# The Bruhat Order of a Finite Coxeter Group and Elnitsky Tilings

Rob Nicolaides and Peter Rowley

###### Abstract

Suppose that $W$ is a finite Coxeter group and $W_{J}$ a standard parabolic
subgroup of $W$. The main result proved here is that for any for any $w\in W$
and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon,
where $m=[W:W_{J}]$. The proof is constructive and draws together the work on
E-embedding in [7] and the deletion order in [9]. Computer programs which
produce such tilings may be downloaded from [10] and here we also present
examples of the tilings for, among other Coxeter groups, the exceptional
Coxeter group $\mathrm{E}_{8}$.

##  1 Introduction

In his PhD thesis, [2] (see also [3]), Elnitsky examined a fascinating
relationship between the reduced words of a selection of Coxeter groups and
certain tilings of polygons. More specifically, for each of the families of
Coxeter groups of types A, B and D, he produces an elegant bijection between
classes of their reduced words and particular tilings of polygons. These, now
referred to as Elnitsky tilings, were produced using the degree $n$
permutation representation for the Coxeter group $\mathrm{A}_{n-1}$ and the
signed permutation representations for types B and D. The methods revealed in
this paper make use of any of the faithful permutation representations on the
cosets of standard parabolic subgroups of any finite Coxeter group, so
yielding an extensive collection of tilings, with the tilings in [2] as a
special case. Our main result is as follows.

######  Theorem 1.1.

Suppose that $W$ is a finite Coxeter group with $W_{J}$ a standard parabolic
subgroup of $W$ which is core-free. Set $m=[W:W_{J}]$. Then for any $w\in W$
and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon.

Theorem 1.2 of [7], shows how certain embeddings (that is, injective
homomorphisms) of Coxeter groups into symmetric groups naturally give rise to
bijections between classes of reduced words and certain tilings of polygons.
These embeddings, called E-embeddings with the E standing for Elnitsky, were
introduced in [7] and are defined as follows.

######  Definition 1.2.

Suppose that $W$ is a Coxeter group and that
$\varphi:W\hookrightarrow\mathrm{Sym}({n})$ is an embedding. Then $\varphi$ is
an E-embedding for $W$ if for all $u,v\in W,$

$u<_{R}v\text{ implies }\varphi(u)<_{B}\varphi(v).$

Here, $<_{R}$ and $<_{B}$ denote the weak right order and the Bruhat order
respectively, their full definitions are recalled in Section 2. We use
$\mathrm{Sym}({n})$ to denote the symmetric group of degree $n$ which we
regard as a Coxeter group with its fundamental reflections being
$\\{(i,i+1)\;|\;i=1,\dots,n-1\\}$. Further we shall write our permutations on
the right of the elements in $\\{1,\dots,n\\}$ and so composition is evaluated
from left to right. For a Coxeter group $W$, $S$ will denote its set of
fundamental (or simple) reflections.

No means for constructing E-embeddings were provided in [7] and so the focus
of this article is to provide a universal construction for all finite Coxeter
groups. This is done by exploring a special subset of the E-embeddings which
we call strong E-embeddings, and will enable us to give explicit constructions
of the tilings.

######  Definition 1.3 (Strong E-embedding).

Suppose that $W$ is a Coxeter group. Let
$\varphi:W\hookrightarrow\mathrm{Sym}({n})$ be an embedding for some positive
integer $n$. Then $\varphi$ is a strong E-embedding if for all $w\in W$ and
$s\in S$,

$w<_{R}ws\text{\quad implies \quad}\varphi(w)<_{B}\varphi(w)t_{i}$

for each $i=1\ldots k$, where $\varphi(s)=t_{1}\ldots t_{k}$ is such that each
$t_{i}$ is a transposition in $\mathrm{Sym}({n})$ and $t_{i},t_{j}$ are
disjoint for all $1\leq i<j\leq k$.

As we will see in Theorems 1.5 and 1.6, in order to construct strong
E-embedding, we need a total order of the Coxeter group $W$ which is a
refinement of the Bruhat order on $W$. There are many possible choices for
total orders which are refinements of the Bruhat order. A particularly easy
type of example may be made by first ordering by element length and among
those of a fixed length then dictating an arbitary total order. The last part
of this recipe is, to say the least, rather unsatisfactory given the richness
of Coxeter groups particularly in connection with the Bruhat order. We prefer
to employ the deletion order, which is canonical, is a refinement of the
Bruhat order and also has many interesting properties (see [9] for more on
this order). One particularly noteworthy feature is that (except in small
cases) the deletion order is not ranked by element length. Moreover the
tilings obtained using the deletion order align with the Elnitsky tilings in
[3] for the Coxeter groups of type A, B and D. In Section 4 where we present
examples of tilings for groups of type
$\mathrm{D}_{5},\mathrm{A}_{2}\times\mathrm{A}_{3}$ band $\mathrm{E}_{8}$, we
shall employ the deletion order. These examples were produced with programs
which are available from [10] so as the reader can produce tilings for other
elements of other Coxeter groups.

In Section 2 we review some standard results we will need about Coxeter
groups, focussing on the Bruhat order and minimal coset representatives for
standard parabolic subgroups. Then Section 3 is devoted to proving all our
main results, starting off by showing that strong E-embeddings are indeed
E-embeddings. We then show that the definition can be reformulated as follows.

######  Proposition 1.4.

Suppose that $W$ is a Coxeter group with $T$ its set of reflections. Let
$\varphi:W\hookrightarrow\mathrm{Sym}({n})$ be an embedding for some positive
integer $n$. Then the following are equivalent:

  1. (i)

$\varphi$ is a strong E-embedding;

  2. (ii)

for all $w\in W$ and for all $t\in T$,

$w<_{B}wt\text{\quad implies \quad}\varphi(w)<_{B}\varphi(w)t_{i}$

for each $i=1\ldots k$, where $\varphi(t)=t_{1}\ldots t_{k}$ is such that each
$t_{i}$ is a transposition in $\mathrm{Sym}({n})$ and $t_{i},t_{j}$ are
disjoint for all $1\leq i<j\leq k$.

The remainder of Section 3 aims to construct strong E-embeddings from Cayley
embeddings equipped with a total ordering. We will use $\ll$ to denote a total
ordering on $W$ meaning for all distinct $u,v\in W$ either $u\ll v$ or $v\ll
u$. We define $L_{\ll}(w)=|\\{u\in W\,|\,u\ll w\\}|$. Then the Cayley
embedding induced from $\ll$ is given by
$\varphi_{\ll}:W\hookrightarrow\mathrm{Sym}({|W|})$ where

$L_{\ll}(u)\varphi_{\ll}(w)=L_{\ll}(uw^{-1})\;\mbox{for all}\;u,v\in W.$

This means that

$\varphi_{\ll}(s)=\prod_{w<_{R}ws}(L_{\ll}(w),L_{\ll}(ws))$

for each $s\in S$.

We ask for which total orderings $\ll$, does $\varphi_{\ll}$ happen to be a
strong E-embedding? The answer is given by our next theorem.

######  Theorem 1.5.

Let $\ll$ be a total ordering of $W$ such that $L_{\ll}(1)=0$. Then the
following are equivalent:

  1. (i)

$\ll$ is a refinement of the Bruhat order;

  2. (ii)

$\varphi_{\ll}$ is a strong E-embedding.

For $\ll$ to be a refinement of the Bruhat order, it needs to satisfy the
condition that $u<_{B}v$ implies $u\ll v$ for all distinct $u,v\in W$. Theorem
1.6 gives a more general result than Theorem 1.5. It relaxes the condition
that $\varphi_{\ll}$ needs to be a Cayley embedding where instead, we can take
any $J\subseteq S$ and consider permutations restricted to the minimal coset
representatives, $W^{J}$, of $W_{J}=\,<J>$ (precise details are given in
Section 3).

######  Theorem 1.6.

Let $J\subseteq S$ and $\ll^{J}$ be a total ordering of $W^{J}$ such that
$L_{\ll^{J}}(1)=0$. Then the following are equivalent:

  1. (i)

$\varphi_{\ll}^{J}$ is a strong E-embedding;

  2. (ii)

$\ll^{J}$ is a total refinement of $<_{B}$, restricted to $W^{J}$.

For further material on tilings the reader may consult [4], [5], [8] [11] and
[12].

##  2 Background

A group, $W$, is a _(finitely generated) Coxeter group_ if it admits a
presentation over some finite set $S=\\{s_{1},\ldots,s_{n}\\}$, satisfying

$W=\langle\,S\,\,|\,\,(s_{i}s_{j})^{m_{ij}}=1\,\rangle$

Here, for each $i\neq j$, $m_{ij}$ is a positive integer such that
$m_{ij}=m_{ji}>1$ and $m_{ii}=1$. We call the pair $(W,S)$ a _Coxeter system_
and the elements of $S$ the fundamental reflections of $W$. The set of
_reflections_ of a Coxeter system is the set $T:=S^{W}=\\{ws_{i}w^{-1}\;|\;\
w\in W,s_{i}\in S\\}$.

Let $\Gamma(W)$ be the graph whose nodes are $S$ and whose set of edges is
$\\{\\{s_{i},s_{j}\\}\,|\,m_{ij}\neq 2\\}$. Sometimes $\Gamma(W)$ is called
the Dynkin diagram of $W$. A Coxeter group is called _irreducible_ if
$\Gamma(W)$ is connected. For the remainder of this paper, we assume the
Coxeter groups we consider are finite.

An example, as already mentioned, of a Coxeter group is the symmetric group.
Take $S=\\{s_{1},\ldots,s_{n}\\}$ and consider

$W=\langle\,S\,\,|\,\,(s_{i}s_{j})^{m_{ij}}=1\,\rangle$

such that $m_{ij}=3$ if $|i-j|=1$ and $m_{ij}=2$ otherwise for $i\neq j$. The
map sending $s_{i}$ to the adjacent transposition
$(i,i+1)\in\mathrm{Sym}({n+1})$, extends uniquely to an isomorphism from $W$
to $\mathrm{Sym}({n+1})$.

For a general Coxeter group, $W$, and $w\in W$, a _word_ for $w$ over $S$ is
an expressions of the form $w=s_{i_{1}}s_{i_{2}}\ldots s_{i_{k}}$ where
$s_{i_{j}}\in S$ for each $j=1,\ldots,k$. A word is called _reduced_ if it is
of minimal length and we denote that length by $\ell(w)$.

Two important posets related to the reduced words of Coxeter groups are the
_weak (right) order_ , denoted $<_{R}$, and the _Bruhat order_ , denoted
$<_{B}$. The weak order is defined as the reflexive, transitive closure of the
covering relations $w\triangleleft_{R}ws$ if and only if $\ell(ws)=\ell(w)+1$
where $w\in W$ and some $s\in S$. Similarly, the Bruhat order is defined by
the covering relations $w\triangleleft_{B}wt$ if and only if
$\ell(wt)=\ell(w)+1$, where $w\in W$ and some $t\in T$. The weak order can be
characterized in terms of prefixes of reduced words (Proposition 3.1.2 of
[1]): for all $u,v\in W$, $u<_{R}v$ if and only if there is a reduced word for
$u$ that is a prefix of some reduced word for $v$. Less obviously, the Bruhat
order can also be characterized in terms of subwords of reduced words as
stated in Lemma 2.1.

######  Lemma 2.1.

For $u,v\in W$, the following are equivalent:

  1. $(i)$

$u<_{B}w$;

  2. $(ii)$

for every reduced word of $w$, there exists a subword that is a reduced word
for $u$; and

  3. $(iii)$

some reduced word of $w$ has a subword that is a reduced word for $u$.

###### Proof.

This is Corollary 2.2.3 of [1]. ∎

For the symmetric group, the Bruhat order has the following description in
terms of its usual group action.

######  Lemma 2.2.

Let $w,t\in\mathrm{Sym}({n})$ for some positive integer $n$ such that $t$ is a
transposition $t=(a,b)$ with $1\leq a<b\leq n$. Then

$w<_{B}wt$

if and only if $(a)w^{-1}<(b)w^{-1}$.

###### Proof.

This is a direct consequence of Lemma 2.1.4 of [1] and taking into account
that we multiply permutations from left to right. ∎

######  Lemma 2.3.

Let $u,v\in W$ and $t\in T$ be such that $u<_{B}ut$ and $v<_{B}tv$, then
$uv<_{B}utv$.

###### Proof.

This is Lemma 2.2.10 of [1]. ∎

Let $J\subseteq S$, then we define $W_{J}=\langle J\rangle$. These subgroups
are known as the _standard parabolic subgroups_. Each left (or right) coset of
$W_{J}$ has a unique element of minimal length. The set of these minimal left
coset representatives is denoted by $W^{J}$. This leads to a unique
decomposition property.

