The NP right-chiral $CC$ coupling constant estimation in neutrino oscillation experiments

TL;DR

This paper evaluates the error probability in distinguishing the Standard Model with massive neutrinos from its new physics extension in muon neutrino oscillation experiments, demonstrating a very low error bound of about 2.3×10⁻⁶.

Contribution

It introduces a method to estimate the NP right-chiral CC coupling constant in neutrino oscillation experiments and assesses the robustness and stability of this estimation.

Findings

Error probability of model discrimination is approximately 2.3×10⁻⁶.

Estimation of the NP coupling constant is stable and robust.

Upper bound on misidentification error is significantly small.

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 $\varepsilon_{R}$ 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 $2.3 \times 10^{-6}$.

Equations

$$|A_i^{\mu }{}^{+1;+1}(\mathrm{p}){|}^2=|A_i^{\mu }{}^{-1;-1}(\mathrm{p}){|}^2\frac{|{\epsilon }_R{|}^2|U_{\mu i}^R{|}^2}{|{\epsilon }_L{|}^2|U_{\mu i}^L{|}^2},$$

$${\varrho }_{-1;-1}^{\mathrm{P}\mu i;i^{\prime }}=\frac{A_{{\epsilon }_L^2}}{\text{N}}|{\epsilon }_L{|}^2{U}_{\mu i}^{L\ast }{U}_{\mu i^{\prime }}^L,{\varrho }_{+1;+1}^{\mathrm{P}\mu i;i^{\prime }}=\frac{A_{{\epsilon }_R^2}}{\text{N}}|{\epsilon }_R{|}^2{U}_{\mu i}^{R\ast }{U}_{\mu i^{\prime }}^R,$$

$${\varrho }_{-1;-1}^{\mathrm{P}\mu i;i^{\prime }}=\frac{|{\epsilon }_L{|}^2{U}_{\mu i}^{L\ast }{U}_{\mu i^{\prime }}^L}{|{\epsilon }_R{|}^2+|{\epsilon }_L{|}^2},{\varrho }_{+1;+1}^{\mathrm{P}\mu i;i^{\prime }}=\frac{|{\epsilon }_R{|}^2{U}_{\mu i}^{R\ast }{U}_{\mu i^{\prime }}^R}{|{\epsilon }_R{|}^2+|{\epsilon }_L{|}^2}.$$

$${\rho }^{\mu }(t=0)\to {\rho }^{\mu }(t)=e^{-i\mathcal{H}t}{\rho }^{\mathrm{P}\mu }(t=0)e^{i\mathcal{H}t},$$

$$\mathcal{H}=\mathcal{M}+{\mathcal{H}}_{int},$$

$${\varrho }^{\mu }(z)=\sum _{j=1}^{\aleph }{p}^j(z)|w_j^{\mu }(z)⟩⟨w_j^{\mu }(z)|,$$

$$p_{SM}^j=p_{SM}^j(z=0)=p_{SM}^j(z=\mathrm{L}),$$

$$p_{NP}^j=p_{NP}^j(z=0)=p_{NP}^j(z=\mathrm{L}),j=1,2,...,\aleph =6.$$

$$p_{NP}^4\approx |{\epsilon }_R{|}^2-|{\epsilon }_R{|}^4+1.4911|{\epsilon }_R{|}^6\mathrm{for}\mathcal{R}<0.045$$

$$P_E({\varrho }_{SM}^{\mu },{\varrho }_{NP}^{\mu })=e^{-NS({\varrho }_{SM}^{\mu }\Vert {\varrho }_{NP}^{\mu })},$$

$$S({\varrho }_{SM}^{\mu }\Vert {\varrho }_{NP}^{\mu }):=\mathrm{Tr}[{\varrho }_{SM}^{\mu }(\ln {\varrho }_{SM}^{\mu }-\ln {\varrho }_{NP}^{\mu })]$$

$$S(p_{SM}\Vert p_{NP})\le S_1({\varrho }_{SM}^{\mu }\Vert {\varrho }_{NP}^{\mu })\le S({\varrho }_{SM}^{\mu }\Vert {\varrho }_{NP}^{\mu }).$$

$$S(p_{SM}\Vert p_{NP}):=\sum _{j=1}^6{p}_{SM}^j\ln \frac{p_{SM}^j}{p_{NP}^j}$$

$$P_E({\varrho }_{SM}^{\mu },{\varrho }_{NP}^{\mu })\le P_E(p_{SM},p_{NP})=e^{-NS(p_{SM}\Vert p_{NP})}.$$

