
solid and dotted lines represent the total and photospheric emission,
respectively.

Figure 3: Fe II $\lambda$5169 absorption velocity evolution of SN 2020bvc and
the best-fitting photospheric velocity evolutions for SN models with (left)
and without (right) $r$-process material. Data are taken from Rho et al. 2021.

Table 1 shows the best-fitting parameters and their corresponding 1$\sigma$
uncertainties from both models. The reduced $\chi^{2}$ derived by both models
are large. $\chi^{2}_{\rm tot}/{\rm dof}\approx 55$ for $r$-process-enriched
model is slightly smaller than $\chi^{2}_{\rm tot}/{\rm dof}\approx 59$ for
$r$-process-free model. When only counting multi-band light curves,
$\chi^{2}_{\rm lc}/{\rm dof}\approx 56$ and $\chi^{2}_{\rm lc}/{\rm
dof}\approx 59$ for $r$-process-enriched and $r$-process-free models. When
only considering velocity evolution, $\chi^{2}_{\rm vel}/{\rm dof}\approx 3$
and $\chi^{2}_{\rm vel}/{\rm dof}\approx 70$ for $r$-process-enriched and
$r$-process-free models, respectively. This suggests that the $r$-process-
enriched model has almost no advantage over the $r$-process-free model in
explaining the multi-band light curves of SN 2020bvc, but the $r$-process-
enriched model is significantly better than the $r$-process-free model in
explaining the velocity evolution. To verify whether the large $\chi^{2}$ is
related to the dispersion of the data, we divided the dataset into two groups
based on the evolution of the multi-band light curves: the flatter evolving
dataset A1, and the steeper evolving dataset A2, as shown in Figure 6 in
Appendix A. After fitting the $r$-process-enriched model to data sets A1 and
A2, we derived $\chi^{2}_{\rm tot}/{\rm dof}\approx 45$ and $\chi^{2}_{\rm
tot}/{\rm dof}\approx 58$, respectively. This suggests that the $r$-process-
enriched model is more compatible with data set A1, and that the dispersion of
the data is only responsible for a fraction of the large $\chi^{2}$.

The best-fitting results from both models are similar. We find that in order
to explain the main peak requires a total kinetic energy of $E_{\rm SN}\approx
2\times 10^{52}~{}\rm erg$, a total ejecta mass of $M_{\rm ej}\approx
5.7-5.9~{}M_{\odot}$, and a radioactive nickel mass of $M_{\rm Ni}\approx
0.6-0.8~{}M_{\odot}$. The maximum dimensionless mixing radius $x_{\rm
Ni}\approx 1.0$ suggests that the 56Ni is almost completely mixed with the
entire ejecta. In order to explain the first peak of the light curves, an
extended material with $M_{e}\approx 0.05~{}M_{\odot}$, $R_{e}\approx
130-180~{}R_{\odot}$ is needed. The estimated total mass of the extended
material is lower than other type Ic SNe (e.g., Piro 2015; Taddia et al.
2018), but optically thick enough to ensure continuous propagation of the
shock (e.g. Equation 1 from Piro 2015). In Figure 1, we plot the best-fitting
$r$-process-free nebular SEDs as well as the late-time photometry of several
SNe. The spectral shape are similar to those of the late-time photometry of
type Ib/c SNe, with a quite flat profile between the $B$ and $z$ bands. For
$r$-process-enriched model, the intermediate opacity $\kappa_{\rm rp}\approx
4~{}\rm cm^{2}~{}g^{-1}$ derived from the fitting results suggests that the
outflows may have only undergone the $r$-process extending to the second
abundance peak ($A\sim 130$), with a small or zero mass fraction of
lanthanides. A total $r$-process mass of $M_{\rm rp}\approx 0.36~{}M_{\odot}$
and a maximum mixing radius of $x_{\rm rp}\approx 0.5$ are needed to account
for the photospheric velocity evolution. The velocity evolution of the
$r$-process-enriched model is comparable to that of the $r$-process-free model
until about 30 days, when the $r$-process material is still deeply buried
within the photosphere. Thereafter, $r$-process material begins to emerge,
slowing the recession of the photosphere due to its great opacity, keeping the
photoshperic velocity at a high level for a long time. In Appendix B, we
display the corner plots showing the results of our MCMC parameter estimation.
We note that several parameters show degeneracies, as it can be seen from the
correlations in Figure 8 and 9. This means that there are many combinations of
parameters that are compatible with the observations. In particular, there is
a strong anti-correlation between $M_{\rm rp}$ and $\kappa_{\rm rp}$. It can
be expected that outflows with high opacity/low mass or low opacity/high mass
combinations can also explain the observed multi-band light curves and
velocity evolution.

In Figure 4, we plot the luminosity evolution of the different components. The
mixing of the $r$-process material redistributes the energy between the
luminosity components. The late-time photosphere luminosity $L_{\rm ph}$ is
comparable to the $r$-process-free nebular luminosity $L_{\rm neb}^{\rm sn}$,
in contrast to the late-time dominance of $L_{\rm neb}^{\rm sn}$ in the
$r$-process-free model. Within 80 days, however, the contribution of the
$r$-process nebular luminosity $L_{\rm neb}^{\rm rp}$ is not dominant. This
implies that the uncertainty in the $r$-process nebular SED has little effect
on the fitting results.

Figure 4: Photospheric luminosity ($L_{\rm ph}$), $r$-process nebular
luminosity ($L_{\rm neb}^{\rm rp}$), and $r$-process-free nebular luminosity
($L_{\rm neb}^{\rm sn}$) evolutions dervied from the best-fitting results for
SN models with (left) and without (right) $r$-process material.  Table 1: Free
parameters, priors, and best-fitting results in our model.

Parameter | Prior A | Result A | Prior B | Result B
---|---|---|---|---
| (with $r$-process) | (without $r$-process)
$E_{\rm SN}~{}(10^{51}~{}{\rm erg})$ | $[0,100]$ | $21.10^{+1.44}_{-1.81}$ | $[0,100]$ | $20.43^{+2.02}_{-2.25}$
$M_{\rm ej}~{}(M_{\odot})$ | $[0,10]$ | $5.88^{+0.21}_{-0.27}$ | $[0,10]$ | $5.72^{+0.66}_{-0.75}$
$M_{\rm Ni}~{}(M_{\odot})$ | $[0,1]$ | $0.77^{+0.03}_{-0.04}$ | $[0,1]$ | $0.61^{+0.03}_{-0.03}$
$M_{\rm rp}~{}(M_{\odot})$ | $[0,1]$ | $0.36^{+0.26}_{-0.21}$ | - | -
$\kappa_{\rm rp}~{}(\rm cm^{2}~{}g^{-1})$ | $[1,30]$ | $4.29^{+2.09}_{-2.28}$ | - | -
$x_{0}$ | $[0,1]$ | $0.66^{+0.00}_{-0.00}$ | $[0,1]$ | $0.64^{+0.00}_{-0.00}$
$x_{\rm Ni}$ | $[0,1]$ | $0.98^{+0.01}_{-0.01}$ | $[0,1]$ | $0.97^{+0.01}_{-0.01}$
$x_{\rm rp}$ | $[0,1]$ | $0.54^{+0.03}_{-0.02}$ | - | -
$t_{\rm shift}~{}(\rm day)$ | $[0,0.67]$ | $0.00^{+0.00}_{-0.00}$ | $[0,0.67]$ | $0.00^{+0.00}_{-0.00}$
$M_{e}~{}(M_{\odot})$ | $[0,1]$ | $0.05^{+0.02}_{-0.01}$ | $[0,1]$ | $0.05^{+0.02}_{-0.03}$
$R_{e}~{}(R_{\odot})$ | $[50,5000]$ | $180.51^{+49.56}_{-59.77}$ | $[50,5000]$ | $130.96^{+65.22}_{-29.05}$
$\log[f_{a}]$ | $[-50,0]$ | $-44.12^{+0.07}_{-0.06}$ | $[-50,0]$ | $-44.39^{+0.04}_{-0.03}$
$\log[f_{h}]$ | $[13,15]$ | $14.66^{+0.01}_{-0.01}$ | $[13,15]$ | $14.67^{+0.01}_{-0.01}$
$\log[f_{k}]$ | $[-20,0]$ | $-15.01^{+0.04}_{-0.04}$ | $[-20,0]$ | $-15.29^{+0.03}_{-0.02}$
$\chi^{2}_{\rm lc}/\rm{dof}$ | - | $20533/365$ | - | $21775/368$
$\chi^{2}_{\rm vel}/\rm{dof}$ | - | $21/7$ | - | $492/7$
$\chi^{2}_{\rm tot}/\rm{dof}$ | - | $20554/372$ | - | $22267/375$

Note. — The uncertainties of the best-fitting parameters are measured as
$1\sigma$ confidence ranges.

##  4 Discussion & Conclusions

In Figure 5, we compare the Fe II $\lambda$5169 absorption velocity evolution
of SN 2020bvc with a large sample of SNe Ic and SNe Ic-BL (Modjaz et al.,
2016; Liu et al., 2017). The early high velocities of SN 2020bvc are well
consistent with those of SNe Ic-BL, but significantly higher than those of SNe
Ic. At the $V$-band maximum, the weighted average absorption velocities in SNe
Ic-BL and SNe Ic are $18,500\pm 7400~{}\rm km~{}s^{-1}$ and $8000\pm
1400~{}\rm km~{}s^{-1}$ (Liu et al., 2017), respectively, while the absorption
velocity for SN 2020bvc is $\sim 21,000~{}\rm km~{}s^{-1}$. Thereafter, the
average velocities of SNe Ic/Ic-BL continued to decrease to $\lesssim
10,000~{}\rm km~{}s^{-1}$, which contrasts with the unusually slow decreasing
velocities of 2020bvc. The slow decay of the velocities of SN 2020bvc is
followed by a long plateau phase, which has also been observed in other SNe
Ic/Ic-BL, but the plateau of SN 2020bvc is much higher ($\sim 16,000~{}\rm
km~{}s^{-1}$). Previous radiation transfer simulations of stripped-envelope
SNe indeed show a late flat evolution of the photospheric velocity, which is
similar to the observed velocity evolution of SNe Ic/Ic-BL and may be
attributed to recombination of ejecta and/or other radiative transfer effects
(e.g. Dessart et al. 2012; Moriya et al. 2020). However, modelling such a high
velocity plateau for SN 2020bvc based on radiative transfer models is a
challenge, and a wide parameter space needs to be explored to determine
whether observations can be interpreted without invoking the $r$-process.

