Neutrino oscillation induced by horizontal symmetry

To obtain the interactions which cause neutrino flavor conversion, we introduce a horizontal symmetry into the standard model (SM) and propose the hypothesis that new interactions generated by the horizontal symmetry lead to neutrino flavor conversion and oscillation. To support our hypothesis, we evaluate the flavor conversion probability by new interactions by utilizing the definition of cross section, and the prediction is consistent to experimental data. From our hypothesis, neutrino oscillation is fluctuation of flavor distribution before arriving at equilibrium.


Introduction
The phase evolution can depict phenomenologically neutrino oscillation [1] very well, and also manifest neutrinos massive. But the interactions which cause neutrino flavor conversion remain puzzling [2,3], which we will attempt to investigate in this paper.
No interactions implied in the SM can induce neutrino flavor conversion, while a horizontal symmetry added into SM will provide these interactions [4] and also the masses of neutrinos [5]. Therefore we extend the SM and propose a hypothesis that new interactions from horizontal symmetry lead to neutrino flavor conversions. And neutrino oscillation is the macro phenomenon of the sum of all interactions before equilibrium, which will results in a mixing matrix of anarchy [6,7].
To support our hypothesis, we will calculate the flavor conversion probability by new interactions and compare it to the experimental data. In these calculation, we will assume neutrinos collide due to 'Brown movement' with their moments of all possible directions. We will uncover that it is their mass differences that lead to three different neutrino mixing angles. This paper will be constructed as follows. We will introduce the model of SU(2) L ⊗ U(1) Y ⊗ SU(2) N , which is an extension of SM by adding a horizontal symmetry SU(2) N . We will evaluate the flavor conversion probability induced by new interactions and compare it with experimental data. We will demonstrate the resultant phenomenon of neutrino oscillation.
The model of SU(2) L ⊗ U(1) Y ⊗ SU(2) N For simplicity, we will take two-flavor frame. Thus, our model contains four leptons v e , e, v µ , µ and four quarks with neglecting the color degree of freedom. (For anomaly free, quark sector is assigned the same as SM and we will not discuss them. And we will not discuss the source of neutrino mass in this paper either.) The total group is SU(2) L ⊗ U(1) Y ⊗ SU(2) N . Considering suppression of e ↔ µ in experiments, the horizontal flavor symmetry SU(2) N only works between neutrinos. Thus the up sectors of SM doublets are under the horizontal symmetry SU(2) N , while the lower sectors are not. We will assign the scale which breaking SU(2) N just a little higher than the scale of SM. After SU(2) N broken, the model should return to SM. For 2 × 2 representation of field like ψ ,the transformation under the group SU(2) L ⊗ U(1) Y ⊗ SU(2) N can be equivalent to transformation under group SU(2) L , U(1) Y and SU(2) N in the meantime. (1) is the transformations of ψ under group SU(2) L , U(1) Y and SU(2) N respectively: In (1), τ w , τ F are generators associated with SU(2) L and SU(2) N respectively,Ŷ is the weak hypercharge operator related to U(1) Y . The covariant derivatives related with transformation of ψ under SU(2) L , U(1) Y and SU(2) N respectively are In our model, two neutrino flavors are assigned into the doublet representation of SU(2) N as follows v e v µ (2, 2, −1) The numbers within parentheses stand for SU(2) L , SU(2) N and U(1) Y quantum numbers ( 2I + 1, 2I N + 1, Y ) where I, I N are isospins of the subgroups SU(2) L , SU(2) N respectively, and Y is the U(1) -hypercharge. The and the lower sectors e L , µ L are assigned as (2, 1, −1) . We assign all leptons in a left-handed 2 × 2 matrix v e v µ e µ L and two right handed singlets The electric charge formula is given by Thus, there are seven vector bosons W i α , F i α (i = 1 ∼ 3) and Y α , associated with the subgroups SU(2) L , SU(2) N , U(1) Y respectively.
The fermion dynamical Lagrangian is where 1 2 The difference of (10) from [4] is that τ i F only act on neutrino fields and vector bosons Though our horizontal symmetry exists just between neutrinos, the masses of vector bosons are given by Higgs scalars vacuum expectation value (VEV) and the mass matrix (13) for W 3 , F 3 , Y and the diagonal vectors (14) in [4] are also suitable for our model The diagonal neutral vectors are For conveniently comparing with the Weinberg angle in SM, we assume The interaction Lagrangian can then be written as The coupling constants in (16) compared with the ones in SM will yield Comparing the third formula in (15) and the second formula in (17), we can deduce that g w in our model is just the one in SM. Combined with the first formula in (17), we can deduce that φ is just the Weinberg angle θ W in SM. The maximal mixture between neutrinos implies the coupling constant g F for interactions between different neutrinos is similar to g Z in SM for the same flavors. Then we assume the magnitude of g F is the same as g Z in SM.

