\section{Introduction}\label{sec1}

The phenomenon of neutrino oscillation indicates that neutrinos have nonzero masses and there are mass splttings between different neutrino species \cite{Lesgourgues:2006nd,Agashe:2014kda}. The neutrino oscillation experiments can provide the information about the squared mass differences between the neutrino mass eigenstates. Specifically, the solar and reactor experiments give the result of $\Delta m_{21}^{2} \simeq 7.5 \times 10^{-5}$ $\rm eV^{2}$, and the atmospheric and accelerator beam experiments give the result of $|\Delta m_{31}^{2}| \simeq 2.5 \times 10^{-3}$ $\rm eV^{2}$ \cite{Agashe:2014kda,Xing:2019vks}. Therefore, we can get two possible mass hierarchies of the neutrino mass spectrum, i.e., the normal hierarchy (NH) with $m_{1} < m_{2} \ll m_{3}$ and the inverted hierarchy (IH) with $m_{3} \ll m_{1} < m_{2}$, where $m_{1}$, $m_{2}$, and $m_{3}$ denote the masses of neutrinos for the three mass eigenstates. However, the absolute masses of neutrinos are still unknown.

In principle, laboratory experiments of particle physics can directly measure the absolute masses of neutrinos, but these experiments have always been facing great challenges \cite{Osipowicz:2001sq, KlapdorKleingrothaus:2002ip, KlapdorKleingrothaus:2004wj, Kraus:2004zw, Otten:2008zz, Wolf:2008hf, Huang:2016qmh, Zhang:2017ljh, Betts:2013uya}. Compared with these particle physics experiments, cosmological observations are more prone to be capable of measuring the absolute masses of neutrinos \cite{Valle:2005ai,Hannestad:2010kz,Lesgourgues:2012uu}, since massive neutrinos can leave rich signatures on the cosmic microwave background (CMB) anisotropies and the large-scale structure (LSS) formation at different epochs of the cosmic evolution \cite{Abazajian:2013oma}. Thus, we can extract useful information on neutrinos from these available cosmological observations.

Recently, the issue of cosmological constraints on the total neutrino mass with the consideration of mass hierarchy using the latest observational data has been discussed in Ref. \cite{Zhang:2020mox}. In Ref. \cite{Zhang:2020mox}, the authors discussed the constraints on neutrino mass in several typical dark energy models, e.g., the $\Lambda$ cold dark matter ($\Lambda$CDM), $w$CDM, Chevallier-Polarski-Linder (CPL), and holographic dark energy (HDE) models. It was found that, compared to the $\Lambda$CDM+$\sum m_{\nu}$ model, larger neutrino masses are favored in the $w$CDM+$\sum m_{\nu}$ and CPL+$\sum m_{\nu}$ models, and the most stringent upper limits are obtained in the HDE+$\sum m_{\nu}$ model. Moreover, in Ref. \cite{Zhang:2020mox}, it was also confirmed that the NH case is more favored by current cosmological observations than the IH case. For more relevant studies on constraining the total neutrino mass by using cosmological observations, see e.g., Refs. \cite{Hu:1997mj,Reid:2009nq,Thomas:2009ae,Carbone:2010ik,Li:2012vn,Wang:2012vh,Wang:2012uf,Audren:2012vy,Riemer-Sorensen:2013jsa,Font-Ribera:2013rwa,Zhang:2014dxk,Zhang:2014nta,Zhang:2014ifa,Palanque-Delabrouille:2014jca,Geng:2014yoa,Li:2015poa,Ade:2015xua,Zhang:2015rha,Geng:2015haa,Chen:2015oga,Allison:2015qca,Cuesta:2015iho,Chen:2016eyp,Moresco:2016nqq,Lu:2016hsd,Kumar:2016zpg,Xu:2016ddc,Vagnozzi:2017ovm,Zhang:2017rbg,Lorenz:2017fgo,Zhao:2017jma,Vagnozzi:2018jhn,Wang:2018lun,Li:2017iur,Wang:2017htc,Feng:2017usu,Zhao:2016ecj,Zhang:2015uhk,Huang:2015wrx, Wang:2016tsz,Vagnozzi:2019utt,Vagnozzi:2019jzi,Giusarma:2016phn,Gariazzo:2018pei,Liu:2020vgn,Choudhury:2018byy,Allahverdi:2016fvl,Han:2017wnk,Zhou:2014fva,Huo:2011ve,Zhang:2019ipd,DiazRivero:2019ukx}.


Furthermore, the impacts of interaction between dark energy (DE) and cold dark matter (CDM) on constraining neutrino mass have also been considered. For example, in the scenario of vacuum energy interacting with cold dark matter, which is abbreviated as the I$\Lambda$CDM scenario in this work, the constraint on $\sum m_{\nu}$ becomes $\sum m_{\nu} < 0.10$ eV (2$\sigma$) for $Q=\beta H \rho_{\rm de}$, $\sum m_{\nu} < 0.20$ eV (2$\sigma$) for $Q=\beta H \rho_{\rm c}$ \cite{Guo:2017hea}, and $\sum m_{\nu} < 0.214$ eV (2$\sigma$) for $Q=\beta H_{0} \rho_{\rm c}$ \cite{Feng:2019jqa}. When the mass hierarchies of neutrinos are considered in the I$\Lambda$CDM model \cite{Guo:2018gyo, Feng:2019mym}, the results showed that the degenerate hierarchy (DH) case gives the smallest upper limit of the neutrino mass and the NH case is more favored over the IH case. In the present work, we will revisit the constraints on the total neutrino mass in the I$\Lambda$CDM scenario after the Planck 2018 data release. We will consider more forms of interaction term $Q$, and also adopt the mass hierarchies of neutrinos in this work.

