Two-component Dark Matter with co-genesis of Baryon Asymmetry of the Universe

We discuss the possibility of realising a two-component dark matter (DM) scenario where the two DM candidates differ from each other by virtue of their production mechanism in the early universe. One of the DM candidates is thermally generated in a way similar to the weakly interacting massive particle (WIMP) paradigm where the DM abundance is governed by its freeze-out while the other candidate is produced only from non-thermal contributions similar to freeze-in mechanism. We discuss this in a minimal extension of the standard model where light neutrino masses arise radiatively in a way similar to the scotogenic models with DM particles going inside the loop. The lepton asymmetry is generated at the same time from WIMP DM annihilations as well as partially from the mother particle for non-thermal DM. This can be achieved while satisfying the relevant experimental bounds, and keeping the scale of leptogenesis or the thermal DM mass as low as 3 TeV, well within present experimental reach. In contrast to the TeV scale thermal DM mass, the non-thermal DM can be as low as a few keV, giving rise to the possibility of a sub-dominant warm dark matter (WDM) component that can have interesting consequences on structure formation. The model also has tantalizing prospects of being detected at ongoing direct detection experiments as well as the ones looking for charged lepton flavour violating process like $\mu \rightarrow e \gamma$.


I. INTRODUCTION
There have been irrefutable amount of evidences suggesting the presence of a mysterious, non-luminous, collisionless and non-baryonic form of matter in the present universe [1]. The hypothesis for existence of this form of matter, more popularly known as dark matter (DM) due to its non-luminous nature, is strongly backed by early galaxy cluster observations [2], observations of galaxy rotation curves [3], the more recent observation of the bullet cluster [4] and the latest cosmological data provided by the Planck satellite [5]. The latest data from the Planck satellite suggest that around 27% of the present universe's energy density is in the form of dark matter. In terms of density parameter Ω and h = (Hubble Parameter)/(100 kms −1 Mpc −1 ), the present dark matter abundance is conventionally reported as Ω DM h 2 = 0.120 ± 0.001 (1) at 68% CL [5]. While such astrophysics and cosmology based experiments are providing such evidence suggesting the presence of dark matter at regular intervals in the last several decades, there is hardly anything known about the particle nature of it. The requirements which a particle dark matter candidate has to satisfy, as pointed out in details by the authors of [6] rule out all the standard model (SM) particles from being DM candidates. While the neutrinos in the SM come very close to satisfying these requirements, they have tiny abundance in the present universe. Apart from that, they have a large free streaming length (FSL) due to their relativistic nature and give rise to hot dark matter (HDM), ruled out by observations. This has led to a plethora of beyond standard model (BSM) scenarios proposed by the particle physics community to account for dark matter in the universe. Most of these BSM scenarios are based on a popular formalism known as the weakly interacting massive particle (WIMP) paradigm. In this formalism, a particle dark matter candidate having mass around the electroweak scale and having electroweak type couplings to SM particles can give rise to the correct relic abundance in the present epoch, a remarkable coincidence often referred to as the WIMP Miracle [7]. Since the mass is around the electroweak corner and couplings to the SM particles are sizeable, such DM candidates are produced thermally in the early universe followed by its departure from chemical equilibrium leading to its freeze-out. Such DM candidates typically become non-relativistic shortly before the epoch of freeze-out and much before the epoch of matter-radiation equality. Such DM candidates are also categorised as cold dark matter (CDM).
The CDM candidates in the WIMP paradigm have very good direct detection prospects due to its sizeable interaction strength with SM particles and hence can be observed at ongoing and future direct search experiments [8][9][10][11][12][13][14][15]. However, no such detection has yet been done casting doubts over the viability of such DM paradigms. This has also motivated the particle physics community to look for other alternatives to WIMP paradigm. Although such null results could indicate a very constrained region of WIMP parameter space, they have also motivated the particle physics community to look for beyond the thermal WIMP paradigm where the interaction scale of DM particle can be much lower than the scale of weak interaction i.e. DM may be more feebly interacting than the thermal WIMP paradigm. This falls under the ballpark of non-thermal DM [16]. In this scenario, the initial number density of DM in the early Universe is negligible and it is assumed that the interaction strength of DM with other particles in the thermal bath is so feeble that it never reaches thermal equilibrium at any epoch in the early Universe. In this set up, DM is mainly produced from the out of equilibrium decays of some heavy particles in the plasma. It can also be produced from the scatterings of bath particles, however if same couplings are involved in both decay as well as scattering processes then the former has the dominant contribution to DM relic density over the latter one [16]. The production mechanism for non-thermal DM is known as freezein and the candidates of non-thermal DM produced via freeze-in are often classified into a group called Freeze-in (Feebly interacting) massive particle (FIMP). For a recent review of this DM paradigm, please see [17]. Interestingly, such non-thermal DM candidates can have a wide range of allowed masses, well beyond the typical WIMP regime. was also considered in some recent works [22,23]. However, our model is not restrictive to such combinations as we show that the non-thermal DM candidate can have masses in the keV-GeV range as well.
