Neutrino CP Violating Phase Numerology

The PMNS matrix which governs the likelihood that neutrinos will oscillate into other kinds of neutrinos has four parameters, three mixing angles, each of which has been measured with some accuracy, and one CP violating phase (the analogous matrix for quarks is called the CKM matrix).  The values considered state of the art in an April 3, 2013 paper from the T2K collaboration on their latest efforts to compare their measurements to prior measurements of Ɵ₁₃ prior to their latest experiment were as follows (their paper concluded that this value was confirmed by the latest experiment):

Ɵ₁₂=34° (per the Particle Data Group, the central value is 34.1 and the one sigma CI is 33.2 to 35.2)
Ɵ₂₃=45° (per the Particle Data Group, the central value is greater than 36.8° at a 90% CI)
Ɵ₁₃=9.1±0.6°
ƍ_CP=Unknown

A crude but fairly solid estimate for the CP violating phase of the PMNS matrix should be available within a few years for most of the possible values of this parameter (experiments are more sensitive to some values than to others) given the nature of the neutrino experiments that are currently in progress. In principle, any value between zero and 360° (i.e. 2π radians), with the endpoints being indistinguishable, is possible. A value close to zero would be hard to distinguish from zero itself with sufficient statistical power until considerably later.

Possible CP violating phase values in the four parameter PMNS matrix considered

In this post, I review a few of the interesting hints, some experimental and some little more than numerology, regarding that possible values of the CP violating phase of the PMNS matrix from smallest to greatest.

Keep in mind that error margins of one or two sigma (i.e. standard deviations from the mean value experimentally measured), are routine in physics and three sigma deviations "go away" with improved experimental measurements about half of the time (which is proof positive that error bars are almost always underestimated in physics).  An experimentally measured values that differs by less than two sigma from other experimentally measured value are commonly said to be "consistent" with each other.

It is also worth noting that according to a recent paper:

The recent measurement of the third lepton mixing angle, \theta_{13}, has shown that, although small compared to \theta_{12} and \theta_{23}, it is much larger than anticipated in schemes that generate Tri-Bi-Maximal (TBM) or Golden Ratio (GR) mixing. . . . For comparison we determine the predictions for Bi-Maximal mixing corrected by charged lepton mixing and we discuss the accuracy that will be needed to distinguish between the various schemes.

Likewise, simple versions of "quark-lepton complementarity" which had the corresponding 12 and 23 mixing angles of the CKM matrix and PMNS matrix each sum to forty-five degrees is also disfavored by current experimental data at more than a two sigma significance in the case of the theta 12 angles, although the fact that there are multiple ways to parameterize the respective matrixes somewhat mutes the importance of this observation.

* A zero value would mean no CP violation in neutrino oscillations and fewer than the full four possible degrees of freedom in the PMNS matrix. While this is in some sense the null hypothesis and default assumption, I would honestly be quite surprised if CP violation was truly zero for leptons, while being non-zero for quarks. Note that the data would be consistent with the three mixing angles having a combined value of 90° and a zero CP violating phase as well at the one sigma level.

* If the sum of the four PMNS parameters had their current best fit values and were equal 90° then the value would be 1.9° (0.033 radians) and might not be distinguishable from zero to within margin of error for some time.

* A sum of all eight PMNS and CKM matrix parameters equal to 180° would imply a ƍ_CP PMNS = 7.5±7.8°

* If ƍ_CP PMNS + ƍ_CP CKM = 90°, then ƍ_CP PMNS is approximately 21.2° but could be as low as 15.8° if the sum of the four CKM parameterswere actually 90° rather than their current best fit values. This would be a 1.2 sigma difference from the measured value of the CP violating parameter in the CKM matrix which would appear mostly via a CP violating phase of closer to 74.2° (1.295 radians), rather than the current best fit value discussed below.

* This paper predicts a value of 33° (0.57 radians) or less.

* This paper thinks that the Daya Bay data predicts a CP violation phase of a approximately 45° (π/4 radians).

* A value of 68.8±4.6°(1.20±0.08 radians) which would be identical to the CP violation phase in the quark sector is plausible. There is no particularly good a priori reason for CP violation to be different for leptons than it is for quarks even though there is no particularly compelling reason for them to be the same either. It isn't obvious that the mechanism through which the non-CP violating mixing matrix angles are generated which appears to be intimately related to the mass matrix values for fermions, has any relationship to the CP violating phase of either the CKM or PMNS matrix.

* A value of 90° (π/2 radians) would be halfway between minimal and maximal CP violation in neutrino oscillations in phase space.

* Slightly less than 180° (i.e. slightly less than π radians). This would be a scenario with almost, but not quite, maximal CP violation in neutrino oscillations. There are weak hints, referenced below, of high levels of CP violation in neutrinos, but none of the Standard Model mixing angles in the CKM or PMNS matrixes (with the possible exception of Ɵ₂₃ or CP violating phases already known are exact fits to any "round" number and the hints that the estimate of that phase are a bit high are also weakly present).

