# Doubly-charmed baryon production in $Z$ boson decay

Xuan Luo  Ying-Zhao Jiang  Gui-Yuan Zhang  Zhan Sun<EMAIL_ADDRESS>Department of Physics, Guizhou Minzu University, Guiyang 550025, People’s
Republic of China.

###### Abstract

In this paper, we carry out a detailed study of doubly-charmed baryon
production in $Z$ boson decay, on the basis of the nonrelativistic QCD
factorization. With the inclusion of the di-quark states
$(cc)[^{3}S_{1}]_{\bar{\textbf{3}}}$ and $(cc)[^{1}S_{0}]_{{\textbf{6}}}$, the
branching ratio of $\mathcal{B}_{Z\to\Xi_{cc}+X}$ is predicted to be of the
$10^{-5}$ order, indicating its experimental measurability. By comparing to
the $\Lambda^{+}_{c}$ yield in $Z$ decay, we predict
$\mathcal{R}_{\Xi_{cc}^{+}}(=\frac{\Gamma(Z\to\Xi^{+}_{cc})\times\mathcal{B}(\Xi^{+}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})})=(0.85^{+0.10}_{-0.07})\times
10^{-4}$ and
$\mathcal{R}_{\Xi_{cc}^{++}}(=\frac{\Gamma(Z\to\Xi^{++}_{cc})\times\mathcal{B}(\Xi^{++}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})})=(1.70^{+0.20}_{-0.14})\times
10^{-4}$, which are at clear variance with the SELEX measurements but
comparable with the values given by the LHCb and Belle collaborations.

###### pacs:

12.38.Bx, 12.39.Jh, 13.38.Dg

##  I Introduction

The doubly-charmed baryon (labeled as $\Xi_{cc}$), which is assumed to contain
two $c$ quarks and a light quark $q$ ($q=u,d,s$) based on the quark model
quark model 1 ; quark model 2 ; quark model 3 ; quark model 4 ; quark model 5
, can provide unique test for the quantum chromodynamics (QCD). The past
decades have seen the rapid developments of the $\Xi_{cc}$ related studies,
including both experimental and theoretical aspects.

By reconstructing $\Xi_{cc}^{+}$ ($ccu$) via its decay into
$\Lambda_{c}^{+}K^{-}\pi^{+}$, the SELEX collaboration reported a large
production rate of $\Xi_{cc}^{+}$ (i.e.
$\mathcal{R}_{\Xi_{cc}^{+}}=\frac{\sigma(\Xi^{+}_{cc})\times\mathcal{B}(\Xi^{+}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+})}{\sigma(\Lambda_{c}^{+})}=9\%$)
SELEX:2002wqn , which, however, was even not confirmed by the FOCUS
collaboration Ratti:2003ez  that is at the same collider of SELEX. In 2013,
the LHCb Collaboration performed its first search for $\Xi_{cc}^{+}$,
reporting the upper limit of $\mathcal{R}_{\Xi_{cc}^{+}}$ is from $1.5\times
10^{-2}$ (corresponding to the $\Xi_{cc}^{+}$ lifetime of 100 fs) to
$3.9\times 10^{-4}$ (400 fs) LHCb:2013hvt , which is recently updated to be
from $6.5\times 10^{-3}$ (40 fs) to $9\times 10^{-4}$ (160 fs) for
$\sqrt{s}=8$ TeV, and from $4.5\times 10^{-4}$ (40 fs) to $1.2\times 10^{-4}$
(160 fs) for $\sqrt{s}=13$ TeV LHCb:2019gqy . The LHCb data appear to be at
clear variance with the SELEX-measured $\mathcal{R}_{\Xi_{cc}^{+}}$. Comparing
to $\Xi_{cc}^{+}$, $\Xi_{cc}^{++}$ ($uud$) has a much longer lifetime and is
then much easier to be tagged from the LHC background Wu:2019gta . In 2017,
the LHCb collaboration firstly detected the decay channel
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ with $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$ LHCb:2017iph , and then observed the decay channels of
$\Xi_{cc}^{++}\to\Xi_{cc}^{+}\pi^{+}$ LHCb:2018pcs  and
$\Xi_{cc}^{++}\to\Xi_{cc}^{+}\pi^{+}\pi^{+}$ LHCb:2018zpl . In 2019, LHCb
achieved the first measurement of the $\Xi_{cc}^{++}$ production in proton-
proton collision LHCb:2019qed , reporting
$\mathcal{R}_{\Xi_{cc}^{++}}(=\frac{\sigma(\Xi^{++}_{cc})\times\mathcal{B}(\Xi^{++}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})}{\sigma(\Lambda_{c}^{+})})=(2.22\pm
0.27\pm 0.29)\times 10^{-4}$. Besides the hadroproduction, the $\Xi_{cc}$
production in $e^{+}e^{-}$ annihilation has also been measured BaBar:2006bab ;
Belle:2006edu ; Belle:2013htj . The upper limits of
$\sigma(e^{+}e^{-}\to\Xi_{cc}^{+(++)}+X)$ given by the Belle collaboration are
comparable with the theoretical predictions; the values of
$\mathcal{R}_{\Xi_{cc}^{+(++)}}$ are measured to be $\sim 10^{-4}$.

In addition to the direct productions direct 1 ; direct 2 ; direct 3 ; direct
4 ; direct 5 ; direct 6 ; direct 7 ; direct 8 ; direct 9 ; direct 10 ; direct
11 ; direct 12 ; direct 13 ; direct 14 ; direct 15 ; direct 16 ; direct 17 ;
direct 18 ; direct 19 ; direct 20 ; direct 21 ; direct 22 ; direct 23 ; direct
24 ; direct 25 ; direct 26 ; direct 27 ; direct 28 ; direct 29 ; direct 30 ;
direct 31 ; direct 32 ; direct 33 ; direct 34 ; direct 35 ; direct 36 ; direct
37 , such as the hadro-, photo-, and electroproductions, the indirect
$\Xi_{cc}$ yield in decays indirect 1 ; indirect 2 ; indirect 3 ; indirect 4 ,
which is indeed very complementary to the direct case, is also of great
interest to study the doubly-charmed baryon. For example, Niu. $et~{}al.$
pointed out about $10^{3}$ $\Xi_{cc}$ events can be accumulated through higgs
decay in one running year at the proposed HL-LHC; Li. $et~{}al.$ indicated the
branching ratio of $\mathcal{B}_{\Upsilon(1S)\to\Xi_{cc}+X}$ can be
significant and can be well measured as the $\Upsilon(1S)$ decay to
$J/\psi+c+\bar{c}$+X. In addition to these decay processes, the $Z$ boson
decay could also provide a uniquely good chance for the $\Xi_{cc}$-related
study. At the LHC, $\sim 10^{9}$ $Z$ events are expected to be generated per
year Liao:2015vqa , which would be largely increased by the HE(L)-LHC upgrade
program. The proposed future $e^{+}e^{-}$ collider, CEPC CEPC , equipped with
$``\textrm{clean}"$ background and enormous $Z$ production events ($\sim
10^{12}$/year), would also be beneficial to hunt $\Xi_{cc}$ yield through $Z$
decay. Thus, it appears promising to measure $Z$ decaying into inclusive
$\Xi_{cc}$. Moreover, heavy-quarkonium production in $Z$ decay (such as $Z\to
J/\psi+X$), which is analogue to our concerned $\Xi_{cc}$ case, has triggered
increasing attentions and has accumulated abundant experimental data
Zyla:2020zbs . Taken together, we, in this paper, would perform a detailed
study of $Z\to\Xi_{cc}+X$, presenting the estimations of
$\mathcal{R}_{\Xi_{cc}^{+(++)}}$ in the course of $Z$ decay.

The rest of the paper is organized as follows: In Sec. II, we give a
description on the calculation formalism. In Sec. III, the phenomenological
results and discussions are presented. Section IV is reserved as a summary.

##  II Calculation Formalism

###  II.1 General Formalism

The production of the doubly-charmed baryon is often assumed to be factorized
into two steps direct 1 ; direct 11 ; direct 27 . The first procedure is to
produce a $c$-quark pair $(cc)[n]$ by the perturbative calculable hard
processes, with subsequent nonperturbative transition into a bounding diquark
$\langle cc\rangle[n]$ that can be described by a matrix element, $h_{[n]}$.
The next step is the hadronization of $\langle cc\rangle[n]$ into a physical
colorless baryon $\Xi_{ccq}$ ($q=u,d,s$) by grabbing a light quark with
possible soft gluons from the hadron; during the hadronization, the
$``\textrm{direct~{}evolution~{}mechanism}"$ assumes the total evolving
probability to be $100\%$, among which the fragmentation probabilities into
$\Xi_{ccu}$, $\Xi_{ccd}$, and $\Xi_{ccs}$ account for $43\%$, $43\%$, and
$14\%$, respectively Wu:2019gta ; direct 30 ; Sjostrand:2006za .

Within the nonrelativistic QCD (NRQCD) framework NRQCD , the differential
decay width of $Z\to\Xi_{cc}+X$ can be expressed as

$\displaystyle
d\Gamma=d\hat{\Gamma}_{Z\to(cc)[n]+X}\langle\mathcal{O}^{\Xi_{cc}}(n)\rangle,$
(1)

where $\hat{\Gamma}_{Z\to(cc)[n]+X}$ is the perturbative calculable short
distance coefficients (SDCs), representing the production of a configuration
of the $(cc)[n]$ intermediate state. At the leading order of $v_{c}$ (the
relative velocity of the two constituent $c$ quarks in the diquark),111The
contributions of the $P$-wave processes would at least be $v_{c}^{2}$
suppressed to that of the $S$-wave processes. For example, the $P$-wave
contributions just account for about $3\%-5\%$ of the total cross sections of
the hadroproduced $\Xi_{cc}$ Berezhnoy:2020aox . $n$ takes on either
$[^{3}S_{1}]_{\bar{\textbf{3}}}$ or $[^{1}S_{0}]_{{\textbf{6}}}$, due to the
symmetry of identical particles in the diquark state. The subscripts
$\bar{\textbf{3}}({\textbf{6}})$ (as will be depicted in later section) denote
the color state of the diquark. The universal long distance matrix element
$\langle\mathcal{O}^{\Xi_{cc}}(n)\rangle$ stands for the transition
probability of the $c$-quark pair into the diquark multiplied by the
subsequent fragmentation probability into $\Xi_{cc}$, i.e.,
$h_{([^{3}S_{1}]_{\bar{\textbf{3}}},[^{1}S_{0}]_{{\textbf{6}}})}\times 100\%$
based on the “direct evolution mechanism”.

###  II.2 Amplitudes

The SDCs in Eq. (1) can be written as

$\displaystyle\hat{\Gamma}_{Z\to(cc)[n]+X}=|\mathcal{M}|^{2}d\Phi_{3},$ (2)

where $|\mathcal{M}|^{2}$ and $d\Phi_{3}$ are the squared matrix element and
the standard 3-body phase space, respectively. For
$Z\to(cc)[n]+\bar{c}+\bar{c}$, there are in total 8 Feynman diagrams, half of
which are shown in Fig. 1. The other 4 ones can be obtained by exchanging the
two identical $c$-quark lines inside the diquark. Because we have set
$v_{c}=0$, the two portions of 4 diagrams contribute identically; in this
case, we only need to calculate the 4 diagrams in Figure 1, and multiply a
factor of $2^{2}$. Simultaneously we should introduce an additional factor of
$1/(2!2!)$ to deal with the identities of the two constituent $c$ quarks
inside the diquark and the two final-state $\bar{c}$ quarks.

