# Production of the newly observed $\bar{T}_{c\bar{s}0}$ by kaon-induced
reactions on a proton/neutron target

Yin Huang<EMAIL_ADDRESS>School of Physical Science and Technology,
Southwest Jiaotong University, Chengdu 610031,China  Hao Hei  School of
Physical Science and Technology, Southwest Jiaotong University, Chengdu
610031,China  Jing-wen Feng  School of Physics and Electronic Engineering,
Sichuan Normal University, Chengdu 610101, China  Xurong Chen  Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing 100049, China  Rong Wang  Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China

###### Abstract

Recently, a new doubly charged tetraquark $T^{a}_{c\bar{s}0}(2900)^{++}$ and
its neutral partner $T^{a}_{c\bar{s}0}(2900)^{0}$ at the invariant mass
spectrum of $\pi{}D_{s}$ were observed by the LHCb Collaboration. According to
its properties, such as the mass and decay width, the
$T^{a}_{c\bar{s}0}(2900)^{++/0}$ have been suggested to be a compact multi-
quark state or a hadron molecule. In order to distinguish the various
interpretations of the $T^{a}_{c\bar{s}0}(2900)^{++/0}$, we investigate the
possibility to study the $\bar{T}^{a}_{c\bar{s}0}(2900)$ [the antiparticle of
$T^{a}_{c\bar{s}0}(2900)$] by kaon-induced reactions on a proton target in an
effective Lagrangian approach. The production mechanism is characterized by
the $t$-channel $D$ meson exchange. Our theoretical approach is based on the
assumption that $\bar{T}^{a}_{c\bar{s}0}(2900)$ can be either a
$K^{*}D^{*}-D_{s}^{*}\rho$ molecule or a compact tetraquark state. Using the
coupling constants of the $\bar{T}^{a}_{c\bar{s}0}(2900)$ to $KD$ channel
obtained from molecule or compact tetraquark picture of the
$\bar{T}^{a}_{c\bar{s}0}(2900)$, we compute the cross-sections for the process
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$. The $\bar{K}N$
initial state interaction mediated by Pomeron and Reggeon exchanges is also
included, which reduces the production of the $\bar{T}^{a}_{c\bar{s}0}(2900)$.
Our calculations show that whether $\bar{T}^{a}_{c\bar{s}0}(2900)$ is a
molecule or a compact tetraquark state, the cross-sections for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction are of
similar magnitude, ranging from approximately 0.075 nb to 0.270 nb. However,a
clearer comparison can be made by computing the cross-section of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reaction. The results indicates that the cross-section for the molecule
assignment of $\bar{T}^{a}_{c\bar{s}0}(2900)$ can reach up to
$1.83\times{}10^{-3}$ nb, which is significantly smaller than that of 0.122 nb
by assuming $\bar{T}^{a}_{c\bar{s}0}(2900)$ as a compact tetraquark state.
Those results can be measured in future experiments, and can be used to test
the nature of the $\bar{T}^{a}_{c\bar{s}0}(2900)$. Last, we also propose to
search for the unreported charged tetraquark $T^{a}_{c\bar{s}0}(2900)^{+}$ in
the $K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ reaction.

##  I Introduction

Studying hadrons with more complex internal structures than quark states,
where mesons are composed of quark-antiquark pairs Godfrey:1985xj , and
baryons are constructed from three quarks Koniuk:1979vy ; Isgur:1978wd , is a
prominent topic in particle physics. We call them exotic hadrons. For an
extended period, little noteworthy advancement has been made in the
exploration of exotic states. Only a few phenomena suggesting that quarks
$u/d/s$ can form exotic hadrons. For example, In constituent quark models, the
mass of the strange quark is approximately 50% heavier than that of the $u/d$
quarks. This leads to questions regarding why the $\Lambda(1405)$ has a
significantly lower mass than the $N(1535)$. Moreover, it is puzzling to
observe that the $N(1440)$, as a $N=2$ baryon, is much lighter than the
nucleon resonance $N(1535)$ with $N=1$, where $N$ is the main quantum number.
These issues led Zou and his collaborators to propose the existence of
significant five-quark components in the nucleon and its resonances Zou:2005xy
; Riska:2005bh ; An:2005cj . The mass inversion problem could be easily
understood if there are substantial five-quark $uuds\bar{s}$ components in the
$N(1535)$ Zou:2010tc ; Zou:2009zz . Furthermore, the five-quark configurations
also provide a natural explanation for its large strange decay Liu:2005pm ;
Xie:2007qt .

However, it was in 2003 when the LHCb collaboration observed the $X(3872)$ in
the $\pi^{+}\pi^{-}J/\psi$ mass spectra Belle:2003nnu , that this field
entered a new era. From its observed decay mode, $X(3872)$ is known to consist
of a pair of hidden-charm quarks and two pairs of light quarks. Subsequent
discoveries of several hidden-charm pentaquark states, including
$P_{c}(4380)$, $P_{c}(4440)$, $P_{c}(4457)$, $P_{c}(4312)$, $P_{cs}(4338)$,
and $P_{cs}(4459)$, have further strengthened the belief in significant
progress in the research of exotic hadrons LHCb:2015yax ; LHCb:2016ztz ;
LHCb:2016lve ; LHCb:2019kea ; LHCb:2020jpq ; LHCb:2022ogu . The recent
observation of a doubly charged tetraquark and its neutral partner by the LHCb
collaboration in the analysis of the $B^{0}\to\bar{D}^{0}D^{+}_{s}\pi^{-}$ and
$B^{-}\to\bar{D}^{-}D^{+}_{s}\pi^{-}$ reactions LHCb:2022sfr  marks an another
significant advancement in the study of exotic hadrons. Their masses and
widthes were measured to be

$\displaystyle T^{a}_{c\bar{s}0}(2900)^{0}:~{}~{}~{}M$
$\displaystyle=2.892\pm{}0.014\pm{}0.015~{}~{}~{}~{}~{}{\rm GeV},$
$\displaystyle\Gamma$ $\displaystyle=0.119\pm 0.026\pm
0.013~{}~{}~{}~{}~{}{\rm GeV},$ $\displaystyle
T^{a}_{c\bar{s}0}(2900)^{++}:~{}~{}~{}M$
$\displaystyle=2.921\pm{}0.017\pm{}0.020~{}~{}~{}~{}~{}{\rm GeV},$
$\displaystyle\Gamma$ $\displaystyle=0.137\pm 0.032\pm
0.017~{}~{}~{}~{}~{}{\rm GeV},$ (1)

respectively. Supposing the states belong to the same isospin triplet, the
experiment also gave the shared mass and width,

$\displaystyle T_{c\bar{s}0}:~{}~{}~{}M$
$\displaystyle=2908\pm{}23~{}~{}~{}~{}~{}{\rm MeV},$ $\displaystyle\Gamma$
$\displaystyle=136\pm 25~{}~{}~{}~{}~{}~{}{\rm MeV},$ (2)

Similar to the challenges faced with other exotic hadrons, the true internal
structure of these two newly observed mesons cannot be completely determined
based on existing experimental data. Due to the close proximity of the mass of
$T^{a}_{c\bar{s}0}(2900)$ to the $DK$ threshold, the authors of
Ref.Chen:2022svh  proposed a novel interpretation for the recently discovered
$T^{a}_{c\bar{s}0}(2900)$. They suggest that these two states could be an
isovector $D^{*}K^{*}$ molecular state with quantum numbers
$I(J^{P})=1(0^{+})$. In an independent study conducted by the authors of
Ref.Yue:2022mnf , it was found that if the $T^{a}_{c\bar{s}0}(2900)^{0}$ is
indeed a molecular state formed by $D^{*0}K^{*0}$, its primary decay mode
would likely be into $D^{0}K^{0}$, rather than the observed experimental decay
channel $D_{s}^{+}\pi^{-}$. The authors in Ref. Agaev:2022eyk  support the
$T^{a}_{c\bar{s}0}(2900)$ as a $D^{*}K^{*}$ molecule based on the analysis of
the mass spectrum and the partial widths using the QCD light-cone sum rule
approach and soft-meson approximation. The mass of the $T^{a}_{c\bar{s}0}$ was
studied in the coupled-channel approach and it was shown that $T_{c\bar{s}0}$
might be $D^{*}K^{*}-D_{s}^{*}\rho$ couple molecular state Duan:2023lcj .
However, Ref. Ke:2022ocs  reached a strikingly different conclusion, arguing
that $T^{a}_{c\bar{s}0}$ should ont be considered as a $D^{*}K^{*}$ bound
state, but instead might be compact tetraquarks.

Indeed, the newly observed two mesons can be assigned to be the lowest $1S$
-wave tetraquark states Liu:2022hbk  within the framework of a nonrelativistic
potential quark model. Their analysis indicates that the dominant decay mode
is the $D_{s}^{*}\rho$. Furthermore, the compact tetraquark explanation of
$T^{a}_{c\bar{s}0}$ are also supported by estimates obtained from the
multiquark color flux-tube model Wei:2022wtr . QCD sum rules, informed by the
examination of mass spectrum and the two-body strong decays, have lead Refs.
Yang:2023evp ; Lian:2023cgs  to classify $T^{a}_{c\bar{s}0}$ as compact
tetraquark states. And the studies also reveal that the primary decay models
for $T^{a}_{c\bar{s}0}$ involve $D_{s}\pi$ and $DK$ channels Lian:2023cgs .
The compact tetraquark candidates for $T^{a}_{c\bar{s}0}$ gains additional
support from the Refs. Jiang:2023rcn ; Dmitrasinovic:2023eei ; Ortega:2023azl
. We note that the threshold effect from the interaction between the
$D^{*}K^{*}$ and $D_{s}^{*}\rho$ channels and the kinetic effect from a
triangle singularity for $T^{a}_{c\bar{s}0}$ are also proposed in Ref.
Molina:2022jcd  and Ref. Ge:2022dsp , respectively.

In addition to analyzing the mass spectrum and decay width, exploring the
production mechanism provides a more effective approach to evaluating the
nature of $T^{a}_{c\bar{s}0}$. This is mainly due to the production process
strong dependence on the internal structure of $T^{a}_{c\bar{s}0}$. We find
that whether $T^{a}_{c\bar{s}0}$ is a molecular state or a compact multi-quark
state, it exhibits a significant $KD$ decay width. This motivates our quest to
search for $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reactions. Notably, high-energy kaon beams are available at OKA@U-70
Obraztsov:2016lhp , SPS@CERN Velghe:2016jjw , CERN/AMBER Quintans:2022utc ,
and potential upgrades to the J-PARC kaon beam, enabling us to reach the
necessary energy range for $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ production
Nagae:2008zz . Consequently, the searching for
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reactions becomes feasible. This approach facilitates a straightforward
differentiation between molecular and compact tetraquark states through the
production process.

In this study, we examine the recently observed
$\bar{T}^{a}_{c\bar{s}0}(2900)$ production in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reactions by considering $T^{a}_{c\bar{s}0}$ as a molecular state and a
compact multiquark state, respectively. A conclusive determination of the
inner structure of $T^{a}_{c\bar{s}0}$ can be attained by comparing the
obtained cross-section with future experimental data. To enhance the
reliability of our predictions, the effect from the $\bar{K}N$ initial state
interaction (ISI) must be taken into account due to there exist plenty of
experimental information about the $\bar{K}N$ elastic interaction in the
considering energy region. Moreover, we also proposed to search for its
isospin partner $\bar{T}^{a}_{c\bar{s}0}(2900)^{-}$ in the
$K^{-}p\to\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ reaction. This
paper is organized as follows. In Sec. II, we will present the theoretical
formalism. In Sec. III, the numerical result will be given, followed by
discussions and conclusions in the last section.

##  II FORMALISM AND INGREDIENTS

In this study, we investigate the feasibility of measuring
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction. We
consider two scenarios for $T^{a}_{c\bar{s}0}$: one as a molecular state and
the other as a compact multi-quark state. The considered Feynman diagrams are
illustrated in Fig. (1), which includes only the $t$-channel $D^{-}$ meson
exchange diagram. And the ISI is represented by the red circle. This
production process differs from the complex proton-proton collisions
LHCb:2022sfr , as $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ production in
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ occurs more
simply. The reason lies in the significantly lower required center-of-mass
energies compared to proton-proton collisions. At these lower energies, we can
neglect contributions from the $s$\- and $u$-channels, which involve the
creation of an additional $c\bar{c}$ quark pair in kaon-induced production and
are typically strongly suppressed. Hence, the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction is
expected to be primarily governed by Born terms through the $t$-channel
$D^{-}$ exchanges, resulting in minimal background interference.

Figure 1: Feynman diagram for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction. The
contributions from the $t$-channel $D^{-}$ meson exchange. We also show the
definition of the kinematics ($p_{1},p_{2},q_{1},q_{2}$) that we use in the
present calculation.

