Arcadian Pseudofunctor

Last Post

2012-02-08T11:48:00.000+13:00

Kea hereby declares Arcadian Pseudofunctor closed.

Conferences Coming

2012-02-04T15:18:00.005+13:00

Amongst the many interesting, upcoming conferences that I am unable to attend is Cosgrav 12: International Conference on Modern Perspectives of Cosmology and Gravitation. Oh, look. The invited speakers all seem to be men, yet again. Perhaps I won't be missing much.

Further away, there is Moriond in March, and a workshop on Scattering Amplitudes at Cambridge in April. Anastasia Volovich is of course the token woman there, and participation is by invitation only. Meanwhile, the icHEP12 website has a cute disclaimer:

In particular, no bona fide scientist will be excluded from participation on the grounds of national origin, nationality, or political considerations unrelated to science.

Yes, we need that qualification there, don't we?

Tevatron Strikes Back

2012-02-03T10:23:00.006+13:00

A Fairy Update! A new CDF analysis of $10$ inverse femtobarns in the $\gamma \gamma$ channel. No fairies.

Does that $125$ GeV Higgs look a little more uncertain now? Not really, since this is at ten times the SM $\sigma$. Only CMS and ATLAS can tell us the answer.

Color Z News

2012-02-01T12:45:00.004+13:00

As discussed a few years ago around the physics blogosphere, one possibility for new physics at the LHC was a $720$ GeV state that had something to do with $Z'$ bosons. Evidence for $Z'$ type anomalies, of the non standard kind, has grown in recent years. Now we hear that the only possible new physics seen by CMS is in fact a three event HSCP signal somewhere near $720$ GeV.

Not much to speak of, three events, but definitely something to ask ATLAS about.

Quote of the Month

2012-01-31T10:32:00.002+13:00

The truth is that sustainable development is fundamentally a question of people’s opportunities to influence their future, claim their rights and voice their concerns. Democratic governance and full respect for human rights are key prerequisites for empowering people to make sustainable choices. The peoples of the world will simply not tolerate continued environmental devastation or the persistent inequality which offends deeply held universal principles of social justice. Citizens will no longer accept governments and corporations breaching their compact with them as custodians of a sustainable future for all.

From the new UN report: Resilient People, Resilient Planet: A Future Worth Choosing

Quote of the Week

2012-01-31T09:00:00.001+13:00

I'm vehemently against copyright infringement: the files I lost were created & owned by me for my job.

One of 150 million legitimate Megaupload users, whose data has been stolen and censored by the US government.

Dark Matters

2012-01-27T10:33:00.003+13:00

Recent dark musings now summarised on vixra.

More Than Men

2012-01-26T10:51:00.000+13:00

From the More Than Men project:

I can sense the comments coming in already. ”But Sasha,” you ask, “isn’t that just establishing quotas? Aren’t we ignoring qualified dudes because we’re stocking our conferences with woman and minority speakers?” My answer is “No, don’t be so ridiculous and offensive by assuming that trying to create a diverse group of speakers means choosing less-qualified speakers in order to have more than white men talking.

Let me say that again, very clearly: If you think that ensuring a diverse representation of genders, races, and physical abilities in your conference speakers means creating a conference stocked with unqualified “tokens” you are making a prejudiced assumption. Your internalized bias is showing. Don’t freak out, it happens to the best of us. So take a deep breath, maybe have a laugh with yourself for freaking out like that and then look a little deeper for those experts who are just too busy doing work and making a name for themselves that you haven’t heard of them yet. Or maybe you have heard of them but didn’t think of them because our brains are lazy and go for the comfortable and familiar rather than stretch a little.

Year of the Dragon

2012-01-22T11:06:00.002+13:00

Quote of the Week

2012-01-19T09:48:00.002+13:00

Don't take our generation lightly

Arfa Karim Randhawa (1995 - 2012)
(and at the BBC)

Witten on Yang Mills

2012-01-18T14:42:00.008+13:00

Witten speaks at the latest Simons Center conference on mass gaps in Yang Mills theories. In four dimensions, he stresses that the role of the mass gap is to hide quantum phenomena at large distances, and that no simple perturbation theory can really see the mass gap. A theme of the conference seems to be a renewal of interest in constructive approaches, starting with tractable questions.

So the talk focuses on three dimensional theories, starting with the pure Yang Mills action for $R^3$. Here a mass gap exists on dimensional grounds, with $m = w g^2$ for $w$ some constant and $g$ the coupling. But what happens at long distances in three dimensions? Several approaches are briefly reviewed, including (i) a string tension Hamiltonian approach, and (ii) Feynman and Singer's Hamiltonian for the space $R^2$. In (ii) there is the usual (infinite dimensional) space of connections $A$ and gauge transformations $G$. Witten writes down the basic Hamiltonian

$H = g^2 \int \textrm{Tr} E^2 + \frac{1}{g^2} \int \textrm{Tr} B^2$

noting that the first term dominates in the limit $g^2 \rightarrow \infty$, and that the eigenvalues of $H$ should be basic numbers, like $\zeta (2)$.

But he appears to believe that a currently tractable approach to mass gaps would look at non-trivial CFTs, for the following reasons. First observe that (in three dimensions) if we add a few essentially massless fermions (like up and down quarks) there is no mass gap. Now we can add a Chern-Simons term, as a mass gap term, to the usual Yang-Mills action, where $m$ is proportional to $k$, the CS integral coefficient (the fourth dimension is irrelevant). If this (CS) theory is rigorously defined in three dimensions, there should be a theory for any $3$-manifold $M$ with any Riemannian metric. Scale the metric by a factor $t \rightarrow \infty$. That is, we imagine that human experiments are limited by the characteristic scale of the manifold $M$, and at long distances there is the TFT (IR limit). This is of course the context of Jones polynomials and knots.

The Usual

2012-01-18T14:20:00.002+13:00

As expected, the motives paper was summarily rejected by the one referee who looked at it, with some brief comments about my suspicious use of (now) trendy terms. Why this took over six months is beyond me. I also receive about one automated postdoc rejection each day at this time of year (not everyone bothers sending out rejections). I know they are automated, because they usually include a phrase like you and other excellent candidates or we considered your application carefully.

Internet Freedom

2012-01-18T12:23:00.002+13:00

We support the Sopa protest.

Theory Update 136

2012-01-16T17:22:00.005+13:00

We can draw the Jacobi rule of knotty gauge theory a slightly different way.

Recall the connection to knots in renormalisation theory. The loop formed by the traced braid diagram forms the loop of a chord diagram. The example shown on the right is a trefoil knot, with zeta value $\zeta (3)$. A double copy of the right hand side of the Jacobi rule appears in the 4T relation. Now the trivalent vertex on the left of the Jacobi rule is interpreted as a resolution (into trivalent vertices) of the diagram that is half way between the two diagrams on the right (namely the $4$-valent diagram of the node cross over). The pair of Jacobi rules gave us the color kinematic duality of $N = 8$ supergravity.

Kea At Work

2012-01-15T11:21:00.004+13:00

far away

Theory Update 135

2012-01-12T14:19:00.006+13:00

A fascinating new paper by Yang Jiao and Salvatore Torquato defines three dimensional packings of truncated tetrahedra, using a lattice vector deformation parameter $g$ that takes values in the interval $[0,2/9]$.

A trucated tetrahedron has four hexagonal faces and four triangular ones. The maximal $g = 2/9$ corresponds to the densest known packing of truncated tetrahedra, with packing fraction $207/208$. The packing has tetrahedra shaped holes. With Conway they have also constructed a packing with fraction $23/24$, and considered packings involving octahedra.

Theory Update 134

2012-01-11T15:01:00.007+13:00

In the new lectures, Arkani-Hamed reviews basic twistor geometry. As usual we work with the $N = 4$ theory, because that is the amount of supersymmetry in the complex numbers $C$. The bioctonion case of $C \cdot C$ allows us to discuss $N = 8$ supergravity. In the lectures, traditional supersymmetry is used to obscure the categorical structure of the combinatorics. Recall that a trivalent vertex may be used to represent a point associahedron, and comes in two allowed helicity triplets.
The counting of $\textrm{N}^{k}\textrm{MHV}$ terms for $n$ legs is given by the Catalan number of the appropriate associahedron. The distinct trivalent vertices may then define a directed associahedron edge, via their representation as points.

Arkani-Hamed stresses the importance of Grothendieck's mathematics and other motivic mathematics, such as the Connes-Kreimer approach to renormalisation. However, there still appears to be a belief that this will all work out without any abstract airy fairy category theory. Although he admits a previous long term allergy to all things motivic, there is obvious excitement about the new connections to the interests of mathematicians like Deligne. Unofficially, the lectures are about the positive Grassmannian (which is, of course, all about generalised associahedra).

