arXiv:2108.07170v1 [physics.gen-ph] 27 Jul 2021 atclrgahocrigwti h rm fteSadr Model neutrinos. Standard of the description of proce the frame of [2] the modification oscillations minor within neutrino also occurring that graph paper particular present the by shown rce i o rp oiae yteecag fthe of exchange the by dominated graph box a via proceed ytepiayfr fteLgaga.I ossso h olwn t following the of consists implied It are Lagrangian. oscillations the Neutrino of [6]. form by m primary fi given containing the Higgs is by [4], 4-component derivation the Lagrangian this and the of bosons of summary gauge form less mass primary fermions, a less from derived be can novn nitreit tt.Frexample For state. intermediate an involving h Lagrangian The siltosbtenqatmsae a rce i mechanism a via proceed may states quantum between Oscillations etiooclain;Sadr oe;Hgsfield Higgs Model; Standard oscillations; Neutrino state. a intermediate handed proc as oscillations left field Neutrino a Higgs of fermion. 4-component composed less mass is handed neutrino right a that assuming Model, etioOclain eitdb the by mediated Oscillations Neutrino etiooclain cu ihntefaeo h Standar the of frame the within occur oscillations Neutrino AS1.0Dm 12.10 PACS L L = fteSadr oe o h bevbeprils[3] particles observable the for Model Standard the of L .Schmidt-Parzefall W. ψ Universit¨at Hamburg + uut1,2021 17, August ig field Higgs L χ + Abstract L φ 1 + L Y + L W B + B ¯ L eo siltos[1] oscillations meson B . e i the via eed t ur.I is It quark. da nd l 5.A [5]. eld dvaa via ed erms d fe a after ass (1) The terms of the primary form of are L L = ψ iγµ(∂ U −1∂ U )ψ ψ iR µ − ψ µ ψ iR Xi µ −1 Lχ = χiL iγ (∂µ Uχ ∂µUχ)χiL i − X 2 2 1 −1 † −1 mH † mH † 2 Lφ = (∂µ Uφ ∂µUφ)φ (∂µ Uφ ∂µUφ)φ + φ φ 2 (φ φ) 2 − − 2 − 4v !   L = C ψ φ†χ + χ φ ψ Y − i iR iL iL iR Xi   1 L = W Wµν W −4 µν 1 L = F F µν. (2) B −4 µν

ψiR is a mass less right handed fermion turning into the right handed component of an observable fermion with mass. It carries the weak isospin t3 = 0 and the U(1) charge g y . It has U(1) symmetry and interacts with ′ ψ the U(1) potential Bµ by the interaction transformation

µ Uψ = exp i g yψBµdx . (3) − Z ′  χiL is a mass less left handed fermion turning into the lefthanded compo- 1 nent of an observable fermion with mass. It carries the weak isospin t3 = 2 and the U(1) charge g y . It has SU(2) U(1) symmetry and interacts with± ′ χ × the three Wµ potentials and the Bµ potential by the interaction transforma- tion

g + − 3 µ Uχ = exp i (T+Wµ + T−Wµ )+ gt3Wµ + g yχBµ dx , (4) − Z √2 ′ ! ! where the T± are the weak isospin rising or lowering operators, and g is the universal SO(3) charge.

φ is a 4-component scalar field. mH 125GeV is the mass of the observable 1-component Higgs particle, but≈ the sign of the mass term of φ is opposite to the sign describing an observable particle. v 246GeV is the electroweak scale. ≈

The two mass less fermions ψiR and χiL forming together a fermion with mass interact with each other via the field φ by a Yukawa interaction represented by the Lagrangian LY . The Ci are coupling constants.

