\section{Introduction}

Long before the discovery of the Higgs boson in 2012, vacuum decay had been used as a theoretical tool to constrain the masses of the Standard Model (SM) particles~\cite{isidori2001metastability, casas1995improved, espinosa1995improved, altarelli1994lower, sher1993precise}. The critical reason for vacuum instability in the SM is that with loop corrections resummed via the renormalization group, the quartic coefficient of the effective potential becomes negative as the renormalization scale approaches about $10^{10}$ \ensuremath{\,\text{GeV}}\xspace \cite{buttazzo2013investigating}. This means that the effective potential must be ultraviolet completed such that it is bounded from below and a second, deeper minimum arises~\cite{sher1989electroweak, krive1976vacuum}. Fortunately, detailed calculation shows that the tunneling time of the desired electroweak symmetry breaking~(EW-breaking) vacuum into this deeper vacuum is longer than the age of the universe~\cite{andreassen2018scale}.

In beyond the SM (BSM) scenarios, like supersymmetric models, extra scalar fields are usually introduced, which can get non-zero vacuum expectation values and destabilise the EW-breaking vacuum. If another lower energy minimum exists, the EW-breaking vacuum could transition into such a vacuum via quantum tunneling or thermal excitation. Since the EW-breaking vacuum is compatible with experimental results, we naturally demand that the transition time is longer than the age of the universe, which constrains the model to describe Nature. In simple models such constraints for stability at zero temperature can be derived analytically
~\cite{ferreira2004stability, barroso2007neutral, barroso2013metastability}. To deal with multi-fields or high temperature effects, several numerical tools were developed ~\cite{camargo2013vevacious, masoumi2017efficient, sato2021simplebounce}. 
Both analytic and numerical calculations assume that the set of considered parameters ensure the realization of the EW-breaking vacuum during the evolution of the universe. 

This assumption, however, is not always correct from the perspective of finite temperature field theory. Previously, Cline \emph{et al.}~\cite{cline1999electroweak} investigated whether the transition process from a color-breaking vacuum to the EW-breaking vacuum can happen and found that the nucleation rate is many orders of magnitude too small even if the parameters are optimized. Subsequently, Sebastian \emph{et al.}~\cite{baum2021nucleation} stressed the importance of nucleation temperature in determining whether the tunneling process can happen and also pointed out that the two-step phase transition will cease due to lack of nucleation temperature and hence the vacuum will be trapped at the trivial minimum. Meanwhile, Thomas \emph{et al.} also found that in the 2HDM~\cite{2hdmtrapped} or N2HDM~\cite{biekotter2021fate} the transition from the electroweak symmetry restored (EW-restored) minimum to the desired EW-breaking vacuum may not be achieved. These studies strongly indicate that when considering the vacuum stability problem, the thermal history may play an important role in addition to traditional analysis. Besides, there is a case that the non-desired vacuum is always the global minimum during the evolution of the universe, which is also need to be excluded. 

In this letter, we study the vacuum stability problem incorporating cosmological evolution, and show that it can provide a new limit on new physics models. By comparing results in the singlet extension of the SM (xSM), we reveal that checking the evolution of vacuum from high temperature to low temperature is more reasonable than that from bottom up. Additionally, our method could be computationally economic for some models. 

\section{Vacuum stability analysis with cosmological history}

\begin{figure*}[!ht]
\centering 
\includegraphics[width=0.49\textwidth]{figs/No_Tc1.pdf}
\includegraphics[width=0.49\textwidth]{figs/No_Tc2.pdf}
\caption{Situations in which the universe always stays in the non-desired vacuum during the thermal history. The solid green ball indicates the vacuum where the universe stays at the corresponding temperature, while the hollow gray ball stands for the SM EW-breaking vacuum at zero-temperature. The arrow with cross means that the transition can not take place. They can be realized in the xSM using parameters shown by the labels. The scaled effective potential ($\Delta V = [V(h,s)-V(0,0)]$) is along the direct path between the two minimums $(0,v_s)$ and $(v_h,0)$.} 
\label{fig:bk1}
\end{figure*}

Traditionally, focusing on the effective potential of today's universe, the EW-braking vacuum of new physics models can be divided into two categories, as illustrated by the blue lines in Fig.\ref{fig:bk1}. In the left frame, the desired EW-breaking vacuum of zero-temperature is indicated by gray circle, which is a global minimum. Thus, without considering thermal effect, this desired vacuum is completely safe. A meta-sable desired vacuum is shown in the right frame. If the tunneling time from it to the global minimum, indicated by a green dot, is longer than the age of the universe, we can still retain this model. 