Figure 5: Fe II $\lambda 5169$ absorption velocities of SN 2020bvc (black)
compared to the weighted moving average velocities of SNe Ic (blue) and SNe
Ic-BL (orange) from Liu et al. (2017).

The largest uncertainty in our emission model arises from the uncertainty in
the $r$-process-free nebular SED. Barnes & Metzger (2022) fitted the
photometric data from $t\approx 120$ days after B-band maximum of type Ic SN
2007gr with a spline and adopted it as the $r$-process-free nebular SED.
However, it is questionable whether the late-time SED of SN 2007gr can act as
a representative of standard $r$-process-free SNe. As shown in Figure 1, the
late-time photometry of type Ib/c SNe exhibits diversity. Even for the same SN
(SN 2007gr), the late-time photometry is different at different epochs. We
therefore loosen the restrictions on the $r$-process-free nebular SED by
including an empirical function with three free parameters in our fitting.
Nevertheless the $r$-process-free nebular SED will only have an effect on the
late-time multi-band light curves, not on the early-time light curves and the
photospheric velocity evolution. This is crucial as the properties of the
explosion and ejecta are mainly determined by the main peak and the early-time
photospheric velocity. This conclusion is supported by the fact that the best-
fitting parameters derived from both models are almost identical, although
their $r$-process-free nebular SEDs are not the same. Therefore, the
uncertainty in the nebular SED has little impact on the properties of ejecta
and explosions. On the other hand, the photospheric velocity is mainly
determined by the photospheric location, which in turn is governed by the
density and opacity of the ejecta. The density profile of the ejecta is
related to the total mass of the ejecta. As the fitting result for
$r$-process-free model shows, for a certain ejecta mass (determined by the
early-time light curves), ejecta with typical density profile and opacity
cannot reproduce the late-time photospheric velocity evolution of SN 2020bvc.
The plausible explanation for the consistently high photosperic velocity is
that the exceptionally high opacity of the material in the inner layers of the
ejecta prevents the recession of the photosphere. In our model, this high
opacity material originates from the neutron-rich disk outflows of the
collaspar that have undergone a $r$-process nucleosynthesis. The derived
opacity of $r$-process material $\kappa_{\rm rp}\approx 4~{}\rm
cm^{2}~{}g^{-1}$ suggests the electron fraction of the outflows to be
$Y_{e}\approx 0.25-0.35$. The required mass of $r$-process material $M_{\rm
rp}\approx 0.36~{}M_{\odot}$ suggests a total accreted mass of $\gtrsim
1.2~{}M_{\odot}$ based on the numerical simulations (Siegel & Metzger, 2017,
2018; Fernández et al., 2019). In addition, the possible association of SN
2020bvc with a low luminosity GRB also hints at the origin of the collapsar.

In this paper, we investigate the multi-band light curves and photospheric
velocity evolution of SN 2020bvc based on a semi-analytic emission model with
and without $r$-process matter. Although the $r$-process-enriched model has no
advantage over the $r$-process-free model in explaining the multi-band light
curves of SN 2020bvc, the $r$-process-enriched model can reproduce the
photospheric velocity evolution significantly better than the $r$-process-free
model. The fitting results are robust and weakly dependent on the $r$-process
and $r$-process-free nebular SEDs. We therefore conclude that SN2020bvc is
likely to have originates from a collapsar. A moderate amount of $r$-process
material was produced in the disk outflows of this collapsar and partially
mixed with the ejecta to significantly reshape the photospheric velocity
evolution of SN 2020bvc. Future statistical analysis of a sample of type Ic-BL
SNe with both multi-band photometry and rich and detailed spectroscopic
information will help us to understand how much collapsars can contribute to
the abundance of the $r$-process abundance in the Universe.

We thank the anonymous referee for the helpful suggestions that allowed us to
improve our manuscript. L.L. and S.Q.Z. acknowledge support from the National
Natural Science Foundation of China (grant No. 12247144) and China
Postdoctoral Science Foundation (grant Nos. 2021TQ0325 and 2022M723060).
Z.G.D. is supported by the National Key Research and Development Program of
China (grant No. 2017YFA0402600), National SKA Program of China (grant No.
2020SKA0120300), and National Natural Science Foundation of China (grant No.
11833003).

##  Appendix A Best-fitting results for data sets A1 and A2

We show the best-fitting multi-band light curves and photospheric velocity
evolution derived from fitting the $r$-process-enriched model to data sets
A1/A2 in Figures 6 and 7, respectively. Table 2 shows the corresponding best-
fitting parameters and their $1\sigma$ uncertainties.

Figure 6: Multi-band light curves for 2020bvc from data sets A1 (left) and A2
(right), and the best-fitting results based on $r$-process-enriched model.
Data for LCO (filled circles), Konkoly (open circles), and ZTF (filled
Triangles) are from Rho et al. 2021. Early photometric data from Ho et al.
2020 are represented by open squares. The time zero point is set to the most
recent nondetection before the first detection (MJD = 58,882.67; Ho et al.
2020). The solid and dotted lines represent the total and photospheric
emission, respectively.

Figure 7: Fe II $\lambda$5169 absorption velocity evolution of SN 2020bvc and
the best-fitting photospheric velocity evolution derived from fitting the
$r$-process-enriched model to data sets A1 (left) and A2 (right). Data are
taken from Rho et al. 2021.  Table 2: Free parameters, priors, and best-
fitting results in our model.

Parameter | Prior A | Result A1 | Result A2
---|---|---|---
$E_{\rm SN}~{}(10^{51}~{}{\rm erg})$ | $[0,100]$ | $20.58^{+1.49}_{-1.78}$ | $21.93^{+2.20}_{-2.79}$
$M_{\rm ej}~{}(M_{\odot})$ | $[0,10]$ | $5.33^{+0.26}_{-0.26}$ | $5.89^{+0.53}_{-0.15}$
$M_{\rm Ni}~{}(M_{\odot})$ | $[0,1]$ | $0.70^{+0.02}_{-0.03}$ | $0.74^{+0.02}_{-0.02}$
$M_{\rm rp}~{}(M_{\odot})$ | $[0,1]$ | $0.21^{+0.19}_{-0.11}$ | $0.15^{+0.07}_{-0.07}$
$\kappa_{\rm rp}~{}(\rm cm^{2}~{}g^{-1})$ | $[1,30]$ | $4.26^{+2.26}_{-2.35}$ | $5.98^{+1.05}_{-1.31}$
$x_{0}$ | $[0,1]$ | $0.61^{+0.01}_{-0.00}$ | $0.44^{+0.00}_{-0.00}$
$x_{\rm Ni}$ | $[0,1]$ | $0.99^{+0.01}_{-0.01}$ | $0.98^{+0.01}_{-0.02}$
$x_{\rm rp}$ | $[0,1]$ | $0.50^{+0.03}_{-0.02}$ | $0.40^{+0.01}_{-0.02}$
$t_{\rm shift}~{}(\rm day)$ | $[0,0.67]$ | $0.00^{+0.00}_{-0.00}$ | $0.00^{+0.00}_{-0.00}$
$M_{e}~{}(M_{\odot})$ | $[0,1]$ | $0.05^{+0.01}_{-0.00}$ | $0.04^{+0.01}_{-0.01}$
$R_{e}~{}(R_{\odot})$ | $[50,5000]$ | $130.32^{+60.20}_{-53.87}$ | $131.25^{+66.55}_{-79.03}$
$\log[f_{a}]$ | $[-50,0]$ | $-44.39^{+0.05}_{-0.05}$ | $-44.17^{+0.04}_{-0.05}$
$\log[f_{h}]$ | $[13,15]$ | $14.70^{+0.01}_{-0.01}$ | $14.70^{+0.01}_{-0.01}$
$\log[f_{k}]$ | $[-20,0]$ | $-15.25^{+0.03}_{-0.04}$ | $-15.15^{+0.03}_{-0.03}$
$\chi^{2}_{\rm lc}/\rm{dof}$ | - | $12761/280$ | $17368/292$
$\chi^{2}_{\rm vel}/\rm{dof}$ | - | $19/7$ | $36/7$
$\chi^{2}_{\rm tot}/\rm{dof}$ | - | $12780/287$ | $17404/299$

Note. — The uncertainties of the best-fitting parameters are measured as
$1\sigma$ confidence ranges.

##  Appendix B Corner plot

We show the corner plots for the $r$-process-enriched and $r$-process-free
models in Figure 8 and 9, respectively.

Figure 8: Corner plots showing the one and two dimensional projections of the
posterior probability distributions of parameters for the $r$-process-enriched
model.  Figure 9: Corner plots showing the one and two dimensional projections
of the posterior probability distributions of parameters for the $r$-process-
free model.

