CP violation and circular polarisation in neutrino radiative decay (1910.08558v2)

Abstract: The radiative decay of neutral fermions has been studied for decades but $CP$ violation induced within such a paradigm has evaded attention. $CP$ violation in these processes can produce an asymmetry between circularly polarised directions of the radiated photons and produces an important source of net circular polarisation in particle and astroparticle physics observables. The results presented in this work outlines the general connection between $CP$ violation and circular polarisation for both Dirac and Majorana fermions and can be used for any class of models that produce such radiative decays. The total $CP$ violation is calculated based on a widely studied Yukawa interaction considered in both active and sterile neutrino radiative decay scenarios as well as searches for dark matter via direct detection and collider signatures. Finally, the phenomenological implications of the formalism on keV sterile neutrino decay, leptogenesis-induced right-handed neutrino radiative decay and IceCube-driven heavy dark matter decay are discussed.

• The paper establishes a formalism linking CP violation in neutrino radiative decays to the generation of circularly polarized photons.
• It employs interference between Standard Model and New Physics loop diagrams, highlighting the role of complex phases and on-shell intermediate states.
• The study explores applications to keV sterile neutrino dark matter, seesaw neutrinos, and PeV-scale decaying dark matter with astrophysical detection prospects.

This paper investigates CP violation in the radiative decay of neutral fermions, specifically neutrinos ($\nu_i \to \nu_j + \gamma$), and establishes a direct connection between this CP violation and the generation of circularly polarised photons in the final state (1910.08558). This provides a potential probe for CP violation in the neutrino and dark matter sectors through astrophysical observations of photon polarization.

General Framework

The core idea relies on parameterizing the decay amplitude using electromagnetic transition dipole moments. For a Dirac neutrino decay $\nu_i \to \nu_j + \gamma$, the amplitude involves form factors $f^L_{ij}$ and $f^R_{ij}$ associated with left-handed and right-handed final photons ($\gamma_+$ and $\gamma_-$ respectively, in the rest frame of $\nu_i$).

• The matrix element is given by:

  $$i \mathcal{M} (\nu_i \to \nu_j + \gamma_{\pm}) = i \bar{u}(p_j) \Gamma_{ij}^\mu(q^2) u(p_i) \varepsilon^*_{\pm,\mu}(q)$$

• For on-shell photons ($q^2=0$) and neutral neutrinos, the relevant part of the vertex function $\Gamma_{ij}^\mu$ is:

  $$\Gamma^\mu_{ij} (0) = i \sigma^{\mu \nu} q_\nu [f^L_{ij} P_L + f^R_{ij} P_R]$$

  where $f^{L,R}_{ij} = - f^M_{ij} \pm i f^E_{ij}$ are combinations of magnetic ($f^M$) and electric ($f^E$) dipole form factors, and $P_{L,R} = (1 \mp \gamma_5)/2$.

• The amplitudes for producing right-handed ($\gamma_+$) and left-handed ($\gamma_-$) photons are proportional to $f^L_{ij}$ and $f^R_{ij}$, respectively:

  $$\mathcal{M}(\nu_i \to \nu_j + \gamma_+) \propto f^L_{ij}$$

  $$\mathcal{M}(\nu_i \to \nu_j + \gamma_-) \propto f^R_{ij}$$

• For the CP conjugate process (antineutrino decay $\bar{\nu}_i \to \bar{\nu}_j + \gamma$), the amplitudes are proportional to $f^L_{ji}$ and $f^R_{ji}$. CPT invariance implies $\bar{f}^{L,R}_{ij} = -f^{L,R}_{ij}$.

CP asymmetries are defined by comparing decay rates:

• $$\Delta_{CP,+} = \frac{\Gamma(\nu_i \to \nu_j \gamma_+) - \Gamma(\bar{\nu}_i \to \bar{\nu}_j \gamma_-)}{\text{Total Rate}} \propto |f^L_{ij}|^2 - |f^R_{ji}|^2$$
• $$\Delta_{CP,-} = \frac{\Gamma(\nu_i \to \nu_j \gamma_-) - \Gamma(\bar{\nu}_i \to \bar{\nu}_j \gamma_+)}{\text{Total Rate}} \propto |f^R_{ij}|^2 - |f^L_{ji}|^2$$

The net circular polarization (asymmetry between $\gamma_+$ and $\gamma_-$ production, assuming equal initial numbers of $\nu$ and $\bar{\nu}$) is:

• $$\Delta_{+-} = \frac{\Gamma_{tot}(\gamma_+) - \Gamma_{tot}(\gamma_-)}{\text{Total Rate}} \propto (|f^L_{ij}|^2 + |f^L_{ji}|^2) - (|f^R_{ij}|^2 + |f^R_{ji}|^2)$$

Crucially, these are related:

$\Delta_{+-} = \Delta_{CP,+} - \Delta_{CP,-}$

This shows that a net circular polarization $\Delta_{+-}$ requires CP violation ($\Delta_{CP,+} \neq \Delta_{CP,-}$). Non-zero CP violation arises from the interference of different loop contributions ($l, l'$) where both the coupling combinations ($C_l C_{l'}^*$) and the kinematic loop factors ($K^L_l K^{L*}_{l'}$ or $K^R_l K^{R*}_{l'}$) have imaginary parts. The imaginary parts of loop factors typically arise when intermediate particles can go on-shell, as dictated by the optical theorem.

