Coscattering in the Extended Singlet-Scalar Higgs Portal

The cBE for $Y_{1}$ and $Y_{2}$ reads

where $H$ denotes the Hubble rate, $0$ stand for any SM particles, and $1,2$ for $S_{1}$ and $S_{2}$ respectively. The equilibrium yields are given by

where $K_{2}(x)$ is the modified Bessel function of the second kind, $g_{*S}(x)$ the number of effective degrees of freedom associated to the entropy density $s=\frac{2\pi^{2}}{45}g_{*S}(T)T^{3}$. In contrast to the cBE for coannihilation, eqs. (2) explicitly contains the DM conversion rate

where $k$ and $l$ denote light SM states. The calculation of the relevant conversion cross sections together with their thermal average is presented in App. B. The second important conversion process is given by decays of the unstable partner $S_{2}$ with the thermally averaged decay rate Garny:2017rxs

We solve the above cBE using micrOMEGAs 5.3.41 Belanger:2001fz ; Alguero:2022inz , considering three separate sectors: i) the SM, ii) the DM candidate $S_{1}$, and iii) a dark sector for $S_{2}$. micrOMEGAs solves all the relevant average cross sections, including the two and three-body decay widths of $S_{2}$ considering Lorentz time effects. To quantify the impact of coscattering and compare the results obtained from the full cBE to the results assuming CE we use Alguero:2022inz

where $\Omega h^{2}$(1 sector) is obtained using the darkOmega function of micrOMEGAs and $\Omega h^{2}$(2 sectors) is obtained from darkOmegaN¹¹1In the present paper we did not make explicit use of the function $\Delta_{2s}^{\Omega}$ defined in Alguero:2022inz , although part of the analysis in this section contemplates the information that could be obtained with that function.. The scaling of each process with the model parameters are listed in Table 1.

The basic characteristic of the conversion-driven freeze-out in the two scalar Higgs portal are:

1. 1.

   $S_{1}$ remains in CE with $S_{2}$ only through either (inverse) decays or coscattering processes $10\leftrightarrow 20$.

2. 2.

   Annihilation processes $S_{1}S_{i}\rightarrow XX$ involving $S_{1}$ can be neglected.

The first condition requires that $\lambda_{12}$ is non-vanishing but small enough for the conversion processes not to surpass the Hubble expansion rate at $T\lesssim m_{1}$. On the other hand, to prevent an early freeze-out and overabundance of DM it is required that $S_{2}$ couples strongly with the Higgs $\lambda_{H2}\sim 1$. The second condition is fulfilled only when in addition to $\lambda_{12}$, also $\lambda_{H1}\ll 1$. In case of on-shell (inverse) decays of $S_{2}$, the dark sector can stay in CE for much smaller couplings compared to the case of off-shell decays, however, we have checked that in both cases conversion-driven freeze-out is possible (in contrast to DAgnolo:2017dbv who assumed that 2-body decays are forbidden). In the last part of this section we analyse this point in more detail. Lastly, we note that the contact interaction terms in eq. (1) can not be arbitrarily large, as otherwise they will recover CE between $S_{1}$ and $S_{2}$. The impact of $\lambda_{22},\lambda_{13}$ and $\lambda_{31}$ is discussed in more detail at the end of this section. In Table 1 we show the parameter dependence for each process that enters in eqs. 2.

In order to simplify the discussion and exploration of the parameter space of the model in the coscattering framework, we define the simplest benchmark scenario (SBS) considering $\lambda_{H1}=\lambda_{22}=0$, and the relevant parameters as

Deviations from the SBS will be explicitly shown in some parts of the paper. As a warm up example of the features of coscattering in the SBS, in Fig. 1 we show a typical evolution of the DM yield in the coscattering regime fulfilling the correct relic abundance $\Omega h^{2}=0.12$, for $(m_{1},m_{2})=(500,505)$ GeV, and $(\lambda_{12},\lambda_{H2})=(2.6\times 10^{-5},1)$. Notice that $Y_{1}$ deviates from its equilibrium already near $x\approx 12$, whereas $Y_{2}$ stays in equilibrium for longer. This behavior is characteristic of the coscattering regime. In the right plot, we compare the reaction rates with the Hubble expansion, where $\Gamma_{ij}\equiv\frac{\gamma_{ij\rightarrow kl}}{n_{1e}}$ and $\gamma_{ij\rightarrow kl}$ denotes the reaction density. In particular it can be seen that the DM conversion rate $1020$ (yellow line) drops below the Hubble expansion at the same time as $S_{1}$ starts to freeze out from the thermal bath. Note that in the SBS scenario, decays and coannihilations are well below the Hubble rate and are completely negligible during the freeze-out process.