* A value of 180° (π radians) would be maximal CP violation in neutrino oscillations.  There have been some weak hints of near maximal CP violation, but nothing at all solid. The best fit value at one sigma is consistent with maximal CP violation, but at even slightly less than two sigma from the best fit value, all possible values of the CP violating phase are permitted.

* If one wanted to have a fit in which the sum of the two CP violation phases was equal to 90° modulo 180°(π/2 modulo π radians) and also wanted the sum of the three CKM mixing angles and its CP violation phase to equal 90° (π/2 radians), then the PMNS CP violation phase would have to be about 195.8° (1.088π radians) which is very close to the best global fit value of PMNS CP violation with a normal hierarchy. The experimentally allowed one sigma range without adding in quadrature for the sum of the three PMNS matrix mixing angles is 90°+6.4°-9.8° neither of which is easily squared with a 15.8° deviation from 180° in the CP violation phase without a further adjustment for two Majorana phases that weren't quite equal to a neat multiple of 180° of either 9.0° (0.05π radians) or 22.6°(0.126π radians) depending on the signs of the terms.

* A more recent effort to make a global fit found a best fit value of 306° (not a typo for 360°, and equivalent to 1.70 radians) with a one sigma range of 162°-365.4° ("0.90 to 2.03π radians", I put this in quotes because the range of permitted values for this parameter is [0-2π] so a 2.03π radians value isn't allowed). All possible values of the CP violating phase of the PMNS matrix, however, are consistent with that global fit of the other data at something less than the three sigma level, which is to say that we really don't know anything with any confidence about the CP violating phase of the PMNS matrix at this point.

* UPDATE (April 5, 2013): One of the more interesting models by Barr and Chen at the University of Delaware (September 28, 2012) inspired by SU(5) GUT concepts, develops a formula for the neutrino masses and mixings based on the mass and mixing matrixes of the quarks and charged leptons with parameters for the overall mass scale of the of the neutrinos and two more conceptualized as being related to Majorana CP violation phases but realized simply as two fitting constants each of which are a combination of a Majorana CP violation phase and another independent physical constant (they are q*e^(i*beta) and p*e^(i*alpha)).

The allocation of theses two fitting constants between the complex exponent and other fitting constant of each is not explained in this paper. The value they fit for q and beta has a value for beta equal to -2π/18 (with the eighteen and choice of fitting constant "q" suggestive of an eighteen value chosen based on there being six quark types of three color charges each, a number also relevant in weak force decay frequencies for all quarks combined). The value they fit of alpha is very close to 8/18th (perhaps for the number of gluon types relative to the number of quark types, or simply to make the combined values of alpha and beta Majorana CP violating phases equal to -π radians).

These three constants (together with the known quark and charged lepton values) are given best fit values based upon the three neutrino mixing angles and two mass differences known to predict the remaining unknown components of the entire neutrino mass and mixing matrix (i.e. the electron neutrino mass (and implicitly the neutrino hierarchy) and the PMNS matrix CP violation phase).  The best fit process also massages the experimental data by choosing particular values other than the isolated measurement best fits for those experimental values.
In the quark matrix, to make the fits work, the ratio of the strange quark mass to the down quark mass which experimentally is in a range of 17 to 22, is fixed at a slightly low value of 19 rather than the mean value of 20.5; the CKM CP violation phase for which the experimentally measured value is 1.187 +0.175-0.192 radians is set at a value towards the high end of that range at 74.5° (1.30 radians), PMNS theta12 is set at 34.1° (which is the current mean value but was higher than the published mean value in their reference at the time), PMNS theta23 is set at 40° rather than 45° +/- 6.5° (which is a decent estimate of what the most recent experimental evidence is trending towards showing), and tweaked the square of the difference between first and second neutrino mass value which an experimentally measured mean of 7.5 +/- 0.2 * 10^-5 eV^2 to a value of the initial number to 7.603.  All of these tweaks are within one sigma of the experimentally measured values.

While it is presented as an output of the model rather than in input, this fitting, which predicts a neutrino CP violation parameter in the four parameter PMNS model of 207° (1.15π radians) was done knowing that the experimentally measured Daya Bay one sigma value for the neutrino CP violation was 198° (1.1π +0.3π-0.4π radians).  Their model also makes a prediction for the electron neutrino mass of 0.0020 eV, a number that is quite in line with what one might expect simply assuming a normal and non-degenerate neutrino mass hierarchy (which while few published papers come out and say so, it the natural favorite since it is what we see in the quarks and charged leptons, and is capable of being consistent with data showing a much larger mass gap between the second and third neutrino masses and the first and second neutrino masses).

