Abstract
We explore the possibility that high energy astrophysical neutrinos can interact with the dark matter on their way to Earth. Keeping in mind that new physics might leave its signature at such energies, we have considered all possible topologies for effective interactions between neutrino and dark matter. Building models, that give rise to a significant flux suppression of astrophysical neutrinos at Earth, is rather difficult. We present a mediated model in this context. Encompassing a large variety of models, a wide range of dark matter masses from eV up to a TeV, this study aims at highlighting the challenges one encounters in such a model building endeavour after satisfying various cosmological constraints, collider search limits and electroweak precision measurements.
Interactions of Astrophysical Neutrinos with Dark Matter: A model building perspective
Sujata Pandey^{*}^{*}*Email: , Siddhartha Karmakar^{†}^{†}†Email: and Subhendu Rakshit^{‡}^{‡}‡Email:
[2mm]
Discipline of Physics, Indian Institute of Technology Indore,
Khandwa Road, Simrol, Indore  453 552, India
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I Introduction
IceCube has been to designed to detect high energy astrophysical neutrinos of extragalactic origin. Beyond neutrino energies of TeV the background of atmospheric neutrinos get diminished and the neutrinos of higher energies are attributed to extragalactic sources Denton:2017csz . However, there is a paucity of high energy neutrino events observed at IceCube for neutrino energies greater than TeV Aartsen:2017mau . There are a few events around PeV or higher, whose origin perhaps can be described by the decay or annihilation of very heavy new particles Boucenna:2015tra ; Borah:2017xgm ; Zavala:2014dla ; Dev:2016qeb ; Hiroshima:2017hmy ; Lambiase:2018yql ; Murase:2015gea ; Dhuria:2017ihq or even without the help of any new physics Murase:2016gly ; Chen:2013dza ; Murase:2015ndr . In the framework of standard astrophysics, high energy cosmic rays of energies up to eV have been observed, which leads to the prediction of the existence of neutrinos of such high energies as well Aab:2017tyv ; Bahcall:1999yr ; Waxman:1998yy . In this context, it is worth exploring whether the flux of such neutrinos can get altered due to their interactions with DM particles. However, it is challenging to build such models given the relic abundance of dark matter. Few such attempts have been made in literature but these models also suffer from cosmological and collider constraints. Hence, in this paper, we take a model building perspective to encompass a large canvas of such interactions that can lead to appreciable flux suppression at IceCube.
In presence of neutrinoDM interaction, the flux of astrophysical neutrinos passing through isotropic DM background is attenuated by a factor . Here denotes number density of DM particles, is the distance traversed by the neutrinos in the DM background and represents the crosssection of neutrinoDM interaction. The neutrinoDM interaction can produce appreciable flux suppression only when the number of interactions given by is . For lower masses of DM, the number density is significant. But the crosssection depends on both the structure of the neutrinoDM interaction vertex and the DM mass. The neutrinoDM crosssection might increase with DM mass for some particular interactions. Hence, it is essentially the interplay between DM number density and the nature of the neutrinoDM interaction, which determines whether a model leads to a significant flux suppression. As a prefilter to identify such cases we impose the criteria that the interactions must lead to at least suppression of the incoming neutrino flux. For the rest of the paper, a flux suppression of less than has been addressed as ‘not significant’. While checking an interaction against this criteria, we consider the entire energy range of the astrophysical neutrinos. If an interaction leads to change in neutrino flux after considering the relevant collider and cosmological constraints in any part of this entire energy range, it passes this empirical criteria. We explore a large range of DM mass ranging from subeV regimes to WIMP scenarios. In the case of subeV DM, we investigate the ultralight scalar DM which can exist as a BoseEinstein condensate in the present Universe.
In general, various aspects of the neutrinoDM interactions have been addressed in the literature Boehm:2013jpa ; Campo:2017nwh ; Wilkinson:2014ksa ; Escudero:2015yka ; Barranco:2010xt ; Reynoso:2016hjr ; Arguelles:2017atb ; deSalas:2016svi ; Huang:2018cwo . The interaction of astrophysical neutrinos with cosmic neutrino background can lead to a change in the flux of such neutrinos as well Ng:2014pca ; DiFranzo:2015qea ; Araki:2015mya ; Mohanty:2018cmq ; Chauhan:2018dkd ; Kelly:2018tyg ; Cherry:2016jol ; Shoemaker:2015qul ; Ibe:2014pja . But it is possible that the dark matter number density is quite large compared to the number density of the relic neutrinos, leading to more suppression of the astrophysical neutrino flux.