######  Lemma 2.4.

For all $J\subseteq S$ and $w\in W$, there exists unique $w^{J}\in W^{J}$,
$w_{J}\in W_{J}$ such that $w=w^{J}w_{J}$. Moreover,
$\ell(w)=\ell(w^{J})+\ell(w_{J})$.

###### Proof.

This is Proposition 2.2.4 of [1]. ∎

If $w\in W$, we will use $w^{J}$ and $w_{J}$ to denote its unique factors as
given in Lemma 2.4 from $W^{J}$ and $W_{J}$, respectively.

######  Lemma 2.5.

Let $J\subseteq S$ and $u,v\in W$. If $u<_{B}v$, then $u^{J}<_{B}v^{J}$.

###### Proof.

This is Lemma 2.5.1 of [1]. ∎

We note that if $I\subseteq J\subseteq S$, then $W^{J}\subseteq W^{I}$. To see
this, note that $W_{I}\leq W_{J}$ and so each coset of $W_{J}$ is partitioned
by those of $W_{I}$ by Lagrange’s theorem. Since each coset of a standard
parabolic subgroup has a unique minimal representative, any such member of a
coset of $W_{J}$ must also be the minimal element of the coset of $W_{I}$
containing it.

For each $J\subseteq S$, we may equip $W^{J}$ with a total ordering. We will
denote such an ordering by $\ll^{J}$ and define
$L_{\ll}^{J}(w^{J})=|\\{u^{J}\in W^{J}\,|\,u^{J}\ll^{J}w^{J}\\}|$. Analogous
to $\varphi_{\ll}$, we may define
$\varphi_{\ll}^{J}:W\hookrightarrow\mathrm{Sym}({|W^{J}|})$ by

$(L_{\ll}^{J}(u^{J}))\varphi_{\ll}^{J}(w)=L_{\ll}^{J}((u^{J}w^{-1})^{J})$

for all $u^{J}\in W^{J}$ and $w\in W$. For all $w\in W^{J}$ such that $ws\in
W^{J}$ also, we have

$\varphi_{\ll}^{J}(s)=\prod_{w<_{R}ws}(L_{\ll}^{J}(w),L_{\ll}^{J}(ws)).$

These maps will be embeddings provided $W_{J}$ is a core-free subgroup of $W$.

When $J=\emptyset$, the definitions of $\ll^{\emptyset},L_{\ll}^{\emptyset}$
and $\varphi_{\ll}^{\emptyset}$ coincide with those of $\ll,L_{\ll}$ and
$\varphi$ respectively. For convenience then, we omit the appearance of
$J=\emptyset$ in their notation.

##  3 Strong E-embeddings

Our first aim of this section is to prove that strong E-embeddings are indeed
E-embeddings. This is done in Proposition 3.2 with Lemma 3.1 acting as an
auxiliary lemma.

######  Lemma 3.1.

Let $t_{1},\ldots,t_{k}\in\mathrm{Sym}({n})$ be mutually disjoint
transpositions for some positive integer $n$. For all $w\in\mathrm{Sym}({n})$,
if $w<_{B}wt_{i}$ for each $i=1,\ldots,k$, then $w<_{B}wt_{1},\ldots t_{k}$.

###### Proof.

Write $t_{i}=(a_{i},b_{i})$ for each $i=1,\ldots,k$. Since
$t_{1},\ldots,t_{k}$ are disjoint, $a_{i}$ and $b_{i}$ are moved by only
$t_{i}$. Equivalently, $\\{a_{i},b_{i}\\}\cap\\{a_{j},b_{j}\\}=\emptyset$ for
all distinct pairs $i,j=1,\ldots,k$. We may now iteratively apply Lemma 2.2 to
achieve the chain $w<_{B}wt_{1}<_{B}wt_{1}t_{2}<_{B}\ldots<_{B}wt_{1},\ldots
t_{k}$. ∎

######  Proposition 3.2.

Suppose $W$ is a Coxeter group and let
$\varphi:W\hookrightarrow\mathrm{Sym}({n})$ be an embedding for some positive
integer $n$. If $\varphi$ is a strong E-embedding, then $\varphi$ is an
E-embedding.

###### Proof.

Suppose $\varphi$ is a strong E-embedding and consider $w\in W$ and $s\in S$
such that $w<_{R}ws$. If $\varphi(s)=t_{1}\ldots t_{k}$ as the product of
disjoint transpositions, then, since $\varphi$ is a strong E-embedding,
$\varphi(w)<_{B}\varphi(w)t_{i}$ for each $i=1,\ldots,k$. By Lemma 3.1,
$\varphi(w)<_{B}\varphi(w)t_{1}\ldots t_{k}=\varphi(ws)$. ∎

Now we prove Proposition 1.4 where we show that strong E-embeddings can be
reformulated in terms of the Bruhat order only.

###### Proof of Proposition 1.4.

Since $<_{B}$ is a refinement of $<_{R}$, $(ii)$ necessarily implies $(i)$.

Now suppose that $\varphi$ is a strong E-embedding and take $w\in W$, $t\in T$
such that $w\triangleleft_{B}wt$. Since $w\triangleleft_{B}wt$, Lemma 2.1
informs us that there exists $u,v\in W$ and $s\in S$ such that $wt=usv^{-1}$,
$w=uv^{-1}$. Some consequences of this are:

  * •

$\ell(wt)=\ell(u)+\ell(s)+\ell(v^{-1})$;

  * •

$u<_{R}us$ and $v<_{R}vs$; and

  * •

$t=s^{v}=vsv^{-1}$.

We may write $\varphi(t)$ and $\varphi(s)$ as products of disjoint
transpositions thus $\varphi(t)=t_{1}\dots t_{k}$ and $\varphi(s)=s_{1}\dots
s_{k}$, where $t_{i}=(a_{i},b_{i})$ and $s_{i}=(\alpha_{i},\beta_{i})$ for
$i=1,\dots,k$. Since $t=s^{v}=vsv^{-1}$, conjugation in $\mathrm{Sym}({n})$
ensures that both $\varphi(t)$ and $\varphi(s)$ are products of the same
number of transpositions, $k$, and we may index in such a way that
$(\alpha_{i})\varphi(v^{-1})=a_{i}$ and $(\beta_{i})\varphi(v^{-1})=b_{i}$.

It remains to show that $\varphi(w)<_{B}\varphi(w)t_{i}$ for each
$i=1,\ldots,k$. By Lemma 2.2, $\varphi(w)<_{B}\varphi(w)t_{i}$ if and only if
$(a_{i})\varphi(w)^{-1}<(b_{i})\varphi(w)^{-1}$. Now the following are
equivalent.

$\displaystyle\varphi(w)$ $\displaystyle<_{B}\varphi(w)t_{i}$
$\displaystyle(a_{i})\varphi(w)^{-1}$ $\displaystyle<(b_{i})\varphi(w)^{-1}$
$\displaystyle(\alpha_{i})\varphi(v^{-1})\varphi(w)^{-1}$
$\displaystyle<(\beta_{i})\varphi(v^{-1})\varphi(w)^{-1}$
$\displaystyle(\alpha_{i})\varphi(v^{-1})\varphi(uv^{-1})^{-1}$
$\displaystyle<(\beta_{i})\varphi(v^{-1})\varphi(uv^{-1})^{-1}$
$\displaystyle(\alpha_{i})\varphi(v^{-1}vu^{-1})$
$\displaystyle<(\beta_{i})\varphi(v^{-1}vu^{-1})$
$\displaystyle(\alpha_{i})\varphi(u)^{-1}$
$\displaystyle<(\beta_{i})\varphi(u)^{-1}$ $\displaystyle\varphi(u)$
$\displaystyle<_{B}\varphi(u)s_{i}.$

Since $u<_{R}us$, $\varphi(u)<_{B}\varphi(u)s_{i}$ follows from the fact that
$\varphi$ is a strong E-embedding, which completes the proof. ∎

Now we focus on proving Theorem 1.6 using the auxiliary lemma, Lemma 3.3.

######  Lemma 3.3.

Suppose $I\subseteq J\subseteq S$ and that $\ll^{I}$ and $\ll^{J}$ are total
refinements of $<_{B}$ restricted to $W^{I}$ and $W^{J}$ respectively. Suppose
further that $\ll^{I}$ and $\ll^{J}$ agree on $W^{J}$: for all $u,v\in W^{J}$,
$u\ll^{I}v$ if and only if $u\ll^{J}v$. Then if $\varphi_{\ll}^{I}$ is a
strong E-embedding, so is $\varphi_{\ll}^{J}$.

###### Proof.

Recall that since $I\subseteq J$, it follows that $W^{J}\subseteq W^{I}$. Now
suppose that $\varphi_{\ll}^{I}$ is a strong E-embedding and take some $w\in
W$ and $s\in S$ such that $w<_{R}ws$. Since $\varphi_{\ll}^{I}$ is an
E-embedding,

$\varphi_{\ll}^{I}(w)<_{B}\varphi_{\ll}^{I}(w)(L_{\ll}^{I}(u),L_{\ll}^{I}(us))$

for all $u,us\in W^{I}$ such that $u^{I}\ll^{I}(us)^{I}$. Equivalently, by
Lemma 2.2, $(uw^{-1})^{I}\ll^{I}(usw^{-1})^{I}$. But since $\ll^{I}$ and
$\ll^{J}$ agree on $W^{J}\subseteq W^{I}$, it’s true that
$(uw^{-1})^{J}\ll^{J}(usw^{-1})^{J}$ also for all $u,us\in W^{J}$ such that
$u^{J}\ll^{J}(us)^{J}$. Therefore $\varphi_{\ll}^{I}$ is also a strong
E-embedding. ∎

###### Proof of Theorem 1.6..

First, we show that total refinements of the Bruhat order give rise to strong
E-embeddings. We focus on the case when $J=\emptyset$ before extending it to
all $J\subseteq S$.

For all $w\in W$ and $t\in T$ such that $w<_{B}wt$, we need to show that

$\varphi(w)<_{B}\varphi(w)(L_{\ll}(u),L_{\ll}(ut))$

for all $u\in W$ such that $u<_{B}ut$. Since $\ll$ is a total refinement of
$<_{B}$, $L_{\ll}(u)<L_{\ll}(ut)$ and thus proving
$\varphi(w)<\varphi(w)(L_{\ll}(u),L_{\ll}(ut))$ is equivalent to showing
$uw^{-1}\ll utw^{-1}$. But by Lemma 2.3, $uw^{-1}<_{B}utw^{-1}$ and so
$uw^{-1}\ll utw^{-1}$.

Now we extend the results to all $J\subseteq S$. Every total refinement of the
Bruhat order on $W^{J}$ is a restriction of some total refinement of the
Bruhat order on $W$. The implication now follows from Lemma 3.3.

Conversely, suppose for contradiction, that $\varphi_{\ll}^{J}$ is a strong
E-embedding but $\ll^{J}$ is not a refinement of the Bruhat order restricted
to $W^{J}$. Then there exists some $w\in W^{J}$, $t\in T$ such that $wt\in
W^{J}$, $w<_{B}wt$ but $wt\ll^{J}w$. Since $\varphi_{\ll}^{J}$ is a strong
E-embedding, $w<_{B}wt$ implies that

$\varphi_{\ll}^{J}(w)<_{B}\varphi_{\ll}^{J}(w)(L_{\ll}^{J}(wt),L_{\ll}^{J}(w)).$

But by Lemma 2.2, this is equivalent to $(wtw^{-1})^{J}\ll(ww^{-1})^{J}$. But
this contradicts the fact that $L_{\ll}^{J}(1)=0$. ∎

Theorem 1.5 follows from Theorem 1.6 on setting $J=\emptyset$.

##  4 Constructing Elnitsky’s Tilings

Using Theorem 1.6 we now have a construction for producing new E-tilings for
any reduced word from any finite Coxeter group, thus proving Theorem 1.1

###### Proof.

The construction works as follows:

  1. $(i)$

Choose some parabolic subgroup $W_{J}<W$ of our Coxeter group $W$ which is
core-free (and so $J\neq S$);

  2. $(ii)$

Produce a total order $\ll$ of $W^{J}$ that is a refinement of the Bruhat
order. A particular method of doing this is outlined in [9];

  3. $(iii)$

Construct $\phi_{\ll}^{J}$ as defined in Section 2. By Theorem 1.6 this is a
strong E-embedding;

  4. $(iv)$

By Proposition 3.2, $\phi_{\ll}^{J}$ is an E-embedding, and therefore by
Theorem 1.2 of [7], for any reduced word of any element of $W$, there exists
an associated E-tiling.

∎

We stress that the proof of Theorem 1.1 does not suggest that all E-embeddings
must be constructed in this way, although we are yet to find a counterexample.

In the remainder of this section we apply the proof of Theorem 1.1 in three
scenarios. First, we re-examine Elnitsky’s original tilings, realising them as
strong-E-Embeddings. Next, we concentrate on examples of reducible finite
Coxeter groups. Finally, we give two new tilings for the longest element of
$\mathrm{E}_{8}$.