In the so-called ``interacting dark energy" (IDE) scenario, some direct, non-gravitational coupling between dark energy and dark matter is assumed and its cosmological consequences have been widely studied \cite{Amendola:1999er,Amendola:2001rc,Comelli:2003cv, Cai:2004dk, Zhang:2005rg, Zimdahl:2005bk, Zhang:2004gc, Wang:2006qw, Guo:2007zk, Bertolami:2007zm, Zhang:2007uh, Boehmer:2008av, Valiviita:2008iv, He:2008tn, He:2009mz, He:2009pd, Koyama:2009gd, Xia:2009zzb, Li:2009zs, Zhang:2009qa, Wei:2010cs, Li:2010ak, He:2010im, Li:2011ga, Fu:2011ab, Zhang:2012uu, Zhang:2013lea, Li:2013bya, Geng:2015ara,Yin:2015pqa, Murgia:2016ccp,Wang:2016lxa, Pourtsidou:2016ico, Costa:2016tpb, Sola:2016ecz, Feng:2016djj, Xia:2016vnp, vandeBruck:2016hpz, Sola:2016zeg, Kumar:2017dnp, Sola:2017jbl}. Theoretically speaking, the consideration of such an interaction is helpful in solving the cosmic coincidence problem \cite{Comelli:2003cv,Cai:2004dk,Zhang:2005rg,He:2008tn,He:2009pd}, but actually what is more important is to detect such an interaction using the cosmological observations. The impacts of interactions between dark energy and dark matter on the CMB \cite{He:2009pd,Pourtsidou:2016ico} and LSS \cite{Amendola:2001rc,Bertolami:2007zm,He:2008tn,Koyama:2009gd,Li:2013bya,Pourtsidou:2016ico} have been studied in-depth.




In this paper, we only consider the simplest class of models in the IDE scenario, i.e., the I$\Lambda$CDM models, in which the vacuum energy with $w=-1$ serves as dark energy. In this scenario, the energy conservation equations of the vacuum energy and the cold dark matter satisfy
\begin{equation}\label{conservation1}
\dot{\rho_{\rm de}} = Q,
\end{equation}
\begin{equation}\label{conservation2}
\dot{\rho_{\rm c}} = -3 H \rho_{\rm c}-Q,
\end{equation}
where $\rho_{\rm de}$ and $\rho_{\rm c}$ represent the densities of dark energy (namely, vacuum energy) and cold dark matter, respectively, $H$ is the Hubble parameter, the dot represents the derivative with respect to the cosmic time $t$, and $Q$ is the energy transfer rate. Usually, the form of $Q$ is assumed to be proportional to the density of dark energy or dark matter, i.e., $Q=\beta H \rho_{\rm de}$ or $Q=\beta H \rho_{\rm c}$, where the appearance of $H$ is only for mathematical convenience. In the research area of interacting dark energy, another perspective is to consider $Q=\beta H_{0} \rho_{\rm de}$ or $Q=\beta H_{0} \rho_{\rm c}$ \cite{Valiviita:2008iv}, where the appearance of $H_{0}$ is only for a dimensional consideration. From Eqs.~(\ref{conservation1}) and ~(\ref{conservation2}), it is known that $\beta > 0$ means cold dark matter decaying into dark energy, $\beta < 0$ means dark energy decaying into cold dark matter, and $\beta = 0$ indicates no interaction between vacuum energy and cold dark matter.

Different phenomenological models of I$\Lambda$CDM can be built by assuming different forms of $Q$. In this work, we will collect the popular forms of $Q$ in the current literature and then focus on the impacts of different forms of $Q$ on constraining the total neutrino mass after the Planck 2018 data release. We will consider the four typical forms of $Q$: $Q=\beta H \rho_{\rm de}$, $Q=\beta H \rho_{\rm c}$, $Q=\beta H_{0} \rho_{\rm de}$ and $Q=\beta H_{0} \rho_{\rm c}$. The mass hierarchies of neutrinos are also considered in this work. In addition, we also wish to see whether some hint of the existence of nonzero interaction can be found in these I$\Lambda$CDM models by using the latest observational data.


This paper is organized as follows. In Sec.~\ref{sec2}, we introduce the cosmological observations used in this work and briefly describe the analysis method. In Sec.~\ref{sec3}, we report the constraint results and then make some relevant discussions. The issue of $H_{0}$ tension will also be discussed in this section. Conclusion is given in Sec.~\ref{sec4}.



\section{Method and data}\label{sec2}


In the I$\Lambda$CDM model, there are seven basic cosmological parameters  $\{\omega_b,~\omega_c,~100\theta_{\rm MC},~\tau,~n_s,~\ln (10^{10}A_s),\beta\}$, where $\omega_b$ is the present density of baryons, $\omega_c$ is the present density of cold dark matter, $\theta_{\rm MC}$ is the ratio between the sound horizon to the angular diameter distance at the decoupling epoch, $\tau$ is the Thomson scattering optical depth to reionization, $n_s$ is the scalar spectral index, $A_s$ is the amplitude of primordial scalar perturbation power spectrum, and $\beta$ is the dimensionless coupling constant describing the coupling strength between vacuum energy and dark matter.

For the I$\Lambda$CDM model there is a problem of early-time perturbation instability, because in the IDE models, the cosmological perturbations of dark energy will be divergent in a part of the parameter space, which ruins the IDE cosmology in the perturbation level. The origin of the difficulty is that we know little about the nature of dark energy, so we do not know how to treat the spread of sounds in dark energy fluid which has a negative equation of state (EoS). To overcome the problem of perturbation instability, in 2014, Yun-He Li, Jing-Fei Zhang, and Xin Zhang established an effective theoretical framework for IDE cosmology based on the extended version of the parameterized post-Friedmann (PPF) approach, which can safely calculate the cosmological perturbations in the whole parameter space of an interacting dark energy model. About the extended PPF method, see Refs. \cite{Li:2014eha,Li:2014cee,Li:2015vla,Zhang:2017ize,Feng:2018yew}, and the original PPF method is introduced in Refs. \cite{Hu:2008zd,Fang:2008sn}. In this work, we will employ the extended PPF method \cite{Li:2014eha,Li:2014cee,Li:2015vla,Zhang:2017ize,Feng:2018yew} to calculate the cosmological perturbations in the I$\Lambda$CDM model.

