#  Productions of $X(3872)$/$Z_{c}(3900)$ and $X_{2}(4013)$/$Z_{c}(4020)$ in
$Y(4220)$ and $Y(4360)$ decays

Ming-Zhu Liu  Frontiers Science Center for Rare isotopes, Lanzhou University,
Lanzhou 730000, China  School of Nuclear Science and Technology, Lanzhou
University, Lanzhou 730000, China  Xi-Zhe Ling  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
102206, China  Beijing Key Laboratory of Advanced Nuclear Materials and
Physics, Beihang University, Beijing, 102206, China  Peng Huanwu Collaborative
Center for Research and Education, Beihang University, Beijing 100191, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China

###### Abstract

The two excited vector charmonium states $Y(4220)$ and $Y(4360)$ are difficult
to be understood as pure $c\bar{c}$ charmonium states. Since they are located
close to the mass thresholds of $\bar{D}D_{1}$ and $\bar{D}^{*}D_{1}$, they
can be viewed as $\bar{D}D_{1}$ and $\bar{D}^{*}D_{1}$ molecules. Furthermore,
recent studies indicated that the exotic states $X(3872)$/$Z_{c}(3900)$ and
$X_{2}(4013)$/$Z_{c}(4020)$ are the isoscalar/isovector $\bar{D}D^{*}$ and
isoscalar/isovector $\bar{D}^{*}D^{*}$ molecules, respectively. In this work,
in the molecular picture, we employ the triangle diagram mechanism to study
the productions of $Z_{c}(3900)$ and $X(3872)$ in the pionic and radiative
decays of $Y(4220)$, as well as their heavy-quark spin symmetry (HQSS)
partners, i.e., the productions of $Z_{c}(4020)$ and $X_{2}(4013)$ in the
pionic and radiative decays of $Y(4360)$. Using the effective Lagrangian
approach, we obtain the ratios of the branching fractions
$\mathcal{B}[Y(4360)\to Z_{c}(4020)\pi]/\mathcal{B}[Y(4220)\to
Z_{c}(3900)\pi]=1.2$ and $\mathcal{B}[Y(4360)\to
X_{2}(4013)\gamma]/\mathcal{B}[Y(4220)\to X(3872)\gamma]=0.5$, almost
independent of model parameters, which indicate that the productions of
$X_{2}(4013)$ and $Z_{c}(4020)$ in the radiative and pionic decays of
$Y(4360)$ are likely to be measured in the future. The experimental studies of
the predicted decay modes will help verify the molecular nature of $X(3872)$,
$Z_{c}(3900)$, and $Y(4220)$. We hope the present work can stimulate
experimental and further theoretical studies on these decay modes.

##  I Introduction

In 2003, the Belle Collaboration observed a charmonium-like state $X(3872)$ in
the $J/\psi\pi\pi$ mass distribution of the $B$ decay Choi et al. (2003),
which was later confirmed by several other experiments Aubert et al. (2004);
Acosta et al. (2004); Abazov et al. (2004); Chatrchyan et al. (2013); Aaij et
al. (2012); Ablikim et al. (2014). In 2013, the LHCb Collaboration concluded
that the quantum numbers of $X(3872)$ are $J^{PC}=1^{++}$ Aaij et al. (2013).
The mass of $X(3872)$ is lower than the Goldfrey-Isgur model prediction by 90
MeV Godfrey and Isgur (1985), and the ratios of $\mathcal{B}[X(3872)\to
J/\psi\pi\pi\pi]/\mathcal{B}[X(3872)\to J/\psi\pi\pi]\approx 1$ Abe et al.
(2005); del Amo Sanchez et al. (2010); Ablikim et al. (2019a) and
$\mathcal{B}[B^{+}\to X(3872)K^{+}]/\mathcal{B}[B^{0}\to X(3872)K^{0}]\approx
0.5$ Choi et al. (2011) hint at large isospin breaking if $X(3872)$ is
regarded as a conventional charmonium $\chi_{c1}(2P)$. Due to these anomalous
properties, $X(3872)$ is widely viewed as an exotic state, triggering many
theoretical and experimental studies. Identified as a $\bar{D}D^{\ast}$
molecule, the mass and isospin breaking decay modes of $X(3872)$ can be well
described Swanson (2004); Voloshin (2004); AlFiky et al. (2006); Liu et al.
(2008); Sun et al. (2011); Nieves and Valderrama (2012); Guo et al. (2013a);
Karliner and Rosner (2015); Liu et al. (2019a), which imply that the $X(3872)$
contains a sizable $\bar{D}D^{\ast}$ molecular component. Considering the
heavy-quark spin symmetry(HQSS), the contact potentials of $J^{PC}=1^{++}$
$\bar{D}D^{\ast}$ and $J^{PC}=2^{++}$ $\bar{D}^{*}D^{\ast}$ systems are the
same Liu and Geng (2021). Therefore, assuming $X(3872)$ as a $J^{PC}=1^{++}$
$\bar{D}D^{\ast}$ bound state, it is natural to expect a $J^{PC}=2^{++}$
$\bar{D}^{*}D^{\ast}$ bound state (denoted by $X_{2}(4013)$), in agreement
with Refs. Nieves and Valderrama (2012); Guo et al. (2013a); Baru et al.
(2016). The BESIII Collaboration did not find evidence for $X_{2}(4013)$ by
studying the process $e^{+}e^{-}\to\rho^{0}X_{2}(4013)$ following the decay
$X_{2}(4013)\to D\bar{D}$ Ablikim et al. (2019b), and the Belle Collaboration
discovered a similar state named as $X_{2}(4014)$ in the $\gamma\psi(2S)$ mass
distribution of the $\gamma\gamma\to\gamma\psi(2S)$ process with a low
statistical significance Wang et al. (2022).

In 2013, the BESIII Collaboration and Belle Collaboration discovered a charged
charmonium-like state $Z_{c}(3900)$ in the $J/\psi\pi^{\pm}$ mass distribution
of $e^{+}e^{-}\to J/\psi\pi^{+}\pi^{-}$ Ablikim et al. (2013a); Liu et al.
(2013), which is above the $\bar{D}D^{\ast}$ mass threshold. As a result, the
$Z_{c}(3900)$ can be explained as a $\bar{D}D^{\ast}$ resonant state. In terms
of HQSS, the contact-range potentials of $J^{PC}=1^{+-}$ $\bar{D}D^{\ast}$ and
$J^{PC}=1^{+-}$ $\bar{D}^{\ast}D^{\ast}$ are the same Yang et al. (2021); Wu
et al. (2024), and therefore it is natural to expect the existence of a
$\bar{D}^{\ast}D^{\ast}$ resonant state, which may be associated with the
$Z_{c}(4020)$ state discovered by the BESIII Collaboration in the
$\pi^{\pm}h_{c}$ mass distribution of $e^{+}e^{-}\to h_{c}\pi^{+}\pi^{-}$
Ablikim et al. (2013b). In Ref. Wang et al. (2020), Wang et al. concluded that
$Z_{c}(3900)$ and $Z_{c}(4020)$ are $\bar{D}D^{\ast}$ and
$\bar{D}^{\ast}D^{\ast}$ resonant states related by the HQSS. This molecular
doublet picture was reinforced by a series of studies after $Z_{cs}(3985)$
Ablikim et al. (2021), the SU(3) flavor partner of $Z_{c}(3900)$, was
discovered Yang et al. (2021); Meng et al. (2020); Baru et al. (2022); Yan et
al. (2021); Du et al. (2022).

The vector charmonium states can be produced directly in $e^{+}e^{-}$
collisions. Among them, $Y(4260)$ has been intensively studied, which was
first discovered by the BaBar Collaboration in the $J/\psi\pi\pi$ mass
distribution of the $e^{+}e^{-}\to\gamma_{ISR}\pi^{+}\pi^{-}J/\psi$ process
Aubert et al. (2005), and then confirmed by the CLEO Collaboration Coan et al.
(2006) and Belle Collaboration Yuan et al. (2007). Later, with larger data
samples, the BESIII Collaboration found that the $Y(4260)$ split into two
states $Y(4220)$ and $Y(4320)$ Ablikim et al. (2017), in agreement with the
previous observations of $Y(4260)$ and $Y(4360)$, where $Y(4360)$ is
discovered in the process of $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ Aubert et
al. (2007). Referring to the Review of Particle Physics(RPP) Zyla et al.
(2020), the masses and widths of $Y(4220)$ and $Y(4360)$ are $(4222,48)$ MeV
and $(4372,115)$ MeV, respectively. From the measured $R$ value of the BESIII
Collaboration, the peaks at the energies of 4220 MeV and 4360 MeV are not
pronounced Ablikim et al. (2007), which indicate that these two states can not
be easily interpreted as conventional charmonium states. Since $Y(4220)$ and
$Y(4360)$ are near the mass thresholds of $\bar{D}D_{1}$ and
$\bar{D}^{*}D_{1}$, they can be viewed as $\bar{D}D_{1}$ and
$\bar{D}^{*}D_{1}$ molecules. Recently, Ji et al. employed the meson exchange
theory to assign $Y(4220)$, $Y(4360)$, and $Y(4415)$ as $\bar{D}D_{1}$,
$\bar{D}^{*}D_{1}$, and $\bar{D}^{*}D_{2}$ molecules Ji et al. (2022),
confirmed later by a similar approach Wang et al. (2023). In addition to the
molecular interpretation for the $XYZ$ states, there exist other
interpretations such as compact tetraquark states, mixing states, and
kinematic effect, see, e.g., Refs. Chen et al. (2016); Hosaka et al. (2017);
Lebed et al. (2017); Oset et al. (2016); Esposito et al. (2017); Dong et al.
(2017); Guo et al. (2018); Olsen et al. (2018); Ali et al. (2017); Karliner et
al. (2018); Guo et al. (2020); Brambilla et al. (2020); Liu et al. (2019b);
Meng et al. (2023); Liu et al. (2024).

It should be noted that the production of $X$ and $Z$ states are correlated
with the excited $Y$ states. In Ref. Ablikim et al. (2014), the BESIII
Collaboration found that $X(3872)$ can be produced through the radiative decay
of $Y(4260)$. Moreover, the $Z_{c}(3900)$ can be produced in the decay of
$Y(4260)$ Ablikim et al. (2013a); Liu et al. (2013). The experimental
measurements have motivated theoretical studies on the production of $XZ$
state in the $Y$ decays. In Refs. Wang et al. (2013); Chen et al. (2013); Liu
and Li (2013), the authors described the line shape of $J/\psi\pi$ and
$\pi\pi$ mass distribution and reproduced the $Z_{c}(3900)$ peak in the decay
of $Y(4260)\to J/\psi\pi\pi$ with the assumption that $Y(4260)$ is strongly
coupled to a pair of ground and excited charmed mesons. In Ref. Guo et al.
(2013b), Guo et al. proposed to produce $X(3872)$ in the most promising
channel $Y(4260)\to X(3872)\gamma$, where $Y(4260)$ and $X(3872)$ are proposed
as $\bar{D}D_{1}$ and $\bar{D}D^{*}$ molecules. In the molecular picture, the
decay width of $Y(4260)\to X(3872)\gamma$ is estimated to be tens of keV
through the effective Lagrangian approach Dong et al. (2014a). Later, Guo et
al. investigated the production of $X_{2}(4013)$ in the $Y(4260)$ radiative
decay Guo et al. (2015). Chen et al. estimated the partial decay widths of
$Y(4390)$ assuming $Y(4390)$ as a $\bar{D}^{*}D_{1}$ molecule, where the
$Z_{c}(4020)$ is produced in the $Y(4390)$ decay Chen et al. (2017).

The productions of $X(3872)$ and $Z_{c}(3900)$ in the decays of $Y(4260)$
inspire us to study the relationship among the $XYZ$ states. As indicated in
Ref. Zhu (2021), there is some relation among $X(3872)$, $Z_{c}(3900)$, and
$Y(4220)$ as tetraquark states that can be seen in the triangle diagram
mechanism, where $X(3872)$ and $Z_{c}(3900)$ are produced in the radiative and
pionic decays of $Y(4220)$. Two other sets of $XYZ$ states can be connected
via the triangle diagram mechanism with a similar relationship, i.e.,
$Y(4360)$, $X(4012)$, and $Z_{c}(4020)$ as well as $Y(4590)$, $X(4232)$, and
$Z_{cs}(4260)$. The former three masses can be obtained by shifting the masses
of $X(3872)$, $Z_{c}(3900)$, and $Y(4220)$ up by $140$ MeV and the latter by
$360$ MeV. We note that the so-obtained masses of the former three agree more
with the corresponding experimental data. Therefore, in this work, we only
focus on the $X(3872)$, $Z_{c}(3900)$, and $Y(4220)$ and $X_{2}(4013)$,
$Z_{c}(4020)$, and $Y(4360)$, assuming $Y(4220)$ and $Y(4360)$ as
$\bar{D}D_{1}$ and $\bar{D}^{*}D_{1}$ bound states, $X(3872)$ and
$X_{2}(4013)$ as $\bar{D}D^{*}$ and $\bar{D}^{*}D^{*}$ bound states, and
$Z_{c}(3900)$ and $Z_{c}(4020)$ as $\bar{D}D^{*}$ and $\bar{D}^{*}D^{*}$
resonant states. More specifically, we explore their relationship via the
radiative and pionic decays of the $Y$ states as shown in Fig. 1, where two
sets of $XYZ$ states are related to each other by HQSS.

\begin{overpic}[scale={0.45}]{hqss.pdf} \end{overpic} Figure 1:  Decays of
$Y(4220)\to Z_{c}(3900)\pi/X(3872)\gamma$ and decays of their HQSS partners
$Y(4360)\to Z_{c}(4020)\pi/X_{2}(4013)\gamma$.

