Interpreting dark matter solution for B − L gauge symmetry

It is shown that the solution for B − L gauge symmetry with B − L = − 4 , − 4 , +5 assigned for three right-handed neutrinos respectively, reveals a novel scotogenic mechanism with implied matter parity for neutrino mass generation and dark matter stability. Additionally, the world with two-component dark matter is hinted.

Ref. [12] discovered a simple option of Dirac neutrino mass as suppressed by a potential with accidental ν 3R dark matter.Ref. [13] investigated radiative neutrino mass and ν 3R dark matter mass by introducing an extra vectorlike lepton S with B − L = 8 besides ν R 's, whereas Ref. [14] proposed a scotogenic scheme in a variant with seven extra neutral leptons alternative to ν R 's.Ref. [15] examined radiative Dirac neutrino mass and dark matter by imposing extra lepton doublets and Z 2 for dark matter stability.
As a next attempt to the above process, I argue that the second solution provides naturally both dark matter and neutrino mass, without requiring any extra fermion and extra symmetry.It is indeed the first scotogenic mechanism realized for minimal right-handed neutrino content with B − L = −4, −4, +5 and a residual matter parity, alternative to [16].
In a period, the matter parity which stabilizes dark matter has been found usefully in supersymmetry.I argue that the matter parity naturally arises from the second solution for B − L gauge symmetry, without necessity of supersymmetry.
Field presentation content of the model.
for consistency with the standard model.Additionally, the scalar χ couples l aL to ν αR , while the scalar η couples χ to Hϕ 1 as well as to ϕ 2 , which radiatively generates neutrino mass (see Fig. 1).The fields χ, η have vanished VEVs, preserved by the matter parity conservation.
This realizes a scotogenic scheme with automatic matter parity by the model itself, which stabilizes dark matter candidates ν αR , χ 0 , η, opposite to [16] for which a Z 2 is ad hoc input.
Matter parity.-AB − L transformation has the form, P = e ix(B−L) , where x is a parameter.P conserves both the vacua w 1,2 , i.e.P w 1 = w 1 and P w 2 = w 2 , given that e i8x = 1 and e −i10x = 1.It leads to x = kπ, thus P = (−1) k(B−L) , for k integer.Acting P on every field, I derive P = 1 for minimal |k| = 6, except the identity with k = 0.This defines a residual group Z 6 = {1, p, p 2 , p 3 , p 4 , p 5 }, where p = (−1) B−L and p 6 = 1.I factorize to be matter parity similar to that in supersymmetry, governing this model. 1The matterparity group M = {1, P } instead of Z 2 has two irreducible representations 1 and 1 ′ according to P = 1 and P = −1 respectively, collected in Tab.I for every field.The lightest of odd fields ν αR , η, χ is absolutely stabilized by the matter parity conservation, providing a dark matter candidate.However, since ν 3R does not singly couple to standard model fields at renormalizable level similar to proton, ν 3R has a lifetime bigger than the universe age (see below), supplying an alternative dark matter candidate, kind of minimal dark matter.
Scalar potential and mass splitting.-Iwrite the scalar potential , where the first part includes only the fields that induce breaking, while the second part is relevant to η, χ and mixed terms with breaking fields, The trivial η, χ and nontrivial H, ϕ 1,2 vacua acquire µ 2 1,2 > 0, κ 2 < 0, and κ 2 1,2 < 0. Additionally, the potential bounded from below demands that c > 0, c 1,2 > 0, and λ 1,2 > 0, which are derived from V > 0 when H, ϕ 1 , ϕ 2 , η, and χ separately tending to infinity.Further, V > 0 applies when every two of these fields simultaneously tend to infinity, yielding , where Θ is the Heaviside step function.Notice that V > 0 for every three (every four, every five) of scalar fields simultaneously tending to infinity will supply extra, complicated conditions for scalar self-couplings. 2Furthermore, constraints of physical scalar masses squared to be positive might exist, but most of which would be equivalent to the given conditions. Let , which all are at least at w 1,2 scale.The field χ ± is a physical field by itself with mass m 2 χ ± = M 2 2 + λ 10 2 v 2 .The fields R, R 1 and I, I 1 mix in each pair, such as I define two mixing angles, The physical fields are with respective masses, 2 All such conditions ensure the quartic coupling matrix to be co-positive responsible for the vacuum stability, which can be derived with the aid of [18].
where the approximations come from |θ R,I | ≪ 1 due to v ≪ w 1,2 , and it is clear that the R, I and R 1 , I 1 masses are now separated.
Neutrino mass.-TheYukawa Lagrangian relevant to neutral fermions is When ϕ 1,2 develop VEVs, ν R 's obtain Majorana masses, such as where I assume t αβ to be flavor diagonal, i.e. ν 1,2,3R are physical fields by themselves.This Yukawa Lagrangian combined with the above scalar potential, i.e.
1 , up to kinetic terms yields necessary features for the diagram in Fig. 1 in mass basis.That said, the loop is propagated by physical fermions ν 1,2R and physical scalars R It is noteworthy that the divergent parts arising from individual one-loop contributions by , as well as the mixing angles θ 2 R,I ∼ λ 2 v 2 /w 2 1,2 , the resultant neutrino mass in ( 12) manifestly achieved, proportional to m ν ∼ λ 2 h 2 v 2 /32π 2 w 1,2 is small, as expected.
The field ν 3R communicates with normal matter through the Z ′ B−L and ϕ 2 portals only, unlike the scotogenic fields that interact directly with usual leptons and Higgs field additionally.Obviously the ϕ 2 portal couples to normal matter only through a mixing with the usual Higgs field, giving a small contribution to dark matter observables.The gauge portal dominantly contributes to dark matter annihilation to normal matter via s-channel  relevant mass resonances m ν 1R = 1 2 m Z ′ B−L (as vertical line) and m ν 3R = 1 2 m Z ′ B−L (as horizontal line) are crucial to set the correct relic density Ω DM h 2 = 0.12 as the density curve is based/distributed around these resonant lines.Additionally, if the mass resonance occurs at ν 1R then its partner ν 3R mainly contributes to the density, and vice versa.Lastly, as ν 3R in previous scenario, both ν 1,3R in two-component dark matter scheme possess a negligible scattering cross-section with nuclei in direct detection, appropriate to observation.