The flavor conversion probability per unit time
The cross section σ and the scattering probability per unit time at the start of neutrino flight We assume neutrinos collide in beam due to their 'Brownian movement' with their moments of all possible directions. According to our model, neutrino flavor conversions are caused by interactions intermediated by F µ . At the start of neutrino's flight, all neutrino-neutrino interactions can be shown in one Feynman diagram as (18 v α is the initial neutrino and N is vector boson which intermediates neutrinoneutrino interactions such as Z, F ± . We assume these vector bosons have masses similar to Z 0 in SM. For small mass of neutrino, our following calculation will ignore neutrino mass. Thus, the scattering amplitude of (18) is As is mentioned, the magnitude of g F ∼ g Z (inSM) and then Not that (21) is obtained with the convention that the impact parameter b ∼ 0 . The neutrino conversion probability generally is obtained on axis, so (21) makes sense.
In center of mass frame , we have And we get scattering cross section in center of mass frame as below Let We can obtain the expression about s of section σ Taking the rest frame of the initial neutrino v α (p) before interaction, we have Then we have scattering cross section as below After some calculation of natural units conversion (which we will show in Appendix B), we get Here we assume m v α = m v e ∼ 1eV and the energy of neutrino E = 1GeV . Generally, the masses of the former two neutrino families v e , v µ are considered far lighter than the third v τ . Thus when v τ participates in interactions, its mass in (22) can not be ignored and (27) is not suitable for these interactions and therefore the cross section of these interactions should obtained from (22) directly. In this paper, we only want to know whether our predicting conversion probability is consistent to experiments. For simplicity and without loss of generality, we only need to evaluate the order of predicting P(v e → v µ ) .
Following, we will show how to evaluate conversion probability by cross section σ . Consider the definition of σ : In equation (28), n B denotes current density of incident particle B; N A denotes particle number of target particle A; N denotes all scattering events in unit time.
From (28), if beam section is unit area, the value of current density n B equals incident particle number in unit time. In neutrino beam , target particle is also particles in beam of the same section and the number of target particle N A = n B . Thus , according to (28), the value of σ equals the average scattering probability per neutrino in unit time. That means the value of σ equals the value of average conversion probability per unit time of one initial neutrino. Because the cross section formula (21) is established with the convention b ∼ 0 , we will choose the unit area 1cm 2 . Then the magnitude of neutrino-neutrino scattering probability per unit time is The factor 2 is because 2 initial neutrinos.
Thus our predicting neutrino flavor conversion probability per unit time is

The actual conversion probability per unit time at the start of flight
We can obtain the conversion probability per time (32) by taking a derivative of the empirical formula (31) with respect to t: The last term of (32) is because t ≪ L 0 at the start of flight.
In the first second of flight, To estimate the magnitude of (33), we take a special example as the conversion of v e → v µ . According to experimental data, we take values of (34) into (33): So, the conversion probability per unit time at the start of neutrino flight is Apparently, our prediction (30) is very close to the actual value(35).

Neutrino oscillation
According to our model, neutrino-neutrino interaction is As is mentioned before, we assume then the whole approximate effective Hamiltonian can be written in one term as follows Here v ′ is not a pure state and is some mixture of several states. While v iL , v jL are initial states. Thus the neutrino mixture is just the macro result of the sum of all interactions. And we also can conclude that the mixture of neutrino flavors will not be constant until all neutrino-neutrino interactions arrive at balance.
We can see that there is a equilibrium state for these interactions. There must have some oscillations before neutrino probability distribution arrive at equilibrium distribution.

Conclusions
For interactions responsible for neutrino flavor conversions, we constructed the extension of SM including a horizontal flavor symmetry i.e. SU(2) L ⊗ U(1) Y ⊗ SU(2) N . The horizontal symmetry SU(2) N introduces interactions between different neutrino flavors and leads to neutrino flavor conversions.
To testify our idea, we have evaluated the flavor conversion probability induced by new interactions by utilizing the definition of cross section. In this calculation, we assume neutrinos collide due to 'Brown movement' and the direction of individual neutrino instantaneous moment in beam is not definite and generally not consistent to the direction of neutrino flux. Fortunately, the prediction is consistent with experimental data. And we also demonstrate neutrino oscillation as phenomenon before all neutrino-neutrino interactions arriving at balance. meantime: α, β, ρ is infinitesimal, τ w , τ F are generators associated with SU(2) L and SU(2) N respectively,Ŷ is the weak hypercharge operator related to U(1) Y . We introduce gauge fields W i α , Y µ , F i α to relate groups SU(2) L , U(1) Y and SU(2) N respectively.
The covariant derivatives related with transformation under SU(2) L , U(1) Y and SU(2) N respectively are Define the gauge transformation of gauge fields W i α , Y µ , F i α as follows The fermion dynamical Lagrangian invariant under the group SU(2) L ⊗ In (A.4), for left-handed lepton, the value of the hypercharge Y is -1 and for right-handed lepton R, the value of the hypercharge Y is -2.
With the transformation of these gauge fields (42), the fermion dynamical Lagrangian (A.4) will be invariant under the group of SU(2) L ⊗ U(1) Y ⊗ SU(2) N . We will prove it term by term.

Proof:
1. The first two terms in (A.4) The covariant derivative is We only need to prove Omitting the higher order of O(gβ) , we can obtain

The third and fourth term
We will prove the invariance of these two terms by prove invariance under group SU(2) L , U(1) Y and SU(2) N respectively.
The transformations of the gauge fields are Which we need to prove is that is Left of (A.6) = The right of (A.6) is Thus (A.6) can be established.
The right of (A.7)= Thus (A.7) can be established.
Taking above values into the cross section formula, we will get Because the cross section formula (B.1) is obtained with the convention that the impact parameter b ∼ 0 , we adopt the unit cm 2 .