To explore large categories of models with neutrinoDM interactions, we take into account the renormalisable as well as the nonrenormalisable models. In case of nonrenormalisable models, we consider neutrinoDM effective interactions up to dimensioneight. However, it is noteworthy that for a wide range of DM mass the centreofmass energy of the neutrinoDM scattering can be such that the effective interaction scale can be considered to be as low as MeV. We discuss relevant collider constraints on both the effective interactions and renormalisable models. We consider thermal DM candidates with masses ranging in MeVTeV range as well as nonthermal ultralight DM with subeV masses. For the thermal DM candidates, we demonstrate the interplay between constraints from relic density, collisional damping and the effective number of light neutrinos on the respective parameter space. Only for a few types of interactions, one can obtain significant flux suppressions. For the renormalisable interaction leading to flux suppression, we present a UVcomplete model taking into account anomaly cancellation, collider constraints and precision bounds.
In Sec. II we discuss the nature of the DM candidates that might lead to flux suppression of neutrinos. In Sec. III we present the nonrenormalisable models, i.e., the effective neutrinoDM interactions categorised into four topologies. In Sec. IV we present three renormalisable neutrinoDM interactions and the corresponding crosssections in case of thermal as well as nonthermal ultralight scalar DM. In Sec. V we present a UVcomplete model mediated by a light which leads to a significant flux suppression. Finally in Sec. VI we summarise our key findings and eventually conclude.
Ii Dark Matter Candidates
In this section, we systematically narrow down the set of DM candidates we are interested in considering a few cosmological and phenomenological arguments.
The Lambda cold dark matter (CDM) model explains the anisotropies of cosmic microwave background (CMB) quite well. The weakly interacting massive particles (WIMP) are interesting candidates of CDM, mostly because they appear in wellmotivated BSM theories of particle physics. Nevertheless, CDM with subGeV masses are also allowed. The most stringent lower bound on the mass of CDM comes from the effective number of neutrinos () implied by the CMB measurements from the Planck satellite. For complex and real scalar DM as well as Dirac and Majorana fermion DM, this lower bound comes out to be MeV Boehm:2013jpa ; Campo:2017nwh . Thermal DM with masses lower than MeV are considered hot and warm DM candidates and are allowed to make up only a negligible fraction of the total dark matter abundance Bode:2000gq . The ultralight nonthermal BoseEinstein condensate (BEC) dark matter with mass eV is also a viable cold dark matter candidate Boehm:2003hm . In the rest of this paper, unless mentioned otherwise, by ultralight DM we refer to the nonthermal ultralight BEC DM.
Numerical simulations with the CDM model show a few tensions with cosmological observations at small, i.e., galactic scales Salucci:2002nc ; Tasitsiomi:2002hi ; Blok:2002tr . It predicts too many subhalos of DM in the vicinity of a galactic DM halo, thus predicting the existence of many satellite galaxies which have not been observed. This is known as the missing satellite problem Klypin:1999uc . It also predicts a ‘cusp’ nature in the galactic rotational curves, i.e., a density profile that is proportional to near the centre, with being the radial distance from the centre of a galaxy. On the contrary, the observed rotational curves show a ‘core’, i.e., a constant nature. This is known as the cusp/core problem Navarro:1995iw . Ultralight scalar DM provides an explanation to such smallscale cosmological problems. In such models, at small scales, the quantum pressure of ultralight bosons prevent the overproduction of subhalos and dwarf satellite galaxies Alcubierre:2001ea ; Hu:2000ke ; Harko:2011jy . Also, choosing suitable boundary condition while solving the Schrödinger equation for the evolution of ultralight DM wavefunction can alleviate the cusp/core problem Hu:2000ke ; Peebles:2000yy ; Matos:2007zza ; Su:2010bj , making ultralight scalar an interesting, even preferable alternative to WIMP. Ultralight DM form BEC at an early epoch and acts like a “cold” species in spite of their tiny masses Sikivie:2009qn . Numerous searches of these kinds for DM are underway, namely ADMX Duffy:2006aa , CARRACK Tada:1999tu etc. It has been recently proposed that gravitational waves can serve as a probe of ultralight BEC DM as well Dev:2016hxv . But the ultralight fermionic dark matter is not a viable candidate for CDM, because it can not form such a condensate and is, therefore “hot”. The case of ultralight vector dark matter also has been studied in the literature Graham:2015rva .
The scalar DM can transform under as a part of any multiplet. In the case of a doublet or higher representations, the DM candidate along with other degrees of freedom in the dark sector couple with bosons at the tree level. This leads to stringent bounds on their masses as light DM candidates can heavily contribute to the decay width of SM gauge bosons, and hence, are ruled out from the precision experiments. Moreover, Higgsportal WIMP DM candidates with are strongly constrained from the Higgs invisible decay width as well. The failure of detecting DM particles in collider searches and the direct DM detection experiments rule out a vast range of parameter space for WIMPs. In light of current LUX and XENON data, amongst low WIMP DM masses, only a narrow mass range near the Higgs funnel region, i.e., GeV, survives Aprile:2015uzo ; Akerib:2016vxi ; Athron:2017kgt . As alluded to earlier, the ultralight scalar DM can transform only as a singlet under because of its tiny mass.
