Jarlskog Invariant

Question

Jarlskog Invariant is directly proportional to CP violation. I want to know why it's called "an invariant". What is the nomenclature of the Jarlskog Invariant?

Answer 1

There is a large rephasing freedom in the SM formalism for fermions in CP violating amplitudes: these are based on the existence of at least one nonremovable non-vanishing phase in the CKM (and PMNS) matrix.

That is, variation of the bases/phases of the fermions may suffice to remove CP violation for two generations, but not for three such, the Kobayashi-Maskawa insight. Thus, the most general allowable CKM matrix must contain an invariant phase δ, in principle, invariant under such rephasings, and non-removable: it is there to stay and violate CP.

The robustness/non-removability of this phase is, however, predicated on the nontrivial coupling of the third generation to the other (lighter) two; even if, superficially, they are solely involved in some CP violating phenomena: they are not!

The third generation is instrumental in intermediate states. If this generation were decoupled from the other two, e.g. by a vanishing angle and hence sine ($s_{13}$, or $s_{23}$) to the two lighter ones, the phase δ could be rephased away! (Check this.)

The rephasing-invariant combination $$ J = c_{12}\ c_{13}^2\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta\ \approx\ 0.00003 $$ encodes this feature and further reminds us of the needed ingredients for a CP-violating amp, as well as its smallness. The J invariant reflects the inability of our minds to readily parse out CKM 3×3 matrices by inspection!