[backgroundcolor=gray!20, hidealllines=true]  Category: analogy Perturbation
Type: FALSE PREMISE Target: A strong defense is essential for success in both
football and military. Gold Premises: A football team needs a strong defense
to win games. A military needs a strong defense to win battles. Rank: 1 Rank
(G) 1: a football team needs a strong defense to win games. a military needs a
strong defense to win battles. Rank 2: a football team needs a strong defense
to win games. all militaries without a strong defense always win battles. Rank
3: a military needs a strong defense to win battles. football teams with weak
defenses always win games. Rank 4: all militaries without a strong defense
always win battles. football teams with weak defenses always win games.

[backgroundcolor=gray!20, hidealllines=true]  Category: analogy Perturbation
Type: IRRELEVANT FACT Target: A strong defense is essential for success in
both football and military. Gold Premises: A football team needs a strong
defense to win games. A military needs a strong defense to win battles. Rank:
1 Rank (G) 1: a military needs a strong defense to win battles. a football
team needs a strong defense to win games. Rank 2: a football team needs a
strong defense to win games. military commanders who prefer coffee over tea
are more successful in battle. Rank 3: a football team needs a strong defense
to win games. military units that have a team song tend to have higher morale.
Rank 4: military generals who have mustaches win more battles. a football team
needs a strong defense to win games. Rank 5: a military needs a strong defense
to win battles. football teams that wear black uniforms win more games. Rank
6: a military needs a strong defense to win battles. football teams that have
a team mascot tend to win more games. Rank 7: a military needs a strong
defense to win battles. football teams with more instagram followers win more
games. Rank 8: football teams that wear black uniforms win more games.
military commanders who prefer coffee over tea are more successful in battle.
Rank 9: football teams that wear black uniforms win more games. military units
that have a team song tend to have higher morale. Rank 10: football teams that
have a team mascot tend to win more games. military commanders who prefer
coffee over tea are more successful in battle. Rank 11: football teams with
more instagram followers win more games. military units that have a team song
tend to have higher morale. Rank 12: football teams with more instagram
followers win more games. military commanders who prefer coffee over tea are
more successful in battle. Rank 13: football teams that wear black uniforms
win more games. military generals who have mustaches win more battles. Rank
14: football teams that have a team mascot tend to win more games. military
units that have a team song tend to have higher morale. Rank 15: military
generals who have mustaches win more battles. football teams that have a team
mascot tend to win more games. Rank 16: football teams with more instagram
followers win more games. military generals who have mustaches win more
battles.

[backgroundcolor=gray!20, hidealllines=true]  Category: analogy Perturbation
Type: Incorrect Quantifier Target: A strong defense is essential for success
in both football and military. Gold Premises: A football team needs a strong
defense to win games. A military needs a strong defense to win battles. Rank:
1 Rank (G) 1: a military needs a strong defense to win battles. a football
team needs a strong defense to win games. Rank 2: a military needs a strong
defense to win battles. some football teams need a strong defense to win
games. Rank 3: a football team needs a strong defense to win games. none of
the militaries need a strong defense to win battles. Rank 4: none of the
militaries need a strong defense to win battles. some football teams need a
strong defense to win games.

[backgroundcolor=gray!20, hidealllines=true]  Category: analogy Perturbation
Type: NEGATED Target: A strong defense is essential for success in both
football and military. Gold Premises: A football team needs a strong defense
to win games. A military needs a strong defense to win battles. Rank: 1 Rank
(G) 1: a football team needs a strong defense to win games. a military needs a
strong defense to win battles. Rank 2: not all militaries need a strong
defense to win battles. a football team needs a strong defense to win games.
Rank 3: a military needs a strong defense to win battles. not all football
teams need a strong defense to win games. Rank 4: not all militaries need a
strong defense to win battles. not all football teams need a strong defense to
win games.

[backgroundcolor=gray!20, hidealllines=true]  Category: categorical syllogism
Perturbation Type: FALSE PREMISE Target: Whiskers is an animal. Gold Premises:
All cats are animals. Whiskers is a cat. Rank: 1 Rank (G) 1: all cats are
animals. whiskers is a cat. Rank 2: some cats are not mammals. whiskers is a
cat.

[backgroundcolor=gray!20, hidealllines=true]  Category: categorical syllogism
Perturbation Type: IRRELEVANT FACT Target: Whiskers is an animal. Gold
Premises: All cats are animals. Whiskers is a cat. Rank: 4 Rank 1: whiskers
has a white spot on its paw. whiskers is a cat. Rank 2: whiskers is a siamese
cat. whiskers is a cat. Rank 3: whiskers is a cat. whiskers has a collar with
a bell on it. Rank (G) 4: all cats are animals. whiskers is a cat. Rank 5:
whiskers is a siamese cat. whiskers is a rescue cat. Rank 6: whiskers has a
white spot on its paw. whiskers is a rescue cat. Rank 7: whiskers is a siamese
cat. whiskers likes to climb trees. Rank 8: whiskers is a rescue cat. whiskers
has a collar with a bell on it. Rank 9: whiskers is a siamese cat. whiskers
likes to sleep in the sun. Rank 10: whiskers has a white spot on its paw.
whiskers likes to climb trees. Rank 11: all cats are animals. whiskers is a
rescue cat. Rank 12: whiskers likes to sleep in the sun. whiskers has a white
spot on its paw. Rank 13: whiskers likes to climb trees. whiskers has a collar
with a bell on it. Rank 14: whiskers likes to sleep in the sun. whiskers has a
collar with a bell on it. Rank 15: all cats are animals. whiskers likes to
climb trees. Rank 16: whiskers likes to sleep in the sun. all cats are
animals.

[backgroundcolor=gray!20, hidealllines=true]  Category: categorical syllogism
Perturbation Type: Incorrect Quantifier Target: Whiskers is an animal. Gold
Premises: All cats are animals. Whiskers is a cat. Rank: 1 Rank (G) 1:
whiskers is a cat. all cats are animals. Rank 2: whiskers is a cat. some cats
are mammals. Rank 3: some cats are mammals. whiskers is not a cat.

[backgroundcolor=gray!20, hidealllines=true]  Category: categorical syllogism
Perturbation Type: NEGATED Target: Whiskers is an animal. Gold Premises: All
cats are animals. Whiskers is a cat. Rank: 1 Rank (G) 1: all cats are animals.
whiskers is a cat. Rank 2: whiskers is a cat. some cats are not animals. Rank
3: all cats are animals. whiskers is not a cat. Rank 4: whiskers is not a cat.
some cats are not animals.

[backgroundcolor=gray!20, hidealllines=true]  Category: causal reasoning
Perturbation Type: FALSE PREMISE Target: John may become obese. Gold Premises:
Eating too much junk food causes obesity. John eats a lot of junk food. Rank:
3 Rank 1: eating too much healthy food causes obesity. john eats a lot of junk
food. Rank 2: eating too much junk food causes obesity. john eats a lot of
healthy food. Rank (G) 3: eating too much junk food causes obesity. john eats
a lot of junk food. Rank 4: eating too much healthy food causes obesity. john
eats a lot of healthy food.

[backgroundcolor=gray!20, hidealllines=true]  Category: causal reasoning
Perturbation Type: IRRELEVANT FACT Target: John may become obese. Gold
Premises: Eating too much junk food causes obesity. John eats a lot of junk
food. Rank: 1 Rank (G) 1: john eats a lot of junk food. eating too much junk
food causes obesity. Rank 2: john eats a lot of junk food. john wears glasses.
Rank 3: john is allergic to peanuts. john eats a lot of junk food. Rank 4:
john eats a lot of junk food. john has a cat named whiskers. Rank 5: junk food
is often advertised during sporting events. john wears glasses. Rank 6: junk
food is often sold in convenience stores. john wears glasses. Rank 7: junk
food is often sold in brightly colored packaging. john wears glasses. Rank 8:
junk food is often sold in brightly colored packaging. john is allergic to
peanuts. Rank 9: john is allergic to peanuts. junk food is often advertised
during sporting events. Rank 10: john has a cat named whiskers. junk food is
often advertised during sporting events. Rank 11: junk food is often sold in
convenience stores. john is allergic to peanuts. Rank 12: junk food is often
sold in brightly colored packaging. john has a cat named whiskers. Rank 13:
junk food is often sold in convenience stores. john has a cat named whiskers.
Rank 14: eating too much junk food causes obesity. junk food is often
advertised during sporting events. Rank 15: junk food is often sold in
brightly colored packaging. eating too much junk food causes obesity. Rank 16:
junk food is often sold in convenience stores. eating too much junk food
causes obesity.

[backgroundcolor=gray!20, hidealllines=true]  Category: causal reasoning
Perturbation Type: Incorrect Quantifier Target: John may become obese. Gold
Premises: Eating too much junk food causes obesity. John eats a lot of junk
food. Rank: 1 Rank (G) 1: john eats a lot of junk food. eating too much junk
food causes obesity. Rank 2: john eats some junk food. eating too much junk
food causes obesity. Rank 3: john eats a lot of junk food. all junk food
causes obesity. Rank 4: john eats some junk food. all junk food causes
obesity.

[backgroundcolor=gray!20, hidealllines=true]  Category: causal reasoning
Perturbation Type: NEGATED Target: John may become obese. Gold Premises:
Eating too much junk food causes obesity. John eats a lot of junk food. Rank:
1 Rank (G) 1: eating too much junk food causes obesity. john eats a lot of
junk food. Rank 2: eating too much junk food causes obesity. john doesn’t eat
a lot of junk food. Rank 3: not all junk food causes obesity. john eats a lot
of junk food. Rank 4: not all junk food causes obesity. john doesn’t eat a lot
of junk food.