$$g_{{\epsilon }_R{\epsilon }_R}^{\mathrm{c}}=\sum _{j=1}^6{p}_{NP}^j\frac{\partial \ln p_{NP}^j}{\partial {\epsilon }_R}\frac{\partial \ln p_{NP}^j}{\partial {\epsilon }_R},$$

$$g_{{\epsilon }_R{\epsilon }_R}^{\mathrm{c}}({\epsilon }_R)\approx 4(1-2|{\epsilon }_R{|}^2+5.4556|{\epsilon }_R{|}^4)\mathrm{for}\mathcal{R}<\mathrm{0.045.}$$

$${\sigma }^2({\hat{\epsilon }}_R)$$

$$D_{Rao}(p_{SM},p_{NP})$$

$$D(p_{SM},p_{NP})$$

$$D_{DO}({\varrho }_{SM}^{\mu },{\varrho }_{NP}^{\mu })\ge D(p_{SM},p_{NP})\ge D_{Rao}(p_{SM},p_{NP}).$$

$${\sigma }^2({\tilde{\epsilon }}_R^{\mathrm{c}})=\frac{1}{N{g}_{{\epsilon }_R{\epsilon }_R}^{\mathrm{c}}}\ge {\sigma }^2({\tilde{\epsilon }}_R):=\frac{1}{N{g}_{{\epsilon }_R{\epsilon }_R}^{DO}}.$$

$${\varrho }_{\sigma ;{\sigma }^{\prime }}^{\mu n;n^{\prime }}(t=T)=\sum _{i\lambda }\sum _{i^{\prime }{\lambda }^{\prime }}\sum _{l,l^{\prime }}{W}_{n\sigma ;l}{W}_{i\lambda ;l}^{\ast }{\varrho }_{\lambda ;{\lambda }^{\prime }}^{\mathrm{P}\mu i;i^{\prime }}(t=0)e^{-i\Delta E_{l{l}^{\prime }}T}{W}_{i^{\prime }{\lambda }^{\prime };l^{\prime }}{W}_{n^{\prime }{\sigma }^{\prime };l^{\prime }}^{\ast },\text{vspace}-4mm$$

Full text

1 The muon neutrino density matrix

The well known modelling of the chiral right-handed currents is connected with left-right symmetric extensions of the Standard Model (SM) Siringo ; Mohapatra-Pati ; Zuber . There are also effective-Lagranganian SM extensions which can be used to inspect the existence of the chiral right-handed interactions Bergmann-Grossman-Nardi ; A-B-Sz-Wudka-Z ; AZS . This paper follows this path. Let the muon neutrino $\nu_{\mu}$ be produced in the decay $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ of pion to muon and the muon Dirac neutrino Giunti-Kim . The neutrino $\nu_{\mu}$ produced in this process is the relativistic one. The muon flavour neutrino state $|\nu_{\mu}\rangle$ is a superposition of the stationary states $|\nu_{i}\rangle_{\lambda}\equiv|{\rm p},\lambda,i\rangle\,$ Giunti-Kim of definite masses $m_{i}$, $i=1,2,3$, helicities $\lambda=-1$ or $+1$ and four-momentum ${\rm p}$ OSZ . By including new physics (NP) interactions Kuno-Okada , e.g., the chiral right-handed interactions AZS , this superposition composes the mixed state OSZ ; ZZS . The other reason of the departure from the pure state can be connected, e.g., with the existence of scalar interactions OSZ . From the $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ decay experiments we know that the fraction of the right-handed $N_{\nu_{+1}}$ to the left-handed $N_{\nu_{-1}}$ neutrinos fulfils the constraint $N_{\nu_{+1}}/N_{\nu_{-1}}<0.002$ PDG_epsR_2 ; PDG_epsR_3 . Let us assume that the pion decays effectively both in the left ($L$) and right ($R$) chiral charged current ($CC$) interactions ZZS via the exchange of the SM $W$-boson only. Then, at the $W$-boson energy scale, the $R$ and $L$ chiral pion decay constants Berman-Kinoshita-1 ; Berman-Kinoshita-2 ; Berman-Kinoshita-3 are equal and due to its smallness the (pseudo)scalar correction can be neglected C-M ; EGPR . The invariant amplitudes $A_{i}^{\mu}{{}^{\;\lambda;\lambda_{\mu}}}({\rm p})\,$ ZZS in the decay $\pi^{+}\rightarrow\mu^{+}+\nu_{i,\lambda}$, where $\lambda_{\mu}=-1$ or $+1$ is the muon helicity, are related as follows:

[TABLE]

where $U_{\alpha i}^{L}$ and $U_{\alpha i}^{R}$ are the $L$ and $R$ chiral neutrino mixing matrices, which enter into the $CC$ Lagrangian in the products with the coupling constants $\varepsilon_{L}$ and $\varepsilon_{R}$, respectively ZZS . $U_{\alpha i}^{L}$ is the Maki-Nakagawa-Sakata-Pontecorvo neutrino mixing matrix MNSP-1 ; MNSP-2 . For relativistic neutrinos, the dependance of the production process on the neutrino masses can be neglected Giunti-Kim . Then, in the production (P) process, in the center of mass (CM) frame and in $|\nu_{i}\rangle_{\lambda}$ basis, the elements of the general form of the $3\times 3$-dimensional nonzero muon neutrino reduced mass-helicity density matrix (obtained from the full density matrix by tracing out the other degrees of freedom) are as follows OSZ ; ZZS :

[TABLE]

where $\texttt{N}=A_{\varepsilon_{R}^{2}}|\varepsilon_{R}|^{2}+A_{\varepsilon_{L}^{2}}|\varepsilon_{L}|^{2}\,$ is the normalization constant and $A_{\varepsilon_{R}^{2}}=A_{\varepsilon_{L}^{2}}$. The functions $A_{\varepsilon_{L}^{2}}$ and $A_{\varepsilon_{R}^{2}}$ are the amplitudes for the $CC$ vector-axial processes, i.e., V-A and V$+$A, respectively. They depend on the energies and momenta of the particles in the production process of the neutrino. Thus, the density matrix elements are as follows:

[TABLE]

They constitute the muon neutrino $6\times 6$-dimensional block diagonal density matrix $\rho^{{\rm P}\mu}\equiv(\varrho^{{\rm P}\mu}_{\lambda;\,\lambda^{\prime}})={\rm diag}((\varrho^{{\rm P}\mu\;i;\,i^{\prime}}_{-1;\,-1}),(\varrho^{{\rm P}\mu\;i;\,i^{\prime}}_{+1;\,+1}))$. We choose $U_{\alpha i}^{R}=U_{\alpha i}^{L}$, otherwise there is not only the neutrino helicity mixing but also the mass mixing OSZ . Since the density matrix elements (Eq.(3)) depend on the norms of $\varepsilon_{L}$ and $\varepsilon_{R}$, and not on their phases, we assume in the analysis that these coupling constants are real. The NP values of $\varepsilon_{L}$ and $\varepsilon_{R}$ can deviate slightly from the SM values 1 and 0, respectively. When the constraint $N_{\nu_{+1}}/N_{\nu_{-1}}<0.002$ is used, from Eq.(1) the bound on the ratio ${\cal R}=|\varepsilon_{R}/\varepsilon_{L}|<0.0447\approx 0.045$ results. Since the Fermi constant constraint $\varepsilon_{L}^{4}+\varepsilon_{R}^{4}=1$ should be also held, we obtain (due to the 4th power) $|\varepsilon_{R}|<0.0447\approx 0.045$ which constraints the density matrix $(\varrho^{{\rm P}\mu\;i;\,i^{\prime}}_{+1;+1})$ of the initial neutrino.

The muon neutrino $\nu_{\mu}$ produced in the process $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ is the relativistic one (the neutrino energy $>$ 100 MeV in the Laboratory (L) frame). Thus, the effect of the helicity Wigner rotation is negligible OSZ resulting in $\varrho_{\rm{\bf L}}^{{\rm P}\mu}(\vec{{\rm p}}_{\rm{\bf L}})=\varrho^{{\rm P}\mu}(\vec{{\rm p}})$ for the density matrix in the L frame, where $\vec{{\rm p}}_{\rm{\bf L}}$ and $\vec{{\rm p}}$ are the neutrino momenta in the L and CM frames, respectively. Only the neutrino produced in the L frame in the forward direction along the $z$-axis reaches the detector and this axis is chosen as the quantization one Giunti-Kim . After production, the neutrino $\nu_{\mu}$ propagates in matter and we assume that this is the non-dissipative DLS homogeneous medium. By virtue of quantum mechanical unitarity of the muon-environment time evolution, the interactions of the entangled muons with their environment cannot affect, in any experiment, the probability of neutrino oscillation that follows the pion decay Jones . Thus, in the relativistic case, when the distance $z$ and the propagation time $t$ approach the relation $z=t$ (see, e.g., Giunti-Kim for the so-called light-ray approximation and Appendix), the evolution rule for the neutrino density matrix is as follows:

[TABLE]

where $\rho^{{\rm P}\mu}$ is the initial density matrix (3) and ${\cal{H}}$ is the effective Hamiltonian. Although the coherence properties of the neutrino beam resulting from pion decay are influenced OSZ by the initial pion state, for standard neutrinos no coherence loss is expected on terrestrial scales Jones . Under the above assumptions, the oscillation probability from $\mu$ to $\beta$ flavour at the detection (D) point at $z={\rm L}$ is equal to $P_{\mu\rightarrow\beta}({\rm L})={\rm Tr}\left[\rho^{\mu}({\rm L})\hat{P}^{\,\beta}\right]$. Here $\hat{P}^{\,\beta}\equiv(\hat{P}^{\,\beta}_{i,i^{\prime}})={\rm diag}((U_{\beta i}^{L}U^{L\ast}_{\beta i^{\prime}}),\,(U_{\beta i}^{R}U^{R\ast}_{\beta i^{\prime}}))$ is a $6\times 6$-dimensional block diagonal projection operator to the $\beta$ flavour direction in the neutrino flavour space Sz-Z .

With three massive and two helicity neutrino states, ${\cal{H}}$ has the $6\times 6$-dimensional representation (see, e.g., AZS ; ZZS ; Dziekuje-za-neutrino-faza ):

[TABLE]

where the $6\times 6$-dimensional diagonal matrix $\mathcal{M}$ is the mass term AZS ; Giunti-Kim . Here, ${\cal{H}}_{int}$ is the $6\times 6$ matrix representation of the interaction Hamiltonian ZZS ; Dziekuje-za-neutrino-faza for the coherent neutrino scattering inside the non-dissipative homogeneous medium AZS ; DLS ; Dziekuje-za-neutrino-faza ; Dziekuje-za-magnetyzacje . We will see that, under the above conditions, data obtained in all earth’s oscillation experiments in which the muon neutrinos are produced in the process $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ fall into one category of results that together enable the discrimination of the SM from the NP model (expressed by Eq.(3).)

Note. Usually, the precise knowledge of the evolution of the neutrino density matrix during oscillation experiments AZS ; OSZ ; ZZS ; Kim_Pevsner ; Bekman-2002 ; AZS-conf (Appendix), ruled by the particular form of the Hamiltonian ${\cal{H}}$ is necessary. It is the case, for example, in the consistency analysis Dziekuje-za-neutrino-faza of the values of parameters of $U_{\alpha i}^{L}$ with the predictions of (the type of) the Aharonov-Anandan neutrino geometric phase considerations sjuk2 .

2 The SM and NP model discrimination

The discrimination of the SM from the NP model presented below takes into account both the problem of its sensitivity and the stability of the $\varepsilon_{R}$ estimation. The value of the departure of the purity of the quantum state ${\rm Tr}\left[(\varrho^{\mu})^{2}\right]$ from 1 Bengtsson_Zyczkowski , is a second order effect in the NP parameter $\varepsilon_{R}$ OSZ . In order to find the distance in the statistical space of distributions, it is convenient to represent the density operator in the spectral-decomposition form, i.e.:

[TABLE]

where $p^{j}(z)\geq 0$ and $|w_{j}^{\mu}(z)\rangle$ are the eigenvalues and (normalized) eigenvectors of $\varrho^{\mu}(z)$, respectively, and $\sum_{j=1}^{\aleph}\,p^{j}(z)=1$, while the maximal rank of $\varrho^{\mu}$ is equal to $\aleph$. The NP and SM neutrino quantum states are given by the density matrices $\varrho^{\mu}_{NP}$ and $\varrho^{\mu}_{SM}$, respectively. The spectral decomposition of the density matrix is unique in the sense that for it the von Neumann entropy $S(\varrho^{\mu}):=-{\rm Tr}(\varrho^{\mu}\ln\varrho^{\mu})$ is equal to the Shannon entropy $S(p)=-\sum_{j=1}^{\aleph}\,p^{j}\ln p^{j}$ of the probability distribution $p\equiv\{p^{j}\}_{j=1}^{\aleph}$ Bengtsson_Zyczkowski . The advantage of the spectral-decomposition form is that via Fisher-Rao metric (which is related to Shannon entropy Bengtsson_Zyczkowski ), the eigenvalues $p^{j}$ enter into the calculation of the classical lower bound for the variance of the unbiased estimator of a parameter.

