Interplay between dark matter and leptogenesis in a common framework

The observed baryon asymmetry and dark matter relic density in the universe Planck:2018vyg are two long-standing problems that can provide us with new hints for new physics. Moreover, the baryon asymmetry can be obtained through the leptogenesis mechanism Fukugita:1986hr ; Luty:1992un ; Plumacher:1996kc ; Buchmuller:2004nz ; Davidson:2008bu , which is also related to the problem of tiny neutrino masses. As for the dark matter, the correct relic density can be produced conservatively by Freeze-out Chiu:1966kg or Freeze-in Hall:2009bx depending on the strength of the interactions associated with dark matter and the initial density of dark matter in the early universe.

A common framework for unified dark matter as well as leptogenesis can be found in Ref. Josse-Michaux:2011sjn , which can explain quantitatively both the observed baryon asymmetry of our universe and dark matter relic density. However, this extension is difficult to be falsified. On the other hand, according to the fact of $\Omega h^{2}\approx 5\Omega_{b}h^{2}$ Planck:2018vyg , where $\Omega h^{2}\approx 0.12$ is the observed dark matter relic density and $\Omega_{b}h^{2}$ is the baryon density, a shared mechanism to generate baryon asymmetry and dark matter simultaneously will be much more attractive for the quantitive relationship, and related discussions can be found in Refs. Chu:2021qwk ; Cui:2020dly ; Cui:2016rqt ; Dasgupta:2016odo ; Cui:2012jh . Especially, when comes to leptogenesis, the right-handed neutrinos may not only couple to the Standard Model (SM) but also a hidden sector where the dark matter resides, and a lepton asymmetry as well as dark matter production can be both generated by the out-of-equilibrium decay of the right-handed neutrinos Falkowski:2011xh ; Falkowski:2017uya . However, such scenarios often indicate asymmetric dark matter (ADM) Nussinov:1985xr ; Kaplan:1991ah ; Zurek:2013wia ; Graesser:2011wi , where relic dark matter is not determined by the annihilation cross section but asymmetry between particle-antiparticle number densities of dark matter.

From the point of the standard leptogenesis and WIMP (weakly interacting massive particles) DM scenario, it seems that there is little connection between leptogenesis and dark matter. On the one hand, a lepton asymmetry is generated by the decay of right-handed neutrino (RHN) out of equilibrium, then the lepton asymmetry is converted into baryon asymmetry by the sphaleron process. Such a process always happens at a high scale and demands the right neutrino mass larger than $10^{9}$ GeV with Davidson-Ibarra bound Davidson:2002qv while dark matter is still in thermal equilibrium. On the other hand, dark matter will make little difference in baryon asymmetry result in the case of Freeze-in due to the weak interaction. Alternatively, low-scale leptogenesis will be more attractive because high-scale leptogenesis is difficult to be tested at colliders. The degenerate right-handed neutrino mass can contribute to a resonant enhancement to the CP violation so that one can obtain the baryon asymmetry with right-handed mass at the TeV scale, which is the so-called resonant leptogenesis Pilaftsis:2003gt . Other low-scale leptogenesis scenarios such as the Akhmedov-Rubakov-Smirnov (ARS) mechanism can be found in Hambye:2016sby ; Alanne:2018brf ; Liu:2020mxj . Although we can decrease right-handed neutrino mass to a low scale, dark matter may still be hardly relevant to leptogenesis since the dark matter will freeze out earlier than leptogenesis in the case of dark matter mass much larger than right-handed neutrino mass.

In fact, the symmetric WIMP DM can be connected with baryon asymmetry via the so-called WIMPy miracle Cui:2011ab , where the WIMP DM annihilation is directly responsible for baryogenesis. In the WIMPy baryogenesis, the Sakharov conditions are satisfied with: baryon number violation, CP violation and departure from thermal equilibrium, then a non-zero baryon number asymmetry can be generated from dark matter annihilation, and one can obtain the observed baryon asymmetry as well as dark matter relic density simultaneously. Furthermore, the WIMPy leptogenesis has also been discussed in Kumar:2013uca , in which the lepton asymmetry arises from dark matter annihilation processes which violate CP and lepton number, and the lepton asymmetry is then converted into baryon asymmetry by electroweak sphalerons.