## References

  * Abbott et al. (2017a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, Phys. Rev. Lett., 119, 161101, doi: 10.1103/PhysRevLett.119.161101
  * Abbott et al. (2017b) —. 2017b, ApJ, 848, L12, doi: 10.3847/2041-8213/aa91c9
  * Anand et al. (2023) Anand, S., Barnes, J., Yang, S., et al. 2023, arXiv e-prints, arXiv:2302.09226, doi: 10.48550/arXiv.2302.09226
  * Arcavi et al. (2017) Arcavi, I., Hosseinzadeh, G., Howell, D. A., et al. 2017, Nature, 551, 64, doi: 10.1038/nature24291
  * Arnett (1980) Arnett, W. D. 1980, ApJ, 237, 541, doi: 10.1086/157898
  * Arnett (1982) —. 1982, ApJ, 253, 785, doi: 10.1086/159681
  * Arnould et al. (2007) Arnould, M., Goriely, S., & Takahashi, K. 2007, Phys. Rep., 450, 97, doi: 10.1016/j.physrep.2007.06.002
  * Barnes & Kasen (2013) Barnes, J., & Kasen, D. 2013, ApJ, 775, 18, doi: 10.1088/0004-637X/775/1/18
  * Barnes & Metzger (2022) Barnes, J., & Metzger, B. D. 2022, ApJ, 939, L29, doi: 10.3847/2041-8213/ac9b41
  * Bianco et al. (2014) Bianco, F. B., Modjaz, M., Hicken, M., et al. 2014, ApJS, 213, 19, doi: 10.1088/0067-0049/213/2/19
  * Branch et al. (2002) Branch, D., Benetti, S., Kasen, D., et al. 2002, ApJ, 566, 1005, doi: 10.1086/338127
  * Chatzopoulos et al. (2012) Chatzopoulos, E., Wheeler, J. C., & Vinko, J. 2012, ApJ, 746, 121, doi: 10.1088/0004-637X/746/2/121
  * Chevalier (1982) Chevalier, R. A. 1982, ApJ, 258, 790, doi: 10.1086/160126
  * Chevalier & Soker (1989) Chevalier, R. A., & Soker, N. 1989, ApJ, 341, 867, doi: 10.1086/167545
  * Chornock et al. (2017) Chornock, R., Berger, E., Kasen, D., et al. 2017, ApJ, 848, L19, doi: 10.3847/2041-8213/aa905c
  * Colgate & McKee (1969) Colgate, S. A., & McKee, C. 1969, ApJ, 157, 623, doi: 10.1086/150102
  * Colgate et al. (1980) Colgate, S. A., Petschek, A. G., & Kriese, J. T. 1980, ApJ, 237, L81, doi: 10.1086/183239
  * Côté et al. (2019) Côté, B., Eichler, M., Arcones, A., et al. 2019, ApJ, 875, 106, doi: 10.3847/1538-4357/ab10db
  * Coulter et al. (2017) Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, Science, 358, 1556, doi: 10.1126/science.aap9811
  * Cowan et al. (2021) Cowan, J. J., Sneden, C., Lawler, J. E., et al. 2021, Reviews of Modern Physics, 93, 015002, doi: 10.1103/RevModPhys.93.015002
  * Cowan et al. (1991) Cowan, J. J., Thielemann, F.-K., & Truran, J. W. 1991, Phys. Rep., 208, 267, doi: 10.1016/0370-1573(91)90070-3
  * Cowperthwaite et al. (2017) Cowperthwaite, P. S., Berger, E., Villar, V. A., et al. 2017, ApJ, 848, L17, doi: 10.3847/2041-8213/aa8fc7
  * Curtis et al. (2019) Curtis, S., Ebinger, K., Fröhlich, C., et al. 2019, ApJ, 870, 2, doi: 10.3847/1538-4357/aae7d2
  * Dessart et al. (2012) Dessart, L., Hillier, D. J., Li, C., & Woosley, S. 2012, MNRAS, 424, 2139, doi: 10.1111/j.1365-2966.2012.21374.x
  * Dessart et al. (2015) Dessart, L., Hillier, D. J., Woosley, S., et al. 2015, MNRAS, 453, 2189, doi: 10.1093/mnras/stv1747
  * Drout et al. (2017) Drout, M. R., Piro, A. L., Shappee, B. J., et al. 2017, Science, 358, 1570, doi: 10.1126/science.aaq0049
  * Evans et al. (2017) Evans, P. A., Cenko, S. B., Kennea, J. A., et al. 2017, Science, 358, 1565, doi: 10.1126/science.aap9580
  * Fernández et al. (2019) Fernández, R., Tchekhovskoy, A., Quataert, E., Foucart, F., & Kasen, D. 2019, MNRAS, 482, 3373, doi: 10.1093/mnras/sty2932
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fujibayashi et al. (2022) Fujibayashi, S., Sekiguchi, Y., Shibata, M., & Wanajo, S. 2022, arXiv e-prints, arXiv:2212.03958, doi: 10.48550/arXiv.2212.03958
  * Goldstein et al. (2017) Goldstein, A., Veres, P., Burns, E., et al. 2017, ApJ, 848, L14, doi: 10.3847/2041-8213/aa8f41
  * Hansen et al. (2018) Hansen, T. T., Holmbeck, E. M., Beers, T. C., et al. 2018, ApJ, 858, 92, doi: 10.3847/1538-4357/aabacc
  * Haynes & Kobayashi (2019) Haynes, C. J., & Kobayashi, C. 2019, MNRAS, 483, 5123, doi: 10.1093/mnras/sty3389
  * Ho et al. (2020a) Ho, A. Y. Q., Cenko, B., Perley, D., Corsi, A., & Brightman, M. 2020a, Transient Name Server AstroNote, 2020-45, 1
  * Ho et al. (2020b) Ho, A. Y. Q., Kulkarni, S. R., Perley, D. A., et al. 2020b, ApJ, 902, 86, doi: 10.3847/1538-4357/aba630
  * Hunter et al. (2009) Hunter, D. J., Valenti, S., Kotak, R., et al. 2009, A&A, 508, 371, doi: 10.1051/0004-6361/200912896
  * Izzo et al. (2020) Izzo, L., Auchettl, K., Hjorth, J., et al. 2020, A&A, 639, L11, doi: 10.1051/0004-6361/202038152
  * Just et al. (2022a) Just, O., Aloy, M. A., Obergaulinger, M., & Nagataki, S. 2022a, ApJ, 934, L30, doi: 10.3847/2041-8213/ac83a1
  * Just et al. (2022b) Just, O., Goriely, S., Janka, H. T., Nagataki, S., & Bauswein, A. 2022b, MNRAS, 509, 1377, doi: 10.1093/mnras/stab2861
  * Kasen & Bildsten (2010) Kasen, D., & Bildsten, L. 2010, ApJ, 717, 245, doi: 10.1088/0004-637X/717/1/245
  * Kasliwal et al. (2017) Kasliwal, M. M., Nakar, E., Singer, L. P., et al. 2017, Science, 358, 1559, doi: 10.1126/science.aap9455
  * Korobkin et al. (2012) Korobkin, O., Rosswog, S., Arcones, A., & Winteler, C. 2012, MNRAS, 426, 1940, doi: 10.1111/j.1365-2966.2012.21859.x
  * Li & Paczyński (1998) Li, L.-X., & Paczyński, B. 1998, ApJ, 507, L59, doi: 10.1086/311680
  * Liu et al. (2017) Liu, Y.-Q., Modjaz, M., & Bianco, F. B. 2017, ApJ, 845, 85, doi: 10.3847/1538-4357/aa7f74
  * MacFadyen & Woosley (1999) MacFadyen, A. I., & Woosley, S. E. 1999, ApJ, 524, 262, doi: 10.1086/307790
  * Martínez-Pinedo et al. (2012) Martínez-Pinedo, G., Fischer, T., Lohs, A., & Huther, L. 2012, Phys. Rev. Lett., 109, 251104, doi: 10.1103/PhysRevLett.109.251104
  * Matzner & McKee (1999) Matzner, C. D., & McKee, C. F. 1999, ApJ, 510, 379, doi: 10.1086/306571
  * Metzger et al. (2010) Metzger, B. D., Martínez-Pinedo, G., Darbha, S., et al. 2010, MNRAS, 406, 2650, doi: 10.1111/j.1365-2966.2010.16864.x
  * Miller et al. (2020) Miller, J. M., Sprouse, T. M., Fryer, C. L., et al. 2020, ApJ, 902, 66, doi: 10.3847/1538-4357/abb4e3
  * Modjaz et al. (2016) Modjaz, M., Liu, Y. Q., Bianco, F. B., & Graur, O. 2016, ApJ, 832, 108, doi: 10.3847/0004-637X/832/2/108
  * Molero et al. (2021) Molero, M., Simonetti, P., Matteucci, F., & della Valle, M. 2021, MNRAS, 500, 1071, doi: 10.1093/mnras/staa3340
  * Moriya et al. (2013) Moriya, T. J., Maeda, K., Taddia, F., et al. 2013, MNRAS, 435, 1520, doi: 10.1093/mnras/stt1392
  * Moriya et al. (2020) Moriya, T. J., Suzuki, A., Takiwaki, T., Pan, Y.-C., & Blinnikov, S. I. 2020, MNRAS, 497, 1619, doi: 10.1093/mnras/staa2060
  * Nakar & Piro (2014) Nakar, E., & Piro, A. L. 2014, ApJ, 788, 193, doi: 10.1088/0004-637X/788/2/193
  * Nicholl et al. (2017) Nicholl, M., Berger, E., Kasen, D., et al. 2017, ApJ, 848, L18, doi: 10.3847/2041-8213/aa9029
  * Nishimura et al. (2017) Nishimura, N., Sawai, H., Takiwaki, T., Yamada, S., & Thielemann, F. K. 2017, ApJ, 836, L21, doi: 10.3847/2041-8213/aa5dee
  * Nishimura et al. (2015) Nishimura, N., Takiwaki, T., & Thielemann, F.-K. 2015, ApJ, 810, 109, doi: 10.1088/0004-637X/810/2/109
  * Perley et al. (2020) Perley, D., Schulze, S., & Bruch, R. 2020, Transient Name Server AstroNote, 2020-37, 1
  * Pian et al. (2017) Pian, E., D’Avanzo, P., Benetti, S., et al. 2017, Nature, 551, 67, doi: 10.1038/nature24298
  * Piro (2015) Piro, A. L. 2015, ApJ, 808, L51, doi: 10.1088/2041-8205/808/2/L51
  * Qian & Wasserburg (2007) Qian, Y. Z., & Wasserburg, G. J. 2007, Phys. Rep., 442, 237, doi: 10.1016/j.physrep.2007.02.006
  * Qian & Woosley (1996) Qian, Y. Z., & Woosley, S. E. 1996, ApJ, 471, 331, doi: 10.1086/177973
  * Rho et al. (2021) Rho, J., Evans, A., Geballe, T. R., et al. 2021, ApJ, 908, 232, doi: 10.3847/1538-4357/abd850
  * Roberts et al. (2011) Roberts, L. F., Kasen, D., Lee, W. H., & Ramirez-Ruiz, E. 2011, ApJ, 736, L21, doi: 10.1088/2041-8205/736/1/L21
  * Roberts et al. (2012) Roberts, L. F., Reddy, S., & Shen, G. 2012, Phys. Rev. C, 86, 065803, doi: 10.1103/PhysRevC.86.065803
  * Roederer et al. (2014) Roederer, I. U., Preston, G. W., Thompson, I. B., et al. 2014, AJ, 147, 136, doi: 10.1088/0004-6256/147/6/136
  * Rosswog et al. (2018) Rosswog, S., Sollerman, J., Feindt, U., et al. 2018, A&A, 615, A132, doi: 10.1051/0004-6361/201732117
  * Savchenko et al. (2017) Savchenko, V., Ferrigno, C., Kuulkers, E., et al. 2017, ApJ, 848, L15, doi: 10.3847/2041-8213/aa8f94
  * Siegel (2022) Siegel, D. M. 2022, Nature Reviews Physics, 4, 306, doi: 10.1038/s42254-022-00439-1
  * Siegel et al. (2019) Siegel, D. M., Barnes, J., & Metzger, B. D. 2019, Nature, 569, 241, doi: 10.1038/s41586-019-1136-0
  * Siegel & Metzger (2017) Siegel, D. M., & Metzger, B. D. 2017, Phys. Rev. Lett., 119, 231102, doi: 10.1103/PhysRevLett.119.231102
  * Siegel & Metzger (2018) —. 2018, ApJ, 858, 52, doi: 10.3847/1538-4357/aabaec
  * Smartt et al. (2017) Smartt, S. J., Chen, T. W., Jerkstrand, A., et al. 2017, Nature, 551, 75, doi: 10.1038/nature24303
  * Sneden et al. (2008) Sneden, C., Cowan, J. J., & Gallino, R. 2008, ARA&A, 46, 241, doi: 10.1146/annurev.astro.46.060407.145207
  * Soares-Santos et al. (2017) Soares-Santos, M., Holz, D. E., Annis, J., et al. 2017, ApJ, 848, L16, doi: 10.3847/2041-8213/aa9059
  * Stanek (2020) Stanek, K. Z. 2020, Transient Name Server Discovery Report, 2020-50, 1
  * Taddia et al. (2018) Taddia, F., Sollerman, J., Fremling, C., et al. 2018, A&A, 609, A106, doi: 10.1051/0004-6361/201629874
  * Takahashi et al. (1994) Takahashi, K., Witti, J., & Janka, H. T. 1994, A&A, 286, 857
  * Tanaka et al. (2020) Tanaka, M., Kato, D., Gaigalas, G., & Kawaguchi, K. 2020, MNRAS, 496, 1369, doi: 10.1093/mnras/staa1576
  * Tanvir et al. (2017) Tanvir, N. R., Levan, A. J., González-Fernández, C., et al. 2017, ApJ, 848, L27, doi: 10.3847/2041-8213/aa90b6
  * Thielemann et al. (2011) Thielemann, F. K., Arcones, A., Käppeli, R., et al. 2011, Progress in Particle and Nuclear Physics, 66, 346, doi: 10.1016/j.ppnp.2011.01.032
  * Thompson et al. (2001) Thompson, T. A., Burrows, A., & Meyer, B. S. 2001, ApJ, 562, 887, doi: 10.1086/323861
  * van de Voort et al. (2020) van de Voort, F., Pakmor, R., Grand, R. J. J., et al. 2020, MNRAS, 494, 4867, doi: 10.1093/mnras/staa754
  * Watson et al. (2019) Watson, D., Hansen, C. J., Selsing, J., et al. 2019, Nature, 574, 497, doi: 10.1038/s41586-019-1676-3
  * Wehmeyer et al. (2015) Wehmeyer, B., Pignatari, M., & Thielemann, F. K. 2015, MNRAS, 452, 1970, doi: 10.1093/mnras/stv1352
  * Woosley et al. (1994) Woosley, S. E., Wilson, J. R., Mathews, G. J., Hoffman, R. D., & Meyer, B. S. 1994, ApJ, 433, 229, doi: 10.1086/174638


