# Two dynamical generated $a_{0}$ resonances by interactions between vector
mesons

Zheng-Li Wang1,2,111Email address<EMAIL_ADDRESS>and Bing-Song
Zou1,2,3222Email address<EMAIL_ADDRESS>
1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190,China
2School of Physics, University of Chinese Academy of Sciences (UCAS), Beijing
100049, China
3School of Physics and Electronics, Central South University, Changsha 410083,
China

###### Abstract

We study dynamically generated $a_{0}$ resonances by interactions between
vector mesons including their coupling to channels of pseudoscalar mesons
within coupled-channel approach. Both vector and pseudoscalar mesons are
considered as $t$-channel exchanged mesons for calculating interactions
between vector mesons. Analogous to $a_{0}(980)$ as a $K\bar{K}-\pi\eta$
dynamically generated state, there is an $a_{0}(1710)$ as a coupled channel
dynamically generated state near $K^{*}\bar{K}^{*}$ threshold. The channels
involved are $\rho\phi,K^{*}\bar{K}^{*},\rho\omega,K\bar{K},\pi\eta$. This
$a_{0}$ mainly decays to $\rho\omega$, $\pi\eta$ and $K\bar{K}$. This pole is
much tied to the coupled channel effect. If we turn off either $\rho\phi$ or
$K^{*}\bar{K}^{*}$, the pole disappears. In addition, it is found that
$a_{0}(1450)$ may also be dynamically generated by $\rho\omega$ interactions
due to $\pi$ and $\eta$ exchange.

##  1 Introduction

More and more hadron resonances have been proposed to be hadronic molecules
[1] with much more predicted ones to be searched for [2, 3]. Among various
approaches for studying hadronic molecules, a quite popular one is the unitary
extension of chiral perturbation theory, which has been successfully to study
the meson-baryon and meson-meson interactions at low energy [4, 5, 6, 7, 8, 9,
10, 11]. A well-known example is the $\Lambda(1405)$ [12], which can be
dynamically generated in the vicinity of the $\pi\Sigma$ and $K^{-}p$
thresholds. The another example is $f_{0}(980)$ [9, 13], which is considered
to arise due to $\pi\pi$ and $K\bar{K}$ coupled channel interaction. Some
recent works [14, 15] studied the interaction of the nonet of vector mesons
themselves and found a pole with quantum number $I^{G}(J^{PC})=1^{-}(0^{++})$
mainly coupling to $\bar{K}^{*}K^{*}$ channel. No such $a_{0}$ around
$\bar{K}^{*}K^{*}$ threshold is listed in PDG [16]. However, recently both
Babar Collaboration [17] and BESII Collaboration [18] have reported strong
evidence for a new $a_{0}(1710)$ resonance with mass nearly degenerate with
$f_{0}(1710)$. The claimed mass is smaller than predicted ones of Refs.[14,
15] which have ignored the coupled channels of pseudoscalar mesons. In this
paper, we extend the previous study [15] of this resonance by including its
coupling to channels of pseudoscalar mesons in addition to vector mesons to
see how these more coupled channels influence the pole and result in
corresponding partial decay widths to these channels. For the $t$-channel
meson exchanges, besides vector mesons considered in [14, 15], we also include
pseudoscalar mesons. In addition to the dynamically generated $a_{0}(1710)$
close to the $\bar{K}^{*}K^{*}$ threshold, it is found that $a_{0}(1450)$ may
also be dynamically generated by $\rho\omega$ interactions due to $\pi$ and
$\eta$ exchange.

In the following, we first outline the formalism to the coupled-channel
interaction [19] in Sect. 2, then in Sect. 3, we give our numerical results
and discussion, with a brief summary at the end.

##  2 Formalism

The interaction Lagrangian among vector mesons and pseudoscalar mesons is
given by Refs.[20, 21] as the following

$\displaystyle\mathcal{L}_{VPP}=-ig\langle V^{\mu}[P,\partial_{\mu}P]\rangle,$
(1)
$\displaystyle\mathcal{L}_{VVP}=\frac{G^{\prime}}{\sqrt{2}}\epsilon^{\mu\nu\alpha\beta}\left\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\right\rangle.$
(2)

with

$G^{\prime}=\frac{3g^{\prime 2}}{4\pi^{2}f_{\pi}}\qquad
g^{\prime}=-\frac{G_{V}M_{V}}{\sqrt{2}f^{2}_{\pi}},$ (3)

where the symbol $\langle\ldots\rangle$ stands for the trace in the $SU(3)$
space and the coupling constant $g=M_{V}/2f_{\pi}$ with $M_{V}=845.66MeV$ the
$SU(3)$-averaged vector-meson mass, $G_{V}\simeq 55MeV$ and $f_{\pi}=93MeV$
the pion decay constant. The vector field $V^{\mu}$ is

$V^{\mu}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right)^{\mu},$ (4)

and the pseudoscalar field $P$ is

$P=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\\\ \end{array}\right).$ (5)

With the Lagrangian given in Eq. (1), we are able to calculate the vector-
vector to pseudoscalar-pseudoscalar scattering amplitudes. The Feynman
diagrams needed are shown in Fig. 1, where $V$ means vector meson and $P$
means pseudoscalar meson.

Figure 1: The $t$\- and $u$-channel Feynman diagrams

The amplitudes with isospin-$1$ for the processes $V(p_{1})V(p_{2})\to
P(p_{3})P(p_{4})$ are listed in Table 1.

$\begin{array}[]{cc}\hline\cr\text{Channel}&T^{(1)}\\\
\hline\cr\rho\phi\xrightarrow{K,K^{*}}K\bar{K}&(VP_{t}^{K}+VP_{u}^{K})-(VV_{t}^{K^{*}}+VV_{u}^{K^{*}})\\\
K^{*}\bar{K}^{*}\xrightarrow{K,K^{*}}\pi\eta&-\frac{\sqrt{6}}{2}(VP_{t}^{K}+VP_{u}^{K})+\frac{1}{\sqrt{6}}(VV_{t}^{K^{*}}+VV_{u}^{K^{*}})\\\
K^{*}\bar{K}^{*}\xrightarrow{\pi,\eta,\rho,\omega,\phi}K\bar{K}&(-\frac{1}{2}VP_{t}^{\pi}+\frac{3}{2}VP_{t}^{\eta})+(-\frac{1}{2}VV_{t}^{\rho}+\frac{1}{2}VV_{t}^{\omega}+VV_{t}^{\phi})\\\
\rho\omega\xrightarrow{\rho,\omega}\pi\eta&\frac{2}{\sqrt{3}}(VV_{t}^{\rho}+VV_{t}^{\omega})\\\
\rho\omega\xrightarrow{K,K^{*}}K\bar{K}&-\frac{1}{\sqrt{2}}(VP_{t}^{K}+VP_{u}^{K})-\frac{1}{\sqrt{2}}(VV_{t}^{K^{*}}+VV_{u}^{K^{*}})\\\
\hline\cr\end{array}$ Table 1: The potential of each channel with isospin-$1$

The convention used to relate the particle basis to the isospin basis is

$\begin{split}|\pi^{+}\rangle&=-|1,1\rangle\qquad|K^{+}\rangle=-|\frac{1}{2},\frac{1}{2}\rangle,\\\
|\rho^{+}\rangle&=-|1,1\rangle\qquad|K^{*+}\rangle=-|\frac{1}{2},\frac{1}{2}\rangle.\end{split}$
(6)

The $VP_{t(u)}$ and $VV_{t(u)}$ correspond to the $t(u)$-channel diagrams with
pseudoscalar meson and vector meson exchange, respectively. The superscript is
the particle exchanged. Here, $t=(p_{1}-p_{3})^{2}$ and $u=(p_{1}-p_{4})^{2}$
are the usual Mandelstam variables. The potential is given by

$\displaystyle VP_{t(u)}^{ex}=\frac{-4g^{2}}{t(u)-m^{2}_{ex}}\epsilon_{1}\cdot
p_{3}\epsilon_{2}\cdot p_{4},$ (7) $\displaystyle
VV_{t(u)}^{ex}=-\frac{G^{\prime
2}}{2}\frac{1}{t(u)-m^{2}_{ex}}\varepsilon_{\mu\nu\alpha\beta}p_{1}^{\mu}\epsilon_{1}^{\nu}q^{\alpha}\varepsilon_{\lambda\tau\gamma\delta}p_{2}^{\lambda}\epsilon_{2}^{\tau}q^{\gamma}\left(-g^{\beta\delta}+\frac{q^{\beta}q^{\delta}}{q^{2}}\right).$
(8)

where the $\epsilon_{i}$ is the $i$-th polarization vector of the incoming
vector meson. The polarization vector can be characterized by its three-
momentum $\mathbf{p}_{i}$ and the third component of the spin in its rest
frame, and the explicit expression of the polarization vectors can be found in
Appendix A of Ref. [22]. At the threshold, $s_{th}=(m_{1}+m_{2})^{2}$, the
potential is

$VP_{t}^{ex}=-VV_{t}^{ex}=\frac{4p^{2}_{f}}{\sqrt{3}(m_{1}m_{2}-(m_{2}m_{3}^{2}+m_{1}m_{4}^{2})/\sqrt{s_{th}}+m_{ex}^{2})}g^{2}$
(9)

where $p_{f}$ is the on-shell three momentum of the final state.

