New Physics Right-Chiral CC Coupling Constant Estimation in Neutrino Oscillation Experiments

Abstract

The error probability of the discrimination of the Standard Model (SM) with massive neutrinos and its new physics (NP) model extension in experiments of the muon neutrino oscillation, following the pion decay $\pi ^{+} \rightarrow \mu ^{+} + \nu _{\mu }$, is calculated. The stability of the estimation of the NP charged current coupling constant ε_R is analysed and the robustness of this estimation is checked. It is shown that the upper bound on the error probability of erroneous identification of the Standard Model with its NP model extension has reached the significantly small value of approximately 2.3 × 10^− 6.

Appendix: The density matrix at the detection point

The effective Hamiltonian ${{\mathscr{H}}}$, (5), and neutrino density matrix ρ^Pμ, (3), have the 6 × 6-dimensional matrix representations. The diagonalisation of ${{\mathscr{H}}}$ [11, 31, 32] gives ${{\mathscr{H}}} = \frac {1}{2 E_{\nu }} W \text {diag} (\tilde {m}_{l}^{2}) W^{\dagger }$, where W is the diagonalising unitary matrix defined by the eigenvectors of $2 E_{\nu } {{\mathscr{H}}}$, and the corresponding real eigenvalues $\tilde {m}_{l}^{2}$, l = 1,..., 6, are the neutrino effective squared masses. E_ν is the neutrino energy, neglecting the mass contribution. W ≡ (W_iλ;l) ≡ (_λ〈ν_i|l〉) defines the transformation from the helicity-mass basis |ν_i〉_λ ≡|p,λ, i〉 to the eigenvector basis |l〉 of ${{\mathscr{H}}}$. For the relativistic neutrino ν_μ and in the non-dissipative homogeneous medium, from (4) it follows that in |ν_i〉_λ basis the density matrix at the point z = L of ν_μ detection is [15]:

$$ \begin{array}{@{}rcl@{}} \varrho^{\mu n; n^{\prime}}_{\sigma; \sigma^{\prime}} (t = T) = \sum\limits_{i \lambda} \sum\limits_{i^{\prime} \lambda^{\prime}} \sum\limits_{l,l^{\prime}} W_{n \sigma; l} W_{i \lambda; l}^{*} \varrho^{\mathrm{P} \mu i; i^{\prime}}_{\lambda; \lambda^{\prime}} (t = 0) e^{- i {\Delta} E_{ll^{\prime}} T} W_{i^{\prime} \lambda^{\prime}; l^{\prime}} W_{n^{\prime} \sigma^{\prime}; l^{\prime}}^{*} , \end{array} $$

(22)

where T is the time between neutrino production and detection, ${\Delta } E_{ll^{\prime }} \equiv E_{l} - E_{l^{\prime }} = \frac {\Delta \tilde {m}_{ll^{\prime }}^{2}}{2 E_{\nu }} = \frac {\tilde {m}_{l}^{2} - \tilde {m}_{l^{\prime }}^{2}}{2 E_{\nu }}$. The equality $\varrho _{\text {\textbf {L}}}^{\mu }(\vec {\mathrm {p}}_{\text {\textbf {L}}}) = \varrho ^{\mu }(\vec {\mathrm {p}})$ of the density matrices in the L frame and CM frame is assumed [13]. Because of the W matrix unitarity, the L frame neutrino density matrix at the detection point is normalized, i.e., Tr[ϱ^μ(t = T)] = 1. Equation (22) is valid in the so-called light-ray approximation T = L [12]. The deviation of T from the relation T = L is experimentally significant if some corrections $\varepsilon _{ll^{\prime }}$ [12] to the oscillation phases ${\Delta } \phi _{ll^{\prime }} = |{\Delta } E_{ll^{\prime }}| T$ are also significant. As $\varepsilon _{ll^{\prime }}$ are functions of ${\Delta } \phi _{ll^{\prime }}$, this would require ${\Delta } \phi _{ll^{\prime }} \gg 1$ [12]. However, for the oscillations to be measurable at all, it is necessary that ${\Delta } \phi _{ll^{\prime }} \sim 1$, in which case the corrections $\varepsilon _{ll^{\prime }}$ to ${\Delta } \phi _{ll^{\prime }}$ can be neglected [12], validating the light-ray approximation.

Finally, using ϱ^μ(t = T), (22), one can also calculate, e.g., the geometric phase of the μ flavour neutrino state [29] or the cross section $\sigma _{\mu \rightarrow \beta }$ for the detection of the β flavour neutrino in the L frame [15, 33].