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Basic oscillation measurables in the neutrino pair beam

It is shown that the vector current contribution of neutrino interaction with electrons in ion gives rise to oscillating component, which is absent for the axial-vector contribution, when a single neutrino is detected in the recently proposed neutrino pair beam. CP violation measurements are thus possible with high precision along with determination of mass hierarchical patterns.

Introduction A new strong source of neutrinos consisting of all flavor pairs of ν a andν a (a = e, µ, τ ) was recently proposed to further accelerate neutrino physics experiments [1]. It was however pointed out in [2] that a single neutrino detection in the neutrino pair beam does not exhibit oscillation pattern, making detection of oscillation harder due to large backgrounds in double event detection. This disappearance of oscillation is based on (i) the unitarity of the 3 × 3 neutrino mixing matrix and (ii) the equality of pair emission amplitude squared that holds for the dominant axial vector contribution of light ions.
In the present work we show that the second condition, the equality of pair emission amplitude squared, does not hold in the vector current contribution (sub-dominant, when ionic electrons move with nonrelativistic velocities, but may be comparable to the axial vector contribution in heavy ions) of pair emission amplitude, hence the emergence of oscillation patterns occurs from the vector contribution. When neutrino oscillation is made possible this way, the CP violating (CPV) parameter determination (the CPV phase δ common to both Dirac and Majorana cases) becomes possible.
We derive basic formulas for the three-flavor neutrino scheme including the earth matter effect and present numerical outputs of quantities for new experiments using the neutrino pair beam. It is found that oscillation patterns appear in all ν a ,ν a , a = e, µ, τ , but determination of CPV parameter is possible only by detection of ν µ ,ν µ and tau-neutrinos. Electron neutrinos do not allow CPV determination. If the accelerator ring is placed in the underground of depth d, the neutrino pair beam appear on earth at a distance ∼ √ 2dR with R the radius of the earth. This distance is ∼ 35 km for d = 100 m. It is found that one can do sensitive CPV measurements at this distance.
Throughout this work we use the natural unit ofh = c = 1.
How oscillation pattern appears in single neutrino detection We shall follow notations of [2].
The probability amplitude of the entire process consists of three parts: the production, the propagation, and the detection due to charged current interaction (neutral current interaction is much smaller, hence not considered here), each to be multiplied at the amplitude level. Thus, one may write the probability for the ν a neutrino quasi-elastic scattering (with J α the nucleon weak current) as where H (H) is the hamiltonian for propagation of neutrino (antineutrino) including earth-induced matter effect [4], [5], [6], which is in the flavor basis where U ai is the neutrino mixing matrix with |a = i U * ai |i , a = e, µ, τ, i = 1, 2, 3, and n e is the number density of electrons in the earth.H can be obtained by replacing U → U * and changing the sign in the second term ∝ G F .
We shall denote three eigenvalues by λ i for neutrinos, andλ i for anti-neutrinos. Let V (∼ U ) andV are unitary 3 × 3 matrices that diagonalize the hamiltonian H for neutrino andH for anti-neutrino, including the earth matter effect. The propagation amplitude is then The factor c b arises from the production amplitude Mb b (1, 2) and it is (c A b ) = 1 2 (1, −1, −1) for the axial vector contribution and for the vector contribution, with the weak mixing angle θ w . The precise relation between neutrino and anti-neutrino eigenvalue problem is given byλ The rate of neutrino ν a detected andν c undetected contains the squared propagation factor, When (|c A b | 2 ) = (1, 1, 1)/4 ∝ 1 for the axial vector contribution, p jl = δ jl /4 and the detection probability becomes 1/4, hence no oscillation pattern exits.
The relevant weak amplitude for the vector part gives oscillating components. Candidate ions for circulation that contribute to the vector current interaction are Be-like heavy ions of 2p2s 3 P − 1 and Ne-like heavy ions of 2p + 3s 3 P − 1 (electron-hole system). Basic measurable quantities in neutrino pair beam We first note The detection probability of ν a (when the other neutrino of the pair is undetected) is given by the oscillation formula based on the vector part of weak current, with sin 2 θ w ∼ 0.231. The formula (9) is valid when the earth matter effect is neglected. When the earth matter effect is included, one replaces U → V , m 2 j /2E → λ j . The quantity P a (E, L; m i , δ) is the normalized probability: a P a (E, L; m i , δ) = 1. The oscillating component in eq. (9) is equivalent to the ν e → ν a , a = µ, τ appearance probability multiplied by Thus, the constant off-set term ∝ (1 − 4 sin 2 θ w ) 2 in eq. (9) is very small. In the limit of sin 2 θ w = 1/4 there is no contribution to the vector part from Z-boson exchange. Due to the dominance of ν e → ν a , a = µ, τ in the oscillating term, oscillation patterns in the pair beam have similarities to the β [7] and β ± beam [8].
The most striking feature of the neutrino pair beam is that circulating quantum ions produce coherent pairs of all flavors, ν aνa , a = e, ν, τ . When these pairs propagate, all mass eigen-states get involved, and relevant oscillation extrema at L/E = 2π/δm 2 ij , (ij) = (12), (23), (13) may become relevant, making short baseline experiments a feasible approach.
Experimentally, this quantity may be derived from measurements of both ν a andν a events.
We note a few important results: (i) CPV δ measurement is impossible for ν e ,ν e events, because the oscillation probability appearing in eq. (9) depends on the quantity |U ej | 2 , hence is insensitive to δ. (ii)  To incorporate the earth matter effect, we numerically diagonalize the effective hamiltonian (2) [13].
The oscillation patterns including the earth matter effect are illustrated in Figs. 4 and 5. We took a pure SiO 2 model with density 2.8 g/cm 3 for the earth matter.

