#  $2$-class groups of cyclotomic towers of imaginary biquadratic fields and
applications

MOHAMED MAHMOUD CHEMS-EDDIN  Mohamed Mahmoud Chems-Eddin: Mohammed First
University, Mathematics Department, Sciences Faculty, Oujda, Morocco
<EMAIL_ADDRESS>and  KATHARINA MÜLLER  Katharina Müller: Georg-August-
Universität, Göttingen, Germany<EMAIL_ADDRESS>goettingen.de

###### Abstract.

Let $d$ be an odd positive square-free integer. In this paper we shall
investigate the structure of the $2$-class group of the cyclotomic
$\mathbb{Z}_{2}$-extension of the imaginary biquadratic number field
$\mathbb{Q}(\sqrt{d},\sqrt{-1})$ if $d$ is of specifiic form. Furthermore, we
deduce the structure of the $2$-class group of the cyclotomic
$\mathbb{Z}_{2}$-extension of $\mathbb{Q}(\sqrt{-d})$.

###### Key words and phrases:

Cyclotomic $Z_{p}$-extension, $2$-rank; $2$-class group.

###### 2010 Mathematics Subject Classification:

11R29; 11R23; 11R18; 11R20.

##  1\. Introduction

Let $p$ be a prime number and $k$ be a number field. Denote by $k_{\infty}$
the cyclotomic $\mathbb{Z}_{p}$-extension of $k$. The field $k_{\infty}$
contains a unique cyclic subfield $k_{n}$ of degree $p^{n}$ over $k$. The
field $k_{n}$ is called the $n$-th layer of the $\mathbb{Z}_{p}$-extension of
$k$. In $1959$, the study of $p$-class numbers of number fields with large
degree led to a spectacular result due to Iwasawa, that we shall recall here
and use later (for $p=2$). Denote by $e_{n}$ the highest power of $p$ dividing
the class number of $k_{n}$. Then there exist integers $\lambda$, $\mu\geq 0$
and $\nu$, all independent of $n$, and an integer $n_{0}$ such that:

$\displaystyle e_{n}=\lambda n+\mu p^{n}+\nu,$ (1)

for all $n\geq n_{0}$. The integers $\lambda$, $\mu\geq 0$ and $\nu$ are
called the Iwasawa invariants of $k_{\infty}$(cf. [10]).

Thereafter, the study of cyclotomic $\mathbb{Z}_{2}$-extensions of
$\mathrm{CM}$-Fields was the subject of many papers and is still of huge
interest in algebraic number theory. Let us mention a few papers related to
the subject of this paper. In 1980, Kida studied the Iwasawa
$\lambda^{-}$-invariants and the 2-ranks of the narrow ideal class groups in
the 2-extensions of $\mathrm{CM}$-fields (cf. [12]). In $2018$, Atsuta (cf.
[5]) studied the maximal finite submodule of the minus part of the Iwasawa
module attached to $k_{\infty}$, while the second named author worked on the
capitulation in the minus-part in the steps of the cyclotomic
$\mathds{Z}_{p}$-extension of a $\mathrm{CM}$-field $k$ (cf. [19]).

In this paper we will concentrate on $\mathrm{CM}$-fields of the following
form: Let $n\geq 0$ be a natural number, $d$ be an odd positive square-free
integer and $L_{n,d}:=\mathbb{Q}(\zeta_{2^{n+2}},\sqrt{d})$. In $2019$, the
first named author, Azizi and Zekhnini, computed the rank of the $2$-class
group of $L_{n,d}$, the layers of the $\mathbb{Z}_{2}$-extension of some
special Dirichlet fields of the form $L_{0,d}=\mathbb{Q}(\sqrt{d},\sqrt{-1})$
(cf. [2, 7, 6]). Li, Ouyang, Xu and Zhang computed the $2$-class groups of
these fields for $d$ being a prime congruent to $3\pmod{8}$, $5\pmod{8}$ and
$7\pmod{16}$ (cf. [16]).

In the present work we consider some different infinite families of
biquadratic fields $L_{0,d}$ and determine the structure of the $2$-class
group of the $n$-th layer of their cyclotomic $\mathds{Z}_{2}$-extensions. Let
$h_{2}(d)$ denote the $2$-class number of the quadratic field
$\mathbb{Q}(\sqrt{d})$. The main aim of this paper is to prove the following
Theorem using some new techniques based on Iwasawa theory.

######  Theorem 1.

Let $d$ be a positive square-free integer and $n\geq 1$ be an integer.

  1. 1.

Assume $d$ has one of the following forms:

    1. $\bullet$

$d=p$, for a prime $p\equiv 9\pmod{16}$ such that $(\frac{2}{p})_{4}=1$,

    2. $\bullet$

$d=pq$, for two primes $p\equiv q\equiv 3\pmod{8}$.

Then the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathds{Z}/2^{n+r-2}\mathds{Z}\times\mathds{Z}/2\mathds{Z}$, for a constant
$r$ such that $2^{r}=h_{2}(-2d)$.

  2. 2.

Let $d=pq$, for two primes $p$ and $q$ such that $p\equiv-q\equiv 5\pmod{8}$.
Then the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathds{Z}/2^{n+r-2}\mathds{Z}$, for a constant $r$ such that and
$2^{r}=2\cdot h_{2}(-pq)$. Further, Greenberg’s Conjecture holds for the field
$L_{n,d}^{+}$, i.e., the $2$-class number of $L_{n,d}^{+}$ is uniformly
bounded.

The plan of this paper is the following: In section 2 we will summarize some
results on minus parts of $2$-class groups of $\mathrm{CM}$-fields. In section
3 we collect results on the rank of the $2$-class groups of the fields
$L_{n,d}$ and prove some of the main ingredients for the proof of the main
Theorem. Section 4 contains the proof of the main Theorem (cf. Theorems 8 and
9) and finally in section 5, we apply our main results to give the $2$-class
groups of the layers of the cyclotomic $\mathds{Z}_{2}$-extension of some
imaginary quadratic fields. The cyclotomic $\mathds{Z}_{2}$-extension of
imaginary quadratic fields were already investigated by Mizusawa in [15]. As
applications of our first above result we give a more precise description of
the structure of the $2$-class groups of the cyclotomic
$\mathds{Z}_{2}$-extensions for certain families of imaginary quadratic fields
(cf. Theorem 11 and Theorem 12):

######  Theorem 2.

Let $d$ be a positive square-free integer and $n\geq 1$. Let
$K_{0,d}=\mathbb{Q}(\sqrt{-d})$ and define the field $K_{n,d}$ as the $n$-th
layer of the cyclotomic $\mathds{Z}_{2}$-extension of $K_{0,d}$.

  1. 1.

Let $d$ have one of the following forms

    1. $\bullet$

$d=pq$ for two primes $p\equiv q\equiv 3\pmod{8}$,

    2. $\bullet$

$d=p$ for a prime $p\equiv 9\pmod{16}$ such that $(\frac{2}{p})_{4}=1$.

Let $2^{r}=h_{2}(-2d)$. Then the $2$-class group of $K_{n,d}$ is isomorphic to
$\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2^{n+r-1}\mathds{Z}$.

  2. 2.

Assume that $d=pq$ is the product of two primes $p\equiv-q\equiv 5\pmod{8}$
and let $2^{r}=2\cdot h_{2}(-pq)$. Then the $2$-class group of $K_{n,d}$ is
isomorphic to $\mathds{Z}/2^{n+r-1}\mathds{Z}$.

## Notations

Let $k$ be a $\mathrm{CM}$ number field. The next notations will be used for
the rest of this article:

  * •

$d$: An odd positive square-free integer,

  * •

$n$: An integer $\geq 0$,

  * •

$K_{n}=\mathbb{Q}(\zeta_{2^{n+2}})$,

  * •

$K_{n}^{+}$: The maximal real subfield of $K_{n}$,

  * •

$L_{n,d}=K_{n}(\sqrt{d})$,

  * •

$L_{n,d}^{+}$: The maximal real subfield of $L_{n,d}$,

  * •

$\tau$: A topological generator of $Gal(L_{\infty,d}/L_{0,d})$,

  * •

$\Lambda=\mathds{Z}_{p}[[T]]$ for $T=\tau-1$,

  * •

$\omega_{n}=(T+1)^{2^{n}}-1$,

  * •

$\nu_{n,m}=\omega_{n}/\omega_{m}$ for $n>m\geq 0$,

  * •

$\mu(M)$, $\lambda(M)$: The Iwasawa invariants introduced in (1) for a
$\Lambda$-torsion module $M$,

  * •

$\lambda(k)=\lambda(A_{\infty}(k))$ (a precise definition of $A_{\infty}$ for
the base field $k$ is given in Section 2), if $k=L_{0,d}$ we will also write
$\lambda$ for $\lambda(L_{0,d})$,

  * •

$\lambda^{-}(k)=\lambda(A_{\infty}^{-}(k))$ (a precise definition of
$A_{\infty}^{-}$ for the base field $k$ is given in Section 2), if $k=L_{0,d}$
we will also write $\lambda^{-}$ for $\lambda^{-}(L_{0,d})$,

  * •

$\mu^{-}(k)=\mu(A^{-}_{\infty}(k))$,

  * •

$\mathcal{N}$: The application norm for the extension $L_{n,d}/K_{n}$,

  * •

$E_{k}$: The unit group of $k$,

  * •

$\mathrm{Cl}(k)$: The class group of $k$,

  * •

$\mu_{k}$: The number of roots of unity contained in $k$,

  * •

$W_{k}$: The group of roots of unity in $k$,

  * •

$h_{2}(d)$: The $2$-class number of the quadratic field
$\mathbb{Q}(\sqrt{d})$,

  * •

$\genfrac{(}{)}{}{0}{a}{p}_{4}$: The biquadratic residue symbol,

  * •

$\left(\frac{\alpha,d}{\mathfrak{p}}\right)$: The quadratic norm residue
symbol for $L_{n,d}/K_{n}$,

  * •

$Q_{k}$: Hasse’s unit index of a CM-field $k$,

  * •

$q(L_{1,d}):=(E_{L_{1,d}}:\prod_{i}E_{k_{i}})$, with $k_{i}$ are the quadratic
subfields of $L_{1,d}$.

##  2\. Some preliminary results on the minus part of the $2$-class group

Let $p$ be a prime and $k$ be an arbitrary $\mathrm{CM}$-field containing the
$p$-th roots of unity (the $4$-th roots of unity if $p=2$). Consider the
cyclotomic $\mathds{Z}_{p}$-extension of $k$, denoted by $k_{\infty}$. The
complex conjugation of $k$, denoted by $j$, acts on $A_{n}$, the $p$ part of
the class group of the intermediate fields $k_{n}$, as well as on the
projective limit $A_{\infty}=\lim_{\infty\leftarrow n}A_{n}$. Usually one
defines the minus part of the class group as $\widehat{A^{-}_{n}}=\\{a\in
A_{n}\mid ja=-a\\}$ and the plus part as $\widehat{A^{+}_{n}}=\\{a\in
A_{n}\mid ja=a\\}$. For $p\neq 2$ this yields a direct decomposition of
$A_{n}=\widehat{A_{n}^{-}}\oplus\widehat{A_{n}^{+}}$. Further, it is well
known that there is no capitulation on the minus part for $p\neq 2$. For $p=2$
this is in general not true. To avoid this problem we define $A_{n}^{+}$ as
the group of strongly ambiguous classes with respect to the extension
$k_{n}/k_{n}^{+}$ and $A_{n}^{-}=A_{n}/A_{n}^{+}$. Note that
$A_{\bullet}^{+}=\widehat{A_{\bullet}^{+}}$ and
$A_{\bullet}^{-}\cong\widehat{A_{\bullet}^{-}}$ for $p\neq 2$ (see [19]).For
the rest of the paper we will study only cyclotomic
$\mathds{Z}_{2}$-extensions.

