In the $\textit{ν}$SM extended by adding an eV-scale sterile state, the (3 + 1) model, the sterile-active level crossing entails the MSW resonance, here referred as the sterile-active (SA) resonance. In this paper, we construct an effective theory of SA resonance which involves only the sterile-active mixing angles and ∆ $$ {m}_{41}^2 $$ , thanks to the given environment of high matter potential which freezes the $\textit{ν}$SM oscillations. We give our first attempt at an analytic treatment of the effective theory to illuminate the global picture of the SA resonance at a glance. We formulate a perturbative framework in which the structure of “texture zeros” of the $\textit{S}$ matrix in the flavor space and the suppression by the small parameters sin $\textit{θ}$ ($\textit{j}$ = 1$\textit{,}$ 2$\textit{,}$ 3) allows us to reveal the flavor–event-type hierarchy of the resonance-effect strength in the probabilities. We have shown that the cascade events dominantly comes from the three paths through $\textit{P}$($\textit{ν}$ → $\textit{ν}$ ), $\textit{P}$( $$ {\overline{\nu}}_e $$ → $$ {\overline{\nu}}_e $$ ), and $\textit{P}$( $$ {\overline{\nu}}_{\mu } $$ → $$ {\overline{\nu}}_{\tau } $$ ), and a three-component fit is suggested to disentangle the SA resonance generation mechanisms.

United Kingdom

Mark Brettell

Ivan Martinez-Soler

United States

Hisakazu Minakata

Sterile-active resonance: A global qualitative picture

The S matrix rephasing invariance is one of the fundamental principles of quantum mechanics that originates in its probabilistic interpretation. For a given S matrix which describes neutrino oscillation, one can define the two different rephased amplitudes   <math> <msubsup> <mi>S</mi> <mi>αβ</mi> <mrow> <mtext>Reph</mtext> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>≡</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mfenced> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>/</mo> <mn>2</mn> <mi>E</mi> </mrow> </mfenced> <mi>x</mi> </mrow> </msup> <msub> <mi>S</mi> <mi>αβ</mi> </msub> </math>  $$ {S}_{\alpha \beta}^{\textrm{Reph}-1}\equiv {e}^{i\left({\lambda}_1/2E\right)x}{S}_{\alpha \beta} $$  and   <math> <msubsup> <mi>S</mi> <mi>αβ</mi> <mrow> <mtext>Reph</mtext> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>≡</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mfenced> <mrow> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>/</mo> <mn>2</mn> <mi>E</mi> </mrow> </mfenced> <mi>x</mi> </mrow> </msup> <msub> <mi>S</mi> <mi>αβ</mi> </msub> </math>  $$ {S}_{\alpha \beta}^{\textrm{Reph}-2}\equiv {e}^{i\left({\lambda}_2/2E\right)x}{S}_{\alpha \beta} $$ , which are physically equivalent to each other, where λ  k  /2E denotes the energy eigenvalue of the k-th mass eigenstate. We point out that the transformation of the reparametrization (Rep) symmetry obtained with “Symmetry Finder” maps   <math> <msubsup> <mi>S</mi> <mi>αβ</mi> <mrow> <mtext>Reph</mtext> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math>  $$ {S}_{\alpha \beta}^{\textrm{Reph}-1} $$  to   <math> <msubsup> <mi>S</mi> <mi>αβ</mi> <mrow> <mtext>Reph</mtext> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </math>  $$ {S}_{\alpha \beta}^{\textrm{Reph}-2} $$ , and vice versa, providing a local and manifest realization of the S matrix rephasing invariance by the Rep symmetry of the 1–2 state exchange type. It is strongly indicative of quantum mechanical nature of the Rep symmetry. The rephasing and Rep symmetry relation, though its all-order treatment remains incomplete, is shown to imply absence of the pure 1–3 exchange symmetry in Denton et al. perturbation theory. It then triggers a study of convergence of perturbation series.

