# Probing secret interactions of eV-scale sterile neutrinos with the diffuse supernova neutrino background

###### Abstract

Sterile neutrinos with mass in the eV-scale and large mixings of order could explain some anomalies found in short-baseline neutrino oscillation data. Here, we revisit a neutrino portal scenario in which eV-scale sterile neutrinos have self-interactions via a new gauge vector boson . Their production in the early Universe via mixing with active neutrinos can be suppressed by the induced effective potential in the sterile sector. We study how different cosmological observations can constrain this model, in terms of the mass of the new gauge boson, , and its coupling to sterile neutrinos, . Then, we explore how to probe part of the allowed parameter space of this particular model with future observations of the diffuse supernova neutrino background by the Hyper-Kamiokande and DUNE detectors. For keV and , as allowed by cosmological constraints, we find that interactions of diffuse supernova neutrinos with relic sterile neutrinos on their way to the Earth would result in significant dips in the neutrino spectrum which would produce unique features in the event spectra observed in these detectors.

IFIC/18-12

## 1 Introduction

The past two decades of neutrino oscillation data have proven to be a rich source of information about the masses and mixing in the leptonic sector Patrignani:2016xqp (). The standard model three-flavor framework has left-handed neutrinos with mass-squared differences much less than 1 eV. However, short baseline oscillation experiments like LSND Athanassopoulos:1995iw (); Athanassopoulos:1996jb (); Athanassopoulos:1996wc (); Aguilar:2001ty () and MiniBooNE AguilarArevalo:2007it (); AguilarArevalo:2010wv (); Aguilar-Arevalo:2013pmq () and some reactor neutrino experiments Mention:2011rk () show anomalies from and disappearance, respectively. These anomalous results can be interpreted as pointing to the existence of another species of (sterile) neutrinos with mass and a mixing angle with an active neutrino species of Kopp:2013vaa (); Giunti:2013aea (). Indeed, in a framework with three sub-eV active plus one sterile neutrino, global analyses including data from disappearance and appearance experiments obtain as best fit values eV and , and Collin:2016aqd (); Gariazzo:2017fdh (); Dentler:2017tkw (); Gariazzo:2018mwd ().

With such large mixing angles with active neutrinos, sterile neutrinos would fully thermalize in the bath of Standard Model (SM) particles Barbieri:1989ti (); Kainulainen:1990ds (); Barbieri:1990vx (); Enqvist:1991qj (); Shi:1993hm (); Kirilova:1997sv (); Abazajian:2001nj (); DiBari:2001ua (); Dolgov:2003sg (); Melchiorri:2008gq (); Hannestad:2012ky (); Mirizzi:2012we (); Mirizzi:2013gnd (); Saviano:2013ktj (); Jacques:2013xr (); Hannestad:2015tea (); Bridle:2016isd () accounting for one extra neutrino at the time of big bang nucleosynthesis (BBN) and the cosmic microwave background (CMB), in tension with current data Patrignani:2016xqp (). For ( is the mass of the mostly sterile neutrino), sterile neutrinos would be relativistic, and their contribution to the relativistic energy density can be parameterized by the effective number of neutrinos, . For instance, different cosmological observations, including BBN data, constrain the effective number of fully thermalized neutrinos to be and the mass of the (mostly) sterile neutrinos to be eV (the exact limits depending on the particular data set considered) Giusarma:2014zza () (see also, e.g., Refs. Hamann:2010bk (); Hamann:2011ge (); Mirizzi:2013gnd (); DiValentino:2013qma (); Bergstrom:2014fqa ()). This bounds can be satisfied if sterile neutrinos are not in equilibrium or are only partially thermalized, so their contribution to at the relevant epochs is small.