To calculate the diagrams depicted in Fig. (1), it is necessary to determine
the effective Lagrangian densities corresponding to the relevant interaction
vertices. In the case of $\Lambda_{c}ND$ coupling, we adopt the Lagrangian
densities employed in Ref.Dong:2014ksa ; Zhu:2019vnr

$\displaystyle{\cal{L}}_{\Lambda_{c}ND}$
$\displaystyle=ig_{\Lambda_{c}ND}\bar{\Lambda}_{c}\gamma_{5}ND+H.c.,$ (3)

where the coupling constant $g_{\Lambda_{c}ND}=-13.98$ is established from the
SU(4) invariant Lagrangians Dong:2010xv  in terms of $g_{\pi{}NN}=13.45$ and
$g_{\rho{}NN}=6.0$. $N$, $D$, and $\Lambda_{c}$ are the nucleon, $D$ meson,
and $\Lambda_{c}^{+}$ baryon fields, respectively.

To calculate the diagrams depicted in Fig. (1), it is also to determine the
effective Lagrangian densities for the interaction vertex involving
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}K^{-}D^{-}$. Since the spin-parity of
$T^{a}_{c\bar{s}0}$ is established as $I(J^{P})=1(0^{+})$, the coupling
between $T^{a}_{c\bar{s}0}$ and $KD$ predominantly occurs through $S$\- and
$D$-wave interactions. Given our focus on studying the production rate of
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ near the threshold region, the
contribution from the lowest angular momentum state is most significant. This
phenomenon can be attributed to the higher energy requirement for $D$-wave
production cross-section than that of the $S$-wave production cross-section.
Thus, in this study, we will employ the effective Lagrangian densities
corresponding to the $S$-wave $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}K^{-}D^{-}$
interaction vertex. It is important to highlight that $S$-wave effective
Lagrangians are always characterized by fewer derivatives. This leads us to
express the Lagrangian densities for the $S$-wave coupling between
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ and $K^{-}D^{-}$ as follows Zhu:2019vnr

$\displaystyle{\cal{L}}_{\bar{T}_{c\bar{s}0}}$
$\displaystyle=g_{\bar{T}_{c\bar{s}0}}\bar{K}\vec{\tau}\cdot\bar{T}_{c\bar{s}0}\bar{D},$
(4)

where the $\tau$ is the corresponding pauli matrix reflecting the isospin of
the $\bar{T}^{a}_{c\bar{s}0}(2900)$. Please note that we have

$\displaystyle\vec{\tau}\cdot\vec{T}^{a}_{c\bar{s}0}=\left(\begin{array}[]{cc}T^{a}_{c\bar{s}0}(2900)^{+}&\sqrt{2}T^{a}_{c\bar{s}0}(2900)^{++}\\\
\sqrt{2}T^{a}_{c\bar{s}0}(2900)^{0}&-T^{a}_{c\bar{s}0}(2900)^{+}\end{array}\right)$
(7)

where the state $T^{a}_{c\bar{s}0}(2900)^{+}$ has not yet been discovered. It
only indicates the signal of $T^{a}_{c\bar{s}0}(2900)^{+}$ in the
$B^{0}\to{}D^{-}D^{0}K^{+}$ reaction Duan:2023qsg .

In Eq. (4), the coupling constant $g_{\bar{T}_{c\bar{s}0}}$ is determined from
the partial decay width of $T^{++}_{c\bar{s}0}\to{}K^{+}D^{+}$, which is
obtained as follow,

$\displaystyle\Gamma_{T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}}=\frac{g^{2}_{T_{c\bar{s}0}}}{4\pi}\frac{|\vec{p}^{c.m}_{D^{+}}|}{m^{2}_{T^{++}_{c\bar{s}0}}},$
(8)

where $\vec{p}^{c.m}_{D^{+}}$ is the three-vector momentum of the $D^{+}$ in
the $T^{a}_{c\bar{s}0}(2900)^{++}$ meson rest frame. Unfortunately, there is
no experimental information on the decay widths for
$\Gamma{(T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+})}$, as this is very
difficult to determine. Thus, it is necessary to rely on theoretical
predictions, such as those of Refs. Liu:2022hbk ; Lian:2023cgs . Assuming the
$T^{a}_{c\bar{s}0}(2900)^{++}$ as a compact multi-quark state, the partial
decay width of the $T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$ is predicted
to be $\Gamma({T^{++}_{c\bar{s}0}\to{}K^{+}D^{+}})=56.8\pm{}33.4$ MeV
Lian:2023cgs . Using the corresponding experimental masses of the relevant
particles given in Ref. ParticleDataGroup:2022pth , we obtain
$g_{T_{c\bar{s}0}}=2.836_{-1.016}^{+0.739}$. Note that the partial decay width
of the $T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$ is also evaluated in Ref.
Liu:2022hbk , adopting the compact multi-quark state assignment for
$T^{a}_{c\bar{s}0}(2900)^{++}$, and found that the obtained partial decay
width falls within the range reported in Ref. Lian:2023cgs .

Figure 2: Feynman diagram contributing to
$T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$. We also show the definition of
the kinematics ($q,p_{1},p_{2},q_{1},q_{2}$) that we use in the present
calculation.

Considering $T^{a}_{c\bar{s}0}(2900)^{++/0}$ as an $S$-wave $DK-D_{s}^{*}\rho$
molecule Duan:2023lcj , we analyze the partial decay width of
$T^{a}_{c\bar{s}0}(2900)^{++}$ into the $K^{+}D^{+}$ final state through
hadronic loops with the help of the effective Lagrangians. The loop diagrams
are depicted in Fig. (2), and the resulting partial decay widths are provided
in Table. 1 (additional details can be found in the Appendix. Appendix:
PARTIAL DECAY WIDTHS $T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$). Utilizing
these decay widths, coupling constants are evaluated and collected in Table.
1.

Table 1: Coupling constants $g_{T_{\bar{c}s0}}$ and the $K^{+}D^{+}$ decay width (in units of MeV) of the $T^{a}_{c\bar{s}0}(2900)^{++/0}$. The pole positions and effective couplings evaluated in Ref. Duan:2023lcj . $\sqrt{s}_{pole}$ | $|g_{D^{*}K^{*}}|$ | $|g_{D_{s}^{*}\rho}|$ | $\Gamma({T^{++}_{c\bar{s}0}\to{}K^{+}D^{+}})$ |  $g_{T_{\bar{c}s0}}$
---|---|---|---|---
2885 | 5531 | 5379 | 48.72-54.59 | 2.647-2.802
2887 | 2198 | 2082 | 7.37-8.25 | 1.029-1.089

Due to the hadrons are not pointlike particles, it becomes imperative to
incorporate form factors when evaluating the scattering amplitudes of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction. For
the $t$-channel $D^{-}$ meson exchange diagram, we adopt a widely used
approach found in many previous studies Zhu:2019vnr ; He:2011jp ; Xie:2015zga
with the expression

$\displaystyle{\cal{F}}_{D^{-}}(q^{2}_{D^{-}},m_{D^{-}})=\frac{\Lambda_{D^{-}}^{2}-m_{D^{-}}^{2}}{\Lambda_{D^{-}}^{2}-q^{2}_{D^{-}}},$
(9)

where $q^{2}_{D^{-}}$ and $m_{D^{-}}$ represent the four-momentum and mass of
the exchanged $D^{-}$ meson, respectively. The parameter $\Lambda_{D^{-}}$
serves as a hard cutoff, directly linked to the size of the hadron.
Empirically, $\Lambda_{D^{-}}$ should exceed the $m_{D^{-}}$ mass by several
hundred MeV at least. Therefore, we choose
$\Lambda_{D^{-}}=m_{D^{-}}+\alpha\Lambda_{QCD}$, following the precedent set
by prior works Zhu:2019vnr ; He:2011jp ; Xie:2015zga . The parameter $\alpha$
reflects the nonperturbative property of QCD at the low-energy scale, which
will be taken as a parameter and discussed later.

In this study, we will examine the impact of the $K^{-}n$ initial state
interaction on the cross-section of $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$
production in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction. A
straightforward approach utilized in Ref. Lebiedowicz:2011tp  yields a
satisfactory representation of the existing experimental data for $K^{-}p$ and
$K^{-}n$ scattering at high energies. Consequently, we employ this methodology
to estimate the initial state interaction (ISI) in the $K^{-}n\to K^{-}n$
reaction at high energies.

Figure 3: Feynman diagram for the mechanism of the initial state interaction
of the $K^{-}n\to{}K^{-}n$.

The pertinent Feynman diagram depicting the ISI for $K^{-}N\to{}K^{-}N$
reaction is illustrated in Fig. (3), where exchanges involving the Pomeron and
$f_{2}$, $a_{2}$, $\rho$, and $\omega$ Reggeons (IR) are considered. The total
amplitude ${\cal{T}}_{K^{-}N\to{}K^{-}N}$, incorporating Pomeron and Reggeon
exchanges, can be expressed as a sum of individual contributions, as indicated
by Lebiedowicz:2011tp

$\displaystyle{\cal{T}}_{K^{-}N\to{}K^{-}N}(s,t)$
$\displaystyle={\cal{A}}_{IP}(s,t)+{\cal{A}}_{f_{2}}(s,t)\pm{\cal{A}}_{a_{2}}(s,t)$
$\displaystyle+{\cal{A}}_{\omega}(s,t)\pm{\cal{A}}_{\rho}(s,t),$ (10)

where $s=(p_{1}+p_{2})^{2}$ and $t=(p_{1}-q_{1})^{2}$. The $(+)$ and $(-)$ are
for the $K^{-}p\to{}K^{-}p$ and $K^{-}n\to{}K^{-}n$ interactions,
respectively. For large center-of-mass energies $\sqrt{s}$, the individual
contribution to the $\bar{K}N\to{}\bar{K}N$ amplitude can be parameterized as
follows

$\displaystyle{\cal{A}}_{i}(s,t)=\eta_{i}s{\cal{C}}_{i}^{KN}(\frac{s}{s_{0}})^{\alpha_{i}(t)-1}\exp(\frac{{\cal{B}}^{i}_{KN}}{2}t),$
(11)

where $i=IP$ represents the Pomeron and $f_{2}$, $a_{2}$, $\omega$, and $\rho$
Reggeons. The energy scale is $s_{0}=1$ GeV2. The coupling constants
${\cal{C}}_{i}^{\bar{K}N}$, the parameters of the Regge linear trajectories
$\alpha_{i}(t)=\alpha_{i}(0)+\alpha_{i}^{{}^{\prime}}t$, the signature factors
$\eta_{i}$, and the ${\cal{B}}^{i}_{\bar{K}N}$ utilized in
Ref.Lebiedowicz:2011tp  offer a suitable description of the experimental data.
The parameters determined in Ref.Lebiedowicz:2011tp  are outlined in Table 2.

Table 2: Parameters of Pomeron and Reggeon exchanges determined from elastic and total cross sections in Ref. Lebiedowicz:2011tp . i | $\eta_{i}$ | $\alpha_{i}(t)$ |  ${\cal{C}}_{i}^{KN}$(mb) |  ${\cal{B}}_{i}^{KN}$(GeV-2)
---|---|---|---|---
$IP$ | $i$ | 1.081+(0.25 GeV-2)t | $11.82$ | $5.5$
$f_{2}$ | $-0.861+i$ | 0.548+(0.93 GeV-2)t | $15.67$ | $4.0$
$\rho$ | $-1.162-i$ | 0.548+(0.93 GeV-2)t | $2.05$ | $4.0$
$\omega$ | $-1.162-i$ | 0.548+(0.93 GeV-2)t | $7.055$ | $4.0$
$a_{2}$ | $-0.861+i$ | 0.548+(0.93 GeV-2)t | $1.585$ | $4.0$

Using the effective Lagrangians mentioned above and taking the ISI of the
$K^{-}p$ system into account, the full amplitude of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction can be
derived as

$\displaystyle{\cal{M}}^{full}={\cal{M}}^{Born}+{\cal{M}}^{K^{-}n-ISI},$ (12)

where the Born amplitude is written as

$\displaystyle{\cal{M}}^{Born}$
$\displaystyle=if_{I}g_{\bar{T}_{c\bar{s}0}}g_{\Lambda_{c}ND}\bar{u}(q_{2},s_{\Lambda^{+}_{c}})\gamma_{5}u(p_{2},s_{p})$
$\displaystyle\times\frac{i}{(q_{1}-p_{1})^{2}-m^{2}_{D^{-}}}{\cal{F}}_{D^{-}}[(q_{1}-p_{1})^{2}_{D^{-}},m_{D^{-}}],$
(13)

and the corrections to the Born amplitude due to $K^{-}n$ interactions were
taken into account in Lebiedowicz:2011tp ; Lebiedowicz:2011nb  as

$\displaystyle{\cal{M}}^{K^{-}n-ISI}=\frac{i}{16\pi^{2}s}\int{}d^{2}\vec{k}_{t}{\cal{T}}_{K^{-}n\to{}K^{-}n}(s,k^{2}_{t}){\cal{M}}^{Born}.$
(14)

where $k_{t}$ is the momentum transfer in the $K^{-}n\to{}K^{-}n$ reaction.
$\bar{u}(q_{2},s_{\Lambda^{+}_{c}})$ and $u(p_{2},s_{p})$ are the Dirac
spinors, with $s_{\Lambda^{+}_{c}}(q_{2})$ and $s_{p}(p2)$ being the spins
(the four-momenta) of the outgoing $\Lambda^{+}_{c}$ and the initial proton,
respectively.