Neutrino amplitude decomposition, S matrix rephasing invariance, and reparametrization symmetry

Recently, we have developed a method called “Symmetry Finder” (SF) for hunting the reparametrization symmetry in the three-neutrino system in matter. Here, we apply SF to the Denton <span class="it">et al.</span> (DMP) perturbation theory extended by including unitarity violation (UV), a possible low-energy manifestation of physics beyond the $\nu $SM. Implementation of UV into the SF framework yields the additional two very different constraints, which nonetheless allow for remarkably consistent solutions, the eight DMP-UV symmetries. Treatment of one of the constraints, the genuine non-unitary part, leads to the key identity which entails the UV $\alpha $ parameter transformation only by rephasing, which innovates the invariance proof of the Hamiltonian. The quantum mechanical nature of the symmetry dictates the both $\nu $SM and UV variables to transform jointly, through which the response of the two sectors are related to reveal their interplay. Thus, the symmetry can serve as a tool for diagnostics, probing the interrelation between the $\nu $SM and a low-energy description of new physics. The problem of SF symmetry in vacuum is revisited to complete eight symmetries akin to DMPs.

Symmetry in Neutrino Oscillation in Matter with Non-unitarity

A new picture of “one-resonance–one-symmetry” has been proposed recently to reveal nature of the reparametrization symmetry in neutrino oscillation in matter and where it resides: Symmetry of $i \leftrightarrow j$ state-exchange-type exists at around a resonance, with $i$ and $j$ being the states which participate in the level crossing. Consistently, the 1–2 and 1–3 state-exchange symmetries are identified at around the solar and atmospheric resonances, respectively, in the locally-valid frameworks. On the other hand, the Denton <span class="it">et al.</span> (DMP) perturbation theory, a globally-valid framework, has the 1–2 exchange symmetry which is akin to the one in the aforementioned solar-resonance perturbation (SRP) theory. In our picture, the symmetry must be associated with the resonance, not the framework, and if so, these two 1–2 symmetries must be identical to each other. We conduct a comparative study of the 1–2 symmetries possessed by SRP and DMP to confirm their identity. An almost identity is verified, but in a highly nontrivial way.

Comparative Study of the 1–2 Exchange Symmetries in Neutrino Frameworks with Global and Local Validities

We argue that neutrino oscillations at JUNO offer a unique opportunity to study Sorkin’s triple path interference, which is predicted to be zero in canonical quantum mechanics by virtue of the Born rule. In particular, we compute the expected bounds on triple path interference at JUNO and demonstrate that they are comparable to those already available from electromagnetic probes. Furthermore, the neutrino probe of the Born rule is much more direct due to an intrinsic independence from any boundary conditions, whereas such dependence on boundary conditions is always present in the case of electromagnetic probes. Thus, neutrino oscillations present an ideal probe of this aspect of the foundations of quantum mechanics.

Patrick Huber

Djordje Minic

Rebekah Pestes

Tatsu Takeuchi

Neutrino oscillations at JUNO, the Born rule, and Sorkin’s triple path interference

Abstract We discuss correlations between the neutrino-mass-embedded Standard Model CP phase δ and the phases that originate from new physics which cause neutrino-sector unitarity violation (UV) at low energies. This study aims to provide one of the building blocks for machinery to diagnose non-unitarity, our ultimate goal. We extend the perturbation theory of neutrino oscillation in matter proposed by Denton et al. (DMP) to include the UV effect expressed by the α parametrization. By analyzing the DMP-UV perturbation theory to first order, we are able to draw a complete picture of the δ–UV phase correlations in the whole kinematical region covered by terrestrial neutrino experiments. Two regions exist with characteristically different patterns of the correlations: (i) the chiral-type  (Particle Data Group convention) correlation in the entire high-energy region  GeV, and (ii) (blobs of the α parameters)–e$^{±iδ}$ correlation anywhere else. Some relevant aspects for the measurement of the UV parameters, such as the necessity of determining all the α$_{βγ}$ elements at once, are also pointed out. Subject Index: B52, B54

Toward diagnosing neutrino non-unitarity through CP phase correlations

In a previous paper, Symmetry Finder (SF) method is proposed to find the reparametrization symmetry of the state-exchange type in neutrino oscillation in matter. It has been applied successfully to the 1–2 state exchange symmetry in the DMP perturbation theory, yielding the eight symmetries. In this paper, we apply the SF method to the atmospheric-resonance perturbation theory to uncover the 1–3 state relabeling symmetries. The pure 1–3 state symmetry takes the unique position that it is practically impossible to formulate in vacuum under the conventional choice of the flavor mixing matrix. In contrast, our SF method produces the sixteen 1–3 state exchange symmetries in matter. The relationship between the symmetries in the original (vacuum plus matter) Hamiltonian and the ones in the diagonalized system is discussed.