News from Fermi

2012-01-11T13:51:00.002+13:00

From Fermi LAT:

One example is the well-known radio galaxy NGC 1275, which is a bright, isolated source below 10 GeV. At higher energies it fades appreciably and another nearby source begins to appear. Above 100 GeV, NGC 1275 becomes undetectable by Fermi, while the new source, the radio galaxy IC 310, shines brightly.

Quote of the Month

2012-01-10T08:52:00.002+13:00

From Woity Toity:

As usual, Arkani-Hamed makes wildly enthusiastic claims. In this case, he claims to have finally found a remarkable new understanding of the subject, based on some combinatorial objects and dramatic new mathematical ideas. He does note that this still hasn’t been written up, and that it’s the third time in the past year that he has thought he had things understood, with the last two times not working out.

Women A Mystery

2012-01-09T16:16:00.002+13:00

It seems the biggest news of the day, around the blogs and in the press, is that Hawking said women were a complete mystery, which is not surprising for someone who never actually had to work with any women. So I scoured the bemused stories, looking for any mention of sexism or feminism, with regard to any of the parties involved, but of course to no avail. Perhaps there is one somewhere. Only in the comments sections did I find some relief. Melodie Benford summed it all up:

I find this article so freakin sexist ...

More Quick Neutrinos

2012-01-09T12:29:00.002+13:00

I was told that New Scientist had yet another story on the impossibility of superluminal neutrinos, but it turns out to be yet another case of imposing a classical concept of energy on a phenomenomen that quite clearly won't be adequately explained by such a theory, if true. Weak logic makes poor news. Back in OPERA month, we mentioned classical tachyons, whose low energy states move at the fastest speed, but this old rule could easily be inverted in quantum gravity, so that tachyons travelling just faster than $c$ are the black sheep.

A more interesting article from this edition is a note on dark matter in globular clusters.

Quick Neutrino Review

2012-01-05T16:44:00.003+13:00

Some years ago we were wondering about the relation between neutrino physics and the McKay correspondence, for the exceptional groups $E_6$, $E_7$ and $E_8$. Much has happened since then, in the twistor world, in the Jordan algebra world, and in the cat world. It is clear that exceptional structures play an important role in gravity.

Dark Matter Update

2012-01-05T15:27:00.013+13:00

Let us recall the basic dark matter fractions in the quark gluon plasma. That's $3 / 4 \pi$ for the dark matter fraction and $9/ 4 \pi$ for the (so called) dark energy, with the baryonic fraction $\Omega_{b} = 1 - 3/ \pi$, as confirmed by WMAP.

In the information theory, we need two kinds of dimension. First, path bases define an integral dimension, $27$ for the Jordan algebra. But complex matrix entries tend to be characters for finite fields, so that the root of unity corresponding to a phase $2 \pi /27$ would specify the cardinality of a field with $27$ elements. These two dimension types should be related by the mirror physics that explains the dark matter fractions.

A dark matter ansatz: the total dark sector density fraction $\Omega_{d}$, as a fraction of the character dimension for $27$ paths, equals the observable universal fraction of the integral dimension. In other words, we would like to write

$ \frac{3}{\pi} \frac{2 \pi}{27} = \frac{2}{9} $

so that $\Omega_{d} = 3/ \pi$. Now observe that we can split the dark sector into quark and lepton components. The leptonic dark matter fraction $3/ 4 \pi$ now corresponds to the fraction $1/18$, a single quark path from the quark set. Similarly, the dark energy fraction $9/ 4 \pi$ corresponds to the fraction $1/6$, a single lepton path. The total $3/ \pi$ is then the only solution obeying this quark lepton complementarity.

In the quark gluon plasma, the minimal viscosity to entropy ratio is $1/ 4 \pi$. As a character dimension fraction, $1/ 4 \pi$ corresponds to $1/54$, a single basis path in the full bioctonion algebra.

Auld Lang Syne

2012-01-03T12:51:00.004+13:00

As Kitaev will tell you, quantum codes are about things like ribbon categories. Ribbons and knots, knots and ribbons.

Back in the mid '90s, everyone was into quantum field theories with knots. I was supposed to be working in standard lattice QCD, in a Physics department, but being enamoured of knots I instead found myself lost amongst the deserted stacks of the mathematics library, and chatting to mathematicians like Bai-Ling Wang and Alan Carey. At that time, classical geometry was still boss, and everywhere I looked there were manifolds, manifolds, manifolds. The mathematicians weren't at all bothered by all the manifolds, but as a physicist who haunted the library stacks, I was looking for something more, let's say, background independent. Alan gave me some excellent advice, and told me to work on my thesis and forget about the knots (but naturally, the advice was bound to be ignored).

Anyway, a standard gadget in the knotty manifold world was a framed knot, where one thickened a knot into a kind of twisted torus so that it sat in a nice way in a larger space. By drawing lines along the torus, one could get ribbon diagrams. For example, quantum mathematicians liked symplectic spaces, such as $T^{*}S^3$, the cotangent bundle of the three dimensional sphere $S^3$. Such a six dimensional space has three dimensional Lagrangian submanifolds, good for putting knots in, and with which mathematicians can do some knot wizardry.

In time, the trend moved towards combinatorial alternatives. The only constant was the universal obsession with knots and ribbons. Six strands for three ribbons. I could no longer afford to listen to mathematics lectures in Princeton or Canberra, and nobody was willing to supervise my thesis proposal on a lovely $7$ dimensional Chern-Simons theory, so I went to Wanaka and bumped chairs and served chips to hungry skiers, and worried instead about the bottletops that my flatmate insisted on leaving all over the floor.

Higgsy Codes IV

2012-01-01T14:41:00.006+13:00

The cubed root symbol $\omega$ may be thought of as a non zero element in the four element field. In the hexacode description of the miracle octad there are $3 \times 6$ arrays, that is, three length $6$ words in four letters. A key three word matrix is built from the $3 \times 3$ identity and the Fourier operator,
in analogy to the (gauge fixed) $2 \times 4$ arrays that create a Grassmannian Minkowski space in the $2 \times 2$ twistor formalism. Conway has a similar construction for the ternary code, known as miniMOG. The hexacode is used to specify five points from which a Steiner octad is created.

Higgsy Codes III

2012-01-01T12:47:00.004+13:00

On Cullinane's page on the miracle octad generator we find a beautiful note by Robert Wilson, the Wilson of the new Leech lattice, for R. T. Curtis' 60th birthday in 2007. The note looks at Lie algebra root systems in terms of integral quaternions and octonions. He defines an order $24$ group

$\omega \equiv \frac{1}{2} (-1 + i + j + k)$

with elements $\{ \pm 1, \pm i, \pm j, \pm k \}$ and $\{ 1, \omega, \overline{\omega} \}$, and the neat relation $i^{\omega} = j$. This gives a root system for $D_4$ and is extended using norm $2$ elements to a root system for $F_4$. Of course, with the octonions, we get $E_8$. Morally, the octonions are responsible for all exceptional Lie algebras. The exceptional algebras $G_2$, $F_4$ and $E_8$ have orders $14$, $52$ and $248$, with respective prime factors

$7 = 1 + 2 + 2^2$
$13 = 1 + 3 + 3^2$
$31 = 1 + 5 + 5^2$.

Wilson spots the combinatorics of finite projective planes, but then notes we can also write the triality

$14 = 2 (2^3 - 1) = 16 - 2$
$52 = 2 (3^3 - 1) = 54 - 2$
$248 = 2 (5^3 - 1) = 250 - 2$

where, as kneemo keeps saying, these numbers appear in qutrit symmetry groups. Recall that $14$ also counts the vertices of the three dimensional associahedron, and $16$ the extension to crossing partitions, while $54$ is the dimension of the bioctonion algebra. Speaking of trialities, the largest Heegner number is also written as

$163 = 1 + 1 + 1 + 2^3 + 3^3 + 5^3 = 54 + 54 + 54 + 1$

where $2^3$ counts vertices on a parity cube, $3^3$ paths for the tetractys, and $5^3$ paths for $5$-valued states.

Higgsy Codes II

2012-01-01T03:01:00.003+13:00

One more pertinent fact about the number $28 = 27 + 1$: it counts the number of bitangents to a quartic curve, or pairs of lines from the $56$ lines on a (degree $2$) del-Pezzo surface. Thus it gives the number of (odd) theta characteristics. For us, $56 = 2 \times 28$ will always be the dimension of the FTS for the $3 \times 3$ octonion Jordan algebra, or the $56$ triangles of the genus $3$ Klein quartic, which has an automorphism group of order $168 = 3 \times 56$ $= 7 \times 24$, as there are $7$ sides on the heptagon tiles.