We use the modified version of the publicly available Markov-Chain Monte Carlo package CosmoMC \cite{Lewis:2002ah} to constrain the neutrino mass and other cosmological parameters. We monitor the convergence of the generated MCMC chains by using the Gelman-Rubin parameter $R$ \cite{Gelman:1992zz}, requiring $R-1< 0.01$ for our MCMC chains to be considered as converged. When considering the neutrino mass splitting, we should note the following rules. For the NH case, the neutrino mass spectrum is
\begin{equation}\label{neuNH}
(m_{1},m_{2},m_{3})=(m_{1},\sqrt{m_{1}^{2}+\Delta m_{21}^{2}},\sqrt{m_{1}^{2}+|\Delta m_{31}^{2}|})
\end{equation}
where $m_{1}$ is a free parameter; for the IH case, the neutrino mass spectrum is
\begin{small}
\begin{equation}\label{neuIH}
(m_{1},m_{2},m_{3})=(\sqrt{m_{3}^{2}+|\Delta m_{31}^{2}|},\sqrt{m_{3}^{2}+|\Delta m_{31}^{2}|+\Delta m_{21}^{2}},m_{3})
\end{equation}
\end{small}
where $m_{3}$ is a free parameter; for comparison, the DH case is also considered, in which the neutrino mass spectrum is
\begin{equation}\label{neuDH}
m_{1}=m_{2}=m_{3}=m
\end{equation}
where $m$ is a free parameter. Note also that the input lower bounds of $\sum m_{\nu}$ are 0.06 eV for the NH case, 0.10 ev for the IH case, and 0 eV for the DH case, respectively.

The current observational data sets we used in this paper include CMB, BAO, SN and $H_{0}$. For the CMB data, we use the Planck TT, TE, EE spectra at $\ell \geq 30$, the low-$\ell$ temperature Commander likelihood, and the low-$\ell$ SimAll EE likelihood, from the Planck 2018 data release \cite{Aghanim:2018eyx}. For the BAO data, we consider the measurements from 6dFGS ($z_{\rm eff}=0.106$) \cite{Beutler:2011hx}, SDSS-MGS ($z_{\rm eff}=0.15$) \cite{Ross:2014qpa}, and BOSS DR12 ($z_{\rm eff}=0.38$, 0.51, and 0.61) \cite{Alam:2016hwk}. For the SN data, we employ the latest Pantheon sample, which is comprised of 1048 data points from the Pantheon compilation \cite{Scolnic:2017caz}. For the $H_{0}$ data, we use the 2019 local distance ladder measurement of the Hubble constant $H_{0}=74.03\pm1.42~ \rm{km~ s^{-1} Mpc^{-1}}$ from the Hubble Space Telescope \cite{Riess:2019cxk}. In our analysis, we will use two data combinations, i.e., CMB+BAO+SN and CMB+BAO+SN+$H_{0}$, to constrain the cosmological parameters.

\begin{figure*}[!htp]
\includegraphics[scale=0.55]{hpde.pdf}\ \hspace{1cm}
\includegraphics[scale=0.55]{2hpde.pdf}\ \hspace{1cm}
\centering \caption{\label{fig1} Observational constraints (68.3\% and 95.4\% confidence level) on the I$\Lambda$CDM1+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm de}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+$H_{0}$ (right) data combinations, respectively.}
\end{figure*}
\begin{figure*}[!htp]
\includegraphics[scale=0.55]{hpc.pdf}\ \hspace{1cm}
\includegraphics[scale=0.55]{2hpc.pdf}\ \hspace{1cm}
\centering \caption{\label{fig2} Observational constraints (68.3\% and 95.4\% confidence level) on the I$\Lambda$CDM2+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm c}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+$H_{0}$ (right) data combinations, respectively.}
\end{figure*}
\begin{figure*}[!htp]
\includegraphics[scale=0.55]{h0pde.pdf}\ \hspace{1cm}
\includegraphics[scale=0.55]{2h0pde.pdf}\ \hspace{1cm}
\centering \caption{\label{fig3} Observational constraints (68.3\% and 95.4\% confidence level) on the I$\Lambda$CDM3+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm de}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+$H_{0}$ (right) data combinations, respectively.}
\end{figure*}
\begin{figure*}[!htp]
\includegraphics[scale=0.55]{h0c.pdf}\ \hspace{1cm}
\includegraphics[scale=0.55]{2h0c.pdf}\ \hspace{1cm}
\centering \caption{\label{fig4} Observational constraints (68.3\% and 95.4\% confidence level) on the I$\Lambda$CDM4+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm c}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+$H_{0}$ (right) data combinations, respectively.}
\end{figure*}

\begin{figure*}[!htp]
\includegraphics[scale=0.3]{NH.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{IH.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{DH.pdf}\ \hspace{0.6cm}
\centering \caption{\label{fig5} The one-dimensional posterior distributions for the parameter $H_{0}$ for the $\Lambda$CDM+$\sum m_{\nu}$ (NH, IH, and DH) model and I$\Lambda$CDM1+$\sum m_{\nu}$ (NH, IH, and DH) model with $Q=\beta H \rho_{\rm de}$ by using the CMB+BAO+SN data combination. The result of the local measurement of Hubble constant ($H_{0}=74.03\pm1.42~ \rm{km~ s^{-1} Mpc^{-1}}$) is shown by the hotpink band.}
\end{figure*}