Note that the projective limit $A_{\infty}^{-}=\lim_{\infty\leftarrow
n}A_{n}^{-}$ is a finitely generated $\Lambda$-torsion module.

######  Lemma 1.

Assume that $\mu(A_{\infty}^{-})=0$. Then there exists some $n_{0}\geq 0$ such
that we have $\lambda(A_{\infty}^{-})\geq 2\textup{-rank}(A_{n}^{-})$ for all
$n\geq n_{0}$.

###### Proof.

By [19, Theorem 2.5] there is no finite submodule in $A_{\infty}^{-}$. So if
$\mu(A_{\infty}^{-})=0$ the $2$-rank and $\lambda$-invariant of
$A_{\infty}^{-}$ are equal. Thus, the claim is immediate for $n_{0}$ being the
index such that all primes above $2$ are totally ramified in
$K_{\infty}/K_{n_{0}}$. ∎

######  Remark 1.

If $K=L_{0,d}=\mathbb{Q}(\sqrt{-1},\sqrt{d})$, then $K_{\infty}/K$ is totally
ramified and $n_{0}=0$.

######  Lemma 2.

Assume that $\mu(A_{\infty}^{-})=0$. Then
$\lambda(A^{-}_{\infty})=\lambda(\widehat{A^{-}_{\infty}})$.

###### Proof.

Note that $2A_{\infty}\subset(1+j)A_{\infty}+(1-j)A_{\infty}\subset
A_{\infty}$. Clearly, all elements in $(1+j)A_{\infty}$ are strongly
ambiguous. Thus, if we consider the projection

$\pi:A_{\infty}\to A_{\infty}^{-}$

we see that $(1+j)A_{\infty}$ lies in the kernel of $\pi$. On the other hand
$j(1-j)a=-(1-j)a$. So if a class in $(1-j)A_{\infty}$ is strongly ambiguous
then it is of order dividing $2$. As $\mu=0$ we obtain that $(1-j)A_{\infty}$
intersects the kernel of $\pi$ only in a finite submodule. It follows that

$\lambda(A^{-}_{\infty})=\lambda((1-j)A_{\infty}).$

Note that
$2\widehat{A^{-}_{\infty}}\subset(1-j){A^{-}_{\infty}}\subset\widehat{A^{-}_{\infty}}$.
Hence, we see that

$\lambda(A_{\infty}^{-})=\lambda(\widehat{A^{-}_{\infty}}).$

∎

##  3\. Preliminaries on the fields $L_{n,d}$ and $L_{n,d}^{+}$

To determine the structure of the $2$-class group along a cyclotomic tower the
$\lambda$-invariants of $A_{n}$ are of particular interest. Kida proved the
following formula.

######  Theorem 3.

[12, Theorem 3] Let $F$ and $K$ be $\mathrm{CM}$-fields and $K/F$ a finite $2$
extension. Assume that $\mu^{-}(F)=0$. Then

$\lambda^{-}(K)-\delta(K)=[K_{\infty}:F_{\infty}]\left(\lambda^{-}(F)-\delta(F)\right)+\sum(e_{\beta}-1)-\sum(e_{\beta^{+}}-1),$

where $\delta(k)$ takes the values $1$ or $0$ according to whether
$k_{\infty}$ contains the fourth roots of unity or not. The $e_{\beta}$ is the
ramification index of a prime $\beta$ in $K_{\infty}/F_{\infty}$ coprime to
$2$ and $e_{\beta^{+}}$ is the ramification index for a prime coprime to $2$
in $K_{\infty}^{+}/F_{\infty}^{+}$.

Note that Kida proves results for $\lambda(\widehat{A^{-}})$. But due to Lemma
2 this $\lambda$-invariant equals the $\lambda$-invariant of $A^{-}$.

######  Theorem 4.

Assume that $d$ is the product of $r$ primes congruent to $7$ or $9\pmod{16}$
and $s$ primes congruent to $3$ or $5\pmod{8}$. Then

$\lambda^{-}(L_{0,d})=2r+s-1.$

###### Proof.

Let $K=L_{0,d}=\mathbb{Q}(\sqrt{d},\sqrt{-1})$ and $F=\mathbb{Q}(\sqrt{-1})$.
Then $\delta(F)=\delta(K)=1$ and $\lambda^{-}(F)=0$. Every prime congruent to
$7$ or $9$ modulo $16$ splits into $4$ primes in $K_{n}$ for $n$ large enough,
while it splits only into $2$ primes in $K^{+}_{n}$ (see [6, Proposition 1]).
Primes congruent to $3$ or $5$ modulo $8$ decompose into $2$ primes in
$K_{n}$, while $K_{n}^{+}$ contains only one prime above $p$ (see [7,
Proposition 2]). As
$[K_{\infty}:F_{\infty}]=[K_{\infty}^{+}:F_{\infty}^{+}]=2$ all the non
trivial terms satisfy $e_{\beta}=e_{\beta^{+}}=2$. Plugging all of this into
Kida’s formula we obtain

$\lambda^{-}(L_{0,d})-1=2(0-1)+4r+2s-2r-s=2r+s-2$

and the claim follows. ∎

The above result gives $\lambda^{-}(L_{0,d})$ of some fields $L_{0,d}$. Noting
that $\lambda^{+}(L_{0,d})$ is related to the class numbers of the real fields
$L_{n,d}^{+}$, we need the following theorem:

######  Theorem 5.

Let $d$ be an odd positive square-free integer and $n\geq 1$. Then, the class
number of $L_{n,d}^{+}$ is odd if and only if $d$ takes one of the following
forms

  1. 1.

$d=q_{1}q_{2}$ with $q_{i}\equiv 3\pmod{4}$ and $q_{1}$ or $q_{2}\equiv
3\pmod{8}$.

  2. 2.

$d$ is a prime $p$ congruent to $3\pmod{4}$.

  3. 3.

$d$ is a prime $p$ congruent to $5\pmod{8}$.

  4. 4.

$d$ is a prime $p$ congruent to $1\pmod{8}$ and
$(\frac{2}{p})_{4}(\frac{p}{2})_{4}=-1$.

###### Proof.

The extension $L_{n+1,d}^{+}/L_{n,d}^{+}$ is a quadratic extension that
ramifies at the prime ideals of $L_{n,d}^{+}$ lying over $2$ and is unramified
elsewhere for all $n\geq 1$. Let $H(L_{n,d}^{+})$ be the $2$-Hilbert class
field of $L_{n,d}^{+}$ and $X_{n}$ its Galois group over $L_{n,d}^{+}$. Let
$Y$ be the $\Lambda$-submodule of $X_{\infty}=\lim_{\infty\leftarrow n}X_{n}$
such that $X_{0}\cong X_{\infty}/Y$. Then $X_{n}\cong X_{\infty}/\nu_{n,0}Y$
[21, Lemma 13.18]. In particular, if $X_{n}$ is trivial then
$X_{\infty}=\nu_{n,0}X_{\infty}$ and $X_{\infty}$ is trivial by Nakayama’s
Lemma. Hence, the class number of $L_{1,d}^{+}$ being odd implies that the
class number of $L_{n,d}^{+}$ is odd. The converse follows from [21, Theorem
10.1] and the fact that the extension $L_{n+1,d}^{+}/L^{+}_{n,d}$ is totally
ramified. Hence, the class number of $L_{n,d}^{+}$ is odd if and only if the
class number of $L_{1,d}^{+}=\mathbb{Q}(\sqrt{2},\sqrt{d})$ is odd. See [8,
pp. 155, 157] and [9, p. 78] for the rest. ∎

######  Theorem 6.

Let $d>2$ be an odd square-free integer and $n\geq 2$ a positive integer. Then
the $2$-class group of $L_{n,d}$ is cyclic non-trivial if and only if $d$
takes one of the following forms:

  1. 1.

$d$ is a prime congruent to $7\pmod{16}$,

  2. 2.

$d=pq$, where $p$ and $q$ are two primes such that $q\equiv 3\pmod{8}$ and
$p\equiv 5\pmod{8}$.

###### Proof.

By [7, Theorem 6], it suffices to check the case when $d=p$ is a prime
congruent to $7\pmod{8}$. We shall distinguish two cases.

  1. $\bullet$

Suppose that $p$ is congruent to $15\pmod{16}$ and let $\sigma$ denote its
Frobenius homomorphism in $Gal(\mathbb{Q}(\zeta_{16})/\mathbb{Q})$. Then
$\sigma(\zeta_{16})=\zeta_{16}^{p}$ by the definition of the Frobenius
homomorphism. Let $H$ be the group generated by $\sigma$. Then $p$ is totally
split in $\mathbb{Q}(\zeta_{16})^{H}/\mathbb{Q}$. Since $p\equiv 15\mod{16}$,
$\sigma$ is the complex conjugation. Hence, $p$ is totally split in
$\mathbb{Q}(\zeta_{16})^{+}/\mathbb{Q}$ and inert in
$\mathbb{Q}(\zeta_{16})/\mathbb{Q}(\zeta_{16})^{+}$.

In particular, there are $4$ primes of $K_{2}$ lying over $p$. Then, by the
ambiguous class number formula (cf. [23, Lemma 2.4])
$2$-rank$(\mathrm{Cl}(L_{2,d}))=4-1-e$, where $e$ is defined by
$2^{e}=[E_{K_{2}}:E_{K_{2}}\cap\mathcal{N}(L_{2,d}^{*})]$. The unit group of
$K_{2}$ is given by
$E_{K_{2}}=\langle\zeta_{16},\xi_{3},\xi_{5},\xi_{7}\rangle$, where
$\xi_{k}=\zeta_{16}^{(1-k)/2}\frac{1-\zeta_{16}^{k}}{1-\zeta_{16}}$. Let
$\mathcal{N}^{\prime}$ be the norm form $K_{2}$ to $K_{2}^{+}$. Since $p$ is
inert in $K_{2}/K_{2}^{+}$ we obtain for $k=3,5$ or $7$

$\left(\frac{\xi_{k},p}{\mathfrak{p}_{K_{2}}}\right)=\left(\frac{\mathcal{N}^{\prime}(\xi_{k}),p}{\mathfrak{p}_{K_{2}^{+}}}\right)=\left(\frac{\xi_{k}^{2},p}{\mathfrak{p}_{K_{2}^{+}}}\right)=1.$

Then $e$ is at most $1$. So $2$-rank$(\mathrm{Cl}(L_{2,d}))\geq 4-1-1=2$.
Hence, the $2$-class group of $L_{n,d}$ is not cyclic.