With the scattering amplitudes of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction
obtained in the previous section, the differential cross section in the center
of mass (cm) frame for the process
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ can be
calculated ParticleDataGroup:2022pth

$\displaystyle\frac{d\sigma}{d\cos\theta}=\frac{m_{N}m_{\Lambda_{c}^{+}}}{8\pi{}s}\frac{|\vec{q}_{1cm}|}{|\vec{p}_{1cm}|}(\frac{1}{2}\sum_{s_{n},s_{\Lambda_{c}^{+}}}|{\cal{M}}|^{2}),$
(15)

where the $\theta$ is the scattering angle of the outgoing
$\bar{T}^{a}_{c\bar{s}0}$ meson relative to the beam direction, while
$\vec{p}^{1cm}$ and $\vec{q}^{1cm}$ are the $K^{-}$ and
$\bar{T}^{a}_{c\bar{s}0}$ meson three momenta in the center-of-mass frame,
respectively, which are

$\displaystyle|\vec{p}_{1cm}|=\frac{\lambda^{1/2}(s,m_{K^{-}},m_{n})}{2\sqrt{s}};~{}|\vec{q}_{1cm}|=\frac{\lambda^{1/2}(s,m_{T^{a}_{c\bar{s}0}},m_{\Lambda_{c}^{+}})}{2\sqrt{s}}.$
(16)

Here $\lambda(x,y,z)=(x-y-z)^{2}-4yz$ is the Källen function.

##  III RESULTS AND DISCUSSIONS

With the formalism and ingredients given above, the cross-section as a
function of the beam momentum $P_{K^{-}}$ for
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction can be
easily obtained. Before presenting the results, it is important to discuss the
parameter $\alpha$ that relates to the form factor. This is because the value
of the cross-section is highly sensitive to the model parameter $\alpha$.
However, determining the value of $\alpha$ from first principles is currently
not feasible. Instead, it can be better determined from the experimental data.
Indeed, it has been established that the free parameter $\alpha=1.5$ or $1.7$
were fixed by fitting the experimental data of the processes
$e^{+}e^{-}\to{}D\bar{D}$ Belle:2007qxm  and
$e^{+}e^{-}\to\gamma{ISR}D\bar{D}$ BaBar:2006qlj . The procedures for this
fitting is outlined in Ref.Guo:2016iej . For this study, we adopt the values
$\alpha=1.5$ or $1.7$, as they have been determined from the experimental data
of Refs. Belle:2007qxm ; BaBar:2006qlj  using the same $D$ form factors
employed in our current work.

Figure 4: The cross-section for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction as a
function of the beam momentum $P_{K^{-}}$ with assuming
$\bar{T}^{a}_{c\bar{s}0}$ as a compact tetraquarks state. The (a)
corresponding to the coupling constant $g_{T_{c\bar{s}0}}=2.836$, and the cyan
bands denote the cross-sections for different
$\Gamma(T^{++}_{c\bar{s}0}\to{}K^{+}D^{+})$ values Lian:2023cgs .

With the obtained $\alpha$ value, the cross-section for
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction is
evaluated by treating $T_{c\bar{s}0}^{a}$ as a compact multi-quark state. The
theoretical results obtained with a cutoff $\alpha=1.5$ or $1.7$ for the beam
energy from near threshold up to 27.5 GeV are shown in Fig. 4. We can find
that the cross-section exhibits a sharp increase near the threshold of
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$, an effect attributed to
the opening of phase space at that energy. Following this, the cross-section
continues to increase, albeit at a comparatively slower rate compared to the
threshold region. However, a modest decline in the cross-section is observed
as the beam energy $P_{K^{-}}$ is varied from 23.1 to 27.5 GeV. For deeper
insight, we illustrate the obtained total cross-section behavior, ranging from
approximately 0.134 nb to 0.127 nb for $\alpha=1.5$, with varying beam
momentum from 23.25 GeV to 26.80 GeV. Within the same energy range, but
adopting $\alpha=1.7$, the cross-section spans from 0.169 nb to 0.160 nb.
These outcomes suggest that the value of the cross-section is not very
sensitive to the model parameter $\alpha$ when varying the cutoff parameter
$\alpha$ from 1.5 to 1.7.

In addition to showing the central values of the cross-sections corresponding
to $g_{T_{c\bar{s}0}}=2.836$, we also present the variation of the cross-
sections for different $g_{T_{c\bar{s}0}}$ values, which are determined based
on the theoretically predicted partial decay width of
$T^{++}_{c\bar{s}0}\to{}K^{+}D^{+}$ Lian:2023cgs . We depict the results for
the cutoffs $\alpha=1.5$ and $\alpha=1.7$ in the Fig.4 (b) and Fig. 4 (c),
respectively. Remarkably, a significant variations in the cross-sections is
observed. For $\alpha=1.5$, the obtained cross-section ranges from 0.052 nb to
0.201 nb, and for $\alpha=1.7$, it ranges from 0.066 nb to 0.253 nb, both at
an example energy of approximately $P_{K^{-}}=26.80$ GeV. This suggests that
the cross-section for the maximum value is about four times larger than that
of the minimum value. This discrepancy can be attributed to the fact that the
ratio of the coupling constant for the maximum value to that of the minimum
value is around 2.0. And the cross-section is proportional to the square of
the coupling constant.

We now shift our focus to the cross-section of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction,
considering $T^{a}_{c\bar{s}0}(2900)$ as a $K^{*}D^{*}-D_{s}^{*}\rho$
molecule. The cross-section, varying with the beam energy $P_{K^{-}}$ from
just above the threshold up to 27.5 GeV, is depicted in Fig. 5. We clearly
observe that the line shapes of the cross-sections mirror those obtained by
considering $T^{a}_{c\bar{s}0}(2900)$ as a compact multi-quark state. That
means the cross-section also increases sharply near the threshold, followed by
a gradual and sustained increase at higher energies, concluding with a gradual
decrease.

Figure 5: The cross-section for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction as a
function of the beam momentum $P_{K^{-}}$ with assuming
$\bar{T}^{a}_{c\bar{s}0}$ as a $K^{*}D^{*}-D_{s}^{*}\rho$ molecule. The color
bands denote the cross-sections for different
$\Gamma(T^{++}_{c\bar{s}0}\to{}K^{+}D^{+})$ values, and $m$ is the masses of
the bound states $T^{a}_{c\bar{s}0}(2900)$ that obtained in Ref. Duan:2023lcj
.

The results in Fig. 5 also tell us that the
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ production cross-section for the cutoff
$\alpha=1.5$ is slightly smaller than that for cutoff $\alpha=1.7$. The
variation of the cross-sections for different
$\Gamma(T^{++}_{c\bar{s}0}\to{}K^{+}D^{+})$ values is very small. To see how
much the cross-section depends on the
$\Gamma(T^{++}_{c\bar{s}0}\to{}K^{+}D^{+})$ decay widths and the cutoff
$\alpha$, we take the cross-section at a beam momentum of about
$P_{K^{-}}=25.0$ GeV and the mass of the bound state $m=2885$ MeV as an
example. The so-obtained cross-section ranges from 0.156 nb to 0.175 nb for
$\alpha=1.5$ and from $0.197$ to 0.221 nb for $\alpha=1.7$. We also find that
if the $\bar{T}^{a}_{c\bar{s}0}(2900)$ is produced as a
$K^{*}D^{*}-D_{s}^{*}\rho$ molecule with mass $m=2885$ MeV, the cross-section
is significantly larger than the results obtained by assuming
$\bar{T}^{a}_{c\bar{s}0}(2900)$ as a $KD-D_{s}^{*}\rho$ molecule with mass
$m=2887$ MeV, by about $6-8$ times. In other words, the cross-section is
heavily depends on the masses of the bound states.

By comparing the cross-sections depicted in Fig.4 and Fig.5, we observe that
if the $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ is a compact tetraquark state, its
production cross-sections match the results predicted by considering the
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ as a $K^{*}D^{*}-D_{s}^{*}\rho$ molecule
with a mass of $m=2885$ MeV. Specifically, the cross-sections for these two
assignments can reach 0.269 nb (for compact tetraquark state) and 0.227 nb
(for $K^{*}D^{*}-D_{s}^{*}\rho$ molecule), respectively. These results suggest
that if the $\bar{T}^{a}_{c\bar{s}0}(2900)$ is a $K^{*}D^{*}-D_{s}^{*}\rho$
molecule with a mass of $m=2887$ MeV, distinguishing its
$K^{*}D^{*}-D_{s}^{*}\rho$ molecular nature from the compact tetraquark state
is challenging through the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction.
However, when considering a smaller cross-section obtained from the assumption
that the $\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$ is a $K^{*}D^{*}-D_{s}^{*}\rho$
molecule with a mass of $m=2887$ MeV, the cross-section is limited to 0.034
nb. This discrepancy magnifies the difference between the results of the two
scenarios by a factor of about 8.0. This indicates that if the mass of the
$K^{*}D^{*}-D_{s}^{*}\rho$ molecule is $m=2887$ MeV, a clear conclusion about
the the nature of the $\bar{T}^{a}_{c\bar{s}0}(2900)$ can be easily obtained
by comparing the obtained cross-section with future experimental data.

Figure 6: The cross-sections for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reaction as a function of the beam momentum $P_{K^{-}}$ with assuming
$\bar{T}^{a}_{c\bar{s}0}$ as a $K^{*}D^{*}-D_{s}^{*}\rho$ molecule (Black
solid line) or compact tetraquark state (red dash line). Here, we display
results associated with $\alpha=1.7$, the bound state mass $m=2885$ MeV, and
the maximum value of the coupling constants $g_{T^{a}_{c\bar{s}0}}$.

However, a more distinct comparison can be drawn from the production of
$\bar{T}^{a}_{c\bar{s}0}(2900)$ in the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reaction, and the results of this comparison are shown in Fig. 6. We can find
that the cross-section for the molecule assignment of
$\bar{T}^{a}_{c\bar{s}0}(2900)$ can reach up to $1.83\times{}10^{-3}$ nb. This
value is significantly smaller than that of 0.122 nb at the same beam energy.
And the larger cross-section was derived from considering the
$\bar{T}^{a}_{c\bar{s}0}(2900)$ as a compact tetraquark state. The significant
difference between the results in these two pictures arises from considering
$T^{a}_{c\bar{s}0}(2900)$ in two different ways. If it’s seen as a compact
tetraquark state, the calculated partial decay width of
$T^{a}_{c\bar{s}0}(2900)^{++}\to{}\pi^{+}D^{+}_{s}$ is $48.5\pm{}30.0$ MeV
Lian:2023cgs (we use 78.5 MeV). This contrasts sharply with the range of
0.132-1.167 MeV (we use 1.167 MeV) obtained by treating
$T^{a}_{c\bar{s}0}(2900)$ as a $K^{*}D^{*}-D_{s}^{*}\rho$ molecule with a mass
of $m$=2885 MeV. Importantly, the partial decay width of
$T^{a}_{c\bar{s}0}(2900)^{++}\to{}\pi^{+}D^{+}_{s}$ from the $D_{s}^{*}\rho$
channel contribution is prohibited, and the necessary amplitudes required for
the $T^{a}_{c\bar{s}0}(2900)^{++}\to{}\pi^{+}D^{+}_{s}$ reaction in our work
are available in Ref. Yue:2022mnf . Note that the cross-section for the
$K^{-}n\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$ reaction are computed from the
following differential cross-section Kim:2017nxg

$\displaystyle\frac{d\sigma_{K^{-}n\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}}}{d{\cal{M}}_{\pi^{-}D_{s}^{-}}}$
$\displaystyle\approx{}\frac{2m_{\bar{T}^{a}_{c\bar{s}0}}{\cal{M}}_{\pi^{-}D_{s}^{-}}}{\pi}$
$\displaystyle\times\frac{\sigma_{K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}}\Gamma_{\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\to{}\pi^{-}D^{-}_{s}}}{({\cal{M}}^{2}_{\pi^{-}D_{s}^{-}}-m^{2}_{\bar{T}^{a}_{c\bar{s}0}})^{2}+m^{2}_{\bar{T}^{a}_{c\bar{s}0}}\Gamma^{2}_{\bar{T}^{a}_{c\bar{s}0}}},$
(17)

where $m_{\bar{T}^{a}_{c\bar{s}0}}$ and $\Gamma_{\bar{T}^{a}_{c\bar{s}0}}$ are
the mass and width of the $\bar{T}^{a}_{c\bar{s}0}$, respectively.
${\cal{M}}_{\pi^{-}D_{s}^{-}}$ is in the range of the
$(m_{\pi^{-}}+m_{D_{s}^{-}})-(\sqrt{s}-m_{\Lambda_{c}^{+}})$.