###  4.1 Tilings for $\mathrm{D}_{5}$

In [7], it is shown how each E-embedding has an associated tiling system.
Coincidentally, all of Elnitsky’s original tilings correspond to strong-E-
embeddings that are actions on the distinguished elements of specific
parabolic subgroups. We showcase this for the group $\mathrm{D}_{5}$ but note
that the details for realizing his remaining tiling systems as strong-E-
embeddings are much the same.

Let $W$ be the Coxeter group $\mathrm{D}_{5}$ with $\Gamma(W)$ given by

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$

We give a very brief overview (with minor adjustments for convenience) of
Elnitsky’s tiling system for $\mathrm{D}_{5}$, valuing brevity over depth. For
a more thorough and comprehensive introduction, we point the reader to [2],
[3], [8], and [11].

Consider the embedding of $\mathrm{D}_{5}$ into $\mathrm{Sym}({10})$ where
$s_{1}=(5,7)(4,6)$ and $s_{i}=(6-i,7-i)(4+i,5+i)$ for each
$i\in\\{2,\dots,5\\}$. As a permutation representation, it’s represented by
the graph in Figure 1.

$5$$6$$7$$8$$9$$10$$4$$3$$2$$1$$s_{1}$$s_{1}$$s_{2}$$s_{2}$$s_{3}$$s_{3}$$s_{4}$$s_{4}$$s_{5}$$s_{5}$
Figure 1: Permutation representation of $\mathrm{D_{5}}$ in
$\mathrm{Sym}({10})$.

For all $w\in\mathrm{D}_{5}$, define $\mathrm{P}(w)$ to be the (possibly
deformed) 20-gon made like so:

  1. $(i)$

Let $\mathrm{L}$ be the lower-most vertex of our $20$-gon.

  2. $(ii)$

Let the first $10$ edges clockwise from $\mathrm{L}$ be those of the regular
$20$-gon with unit length edges, labelling them with $\\{1,2,\ldots,10\\}$
respectively. Let $\mathrm{X}$ denote the horizontal line passing through the
$5^{\text{th}}$ vertex clockwise from $\mathrm{L}$ in this way.

  3. $(iii)$

Construct and label the $10$ edges anti-clockwise from $\mathrm{L}$ so that
the $i^{th}$ edge from $\mathrm{L}$ in this direction is a unit length edge
parallel to the edge labelled $(i)w^{-1}$ from step $(ii)$.

We present an example in Figure 2 for $w=(1,10)(2,9)(3,8)(4,7)$.

Figure 2: The polygon $\mathrm{P}((1,10)(2,9)(3,8)(4,7))$.

For each $w\in W$, Elnitsky is interested in tilings of $\mathrm{P}(w)$ that
are made by a restricted set of tiles satisfying certain rules. The tiles
placed can either be rhombi or a so-called megatile (described in detail
below) but the tiling must be symmetric about $\mathrm{X}$.

A megatile is an octagonal tile satisfying the following properties:

  1. $(i)$

Its edges are unit length and it must be placed so that it is symmetric
through $\mathrm{X}$;

  2. $(ii)$

Its upper-most vertex, $\mathrm{U}_{0}$, and lower-most vertex,
$\mathrm{L}_{0}$, must lie on a vertical line;

  3. $(iii)$

Let $E_{0}$ denote the first four edges anti-clockwise from $\mathrm{U}_{0}$,
then the remaining four edges can be obtained like so: reflect $E_{0}$ through
the vertical line passing through $\mathrm{U}_{0}$ and $\mathrm{L}_{0}$ before
transposing the resulting first and second pair of edges, and the third and
fourth pair of edges.

We present two such tilings for the same polygon in Figure 3.

Figure 3: Two Elnitsky tilings for $\mathrm{P}((1,10)(2,9)(3,8)(4,7))$.

Theorem 7.1 of [3] gives a bijection between classes of the reduced words of
$\mathrm{D}_{n}$ and tilings of the kind described above. To produce a tiling
from a reduced word, the word is read from left to right with each subsequent
generator corresponding to instructions for placing the next set of tiles.
Reading the generator $s_{1}$ compels us to place a megatile with the edges
$E_{0}$ of the megatile are formed by taking the right-most edges that are
currently the $4^{th},5^{th},6^{th}$ and $7^{th}$ edges from $\mathrm{L}$
respectively. For $i>1$, $s_{i}$ is associated with placing a rhombic tile
sharing the right-most edges placed in the $i^{th}$ and $(i-1)^{th}$ position
from $\mathrm{L}$ and its mirror image through $\mathrm{X}$. The two Elnitsky
tiling s of Figure 1 are derived from the reduced words

$s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}s_{2}s_{1}s_{5}s_{4}s_{3}s_{2}s_{1}$

and

$s_{5}s_{3}s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{5}s_{4}s_{5}s_{3}s_{4}s_{2}s_{1}s_{3}s_{4}s_{2},$

respectively. We give an explicit step-by-step illustration of how to produce
the tiling for
$s_{5}s_{3}s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{5}s_{4}s_{5}s_{3}s_{4}s_{2}s_{1}s_{3}s_{4}s_{2}$
in Figure 4.

Figure 4: A step-by-step illustration of the intermediate tilings associated
to the reduced word
$s_{5}s_{3}s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{5}s_{4}s_{5}s_{3}s_{4}s_{2}s_{1}s_{3}s_{4}s_{2}$
using Elnitsky’s method (read from left to right, top to bottom).

We now pivot to reconstructing the same tilings from the perspective of
strong-E-embeddings and the Bruhat order. Start by considering
$\mathrm{D}_{4}$ as the index 10, standard parabolic subgroup of
$\mathrm{D}_{5}$. That is, let $J=\\{s_{1},s_{2},s_{3},s_{4}\\}$ so
$W_{J}=\langle s_{1},s_{2},s_{3},s_{4}\rangle$. We display the Hasse Diagram
of the Bruhat order restricted to $W^{J}$ in Figure 5: one element lies
vertically above another if and only if the higher element is greater than the
lower in the Bruhat order (whilst elements of the same height are incomparable
in the Bruhat Order).

1$s_{5}$$s_{5}s_{4}$$s_{5}s_{4}s_{3}$$s_{5}s_{4}s_{3}s_{2}$$s_{5}s_{4}s_{3}s_{1}$$s_{5}s_{4}s_{3}s_{1}s_{2}$$s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}$$s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}$$s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}s_{5}$
Figure 5: The Hasse diagram of the Bruhat Order restricted to the minimal
right cosets representatives of $\mathrm{D}_{4}$ in $\mathrm{D}_{5}$.

Now we need to choose some total ordering of our minimal representatives that
is a refinement of the Bruhat order. There are exactly two such refinements:

  1. $(i)$

$1\ll s_{5}\ll s_{5}s_{4}\ll s_{5}s_{4}s_{3}\ll s_{5}s_{4}s_{3}s_{2}\ll
s_{5}s_{4}s_{3}s_{1}\ll s_{5}s_{4}s_{3}s_{1}s_{2}$ $\ll
s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}\ll s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}\ll
s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}s_{5}$; and

  2. $(ii)$

$1\ll s_{5}\ll s_{5}s_{4}\ll s_{5}s_{4}s_{3}\ll s_{5}s_{4}s_{3}s_{1}\ll
s_{5}s_{4}s_{3}s_{2}\ll s_{5}s_{4}s_{3}s_{1}s_{2}$ $\ll
s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}\ll s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}\ll
s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}s_{5}$.

For this small example, finding and choosing a refinement is a manageable
task. For larger examples, however, this proves to be a more difficult task to
do satisfactorily. Fortunately, we can always use an instance of Prim’s greedy
algorithm to create such a refinement, namely the deletion order. Details of
this are given in [9]. Choosing the first of the two refinements to be $\ll$,
we can make $\varphi_{\ll}^{J}$ as in Section 2. Explicitly,

$\displaystyle\varphi_{\ll}^{J}(s_{1})$
$\displaystyle=(L_{\ll}^{J}(s_{5}s_{4}s_{3}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}))(L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{2}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}))$
$\displaystyle=(4,6)(5,7)$ $\displaystyle\varphi_{\ll}^{J}(s_{2})$
$\displaystyle=(L_{\ll}^{J}(s_{5}s_{4}s_{3}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{2}))(L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}))$
$\displaystyle=(4,5)(6,7);$ $\displaystyle\varphi_{\ll}^{J}(s_{3})$
$\displaystyle=(L_{\ll}^{J}(s_{5}s_{4}),L_{\ll}^{J}(s_{5}s_{4}s_{3}))(L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}))$
$\displaystyle=(3,4)(7,8);$ $\displaystyle\varphi_{\ll}^{J}(s_{4})$
$\displaystyle=(L_{\ll}^{J}(s_{5}),L_{\ll}^{J}(s_{5}s_{4}))(L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}))$
$\displaystyle=(2,3)(8,9);$ $\displaystyle\varphi_{\ll}^{J}(s_{4})$
$\displaystyle=(L_{\ll}^{J}(1),L_{\ll}^{J}(s_{5}))(L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}),L_{\ll}^{J}(s_{5}s_{4}s_{3}s_{1}s_{2}s_{3}s_{4}s_{5}))$
$\displaystyle=(1,2)(9,10).$

The permutation representation graph of $\varphi_{\ll}^{J}$ is therefore the
same as that used in Elnitsky’s construction in Figure 1. By Theorem 1.5,
$\varphi_{\ll}^{J}$ is a strong-E-embedding and thus, by Proposition 3.2, an
E-embedding. Using the methods in [7], we can derive the tiling system
associated to $\varphi_{\ll}^{J}$ that transforms reduced words into
E-tilings. This system is exactly the same as Elnitsky’s tiling method for
type $\mathrm{D}$.

Elnitsky’s other tiling systems can be realized using the standard parabolic
subgroups $\mathrm{A}_{n-1}$ in $\mathrm{A}_{n}$ and $\mathrm{A}_{n-1}$ in
$\mathrm{B}_{n}$.

###  4.2 Tilings of finite reducible Coxeter groups

Here we show how the steps outlined in the proof of Theorem 1.1 work equally
well for the finite reducible Coxeter groups as opposed to their irreducible
counterparts. By way of example, we consider the reducible Coxeter group
$W=\mathrm{A}_{2}\times\mathrm{A}_{3}$. We label and order the generators like
so:

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$

Taking $J=\\{s_{1},s_{3},s_{4}\\}$, the Bruhat order restricted to $W^{J}$ is
depicted in Figure 6.

$1$$s_{2}$$s_{2}s_{1}$$s_{5}$$s_{2}s_{5}$$s_{2}s_{1}s_{5}$$s_{5}s_{4}$$s_{2}s_{5}s_{4}$$s_{2}s_{1}s_{5}s_{4}$$s_{5}s_{4}s_{3}$$s_{2}s_{5}s_{4}s_{3}$$s_{2}s_{1}s_{5}s_{4}s_{3}$
Figure 6: A Hasse Diagram for the Bruhat order restricted to $W^{J}$ for
$\mathrm{A}_{2}\times\mathrm{A}_{3}$.

Taking the refinement

$1\ll s_{2}\ll s_{2}s_{1}\ll s_{5}\ll s_{2}s_{5}\ll s_{2}s_{1}s_{5}\ll
s_{5}s_{4}\ll s_{2}s_{5}s_{4}$ $\ll s_{2}s_{1}s_{5}s_{4}\ll s_{5}s_{4}s_{3}\ll
s_{5}s_{4}s_{3}s_{2}\ll s_{5}s_{4}s_{3}s_{2}s_{1},$

say, we create the strong E-embedding $\varphi^{J}_{\ll}$ such that

$\displaystyle\varphi^{J}_{\ll}(s_{1})$
$\displaystyle=(2,3)(5,6)(8,9)(11,12);$
$\displaystyle\varphi^{J}_{\ll}(s_{2})$
$\displaystyle=(1,2)(4,5)(7,8)(10,11);$
$\displaystyle\varphi^{J}_{\ll}(s_{3})$ $\displaystyle=(7,10)(8,11)(9,12);$
$\displaystyle\varphi^{J}_{\ll}(s_{4})$ $\displaystyle=(4,7)(5,8)(6,9);$
$\displaystyle\varphi^{J}_{\ll}(s_{5})$ $\displaystyle=(1,4)(2,5)(3,6).$

Now we can use $\varphi^{J}_{\ll}$ to make E-tilings as Figure 7 shows.

Figure 7: An E-tiling for the reduced word
$s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{4}$ in the Coxeter group
$\mathrm{A}_{2}\times\mathrm{A}_{3}$.

###  4.3 New tilings for $\mathrm{E}_{8}$

Let $W$ be the Coxeter group $\mathrm{E}_{8}$ and let $\omega_{0}$ be the
longest element of $W$. We label $\Gamma(W)$ as on page 16 of [6] and we order
the elements of $S$ thus $s_{1}<s_{2}<\ldots<s_{8}$. We follow our general
procedure to produce new tilings for $\mathrm{E}_{8}$ using the standard
parabolic subgroup $\mathrm{E}_{7}$.