  2. $\bullet$

Suppose now that $p$ is congruent to $7\pmod{16}$, then by Theorem 5, the
class number of $L_{n,d}^{+}$ odd. Hence, $\lambda=\lambda^{-}$. Since the
primes above $2$ are unramified in $L_{n,p}/L_{n,p}^{+}$ for $n$ large enough
all strongly ambigous ideals in $L_{n,d}$ are actually ideals from
$L_{n,d}^{+}$ and the $2$-rank of $A_{n}$ is bounded by $\lambda$. By [2,
Theorem 4.4], the $2$-class group of $L_{1,p}$ is cyclic non-trivial and by
Theorem 4 $\lambda^{-}=1$. Which completes the proof.

∎

######  Theorem 7.

Assume that $d$ takes one of the forms of Theorem 6. Then $\lambda(L_{0,d})=1$
and Greenberg’s conjecture holds for $L_{n,d}^{+}$.

###### Proof.

By Theorem 6 the $2$-class group of $L_{n,d}$ is cyclic. By Theorem 4
$\lambda^{-}(L_{0,d})=1$. Thus, $\lambda(L_{0,d})=\lambda^{-}(L_{0,d})=1$ and
the first claim follows. Recall that
$\lambda(\widehat{A^{-}_{\infty}})=\lambda(A^{-}_{\infty})$. Note that the
groups $\widehat{A^{-}_{n}}\cap A^{+}_{n}$ are of exponent $2$. So if we know
that the $2$-class group of $L_{n,d}$ is cyclic and
$\lambda(\widehat{A^{-}_{\infty}})=1$, then $A_{n}^{+}$ contains at most $2$
elements. As the capitulation kernel $A_{n}(L_{n,d}^{+})\to A_{n}(L_{n,d})$
contains at most $2$ elements due to [21, Theorem 10.3], we see that the
$2$-class group of $L_{n,d}^{+}$ is uniformly bounded. ∎

We will also need [7, Theorem 5] and [6, Theorem 1] which are summarized in
the following Theorem.

######  Theorem 8.

Let $n\geq 1$ and assume that $d$ takes one of the following forms:

  1. 1.

$d=pq$, for two primes $p$ and $q$ congruent to $3\pmod{8}$.

  2. 2.

$d=p$, is a prime congruent to $9\pmod{16}$.

Then the rank of the $2$-class group of $L_{n,d}$ is $2$.

##  4\.  The Main results

######  Lemma 3.

Let $d$ be a square-free integer. We have:

  1. $1.$

$h_{2}(L_{1,d})=2\cdot h_{2}(-d)$, if $d=pq$, for two primes $p\equiv
5\pmod{8}$ and $q\equiv 3\pmod{8}$.

  2. $2.$

$h_{2}(L_{1,d})=h_{2}(-2d)$, if $d=pq$, for two primes $p\equiv q\equiv
3\pmod{8}$ or $d=p$ for a prime $p$ such that $p\equiv 9\pmod{16}$ and
$(\frac{2}{p})_{4}=1$.

###### Proof.

  1. $1.$

Suppose that $d$ takes the first form of the lemma. By [8, Corollary 19.7]
$h_{2}(pq)=h_{2}(2pq)=2$ and by [11, p. 353] $h_{2}(-2qp)=4$. Denote by
$\varepsilon_{2pq}$ the fundamental unit of the quadratic field
$\mathbb{Q}(\sqrt{2pq})$. We have $\varepsilon_{2pq}=x+y\sqrt{2pq}$, for some
integers $x$ and $y$. Since $\varepsilon_{2pq}$ has a positive norm we obtain
$x^{2}-2pqy^{2}=1$. Thus $x^{2}-1=2pqy^{2}$. Put $y=y_{1}y_{2}$ for
$y_{i}\in\mathbb{Z}$. For a moment, let us suppose that we can write

$\left\\{\begin{array}[]{ccc}x\pm 1&=&y_{1}^{2}\\\ x\mp
1&=&2pqy_{2}^{2}.\end{array}\right.$

Hence $1=\left(\frac{y_{1}^{2}}{p}\right)=\left(\frac{x\pm
1}{p}\right)=\left(\frac{x\mp 1\pm 2}{p}\right)=\left(\frac{\pm
2}{p}\right)=\left(\frac{2}{p}\right)=-1,$ which is impossible. So $x\pm 1$ is
not square in $\mathbb{N}$. So from the third and the fourth item of [4,
Proposition 3.3], we deduce that $q(L_{1,d})=4$. By the class number formula
$($cf. [20, p. 201]$)$, we have

$\displaystyle h_{2}(L_{1,d})$ $\displaystyle=$
$\displaystyle\frac{1}{2^{5}}q(L_{1,d})h_{2}(pq)h_{2}(-pq)h_{2}(2pq)h_{2}(-2qp)h_{2}(2)h_{2}(-2)h_{2}(-1)$
$\displaystyle=$
$\displaystyle\frac{1}{2^{5}}q(L_{1,d})h_{2}(pq)h_{2}(-pq)h_{2}(2pq)h_{2}(-2qp)$
$\displaystyle=$ $\displaystyle\frac{1}{2^{5}}\cdot 4\cdot 2\cdot
h_{2}(-pq)\cdot 2\cdot 4$ $\displaystyle=$ $\displaystyle 2\cdot h_{2}(-pq).$

  2. $2.$

Suppose now that $d$ takes one of the forms in the second item. Then we have
the result by [3, Corollary 2] and [3, The proof of Theorem 1, p. 7].

∎

######  Theorem 9.

Let $d$ be in one of the following cases:

  * •

$d=p$ be a prime congruent to $9\pmod{16}$ and assume that
$(\frac{2}{p})_{4}=1$.

  * •

$d=pq$ for two primes congruent to $3\pmod{8}$.

Let $2^{r}=h_{2}(-2d)$.Then for $n\geq 1$ the $2$-class group of $L_{n,d}$ is
isomorphic to the group
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{n+r-2}\mathbb{Z}$. In the
projective limit we obtain $\mathds{Z}_{2}\times\mathds{Z}/2\mathds{Z}$.

###### Proof.

By Theorem 8 we know that the $2$-rank of the $2$-class group of $L_{n,d}$
equals $2$ for $n\geq 1$. Further $\lambda^{-}(L_{0,d})=1$ due to Theorem 4
and $h_{2}(L_{1,d})=2^{r}$ by Lemma 3. By Theorem 5 the class number of
$L^{+}_{n,d}$ is odd for all $n$. As there is no capitulation in $A_{n}^{-}$
(see [19, Lemma 2.2]) and $\lambda^{-}(L_{0,d})=1$ we see that $A_{n}^{-}$ has
rank one for $n$ large enough (see also Lemma 1). That implies that the second
generator of the $2$-class group of $L_{n,d}$ is a class of a ramified prime
in $L_{n,d}/L^{+}_{n,d}$. As the class number of $L_{n,d}^{+}$ is odd these
ramified classes have order $2$ and we obtain that the $2$-class group of
$L_{n,d}$ is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{l_{n}}\mathbb{Z}$.

Let $E$ be the elementary $\Lambda$-module associated to $A_{\infty}$. Then
according to [21, page 282-283] $\nu_{n,0}E=2\nu_{n-1,0}E$ for all $n\geq 2$:
Indeed, $\nu_{n,0}=\nu_{n,n-1}\nu_{n-1,0}$. As $E$ has $\mathds{Z}_{2}$-rank
$1$ we know that $T$ acts as $2v$ on $E$ for some $v\in\mathds{Z}_{p}$. For
$n\geq 2$ we have $\nu_{n,n-1}=(T+1)^{2^{n-1}}-1+2$. The term
$(T+1)^{2^{n-1}}-1=T^{2^{n-1}}+O(2T)$ acts as $4v^{\prime}$ on $E$ for some
$v^{\prime}\in\mathds{Z}_{2}$. Hence, $\nu_{n,n-1}$ acts as $2u$ on $E$ for
some unit $u\in\mathds{Z}_{2}$ and all $n\geq 2$. Hence,

$|E/\nu_{n,0}E|=|E/2^{n-1}E||E/\nu_{1,0}E|=2^{n-1+c^{\prime}}$

for $n\geq 1$ and some constant $c^{\prime}\geq 1$ independent of $n$. Note
that we can rewrite this as $|E/\nu_{n,0}E|=2^{n+c}$. As $E$ has only one
$\mathds{Z}_{2}$-generator we can assume that the pseudoisomorphism
$\phi:A_{\infty}\to E$ is surjective. The maximal finite submodule of
$A_{\infty}$ is generated by the classes $(c_{n})_{n\in\mathbb{N}}$ of the
ramified primes above $2$. Let $\tau$ be a generator of
$Gal(L_{d,\infty}/L_{0,d})$. Then $\tau(c_{n})=c_{n}$ as the primes above $2$
are totally ramified in $L_{\infty,d}/\mathbb{Q}(\sqrt{d})$. It follows that
$Tc_{n}=0$. Hence, for every $n\geq 1$ the kernel of
$\overline{\phi}:A_{\infty}/\nu_{n,0}A_{\infty}\to E/\nu_{n,0}E$ is isomorphic
to the maximal finite submodule in $A_{\infty}$ and contains $2$ elements. Let
$Y$ be such that $A_{\infty}/Y\cong A_{0}$. Then $A_{n}\cong
A_{\infty}/\nu_{n,0}Y$ [21, page 281]. Then we obtain

$|A_{n}|=|A_{\infty}/\nu_{n,0}Y|=|A_{\infty}/\nu_{n,0}A_{\infty}||\nu_{n,0}A_{\infty}/\nu_{n,0}Y|=2^{n+c+1}|\nu_{n,0}A_{\infty}/\nu_{n,0}Y|\textup{
for }n\geq 1.$

As the maximal finite submodule in $A_{\infty}$ is annihilated by $\nu_{n,0}$
we see that the size of $\nu_{n,0}A_{\infty}/\nu_{n,0}Y$ is constant
independent of $n$. Hence, we obtain that the $2$-class group of $L_{n,d}$ is
of size $2^{n+\nu}$ for all $n\geq 1$. Using that $h_{2}(L_{1,d})=2^{r}$ we
obtain $\nu=r-1$. This yields $2\cdot 2^{l_{n}}=2^{n+r-1}$ and we obtain
$l_{n}=n+r-2$. Noting that $L_{n,d}$ is the $n$-th step of the field $L_{0,d}$
finishes the proof of the first claim. As the direct term
$\mathds{Z}/2\mathds{Z}$ is norm coherent the second claim is immediate. ∎

######  Corollary 1.

Let $d$ be in one of the following cases:

  1. $\bullet$

$d=p$ a prime congruent to $9\pmod{16}$ and assume that $(\frac{2}{p})_{4}=1$,

  2. $\bullet$

$d=pq$ for two primes congruent to $3\pmod{8}$.

If $d$ takes the first form set $p=u^{2}-2v^{2}$ where $u$ and $v$ are two
positive integers such that $u\equiv 1\mod 8$.
If $d$ takes the second form set $\left(\frac{p}{q}\right)=1$ and let the
integers $X,Y,k,l$ and $m$ be such that $2q=k^{2}X^{2}+2lXY+2mY^{2}$ and
$p=l^{2}-2k^{2}m$ $($see [11, p. 356] for their existence$)$. Let
$2^{r}=h_{2}(-d)$. For all $n\geq 1$, we have:

  1. 1.

If $d$ takes the first from, then the $2$-class group of $L_{n,d}$ is
isomorphic to $\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2^{n+1}\mathds{Z}$ if
and only if $\left(\frac{u}{p}\right)_{4}=-1$.
Elsewhere, it is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{n+r-2}\mathbb{Z}$, where $r\geq 4$
was defined in Theorem 9.