\begin{figure*}[!htp]
\includegraphics[scale=0.3]{NH0.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{IH0.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{DH0.pdf}\ \hspace{0.6cm}
\centering \caption{\label{fig6} The one-dimensional posterior distributions for the parameter $H_{0}$ for the $\Lambda$CDM+$\sum m_{\nu}$ (NH, IH, and DH) model and I$\Lambda$CDM1+$\sum m_{\nu}$ (NH, IH, and DH) model with $Q=\beta H \rho_{\rm de}$ by using the CMB+BAO+SN+$H_{0}$ data combination. The result of the local measurement of Hubble constant ($H_{0}=74.03\pm1.42~ \rm{km~ s^{-1} Mpc^{-1}}$) is shown by the hotpink band.}
\end{figure*}

\begin{figure*}[!htp]
\includegraphics[scale=0.3]{N.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{I.pdf}\ \hspace{0.6cm}
\includegraphics[scale=0.3]{D.pdf}\ \hspace{0.6cm}
\centering \caption{\label{fig7} The one-dimensional posterior distributions for the parameter $H_{0}$ for the I$\Lambda$CDM1+$\sum m_{\nu}$ (NH, IH, and DH) model with $Q=\beta H \rho_{\rm de}$ by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively. The result of the local measurement of Hubble constant ($H_{0}=74.03\pm1.42~ \rm{km~ s^{-1} Mpc^{-1}}$) is shown by the hotpink band.}
\end{figure*}

\begin{table*}\small
\setlength\tabcolsep{2pt}
\renewcommand{\arraystretch}{1.5}
\centering
\caption{\label{tab1}Fitting results for the $\Lambda$CDM+$\sum m_{\nu}$ model by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.}
\begin{tabular}{ccccccccccc}


\hline Data &\multicolumn{3}{c}{CMB+BAO+SN}&&\multicolumn{3}{c}{CMB+BAO+SN+$H_{0}$}\\
           \cline{2-4}\cline{6-8}
       Model  & $\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$ &&$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$\\

\hline
$\Omega_{\rm m}$                         &$0.3126\pm0.0063$
                                         &$0.3150\pm0.0060$
                                         &$0.3097\pm0.0063$
                                         &&$0.3044\pm0.0056$
                                         &$0.3069\pm0.0056$
                                         &$0.3015\pm0.0056$\\

$H_0\,[{\rm km}/{\rm s}/{\rm Mpc}]$      &$67.48\pm0.47$
                                         &$67.26\pm0.45$
                                         &$67.75\pm0.49$
                                         &&$68.11\pm0.43$
                                         &$67.88\pm0.43$
                                         &$68.40\pm0.44$\\


$\sigma_{8}$                             &$0.801^{+0.011}_{-0.008}$
                                         &$0.793^{+0.010}_{-0.008}$
                                         &$0.812^{+0.013}_{-0.008}$
                                         &&$0.801^{+0.009}_{-0.008}$
                                         &$0.792^{+0.009}_{-0.008}$
                                         &$0.813^{+0.010}_{-0.008}$\\

$\sum m_{\nu}$ [eV]                                      &$<0.156$
                                         &$<0.185$
                                         &$<0.123$
                                         &&$<0.125$
                                         &$<0.160$
                                         &$<0.082$\\


\hline
\end{tabular}

\end{table*}

\begin{table*}\small
\setlength\tabcolsep{2pt}
\renewcommand{\arraystretch}{1.5}
\centering
\caption{\label{tab2}Fitting results for the I$\Lambda$CDM1+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm de}$) model by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.}
\begin{tabular}{ccccccccccc}


\hline Data &\multicolumn{3}{c}{CMB+BAO+SN}&&\multicolumn{3}{c}{CMB+BAO+SN+$H_{0}$}\\
           \cline{2-4}\cline{6-8}
       Model  & I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$ &&I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$\\

\hline
$\Omega_{\rm m}$                         &$0.285\pm0.029$
                                         &$0.279\pm0.030$
                                         &$0.292^{+0.029}_{-0.030}$
                                         &&$0.235\pm0.027$
                                         &$0.229\pm0.027$
                                         &$0.243^{+0.028}_{-0.027}$\\

$H_0\,[{\rm km}/{\rm s}/{\rm Mpc}]$      &$68.08\pm0.81$
                                         &$68.08^{+0.83}_{-0.82}$
                                         &$68.14\pm0.83$
                                         &&$69.64^{+0.73}_{-0.72}$
                                         &$69.62^{+0.75}_{-0.73}$
                                         &$69.67^{+0.74}_{-0.75}$\\

$\beta$                                  &$0.10^{+0.10}_{-0.11}$
                                         &$0.13\pm0.11$
                                         &$0.07^{+0.10}_{-0.11}$
                                         &&$0.257^{+0.096}_{-0.097}$
                                         &$0.286\pm0.098$
                                         &$0.215\pm0.099$\\

$\sigma_{8}$                             &$0.870^{+0.058}_{-0.088}$
                                         &$0.884^{+0.061}_{-0.094}$
                                         &$0.859^{+0.056}_{-0.085}$
                                         &&$1.013^{+0.076}_{-0.123}$
                                         &$1.031^{+0.081}_{-0.131}$
                                         &$0.987^{+0.073}_{-0.117}$\\

$\sum m_{\nu}$ [eV]                                      &$<0.189$
                                         &$<0.218$
                                         &$<0.151$
                                         &&$<0.177$
                                         &$<0.209$
                                         &$<0.138$\\


\hline
\end{tabular}

\end{table*}
\begin{table*}\small
\setlength\tabcolsep{2pt}
\renewcommand{\arraystretch}{1.5}
\centering
\caption{\label{tab3}Fitting results for the I$\Lambda$CDM2+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm c}$) model by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.}
\begin{tabular}{ccccccccccc}


\hline Data &\multicolumn{3}{c}{CMB+BAO+SN}&&\multicolumn{3}{c}{CMB+BAO+SN+$H_{0}$}\\
           \cline{2-4}\cline{6-8}
       Model  & I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$ &&I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$\\

\hline
$\Omega_{\rm m}$                         &$0.3085\pm0.0080$
                                         &$0.3092\pm0.0081$
                                         &$0.3077\pm0.0081$
                                         &&$0.2953\pm0.0071$
                                         &$0.2960^{+0.0071}_{-0.0072}$
                                         &$0.2946^{+0.0071}_{-0.0072}$\\