A possible solution for this problem was suggested long ago in the context of Majoron models Babu:1991at (); Cline:1991zb () and has been recently revised in scenarios with sterile neutrino self interactions, mediated by a boson Hannestad:2013ana (); Dasgupta:2013zpn (), sometimes called secret interactions. The new interaction term in the sterile sector would induce an effective potential, which could suppress the large mixing angle in vacuum with active neutrinos and hence, prevent equilibration of sterile neutrinos, and avoiding the bounds from the effective number of relativistic degrees of freedom at BBN. Thus, the production of massive (mostly) sterile neutrinos is suppressed at the epoch of BBN. Even if they recouple with, and again decouple from, active neutrinos before the recombination time, they would satisfy current bounds from both BBN and CMB data Hannestad:2013ana (); Dasgupta:2013zpn (); Saviano:2014esa (); Mirizzi:2014ama (); Cherry:2014xra (); Tang:2014yla (); Chu:2015ipa (); Archidiacono:2015oma (); Cherry:2016jol (); Archidiacono:2016kkh (). In recent years, more detailed analyses of cosmological data have been performed and different phenomenological consequences of the idea have been studied Bringmann:2013vra (); Ko:2014nha (); Archidiacono:2014nda (); Kopp:2014fha (); Saviano:2014esa (); Mirizzi:2014ama (); Cherry:2014xra (); Tang:2014yla (); Forastieri:2015paa (); Chu:2015ipa (); Archidiacono:2015oma (); Cherry:2016jol (); Archidiacono:2016kkh (). The conclusion is that as a result of these new interactions, BBN and CMB limits permit a relic density of sterile neutrinos with a mass of about 1 eV.

On the other hand, the coupling of neutrinos to a mediator could give rise to an attenuated spectrum of cosmic neutrinos due to the resonance production of this mediator in the relic neutrino background PalomaresWeiler (); Weilertalk (); PalomaresRuiztalk07 (); PalomaresRuiztalk11 (); Hooper:2007jr (); Ioka:2014kca (); Ng:2014pca (), an idea similar to the attenuation of the flux of ultra-high energy neutrinos due to the resonant interaction with cosmic relic neutrinos at the -pole Weiler:1982qy (); Weiler:1983xx (); Roulet:1992pz (); Yoshida:1996ie (); Eberle:2004ua (); Barenboim:2004di (). Similarly, the diffuse supernova neutrino background (DSNB) flux could also experience distortions en route to Earth, either because of interactions with the relic neutrino background in models with additional gauge bosons coupled to neutrinos Goldberg:2005yw (); Baker:2006gm () or because of interactions with dark matter particles Farzan:2014gza () in models with radiatively-generated neutrino masses Boehm:2006mi (); Farzan:2009ji (); Farzan:2010mr (); Farzan:2011tz (); Farzan:2012ev ().

In the scenario discussed in this paper (i.e., an extra eV-scale sterile neutrino with self interactions mediated by a vector boson ), the relic density of sterile neutrinos could act as the target for the DSNB flux, which would resonantly produce the vector boson . Thus, we would also expect a dip in the event spectrum from the DSNB if the resonant energy of this interaction lies in the relevant range for supernova (SN) neutrinos (i.e., tens of MeV). For targets with eV mass, absorption features could show up for mediators with masses in the keV range. Moreover, note that for masses in the MeV range, these secret interactions in the sterile neutrino sector could also produce dips in the cosmic neutrino spectrum Cherry:2014xra (); Cherry:2016jol ().

Here we focus on the expected signals by the DSNB in the liquid argon (LAr) detector planned for the Deep Underground Neutrino Experiment (DUNE) Acciarri:2015uup (); Acciarri:2016crz () and in the water-Čerenkov Hyper-Kamiokande (HK) detector HK (), in scenarios with self interactions of sterile neutrinos with eV masses and relatively large vacuum mixing with active neutrinos ().

The structure of the paper is as follows. In Section 2, we describe the low-energy Lagrangian in the sterile neutrino sector and its interactions with a new vector boson. We introduce the sterile neutrino production rate in the early Universe and describe its ingredients in detail: the collision rate and the average probability for active-sterile neutrino conversions in the medium, which depend on the effective potential induced by these new interactions and on the quantum damping rate. We provide a compendium of the relevant cross sections, including thermal averaging, relevant to the cosmological constraints reviewed and discussed in Section 3. The different models we consider for the DSNB flux are described in Section 4, as well as the signals expected in future DUNE and HK detectors with and without self interactions of sterile neutrinos. In Section 5 we discuss our results and draw our conclusions. Finally, in Appendix A we include the details of the calculation of the effective potential due to the new interactions in the sterile neutrino sector.