We select two reduced words of $\omega_{0}$ to make our specific tilings. Here
$w_{0}$ denotes the longest element of $\mathrm{E}_{8}$. The first is made by
taking the concatenation of 15 copies of
$c=s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{8}$ (a Coxeter element of $W$). One
can prove that $\omega_{0}=c^{15}$ and that this is indeed a reduced word. The
tiling for this is given in Figure 8. The second tiling is produced by taking
the least reduced word of $\omega_{0}$ with respect to the lexicographic (from
right to left) ordering induced from the ordering of $S$.

Figure 8: E-tiling for the the reduced exprssion $c^{15}$ of $w_{0}$.

Figure 9: E-tiling for the reduced expression least reduced word of
$\omega_{0}$ with respect to the right to left lexicographic ordering.

## References

  * [1] Björner, Anders; Brenti, Francesco. Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.
  * [2] Elnitsky, Serge. Rhombic tilings of polygons and classes of reduced words in Coxeter groups. PhD dissertation, University of Michigan, 1993.
  * [3] Elnitsky, Serge. Rhombic tilings of polygons and classes of reduced words in Coxeter groups. J. Combin. Theory Ser. A 77 (1997), no. 2, 193–221.
  * [4] Escobar, Laura; Pechenik, Oliver; Tenner, Bridget Eileen; Yong, Alexander. Rhombic tilings and Bott-Samelson varieties. Proc. Amer. Math. Soc. 146 (2018), no. 5, 1921–1935.
  * [5] Hamanaka, H.; Nakamoto, A.; Suzuki, Y.. Rhombus Tilings of an Even-Sided Polygon and Quadrangulations on the Projective Plane. Graphs and Combinatorics 36 (2020), 561–571.
  * [6] Geck, Meinolf; Pfeiffer, Götz. Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. London Mathematical Society Monographs New Series, 21, Clarendon Press, Oxford, 2000.
  * [7] Nicolaides, Robert; Rowley, Peter. Finite Coxeter Groups and Generalized Elnitsky Tilings. arXiv:2111.10669.
  * [8] Nicolaides, Robert; Rowley, Peter. Subtilings of Elnitsky’s Tilings for Finite Irreducible Coxeter Groups. arXiv 2106.03173.
  * [9] Nicolaides, Robert; Rowley, Peter. The Deletion Order and Coxeter Groups. arXiv:2407.07881.
  * [10] Nicolaides Robert; Rowley Peter. Elnitsky Tiling Code Repository. https://github.com/robnico/Elnitsky-Tilings-Mathematica.git
  * [11] Tenner, Bridget Eileen. Enumerating in Coxeter Groups (Survey) Advances in Mathematical Sciences, Association for Women in Mathematics Series, Springer (2020), no. 21, 75-82.
  * [12] Tenner, Bridget Eileen. Tiling-based models of perimeter and area. Adv. in Appl. Math. 119 (2020), no. 29


# Ground and Excited state masses of $\Omega_{c}^{0}$, $\Omega_{cc}^{+}$ and
$\Omega_{ccc}^{++}$ baryons

Juhi Oudichhya  Keval Gandhi<EMAIL_ADDRESS>Ajay Kumar Rai
Department of Physics, Sardar Vallabhbhai National Institute of Technology,
Surat, Gujarat-395007, India

###### Abstract

The masses of ground and excited states of $\Omega_{c}^{0}$, $\Omega_{cc}^{+}$
and $\Omega_{ccc}^{++}$ baryons are calculated within the framework of Regge
phenomenology. Here our attempt is to assign a possible spin-parity to these
obtained masses. Our calculated masses in $(J,M^{2})$ plane with both natural
($J^{P}$ = $1/2^{+}$, $3/2^{-}$, $5/2^{+}$, ….) and unnatural ($J^{P}$ =
$3/2^{+}$, $5/2^{-}$, $7/2^{+}$, ….) spin-parities are in agreement with the
experimental observations where available and reasonably close to other
theoretical predictions. Further we fix the slope and intercept of each Regge
line in $(n,M^{2})$ plane and estimate the masses of higher excited states of
these baryons. The obtained mass relations and the mass value predictions
could provide an useful information in future experimental searches and the
spin-parity assignment of these states.

††preprint: APS/123-QED

##  I  Introduction

In 2017, the LHCb collaboration observed five new narrow excited
$\Omega_{c}^{0}$ states such as $\Omega_{c}(3000)^{0}$,
$\Omega_{c}{(3050)}^{0}$, $\Omega_{c}{(3066)}^{0}$, $\Omega_{c}{(3090)}^{0}$
and $\Omega_{c}{(3119)}^{0}$ decaying into $\Xi^{+}K^{-}$ [1]. Except for
$\Omega_{c}{(3119)}^{0}$ other four states were further confirmed by Belle [2]
in 2018. Frequent questions have been arises with this discovery, for instance
(i) why only five states were observed ? (ii) why they are so narrow ?. Many
authors have been addressed these questions with recent studies given in Refs.
[4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Since $\Omega_{c}^{0}$ is
composed of two strange quarks $(ss)$ and one charm quark $(c)$, if $ss$
diquark remains in its color triplet it has spin-1, as symmetry rules do not
allow a spin-0 diquark. This spin combined with spin 1/2 of $c$ quark, we have
$S=1/2$ or $3/2$. Now consider states with relative orbital momentum $L=1$ and
combining with $S=1/2$ and $3/2$ gives the total spin $J=1/2$, $3/2$ and
$J=1/2$, $3/2$, $5/2$, respectively. All these five states have negative
parity. Also these states are extremely narrow which indicates that it is
difficult to pull apart the two $s$ quarks in a diquark [4].

The study of heavy-light bound state provides an ideal platform to understand
the dynamics of quantum chromodynamics (QCD) at low energy regime. The strong
interaction coupling is small for the hadrons containing heavy quark ($c$ or
$b$) which makes it easier to understand the QCD perturbatively than the
system containing only light quark [16]. Different experimental facilities
have provided new information in the sector of hadron spectroscopy like
masses, decay width, branching ratios, isospin mass splittings, apin, parity,
polarisation amplitude, etc. It is very crucial to assign the spin-parity of
hadrons which facilitate the determination of experimental properties.
Experimentally only the ground state $\Omega_{c}^{0}$ and
$\Omega_{c}(2770)^{0}$ are observed with their quantum numbers, $J^{P}$ =
$\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ respectively [17], where $J$ is the
total spin and $P$ denotes the parity. The $J^{P}$ values of excited state
$\Omega_{c}^{0}$ baryons are still missing as shown in Table 1.

Upto the date there are different research aspects to study the hadron
spectroscopy, one of the prominent way to determine the properties of hadronic
states is mass spectrum analysis. So far there are various method to depict
the mass spectrum. The authors of Refs. [18, 19, 20, 21] calculate the excited
state masses of singly, doubly and triply heavy baryons using non-relativistic
approach of hypercentral constituent quark model (hCQM). In Ref. [22] mass
analysis is carried out for the newly excited $\Omega_{c/b}$ baryons by
proposing a new scheme of state classification called $Jls$ mixing coupling
which goes beyond $jj$ coupling. The results favor to predict the spin-parity
of these states. The ground state and excited state spectra of
$\Omega_{c}^{0}$ baryon are estimated from lattice QCD with dynamical quark
field in Ref. [5] and results predicted the $J^{P}$ assignments of the states
$\Omega_{c}(3000)^{0}$, $\Omega_{c}{(3050)}^{0}$ and $\Omega_{c}{(3066)}^{0}$,
$\Omega_{c}{(3090)}^{0}$ and $\Omega_{c}{(3119)}^{0}$ with $1/2^{-}$,
$3/2^{-}$ and $5/2^{-}$, respectively. The authors of Ref. [4] predicts the
existence of five states with negative parity by assuming these baryonic
states as bound state of $c$-quark and a $P$ wave $ss$-diquark. Ebert et al.
[23] determine mass spectra of excited heavy baryons consisting of two light
$(u,d,s)$ and one heavy $(c,b)$ quarks, in the framework of QCD-motivated by
relativistic quark model. Robert et al.[24] used non-relativistic hamiltonian
quark model to calculate the masses of heavy baryons. Ref. [25] estimate the
masses and residues of these newly observed $\Omega_{c}^{0}$ baryon states
within QCD sum rules and try to determine the nature of these states. So far
none of the doubly/triply $\Omega$ baryons have been experimently detected.
Although many authors have predicted the masses of $\Omega_{cc}$ and
$\Omega_{ccc}$ baryons using various theoretical and phenomenological models,
like Relativistic quark model [26] , Salpeter model [27], Lattice QCD [28,
29], the Hamiltonian model [30], the Hypercentral constituent quark model [18,
19] etc.

Wei et al. [31] derive some useful mass relations such as linear mass
inequalities, quadratic mass inequalities and quadratic mass equalities for
hadrons using quasilinear Regge trejectory ansatz. Based on these relations
they determine mass ranges of some mesons and baryons and predict $J^{P}$
assignments of $\Xi_{c}$(2980), $\Xi_{c}$(3055), $\Xi_{c}$(3077) and
$\Xi_{c}$(3123) baryons. Ref. [32] the same group extended this methodology to
compute the masses of ground state and excited state of other doubly and
triply charmed baryons such as $\Omega_{cc},\Xi_{cc}^{*},\Omega_{cc}^{*}$ and
$\Omega_{ccc}$. In the present work we have used the same approach of Regge
phenomenology with the assumption of linear Regge trejectories. We obtained
relations between intercept, slope ratios and baryon masses in both
($J,M^{2}$) and ($n_{r},M^{2}$) planes. Using these relations we determine the
mass spectra of $\Omega_{c}$ baryon and try to assign a possible spin-parity
to these recently observed five baryonic states. Further we apply the same
scheme for doubly and triply charmed omega baryons. Since these baryons are
yet to be observed by experiments and therefore they are not listed in PDG
[17] yet. This motivate us to study the mass spectra of $\Omega_{c}^{0}$,
$\Omega_{cc}^{+}$ and $\Omega_{ccc}^{++}$ baryons.

Table 1:  Masses, width and $J^{P}$ value of $\Omega_{c}^{0}$ baryons reported by LHCb [1].  Resonance | Mass(MeV) | Width(MeV) | $J^{P}$
---|---|---|---
$\Omega_{c}^{0}$ | 2695.2$\pm 1.7$ | - | $\frac{1}{2}^{+}$
$\Omega_{c}(2770)^{0}$ | 2765.9$\pm 2.0$ | - | $\frac{3}{2}^{+}$
$\Omega_{c}(3000)^{0}$ | 3000.4$\pm 0.22\pm 0.1$ | $4.5\pm 0.6\pm 0.3$ | $?^{?}$
$\Omega_{c}(3050)^{0}$ | 3050.2$\pm 0.1\pm 0.1$ | $0.8\pm 0.2\pm 0.1$ | $?^{?}$
$\Omega_{c}(3065)^{0}$ | 3065.6$\pm 0.1\pm 0.3$ | $3.5\pm 0.4\pm 0.2$ | $?^{?}$
$\Omega_{c}(3090)^{0}$ | 3090.2$\pm 0.3\pm 0.9$ | 8.7$\pm 0.1\pm 0.8$ | $?^{?}$
$\Omega_{c}(3120)^{0}$ | 3119$\pm 0.3\pm 0.9$ | 1.1$\pm 0.8\pm 0.4$ | $?^{?}$

The remainder of this paper is organised as follows. After introduction, in
sec II we briefly explain the linear Regge trejectories and extract the
quadratic mass relation in $(J,M^{2})$ plane to calculate the ground state and
excited state masses of $\Omega_{c}$ baryon. Further we extend this work and
again derive some mass equations to evaluate the mass spectra for
$\Omega_{cc}$ and $\Omega_{ccc}$ baryons. In addition, we estimate the masses
of these baryons in $(n,M^{2})$ plane by calculating Regge slopes and
intercepts of particular regge lines. The discussions of our results appear in
sec.III.