  2. 2.

If $d$ takes the second from, then the $2$-class group of $L_{n,d}$ is
isomorphic to $\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2^{n+1}\mathds{Z}$ if
and only if $\left(\frac{-2}{|X+lY|}\right)=-1$.

Elsewhere, it is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{n+r-2}\mathbb{Z}$, where $r\geq 4$
was defined in Theorem 9.

###### Proof.

By Lemma 3 we know $h_{2}(L_{1,d})=h_{2}(-2d)$. Since the $2$-rank of
$\mathrm{Cl}(L_{1,d})$ equals $2$ and $|\mathrm{Cl}(L_{n,d})|\neq 4$ (see [2,
Theorem 5.7]) it follows that $h_{2}(-2d)$ is divisible by $8$. Thus [14,
Theorem 2] (resp. [11, pp. 356-357]) gives the first (resp. second) item. ∎

We give the following numerical examples that illustrating the above
corollary:

  1. (1)

Set $p=89$, $u=17$ and $v=10$. We have $p=u^{2}-2v^{2}$ and
$\left(\frac{2}{p}\right)_{4}=-\left(\frac{u}{p}\right)_{4}=1$. So the
$2$-class group of $L_{n,d}$ is isomorphic to
$\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2^{n+1}\mathds{Z}$, for all $n\geq 1$.

  2. (2)

Let $p=11$, $q=19$, $k=1$, $l=3$, $m=-1$, $X=4$ and $Y=1$. We have :
$p=l^{2}-2k^{2}m$ and $2q=k^{2}X^{2}+2lXY+2mY^{2}$. Since
$\left(\frac{-2}{\left|X+lY\right|}\right)=\left(\frac{-2}{7}\right)=-1$, So
the $2$-class group of $L_{n,p}$ is isomorphic to
$\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2^{n+1}\mathds{Z}$, for all $n\geq 1$.

######  Theorem 10.

Assume that $d=pq$ is the product of two primes $p\equiv-q\equiv 5\pmod{8}$
and $2^{r}=2\cdot h_{2}(-pq)$. Then for $n\geq 1$ the $2$-class group of
$L_{n,d}$ is isomorphic to $\mathbb{Z}/2^{n+r-1}\mathbb{Z}$.

###### Proof.

We know already from Theorem 6 that the $2$-class group of $L_{n,d}$ is cyclic
and by Theorem 4 that $\lambda(L_{0,d})=1$. In particular, the module
$A_{\infty}$ does not contain a finite submodule and is hence isomorphic to
its elementary module $E$. Let $Y$ as in the proof of Theorem 9, then there is
no $\nu_{n,0}$-torsion and we obtain that the size of
$\nu_{n,0}A_{\infty}/\nu_{n,0}Y$ equals a constant independent of $n$. As
before we obtain
$|A_{n}|=|A_{\infty}/\nu_{n,0}A_{\infty}||\nu_{n,0}A_{\infty}/\nu_{n,0}Y|=2^{n+d}.$
In particular, Iwasawa’s formula holds for all $n\geq 1$. Hence,
$h_{2}(L_{1,d})=2^{r}=2^{1+\nu}$ and $\nu=r-1$ and the claim follows. ∎

######  Corollary 2.

Let $d=pq$ be the product of two primes $p$ and $q$ such that $p\equiv-q\equiv
5\pmod{8}$. Then for all $n\geq 1$, we have

  1. 1.

If $\left(\frac{p}{q}\right)=-1$, then, the $2$-class group of $L_{n,d}$ is
isomorphic to $\mathds{Z}/2^{n+1}\mathds{Z}$.

  2. 2.

If $\left(\frac{p}{q}\right)=1$ and $\left(\frac{q}{p}\right)_{4}=1$, then,
the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathds{Z}/2^{n+2}\mathds{Z}$.

Elsewhere, the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathds{Z}/2^{n+r-1}\mathds{Z}$, where $r\geq 4$ was defined in Theorem 10.

###### Proof.

By the previous theorem, the first item is direct from [8, 19.6 Corollary] and
also the rest is direct from [22, Theorem 3.9] and its proof. ∎

We give the following numerical examples illustrating the above corollary:

  1. (1)

Let $d=13\cdot 19$. We have $\left(\frac{13}{19}\right)=-1$. So the $2$-class
group of $L_{n,p}$ is isomorphic to $\mathds{Z}/2^{n+1}\mathds{Z}$, for all
$n\geq 1$.

  2. (2)

Let $d=5\cdot 11$. We have $\left(\frac{5}{11}\right)=1$ and
$\left(\frac{11}{5}\right)_{4}=1$. The $2$-class group of $L_{n,d}$ is
isomorphic to $\mathds{Z}/2^{n+2}\mathds{Z}$, for all $n\geq 1$.

Let now $X^{\prime}$, $Y^{\prime}$ and $Z$ three positive integers verifying
the Legendre equation

$\displaystyle pX^{\prime 2}+qY^{\prime 2}-Z^{2}=0$ (2)

And satisfying

$\displaystyle(X^{\prime},Y^{\prime})=(Y^{\prime},Z)=(Z,X^{\prime})=(p,Y^{\prime}Z)=(q,X^{\prime}Z)=1,$
(3)

and

$\displaystyle X^{\prime}\text{ odd},Y^{\prime}\text{ even and }Z\equiv
1\pmod{4}.$ (4)

(see [13] for more details)

######  Corollary 3.

Let $d=pq$ be the product of two primes $p$ and $q$ satisfying
$p\equiv-q\equiv 5\pmod{8}$, $\left(\frac{p}{q}\right)=1$ and
$\left(\frac{-q}{p}\right)_{4}=1$. Let $X^{\prime}$, $Y^{\prime}$ and $Z$ be
three positive integers satisfying the equation $(\ref{eq legendre})$ and the
conditions $(\ref{cond 1})$ and $(\ref{cond 2})$. If
$\left(\frac{Z}{p}\right)_{4}\not=\left(\frac{2X^{\prime}}{Z}\right)$, then
the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathbb{Z}/2^{n+3}\mathbb{Z}$. Elsewhere, it is isomorphic to
$\mathbb{Z}/2^{n+r-1}\mathbb{Z}$, for $r\geq 5$ defined as in Theorem 10.

###### Proof.

It is immediate that the assumptions of Corollary 2 are not satisfied. Hence,
$r\geq 4$. The rest follows directly from Theorem 10 and [13, Theorem 2]. ∎

Now we close this section with some numerical examples illustrating the above
corollary:

  1. (1)

Let $p=5$, $q=19$ and $d=-pq$. Then $X^{\prime}=1$, $Y^{\prime}=2$ and $Z=9$
are solutions of the equation $(\ref{eq legendre})$ verifying the condition
$(\ref{cond 1})$ and $(\ref{cond 2})$. Furthermore,
$\left(\frac{9}{5}\right)_{4}=-\left(\frac{2}{9}\right)=-1$. Thus, the
$2$-class group of $L_{n,d}$ is isomorphic to $\mathbb{Z}/2^{n+3}\mathbb{Z}$.

  2. (2)

Let $p=37$, $q=11$ and $d=-pq=-407$. Then $X^{\prime}=1$, $Y^{\prime}=56518$
and $Z=187449$ are solutions of the equation $(\ref{eq legendre})$ verifying
the condition $(\ref{cond 1})$ and $(\ref{cond 2})$. Furthermore,
$\left(\frac{187449}{37}\right)_{4}=\left(\frac{2}{187449}\right)=1$. Thus,
the $2$-class group of $L_{n,d}$ is isomorphic to
$\mathbb{Z}/2^{n+r-1}\mathbb{Z}$, for some $r\geq 5$. Indeed with these
settings $r=5$ (see [13, p. 230]).

##  5\. Applications

Note that [19, Theorem 2.5] holds for CM-fields containing the fourth root of
unity $i$. Therefore we cannot compute the $2$-class groups of layers of the
cyclotomic $\mathds{Z}_{2}$-extension of imaginary quadratic fields with the
same techniques used in the previous sections. Therefore as applications of
our above results and using class number formulas and some computations on
Hasse’s unit index, we deduce the structure of the $2$-class groups of the
cyclotomic $\mathds{Z}_{2}$-extension of some imaginary quadratic fields.

######  Theorem 11.

Let $d$ be an odd positive square free integer and $r$ such that
$2^{r}=h_{2}(-2d)$. Let $K_{0,d}=\mathbb{Q}(\sqrt{-d})$ and denote by
$K_{n,d}$ the $n$-th step of the cyclotomic $\mathbb{Z}_{2}$-extension of
$K_{0,d}$. Suppose that $d$ takes one of the following forms:

  * •

$d=pq$, for two primes $p$ and $q$ congruent to $3\pmod{8}$.

  * •

$d=p$, for a prime $p\equiv 9\pmod{16}$ such that that $(\frac{2}{p})_{4}=1$.

Then for all $n\geq 1$ the $2$-class group of $K_{n,d}$ is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{n+r-1}\mathbb{Z}$. In the
projective limit we obtain $\mathds{Z}_{2}\times\mathds{Z}/2\mathds{Z}$.

###### Proof.

Let $K_{n}=\mathbb{Q}(\zeta_{2^{n+2}})$ and
$K_{n,d}=\mathbb{Q}(\zeta_{2^{n+2}}+\zeta_{2^{n+2}}^{-1},\sqrt{-d})=K_{n}^{+}(\sqrt{-d})$.
We have the following field diagram (see Figure 1):

---
$\textstyle{L_{n,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{n,d}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{n,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{n}^{+}}$
Figure 1. Subfields of $L_{n,d}/K_{n}^{+}$.

So by class number formula (cf. [17, Proposition 3 and equation (1)]) we have:

$\displaystyle h_{2}(L_{n,d})$ $\displaystyle=$
$\displaystyle\frac{Q_{L_{n,d}}}{Q_{K_{n}}Q_{K_{n,d}}}\cdot\frac{\mu_{L_{n,d}}}{\mu_{K_{n}}\mu_{K_{n,d}}}\cdot\frac{h_{2}(K_{n})h_{2}(K_{n,d})h_{2}(L_{n,d}^{+})}{h_{2}(K_{n}^{+})^{2}}$

It is known that $h_{2}(K_{n})=1$ and by Theorems 5 and 9 we respectively have
$h_{2}(L_{n,d}^{+})=1$ and $h_{2}(L_{n,d})=2^{n+r-1}$. Therefore

$\displaystyle 2\cdot 2^{n+r-1}=\frac{Q_{L_{n,d}}}{Q_{K_{n}}Q_{K_{n,d}}}\cdot
h_{2}(K_{n,d})$ (5)

It is known that $Q_{K_{n}}=1$. Let $k=\mathbb{Q}{(i,\sqrt{d})}$. As the
natural norm $N_{L_{1,d}/k}:W_{L_{1,d}}/W_{L_{1,d}}^{2}\rightarrow
W_{k}/W_{k}^{2}$ is onto, we obtain $Q_{L_{1,d}}$ divides $Q_{k}$ (cf. [17,
Proposition 1]). Since $Q_{k}=1$ (cf. [1, p. 19] and the proof of [3, Lemma
4]), then $Q_{L_{1,d}}=1$. Since
$N_{L_{n,d}/L_{n-1,d}}:W_{L_{n,d}}/W_{L_{n}}^{2}\rightarrow
W_{L_{n-1}}/W_{L_{n-1}}^{2}$ is onto, it follows that $Q_{L_{n,d}}$ divides
$Q_{L_{n-1,d}}$. Thus, by induction $Q_{L_{n,d}}=1$.