# Near-threshold scaling of resonant inelastic collisions at ultralow
temperatures

Rebekah Hermsmeier1, Adrien Devolder2, Paul Brumer2, and Timur V. Tscherbul1
1Department of Physics, University of Nevada, Reno, NV, 89557, USA
2Chemical Physics Theory Group, Department of Chemistry, and Center for
Quantum Information and Quantum Control, University of Toronto, Toronto,
Ontario, M5S 3H6, Canada

###### Abstract

We show that the cross sections for a broad range of resonant inelastic
processes accompanied by excitation exchange (such as spin-exchange, Förster
resonant, or angular momentum exchange) exhibit an unconventional near-
threshold scaling $E^{\Delta m_{12}}$, where $E$ is the collision energy,
$\Delta m_{12}=m_{1}^{\prime}+m_{2}^{\prime}-m_{1}-m_{2}$, and $m_{i}$ and
$m_{i}^{\prime}$ are the initial and final angular momentum projections of the
colliding species ($i=1,\,2$). In particular, the inelastic cross sections for
$\Delta m_{12}=0$ transitions display an unconventional $E^{0}$ scaling
similar to that of elastic cross sections, and their rates vanish as
$T^{\Delta m_{12}+1/2}$. For collisions dominated by even partial waves (such
as those of identical bosons in the same internal state) the scaling is
modified to $\sigma_{\text{inel}}\propto E^{\Delta m_{12}+1}$ if $\Delta
m_{12}$ is odd. We present accurate quantum scattering calculations that
illustrate these modified threshold laws for resonant spin exchange in
ultracold Rb + Rb and O2 \+ O2 collisions. Our results illustrate that the
threshold scaling of collision cross sections is determined only by the
energetics of the underlying process (resonant vs. exothermic) rather than by
whether the internal states of colliding particles is changed in the
collision.

Introduction. The unique controllability of ultracold atomic and molecular
collisions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] gives rise to numerous applications
of ultracold quantum gases in quantum information science and on precision
tests of foundations of chemical and statistical physics [5, 11]. Further,
much progress has been achieved over the last few years in our ability to
observe the reactants and products of molecular collisions and chemical
reactions in single, well-controlled quantum states [6, 7, 8, 9, 11]. Recent
examples include precision studies of photoexcited reaction complexes [9] and
product state distributions [11] in the ultracold chemical reaction KRb + KRb
$\to$ K2 \+ Rb2, electric field-induced dipolar exchange resonances in KRb +
KRb collisions [7], and microwave shielding of NaK + NaK collisions [8]. In
addition, observations have been made of ultracold atom-molecule collisions in
an optically trapped Na - NaLi(${}^{3}\Sigma$) mixture, of CaF$(^{2}\Sigma)$
\+ CaF$(^{2}\Sigma)$ collisions in an optical dipole trap [12] and of cold
O${}_{2}(^{3}\Sigma)$ \+ O${}_{2}(^{3}\Sigma)$ collisions in a magnetically
trapped gas of oxygen molecules [13].

Ultracold atoms and molecules are typically prepared and trapped in quantum
states with a fixed value of the angular momentum projection $m$ on a space-
fixed quantization axis (typically fixed by an external electromagnetic
field). Angular momentum projection-changing collisions play a key role in
ultracold chemistry. If such collisions are exothermic, they convert the
intramolecular energy into translational energy of the relative motion,
leading to undesirable trap loss and heating. In contrast, resonant excitation
exchange (EE) collisions conserve the internal energy but lead to a coherent
transfer of excitations (such as spin polarization) from one particle to
another. Examples of resonant EE processes include spin-exchange in atomic,
molecular, and ionic collisions [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25], Förster resonances in collisions of Rydberg atoms [26, 27, 28, 29, 30,
31], atom-dimer exchange chemical reactions [32, 20], excitation exchange
between identical atoms or molecules [33], charge transfer in cold ion-atom
collisions [34, 35, 36], and rotational angular momentum projection-changing
collisions [10].

Spin-exchange collisions, in particular, play an important role in a wide
array of research fields, ranging from spin-exchange optical pumping [14, 15,
16, 17], cold chemistry [32, 20, 21], precision magnetometry [18], and
astrophysics [37, 38, 39, 40, 41] to quantum many-body physics [22, 23, 24,
25] and quantum information processing, where they are used to generate
entangled states [42, 43] and to operate quantum logic gates [44, 45, 46, 47,
48]. Recent experiments observed spin-exchange collisions of ultracold alkali-
metal atoms [49, 50], Förster resonant energy transfer in Rydberg atom
collisions [28, 29, 30], and electric field-induced resonances in collisions
of rotationally excited KRb molecules [7]. We have recently found that
ultracold EE collisions are amenable to extensive coherent control [10] due to
the suppression of $m_{12}$-changing collisions at ultralow temperatures.

