# Higgs sector extension of the neutrino minimal standard model with thermal freeze-in production mechanism

###### Abstract

The neutrino minimal Standard Model (MSM) is the minimum extension of the standard model. In this model, the Dodelson-Widrow mechanism (DW) produces keV sterile neutrino dark matter (DM) and the degenerate GeV heavy Majorana neutrinos lead to leptogenesis. However, the DW mechanism has been ruled out by Lyman- bounds and X-ray constraints. An alternative scenario that evades these constraints has been proposed, where the sterile neutrino DM is generated by the thermal freeze-in mechanism via a singlet scalar. In this paper, we consider a Higgs sector extension of the MSM to improve dark matter sectors and leptogenesis scenarios, focusing on the thermal freeze-in production mechanism. We discuss various thermal freeze-in scenarios for the production of keV-MeV sterile neutrino DM with a singlet scalar, and reinvestigate the Lyman- bounds and the X-ray constraints on the parameter regions. Furthermore, we propose thermal freeze-in leptogenesis scenarios in the extended MSM. The singlet scalar needs to be TeV scale in order to generate the observed DM relic density and baryon number density with the thermal freeze-in production mechanism.

^{†}

^{†}preprint: KEK-TH-1798

## I Introduction

The standard model of particle physics (SM) has demonstrated great success in high energy physics. However, it is not the complete theory because the Standard Model cannot treat gravity consistently nor explain several observed phenomena, such as neutrino oscillations, cosmological dark matter, the baryon asymmetry of the universe, the horizon and flatness problems, etc. In the dark matter sector, TeV-scale supersymmetry provides weakly interacting massive particles (WIMPs) as leading dark matter candidates Vecchi (2013). However, the first run of the LHC experiment excludes a significant region of parameter space for the weak super-partners, and the recent results from LUX Faham (2014); Akerib et al. (2014) and XENON100 Lavina (2013); Aprile et al. (2013, 2012) have severely restricted the WIMP cross section. This situation is the same in other beyond standard models. Feebly Interacting Massive Particles (FIMPs) Hall et al. (2010); Blennow, Fernandez-Martinez, and Zaldivar (2014); Yaguna (2012); Bhupal Dev, Mazumdar, and Qutub (2014) are not constrained by such direct detection experiments due to their much smaller couplings. Therefore, FIMPs are an interesting candidate for dark matter.

A sterile neutrino can be a FIMP. In particular, keV sterile neutrinos are a candidate for warm dark matter. The neutrino minimal Standard Model (MSM) Asaka, Blanchet, and Shaposhnikov (2005); Asaka and Shaposhnikov (2005); Canetti et al. (2013) has three right-handed neutrinos below the electroweak-scale and the lightest right-handed neutrino may have a mass around the keV scale. The lightest right-handed neutrino becomes the keV sterile neutrino dark matter and is produced via active-sterile neutrino oscillation, which is called the Dodelson-Widrow mechanism Dodelson and Widrow (1994). The other heavy right-handed neutrinos and lead to leptogenesis via CP-violating oscillations Akhmedov, Rubakov, and Smirnov (1998); Canetti, Drewes, and Shaposhnikov (2013), where the heavy right-handed neutrinos should satisfy . Furthermore, it is possible realise Higgs inflation by introducing a non-minimal coupling between the Higgs and gravity Bezrukov and Shaposhnikov (2008); Bezrukov et al. (2011). Thus, the MSM can explain a large number of phenomena with a minimum number of parameters.

However, the MSM is severely constrained by recent observations. In particular, there are severe constraints on sterile neutrino DM. The Dodelson-Widrow mechanism is known to be excluded by Lyman- bounds and X-ray constraints Abazajian and Koushiappas (2006). To evade these constraints, the production of sterile neutrino DM by the thermal freeze-in mechanism has been considered in Refs.Shaposhnikov and Tkachev (2006); Kusenko (2006); Petraki and Kusenko (2008); Merle, Niro, and Schmidt (2014); Adulpravitchai and Schmidt (2014); Roland, Shakya, and Wells (2014); Merle and Schneider (2014); Merle and Totzauer (2015) ^{1}^{1}1Ref.Merle and Totzauer (2015) gives a comprehensive study of keV sterile neutrino DM via a singlet scalar, but our purpose is to estimate the scale in the extended MSM to improve the dark matter sectors and the leptogenesis scenarios rather than such a generic study of the sterile neutrino DM..
Ref.Shaposhnikov and Tkachev (2006) considers a scenario in which the inflaton decays into sterile neutrino DM. Refs.Kusenko (2006); Petraki and Kusenko (2008) show that a GeV-scale singlet scalar produces the sterile neutrino DM. Refs.Merle, Niro, and Schmidt (2014); Adulpravitchai and Schmidt (2014) consider the non-thermal production via the decay of a singlet scalar. Furthermore, the singlet scalar can improve the electroweak vacuum stability or the Higgs inflation as well as the dark matter sectors.