Concerning collider limits.-Z ′
B−L couples to both leptons and quarks, presenting promising signals at colliders.The LEPII experiment [23]

1 FIG. 1 .
FIG. 1. Neutrino mass generation induced by dark matter solution of B − L gauge symmetry.

+ 100w 2 2 , it correspondingly limits 64w 2 1 + 100w 2 2 > 6 > ∼ 5
searched for such a new gauge boson through process e + e − → f f for f = µ, τ , described by the effective LagrangianL eff ⊃ (g B−L /m Z ′ B−L ) 2 (ēγ µ e)( f γ µ f ), making a bound m Z ′ B−L /g B−L > 6 TeV.Since m Z ′ B−L = g B−L 64w 2 1 TeV; particularly, w 1 ∼ w 2 >∼ 0.5 TeV if the two scales are equivalent.Alternatively, the LHC experiment[24,25] looked for dilepton signals via process pp→ Z ′ B−L → f f for f = e, µ, yielding a Z ′ B−L mass bound roundly m Z ′ B−L TeV for Z ′ B−L coupling relative to that of Z, such as g B−L = 5/8s W g Z ≃ 0.28.This converts to 64w 2 1 + 100w 2 2 = m Z ′ B−L /g B−L > ∼ 17.85 TeV, thus w 1 ∼ w 2 > ∼ 1.39 TeV, which is radically bigger than the LEPII.The last bound is appropriate to those imposed for neutrino mass and dark matter, as desirable.Concluding remarks.-Thedark side of the B − L gauge symmetry is perhaps associated