Figure 1:  Typical Feynman diagrams of $Z\to(cc)[n]+\bar{c}+\bar{c}$.

According Fig. 1, one can obtain

$\displaystyle\mathcal{M}_{1}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{12}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{2}}}\right)\bar{u}\left({{p_{11}}}\right)\left({-i{\gamma^{\nu}}}\right)\left({{m_{c}}+{p\\!\\!\\!/_{1}}+{p\\!\\!\\!/_{2}}}\right)\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})v\left({{p_{3}}}\right)}}{{{{\left({{p_{12}}+{p_{2}}}\right)}^{2}}\left[{{{\left({{p_{1}}+{p_{2}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{2}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{12}}}\right)\left({-i{\gamma^{\nu}}}\right)\left({{m_{c}}+{p\\!\\!\\!/_{1}}+{p\\!\\!\\!/_{3}}}\right)\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})v\left({{p_{2}}}\right)\bar{u}\left({{p_{11}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{11}}+{p_{3}}}\right)}^{2}}\left[{{{\left({{p_{1}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{3}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{12}}}\right)\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})\left({{m_{c}}-{p\\!\\!\\!/_{11}}-{p\\!\\!\\!/_{2}}-{p\\!\\!\\!/_{3}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{2}}}\right)\bar{u}\left({{p_{11}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{11}}+{p_{3}}}\right)}^{2}}\left[{{{\left({{p_{11}}+{p_{2}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{4}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{12}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{2}}}\right)\bar{u}\left({{p_{11}}}\right)\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})\left({{m_{c}}-{p\\!\\!\\!/_{12}}-{p\\!\\!\\!/_{2}}-{p\\!\\!\\!/_{3}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{12}}+{p_{2}}}\right)}^{2}}\left[{{{\left({{p_{12}}+{p_{2}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$

where
$\kappa=-\mathcal{C}\frac{{ieg_{s}}^{2}}{4\cos\theta_{\textrm{w}}\sin\theta_{\textrm{w}}}$
with $\mathcal{C}$ denoting the color factor. $\epsilon(p_{0})$ is the
polarization vector of the initial $Z$ boson. The coefficient $c$ reads

$\displaystyle c=\frac{8}{3}\sin^{2}\theta_{\textrm{w}}-1.$ (4)

The momenta of the constituent quarks in the diquark follow as

$\displaystyle
p_{11}=\frac{m_{c}}{M_{cc}}p_{1}+q~{}~{}\textrm{and}~{}~{}p_{12}=\frac{m_{c}}{M_{cc}}p_{1}-q,$
(5)

where $m_{c}=M_{cc}/2$ is implicitly adopted to ensure the gauge invariance of
the hard scattering amplitude; $q(\simeq 0)$ is the relative momentum between
the two constituent $c$ quarks inside the diquark.

By inserting the charge conjugate matrix $C=-i\gamma^{2}\gamma^{0}$ that
satisfies the following equations direct 27 ,

$\displaystyle
CC^{-1}=1,~{}~{}~{}v^{T}(p)C=-\bar{u}(p),~{}~{}~{}C^{-}\bar{u}\left(p\right)^{T}=v\left(p\right),$
$\displaystyle
C^{-}(\gamma^{\mu})^{T}C=-\gamma^{\mu},~{}~{}~{}C^{-}(\gamma^{\mu}\gamma^{5})^{T}C=\gamma^{\mu}\gamma^{5},$
$\displaystyle C^{-}s^{T}_{f}(k,m)C=s_{f}(-k,m),$ (6)

the amplitudes in Eq. (LABEL:ori_amp) can be rewritten as

$\displaystyle\mathcal{M}_{1}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{2}}}\right)\left({-i{\gamma^{\nu}}}\right)\Pi^{[n]}_{p_{1}}\left({-i{\gamma^{\nu}}}\right)\left({{m_{c}}+{p\\!\\!\\!/_{1}}+{p\\!\\!\\!/_{2}}}\right)\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})v\left({{p_{3}}}\right)}}{{{{\left({{p_{12}}+{p_{2}}}\right)}^{2}}\left[{{{\left({{p_{1}}+{p_{2}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{2}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{2}}}\right)\left({-i{\gamma^{\nu}}}\right)\left({{m_{c}}-{p\\!\\!\\!/_{1}}-{p\\!\\!\\!/_{3}}}\right)\epsilon\\!\\!\\!/(p_{0})(c-\gamma^{5})\Pi^{[n]}_{p_{1}}\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{11}}+{p_{3}}}\right)}^{2}}\left[{{{\left({{p_{1}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{3}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{2}}}\right)\epsilon\\!\\!\\!/(p_{0})(c-\gamma^{5})\left({{m_{c}}+{p\\!\\!\\!/_{11}}+{p\\!\\!\\!/_{2}}+{p\\!\\!\\!/_{3}}}\right)\left({-i{\gamma^{\nu}}}\right)\Pi^{[n]}_{p_{1}}\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{11}}+{p_{3}}}\right)}^{2}}\left[{{{\left({{p_{11}}+{p_{2}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
$\displaystyle\mathcal{M}_{4}$ $\displaystyle=$
$\displaystyle-\kappa\frac{{\bar{u}\left({{p_{2}}}\right)\left({-i{\gamma^{\nu}}}\right)\Pi^{[n]}_{p_{1}}\epsilon\\!\\!\\!/(p_{0})(c+\gamma^{5})\left({{m_{c}}-{p\\!\\!\\!/_{12}}-{p\\!\\!\\!/_{2}}-{p\\!\\!\\!/_{3}}}\right)\left({-i{\gamma^{\nu}}}\right)v\left({{p_{3}}}\right)}}{{{{\left({{p_{12}}+{p_{2}}}\right)}^{2}}\left[{{{\left({{p_{12}}+{p_{2}}+{p_{3}}}\right)}^{2}}-m_{c}^{2}}\right]}},$
(7)

where $\Pi^{[n]}_{q}$ denotes the spin projector operators Petrelli:1997ge ,

$\displaystyle\Pi^{[^{1}S_{0}]}_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{8m_{c}^{3}}}\left(\frac{q\\!\\!\\!/}{2}-m_{c}\right)\gamma^{5}\left(\frac{q\\!\\!\\!/}{2}+m_{c}\right),$
$\displaystyle\Pi^{[^{3}S_{1}]}_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{8m_{c}^{3}}}\left(\frac{q\\!\\!\\!/}{2}-m_{c}\right)\epsilon\\!\\!\\!/_{q}\left(\frac{q\\!\\!\\!/}{2}+m_{c}\right).$
(8)

###  II.3 Color Factor

According to Fig. 1, the color factor $\mathcal{C}$ can be expressed as

$\displaystyle{\mathcal{C}}=T^{a}_{im}T^{a}_{jn},$ (9)

where $a=1\cdots 8$ is the color indices of the incoming gluon; $i,j=1,2,3$
and $m,n=1,2,3$ denote the color indices of the two constituent $c$ quarks in
the diquark and that of the two final-state $\bar{c}$ quarks, respectively. By
the fact that $3\otimes 3=\bar{\textbf{3}}\oplus\textbf{6}$ in
$\textrm{SU}_{\textrm{c}}(3)$ group, the diquark can be either in anti-triplet
$\bar{\textbf{3}}$ or in sextuplet 6 color state; in this case, we introduce
the function $\frac{G_{ijk}}{N}$ to describe the diquark color, $k=3$ and
$N=\sqrt{2}$ being the color indices of the diquark and the normalized factor,
respectively. $G_{ijk}$ is identical to the antisymmetric $\varepsilon_{ijk}$
($\bar{\textbf{3}}$) or the symmetric $f_{ijk}$ (6), which satisfies the
following equations

$\displaystyle\varepsilon_{ijk}\varepsilon_{i^{{}^{\prime}}j^{{}^{\prime}}k}$
$\displaystyle=$
$\displaystyle\delta_{ii^{{}^{\prime}}}\delta_{jj^{{}^{\prime}}}-\delta_{ji^{{}^{\prime}}}\delta_{ij^{{}^{\prime}}},$
$\displaystyle f_{ijk}f_{i^{{}^{\prime}}j^{{}^{\prime}}k}$ $\displaystyle=$
$\displaystyle\delta_{ii^{{}^{\prime}}}\delta_{jj^{{}^{\prime}}}+\delta_{ji^{{}^{\prime}}}\delta_{ij^{{}^{\prime}}}.$
(10)

##  III Phenomenological results

The input parameters in our calculations are set as

$\displaystyle m_{Z}$ $\displaystyle=$ $\displaystyle
91.1876~{}\textrm{GeV},~{}~{}~{}m_{c}=1.8\pm 0.05~{}\textrm{GeV},$
$\displaystyle\sin^{2}\theta_{\textrm{w}}$ $\displaystyle=$ $\displaystyle
0.23116,~{}~{}~{}\alpha=1/137.$ (11)

According to the velocity scaling rule of NRQCD, we adopt the usual assumption
that the matrix elements $h_{[^{3}S_{1}]_{\bar{\textbf{3}}}}$ and
$h_{[^{1}S_{0}]_{{\textbf{6}}}}$ have the equal values direct 1 ; direct 11 ;
direct 27 , which are given by the wave function at the origin direct 5 ;
direct 27

$\displaystyle
h_{[^{3}S_{1}]_{\bar{\textbf{3}}}}=h_{[^{1}S_{0}]_{{\textbf{6}}}}=\big{|}\Psi_{cc}(0)\big{|}^{2}=0.039~{}\textrm{GeV}^{3}.$
(12)

Table 1: Decay widths (in units of KeV) of $Z\to\Xi_{cc}+X$. $\mu_{r}$ |  $m_{c}$ (GeV) | $~{}~{}~{}\Gamma_{[^{3}S_{1}]_{\bar{\textbf{3}}}}~{}~{}~{}$ | $~{}~{}~{}\Gamma_{[^{1}S_{0}]_{{\textbf{6}}}}~{}~{}~{}$ | $~{}~{}~{}\Gamma_{\textrm{Total}}~{}~{}~{}$ | $~{}~{}~{}\mathcal{B}(\times 10^{-5})~{}~{}~{}$
---|---|---|---|---|---
| $1.75$ | $16.36$ | $58.87$ | $75.23$ | $3.015$
$2m_{c}$ | $1.80$ | $14.59$ | $52.51$ | $67.10$ | $2.689$
| $1.85$ | $13.33$ | $48.01$ | $61.34$ | $2.458$
| $1.75$ | $4.79$ | $17.21$ | $22.00$ | $0.882$
$m_{Z}/2$ | $1.80$ | $4.40$ | $15.82$ | $20.22$ | $0.810$
| $1.85$ | $4.05$ | $14.57$ | $18.62$ | $0.746$

We summarize the predicted decay widths of $Z\to\Xi_{cc}+X$ in Tab. 1.
Inspecting the data, one can find the branching ratio of $Z\to\Xi_{cc}+X$
amounts to $\sim 10^{-5}$, which is comparable with the color-singlet
predictions of $\mathcal{B}_{Z\to J/\psi+X}$ Barger:1989cq ; Braaten:1993mp .
In the predictions of the total decay width, the state of
$(cc)[^{3}S_{1}]_{\bar{\textbf{3}}}$ plays the leading role, more than three
times bigger in magnitude than that of $(cc)[^{1}S_{0}]_{{\textbf{6}}}$.
Varying $m_{c}$ around the default value of $1.8$ GeV by $\pm 0.05$ GeV would
arouse a $10\%$ variation of the decay width.