In this paper, we do not discuss the theoretical merits of the singlet scalar in the MSM such as the electroweak vacuum stability Chen and Tang (2012); Lebedev (2012), Higgs inflation Giudice and Lee (2011) and scale invariance Kang (2014), although we are motivated by these theoretical aspects. Instead, we concentrate on estimating the scale of the singlet scalar to improve the dark matter sector and leptogenesis scenarios. We discuss various thermal freeze-in production scenarios for keV-MeV sterile neutrinos in the extended MSM with a singlet scalar. In particular, we revisit the Lyman- bounds and the X-ray constraints and show that the singlet scalar cannot be heavier than the TeV scale. We also discuss thermal freeze-in leptogenesis scenarios, which are able to produce a larger lepton asymmetry than is produced in thermal leptogenesis due to the contribution from the singlet scalar. In the leptogenesis scenarios, the singlet scalar needs to be lighter than 1 TeV in order to generate the observed baryon asymmetry.

This paper is organized as follows. In section II, we review the scalar singlet extension of the MSM. In section III, we consider two thermal freeze-in scenarios, one utilizing a thermal singlet scalar and the other a non-thermal singlet scalar. In section IV, we review the X-ray constraints and the lifetime bounds on the sterile neutrino dark matter. In section V, we investigate the free streaming horizon and the Lyman- constraints in our scenarios. In section VI, we discuss thermal freeze-in leptogenesis with the singlet scalar. Section VII is devoted to discussion and conclusions.

## Ii The scalar singlet extensions of the Msm

In this section, we review the extended MSM which contains three right-handed sterile neutrinos () and one real singlet scalar ^{2}^{2}2Here we do not consider a complex singlet scalar because in that case a light Nambu-Goldstone boson appears with breaking. The presence of such light bosons would make the sterile neutrinos unstable..
In this model, the vacuum expectation value (VEV) of generates a Majorana mass for the right-handed neutrino . The Lagrangian is given as follows,

(1) |

where are the lepton doublets, is the Higgs doublet, and are the Yukawa couplings. is the Higgs potential. After spontaneous symmetry breaking, the Higgs doublet and the scalar singlet develop the VEVs and , respectively, where = 247 GeV and . The right-handed neutrinos acquire the Majorana masses . Without loss of generality, we can choose the mass basis where the Majorana mass term is diagonal. The Lagrangian is written as follows,

(2) |

If the Dirac masses are much smaller than the Majorana masses, then, as a result of the type I seesaw mechanism, the left-handed neutrino masses can be expressed as follows,

(3) | |||

(4) |

For the scalar potential , we impose the softly broken discrete symmetry , where the scalar singlet is -odd () and all the other fields are -even. We can then construct the following scalar potential with even powers,

(5) |

where is a soft breaking term. The spontaneous breaking of the discrete symmetries could produce domain walls Zeldovich, Kobzarev, and Okun (1974). The soft breaking term makes the vacua of the singlet scalar degenerate so that the domain wall problem is evaded Zeldovich, Kobzarev, and Okun (1974); Vilenkin (1985); Kibble (1976). The minima of the scalar potential are given by the following equations,

(6) |

The mass eigenstates of the Higgs and singlet scalar are and , where approximately corresponds to the SM higgs boson. The physical masses of and are given by,

(7) | |||

(8) |

The Higgs portal coupling induces doublet-singlet mixing.
In this paper, we consider a TeV-scale singlet scalar which decays into sterile neutrino DM and heavy Majorana neutrinos. Therefore, there is essentially no constraint on the coupling . However, the size of the coupling still affects the thermal history of . The references Petraki and Kusenko (2008); McDonald (1994) show that is out of thermal equilibrium if the Higgs portal coupling satisfies .
In this paper, we assume that the singlet scalar is out of thermal equilibrium for and concentrate on the thermal freeze-in production mechanism ^{3}^{3}3If the singlet scalar is not directly produced from inflatons and the reheating temperature is low enough, the singlet scalar can be out of thermal equilibrium even if ..

