Electroweak Baryogenesis (EWBG) is a compelling scenario for explaining the
matter-antimatter asymmetry in the universe. Its connection to the electroweak
phase transition makes it inherently testable. However, completely excluding this
scenario can seem difficult in practice, due to the sheer number of proposed models.
We investigate the possibility of postulating a “no-lose” theorem for testing EWBG in
future
Article funded by SCOAP3
0.6 - 1.6. \end{equation} The right-hand side is conventionally taken to be 1.0, but we consider the shown numerical range to reflect unknown details of the baryon number generation mechanism during the phase transition~\cite{Patel-ml-2011th}. There are a number of possible ways to achieve this phase transition by introducing new particles which couple to the higgs, modifying its potential by a variety of mechanisms (see~\cite{Chung-ml-2012vg} for a categorization of phase transitions and their correlation to higgs observables). Therefore, moving down the axis of difficulty in testing sources of a strong phase transition is relatively straightforward. The most-hidden particles that can increase the strength of the EW phase transition are SM singlet scalars. SM singlets that couple to the higgs to achieve EWBG have been studied in great detail, both by themselves~\cite{Profumo-ml-2007wc, Ashoorioon-ml-2009nf, Damgaard-ml-2013kva, Barger-ml-2007im, Espinosa-ml-2011ax, Noble-ml-2007kk, Cline-ml-2012hg, Cline-ml-2013gha, Alanne-ml-2014bra, Espinosa-ml-2007qk, Profumo-ml-2014opa, Fuyuto-ml-2014yia, Fairbairn-ml-2013uta} and in the context of supersymmetry~\cite{Pietroni-ml-1992in, Davies-ml-1996qn, Huber-ml-2006wf, Menon-ml-2004wv, Huber-ml-2006ma, Huang-ml-2014ifa, Kozaczuk-ml-2014kva}. In this paper we investigate the maximally hidden singlet scalar model, find where a strong phase transition can occur, and then correlate this with the reach of experimental probes. The basic setup for this ``nightmare scenario'' is as follows. We introduce a real singlet field that couples to the higgs and has a $\mathbb Z_2$ symmetry to forbid higgs-singlet mixing~\cite{Barger-ml-2007im, Noble-ml-2007kk, Espinosa-ml-2011ax, Cline-ml-2012hg, Cline-ml-2013gha, Alanne-ml-2014bra, Espinosa-ml-2007qk}. This rules out electroweak precision tests and higgs coupling modifications as experimental probes. We then set $m_S > m_h/2$ to avoid modified higgs decays, in particular an exotic higgs decay mode which would be relatively easy to discover at future colliders (see~\cite{Curtin-ml-2013fra} for a review). This nightmare scenario, while difficult to test, still has a number of potential experimental signatures. For instance, colliders can probe the direct production of the singlet states, as well as shifts in the triple higgs couplings and $Zh$ cross section. Furthermore, the presence of the $\mathbb Z_2$ symmetry has implications for dark matter searches. One could, in principle, make the above setup even more difficult to discover by including extra singlets that decrease some of the experimental signatures while leaving the phase transition intact. However, as we show in this paper, excluding even this basic nightmare scenario \emph{requires} at least a 100\,TeV hadron collider, such as the proposed SPPC/FCC. A higgs factory like CEPC, ILC, or TLEP is not sufficient, based on existing studies for precision measurements of higgs self-couplings. Remarkably, the fact that this scenario \emph{is} testable at the SPPC/FCC demonstrates that it may be possible to postulate a ``no-lose'' theorem for EWBG with future colliders. Our paper is organized as follows. In \sref{scenario}, we define the $\mathbb{Z}_2$ symmetric singlet scalar model and the two-dimensional parameter plane that illustrates its entire phenomenology. Section~\ref{s.ewpt} contains our analyses of the one-step and two-step phase transitions which enable EWBG in this model. Sections~\ref{s.collider} and~\ref{s.indirect} examine direct and indirect signatures of the singlet scalar at colliders, and show how the discovery potential overlaps with the EWBG-favored regions of parameter space. We consider cosmological constraints on the singlet in \sref{dm} and show that, under certain assumptions, the entire parameter space can be excluded by future direct detection experiments. Renormalization group (RG) evolution and the implications of strong couplings are discussed in \sref{strongcouplings}. We summarize our findings and discuss implications in \sref{discussion}. ]]>