Based on the aforementioned capability of hunting $Z$ boson at LHC and CEPC,
about $10^{4}$ (LHC) and $10^{7}$ (CEPC) $\Xi_{cc}$ production events arisen
from $Z$ decay would be collected in one running year at the two colliders. By
further considering the decay chains of
$\Xi_{cc}^{+}\to\Lambda_{c}^{+}K^{-}\pi^{+}$ ($\simeq 5\%$ LHCb:2013hvt ) or
$\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ ($\simeq 10\%$
LHCb:2017iph ; Yu:2017zst ) with the cascade decay $\Lambda_{c}^{+}\to
pK^{-}\pi^{+}$($\simeq 5\%$ LHCb:2013hvt ), we would accumulate about $10$ and
$10^{4}$ reconstructed $\Xi^{+(++)}_{cc}$ events at LHC and CEPC,
respectively. Note that, the proposed HL(E)-LHC upgrade program may largely
increase the $\Xi_{cc}$ yield events.

We are now in a position to estimate the ratio of the production rate of
$\Xi_{cc}^{+(++)}$ in $Z$ decay to that of $\Lambda_{c}^{+}$. According to
$\mathcal{B}_{Z\to\Lambda_{c}^{+}}=\mathcal{B}_{Z\to c\bar{c}}\times
f_{c\to\Lambda_{c}^{+}}=0.12\times 0.057=6.84\times 10^{-3}$, Zyla:2020zbs ;
Gladilin:2014tba  we have

$\displaystyle\mathcal{R}_{\Xi_{cc}^{+}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z\to\Xi^{+}_{cc})\times\mathcal{B}(\Xi^{+}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})}=0.85^{+0.10}_{-0.07}\times
10^{-4},$ $\displaystyle\mathcal{R}_{\Xi_{cc}^{++}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z\to\Xi^{++}_{cc})\times\mathcal{B}(\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})}=1.70^{+0.20}_{-0.14}\times
10^{-4},$ (13)

where $\mu_{r}=2m_{c}$ and the uncertainties are caused by $m_{c}=1.8\pm 0.05$
GeV. From the ratios one can perceive the predicted
$\mathcal{R}_{\Xi_{cc}^{+(++)}}$ in $Z$ decay is comparable with the
measurements of LHCb (13 TeV) LHCb:2019gqy ; LHCb:2019qed  and Belle
collaborations BaBar:2006bab , while still significantly below the SELEX-
reported $\mathcal{R}_{\Xi_{cc}^{+}}$ SELEX:2002wqn .

Figure 2:  $\Xi_{cc}$ energy distributions in $Z\to\Xi_{cc}+X$ with $z$
defined as $\frac{2E_{\Xi_{cc}}}{m_{Z}}$.

At last, we give the predictions of the $\Xi_{cc}$ energy distributions in
Fig. 2. The peak of
$\frac{d\Gamma}{dz}\big{|}_{(cc)[^{3}S_{1}]_{\bar{\textbf{3}}}}$ is around
$z=0.75$ and $\frac{d\Gamma}{dz}\big{|}_{(cc)[^{1}S_{0}]_{\textbf{6}}}$ peaks
near $z=0.7$. That the peak of $\Xi_{cc}$ energy distribution in
$Z\to\Xi_{cc}+X$ lies in the large $z$ region can primarily be attributed to
the dominance of the $c$-quark fragmentation mechanism.

##  IV Summary

In this manuscript, we apply the NRQCD factorization to study the $Z$ boson
decaying into inclusive doubly-charmed baryon. By including the contributions
of the di-quark states $(cc)[^{3}S_{1}]_{\bar{\textbf{3}}}$ and
$(cc)[^{1}S_{0}]_{{\textbf{6}}}$, the branching ratio of
$\mathcal{B}_{Z\to\Xi_{cc}+X}$ is predicted to be $\sim 10^{-5}$, following
which as high as $10^{4(7)}$ $\Xi_{cc}$ events from $Z$ decay are expected to
be accumulated at LHC (CEPC) per year. By comparing to the measurements on
$\mathcal{B}_{Z\to\Lambda^{+}_{c}+X}$, we predict
$\mathcal{R}_{\Xi_{cc}^{+}}(=\frac{\Gamma(Z\to\Xi^{+}_{cc})\times\mathcal{B}(\Xi^{+}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})})=(0.85^{+0.10}_{-0.07})\times
10^{-4}$ and
$\mathcal{R}_{\Xi_{cc}^{++}}(=\frac{\Gamma(Z\to\Xi^{++}_{cc})\times\mathcal{B}(\Xi^{++}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+})}{\Gamma(Z\to\Lambda_{c}^{+})})=(1.70^{+0.20}_{-0.14})\times
10^{-4}$, where the uncertainties are estimated using alternative choices of
the $c$-quark mass. Our prediction of $\mathcal{R}_{\Xi_{cc}^{+(++)}}$ in the
course of $Z$ decay is still inconsistent with the SELEX data but compatible
with the LHCb- and Belle-measured values.

##  V Acknowledgments

Acknowledgments: This work is supported in part by the Natural Science
Foundation of China under the Grant No. 12065006, and by the Project of
GuiZhou Provincial Department of Science and Technology under Grant No.
QKHJC[2020]1Y035. and No. QKHJC[2019]1167.

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#  On $p-$Rationality of Number Fields and Greenberg’s Generalized Conjecture

Hang Li and Derong Qiu
(School of Mathematical Sciences, Capital Normal University,
Beijing 100048, P.R.China)  Corresponding author. E-mail addresses:
<EMAIL_ADDRESS>(H.Li<EMAIL_ADDRESS>(D.Qiu)

Abstract In this paper, a new criteria is given to determine the
$p-$rationality of number fields, some illustrated examples are constructed,
the relations between Greenberg’s Generalized Conjecture (GGC, for short),
generalized $abc-$conjecture and the $p-$rationality are discussed, from which
some explicit fields satisfying GGC are obtained.

Keywords: $p-$rationality, Greenberg’s Generalized Conjecture (GGC),
generalized $abc-$conjecture, class group.

Mathematics Subject Classification 2020 : 11R23 , 11R29.

1\. Introduction and Main Results

Let $F$ be a number field, $p$ a prime number. Let $S_{p}$ denote the set of
all places of $F$ above $p,\ M_{S_{p}}(F)$ the maximal pro-$p$ extension
unramified outside $S_{p},$ and $F_{S_{p}}$ the maximal Abelian pro-$p$
extension unramified outside $S_{p}.$ Write
$G_{S_{p}}(F):=\text{Gal}(M_{S_{p}}(F)/F),$ then
$G_{S_{p}}(F)^{ab}=\text{Gal}(F_{S_{p}}/F).$ In the following, we denote by
$\mathcal{T}_{F}=\text{tor}(G_{S_{p}}(F)^{ab})$ the maximal
${\mathbb{Z}}_{p}-$torsion submodule of $G_{S_{p}}(F)^{ab}.$
$F$ is called $p-$rational if $G_{S_{p}}(F)$ is a free pro-$p$ group (see
[Mov1, 2]).
It is known ([Gr2],section 3) that a number field $F$ is $p-$rational if and
only if the following two conditions hold
(1) $\text{rank}_{{\mathbb{Z}}_{p}}(G_{S_{p}}(F)^{ab})=r_{2}(F)+1;$
(2) $G_{S_{p}}(F)^{ab}$ is a torsion free ${\mathbb{Z}}_{p}-$module, i.e.,
$\mathcal{T}_{F}=\\{1\\}.$
Condition (1) above is equivalent to say that Leopoldt’s conjecture holds for
$(F,p),$ while condition (2) means that $F_{S_{p}}=\widetilde{F},$ the
composite field of all the ${\mathbb{Z}}_{p}-$extension of $F.$
The $p-$rationality of number fields has been extensively studied, one of its
important application is on constructing non-abelian number fields satisfying
Leopoldt’s conjecture (e.g. [Mov1, 2]). Besides, much questions about it are
still open, such as
Conjecture 1.1 (Gras [G3, Conjecture 8.11]). Every number field $F$ is
$p-$rational if $p$ is sufficiently large.
Conjecture 1.2 (Greenberg [Gr2]). For any odd prime number $p$ and positive
integer $t,$ there exists a Galois extension $F$ of the rational number field
${\mathbb{Q}}$ such that $F$ is $p-$rational and
$\text{Gal}(F/{\mathbb{Q}})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{t}.$
When $F$ is a quadratic number field or some of their composite, much progress
have been made on the Greenberg’s conjecture above (see e.g., [AB],[BM], [Ko],
[CLS]).
There are a series of work on determining the $p-$rationality of totally real
number fields and quadratic imaginary number fields (see e.g., [Gr2], [Li]).
In this paper, we will study similar questions on some cases that $F$ is
neither totally real nor quadratic, more precisely, we consider the
$p-$rationality of complex cubic and pure imaginary quartic Number Fields, and
the connection of $p-$rationalality of certain CM number fields with GGC. Our
main results are the following

Theorem 1.3. Let $K$ be a complex cubic or pure imaginary quartic number
field, $p$ a prime number (when $p=3,$ assume it is unramified in
$K/{\mathbb{Q}},$ when $p=5$ and $K$ is quartic, assume $5$ is not totally
ramifed in $K$).
Then $K$ is $p-$rational if and only if the following two conditions hold:
(1) $H_{p}(K)\subseteq\widetilde{K},$ where $H_{p}(K)$ is the Hilbert
$p-$class field of $K,$ and $\widetilde{K}$ is the composite of all
${\mathbb{Z}}_{p}-$extension of $K.$
(2) There is a prime ideal $\mathfrak{p}$ of $K$ lying over $p$ such that
$\varepsilon^{p^{f}-1}\not\equiv 1(\text{mod}\ \mathfrak{p}^{e+1})$ for any
fundamental unit $\varepsilon$ of $K,$ where $e$ and $f$ are the ramification
index and residue degree of $\mathfrak{p}$ in $K/{\mathbb{Q}},$ respectively.

