Saturday, September 16, 2017

particle physics - 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(VusVcbVubVcs) 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(VusVcbVubVcs), and how do I generalize this to arbitrary n×n unitary matrices ?

  2. The Jarlskog invariant is invariant under a change of basis. What is the elegant way of showing this?




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