Apart from the mysterious 27% of the universe in the form of unknown DM, the visible sector making up to 5% of the universe also creates a puzzle. This is due to the asymmetric nature of the visible sector. The visible or baryonic part of the universe has an abundance of baryons over anti-baryons. This is also quoted as baryon to photon ratio (n B − nB)/n γ ≈ 10 −10 which is rather large keeping in view of the large number density of photons. If the universe is assumed to start in a symmetric manner at the big bang epoch which is a generic assumption, there has to be a dynamical mechanism that can lead to a baryon asymmetric universe at present epoch. The requirements such a dynamical mechanism needs to satisfy were put forward by Sakharov more than fifty years ago, known as the Sakharov's conditions [24]: baryon number (B) violation, C and CP violation and departure from thermal equilibrium. Unfortunately, all these requirements can not be fulfilled in the required amount within the framework of the SM, again leading to several BSM scenarios.
out of equilibrium decay of a heavy particle leading to the generation of baryon asymmetry has been a very well known mechanism for baryogenesis [25,26]. One interesting way to implement such a mechanism is leptogenesis [27] where a net leptonic asymmetry is generated first which gets converted into baryon asymmetry through B + L violating EW sphaleron transitions. The interesting feature of this scenario is that the required lepton asymmetry can be generated within the framework of the seesaw mechanism [28][29][30][31][32][33] that explains the origin of tiny neutrino masses [1], another observed phenomena which the SM fails to address.
Although the explanation for dark matter, baryon asymmetry of the universe and origin of neutrino mass can arise independently in different BSM frameworks, it is interesting, economical and predictive to consider a common framework for their origin. In fact a connection between DM and baryons appears to be a natural possibility to understand their same order of magnitude abundance Ω DM ≈ 5Ω B . Discarding the possibility of any numerical coincidence, one is left with the task of constructing theories that can relate the origin of these two observed phenomena in a unified manner. There have been several proposals already which mainly fall into two broad categories. In the first one, the usual mechanism for baryogenesis is extended to apply to the dark sector which is also asymmetric [34][35][36][37].
The second one is to produce such asymmetries through annihilations [38][39][40] where one or more particles involved in the annihilations eventually go out of thermal equilibrium in order to generate a net asymmetry. The so-called WIMPy baryogenesis [41][42][43] belongs to this category, where a dark matter particle freezes out to generate its own relic abundance and then an asymmetry in the baryon sector is produced from DM annihilations. The idea extended to leptogenesis is called WIMPy leptogenesis [44][45][46][47]. Motivated by all these, we This paper is organised as follows. In section II we discuss our model followed by the origin of neutrino mass in section III. In section IV we describe the co-genesis of WIMP,

II. THE MODEL
We consider a minimal extension of the SM by two different types of singlet fermions and three different types of scalar fields shown in table I, II respectively. To achieve the desired interactions of these new fields among themselves as well as with the SM particles, we consider additional discrete symmetries Z 2 × Z 2 . While one such Z 2 symmetry is enough to accommodate DM, radiative neutrino mass as well as generation of lepton asymmetry from DM annihilation in a way similar to what we achieve in a version of scotogenic model [47], the other discrete symmetry Z 2 is required in order to have the desired couplings of

FIMP DM. To prevent tree level interaction between FIMP DM and SM leptons through
LHχ (needed to avoid the decay of χ to light SM particles), we have introduced this another discrete symmetry Z 2 under which χ, φ and another singlet scalar φ are odd. If φ acquires a non-zero vacuum expectation value (vev), it can lead to one loop mixing between neutrinos and non-thermal DM. This possibility is shown in table I, II.
