Universal Scotogenic Fermion Masses in Left-Right Gauge Model

In the conventional left-right gauge model, if the Higgs scalar sector consists only of an SU(2)L doublet and an SU(2)R doublet, fermion masses are zero at tree level. There have been many studies on how they would become massive. A recent new idea is extended to show how a dark sector with U(1)D gauge symmetry may allow all standard-model fermions to acquire realistic masses radiatively. In this context, the particle content of the model also implies the automatic conservation of baryon number B and lepton number L as in the standard model. ar X iv :2 01 2. 03 12 8v 1 [ he pph ] 5 D ec 2 02 0 Introduction : In the standard SU(3)C × SU(2)L × U(1)Y gauge model (SM) of quarks and leptons, the one scalar Higgs doublet serves two purposes. It breaks SU(2)× U(1)Y to the electromagnetic gauge symmetry U(1)Q, and renders all fermions massive at tree level, except the left-handed doublet neutrino, unless it has a right-handed singlet counterpart. In the canonical left-right extension to SU(3)C × SU(2)L × SU(2)R × U(1)B−L, left-handed fermions are SU(2)L doublets, right-handed fermions are SU(2)R doublets. The breaking of SU(2)L × SU(2)R × U(1)B−L to U(1)Q may be achieved by an SU(2)L Higgs doublet and an SU(2)R Higgs doublet, but they do not render the fermions massive at tree level. A scalar bidoublet under SU(2)L × SU(2)R would be needed, but is being withheld on purpose. There have been many studies on how quarks and leptons may acquire masses in this situation [1, 2, 3, 4, 5, 6]. They are usually not applicable to the top quark, because mt = 173 GeV is of order the electroweak breaking scale v = √ 2〈φ0〉 = 246 GeV, and conventional wisdom would insist that it be accorded a tree-level mass [7]. In particular, if a radiative mt is desired, then it ought to be proportional to v, but suppressed by the typical loop factor of 16π. Hence very large couplings to new particles are required and perturbative calculations become unreliable. In this work, it will be shown how this objection may be overcome, and all SM fermion masses may be generated radiatively from a dark sector (scotogenic) with a U(1)D gauge symmetry in the left-right context. Phenomenological consequences will be discussed. Outline of Model : The particle content of the proposed model is listed in Table 1. There are three families of quarks and leptons, as well asNL,R. There is only one copy each of the scalars ΦL,ΦR, ζL,R, ηL,R, σ. The dark U(1)D gauge symmetry is broken by three units through the complex singlet scalar σ. This allows a global D symmetry to remain and prevents NL,R as well as νL,R to acquire Majorana masses, as pointed out first in Ref. [8] and applied to Dirac neutrinos using B − L in Ref. [9]. From the allowed Yukawa couplings between the

A scalar bidoublet under SU (2) L × SU (2) R would be needed, but is being withheld on purpose. There have been many studies on how quarks and leptons may acquire masses in this situation [1,2,3,4,5,6]. They are usually not applicable to the top quark, because m t = 173 GeV is of order the electroweak breaking scale v = √ 2 φ 0 = 246 GeV, and conventional wisdom would insist that it be accorded a tree-level mass [7]. In particular, if a radiative m t is desired, then it ought to be proportional to v, but suppressed by the typical loop factor of 16π 2 . Hence very large couplings to new particles are required and perturbative calculations become unreliable. In this work, it will be shown how this objection may be overcome, and all SM fermion masses may be generated radiatively from a dark sector (scotogenic) with a U (1) D gauge symmetry in the left-right context. Phenomenological consequences will be discussed.
Outline of Model : The particle content of the proposed model is listed in Table 1. There are three families of quarks and leptons, as well as N L,R . There is only one copy each of the scalars The dark U (1) D gauge symmetry is broken by three units through the complex singlet scalar σ. This allows a global D symmetry to remain and prevents N L,R as well as ν L,R to acquire Majorana masses, as pointed out first in Ref. [8] and applied to Dirac neutrinos using B − L in Ref. [9]. From the allowed Yukawa couplings between the fermion/scalar  Origin of Large Radiative Top Mass : The one-loop diagram for a quark with charge 2/3 is given in Fig. 1. The scalars ζ L,R mix to form mass eigenstates ζ 1 = cos θζ L − sin θζ R , Figure 1: Scotogenic u quark mass. ζ 2 = cos θζ R + sin θζ L , with masses m 1,2 . The diagram is then easily calculated to be which is of the same form as that of the original scotogenic model [10] for Majorana neutrino mass. The usual assumption is that m 2 2 − m 2 1 is small compared to m 2 0 = (m 2 2 + m 2 1 )/2 and m 0 << m N , in which case a seesaw radiative mass is obtained, i.e.
This clearly makes m u very small. Another choice [11,12] is m 1,2 >> m N , in which case This formula was applied [12] to neutrinos where m N is of order keV to act as warm dark matter, but it is obvious that m N is actually arbitrary and may be chosen large enough to allow m t = 173 GeV as a radiative effect.