With this simple picture in mind, we now vary $m_{2}$ and $\lambda_{12}$ and study their impact on the relic abundance. We have performed a grid scan over $\lambda_{12}\in[10^{-5},10]$ and $m_{2}\in[500,630]$ GeV, keeping $m_{1}=500$ GeV and $\lambda_{H2}=1$ fixed. The results are shown in Fig. 2, where the red curves correspond to the solutions of the full cBE obtained with darkOmegaN, the blue curves where obtained using darkOmega and the orange curves where obtained ignoring the conversion processes $1020$. While the relic abundance shows a similar behavior when varying $\lambda_{12}$ for different values of $\Delta m$, the predicted relic abundance differs very strongly. This is due to the fact that the effective annihilation rate determining the point of freeze-out $e^{-2x\Delta m/m_{1}}\expectationvalue{\sigma_{2200}v}$ is exponentially suppressed for large $\Delta m$. This suppression leads to a smaller effective cross section which implies a faster freeze-out and larger relic abundance, as can be seen in Fig. 2.

As an example to better understand the dependence of $\Omega h$ on $\lambda_{12}$, we consider the case $\Delta m=30$ GeV (dot-dashed line). In Fig. 2 we have highlighted three distinct regions for the behaviour of the relic abundance. The coscattering mechanism is only active in region I where $\lambda_{12}$ is small enough so that the $1020$ conversion processes freeze-out quickly. As the coupling increases, CE is recovered and the relic abundance becomes insensitive to $\lambda_{12}$ in region II. In this case the relic abundance is mainly determined by $S_{2}$ annihilation, which is also called mediator freeze-out regime Junius:2019dci . Finally, in region III for $\lambda_{12}\gtrsim 0.1$ coannihilations between $S_{1}$ and $S_{2}$ become relevant and the relic abundance again depends on $\lambda_{12}$.

For each $\Delta m$ in Fig. 2 we have also included the corresponding relic abundance obtained from darkOmegaN when neglecting the processes 1020, but keeping decays²²2In micrOMEGAs this is achieved using the option Excluding2010.. The resulting abundances are plotted as the orange lines, highlighting the fact that for small values of $\lambda_{12}$ decays are not able to support CE in the absence of processes of the type 1020. In the case of on-shell decays (solid orange), where the decay rates are much larger, CE is maintained also at small $\lambda_{12}$. In this case the orange and red lines overlap in the whole range of small couplings.

We have also included the results for the relic abundance calculated using the function darkOmega of micrOMEGAs (blue lines). This function assumes that that CE between $S_{1}$ and $S_{2}$ is maintained during the entire evolution of the DM yield. In case of $\Delta m=1$ and $30$ GeV, the relic abundance obtained with the functions darkOmega and darkOmegaN agree very well in regions II and III, indicating that CE is present. In case of $\Delta m=120$ GeV, the results assuming CE are larger by roughly a factor of two, which further increases for larger mass differences. We have checked that in these cases the rate of (inverse) decays remains above the Hubble expansion, ensuring CE. The correct relic abundance is therefore obtained from darkOmega, while darkOmegaN assumes separate CE of the different sectors, which is unrealistic, particularly when on-shell decays are present.

From the case with $m_{2}=630$ GeV shown in Fig. 2, we have seen that on-shell decays maintain CE for much smaller $\lambda_{12}$ values. To further illustrate this fact, in Fig. 3 (left) we show the early departure from CE of the yield $Y_{1}$ in the off-shell case, $m_{2}=620$ GeV, compared to the on-shell case, $m_{2}=630$ GeV, for a fixed value of $\lambda_{12}$. In Fig. 3 (right) we show that, once on-shell decays are present, small enough values of $\lambda_{12}$ also break the CE³³3We have checked this in the case of on-shell decays, coscattering appears in the ballpark of $\mathcal{O}(\lambda_{12})\propto 10^{-6}$, but as a strong overabundance is obtained in this parameter space region, we do not focus on this case in this paper.. In other words, as $\Delta m$ increases, the relevance of decays in maintaining CE becomes stronger, requiring smaller $\lambda_{12}$ for coscattering.