However, despite the ever-increasing interest in ultracold EE collisions,
there remains a considerable uncertainty about the scaling of their integral
cross sections (ICS) near collision thresholds [51, 52]. It is commonly
believed that the inelastic ICSs scale as $k^{-1}$ with the incident collision
wavevector $k$ in the limit $k\to 0$ [53, 54]. While some computational
studies did observe the $k^{-1}$ scaling [16, 39, 21] others reported that the
ICS for atomic EE collisions approach a constant value [40, 41, 35, 38], a
highly unexpected observation that has, to our knowledge, remained
unexplained. Several authors attributed the $k^{0}$ scaling to a deficiency of
the elastic approximation [55, 56, 57], which effectively neglects the
hyperfine structure of colliding atoms [38, 39, 19]. More recently, the
constant near-threshold scaling of spin-exchange cross sections was claimed to
be incorrect [39], further deepening the existing controversy surrounding the
threshold behavior of EE collisions.

Krems and Dalgarno presented a derivation of the threshold laws for
$m$-changing collisions of an open-shell atom (or rotating molecule) with a
spherically symmetric particle [58]. However, useful as it is, their analysis
could not be applied to describe EE processes, where both species exchange
their angular momenta. As such, a comprehensive understanding of the threshold
behavior of EE processes is still lacking, limiting our ability to control a
wide range of ultracold EE collisions that are key to implementing quantum
logic gates [44, 45, 46, 47, 48] and to generating entanglement [42, 43] and
strong correlations [22, 23, 25, 24] in quantum many-body systems.

Here, we resolve the long-standing controversy regarding the threshold
behavior of EE cross sections. By deriving the threshold laws for ultracold EE
collisions of two species both of which possess internal angular momenta, we
show that the ICS for these processes display a collision energy scaling
$\sigma_{\text{EE}}\propto E^{|\Delta m_{12}|}$. Here, $\Delta
m_{12}=m_{1}^{\prime}+m_{2}^{\prime}-m_{1}-m_{2}$, and $m_{i}$ and
$m_{i}^{\prime}$ are the initial and final angular momentum projections of the
colliding species ($i=1,\,2$). In the important particular case $\Delta
m_{12}=0$, the $s$-wave inelastic ICS approaches a constant value and the
corresponding rate vanishes as $T^{1/2}$ as typically expected of elastic
collisions [54]. To our knowledge, these results provide the first conclusive
explanation for the unconventional $k^{0}$ near-threshold scaling of the
resonant spin-exchange and charge exchange processes observed in a number of
previous quantum scattering calculations [38, 35]. To illustrate our findings,
we present numerically exact quantum scattering calculations of ultracold O2
\+ O2 and Rb + Rb collisions, which confirm our analytic results and provide a
blueprint for observing the unconventional threshold scaling of inelastic
cross sections in ultracold atomic and molecular collision experiments. (For
clarification, note that inelastic refers to any collision where the nature or
internal state of a colliding partner changes [59, 60]).

Theory. Consider a binary collision of two atoms and/or molecules, each
possessing total internal angular momenta $\mathbf{j}_{1}$ and
$\mathbf{j}_{2}$. The nature of the operators $\mathbf{j}_{i}$ is set by the
details of the internal structure of the colliding species, which is
immaterial for the discussion below. We focus on the case of weak external
fields, where $j_{i}$ remain good quantum numbers, and consider as numerical
examples, collisions of two open-shell 17O${}_{2}(^{3}\Sigma)$ molecules
($\mathbf{j}=\mathbf{S}+\mathbf{N}$, where $S$ is the electron spin and $N$ is
the rotational angular momentum) and of two 87Rb atoms
($\mathbf{j}=\mathbf{S}+\mathbf{I}$, where $\mathbf{I}$ is the atomic nuclear
spin).

The starting point for our discussion is the expression for the integral cross
section for two particles initially colliding in well-defined angular momentum
eigenstates $|j_{i}m_{i}\rangle$ (the subscripts $j_{1}$ and $j_{2}$ will be
omitted for brevity unless stated otherwise)

$\sigma_{m_{1}m_{2}\to
m_{1}^{\prime}m_{2}^{\prime}}=\frac{\pi}{k^{2}}\sum_{l,m_{l}}\sum_{l\textquoteright,m\textquoteright_{l}}|T_{m_{1}m_{2},lm_{l}\to
m_{1}^{\prime}m_{2}^{\prime},l\textquoteright m\textquoteright_{l}}|^{2}$ (1)

where $|lm_{l}\rangle$ and $|l^{\prime}m_{l}^{\prime}\rangle$ are the
eigenstates of the orbital angular momentum $\hat{L}^{2}$ of the collision
complex and its projection $\hat{L}_{z}$ on the space-fixed quantization axis.
We are interested in the threshold scaling of the ICS with the collision
energy $E$, which is determined by that of the transition $T$-matrix elements
$T_{m_{1}m_{2},lm_{l}\to m_{1}^{\prime}m_{2}^{\prime},l\textquoteright
m\textquoteright_{l}}$. To make the threshold scaling more explicit, we
rewrite the $T$-matrix in the total angular momentum representation [61, 62]

$T_{j_{1}m_{1}j_{2}m_{2},lm_{l}\to
j_{1}^{\prime}m_{1}^{\prime}j_{2}^{\prime}m_{2}^{\prime},l\textquoteright
m\textquoteright_{l}}=\sum_{J,M}\sum_{j_{12},m_{12}}\sum_{j\textquoteright_{12},m\textquoteright_{12}}(2J+1)\\\
\sqrt{(2j_{12}+1)(2j\textquoteright_{12}+1)}(-1)^{l\textquoteright+l}T^{J}_{j_{1}j_{2}j_{12}l\to
j_{1}^{\prime}j_{2}^{\prime}j\textquoteright_{12}l\textquoteright}\\\
(-1)^{j_{12}+j\textquoteright_{12}+m_{12}+m\textquoteright_{12}}\begin{pmatrix}j_{1}&j_{2}&j_{12}\\\
m_{1}&m_{2}&-m_{12}\end{pmatrix}\begin{pmatrix}j_{12}&l&J\\\
m_{12}&m_{l}&-M\end{pmatrix}\\\
(-1)^{j_{1}+j\textquoteright_{1}-j_{2}-j\textquoteright_{2}}\begin{pmatrix}j\textquoteright_{1}&j\textquoteright_{2}&j\textquoteright_{12}\\\
m\textquoteright_{1}&m\textquoteright_{2}&-m\textquoteright_{12}\end{pmatrix}\begin{pmatrix}j\textquoteright_{12}&l\textquoteright&J\\\
m\textquoteright_{12}&m\textquoteright_{l}&-M\end{pmatrix}$ (2)

where $j_{12}=|\mathbf{j}_{12}|=|\mathbf{j}_{1}+\mathbf{j}_{2}|$ is the total
internal angular momentum, and
$J=|\mathbf{J}|=|\mathbf{j}_{1}+\mathbf{j}_{2}+\mathbf{l}|$ is the total
angular momentum of the collision complex, which is conserved in the absence
of external fields.

The advantages of using Eq. (2) are two-fold. First, the threshold behavior of
the $T$-matrix elements $T^{J}_{j_{1}j_{2}j_{12}l\to
j_{1}^{\prime}j_{2}^{\prime}j\textquoteright_{12}l\textquoteright}$ only
depends on $l$ and $l^{\prime}$ through the Wigner threshold law [51, 52]

$T_{\gamma l,\gamma^{\prime}l^{\prime}}\propto
k^{l+1/2}(k^{\prime})^{l^{\prime}+1/2},$ (3)

where $k$ and $k^{\prime}$ are the incident and final wavevectors [51, 63, 52]
and $\gamma$ stands for the internal quantum numbers $j_{1}$, $j_{2}$, and
$j_{12}$ (i.e. all quantum numbers other than $l$). We note that Eq. (3)
assumes the absence of near-threshold resonance, bound, and virtual states. A
different $k$ scaling would result if such states were present [52, 63, 64].
In the case of resonant scattering in the absence of external fields
considered below, the initial and final states are degenerate ($k=k^{\prime}$)
and the near-threshold dependence of $T$-matrix elements takes the form

$T_{\gamma l,\gamma^{\prime}l^{\prime}}\propto k^{l+l^{\prime}+1}.$ (4)

In the limit of zero collision energy ($k\to 0$) it follows from Eq. (4) that
the $T$-matrix elements with the lowest $l$ and $l\textquoteright$ provide the
dominant contributions to the sum in Eq. (2) and hence to the ICS (1).

Second, the 3$j$-symbols $(:::)$ in Eq. (2) make explicit the rotational
symmetry restrictions on the possible values of $l,m_{l},l^{\prime}$, and
$m^{\prime}_{l}$. The 3-$j$ symbols must satisfy the selection rule
$m_{12}+m_{l}=M=m\textquoteright_{12}+m\textquoteright_{1}$,$m_{1}+m_{2}=m_{12}$,
and $m\textquoteright_{12}=m\textquoteright_{1}+m\textquoteright_{2}$, which
implies conservation of the total angular momentum projection,
$M=m_{1}+m_{2}+m_{l}=m_{1}^{\prime}+m\textquoteright_{2}+m\textquoteright_{l}$.
The values of $m_{l}$ and $m^{\prime}_{l}$ are then restricted by the change
of the internal projection $\Delta
m_{12}=m_{12}-m_{12}^{\prime}=m_{l}^{\prime}-m_{l}$. To illustrate the
restrictions on $l$ and $l^{\prime}$, consider the $s$-wave scattering
($l=m_{l}=0$) of two distinguishable particles that changes
$m_{12}=m_{1}+m_{2}$ by 1. As stated above, since $M$ is conserved, we have
$\Delta m_{12}=m_{12}-m_{12}^{\prime}=m_{l}^{\prime}-m_{l}=\pm 1$ and thus
$m_{l}^{\prime}=\pm 1$. In the absence of additional symmetry restrictions on
the lowest possible values of $l$ or $l^{\prime}$ in Eq. (1), the dominant
partial wave contributions are $l=0$ and $l^{\prime}=|\Delta{m_{l}}|=|\Delta
m_{12}|$. Substituting these values into Eq. (4) we find $T_{\gamma
l,\gamma^{\prime}l^{\prime}}\propto k^{|\Delta m_{12}|+1}$.