Here the choice m 2 2 >> m 2 N 3 >> m 2 1 is made, allowing different choices for N 1,2 , to be discussed later. Hence As an example, let m 2 = 50 TeV, m N 3 = 15 TeV, m 1 = 1 TeV, then m t = 173 GeV is obtained for f L f R sin θ cos θ = 0.682.
The above may be achieved in the SM, but not in a very natural way. First, the treelevel Higgs coupling to fermions must be forbidden by a new symmetry, say Z 2 under which all right-handed fermions are odd. This is often used for example in models where a small Dirac neutrino mass is desired [13]. Another is to postulate a non-Abelian discrete family symmetry, such as A 4 [14], and assign left-handed and right-handed fermions differently so that they do not couple to the SM Higgs boson. However, such symmetries must be softly broken appropriately to allow radiative fermion masses to appear [15]. To obtain m t = 173 GeV, this requires the corresponding soft breaking trilinear scalar term to have a coupling of order 10 7 GeV. In contrast, it is here naturally derived from the breaking of SU (2) R at a high scale.
Dark Scalar Sector : The 2 × 2 mass-squared matrix spaning (ζ L , ζ R ) is where the off-diagonal term comes from the quartic coupling This means that v R has to be very large for m 2 = 50 TeV, say of order 10 7 GeV, so that the SU (2) R gauge bosons are out of reach at the Large Hadron Collider (LHC).
In the d quark sector, the corresponding diagram is given in Fig. 2. Hence the off- Whereas only N 3 contributes to m tt as in Eq. (4), N 2 contributes to the 2 × 2 submatrix spanning (c, t) of the form and N 1 contributes to the entire 3 × 3 matrix of the form It is clear that the × relative velocity is given by This should be multiplied by a factor of 8 to account for the 3 colors of c plus those of s, as well as µ and ν µ , assuming these other contributions are the same in magnitude, and set equal to the canonical value of 3 × 10 −26 cm 3 /s. For m N 2 = 800 GeV and m 1 = 1 TeV, is obtained. As an example, sin θ = cos θ = 1/ √ 2 yields (f 4 L + f 4 R ) 1/4 = 0.72. Since ζ 1 is analogous to a scalar quark in supersymmetry and N 2 analogous to a neutralino, they are subject to search limits at the LHC. The updated ATLAS result [16] shows that for m 1 = 1 TeV, m N 2 = 800 GeV is just at the edge of the allowed region.
Higgs and Gauge Sectors : The Higgs sector consists of the scalars Φ L,R and σ. Their potential is given by 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 Since v R ∼ 10 7 GeV, and v D should be at least a few TeV, h L acts to all intents and purposes as the SM Higgs boson in this model. The heavier scalars h R and h D decay quickly to h L h L through λ LR and λ Lσ respectively.
In the gauge sector, the Z D boson gets a mass equal to 3g D v D . It should be heavier than 2m N 2 , so it decays at least to N 2N2 . 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 Ref. [18]. There are three contributions, the first being the λ LR (ζ L φ 0 L )(ζ R φ 0 R ) * coupling, the others from |φ 0 L | 2 |ζ L | 2 and |φ 0 L | 2 |ζ R | 2 . The latter are suppressed by the ratio v L /v R , and will be neglected. The h L coupling tott is then given by [18] f Using the expression for m t from Eq. (1), where the example of m N = 15 TeV, m 1 = 1 TeV, and m 2 = 50 TeV has been used as previously. Thus f t is predicted to be greater than that of the SM, which is a generic result [18]. If sin θ = 0.1 is assumed, then the ratio of f t to that of the SM is 1.17. The above may also be applied to f b with same m L,R but different θ. The present measurements of Higgs production and decay from collider data [17] [19]. The usual seesaw mechanism applies and neutrinos obtain naturally small Majorana masses. Note that ∆ R does not affect the generic radiative mechanism for the Dirac fermion masses.
Conclusion : In the context of left-right gauge symmetry, a natural scenario exists where all fermions obtain radiative masses. This is enforced by a very simple Higgs sector, consisting of one SU (2) L doublet and one SU (2) R doublet. The absence of a scalar bidoublet means that all quarks and leptons are massless at tree level. A dark sector is then assumed with a gauge U (1) D symmetry as shown in Table 1. It is spontaneously broken by a complex singlet scalar with D = 3. The resulting theory conserves global D, as well as global baryon number B and lepton number L.
The SU (2) R gauge symmetry is broken at a high scale, say of order 10 7 GeV, allowing m t = 173 GeV to be radiatively generated perturbatively in one loop. This goes against the conventional wisdom that m t must be a tree-level mass, based on knowing that the electroweak breaking scale is v L = 246 GeV. Anomalous Higgs couplings to all fermions are predicted. This will affect both the production and decay of the observed 125 GeV Higgs boson at the LHC. It accentuates the importance of measuring all Higgs properties precisely in the future.
Dark matter is now intimately related to fermion mass generation. It is a gauge singlet Dirac fermion N 2 . It couples mainly to the second family of quarks and leptons. It annihilates through dark scalar exchange to c, s, µ, ν µ , but does not interact with nuclei significantly.
Because of the dark color scalars which may be produced copiously in pairs at the LHC, N 2 may be observed as large missing momentum in their decays.