We investigate the scenarios of scalar dark matter, both thermal and ultralight, as possible candidates to cause flux suppression of the high energy astrophysical neutrinos. Such a suppression depends on the length of the path the neutrino travels in the isotropic DM background and the mean free path of neutrinos, which depends on the crosssection of neutrinoDM interaction and the number density of DM particles. We take the length traversed by neutrinos to be Mpc, the distance from the nearest group of quasars Schawinski:2010up , which yields a conservative estimate for the flux suppression. Moreover, we consider the density of the isotropic DM background to be Agashe:2014kda . Comparably, in the case of WIMP DM, the number density is much smaller, making it interesting to investigate whether the crosssection of neutrinoDM interaction in these cases can be large enough to compensate for the smallness of DM number density. This issue will be addressed in a greater detail in Sec. IV.
Iii Effective Interactions
In order to exhaust the set of higher dimensional effective interactions contributing to the process of neutrino scattering off scalar DM particles, we consider four topologies of diagrams representing all the possibilities as depicted in fig. 1. Topology I represents a contact type of interaction. In case of topologies II, III, and IV we consider higher dimensional interaction in one of the vertices while the neutrinoDM interaction is mediated by either a vector, a scalar or a fermion, whenever appropriate.
DM DM effective interactions can arise from higher dimensional gaugeinvariant interactions as well. In this case, the bounds on such interactions may be more restrictive than the case where the mediators are light and hence, are parts of the low energy spectrum. In general low energy neutrinoDM effective interactions need not reflect explicit gauge invariance.
We discuss the bounds on the effective interactions based on LEP monophoton searches and the measurement of the decay width. The details of our implementation of these two bounds are as follows:
Bounds from LEP monophoton searches
For explicitly gaugeinvariant effective interactions, DM DM interactions come along with DM DM interactions. DM DM interactions can be constrained from the channel using FEMC data at DELPHI detector in LEP for GeV GeV. To extract a conservative estimate on the interaction, we assume that the new contribution saturates the error in the measurement of the crosssection pb at 1 Abdallah:2003np . By the same token, we consider only one effective interaction at a time. The corresponding kinematic cuts on the photon at the final state were imposed in accordance with the FEMC detector: , and . Here stands for the energy of the outgoing photon and is its angle with the beam axis. We use FeynRules2.3.32 Christensen:2008py , CalcHEP3.6.27 Belyaev:2012qa and MadGraph2.6.1 Alwall:2014hca for computations.
Although here we considered gaugeinvariant interactions, DM DM interactions can be directly constrained from the monophoton searches due to the existence of the channel DM DM via a boson. But such bounds are generally weaker than the bounds obtained from decay which we are going to consider next.
DM DM interactions can contribute to the muon decay width which is measured with an error of %. However, the partial decay width of the muon via DM DM channel is negligible compared to the error. Hence, these interactions are essentially unbounded from such considerations. The percentage error in the decay width for tauon is even larger and hence, the same is true for DM DM interactions.
Bounds from the leptonic decay modes of the boson
The effective DM DM interactions can be constrained from the invisible decay width of the boson which is measured to be GeV Agashe:2014kda . When the gaugeinvariant forms of such effective interactions are taken into account, DM DM interactions may be constrained from the experimental error in the partial decay width of the channel : MeV for at 1 Agashe:2014kda . To extract conservative upper limits on the strength of such interactions, one can saturate this error with the partial decay width DM DM).
If such interactions are mediated by some particle, say a light , then a stringent bound can be obtained by saturating with . Similar considerations hold true for DM DM mediated by a . We note in passing that such constraints from decay measurements are particularly interesting for light DM candidates.
iii.1 Topology I
In this subsection effective interactions up to dimension 8 have been considered which can give rise to neutrinoDM scattering. The phase space factor for the interaction of the high energy neutrinos with DM can be found in appendix A.1.

A sixdimensional interaction term leading to neutrinoDM scattering can be written as,
(1) where is SM neutrino, is the scalar DM and is the effective interaction scale.
