The SM extensions with additional light scalar singlet, nonrenormalizable Yukawa interactions and $(g-2)_{\mu}$

We consider the SM extension with additional light real singlet scalar, right-handed neutrino and nonrenormalizable Yukawa interaction for the first two generations. We show that the proposed model can explain the observed $(g - 2)$ muon anomaly. Phenomenological consequences as flavour violating decays $\tau \rightarrow \mu\mu\mu, \mu \mu e, \mu e e $ are briefly discussed. We also propose the $U_R(1)$ gauge generalization of the SM with complex scalar singlet and nonzero right-handed charges for the first two generations.


Introduction
The discovery of the neutrino oscillations [1,2] means that at least two neutrino have nonzero masses. The minimal extension of the SM with nonzero neutrino masses is the νMSM [3,4]. In this model one adds to the SM three additional massive Majorana(righthanded) fermions ν Ri , i = 1, 2, 3. Due to seesaw mechanism [3,5] [4].
In this report which is based mainly on Refs. [6] we consider the extension of the νMSM with additional scalar field and nonrenormalizable Yukawa interaction for the first two generations. We show that the SM extension with additional light real singlet field can explain the (g − 2) muon anomaly. Phenomenological consequences of the proposed model as flavour violating decays τ → µµµ, µee are brifly discussed. We also propose the U R (1) gauge generalization of the SM with complex scalar singlet and nonzero righthanded charges for the first two generations.

2
The νMSM extension with additional real scalar isosinglet and nonrenormalizable Yukawa interaction In this section we consider the extension of the νMSM with additional scalar field and nonrenormalizable Yukawa interaction for the first two generations [6]. The Lagrangian of the model has the form Here where L SM is the SM Lagrangian, ν Rj -are the Majorana neutrinos and L 1 = (ν eL , e L ), The main peculiarity of the model (1-6) is the use of nonrenormalizable Yukawa interactions (2-4) 2 . Here we consider the particular case of the general model (1 -6) with nonzero renormalizable Yukawa interaction only for the third fermion generation. We assume that the masses of the first two light generations arise due to nonrenormalizable interactions (2)(3)(4). We impose the discrete symmetry e Rk → −e Rk (k = 1, 2) , The discrete symmetry (7 -9) restricts the form of nonrenormalizable interactions (2)(3)(4), h Leφ,i3 = 0 , As a consequence of the (7-9) the renormalizable SM Yukawa interaction with the first two quark and lepton generations vanishes and the fermions of the first two generations acquire masses only due to nonrenormalizable interactions (2)(3)(4).
We can consider the nonrenormalizable interactions (2 -4) as some effective interactions arising from renormalizable interactions. For instance, the interaction (2) can be realized in renormalizable extension of the SM with additional scalar field φ and new massive quark SU(2) L singlet fields D R , D L with a mass M D and U(1) hypercharges The interaction of new quark fields D R , D L with ordinary quarks and the neutral scalar field φ is In the heavy D-quark mass limit M D → ∞ we obtain the effective interaction (2) with Analogously we can consider nonrenormalizable interaction (4) as an effective interaction which arises in renormalizable extension of the SM with additional scalar field and new The interaction of new lepton fields E R , E L with ordinary quarks and the neutral scalar field φ is In the heavy E-lepton mass limit M E → ∞ we obtain the effective interaction (4) with After the spontaneous SU(2) L ⊗ U(1) electroweak symmetry breaking the Yukawa interaction of the scalar field with charged leptons takes the form whereh i = 1, 2, 3, k = 1, 2 and e L1 = e L , e L2 = µ L , e L3 = τ L , < H >= 174 GeV . Nonzero vacuum expectation value for the real field φ generates nonzero lepton masses for electrons and muons while the mass of τ -lepton arises due to renormalizable Yukawa coupling. The lepton mass matrix and the Yukawa lepton φ ' = φ − < φ > interactions are where i = 1, 2, 3 and k = 1, 2. The mass terms (18) and the interaction (19) have different flavour structure that leads to the tree level flavour changing transitions like τ → µ + φ, τ → e + φ and as a consequence to the flavour violating decays like At present state of art we can't predict the value of flavour violating Yukawa couplings The interaction (19) leads, in particular, to the additional one loop contribution to muon magnetic moment due to φ ' scalar exchange, namely [8] ∆a µ = h 2 Le,22 where λ = mµ m φ . In the limit M φ >> m µ The precise measurement of the anomalous magnetic moment of the positive muon from the Brookhaven AGS experiment [9] gives a result which is 3.6σ higher than the Standard Model (SM) prediction where a µ ≡ gµ−2 2 . Using the formulae (25, 26) we find that for m Φ = (100, 10, 1, 0.5) GeV the muon g − 2 anomaly can be explained for For the opposite limit m µ ≫ m φ ∆a µ = 3h 2 Le,22 and as a consequence of (26, 31) we find that The nonrenormalizable Yukawa interactions of the first and second generation fermions in full analogy with the (2-4) interactions take the form Note that proposed model is free from γ 5 -anomalies and we can consider the origin of the U R (1) gauge group as a result of the gauge symetry breaking SU L (2) ⊗ SU R (2) ⊗ U(1) → SU L (2) ⊗ U R (1) ⊗ U(1). We assume that in the considered model < Φ > = 0 that leads to nonzero X gauge boson mass and nonzero fermion masses for the first and second generations. In the unitaire gauge Φ = φ + < Φ >, where φ = φ * is real scalar field as in previous section plus we have massive vector boson X. So in this model after U R (1) gauge symmetry breaking in addition to the νMSM spectrum we have both scalar and 4 In Refs. [12] new light vector boson interacting with the L µ − L τ current has been proposed for (g − 2) µ anomaly explanation, see also Ref. [13] where the model with new light gauge boson interacting with the SM electromagnetic current has been proposed for the (g − 2) µ anomaly explanation and Ref. [14] where the interaction of light gauge boson with (B − L) + xY current has been considered.
vector particles. The one loop contribution to the anomalous muon(electron) magnetic moment due to the φ and X exchanges is where the ∆a µ (φ) contribution is given by the formulae (24,25,31) and the vector X boson contribution for (V + A) right-handed coupling 5 with fermions is [8] The X boson contribution (41) to the (g − 2) is negative. For instance, for α X =

Conclusion
The νMSM with additional scalar field and nonrenormalizable interaction for the first two generations can explain the observed muon (g − 2) anomaly. The model predicts the existence of flavour violating quark and lepton decays like τ → µµµ, µµe, µee. Besides the U(1) gauge generalization of the model with real isosinglet scalar field is also able to explain muon (g − 2) anomaly.
We are indebted to colleagues from INR theoretical department for discussions and comments. 5 The interaction Lagrangian of the vector X field with muons is L Xμµ = g X X νμ γ ν (1 + γ 5 )µ.