m_h/2$ (our nightmare scenario). \emph{Left:} the red shaded region indicates when $\mu^2$ is negative. The dotted red contours indicate $\mathrm{Sign}(\mu_S^2)|\mu_S|$. The blue contours show the minimum $S^4$ quartic coupling $\lambda_S$ required for the electroweak symmetry breaking (EWSB) vacuum to be the ground state of the universe, while the green contours show the minimum $\lambda_S$ to avoid negative runaways. \emph{Right:} gray regions indicate where theoretical control is lost due to non-perturbative $\lambda_S$. Perturbative analysis of the phase transition breaks down in the blue shaded regions, see \sref{ewpt}. The red and white regions are the possible parameter space of this nightmare scenario.]]>
0. \end{equation} The other parameter which dictates the phenomenology of the singlet is its coupling to our sector through the higgs, the $hSS$ coupling. This coupling determines singlet production and annihilation cross sections and is given by $\lambda_{HS}$.\footnote{When discussing the effective potential at one-loop in \sref{ewpt} we choose a scheme in which the tree-level parameter $\lambda_{HS}$ corresponds to the physical $hSS$ coupling $\mathcal{L_\mathrm{eff}} \supset - v \lambda_{HS} h SS$.} The singlet self interaction, $\lambda_S$, is important when discussing regions with a possible phase transition, but does not play a direct role in the phenomenology of this model. Thus, \emph{all} the relevant features of our nightmare scenario can be shown in the $(m_S, \lambda_{HS})$ plane. The $(m_S, \lambda_{HS})$ plane can be divided into regions where all couplings are under perturbative control or not, and further divided based on the sign of $\mu_S^2$. This division has consequences for the vacuum structure of the theory, and hence the qualitative mechanisms at play to produce strong phase transitions. If all the quartics are positive, then for positive $\mu_S^2$ the only minimum is the EWSB vacuum at $(h,S) = (v,0)$. When $\lambda_{HS} > m_S^2/v^2$, $\mu_S^2$ is negative. This region is shaded red in \fref{SMSparamspace} (left). In this case, there are two local minima: the EWSB vacuum and a ``singlet-VEV vacuum'' at $(h,S)=(0,w)$. A surviving $\mathbb{Z}_2$ symmetry prevents higgs-singlet mixing in both vacua. For the scenario with negative $\mu_S^2$, we can ensure that our universe ends up in the correct EWSB vacuum by requiring that the potential $V_0(h,s)$ satisfies $V_0(0,w) > V_0(v, 0)$. It is clear that this requires a minimum value of $\lambda_S$ which depends on the choice of $m_S$ and $\lambda_{HS}$: \begin{equation} \label{e.lambdaSmin} \lambda_S^\mathrm{min} = \lambda \frac{\mu_S^4}{\mu^4} = \frac{2(m_S^2 - v^2 \lambda_{HS})^2}{m_h^2 v^2} \end{equation} The blue contours in \fref{SMSparamspace} (left) show this minimum $\lambda_S$ at tree-level, which rapidly becomes non-perturbative as we move deeper into the shaded red region. Requiring $\lambda_S < 8$ excludes the gray region in the top corner of \fref{SMSparamspace} (right) from being part of the viable parameter space. In the remaining red strip, it is possible to choose a quartic coupling $\lambda_S$ to ensure the universe eventually ends up in the EWSB vacuum. There are additional constraints on $\lambda_S$ that come from avoiding runaway directions in the potential at large field values. Avoiding a negative runaway\footnote{In the presence of negative runaways the tree level potential has local minima at $h, S \neq 0$. However, by the positivity assumption of \eref{mSexpression}, these local minima are always at higher potential than the electroweak breaking minimum.