This work is organized as follows. We briefly introduce the triangle diagram
mechanism for the $Y(4220)/Y(4360)$ as the $\bar{D}D_{1}/\bar{D}^{*}D_{1}$
molecules decaying into $Z_{c}(3900)/Z_{c}(4020)$ as the
$\bar{D}D^{*}/\bar{D}^{*}D^{*}$ molecules and $\pi(\gamma)$ as well as
$X(3872)/X_{2}(4013)$ together with $\pi$ and $\gamma$, and the effective
Lagrangian approach in Sec. II. The numerical results and discussions are
given in Sec. III, followed by a summary in the last section.

##  II Theoretical formalism

First, we explain how to construct the triangle diagrams to study the
radiative and pionic decays of $Y(4220)$ and $Y(4360)$ to
$Z_{c}(3900)/X(3872)$ and $Z_{c}(4020)/X_{2}(4013)$. In the molecular picture,
the triangle diagrams for the radiative and pionic decays of $Y(4220)$ and
$Y(4360)$ are shown in Figs. 2 and 3, where the excited charmed meson $D_{1}$
first decays into the ground-state mesons ${D}^{(\ast)}$ and $\pi(\gamma)$,
and then the $\bar{D}^{(\ast)}$ and ${D}^{(\ast)}$ interactions dynamically
generate the molecules $X(3872)$, $X_{2}(4013)$, $Z_{c}(3900)$, and
$Z_{c}(4020)$. One can see that the initial and final molecules in Figs. 2
(a-b) and Figs. 2 (c-d) as well as in Figs. 3 (a-b) and Figs. 3 (c-d) are
related to each other by the HQSS. Therefore, using the decays of $Y(4220)\to
X(3872)\gamma$ and $Y(4220)\to Z_{c}(3900)\pi$ already measured experimentally
as a benchmark, we can predict the decays of their HQSS partners, i.e.,
$Y(4360)\to X(4012)\gamma$ and $Y(4360)\to Z_{c}(4020)\pi$.

\begin{overpic}[scale={0.4}]{Y4220decay.pdf} \end{overpic} Figure 2: Triangle
diagrams accounting for $Y(4220)/Y(4360)\to Z_{c}(3900)/Z_{c}(4020)\pi/\gamma$
in the molecular picture.  \begin{overpic}[scale={0.45}]{Y4360decay.pdf}
\end{overpic} Figure 3: Triangle diagrams accounting for $Y(4220)/Y(4360)\to
X(3872)/X_{2}(4013)\pi/\gamma$ in the molecular picture.

###  II.1  Effective Lagrangians

In this work, we employ the effective Lagrangian approach to calculate the
Feynman diagrams of Fig. 2 and Fig. 3. The Lagrangians that describe each
interaction vertex in the triangle diagrams can be classified into two
categories. The first category involves the radiative and pionic decays of the
excited charmed meson, i.e., $D_{1}\to D^{(\ast)}\gamma$ and $D_{1}\to
D^{(\ast)}\pi$. The orbital angular momentum allowed for the decay $D_{1}\to
D^{\ast}\pi$ is either $L=0$ or $L=2$. Being the HQSS partner of $D_{1}$,
$D_{2}$ can only decay into $D^{*}\pi$ via $D-$wave, and the decay of
$D_{1}\to D^{\ast}\pi$ only proceeds via $D-$wave as well, which is the main
reason why $D_{1}$ and $D_{2}$ are narrow.

The Lagrangians describing the decays of $D_{1}\to D^{\ast}\pi$ and $D_{1}\to
D^{(\ast)}\gamma$ read Ding (2009); Li and Liu (2013); Liu and Li (2013); Dong
et al. (2014a)

$\displaystyle\mathcal{L}_{D_{1}D^{*}\pi}$ $\displaystyle=$ $\displaystyle
g_{D_{1}D^{*}\pi}(3D_{1}^{\mu}\partial_{\mu}\partial_{\nu}\pi
D^{\ast\nu}-D_{1}^{\mu}\partial^{\nu}\partial_{\nu}\pi D_{\mu}),$ (1)
$\displaystyle\mathcal{L}_{D_{1}D^{*}\gamma}$ $\displaystyle=$ $\displaystyle
eg_{D_{1}D^{*}\gamma}\varepsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu}D_{1\alpha}D^{\ast}_{\beta},$

where the couplings $g_{D_{1}D^{*}\pi}$ and $g_{D_{1}D^{*}\gamma}$ can be
determined by saturating their partial widths Cleven et al. (2014). We assume
that the decay mode $D_{1}\to D^{*}\pi$ saturates the width of $D_{1}$.
Therefore, one can determine the coupling $g_{D_{1}D^{*}\pi}=12.67$ GeV-1,
consistent with Refs. Li and Liu (2013); Wang and Liu (2021). On the other
hand, $D_{1}(2420)$, viewed as a conventional ${}^{1}P_{1}$ state, decays into
$D\gamma$ via the electromagnetic transition. As for the other state with the
same angular momentum, the $D_{1}(2430)$, viewed as a conventional
${}^{3}P_{1}$ state, only decays into $D^{*}\gamma$. However, the physical
$D_{1}(2420)$ and $D_{1}(2430)$ states are mixtures of ${}^{3}P_{1}$ and
${}^{1}P_{1}$ components with a mixing angle in the range of
$-25.7^{\circ}\sim-54.7^{\circ}$ Godfrey and Moats (2016); Li and Chen (2022).
In this work, we take the average value of $D_{1}(2420)$ radiative decay width
obtained in Refs. Godfrey and Moats (2016); Li and Chen (2022) to fix the
couplings. Then, with the widths of $\Gamma_{D_{1}^{0}\to D^{\ast
0}\gamma}=187$ keV and $\Gamma_{D_{1}^{+}\to D^{\ast+}\gamma}=19$ keV, we
obtain the couplings: $g_{D_{1}^{0}D^{\ast 0}\gamma}=1.82$ and
$g_{D_{1}^{+}D^{\ast+}\gamma}=0.58$, with $e=0.303$.

The second category involves the interactions between the hadronic molecules
and their constituents. The molecules’ interactions with the corresponding
constituents are described by the following Lagrangians Li and Liu (2013);
Dong et al. (2013); Chen et al. (2017)

$\displaystyle\mathcal{L}_{Y(4220)D_{1}\bar{D}}$ $\displaystyle=$
$\displaystyle g_{Y(4220)D_{1}\bar{D}}Y^{\nu}(4220)D_{1\mu}\bar{D},$ (2)
$\displaystyle\mathcal{L}_{Y(4360)D_{1}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
g_{Y(4360)D_{1}\bar{D}^{*}}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}Y_{\nu}(4360)D_{1\alpha}\bar{D}^{*}_{\beta},$
$\displaystyle\mathcal{L}_{Z_{c}(3900)D\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
g_{Z_{c}(3900)D\bar{D}^{*}}Z_{c}^{\mu}(3900)D\bar{D}^{*}_{\mu},$
$\displaystyle\mathcal{L}_{Z_{c}(4020)D^{*}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
ig_{Z_{c}(4020)D^{*}\bar{D}^{*}}\varepsilon_{\mu\nu\alpha\beta}\partial^{\mu}Z_{c}^{\nu}(4020)D^{\ast\alpha}\bar{D}^{\ast\beta},$
$\displaystyle\mathcal{L}_{X(3872)D\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle g_{X(3872)D\bar{D}^{*}}X^{\mu}(3872)D\bar{D}^{*}_{\mu},$
$\displaystyle\mathcal{L}_{X_{2}(4013)D^{*}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle
g_{X_{2}(4013)D^{*}\bar{D}^{*}}X_{2}^{\mu\nu}(4013)D^{*}_{\mu}\bar{D}^{*}_{\nu},$

where $g$ with the specific subscript denotes the molecule’s couplings to
their constituents, estimated in the contact-range approach.

###  II.2  Contact-range effective field theory

These couplings are estimated from the residues of poles obtained by solving
the Lippmann-Schwinger equation Liu et al. (2019c)

$\displaystyle T=(1-VG)^{-1}V,$ (3)

where $V$ is the hadron-hadron potential determined by the contact EFT
approach described below, and $G$ is the two-body propagator. In evaluating
the loop function $G$, we introduce a regulator of Gaussian form
$e^{-2q^{2}/\Lambda^{2}}$ in the integral as

$\displaystyle
G(s)=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{e^{-2q^{2}/\Lambda^{2}}}{{\sqrt{s}}-m_{1}-m_{2}-q^{2}/(2\mu_{12})+i\varepsilon},$
(4)

where $\sqrt{s}$ is the total energy in the c.m.frame of $m_{1}$ and $m_{2}$,
$\mu_{12}=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the reduced mass, and $\Lambda$
is the momentum cutoff. Following our previous works Liu et al. (2019c); Xie
et al. (2022), we take $\Lambda=1$ GeV in the present work. As the poles are
dynamically generated in the unphysical sheet, the loop function of Eq. (4)
becomes

$\displaystyle
G^{II}(s,m_{1},m_{2})=G^{I}(s,m_{1},m_{2})+i\mu_{12}\frac{p}{2\pi}{e^{-2p^{2}/\Lambda^{2}}},$
(5)

where the c.m. momentum $p$ is

$\displaystyle p=\sqrt{2\mu_{12}\left(\sqrt{s}-m_{1}-m_{2}\right)}.$ (6)

In the heavy-quark limit, the $\bar{D}^{(*)}D^{(*)}$ contact potentials are
characterized by two parameters $C_{a}$ and $C_{b}$ ( see Ref. Liu and Geng
(2021) for details), and the $\bar{D}^{(*)}D_{1}$ contact potentials are
characterised by four parameters $C_{a}$, $C_{b}$, $D_{a}$, and $D_{b}$ (see
Ref. Peng et al. (2022) for details). Here, we only present the potentials to
be used in the present work

$\displaystyle V(J^{PC}=1^{++}~{}\bar{D}^{*}D)$ $\displaystyle=$
$\displaystyle C_{a}+C_{b},$ (7) $\displaystyle
V(J^{PC}=2^{++}~{}\bar{D}^{*}D^{*})$ $\displaystyle=$ $\displaystyle
C_{a}+C_{b},$ $\displaystyle V(J^{PC}=1^{+-}~{}\bar{D}^{*}D)$ $\displaystyle=$
$\displaystyle C_{a}-C_{b},$ $\displaystyle
V(J^{PC}=1^{+-}~{}\bar{D}^{*}D^{*})$ $\displaystyle=$ $\displaystyle
C_{a}-C_{b},$ $\displaystyle V(J^{PC}=1^{--}~{}\bar{D}D_{1})$ $\displaystyle=$
$\displaystyle C_{a}-\frac{2}{3}D_{b},$ $\displaystyle
V(J^{PC}=1^{--}~{}\bar{D}^{*}D_{1})$ $\displaystyle=$ $\displaystyle
C_{a}-\frac{5}{4}C_{b}-\frac{1}{6}D_{b}-\frac{5}{4}D_{c}.$

With the potentials above, we search for poles in the vicinity of
$\bar{D}^{(*)}D^{(*)}$ and $\bar{D}^{(*)}D_{1}$ mass thresholds and then
determine the couplings between the molecular states and their constituents
from the residues of the corresponding poles,

$\displaystyle
g_{i}g_{j}=\lim_{{\sqrt{s}}\to{\sqrt{s_{0}}}}\left({\sqrt{s}}-{\sqrt{s_{0}}}\right)T_{ij}(\sqrt{s}),$
(8)

where $g_{i}$ denotes the coupling of channel $i$ to the dynamically generated
state and ${\sqrt{s_{0}}}$ is the pole position.

###  II.3  Decay amplitudes

Utilizing all the relevant Lagrangians, the amplitudes of the decays
$Y(4220)[k_{0},\varepsilon(k_{0})]\to
Z_{c}(3900)[p_{2},\varepsilon(p_{2})]\pi(p_{1})$ and
$Y(4360)[k_{0},\varepsilon(k_{0})]\to
Z_{c}(4020)[p_{2},\varepsilon(p_{2})]\pi(p_{1})$ in Fig. 2 can be expressed as

$\displaystyle i\mathcal{M}_{1a,1b}^{\pi}$ $\displaystyle=$ $\displaystyle
g_{Y(4220)D_{1}\bar{D}}g_{D_{1}D^{*}\pi}g_{D^{*}\bar{D}Z_{c}(3900)}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q_{1}^{2}-m_{D_{1}}^{2}}\frac{1}{q_{2}^{2}-m_{\bar{D}}^{2}}\frac{1}{q^{2}-m_{D^{*}}^{2}}\varepsilon^{\mu}(k_{0})$
$\displaystyle(-g^{\mu\nu}+\frac{q_{1}^{\mu}q_{1}^{\nu}}{m_{D_{1}}^{2}})(3p_{1}^{\nu}p_{1}^{\alpha}-g^{\nu\alpha}p_{1}^{2})(-g^{\alpha\beta}+\frac{q^{\alpha}q^{\beta}}{m_{D^{*}}^{2}})\varepsilon^{\beta}(p_{2})F(q^{2}),$
$\displaystyle i\mathcal{M}_{1c,1d}^{\pi}$ $\displaystyle=$ $\displaystyle
g_{Y(4360)D_{1}^{*}\bar{D}}g_{D_{1}D^{*}\pi}g_{D^{*}\bar{D}^{*}Z_{c}(4020)}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q_{1}^{2}-m_{D_{1}}^{2}}\frac{1}{q_{2}^{2}-m_{\bar{D}^{*}}^{2}}\frac{1}{q^{2}-m_{D^{*}}^{2}}\varepsilon^{\mu\nu\alpha\beta}k_{0\mu}\varepsilon_{\nu}(k_{0})$
$\displaystyle(-g^{\alpha\theta}+\frac{q_{1}^{\alpha}q_{1}^{\theta}}{m_{D_{1}}^{2}})(3p_{1}^{\theta}p_{1}^{\tau}-g^{\theta\tau}p_{1}^{2})(-g^{\tau\rho}+\frac{q^{\tau}q^{\rho}}{m_{D^{*}}^{2}})(-g^{\beta\phi}+\frac{q_{2}^{\beta}q_{2}^{\phi}}{m_{D^{*}}^{2}})\varepsilon^{\omega\eta\phi\rho}p_{2\omega}\varepsilon_{\eta}(p_{2})F(q^{2}),$

where $q_{1}$, $q_{2}$, and $q$ denote the momenta of $D_{1}$,
$\bar{D}^{(\ast)}$, and ${D}^{\ast}$ in the triangle diagrams, and
$\varepsilon^{\mu}$ and $\varepsilon^{\mu\nu}$ represent the polarization
vectors for the states of spin $S=1$ and spin $S=2$, respectively. The tensor
polarization vector satisfies the following relationships

$\displaystyle\varepsilon^{\mu\nu}_{(\lambda)}(q,m)q_{\mu}$ $\displaystyle=$
$\displaystyle 0$ (10)
$\displaystyle\varepsilon^{\mu\nu}_{(\lambda)}(q,m)g_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\sum_{\lambda=0,\pm 1,\pm
2}\varepsilon^{\mu\nu}_{(\lambda)}(q,m)\varepsilon^{\mu^{\prime}\nu^{\prime}}_{(\lambda)}(q,m)$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\bar{g}_{\mu\mu^{\prime}}\bar{g}_{\nu\nu^{\prime}}+\bar{g}_{\mu\nu^{\prime}}\bar{g}_{\nu\mu^{\prime}})-\frac{1}{3}\bar{g}_{\mu\nu}\bar{g}g_{\mu^{\prime}\nu^{\prime}},$

with $\bar{g}^{\mu\nu}=-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}$.