The relevant part of the Yukawa Lagrangian is The scalar potential is Since we require the SM Higgs and the singlet scalar φ to acquire non-zero vev as we minimise the above scalar potential with respect to these two fields and find the following minimisation conditions.
The corresponding mass squared matrix is This will give rise to a mixing between the SM like Higgs and a singlet scalar given by where in the last step we have assumed a hierarchy u v. The mass eigenstates corresponding to the charged and pseudo-scalar components of η are The neutral scalar component of η and φ mix with each other resulting in the following mass squared matrix.
which can be diagonalized by a 2 × 2 unitary matrix with the mixing angle given by Considering |m η −m φ | < m χ , we can prevent the three body decays η → φχν or φ → ηχν.
Even if we allow such three body decays, they will be phase space suppressed compared to two body decays which contribute to the production of non-thermal DM χ which we will discuss shortly. We also consider the mixing between η−φ to be non-zero so that the thermal DM is an admixture of singlet and doublet scalars 1 . This has crucial implications for DM phenomenology as well as leptogenesis as we discuss below. Assuming m χ ∼ keV-GeV, we consider its production mechanisms which also have the potential to produce a lepton asymmetry. The relevant diagrams for producing lepton asymmetry and non-thermal DM are shown in FIG. 1 and 2, respectively. we implement the model in SARAH 4 [48] and extract the thermally averaged annihilation rates from micrOMEGAs 4.3 [49] to use while solving the relevant Boltzmann equations to be discussed below.
Feynman diagrams corresponding to the production of non-thermal DM χ.

III. NEUTRINO MASS
As can be noticed from the particle content of the model and the Yukawa Lagrangian  [50]. The one-loop expression for neutrino mass is where M k is the right handed neutrino mass. The above Eq. (13) equivalently can be written where Λ can be defined as, In order to incorporate the constraints from neutrino oscillation data on three mixing angles and two mass squared differences, it is often useful to express these Yukawa couplings in terms of light neutrino parameters. This is possible through the Casas-Ibarra (CI) parametrisation [51] extended to radiative seesaw model [52] which allows us to write the Yukawa couplings as Here m diag ν = diag(m 1 , m 2 , m 3 ) is the diagonal light neutrino mass matrix and R can be a complex orthogonal matrix in general with RR T = 1 which we have taken it to be a general, this 3 × 3 orthogonal matrix R can be parametrised by three complex parameters of type In general, the orthogonal matrix R for n flavours can be product of n C 2 number of rotation matrices of type with rotation in the α − β plane and dots stand for zero. For example, taking α = 1, β = 2 we have 2 For some more discussions on different possible structure of this matrix and implications on a particular leptogenesis scenario in this model, we refer to the recent work [54].
We see that CP phases in U do not contribute to N i η given in eq.(24), but complex variables in the orthogonal matrix R can lead to non-vanishing value of N i η . This is similar to leptogenesis from pure decay in this model [55] where, in the absence of flavour effects, the orthogonal matrix R played a crucial role. The matrix denoted by U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix If the charged lepton mass matrix is diagonal or equivalently, U L = 1, then the PMNS mixing matrix is identical to the diagonalising matrix of neutrino mass matrix. The PMNS mixing matrix can be parametrised as where c ij = cos θ ij , s ij = sin θ ij and δ is the leptonic Dirac CP phase. The diagonal matrix For such tiny couplings, the mixing between non-thermal DM and light neutrinos will be too small to have any observable consequences like monochromatic lines in X-ray or Gammaray spectrum. We leave such exploration of detection prospects for such non-thermal DM candidates to future studies.

IV. ANALYSIS OF CO-GENESIS
In order to do the entire analysis we need to solve the coupled differential equations  For this scenario the Boltzmann equations for the Z 2 odd particles take the following form: where show the details of vanilla leptogenesis here, but include that contribution into account for the final lepton asymmetry. Instead, we highlight the other feature from WIMP DM annihilation, as it connects the source of baryon asymmetry to the dark matter sector. For the details of vanilla leptogenesis, we refer to the above references where vanilla leptogenesis was studied in the context of type I seesaw as well as minimal scotogenic model. It should be noted that typically, the lowest scale of lepton number violation is more effective in creating lepton asymmetry. This scale, in our case is the scale of WIMP DM freeze-out, which lies below the right handed neutrino masses. However, if right handed neutrino masses are not very heavy compared to WIMP DM mass, then both of them can have some sizeable contribution to the origin of lepton asymmetry. We will show more details of our hybrid source of leptogenesis in a companion paper.