Theorem 1.4. Let $K$ be a CM field, $K^{+}$ be its maximal real subfield, let
$p$ be an odd prime number which splits completely in $K.$ Let
$K_{\infty}^{+}$ be the cyclotomic ${\mathbb{Z}}_{p}-$extension of $K^{+}.$
Assume $p\nmid h(K),$ where $h(K)$ is the class number of $K.$ Then
(1) The Iwasawa invariants $\lambda,\mu,\nu$ of $K_{\infty}^{+}/K^{+}$ all
vanish and the Leopoldt’s conjecture holds for $(K^{+},p)$ if and only if
$K^{+}$ is $p-$rational.
(2) If $K^{+}$ is $p-$rational, then GGC holds for $(K,p).$

Theorem 1.5. Let $K\subset{\mathbb{R}}$ be a complex cubic number field,
$\varepsilon>0$ be its fundamental unit,
$f(x)=x^{3}-a_{2}x^{2}-a_{1}x-a_{0}\in{\mathbb{Z}}[x]$ be the minimal
polynomial of $\varepsilon$ over ${\mathbb{Q}}.$ Let $p$ be an odd prime
number with $p\nmid d(f)$ (the discriminant of $f(x)$). Define the arithmetic
third order sequence of rational integers
$F_{n+3}=a_{2}F_{n+2}+a_{1}F_{n+1}+a_{0}F_{n}\ (n\geq 0),\
F_{0}=F_{1}=0,F_{2}=1.$
Then $p$ is unramified over $K/{\mathbb{Q}}.$ Furthermore,
(1) if $p$ splits completely over $K/{\mathbb{Q}},\ F_{p-1}\not\equiv
0(\text{mod}\ p^{2}),$ then there is a prime ideal $\mathfrak{p}$ of $K$ lying
over $p$ such that $\varepsilon^{p-1}\not\equiv 1(\text{mod}\
\mathfrak{p}^{2}).$
(2) if $p\mathcal{O}_{K}=\mathfrak{p}_{1}\mathfrak{p}_{2}$ for two different
prime ideals in $K$ with $f(\mathfrak{p}_{1}/p)=1$ and
$f(\mathfrak{p}_{2}/p)=2,$ then
$F_{p^{2}-1}\not\equiv 0(\text{mod}\
p^{2})\Rightarrow\varepsilon^{p-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{1}^{2})$ or $\varepsilon^{p^{2}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{2}^{2}).$
(3) if $p$ is inertia over $K/{\mathbb{Q}},$ i.e.,
$p\mathcal{O}_{K}=\mathfrak{p}$ is a prime ideal in $K,$ then
$F_{p^{3}-1}\not\equiv 0(\text{mod}\
p^{2})\Rightarrow\varepsilon^{p^{3}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}^{2}).$

Theorem 1.6. Let $K$ be a complex cubic or pure imaginary quartic number
field. If the generalized $abc-$conjecture holds for $K,$ then as
$x\rightarrow\infty,$ we have

$\sharp\\{\text{prime numbers}\ p\leq x,K\ \text{is}\ p-\text{rational}\\}\geq
c\log x$

for some constant $c>0$ depending only on $K.$

Remark 1.7. Under the assumption of Theorem 1.4 above, by the proof of
Prop.2.14 of [BM], one can obtain that, $K^{+}$ is $p-$rational
$\Leftrightarrow$ $K$ is $p-$rational. So the conclusion (2) of Theorem 1.4
above can also be stated as: If $K$ is $p-$rational, then GGC holds for
$(K,p).$ This partially answers the following open question in [ABo]:
Do $p-$rational number fields satisfy Greenberg’s generalized conjecture ?

In the following, for a number field $F,$ as usual, we denote by $r_{1}$ and
$r_{2}$ the number of real and complex infinite places, respectively. So
$r_{1}+2r_{2}=[F:{\mathbb{Q}}].$ Let $\mathcal{O}_{F}$ be the ring of integers
of $F$ and $E_{F}=\mathcal{O}_{F}^{\times}$ denote the group of units. For any
place $v$ of $F,$ let $F_{v}$ denote the completion of $F$ at $v,$ and
$U_{v}^{(1)}$ denote the group of principal units of $F_{v}.$ Also, we let
$h(F)$ denote the class number of $F.$
In general, for a finitely generated ${\mathbb{Z}}_{p}-$module $M,$ let
$\text{tor}(M)$ denote its maximal torsion ${\mathbb{Z}}_{p}-$submodule.

2\. Proof of Theorem 1.3

Let $K$ be a complex cubic number field, then $[K:{\mathbb{Q}}]=3$ and
$r_{1}=r_{2}=1.$ Let $\varepsilon>1$ be its fundamental unit. For any prime
number $p,$ it is easy to see that the Leopoldt’s conjecture ([Ka]) holds for
$(K,p).$

Proposition 2.1. Let $K$ be a complex cubic number field, $p\geq 3$ a prime
number which splits completely in $K/{\mathbb{Q}}.$ Let
$\mathfrak{p}_{i}(i=1,2,3)$ be all the primes of $K$ lying over $p.$ Then

$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i=1}^{3}U_{i}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}\
\Leftrightarrow\ \varepsilon^{p-1}\not\equiv 1(\text{mod}\
p^{2}\mathcal{O}_{K}),\quad(2.1)$

where $U_{i}^{(1)}$ is the group of principal units of the local field
$K_{\mathfrak{p}_{i}},E^{\prime}=\\{\varepsilon\in E_{K}:\varepsilon\equiv
1(\text{mod}\ \mathfrak{p}_{i})(i=1,2,3)\\},$ and $\overline{E^{\prime}}$ is
the closure of the image of $E^{\prime}$ in $\prod_{i=1}^{3}U_{i}^{(1)}$ via
the diagonal imbedding.

Proof. To begin with, we have the exact sequence ([G1, Chapt.3, Lemma4.2.4])

$1\rightarrow\frac{\text{tor}_{{\mathbb{Z}}_{p}}(\prod_{i=1}^{3}U_{i}^{(1)})}{\text{tor}_{{\mathbb{Z}}_{p}}(\overline{E^{\prime}})}\rightarrow\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i=1}^{3}U_{i}^{(1)}}{\overline{E^{\prime}}})\stackrel{{\scriptstyle\text{log}_{p}}}{{\rightarrow}}\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})})\rightarrow
0.\quad(2.2)$

By assumption,
$E_{K}=<-1>\times<\varepsilon>,e(\mathfrak{p}_{i}/p)=1,i=1,2,3.$ So
$\frac{e}{p-1}=\frac{1}{p}<1.$ By [N, Chapt2, prop.5.5],
$\log_{p}:U_{i}^{(1)}\rightarrow\mathfrak{p}_{i}$ are isomorphisms, so each
$U_{i}^{(1)}$ is ${\mathbb{Z}}_{p}-$torsion free $(i=1,2,3),$ and then
$\frac{\text{tor}_{{\mathbb{Z}}_{p}}(\prod_{i=1}^{3}U_{i}^{(1)})}{\text{tor}_{{\mathbb{Z}}_{p}}(\overline{E^{\prime}})}=\\{1\\}.$
Therefore, to show (2.1), we only need to show that
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})})=\\{0\\}\Leftrightarrow\varepsilon^{p-1}\not\equiv
1(\text{mod}\ p^{2}\mathcal{O}_{K}).$
Note that $-1\notin E^{\prime}$ because $p>2.$ Let $i_{p}:\
E^{\prime}\rightarrow U_{1}^{(1)}\times U_{2}^{(1)}\times
U_{3}^{(1)},x\mapsto(i_{1}(x),i_{2}(x),i_{3}(x))$ be the diagonal imbedding.
Then
$\text{rank}_{{\mathbb{Z}}}(i_{p}(E^{\prime}))\leq\text{rank}_{{\mathbb{Z}}}(E^{\prime})=1.$
On the other hand, $p$ splits completely in $k/{\mathbb{Q}},$ so
$\varepsilon^{p-1}\equiv 1(\text{mod}\ \mathfrak{p}_{i}),i=1,2,3.$ hence
$i_{s}(\varepsilon^{p-1})\in U_{s}^{(1)}(s=1,2,3),$ i.e.,
$\varepsilon^{p-1}\in E^{\prime},i_{p}(\varepsilon^{p-1})\in
i_{p}(E^{\prime}).$ So $<i_{p}(\varepsilon^{p-1})>\subset i_{p}(E^{\prime}),$
which implies $\text{rank}_{{\mathbb{Z}}}(i_{p}(E^{\prime}))\geq 1,$ and then
$\text{rank}_{{\mathbb{Z}}}(i_{p}(E^{\prime}))=1.$ Since each element in
$i_{p}(E^{\prime})$ is of the form $i_{p}(\varepsilon^{t}),\varepsilon^{t}\in
E^{\prime},$ we may take a smallest positive integer $t_{0}$ such that
$i_{p}(E^{\prime})=<i_{p}(\varepsilon^{t_{0}})>.$ Then $t_{0}\mid(p-1).$ Let
$\overline{E^{\prime}}$ be the closure of $i_{p}(E^{\prime})$ in
$\prod_{i=1}^{3}U_{i}^{(1)}.$ Then
$\text{rank}_{{\mathbb{Z}}_{p}}(\overline{E^{\prime}})=1,$ and as a
${\mathbb{Z}}_{p}-$module, $\overline{E^{\prime}}$ is generated by
$i_{p}(\varepsilon^{t_{0}}),$ so $\log_{p}(\overline{E^{\prime}})$ is
generated by $\log_{p}(i_{p}(\varepsilon^{t_{0}})).$ Hence for any
$(a,b,c)\in\log_{p}(\overline{E^{\prime}})\subset\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)}),$
there is $d\in{\mathbb{Z}}_{p}$ such that
$(a,b,c)=d\cdot(\log_{p}(i_{1}(\varepsilon^{t_{0}})),\log_{p}(i_{2}(\varepsilon^{t_{0}})),\log_{p}(i_{3}(\varepsilon^{t_{0}}))).$
On the other hand, since $t_{0}\mid p-1,$ we may let $p-1=r_{0}t_{0}.$ Then
$(r_{0},p)=1$ and $\frac{1}{r_{0}}\in{\mathbb{Z}}_{p}.$ Hence
$d\cdot\log_{p}(i_{s}(\varepsilon^{t_{0}}))=\frac{d}{r_{0}}\cdot\log_{p}(i_{s}(\varepsilon^{p-1}))\in{\mathbb{Z}}_{p}\cdot\log_{p}(i_{s}(\varepsilon^{p-1})),$
which implies
$(a,b,c)\in\prod_{s=1}^{3}{\mathbb{Z}}_{p}\cdot\log_{p}(i_{s}(\varepsilon^{p-1})).$
Now we come to show the assertion (2.1) above.
$\Rightarrow:$ Suppose
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})})=\\{0\\}$
but $\varepsilon^{p-1}\equiv 1(\text{mod}\ p^{2}\mathcal{O}_{K}).$ Then
$\varepsilon^{p-1}\equiv 1(\text{mod}\ \mathfrak{p}_{s}^{2})(s=1,2,3)$ because
$p^{2}\mathcal{O}_{K}=\mathfrak{p}_{1}^{2}\mathfrak{p}_{2}^{2}\mathfrak{p}_{3}^{2}.$
So $i_{s}(\varepsilon^{p-1})\in U_{s}^{(2)}=(U_{s}^{(1)})^{p}\ (s=1,2,3)$
because $\frac{e}{p-1}=\frac{1}{p-1}<1$ (see [S, chat.14,prop9]). Hence there
is $(x,y,z)\in U_{1}^{(1)}\times U_{2}^{(1)}\times U_{3}^{(1)}$ such that
$(i_{1}(\varepsilon^{p-1}),i_{2}(\varepsilon^{p-1}),i_{3}(\varepsilon^{p-1}))=(x^{p},y^{p},z^{p})\in
U_{1}^{(2)}\times U_{2}^{(2)}\times U_{3}^{(2)}.$ If
$\log_{p}x\in{\mathbb{Z}}_{p}\cdot\log_{p}(i_{1}(\varepsilon^{p-1})),$ then
there exists $a\in{\mathbb{Z}}_{p}$ such that
$\text{log}_{p}(i_{1}(\varepsilon^{p-1}))=p\log_{p}x=pa\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1})),$
so
$\text{log}_{p}(i_{1}(\varepsilon^{p-1}))=pa\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1})),$
which implies that $pa=1$ (notice that
$\text{log}_{p}(i_{1}(\varepsilon^{p-1}))\neq 0$ because
$i_{1}(\varepsilon^{p-1})\neq 1$), hence $p$ is invertible in
${\mathbb{Z}}_{p},$ a contradiction! Similarly for $y,z.$ So we have
$(\log_{p}x,\log_{p}y,\log_{p}z)\notin\prod_{s=1}^{3}{\mathbb{Z}}_{p}\cdot\log_{p}(i_{s}(\varepsilon^{p-1})),$
but
$p\cdot(\log_{p}x,\log_{p}y,\log_{p}z)=(p\log_{p}x,p\log_{p}y,p\log_{p}z)=(\log_{p}(i_{1}(\varepsilon^{p-1})),\log_{p}(i_{2}(\varepsilon^{p-1})),\log_{p}(i_{3}(\varepsilon^{p-1})))\in\log_{p}(\overline{E^{\prime}}).$
So $(\log_{p}x,\log_{p}y,\log_{p}z)\ \text{mod}$
$\log_{p}(\overline{E^{\prime}})$ is a torsion element in
$\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})},$
i.e.,
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})})\neq
0,$ contradicts to the assumption!
$\Leftarrow:$ Suppose
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\log_{p}(\prod_{i=1}^{3}U_{i}^{(1)})}{\log_{p}(\overline{E^{\prime}})})\neq\\{0\\}.$
Then there is an element $(\log_{p}x_{1},\log_{p}x_{2},\log_{p}x_{3})$
$\in\prod_{i=1}^{3}\log_{p}(U_{i}^{(1)})\setminus\log_{p}(\overline{E^{\prime}})$
such that
$p\cdot(\log_{p}x_{1},\log_{p}x_{2},\log_{p}x_{3})\in\log_{p}(\overline{E^{\prime}}).$
We assert that
$\log_{p}x_{s}\notin{\mathbb{Z}}_{p}\text{log}_{p}(i_{s}(\varepsilon^{p-1}))$
$(s=1,2,3).$ If otherwise, then there are two cases discussed as follows:
(1) $\log_{p}x_{s}\in{\mathbb{Z}}_{p}\text{log}_{p}(i_{s}(\varepsilon^{p-1}))\
(s=1,2,3).$ Then
$\log_{p}x_{s}=a_{s}\cdot\text{log}_{p}(i_{s}(\varepsilon^{p-1}))$ for some
$a_{s}\in{\mathbb{Z}}_{p}(s=1,2,3).$ Since
$p\cdot(\log_{p}x_{1},\log_{p}x_{2},\log_{p}x_{3})\in\log_{p}(\overline{E^{\prime}}),$
there is $d\in{\mathbb{Z}}_{p}$ such that
$p\cdot(\log_{p}x_{1},\log_{p}x_{2},\log_{p}x_{3})=(d\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1}))),d\cdot\text{log}_{p}(i_{2}(\varepsilon^{p-1}))),d\cdot\text{log}_{p}(i_{s}(\varepsilon^{p-1}))),$
from which we get $pa_{1}=pa_{2}=pa_{3},$ so $a_{1}=a_{2}=a_{3}$ and then
$(\log_{p}x_{1},\log_{p}x_{2},\log_{p}x_{3})\in\log_{p}(\overline{E^{\prime}}).$
A contradiction!
(2) For the remained case, we may as well assume that
$\log_{p}x_{1}\notin{\mathbb{Z}}_{p}\text{log}_{p}(i_{1}(\varepsilon^{p-1})),$
but
$\log_{p}x_{2}\in{\mathbb{Z}}_{p}\text{log}_{p}(i_{2}(\varepsilon^{p-1})).$
Let $\log_{p}x_{2}=b\cdot\text{log}_{p}(i_{2}(\varepsilon^{p-1}))\
(b\in{\mathbb{Z}}_{p}).$ As discussed in case (1) above,
$p\cdot\log_{p}x_{s}=d\cdot\text{log}_{p}(i_{s}(\varepsilon^{p-1}))(s=1,2).$
Thus $d=pb,$ so
$p\cdot\log_{p}x_{1}=pb\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1})).$ We get
$\log_{p}x_{1}=b\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1}))\in{\mathbb{Z}}_{p}\text{log}_{p}(i_{1}(\varepsilon^{p-1})).$
A contradiction!
This shows the above assertion, i.e.,
$\log_{p}x_{s}\notin{\mathbb{Z}}_{p}\text{log}_{p}(i_{s}(\varepsilon^{p-1}))\
(s=1,2,3).$ As discussed above,
$p\cdot\log_{p}x_{1}=b\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1}))$ for some
$0\neq b\in{\mathbb{Z}}_{p}.$ Let $b=p^{n}\cdot u,n\in{\mathbb{Z}}_{\geq
0},u\in{\mathbb{Z}}_{p}^{\times}.$ If $n>0,$ then
$p\cdot\log_{p}x_{1}=p\cdot(p^{n-1}\cdot
u\cdot\text{log}_{p}(i_{1}(\varepsilon^{p-1})))$ which implies
$\log_{p}x_{1}=p^{n-1}\cdot
u\text{log}_{p}(i_{1}(\varepsilon^{p-1}))\in{\mathbb{Z}}_{p}\text{log}_{p}(i_{1}(\varepsilon^{p-1})),$
a contradiction! If $n=0,$ then
$p\cdot\log_{p}x_{1}=u\text{log}_{p}(i_{1}(\varepsilon^{p-1})),$ so
$i_{1}(\varepsilon^{p-1})=\text{exp}_{p}(u^{-1}\text{log}_{p}(x_{1}^{p}))=(\text{exp}_{p}(u^{-1}\text{log}_{p}(x_{1})))^{p}\in(U_{1}^{(1)})^{p}=U_{1}^{(2)}.$
By similar discussion for $x_{2},x_{3}$ above, we also get
$i_{s}(\varepsilon^{p-1})\in U_{s}^{(2)}\ (s=2,3),$ which implies
$\varepsilon^{p-1}\equiv 1(\text{mod}\ p^{2}\mathcal{O}_{K}),$ and the proof
of Proposition.2.1 is completed. $\Box$