[backgroundcolor=gray!20, hidealllines=true]  Category: classification
Perturbation Type: FALSE PREMISE Target: Apples, bananas, oranges, and grapes
are all types of fruit. Gold Premises: Apples and bananas are types of fruit.
Oranges and grapes are types of fruit. Rank: 1 Rank (G) 1: oranges and grapes
are types of fruit. apples and bananas are types of fruit. Rank 2: bananas and
apples are not types of food. apples and bananas are types of fruit. Rank 3:
oranges and grapes are types of fruit. oranges and grapes are not fruits. Rank
4: bananas and apples are not types of food. oranges and grapes are not
fruits.

[backgroundcolor=gray!20, hidealllines=true]  Category: classification
Perturbation Type: IRRELEVANT FACT Target: Apples, bananas, oranges, and
grapes are all types of fruit. Gold Premises: Apples and bananas are types of
fruit. Oranges and grapes are types of fruit. Rank: 1 Rank (G) 1: apples and
bananas are types of fruit. oranges and grapes are types of fruit. Rank 2:
apples and bananas are types of fruit. grapes can be used to make wine. Rank
3: apples are a common ingredient in apple pie. oranges and grapes are types
of fruit. Rank 4: oranges and grapes are types of fruit. apples are often used
to make cider. Rank 5: apples and bananas are types of fruit. bananas are
often peeled before eaten. Rank 6: oranges are a good source of vitamin c.
oranges and grapes are types of fruit. Rank 7: apples and bananas are types of
fruit. bananas can be peeled from the top or bottom. Rank 8: grapes can be
used to make wine. oranges are a good source of vitamin c. Rank 9: oranges are
a good source of vitamin c. bananas are often peeled before eaten. Rank 10:
bananas can be peeled from the top or bottom. oranges are a good source of
vitamin c. Rank 11: bananas can be peeled from the top or bottom. apples are a
common ingredient in apple pie. Rank 12: grapes can be used to make wine.
apples are a common ingredient in apple pie. Rank 13: apples are a common
ingredient in apple pie. bananas are often peeled before eaten. Rank 14:
bananas can be peeled from the top or bottom. apples are often used to make
cider. Rank 15: bananas are often peeled before eaten. apples are often used
to make cider. Rank 16: grapes can be used to make wine. apples are often used
to make cider.

[backgroundcolor=gray!20, hidealllines=true]  Category: classification
Perturbation Type: Incorrect Quantifier Target: Apples, bananas, oranges, and
grapes are all types of fruit. Gold Premises: Apples and bananas are types of
fruit. Oranges and grapes are types of fruit. Rank: 2 Rank 1: all apples and
bananas are types of fruit. oranges and grapes are types of fruit. Rank (G) 2:
oranges and grapes are types of fruit. apples and bananas are types of fruit.
Rank 3: none of the oranges and grapes are types of fruit. apples and bananas
are types of fruit. Rank 4: all apples and bananas are types of fruit. none of
the oranges and grapes are types of fruit.

[backgroundcolor=gray!20, hidealllines=true]  Category: classification
Perturbation Type: NEGATED Target: Apples, bananas, oranges, and grapes are
all types of fruit. Gold Premises: Apples and bananas are types of fruit.
Oranges and grapes are types of fruit. Rank: 1 Rank (G) 1: oranges and grapes
are types of fruit. apples and bananas are types of fruit. Rank 2: not all
oranges and grapes are types of fruit. apples and bananas are types of fruit.
Rank 3: oranges and grapes are types of fruit. not all apples and bananas are
types of fruit. Rank 4: not all oranges and grapes are types of fruit. not all
apples and bananas are types of fruit.

[backgroundcolor=gray!20, hidealllines=true]  Category: comparative reasoning
Perturbation Type: FALSE PREMISE Target: The blue car is faster than the
yellow car. Gold Premises: The blue car is faster than the red car. The red
car is faster than the yellow car. Rank: 1 Rank (G) 1: the red car is faster
than the yellow car. the blue car is faster than the red car. Rank 2: the
yellow car has a slower acceleration than the red car. the blue car is faster
than the red car. Rank 3: the blue car has a smaller engine than the red car.
the red car is faster than the yellow car. Rank 4: the blue car has a smaller
engine than the red car. the yellow car has a slower acceleration than the red
car.

[backgroundcolor=gray!20, hidealllines=true]  Category: comparative reasoning
Perturbation Type: IRRELEVANT FACT Target: The blue car is faster than the
yellow car. Gold Premises: The blue car is faster than the red car. The red
car is faster than the yellow car. Rank: 1 Rank (G) 1: the red car is faster
than the yellow car. the blue car is faster than the red car. Rank 2: the blue
car has a sunroof. the red car is faster than the yellow car. Rank 3: the
yellow car has a sunroof. the blue car is faster than the red car. Rank 4: the
yellow car has leather seats. the blue car is faster than the red car. Rank 5:
the yellow car has a gps system. the blue car is faster than the red car. Rank
6: the blue car has a full tank of gas. the red car is faster than the yellow
car. Rank 7: the blue car was once owned by a famous celebrity. the red car is
faster than the yellow car. Rank 8: the yellow car has a sunroof. the blue car
has a full tank of gas. Rank 9: the yellow car has a gps system. the blue car
has a sunroof. Rank 10: the blue car has a full tank of gas. the yellow car
has leather seats. Rank 11: the blue car has a full tank of gas. the yellow
car has a gps system. Rank 12: the yellow car has leather seats. the blue car
has a sunroof. Rank 13: the yellow car has a gps system. the blue car was once
owned by a famous celebrity. Rank 14: the yellow car has a sunroof. the blue
car was once owned by a famous celebrity. Rank 15: the yellow car has leather
seats. the blue car was once owned by a famous celebrity. Rank 16: the yellow
car has a sunroof. the blue car has a sunroof.

[backgroundcolor=gray!20, hidealllines=true]  Category: comparative reasoning
Perturbation Type: Incorrect Quantifier Target: The blue car is faster than
the yellow car. Gold Premises: The blue car is faster than the red car. The
red car is faster than the yellow car. Rank: 1 Rank (G) 1: the red car is
faster than the yellow car. the blue car is faster than the red car. Rank 2:
the red car is faster than the yellow car. all blue cars are faster than red
cars. Rank 3: the blue car is faster than the red car. some yellow cars are
slower than red cars. Rank 4: some yellow cars are slower than red cars. all
blue cars are faster than red cars.

[backgroundcolor=gray!20, hidealllines=true]  Category: comparative reasoning
Perturbation Type: NEGATED Target: The blue car is faster than the yellow car.
Gold Premises: The blue car is faster than the red car. The red car is faster
than the yellow car. Rank: 1 Rank (G) 1: the blue car is faster than the red
car. the red car is faster than the yellow car. Rank 2: the blue car is faster
than the red car. the red car is not faster than the yellow car. Rank 3: the
red car is faster than the yellow car. the blue car is not faster than the red
car. Rank 4: the red car is not faster than the yellow car. the blue car is
not faster than the red car.

[backgroundcolor=gray!20, hidealllines=true]  Category: composition
Perturbation Type: FALSE PREMISE Target: The salad is healthy. Gold Premises:
Every ingredient in a salad is healthy. A salad is made up of these
ingredients. Rank: 2 Rank 1: a salad is made up of these unhealthy
ingredients. every ingredient in a salad is healthy. Rank (G) 2: a salad is
made up of these ingredients. every ingredient in a salad is healthy. Rank 3:
a salad is made up of these ingredients. every ingredient in a salad is
unhealthy. Rank 4: a salad is made up of these unhealthy ingredients. every
ingredient in a salad is unhealthy.

[backgroundcolor=gray!20, hidealllines=true]  Category: composition
Perturbation Type: IRRELEVANT FACT Target: The salad is healthy. Gold
Premises: Every ingredient in a salad is healthy. A salad is made up of these
ingredients. Rank: 4 Rank 1: every ingredient in a salad is healthy. the salad
contains at least one type of fruit. Rank 2: a salad can be served as an
appetizer or a main course. every ingredient in a salad is healthy. Rank 3: a
salad contains at least 3 different colors. every ingredient in a salad is
healthy. Rank (G) 4: every ingredient in a salad is healthy. a salad is made
up of these ingredients. Rank 5: the salad contains at least one type of
fruit. every ingredient in a salad is organic. Rank 6: a salad can be served
as an appetizer or a main course. every ingredient in a salad is organic. Rank
7: a salad is made up of these ingredients. every ingredient in a salad is
organic. Rank 8: a salad contains at least 3 different colors. every
ingredient in a salad is organic. Rank 9: every ingredient in a salad is grown
outdoors. the salad contains at least one type of fruit. Rank 10: every
ingredient in a salad is available at the grocery store. the salad contains at
least one type of fruit. Rank 11: every ingredient in a salad is grown
outdoors. a salad can be served as an appetizer or a main course. Rank 12:
every ingredient in a salad is available at the grocery store. a salad can be
served as an appetizer or a main course. Rank 13: every ingredient in a salad
is grown outdoors. a salad contains at least 3 different colors. Rank 14:
every ingredient in a salad is grown outdoors. a salad is made up of these
ingredients. Rank 15: every ingredient in a salad is available at the grocery
store. a salad is made up of these ingredients. Rank 16: a salad contains at
least 3 different colors. every ingredient in a salad is available at the
grocery store.

[backgroundcolor=gray!20, hidealllines=true]  Category: composition
Perturbation Type: Incorrect Quantifier Target: The salad is healthy. Gold
Premises: Every ingredient in a salad is healthy. A salad is made up of these
ingredients. Rank: 3 Rank 1: some ingredients in a salad are healthy. a salad
is made up of only these healthy ingredients. Rank 2: a salad is made up of
only these healthy ingredients. every ingredient in a salad is healthy. Rank
(G) 3: a salad is made up of these ingredients. every ingredient in a salad is
healthy. Rank 4: a salad is made up of these ingredients. some ingredients in
a salad are healthy.