Now, for this interaction, the constraint from invisible decay reads GeV. The bounds from the measurements of the channel are dependent on the lepton flavours, and are found to be: GeV, GeV and GeV. The gaugeinvariant form of this effective interaction leads to a fivepoint vertex of , which in turn leads to a new fourbody decay channel of the boson. Due to the existence of such a vertex, the bound on this interaction from the decay width reads GeV. The electronDM effective interactions can be further constrained from the measurements of , leading to GeV. It can be seen that for the effective interaction with electrons, the bound from the measurement of the crosssection in the channel can be quite stringent even compared to the bounds coming from the decay width. Among all the constraints pertaining to such different considerations, if one assumes the least stringent bound, the interaction still leads to only flux suppression. The renormalisable model discussed in Sec. IV.1.1 is one of the scenarios that leads to the effective interaction as in eq. (1).

Another sixdimensional interaction is given as:
(2) The constraint from the measurement of the decay width in the channel reads GeV for light DM. The bounds on the gaugeinvariant form of the interaction in eq. (2) from the measurement of reads GeV, GeV and GeV. The bound from the channel reads GeV. Even with the value GeV, such an effective interaction does not give rise to an appreciable flux suppression due to the structure of the vertex.

Another five dimensional effective Lagrangian for the neutrinoDM fourpoint interaction is given by:
(3) The above interaction gives rise to neutrino mass at the looplevel which is proportional to . This, in turn, leads to a bound on the effective interaction due to the smallness of neutrino mass,
(4) up to a factor of . In the ultralight regime mass of DM eV. Hence eq. (4) does not lead to any useful constraint on . The constraint from invisible decay on this interaction reads GeV, independent of neutrino flavour. The gaugeinvariant form of this interaction does not contain additional vertices involving the charged leptons and hence leads to no further constraints. For such a value of coupling, there can be a significant flux suppression for the entire range of ultralight DM mass, independent of the energy of the incoming neutrino as shown in fig. 2.
In passing, we note that the interaction can be written in a gaugeinvariant manner at the treelevel only when , a triplet with hypercharge , is introduced. The resulting gaugeinvariant term goes as . When obtains a vacuum expectation value , the above interaction represents an effective interaction between neutrinos and DM as in eq. (3). Such an interaction can arise from the mediation of another scalar triplet with mass . The LEP constraint on the mass of the neutral scalar other than the SMlike Higgs, arising from such a Higgs triplet reads GeV Schael:2006cr . Furthermore, theoretical bounds, constraints from parameter and Higgs signal strength in the diphoton channel dictate that GeV Das:2016bir for GeV. For smaller values of , such as GeV, the bound can be even stronger, GeV. Moreover, the corresponding Wilson coefficient should be perturbative, . These two facts together lead to GeV for GeV. Such values of are rather small to lead to any significant flux suppression. While this is true for a treelevel generation of this interaction via a triplet scalar exchange, such interactions can be generated at the looplevel or by some new dynamics.

There can also be a dimensionseven effective interaction vertex for neutrinoDM scattering:
(5) Bound on this interaction comes from invisible decay width and reads GeV. There is no counterpart of such an interaction involving the charged leptons. Thus the gaugeinvariant form of this vertex does not invite any tighter bounds. Such a bound dictates that this interaction does not lead to any considerable flux suppression.

Another sevendimensional interaction can be given as:
(6) From invisible decay width the constraint on the coupling reads GeV. The measurement of or LEP monophoton searches does not invite any further constraint on this interaction due to the same reasons as in case of eq. (3) and (5). Due to such a constraint, no significant flux suppression can take place in presence of this interaction.

Another neutrinoDM interaction of dim8 can be written as follows:
(7) The coupling of interaction given by eq. (7) is constrained from invisible decay width as GeV. The constraint on the gaugeinvariant form of this interaction reads GeV, which is similar for all three charged leptons. The gaugeinvariant form of the above effective interaction also gives rise to fivepoint vertices involving the boson. These lead to bounds from the observations of and which read GeV and GeV respectively. The bound from the process reads GeV. Even with the least stringent constraint among the different considerations stated above, such an interaction does not lead to any significant flux suppression of the astrophysical neutrinos.
iii.2 Topology II

We consider a vector mediator , with couplings to neutrinos and DM given by:
(8) This interaction has the same form of interaction as in eq (7) of Topology I. Bound on this interaction from invisible decay reads GeV. The constraints on the gaugeinvariant form of such interactions are GeV, GeV. The bound on the process reads GeV.
For this interaction, the vertex from eq. (8) takes the form,
where and are the fourmomenta of the incoming and outgoing DM respectively. In light of the constraints from decay, the factor is much smaller than unity when the dark matter is ultralight, i.e., eV and incoming neutrino energy PeV. The rest of the Lagrangian is identical to the renomalisable vectormediated process discussed in Sec. IV.3.3 and Sec. IV.2. Further the charged counterpart of the second term in eq. (8) contributes to of charged leptons and also leads to new threebody decay channels of . As mentioned in Sec. IV.1.3, the bounds on the these couplings read , and for MeV. So among the constraints from different considerations, even the least stringent one ensures that no significant flux suppression takes place with this interaction in case of ultralight DM.