Figure 7: The cross-section for the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ reaction as a
function of the beam momentum $P_{K^{-}}$ with assuming
$\bar{T}^{a}_{c\bar{s}0}$ as a $K^{*}D^{*}-D_{s}^{*}\rho$ molecule (red and
black bands) or compact tetraquark (cyan bands). The color bands denote the
cross-sections for different $\Gamma(T^{+}_{c\bar{s}0}\to{}K^{+}D^{0})$
values. The red and black bands corresponding to the bound state mass $m=2885$
MeV and $M$=2887 MeV, respectively, that obtained in Ref. Duan:2023lcj . The
cyan bands is the cross-section for the mass of the bound state $m=2921$ MeV
LHCb:2022sfr .

Considering isospin symmetry, the decay width of $K^{+}D^{0}$ for the
unreported $T^{a}_{c\bar{s}0}(2900)^{+}$ state is expected to be half of the
partial decay width of $T^{++}_{c\bar{s}0}\to{}K^{+}D^{+}$ (as seen in Eq. 7).
If we regard the $T^{a}_{c\bar{s}0}(2900)^{+}$ as a $K^{*}D^{*}-D_{s}^{*}\rho$
molecule, the predicted partial decay widths for the bound state with masses
$m=2885$ MeV and $m=2887$ MeV are
$\Gamma({T^{+}_{c\bar{s}0}\to{}K^{+}D^{0}})=24.36-27.30$ MeV and
$\Gamma({T^{+}_{c\bar{s}0}\to{}K^{+}D^{0}})=2.69-4.13$ MeV, respectively.
Furthermore, the decay width $\Gamma({T^{+}_{c\bar{s}0}\to{}K^{+}D^{0}})$ is
estimated to be within the range of 11.7-45.1 MeV when considering the
$T^{+}_{c\bar{s}0}$ as a compact tetraquark state. These predictions open up
an opportunity to search for the $\bar{T}^{a}_{c\bar{s}0}(2900)^{-}$
[$\bar{T}^{a}_{c\bar{s}0}(2900)^{-}$ is the antiparticle of
$T^{a}_{c\bar{s}0}(2900)^{+}$] in the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reaction. The
cross-section for the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reaction, with a
cutoff $\alpha=1.7$ and beam momentum $P_{K^{-}}$ ranging from near threshold
to 27.5 GeV, are presented in Fig. 7. The findings reveal that the maximum
value of the cross-section for $\bar{T}^{a}_{c\bar{s}0}(2900)^{-}$ production
in the $K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reaction
is approximately 0.114 nb, which is bigger than 0.075 nb that is the maximum
values by assigning the $\bar{T}^{a}_{c\bar{s}0}(2900)^{-}$ as a compact
tetraquark state.

Figure 8: The cross-sections for the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ and
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reactions as a
function of the beam momentum $P_{K^{-}}$ with assuming
$\bar{T}^{a}_{c\bar{s}0}$ as a compact tetraquark state. The black solid lines
and dash red lines represent the cross-section that obtained with and without
ISI, respectively.

It is worth noting that the contributions to the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reaction is
mediated by the exchange of $D$ meson in the $t$-channel, which is identical
to the production process of its isospin partner
$\bar{T}^{a}_{c\bar{s}0}(2900)^{--}$. The only difference is the influence of
$\bar{K}N$ ISI on their product cross-sections. To show the effect from the
$\bar{K}N$ ISI, we compare the cross-sections obtained with and without ISI
for the cutoff $\alpha=1.7$ and $g_{T_{c\bar{s}0}}=2.836$ in Fig. 8, for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda_{c}^{+}$ [Fig. 8(a)] and
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ [Fig. 8(b)]
reactions, respectively. In Fig. 8, the black solid line are the pure Born
amplitude contribution, while the dash red line are the full results. We can
find that the presence of $\bar{K}N$ ISI leads to a reduction in the cross-
section of the $K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$
and $K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda_{c}^{+}$ reactions by
approximately 20%. This suggest that the $\bar{K}N$ ISI have a significant
impact on the search for $\bar{T}^{a}_{c\bar{s}0}(2900)$ in the
$K^{-}N\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-/--}\Lambda_{c}^{+}$ reactions.
Similar conclusions regarding the reduction in cross-section could be drawn if
one were to consider the $\bar{T}^{a}_{c\bar{s}0}(2900)$ as a $KD-
D_{s}^{*}\rho$ molecule, and we here do not discuss it in details.

##  IV Summary

Theoretical investigations on the production processes will be helpful to
distinguish which inner structure of the$T^{a}_{c\bar{s}0}(2900)$ is possible.
This is because the different production mechanisms of the
$T^{a}_{c\bar{s}0}(2900)$ rely on its structure assignments. In this work, we
examine the recently observed $\bar{T}^{a}_{c\bar{s}0}(2900)$ production in
the $K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reactions by considering $T^{a}_{c\bar{s}0}$ as a molecular state and a
compact multiquark state, respectively. The $T^{a}_{c\bar{s}0}(2900)$ can be
produced though the exchange of $D$\- meson in the $t$-channel.

Using the coupling constants of the $\bar{T}^{a}_{c\bar{s}0}(2900)$ to $KD$
channel obtained from molecule or compact tetraquark picture of the
$\bar{T}^{a}_{c\bar{s}0}(2900)$, we compute the cross-sections for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reactions, respectively. The numerical results reveal that whether
$\bar{T}^{a}_{c\bar{s}0}(2900)$ is categorized as a molecule or a compact
tetraquark state, the cross-sections for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}$ reaction
exhibit similar magnitudes, spanning roughly from 0.06 nb to 0.3 nb.
Nevertheless, a more distinct comparison can be drawn by computing the cross-
section of the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}c\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
process. The results indicate that, assuming the molecule assignment for
$\bar{T}^{a}_{c\bar{s}0}(2900)$, the cross-section for the
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}c\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$
reaction could peak at $1.83\times{}10^{-3}$ nb, notably smaller than the
0.122 nb obtained by presuming $\bar{T}^{a}_{c\bar{s}0}(2900)$ to be a compact
tetraquark state. These findings are poised for future experimental
measurement and can serve as tests to discern the nature of
$\bar{T}^{a}_{c\bar{s}0}(2900)$. Lastly, we suggest a quest for the unreported
charged tetraquark $T^{a}_{c\bar{s}0}(2900)^{+}$ in the
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ reaction due to
its production cross-section can reach up to 0.114 nb.

A rough estimation finds that the production cross-section of the
$\bar{T}^{a}_{c\bar{s}0}(2900)$ through high-energy proton-proton collisions
is approximately 17 fb (see Fig.3 in Ref. LHCb:2022sfr ). This value is about
$10^{4}$ times smaller than our calculated results. This difference is why we
propose to search for the $\bar{T}^{a}_{c\bar{s}0}(2900)$ in the reactions
$K^{-}p\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{-}\Lambda^{+}_{c}$ and
$K^{-}n\to{}\bar{T}^{a}_{c\bar{s}0}(2900)^{--}\Lambda^{+}_{c}\to{}\pi^{-}D^{-}_{s}\Lambda^{+}_{c}$.
Furthermore, kaon beams with momenta ranging from 50 to 280 GeV/c and an RMS
below 5% are available from the M2 beamline at AMBERQuintans:2022utc . AMBER
is a new fixed-target experiment at CERN that began its data-taking in 2023,
providing an excellent platform for the search for
$\bar{T}^{a}_{c\bar{s}0}(2900)$.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant No.12005177.

## Appendix: PARTIAL DECAY WIDTHS
$T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$

In this Appendix, we show how to compute the partial decay width of
$T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+}$ reaction. The corresponding
Feynman diagrams are shown in Fig. (2). To compute the diagrams, we require
the effective Lagrangian densities for the relevant interaction vertices.
Since $T^{a}_{c\bar{s}0}(2900)^{++}$ resonance can be identified as an
$S$-wave $D^{*}K^{*}-\rho{}D_{s}^{*}$ molecule Duan:2023lcj , the Lagrangian
densities for $T^{a}_{c\bar{s}0}(2900)^{++}{}D^{*}K^{*}$ and
$T^{a}_{c\bar{s}0}(2900)^{++}{}\rho{}D_{s}^{*}$ vertices can be written down
as Duan:2023qsg

$\displaystyle{\cal{L}}_{T^{a}_{c\bar{s}0}K^{*}D^{*}}$
$\displaystyle=g_{T^{a}_{c\bar{s}0}K^{*}D^{*}}D^{*\mu}\vec{\tau}\cdot\vec{T}^{a}_{c\bar{s}0}K^{*}_{\mu},$
(18) $\displaystyle{\cal{L}}_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}$
$\displaystyle=g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}D_{s}^{*\mu}\vec{\rho}_{\mu}\cdot\vec{T}^{a}_{c\bar{s}0},$
(19)

where the coupling constants $g_{T^{a}_{c\bar{s}0}K^{*}D^{*}}=2.198-5.531$ GeV
and $g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}=2.082-5.379$ GeV, which correspond
to the physical sheet Duan:2023lcj .

Considering the heavy quark limit and chiral symmetry, the relevant
phenomenological Lagrangians for ${\cal{D}}^{*}{\cal{D}}{\cal{P}}$ and
${\cal{D}}^{*}{\cal{D}}{\cal{V}}$ vertices are Yue:2022mnf ; Casalbuoni:1996pg
; Altenbuchinger:2013vwa

$\displaystyle{\cal{L}}_{{\cal{D}}^{*}{\cal{D}}{\cal{P}}}$
$\displaystyle=ig\langle{\cal{D}}^{*}_{\mu}u^{\mu}{\cal{D}}^{\dagger}-{\cal{D}}u^{\mu}{\cal{D}}^{*\dagger}_{\mu}\rangle,$
(20) $\displaystyle{\cal{L}}_{{\cal{D}}^{*}{\cal{D}}{\cal{V}}}$
$\displaystyle=-2f_{D^{*}D{\cal{V}}}\epsilon_{\mu\nu\alpha\beta}(\partial^{\mu}{\cal{V}}^{\nu})^{i}_{j}$
$\displaystyle\times({\cal{D}}_{i}\overleftrightarrow{\partial}_{\alpha}{\cal{D}}^{*\beta{}j\dagger}-{\cal{D}}_{i}^{*\beta}\overleftrightarrow{\partial}_{\alpha}{\cal{D}}^{j\dagger}),$
(21)

where the $\langle...\rangle$ denotes trace in the SU(3) flavor space and
$\epsilon^{0123}=1$. ${\cal{P}}$ and ${\cal{V}}^{\mu}$ are the SU(3)
pseudoscalar meson and vector meson matrices, respectively,

$\displaystyle{\cal{P}}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\\\ \end{array}\right),$ (25)
$\displaystyle{\cal{V}}_{\mu}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}(\rho^{0}+\omega)&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{1}{\sqrt{2}}(-\rho^{0}+\omega)&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right)_{\mu},$ (29)

and the ${\cal{D}}^{(*)}=(D^{(*)0},D^{(*)+},D_{s}^{*(+)})$. $u^{\mu}$ is the
axial vector combination of the pseudoscalar-meson fields and at the lowest
order $u^{2}=-\sqrt{2}\partial^{\mu}{\cal{P}}/f_{0}$ with $f_{0}=92.4$ MeV.
The coupling constants
$f_{D^{*}D{\cal{V}}}=\lambda{}m_{\rho}/(\sqrt{2}f_{\pi})$ with $\lambda=0.56$
GeV-1 ,$f_{\pi}$=132 MeV Casalbuoni:1996pg , and the $m_{\rho}$ is the mass of
the $\rho$ meson. The coupling constant $g=1.097$ is determined from the
strong decay width $\Gamma(D^{*+}\to{}D^{0}\pi^{+})=56.46\pm{}1.22$ keV,
together with the branching ratio
$BR(D^{*+}\to{}D^{0}\pi^{+})=(67.7\pm{}0.5)\%$.