The Mathieu group $M_{24}$ can also be constructed using the Klein quartic symmetries, along with an extra permutation associated to the (Leech lattice's) small cubicuboctahedron, as mentioned recently by kneemo. If we can have this much fun with classical codes, just imagine how much fun we can have with braids!

Higgsy Codes

2012-01-01T02:01:00.019+13:00

Dharwadker's four color theorem uses the Steiner system $S(5,8,24)$, otherwise known as the Witt design. To construct the Steiner system, one starts with the extended binary Golay code. This has $2^{12} = 4096$ length $24$ words, selected from all possible binary words by writing a list of length $24$ binary words so that each word differs from the previous ones in at least $8$ places. It turns out that the code words are partitioned

$4096 = 1 + 759 + 2576 + 759 + 1$

into words with no $1$-bits, words with $8$ $1$-bits, words with $12$ $1$-bits, $16$ $1$-bits and $24$ $1$-bits, respectively. The $729$ words with $8$ $1$-bits form the Steiner system, which is defined as a set of $24$ points, along with the $759$ special subsets of size $8$, called blocks, such that every $5$ points is contained in one of the blocks. The number $759$ is very interesting. It can be partitioned into

$759 = 1 + 280 + 448 + 30$

which counts, respectively, the number of subsets that intersect a given subset in $8$, $4$, $2$ or zero elements. That is, there are $30$ blocks with no intersection with a given block, and $729 = 27^2$ with a non empty intersection. We write $280 = 4 \times 70$, $448 = 16 \times 28$ and $30 = 30 \times 1$, where $70$, $28$ and $1$ are weights for the Reed-Muller code $RM(2,3)$. The $4$, $16$ and $30$ appear in the bottom row of the Pascal triangle below. The number 759 is the summit of this triangle, which counts the number of blocks that (i) contain a point set with $n$ elements and (ii) are disjoint from another set with $m$ elements. The union cardinality $m + n$ is fixed in each row. So there are $253$ occurrences of any given object, and a complementary $506$ blocks that do not contain it.

There are $42504$ ways to choose $5$ objects from $24$, and $42504 = 759 \times 56$. To obtain the $8$ bits, after choosing the first $5$, there are $969$ possibilities (namely $3$ out of $19$) and we see that $969 \times 759$ is indeed the binomial coefficient $B(24,8) = 735471$. Why the redundancy factor of $56$, the FTS dimension? This counts the number of ways of choosing $3$ objects from $8$, namely the $3$ that are omitted from the set of $5$.

The Mathieu group $M_{24}$ has a 729 dimensional permutation representation with suborbit lengths given by the $1$, $30$, $280$ and $448$. Similarly, the Mathieu group $M_{12}$ is associated to the extended ternary Golay code. The two binary and two ternary Golay codes, with lengths $11$, $12$, $23$ and $24$, are the only binary and ternary codes with suitably nice properties. There are $729 = 27^2$ words in the ternary code of length $11$, and the extended ternary code also forms a Steiner system, on $12$ objects. The $12$ objects are thought of as a projective line over the $11$ element field $F_{11}$ with the addition of $\infty$. The blocks are built using fractional linear transformations, which we know how to cover with Jordan algebra braids.

2012

2012-01-01T01:01:00.000+13:00

Judgement Year.

Higgs Koide Triplets

2011-12-19T14:38:00.003+13:00

One more note before I leave. Recall that the Koide matrix has a real and imaginary part. Observe that the real part gives a $(W,W,Z)$ type triplet, whereas the purely imaginary part gives an annihilation triplet containing the photon. Thus all four electroweak bosons are accounted for in a Koide operator.

Summer Time

2011-12-18T09:56:00.004+13:00

Newton's birthday is fast approaching and so I must soon head into the mountains for a week. There is a fine spell of weather ahead. Still a lot of snow up high, even after a hot November, which should make the pass to the glacier easy going. Until 2012 then.

The Condensate Scale II

2011-12-17T19:08:00.017+13:00

Yesterday we didn't write out the $(W^{+}, W^{-}, Z)$ bosons as a Koide triplet, so let us do that now. This triplet is clearly of the form

$(1,1, \frac{\alpha}{2})$

where $\alpha \cos \theta_{W} = 2$, for the Weinberg angle $\theta_{W}$. The eigenvalue set at the Koide phase $\phi = 0$ is

$\lambda \in \{ 1 + \alpha, 1 - \frac{\alpha}{2} \}$

with multiplicity $2$ for the second eigenvalue. At $\theta_{W} = \pi /6$ we obtain

$\lambda \in \{ 1 + \frac{4}{\sqrt{3}}, 1 - \frac{2}{\sqrt{3}} \}$.

Squaring and summing the rest mass triplet for the Higgs, we obtain $m_{H} = 11/2$ exactly, in the units chosen, which just happen to be the Koide scale for the top quark triplet.

The Condensate Scale

2011-12-16T14:07:00.003+13:00

Now that kneemo has also spotted the mathematical sophistication of the four colour Higgs, let us look once again at the corresponding electroweak scale $M = 2m_{H}$. Since the $W^{\pm}$ bosons have equal rest mass, we write

$M = m_{Z} + 2m_{W}$

or rather

$M = m_{Z} + 2 m_{Z} \cos \theta_{W} = m_{Z} (1 + 2 \cos \theta_{W} )$

where $\theta_{W}$ is the Weinberg angle. Looks like a Koide eigenvalue, doesn't it? Perhaps it suggests that the ratio of $M / m_{Z}$ is the square root of some other critical number (which equals roughly $7.5$). The $r$ parameter is $1$, which is one of the neutrino mixing parameters, interpretable as the phase $\pi /4$. If $\theta_{W} = \pi /6$, as suggested by the four colour paper, we have a triplet containing

$M/m_{Z} = 1 + \sqrt{3}$

along with rotated eigenvalues $1 - \sqrt{3}$ (which gives around $67$ GeV) and $1$ (for $m_{Z}$). Since all the massive bosons are created from the color $Z$ ribbons, this seems interesting.

All Quiet On The Front

2011-12-16T08:08:00.003+13:00

Pretty quiet around the physics blogosphere, isn't it? Ah, yes, everyone is locked up in the office again, furiously writing about the four color theorem as if it was their idea, even though they only heard about it for the first time yesterday. We should see another arxiv paper onslaught this week, probably triple the size of the superluminal neutrino theory burst.

So, A Condensate Higgs

2011-12-15T17:51:00.004+13:00

So the observed Higgs mass simply agrees with the condensate formula

$2 m_{H} = m_{W} + m_{W} + m_{Z}$,

where a pair of Higgs form a Cooper pair. Anyway, a Standard Model Higgs was always essentially a condensate. But if we can elaborate further on the structure of this condensate, perhaps with our zoo of mirror particles, then in what sense does the Higgs exist? It exists because it reproduces the SM cross section correctly, as observed at the LHC. That's what matters. After all these decades, the Standard Model finally finds its home.

Through The Looking Glass IV

2011-12-15T09:42:00.013+13:00

I glanced at the Higgs paper a year or two ago, but since I was not familiar with the proof of the four colour theorem, and I did not believe in localised Higgs states, I put it aside and forgot about it. Here is an old blog post by the authors.

Now we see that the construction looks much like the $12 + 12 = 24$ dimensions of a tetractys core. There are $12$ sectors on each sheet of their particle frame, left and right. Like the $B_3$ braids, each sector has a triplet structure.