However, we find that this analysis is not sufficient. Fig.~\ref{fig:bk1} presents two situations in which today's desired EW-breaking vacuum is stable or long-lived but the universe is trapped in a non-desired vacuum. In the early hot universe, the global minimum (green dots) of effective potential (red lines) is EW-restored. With decreasing temperature a new minimum appears in the potential (orange lines). In the left frame, the new minimum turns into the desired EW-breaking vacuum and global minimum at zero-temperature, but the transition from the origin to it may not happen. In the right frame, the new minimum also evolves into the desired EW-breaking vacuum but always has higher free energy than the origin, so there is no transition at all. Therefore, the correct way to checking the vacuum of today's universe to follow the thermal history, starting from high temperature, calculating every possible transition
and tracing the minimums of potential as they evolve to zero-temperature. 

In Ref.~\cite{camargo2013vevacious} thermal transitions are taken into account in the study of vacuum stability, but in the traditional view. For the right frame of Fig.~\ref{fig:bk1}, with increasing temperature, the transition probability of the desired EW-breaking vacuum increases, which makes it a short-lived vacuum. Thus, this method can also correctly exclude this situation. However, if there are more than one minima that the desired EW-breaking vacuum can transition into at finite temperature, this method becomes invalid. Meanwhile, as shown later, the calculation of transition probability is fairly time consuming, while our new method only needs to trace the free energies of the minima for this situation, which precedes calculating transition probability. 


An important and time-consuming part to obtain our new limit is determining the occurrence of the thermal transition. Previous studies~\cite{baum2021nucleation, 2hdmtrapped, biekotter2021fate} of the situation shown in the left frame of Fig.~\ref{fig:bk1} adopted the existence of nucleation temperature as a criterion. 
The transition process is identified with the nucleation process, in which the surface tension of the vacuum bubble wall and the energy of the bubble expansion compete against each other. As the universe cools, the thermal transition rate increases dramatically. The transition can be regarded as happening once the probability to nucleate one critical bubble in the Hubble volume is of order one, at the so-called nucleation temperature $T_n$, which can be roughly estimated from $S(T_{n})/T_{n} \sim 140$ \ensuremath{\,\text{GeV}}\xspace. However, the nucleation temperature may be ill-defined for slow or supercooled transitions~\cite{cai2017gravitational, Athron:2022mmm}.
Here we calculate the transition probability following \texttt{Vevacious}~\cite{camargo2014constraining}.



The transition rate, per unit time per unit volume, of the meta-stable vacuum at finite temperature can be expressed as~\cite{linde1983decay} 
\begin{equation}
    \frac{\Gamma}{V} = B(T)e^{-S_{3}(T)/T} ,
\end{equation}
where $S_{3}$ denotes the Euclidean bounce action over three dimensions and $B(T)$ is a dimensionful parameter, which is often estimated as $T^4$.
The probability $P(T_i, T_f)$ of non-transitioning (surviving) during the period from an initial temperature $T_i$ to a final temperature $T_f$ can be obtained by trading time $t$ with temperature $T$ in the differential equation
\begin{equation}
    P(t+{\rm d}t) = P(t)(1-\Gamma {\rm d}t).
\end{equation}
As a result, we obtain 
\small
\begin{equation}
    P(T_i, T_f) = \mathrm{exp}\left[-\int^{T_{f}}_{T_{i}}\frac{{\rm d}t}{{\rm d}T}V(T)B(T)e^{-S_{3}(T)/T}{\rm d}T\right],
\end{equation}
\normalsize
where $V(T)$ is the observable volume. 
Assuming the universe is radiation dominated during the evolution from $T_i$ to $T_f$, entropy is approximately conserved and $S_{3}(T)$ increase monotonically with $T$ increasing. The upper bound of $P(T_i, T_f)$ is then expressed as 
\small
\begin{eqnarray} \label{eq:vevacious_P}
    P(T_i = T, T_f = 0) \leq  \mathrm{exp}(-1.581 \times 10^{106} e^{-S_{3}/T}/S_{3}).
\end{eqnarray}
\normalsize
A temperature $T_{opt}$ to maximize the right-hand side of Eq.~(\ref{eq:vevacious_P}) can be found by fitting the thermal action to $b/(T-T_c)^2$, where $T_{c}$ is the critical temperature at which the two minima have degenerate free energy. \texttt{Cosmotranstions}~\cite{wainwright2012cosmotransitions} is used to calculate the action. Then we can set a threshold, $20\%$ in this work, on the maximal surviving probability to judge whether the tunneling process occurs or not. The results are not particularly sensitive to the value of the threshold. 