For the $\rho{}DD$ and $KD^{*}D_{s}^{*}$ vertices, the following effective
Lagrangian are needed Oh:2000qr ; Lin:1999ad ; Azevedo:2003qh

$\displaystyle{\cal{L}}_{DD\rho}$
$\displaystyle=ig_{DD\rho}(D^{\dagger}\vec{\tau}\cdot\vec{\rho}^{\mu}\partial_{\mu}\bar{D}-\partial_{\mu}D^{\dagger}\vec{\tau}\cdot\vec{\rho}^{\mu}\bar{D}),$
(30) $\displaystyle{\cal{L}}_{KD_{s}^{*}D^{*}}$
$\displaystyle=-g_{KD_{s}^{*}D^{*}}\epsilon^{\mu\nu\alpha\beta}(\partial_{\mu}\bar{D^{*\dagger}}_{\nu}\partial_{\alpha}D_{s\beta}^{*}\bar{K}+\partial_{\mu}D^{*\dagger}_{\nu}\partial_{\alpha}\bar{D^{*}}_{s\beta}K),$
(31)

where the coupling constant $g_{DD\rho}=2.52$ Lin:1999ad  was derived from the
D-meson electric form factor in the standard framework of the vector meson
dominance model. $g_{KD_{s}^{*}D^{*}}=7.0$ GeV-1 was computed from the SU(4)
relations Azevedo:2003qh . The charm and $K$ mesons isodoublets were defined
as

$\displaystyle\bar{D}^{(*)\dagger}$
$\displaystyle=\left(\begin{array}[]{cc}\bar{D}^{(*)0}&D^{(*)-}\\\
\end{array}\right),~{}~{}~{}D^{(*)}=\left(\begin{array}[]{cc}D^{(*)0}\\\
D^{(*)+}\\\ \end{array}\right),$ (35) $\displaystyle\bar{K}^{(*)\dagger}$
$\displaystyle=\left(\begin{array}[]{cc}K^{(*)-}&\bar{K}^{(*)0}\\\
\end{array}\right),~{}~{}~{}K^{(*)}=\left(\begin{array}[]{cc}K^{(*)+}\\\
K^{(*)0}\\\ \end{array}\right),$ (39)

In addition to the vertices described above, we also need the following
effective Lagrangians Yue:2022mnf

$\displaystyle{\cal{L}}_{{\cal{P}}{\cal{P}}{\cal{V}}}$
$\displaystyle=-ig_{h}\langle[{\cal{P}},\partial^{\mu}{\cal{P}}]{\cal{V}}_{\mu}\rangle,$
(40) $\displaystyle{\cal{L}}_{K^{*}K{\cal{V}}^{{}^{\prime}}}$
$\displaystyle=-g_{K^{*}K{\cal{V}}^{{}^{\prime}}}\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}\bar{K}^{*}_{\beta}\partial_{\mu}{\cal{V}}_{\nu}^{{}^{\prime}}K+H.c.,$
(41)

where the ${\cal{V}}^{{}^{\prime}}$ is meson matrices,

$\displaystyle{\cal{V}}^{{}^{\prime}}_{\mu}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}(\rho^{0}+\omega)&\rho^{+}\\\
\rho^{-}&\frac{1}{\sqrt{2}}(-\rho^{0}+\omega)\\\ \end{array}\right)_{\mu}.$
(44)

The coupling constants
$g_{K^{*}K{\cal{V}}^{{}^{\prime}}}=3g_{h}^{2}/(64\pi^{2}f_{\pi})$ with the
$g_{h}$ is determined via measured width of $K^{*}\to\pi{}K$. With the help of
Eq. (40), the two-body decay width $K^{*}\to{}K\pi$ is related to $g_{h}$ as

$\displaystyle\Gamma(K^{*+}\to{}K^{0}\pi^{+})=\frac{g^{2}}{24\pi{}m^{2}_{K^{*+}}}{\cal{P}}^{3}_{\pi{}K^{*}}=\frac{2}{3}\Gamma_{K^{*+}},$
(45)

where ${\cal{P}}_{\pi{}K^{*}}$ is the three-momentum of the $\pi$ in the rest
frame of the $K^{*}$. Using the experimental strong decay width
($\Gamma_{K^{*+}}=50.3\pm{}0.8$ MeV) and the masses of the particles shown in
Ref. ParticleDataGroup:2022pth  we obtain $g_{h}=9.11$.

Putting all the pieces together, we obtain the following strong decay
amplitudes,

$\displaystyle{\cal{M}}_{a}^{\pi^{0}}$
$\displaystyle=-i\frac{gg_{h}g_{T^{a}_{c\bar{s}0}}}{f_{0}}\int\frac{d^{4}q}{(2\pi)^{4}}q_{\mu}\frac{-g^{\mu\nu}+q_{1}^{\mu}q_{1}^{\nu}/m^{2}_{D^{*+}}}{q_{1}^{2}-m^{2}_{D^{*+}}}$
$\displaystyle\times{}\frac{-g^{\nu\sigma}+q_{2}^{\nu}q_{2}^{\sigma}/m^{2}_{K^{*+}}}{q_{2}^{2}-m^{2}_{K^{*+}}}(q_{\sigma}+p_{2\sigma})\frac{1}{q^{2}-m^{2}_{\pi^{0}}},$
(46) $\displaystyle{\cal{M}}_{a}^{\eta}$
$\displaystyle=i\frac{gg_{h}g_{T^{a}_{c\bar{s}0}}}{3f_{0}}\int\frac{d^{4}q}{(2\pi)^{4}}q_{\mu}\frac{-g^{\mu\nu}+q_{1}^{\mu}q_{1}^{\nu}/m^{2}_{D^{*+}}}{q_{1}^{2}-m^{2}_{D^{*+}}}$
$\displaystyle\times{}\frac{-g^{\nu\sigma}+q_{2}^{\nu}q_{2}^{\sigma}/m^{2}_{K^{*+}}}{q_{2}^{2}-m^{2}_{K^{*+}}}(q_{\sigma}+p_{2\sigma})\frac{1}{q^{2}-m^{2}_{\eta}},$
(47) $\displaystyle{\cal{M}}_{b}^{\rho^{0}}$
$\displaystyle=-i\sqrt{2}f_{D^{*}D\rho}g_{T^{a}_{c\bar{s}0}}g_{K^{*}K\rho}\int\frac{d^{4}q}{(2\pi)^{4}}\epsilon_{\mu\nu\alpha\beta}q^{\mu}$
$\displaystyle\times(p^{\alpha}_{1}+q_{1}^{\alpha})\frac{-g^{\beta\sigma}+q_{1}^{\beta}q_{1}^{\sigma}/m^{2}_{D^{*+}}}{q_{1}^{2}-m^{2}_{D^{*+}}}\frac{-g^{\sigma\eta}+q_{2}^{\sigma}q_{2}^{\eta}/m^{2}_{K^{*+}}}{q_{2}^{2}-m^{2}_{K^{*+}}}$
$\displaystyle\times\epsilon_{\tau\lambda\kappa\eta}q_{2}^{\kappa}q^{\tau}\frac{-g^{\lambda\nu}+q^{\lambda}q^{\nu}/m^{2}_{\rho^{0}}}{q^{2}-m^{2}_{\rho^{0}}},$
(48) $\displaystyle{\cal{M}}_{b}^{\omega}$
$\displaystyle=i\sqrt{2}f_{D^{*}D\omega}g_{T^{a}_{c\bar{s}0}}g_{K^{*}K\omega}\int\frac{d^{4}q}{(2\pi)^{4}}\epsilon_{\mu\nu\alpha\beta}q^{\mu}$
$\displaystyle\times(p^{\alpha}_{1}+q_{1}^{\alpha})\frac{-g^{\beta\sigma}+q_{1}^{\beta}q_{1}^{\sigma}/m^{2}_{D^{*+}}}{q_{1}^{2}-m^{2}_{D^{*+}}}\frac{-g^{\sigma\eta}+q_{2}^{\sigma}q_{2}^{\eta}/m^{2}_{K^{*+}}}{q_{2}^{2}-m^{2}_{K^{*+}}}$
$\displaystyle\times\epsilon_{\tau\lambda\kappa\eta}q_{2}^{\kappa}q^{\tau}\frac{-g^{\lambda\nu}+q^{\lambda}q^{\nu}/m^{2}_{\omega}}{q^{2}-m^{2}_{\omega}},$
(49) $\displaystyle{\cal{M}}_{c}^{D^{0}}$
$\displaystyle=-i\frac{2gg_{DD\rho}g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}}{f_{0}}\int\frac{d^{4}q}{(2\pi)^{4}}p_{2}^{\mu}\frac{-g^{\mu\nu}+q_{1}^{\mu}q_{1}^{\nu}/m^{2}_{D_{s}^{*+}}}{q_{1}^{2}-m^{2}_{D_{s}^{*+}}}$
$\displaystyle\times{}\frac{-g^{\nu\sigma}+q_{2}^{\nu}q_{2}^{\sigma}/m^{2}_{\rho^{+}}}{q_{2}^{2}-m^{2}_{\rho^{+}}}(q_{\sigma}+p_{1\sigma})\frac{1}{q^{2}-m^{2}_{D^{0}}},$
(50) $\displaystyle{\cal{M}}_{c}^{D^{*0}}$
$\displaystyle=i2g_{KD_{s}^{*}D^{*}}g_{D^{*}D\rho}g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}\int\frac{d^{4}q}{(2\pi)^{4}}\epsilon_{\mu\nu\alpha\beta}q^{\mu}q_{1}^{\alpha}$
$\displaystyle\times\frac{-g^{\beta\sigma}+q_{1}^{\beta}q_{1}^{\sigma}/m^{2}_{D_{s}^{*+}}}{q_{1}^{2}-m^{2}_{D_{s}^{*+}}}\frac{-g^{\sigma\eta}+q_{2}^{\sigma}q_{2}^{\eta}/m^{2}_{\rho^{+}}}{q_{2}^{2}-m^{2}_{\rho^{+}}}$
$\displaystyle\times{}\epsilon_{\tau\eta\lambda\kappa}q_{2}^{\tau}(q^{\lambda}+p_{1}^{\lambda})\frac{-g^{\nu\kappa}+q^{\nu}q^{\kappa}/m^{2}_{D^{*0}}}{q^{2}-m^{2}_{D^{*0}}},$
(51) $\displaystyle{\cal{M}}_{d}^{K^{0}}$
$\displaystyle=i\frac{\sqrt{2}gg_{h}g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}}{f_{0}}\int\frac{d^{4}q}{(2\pi)^{4}}p_{1}^{\mu}$
$\displaystyle\times\frac{-g^{\mu\nu}+q_{1}^{\mu}q_{1}^{\nu}/m^{2}_{D_{s}^{*+}}}{q_{1}^{2}-m^{2}_{D_{s}^{*+}}}\frac{-g^{\nu\eta}+q_{2}^{\nu}q_{2}^{\eta}/m^{2}_{\rho^{+}}}{q_{2}^{2}-m^{2}_{\rho^{+}}}$
$\displaystyle\times{}(p_{2\eta}+q_{\eta})\frac{1}{q^{2}-m^{2}_{K^{0}}},$ (52)
$\displaystyle{\cal{M}}_{d}^{K^{*0}}$
$\displaystyle=i2g_{K^{*}D_{s}^{*}D}g_{K^{*}K\rho}g_{T^{a}_{c\bar{s}0}\rho{}D_{s}^{*}}\int\frac{d^{4}q}{(2\pi)^{4}}\epsilon_{\mu\nu\alpha\beta}q^{\mu}(q_{1}^{\alpha}+p_{1}^{\alpha})$
$\displaystyle\times\frac{-g^{\beta\sigma}+q_{1}^{\beta}q_{1}^{\sigma}/m^{2}_{D_{s}^{*+}}}{q_{1}^{2}-m^{2}_{D_{s}^{*+}}}\frac{-g^{\sigma\eta}+q_{2}^{\sigma}q_{2}^{\eta}/m^{2}_{\rho^{+}}}{q_{2}^{2}-m^{2}_{\rho^{+}}}$
$\displaystyle\times{}\epsilon_{\tau\eta\lambda\kappa}q_{2}^{\tau}q^{\lambda}\frac{-g^{\nu\kappa}+q^{\nu}q^{\kappa}/m^{2}_{K^{*0}}}{q^{2}-m^{2}_{K^{*0}}},$
(53)

where $m_{D_{s}^{*}}$, $m_{D}$, $M_{D^{*}}$, $m_{K^{*}}$, and $m_{K}$ are the
masses of the $D_{s}^{*}$, $D$, $D^{*}$, $K^{*}$, and $K$ mesons,
respectively. It is evident that these amplitudes suffer from ultraviolet (UV)
divergence. Nevertheless, even the loops that are UV finite receive
contributions from short distances when integrated over the entire momentum
space. To address this, we will utilize a UV regulator, as described in
Lin:2017mtz , which effectively suppresses the short-distance contributions,
thereby rendering the amplitudes UV finite. The UV regulator takes the form

$\displaystyle\tilde{\Phi}(p_{E}^{2}/\Lambda^{2})\equiv\exp{(-p_{E}^{2}/\Lambda^{2})},$
(54)

where $P_{E}$ is the Euclidean Jacobi momentum defined as
$P_{E}=m_{i}p_{j}/(m_{i}+m_{j})-m_{j}p_{i}(m_{i}+m_{j})$ for the $(ij)$
molecule.

