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Neutrino Portal via Loops

One loop correction to the dark matter (DM) annihilation cross section is usually assumed to be sub-dominate in calculating the DM relic density. In this paper, we propose a scenario where DM freezes out mainly via loops in the neutrino portal. For a scalar DM annihilating into neutrino via a triangle diagram, or a Majorana DM annihilating into neutrino via a box diagram, the observed DM relic density can be generated if the mediator is Majorana fermion. We further work out the neutrino Debye mass and neutrino oscillation probabilities in the dark halo. Numerical results show that impactions of DM to these physical observables are negligible on the Earth, however the analytical results can be applied to evaluate neutrino properties in a dense DM environment.


I. INTRODUCTION
Astrophysical observations have confirmed the existence of dark matter (DM) [1], which cannot be addressed by the minimal Standard Model (SM) of particle physics. Of all the SM particles, neutrinos are most similar to DM in nature. In fact, neutrinos themselves are candidates for hot DM, which however cannot explain the formation of individual galaxies after the big bang. To explain the small-scale structure in the early Universe, we must rely on cold DM. What is the mass of the cold DM? how it couples to the SM particles? and what is the thermal history of the cold DM in the early Universe? are three unsolved mysteries. For the past decades, many DM models have been created with mass ranging from 10 −20 eV to 10 55 GeV, of which, weakly interacting massive particle [2][3][4][5][6][7], sub-GeV DM [8], axion [9][10][11][12] and primordial black hole [13] are typical DM candidates with abundant physical signals.
In this paper, we focus on the thermal history of the cold DM in the early Universe.
It is well-known that DM can be produced thermally via the freeze-out or freeze-in mechanism [14], or non-thermally via the misalignment mechanism [10][11][12], depending on the dynamics of the hidden sector. Non-thermal production mechanism is usually related to the phase transition, during which epoch the energy stored in the scalar potential false vacuum can be transferred to the DM. In the freeze-in scenario, DM is produced by the annihilation or decay of heavier particle, but the DM interaction rate is too small to thermalize. The freeze-out scenario requires a large enough interaction rate. If DM freezes out relativistically, then its relic density is simply Ωh 2 = (78/g * s )(m DM /KeV) [15,16], with g s * the number of degrees of freedom contributing to entropy at DM decoupling. Alternatively, if DM freeze-out non-relativistically, the relic density depends only on the annihilation cross section, which takes the value σv ∼ 10 −26 cm 3 · s −1 [7].
We propose an interesting possibility where DM is thermalized via tree-level interactions but non-relativistically freeze-out via high order processes. Usually these loop effects are negligibly small and cannot solely fit to a correct relic abundance when the tree-level annihilation process is kinematically forbidden. We find that the one-loop contribution can be enhanced in the neutrino portal whenever the mediator is a Majorana fermion. To illustrate this point, we propose two sub-GeV DM models: a scalar DM model, which freezes out via a triangle diagram, and a Majorana DM which freezes out via a box diagram. In both scenarios, the loop integrals are finite and large enough give rise to a correct relic abundance. Considering that thermal correction to the relic density is only proportional to O(T 4 ) [17,18], the result of loop calculation at the zero-temperature is credible.
It has been shown that active neutrino may acquire a sizable nonzero mass from a long range potential sourced by DM distributions [19,20]. We evaluate the neutrino Majorana mass induced by the scalar DM. It shows that the induced neutrino mass in the loop-portal is proportional to the DM density, and is negligibly small on the Earth as the local DM density is too low. We further investigate the matter effect of neutrino oscillations induced by the DM. For the first time, we derive the formulae of DM effect in neutrino-antineutrino oscillation probability, which is applicable to the investigation of neutrino oscillations with large Debye mass. For loop-portal scenarios, it is unlikely to test the DM effect in neutrino oscillation experiments in the solar system. Our results, in addition to the conclusions of Refs. [21][22][23] (and these from references cited therein), clarify the physical picture of neutrino oscillations in the dark halo.
The remaining of the paper is organized as follows: In section II we introduce the loopportal in detail and calculate the DM relic density. Section III is devoted to the study of neutrino mass induced by the DM. In section IV, we study the DM effect in neutrino oscillations. The last part is concluding remarks.