In term of these amplitudes with isospin-$1$, we can get the $S$-wave
potential via [22]

$\begin{split}T^{(JI)}_{\ell
S;\bar{\ell}\bar{S}}(s)&=\frac{Y^{0}_{\bar{\ell}}(\hat{\mathbf{z}})}{\sqrt{2}^{N}(2J+1)}\sum_{\begin{subarray}{c}\sigma_{1},\sigma_{2},\bar{\sigma}_{1}\\\
\bar{\sigma}_{2},m\end{subarray}}\int\mathrm{d}\hat{\mathbf{p}}^{\prime\prime}Y^{m}_{\ell}(\mathbf{p}^{\prime\prime})^{*}(\sigma_{1}\sigma_{2}M|s_{1}s_{2}S)\\\
&\times(mM\bar{M}|\ell
SJ)(\bar{\sigma}_{1}\bar{\sigma}_{2}\bar{M}|\bar{s}_{1}\bar{s}_{2}\bar{S})(0\bar{M}\bar{M}|\bar{\ell}\bar{S}J)\\\
&\times
T^{(I)}(p_{1},p_{2},p_{3},p_{4};\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}).\end{split}$
(10)

with $s=(p_{1}+p_{2})^{2}$ the usual Mandelstam variable,
$M=\sigma_{1}+\sigma_{2}$ and $\bar{M}=\bar{\sigma}_{1}+\bar{\sigma}_{2}$. And
$N$ accounts for the identical particles, for example

$\displaystyle N$ $\displaystyle=2\text{ for }\rho\rho\to\pi\pi,$ (11)
$\displaystyle N$ $\displaystyle=1\text{ for }\rho\rho\to K\bar{K},$ (12)
$\displaystyle N$ $\displaystyle=0\text{ for }\omega\phi\to K\bar{K}.$ (13)

Like vector scattering $VV\to VV$, the partial wave projection Eq. (10) for a
$t$-channel exchange amplitude of $VV\to PP$ would also develop a left-hand
cut via [15]

$\begin{split}\frac{1}{2}&\int_{-1}^{+1}\mathrm{d}\cos\theta\frac{1}{t-m^{2}_{ex}+i\epsilon}=-\frac{s}{\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})\lambda(s,m_{3}^{2},m_{4}^{2})}}\\\
&\times\log\frac{m_{1}^{2}+m_{2}^{2}-\frac{(s+m_{1}^{2}-m_{2}^{2})(s+m_{3}^{2}-m_{4}^{2})}{2s}-\frac{\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})\lambda(s,m_{3}^{2},m_{4}^{2})}}{2s}-m_{ex}^{2}+i\epsilon}{m_{1}^{2}+m_{2}^{2}-\frac{(s+m_{1}^{2}-m_{2}^{2})(s+m_{3}^{2}-m_{4}^{2})}{2s}+\frac{\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})\lambda(s,m_{3}^{2},m_{4}^{2})}}{2s}-m_{ex}^{2}+i\epsilon}.\end{split}$
(14)

with $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ac$ the Källén function. In
vector scattering $VV\to VV$, left-hand cuts are smoothed by the $N/D$ method
[23, 24]. As for the scattering $VV\to PP$, all left-hand cuts are located
below the $PP$ threshold, which are far away from the energy region we are
interested in, so we do not deal with these cuts.

The basic equation to obtain the unitarized $T$-matrix is

$T^{(JI)}(s)=\left[1-V^{(JI)}(s)\cdot G(s)\right]^{-1}\cdot V^{(JI)}(s).$ (15)

Here $V^{(JI)}$ denotes the partial-wave amplitudes and $G(s)$ is a diagonal
matrix made up by the two-point loop function $g_{i}(s)$,

$g_{i}(s)=i\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\frac{1}{(q^{2}-m^{2}_{i1}+i\epsilon)((P-q)^{2}-m^{2}_{i2}+i\epsilon)}.$
(16)

with $P^{2}=s$ and $m_{i1,2}$ the masses of the particles in the $i$-th
channel. The pole position is at the zeros of determinant

$\text{Det}\equiv\text{det}\left[1-V^{(JI)}(s)\cdot G(s)\right].$ (17)

The above loop function is logarithmically divergent and can be calculated
with a once-subtracted dispersion relation or using a regularization
$f_{\Lambda}(q)$

$g_{i}(s)=i\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\frac{f^{2}_{\Lambda}(q)}{(q^{2}-m^{2}_{i1}+i\epsilon)((P-q)^{2}-m^{2}_{i2}+i\epsilon)}.$
(18)

after the $q^{0}$ integration is performed by choosing the contour in the
lower half of the complex plane, we get

$g_{i}(s)=\int_{0}^{\infty}\frac{|\mathbf{q}|^{2}\mathrm{d}|\mathbf{q}|}{(2\pi)^{2}}\frac{\omega_{i1}+\omega_{i2}}{\omega_{i1}\omega_{i2}(s-(\omega_{i1}+\omega_{i2})^{2}+\mathrm{i}\epsilon)}f_{\Lambda}^{2}(|\mathbf{q}|).$
(19)

where $\mathbf{q}$ is the three-momentum and
$\omega_{i1,2}=\sqrt{\mathbf{q}^{2}+m^{2}_{i1,2}}$. In order to proceed we
need to determine $f_{\Lambda}(\mathbf{q})$. There are two kinds of choices,
sharp cutoff and smooth cutoff, typically:

$f_{\Lambda}(\mathbf{q})=\left\\{\begin{array}[]{l}\Theta(\Lambda^{2}-\mathbf{q}^{2})\\\
\exp\left[-\frac{\mathbf{q}^{2}}{\Lambda^{2}}\right]\\\ \end{array}\right.$
(20)

In order to compare with the previous results of coupled channel approach
[15], the same sharp cutoff is used in this paper when channels with
pseudoscalar mesons are included in addition. To explore the position of the
poles we need to take into account the analytical structure of these
amplitudes in the different Riemann sheets. By denoting $q_{on}$ for the CM
tri-momentum of the particles $1$ and $2$ in the $i$-th channel

$q_{i}^{\text{on}}=\frac{\sqrt{(s-(m_{i1}-m_{i2})^{2})(s-(m_{i1}+m_{i2})^{2})}}{2\sqrt{s}}.$
(21)

As the quantity is two-valued itself [25], we need to distinguish the two
Riemann sheets of $q_{i}^{\text{on}}$ uniquely according to

$q_{i}^{\text{on}>}=\left\\{\begin{array}[]{cl}-q_{i}^{\text{on}}&\text{if
}\text{Im}q_{i}^{\text{on}}<0\\\ q_{i}^{\text{on}}&\text{else}\\\
\end{array}\right.$ (22)

And the analytic continuation to the second Riemann sheet is given by

$g^{(2)}_{i}(s)=g_{i}(s)+\frac{i}{4\pi}\frac{q_{i}^{\text{on}>}}{\sqrt{s}}.$
(23)

##  3 Numerical results and discussion

Firstly, within the isospin formalism the $\rho\rho$ states obey Bose-Einstein
statistics, so that only states with even $l+S+I$ are allowed, which means
that in our case $\rho\rho$ channel is ruled out.

For $K^{*}\bar{K}^{*}$ single channel, we already know that there is a bound
state for $I=0$ sector, and the potential is about
$V_{11}^{(0)}\simeq-7.7g^{2}$ [15]. We label the $K^{*}\bar{K}^{*}$ as channel
$1$, and the remain channel indices are listed in Table 2. While for the $I=1$
sector, the potential is about $V_{11}^{(1)}\simeq-0.98g^{2}$, which is too
weak to produce a bound state.