## 2 Sterile neutrino interactions

Here we consider a scenario with one extra sterile neutrino which has self-interactions mediated by a new keV-MeV scale vector boson () and has interactions with the SM sector only via mixing with active neutrinos. In this section, we review the interaction rates considered in different detail in the literature Hannestad:2013ana (); Dasgupta:2013zpn (); Cherry:2014xra (); Chu:2015ipa (); Cherry:2016jol (). We include thermal averaging of the cross sections, study the impact of the resonant scattering and of the effective potential , that enters the sterile-active mixing in the medium. The interaction rates and effective mixing angle are inputs to determine cosmologically allowed regions in the coupling constant–vector boson mass parameter space.

In this work, cosmologically relevant interactions in the sterile sector are assumed to occur between sterile neutrinos and a vector boson , described by the interaction term Dasgupta:2013zpn (); Chu:2015ipa ()

(1) |

where . The coupling keeps the number of degrees of freedom for sterile neutrinos plus antineutrinos the same as for active neutrinos, namely, .^{1}^{1}1We use to denote the coupling and for the number of degrees of freedom of the sterile neutrino. We also assume that this sector ( and ) decouples from SM particles above the TeV scale, when the effective number of degrees of freedom is on the order of Kolb:1990vq (). If particles in the sterile sector do not recouple to SM particles (e.g., via mixing) before BBN, the early decoupling ensures that the number density of sterile neutrinos does not equal that of active neutrinos (due to the entropy release in the SM sector), so that the constraint on the effective number of extra neutrinos during the BBN epoch, namely that is a fraction of a SM neutrino species at MeV, is satisfied Cyburt:2015mya (); Pitrou:2018cgg () (see also Ref. Cooke:2017cwo () for a less constraining limit).

With these assumptions, neglecting the impact of the active-sterile mixing, the ratio of the sterile neutrino temperature to the active neutrino temperature , at MeV depends on whether or not ’s are present.

If the ’s are relativistic at BBN^{2}^{2}2Given that we are not solving the Boltzmann equations, the exact value for the transition from relativistic to non-relativistic is set by matching the constraints presented below in the two regimes. (i.e., MeV), they are easily produced with a temperature equal to that of sterile neutrinos. Therefore, the temperature ratio and the effective number of neutrino species during BBN are

(2) | |||||

(3) |

where we have used the SM value, Mangano:2001iu (); Mangano:2005cc () and .

If ’s are non-relativistic (i.e., MeV), they would have decayed away into sterile neutrinos by the BBN epoch and thus,

(4) | |||||

(5) |

where the second factor in accounts for the ‘heating’ of sterile neutrinos from decays. Note that this factor is not present in Eq. (2), where the temperature ratio only corresponds to the SM entropy release between high temperatures ( TeV) and BBN temperatures ( MeV), with no ‘heating’ in the sterile sector. Therefore, these results represent the two limiting cases for different . As we can see, regardless the value of , the BBN constraint Cyburt:2015mya (); Pitrou:2018cgg (); Cooke:2017cwo () is always satisfied if the sterile sector does not recouple to the SM sector before MeV. If recoupling occurs before that time, the bounds from BBN would be violated. We will use the ratio of temperatures given in Eq. (2) or (4), for MeV or MeV, respectively, in our evaluation of cosmological constraints on the sterile sector. Obviously, this is a rough approximation, although the changes in the results are not qualitatively important.

Large mixing between sterile and active neutrinos would drive the former to reach thermal equilibrium in the early Universe, i.e., to recouple and thus, there would be an extra contribution to the number of neutrino degrees of freedom, violating cosmological (BBN and CMB) bounds. Active-sterile mixing in the medium depends on the vacuum mixing angle and on the effective potential Notzold:1987ik () (see below). In the case of no extra new interaction in the sterile sector, the SM weak potential is negligible at MeV temperatures or below, so the mixing angle is the one in vacuum. Nevertheless, if a term like Eq. (1) is present, there is an extra contribution to the sterile neutrino self-energy, which can dominate over SM matter effects, even for small values of . This is so because the mass of the under consideration here is much smaller than the mass.

The production rate of sterile neutrinos, , is given by the product of half the total interaction rate Stodolsky:1986dx (); Thomson:1991xq () times the thermal average of the active-sterile neutrino conversion probability Foot:1996qc ()

(6) |

The average probability for active-sterile neutrino conversions in the medium (in the adiabatic limit) can be written as Foot:1996qc (); Volkas:2000ei ()

(7) |

where the overall factor of is the result of averaging the oscillatory term, is the vacuum mixing angle and is the (mostly) active-sterile neutrino mass difference squared. In what follows, we take and eV.