Symmetry Finder applied to the 1–3 mass eigenstate exchange symmetry

We propose a unitarity method for determining <math><mrow><mi>τ</mi></mrow></math> neutrino appearance probability <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>)</mo></mrow></math> in long-baseline (LBL) accelerator experiments and atmospheric neutrino observations. When simultaneous in situ measurements of <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>)</mo></mrow></math> and <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>)</mo></mrow></math> proceed, as is typical in the LBL experiments, one can use unitarity to “measure” <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>)</mo></mrow></math>. A theorists’ toy analysis for the model-independent determination of <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>)</mo></mrow></math> and <math><mrow><mi>P</mi><mo>(</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>)</mo></mrow></math> is presented by using the NOvA data. It is shown in our analysis that <math><mrow><mo>≲</mo><mn>5</mn><mo>%</mo></mrow></math> (8%) measurement of <math><mi>τ</mi></math> neutrino appearance probability in neutrino (antineutrino) mode is possible in the peak region <math><mrow><mn>1.5</mn><mo>≲</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>≲</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. The <math><mrow><mi>ν</mi><mi>SM</mi></mrow></math>-independent nature of determination of the probabilities is emphasized.

Puerto Rico

Measuring tau neutrino appearance probability via unitarity

Symmetry in neutrino oscillation serves for a better understanding of the physical properties of the phenomenon. We present a systematic way of finding symmetry in neutrino oscillation, which we call symmetry finder (SF). By extending the known framework in vacuum into a matter environment, we derive the SF equation, a powerful machinery for identifying symmetry in the system. After learning lessons on symmetry in the Zaglauer-Schwarzer system with matter equivalent to the vacuum symmetry, we apply the SF method to the [P. B. Denton et al., Compact perturbative expressions for neutrino oscillations in matter, J. High Energy Phys. 06 (2016) 051.] (DMP) perturbation theory to first order. We show that the method is so powerful that we uncover the eight reparametrization symmetries with the <math><mn>1</mn><mo>↔</mo><mn>2</mn></math> state exchange in DMP, denoted as IA, IB, <math><mo>…</mo></math>, IVB, all new except for IA. The transformations consist of both fundamental and dynamical variables, indicating their equal importance. It is also shown that all the symmetries discussed in this paper can be understood as the Hamiltonian symmetries, which ensures their all-order validity and applicability to varying density matter.

Symmetry finder: A method for hunting symmetry in neutrino oscillation

Observation of the interference between the atmospheric-scale and solar-scale oscillations is one of the challenging and tantalizing goals of the ongoing and upcoming neutrino experiments. An inevitable first step required for such analyses is to establish the way the oscillation <math><mi>S</mi></math> matrix can be decomposed into the atmospheric and solar waves, the procedure dubbed as the amplitude decomposition. In this paper, with the use of the perturbative framework proposed by Denton et al. (DMP), we establish the prescription for amplitude decomposition which covers the whole kinematical region of the terrestrial neutrino experiments. We analyze the limits to the atmospheric- and solar-resonance regions to argue that the dynamical two modes of the DMP decomposition can be interpreted as the matter-dressed atmospheric and solar oscillations. The expressions of the oscillation probability, which are decomposed into the noninterference and interference terms, are derived for all the relevant flavor oscillation channels. Through construction of the DMP decomposition, we reveal the nature of <math><mi>ψ</mi></math> (<math><msub><mi>θ</mi><mn>12</mn></msub></math> in matter) symmetry as due to the <math><mi>S</mi></math> matrix rephasing invariance. A new picture of the DMP perturbation theory emerged, a unified perturbative framework for neutrino oscillation in earth matter.