Leptonic Unitarity Triangles Research Articles

A geometrical look at neutrino oscillation probabilities with non-standard interactions

In this talk, we discuss an alternative description of neutrino flavor oscillations using leptonic unitarity triangles (LUT) which allows for geometric visualization of neutrino oscillations. After a brief review of the LUT description of neutrino oscillations for propagation in vacuum and in matter with standard interactions, we discuss how we can extend this approach to describe neutrino oscillations in presence of new physics. We consider non-standard neutrino interactions (NSI) as the toy model of new physics. The content presented in this talk is based on Ref. [1].

Non-standard neutrino oscillations: perspective from unitarity triangles

We formulate an alternative approach based on unitarity triangles to describe neutrino oscillations in presence of non-standard interactions (NSI). Using perturbation theory, we derive the expression for the oscillation probability in case of NSI and cast it in terms of the three independent parameters of the leptonic unitarity triangle (LUT). The form invariance of the probability expression (even in presence of new physics scenario as long as the mixing matrix is unitary) facilitates a neat geometric view of neutrino oscillations in terms of LUT. We examine the regime of validity of perturbative expansions in the NSI case and make comparisons with approximate expressions existing in literature. We uncover some interesting dependencies on NSI terms while studying the evolution of LUT parameters and the Jarlskog invariant. Interestingly, the geometric approach based on LUT allows us to express the oscillation probabilities for a given pair of neutrino flavours in terms of only three (and not four) degrees of freedom which are related to the geometric properties (sides and angles) of the triangle. Moreover, the LUT parameters are invariant under rephasing transformations and independent of the parameterization adopted.

Leptonic unitarity triangles

We present a comprehensive analysis of leptonic unitarity triangles, using both current neutrino oscillation data and projections of next-generation oscillation measurements. Future experiments, sensitive to the degree of CP violation in the lepton sector, will enable the construction of precise triangles. We show how unitarity violation could manifest in the triangles and discuss how they serve as unitarity tests. We also propose the use of Jarlskog factors as a complementary means of probing unitarity. This analysis highlights the importance of testing the unitarity of the leptonic mixing matrix, an understanding of which is crucial for deciphering the nature of the neutrino sector.

Leptonic unitarity triangles: RGE running effects and μ−τ reflection symmetry breaking

There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale $\Lambda_{\rm H}^{}$ to the electroweak scale $\Lambda_{\rm EW}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the description of the RGE-induced $\mu$-$\tau$ reflection symmetry breaking analytically and numerically.

Connecting the leptonic unitarity triangle to neutrino oscillation with CP violation in the vacuum and in matter

Leptonic unitarity triangle (LUT) provides fundamental means to geometrically describe CP violation in neutrino oscillation. In this work, we use LUT to present a new geometrical interpretation of the vacuum oscillation probability, and derive a compact new oscillation formula in terms of only 3 independent parameters of the corresponding LUT. Then, we systematically study matter effects in the geometrical formulation of neutrino oscillation with CP violation. Including nontrivial matter effects, we derive a very compact new oscillation formula by using the LUT formulation. We further demonstrate that this geometrical formula holds well for applications to neutrino oscillations in matter, including the long baseline experiments T2K, MINOS, NOvA, and DUNE.

Leptonic unitarity triangles and effective mass triangles of the Majorana neutrinos

Given the best-fit results of six neutrino oscillation parameters, we plot the Dirac and Majorana unitarity triangles (UTs) of the 3×3 lepton flavor mixing matrix to show their real shapes in the complex plane. The connections of the three Majorana UTs with neutrino–antineutrino oscillations and neutrino decays are explored, and the possibilities of right or isosceles UTs are discussed. In the neutrino mass limit of m1→0 or m3→0, which is definitely allowed by current data, we show how the six triangles formed by the effective Majorana neutrino masses 〈m〉αβ (for α,β=e,μ,τ) and their corresponding component vectors look like in the complex plane. The relations of such triangles to the Majorana phases and to the lepton-number-violating decays H++→α+β+ in the type-II seesaw mechanism are also illustrated.