There are two important terms in Eq. (7) which are present because oscillations take place in a medium: is the effective potential induced by neutrino forward scattering in the thermal bath (both from SM interactions and from the new sterile sector interactions) and is the quantum damping rate and accounts for the loss of coherence due to collisions Stodolsky:1986dx ().

The effective potential receives finite temperature contributions from both SM and new sector interactions. Due to mixing between active and sterile neutrinos, SM interactions also contribute to the effective potential of (mostly) sterile neutrinos, although suppressed by four powers of the mixing angle, . Similarly, interactions in the sterile sector, Eq. (1), also contribute to the effective potential of active neutrinos. Therefore, the effective potential appearing in Eq. (7) can be written as

(8) |

The SM contribution to the effective potential at temperatures below the mass of the SM gauge bosons was computed three decades ago and is given by Notzold:1987ik ()

(9) | |||||

where, at the temperatures MeV, for and for and (as there are no or leptons at MeV). For numerical computations we use the electron neutrino case, although the differences do not affect our discussion.

Although we use the full expression, Eq. (A) in Appendix A, the effective potential from interactions in the sterile sector, Eq. (1), can be analytically computed in the low- and high-temperature limits Dasgupta:2013zpn (),

(10) |

where is the sterile neutrino energy (which has to coincide with that of active neutrinos so that the two states can oscillate) and equal distributions for active neutrinos and antineutrinos are assumed. In the Appendix, we include the expression for corresponding to light sterile neutrinos, which is also applicable near the resonant production of . When the effective potential is more important than the vacuum term, the probability for the active-sterile neutrino conversion is suppressed, preventing the equilibration of sterile and active neutrinos and preserving the consistency with the BBN limit on effective number of extra neutrinos Hannestad:2013ana (); Dasgupta:2013zpn ().

The total interaction rate, which appears in the definition of the sterile neutrino production rate, , and in the damping rate, , can be written as

(11) |

where , and are the number densities and , and are the thermal average of the cross section times the Möller velocity (equal to the relative velocity in the lab or center-of-mass frames), corresponding to active and sterile neutrinos and bosons, respectively. The total interaction rate of sterile neutrinos, , has three contributions: the usual one from SM interactions of active neutrinos, the ones from collisions of sterile neutrinos or between sterile neutrinos and bosons due to the new interaction term. In the first case, with SM interactions, sterile neutrino production proceeds via active-active neutrino interactions and then active-sterile neutrino mixing, so the temperature of the final sterile neutrino state is that of active neutrinos. In the second and third cases, with new sterile neutrino interactions, sterile neutrinos are produced via active-sterile neutrino mixing and then sterile-sterile neutrino and sterile neutrino- boson interactions, respectively, so the temperatures of the distributions of the incoming active and sterile neutrinos are, in principle, different, . Sterile neutrino- interactions do not change our conclusions here, so for simplicity, we do not discuss them further here.

The contribution from active-active neutrino interactions, in the case of active-sterile neutrino oscillations, is given by Enqvist:1991qj ()

(12) |

where for and for . For numerical computations we consider the electron neutrino case, although very similar results are obtained otherwise.

For , the contribution to the total interaction rate arising entirely from the sterile neutrino sector, we have to consider both elastic and inelastic interactions of sterile neutrinos induced by the new term in the Lagrangian, Eq. (1),

(13) | |||||

(14) | |||||

(15) |

and similarly for .

The first process is equivalent to Bhabha scattering of electrons, with the substitution of couplings of a massive boson for the vector couplings of the photon. The cross section has both - and -channel contributions to the matrix element squared, where and are the Mandelstam variables. Although in our calculations we use the full expression, in various limits, the cross section for and decay width read

(16) | |||||

(17) |

Note that in the low-mass limit, , the -channel cross section depends on , instead of , as considered in Refs. Archidiacono:2014nda (); Mirizzi:2014ama (); Chu:2015ipa (). This had already been noted in Refs. Cherry:2014xra (); Cherry:2016jol ().