Note that the extensions $K_{n,d}$ are essentially ramified (cf. [17, p. 349])
for all $n\geq 1$. Since $\mu_{K_{n,d}}=2$ we obtain by [17, Theorem 1]
$Q_{K_{n,d}}=1$. Hence, $h_{2}(K_{n,d})=2^{n+r}$ and this is for all $n\geq
1$. Since the rank of the $2$-class group of $K_{1,d}$ equals $2$ (cf. [18,
Proposition 4]) and the $2$-class group of $K_{n,d}$ is of type
$(2,2^{\bullet})$ for $n$ large enough (cf [15, p. 119]), we get the result.

∎

Now using second main theorem of the previous section we will show the next
result.

######  Theorem 12.

Assume that $d=pq$ is the product of two primes $p\equiv-q\equiv 5\pmod{8}$
and $2^{r}=2\cdot h_{2}(-pq)$. Let $K_{0,d}=\mathbb{Q}(\sqrt{-d})$ and denote
for $n\geq 1$ by $K_{n,d}$ the $n$-th step of the cyclotomic
$\mathbb{Z}_{2}$-extension of $K_{0,d}$. Then for $n\geq 1$ the $2$-class
group of $K_{n,d}$ is isomorphic to $\mathbb{Z}/2^{n+r-1}\mathbb{Z}$.

###### Proof.

We keep similar notations and proceed as in the proof of Theorem 11. Note that
by [3, Proposition 3], we have $h_{2}(L_{n,d}^{+})=2$. So as above we show
that:

$\displaystyle h_{2}(L_{n,d})$ $\displaystyle=$
$\displaystyle\frac{Q_{L_{n,d}}}{Q_{K_{n}}Q_{K_{n,d}}}\cdot\frac{\mu_{L_{n,d}}}{\mu_{K_{n}}\mu_{K_{n,d}}}\cdot\frac{h_{2}(K_{n})h_{2}(K_{n,d})h_{2}(L_{n,d}^{+})}{h_{2}(K_{n}^{+})^{2}}\cdot$

Thus

$\displaystyle 2^{n+r-1}$ $\displaystyle=$ $\displaystyle\frac{1}{1\cdot
1}\cdot\frac{2^{n}}{2^{n}\cdot 2}\cdot\frac{1\cdot h_{2}(K_{n,d})\cdot
2}{1}\cdot$

Thus, $h_{2}(K_{n,d})=2^{n+r-1}$, for all $n$. Since $L_{n,d}/K_{n,d}$ is
ramified, then $2$-rank$(\mathrm{Cl}(K_{n,d}))\leq$
$2$-rank$(\mathrm{Cl}(L_{n,d}))=1$ (Theorem 10). Which completes the proof. ∎

######  Remark 2.

One can easily deduce analogous corollaries as in the previous section.

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# Investigation of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons in Regge
phenomenology

Juhi Oudichhyaa<EMAIL_ADDRESS>Keval Gandhib and Ajay Kumar Raia
Department of Physics, Sardar Vallabhbhai National Institute of Technology,
Surat, Gujarat-395007, India  Department of Computer Sciences and Engineering,
Institute of Advanced Research, Gandhinagar, Gujarat 382426, India

###### Abstract

Triply heavy baryons with quark content $ccb$ and $cbb$ are investigated
within the framework of Regge phenomenology. With the assumption of linear
Regge trajectories, we have extracted the relations between Regge parameters
and baryon masses. Using these relations, we compute the ground state masses
of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons. Further, the Regge slopes and
intercepts are estimated for these baryons to obtain the excited state masses
in the $(J,M^{2})$ and $(n,M^{2})$ planes. Also, using the obtained results we
calculate the other properties like magnetic moment and radiative decay width
of these triply heavy baryons. We compare our evaluated results with those
obtained by the other theoretical approaches, and our results show a general
agreement with them. The present study and our predictions will provide
significant clues for future experimental research of these unseen triply
heavy baryons.

###### keywords:

Regge Phenomenology, hadron spectroscopy, triply heavy baryons

###### pacs:

24.10.-i, 25.70.Jj, 25.70.Gh

##  1 Introduction

The internal dynamics of heavy baryons and the nature of their excitation
mechanisms can be revealed by studying their excited states. After, the
significant experimental progress in discovering the singly and doubly heavy
baryons [1, 2, 3, 4, 5, 6, 7], now the next significant step ahead is the
identification of triply heavy baryons. Triply heavy baryons are composed of
three heavy quarks ($c$ or $b$). There are four baryons having three heavy
quarks, namely $\Omega_{ccc}$, $\Omega_{bbb}$, $\Omega_{ccb}$, and
$\Omega_{cbb}$. The mass spectra of $\Omega_{ccc}$ and $\Omega_{bbb}$ have
been already calculated in our previous work [8, 9]. Since none of these
triply heavy baryons have yet been reported experimentally, their production
is highly challenging. Baranov et. al. predicted that the baryon with three
charm quarks might not be seen in $e^{+}e^{-}$ collisions and the predictions
for the baryon composed of three bottom quarks would be substantially worse
[10]. First estimations of the production cross-section of the baryons with
three heavy quarks at the LHC were assessed in the Refs. [11, 12, 13]. In
2011, it was estimated that around $10^{4}-10^{5}$ events of $\Omega_{ccc}$
and $\Omega_{ccb}$ baryons, could be accumulated for 10 $fb^{-1}$ integrated
luminosity at LHC [14]. In Refs. [15, 16, 17] certain analyses of the
production rate of multi-charmed hadrons in heavy-ion collisions at high
energies are discussed. This indicates that there is a good chance of triply
charmed baryon to be observed at LHC.

In the last few years, many authors have studied the properties of triply
heavy baryons using different theoretical approaches. In a recent study [18],
the authors obtained the ground as well as excited state masses of triply
heavy baryons by employing the relativistic quark model based on the quark-
diquark picture in the quasipotential approach in Quantum chromodynamics. Shah
et. al. [19, 20] determined the radial and orbital excited state masses of the
triply heavy baryons within the framework of the hypercentral constituent
quark model with the addition of the first order correction in the potential
term of the Hamiltonian. Further, the Regge trajectories for these baryons are
also drawn. The authors of Ref. [21] computed the mass spectra of triply heavy
baryons using the constituent quark model. The Gaussian expansion method is
used to solve the non-relativistic three-body bound state problem. The
nonrelativistic quark model with the harmonic oscillator wave functions was
used to calculate the triply heavy baryon masses in the Ref. [22]. In Ref.
[23] the authors computed the masses of triply heavy baryons and made
reasonable predictions using QCD sum rules. The ground state $\frac{1}{2}^{+}$
and $\frac{3}{2}^{+}$ masses of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons are
evaluated by employing the QCD sum rules in the Refs. [24, 25].

Also, there have been other theoretical mass determinations including non-
relativistic quark model [26, 27], relativistic quark model [28, 29], QCD sum
rule [30], continuum approach to QCD based on the Dyson-Schwinger equations
[31, 32], lattice gauge theories [33, 34], Bag model [35, 36], and potential
non-relativistic QCD (pNRQCD) [37, 38]. Since various theoretical approaches
give different predictions and a wide range of mass differences can be seen in
the outcomes of different models for these triply heavy baryons. Therefore,
more comparisons of computations is required in the family of triply heavy
$\Omega$, which motivates us to study these baryons. In the present work, the
spectroscopic study of triply heavy baryons with $ccb$ and $cbb$ quark content
is done. The ground as well as excited state masses are obtained within the
framework of Regge phenomenology. With the assumption of quasi-linear Regge
trajectories, the relations between Regge parameters and baryon masses have
been derived. With the help of these extracted relations the ground state
masses of these unseen triply heavy baryons are estimated. Further, the Regge
parameters are computed to obtain the orbital and radial excited state masses
of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons.

The present article is arranged as follows. After briefing the various
theoretical approaches studies the triply heavy baryons, in Sec 2 we explain
the complete Regge theory and derive certain relations in terms of baryon
masses, Regge slopes, and intercepts. The ground state masses of these unseen
triply heavy baryons; $\Omega_{ccb}$ and $\Omega_{cbb}$ have been evaluated
with the aid of the extracted relations. After that, Regge parameters are
estimated and with the help of the values of Regge slopes and intercepts, the
excited state masses are obtained in the $(J,M^{2})$ and $(n,M^{2})$ planes.
Sec. 3 deals with the magnetic moment followed by the calculation of the
radiative decay widths of these baryons in Sec. 4. Sec. 5 provides a detailed
description of our obtained results. Finally, we concluded our work in Sec. 6.

##  2 Theoretical Framework

In 1978, Nambu put forth the simplest hypothesis to account for the linear
Regge trajectories [39, 40]. The Regge theory has been successfully used in
our previous work to obtain the mass spectra of light as well as heavy baryons
[8, 9, 41, 42]. We use the same phenomenology in the present work to study the
triply heavy baryons. Since, there is no experimental evidence is found in the
field of triply heavy baryons, but the various theoretical approaches and the
predicted results shows that the baryons ranging from light to singly, doubly,
and triply heavy follows the linear behaviour [18, 44, 45, 47, 46].Hence, we
have taken the quasi-linear Regge trajectories for these triply heavy baryons.
The general form of linear Regge trajectories can be expressed as [8, 9, 48],

$J=\alpha(M)=a(0)+\alpha^{{}^{\prime}}M^{2},$ (1)

where $a(0)$ and $\alpha^{{}^{\prime}}$ denotes the intercept and slope of the
Regge trajectory respectively. These Regge intercepts and slopes for different
flavors of a baryon multiplet can be related by two relations:
the additivity of intercepts;

$a_{ppr}(0)+a_{qqr}(0)=2a_{pqr}(0),$ (2)

the additivity of inverse slopes;

$\displaystyle\frac{1}{{\alpha^{{}^{\prime}}}_{ppr}}+\frac{1}{{\alpha^{{}^{\prime}}}_{qqr}}=\frac{2}{{\alpha^{{}^{\prime}}}_{pqr}},$
(3)

where $p,q,$ and $r$ represent the quark flavors. These two relations were
derived in a $q\overline{q}$ string picture of hadrons [49] and are
generalised for baryons as well by assuming the baryon as a quark-diquark
string object [50, 51, 52, 53]. Also, the ref. [51] showed that the additivity
of inverse Regge slopes is consistent with the formal chiral and heavy quark
limits for both mesons and baryons. For a detailed description of these two
relations, see [50, 48] and references therein.

