A Portalino to the Dark Sector

"Portal"models that connect the Standard Model to a Dark Sector allow for a wide variety of scenarios beyond the simplest WIMP models. Kinetic mixing of gauge fields in particular has allowed a broad range of new ideas. However, the models that evade CMB constraints are often non-generic, with new mass scales and operators to split states and suppress indirect detection signals. Models with a"portalino", a neutral fermion that marries a linear combination of a standard model neutrino and dark sector fermion and carries a conserved quantum number, can be simpler. This is especially interesting for interacting dark sectors; then the unmarried linear combination which we identify as the standard model neutrino inherits these interactions too, and provides a new, effective interaction between the dark sector and the standard model. These interactions can be simple $Z'$ type interactions or lepton-flavor changing. Dark matter freezes out into neutrinos, thereby evading CMB constraints, and conventional direct detection signals are largely absent. The model offers different signals, however. The"portalino"mechanism itself predicts small corrections to the standard model neutrino couplings as well as the possibility of discovering the portalino particle in collider experiments. Possible cosmological and astroparticle signatures include monochromatic neutrino signals from annihilation, spectral features in high energy CR neutrinos as well as conventional signals of additional light species and dark matter interactions.

them with new scalar fields.
The neutrino portal has been well studied mostly in the context of sterile neutrino dark matter e.g., [10][11][12]. However, it, too, can yield thermal models such as via a new particle for dark matter to annihilate into [13][14][15][16][17]. The right handed neutrino, given a Majorana mass, can also be integrated out, yielding non-renormalizable mass-mixing operators with charged dark sectors, e.g., [18]. However, in these cases the light neutrino mass is corrected by an amount δm ν ≈ sin 2 θm heavy , meaning either the mixing must be very small, or the heavy state must be eV in scale.
In this note, we show that the neutrino portal can also yield scenarios that are structurally as simple as kinetic mixing models, but instead have a dark sector that dominantly interacts with neutrinos, rather than charge. The effect arises from the inclusion of a "portalino," a gauge-neutral fermion with an exact or nearly exact global quantum number. The scenarios we arise at naturally have potentially sizable (g ν ∼ 10 −2 ) new interactions for neutrinos, and dark matter freezeout into neutrinos. The dark matter in these scenarios can be light (m χ > ∼ 10 MeV), without needing to turn off annihilation channels because the only coupling to SM particles is to neutrinos which do not strongly affect the CMB.

II. THE PORTALINO
The neutrino portal is typically thought of as a relatively benign interaction. It produces a Dirac mass with a SM singlet fermion. If the singlet fermion has a large Majorana mass, the physical mass is suppressed by the seesaw mechanism and the Dirac mass can be sizable. If there is a lepton number symmetry, the Dirac mass sets the scale for the neutrino mass and thus must be small enough to be consistent with terrestrial and cosmological measurements.
However, this latter case assumes that there are no other particles involved. It is this possibility that is our focus.
If one extends the model simply by adding a second singlet fermion ψ the physical consequences are significant. If ψ has a Dirac mass m n with the first singlet fermion, then there is a massless field in the spectrum. Specifically, there is a field we would identify as a neutrino ν = c θ ν L + s θ ψ, (2) with a massive partner Here tan θ = m D /m n , and the heavy mass eigenstate has mass m = m 2 d + m 2 n . ν has its interactions suppressed compared to the standard model by c θ which leads to strong constraints on the mixing angle s θ < ∼ 10 −1 − 10 −3 from precision measurements of weak decays with neutrinos. However, if the neutrino in question is ν τ , there is far less precision information on its couplings and mixing angles as large as s θ ∼ 0.3 are possible. Importantly, the Dirac mass m d can naturally be as large as charged lepton masses, even if the heavy neutrino state is still weak scale.
Let us complicate the situation further -if we imagine the state ψ has interactions of its own, whether scalar or vector, the light mass eigenstate will inherit those interactions, with a coupling suppressed by powers of s θ . As will be our principal focus, let us suppose a coupling of ψ to a new massive vector boson ω In terms of mass eigenstates, this interaction becomes L ⊃ s 2 θ ν †σ µ ν ω µ + s θ c θ ν †σ µ n ω µ + s θ c θ n †σ µ ν ω µ + c 2 θ n †σ µ n ω µ .
Thus, the light neutrino mass eigenstate, which we identify as the physical neutrino, carries a residual vector interaction as well. This "effective Z " process [19] [21][22][23], often described as an effective theory. Direct mass mixings of the form ψφhl/M have been studied as a means to induce interactions for neutrinos [18,24,25], including in chiral models [26], but, absent tuning, these tend to require either small mixings or light sterile neutrinos. More closely related to dark matter, [27] studied an effective Z model, where charged fermions marry the neutrino by extending the SM with a second Higgs doublet charged under a new U (1). [13,17] showed how an inverse seesaw could yield a large DM − ν − φ Yukawa interaction, where φ is some new lepton number carrying force carrier. These last two are closest in content to the scenario described here.