Consider a scalar mediator with a momentumdependent coupling with DM,
(9) Here can be realised as the neutral component of a triplet scalar with . A Majorana neutrino mass term with also exists along with the second term of eq. (9), where is the vev of the neutral component of the triplet scalar. The measurement of the parameter dictates, GeV Agashe:2014kda . For GeV, the smallness of neutrino mass constrains the coupling at . The mass of the physical scalar is constrained to be GeV Das:2016bir for GeV. For and GeV, such an interaction does not give rise to an appreciable flux suppression for ultralight DM.
iii.3 Topology III
We consider the vector boson mediating the neutrinoDM interaction, with a renormalisable vectorlike coupling with the DM, but a nonrenormalisable dipoletype interaction in the vertex. The interaction terms are given as,
(10) 
This interaction can be constrained from the measurement of the invisible decay width of . The flavourindependent bound on the coefficient reads, GeV. The interactions in eq. (10) can be realised as the renormalisable description of the effective Lagrangian as mentioned in eq. (5).
From fig. 3 it can be seen that, for MeV, such an interaction leads to a significant flux suppression of neutrinos with energy PeV for DM mass in the range eV and eV respectively.
iii.4 Topology IV
We consider the fermionic field mediating the neutrinoDM interaction with
(11) 
In eq. (11), after the Higgs acquires vacuum expectation value (vev), the first term reduces to the second term up to a further suppression of (). Following the discussion in Sec. IV.1.1, such interactions do not lead to a significant flux suppression.
Effective interactions with thermal DM
So far we have mentioned the constraints on several neutrinoDM interactions in case of ultralight DM and whether such interactions can lead to any significant flux suppression.
Here we discuss such effective interactions of neutrinos with thermal DM with mass MeV.
In case of thermal DM, bounds on the effective interactions considered above can come from the measurement of the relic density of DM, collisional damping and the measurement of the effective number of neutrinos, discussed in detail in Sec. IV.2.
As mentioned earlier, the case of thermal DM becomes interesting in cases where the crosssection of neutrinoDM scattering increase with DM mass. For example, in topology II with the interaction given by eq. (8), the neutrinoDM scattering crosssection is proportional to which increases with DM mass. However, considering the bound on from decay, the relic density and thus the number density of the DM with such an interaction comes out to be quite small, leading to no significant flux suppression. The following argument holds for all effective interactions considered in this paper for neutrino interactions with thermal DM.
The thermallyaveraged DM annihilation crosssection is given by , where for five, six, seven and eightdimensional effective interactions respectively.
In order to have sufficient number density, the DM should account for the entire relic density, i.e., .
To comply with the measured relic density, the required values of come out to be rather large leading to small cross section.
Iv The Renormalisable Models
iv.1 Description of the models
Here we have considered three cases of neutrinos interacting with scalar dark matter at the treelevel via a fermion, a vector, and a scalar mediator.
iv.1.1 Fermionmediated process
In this case, the Lagrangian which governs the interaction between neutrinos and DM is given by:
(12) 
Here and stand for SM lepton doublet and singlet respectively. are the mediator fermions. As it was discussed earlier, a scalar DM of ultralight nature can only transform as a singlet under the SM gauge group. So, the new fermions and should transform as singlets and doublets under respectively. In such cases, the LEP search for exotic fermions with electroweak coupling lead to the bound on the masses of these fermions as, GeV Achard:2001qw . A scalar DM candidate can be both selfconjugate and nonselfconjugate. The stability of such DM can be ensured by imposing a discrete symmetry, for example, a symmetry. A nonselfconjugate DM can be stabilised by imposing a continuous symmetry as well. For selfconjugate DM, the neutrinoDM interaction takes place via  and channel processes and such contributions tend to cancel each other in the limit Boehm:2003hm . In contrary, for nonselfconjugate DM the process is mediated only via the channel and leads to a larger crosssection compared to the former case. In this paper, we only concentrate on the nonselfconjugate DM in this scenario.