The Boltzmann equations responsible for the DM density is given by eq.(21) and leptonic asymmetry is given as follows: And the CP asymmetry which is arising from the interference between tree and 1-loop diagrams in Fig. 1 can be estimated as It should be noted that in the above expression always (1 ≤ r j ≤ r i ) where j stands for N j inside the loop while i stands for N i as one of the initial state particles, shown in Fig.  1. This is simply to realise the "on-shell" -ness of the loop particles in order to generate the required CP asymmetry. In the above Boltzmann equation we see that along with the process which produces the asymmetry i.e σv N i η→XL and Γ N i →Lη we have washout terms coming from three kinds of processes: 1)the process σv ηη→LL poses as one of the wash out along with 2) σv ηL→ηL . Now, according to Cui et. al [41] if we need to achieve asymmetry through dark matter annihilation then the wash-out processes σv N i η→XL should freeze-out before the WIMP freeze-out. In order to do that one has to keep the following ratio below So, in our case σ ann v is similar to that of the standard Inert Doublet Model WIMP annihilation channel (ηη → W + W − ) which is naturally stronger than the σ wash−out v which in our case is for (N i η → XL). Further details of the asymmetry generated through such t channel annihilations of dark matter are shown in [47].
Non-thermal DM can be produced in a way similar to the FIMP scenario mentioned above.
In such a case, the initial abundance is assumed to be zero or negligible and its interaction rate with the standard model particles or thermal bath is so feeble that thermal equilibrium is never attained. In such a case, non-thermal can be produced by out of equilibrium decays or scattering from particles in the thermal bath while the former typically dominates if same type of couplings is involved in both the processes. Further details of this mechanism for keV scale sterile neutrinos can be found in [65][66][67] as well as the review on keV sterile neutrino DM [18] 4 . For a general review of FIMP DM paradigm, please see [17], as mentioned earlier.
Using the FIMP prescription described in the above-mentioned works, we can write down the corresponding Boltzmann equation for χ, the FIMP candidate as Here the first contribution on the right hand side is from the decay process N → φχ while the second one is from annihilation N N → χχ. The fact that χ was never produced in equilibrium requires the Yukawa coupling governing the interaction among N, φ, χ to be very small, as we mention below. Since the same Yukawa coupling appears twice in the annihilation process N N → χχ, the two body decay will dominate the production. Another dominant contribution can come from the s-channel annihilation process of (ν, l ± ), (η 0 , η ± ) → φχ that appears in the third term on the right hand side of the above equation. The dominant production processes of χ in our work are shown in FIG. 2.

V. DIRECT DETECTION AND LEPTON FLAVOUR VIOLATION
Although the detection prospects of FIMP candidate are very limited, the WIMP can have very good direct detection signatures that can be probed at direct detection experiments like LUX [11], PandaX-II [8,9] and Xenon1T [10,72]. Since the WIMP is a scalar, we can have Higgs mediated spin independent elastic scattering of DM off nucleons. This direct detection cross section can be estimated as [73] σ SI = λ 2 L f 2 4π where µ = m n m DM /(m n + m DM ) is the DM-nucleon reduced mass and λ L is the quartic coupling involved in DM-Higgs interaction. For WIMP, an admixture of scalar doublet and scalar singlet given by η 1 = cos θ 2 η R + sin θ 2 φ, the Higgs-DM coupling will be λ L = cos θ 2 (λ 3 + λ 4 + λ 5 ) + sin θ 2 λ 7 /2. A recent estimate of the Higgs-nucleon coupling f gives f = 0.32 [74] although the full range of allowed values is f = 0.26 − 0.63 [75]. Since DM has a doublet component in it, there arises the possibility of tree level Z boson mediated processes η R n → η I n, n being a nucleon. This process, if allowed, can give rise to a very large direct detection rate ruled out by experimental data. However, due to the inelastic nature of the process, one can forbid such scattering if δ = m η I − m η R > 100 keV, typical kinetic energy of DM particle.