In this paper, we will not focus on the new mechanism to generate the baryon asymmetry and dark matter relic density but focus on the interplay between dark matter and baryon asymmetry. Therefore, we discuss them in a conservative choice that dark matter relic density is generated by the Freeze-out mechanism and baryon asymmetry is obtained via resonant leptogenesis, and the latter means right-handed neutrino masses are degenerate. Although baryon asymmetry is not generated by dark matter annihilation with the WIMPy miracle, the existence of dark matter can affect the baryon asymmetry result as long as dark matter production is related to right-handed neutrinos. We consider the case that right-handed neutrino masses are approximate to the dark matter mass. Such a region will be much more interesting since either dark matter or right-handed neutrinos are out-of-equilibrium and dark matter Freeze-out as well as leptogenesis can occur nearly at the same time. Generally speaking, new processes related to right-handed neutrinos will keep right-handed neutrino number density close to equilibrium so that dilute the baryon asymmetry. However, dark matter can also annihilate into a pair of right-handed neutrinos therefore baryon asymmetry will be strengthened. Notice that right-handed mass can be at the TeV scale in the case of successful resonant leptogenesis, and we demand the dark matter mass also at the TeV scale.

We will not give a UV-completion model in this work, but a minimal scenario including right-handed neutrino and dark sector. For the dark sector, we introduce a singlet scalar $S$ and a fermion $\chi$, where $\chi$ carries $i$ charge and $S$ with $-1$ charge under discrete $Z_{4}$ symmetry. $S$ obtains vaccum expectation value and $\chi$ acquires mass after spontaneously breaking. For the visible sector, we introduce three right-handed neutrinos $N_{j}$ ($j=1,2,3$) as well as a singlet scalar $\phi$ to the Standard Model (SM). The singlet $\phi$ also obtains the non-zero vev and couples to the three right-handed neutrinos with $\phi N_{i}N_{j}$. As for the scalar $\phi$, we have the following comments. Firstly, the singlet-neutrino couplings break the accidental global lepton-number symmetry of the SM, which is similar to the right-handed neutrino Majorana masses in the seesaw Lagrangian. As mentioned in Ref. Alanne:2018brf , this indicates such Yukawa couplings are possible to present in any low energy effective theory that contains the singlet scalar $\phi$ and a set of sterile neutrinos. Then, we assume the singlet $\phi$ is odd under a $Z_{2}$ symmetry of the scalar potential, such $Z_{2}$ symmetry can be the resident symmetry of a new gauge symmetry such as $U(1)_{B-L}$ after spontaneously breaking and the $Z_{2}$ symmetry can limit the couplings in the scalar potential. On the other hand, the $Z_{2}$ symmetry in the scalar potential may be at most an approximate symmetry that is explicitly broken by the singlet-neutrino couplings due to the small couplings. Last but not least, the introduction of the $Z_{2}$ symmetry can decrease the input parameters in the model and simplify our discussion. $\phi$ does not couple to $\chi$ directly because of the different discrete symmetry and we introduce $S$ to the model. As a result, the mixings of the scalars can induce processes related to dark matter and right-handed neutrinos. We consider the decoupling limit that both mixing angles of the singlet scalars with SM Higgs can be negligible so that contribution of the SM Higgs to the dark matter production can be ignored, and dark matter relic density is related to the mixing of the singlet scalars.

The paper is arranged as followed, we describe our framework in section 2. In section 3, we give the Blotzman equations related to baryon asymmetry and dark matter. In section 4, we give the evolution of dark matter with different parameters. In section 5, we scan the parameter space the satisfying dark matter relic density constraint as well as baryon asymmetry constraint and discuss the relationship between dark matter and baryon asymmetry. We give a summary in the last part.