Proposition 2.2. Let $K$ be a complex cubic number field, $p$ an odd prime
number (when $p=3,$ assume it is unramified in $K/{\mathbb{Q}}$). If there is
only one prime $\mathfrak{p}$ of $K$ over $p,$ then

$\text{tor}(\frac{U_{\mathfrak{p}}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}\
\Leftrightarrow\ \varepsilon^{p^{f}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}^{e+1}),\quad(2.2)$

where $e=e(\mathfrak{p}/p),f=f(\mathfrak{p}/p)$ are the ramification index and
residue degree of $\mathfrak{p}$ in $K/{\mathbb{Q}},$ respectively.

Proof. It is easy to see that, as a ${\mathbb{Z}}_{p}-$submodule of
$U_{\mathfrak{p}}^{(1)},\ \overline{E^{\prime}}$ is generated by
$i_{p}(\varepsilon^{p^{f}-1}).$ Also $\frac{e(\mathfrak{p}/p)}{p-1}<1,$ so
$(U_{\mathfrak{p}}^{(1)})^{p}=U_{\mathfrak{p}}^{(e+1)}.$ Since
$U_{\mathfrak{p}}^{(1)}$ is ${\mathbb{Z}}_{p}-$torsion free, similar to the
discussion in the proof of Prop.2.1 above, we are done, and the proof is
completed. $\Box$

Proposition 2.3. Let $K$ be a complex cubic number field, $p\geq 5$ a prime
number. If $p\mathcal{O}_{K}=\mathfrak{p}_{1}^{2}\mathfrak{p}_{2}$ with prime
ideals $\mathfrak{p}_{1}\neq\mathfrak{p}_{2}$ in $K.$ Then

$\text{tor}(\frac{U_{\mathfrak{p}_{1}}^{(1)}\times
U_{\mathfrak{p}_{2}}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}\ \Leftrightarrow\
\varepsilon^{p-1}\not\equiv 1(\text{mod}\ \mathfrak{p}_{1}^{3})\ \text{or}\
\varepsilon^{p-1}\not\equiv 1(\text{mod}\ \mathfrak{p}_{2}^{2}).\quad(2.3)$

Proof. The proof is similar to that of Prop.2.2 above. $\Box$

Proposition 2.4. Let $K$ be a complex cubic number field, $p\geq 3$ a prime
number. If $p\mathcal{O}_{K}=\mathfrak{p}_{1}\mathfrak{p}_{2}$ with residue
degree $f(\mathfrak{p}_{1}/p)=2,f(\mathfrak{p}_{2}/p)=1$ for prime ideals
$\mathfrak{p}_{1}\neq\mathfrak{p}_{2}$ in $K.$ Then

$\text{tor}(\frac{U_{\mathfrak{p}_{1}}^{(1)}\times
U_{\mathfrak{p}_{2}}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}\ \Leftrightarrow\
\varepsilon^{p^{2}-1}\not\equiv 1(\text{mod}\ \mathfrak{p}_{1}^{2})\
\text{or}\ \varepsilon^{p-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{2}^{2}).\quad(2.4)$

Proof. Similar to the proof of Prop.2.1 above, we can show that, as
${\mathbb{Z}}_{p}-$modules, $\overline{E^{\prime}}$ is generated by
$i_{p}(\varepsilon^{p^{2}-1}),$ both $U_{\mathfrak{p}_{1}}^{(1)}$ and
$U_{\mathfrak{p}_{2}}^{(1)}$ are ${\mathbb{Z}}_{p}-$torsion free, as well as

$\text{tor}(\frac{U_{\mathfrak{p}_{1}}^{(1)}\times
U_{\mathfrak{p}_{2}}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}\ \Leftrightarrow\
\varepsilon^{p^{2}-1}\not\equiv 1(\text{mod}\ \mathfrak{p}_{1}^{2})\
\text{or}\ \varepsilon^{p^{2}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{2}^{2}).$

Notice that for the prime $\mathfrak{p}_{2},$ we have

$\varepsilon^{p-1}\equiv 1(\text{mod}\ \mathfrak{p}_{2}^{2})\ \Leftrightarrow\
\varepsilon^{p^{2}-1}\equiv 1(\text{mod}\ \mathfrak{p}_{2}^{2}).$

In fact, the necessity is obvious. Conversely,
$\varepsilon^{p-1}=\varepsilon^{p^{2}-1}/(\varepsilon^{p-1})^{p}\in
U_{\mathfrak{p}_{2}}^{(2)}$ because
$(\varepsilon^{p-1})^{p}\in(U_{\mathfrak{p}_{2}}^{(1)})^{p}=U_{\mathfrak{p}_{2}}^{(2)}.$
The proof is completed. $\Box$

Now we come to prove our Theorem 1.3.