A. A Simple Model
Taking the above discussion and translating into a full Lagrangian is straightforwardone must simply cancel gauge anomalies (by adding a conjugate ψ c ) and ensuring that there are no additional massless states (which are constrained by the CMB).
The simplest example is where l and h are the usual Standard Model fields, n c and x c are gauge singlets and n + and x − are two SM singlets which are charged under the dark U (1) d . We use the x and n labels to distinguish the mass eigenstates after U (1) d breaking. Connecting to the previous section, n c is our portalino, n + ↔ ψ, and we extend the model with additional fields to cancel the gauge anomalies and provide masses for all new states. Mass terms n c x c and n + x − would be allowed by the gauge symmetries, but are easily forbidden with global symmetries.
Assuming that the uneaten fields in the scalars h and φ are heavy enough to be ignored we replace them by their VEVs and we arrive at two (Dirac) mass terms There are two massive Dirac particles: x c pairs up with x − , and n c combines with linear combination (n) of ν L and n + , the other linear combination remains massless and is what is identified as the "usual" neutrino ν. Note that we have included only one portalino field in this simple model. It couples to a linear combination of the SM lepton doublets yl → i y i l i in Eq. 6. Generalizations with multiple portalinos are straightforward, see Section II B.
The fields with couplings to gauge bosons in the mass eigenstate basis are with couplings Z µ : In this model, n acts as an unstable heavy neutrino, the portalino, while x is stable. As we shall discuss in Section III, x provides a natural dark matter candidate. The details of the model's dark phenomenology depend on the ordering of the massive particles, n, x and ω.
All three obtain their masses from the symmetry breaking vev v φ proportional to coupling constants. As a benchmark scenario we envision the ω as the heaviest particle with a mass of order GeV, the dark matter x with a mass in the 10 to 100 MeV range and the portalino n somewhere in-between. With this ordering of the spectrum portalinos decay invisibly to dark matter particles, and dark matter freeze-out occurs via annihilation into neutrinos.
Remarkably, all other orderings also produce viable models of thermal dark matter.

B. Non Abelian Models and generalizations
The U (1) model is a very simple example of the portalino mechanism with an automatic dark matter candidate. However, there are a number of variants which we might also expect.

Multiple Portalinos
The simplest extension of the above scenario is to enlarge n P , the total number of gauge singlet portalinos. As we do so, we should simultaneously consider enlarging n D , the number of dark gauge-charged fermions ψ. In the most straightforward examples n P = n D and there are no new massless degrees of freedom.
However, even in this simple modification, there are important differences. The first critical difference is that the heavier portalinos will dominantly decay invisibly, via an offshell ω, irrespective of the ordering of the mass spectrum of n, x and ω. This will be critical as we discuss the experimental constraints on this scenario in Section IV.
The second difference is that the mixing angle of the lightest portalino is not necessarily the largest mixing angle. Thus, the massless "neutrinos" can have larger couplings to ω, only constrained by the properties of a heavier, invisibly decaying portalino.
We can also consider moving away from n P = n D . If n P > n D , we will require some state to have extremely small (< 10 −12 ) Yukawas, lest the SM neutrinos have too large Dirac masses. Since our premise is that all these couplings should be more comparable to ordinary Yukawas, this moves the scenario in a qualitatively different direction.
If we take n D > n P , we will have new, massless degrees of freedom. As we will discuss later, there are well-known and generic constraints on new light degrees of freedom. There are unavoidable production processes for these states as well. Just as the portalino can decay to SM neutrinos, it will be able to decay to these new states as well. Four-fermi operators will allow production of these states via νν → ψψ. If these states are charged under the ω gauge group, the rate of production will be large. If they are charged under some other gauge group, they could be produced by processes mediated by that gauge boson. However, even if those processes are suppressed, they will still be produced with a rate T 5 sin θ 4 where θ ν , θ ψ are the mixings of ν and ψ with the portalino, respectively. Terminating this process by T ∼ GeV, requires m ω > ∼ 10 5 GeV × sin θ ψ sin θ ν . Precise constraints depend on the details of the new hidden sectors.