Now using Eqs. (1) and (2) together with Eq. (3), we get two pairs of
solutions expressed in terms of slope ratios and baryon masses,

$\displaystyle\frac{\alpha^{{}^{\prime}}_{qqr}}{\alpha^{{}^{\prime}}_{ppr}}$
$\displaystyle=$
$\displaystyle\frac{1}{2M^{2}_{qqr}}\times[(4M^{2}_{pqr}-M^{2}_{ppr}-M^{2}_{qqr})$
(4) $\displaystyle\pm$
$\displaystyle\sqrt{{{(4M^{2}_{pqr}-M^{2}_{ppr}-M^{2}_{qqr}})^{2}}-4M^{2}_{ppr}M^{2}_{qqr}}],$

and,

$\displaystyle\frac{\alpha^{{}^{\prime}}_{pqr}}{\alpha^{{}^{\prime}}_{ppr}}$
$\displaystyle=$
$\displaystyle\frac{1}{4M^{2}_{pqr}}\times[(4M^{2}_{pqr}+M^{2}_{ppr}-M^{2}_{qqr})$
(5) $\displaystyle\pm$
$\displaystyle\sqrt{{{(4M^{2}_{pqr}-M^{2}_{ppr}-M^{2}_{qqr}})^{2}}-4M^{2}_{ppr}M^{2}_{qqr}}].$

The above obtained Eq. (4) can also be expressed as,

$\frac{\alpha^{{}^{\prime}}_{qqr}}{\alpha^{{}^{\prime}}_{ppr}}=\frac{\alpha^{{}^{\prime}}_{kkr}}{\alpha^{{}^{\prime}}_{ppr}}\times\frac{\alpha^{{}^{\prime}}_{qqr}}{\alpha^{{}^{\prime}}_{kkr}},$
(6)

here $k$ can be any quark flavor. Thus we have [48],

$\displaystyle\frac{[(4M^{2}_{pqr}-M^{2}_{ppr}-M^{2}_{qqr})+\sqrt{{{(4M^{2}_{pqr}-M^{2}_{ppr}-M^{2}_{qqr}})^{2}}-4M^{2}_{ppr}M^{2}_{qqr}}]}{2M^{2}_{qqr}}$
$\displaystyle=\frac{[(4M^{2}_{pkr}-M^{2}_{ppr}-M^{2}_{kkr})+\sqrt{{{(4M^{2}_{pkr}-M^{2}_{ppr}-M^{2}_{kkr}})^{2}}-4M^{2}_{ppr}M^{2}_{kkr}}]/2M^{2}_{kkr}}{[(4M^{2}_{qkr}-M^{2}_{qqr}-M^{2}_{kkr})+\sqrt{{{(4M^{2}_{qkr}-M^{2}_{qqr}-M^{2}_{kkr}})^{2}}-4M^{2}_{qqr}M^{2}_{kkr}}]/2M^{2}_{kkr}}.$
(7)

This is the significant relationship we have derived between the masses of
different flavors of baryons. This relation can be used to estimate the mass
of any baryon state if all other masses are known. In the present work the
ground state masses of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons are
calculated with the aid of this above extracted relation.

###  2.1 Ground state masses of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons

In this section, we evaluate the ground state $\frac{1}{2}^{+}$ and
$\frac{3}{2}^{+}$ masses of unseen triply heavy $\Omega_{ccb}$ and
$\Omega_{cbb}$ baryons using Eq. (2). With the quark composition of the
$\Omega_{ccb}$ baryon which consists of two charm quarks ($c$) and one bottom
quark ($b$), we put $p=n(u$ or $d)$, $q=c$, $r=b$, and $k=n(u$ or $d)$ in Eq.
(2), and after simplifying we obtain a quadratic relation in terms of
different flavors of baryon masses which is expressed as,

$\displaystyle\frac{4M^{2}_{\Sigma_{b}}M^{2}_{\Omega_{ccb}}}{(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{b}}-M^{2}_{\Omega_{ccb}})}-(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{b}}-M^{2}_{\Omega_{ccb}})$
$\displaystyle=\sqrt{M^{2}_{\Sigma_{b}}(M^{2}_{\Sigma_{b}}-2M^{2}_{\Omega_{ccb}}-8M^{2}_{\Xi_{bc}})+M^{2}_{\Omega_{ccb}}(M^{2}_{\Omega_{ccb}}-M^{2}_{\Xi_{bc}})+16M^{2}_{\Xi_{bc}}}$
(8)

similarly, for $\Omega_{cbb}$ baryon which is made up of two bottom quarks and
one charm quark, we put $p=n(u$ or $d)$, $q=b$, $r=c$, and $k=n(u$ or $d)$ in
Eq. (2), we have

$\displaystyle\frac{4M^{2}_{\Sigma_{c}}M^{2}_{\Omega_{cbb}}}{(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{cbb}})}-(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{cbb}})$
$\displaystyle=\sqrt{M^{2}_{\Sigma_{c}}(M^{2}_{\Sigma_{c}}-2M^{2}_{\Omega_{cbb}}-8M^{2}_{\Xi_{bc}})+M^{2}_{\Omega_{cbb}}(M^{2}_{\Omega_{cbb}}-M^{2}_{\Xi_{bc}})+16M^{2}_{\Xi_{bc}}}$
(9)

after inserting the masses of $\Sigma_{b}$ and $\Sigma_{c}$ from PDG [54], and
$\Xi_{bc}$ calculated in the previous work [41] for $J^{P}=\frac{3}{2}^{+}$,
into Eqs. (2.1) and (2.1), we get the ground state masses of $\Omega_{ccb}$
and $\Omega_{cbb}$ as 8.223 GeV and 11.541 GeV respectively for
$J^{P}=\frac{3}{2}^{+}$. Similarly we can obtain the masses for
$J^{P}=\frac{1}{2}^{+}$ as well. The calculated results for ground state
masses of $\Omega_{ccb}$ and $\Omega_{bbc}$ baryons are shown in table 1 along
with the predicted outcomes of various theoretical approaches. Also, we have
shown graphically the comparison of our obtained ground state masses with the
results of other theories in Figs 1-4.

Table 1:  Ground state $J^{P}=\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ masses of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons (in GeV). |  $\Omega_{ccb}$ |  $\Omega_{cbb}$
---|---|---
$J^{P}$ |  $\frac{1}{2}^{+}$ | $\frac{3}{2}^{+}$ |  $\frac{1}{2}^{+}$ |  $\frac{3}{2}^{+}$
Our | 8.192 | 8.223 | 11.526 | 11.541
[22] | 8.245 | 8.265 | 11.535 | 11.554
[19] | 8.005 | 8.049 | 11.231 | 11.296
[23] | 8.230 | 8.230 | 11.500 | 11.490
[21] | 8.004 | 8.023 | 11.220 | 11.221
[55] | 8.301 | 8.301 | 11.218 | 11.218
[34] | 8.005 | 8.026 | 11.194 | 11.195
[57] | 8.007 | 8.037 | 11.218 | 11.229
[29] | 8.018 | 8.025 | 11.280 | 11.287
[24] | 8.500 |  | 11.730 |
[31] | 8.190 | 8.190 | 11.370 | 11.380
[18] | 7.984 | 7.999 | 11.198 | 11.217
[58] |  | 8.200 |  | 11.480
Figure 1: Comparison between the computed ground state $J^{P}=\frac{1}{2}^{+}$
mass of $\Omega_{ccb}$ baryon with other theoretical approaches. Figure 2:
Comparison between the computed ground state $J^{P}=\frac{3}{2}^{+}$ mass of
$\Omega_{ccb}$ baryon with other theoretical approaches. Figure 3: Comparison
between the computed ground state $J^{P}=\frac{1}{2}^{+}$ mass of
$\Omega_{cbb}$ baryon with other theoretical approaches. Figure 4: Comparison
between the computed ground state $J^{P}=\frac{3}{2}^{+}$ mass of
$\Omega_{cbb}$ baryon with other theoretical approaches.

###  2.2 Masses in the $(J,M^{2})$ plane

In this section, we determine the orbitally excited state masses of
$\Omega_{ccb}$ and $\Omega_{cbb}$ baryons after calculating their ground state
masses. The Regge slopes ($\alpha^{{}^{\prime}}$) for $\frac{1}{2}^{+}$ and
$\frac{3}{2}^{+}$ trajectories are evaluated using the relation (4). For
instance, to extract the value of Regge slope for $\Omega_{ccb}$
($\alpha^{{}^{\prime}*}_{\Omega_{ccb}}$) for $\frac{3}{2}^{+}$ trajectory,
again we insert $p=u$, $q=c$, and $r=b$ into Eq. (4) and we get,

$\displaystyle\frac{\alpha^{{}^{\prime}*}_{\Omega_{ccb}}}{\alpha^{{}^{\prime}*}_{\Sigma_{b}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2M^{2}_{\Omega_{ccb}}}\times[(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{b}}-M^{2}_{\Omega_{ccb}})$
(10) $\displaystyle+$
$\displaystyle\sqrt{{{(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{b}}-M^{2}_{\Omega_{ccb}}})^{2}}-4M^{2}_{\Sigma_{b}}M^{2}_{\Omega_{ccb}}}],$

by inserting the masses of $\Sigma_{b}$ from PDG [54], $\Xi_{bc}$ from [41],
and $\Omega_{ccb}$ (calculated above) into Eq. (12), we obtain the value of
$\alpha^{{}^{\prime}*}_{\Omega_{ccb}}$ as 0.2134 GeV-2 for $\frac{3}{2}^{+}$
trajectory. Here we have taken $\alpha^{{}^{\prime}*}_{\Sigma_{b}}$ = 0.2906
GeV-2 from our previous work [9]. In the same manner we can obtain the value
of $\alpha^{{}^{\prime}}_{\Omega_{ccb}}$ = 0.1978 GeV-2 for $\frac{1}{2}^{+}$
trajectory.

Similarly, we can get the expression to calculate the Regge slope for
$\Omega_{ccb}$ baryon as well by inserting the values of $p,q,$ and $r$ in the
relation (4) according to its quark composition. The derived relation is
expressed as,

$\displaystyle\frac{\alpha^{{}^{\prime}*}_{\Omega_{cbb}}}{\alpha^{{}^{\prime}*}_{\Sigma_{c}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2M^{2}_{\Omega_{cbb}}}\times[(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{cbb}})$
(11) $\displaystyle+$
$\displaystyle\sqrt{{{(4M^{2}_{\Xi_{bc}}-M^{2}_{\Sigma_{c}}-M^{2}_{\Omega_{cbb}}})^{2}}-4M^{2}_{\Sigma_{c}}M^{2}_{\Omega_{cbb}}}],$

again putting the masses of $\Sigma_{b}$ from PDG [54], $\Xi_{bc}$ from [41],
and $\Omega_{cbb}$ (calculated above) in the above relation, and after
simplifying we can have the value of $\alpha^{{}^{\prime}*}_{\Omega_{cbb}}$ as
0.1279 GeV-2 for $\frac{3}{2}^{+}$ trajectory. Here also, the
$\alpha^{{}^{\prime}*}_{\Sigma_{c}}$ = 0.5688 GeV-2 is calculated in the same
manner as $\alpha^{{}^{\prime}*}_{\Sigma_{b}}$. Also, for $\frac{1}{2}^{+}$
trajectory we have $\alpha^{{}^{\prime}}_{\Omega_{cbb}}$ = 0.1188 GeV-2.

Now, another expression can be obtained using the Eq. (1) in terms of Regge
slope and masses, which is expressed as,

$M_{J+1}=\sqrt{M_{J}^{2}+\frac{1}{\alpha^{{}^{\prime}}}}.$ (12)

With the aid of the above relation and the values of Regge slopes,
$\alpha^{{}^{\prime}}_{\Omega_{ccb}}$ and
$\alpha^{{}^{\prime}}_{\Omega_{bbc}}$ extracted above, we can calculate the
orbitally excited state masses for triply heavy $\Omega$ baryons for natural
parity states ( $J^{P}$ = $1/2^{+}$, $3/2^{-}$, $5/2^{+}$, ….) and unnatural
parity states ( $J^{P}$ = $3/2^{+}$, $5/2^{-}$, $7/2^{+}$, ….) lying on that
Regge trajectories. The calculated results are shown in tables 2 and 3 along
with the predictions of other theoretical models.