For the special case of spin-exchange transitions, which only change the
projections of the internal angular momenta ($j_{1}=j^{\prime}_{1}$ and
$j_{2}=j^{\prime}_{2}$), the conservation of the total inversion parity
$p=(-1)^{j_{1}+j_{2}+l}$ implies that $l$ and $l^{\prime}$ must have the same
parity. For ultracold collisions $l=0$ and thus $l^{\prime}$ must be even. The
relation $l^{\prime}=|\Delta m_{12}|$ still holds for even $|\Delta m_{12}|$
while for odd values of $|\Delta m_{12}|$, it must be replaced by
$l^{\prime}=|\Delta m_{12}|+1$.

For collisions of identical bosons (fermions) in the same internal state, the
permutation symmetry must also be taken into account. Only even (odd) partial
waves will be present in Eq. (1) [65, 16]. For ultracold collisions of
identical bosons $l=0$, and all permissible values of $l^{\prime}$ are even as
the case of $j_{i}$-conserving spin-exchange transitions. For ultracold
collisions of identical fermions, $l=1$ and all permissible values of
$l^{\prime}$ are odd. The relation $l^{\prime}=|\Delta m_{12}|$ holds for odd
$|\Delta m_{12}|$ while for even values of $|\Delta m_{12}|$, it must be
replaced by $l^{\prime}=|\Delta m_{12}|-1$. Again, we find the same near-
threshold scaling $T_{\gamma l,\gamma^{\prime}l^{\prime}}\propto k^{|\Delta
m_{12}|+1}$ for even $|\Delta m_{12}|$ and $T_{\gamma
l,\gamma^{\prime}l^{\prime}}\propto k^{|\Delta m_{12}|+2}$ for odd $|\Delta
m_{12}|$. Note that $|\Delta m_{12}|=0$ is a special case: the lowest value of
$l^{\prime}$ is 1 and $T_{\gamma l,\gamma^{\prime}l^{\prime}}\propto k^{3}$.
Thus, ultracold inelastic collisions of identical fermions with $|\Delta
m_{12}|=0,1$ and $2$ all follow a near-threshold scaling identical to that of
$p$-wave elastic scattering, i.e., $T_{\gamma
l,\gamma^{\prime}l^{\prime}}\propto k^{3}$.

Summarizing the above discussion, the threshold behavior of the $T$-matrix
elements in the absence of $l$-restricting symmetries is given by $T_{\gamma
l,\gamma^{\prime}l^{\prime}}\propto k^{|\Delta m_{12}|+1}$. By taking the
absolute magnitude squared of the $T$-matrix elements and dividing by
$k^{2}=2\mu E$, where $\mu$ is the reduced mass of the collision complex [see
Eq. (1)], we obtain the threshold scaling of the ICS for collision-induced EE
processes in the general case

$\displaystyle\sigma_{j_{1}m_{1}j_{2}m_{2}\to
j_{1}^{\prime}m_{1}^{\prime}j_{2}^{\prime}m_{2}^{\prime}}$
$\displaystyle\simeq E^{|\Delta m_{12}|}.$ (5)

A modified threshold scaling applies in the case of even partial wave
scattering (spin-exchange transitions with $j_{1}=j_{1}^{\prime}$ and
$j_{2}=j_{2}^{\prime}$ or those between identical bosons in the same internal
states)

$\displaystyle\sigma_{j_{1}m_{1}j_{2}m_{2}\to
j_{1}^{\prime}m_{1}^{\prime}j_{2}^{\prime}m_{2}^{\prime}}$
$\displaystyle\simeq E^{|\Delta m_{12}|}\quad\,(|\Delta
m_{12}|\,\,\text{even}),$ $\displaystyle\simeq E^{|\Delta m_{12}|+1}\,(|\Delta
m_{12}|\,\,\text{odd}).$ (6)

Significantly, unlike in the simplified case considered by Krems and Dalgarno
[58], $m_{i}$ can be nonzero for both of the colliding species, and thus
$|\Delta m_{12}|=0$ can correspond to inelastic as well as to elastic
scattering. Therefore, remarkably, these expressions show that the threshold
scaling of the ICS for an inelastic EE process that conserves $m_{12}$ (such
as flip-flop spin-exchange collisions) approach a constant value in the limit
of ultralow collision energies. This behavior stands in contrast with the
expected $E^{-1/2}$ scaling of the inelastic ICSs in the limit $E\to 0$ [54,
4], and was reported without explanation in several previous quantum
scattering calculations of near-resonant charge exchange [35] and spin-
exchange in cold H + H collisions [38].

The physical origin of the unconventional Wigner threshold scaling can be
traced back to the threshold behavior of the $T$-matrix elements, which
depends only on the initial and final values of collision wavevectors $k$ and
$k^{\prime}$ and on $l$ and $l^{\prime}$ (see above and Eq. (3) [51]). As a
result, as long as a collision process remains resonant ($k=k^{\prime}$) it
will be governed by the same threshold dependence, regardless of whether the
process is elastic or inelastic. Thus, as shown above, both the elastic and
inelastic collisions that conserve $k$ and $\Delta m_{12}$ have the same
constant near-threshold scaling, usually characteristic of the elastic ICS.

Note that the above discussion can be straightforwardly extended to exothermic
(relaxation) collisions. Assuming that $k^{\prime}$ is outside of the
threshold regime, the dependence of the $T$-matrix element on $k^{\prime}$ is
negligibly weak, and $T_{\gamma l,\gamma^{\prime}l^{\prime}}\propto
k^{l+1/2}$. As a result, the relaxation ICSs assume their conventional
$k^{-1}$ scaling with relaxation rates approaching constant values in the
$T\to 0$ limit.

Numerical examples: Spin exchange in ultracold O2 \+ O2 and Rb + Rb
collisions. To verify the unconventional Wigner threshold scaling of inelastic
EE processes derived above, we carried out numerically exact quantum
scattering calculations of spin exchange in ultracold O2 \+ O2 and Rb + Rb
collisions.

We first consider ultracold collisions of identical O${}_{2}(X^{3}\Sigma^{-})$
molecules initially in their ground rovibrational states
$|N_{i}=0,j_{i}=1,m_{i}\rangle$, where
$\mathbf{j}_{i}=\mathbf{N}_{i}+\mathbf{S}_{i}$ ($i=1,2$) and $m_{i}$ is the
eigenvalue of ${j}_{i_{z}}$. Since all $j_{i}>1$ states are energetically
unavailable at ultralow collision energies, the initial and final two-molecule
states can be denoted $|m_{1}m_{2}\rangle$ with $m_{i}=-1,0,1$ (here, we
neglect the molecular hyperfine structure for simplicity). Additionally, the
initial states for identical bosons (such as 17O2) must be symmetrized to
account for identical particle permutation symmetry [16]

$\ket{{m_{1},m_{2}}}=\frac{1}{\sqrt{2(1+\delta_{m_{1},m_{2}})}}\left[\ket{m_{1}}\ket{m_{2}}+\ket{m_{2}}\ket{m_{1}}\right].$
(7)

where $m_{2}\geq m_{1}$.

Figure 1(a) shows the ICS for spin exchange in 17O2 \+ 17O2 collisions
calculated as a function of collision energy by integrating the coupled-
channel (CC) Schrödinger equations [66] on the radial grid extending out to
$R_{\text{max}}=200a_{0}$. The other computation parameters are identical to
those employed in Ref. [10]. The ICS for the $|\Delta m_{12}|$ conserving
$|-1,+1\rangle\to|00\rangle$ transition is seen to approach a constant value
in the limit $E\to 0$ as predicted by Eq. (Near-threshold scaling of resonant
inelastic collisions at ultralow temperatures). The corresponding inelastic
rate tends to zero as $T^{1/2}$ as illustrated in Fig. 1(b). Note that for
$j=1$, the only transition with $|\Delta m_{12}|=0$ is between the states
$\ket{{0,0}}$ and $\ket{{-1,+1}}$.

In Fig. 1(a) we plot the ICS for $|\Delta m_{12}|$ changing transitions for
two identical bosons in the same internal state
($\ket{0,0}\rightarrow\ket{+1,+1}$ and $\ket{0,0}\rightarrow\ket{0,+1}$), and
for two identical bosons in different internal states
($\ket{-1,+1}\rightarrow\ket{0,+1}$). As follows from Eq. (Near-threshold
scaling of resonant inelastic collisions at ultralow temperatures), these
three transitions must have the same $E^{2}$ scaling, as is indeed observed.
The corresponding inelastic rate tends to zero as $T^{5/2}$ as illustrated by
the fit in Fig. 1(b).

Figure 1: (a) State-to-state inelastic ICSs for ultracold O2 \+ O2 collisions
plotted as a function of collision energy. The initial and final values of
$m_{i}$ for the O2 molecules are indicated in the legend as $m_{1}m_{2}$ to
$m_{1}^{\prime}m_{2}^{\prime}$. The fits as $E^{0}$ and $E^{2}$ are
represented by dashed lines. (b) Same as panel (a) but for the temperature
dependence of state-to-state inelastic collision rates $K_{m_{1}m_{2}\to
m_{1}^{\prime}m_{2}^{\prime}}$. The fits as $T^{1/2}$ and $T^{5/2}$ are
represented by dashed lines.