Similarly, we express the amplitudes of the decays
$Y(4220)[k_{0},\varepsilon(k_{0})]\to
X(3872)[p_{1},\varepsilon(p_{1})]\,\gamma[p_{2},\varepsilon(p_{2})]$ and
$Y(4360)[k_{0},\varepsilon(k_{0})]\to
X(4013)[p_{1},\varepsilon(p_{1})]\,\gamma[p_{2},\varepsilon(p_{2})]$ of Fig. 3

$\displaystyle i\mathcal{M}_{2a,2b}^{\gamma}$ $\displaystyle=$ $\displaystyle
g_{Y(4220)D_{1}\bar{D}}g_{D_{1}D^{*}\gamma}g_{D^{*}\bar{D}X(3872)}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q_{1}^{2}-m_{D_{1}}^{2}}\frac{1}{q_{2}^{2}-m_{\bar{D}}^{2}}\frac{1}{q^{2}-m_{D^{*}}^{2}}\varepsilon_{\mu}(k_{0})$
$\displaystyle(-g^{\mu\nu}+\frac{q_{1}^{\mu}q_{1}^{\nu}}{m_{D_{1}}^{2}})\varepsilon^{\alpha\beta\nu\theta}p_{1\alpha}\varepsilon_{\beta}(p_{1})(-g^{\theta\phi}+\frac{q^{\theta}q^{\phi}}{m_{D^{*}}^{2}})\varepsilon_{\phi}(p_{2})F(q^{2}),$
$\displaystyle i\mathcal{M}_{2c,2d}^{\gamma}$ $\displaystyle=$ $\displaystyle
g_{Y(4360)D_{1}\bar{D}^{*}}g_{D_{1}D^{*}\gamma}g_{D^{*}\bar{D}^{*}X(4013)}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q_{1}^{2}-m_{D_{1}}^{2}}\frac{1}{q_{2}^{2}-m_{\bar{D}^{*}}^{2}}\frac{1}{q^{2}-m_{D^{*}}^{2}}\varepsilon^{\mu\nu\alpha\beta}$
$\displaystyle
k_{0\mu}\varepsilon_{\nu}(k_{0})(-g^{\alpha\phi}+\frac{q_{1}^{\alpha}q_{1}^{\phi}}{m_{D_{1}}^{2}})\varepsilon^{\omega\eta\phi\rho}p_{1\omega}\varepsilon_{\eta}(p_{1})(-g^{\rho
c}+\frac{q^{\rho}q^{c}}{m_{D^{*}}^{2}})(-g^{\beta
d}+\frac{q_{2}^{\beta}q_{2}^{d}}{m_{D^{*}}^{2}})\varepsilon_{cd}F(q^{2}).$

In addition, to eliminate the ultraviolet divergence of the above amplitudes,
we supplement the relevant vertices of ${D}^{*}$ exchange with the following
monopolar form factor $F(q^{2})$

$\displaystyle F(q^{2})=(\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-q^{2}})^{2},$
(12)

which reflects the internal structure of hadrons, similar to the OBE model Liu
et al. (2018). An additional parameter $\Lambda$ is parameterised as a
dimensionless parameter $\alpha$, i.e., $\Lambda=m+\alpha\Lambda_{QCD}$, where
$\Lambda_{QCD}\sim 200-300$ MeV, and $\alpha$ is around $1$. In this work, we
vary $\alpha$ from $0.5$ to $1.5$ to study the result’s dependence on the
cutoff.

For the radiative decays, gauge symmetry must be satisfied. If the quantum
number of the initial state is $1^{-}$ and that of the final state is $1^{+}$,
the covariant amplitude of the radiative decay for the triangle diagram is
written as

$\displaystyle\mathcal{M}=\varepsilon_{\mu}(k_{0})\varepsilon_{\beta}(p_{1})\varepsilon_{\phi}(p_{2})M_{loop}^{\mu\beta\phi},$
(13)

where $M_{loop}^{\mu\beta\phi}$ contains the following independent terms

$\displaystyle
M_{loop}^{\mu\beta\phi}=\varepsilon^{p_{2}p_{1}\mu\phi}p_{1}^{\beta}g_{1}^{Tri}(q^{2})+\varepsilon^{p_{2}p_{1}\beta\phi}p_{2}^{\mu}g_{2}^{Tri}(q^{2}),$
(14)

with $g^{Tri}(q^{2})$ representing the loop integrals Dubnicka et al. (2011);
Dong et al. (2014a); Ganbold et al. (2021).

If the quantum number of the initial state is $1^{-}$ and that of the final
state is $2^{+}$, the corresponding amplitude is expressed as

$\displaystyle\mathcal{M}=\varepsilon_{\nu}(k_{0})\varepsilon_{\eta}(p_{1})\varepsilon_{cd}(p_{2})M_{loop}^{\nu\eta
cd},$ (15)

where $M_{loop}^{\nu\eta cd}$ is written as the following independent terms
Ganbold et al. (2021)

$\displaystyle M_{loop}^{\nu\eta cd}$ $\displaystyle=$ $\displaystyle
g_{1}^{Tri}(q^{2})\\{g^{c\nu}[g^{\eta d}(p_{1}\cdot
p_{2})-p_{1}^{d}p_{2}^{\eta}]+g^{d\nu}[g^{\eta c}(p_{1}\cdot
p_{2})-p_{1}^{c}p_{2}^{\eta}]\\}$ $\displaystyle+$ $\displaystyle
g_{2}^{Tri}(q^{2})\\{g^{\eta\nu}[p_{1}^{c}p_{2}^{d}+p_{1}^{d}p_{2}^{c}]-g^{c\eta}p_{1}^{\nu}p_{2}^{d}-g^{d\eta}p_{1}^{\nu}p_{2}^{c}\\}.$

With the amplitudes of pionic and radiative decays of the $Y$ states to the
$X$ and $Z$ states determined, one can obtain the corresponding partial decay
width as

$\displaystyle\Gamma=\frac{1}{2J+1}\frac{1}{8\pi}\frac{|\vec{p}|}{m_{Y}^{2}}\bar{|\mathcal{M}|}^{2},$
(17)

where $J$ is the total angular momentum of the initial $Y$ state, the overline
indicates the sum over the polarization vectors of the final states, and
$|\vec{p}|$ is the momentum of either final state in the rest frame of the $Y$
state.

##  III Numerical Results and Discussions

Table 1: Masses and quantum numbers of the mesons relevant to the present work. Zyla et al. (2020).  Meson | $I(J^{P})$ | M (MeV) | Meson | $I(J^{P})$ | M (MeV)
---|---|---|---|---|---
$\pi^{0}$ | $1(0^{-})$ | $134.98$ | $\pi^{\pm}$ | $1(0^{-})$ | $139.57$
$D^{0}$ | $\frac{1}{2}(0^{-})$ | $1864.84$ | $D^{\pm}$ | $\frac{1}{2}(0^{-})$ | $1869.66$
$D^{\ast 0}$ | $\frac{1}{2}(1^{-})$ | $2006.85$ | $D^{\ast\pm}$ | $\frac{1}{2}(1^{-})$ | $2010.26$
$D_{1}^{0}$ | $\frac{1}{2}(1^{+})$ | $2422.06$ | $D_{1}^{\pm}$ | $\frac{1}{2}(1^{+})$ | $2426.06$
$Y(4220)$ | $0(1^{--})$ | $4222.7$ | $Y(4360)$ | $0(1^{--})$ | $4372$
$Z_{c}(3900)$ | $1(1^{+-})$ | $3887.1$ | $Z_{c}(4020)$ | $1(1^{+-})$ | $4024.1$
$X(3872)$ | $0(1^{++})$ | $3871.65$ | $X(4012)$ | $0(2^{++})$ | $4013$

In Table 1, we tabulate relevant particles’ masses and quantum numbers. First,
we determine the couplings between the molecules and their constituents in the
contact-range EFT approach. Following Refs. Pan et al. (2023); Wu et al.
(2024), we determine the unknown parameters of the contact-range potential by
fitting to the experimental data, and then the molecules’ coupling to their
constituents are derived from the residues of the poles. For the
$\bar{D}^{(\ast)}{D}^{(\ast)}$ system, we can see that the $J^{PC}=1^{++}$
$\bar{D}D^{\ast}$ potential is the same as the $J^{PC}=2^{++}$
$\bar{D}^{\ast}D^{\ast}$ potential. Identifying the $X(3872)$ as a
$J^{PC}=1^{++}$ $\bar{D}D^{\ast}$ bound state, we determine the value of
$C_{a}+C_{b}=-20.25$ GeV-2, and then obtain the coupling
$g_{X\bar{D}D^{\ast}}=9.16$ GeV. Considering the HQSS, we find a
$\bar{D}^{*}D^{*}$ bound state with a mass of $4013$ MeV denoted by
$X_{2}(4013)$, and then its coupling is determined as
$g_{X_{2}\bar{D}^{*}D^{\ast}}=10.87$ GeV. Using the Weinberg compositeness
theorem Wang et al. (2013), one can calculate these couplings as
$g_{X\bar{D}^{*}D}=8.29$ GeV and $g_{X_{2}\bar{D}^{*}D^{\ast}}=9.00$ GeV,
consistent with our estimations.

For the $\bar{D}^{(\ast)}D_{1}$ system, assuming $Y(4360)$ and $Y(4220)$ as
$J^{PC}=1^{--}$ $\bar{D}^{\ast}D_{1}$ and $J^{PC}=1^{--}$ $\bar{D}D_{1}$ bound
states, we determine the $D_{1}\bar{D}^{*}$ and $D_{1}\bar{D}$ contact
potentials as
$C_{a}-\frac{5}{4}C_{b}-\frac{1}{6}D_{b}-\frac{5}{4}D_{c}=-32.27$ GeV-2 and
$C_{a}-\frac{2}{3}D_{b}=-34.65$ GeV-2, which to some extent reflect the HQSS
between the $D_{1}\bar{D}^{*}$ and $D_{1}\bar{D}$ systems. With the obtained
contact-range potentials, we determine the couplings of
$g_{Y(4360)D_{1}\bar{D}^{*}}=6.97$ and $g_{Y(4220)D_{1}\bar{D}}=31.32$ GeV. In
terms of the Weinberg compositeness theorem, one can obtain the couplings of
$g_{Y(4360)D_{1}\bar{D}^{*}}=4.11$ and $g_{Y(4220)D_{1}\bar{D}}=18.29$ GeV Li
and Liu (2013); Chen et al. (2017), which are smaller than those of our
estimations. We note that the Weinberg compositeness theorem is only
applicable to weakly bound states, while the binding energies of the
$\bar{D}^{(\ast)}D_{1}$ molecules are about 70 MeV.

The leading-order contact potentials can only generate bound or virtual poles
Dong et al. (2021). To generate resonance states, one has to add higher-order
momentum-dependent contact potentials Zhai et al. (2022). In the heavy-quark
limit, the contact potentials of $J^{PC}=1^{+-}$ $\bar{D}D^{\ast}$ and
$J^{PC}=1^{+-}$ $\bar{D}^{\ast}D^{\ast}$ are expressed as
$C_{LO}+C_{NLO}~{}q^{2}$, where $C_{LO}=C_{a}-C_{b}$ and $q$ is the c.m.
momentum. By reproducing the mass and width of $Z_{c}(3900)$, one can obtain
the value of $C_{LO}=-7.7$ GeV-2 and $C_{NLO}=-211$ GeV-4, and then determine
the coupling $g_{Z_{c}(3900)\bar{D}D^{*}}=7.10$ GeV. With the obtained values
of $C_{LO}$ and $C_{NLO}$, we predict the mass and width of the
$\bar{D}^{*}D^{*}$ molecule, i.e., (4028,13) MeV, consistent with the RPP, and
then determine the coupling $g_{Z_{c}(4020)\bar{D}^{*}D^{*}}=1.77$.