Constraining astrophysical neutrino flavor composition from leptonic unitarity

The recent IceCube observation of ultra-high-energy astrophysical neutrinos has begun the era of neutrino astronomy. In this work, using the unitarity of leptonic mixing matrix, we derive nontrivial unitarity constraints on the flavor composition of astrophysical neutrinos detected by IceCube. Applying leptonic unitarity triangles, we deduce these unitarity bounds from geometrical conditions, such as triangular inequalities. These new bounds generally hold for three flavor neutrinos, and are independent of any experimental input or the pattern of lepton mixing. We apply our unitarity bounds to derive general constraints on the flavor compositions for three types of astrophysical neutrino sources (and their general mixture), and compare them with the IceCube measurements. Furthermore, we prove that for any sources without ντ neutrinos, a detected νμ flux ratio < 1/4 will require the initial flavor composition with more νe neutrinos than νμ neutrinos.

Updated fit to three neutrino mixing: status of leptonic CP violation

We present a global analysis of solar, atmospheric, reactor and accelerator neutrino data in the framework of three-neutrino oscillations based on data available in summer 2014. We provide the allowed ranges of the six oscillation parameters and show that their determination is stable with respect to uncertainties related to reactor neutrino and solar neutrino flux predictions. We find that the maximal possible value of the Jarlskog invariant in the lepton sector is $0.0329 \pm 0.0009$ ($\pm 0.0027$) at the $1\sigma$ ($3\sigma$) level and we use leptonic unitarity triangles to illustrate the ability of global oscillation data to obtain information on CP violation. We discuss "tendencies and tensions" of the global fit related to the octant of $\theta_{23}$ as well as the CP violating phase $\delta_\mathrm{CP}$. The favored values of $\delta_\mathrm{CP}$ are around $3\pi/2$ while values around $\pi/2$ are disfavored at about $\Delta\chi^2 \simeq 6$. We comment on the non-trivial task to assign a confidence level to this $\Delta\chi^2$ value by performing a Monte Carlo study of T2K data.

Connecting leptonic unitarity triangle to neutrino oscillation

The leptonic unitarity triangle (LUT) provides a geometric description of $CP$ violations in the lepton-neutrino sector and is directly measurable in principle. In this paper, we reveal that the angles in the LUT have definite physical meaning, and demonstrate the exact connection of the LUT to neutrino oscillations. For the first time, we prove that these leptonic angles act as phase shifts in neutrino oscillations, by shifting $\mathrm{\ensuremath{\Delta}}{m}^{2}L/2E$ to $\mathrm{\ensuremath{\Delta}}{m}^{2}L/2E+\ensuremath{\alpha}$, where $(L,E,\ensuremath{\alpha})$ denote the baseline length, neutrino energy and corresponding angle of the LUT. Each LUT has three independent parameters and contains only partial information of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. We demonstrate that the partial information in each LUT can describe the corresponding neutrino oscillation. Hence, for the first time, we uncover that any given kind of neutrino oscillations contains at most three (rather than four) independent degrees of freedom from the PMNS matrix, and this may provide a cleaner way for fitting the corresponding oscillation data.

Dirac leptonic angle matrix versus Majorana leptonic angle matrix and their renormalization group running behaviors

Enlightened by the idea of the 3 times 3 CKM angle matrix proposed recently by Harrison et al., we introduce the Dirac angle matrix Phi and the Majorana angle matrix Psi in the lepton sector for Dirac and Majorana neutrinos respectively. We show that in presence of the CP violation, the angle matrix Phi or Psi is entirely equivalent to the complex MNS matrix V itself, but has the advantage of being real, phase rephasing invariant, directly associated to the leptonic unitarity triangles (UTs) and do not depend on any particular parametrization of V. In this paper, we further analyzed how the angle matrices evolve with the energy scale. The one-loop Renormalization Group Equations (RGEs) of Phi, Psi and some other rephasing invariant parameters are derived and the numerical analysis is performed to compare between the case of Dirac and Majorana neutrinos. Different neutrino mass spectra are taken into account in our calculation. We find that apparently different from the case of Dirac neutrinos, for Majorana neutrinos the RG-evolutions of Phi, Psi and the Jarlskog strongly depend on the Majorana-type CP-violating parameters and are quite sensitive to the sign of Delta m^{2}_{31}. They may receive significant radiative corrections in the MSSM if three neutrino masses are nearly degenerate.