Furthermore, we adopt the dipole form factor
${\cal{F}}(q^{2})=(\Lambda^{2}-m^{2})/(\Lambda^{2}-q^{2})$ to account for the
off-shell effect of the exchanged mesons. In this expression, $m$ and $q$
represent the mass and four-momentum of the exchanged mesons, respectively.
The parameter $\Lambda$ is typically parameterized as
$\Lambda=m+\alpha\Lambda_{QCD}$, where $\Lambda_{QCD}=220$ MeV. The value of
$\alpha$ is chosen to be approximately 1.0 to ensure that $\Lambda$ closely
aligns with the mass of the exchanged mesons. In this study, we consider a
range of $\alpha$ values within $0.91\leq\alpha\leq 1.0$, a range derived from
experimental data Dong:2017rmg . Then we have

$\displaystyle{\cal{M}}_{total}=\sum_{i=a,b,c,d}{\cal{M}}_{i}\tilde{\Phi}(p_{E}^{2}/\Lambda^{2}){\cal{F}}^{2}(q^{2}).$
(55)

Once the amplitudes are determined, the corresponding partial decay widths can
be obtained, which read,

$\displaystyle\Gamma(T^{a}_{c\bar{s}0}(2900)^{++}\to{}K^{+}D^{+})=\frac{1}{8\pi}\frac{|\vec{p}_{K^{+}}|}{m^{2}_{T^{a}_{c\bar{s}0}(2900)^{++}}}\overline{|{\cal{M}}|^{2}},$
(56)

where the $\vec{p}_{K^{+}}$ is the three-momenta of the decay products in the
center of mass frame, the overline indicates the sum over the polarization
vectors of the final hadrons.

## References

  * (1) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189-231 (1985).
  * (2) R. Koniuk and N. Isgur, Phys. Rev. D 21, 1868 (1980) [erratum: Phys. Rev. D 23, 818 (1981)].
  * (3) N. Isgur and G. Karl, Phys. Rev. D 19, 2653 (1979) [erratum: Phys. Rev. D 23, 817 (1981)].
  * (4) B. S. Zou and D. O. Riska, Phys. Rev. Lett. 95, 072001 (2005).
  * (5) D. O. Riska and B. S. Zou, Phys. Lett. B 636, 265-269 (2006).
  * (6) C. S. An, D. O. Riska and B. S. Zou, Phys. Rev. C 73, 035207 (2006).
  * (7) B. S. Zou, Nucl. Phys. A 835, 199-206 (2010).
  * (8) B. S. Zou, Nucl. Phys. A 827, 333C-335C (2009).
  * (9) B. C. Liu and B. S. Zou, Phys. Rev. Lett. 96, 042002 (2006).
  * (10) J. J. Xie, B. S. Zou and H. C. Chiang, Phys. Rev. C 77, 015206 (2008).
  * (11) S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003).
  * (12) R. Aaij et al. [LHCb], Phys. Rev. Lett. 115, 072001 (2015).
  * (13) R. Aaij et al. [LHCb], Phys. Rev. Lett. 117, 082002 (2016).
  * (14) R. Aaij et al. [LHCb], Phys. Rev. Lett. 117, 082003 (2016).
  * (15) R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, 222001 (2019).
  * (16) R. Aaij et al. [LHCb], Sci. Bull. 66, 1278-1287 (2021).
  * (17) R. Aaij et al. [LHCb], Phys. Rev. Lett. 131,031901 (2023).
  * (18) R. Aaij et al. [LHCb], Phys. Rev. Lett. 131, 041902 (2023).
  * (19) R. Chen and Q. Huang, [arXiv:2208.10196 [hep-ph]].
  * (20) Z. L. Yue, C. J. Xiao and D. Y. Chen, Phys. Rev. D 107, 034018 (2023).
  * (21) S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D 107, 094019 (2023).
  * (22) M. Y. Duan, M. L. Du, Z. H. Guo, E. Wang and D. Y. Chen, [arXiv:2307.04092 [hep-ph]].
  * (23) H. W. Ke, Y. F. Shi, X. H. Liu and X. Q. Li, Phys. Rev. D 106,114032 (2022).
  * (24) F. X. Liu, R. H. Ni, X. H. Zhong and Q. Zhao, Phys. Rev. D 107, 096020 (2023).
  * (25) J. Wei, Y. H. Wang, C. S. An and C. R. Deng, Phys. Rev. D 106, 096023 (2022).
  * (26) X. S. Yang, Q. Xin and Z. G. Wang, Int. J. Mod. Phys. A 38, 2350056 (2023).
  * (27) D. K. Lian, W. Chen, H. X. Chen, L. Y. Dai and T. G. Steele, [arXiv:2302.01167 [hep-ph]].
  * (28) C. Jiang, Y. Jin, S. Y. Li, Y. R. Liu and Z. G. Si, Symmetry 15, 695 (2023).
  * (29) V. Dmitrašinović, [arXiv:2301.05471 [hep-ph]].
  * (30) P. G. Ortega, D. R. Entem, F. Fernandez and J. Segovia, [arXiv:2305.14430 [hep-ph]].
  * (31) R. Molina and E. Oset, Phys. Rev. D 107,056015 (2023).
  * (32) Y. H. Ge, X. H. Liu and H. W. Ke, Eur. Phys. J. C 82,955 (2022).
  * (33) V. Obraztsov [OKA], Nucl. Part. Phys. Proc. 273-275, 1330-1333 (2016).
  * (34) B. Velghe [NA62-RK and NA48/2], Nucl. Part. Phys. Proc. 273-275, 2720-2722 (2016).
  * (35) C. Quintans [AMBER], Few Body Syst. 63, 72 (2022).
  * (36) T. Nagae, Nucl. Phys. A 805, 486-493 (2008).
  * (37) Y. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 90, 094001 (2014).
  * (38) H. Zhu and Y. Huang, Phys. Rev. D 100,054031 (2019)..
  * (39) Y. Dong, A. Faessler, T. Gutsche, S. Kumano and V. E. Lyubovitskij, Phys. Rev. D 82, 034035 (2010).
  * (40) M. Y. Duan, E. Wang and D. Y. Chen, [arXiv:2305.09436 [hep-ph]].
  * (41) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022).
  * (42) H. Zhu and Y. Huang, Phys. Rev. D 100, 054031 (2019).
  * (43) J. He, Z. Ouyang, X. Liu and X. Q. Li, Phys. Rev. D 84, 114010 (2011).
  * (44) J. J. Xie, Y. B. Dong and X. Cao, Phys. Rev. D 92,034029 (2015).
  * (45) P. Lebiedowicz and A. Szczurek, Phys. Rev. D 85, 014026 (2012).
  * (46) P. Lebiedowicz, R. Pasechnik and A. Szczurek, Phys. Lett. B 701, 434-444 (2011).
  * (47) G. Pakhlova et al. [Belle], Phys. Rev. D 77, 011103 (2008).
  * (48) B. Aubert et al. [BaBar], Phys. Rev. D 76, 111105 (2007).
  * (49) X. D. Guo, D. Y. Chen, H. W. Ke, X. Liu and X. Q. Li, Phys. Rev. D 93,054009 (2016).
  * (50) R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145-238 (1997).
  * (51) M. Altenbuchinger, L. S. Geng and W. Weise, Phys. Rev. D 89, no.1, 014026 (2014).
  * (52) Y. s. Oh, T. Song and S. H. Lee, Phys. Rev. C 63, 034901 (2001).
  * (53) Z. w. Lin and C. M. Ko, Phys. Rev. C 62, 034903 (2000).
  * (54) R. S. Azevedo and M. Nielsen, Phys. Rev. C 69, 035201 (2004).
  * (55) Y. H. Lin, C. W. Shen, F. K. Guo and B. S. Zou, Phys. Rev. D 95,114017 (2017).
  * (56) Y. Dong, A. Faessler, T. Gutsche, Q. Lü and V. E. Lyubovitskij, Phys. Rev. D 96,074027 (2017).
  * (57) S. H. Kim, S. i. Nam, D. Jido and H. C. Kim, Phys. Rev. D 96,014003 (2017).


# Runaway electron current reconstitution after a non-axisymmetric
magnetohydrodynamic flush

Christopher J. McDevitt  Nuclear Engineering Program, University of Florida,
Gainesville, FL 32611  Xian-Zhu Tang  Theoretical Division, Los Alamos
National Laboratory, Los Alamos, NM 87545

(August 27, 2024)

###### Abstract

Benign termination of mega-ampere (MA) level runaway current has been
convincingly demonstrated in recent JET and DIII-D experiments, establishing
it as a leading candidate for runaway mitigation on ITER. This comes in the
form of a runaway flush by parallel streaming loss along stochastic magnetic
field lines formed by global magnetohydrodynamic instabilities, which are
found to correlate with a low-Z injection that purges the high-Z impurities
from a post-thermal-quench plasma. Here we show the competing physics that
govern the post-flush reconstitution of the runaway current in a ITER-like
reactor where significantly higher current is expected. The trapped “runaways”
are found to dominate the seeding for runaway reconstitution, and the
incomplete purge of high-Z impurities helps drain the seed but produces a more
efficient avalanche, two of which compete to produce a 2-3 MA step in current
drop before runaway reconstitution of the plasma current.

The generation and evolution of runaway electrons (RE) have been extensively
studied in a variety of contexts including atmospheric plasmas Dwyer and Uman
(2014), solar flares Holman (1985); Aschwanden (2002) and magnetic fusion
devices Knoepfel and Spong (1979). These highly relativistic electrons have
recently emerged as a topic of particular interest and importance to the
magnetic fusion community. This is due to the possibility that a large
population of REs may be inadvertently generated during a tokamak disruption
Hender et al. (2007a). Due to their high energy, often in excess of
$10\;\text{MeV}$, such electrons have the potential to impart substantial
damage to plasma facing components (PFC) Hender et al. (2007a).

A major step toward mitigating the threat posed by REs has recently been
taken. Specifically, recent experiments on JET Reux et al. (2021) and DIII-D
Paz-Soldan et al. (2021) have shown that (1) massive injection of deuterium
into a post-thermal-quench plasma can purge the high-Z impurities and
subsequently trigger large scale magnetohydrodynamic (MHD) modes, (2)
independent of the specific MHD modes involved, which are different in DIII-D
and JET experiments, parallel streaming loss in the resulting stochastic
magnetic field lines leads to the expulsion of a preformed beam of REs, and
(3) the spread of the escaping runaways on the PFCs is sufficiently broad such
that no appreciable localized heat load is observed. A most striking feature
of this scheme is its compatibility with a thermal quench in which plasma
energy loss is dominated by impurity radiation. The high-Z impurities could be
introduced into the plasma accidentally, for example, in the form of a
tungsten flake, or deliberately, for example, through pellet injection, to
mitigate the thermal load on the PFC. The byproduct of such strong radiative
cooling is a robust Ohmic-to-runaway current conversion. On ITER, a 15 mega-
ampere (MA) plasma current discharge could produce a post-thermal-quench
plasma of over 10 MA RE current on Disrup MHD and Editors (1999); Hender et
al. (2007b); Martín-Solís et al. (2017); Vallhagen et al. (2020). Safely
terminating such a large runaway current has been a particularly difficult
challenge, for which the 3D MHD flush of REs associated with high-Z impurity
purge by massive deuterium injection offers an attractive solution.

An issue that is anticipated, but has not materialized with certainty in
experiments to date, is the RE current reconstitution after the spontaneous
expulsion of REs by the 3D magnetic fields. As long as the flux surfaces
reheal after the self-excited 3D MHD event, the Ohmic plasma current after the
runaway flush is similarly susceptible to Ohmic-to-runaway conversion, just
like the plasma after the initial radiative thermal quench. One difference is
the reduced impurity content after the purge due to deuterium injection. If
the purge is sufficiently complete and the remnant deuterium density is not
too high, Ohmic heating can offset the radiative and transport losses, and
reheats the plasma so the parallel electric field $E_{\parallel\eta}=\eta
j_{\|}$ can drop below the runaway avalanche threshold $E_{AV}.$ If this could
be maintained over the rest of the current quench, effective runaway
“avoidance” would have been achieved. The primary challenge in that scenario
becomes a goldilock requirement on ion densities such that radiative cooling
and Ohmic heating would offset each other to lock the plasma to a temperature
McDevitt et al. (2022) that is consistent with an Ohmic current decay time in
the range of 50-150 ms for ITER. Lehnen et al. (2015); Hollmann et al. (2015)

In the more conservative and perhaps more likely scenario that the impurity
purge is inadequately complete and electron reheating is insufficient to reach
$E_{\parallel\eta}<E_{AV},$ the same avalanche physics can drive runaway
current reconstitution. The key question becomes how much the plasma current
would drop before another Ohmic-to-runaway current conversion is completed.
The answer to this question would dictate the issues one must face in the
post-flush mitigation designs.