The Higgs particles are supposed to form Cooper pairs, which must be able to generate the mass of the electroweak bosons, hence their mass formula. Of course, this formula is then independent of the construction that they happen to use, and may be considered evidence for a condensate. Anyway, picture examples of particle assignments in the paper include:

Through The Looking Glass III

2011-12-15T06:56:00.011+13:00

In 1995 I was in Adelaide attending mathematics workshops on topological field theories. The excellent speakers included Lisa Jeffrey and Frances Kirwan. One day, a few of us went to the pub. Vaughan Jones was talking about knots and subfactors, and in particular the values in the subfactor index series that are less than $4$, given by

$4 \cos^{2} \frac{\pi}{n}$

for $n \geq 3$ an ordinal. I was sitting next to Professor Jones at the pub, and very keen to understand knots, so I asked him where the coefficient of $4$ came from. A local mathematician immediately scoffed at my stupid question, but Professor Jones frowned and thought for a minute, and finally he said: I think the $4$ is the $4$ of the four color theorem. So let us look once again at the successful Higgs mass prediction by Dharwadker and Khachatryan,

$M = \frac{1}{2}(m_{W^{+}} + m_{W^{-}} + m_{Z})$

Observe that $2 M$ is $250$ GeV, the electroweak scale. Fourier twisting the $(W^{+}, W^{-}, Z)$ triplet gives a $(Z,Z,Z)$ color triplet, and dualising this triplet we get the right handed $(e^{+}, e^{-}, \nu)$. The left handed triplet comes from $(W^{+}, W^{-}, \gamma)$, which we can send to the $(e^{+}, e^{-}, \gamma)$ of creation and annihilation with a mixture of Fourier duals. Three massive (weak interaction) bosons and three (electromagnetic) particles, two sides of a coin.

The Higgs Hypothesis

2011-12-14T15:08:00.003+13:00

Let us now consider the (possibly an effective) Higgs hypothesis. Since it need not resemble the spin $1/2$ and spin $1$ braids, this one extra state could arise as a three dimensional state formed by symmetrizing the unused tetractys diagonal,

$XXX \oplus YYY \oplus ZZZ$

for some ill defined $\oplus$. Recall that the $XXX$, $YYY$ and $ZZZ$ paths define the axes of the Koide space. Since planes in this space are the basis of triality, we could draw this product as a trivalent vertex with an $H$ at the centre point. This vertex closes the doubled tetractys into a tetrahedron, now with $56 = 2 \times 28$ labels.

Note that Tony Smith's original 1993 calculation of the (so called) truth quark mass, based on triality, gave an estimate of $130$ GeV, not far from the observed value.

Higgs Update

2011-12-14T12:50:00.007+13:00

The combination of results does look like very good evidence for a SM Higgs at $125$ GeV. I will be taking it seriously. However, on looking at the ATLAS and CMS data, the situation is far murkier than I would have thought possible with $5$ $\textrm{fb}^{-1}$ of data. The ATLAS presentation is especially beautiful. Here is a background-only hypothesis analysis, showing the $125$ GeV peak very clearly against the falling expectation curve.

In the $ZZ$ channel:

And CMS gives us a nice $ZZ$ channel summary:

A Gibbs Combination Plot

2011-12-14T07:08:00.001+13:00

Thanks to the incredible (live time) work of Phil Gibbs.

A Four Color SM Higgs

2011-12-14T06:32:00.004+13:00

Yesterday, Mitchell mentioned a Higgs mass formula by Dharwadker et al, based on the four colour theorem. This formula correctly predicted the Higgs mass. It is

$m_{H} = \frac{1}{2} (m_{W^{+}} + m_{W^{-}} + m_{Z})$

I'm not yet sure what to make of this, but since it must come from the tetractys diagonal, it suggests something like a degenerate boson Koide triplet $(W^{+}, W^{-}, Z)$. On taking the twisted Fourier dual, this is a $(e^{+}, e^{-}, \nu_{R})$ triplet. Note that it includes no generation multiplicity, but it seems to be about quark lepton mixing. If we include mirror states, we could interpret this triplet as $(e^{+}, e^{-}, \gamma)$, a basic annihilation triplet.

Recall that there are three ways to obtain each of these bosons via composition of $Z$ color states. Thus the boson triplet comes from three copies of the $Z$ color triplet.

Do Fairies Exist?

2011-12-14T05:49:00.014+13:00

Well, it seems I was wrong about that one. The excess may disappear next year, but that seems extremely unlikely. Then again, Strassler and others, who actually understand operations at CERN, remain skeptical that any conclusions can be drawn at this point. Is it an SM or MSSM fairy?

As reported on many blogs, both CMS and ATLAS report a significant excess at $125$ GeV. ATLAS claim a whopping $3.6$ sigma, but the CMS excess is much smaller. Until further data is analysed, the result remains inconclusive, but it certainly looks real, even if the number of events is rather small. The excess is seen in two channels, and strangely exhibits a cross section much larger than a fairy field would. This is considered a lucky fluke, which it probably is.

Congratulations to both teams for the most outstanding work and for showing us the way forward.

Through The Looking Glass II

2011-12-13T12:25:00.006+13:00

Let us say this as simply as possible. Here we use the ordinary quantum Fourier transform, not the twisted version (which sends one neutrino to the photon).

In passing the looking glass, the neutral particles are special. Recall that the set of charged leptons and quarks can be sent through a mirror set in two ways. One can (i) flip charges on all ribbons, by flipping the $B_2$ crossings, or (ii) flip the two crossings in the $B_3$ braid. In case (i) the neutrino sector is unaltered, but the $Z$ color triplets cross the mirror. In case (ii) the $Z$ triplets are unaltered, because they are bosons, but the neutrino sector is swapped.

In both cases, the photon is fixed at the mirror. The fun Fourier transform sends neutrino operators to either photons or $Z$ bosons, and vice versa. Thus (i) and (ii) are just two ways of looking through the glass. So we expect the relation between $Z$ bosons and photons to resemble the relation between neutrinos, mirror neutrinos and photons.

Altogether there are $12$ neutrino states, including generations, and $12$ $Z$ bosons, including color. This defines a $24$ dimensional space with Fourier supersymmetry. The $24$th roots of unity ($\pm \pi /12$) denote this arithmetic dimension. They can only appear as Koide phases for the neutrinos, because $Z$ bosons are diagonal. The left right pairing of braids to create rest mass is the color kinematic duality of Bern et al.

Science Awaits

2011-12-13T10:51:00.002+13:00

The BBC's Pallab Ghosh appears to have the only balanced coverage of tomorrow's fairy funeral. He writes:

But there are two other scenarios. It is possible that neither experiment has detected any signal across the range where the Higgs is expected to be found. This too is incredibly interesting; far more interesting to physicists than finding the Higgs, in fact.

Unfortunately, the science section doesn't admit the possibility that fairies do not exist:

Prof Stefan Soldner-Rembold, from the University of Manchester, called the quality of the LHC's results "exceptional" ... The Higgs particle would, of course, be a great discovery, but it would be an even greater discovery if it didn't exist where theory predicts it to be."

Perhaps it doesn't exist anywhere, Stefan. Meanwhile, Scientific American writes:

Even before the data are out, theoretical physicists around the world are working out the possible implications. Some have pointed out that a value of $125$ GeV would be good news for supersymmetry, a theory that predicts that each particle would have a heavier partner known as a superparticle (at least for particles within the framework of the Standard Model of particle physics, the currently accepted description of the subatomic world).

Yawn.

Through The Looking Glass

2011-12-12T16:24:00.020+13:00

When Graham Dungworth and I wrote about mirror neutrinos, I was rather fixated on the appearance of the last braid in a natural set of $B_3$ ribbons. Let us look once more at the Z boson, which comes in two forms, from the Fourier transform $F$ of the mass splitting circulants.

A pair of Koide phases, $\phi$ and $\overline{\phi}$, also result in a conjugate pair of diagonals, each a rotation of the cubed roots in the plane.
We use minus signs to denote a mirror pair of circulants, giving a triplet of sixth roots on the $Z$ boson side. Altogether there are $12$ neutrinos, including generation, and $12$ $Z$ bosons, including color. Recall that the mirror neutrino $- \pi/12$ phase results in the CMB temperature. For the charged leptons, this mirror phase component essentially vanishes, or equals $2 m \pi$. Thus we associate the interval $[- \phi, + \phi]$ with neutral (or partially neutral) mixing.

The CMB suggests not one, but a vast collection of particles behind the mirror. Infinity meets small ordinals, as in the contrast between the analysis of the CMB and the quark gluon plasma. Across the mirror there are three mirror neutrino temperatures: the past, the present and the future.

When the phase $\phi$ is tripled, it splits $[- 3\phi, + 3\phi]$ into six segments, shrinking the six segments formed by the basic sixth roots of unity. The phase shrinkage of $2 \pi$ to $2 \phi$ is reminiscent of the conical singularities in topological gravity, where one glues the two edges of a phase interval to create a cone, defining a point mass at the vertex.

Theory Update 133

2011-12-12T09:18:00.008+13:00

If we start with two charged quark hexagons, the third hexagon in the lattice could become a $(b,c,s)$ triplet.
Recall that the phase for the $(b,c,s)$ triplet is $18/27$, three times the charged lepton phase. The phase for a charged triplet is $2/27$ or $4/27$. Three hexagon tiles form the core of a honeycomb dual to a four qutrit extended tetractys. Note that the outer polygons here are rather asymmetrical. An alternative uses three charged triplets, including one non standard configuration:
This choice maintains (charge) triality on the trivalent subdiagrams.