In this part, we give the framework including dark matter and right-handed neutrino. For the dark sector, we introduce one singlet scalar $S$ and a fermion $\chi$, where $\chi$ as the dark matter carries $i$ charge under a discrete $Z_{4}$ symmetry and $S$ charge is $-1$. $S$ can obtain non-zero vev $v_{s}$ after spontaneously symmetry breaking (SSB) so that $\chi$ will acquire mass. We also introduce three right-handed neutrinos and another singlet scalar $\phi$ to the Standard Model. The additional Lagrangian is given as followed,

suppressing the generation indexes, where $L$ is the SM leptons and $H$ is the SM Higgs doublet. The term $\mathcal{V}(S,H,\phi)$ in Eq. 1 involves Higgs potential and other terms with the singlet scalar $S$ as well as $\phi$. We assume the singlet $\phi$ is $Z_{2}$ odd in the scalar potential and obtain the vaccum expectation value $v_{b}$ after SSB. Therefore, the term $\mathcal{V}(S,H,\phi)$ can be given by,

Note that we have no terms such as $H^{2}S\phi$ because $S$ and $\phi$ carry different charges. In the unitary gauge, $H$, $S$ and $\phi$ can be given by,

where $v=246$ GeV is the SM vaccum expectation value. We consider the decoupling limit that $\lambda_{hp},\lambda_{hs}\ll 1$ so that the mass matrix $M_{h}$ for scalars after SSB can be simplified by,

The mass matrix $M_{h}$ can be diagonalized by the matrix $U$ with

where $\theta$ is the mixing angle of the two singlet scalars. Correspondingly, after rotation from the flavor eigenstate to the mass eigenstate, we have three Higgses defined by,

where $h_{0}$ is the SM Higgs, and $h_{1,2}$ are the new Higgs particles in the model. Particularly, $h_{1}$ and $h_{2}$ can mix with each other, and the mixing angle $\theta$ will also determine dark matter relic density. Furthermore, we choose Higgs masses as well as the mixing angle as the inputs so that parameters related to singlet scalars in the model can be expressed as follows,

where $m_{1,2}$ are the $h_{1,2}$ mass. To obtain a stable potential, we have

known as co-positivity constraints Kannike:2012pe , and the perturbativity constraints are given by $\lambda_{sp,s,p}<4\pi$.

We consider a conservative choice of the decoupling limit that the mixings of singlets with SM doublets are negligible. Therefore, the contribution of SM particles to the dark matter production such as $ff\to\chi\chi$ are highly suppressed where $f$ represents the SM fermions, and the left relevant processes with dark matter are the right-handed neutrinos as well as the $h_{1,2}$. It is worth stressing that the most stringent bounds on the mixing angle $\alpha$ of the SM Higgs arise from the $W$ boson mass correction Lopez-Val:2014jva , and the current constraint is given by $|sin\alpha|\lesssim 0.24$ at $95\%$ C.L. according to Papaefstathiou:2022oyi .

In this work, we assume the vaccum expectation value $v_{b}$ as a free parameter and both $v_{b}$ and $v_{s}$ are beyond EWPT (electroweak phase transition) scale. For simplicity, we fix $v_{b}=2$ TeV as a constant in the following discussion. After SSB, dark matter mass $m_{\chi}$ can be given by $m_{\chi}=\lambda_{sx}v_{s}/2\sqrt{2}$, the term $(M_{n}+\lambda_{mn}v_{b}/\sqrt{2})/2$ gives the Majarona mass matrix of the right-handed neutrino with $M_{n}$ being the bare mass term of right-handed neutrinos, and the dynamical origin of $M_{n}$ is left unspecified for our work. The matrix $\lambda_{mn}$ characterizes the strength of the Yukawa coupling of $\phi NN$ while $\lambda_{sx}$ describes the strength of singlet-DM Yukawa interaction.