Proof of Theorem 1.3. We only prove the case that $K$ is a complex cubic
number field, the other case is similar. Then Leopoldt’s conjecture holds for
$K.$ So $K$ is $p-$rational if and only if $\mathcal{T}_{K}=\\{1\\}.$ By [G1],
Chap.3, Cor.2.6.1,

$\mid\mathcal{T}_{K}\mid=\mid\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i}U_{i}^{(1)}}{\overline{E^{\prime}}})\mid\times\frac{\mid\text{cl}(K)_{p}\mid}{({\mathbb{Z}}_{p}\text{log}_{p}(I_{p}):{\mathbb{Z}}_{p}\text{log}_{p}(P_{p}))}.$

Hence, $K$ is $p-$rational if and only if
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i}U_{i}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}$
and
$\mid\text{cl}(K)_{p}\mid=({\mathbb{Z}}_{p}\text{log}_{p}(I_{p}):{\mathbb{Z}}_{p}\text{log}_{p}(P_{p})).$
By Prop.2.1$\sim$2.4,
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i}U_{i}^{(1)}}{\overline{E^{\prime}}})=\\{1\\}$
if and only if there exists a prime $\mathfrak{p}$ of $K$ over $p$ such that
$\varepsilon^{p^{f}-1}\not\equiv 1(\text{mod}\ \mathfrak{p}^{e+1}).$ Lastly,
by [G1],
$\mid\text{cl}(K)_{p}\mid=({\mathbb{Z}}_{p}\text{log}_{p}(I_{p}):{\mathbb{Z}}_{p}\text{log}_{p}(P_{p}))\Leftrightarrow
H_{p}(K)\cap\widetilde{K}=H_{p}(K)\Leftrightarrow
H_{p}(K)\subset\widetilde{K},$ and the proof of Theorem 1.3 is completed.
$\Box$

3\. Proof of Theorem 1.4

Let $F$ be a number field, $p$ a prime number. Let $\widetilde{F}$ denote the
composite of all ${\mathbb{Z}}_{p}-$extension of $F,$ and write
$\Lambda:={\mathbb{Z}}_{p}[[\text{Gal}(\widetilde{F}/F)]].$ Denote by
$H(\widetilde{F})$ the maximal unramified Abelian pro-$p$ extension of
$\widetilde{F},$ and write
$X(\widetilde{F})=\text{Gal}(H(\widetilde{F})/\widetilde{F}).$ Then
$X(\widetilde{F})$ is a $\Lambda-$module.

Cojecture 3.1(Greenberg [Gr1]). The $\Lambda-$module $X(\widetilde{F})$ is
pseudo-null.

This is the well-known Greenberg’s generalized conjecture (GGC, for short)(see
e.g., [F],[It],[Mi], [T]).

Proof of Theorem 1.4. By definition, $K$ is $p-$rational if and only if
$\text{Gal}(K_{S}^{+})/K^{+}$ contains no ${\mathbb{Z}}_{p}-$torsion elements
and the Leopoldt’s conjecture holds for $K^{+}.$ So by Lemma 8 of [F],
conclusion (1) holds, and then conclusion (2) follows from Theorem 1 in [F].
The proof of Theorem 1.4 is completed. $\Box$

4\. Proof of Theorem 1.5

Proof of Theorem 1.5. We have $K={\mathbb{Q}}(\varepsilon),d_{K}\mid d(f),$
where $d_{K}$ is the discriminant of $K.$ So by assumption $p$ is unramified
in $K.$
let $\sigma_{1}=\text{id},\sigma_{2},\sigma_{3}$ be the three imbedding of $K$
into $\overline{{\mathbb{Q}}}$ over ${\mathbb{Q}}.$ Then
$f(x)=(x-\varepsilon)(x-\sigma_{2}(\varepsilon))(x-\sigma_{3}(\varepsilon)).$
By the general term formula,

$\begin{array}[]{l}F_{n}=A\varepsilon^{n}+B\sigma_{2}(\varepsilon)^{n}+C\sigma_{3}(\varepsilon)^{n},\
\text{where}\\\
A=[(\sigma_{1}(\varepsilon)-\sigma_{2}(\varepsilon))(\sigma_{1}(\varepsilon)-\sigma_{3}(\varepsilon))]^{-1},\\\
B=[(\sigma_{2}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{2}(\varepsilon)-\sigma_{3}(\varepsilon))]^{-1},\\\
C=[(\sigma_{3}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{3}(\varepsilon)-\sigma_{2}(\varepsilon))]^{-1}.\end{array}$

Notice that $A+B+C=0,$ so
$F_{n}=B(\sigma_{2}(\varepsilon)^{n}-\varepsilon^{n})+C(\sigma_{3}(\varepsilon)^{n}-\varepsilon^{n}),$
and so

$\begin{array}[]{l}(\sigma_{2}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{2}(\varepsilon)-\sigma_{3}(\varepsilon))(\sigma_{3}(\varepsilon)-\sigma_{1}(\varepsilon))F_{n}\\\
=(\sigma_{3}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{2}(\varepsilon)^{n}-\varepsilon^{n})-(\sigma_{2}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{3}(\varepsilon)^{n}-\varepsilon^{n}).\end{array}$

Let
$D=(\sigma_{2}(\varepsilon)-\sigma_{1}(\varepsilon))(\sigma_{2}(\varepsilon)-\sigma_{3}(\varepsilon))(\sigma_{3}(\varepsilon)-\sigma_{1}(\varepsilon)),$
then $D^{2}=d(f).$ Now we come to show the results (1), (2) and (3) of the
Theorem. Our way of proof is via contradiction.
(1) Let $L$ be the normal closure of $K/{\mathbb{Q}},$ each
$\sigma_{i}(i=1,2,3)$ can be extended on $L,$ and denoted by the same
notations. If we suppose that $\varepsilon^{p-1}\equiv 1(\text{mod}\
p^{2}\mathcal{O}_{K}),$ then $\varepsilon^{p-1}\equiv 1(\text{mod}\
p^{2}\mathcal{O}_{L}).$ So $\sigma_{i}(\varepsilon^{p-1})\equiv 1(\text{mod}\
\sigma_{i}(p^{2}\mathcal{O}_{L})=p^{2}\mathcal{O}_{L})(i=1,2,3).$ Hence
$\varepsilon^{p-1}\equiv\sigma_{2}(\varepsilon^{p-1})\equiv\sigma_{3}(\varepsilon^{p-1})(\text{mod}\
p^{2}\mathcal{O}_{L}),$ which implies that $DF_{p-1}\in p^{2}\mathcal{O}_{L},$
and so $F_{p-1}\in p^{2}\mathcal{O}_{L}$ because $p\nmid d(f)=D^{2}.$
Therefore $p^{2}\mid F_{p-1}$ because $F_{p-1}\in{\mathbb{Z}}$ is an integer.
This contradicts to the assumption. This proves (1).
(2) and (3) can be similarly done, and the proof is completed. $\Box$

5\. Proof of Theorem 1.6

Proof of Theorem 1.6. Our proof is in the same spirit of ([MR], second part).
For the generalized $abc-$conjecture, see e.g., [Ic], [MR]. Let $\varepsilon$
be the fundamental unit of $K.$ Then $\exists\ n_{0}\in{\mathbb{Z}}_{>0}$ such
that for every integer $n\geq n_{0}$ with $\varphi(n)\geq n/2$ ($\varphi$ is
the Euler function), there is a prime ideal $\mathfrak{p}_{n}$ of
$\mathcal{O}_{K}$ satisfying
(1) $\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2});$
(2) $\mathfrak{p}_{n}\neq\mathfrak{p}_{n^{\prime}}$ if $n\neq n^{\prime};$
(3) there is a real number $r>0$ depending only on $K,$ such that
$N\mathfrak{p}_{n}\leq r^{n}.$ Here $N:K\rightarrow{\mathbb{Q}}$ is the norm
map.
Let $x\geq 1,$ for the above real number $r,$ take $n_{1}=[\log_{r}x].$ Let
$x$ be sufficiently large such that $n_{1}\geq n_{0}.$ Then for any integer
$n:n_{0}\leq n\leq n_{1}$ with $\varphi(n)\geq n/2,$ there exists a prime
ideal $\mathfrak{p}_{n}$ of $\mathcal{O}_{K}$ such that
$\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2}).$ Let
$\mathfrak{p}_{n}\cap{\mathbb{Z}}=p_{n}{\mathbb{Z}}$ for some prime number
$p_{n}.$ Then $p_{n}\leq N\mathfrak{p}_{n}\leq r^{n}\leq r^{n_{1}}\leq x.$ So
$\frac{1}{m}\cdot\sharp\\{n:n_{0}\leq n\leq n_{1},\varphi(n)\geq
n/2\\}\leq\sharp\\{\text{primes}\ p_{n}\leq x:\exists\ \text{prime ideal}\
\mathfrak{p}_{n}\lhd\mathcal{O}_{K},\mathfrak{p}_{n}\mid
p_{n},\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2})\\},$ where $m=[K:{\mathbb{Q}}].$ By Lemma 6 of [Sil],
$\sharp\\{n\leq x:\varphi(n)\geq
n/2\\}\geq(\frac{6}{\pi^{2}}-\frac{1}{2})x+O(\log x).$ Therefore, there exists
a real number $c>0$ depending only on $K,$ such that, as $x\rightarrow\infty,\
\sharp\\{\text{primes}\ p_{n}\leq x:\exists\ \text{prime ideal}\
\mathfrak{p}_{n}\lhd\mathcal{O}_{K},\mathfrak{p}_{n}\mid
p_{n},\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2})\\}\geq c\cdot\log x.$ Let $A=\\{\text{prime}\ p:p\mid
h(K)\ \text{or}\ p\ \text{is ramified in}\ K\\}(h(K))$ is the class number of
K). Then $\sharp A<\infty.$ By Theorem 1.3 above, as $x\rightarrow\infty,\
\sharp\\{\text{primes}\ p\leq x:K\ \text{is}\
p-\text{rational}\\}\geq\sharp(\\{\text{primes}\ p_{n}\leq x:\exists\
\text{prime ideal}\ \mathfrak{p}_{n}\lhd\mathcal{O}_{K},\mathfrak{p}_{n}\mid
p_{n},\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2})\\}\setminus A)\geq\sharp\\{\text{primes}\ p_{n}\leq
x:\exists\ \text{prime ideal}\
\mathfrak{p}_{n}\lhd\mathcal{O}_{K},\mathfrak{p}_{n}\mid
p_{n},\varepsilon^{N\mathfrak{p}_{n}-1}\not\equiv 1(\text{mod}\
\mathfrak{p}_{n}^{2})\\}-\sharp A\geq c\log x-\sharp A\geq c^{\prime}\log x\
(x\rightarrow\infty).$ The proof of Theorem 1.6 is completed. $\Box$

