Leptonic CP Violation

• Leptonic CP violation is the breakdown of charge-parity symmetry in the lepton sector, impacting neutrino oscillations and signaling physics beyond the Standard Model.
• It is investigated through neutrino oscillation experiments and neutrinoless double beta decay, which probe the Dirac and Majorana phases of the PMNS matrix.
• Theoretical frameworks like the seesaw mechanism link low-energy CP violations to leptogenesis, offering explanations for the cosmic matter-antimatter imbalance with collider and rare decay signals.

Leptonic CP violation characterizes the breakdown of the combined charge-conjugation (C) and parity (P) symmetry in interactions involving leptons, most notably in the context of neutrino masses and their mixing facilitated by a nontrivial Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix. The experimental observation of nonzero neutrino masses and flavor oscillations necessitates physics beyond the Standard Model (SM) and introduces a rich structure for leptonic CP-violating phenomena, both at low energy (oscillations, lepton-number violation) and at high scales (origin of the cosmic baryon asymmetry). Theoretical models, especially variants of the seesaw mechanism, provide the framework for relating observed CP-odd effects to fundamental flavor parameters.

1. Neutrino Masses, Mixing, and the PMNS Matrix

Nonzero neutrino masses, as inferred from oscillation experiments, imply that the SM must be extended with new mass terms. The lepton sector charged-current Lagrangian, after diagonalization of charged-lepton and neutrino mass matrices, yields the unitary PMNS mixing matrix $U$: $U = V K$ where $V$ is conventionally parametrized with three rotation angles $(\theta_{12}, \theta_{23}, \theta_{13})$ and a Dirac-type CP phase $\delta$, and

$$K = \operatorname{diag}(1, e^{i\alpha_1/2}, e^{i\alpha_2/2})$$ includes two Majorana phases $\alpha_{1,2}$. The explicit form of $V$ (using sines $s_{ij} \equiv \sin\theta_{ij}$, cosines $c_{ij} \equiv \cos\theta_{ij}$) is: $$V = \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \ -s_{12} c_{23}-c_{12} s_{23} s_{13} e^{i\delta} & c_{12} c_{23}-s_{12} s_{23} s_{13} e^{i\delta} & s_{23} c_{13} \ s_{12} s_{23}-c_{12} c_{23} s_{13} e^{i\delta} & -c_{12} s_{23}-s_{12} c_{23} s_{13} e^{i\delta} & c_{23} c_{13} \end{pmatrix}$$

The physical observables include the mixing angles and the Dirac phase $\delta$ (relevant for neutrino oscillations), while the Majorana phases $\alpha_{1,2}$ affect only lepton-number–violating processes.

2. Theoretical Framework: Seesaw Mechanisms and Sources of CP Violation

Because neutrinos are anomalously light, the seesaw mechanism provides a natural explanation for a suppressed scale of their masses. The three principal types are:

• Type I seesaw: SM extended by heavy singlet Majorana neutrinos; the mass matrix is

$m_\nu = -v^2 Y^\nu m_R^{-1} (Y^\nu)^T$

where $Y^\nu$ is the Dirac neutrino Yukawa coupling matrix, $m_R$ the heavy neutrino Majorana mass matrix, and $v$ the Higgs vacuum expectation value.

• Type II seesaw: Introduction of heavy scalar $SU(2)_L$-triplet(s) $\Delta$,

$m_\nu = \frac{\mu\,v^2}{M_\Delta} Y^\Delta$

with $Y^\Delta$ symmetric and $\mu$ (in general complex) controlling lepton number violation.

• Type III seesaw: Addition of heavy fermion triplets with SU(2) gauge interactions.

CP-violating phases generally originate from complex Yukawa or scalar couplings in these frameworks. These phases not only affect low-energy effective neutrino masses and mixing (leading to observable CP-violating effects in oscillations and neutrinoless double beta decay) but also modify the decay properties of the heavy seesaw mediators, making them relevant for leptogenesis.

3. Experimental Manifestations and Observables of Leptonic CP Violation

a) Neutrino Oscillations

CP violation is directly measurable in long-baseline oscillation experiments via asymmetry in transition probabilities: $$\Delta P_{\alpha\beta} = P(\nu_\alpha \to \nu_\beta) - P(\bar{\nu}_\alpha \to \bar{\nu}_\beta) = 4 \sum_{k>j} \mathcal{J}_{\alpha\beta}^{kj} \sin \Delta_{kj}$$ with

$$\mathcal{J}_{e\mu}^{21} = \frac{1}{8} \sin(2\theta_{12}) \sin(2\theta_{13}) \sin(2\theta_{23}) \sin \delta$$

and $\Delta_{kj} = \Delta m^2_{kj} L / (2E)$. The magnitude of the effect is controlled by the size of $\theta_{13}$ and $\sin\delta$. Matter effects introduce additional CP-odd signals and strategies such as using different baselines or "magic" configurations are necessary to isolate genuine leptonic CP violation.

b) Neutrinoless Double Beta Decay ($0\nu\beta\beta$)