$H_0\,[{\rm km}/{\rm s}/{\rm Mpc}]$      &$67.83\pm0.64$
                                         &$67.74\pm0.65$
                                         &$67.92\pm0.65$
                                         &&$68.92\pm0.60$
                                         &$68.83^{+0.61}_{-0.60}$
                                         &$69.01^{+0.61}_{-0.60}$\\

$\beta$                                  &$0.0011^{+0.0013}_{-0.0012}$
                                         &$0.0014^{+0.0012}_{-0.0013}$
                                         &$0.0005\pm0.0013$
                                         &&$0.0024\pm0.0012$
                                         &$0.0028\pm0.0012$
                                         &$0.0019\pm0.0012$\\

$\sigma_{8}$                             &$0.806^{+0.014}_{-0.012}$
                                         &$0.800^{+0.014}_{-0.012}$
                                         &$0.814^{+0.014}_{-0.013}$
                                         &&$0.815^{+0.013}_{-0.012}$
                                         &$0.809^{+0.013}_{-0.012}$
                                         &$0.824^{+0.014}_{-0.012}$\\

$\sum m_{\nu}$ [eV]                                      &$<0.190$
                                         &$<0.223$
                                         &$<0.149$
                                         &&$<0.170$
                                         &$<0.202$
                                         &$<0.126$\\


\hline
\end{tabular}

\end{table*}
\begin{table*}\small
\setlength\tabcolsep{2pt}
\renewcommand{\arraystretch}{1.5}
\centering
\caption{\label{tab4}Fitting results for the I$\Lambda$CDM3+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm de}$) model by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.}
\begin{tabular}{ccccccccccc}


\hline Data &\multicolumn{3}{c}{CMB+BAO+SN}&&\multicolumn{3}{c}{CMB+BAO+SN+$H_{0}$}\\
           \cline{2-4}\cline{6-8}
       Model  & I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$ &&I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$\\

\hline
$\Omega_{\rm m}$                         &$0.282^{+0.036}_{-0.035}$
                                         &$0.276\pm0.035$
                                         &$0.291\pm0.036$
                                         &&$0.223^{+0.032}_{-0.031}$
                                         &$0.217\pm0.033$
                                         &$0.233\pm0.032$\\

$H_0\,[{\rm km}/{\rm s}/{\rm Mpc}]$      &$68.03\pm0.81$
                                         &$67.96^{+0.79}_{-0.80}$
                                         &$68.10\pm0.83$
                                         &&$69.57\pm0.72$
                                         &$69.50\pm0.74$
                                         &$69.63^{+0.73}_{-0.72}$\\

$\beta$                                  &$0.14\pm0.16$
                                         &$0.18\pm0.16$
                                         &$0.08^{+0.16}_{-0.17}$
                                         &&$0.37^{+0.13}_{-0.14}$
                                         &$0.40\pm0.14$
                                         &$0.31\pm0.14$\\

$\sigma_{8}$                             &$0.886^{+0.069}_{-0.118}$
                                         &$0.899^{+0.072}_{-0.122}$
                                         &$0.868^{+0.068}_{-0.113}$
                                         &&$1.072^{+0.095}_{-0.172}$
                                         &$1.10^{+0.10}_{-0.19}$
                                         &$1.036^{+0.091}_{-0.156}$\\

$\sum m_{\nu}$ [eV]                                      &$<0.179$
                                         &$<0.208$
                                         &$<0.140$
                                         &&$<0.166$
                                         &$<0.198$
                                         &$<0.128$\\


\hline
\end{tabular}

\end{table*}
\begin{table*}\small
\setlength\tabcolsep{2pt}
\renewcommand{\arraystretch}{1.5}
\centering
\caption{\label{tab5}Fitting results for the I$\Lambda$CDM4+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm c}$) model by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.}
\begin{tabular}{ccccccccccc}


\hline Data &\multicolumn{3}{c}{CMB+BAO+SN}&&\multicolumn{3}{c}{CMB+BAO+SN+$H_{0}$}\\
           \cline{2-4}\cline{6-8}
       Model  & I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$ &&I$\Lambda$CDM+$\sum m_{\nu}^{\rm NH}$&I$\Lambda$CDM+$\sum m_{\nu}^{\rm IH}$ &I$\Lambda$CDM+$\sum m_{\nu}^{\rm DH}$\\

\hline
$\Omega_{\rm m}$                         &$0.299\pm0.016$
                                         &$0.297\pm0.016$
                                         &$0.302^{+0.016}_{-0.017}$
                                         &&$0.272\pm0.013$
                                         &$0.269\pm0.013$
                                         &$0.275\pm0.013$\\

$H_0\,[{\rm km}/{\rm s}/{\rm Mpc}]$      &$68.05\pm0.80$
                                         &$68.07^{+0.83}_{-0.82}$
                                         &$68.07\pm0.81$
                                         &&$69.58\pm0.72$
                                         &$69.58\pm0.73$
                                         &$69.58^{+0.71}_{-0.72}$\\

$\beta$                                  &$0.043\pm0.047$
                                         &$0.058^{+0.047}_{-0.048}$
                                         &$0.024^{+0.047}_{-0.048}$
                                         &&$0.111\pm0.043$
                                         &$0.128\pm0.043$
                                         &$0.092\pm0.044$\\

$\sigma_{8}$                             &$0.814^{+0.019}_{-0.021}$
                                         &$0.812\pm0.019$
                                         &$0.820^{+0.019}_{-0.020}$
                                         &&$0.840\pm0.019$
                                         &$0.837\pm0.019$
                                         &$0.845\pm0.019$\\

$\sum m_{\nu}$ [eV]                                      &$<0.202$
                                         &$<0.235$
                                         &$<0.156$
                                         &&$<0.202$
                                         &$<0.239$
                                         &$<0.162$\\