6\. Examples

Example 6.1. Let $p\geq 5$ be a prime number, $K={\mathbb{Q}}(\alpha)$ with
$\alpha=\sqrt[3]{p^{3}-1}.$ Then $p-\alpha$ is a fundamental unit of $K$ (see
[St]). Let $\varepsilon=\frac{1}{p-\alpha}=p^{2}+pa+\alpha^{2},$ then
$\varepsilon>1$ is also a fundamental unit of $K.$ The minimal polynomial of
$\alpha$ on ${\mathbb{Q}}$ is $f(x)=x^{3}+1-p^{3}$ with discriminant
$d(f)=-27(1-p^{3})^{2}.$ It can be easily verified that (see e.g. [N],Chapt 1,
Prop.8.3),
(R1) if $p\equiv 1(\text{mod}\ 3),$ then $p$ splits completely in
$K/{\mathbb{Q}},$ i.e.,
$p\mathcal{O}_{K}=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}$ for three
deferent prime ideals in $K;$
(R2) if $p\equiv 2(\text{mod}\ 3),$ then
$p\mathcal{O}_{K}=\mathfrak{p}\mathfrak{q}$ for two different prime ideals in
$K$ with residue degree $f(\mathfrak{p}/p)=1,f(\mathfrak{q}/p)=2.$
For (R1), $f=f(\mathfrak{p}_{i}/p)=1(i=1,2,3).$ We have
$\varepsilon^{p-1}=(p^{2}+pa+\alpha^{2})^{p-1}=(p(p+\alpha)+\alpha^{2})^{p-1}\equiv\alpha^{2p-2}+(p-1)\alpha^{2p-4}p(p+\alpha)\equiv(\alpha^{3})^{\frac{2(p-1)}{3}}+(p-1)\alpha^{2p-4}p\alpha\equiv(p^{3}-1)^{\frac{2(p-1)}{3}}-p\alpha^{2p-3}\equiv
1-p\alpha^{2p-3}\not\equiv 1(\text{mod}\ p^{2}\mathcal{O}_{K}).$ Hence $(K,p)$
satisfies the condition (2) in Theorem 1.3 above. So does for (R2), too.
By calculating class number via PARI, for $5\leq p\leq 10000$ except
$p=2791\\\ (h({\mathbb{Q}}(\sqrt[3]{2791^{3}-1}))=31876011),$ we have $p\nmid
h({\mathbb{Q}}(\sqrt[3]{p^{3}-1})).$ So the $p-$Hilbert class field
$H_{p}(K)=K\subseteq\widetilde{K},$ i.e., $(K,p)$ satisfies the condition (1)
in Theorem 1.3 above. Therefore by the above discussion and Theorem 1.3,
${\mathbb{Q}}(\sqrt[3]{p^{3}-1})$ is $p-$rational for all primes $p:5\leq
p\leq 10000,p\neq 2791.$ On the other hand, by Gras’ method on $p-$rationality
using ray class group [G2], via calculating with computer, one can know that
${\mathbb{Q}}(\sqrt[3]{p^{3}-1})$ are $p-$rational for all primes $p:p\leq
10000$ including $p=2791.$
Furthermore, it is easy to know that the normal closure of
${\mathbb{Q}}(\sqrt[3]{p^{3}-1})$ is
$K={\mathbb{Q}}(\sqrt[3]{p^{3}-1},\sqrt{-3}),\ \text{Gal}(K/{\mathbb{Q}})\cong
S_{3}.$ So by the above discussion and Prop.3.8 of [Gr2], we can obtain that
the sextic number fields $K={\mathbb{Q}}(\sqrt[3]{p^{3}-1},\sqrt{-3})$ are
$p-$rational for all primes $p:5\leq p\leq 10000.$
More generally, we have the following conclusion:
Let $K$ be a normal closure of a complex cubic number field $k,F$ the unique
quadratic subfield of $K,p\geq 5$ a prime number. Then $K$ is $p-$rational if
and only if both $k$ and $F$ are $p-$rational. In particular, if $k$ is a pure
cubic number field, then $F={\mathbb{Q}}(\sqrt{-3}),$ and then $K$ is
$p-$rational if and only if $k$ is $p-$rational.

Example 6.2. Here we come to construct a $3-$rational complex cubic number
field with non-trivial Hilbert class group. Our calculation need using
Computer PROCESS PARI/GP.
Let $\alpha$ be a root of $f(x)=x^{3}-4x+27,k={\mathbb{Q}}(\alpha),p=3.$ By
PARI/GP, one can calculate the class number $h(k)=3.$ The discriminant
$d(f)=-19427.$ Notice that $19427$ is a prime, the integral ring
$\mathcal{O}_{k}={\mathbb{Z}}[\alpha]$ with discriminant $d_{k}=d(f).$ Since
$f(x)\equiv x(x-1)(x+1)(\text{mod}3),3$ splits completely in
$k:3\mathcal{O}_{k}=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3},$ where
$\mathfrak{p}_{1}=(3,\alpha),\mathfrak{p}_{2}=(3,\alpha-1),\mathfrak{p}_{3}=(3,\alpha+1).$
Similarly, one has $2\mathcal{O}_{k}=PQ$ with
$P=(2,\alpha+1),Q=(2,\alpha^{2}-\alpha+1),$ the inertia index
$f(P/2)=1,f(Q/2)=2.$ By PARI/GP, one can determine that $P$ is not principal
in $k,$ so $[P]\in\text{cl}(k)$ is of order $3,$ and $[P]^{3}$ is generated by
the element $g=-835+265\alpha-77(\alpha^{2}-3).$ Now
$g^{2}=2027557-643443\alpha+186957(\alpha^{2}-3)\equiv
1+3\alpha(\text{mod}9\mathcal{O}_{k}).$ Let $I_{3}$ denote the group consists
of all fractional ideals of $k$ which are prime to each
$\mathfrak{p}_{i}(i=1,2,3),P_{3}$ the subgroup consists of all principal
ideals of $I_{3}.$ By the above discussion, $\text{cl}(k)=<[P]>,$ So
$I_{3}=<P>P_{3}.$ Consider the map ([G1], Chap.3)

$\text{Log}_{T_{3}}:I_{3}\rightarrow k_{v_{1}}\times k_{v_{2}}\times
k_{v_{3}}/{\mathbb{Q}}_{3}(\text{log}_{T_{3}}(E)),\mathfrak{a}\mapsto\frac{1}{m}(\text{log}_{v}(\alpha))_{v\mid
3}\text{mod}{\mathbb{Q}}_{3}(\text{log}_{T_{3}}(E)),$

where $\mathfrak{a}^{m}=(\alpha),\alpha\in k_{T_{3}}^{\times}=\\{\beta\in
k^{\times}:(\beta,v)=1,\forall v\in T_{3}\\},$ here
$T_{3}=\\{\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{3}\\}.$ Then
${\mathbb{Z}}_{3}\text{Log}_{T_{3}}(I_{3})={\mathbb{Z}}_{3}\text{Log}_{T_{3}}(P)+{\mathbb{Z}}_{3}\text{Log}_{T_{3}}(P_{3}).$
By definition, we have
${\mathbb{Z}}_{3}\text{Log}_{T_{3}}(P_{3})=\oplus_{v\in
T_{3}}\text{log}_{3}(U_{v}^{(1)})\text{mod}{\mathbb{Q}}_{3}(\text{log}_{T_{3}}(E))=3{\mathbb{Z}}_{3}\oplus
3{\mathbb{Z}}_{3}\oplus
3{\mathbb{Z}}_{3}\text{mod}{\mathbb{Q}}_{3}(\text{log}_{T_{3}}(E)).$
On the other hand, by definition,

$\text{Log}_{T_{3}}(P)=\frac{1}{3}(\text{log}_{3}(g_{1}),\text{log}_{3}(g_{2}),\text{log}_{3}(g_{3}))=\frac{1}{6}(\text{log}_{3}(g_{1}^{2}),\text{log}_{3}(g_{2}^{2}),\text{log}_{3}(g_{3}^{2})),$

where $g_{i}=-835+265\alpha_{i}-77(\alpha_{i}^{2}-3),\alpha_{i}$ is the image
of the natural embedding of $k$ into $k_{v_{i}},i=1,2,3.$ Since $g^{2}\equiv
1+3\alpha(\text{mod}9\mathcal{O}_{k}),$ we have $g_{i}^{2}\equiv
1+3\alpha_{i}(\text{mod}9\mathcal{O}_{k_{v_{i}}}),i=1,2,3.$ So
$v_{3}(\text{log}_{3}(g_{i}^{2}))=v_{3}(g_{i}^{2}-1),i=1,2,3.$ (here
$v_{3}(3)=1.$) Hence by $\alpha_{i}\equiv i-1(\text{mod}3),$ we get
$g_{i}^{2}\equiv 3i-2(\text{mod}9\mathcal{O}_{k_{v_{i}}}),i=1,2,3.$ Write
$u_{i}=\frac{1}{6}\text{log}_{3}(g_{i}^{2}),i=1,2,3.$ Then $v_{3}(u_{1})\geq
1,v_{3}(u_{2})=v_{3}(u_{3})=0,$ which shows that
$\text{Log}_{T_{3}}(P)=(u_{1},u_{2},u_{3})\notin 3{\mathbb{Z}}_{3}\oplus
3{\mathbb{Z}}_{3}\oplus 3{\mathbb{Z}}_{3},$ so
$(\text{Log}_{T_{3}}(I_{3}):\text{Log}_{T_{3}}(P_{3}))=3.$ Therefore,
$[H_{3}(k)\cap\widetilde{k}:k]=3,$ and then $H_{3}(k)\subset\widetilde{k}$
because $h(k)=3.$ By PARI/GP again, one can calculate the fundamental unit of
$k,$ i.e., $\epsilon=-729\alpha^{2}-3462\alpha-3280,$ so
$\epsilon^{2}=-68846348+28552197\alpha+18893448(\alpha^{2}-3)\equiv
1+3(\alpha+2)(\text{mod}9\mathcal{O}_{k}).$ Obviously $\alpha+2\notin
3\mathcal{O}_{k}.$ Therefore, $\epsilon^{2}\not\equiv
1(\text{mod}9\mathcal{O}_{k}).$ To sum up, by Theorem 1.3 above,
$k={\mathbb{Q}}(\alpha)$ is $p-$rational.

Example 6.3. $f(x)=x^{4}-2x^{2}+3$ has four roots $\pm\sqrt{1\pm\sqrt{-2}}.$
Take $\alpha=\sqrt{1+\sqrt{-2}}.$ Then $k={\mathbb{Q}}(\alpha)$ is a pure
imaginary quartic number field, $r_{2}(k)=2.$ Calculating by PARI/GP, the
class number $h(k)=1,\varepsilon=\alpha^{3}+\alpha^{2}-\alpha-2$ is a
fundamental unit of $k,$ and $\\{1,\alpha,\alpha^{2},\alpha^{3}\\}$ is an
integral base of $k,$ Take $p=5.$ It is easy to verify that
$5\nmid[\mathcal{O}_{k}:{\mathbb{Z}}[\alpha]],$ and $f(x)\text{mod}5$ is
irreducible over ${\mathbb{F}}_{5}.$ Hence $5$ is inertia in $k$ with residual
degree 4. Also by calculating, $\varepsilon^{5^{4}-1}\equiv
1+5\alpha+15\alpha^{3}\not\equiv 1(\text{mod}25\mathcal{O}_{k}).$ Hence by
Theorem 1.3 above, ${\mathbb{Q}}(\sqrt{1+\sqrt{-2}})$ is $5-$rational.

Example 6.4. Let $p$ be a prime number, $K={\mathbb{Q}}(\sqrt{p^{2}-1}),$ the
fundamental unit of $K$ is $\varepsilon=p+\sqrt{p^{2}-1}$ (see [Ku], Thm.1).
All such $K$ are $p-$rational (see [BR]). Let
$L=K(\sqrt{-1})={\mathbb{Q}}(\sqrt{p^{2}-1},\sqrt{-1}).$ We come to show that,
there are infinitely many prime number $p$ such that GGC holds for
$({\mathbb{Q}}(\sqrt{p^{2}-1},\sqrt{-1}),p).$
To see this, let
$K_{1}=K,K_{2}={\mathbb{Q}}(\sqrt{1-p^{2}}),K_{3}={\mathbb{Q}}(\sqrt{-1})$ be
the three non-trivial intermediate subfields of $L/{\mathbb{Q}}$ as the
following diagram
We will discuss in the following steps
(Step1) If $p\nmid h(L),$ then GGC holds for $(L,p).$
In fact, if $p\equiv 1(\text{mod}\ 4),$ then by our assumption, $p$ splits
completely in $L/{\mathbb{Q}}.$ Since $K_{1}$ is $p-$rational and is the
maximal real subfield of $L,$ by Theorem 1.4, GGC holds for $(L,p).$ If
$p\equiv 3(\text{mod}\ 4),$ then $p$ splits in $K_{2}/{\mathbb{Q}},$ and each
prime of $K_{2}$ above $p$ is inertia in $L/K_{2}.$ So by ([Mi], Prop.3.B,
Remark3) we know that GGC holds for $(L,p).$
(Step2) Now we consider the class number. By Kuroda’s class number formula
[Le],

$h(L)=\frac{1}{2}q(L)\frac{h(K_{1})h(K_{2})h(K_{3})}{h({\mathbb{Q}})}=\frac{1}{2}q(L)h(K_{1})h(K_{2}),\quad(4.1)$

where $q(L)=(E_{L}:E_{K_{1}}E_{K_{2}}E_{K_{3}}).$ We have $q(L)\leq 2$ because
$(E_{L}:W(L)E_{K})\leq 2$ (see [W],Thm.4.12), $W(L)$ is the group of roots of
unity of $L.$ On the other hand, by the relation of class number and
$p-$rationality ([Gr2], Prop.4.1), $p\nmid h(K_{1}).$ Therefore, by formula
(4.1) above, we have that $p\nmid h(L)\ \Leftrightarrow\ p\nmid h(K_{2}).$
(Step3) We need the following two Lemmas, the proofs are based on the method
of [Ko].