$\begin{array}[]{ccc}\hline\cr\text{Channel
index}&\text{Channel}&\text{Threshold}~{}(GeV)\\\ \hline\cr
1&K^{*}\bar{K}^{*}&1.784\\\ 2&\rho\omega&1.552\\\ 3&\rho\phi&1.79\\\
4&K\bar{K}&0.99\\\ 5&\pi\eta&0.686\\\ \hline\cr\end{array}$ Table 2: Channel
indices and threshold energies

Then we turn on additional channels to study their influence on the mass and
width of the resonance. The $T$-matrix for a single channel $a$ is given by

$T_{aa}=\frac{V_{aa}}{1-V_{aa}g_{a}}.$ (24)

If we turn on another channel $b$, then the $T$-matrix for $a\to a$ becomes to
be

$T_{aa}=\frac{V_{aa}+\frac{V_{ab}^{2}g_{b}}{1-V_{bb}g_{b}}}{1-(V_{aa}+\frac{V_{ab}^{2}g_{b}}{1-V_{bb}g_{b}})g_{a}}.$
(25)

Compared to the single channel, $V_{aa}$ is replaced by

$V_{eff}=V_{aa}+\frac{V_{ab}^{2}g_{b}}{1-V_{bb}g_{b}}.$ (26)

Denoting the second term as

$V^{\prime}=\frac{V_{ab}^{2}g_{b}}{1-V_{bb}g_{b}},$ (27)

then $T_{aa}$ can be written as

$T_{aa}=\frac{V_{aa}+V^{\prime}}{1-(V_{aa}+V^{\prime})g_{a}}=\frac{(1+\alpha)V_{aa}}{1-(1+\alpha)V_{aa}g_{a}}.$
(28)

with $\alpha=V^{\prime}/V_{aa}$. For calculating the loop integral $g_{a}$ for
channels of vector mesons, we use the same sharp cutoff as in the previous
study [15]. However, if we turn on the channels of pseudo-scalar mesons, the
t-channel exchanged pseudo-scalar meson in $VV\to PP$ is mostly off shell for
the calculation of $V^{\prime}$, implementation of some off-shell form factors
is necessary [32]. Adjustment of the sharp cutoff parameter $q_{max}$ for the
loop integration has no influence for the imaginary part of $g_{b}$, which
corresponds to two on-shell pseudo-scalar mesons with an off-shell t-channel
exchanged pseudo-scalar meson. To take into account this off-shell effect of
the t-channel exchanged meson, the same kind of monopole form factor as in our
triangle loop approach as well as in Ref.[32]

$F=\frac{\Lambda^{2}-m^{2}_{ex}}{\Lambda^{2}-q^{2}}$ (29)

is included at each $VPP$ vertex for the exchanged pseudo-scalar meson with
momentum $q$. For the $VV\to PP$ process, the t-channel exchange meson is much
more off-shell than corresponding $VV\to VV$ case.

Firstly, we turn on the $\rho\phi$ channel, it has a contribution about
$V_{13}^{2}g_{3}(s)\simeq-4g^{2}$ near the $K^{*}\bar{K}^{*}$ threshold,
together with $V_{11}\simeq-0.98g^{2}$, $V_{eff}\simeq-5g^{2}$ and there will
be a bound state below the $K^{*}\bar{K}^{*}$ threshold. We find that the
coupled $K^{*}\bar{K}^{*}$ and $\rho\phi$ channels are necessary to produce
the bound state. There would be no bound state near the $K^{*}\bar{K}^{*}$
threshold for ether $K^{*}\bar{K}^{*},\rho\omega,K\bar{K},\pi\eta$ coupled
system or $\rho\phi,\rho\omega,K\bar{K},\pi\eta$ coupled system .

Then we turn on other channel one by one. For
$\rho\phi-K^{*}\bar{K}^{*}-\rho\omega$ system, the pole position is about
$1.74-i0.03(GeV)$. For the the $\rho\phi-K^{*}\bar{K}^{*}-K\bar{K}$ system,
the pole position is about $1.78-i0.07(GeV)$. For the
$\rho\phi-K^{*}\bar{K}^{*}-\pi\eta$ system, the pole position is about
$1.78-i0.02(GeV)$.

Finally, we turn on all channels

Figure 2: $|T_{11}|^{2}$ for $\Lambda=1.0GeV$ and $q_{max}=1.0GeV$ in 2-dim
and 3-dim

and show the $|T_{11}|^{2}$ in 2-dim and 3-dim with $q_{max}=1.0GeV$ in Fig.
2. The pole position of the resonance is listed in Table 3 for different
cutoffs.

$\begin{array}[]{cccccc}\hline\cr q_{max}(GeV)&0.9&1.0&1.1\\\ \hline\cr
Pole(GeV)&1.76-0.09i&1.72-0.10i&1.69-0.11i\\\ \hline\cr\end{array}$ Table 3:
The resonance pole for different cutoffs

We find that the real part of the resonance is about $1720MeV$ and the width
is about $200MeV$ with $q_{max}=1.0GeV$. With the effective coupling constant
of $a_{0}\bar{K}^{*}K^{*}$ determined by its binding energy, the partial decay
width of $a_{0}\to\bar{K}^{*}K^{*}\to K\pi\bar{K}\pi$ can be calculated
straightforwardly to be around 21 MeV, together with the two body decays, the
total decay width of $a_{0}$ is about $220MeV$, larger than $f_{0}(1710)$.
Compared with previous results of Refs.[14, 15] which ignored the couplings of
vector mesons to pseudoscalar mesons, the mass of the resonance gets smaller
to be nearly degenerate with its isoscalar partner $f_{0}(1710)$ and fits in
the experimental claimed one $a_{0}(1710)$ very well.

For the unitary coupled channel approach, we can calculate each partial decay
width via [16]

$\Gamma_{R\to a}=\frac{|\tilde{g}_{a}|^{2}}{M_{R}}\rho_{a}(M^{2}_{R}).$ (30)

with $\tilde{g}_{a}=\mathcal{R}_{ba}/\sqrt{\mathcal{R}_{bb}}$ and $\rho_{a}$
the two-body phase space. The residues may be calculated via an integration
along a closed contour around the pole using

$\mathcal{R}_{ba}=-\frac{1}{2\pi i}\oint ds\mathcal{M}_{ba}.$ (31)

The partial decay widths obtained this way are 4

$\begin{array}[]{cccccc}\hline\cr\Gamma(K^{*}\bar{K}^{*}\to\rho\omega)&\Gamma(K^{*}\bar{K}^{*}\to
K\bar{K})&\Gamma(K^{*}\bar{K}^{*}\to\pi\eta)\\\ \hline\cr
61.0MeV&74.4MeV&66.9MeV\\\
\hline\cr\Gamma(\rho\phi\to\rho\omega)&\Gamma(\rho\phi\to
K\bar{K})&\Gamma(\rho\phi\to\pi\eta)\\\ \hline\cr 60.8MeV&74.2MeV&66.6MeV\\\
\hline\cr\end{array}$ Table 4: The partial decay widths

Note that there is no tree diagram for $\rho\phi\to\rho\omega$, which means
that $V_{34}=0$, but $T_{34}$ is not $0$ due to the strongly coupled channels
of $\rho\phi$ and $K^{*}\bar{K}^{*}$ with nearly degenerate thresholds. In
both cases the total widths are about $200MeV$. The relative ratio of the
resonance decaying to $\rho\omega,K\bar{K},\pi\eta$ is about $1:1:1$. While in
Ref. [33], the ratio is about $2:1:1$, since for $VV\to PP$ we include also
vector meson exchange in addition to pseudoscalar meson exchange considered in
Ref. [33], the decay widths to pseudoscalar channel become larger. The
residues of channel $K^{*}\bar{K}^{*}$ is about $8731-i2200(MeV)$ and
$\rho\phi$ is about $6013-i1785(MeV)$, which means that the coupling of the
resonance to $\rho\phi$ cannot be neglected. Indeed, if we turn off the
$\rho\phi$, the resonance disappears.