This Letter lays out the fundamental physics considerations underlying the
answer to the question of runaway reconstitution after an MHD flush, which are
of critical importance to a tokamak reactor like ITER. The interesting finding
is that the runaway current reconstitution follows the same avalanche growth
physics as the initial runaway current formation, but the runaway seeding
takes place via a new route that makes the runaway reconstitution a far more
robust process than the initial runaway plateau formation immediately
following the plasma thermal quench. The new feature is the “trapped runaway”
population in the runaway plateau phase that is greatly enhanced by the high-Z
impurities before their purge by massive deuterium injection. In a mitigated
post-TQ plasma, the electron temperature is radiatively clamped to a very low
temperature, possibly in the few eV range. It is due to the partial screening
effect that weakly ionized high-atomic-number impurities can efficiently
scatter the (passing) runaways into the trapped region. The radial loss of
trapped electrons thus formed, in sharp contrast to that of passing runaways,
are insensitive to the stochastic magnetic fields in an MHD event.
Furthermore, because of their high energies, the trapped runaways are
resilient against both collisional slowing down and collisional detrapping
during the transition period from the purge to the eventual flux surface
rehealing. This plants the seed for a robust runaway reconstitution of the
plasma current once the flux surfaces are rehealed after the 3D MHD event.

In a post-flush plasma with $E_{\parallel\eta}>E_{AV},$ the runaway current
reconstitution follows the same avalanche physics as during the initial
formation of the RE plateau, i.e.

$I^{(max)}_{RE}=2\pi\int^{a}_{0}drrj^{\left(seed\right)}_{RE}\left(r\right)10^{\left|\psi\left(r\right)/\psi_{10}\left(r\right)\right|},$
(1)

which we have written for a simple geometry with circular flux surfaces, $r$
is the radial variable, and $a$ is the minor radius. The amount of poloidal
flux required for an order of magnitude increase in runaway population is
labeled as $\psi_{10},$ itself a function of minor radius $r.$ Eq. (1)
expresses the amount of RE current that could be generated if all of the
available poloidal flux were used to amplify the RE seed, and thus corresponds
to an upper bound on the amount of RE current that can be generated for a
given RE seed. To minimize runaway current reconstitution, one aims for a
higher flux consumption rate $\psi_{10}(r)$ and a smaller runaway seed
$j_{RE}^{(seed)}.$ A post-flush plasma appears to be favorable on both
accounts: (1) reduced high-Z impurity density due to its purge by hydrogen
injection decreases $\psi_{10};$ (2) a lower $T_{e}$ implies ineffectiveness
of both the Dreicer flux and hot tail formation in seeding the runaways, which
points to the most optimistic scenario in which tritium decay and Compton
scattering set the minimal runaway seeding in the nuclear phase of ITER. The
reality tilts toward the opposite, and the spoiler turns out to be magnetic
trapping of high energy electrons.

Figure 1: Average loss time of electrons in an imposed 3D magnetic field. The
dashed red lines indicate the location of the trapped-passing boundary. The
imposed perturbation has the form
$\delta\mathbf{B}=\nabla\times\left(\alpha\mathbf{B}_{eq}\right)$, where
$\mathbf{B}_{eq}$ is the equilibrium magnetic field,
$\alpha=\sum_{m,n}\alpha_{m,n}\left(r\right)\cos\left(m\theta-n\varphi+\delta_{m,n}\right)$,
with $n=\left[1,6\right]$, $m=\left[1,24\right]$, and the magnitude of
$\alpha_{m,n}\left(r\right)$ was increased until a magnetic field without any
detectable integrable regions was achieved. The electrons were all initialized
at $r/a=0.5$. The plasma parameters were taken to be $n_{D}=5\times
10^{20}\;\text{m}^{-3}$, $E/E_{c}=50$, an ITER like plasma with
$B_{0}=5.3\;\text{T}$ and minor radius of $a=200\;\text{cm}$ was assumed.

Magnetic trapping in a stochastic field.— Fig. 1 shows the average loss time
of electrons as a function of their phase space location in a globally
stochastic magnetic field. Here the electron losses are either through spatial
transport to the vessel wall or collisional slowing down to the bulk plasma.
It is evident that passing electrons with $\left|\xi\right|\gtrsim 0.5$ are
rapidly lost along the open magnetic field lines, but electrons initially
located within the trapped region of momentum space can remain confined for a
far longer period of time. This is due to magnetically trapped electrons only
following a magnetic field line for a short distance before being reflected
and retracing the same magnetic field line, thus sharply limiting their
spatial transport. The loss time for such magnetically trapped electrons is
determined by their detrapping rate. Since the collisional detrapping rate
decreases rapidly with the electron’s energy, relativistic trapped electrons
can remain confined in the plasma for an extended period, even for a fully
stochastic magnetic field. This population thus provides a remnant seed
capable of surviving global MHD instabilities. A primary aim of this work will
be to evaluate the magnitude and decay rate of this seed across a range of
plasma conditions, thus providing guidance on what experimental conditions
must be achieved to reduce this seed to a specific level.

Self-Consistent RE Evolution.— Our simulation model includes a drift kinetic
description of runaway electron evolution McDevitt et al. (2019a), a power
balance equation, and a flux diffusion equation, all in a toroidal plasma with
nested circular flux surfaces. Specifically, for the poloidal magnetic flux
$\psi,$

$\left.\frac{\partial\psi}{\partial
t}\right|_{r}=\frac{\eta}{\mu_{0}}\frac{1}{\left\langle
R^{-2}\right\rangle}\frac{1}{r}\frac{\partial}{\partial r}\left[r\left\langle
R^{-2}\right\rangle\frac{\partial\psi}{\partial
r}\right]-\frac{\eta}{B_{0}R_{0}}\frac{\left\langle\mathbf{j}_{RE}\cdot\mathbf{B}\right\rangle}{\left\langle
R^{-2}\right\rangle},$ (2)

with $\eta$ the background plasma resistivity and
$\left\langle\cdots\right\rangle$ a flux surface average. Ohm’s law takes the
form $E_{\|}=\eta\left(j_{\|}-j_{RE}\right)$ with $j_{RE}$ the runaway current
density. A conducting wall boundary condition $\psi(r=a)=0$ is imposed. The
power balance for background plasma follows

$\frac{3}{2}\frac{\partial p}{\partial t}=\frac{1}{r}\frac{\partial}{\partial
r}\left(rn\chi\frac{\partial T_{e}}{\partial
r}\right)-S_{rad}\left(n_{e},T_{e}\right)+S_{RE}+\frac{E^{2}_{\|}}{\eta}.$ (3)

Here electrons, ions and neutrals are assumed to have the same temperature,
$p=n_{e}T+T\sum_{j}n_{j}$ is the total background plasma pressure with the sum
over all ion and neutral species, $\chi$ is the heat diffusivity taken to be
$\chi=1\;\text{m}^{2}/\text{s}$, $S_{rad}$ describes radiative losses,
$S_{RE}$ is the energy gained by the bulk plasma due to REs slowing down
against free electrons, and the last term describes Ohmic heating. The charge
state and $S_{rad}$ are evaluated using data generated by the collisional
radiative code FlyChk Chung et al. (2005) under the assumption of steady
state.

Simulation set up.—The first step of the simulation study is the preparation
of a runaway current plateau. This is obtained through an idealized thermal
quench that imposes a cooling history of
$T(r,t)=\left[T_{init}\left(r\right)-T_{final}\left(r\right)\right]\exp\left(-t/\Delta
t_{TQ}\right)+T_{final}\left(r\right)$, with the final temperature
$T_{final}\left(r\right)=T_{f}\left[1-0.7\left(r/a\right)^{2}\right]$ and
$T_{f}$ the on-axis temperature after the thermal quench. The initial
temperature and density profiles follow
$T_{init}\left(r\right)=T_{0}\left[1-0.7\left(r/a\right)^{2}\right]^{2}$ and
$n_{e,init}\left(r\right)=n_{D,init}\left(r\right)=n_{D0}\left[1-0.9\left(r/a\right)^{2}\right]^{2/3}$,
where $T_{0}$ and $n_{D0}$ are the on-axis temperature and deuterium density,
respectively.

For all cases considered, we will assume
$T_{0}=3.1\;\text{keV},T_{f}=10\;\text{eV}$, and $n_{D0}=2.8\times
10^{13}\;\text{cm}^{-3}$. Once the hot tail seed has formed, it is amplified
by the avalanche mechanism leading to the formation of a RE plateau, where the
amount of RE current is controlled by varying the amount of initial plasma
current. After the Ohmic to RE current conversion is complete, large
quantities of material are injected into the plasma at two different times.
During the first injection, a variable amount of neon is injected into the RE
beam. By injecting the neon into an existing RE beam, we have the freedom to
vary the amount of RE current and injected neon independently.

Once the neon enters the plasma, the temperature is evaluated via the power
balance equation [Eq. (3)]. A second injection composed entirely of deuterium
is made later in the simulation, resulting in a factor of ten increase in the
deuterium density. Shortly after the deuterium injection, a fraction
$f_{purge}$ of the initial neon is removed from the plasma in order to
describe the purge of high-Z material, a phenomenon observed across a range of
experiments Shiraki et al. (2018); Pautasso et al. (2020); Hollmann et al.
(2020); Reux et al. (2022).

Figure 2: (a) The evolution of plasma current versus time. The solid curve
indicates the total current, whereas the dashed curved indicated the non-
thermal current. Time slices of the momentum distribution of REs before
[$t-t_{TQ}\approx 33\;\text{ms}$, panel (b)], during [$t-t_{TQ}\approx
39\;\text{ms}$, panel (c)] and after [$t-t_{TQ}\approx 47\;\text{ms}$, panel
(c)] the MHD instability. The plasma was assumed to initially carry
$I_{p}\approx 8\;\text{MA}$, minor radius $a=200\;\text{cm}$, an on-axis
magnetic field $B_{0}=3\;\text{T}$, $f_{purge}=0.9$, and $\Delta
t_{open}\approx 1.43\;\text{ms}$.

Runaway Flush & Remnant Seed.—The RE plateau thus formed provides the initial
condition for investigating the 3D MHD flush of runaways. Consistent with the
toroidally-averaged formulation of Eq. (2), we model the enhanced loss of
runaways in a globally stochastic magnetic field by a spatial diffusivity of
the form,

$D^{kin}_{RE}\left(\gamma,\xi,r,\theta\right)=\frac{\left|v_{\|}\right|}{c}\Theta\left[\left|\xi\right|-\xi_{trap}\left(r,\theta\right)\right]D_{RE},$
(4)

where $\xi_{trap}\left(r,\theta\right)=1-B\left(r,\theta\right)/B_{Max}$,
$B\left(r,\theta\right)$ is the magnetic field strength at the electron’s
current location, $B_{Max}$ is the maximum value of the magnetic field on the
flux surface, and
$D_{RE}=\left(1/4\right)\left(1/N_{t}\right)\left(a^{2}/\tau_{transit}\right)$.
Here, $\tau_{transit}=2\pi R_{0}/c$ is the time it takes a relativistic
electron to make one toroidal transit, $N_{transit}$ is the number of toroidal
transits made by a stochastic magnetic field line while traversing the plasma,
where this latter quantity is used to parameterize the strength of the spatial
transport. In order to describe the increase in RE transport induced by the
large, though transient MHD instability, the diffusivity will be assumed to
follow
$D_{RE}\left(t\right)=D^{kin}_{RE}\exp\left[-\left(t-t_{open}\right)^{2}/\Delta
t^{2}_{open}\right]$, where $t_{open}$ is the time at which the RE transport
is the largest, and $\Delta t_{open}$ sets the duration of the MHD event.

An example of the impact of a global MHD instability on the phase space
distribution of REs is shown in Fig. 2. Here, the MHD instability reaches its
peak amplitude at $t\approx 35.8\;\text{ms}$, resulting in the loss of nearly
the entire RE current. From Figs. 2(b) and 2(c), it is evident that the
expulsion of the relativistic electron population is not complete. In
particular, a significant number of trapped relativistic electrons remain
confined, providing a seed RE population after the MHD instability ceases.
This relativistic population of trapped electrons emerges due to the strong
pitch-angle scattering Hesslow et al. (2017) coinciding with the presence of a
high-Z material such as neon during the current plateau McDevitt et al.
(2019b); Liu et al. (2021). Once the flux surfaces reheal, the detrapping of
this remnant population of trapped electrons (typically less than one percent
of the initial RE population), allows for a sizable RE seed to robustly form
[see Fig. 2(d)]. This seed is subsequently amplified by the avalanche
mechanism allowing for the partial reformation of the RE plateau. In the
following we will identify key parameters that influence $j^{(seed)}_{RE}$ and
$\psi_{10}$, and thus runaway reconstitution.

Figure 3: Panel (a) shows the evolution of the number of energetic electrons
for $\Delta t_{open}\approx 1.79\times 10^{-4}\;\text{s}$ (blue curve) and
$\Delta t_{open}\approx 1.43\times 10^{-3}\;\text{s}$ (green curve). Panel (b)
shows the fraction of surviving RE electrons versus the duration of the MHD
instability $\Delta t_{open}$ normalized to the transport time scale
$\tau_{trans}\equiv a^{2}/D_{RE}$. The other parameters are $I_{p}\approx
8\;\text{MA}$, minor radius $a\approx 200\;\text{cm}$, $B_{0}=3\;\text{T}$,
$n_{Ne}=2n_{D0}$, and $f_{purge}=0.9$.