Table 2:  Values of the $XYZ$ states couplings to their constituents.  |  $X(3872)$ |  $X_{2}(4013)$ |  $Z_{c}(3900)$ |  $Z_{c}(4020)$ |  $Y(4220)$ |  $Y(4360)$
---|---|---|---|---|---|---
g |  $9.16$ GeV |  $10.87$ GeV |  $7.10$ GeV |  $1.77$ |  $31.32$ GeV |  $6.97$

In Table 2, we collect the values of the $XYZ$ molecular candidates couplings
to their constituents, which are determined in the isospin basis. However, the
couplings in the particle basis shown in Fig. 2 and Fig. 3 are necessary for
our study. Working in the isospin limit, we present the ratios of the
couplings in the particle basis to those in the isospin basis in Table 3,
which help us determine the values of the molecules couplings to their
constituents in the particle basis. Moreover, the signs of ratios given in
Table 3 are consistent with the signs in the triangle diagrams, which
determine the signs between the triangle diagrams. In the isospin limit, the
amplitudes for the decays $Y(4220)\to X(3872)\pi$ and $Y(4220)\to
Z_{c}(3900)\gamma$ as well as their HQSS partners $Y(4360)\to X_{2}(4013)\pi$
and $Y(4360)\to Z_{c}(4020)\gamma$ illustrated by the triangle diagrams are
zero, and therefore we focus on the decays of $Y(4220)\to X(3872)\gamma$ and
$Y(4220)\to Z_{c}(3900)\pi$ as well as the decays of their HQSS partners
$Y(4360)\to X_{2}(4013)\gamma$ and $Y(4360)\to Z_{c}(4020)\pi$ in this work.

Table 3: Ratios of the couplings in the particle basis to the coupling in the isospin basis. Molecules |  ${D}^{\ast+}D^{-}$ |  ${D}^{+}D^{\ast-}$ |  ${D}^{\ast 0}\bar{D}^{0}$ |  ${D}^{0}\bar{D}^{\ast 0}$
---|---|---|---|---
$X(3872)$ |  $1/2$ |  $-1/2$ |  $1/2$ |  $-1/2$
$Z_{c}(3900)$ |  $1/2$ |  $1/2$ |  $-1/2$ |  $-1/2$
Molecules |  ${D}^{+}_{1}D^{(\ast)-}$ |  ${D}^{(\ast)+}D_{1}^{-}$ |  ${D}^{0}_{1}\bar{D}^{(\ast)0}$ |  ${D}^{(\ast)0}\bar{D}^{0}_{1}$
$Y(4220)$ |  $-1/2$ |  $1/2$ |  $-1/2$ |  $1/2$
$Y(4360)$ |  $-1/2$ |  $1/2$ |  $-1/2$ |  $1/2$
Molecules |  ${D}^{\ast+}D^{\ast-}$ |  ${D}^{\ast 0}\bar{D}^{\ast 0}$ |  |
$X_{2}(4013)$ |  $1/\sqrt{2}$ |  $1/\sqrt{2}$ |  |
$Z_{c}(4020)$ |  $1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  |

In our theoretical framework, an unknown dimensionless parameter $\alpha$
exists in the form factor. The value of $\alpha$ is assumed to be around 1,
and therefore, $\alpha$ is varied from 0.5 to 1.5 to reflect the uncertainties
induced by the form factors. Moreover, the uncertainties of the couplings in
the three vertices of the triangle diagrams lead to uncertainties in the final
results. Following Ref. Wu et al. (2024), the molecule’s couplings to their
constituents lead to a $10\%$ uncertainty as the cutoff $\Lambda$ varies from
$1$ GeV to $2$ GeV. The HQSS is adopted to estimate the coupling
$g_{D_{1}D^{*}\pi}$, resulting in a $15\%$ uncertainty Pan et al. (2020). The
coupling $g_{D_{1}D^{*}\gamma}$ likely leads to about a $30\%$ uncertainty due
to averaging the widths of Ref. Godfrey and Moats (2016) and Ref. Li and Chen
(2022). Finally, we obtain the uncertainties for the partial decay widths
originating from the uncertainties of these couplings via a Monte Carlo
sampling within their $1\sigma$ intervals.

---

\begin{overpic}[scale={.35}]{ytoz.pdf} \end{overpic}

\begin{overpic}[scale={.35}]{ytox.pdf} \end{overpic}

Figure 4: Partial decay widths of $Y(4220)\to Z_{c}(3900)\pi$ and $Y(4360)\to
Z_{c}(4020)\pi$ (a), and $Y(4220)\to X(3872)\gamma$ and $Y(4360)\to
X_{2}(4013)\gamma$ (b) as a function of $\alpha$. The lines and bands
represent their central values and corresponding uncertainties.

In Fig. 4(a), we present the partial decay widths of $Y(4220)\to
Z_{c}(3900)\pi$ and $Y(4360)\to Z_{c}(4020)\pi$ as a function of $\alpha$, and
their corresponding partial decay widths are in the range of $3.7\sim 6.5$ MeV
and $10.3\sim 18.9$ MeV, respectively, in agreement with Refs. Dong et al.
(2014b); Chen et al. (2017). Considering the widths of $Y(4220)$ and
$Y(4360)$, their decaying branching fractions are found to be $0.11$ and
$0.13$ for $\alpha=1$. Referring to the RPP, the partial decay width of
$Y(4220)\to J/\psi\pi\pi$ is around 0.29 MeV and the ratio of the branching
fraction $\mathcal{B}[Y(4220)\to Z_{c}(3900)\pi\to
J/\psi\pi\pi]/\mathcal{B}[Y(4220)\to J/\psi\pi\pi]=0.215$, which imply that
the upper limit of the branching fraction $\mathcal{B}[Z_{c}(3900)\to
J/\psi\pi]$ is 0.01 according to our analysis.

For radiative decays, gauge symmetry must be satisfied. The loop integrals of
Eqs. (II.3) can be decomposed into a series of terms in Eq. (14) and Eq.
(II.3). In addition, we checked that the loop integrals of Eqs. (II.3) satisfy
gauge symmetry, i.e., $p_{1}\mathcal{M}_{2a,2b}^{\gamma}=0$ and
$p_{1}\mathcal{M}_{2c,2d}^{\gamma}=0$. In Fig. 4(b), we present the partial
decay widths of $Y(4220)\to X(3872)\gamma$ and $Y(4360)\to X_{2}(4013)\gamma$
as a function of $\alpha$, and their partial decay widths are $9\sim 24$ keV
and $11\sim 30$ keV, respectively. In Ref. Dong et al. (2014a), Dong et al.
estimated the partial decay width of $Y(4220)\to X(3872)\gamma$ to be
$23.2\sim 48.6$ keV, a bit larger than ours. We note that the mass of the
initial state of this work and that of Ref. Dong et al. (2014a) is a bit
different. Considering the widths of $Y(4220)$ and $Y(4360)$, we obtain the
decay branching fractions for $\alpha=1$, i.e., $\mathcal{B}[Y(4220)\to
X(3872)\gamma]=3.67\times 10^{-4}$ and $\mathcal{B}[Y(4360)\to
X_{2}(4013)\gamma]=1.89\times 10^{-4}$, which are smaller than those of the
above pionic decays by two orders of magnitude.

The uncertainties in calculating the absolute branching fractions can be
reduced for ratios. In Fig. 5(a), we show the ratios of the decay widths of
$Y(4360)\to Z_{c}(4020)\pi$ to those of $Y(4220)\to Z_{c}(3900)\pi$ as a
function of $\alpha$. One can see that the result is weakly dependent on the
parameter $\alpha$, which is always around $2.8$. Combing the widths of
$Y(4220)$ and $Y(4360)$, we have the ratio $\mathcal{B}[Y(4360)\to
Z_{c}(4020)\pi]/\mathcal{B}[Y(4220)\to Z_{c}(3900)\pi]=1.2$, indicating that
the production rate of $Z_{c}(4020)$ in the $Y(4360)$ decay is a bit larger
than that of $Z_{c}(3900)$ in the $Y(4220)$ decay. Similarly, the ratio of the
partial decay width of $Y(4360)\to X_{2}(4013)\gamma$ to that of $Y(4220)\to
X(3872)\gamma$ as a function of $\alpha$ is shown in Fig. 5(b), which is
around $1.2$. Furthermore, we obtain the ratio $\mathcal{B}[Y(4360)\to
X_{2}(4013)\gamma]/\mathcal{B}[Y(4220)\to X(3872)\gamma]=0.5$, implying that
the production rates of $X_{2}(4013)$ in the $Y(4360)$ decay is lower than
that of $X(3872)$ in the $Y(4220)$ decay. The ratios predicted in this work
can help verify or refute the molecular interpretation of the $XYZ$ states.

---

\begin{overpic}[scale={.35}]{ratio-strong.pdf} \end{overpic}

\begin{overpic}[scale={.35}]{ratio-radiative.pdf} \end{overpic}

Figure 5:  Ratios of $\Gamma[Y(4360)\to Z_{c}(4020)\pi]/\Gamma[Y(4220)\to
Z_{c}(4020)\pi]$ (a) and $\Gamma[Y(4360)\to
X_{2}(4013)\gamma]/\Gamma[Y(4220)\to X(3872)\gamma]$ (b) as a function of
$\alpha$. The lines and bands represent their central values and corresponding
uncertainties.

One should note that the $XYZ$ states are regarded as pure hadronic molecules
in this work. However, recent studies show these $XYZ$ states can not be pure
hadronic molecules. In terms of the mass distribution of $X(3872)$, the
$\bar{D}^{*}D$ molecular component accounts for $82\%$ of the $X(3872)$ total
wave function according to the BESIII Collaboration Ablikim et al. (2023) and
$85\%$ according to the LHCb Collaboration Aaij et al. (2020). Very recently,
the $Y(4220)$ is assigned as the $D_{1}\bar{D}$ hadronic molecule by analyzing
its mass distributions in $e^{+}e^{-}$ collisions in the energy region of 4.2
GeV-4.35 GeV von Detten et al. (2024). As for the $Z_{c}(3900)$, the
$\bar{D}^{*}D$ molecular component accounts for around $40\%$ of its total
wave function by analyzing the BESIII data and the lattice QCD simulations Yan
et al. (2024). Given these results, our predictions for the partial decay
widths would decrease if other minor components were considered.

##  IV Summary and Discussion

Since $X(3872)$ was discovered by the Belle Collaboration in 2003, many
charmonium-like states beyond the conventional quark model picture (often
named $XYZ$ states) have been found experimentally, most of which are close to
the mass threshold of a pair of charmed mesons. Recent studies have shown that
these $XYZ$ states have strong couplings to a pair of charmed mesons to the
extent that they can be regarded as hadronic molecules. In this work, assuming
$X(3872)$, $Z_{c}(3900)$, and $Y(4220)$ as the $\bar{D}^{*}D$, $\bar{D}^{*}D$,
and $\bar{D}_{1}D$ hadronic molecules, we predicted their HQSS partners in the
contact range EFT, likely corresponding to $X_{2}(4013)$, $Z_{c}(4020)$, and
$Y(4360)$. We then employed the triangle diagram mechanism to describe the
productions of the $X$ and $Z_{c}$ states in the radiative and pionic decays
of the $Y$ states. Finally, we adopted the effective Lagrangian approach to
calculate the partial decay widths.

Our results showed that the branching fractions are $\mathcal{B}[Y(4220)\to
Z_{c}(3900)\pi]=0.11$, $\mathcal{B}[Y(4220)\to X(3872)\gamma]=3.67\times
10^{-4}$, $\mathcal{B}[Y(4360)\to Z_{c}(4020)\pi]=0.13$, and
$\mathcal{B}[Y(4360)\to X_{2}(4013)\gamma]=1.89\times 10^{-4}$, which are
dependent on the unknown parameters $\alpha$ in the form factor. In
particular, we showed the ratios of the branching fractions
$\mathcal{B}[Y(4360)\to Z_{c}(4020)\pi]/\mathcal{B}[Y(4220)\to
Z_{c}(3900)\pi]=1.2$ and $\mathcal{B}[Y(4360)\to
X_{2}(4013)\gamma]/\mathcal{B}[Y(4220)\to X(3872)\gamma]=0.5$, which are
weakly dependent on $\alpha$. From our study, one can conclude that the
$Z_{c}(4020)$ and $X_{2}(4013)$ can be produced in the pionic and radiative
decays of $Y(4360)$, which will serve as a model-independent verification of
the molecular nature of the $XYZ$ states if observed. Our studies demonstrated
that studying the radiative and pionic decays of the HQSS doublets can deepen
one’s understanding of the nature of the $XYZ$ states.

##  V Acknowledgments

This work is partly supported by the National Key R&D Program of China under
Grant No. 2023YFA1606703. M.Z.L acknowledges support from the National Natural
Science Foundation of China under Grant No.12105007. X.Z.L acknowledges
support from the National Natural Science Foundation of China under Grant No.
12247159 and China Postdoctoral Science Foundation under Grant No.
2022M723149.

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# Orbits in symmetric spaces, II

N. J. Kalton  Department of Mathematics, University of Missouri-Columbia,
Columbia, Missouri 65211, U.S.A<EMAIL_ADDRESS>,  F. A. Sukochev
School of Mathematics and Statistics, University of New South Wales, Sydney,
NSW 2052, Australia<EMAIL_ADDRESS>and  D. Zanin  School of Computer
Science, Engineering and Mathematics, Flinders University, Adelaide, SA 5042,
Australia<EMAIL_ADDRESS>
###### Abstract.

Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or
$(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and
sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed
convex hull of its extreme points. We also give an application to
symmetrically normed ideals of compact operators on a Hilbert space.