##  II Theoretical Framework

Regge theory is one of the simplest and effective phenomenological approach to
study the hadron spectroscopy. Various theories were developed to understand
the regge trejectory. Among them, one of the straightforward explaination for
linear regge trejectories were proposed by Nambu [33, 34]. He assumed that the
uniform interaction of quark and antiquark pair form a strong flux tube and at
the end of the tube the light quarks rotating with the speed of light at
radius $R$. The mass originating in this flux tube is estimated as [35],

$M=2\int_{0}^{R}\dfrac{\sigma}{\sqrt{1-\nu^{2}(r)}}dr=\pi\sigma R,$ (1)

where $\sigma$ represent the string tension, i.e the mass density per unit
length. Likewise, the angular momentum of this flux tube is evaluated as,

$J=2\int_{0}^{R}\dfrac{\sigma r\nu(r)}{\sqrt{1-\nu^{2}(r)}}dr=\dfrac{\pi\sigma
R^{2}}{2}+c^{{}^{\prime}},$ (2)

one can also write,

$J=\dfrac{M^{2}}{2\pi\sigma}+c^{{}^{\prime\prime}},$ (3)

where $c^{{}^{\prime}}$ and $c^{{}^{\prime\prime}}$ are the constants of
integration. Hence we can say that, $J$ and $M^{2}$ are linearly related to
each other. The plots of Regge trejectories of hadrons in the $(J,M^{2})$
plane are usually called Chew-frautschi plots [36]. Also hadrons lying on the
same Regge line possess the same internal quantum numbers. From Eq. (3), the
most general form of linear regge trejectories can be expressed as [31],

$J=\alpha(M)=a(0)+\alpha^{{}^{\prime}}M^{2},$ (4)

where $a(0)$ and $\alpha^{{}^{\prime}}$ represents the intercept and slope of
the trejectory respectively. These parameters for different quark constituents
of a baryon multiplet can be related by two relations [31, 37, 38, 39, 40],

$a_{iiq}(0)+a_{jjq}(0)=2a_{ijq}(0),$ (5)

$\dfrac{1}{{\alpha^{{}^{\prime}}}_{iiq}}+\dfrac{1}{{\alpha^{{}^{\prime}}}_{jjq}}=\dfrac{2}{{\alpha^{{}^{\prime}}}_{ijq}},$
(6)

where $i,j,q$ represent quark flavors, $m_{i}<m_{j}$ and $q$ denotes an
arbitrary light or heavy quark. Using Eqs. (4) and (5) we obtain,

$\alpha^{{}^{\prime}}_{iiq}M^{2}_{iiq}+\alpha^{{}^{\prime}}_{jjq}M^{2}_{jjq}=2\alpha^{{}^{\prime}}_{ijq}M^{2}_{ijq}.$
(7)

Combining the relations (6) and (7) and after solving the quadratic equation,
we obtain a pair of solutions as,

$\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{1}{2M^{2}_{jjq}}\times[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})\pm\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}].$
(8)

This is the important relation we obtained between slope ratios and baryon
masses. Eq. (8) can also be expressed in terms of baryon masses by introducing
$k$, where $k$ can be any quark flavor,

$\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{\alpha^{{}^{\prime}}_{kkq}}{\alpha^{{}^{\prime}}_{iiq}}\times\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{kkq}},$
(9)

we have,

$\begin{split}\dfrac{[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})+\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}]}{2M^{2}_{jjq}}\\\
=\dfrac{[(4M^{2}_{ikq}-M^{2}_{iiq}-M^{2}_{kkq})+\sqrt{{{(4M^{2}_{ikq}-M^{2}_{iiq}-M^{2}_{kkq}})^{2}}-4M^{2}_{iiq}M^{2}_{kkq}}]/2M^{2}_{kkq}}{[(4M^{2}_{jkq}-M^{2}_{jjq}-M^{2}_{kkq})+\sqrt{{{(4M^{2}_{jkq}-M^{2}_{jjq}-M^{2}_{kkq}})^{2}}-4M^{2}_{jjq}M^{2}_{kkq}}]/2M^{2}_{kkq}}.\end{split}$
(10)

This is the general relation we have derived in terms of baryon masses which
can be used to predict the mass of any baryon state by knowing all other
masses.

###  II.1 Masses of $\Omega_{c}^{0}$ baryon

In the present work we used Eq. (10) to evaluate the ground state masses of
$\Omega_{c}^{0}$ baryon. Since $\Omega_{c}^{0}$ is composed of two strange
quark and one charmed quark $(ssc)$. So keeping this configuration in mind we
put $i=u$, $j=s$, $q=c$, $k=u$ in Eq. (10) and we get,

$\left[2M_{\Xi_{c}^{+}}(M_{\Omega_{c}^{0}}-2M_{\Xi_{c}^{+}})-M^{2}_{\Omega_{c}^{0}}-M^{2}_{\varSigma_{c}^{++}}\right]=\sqrt{(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\varSigma_{c}^{++}}-M^{2}_{\Omega_{c}^{0}})^{2}-4M^{2}_{\Omega_{c}^{0}}M^{2}_{\Xi_{c}^{+}}},$
(11)

substituting the masses of $\Xi_{c}^{+}$ $(J^{P}=1/2^{+},3/2^{+})$ and
$\varSigma_{c}^{++}$ $(J^{P}=1/2^{+},3/2^{+})$ baryon from PDG [17] into
relation (11), we obtain the ground state masses of $\Omega_{c}^{0}$ baryon
such as 2.702 GeV for $J^{P}=\frac{1}{2}^{+}$ and 2.772 GeV for
$J^{P}=\frac{3}{2}^{+}$. From Eq. (8) we can write ,

$\begin{split}\dfrac{\alpha^{{}^{\prime}}_{ssc}}{\alpha^{{}^{\prime}}_{uuc}}=\dfrac{1}{2M^{2}_{\Omega_{c}^{0}}}\times[(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\varSigma_{c}^{++}}-M^{2}_{\Omega_{c}^{0}})\\\
+\sqrt{{{(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\varSigma_{c}^{++}}-M^{2}_{\Omega_{c}^{0}}})^{2}}-4M^{2}_{\varSigma_{c}^{++}}M^{2}_{\Omega_{c}^{0}}}].\end{split}$
(12)

Putting the values of masses in above equation, we get
$\alpha^{{}^{\prime}}_{ssc}/\alpha^{{}^{\prime}}_{uuc}$. We can determine the
slope by using Eq. (4),

$\alpha^{{}^{\prime}}=\dfrac{(J+1)-J}{M^{2}_{J+1}-M^{2}_{J}},$ (13)

for $\varSigma_{c}^{++}$$(uuc)$ it will be,

$\alpha^{{}^{\prime}}_{uuc}=\dfrac{1}{M^{2}_{\varSigma^{++}_{c}(\frac{3}{2}^{-})}-M^{2}_{\varSigma^{++}_{c}(\frac{1}{2}^{+})}},$
(14)

we get $\alpha^{{}^{\prime}}_{ssc}=$ 0.54835 $GeV^{-2}$ for
$J^{P}=\frac{1}{2}^{+}$. Similarly we get $\alpha^{{}^{\prime}}_{ssc}=$
0.66524 $GeV^{-2}$ for $J^{P}=\frac{3}{2}^{+}$.

From Eq. (4) one can have,

$M_{J+1}=\sqrt{M_{J}^{2}+\dfrac{1}{\alpha^{{}^{\prime}}}}.$ (15)

Using this equation we calculate the masses of excited states of
$\Omega_{c}^{0}$ baryon, which are presented in Table 2 and Table 3 for
natural and unnatural parity states, respectively. We compared our results
with other theoritical predictions and experimental observations where
available.

Table 2:  Masses of excited states of $\Omega_{c}^{0}$ baryon (in GeV) in $(J,M^{2})$ plane with natural parities $P=(-1)^{J-\frac{1}{2}}$.  States | $1^{2}S_{\frac{1}{2}}$ | $1^{2}P_{\frac{3}{2}}$ | $1^{2}D_{\frac{5}{2}}$ | $1^{2}F_{\frac{7}{2}}$ | $1^{2}G_{\frac{9}{2}}$ | $1^{2}H_{\frac{11}{2}}$
---|---|---|---|---|---|---
Present | 2.702 | 3.049 | 3.360 | 3.645 | 3.871 | 4.156
LHCb [1] |  | 3.050 |  |  |  |
Belle [2] |  | 3.050 |  |  |  |
PDG [17] | 2.695 | 3.050 |  |  |  |
Ref. [20] | 2.696 | 2.968 | 3.251 | 3.519 | 3.783 |
Ref. [21] | 2.695 | 3.024 | 3.299 | 3.565 |  |
Ref. [23] | 2.698 | 3.054 | 3.297 | 3.514 | 3.705 |
Ref. [24] | 2.718 | 2.977 | 3.196 |  |  |
Ref. [30] | 2.731 | 3.033 |  |  |  |
Ref. [41] | 2.698 | 3.026 | 3.218 |  |  |
Ref. [42] | 2.718 | 3.056 | 3.273 |  |  |
Table 3:  Masses of excited states of $\Omega_{c}^{0}$ baryon (in GeV) in $(J,M^{2})$ plane with unnatural parities $P=(-1)^{J+\frac{1}{2}}$.  States | $1^{4}S_{\frac{3}{2}}$ | $1^{4}P_{\frac{5}{2}}$ | $1^{4}D_{\frac{7}{2}}$ | $1^{4}F_{\frac{9}{2}}$ | $1^{4}G_{\frac{11}{2}}$ | $1^{4}H_{\frac{13}{2}}$
---|---|---|---|---|---|---
Present | 2.772 | 3.055 | 3.314 | 3.555 | 3.779 | 3.991
PDG[17] | 2.765 |  |  |  |  |
Ref. [20] | 2.766 | 2.962 | 3.241 | 3.503 | 3.756 |
Ref. [21] | 2.765 | 3.010 | 3.276 | 3.532 |  |
Ref. [23] | 2.776 | 3.051 | 3.283 | 3.485 | 3.665 |
Ref. [24] | 2.776 | 3.014 | 3.206 |  |  |
Ref. [30] | 2.779 | 3.057 |  |  |  |
Ref. [41] | 2.765 | 3.022 | 3.237 |  |  |
Ref. [42] | 2.766 | 3.014 |  |  |  |

###  II.2 Masses of $\Omega_{cc}$ and $\Omega_{ccc}$

The values of $\alpha^{{}^{\prime}}_{jjq}/\alpha^{{}^{\prime}}_{iiq}$ from Eq.
(8) should be a real number. So we can write Eq. (8) as,

$|4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}|\geqslant 2M_{iiq}M_{jjq},$ (16)

this inequality relation can be simplified to,

$2M_{ijq}\geqslant(M_{iiq}+M_{jjq}).$ (17)

Many theoretical studies shows that the slopes of Regge trejectories decreases
with increasing quark masses [37, 38, 43, 44, 45, 46]. For $m_{j}>m_{i}$, we
can write $\alpha^{{}^{\prime}}_{jjq}/\alpha^{{}^{\prime}}_{iiq}<1$. Therefore
Eq. (8) is,

$\begin{split}\dfrac{\alpha^{{}^{\prime}}_{jjq}}{\alpha^{{}^{\prime}}_{iiq}}=\dfrac{1}{2M^{2}_{jjq}}\times[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})\\\
+\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^{2}}-4M^{2}_{iiq}M^{2}_{jjq}}]<1,\end{split}$
(18)

the above relation gives another inequality equation as,

$2M^{2}_{ijq}<M^{2}_{iiq}+M^{2}_{jjq},$ (19)

so from Eqs. (17) and (19) we have,

$\dfrac{M_{iiq}+M_{jjq}}{2}<M_{ijq}<\sqrt{\dfrac{M^{2}_{iiq}+M^{2}_{jjq}}{2}}.$
(20)

This relation can be used to calculate the upper and lower limit for baryon
masses $M_{ijq}$. Further to estimate the deviation of relation (20), we
introduce a parameter called $\delta$ which is used to replace the signs of
inequalities to equal signs. For baryons it is denoted by $\delta^{b}_{ij,q}$
and given by,

$\delta^{b}_{ij,q}=M^{2}_{iiq}+M^{2}_{jjq}-2M^{2}_{ijq},$ (21)

here also $i,j$ and $q$ represents the arbitrary light or heavy quarks.