###  2.3 Masses in the $(n,M^{2})$ plane

Table 2: Excited state masses of $\Omega_{ccb}$ baryon in the ($J,M^{2}$) plane (in GeV). States | Our | [22] | [21] | [19] | [55] | [18]
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ |  |  |  |  |  |
$1^{2}S_{\frac{1}{2}}$ | 8.192 | 8.245 | 8.004 | 8.005 | 8.301 | 7.984
$1^{2}P_{\frac{3}{2}}$ | 8.495 | 8.420 |  |  |  | 8.262
$1^{2}D_{\frac{5}{2}}$ | 8.787 | 8.568 |  |  |  | 8.473
$1^{2}F_{\frac{7}{2}}$ | 9.070 |  |  |  |  |
$1^{2}G_{\frac{9}{2}}$ | 9.344 |  |  |  |  |
$1^{2}H_{\frac{11}{2}}$ | 9.611 |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 8.223 | 8.265 | 8.023 | 8.049 | 8.301 | 7.999
$1^{4}P_{\frac{5}{2}}$ | 8.503 | 8.432 | 8.311 | 8.365 | 8.491 | 8.267
$1^{4}D_{\frac{7}{2}}$ | 8.774 | 8.568 | 8.538 | 8.780 | 8.647 | 8.473
$1^{4}F_{\frac{9}{2}}$ | 9.037 |  |  | 9.184 |  |
$1^{4}G_{\frac{11}{2}}$ | 9.293 |  |  |  |  |
$1^{4}H_{\frac{13}{2}}$ | 9.542 |  |  |  |  |
Table 3: Excited state masses of $\Omega_{cbb}$ baryon in the ($J,M^{2}$) plane (in GeV). States | Our | [22] | [21] | [20] | [18] | [23]
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ |  |  |  |  |  |
$1^{2}S_{\frac{1}{2}}$ | 11.526 | 11.535 | 11.200 | 11.231 | 11.198 | 11.500
$1^{2}P_{\frac{3}{2}}$ | 11.885 | 11.711 |  |  | 11.424 |
$1^{2}D_{\frac{5}{2}}$ | 12.234 | 11.823 |  |  | 11.632 |
$1^{2}F_{\frac{7}{2}}$ | 12.573 |  |  |  |  |
$1^{2}G_{\frac{9}{2}}$ | 12.903 |  |  |  |  |
$1^{2}H_{\frac{11}{2}}$ | 13.225 |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 11.541 | 11.554 | 11.221 | 11.296 | 11.217 | 11.490
$1^{4}P_{\frac{5}{2}}$ | 11.886 | 11.762 | 11.569 | 11.558 | 11.543 |
$1^{4}D_{\frac{7}{2}}$ | 12.220 | 11.810 | 11.688 | 11.909 | 11.635 |
$1^{4}F_{\frac{9}{2}}$ | 12.546 |  |  | 12.249 |  |
$1^{4}G_{\frac{11}{2}}$ | 12.863 |  |  |  |  |
$1^{4}H_{\frac{13}{2}}$ | 13.173 |  |  |  |  |

After computing the excited orbital states of $\Omega_{ccb}$ and
$\Omega_{cbb}$ baryons using the derived relations in the $(J,M^{2})$ plane,
in this section we extend our theory to determine the radial excitations of
these baryons in the $(n,M^{2})$ plane. The similar equation for the linear
Regge trajectories in the ($n,M^{2}$) plane can be expressed as,

$n=\beta_{0}+\beta M^{2},$ (13)

where $n$ = 1, 2, 3…. is the principal quantum number, $\beta_{0}$, and
$\beta$ are the intercept and slope of the trajectories. Since the baryon
multiplets lying on the Regge line have the same Regge parameters [48], hence
to obtain the radially excited state masses firstly we evaluate the values of
$\beta$ and $\beta_{0}$ for $S$, $P$, and $D$-waves. For $\Omega_{ccb}$
baryon, with the help of the slope equation, we can have
$\beta_{(S)}=1/(M^{2}_{\Omega_{ccb}(2S)}-M^{2}_{\Omega_{ccb}(1S)})$, where
$M_{\Omega_{ccb}(1S)}$ = 8.192 GeV (calculated above). Now since due to the
unavailability of experimental masses of triply heavy baryons, to determine
the values of Regge parameters we have taken the theoretical predictions as
input masses. Hence for $n=2$ the $M_{\Omega_{ccb}(2S)}$ = 8.621 GeV is taken
from [19]. We get $\beta_{(S)}$ = 0.1386 GeV-2 for $S$-states for the
$1/2^{+}$ trajectory. Now from Eq. (13) we can write,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\beta_{0(S)}+\beta_{(S)}M^{2}_{\Omega_{ccb}(1S)},$
$\displaystyle 2$ $\displaystyle=$
$\displaystyle\beta_{0(S)}+\beta_{(S)}M^{2}_{\Omega_{ccb}(2S)},$ (14)

Using the above relations we can determine the value for Regge intercept as
$\beta_{0(S)}$ = -8.3041 for spin $S$ = 1/2. Similarly, we can express these
relations for $P$ and $D$-wave as,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\beta_{0(P)}+\beta_{(P)}M^{2}_{\Omega_{ccb}(1P)},$
$\displaystyle 2$ $\displaystyle=$
$\displaystyle\beta_{0(P)}+\beta_{(P)}M^{2}_{\Omega_{ccb}(2P)},$
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\beta_{0(D)}+\beta_{(D)}M^{2}_{\Omega_{ccb}(1D)},$ (15)
$\displaystyle 2$ $\displaystyle=$
$\displaystyle\beta_{0(D)}+\beta_{(D)}M^{2}_{\Omega_{ccb}(2D)},$

Now, using the above relations we can extract the values of Regge parameters
for $\Omega_{ccb}$ baryon for each Regge line and with the aid of these
evaluated Regge slopes and intercepts, the radially excited state masses for
$n=3,4,5..$ are calculated. The same procedure is follows to obtain the mass
spectra for $\Omega_{cbb}$ baryon also. The predicted numerical values are
shown in Tables 4 and 5 along with the outcomes of other theoretical models.

Table 4: Excited state masses of $\Omega_{ccb}$ baryon in the ($n,M^{2}$) plane. The numbers in the boldfaced are the inputs taken from [19] (in GeV). States | Our | [22] | [21] | [55] | [18] | [56]
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ |  |  |  |  |  |
$1^{2}S_{1/2}$ | 8.192 | 8.245 | 8.004 | 8.301 | 7.984 | 8.019
$2^{2}S_{1/2}$ | 8.621 | 8.537 | 8.455 | 8.600 | 8.361 | 8.450
$3^{2}S_{1/2}$ | 9.030 |  |  |  |  |
$4^{2}S_{1/2}$ | 9.420 |  |  |  |  |
$5^{2}S_{1/2}$ | 9.796 |  |  |  |  |
$6^{2}S_{1/2}$ | 10.157 |  |  |  |  |
$1^{4}S_{3/2}$ | 8.223 | 8.265 | 8.023 | 8.301 | 7.999 | 8.056
$2^{4}S_{3/2}$ | 8.637 | 8.553 | 8.468 | 8.600 | 8.366 | 8.465
$3^{4}S_{3/2}$ | 9.032 |  |  |  |  |
$4^{4}S_{3/2}$ | 9.410 |  |  |  |  |
$5^{4}S_{3/2}$ | 9.774 |  |  |  |  |
$6^{4}S_{3/2}$ | 10.125 |  |  |  |  |
$1^{4}P_{5/2}$ | 8.503 | 8.432 | 8.311 | 8.491 | 8.267 | 8.331
$2^{4}P_{5/2}$ | 8.955 |  | 8.667 |  |  | 8.589
$3^{4}P_{5/2}$ | 9.385 |  |  |  |  |
$4^{4}P_{5/2}$ | 9.797 |  |  |  |  |
$5^{4}P_{5/2}$ | 10.191 |  |  |  |  |
$1^{4}D_{7/2}$ | 8.774 | 8.568 | 8.538 | 8.647 | 8.473 | 8.528
$2^{4}D_{7/2}$ | 9.368 |  | 8.839 |  |  | 8.762
$3^{4}D_{7/2}$ | 9.926 |  |  |  |  |
$4^{4}D_{7/2}$ | 10.455 |  |  |  |  |
$5^{4}D_{7/2}$ | 10.959 |  |  |  |  |
Table 5: Masses for $\Omega_{cbb}$ baryon in the ($n,M^{2}$) plane. The numbers in the boldfaced are the inputs taken from [20] (in GeV). States | Our | [22] | [21] | [55] | [18] | [56]
---|---|---|---|---|---|---
$N^{2S+1}L_{J}$ |  |  |  |  |  |
$1^{2}S_{1/2}$ | 11.526 | 11.535 | 11.200 | 11.218 | 11.198 | 11.217
$2^{2}S_{1/2}$ | 11.757 | 11.787 | 11.607 | 11.585 | 11.507 | 11.625
$3^{2}S_{1/2}$ | 11.984 |  |  |  |  |
$4^{2}S_{1/2}$ | 12.206 |  |  |  |  |
$5^{2}S_{1/2}$ | 12.424 |  |  |  |  |
$6^{2}S_{1/2}$ | 12.639 |  |  |  |  |
$1^{4}S_{3/2}$ | 11.541 | 11.554 | 11.221 | 11.218 | 11.217 | 11.251
$2^{4}S_{3/2}$ | 11.779 | 11.798 | 11.622 | 11.585 | 11.515 | 11.643
$3^{4}S_{3/2}$ | 12.012 |  |  |  |  |
$4^{4}S_{3/2}$ | 12.241 |  |  |  |  |
$5^{4}S_{3/2}$ | 12.466 |  |  |  |  |
$6^{4}S_{3/2}$ | 12.686 |  |  |  |  |
$1^{4}P_{5/2}$ | 11.886 | 11.762 | 11.569 | 11.601 | 11.543 | 11.598
$2^{4}P_{5/2}$ | 12.049 |  | 11.888 |  |  | 11.899
$3^{4}P_{5/2}$ | 12.210 |  |  |  |  |
$4^{4}P_{5/2}$ | 12.369 |  |  |  |  |
$5^{4}P_{5/2}$ | 12.525 |  |  |  |  |
$1^{4}D_{7/2}$ | 12.220 | 11.810 | 11.688 | 11.626 | 11.635 | 11.718
$2^{4}D_{7/2}$ | 12.396 |  | 11.963 |  |  | 11.986
$3^{4}D_{7/2}$ | 12.569 |  |  |  |  |
$4^{4}D_{7/2}$ | 12.741 |  |  |  |  |
$5^{4}D_{7/2}$ | 12.910 |  |  |  |  |

##  3 Magnetic moment

The study of electromagnetic properties of baryons is a significant topic for
theoretical as well as experimental work. The Magnetic moment is an intrinsic
property that aids in understand the shape and other dynamics of transitions
in decay modes. The magnetic moment expression can be obtained by operating
the spin-flavour wave function to the $z$-component of the magnetic moment
operator. The generalised form of magnetic moment is given as [26, 59].