II. LOOP-PORTAL
In this section we present two models of neutrino-portal via loops: the scenario A for Majorana DM χ and the scenario B for scalar DM ϕ. They are both two-component DM models, of which the heavier DM component will dominantly annihilate into the lighter one resulting in negligible relic density. We assume the heavier DM component is a complex scalar Φ, which couples to the right-handed Majorana neutrino N R and the left-handed active   where m χ and m ϕ are masses of χ and ϕ respectively, Y χ is the Yukawa coupling and λ is the quartic coupling. Interactions between the SM Higgs and new scalar singlets, as well as quartic coupling between ϕ 2 and Φ † Φ bilinears, are neglected for simplicity. Throughout this paper we focus on the thermal dynamics of sub-GeV DMs and neglect the active neutrino mass.
In scenario A, both χ and Φ are DM candidates stabilized by a Z 2 symmetry. Assuming the mass hierarchy m N > m Φ > m χ , Φ annihilates into χ in the early Universe, resulting in negligible relic abundance for a sizable Yukawa coupling Y χ , while χ can only annihilate into neutrinos via the Box diagram given in the left-panel of the Fig. 1. The thermal average of the reduced annihilation cross section can then be written as where I is the four point loop function proportional to the Passarino-Veltman scalar integral D 0 [24][25][26]. To the leading order(neglecting the momentum of DM and neutrino in the propagators of heavy neutrino), I takes the form: The loop integral is calculated and simplified with the help of the Package-X [42,43]. There are neutrino-number-violating box diagrams, but their contribution to the annihilation are automatically cancelled.
As an illustration, we show in the left-panel of the Fig. 2 contours of the relic abundance of χ in the m χ − m Φ plane by setting Y χ = 1, Y N = 0.2 and m N = 10 GeV. Contours from the top to the bottom correspond to Ωh 2 = 0.01, 0.12 and 0.5, respectively. The lore from Tremaine and Gunn [44], which is based on the Pauli exclusion principle, sets a lower bound m DM > 100 eV on the mass of fermionic dark matter. Besides, a lower bound for a cold DM mass is about 1 keV [27]. We take m χ > 10 MeV in the numerical simulation. Yukawa interactions in this scenario may contribute to active neutrino mass as well as neutrino-DM mixing through the so-called scotogenic mechanism [45] whenever there is a mass splitting between the CP-even and the CP-odd components of Φ. Alternatively, this constraint can be avoided for a degenerate Φ.
In scenario B, both Φ and ϕ are DM candidates, while the relic density of Φ can be small for a sizable quartic coupling λ. ϕ freezes out non-relativistically via a triangle diagram, as shown in the right-panel of the Fig. 1, in which both neutrinos go out and the neutrino number is thus violated. The thermal average of the reduced annihilation cross section takes the form The relic abundance of ϕ is illustrated in the right-panel of Alternatively, if Φ possesses the Higgs portal interaction, the DM-nucleon scattering may arise at the one-loop level, which has been shown to be small as compared with current constraints put by DM direct detection experiments [28,29]. High luminosity fixed target  [32], which sets a lower bound on the DM mass, m DM > 3.7 MeV at 2σ [33]. This constraint is consistent with our parameter settings. For further discussions of impact of light DM on the Big Bang Nucleosynthesis and CMB, we refer the reader to Refs. [33][34][35][36] and references cited therein.