Back to Koide V

2011-12-10T10:47:00.004+13:00

A brief note for those who are following proceedings. In allowing paths through all eight quadrants in three dimensional $XYZ$ space, we can see the geometry of $(b,c,s)$ type quark triplets (with the correct Koide eigenvalue signs). We can also split the central (six path) vertex on a tetractys into convenient subsets. For instance, take three separated $1$-circulant paths ($XYZ$, $YZX$ and $ZXY$) in the quadrant $+++$, and the three separated $2$-circulants in another quadrant. Recall that although we start with diagonal ($\sqrt{m}$) space, the Fourier transform will take us to the circulant space and back again.

Theory Update 132

2011-12-09T17:16:00.003+13:00

Letting the quadrant signs determine the qubit component of the $216 = 2^3 3^3$ dimensions, we observe a parity cube, defined by the eight tetractys centres. The hexagon cores of the eight tetractys faces on the octahedron form the eight hexagonal faces of a $24$ vertex permutohedron. Both cubes and permutohedra tile three dimensional space. The six squares of the permutohedron may be labeled by the planes they lie in, as in $XY^{\pm}$. Each vertex of the permutohedron carries $3$ quark paths from the (signed) tetractys.

Quote of the Week

2011-12-08T15:37:00.002+13:00

It is really the case that there are brilliant loners out there and that there is some kind of conspiracy by the physics “establishment” to prevent their voices being heard?

Mike Duff, Imperial College
(warning: paper contains graphic language, but is mostly amusing)

Theory Update 131

2011-12-07T13:08:00.008+13:00

Recall that the set of qutrit paths was considered by Kapranov in the context of categorical noncommutative Fourier transforms. Kapranov, an author of the green book, is one of few mathematicians with a real feel for the pulse of contemporary physics. Lately he has been considering matrix valued energy functions, in a paper linking thermodynamics to the tropical geometry of tetractys diagrams.

Moment maps typically take values in the dual of a Lie algebra. In particular, the dual of energy space $R^{3}$ is a thermodynamic $\beta$ space $R^{3}$, where as usual $\beta = 1/ kT$. Integer lattices in $R^{n}$ are associated to toric varieties. In this case, moment maps can generate convex polytopes.

Theory Update 130

2011-12-06T13:15:00.005+13:00

Let's get back to the basic reason that String Theory is obviously wrong. As neatly summarised in Furey's paper, we can list all particle states using the normed division algebras from triality: the reals, complex numbers, quaternions and octonions. Let us call this large collection of numbers $RCHO$. Note that dimensionally, $RCHO$ resembles $O \cdot O$, octonions with octonion coefficients. M theorists like kneemo have been studying Jordan algebras over such large collections.

Consider $J_{3}(O \cdot O)$, the algebra of $3 \times 3$ matrices over $O \cdot O$. Since $O \cdot O$ has (real) dimension $64$, $J_{3}(O \cdot O)$ has dimension $216$, from three copies of $64$ off the diagonal and $3 \times 8$ along the diagonal. Note that $216 = 2^3 3^3$, the information dimension for three qubits and three qutrits. The simple prime factorization allows us to parse matrix components in many ways, for both $2 \times 2$ and $3 \times 3$ matrices. For instance, we could use a $3 \times 3$ pseudo-algebra built with off diagonal copies of $J_{3}(C \cdot O)$, the $54$ dimensional bioctonion algebra of the doubled tetractys. The diagonal would then require objects of dimension $18$, twice the dimension for the two qutrit hexagon path diagram.

Compare this with particles. There are $54$ left and right handed leptons and quarks, including color and generation. Then $216 = 4 \times 54$, allowing for fermions, antifermions and mirror versions. This full list is easily expressed in terms of ribbon diagrams with three strands. Fourier supersymmetry creates all bosonic states. No fairies or zombies appear anywhere.

Back to Koide IV

2011-12-05T14:22:00.004+13:00

Today Alejandro Rivero writes about the new Koide triplets, where the two charged quark triplets are interwoven to create a sequence of six quarks, including the special $(b,c,s)$ triplet.

This brings my attention to Koide's 2000 paper on Fourier operations in mixing. Here Koide perturbs the basic circulant parameters to fit masses for charged leptons and charged quark triplets. In this case, he observes that one can use approximate phases

$\cos^2 \theta_{l} = \frac{11}{24}$, $\cos^2 \theta_{d} = \frac{9}{24}$, $\cos^2 \theta_{u} = \frac{7}{24}$

But this looks familiar, doesn't it? The numerators sum to the tetractys total of $27$. The $24$ is the number of leptons and quarks without the three mirror neutrinos. The complementarity $27 = 11 + 16$ appears often in M Theory. The paper also discusses fixing CKM parameters using the fitted rest masses. For example, $\sqrt{m_t / m_u}$ gives a small parameter $V_{\textrm{ub}}$, close to $0.0035$.

Fairies Still Fairies

2011-12-04T16:34:00.006+13:00

This week some people have been convinced, through the usual entanglement of prejudice and random internet rumour, that an upcoming December 13 CERN seminar will announce evidence for fairies. We should only believe rumours that make physical sense, like superluminal neutrinos. There are no fairies.

The 20th century insistence on manifest locality in particle physics hearkens back to the materialist age, and to the inadequacy of classical methods. Today we know that locality is an extra condition, emerging from more fundamental categorical rules. Geometry is not always commutative. The tetractys is but a commutative shadow of three qutrit paths, and a doubling of the tetractys is obtained on considering two distinct bracketings for nonassociative paths.
Rest mass cannot be due to fairies, because rest mass must reside in the localization rules. As the successful Koide formula has shown us, these rules go beyond classical geometry, classical symmetry and the ubiquitous Lagrangian. The particles we observe are braids, and there are no braids for fairies. The $54$ dimensions of the bioctonion algebra account for $27$ (left handed) particles: $3$ electrons, $6$ neutrinos, $18$ quarks. The dark sector will be explained by the non local properties of this spectrum. New matter is a matter for engineering, not for fairy tales.

CDF on Charm CP

2011-12-03T16:35:00.004+13:00

Thanks to Resonaances for a report on the new CDF paper about $D^0$ decays and the (CP) asymmetry recently measured at LHCb. From the abstract:

We report on a measurement of CP violating asymmetries (Acp) in the Cabibbo suppressed $D^0$ $\rightarrow$ $\pi^{+} \pi^{+}$ and $D^0$ $\rightarrow$ $K^{+} K^{-}$ decays ... We measure Acp($\pi^{+} \pi^{-}$) = $+0.22 \pm 0.24$ (stat) $\pm 0.11$ (syst) (percent) and Acp($K^{+} K^{-}$) = $-0.24 \pm 0.22$ (stat) $\pm 0.09$ (syst), in agreement with CP conservation ... Under the assumption of negligible direct CP violation in both decays, the results provide an upper limit to the CP violating asymmetry in $D^0$ mixing, $|\textrm{Acp}^{\textrm{ind}}| < 0.13$, at the 90 percent confidence level.

Impossible Dream

2011-12-03T12:43:00.006+13:00

One day someone will employ a female feminist theoretical physicist, and she will be able to work without persecution. People might even talk to her. She will have a real desk, in her own office. She won't be expected to cook, or clean or look after things. She will have a whiteboard to use, with coloured marker pens, plenty of good food to eat, and students whose careers won't be ruined for talking to her.

One day.

Old Symmetry

2011-12-02T14:50:00.008+13:00

As in the tetractys qutrit path count, the hexagonal symmetries of 20th century physics include a doubling of points at the centre of the hexagon. Although clearly discrete, such hexagons are usually interpreted in terms of the continuum representation theory.

Observe how the quantum numbers $q$ and $s$ take values $-1$, $0$, $1$. The charge $q = \pm 1$ is now associated to a triplet of ribbon twists. Similarly, rotation of a Furey hexagon (Fano plane) gives color, and a tripled hexagon gives generations. There are four index hexagons in the tetractys: three at the corners and one in the centre.

One easily triples the state count on the baryon octet, accounting for quarks. This octet count now agrees with the tetractys.

Neutrino Browsing

2011-12-02T13:29:00.004+13:00

Two notable 2009 papers connecting neutrinos to Dark Energy are here and here. The MIT superluminal neutrino page has a long list of links. These include viXra papers, such as Ervin's short note on the Fourier neutrino photon doublet. One good find is this 2006 note by Klinkhamer on neutrinos and emergent symmetries.