EXPLORING THE LIKELIHOOD OF CP VIOLATION IN NEUTRINO OSCILLATIONS

In view of the latest T2K and MINOS observations regarding the mixing angle s13, we have explored the possibility of the existence of CP violation in the leptonic sector. Using hints from the construction of the "db" unitarity triangle in the quark sector, we have made an attempt to construct the "ν1⋅ν3" leptonic unitarity triangle, suggesting a good possibility of having nonzero CP violation.

Constructing the leptonic unitarity triangle

Following the analogy of the ''db'' triangle in the quark-mixing case, we have explored the construction of the ''{nu}{sub 2}.{nu}{sub 3}'' leptonic unitarity triangle using the Maki-Nakagawa-Sakata matrix obtained by Bjorken et al. through generalization of the tribimaximal scenario. In particular, for the U{sub e3} range 0.05-0.15, the existence of the leptonic unitarity triangle indicates a fairly good possibility of having nonzero CP violation.

Cosmological family asymmetry andCPviolation

We discuss how the cosmological baryon asymmetry can be achieved by the lepton family asymmetries of heavy Majorana neutrino decays and how the lepton family asymmetries are related to $CP$ violation in neutrino oscillation, in the minimal seesaw model with two heavy Majorana neutrinos. We derive the most general formula for $CP$ violation in neutrino oscillation in terms of the heavy Majorana masses and Yukawa mass term. It is shown that the formula is very useful to classify several models in which $e$, $\ensuremath{\mu}$, and $\ensuremath{\tau}$ leptogenesis can be separately realized and to see how they are connected with low energy $CP$ violation. To make the models predictive, we take texture with two zeros in the Dirac matrix. In particular, we find some interesting cases in which $CP$ violation in neutrino oscillation can happen while lepton family asymmetries do not exist at all. On the contrary, we can find $e$-, $\ensuremath{\mu}$-, and $\ensuremath{\tau}$-leptogenesis scenarios in which the cosmological $CP$ violation and low energy $CP$ violation measurable via neutrino oscillations are very closely related to each other. By determining the allowed ranges of the parameters in the models, we predict the sizes of $CP$ violation in neutrino oscillation and $|{V}_{e3}^{\mathrm{MNS}}|$. Finally, the leptonic unitarity triangles are reconstructed.

Reconstruction of the neutrino mixing matrix and leptonic unitarity triangles from long-baseline neutrino oscillations

We derive a new set of sum rules between the neutrino mass and mixing parameters in vacuum and their effective counterparts in matter. The moduli of nine genuine lepton mixing matrix elements can then be calculated in terms of the matter-corrected ones, and the latter can directly be determined from a variety of long-baseline neutrino oscillations. We show that it is possible to reconstruct the leptonic unitarity triangles and CP violation in a similar parametrization-independent way.

Leptonic unitarity triangles in matter

We present a geometric description of lepton flavor mixing and CP violation in matter by using the language of leptonic unitarity triangles. The exact analytical relations for both sides and inner angles are established between every unitarity triangle in vacuum and its effective counterpart in matter. The typical shape evolution of six triangles with the terrestrial matter density is illustrated for a realistic long-baseline neutrino oscillation experiment.

Leptonic unitarity triangle andCPviolation

The area of the unitarity triangle is a measure of $\mathrm{CP}$ violation. We introduce the leptonic unitarity triangles and study their properties. We consider the possibility of reconstructing the unitarity triangle in future oscillation and nonoscillation experiments. A set of measurements is suggested which will, in principle, allow us to measure all sides of the triangle, and consequently to establish $\mathrm{CP}$ violation. For different values of the $\mathrm{CP}$-violating phase ${\ensuremath{\delta}}_{D},$ the required accuracy of measurements is estimated. The key elements of the method are the determination of $|{U}_{e3}|$ and studying the ${\ensuremath{\nu}}_{\ensuremath{\mu}}\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\mu}}$ survival probability in the oscillations driven by the solar mass splitting $\ensuremath{\Delta}{m}_{\mathrm{sun}}^{2}.$ We suggest additional astrophysical measurements which may help to reconstruct the triangle. The method of the unitarity triangle is complementary to the direct measurements of $\mathrm{CP}$ asymmetry. It requires mainly studies of the survival probabilities and processes where oscillations are averaged or the coherence of the state is lost.