Now from Eqs. (5) and (6) we can write,

$a_{iiq}(0)-a_{ijq}(0)=a_{ijq}(0)-a_{jjq}(0),$ (22)

$\frac{1}{\alpha_{iiq}^{{}^{\prime}}}-\frac{1}{\alpha_{ijq}^{{}^{\prime}}}=\frac{1}{\alpha_{ijq}^{{}^{\prime}}}-\frac{1}{\alpha_{jjq}^{{}^{\prime}}},$
(23)

based on these equations we introduce two parameters,

$\lambda_{x}=a_{nnn}(0)-a_{nnx}(0),\gamma_{x}=\frac{1}{\alpha^{{}^{\prime}}_{nnx}}-\frac{1}{\alpha^{{}^{\prime}}_{nnn}};$
(24)

where $n$ represents light nonstrange quark ($u$ or $d$) and $x$ denotes $i,j$
or $q$. From Eqs. (22-24)we have,

$\begin{split}a_{ijq}(0)=a_{nnn}(0)-\lambda_{i}-\lambda_{j}-\lambda_{q},\\\
\frac{1}{\alpha_{ijq}^{{}^{\prime}}}=\frac{1}{\alpha_{nnn}^{{}^{\prime}}}+\gamma_{i}+\gamma_{j}+\gamma_{q}.\end{split}$
(25)

Since in baryon multiplets for $nnn$ and $ijq$ states we can write from Eq.
LABEL:(eq:4),

$\begin{split}J=a_{nnn}(0)+\alpha_{nnn}^{{}^{\prime}}M^{2}_{nnn},\\\
J=a_{ijq}(0)+\alpha^{{}^{\prime}}_{ijq}M^{2}_{ijq},\end{split}$ (26)

solving Eqs. (25) and (26) we have,

$M^{2}_{ijq}=(\alpha^{{}^{\prime}}_{nnn}M^{2}_{nnn}+\lambda_{i}+\lambda_{j}+\lambda_{q})\left(\frac{1}{\alpha^{{}^{\prime}}_{nnn}}+\gamma_{i}+\gamma_{j}+\gamma_{q}\right).$
(27)

Therefore from Eqs.(21) and (27), we have

$\displaystyle\delta^{b}_{ij,q}$ $\displaystyle=$ $\displaystyle
M^{2}_{iiq}+M^{2}_{jjq}-2M^{2}_{ijq}$ (28) $\displaystyle=$
$\displaystyle(\alpha^{{}^{\prime}}_{nnn}M^{2}_{nnn}+2\lambda_{i}+\lambda_{q})\left(\frac{1}{\alpha^{{}^{\prime}}_{nnn}}+2\gamma_{i}+\gamma_{q}\right)+(\alpha^{{}^{\prime}}_{nnn}M^{2}_{nnn}+2\lambda_{j}+\lambda_{q})\left(\frac{1}{\alpha^{{}^{\prime}}_{nnn}}+2\gamma_{j}+\gamma_{q}\right)$
$\displaystyle-$ $\displaystyle
2(\alpha^{{}^{\prime}}_{nnn}M^{2}_{nnn}+\lambda_{i}+\lambda_{j}+\lambda_{q})\left(\frac{1}{\alpha^{{}^{\prime}}_{nnn}}+\gamma_{i}+\gamma_{j}+\gamma_{q}\right)$
$\displaystyle=$ $\displaystyle
2(\lambda_{i}-\lambda_{j})(\gamma_{i}-\gamma_{j}).$

So, we can say that $\delta^{b}_{ij,q}$ is independent of quark flavor $q$.
Hence by keeping $i$ and $j$ fixed and changing the $q$ quark, using Eq. (21)
we have some following relations for $\frac{3}{2}^{+}$ states :
(I) $i=u$, $j=s$, $q=u$, $c$

$\begin{split}\delta_{us}^{(3/2)^{+}}=M^{2}_{\Delta}+M^{2}_{\Xi^{*}}-2M^{2}_{\Sigma^{*}}\\\
=M^{2}_{\Sigma_{c}^{*}}+M^{2}_{\Omega_{c}^{*}}-2M^{2}_{\Xi_{c}^{*}};\end{split}$
(29)

(II) $i=u$, $j=c$, $q=u$ ,$s$

$\begin{split}\delta_{uc}^{(3/2)^{+}}=M^{2}_{\Delta}+M^{2}_{\Xi^{*}_{cc}}-2M^{2}_{\Sigma^{*}_{c}}\\\
=M^{2}_{\Sigma^{*}}+M^{2}_{\Omega_{cc}^{*}}-2M^{2}_{\Xi_{c}^{*}};\end{split}$
(30)

(III) $i=s$, $j=c$, $q=u$ ,$c$

$\begin{split}\delta_{sc}^{(3/2)^{+}}=M^{2}_{\Xi^{*}}+M^{2}_{\Xi^{*}_{cc}}-2M^{2}_{\Xi^{*}_{c}}\\\
=M^{2}_{\Omega_{c}^{*}}+M^{2}_{\Omega_{ccc}^{*}}-2M^{2}_{\Omega_{cc}^{*}};\end{split}$
(31)

solving Eqs. (29) and (30) we have,

$(M^{2}_{\Omega_{cc}^{*}}-M^{2}_{\Xi^{*}_{cc}})+(M^{2}_{\Xi^{*}}-M^{2}_{\Sigma^{*}})=(M^{2}_{\Omega_{c}^{*}}-M^{2}_{\Sigma_{c}^{*}}),$
(32)

similarly, its corresponding relation for $\frac{1}{2}^{+}$ baryonic states
is,

$(M^{2}_{\Omega_{cc}}-M^{2}_{\Xi_{cc}})+(M^{2}_{\Xi}-M^{2}_{\Sigma})=(M^{2}_{\Omega_{c}}-M^{2}_{\Sigma_{c}}),$
(33)

using Eqs. (11) and (33) we have the expression,

$\begin{split}\left(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\Omega_{cc}^{+}}+M^{2}_{\Xi_{cc}^{++}}-2M^{2}_{\Sigma_{c}^{++}}-M^{2}_{\Xi^{0}}+M^{2}_{\Sigma^{+}}\right)-2M_{\Sigma_{c}^{++}}\sqrt{M^{2}_{\Omega_{cc}^{+}}-M^{2}_{\Xi_{cc}^{++}}+M^{2}_{\Xi^{0}}-M^{2}_{\Sigma^{+}}+M^{2}_{\Sigma_{c}^{++}}}\\\
=-\sqrt{\left(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\Omega_{cc}^{+}}+M^{2}_{\Xi_{cc}^{++}}-2M^{2}_{\Sigma_{c}^{++}}-M^{2}_{\Xi^{0}}+M^{2}_{\Sigma^{+}}\right)^{2}-4M^{2}_{\Sigma_{c}^{++}}\left(M^{2}_{\Omega_{cc}^{+}}-M^{2}_{\Xi_{cc}^{++}}+M^{2}_{\Xi^{0}}-M^{2}_{\Sigma^{+}}+M^{2}_{\Sigma_{c}^{++}}\right)}.\end{split}.$
(34)

Using the above expression we have calculated the ground state masses of
$\Omega_{cc}^{+}$ baryon. Inserting the masses $M_{\Xi_{c}^{+}}$,
$M_{\Sigma_{c}^{++}}$, $M_{\Xi_{cc}}$ and $M_{\Xi}$, $M_{\Sigma}$ from PDG[17]
and $M_{\Omega_{c}^{0}}$ (calculated above) in Eq. (34), we get
$M_{\Omega_{cc}^{+}}$=3.752 GeV for $J^{P}$=$\frac{1}{2}^{+}$. Similarly we
get $M_{\Omega_{cc}^{+}}$=3.816 GeV for $J^{P}$=$\frac{3}{2}^{+}$. Since the
quark configuration of $\Omega_{cc}^{++}$ is scc, hence using relation (8) we
have,

$\begin{split}\dfrac{\alpha^{{}^{\prime}}_{scc}}{\alpha^{{}^{\prime}}_{uus}}=\dfrac{1}{2M^{2}_{\Omega_{cc}^{+}}}\times[(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\varSigma^{+}}-M^{2}_{\Omega_{cc}^{+}})\\\
+\sqrt{{{(4M^{2}_{\Xi_{c}^{+}}-M^{2}_{\varSigma^{+}}-M^{2}_{\Omega_{cc}^{+}}})^{2}}-4M^{2}_{\varSigma^{+}}M^{2}_{\Omega_{cc}^{+}}}].\end{split}.$
(35)

Further using Eqs. (35) and (15) we calculate the excited state masses of
$\Omega_{cc}^{+}$ baryon in the same way we have done for $\Omega_{c}^{0}$.
Our predicted masses are labeled with ”Present” for both natural and unnatural
parity states presented in Table 4 and Table 5 respectively, where our
predictions are reasonably close to other theoretical predictions.

Table 4:  Masses of excited states of $\Omega_{cc}$ baryon (in GeV) in $(J,M^{2})$ with natural parities $P=(-1)^{J-\frac{1}{2}}$ .  States | $1^{2}S_{\frac{1}{2}}$ | $1^{2}P_{\frac{3}{2}}$ | $1^{2}D_{\frac{5}{2}}$ | $1^{2}F_{\frac{7}{2}}$ | $1^{2}G_{\frac{9}{2}}$ | $1^{2}H_{\frac{11}{2}}$
---|---|---|---|---|---|---
Present | 3.752 | 3.975 | 4.186 | 4.387 | 4.579 | 4.763
Ref. [18] | 3.650 | 3.972 | 4.141 | 4.296 |  |
Ref. [24] | 3.815 | 4.052 | 4.202 |  |  |
Ref. [26] | 3.719 |  |  |  |  |
Ref. [30] | 3.832 | 4.086 | 4.264 |  |  |
Ref. [31] | 3.650 |  | 4.174 |  |  |
Ref. [32] | 3.650 | 3.910 | 4.153 | 4.383 |  |
Ref. [47] | 3.778 | 4.102 |  |  |  |
Ref. [48] | 3.710 |  |  |  |  |
Ref. [49] | 3.637 |  |  |  |  |
Ref. [50] | 3.732 | 3.986 |  |  |  |
Ref. [51] | 3.804 |  |  |  |  |
Table 5:  Masses of excited states of $\Omega_{cc}$ baryon (in GeV) in $(J,M^{2})$ with unnatural parities $P=(-1)^{J+\frac{1}{2}}$.  states | $1^{4}S_{\frac{3}{2}}$ | $1^{4}P_{\frac{5}{2}}$ | $1^{4}D_{\frac{7}{2}}$ | $1^{4}F_{\frac{9}{2}}$ | $1^{4}G_{\frac{11}{2}}$ | $1^{4}H_{\frac{13}{2}}$
---|---|---|---|---|---|---
Present | 3.816 | 4.094 | 4.354 | 4.599 | 4.832 | 5.055
Ref. [18] | 3.810 | 3.958 | 4.122 | 4.274 |  |
Ref. [24] | 3.876 | 4.152 | 4.230 |  |  |
Ref. [29] | 3.822 |  |  |  |  |
Ref. [31] | 3.808 |  | 4.313 |  |  |
Ref. [32] | 3.809 | 4.058 | 4.294 | 4.516 |  |
Ref. [48] | 3.760 |  |  |  |  |
Ref. [49] | 3.762 |  |  |  |  |
Ref. [50] | 3.765 |  |  |  |  |
Ref. [51] | 3.850 |  |  |  |  |
Ref. [52] | 3.730 | 4.134 | 4.204 |  |  |
Ref. [53] | 3.800 |  |  |  |  |

Similarly for triply charmed omega baryon ($\Omega_{ccc}$), using relations
(31) and (34) we have,

$\begin{split}\left(3M^{2}_{\Xi_{cc}}-M^{2}_{\Omega_{ccc}}+6M^{2}_{\Xi_{c}}-4M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{c}}+M^{2}_{\Xi}+2M^{2}_{\varSigma}\right)-2M_{\Sigma_{c}^{++}}\sqrt{M^{2}_{\Omega_{ccc}}-3M^{2}_{\Xi_{cc}}+M^{2}_{\Omega_{c}}+M^{2}_{\Xi}+2\left(M^{2}_{\Xi_{c}}+M^{2}_{\varSigma_{c}}-M^{2}_{\varSigma}\right)}\\\
=-\sqrt{\left(3M^{2}_{\Xi_{cc}}-M^{2}_{\Omega_{ccc}}+6M^{2}_{\Xi_{c}}-4M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{c}}+M^{2}_{\Xi}+2M^{2}_{\varSigma}\right)^{2}-8M^{2}_{\Sigma_{c}^{++}}\left[M^{2}_{\Omega_{ccc}}-3M^{2}_{\Xi_{cc}}+M^{2}_{\Omega_{c}}+M^{2}_{\Xi}+2\left(M^{2}_{\Xi_{c}}+M^{2}_{\varSigma_{c}}-M^{2}_{\varSigma}\right)\right]}.\end{split}$
(36)

Since the ground state $(J^{P}$=$\frac{3}{2}^{+})$ mass of $\Xi_{cc}^{++}$ is
not confirmed by PDG yet, so we take $M_{\Xi_{cc}}^{*}$= 3.695 GeV [32] and
inserting all other masses from latest PDG [17] into Eq. (36), we obtain
$M_{\Omega_{ccc}^{*}}$=4.841 GeV for $J^{P}$=$\frac{3}{2}^{+}$. Excited state
masses for ${\Omega_{ccc}}$ baryon are calculated in the similar manner as we
have calculated for $\Omega_{c}^{0}$ and $\Omega_{cc}^{+}$, only by changing
the quark constituents i.e. $i$ ,$j$ and $q$. Our calculated masses of excited
$\Omega_{ccc}$ are presented in Table 6. Our results are in accordance with
other theoritical predictions.