Up to now, for the $I=0$ sector, there are three hadronic molecules calimed to
be dynamically generated from the $K\bar{K},\rho\rho,K^{*}\bar{K}^{*}$
interaction through vector meson exchange potentials, which are assigned to
$f_{0}(980),f_{0}(1500),f_{0}(1710)$, respectively [9, 34, 35]. For the $I=1$
sector, the attractive potentials for the $K\bar{K}$ and $K^{*}\bar{K}^{*}$
channels are much weaker than the corresponding isoscalar cases. No bound
states can be formed for each single channel. The corresponding $a_{0}(980)$
and $a_{0}(1710)$ can only be dynamically generated through strong coupled
channel effects. The $a_{0}(980)$ arises due to the interaction in the
$K\bar{K}-\pi\eta$ system, while $a_{0}(1710)$ close to $K^{*}\bar{K}^{*}$
threshold arises due to interaction in the
$\rho\phi-K^{*}\bar{K}^{*}-\rho\omega-K\bar{K}-\pi\eta$ system. Then
corresponding to the $f_{0}(1500)$ as the isoscalar $\rho\rho$ bound state, we
also examine whether $a_{0}(1450)$ can be dynamically generated by the
$\rho\omega-K\bar{K}-\pi\eta$ interaction.

For the $\rho\omega$ interaction, no $t$-channel vector meson exchange is
allowed. Therefore no dynamically generated $a_{0}$ is found around
$\rho\omega$ threshold in the previous studies [14, 15]. However, when taking
into account the pseudoscalar meson exchange force, the potential of
$\rho\omega\xrightarrow{\pi,\eta}\rho\omega$ is shown in Fig. 3

Figure 3: Potential of $\rho\omega\to\rho\omega$, the left one is $\pi$
exchange and the right one is $\eta$ exchange. The blue line is real part of
the potential and the yellow line is imaginary part

The potential is $0$ at the threshold and the left hand cut is close to the
threshold. The $\pi$ exchange is very steep at the threshold, so when the
$N/D$ method is used, the potential is strong enough to produce a pole below
the $\rho\omega$ threshold. However, $\eta$ exchange is very weak. When
scanning on the complex plane, we find the pole position is about
$1.5-i0.015(GeV)$, and $\Gamma(a_{0}(1450)\to K\bar{K})=13.2MeV$,
$\Gamma(a_{0}(1450)\to\pi\eta)=14.7MeV$, which gives a relative ratio of 0.9
in consistent with the PDG value. Then we calculate the three body decay
$a_{0}(1450)\to\omega\pi\pi$ as shown in Fig.4. The three-body decay width is
about $100MeV$, and results in the total decay width to be around 128 MeV,
smaller than its PDG average value. But different experiments report quite
different values for both its mass and width. Some of them are in fact
consistent with our theoretical values.

Figure 4: Three body decay $a_{0}(1450)\to\omega\pi\pi$

In summary, we extend the coupled channel interaction of vector mesons by
including channels of pseudo-scalar mesons in addition, using the unitary
coupled-channel approach in the $I=1$ sector. The pole near the
$K^{*}\bar{K}^{*}$ threshold remains to be there, but with its mass lower to
be nearly degenerate with its isoscalar partner $f_{0}(1710)$. The mass is
consistent with the $a_{0}(1710)$ newly reported by Babar Collaboration [17]
and BESII Collaboration [18]. The ratio of its decays to
$\rho\omega,K\bar{K},\pi\eta$ is predicted to be about $1:1:1$. With pseudo-
scalar meson exchange included, another pole is found to be just below
$\rho\omega$ threshold, with its mass nearly degenerate with isoscalar
$\rho\rho$ bound state $f_{0}(1500)$ and in consistent with $a_{0}(1450)$
within experimental uncertainties. Its dominant decay mode is predicted to be
$\omega\pi\pi$.

Previously, both $f_{0}(1500)$ and $f_{0}(1710)$ have been considered as
glueball candidates and extensively studied within the quarkonia-guleball
mixing picture [36, 37, 38, 39, 40, 41, 42]. With the success of the new
possible configuration as hadron molecule to explain their properties [34,
35], to pin down their nature, it is crucial to study also their relevant iso-
vector $a_{0}$ mesons. While scalar glueballs are not ecpected to have iso-
vector partners, the $f_{0}(1500)$ and $f_{0}(1710)$ as scalar hadronic
molecules should have nearly degenerate iso-vector partners, $a_{0}(1450)$ and
$a_{0}(1710)$, respectively, just as nearly degenerate $f_{0}(980)$ and
$a_{0}(980)$ as $\bar{K}K$ molecules. The $a_{0}(1710)$ is still not listed in
PDG [16] and the dominant decay mode of $a_{(}1450)$ is still not identified.
To establish these two iso-vector hadronic molecules, it would be very useful
to study $\gamma\gamma\to\eta\pi$, $\bar{K}K$, $\omega\pi\pi$ at Belle2
experiment.

Recently, the newly observed $J^{PC}=1^{-+}$ hybrid candidate $\eta_{1}(1855)$
[43] is also explained as a hadronic molecule [44]. It is very likely that the
unquenching dynamics leading to hadonic molecules would be the dominant
excitation mechanism for hadrons and make it impossible to form glueballs or
hybrids.

## Acknowledgments

We thank useful discussions and valuable comments from Feng-Kun Guo, Ulf-G.
Meißner and Jia-Jun Wu. This work is supported by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 - TRR 110), by the NSFC Grant No.11835015, No.12047503,
and by the Chinese Academy of Sciences (CAS) under Grant No.XDB34030000.