\hline
\end{tabular}

\end{table*}

\section{Results and discussion}\label{sec3}

In this section, we report the constraint results of cosmological parameters for these I$\Lambda$CDM+$\sum m_{\nu}$ models. The fitting results are listed in Table \ref{tab1} for the $\Lambda$CDM+$\sum m_{\nu}$ model and Tables \ref{tab2}--\ref{tab5} and Figs.~\ref{fig1}--\ref{fig4} for the four I$\Lambda$CDM+$\sum m_{\nu}$ models. For convenience, the I$\Lambda$CDM models with the interaction terms $Q=\beta H \rho_{\rm de}$, $Q=\beta H \rho_{\rm c}$, $Q=\beta H_{0} \rho_{\rm de}$ and $Q=\beta H_{0} \rho_{\rm c}$ are denoted as ``I$\Lambda$CDM1", ``I$\Lambda$CDM2", ``I$\Lambda$CDM3", and ``I$\Lambda$CDM4", respectively. In these tables, we show the best fit values with $\pm 1 \sigma$ errors of the cosmological parameters, but for the total neutrino mass $\sum m_{\nu}$, which cannot be well constrained, the $2\sigma$ upper limits are given.


\subsection{Neutrino mass}\label{seca}


Firstly, we use the CMB+BAO+SN data combination to constrain these models. In the $\Lambda$CDM+$\sum m_{\nu}$ model, we obtain $\sum m_{\nu}<0.156$ eV for the NH case, $\sum m_{\nu}<0.185$ eV for the IH case, and $\sum m_{\nu}<0.123$ eV for the DH case, as shown Table \ref{tab1}. In the I$\Lambda$CDM1+$\sum m_{\nu}$ model, the constraint results are $\sum m_{\nu}<0.187$ eV for the NH case, $\sum m_{\nu}<0.218$ eV for the IH case, and $\sum m_{\nu}<0.151$ eV for the DH case (see Table \ref{tab2}); in the I$\Lambda$CDM2+$\sum m_{\nu}$ model, the results are $\sum m_{\nu}<0.190$ eV for the NH case, $\sum m_{\nu}<0.223$ eV for the IH case, and $\sum m_{\nu}<0.149$ eV for the DH case (see Table \ref{tab3}); in the I$\Lambda$CDM3+$\sum m_{\nu}$ model, we get $\sum m_{\nu}<0.179$ eV for the NH case, $\sum m_{\nu}<0.208$ eV for the IH case, and $\sum m_{\nu}<0.140$ eV for the DH case (see Table \ref{tab4}); in the I$\Lambda$CDM4+$\sum m_{\nu}$ model, the constraint results become $\sum m_{\nu}<0.202$ eV for the NH case, $\sum m_{\nu}<0.235$ eV for the IH case, and $\sum m_{\nu}<0.156$ eV for the DH case (see Table \ref{tab5}). We find that, the constraint results of $\sum m_{\nu}$ are looser in the four I$\Lambda$CDM+$\sum m_{\nu}$ models than those in the $\Lambda$CDM+$\sum m_{\nu}$ model. When considering the three mass hierarchies, we find that the constraint results of $\sum m_{\nu}$ are tightest in the DH case and loosest in the IH case (see the left panels in Figs.~\ref{fig1}--\ref{fig4}).

Then, we consider the data combination involving the latest local measurement of the Hubble constant $H_{0}$ to constrain these models. By using the CMB+BAO+SN+$H_{0}$ data combination, we find that the constraint results of $\sum m_{\nu}$ are looser in the four I$\Lambda$CDM+$\sum m_{\nu}$ models than those in the $\Lambda$CDM+$\sum m_{\nu}$ model, and when considering the three mass hierarchies, the constraint results of $\sum m_{\nu}$ are tightest in the DH case and loosest in the IH case. These conclusions are consistent with the case using the CMB+BAO+SN data combination. Additionally, we also find that the constraints on $\sum m_{\nu}$ become slightly tighter for using CMB+BAO+SN+$H_{0}$ than CMB+BAO+SN.

\subsection{Coupling parameter}\label{secb}


In this subsection, we discuss the fitting results of the coupling parameter $\beta$ in these four I$\Lambda$CDM+$\sum m_{\nu}$ models by using the CMB+BAO+SN and CMB+BAO+SN+$H_{0}$ data combinations, respectively.

First, we constrain the I$\Lambda$CDM1+$\sum m_{\nu}$ model (see Table \ref{tab2}) using the CMB+BAO+SN data combination, and we obtain $\beta=0.10^{+0.10}_{-0.11}$ for the NH case, $\beta=0.13\pm0.11$ for the IH case, and $\beta=0.07^{+0.10}_{-0.11}$ for the DH case. It is shown that a positive value of $\beta$ is favored and $\beta>0$ is at the $0.91\sigma$, $1.18\sigma$, and $0.64\sigma$ levels for the three mass hierarchy cases, respectively. Furthermore, we constrain this model by using the CMB+BAO+SN+$H_{0}$ data combination, and we obtain $\beta=0.257^{+0.096}_{-0.097}$ for the NH case, $\beta=0.286\pm0.098$ for the IH case, and $\beta=0.215\pm0.099$ for the DH case. Now, $\beta>0$ is obtained at the $2.65\sigma$, $2.92\sigma$, and $2.17\sigma$ levels, respectively. This indicates that cold dark matter decaying into dark energy is favored when using the CMB+BAO+SN+$H_{0}$ data combination.