$\mu_{B}=\sum_{i}\langle\phi_{sf}|\mu_{iz}|\phi_{sf}\rangle,$ (16)

where, $i$ = $c$ or $b$. $\phi_{sf}$ represents the spin-flavour wave function
of the baryons. The magnetic moment of the individual quark appears as,

$\mu_{iz}=\dfrac{e_{i}}{2m_{i}^{eff}}\sigma_{iz},$ (17)

here $e_{i}$ is the quark charge, $\sigma_{iz}$ is the spin of the respective
constituent quark corresponding to the spin flavour wave-function of the
baryonic state and $m_{i}^{eff}$ is the effective mass of the constituent
quarks [45, 59]. Since we have not used the constituent quark masses in our
formalism to predict the mass spectra. Here we have taken the values of these
quark masses, $m_{c}$ = 1.275 GeV and $m_{b}$ = 4.250 GeV, constituent masses
of the charm and bottom quark respectively, from Particle data group (PDG)
[54]. Using the above equations we have calculated the ground state magnetic
moments of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons. The final spin-flavor
wave function and the evaluated results of magnetic moment along with the
predictions of other theoretical approaches are shown in table 6.

Table 6: Magnetic moments (in $\mu_{N}$) of $J^{P}$ = $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons. Spin | Baryon | $\sigma_{iz}$ |  $\mu$ (Ours) | [59] | [60] | [26]
---|---|---|---|---|---|---
$\frac{1}{2}^{+}$ | $\Omega_{ccb}$ |  $\frac{4}{3}$ $\mu_{c}-\frac{1}{3}\mu_{b}$ | 0.5632 | 0.5261 | 0.540 | 0.502
$\frac{3}{2}^{+}$ | $\Omega_{ccb}^{*}$ | $\mu_{b}+2\mu_{c}$ | 0.7503 | 0.6952 | 0.720 | 0.651
$\frac{1}{2}^{+}$ | $\Omega_{cbb}$ | $\frac{4}{3}\mu_{b}+\frac{1}{3}\mu_{c}$ | -0.2218 | -0.2096 | -0.210 | -0.203
$\frac{3}{2}^{+}$ | $\Omega_{cbb}^{*}$ | $2\mu_{b}+\mu_{c}$ | 0.2908 | 0.2558 | 0.270 | 0.216

##  4 Radiative decay width

Radiative decay is the process of $\gamma$ emission occurring as the spin
state changes from $\frac{3}{2}$ to $\frac{1}{2}$. For the better
understanding of the intrinsic properties of heavy baryons, the experimental
determination of the radiative decay width is important. The expression for
radiative decay width is given by [59],

$\Gamma_{R}=\dfrac{k^{3}}{m_{p}^{2}}\dfrac{2}{2J+1}\dfrac{e^{2}}{4\pi}|\mu_{B\rightarrow
B^{{}^{\prime}}}|^{2},$ (18)

where,

$k=\dfrac{M^{2}_{B}-M^{2}_{B^{{}^{\prime}}}}{2M_{B}}$ (19)

where $k$ is the photon energy, $M_{B}$ and $M_{B^{{}^{\prime}}}$ are the
masses of initial and final baryon state, respectively. $m_{p}$ is the photon
mass, and J is the initial angular momentum. Also, $\mu_{B\rightarrow
B^{{}^{\prime}}}$ is the transition magnetic moment which is expressed as,

$\mu_{B\rightarrow B^{{}^{\prime}}}=\langle\phi_{B}|\mu_{B\rightarrow
B^{{}^{\prime}}}|\phi_{B^{{}^{\prime}}}\rangle$ (20)

Here, $\phi_{B}$ and $\phi_{B^{{}^{\prime}}}$ are the wave functions of
initial and final baryonic states respectively. Table 7 shows the calculated
transition magnetic moment and radiative decay widths of $\Omega_{ccb}$ and
$\Omega_{cbb}$ baryons.

Table 7:  The Radiative decay widths (in keV) of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons. Decay | Wave-function | Transition magnetic |  $\Gamma_{R}$ (Ours) |  $\Gamma_{R}$ [59] |  $\Gamma_{R}$ [61]
---|---|---|---|---|---
|  | moment ($\mu_{N}$) | (keV) |  |
$\Omega_{ccb}^{*+}\rightarrow\Omega_{ccb}^{+}\gamma$ | $\frac{2\sqrt{2}}{3}(\mu_{c}-\mu_{b})$ | 0.441 | 0.0238 | 0.0192 | 0.004
$\Omega_{cbb}^{*0}\rightarrow\Omega_{cbb}^{+}\gamma$ | $\frac{2\sqrt{2}}{3}(\mu_{b}-\mu_{c})$ | -0.451 | 0.0028 | 0.0282 | 0.005

##  5 Results and Discussion

The spectroscopic study of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryon sectors is
successfully obtained in the present work using the Regge phenomenology. We
have evaluated the ground and the excited state masses of these triply heavy
baryons. With no experimental information available on these systems, we have
compared our estimated results with recent theoretical predictions. Table 1.
shows the calculated ground state masses of $\Omega_{ccb}$ and $\Omega_{cbb}$
baryons along with the comparison of the results obtained from different
theoretical approaches. Tables 2 and 3 represents the orbitally excited state
masses evaluated in the $(J,M^{2})$ plane for these triply heavy baryons.
Further, our calculated results for the radially excited states in the
$(n,M^{2})$ plane are summarised in tables 4 and 5. The properties like
magnetic moment and radiative decay for $\Omega_{cbb}$ baryon have also been
calculated. The obtained results along with the comparison of the predictions
from other appproaches are shown in table 6 and 7.

###  5.1 The $\Omega_{ccb}$ baryon sector

The ground state masses obtained in the present work for $\Omega_{ccb}$ baryon
are 8.192 GeV and 8.223 GeV for $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ states
respectively. The ground state masses predicted in the recent studies by the
different theoretical approaches vary in the range of 8.0 - 8.5 GeV and our
calculated masses fall within this range (shown in table 1). The estimated
masses for $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ states are in well
agreement with the predicted masses obtained in the non-relativistic quark
model [22] and seem to be very close to the results obtained using QCD sum
rules [23] with a mass difference of few MeV. In contrast, some studies [19,
21, 57, 29] show a quite large mass difference of around 170-200 MeV with our
predicted masses. This can also be seen in graphical representation where our
calculated masses are presented by ”Ours” which lies in between the spreaded
results of other theoretical predictions (shown in Figs. 1 and 2). Further,
the calculated masses for orbitally excited states of $\Omega_{ccb}$ baryons
are shown in table 2. Again we compared our results with other theoretical
predictions and a general agreement can be seen with the results of some
references. The predicted mass for $1^{2}P_{\frac{3}{2}}$ state is close the
results of [22] and shows a large mass difference with the prediction of [18].
For the $1^{4}P_{\frac{5}{2}}$ state, the calculated mass is very close to the
result obtained in [55] with a mass difference of 12 MeV and also in agreement
with the prediction of [22]. The results presented in Refs. [21, 19, 18] are
found to be somewhat lower than our obtained mass. Similarly, the masses
estimated for $1^{4}D_{\frac{7}{2}}$ and $1^{4}F_{\frac{9}{2}}$ states are
close to the results of [19] and comparatively higher than the results
obtained in Refs. [22, 21, 18]. Table 4 shows radially excited state masses
obtained in the $(n,M^{2})$ plane. Very few results are available from
previous theoretical predictions. The masses reported in the Refs. [22, 55]
shows a general agreement with our predicted masses with a mass difference of
few MeV.

Further using the obtained results we calculate other properties like the
magnetic moment of ground states and the radiative decay width. We compared
our results with the predictions of various other theoretical approaches [59,
60, 26] and they are in consistent with our results.

###  5.2 The $\Omega_{cbb}$ baryon sector

Similar behaviour can be seen in the case of $\Omega_{cbb}$ baryon. The ground
state masses predicted in other references vary in the range of 11.2-11.7 GeV
and our estimated masses of 11.526 GeV and 11.541 GeV for $\frac{1}{2}^{+}$
and $\frac{3}{2}^{+}$ states respectively lie in this range. The predicted
ground state masses in Refs. [22, 23, 58] are very close to our obtained
masses with a mass difference of few MeV but the results obtained in other
refs. [20, 21, 55, 29] shows a bit large mass difference of 0.2-0.3 GeV with
our predicted masses as shown in table 1. Figs 3 and 4 which shows the
comparison of our obtained ground state masses with other theoretical results
shows a clear picture that our calculated masses fall within the range of the
masses obtained from various theories. Tables 3 and 5 represent the orbital
and radial excited state masses calculated in the $(J,M^{2})$ and $(n,M^{2})$
planes respectively. We compared our computed results with various theoretical
predictions. For low-lying states, the predicted states reported in [22] agree
with our calculated masses. As we go for higher excited states, quite
different results with large uncertainties can be seen in the masses predicted
by different theoretical models. There is no clear consensus between the
different approaches for the excited state masses of $\Omega_{cbb}$ baryon.
The masses reported in the Ref. [22, 56] agrees well with our predicted masses
and the predicted states reported in Refs. [20, 18, 21] are lower than ours.

The other properties such as magnetic moment and radiative decay width of are
evaluated. The summerized results along with the comparsion from various
theoretical predictions are shown in tables 6 and 7. We compared our
calculated magnetic moment values with the results obtained in [59, 60, 61]
and they are in agreement with them. The obtained rediative decay width shows
variation with the predictions of [59, 61], it may be due to the large
variation in the ground state masses of $\Omega_{cbb}$ baryon.

##  6 Conclusion

In this paper, we applied the Regge phenomenology for the calculation of the
mass spectra of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons. Different results
with large uncertainties are found in different theoretical approaches. The
ground state masses of $\Omega_{ccb}$ and $\Omega_{cbb}$ baryons obtained from
various studies vary in the range of 8.0-8.5 GeV and 11.2-11.7 GeV
respectively and our obtained masses fall within this range. The excited
states of these triply heavy baryons have been less explored theoretically. We
compared our obtained results with the masses of various theoretical
approaches. Here, we reached the conclusion that the low lying masses are in
well agreement with the predictions of some theories. However discrepancies
are found in excited states. Since different approaches gives different
predictions, hence more comparison of computations with experimental
measurements is required. Also, the mass values which are taken as inputs in
the calculation are the experimental masses where available and the
theoretical masses which we have calculated in our previous work. The model’s
reliability depend on the availability of experimental data that we have taken
as a input mass, hence whatever the mass value predictions we have obtained in
the present work follows a general trend with the results of other theoretical
models. Also, there are no fitting parameters involved in our formalism, the
Regge parameters are calculated from the experimental masses in the present
work. We have also seen that the results obtained in some theoretical
approaches like hCQM [19, 20], does not follow the general trend. For example,
the highest $1P$ state has the lower mass with respect to the lower $1P$
state.

Further, the other properties such as magnetic moment and radiative decay
widths have been obtained using the mass value predictions. They also shown a
general agreement with other the theoretical predictions, whereas some results
shows slight discrepancies. Since, there is no experimental data is available
related to triply heavy baryons. Hence, our predicted mass values, magnetic
moment and decay widths  could be extremely valuable in future experimental
searches.

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