Such interactions contribute to the anomalous magnetic moment, , of the charged SM leptons, which in turn constrains the value of the coefficients . The contribution of the interaction in eq. (12) to the anomalous dipole moment of SM charged lepton of flavour is given by Leveille:1977rc :
(13) 
where is the mass of the corresponding charged lepton. In the limit , the anomalous contribution due to new interaction reduces to,
(14) 
For electron and muon the bound on the ratio reads GeV and GeV respectively. There is no such bound for the tauon.
iv.1.2 Scalarmediated process
The Lagrangian for the scalarmediated neutrinoDM interaction can be written as:
(15) 
where are the SM lepton doublets and is the triplet with hypercharge . When acquires a vev , the first term in eq. (15) leads to a nonzero neutrino mass . For GeV and mass of the neutrino eV the constraint on the coupling reads . The second term in eq. (15) contributes to DM mass . In case the DM mass is solely generated from such a term, the upper bound on dictated by the measurement of parameter, implies a lower bound on . The mass term for DM might also arise from some other mechanisms, for example, by vacuum misalignment in case of ultralight DM. In such a scenario, for a particular value of and there exists an upper bound on the value of .
The lower bound on the mass of the heavy CPeven neutral scalar arising from the triplet is GeV for GeV Das:2016bir , which comes from the theoretical criteria such as perturbativity, stability and unitarity, as well as the measurement of the parameter and .
iv.1.3 Light mediated process
The interaction of a scalar DM with a new gauge boson is given by the Lagrangian,
(16) 
Here, are the couplings of the kind of neutrinos with the new boson , while is the coupling between the dark matter and the mediator. can be constrained from the measurements. Due to the same reason as in the fermionmediated case, the coupling of with flavoured neutrinos is not constrained from measurements. Constraints for this case from the decay width of boson will be discussed in Sec V.
For the mass of the SM charged lepton, and the boson, , the anomalous contribution to the takes the form Leveille:1977rc :
(17) 
We have considered vectorlike coupling between the and charged leptons. For electrons and muons we find the constraints on couplingstomediator mass ratio to be rather strong Agashe:2014kda ,
(18) 
From the measurement of the lower bound on the mass of a light interacting with SM neutrinos at the time of nucleosynthesis reads MeV Huang:2017egl .
iv.2 Thermal Relic Dark Matter
In this scenario, the DM is initially in thermal equilibrium with other SM particles via its interactions with the neutrinos. For models with thermal dark matter interacting with neutrinos, three key constraints come from the measurement of the relic density of DM, collisional damping and the measurement of the effective number of neutrinos. These three constraints are briefly discussed below.
Relic density
If the DM is thermal in nature, its relic density is set by the chemical freezeout of this particle from the rest of the primordial plasma. The observed value of DM relic density is Agashe:2014kda , which corresponds to the annihilation crosssection of the DM into neutrinos In order to ensure that the DM does not overclose the Universe, we impose
(19) 
Collisional damping
NeutrinoDM scattering can change the observed CMB as well as the structure formation. In presence of such interactions, neutrinos scatter off DM, thereby erasing small scale density perturbations, which in turn suppresses the matter power spectrum and disrupts large scale structure formation. The crosssection of such interactions are constrained by the CMB measurements from Planck and Lyman observations as Wilkinson:2014ksa ; Escudero:2015yka ,
(20) 
Effective number of neutrinos
In standard cosmology, neutrinos are decoupled from the rest of the SM particles at a temperature MeV and the effective number of neutrinos is evaluated to be deSalas:2016ztq . For thermal DM in equilibrium with the neutrinos even below , entropy transfer takes place from dark sector to the neutrinos, which leads to the bound MeV from the measurement of . It can be understood as follows. In presence of species with thermal equilibrium with neutrinos, the change in is encoded as Boehm:2013jpa ,
(21) 
where,
(22) 
Here, the effective number of relativistic degrees of freedom in thermal equilibrium with neutrinos is given as
(23) 
In eqs. (4.10) and (4.12), denotes the number of species in thermal equilibrium with neutrinos, for fermions (bosons) and the functions and can be found in ref. Boehm:2013jpa . For a DM in thermal equilibrium with neutrinos and MeV, the contribution of to is quite large, and such values of DM mass can be ruled out from Ade:2015xua , obtained from the CMB measurements.
We implement the above constraints in cases of the renormalisable models discussed in Sec IV. We present the thermallyaveraged annihilation crosssection and the crosssection for elastic neutrinoDM scattering for the respective models in table 1. The notations for the couplings and masses follow that of Sec IV. In the expressions of , can be further simplified as where is the virial velocity of DM in the galactic halo Campo:2017nwh . In the expressions of , represents the energy of the incoming relic neutrinos which can be roughly taken as the CMB temperature of the present Universe.
Two of the three renormalisable interactions discussed in this paper, namely the cases of fermion and vector mediators have been discussed in the literature in light of the cosmological constraints, i.e., relic density, collisional damping and Campo:2017nwh . For a particular DM mass, the annihilation crosssection decreases with increasing mediator mass. Thus, in order for the DM to not overclose the Universe, there exists an upper bound to the mediator mass for a particular value of . With mediator mass less than such an upper bound, the relic density of the DM is smaller compared to the observed relic density, leading to a smaller number density.