Lemma A. Given a real number $T>0,$ there are infinitely many prime $p\equiv
1(\text{mod}\ 4)$ satisfying the following condition:
there are $m,n\in{\mathbb{Z}}_{>0}$ such that $n^{2}|p-1,m^{2}|p+1,$ and
$m,n>(\log p)^{T}.$

Proof. We only take $p\pm 1$ instead of the $p\pm 2$ in [Ko]. For the
convenience of readers, we write the detail here. For integers $D>0,l$ with
$(D,l)=1,$ write $\pi(x,D,l)=\\{\text{prime}p\leq x:p\equiv l(\text{mod}\
D)\\}.$ By ([E], Lemma 2.9), given a real number $c,$ for all
$l,D:(l,D)=1,1\leq D\leq(\log x)^{c},$ we have
$\pi(x,D,l)=\frac{1}{\varphi(D)}\int_{2}^{x}\frac{dt}{\log
t}+O(xe^{-c_{1}\sqrt{\log x}})$ (here the implicit constant $O(\cdot)$ is
independent of the choice of $l,D$). Take $c=4T+1,$ we have (for $l,D$ above)
$\pi(x,D,l)-\pi(\frac{x}{2},D,l)=\frac{1}{\varphi(D)}\int_{x/2}^{x}\frac{dt}{\log
t}+O(xe^{-c_{1}\sqrt{\log x}}).$ For any $D$ satisfying $1\leq
D\leq(\text{log}(x))^{c},$ we have

$\frac{1}{\varphi(D)}\int_{x/2}^{x}\frac{dt}{\log
t}\geq\frac{1}{\varphi(D)}\frac{x}{2\log
x}\geq\frac{x}{2\log(x)^{c+1}}\gg\frac{x}{e^{c_{1}\sqrt{\log x}}}.$

So for sufficiently large $x,\ \pi(x,D,l)-\pi(\frac{x}{2},D,l)>0$ holds for
all $l,D$ satisfying $(l,D)=1$ and $1\leq D\leq(\text{log}(x))^{c}.$ Now let
$x$ be sufficiently large such that there exist integers $n,m=n+1$ satisfying
$2\mid n$ and $(\log x)^{T}<n,m<2(\log x)^{T}.$ Obviously, $(m,n)=1.$ By
Chinese remainder theorem, there an integer $l$ such that

$\left\\{\begin{array}[]{l}y\equiv 1(\text{mod}\ n^{2})\\\
y\equiv-1(\text{mod}\ m^{2})\end{array}\right.$

and $y\equiv l(\text{mod}\ D)$ have the same solutions of integers. Here
$D=m^{2}n^{2}\leq 16(\log x)^{4T}\leq(\log x)^{c}\ (c=4T+1).$ By the
discussion above, for sufficiently large $x,$ there exist primes $p$
satisfying $\frac{x}{2}<p\leq x$ and $p\equiv l(\text{mod}\ D),$ then every
such $p$ satisfies $p\equiv 1(\text{mod}\ n^{2})$ and $p\equiv-1(\text{mod}\
m^{2}).$ Since $n$ is even, we have $p\equiv 1(\text{mod}\ 4).$ let $x$ become
more and more large, one can obtain infinitely many primes $p$ satisfying the
conditions of Lemma A. $\Box$

Lemma B([Lou],Section1, Prop.2). Let $K$ be an imaginary quadratic field with
discriminant $-d_{K}(d_{K}>0).$ Then

$h(K)\leq\frac{\omega_{K}\cdot\sqrt{d_{K}}}{4\pi}(\log(d_{K})+2+\gamma-\log\pi),$

where $\omega_{K}$ is the number of roots of unity in $K,\gamma$ is the Euler
constant.
Take $T=1$ in Lemma A above, let $\mathcal{P}$ be the set consisting of all
the primes $p$ satisfying the conditions of Lemma A. For $p\in\mathcal{P},$
let $L={\mathbb{Q}}(\sqrt{1-p^{2}}),$ then $|d_{L}|\leq 4\frac{p^{2}-1}{(\log
p)^{4}},$ so $h(L)\leq\frac{2}{4\pi}\sqrt{4\frac{p^{2}-1}{(\log
p)^{4}}}(\log(4\frac{p^{2}-1}{(\log p)^{4}})+2+\gamma-\log\pi)\ll\frac{p}{\log
p}(p\rightarrow\infty,p\in\mathcal{P}).$ Hence there are infinitely many
primes $p$ such that $h({\mathbb{Q}}(\sqrt{1-p^{2}}))<p.$ In particular, such
$p$ satisfying $p\nmid h({\mathbb{Q}}(\sqrt{1-p^{2}})),$ hence GGC holds for
the corresponding pair $({\mathbb{Q}}(\sqrt{p^{2}-1},\sqrt{-1}),p).$

7\. Tables

Table 1: Table 1 $T(|d_{K}|<3000)$ | $5\leq p\leq 100$
---|---
$p|h(K)$ | $\text{tor}(\frac{\prod_{i}U_{i}^{(1)}}{\overline{E^{\prime}}})\neq\\{1\\}$ | not $p$-rational
$x^{3}-x^{2}+x-9$ | $\emptyset$ | 13 | 13
$x^{3}-x^{2}+5*x+1$ | $\emptyset$ | 17 | 17
$x^{3}-x^{2}-2*x+6$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}+x+5$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}+5*x+2$ | $\emptyset$ | 11 | 11
$x^{3}-6*x-12$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}-x+13$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}-x-6$ | $\emptyset$ | 11 | 11
$x^{3}-x^{2}+5*x+11$ | $\emptyset$ | 11 | 11
$x^{3}-x^{2}+7*x-2$ | $\emptyset$ | 5 | 5
$x^{3}-8*x-11$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}-4*x+9$ | $\emptyset$ | 19 | 19
$x^{3}-x^{2}+7*x-6$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}+x+15$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}+x-24$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}+4*x-9$ | $\emptyset$ | 13 | 13
$x^{3}-x^{2}-6*x-16$ | $\emptyset$ | 5 | 5
$x^{3}+10*x-12$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}+10*x-16$ | 5 | $\emptyset$ | 5?
$x^{3}-26$ | $\emptyset$ | 11 | 11
$x^{3}-x^{2}-8*x-10$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}-x-26$ | $\emptyset$ | 61 | 61
$x^{3}-x^{2}+13*x-1$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}-3*x-17$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}+7*x-19$ | $\emptyset$ | 31 | 31
$x^{3}-x^{2}-11*x+21$ | $\emptyset$ | 11 | 11
$x^{3}-11*x-17$ | $\emptyset$ | 5 | 5
$x^{3}-x^{2}+6*x-10$ | $\emptyset$ | 23 | 23
$x^{3}-x^{2}-10*x-20$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}-11*x-21$ | $\emptyset$ | 7 | 7
$x^{3}-2*x-20$ | 5 | $\emptyset$ | 5?
$x^{3}+2*x-10$ | $\emptyset$ | 7 | 7
$x^{3}+4*x-20$ | $\emptyset$ | 13 | 13
$x^{3}-x^{2}+5*x-32$ | 5 | $\emptyset$ | 5?
$x^{3}-x^{2}+9*x-21$ | 7 | $\emptyset$ | 7?
Table 2: Table 2 $T(d_{K}<3000)$ | $5\leq p\leq 100$
---|---
5 totally ramified | $p|h(K)$ | $\text{tor}(\frac{\prod_{i}U_{i}^{(1)}}{\overline{E^{\prime}}})\neq\\{1\\}$ | not $p$-rational
$x^{4}-x^{3}+x^{2}-x+1$ | Yes | $\emptyset$ | $\emptyset$ | -
$x^{4}+1$ | No | $\emptyset$ | 13,31 | 13,31
$x^{4}-2*x^{2}+4$ | No | $\emptyset$ | 7 | 7
$x^{4}+2*x^{2}+4$ | No | $\emptyset$ | 13,31 | 13,31
$x^{4}-2*x^{3}-2*x+5$ | No | $\emptyset$ | 11 | 11
$x^{4}-x^{3}-4*x^{2}+4*x+7$ | No | $\emptyset$ | 23 | 23
$x^{4}-2*x^{3}+5*x^{2}-4*x+2$ | No | $\emptyset$ | 13,31 | 13,31
$x^{4}-x^{3}-2*x^{2}-3*x+9$ | No | $\emptyset$ | 29,37 | 29,37
$x^{4}-2*x^{3}-4*x^{2}+5*x+7$ | No | $\emptyset$ | 5 | 5
$x^{4}-2*x^{3}-3*x^{2}+4*x+5$ | No | $\emptyset$ | 11 | 11
$x^{4}+4*x^{2}+2$ | No | $\emptyset$ | 13,31 | 13,31
$x^{4}+9$ | No | $\emptyset$ | 7 | 7

In the above two tables, we use LMFDB data: In Table 1, there are totally
$419$ number of complex cubic number fields $K$ of which the absolute values
of discriminants $<3000.$ We take all the $23$ number of prime numbers
$p:5\leq p\leq 100.$ For the $419\times 23=9637$ pair of $(K,p),$ by our
criteria, except the cases of the above table, all the remainder are
$p-$rational. In the table, we use a polynomial $T$ of degree $3$ to indicate
the cubic number field which is generated by a root of $T.$ For the case that
$p\mid h(K)$ with
$\text{tor}_{{\mathbb{Z}}_{p}}(\frac{\prod_{i=1}^{3}U_{i}^{(1)}}{\overline{E^{\prime}}})=\\{1\\},$
to determine the $p-$rationality, one need to further determine whether
$H_{p}(K)\subset\widetilde{K}.$ At present, we don’t know whether there are
such effective algorithms.
In Table 2, there are totally $147$ number of pure imaginary quartic number
fields $K$ of which the absolute values of discriminants $<3000.$ We take all
the $23$ number of prime numbers $p:5\leq p\leq 100.$ For the $147\times
23=3381$ pair of $(K,p),$ by our criteria, except the cases of the above
table, all the remainder are $p-$rational. In the table, we use a polynomial
$T$ of degree $4$ to indicate the quartic number field which is generated by a
root of $T.$ We need to point out, at present, for the case that $5$ is
totally ramified in the quartic number field, we can not use the result of
Theorem 1.3 above to determine the $5-$rationality.
For the determined cases, the calculated results in our tables are consistent
with the ones done by the algorithm of Gras in [G2].

Acknowledgments  We would like to thank Georges Gras for sending us his
several papers and giving helpful suggestions. We would like to thank Zouhair
Boughadi for informing us the relation of an open question in [ABo] and our
Theorem 1.4 as in Remark 1.7.

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