Friday, December 16, 2011

The Condensate Scale

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.

8 comments:

  1. I'd like to mention a bit about the octonionic Leech lattice construction. This may shed some light on the Higgs mass relation you mention. One can construct the E8 root lattice as the lattice L spanned by 240 octonions of the form: $\pm e_i \pm e_j$ (i,j distinct, 112 total) and $\frac{1}{2}(\pm 1 \pm e_1 \pm ... \pm e_7)$ (odd number of minus signs, 128 total).

    Using L one can construct the Leech lattice as the set of octonionic triples (x,y,z) with norm $N(x,y,z)=\frac{1}{2}(x\overline{x}+y\overline{y}+z\overline{z})$, where x,y,z are in L, x+y, x+z, y+z are in $L\overline{s}$ and x+y+z is in $Ls$. Knowing this, the Higgs mass relation you mention might arise from the norm condition from a triple where the $W^{\pm},Z$ masses are the norms of its components.

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  2. Sounds reasonable, kneemo. After all, $N(x,y,z)$ looks exactly the same, doesn't it, and we already have the octonion tetractys for bosons, via a Fourier transform of the fermion one.

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  3. So then we can use colored $Z$ operators for the $x,y,z$, as planned.

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  4. Robert Wilson has shown automorphisms of the Leech lattice can be written in terms of 3x3 octonionic matrices. Even better, for each of the four types, I checked and all such matrices are unitary, i.e., they are F4 transformations over the integral octonions. (Indeed, F4(Z) is in E6(Z), so the scent is that of U-duality in some 24-dimensional integral charge space.)

    Looking back at the triples $v=(x,y,z)$, one can always turn these into projectors via $P=vv^{\dagger}$. This allows one to map the Leech lattice to an integral Cayley plane. This seems reasonable, given that F4 is the isometry group of the Cayley plane, and automorphisms of the Leech lattice are also F4 transformations.

    One can even construct Hermitian operators with spectral decomposition given in terms of projectors made from Leech lattice vectors. This is where one can perhaps touch base with the Koide relation, by attempting to construct Hermitian mass matrices and unitary transformations out of Leech lattice points $v=(x,y,z)$.

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  5. And the bioctonions? Surely we need the 'mirror' pairing for the Koide operators? It would be fine if the Leech construction alleviated this need, but I worry about the extension to braids.

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  6. Yes, I'm thinking about constructing integral bioctonions formed from pairs of E8 lattice octonions. This would give a larger lattice that contains the Leech. E6(Z) would act on this. Either way, this definitely smells nonperturbative.

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