For the I$\Lambda$CDM2+$\sum m_{\nu}$ model (see Table \ref{tab3}), we obtain $\beta=0.0011^{+0.0013}_{-0.0012}$ for the NH case, $\beta=0.0014^{+0.0012}_{-0.0013}$ for the IH case, and $\beta=0.0005\pm0.0013$ for the DH case, by using the CMB+BAO+SN data combination. Thus, $\beta>0$ is favored at the $0.92\sigma$, $1.08\sigma$, and $0.38\sigma$ levels, respectively. When using the CMB+BAO+SN+$H_{0}$ data combination, we obtain $\beta=0.0024\pm0.0012$ for the NH case, $\beta=0.0028\pm0.0012$ for the IH case, and $\beta=0.0019\pm0.0012$ for the DH case, which indicates that a positive value of $\beta$ can be detected at the $2.00\sigma$, $2.33\sigma$, and $1.58\sigma$ levels, respectively.

As for the I$\Lambda$CDM3+$\sum m_{\nu}$ model (see Table \ref{tab4}), we obtain $\beta=0.14\pm0.16$ for the NH case, $\beta=0.18\pm0.16$ for the IH case, and $\beta=0.08^{+0.16}_{-0.17}$ for the DH case, by using the CMB+BAO+SN data combination. Therefore, the positive values of $\beta$ are favored and $\beta>0$ is preferred at the $0.88\sigma$, $1.13\sigma$, and $0.50\sigma$ levels, respectively. When using the CMB+BAO+SN+$H_{0}$ data combination, we obtain $\beta=0.37^{+0.13}_{-0.14}$ for the NH case, $\beta=0.40\pm0.14$ for the IH case, and $\beta=0.31\pm0.14$ for the DH case, respectively. And $\beta>0$ is detected at the $2.64\sigma$, $2.90\sigma$, and $2.21\sigma$ levels, respectively, which indicates cold dark matter decaying into dark energy.

Finally, we show the constraint results of I$\Lambda$CDM4+$\sum m_{\nu}$ model (see Table \ref{tab5}). We obtain $\beta=0.043\pm0.047$ for the NH case, $\beta=0.058^{+0.047}_{-0.048}$ for the IH case, and $\beta=0.024^{+0.047}_{-0.048}$ for the DH case, by using the CMB+BAO+SN data combination. So, a positive value of $\beta$ is favored and $\beta>0$ is at the $0.91\sigma$, $1.21\sigma$, and $0.50\sigma$ levels, respectively. When using the CMB+BAO+SN+$H_{0}$ data combination, we obtain $\beta=0.111\pm0.043$ for the NH case, $\beta=0.128\pm0.043$ for the IH case, and $\beta=0.092\pm0.044$ for the DH case. Now, $\beta>0$ is preferred at the $2.58\sigma$, $2.98\sigma$, and $2.09\sigma$ levels, respectively. The conclusion is the same as the above three cases, i.e., cold dark matter decaying into dark energy is supported by the CMB+BAO+SN+$H_{0}$ data combination.


In summary, for all the I$\Lambda$CDM+$\sum m_{\nu}$ models considered in this paper, the values of $\beta$ are greater by using the CMB+BAO+SN+$H_{0}$ data combination than using CMB+BAO+SN data combination. We can also intuitively obtain this conclusion by comparing the left and right panels of Figs.~\ref{fig1}--\ref{fig4}. Additionally, when using CMB+BAO+SN+$H_{0}$ data combination, $\beta>0$ is favored at more than 1$\sigma$ level in all the I$\Lambda$CDM+$\sum m_{\nu}$ models, which indicates that cold dark matter decaying into dark energy is supported in these models.



\subsection{The $H_{0}$ tension}\label{secc}

In this subsection, we discuss the issue of $H_{0}$ tension between the Planck observation of the CMB power spectra and the local measurement based on the method of distance ladder. The detailed fitting results are given in Tables \ref{tab1}--\ref{tab5} and Figs.~\ref{fig5}--\ref{fig7}.

In Table \ref{tab1}, we show the constraint results of the $\Lambda$CDM+$\sum m_{\nu}$ model by using the CMB+BAO+SN data combination. We obtain $H_{0}=67.48\pm0.47~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=67.26\pm0.45~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=67.75\pm0.49~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case, which are $4.38 \sigma$ $4.54 \sigma$ and $4.18 \sigma$ lower than the direct measurement of the Hubble constant ($H_{0}=74.03\pm1.42~ \rm{km~ s^{-1} Mpc^{-1}}$). So, we investigate whether the $H_{0}$ tension can be solved or relieved in the interacting dark energy scenario. In Tables \ref{tab2}--\ref{tab5}, we show the constraint results of the I$\Lambda$CDM1+$\sum m_{\nu}$, I$\Lambda$CDM2+$\sum m_{\nu}$, I$\Lambda$CDM3+$\sum m_{\nu}$, and I$\Lambda$CDM4+$\sum m_{\nu}$ models from the CMB+BAO+SN data combination. In the I$\Lambda$CDM1+$\sum m_{\nu}$ model, we obtain $H_{0}=68.08\pm0.81~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=68.08^{+0.83}_{-0.82}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=68.14\pm0.83~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM2+$\sum m_{\nu}$ model, we obtain $H_{0}=67.83\pm0.64~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=67.74\pm0.65~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=67.92\pm0.65~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM3+$\sum m_{\nu}$ model, we obtain $H_{0}=68.03\pm0.81~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=67.96^{+0.79}_{-0.80}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=68.10\pm0.83~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM4+$\sum m_{\nu}$ model, we obtain $H_{0}=68.05\pm0.80~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=68.07^{+0.83}_{-0.82}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=68.07\pm0.81~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case. For these cases, the tensions with the Hubble constant direct measurement are at the $3.64 \sigma$ level, $3.62 \sigma$ level, $3.58 \sigma$ level, $3.98 \sigma$ level, $4.03 \sigma$ level, $3.91 \sigma$ level, $3.67 \sigma$ level, $3.74 \sigma$ level, $3.61 \sigma$ level, $3.67 \sigma$ level, $3.62 \sigma$ level, and $3.65 \sigma$ level, respectively.