###### Key words and phrases:

Fully symmetric spaces, Hardy-Littlewood majorization, orbits

###### 1991 Mathematics Subject Classification:

46E30, 46B70, 46B20

The first author acknowledges support from NSF grant DMS-0555670; the second
and third authors acknowledge support from the ARC

##  1\. Introduction

Let $I$ be either the interval $(0,1)$ or the semi-axis $(0,\infty)$ and
suppose $f\in L_{1}(I)+L_{\infty}(I).$ We define the orbit $\Omega(f)$ of $f$
to be set of $Tf$ where $T:L_{1}+L_{\infty}\to L_{1}+L_{\infty}$ is an
operator with $\|T\|_{L_{1}\to L_{1}},\|T\|_{L_{\infty}\to L_{\infty}}\leq 1$
(see [Ryff1965, KreinPetuninSemenov1982]). Then it follows from the Calderón-
Mitjagin Theorem [Calderon1966, Mitjagin1965, BennettSharpley1982,
KreinPetuninSemenov1982] that $\Omega(f)$ can be characterized as the set of
$g\in L_{1}+L_{\infty}$ such that

(1) $\int_{0}^{t}g^{*}(s)\,ds\leq\int_{0}^{t}f^{*}(s)\,ds,\qquad 0<t<\infty$

where as usual $f^{*}$ is the decreasing rearrangement of $|f|$ (see §2 for
definitions). This may be written $g\preceq f$ where $\preceq$ is the Hardy-
Littlewood-Polya ordering. Thus $E$ is an exact interpolation space if and
only if it is fully symmetric (see §2).

The extreme points of $\Omega(f)$, which we denote $\partial_{e}\Omega(f)$
were obtained in [Ryff1967] (for case of spaces on $(0,1)$) and
[ChilinKryginSukochev1992] (for the general case). Except in the special case
when $I=(0,\infty)$ and $E\supset L_{\infty}$ these are given by
$\partial_{e}\Omega(f)=\\{g:\ g^{*}=f^{*}\\}$ (see §2 for full details; in the
exceptional cases the extreme points from a subset of this set). Let
$\mathcal{Q}(f)$ be the convex hull of the set $\\{g:\ g^{*}\leq f^{*}\\}.$
Then it is clear that if $E$ is fully symmetric and $f\in E$ the closure
$\mathcal{Q}_{E}(f)$ of $\mathcal{Q}(f)$ in $E$ coincides with the closed
convex hull of $\partial_{e}\Omega(f).$

In [Ryff1965] it was shown for the case of $I=(0,1)$ that the orbit
$\Omega(f)$ is always weakly compact in $L_{1}(0,1).$ It follows from results
in [DoddsSukochevSchluchtermann2001] that if $E$ is an order-continuous
(equivalently, separable) symmetric function space (which is necessarily fully
symmetric) and $f\in E$ then $\Omega(f)$ is weakly compact in $E$. Thus it is
an immediate consequence of the Krein-Milman theorem that $\mathcal{Q}_{E}(f)$
coincides with $\Omega(f).$

For the case of non-separable fully symmetric spaces the situation is less
clear. The example $E=L_{\infty}$ and $f=1$ shows that $\mathcal{Q}_{E}(f)$
may still coincide with $\Omega(f).$ This problem was first investigated by
Braverman and Mekler [BravermanMekler1977] for the unit interval i.e.
$I=(0,1)$. They gave a sufficient condition for $\Omega(f)=\mathcal{Q}_{E}(f)$
in terms of the behavior of the dilation operators $\sigma_{\tau}$ (see §2 for
the appropriate definitions). Precisely they showed that if $E$ is a fully
symmetric Banach function space on $(0,1)$ such that

$\lim_{\tau\to\infty}\frac{\|\sigma_{\tau}\|_{E\to E}}{\tau}=0$

then $\Omega(f)=\mathcal{Q}_{E}(f)$ for every $f\in E.$ This condition is,
however, not necessary since it may fail in separable symmetric spaces (e.g.
$E=L_{1}$).

Recently two of the current authors [SukochevZanin2009] found a necessary and
sufficient condition for the similar problem concerning the positive part of
the orbit. If $f\geq 0$ we denote by $\Omega_{+}(f)$ the set $\\{g:\
g\in\Omega(f),\ g\geq 0\\}.$ In [SukochevZanin2009] it was shown that for a
fully symmetric Banach function space $E$ with a Fatou norm (sometimes called
a weak Fatou property) that if $f\in E_{+}$ then $\Omega_{+}(f)$ coincides
with the closed convex hull of its extreme points if and only if a local
Braverman-Mekler type condition holds. If $I=(0,1)$ or if $I=(0,\infty)$ and
$E$ is not contained in $L_{1}(0,\infty)$ this condition takes the form

(2) $\lim_{\tau\to\infty}\frac{\|\sigma_{\tau}(f^{*})\|_{E}}{\tau}=0.$

If $I=(0,\infty)$ and $E\subset L_{1}$ we must replace (2) by

(3)
$\lim_{\tau\to\infty}\frac{\|\chi_{(0,1)}\sigma_{\tau}(f^{*})\|_{E}}{\tau}=0.$

The results of [SukochevZanin2009] imply that under the same hypotheses on $E$
(full symmetricity and a Fatou norm) that (2) and (3) are sufficient for
$\mathcal{Q}_{E}(f)=\Omega(f).$

Our main result in this paper is to show that, indeed, if $E$ is a fully
symmetric Banach space with a Fatou norm on $(0,1)$ or $(0,\infty)$, (2) and
(3) are necessary and sufficient for $\Omega(f)=\mathcal{Q}_{E}(f).$ These
results are Theorems 4.1, 4.2 and 4.3 below. We also establish the
corresponding result for sequence spaces in Theorem 4.5; sequence spaces were
not covered in [SukochevZanin2009] so we are also able to complete the picture
for the positive part of the orbit.

We conclude the paper with an application to orbits in symmetrically normed
ideals of compact operators on a Hilbert space.

##  2\. Preliminaries

In this section we present some definitions from the theory of symmetric
spaces. For more details on the latter theory we refer to
[KreinPetuninSemenov1982, LindenstraussTzafriri1979, BennettSharpley1982].

Let $I$ denote either $(0,1)$ or on $(0,\infty)$ with Lebesgue measure $\mu$.
If $f\in L_{1}(I)+L_{\infty}(I)$ we denote by $f^{*}$ the decreasing
rearrangement of $f$, i.e.

$f^{*}(t)=\inf_{\mu A=t}\sup_{s\in I\setminus A}|f(s)|.$

If $f,g$ are functions in $L_{1}+L_{\infty}$ we write $g\preceq f$ if

$\int_{0}^{t}g^{*}(s)\,ds\leq\int_{0}^{t}f^{*}(s)\,ds,\qquad t\in I.$

This defines the Hardy-Littlewood-Polya ordering.

A symmetric Banach function space $E$ on $I$ is a linear space with $L_{1}\cap
L_{\infty}\subset E\subset L_{1}+L_{\infty}$, with an associated norm
$\|\cdot\|_{E}$ such that $(E,\|\cdot\|_{E})$ is complete and if $f\in E,\
g\in L_{1}+L_{\infty}$ with $g^{*}\leq f^{*}$ then $g\in E$ and
$\|g\|_{E}\leq\|f\|_{E}.$ We will use $E_{+}$ to denote the positive cone of
$E$ i.e. $\\{f:\ f\in E,\ f\geq 0\text{ a.e.}\\}.$ We will also assume the
normalization that $\|\chi_{(0,1)}\|=1.$ Let
$\varphi_{E}(t)=\|\chi_{(0,t)}\|_{E}$ be the fundamental function of $E.$

$E$ is said to have a Fatou norm if for every sequence
$(f_{n})_{n=1}^{\infty}$ of nonnegative functions such that $f_{n}\uparrow f$
a.e. with $f\in E$ we have $\lim_{n\to\infty}\|f_{n}\|_{E}=\|f\|_{E}.$

A symmetric Banach function space $E$ is said to be fully symmetric if and
only if $f\in E,\ g\in L_{1}+L_{\infty}$ with $g\preceq f$, then $g\in E$ and
$||f||_{E}\leq||g||_{E}.$ $E$ is fully symmetric precisely when $E$ is an
exact interpolation space for the couple $(L_{\infty}(I),L_{1}(I))$ by the
Calderón-Mitjagin theorem [Mitjagin1965, Calderon1966]. In this paper we will
only consider fully symmetric Banach function spaces.

We will need the following inequality can be found in
[KreinPetuninSemenov1982], Theorem II.3.1. If $f,g\in L_{1}+L_{\infty},$ then

(4) $(f^{*}-g^{*})\preceq(f-g)^{*}.$

As a consequence if $E$ is fully symmetric and $f,g\in E$ we have

(5) $\|f^{*}-g^{*}\|_{E}\leq\|f-g\|_{E}.$

If $E$ is a fully symmetric Banach function space and $f\in E$ we define the
orbit of $f$ by $\Omega(f)=\\{g:\ g^{*}\preceq f^{*}\\}\subset E.$ The set of
the extreme points of the set $\Omega(f)$ is well-known (see [Ryff1967,
ChilinKryginSukochev1992]) and, if $I=(0,1)$ or $I=(0,\infty)$, and $E$ does
not contain $L_{\infty}$ it is given by

$\partial_{e}(\Omega(f))=\\{g\in L_{1}+L_{\infty}:\ f^{*}=g^{*}\\}.$

If $I=(0,\infty)$ and $E$ contains $L_{\infty}$ we must make a small
correction:

$\partial_{e}(\Omega(f))=\\{g\in L_{1}+L_{\infty}:\ f^{*}=g^{*},\
|g(t)|\geq\lim_{s\to\infty}f^{*}(s)\text{ a.e. }\\}.$

We define $\mathcal{Q}(f)$ to be the convex hull of the set $\\{g\in
L_{1}+L_{\infty}:\ g^{*}\leq f^{*}\\}.$ We will denote by $\mathcal{Q}_{E}(f)$
the closure in $E$ of $\mathcal{Q}(f).$ This is easily seen to coincide with
the closed convex hull of $\partial_{e}\Omega(f).$ Thus
$\mathcal{Q}_{E}(f)\subset\Omega(f).$

We next define the dilation operators on $E$. If $\tau>0$ and $I=(0,\infty)$
the dilation operator $\sigma_{\tau}$ is defined by setting

$(\sigma_{\tau}(f))(s)=f({s}/{\tau}),\qquad s>0.$

In the case of the interval $(0,1)$ the operator $\sigma_{\tau}$ is defined by

$(\sigma_{\tau}f)(s)=\begin{cases}f(s/\tau),&s\leq\min\\{1,\tau\\}\\\
0,&\tau<s\leq 1.\end{cases}$

The operators $\sigma_{\tau}$ ($\tau\geq 1$) satisfy semi-group property
$\sigma_{\tau_{1}}\sigma_{\tau_{2}}=\sigma_{\tau_{1}\tau_{2}}.$ If $E$ is a
symmetric space and if $\tau>0,$ then the dilation operator $\sigma_{\tau}$ is
a bounded operator on $E$ and

$||\sigma_{\tau}||_{E\to E}\leq\max\\{1,\tau\\}.$

If $E$ is a fully symmetric function space on $(0,\infty)$ then $E+L_{\infty}$
is also a fully symmetric function space under the norm

$\|f\|_{E+L_{\infty}}=\|f^{*}\chi_{(0,1)}\|_{E}.$

The next Lemma will be used later.

######  Lemma 2.1.

Let $E$ be a symmetric function space on $(0,\infty)$, such that $E\setminus
L_{1}\neq\emptyset,$ and suppose $f\in L_{1}\cap E.$ Then

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E+L_{\infty}}.$

###### Proof.

We may suppose $f$ is nonnegative and decreasing. Let $\varphi=\varphi_{E}$ be
the fundamental function of $E$ and let $\psi$ be its least concave majorant
of $\varphi.$ Since $E\setminus L_{1}\neq\emptyset$ we have
$\lim_{t\to\infty}\psi^{\prime}(t)=0.$ For any $\tau>1$ we have, using Theorem
II.5.5 of [KreinPetuninSemenov1982]

$\displaystyle\|(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}$
$\displaystyle\leq\|f(\tau^{-1})\chi_{(0,1)}+(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}$
$\displaystyle\leq
f(\tau^{-1})\int_{0}^{1}\psi^{\prime}(s)\,ds+\int_{1}^{\infty}\psi^{\prime}(s)f(\tau^{-1}s)\,ds$
$\displaystyle\leq\psi(1)f(\tau^{-1})+\tau\int_{\tau^{-1}}^{\infty}\psi^{\prime}(\tau
s)f(s)\,ds.$

Now we have, since $f\in L_{1},$

$\lim_{\tau\to\infty}\tau^{-1}f(\tau^{-1})=0$

and by the Dominated Convergence Theorem,

$\lim_{\tau\to\infty}\int_{\tau^{-1}}^{\infty}\psi^{\prime}(\tau
s)f(s)\,ds=\lim_{\tau\to\infty}\int_{0}^{\infty}\chi_{(\tau^{-1},\infty)}(s)\psi^{\prime}(\tau
s)f(s)\,ds=0.$

Hence

$\lim_{\tau\to\infty}\tau^{-1}\|(\sigma_{\tau}f)\chi_{(1,\infty)}\|_{E}=0$

and the Lemma follows. ∎

We next discuss the corresponding notions for sequence spaces. If
$\xi=(\xi_{n})_{n=1}^{\infty}$ is a sequence then $\xi^{*}$ denotes its
decreasing rearrangement:

$\xi^{*}_{n}=\inf_{|\mathbb{A}|=n-1}\sup_{k\in\mathbb{N}\setminus\mathbb{A}}|\xi_{k}|.$

A Banach sequence space $E$ is called symmetric if $\xi\in E$ and
$\eta^{*}\leq\xi^{*}$ implies that $\eta\in E$ and
$\|\eta\|_{E}\leq\|\xi\|_{E}.$ We write $\eta\preceq\xi$ if

$\sum_{k=1}^{n}\eta_{k}^{*}\leq\sum_{k=1}^{n}\xi_{k}^{*},\qquad
n\in\mathbb{N}.$

$E$ is called fully symmetric if $\xi\in E$ and $\eta\preceq\xi$ implies that
$\eta\in E$ and $\|\eta\|_{E}\leq\|\xi\|_{E}.$ If $\xi$ is any bounded
sequence we define its orbit $\Omega(\xi)=\\{\eta:\,\eta\preceq\xi\\}.$

In this context, we define the dilation operators $\sigma_{m}$ only for
$m\in\mathbb{N}.$ Then

$\sigma_{m}(\xi)=(\xi_{1},\ldots,\xi_{1},\xi_{2},\ldots,\xi_{2},\xi_{3}\ldots)$

where each $\xi_{j}$ is repeated $m$ times.

