Leptonic unitary triangles and boomerangs

We review the idea of leptonic unitary triangles and extend the concept of the recently proposed unitary boomerangs to the lepton sector. Using a convenient parametrization of the lepton mixing, we provide approximate expressions for the side lengths and the angles of the six different triangles and give examples of leptonic unitary boomerangs. Possible applications of the leptonic unitary boomerangs are also briefly discussed.

Figure 1

The $\underline{\mu \tau }$ triangle with best-fit values for the mixing parameters (see Table ) and $\delta =\pi /2$. The angles are ${\alpha }_{\mu \tau }\simeq 19.0°$, ${\beta }_{\mu \tau }\simeq 8.9°$, and ${\gamma }_{\mu \tau }\simeq 152.1°$, the side lengths $|U_{\mu 2}^{*}{U}_{\tau 2}|\simeq 0.34$, $|U_{\mu 1}^{*}{U}_{\tau 1}|\simeq 0.16$, and $|U_{\mu 3}^{*}{U}_{\tau 3}|\simeq 0.49$. The area is $A_{\underline{\mu \tau }}=1/2|U_{\mu 2}^{*}{U}_{\tau 2}||U_{\mu 3}^{*}{U}_{\tau 3}|\sin {\beta }_{\mu \tau }\simeq 0.013$.

Figure 2

The $\underline{12}$ triangle with best-fit values for the mixing parameters and $\delta =\pi /2$. The angles are ${\alpha }_{12}\simeq 13.9°$, ${\beta }_{12}\simeq 152.1°$, and ${\gamma }_{12}\simeq 13.9°$, the side lengths $|U_{\mu 1}^{*}{U}_{\mu 2}|\simeq 0.24$, $|U_{e1}^{*}{U}_{e2}|\simeq 0.46$, and $|U_{\tau 1}^{*}{U}_{\tau 2}|\simeq 0.24$. The area is $A_{\underline{12}}=1/2|U_{\mu 1}^{*}{U}_{\mu 2}||U_{e1}^{*}{U}_{e2}|\sin {\gamma }_{12}\simeq 0.013$. The $\underline{\mu \tau }$ triangle and the $\underline{12}$ triangle (Fig. 1) have one common angle: ${\gamma }_{\mu \tau }={\beta }_{12}$.

Figure 3

By overlapping the $\underline{\mu \tau }$ triangle from Fig. 1 and the $\underline{12}$ triangle from Fig. 2 at their common angle we get the unitary $\underline{\mu \tau }$-$\underline{12}$ boomerang.