} at tree-level requires \begin{equation} \lambda > 0 \ \,, \ \ \ \lambda_S > 0 \ \,, \ \ \ \lambda_{HS} > - \sqrt{\lambda \lambda_S}. \end{equation} For a given negative $\lambda_{HS}$, which in our scenario implies $\mu_S^2 > 0$, this requirement leads to a minimum value of $\lambda_S$: \begin{equation} \label{e.lambdaSmin2} \lambda_S^\mathrm{min'} = \frac{\lambda^2_{HS}}{\lambda} = \lambda_{HS}^2 \frac{2 v^2}{m_h^2} \end{equation} which is indicated by the green contour lines in \fref{SMSparamspace} (left). Again, the required quartic coupling $\lambda_S$ becomes non-perturbative as we move to larger negative $\lambda_{HS}$. Applying the same $\lambda_S < 8$ cutoff as before excludes the lower gray shaded region in \fref{SMSparamspace} (right). This corresponds to the requirement that \begin{equation} \lambda_{HS} \gtrsim - 1.0. \end{equation} In allowing $\lambda_S$ to be as big as 8 we are being somewhat generous --- theoretical control could break down at smaller couplings. However, the purpose of this demarcation of parameter space is to identify regions that we would need to probe, with either direct or indirect measurements, to exclude this model as a viable EWBG scenario. It is therefore sensible to charitably assess theoretical control and slightly over-estimate the size of parameter regions with a strong phase transition. This ensures that no viable EWBG scenarios are missed. In particular, as we will show in future sections, the region that is explored with a more optimistic definition of perturbative control is always easier to probe directly or indirectly, thereby not changing the conclusions of our study. The constraints that excise certain regions of parameter space thus far are based on tree-level considerations requiring couplings at non-perturbative values. There are additional perturbativity constraints from quantum effects. In \sref{ewpt}, we will demonstrate that a one-loop perturbative analysis of the phase transition breaks down for $\lambda_{HS} \gtrsim 6$ for $\mu_S^2 < 0$ and for $\lambda_{HS} \gtrsim 5$ for $\mu_S^2 > 0$. (This roughly coincides with regions where the quartic couplings develop Landau Poles below 10--100\,TeV, as we discuss in \sref{strongcouplings}.) These regions, which are meant to be \emph{approximate indications} of where perturbative calculations become very unreliable, are shaded blue in the top right corner of \fref{SMSparamspace} (right). The viable parameter space of the nightmare scenario is therefore the red and white regions in \fref{SMSparamspace} (right). As we will see, these two regions behave very differently with regards to EWBG as well as their signals for direct and indirect measurements. The phenomenology of regions with large couplings is further discussed in \sref{strongcouplings}. ]]>
0$. In this case, the phase transition occurs purely along the higgs direction. However, if the singlet is sufficiently strongly coupled to the higgs, its zero-temperature loop corrections to the higgs potential can be big enough to allow SM thermal effects to trigger a strong phase transition. On the other hand, if $\mu_S^2$ is negative there can be two vacua. The universe can then undergo a two-step phase transition, first to a singlet VEV vacuum, and then to the true EWSB vacuum. This tree-level modification of the higgs potential can result in an arbitrarily strong phase transition by adjusting the potential difference between the two vacua via the choice of $\lambda_S$. If $\mu_S^2 \lesssim - (100 \gev)^2$ and the singlet self- and scalar-couplings are just right, it is also possible for a one-step transition to occur via thermal effects, akin to the MSSM light stop scenario. This is only realized for a small region of parameter space, entirely contained within the two-step phase transition region of the $(m_S, \lambda_{HS})$ plane. It is therefore clear that the two different regions of parameter space delineated in \sref{scenario} also realize strong phase transitions differently: one with relatively low $m_S$ and negative $\mu_S^2$, and one with large $m_S$ and stronger singlet-higgs coupling. In what follows, we review the phase transition calculation in detail and define the regions of parameter space which can realize EWBG. It is important to note that while these regions are very distinct from the point of view of the phase transition, they are continuously connected in the phenomenological parameter space $(m_S, \lambda_{HS})$. ]]>
0$ and without negative runaways, the singlet never attains a VEV, and there are no tree-level effects to enhance the phase transition. However, it is still possible to induce a strong electroweak phase transition via sizable one-loop zero-temperature corrections to the SM higgs potential. ]]>
0$ is almost entirely determined by the two parameters $(m_S, \lambda_{HS})$. ]]>