##  3\. Approximation of the orbit

Our first proposition gives a simple criterion which will enable us to check
when $\mathcal{Q}_{E}(f)=\Omega(f).$

######  Proposition 3.1.

Let $E$ be a fully symmetric Banach space on $(0,\infty).$ Suppose $f,g$ are
nonnegative decreasing functions in $E$. Then $g\in\mathcal{Q}_{E}(f)$ if and
only if, given $\epsilon>0,$ there exists a nonnegative decreasing function
$h\in E$ and an integer $p$ such that $0\leq h\leq g$ and

(6) $\|g-h\|_{E}<\epsilon$

and

(7) $\int_{pa}^{b}h(t)\,dt\leq\int_{a}^{b}f(t)\,dt,\qquad 0<pa<b<\infty.$

###### Proof.

Suppose first $g\in\mathcal{Q}_{E}(f).$ Then given $\epsilon>0$ there exist
$f_{1},\ldots,f_{p}\in E$ such that $f_{j}^{*}\leq f$ for $1\leq j\leq p$ and

$\|g-\frac{1}{p}(f_{1}+\cdots+f_{p})\|_{E}<\epsilon.$

Let

$u=\frac{1}{p}(f_{1}+\cdots+f_{p}),\quad
v=\frac{1}{p}(|f_{1}|+\cdots+|f_{p}|).$

Then if $h=g\wedge v^{*},$ using (5)

$\|g-h\|_{E}\leq\|g-g\wedge
u^{*}\|_{E}\leq\|g-u^{*}\|_{E}\leq\|g-u\|_{E}<\epsilon.$

It remains to observe that (7) holds by Lemma 4.1 of [KaltonSukochev2008].

The converse is easy. If $h$ satisfies (6) and (7) then
$h\in\alpha\mathcal{Q}(f)$ for every $\alpha>1$ by Theorem 6.3 of
[KaltonSukochev2008]. Hence $h\in\mathcal{Q}_{E}(f)$ and so
$d(g,\mathcal{Q}_{E}(f))<\epsilon.$ ∎

The next Lemma is surely well-known but we use it in the main result and
include a proof for completeness.

######  Lemma 3.2.

Let $F$ be a continuous nonnegative increasing concave function on
$[0,\infty)$ with $F(0)=0.$ Let us suppose that
$(\alpha_{n})_{n\in\mathbb{Z}}$ is an increasing doubly infinite sequence of
distinct positive reals with

$\lim_{n\to-\infty}\alpha_{n}=0,\qquad\lim_{n\to\infty}\alpha_{n}=\infty.$

Suppose that $(\beta_{n})_{n\in\mathbb{Z}}$ is any sequence with

$0\leq\beta_{n}\leq F(\alpha_{n}),\qquad n\in\mathbb{Z}.$

(i) There is a least concave function $G$ on $[0,\infty)$ such that $G(0)\geq
0,$ and $G(\alpha_{n})\geq\beta_{n}$ for $n\in\mathbb{Z}.$ $G$ is continuous
nonnegative and increasing and $G(0)=0.$
(ii) Furthermore if $n\in\mathbb{Z}$ then either

$G(t)=G(\alpha_{n})t/\alpha_{n},\qquad 0\leq t\leq\alpha_{n}$

or there exists $m<n$ so that

$G(t)=\beta_{m}+\frac{G(\alpha_{n})-\beta_{m}}{\alpha_{n}-\alpha_{m}}(t-\alpha_{m}),\qquad\alpha_{m}\leq
t\leq\alpha_{n}.$

###### Proof.

(i) is almost immediate. $G$ is defined as the infimum of the collection
$\mathcal{C}$ of all increasing concave functions $H$ on $[0,\infty)$ such
that $H(\alpha_{n})\geq\beta_{n}$ for all $n\in\mathbb{Z}$ and $H(0)\geq 0.$
This collection is non-empty since $F\in\mathcal{C}.$ $G$ is affine on each
interval $[\alpha_{n},\beta_{n+1}]$ and since $G\leq F$, $G$ is continuous at
$0.$

For (ii), assume $G$ is not affine on $[0,\alpha_{n}]$. Then there exists a
least $p<n$ so that $g$ is affine on $[\alpha_{p},\alpha_{n}].$ Let $G_{0}$ be
the function equal to $G$ on $[0,\alpha_{p-1}]$ and $[\alpha_{n},\infty)$ and
affine on $[\alpha_{p-1},\alpha_{n}].$ Then for any $0<\lambda<1$ we have
$(1-\lambda)G+\lambda G_{0}\notin\mathcal{C}.$ Hence there exists
$k(\lambda)\in\\{p,p+1,\ldots,n-1\\}$ so that

$(1-\lambda)G(\alpha_{k(\lambda)})+\lambda
G_{0}(\alpha_{k(\lambda)})<\beta_{k(\lambda)}.$

Letting $\lambda\to 0$ through as a suitable sequence where $k(\lambda)=m<n$
is constant we obtain $G(\alpha_{m})=\beta_{m}$ and the second alternative
holds.∎

We now prove our main result.

######  Theorem 3.3.

Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with Fatou
norm. Suppose $f\in E_{+}\setminus L_{1}$ is such that
$\Omega(f)=\mathcal{Q}_{E}(f).$ Then

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$

###### Proof.

We may suppose that $f$ is decreasing. We let

$F(t)=\int_{0}^{t}f(s)\,ds.$

Let us define a doubly infinite sequence $(a_{n})_{n\in\mathbb{Z}}$ by
$F(a_{n})=(5/4)^{n}.$

We next introduce the family $\mathcal{K}$ of doubly infinite sequences
$\kappa=(\kappa_{n})_{n\in\mathbb{Z}}$ such that either
$\kappa_{n}\in\mathbb{N}$ with $1\leq\kappa_{n}<a_{n+1}/a_{n}$ or
$\kappa_{n}=\infty.$ Then $\mathcal{K}$ is a complete lattice under the order
$\kappa\leq\kappa^{\prime}$ if $\kappa_{n}\leq\kappa^{\prime}_{n}$ for all
$n$. We may define the lattice operations
$(\kappa\vee\kappa^{\prime})_{n}=\max(\kappa_{n},\kappa^{\prime}_{n})$ and
$(\kappa\wedge\kappa^{\prime})_{n}=\min(\kappa_{n},\kappa^{\prime}_{n}).$

For each $\kappa\in\mathcal{K}$ we define $\psi_{\kappa}\in E$ as follows. Let
$\Psi(t)=\Psi_{\kappa}(t)$ be the least increasing concave function such that
$\Psi(0)\geq 0,$

$\Psi(\kappa_{n}a_{n})\geq F(a_{n}),\qquad\text{ if }\kappa_{n}<\infty,$

and

$\Psi(a_{n})\geq 0\qquad\text{ if }\kappa_{n}=\infty.$

The existence and properties of $\Psi$ are guaranteed by applying Lemma 3.2
when $\alpha_{n}=\kappa_{n}a_{n}$ if $\kappa_{n}<\infty$ and
$\alpha_{n}=a_{n}$ if $\kappa_{n}=\infty$ and $\beta_{n}=F(a_{n})$ if
$\kappa_{n}<\infty$ and $\beta_{n}=0$ if $\kappa_{n}=\infty.$ Since
$F(a_{n})\leq F(\kappa_{n}a_{n})$ it is clear from Lemma 3.2 that $\Psi$
exists and $\Psi\leq F.$ Furthermore $\Psi$ is piecewise affine on
$(0,\infty)$ and we may define $\psi_{\kappa}=\Psi^{\prime}$ is a nonnegative
piecewise constant decreasing function on $(0,\infty)$. Clearly
$\psi_{\kappa}\in\Omega(f)\subset E.$

We note some elementary properties of the map $\kappa\to\psi_{\kappa}.$

######  Lemma 3.4.

(8) $\psi_{\kappa}\preceq\psi_{\kappa^{\prime}},\qquad\text{ if
}\kappa^{\prime}\leq\kappa$ (9)
$\psi_{\kappa\wedge\kappa^{\prime}}\preceq\psi_{\kappa}\vee\psi_{\kappa^{\prime}},\qquad\kappa,\kappa^{\prime}\in\mathcal{K}.$

###### Proof.

(8) is quite trivial.

To see (9) note that

$\int_{0}^{t}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\max(\Psi_{\kappa}(t),\Psi_{\kappa^{\prime}}(t)).$

Now if $\kappa_{n}\wedge\kappa^{\prime}_{n}<\infty$ and
$\kappa_{n}\leq\kappa^{\prime}_{n}$ we have

$\int_{0}^{\kappa_{n}a_{n}}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\Psi_{\kappa}(\kappa_{n}a_{n})\geq
F(a_{n})$

and with a similar inequality when $\kappa^{\prime}_{n}<\kappa_{n}$ we obtain,
from the definition of $\Psi_{\kappa\wedge\kappa^{\prime}}$,

$\int_{0}^{t}\max(\psi_{\kappa}(s),\psi_{\kappa^{\prime}}(s))\,ds\geq\Psi_{\kappa\wedge\kappa^{\prime}}(t),\qquad
0\leq t<\infty.$

This proves (9). ∎

######  Lemma 3.5.

Suppose $\kappa\in\mathcal{K}$ satisfies

(10) $\max(\kappa_{n},\kappa_{n+1})=\infty,\qquad n\in\mathbb{Z}.$

Then for any $n\in\mathbb{Z}$ such that $\kappa_{n}<\infty$ we have:

(11) $\psi_{\kappa}(t)\geq\frac{9F(a_{n})}{25\kappa_{n}a_{n}}\qquad a_{n}\leq
t\leq\kappa_{n}a_{n}.$

###### Proof.

If $f$ is not identically zero then $\psi_{\kappa}$ is only identically zero
when $\kappa$ is identiclaly $\infty;$ we exclude this case so that
$\Psi_{\kappa}(t)>0$ for $t>0.$ Observe first that $\psi_{\kappa}$ is constant
on $(a_{n},\kappa_{n}a_{n}).$ If for every $m<n$ such that $\kappa_{m}<\infty$
we have $\Psi_{\kappa}(\kappa_{m}a_{m})>F(a_{m})$ then

$\psi_{\kappa}(t)\geq\frac{F(a_{n})}{\kappa_{n}a_{n}},\qquad
0<t\leq\kappa_{n}a_{n}.$

Otherwise, since $\Psi_{\kappa}(t)>0$ for all $t>0,$ we have that, by Lemma
3.2, there exists $m<n$ so that $\kappa_{m}<\infty$ and

$\psi_{\kappa}(t)=\frac{\Psi_{\kappa}(\kappa_{n}a_{n})-F(a_{m})}{\kappa_{n}a_{n}-\kappa_{m}a_{m}},\qquad\kappa_{m}a_{m}<t<\kappa_{n}a_{n}.$

Then

$\psi_{\kappa}(t)\geq\frac{F(a_{n})-F(a_{m})}{\kappa_{n}a_{n}-\kappa_{m}a_{m}},\qquad
a_{n}\leq t\leq\kappa_{n}a_{n}.$

Noting that $m\leq n-2$ by (10) so that $F(a_{m})\leq(4/5)^{2}F(a_{n})$ this
implies that (11) holds for either alternative. ∎

For $\kappa\in\mathcal{K}$ and $r\in\mathbb{N}$ we will define a
$\kappa^{[r]}\geq\kappa$ by suppressing the values of $\kappa$ which are less
than $r$. Precisely:

$\kappa^{[r]}_{n}=\begin{cases}\kappa_{n},\qquad\text{if }\kappa_{n}\geq r\\\
\infty\qquad\text{ if }\kappa_{n}<r.\end{cases}$

We next prove the following Lemma, which is the heart of the argument for
Theorem 3.3:

######  Lemma 3.6.

Under the hypotheses of the theorem, we have that for any
$\kappa\in\mathcal{K}$

$\lim_{r\to\infty}\|\psi_{{\kappa}^{[r]}}\|_{E}=0.$

###### Proof.

We will first prove the Lemma under the additional assumption that (10) holds.
Since $\|\psi_{{\kappa}^{[r]}}\|_{E}$ is decreasing in $r$ (by (8)) it
suffices to show that for given $\epsilon>0$ we can find $r$ so that
$\|\psi_{{\kappa}^{[r]}}\|_{E}<\epsilon.$ By Proposition 3.1 for any
$\epsilon>0$ we can find a nonnegative decreasing function
$h\leq\psi_{\kappa}$ and an integer $p$ so that

(12) $\int_{pa}^{b}h(t)\,dt\leq\int_{a}^{b}f(t)\,dt,\qquad 0<pa<b<\infty,$

and

(13) $\|\psi_{\kappa}-h\|_{E}<\epsilon/10.$

We shall take $r=36p.$ Let $v=10(\psi_{\kappa}-h).$ We will show that
$\psi_{\kappa^{[r]}}\preceq v.$ In order to do this we must show that if
$\kappa_{n}^{[r]}<\infty$ we have

(14) $F(a_{n})\leq\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt.$

If $\kappa_{n}^{[r]}<\infty$

$\displaystyle\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt$
$\displaystyle\geq\int_{pa_{n}}^{\kappa_{n}a_{n}}v(t)\,dt$
$\displaystyle=10\left(\int_{pa_{n}}^{\kappa_{n}a_{n}}\psi_{\kappa}(t)\,dt-\int_{pa_{n}}^{\kappa_{n}a_{n}}h(t)\,dt\right)$
$\displaystyle\geq
10\left(\int_{pa_{n}}^{\kappa_{n}a_{n}}\psi_{\kappa}(t)\,dt-\int_{a_{n}}^{\kappa_{n}a_{n}}f(t)\,dt\right),$

by (13). Hence by (11) of Lemma 3.5

$\displaystyle\int_{0}^{\kappa_{n}a_{n}}v^{*}(t)\,dt$ $\displaystyle\geq
10\left(\frac{9(\kappa_{n}a_{n}-pa_{n})F(a_{n})}{25\kappa_{n}a_{n}}-\int_{a_{n}}^{a_{n+1}}f(t)\,dt\right)$
$\displaystyle\geq
10\left(\frac{35}{36}\frac{9}{25}F(a_{n})-\frac{1}{4}F(a_{n})\right)$
$\displaystyle=F(a_{n}).$

This show that (14) holds and so $\psi_{\kappa^{[r]}}\preceq v$ and
$\|\psi_{\kappa^{[r]}}\|_{E}<\epsilon.$ This completes the proof when (10)
holds.

