Leptonic Source of Dark Matter and Radiative Majorana or Dirac Neutrino Mass

The notion of U(1) lepton number (which may only be softly broken) is applied to models of dark matter which interacts with leptons. Previous scotogenic models of Majorana or Dirac neutrino mass are shown to be derivable in this framework without additional symmetries. Only complete renormalizable theories are considered. An explicit class of models with $Z_n$ ($n \geq 5$) lepton and dark symmetry for Dirac neutrinos is derived, as well as an example of $Z_3$ dark symmetry.

Introduction : Two outstanding fundamental issues in particle physics and astroparticle physics are neutrinos and dark matter. They have been shown [1] to be intimately connected in all simple models of dark matter, with dark parity π D derivable from lepton parity π L with the factor (−1) 2j for a particle of spin j.
To explore further this connection, it is assumed that lepton number may be imposed as a global U (1) L symmetry in dark-matter extensions of the standard model (SM) of quark and lepton interactions. The extended field theory is required to be renormalizable and the U (1) L symmetry be respected by all dimension-four terms in the Lagrangian, whereas the soft dimension-three and dimension-two terms are allowed to break U (1) L to Z N . In particular, the U (1) L is used to forbid a tree-level Majorana or Dirac neutrino mass, whereas its soft breaking will usher in a radiative Majorana or Dirac neutrino mass through dark matter, i.e. the scotogenic mechanism. Because of the chosen particle content and its original U (1) L assignments, the resulting theory conserves either lepton parity for Majorana neutrinos, or (redefined) lepton number for Dirac neutrinos. At the same time, a dark symmetry also emerges.
. The choice of ±y = 0 is to distinguish Φ from η 1,2 and to allow the quartic coupling (Φ † η 1 )(Φ † η 2 ) as first proposed in Ref. [2]. In the original scotogenic model [3], η 1 = η 2 and is distinguished from Φ by lepton parity [1]. In the present framework, there are four variations as shown in Fig. 1. x 1 = −y − 1, x 2 = −y + 1; x 2 = y + 1 = −y + 1; x 1 = y − 1 = −y − 1. The last two options require y = 0 which is ruled out. The first two options requires x 2 − x 1 = 2. This means that the soft termN L N R must break U (1) L by two units. At the same time, the soft term N L N L breaks it by 2y − 2 or 2y + 2 units, whereas the soft term N R N R does it by 2y + 2 or 2y − 2 units. Furthermore, the soft terms Φ † η 1,2 would break U (1) L and allow η 0 1,2 to couple toν L N R through φ 0 , so they must be forbidden. To do so, y should be odd, because thē N L N R breaking is even (2 units) and these terms will not be generated if they are assumed to be absent in the beginning.
If y = 1, then x 1 = 0 and x 2 = 2. The resulting symmetry is just lepton parity, i.e. η 1,2 are odd and N L,R are even. If this symmetry was imposed in the beginning, then η 1 = η 2 and N L =N R may be assumed, and the original scotogenic model [3] is recovered. However, with soft breaking U (1) L , η 1 = η 2 and N L =N R . Nevertheless, lepton parity π L is still conserved, hence also dark parity π D = π L (−1) 2j as remarked earlier. Note that this happens for any odd y, pointing to the generality of the U (1) L approach.
In the above example, the dark symmetry is π D . If U (1) D is desired, then it has to be imposed as in Ref. [2]. To insist on obtaining U (1) D without imposing it from the beginning, using only U (1) L , the following variation may be considered, as shown in Table 2. The  mass term by two units, but now a dark U (1) D symmetry remains for η 1 , E, η † 2 . On the other hand, η 0 1 is unsuitable as a dark-matter candidate because it couples to the Z boson and thus ruled out by underground direct search experiments. This model (without χ 0 ) was considered previously in Ref. [4], but dark parity was imposed there and lepton number was said to be broken by the dimension-four term (Φ † η 1 )(Φ † η 2 ), without realizing that its correct implemention is softly broken U (1) L and that a dark U (1) D symmetry remains. Here, the added complex scalar singlet χ 0 is a viable dark-matter candidate, providing that the coupling is suitably small. Again, any odd y works, with x 1 = y − 1, x 2 = y + 1. In the special case y = 1, the additional term χ 0Ē L e R is allowed, with further possible interesting phenomenology. This second example shows the power of U (1) L in combination of the chosen particle content in acquiring radiative Majorana neutrino mass together with a dark symmetry.
