Wednesday, May 28, 2014

An Alternative CKM Matrix Parameterization?


The Cabibbo-Kobayashi-Mashkawa (CKM) matrix is the matrix that sets forth the empirically determined square root of the probability of a quark of one type changing into a quark of another type when it emits a W boson.

Any up type quark can give rise to any of the three down type quarks.  Any down type quark can give rise to any of the three up type quarks.  It is trust as a matter of definition, that the sum of the probabilities of a particular quark, say, a charm quark, changing into each of the possible down type quarks is exactly 100%.  It is empirically true, however, that the matrix is unitary.  Thus, the sum of the squares of the entries in the columns of the matrix as well as the sum of the squares of the entries in rows of the matrix equal 100%.

The probability of a particular up type quark transforming into a particular down type quark is the same as the probability of that down type quark transforming into that up type quark, except to the extent of CP violation in the matrix, which has complex number valued entries.

Like any unitary matrix, the CKM matrix can be parameterized in an infinite number of ways using four parameters (because a unitary three by three matrix has four degrees of freedom).

The magnitude of the entries is as follows:*

One interesting parameterization of the matrix, because it suggests a hidden structure to its entries, is the Wolfenstein parameterization (originally proposed in a 1983 paper by Lincoln Wolfenstein). To order λ3, it is:

Using the values of the previous section for the CKM matrix, the best determination of the Wolfenstein parameters is:
λ = 0.2257+0.0009 −0.0010
A = 0.814+0.021 −0.022
ρ = 0.135+0.031 −0.016, and
η = 0.349+0.015 −0.017.

* The material from the * to this explanatory note is from Wikipedia.

Up to adjustments for CP violation, this parameterization suggests that the probability of a first to second generation transition (or second to first generation transition) is λ, that the probability of a second to third generation (or third to second generation) transition is Aλ2, and that the probability of a first to third generation (or third to first generation) transition is λ*Aλ(i.e. product of the probability of making first one of the single generation step transitions and then the second).  The probability of transitioning to a quark of the same generation is the residual probability after the probability of the other two options is subtracted out.

Higher order and variant Wolfenstein parameterizations are discussed in a 2011 paper.  This suggests that some tweaks to the original Wolfenstein parameterization are necessary to fit the data, while being motivated by the same principles.

A 1994 paper, expands the Wolfenstein parameterization to a higher order of lambda and pointing out that a symmetric CKM matrix of the kind originally envisioned is almost ruled out by the data.  A 2014 paper attempted to generalize this parameterization to leptons.


Nothing mathematically requires that it be possible to parameterize the CKM matrix with three rather than four empirically determined constants.  But, there is a way to do so that is consistent with empirical evidence.

2 is equal to (2λ)at the 0.1 sigma level of precision, and there is no place in the Wolfenstein parameterization of the CKM matrix where this substitution cannot be made.

Thus, Vcb becomes approximately (2λ)4, Vts becomes approximately -(2λ)4, Vub becomes approximately (2λ)4*λ*(ρ-iη) and Vtd becomes approximately (2λ)4*λ*(1-ρ-iη).

Moreover if we take ρ-iη to be a single complex number C, and C* to be the complex conjugate of C, then we can state that: Vub becomes approximately (2λ)4*λ*C and Vtd becomes approximately (2λ)4*λ*(1-C*).

Thus, the entire CKM matrix is a function of one empirically determined real number, λ, and one complex number pertinent only to CP violation, C.  This approach thus suggests that there is even more method to the apparent randomness of the CKM matrix than the Wolfenstein parameterization would already suggest.

This variation on the Wolfenstein parameterization of the CKM matrix suggests that we should be looking for physics explaining the second to third generation transition that are basically a power of four different than the first to second generation transition, rather than a power of two different.  For example, the power of four might have some physical or geometrical relationship to the four dimensions of space-time.

To be clear, I'm not by any means the first person to see the possibility of expressing the CKM matrix with fewer than four parameters.  A very different proposed single constant parameterization of both the CKM matrix and PMNS matrix, for example, can be found here.


Anonymous said...

It's certainly interesting that it's a simple formula relating A and lambda. As I'll discuss below, there are likely only two free variables in each of the CKM and PMNS matrices.

