I've been trying to find out some info about CP violation in the lepton sector at a basic (ie. a fresh postgraduate) level. We can take the neutrino mixing matrix U in its standard parametrization:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \left( \begin{array}{ccc} 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{array} \right) [/tex]

where [itex]c_{ij}[/itex] means [itex]cos(\theta_{ij})[/itex], and s for sine. One may cross off a row r and a column s, and from the remaining 2x2 matrix define a quantity called the Jarlskog invariant [itex] J = (-1)^{r+s} Im (U_{ij}U_{lk}U_{ik}^*U_{lj}^*) [/itex], which in this case is [itex] J = c_{12}c_{13}^2c_{23}s_{12}s_{13}s_{23}sin(\delta) [/itex].

Now I'm reading that leptogenesis is the term for the imbalance of leptonic matter over antimatter, and that it requires CP violation to have happened. Also, apparently J is a "measure of CP violation", but I'm struggling to find an example of where it is actually used in this manner.

I mean, say in a neutrino oscillation experiment between states [itex]\nu_\alpha \rightarrow \nu_\beta[/itex], CP violation would cause [itex]P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\overline{\nu_\alpha} \rightarrow \overline{\nu_\beta}) [/itex]. These experiments are being done to measure the values of the 4 parameters of the matrix U, namely [itex] \theta_{12}, \theta_{13}, \theta_{23} [/itex] and [itex] \delta [/itex], where a non-zero [itex]\delta[/itex] allows CP violation by causing [itex] U \neq U^{\dagger} [/itex], which is the reason for [itex]P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\overline{\nu_\alpha} \rightarrow \overline{\nu_\beta}) [/itex].

My question is, I keep seeing this Jarlskog invariant being mentioned a fair bit, but I'm struggling to see what the point of defining it is? What does this J allow us to do? What is J telling us about CP violation exactly? Is it something like not being able to directly measure [itex]\delta[/itex] or something?

Can it be used to calculate how many more leptons than antileptons there should be in the universe or something like that?

Sorry if I haven't asked this question very well, but I'm a bit confused at this moment.

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# The Jarlskog invariant and leptogenesis?

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