0.6$ where EWBG could proceed with an efficient baryon number generation mechanism. However, the exact choice of the minimum $v_c/T_c$ does not qualitatively affect the definition of the EWBG-compatible region. Strong couplings $\lambda_{HS} \gtrsim 2$ are needed. The critical temperature decreases with increasing coupling and is in the range of $T_c \sim 130 - 160 \gev$. To understand this mechanism for generating a strong phase transition, we repeat the above calculation with various contributions to the total effective potential \eref{Veff} switched off. We find that using \begin{eqnarray*} V_\mathrm{eff} &=& V_0 \ \ \ \mbox{(tree-level potential)} \\ && + \ \mbox{(only \emph{singlet} contributions to $V_0^\mathrm{CW}$)}\\ && + \ \mbox{(only \emph{SM} thermal contributions)} \end{eqnarray*} gives a very similar result, with the $\lambda_{HS}$ necessary for a strong phase transition underestimated by about 10\%. This implies that sizable zero-temperature one-loop higgs potential contributions from the singlet reduce the potential difference between the EWSB vacuum and the origin, which then makes it easier for SM thermal contributions to generate an energy barrier between the two degenerate local minima at some $T = T_c$. This is illustrated in \fref{Vcomparisonplot}. For very strong coupling, the one-loop effects create an energy barrier even at zero temperature. This is the case above the dashed green line in \fref{vcoverTcplot}. However, as we discuss in the next subsection, our one-loop analysis may not be valid for such high coupling. \begin{figure} \centering \begin{tabular}{c} \includegraphics[width=0.472\textwidth]{EWBGvcoverTcplot} \end{tabular} ]]>
0$. The orange shaded region indicates $v_c/T_c > 0.6$, where EWBG occurs via a loop-induced strong one-step PT. Above the green dashed line, singlet loop corrections generate a barrier between $h = 0$ and $h = v$ even at $T = 0$, but results in the dark shaded region might not be reliable, see \sssref{perturbativereliability}. ]]>
1$. The one-loop contribution of the singlet reduces the potential difference between the origin and the EWSB vacuum. ]]>
0$ such that there are two local minima, one at $h = 0, S = w$ and a deeper one at $h = v, S = 0$ (at zero temperature). When the universe is very hot $T \gg 100 \gev$, thermal contributions stabilize both fields at the origin. Since the singlet couples to fewer degrees of freedom, its thermal mass is lower than that of the SM higgs. Therefore, as the universe cools, the singlet gets destabilized before the higgs (see e.g.~\cite{Cline-ml-2012hg}). The electroweak phase transition then starts in the singlet-VEV minimum and ends in the EWSB minimum. As outlined in \ssref{couplingplane}, we can always choose a $\lambda_S$ to make the two local minima degenerate. This corresponds to zero critical temperature, i.e.\ the universe never transitions from the singlet-VEV minimum to the EWSB minimum. For any given point $(m_S, \lambda_{HS})$ in the red region of \fref{SMSparamspace} (right) one can then imagine taking $\lambda_S$ a little bit larger than $\lambda_S^\mathrm{min}$ in \eref{lambdaSmin}. This gives an arbitrarily low $T_c$, and hence an arbitrarily large ratio $v_c/T_c$, easily satisfying the baryon number preservation criterion \eref{bnpc}, while ensuring the singlet-VEV vacuum is short-lived.\footnote{The tunneling rate from the singlet-VEV minimum to the EWSB minimum is $\Gamma \sim e^{-S_E}$, where $S_E$ is the finite-temperature bounce action~\cite{Coleman-ml-1977py}. For $S_E \sim 100$, the false vacuum decays quickly in the early universe~\cite{Delaunay-ml-2007wb}. We computed the \emph{zero temperature} bounce action $B$ in the triangle potential barrier approximation~\cite{Duncan-ml-1992ai} and found that $B < 100$ for some range of $\lambda_S < 8$ in most of the red shaded region of \fref{SMSparamspace}. Thermal fluctuations greatly enhance the tunneling rate, $S_E < B$. Therefore, the transition between the two minima can be sufficiently fast to ensure a viable thermal history for the universe.} The above discussion may be modified slightly by loop and thermal effects. By and large, however, in (or close to) the red shaded region of \fref{SMSparamspace} (right) and \fref{vcoverTcplot}, EWBG is possible via a strong two-step phase transition, which is induced by tree-level modifications to the higgs potential. ]]>