Notice that the existence of the singlet scalar $\phi$ can also induce a successful low-scale leptogenesis by the scalar-singlet-mediated one-loop diagrams due to the Yukawa coupling with right-handed neutrinos Alanne:2018brf only if $M_{n}$ and $\lambda_{mn}$ can not be diagonalized simultaneously, and we ignore the effect of $\phi$ on leptogenesis in our work.

In this work, we consider the interplay between dark matter and leptogenesis in a minimal framework, we assume dark matter relic density is generated via the Freeze-out mechanism, and dark matter production processes related to SM particles are highly suppressed under the decoupling limit, while almost contribution to DM relic density comes from the new Higgses as well as right-handed neutrinos. On the other hand, we consider the observed baryon asymmetry is generated by resonant leptogenesis, and annihilation of right-handed neutrinos to dark matter may keep the neutrino number close to thermal equilibrium which dilutes the baryon asymmetry. Inversely, more right-handed neutrinos can be generated by the inverse process so that the baryon asymmetry will be strengthened. In both cases, the final baryon asymmetry can be influenced by the introduction of dark matter.

In this section, we discuss the Boltzmann equations of $N$ and dark matter. In our work, we consider the decoupling limit that the mixings of both singlets with SM doublets are negligible. Under such consideration, dark matter production is mainly related to right-handed neutrinos as well as $h_{1,2}$, while the contribution from SM particles is highly suppressed. Extra contribution to the leptogenesis can arise from the channels $NN\to\chi\chi$, which is also related to dark matter. Notice that DM can annihilate into pairs of scalars that are heavier than the DM particles, which is similar with the so-called forbidden DM scenario DAgnolo:2015ujb . On the other hand, we assume $m_{N}<m_{1,2}$ and ignore the contribution of $NN\to h_{1,2}h_{1,2}$ to the BAU.

The Boltzmann equations for the $N$ abundance $Y_{N}$, dark matter abundance $Y_{X}$ and the total $(B-L)$ asymmetry $Y_{B-L}$ are given by Liu:2021akf ,

where $\epsilon_{CP}$ is the CP asymmetry parameter. The CP asymmetry $\epsilon_{i}$ can be given by Iso:2010mv :

where

are respectively the vertex correction and RHN self-energy correction to the decay process with $m_{N_{i}}$ the $N_{i}$ mass and $\Gamma_{N_{i}}$ the $N_{i}$ decay width. On the other hand, the tiny left-handed neutrino masses $m_{\nu}$ are generated via the Type-I seesaw mechanism with Liu:2021akf :

For the $\mathcal{O}(\mathrm{TeV})$ leptogenesis, we have $y\sim 10^{-6}$, and this makes the contribution of terms $\gamma_{hs,ht,N,Nt}$ rather small since they are proportional to $y^{2}$ or $y^{4}$.

The BAU obtains contributions from three right-handed neutrinos when we assume the right-handed neutrinos are degenerate. If the decay widths of the three right-handed neutrinos are comparable, then the generated BAU should be three times that mere one right-handed neutrino decay. On the other hand, if the decay widths have a hierarchy, the CP asymmetries will also do so and the generated BAU can be dominated by one of the right-handed neutrinos Iso:2010mv , so that a one-flavor discussion will be sufficient, and we consider such a scenario in this work and we have $\epsilon_{CP}=\epsilon_{1}$. For simplicity, we assume that dark matter is also determined by mere one right-handed neutrino, and we will have a similar conclusion when we consider three right-handed neutrinos. We denote $N$ as $N_{1}$ and $\lambda_{mn}$ as the Yukawa coupling of $\phi N_{1}N_{1}$ for simplicity. $z$ is defined by $z=m_{N}/T$ with $T$ being the temperature. $H_{N}$ and $s_{N}$ correspond to the Hubble rate and entropy density respectively. $Y_{Neq}$ as well as $Y_{Xeq}$ correspond to the abundance of right-handed neutrino and dark matter at thermal equilibrium respectively, which can be given by,

and $Y_{Leq}$ is the lepton abundance at thermal equilibrium with

where $g_{*}=106.75$ is the effective degree of freedom, $\zeta(x)$ is the Riemann zeta function and $K_{2}(x)$ is the Bessel function.