We next consider ultracold spin-exchange collisions of 87Rb atoms, which have
been the subject of much experimental study (for a review, see [2]). The two-
atom threshold states
$|j_{1}m_{1}\rangle|j_{2}m_{2}\rangle=|F_{1}m_{F_{1}}\rangle|F_{2}m_{F_{2}}\rangle$,
where $|j_{i}m_{F_{i}}\rangle=F_{i}m_{F_{i}}\rangle$ are the atomic hyperfine
states and $\mathbf{j}_{i}=\mathbf{S}_{i}+\mathbf{I}_{i}$ are the total atomic
angular momenta, which are vector sums of the electron and nuclear spins of
the $i$-th atom ($i=1,2$). The quantum scattering problem for 87Rb + 87Rb is
solved using the standard CC approach as described in, e.g., Ref. [67] by
integrating the CC equations on a grid of $R$ values from 2 to 400$a_{0}$ with
a step size of $5\times 10^{-3}a_{0}$.

The $s$-wave ICSs for 87Rb + 87Rb spin-exchange collisions are plotted as a
function of collision energy in Fig. 2(a) for three representative hyperfine
transitions $|10\rangle|10\rangle\to|1,-1\rangle|1,+1\rangle$,
$|20\rangle|20\rangle\to|2,-1\rangle|2,+1\rangle$, and
$|20\rangle|20\rangle\to|2,-2\rangle|2,+2\rangle$. As all of these transitions
have $\Delta m_{12}=0$, Eq. (Near-threshold scaling of resonant inelastic
collisions at ultralow temperatures) establishes that their ICSs should scale
as $E^{0}$ in the limit of zero collision energy, and their rates should
decrease as $T^{1/2}$ as $T\to 0$. The predicted trends are clearly observable
in Fig. 2.

Figure 2: (a) State-to-state inelastic ICSs for ultracold $s$-wave 87Rb + 87Rb
collisions plotted vs. collision energy at a magnetic field of 0.01 G. The
initial and final hyperfine states of the Rb atoms $F_{i}m_{F_{i}}$ are
indicated in the legend.The fits as $E^{0}$ are represented by green dots. (b)
Same as panel (a) but for the state-to-state inelastic collision rates
$K_{m_{F_{1}}m_{F_{2}}\to m_{F_{1}}^{\prime}m_{F_{2}}^{\prime}}$ as a function
of temperature. The fits as $T^{1/2}$ are represented by dashed lines.

In conclusion, we have shown that the near-threshold scaling of the ICS for
resonant inelastic EE processes (such as spin exchange) is given by
$\sigma_{\text{inel}}\propto E^{\Delta m_{12}}$ and only depends on the
difference between the combined angular momentum projections $\Delta m_{12}$
in the incident and final collision channels. For $\Delta m_{12}=0$ the
scaling of the inelastic ICS is the same as that of the elastic ICS, i.e.,
$\sigma_{\text{inel}}\propto E^{0}$. We have presented rigorous quantum
scattering calculations to demonstrate the scaling for ultracold O2 \+ O2
collisions in the experimentally accessible regimes.

Our work resolves the long-standing controversy concerning the threshold
behavior of EE cross sections and confirms the correctness of the calculations
reported in Refs. [40, 41]. More generally, it suggests that it is the
energetics of a collisional EE process (resonant vs. exothermic) rather than
the change in the internal states of collision partners, that determines the
process’ threshold behavior. Our results demonstrate a universal $T^{\Delta
m_{12}+1/2}$ suppression of a wide class of resonant EE processes at ultralow
temperatures, which could be observed experimentally for, e.g., spin-exchange
atom-atom collisions in optical tweezers [49], atom-ion collisions in an
optical lattice setup, which allows for high collision energy resolution [68],
as well as in cold and ultracold collisions of Yb atoms [69], Ti atoms [70],
Rydberg atoms [28, 29] and polar molecules [7, 11].

We are grateful to Bernard Zygelman for bringing Ref. [38] to our attention.
This work was supported by the U.S. Air Force Office for Scientific Research
(AFOSR) under Contract No. FA9550-19-1-0312 and partially by the NSF (Grant
No. PHY-1912668).