A third example uses a scalar triplet and a singlet, so that the soft bilinear (and trilinear) scalar terms are absent to begin with. Hence there is no constraint on y at this stage.
However, the dark fermions must now be doublets and they can form bilinear terms with the SM lepton doublets. To prevent their existence, y must again be odd as shown below. This model was one of the compilations considered in Refs. [5,6]. Again their basic assumption was to have dark parity to begin with, in which case the scalar triplet ρ may be chosen real. Here only U (1) L is used, which requires x 1 = y − 1, x 2 = y + 1 as in the previous two examples. The soft termĒ L E R again breaks U (1) L by two units, whereas the soft termē L E R would break it by y, so again y must be odd to allow the latter to be absent. Since ρ 0 has no coupling to Z, it is a viable dark-matter candidate in this case and the dark symmetry U (1) D emerges as a consequence of softly broken U (1) L together with the chosen particle content.
Scotogenic Dirac Neutrino Mass : The same idea of using U (1) L may be applied to Dirac neutrinos. Assuming that U (1) L comes from gauged B − L, there have been three recent studies [7,8,9]. The following analysis shares many of their methods and results, but with an important difference. Whereas they consider only dimension-five operators for obtaining radiative Dirac neutrino masses, the adopted procedure here is to use dimension-four operators with softly broken U (1) L , i.e. the imposition of U (1) L to forbid the tree-level mass, but to allow a radiative mass to appear from dimension-two and/or dimension-three terms which break U (1) L , so that a dark symmetry emerges as well.
To have a Dirac neutrino mass, the right-handed singlet neutrino ν R must exist. It should pair up with ν L through the SM Higgs boson φ 0 . Hence it should have L = 1 under U (1) L .
In that case, a tree-level Yukawa coupling is allowed which must however be very small to account for the observed neutrino mass limit of 1.1 eV [10]. To forbid this tree-level coupling, a symmetry is routinely applied to distinguish ν R from the other SM particles, but L = 1 is retained. For a short review, see Ref. [11]. A generic one-loop diagram is depicted in Fig. 3, with its particle content shown in Table 4.  Here x = 1 is imposed so that ν R does not couple to ν L at tree level. To connect them in one loop, the trilinearη 0 φ 0 χ 0 term must break U (1) L softly by x − 1. Now N L and N R are assumed to have the same U (1) L charge, i.e. y, so that y = ±1 and y = ±x are required. Let x = −n + 1, then U (1) L breaks to Z n . If for example n = 3 [2], it would be impossible for neutrinos to be Majorana, i.e. they must remain Dirac as shown in Ref. [12].
The structure of Fig. 3 is well-known [13,14]. It is realized conventionally by 3 symmetries: (A) conventional lepton number, where ν L,R , N L,R have L = 1, and Φ, η, χ have L = 0, which is strictly conserved; (B) dark Z 2 symmetry, under which N L,R , η, χ are odd and others even, which is strictly conserved; and (C) an ad hoc Z 2 symmetry under which ν R , χ are odd and all others even, which is softly broken by the η † Φχ term. In previous applications, χ is assumed to be a real neutral scalar singlet for simplicity. Here it is crucial that it is complex to carry the nonzero U (1) L charge y − x.
In Table 4, in the column denoted by * , the U (1) L charges are chosen explicitly. The termsN L ν R and χ 0 χ 0 have U (1) L charges −1 and 2. They are not zero or divisible by n ≥ 3.
The terms ν R ν R , N R ν R and N R N R should also be absent, their U (1) L charges divided by n are (2/n) − 2, (3/n) − 2 and (4/n) − 2, hence n = 3, 4 are ruled out. All higher values of n are acceptable. The resulting theory allows two related symmetries: (I) Z L n lepton symmetry under which ν L,R , e L,R , η, χ ∼ ω and N L,R ∼ ω 2 , where ω n = 1; (II) Z D n dark symmetry, derivable from lepton symmetry by multiplying the latter by ω −2j where j is the particle's spin. As a result, ν L,R , e L,R ∼ 1 and N L,R , η, χ ∼ ω. This is the Dirac generalization of the Majorana case of the derivation of dark parity π D from lepton parity π L first pointed out in Ref. [1].