Theory Update 129

2011-12-01T13:05:00.003+13:00

It is simpler to describe the secondary polytope for the Fano plane than for the tetractys, since the Fano plane is smaller. Recall first the cover example of the green book. An associahedron edge is extended to a triangle ($2$-simplex) with the addition of chords containing the centre point of the index square. For the three dimensional associahedron, which is indexed as always by the outer hexagon of the Fano plane, there are three ways to draw point centred chords, and these new faces eminate from the three squares of the associahedron. Using a basic dimensional argument, the extra structure resembles a cube, with three source squares on the associahedron and three others reaching out to the target vertex, which is marked by the three intersecting chords.

For the tetractys point configuration, we start with the associahedron in $6$ dimensions, indexed by the $9$-gon.

From Neutrinos to Pions

2011-12-01T12:06:00.007+13:00

If neutrinos can be superluminal, what happens with the pions? Neutral pions are built from quarks on the sides of the tetractys. Recall that the honeycomb dual to the tetractys becomes a triangle diagram for the Fano plane, when the up and down quarks are merged, as in CKM mixing.

The honeycomb is a rule on a set of three $3 \times 3$ matrices. Thus the triangle diagram looks like a rule for three $2 \times 2$ matrices, just like the triality of mixing. Moreover, the color kinematic duality of Bern et al suggests a special role for the gluon preons in perturbative gravity. These are obtained (as non braid objects) from the Fourier transform of up and down quarks, just as the photon arises as the transform of the (left handed) neutrino.

If gluon preons are essentially superluminal, what does this say about the pions? A natural hypothesis is that all the local aspects of mixing, for quarks and neutrinos, are governed by underlying superluminal phenomena. Pions might travel at close to but slightly above $c$. However, for now we will stick to the hypothesis that the small rest masses of neutrinos are responsible for their superluminal behavour, in line with the surface of last scattering as a honeycomb boundary that separates the neutrinos from all other localised states. Nonetheless, pion triangles are also a crucial piece of the mixing puzzle.

Cat Neutrinos

2011-11-30T13:54:00.010+13:00

Recall that Ma's finite group $A_4$ is a subgroup of the permutation group $S_4$ on four letters. Thus it labels $12$ of the $24$ vertices of the permutohedron polytope in three dimensions, and contains the Klein four group (using $2143, 3412$ and $4321$). The reduction of the permutohedron to the associahedron, which itself exhibits triality, suggests a selection of $7$ vertices out of the $14$, which happens to be the number of points (or lines) in the Fano plane.

The neutrino literature tends to associate these finite groups directly with larger continuous symmetry groups, but this really misses the fundamental nature of arithmetic structure in emergent geometry. When the neutrino set is doubled to include mirror neutrinos, we can presumably characterise mixing with the full set of vertices on operad polytopes. Similarly, Fourier circulants can arise from a Fano star, or double tetractys. Since the tetractys indexes a much larger (secondary) polytope, based on the central point, the polytopes provide more information than the basic finite groups.

Medusa, The Internet

2011-11-29T18:22:00.005+13:00

Mitchell Porter, alone of all the people in the world, saw fit to show interest in my work in a thread on the PhysicsForums site. After eight months, the forum moderators suddenly decided that the thread did not fit their commenting policy, and the thread was promptly deleted. Oh, Medusa, there are still those who have not seen your strength. Of course now Mitchell has posted the thread on a new blog.

Tachyon Theory Update

2011-11-28T18:46:00.007+13:00

Other categorical mathematicians will also be busy studying neutrino physics. In August, Baez et al wrote about $2$-groups and teleparallel classical gravity. Unsurprisingly, the late 2011 neutrino theory arXiv onslaught contains a consideration of teleparallel ideas, linking torsion to superluminal neutrinos. There is also a 2007 model of a teleparallel neutrino. This is a long way from deriving GR via teleparallelism, but we should appreciate the growing web of support for tachyons arising in Fourier duality.

Cat Conferences II

2011-11-28T16:50:00.007+13:00

Don't forget the upcoming Paris conference on Categories in Physics! They now appear to have two women, one of whom will speak on Topos Gravity and the other on topos mathematics. On the schedule it says that Louis Crane will speak on Neutrino Physics. Here is Crane's (pre OPERA) paper from May 2011. Although my name is on the participant list, I am of course not attending. I cannot afford a new pair of shoes, let alone an airfare to Paris.

Theory Update 128

2011-11-24T09:24:00.005+13:00

This tells us that there is another way to organise $57$ dimensions into a $3 \times 3$ array, where each lepton (including generation) is counted three times, each quark is counted twice and each mirror neutrino is counted only once. When we place mirror neutrinos on the diagonal, we have no need for gravitons, gravitinos or fairies.

Theory Update 127

2011-11-22T17:04:00.011+13:00

Dolan's triality studies conformal field theories at $c = 24 = 8 \times 3$, where an $8$ is associated to the qubit edge of a tetractys. It should not surprise us then that massive theories contain trialities using the centre of the tetractys. As it happens, the four largest Heegner numbers all equal $1$ mod $6$.

These are the numbers $d$ that generate near integers using the j-invariant value

$j( \frac{1 + \sqrt{-d}}{2} ) = k^3$

for $k$ an integer. The number $k^3$ always has a factor of $27 \times 8$. The Euler primes that generate the Heegner numbers are supersingular. Observe that the largest Heegner number uses the dimension of the bioctonion Jordan algebra.
Strangely, if we restrict to the massless dimensions, such as $24$ in $27$, we obtain the following.

Womanspace Gate

2011-11-22T13:00:00.003+13:00

Contemplative Mammoth provides the links. There isn't much I can add, except to say there had to be some reason that I don't read the Boys Club journal Nature.

Theory Update 126

2011-11-21T09:15:00.003+13:00

I cannot access it myself, but this old paper by Louise Dolan et al is clearly relevant to the dimension $24$. From the abstract:

In the case where $C$ is the Golay code, $Λ_C$ is the Leech lattice and the induced triality is the extra symmetry necessary to generate the Monster group from (an extension of) Conway's group. Thus it is demonstrated that triality is a generic symmetry.

The Central Fire

2011-11-20T18:36:00.003+13:00

Copernicus was inspired by the cosmology of the ancient Pythagorean, Philolaus, who lived a century after Pythagoras himself. Philolaus said that the world came from both the limited and the unlimited, and he placed the ten points of the tetractys on spheres around the central fire of the cosmos. The Earth sat on the inner sphere, rotating in $24$ hours around its fiery centre. In one conveniently lost work of Aristotle, there is an association between (i) the breath of fire and cosmic time and (ii) orbits and unlimited time.

Neutrino Theta Update

2011-11-20T14:16:00.009+13:00

The biggest problem with being an intelligent woman in physics is that nobody ever tells her anything. It is one of many, many means to slow her down.

Anyway, here is a new combined constrained range for a non zero $\theta_{13}$ angle, favouring the Chooz result and beginning to disfavour the low estimate of MINOS. Below are the predicted limits for a year from now. At present, a $\sin^2 2 \theta_{13}$ of $0.04$ sits at the lower end of the range.

Theory Update 125

2011-11-20T09:32:00.005+13:00

Recall that in the $27$ paths of the tetractys, $2/9$ were lepton paths and $6/9$ were quark paths. These happen to be the Koide phases that give a neat set of rest masses. In 2009 we saw how the tetractys cardinals were associated to Louise Riofrio's fractions for the dark sector. As the poem says:

The Tetrad completes the arrangement
Of the Soul and what is seen.
Ancient Tetraktys, Pythagoras' vision divine,
The Decad, a perfect Limit, and Cosmic Paradigm.

Koide Quarks III

2011-11-19T18:19:00.004+13:00

On the new Koide thread, Alejandro mentions the Georgi-Jarlskog mass relation. This 1979 relation, which is supposed to hold at high energy, simply states that

$m_b = m_{\tau}$, $m_{\mu} = 3 m_s$, $m_d = 3 m_e$

which uses the original charge quark triplet at $r = 1.76$. This relation arises from (relatively nice) GUT models.

Koide Quarks II

2011-11-19T16:10:00.016+13:00

To reiterate the Koide situation for quarks ...

