M Theory Lesson 320

Consider the triple product of three such scaled $R_2$ factors, where $a$, $b$ and $c$ are real. By design, the matrix $M$ is the sum of a real $1$-circulant and an imaginary $2$-circulant. It follows that it is a doubly magic matrix, determined by a $2 \times 2$ block. Consider the block $B$ of the norm squares of $M$ given by After renormalising, this is fitted to the canonical $2 \times 2$ block of the CKM matrix (norm squares): The full CKM matrix follows from the magic properties. I used Maxima for this, and a row sum parameter for $|M|^2$ of $192.9986$, which turns out to be close to the natural normalisation of $M$ given by
$(a^2 + 1)(b^2 + 1)(c^2 + 1)$

M Theory Lesson 319

Let us consider Brannen's circulant sum form for the tribimaximal mixing matrix. There are a few ways to do this, but we will start by mixing up $1$-circulant and $2$-circulant parts to obtain: Note that the $2 \times 2$ block is not $F_2$, but rather the MUB alternative $R_2$. This gives a purely two dimensional product form for the tribimaximal matrix: in terms of $R_2$ type operators, with $4$ parameters $A$, $B$, $X$ and $Y$. Note that it doesn't matter much if we swap $\sqrt{2}^{-1}$ for $\sqrt{2}$. There are alternative forms of this result, again making the tribimaximal matrix robust.

Quote of the Month

Promotions are based on merit at universities; however, it has been interesting to note that at Canterbury University the number of women in the position of associate professor has recently more than doubled following an internal complaint by female staff members to the Human Rights Commission. The complaint regarded a ‘male-dominated university culture’ where women struggled to progress. Eight female promotions to associate professor took the total of women in that position to 14.

From an NZ Ministry response to an OECD 2005 questionnaire on women in scientific careers.

Go LHCb

This is so beautiful that I simply had to steal it and put it on AP. Courtesy of LHCb, from a real $7$TeV event reconstruction for a $B^{+}$ vertex, the invariant mass distribution for $\mu^{\pm}$ pairs associated to the the $J/ \Psi$ particle:

M Theory Lesson 318

In looking at this parameterisation of Koide matrices, we are shifting the mass scale for each triplet, but there is something natural about doing this. That is, for a mass scale set by the triplet itself, the mass gaps are simply described in terms of the eigenvalues for the given $\phi$. For the record, there are only two angles that occur in the analysed masses, and these have zero sum eigenvalue sets:
1. neutrino, $\pi/12 + 2/9: -0.0791, -1.6911, 1.7703$
2. electron, $2/9: -0.5937, -1.3572, 1.9508$
which give fixed mass gap ratios of $2.147$ and $4.333$ respectively. Anyway, let us look at the first set of meson data in Carl's hadron paper.

In shifting scales we want to look at ratios $s/v$, in terms of Carl's variables. For the q-qbar mesons this gives us the eight values: $0.2647$, $0.2322$, $0.1024$, $0.03172$, $0.1041$, $0.0807$, $0.0356$, $0.1874$. Now it turns out that the $0.1041$ makes things messy, so without any guilt whatsoever I am going to cheat and leave it out, giving $6$ mass gaps for this set of mesons. Arranging them in order and plotting against the counting number $n$ (in a Maxima graph) we find: Oops, that's only $5$ data points. I haven't tried the other data sets, but that could be fun too!

M Theory Lesson 317

The Koide mass matrices are written in terms of two parameters, which we will call $n$ (placed on the diagonal) and $\phi$ (the phase angle). The eigenvalues of such a matrix may be expressed as $\lambda + n$ for an eigenvalue $\lambda$ of since adding $n \mathbf{1}$ to the Koide matrix leaves the eigenvectors intact. Note that the eigenvalue set for $K$ always sums to zero, so these three $\lambda$ describe the mass differences within a triplet. For example, at $\phi = 2/9$ the eigenvalues of $K$ are $\{ -0.59367, -1.35715, 1.9508 \}$. At a cubed root, the eigenvalues are $\{ -1, -1, 2 \}$, like the entries of the idempotent $S$. All possible zero mass triplets are thus described by the $U(1)$ of the variable $\phi$. As $\phi$ varies, a given eigenvalue varies between $-2$ and $+2$.

Modular Fun

After Einstein and Grossmann developed relativity, physicists grew to love Riemannian geometry. After Heisenberg rediscovered matrices, physicists grew to love linear algebra, and with the 20th century development of particle physics they came to absolutely love groups, especially continuous ones.

When they use matrices to describe groups, it is usually in the context of representation theory, where the matrices are linear maps between vector spaces over a familiar continuum, such as the real or complex numbers. For the field with one element, however, matrices are no such thing. But we still like to multiply matrices, and to study interesting finitely generated sets of matrices.

For example, consider allowing a zero element to be attached to a group like set. Groups themselves don't have zeroes, because a zero has no inverse. Let all other members have inverses, but note that members may be zero divisors. Although this is not a ring, because there are only products, we love the zero divisors. So taking two familiar matrix generators: we have an infinite example analogous to the $12$ element group $A_4$ (which has generators $S$ and $T$ such that $S^2 = 1$ and $T^3 = 1$). Here the relations are $S^2 = S$, $T^3 = 1$ and $(ST)^3 = (STT)^3 = (TS)^3 = 0$. The list consists of all words in $S$ and $T$ such that $S$ occurs singly and $T$ singly or doubly. Most strings collapse or terminate eventually, but $(ST^2 ST)^n$ (which is circulant) just keeps on going. Everything has an inverse and everything is a zero divisor. Idempotents like $S$ are useful for creating zero divisors.

In summary, this object has a zero, an identity and multiplication, just like the field characters that we like to put into matrices. It is a bit like a noncommutative semiring, except that semirings have addition, and it is certainly a monoid and a semigroup, but these don't usually involve zeroes. Unromantic, it is called a semigroup with zero.