## References

  * Krems [2019] R. V. Krems,  _Molecules in Electromagnetic Fields_ (Wiley-VCH, 2019).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Carr  _et al._ [2009] L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, Cold and ultracold molecules: science, technology and applications, New J. Phys 11, 055049 (2009).
  * Balakrishnan [2016] N. Balakrishnan, Perspective: Ultracold molecules and the dawn of cold controlled chemistry, J. Chem. Phys. 145, 150901 (2016).
  * Bohn  _et al._ [2017] J. L. Bohn, A. M. Rey, and J. Ye, Cold molecules: Progress in quantum engineering of chemistry and quantum matter, Science 357, 1002 (2017).
  * Wolf  _et al._ [2017] J. Wolf, M. Deiß, A. Krükow, E. Tiemann, B. P. Ruzic, Y. Wang, J. D’Incao, P. S. Julienne, and J. H. Denschlag, State-to-state chemistry for three-body recombination in an ultracold rubidium gas, Science 358, 921 (2017).
  * Matsuda  _et al._ [2020] K. Matsuda, L. De Marco, J.-R. Li, W. G. Tobias, G. Valtolina, G. Quéméner, and J. Ye, Resonant collisional shielding of reactive molecules using electric fields, Science 370, 1324 (2020).
  * Yan  _et al._ [2020] Z. Z. Yan, J. W. Park, Y. Ni, H. Loh, S. Will, T. Karman, and M. Zwierlein, Resonant dipolar collisions of ultracold molecules induced by microwave dressing, Phys. Rev. Lett. 125, 063401 (2020).
  * Liu  _et al._ [2020a] Y. Liu, M.-G. Hu, M. A. Nichols, D. D. Grimes, T. Karman, H. Guo, and K.-K. Ni, Photo-excitation of long-lived transient intermediates in ultracold reactions, Nat. Phys. 16, 1132 (2020a).
  * Devolder  _et al._ [2021] A. Devolder, P. Brumer, and T. V. Tscherbul, Complete quantum coherent control of ultracold molecular collisions, Phys. Rev. Lett. 126, 153403 (2021).
  * Liu  _et al._ [2020b] Y. Liu, M.-G. Hu, M. A. Nichols, D. Yang, D. Xie, H. Guo, and K.-K. Ni, Precision test of statistical dynamics with state-to-state ultracold chemistry, arXiv:2012.15842 (2020b).
  * Cheuk  _et al._ [2020] L. W. Cheuk, L. Anderegg, Y. Bao, S. Burchesky, S. S. Yu, W. Ketterle, K.-K. Ni, and J. M. Doyle, Observation of collisions between two ultracold ground-state CaF molecules, Phys. Rev. Lett. 125, 043401 (2020).
  * Segev  _et al._ [2019] Y. Segev, M. Pitzer, M. Karpov, N. Akerman, J. Narevicius, and E. Narevicius, Collisions between cold molecules in a superconducting magnetic trap, Nature 572, 189 (2019).
  * Walker and Happer [1997] T. G. Walker and W. Happer, Spin-exchange optical pumping of noble-gas nuclei, Rev. Mod. Phys. 69, 629 (1997).
  * Walker [2011] T. G. Walker, Fundamentals of spin-exchange optical pumping, J. Phys. C 294, 012001 (2011).
  * Tscherbul  _et al._ [2009a] T. V. Tscherbul, P. Zhang, H. R. Sadeghpour, and A. Dalgarno, Collision-induced spin exchange of alkali-metal atoms with ${}^{3}\text{H}\text{e}$: An ab initio study, Phys. Rev. A 79, 062707 (2009a).
  * Tscherbul  _et al._ [2011] T. V. Tscherbul, P. Zhang, H. R. Sadeghpour, and A. Dalgarno, Anisotropic hyperfine interactions limit the efficiency of spin-exchange optical pumping of ${}^{3}\mathrm{He}$ nuclei, Phys. Rev. Lett. 107, 023204 (2011).
  * Savukov and Romalis [2005] I. M. Savukov and M. V. Romalis, Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields, Phys. Rev. A 71, 023405 (2005).
  * Sikorsky  _et al._ [2018a] T. Sikorsky, M. Morita, Z. Meir, A. A. Buchachenko, R. Ben-shlomi, N. Akerman, E. Narevicius, T. V. Tscherbul, and R. Ozeri, Phase locking between different partial waves in atom-ion spin-exchange collisions, Phys. Rev. Lett. 121, 173402 (2018a).
  * Rui  _et al._ [2017] J. Rui, H. Yang, L. Liu, D.-C. Zhang, Y.-X. Liu, J. Nan, Y.-A. Chen, B. Zhao, and J.-W. Pan, Controlled state-to-state atom-exchange reaction in an ultracold atom–dimer mixture, Nat. Phys. 13, 699 (2017).
  * Liu  _et al._ [2019] Y.-X. Liu, J. Nan, D.-C. Zhang, L. Liu, H. Yang, J. Rui, B. Zhao, and J.-W. Pan, Observation of a threshold behavior in an ultracold endothermic atom-exchange process involving Feshbach molecules, Phys. Rev. A 100, 032706 (2019).
  * Kawaguchi and Ueda [2012] Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep. 520, 253 (2012).
  * Stamper-Kurn and Ueda [2013] D. M. Stamper-Kurn and M. Ueda, Spinor Bose gases: Symmetries, magnetism, and quantum dynamics, Rev. Mod. Phys. 85, 1191 (2013).
  * Nishida [2013] Y. Nishida, SU(3) orbital Kondo effect with ultracold atoms, Phys. Rev. Lett. 111, 135301 (2013).
  * Bauer  _et al._ [2013] J. Bauer, C. Salomon, and E. Demler, Realizing a Kondo-correlated state with ultracold atoms, Phys. Rev. Lett. 111, 215304 (2013).
  * Safinya  _et al._ [1981] K. A. Safinya, J. F. Delpech, F. Gounand, W. Sandner, and T. F. Gallagher, Resonant Rydberg-Atom-Rydberg-Atom collisions, Phys. Rev. Lett. 47, 405 (1981).
  * Zanon  _et al._ [2002] R. A. D. S. Zanon, K. M. F. Magalhaes, A. L. de Oliveira, and L. G. Marcassa, Time-resolved study of energy-transfer collisions in a sample of cold rubidium atoms, Phys. Rev. A 65, 023405 (2002).
  * Ryabtsev  _et al._ [2010] I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov, and V. M. Entin, Observation of the Stark-tuned Förster resonance between two Rydberg atoms, Phys. Rev. Lett. 104, 073003 (2010).
  * Nipper  _et al._ [2012] J. Nipper, J. B. Balewski, A. Krupp, B. Butscher, R. Low, and T. Pfau, Highly resolved measurements of Stark-tuned forster resonances between Rydberg atoms, Phys. Rev. Lett. 108, 113001 (2012).
  * Ravets  _et al._ [2014] S. Ravets, H. Labuhn, D. Barredo, L. Béguin, T. Lahaye, and A. Browaeys, Coherent dipole–dipole coupling between two single Rydberg atoms at an electrically-tuned Förster resonance, Nat. Phys. 10, 914 (2014).
  * Win  _et al._ [2012] A. L. Win, W. D. Williams, T. J. Carroll, and C. I. Sukenik, Catalysis of Stark-tuned interactions between ultracold Rydberg atoms, Phys. Rev. Lett. 108, 032703 (2012).
  * Sikorsky  _et al._ [2018b] T. Sikorsky, Z. Meir, R. Ben-shlomi, N. Akerman, and R. Ozeri, Spin-controlled atom–ion chemistry, Nat. Commun. 9, 920 (2018b).
  * Bouledroua  _et al._ [2001] M. Bouledroua, A. Dalgarno, and R. Côté, Diffusion and excitation transfer of excited alkali atoms, Phys. Rev. A 65, 012701 (2001).
  * Côté and Dalgarno [2000] R. Côté and A. Dalgarno, Ultracold atom-ion collisions, Phys. Rev. A 62, 012709 (2000).
  * Bodo  _et al._ [2008] E. Bodo, P. Zhang, and A. Dalgarno, Ultra-cold ion–atom collisions: near resonant charge exchange, New J. Phys. 10, 033024 (2008).
  * Zhang  _et al._ [2009] P. Zhang, A. Dalgarno, and R. Côté, Scattering of Yb and Yb+, Phys. Rev. A 80, 030703 (2009).
  * Zygelman [2005] B. Zygelman, Hyperfine level–changing collisions of hydrogen atoms and tomography of the dark age Universe, Astrophys. J. 622, 1356 (2005).
  * Zygelman [2010] B. Zygelman, Electronic spin-flipping collisions of hydrogen atoms, Phys. Rev. A 81, 032506 (2010).
  * Li and Gao [2015] M. Li and B. Gao, Proton-hydrogen collisions at low temperatures, Phys. Rev. A 91, 032702 (2015).
  * Glassgold  _et al._ [2005] A. E. Glassgold, P. S. Krstić, and D. R. Schultz, H+ \+ H scattering and ambipolar diffusion heating, Astrophys. J. 621, 808 (2005).
  * Furlanetto and Furlanetto [2007] S. R. Furlanetto and M. R. Furlanetto, Spin exchange rates in proton–hydrogen collisions, Mon. Not. R. Astron. Soc. 379, 130 (2007).
  * Duan  _et al._ [2002] L.-M. Duan, J. I. Cirac, and P. Zoller, Quantum entanglement in spinor Bose-Einstein condensates, Phys. Rev. A 65, 033619 (2002).
  * Chang  _et al._ [2005] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M. S. Chapman, Coherent spinor dynamics in a spin-1 Bose condensate, Nat. Phys. 1, 111 (2005).
  * Jaksch  _et al._ [1999] D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, Entanglement of atoms via cold controlled collisions, Phys. Rev. Lett. 82, 1975 (1999).
  * Calarco  _et al._ [2000] T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps, Phys. Rev. A 61, 022304 (2000).
  * Hayes  _et al._ [2007] D. Hayes, P. S. Julienne, and I. H. Deutsch, Quantum logic via the exchange blockade in ultracold collisions, Phys. Rev. Lett. 98, 070501 (2007).
  * Jensen  _et al._ [2019] J. H. M. Jensen, J. J. Sørensen, K. Mølmer, and J. F. Sherson, Time-optimal control of collisional $\sqrt{\mathrm{swap}}$ gates in ultracold atomic systems, Phys. Rev. A 100, 052314 (2019).
  * Ni  _et al._ [2018] K.-K. Ni, T. Rosenband, and D. D. Grimes, Dipolar exchange quantum logic gate with polar molecules, Chem. Sci. 9, 6830 (2018).
  * Schmidt  _et al._ [2019] F. Schmidt, D. Mayer, Q. Bouton, D. Adam, T. Lausch, J. Nettersheim, E. Tiemann, and A. Widera, Tailored single-atom collisions at ultralow energies, Phys. Rev. Lett. 122, 013401 (2019).
  * Fang  _et al._ [2020] F. Fang, J. A. Isaacs, A. Smull, K. Horn, L. D. Robledo-De Basabe, Y. Wang, C. H. Greene, and D. M. Stamper-Kurn, Collisional spin transfer in an atomic heteronuclear spinor Bose gas, Phys. Rev. Research 2, 032054 (2020).
  * Wigner [1948] E. P. Wigner, On the behavior of cross sections near thresholds, Phys. Rev. 73, 1002 (1948).
  * Sadeghpour  _et al._ [2000] H. R. Sadeghpour, J. L. Bohn, M. Cavagnero, B. D. Esry, I. I. Fabrikani, J. H. Macek, and A. R. P. Rau, Collisions near threshold in atomic and molecular physics, J. Phys. B 33, R39 (2000).
  * Balakrishnan  _et al._ [1997] N. Balakrishnan, V. Kharchenko, R. C. Forrey, and A. Dalgarno, Complex scattering lengths in multi-channel atom–molecule collisions, Chem. Phys. Lett. 280, 5 (1997).
  * Krems [2005] R. V. Krems, Molecules near absolute zero and external field control of atomic and molecular dynamics, Int. Rev. Phys. Chem. 24, 99 (2005).
  * Dalgarno and Bates [1961] A. Dalgarno and D. R. Bates, Spin-change cross-sections, Proc. R. Soc. London. Ser. A. 262, 132 (1961).
  * Dalgarno  _et al._ [1965] A. Dalgarno, M. R. H. Rudge, and D. R. Bates, Spin-change cross-sections for collisions between alkali atoms, Proc. R. Soc. London. Ser. A. 286, 519 (1965).
  * Moerdijk  _et al._ [1996] A. J. Moerdijk, B. J. Verhaar, and T. M. Nagtegaal, Collisions of dressed ground-state atoms, Phys. Rev. A 53, 4343 (1996).
  * Krems and Dalgarno [2003] R. V. Krems and A. Dalgarno, Threshold laws for collisional reorientation of electronic angular momentum, Phys. Rev. A 67, 050704 (2003).
  * Cohen-Tannoudji  _et al._ [1977] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977) p. 951.
  * Weiner  _et al._ [1999] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Experiments and theory in cold and ultracold collisions, Rev. Mod. Phys. 71, 1 (1999).
  * Arthurs and Dalgarno [1960] A. M. Arthurs and A. Dalgarno, The theory of scattering by a rigid rotator, Proc. R. Soc. London. Ser. A. 256, 540 (1960).
  * Rowe and McCaffery [1979] M. D. Rowe and A. J. McCaffery, Transfer of state multipoles in excited A ${}^{1}\Sigma^{+}_{u}$ 7Li2 following rotationally inelastic collisions with He: Experiment and theory, Chem. Phys. 43, 35 (1979).
  * Ross and Shaw [1961] M. H. Ross and G. L. Shaw, Multichannel effective range theory, Ann. Phys. 13, 147 (1961).
  * Simbotin  _et al._ [2014] I. Simbotin, S. Ghosal, and R. Côté, Threshold resonance effects in reactive processes, Phys. Rev. A 89, 040701 (2014).
  * Green [1975] S. Green, Rotational excitation in H2-H2 collisions: Close-coupling calculations, J. Chem. Phys. 62, 2271 (1975).
  * Tscherbul  _et al._ [2009b] T. V. Tscherbul, Y. V. Suleimanov, V. Aquilanti, and R. V. Krems, Magnetic field modification of ultracold molecule–molecule collisions, New J. Phys. 11, 055021 (2009b).
  * Li  _et al._ [2008] Z. Li, S. Singh, T. V. Tscherbul, and K. W. Madison, Feshbach resonances in ultracold ${}^{85}\mathrm{Rb}\text{$-$}^{87}\mathrm{Rb}$ and ${}^{6}\mathrm{Li}\text{$-$}^{87}\mathrm{Rb}$ mixtures, Phys. Rev. A 78, 022710 (2008).
  * Ben-shlomi  _et al._ [2021] R. Ben-shlomi, M. Pinkas, Z. Meir, T. Sikorsky, O. Katz, N. Akerman, and R. Ozeri, High-energy-resolution measurements of an ultracold-atom–ion collisional cross section, Phys. Rev. A 103, 032805 (2021).
  * Uetake  _et al._ [2012] S. Uetake, R. Murakami, J. M. Doyle, and Y. Takahashi, Spin-dependent collision of ultracold metastable atoms, Phys. Rev. A 86, 032712 (2012).
  * Lu  _et al._ [2009] M.-J. Lu, V. Singh, and J. D. Weinstein, Inelastic titanium-titanium collisions, Phys. Rev. A 79, 050702 (2009).