## References

  * [1] F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018) doi:10.1103/RevModPhys.90.015004 [arXiv:1705.00141 [hep-ph]].
  * [2] X. K. Dong, F. K. Guo and B. S. Zou, Progr. Phys. 41, 65-93 (2021) doi:10.13725/j.cnki.pip.2021.02.001 [arXiv:2101.01021 [hep-ph]].
  * [3] X. K. Dong, F. K. Guo and B. S. Zou, Commun. Theor. Phys. 73, no.12, 125201 (2021) doi:10.1088/1572-9494/ac27a2 [arXiv:2108.02673 [hep-ph]].
  * [4] J. Oller, E. Oset and J. Pelaez, Phys. Rev. D 62, 114017 (2000) doi:10.1103/PhysRevD.62.114017 [arXiv:hep-ph/9911297 [hep-ph]].
  * [5] J. Oller and U. G. Meißner, Phys. Lett. B 500, 263-272 (2001) doi:10.1016/S0370-2693(01)00078-8 [arXiv:hep-ph/0011146 [hep-ph]].
  * [6] A. Dobado and J. Pelaez, Phys. Rev. D 56, 3057-3073 (1997) doi:10.1103/PhysRevD.56.3057 [arXiv:hep-ph/9604416 [hep-ph]].
  * [7] J. Oller and E. Oset, Phys. Rev. D 60, 074023 (1999) doi:10.1103/PhysRevD.60.074023 [arXiv:hep-ph/9809337 [hep-ph]].
  * [8] J. Oller, E. Oset and J. Pelaez, Phys. Rev. D 59, 074001 (1999) doi:10.1103/PhysRevD.59.074001 [arXiv:hep-ph/9804209 [hep-ph]].
  * [9] J. Oller and E. Oset, Nucl. Phys. A 620, 438-456 (1997) doi:10.1016/S0375-9474(97)00160-7 [arXiv:hep-ph/9702314 [hep-ph]].
  * [10] J. Oller, E. Oset and A. Ramos, Prog. Part. Nucl. Phys. 45, 157-242 (2000) doi:10.1016/S0146-6410(00)00104-6 [arXiv:hep-ph/0002193 [hep-ph]].
  * [11] R. Molina and E. Oset, Phys. Lett. B 811, 135870 (2020) doi:10.1016/j.physletb.2020.135870 [arXiv:2008.11171 [hep-ph]].
  * [12] D. Jido, J. Oller, E. Oset, A. Ramos and U. Meissner, Nucl. Phys. A 725, 181-200 (2003) doi:10.1016/S0375-9474(03)01598-7 [arXiv:nucl-th/0303062 [nucl-th]].
  * [13] G. Janssen, B. Pearce, K. Holinde and J. Speth, Phys. Rev. D 52, 2690-2700 (1995) doi:10.1103/PhysRevD.52.2690 [arXiv:nucl-th/9411021 [nucl-th]].
  * [14] L. S. Geng and E. Oset, Phys. Rev. D 79, 074009 (2009) doi:10.1103/PhysRevD.79.074009 [arXiv:0812.1199 [hep-ph]].
  * [15] M. L. Du, D. Gülmez, F. K. Guo, U. G. Meißner and Q. Wang, Eur. Phys. J. C 78, no.12, 988 (2018) doi:10.1140/epjc/s10052-018-6475-8 [arXiv:1808.09664 [hep-ph]].
  * [16] P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) doi:10.1093/ptep/ptaa104
  * [17] J. P. Lees et al. [BaBar], Phys. Rev. D 97, no.11, 112006 (2018) doi:10.1103/PhysRevD.97.112006 [arXiv:1804.04044 [hep-ex]].
  * [18] M. Ablikim et al. [BESIII], [arXiv:2110.07650 [hep-ex]].
  * [19] J. A. Oller, Prog. Part. Nucl. Phys. 110, 103728 (2020) doi:10.1016/j.ppnp.2019.103728 [arXiv:1909.00370 [hep-ph]].
  * [20] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985) doi:10.1103/PhysRevLett.54.1215
  * [21] M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217-314 (1988) doi:10.1016/0370-1573(88)90019-1
  * [22] D. Gülmez, U. G. Meißner and J. A. Oller, Eur. Phys. J. C 77, no.7, 460 (2017) doi:10.1140/epjc/s10052-017-5018-z [arXiv:1611.00168 [hep-ph]].
  * [23] G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467-477 (1960) doi:10.1103/PhysRev.119.467
  * [24] J. Bjorken, Phys. Rev. Lett. 4, 473-474 (1960) doi:10.1103/PhysRevLett.4.473
  * [25] M. Doring, C. Hanhart, F. Huang, S. Krewald and U. G. Meißner, Nucl. Phys. A 829, 170-209 (2009) doi:10.1016/j.nuclphysa.2009.08.010 [arXiv:0903.4337 [nucl-th]].
  * [26] M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meißner, J. A. Oller and Q. Wang, Phys. Rev. Lett. 124, no.7, 072001 (2020) doi:10.1103/PhysRevLett.124.072001 [arXiv:1910.11846 [hep-ph]].
  * [27] Y. H. Lin and B. S. Zou, Phys. Rev. D 100, no.5, 056005 (2019) doi:10.1103/PhysRevD.100.056005 [arXiv:1908.05309 [hep-ph]].
  * [28] F. K. Guo, H. J. Jing, U. G. Meißner and S. Sakai, Phys. Rev. D 99, no.9, 091501 (2019) doi:10.1103/PhysRevD.99.091501 [arXiv:1903.11503 [hep-ph]].
  * [29] Y. H. Lin, C. W. Shen, F. K. Guo and B. S. Zou, Phys. Rev. D 95, no.11, 114017 (2017) doi:10.1103/PhysRevD.95.114017 [arXiv:1703.01045 [hep-ph]].
  * [30] R. Machleidt, Adv. Nucl. Phys. 19, 189-376 (1989)
  * [31] A. I. Titov, B. Kampfer and B. L. Reznik, Eur. Phys. J. A 7, 543-557 (2000) doi:10.1007/s100500050427 [arXiv:nucl-th/0001027 [nucl-th]].
  * [32] R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D 78, 114018 (2008) doi:10.1103/PhysRevD.78.114018 [arXiv:0809.2233 [hep-ph]].
  * [33] E. Oset, L. S. Geng and R. Molina, J. Phys. Conf. Ser. 348, 012004 (2012) doi:10.1088/1742-6596/348/1/012004
  * [34] Z. L. Wang and B. S. Zou, Phys. Rev. D 99, no.9, 096014 (2019) doi:10.1103/PhysRevD.99.096014 [arXiv:1901.10169 [hep-ph]].
  * [35] Z. L. Wang and B. S. Zou, Phys. Rev. D 104, no.11, 114001 (2021) doi:10.1103/PhysRevD.104.114001 [arXiv:2107.14470 [hep-ph]].
  * [36] C. Amsler and F. E. Close, Phys. Rev. D 53, 295-311 (1996) doi:10.1103/PhysRevD.53.295 [arXiv:hep-ph/9507326 [hep-ph]].
  * [37] M. Albaladejo and J. A. Oller, Phys. Rev. Lett. 101, 252002 (2008) doi:10.1103/PhysRevLett.101.252002 [arXiv:0801.4929 [hep-ph]].
  * [38] F. E. Close, G. R. Farrar and Z. P. Li, Phys. Rev. D 55, 5749-5766 (1997) doi:10.1103/PhysRevD.55.5749 [arXiv:hep-ph/9610280 [hep-ph]].
  * [39] F. E. Close and Q. Zhao, Phys. Rev. D 71, 094022 (2005) doi:10.1103/PhysRevD.71.094022 [arXiv:hep-ph/0504043 [hep-ph]].
  * [40] F. Giacosa, T. Gutsche, V. E. Lyubovitskij and A. Faessler, Phys. Rev. D 72, 094006 (2005) doi:10.1103/PhysRevD.72.094006 [arXiv:hep-ph/0509247 [hep-ph]].
  * [41] M. Chanowitz, Phys. Rev. Lett. 95, 172001 (2005) doi:10.1103/PhysRevLett.95.172001 [arXiv:hep-ph/0506125 [hep-ph]].
  * [42] K. T. Chao, X. G. He and J. P. Ma, Phys. Rev. Lett. 98, 149103 (2007) doi:10.1103/PhysRevLett.98.149103 [arXiv:0704.1061 [hep-ph]].
  * [43] M. Ablikim et al. [BESIII], [arXiv:2202.00621 [hep-ex]].
  * [44] X. K. Dong, Y. H. Lin and B. S. Zou, [arXiv:2202.00863 [hep-ph]].


# The state of CO and CO2 ices in the Kuiper belt as seen by JWST

Michael E. Brown Division of Geological and Planetary Sciences
California Institute of Technology
Pasadena, CA 9125, USA  Wesley C. Fraser Herzberg Astronomy and Astrophysics
Research Centre, National Research Council, 5071 W. Saanich Rd. Victoria, BC,
V9E 2E7, Canada  Department of Physics and Astronomy, University of Victoria,
Elliott Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

###### Abstract

JWST has shown that CO2 and CO are common on the surfaces of objects in the
Kuiper belt and have apparent surface coverages even higher than that of water
ice, though water ice is expected to be significantly more abundant in the
bulk composition. Using full Mie scattering theory, we show that the high
abundance and the unusual spectral behaviour around the 4.26 $\mu$m $\nu_{1}$
band of CO2 can be explained by a surface covered in a few $\mu$m thick layer
of $\sim 1-2$ $\mu$m CO2 particles. CO is unstable at the temperatures in the
Kuiper belt, so the CO must be trapped in some more stable species. While
hydrate clathrates or amorphous water ice are often invoked as a trapping
mechanism for outer solar system ices, the expected spectral shift of the
absorption line for a CO hydrate clathrates or trapping in amorphous ice is
not seen, nor does the H2O abundance appear to be high enough to explain the
depth of the CO absorption line. Instead, we suggest that the CO is created
via irradiation of CO2 and trapped in the CO2 grains during this process. The
presence of a thin surface layer of CO2 with embedded CO suggests volatile
differentiation driving CO2 from the interior as a major process driving the
surface appearance of these mid-sized Kuiper belt objects, but the mechanisms
that control the small grain size and depth of the surface layer remain
unclear.

††software: miepython
(http://miepython.readthedocs.io)

##  1 Introduction

Observations from JWST have show that CO2 and CO ices are ubiquitous on the
surfaces of small bodies in the outer solar system (Pinilla-Alonso et al.,
submitted, Hénault et al. 2023) CO and CO2 are the most abundant volatiles in
comets after H2O, at a $\sim$10% mixing ratio, and presumably major components
of icy bodies in the outer solar system (Bockelée-Morvan & Biver, 2017), but
the difficulty of observing these species in their solid form means that
little has been known of their distribution and abundance in the small body
population today.

At the surface temperatures of the Kuiper belt and even throughout some of the
Centaur range, CO2 should be stable, thus its presence in the outer solar
system should come as little surprise. With its lower abundance than H2O in
comets, it should be expected to be a minor surface component on bodies
wherever H2O is exposed at the surface. CO, in contrast, is unstable to
sublimation anywhere inside $\sim$200 AU, and thus would not be expected to
appear on any small bodies in the Kuiper belt (Sekanina, 1992).