[backgroundcolor=gray!20, hidealllines=true]  Category: composition
Perturbation Type: NEGATED Target: The salad is healthy. Gold Premises: Every
ingredient in a salad is healthy. A salad is made up of these ingredients.
Rank: 2 Rank 1: a salad is not made up of these healthy ingredients. every
ingredient in a salad is healthy. Rank (G) 2: a salad is made up of these
ingredients. every ingredient in a salad is healthy. Rank 3: not every
ingredient in a salad is healthy. a salad is made up of these ingredients.
Rank 4: not every ingredient in a salad is healthy. a salad is not made up of
these healthy ingredients.

[backgroundcolor=gray!20, hidealllines=true]  Category: divisions Perturbation
Type: FALSE PREMISE Target: The individual players are skilled at playing
offense and defense. Gold Premises: The football team is skilled at playing
offense and defense. The individual players are part of the football team.
Rank: 1 Rank (G) 1: the individual players are part of the football team. the
football team is skilled at playing offense and defense. Rank 2: each player
has never played football before. the football team is skilled at playing
offense and defense. Rank 3: the football team has never won a game before.
the individual players are part of the football team. Rank 4: the football
team has never won a game before. each player has never played football
before.

[backgroundcolor=gray!20, hidealllines=true]  Category: divisions Perturbation
Type: IRRELEVANT FACT Target: The individual players are skilled at playing
offense and defense. Gold Premises: The football team is skilled at playing
offense and defense. The individual players are part of the football team.
Rank: 1 Rank (G) 1: the football team is skilled at playing offense and
defense. the individual players are part of the football team. Rank 2: the
football team is skilled at playing offense and defense. each player has a
favorite video game. Rank 3: each player has a favorite food. the football
team is skilled at playing offense and defense. Rank 4: the football team is
skilled at playing offense and defense. each player has a favorite type of
sandwich. Rank 5: the individual players are part of the football team. the
football team practices on a grass field. Rank 6: the football team practices
at the local high school. the individual players are part of the football
team. Rank 7: the football team practices on a grass field. each player has a
favorite video game. Rank 8: the football team practices at the local high
school. each player has a favorite video game. Rank 9: each player has a
favorite food. the football team practices on a grass field. Rank 10: the
football team practices at the local high school. each player has a favorite
food. Rank 11: each player has a favorite type of sandwich. the football team
practices on a grass field. Rank 12: the individual players are part of the
football team. the football team has a team mascot. Rank 13: the football team
practices at the local high school. each player has a favorite type of
sandwich. Rank 14: each player has a favorite video game. the football team
has a team mascot. Rank 15: each player has a favorite food. the football team
has a team mascot. Rank 16: each player has a favorite type of sandwich. the
football team has a team mascot.

[backgroundcolor=gray!20, hidealllines=true]  Category: divisions Perturbation
Type: Incorrect Quantifier Target: The individual players are skilled at
playing offense and defense. Gold Premises: The football team is skilled at
playing offense and defense. The individual players are part of the football
team. Rank: 2 Rank 1: all football teams are skilled at playing offense and
defense. the individual players are part of the football team. Rank (G) 2: the
football team is skilled at playing offense and defense. the individual
players are part of the football team. Rank 3: the football team is skilled at
playing offense and defense. some players are not part of the football team.
Rank 4: all football teams are skilled at playing offense and defense. some
players are not part of the football team.

[backgroundcolor=gray!20, hidealllines=true]  Category: divisions Perturbation
Type: NEGATED Target: The individual players are skilled at playing offense
and defense. Gold Premises: The football team is skilled at playing offense
and defense. The individual players are part of the football team. Rank: 1
Rank (G) 1: the individual players are part of the football team. the football
team is skilled at playing offense and defense. Rank 2: each player is not
part of the football team. the football team is skilled at playing offense and
defense. Rank 3: the individual players are part of the football team. the
football team is not skilled at playing offense and defense. Rank 4: the
football team is not skilled at playing offense and defense. each player is
not part of the football team.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus ponens
Perturbation Type: FALSE PREMISE Target: The temperature is warm. Gold
Premises: If it’s sunny, the temperature is warm. It’s sunny. Rank: 1 Rank (G)
1: it’s sunny. if it’s sunny, the temperature is warm. Rank 2: all players are
part of the football team. if it’s sunny, the temperature is warm. Rank 3:
it’s sunny. all football teams are skilled at playing offense and defense.
Rank 4: all football teams are skilled at playing offense and defense. all
players are part of the football team.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus ponens
Perturbation Type: IRRELEVANT FACT Target: The temperature is warm. Gold
Premises: If it’s sunny, the temperature is warm. It’s sunny. Rank: 1 Rank (G)
1: it’s sunny. if it’s sunny, the temperature is warm. Rank 2: each player has
a favorite color. if it’s sunny, the temperature is warm. Rank 3: each player
has a favorite type of car. if it’s sunny, the temperature is warm. Rank 4:
each player has a favorite type of candy bar. if it’s sunny, the temperature
is warm. Rank 5: it’s sunny. the football team has a popular pre-game ritual.
Rank 6: the football team has a new coach this season. it’s sunny. Rank 7: the
football team has a cheerleading squad. it’s sunny. Rank 8: the football team
has a new coach this season. each player has a favorite color. Rank 9: the
football team has a new coach this season. each player has a favorite type of
car. Rank 10: each player has a favorite color. the football team has a
popular pre-game ritual. Rank 11: each player has a favorite color. the
football team has a cheerleading squad. Rank 12: each player has a favorite
type of car. the football team has a popular pre-game ritual. Rank 13: the
football team has a new coach this season. each player has a favorite type of
candy bar. Rank 14: each player has a favorite type of car. the football team
has a cheerleading squad. Rank 15: the football team has a popular pre-game
ritual. each player has a favorite type of candy bar. Rank 16: the football
team has a cheerleading squad. each player has a favorite type of candy bar.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus ponens
Perturbation Type: Incorrect Quantifier Target: The temperature is warm. Gold
Premises: If it’s sunny, the temperature is warm. It’s sunny. Rank: 1 Rank (G)
1: it’s sunny. if it’s sunny, the temperature is warm. Rank 2: no player is
part of the football team. if it’s sunny, the temperature is warm. Rank 3:
some football teams are skilled at playing offense and defense. it’s sunny.
Rank 4: some football teams are skilled at playing offense and defense. no
player is part of the football team.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus ponens
Perturbation Type: NEGATED Target: The temperature is warm. Gold Premises: If
it’s sunny, the temperature is warm. It’s sunny. Rank: 1 Rank (G) 1: it’s
sunny. if it’s sunny, the temperature is warm. Rank 2: a player is not part of
the football team. if it’s sunny, the temperature is warm. Rank 3: not all
football teams are skilled at playing offense and defense. it’s sunny. Rank 4:
not all football teams are skilled at playing offense and defense. a player is
not part of the football team.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus tollens
Perturbation Type: FALSE PREMISE Target: John is not at work. Gold Premises:
If John is at work, he is not at home. John is at home. Rank: 2 Rank 1: john
is not at home. if john is at work, he is not at home. Rank (G) 2: john is at
home. if john is at work, he is not at home. Rank 3: john is not at home. all
those who work are at the office. Rank 4: john is at home. all those who work
are at the office.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus tollens
Perturbation Type: IRRELEVANT FACT Target: John is not at work. Gold Premises:
If John is at work, he is not at home. John is at home. Rank: 1 Rank (G) 1:
john is at home. if john is at work, he is not at home. Rank 2: john wears
glasses. if john is at work, he is not at home. Rank 3: if john is at work, he
is not at home. john has a cat named whiskers. Rank 4: it is monday today. if
john is at work, he is not at home. Rank 5: john is at home. john is wearing a
blue shirt. Rank 6: john is at home. john ate a sandwich for lunch. Rank 7:
john drives a blue car. john is at home. Rank 8: john has a cat named
whiskers. john is wearing a blue shirt. Rank 9: john wears glasses. john is
wearing a blue shirt. Rank 10: john wears glasses. john ate a sandwich for
lunch. Rank 11: it is monday today. john is wearing a blue shirt. Rank 12:
john ate a sandwich for lunch. john has a cat named whiskers. Rank 13: john
wears glasses. john drives a blue car. Rank 14: john drives a blue car. john
has a cat named whiskers. Rank 15: it is monday today. john ate a sandwich for
lunch. Rank 16: it is monday today. john drives a blue car.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus tollens
Perturbation Type: Incorrect Quantifier Target: John is not at work. Gold
Premises: If John is at work, he is not at home. John is at home. Rank: 1 Rank
(G) 1: if john is at work, he is not at home. john is at home. Rank 2: if john
is at work, he is not at home. some people at home are not on vacation. Rank
3: john is at home. all who work are at the office. Rank 4: some people at
home are not on vacation. all who work are at the office.

[backgroundcolor=gray!20, hidealllines=true]  Category: modus tollens
Perturbation Type: NEGATED Target: It is not a mammal. Gold Premises: If it is
a mammal, it has fur. The animal has no fur. Rank: 1 Rank (G) 1: if it is a
mammal, it has fur. the animal has no fur. Rank 2: not all mammals have fur.
the animal has no fur. Rank 3: not all mammals have fur. the animal has fur.
Rank 4: if it is a mammal, it has fur. the animal has fur.