## Iii Sterile neutrino dark matter from the thermal freeze-in production mechanism

In this section, we consider various scenarios for the production of sterile neutrino DM in the extended MSM with a singlet scalar. Sterile neutrinos could be produced by thermal freeze-out, thermal freeze-in or non-thermal decay. These scenarios also depend on whether the singlet scalar is generated by freeze-out or freeze-in. In addition, the Dodelson-Widrow mechanism can produce sterile neutrinos via active-sterile oscillations.

It is possible to constrain these scenarios using the mass relation of the seesaw mechanism. For simplicity, we assume that the lightest right-handed neutrino is sterile neutrino DM, with a mass of about keV. We will see later that a sterile neutrino with mass above 1 MeV is not favored by X-ray constraints and lifetime bounds. If the Yukawa coupling of the singlet scalar and the right-handed neutrino is and the vacuum expectation value of the singlet scalar is TeV, then the following relations can be derived from the seesaw mechanism,

(9) | |||

(10) |

The Yukawa couplings , are very small and . If the reheating temperature satisfies , the sterile neutrino DM does not come into thermal equilibrium for Vilja (1994); Enqvist, Kainulainen, and Vilja (1993). Therefore, we may regard the sterile neutrino DM as non-thermal particles in the early universe. In such a case, we find only two realistic dark matter scenarios to realize keV-MeV-scale sterile neutrinos. We will now proceed to describe these two scenarios.

### iii.1 The singlet scalar is in thermal equilibrium

If the Higgs portal coupling is relatively large , the singlet scalar enters into thermal equilibrium and the sterile neutrino DM can be produced via the thermal freeze-in of . In addition, couples to with suppressed coupling, so after the EW symmetry breaking there is a small mixing between and . To check the effect of this mixing we consider the contribution to sterile neutrino production as well. The production by the singlet scalar has been considered in Ref.Kusenko (2006); Petraki and Kusenko (2008). The thermal freeze-in production is caused by the Yukawa interaction of and or and . Under the assumption , as the universe is expanding the temperature becomes low and disappears first. The Higgs boson , however, is still in thermal equilibrium and thermal freeze-in production by is effective until .

The dark matter yield can be calculated by solving the Boltzmann equations. In this scenario, the relevant Boltzmann equations for are given as follows,

(11) |

where is from () decays respectively.

In FIG.1(a), FIG.1(b) and FIG.1(c), we show numerical results for the evolution of the sterile neutrino yield and the singlet scalar yield for various thermal freeze-in mechanisms. Sterile neutrino DM is generated by the thermal freeze-in production of in FIG.1(a), the thermal freeze-in production of in FIG.1(b), and the non-thermal decay production of in FIG.1(c).

Now, is obtained from the following calculation. The Boltzmann equation for the sterile neutrino number density involving is written as,

(12) | |||||

We assume that the initial abundance of sterile neutrinos can be neglected and the singlet scalar enters into thermal equilibrium, such that,

(13) | |||||

The sterile neutrino yield can be obtained from the entropy density , and satisfies the following equation,

(14) | |||||

Similarly, also produces as the following,

(15) | |||||

where is the planck mass, and are the effective number of degrees of freedom for energy and entropy and is the modified Bessel function of the second kind. The equilibrium yields are expressed as,

(16) | |||

(17) |

The partial decay width of into is obtained as,

(18) |

and the partial decay width of into is given as,

(19) |

In order to estimate the yield, we analytically integrate the relevant Boltzmann equations. The yield at the temperature of the universe today is given for as,

(20) | |||||

where we assume . The relic density of the sterile neutrino DM can be obtained as,

(21) | |||||

The DM relic density observed by Planck+WP Ade et al. (2014) is estimated as,

(22) |

The sterile neutrino mass required to explain the observed DM relic density is thus for and for . FIG.2 shows the relic density of sterile neutrinos as a function of for different values of .

Similarly, we can integrate the Boltzmann equation of thermal freeze-in production via , obtaining,

(23) | |||||

with the relic density of given as,

(24) |

Finally, sterile neutrino DM is also produced by the thermal background of active neutrinos via coherent scattering (Dodelson-Widrow mechanism). The dark matter relic density is found to be given as Abazajian (2006),

(25) |

In the case of keV-MeV-scale sterile neutrino DM, the contribution to given in Eq.(25) is larger than that from the thermal freeze-in production via given in Eq.(24). In fact, there are additional contributions of the same order as those given in Eq.(24) that come from the decay of Z-bosons or W-bosons due to neutrino mixing, but we can safely ignore these contributions in the mass region under consideration. Altogether, the total sterile neutrino DM relic density is given as,

(26) | |||||

When the mass and the VEV of the singlet scalar are of order 1 TeV, the dominant mechanism for production of keV-MeV-scale sterile neutrinos is via the thermal freeze-in of the singlet scalar.

