This parameter, together with theta12, theta23, and a CP violating phase, in a standard parameterization define the PMNS matrix (the leptonic answer to the CKM matrix for quarks which explains how likely a quark of one kind is to turn into another kind of quark in beta decay-like weak force interactions). The PMNS matrix comes up with particular relevance in neutrino oscillation measurements.

**The Status Quo**

The most accurately measured values of the corresponding CKM matrix angles are: θ12 = 13.04±0.05°, θ13 = 0.201±0.011°, θ23 = 2.38±0.06°, and δ13 = 1.20±0.08.

Some previously measured values for the PMNS matrix angles were approximately Θ12=45 degrees, Θ23=34 degrees, and Θ13=4.4 degrees, and estimate that has now been doubled. But, there is far less accuracy in these mesaurements.

A recitation of the experimental data in another form states:

* sin2(2θ13) = 0.092±0.017

* tan2(θ12) = 0.457+0.040−0.029. This corresponds to θ12 ≡ θsol = 34.06+1.16−0.84° ("sol" stands for solar)

* sin2(2θ23) > 0.92 at 90% confidence level, corresponding to θ23 ≡ θatm = 45±7.1° ("atm" stands for atmospheric)

* Δm2 21 ≡ Δm2 sol = 7.59+0.20−0.21×10−5 eV2

* |Δm2 31| ≈ |Δm2 32| ≡ Δm2 atm = 2.43+0.13−0.13×10−3 eV2

* δ, α1, α2, and the sign of Δm2 32 are currently unknown.

As the Wikipedia article that is the source of the data above explains:

The phase factors α1 and α2 are physically meaningful only if neutrinos are Majorana particles — i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown) — and do not enter into oscillation phenomena regardless. If neutrinoless double beta decay occurs, these factors influence its rate. The phase factor δ is non-zero only if neutrino oscillation violates CP symmetry. This is expected, but not yet observed experimentally. If experiment shows this 3×3 matrix to be not unitary, a sterile neutrino or some other new physics is required.

If the paramter "δ" in hte PMNS matrix is non-zero, oscillations of neutrinos might occur at different rates for a "forward" process, and the "reverse process" which is also flipped in anti-particle character. CP violation hasn't been observed in neutrinos, but this may be because it is much harder to measure for neutrinos than for charged particles since it is much harder to distinguish a neutrino from an antineutrino directly. Generally, one has to figure out where the process should be generating neutrinos or antineutrinos based upon the other particles that generate them, and then observe the rates of neutrinos produced in comparable neutrino generating and antineutrino generating processes and then comparing those rates. Since one uses separate experiments with different processees in that kind of analysis, however, one can't rely of symmetry or identical particle sources to cleverly reduce systemmic errors - one needs raw precision in both experimental set up and in theoretical predictions of the outcomes in both cases in a highly theory specific analysis.

The theta13 determination may indirectly influence experimental evidence for a value of the CP violating phase in the PMNS matrix because, according to a 2007 paper on the Daya Bay reactor neutrino experiment:

As shown in the parameterized Pontecorvo-Maki-Nakagawa-Sakata matrix, accurate measurement of θ13 may enable the study of CP violation in the lepton sector, in addition to supporting our understanding of neutrino oscillation arising from the mixing of mass eigenstates. . . . The PMNS matrix reveals that the CP violating phase factor, δ, is directly proportional to the square of the sine of the oscillation angle θ13; if θ13 proves to be very small or zero, this term will subsequently vanish, adding to the motivation for measuring θ13 as its magnitude will determine whether or not CP violation in the lepton sector is observable. . . . Daya Bay has the potential to reach a sensitivity of 1% in measuring sin22θ13. While on route to that sensitivity, Daya Bay may pin-point the value of the θ13 oscillation angle which, depending on its magnitude, may provide valuable information regarding the search for leptonic CP violation.

The relationship flows from the fact that in the relevant parameterization, the CP violating phase appears both in two elements of one of the matrixes that are combined to produce the PMNS 3x3 matrix: in one the element in the cell is -sin(theta13)*e^(i*δ), while in the other it is sin(theta13)*e^(-1*i*δ). The 1-3 and 3-1 neutrino oscillations are functions of both theta12 and the CP violating phase parameter δ in these terms, but the value of theta13 also contributes to another matrix element (with a cos(theta13 term) allowing the theta13 value and the CP violating phase to be distinguished from each other, at least in principle, with sufficiently accurate experimental data.