V_\mathrm{eff}(v,0,T=0). \end{equation} Requiring $\lambda_S^\mathrm{min}(m_S, \lambda_{HS}) < 8$ at tree-level was used in \ssref{couplingplane} to define the viable $\mu_S^2 < 0$ region of parameter space, shaded red in figures~\ref{f.SMSparamspace}~(right) and~\ref{f.vcoverTcplot}. We find that the definition of this region does not change significantly when including loop corrections, except for the fact that $\lambda_{S}^\mathrm{min} > 8$ for all $\mu_S^2 < 0$ when $\lambda_{HS} \gtrsim 6$. Therefore we regard any calculation in the $\mu_S^2 < 0$ region with larger $\lambda_{HS}$ coupling as unreliable, indicated with the blue shading above $\lambda_{HS} = 6$ in \fref{SMSparamspace} (right) and all following figures. \sloppy{A given choice of $(m_S, \lambda_{HS})$ and $\lambda_S > \lambda_S^\mathrm{min}(m_S,\lambda_{HS})$ completely defines the temperature-dependent effective potential, and it is straightforward to analyze the two-step phase transition. We will focus on values of $\lambda_S = \lambda_S^\mathrm{min} + \mathcal{O}(0.1)$.} \looseness=-1 At very high temperature both fields $(h,S)$ are stabilized at the origin.\footnote{This is not the case for large $\lambda_S$ or $\lambda_{HS}$, since the singlet thermal potential develops a high-temperature instability if $\frac{1}{3}\lambda_{HS} + \frac{1}{3}\lambda_S > \frac{\pi^2}{9} \approx 1.1$. At high temperature, the singlet then has nonzero VEV. This does not affect our argument for a strong phase transition, since it essentially corresponds to $T_{c1} \gg T_{c2}$. It also does not affect the one-step phase transition for $\mu_S^2 > 0$.} As the universe cools, the singlet transitions to a nonzero VEV first, in a second-order phase transition at temperature $T_{c1} \sim $ few $100 \gev$. The EWSB minimum eventually drops below the singlet-VEV minimum at temperature $T_{c2} < T_{c1}$. Since the tree-level barrier between the two minima survives at $T_{c2}$, the universe undergoes a first-order phase transition to $(v_{c2},0)$, where $v_{c2} \equiv v(T_{c2})$. $T_{c2}$ varies in the 2-step phase transition region, increasing as we take $\mu_S^2 \rightarrow 0$. \begin{itemize} \item For $\lambda_{S} = \lambda_S^\mathrm{min}$, $T_{c2} < 45 \gev$ in the entire two-step region. The phase transition is very strong, $v_{c2}/T_{c2} \sim 4$ near $\mu_S^2 \to 0$ (outer boundary of 2-step phase transition region) and $\gtrsim 8$ as $\lambda_{S}^\mathrm{max}$ approaches its maximum allowed value of 8 (inner boundary of 2-step phase transition region). \item For $\lambda_S = \lambda_S^\mathrm{min} + 0.1$, $T_{c2} \sim 30 - 100 \gev$ in the entire 2-step phase transition region, with $v_{c2}/T_{c2} > 2$. \end{itemize} Clearly a relatively small increase in $\lambda_S$ compared to its minimum value guarantees that the phase transition takes place at a cosmologically safe temperature, and the singlet-VEV vacuum is short-lived. This calculation demonstrates the validity of the tree-level arguments in \sssref{twostep_tree}. A strong two-step phase transition can be achieved in the entire viable $\mu_S^2 < 0$ region, shaded red in all our plots. The loop-level analysis reveals perturbativity is lost for $\lambda_{HS} \gtrsim 6$, which is shaded blue in all our plots. ]]>
\lambda_S^\mathrm{min}$ and $|\mu_S| \lesssim 100 \gev$, the singlet bare mass cancels its thermal mass and generates a negative cubic term in the finite-temperature higgs potential, while also stabilizing the singlet at the origin for $T \geq T_c$. This replicates the well-known mechanism for a strong phase transition realized, for example, by the Light Stop Scenario in the MSSM~\cite{ Huet-ml-1995sh, Carena-ml-1996wj, Laine-ml-1998qk, Laine-ml-2000xu, Espinosa-ml-1996qw, Carena-ml-1997ki, Huber-ml-2001xf, Cline-ml-2000kb, Carena-ml-2002ss, Lee-ml-2004we, Cirigliano-ml-2009yd, Carena-ml-2008rt, Carena-ml-2008vj, Quiros-ml-1999tx, Delepine-ml-1996vn}. However, since the thermally driven phase transition only occurs for some finely tuned $\lambda_S$ in a very small part of the two-step phase transition region, we do not miss any EWBG-viable regions by only discussing the $\mu_S^2 > 0$ one-step and $\mu_S^2 < 0$ two-step phase transition regions in the phenomenological analysis of the following sections. ]]>
