Jarlskog Invariant and its mathematical origin

CP violation is present in the weak interactions if

1. There are no degeneracies in the up-quark/down-quark matrices
2. The Jarlskog invariant $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$ is nonvanishing

Furthermore, all CP violating effects are proportional to $J$.

I am getting stuck on showing how all CP violating effects are proportional to $J$. Also, is the Jarlskog invariant a well-known mathematical property of a unitary matrix? What is it quantifying? I'd like to know this to the extent I can generalize this to larger CKM matrices.

Edit: I did a bad job writing my question. I rewrite it here:

Question:

1. How do I constructively derive $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$, and how do I generalize this to arbitrary $n\times n$ unitary matrices ?
2. The Jarlskog invariant is invariant under a change of basis. What is the elegant way of showing this?
• Oct 21 '17 at 20:08
• Xref: I posted this related question on MathOverflow. Jul 29 '18 at 15:09

Cecilia Jarlskog proposed this invariant already in 1973 and it was mentioned in the original Kobayashi-Maskawa paper.

For three families, it's easy to see why it is nonzero iff the unitary matrix in $U(3)$ can't be brought to the real, orthogonal i.e. $O(3)$ form. It's because after the 5 phase redefinitions of the up-type-quark and down-type-quark eigenstates, every $SU(3)$ matrix may be brought to the form of an $SO(3)$ matrix expressed by 3 real angles $\theta_{ij}$ and a single extra complex phase $\delta$, well, I mean $\exp(i\delta)$, added to a matrix element.

In that parameterization of the $SU(3)$ matrix, the invariant is simply $$J = c_{12}c_{13}^2 c_{23}s_{12}s_{13}s_{23}\sin \delta.$$ Note that it vanishes exactly if at least one of the factors is zero which means either if the complex phase $\delta$ is zero or $\pi$ mod $2\pi$ – then the matrix is explicitly real orthogonal and CP-preserving – or if any of the sines or cosines of the angles vanish in which case it's also possible to bring the matrix into a real form.

See

especially pages 7, 8, 11, 12 for some details and formulae. In particular, the first "standard" formula on page 7 makes it clear that the $SU(3)$ matrix is real – or can be made real - whenever one of the factors in $J$ vanishes.

EDIT:

The added questions have nothing whatsoever to do with the original one but they may be answered, too. There is no "constructive way" to derive the Jarlskog invariant. It was a clever guess, a proposed convention. A quantity that is zero whenever it should be is clearly not defined uniquely.

Also, it is incorrect to expect a canonical generalization to larger unitary matrices. Moreover, larger matrices actually have several independent sources of CP-violation, in the same sense as 2 x 2 matrix for 2 families has none. So it would be more natural to have several invariants for larger matrices and say that CP is preserved if all of them are zero. But once again, those invariants wouldn't be unique in any sense.

Concerning the third question, independence on bases, it's trivial to see. The CKM matrix $V$ is the transition matrix mapping three particular mass eigenstates to the $SU(2)$ partners of three other particular eigenstates. All these six eigenstates are determined uniquely, up to a phase (assuming they're normalized).

But it's easy to see that $J$ is invariant under these six changes of phases. For example, change the phase of the $b$ eigenvector by $\exp(i\beta)$. This phase gets cancelled in $J$ because $J$ depends on this phase only via $V_{cb}$ and $V_{ub}^*$ factors: in both of them, $b$ is the second index so the dependence on $\beta$ is the same but the second matrix element is complex conjugated so the phase cancels. Similarly one may verify the cancelation of the five other possible phases and that proves the independence on the basis.

• Thanks for the answer. But, I already knew most of everything you wrote, which means I asked the wrong question. I have rewritten my question in the edit. Would you please check it again? Thanks! Sep 9 '12 at 14:52
• Come on, the new questions have nothing to do with the old ones. Jarlskog didn't "derive" it constructively - it was a clever guess, a convention (we're only looking for a quantity that is nonzero when it should be, and the exact value of such a quantity is clearly not unique) and there is no "canonical" generalization of it to larger matrices. For larger matrices, there are many CP-violating angles, not just one, so you should have several invariants, too. Sep 10 '12 at 5:37
• Of course my new questions have nothing to do with the original ones-- that's what happens when the original questions weren't the right ones to ask. Ok, now my follow up question to your response is: what is the strategy to follow to construct the different invariants in--say--the 4x4 case Sep 10 '12 at 19:43
• I don't understand your comment, user. Your comment is exactly equivalent to the original question and my answer was written in order to answer this question, so why do you ask again? Obviously, CP-violation is present if and only if the matrix cannot be brought into a real form. So even if $s_{13}=0$ but some of the other sines and cosines entering $J$ are zero so that $J=0$, then it is possible to bring the matrix into a real form. The transformation needed to do so is different than those you probably have in mind but it exists. Oct 12 '14 at 10:34
• But the standard parameterization isn't the only parameterization. If the matrix is complex in the standard parameterization, it doesn't mean that it can't be brought to a real form. Oct 12 '14 at 16:36