The thermal average of the cross section times relative velocity is given by Gondolo:1990dk ()

(18) |

where is the distribution function of species . We assume neutrinos and antineutrinos are equally distributed and thus, . The Möller velocity, , is defined as

(19) |

which is approximated by in the relativistic limit (), valid for most of our discussion since all neutrino masses are of the order of 1 eV or below. Thermal averaging with Eq. (18) is essential for the proper treatment of the resonance behavior of the annihilation cross section. From the cross sections in Eq. (16), the analytic expression of thermal average for the high- and low-energy limits is

(20) |

where and are the temperatures of the two neutrino distributions, which can be different due to active-neutrino mixing. For , the thermally averaged (-channel) cross section increases rapidly. A simple form in this case is not available. We use the numerical results below.

The corresponding result for can be obtained by using crossing symmetry. In the high- and low-energy limits, it is given by

(21) |

For the process, the cross section is

(22) |

with

The cross section can be obtained in a similar way, although collinear divergences have to be taken care of. In any case, both and are subdominant processes at all temperatures.

We show in Fig. 1 the thermal averaged annihilation and scattering cross sections, as a function of the standard model neutrino temperature, for and keV. The cross sections are relevant to the cosmological constraints when a sterile neutrino scatters with an active neutrino, via mixing. The thermally averaged total cross section for process (upper red solid curve) results from the sum of the -channel and -channel plus interference term. The scattering process is shown by the blue dashed curve (-channel). We also show the sterile neutrino annihilation into (green dot-dashed curve). As can be seen from the figure, only the cross sections for and are relevant to determine whether sterile neutrinos are in thermal equilibrium or not.

## 3 Cosmological constraints on

As discussed above, we assume the sterile sector particles are present in the early Universe, but that they decouple from active neutrinos at temperatures well above the electroweak scale. In addition to the primordial population, sterile neutrinos can be produced from interactions with active neutrinos via mixing. When the production rate is higher than the Hubble expansion rate, thermal equilibrium is established between sterile and active neutrinos. If equilibration is reached before BBN, this implies an extra neutrino degree of freedom, which would be in tension with current data Patrignani:2016xqp (). At the recombination epoch, most neutrinos have to be free streaming to agree with the temperature and polarization CMB data and this imposes additional constraints on the parameter space of the new interactions Hannestad:2004qu (); Trotta:2004ty (); Bell:2005dr (); Cirelli:2006kt (); Friedland:2007vv (); Basboll:2008fx (); Cyr-Racine:2013jua (); Archidiacono:2013dua (); Forastieri:2015paa (); Lancaster:2017ksf (); Oldengott:2017fhy (); Koksbang:2017rux ().

As we discussed in the previous section, the production rate of the sterile neutrinos, , is equal to the product of the damping rate and the thermally averaged conversion probability. On the other hand, the Hubble expansion rate at radiation-dominated epochs, in terms of the photon temperature , is

(23) |

with the Planck mass MeV. The temperature of the sterile sector (relative to the one of active neutrinos) and the number of relativistic degrees of freedom at BBN depend on the mass of the new boson , as discussed above. Therefore, constraints from BBN data depend on , too. At MeV, for MeV, , with , and . whereas for MeV, and . At later times (lower temperatures), sterile neutrino recoupling would make the active and sterile neutrino temperatures equal.

Before discussing in more detail the allowed regions in the parameters space of the hidden sector, (fixing and ), we illustrate in Fig. 2 how BBN and CMB cosmological constraints apply. We show the ratio of the production rate of sterile neutrinos to the Hubble expansion rate, , as a function of the photon temperature, for several representative pairs of (excluded) values of . We also show this ratio for the default values we consider in Section 4 (black solid curve), keV and . In this case, for MeV and eV, so in these temperature regimes, sterile neutrinos are not in equilibrium with active neutrinos and both BBN and CMB constraints are satisfied. Nevertheless, for the other parameter sets presented in the figure, at either or . As a consequence, these parameter choices are in conflict with the constraints on the number of effective neutrino species or the condition of free-streaming at recombination (see below). In the following, we investigate the excluded/allowed regions of the parameter space in more detail.