For Majorana neutrinos, the particle is its own antiparticle. The asymmetry becomes:

• $$\Delta_{+-}^M = \frac{\Gamma^M(\nu_i \to \nu_j \gamma_+) - \Gamma^M(\nu_i \to \nu_j \gamma_-)}{\Gamma^M(\nu_i \to \nu_j \gamma)} \propto |f^L_{ij} - f^L_{ji}|^2 - |f^R_{ij} - f^R_{ji}|^2$$ In this case, $\Delta_{CP,+}^M = -\Delta_{CP,-}^M = \Delta_{+-}^M$.

Calculating CP Violation: Example Model

The paper calculates CP violation in the decay $\nu_s \to \nu_i + \gamma$ (sterile to active neutrino) using a simplified model adding a charged fermion $\psi$ and a charged scalar $\phi$ with Yukawa couplings $\lambda_m$ to neutrinos $\nu_m$:

$$-\mathcal{L}_{\rm NP} \supset \sum_{m=i,s} \lambda_{m} \bar{\psi} \phi^* P_L \nu_m + h.c.$$

• SM Contribution: The one-loop SM contribution (via W boson and charged leptons) does not generate CP violation for $m_s < m_W$ because the loop functions are real.
• NP Contribution: Two one-loop diagrams involving $\psi$ and $\phi$ contribute.
• Interference: CP violation arises primarily from the interference between the SM loop and the NP loops. An imaginary part in the NP loop functions ($K^{L,R}_{NP}$) is required. This occurs if the intermediate $\psi$ and $\phi$ can be produced on-shell, i.e., if $m_s > m_\psi + m_\phi$.
• Calculation: Assuming $m_i \ll m_s$, the $f^L$ terms are suppressed. The calculation yields:

  $$|f^R_{is}|^2 - |f^L_{si}|^2 \propto \sum_{\alpha=e,\mu,\tau} \text{Im}(C_{\alpha,SM} C^*_{NP}) K^R_{\alpha,SM} \text{Im}(K^R_{NP})$$

  where $C_{\alpha,SM} \propto U_{\alpha i}^* U_{\alpha s}$ (from SM loop) and $C_{NP} \propto \lambda_s \lambda_i^*$ (from NP loop). $K^R_{NP} = K^R_{1,NP} - K^R_{2,NP}$.

• The imaginary part $\text{Im}(K^R_{NP})$ depends on $m_s, m_\phi, m_\psi$ and is calculated explicitly. It involves logarithmic functions and is non-zero only when $m_s > m_\phi + m_\psi$.
• The resulting asymmetries for Dirac neutrinos (in the $m_i \ll m_s$ limit) are:

  $$\Delta_{CP,-} \approx -\Delta_{+-} \propto \frac{\sum_\alpha \text{Im}(U_{\alpha s} U_{\alpha i}^* \lambda_i \lambda_s^*) F_\alpha I_{\phi\psi}}{|f^L_{is}|^2 + |f^R_{is}|^2}$$

  where $F_\alpha$ is the SM loop function and $I_{\phi\psi}$ encapsulates the NP loop kinematics giving the imaginary part. A similar result holds for $\Delta_{+-}^M$ in the Majorana case.

Phenomenological Applications

1. keV Sterile Neutrino DM: If $\nu_s$ is keV DM, the NP particles $\phi, \psi$ must be light. To avoid rapid DM decay via $\nu_s \to \psi \phi$, the coupling $|\lambda_s|$ must be tiny. $\phi, \psi$ might need to be millicharged to evade QED constraints. The CP asymmetry $\Delta_{+-}$ is suppressed by the millicharge $Q$ and potentially small couplings but could be $\sim 10^{-5}$ for $Q \sim 10^{-4}e$ and moderate parameters.
2. Seesaw & Leptogenesis: Applying the framework to heavy Majorana neutrinos $N_I$ in Type-I seesaw ($N_I \to N_J \gamma$). Here, the loops involve SM leptons ($\psi \equiv \ell_\alpha$) and Higgs/W bosons ($\phi \equiv H^+/W^+$). The necessary complex phases come from the standard neutrino Yukawa couplings $\lambda_{\alpha I}$. CP violation occurs via interference between diagrams with different leptons ($\alpha, \beta$) in the loop. The asymmetry $\Delta_{+-}^M$ is non-zero if $M_I > M_J + m_W + m_\ell$. However, it's suppressed by $(m_\tau / M_I)^2$ compared to standard leptogenesis asymmetries and likely washed out in the early Universe. Low-scale seesaw models might offer better prospects but require higher-loop calculations.
3. Heavy DM & IceCube: PeV-scale decaying DM ($N_{DM} \to \nu \gamma$) could explain IceCube signals. The radiative decay can exhibit CP violation if couplings are complex, but the asymmetry is again suppressed by $(m_\tau / M_{DM})^2 \lesssim 10^{-6}$, making observation extremely difficult.

Conclusion

The paper provides a general formalism linking CP violation in neutral fermion radiative decays to the circular polarization of the emitted photons. It demonstrates how to calculate this effect within a specific NP model featuring Yukawa interactions, highlighting the requirement for interfering amplitudes with complex phases and on-shell intermediate states ($$m_{decaying} > m_{intermediate1} + m_{intermediate2}$$) for one-loop CP violation. While potentially significant for keV DM searches via X-ray polarization, the effect appears heavily suppressed for typical high-scale seesaw or PeV DM scenarios.