The NP effects change the neutrino state in the course of the oscillation in a different way than the SM Dziekuje-za-neutrino-faza . Yet, due to the unitarity of the evolution given by Eq.(4), the eigenvalues of the density matrices $\varrho^{\mu}_{SM}$ and $\varrho^{\mu}_{NP}$, which we denote as $p^{j}_{SM}$ and $p^{j}_{NP}$, respectively, do not vary with $z$. This happens only if the neutrino evolution remains unitary, as it is, e.g., in the case of the (constant density) slab approximation Giunti-Kim . Thus

[TABLE]

It means that, from the moment of the neutrino production at $z=0$ up to the point of its detection at $z={\rm L}$, the NP v.s. SM discrimination reflected in the probability distributions $p_{NP}\!\equiv\!\{p^{j}_{NP}\}$ and $p_{SM}\!\equiv\!\{p^{j}_{SM}\}$ does not change during its propagation. Therefore, $p^{j}_{NP}$ and $p^{j}_{SM}$ are the invariants of the neutrino oscillation phenomenon. This is the reason why for neutrinos $\nu_{\mu}$ produced in $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ all $\nu_{\mu}\rightarrow\nu_{\mu}$ survival experiments are integral with each other and form one general experiment from which all data can be taken simultaneously. The limits on the unitary evolution can appear when, e.g., the sterile neutrinos Rasmussen , heavy neutrinos Bekman-2002 , decoherence and dissipation DLS , or other phenomena Rasmussen , are included, violating the result given by Eq.(2). In the SM, the produced muon neutrino is in the pure state with helicity $\lambda=-1$, i.e., only one eigenvalue of $\varrho^{\mu}_{SM}$ is nonzero, say $p^{1}_{SM}=1$, and $p^{j}_{SM}=0$ for $j=2,...,6$. In the NP case, the muon neutrino is in the mixture of two helicity states $\lambda=-1$ and $+1$, i.e., two eigenvalues of $\varrho^{\mu}_{NP}$ are nonzero, say $p^{1}_{NP}$ and $p^{4}_{NP}$ and the others are $p^{j}_{NP}=0$, $j=2,3,5,6$. From the spectral-decomposition of $\rho^{{\rm P}\mu}$, Eq.(3), we obtain numerically (with the accuracy of the expansion coefficients up to the forth decimal place) the truncated series expansion in $\varepsilon_{R}$ parameter:

[TABLE]

and $p_{NP}\rightarrow p_{SM}$ for $\varepsilon_{R}\rightarrow 0$. Through the work, the symbol of approximate equality will appear as a consequence of the approximation occurring in Eq.(8).

The sensitivity problem is defined as follows: We are unaware whether we are sampling from the SM or NP model distribution. The sensitivity problem for discrimination of two probability models is connected with the erroneous identification of the probability distribution in the $N$-dimensional sampling. This identification results in a type II error of selection of the SM (${\rm H_{0}}$ hypothesis of the statistical test,) although it is the NP model (${\rm H_{1}}$ hypothesis) which is true. The achievable infimum $P_{E}$ of the probability $\beta$ of this error, for distributions generated by two density matrices, here $\varrho^{\mu}_{NP}$ and $\varrho^{\mu}_{SM}$, was found by Hiai and Petz H-P ; O-N :

[TABLE]

where the number $N$ of quantum copies of the system is very large (in principle infinite) and

[TABLE]

is the always nonnegative Umegaki quantum relative entropy Umegaki-1 ; Umegaki-2 . $S(\varrho^{\mu}_{SM}\|\varrho^{\mu}_{NP})$ is the measure of how far from each other are the NP and SM neutrino quantum states. By the monotonicity of the relative entropy it was proven that Lindblad

[TABLE]

Here

[TABLE]

is the classical Kullback-Leibler relative entropy and $S_{1}(\varrho^{\mu}_{SM}\|\varrho^{\mu}_{NP})$ is the quantum relative entropy that takes the supremum over all possible Positive Operator Valued Measures Bengtsson_Zyczkowski . It should be stressed that from Eq.(2) it follows that the relative entropy $S(p_{SM}\|p_{NP})$ does not vary with $z$, having therefore the same value at the points of the neutrino production and detection. By using the relative entropy $S(p_{SM}\|p_{NP})$ in Eq.(9) instead of $S(\varrho^{\mu}_{SM}\|\varrho^{\mu}_{NP})$, the significance of the difference between the NP and SM states is underestimated. However it gives the operationally easier (classical) bound $P_{E}(p_{SM},p_{NP})$ for the calculation of the error probability Bengtsson_Zyczkowski :

[TABLE]

The relations (9) and (13) are asymptotically strict.

The dependance of $P_{E}(p_{SM},p_{NP})$ on $\varepsilon_{R}$ for various sample size $N$ is presented in Figure 1. Since $P_{E}(p_{SM},p_{NP})\rightarrow 1$ for $\varepsilon_{R}\rightarrow 0$, thus the smaller the $\varepsilon_{R}$, the easier the erroneous identification of the two models. To prevent $P_{E}(p_{SM},p_{NP})$ from increasing with the decrease of $\varepsilon_{R}$, the sample size $N$ has to rise.

