Leptonic $CP$ violation

Several topics on $CP$ violation in the lepton sector are reviewed. A few theoretical aspects concerning neutrino masses, leptonic mixing, and $CP$ violation will be covered, with special emphasis on seesaw models. A discussion is provided on observable effects which are manifest in the presence of $CP$ violation, particularly, in neutrino oscillations and neutrinoless double beta decay processes, and their possible implications in collider experiments such as the LHC. The role that leptonic $CP$ violation may have played in the generation of the baryon asymmetry of the Universe through the mechanism of leptogenesis is also discussed.

Figure 1

Majorana unitarity triangle $T_{12}$. The arrows indicate the orientation of the triangle, which is determined by the Majorana phases and cannot be rotated in the complex plane.

Figure 2

Canonical seesaw mechanisms for neutrino mass generation. The left diagram corresponds to type I and type III seesaw masses, mediated by the tree-level exchange of singlet (${\nu }_R$) and triplet ($\Sigma$) fermions, respectively. The right diagram leads to type II seesaw neutrino masses via the exchange of a triplet scalar $\Delta$.

Figure 3

Favored and excluded regions of neutrino mass-squared differences and mixing angles taken into account the data of several neutrino experiments. From 338.

Figure 4

Transformations of the different flavor-transition channels under $CP$, $T$, and $CPT$.

Figure 5

Top panel: An illustrative example of the eightfold degeneracy in terms of the bi-probability plot in $P_{\mu e}\mathrm{\text{−}}{\overline{P}}_{\mu e}$ space (319). Bottom panel: Values of (${sin}^{2}2{\theta }_{13}$, $\delta$) for the true solutions and the clone solutions II–VIII in ${sin}^{2}2{\theta }_{13}\mathrm{\text{−}}\delta$ space. The correspondence between the ellipses (top panel) and the solution labels are made manifest by using the same color lines and/or symbols in both panels. From 322.

Figure 6

Top: Time evolution of the sensitivity of ${\theta }_{13}$ at 90% C.L., defined as the limit which is obtained if the true value of ${\theta }_{13}$ is zero. Center and bottom: Discovery potential of ${\theta }_{13}$ at $3\sigma$ as a function of time (given as the smallest value of ${\theta }_{13}$, which can be distinguished from zero), for a NH (center) and an IH (bottom) neutrino mass spectrum. From 253.

Figure 7

SPL and T2HK CPV discovery potential. Inside the regions delimited by the solid (dashed) lines, $CP$ conservation can be excluded at the $3\sigma$ level, considering systematic errors of 2% (5%). From 130; 57.

Figure 8

$3\sigma$ sensitivity to CPV for the three $\beta$-beam configurations: $\mathrm{LE}\beta \beta$, $\mathrm{HE}\beta \beta$-a, and $\mathrm{HE}\beta \beta$-b (see text for more details). From 57.

Figure 9

Fits for ${\theta }_{13}$ and $\delta$ in an electron-capture experiment with a baseline $L=130\mathrm{km}$ (CERN-Fréjus) and a 440 kton water Čerenkov detector. Two 5 yr-running periods with $\gamma =195$ and $\gamma =90$ have been considered. From 77.

Figure 10

$5\sigma$ discovery reach of CPV (left), NMH (center), and ${\theta }_{13}$ (right) for several NF setups: “$50\mathrm{kt}+50\mathrm{kt}$” refers to a combination of two 50 kton MINDs at $L_1=4000\mathrm{km}$ and $L_2=7500\mathrm{km}$, “100 kt only” to a 100 kton MIND at $L=4000\mathrm{km}$, “$100\mathrm{kt}+50\mathrm{kt}$” to a 100 kton MIND at $L_1=4000\mathrm{km}$ and a 50 kton MIND at $L_2=7500\mathrm{km}$, and “IDS-NF 1.0” to the IDS-NF setup. From 12.

Figure 11

Dependence of $m_{ee}$ on the lightest neutrino mass $m_0$ for a normal (HI) and inverted (IH) neutrino mass spectrum. The yellow (light blue) region corresponds to $3\sigma$ intervals of the STV for a normal (inverted) neutrino mass spectrum. The Majorana phases ${\alpha }_{1,2}$ are varied in the interval $[0,2\pi ]$. The regions disfavored by kinematical searches and cosmology are delimited by the vertical shaded bands. The future Katrin and Planck satellite sensitivities are indicated by the vertical dashed line. The horizontal purple band refers to the $m_{ee}$ region disfavored by the $0\nu \beta \beta$ Heidelberg-Moscow experiment. In turn, the dash-dotted horizontal line at $m_{ee}=0.01\mathrm{eV}$ illustrates the sensitivity of future $0\nu \beta \beta$ experiments. The $m_{ee}$ allowed region delimited by the solid black lines is obtained when the best-fit values of the STV global neutrino data analysis are considered (see Table ).