Another interesting observational prospect of our model is the area of charged lepton Here we use the SPheno 3.1 interface to check the constraints from cLFV data. We particularly focus on three such cLFV decays namely, µ → eγ, µ → 3e and µ → e (Ti) conversion that not only are strongly constrained by present experiments but also have tantalising future prospects [52]. The present bounds are: BR(µ → eγ) < 4.2 × 10 −13 [77], BR(µ → 3e) < 1.0 × 10 −12 [78], CR(µ, Ti → e, Ti) < 4.3 × 10 −12 [79]. It may be noted that the sensitivities of the first two processes will be improved by around one order of magnitude compared the present upper limit on branching ratios. On the other hand, the µ to e conversion (Ti) sensitivity will be increased by six order of magnitudes [52] making it a highly promising test of different new physics scenarios around the TeV corner.

VI. RESULTS AND DISCUSSION
Since we have a large parameter space, we first choose the benchmark points in a way that gives rise to the desired phenomenology. Also, we choose three different masses of FIMP DM namely, 1 keV, 1 MeV and 1 GeV and choose the parameters in such a way that all these three cases correspond to 50% contribution of FIMP to total DM abundance. The WIMP DM mass is kept fixed at 3 TeV in our analysis. We find that this relative contribution of As we can see from these plots, the WIMP as well as the lightest right handed neutrino are in equilibrium initially followed by WIMP freeze-out and right handed neutrino decay  asymmetry as well as FIMP generation respectively, one can see the yield in ∆L and FIMP by the epochs of WIMP freeze-out and right handed neutrino decay. It can be seen from these plots that the required asymmetry along with WIMP-FIMP relative abundance can be achieved simultaneously leading to a successful co-genesis. In order to get the leptonic asymmetry we need the yukawa coupling y ij to be of O(1) which would be fulfilled if we take the λ 5 to be very less, to be in agreement with light neutrino masses discussed above.
In doing so we would be compromising the mass difference between the η R and η I . The decreasing of the mass difference opens up the inelastic channel η R , (n, p) → η I , (n, p) which is ruled out, as mentioned earlier. This is where the singlet scalar φ comes to rescue as it relaxes the tension among neutrino data, dark matter direct detection and generating correct lepton asymmetry This was also noted in a recent work [47]. The mixing between the doublet and singlet scalars through λ 9 helps in evading the Direct-Detection bound as is enters the effective Z-coupling to scalar WIMP. All these cases shown in FIG. 5, 6 It should be noted that the FIMP DM with mass in the keV scale can face constraints from structure formation data. As noted in [20], Lyman-α bounds restrict the keV fermion mass to be above 8 keV if it is non-resonantly produced (similar to our model) and contributes 100% to the total DM abundance. However, for less than 60% contribution to total DM, such strict mass bounds do not apply. Therefore, our benchmark value of 1 keV FIMP mass in one of the cases mentioned above remains safe from such bounds.
In table V we show the other parameters of the model for a chosen benchmark point (BP) giving 50% − 50% WIMP-FIMP proportion along with successful leptogenesis. We will compare our subsequent results with respect to this BP that satisfies all our criteria.
We will see that this BP remains sensitive to LFV as well as direct detection experiments.  In FIG. 8 we have shown the scatter plot for LFV branching ratios by varying the key parameters affecting them, as shown in table VI. In all these plots we have not taken any constraint from the relic, but the neutrinos mass constraints are being taken care of by the Casas-Ibarra parametrsation which in turn fixes the Yukawa's. The benchmark point that satisfies all relevant bounds from WIMP-FIMP as well as correct lepton asymmetry is also indicated as BP. For the same range of parameters we also show the WIMP direct detection rates in FIG. 9 where the BP is also indicated. It is clear that our BP is very sensitive to the current experimental upper bounds on µ → eγ as well as direct detection rates, keeping the detection prospects very much optimistic. The WIMP annihilations also produce a non-zero lepton asymmetry in a way similar to WIMPy leptogenesis scenarios. The WIMP is an admixture of a scalar doublet's neutral component and a scalar singlet to satisfy the criteria of neutrino mass, dark matter relic, direct detection and leptogenesis simultaneously. Interestingly, the particles which assist in the production of FIMP also partially contribute to the origin of lepton asymmetry resulting in a hybrid setup. We outline such a hybrid co-genesis of multi-component DM, lepton asymmetry in this work for some benchmark scenarios leaving a more detailed analysis for an upcoming work. We also find that our benchmark point satisfying the required abundance of WIMP-FIMP and baryon asymmetry also remains sensitive to dark matter direct detection as well charged lepton flavour violation like µ → eγ.