Given that neutrinos mix into each other, scientists have indirectly shown that neutrinos have mass. (Other indirect evidence for neutrino mass is that neutrinos from the 1987 supernova explosion arrived later than photons from the explosion...and the less energetic neutrinos on average arrived later than the more energetic neutrinos...in line with the special theory of relativity.) So, it's now been obvious that the MSM is not a valid theory of the universe (and this is not news.) But the MSM seems to do quite well at predicting the rates of interaction between quarks and leptons. So, we are looking into how to tweak the MSM to account for neutrino mixing, but without messing up the parts of the MSM that work extraordinarily well.

In this post, I want to discuss the similarities and the differences between the CKM and PMNS matrices because this are the two matrixes (that we know of so far) in which there is CP-symmetry violating physics (and hence T-symmetry violating physics. ) There is an eerily close resemblance between the CKM matrix and the PMNS matrix, but the eigenvalues of the matrices are not exactly the same. It is the PMNS that shows the strongest CP-violation, but it is also the matrix that has the most uncertainty in the values. Hence, it is crucially important that we decease the uncertainty in the values of the PMNS matrix by creating more precise measurements of neutrinos. Only by decreasing the uncertainyty will it be clear what is the strength of the CP-violating term in the matrix, and perhaps whether we need to include sterile neutrinos into the PMNS matrix.

For these reasons, I'll start with an analysis of the PMNS matrix.

To visualize the 3by3 PMNS matrix, I suggest that you input the following code into Wolfram Alpha or the following table into Excel:

Link to Wolfram with code already used as input

{{0.822,0.547,(-0.15+0.0405i)},{(-0.355+0.0211i),(0.702+0.014i),0.616},{(0.443+0.0278i),(-0.454+0.0176i),0.772}

PMNS Matrix with complex values

0.822 | 0.547 | -0.15+0.0405i |

-0.355+0.0211i | 0.702+0.014i | 0.616 |

0.443+0.0278i | -0.454+0.0176i | 0.772 |

Here's what we can determine so far from the PMNS matrix (Note that there is still some uncertain in the values and the number of significant digits listed below is not indicative of the level of precision in the measurements):

Determinant = 0.998695+0.0005944i (Very close to 1, but not exact)

Trace = 2.296+0.014i (Well below a value of 3.0, which would indicate no flavour mixing.)

Eigenvalues = 0.633668 -0.773526i = 0.99994*exp(-0.884i)

0.998795 + 0.0405336i = 0.999617*exp(+0.0406i)

0.663537 + 0.746992i = 0.99914*exp(+0.844i)

(Note that one of the eigenvalues is almost exactly equal to a value of 1. If there were no mixing between the three types of active neutrinos, then there would be three degenerate eigenvalues of value equal to 1. However, in this case, there is only one eigenvalue of 1. The eigenvector roughly corresponds to the case of 50% electron neutrinos, 15% muon neutrinos, and 35% tau neutrinos. It seems that if you have exactly this amount of each of the neutrinos, then you will remain with this amount of neutrinos.)

The sum of the amplitude
squared of each of the elements in a column is equal to 0.9992 for column1,
0.9986 for column2, and 0.9996 for column3. This is another way as saying that
the PMNS matrix multiplied by the transposed conjugate of itself is equal to
the identity matrix. This is a good sign that there are only 3 light neutrinos (less than ~eV in mass.) However, given that all of these values are less than
1.0000, it is still entirely possible that the neutrinos can convert into a 4th
neutrino, provided that the mass of this neutrino were much greater than the
mass of the light neutrinos. We need more precise experimental measurements before we can
rule out mixing into keV sterile neutrinos Bulbul

*et al*. and Abazajian.){{0.97423,0.2255,(0.00127-0.00327i)},{(-0.22538-0.00013i),(0.97339-0.00003i),0.041506},{(0.00812-0.00318i),(-0.04015-0.000736i),0.999132}

Link to Wolfram Alpha site with the CKM matrix

CKM matrix with complex values

0.97423 | 0.22552 | 0.00127-0.00327i |

-0.22537-0.00013i | 0.97339-0.00003i | 0.04151 |

0.00812-0.00318i | 0.04015-0.00074i | 0.99913 |

Here's what we can determine so far from the CKM matrix (Note that there is still some uncertain in the values and the number of significant digits listed below is not indicative of the level of precision in the measurements):

Determinant = 0.999973+0.000005i (Very very close to 1, but not exact)

Trace = 2.94675-0.00003i (Close to, but below a value of 3.0, which would indicate no mixing between quarks.)

Eigenvalues = 1.00001 - 0.00114175i = 1.00001*exp(-0.0011i) ≈ 1+0i

0.973504 + 0.228598i = 0.99998*exp(-0.2306i)

0.973242 + 0.229709i = 0.99998*exp(+0.2318i)

The absolute value of the eigenvalues is essentially equal to one, which means that the eigenvalues lie on the unit circle. This is required if the matrix is unitary. But what's interesting here is that the eigenvalues are very nearly exactly complex conjugates of each other. But since the trace is not exactly equal to one, the characteristic equation for the matrix has complex values, which means that the eigenvalues do not need to be mirror symmetric about the real axis (i.e. complex conjugates of each other.)

In the plot at the beginning of the post (repeated below), I'm graphing the values of each the three eigenvalues for the CKM and the PMNS matrices. Note that all of the values are very close to the unit circle, and note also that the eigenvalues for CKM are much more symmetric than are the values of the PMNS matrix. This might go away when we measure these the values of the PMNS matrix with better accuracy. Though, it could also be pointing to the fact that there is a heavier neutrino that needs to be included in the PMNS matrix.

(making it a 4by4 or larger matrix.) The fact that the eigenvalues of the PMNS matrix are not symmetric w.r.t. to the real axis is evidence for CP violation. It is much easier to see this CP violation than it is to see the CP violation in the eigenvalues of the CKM matrix, but there is a slight asymmetry in the eigenvalues. This asymmetry is greater than the uncertainty in the measurements.

What's also interesting is that, so far, the only CP violating terms in the MSM and the vMSM Hamiltonian can be represented using these two matrices. CP violation can originate in the MSM and the vMSM by two sources: (1) explicit symmetry breaking (i.e. there is a term in the Hamiltonian that is explicitly CP violating. If this is the case, then all parameters of the Hamiltonian that can be complex will be complex.) (2) spontaneous symmetry breaking (i.e. the Hamiltonian is CP_symmetric, however, the underlying symmetry of the physical vacuum is not CP symmetric.)

My somewhat educated guess is that case#2 is what is happening in our universe. In my next post, I'll discussing possible ways in which spontaneous symmetry breaking (at the start of the Big Bang) could have created the CP violating terms in the Hamiltonian that were discussed in this post.

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