On the other hand, the contact terms proportional to the couplings $\lambda_{22},\lambda_{13}$ and $\lambda_{31}$ can have a strong impact on the relic density. In particular, when they take sizable values, i.e. either $\lambda_{22},\lambda_{13}$ or $\lambda_{31}\gtrsim 0.1$, they bring $S_{1}$ and $S_{2}$ back into CE. To quantify this, in Fig. 4 (left) we show the effect of each separate coupling on the relic abundance for two set of masses. In each case, the value of $\lambda_{12}$ was fixed in order to obtain the correct relic abundance by darkOmegaN in the limit of vanishing contact terms. As the contact couplings get sizable values, they start to affect the relic abundance calculation with darkOmegaN  as they tend to establish partial CE between $S_{1}$ and $S_{2}$. Once the contact couplings are big enough, the CE is established, such that the calculation using darkOmega and darkOmegaN agree with each other⁴⁴4It is interesting to remark that for the case in which $\lambda_{22}$ takes sizable values, and $\lambda_{12}$ remains sufficiently small to not maintain CE between $S1$ and $S_{2}$, one recover the yield dynamic of two stable DM, known as assisted freeze-out DiazSaez:2021pfw (also see Belanger:2011ww ; Maity:2019hre ). As in the present framework $S_{2}$ is unstable, after the breaking of its CE with $S_{1}$, $Y_{2}$ will continuously decrease as $x$ increases.. As we focus on the coscattering, we do not include deviations induced by the contact terms of this Higgs portal scenario, therefore in the rest of the paper we assume they are sufficiently small to not deviate from the relic abundance calculation with darkOmegaN.

To end this section, we comment about the low mass regime $m_{1}<m_{h}/2$, which turns out to be disfavored by LHC data. From the above discussion we found that coscattering requires nearly degenerate masses of the new scalars, i.e. $m_{1}\lesssim m_{2}<m_{h}/2$. On the other hand, we have checked numerically that in order to obtain the correct relic abundance $\lambda_{12}$ would be large enough to also recover CE within the dark sector, making coscattering ineffective unless $\lambda_{H2}\gtrsim 1$. However, searches of Higgs to invisible at the LHC have set limits on $\Gamma(h\rightarrow S_{2}S_{2})$ ATLAS:2022yvh , that translate into $\lambda_{H2}\lesssim 10^{-2}$. We have also checked that the inclusion of the contact terms does not change this result.

To summarize, we have presented the cBE for the system of $S_{1}$ and $S_{2}$, and we have solve them making use of the micrOMEGAs code. The three regimes that we have distinguished, coscattering, mediator FO, and (co)annihilations, depend strongly on the parameters $\Delta m$ and $\lambda_{12}$, with coscattering favoring $\Delta m\ll(m_{1},m_{2})$ and $\lambda_{12}\ll 1$. Besides, on-shell (inverse) decay rates of $S_{2}$ are very efficient to maintain CE for much smaller values of $\lambda_{12}$ than in the case of off-shell decays. Contact terms are not essential to have coscattering, and we have seen that light DM is ruled-out by LHC bounds.

In the coscattering regime, the coupling $\lambda_{12}$ which determines the decay width of $S_{2}$ is very small, while simultaneously $\Delta m\ll(m_{1},m_{2})$. Therefore, the dark partner $S_{2}$ typically constitutes a long-lived particle (LLP) with a wide range of possible lifetimes in different regions of the parameter space. While single production of $S_{2}$ (like production of the DM relic $S_{1}$) at colliders is suppressed by $\lambda_{12}$, pair production of $S_{2}$ through an intermediate Higgs boson must be sizable, via the chain Craig:2014lda

with $X$ being other states not relevant for the discussion. The goal of this section is to compare a few $S_{2}$ lifetime estimations predicted by the extended Higgs-singlet scenario that could be in the reach of present and future experiments, specially when the production mechanism is motivated by coscattering. In our knowledge, as LLP in Higgs portals have only been considered for mediator masses $\lesssim m_{h}/2$ Curtin:2018mvb ; Alimena:2019zri , the results presented here could motivate the search of heavier scalars through the Higgs portal.

In Fig. 6, we show the results for the lifetime of $S_{2}$ as a function of its mass for $\lambda_{H1}=0$ and fixed $\lambda_{H2}=0.5$ (left), $1$ (middle) and $\pi$ (right). The row of points from top to bottom corresponds to $\Delta m=$ 1, 5, 10 and 20 GeV, while the color of each point indicates $\Delta_{1s}^{\Omega}$. The variation in $c\tau$ depends strongly on scalar mass difference, with small values of $\Delta m$ favoring the coscattering regime ($\Delta_{1s}^{\Omega}\lesssim 1$), and in turn giving rise to enormous lifetimes of $S_{2}$, with some of the points well beyond earth size experiments, thereby confronting bounds coming from BBN. As $\Delta m$ increases, the values of $c\tau$ decrease to the point of reaching typical decay lengths for future experiments such as MATHUSLA Curtin:2018mvb . Notice that the reach of the latter may not only test particles that were produced in the coscattering regime, but also probe the other two regimes that we studied in Sec 3.2 (blue points). There are also model predictions in the reach of displaced vertex (DV) for ATLAS or CMS Lee:2018pag (grey band in each plot). Finally, the blue points in each plot are not unique for the corresponding chosen parameter space points shown in Fig. 6, since as some of them belong to the mediator freeze-out regime (see region II in Fig. 2), there is a range of $\lambda_{12}$ values fulfilling the measured relic abundance, then varying in orders of magnitude their corresponding $c\tau$ value.