The theoretical and experimental agenda for measuring CP violation in the neutrino sector at Daya Bay (the source of the most nine degree estimate for theta13) is set forth in the linked pdf slides. Existing experimental methods are only sensitive enough to even conceivably have a chance of measuring the CP violating phase of the PMNS matrix when theta13 has a value of more than two degrees.

To say that the sign of Δm2 32 is unknown is a fancy way of saying that we don't know for sure if the third neutrino mass eigenvalue is lighter or heavier than the second neutrino mass eigenvalue. If the third is heavier than the second, we call it a "normal" mass hierarchy. If the third generation is lighter than the second, we call it an "inverted" mass hiearchy. The "delta m" numbers, in general, refer to the difference between the 1st and 2nd, the 2nd and 3rd and the first and 3rd eigenvalues of neutrino mass (neutrino "generation" and neutrino mass eigenvalues do not correspond precisely in the way that they do for other fermions - neutrino mass eigenvalues can be understood as the dominant linear combinatioons of neutrinos of different generations, just as some of the electroweak bosons and some of the hadrons and gluon color combinations are viewed of linear combinations).

We don't know the absolute mass of any of the neutrino mass eigenvalues, subject only to very gross limitations much greater in magnitude than the comparatively precisely measured differences in masses between neutrino mass eigenvalues.

The most common theoretical expectation regarding neutrino masses is that neutrino masses from the neutrino mass eigenstate mass difference measurements to date is that the hierarchy is normal and that the absolute values of mass eigenvalue states is on the same order of magnitude of the differences between states, even though inverted hierachies and a significant base mass of all neutrinos from which each mass eigenvalue differs only slightly is not ruled out by experiment:

m1 is on the order of 7.59×10^−5 eV^2

m2 is on the order of 2.43x10^-3 eV^2 and that

m3 is heavier than m2 by something on the order of magnitude of 7.59x10^-5 ev^2.

If, like me, you think the neutrino mass is Dirac and Majorana, then you aren't worried about a lack of measured values for α1 and α2, because they aren't part of your model. The main experiment bound on these two value comes from the searches for neutrinoless double beta decay, which is only possible in the Standard Model if these parameters are non-zero. To date, there has been no widely accepted experimental evidence showing a detection of neutrinoless double beta decay, so the values must be small and experimental data is consistent with them being zero.

**What Does A Larger Mixing Angle Value Mean?**

Larger mixing angles suggest more common shifts from one particle generation to another, while smaller ones suggest a strong tendency for weak force interactions to product particles in the same generation. Neutrino mixing angles, in general, tend to be much larger than quark mixing angles. At this point, no CP violating phase in neutrino mixing has been observed.

A 10 degree value for theta13 in the PMNS matrix is not terribly encouraging for the notion of quark-lepton complementarity. QLC observed that the sum of theta12 CKM, and theta12 PMNS was about 45 degrees, and that the sum of theta23 CKM and theta23 PMNS was about 45 degrees by some measurements. But, increasingly, the numbers just aren't adding up and the theta13 case with a sum of about ten or eleven degrees doesn't help.

Of course, mankind's search for patterns is never ending, as the first comment to the linked blog post notes that in the PMNS matrix that theta12+theta12=theta23=45 degrees.

Another comment at the post cited “Neutrino mass anarchy” L.J. Hall, H. Murayama, and N. Weiner, Phys. ReV. Lett. 84, 2572 (2000)

http://arxiv.org/abs/hep-ph/9911341 as making interesting predictions regarding neutrino mixing angles.

**BSM Theory Implications**

Predictions of a variety of beyond the standard model theoretical predictions for theta13 were compiled in 2007. Among the theories whose predictions inconsistent with the result announced this month are:

Orbifold SO(10), SO(10)+Texture, 3x2 see-saw, Renormalization Group Enhancement, and M-Theory Model. The results are, barely, consistent with a Minimal SO(10) model and "Anarchy." Several other theories made predictions that were so vague that they covered the entire experimental range of possibility as of 2007 and hence were uniformitive predictions.

## 1 comment:

If the sum of all eight parameters with the CP violating number converted from radians to degrees need to total 180, then the CP violating parameter for neutrinos is about 76 degrees which is about 1.32, which is within margins of error of the CP violating parameter being identical in both the CKM and PMNS matrixes.

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