Figure 12

Allowed regions for $R_{\tau \mu }$ (upper plots) and $R_{\tau e}$ (lower plots) defined in Eqs. (3.40, 3.42) as a function of $s_{13}$ and $\delta$, for both the NO (left plots) and the IO (right plots) neutrino mass spectra. In dark (light) shading we show the $3\sigma$ (best-fit) allowed regions obtained by varying the $CP$-violating phase $\delta$ in the interval $[0,2\pi ]$ and using the neutrino data displayed in Table . The solid, dashed, and dash-dotted lines delimits the $3\sigma$ region for $\delta =0$, $\delta =\pi$, and $\delta =\pi /4$, respectively. Adapted from 264.

Figure 13

Density plot of the ratio $R_{\tau \mu \mu \mu }$ defined in Eq. (3.41) as a function of the Majorana phases ${\alpha }_{1,2}$ for a QD neutrino mass spectrum with $m_0=0.1\mathrm{eV}$, $s_{13}=0.1$, and $\delta =\pi /2$.

Figure 14

Variation of ${\mathrm{BR}}_{\Delta ee}$ (top), ${\mathrm{BR}}_{\Delta \mu e}$ (center), and ${\mathrm{BR}}_{\Delta \mu \mu }$ (bottom) in the ${\alpha }_1–{\alpha }_2$ parameter space for a quasidegenerate neutrino mass spectrum with $m_0=0.1\mathrm{eV}$, $s_{13}=0.1$, and $\delta =\pi /2$. The remaining neutrino parameters are taken at the best-fit values of the STV analysis (see Table ).

Figure 15

(a) to (c) Scatter plots for $|A_{e\mu }|$, $|A_{\mu \tau }|$ and $|A_{e\tau }|$, respectively, as a function of the Dirac phase $\delta$. The neutrino parameters are fixed at $s_{13}=0.1$, $s_{12}^2=1/3$, ${\theta }_{23}=\pi /4$, $\Delta m_{21}^2=8\times {10}^{-5}{\mathrm{eV}}^2$, and $\Delta m_{31}^2=2.4\times {10}^{-3}{\mathrm{eV}}^2$. The results are presented considering several experimental setups, namely, $\mathrm{HE}\beta \mathrm{B}$ (high-energy beta beam with $E=1\mathrm{GeV}$ and $L=732\mathrm{km}$), the upgraded T2K, T2HK ($E=0.75\mathrm{GeV}$ and $L=295\mathrm{km}$), the CERN superbeam project SPL ($E=0.3\mathrm{GeV}$ and $L=130$ or 732 km), and neutrino factories ($\mathrm{NF}@\mathrm{L}$) with $E=35$ and 30 GeV in panels (b) and (c), respectively.  (d) $A_{e\tau }$ as a function of $A_{e\mu }$ considering a baseline $L=1500\mathrm{km}$ and $E=30\mathrm{GeV}$. The neutrino parameters are the same as in the previous panels. In all cases, the MUV parameters are varied in their allowed ranges and the solid lines correspond to the standard unitary limit. From 27.

Figure 16

Diagrams contributing to the $CP$ asymmetry ${ϵ}_i^{\alpha }$ in type I seesaw leptogenesis. The last diagram corresponds to the wave function corrections: the one with an internal ${\ell }_{\beta }$ is lepton flavor and lepton-number violating, while the one with an internal ${\overline{\ell }}_{\beta }$ is lepton flavor violating but lepton-number conserving, thus giving no contribution to the unflavored $CP$ asymmetry.

Figure 17

Tree-level diagrams for the scalar triplet decays and one-loop diagram contributing to the $CP$ asymmetry ${ϵ}_i^{\alpha \beta }$ in type II seesaw leptogenesis.

Figure 18

Diagrams contributing to the $CP$ asymmetry ${ϵ}_i^{\alpha }$ in type III seesaw leptogenesis. As in the type I seesaw case, the last diagram involves two graphs, one which is lepton flavor and lepton-number violating and another which is lepton flavor violating but does not give contribution to the unflavored $CP$ asymmetry.

Figure 19

The $CP$ asymmetries ${ϵ}_1$ and ${ϵ}_2$ as functions of $\sin {\theta }_{13}$ generated in a minimal radiative leptogenesis scenario. The curves correspond to the approximate analytic expressions given in Eqs. (4.61).