Recall that there are $6$ basic leptons and $18$ quarks on the hexagon in the tetractys. This factor of $3$, due to color, now appears in the Koide scale for the new quark triplet. So if the charged leptons use a natural scale $\mu_{l} = 1 \equiv 313.8$ MeV, then the $(b,c,s)$ triplet uses a scale $\mu_{q} = 3$ (ie. the proton mass). The phases are $\theta_{l} = 2/9$ and $\theta_{q} = 6/9$. Both triplets, and the neutrino triplets, use the same central $r$ parameter of $\sqrt{2}$. The neutrinos require the phase adjustment of $\pm \pi /12$ and a much smaller scale, which is $\mu_{\nu} = 3.14 \times 10^{-10}$. This determines all quark, lepton and mirror masses, via the Koide formula

$m_k = \mu (1 + \sqrt{2} \cos (\theta + 2 \pi k/3))^2$

for $k = 1,2,3$, and using a simple extension to six dimensions in order to include the other quarks. This extension replaces the $\pi /4$ mixing phase with a $\pi /3$ phase, as expected for a higher dimensional information space.

You are all free to check this yourselves.

Fairy Update

2011-11-19T08:48:00.002+13:00

Varying Rest Mass

2011-11-18T16:59:00.008+13:00

We have at various times discussed the mass differences between standard Koide triplets and mirror ones. Some suggest uncertain masses, and the latest MINOS results for the antineutrino masses certainly show a large uncertainty. But rest mass is of course a quantum number. We can say that the antineutrino beam contains both ordinary $\overline{\nu}$ mass states and mirror ones. Can we use this simple double state hypothesis to fit the MINOS results? Those damned reconstructed energy curves never seem to go through the data points!
The varying mass hypothesis would extrapolate the neutrino triplet straight through the charged lepton triplet on its way to the mirror phase.

OPERA Confirms Tachyons

2011-11-18T15:03:00.007+13:00

I can't yet find a link, but I must beat Jester at posting by linking Tommaso's report. No wait, duh, OPERA have posted an updated paper, which includes the short pulse data that confirms the superluminal neutrino speed at event level. The new events have a notable spread of delay times.

God Help US

2011-11-18T11:09:00.001+13:00

Internet censorship alert. Arbitrary website blocking likely to be permitted under new laws.

Perturbative Dark Matter

2011-11-17T16:30:00.008+13:00

The $N = 8$ discussion on the dark sector reminds us that dark matter is empirically described at galactic scales by MOND. Recall that we required duality in order to justify the existence of a stable $1/r$ law, and abstract duality does indeed underlie the $2 \times 2$ Jordan structure of the $N = 8$ theory, which may be expressed in braid diagrams.

What about galaxy cluster scales? Assumptions regarding perturbative locality must break down somewhere. For a galaxy, we correlate the so called dark force with the acceleration of outer bodies. This empirical observation alone strongly suggested a tight connection between the dark energy and the dark matter.

Koide Quarks

2011-11-17T08:48:00.008+13:00

These new quark Koide triplets of Alejandro Rivero's are super cool! So instead of the original charge based triplets, we split the six quarks into two triplets in other ways. Alejandro notes that the $(b,c,s)$ triplet works for Koide parameters

$r = \sqrt{2}$ (the basic lepton parameter)
$\theta = 0.66642$

at a scale of $939.65$ MeV (roughly the proton mass). This gives the mass set ($4189.9$,$92.01$,$1356.0$) MeV. The strange quark eigenvalue comes with a minus sign. Now to check the $(t,u,d)$ triplet ...

Quarks and Neutrinos

2011-11-16T19:30:00.003+13:00

Recall that some time ago we were thinking about quark lepton scale complementarity, since the Koide charged lepton scale of $313.8$ MeV happens to equal the dynamical quark mass. Today, Alejandro adds another quark lepton coincidence for the proton mass, using strange quarks. Note that our Koide fit for the quarks resembles the neutrinos in having one minus sign out of the six eigenvalue signs, which is for the down quark, the way I parameterised it. Alejandro, please provide references for your comments!

Mitchell's Slides

2011-11-16T19:09:00.003+13:00

It seems that Mitchell has prepared a very neat set of slides about the so called $N = 8$ cosmology. Mitchell, did you actually give this seminar? Were there any questions?

Flavor Physics Year

2011-11-16T15:21:00.008+13:00

Recall that there is already good evidence for non standard CKM physics around the electroweak scale. The few models that attack the non standard phase problem tend to correlate the appropriate $\sin 2 \beta$ parameter with the measured asymmetry, $a$. For example, a four generation model gives the ($B$ physics) result below.

For one phase of current interest, namely

$\beta \equiv \frac{V_{cb}^{*} V_{ub}}{V_{us}^{*} V_{cs}}$,

the naive triality value is $2 \beta = - 0.096$. If such new phases are the right way to view these anomalous asymmetries, we must understand the evolution of the phases from standard CKM values (small) to the new values. Triality arguments from M Theory at least have the advantage that the $(a,b,c)$ parameters are intimately related to physical scale, although this is not yet well understood.

LHCb RULES!

2011-11-15T08:15:00.007+13:00

Reported at Resonaances: an anomalous asymmetry at LHCb in $D^{0}$ decays (to kaons and pions)! Excellent! It is not clear how to estimate the measured quantity with the Standard Model, so some will say the large asymmetry is due to incorrect assumptions, rather than new physics. In a sense, this is correct, but a reformulated Standard Model based on non perturbative physics would be new physics, by my definition. At present the result is at $3.5 \sigma$, so now LHCb will analyse the result with further data, eventually reaching $5$ sigma if the effect is real. The result:

Quote of the Week

2011-11-13T09:24:00.002+13:00

Only a true philosopher could encapsulate academia today so brilliantly:

I think it worth emphasizing that the most important thing for everyone involved ... is to ensure, by all means possible, that they bend over backwards so as to make sure that there is never any possibility that some Anonymous Internet Person might conceivably be offended at the suggestion that conference organizers anywhere - let alone conference organizers at an institution such as Oxford, whose commitment to gender equity and rejection of male privilege in education runs as far back as the High Middle Ages, I’m sorry, I mean 1974 — should risk feeling any twinge of private or, Heaven forfend, public embarrassment in the face of some no doubt imagined tendency to repeatedly organize conferences that feature only men on the program.

Another Glorious Rumour

2011-11-12T12:47:00.005+13:00

On Woit's latest fairy post, blogger Jester starts a glamorous rumour! That is, at HCP 2011 next week, LHCb will release a large CP asymmetry result! Will it match the known like-sign dimuon anomaly? This would be interpretable with a non standard $\beta$ phase for the CKM matrix. Jester's rumour mentions a difference between decays to kaons and pions, but we won't really know what it is until next week. Stay tuned!

Nagging Neutrinos

2011-11-12T09:47:00.010+13:00

With improved evidence for a non zero $\theta_{13}$ (assuming for now that neutrinos and antineutrinos share the same mixing parameters) we must ask once again: why, oh why, does the tribimaximal matrix fit so nicely with the charged lepton triplet? Perhaps quark lepton complementarity provides a clue.

Observe that with mirror neutrinos, quark color reflects the three lepton sectors: $e^{\pm}$, $\nu$ and $\nu^{m}$. And in perturbative gravity at least, color and generation are closely associated. Each $R_2$ factor in a mixing matrix is thus associated to either generation or color. So the zero parameter in the tribimaximal matrix could be associated to the lack of mixing for charged leptons, forming one third of the color spectrum. Similiarly, the closeness to the identity of the CKM quark mixing matrix reflects the appearance of all three colors in the quark set. Moreover, a (down) quark braid is somehow a mixture of $(e^{\pm}, \nu, \nu^{m})$ strands, meaning that there exist mirror quarks formed through a mirroring of neutrino components. Recall that the CKM matrix is similar to a reordering of a three parameter neutrino matrix.

What would that mean for neutrino and mirror neutrino mixing? We can create a neutrino braid with, for instance, one strand from each of a $(u,d,d)$ triplet, by cancelling the charges. Thus conversely, the neutrino matrices come from the quark ones.

Chooz Weighs In

2011-11-10T18:24:00.005+13:00

And the latest news from Double Chooz is a neutrino mixing $\theta_{13}$ parameter given by

$\sin^{2} 2 \theta_{13} = 0.085 \pm 0.029 \pm 0.042$

in agreement with the non zero value from MINOS and T2K! Wow! This appears to strengthen the non zero $\theta_{13}$ hypothesis, bringing the neutrinos in line with the three parameter CKM mixing. Note that the central MINOS value of $0.04$ is well within experimental limits. However, all $\theta_{13}$ measurements are still roughly consistent with zero, and the count of $\overline{\nu}_{e}$ events is open to alternative interpretations.