### iii.2 The singlet scalar is out of thermal equilibrium

In this case and both and never enter into thermal equilibrium in the early universe. Sterile neutrino DM is generated via the thermal freeze-in of . The singlet scalar is also generated by thermal freeze-in production, and then proceeds to decay efficiently into . In this section, we assume , so that cannot decay into and .

To calculate the yield of sterile neutrinos, we have to solve the Boltzmann equations given by the following two interrelated equations,

(28) |

Standard model particles in thermal equilibrium can annihilate into a singlet scalar. For simplicity, we concentrate on thermal Higgs annihilation as the dominant production mechanism of singlet scalars and ignore the other standard model effects. The Boltzmann equation for the annihilation process can be expressed as follows,

(29) |

We integrate Eq.(III.2) to estimate the yield of sterile neutrinos . There is no initial yield () and no final yield (), and therefore, the following equation can be obtained,

(30) |

The yield of sterile neutrinos at today’s temperature can be obtained using Eq.(29) and Eq.(30),

(31) | |||||

Therefore, the yield at today’s temperature is given as,

(32) | |||||

The sterile neutrino DM relic density resulting from the non-thermal decay mechanism is obtained as,

(33) |

FIG.3 shows the relic density of sterile neutrinos as a function of for different values of the Higgs portal coupling and the sterile neutrino mass .

In this scenario, the Dodelson-Widrow mechanism can also produce sterile neutrino DM. Therefore, the total relic density is obtained as,

(34) | |||||

The relic density formula depends on the Higgs portal coupling . The coupling is bounded as so that does not come into thermal equilibrium. Therefore, can not be heavier than the TeV scale in order to produce keV-MeV sterile neutrino dark matter via the non-thermal decay production mechanism.

## Iv X-ray constraints and lifetime bounds on sterile neutrino dark matter

In this section, we will review the X-ray bounds and the lifetime bounds on sterile neutrino DM. Sterile neutrinos can decay into standard model particles through active-sterile neutrino mixings. In the keV-MeV mass range, sterile neutrinos decay mainly into the three active neutrinos Fuller, Kishimoto, and Kusenko (2011); Abazajian, Fuller, and Patel (2001); Abazajian, Fuller, and Tucker (2001). For the three-neutrino decay channel, the decay lifetime is expressed as,

(35) |

Their lifetime must be longer than the age of the universe () if sterile neutrinos are to constitute dark matter, which constrains the mixing angle and the sterile neutrino mass as follows,

(36) |

If sterile neutrinos constitute dark matter, their radiative decay () would lead to a cosmic X-ray background. We have not seen such an X-ray excess except for the recent observation of a 3.5 keV signal Bulbul et al. (2014); Boyarsky et al. (2014) in galactic clusters. This puts an upper limit on the neutrino mixing angle for a given sterile neutrino mass. From the diffuse X-ray background observations XMM-Newton Lumb et al. (2002); Read and Ponman (2003) and HEAO-1 Gruber et al. (1999), the authors of Ref.Abazajian and Koushiappas (2006); Boyarsky et al. (2006a) obtain the simple empirical formula,

(37) |

The XMM-Newton observations of the Virgo and Coma galaxy clusters present the more stringent constraints Abazajian and Koushiappas (2006); Boyarsky et al. (2006b),

(38) |

More precise X-ray constraints have been reported in Ref.Abazajian and Koushiappas (2006). Note that these bounds are given for sterile neutrino DM which explains the current dark matter density. If sterile neutrino DM only constitutes part of the total dark matter, the X-rays bounds become weaker.

## V The free streaming horizon and Lyman- constraints

Recent observations such as the WMAP and Planck missions have proven that the CDM model, which contains cold dark matter, is an extremely successful cosmological model Ade et al. (2014). However the CDM can not solve the small-scale crises Weinberg et al. (2013), including the missing satellite problem and the cuspy halo problem. Warm dark matter (WDM), which has an adequate free streaming horizon and suppresses the structure of dwarf galaxies size, may solve the problem. The upper bound on the free streaming scale of WDM is obtained from the observed Lyman- forest, which refers to the absorption lines of intergalactic neutral hydrogen in the spectra of distant quasars and galaxies.

