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.
16 years ago
Showing posts with label theory. Show all posts
Showing posts with label theory. Show all posts
Friday, February 3, 2012
Tevatron Strikes Back
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.
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.
Wednesday, February 1, 2012
Color Z News
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.
Not much to speak of, three events, but definitely something to ask ATLAS about.
Friday, January 27, 2012
Wednesday, January 18, 2012
Witten on Yang Mills
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.
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.
Monday, January 16, 2012
Theory Update 136
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.
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.
Thursday, January 12, 2012
Theory Update 135
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.
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.
Wednesday, January 11, 2012
Theory Update 134
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).
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).
Tuesday, January 10, 2012
Quote of the Month
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.
Monday, January 9, 2012
More Quick Neutrinos
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.
A more interesting article from this edition is a note on dark matter in globular clusters.
Thursday, January 5, 2012
Quick Neutrino Review
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
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.
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.
Sunday, January 1, 2012
Higgsy Codes IV
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.
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
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.
$\omega \equiv \frac{1}{2} (-1 + i + j + k)$
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
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!
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
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.
$4096 = 1 + 759 + 2576 + 759 + 1$
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.
Monday, December 19, 2011
Higgs Koide Triplets
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.
Saturday, December 17, 2011
The Condensate Scale II
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.
$(1,1, \frac{\alpha}{2})$
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.
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.
$M = m_{Z} + 2m_{W}$
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.
Thursday, December 15, 2011
So, A Condensate Higgs
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.
$2 m_{H} = m_{W} + m_{W} + m_{Z}$,
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
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:
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:
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