[backgroundcolor=gray!20, hidealllines=true]  Category: definition
Perturbation Type: FALSE PREMISE Target: Basil is a plant used for flavoring,
medicine, or fragrance. Gold Premises: An herb is a plant used for flavoring,
medicine, or fragrance. Basil is an herb. Rank: 1 Rank (G) 1: basil is an
herb. an herb is a plant used for flavoring, medicine, or fragrance. Rank 2:
basil is a type of insect. an herb is a plant used for flavoring, medicine, or
fragrance. Rank 3: a herb is a type of animal. basil is an herb. Rank 4: a
herb is a type of animal. basil is a type of insect.

[backgroundcolor=gray!20, hidealllines=true]  Category: definition
Perturbation Type: IRRELEVANT FACT Target: Basil is a plant used for
flavoring, medicine, or fragrance. Gold Premises: An herb is a plant used for
flavoring, medicine, or fragrance. Basil is an herb. Rank: 1 Rank (G) 1: an
herb is a plant used for flavoring, medicine, or fragrance. basil is an herb.
Rank 2: an herb is a plant used for flavoring, medicine, or fragrance. basil
is native to tropical regions. Rank 3: basil is an herb. basil is used in many
cuisines around the world. Rank 4: basil is used in many cuisines around the
world. herbs have been used for medicinal purposes since ancient times. Rank
5: basil is an herb. the scientific name for basil is ocimum basilicum. Rank
6: basil is used in many cuisines around the world. herbs are often used to
treat minor ailments. Rank 7: basil is used in many cuisines around the world.
basil is native to tropical regions. Rank 8: basil is an herb. herbs are
commonly used in italian cuisine. Rank 9: herbs are commonly used in italian
cuisine. basil is native to tropical regions. Rank 10: basil is native to
tropical regions. the scientific name for basil is ocimum basilicum. Rank 11:
an herb is a plant used for flavoring, medicine, or fragrance. herbs have been
used for medicinal purposes since ancient times. Rank 12: an herb is a plant
used for flavoring, medicine, or fragrance. herbs are often used to treat
minor ailments. Rank 13: herbs have been used for medicinal purposes since
ancient times. the scientific name for basil is ocimum basilicum. Rank 14:
herbs are often used to treat minor ailments. the scientific name for basil is
ocimum basilicum. Rank 15: herbs are commonly used in italian cuisine. herbs
have been used for medicinal purposes since ancient times. Rank 16: herbs are
commonly used in italian cuisine. herbs are often used to treat minor
ailments.

[backgroundcolor=gray!20, hidealllines=true]  Category: definition
Perturbation Type: Incorrect Quantifier Target: Basil is a plant used for
flavoring, medicine, or fragrance. Gold Premises: An herb is a plant used for
flavoring, medicine, or fragrance. Basil is an herb. Rank: 1 Rank (G) 1: basil
is an herb. an herb is a plant used for flavoring, medicine, or fragrance.
Rank 2: all plants used for flavoring, medicine, or fragrance are herbs. basil
is an herb. Rank 3: some basil is not an herb. an herb is a plant used for
flavoring, medicine, or fragrance. Rank 4: some basil is not an herb. all
plants used for flavoring, medicine, or fragrance are herbs.

[backgroundcolor=gray!20, hidealllines=true]  Category: definition
Perturbation Type: NEGATED Target: Basil is a plant used for flavoring,
medicine, or fragrance. Gold Premises: An herb is a plant used for flavoring,
medicine, or fragrance. Basil is an herb. Rank: 1 Rank (G) 1: basil is an
herb. an herb is a plant used for flavoring, medicine, or fragrance. Rank 2:
basil is not a herb. an herb is a plant used for flavoring, medicine, or
fragrance. Rank 3: not all plants used for flavoring, medicine, or fragrance
are herbs. basil is an herb. Rank 4: basil is not a herb. not all plants used
for flavoring, medicine, or fragrance are herbs.

[backgroundcolor=gray!20, hidealllines=true]  Category: temporal reasoning
Perturbation Type: FALSE PREMISE Target: I should leave at 7 Gold Premises:
The party starts at 8 PM. It takes 30 minutes to get there. Rank: 3 Rank 1: it
takes 30 minutes to get there. the party starts at 7 pm. Rank 2: it takes 15
minutes to get there. the party starts at 7 pm. Rank (G) 3: it takes 30
minutes to get there. the party starts at 8 pm. Rank 4: it takes 15 minutes to
get there. the party starts at 8 pm.

[backgroundcolor=gray!20, hidealllines=true]  Category: temporal reasoning
Perturbation Type: IRRELEVANT FACT Target: I should leave at 7 Gold Premises:
The party starts at 8 PM. It takes 30 minutes to get there. Rank: 1 Rank (G)
1: the party starts at 8 pm. it takes 30 minutes to get there. Rank 2: it
takes 30 minutes to get there. the party is a costume party. Rank 3: it takes
30 minutes to get there. the party has a live band. Rank 4: the party starts
at 8 pm. the host is left-handed. Rank 5: the party starts at 8 pm. the host
is a teacher. Rank 6: it takes 30 minutes to get there. the party has a pool.
Rank 7: the party starts at 8 pm. the host is allergic to peanuts. Rank 8: the
host is left-handed. the party is a costume party. Rank 9: the host is a
teacher. the party is a costume party. Rank 10: the party has a live band. the
host is left-handed. Rank 11: the party has a live band. the host is a
teacher. Rank 12: the host is left-handed. the party has a pool. Rank 13: the
host is allergic to peanuts. the party is a costume party. Rank 14: the host
is a teacher. the party has a pool. Rank 15: the host is allergic to peanuts.
the party has a live band. Rank 16: the host is allergic to peanuts. the party
has a pool.

[backgroundcolor=gray!20, hidealllines=true]  Category: temporal reasoning
Perturbation Type: Incorrect Quantifier Target: I should leave at 7 Gold
Premises: The party starts at 8 PM. It takes 30 minutes to get there. Rank: 1
Rank (G) 1: it takes 30 minutes to get there. the party starts at 8 pm. Rank
2: it takes 30 minutes to get there. all parties start at 8 pm. Rank 3: some
trips take 30 minutes to get there. the party starts at 8 pm. Rank 4: some
trips take 30 minutes to get there. all parties start at 8 pm.

[backgroundcolor=gray!20, hidealllines=true]  Category: temporal reasoning
Perturbation Type: NEGATED Target: I should leave at 7 Gold Premises: The
party starts at 8 PM. It takes 30 minutes to get there. Rank: 1 Rank (G) 1: it
takes 30 minutes to get there. the party starts at 8 pm. Rank 2: the party
does not start at 8 pm. it takes 30 minutes to get there. Rank 3: it does not
take 30 minutes to get there. the party starts at 8 pm. Rank 4: it does not
take 30 minutes to get there. the party does not start at 8 pm.

[backgroundcolor=gray!20, hidealllines=true]  Category: propositional logic
Perturbation Type: FALSE PREMISE Target: The streets are wet. Gold Premises:
It is raining outside. If it is raining, then the streets are wet. Rank: 2
Rank 1: it is raining outside. if it is snowing, then the streets are wet.
Rank (G) 2: if it is raining, then the streets are wet. it is raining outside.
Rank 3: if it is raining, then the streets are wet. it is snowing outside.
Rank 4: if it is snowing, then the streets are wet. it is snowing outside.

[backgroundcolor=gray!20, hidealllines=true]  Category: propositional logic
Perturbation Type: IRRELEVANT FACT Target: The streets are wet. Gold Premises:
It is raining outside. If it is raining, then the streets are wet. Rank: 1
Rank (G) 1: if it is raining, then the streets are wet. it is raining outside.
Rank 2: if it is raining, then the streets are wet. there is a car parked
outside. Rank 3: if it is raining, then the streets are wet. my car needs new
brakes. Rank 4: if it is raining, then the streets are wet. my neighbor has a
dog. Rank 5: i like to wear rain boots on cloudy days. it is raining outside.
Rank 6: it is raining outside. i like to drink coffee when it rains. Rank 7:
there is a car parked outside. i like to wear rain boots on cloudy days. Rank
8: there is a car parked outside. i like to drink coffee when it rains. Rank
9: i like to wear a hat when it’s cold. it is raining outside. Rank 10: my car
needs new brakes. i like to drink coffee when it rains. Rank 11: i like to
wear rain boots on cloudy days. my car needs new brakes. Rank 12: i like to
drink coffee when it rains. my neighbor has a dog. Rank 13: there is a car
parked outside. i like to wear a hat when it’s cold. Rank 14: i like to wear
rain boots on cloudy days. my neighbor has a dog. Rank 15: i like to wear a
hat when it’s cold. my car needs new brakes. Rank 16: i like to wear a hat
when it’s cold. my neighbor has a dog.

[backgroundcolor=gray!20, hidealllines=true]  Category: propositional logic
Perturbation Type: Incorrect Quantifier Target: The streets are wet. Gold
Premises: It is raining outside. If it is raining, then the streets are wet.
Rank: 1 Rank (G) 1: if it is raining, then the streets are wet. it is raining
outside. Rank 2: if it is raining, then the streets are wet. all precipitation
is rain.

[backgroundcolor=gray!20, hidealllines=true]  Category: propositional logic
Perturbation Type: NEGATED Target: The streets are wet. Gold Premises: It is
raining outside. If it is raining, then the streets are wet. Rank: 1 Rank (G)
1: it is raining outside. if it is raining, then the streets are wet. Rank 2:
if it is raining, then the streets are wet. it is not raining outside. Rank 3:
if it is not raining, then the streets are not wet. it is raining outside.
Rank 4: if it is not raining, then the streets are not wet. it is not raining
outside.

[backgroundcolor=gray!20, hidealllines=true]  Category: quantification logic
Perturbation Type: FALSE PREMISE Target: Some friendly dogs are black. Gold
Premises: Some dogs are black. All black dogs are friendly. Rank: 1 Rank (G)
1: some dogs are black. all black dogs are friendly. Rank 2: no dogs are
black. all black dogs are friendly.