Fermionmediated  Scalarmediated  Vectormediated  

As discussed earlier, the measurement of places a lower bound on DM mass MeV. DM number density is proportional to the relic abundance and inversely proportional to the DM mass. Thus the most ‘optimistic’ scenario in context of flux suppression is when MeV and the masses of the mediators are chosen in such a way that those correspond to the entire relic density in fig. 5. Such a choice leads to the maximum DM number density while satisfying the constraint of relic density and . As it can be seen from fig. 5, such values of mediator and DM mass satisfies the constraint from collisional damping as well. For example, as fig. 5(a) suggests, MeV and GeV correspond to the upper boundary of the blue region, which represents the point of highest relic abundance. Similarly for the scalar and vector mediated case, the values of mediator masses come out to be MeV and GeV respectively for MeV.
With the abovementioned values of the DM and mediator masses, the neutrinoDM scattering crosssection for the entire range of energy of astrophysical neutrinos fall short of the crosssection required to produce flux suppression, by many orders of magnitude. The key reason behind this lies in the fact that for the range of allowed DM mass, corresponding number density is quite small and the neutrinoDM scattering crosssection cannot compensate for that. The crosssection in the fermion and scalar mediated cases decrease with energy in the relevant energy range. Such a fall in crosssection is much more faster in the scalar case compared to the fermion one. Though in the vectormediated case the crosssection remains almost constant in the entire energy range under consideration. The crosssection in the fermion, scalar and vectormediated cases are respectively , and orders smaller than the required crosssection in the energy range of TeV  PeV. Thus we conclude that the three renormalisable interactions stated above do not lead to any significant flux suppression of astrophysical neutrinos in case of cold thermal dark matter.
iv.3 Ultralight Scalar Dark Matter
Here we consider the DM to be an ultralight BEC scalar with mass eV. The centreofmass energy for the neutrinoDM interaction in this case always lies between eV to MeV for incoming neutrino of energy PeV depending on DM mass. We consider below the models described in Sec. IV to calculate the crosssection of neutrinoDM interaction and compare those to the crosssection required for a flux suppression at IceCube.
iv.3.1 Fermionmediated process
The crosssection for neutrinoDM scattering through a fermionic mediator in case of ultralight scalar DM is given as
where are the mass and energy of the incoming neutrino respectively, is the mass of the ultralight DM, and is the mass of the heavy fermionic mediator. As the mass of the DM is quite small, at lower neutrino energies and hence, the crosssection remains constant. As the energy increases, the term becomes more dominant and eventually, the crosssection increases with energy.
Such an interaction has been studied in literature in case of ultralight DM Barranco:2010xt . This analysis was improved with the consideration of nonzero neutrino mass in ref. Reynoso:2016hjr . For example, from fig. 6(a) it can be seen that the crosssection for eV is larger compared to that for eV. In fig. 6(b), with eV, it is shown that no significant flux suppression takes place for a DM heavier than eV for GeV. However, it has been shown that the quantum pressure of the particles of mass eV suppresses the density fluctuations relevant at small scales Mpc, which is disfavoured by the Lyman observations of the intergalactic medium Irsic:2017yje ; Armengaud:2017nkf . Also, the constraint on the mass of such a mediator fermion, which couples to the boson with a coupling of the order of electroweak coupling, reads GeV Achard:2001qw . These facts together suggest that eV and GeV, as considered in ref. Reynoso:2016hjr , are in tension with Lyman observations and LEP searches for exotic fermions respectively. If we consider eV along with GeV, it leads to a larger crosssection compared to that with eV, which is still smaller compared to the crosssection required to induce a significant flux suppression. Thus, taking into account such constraints, the interaction in eq. (12) does not lead to any appreciable flux suppression in case of ultralight DM.
iv.3.2 Scalarmediated process
As mentioned in Sec. IV.1.2, the bound on the coupling of a scalar mediator with neutrinos is quite stringent, eV. Moreover, the mass of such a mediator are constrained as GeV Das:2016bir . In this case, the crosssection of neutrinoDM scattering is independent of the DM as well as the neutrino mass for neutrino energies under consideration. As fig. 7 suggests, the neutrinoDM crosssection in this case never reaches the value of crosssection required to induce a significant suppression of the astrophysical neutrino flux for eV. As mentioned earlier, DM of mass smaller than eV are disfavoured from Lyman observations.
iv.3.3 Vectormediated process
As it has been discussed in Sec. IV.1.3, the couplings of electron and muonflavoured neutrinos to the are highly constrained, . However, as it will be discussed in Sec. V, for the tauneutrinos such a coupling is less constrained, . From fig. 8(a) it can be seen that, in presence of such an interaction, an appreciable flux suppression can take place for TeV, with MeV, and eV. Instead, if we fix PeV, it can be seen from fig. 8(b) that the entire range of DM mass in the ultralight regime, i.e., eV to eV, can lead to an appreciable flux suppression. In the next section, we present a UVcomplete model that can provide such a coupling between the mediator and neutrinos in order to obtain a crosssection which leads to an appreciable flux suppression.