0$, zero-temperature loop effects from the singlet raise the EWSB minimum, which allows SM thermal contributions to generate a sizable energy barrier. This makes EWBG possible in the orange-shaded region. For $\lambda_{SH} \lesssim 5$ $(6)$ in the one (two) step phase transition region our analysis is perturbatively reliable, with untrustworthy regions indicated by the blue shading in all our figures. \looseness=-1 This establishes the regions of the $(m_S, \lambda_{HS})$ plane where EWBG could occur. We now move on to discuss ways of directly and indirectly detecting signatures of the strong phase transition. ]]>
150$\,GeV to cut down on QCD background. Shading identical to figures~\ref{f.vcoverTcplot} and~\ref{f.perturbativevalidityplot}. ]]>
40$\,GeV, $|\eta_{j_{1,2}}| < 5$ \item $\met>$ 150\,GeV, \item $\Delta\eta_{jj}=|\eta_{j_1}-\eta_{j_2}|>3.5$ and $|\eta_{j_{1,2}}| > 1.8$, \item $M_{jj}>800$\,GeV. \item reject events with leptons satisfying $|\eta| < 2.5$ and $p_T > 15 \gev$. \end{itemize} \looseness=-1 We consider $(Z\to \nu \nu) + jj$ background from both Drell-Yan and VBF production. We use \texttt{MadGraph5 v1.5.12}~\cite{Alwall-ml-2011uj} evaluated with the \texttt{CTEQ6l}~\cite{Pumplin-ml-2002vw,Nadolsky-ml-2008zw} parton distribution functions and \texttt{Pythia8}~\cite{Sjostrand-ml-2006za,Sjostrand-ml-2007gs} showering \& hadronization to generate the signal events. For detector simulation, we use \texttt{Delphes 3.1.2}~\cite{deFavereau-ml-2013fsa} with the same detector card as the 100\,TeV Snowmass Studies~\cite{Anderson-ml-2013kxz,Avetisyan-ml-2013onh,Avetisyan-ml-2013dta}. For the background, we used pre-computed \texttt{Bj-4p} and \texttt{Bjj-vbf} event samples without pile-up from the Snowmass database~\cite{snowmassbkgd}. Pile-up was neglected. Figure~\ref{f.SB100TeV} shows the resulting $S/\sqrt{B}$ contours in the $(m_S, \lambda_{HS})$ plane for a 100\,TeV $pp$ collider with $3000 \ifb$ of data. Our naive estimate suggests $S/\sqrt{B}$ is order unity in the entire two-step phase transition region. The actual sensitivity will depend on the detector capabilities and total luminosity of the potential future 100\,TeV collider program, but probing the entire two-step region via singlet VBF production may be possible. More sophisticated search and background reduction techniques may improve on these estimates. This search will be challenging in practice due to its sensitivity to systematic errors. However, there are potential data-driven methods for addressing this. For example, the $(Z\to\nu\nu)jj$ background is kinematically identical to the $(Z\to\ell\ell)jj$ background under the replacement of $p^T_{\ell\ell} \ \to\ \met$. This suggests a very statistically precise background template could be derived from data, greatly reducing systematics compared to a naive estimate. Most of the parameter space for the strong one-step phase transition seems entirely out of reach by direct detection. However, as we see below, indirect measurements can be sensitive to the rest of the relevant parameter space. ]]>
0.6$ (1.0) implies $\lambda_3/\lambda_3^\mathrm{SM} > 1.2$ $(1.3)$. Such a sizable deviation makes it possible to exclude this type of strong phase transition. \begin{figure} \centering \includegraphics[width=0.472\textwidth]{lambda3} ]]>
100 \gev$. In the two-step region, $T_f < 22 \gev$, and we find in \sssref{twostep_full} that even $\lambda_S = \lambda_S^\mathrm{min} + 0.1$ results in $T_c = T_{c2} > 30 \gev$ for almost the entire two-step region. In this case, if $S$ is stable then the bounds calculated in this section apply to our model. However, it is possible to tune $\lambda_S \to \lambda_S^\mathrm{min}$ and achieve $T_c < T_f$. For this case, there are two possibilities for singlet freeze-out in the two-step phase transition region: \begin{itemize} \item The singlet freezes out in the unbroken phase at temperature $T_f^\mathrm{h=0}$. Since the universe resides in the singlet-VEV vacuum before the phase transition, the singlet can decay via $S \to hh$. This could deplete the singlet density to values much lower than indicated in \fref{singletdm} (left). \item The singlet is in thermal equilibrium just before the phase transition at $T_c < 22 \gev$. If the singlet becomes lighter, it remains in thermal equilibrium and our above freeze-out estimate should apply. If it becomes heavier, it likely