However, I think that this whole numerology thing can often be taken way too far...especially in the paper you link to from a professor at Yaroslavl University. (I have nothing again the city, and in fact it's my favorite city in Russia.)

In that paper, the professor tries to extend a 1-parameterization to both the CKM and PMNS matrices. This is too much because there's still a good chance that there's a fourth type of neutrino (or semi-sterile version of the existing three.) The determinant and the square of the columns of the PMNS matrix, using currently measured best fits, are all slightly less than 1. See link below

I think that it's more important to talk about the eigenvalues of the matrix rather than the 4 parameters because the eigenvalues remain the same no matter how the matrix is parameterized.

The key thing (as far as CP-violation) is that there is no mirror plane along the x-axis, about which the eigenvalues remain the same after a mirror flip.

The "CP-violating eigenvalue angle" one measures for the PMNS matrix is 0.04 radian (which is the angle by which two of the three eigenvalues need to be moved on the unit circle in order for CP-violation to go away.)

The same CP-violating angle one measures for the CKM matrix is 0.001 radian. The CP-violating term for the PMNS matrix is ~40 times larger than the CKM matrix.

So, this much I agree with you.
It appears that you only need two parameters to describe the CKM matrix (if assuming that the eigenvalues are on the unit circle and there aren't more columns in the matrix.)
eig#1 = 1*exp(-a)
eig#2 = 1*exp(-b+a/2)
eig#3 = 1*exp(+b+a/2)
where a = 0.001 radian and b = 0.02312

For the PMNS matrix
eig#1 = 0.99994*exp(-a')
eig#2 = 0.99962*exp(-b'+a'/2)
eig#3 = 0.99914*exp(+b'+a'/2)
where a' = 0.04 radian and b' = 0.086

So, to summarize, there's no point on working on numerology between the 4-parameters. If you're going to work on numerology, I'd try to understand why a' is 40 times larger than a, while b' is only roughly 4 times larger than b. Is there any theory that predicts from first principles that a' is 40 times larger than a?

andrew said...

I certainly don't assert that the numerology is definitely meaningful, but the more ways of looking at a problem you have floating around in your head, the more likely it is that the pattern recognition that humans are so good at will kick in to connect other dots in an appropriate way.

Numerology, done well, is basically an attempt to reverse engineer nature, rather than a hypothesis testing exercise. The notion is to have some sense of what kind of form the true relationships must hold in an effort to gain some insight into what kind of processes you should be looking for.

In the case of CP violation, the margins of error are sufficiently large (absolutely in the PMNS case and even in the CKM case), that I think it is entirely possible that conventional parameterizations have the functional form of this impact wrong and/or that in a truly enlightened understanding that it is possible to disentangle CP violation and other CKM/PMNS matrix parameter from each other almost entirely as independent phenomena that are simply hard to observe experimentally in isolation.

By analogy, the "core" CKM/PMNS matrixes may be analogous to Newton's laws of motion and gravity, and CP violation may be analogous to friction. Experimenters conducting real world experiments made without understanding that air resistance and contacts with surfaces lead to friction would find it very hard to devise an experiment that is frictionless and measures only Newton's laws of motion and gravity. But, analytically, Newton's laws of motion and gravity are functionally very simple and usually our starting point, while friction is usually modeled phenomenologically as an independent process that greatly simplifies a dramatically more complex analysis at the fundamental level.

I could easily imagine that the "core" CKM and PMNS matrixes can both combined be derived from one or two empirically measured constants, while CP violation has an analytically distinct cause that is different in nature and more complex that is hard to distangle because it is operating at a different scale relative to the magnitude of the "core" matrix elements.

andrew said...

"If you're going to work on numerology, I'd try to understand why a' is 40 times larger than a, while b' is only roughly 4 times larger than b. Is there any theory that predicts from first principles that a' is 40 times larger than a?"

a and a', in radians, are formulations that are absolute dimensionful scale independent. It could be that the 40x factor is a hint that a scale independent way of formulating this parameter is not illuminating, and that some scale setting, dimensionful formulation would provide more insight.

For example, the 40x is on the same order of magnitude as the fourth root of the mass of the down quark over the mass of the electric neutrino mass. Also CP terms become significant in Wolfenstein-like parameterizations like the one I suggest in the original post in terms with lambda to the fourth power in them.