[backgroundcolor=gray!20, hidealllines=true]  Category: quantification logic
Perturbation Type: IRRELEVANT FACT Target: Some friendly dogs are black. Gold
Premises: Some dogs are black. All black dogs are friendly. Rank: 1 Rank (G)
1: some dogs are black. all black dogs are friendly. Rank 2: some dogs are
black. the friendly black dogs weigh more than the unfriendly ones. Rank 3:
all black dogs are friendly. some dogs have brown eyes. Rank 4: some dogs have
brown eyes. the friendly black dogs weigh more than the unfriendly ones. Rank
5: all black dogs are friendly. the basket is made of wicker. Rank 6: some
dogs are black. some of the apples in the basket were grown organically. Rank
7: all black dogs are friendly. the apples were picked in the morning. Rank 8:
the basket is made of recycled materials. some dogs are black. Rank 9: the
basket is made of wicker. the friendly black dogs weigh more than the
unfriendly ones. Rank 10: the friendly black dogs weigh more than the
unfriendly ones. the apples were picked in the morning. Rank 11: some of the
apples in the basket were grown organically. some dogs have brown eyes. Rank
12: the basket is made of recycled materials. some dogs have brown eyes. Rank
13: the basket is made of recycled materials. the apples were picked in the
morning. Rank 14: some of the apples in the basket were grown organically. the
basket is made of wicker. Rank 15: some of the apples in the basket were grown
organically. the apples were picked in the morning. Rank 16: the basket is
made of recycled materials. the basket is made of wicker.

[backgroundcolor=gray!20, hidealllines=true]  Category: quantification logic
Perturbation Type: Incorrect Quantifier Target: Some friendly dogs are black.
Gold Premises: Some dogs are black. All black dogs are friendly. Rank: 1 Rank
(G) 1: all black dogs are friendly. some dogs are black. Rank 2: all black
dogs are friendly. some dogs are not black. Rank 3: not all black dogs are
friendly. some dogs are not black.

[backgroundcolor=gray!20, hidealllines=true]  Category: quantification logic
Perturbation Type: NEGATED Target: Some friendly dogs are black. Gold
Premises: Some dogs are black. All black dogs are friendly. Rank: 1 Rank (G)
1: all black dogs are friendly. some dogs are black. Rank 2: not all black
dogs are friendly. some dogs are black. Rank 3: not all dogs are black. all
black dogs are friendly. Rank 4: not all dogs are black. not all black dogs
are friendly.

[backgroundcolor=gray!20, hidealllines=true]  Category: spatial reasoning
Perturbation Type: FALSE PREMISE Target: The museum is northeast of the
library. Gold Premises: The library is north of the park. The museum is east
of the park. Rank: 1 Rank (G) 1: the museum is east of the park. the library
is north of the park. Rank 2: the museum is west of the park. the library is
north of the park. Rank 3: the library is south of the park. the museum is
east of the park. Rank 4: the library is south of the park. the museum is west
of the park.

[backgroundcolor=gray!20, hidealllines=true]  Category: spatial reasoning
Perturbation Type: IRRELEVANT FACT Target: The museum is northeast of the
library. Gold Premises: The library is north of the park. The museum is east
of the park. Rank: 1 Rank (G) 1: the library is north of the park. the museum
is east of the park. Rank 2: the library has a red brick exterior. the museum
is east of the park. Rank 3: the library has a book drop on the side of the
building. the museum is east of the park. Rank 4: the library has a statue of
a famous author in front of it. the museum is east of the park. Rank 5: the
library is north of the park. the museum has a café on the first floor. Rank
6: the library is north of the park. the museum has a gift shop that sells
postcards and souvenirs. Rank 7: the library is north of the park. the museum
has a volunteer program for high school students. Rank 8: the library has a
red brick exterior. the museum has a gift shop that sells postcards and
souvenirs. Rank 9: the museum has a café on the first floor. the library has a
red brick exterior. Rank 10: the library has a book drop on the side of the
building. the museum has a café on the first floor. Rank 11: the library has a
book drop on the side of the building. the museum has a gift shop that sells
postcards and souvenirs. Rank 12: the library has a statue of a famous author
in front of it. the museum has a café on the first floor. Rank 13: the museum
has a volunteer program for high school students. the library has a red brick
exterior. Rank 14: the library has a statue of a famous author in front of it.
the museum has a gift shop that sells postcards and souvenirs. Rank 15: the
library has a book drop on the side of the building. the museum has a
volunteer program for high school students. Rank 16: the library has a statue
of a famous author in front of it. the museum has a volunteer program for high
school students.

[backgroundcolor=gray!20, hidealllines=true]  Category: spatial reasoning
Perturbation Type: Incorrect Quantifier Target: The museum is northeast of the
library. Gold Premises: The library is north of the park. The museum is east
of the park. Rank: 1 Rank (G) 1: the museum is east of the park. the library
is north of the park. Rank 2: some museums are east of the park. the library
is north of the park. Rank 3: the museum is east of the park. all books in the
library are north of the park. Rank 4: all books in the library are north of
the park. some museums are east of the park.

[backgroundcolor=gray!20, hidealllines=true]  Category: spatial reasoning
Perturbation Type: NEGATED Target: The museum is northeast of the library.
Gold Premises: The library is north of the park. The museum is east of the
park. Rank: 1 Rank (G) 1: the library is north of the park. the museum is east
of the park. Rank 2: the museum is east of the park. the library is not north
of the park. Rank 3: the library is north of the park. the museum is not east
of the park. Rank 4: the museum is not east of the park. the library is not
north of the park.


# The $\Omega(2012)$ as a hadronic molecule

Ju-Jun Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Southern
Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese
Academy of Sciences, Huizhou 516000, 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

Recently, a new excited baryon state, $\Omega(2012)$, was observed in the
invariant mass spectra of $K^{-}\Xi^{0}$ and $K^{0}_{S}\Xi^{-}$ by the Belle
Collaboration. This state has a narrow width ($\sim 6$ MeV) compared to other
baryon states with a similar mass. In this paper, we provide a mini-review on
the $\Omega(2012)$ state from the molecular perspective, where it appears to
be a dynamically generated state with spin-parity $3/2^{-}$ from the coupled-
channels interactions of the $\bar{K}\Xi(1530)$ and $\eta\Omega$ in $s$-wave
and $\bar{K}\Xi$ in $d$-wave. Additionally, alternative explanations for the
$\Omega(2012)$ resonance are also discussed.

##  I Introduction

In the past decades, hadron physicists showed a great interest in hunting for
evidences of exotic states containing some properties that cannot be easily
explained by the classical quark models. These exotic states are of special
importance in the hadron family. In 2018, a new excited hyperon with
strangeness $S=-3$ and isospin $I=0$, the $\Omega(2012)$ state, was discovered
by the Belle collaboration in the $K^{-}\Xi^{0}$ and $K^{0}_{S}\Xi^{-}$
invariant mass distributions in $\Upsilon(1S)$, $\Upsilon(2S)$, and
$\Upsilon(3S)$ decays Yelton:2018mag . Its measured mass and width are
$M=2012.4\pm 0.7({\rm stat})\pm 0.6({\rm syst})$ MeV and
$\Gamma=6.4^{+2.5}_{-2.0}({\rm stat})\pm 1.6({\rm syst})$ MeV. The
$\Omega(2012)$ is compiled in the Particle Data Group (PDG)
ParticleDataGroup:2022pth  as a three-star excited $\Omega$ state. In
addition, the $\Omega(2012)$ has a narrow width and is the first $\Omega$
excited state with preferred negative parity.

The $\Omega$ state contains only three strange quarks, making its excited
states challenging to detect. Before the observation of $\Omega(2012)$, most
information on the $\Omega$ excited states were extracted from the bubble
chamber experiments in the 1980s, and there was only one three-star $\Omega$
excited state, $\Omega(2250)$, with spin-parity ($J^{P}$) unknown. Meanwhile
the other two higher two-star states, $\Omega(2380)$ and $\Omega(2470)$, are
very badly known, and they were omitted from the summary table in the PDG
ParticleDataGroup:2022pth . The $\Omega(2012)$ resonance has a mass about
$340$ MeV higher than the ground state $\Omega(1672)$, and $198.5$ MeV higher
than the $\bar{K}\Xi$ mass threshold ($m_{\bar{K}}+M_{\Xi}=1813.9$ MeV). But,
as shown in Fig. 1, the mass of $\Omega(2012)$ is just $16.6$ MeV below the
$\bar{K}\Xi(1530)$ mass threshold ($m_{\bar{K}}+M_{\Xi(1530)}=2029$ MeV), and
it is far from the thresholds of $\eta\Omega$ and $\bar{K}\Xi$, which
indicates that it could be a possible $\bar{K}\Xi(1530)$ molecule state
Guo:2017jvc . In addition, its narrow width implies that its quantum numbers
$J^{P}=3/2^{-}$ identification is more likely.

Figure 1: The comparision of the experimental mass of $\Omega(2012)$ and the
mass thresholds of $\bar{K}\Xi$ and $\bar{K}\Xi(1530)$.

In fact, such $\Omega$ excited states have been theoretic studied before the
Belle collaboration published their results. For example, using the chiral
unitary approach considering the coupled channels interactions of the
$\bar{K}\Xi(1530)$ and $\eta\Omega$ in $s$-wave, the $\Omega$ excited states
were investigated Kolomeitsev:2003kt ; Sarkar:2004jh ; GarciaRecio:2006bk ;
Si-Qi:2016gmh . A $J^{P}=3/2^{-}$ $\Omega$ excited state with mass around
2012.7 MeV can be dynamically generated Si-Qi:2016gmh  with a reasonable value
of the subtraction constant: $a_{\mu}=-2.5$ and $\mu=700$ MeV. However, the
obtained width for $\Omega(2012)$ in Ref. Si-Qi:2016gmh  is zero, since no
open channel (for example, the $\bar{K}\Xi$) was included.