The free-streaming horizon corresponds to the average distance travelled by DM particles and is a good measure to classify CDM, WDM and HDM. The free streaming horizon is given as,

(39) |

where is the DM production time, is the current time, is the average thermal velocity of the DM particles, and is the scale factor. In this paper we assume that the free-streaming scale of CDM, WDM and HDM satisfy , and , respectively. This is not an accurate definition, but gives a useful criteria to classify the thermal property of DM. Note that HDM is excluded by observations of the Lyman- forest.

In order to determine the free-streaming horizon, we now consider the average thermal velocity of the sterile neutrino DM . We define as the time when becomes non-relativistic, which we take to be when the equality is satisfied. The approximate average thermal velocity is then given as follows,

(40) |

The non-relativistic thermal velocity is expressed in terms of the average thermal momentum, which can be extracted from the distribution function and depends on the DM production mechanism. In this section we will consider the average thermal momentum and the free streaming horizon when production is via thermal freeze-in of the singlet scalar, via the Dodelson-Widrow mechanism and via the non-thermal singlet scalar. Finally, we determine the Lyman- constraints and the allowed parameter region for each production mechanisms.

### v.1 Production via thermal freeze-in of the singlet scalar

For production via the thermal freeze-in of the singlet scalar boson, the momentum distribution of sterile neutrino DM Kamada et al. (2013); Boyanovsky (2008) is given by,

(41) |

where

(42) |

The normalization factor is determined by the Yukawa coupling and the singlet scalar mass , with . The average thermal momentum can be calculated as,

(43) |

This average thermal momentum leads to the average thermal velocity ,

(44) |

The time when DM particles become non-relativistic is sec. The free streaming horizon is calculated as,

(45) | |||||

To obtain the last line, we neglect the third and the last terms of the third line.

In this production mechanism, the DM is produced at high temperatures, , and entropy dilution affects the free streaming horizon. The effect of entropy dilution can be estimated by the factor which is given by,

(46) |

Now we assume that both and contribute to the effective number of degrees of freedom and ignore the tiny effect of the other heavy right-handed neutrinos and . Taking entropy dilution into account and using the conversion factor , the final expression is given as,

(47) |

The Lyman- bound on is given by,

(48) |

The range of sterile neutrino mass corrsponding to WDM is obtained as,

(49) |

In FIG.4, we show the X-ray bounds and the HDM, WDM and CDM regions for sterile neutrino production via the thermal freeze-in of the singlet scalar. In this figure, we assume that is larger than but smaller than (a) and (b) . We also show the parameter region where more than 1 of the DM is produced by the DW mechanism. When is (), the sterile neutrino DM is warm (cold). However, the scenario with suffers from the X-ray constraints.

### v.2 Production via non-thermal decay of the singlet scalar

If the Higgs portal coupling is small and the singlet scalar is out of thermal equilibrium, it decays into the sterile neutrino. The free streaming horizon was considered in Merle, Niro, and Schmidt (2014); Adulpravitchai and Schmidt (2014). The momentum distribution of the sterile neutrino DM is given as Kaplinghat (2005); Hisano, Kohri, and Nojiri (2001); Strigari, Kaplinghat, and Bullock (2007); Aoyama et al. (2011),

(50) |

where is a normalization factor and the DM temperature is . The average thermal momentum is given as,

(51) |

From Eq.(51), the average thermal velocity is expressed as,

(52) |

Now, we assume that the production time is , where is the freeze-in time of and is given as sec, where is the freeze-in temperature, and the lifetime of is . The time at which the sterile neutrinos become non-relativistic is given by sec and the time of matter-radiation equality is . We estimate the free-streaming horizon of the DM sterile neutrinos using the formula,

(53) |

For and , the Lyman- bound on is obtained as,

(54) |

The WDM sterile neutrino mass can be constrained as

(55) |

For and , the Lyman- bound is given as,

(56) |

The WDM sterile neutrino mass range is obtained as

(57) |

Therefore, in this scenario, the singlet scalar can not be heavier than the TeV scale ^{4}^{4}4Ref.Merle and Totzauer (2015) presents more detailed calculations in this scenario, solving numerically the system of Boltzmann equations. The singlet scalar mass could be more tightly restricted.
This constraint is tighter than that in the DW mechanism