Truly Minimal Left-Right Model of Quark and Lepton Masses

We propose a left-right model of quarks and leptons based on the gauge group SU(3)C ×SU(2)L×SU(2)R×U(1)B−L, where the scalar sector consists of only two doublets: (1,2,1,1) and (1,1,2,1). As a result, any fermion mass, whether it be Majorana or Dirac, must come from dimension-five operators. This allows us to have a common view of quark and lepton masses, including the smallness of Majorana neutrino masses as the consequence of a double seesaw mechanism. In the standard model of electroweak interactions, neutrinos are massless. On the other hand, recent experimental data on atmospheric [1] and solar [2] neutrinos indicate strongly that they are massive and mix with one another [3]. To allow neutrinos to be massive theoretically, the starting point is the observation of Weinberg [4] over 20 years ago that a unique dimension-five operator exists in the standard model, i.e. LΛ = fij 2Λ (νiφ 0 − eiφ)(νjφ − ejφ) +H.c. (1) which generates a Majorana neutrino mass matrix given by (Mν)ij = fijv 2 Λ , (2) where v is the vacuum expectation value of φ. This also shows that whatever the underlying mechanism for the Majorana neutrino mass, it has to be “seesaw” in character, i.e. v divided by a large mass [5]. 2 If the particle content of the standard model is extended to include leftright symmetry [6], then the gauge group becomes GLR ≡ SU(3)C×SU(2)L× SU(2)R×U(1)B−L, whose diagonal generators satisfy the charge relationship Q = T3L + T3R + (B − L) 2 = T3L + Y 2 . (3) Quarks and leptons transform as qL = (u, d)L ∼ (3, 2, 1, 1/3), (4) qR = (u, d)R ∼ (3, 1, 2, 1/3), (5) lL = (ν, e)L ∼ (1, 2, 1,−1), (6) lR = (N, e)R ∼ (1, 1, 2,−1), (7) where a new fermion, i.e. NR, has been added in order that the left-right symmetry be maintained. In all previous left-right models, a scalar bidoublet transforming as (1, 2, 2, 0) is then included for the obvious reason that we want masses for the quarks and leptons. Suppose however that we are only interested in the spontaneous breaking of SU(2)L×SU(2)R×U(1)B−L to U(1)em with vR >> vL, then the simplest way is to introduce two Higgs doublets transforming as ΦL = (φ + L , φ 0 L) ∼ (1, 2, 1, 1), (8) ΦR = (φ + R, φ 0 R) ∼ (1, 1, 2, 1). (9) 3 Suppose we now do not admit any other scalar multiplet. This is analogous to the situation in the standard model, where SU(2)L×U(1)Y is spontaneously broken down to U(1)em by a Higgs doublet and we do not admit any other scalar multiplet. In that case, we find that quark and charged-lepton masses are automatically generated by the existing Higgs doublet, but neutrinos obtain Majorana masses only through the dimension-five operator of Eq. (1). In our case, in the absence of the bidoublet, all fermion masses, be they Majorana or Dirac, must now have their origin in dimension-five operators, as shown below. Using Eqs. (4) to (9), it is clear that (lLΦL) = νLφ 0 L − eLφL (10) and (lRΦR) = NRφ 0 R − eRφR (11) are invariants under GLR. Hence we have the dimension-five operators given by LM = f ij 2ΛM (liLΦL)(ljLΦL) + f ij 2ΛM (liRΦR)(ljRΦR) +H.c., (12) which will generate Majorana neutrino masses proportional to v L/ΛM for νL and v 2 R/ΛM for NR. (The different possible origins of this operator are 4 explained fully in Ref.[5].) In addition, we have LD = f ij ΛD (l̄iLΦ ∗ L)(ljRΦR) +H.c. (13) and the corresponding dimension-five operators which will generate Dirac masses for all the quarks and charged leptons. From Eq. (13), it is clear that (mD)ij = f ij vLvR ΛD , (14) hence νL gets a double seesaw [7] mass of order mD mN ∼ v 2 Lv 2 R ΛD ΛM v R = v LΛM ΛD , (15) which is much larger than v L/ΛM if ΛD << ΛM . Take for example ΛM to be the Planck scale of 10 GeV and ΛD to be the grand-unification scale of 10 GeV, then the neutrino mass scale is 1 eV (for vL of order 100 GeV). The difference between ΛM and ΛD may be due to the fact that if we assign a global fermion number F to lL and lR, then LM has F = ±2 but LD has F = 0. Since the Dirac masses of quarks and charged leptons are also given by Eq. (14), vR cannot be much below ΛD. This means that SU(2)R ×U(1)B−L is broken at a very high scale to U(1)Y , and our model at low energy is 5 just the standard model. We do however have the extra singlet neutrinos NR with masses of order v 2 R/ΛM , i.e. below 10 13 GeV, which are useful for leptogenesis, as is well-known [8]. Formt = 174.3±5.1 GeV, we need vR/ΛD to be of order unity in Eq. (14). One may wonder in that case whether we can still write Eq. (13) as an effective operator. The answer is yes, as can be seen with the following specific example [9]. Consider the singlets UL, UR ∼ (3, 1, 1, 4/3), (16) with invariant mass MU of order ΛD, then the 2 × 2 mass matrix linking (t̄L, ŪL) to (tR, UR) is given by MtU = 