Theory Update 124

2011-11-10T13:48:00.007+13:00

Recall that the Euler characteristic for a genus zero surface is given by $2 = V - E + F$. By adding a central vertex, which might create simplices for the three dimensional ball, we get a shifted invariant $\chi = 3$.

Now it may be that all of $V$, $E$ and $F$ are divisible by $3$, as in the examples shown. A counterexample is the $\chi = 3$ tetrahedron, which has $4$ faces and $5$ vertices. Counting $\chi = 3$ is a ternary analogue of the $\chi = 2$ Poincare paradigm. For a polygon in the plane, the characteristic $V - E$ is usually zero, but this can now be amended to $1$.

Theory Update 123

2011-11-10T11:42:00.009+13:00

With this way of drawing the double tetractys, there are six associahedron index hexagons inside the diagram (see the green book).

The mirror neutrinos are required to provide a total of $18$ leptons against $36$ quarks in the quark lepton complementarity associated to triality for the associahedron.

Greek Trinity III

2011-11-08T13:51:00.005+13:00

Many writers today will criticise the Neoplatonists for lending dogma to religion, since Saint Augustine himself was educated in this philosophy. But it seems that Augustine, despite his remarkable brilliance, had very little appreciation for the centrality of mathematics to the trinity, which is thus corrupted. Neoplatonism made a notable comeback in medieval Europe. However, it was crushed in the south, and its role in the reformation of the north appears to be largely forgotten today.

Agrippa ... taught the three-fold nature of the Universe, the identity of the Macrocosm and the Microcosm, and traced out the lines of correspondence between them. Space, he said, is threaded with invisible lines of magnetic force, which unite men's principles with the occult forces of Nature.

I can't help but note that Agrippa, in 1529, wrote a text with the title Declamation on the Nobility and Preeminence of the Female Sex, which was distributed with the aid of that marvellous new invention, the printing press.

Theory Update 122

2011-11-08T09:46:00.005+13:00

Thus Bruno's double tetractys accounts for $48$ leptons and quarks (charged and neutral) including mass triplets. Note that there is no triple $t$ quark state, as in Tony Smith's LHC predictions. There are $6$ mirror neutrino states on the outside. These are the $54$ dimensions of the bioctonion Jordan algebra.

Greek Trinity II

2011-11-06T15:35:00.006+13:00

After the destruction of the pagan temples, a more unified Church succeeded in quelling Neoplatonic thought, for a while. In Europe, a notable Renaissance Neoplatonist was Giordano Bruno, burned at the stake in 1600. The charges against him spoke to the Neoplatonic elements of his cosmology, with its plurality of worlds of the mind, and so called erroneous opinions about the Trinity. At his sentencing, Bruno said,

Perhaps you, my judges, pronounce this sentence against me with greater fear than I receive it.

As the modern Theosophists say,

Modern science and philosophy are said to have been born in this century. Any one, however, who is acquainted with the scientific and philosophical concepts of the ancient Greeks will discover that these "modern" ideas are but warmed-over dishes covered, in most cases, with a thick sauce of crass materialism.

Thus many a modern scientist has failed to understand Bruno's words, and Galileo's relatively unthreatening views. Bruno's infinite aether was not pre-ordained by an external observer, but was the substance of which the mind created matter, and the logos was the Word of the Trinity.

Greek Trinity

2011-11-06T10:09:00.005+13:00

The Christian doctrine of Holy Trinity comes not from the bible, but from the development of the religion in the third and forth centuries AD. A notable influence was the school of Neoplatonism founded by Plotinus, who held that the cosmos emerged from the One, the Intelligence, and the Soul. The influential Christian bishop, Synesius of Cyrene, was a devoted student of the great neoplatonist Hypatia of Alexandria. It is thought that Synesius did not live long enough to hear of her gruesome death at the hands of a Christian mob. In a hymn, he writes:

I sing to Thee, Unity, I sing to Thee, Trinity;
Thou art One being Three, art Three being One;
and the intelligible segment holds what has been divided still indivisible.

Job of the Year

2011-11-03T16:54:00.001+13:00

Minorities and women are encouraged to apply. Brown University is an Equal Opportunity/Affirmative Action employer.

From a typical postdoc ad

Theory Update 121

2011-11-02T12:37:00.005+13:00

As noted in McElrath's 2008 paper, gravitational condensates involving neutrinos must also consider the electroweak bosons. From the braid diagrams, one obtains the mirror sector by flipping all twisted ribbon charges. The neutrinos, mirror neutrinos and photons are the only particles invariant under this process. The $Z$ boson, although neutral, consists of a colored triplet of $(+,-,0)$ ribbons. This triplet is the Fourier dual of the right handed leptons. Observe that a $Z$ boson is the only braid object composing with its mirror braid to form a massless photon, a dual to the left handed neutrino.

Quote of the Month

2011-10-30T11:23:00.001+13:00

"Our work has shown that stars have no problem making complex organic compounds under near-vacuum conditions,” says Kwok. “Theoretically, this is impossible, but observationally we can see it happening.”

via Physics World

Theory Update 120

2011-10-27T15:54:00.014+13:00

Now we should start applying quark lepton complementarity to the MNS and CKM mixing matrices, starting with the $\theta_{13} = 0$ tribimaximal form. Today's observation is that, using the $R_2$ factor parameters

CKM: $(-0.231,24,0.0035)$
MNS: $(1,\sqrt{2},0)$

the resulting matrices have very similar eigenvalues. That is,

CKM: $-0.9651 - 0.2621 i$, $0.4813 \pm 0.8765 i$
MNS: $-0.9856 + 0.1691 i$, $0.4928 \pm 0.8701 i$

where the first eigenvalue is the determinant, and the real and imaginary parts of the determinant for the MNS matrix are exactly $(\pm \sqrt{2} - 1)/ \sqrt{6}$.

Using the known CKM parameters, and matching the determinants, we could then determine a small $\theta_{13}$ for the neutrino mixing matrix.

Theory Update 119

2011-10-26T18:29:00.004+13:00

Thus quark lepton complementarity utilises the three map colors of the honeycomb dual to the triangulated tetractys.

Theory Update 118

2011-10-26T13:10:00.017+13:00

Recall that Furey's Fano plane mixes the charged leptons and neutrinos, as in the projection of a charge cube on the $27$ qutrit paths.

The $21$ Fano paths remind us of the (triality labeled) 21 edges in the three dimensional associahedron polytope. Recall also how 16 paths (out of $27$) are placed on the associahedron. Space really wants to start out being three dimensional. Ribbon charge is specified using the quark path mixtures of the quandle plane. An $XXX$ charged lepton path obtains a unit charge from $1/3 + 2/3$. The six cyclic permutations of the neutrino source vertex themselves represent the hexagon of $6$ paths from source to target on a cube. The neutrino hexagon appears in the honeycomb dual to the tetractys brane, and may thus be considered as a surface of last scattering.

Dark Matter at the LHC III

2011-10-25T11:09:00.020+13:00

Now we see what this whole AdS/CFT business is about: the two sides of the correspondence are actually exactly the same thing, appearing distinct only through the use of a wrong language and wrong physics. Quark lepton complementarity informs us about electroweak scale physics, as we extend the mirror neutrino CMB correspondence to the Koide masses of the quarks.

$\bullet$ mirror $\nu$: cosmic $T \in (1.40, 2.73, 135.9)$ K (future, present, past)
$\bullet$ u quarks: $T \in (4.8 \times 10^{9}, 3.0 \times 10^{12}, 4.1 \times 10^{14})$ K
$\bullet$ d quarks: $T \in (1.3 \times 10^{10},1.8 \times 10^{11},1.1 \times 10^{13})$ K

Observe that the $(u,d,s)$ triplet have temperatures below the QCD critical point at $4 \times 10^{11}$ K, while the $(c,b,t)$ triplet has temperatures above this point. The minimal quark temperature of $T_{u} = 5 \times 10^{9}$ K corresponds to the quark to lepton transition in the early universe. In our time, the RHIC and ALICE ion collision experiments create temperatures of trillions of degrees in the laboratory, and the other LHC experiments also observe a quark gluon plasma. In the next few years, we should learn a great deal about EW symmetry breaking.

Dark Matter at the LHC II

2011-10-24T17:39:00.018+13:00

There are many ways to parse the bounds on $\eta /s$, but Riofrio's fractions suggest
recalling that the baryon fraction $\Omega_{b} = 1 - 3/ \pi$. The lower bound, a dipole measure, originally comes from the pioneering QGP paper on the strongly coupled $N=4$ SYM theory. As we know, this theory is commensurate with braid physics. Triality is presumably responsible for the factor of $1/3$, since dipoles are the only polygons without area.