The term $\gamma_{D}$ is the reaction rate for $N\to HL$, $\gamma_{hs}$ and $\gamma_{ht}$ are the reaction rate of s-channel $NL\to qt$ and t-channel $Nt\to Lq$ mediated by Higgs, where $t$ is the top quark. The terms $\gamma_{N}$ and $\gamma_{Nt}$ in the $(B-L)$ asymmetry equations are the s-channel and t-channel contributions of $LH\to\bar{L}H$, which can wash out the baryon asymmetry. We follow the results of Iso:2010mv , and the analytic expressions for these terms can be found in Iso:2010mv . We have new terms involving dark matter as well as RHN. The term $\gamma_{N\chi}$ represents the reaction rate of $NN\to\chi\chi$, which is defined by,

where $K_{1}(x)$ is the modified Bessel function, $x$ is the squared center-of-mass energy and $\hat{\sigma_{N\chi}}(x)$ is the reduced cross section of $NN\to\chi\chi$, which is given by,

where $\lambda(a,b,c)=\sqrt{(a-b-c)^{2}-4bc}$, $\theta(\sqrt{x}-2m_{N})$ is the theta function. Note that the contribution of right-handed neutrinos to dark matter can be ignored in the limit of $\sin 2\theta\to 0$ so that dark matter relic density is determined by the new Higgses, and dark matter will make no difference in the baryon asymmetry.

The term $\gamma_{\chi h}$ represents the sum of the reaction rate for $\chi\chi\to h_{1}h_{1},h_{2}h_{2}$ and $h_{1}h_{2}$. Relevant processes are given in Fig. 1, we have the following results of the reduced cross section,

where $\sigma_{ij}$ corresponds to the cross section of $\chi\chi\to h_{i}h_{j}$ with $i,j=1,2$. The cross sections related to the processes of Fig. 1 are calculated with Calchep Belyaev:2012qa , and the expressions involving only s-channels are given as follows, while the complete expressions can be found in App. A.

and

For the reaction rate of $\chi\chi\to h_{1}h_{1},h_{1}h_{2}$ and $h_{2}h_{2}$, we have

Note that we have ignored the contribution of $\chi\chi\to\nu\nu$ and $\chi\chi\to N\nu$ due to the tiny heavy-light neutrino mixing angle, where $\nu$ represents neutrino. On the other hand, the interactions involving dark matter as well as $h_{1,2}$ do not enter the baryon asymmetry equation at this order and therefore can not wash out the asymmetry.

We assume our universe started with the total $(B-L)$ charge zero, and the non-zero baryon asymmetry $Y_{B}$ can be dynamically generated above the sphaleron decoupling temperature $T_{sph}=131.7$ GeV DOnofrio:2014rug , which is given by $Y_{B}=\frac{28}{79}Y_{B-L}$ Pilaftsis:2003gt . Alternatively, we focus on the strong wash-out regime that $\tilde{m}/m_{*}\gg 1$, where $\tilde{m}$ is the effective neutrino mass defined by $v^{2}(yy^{\dagger})/m_{N}$ and $m_{*}\approx 1.08\times 10^{-3}$ eV is the equilibrium neutrino mass. Therefore, flavor effects in the charged-lepton sector are not expected to have a major influence on our results Hambye:2016sby .

Technically, we implement the model with Feynrules Alloul:2013bka , and calculate the relic density with MicrOMEGAs Belanger:2013oya numerically. We give the evolution of dark matter relic density with dark matter mass $m_{\chi}$ in Fig. 2 and Fig. 3 corresponding to $\sin\theta=0.01$ and $\sin\theta=0.9$ respectively, where dark matter mass is set within $[400$ GeV, 2000 GeV]. In both figures, the black line is the observed value with $\Omega h^{2}=0.12$ Planck:2015ica . The blue line is the benchmark line we choose $m_{2}=1500$ GeV, $m_{1}=1000$ GeV, $\lambda_{sx}=0.1$, $\lambda_{mn}=0.1$ and $m_{N}=800$ GeV, and other colored lines correspond to the case varying one of the parameters.