The observation of $\Omega(2012)$ prompted many theoretical investigations to
explain its properties with different theoretical models, such as the quark
model and the molecular picture on the meson-baryon interaction. For example,
its hadronic molecule nature of $\bar{K}\Xi(1530)$ was further investigated in
Ref. Valderrama:2018bmv  within the one-boson-exchange model and in Refs.
Lin:2018nqd ; Huang:2018wth ; Pavao:2018xub ; Lin:2019tex ; Lu:2020ste  within
the chiral unitary approach. It was found that the $\Omega(2012)$ can be
dynamically generated from the interactions of $\bar{K}\Xi(1530)$ and
$\eta\Omega$ in $s$-wave and $\bar{K}\Xi$ in $d$-wave, and the
$\bar{K}\Xi(1530)$ channel is dominant. The measured width of $\Omega(2012)$
can also be reproduced, since the open channel $\bar{K}\Xi$ is also taken into
account. However, one should note that the $\eta\Omega$ channel is crucial to
produce the $\Omega(2012)$, even its mass threshold
($m_{\eta}+M_{\Omega}=2220.3$ MeV) is much higher. Without the $\eta\Omega$
channel, there will be no dynamically generated state because the interaction
in the $\bar{K}\Xi(1530)$ and $\eta\Omega$ coupled channels is off diagonal in
the chiral unitary approach, which means that, at the leading order, the
transition potentials $V_{\bar{K}\Xi(1530)\to\bar{K}\Xi(1530)}$ and
$V_{\eta\Omega\to\eta\Omega}$ are zero.

The molecular structure of the $\Omega(2012)$, including five quarks, has its
replica in the charm sector. In Ref. Ikeno:2023uzz  it is shown that the
$\Omega(2012)$ is the analog state of the $\Omega_{c}(3120)$ discovered by the
LHCb Collaboration LHCb:2017uwr .

##  II Three-body decay of
$\Omega(2012)\to\bar{K}\Xi^{*}(1530)\to\bar{K}\pi\Xi$

In the $\bar{K}\Xi(1530)$ molecular picture, as shown in Fig. 2, the three-
body decay mode of $\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$ is
important, and the three-body decay width for $\Omega(2012)\to\bar{K}\pi\Xi$
is comparable with the one for $\Omega(2012)\to\bar{K}\Xi$. The $\pi\Xi$
invariant mass distributions can be written as

$\displaystyle\frac{d\Gamma_{\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi}}{dM_{\pi\Xi}}\propto\frac{1}{M}\frac{p_{\bar{K}}p^{*}_{\pi}\Gamma_{\Xi^{*}}}{(M_{\pi\Xi}-M_{\Xi^{*}})^{2}+\Gamma_{\Xi^{*}}^{2}/4},$
(1)

where $M_{\Xi^{*}}$ and $\Gamma_{\Xi^{*}}$ are the mass and width of
$\Xi(1530)$, and $M_{\pi\Xi}$ is the invariant mass of the $\pi$ and $\Xi$
system. While $p_{\bar{K}}$ and $p^{*}_{\pi}$ are the three-momentum of
$\bar{K}$ and $\pi$ in the $\Omega(2012)$ rest frame and the $\pi$ and $\Xi$
center of mass frame, respectively. Then the invariant $\pi\Xi$ mass
distributions can be easily obtained as shown by the black curve in Fig. 3.
The numerical results are calculated with averaged values for the masses of
those involved particles, and we take $\Gamma_{\Xi^{*}}=9.5$ MeV.

In the molecular picture, the production of the $\Omega(2012)$ resonance in
the nonleptonic weak decays of
$\Omega_{c}^{0}\to\pi^{+}\bar{K}\Xi(1530)(\eta\Omega)\to\pi^{+}(\bar{K}\Xi)^{-}$
and $\pi^{+}(\bar{K}\Xi\pi)^{-}$ via final-state interactions of the
$\bar{K}\Xi(1530)$ and $\eta\Omega$ pairs was investigated Zeng:2020och ;
Ikeno:2022jpe . Later on, the theoretical results of Ref. Zeng:2020och  were
confirmed by the Belle Collaboration Belle:2021gtf . The significance of the
$\Omega(2012)$ signal in the
$\Omega^{0}_{c}\to\pi^{+}\Omega^{-}(2012)\to\pi^{+}(\bar{K}\Xi)^{-}$ is
$4.2\sigma$, including the systematic uncertainties. Further experimental
studies about this decay channel are most welcome.

Figure 2: Feynman diagram for the three-body decay of
$\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$.

On the other hand, because of the small phase space [the mass of
$\Omega(2012)$ is below the mass threshold of $\bar{K}\Xi(1530)$ and the small
decay width ($\sim 10$ MeV) of $\Xi(1530)$], it is difficult to analyze the
three-body decay of $\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$. To see
this, we define

$\displaystyle\frac{d|T|^{2}_{\pi\Xi\to\Xi(1530)\to\pi\Xi}}{dM_{\pi\Xi}}\propto\frac{M_{\Xi^{*}}\Gamma_{\Xi^{*}}}{(M_{\pi\Xi}-M_{\Xi^{*}})^{2}+\Gamma_{\Xi^{*}}^{2}/4}.$
(2)

The obtained $\pi\Xi$ invariant mass distribution is also shown in Fig. 3 by
the red curve. One can clearly see that the two peaks (we have normalized them
to the same value) are much separated and the overlapping part of the two line
shapes is very narrow, then the
$\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$ decay will be suppressed due
to the highly off-shell effect of the $\Xi(1530)$ propagator, though the
coupling of $\Omega(2012)$ to the $\bar{K}\Xi(1530)$ channel is strong
Lu:2020ste ; Arifi:2022ntc . Therefore, in the real analysis, the physical
masses for the $\Xi(1530)$, $\bar{K}$, $\pi$, and $\Xi$ with different charges
should be taken into account explicitly Huang:2018wth .

Figure 3: The $\pi\Xi$ invariant mass distributions. The black curve
represents the $\pi\Xi$ invariant mass distributions of the
$\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$ decay, while the red curve
represents the module squared of $\pi\Xi\to\Xi(1530)\to\pi\Xi$ as a function
of the invariant mass $M_{\pi\Xi}$ (see Eq. (2)).

It is worth mentioning that in an early search for the
$\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi$ decay by the Belle
collaboration Belle:2019zco , it did not find the signal and set an upper
limit at $90\%$ confidence level on the branching fraction ratio

${\cal R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}=\frac{{\rm
Br}[\Omega(2012)\to\bar{K}\Xi(1530)\to\bar{K}\pi\Xi]}{{\rm
Br}[\Omega(2012)\to\bar{K}\Xi]}<11.9\%.$ (3)

While, in the very recent measurement of the Belle Collaboration Belle:2022mrg
, the ratio ${\cal R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}$ was updated with improved
selection criteria, accounting for the $\Omega(2012)$ being below the
$\bar{K}\Xi(1530)$ mass threshold. In this new analysis, $M_{\pi\Xi}<1.517$
GeV is required and the signal shape of $\Omega(2012)$ was parameterized with
a Flatté-like function to account for the allowed phase space. Such selection
criteria were not included in the previous analysis in Ref. Belle:2019zco ,
and they improve the signal-background sepration and hence increase the signal
yield. The new result of the ratio ${\cal R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}$ is
$0.97\pm 0.24\pm 0.07$. This is consistent with the molecular interpretation
for the $\Omega(2012)$ as proposed in Refs. Lin:2018nqd ; Huang:2018wth ;
Pavao:2018xub ; Lin:2019tex ; Lu:2020ste . In the molecular picture for
$\Omega(2012)$, there are two free model parameters for the $d$-wave
interaction. By adjusting these two model parameters, it is found that the
ratio ${\cal R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}$ reported recently by the Belle
Collaboration can be also easily accommodated.

##  III Two-body decay of $\Omega(2012)\to\bar{K}\Xi$

In Ref. Lu:2020ste , the $\Omega(2012)$ state was investigated from the
molecular perspective in which the resonance is dynamically generated from the
interactions of $\bar{K}\Xi^{*}(1530)$, $\eta\Omega$ and $\bar{K}\Xi$ in
coupled channels, where the interactions of $\bar{K}\Xi^{*}(1530)$ and
$\eta\Omega$ are in $s$-wave, and the $\bar{K}\Xi$ interaction is in $d$-wave.
One has to include further parameters for building the transition potentials
of $V_{\bar{K}\Xi(1530)\to\bar{K}\Xi}$ and $V_{\eta\Omega\to\bar{K}\Xi}$.
Fortunately, these parameters can be determined from the measured mass and
width of the $\Omega(2012)$ state. Furthermore, for the effective
$\Omega(2012)\bar{K}\Xi$ vertex, it can be obtained from the $s$-wave
$\Omega(2012)\bar{K}\Xi^{*}(1530)$ and $\Omega(2012)\eta\Omega$ interactions
and the re-scattering of $\bar{K}\Xi^{*}(1530)$ and $\eta\Omega$ pairs, which
proceed as shown in Fig. 4. In this respect, the final state interactions of
$\bar{K}\Xi\to\bar{K}\Xi$ is already taken into account in the effective
coupling of the $\Omega(2012)\bar{K}\Xi$ vertex.

Figure 4: Feynman diagram for the effective two-body decay of
$\Omega(2012)\to\bar{K}\Xi$ through the re-scattering of $\bar{K}\Xi(1530)$
and $\eta\Omega$ in coupled channels.