Here we use JWST spectra in the 0.8-5.2 $\mu$m range to examine the state and
abundance of CO2 and of CO on objects in the Kuiper belt. We show that,
contrary to expectation, CO2 appears significantly more abundant on the
surface of KBOs than does water ice and that CO is retained at temperatures
significantly higher than its sublimation temperature. We find that both of
these properties can be explained as a consequence of a thin layer of fine-
grained CO2 frost on these objects.

##  2 Observations and data reduction

A large selection of KBOs and Centaurs is being observed by JWST program 2418.
An overview of the observations and first results is presented in Pinilla-
Alonso et al. (submitted). CO2 and CO appear in many of the spectra, and the
spectral appearance of the CO2 can be roughly sorted by the shape of the
spectrum near the 4.26 $\mu$m $\nu_{3}$ absorption feature (Hénault et al.,
2023). We examine two spectra, representative of the range of appearance of
the CO2 feature, in detail to try to understand the state and abundance of CO2
and CO on these objects. Table 1 lists the two objects and their known
characteristics, including diameter and albedo from Vilenius et al. (2012) and
Fornasier et al. (2013) and orbital elements from JPL
Horizons111ssd.jpl.nasa.gov.

Table 1: Properties of the observed KBOs Name | 2005 RN43 | 2002 TC302
---|---|---
diameter (km) | 679${}^{+55}_{-73}$ | 584${}^{+106}_{-88}$
absolute magnitude (H) | 3.89$\pm$0.05 | 4.17$\pm$0.1
albedo | 0.11${}^{+.03}_{-.02}$ | 0.11${}^{+0.05}_{-0.03}$
semimajor axis (AU) | 41.8 | 55.7
eccentricity | 0.03 | 0.30
inclination (deg) | 19.2 | 35.0
date of observation | 3 Nov 2022 | 2 Jan 2023
heliocentric distance (AU) | 40.6 | 42.9

Each object is observed using the NIRSPEC integral field unit (IFU) mode with
the prism disperser (Jakobsen et al., 2022; Böker et al., 2022), giving a
spectral resolution of $R\sim 30-300$ from 0.6 to 5.2 $\mu$m and a spatial
scale of 0.1 arcseconds square within a 3 arcsecond square box. Each of the
KBOs appears as a point source. We have experimented with multiple methods for
spectral extraction, and we find that a modification of the standard JWST
pipeline (version #1.8.5) currently gives the best results. We begin our data
analysis with the Level 2 “rate” files, which are the calibrated two-
dimensional detector images showing individual spectra from all 30 of the IFU
slits. Examination of these files readily shows the effects of $1/f$ signal
modulation caused by electronic drifts during the detector readout. Many of
the detector pixels are not exposed to sky and so can be used to measure and
remove this modulation. We use the Level 2 “cal” files to find unilluminated
pixels, and, for each pixel in the “rate” image, we take the median of all
unilluminated pixels in a column within $\pm$150 rows to define the modulation
and subtract it from the “rate” image. The standard pipeline is then run on
the new “rate” file, including the AssignWcs, MSAFlagOpen, SourceType,
FlatField, PathLoss, and Photom steps. The final step in the standard pipeline
is the CubeBuild, which converts the two-dimensional spectral images into
three-dimensional spectral cubes. We circumvent as much of the interpolation
required for mapping the slits to on-sky positions by forcing the pipeline to
build the cubes with one axis parallel to the slits and one perpendicular to
the slits (using the “internal_cal” geometry in CubeBuild). A single column of
these cubes corresponds to a single slit.

Significant background appears in the non-target parts of the spectral cubes,
particularly at the shortest and longest wavelengths. We remove this
background by taking the median of all of the pixels at each wavelength that
are 5 pixels or more distant from the center of the target. Dithered images
could be used for background subtraction, but the dither positions are often
too closely spaced to use effectively. Significant numbers of bad pixels
appear in the image cubes, suggesting that PSF fitting with robust outlier
rejection could be a profitable method of spectral extraction. Unfortunately,
fitting using calculated PSFs (from, i.e.,
WebbPSF222https://www.stsci.edu/jwst/science-planning/proposal-planning-
toolbox/psf-simulation-tool) suffers from the undersampling of the PSF by the
detector, even with these minimally resampled cubes. In these data, the
undersampling of the PSF causes spectral ripples in individual pixels of the
image cube. The ripples are caused as the peak of the PSF slowly moves from
the center of pixel, to the edge, to an adjacent pixel as a function of
wavelength. The frequency of these ripples thus corresponds to the frequency
of the spectral trace moving from pixel-to-pixel, which occurs once every
$\sim$200 spectral pixels. The PSF as sampled by the detector thus also
changes over this pixel scale in a manner not currently captured by any of the
available PSF modeling tools. We circumvent this difficulty by constructing an
empirical PSF at each wavelength from the median of the spectral images within
25 pixels of that wavelength normalized to a total value if unity. We then
divide the spectral image by the empirical PSF at that wavelength and take the
median of this division within a 3 pixel radius as the brightness at that
pixel. The final spectrum is thus free of the pixel-to-pixel ripples
introduced by the PSF undersampling.

To obtain the relative reflectance of the object, we divide the extracted
spectrum by an identically reduced spectrum of a PRISM observation of the
solar analog 2MASS J16194609+5534178 (Gordon et al., 2022) from Program #
1128. This division removes any wavelength dependent calibration issues in the
JWST pipeline, corrects for the wavelength dependent PSF changes, and converts
the flux to a relative reflectance. We normalize each spectrum to have a
median value of unity between 1.5 and 2.5 $\mu$m. The final spectra of our two
example objects are shown in Figure 1.

Figure 1: The spectra of two representative KBOs from the sample. Each shows
subtle 1.5 and 2.0 $\mu$ absorption features indicative of a modest exposure
of water ice, deep 3 $\mu$m absorption consistent with a tholin-like processed
organic surface, a broad 3.4-3.6$\mu$m absorption owing to organic materials,
a narrow 4.6 $\mu$m absorption from CO, and numerous and variable features
from 4.0 to 4.4 $\mu$m all attributably to CO2.

##  3 Results and modeling

The features surrounding the deep 4.26$\mu$m CO2 absorption on 2005 RN43 and
2002 TC302 are unlike any reflectance spectrum previously seen. A 4.26$\mu$m
absorption has been seen on the icy satellites of Jupiter (Hibbitts et al.,
2003) and Saturn (Cruikshank et al., 2010), but in all cases the absorption is
modest and shows no additional absorptions nearby. In these cases, however,
CO2 is not in pure form but must be trapped as a minor constituent in a
background material to allow the CO2 to be stable at the elevated temperatures
of these satellites. These new KBO observations are the first at temperatures
where pure CO2 is stable and potentially dominating the spectral region around
the 4.26$\mu$m line. We thus carefully examine the reflectivity of pure CO2
through this region.

Between 4.0 and 4.5 $\mu$m, CO2 has not just a large change in absorptivity,
but its real index of refraction goes from its typical value of $\sim$1.3 at
shorter wavelengths, approaches zero near 4.19$\mu$m, climbs above a value of
5 at 4.27$\mu$m and then drops to 1.5 at longer wavelengths (Gerakines &
Hudson, 2020). Dramatic changes such as these in the real index of refraction
alongside large changes in the imaginary index of refraction (related to
absorption) cause complicated spectral behaviour. In Figure 2 we plot the
spectrum of both objects, along with the real and imaginary indices of
refraction. As can be seen, the most anomalous-appearing spectral behaviour of
2005 RN43 appears at precisely the locations where the real index of
refraction goes through unity, where it approaches zero, where the imaginary
index peaks, and where the imaginary index falls back to zero but the real
index changes rapidly.

Figure 2: The real ($n$) and imaginary ($k$) indecies of refraction of CO2
compared to the spectra of 2005 RN43 and 2002 TC302, which are both offset for
clarity with their zero points shown as blue and red horizontal dashed lines,
respectively. Major spectral features on both objects occur at the locations
marked by dashed lines, including when the real index passes through unity,
when it approaches zero, where the imaginary index peaks, and where the
imaginary index returns to near zero while the real index changes rapidly.

Because of the complexity of the behaviour in this region, we resort to full
Mie theory to understand the reflectivty across this wavelength region. We use
the Python package miepython, which implements and has been validated against
the Wiscombe (1979) algorithm, to calculate the single-scattering albedo of
spherical CO2 particles from very small 0.1 $\mu$m particles, in the Rayleigh
scattering regime, to 100 $\mu$m particles, in the geometric optics regime.
Using the single-scattering albedo, we calculate the geometric albedo assuming
a uniform surface of isotropic scatters using the formulation of Hapke (1993),
equation (10.37).