Non-Abelian Models
While the simplest model from Section II A is based on a U (1) gauge group, the charged fields can be charged under any other group so long as a gauge singlet fermion operator is present. Simple extensions would be SU (2)

III. DARK MATTER FREEZEOUT
Since the field ψ is charged under a hidden gauge symmetry it is expected that there must be some additional field ψ c to cancel the gauge anomalies. It is natural (although not required) that due to gauge or global charges of ψ c no ψ c ψ Dirac mass term is present. If ψ c acquires a mass by marrying a different field, it forms a natural candidate for dark matter.
This is simply illustrated within the context of the U (1) model of Section II A where x forms a natural dark matter candidate.

IV. PORTALINO PHENOMENOLOGY AND NEUTRINO CONSTRAINTS
As these scenarios can yield O(1) corrections to the SM neutrino couplings, it is clear that constraints arise from a number of sources. Some constraints are modified in the presence of multiple portalinos, and we shall review them here. Our bounds are adapted from the excellent review by de Gouvea and Koblach [28], the SHIP white paper [29], and [13,17]  The dotted line bounds (orange shading) are more model dependent as they rely on portalino decays to visible SM particles. For details, see text and references [13,28,29].
whether it decays into dark sector or SM states. If the portalino is the lightest dark sector particle, it can still decay as n → 3ν via an off-shell ω with Γ n ≈ 1 10 −2 sec Thus even for portalinos which can only decay to SM particles the decay is sufficiently rapid.
When there is a dark sector state that is lighter than the portalino under consideration (for example, if there is a lighter portalino or if the dark matter particle is lighter than the portalino) then the lifetime for three body decays is much shorter with sin 2 θ max → 1. If the portalino is heavier than m ω then the two body decay n → ων becomes dominant. Thus even portalinos as light as 10 MeV can decay promptly before BBN under a wide range of circumstances.
In determining the 10 MeV lower bound for the portalino mass, the dominant constraint comes from CMB bounds on light species. After chemical decoupling of the neutrinos from the electrons, portalinos can effectively produce chemical potentials for the neutrinos via either annihilation or decay, but the final state products can be thermalized. After kinetic decoupling of the neutrinos, the entire energy of any portalinos which are still present is absorbed by the neutrinos, thus potentially increasing the temperature.
Assuming each portalino is converted into a single neutrino via the semi-annihilation pro- we have ∆N ef f = n n /n ν < 0.33 (where n ν is the number density of a single neutrino species).
This results in a bound of m n > 9.2, 15.2 MeV for e and µ/τ , respectively. Requiring that the total energy in the portalino is less than .33 of that of a neutrino species at kinetic decoupling of the neutrinos yields the slightly weaker bounds m n > 7.1, 11.6 MeV. Both of these bounds assume that the portalinos cannot deposit their energy directly into SM particles other than the neutrinos, and, in the latter case, that the ν µ,τ cannot rethermalize with ν e via the new ω interactions (in which case the lower ν e bound applies).

A. Light Species
In this scenario, it is quite common that there are additional light species present. This can be because the gauge sector has a residual unbroken component, or non-Abelian partners of ψ have small or zero masses. Thus, it is worth considering what constraints on N ef f imply for such portalino scenarios.
Assuming that the hidden sector decouples from the SM at a temperature T dec , then the effective number of neutrinos contributed by the particles in the hidden sector is 7 g * D g dec * D g * SM g * D g dec * SM where g * D , g * SM , g dec * D , g dec * SM are the effective number of degrees of freedom in the dark sector and the SM at low energies, and in the dark sector and the SM at decoupling, respectively.
Specifically, g * SM = 10.75, since we are anchoring to the last point when neutrinos have their entropy increased. For simplicity, we can take g * D = g dec * D and assume T dec > ∼ 1GeV at which time g dec * SM = 61.75, yielding Taking the Planck limit (95% confidence) of ∆N ef f < 0.33 [34] we find the relatively mild g * D < 6. On the other hand, If T dec < ∼ Λ QCD at which time g dec * SM = 17.25, this becomes a more stringent requirement g * D < 1.1.