At the pole position of $\Omega(2012)$, one can extract the effective strong
coupling of $\Omega(2012)$ to the $\bar{K}\Xi$ channel,
$g_{\Omega(2012)\bar{K}\Xi}$, from the unitarized scattering amplitude
$T_{\bar{K}\Xi\to\bar{K}\Xi}$, which was obtained by solving the Bethe-
Salpeter equation and considering the re-scattering of all the three coupled
channels. Close to the pole at $z_{R}=M_{R}-i\Gamma_{R}/2$, the scattering
amplitude $T_{\bar{K}\Xi\to\bar{K}\Xi}$ can be written as

$\displaystyle
T_{\bar{K}\Xi\to\bar{K}\Xi}=\frac{|g_{\Omega(2012)\bar{K}\Xi}|^{2}}{\sqrt{s}-z_{R}}.$
(4)

Then the partial decay width of the $\Omega(2012)\to\bar{K}\Xi$ can be easily
obtained with

$\Gamma_{\Omega(2012)\rightarrow\bar{K}\Xi}=\frac{|g_{\Omega(2012)\bar{K}\Xi}|^{2}}{2\pi}\frac{M_{\Xi}}{M}q_{\bar{K}},$
(5)

where $g_{\Omega(2012)\bar{K}\Xi}$ is the effective coupling constant of the
$\Omega(2012)\bar{K}\Xi$ vertex obtained as explained above, and $q_{\bar{K}}$
is the momentum of $\bar{K}$ in the rest frame of $\Omega(2012)$, which is

$\displaystyle
q_{\bar{K}}=\frac{\sqrt{[M^{2}-(m_{\bar{K}}+M_{\Xi})^{2}][M^{2}-(m_{\bar{K}}-M_{\Xi})^{2}]}}{2M}.$
(6)

In general, since the $\Omega(2012)\to\bar{K}\Xi$ decay is in $d$-wave, the
$\Gamma_{\Omega(2012)\rightarrow\bar{K}\Xi}$ should be proportional to
$q^{5}_{\bar{K}}$, but the transition amplitudes
$T_{\bar{K}\Xi(1530)\to\bar{K}\Xi}$ and $T_{\eta\Omega\to\bar{K}\Xi}$
potentials incorporate the four extra powers of $q_{\bar{K}}$. (More details
can be found in Ref. Pavao:2018xub  and in Refs. Sarkar:2005ap ; Roca:2006sz
for the case of the $\Lambda(1520)$ resonance where the interactions of
$\bar{K}N$ and $\pi\Sigma$ in $d$-wave are included.) In this way, the
obtained effective coupling constant $g_{\Omega(2012)\bar{K}\Xi}$ is
dimensionless. Assuming that only the three body and two-body strong decays
contribute to the total decay width of $\Omega(2012)$, and with the new
measurements for the ratio ${\cal R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}$, we obtain

$\displaystyle|g_{\Omega(2012)\bar{K}\Xi}|=0.28\pm 0.14,$ (7)

with the error obtained from the uncertainties of ${\cal
R}^{\bar{K}\pi\Xi}_{\bar{K}\Xi}$ and the total decay width of $\Omega(2012)$.

##  IV Other explanations of the $\Omega(2012)$

On the other hand, although the $\Omega(2012)$ resonance was proposed to be a
dynamically generated state, it is difficult to be clearly distinguished from
the classical quark model state due to the possible large mixing of molecular
component and the $sss$ configuration. In fact, the $\Omega$ excited states
were also investigated in classical quark models Capstick:1986bm ;
Loring:2001ky ; Pervin:2007wa ; Chen:2009de ; Faustov:2015eba  and in the
five-quark picture Yuan:2012zs ; An:2013zoa ; An:2014lga . In a pioneering
study Chao:1980em , the $\Omega$ excited states were predicted employing a
quark model with ingredients suggested by QCD, and the mass of one predicted
state with $J^{P}=3/2^{-}$ is about $2020$ MeV.

In the framework of the chiral quark model, the authors of Refs. Liu:2020yen ;
Hu:2022pae  performed dynamical calculations of pentaquark systems with
$sssu\bar{u}$ and $sss\bar{d}d$, and conclude that the $\Omega(2012)$ can be
interpreted as a $\bar{K}\Xi(1530)$ molecular state with isospin zero and
spin-parity $J^{P}=3/2^{-}$. The mass and two-body strong decays of the
$\Omega(2012)$ were also studied within the quantum chromodynamics sum rules
Aliev:2018yjo ; Aliev:2018syi ; Su:2024lzy  and constituent quark model
Xiao:2018pwe ; Wang:2018hmi ; Liu:2019wdr ; Wang:2022zja ; Zhong:2022cjx , and
these studies showed that the $\Omega(2012)$ could be a good candidate of the
first orbital $(1P)$ excitations of the ground-state $\Omega$ baryon with
$J^{P}=3/2^{-}$. A spin-orbit partner with $J^{P}=1/2^{-}$ of $\Omega(2012)$
was naturally predicted in Ref. Arifi:2022ntc  within the constituent quark
model. In addition, within a nonretivistic constituent quark potential model,
a $J^{P}=1/2^{-}$ state $\Omega(1^{2}P_{1/2^{-}})$ with mass around 1950 MeV
and width about 12 MeV was also predicted Liu:2019wdr . Such a $J^{P}=1/2^{-}$
state is hard to be generated in the molecular scenario of $\Omega(2012)$.
Therefore, the discovery of the spin-orbit partner with $J^{P}=1/2^{-}$ would
be crucial in distinguishing between the three-quark model and the hadronic
molecular nature for the $\Omega(2012)$ state.

Meanwhile, in Ref. Lu:2022puv , a systematic analysis of the mass and strong
decays of the $\Omega(2012)$ resonance was performed in a coupled-channel
approach of a bare three-quark state and a composite meson-baryon state.
Besides the coupling of the bare state and $\bar{K}\Xi(1530)$ channel, the
effective meson-baryon interactions are also included. It was found that both
the three-quark core and $\bar{K}\Xi(1530)$ channel are essential for the
description of $\Omega(2012)$ resonance.

##  V Propects for the obeservation of $\Omega(2012)$ from its radiative decay
of $\Omega(2012)\to\gamma\Omega$

Within the three quark model, in Ref. Liu:2019wdr , it was found that the
$\Omega(2012)$ resonance is most likely to be a $J^{P}=3/2^{-}$ 1P-wave state
$\Omega(1^{2}P_{3/2^{-}})$, and it was pointed out that the $\Omega(2012)$ has
a large potential to be observed in the $\gamma\Omega$ channel. While, within
the molecular picture, the ratiative decay of $\Omega(2012)\to\gamma\Omega$
can also be proceed with the triangle diagram as shown in Fig. 5. The
radiative decay of $\Omega(2012)$ is difficult to be measured by the Belle
Collaboration, but, it could be done by the BESIII Collaboration and the
planed high-luminosity electron-positron collision machine Super $\tau$-Charm
Facility Guo:2022kdi .

Figure 5: The radiative decay of $\Omega(2012)\to\gamma\Omega$ through the
triangle diagram.

Until now, only the mass and width of the $\Omega(2012)$ resonance were
measured, more experimental information is really needed. For example, the
angular distributions for its two-body strong decay of
$\Omega(2012)\to\bar{K}\Xi$, from where its $J^{P}$ quantum numbers can be
determined. Furthermore, both on theoretical and experimental sides, such
studies on the radiative decay of $\Omega(2012)\to\gamma\Omega$ are mostly
welcome, and they will be help to further understand the nature of the
$\Omega(2012)$ state.

##  VI Summary

The discovery of $\Omega(2012)$ state has attracted many theoretical works,
trying to understand its nature with different models. In the molecular
picture, the $\Omega(2012)$ state appears to be dynamically generated from the
coupled channels interactions of the $\bar{K}\Xi(1530)$ and $\eta\Omega$ in
$s$-wave and $\bar{K}\Xi$ in $d$-wave. In such a scenario, the $\Omega(2012)$
is interpreted as a $3/2^{-}$ molecule state, and most of its properties can
be well accommodated. More precise experimental measurements on the strong and
radiative decays of the $\Omega(2012)$ state are expected to provide valuable
insights into its nature. These measurements can be carried out not only by
the Belle collaboration but also by the Belle II, LHCb, and BESIII
Collaborations in the near future, as well as by the next generation
facilities KLF:2020gai ; Aoki:2021cqa . Searching for its symmetry and three-
body counterparts as well as studying the $\eta\Omega$ and $\bar{K}\Xi(1530)$
interactions using the femtoscopic technique will also be useful in verifying
the molecular nature of $\Omega(2012)$ Liu:2024uxn .

## Acknowledgments

We would like to thank Profs. Eulogio Oset, Bing-Song Zou, and Cheng-Ping Shen
for fruitful discussions. This work is partly supported by the National Key
R&D Program of China under Grant No. 2023YFA1606703 and by the National
Natural Science Foundation of China under Grant Nos. 12075288 and 12361141819.
It is also supported by the Youth Innovation Promotion Association CAS.