To learn about the stability of the estimation of $\varepsilon_{R}$, the lower bound on the variance of its estimator $\hat{\varepsilon}_{R}$ has to be found. The relationship between two lower bounds, classical and quantum, will be determined. It will be shown that the classical lower bound is not smaller than the quantum one, therefore, from the experimental point of view, the classical bound (which needs the bigger sample) is more restrictive than the quantum bound.

The classical lower bound is defined as follows: In the classical (c) approach it is the Fisher information on $\varepsilon_{R}$ parameter that has to be calculated. In general, a probability distribution $p_{\xi}$ is parameterized by a $n$-dimensional parameter $\xi=(\xi^{a})_{a=1}^{n}\in\Xi$, where $\Xi$ is a subset of $\mathbb{R}^{n}$. The Riemannian metric $g_{ab}^{\rm c}$ of the statistical model ${\cal S}=\{p_{\xi}|\,\xi=(\xi^{a})_{a=1}^{n}\in\Xi\}$ is called the Fisher-Rao metric Amari-Nagaoka-book . In this paper $\xi$ is reduced to the scalar NP parameter $\varepsilon_{R}$ and the $n=1$-dimensional manifold ${\cal S}=\{p_{NP}(\varepsilon_{R})|\,\varepsilon_{R}\in\langle 0,1)\}$ is coordinatized by the parameter $\varepsilon_{R}$. Then, the Fisher-Rao metric consists of one component only:

[TABLE]

which, for the distribution $p_{NP}$ given by Eq.(8), is equal to:

[TABLE]

The Fisher information on $\varepsilon_{R}$ in the $N$-dimensional sample is equal to $I_{F}(\varepsilon_{R})=N\,g_{\varepsilon_{R}\varepsilon_{R}}^{\rm c}$ Amari-Nagaoka-book , and from the scalar Cramér-Rao inequality Amari-Nagaoka-book we obtain, in the classical approach, the lower bound $\sigma^{2}(\tilde{\varepsilon}_{R}^{\rm c})$ on the variance of any unbiased estimator $\hat{\varepsilon}_{R}$ of $\varepsilon_{R}$:

[TABLE]

Thus, the standard error $\sigma(\hat{\varepsilon}_{R})=\sqrt{\sigma^{2}(\hat{\varepsilon}_{R})}\geq\sigma(\tilde{\varepsilon}_{R}^{\rm c})=\sqrt{\sigma^{2}(\tilde{\varepsilon}_{R}^{\rm c})}$ $\approx(1+|\varepsilon_{R}|^{2}-1.2278\,|\varepsilon_{R}|^{4}\,)/(2\sqrt{N})$. The values of the lower bound $\sigma(\tilde{\varepsilon}_{R}^{\rm c})$ for the standard error $\sigma(\hat{\varepsilon}_{R})$ as the function of $\varepsilon_{R}$ for some $N$ are shown in Figure 2.

Finally, in the classical approach, the Rao distance between the distributions $p_{NP}$ and $p_{SM}$ in the statistical model ${\cal S}$ (after applying Eq.(15)), is equal to

[TABLE]

The bound to the quantum lower bound is as follows. Let us consider the distance function $D(p_{SM},p_{NP}):=2\,\sqrt{S(p_{SM}\|p_{NP})}$, which for $\varepsilon_{R}\rightarrow 0$ is consistent with the Rao distance Bengtsson_Zyczkowski . From Eqs.(8), (12) it follows that

[TABLE]

The quantum density-operator (DO) distance between the NP and SM neutrino states based on $S(\varrho^{\mu}_{SM}\|\varrho^{\mu}_{NP})$ is given by $D_{DO}(\varrho^{\mu}_{SM},\varrho^{\mu}_{NP}):=2\,\sqrt{S(\varrho^{\mu}_{SM}\|\varrho^{\mu}_{NP})}$. Applying the above results and Eq.(11), we obtain

[TABLE]

Thus, the quantum DO metric $g^{DO}_{\varepsilon_{R}\varepsilon_{R}}$ (in the square of the line element $dD_{DO}^{2}(\varrho^{\mu}_{SM},$ $\varrho^{\mu}_{NP})=g^{DO}_{\varepsilon_{R}\varepsilon_{R}}d\varepsilon_{R}^{2}$), fulfils the inequality $g^{DO}_{\varepsilon_{R}\varepsilon_{R}}(\varrho^{\mu}_{SM},\varrho^{\mu}_{NP})\geq g_{\varepsilon_{R}\varepsilon_{R}}^{\rm c}$ and therefore Braunstein-Caves ; Majtey-Lamberti-Prato :