So, maybe a and a' are almost linear to the fourth power/root of the masses of the particles involved rather than being most fruitfully described in the conventional way as mixing angles that are scale invariant, perhaps related to the existence of four dimensions of space-time.

Tienzen said...

andrew: " Numerology, done well, is basically an attempt to reverse engineer nature, rather than a hypothesis testing exercise."

Excellent description about the numerology (the reverse engineering). In fact, for every given target (a number or a value), we can ‘always’ reversely engineer a ‘bow and arrows’ to hit that target. Thus, it is very easy to tell whether a formula is a ‘numerology’ or not with two criteria.

Criterion one: does the formula has a ‘preset’ framework which is not ‘directly’ connected to the target(s).

Criterion two: a bow/arrows set which hits one target in a ‘field’ must be able to hit the other targets in that same field (field as discipline, such as, physics, chemistry, biology, etc.).

With these two criteria clearly defined, we can now review some archery acrobatics and to see whether they are numerologies or not.

Target oen: Planck data (dark energy = 69.2; dark matter = 25.8; and visible matter = 4.82).

Bow/arrow set one: the Pimple model (or house-address name model) ---
1. There are 48 mass-particles, and each of these 48 has its own ‘measured’ mass-value which is different from all other mass-particles.
2. For every mass-particle, its measured mass is only a ‘name-tag (or a Pimple)’ for its house which also encompasses all 47 other mass-particles. That is, for every visible particle, it is only a ‘part’ of the house which houses the entire 48 particles.

With this bow/arrow, only 7 of the 48 mass-particles [the first generation matter (not anti-matter)] gives out lights (excluding e-neutrino). Thus, the dark mass/visible mass ratio = [41 (100 – w)% / 7] . The *w* is the percentage of the dark matter which does give out lights. According to the AMS02 data, it is between 8 to 10%. By choosing w = 9, the d/v ratio = 5.33 (while the Planck data shows d/v ratio = 25.8/4.82 = 5.3526). Details, (see ).

This bow/arrow is ‘preset’, no reverse-engineering of any kind in it. One free-parameter of ‘w’ is a ‘physics’ parameter, not an ad hoc numerology. That is, the ‘w’ should work in some other bow/arrow set in addition to this target only.

For dark energy, the bow/arrow is ‘preset’ as an iceberg model --- that is the Time, Space and Mass (dark + visible) form an iceberg system, while the mass is the iceberg. And, they three take the *equal* share.
So, the dark mass = [(33.3 – 4.82) x (100 -w)%] = 25.91 (while the Planck data is 25.8), with d/v ratio = 5.37. The w = 9% here is the melting ratio from the dark matter. Thus, the dark energy = 66.66 + [(33.3 – 4.82) x w%] = 66.66 + 2.56 = 69.22 (while the Planck data is 69.2).

There are three points on this.
a. The dark-mass target was hit twice on the bull’s-eye by two different sets of bow/arrow.
b. The free-parameter ‘w’ works for both sets of bow/arrow.
c. The ‘w’ is able to encompass the small (in 10%) adjustment in Planck data.

Target two: (1/Alpha) = 137.0359 …
Bow/arrow: the structure ‘number’ model --- the structure of this universe is 100% decided with two numbers (64, 48), see

Target three: Weinberg angle (from 28 to 30 degrees)
Bow/arrow: the same as for the Alpha calculation. See .

Target four: Cabibbo angle (about 13 degree)
Bow/arrow: the same as for the Weinberg angle.

Target five: the r-ratio of B-mode in the CMS. The value is currently unknown but claimed by BICEP2 as r = 0.2
Bow/arrow: the same set as for Cabibbo/Weinberg angles and Alpha, the structure ‘number’ model --- the structure of this universe is 100% decided with two numbers (64, 48). The 48 forms the mass-field, encompassing 48 mass-particles. The 16 forms the energy-field (the dark energy). Thus, the energy/mass ratio = (64 – 48)/64 = 0.25

In fact, those five targets are hit by the same bow/arrow.

andrew said...

An article on CKM and PMNS matrix mixing angles from e-mail from a reader:

andrew said...

The latest global fit shows about a 5% difference between the results, which is about 2.5 sigma and probably means that the three parameter fit is false.

0.0424446 v. 0.0408221 diff 0.0016225.


andrew said...


andrew said...

Last link in paper is dead. New Link is