2 LY describes the reactions

ψ + φ χ and χ ψ + φ. (5) iR → iL iL → iR Since weak isospin and the U(1) charge are conserved by these reactions, the interaction transformation of φ is

−1 Uφ = Uψ Uχ. (6)

The mass less fermions χiL and ψiR carry electric charge. The electric charge q is a linear combination of the weak charge gt3 and the U(1) charge g y, ◦ ′ realised by a rotation about the electroweak angle θW 28.6 , which is defined by the condition, that the two mass less fermions forming≈ together a fermion with mass, carry the same electric charge,

q = sin θ g t3 + cos θ g y = cos θ g y = q . (7) χ W W ′ χ W ′ ψ ψ The electric charge of φ is

q = q q =0. (8) φ χ − ψ The charges of the mass less fermions forming leptons are plotted in Figure 1, including the corresponding antiparticle states.

g y ′ q ✻ ✕ ψ1L ✉ e χ0 χ1 R ✉ g′ ✉ R 2 θW ψ 0L ✲ ✉✉ g t3 g ψ g − 2 0R 2 ✉ g′ ✉ χ1L − 2 χ0L e − ✉ ψ1R

Figure 1: The charges of the mass less fermions forming leptons approximate an SU(3) octet. The two mass less fermions χiL and ψiR forming together a lepton are connected by a dotted line.

3 Deviating from the historical Standard Model, here a neutrino is assumed to be composed of a left handed and a right handed mass less fermion, like a charged lepton. The two constituents of a neutrino χL and ψR interact with the Higgs field φ by a Yukawa interaction and form a massive fermion, obtaining its mass mi = Civ by a term of the Lagrangian LY .

The interactions implied by the Lagrangian LY are described by the Feynman diagrams Figure 2.

❍ χL ψR✟✟ ψR χL ❍❍❥ ✟ ❆ ✁ ❍❍ ✟✯✟ ❆ ❍❍✟✟ ❆❯ ✁ ❆ ✁ ❆ ✁✕ φ ✁ νj φ νj νj ❆ ✁ νk ✁ ❆ ✁ ❆ ❆❯❆ ✟✟❍❍ ✁✕✁ ✟ ❍❥❍ ❆ ✟✟✯✟ ❍ ✁ ✟ ψR χL ❍❍ ✁χL ψR❆ 2a 2b

Figure 2: The Feynman diagrams corresponding to the Lagrangian LY for the Yukawa interaction of the constituents χL and ψR of a neutrino. The arrows denote ingoing and outgoing states. The diagram 2a is explicitly implied by LY . The diagram 2b occurs due to crossing symmetry. It transforms a νJ into a νk, j, k = e, µ, τ and thus represents the mechanism for neutrino oscillations.

Actually, only the reaction corresponding to the diagram 2a is explicitly implied by the Lagrangian LY . In order to describe the observed neutrino oscillations within the frame of the Standard Model, the assumption is made, that there is crossing symmetry [7], and therefore also the reaction of the rotated diagram 2b occurs. As a justification the similarity with the electromagnetic interaction is considered, where the e+e− Bhabha scattering proceeds via two diagrams related by crossing symmetry.

Conservation of weak isospin and U(1) charge requires for the reaction 2b that the interaction transformation of the field φ is

Uφ = UψUχ. (9)

−1 In addition, the relation (6) Uφ = Uψ Uχ is imposed by diagram 2a. Both

4 conditions together result in the requirement

Uψ =1. (10)

This is fulfilled for neutrinos, but not for charged leptons or quarks. The ψR component of neutrinos does not carry any charge. Therefore, the constituents of neutrinos interact not only according to diagram 2a but also according to diagram 2b.

By both reactions, 2a and 2b a neutrino is transformed into a neutrino. By reaction 2a a neutrino is only transformed into itself. However, by reaction 2b a neutrino can also be transformed into a neutrino of a dif- ferent generation. This results in neutrino oscillations, proceeding via the 4-component Higgs field φ as an intermediate state.

In summary, the observed neutrino oscillations can be explained within the frame of the Standard Model, assuming that a neutrino is composed of a left handed and a right handed mass less fermion, and assuming crossing symmetry for the reactions implied by the Yukawa interaction of the mass less fermions with the 4-component Higgs field. This picture allows to compute the neutrino oscillation rates, once the neutrino masses are known.

5 References

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