Canonical seesaw implication for two-component dark matter

We show that the canonical seesaw mechanism implemented by the U(1)B−L gauge symmetry provides two-component dark matter naturally. The seesaw scale that breaks B−L defines a residual gauge symmetry to be Z6 = Z2 ⊗ Z3, where Z2 leads to the usual matter parity, while Z3 is newly recognized, transforming quark fields nontrivially. The dark matter component that is odd under the matter parity has a mass ranging from keV to TeV. Another dark matter component that lies in a nontrivial representation of Z3 can gain a mass in the range similar to the former component, in spite of the fact that it can be heavier the light quarks u, d. This two-component dark matter setup can address the XENON1T anomaly recently observed.

while the heavy neutrinos (∼ ν aR ) obtain large masses at the B − L breaking scale, M ∼ Λ.
Residual symmetry and dark matter. The symmetry breaking scheme is obtained as Here the electric charge is related to the isospin and hypercharge as Q = T 3 + Y . R is a residual symmetry of U (1) B−L that conserves the χ vacuum, although this vacuum χ = Λ/ √ 2 = 0 breaks B − L by two unit. As being a U (1) B−L transformation, R = e iα(B−L) where α is a transforming parameter. The vacuum conservation condition R χ = χ leads to e iα(2) = 1, or equivalently α = kπ for k integer. Hence, the residual symmetry is It is noted that the transformation with k is conjugated to that with −k, i.e. R † = (e iπ(B−L) ) −k = R −1 . The R values of all fields are collected in Table I. From this table, we derive that R = 1 for the minimal value of |k| = 6 and for every field, except the identity k = 0. Hence, the residual symmetry R is automorphic to where p ≡ e iπ(B−L) and p 6 = 1. Further, we factorize where is the invariant (or normal) subgroup of Z 6 , while is the quotient group of Z 6 by Z 2 . Thus, the theory automatically conserves both residual symmetries Z 2 and Z 3 after symmetry breaking. The field representations under Z 2 and Z 3 are computed in Table II, where w ≡ e i2π/3 is the cube root of unity. Here note that Z 2 has two irreducible representations, 1 according to p 3 = 1 and 1 according to p 3 = −1, whereas Z 3 has three irreducible representations, 1 according to (p 2 , p 5 ) = (1, 1) or (1, −1), 1 according to (p 2 , p 5 ) = (w, w) or (w, −w), and 1 according to (p 2 , p 5 ) = (w 2 , w 2 ) or (w 2 , −w 2 ), which are independent of p 3 values, 1 or −1, that identify Z 6 elements in a coset of the quotient group, respectively. 1 The representation 1 is not presented for the existing fields, but the antiquarks (u c , d c ) belong to 1 under Z 3 .
For brevity, the quotient group can be defined as where each (coset) element [x] consists of two elements of Z 6 , the characteristic x and the other p 3 x, as multiplied by p 3 . Hence, That said, the Z 3 irreducible representations 1, 1 , and 1 are simply determined by [ consists of r and ±r as multiplied by p 3 = ±1 respectively, which are homomorphic to that of Z 3 , [r] = {r, ±r} → r.
Since the spin parity P S = (−1) 2s is always conserved by the Lorentz symmetry, we can conveniently multiply the residual symmetry R = Z 2 ⊗ Z 3 with spin-parity group S = {1, P S } to perform where Z 3 is retained as the quotient group. The new invariant subgroup Z 2 ⊗ S defines a matter parity analogous to the R-parity in supersymmetry. Because of P 2 M = 1, we have P = {1, P M } to be a group of matter-parity symmetry by itself, which is an invariant subgroup of Z 2 ⊗ S. Therefore, we factorize Here ( Therefore, instead of R ⊗ S, we can consider an alternative residual symmetry which is contained in where the quotient group (Z 2 ⊗ S)/P is neglected, since the theory automatically preserves it. Of course, the theory conserves both P and Z 3 , under which the representations under these groups are given in Table III.
respectively. Upon P ⊗ Z 3 , let us assume the simplest dark matter candidates, as summarized in Table IV. Note that B − L charge of each dark field can deviate from the supplied value by an arbitrary even number that does not change the representations, because of the cyclic property of the residual symmetries. 2 Here, F and Φ mean fermion and scalar dark fields, respectively. Further, we assume the net mass of F 1 and F 2 is smaller than that of Φ.
The dark matter component stabilized by P (i.e., F 1 ) can have an arbitrary mass. This is also valid for the dark matter component stabilized by Z 3 (i.e., F 2 ), even though this component may be heavier than the light quarks u, d, which all transform nontrivially under Z 3 . Indeed, the Z 3 dark matter component must be color neutral, hence cannot decay to any colored final state, such as single quarks, because of the color conservation. This color conservation requires a color-neutral final state, if it comes from a dark matter decay. Obviously, the colorneutral final state if containing quarks must take only combinations of q c q and/or qqq. It follows that the final state is invariant (i.e. singlet) under Z 3 too, hence cannot be the product of any Z 3 dark matter decay, because of the Z 3 conservation. In other words, the SU (3) C and Z 3 symmetries jointly suppress the decay of Z 3 dark matter component (i.e. stable), even if this component has a mass larger than that of quark. With this proposal, we have the novel, simplest model for two-component dark matter based upon F 1 and F 2 self-interacting through a heavier dark field Φ, which is of course implied by the residual symmetry P ⊗ Z 3 , thus the canonical seesaw. [We can have other scenarios for two-component dark matter, if more dark fields are introduced, but they are complicated and suppressed.] Notice that since F 1,2 and Φ are the standard model singlets, the U (1) B−L dynamics is crucially/sufficiently governing the dark matter observables, besides the known consequences of neutrino mass and baryon asymmetry [23].
