One-Loop Electron Mass and Three-Loop Dirac Neutrino Masses

In the context of a left-right extension of the standard model of quarks and leptons with the addition of a gauged $U(1)_D$ dark symmetry, it is shown how the electron may obtain a radiative mass in one loop and two Dirac neutrinos obtain masses in three loops.

Introduction : Neutrino masses are very small. In the context of the minimal standard model (SM) of particle interactions, it has a natural explanation because they must come from a dimension-five operator [1], i.e.
where (ν, l) 1,2,3 are the left-handed lepton doublets of the three families, and (φ + , φ 0 ) = Φ is the one Higgs scalar doublet. Neutrino masses are thus Majorana and inversely proportional to the large scale Λ, hence the name "seesaw". There are exactly three ways [2] to realize L 5 at tree level, establishing the nomenclature Type I,II,III seesaw, as well as three ways to realize it radiatively in one loop [2]. One particular application is to let dark matter [3] be the origin of radiative neutrino mass, as in the "scotogenic" model [4].
If neutrinos are Dirac particles, thereby requiring the existence of ν R and the maintenance of a conserved lepton number L, the SM is not a natural accommodating framework. First, Second, the L symmetry must be imposed to forbid the otherwise allowed ν R Majorana mass. Third, the Yukawa coupling linking ν L to ν R through φ 0 must be chosen to be of order 10 −12 to account for the data on neutrino masses. To alleviate this last shortcoming, a symmetry is often employed to forbid the offending Yukawa term, say Z 2 under which ν R is odd. At the same time, L is still assumed to be exact, wheras Z 2 will be broken softly by the addition of other fermions and scalars. This procedure [5] has been studied in a variety of models.
Consider now the left-right extension of the SM. The existence of ν R is required as part of an SU (2) R doublet. Neutrinos obtain Dirac masses in the same way as the other fermions through a scalar bidoublet. If SU (2) L,R doublets Φ L,R are also added to break then L remains a global symmetry. The first and second shortcomings of the SM for Dirac neutrinos no longer apply, but the third remains. A well-known approach [6] is to keep only Φ L,R without the scalar bidoublet. Hence all fermion masses are zero at this point. By adding heavy singlet fermions, the known quarks and leptons may now obtain seesaw masses [7]. Neutrinos may also be chosen to have radiative Dirac masses [8,9,10,11]. Recently, it has been shown [12] that all fermion masses could be radiative if the SU (2) R breaking scale is high enough. In this work, a new and different scenario is envisioned: the emergence of three-loop neutrino masses in the presence of a one-loop electron mass, with the help of a dark gauged U (1) D symmetry.
Description of Model : The particle content of the proposed model is listed in Table 1. The one-loop diagram for the electron mass is given in Fig. 1, with the understanding that there Figure 1: Scotogenic electron mass.
are two additional insertions linking N L to N R as shown in Fig. 2. Figure 2: Additional insertions to electron mass.
Assuming that N is the heaviest particle in the loop, the electron mass is then approximately given by where f is a typical Yukawa coupling and v L,R / √ 2 are the vacuum expectation values of φ 0 L,R . Note that lepton number L may be defined to be transmitted by χ − across the loop.
Note also that neutrinos remain massless in one loop because the corresponding transmitter  Figure 3: Scotogenic Dirac neutrino mass.
A very rough estimate of the neutrino mass is then where λ is the quartic coupling of (χ + χ − )(η + η − ). Hence m ν /m e ∼ λf 2 /(16π 2 ) 2 which is naturally of order 10 −7 or so, in agreement with data. Since there are two copies each of (N, E), S L,R , and (N , E ), two neutrinos will get mass. The third neutrino gets a negligible mass from W L,R exchange, in analogy to the 2 − W exchange [13] in the SM for Majorana neutrinos. Other fermions may acquire seesaw Dirac masses [7] or radiatively [11,12].
Whereas Fig. 3 represents the realization of the dimension-five operator

Fig. 1 corresponds to
These are the left-right analogs of Eq. (1). In previous applications, the electron mass often comes from a tree-level realization of L e 5 , and only L ν 5 is radiative in origin. Recently [11,12], both operators are derived in one loop, in which case there is no real understanding for why the Dirac neutrino masses are so much smaller than the electron mass. Here the first example of one-loop electron mass and three-loop Dirac neutrino masses is presented. Note also that Fig. 3 is a close analog to that [14] for a Majorana neutrino, known already many years ago.
Higgs and Gauge Sectors : The Higgs sector is identical to that of Ref. [12], consisting of scalars Φ L,R and σ. Although σ now has D = 2 instead of D = 3, it does not affect the resulting Higgs potential, i.e.
After the spontaneous breaking of SU (2) L × SU (2) R × U (1) B−L × U (1) D , the only physical scalars left are the real parts of φ 0 L,R and σ. Let then the 3 × 3 mass-squared matrix spanning (h L , h R , h D ) is In the gauge sector, the Z D boson gets a mass equal to 2g D v D . The charged W ± L,R masses are g L v L and g R v R . The Z, Z mass-squared matrix is where e −2 = g −2 L +g −2 R +g −2 B , and g L = g R with x = sin 2 θ W . The Z −Z mixing is then about Assuming f L = f R = f and neglecting the N − S and N − S mixings for now, this yields two Majorana fermions of masses |m S ± f v D |, each coupling to h D . Let S 1 be the lighter, i.e. the dark-matter candidate, with mass m annihilate to h D , as shown in Fig. 4. The first diagram is also accompanied by its u−channel counterpart, which has the same amplitude in the limit that S 1 is at rest. Let x = m h D /m S 1 and using m 2 h D = λ σ v 2 D , this cross section at rest multiplied by relative velocity is