The Jarlskog invariant and leptogenesis?

[Doofy]

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:
$$\left( \begin{array}{ccc} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\ -s_{12}c_{23} -c_{12}s_{23}s_{13}e^{-i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{-i\delta} & s_{23}c_{13} \\ s_{12}s_{23} -c_{12}c_{23}s_{13}e^{-i\delta} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{-i\delta} & c_{23}c_{13} \end{array} \right)$$
where $c_{ij}$ means $cos(\theta_{ij})$, 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 $J = (-1)^{r+s} I am (U_{ij}U_{lk}U_{ik}^*U_{lj}^*)$, which in this case is $J = c_{12}c_{13}^2c_{23}s_{12}s_{13}s_{23}sin(\delta)$.

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 $\nu_\alpha \rightarrow \nu_\beta$, CP violation would cause $P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\overline{\nu_\alpha} \rightarrow \overline{\nu_\beta})$. These experiments are being done to measure the values of the 4 parameters of the matrix U, namely $\theta_{12}, \theta_{13}, \theta_{23}$ and $\delta$, where a non-zero $\delta$ allows CP violation by causing $U \neq U^{\dagger}$, which is the reason for $P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\overline{\nu_\alpha} \rightarrow \overline{\nu_\beta})$.

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 $\delta$ 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.

 

[fzero]

You have nearly all of the ingredients to answer the question. CP violation will occur if $U=U^\dagger$, which requires $\delta\neq 0$. Also, it's clear that $J$ vanishes if $U$ is real. The missing piece is the expression of $P(\alpha\rightarrow \beta)$ in terms of $U$, which is given a few lines down at http://en.wikipedia.org/wiki/Neutrino_oscillation#Propagation_and_interference. This is a useful formula, because we could conceive of doing an experiment where we could measure the coefficients of the $\sin^2$ and $\sin$ terms. But the coefficient of the $\sin$ term is directly proportional to $J$. So if we did a typical experiment, then $J$, rather than $U$, is what we would directly measure.

 

[Doofy]

You have nearly all of the ingredients to answer the question. CP violation will occur if $U=U^\dagger$, which requires $\delta\neq 0$. Also, it's clear that $J$ vanishes if $U$ is real. The missing piece is the expression of $P(\alpha\rightarrow \beta)$ in terms of $U$, which is given a few lines down at http://en.wikipedia.org/wiki/Neutrino_oscillation#Propagation_and_interference. This is a useful formula, because we could conceive of doing an experiment where we could measure the coefficients of the $\sin^2$ and $\sin$ terms. But the coefficient of the $\sin$ term is directly proportional to $J$. So if we did a typical experiment, then $J$, rather than $U$, is what we would directly measure.

ah right, thanks, but now I have another question. Say we have $J = c_{12} c_{13}^2 c_{23} s_{12} s_{13} s_{23} sin \delta$.

Now assuming we already know the mixing angles $\theta_{12}, \theta_{13}, \theta_{23}$, if we want to calculate $\delta$ then we need to know what J is. I thought that neutrino oscillation experiments were about measuring $\theta_{12}, \theta_{13}, \theta_{23}$ rather than J. How is J actually determined?

* In fact, I don't yet understand how people even manage to weedle out the mixing angles (and also mass^2 differences) from the data they measure at these detectors, which if I'm not mistaken is essentially just
$$P(\alpha \rightarrow \beta) = \frac{measured \hspace{1mm} flux \hspace{1mm} of \hspace{1mm} \nu_{\alpha}}{expected \hspace{1mm} flux \hspace{1mm} of \hspace{1mm} \nu_{\alpha} \hspace{1mm} without \hspace{1mm} oscillation \hspace{1mm} \nu_{\alpha} \rightarrow \nu_{\beta}}$$
because it seems to me that you end up with a single equation for the probability (that link you gave me) that depends on like 3 or 4 variables.

 

[fzero]

ah right, thanks, but now I have another question. Say we have $J = c_{12} c_{13}^2 c_{23} s_{12} s_{13} s_{23} sin \delta$.

Now assuming we already know the mixing angles $\theta_{12}, \theta_{13}, \theta_{23}$, if we want to calculate $\delta$ then we need to know what J is. I thought that neutrino oscillation experiments were about measuring $\theta_{12}, \theta_{13}, \theta_{23}$ rather than J. How is J actually determined?

* In fact, I don't yet understand how people even manage to weedle out the mixing angles (and also mass^2 differences) from the data they measure at these detectors, which if I'm not mistaken is essentially just
$$P(\alpha \rightarrow \beta) = \frac{measured \hspace{1mm} flux \hspace{1mm} of \hspace{1mm} \nu_{\alpha}}{expected \hspace{1mm} flux \hspace{1mm} of \hspace{1mm} \nu_{\alpha} \hspace{1mm} without \hspace{1mm} oscillation \hspace{1mm} \nu_{\alpha} \rightarrow \nu_{\beta}}$$
because it seems to me that you end up with a single equation for the probability (that link you gave me) that depends on like 3 or 4 variables.

This is not a subject that I'm very familiar with, so I would suggest a review like http://inspirehep.net/record/748589?ln=en. I took a quick look through there and it seems like extracting parameters relies on exploiting the different physics that is relevant to specific types of events at specific detectors.

In particular, the neutrino source is important. The Sun is a source of mostly electron neutrinos, while beam lines from Earth-based accelerators can be mostly muon neutrinos. Measuring the fluxes of electron and muon neutrinos at the detectors gives independent information.

 

[AdrianTheRock]

You have nearly all of the ingredients to answer the question. CP violation will occur if $U=U^\dagger$, which requires $\delta\neq 0$...

Are you sure that's quite right? If we are accepting the PDG parameterisation as used for the CKM matrix, then according to the SM textbook I have CP violation only requires that U cannot be made real by absorbing phases into wavefunction redefinitions, ie $U≠U^*$. Making U real is possible iff $J=0$.

Also, with the PDG parameterisation, $U$ is unitary, hence $U^\dagger = U^{-1}$, which is the form used when transforming mass eigenstates back into flavour ones.

 

[fzero]

Are you sure that's quite right? If we are accepting the PDG parameterisation as used for the CKM matrix, then according to the SM textbook I have CP violation only requires that U cannot be made real by absorbing phases into wavefunction redefinitions, ie $U≠U^*$. Making U real is possible iff $J=0$.

Also, with the PDG parameterisation, $U$ is unitary, hence $U^\dagger = U^{-1}$, which is the form used when transforming mass eigenstates back into flavour ones.

Yes, I meant $\mathrm{Im}(U)\neq 0$. I was typing so carelessly I didn't even get the $\neq$ correct.

 

[Doofy]

thanks for your replies guys, but I'm still not quite clear on this J thing. Let's just ignore the details of how the angles and mass^2 differences are measured - it seems the ways they arrive at these values are quite varied and complicated, and possibly unique to each individual experiment.

Assuming we know what the angles are though, is anyone able to explain to me what the basic premise of an experiment to measure J would be, so that $\delta$ could be calculated ?

 

[fzero]

In principle, if you could measure $P(\alpha\rightarrow\beta)$ for a range of neutrino energies $E$ and baselines $L$, you could separate the $\sin$ term from the $\sin^2$ term in the expression from http://en.wikipedia.org/wiki/Neutrino_oscillation#Propagation_and_interference. This would be a measurement of $J$. That big review of Gonzalez-Garcia and Maltoni comments in Ch 7 about this in a few places (for example item (iv) on page 127), but this might be difficult depending on the values of the other angles (more about this on pages 128-129).