LOG#136. Flavor ν mixtures.




Neutrino oscillations are one of the most surprising “sounds” in the whole Universe. Since neutrinos do oscillate/mix, they are massive. And due to mass, they can experiment “mixing” or “changes” of flavor (mass and flavor basis are different!). Even more, if some neutrino species were provided to experiment “decay”, we would have a very striking analogy with classical mechanics and sound waves: sound waves experiment frequency (mass) oscillations and damping or attenuation when “air” vibrates. Of course, the analogy is not full or complete since we do not know what is the vibrating medium (of course, in field theory, we assume that is the neutrino field what oscillates) and the decay of neutrino, if it exists, is either impossible or a very uncommon phenomenon. Anyway, we can study these effects and this analogy (with the pictures above) should be helpful for eager readers.

We can implement the celebrated density matrix approach in the description of the flavor mixture for neutrinos. It provides a very convenient tool for further extensions, such as the inclusion of neutrino decay. Neutrino decay is important because it is a possibility we have when we consider BSM (Beyond Standard Model) theories and/or some interesting cosmological applications.

Let us define some weights \omega_\alpha, with \alpha=e, \mu, \tau,\ldots, so

\displaystyle{\sum_\beta \omega_\beta =1}

In the initial time t=0 the system is described by a pure diagonal density matrix (pure state):

\boxed{\displaystyle{\rho (0)=\rho (t=0)=\rho_0=\sum_\beta \omega_\beta \vert \nu_\beta \rangle \langle \nu_\beta\vert}}

and it is normalized under the condition


Using the common PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix for the neutrino mixing matrix,

\boxed{\vert \nu_\beta\rangle=U_{\beta j}\vert \nu_j\rangle}

Then, we also have

U_{\beta j}=\langle \nu_j\vert \nu_\beta\rangle=\langle \nu_\beta \vert \nu_j\rangle^ *

The time evolution of the density matrix is determined by the following equation

\displaystyle{\rho (t)=\sum_\beta \omega_\beta\sum_j \left[e^{-\frac{im_j^2t}{2E}}e^{-\frac{\Gamma_jt}{2}}U_{\beta j}\vert \nu_j\rangle\right]\sum_k\left[\langle\nu_k\vert U^*_{\beta k }e^{+\frac{im_k^2t}{2E}}e^{-\frac{\Gamma_k t}{2}}\right]}

Equivalently, we can write it as follows

\displaystyle{\rho (t)=\sum_\beta \omega_\beta\sum_{j,k} U_{\beta j}U^*_{\beta k}e^{+\frac{i\Delta m_{kj}^2t}{2E}}e^{-\frac{\Gamma_j+\Gamma_k }{2} t}\vert \nu_j\rangle\langle\nu_k\vert}

and where

\Delta m_{kj}^2=m_k^2-m_j^2

In general, this density matrix will not be diagonal, and here there


is the annihilation rate (or decay width). Note that \Gamma_k, using the damping term


accounts for the absorption, decay or instability of some particular neutrino species! The probability for some flavor state \alpha to detect the state \nu_\alpha is giving by the next quantum result (remember that with no neutrino masses and complete stability of all neutrino species we get some uninteresting result):

\boxed{\displaystyle{P_{\nu_\alpha}(det)=\langle\nu_\alpha\vert\rho (t)\vert\nu_\alpha\rangle=\sum_\beta\omega_\beta\sum_{j,k}U_{\beta j}U_{\beta k}^*U_{\alpha k}U_{\alpha j}^*e^{-\frac{i\Delta m_{kj}^2 t}{2}}e^{-\frac{\Gamma_j+\Gamma_k}{2}t}}}

We can also rewrite this deep result separating the diagonal and non diagonal part (due to quantum interference!). The final formula reads

\boxed{\displaystyle{P_{\nu_\alpha}(det)=\sum_\beta\omega_\beta\sum_{j}\vert U_{\alpha j}\vert^2\vert U_{\beta j}\vert^2e^{-\Gamma t}+2\sum_\beta\omega_\beta\sum_{j<k}e^{-\frac{\Gamma_j+\Gamma_k}{2}t }\left[\mathcal{A}(U,m_{kj},E,t)\right]}}


\mathcal{A}(U,m_{kj},E,t)=\mathcal{R}(U_{\beta j}U^*_{\beta k}U_{\alpha k}U^*_{\alpha j})\cos \left(\dfrac{\Delta m_{kj}^2 t}{2E}\right)-\mathcal{I}(U_{\beta j}U^*_{\beta k}U_{\alpha k}U^*_{\alpha j})\sin \left(\dfrac{\Delta m_{kj}^2t}{2E}\right)

Averaging with this formula, we get that for detected neutrinos in the flavor state \alpha (remember that due to mass, mass eigenstates are different to flavor eigenstates):

\boxed{\displaystyle{\langle P_{\nu_\alpha}(det)\rangle=\sum_j \vert U_{\alpha j}\vert^2\sum_\beta \omega_\beta \vert U_{\beta j}\vert^2}}

Cosmic neutrino detectors (neutrino telescopes) are unlikely to be capable of neutrino flavor tagging. Therefore, summing over all neutrino species

\boxed{\displaystyle{P_{\nu}(all)=\sum_\alpha \langle P_{\nu_\alpha}(det)\rangle=\dfrac{1}{3}\sum_j e^{-\Gamma_j t}}}

and where we have supposed the common assumption that under “a long journey”, and with 3 active neutrino species, we would expect equiprobability, i.e., equal weights



\displaystyle{\sum_\beta\omega_\beta \vert U_{\beta j}\vert^2=\dfrac{1}{3}}

for all j=1,2,3 and \beta=e,\mu,\tau.

This density matrix formalism is very useful when one tries to include the redshift effect on neutrino oscillations and it makes us to wonder if quantum entanglement with neutrino beams could be observed in the near future. Indeed, the potential applications of neutrino entanglement are yet to be unveiled. Perhaps, neutrino entanglement could be useful for medicine, security, encryption and decode of information, and fascinating new technologies…

See you in my next TSOR post!!!!!

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