VI. DISCUSSION
Dark matter that primarily interacts with neutrinos is a challenging scenario to test.
Mixing of the portalino with SM neutrinos is a crucial component of the scenario and can be tested by precision measurements of the SM neutrino couplings. But there are also several more model-dependent possible signals which depend on the details of the hidden sector.
• In addition to the large interaction of the ω with neutrinos, it may have small interactions with other SM particles. This could arise, for instance, from a small kinetic mixing with the SM photon. Alternatively, we could identify it as the gauge boson of some other SM symmetry, such as baryon number, µ − τ , or some effective Z , although this would require either small couplings or m ω > ∼ 100GeV. In addition to the well-studied phenomena associated with those forces, this would also yield signals of enhanced ν − SM interactions, especially at energies comparable to m ω .
• The portalino couplings may also violate lepton flavor. Then W loops in association with flavor violating portalino coupling insertions can give rise to processes like µ → eγ and µ to e conversion in the background of a nucleus. The Mu2e experiment [35] at Fermilab will look for µ to e conversion in the field of an Aluminum nucleus, and is expected to improve the limits on |U νµn U νen | by two orders of magnitude.
• A clear signature of this scenario would be the detection of monochromatic neutrinos from dark matter annihilation. Current limits on this from the galactic center are roughly 100 − 1000 times thermal from 10 GeV < ∼ m χ < ∼ TeV [36,37], making a straightforward detection of the scenarios described above challenging. However, it is quite simple to employ the portalino to yield models that could be detected.
In particular, one can envision a scenario with m ω < 2m n and dark matter a vectorlike state with ∼ TeV mass, charged under ω. This allows one to straightforwardly adopt the construction of [7], and consider χχ → ωω, with ω → νν. For m ω ∼ GeV a sizable Sommerfeld enhancement would boost the signal into the detectable regime.
For m ω > 2m n the signals could become partially visible at the level of the visible BR of the n, yielding other signatures, including CMB constraints.
Thus, for σ ∼ sin 4 θ/m 2 ω ∼ (10 −3 ) 2 /(100MeV) 2 ∼ (100 GeV) −2 (which is the approximate size of the cross section on resonance), one can reasonably have a universe that is somewhat opaque to neutrinos at the resonance energy. This would lead to distortions of the cosmic ray neutrino spectrum which could be detectable at IceCube. [23,40] • Self-interactions of the dark matter may thermalize the cores of dark matter halos, potentially resolving small scale anomalies [41][42][43][44]. The interactions between the dark matter and neutrinos or between dark matter and dark radiation in our models can also leave a measurable imprint on the large scale structure of the universe [45,46] The portalino could naturally find itself embedded in other scenarios, such as solutions to the hierarchy problem. In a SUSY model, the mass scale could arise radiatively, analogously to kinetic mixing scenarios [47][48][49]. In a Twin Higgs scenario, portalinos could serve as a means to marry off just three of the six neutrinos, ameliorating the massless degree of freedom problem in those models. We leave a detailed study of these possibilities for future work.
We should also note that while we have focused on the portalino coupling to the neutrino and carrying an effective lepton number, it is also possible to write the non-renormalizable operator uddn/M 2 , replacing lepton number with neutron number. This would yield a small n-neutron mass mixing, and by analogy with the neutrino mixing scenario, would lead to neutron-specific effective interactions. Given the tremendous questions of flavor and collider limits, a full discussion warrants further study.
The astute reader will have noticed that we have not said anything about neutrino masses. This is because the portalino is compatible with many different ideas for neutrino mass generation. One scenario that appears particularly intriguing is radiative neutrino mass generation via n number violation. Or one could envision an inverse seesaw due to a small Majorana mass for ψ.
While we have illustrated a variety of interesting scenarios, we have only scratched the surface of possible models. In particular, chiral models, non-Abelian models, and a thorough exploration of dark matter scenarios is warranted.