Seesaw implication for the XENON1T excess. The XENON1T experiment has recently reported an excess in electronic recoil energy ranging from 1 keV to 7 keV, peaked around 2.4 keV, having a local statistical significance above 3σ [14]. Such signal of electron recoils seems to reveal the existence of a structured dark sector . Indeed, the dark matter component that scatters off electrons should be fast moving v ∼ 0.03-0.25 for the dark matter mass m 2 ∼ 0.1 MeV to 10 GeV, which exceeds the velocity of cold dark matter v ∼ 10 −3 (cf. [25]).
This fast dark matter component (F 2 ) may be generated locally as a boosted dark matter from the annihilation or semi-annihilation of the heavier dark matter component (F 1 ), which is nicely implicated by our model. As a matter of the fact, the heavier dark matter component F 1 ∼ (1, 1, 0, 0) which interacts with normal matter only via gravity would dominate the cold dark matter, set by its annihilation or co-annihilation to the lighter dark matter component F 2 . The lighter dark matter component F 2 sub-dominates the dark matter abundance since it strongly couples to normal matter via the Z portal.
The relevant Lagrangian terms are where D µ = ∂ µ + ig B−L (B − L)Z µ and the dark matter masses obey m 0 > m 1 + m 2 and m 1 > m 2 . Since the B − L charge of F 1 is fixed, the remaining dark fields can possess more general B − L charges, for n = 0, ±1, ±2, · · · , as mentioned. 1. Annihilation (left) and co-annihilation (right) processes of F1 that set the cold dark matter density.
The relic density of F 1 is governed by Feynman diagrams in Fig. 1. The co-annihilation process is only enhanced when the masses of F 1 and Φ are highly degenerate. However, this work signifies m 0 > m 1 + m 2 > m 1 such that the co-annihilation contribution is negligible. The dark matter abundance is given by the F 1 annihilation in the left diagram.
Applying the Feynman rules, we obtain the thermal average cross-section times relative velocity as which relates to the F 1 abundance, Ωh 2 0.1 pb/ σv rel , where h is the reduced Hubble parameter without confusion. Using the experimental data Ωh 2 0.12 [1] and the fact that m 0 > m 1 > m 2 > m e , we get the constraint of the dark matter self-coupling to be |h| 0.015 Of course, at present, F 2 is locally generated by the left diagram in Fig. 1 which subsequently scatters off electrons in the XENON1T experiment through the diagram in Fig. 2. In the limit of mediator mass m Z to be much larger than the momentum transfer, the F 2 -electron scattering cross-section can be written as [57] σ e = g 4 This leads to the number of the signal events as related to the scattering cross-section by [35,58] N sig 100 = 1.6 × σ e 3 × 10 5 pb We require the number of the signal events about 100/ton/year in order to explain the XENON1T excess. This yields the mass of dominant dark matter F 1 as related to the U (1) B−L breaking scale, Since the dominant dark matter F 1 is thermally produced, its mass should obey m 1 > m 2 > m e as given. This suggests an upper bound on Λ to be Λ < 1.5 |1 + 6n| GeV.
Additionally, the new physics scale must satisfy Λ > O(1 TeV) in order for the seesaw mechanism properly working. The free parameter n that relates to the B − L charges of F 2 and Φ obeys |n| > 0.74 × 10 5 .
Further, one demands a perturbative condition for the U (1) B−L gauge interaction, i.e. |1/3 + 2n|g B−L < √ 4π, which along with the above result implies corresponding to the Z mass bounded as It is verified that the F 2 relic density is negligible, where F 2 completely annihilates to the standard model particles via the (s-channel) Z portal.
The small coupling and the mass of Z obviously satisfy the low energy constraints from the electron-positron colliding experiment KLOE2 [59], NA64 experiment [60], TEXONO experiment [61], or (g − 2) µ,e [62] Conclusion. We have discovered a seminal consequence of the canonical seesaw mechanism in addition to the known result of leptogenesis, such that this neutrino mass generation scheme with B − L gauge completion manifestly resolves the long-standing hypothesis of structured dark matter stability. The seesaw scale has a nontrivial physical vacuum that preserves two residual B − L symmetries related to the usual matter parity P M = (−1) 3(B−L)+2s and the new Z 3 quotient generator [p 2 ] = [w 3(B−L) ], respectively. This yields a novel scenario of two-component dark matter appropriate to the recent XENON1T experiment, where the cold dark matter F 1 has B − L = 0, while the boosted dark matter F 2 has a B − L charge deviating from 1/3 by five order of magnitude, which is allowed by the cyclic property of the residual generators. F 1,2 possess masses beyond the electron mass, while the B −L gauge boson has the gauge coupling and mass limited below 2.4 × 10 −5 and 50 MeV, respectively. If the XENON1T anomaly is relaxed, this setup can provide a generic scenario of two-component dark matter weakly interacting with normal matter.