Dark $SU(2)_D$ Gauge Symmetry and Scotogenic Dirac Neutrinos

Dark matter is postulated to transform under the non-Abelian $SU(2)_D$ gauge symmetry. Its connection to Dirac neutrino masses is explored.

Introduction : The existence of dark matter [1] is well established, but its nature is still unknown. It is however very likely to be stabilized by an unbroken symmetry, such as Z 2 dark parity, under which it is odd and all particles of the Standard Model (SM) are even. If it has interactions with Majorana neutrinos, it is also derivable [2] from lepton parity, i.e. π D = π L (−1) 2j , where π L = (−1) L and j is the intrinsic spin of the particle. If the dark symmetry is global U (1) D , which may be the remnant [3] of an U (1) gauge symmetry, it is possibly connected to Dirac neutrinos [4,5].
To explore further the origin of dark symmetry, the non-Abelian gauge symmetry SU (2) D may be considered [6,7]. In this paper, a model is studied where SU (2) D is spontaneously broken to retain a global SU (2) symmetry, with additional fermions and scalars which link them to the leptons of the Standard Model (SM) of particle interactions. Two Dirac neutrino masses are generated in one loop from these SU (2) D dark fermions and scalars.

SU
(2) D fermion doublets as in this model, the invariantψ L1 ψ R1 +ψ L2 ψ R2 is a nonzero mass term. If there are three chiral SU (2) D fermion doublets, it is easily shown that two will pair up as in this model, and the other will remain massless.
As shown in Table I and II, lepton number L is imposed and required to be conserved by all terms in the Lagrangian. The Z 2 symmetry applies to all dimension-four terms of the Lagrangian, but is broken by soft dimension-three terms. It serves the purpose of forbidding the coupling of the right-handed neutrino ν R to ν L through φ 0 . Hence neutrinos are massless at tree level. They will pick up Dirac masses in one loop through the soft breaking of Z 2 to be discussed. Whereas there are three families of ν L , there are only two of ν R . If more than two are assumed, only two linear combinations will pick up masses because only two SU (2) D fermion doublets are postulated. Therefore, one neutrino must be massless in this model, which is of course allowed by neutrino oscillations where only mass-squared differences are measured.
The spontaneous breaking of the SU (2) D gauge symmetry comes from the SU (2) D scalar doublet ζ. Without loss of generality, ζ 2 = v D may be chosen. As a result, all three SU (2) D gauge bosons W D acquire the same mass m W D = g D v D / √ 2, and the one physical dark Higgs . The Higgs potential consisting of Φ and ζ is The other scalars, i.e. the bidoublet (ν L1 ,l L1 ;ν L2 ,l L2 ) and the doublet (ν R1 ,ν R2 ) have L = 1 and will not develop vacuum expectation values. Their quartic interactions with Φ and ζ are products of bilinear invariants, which are easily written out and are not very illuminating or relevant. There are however trilinear terms which break Z 2 softly and are important for generating one-loop Dirac neutrino masses to be discussed.
The only physical scalars are h D and the SM h = √ 2Re(φ 0 ), resulting in Hence h D mixes with h according to and may decay to SM fermions through the h Yukawa couplings. Note that h D does not couple directly toψψ.
Scotogenic Dirac Neutrino Mass : As pointed out earlier, the Z 2 symmetry of Table I, II is required for all dimension-four couplings. Hence ν R cannot couple to ν L through φ 0 , but they do couple to the dark fermions with L = 0 and dark scalars with L = 1, i.e.
Thus there are two connections linking ν L to ν R , as shown in Fig. 1. It is understood that they are accompanied by two more diagrams which render the vertices invariant under SU (2) D as given in L Y . The Z 2 symmetry is broken softly by the trilinear term (ν * R1ν L1 + ν * R2ν L2 )φ 0 . Note also that the lepton number L assignments of Table I, II are mandatory because of the SU (2) D gauge symmetry which forces the ψ fermions to have L = 0.
The scalarν L andν R doublets mix through their coupling to φ 0 . Let their mass eigenstates be then the radiative Dirac neutrino mass matrix is given by where F (a, b) = a ln(a/b)/(a − b). Thus two neutrinos are massive and one is massless in this model, allowing two mass-squared differences as observed in neutrino oscillations.
Freeze-Out Scenario : The dark sector consists of the three W D gauge bosons, the fermion The respective cross sections × relative velocity are [8] σ and [9] where In order to discuss the relic density of two-component dark matter [10][11][12][13][14] quantitatively, we first write the coupled Boltzmann equations (BEs) for comoving number densities of W D and ψ respectively. Choosing the variable to be respectively, the BEs in the limit of two stable dark matter candidates with the dominant annihilations specified above, can be written as where and H is the Hubble parameter. The respective cross-sections (σv rel ) are given above while σv rel is the thermally averaged annihilation cross-section [15]. The first term of the right hand side of Eq. (11) represents the effect of the annihilation of two W D to two dark Higgs (h D ) whereas the second term shows the conversion process between ψ and W D . On the other hand, the right hand side of Eq. (12), contains the conversion term only as ψ dominantly annihilates into the other dark matter, namely W D . In order to calculate the thermally averaged annihilation cross-sections and solve the above BEs numerically, we use micrOMEGAs clearly notice that the ψ contributes more to the relic density than the W D . This is because, being the heaviest dark sector particle with only one annihilation channel affecting its relic density, ψ decouples earlier than W D with the latter being in thermal equilibrium for longer duration by virtue of its multiple annihilation channels. The efficient annihilation of W D till late epochs leads to a lower relic density. The right panel of Fig. 5 shows total relic density as function of M ψ where the other parameters have been fixed as shown in the figure itself. As the mass difference between different dark sector particles is reduced, the total relic increases due to inefficient annihilation rates as expected.