Mie scattering calculations are only strictly accurate for well-separated
spheres, which is clearly not the case on a planetary surface. Mustard & Hays
(1997) investigated the spectra of olivine and quartz powders with grain sizes
near and below the wavelength of spectral features and found that a combined
Mie theory/Hapke modeling approach as used above provides the best modeling
match to laboratory data. Some limitations to this approach are discussed
below.

Figure 3 shows the geometric albedo as a function of the particle size of the
scatterers for the 5 key wavelengths shown in Figure 2. One point is
immediately apparent from this Figure. For the geometric albedo at 4.31 $\mu$m
to rise above that at the 4.19 $\mu$m peak or at the 4.5$\mu$m continuum
beyond that, the grains must be smaller than about 2 $\mu$m. It is also clear
that as particles approach the Rayleigh scattering regime the contrast between
the peak at 4.19 $\mu$m and the trough at 4.26$\mu$m becomes increasingly
smaller while the overall albedo declines quickly.

Figure 3: The geometric albedo from a collection of Mie scattering particles
from 0.1 to 10 $\mu$m in diameter. The colors of the lines correspond to the
scattering behavior at the different wavelengths of the same colors of the
dashed lines in Figure 2. The strong changes in both the real and imaginary
index of refraction across this spectral region makes the spectral behaviour
an extremely complicated and sensitive function of particle size.

To understand the full spectrum through this region we show, in Figure 4,
model spectra of a uniform isotropic surface of Mie scattering particles of
various sizes. For 0.5 $\mu$m particles the 4.26$\mu$m absorption no longer
appears distinct, while at sizes of 1-2 $\mu$m the doublet in that absorption
band – which is clear in the data – emerges. We conclude that the main
structures seen in the spectrum of 2005 RN43 around 4.26 $\mu$m, namely the
absorption near 4.10 $\mu$m, the narrow peak at 4.19 $\mu$m, the doublet
absorption around 4.26 $\mu$m, and the broader strong peak at 4.31 $\mu$m can
all be explained if this object is covered in CO2 grains between about 1 and 2
$\mu$m in diameter. The behaviour seen in 2002 TC302, with a single peak
redward of the deep CO2 absorption is not seen in these models.

Figure 4: Full spectral models of a collection of Mie-scattering CO2 particles
of 4 particle sizes. The prominent peaks at both 4.19 and 4.34 $\mu$m are only
seen for particle diameters of 1-2 $\mu$m

One feature prevalent in all of the model spectra with small CO2 grains is
that the albedo is a minimum near 4.1 $\mu$m, where the real index of
refraction passes through unity. The imaginary index of refraction is small
but non-zero at this wavelength, so light will pass through all grains
unaffected by refraction or reflection, and simply be absorbed after passage
through a large number of grains. The albedo will thus stay near zero. In the
real spectra of the KBOs observed in the full program, however, the albedo is
always lowest in the 4.23/4.26 $\mu$m absorption doublet (Hénault et al.,
2023). Such a spectral shape can be a simple consequence of reflection from a
layer of CO2 particles covering a reflective medium. At 4.1 $\mu$m the CO2 is
essentially transparent, so it will propagate through the CO2 layer and
reflect off of the underlying surface. At 4.23/4.26 $\mu$m the light can –
depending on the optical depth of the layer – still be fully absorbed within
the CO2 layer, leading to deeper absorption lines. We model this behaviour
using the formalism for layered media of Hapke (1993) (equation 9.14). We
assume a thin layer of CO2 particles with a depth of $l$ and particle diameter
of $d$ overlying a surface with a constant geometric albedo of $a$. Figure 5
shows three spectral models with only small changes in these parameters. These
models reproduce the basic behaviour of both the 2005 RN43-type of CO2
spectrum, with sharp lines on either side of the main absorption, and the 2002
TC302-type spectrum, with a single line redward of the CO2 absorption. In the
models shown here, the CO2 layer is between 1 (orange and green lines) and 6
(blue line) particle diameters thick.

The contrast between the continuum and the peaks seen in these modeled spectra
is higher than that seen in the real KBOs. Interestingly, the same behaviour
is seen by Mustard & Hays (1997) in the laboratory vs. modeled spectra of
quartz, which has similar values of the both the real and imaginary index of
refraction in the 9 $\mu$m region as CO2 does in the 4.3 $\mu$m region. We
thus suspect that the too-high contrast of the models is a consequence of the
difficulties of using Mie theory to model tight packed grains such as these
and that we are indeed observing a surface covered with such small grains.

Figure 5: Spectral models of a thin layer of CO2 particles overlying a
reflective medium. Models such as these are needed to correctly reproduce the
non-zero albedo at 4.1 $\mu$m where CO2 becomes essentially transparent.
Slight changes in grain sizes and CO2 layer thickness give rise to many
different apparent spectral types, allowing the full range of CO2 detected in
the Kuiper belt to be explained.The three models shown have background albedo,
CO2 grain diameter, and CO2 layer thickness of 0.05, 2$\mu$m, 2$\mu$m
(orange); 0.16, 0.7$\mu$m, 10 $\mu$m (blue); and 0.23, 0.5$\mu$m, 1$\mu$m
(green).

An interesting consequence of the highly variable nature of the spectra from
4.0 to 4.4 $\mu$m is the difficulty of quantitatively interpreting the 13CO2
line at 4.38 $\mu$m. In Figure 5, for example, the 13CO2 abundance is just the
terrestrial value and is seen in the measured values of $n$ and $k$ simply
because of natural contamination of the laboratory sample of CO2. In each
modeled spectrum, however, though the 12C/13C ratio is identical, the depth of
the 13CO2 line is variable. The variability of this line in these models
highlights the difficulty of the using these observations to make precise
quantitative statements about the 12C/13C ratio in these objects. Nonetheless,
the observations of the 13CO2 line on these objects does not appear strongly
inconsistent with the strength seen in the spectral models, suggesting that
the isotopic ratio is not strongly inconsistent with the terrestrial value of
89. Such a finding is not surprising given the measurements of the 12C/13C in
cometary C2, CN, and HCN, which also give values close to the terrestrial
value (Bockelée-Morvan et al., 2015).

##  4 CO

The presence of CO on small bodies in the Kuiper belt requires that the CO be
trapped to prevent sublimation at these temperatures. Trapping of CO in
amorphous water ice or as a clathrate hydrate is often invoked as a possible
process important in the outer solar system (Bar-Nun et al., 1985; Gautier &
Hersant, 2005). The $\nu_{1}$ absorption feature of enclathrated CO is
spectrally shifted from its position at 4.675 $\mu$m in pure CO ice (Dartois,
2011). In Figure 6 we show a detail of the spectrum of 2005 RN43 near the CO
region.

Measuring the precise wavelength of the CO feature is complicated by the
uncertainty that still remains in the wavelength scale of the PRISM mode
observations. To understand possible wavelength uncertainties, we examined
wavelengths of known emission lines in a star forming region of NGC 7319
(Program 2732). After correcting for the redshift, we find that beyond 4
$\mu$m, the wavelength scale is off by about half a pixel ($\sim$0.003 $\mu$m)
and that the wavelength scale between 4 and 5 $\mu$m likewise changes by about
half a pixel. To account for his effect as best as possible, we set the
wavelength scale by forcing the 13CO2 absorption to occur precisely at 4.381
$\mu$m (which involves a 0.005 $\mu$m shift from the nominal wavelength
solution). While trapped CO2 will have a wavelength shift from this nominal
wavelength (for example on Ganymede; Hibbitts et al., 2003), the high
abundance of CO2 ensures that we are not seeing a small trapped population
here. Thus, based on the analysis of NGC 7319, the wavelength solution at the
location of the CO line should be accurate to within at least a 0.005 $\mu$m
pixel.

With this wavelength solution, the CO line on 2005 RN43 appears at the 4.675
$\mu$m wavelength of pure CO, and not at the 4.686 $\mu$m wavelength of CO
clathrate nor the 4.682 $\mu$m wavelength measured for trapping on amorphous
water ice at 50 K (Sandford & Allamandola, 1988). This 0.007 - 0.011 $\mu$m
difference is well above any expected wavelength uncertainty. Lack of evidence
for a water ice CO clathrate or trapping in amorphous water ice is perhaps not
surprising given the weak 1.5 and 2.0 $\mu$m absorption lines of water ice.