If neutrinos are Majorana particles, $0\nu\beta\beta$ becomes allowed with a rate sensitive to

$$m_{ee} = | c_{13}^2 (m_1 c_{12}^2 + m_2 e^{-i\alpha_1} s_{12}^2) + m_3 e^{-i\alpha_2} s_{13}^2 |$$

which depends critically on the Majorana phases $\alpha_{1,2}$. Extraction of these phases is complicated by theoretical uncertainties in nuclear matrix elements but remains a central test of the Majorana nature and leptonic CP violation.

c) Collider and Charged-Lepton Flavor-Violating Signatures

Seesaw mediators with masses near the TeV scale (especially in type II) can be produced at high-energy colliders (e.g., the LHC), for instance via $pp \to \Delta^{++}$. The decay pattern

$$\Delta^{++} \to \ell_i^+ \ell_j^+,\qquad \mathrm{BR}_{\Delta ij} \propto |(m_\nu)_{ij}|^2$$

reflects both mixing angles and potentially CP-violating phases. Additionally, charged-lepton flavor violating decays $\mu \to e\gamma$, $\tau \to 3\ell$ may be enhanced if new physics related to the seesaw exists. Correlations among flavor structure and CP phases can manifest in decay distributions.

4. Leptogenesis: Connecting CP Violation to the Matter–Antimatter Asymmetry

Leptonic CP violation supplies a central ingredient for scenarios explaining the observed cosmic baryon asymmetry via leptogenesis. The Sakharov conditions are fulfilled as follows:

• Lepton number violation: Heavy Majorana neutrino decays violate lepton number.
• CP violation: Decay asymmetries require complex Yukawa couplings, computed as

$$\epsilon_i^\alpha = \frac{\Gamma(N_i \to \phi \ell_\alpha) - \Gamma(N_i \to \phi^\dagger \bar\ell_\alpha)} { \sum_\beta \left[\Gamma(N_i \to \phi \ell_\beta) + \Gamma(N_i \to \phi^\dagger \bar\ell_\beta)\right] }$$

with explicit form (summing over flavors)

$$\epsilon_i = \frac{1}{8\pi} \frac{1}{(Y^{\nu\dagger} Y^\nu)_{ii}} \sum_{j \neq i} \mathrm{Im}\left[ ((Y^{\nu\dagger}Y^\nu)_{ij})^2 \right] \left[ f\left(\frac{M_j^2}{M_i^2}\right) + g\left(\frac{M_j^2}{M_i^2}\right) \right]$$

• Departure from equilibrium: Satisfied naturally if the heavy neutrinos decay as the universe cools.

The resulting lepton asymmetry is partially converted to a baryon asymmetry by sphaleron transitions, with the observed ratio

$$\eta_B \equiv \frac{n_B - n_{\bar{B}}}{n_\gamma} \sim 6.2 \times 10^{-10}$$

"Resonant leptogenesis" allows enhancements for quasi-degenerate heavy states.

5. High–Low Energy CPV Connection and Model Dependence

A central problem is relating high-energy CP violation, responsible for leptogenesis, to low-energy phases accessible in oscillation or $0\nu\beta\beta$ experiments. In type I seesaw, the freedom encoded in the complex orthogonal matrix $R$ (Casas–Ibarra parametrization: $Y^\nu = v^{-1} U^* d_m^{1/2} R d_M^{1/2}$) means that generally, the high-scale phases cannot be reconstructed from low-energy data. Imposing further structure—texture zeros, flavor symmetries, or $R$ real—can create correlations, possibly connecting the sign or magnitude of the baryon asymmetry to the Dirac or Majorana phases in $U$.

CP-odd weak-basis invariants (e.g., $$\operatorname{Tr}([ m_u m_u^\dagger,\, m_d m_d^\dagger ]^3 )$$ for quarks; analogous invariants for leptons) provide a basis-independent diagnosis of CP violation and can be constructed for both low- and high-energy sectors.

6. Experimental Status and Future Prospects

With the measurement of $\sin^2(2\theta_{13}) \approx 0.09$ by T2K and Daya Bay, the leptonic sector has entered a phase where CP violation is accessible in oscillation experiments. Near-term and planned facilities (NO$\nu$A, T2K, proposed neutrino factories) will further constrain or measure $\delta$. $0\nu\beta\beta$ decay experiments probe the effective mass $m_{ee}$; detection would confirm the Majorana nature and the relevance of associated CP phases. Collider searches for type II seesaw partners and precision measurements of lepton flavor violation offer complementary sensitivity.

7. Conclusions and Outlook

Leptonic CP violation remains a major focus of research, with open questions on both its magnitude and role in the origin of cosmic matter. Precision measurement of $\theta_{13}$ heralds an era where the Dirac phase $\delta$ can be sought, and $0\nu\beta\beta$ searches may expose the Majorana phases. A direct connection between observable CPV at low energy and that required for leptogenesis is only guaranteed in highly constrained models. The detection of signals at high-energy colliders, rare decay processes, and non-standard effects in oscillation observables would further clarify the flavor and CP structure of the lepton sector, with implications for understanding baryogenesis and the symmetry structure of fundamental interactions.