\section{Example Toy Model}

The situations displayed in Fig.\ref{fig:bk1} can be realized in a very simple model, the $Z_{2}$ symmetric real scalar extension of the SM (xSM).  In this section we use this model to demonstrate the above generic discussion. 

The tree-level effective potential of the xSM can be written in the form of
\begin{equation}\label{eq:xsm_tree}
     V_{0}(h,s) = -\frac{\mu_{h}^{2}}{2}h^2 + \frac{\lambda_{h}}{4}h^4 - \frac{\mu_{s}^2}{2}s^2 + \frac{\lambda_{s}}{4}s^4 + \frac{\lambda_{hs}}{4}h^2s^2,
\end{equation}
where $h$ and $s$ are background field configurations for the Higgs and extra scalar fields, respectively. We build the one-loop effective potential $V(h,s;T)$ in Landau gauge using the on-shell-like renormalization scheme and the Parwani method for daisy resummation~\cite{parwani1992resummation}. See Ref.~\cite{athron2022arbitrary} for detailed expressions and uncertainties induced by these choices. We set $m_h=125~\ensuremath{\,\text{GeV}}\xspace$ and $v_{\rm EW}=246~\ensuremath{\,\text{GeV}}\xspace$, and fix the Lagrangian parameters by tadpole conditions. The remaining free parameters are chosen to be $\lambda_{s}$, $\lambda_{hs}$ and the singlet mass $m_s$. 


Fig.\ref{fig:bk1} was, in fact, generated using two benchmark points of the xSM, which have $m_{s} = 234$ and $220~\ensuremath{\,\text{GeV}}\xspace$ respectively, together with fixed $\lambda_{hs}=2.0$ and $\lambda_{s} = 0.2$. To visualize the two-dimensional potential in one-dimension, we only show the potential along the straight line between the two minima in the $(h,s)$ plane. 

For the benchmark point in the left frame of Fig.\ref{fig:bk1}, when $T>267.9~\ensuremath{\,\text{GeV}}\xspace$ there is only one minimum in the potential located at the origin. Below that temperature 
the singlet field develops a non-zero expectation value smoothly increasing from $0$ to $168.3~\ensuremath{\,\text{GeV}}\xspace$ until zero-temperature.
A new minimum of $(v_h=54.7~\ensuremath{\,\text{GeV}}\xspace,v_s=0)$ appears at $T=147.6~\ensuremath{\,\text{GeV}}\xspace$ and evolves into the desired EW-breaking vacuum $(v_h=246~\ensuremath{\,\text{GeV}}\xspace,v_s=0)$ as the temperature goes to zero. The two minima are degenerated at $T_c=120.9~\ensuremath{\,\text{GeV}}\xspace$, so after that the universe may transition from $(0,v_s)$ to $(v_h,0)$. We find, however, that the maximal probability of non-transitioning $P(T_{opt},0)$ is approximately 100\%. Therefore, it is unlikely that the universe will evolve into the desired EW-breaking vacuum for this benchmark point. Just looking at the zero-temperature potential, however, we see a stable EW-breaking vacuum which has no stability problem. This demonstrates the importance of calculating transitions in the cosmological history.   

The benchmark point of the right frame of Fig.\ref{fig:bk1} has similar minimum structure as above. The difference is that there is no critical temperature between the two minima $(0,v_s)$ and $(v_h,0)$, i.e. the free energy of $(0,v_s)$ is larger than that of $(v_h, 0)$ throughout. 
As a result, the universe always stays in the EW-restored minimum. In the zero-temperature potential, the desired EW-breaking vacuum is meta-stable with a lifetime much longer than the universe's age, so it is acceptable in the traditional method. For this kind of situation, checking the history to exclude it is time-saving, because it only need to trace the position and free energy of minima, no transition calculation is needed. 

Thus, it is necessary to modify the procedure for accessing vacuum stability.  Below, we assume that this is the case for even more complicated models, and the universe is located at the EW-restored vacuum at some high temperature.  Then, we can assess vacuum stability in following steps staring from the EW-restored vacuum.
\begin{itemize}
\item[(1)] Map out the position and free energy of the minima of the effective potential as the function of temperature (e.g. using numerical tools~\cite{wainwright2012cosmotransitions,Athron:2020sbe}).
\item[(2)] As the temperature is lowered,
whenever degenerate minima occur, i.e. if there is a critical temperature $T_{c}$, calculate the transition probability and transition temperature below $T_c$. 
\item[(3)] If the transition probability is larger than a certain threshold, set the vacuum of the universe to be the new minimum, and go to Step (2) until $T=0$. 
\item[(4)] Check whether the vacuum of the universe at $T=0$ is the desired EW-breaking vacuum. If not, this point is excluded. If yes, it is allowed when the vacuum is stable or long-lived, and is excluded when the vacuum is short-lived.
\end{itemize} 