In the standard cosmology neutrinos thermally decouple from electrons, and thus from photons, near MeV. Ultralight DM with mass forms a BoseEinstein condensate below a critical temperature eV, where is the scale factor of the particular epoch Das:2014agf . When the temperature of the Universe is , MeV for eV, i.e., the ultralight DM exists as a BEC. In order to check whether the benchmark scenario presented in fig. 8(a) leads to late kinetic decoupling of neutrinos, we verify if . Here, and are the density of relic neutrinos and the Hubble rate at temperature respectively,
(24) 
For eV, mediator mass MeV and neutrinoDM coupling , cm. Thus, at , eV with . This reflects that the neutrinoDM interaction in our benchmark scenario does not cause late kinetic decoupling of neutrinos. However, as fig. 8(b) suggests, for a particular neutrino energy the neutrinoDM crosssection is sizable for higher values of , that can lead to late neutrino decoupling and we do not consider such values of . It was also pointed out that a strong neutrinoDM interaction can degrade the energies of neutrinos emitted from core collapse Supernovae and scatter those off by an significant amount to not be seen at the detectors Mangano:2006mp ; Fayet:2006sa ; Bertoni:2014mva . This imposes the following constraint on the neutrinoDM crosssection Boehm:2013jpa ; Mangano:2006mp : cmMeV) for MeV. It can seen from fig. 8(a) that such a constraint is comfortably satisfied in our benchmark scenario.
V A UVcomplete model for vectormediated ultralight scalar DM
Here we present a UVcomplete scenario which accommodates an ultralight scalar DM as well as a with mass MeV. The mediates the interaction between the DM and neutrinos.
The coupling of such a with the first two generations of neutrinos cannot be significant because of the stringent constraints on the couplings of the with electron and the muon. As it was discussed in Sec. IV.1.3, those couplings have to be for MeV. Thus, only the couplings to the third generation of leptons can be sizable. However, the coupling of the with the quark is also constrained from the invisible decay width of . The bound from such invisible decay width dictates , where and stand for coupling with DM and the quarks respectively Fayet:2009tv . Thus we construct a model such that the couples only to the third generation of leptons among the SM particles.
The boson is realised as the gauge boson corresponding to a gauge group, which gets broken at MeV due to the vev of the real component of a complex scalar transforming under the . As the third generation of SM leptons are also charged under , in order to cancel the chiral anomalies it is necessary to include another generation of heavy chiral fermions to the spectrum Frampton:1999xi . The cancellation of chiral anomalies in presence of the fourth generation of chiral fermions under is discussed in appendix B. If the exotic fermions obtain masses from the vev of the scalar which is also responsible for the mass of , the mass of the exotic fermion is related to the gauge coupling of in the following manner DiFranzo:2015qea ; Dobrescu:2014fca ,
(25) 
Here, is gauge coupling of and is the charge of the scalar . It is clear from eq. (25) that, in order to satisfy the collider search limit on the masses of exotic leptons GeV, the gauge coupling of has to be rather small. Such a constraint can be avoided if the exotic fermions obtain masses from a scalar other than . This scalar cannot be realised as the SM Higgs, because then the effect of the heavy fourth generation fermions do not decouple in the loopmediated processes like , etc. To evade both these constraints we consider that the exotic fermions get mass from a second Higgs doublet.
1  1  0 
In order to avoid Higgsmediated flavourchanging neutral current at the treelevel, it is necessary to ensure that no single type of fermion obtains mass from both the doublets . Hence, we impose a symmetry to secure the above arrangement under which the fields transform as it is mentioned in table 2. After electroweak symmetry breaking, the spectrum of physical states of this model will contain two neutral CPeven scalars and , a charged scalar , and a pseudoscalar . The Yukawa sector of this model looks like,
(26) 
with,
(27) 
Here, and are the coupling multipliers of the SM and exotic fermions to the neutral scalars respectively. It can be seen that the couplings of the Higgses with SM fermions in this model are the same as in a TypeI 2HDM. is the mixing angle between the neutral CPeven Higgses and quantifies the ratio of the vevs of the two doublets, . The coupling of the SMlike Higgs to the exotic fermions tend to zero as