### 3.1 Big Bang Nucleosynthesis constraints

If sterile neutrinos recouple with the SM sector before the BBN time, they would violate observational data and hence this constrains the parameter space . The equilibration between sterile and active neutrinos would occur when the production rate of sterile neutrinos, , exceeds the expansion rate of the Universe. The temperature when equilibration is reached is the recoupling temperature, so it is defined as . In order satisfy the BBN bound on , we find the constrained parameter space for by demanding the recoupling temperature to be lower than temperatures below which BBN can be affected,

(24) |

At the BBN epoch, . Using the expressions for the interaction rates from the previous section, the resulting constraints on the coupling , as a function of , are depicted in Fig. 3. The excluded values of are represented by the gray region, discussed below. In the region of the parameters space where the effective potential due to sterile-sterile neutrino interactions is smaller than the vacuum term, , the mixing angle is the one in vacuum. Thus, the lower the temperature, the more constraining this condition is (for high temperatures, the region moves towards large masses and couplings). For MeV, this is represented by the cross-hatched area, which is excluded due to SM interactions, given the very large mixing we consider Barbieri:1989ti (); Kainulainen:1990ds (); Barbieri:1990vx (); Enqvist:1991qj (); Shi:1993hm (); Kirilova:1997sv (); Abazajian:2001nj (); DiBari:2001ua (); Dolgov:2003sg (); Melchiorri:2008gq (); Hannestad:2012ky (); Mirizzi:2012we (); Mirizzi:2013gnd (); Saviano:2013ktj (); Jacques:2013xr (); Hannestad:2015tea (); Bridle:2016isd ().

We now turn to the BBN constraints in three limiting cases: the limit of , in the limit of , and the case in which equilibration can potentially happen before BBN.

#### 3.1.1 Low-mass limit ()

When bosons are relativistic at the BBN epoch, i.e., , the temperature of the sterile sector is given by Eq. (2), i.e., (and ), and the effective potential (only from interactions in the sterile sector, i.e., neglecting the SM contribution) is given by, (see Eq. (10)). In order to obtain the constraints on the parameter space, it is useful to determine different regimes. It is useful to consider the regimes such that:

(25) | |||||

(26) | |||||

(27) |

When the sterile neutrino cross section is -channel dominated,

(28) | |||||

(29) | |||||

(30) |

while for the -channel dominated sterile neutrino cross section,

(31) | |||||

(32) | |||||

(33) |

where we have used eV, and . These inequalities show that dominates for most of the parameter space under discussion.

We first consider the case where the -channel is the most important contribution to the sterile neutrino interaction cross section, i.e., Eq. (27), which roughly represents half of the low mass region ( MeV) depicted in Fig. 3. Using Eq. (20), the total interaction rate in the sterile sector is given by

(34) |

where the first parenthesis corresponds to the thermal average of the sum of cross sections for and , whereas the second term is the equilibrium number density of sterile neutrinos, .

In the region of the parameter space where and (-channel), i.e., , the thermal average of the probability of active-sterile neutrino conversion can be approximated by

(35) |

and thus, from the condition in Eq. (24), the excluded region is given by

(36) |

Hence, in the limit in which the effective potential suppresses oscillations and the interaction rate is dominated by the -channel cross section, the production rate does not depend on the coupling , because both and the (-channel) cross section are proportional to . Accordingly, only constraints on can be set within this region Cherry:2016jol (). This is represented by the vertical line in the left of Fig. 3, marking the boundary of the dark gray region.

In the region of the parameter space where and , i.e., and , interactions interrupt oscillations; they act as a ‘measurement’ of the neutrino state (Turing’s or Zeno’s paradox) Harris:1980zi (). In this situation, the average of the conversion probability can be approximated as

(37) |

which results in the excluded region

(38) |

This excludes boson masses in the range for , which is complementary to the exclusion region represented by Eq. (36).

In the low-mass region where the -channel dominates the total interaction rate, there is only a very small corner for which the mixing angle is approximately that of vacuum, . In that case, , and so, equilibration is even more effective than in the standard scenario of sterile neutrino production in the early Universe. Therefore, for the large mixing angle we consider, full equilibration of sterile neutrinos is achieved and BBN constraints are violated.