Table 6:  Masses of excited states of $\Omega_{ccc}$ baryon (in GeV) in $(J,M^{2})$ plane with unnatural parities $P=(-1)^{J+\frac{1}{2}}$.  states | $1^{4}S_{\frac{3}{2}}$ | $1^{4}P_{\frac{5}{2}}$ | $1^{4}D_{\frac{7}{2}}$ | $1^{4}F_{\frac{9}{2}}$ | $1^{4}G_{\frac{11}{2}}$ | $1^{4}H_{\frac{13}{2}}$
---|---|---|---|---|---|---
Present | 4.841 | 5.081 | 5.310 | 5.530 | 5.741 | 5.945
Ref. [24] | 4.965 |  | 5.331 |  |  |
Ref. [26] | 4.803 |  |  |  |  |
Ref. [31] | 4.818 |  | 5.302 |  |  |
Ref. [32] | 4.834 |  | 5.301 |  |  |
Ref. [50] | 4.773 |  |  |  |  |
Ref. [51] | 4.930 |  |  |  |  |
Ref. [54] | 4.827 |  |  |  |  |
Ref. [55] | 4.758 |  | 5.300 |  |  |
Ref. [56] | 4.761 |  | 5.396 |  |  |
Ref. [57] | 4.880 |  |  |  |  |

###  II.3 Masses of $\Omega_{c}^{0}$, $\Omega_{cc}^{+}$ and
$\Omega_{ccc}^{++}$ baryons in $(n,M^{2})$ Plane.

The general equation for linear Regge trejectories in $(n,M^{2})$ plane can be
expressed as,

$n=\beta_{0}+\beta M^{2},$ (37)

where $n$ = 1,2,3….is the radial principal quantum number, $\beta$ and
$\beta_{0}$ are the slope and intercept of the trejectories. The Regge slope
($\beta$) and Regge intercept ($\beta_{0}$) are assumed to be same for all
baryon multiplets lying on the single Regge line. So first of all these
parameters ($\beta$ and $\beta_{0}$) are calculated, and with the help of them
the excited states masses ($N^{2S+1}L_{J}$, where $N$, $L$, $S$ denote the
radial excited quantum number, the orbital quantum number and the intrinsic
spin, respectively) of $\Omega_{c}^{0}$, $\Omega_{cc}^{+}$ and
$\Omega_{ccc}^{++}$ baryons are estimated lying on these Regge trejectories.
For $\Omega_{c}^{0}$ baryon, using the slope equation
$\beta_{(S)}=1/(M^{2}_{\Omega_{c}(2S)}-M^{2}_{\Omega_{c}(1S)})$, where
$M_{\Omega_{c}(1S)}$=2.702 GeV (calculated above) and taking
$M_{\Omega_{c}(2S)}$=3.164 GeV from [21] for $J^{P}$ = $1/2^{+}$ state, we get
$\beta_{(S)}$ = 0.3689 $GeV^{-2}$. Using the following equations,

$\begin{split}1=\beta_{0(S)}+\beta_{(S)}M^{2}_{\Omega_{c}(1S)},\\\
2=\beta_{0(S)}+\beta_{(S)}M^{2}_{\Omega_{c}(2S)},\end{split}$ (38)

we have $\beta_{0(S)}$ = -1.6939. In the same way we can write,

$\begin{split}1=\beta_{0(P)}+\beta_{(P)}M^{2}_{\Omega_{c}(1P)},\\\
2=\beta_{0(P)}+\beta_{(P)}M^{2}_{\Omega_{c}(2P)},\\\
1=\beta_{0(D)}+\beta_{(D)}M^{2}_{\Omega_{c}(1D)},\\\
2=\beta_{0(D)}+\beta_{(D)}M^{2}_{\Omega_{c}(2D)}.\end{split}$ (39)

With the help $\beta_{(S)}$ and $\beta_{0(S)}$, we calculate the masses of
excited $\Omega_{c}^{0}$ baryon for n=3,4,5… Tables 7 and 8 shows our
calculated radial and orbital excited state masses of $\Omega_{c}$ baryon in
$(n,M^{2})$ plane for natural and unnatural parity states respectively. In the
similar manner we estimate the mass spectra for $\Omega_{cc}$ (see Tables 9
and 10) and $\Omega_{ccc}$(see Table 11) baryons also in $(n,M^{2})$ plane.

Table 7:  Masses of excited states of $\Omega_{c}^{0}$ baryon (in GeV) in $(n,M^{2})$ plane with natural parity $P=(-1)^{J-\frac{1}{2}}$. The masses from [21] are taken as input.  $N^{2S+1}L_{J}$ | Present | [23] | [20] | [24] | [55] | [30]
---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 2.702 | 2.698 | 2.696 | 2.718 | 2.699 | 2.731
$2^{2}S_{\frac{1}{2}}$ |  3.164 [21] | 3.088 | 3.165 | 3.152 | 3.159 | 3.227
$3^{2}S_{\frac{1}{2}}$ | 3.566 | 3.489 | 3.540 |  |  | 3.292
$4^{2}S_{\frac{1}{2}}$ | 3.928 | 3.814 | 3.895 |  |  |
$5^{2}S_{\frac{1}{2}}$ | 4.259 | 4.102 | 4.238 |  |  |
$6^{2}S_{\frac{1}{2}}$ | 4.566 |  | 4.569 |  |  |
$1^{2}P_{\frac{3}{2}}$ | 3.049 | 3.054 | 2.968 | 2.986 |  | 3.033
$2^{2}P_{\frac{3}{2}}$ |  3.408 [21] | 3.433 | 3.334 |  |  | 3.056
$3^{2}P_{\frac{3}{2}}$ | 3.732 | 3.752 | 3.687 |  |  |
$4^{2}P_{\frac{3}{2}}$ | 4.031 | 4.036 | 4.029 |  |  |
$5^{2}P_{\frac{3}{2}}$ | 4.309 |  | 4.362 |  |  |
$1^{2}D_{\frac{5}{2}}$ | 3.360 | 3.297 | 3.251 |  |  |
$2^{2}D_{\frac{5}{2}}$ |  3.680 [21] | 3.626 | 3.604 |  |  |
$3^{2}D_{\frac{5}{2}}$ | 3.974 | 3.752 | 3.947 |  |  |
$4^{2}D_{\frac{5}{2}}$ | 4.248 |  | 4.282 |  |  |
$5^{2}D_{\frac{5}{2}}$ | 4.505 |  |  |  |  |
Table 8:  Masses of excited states of $\Omega_{c}^{0}$ baryon (in GeV) in $(n,M^{2})$ plane with unnatural parity $P=(-1)^{J+\frac{1}{2}}$. The masses from [21] are taken as input.  $N^{2S+1}L_{J}$ | Present | [23] | [20] | [24] | [55] | [30]
---|---|---|---|---|---|---
$1^{4}S_{\frac{3}{2}}$ | 2.772 | 2.768 | 2.766 | 2.776 | 2.768 | 2.779
$2^{4}S_{\frac{3}{2}}$ |  3.197 [21] | 3.123 | 3.208 | 3.190 | 3.202 | 3.257
$3^{4}S_{\frac{3}{2}}$ | 3.571 | 3.510 | 3.564 |  |  | 3.285
$4^{4}S_{\frac{3}{2}}$ | 3.910 | 3.830 | 3.911 |  |  |
$5^{4}S_{\frac{3}{2}}$ | 4.222 | 4.114 | 4.248 |  |  |
$6^{4}S_{\frac{3}{2}}$ | 4.513 |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 3.055 | 3.051 | 2.962 | 3.041 |  | 3.057
$2^{4}P_{\frac{5}{2}}$ |  3.393 [21] | 3.427 | 3.328 |  |  | 3.477
$3^{4}P_{\frac{5}{2}}$ | 3.700 | 3.744 | 3.681 |  |  | 3.620
$4^{4}P_{\frac{5}{2}}$ | 3.983 | 4.028 | 4.023 |  |  |
$5^{4}P_{\frac{5}{2}}$ | 4.248 |  |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 3.314 | 3.283 | 3.241 |  |  |
$2^{4}D_{\frac{7}{2}}$ |  3.656 [21] | 3.611 | 3.594 |  |  |
$3^{4}D_{\frac{7}{2}}$ | 3.968 |  | 3.938 |  |  |
$4^{4}D_{\frac{7}{2}}$ | 4.258 |  | 4.273 |  |  |
$5^{4}D_{\frac{7}{2}}$ | 4.529 |  | 4.594 |  |  |
Table 9:  Masses of excited states of $\Omega_{cc}^{+}$ baryon (in GeV) in $(n,M^{2})$ plane with natural parity $P=(-1)^{J-\frac{1}{2}}$. The masses from [18] are taken as input.  $N^{2S+1}L_{J}$ | Present | [47] | [30] | [24] | [32] |
---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 3.752 | 3.778 | 3.832 | 3.815 | 3.650 |
$2^{2}S_{\frac{1}{2}}$ |  4.028 [18] | 4.075 | 4.227 | 4.180 |  |
$3^{2}S_{\frac{1}{2}}$ | 4.310 | 4.321 | 4.295 |  |  |
$4^{2}S_{\frac{1}{2}}$ | 4.564 |  |  |  |  |
$5^{2}S_{\frac{1}{2}}$ | 4.804 |  |  |  |  |
$6^{2}S_{\frac{1}{2}}$ | 5.033 |  |  |  |  |
$1^{2}P_{\frac{3}{2}}$ | 3.975 | 4.102 | 4.086 | 4.052 | 3.910 |
$2^{2}P_{\frac{3}{2}}$ |  4.259 [18] | 4.345 | 4.201 | 4.140 |  |
$3^{2}P_{\frac{3}{2}}$ | 4.525 |  |  |  |  |
$4^{2}P_{\frac{3}{2}}$ | 4.776 |  |  |  |  |
$5^{2}P_{\frac{3}{2}}$ | 5.015 |  |  |  |  |
$1^{2}D_{\frac{5}{2}}$ | 4.186 |  | 4.264 | 4.202 | 4.153 |
$2^{2}D_{\frac{5}{2}}$ |  4.407 [18] |  |  |  |  |
$3^{2}D_{\frac{5}{2}}$ | 4.617 |  |  |  |  |
$4^{2}D_{\frac{5}{2}}$ | 4.818 |  |  |  |  |
$5^{2}D_{\frac{5}{2}}$ | 5.011 |  |  |  |  |
Table 10:  Masses of excited states of $\Omega_{cc}^{+}$ baryon (in GeV) in $(n,M^{2})$ plane with unnatural parity $P=(-1)^{J+\frac{1}{2}}$. The masses from [18] are taken as input.  $N^{2S+1}L_{J}$ | Present | [47] | [30] | [24] | [58] |
---|---|---|---|---|---|---
$1^{4}S_{\frac{3}{2}}$ | 3.816 | 3.872 | 3.883 | 3.876 | 3.824 |
$2^{4}S_{\frac{3}{2}}$ |  4.096 [18] | 4.174 | 4.263 | 4.188 | 4.163 |
$3^{4}S_{\frac{3}{2}}$ | 4.358 |  | 4.265 |  |  |
$4^{4}S_{\frac{3}{2}}$ | 4.605 |  |  |  |  |
$5^{4}S_{\frac{3}{2}}$ | 4.839 |  |  |  |  |
$6^{4}S_{\frac{3}{2}}$ | 5.063 |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 4.094 |  | 4.220 | 4.152 |  |
$2^{4}P_{\frac{5}{2}}$ |  4.247 [18] |  |  |  |  |
$3^{4}P_{\frac{5}{2}}$ | 4.394 |  |  |  |  |
$4^{4}P_{\frac{5}{2}}$ | 4.537 |  |  |  |  |
$5^{4}P_{\frac{5}{2}}$ | 4.676 |  |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 4.354 |  |  |  |  |
$2^{4}D_{\frac{7}{2}}$ |  4.391 [18] |  |  |  |  |
$3^{4}D_{\frac{7}{2}}$ | 4.427 |  |  |  |  |
$4^{4}D_{\frac{7}{2}}$ | 4.464 |  |  |  |  |
$5^{4}D_{\frac{7}{2}}$ | 4.500 |  |  |  |  |
Table 11:  Masses of excited states of $\Omega_{ccc}^{++}$ baryon (in GeV) in $(n,M^{2})$ plane with unnatural parity $P=(-1)^{J+\frac{1}{2}}$. The masses from [19] are taken as input.  $N^{2S+1}L_{J}$ | Present | [24] | [32] | [31] | [55]
---|---|---|---|---|---
$1^{4}S_{\frac{3}{2}}$ | 4.841 | 4.965 | 4.834 | 4.818 | 4.758
$2^{4}S_{\frac{3}{2}}$ |  5.300 [19] | 5.313 |  |  |
$3^{4}S_{\frac{3}{2}}$ | 5.722 |  |  |  |
$4^{4}S_{\frac{3}{2}}$ | 6.115 |  |  |  |
$5^{4}S_{\frac{3}{2}}$ | 6.484 |  |  |  |
$6^{4}S_{\frac{3}{2}}$ | 6.834 |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 5.081 |  |  |  |
$2^{4}P_{\frac{5}{2}}$ |  5.553 [19] |  |  |  |
$3^{4}P_{\frac{5}{2}}$ | 5.987 |  |  |  |
$4^{4}P_{\frac{5}{2}}$ | 6.393 |  |  |  |
$5^{4}P_{\frac{5}{2}}$ | 6.774 |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 5.310 | 5.331 | 5.301 | 5.302 | 5.300
$2^{4}D_{\frac{7}{2}}$ |  5.961 [19] |  |  |  |
$3^{4}D_{\frac{7}{2}}$ | 6.547 |  |  |  |
$4^{4}D_{\frac{7}{2}}$ | 7.085 |  |  |  |
$5^{4}D_{\frac{7}{2}}$ | 7.585 |  |  |  |

##  III Result and discussion

In the framework of regge phenomenology , we have obtained the radial and
orbital excited states of singly, doubly and triply charmed omega baryons. In
the present work we have focused on studying the masses of $\Omega_{c}^{0}$
baryon by obtaining the mass relations derived from regge phenomenology.
Ground state as well as excited state masses are obtained successfully in both
$(J,M^{2})$ and $(n,M^{2})$ planes for the following baryons.