In Fig. 2, the small $\sin\theta$ value limits the contribution of right-handed neutrinos to dark matter, and new Higgses play a more important role in determining DM relic density. The relic density increases with the increase of $m_{\chi}$ but we have a peak in the case of $m_{\chi}\approx 1/2m_{1}$ due to the s-channel resonant-enhanced processes of $\chi\chi\to NN$. However, such a resonant-enhanced effect does not decrease relic density a lot due to the small $\sin\theta$ as we mentioned above. On the other hand, the small dark matter-scalar Yukawa coupling $\lambda_{sx}$ induces a small annihilation cross section so that dark matter is over-abundant, and dark matter relic density much decreases in the case of $\lambda_{sx}=1$, where the processes of dark matter annihilation into new Higgses are dominant and the peak arising from s-channel enhanced resonant is not obvious.

According to Fig. 3, we have different results in the case of $\sin\theta=0.9$ since right-handed neutrinos can also play important role in determining dark matter relic density. We have a resonant region at about $m_{\chi}=1/2m_{1}$, where the relic density drops sharply, and intersect with the relic density constraint curve. What’s more, we have another peak in the case of $m_{\chi}\approx m_{1}$, where the t-channel Higgs-mediated processes open so that decreasing the relic density. In both figures, the lines correspond to different $\lambda_{mn}$ and $m_{N}$ are almost coincide with the benchmark line, which indicates $m_{N}$ and small $\lambda_{mn}$ can make little difference on relic density.

Aside from the observed relic density constraint, direct detection for dark matter puts the most stringent limit on the dark matter parameter space. Concretely speaking, for dark matter mass $m_{\chi}\gtrsim 2$ GeV, the direct detection experiments give the most stringent constraint on the spin-independent DM matter scattering with nucleon, and XENON1T XENON:2020gfr gives a stringent bound for $m_{\chi}>$ 6 GeV. On the other hand, considering neutrino floor limit Billard:2021uyg ; Billard:2013qya on the current probe sensitive to dark matter, WIMP dark matter is facing a serious crisis since there is no evidence for the existence of dark matter. Fortunately, in our work, direct detection constraint is much weak since processes involving dark matter are new Higgses as well as right-handed neutrinos but SM particles are almost irrelevant.

In this work, we discuss the interplay between dark matter and leptogenesis in a common framework. We do not give a UV-completion model but consider a minimal scenario including right-handed neutrino and a fermion dark matter. Dark matter and right-handed neutrino are connected by the mixing of two singlet scalar fields. We consider the decoupling limit that mixings of SM doublets with the singlet fields negligible, and dark matter relic density is determined by the two new Higgs particles and right-handed neutrinos. On the other hand, we consider the right-handed neutrino masses are degenerate at the TeV level and the baryon asymmetry is generated by the resonant leptogenesis. As for the two singlet scalar fields, we consider two cases with $m_{2}=1.5m_{1}$ and $m_{2}=0.8m_{1}$ for simplicity. We scan the parameter space satisfying relic density constraint in the case of $\sin\theta=0.9$ and $\sin\theta=0.01$, and found a more flexible parameter space in the case of $\sin\theta=0.9$, where $NN\to\chi\chi$ can also play an important role in determining dark matter relic density. Then, we discuss the relationship between dark matter and leptogenesis in the case of $\sin\theta=0.9$ since dark matter will make little difference in the leptogenesis in the case of small $\sin\theta$. We generate the baryon asymmetry via the resonant leptogenesis, and the existence of dark matter in the model can not only dilute the baryon asymmetry result but may also strengthen the baryon asymmetry since more right-handed neutrnios can be generated via the process of $\chi\chi\to NN$, and both the enhanced effect and dilution effect can occur in either case of $m_{\chi}>m_{N}$ and $m_{\chi}<m_{N}$.