\begin{figure}[t]
\centering 
\includegraphics[width=0.5\textwidth]{figs/scan.pdf}
\caption{Classifications of vacuum structure in parameter space of the xSM. In the red region, the desired EW-breaking vacuum is meta-stable with a life-time shorter than the age of the universe. In the orange region, the desired vacuum is long-lived but always has higher free energy than the EW-restored vacuum in the cosmological history. The desired vacuum in the yellow region is stable but the early universe can not transition to it. Only in the green region the universe can safely reach to the stable desired SM vacuum. 
Therefore, the dotted and solid curves represent respectively limits from checking minima in the zero-temperature potential and the evolution history of the universe. }
\label{fig:bk3}
\end{figure}

Following this procedure, we explore the parameter space of the xSM. For instance, the result of fixing  $\lambda_{s} = 0.2$ is displayed in Fig.\ref{fig:bk3}, where we compare the limit from only checking zero-temperature potential and the limit from considering  evolution history of the universe.  The parameter space is split by the dashed curve, at which the two minima $(v_h,0)$ and $(0,v_s)$ are degenerate at zero-temperature, and the parameters satisfy 
\begin{equation}
    \lambda_{hs} = \frac{v_{h}^2}{2} \left(m_{s}^2 + \mu_{h} \sqrt{\frac{\lambda_{s}}{\lambda_{h}}} \right)
\end{equation}
in the on-shell-like renormalization scheme. The yellow and orange regions correspond the situations described in the left and right frames of Fig.~\ref{fig:bk1}, respectively.

Above the dashed curve, the EW-restored vacuum is the  global minimum at zero-temperature, as well as at finite temperature. The energy gap between the two vacua increases with $\lambda_{hs}$ increasing. Without considering the cosmological history, the desired EW-breaking vacuum is meta-stable, but can be long-lived, such as in the orange region of Fig.~\ref{fig:bk3}. Therefore, the transitional limit is the upper bound of the orange region. Thermal history tells us that the universe starts from the EW-restored vacuum and remains there as it is global minimum throughout. So the orange region is actually ruled out. 

Below the dashed curve, the desired EW-breaking vacuum at zero-temperature is the global minimum and thus stable. In the yellow region, the energy gap between the vacua is not large enough to achieve the transition that breaks electroweak symmetry.  Only in the green region, the early universe transits from $(0,v_s)$ or $(0,0)$ to $(v_h,0)$ and then evolves to $(246~\ensuremath{\,\text{GeV}}\xspace,0)$ at $T=0$. The $\lambda_{hs}$ of the new limit~(pink solid curve) is about 0.2 and 0.9 lower than that of the dashed curve and the old limit~(blue dotted curve).

We see that the traditional method accepts some regions of meta-stable EW-breaking vacuum, but checking the cosmological history proves that incorrect. The exclusion of such regions is computationally fast, because it only needs to trace the minima of the potential with temperature. On the other hard, it is more time-consuming to exclude the yellow region, as it involves calculation of transition probability at finite temperature. 

The improvement of the new limit on vacuum stability is significant in the xSM, let alone in complicated models. With more than one first-order phase transitions in the thermal history, such as in a supersymmetric model~\cite{Athron:2019teq}, every transition has to be examined carefully. The relationship between the old and new limits may be tangled, and deserve case by case investigation. 





\section{Conclusion}

We argue that the study of vacuum stability should involve full consideration of the cosmological history, which is typically ignored. With the experimental discovery of gravitational waves, the impact of BSM on the cosmological history draws attention because a strong first order electroweak phase transition may generate detectable gravitational wave signals. Our study shows that there are parameter space regions where the transition to the desired EW-breaking vacuum is of low probability or does not even exist, which misleads the traditional vacuum stability analysis. 

In this letter we studied vacuum stability taking into account the thermal history of the universe. After demonstrating the lack of transition to the desired EW-breaking vacuum in the xSM, we gave a general procedure which incorporates cosmological history in the vacuum stability study, and compared the traditional and new limits. We found that checking the cosmological history can provide a much stringent and sometimes computationally cheaper limit of vacuum stability. In turn, this motivates further studies of phase transitions in new physics models.

{\em Acknowledgments:}
This work was supported by the National Natural Science Foundation of China (2105248, 11821505, 12075300), the Australian Research Council (DP210101636),  Peng-Huan-Wu Theoretical Physics Innovation Center (12047503),
Key R\&D Program of Ministry of Science and Technology (2017YFA0402204), and by the Key Research Program of the Chinese Academy of Sciences (XDPB15).