For the general case let us introduce $\kappa(0)_{n}=\kappa_{n}$ if $n$ is
even and $\kappa(0)_{n}=\infty$ if $n$ is odd. Similarly
$\kappa(1)_{n}=\kappa_{n}$ if $n$ is odd and $\kappa(1)_{n}=\infty$ if $n$ is
even. Both $\kappa(0)$ and $\kappa(1)$ satisfy (10) Then for an arbitrary
$\kappa$ we have $\kappa^{[r]}=\kappa(0)^{[r]}\wedge\kappa(1)^{[r]}$ and so by
(9),

$\limsup_{r\to\infty}\|\psi_{\kappa^{[r]}}\|_{E}\leq\limsup_{r\to\infty}\|\psi_{\kappa(0)^{[r]}}\|_{E}+\limsup_{r\to\infty}\|\psi_{\kappa(1)^{[r]}}\|_{E}=0.$

∎

Next for any integer $p$ we define $\gamma^{p}_{n}=p$ if $pa_{n}<a_{n+1}$ and
$\gamma^{p}_{n}=\infty$ otherwise. For each $q>p$ we define
$\gamma^{p,q}_{n}=p$ if $pa_{n}<a_{n+1}$ and $|n|\leq q$ and
$\gamma^{p,q}_{n}=\infty$ otherwise. Let $\psi_{p}=\psi_{\gamma^{p}}$ and
$\psi_{p,q}=\psi_{\gamma^{p,q}}.$

######  Lemma 3.7.

Under the hypotheses of the Theorem,

$\lim_{p\to\infty}\|\psi_{p}\|_{E}=0.$

###### Proof.

Clearly $\|\psi_{p}\|_{E}$ is decreasing in $p$. Assume
$\|\psi_{p}\|_{E}>\epsilon>0$ for all $p\in\mathbb{N}.$ Since $E$ has a Fatou
norm for each $p$ there exists $q(p)>p$ so that
$\|\psi_{p,q(p)}\|_{E}>\epsilon.$ Let

$\kappa=\wedge_{p}\gamma^{p,q(p)}.$

Thus $\kappa$ is given by the formula

$\kappa_{n}=\inf\\{p:\ p<a_{n+1}/a_{n},\ |n|\leq q(p)\\}$

and $\kappa$ has the properties that $\kappa\leq\gamma_{p,q(p)}$ for all $p$
and $\lim_{|n|\to\infty}\kappa_{n}=\infty.$

By Lemma 3.6 there exists $r\in\mathbb{N}$ so that
$\|\psi_{\kappa^{[r]}}\|_{E}<\epsilon.$ But then the set
$\\{n:\kappa_{n}<r\\}$ is finite and so there is a choice of $p$ such that
$p>a_{n+1}/a_{n}$ whenever $\kappa_{n}<r.$ Thus $\gamma^{p}_{n}=\infty$ if
$\kappa_{n}<r.$ Thus

$\kappa^{[r]}\leq\gamma^{p,q(p)}$

and so by (8),

$\|\psi_{p,q(p)}\|_{E}<\epsilon$

which gives a contradiction. ∎

We now can complete the proof of the Theorem. We will show that if
$p\in\mathbb{N},$

(15)
$F(t)\leq\frac{4}{5}F(p^{2}t)+\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds,\qquad
0<t<\infty.$

Indeed if (15) fails for some $t$ we can assume $a_{n}\leq t<a_{n+1}$ for some
$n\in\mathbb{Z}.$ We first argue that $a_{n+1}\leq pa_{n}$. Suppose, on the
contrary, that $a_{n+1}>pa_{n}$. Then we have

$\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds\geq\frac{5}{4}\int_{0}^{pa_{n}}\psi_{p}(s)\,ds\geq\frac{5}{4}F(a_{n})=F(a_{n+1})\geq
F(t),$

which contradicts our hypothesis. Next we show that $a_{n+2}\leq pa_{n+1}.$
Indeed if $a_{n+2}>pa_{n+1},$ then $p^{2}t\geq pa_{n+1}$ and

$\frac{5}{4}\int_{0}^{p^{2}t}\psi_{p}(s)\,ds\geq\frac{5}{4}\int_{0}^{pa_{n+1}}\psi_{p}(s)\,ds\geq\frac{5}{4}F(a_{n+1})>F(t).$

But then $a_{n+2}\leq p^{2}a_{n}$ and so

$\frac{4}{5}F(p^{2}t)\geq\frac{4}{5}F(a_{n+2})=F(a_{n+1})>F(t)$

and we have a contradiction. This establishes (15).

Now if $\tau\geq 1$ we replace $t$ in (15) by $t/\tau$ and interpret the
inequality in the form:

$\frac{1}{\tau}\sigma_{\tau}f\preceq\frac{1}{p^{-2}\tau}\sigma_{p^{-2}\tau}((4/5)f+(5/4)\psi_{p}).$

Hence

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}\leq(4/5)\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}+(5/4)\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}\psi_{p}\|_{E}$

so that

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{E}\leq(5^{2}/4)\|\psi_{p}\|_{E}.$

Combining with Lemma 3.7 we obtain the theorem. ∎

The case when $f\in L_{1}$ is handled by reduction to the previous case:

######  Theorem 3.8.

Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm. Suppose $f$ is a decreasing nonnegative function such that $f\in
E_{+}\cap L_{1}$ and $\Omega(f)=\mathcal{Q}_{E}(f).$ Then

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E+L_{\infty}}=0.$

###### Proof.

An easy computation shows that
$\mathcal{Q}(f+1)=\mathcal{Q}(f)+\mathcal{Q}(1).$ Hence
$\mathcal{Q}_{E+L_{\infty}}(f+1)\supset\mathcal{Q}_{E}(f)+\mathcal{Q}_{L_{\infty}}(1)=\Omega(f)+\Omega(1).$
If $0\leq g\in\Omega(f+1)$ then $g-g\wedge 1\in\Omega(f)$ and $g\wedge
1\in\Omega(1)$ so that $\Omega(f)+\Omega(1)=\Omega(f+1).$ Hence
$\mathcal{Q}_{E+L_{\infty}}(f+1)=\Omega(f+1)$ and we can apply Theorem 3.3. ∎

##  4\. The main results

We can next state our main results:

######  Theorem 4.1.

Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm, and such that $E\setminus L_{1}\neq\emptyset.$ Suppose $f\in E.$
Then $\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$

###### Proof.

If $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0$ then
$\Omega_{+}(f)\subset\mathcal{Q}_{E}(f)$ by Theorem 25 of [SukochevZanin2009];
thus $\Omega(f)=\mathcal{Q}_{E}(f).$ Conversely if
$\Omega(f)=\mathcal{Q}_{E}(f)$ we have either

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=0$

(when $f\notin L_{1}$ by Theorem 3.3) or

$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E+L_{\infty}}=0$

(when $f\in L_{1}$ by Theorem 3.8). Then Lemma 2.1 shows that in both cases we
have $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f)\|_{E}=0$.∎

######  Theorem 4.2.

Let $E$ be a fully symmetric Banach function space on $(0,\infty)$ with a
Fatou norm, and such that $E\subset L_{1}$. If $f\in E$ then
$\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E+L_{\infty}}=\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\chi_{(0,1)}\|_{E}=0.$

###### Proof.

The proof is very similar to that of Theorem 4.1 using instead Theorem 24 of
[SukochevZanin2009]. ∎

We first give the extension to function spaces on $(0,1).$

######  Theorem 4.3.

Let $E$ be a fully symmetric Banach function space on $(0,1)$ with a Fatou
norm. Suppose $f\in E.$ Then $\Omega(f)=\mathcal{Q}_{E}(f)$ if and only if
$\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}(f^{*})\|_{E}=0.$

###### Proof.

We define $F$ to be the function space on $(0,\infty)$ defined by $f\in F$ if
and only if $f^{*}\chi_{(0,1)}\in E$ and $f\in L_{1}$ with the norm

$\|f\|_{F}=\max(\|f^{*}\chi_{(0,1)}\|_{E},\|f^{*}\|_{L_{1}}).$

Suppose $f\in E$ is nonnegative and decreasing. We will show that, regarding
$f$ as an member of $F,$ we have $\Omega(f)=\mathcal{Q}_{F}(f).$ Note that the
hypothesis $\Omega(f)=\mathcal{Q}_{E}(f)$ on $(0,1)$ implies only that if
$g\in F$ and $g\in\Omega(f)$ then $g\in\mathcal{Q}_{F}(f)$ provided $\mu({\rm
supp}\,g)\leq 1.$ We will show, however, that $\Omega(f)=\mathcal{Q}_{F}(f)$
and then the Theorem follows.

We will need the following Lemma:

######  Lemma 4.4.

Let $h\in F$ be nonnegative and decreasing. Suppose $g\in F$ is nonnegative
and decreasing and satisfies the conditions such that $g\preceq h$ and
$g(x)=0$ for some $0<x<\infty.$ If there exists $c>0$ so that $g(t)\leq h(t)$
for $0<t\leq c$ then $g\in\mathcal{Q}_{F}(h).$

###### Proof.

For any $\theta>1$ we may pick $p>x/c$ so that

$\int_{0}^{c}h(s)\,ds\leq\theta\int_{x/p}^{c}h(s)\,ds.$

Then if $0<pa<b<\infty$ with $c\leq b\leq x$ we have

$\int_{pa}^{b}g(s)\,ds\leq\int_{0}^{b}h(s)\,ds\leq\theta\int_{x/p}^{b}h(s)\,ds\leq\theta\int_{pa}^{b}h(s)\,ds.$

The same inequality holds trivially if $b>x$ or $b<c.$ Thus by Theorem 6.3 of
[KaltonSukochev2008] we have $g\in\lambda Q(h)$ for any $\lambda>1$ and the
Lemma follows.∎

We continue the proof of the theorem. We will assume, without loss of
generality that $\int_{0}^{1}f(t)\,dt=1.$ First suppose $g\in\Omega(f)$ is
nonnegative, not identically zero, and decreasing and satisfies $g(x)=0$ for
some $0<x<\infty.$ Given $\epsilon>0$ we may find $c_{0}>0$ so that

$\int_{0}^{c_{0}}f(s)\,ds<\epsilon/2.$

Let

$\alpha=\sup_{0<t\leq
c_{0}}\frac{\int_{0}^{t}g(s)\,ds}{\int_{0}^{t}f(s)\,ds}.$

We have $\alpha>0$ and we may pick $0<\beta<\alpha$ with
$(\alpha-\beta)<\epsilon/2$ and then $0<c<c_{0}$ with

$g(c)>\beta f(c).$

Let $c^{\prime}\geq c$ be the least solution of

$\alpha\int_{0}^{c^{\prime}}f(s)\,ds=\int_{0}^{c}g(s)\,ds+(c^{\prime}-c)g(c).$

We now define

$h(t)=\begin{cases}g(t)+(1-\alpha)f(t),\qquad 0<t\leq c\\\
g(c)+(1-\alpha)f(t),\qquad c<t\leq\min(c^{\prime},1)\\\
f(t),\qquad\qquad\qquad\qquad\min(c^{\prime},1)<t\leq 1\\\
0\qquad\qquad\qquad\qquad\qquad t\geq 1.\end{cases}$

From the construction we have $h\in\Omega(f).$ Thus $h\in\mathcal{Q}_{F}(f).$
For any $t\leq\min(c^{\prime},1)$ we have

$\int_{0}^{t}g(s)\,ds\leq\int_{0}^{t}h(s)\,ds.$

If $c^{\prime}<1$ then

$\int_{0}^{t}h(s)\,ds=\int_{0}^{t}f(s)\,ds\geq\int_{0}^{t}g(s)\,ds,\qquad
t>c^{\prime}.$

If $c^{\prime}\geq 1$ then

$\displaystyle\int_{0}^{1}h(s)\,ds$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\int_{0}^{c}g(s)\,ds+g(c)(1-c)$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\beta\int_{0}^{c}f(s)\,ds-\epsilon/2+\beta
f(c)(1-c)$
$\displaystyle\geq(1-\alpha)\int_{0}^{1}f(s)\,ds+\beta\int_{0}^{1}f(s)\,ds-\epsilon/2$
$\displaystyle\geq 1-\epsilon.$

Hence $(1-\epsilon)g\preceq h$ and by Lemma 4.4 we have
$(1-\epsilon)g\in\mathcal{Q}_{F}(h)$. Since $\epsilon>0$ is arbitrary we have
$g\in\mathcal{Q}_{F}(h)\subset\mathcal{Q}_{F}(f).$

Finally let us note for general nonnegative decreasing $g\in F$ we have
$\lim_{m\to\infty}\|g-g\chi_{(0,m)}\|_{F}=0$ so that we have
$\Omega_{F}(f)=\mathcal{Q}_{F}(f).$

Now the result reduces to Theorems 4.1 and 4.2. ∎

The extension to sequence spaces requires similar type of argument:

######  Theorem 4.5.

Let $E$ be a fully symmetric Banach sequence space with a Fatou norm and such
that $E\setminus\ell_{1}\neq\emptyset$. Suppose $\xi\in E.$ Then
$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$ if and only if
$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(\xi^{*})\|_{E}=0.$

###### Proof.

We consider the Banach function space $F$ of all bounded functions such that
$(f^{*}(0),f^{*}(1),\ldots)\in E$ with the norm

$\|f\|_{F}=f^{*}(0)+\|(a_{n})_{n=1}^{\infty}\|_{E},$

where $f^{*}(0)=\|f\|_{L_{\infty}}$ and $a_{n}:=\int_{n-1}^{n}f^{*}(s)ds$,
$n\geq 1$. Then let $F(\mathbb{N})$ be the subspace of $F$ of all functions
$f$ which are constant on each interval $(n-1,n]$. Clearly, the Banach spaces
$(F(\mathbb{N}),\|\cdot\|_{F})$ and $(E,\|\cdot\|_{E})$ are linearly
isomorphic, in particular

$\|\xi\|_{E}\leq\|\xi\|_{F}\leq 2\|\xi\|_{E},\quad\forall\xi\in
E=F(\mathbb{N}).$

Let $\mathbb{E}$ denote the conditional expectation operator

$\mathbb{E}f=\sum_{n\in\mathbb{N}}\chi_{(n-1,n]}\int_{n-1}^{n}f(t)\,dt.$

Suppose $\xi$ is a nonnegative decreasing sequence and let

$f=\sum_{j=1}^{\infty}\xi_{j}\chi_{(j-1,j]}\in F.$

The result will follow from:

######  Theorem 4.6.

$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$ if and only if
$\Omega(f)=\mathcal{Q}_{F}(f).$

###### Proof.

Let us suppose that $\Omega(\xi)=\mathcal{Q}_{E}(\xi)$. We may suppose $\xi$
has infinite support. Suppose $g\in\Omega(f)$ is nonnegative and decreasing;
we will show that $g\in\mathcal{Q}_{F}(f)$ and then it follows that
$\mathcal{Q}_{F}(f)=\Omega(f).$

Suppose $\epsilon>0$. Then we may pick an integer $m\in\mathbb{N}$ so that

$g^{*}(m)-\lim_{n\to\infty}g^{*}(n)<\epsilon/4.$

Now $\mathbb{E}g\in\mathcal{Q}_{F}(f)$ since
$\Omega(\xi)=\mathcal{Q}_{E}(\xi)$. Hence, by Proposition 3.1 there is a
nonnegative decreasing function $h$ with $0\leq h\leq\mathbb{E}g$ such that
$\|\mathbb{E}g-h\|_{E}<\epsilon/4$ and such that for some $p\in\mathbb{N},$

$\int_{pa}^{b}h(s)\,ds\leq\int_{a}^{b}f(s)\,ds\qquad 0<pa<b<\infty.$

Next we define

$\varphi(s)=\begin{cases}g(s),\qquad 0<s\leq m\\\ h(s),\qquad
m<s<\infty.\end{cases}$

Note that $0\leq\varphi\preceq g\preceq f.$ We show that
$\varphi\in\mathcal{Q}_{F}(f).$ Let us suppose $r>p$ and that $0<ra<b.$ Then
if $m\leq ra$ we clearly have

$\int_{ra}^{b}\varphi(s)\,ds\leq\int_{a}^{b}f(s)\,ds.$

On the other hand if $0<ra<m$, let $c=\min(b,m).$ Then

$\displaystyle\int_{ra}^{b}\varphi(s)\,ds$
$\displaystyle\leq\int_{0}^{b}f(s)\,ds$
$\displaystyle\leq\int_{a}^{b}f(s)\,ds+c\xi_{1}/r$
$\displaystyle\leq\int_{a}^{b}f(s)\,ds+\frac{\xi_{1}}{(r-1)\xi_{m}}\int_{c/r}^{c}f(s)\,ds$
$\displaystyle\leq\left(1+\frac{\xi_{1}}{(r-1)\xi_{m}}\right)\int_{a}^{b}f(s)\,ds.$

Since $r$ is arbitrary these estimate show that
$\varphi\in\lambda\mathcal{Q}(f)$ for every $\lambda>1$ (Theorem 6.4 of
[KaltonSukochev2008]) and hence $\varphi\in\mathcal{Q}_{F}(f).$

Now

$\|g-\varphi\|_{F}=\|(g-\varphi)\chi_{(m,\infty)}\|_{F}\leq\|(g-\mathbb{E}g)\chi_{(m,\infty)}\|_{F}+\|\mathbb{E}g-h\|_{F}.$

However,

$\|(g-\mathbb{E}g)\chi_{(m,\infty)}\|_{F}\leq\sum_{j=m}^{\infty}2(g(j)-g(j+1))<\epsilon/2.$

Hence

$d(g,\mathcal{Q}_{E}(f))\leq\|g-\varphi\|_{F}<\epsilon.$

Since $\epsilon>0$ is arbitrary we have $g\in\mathcal{Q}_{F}(f).$ This shows
that $\Omega(f)=\mathcal{Q}_{F}(f).$

We next turn to the converse. Assume $\mathcal{Q}_{E}(f)=\Omega(f)$ and that
$\eta\in\Omega(\xi)$ is a decreasing sequence. Let
$g=\sum_{n\in\mathbb{N}}\eta_{n}\chi_{(n-1,n]}$. Then $g\in\Omega(f)$ and so,
by Proposition 3.1, given $\epsilon>0$, there exists a decreasing $0\leq h\leq
g$ with $\|g-h\|_{F}<\epsilon$ and such that for some $p\in\mathbb{N}$ we have

$\int_{pa}^{b}h(s)\,ds\leq\int_{a}^{b}f(s)\,ds,\qquad 0<pa<b<\infty.$

Let $\zeta\in E$ be defined by $\zeta_{n}=\int_{n-1}^{n}h(s)\,ds.$ Then for
$0\leq pm\leq n$ we have

$\sum_{k=pm+1}^{n}\zeta_{k}=\int_{pm}^{n}h(s)\,ds\leq\int_{m}^{n}f(s)\,ds=\sum_{k=m+1}^{n}\xi_{k}.$

Hence $\zeta\in\lambda\mathcal{Q}(\xi)$ for every $\lambda>1$, by Theorem 5.5
of [KaltonSukochev2008], so that $\zeta\in\mathcal{Q}_{E}(\xi).$ Furthermore,

$\|\eta-\zeta\|_{E}\leq\|g-\mathbb{E}h\|_{F}\leq\|g-h\|_{F}<\epsilon.$

It now follows that $\eta\in\mathcal{Q}_{E}(\xi)$ and the proof of the Lemma
is complete. ∎

Theorem 4.5 now follows directly from Theorem 4.6. ∎

Let us observe that the argument of Theorem 4.6 allows us to complete the
picture for positive orbits in [SukochevZanin2009]:

######  Theorem 4.7.

Let $E$ be a fully symmetric sequence space with Fatou norm. Then for any
$\xi\in E_{+}$ the set $\Omega_{+}(\xi)=\Omega(\xi)\cap E_{+}$ coincides with
the closed convex hull of its extreme points if and only if

$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(\xi)\|_{E}=0.$

In fact we can prove by the same argument as in Theorem 4.6 that
$\Omega_{+}(\xi)=\mathcal{Q}_{E}(\xi)\cap E_{+}$ if and only if
$\Omega_{+}(f)=\mathcal{Q}_{F}(f)\cap F_{+}.$

We remark that in [SukochevZanin2009] some examples of Marcinkiewicz spaces
and Orlicz spaces are discussed in the context of Theorems 4.1, 4.2, 4.3 and
4.5. We refer the reader to [SukochevZanin2009] for details. We take the
opportunity to improve Proposition 33 of [SukochevZanin2009]:

######  Proposition 4.8.

Let $M$ be an Orlicz function. Then for any $f\in L_{M}(0,\infty)$ we have
$\Omega(f)=\mathcal{Q}_{L_{M}}(f).$ Similarly for for any $\xi\in\ell_{M}$ we
have $\Omega(\xi)=\mathcal{Q}_{\ell_{M}}(\xi).$

###### Proof.

We give the proof only for $L_{M}(0,\infty).$ Suppose first that $M(t)=o(t)$
when $t\to 0.$ We show that
$\|\lim_{\tau\to\infty}\tau^{-1}\sigma_{\tau}f\|_{L_{M}}=0$ whenever
$f\in(L_{M})_{+}.$ Suppose $\alpha>0.$

$\int_{0}^{\infty}M\left(\frac{\alpha
f(s/\tau)}{\tau}\right)ds=\int_{0}^{\infty}\tau M\left(\frac{\alpha
f(s)}{\tau}\right)\,ds$

for any $\tau>1.$ Since $f\in L_{M}$ there exists $\tau_{0}$ so that

$\int_{0}^{\infty}\tau_{0}M\left(\frac{\alpha
f(s)}{\tau_{0}}\right)\,ds<\infty.$

Now letting $\tau\to\infty$ we obtain from the Dominated Convergence Theorem
that

$\lim_{\tau\to\infty}\int_{0}^{\infty}\tau M\left(\frac{\alpha
f(s)}{\tau}\right)\,ds=0$

so that $\lim_{\tau\to\infty}\tau^{-1}\|\sigma_{\tau}f\|_{L_{M}}=0$ and we can
apply Theorem 4.1.

Now if $M(t)\geq ct$ for all $t>0$ where $c>0$ we have $L_{M}\subset L_{1}.$
For any $\alpha>0$ and $\tau>1$

$\int_{0}^{1}M\left(\frac{\alpha
f^{*}(s/\tau)}{\tau}\right)ds=\int_{0}^{1}\tau\chi_{(0,\tau^{-1})}(s)M\left(\frac{\alpha
f^{*}(s)}{\tau}\right)\,ds.$

As before the right-hand side is integrable for some $\tau=\tau_{0}$ and we
can apply the Dominated Convergence Theorem to deduce that
$\tau^{-1}\|(\sigma_{\tau}f^{*})\chi_{(0,1)}\|_{L_{M}}$ tends to $0$ as $\tau$
approaches infinity. Now one can apply Theorem 4.2. ∎

##  5\. A noncommutative analog

Let $\mathcal{H}$ be a separable complex Hilbert space. We denote by
$\mathcal{B}(\mathcal{H})$ the space of bounded operators on $\mathcal{H}$ and
by $\mathcal{K}(\mathcal{H})$ the ideal of compact operators on $\mathcal{H}.$
For any $T\in\mathcal{B}(\mathcal{H})$ we define the singular values

$s_{n}(T)=\inf\\{\|T(I-P)\|,$

where the infimum is taken over all orthogonal projections $P$ such that ${\rm
rank}(P)<n\\}.$

If $E$ is a symmetric sequence space then we can define a Banach ideal of
compact operators on $\mathcal{H}$ by $T\in\mathcal{S}_{E}$ if and only if
$(s_{k}(T))_{k=1}^{\infty}\in E$ and then the norm is given by
$\|T\|_{E}=\|(s_{k}(T))_{k=1}^{\infty}\|_{E}.$ For fully symmetric spaces this
is well-known (e.g. see [GohbergKrein1969] but for symmetric spaces it follows
from [KaltonSukochev2008]).

Let $\mathcal{H}$ be a separable Hilbert space and suppose
$T\in\mathcal{K}(\mathcal{H}).$ Let $\mathcal{Q}(T)$ be the convex hull of the
set $\\{ATB;\ \|A\|,\|B\|\leq 1\\}.$ We define its orbit $\Omega(T)$ to be the
closure of $\mathcal{Q}(T)$ in $\mathcal{K}(\mathcal{H}).$ It is easy to check
from the definition that $R\in\Omega(T)$ if and only if

$\sum_{k=1}^{n}s_{k}(R)\leq\sum_{k=1}^{n}s_{k}(T),\qquad n=1,2,\ldots$

For any symmetric Banach sequence space $E$ we may define $\mathcal{Q}_{E}(T)$
to be the closure of $\mathcal{Q}(T)$ in $\mathcal{S}_{E}.$

######  Theorem 5.1.

Let $\mathcal{E}$ be a fully symmetric sequence space with a Fatou norm.
Suppose $\mathcal{S}_{E}$ is the corresponding ideal of compact operators.
Then for $T\in\mathcal{S}_{E}$ we have $\Omega(T)=\mathcal{Q}_{E}(T)$ if and
only if

$\lim_{m\to\infty}m^{-1}\|\sigma_{m}(s_{k}(T))_{k=1}^{\infty}\|_{E}=0.$

###### Proof.

Let $\xi=(s_{k}(T))_{k=1}^{\infty}.$ Let $R\in\mathcal{K}(\mathcal{H})$ and
let $\eta=(s_{k}(R))_{k=1}^{\infty}$. If $R\in\mathcal{Q}(T)$ then it follows
from Proposition 8.6 and Theorem 5.5 of [KaltonSukochev2008] that
$\eta\in\lambda\mathcal{Q}(\xi)$ for every $\lambda>1.$

First suppose that $\Omega(T)=\mathcal{Q}_{E}(T).$ If $S\in\Omega(T)$ then
given $\epsilon>0$ there exists $R\in\mathcal{Q}(T)$ with
$\|R-S\|_{E}<\epsilon.$ Let $\zeta=(s_{k}(S))_{k=1}^{\infty}.$ Then by the
submajorization inequality of [DoddsDoddsdePagter1989a],

$\eta-\zeta\preceq(s_{k}(R-S))_{k=1}^{\infty}$

so that $\|\eta-\zeta\|_{E}<\epsilon.$ Since $\eta\in\mathcal{Q}_{E}(\xi)$ and
$\epsilon>0$ is arbitrary, this implies that $\zeta\in\mathcal{Q}_{E}(\xi)$
and so $\mathcal{Q}_{E}(\xi)=\Omega(\xi).$ Theorem 4.5 can then be applied.

The converse direction is immediate. ∎

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