[TABLE]

From Eq.(20) we see that as the classical lower bound $\sigma^{2}(\tilde{\varepsilon}_{R}^{\rm c})$ on $\sigma^{2}(\hat{\varepsilon}_{R})$ is bigger than the quantum lower bound $\sigma^{2}(\tilde{\varepsilon}_{R})$, hence the quantum estimation is more effective. Yet, since $\sigma^{2}(\tilde{\varepsilon}_{R}^{\rm c})$ is calculated from Eq.(14) with the eigenvalues $p^{j}_{NP}$, Eq.(8), thus, unlike $\sigma^{2}(\tilde{\varepsilon}_{R})$, the classical bound depends neither on the relativistic neutrino energy nor on the baseline of the experiment. Finally, let us note that, from a practical point of view, it appears that if the classical lower bound is experimentally satisfactory (see text below) then the quantum one, though not designated, is even more powerful.

3 Conclusions

Two model characteristics were evaluated in this paper: (i) the model selection one for the sensitivity of the NP-SM discrimination with the change of the NP right-chiral $CC$ coupling constant $\varepsilon_{R}$ based on the upper bound $P_{E}(p_{SM},p_{NP})$ of the erroneous identification of the model probability distribution $P_{E}(\varrho_{SM},\varrho_{NP})$ and (ii) the one of the stability of the NP model estimator $\hat{\varepsilon}_{R}$ with the change of $\varepsilon_{R}$ based on classical Fisher-Rao metric. The decay $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ of the high energy pion followed by the relativistic neutrino unitary propagation constituted the background for the considerations.

Due to the unitarity of the evolution of the density matrix $\varrho_{NP}$, its eigenvalues $p^{j}_{NP}$ (Eq.(8)) depend neither on the relativistic neutrino energy nor on the baseline of the experiment. From this point of view, all $\nu_{\mu}\rightarrow\nu_{\mu}$ survival experiments form one general class. With these $p^{j}_{NP}(\varepsilon_{R})$, the classical lower bound $\sigma^{2}(\tilde{\varepsilon}_{R}^{c})$ (Eq.(16)) on the variance of $\hat{\varepsilon}_{R}$ (which is bigger than the quantum lower bound $\sigma^{2}(\tilde{\varepsilon}_{R})$, Eq.(20)) was calculated. Thus, with Eq.(16) the analysis of the robustness of the estimation of $\varepsilon_{R}$ (see text below Eq.(20)) can be performed globally, i.e., for all production-oscillation (PO) experiments taken jointly. To summarize, $N$ in Eq.(16) can be taken as the total size of all samples obtained in all survival experiments.

There exists the upper bound (not of the oscillation experiments origin) on the ratio ${\cal R}\equiv|\varepsilon_{R}/\varepsilon_{L}|<0.045$. It follows from the analysis of $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ decay experiments, in which the polarization of the emitted muon was measured PDG_epsR_2 ; PDG_epsR_3 . Eq.(16) for the PO experiments shows that for $|\varepsilon_{R}|\approx 0.045$ to diminish the standard error $\sigma(\hat{\varepsilon}_{R})\geq\sigma(\tilde{\varepsilon}_{R}^{\rm c})$ below $|\varepsilon_{R}|$ value, i.e., for the robust $\varepsilon_{R}$ estimation, $N\gtrsim 125$ is required. Then also, for $N\approx 125$, $P_{E}(p_{SM},p_{NP})\approx 0.78$, and the probability of the erroneous identification of the NP model with SM would be high. Yet, at the end of 2017 the number $N$ of survival $\nu_{\mu}\rightarrow\nu_{\mu}$ events in all $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ experiments (which form a combination of T2K, N0vA and mainly MINOS observations) was already about 6500 dane-o-survival-1 ; dane-o-survival-2 ; dane-o-survival-3 ; dane-o-survival-4 . For $N=6500$ we obtain $\sigma(\tilde{\varepsilon}_{R}^{\rm c})\approx 0.006$ and simultaneously the probability $P_{E}(p_{SM},p_{NP})\approx 2.32\times 10^{-6}$ is significantly small, leading to good NP-SM discrimination.