From the point of view of predicting something unexpected, this isn't impressive.  It shows that the authors can read the tea leaves in hints from experimental data to fit six Standard Model constants at particular spots within the one sigma ranges of the measured values of those constants and can peg the value of one non-measured constant where it would be expected with a couple of best bet assumptions and a bit of educated guesswork.  The outcomes are essentially equivalent to betting on the favorite in a horse race making conventional assumptions at every step without being too unduly rigid about it, although these predictions do leave some room for falisfiability if the data take an unexpected turn.

Indeed, the paper expressly explains precisely which of the experimentally measured values the fit is most sensitive to with charts illustrating the point graphically for several of them.  For all of the fits to be internally consistent, the model puts considerable pressure on PMNS theta23 to be lower than 41°.

Particularly interesting is that their model needs the CP violating phase in the CKM matrix for quarks to be at the high end of the experimentally measured value at about 74.5°(1.30 radians aka 0.41π radians), which is almost precisely the value that makes the sum of the three CKM mixing matrix angles and the CP violating phase of the CKM matrix equal to 90° (π/2 radians). From the perspective of their model, this seems to be almost a coincidence as this model's best fit value, the sum of the three PMNS matrix mixing angles and the CP violating phase of the PMNS matrix is a not particularly numerologically notable 290.22° (1.61π radians), although one might argue that this needs to be modified by the sum of the two Majorana phases alpha and beta, both of which are negative, which they fit to -176.7° (-0.98π radians) which is suggestive of some possible relationship that would require the two Majorana phases combined to have a value of -π radians, even though the combined value for the three PMNS matrix mixing angles and three CP violating phases in this model of 0.61π isn't very notable (unlike a 0.5π value which would make the combined value of all of the angles in the model equal to π); but since each of their proposed Majorana phases has a second corresponding constant that always appears at the same time, it ought to be possible to fit their model such that the sum of all of the PMNS matrix phases did equal π/2 without too much difficulty.

What does make the paper impressive, however, is that it offers up a coherent, formalized set of equations that relates all of the fermionic masses and mixing matrix parameters of the Standard Model to each other in a coherent analytical framework that is capable of fitting all twenty of these (plus two additional Majorana mass CP violation phases) to the data consistent with all experimental data to date at a one sigma level. As a Within the Standard Model (WSM) effort to look for a deeper relationship between the myriad physical constants that the Standard Model treats are arbitrary, this is a solid accomplishment.

Put another way, this model is asking, and answering, some of the right questions and demonstrating that it is indeed possible to formalize the relationships between these quantities.

Caveats

The four parameter model used here which is the dominant paradigm for experimenters at this point, implicity assumes that neutrinos have Dirac mass rather than Majorana mass, which introduce a couple of additional phases.

Neutrino researchers are also trying to determine the absolute mass of the different kinds of neutrinos, which is irrelevant to determining the PMNS matrix entries. And, they are also trying to determine if the sign (i.e. positive or negative) of one of the know mass differences between two of the kinds of neutrinos in the mass hierarchy. A positive value is a "normal hierarchy" and a negative one is an "inverted hierarchy". The sign of this mass different term, practical purposes, tweaks the estimated value of the mixing angle parameters given a particular set of experimental data by about 1%-2%, which is similar to or smaller than the overall precision of all of the measurements of Standard Model constants being made at this point in the neutrino sector.

UPDATED (April 5, 2013) footnote on Majorana model fits to the PMNS matrix: In general, it is alway possible to fit a three by three unitary matrix, like the PMNS matrix, with four independent parameters.  To prove this by example, assume that you know the value of the first two entries in the bottom and next to bottom row of the matrix.  You can always determine the other five values from those four parameters simply by knowing that the matrix is unitary.  Any model with three mixing angle parameters and one CP violation parameter is a parameterization of the matrix with a minimal number of parameters and there is not a unique four parameter parameterization, or even a unique four parameter parameterization in which each of the parameters is independent of each of the other parameters.  (Note, however, that Majorana neutrino models often have a non-unitarian PMNS matrix.)

It follow from this conclusion that it is impossible to determine uniquely, even in any given parameterization scheme for the PMNS matrix, all three mixing matrix values and all three Majorana neutrino CP violation phases, even if you can measure all nine entries in the PMNS matrix experimentally.  A unique set of neutrino parameters in the Majorana neutrino mass scenario also requires information in addition to the three left handed neutrino mass states and a full understanding of the oscillations of these three classes of neutrinos.  Those measurements inherently conflate the three CP violation phases in a Majorana mass model into a single number.  To decompose those three phases in a Majorana model from each other, you also need two more parameters related in some way to the see-saw mechanism of these mass models.  As some researchers have acknowledged, discerning these parameters experimentally could be very difficult.