III. NEUTRINO MASS
Integrating out the heavy Majorana neutrino N and the complex scalar Φ, one may get the effective Hamiltonian describing DM-neutrino interactions, where Fierz transformation [46] is applied to the first equation. The Hamiltonian in Eq. (7) contributes to neutrino oscillation in the dark halo which will be discussed in the next section. The Hamiltonian in Eq. (8) contributes to neutrino Majorana mass as well as neutrino-antineutrino oscillations. In this section, we discuss the neutrino Majorana masses induced by DM in scenario B.
Assuming the function f (E, T ) is the statistical distribution of the DM with energy E and temperature T , one has d 3 pf (E, T ) = n φ V , where n φ is the DM density of the medium, n Φ V is the total number of the DM. The average of the effective Hamiltonian over the DM background in the rest frame of the medium is where ρ ϕ = n ϕ m ϕ with n ϕ the DM number density and P ϕ ∼ n ϕ T being the pressure associated with the DM. Since T ≪ m ϕ , P ϕ can be safely neglected. There are two main approaches to measure the local DM number density [37,38]: local measures that use the vertical kinematics of stars near the sun; global measures which extrapolate DM density from the rotation curve. The DM density near the sun is about 0.3 ∼ 0.4 GeV/cm 3 [39].
It has been shown in Refs. [19,20] that tiny but non-zero neutrino masses may come from a long range potential sourced by DM distributions. In our scenario B, Majorana neutrino mass δm can be generated in the presence of the constant DM background ϕ, To quantify δm, we consider the DM distribution throughout the MilkyWay Galaxy.
Two popular DM density profiles are used in numerical simulations, the Navarro-Frenk-White (NFW) [40] and Einasto [41]. The distribution of the neutrino mass as a function of the Galactic radius is depicted in Fig. 3. We take r ⊙ = 8. Note that δm is too small to give rise to any significant signals on the Earth.

IV. NEUTRINO OSCILLATION
For scenario A, the effective neutrino potential Hamiltonian in the dark halo with low temperature is H eff = V χ ν L γ 0 ν L with the potential, We assume DM only couples to the muon neutrino, and study the two-flavor neutrino oscillation P (ν e → ν µ ). A straight forward calculation gives P (ν e → ν µ ) = ∆m 2 ∆m 2 2 sin 2 2θ sin 2 ∆m 2 x 4E (12) with where A CC = 2 √ 2EG F n e with n e the number density of electron in the medium and A χ = IEn χ /m 2 Φ , θ and ∆m 2 are the mixing angle and the squared mass difference in the vacuum, respectively.
For scenario B, the DM in the medium may affect the neutrino-antineutrino oscillation probability. We follow the quantum field theory strategy [47] to calculate the DM effect in neutrino-antineutrino oscillations. The coefficient in Eq.(9) plays a rule of classical field and can modify the propagator of neutrinos, where we have defined the coefficient in Eq.(9) as ∆(x) for simplicity. We are unable to find a compact expression for S A , but we can treat ∆(x) as a perturbation. In this case S A can be written as Taking ∆ as a constant, the propagator in the momentum space can be written as iS −1 A (p) = / p − m i − ∆, after the summation. Now we can calculate the neutrino-antineutrino oscillation in the DM medium using quantum field theory approach. The S matrix for the process where g is the weak coupling, U is the so-called PMNS matrix. As a result, neutrinoantineutrino oscillation probability is where S is kinematic factor and is flavor blinded. This probability is same as the result of the νSM [48][49][50][51][52] up to the replacement m i → m i + ∆ i .
Note that the DM density is much smaller than that of the electron on the Earth, so the DM effect is negligible in neutrino oscillation experiments. Here we only estimate the oscillation probability of high energy electron neutrino in a dense DM environment, where the electron number density is negligible. ∆m 2 21 and P (ν e → ν µ ) as the function of A χ /∆m 2 21 are illustrated in the left-panel and right-panel of the Fig. 4, respectively. We have assumed that x/4E = 10 4 eV 2 when making the plot. For neutrino-antineutrino oscillation, the DM effect is negligible as the Debye mass is too tiny ( see Fig. 3 for detail). However our formulae can applied to study neutrino-antineutrino oscillation in a model where there is long range force between DM and neutrino, that results in a huge Debye mass [19] in the dense DM environment.

V. DISCUSSIONS
Neutrino and DM are two of neutral and long-lived elementary particles that constitute the Universe. Neutrino portal has attracted more and more attention in the past decades.
In this paper, we have proposed a new scheme of DM freeze-out from the thermal bath: DM annihilates into neutrinos and freezes-out through a loop. We showed that the annihilation cross section can be large enough to give rise to a correct relic abundance for the case where