In conclusion, if ${\cal R}$ is only slightly smaller than 0.045, then both the significant result for the NP-SM discrimination and robust estimation of the right chiral $CC$ interaction parameter $\varepsilon_{R}$ in neutrino PO experiments have been already reached. It is anticipated that 2026 will be the first year of the beam operations in the DUNE experiment DUNE , which is to result in the observation of more than 7900 $\nu_{\mu}$ survival events over 3.5 years. Therefore, in ten years we will obtain $N=14600$ survival events, and even for ${\cal R}=0.02$ the conventional value $P_{E}(p_{SM},p_{NP})$ $<0.003$ will be reached, suggesting, if not yet observed, the nonexistence of the right chiral $CC$ neutrino interactions. Indeed, on the condition that the NP model (hypothesis $H_{1}$) is true and due to Eqs.(9) and (13) it follows: The probability $\beta$ (of erroneous recognition of the number $N$ of survival $\nu_{\mu}\!\rightarrow\!\nu_{\mu}$ events as being predicted by the SM transition rate formula ZZS (hypothesis $H_{0}$)) is not bigger than $P_{E}(p_{SM},p_{NP})$. Therefore, even for $N\!=\!14600$ the selection efficiency $1-\beta$ for the NP discovery will be close to $1-P_{E}(\varrho^{\mu}_{SM},\varrho^{\mu}_{NP})$ $\geq 1-0.003=0.997$. This would mean (unless the NP right chiral $CC$ neutrino interactions are noticed) the NP nonexistence, or at least point to the oddly small value of $\varepsilon_{R}$.

Acknowledgements.

This work has been supported by L.J.Ch..

It has been also supported by the Institute of Physics, University of Silesia.

Appendix: The

density matrix at the detection point

The effective Hamiltonian ${\cal{H}}$, Eq.(5), and neutrino density matrix $\rho^{{\rm P}\mu}$, Eq.(3), have the $6\times 6$-dimensional matrix representations. The diagonalisation of ${\cal{H}}\;$ AZS ; Kim_Pevsner ; Bekman-2002 gives ${\cal{H}}=\frac{1}{2E_{\nu}}W\,{\rm diag}(\tilde{m}_{l}^{2})\,W^{\dagger}$, where $W$ is the diagonalising unitary matrix defined by the eigenvectors of $2E_{\nu}{\cal{H}}$, and the corresponding real eigenvalues $\tilde{m}_{l}^{2}$, $l=1,...,6$, are the neutrino effective squared masses. $E_{\nu}$ is the neutrino energy, neglecting the mass contribution. $W\equiv(W_{i\,\lambda;\,l})\equiv(\,{}_{\lambda}\langle\nu_{i}|l\rangle\,)$ defines the transformation from the helicity-mass basis $|\nu_{i}\rangle_{\lambda}$ $\equiv|{\rm p},\lambda,i\rangle$ to the eigenvector basis $|l\rangle$ of ${\cal{H}}$. For the relativistic neutrino $\nu_{\mu}$ and in the non-dissipative homogeneous medium, from Eq.(4) it follows that in $|\nu_{i}\rangle_{\lambda}$ basis the density matrix at the point $z={\rm L}$ of $\nu_{\mu}$ detection is ZZS :

[TABLE]

where $T$ is the time between neutrino production and detection, $\Delta E_{ll^{\prime}}\equiv E_{l}-E_{l^{\prime}}=\frac{\Delta\tilde{m}_{ll^{\prime}}^{2}}{2E_{\nu}}=\frac{\tilde{m}_{l}^{2}-\tilde{m}_{l^{\prime}}^{2}}{2E_{\nu}}$. The equality $\varrho_{\rm{\bf L}}^{\mu}(\vec{{\rm p}}_{\rm{\bf L}})=\varrho^{\mu}(\vec{{\rm p}})$ of the density matrices in the L frame and CM frame is assumed OSZ . Because of the $W$ matrix unitarity, the L frame neutrino density matrix at the detection point is normalized, i.e., ${\rm Tr}[\varrho^{\mu}(t=T)]=1$. Eq.(21) is valid in the so-called light-ray approximation $T={\rm L}$ Giunti-Kim . The deviation of $T$ from the relation $T={\rm L}$ is experimentally significant if some corrections $\varepsilon_{ll^{\prime}}$ Giunti-Kim 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 $\Delta\phi_{ll^{\prime}}\gg 1$ Giunti-Kim . 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 Giunti-Kim , validating the light-ray approximation.

Finally, using $\varrho^{\mu}(t=T)$, Eq.(21), one can also calculate, e.g., the geometric phase of the $\mu$ flavour neutrino state Dziekuje-za-neutrino-faza or the cross section $\sigma_{\mu\rightarrow\beta}$ for the detection of the $\beta$ flavour neutrino in the L frame ZZS ; AZS-conf .