Neutrino mass from bremsstrahlung endpoint in coherent scattering on nuclei

We calculate the coherent bremsstrahlung process $\nu +\mathcal{N}\to \mathcal{N}+\nu +\gamma$ off a nucleus $\mathcal{N}$ with the aim of revealing the neutrino mass via the photon endpoint spectrum. Unfortunately, the large required power of a monochromatic neutrino source and/or large detector mass make it difficult to compete with traditional electron-spectrum endpoint measurements in nuclear $\beta$ decay. Our neutral-current process distinguishes between Dirac and Majorana neutrinos, but the change of the photon spectrum is of the order of $m_{\nu }/E_{\nu }$ and thus very small, despite the final-state neutrino coming to rest at the photon endpoint. So the “Dirac-Majorana confusion theorem” remains intact even if $E_{\nu }\gg m_{\nu }$ applies only for the initial state.

Figure 1

Feynman diagrams for coherent neutrino-nucleus scattering with bremsstrahlung. The neutrino is denoted $\nu$ with initial and final momenta $p$ and $p^{\prime }$, the nucleus $\mathcal{N}$ with $q$ and $q^{\prime }$, and photon $\gamma$ with momentum $k$.

Figure 2

Endpoint spectrum of $\nu +\mathcal{N}\to \mathcal{N}+\nu +\gamma$ for monochromatic incident neutrinos (energy $E_{\nu }$) with mass $m_{\nu }$ (blue) and zero mass (green). There is no difference between Majorana and Dirac neutrinos when $E_{\nu }\gg m_{\nu }$. $E_{\nu }-\omega$ is the energy of the outgoing neutrino.

Figure 3

Differential cross section $d\sigma /d\omega$ vs the scaled final neutrino energy $(E_{\nu }-\omega )/m_{\nu }$ for Dirac (green) and Majorana (blue) neutrinos. We show a moderately relativistic initial neutrino energy $E_{\nu }=3m_{\nu }$. We scale by the neutrino momentum $|{\mathbf{p}}^{\prime }|$ to better show the fractional differences.