## References

  * (1) J. Yelton et al. [Belle Collaboration], “Observation of an Excited $\Omega^{-}$ Baryon,” Phys. Rev. Lett. 121, 052003 (2018).
  * (2) R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” Prog. Theor. Exp. Phys. 2022, 083C01 (2022) and 2023 update.
  * (3) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, “Hadronic molecules,” Rev. Mod. Phys. 90, 015004 (2018) [erratum: Rev. Mod. Phys. 94, 029901 (2022)].
  * (4) E. E. Kolomeitsev and M. F. M. Lutz, “On baryon resonances and chiral symmetry,” Phys. Lett. B 585, 243 (2004).
  * (5) S. Sarkar, E. Oset and M. J. Vicente Vacas, “Baryonic resonances from baryon decuplet-meson octet interaction,” Nucl. Phys. A 750, 294 (2005) Erratum: [Nucl. Phys. A 780, 90 (2006)].
  * (6) C. García-Recio, J. Nieves and L. L. Salcedo, “Meson-Baryon $s$-wave Resonances with Strangeness -3,” Eur. Phys. J. A 31, 540 (2007).
  * (7) S. Q. Xu, J. J. Xie, X. R. Chen and D. J. Jia, “The $\Xi^{*}\bar{K}$ and $\Omega\eta$ interaction within a chiral unitary approach,” Commun. Theor. Phys. 65, 53 (2016).
  * (8) M. P. Valderrama, “$\Omega(2012)$ as a hadronic molecule,” Phys. Rev. D 98, 054009 (2018).
  * (9) Y. H. Lin and B. S. Zou, “Hadronic molecular assignment for the newly observed $\Omega^{*}$ state,” Phys. Rev. D 98, 056013 (2018).
  * (10) Y. Huang, M. Z. Liu, J. X. Lu, J. J. Xie and L. S. Geng, “Strong decay modes $\bar{K}\Xi$ and $\bar{K}\Xi\pi$ of the $\Omega(2012)$ in the $\bar{K}\Xi(1530)$ and $\eta\Omega$ molecular scenario,” Phys. Rev. D 98, 076012 (2018).
  * (11) R. Pavao and E. Oset, “Coupled channels dynamics in the generation of the $\Omega(2012)$ resonance,” Eur. Phys. J. C 78, 857 (2018).
  * (12) Y. Lin, Y. H. Lin, F. Wang, B. Zou and B. S. Zou, “Reanalysis of the newly observed $\Omega^{*}$ state in a hadronic molecule model,” Phys. Rev. D 102, 074025 (2020).
  * (13) J. X. Lu, C. H. Zeng, E. Wang, J. J. Xie and L. S. Geng, “Revisiting the $\Omega(2012)$ as a hadronic molecule and its strong decays,” Eur. Phys. J. C 80, 361 (2020).
  * (14) N. Ikeno, W. H. Liang and E. Oset, “Molecular nature of the $\Omega_{c}(3120)$ and its analogy with the $\Omega(2012)$,” Phys. Rev. D 109, 054023 (2024).
  * (15) R. Aaij et al. [LHCb], “Observation of five new narrow $\Omega_{c}^{0}$ states decaying to $\Xi_{c}^{+}K^{-}$,” Phys. Rev. Lett. 118, 182001 (2017).
  * (16) C. H. Zeng, J. X. Lu, E. Wang, J. J. Xie and L. S. Geng, “Theoretical study of the $\Omega(2012)$ state in the $\Omega_{c}^{0}\to\pi^{+}\Omega(2012)^{-}\to\pi^{+}(\bar{K}\Xi)^{-}$ and $\pi^{+}(\bar{K}\Xi\pi)^{-}$ decays,” Phys. Rev. D 102, 076009 (2020).
  * (17) N. Ikeno, W. H. Liang, G. Toledo and E. Oset, “Interpretation of the $\Omega_{c}\to\pi^{+}\Omega(2012)\to\pi^{+}(\bar{K}\Xi)$ relative to $\Omega_{c}\to\pi^{+}\bar{K}\Xi$ from the $\Omega(2012)$ molecular perspective,” Phys. Rev. D 106, 034022 (2022).
  * (18) Y. Li et al. [Belle], “Evidence for the decay $\Omega_{c}^{0}\to\pi^{+}\Omega(2012)^{-}\to\pi^{+}(\bar{K}\Xi)^{-}$,” Phys. Rev. D 104, 052005 (2021).
  * (19) A. J. Arifi, D. Suenaga, A. Hosaka and Y. Oh, “Strong decays of multistrangeness baryon resonances in the quark model,” Phys. Rev. D 105, 094006 (2022).
  * (20) S. Jia et al. [Belle], “Search for $\Omega(2012)\to K\Xi(1530)\to K\pi\Xi$ at Belle,” Phys. Rev. D 100, 032006 (2019).
  * (21) [Belle], “Observation of $\Omega(2012)^{-}\to\Xi(1530)\bar{K}$ and measurement of the effective couplings of $\Omega(2012)^{-}$ to $\Xi(1530)\bar{K}$ and $\Xi\bar{K}$,” [arXiv:2207.03090 [hep-ex]].
  * (22) S. Sarkar, E. Oset and M. J. Vicente Vacas, “Chiral coupled channel dynamics of the $\Lambda(1520)$ and the $K^{-}p\to\pi^{0}\pi^{0}\Lambda$ reaction,” Phys. Rev. C 72, 015206 (2005).
  * (23) L. Roca, S. Sarkar, V. K. Magas and E. Oset, “Unitary coupled channel analysis of the $\Lambda(1520)$ resonance,” Phys. Rev. C 73, 045208 (2006).
  * (24) S. Capstick and N. Isgur, “Baryons in a Relativized Quark Model with Chromodynamics,” Phys. Rev. D 34, 2809 (1986) [AIP Conf. Proc. 132, 267 (1985)].
  * (25) U. Loring, B. C. Metsch and H. R. Petry, “The Light baryon spectrum in a relativistic quark model with instanton induced quark forces: The Strange baryon spectrum,” Eur. Phys. J. A 10, 447 (2001).
  * (26) M. Pervin and W. Roberts, “Strangeness -2 and -3 baryons in a constituent quark model,” Phys. Rev. C 77, 025202 (2008).
  * (27) Y. Chen and B. Q. Ma, “Light flavor baryon spectrum with higher order hyperfine interactions,” Nucl. Phys. A 831, 1-21 (2009).
  * (28) R. N. Faustov and V. O. Galkin, “Strange baryon spectroscopy in the relativistic quark model,” Phys. Rev. D 92, 054005 (2015).
  * (29) S. G. Yuan, C. S. An, K. W. Wei, B. S. Zou and H. S. Xu, “Spectrum of low-lying $s^{3}Q\bar{Q}$ configurations with negative parity,” Phys. Rev. C 87, 025205 (2013).
  * (30) C. S. An, B. C. Metsch and B. S. Zou, “Mixing of the low-lying three- and five-quark $\Omega$ states with negative parity,” Phys. Rev. C 87, 065207 (2013).
  * (31) C. S. An and B. S. Zou, “Low-lying $\Omega$ states with negative parity in an extended quark model with Nambu-Jona-Lasinio interaction,” Phys. Rev. C 89, 055209 (2014).
  * (32) K. T. Chao, N. Isgur and G. Karl, “Strangeness -2 and -3 Baryons in a Quark Model With Chromodynamics,” Phys. Rev. D 23, 155 (1981).
  * (33) X. Liu, H. Huang, J. Ping and D. Chen, “Investigating $\Omega(2012)$ as a molecular state,” Phys. Rev. C 103, 025202 (2021).
  * (34) X. Hu and J. Ping, “Analysis of $\Omega$(2012) as a molecule in the chiral quark model,” Phys. Rev. D 106, 054028 (2022).
  * (35) T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, “Interpretation of the newly discovered $\Omega$(2012),” Phys. Rev. D 98, 014031 (2018).
  * (36) T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, “Nature of the $\Omega(2012)$ through its strong decays,” Eur. Phys. J. C 78, 894 (2018).
  * (37) N. Su, H. X. Chen, P. Gubler and A. Hosaka, “Investigation on the $\Omega(2012)$ from QCD sum rules,” [arXiv:2405.06958 [hep-ph]].
  * (38) L. Y. Xiao and X. H. Zhong, “Possible interpretation of the newly observed $\Omega$(2012) state,” Phys. Rev. D 98, 034004 (2018).
  * (39) Z. Y. Wang, L. C. Gui, Q. F. Lü, L. Y. Xiao and X. H. Zhong, “Newly observed $\Omega(2012)$ state and strong decays of the low-lying $\Omega$ excitations,” Phys. Rev. D 98, 114023 (2018).
  * (40) M. S. Liu, K. L. Wang, Q. F. Lü and X. H. Zhong, “$\Omega$ baryon spectrum and their decays in a constituent quark model,” Phys. Rev. D 101, 016002 (2020).
  * (41) K. L. Wang, Q. F. Lü, J. J. Xie and X. H. Zhong, “Toward discovering the excited $\Omega$ baryons through nonleptonic weak decays of $\Omega_{c}$,” Phys. Rev. D 107, 034015 (2023).
  * (42) H. H. Zhong, R. H. Ni, M. Y. Chen, X. H. Zhong and J. J. Xie, “Further study of $\Omega^{*}|1P_{3/2^{-}}>$ within a chiral quark model,” Chin. Phys. C 47, 063104 (2023).
  * (43) Q. F. Lü, H. Nagahiro and A. Hosaka, “Understanding the nature of $\Omega$(2012) in a coupled-channel approach,” Phys. Rev. D 107, 014025 (2023).
  * (44) F. K. Guo, H. Peng, J. J. Xie and X. Zhou, “Hadron spectroscopy at STCF,” [arXiv:2203.07141 [hep-ph]].
  * (45) M. Amaryan et al. [KLF], “Strange Hadron Spectroscopy with Secondary KL Beam in Hall D,” [arXiv:2008.08215 [nucl-ex]].
  * (46) K. Aoki, H. Fujioka, T. Gogami, Y. Hidaka, E. Hiyama, R. Honda, A. Hosaka, Y. Ichikawa, M. Ieiri and M. Isaka, et al. “Extension of the J-PARC Hadron Experimental Facility: Third White Paper,” [arXiv:2110.04462 [nucl-ex]].
  * (47) M. Z. Liu, Y. W. Pan, Z. W. Liu, T. W. Wu, J. X. Lu and L. S. Geng, [arXiv:2404.06399 [hep-ph]].