Given the large surface coverage of CO2 on these objects, we consider CO
trapped in this species. Simon et al. (2019) examined the trapping of CO in
CO2 and found that such trapping can be even more efficient than trapping in
H2O. The trapped CO was stable at temperatures as high as 80 K, even higher
than the expected surface temperatures of these objects. At 50K, the
wavelength of the absorption from the trapped CO is 4.673 $\mu$m, consistent
with the position of the line seen here. While the wavelength match is good,
the mechanism of gas trapping used in these experiments – mechanical mixing of
the gasses before freezing onto a substrate at 17 K – does not obviously apply
to these objects.

A more promising CO trapping mechanism involves both the production and
trapping of CO in CO2 through irradiation. Martín-Doménech et al. (2015)
showed the UV irradiation of CO2 creates CO which remains stable even at
temperatures as high as 90 K. The trapping mechanism here is different from
either the enclathration or the physical trapping discussed above, so it is
possible that the wavelength shift of CO trapped in this manner is different
from those mechanisms. Unfortunately, the wavelength of the CO absorption at
temperatures relevant for the Kuiper belt was not measured. Physically and
chemically, this explanation for the presence of CO at elevated temperatures
seems most promising, but laboratory measurement of the wavelength of CO
created and trapped at KBO-like temperatures is clearly needed.

Figure 6: A small portion of the spectrum of 2005 RN43 showing both the 13CO2
line, which we use as a wavelength reference, and the CO line. The wavelength
of the CO does not match that of CO clathrate hydrates or CO trapped in
amorphous ice, but is a close match to pure CO (which is unstable at this
temperature), and also to CO trapped in CO2 ice.

##  5 Discussion

The presence of a thin layer of fine-grained CO2 appears common on objects in
the Kuiper belt. CO is likewise common and must be trapped within some
thermally stable material. These two observations lead to a natural model for
the presence of both of these molecules. We posit that CO2 from the interior
of these objects is heated and diffuses toward the surface where it is
retained as a fine-grained frost. Irradiation of the CO2 (UV, solar wind,
and/or cosmic ray) then produces CO within these CO2 grains.

This simple idea hides some difficult complexities. Many of the other KBOs in
this program are seen to have CO2 features manifesting like the two discussed
here (Hénault et al., 2023). Given the high level of sensitivity of the
spectral models to the grain size and layer thickness, the lack of a wider
variety of spectral types is surprising. Hapke (1993) argues that for grain
sizes much smaller than the wavelength of the light the scattered radiance
tends to sample inhomogeneities in the medium that are of order of the
wavelength divided by pi, which is 1.35 $\mu$m in our case and remarkably
close to the preferred range of particle size solutions that we find. In this
case the observations would be indicative of grains a few microns or smaller
without the need to be a specific size, at least partially alleviating the
fine-tuning problem.

The thickness of the CO2 layer also exerts a strong control on the modeled
spectrum, in particular in the albedo at 4.1 $\mu$m where $n=1$ and CO2 is
transparent. Layers larger than a few microns begin to look more like the pure
CO2 spectra of Figure 4 and are hard to reconcile with the observations. The
need to use thin layers in the models is no obvious consequence of
decifiencies of Mie or Hapke theory, so we believe that such thin layers are
indeed intrinsic to the KBOs. How such thin layers are formed and preserved
with such a small thickness range is unclear, but the tight constraints
suggest an ongoing balance between formation and destruction that must be
common to these objects.

This process of outward volatile transport should operate most strongly on the
larger KBOs which have received the most heating. The size range probed in
this program is modest, so smaller objects will have to be observed to see if
this prediction is borne out. No modeling of interior volatile evolution on
objects of these sizes has yet been attempted, though for smaller sizes CO2 is
thought to remain unchanged by the smaller amount of interior heating (De
Sanctis et al., 2001). Understanding the implications of this new body of data
will be a critical next step in piecing together the chemical history of the
outer solar system.

We would like to thank John Stansberry and 2 anonymous reviewers for
interesting comments on the manuscript. This work is based on observations
made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained
from the Mikulski Archive for Space Telescopes at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These
observations are associated with program # 2418 and the specific observations
analyzed can be accessed via https://doi.org/10.17909/tnkm-hq35 (catalog DOI:
10.17909/tnkm-hq35). The authors acknowledge the team led by PI N. Pinilla-
Alonso for developing their observing program with a zero-exclusive-access
period. JWST (NIRSPEC)

## References

  * Bar-Nun et al. (1985) Bar-Nun, A., Herman, G., Laufer, D., & Rappaport, M. L. 1985, Icarus, 63, 317, doi: 10.1016/0019-1035(85)90048-X
  * Bockelée-Morvan & Biver (2017) Bockelée-Morvan, D., & Biver, N. 2017, Philosophical Transactions of the Royal Society of London Series A, 375, 20160252, doi: 10.1098/rsta.2016.0252
  * Bockelée-Morvan et al. (2015) Bockelée-Morvan, D., Calmonte, U., Charnley, S., et al. 2015, Space Sci. Rev., 197, 47, doi: 10.1007/s11214-015-0156-9
  * Böker et al. (2022) Böker, T., Arribas, S., Lützgendorf, N., et al. 2022, A&A, 661, A82, doi: 10.1051/0004-6361/202142589
  * Cruikshank et al. (2010) Cruikshank, D. P., Meyer, A. W., Brown, R. H., et al. 2010, Icarus, 206, 561, doi: 10.1016/j.icarus.2009.07.012
  * Dartois (2011) Dartois, E. 2011, Icarus, 212, 950, doi: 10.1016/j.icarus.2010.12.004
  * De Sanctis et al. (2001) De Sanctis, M. C., Capria, M. T., & Coradini, A. 2001, AJ, 121, 2792, doi: 10.1086/320385
  * Fornasier et al. (2013) Fornasier, S., Lellouch, E., Müller, T., et al. 2013, A&A, 555, A15, doi: 10.1051/0004-6361/201321329
  * Gautier & Hersant (2005) Gautier, D., & Hersant, F. 2005, Space Sci. Rev., 116, 25, doi: 10.1007/s11214-005-1946-2
  * Gerakines & Hudson (2020) Gerakines, P. A., & Hudson, R. L. 2020, ApJ, 901, 52, doi: 10.3847/1538-4357/abad39
  * Gordon et al. (2022) Gordon, K. D., Bohlin, R., Sloan, G. C., et al. 2022, AJ, 163, 267, doi: 10.3847/1538-3881/ac66dc
  * Hapke (1993) Hapke, B. 1993, Theory of reflectance and emittance spectroscopy (Topics in Remote Sensing, Cambridge, UK: Cambridge University Press, —c1993)
  * D (2023) Heńault et al., 2023, submitted. https://www.researchsquare.com/article/rs-2887978.
  * Hibbitts et al. (2003) Hibbitts, C. A., Pappalardo, R. T., Hansen, G. B., & McCord, T. B. 2003, Journal of Geophysical Research (Planets), 108, 5036, doi: 10.1029/2002JE001956
  * Jakobsen et al. (2022) Jakobsen, P., Ferruit, P., Alves de Oliveira, C., et al. 2022, A&A, 661, A80, doi: 10.1051/0004-6361/202142663
  * Martín-Doménech et al. (2015) Martín-Doménech, R., Manzano-Santamaría, J., Muñoz Caro, G. M., et al. 2015, A&A, 584, A14, doi: 10.1051/0004-6361/201526003
  * Mustard & Hays (1997) Mustard, J. F., & Hays, J. E. 1997, Icarus, 125, 145, doi: 10.1006/icar.1996.5583
  * Sandford & Allamandola (1988) Sandford, S. A., & Allamandola, L. J. 1988, Icarus, 76, 201, doi: 10.1016/0019-1035(88)90069-3
  * Sekanina (1992) Sekanina, Z. 1992, in Asteroids, Comets, Meteors 1991, ed. A. W. Harris & E. Bowell, 545
  * Simon et al. (2019) Simon, A., Öberg, K. I., Rajappan, M., & Maksiutenko, P. 2019, ApJ, 883, 21, doi: 10.3847/1538-4357/ab32e5
  * Vilenius et al. (2012) Vilenius, E., Kiss, C., Mommert, M., et al. 2012, A&A, 541, A94, doi: 10.1051/0004-6361/201118743
  * Wiscombe (1979) Wiscombe, W. J. 1979, MIE scattering calculations, advances in technique and fast, vector-shaped computer codes