Gauge $U(1)$ Dark Symmetry and Radiative Light Fermion Masses

A gauge $U(1)$ family symmetry is proposed, spanning the quarks and leptons as well as particles of the dark sector. The breaking of $U(1)$ to $Z_2$ divides the two sectors and generates one-loop radiative masses for the first two families of quarks and leptons, as well as all three neutrinos. We study the phenomenological implications of this new connection between family symmetry and dark matter. In particular, a scalar or pseudoscalar particle associated with this $U(1)$ breaking may be identified with the 750 GeV diphoton resonance recently observed at the Large Hadron Collider (LHC).


Introduction :
In any extension of the standard model (SM) of particle interactions to include dark matter, a symmetry is usually assumed, which distinguishes quarks and leptons from dark matter.
For example, the simplest choice is Z 2 under which particles of the dark sector are odd and those of the visible sector are even. Suppose Z 2 is promoted to a gauge U (1) symmetry, then the usual assumption is that it will not affect ordinary matter. These models all have a dark vector boson which couples only to particles of the dark sector.
New Gauge U (1) D Symmetry : The framework that radiative fermion masses and dark matter are related has been considered previously [3]. Here it is further proposed that families are distinguished by the connecting dark symmetry. In Table 1 we show how they transform under U (1) D as well as the other particles of the dark sector. The U (1) D symmetry is broken spontaneously by the vacuum expectation value σ 1,2 = u 1,2 to an exactly conserved Z 2 which divides the two sectors.  (1) + 3 1 3 The [U (1) D ] 3 anomaly is not zero for either the first or second family, but is cancelled between the two. This is thus a generalization of the well-known anomaly-free L e − L µ gauge symmetry [4] to the difference of B − L − 2Y between the first two families.  Note that at least two copies of (N, N c ) are needed for two charged-lepton masses. The mass Note that the f 1,2,3,4 u 1 terms break lepton number by two units, whereas the f 5,6 u 2 terms do not. Lepton number L = 1 may be assigned to e, µ, τ, N, S and  There are 4 real scalar fields, spanning We denote their mass eigenstates as ρ 0 l with mass m l . In Figs. 1 and 2, let the ν i ψ kη 0 coupling be h ν ik , then the radiative neutrino mass matrix is given by [5] ( where √ 2Re(η 0 ) = l y R l ρ 0 l , √ 2Im(η 0 ) = l y I l ρ 0 l , with l (y R l ) 2 = l (y I l ) 2 = 1, x lk = m 2 l /M 2 k , and the function F is given by There are two charged scalar fields, spanning η ± , χ ± . We denote their mass eigenstates as ρ + r with mass m r . In Fig. 3, let the e L ψ k η + and the e c L ψ k χ − couplings be h e k and h e c k , then where η + = r y η r ρ + r , χ + = r y χ r ρ + r , with r (y η r ) 2 = r (y χ r ) 2 = 1 and r y η r y χ r = 0. A similar expression is obtained for m µ , as well as the light quark masses. where L is the quark charged-current mixing matrix. However, since Z D does not couple to left-handed quarks, and its couplings to right-handed quarks have been chosen to be diagonal in their mass eigenstates, flavor-changing neutral currents are absent in this sector. Of course, they will appear in the scalar sector, and further phenomenological constraints on its parameters will apply. Z D Gauge Boson : As σ 1,2 acquire vacuum expectation values u 1,2 respectively, the Z D gauge boson obtains a mass given by Since σ 1,2 do not transform under the SM, and Φ does not under U (1) D , there is no mixing between Z D and Z. Using Table 1 and assuming that all new particles are lighter than Z D , the branching fraction of Z D to e − e + + µ − µ + is estimated to be 0.07. The c u,d coefficients used in the experimental search [6,7] of Z D are then There are three scalars with integral charges under U (1) D , i.e. Φ and σ 1,2 . Whereas φ 0 = v breaks the electroweak SU (2) L × U (1) Y gauge symmetry as in the SM, σ 1,2 = u 1,2 break U (1) D to Z 2 , with all those particles with half-integral U (1) D charges becoming odd under this exactly conserved dark Z 2 parity. The relevant scalar potential is given by where m 12 has been rendered real by absorbing the relative phase between σ 1,2 . The conditions for v and u 1,2 are 0 = m 2 2 + λ 2 u 2 2 + λ 3 u 2 1 + λ 5 v 2 + m 12 u 2 1 /u 2 .
Dark Matter : The lightest neutral particle with odd Z 2 is a good dark-matter candidate. In this model, it could be the lightest scalar particle in the sector consisting of η 0 = (η R + iη I )/ √ 2 and There are two sectors, the mass-squared matrix spanning η R , χ R is given by and that spanning η I , χ I is where A, B come from the φ 0 η 0 (χ 0 ) * and φ 0 η 0 χ 0 (σ 1 ) * couplings and C from the χ 0 χ 0 (σ 1 ) * coupling. The phenomenology of the lightest particle in this group is similar to that of the so-called inert Higgs doublet model [5,10,11]. For details, see for example recent updates [12,13,14].

Conclusion :
A new idea linking family symmetry to dark symmetry is proposed using a gauge U (1) D symmetry, which breaks to exactly conserved Z 2 . The first and second families of quarks and leptons transform under this U (1) D so that their masses are forbidden at tree level.
They interact with the dark sector in such a way that they acquire one-loop finite masses, together with all three neutrinos. The extra Z D gauge boson may have a mass of order a few TeV, and one particle associated with the breaking of U (1) D may be identified with the 750 GeV diphoton excess recently observed at the LHC.