Apart from the relic density constraints, this model can also be tested in the ongoing and future direct detection experiments [17]. As for direct detection, ψ has no tree-level where m n is the mass of the nucleon, f p and ξ are form factor and dark Higgs-SM Higgs mixing parameter respectively. Such small h D -SM Higgs mixing also keep the invisible decay rate of SM Higgs within control, for light h D below SM-Higgs mass threshold. The latest constraints on such invisible Higgs decay is BR h→inv < 14.6% [19] and BR h→inv < 18% Other relevant bounds on such mixing can arise from electroweak precision data [21][22][23] as well as direct collider searches [24,25]. The chosen value of mixing parameter ξ = 0.001 automatically satisfies these bounds for h D mass above a few tens of GeV. The spin-independent direct detection cross-section for ψ-nucleon elastic scattering at one-loop level can be written as, where f h D is the one-loop effective coupling of ψ with dark Higgs with µ ψn being the reduced mass. In Fig. 6 code indicating the relative contribution of ψ to total dark matter relic density. Clearly, for most part of the parameter space, ψ dominates the total dark matter relic density as noticed in discussions above. Interestingly, some part of the parameter space we have shown above is already ruled out from the present limit of direct detection experiments, keeping the future detection prospects very promising. In Fig. 7, we have shown the spin-independent direct detection cross-section of for both the dark matter components as function of their masses. To calculate the effective spin-independent direct detection cross-section, we have multiplied the individual DM-nucleon scattering rate with the relative number densities of the dark matter particles. As can be seen by comparing with the latest upper limit on spin-independent dark matter nucleon cross-section [17], some part of the parameter space is excluded and the future experiments will be able to probe the parameter space. The scalar mixing is assumed to be 0.001 in this case. While the one-loop direct detection cross-section for ψ remains suppressed (as in the left panel of Fig. 7), the tree level cross-section of W D can be substantially large, in spite of its sub-dominant relic density. Since the direct search limit is on total DM relic density, some part of parameter space in ψ mass plane will also be ruled out. The final allowed parameter space from relic density and direct detection constraints is shown in Fig. 8. Since there are limited annihilation processes and M W D < M ψ , the DM parameter space is limited to a small parameter space for fixed g D . Since ψ dominantly annihilates only into lighter DM W D , its relic density is decided primarily by g D and M ψ .
The lighter DM relic density can be subsequently fixed by appropriately choosing dark scalar parameters. We also check the indirect detection bounds from gamma ray searches [26,27] and find the parameter space shown in Fig. 8 to be allowed for chosen value of scalar mixing. periment N eff = 2.99 +0.34 −0.33 [28], consistent with the SM prediction N SM eff = 3.045. Future CMB experiment CMB Stage IV (CMB-S4) is expected reach a much better sensitivity of ∆N eff = N eff − N SM eff = 0.06 [29], taking it closer to the SM prediction. Some recent works on light Dirac neutrinos and enhancement of N eff can be found in [30][31][32][33][34][35][36][37][38][39]. In our model, the right chiral part of Dirac neutrino can get thermalised by virtue of its Yukawa interactions.
Assuming all three ν R to get thermalised in the early universe and decouple instantaneously above the electroweak scale, simple entropy conservation arguments lead to ∆N eff ≈ 0.14 [30], well within the reach of CMB-S4. Since our DM analysis does not depend on neutrino Yukawa couplings, they can be tuned appropriately to satisfy light neutrino mass criteria while guaranteeing ν R thermalisation in the early universe. A detailed analysis of N eff in this model is beyond the scope of this present work and can be found elsewhere in the context of similar radiative Dirac seesaw models.
Freeze-In Scenario : Under SU (2) D , ζ cannot couple to ψ L,R . However, after the sponta- neous breaking of SU (2) D , the one dark Higgs boson h D obtains a connection toψ L ψ R in one loop through the massive dark gauge bosons W D , as shown in Fig. 9. In this scenario, ψ is assumed to be light, of order GeV, and W D very heavy. The effective h Dψ ψ coupling is then Assuming that h D is much heavier than h, then h decays to ψψ through h − h D mixing. Hence where z = M ψ /m h . The correct dark matter relic density is obtained [40] if f h ∼ 10 −12 z −1/2 .
Assuming M ψ ∼ 2 GeV, λ 3 ∼ 10 −4 , and using M W D = g D v D / √ 2, this is satisfied for TeV. If the reheat temperature of the Universe is a few TeV, then the only production mechanism of ψ is through h decay. Its relic denisty builds up to its present value until h goes out of thermal equilibrium with the other SM particles. It is known as a feebly interacting massive particle (FIMP) [41].
The Boltzmann equation for FIMP dark matter ψ, in terms of its comoving number density, can be written as where x = m h /T and K i is modified Bessel function of i-th order. In Fig. 10, we have shown the non-thermal production of dark matter comoving number density as a function of temperature for different benchmark values of the other parameters as shown in the figure.
The benchmark values of λ 3 are chosen in a way that keeps the dark sector out of equilibrium with thermal bath.
Concluding Remarks : The dark sector is postulated to consist of particles transforming under the gauge symmetry SU (2) D , which is broken by a scalar doublet so that a global SU (2) symmetry remains. Adding fermions and scalars, also transforming as SU (2) D doublets, two Dirac neutrino masses may be obtained radiatively with dark matter in the loop.
The structure of the dark sector allows naturally two dark matter components in a thermal freeze-out scenario. With a different choice of mass parameters, freeze-in production of dark matter may also be realized through Higgs decay.
Acknowledgement : This work was supported in part by the U. S. Department of Energy