  * •

$\Omega_{c}$ baryon :Our calculated masses in $(J,M^{2})$ plane for
$\Omega_{c}^{0}$ baryon are shown in Table 2 and Table 3 for natural and
unnatural parity states respectively. Firstly we compared our predicted ground
state masses with experimently available data [1, 2, 17] and other theoretical
predictions [23, 20, 21, 41, 24, 42, 30]. Our results are very close to [17]
with a slight mass difference of 7-8 MeV and also they are in good agreement
with Refs.[23, 20, 21, 41]. Further the excited state masses are compared with
other theoretical outcomes. For $1^{2}P_{\frac{3}{2}}$ and
$1^{2}P_{\frac{5}{2}}$ states our results are in accordance with Refs.[23, 21,
41, 42, 30] with mass difference of range 5-50 MeV and for
$1^{2}D_{\frac{5}{2}}$-$1^{2}F_{\frac{7}{2}}$ and
$1^{4}D_{\frac{7}{2}}$-$1^{4}F_{\frac{9}{2}}$ states our predictions are
consistent with Refs. [23, 20] with mass difference of 30-65 MeV. Similarly in
$(n,M^{2})$ plane, our calculated results are shown in Tables 7 and 8 for
natural and unnatural parity states respectively. Our predicted masses are in
agreement with Refs.[23, 20, 24]. The experimentally observed state
$\Omega_{c}(3050)^{0}$ having mass 3.050 GeV is close to our prediction 3.049
GeV, so we assigned $\Omega_{c}(3050)^{0}$ as $1P$ state with
$J^{P}$=$3/2^{-}$ for $S$ = $1/2$. Other four states $\Omega_{c}(3000)^{0}$,
$\Omega_{c}{(3066)}^{0}$, $\Omega_{c}{(3090)}^{0}$ and
$\Omega_{c}{(3119)}^{0}$ are not identified in this work (it may belong to
remaining $1P$ states such as $1^{2}P_{\frac{1}{2}}$, $1^{4}P_{\frac{1}{2}}$
and $1^{4}P_{\frac{3}{2}}$ [4]). Since we get masses for
$1^{2}S_{\frac{1}{2}}$, $1^{2}P_{\frac{3}{2}}$, $1^{2}D_{\frac{3}{2}}$,… and
$1^{4}S_{\frac{3}{2}}$, $1^{4}P_{\frac{5}{2}}$, $1^{4}D_{\frac{7}{2}}$,…
states for natural and unnatural parity respectively in $(J,M^{2})$ plane
lying on Regge line.

  * •

$\Omega_{cc}$ baryon : Table 4 and Table 5 shows our estimated results for
natural and unnatural parity states respectively in $(J,M^{2})$ plane. Our
calculated ground state $(1^{2}S_{\frac{1}{2}})$ mass is in good agreement
with the predictions of Refs.[24, 47, 48, 50, 51, 26] with mass difference of
25-65 MeV, and for $1^{2}P_{\frac{3}{2}}$-$1^{2}F_{\frac{7}{2}}$ states our
results are in accordance with Refs.[32, 31, 24, 50] with mass difference of
20-70 MeV (see Table 4). Similarly for $1^{4}S_{\frac{3}{2}}$ state our
predicted mass is very close to the results of Refs.[32, 31, 29, 53] having
slight mass difference of 7-16 MeV, and for
$1^{4}P_{\frac{5}{2}}$-$1^{4}F_{\frac{9}{2}}$ states our results are in
accordance with other theoritical outcomes (see Table 5). In the same manner
we compared our calculated radial and orbital excited state masses evaluated
in $(n,M^{2})$ plane for both natural and unnatural parity states and our
results are consistent with other theoretical and phenomenological studies
(see Tables 9 and 10).

  * •

$\Omega_{ccc}$ baryon : For triply charmed omega baryon, we calculate the
ground state and excited state masses in both $(J,M^{2})$ and $(n,M^{2})$
planes shown in Table 6 and Table 11 respectively. The masses the ground state
$\Omega_{ccc}^{++}$ vary in the range 4.750-4.950 GeV in other theoretical
references (Table 6). Our predicted mass for $(1^{4}S_{\frac{3}{2}})$ state
shows few MeV difference with Refs. [32, 31, 26, 54]. For excited state very
few results are availabe from previous theoretical predictions [24, 31, 32,
55], and our calculate masses are in accordance with them.

We have successfully employed Regge phenomenological approach to calculate the
masses of singly, doubly and triply charmed $\Omega$ baryons. This study will
definitely help future experimental studies at LHCb, Belle II and the upcoming
facility PANDA, to identify these baryonic states from resonances.

###### Acknowledgements.

One of the authors Juhi Oudichhya inspired by the work of Ke-Wei Wei, Bing
Chen, Xin-Heng Guo, De-Min Li, Bing Ma, Yu-Xiao Li, Qian-Kai Yao, Hong Yu on
Regge phenomenology and would like to thank them for their valuabe
contributions to this field.

## References

  * [1] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 118 (2017) 18, 182001.
  * [2] J. Yelton et al. (Belle Collaboration), Phys. Rev. D 97 (2018), 051102
  * [3] B. Chen nd X. Liu, Phys. Rev. D 96, 094015(2017)
  * [4] M. Karliner, J.L. Rosner, Phys. Rev. D 95, 114012(2017)
  * [5] M. Padmanath and N. Mathur, Phys. Rev. Lett. 119, 042001 (2017).
  * [6] W. Wang and R. L. Zhu,Phys. Rev. D 96, 014024(2017)
  * [7] Z. G. WangcitekeyEur. Phys.J. C 77, 325(2017) .
  * [8] S. S. Agaev, K. Azizi, and H. Sundu, Eur. Phys. Lett. 118,61001 (2017)
  * [9] H. X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu, and S.L. Zhu, Phys. Rev. D 95, 094008(2017)
  * [10] H. Y. Cheng and C. W. Chiang, Phys. Rev. D 95,094018(2017)
  * [11] K. L. Wang, L. Y. Xiao, X. H. Zhong, and Q. Zhao, Phys. Rev. D 95, 116010(2017)
  * [12] Z. Zhao, D. D. Ye, and A. Zhang, Phys. Rev. D 95,114024(2017)
  * [13] G. Yang and J. Ping, Phys. Rev. D 97, 034023(2018)
  * [14] H. Huang, J. Ping, and F. Wang, Phys. Rev. D 97, 034027(2018)
  * [15] C. S. An and H. Chen, Phys. Rev. D 96, 034012(2017)
  * [16] A.V. Manohar, M.B. Wise, Heavy Quark Physics (Cambridge Uni-versity Press, Cambridge, 2000)
  * [17] P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020 (2020) 8, 083C01.
  * [18] Z. Shah, K. Thakkar and A. K. Rai,Eur. Phys. J. C. 76,530 (2016)
  * [19] Z. Shah and A.K. Rai, Eur. Phys. J. A, 53, 195 (2017)
  * [20] K.Gandhi, A.K. Rai,Eur. Phys. J Plus 135(2020)
  * [21] Z. Shah, K. Thakkar, A. K. Rai and P.C. Vinodkumar, Eur. Phys. J A52, 313 (2016).
  * [22] D. Jia, J.-H. Pan, and C.-Q. Pang, arXiv:2007.01545v1 [hep-ph] (2020)
  * [23] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 84, 014025 (2011)
  * [24] W. Roberts annd M. Pervin, Int. J. Mod. Phys. A 23, 2817 (2008).
  * [25] T. M. Aliev, S. Bilmis and M. Savci, Mod. Phys. Lett. A 35, (2020)
  * [26] A. P. Martynenko, Phys. Lett. B 663, 317 (2008).
  * [27] F.Giannuzzi, Phys. Rev. D 79,094002 (2009)
  * [28] C. Alexandrou, V. Drach, K. Jansen, C. Kallidonis, G. Koutsou, Phys. Rev. D 90(7),074501 (2014).
  * [29] Z. S. Brown, W. Detmold, S. Meinel, and K. Orginos, Phys. Rev. D 90, 094507 (2014).
  * [30] T. Yoshida, E. Hiyama, A. Hosaka, M. Oka and K. Sadato, Phys. Rev. D 92, 114029 (2015).
  * [31] K.-W. Wei, Bing Chen and X.-H. Guo, Phys. Rev. D 78, , 056005 (2008)
  * [32] K.-W. Wei, Bing Chen and X.-H. Guo, Phys. Rev. D 92, 076008 (2015)
  * [33] Y. Nambu, Physical Review D 10, 4262 (1974)
  * [34] Y. Nambu, Physics Letters B 80, 372 (1979)
  * [35] H.X. Chen et al., Reports on Progress in Physics 80, 076201 (2017)
  * [36] G. F. Chew, S. C. Frautschi, Phys. Rev. Lett. 7 (1961)
  * [37] D.-M. Li, B. Ma, Y.-X. Li, Q.-K. Yao, and H. Yu, Eur. Phys. J. C 37, 323 (2004).
  * [38] L. Burakovsky, T. Goldman, and L. P. Horwitz, Phys. Rev.D 56, 7119 (1997); J. Phys. G 24, 771 (1998).
  * [39] L. Burakovsky and T. Goldman, Phys. Lett. B 434, 251 (1998).
  * [40] A. B. Kaidalov, Z. Phys. C 12, 63 (1982).
  * [41] D. Ebert, R.N. Faustov and V.O. Galkin, Phys. Lett. B 659, 612 (2008).
  * [42] Y. Yamaguchi et al. Phys. Rev. D 91, 034034 (2015).
  * [43] A. Zhang, Phys. Rev. D 72, 017902 (2005); Phys. Lett. B 647, 140 (2007).
  * [44] M. M. Brisudova, L. Burakovsky, and T. Goldman, Phys. Rev. D 61, 054013 (2000).
  * [45] J. L. Basdevant and S. Boukraa, Z. Phys. C 28, 413 (1985); 30, 103 (1986).
  * [46] A. B. Kaidalov, Z. Phys. C 12, 63 (1982).
  * [47] D. Ebert, R.N. Faustov, V.O. Galkin, A.P. Martynenko, Phys. Rev. D 66, 014008 (2002)
  * [48] Z. G. Wang, Eur. Phys. J. C 68, 459(2010)
  * [49] T.W. Chiu and T. H. Hsieh, Nucl. Phys. A755, 471 (2005).
  * [50] S. Migura, D. Merten, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A 28, 41 (2006).
  * [51] L. Burakovsky, T. Goldman, and L. P. Horwitz, Phys. Rev. D 56, 7124 (1997).
  * [52] S. S. Gershtein, V. V. Kiselev, A. K. Likhoded, and A. I. Onishchenko, Phys. Rev. D 62, 054021 (2000); V. V. Kiselev, A. K. Likhoded, O. N. Pakhomova, and V. A. Saleev, Phys. Rev. D 66, 034030 (2002).
  * [53] A. Bernotas and V. Simonis, Heavy hadron spectroscopy and the bag model, Lith. J. Phys. 49, 19 (2009).
  * [54] R. L. Jaffe and J. E. Kiskis, Phys. Rev. D 13, 1355 (1976).
  * [55] A. Valcarce, H. Garcilazo, and J. Vijande, Eur. Phys. J. A37, 217 (2008); J. Vijande, A. Valcarce, and H. Garcilazo, Phys. Rev. D 90, 094004 (2014).
  * [56] M. Padmanath, R. G. Edwards, N. Mathur, and M. Peardon, Phys. Rev. D 90, 074504 (2014).
  * [57] Z. Ghalenovi, A. A. Rajabi, S. x. Qin, and D. H. Rischke, Mod. Phys. Lett. A 29, 1450106 (2014).
  * [58] F. LÃij, K.L. Wang, L.Y. Xiao, X.H. Zhong (2017), 1708.04468