freezes out instantly. \end{itemize} Understanding the consequences of the second possibility would require further study, but it is clear that dark matter relic density may be considerably reduced in the two-step region, resulting in lower relic density and correspondingly weaker direct detection bounds than those shown in \fref{singletdm}. That being said, assuming these direct detection bounds (with $T_f < T_c$ and a stable thermal relic $S$) apply to our model, the nightmare scenario for EWBG is already excluded for $m_S < m_h$ by LUX. Interestingly, the \emph{entire} EWBG-viable parameter space for both a one- and two-step phase transition is excludable at XENON1T. This provides a much earlier discovery possibility than a 100\,TeV collider or a high-energy ILC. ]]>
0$ but $\lambda_{HS} < 0$, the large self-coupling is required to avoid a runaway potential for the singlet. In the absence of full theoretical control, the most conservative approach in examining these two regions is to assume that they maintain the basic vacuum structure implied by a naive classical analysis. Therefore, if the first region were viable, it would simple enlarge the allowed parameter space for a two-step phase transition in the direction of large $\lambda_{HS}$. These strong phase transitions would be much more discoverable (by all experimental avenues) than the cases we have examined, meaning our statements about testability of the strong phase transition remain valid. The non-perturbative region with negative $\lambda_{HS}$ is more difficult to interpret. However, this region is not close to any region with a strong phase transition that is under theoretical control, and is likely not viable due to the appearance of a singlet runaway. \looseness=-1 One may also ask whether there are any interesting effects due to large $\lambda_{HS}$ in regions where a strong phase transition is possible without large $\lambda_S$. Continuing our conservative line of reasoning, increasing $\lambda_{HS}$ would maintain the basic characteristics of the theory (strong phase transition) while making the theory even more testable. New phenomena may also arise in this direction, which has been considered previously in the context of a strongly interacting phase of the MSSM~\cite{Giudice-ml-1998dj}. In such a scenario, the singlets could turn into new composite states bound together by higgs exchange, similar to the stop-balls in~\cite{Giudice-ml-1998dj}. Finally, in regions of parameter space with moderately large but still perturbative couplings, understanding the theory's RG evolution is of critical importance. If the couplings required at low energy for a strong phase transition become non-perturbative at higher energies, it could invalidate our calculation of the universe's thermal history to find regions with acceptable phase transitions. The couplings need not stay perturbative to the GUT or Planck scale, merely in a sufficiently large range to ensure our calculations of the phase transition are trustworthy. To answer this question, we investigate the renormalization group equations (RGEs) of the nightmare scenario below. ]]>
8$ and are not under theoretical control, see \ssref{couplingplane}. \emph{Red} shaded region with red boundary: a strong two-step PT from tree-effects is possible for some choice of $\lambda_S$, see \ssref{onestep}. \emph{Orange} shaded region with orange boundary: a strong one-step PT from zero-temperature loop-effects is possible, see \sssref{ewptvialoop}. Gray-Blue shading in top-right corner indicates the one-loop analysis becomes unreliable for $\lambda_{HS} \gtrsim 5 (6)$ in the one-step (two-step) region, see \sssref{perturbativereliability} and~\ref{sss.twostep_full}. In the \emph{blue} shaded region (demarcated with \emph{blue lines}), higgs triple coupling is modified by more than 16\% compared to the SM, which can be excluded at the $2\sigma$ level by a 100\,TeV collider, see \ssref{triplehiggs}. In the \emph{green} shaded region, our simple collider analysis yields $S/\sqrt{B} = 2, 1, 0.5$ (\emph{green dashed lines} from left to right) for VBF production of $h^*\to S S$ at a 100\,TeV collider, see \sref{collider}. (In both cases assume 3 $\mathrm{ab}^{-1}$ of data.) Note that both EWBG preferred regions are excludable by XENON1T if $S$ is a thermal relic, see \sref{dm}. ]]>