Next, we focus on the parameter region where the channel is the most relevant one in the interaction cross section (of the sterile sector), i.e., . It is interesting to stress, as can be seen from Fig. 1, that as a consequence of the thermal averaging of the cross section, the -channel is the most important contribution over several orders of magnitude in temperature, not just at . Therefore, the constraints for most of the parameter space we consider are a consequence of -channel interactions. In this regime, and using Eq. (20), the total interaction rate in the sterile sector is given by

(39) |

From Eqs. (31) and (33), when the -channel is most important () and bosons are relativistic at the BBN epoch, the effective potential is always larger than the damping term (in the region depicted in Fig. 3). Moreover, the effective potential suppresses the vacuum mixing in the range , Eq. (26). Under these conditions, the excluded region is given by

(40) |

which approximately represents the dark gray region limited by the diagonal line in Fig. 3. For , the low-mass approximation (for the interaction rate and the effective potential) is less accurate and one should use the full numerical result, which produces the shoulder at MeV.

For and , vacuum mixing is recovered. At interactions in the sterile sector are more important than SM collisions between active neutrinos, Eq. (31), in the region shown in Fig. 3, but the equilibration condition is not satisfied for (left-bottom region in Fig. 3). However, in that region of the parameter space and at few MeV, the -channel contribution is again more important than the -channel one. In this case, vacuum mixing is also recovered and , so equilibration between the active and sterile sectors is achieved before the BBN epoch in a similar fashion as in the usually considered active-sterile neutrino mixing scenario.

#### 3.1.2 High-mass limit ()

When bosons are non-relativistic at the BBN epoch, i.e., , the temperature of sterile neutrinos is given by Eq. (4), i.e., (and ), and the effective potential (only from interactions in the sterile sector, i.e., neglecting the SM contribution) is , Eq. (10). In this limit, both - and -channel contributions to the total cross section are relevant to determine constraints in different regimes.

The -channel cross section is important for large masses and couplings and temperatures close to , whereas the -channel contribution is the dominant one to set bounds for small couplings in the entire mass interval considered ( GeV). This can be qualitatively understood as follows. If we were to consider only interactions in the new sterile sector, due to thermal averaging, the -channel would be the dominant one at (see Fig. 1). Thus, for temperatures close to , it is more important for larger masses. Besides, for large couplings the effective potential suppresses the mixing angle in the medium, but and , so the constraints so obtained do not depend on the coupling. For smaller couplings, mixing can be resonantly enhanced around some temperature close to BBN, so the sterile neutrino production rate would be proportional to and hence, the larger the mixing the more effective equilibration would be and the more stringent the constraints would be. This explains the upper right part in Fig. 3. On the other hand, the large enhancement in the interaction cross section produced by the -channel contribution also results in a suppression of the mixing angle. Unlike what happens for the -channel, now , so the sterile neutrino production rate would be proportional to (if no SM interactions were present) and thus, the larger the coupling the smaller the impact on BBN data (equilibration is more difficult to be reached). For small masses, only the -channel contributes to set limits on the parameter space and this bound smoothly connects with the low-mass case discussed above and depicted in Fig. 3. For MeV, interactions in the new sector alone would leave an allowed region; the -channel not being efficient enough and the -channel suppressing too much the sterile-active mixing angle. However, in this region, SM interactions of active neutrinos take over and thermalize sterile neutrinos, excluding that part too.

Although we do not have an analytic expression for the -channel contribution to the total interaction rate, it is illustrative to consider the -channel dominated rate. Analogously to the low-mass limit, it is useful to determine different regimes,

(41) | |||||

(42) | |||||

(43) | |||||

(44) | |||||

(45) |

where we have also used eV, and .

Using Eq. (20), the total interaction rate in the sterile sector for , when it is dominated by the -channel contribution, is given by

(46) |

In this regime, the effective potential is always larger than the damping term^{3}^{3}3This is so even when considering the -channel contribution, except for , due to the behavior of the effective potential., Eq. (44), and it suppresses vacuum mixing for , Eq. (42), which results in an excluded region given by

(47) |

For , this corresponds to the vertical limit of the dark gray region on the right-top part of Fig. 3. If we impose the equilibration condition at higher temperatures and we substitute Eq. (42) into Eq. (47), we get

(48) |

where we take to be constant within the relevant temperature range.

On the other hand, for small couplings, , there is always a temperature , such that . Consequently, mixing is not only unsuppressed, but it is resonantly enhanced before BBN, and the conversion probability is maximal (i.e., ). For , the excluded region is given by^{4}^{4}4Note that Eq. (43) implies Eq. (41) for GeV.