Then, we show the constraint results of these models by using the CMB+BAO+SN+$H_{0}$ data combination (see Tables \ref{tab1}--\ref{tab5}). In the $\Lambda$CDM+$\sum m_{\nu}$ model, we obtain $H_{0}=68.11\pm0.43~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=67.88\pm0.43~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=68.40\pm0.44~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case, which indicates that the tensions with the Hubble constant direct measurement are at the $3.99 \sigma$ level $4.14 \sigma$ level and $3.79 \sigma$ level, respectively. In the I$\Lambda$CDM1+$\sum m_{\nu}$ model, we obtain $H_{0}=69.64^{+0.73}_{-0.72}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=69.62^{+0.75}_{-0.73}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=69.67^{+0.74}_{-0.75}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM2+$\sum m_{\nu}$ model, we obtain $H_{0}=68.92\pm0.60~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=68.83^{+0.61}_{-0.60}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=69.01^{+0.61}_{-0.60}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM3+$\sum m_{\nu}$ model, we obtain $H_{0}=69.57\pm0.72~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=69.50\pm0.74~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=69.63^{+0.73}_{-0.72}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case; in the I$\Lambda$CDM4+$\sum m_{\nu}$ model, we obtain $H_{0}=69.58\pm0.72~ \rm{km~ s^{-1} Mpc^{-1}}$ for the NH case, $H_{0}=69.58\pm0.73~ \rm{km~ s^{-1} Mpc^{-1}}$ for the IH case, and $H_{0}=69.58^{+0.71}_{-0.72}~ \rm{km~ s^{-1} Mpc^{-1}}$ for the DH case. The tensions with the Hubble constant direct measurement are at the $2.75 \sigma$ level, $2.75 \sigma$ level, $2.72 \sigma$ level, $3.31 \sigma$ level, $3.36 \sigma$ level, $3.25 \sigma$ level, $2.80 \sigma$ level, $2.83 \sigma$ level, $2.76 \sigma$ level, $2.80 \sigma$ level, $2.79 \sigma$ level, and $2.80 \sigma$ level, respectively.

From the above constraint results, we find that compared with the $\Lambda$CDM+$\sum m_{\nu}$ model, the $H_{0}$ tension can indeed be relieved in the I$\Lambda$CDM+$\sum m_{\nu}$ model. From Figs.~\ref{fig5}--\ref{fig6} we can clearly see that for whichever neutrino mass hierarchy case, the fitting values of $H_{0}$ in the I$\Lambda$CDM+$\sum m_{\nu}$ models (here, we take I$\Lambda$CDM1+$\sum m_{\nu}$ with $Q=\beta H \rho_{\rm de}$ as an example) are always much larger than those in the $\Lambda$CDM+$\sum m_{\nu}$ model. We also find that, the CMB+BAO+SN+$H_{0}$ data combination (about $2.7-3.4\sigma$ level) is slightly more effective in relieving the $H_{0}$ tension than the CMB+BAO+SN data combination (about $3.6-4.0 \sigma$ level), due to the employment of the $H_{0}$ prior in the data combination. To visually display the result, we also take the I$\Lambda$CDM1+$\sum m_{\nu}$ model as an example to give this result in Fig.~\ref{fig7}. From these figures, we can clearly see that for whichever hierarchy of the neutrino mass spectrum, the values of $H_{0}$ are always much larger when adding the $H_{0}$ data in a cosmological fit. Certainly, the $H_{0}$ tension problem only can be alleviated to some extent in these cases, but cannot be truly solved. For the issue of $H_{0}$ tension, further exploration is needed.

\section{Conclusion}\label{sec4}

In this work, we have investigated the constraints on the total neutrino mass in the scenario of vacuum energy interacting with cold dark matter by using the latest cosmological observations. We consider four typical models, i.e., I$\Lambda$CDM1+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm de}$) model, I$\Lambda$CDM2+$\sum m_{\nu}$ ($Q=\beta H \rho_{\rm c}$) model, I$\Lambda$CDM3+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm de}$) model, and I$\Lambda$CDM4+$\sum m_{\nu}$ ($Q=\beta H_{0} \rho_{\rm c}$) model. For the three-generation neutrinos, we consider the NH, IH, and DH cases. We employ the extended version of the parameterized post-Friedmann approach to calculate the perturbation of dark energy in the IDE cosmology. We use the Planck 2018 CMB data, the BAO measurements, the SN data of Pantheon compilation, and the local measurement of the Hubble constant $H_{0}$ from the Hubble Space Telescope to constrain these models.

We find that, compared with the $\Lambda$CDM+$\sum m_{\nu}$ model, these four I$\Lambda$CDM+$\sum m_{\nu}$ models can provide a much looser constraint on the total neutrino mass $\sum m_{\nu}$. When considering the three mass hierarchies, the upper limits on $\sum m_{\nu}$ are smallest in the DH case and largest in the IH case. We also find that, the constraints on $\sum m_{\nu}$ are slightly tighter by using CMB+BAO+SN+$H_{0}$ than using CMB+BAO+SN. In addition, in all the I$\Lambda$CDM+$\sum m_{\nu}$ models considered in this paper, the fit values of $\beta$ are greater using the CMB+BAO+SN+$H_{0}$ data combination than using the CMB+BAO+SN data combination, and $\beta>0$ is favored at more than 1$\sigma$ level in all the I$\Lambda$CDM+$\sum m_{\nu}$ models when using the CMB+BAO+SN+$H_{0}$ data combination, implying the preference of cold dark matter decaying into dark energy. In addition, we also find that, compared with the $\Lambda$CDM+$\sum m_{\nu}$ model, the $H_{0}$ tension can be alleviated to some extent in the I$\Lambda$CDM+$\sum m_{\nu}$ models.


\begin{acknowledgments}
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11975072, 11875102, 11835009, and 11690021), the Liaoning Revitalization Talents Program (Grant No. XLYC1905011), the Fundamental Research Funds for the Central Universities (Grant No. N2005030), and the Top-Notch Young Talents Program of China (W02070050).


\end{acknowledgments}