First assessing the impact of the duration of the MHD instability $\Delta
t_{open}$ on the size of the remnant seed population, Fig. 3(a) shows the
number of energetic electrons that remain confined in the plasma for two
different values of $\Delta t_{open}$. The shortest value of $\Delta t_{open}$
is chosen to be comparable to the transport time scale
$\tau_{trans}=a^{2}/D_{RE}$ of REs by the 3D magnetic field. For this case, it
is evident that the RE population drops sharply during the period of enhanced
transport, which is a result of the rapid loss of passing REs. In contrast,
considering a case where $\Delta t_{open}\gg\tau_{trans}$, the number of REs
drops sharply initially, but then transitions to a period with a far slower
decay rate. These two different decay rates are linked to the different rates
that passing and trapped relativistic electrons are lost in a stochastic
magnetic field. In particular, while passing electrons can be directly lost
via free streaming along open magnetic field lines, trapped electrons are not
directly lost. Instead magnetically trapped electrons must first be detrapped,
before they may exit the plasma via streaming along open magnetic field lines.
For a relativistic electron population, this detrapping rate is far slower
compared to the time scale associated with transport along open magnetic field
lines, thus resulting in the slow decay rate of the RE population evident in
Fig. 3(a). This transition from a rapid decay rate for $\Delta
t_{open}\lesssim\tau_{trans}$ to a much slower decay rate for $\Delta
t_{open}\gg\tau_{trans}$ is illustrated in Fig. 3(b).

Figure 4: Evolution of the plasma current [panel (a)] and the inferred
$\psi_{10}$ for different purge fractions $f_{purge}$. The blue curves are for
$f_{purge}=0.6$, the black curves are for $f_{purge}=0.9$, and the blue curves
are for $f_{purge}=0.975$. The other parameters are, $I_{p}\approx
8\;\text{MA}$, minor radius $a\approx 200\;\text{cm}$, $B_{0}=3\;\text{T}$,
$n_{Ne}=2n_{D0}$, and $\Delta t_{open}\approx 1.43\;\text{ms}$.

Runaway Reconstitution.—We will also be interested in identifying the impact
of the amount of neon remaining in the plasma after the deuterium injection.
Three cases with different values of $f_{purge}$ are indicated in Fig. 4.
Here, it is apparent that in all three cases a similar amount of RE current is
able to reform. The origin of this apparent insensitivity is the result of two
partially offsetting physical processes. The first is that the amount of
impurities that remain in the plasma increases the detrapping rate of the
remnant population by two distinct mechanisms. The first is due to the higher
neon content increasing the collisional detrapping rate. A more subtle
mechanism is due to the larger inductive electric field present in the higher
neon density case, due to the plasma being radiatively pinned to a lower
temperature. This larger inductive electric field induces a Ware pinch of the
trapped electron population, which convects the electrons inward where they
are more easily detrapped. Hence, the retention of neon in the plasma leads to
a reduction of the remnant seed population. In contrast, the presence of neon
increases the efficiency of the avalanche mechanism. This trend is shown in
Fig. 4(b), where as the quantity of neon is increased, the value of
$\psi_{10}$ decreases, implying less poloidal flux is required to increase the
runaways by an order of magnitude. For the present example, these competing
effects largely offset one another, yielding a weaker than expected
sensitivity to $f_{purge}$.

Figure 5: Current evolution [panel (a)] and the size of the plasma current
drop versus the time scale the flux surfaces remain open [panel (b)] plotted
on a linear scale. The curves correspond to $\Delta t_{open}\approx
0.09\;\text{ms}$ (blue curve), $\Delta t_{open}\approx 0.36\;\text{ms}$ (black
curve), and $\Delta t_{open}\approx 1.43\;\text{ms}$ (cyan curve). The initial
Neon density was $n_{Ne0}=2n_{D0}=5.6\times 10^{13}\;\text{cm}^{-3}$,
$a=200\;\text{cm}$, $B_{0}=3\;\text{T}$ and $\tau_{trans}\equiv
a^{2}/D_{RE}\approx 0.26\;\text{ms}$.

Finally, we will investigate the impact of the duration of the MHD event on
runaway reconstitution. The amount of reformed RE current for three different
values of $\Delta t_{open}$ is shown in Fig. 5(a). When increasing $\Delta
t_{open}$ from $0.09\;\text{ms}$ to $1.43\;\text{ms}$, the amount of reformed
RE current is observed to decrease from $3.9\;\text{MA}$ to $2.0\;\text{MA}$.
Hence, while the time scale that the flux surfaces remain open is increased by
a factor of sixteen, the amount of reformed RE current is only reduced by
roughly a factor of two. This relative insensitivity is due to the slow decay
rate of the trapped remnant energetic electron population, along with the
exponential dependence of the avalanche amplification mechanism on the amount
of poloidal flux consumed. The resulting drop in RE current as a function of
$\Delta t_{open}$ is shown in Fig. 5(b), where it is apparent that the current
drop becomes a weak function of $\Delta t_{open}$ once $\Delta
t_{open}\gg\tau_{trans}$.

Discussion. Runaway reconstitution after a 3D MHD flush is surprisingly robust
because of a new seeding mechanism via “trapped runaways.” While the size of
this seed varies with the amount of neon initially injected into the plasma,
the purge fraction, and the time ($\Delta t_{open}$) that flux surfaces remain
open, its decay is ultimately set by the detrapping from a combination of
pitch-angle scattering and the Ware pinch. For the range of parameters
considered here, the magnitude of this trapped remnant seed ranged from
roughly 1% to 0.1% of the initial RE population. For most cases of interest
$\psi_{10}/\mu_{0}R_{0}\lesssim 1\;\text{MA}$, so runaway reconstitution comes
with a 2-3 MA plasma current drop. Impeding runaway reconstitution depends on
(1) enhancing the detrapping rate and (2) prolonging $\Delta t_{open}.$
Increasing the post-purge $n_{D}$ is a straightforward approach for (1),
although it is constrained by the assimilation of injected deuterium into the
plasma and the density window Paz-Soldan et al. (2021); Battey et al. (2022)
identified for triggering the 3D MHD flush. Experimental studies reveal
significant variation in $\Delta t_{open}$ Paz-Soldan et al. (2021), but the
precise control knob remains to be understood. A more reliable alternative is
a passive runaway coil Boozer (2011); Weisberg et al. (2021); Tinguely et al.
(2021) that could hold the flux surfaces open for a far longer time period.

###### Acknowledgements.

We thank the U.S. Department of Energy Office of Fusion Energy Sciences and
Office of Advanced Scientific Computing Research for support under the Tokamak
Disruption Simulation (TDS) Scientific Discovery through Advanced Computing
(SciDAC) project, and the Base Theory Program, both at Los Alamos National
Laboratory (LANL) under contract No. 89233218CNA000001. This research used
resources of the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy Office of Science User Facility operated under
Contract No. DE-AC02-05CH11231 and the Los Alamos National Laboratory
Institutional Computing Program, which is supported by the U.S. Department of
Energy National Nuclear Security Administration under Contract No.
89233218CNA000001.

## References

  * Dwyer and Uman (2014) J. R. Dwyer and M. A. Uman, Physics Reports 534, 147 (2014).
  * Holman (1985) G. D. Holman, in _Symposium-International Astronomical Union_ (Cambridge University Press, 1985), vol. 107, pp. 191–196.
  * Aschwanden (2002) M. J. Aschwanden, Space Science Reviews 101, 1 (2002).
  * Knoepfel and Spong (1979) H. Knoepfel and D. Spong, Nuclear Fusion 19, 785 (1979).
  * Hender et al. (2007a) T. Hender, J. Wesley, J. Bialek, A. Bondeson, A. Boozer, R. Buttery, A. Garofalo, T. Goodman, R. Granetz, Y. Gribov, et al., Nuclear fusion 47, S128 (2007a).
  * Reux et al. (2021) C. Reux, C. Paz-Soldan, P. Aleynikov, V. Bandaru, O. Ficker, S. Silburn, M. Hoelzl, S. Jachmich, N. Eidietis, M. Lehnen, et al., Physical Review Letters 126, 175001 (2021).
  * Paz-Soldan et al. (2021) C. Paz-Soldan, C. Reux, K. Aleynikova, P. Aleynikov, V. Bandaru, M. Beidler, N. Eidietis, Y. Liu, C. Liu, A. Lvovskiy, et al., Nuclear Fusion 61, 116058 (2021).
  * on Disrup MHD and Editors (1999) I. P. E. G. on Disrup MHD and I. P. B. Editors, Nuclear Fusion 39, 2251 (1999), URL https://doi.org/10.1088/0029-5515/39/12/303.
  * Hender et al. (2007b) T. Hender, J. Wesley, J. Bialek, A. Bondeson, A. Boozer, R. Buttery, A. Garofalo, T. Goodman, R. Granetz, Y. Gribov, et al., Nuclear Fusion 47, S128 (2007b), URL http://stacks.iop.org/0029-5515/47/i=6/a=S03.
  * Martín-Solís et al. (2017) J. Martín-Solís, A. Loarte, and M. Lehnen, Nuclear Fusion 57, 066025 (2017).
  * Vallhagen et al. (2020) O. Vallhagen, O. Embreus, I. Pusztai, L. Hesslow, and T. Fülöp, Journal of Plasma Physics 86, 475860401 (2020).
  * McDevitt et al. (2022) C. McDevitt, X.-Z. Tang, C. Fontes, P. Sharma, and H. Chung, arXiv preprint arXiv:2210.10925 (2022).
  * Lehnen et al. (2015) M. Lehnen, K. Aleynikova, P. Aleynikov, D. Campbell, P. Drewelow, N. Eidietis, Y. Gasparyan, R. Granetz, Y. Gribov, N. Hartmann, et al., Journal of Nuclear materials 463, 39 (2015).
  * Hollmann et al. (2015) E. M. Hollmann, P. B. Aleynikov, T. Fülöp, D. A. Humphreys, V. A. Izzo, M. Lehnen, V. E. Lukash, G. Papp, G. Pautasso, F. Saint-Laurent, et al., Physics of Plasmas 22, 021802 (2015), eprint https://doi.org/10.1063/1.4901251, URL https://doi.org/10.1063/1.4901251.
  * McDevitt et al. (2019a) C. J. McDevitt, Z. Guo, and X. Z. Tang, Plasma Physics and Controlled Fusion 61, 054008 (2019a).
  * Chung et al. (2005) H.-K. Chung, M. Chen, W. Morgan, Y. Ralchenko, and R. Lee, High energy density physics 1, 3 (2005).
  * Shiraki et al. (2018) D. Shiraki, N. Commaux, L. Baylor, C. Cooper, N. Eidietis, E. Hollmann, C. Paz-Soldan, S. Combs, and S. Meitner, Nuclear Fusion 58, 056006 (2018).
  * Pautasso et al. (2020) G. Pautasso, M. Dibon, M. Dunne, R. Dux, E. Fable, P. Lang, O. Linder, A. Mlynek, G. Papp, M. Bernert, et al., Nuclear Fusion 60, 086011 (2020).
  * Hollmann et al. (2020) E. M. Hollmann, I. Bykov, N. Eidietis, J. Herfindal, A. Lvovskiy, R. Moyer, P. Parks, C. Paz-Soldan, A. Y. Pigarov, D. Rudakov, et al., Physics of Plasmas 27, 042515 (2020).
  * Reux et al. (2022) C. Reux, C. Paz-Soldan, N. W. Eidietis, M. Lehnen, P. Aleynikov, S. A. Silburn, V. K. Bandaru, O. Ficker, M. Hoelzl, E. M. Hollmann, et al., Plasma Physics and Controlled Fusion (2022).
  * Hesslow et al. (2017) L. Hesslow, O. Embréus, A. Stahl, T. C. DuBois, G. Papp, S. L. Newton, and T. Fülöp, Phys. Rev. Lett. 118, 255001 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.118.255001.
  * McDevitt et al. (2019b) C. J. McDevitt, Z. Guo, and X.-Z. Tang, Plasma Physics and Controlled Fusion 61, 024004 (2019b).
  * Liu et al. (2021) C. Liu, D. P. Brennan, A. Lvovskiy, C. Paz-Soldan, E. D. Fredrickson, and A. Bhattacharjee, Nuclear Fusion 61, 036011 (2021).
  * Battey et al. (2022) A. Battey et al., _Passive and active re mitigation tools under development at diii-d_ , 9th Runaway Electron Modelling (REM) meeting (2022), URL https://ft.nephy.chalmers.se/?p=confagenda&id=3.
  * Boozer (2011) A. H. Boozer, Plasma Physics and Controlled Fusion 53, 084002 (2011).
  * Weisberg et al. (2021) D. B. Weisberg, C. Paz-Soldan, Y. Liu, A. Welander, and C. Dunn, Nuclear Fusion 61, 106033 (2021).
  * Tinguely et al. (2021) R. Tinguely, V. Izzo, D. Garnier, A. Sundström, K. Särkimäki, O. Embréus, T. Fülöp, R. Granetz, M. Hoppe, I. Pusztai, et al., Nuclear Fusion 61, 124003 (2021).