Rates of neutrino-pair production from quantum ion beam
Calculations in [2] are for the axial-vector contribution. We now repeat calculations based on the vector contribution, following  NH δ=0 w ME δ=0 wo ME δ=π/3 w ME δ=π/3 wo ME Figure 4: ν µ oscillation pattern with and without the earth matter effect (ME). The neutrino energy is fixed at 200 MeV. δ = 0 with ME in solid black, without ME in dashed red, δ = π/3 with ME in dash-dotted blue, and without ME in dotted orange. δ = π/4 neutrino (ν µ ) in solid black, anti-neutrino (ν µ = ν µ b) in dash-dotted blue, δ = π/2 neutrino in dashed red, and anti-neutrino in dotted orange. contribution is thus obtained by a simple replacement from the axial vector contribution: We have replaced the flavor component fraction at production by a simplified result of sin 2 θ w = 1/4, namely (c V b ) 2 = (1, 0, 0). The differential energy spectrum of a single detected neutrino at the forward direction is, in the high energy limit of γ ≫ 1, Here N is the available number of ions, ρ is the radius of the ring, ϕ is the effective angle of the neutrino pair beam.
The angular distribution is readily calculable, and is plotted in Fig. 6. The forward production rates are the most relevant to neutrino oscillation experiments away from the ring. The forward rate is estimated by taking the angular area π/γ 2 times the right hand side of eq. (13). The following figure Fig. 7 illustrates these rates. The forward rates scale with ion parameters ∝ A eg ǫ 5.5 eg , and with the boost factor ∝ γ 3.5 . In order to detect ν µ events, neutrino energies larger than 200 MeV are desired, which gives a constraint on 2γǫ eg .
Thus, if the production rate dΓ dE ∆E presented in the previous section is larger than 10 11 Hz, then events rates at experimental sites are larger than 1 Hz. ∆E is the energy bin taken at each energy. Prospects for high sensitivity experiments are bright.
In summary, we demonstrated that CPV parameter determination with a high precision is possible in a short baseline experiment using the neutrino pair beam. Clearly, both experimental R and D works of quantum coherent ion circulation and theoretical studies of candidate ions are required for further development of this new project. [11] A. Yu. Smirnov and G.T. Zatsepin, Mod. Phys. Lett. A7, 1273 (1992).
[13] We correct the sign mistake of earth matter effect in [4] and [5].