## Tuesday, August 12, 2014

### CKM and PMNS Matrix Numerology

The mixing angles of the CKM matrix which governs weak force flavor changing probabilities for quarks, and the PMNS matrix mixing angles which governs neutrino oscillation, show the following relationship empirically to the limits of current experimental precision:

(θ12PMNS/θ12CKM)*(θ23PMNS/θ23CKM)=(θ13PMNS/θ13CKM).

This tends to imply that the the probability of a first to third generation transition in the PMNS matrix is a function of the probability of a first to second generation transition and the probability of a second to third generation transition, in the same way that it is in the CKM matrix.

This, in turn, suggests that the three mixing angles in each of the matrixes actually involve only two degrees of freedom (i.e. the three mixing angles are not fully independent of each other). If this were true, the number of mixing matrix parameters of the Standard Model would be six instead of eight.

This is also mathematically equivalent to the following:

(θ12PMNS*θ23PMNS)/(θ12CKM*θ23CKM)=(θ13PMNS/θ13CKM).

In other neutrino physics, a new study claims to rule out (light) sterile neutrinos based upon cosmology data. Anonymous said...

Are you sure that this is the case?

I calculated a value of 2.69 for theta12*theta23/theta13 for CKM.

But for PMNS, I calculated a value of 2.87 (using 2014 data) and 2.80 (using 2012 data) for theta12*theta23/theta13.

The values are close, but I'm not sure if they are exactly the same. But I do agree with you that there is likely a connection between the CKM and the PMNS matrix, and that there are less than 4 free variables for each matrix.

Also, as for the "new study" that you link to at the end of the post. The paper does not rule out a 7.1 keV sterile neutrino as long as sinsquaredtheta is less than ~10^-9.
The value for sinsquaredtheta from Bulbul et al. and Boyarsky et al. is ~2*10^-11. For it looks like this 3.55 keV signal could still be half of the energy in the decay of a sterile neutrino into a light active neutrino.

andrew said...

The precision of the PMNS matrix angle measurements is not very precise. Two of the angles have a MOE of roughly +/-5% and the third has a MOE of roughly +/- 3%. The combined MOE from the three measurements is on the order of +/- 7.8%, and when you add the uncertainty in the CKM matrix angles you are closer to +/- 8%. So the ratios coincide to within 1 sigma.