In a renormalizable theory, Z n symmetry is not simply realizable for large n because the Lagrangian admits only terms of dimension four or less. In the above example for n ≥ 5, the Lagrangian cannot admit the term (χ 0 ) n , hence the true symmetry of the theory is a redefined U (1) L , where ν L,R , e L,R , η, χ ∼ 1 and N L,R ∼ 2. The dark symmetry is then U (1) D where it is derived from U (1) L by subtracting from the latter 2j where j is the particle's spin, i.e. ν L,R , e L,R ∼ 0 and N L,R , η, χ ∼ 1. This shows that scotogenic Dirac neutrino mass is derivable from softly broken U (1) L alone with emergent U (1) lepton and dark symmetries.
If Z n symmetry is desired, the scalar sector must be extended. For example, if n = 5, in addition to χ ∼ ω, another scalar σ ∼ ω 3 may be added. The coexisting terms χ 3 σ * and χ 2 σ would then enforce Z D 5 as a dark symmetry, but the lepton symmetry would become global U (1) as conventionally defined, i.e. L = 1 for ν L,R and N L,R . To enforce Z L 5 as a lepton symmetry, a third scalar κ may be added with U (1) L charge = 7. In that case, the terms χσ 2 κ * and κN R ν R would allow ν L,R , χ, η ∼ ω and N L,R , κ ∼ ω 2 under Z L 5 with ω 5 = 1.
Since χ 0 mixes with η 0 , the neutral scalar of the dark sector of this model requires this mixing to be very small and the lighter eigenstate to be mostly χ 0 for it to be a viable dark-matter candidate. Alternatively the lightest N may also be chosen as dark matter. For details, see Ref. [14].
A possible variation of the model is to add a neutral scalar singlet ζ with U (1) L charge n, and require that U (1) L be spontaneously broken only. In that case, the term η † Φχ is replaced with ζ * η † Φχ. Neutrinos obtain radiative Dirac masses as before, with new emergent U (1) lepton and dark symmetries, but now a massless Goldstone boson appears. It is the analog of the majoron which comes from breaking U (1) L spontaneously to Z 2 and is applicable to Majorana neutrinos, whereas here it is the massless diracon [15] which comes from breaking U (1) L spontaneously to Z n (n ≥ 5) and is applicable to Dirac neutrinos.
There is a further use of ζ, if it is allowed to couple anomalously to a pair of exotic quarks (color fermion triplets) or a color fermion octet [16,17]. Then this diracon becomes a QCD (quantum chromodynamics) axion and U (1) L is an extended version of Peccei-Quinn symmetry, as proposed many years ago [18,19] for Majorana neutrinos, and very recently also for Dirac neutrinos [20,21]. In these scenarios, dark matter consists of both the axion and a WIMP (weakly interacting massive particle) [22].
In Fig. 3, the fermion singlets N L,R may be replaced with the doublets (E 0 , E − ) L,R as shown in Table 5. This construction eliminates the existence of many fermion bilinears Hence only 2x and y − 1 must not be zero or divisible by x − 1. Also y = x is required. As a result, it is possible to have Z 3 dark symmetry, i.e.
x = −2 and y = 2, as shown in the column denoted by * * . The analogous one-loop diagram for scotogenic Dirac neutrino mass is shown in Fig. 4. The U (1) L symmetry is broken to Z 3 and transform trivially under Z D 3 . In addition ψ 3 has its own accidental (or predestined) Z 2 symmetry from the chosen particle content of the theory and the imposed U (1) L symmetry.
The Dirac neutrino mass matrix is now 6 × 6 with 3 tree-level masses and 3 one-loop masses, the latter linking only to the left-handed SM neutrinos. However, there could be mixing between the two sectors which may be a source of nonunitarity of the observed 3×3 neutrino mixing matrix. Using the same connection, two examples of scotogenic Dirac neutrino mass have also been described, one with emergent Z n lepton and dark symmetry for n ≥ 5. However, without enlarging the necessary scalar sector, the requirement of renormalizability of these models implies that the true symmetry of the Lagrangian is a redefined U (1) L such that a dark U (1) D is obtained by subtracting the assigned lepton number of a particle by 2j where j is the particle's spin.
The other example chooses a different set of dark fermions so that Z 3 dark symmetry emerges which is maintained explicitly by the renormalizable Lagrangian of the model. It also sustains a conserved lepton number in the conventional way.
These two examples generalize the case of Z 2 lepton and dark parity [1] for Majorana neutrinos to Dirac neutrinos.